Communications In Mathematical Physics - Volume 281

Commun. Math. Phys. 281, 1–21 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0457-x

Communications in

Mathematical Physics

How Smooth is Your Wavelet? Wavelet Regularity via Thermodynamic Formalism Mark Pollicott1 , Howard Weiss2,
1 Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom 2 School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30334, USA.

E-mail: [email protected] Received: 26 April 2006 / Accepted: 3 October 2007 Published online: 6 May 2008 – © Springer-Verlag 2008

Abstract: A popular wavelet reference [W] states that “in theoretical and practical studies, the notion of (wavelet) regularity has been increasing in importance.” Not surprisingly, the study of wavelet regularity is currently a major topic of investigation. Smoother wavelets provide sharper frequency resolution of functions. Also, the iterative algorithms to construct wavelets converge faster for smoother wavelets. The main goals of this paper are to extend, refine, and unify the thermodynamic approach to the regularity of wavelets and to devise a faster algorithm for estimating regularity. The thermodynamic approach works equally well for compactly supported and non-compactly supported wavelets, and also applies to non-analytic wavelet filters. We present an algorithm for computing the Sobolev regularity of wavelets and prove that it converges with super-exponential speed. As an application we construct new examples of wavelets that are smoother than the Daubechies wavelets and have the same support. We establish smooth dependence of the regularity for wavelet families, and we derive a variational formula for the regularity. We also show a general relation between the asymptotic regularity of wavelet families and maximal measures for the doubling map. Finally, we describe how these results generalize to higher dimensional wavelets. 0. Introduction While the Fourier transform is useful for analyzing stationary functions, it is much less useful for analyzing non-stationary cases, where the frequency content evolves over time. In many applications one needs to estimate the frequency content of a nonstationary function locally in time, for example, to determine when a transient event occurred. This might arise from a sudden computer fan failure or from a pop on a music compact disk. The usual Fourier transform does not provide simultaneous time and frequency localization of a function.
The work of the second author was partially supported by a National Science Foundation grant DMS0355180.

2

M. Pollicott, H. Weiss

The windowed or short-time Fourier transform does provide simultaneous time and frequency localization. However, since it uses a fixed time window width, the same window width is used over the entire frequency domain. In applications, a fixed window width is frequently unnecessarily large for a signal having strong high frequency components and unnecessarily small for a signal having strong low frequency components. In contrast, the wavelet transform provides a decomposition of a function into components from different scales whose degree of localization is connected to the size of the scale window. This is achieved by integer translations and dyadic dilations of a single function: the wavelet. The special class of orthogonal wavelets are those for which the translations and dilations of a fixed function, say, ψ form an orthonormal basis of L 2 (R). Examples include the Haar wavelet, the Shannon wavelet, Meyer wavelet, Battle-Lemarié wavelets, Daubechies wavelets, and Coiffman wavelets. A standard way to construct an orthogonal wavelet is to solve the dilation equation ∞ √
φ(x) = 2 ck φ(2x − k),

(0.1)

k=−∞

 with the normalization k ck = 1. Very few solutions of the dilation equation for wavelets are known to have closed form expressions. This is one reason why it is difficult to determine the regularity of wavelets. 1 Provided the solution φ satisfies some additional conditions, the function ψ defined by ψ(x) =

∞


(−1)k c1−k φ(2x − k),

n=−∞

is an orthogonal wavelet, i.e., the set {2 j/2 ψ(2 j x − n) : j, n ∈ Z} is an orthonormal basis for L 2 (R). Henceforth, the term wavelet will mean orthogonal wavelet. Daubechies [Da1] made the breakthrough of constructing smoother compactly supported wavelets. For these wavelets, the wavelet coefficients {ck } satisfy simple recursion relations, and thus can be quickly computed. However, the Daubechies wavelets necessarily possess only finite smoothness.2 Smoother wavelets provide sharper frequency resolution of functions. Also, the regularity of the wavelet determines the speed of convergence of the cascade algorithm [Da1], which is another standard method to construct the wavelet from Eq. (0.1). Thus knowing the smoothness or regularity of wavelets has important practical and theoretical consequences. The main goals of this paper are to extend, refine, and unify the thermodynamic approach to the regularity of wavelets and to devise a faster algorithm for estimating regularity. In Sect. 1 we recall the construction of wavelets using multiresolution analysis. We then discuss how the L p -Sobolev regularity of wavelets is related to the thermodynamic pressure of the doubling map E 2 : [0, 2π ) → [0, 2π ). The latter is the crucial link between wavelets and dynamical systems. Previous authors have noted that for compactly supported wavelets, the transfer operator preserves a finite dimensional subspace and is thus represented by a d × d matrix. Providing the matrix is small, its maximal eigenvalue, which is related to the pressure, 1 We learned in [Da2] (see references) that this dilation equation arises in other areas, including subdivision schemes for computer aided design, where the goal is the fast generation of smooth curves and surface. 2 Frequently, increasing the support of the function leads to increased smoothness.

Wavelet Regularity via Thermodynamic Formalism

3

can be explicitly computed, and thus the L p -Sobolev regularity can be determined by matrix algebra. Cohen and Daubechies identified the L p -Sobolev regularity in terms of the smallest zero of the Fredholm-Ruelle determinant for the transfer operator acting on a certain Hilbert space of analytic functions [CD1,Da2]. Their analysis leads to an algorithm for the Sobolev regularity of wavelets having analytic filters that converges with exponential speed. In Sect. 2, we study the transfer operator acting on the Bergman space of analytic functions and effect a more refined analysis. This leads to a more efficient algorithm which converges with super-exponential speed to the L p -Sobolev regularity of the wavelet. The algorithm involves computing certain orbital averages over the periodic points of the doubling map E 2 . In Sect. 3 we illustrate, in a number of examples, the advantage of this method over the earlier Cohen-Daubechies method. Running on a fast PC, this algorithm provides a highly accurate approximation for the Sobolev regularity in less than two minutes. A feature of this thermodynamic approach is that it works equally well for compactly supported and non-compactly supported wavelets (e.g., Butterworth filters). In practice, for wavelets with large support, the distinction may become immaterial, and the use of our algorithm which applies to arbitrary wavelets may prove quite useful. In Sect. 4, we use this improved algorithm to study the parameter dependence of the L p -Sobolev regularity in several smooth families of wavelets. In particular, we construct new examples of wavelets having the same support as Daubechies wavelets, but with higher regularity. In Sect. 5, we show that the L p -Sobolev regularity depends smoothly on the wavelet, and we provide, via thermodynamic formalism, a variational formula. This problem has been considered experimentally by Daubechies [Da1] and Ojanen [Oj1] for different parameterized families. Our thermodynamic approach, based on properties of pressure, gives a sound theoretical foundation for this analysis. In Sect. 6, we study the asymptotic behavior of the L p -Sobolev regularity of wavelet families. We establish a general relation between this asymptotic regularity and maximal measures for the doubling map, extending work of Cohen and Conze [CC1]. The study of maximal measures is currently an active research area in dynamical systems. In Sect. 7, we describe the natural generalization of these results to arbitrary dimensions. Finally, in Sect. 8, we briefly describe the natural generalization of these results to wavelets whose filters are not analytic. 1. Wavelets: Construction and Regularity Constructing wavelets. We begin by recalling the construction of wavelets via multiresolution analysis. A comprehensive reference for wavelets is [Da1], which also contains an extensive bibliography. The Fourier transform of the scaling equation (0.1) satisfies 1
(ξ ) = m(ξ/2)φ (ξ/2), where m(ξ ) = √ φ ck e−ikξ . (1.1) 2 k The function m is sometimes called the filter defined by the scaling equation. Iterating this expression, and assuming that m(0) = 1, one obtains that (ξ ) = φ

∞  j=1

m(2− j ξ ).

(1.2)

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M. Pollicott, H. Weiss

Conditions (1)–(3) below allow one to invert this Fourier transform and obtain the scaling function φ. The wavelet ψ is then explicitly given by ψ(t) =

√

2

∞


dk φ(2t − k), where

dk = (−1)k−1 c−1−k ,

(1.3)

k=−∞

and thus the family

  ψ j,k = 2 j/2 ψ(2 j t − k)

j,k∈Z

L 2 (R).3

forms an orthonormal basis of A major result in the theory is the following. Theorem 1.1. [Da1,p.186] Consider an analytic 2π -periodic function m with Fourier expansion m(ξ ) = n cn einξ satisfying the following conditions: (1) m(0) = 1; (2) |m(ξ )|2 + |m(ξ + π )|2 = 1; and (3) |m(ξ )|2 has no zeros in the interval [− π3 , π3 ].  defined in (1.2) is the Fourier transform of a function φ ∈ L 2 (R) Then the function φ and the function ψ defined by (1.3) defines a wavelet. (0) = 0, which is necessary to ensure the completeCondition (1) is equivalent to φ ness in the multiresolution defined by φ. Condition (1) is also necessary for the infinite product (1.2) to converge. Condition (2) is necessary for the wavelet to be orthogonal, and condition (3), called the “Cohen condition”, is a sufficient condition for the wavelet to be orthogonal [Da1, Cor. 6.3.2]. If one wishes to construct compactly supported wavelets, then one needs the additional assumption: (4) There exists N ≥ 1 with cn = 0 for |n| ≥ N . If ck = 0 for k ∈ [N1 , N2 ], then the support of φ is also contained in [N1 , N2 ], and the support of ψ is contained in [(N1 − N2 + 1)/2, (N2 − N1 + 1)/2]. It is frequently useful to work with the function q(ξ ) = |m(ξ )|2 . This function is positive, analytic, and even. Condition (1) is equivalent to q(0) = 1 and (2)  is equivalent to q(ξ ) +q(ξ + π ) = 1. A simple exercise shows that (1) is equivalent to k∈Z ck = 1/2 and (2) is equivalent to q(ξ ) = 1/2 + k∈Z c2k+1 cos((2k + 1)ξ ). Thus the set K of positive even analytic functions satisfying both (1) and (2) is a convex subset of the positive analytic functions. Example (Daubechies wavelet family). This family of continuous and compactly supported wavelets φ Daub {N ∈ N}, is obtained by choosing the filter m such that N |m(ξ )|2 = cos2N (ξ/2)PN (sin2 (ξ/2)), where PN (y) =

N −1


 N −1+k  k

yk .

k= 0

The wavelets in this family have no known closed form expression. Note that |m(ξ )|2 vanishes to order 2N at ξ = π . More generally, a trigonometric ξ 2 2 2N polynomial m with |m(ξ )| = cos 2 u(ξ ) satisfies (2) only if u(ξ ) = Q(sin (ξ/2)), where Q(y) = PN (y) + y N R(1/2 − y), and R is an odd polynomial, chosen such that Q(y) ≥ 0 for 0 ≤ y ≤ 1 [Da1, 171]. 3 Engineers often call the family {c , d } a quadratic mirror filter (QMF). k k

Wavelet Regularity via Thermodynamic Formalism

5

Wavelet regularity. Since the Fourier transform is such a useful tool for constructing wavelets, and since wavelets (and other solutions of the dilation equation) rarely have closed form expressions, it seems useful to measure the regularity of wavelets using the Fourier transform. For p ≥ 1, the L p -Sobolov regularity of φ is defined by   f (ξ )| p ∈ L 1 (R) . s p ( f ) = sup s : (1 + |ξ | p )s |  The most commonly studied case is p = 2, where if s2 ( f ) > 1/2, then f ∈ C s2 −1/2− . One can also easily see that s p ( f ) − sq ( f ) ≤

1 1 − , for 0 < p < q. p q

Thermodynamic approach to wavelet regularity. We now discuss a thermodynamic formalism approach to wavelet regularity. We assume that |m(ξ )|2 has a maximal zero of order M at ±π and write |m(ξ )|2 = cos2M (ξ/2)r (ξ ),

(1.4)

where r ∈ C ω ([0, 2π )) is an analytic function with no zeros at ±π . We further assume that r > 0, and thus log r ∈ C ω ([0, 2π )). Given a function g : [0, 2π ) → R and the doubling map E 2 : [0, 2π ) → [0, 2π ) defined by E 2 (x) = 2x(mod 2π ), the pressure of g is defined by  P(g) = sup h(µ) +

2π

g dµ : µ a E 2 -invariant probability measure ,

0

where h(µ) is the measure theoretic entropy of µ. If g is Hölder continuous, then this supremum is realized by a unique E 2 −invariant measure µg , called the equilibrium state for g. The following theorem expresses the relationship between the pressure and the Sobolev regularity s p (φ), and provides the crucial link between wavelets and dynamical systems. Theorem 1.2. The L p -Sobolov regularity satisfies the expression s p (φ) = M −

P( p log r ) . p log 2

Proof. The proof for p = 2 is due to Cohen and Daubechies [CD1] and the general case is due to Eirola [Ei1] and Villemoes [Vi1]. See also [He1,He2,Sun]. In the literature this result is formulated in terms of the spectra radius of the transfer operator on continuous functions; but this is precisely the pressure (see Proposition 2.1).  See Appendix I for a proof of the lower estimate in arbitrary dimensions.

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M. Pollicott, H. Weiss

2. Determinants and Transfer Operators Given h : [0, 2π ) → R we define the semi-norm |h|β = sup x= y

|h(x) − h(y)| , |x − y|β

and the Hölder norm ||h||β = |h|∞ + |h|β , where 0 < β < 1. Given g ∈ C β [0, 2π ), the transfer operator L g : C β ([0, 2π ) → C β ([0, 2π ) is the positive linear operator defined by x

x

x

x

k + exp g +π k +π . L g k(x) = exp g 2 2 2 2 The following properties of pressure are well known [Ru1,PP1]. Proposition 2.1. (1) The spectral radius of L g is exp(P(g)); (2) The pressure P(g) has an analytic dependence on g ∈ C β ([0, 2π )); and  2π (3) The derivative Dg P( f ) = 0 f dµg . The transfer operator L g C β ([0, 2π )) → C β ([0, 2π )) is not trace class on this class of functions. However, following Jenkinson and Pollicott [JP1,JP2], we consider the transfer operator acting on the Bergman space A2 ( ) of square-integrable analytic functions on an open unit disk ⊂ C containing [0, 2π ). In Appendix I we show that for analytic functions g, the transfer operator L g . A2 ( ) → A2 ( ) is trace class and L g has a welldefined Ruelle-Fredholm determinant. For small z this can be defined by det(I − z L g ) =  n k exp(tr(log(I − z L g ))) [Si1]. We can write tr(log(I − z L g )) = ∞ k=1 (z /n)tr(L g ), and k each operator L g is the sum of weighted composition operators having sharp-trace (in the sense of Atiyah and Bott) tr(L kg ) =


exp(Sk g(x)) , 1 − 2−k k

E 2 x=x

Sk (x) = g(x) + g(E 2 x) + · · · + g(E 2k−1 x). The Ruelle-Fredholm determinant for L g can be expressed as ⎛ ⎞ ∞ n

z exp(S g(x)) n ⎠, det(I − z L g ) = exp ⎝− n n 1 − 2−n n=1

E 2 x=x

(2.1)

 and has the “usual determinant property” det(I − z L g ) = ∞ n=1 (1 − zλn ), where λn denotes the n th eigenvalue of L g . Note that this determinant is formally quite similar to the Ruelle zeta function, and has an additional factor of the form 1/(1 − 2−n ) in the inner sum. More generally, L g need not be trace class for expression (2.1) to be well defined for small z (see Sect. 8). For our applications to wavelet regularity, we consider g = p log r (see (1.1)). To compute the determinant, one needs to compute the orbital averages of the function g over all the periodic points of E 2 . Every root of unity of order 2n − 1 is a periodic point

Wavelet Regularity via Thermodynamic Formalism

7

for E 2 , and there are exactly 2n − 1 such roots of unity, hence there are exactly 2n − 1 periodic points of period n for E 2 . If one expands the exponential function in a Taylor series and applies Cauchy’s convolution formula for a power of a Taylor series, one obtains the following formula: d p (z) := det(I − z L plogr ) = 1 +

∞


bn z n ,

n=1

where bn =

n
(−1)k k!

k= 0

and  p

an =

1 1 − 2−n




p

n 1 +···+n k = n

p

an 1 · · · an k , n1 · · · nk

⎡ ⎤p n −1 n−1   2
jk   2 ⎣ ⎦ . r 2n − 1 k= 0

j=0

The next theorem summarizes some important properties of this determinant d p (z). Theorem 2.2. Assume log r ∈ C ω [0, 2π ). Then (1) The function d p (z) can be extended to an entire function of z and a real-analytic function of p; (2) The reciprocal of the exponential of pressure z p := exp(−P( p log r ))) is the smallest zero for d p (z); (3) The Taylor coefficients bn of d p (z) decay to zero at a super-exponential rate, i.e., 2 there exists 0 < A <1 such that bn = O(An ); N (4) Let d p,N (z) = 1 + n=1 bn z n be the truncation of the Taylor series to N terms and consider smallest zero d p,N (z Np ) = 0. Then z Np → z p at a super-exponential 2

rate, i.e., there exists 0 < B < 1 such that |z p − z Np | = O(B N ). In fact, in Theorem 2.2 (3) and (4) we can choose A and B arbitrarily close to 1/2. Parts (1) and (2) follow from [Ru2] and (4) follows from (3) via the argument principle. The essential idea in proving (3) is that one can also express the determinant d p (z) = 1 +

∞
n=1

bn z n =

∞ 

(1 − zλn ) ,

n=1

where λn denotes the n th eigenvalue of L p log r [Ru2]. It follows that the coefficient bn = i1 <···
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M. Pollicott, H. Weiss

For our applications to wavelets, we shall take either g = log |m|2 or g = log r . In fact, both functions lead to equivalent results. The following proposition relates the zeros of det(1 − z L log r ) and det(1 − z L |m|2 ). 4 Proposition 2.3. The determinant det(1 − z L log |m|2 ) has zeros at (1) 1, 2, 4, . . . , 2 M−1 ; and (2) 2 M z, where z is a zero for det(1 − z L log r ). This was observed by Cohen and Daubechies [CD1]. 3. Examples: Computation of Sobolev Regularity Using the Determinant In this section we shall apply Theorem 2.2 to estimate the Sobolev exponent s p (φ) of certain interesting examples. 3.1. Daubechies wavelet family. Example 1: (N = 3 and p = 2). In this case d2 (z) is a polynomial of degree 5 with rational coefficients: d2 (z) = 1 − 2z − 57.1875z 2 − 74.53125z 3 + 210.09375z 4 − 91.125z 5 , which has its first zero at z 0 = 0.111 . . .. By Theorem 1.2 we deduce the precise value s2 (ψ3Daub ) = 3 + log(0.1111 . . .)/2 log 2 = 1.415037 . . ..5 The following table illustrates the remarkable speed of approximation of this algorithm by computing the first few approximations to s2 (ψ3Daub ). The level k approximation to the determinant is obtained by summing over all periodic points with period less than or equal to k. Level 6 approximations take less than a minute on a desktop computer. N

Smallest root of d2

Approx to s2 (ψ3Daub )

1 2 3 4 5 6

0.50000000000000000 0.11590073269851241 0.10935186260699924 0.11120565746767627 0.11111111111111109 0.11111111111111111

2.5000000000000000 1.4454807959406297 1.4035248496765980 1.4156510452962938 1.4150374992788437 1.4150374992788437

Example 2: (N = 2 and p = 4). In this case d4 (z) is a polynomial of degree 5 with rational cofficients: d4 (z) = 1 − 2z − 50.75z 2 − 76.25z 3 + 115z 4 − 32z 5 , which has its first zero at z 0 = 0.115146 . . .. By Theorem 2.1 we deduce that s4 (ψ2Daub ) = 3 + log(0.115146 . . .)/2 log 2 = 1.22038 . . ..6 4 In our method it is desirable to factor out the zeros and work with r because this reduces the amount of computation and may increase the efficiency of estimating the determinant by decreasing C > 0. 5 This compares with the numerical estimate of Cohen-Daubechies of 1.414947, which is accurate to two decimal places. 6 This compares with the estimate of Cohen-Daubechies of 1.220150 . . ., which is accurate to three decimal places.

Wavelet Regularity via Thermodynamic Formalism

9

Example 3: (N = 4 and p = 2). In this case d2 (z) is a polynomial of degree 7 with rational coefficients d2 (z) = 1 − 2z − 433z 2 − 996z 3 + 23572z 4 − 32288z 5 − 68627z 6 − 24414z 7 , which has its first zero at z 0 = 0.045788 . . .. By Theorem 2.1 we deduce that s2 (ψ4Daub ) = 4 + log(0.045788 . . .)/2 log 2 = 1.77557 . . ..7 Based on the same approximation scheme, the following table contains our approximations to s2 (ψ Ndaub ) for N = 1, 2, . . . , 10, and compares them with the values obtained by Cohen and Daubechies in [CD1]. N

New approx to s2 (ψ4Daub )

Previous approx to s2 (ψ4Daub )

1 2 3 4 5 6 7 8 9 10

0.500000 1.000000 1.415037 1.775565 2.096786 2.388374 2.658660 2.914722 3.161667 3.402723

0.338856 0.999820 1.414947 1.775305 2.096541 2.388060 2.658569 2.914556 3.161380 3.402546

Remark. In Examples 1–3 the determinant is a rational function. This is an immediate consequence of the fact that the transfer operator preserves the finite dimensional subspace of functions spanned by {1, cos(ξ ), cos(2ξ ), · · · , cos(N ξ )}. This fact has been observed by several authors, cf., [Oj1]. 3.2. Butterworth filter family. The Butterworth filter is the filter type with flattest pass band and which allows a moderate group delay [OS1]. These filters form an important family of non-compactly supported wavelets and are defined by |m(ξ )|2 =

cos2N (ξ/2) , N ∈ N. sin2N (ξ/2) + cos2N (ξ/2)

We suppress the index n for notational convenience. It is easy to check that |m(ξ )|2 satisfies conditions (1) − (3) of Theorem 1.1. Example 4: (Butterworth filter with N=2, p=2). For this case the determinant d2 (z) is a power series of the form d2 (z) = 1 − 2. z − 2.08 z 2 − 0.479053 z 3 − 0.0124521 z 4 − 0.000086246 z 5 −2.06875 10−9 z 6 − 3.2482 10−13 z 7 − 7.38964 10−13 z 8 + 4.24431 10−12 z 9 −2.94676 10−11 z 10 + · · · , which has its first zero at z 0 = 0.356702 . . ., and thus by Theorem 1.2 we deduce that s2 (ψ2Butt ) = 2 + log(0.356702 . . .)/2 log 2 = 1.25497 . . .. 7 This compares with the estimate of Cohen-Daubechies of 1.775305, which is accurate to three decimal places.

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M. Pollicott, H. Weiss

The following table contains the first few approximations of s2 (ψ2Butt ): N

Root

Approx to s2 (ψ2Butt )

1 2 3 4 5 6 7

0.500000000000000 1.256072268804003 0.356757200613385 0.356702225093496 0.356702089349239 0.356702089348077 0.356702089348077

1.0000000000000000 1.2689757055091953 1.2565072268804003 1.2563960602082473 1.2563957856969186 1.2563957856945698 1.2563957856945698

We can similarly compute s2 (ψ NButt ) for N = 1, 2, . . . , 10 and compare these with the values in [CD1]. N

New approx to s2 (ψ NButt )

Previous approx to s2 (ψ NButt )

1 2 3 4 5 6 7 8 9 10

0.500000 1.256211 2.044109 2.843768 3.648646 4.456118 5.264533 6.072947 6.881125 7.688598

0.338856 1.256395 2.044117 2.843771 3.648817 4.456252 5.264586 6.073034 6.881173 7.688776

4. Compactly Supported Wavelets Smoother than the Daubechies Family We now consider smooth finite dimensional families of wavelets and attempt to find the maximally regular wavelets in the family. Several authors have used this approach to find compactly supported wavelets that are more regular than the Daubechies wavelets with the same support. The most straightforward approach to the problem of maximizing the regularity of compactly supported wavelets in parameterized families reduces to the problem of maximizing the maximal eigenvalue of an N × N matrix. However, except for very small N , there is no algebraic expression for the eigenvalue. In this section, we illustrate the advantages of our fast converging algorithm in seeking more regular wavelets. Example 1. Ojanen [Oj1] experimentally studied the regularity of wavelets corresponding to filters |m(ξ )|2 of width N and which vanish to order M at π . In the case that M ≥ N − 1 he found that the Daubechies polynomials (where M = N ) are apparently the smoothest. For N = 2 and M = 1, say, one can define the family of filters |m(ξ )|2 =

1 (1 + cos ξ ) (1 + 8a cos2 ξ − 8a cos ξ ) .    2

(4.1)

:=r (ξ )

The transfer operator preserves the subspace spanned by {1, cos(ξ ), cos(2ξ )}√and thus can be represented by a 3×3 matrix whose maximal eigenvalue is exp P = 1+ 1 + 16a. In the special case when a = −1/16 one obtains the filter corresponding to ψ2Daub , for which r (π ) = 0. One can easily show that this wavelet is the smoothest in this family. However, for the broader class M ≤ N − 2, this is no longer the case.

Wavelet Regularity via Thermodynamic Formalism

11

Example 2. We consider the two-parameter family of putative wavelets corresponding to filters   1 1 2 − a − b cos(5x). (4.2) |m(x)| = + a cos(x) + b cos(3x) + 2 2 We say putative, because all the conditions in Theorem 1.1 need to be verified for all parameter values. The Daubechies wavelet ψ3Daub corresponds to parameters a = 75/128 and b = −25/256. Here |m(x)|2 has a double zero at x = π and thus one can always take out (at least) a factor of (1 + cos(ξ )) /2. However, it is always necessary to check “by hand” that for each particular choice of a and b expression (4.2) is positive and the Cohen condition holds. It is useful to write (4.2) in the form r (ξ )(1 + cos(ξ ))/2, where r (ξ ) = 1 − 4 (−1 + 2 a + 4 b) cos(ξ ) + 4 (−1 + 2 a + 4 b) cos2 (ξ ) +16 (−1 + 2 a + 2 b) cos3 (ξ ) − 16 (−1 + 2 a + 2 b) cos4 (ξ ). Notice this trigonometric polynomial has the form r (ξ ) = 1 − α cos(ξ ) + α cos2 (ξ ) + β cos3 (ξ ) − β cos4 (ξ ). (A) The transfer matrix approach. We consider the finite dimensional space spanned by {1, cos(ξ ), cos2 (ξ ), cos3 (ξ ), cos4 (ξ )} and write the action of the transfer operator associated to weights of the general form r (ξ ) = 1 − α cos(ξ ) + α cos2 (ξ ) + β cos3 (ξ ) − β cos4 (ξ ). A routine calculation yields that the transfer operator on this space has the following matrix representation: ⎛ ⎞ 2 + α − β2 α−β − β2 0 0 ⎜ ⎟ β ⎜ −α + β −α + β 0 0 ⎟ 2 2 ⎜ ⎟ ⎜ ⎟ β 3β 3β β α α ⎜ L 2 log r = L 2 log r (α, β) = ⎜ 1 + 2 − 4 1 + α − 4 −4 0 ⎟ ⎟. 2 − 4 ⎜ ⎟ 3β β α ⎜ −α + β ⎟ −α + 3β − + 0 ⎝ 2 4 ⎠ 4 2 4 4 1 2

+

α 4

−

β 8

The matrix corresponding to ψ3Daub is

L 2 log r (−7/8, −3/8) =

1+ ⎛ 21 16 ⎜ 11 ⎜ ⎜ 16 ⎜ 21 ⎜ ⎜ 32 ⎜ ⎜ 11 ⎜ 32 ⎝ 21 64

3α 4

−

− 21 1 2 13 32 19 32 17 32

β 1 2 2

3 16 3 − 16 5 − 32 5 32 1 8

+

3α 4

0 0

−

0

3β α 4 4

−

β 2

− β8

⎞

⎟ 0⎟ ⎟ ⎟ 0⎟ ⎟, ⎟ 0⎟ ⎟ ⎠ 3

3 32 3 − 32 1 − 32 64

with eigenvalues 1, 9/16, ±1/4 and 3/64. Then det(I − z L 2 log r ) has a zero at z = 16/9. By a slightly modified version of Theorem 1.2 accounting for the fact that we have not

12

M. Pollicott, H. Weiss

factored out the maximal zero at ±π , we again recover the Sobolev regularity of ψ3Daub : s2 (ψ3Daub ) = 1 − (log(9/16)/(2 log 2) = 1.41504 . . . . By varying the parameters α and β, and always checking that expression (4.2) is positive and satisfies the Cohen condition, one can search for wavelets having maximal L p −Sobolev regularity, and, in particular, having greater regularity than the Daubechies wavelet. The two required conditions make this a difficult constrained optimization problem, which is even difficult numerically. In contrast with Example 1, there appears to be no simple algebraic expression for the eigenvalues for the matrix corresponding to L log r ; one must resort to empirical searches using different choices of α and β. However, it is more illuminating to implement this computation using determinants. (B) The determinant approach. If one computes the determinant det(1 − z L log |m|2 associated with the Daubechies wavelet obtains the power series expansion det(1 − z L log |m|2

Daub

ψ3Daub

Daub

)

using our determinant algorithm, one

) = 1 − 2 z + 1.31904 z 2 − 0.35310 z 3 + 0.03319 z 4 + 0.00120 z 5 − 0.000351 z 6 + · · · .

We observe that: (1) There are “trivial” zeros for det(1 − z L log |m|2 ) at the values z = 1, 2, 4, 8. Daub (2) There is a zero at z = 7.07337 . . . ., which by Proposition 2.3 corresponds to a zero of det(1 − z L log rDaub ) at 7.07337/26 = 0.111 . . . . This agrees with our previous calculation in Sect. 3. By Theorem 1.2, if we change the filter |m|2 to decrease the pressure, then the Sobolev regularity of the associated wavelet increases. One can easily check that if one replaces a = 75/128 by 75/128 − 0.00068, say, and b = −25/256 by −25/256 + 0.00002, say, then the associated filter |m(ξ )2 | remains positive, and by Theorems 1.2 and 2.1, the zero of the determinant increases from 7.1111 to 8.51989. This reflects an increase in the L 2 -Sobolev regularity from 1.415 to 1.545. Thus, this two-parameter family contains wavelets having the same support as ψ3Daub with greater L 2 -Sobolev regularity. The same idea can be applied for other N . Remarks. Other approaches. (a) Daubechies [Da1, p.242] considered the one-parameter family of filters |m(ξ )|2 = cos2(N −1) (ξ/2)(PN −1 (sin2 (ξ/2)) + a cos2N (ξ/2)) cos(ξ ), where a ∈ R. For definiteness, we discuss the case N = 3. For different values of the parameter a we calculate the first zero of the determinant, and thus the Sobolev regularity of the associated wavelet using Theorem 2.2. The following table shows that this zero is monotone increasing as a increases to 3. The zero of the determinant is 0.40344 . . . when a = 3, which corresponds to L 2 Sobolev regularity 1.34443 . . .. It is not necessary to study the case a > 3 since the expansion for |m|2 is no longer positive. To see this, we can expand in terms of ξ − π as |m(ξ )|2 = cos4 (ξ/2)(P2 (sin2 (ξ/2)) + a cos4 (ξ/2)) cos(ξ ) = cos4 (ξ/2 − π/2)(1 + 2 sin2 (ξ/2 − π/2) + a sin6 (ξ/2 − π/2) cos(ξ − π )) = sin4 (ξ/2)(1 + 2 cos2 (ξ/2) − a cos6 (ξ/2) cos(ξ ))

Wavelet Regularity via Thermodynamic Formalism

13

= 1 + 2(1 − (ξ/2)2 + · · · )2 − a(1 − (ξ/2)2 + · · · )6 (1 − ξ 2 + · · · ) = (3 − a) − (1 + 5a/2)ξ 2 + · · · . It is clear this expression is negative values for a > 3. We see that the parameter value a = 3 corresponding to the maximally regular wavelet lies on the boundary of an allowable parameter interval, and thus could not be detected using variational methods. a -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 2.7 2.8 2.9 3.0

Zero of determinant 0.230 0.239 0.250 0.262 0.276 0.294 0.317 0.350 0.367 0.377 0.389 0.403

Sobolev regularity 0.939 0.967 1.000 1.030 1.071 1.116 1.710 1.242 1.276 1.296 1.318 1.344

(b) Daubechies [Da3] also considered the one-parameter family of functions  m(ξ ) =

1 + eiξ 2

2 Q(ξ ),

where |Q(ξ )|2 = P(cos ξ ) and P(x) = 2 − x + a4 (1 − x)2 , with the parameter a chosen such that P(x) > 0 for 0 ≤ x < 1. The value a = 3 corresponds to ψ2Daub , and the associated polynomial has a zero at x = −1. She reparameterizes this wavelet family using the zero at x = −1+δ, and obtains, numerically, a wavelet having the same support as ψ2Daub but with higher regularity. 5. Perturbation Theory and Variational Formulae In this section we observe that the analyticity of pressure implies that the L p Sobolev regularity depends smoothly on the wavelet. It thus seems natural to search for critical points (maximally smooth wavelets) in parametrized families of wavelets by explicitly calculating the derivative of the Sobolev regularity functional. We caution that the example in the previous section shows that maximally regular wavelets can lie on the boundary of allowable parameter intervals. We now restrict |m(ξ )|2 to the subset of analytic functions B satisfying conditions (1) − (3) and let r0 be a tangent vector to this space. It follows from remarks in Sect. 1 that r0 is represented by an analytic function of the form ∞

1
r0 = + d2n+1 cos((2n + 1)ξ ), 2 n= 0

We have the following result.

such that

∞
n= 0

dn =

1 . 2

14

M. Pollicott, H. Weiss

Proposition 5.1. (1) For every p ≥ 1, the regularity mapping K → R defined by |m(ξ )|2 → s p (φ) is analytic; (2) Let |m λ (ξ )|2 = cos2M (ξ/2) (r0 (ξ ) + λ r0 (ξ ) + · · · ) be a smooth parametrized family of filters. Then the change in the Sobolev regularity is given by   1 r0 λ 0 s p (φ ) = s p (φ ) + dµ λ + · · · , p log 2 r0 where µ is the unique equilibrium state for p log r0 . Proof. The analyticity of the Sobolev regularity follows from Proposition 2.1(2), the λ 0 analyticity of the pressure, and the implicit function  theorem. Since log r = log(r + r0 0 r0 + · · · ) = log r + r0 λ + · · · and Dg P(·) = ·dµg , this follows from the chain rule.  If we restrict to finite dimensional spaces we can consider formulae for the second (and higher) derivatives of the pressure, which translate into formulae for the next term in the expansion of s p (φλ ). Also observe that we can estimate the derivatives of the Sobolev exponent of a family s p (φλ ) using the series for d p (φλ , z). More precisely, by the implicit function theorem ds p (φ λ ) d(det(φ λ , z)) = dλ dλ

d(det(φ λ , z)) , dz

which is possible to estimate for periodic orbit data. We can similarly compute the higher derivatives ! d k s p (φ λ ) !! , for k ≥ 1 ! dλk ! λ = λ0

using Mathematica, say. We can then expand the power series for the Sobolev function: ! ! ds p (φ λ ) !! (λ − λ0 )2 d 2 s p (φ λ ) !! λ λ0 s p (φ ) = s p (φ ) + (λ − λ0 ) + + ··· . dλ !λ = λ0 2 dλ2 !λ = λ0 6. Asymptotic Sobolev Regularity The use of the variational principle to characterize pressure (and thus the Sobolev regularity) leads one naturally to the idea of maximizing measures. More precisely, one can associate to g:  2π

Q(g) = sup g dµ, µ an E 2 -invariant Borel probability measure . 0

Unlike pressure, which has analytic dependence on the function, the quantity Q(g) is less regular, but we have the following trivial bound.

Wavelet Regularity via Thermodynamic Formalism

15

Lemma 6.1. For any continuous function g : [0, 2π ) → R, Q(g) ≤ P(g) ≤ Q(g) + log 2. Proof. Since 0 ≤ h(µ) ≤ log 2, this follows from the definitions.



Proposition 6.2. Let φn , n ∈ N be a family of wavelets with filters |m(ξ )|2 = cos2Mn (ξ/2)rn (ξ ), where Mn ≥ 0. Then the asymptotic L p -Sobolev regularity of the family is given by s p (φn ) Mn Q(log rn ) = lim − lim . n→∞ n→∞ n n→∞ n log 2 n lim

Proof. Using Theorem 1.2, Lemma 6.1, and (1.1), one can easily show Q( p log rn ) ≤ p log 2(Mn − s p (φn )) ≤ Q( p log rn ) + log 2. The proposition immediately follows after dividing by n and taking the limit as n → ∞.  We now apply Proposition 6.2 to compute the L p -Sobolev regularity of the Daubechies family of wavelets for arbitrary p ≥ 2. Cohen and Conze [CC1] first proved this result for p = 2. Proposition 6.3. Let p ≥ 2. The asymptotic L p -Sobolev regularity of the Daubechies family is given by s p (ψ NDaub ) log 3 =1− . N →∞ N 2 log 2 lim

Proof. If one applies Proposition 6.2 to the Daubechies filter coefficients (see Sect. I), one immediately obtains " # Q(log PN (sin2 ( 2y ))) s p (φ Daub ) N lim = 1 − lim . N →∞ N →∞ N N log 2 Cohen and Conze [CC1, p. 362] show that y log PN (3/4) , Q log2 PN (sin2 ( )) = 2 2 log 2 and the sup is attained for the period two periodic orbit {1/3, 2/3}. Thus s p (φ Daub ) 1 log PN (3/4) N =1− lim . N →∞ N 2 log 2 N →∞ N  −1 3 N −1 The exponentially dominant term in PN (3/4) is easily seen to be 2N , and N −1 ( 4 ) a routine application of Stirling’s formula yields lim

log PN (3/4) = log 3. N →∞ N lim

The proposition immediately follows.



16

M. Pollicott, H. Weiss

Remark. As briefly discussed in Sect. 3.2, the Butterworth filter family ψ NButt , N ∈ N is the family of non-compactly supported wavelets defined by the filter |m(ξ )|2 =

cos2N (ξ/2) sin2N (ξ/2) + cos2N (ξ/2)

.

Fan and Sun [FS1] computed the asymptotic regularity for this wavelet family; they showed that s p (ψ NButt ) log 3 = . N →+∞ N log 2 lim

This asymptotic linear growth of regularity is similar to that for the Daubechies wavelet family. 7. Regularity of Wavelets in Higher Dimensions The dilation equation (0.1) has a natural generalization to d-dimensions of the form:
φ(x) = det(D) ck φ(Dx − k), (7.1) k∈Zd

where D ∈ G L(d, Z) is a d × d matrix with all eigenvalues having modulus strictly greater than 1. We are interested in solutions φ ∈ L 2 (Rd ). We consider the expanding map T : [0, 2π )d → [0, 2π )d by T (x) = Dx (mod 1). By taking d-dimensional Fourier transformations, (7.1) gives rise to the equation
(ξ ) = m(D −1 ξ )φ (D −1 ξ ), where m(ξ ) = φ ck e−i k,ξ . k∈Zd

 −n (ξ ) = ∞ Thus by iterating this identity we can write φ n=1 m(D ξ ). β d For g ∈ C ([0, 2π ) ), one can define a transfer operator L g : C β ([0, 2π )d ) → β C ([0, 2π )d ) by


L g k(x) = exp g(y) k(y), T y=x

where the sum is over the det(D) preimages of x. We consider the Banach space of periodic analytic functions A in a uniform neighbourhood of the torus [0, π )d and the associated Bergman space of functions. We have that for g ∈ A the operator L g : A → A and its iterates L kg are trace class operators for each k ≥ 1. We can compute tr(L kg ) =


|det(D)|n exp(Sn g(x)) , |det(D n − I )| n

T x =x

where Sn g(x) = g(x) + g(T x) + · · · + g(T n−1 x).

Wavelet Regularity via Thermodynamic Formalism

17

M a(Dx) Assume that we can write m(x) = |det(D)|a(x) r (x), where r (x) > 0.8 The Ruelle-Fredholm determinant of L p log r takes the form: ⎛ ⎞ ∞ n
n exp( pS log r (x))
z |det(D)| n ⎠. det(I − L p log r ) = exp ⎝− n n |det(D n − I )| n=1

T x =x

The statement of Theorem 1.2 has a partial generalization to d-dimensions. We define the L p -Sobolev regularity of p to be (ξ )| p (1 + ||ξ || p )s ∈ Ł1 (Rd )}. s p (φ) := sup{s > 0 : |φ Proposition 7.1. The L p -Sobolev regularity satisfies the inequality s p (φ) ≥

P( p log r ) Mdet(D) − , log λ p log λ

where λ > 1 is smallest modulus of an eigenvalue of D. n (ξ )| ≤ C|det(D)| Mn i=0 Proof. We can write |φ r (D −i ξ ), for some C > 0 and all   n n d ξ ∈ D ([0, 2π ) ). However, we can identify T n D i=0 r (D −i ξ ) p dξ = D L np log r 1(ξ )dξ . Moreover, since L p log r has maximal eigenvalue e P( p log r ) , we can bound this by C  en P( p log r ) , for some C  > 0. In particular, ∞
p p s psn  (ξ )| p dξ |φ (ξ )| (1 + ||ξ || ) dξ ≤ λ |φ Rd

n=1

≤ CC 

T n D−T n−1 D

∞


λ psn |det(D)| M pn en P( p log r ) .

n=1

In particular, the right-hand side converges if λ ps |det(D)| M p e P( p log r ) < 1, from which the result follows.  In certain cases, for example if D has eigenvalues of the same modulus and some other technical assumptions, then this inequality becomes an equality [CD1]. As in Sect. 2, a routine calculation gives that d p (z) := det(I − z L p log r ) = 1 + bn =

n
(−1)k k=0

k!




n=1 p p an 1 · · · an k

n 1 +···+n k =n

n1 · · · nk

and p an

=


|det(D)|n T n x=x

∞




n

T n x=x |det(D n −

bn z n , ,

i=0 r (x)

I )|

where

p .

The simplest case is that of diagonal matrices, but this reduces easily to tensor products (x1 , . . . , xd ) = φ1 (x1 ) · · · φd (xd ) of the one dimensional case. The next easiest example illustrates the “non-separable” case: 8 In particular, if T n x = x we have that n−1 |m(T i x)|2 = 2−n n−1 g(T i x). i =0 i =0

18

M. Pollicott, H. Weiss

Example. For d = 2 we let D =

1

1 1 −1

√  , which has eigenvalues λ = ± 2.

In the d-dimensional setting we have the following version of Theorem 2.2: Theorem 7.2. Assume that log r ∈ A. Then (1) The function d p (z) can be extended to an entire function of z and a real analytic function of p; (2) The exponential of pressure z p := exp(−P( p log r )) is the smallest zero for d p (z); (3) The Taylor coefficients bn of d p (z) decay to zero at a super-exponential rate, i.e., 1+1/d there exists 0 < A <1 such that bn = O(An ); N (4) Let d p,N (z) = 1 + n=1 bn z n be the truncation of the Taylor series to N terms and consider smallest zeros d p,N (z Np ) = 0. Then z Np → z p at a super-exponential rate, i.e., there exists 0 < B < 1 such that |z p − z Np | = O(B N

1+1/d

).

Furthermore, we can choose A and B arbitrarily close to λ−1 . The main difference for d ≥ 2 is that the exponent N 2 is replaced by N 1+1/d in parts 3 and 4. 8. Generalizations to Non Real Analytic Filters m We now consider the more general setting where the filter m is not√ analytic,  and we necessarily obtain weaker results. Recall that the filter m(ξ ) = (1/ 2) k ck e−ikξ , and assume there exists β > 0 such that |cn | = O(|n|−β ). In this case we can only expect that the determinant algorithm for the Sobolevregularity converges with exponential n speed. Recall d p (z) = det(I − z L plogr ) = 1 + ∞ n=1 bn z . Theorem 8.1.

(1) Let 0 < β < 1. If |cn | = O |n|1β , then bn = O(2−nβ(1−) ) for any  > 0;

(2) Let k ≥ 1. If |cn | = O |n|1 k , then bn = O(2−nk(1−) ) for any  > 0. This follows from a standard analysis of transfer operators, and in particular, studying their essential spectral radii. The radius of convergence of d p (z) is the reciprocal of the essential spectral radius of the operators. In the case that 0 < β < 1, this theorem follows from the Hölder theory of transfer operators [PP1]. But for k > 1 this requires a slightly different analysis [Ta1]. The necessary results are summerized in the following proposition. Proposition 8.2. (1) Assume cn = O(|n|−β ). The transfer operator L f has essential spectral radius (1/2)α e P( f ) (and the function d(z) is analytic in a disk |z| < 2β ). (2) Assume cn = O(|n|−k ). The transfer operator L f has essential spectral radius (1/2)k e P( f ) (and the function d(z) is analytic in a disk |z| < 2k ). The first part is proved in [Po1,PP2]. The second part is proved in [Ta1,Ru2]9 . As in Sect. 2, the speed of decay of the coefficients bn immediately translate into the corresponding speed of convergence for the main algorithm. 9 At least when β ∈ N.

Wavelet Regularity via Thermodynamic Formalism

19

Appendix I: Proof of Theorem 2.1 Given a bounded linear operator L : H → H on a Hilbert space H , its i th approximation number (or singular value) 10 si (L) is defined as si (L) = inf{||L − K || : rank(K ) ≤ i − 1}, where K is a bounded linear operator on H . Let r ⊂ C denote the open disk of radius r centered at the origin in the complex plane. The phase space [0, 2π ) for T = E 2 is contained in 2π + for any  > 0, and the two inverse branches T1 (x) = 21 x and T2 (x) = 21 x + 21 have analytic extensions to 2π + satisfying T1 ( 2π + ) ∪ T2 ( 2π + ) ⊂ 2π + 2 . Thus T1 and T2 are strict contractions of 2π + onto 2π + 2 with contraction ratio θ = (2π + )/(2π + 2) < 1. We can choose  > 0 arbitrarily large (and thus θ arbitrarily close to 1/2). Let A2 ( r ) denote  the Bergman Hilbert space of analytic functions on r with inner product f, g := f (z) g(z) d x d y [Ha1]. Lemma A.1. The approximation numbers of the transfer operator L p log r : A2 ( 2π ) → A2 ( 2π ) satisfy s j (L p log r ) ≤

||L p log r || A2 ( 2π ) j θ, 1−θ

for all j ≥ 1. This implies that the operator L p log r is of trace class.  k Proof. Let g ∈ A2 ( 2π + ) and write L p log r g = ∞ k (z) = z . k=−∞ lk (g) pk , where p$ $ π π We can easily check that || pk || A2 ( 2π ) = k+1 (2π )k+1 and || pk || A2 ( 2π + ) = k+1 (2π + )k+1 . The functions { pk }k∈Z form a complete orthogonal family for A2 ( 2π + ), and so L p log r g, pk A2 ( 2π + ) = lk (g)|| pk ||2A2 ( 2π + ) . The Cauchy-Schwarz inequality implies that |lk (g)| ≤ ||L p log r g|| A2 ( 2π + ) || pk ||−1 A2 ( 2π + ) .  j−1

( j)

We define the rank- j projection operator by L p log r (g) = g ∈ A2 ( θ ) we can estimate

k = 0 lk (g) pk .

∞


( j) || L p log r − L p log r (g)|| A2 ( 2π ) ≤ ||L p log r g|| A2 ( 2π ) θ k+1 . k= j

It follows that ( j)

||L p log r − L p log r || A2 ( 2π ) ≤

||L p log r || A2 ( 2π ) j+1 θ , 1−θ

and so s j+1 (L p log r ) ≤ and the result follows.

||L p log r || A2 ( 2π ) j+1 θ , 1−θ



10 We use these terms interchangeably because one can show that s (L) = k

% λk (L ∗ L).

For any

20

M. Pollicott, H. Weiss

We now show that the coefficients of the power series of the determinant decay to zero with super-exponential speed. Lemma A.2. If one writes det(I − z L p log r ) = 1 +  |cm | ≤ B where B =

∞

m=1 (1 − θ

m )−1

||L p log r || A2 ( 2π ) 1−θ

∞

m=1 cm z

m,

then

m θ m(m+1)/2 ,

< ∞.

Proof. By [Si1, Lemma 3.3], the coefficients cn in the power series expansion of the  determinant have the form cm = i 1 <···
|cm | = ≤ = ≤

where B =

∞

! ! !
!
! ! ! ! λ · · · λ si1 (L p log r ) · · · sim (L p log r ) i i m! ≤ 1 ! !i1 <···
m=1 (1 − θ

m )−1 .



Theorem 2.1 follows from Lemma 2.2. The coefficients of det(I − zLs ) = 1 + ∞ n n=1 bn z are given by Cauchy’s Theorem: |bn | ≤

1 sup |det(I − zLs )|, r n |z|= r

for any r > 0.

Using Weyl’s inequality, we can deduce that if |z| = r , then |det(I − zLs )| ≤

∞ 

(1 + |z|λ j ) ≤

"

j=1

≤ 1+ B

∞


∞ 

(1 + |z|s j )

j=1

(r α) θ m

m(m+1) 2

# ,

m=1

where α = ||Ls ||A2 ( 1 ) . If we choose r = r (n) appropriately we obtain the bounds given in the theorem. 

Wavelet Regularity via Thermodynamic Formalism

21

References [CC1] [CD1] [Da1] [Da2] [Da3] [Ei1] [FS1] [GGK1] [Ha1] [He1] [He2] [JP1] [JP2] [Oj1] [OS1] [PP1] [Po1] [Ru1] [Ru2] [Si1] [Sun] [Ta1] [Vi1] [W]

Cohen, A., Conze, J.-P.: Régularité des bases d’ondelettes et mesures ergodiques. Rev. Mat. Iberoamericana 8, 351–365 (1992) Cohen, A., Daubechies, I.: A new technique to estimate the regularity of refinable functions. Rev. Mat. Iberoamericana 12, 527–591 (1996) Daubechies, I.: Ten lectures on wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 61, Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 1992 Daubechies, I.: Using Fredholm determinants to estimate the smoothness of refinable functions. Approximation theory VIII, Vol. 2 (College Station, TX, 1995), Ser. Approx. Decompos. 6 River Edge, NJ: World Sci. Publishing, 1995, pp. 89–112 Daubechies, I.: Orthonormal bases of compactly supported wavelets ii. SIAM J. Math. Anal. 24, 499–519 (1993) Eirola, T.: Sobolev characterization of solutions of dilation equations. SIAM J. Math. Anal. 23, 1015–103 (1992) Fan, A., Sun, Q.: Regularity of butterworth refinable functions. Asian J. Math. 5(3), 433– 440 (2001) Gohberg, I., Goldberg, S., Kaashoek, M.A.: Classes of linear operators. Vol 1, Basel: Birthauser Verlag, 1990 Halmos, P.: Introduction to Hilbert space and the theory of spectral multiplicity. Reprint of the Second (1957) edition, Providence, RI: AMS/Chelsea Publishing, 1998 Hervé, L.: Comportement asymptotique dans l’algorithme de transformée en ondelettes. Rev. Mat. Iberoamericana 11, 431–451 (1985) Hervé, L.: Construction et régularité des fonctions d’´chelle. SIAM J. Math. Anal. 26, 1361–1385 (1995) Jenkinson, O., Pollicott, M.: Computing invariant densities and metric entropy. Commun. Math. Phys. 211, 687–703 (2000) Jenkinson, O., Pollicott, M.: Orthonormal expansions of invariant densities for expanding maps. Adv. in Math. 192, 1–34 (2005) Ojanen, H.: Orthonormal compactly supported wavelets with optimal sobolev regularity. Applied and Computational Harmonic Analysis 10, 93–98 (2001) Oppenheim, A., Shaffer, R.: Digital Signal Processing. Upper Saddie River, NJ: Pearson Higher Education, 1986 Parry, W., Pollicott, M.: Zeta Functions and the Periodic Orbit Structures of Hyperbolic Dynamics. Astérisque 187–188 (1990) Pollicott, M.: Meromorphic extensions for generalized zeta functions. Invent. Math 85, 147– 164 (1986) Ruelle, D.: Thermodynamic Formalism. Reading, MA: Addison-Wesley, 1978 Ruelle, D.: An extension of the theory of fredholm determinants. Inst. Hautes Etudes Sci. Publ. Math. 72, 175–193 (1990) Simon, B.: Trace ideals and their applications. LMS Lecture Note Series, Vol 35, Cambridge: Cambridge University Press, 1979 Sun, Q.: Sobolev exponent estimate and asymptotic regularity of M band Daubechies scaling function. Constructive Approximation 15, 441–465 (1999) Tangerman, F.: Meromorphic continuation of Ruelle zeta functions. Ph.D. Thesis, Boston University, 1988 Villemoes, L.: Energy moments in time and frequency for two-scale difference equation solutions and wavelets. SIAM J. Math. Anal. 23, 1519–1543 (1992) Matlab, Wavelet Toolbox Documentation. www.mathworks.com/access/helpdesk/help/tool-box/ wavelet/

Communicated by J.L. Lebowitz

Commun. Math. Phys. 281, 23–127 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0472-y

Communications in

Mathematical Physics

Noncommutative Finite Dimensional Manifolds II: Moduli Space and Structure of Noncommutative 3-Spheres Alain Connes1,2,3 , Michel Dubois-Violette4 1 2 3 4

Collège de France, 3, rue d’Ulm, Paris, F-75005 France. E-mail: [email protected] I.H.E.S., 35, route deChartres, F-91440 Bures-sur-Yuette, France Mathematics Department, Vanderbitt University, Nashvitte, TN 37235, USA Laboratoire de Physique Théorique, UMR 8627, Université Paris XI, Bâtiment 210, F-91 405 Orsay Cedex, France. E-mail: [email protected]

Received: 31 July 2006 / Accepted: 20 November 2007 Published online: 6 May 2008 – © Springer-Verlag 2008

Abstract: This paper contains detailed proofs of our results on the moduli space and the structure of noncommutative 3-spheres. We develop the notion of central quadratic form for quadratic algebras, and a general theory which creates a bridge between noncommutative differential geometry and its purely algebraic counterpart. It allows to construct a morphism from an involutive quadratic algebra to a C*-algebra constructed from the characteristic variety and the hermitian line bundle associated to the central quadratic form. We apply the general theory in the case of noncommutative 3-spheres and show that the above morphism corresponds to a natural ramified covering by a noncommutative 3-dimensional nilmanifold. We then compute the Jacobian of the ramified covering and obtain the answer as the product of a period (of an elliptic integral) by a rational function. We describe the real and complex moduli spaces of noncommutative 3-spheres, relate the real one to root systems and the complex one to the orbits of a birational cubic automorphism of three dimensional projective space. We classify the algebras and establish duality relations between them. Contents 1. 2.

3.

Introduction . . . . . . . . . . . . . . . . . . . . The Noncommutative 3-Spheres S 3 (
) ⊂ R4 (
) 2.1 Unitary “up to scale” . . . . . . . . . . . . . 2.2 Equation ch1/2 (U ) = 0 . . . . . . . . . . . 2.3 Noncommutative 3-spheres and 4-planes . . The Real Moduli Space M . . . . . . . . . . . . 3.1 M in terms of A3 . . . . . . . . . . . . . . 3.2 Trigonometric parameters ϕ of Sϕ3 . . . . . . 3.3 M in terms of D3 . . . . . . . . . . . . . . 3.4 Roots  = R(G, T) . . . . . . . . . . . . . 3.5 Singular hyperplanes Hα,n . . . . . . . . . .

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. . . . . . . . . . .

25 27 27 27 29 31 32 34 35 37 37

24

A. Connes, M. Dubois-Violette

3.6 Kernel of the exponential map: (T) (nodal group of T) . . . . . . . . 37 3.7 Group of nodal vectors N (G, T) ⊂ (T) . . . . . . . . . . . . . . . . 37 3.8 Affine Weyl group Wa . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.9 Affine Weyl group Wa . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.10 Alcoves and fundamental domain . . . . . . . . . . . . . . . . . . . . 38 4. The Flow F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.1 Compatibility of F with the triangulation by alcoves . . . . . . . . . . 41 4.2 Invariance of R4ϕ under the flow F . . . . . . . . . . . . . . . . . . . . 42 5. The Geometric Data of R4ϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.1 The definition and explicit matrices . . . . . . . . . . . . . . . . . . . 47 5.2 The table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.3 Generic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.4 Face α = n and n even. F1 = {(ϕ
1 , ϕ1 , ϕ3 )} . . . . . . . . . . . . . . 51 5.5 Face α = n and n odd. F2 = { π2 , ϕ2 , ϕ3 } . . . . . . . . . . . . . . . 51
 5.6 Edge α ⊥ β and (n 1 , n 2 ) = (even, odd). L = { π2 , ϕ, ϕ } . . . . . . . 52
 5.7 Edge α − β and (n 1 , n 2 ) = (odd, odd) or (even, odd). L  = { π2 , π2 , ϕ } 53 5.8 Edge α − β and (n 1 , n 2 ) = (even, even). L  = {(ϕ, ϕ, ϕ)} . . . . . . . 53 5.9 Edge α ⊥ β and (n 1 , n 2 ) = (even, even). C+ =
{(ϕ, ϕ, 0)} . . . . . . . 53 5.10 Edge α ⊥ β and (n 1 , n 2 ) = (odd, odd). C− = { π2 + ϕ, π2 , ϕ } . . . . 54 5.11 Vertex P = ( π2 , π2 , π2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.12 Vertex P  = ( π2 , π2 , 0) . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.13 Vertex O = (0, 0, 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6. Isomorphism Classes of R4ϕ and Orbits of the Flow F . . . . . . . . . . . . 55 6.1 Proof in the non-generic case . . . . . . . . . . . . . . . . . . . . . . 55 6.2 Basic notations for elliptic curves . . . . . . . . . . . . . . . . . . . . 57 6.3 The generic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 7. Dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 7.1 Semi-cross product. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 7.2 Application to R4ϕ and Sϕ3 . . . . . . . . . . . . . . . . . . . . . . . . 65 8. The Algebras Calg (R4ϕ ) in the Nongeneric Cases . . . . . . . . . . . . . . . 67 8.1 Uq (sl(2)), Uq (su(2)) and their homogenized versions . . . . . . . . . 67 8.2 The algebras in the nongeneric cases . . . . . . . . . . . . . . . . . . 69 9. The Complex Moduli Space and its Net of Elliptic Curves . . . . . . . . . . 71 9.1 Complex moduli space MC . . . . . . . . . . . . . . . . . . . . . . . 73 9.2 Notations for ϑ-functions . . . . . . . . . . . . . . . . . . . . . . . . 75 9.3 Fiber = characteristic variety, and the birational automorphism σ of P3 (C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 10. The Map from T2η × [0, τ ] to Sϕ3 and the Pairing . . . . . . . . . . . . . . . 82 10.1 Central elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 10.2 The Hochschild cycle ch3/2 (U ) . . . . . . . . . . . . . . . . . . . . . 84 10.3 Elliptic parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 10.4 The sphere Sϕ3 and the noncommutative torus T2η . . . . . . . . . . . . 88 10.5 Pairing with [T2η ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 10.6 Simplifying the ∗-homomorphism ρ˜ . . . . . . . . . . . . . . . . . . . 93 11. Algebraic Geometry and C ∗ -Algebras . . . . . . . . . . . . . . . . . . . . . 96 11.1 Central quadratic forms and generalised cross-products . . . . . . . . . 96 11.2 Positive central quadratic forms on quadratic ∗-algebras . . . . . . . . 99 12. The Jacobian of the Covering of Sϕ3 . . . . . . . . . . . . . . . . . . . . . . 107

Noncommutative 3-Spheres

13. Calculus and Cyclic Cohomology . . . . 14. Appendix 1: The List of Minors . . . . . 15. Appendix 2: The Sixteen Theta Relations References . . . . . . . . . . . . . . . . . . .

25

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113 122 124 126

1. Introduction This paper contains detailed proofs of our results on the moduli space and the structure of noncommutative 3-spheres announced in [17]. Through the analysis of a specific class of noncommutative manifolds which arose as solutions of a simple equation of K -theoretic origin we discovered rather general structures which lie at the intersection of two fundamental aspects of noncommutative geometry, namely • Differential Geometry • Algebraic Geometry. This class of noncommutative manifolds, called noncommutative 3-spheres, has a very rich structure both at the level of the objects themselves as well as at the level of the moduli space which parameterizes these geometric objects. There are two aspects in the geometry of the moduli space: • The real moduli space and its scaling foliation, its link with the alcove structure of the root system D3 and the Morse theory of the character of the signature representation. • The complex moduli space and its net of elliptic curves, its link with the iteration of a cubic transformation of P3 (C). At the level of the structure of the noncommutative 3-spheres our main result is to relate them to very well understood noncommutative nilmanifolds which fall under the framework of the early theory developed in [10] and were analysed in great detail in [1,2]. The core of the paper is to construct the corresponding map of noncommutative spaces and compute its Jacobian. The essence of the work is to extract from very complicated computations the general concepts that allow not only to understand what is going on but also to extend the construction in full generality. Thus at center stage lies the computation of the Jacobian and the gradual simplification of the result which at first was expressed in terms of elliptic functions and the 9th power of the Dedekind η function. We shall reach at the end of the paper (Theorem 13.8) a result of utmost simplicity while the starting point was a computation performed even in the trigonometric case with the help of a computer.1 While we reach a reasonable level of conceptual understanding of the general construction of the map, we believe that a lot remains to be discovered for the abstract construction of the calculus as well as for the general computation of the Jacobian, let alone in the case of higher dimensional spheres which we do not address here. After recalling the basic definitions and properties of the noncommutative 3-spheres in Sect. 2 we analyse the real moduli space in Sect. 3 and exhibit a fundamental domain in terms of alcoves of the root system D3 . In Sect. 4 we define the scaling foliation and show its compatibility with the alcove structure of the real moduli space. We also show that the isomorphism class of the 1 We are grateful to Michael Trott for his help.

26

A. Connes, M. Dubois-Violette

4-spaces R4 (
) remains constant on the leaves of the foliation. In order to prove the converse, i.e. that isomorphism of the 4-spaces R4 (
) implies equality of the leaves we compute in detail in Sect. 5 the geometric data of the quadratic algebra of R4 (
). This allows us to finish the proof of the converse in Sect. 6. We exhibit in Sect. 7 more subtle relations between the 4-spaces R4 (
) given by dualities. At the level of the algebras these are obtained from the general notion of semicross product of graded algebras, a notion used and analysed in [5] where it is referred to as twisting. At the level of the moduli space these dualities shrink the fundamental domain further and that amounts essentially to the transition from the root system D3 to the larger one C3 . We then use these to describe the 4-spaces R4 (
) for degenerate values of the parameter in Sect. 8. In Sect. 9 we analyse the complex moduli space which appears naturally as a net of elliptic curves having eight points in common in P3 (C). We show that in the generic case these elliptic curves are the characteristic varieties of the algebras that their points label. Moreover the canonical correspondence σ is simply the restriction of a globally defined cubic map of P3 (C). This gives in particular a very natural choice of generators for the algebra. As a preparation for the next section we give the natural parameterization of the net of elliptic curves in terms of ϑ-functions. Section 10 is the most technical one and contains the root of the concepts developed in full generality in Sect. 11. In essence what we do first is, starting from unitary representations of the Sklyanin algebra constructed by Sklyanin in his original paper, we derive a one parameter family of ∗-homorphisms from the algebras of R4 (
) in the generic case, to the algebras of noncommutative tori. We then use a suitable restriction to the 3-spheres S 3 (
). After a lot of work on this construction we find that we can eliminate all occurrences of ϑ-functions from the formulas and obtain a purely algebraic formulation of the construction as a morphism to a twisted cross product C ∗ -algebra obtained from the geometric data. Section 11 describes the abstract general construction in the framework of involutive quadratic algebras. The key notions are those of central quadratic form and of positivity for such forms. It is in that section that the interaction between the two above aspects of noncommutative geometry is manifest. In fact we construct a bridge between the purely algebraic notions such as the geometric data of a quadratic algebra and the world of noncommutative geometry including the topological (C ∗ -algebraic) and differential geometric aspects (in the sense of [10,11]). At one end of the bridge one starts with the given involutive quadratic algebra. At the other one has the C ∗ -algebra obtained as the twisted cross product of the charateristic variety by the canonical correspondence. The twisting is effected by an hermitian line bundle and the construction is a special case of a general one due to M. Pimsner. The bridge provides a ∗-algebra morphism. This algebra morphism has a “trivial part” which does not make use of the central quadratic form and lands in a “triangular” subalgebra of the C ∗ -algebra. This part was well-known for quite some time to noncommutative algebraic geometors. The non-triviality of our construction lies in the involved relations coming from the cross terms mixing generators with their adjoints. We compute in Sect. 12 the Jacobian of the above map. We first define what we mean by the jacobian in the sense of noncommutative geometry where Hochschild homology replaces differential forms. The algebraic form of the result then suggests the existence of a calculus of purely algebraic nature allowing to express the cyclic cohomology fundamental class in terms of the algebraic geometry of the characteristic variety and a hermitian structure on the canonical line bundle. This is achieved in Sect. 13 and allows

Noncommutative 3-Spheres

27

us to finally obtain the purest form of the computation of the jacobian in the already mentioned Theorem 13.8. Needless to say this is a paper of highly “computational” nature and we tried to ease the reading by supplying in an appendix the basic factorisations of the minors and the sixteen theta relations which are often used in the text. 2. The Noncommutative 3-Spheres S3 (
) ⊂ R4 (
) We shall recall in this section the basic properties of the noncommutative spheres S 3 (
) and the corresponding 4-spaces R4 (
). 2.1. Unitary “up to scale”. We let A be a unital involutive algebra and first start with a unitary “up to scale” in Mq (A), i.e. U ∈ Mq (A), UU ∗ = U ∗ U ∈ A ⊗ ? ⊂ A ⊗ Mq (C).

(2.1)

Lemma 2.1. Let U ∈ Mq (A) satisfy (2.1) with UU ∗ = U ∗ U = C ⊗ ? ∈ A ⊗ Mq (C), then C is in the center of the ∗-algebra generated by the matrix elements of U . Proof. One has (C ⊗ ?)U = (UU ∗ )U = U (U ∗ U ) = U (C ⊗ ?) and (C ⊗ ?)U ∗ = (U ∗ U )U ∗ = U ∗ (UU ∗ ) = U ∗ (C ⊗ ?) by associativity in Mq (A), which implies the result.   Let

  τµ , µ ∈ 0, . . . , q 2 − 1

(2.2)

be an orthonormal basis of Mq (C) for the scalar product A|B = q1 Trace(A∗ B). Then one has U = τµ z µ , z λ ∈ A, (2.3) where we used the Einstein summation convention on “up-down indices”. The ∗-algebra generated by the matrix elements of U is the ∗-subalgebra of A generated by the z λ , z µ∗ (λ, µ ∈ {0, . . . , q 2 − 1}) and if U is as in the above lemma then one has the equalities   C= z µ z µ∗ = z µ∗ z µ (2.4) µ

µ

for the central element C of the ∗-algebra generated by the z µ . 2.2. Equation ch1/2 (U ) = 0. We now turn to the relation ch1/2 (U ) = 0, in the unreduced complex i.e. using the convention of summation on repeated indices, Uii10 ⊗ (U ∗ )ii10 − (U ∗ )ii01 ⊗ Uii01 = 0,

(2.5)

where the left-hand side belongs to the tensor square A⊗2 = A ⊗ A. Both terms in the left-hand side give sums of the q 2 terms of the form (z µ ⊗ z ν∗ ) (τµ )ii01 (τν∗ )ii10 and similarly for the other. The sum on i 0 , i 1 is thus simply Trace (τµ τν∗ ), i.e. the Hilbert Schmidt inner product (τµ , τν ) = q τν |τµ . This is 0 unless µ = ν and is q if µ = ν. Thus, up to an overall factor of q the equality (2.5) means:  (z µ ⊗ z µ∗ − z µ∗ ⊗ z µ ) = 0. (2.6)

28

A. Connes, M. Dubois-Violette

Lemma 2.2. Equation (2.6) holds iff there exists a unitary symmetric matrix
∈ Mq 2 (C) such that: ν z µ∗ =
µ ν z .

(2.7)

Proof. Let us first assume that the z µ are linearly independent elements of A. Let then ϕµ be linear forms on A with ϕµ (z ν ) = δµν . Applying 1 ⊗ ϕµ to (2.6) we get z µ∗ =



z ν ϕµ (z ν∗ ) =



ν
µ ν z ,

where the matrix
is uniquely prescribed by this relation. Then since the z µ ⊗ z ν are linearly independent in A ⊗ A, the relation (2.6) means, looking at the coefficient of z µ ⊗ z ν on both sides, ν
µ ν =
µ ,

(2.8)

so that the matrix
is symmetric. µ Taking the adjoint of both sides in z µ∗ =
ν z ν one gets ν∗ ν ρ ∗ µ ρ ¯µ ¯µ zµ =
=
ν z ν
ρ z = (

)ρ z

and the linear independence of the z ρ thus shows that:
∗
= 1.

(2.9)

For the general case2 note that Eq. (2.6) is invariant by the transformation z˜ µ = Wνµ z ν , where W ∈ U (q 2 ) is a unitary matrix. Moreover (2.7) implies a similar equation for (˜z µ ) with the matrix ˜ = W¯
W¯ t ,
which is still symmetric and unitary. This allows to assume that the kernel of the linear 2 map from Cq to A determined by eν → z ν is the linear span of a subset I of the basis ν vectors e . In other words the non-zero z ν are linearly independent and the above proof ensures the existence of a matrix fulfilling (2.7) which is symmetric and unitary once extended by the identity on the eν , ν ∈ I .   As pointed out in [15] and as will be explained in Part III, for the study of (2n + 1)dimensional spherical manifolds one should take q = 2n to be coherent in particular with the suspension functor. In the following we shall concentrate on the 3-dimensional case (and the corresponding noncommutative R4 ) which is the lowest dimensional nontrivial case from the noncommutative side and for which ch1/2 (U ) = 0 is the only K -homological condition. Accordingly we take q = 2 in the following. 2 The simplification of the argument of [15] given here is due to G. Skandalis.

Noncommutative 3-Spheres

29

2.3. Noncommutative 3-spheres and 4-planes. We now turn to the case q = 2 and we choose as orthonormal basis of M2 (C) the basis τ0 = ? and τk = iσk , k ∈ {1, 2, 3}, where the σk are the Pauli matrices, i.e.    0 1 0 , σ2 = σ1 = 1 0 i

  −i 1 , σ3 = 0 0

(2.10)

0 −1

 (2.11)

which satisfy σ j∗ = σ j , σ j2 = 1 and σk σ = i εkm σm for any permutation (k, , m) of (1, 2, 3), where ε123 = 1 and ε is totally antisymmetric. This allows to write U = z 0 + i σ1 z 1 + i σ2 z 2 + i σ3 z 3 ,

z µ ∈ A,

(2.12)

and one has U ∗ = z 0∗ − i σ1 z 1∗ − i σ2 z 2∗ − i σ3 z 3∗ . Thus UU ∗ = z 0 z 0∗ + z 1 z 1∗ + z 2 z 2∗ + z 3 z 3∗ + i σ1 (z 1 z 0∗ − z 0 z 1∗ + z 2 z 3∗ − z 3 z 2∗ ) + i σ2 (z 2 z 0∗ − z 0 z 2∗ + z 3 z 1∗ − z 1 z 3∗ ) + i σ3 (z 3 z 0∗ − z 0 z 3∗ + z 1 z 2∗ − z 2 z 1∗ ). Similarly we get, U ∗ U = z 0∗ z 0 + z 1∗ z 1 + z 2∗ z 2 + z 3∗ z 3 + i σ1 (z 0∗ z 1 − z 1∗ z 0 + z 2∗ z 3 − z 3∗ z 2 ) + i σ2 (z 0∗ z 2 − z 2∗ z 0 + z 3∗ z 1 − z 1∗ z 3 ) + i σ3 (z 0∗ z 3 − z 3∗ z 0 + z 1∗ z 2 − z 2∗ z 1 ). Thus Eq. (2.1) is equivalent to 7 relations which are,   z µ∗ z µ , z µ z µ∗ =  εkm z  z m∗ = 0, z k z 0∗ − z 0 z k∗ +  εkm z ∗ z m = 0. z 0∗ z k − z k∗ z 0 +

(2.13) (2.14) (2.15)

We then let S be the space of unitary symmetric matrices, S = {
∈ M4 (C) ;

∗ =
∗
= 1,
t =
}.

(2.16)

We define for
∈ S the algebra Calg (R4 (
)) of coordinates on R4 (
) as the algebra generated by the z µ , z µ∗ with the relations (2.14), (2.15) together with, ν z µ∗ =
µ ν z .

(2.17)

Note that (2.13) follows automatically from (2.17). For S 3 (
) we add the inhomogeneous relation,  z µ∗ z µ = 1. (2.18)  µ∗ µ By Lemma 2.1 the element C = z z is in the center of the involutive algebra Calg (R4 (
)).

30

A. Connes, M. Dubois-Violette

Theorem 2.3. Let A be a unital involutive algebra and U ∈ M2 (A) a unitary such that ch1/2 (U ) = 0. Then there exists
∈ S and a homomorphism ϕ : Calg (S 3 (
)) → A such that: U = ϕ(τµ z µ ). Conversely for any
the unitary U = τµ z µ ∈ M2 (Calg (S 3 (
))) fulfills ch1/2 (U ) = 0. By construction we thus obtain involutive algebras parametrized by
∈ S. They are endowed with a canonical element of H3 (A, A) (in fact of Z 3 ) given by

S 3 (
) = ch3/2 (U ) = Uii10 ⊗Ui∗i2 1 ⊗Uii32 ⊗Ui∗i0 3 −Ui∗i1 0 ⊗Uii21 ⊗Ui∗i3 2 ⊗Uii03. (2.19) The operations U → λ U , U → V1 U V2 , U → U ∗ , for λ ∈ U (1) and V j ∈ SU (2) together with the universality described in Theorem 2.3 give natural isomorphisms between the S 3 (
) as follows, Proposition 2.4. The following define isomorphisms S 3 (
) → S 3 (
 ) (resp. R4 (
) → R4 (
 )) preserving [S 3 ] (resp. [R4 ]) in the first two cases and changing its sign in the last case: 1) For λ ∈ U (1),
 = λ2
, one lets µ

µ

ϕ(z
 ) = λ−1 z
. 2) For V ∈ S O(4),
 = V
V t , one lets µ

ν . ϕ(z
 ) = Vνµ z


3) For
 = ε
−1 ε, one lets µ

µ∗

ϕ(z
 ) = εµ z
, ε0 = 1, εk = −1, εµν = 0 µ = ν. Proof. 1) Let U = U
be the unitary in M2 (Calg (S 3 (
)). Then λ U still fulfills ch1/2 (λ U ) = 0 and ch3/2 (λ U ) = ch3/2 (U ). With z˜ µ = λ z µ one has (˜z µ )∗ = λ¯ z µ∗ = µ µ λ¯
ν z ν = λ¯ 2
ν z˜ ν . This shows that ϕ is a homomorphism ϕ : Calg (S 3 (
 )) → Calg (S 3 (
)) and that ϕ([S 3 (
 )]) = [S 3 (
)]. 2) One has Spin(4) = SU (2) × SU (2) and the covering map π : Spin(4) = SU (2) × SU (2) → S O(4) is given for any (u, v) ∈ SU (2) × SU (2) and ξ ∈ R4 by π(u, v)ξ = η with τµ ηµ = u (τµ ξ µ ) v ∗ .

(2.20) µ

This equality continues to hold for the natural complex linear extension Vν z ν of V = π(u, v) to C4 and it follows that with the notations of assertion 2) one has ν = u U v∗ τµ Vνµ z


with U = U
as above. The unitary u U v ∗ still fulfills ch1/2 (u U v ∗ ) = 0 and ch3/2 (u U v ∗ ) = ch3/2 (U ) since in M2 (C) it is the ordinary product and trace which are

Noncommutative 3-Spheres

31 µ

involved in the formulas for chk/2 . Thus, with z˜ µ = Vν z ν we just have to check that µ

z˜ µ∗ =
ν z˜ ν . One has
 z˜ = V
V t V z = V
z = V z ∗ , and since V has real coefficients this is (V z)∗ = (˜z )∗ . 3) Let U = U
as above, then U ∗ is still unitary and fulfills ch1/2 (U ∗ ) = 0, ch3/2 (U ∗ ) = − ch3/2 (U ). It corresponds to z˜ 0 = z 0∗ , z˜ k = −z k∗ . Then z˜ µ∗ = εµ z µ = µ µ εµ (
−1 )ν z ν∗ = εµ (
−1 )ν εν z˜ ν which gives the value of
 .   Corollary 2.5. For every
∈ S there exists ϕ j ∈ R/ π Z and isomorphisms S 3 (
) → Sϕ3 (resp. R4 (
) → R4ϕ ) where Sϕ3 corresponds to the diagonal matrix   1 0 .
(ϕ) = 0 e−2iϕk Proof. We just need to recall why the matrix
can be diagonalized by a W ∈ S O(4). In fact
is unitary and fulfills
t =
i.e.
∗ = J
J −1 , where J gives the real structure of C4 . An eigenspace E λ = {ξ ∈ C4 ;
ξ = λ ξ } is stable by J since
J −1 ξ = J −1
∗ ξ = J −1 λ¯ ξ = λ J −1 ξ (and J −1 = J ). Thus we can find an orthonormal basis of real vectors: J ξi = ξi which are eigenvectors for
. With ei the standard basis of C4 the map ei → ξi gives an element of O(4) and we can take it in S O(4). Thus
= W D W t with W ∈ S O(4).   The presentation of the algebra of R4ϕ is given by (2.14) (2.15) and the relation (2.7) with
=
(ϕ). This gives z 0∗ = z 0 and z k∗ = e−2iϕk z k so that with x 0 = z 0 , x k = e−iϕk z k we get x µ∗ = x µ , ∀µ ∈ {0, 1, 2, 3}, (2.21) and the six other relations give eiϕk x k x 0 − e−iϕk x 0 x k + eiϕk x 0 x k − e−iϕk xk x 0 +

 

εkm ei(ϕ −ϕm ) x  x m = 0,

(2.22)

εkm e−i(ϕ −ϕm ) x  x m = 0.

(2.23)

This gives, by combining (2.22) and (2.23) the relations sin(ϕk ) [x 0 , x k ]+ = i cos(ϕ − ϕm ) [x  , x m ], cos(ϕk ) [x 0 , x k ] = i sin(ϕ − ϕm ) [x  , x m ]+ ,

(2.24) (2.25)

where we let [a, b]+ = a b + b a be the anticommutator. We shall now study in much greater detail the corresponding moduli space. 3. The Real Moduli Space M We let M be the moduli space of noncommutative 3-spheres. It is obtained from Proposition 2.4 1) and 2) as the quotient M = (U (1) × S O(4))\S,

(3.1)

of S by the action of U (1) × S O(4). This action of U (1) × S O(4) on S is the restriction of the following action of U (4) on S:
∈ S → W
W t , ∀ W ∈ U (4),

(3.2)

32

A. Connes, M. Dubois-Violette

which allows to identify S with the homogeneous space U (4)/O(4)  S.

(3.3)

(Note that any
∈ S can be written as
= V V t for some V ∈ U (4) since it can be diagonalized by an orthogonal matrix.) The presence of U (1) in (3.1) allows to reduce to SU (4) and one obtains this way a first convenient description of M. 3.1. M in terms of A3 . The description of M in terms of the compact Lie group SU (4) is given by: Proposition 3.1. 1) Let N be the normalizer of S O(4) in SU (4). Then one has a canonical isomorphism M  S O(4)\SU (4)/N . (3.4) 2) Let T ⊂ SU (4) be the maximal torus of diagonal matrices, and W ⊂ Aut(T) the corresponding Weyl group. Let D = T ∩ N . Then the above restricts to an isomorphism M  W \T/D. (3.5) 3) The map u → u 2 from T to T induces an isomorphism M  W \T/D  Space of Conjugacy Classes in P SU (4).

(3.6)

Proof. 1) Let Z be the center of SU (4), it is generated by i which has order 4 and contains −1 ∈ S O(4). The normalizer N is generated by S O(4) and the element ⎡

−1

⎢ v= w⎣

0 1 1

0

⎤ ⎥ ⎦,

w = e2πi/8 ,

(3.7)

1

which implements the outer automorphism of S O(4) and whose square v 2 = i generates Z. Let X = U (1)\S considered as an homogeneous space on SU (4) using the action (3.2), i.e.
∈ X → W
W t . Given
∈ S we can find λ ∈ U (1) so that
= λ
1 with Det
1 = 1, thus with S1 = {
∈ S, Det
= 1} one has, X = U (1)\S  w Z \S1  SU (4)/N .

(3.8)

Indeed the first equality follows since the action (3.2) of w is multiplication by i and the second follows by computing the isotropy group K of 1 ∈ wZ \S1 . One has S O(4) ⊂ K and v ∈ K since v t v = v 2 = w 2 . Thus N ⊂ K . Conversely given V ∈ K one has V V t = i N for some N ∈ Z and thus for a suitable power of v one has v k V V t v k = 1, thus v k V ∈ S O(4). This shows that K = N and we get the first statement of the proposition since M  S O(4)\X by construction. Note the standard description of N which is given as follows. One lets θ ∈ Aut(SU (4)) be given by complex conjugation, θ (u) = u¯ = J u J −1

∀ u ∈ SU (4).

(3.9)

Noncommutative 3-Spheres

33

One has θ 2 = 1 and the fixed points of θ give S O(4) = SU (4)θ . The normalizer N of S O(4) is characterized by, u −1 θ (u) ∈ Z , (3.10) where Z is the center of SU (4). 2) Let σ be the map σ (u) = u u t from SU (4) to the space PS1 of classes of elements of S1 modulo the action of Z by multiplication. It follows from 1) that σ is an isomorphism of X = SU (4)/N with PS1 . Given u ∈ SU (4) we can find V ∈ S O(4) such that u ut = V D V t , (3.11) where D is a diagonal matrix. Then σ (u) = σ (V D 1/2 ), where D 1/2 is a diagonal square root of D with determinant equal to 1. Thus every element of the coset space S O(4)\X can be represented by a diagonal matrix, and the natural map given by inclusion W \T/D → S O(4)\X

(3.12)

is surjective. 3) When restricted to T the map σ is simply the squaring u → u 2 . Moreover the equality σ (u 1 ) = σ (u 2 ) in PS1 for u j ∈ T just means that the u 2j define the same element of the maximal torus T/Z of P SU (4). Thus the result follows. Let us check that the group D/Z is (Z/2)3 . The elements of D are the v k u with v as in (3.7) and u in T ∩ S O(4). One has v 2 ∈ Z and modulo Z ∩ S O(4) = ±1 the elements of T ∩ S O(4) form the Klein group H = (Z/2Z)2 . Thus one gets D/Z  (Z/2Z)3 .   We identify the Lie algebra of SU (4) with the Lie algebra of antihermitian matrices with trace 0, Lie (SU (4)) = {T ∈ M4 (C) ; T ∗ = −T, Trace T = 0},

(3.13)

where θ is still acting by complex conjugation. The diagonal matrices D ∈ D form a maximal abelian Lie subalgebra of the eigenspace, (3.14) Lie (SU (4))− = {T, θ (T ) = −T }. The roots α ∈  are given by, αµ,ν (δ) = δµ − δν .

(3.15)

Proposition 3.2. For δ ∈ D one has, eδ ∈ N ⇔ e2δ ∈ Z ⇔ αµ,ν (δ) ∈ i π Z, ∀µ, ν.

(3.16)

Proof. The equivalence between the last two conditions is a general fact for compact Lie groups (cf. [9]). The equivalence between the first two conditions follows from the third statement of Proposition 3.1.   We let  be the lattice  ⊂ D determined by the equivalent conditions (3.16), and T A = D/  be the quotient 3-dimensional torus.

(3.17)

34

A. Connes, M. Dubois-Violette

We let the group W of permutations of 4 elements act on T A by permutations of the δµ . In fact we view it as the Weyl group of the pair (SU (4), N ), i.e. as the quotient, W = N /C

(3.18)

of the normalizer of D in S O(4) by the centralizer of D. Note that v being diagonal is in the centralizer of D so that W does not change in replacing S O(4) by N since v k u normalizes D iff u does. Corollary 3.3. The map σ (δ) = e2δ defines an isomorphism of the quotient of T A by the action of W with the moduli space M = (U (1) × S O(4))\S. Proof. This is just another way to write the second statement of Proposition 3.1.

 

3.2. Trigonometric parameters ϕ of Sϕ3 . We shall now describe a convenient parametrization of the torus T A which gives the corresponding algebras in the form of Corollary 2.5. It is given by the map ϕ = (ϕ1 , ϕ2 , ϕ3 ) ∈ (R/π Z)3 → d(ϕ) = (α0 , α0 − i ϕ1 , α0 − i ϕ2 , α0 − i ϕ3 ), i  ϕj. (3.19) α0 = 4 One has α0,k (d(ϕ)) = i ϕk and the definition of  shows that d is an isomorphism. Also 

e

2d(ϕ)

1  0

0



e−2iϕk

up to multiplication by a scalar. In terms of the parameters ϕ j the twelve roots α ∈  are the following linear forms, (ϕ1 , ϕ2 , ϕ3 ) → i {±ϕ j , ϕk − ϕl }.

(3.20)

The action of the Weyl group W gives the following linear transformations of the ϕ j . Arbitrary permutations of the ϕ j ’s correspond to permutations of the last three αk ’s. The transposition of α0 with α1 corresponds to : T01 (ϕ1 , ϕ2 , ϕ3 ) = (−ϕ1 , ϕ2 − ϕ1 , ϕ3 − ϕ1 ).

(3.21)

The 3-spheres Sϕ3 are parametrized by ϕ = (ϕ1 , ϕ2 , ϕ3 ) ∈ (R/π Z)3

(3.22)

modulo the action of the Weyl group W  S4 generated by the permutation group S3 of the ϕ j ’s and (ϕ j ) → (− ϕ1 , ϕ3 − ϕ1 , ϕ2 − ϕ1 ) = (ϕ j ).

(3.23)

Noncommutative 3-Spheres

35

3.3. M in terms of D3 . To obtain another very convenient parametrization of the torus T A we use the isomorphism A3 ∼ D3 , i.e. SU (4)  Spin (6),

(3.24)

which simply comes from the spin representation of Spin (6). The Clifford algebra Cliff C (R6 ) has dimension 26 and is a matrix algebra Mn (C) with n = 23 = 8. We let γ µ be the corresponding γ -matrices, with γµ∗ = γµ , γµ γν + γν γµ = 2 δµ,ν . We then let σµν =

1 (γµ γν − γν γµ ) 2

which span a real subspace of the Clifford algebra stable under bracket and isomorphic (up to a factor of 2) to the Lie algebra of S O(6) of real antisymmetric 6 by 6 matrices. Proposition 3.4. 1) The following gives a parametrization τ of a maximal torus in Spin(6),  τ θ = (θ j ) ∈ (R/ 2π Z)3 −→ Exp θ j σ2 j−1,2 j ∈ Spin(6) ⊂ Cliff C (R6 ). (3.25) 2) The half Spin representation gives an isomorphism π : Spin(6) → SU (4). 3) One has π(τ (θ )) = eδθ with δθ diagonal given by δθ = i (θ1 + θ2 + θ3 , θ1 − θ2 − θ3 , − θ1 + θ2 − θ3 , − θ1 − θ2 + θ3 ).

(3.26)

Proof. This is a straightforward check, since in the half Spin representation π one has in a suitable basis ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ i 0 i 0 i 0 i −i −i ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ σ12 → ⎣ ⎦, σ34 → ⎣ ⎦, σ56 → ⎣ ⎦, −i i −i 0 −i 0 −i 0 i (3.27) thus π(τ (θ )) = eδθ with δθ given by (3.26).   Thus the trigonometric parameters ϕk are given in terms of the θ ’s by, ϕ1 = 2 (θ2 + θ3 ), ϕ2 = 2 (θ1 + θ3 ), ϕ3 = 2 (θ1 + θ2 ).

(3.28)

The natural parameters for the maximal torus T D of S O(6) are the ψ j = 2 θ j and are defined modulo 2 π Z, i.e. correspond to the Lie algebra element (ψ) = ψ1 β12 + ψ2 β34 + ψ3 β56 ,

(3.29)

where the βi j form the canonical basis of real antisymmetric matrices. The kernel of the covering Spin (6) → S O(6) corresponds to θ j = π so that for S O(6) the torus T D is parametrized by the ψ j ’s defined modulo 2π by ψ ∈ (R/2 π Z)3 → e(ψ) .

(3.30)

36

A. Connes, M. Dubois-Violette

The transition from ψ to ϕ’s is given then by, ϕ j = ψk + ψ , 2 ψ j = ϕk + ϕ − ϕ j ,

(3.31)

as well as ϕ1 − ϕ2 = ψ2 − ψ1 , ϕ2 − ϕ3 = ψ3 − ψ2 , ϕ3 − ϕ1 = ψ1 − ψ3 .

(3.32)

We shall now spellout in great detail the basic Lie group data for D3 and get a description of the moduli space M in these terms. Proposition 3.1 3) gives a natural isomorphism M  Space of Conjugacy Classes in P SU (4) of the moduli space M with the space of conjugacy classes of elements of P SU (4)  P S O(6). The general theory of compact Lie groups provides a natural triangulation of such a space of conjugacy classes in terms of alcoves. The latter are obtained as the connected components of the complement of the union of the singular hyperplanes. Our aim in this section is to describe such a triangulation in our specific case and to exhibit the role of the singular hyperplanes. We shall see in the next section their natural compatibility with the scaling foliation. In terms of the parameters ϕ j of the 3-spheres Sϕ3 the relations which specify the non-generic situations, are all of the form  π ϕ, α(ϕ) = n = G α,n , (3.33) 2 where n is an integer and α is one of the twelve “roots” i.e. α ∈  = ±{ϕ1 , ϕ2 , ϕ3 , ϕ1 − ϕ2 , ϕ2 − ϕ3 , ϕ3 − ϕ1 }. Moreover the periodicity lattice of ϕ is (π Z)3 which is specified by {ϕ, α(ϕ) ∈ π Z, ∀ α ∈ } = ϕ .

(3.34)

We now want to relate more precisely the above situation with canonical objects (root systems, alcoves, chambers, affine Weyl group, nodal vectors . . .) associated to the following data (G, T): G = P S O(6), T = Maximal torus T D / ± 1.

(3.35)

We use the natural parametrization of the Lie algebra Lie (T), (ξ ) = ξ1 β12 + ξ2 β34 + ξ3 β56 .

(3.36)

Since we used the “squaring” u → u 2 in the isomorphism of Proposition 3.1 3), the parameter ψ that appears in the transition ϕ → ψ of Eq. (3.31) is related to ξ by ξ = 2 ψ.

(3.37)

In other words the natural relation between the parameters ϕ and ξ is 2 ϕ j = ξk + ξ , ξ j = ϕk + ϕ − ϕ j .

(3.38)

This accounts for a factor of 21 in (3.33) but does not yet relate it to the equation of singular hyperplanes. To understand this relation more precisely we shall now review briefly the standard ingredients of the theory of alcoves for the specific data (G, T).

Noncommutative 3-Spheres

37

3.4. Roots  = R(G, T). They are by definition the linear forms α on Lie (T) given by eigenvalues X α ∈ Lie G C which fulfill: [ξ, X α ] = α(ξ ) X α

∀ ξ ∈ Lie T ⊂ Lie G.

(3.39)

They are complex valued as defined. In our case they are given by (up to multiplication by i) (± eµ ± eν ) ξ = ± ξµ ± ξν . (3.40) 3.5. Singular hyperplanes Hα,n . They are given by a root α ∈  and n ∈ Z, with Hα,n = {ξ, α(ξ ) = 2πin}.

(3.41)

As we shall explain below while these hyperplanes suffice to obtain a triangulation of the space of conjugacy classes of the simply connected covering of G, we shall need the additional ones with n ∈ 21 Z to describe the space of conjugacy classes in G itself. 3.6. Kernel of the exponential map: (T) (nodal group of T). In our case we are dealing with G = P S O(6) and thus for ξ ∈ Lie T, eξ is 1 in G iff Ad(eξ ) = 1 since the center of G is C(G) = {1}. This means exactly that all eigenvalues of ad(ξ ) belong to Ker (exp) = 2πiZ, thus (T) = {ξ, α(ξ ) ∈ 2πiZ ∀ α ∈ }.

(3.42)

Thus, after applying the change of variables (3.38) ϕ  (T)

(3.43)

which shows that the periodicity lattice we have is the nodal group of T. 3.7. Group of nodal vectors N (G, T) ⊂ (T). This group can be defined in terms of vectors K α which are associated to the roots α ∈  but in our case it is simpler to use the definition as follows: N (G, T) = Kernel of exp : Lie T → G˜ = Universal cover of G.

(3.44)

In our case G˜ = Spin 6 and in terms of γ -matrices the exponential map takes the form,    1 1 1 1 (ξ ) → cos ξ1 + sin ξ1 γ1 γ2 cos ξ2 + sin ξ2 γ3 γ4 2 2 2 2   1 1 (3.45) × cos ξ3 + sin ξ3 γ5 γ6 2 2 whose kernel is given by N (G, T) = {ξ ; ξ j = 2π n j , n j ∈ Z, n j even}.

(3.46)

One has N (G, T) ⊂ (T) and the quotient, of order 4, is generated by ξ = (π, π, π ) ∈ (T).

38

A. Connes, M. Dubois-Violette

3.8. Affine Weyl group Wa . The affine Weyl group Wa is the group generated by the reflexions associated to the hyperplane Hα,n for α ∈  and n ∈ Z. One has [8] (Chapter VI, Prop. 1, p. 173) Wa = N (G, T)
W,

(3.47)

where the Weyl group is generated by the reflexions associated to singular hyperplanes Hα,0 . In our case W = S4 and all its elements are of the form  W = ε σ, σ ∈ S3 , ε = (εi ), εi ∈ ±1, εi = 1, (3.48) where the action on ξ is by permutation of the ξ j for σ (careful that (σ ξ )(i) = ξ(σ −1 (i)) to get a covariant action) and by multiplication by εi for ε. For N (G, T) we can check that the αˇ = K α are simply given by the vectors ± eµ ± eν which correspond to (± 2π, ± 2π, 0) = K ±e1 ± e2 , K α , α = 2.

(3.49)

3.9. Affine Weyl group Wa . It is by definition the semi-direct product, Wa = (T)
W.

(3.50)

What matters is that it still acts on the set of singular hyperplanes. For γ ∈ (T) one has α(h + γ ) = α(h) + α(γ ) and α(γ ) ∈ 2πiZ thus one is just shifting the n in Hα,n . From [9] Prop. 2 (Chapter 9, p. 45) Wa is a normal subgroup of Wa . 3.10. Alcoves and fundamental domain. The alcoves are the connected components of (U Hα,n )c ⊂ Lie T. The chambers are the components of (U Hα,0 )c . By construction the alcoves are intersections of half spaces and are thus convex polyhedra. By [9] the affine Weyl group Wa acts simply transitively on  = the set of alcoves. Since Wa is still acting on  we can identify  with the homogeneous space  = Wa /H X ,

(3.51)

where H X is the finite isotropy group of an alcove X . In our case, we take the following alcove : X = {ξ, ξ1 + ξ2
0, ξ2 − ξ1
0, ξ3 − ξ2
0, ξ2 + ξ3  2π }.

(3.52)

It is a tetrahedron with all 4-faces congruent but 2 long edges and 4 short edges. 

Lemma 3.5. The isotropy subgroup H X ⊂ Wa of X is generated by w1 = 21 , 21 , 21 ,  ε12 σ13 . Proof. For convenience we rescale the ξ j by 2π . The coordinates of the vertices of X are 

then 0, p = 21 , 21 , 21 , q = − 21 , 21 , 21 , p  = (0, 0, 1). One has w1 (0) = p, w1 ( p) = 
p+ε12 σ13 ( p) = p+ − 21 , − 21 , 21 = p  , w1 ( p  ) = p+ε12 σ13 ( p  ) = p+(−1, 0, 0) = q, 
1 1 1
1  w1 (q) = p + ε12 σ13 (q) = 2 , 2 , 2 + − 2 , − 21 , − 21 = 0. 

Noncommutative 3-Spheres

39

Fig. 1. Tiling of the alcove X by 21 -alcoves

P’ Z

Q P

0 Fig. 2. Fundamental domain

We let σ be the orthogonal reflexion around the face ( pqp  ) of X . One has σ ∈ Wa and it is given explicitly by (3.53) σ = ((0, 1, 1), ε23 σ23 ). Indeed, since the face is given by ξ2 +ξ3 = 2π the corresponding nodal vector is (0, 1, 1). One checks that σ 2 = 1 and that σ fixes p, q, p  , (σ ( p  ) = (0, 1, 1) + (0, −1, 0) = (0, 0, 1) = p  ). We let Y = σ (X ) be the reflexion of X along that face which yields the convex pentahedra X ∪ Y . Proposition 3.6. Let X be the alcove given by (3.52) and Y = σ (X ) its reflexion along the face ( p, q, p  ). Then 21 (X ∪ Y ) is a fundamental domain for the action of (T)
W in Lie T. Proof. By the above lemma the action of Wa on the set of alcoves is the same as the action of Wa on Wa /H X . Let Wa = 2
W . The left coset space Wa /Wa is identified with the 8 elements set, /2, where the corresponding map is given by: (γ , w) ∈ Wa → Class of w −1 (γ ) in /2.

(3.54)

40

A. Connes, M. Dubois-Violette

Indeed (0, w −1 )(γ , w) = (w −1 (γ ), 1). We can thus display the action (on the right) of H X as γ → Class of σ13 ε12 ( p + γ ). Let us write this transformation in terms of the ϕ-coordinates. One obtains    1 1 1 , w(ϕ) = (−ϕ3 , ϕ1 − ϕ3 , ϕ2 − ϕ3 ). ϕ→w ϕ+ (3.55) , , 2 2 2 Thus in fact we look at the following transformation of (Z/2)3 ,   1 (3.56) S(a1 , a2 , a3 ) = − a3 , a1 − a3 , a2 − a3 . 2

 



One has S(0) = 21 , 0, 0 , S 21 , 0, 0 = 21 , 21 , 0 , S 21 , 21 , 0 = 21 , 21 , 21 , S 21 , 21 , 21 


1
1  = 0. The other orbit is S 0, , 0 = 2 , 0, 21 , S 21 , 0, 21 = 0, 0, 21 , S(0, 0, 21 ) =
1 1
1 1
1 2  0, 2 , 2 , S 0, 2 , 2 = 0, 2 , 0 . This shows that the double coset space Wa \Wa /H X has cardinality 2 and thus that Wa just has 2 orbits in its action on the set of alcoves. What remains is to show that Y ∈ / Orbit of X . But one has Y = σ (X ) with σ given by (3.53). Thus we just need to determine the double coset of σ , i.e. by (3.54) the class of σ23 ε23 (0, 1, 1) in /2. One just checks that it is in the other orbit.   We thus have a fairly simple fundamental domain for the parameter space of noncommutative 3-spheres. In terms of the ϕ’s a simple translation of the above result gives, Proposition 3.7. 1) The union A ∪ B of the following simplices:     π π π
ϕ1
ϕ2
ϕ3
0 , B = ϕ ; ϕ3 +
ϕ1

ϕ2
ϕ3 , A = ϕ; 2 2 2 (3.57) gives a fundamental domain for the action of W on R3 / ϕ = (R/π Z)3 . 2) The real moduli space M is obtained by glueing the face3 (P  Q Z ) to the face (Z P P  ) by the transformation γ ∈ ϕ
W γ (ϕ1 , ϕ2 , ϕ3 ) = (π − ϕ1 + ϕ2 , π − ϕ1 + ϕ3 , π − ϕ1 ) and crossing each wall, i.e. each face whose supporting hyperplane contains 0, by the corresponding reflexion (in W ). The two simplices A and B have a common face (P Q P  ) supported by the hyperplane ϕ1 = π2 . In order to describe the moduli space one needs to give the identifcations of the boundary components. All faces of A other than (P Q P  ) are walls of chambers and thus crossing them leads to a simple reflexion on that wall. 4. The Flow F While the space M is the natural moduli space for noncommuative 3-spheres, the moduli space of the corresponding 4-spaces R4ϕ is obtained as a space of flow lines for a natural flow F on M. We define this flow F as the gradient flow for the Killing metric on the Lie algebra of S O(6) of the character of the virtual representation given by the “signature”. We first show that F is nicely compatible with the triangulation by alcoves and then check that the isomorphism class of the 4-spaces R4ϕ is constant along the flow 3 We use capital letters P, Q, Q  for the vertices : P = ( π , π , π ), P  = ( π , π , 0), Q = ( π , 0, 0), 2 2 2 2 2 2 Z = P + Q.

Noncommutative 3-Spheres

41

lines. We shall need for the converse to have a complete knowledge of the geometric datas of these quadratic algebras, and this will be obtained in Sect. 5. Thus the converse will be proved later on in Sect. 6. 4.1. Compatibility of F with the triangulation by alcoves. The basic compatibility of the flow F with the structure of the real moduli space M is given by : Proposition 4.1. a) The character of the signature representation is  χ (ξ ) = Trace (∗ π((ξ )) = − 8 sin ξ j .

(4.1)

b) The flow F = ∇ χ (2ψ) is invariant under the action of the Weyl group W and leaves each of the singular hyperplanes Hα,n , n ∈ 21 Z globally invariant. c) In terms of the variables ϕ j one has F=



sin(2ϕ j ) sin(ϕk + ϕ − ϕ j )

∂ . ∂ϕ j

(4.2)

Proof. a) We parametrize the maximal torus T D of S O(6) by the ξ j as in (3.29), i.e. (ξ ) = ξ1 β12 + ξ2 β34 + ξ3 β56 .

 The Killing metric is simply given there (up to scale and sign) by dξ 2j . We identify ∧3 C6 with the linear span of the ei jk = γi γ j γk , card {i, j, k} = 3 in Cliff C (R6 ). We then use γ7 = γ1 γ2 γ3 γ4 γ5 γ6 , which fulfills γ72 = −1 to define the ∗ operation by: ∗ ei jk = γ7 ei jk .

(4.3)

One checks that the matrix of ∗ π((ξ )) restricted to the 12 dim subspace spanned by the ei jk , where two indices belong to one of the 3 subsets {1, 2}, {3, 4}, {5, 6} is off diagonal and hence has vanishing trace. Indeed for instance ∗ π((ξ )) e123 is a linear combination of e356 and e456 which are orthogonal to e123 . On the 8 dimensional subspace of ei jk , i ∈ {1, 2}, j ∈ {3, 4}, k ∈ {5, 6} one gets the product of what happens in the 2-dimensional case with a basis e1 , e2 of ∧1 C2 . One has ∗ e1 = e2 , ∗ e2 = −e1 and the representation is given by e1 → cosξ e1 + sinξ e2 , e2 → cosξ e2 − sinξ e1 so that the trace of ∗ π((ξ )) is −2sinξ . Thus we get, Trace (∗ π((ξ ))) = − 8 sin ξ1 sin ξ2 sin ξ3 .

(4.4)

b) In terms of S O(6) the Weyl group W maps to the permutations of (ψ1 , ψ2 , ψ3 ) and the kernel of this map is the Klein subgroup which is given by the transformation, ψj → εj ψj

3 

εj = 1

(ε j ∈ {±1}).

(4.5)

1

By construction the function χ (ξ ) being a (virtual) character is invariant by these transformations and so is the flow X.  ∂χ ∂ c) Let us rewrite X = ∂ψ j ∂ ψ j in terms of the coordinates ϕ j . One has

  ∂h   ∂h ∂h d ψj, d ϕ j = d ψk + d ψ , d ϕj = + ∂ϕ j ∂ϕk ∂ϕ

∂h ∂h ∂h = + ∂ψ j ∂ϕk ∂ϕ

42

A. Connes, M. Dubois-Violette

and X is given by    ∂χ  ∂ ∂ ∂χ X= + =2 (cos2 ψk sin2 ψ + cos2 ψ sin2 ψk ) sin2 ψ j ∂ψk ∂ψ ∂ϕ j ∂ϕ j  ∂ =2 sin(2 ϕ j ) sin(ϕk + ϕ − ϕ j ) . ∂ϕ j   4.2. Invariance of R4ϕ under the flow F. We let C+ = {(ϕ, ϕ, 0)}, C− =

π 2

, ϕ, ϕ +

π . 2

(4.6)

The critical set of the flow X is described as follows in terms of the action of the Weyl group W : Lemma 4.2. The critical set C of X is given by C = W (C+ ) ∪ W (C− ) ∪ W (P),

P=

π π π  , , . 2 2 2

One checks this directly in the ψ variables. In order to perform a change of variables we use the function,  δ(ϕ) = sinϕ j cos(ϕk − ϕ ) (4.7) and we let

D = {ϕ, δ(ϕ) = 0}.

(4.8)

By construction D is invariant under the group S3 of permutations of the ϕ j ’s. Lemma 4.3. Let ϕ ∈ / C, then there exists g ∈ W such that gϕ ∈ / D. Proof. Let us first show that the conclusion holds if one of the ϕ j vanishes (we always work modulo π and all equalities below have this meaning). Thus assume that ϕ3 = 0, i.e. that ϕ = (ϕ1 , ϕ2 , 0). Then if ϕ1 = ϕ2 one is in C thus we can assume that ϕ1 = ϕ2 . By the transformation (3.23) we get ϕ  = (− ϕ1 , − ϕ1 , ϕ2 − ϕ1 ). If ϕ1 = 0 we treat (0, ϕ2 , 0) ∼ (ϕ2 , 0, 0) by applying (3.23) which gives (−ϕ2 , −ϕ2 , −ϕ2 ) for which δ(ϕ) is 0 only if ϕ2 = 0 in which case we are dealing with the point O =  (0, 0, 0) which is in C. Thus we can assume ϕ1 = 0, ϕ2 − ϕ1 = 0 and we get sinϕ j = 0.  One has ϕk −ϕ ∈ {0, ±ϕ2 } thus the product cos(ϕk −ϕ ) vanishes only if ϕ2 = π2 . 

We are thus dealing with ϕ1 , π2 , 0 (with ϕ1 = 0). We apply (3.23) to π2 , ϕ1 , 0 which gives  π π π . ϕ  = − , − , ϕ1 − 2 2 2  If ϕ1 = π2 then one is in C otherwise sinϕ j = 0. One has ϕk − ϕ ∈ {0, ±ϕ1 } and  since ϕ1 = π2 we get cos(ϕk − ϕ ) = 0. We have thus shown that if ϕ j = 0 for some

Noncommutative 3-Spheres

43

j and  ϕ∈ / C we can find / D. Thus we can now assume that all ϕ j = 0.  g ∈ W with gϕ ∈ Thus sinϕ j = 0. If cos(ϕk − ϕ ) = 0 we can assume ϕ1 − ϕ2 = π2 . We then apply (3.23) and get  π ϕ  = − ϕ1 , ϕ3 − ϕ1 , − . 2 If one of the components of ϕ  is 0 we are back to the previous case, thus we can assume  sinϕ j = 0. The ϕk − ϕ give (up to sign) ϕ3 , ϕ3 − ϕ1 + π2 , ϕ1 − π2 and since ϕ1 = 0  and ϕ2 = ϕ3 − ϕ
1 = 0, cos(ϕk − ϕ ) = 0 can occur only if cos ϕ3 = 0. In that case the original ϕ is ϕ1 , ϕ1 − π2 , π2 which is in C.   Note then that one can always choose the g ∈ W in the Klein subgroup H ⊂ W . Indeed H is a normal subgroup and S3 ⊂ W acts as the group of permutation of the ϕ’s and preserves D. Thus for g = σ k1 , σ ∈ S3 , σ k1 ϕ ∈ / D ⇒ k1 ϕ ∈ / D. Let us now use this lemma to simplify the presentation of the algebra. Lemma 4.4. If δ(ϕ) = 0, there exists 4 non zero scalars ∈ i N R∗ , such that one has and



(sinϕk ) λ λm + cos(ϕ − ϕm ) λ0 λk = 0 λµ = −δ(ϕ).

Proof. Let us choose the square roots 

λ0 = in such a way that



sinϕ j

1/2

⎛

, λk = ⎝sinϕk



⎞1/2 cos(ϕk − ϕ )⎠

 =k

λ j = −δ(ϕ). Note indeed that the product of the squares gives   (sinϕ j )2 (cos(ϕk − ϕ ))2 = δ(ϕ)2 , k<

and thus one can fix the λk and then choose the sign of λ0 so that it fits. Then to prove the 3 equalities one can multiply by λ0 λk which gives (sinϕk )(−δ(ϕ))+cos(ϕ −ϕm ) λ20 λ2k . Thus one needs to check   cos(ϕk − ϕ1 )sinϕk , sinϕk cos(ϕ1 − ϕm 1 ) = cos(ϕ − ϕm ) 1 =k

which is an identity.

 

One then lets

Sµ = λµ x µ , (no summation on µ). Multiplying (2.24) by λ0 λ1 λ2 λ3 one gets [S , Sm ] = i [S0 , Sk ]+

(4.9) (4.10)

while (2.25) gives: cosϕk [S0 , Sk ] = i Note that

λ0 λk sin(ϕ − ϕm ) [S , Sm ]+ . λ λm

 λµ λ0 λk −δ(ϕ) = 2 2 = 2 2 . λ λm λ λm λ λm

(4.11)

44

A. Connes, M. Dubois-Violette

Proposition 4.5. If δ(ϕ) = 0 the quadratic algebra of R4ϕ admits the presentation given by (4.10) and (4.12) [S0 , Sk ] = i Jm [S , Sm ]+ , where the Jm are given by Jm = −tan ϕk tan(ϕ − ϕm ). If ϕk = π2 and ϕ = ϕm the corresponding relation gives 0 = 0. If ϕk = it gives [S , Sm ]+ = 0.

(4.13) π 2

and ϕ = ϕm

Proof. One needs to show that −δ(ϕ) sin(ϕ − ϕm ) = −tan ϕk tan(ϕ − ϕm ). cosϕk λ2 λ2m This amounts to the equality sinϕ sinϕm cos2 (ϕ − ϕm ) cos(ϕ − ϕk )cos(ϕm − ϕk ) = λ2 λ2m , which follows from the definition of the λk ’s.

 

We now find a better way to write the Jm . We first pass in a faithful manner from the ϕ j to π t j = tan ϕ j ∈ R (recall ϕ j = ). (4.14) 2 Note also that t j = 0 since we are on δ(ϕ) = 0. We then introduce the following new parameters, (4.15) sk (ϕ) = 1 + t tm . σ

Lemma 4.6. a) The map ϕ −→ s = (sk (ϕ)) on  = {ϕ | cos(ϕk ) = 0 ∀ k, δ(ϕ) = 0} is a double cover of the open subset of R3 ,   sk = 0 and (sk − 1) > 0. (4.16) b) The map σ is a diffeomorphism of the interior A◦ of A with σ (A◦ ) = {s | 1 < s1 < s2 < s3 }. c) The map σ is a diffeomorphism of B ◦ with σ (B ◦ ) = {s | s3 < s2 < 0, 1 < s1 }. Proof. a) The condition δ(ϕ) = 0 shows that sk = 0, ∀ k. Indeed ϕm = −1  tan ϕ tan means cos(ϕ − ϕm ) = 0. By construction sk = 1 and (sk − 1) = t2 > 0.  2  one gets ( tk ) = (sk − 1) and choosing the sign of the square Knowing the sk ’s root gives p = tk and then tk = p(sk − 1)−1 . Thus one gets a double cover and the range is characterized by the conditions (4.16). The deck transformation is simply ϕ → −ϕ.  b) On A◦ = {ϕ | π2 > ϕ1 > ϕ2 > ϕ3 > 0} one has tk > 0 and thus tk > 0 so that the above map is one to one. One checks that the range σ (A◦ ) is given by σ (A◦ ) = {s | 1 < s1 < s2 < s3 }. c)  On B ◦ = {ϕ | π2 + ϕ3 ϕ1 > π2 > ϕ2 > ϕ3 } one has t1 < 0, t2 > 0, t3 > 0 and thus tk < 0 so that the above map is one to one. One checks that the range σ (B ◦ ) is given by σ (B ◦ ) = {s | s3 < s2 < 0, 1 < s1 }.  

Noncommutative 3-Spheres

45

We define the transformation ρ by s − sm . sk   Lemma 4.7. a) Let sk = 0, then one has (1 + ρ(s)k ) = (1 − ρ(s)k ).   b) If ρ(s) = ρ(s ) there exists λ = 0 with s = λ s. c) The transformation s → s˜ , ρ(s)k =

s˜k =

−sk + s + sm s sm

is involutive and ρ(˜s ) = −ρ(s).

 Proof. a) Both products give, up to the denominator sk , the product of (sk + s − sm ). b) Consider the 3 linear equations (for fixed ρk ) given by s − sm − ρk sk = 0. This corresponds to the 3 × 3 matrix given by ⎡ ρ1 −1 ρ2 M(ρ) = ⎣ 1 −1 1

⎤ 1 −1⎦ . ρ3

(4.17)

(4.18)

The condition a) means exactly that Det (M(ρ)) = 0.  we claim that the rank  Moreover 1 ρ2 which gives 1 + ρ2 and of M(ρ) is 2. Indeed ρ2 appears in the two minors −1 1   −1 1 which gives 1 − ρ2 and one of them is = 0. This shows that the kernel is ρ2 −1 1-dimensional and hence ρ(s) = ρ(s  ) implies s  = λ s for some λ. c) The relation between s and s˜ can be written as s s˜m + sm s˜ = 2.

(4.19)

The determinant of the system is 2 s1 s2 s3 so that (4.19) determines s˜k uniquely. The relation is clearly symmetric. Finally ρ(˜s )1 =

s 3 − s2 s˜2 − s˜3 (− s2 + s1 + s3 ) s2 − (− s3 + s1 + s2 ) s3 = = = −ρ(s)1 . s˜1 s1 (− s1 + s2 + s3 ) s1

  We now get ρ ◦ s (ϕ)k = Jm



ϕk =

 π and δ(ϕ) = 0 . 2

(4.20)

Indeed s − sm tk tm − tk t = = tan ϕk tan(ϕm − ϕ ) = Jm . sk 1 + t tm Lemma 4.8. 1) The flow X fulfills X sk (ϕ) = 4



sin ϕ j sk (ϕ).

(4.21)

46

A. Connes, M. Dubois-Violette

2) Let ϕ, ϕ  ∈ A with ϕ3 > 0, ϕ3 > 0. The following conditions are equivalent: a) Jm (ϕ  ) = Jm (ϕ), ∀k. b) ϕ  belongs to the orbit of ϕ by the flow X . 3) The same statement holds for ϕ, ϕ  ∈ B with ϕ3 + π2 > ϕ1 , ϕ3 + π2 ϕ1 . Proof. 1) One has X (tk ) = sin(2 ϕk )sin(− ϕk + ϕ + ϕn )

∂ tan ϕk = 2 tan ϕk sin(− ϕk + ϕ + ϕm ), ∂ϕk

X (sk ) = X (t ) tm + t X (tm ) = 2 t tm (sin(− ϕ + ϕm + ϕk ) + sin(− ϕm + ϕ + ϕk )), and using sin(a + b) + sin(a − b) = 2sina cosb, one gets X (sk ) = 4 t tm (sinϕk cos(ϕm − ϕ ))   cos(ϕ − ϕ )   m  =4 =4 sinϕ j sinϕ j (1 + tm t ). cosϕm cosϕ 2) In fact 1) shows that the flow X is up to a non-zero change of speed the scaling flow in σ (A). By construction Jm has homogeneity degree zero in sk , thus it is preserved by X and b) ⇒ a). Let us show that a) ⇒ b). We assume first that π2 > ϕ1 . The same then holds for ϕ  uisng a). By Lemma 4.7 the equality a) implies that σ (ϕ  ) = λ σ (ϕ) for some non-zero scalar λ. By Lemma 4.6 the image σ (A◦ ) is convex (as well as its closure) and thus the segment [σ (ϕ), σ (ϕ  )] is contained in σ (A) and its preimage under σ is a segment in a flow line. On the face Y determined by ϕ1 = π2 one has, assuming ϕ2 < π2 , the equalities J12 = − tan ϕ3 tan

π 2

 − ϕ2 = − t3 /t2 ,

 π = t2 /t3 . J31 = − tan ϕ2 tan ϕ3 − 2

Moreover one has X (t2 ) = cos(ϕ2 − ϕ3 ) 2 t2 , X (t3 ) = cos(ϕ2 − ϕ3 ) 2 t3 so that the flow X restricts as the scaling flow (up to a non-zero change of speed) in the parameters t j . Since J23 = ∞ holds iff ϕ1 = π2 for ϕ ∈ A we see that a) then implies ϕ1 = π2 and the proportionality t j = λt j . Since the allowed t j are simply constrained by the inequalities t2 ≥ t1 > 0 the same convexity argument applies. Finally let ϕ with ϕ1 = ϕ2 = π2 , ϕ3 = π2 . Then the only remaining parameter is t3 and one has X (t3 ) = 2sinϕ3 t3 . Thus one is dealing with a single flow line. The proof of 3) is similar.   5. The Geometric Data of R4ϕ In this section we compute the geometric data of the quadratic algebras of functions on R4ϕ for all values of the parameter ϕ. These fall in eleven different classes in each of which one gets further invariants.

Noncommutative 3-Spheres

47

5.1. The definition and explicit matrices. We give a list of the characteristic varieties and correspondences. There are 11 different cases. They are described in terms of the ϕ-coordinates but the result will then be translated in invariant terms using the roots. Let us recall the definition of the geometric data {E, σ, L} for quadratic algebras. Let A = A(V, R) = T (V )/(R) be a quadratic algebra, where V is a finite-dimensional complex vector space and where (R) is the two-sided ideal of the tensor algebra T (V ) of V generated by the subspace R of V ⊗ V . Consider the subset of V ∗ × V ∗ of pairs (α, β) such that ω, α ⊗ β = 0, α = 0, β = 0 (5.1) for any ω ∈ R. Since R is homogeneous, (5.1) defines a subset  ⊂ P(V ∗ ) × P(V ∗ ), where P(V ∗ ) is the complex projective space of one-dimensional complex subspaces of V ∗ . Let E 1 and E 2 be the first and the second projection of  in P(V ∗ ). It is usually assumed that they coincide, i.e. that one has E 1 = E 2 = E ⊂ P(V ∗ ),

(5.2)

and that the correspondence σ with graph  is an automorphism of E, L being the pullback on E of the dual of the tautological line bundle of P(V ∗ ). The algebraic variety E is referred to as the characteristic variety. The algebras we consider have 4 generators and six relations, thus their characteristic variety is obtained as the locus of points where a 4 × 6 matrix has rank less than 4. The various matrices depend upon the choice of the parameters and are listed below. We first give the matrix corresponding to the original quadratic algebra of R4ϕ , ⎡

−cos(ϕ1 ) x1

⎢−cos(ϕ2 ) x2 ⎢ ⎢ ⎢−cos(ϕ3 ) x3 ⎢ ⎢ i sin(ϕ ) x 3 3 ⎢ ⎢ ⎣ i sin(ϕ1 ) x1 i sin(ϕ2 ) x2

cos(ϕ1 ) x0

−i sin(ϕ2 − ϕ3 ) x3

i sin(ϕ1 − ϕ3 ) x3

cos(ϕ2 ) x0

−i sin(ϕ1 − ϕ2 ) x2

−i sin(ϕ1 − ϕ2 ) x1

−cos(ϕ1 − ϕ2 ) x2

cos(ϕ1 − ϕ2 ) x1

i sin(ϕ1 ) x0

−cos(ϕ2 − ϕ3 ) x3

cos(ϕ1 − ϕ3 ) x3

i sin(ϕ2 ) x0

−i sin(ϕ2 − ϕ3 ) x2

⎤

i sin(ϕ1 − ϕ3 ) x1 ⎥ ⎥ ⎥ ⎥ cos(ϕ3 ) x0 ⎥. ⎥ i sin(ϕ3 ) x0 ⎥ ⎥ cos(ϕ2 − ϕ3 ) x2 ⎦

−cos(ϕ1 − ϕ3 ) x1

(5.3) When we pass to the Sklyanin algebra and eliminate the factors i in replacing Z 0 = i S0 , Z k = Sk , we get the following matrix, with α = −J23 , β = −J31 , γ = −J12 , ⎡

z1

⎢z 2 ⎢ ⎢ ⎢z 3 ⎢ ⎢z ⎢ 3 ⎢ ⎣z 1 z2

−z 0

α z3

β z3

−z 0

γ z2

γ z1

z2

−z 1

z0

z3

−z 3

z0

α z2

⎤

β z1⎥ ⎥ ⎥ −z 0 ⎥ ⎥. z0 ⎥ ⎥ ⎥ −z 2 ⎦

(5.4)

z1

The fifteen minors of the matrix (5.3) are listed in factorized form in Appendix 1 and those of the matrix (5.4) in Subsect. 5.3 below (see [29]).

48

A. Connes, M. Dubois-Violette

5.2. The table. We give below the list of the characteristic varieties and correspondences in all cases. Given z ∈ C we let σ (z) be the conjugacy class of semi-simple automorphisms of a curve of genus 0 with eigenvalues {z, z −1 }. By construction σ (z) = σ (z −1 ). The identification of the corresponding algebras Calg (R4ϕ ) in the nongeneric cases (Cases 2 to 11 in the table below) are described in Sect. 8 and may be summarized as follows. In Case 2, Calg (R4ϕ ) is isomorphic to a homogenized version (quadratic) Uq (su(2))hom of Uq (su(2)) with either q ∈ C with |q| = 1 or q ∈ R with 0 < q < 1 or −1 < q < 0; it is important to notice that this is a ∗-isomorphism, (that is the su(2) does really matter in this notation). Case 3 is obtained by duality from Case 2 as explained in Sect. 7. In Case 4, there is a missing relation so Calg (R4ϕ ), which corresponds formally to a version q = 0 of Uq (su(2))hom , is of exponential growth. Case 5 is obtained by α3 -duality (Sect. 7) from Case 6 which is isomorphic to a homogenized version U (su(2))hom of the universal enveloping algebra of su(2) (i.e. q = 1 in Uq (su(2))hom ). Case 7 is the θ -deformation studied in Part I [15] and in [19] while Case 8 (anti θ -deformation) is obtained by α1 -duality (cf. Sect. 7) from Case 7. Case 9 is very singular: 3 relations are missing. Case 10 is obtained by α3 -duality (cf. Sect. 7) from Case 11 which is the ordinary algebra of polynomials with 4 indeterminates C[x 0 , x 1 , x 2 , x 3 ](classical case). Note that the reality conditions above correspond to the hermiticity of the x µ and not of the Sklyanin generators Sµ . It is important to describe the stratification in terms of the roots. The stratas of codimension k correspond to intersections of k singular hyperplanes (up to a factor 21 ) of the form F((α1 , . . . , αk ), (n 1 , . . . , n k )) = ∩

1 H(α j ,n j ) , 2

α j ∈ , n j ∈ Z.

The two dimensional stratas are defined using a single root α ∈  (i.e. k = 1) and since the Weyl group W acts transitively on  only the parity of n matters which gives the two kinds of faces F1 corresponding to n even and F2 to n odd. The one dimensional stratas are defined using two roots α, β ∈  (i.e. k = 2). The roots only matter up to sign and their relative position is described by their angle which (up to the sign) can be π2 in which case we write α ⊥ β, or 2π 3 in which case we write α − β. We thus have the following one dimensional stratas according to the parity of the n j. • α ⊥ β and (n 1 , n 2 ) = (even, odd) gives the line L of Case 4 below. • α − β and (n 1 , n 2 ) = (even, odd) or (n 1 , n 2 ) = (odd, odd) gives the line L  of Case 5 below. • α − β and (n 1 , n 2 ) = (even, even) gives the line L  of Case 6 below. • α ⊥ β and (n 1 , n 2 ) = (even, even) gives the line C+ of Case 7 below. • α ⊥ β and (n 1 , n 2 ) = (odd, odd) gives the line C− of Case 8 below. To double check that the list is complete4 one can use Lemma 4.2 to control the critical set C and then assume by Lemma 4.3 that δ(ϕ) = 0. Then if ϕ ∈ H(α,n) and n is even (resp. odd) the root α is one of the differences ϕk − ϕl (resp. ϕk ). Thus up to permutations of the ϕk one obtains one of the Cases 1)-6). The complete table giving the geometric datas is the following: 4 We are grateful to Marc Bellon for pointing out the subtelty of case 9) which was incomplete in an earlier version.

Noncommutative 3-Spheres

49

Point in M

Case

π 2

Z

1 Generic

α(ϕ) ∈ /

2 Even

ϕ1 = ϕ2 , ϕ2 − ϕ3 ∈ /

3

Odd Face

π 2

ϕj ∈ /

Face

ϕ1 =

π 2

ϕk ∈ /

Z

Correspondence

4 points ∪ Elliptic curve

(id, id, id, id, translation)

2 points, 1 line, 2 conics

  1/2   1/2  i+α i+α id, id, id, σ i−α , σ i−α 1/2 1/2

Z,

α = −J23

, ϕ2 − ϕ3 ∈ / π 2

π 2

Characteristic Variety

π 2

Z

 2 points, 1 line, 2 conics

id, id, −id, exchange with square  1/2 2  i+β σ i−β , β = −J31 1/2

six lines

(id, −id, cyclic permutation of 4 lines (iso, coarse, iso, coarse))

point ∪ P2 (C)

(id, Symmetry of determinant −1)

Z, k = 2, 3

4 α⊥β (0,1)

L=


π

, ϕ, ϕ

5 α−β (0,1)

L =


π

,

6 α−β (0,0)

L  = {(ϕ, ϕ, ϕ)}

point ∪ P2 (C)

(id, id)

7 α⊥β (0,0)

C+ = {(ϕ, ϕ, 0)}

six lines

(id, id, σ (e±2iϕ ))

six lines

(−id, −id, exchanges with square σ (e±4iϕ ))

P3 (C)

Symmetry of determinant − 1 and point → line on a quadric

P3 (C)

Symmetry of determinant 1

P3 (C)

id

8 α⊥β (1,1)

C− =

2

2




9

P=

10

P =

ϕ+


π 2

2

,ϕ

π 2

,

π 2

π 2

,

π 2

,


π

π 2



,

π 2



,ϕ

,0







(in C+ ∩ C− )

11

0 = (0, 0, 0) (in C+ )

The Geometric Data

(5.5)

5.3. Generic case. We now give the detailed description of the geometric data starting with the generic case. This case is defined by  π π  G = ϕ ; ϕj ∈ Z, ϕk − ϕ ∈ Z . (5.6) / / 2 2

50

A. Connes, M. Dubois-Violette

Then the Jk are well defined and are = 0. Let us show that we cannot have J12 = 1, J23 = −1. Indeed tan ϕ3 tan(ϕ1 − ϕ2 ) = −1 means π/2 − ϕ3 = ϕ2 − ϕ1 , while tan ϕ1 tan(ϕ2 −ϕ3 ) = 1 means π2 −ϕ1 = ϕ2 −ϕ3 . This gives π −ϕ1 −ϕ3 = 2 ϕ2 −ϕ1 −ϕ3 and 2 ϕ2 = π which is not allowed by (5.6). We can thus apply the result of Smith-Stafford [29] and get: Proposition 5.1. For ϕ ∈ G the geometric data of R4ϕ is given by 4 points and a nondegenerate elliptic curve E, and σ is identity on the 4 points and a translation of E given explicitly in terms of the parameters α1 = α, α2 = β, α3 = γ with αk = − Jm

(5.7)

by  E = z;

3  0

z 2j

 1−γ 2 1+γ 2 2 z + z + z3 = 0 , = 0, 1+α 1 1−β 2

(5.8)

and ⎡ ⎤ − 2 α β γ z 1 z 2 z 3 − z 0 (− z 02 + β γ z 12 + α γ z 22 + α β z 32 ) ⎡ ⎤ ⎢ ⎥ z0 ⎢ ⎥ 2 α z 0 z 2 z 3 + z 1 (z 02 − β γ z 12 + α γ z 22 + α β z 32 ) ⎥ ⎢z 1 ⎥ σ ⎢ ⎥. ⎣z ⎦ −→ ⎢ 2 2 2 2 ⎢ ⎥ 2 2 β z 0 z 1 z 3 + z 2 (z 0 + β γ z 1 − α γ z 2 + α β z 3 ) ⎣ ⎦ z3 2 γ z 0 z 1 z 2 + z 3 (z 02 + β γ z 12 + α γ z 22 − α β z 32 )

(5.9)

Proof. We rely on Smith-Stafford [29]. The first point is to rewrite the algebra of Proposition (4.5) in the form, [T0 , Tk ]+ = [T , Tm ], [T0 , Tk ] = αk [T , Tm ]+ .

(5.10)

One simply lets T0 = i S0 , Tk = Sk . Then (4.10) means [T , Tm ] = [T0 , Tk ]+ and (4.12) means [T0 , Tk ] = − Jm [T1 , Tm ]+ . Note the crucial − sign in (5.7). By hypothesis on ϕ ∈ G one has α β γ = 0. Moreover we have seen above that for ϕ ∈ G one cannot have J12 = 1, J23 = −1, i.e. α = −1, β = 1 (or any cyclic transformed of that). Now of the α’s, say α in case one  belongs to ±1 the equality α + β + γ + α β γ = 0, i.e. (1 + α) = (1 − α) shows that both +1 and −1 occur and since (−1, 1, x) is excluded the only remaining case is (1, −1, γ ) with γ ∈ / {−1, 0, 1}. Thus the hypotheses of Smith-Stafford are fulfilled and one gets from [29] that besides the 4 points (1, 0, 0, 0) . . . the characteristic variety is the curve in P3 (C) with Eqs. (5.8), the translation σ being given explicity by (5.9).   It will be useful for the computations in the degenerate case to display the list of the 15 minors in the case of the Sklyanin algebra, in the above parameters (α, β, γ ). Their list is given below ([29]).

Noncommutative 3-Spheres

51

{−2 (γ z 1 z 2 − z 0 z 3 ) (z 02 + z 12 + z 22 + z 32 ), 2 (z 0 z 2 − β z 1 z 3 ) (z 02 + z 12 + z 22 + z 32 ), −2 (z 0 z 2 + z 1 z 3 ) (z 02 − γ (β z 12 + z 22 ) + β z 32 ), 2 (z 02 (−(1 + γ ) z 22 + (−1 + β) z 32 ) + z 12 ((−1 + β) γ z 22 + β (1 + γ ) z 32 )), 2 (−z 1 z 2 + z 0 z 3 ) (z 02 − γ (β z 12 + z 22 ) + β z 32 ), 2 (z 0 z 1 − α z 2 z 3 ) (z 02 + z 12 + z 22 + z 32 ), −2 (z 0 z 1 − z 2 z 3 ) (z 02 + γ z 12 − α (γ z 22 + z 32 )), −2 (z 1 z 2 + z 0 z 3 ) (z 02 + γ z 12 − α (γ z 22 + z 32 )),

(5.11)

2 (z 02 ((−1 + γ ) z 12 − (1 + α) z 32 ) + z 22 ((1 + α) γ z 12 + α (−1 + γ ) z 32 )), 2 (z 02 ((1 + β) z 12 − (−1 + α) z 22 ) − ((−1 + α) β z 12 + α (1 + β) z 22 ) z 32 ), −2 (z 0 z 2 − z 1 z 3 ) (z 02 + α z 22 − β (z 12 + α z 32 )), −2 (z 0 z 1 + z 2 z 3 ) (z 02 + α z 22 − β (z 12 + α z 32 )), 2 (z 0 z 2 + β z 1 z 3 ) (z 02 + γ z 12 − α (γ z 22 + z 32 )), 2 (z 0 z 1 + α z 2 z 3 ) (z 02 − γ (β z 12 + z 22 ) + β z 32 ), 2 (γ z 1 z 2 + z 0 z 3 ) (z 02 + α z 22 − β (z 12 + α z 32 ))}. 5.4. Face α = n and n even. F1 = {(ϕ1 , ϕ1 , ϕ3 )}. In that case we have a Sklyanin algebra and with the above notations the parameters are γ = 0 while β = −α = tanϕ1 tan(ϕ3 − ϕ1 ). The list of minors then simplifies as follows: {z 0 z 3 (z 02 + z 12 + z 22 + z 32 ), (z 0 z 2 + α z 1 z 3 ) (z 02 + z 12 + z 22 + z 32 ), −(z 0 z 2 + z 1 z 3 ) (z 02 − α z 32 ), −α z 12 z 32 − z 02 (z 22 + (1 + α) z 32 ), (−z 1 z 2 + z 0 z 3 ) (z 02 − α z 32 ), (z 0 z 1 − α z 2 z 3 ) (z 02 + z 12 + z 22 + z 32 ), −(z 0 z 1 − z 2 z 3 ) (z 02 − α z 32 ), −(z 1 z 2 + z 0 z 3 ) (z 02 − α z 32 ), −α z 22 z 32 − z 02 (z 12 + (1 + α) z 32 ), (−1 + α) (z 12 + z 22 ) (−z 02 + α z 32 ), −(z 0 z 2 − z 1 z 3 ) (z 02 + α (z 12 + z 22 + α z 32 )), −(z 0 z 1 + z 2 z 3 ) (z 02 + α (z 12 + z 22 + α z 32 )), (z 0 z 2 − α z 1 z 3 ) (z 02 − α z 32 ), (z 0 z 1 + α z 2 z 3 ) (z 02 − α z 32 ), z 0 z 3 (z 02 + α (z 12 + z 22 + α z 32 ))}.

(5.12) The detailed analysis shows that the characteristic variety is the union of the two points with coordinates (z 0 , z 1 , z 2 , z 3 ) = (1, 0, 0, 0), (z 0 , z 1 , z 2 , z 3 ) = (0, 0, 0, 1), of the line {(0, z 1 , z 2 , 0)} and of the two conics obtained by intersecting the quadric (z 02 + z 12 + z 22 + z 32 ) = 0 with the two hyperplanes (z 02 − α z 32 ) = 0. The correspondence σ is the identity on the two points, and on the line. It restricts to the two conics and is a rational automorphism of each. It admits two fixed points on each of them and its derivative at the fixed points is given by the following complex numbers: √ √ i− α i+ α √ , √ . i+ α i− α 
5.5. Face α = n and n odd. F2 = { π2 , ϕ2 , ϕ3 }. In that case we have a limiting case of the Sklyanin algebra where the parameter α = ∞ while γ = − β1 . The list of minors

52

A. Connes, M. Dubois-Violette

can be computed directly and gives the following, up to non-zero scalar factors, {(z 1 z 2 + β z 0 z 3 ) (z 02 + z 12 + z 22 + z 32 ), (z 0 z 2 − β z 1 z 3 ) (z 02 + z 12 + z 22 + z 32 ), (−z 0 z 2 − z 1 z 3 ) (β (z 02 + z 12 ) + z 22 + β 2 z 32 ), (−1 + β) (z 02 + z 12 ) (−z 22 + β z 32 ), (−z 1 z 2 + z 0 z 3 ) (β (z 02 + z 12 ) + z 22 + β 2 z 32 ), −z 2 z 3 (z 02 + z 12 + z 22 + z 32 ), (z 0 z 1 − z 2 z 3 ) (−z 22 + β z 32 ), (z 1 z 2 + z 0 z 3 ) (−z 22 + β z 32 ), −β z 02 z 32 − z 22 (z 12 + (1 + β) z 32 ), −β z 12 z 32 − z 22 (z 02 + (1 + β) z 32 ), (−z 0 z 2 + z 1 z 3 ) (z 22 − β z 32 ), (−z 0 z 1 − z 2 z 3 ) (z 22 − β z 32 ), (z 0 z 2 + β z 1 z 3 ) (z 22 − β z 32 ), z 2 z 3 (β (z 02 + z 12 ) + z 22 + β 2 z 32 ), (−z 1 z 2 + β z 0 z 3 ) (z 22 − β z 32 )}.

(5.13)

The detailed analysis shows that the characteristic variety is the union of the two points with coordinates (z 0 , z 1 , z 2 , z 3 ) = (0, 0, 1, 0), (z 0 , z 1 , z 2 , z 3 ) = (0, 0, 0, 1), of the line {(z 0 , z 1 , 0, 0)} and of the two conics obtained by intersecting the quadric (z 02 + z 12 + z 22 + z 32 ) = 0 with the two hyperplanes (z 22 − β z 32 ) = 0. The correspondence σ is the identity on the two points, and is given on the line by σ (z 0 , z 1 , 0, 0) = (z 0 , −z 1 , 0, 0). It exchanges the two conics and is a rational isomorphism of one with the other, moreover the square of σ admits two fixed points on each of them and its derivative at the fixed points is given by the squares of the following complex numbers: √ √ i+ β i− β √ , √ . i− β i+ β
 5.6. Edge α ⊥ β and (n 1 , n 2 ) = (even, odd). L = { π2 , ϕ, ϕ }. From now on we no longer use the change of variables to the Sklyanin algebras but we rely directly on the explicit form of the minors of the original matrix as computed in Appendix 14. Note in particular that the parameters x j are no longer the same as the above z j , but this is irrelevant since we compute intrinsic invariants of the quadratic algebra. In the case at hand the list of minors simplifies (with non-zero scale factors removed) to the following: {(x1 x2 + i x0 x3 ) (−x02 + x12 + x22 + x32 ), (x0 x2 + i x1 x3 ) (x02 − x12 − x22 − x32 ), (x0 x2 + i x1 x3 ) (x02 − x12 + x22 + x32 ), (x02 − x12 ) (x22 + x32 ), (−i x1 x2 +

x0 x3 ) (−x02

+ − − 0, 0, 0}. x12

x22

(5.14)

x32 ), 0, 0, 0, 0, 0, 0, 0,

Thus the characteristic variety consists in the six lines  j given in terms of free parameters x j by {(0, 0, x2 , x3 )}, {(x0 , x1 , 0, 0)}, {(x0 , x0 , x2 , −i x2 )}, {(x0 , x0 , x2 , i x2 )}, {(x0 , −x0 , x2 , i x2 )}, {(x0 , −x0 , x2 , −i x2 )}.

(5.15)

The correspondence σ is the identity on the first line, −1 on the second and permutes cyclically the four others  j . Passing from 4 to 5 or from 6 to 3 one gets the coarse correspondence, while the other maps are rational isomorphisms. The following three lines meet and their point of intersection is mapped by the coarse correspondence to the indicated line: 1 ∩ 4 ∩ 5 → 5 , 1 ∩ 3 ∩ 6 → 3 , 2 ∩ 3 ∩ 4 → 5 , 2 ∩ 5 ∩ 6 → 3 . This is coherent as the restriction of the relevant coarse correspondence.

Noncommutative 3-Spheres

53


 5.7. Edge α − β and (n 1 , n 2 ) = (odd, odd) or (even, odd). L  = { π2 , π2 , ϕ } . In that case the list of minors simplifies (with non-zero scale factors removed) to the following: { x0 x3 (−x02 + x12 + x22 + sin(ϕ)2 x32 ), x1 x3 (−x02 + x12 + x22 + sin(ϕ)2 x32 ), x32 (x0 x2 + i sin(ϕ) x1 x3 ), (x02 − x12 ) x32 , x32 (−i x1 x2 + sin(ϕ) x0 x3 ), x2 x3 (−x02 + x12 + x22 + sin(ϕ)2 x32 ), x32 (x0 x1 − i sin(ϕ) x2 x3 ),

(5.16)

x32 (i x1 x2 + sin(ϕ) x0 x3 ), (x02 − x22 ) x32 , (x12 + x22 ) x32 , x32 (−x0 x2 + i sin(ϕ) x1 x3 ), x32 (x0 x1 + i sin(ϕ) x2 x3 ), x1 x33 , x2 x33 , x0 x33 }. Thus the characteristic variety contains the hyperplane x3 = 0. For x3 = 0 the last three minors show that all other coordinates vanish and this gives an additional point, not in the above hyperplane. The correspondence σ is the symmetry σ (x0 , x1 , x2 , 0) = (−x0 , x1 , x2 , 0). 5.8. Edge α − β and (n 1 , n 2 ) = (even, even). L  = {(ϕ, ϕ, ϕ)}. In that case the list of minors simplifies (with non-zero scale factors removed) to the following: { x0 x3 (sin(ϕ)2 x02 − x12 − x22 − x32 ), x0 x2 (sin(ϕ)2 x02 − x12 − x22 − x32 ), x02 (sin(ϕ) x0 x2 + i x1 x3 ), x02 (x22 + x32 ), x02 (−i x1 x2 + sin(ϕ) x0 x3 ), x0 x1 (sin(ϕ)2 x02 − x12 − x22 − x32 ), x02 (sin(ϕ) x0 x1 − i x2 x3 ), x02 (i x1 x2 + sin(ϕ) x0 x3 ),

(5.17)

x02 (x12 + x32 ), x02 (x12 + x22 ), x02 (sin(ϕ) x0 x2 − i x1 x3 ), x02 (sin(ϕ) x0 x1 + i x2 x3 ), x03 x2 , x03 x1 , x03 x3 }. Thus the characteristic variety contains the hyperplane x0 = 0. For x0 = 0 the last three minors show that all other coordinates vanish and this gives an additional point, not in the above hyperplane. The correspondence σ is the identity. 5.9. Edge α ⊥ β and (n 1 , n 2 ) = (even, even). C+ = {(ϕ, ϕ, 0)}. In that case the list of minors simplifies (with non-zero scale factors removed) to the following: { x0 (x12 + x22 ) x3 , (x12 + x22 ) (−i cos(ϕ) x0 x2 + sin(ϕ) x1 x3 ), (sin(ϕ) x0 x2 + i cos(ϕ) x1 x3 ) (x02 + x32 ), (x02 x22 − x12 x32 ), x1 x2 (x02 + x32 ), (x12 + x22 ) (cos(ϕ) x0 x1 − i sin(ϕ) x2 x3 ), (sin(ϕ) x0 x1 − i cos(ϕ) x2 x3 ) (x02 + x32 ), x1 x2 (x02 + x32 ), (x02 x12 − x22 x32 ), sin(4 ϕ) (x12 + x22 ) (x02 + x32 ), + x22 ) (i sin(ϕ) x0 x2 + cos(ϕ) x1 x3 ), + x22 ) (sin(ϕ) x0 x1 + i cos(ϕ) x2 x3 ), (cos(ϕ) x0 x2 − i sin(ϕ) x1 x3 ) (x02 + x32 ), cos(ϕ) x0 x1 + sin(ϕ) x2 x3 ) (x02 + x32 ), x0 (x12 + x22 ) x3 }. (x12 (x12

(−i

(5.18)

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A. Connes, M. Dubois-Violette

Thus the characteristic variety consists in the six lines  j given in terms of free parameters x j by {(0, x1 , x2 , 0)}, {(x0 , 0, 0, x3 )}, {(x0 , x1 , i x1 , i x0 )}, {(x0 , x1 , −i x1 , i x0 )}, {(x0 , x1 , i x1 , −i x0 )}, {(x0 , x1 , −i x1 , −i x0 )}.

(5.19)

The correspondence σ is the identity on the first two lines. It preserves globally the other  j and induces on each of them the rational automorphism given by multiplication by e±2iϕ .
 5.10. Edge α ⊥ β and (n 1 , n 2 ) = (odd, odd). C− = { π2 + ϕ, π2 , ϕ }. In that case the list of minors simplifies (with non-zero scale factors removed) to the following: {(sin(ϕ)x1 x2 + icos(ϕ)x0 x3 )(x02 − x22 ), x1 x3 (x02 − x22 ), x0 x2 (x12 − x32 ), x02 x32 − x12 x22 , (icos(ϕ)x1 x2 − sin(ϕ)x0 x3 )(x12 − x32 ), (−isin(ϕ)x0 x1 + cos(ϕ)x2 x3 )(x02 − x22 ), (cos(ϕ)x0 x1 − isin(ϕ)x2 x3 )(x02 − x22 ), (cos(ϕ)x1 x2 − isin(ϕ)x0 x3 )(x02 − x22 ), sin(4ϕ)(x02 − x22 )(x12 − x32 ), (x02 x12 − x22 x32 ), x0 x2 (x12 − x32 ), (cos(ϕ)x0 x1 + isin(ϕ)x2 x3 )(x12 − x32 ), x1 x3 (x02 (isin(ϕ)x0 x1 + cos(ϕ)x2 x3 )(x12 − x32 ), (sin(ϕ)x1 x2 − icos(ϕ)x0 x3 )(x12 − x32 )}.

(5.20)

− x22 ),

Thus the characteristic variety consists in the six lines  j given in terms of free parameters x j by {(0, x1 , 0, x3 )}, {(x0 , 0, x2 , 0)}, {(x0 , x1 , x0 , x1 )}, {(x0 , x1 , −x0 , −x1 )}, {(x0 , x1 , x0 , −x1 )}, {(x0 , x1 , −x0 , x1 )}.

(5.21)

The correspondence σ is −1 on the first two lines. It permutes 3 with 4 and its square is the rational automorphism multiplying the ratio x1 /x0 by e4iϕ . It permutes 5 with 6 and its square is the rational automorphism multiplying the ratio x1 /x0 by e−4iϕ . 5.11. Vertex P = ( π2 , π2 , π2 ). In that case all minors vanish identically. Thus the characteristic variety is all projective space. The correspondence σ is given by σ ((x0 , x1 , x2 , x3 )) = (−x0 , x1 , x2 , x3 ), but it degenerates on the quadric Q=



x | x02 −



 xk2 = 0 ,

to a correspondence which assigns to any point p ∈ Q a line ( p) ⊂ Q containing the point σ ( p) and belonging to one of the two families of lines that rule the surface Q.

Noncommutative 3-Spheres

55

5.12. Vertex P  = ( π2 , π2 , 0). In that case all minors vanish identically. Thus the characteristic variety is all projective space. The correspondence σ is given by σ ((x0 , x1 , x2 , x3 )) = (−x0 , x1 , −x2 , x3 ). 5.13. Vertex O = (0, 0, 0). In that case all minors vanish identically and the correspondence σ is the identity. 6. Isomorphism Classes of R4ϕ and Orbits of the Flow F We let as above M be the moduli space of oriented non-commutative 3-spheres and PM its quotient by the symmetry given by Proposition 2.4, 3). For ϕ ∈ M we view the algebra A = Calg (R4ϕ ) as a graded algebra, i.e. we endow it with the one parameter group of automorphisms which rescale the generators xν , θλ ∈ AutA, θλ (xν ) = λ xν , ∀λ ∈ R∗ .

(6.1)

We let PM be the quotient of the real moduli space M by the symmetry of Proposition 2.4, 3). This section will be devoted to prove the following result: Theorem 6.1. Let ϕ j ∈ M; the following conditions are equivalent: a) The graded algebras Calg (R4ϕ j ) are isomorphic. b) ϕ2 ∈ Flow line of ϕ1 in PM. The proof of b) ⇒ a) was given above in Sect. 4.2. The converse is based on the information given by the geometric data which is by construction an invariant of the graded algebra. The proof will be broken up in the non-generic and generic cases.

6.1. Proof in the non-generic case. To analyze the information given by the geometric data we can restrict the parameters ϕ to the fundamental domain A∪ B of Proposition 3.7. The symmetry given by Proposition 2.4, 3) is given explicitly by the transformation ρ(ϕ1 , ϕ2 , ϕ3 ) = (ϕ1 , ϕ1 − ϕ3 , ϕ1 − ϕ2 ).

(6.2)

The identification γ of the face (P  Q Z ) with the face (Z P P  ) (Proposition 3.7) followed by the symmetry ρ (6.2) gives the following symmetry σ = ρ ◦ γ of the face (P  Q Z ), (6.3) σ (ϕ1 , ϕ2 , ϕ3 ) = (π − ϕ1 + ϕ2 , ϕ2 , ϕ2 − ϕ3 ). For vertices we have three elements 0, P, P  modulo the action of 
W . The geometric data allows to separate them. One has a priori nine edges. They fall in five different classes 4) 5) 6) 7) 8). Using γ and ρ one sees that the following four edges are equivalent: [Z P] ∼ [P P  ] ∼ [Q P  ] ∼ [Z Q], and are all in Case 5).

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Similarly, using ρ one sees that the following two edges are equivalent: [O P] ∼ [O Q] and are both in Case 6). The table of geometric datas shows that the geometric data of R4ϕ determines in which of the five Cases 4)-8) one is. Thus when the flow is transitive in the corresponding edge there is nothing to prove. This covers Cases 4), 5), 6). The two Cases 7), 8) correspond to fixed points of the flow. For C+ one gets the edge in [O P  ] ⊂ A¯ given by {ϕ, ϕ, 0} with 0 < ϕ < π2 . The geometric data gives back the set e±2iϕ and this allows to recover ϕ. Thus distinct ϕ give non-isomorphic quadratic algebras. For C− one gets the edge [P  Z ] ⊂ B¯ i.e. {( π2 + ϕ, π2 , ϕ)} whose interior corresponds to 0 < ϕ < π2 . The geometric data gives back the set e±4iϕ . Thus there is an ambiguity ϕ → π2 − ϕ in ϕ knowing the geometric data. To understand it let us note that in fact one checks that σ restricts to the edge [P  Z ] as ϕ → π2 − ϕ. This then accounts for the above ambiguity. We now have to deal with the faces. We start with those which are odd (i.e. Hα,n with n odd). The identification γ of the face (P  Q Z ) with the face (Z P P  ) (Proposition 3.7) shows that we just need to deal with (Z P P  ) and with the face (Q P P  ) which is common to A and B. For the face (Z P P  ) the equation of the supporting hyperplane is ϕ2 = π2 and one is in Case 3) with generic elements of the form (ϕ1 , π2 , ϕ3 ), where ϕ3 + π2 > ϕ1 > π2 > ϕ3 . The geometric data determines the square 2  i + β 1/2 σ , i − β 1/2 1/2

i+β where β = −tanϕ3 /tanϕ1 > 0. Then i−β 1/2 is of modulus one and the geometric data determines β up to the ambiguity given by β → 1/β. But the face (Z P P  ) admits the symmetry given by

γ ◦ ρ(ϕ1 , ϕ2 , ϕ3 ) = (π − ϕ3 , π − ϕ2 , π − ϕ1 ),

(6.4)

whose effect is precisely the transformation β → 1/β. Note that the segment joining P to the middle of [P  Z ] is globally invariant under the flow X . For the face (Q P P  ) the equation of the supporting hyperplane is ϕ1 = π2 and one is in Case 3) with generic elements of the form ( π2 , ϕ2 , ϕ3 ), where π2 > ϕ2 > ϕ3 > 0. The geometric data determines the square 2  i + β 1/2 , σ i − β 1/2 1/2

i+β where β = −tanϕ2 /tanϕ3 < 0. Then i−β 1/2 is real and the geometric data determines β up to the ambiguity given by β → 1/β. But the inequality tanϕ2 > tanϕ3 > 0 shows that in fact β ∈] − ∞, −1[ so that the geometric data determines β uniquely. To summarize we have up to symmetry only two odd faces, the geometric data allows to decide (by |q| = 1 or q ∈ R) in which case one is, and gives back the flow line up to the remaining symmetries. Let us now consider the even faces (i.e. Hα,n with n even). Using ρ we get the equivalence (O P P  ) ∼ (O Q P  ). To be able to use Lemma 4.8 we concentrate on (O P P  )

Noncommutative 3-Spheres

57

on which ϕ3 > 0. The equation of the supporting hyperplane is ϕ1 = ϕ2 and one is in Case 2) with generic elements of the form (ϕ1 , ϕ1 , ϕ3 ), where π2 > ϕ1 > ϕ3 > 0. The geometric data determines   i + α 1/2 σ , i − α 1/2 1/2

i+α where α = tan ϕ1 tan(ϕ1 − ϕ3 ) > 0. Then q = i−α 1/2 is of modulus one. Thus the 1/2 geometric data determines α (since it determines α up to sign). There are however two other even faces namely (O P Q) and (Z P Q). The equation of the supporting hyperplane is the same in both cases and is ϕ2 = ϕ3 and one is in Case 2) with generic elements of the form (ϕ1 , ϕ2 , ϕ2 ), where for (O P Q) one has π π π 2 > ϕ1 > ϕ2 > 0, while for (Z P Q) one gets 2 + ϕ2 > ϕ1 > 2 > ϕ2 . The geometric data determines   i + α 1/2 , σ i − α 1/2 1/2

i+α where α = tan ϕ2 tan(ϕ2 − ϕ1 ) < 0. Then q = i−α 1/2 is real. This first allows to distinguish these faces from the other even faces treated above. Moreover one checks that

α ∈] − 1, 0[, ∀ϕ ∈ (O P Q), α ∈] − ∞, −1[, ∀ϕ ∈ (Z P Q). Thus q > 0 on (O P Q) and q < 0 on (Z P Q) which allows to distinguish these two faces from each other. Finally on each of these faces the geometric data determines α and hence the flow line of ϕ using Lemma 4.8. Thus we get the proof in all cases except the generic case which we shall now treat in detail separately.

6.2. Basic notations for elliptic curves. We recall that given an elliptic curve E viewed as a 1-dimensional complex manifold and choosing a base point p0 ∈ E one gets an isomorphism of the universal cover E˜ of E with base point p0 I E˜  C,

given by the integral  I ( p) =

p

ω,

p0

where ω is a holomorphic (1, 0)-form. This isomorphism is unique up to multiplication by λ ∈ C∗ . Let L ⊂ C be the lattice of periods, then I −1 (L) is the kernel of the covering map π : E˜ → E and one has an isomorphism E ∼ C/L. To eliminate the choice of the base point we let T (E) be the group of translations of E and note that the universal cover T˜ (E) identifies with the additive group C, T˜ (E)  C, T (E)  C/L .

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One can moreover take L of the form L = Z+Zτ , where τ ∈ H/ [1] and H is the upper half plane H = {z ∈ C | Imz > 0}, while in general [n] is the congruence subgroup of level n in S L(2, Z). With H∗ obtained from H by adjoining the rational points of the boundary one has a canonical isomorphism H∗ / [1] → P1 (C) given by Jacobi’s j function. In terms of the elliptic curve E defined by the equation y 2 = 4x 3 − g2 x − g3 , in P2 (C) one has j (E) = 1728

g23 , 

where the discriminant is  = g23 − 27 g32 . One obtains a finer invariant λ(E) if one has the additional structure given by an isomorphism of abelian groups φ:

(Z/2Z)2 →

1 L/L = T2 (E), 2

where T2 (E) is the group of two torsion elements of T (E). This allows to choose the module τ in a finer manner as τ ∈ H/ [2] and one has a canonical isomorphism H∗ / [2] → P1 (C) given by Jacobi’s λ function. In terms of the elliptic curve E defined by the equation  y2 = (x − e j ) in P2 (C) one has λ(E) = Cross Ratio(e1 , e2 ; e3 , e4 ). In more intrinsic terms the labelling of the two torsion T2 (E), ω j ∈ T (E), ω1 = φ(1, 0), ω3 = φ(0, 1), ω2 = −ω1 − ω3

(6.5)

allows to define the following function on the group T (E) of translations of E: Fφ (u) =

℘3 (u) , ℘3 (ω1 )

where ℘3 is defined using a fixed isomorphism T˜ (E)  C, as the sum ⎛ ⎞ ⎛ ⎞   ℘3 (u) = ⎝ y −2 ⎠ − ⎝ y −2 ⎠, π(y)=u

(6.6)

(6.7)

π(y)=ω3

where one defines the sums by restricting y to |y| < R on both sides and then taking the limit. In standard notation with the Weierstrass ℘-function given by   1 1  1 , (6.8) ℘ (z) = 2 + − z (z + )2 2 ∗ ∈L

one gets

℘3 (u) = ℘ (z) − ℘ (w3 ), π(z) = u.

(6.9)

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59

Note that the ratios involved in (6.6) eliminate the scale factor λ in the isomorphism T˜ (E)  C. With these notations one has ℘ (ω2 ) − ℘ (ω3 ) . ℘ (ω1 ) − ℘ (ω3 )

λ(E, φ) = F(w2 ) =

(6.10)

Finally the covering H∗ / [2] → H∗ / [1] is simply given by the algebraic map λ

→

256

(1 − λ + λ2 )3 , λ2 (1 − λ)2

while the group  of deck transformations is the dihedral group generated by the two symmetries u(z) = 1/z v(z) = 1 − z. One has a unique anti-isomorphism w : P S L(2, Z/2Z) →  such that λ(E, φ ◦ α) = w(α)(λ(E, φ)).

(6.11)

Moreover w(t) = v, where t is the transposition of (a, b) → (b, a). Finally one gets, Fφ◦t (u) = 1 − Fφ (u).

(6.12)

6.3. The generic case. We now deal with the generic case. Let s j ∈ R be three real numbers and (α, β, γ ) be given by α=

s3 − s 2 s1 − s3 s2 − s1 , β= , γ = . s1 s2 s3

(6.13)

Let (E, σ ) be the pair of an elliptic curve and a translation associated to (α, β, γ ) by Proposition 5.1. Lemma 6.2. There exists an isomorphism φ : (Z/2Z)2 → T2 (E) such that with e j = s −1 j one has (6.14) λ(E, φ) = Cross Ratio (e2 , e1 ; e3 , ∞), and with Fφ defined by (6.6), Fφ (σ ) =

s1 . s1 − s3

(6.15)

Proof. The proof is straightforward using θ -functions to parametrize the elliptic curve (5.1) but we prefer to give an elementary direct proof. In order to prove (6.14) and (6.15) we start from Proposition 5.1 and replace (α, β, γ ) by their value. The equation of E simplifies to 3 3   xµ2 = 0, sk xk2 = 0. (6.16) 0

1

We then rescale x1 = a X 1 , x2 = b X 2 , x3 = X 3 , where a2 = −

s3 s3 , b2 = − . s1 s2

(6.17)

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A. Connes, M. Dubois-Violette

After this rescaling the second equation of (6.16) gives X 32 = X 12 + X 22 , and one uses the standard rational parametrization of the conic to parametrize the solutions of this equation as x1 = 2 a t, x2 = b(1 − t 2 ), x3 = (1 + t 2 ). (6.18) One then writes the first equation in (5.1) as x02 + 4 a 2 t 2 + b2 (1 − t 2 )2 + (1 + t 2 )2 = 0, and since 1 + b2 = equation

(6.19)

(s2 − s3 )

= 0 this reduces to the elliptic curve defined by the s2 y 2 = t 4 − 2 r t 2 + 1,

(6.20)

where r =−

2 a 2 − b2 + 1 . b2 + 1

Using (6.17) one gets

s1 (s2 + s3 ) − 2 s2 s3 . (6.21) s1 (s3 − s2 ) One checks that r = ±1 since the s j are pairwise distinct. Thus the roots of x 4 − 2 r x 2 + 1 = 0 are all distinct and we write them as ± u, ± v, where uv = 1, u 2 + v 2 = 2r. (6.22) r=

The cross ratio of (−v, u; v, −u) is independent of the above choices and is given by −v − v u − v 4 uv 2 1 + b2 −4 s1 s2 − s3 : =− = − = = = . 2 2 2 2 −v + u u + u (u − v) u +v −2 r −1 1+a s2 s1 − s3 (6.23) one has In other words with e j = s −1 j Cross Ratio(−v, u; v, −u) = Cross Ratio(e2 , e1 ; e3 , ∞).

(6.24)

which transforms (e2 , e1 , e3 , ∞) to (−v, u, v, −u). Let then γ ∈ SL(2, C), γ (X ) = Then for a suitable choice of λ = 0 the transformation, a X +b cX +d

t = γ (X ),

y=

λY , (cX + d)2

gives a birational isomorphism γ˜ of the elliptic curve defined by the equation Y2 =

3 

(X − ei )

(6.25)

1

with the curve (6.20). Choosing the origin of (6.25) as the point at infinity, one gets the point P with coordinates t = −u, y = 0 as the origin of (6.20). One has P = ( p0 , p1 , p2 , p3 ) and p0 = 0 since y = 0. Thus the other coordinates fulfill (6.16) in the simplified form p12 + p22 + p32 = 0, s1 p12 + s2 p22 + s3 p32 = 0, and in homogeneous expressions one can replace pk2 by s − sm .

(6.26)

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61

Let us now determine the translation σ . We just compute the t parameter of σ (P) = (x0 , x1 , x2 , x3 ). The parameter t of (6.18) is recovered in homogeneous coordinates as, t=

b x1 . a x2 + a b x3

(6.27)

One first starts by simplifying (5.9) when applied to P. One has p0 = 0 and up to an overall scale (−α β γ ) one gets using (6.26), i.e. the replacement pk2 → s − sm , −β γ p12 + α γ p22 + α β p32 → − s1 + s2 + s3 , β γ p12 − α γ p22 + α β p32 → s1 − s2 + s3 , β γ p12 + α γ p22 − α β p32 → s1 + s2 − s3 . Thus the coordinates of σ (P) are up to an overall scale x1 = p1 (−s1 + s2 + s3 ), x2 = p2 (s1 − s2 + s3 ), x3 = p3 (s1 + s2 − s3 ),

(6.28)

where the p j are the coordinates of P. These are given by (6.18) taking t = −u, with u, v as above, thus, p1 = −2 a u, p2 = b(1 − u 2 ), p3 = 1 + u 2 .

(6.29)

We then need to compute: t (σ ) =

b(−2 a u)(−s1 + s2 + s3 ) b x1 . (6.30) = a x2 + a b x3 a b(1 − u 2 )(s1 − s2 + s3 ) + a b(1 + u 2 )(s1 + s2 − s3 )

We see that a and b drop out and we remain with t (σ ) =

− u(−s1 + s2 + s3 ) . s1 + (s2 − s3 ) u 2

(6.31)

We just need to compute Cross Ratio(−v, u, v, t (σ )) which is the same as Cross Ratio (e2 , e1 ; e3 , τ ), where τ is ℘ (σ ), up to an affine transformation which transforms the e j to ℘ (ω j ). One has −v − v u−v u − t (σ ) 2v : = −v − t (σ ) u − t (σ ) u − v v + t (σ ) 2 u 2 − u t (σ ) = u 2 − 1 1 + u t (σ )

Cross Ratio(−v, u, v, t (σ )) =

using u v = 1. The u t (σ ) only involves u 2 and one can simplify by its denominator to get (s2 − s3 ) u 4 + (s2 + s3 ) u 2 Cross Ratio(−v, u, v, t (σ )) = 2 . (6.32) (s1 − 2 s3 ) u 4 + 2 s3 u 2 − s1 We then claim that one has: Cross Ratio(−v, u, v, t (σ )) =

s2 − s3 = Cross Ratio(e2 , e1 ; e3 , 0), s1 − s3

(6.33)

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A. Connes, M. Dubois-Violette

i.e. that τ = 0. To see this we replace u 4 by 2 r u 2 − 1 in the expression (6.32) for the cross ratio, which gives Cross Ratio(−v, u, v, t (σ )) =

(2(s2 − s3 ) r + s2 + s3 ) u 2 − (s2 − s3 ) . ((s1 − 2 s3 ) r + s3 ) u 2 − (s1 − s3 )

The computation using (6.21) shows that this fraction is independent of u 2 and equal s2 − s3 . This ends the proof of the lemma since we have shown that, up to an affine to s1 − s3 tranformation, ℘ (ω j ) = e j , ℘ (σ ) = 0, so that Fφ (σ ) =

0 − e3 s1 = . e1 − e3 s1 − s3

  We shall now give the proof of Theorem 6.1 in the generic case. We start with the alcove A. Given an elliptic curve E and a translation σ of E we claim that if there is a labelling φ of T2 (E) such that λ(E, φ) ∈]0, 1[,

Fφ (σ ) < 0,

(6.34)

then this labelling is unique. Indeed the only element of P S L(2, Z/2Z) which preserves the first condition is the transposition t and this does not preserve the second by (6.12). On σ (A) = {s | 1 < s1 < s2 < s3 } one has (Lemma 4.6) Cross Ratio (s2 , s1 ; s3 , 0) ∈]0, 1[,

s1 < 0, s1 − s3

thus by Lemma 6.2 there exists a unique labelling φ of T2 (E) such that (6.34) holds. This s1 s3 gives back both Cross Ratio (s2 , s1 ; s3 , 0) and . The latter gives the ratio =a s1 − s3 s1 s2 − as1 s2 s2 − as1 s1 = , which gives the ratio = b. Thus and the former then gives s1 − as1 s2 (1 − a)s2 s1 we recover the flow line using the convexity of σ (A). Let us now look at the alcove B. One has σ (B) = {s | s3 < s2 < 0, 1 < s1 }. Thus s2−1 < s3−1 < s1−1 . Exactly as above, given an elliptic curve E and a translation σ of E we claim that if there is at most one labelling φ of T2 (E) such that 0 < Fφ (σ ) < λ(E, φ) < 1,

(6.35)

the algebra associated to the s j is unchanged, up to isomorphism, by cyclic permutations of the s j , and the same holds for the associated geometric data. Thus Lemma 6.2 gives a labelling φ such that λ(E, φ) = Cross Ratio (s3 , s2 ; s1 , 0),

Fφ (σ ) =

s2 , s2 − s1

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One checks that it then fulfills (6.35) since with e j = s −1 j one has Cross Ratio(s3 , s2 ; s1 , 0) =

e1 − e3 , e1 − e2

s2 e1 = , s2 − s1 e1 − e2

and e1 − e3 e1 > > 0. e1 − e2 e1 − e2 s1 s3 Thus as above we recover the ratios and and the flow line of ϕ using the convexity s2 s2 of σ (B). Finally note that the conditions (6.34) and (6.35) are exclusive and thus allow to decide using the geometric data whether ϕ ∈ σ (A) or ϕ ∈ σ (B). 1>

7. Dualities We show in this section that there are unexpected dualities between the noncommutative spaces R4ϕ in the following cases of Table 5.5: A ↔ B, 2 ↔ 3, 5 ↔ 6, 7 ↔ 8, 10 ↔ 11. Modulo these dualities the fundamental domain gets reduced from an alcove of the root system D3 to the smaller alcove of the root system C3 . 7.1. Semi-cross product.. Let A be a graded algebra and let α ∈ Aut(A) be a symmetry commuting with the grading (i.e. homogeneous of degree 0). Definition 7.1. The semi-cross product A(α) of A by α is the graded vector space A equipped with the bilinear product .α defined by a .α b = a α n (b), ∀a ∈ An , b ∈ A. It is easily verified that this product is associative and that Am .α An ⊂ Am+n so that A(α) is a graded algebra and that if A is unital then A(α) is also unital with the same unit. As pointed out in the introduction, this notion has been analysed in [5] and used there to reduce and complete the classification of regular algebras of dimension 3 (of [3]) ; the graded algebra A(α) is denoted there by Aα and called the twist of A by α. Some basic properties of the semi-cross product are given by the following proposition. Proposition 7.2. Let A be a graded algebra and let α be an automorphism of degree 0 of A. (i) If β is an automorphism of degree 0 of A which commutes with α, then β is also canonically an automorphism of degree 0 of A(α) and one has A(α)(β) = A(α ◦ β). In particular one has A(α)(α −1 ) = A.

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(ii) If A is a quadratic algebra, then A(α) is also a quadratic algebra and its geometric datas (E  , σ  , L ) are deduced from those (E, σ, L) of A as follows: E  = E, σ  = α t ◦ σ, L = L, where α t is induced by the transposed of (α  A1 ). (iii) If A is an involutive algebra with involution x → x ∗ homogeneous of degree 0 (in short, if A is a graded ∗-algebra) and if α commutes with the involution, then one defines an antilinear antimultiplicative mapping x → x ∗α of A(α) onto A(α −1 ) by setting a ∗α = α −n (a ∗ ) for a ∈ An . In particular if α 2 = 1, then A(α) equipped with the involution x → x ∗α is a graded ∗-algebra. Proof. (i) One has for a ∈ An and b ∈ A, β(a .α b) = β(aα n (b)) = β(a)β(α n (b)) = β(a)α n (β(b)) = β(a) .α β(b), which shows that β is an automorphism of A(α). One has also a .α β n (b) = aα n (β n (b)) = a(α ◦ β)n (b) = a .α◦β b, which implies A(α)(β) = A(α ◦ β). (ii) Assume that A is a quadratic algebra, i.e. A = A(V, R) = T (V )/(R), where V is finite-dimensional and where (R) is the two-sided ideal of the tensor algebra T (V ) of V generated by the subspace R of V ⊗ V . Let m denote the product of A and m denote the product of A(α). Since V = A1 = A(α)1 , one has m = m ◦ (Id ⊗ α) on V ⊗ V and thus m(R) = 0 is equivalent to m ((Id ⊗ α −1 )(R)) = 0 from which it follows easily that A(α) = A(V, R  ) = T (V )/(R  ) with R  = (Id ⊗ α −1 )R so A(α) is quadratic. By definition the graph of σ  is the subset   ⊂ P(V ∗ ) × P(V ∗ ) obtained from the subset of V ∗ × V ∗ of pairs (ω, π ), ω = 0, π = 0 such that ω ⊗ π, r = 0, ∀r ∈ R  . Since R  = (Id ⊗ α −1 )R we thus get σ  = α t ◦ σ . (iii) One has for a ∈ An and b ∈ Am , α −(n+m) ((a .α b)∗ ) = α −(n+m) (α n (b∗ )a ∗ ) = α −m (b∗ )α −m (α −n (a ∗ )), and thus (a .α b)∗α = b∗α .α −1 a ∗α , which implies (iii).

 

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More generally if A is finitely generated in degree 1 and finitely presented, i.e. if A = T (V )/(R) with V finite-dimensional and (R) is the two-sided ideal of T (V ) genN R (R ⊂ V ⊗n ) of T (V ), one has A(α) = erated by the graded subspace R = ⊕n=2 n n N R (α), T (V )/(R(α)) with R(α) = ⊕n=2 n Rn (α) = (Id ⊗ α −1 ⊗ · · · ⊗ α −(n−1) )Rn . In particular A(α) is a N -homogeneous algebra whenever A is N -homogeneous. Let us N recall that an algebra A as above is said to be N -homogeneous iff R = R N ⊂ V ⊗ [6,7]. For these algebras, which generalize the quadratic algebras (N = 2), one has a direct extension of the Koszul duality of quadratic algebras as well as a natural generalization of the notion of Koszulity [6,7]. The stability of the corresponding homological notions with respect to the semi-cross product construction will be studied elsewhere. The terminology semi-cross product of A by α adopted here for A(α) comes from the fact that it can be identified with a subalgebra of the crossed product A
α Z, namely the subalgebra generated by the elements x W, x ∈ A1 , where W denotes the new invertible generator of the crossed product defined by W aW −1 = α(a) for a ∈ A. Indeed one has for a ∈ Am b ∈ A p , aW n bW p = aα n (b) W n+ p . If A is a graded ∗-algebra with α a ∗-homomorphism of degree 0, one endows the crossed product of a structure of ∗-algebra by setting W ∗ = W −1 . The involution of the crossed product induces then, by restriction, the antilinear antimultiplicative mapping ∗α : A(α) → A(α −1 ) of (iii) in the above proposition. For the following application, we shall have α 2 = 1, so A(α) can then be identified with the corresponding subalgebra of the crossed product A
α Z/2Z, and if A is a graded ∗-algebra with α a ∗-homomorphism of degree 0, A
α Z/2Z becomes a ∗-algebra by setting W = W ∗ (= W −1 ) and A(α) is a ∗-subalgebra. 7.2. Application to R4ϕ and Sϕ3 . The above construction allows to give a duality between the following cases of Table 5.5: A ↔ B, 2 ↔ 3, 5 ↔ 6, 7 ↔ 8, 10 ↔ 11, where A and B are the two simplices that together complete Case 1). The explicit transformation on the ϕ-parameters is   π π f 1 (ϕ) = π − ϕ1 , − ϕ1 + ϕ2 , − ϕ1 + ϕ3 , (7.1) 2 2 but there are two others which give relevant additional dualities,  π π f 2 (ϕ) = + ϕ2 − ϕ3 , ϕ2 , − ϕ1 + ϕ2 , 2 2  π π f 3 (ϕ) = − ϕ2 + ϕ3 , − ϕ1 + ϕ3 , ϕ3 . 2 2

(7.2) (7.3)

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P’

Z

Q P

0 Fig. 3. The hyperplanes f j (ϕ) = ϕ

They are all involutions f j2 = 1 and on the fundamental domain A ∪ B one has f 1 (A) = B,

f 1 (B) = A,

f 2 (B) = B,

f 3 (A) = A,

with f 1 (0P Q P  Z ) = (Z P Q P  0), f 2 (P Q P  Z ) = (P Q Z P  ) and f 3 (O P Q P  ) = (P  P Q O). Thus the duality f 2 (resp. f 3 ) operates only in B (resp. A). Moreover the transformations f 2 and f 3 are conjugate under f 1 i.e. f 2 = f 1 ◦ f 3 ◦ f 1 . By Proposition 2.4, 2) any element v of the centralizer C ⊂ SO(4) of the diagonal matrices in SU(4) defines an automorphism of R4ϕ acting by v on the generators. We thus let α j ∈ AutA correspond to the diagonal matrices with respective diagonals given by α1 : (1, 1, −1, −1), α2 : (1, −1, 1, −1), α3 : (1, −1, −1, 1). (7.4) Proposition 7.3. For j ∈ {1, 2, 3} one has canonical ∗-isomorphisms     ρ j : Calg R4f j (ϕ) → Calg R4ϕ (α j ) which preserve the central element S 3f j (ϕ)  Sϕ3 (α j ).



xµ2 and induce corresponding isomorphisms

Proof. One writes explicitly the isomorphisms as follows on the canonical (self-adjoint) µ generators x j of Calg (R4f j (ϕ) ) (which are the canonical noncommutative coordinates of

R4f j (ϕ) ):

ρ1 (x10 ) = x 1 W1 , ρ1 (x11 ) = x 0 W1 , ρ1 (x12 ) = −i x 2 W1 , ρ1 (x13 ) = −i x 3 W1 , ρ2 (x20 ) = x 2 W2 , ρ2 (x21 ) = −i x 3 W2 , ρ2 (x22 ) = x 0 W2 , ρ2 (x23 ) = i x 1 W2 , ρ3 (x30 ) = x 3 W3 , ρ3 (x31 ) = −i x 2 W3 , ρ3 (x32 ) = i x 1 W3 , ρ3 (x33 ) = x 0 W3 . Using the signs in (7.4) one checks that one gets ∗-isomorphisms Calg (R4f j (ϕ) )    µ  Calg (R4ϕ )(α j ) and that one has ρ j ( µ (x j )2 ) = µ (x µ )2 . 

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Corollary 7.4. (1) Let ϕ be generic and (E, σ, L) be the geometric data of Calg (R4ϕ ). Then the geometric data of Calg (R4f j (ϕ) ) is (E, σ + τ j , L), where τ j ∈ T2 (E). (2) The hyperplane f j (ϕ) = ϕ is globally invariant under the flow Z . (3) Let ϕ ∈ A ∪ B be generic, then it belongs to the hyperplane f j (ϕ) = ϕ (with j = 2 on B and j = 3 on A) iff the translation σ fulfills σ ∈ T4 (E). Proof. 1) Using Proposition 7.2 the required equality follows if one shows that the action of α j on E is indeed given by a translation τ j ∈ T2 (E). This can be checked directly using ϑ-functions, i.e. Proposition 9.3.   These new symmetries suggest that one extends the group (T)
W of Sect. 3.3 to include the T j . This amounts to adding the following new roots to the root system  of Sect. 3.3. To the ±ψµ ± ψν we adjoin the ±2ψµ . This is in fact the same as the replacement S O(6) → Sp(3), of the compact group S O(6) by the symplectic group Sp(3) (i.e. of D3 by C3 ). The corresponding Weyl group is now O(3, Z). The description of Sp(3) is obtained as follows using the natural representation ρ : H →   M2 (C) of the field of quaternions H as two by two matrices of the form α β with α, β ∈ C. By definition Sp(3) is the group of three by three matrices −β¯ α¯ Q ∈ M3 (H) whose image ρ(Q) ∈ M6 (C) is unitary. Thus it contains as a subgroup {Q ∈ M3 (H) | ρ(Q) ∈ U (6) ∩ M6 (R)}, which is isomorphic to U (3) and is contained in S O(6) but has only 9 parameters. 8. The Algebras Calg (R4ϕ ) in the Nongeneric Cases In this section we shall describe the algebras Calg (R4ϕ ) in the nongeneric cases using the above dualities. 8.1. Uq (sl(2)), Uq (su(2)) and their homogenized versions. In this section for q ∈ C∗ , Uq (sl(2)) is considered as an associative algebra while for |q| = 1 or q ∈ R∗ , Uq (su(2)) is considered as a ∗-algebra which is a real form of Uq (sl(2)). The coalgebra aspect plays no role in the following and we refer to [23] for a very complete discussion of these topics. It is convenient to start by the homogenized version. For q ∈ C\{−1, 0, 1} we define Uq (sl(2))hom to be the quadratic algebra generated by 4 elements X + , X − , K + , K − with relations K+K− K +X+ K +X− K −X+ K −X−

= K − K +, = q X + K +, = q −1 X − K + , = q −1 X + K − , = q X − K −, (K + )2 − (K − )2 [X + , X − ] = . q − q −1

(8.1) (8.2) (8.3) (8.4) (8.5) (8.6)

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The “classical limit” U (sl(2))hom = U1 (sl(2))hom for q = 1 is obtained by setting q = 1 + ε and K ± = X 0 ± 2ε X 3 . Letting ε → 0, the relations read then [X 0 , X 3 ] = 0,

(8.7)

[X 0 , X + ] = 0,

(8.8)

−

[X , X ] = 0, 0

(8.9)

[X , X ] = 2X X , 3

+

0

−

+

−

[X , X ] = −2X X , 3

−

0

[X , X ] = X X . +

0

3

(8.10) (8.11) (8.12)

To obtain Uq (sl(2)), one notices that relations (8.1) …(8.5) imply that K + K − is central so that one may add the inhomogeneous relation K + K − = ?,

(8.13)

which together with relations (8.1) …(8.6) defines Uq (sl(2)), i.e. Uq (sl(2)) is (as associative algebra) the quotient of Uq (sl(2))hom by the two-sided ideal generated by K + K −−?. Similarily one notices that relations (8.7)…(8.11) imply that X 0 is central and the universal enveloping algebra U (sl(2)) is obtained by adding X 0 = 1 to the relations (8.7) …(8.12). One has of course limq→1 Uq (sl(2)) = U (sl(2)). For q ∈ C\R with |q| = 1 the real version Uq (su(2))hom of Uq (sl(2))hom is obtained by endowing the algebra with the unique antilinear antimultiplicative involution such that (X ± )∗ = X ∓ , ± ∗

∓

(K ) = K ,

(8.14) (8.15)

while for q ∈ R\{−1, 0, 1}, Uq (su(2))hom corresponds to the unique antilinear antimultiplicative involution such that (X ± )∗ = X ∓ , (K ± )∗ = K ± ,

(8.16)

which gives (X ± )∗ = X ∓ , (X 0 )∗ = X 0 ,

(8.17)

(X 3 )∗ = X 3 ,

(8.18)

for the limiting case q = 1, i.e. for U (su(2))hom . These involutions pass to the quotient to define the ∗-algebras Uq (su(2)) for |q| = 1 or q ∈ R∗ . Notice that the involution of Uq (su(2)) is obtained from the one of U (su(2)) 1 3 by setting K ± = q ± 2 X (the relations in terms of X ± and X 3 differ of course).

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8.2. The algebras in the nongeneric cases. We now identify the algebra Calg (R4ϕ ) in the Cases 2 to 11 of Table 5.5. 2. Even face : ϕ1 = ϕ2 = ϕ, ϕ − ϕ3 ∈ π2 Z, ϕk ∈ π2 Z. By setting X ± = x 1 ± i x 2,

(8.19)

the relations (2.24), (2.25) of the algebra read then [x 0 , x 3 ] = 0, cos(ϕ)[x 0 , X + ] = sin(ϕ − ϕ3 )(x 3 X + + X + x 3 ), cos(ϕ − ϕ3 )[x 3 , X + ] = −sin(ϕ)(x 0 X + + X + x 3 ), cos(ϕ)[x 0 , X − ] = −sin(ϕ − ϕ3 )(x 3 X − + X − x 3 ), cos(ϕ − ϕ3 )[x 3 , X − ] = sin(ϕ)(x 0 X − + X − x 0 ), [X + , X − ] = −2sin(ϕ3 )(x 0 x 3 + x 3 x 0 ),

(8.20) (8.21) (8.22) (8.23) (8.24) (8.25)

and since (8.19) implies (X + )∗ = X − , one sees that relations (8.21) and (8.22) are the adjoints of relations (8.23) and (8.24) respectively. We now distinguish the following 2 regions R and R  : R : π2 > ϕ > ϕ3 ≥ 0. By setting K ± = (2tan(ϕ3 ))1/2 ((sin(2ϕ))1/2 x 0 ± i(sin(2(ϕ − ϕ3 )))1/2 x 3 )

(8.26)

the relations (8.20) to (8.25) become (8.1) to (8.6) with q=

1 − i(tan(ϕ − ϕ3 )tan(ϕ))1/2 , 1 + i(tan(ϕ − ϕ3 )tan(ϕ))1/2

(8.27)

so Calg (R4ϕ ) coincides then with Uq (su(2))hom for |q| = 1, q = ±1 as ∗-algebra (one has (K + )∗ = K − ). R  : π2 > ϕ > 0 and π2 + ϕ > ϕ3 > ϕ. By setting K ± = (2tan(ϕ3 ))1/2 ((sin(2ϕ))1/2 x 0 ± (sin(2(ϕ3 − ϕ)))1/2 x 3 ),

(8.28)

the relations (8.20) to (8.25) become (8.1) to (8.6) with q=

1 − (tan(ϕ3 − ϕ)tan(ϕ))1/2 , 1 + (tan(ϕ3 − ϕ)tan(ϕ))1/2

(8.29)

so Calg (R4ϕ ) coincides then with Uq (su(2))hom for q ∈]−1, 0[∪]0, 1[ (one has (K ± )∗ = K ± here). In general for Case 2, it is easy to see that Calg (R4ϕ ) is isomorphic as ∗-algebra either to an algebra of R or of R  . Notice that q > 1, for instance, is the same as q ∈]0, 1[ by exchanging q and q −1 and K + and K − . Thus for Case 2 the Calg (R4ϕ ) are the Uq (su(2))hom . 3. Odd face : ϕ1 = π2 , ϕ2 − ϕ3 ∈ π2 Z, ϕ2 ∈ π2 Z, ϕ3 ∈ π2 Z. Using the analysis of Sect. 7, one sees that the cases corresponding to the plane ϕ1 = π2 are in α3 -duality with the cases corresponding to the plane ϕ2 = ϕ3 . On the other hand, the cases corresponding to the plane ϕ2 = ϕ3 are the same as the cases corresponding to the plane ϕ1 = ϕ2 . So finally (taking into account the forbidden values π2 Z) one sees that Case 3

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(odd face) is obtained by duality (in the sense of Sect. 9) from Case 2 (even face) i.e. from the Uq (su(2))hom . 4. α ⊥ β (0, 1) : ϕ1 = π2 , ϕ2 = ϕ3 = ϕ ∈ π2 Z. This case is singular in the sense that one of the 6 relations is missing, namely: cos(ϕ1 )[x 0 , x 1 ] = isin(ϕ2 − ϕ3 )[x 2 , x 3 ]+ , which gives “0 = 0”. This implies exponential growth. This is a singular limiting case of Uq (su(2))hom for q = 0 which separates the regions 1 < q < 0 and 0 < q < 1 of Case 2. 5. α − β (0, 1) : ϕ1 = π2 , ϕ2 = π2 , ϕ3 ∈ π2 Z. This case is obtained by α3 -duality (Sect. 7) from ϕ1 = ϕ2 = ϕ3 ∈ π2 Z which is Case 6 below. It corresponds to U−1 (su(2))hom . 6. α − β (0, 0) : ϕ1 = ϕ2 = ϕ3 ∈ π2 Z. The relations (2.24), (2.25) reduce in this case to the relations (8.7) to (8.12) by setting X ± = x 1 ± i x 2 , X 3 = 2x 3 and X 0 = −2sin(ϕ1 )x 0 . Thus in this case the ∗-algebra is isomorphic to U (su(2))hom = U1 (su(2))hom . 7. α ⊥ β (0, 0) : ϕ1 = ϕ2 = − 21 θ ∈ π2 Z, ϕ3 = 0. This is the “θ -deformation” studied in [19] and [15]. By setting z 1 = x 0 + i x 3 , z 2 = x 1 + i x 2 , z¯ 1 = (z 1 )∗ , z¯ 2 = (z 2 )∗

(8.30)

the relations (2.24), (2.25) read ⎧ 1 2 z z ⎪ ⎪ ⎨ 1 2 z¯ z z 1 z¯ 2 ⎪ ⎪ ⎩ 1 2 z¯ z¯

= eiθ z 2 z 1 = e−iθ z 2 z¯ 1 = e−iθ z¯ 2 z 1 = eiθ z¯ 2 z¯ 1

(8.31)

and we refer to Part I [15] for more details and generalizations of this algebra. 8. α ⊥ β (1, 1) : ϕ1 = π2 + ϕ, ϕ2 = π2 , ϕ3 = ϕ ∈ π2 Z. This case is obtained from the preceding Case 7 (θ -deformation) by α1 -duality as explained in Sect. 7. 9. ϕ1 = ϕ2 = ϕ3 = π2 . This is the most singular case, 3 relations are missing (i.e. reduce to “0 = 0”) among the 6 relations (2.24), (2.25). The algebra has exponential growth. 10. ϕ1 = ϕ2 = π2 , ϕ3 = 0. This case which is at the intersection of the lines carrying Case 7 and Case 8 is obtained by α3 -duality (Sect. 7) from the next Case 11. 11. ϕ1 = ϕ2 = ϕ3 = 0. This is the “classical case”, the relations (2.24), (2.25) reduce to x µ x ν = x ν x µ for µ, ν ∈ {0, 1, 2, 3} so the algebra reduces to the algebra of polynomials C[x 0 , x 1 , x 2 , x 3 ]. Remark. One can go much further and describe the C ∗ -algebras corresponding to the noncommutative spheres Sϕ3 in all these degenerate cases. It is important in that respect to classify the discrete series besides the obvious continuous series of representations.

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9. The Complex Moduli Space and its Net of Elliptic Curves We first explain in this section a striking coincidence in the generic case between the geometric data of R4ϕ and the fiber of the double cover 4.6. This takes place in the real moduli space and leads us to introduce the complex moduli space in which the equality between the two elliptic curves makes full sense. We then describe the geometric structure of the complex moduli space as a net of elliptic curves in three dimensional projective space and exhibit a presentation of the algebras making the equality “fiber = characteristic” manifest. We showed above in the proof of Lemma 6.2 that the geometric data of R4ϕ can be interpreted as the elliptic curve   3  2 E 2 = (X, Y ) | Y = (X − ei ) (9.1) 1

with e j = s −1 j and endowed with a translation σ sending the point at ∞ to a point of E 2 whose X coordinate vanishes.5 One can write E 2 in the equivalent form,   3  2 E 3 (ϕ) = (X, Y ) | Y = (X s j − 1) . (9.2) 1

In this form this equation is the same as the one involved in the double cover 4.6. More precisely one gets Proposition 9.1. Let ϕ be generic, and Fiber (ϕ) = {ϕ  generic | Jm (ϕ  ) = Jm (ϕ), ∀k}. a) There is a natural isomorphism ϕ : Fiber (ϕ)  E 3 (ϕ) ∩ (R × R∗ ) determined by the equalities  tan(ϕ j ). (9.3) X (ϕ  ) = s j (ϕ  )/s j , ∀ j, Y (ϕ  ) = b) The closure Fiber (ϕ) is the union Fiber (ϕ)∪W (P) and ϕ extends to an isomorphism ϕ : Fiber (ϕ) → E 3 (ϕ) ∩ P2 (R). Proof. a) Let ϕ  ∈ Fiber (ϕ) then by Lemma 4.7 the ratio X = s j (ϕ  )/s j is independent of j. Moreover one has 3 

(X s j − 1) =



tan(ϕ j )

2

,

1

thus the map is well defined. Conversely given (X, Y ) ∈ E 3 (ϕ) ∩ (R × R∗ ) one lets sk = X sk and tk = Y (sk −1)−1 . This determines uniquely the ϕ j such that tan(ϕ j ) = t j . b) Note that Fiber (ϕ) is not connected but falls in four connected components each a flow line of the scaling flow X . These correspond to the four components of E 3 (ϕ) ∩ (R × R∗ ), i.e. of the set of real points of E 3 (ϕ) which are not of two torsion. The inverse map −1 ϕ extends to the four points {∞, e1 , e2 , e3 } of E 3 (ϕ) and one thus obtains a 5 The two choices give isomorphic pairs (E , σ ). 2

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O

S

R

Q

R

S

O

Q

Fig. 4. The elliptic curve E 3 and the fiber for ϕ ∈ A and ϕ ∈ B

compactification of Fiber (ϕ) by adding the four points {P, Q, R, S} in the orbit W (P). The above map ϕ extends continuously to the compactification and the four points {P, Q, R, S} map to the four two torsion points {∞, e1 , e2 , e3 } of E 3 (ϕ) and in that order for ϕ ∈ A as in Fig. 4. The situation is similar on B but the order of e2 and e3 is reversed as well as that of R and S. Moreover the point 0 is now between Q and S instead of being on the left of S as in case ϕ ∈ A (see Fig. 4).   In fact the action of the discrete symmetry given by the Klein group H ⊂ W preserves globally Fiber (ϕ) since it does not alter the Jm (ϕ). In fact this discrete symmetry admits a simple interpretation in terms of translations of order two τ ∈ T2 (E 3 ) as follows, where we let k j ∈ H be given by k1 (ϕ) = (−ϕ1 , ϕ3 − ϕ1 , ϕ2 − ϕ1 ). Proposition 9.2. 1) Given ϕ  ∈ Fiber (ϕ) the map (X, Y ) → (λ X, Y ) for λ = s j (ϕ  )/s j , ∀ j is an isomorphism E 3 (ϕ  )  E 3 (ϕ) compatible with the isomorphisms ϕ and ϕ  of Lemma 9.1. 2) Fiber (ϕ) is globally invariant under k j and under the above isomorphism Fiber (ϕ)  R2 ∩ E 3 (ϕ) the action of k j is given by the translation by the two torsion element (e j , 0) ∈ E 3 (ϕ). Proof. 1) By construction the s j (ϕ  ) are proportional to the s j , so the conclusion follows looking at Definition 9.3 of the identification Fiber (ϕ)  R2 ∩ E 3 (ϕ). 2) Using 1) it is enough to show that the point of E 3 (ϕ) associated to k1 (ϕ) is obtained from (1, t1 t2 t3 ) ∈ E 3 (ϕ) (with t j = tan(ϕ j )) by the translation of order two associated to (e1 , 0) ∈ E 3 (ϕ). One checks that the effect of k1 on the s j (ϕ) is to multiply  tan(ϕ j ) is to replace it by all of them by X  = (1 + t12 )/(s2 s3 ). Its effect on Y = Y  = − t1 (t3 − t1 ) (t2 − t1 )/(s2 s3 ). One then checks by direct computation that the line joining the points (1, t1 t2 t3 ) ∈ E 3 (ϕ) and (X  , −Y  ) ∈ E 3 (ϕ) intersects E 3 (ϕ) in the other point (e1 , 0). The

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73

P

S R Q

Q R S

P Fig. 5. The flow lines

result follows since colinearity of three points A, B, C on the elliptic curve E 3 means A + B + C = 0 in that abelian group, while the opposite of A = (X, Y ) is −A = (X, −Y ).   The above proposition shows that one should give a direct definition of E 3 (ϕ) depending only upon the Jm (ϕ) rather than the s j (ϕ). There is also a strong reason to define in a natural manner the complex points of E 3 (ϕ). Indeed the translation σ corresponds to points where the coordinate X = 0 and the corresponding equation for Y is Y 2 = −1, which does not admit real solutions. 9.1. Complex moduli space MC . We shall show now that there is a natural way of extending the moduli space from the real to the complex domain. The E 3 (ϕ) then appear as a net of degree 4 elliptic curves in P3 (C) having 8 points in common. These elliptic curves will turn out to play a fundamental role and to be closely related to the elliptic curves of the geometric data of the quadratic algebras which their elements label. To extend the moduli space to the complex domain we start with the relations defining the involutive algebra Calg (S 3 (
)) and take for
the diagonal matrix with −1
µ µ := u µ ,

(9.4) µ

where (u 0 , u 1 , u 2 , u 3 ) are the coordinates of u ∈ P3 (C). Using yµ :=
ν z ν one obtains the homogeneous defining relations in the form, u k yk y0 − u 0 y0 yk + u  y ym − u m ym y = 0, u k y0 yk − u 0 yk y0 + u m y ym − u  ym y = 0

(9.5)

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for any cyclic permutation (k, , m) of (1,2,3). The inhomogeneous relation becomes,  (9.6) u µ yµ2 = 1, 3 (u)) only depends upon the class of u ∈ P (C) and the corresponding algebra Calg (SC 3 (see the remark at the end of 9.2). We let Calg (C4 (u)) be the quadratic algebra defined by the six relations (9.5). Taking u µ = e2i ϕµ , ϕ0 = 0, for all µ and x µ := ei ϕµ yµ , we obtain the defining relations of Calg (R4ϕ ) (except for x µ∗ = x µ which allows to pass from C4 (u) to R4ϕ ). Thus the torus T A sits naturally in P3 (C) as

T A = {u ∈ P3 (C) | |u µ | = |u ν |, ∀µ, ν}.

(9.7)

In terms of homogeneous parameters the functions Jm (ϕ) read as Jm (ϕ) = tan(ϕ0 − ϕk )tan(ϕ − ϕm )

(9.8)

for any cyclic permutation (k, , m) of (1,2,3), and extend to the complex domain u ∈ P3 (C) as, Jm (u) =

(u 0 + u  )(u m + u k ) − (u 0 + u m )(u k + u  ) . (u 0 + u k )(u  + u m )

(9.9)

It follows easily from the argument of Proposition 4.5 that for generic values of u ∈ P3 (C) the quadratic algebra Calg (C4 (u)) only depends upon Jk (u). We thus define the complex fiber as F(u) := {v ∈ P3 (C) | Jk (v) = Jk (u)}.

(9.10)

Let then (u) = (a, b, c) = {(u 0 + u 1 )(u 2 + u 3 ), (u 0 + u 2 )(u 3 + u 1 ), (u 0 + u 3 )(u 1 + u 2 )} (9.11)  be the three roots of the Lagrange resolvent of the 4th degree equation (x − u j ) = 0. We view  as a map  : P3 (C)\S → P2 (C), (9.12) where S is the following set of 8 points p0 = (1, 0, 0, 0), p1 = (0, 1, 0, 0), p2 = (0, 0, 1, 0), p3 = (0, 0, 0, 1), (9.13) q0 = (−1, 1, 1, 1), q1 = (1, −1, 1, 1), q2 = (1, 1, −1, 1), q3 = (1, 1, 1, −1). The points q j belong to the torus T A of (9.7), they correspond to the orbit W (P) = (P Q RS). We extend the generic definition (9.10) to arbitrary u ∈ P3 (C)\S and define Fu in general as the union of S with the fiber F(u) of  through u. It can be understood geometrically as follows: Let N be the net of quadrics in P3 (C) which contain S. Given u ∈ P3 (C)\S the elements of N which contain u form a pencil of quadrics with base locus ∩ {Q | Q ∈ N , u ∈ Q} = Yu , (9.14) which is an elliptic curve of degree 4 containing S and u. One has Yu = Fu .

(9.15)

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75

q1 p3

p0 q2

p2

q3

q0 p1

Fig. 6. The Elliptic Curve Fu ∩ P3 (R)

With (u) = (s1 , s2 , s3 ), an explicit isomorphism of the elliptic curve Fu with the elliptic curve  E 3 = {(X, Y ) | Y 2 = (X s j − 1)} of (9.2) is given by (X, Y ) → u, (u k − u 0 )(X sk − 1) − i (u 0 + u k ) Y = 0, ∀k.

(9.16)

Under this isomorphism the point at infinity in E 3 maps to q0 ∈ Fu . 9.2. Notations for ϑ-functions. Let us fix our notations for elliptic ϑ-functions. We fix τ ∈ H a complex number of strictly positive imaginary part and let q = eπiτ , so that |q| < 1. The basic ϑ-function is  2 ϑ3 (z) = q n e2πinz . (9.17) Z

It is periodic in z with period 1 and is (up to scale) the only holomorphic section on E = C/L, L = Z + τ Z of the line bundle associated to the periodicity conditions ξ(z + 1) = ξ(z), ξ(z + τ ) = q −1 e−2πi z ξ(z).

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In particular it is equal (up to scale) to the infinite product ∞ 

(1 + q 2n−1 e2πi z )(1 + q 2n−1 e−2πi z ),

1

and only vanishes at ω2 = 21 + τ2 modulo L. The three other ϑ-functions are deduced from ϑ3 (z) by the translations of order two of E, more precisely one lets,     1 1 1 , ϑ4 (z) = ϑ3 z + . (9.18) iϑ1 (z) = q 4 eπi z ϑ3 (z+ω2 ), ϑ2 (z) = ϑ1 z + 2 2 They all are holomorphic sections on E = C/L, L = Z + τ Z of the line bundles associated to the periodicity conditions of the form ξ(z + 1) = ± ξ(z), ξ(z + τ ) = ± q −1 e−2πi z ξ(z), and it follows that the linear span of their squares ϑ 2j (z) is a complex vector space of dimension two since all are holomorpic sections of a line bundle of degree two. It follows in particular that given any two theta functions the square of any other is a linear combination of the squares of the first two. The relevant coefficients are easy to compute from the values ϑ 2j (0) and if one lets k=

ϑ22 (0) ϑ32 (0)

, k =

ϑ42 (0) ϑ32 (0)

,

one gets ϑ42 (z) = kϑ12 (z)+k  ϑ32 (z), ϑ22 (z) = −k  ϑ12 (z)+k ϑ32 (z), ϑ12 (z) = k ϑ42 (z)−k  ϑ22 (z). (9.19) The λ-function of Jacobi is given by λ(τ ) = k 2 and the ℘-function of Weierstrass by ϑ 2 (z) ℘ (z) = α 42 + β, (9.20) ϑ1 (z) up to irrelevant normalization constants α and β. The main identities we shall use for ϑ-functions are the sixteen theta relations recalled in Appendix 2. 9.3. Fiber = characteristic variety, and the birational automorphism σ of P3 (C). We shall now give, for generic values of (a, b, c) a parametrization of Fu by ϑ-functions. We start with the equations for Fu , (u 0 + u 1 )(u 2 + u 3 ) (u 0 + u 2 )(u 3 + u 1 ) (u 0 + u 3 )(u 1 + u 2 ) = = , a b c

(9.21)

and we diagonalize the above quadratic forms as follows: (u 0 + u 1 ) (u 2 + u 3 ) = Z 02 − Z 12 , (u 0 + u 2 ) (u 3 + u 1 ) = Z 02 − Z 22 , (u 0 + u 3 ) (u 1 + u 2 ) = Z 02 − Z 32 ,

(9.22)

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77

where (Z 0 , Z 1 , Z 2 , Z 3 ) = M.u,

(9.23)

where M is the involution, ⎡

1 1 ⎢1 M := ⎣ 2 1 1

1 1 −1 −1

1 −1 1 −1

⎤ 1 −1 ⎥ . −1 ⎦ 1

(9.24)

In these terms the equations for Fu read Z 2 − Z 22 Z 2 − Z 32 Z 02 − Z 12 = 0 = 0 . a b c

(9.25)

Let now τ ∈ C, Im τ > 0 and η ∈ C be such that one has, modulo projective equivalence,   ϑ2 (0)2 ϑ3 (0)2 ϑ4 (0)2 (a, b, c) ∼ , (9.26) , , ϑ2 (η)2 ϑ3 (η)2 ϑ4 (η)2 where ϑ1 , ϑ2 , ϑ3 , ϑ4 are the theta functions associated as above to τ . More precisely let (a, b, c) be distinct non-zero complex numbers and λ=

a c−b , b c−a

p=

a , a−c

(9.27)

and let τ ∈ C, Im τ > 0 such that λ(τ ) = λ, where λ is Jacobi’s λ function (cf. Subsect. 6.2). Let then η ∈ C be such that with ω1 = 21 , ω3 = τ2 , one has ℘ (η) − ℘ (ω3 ) = p, ℘ (ω1 ) − ℘ (ω3 ) where ℘ is the Weierstrass ℘-function for the lattice L = Z + Zτ ⊂ C. This last equality does determine η only up to sign and modulo the lattice L but this ambiguity does not affect the validity of (9.26) which one checks using the basic properties of ϑ-functions recalled in Subsect. 9.2. Indeed in these terms one has λ(τ ) = k 2 which gives the first equality in (9.26) with (a, b, c) replaced by the right hand side of (9.26). Using (9.20) one gets the second equality since for any z one has ϑ4 (z)2 ℘ (z) − ℘ (ω3 ) = k . ℘ (ω1 ) − ℘ (ω3 ) ϑ1 (z)2 Let then τ and η be fixed by the above conditions; one gets: Proposition 9.3. The following define isomorphisms of C/L with Fu ,   ϑ1 (2z) ϑ2 (2z) ϑ3 (2z) ϑ4 (2z) , , , = (Z 0 , Z 1 , Z 2 , Z 3 ) ϕ(z) = ϑ1 (η) ϑ2 (η) ϑ3 (η) ϑ4 (η) and ψ(z) = ϕ(z − η/2).

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Proof. Up to an affine transformation, ϕ (and ψ are) is the classical projective embedding of C/L in P3 (C). Thus we only need to check that the biquadratic curve Im ϕ = Im ψ is given by (9.25). It is thus enough to check (9.25) on ϕ(z). This follows from the basic relations (9.19) written as ϑ32 (z)ϑ22 (0) = ϑ22 (z)ϑ32 (0) + ϑ12 (z)ϑ42 (0)

(9.28)

ϑ42 (z)ϑ32 (0) = ϑ12 (z)ϑ22 (0) + ϑ32 (z)ϑ42 (0),

(9.29)

and Z 2 −Z 2

Z 2 −Z 2

Z 2 −Z 2

Z 2 −Z 2

which one uses to check 0 a 1 = 0 b 2 and 0 b 2 = 0 c 3 respectively. Let us check the first one using (9.28) to replace all occurrences of ϑ32 (2z) and ϑ32 (η) by the value given by (9.28). One gets Z 02 − Z 22 ϑ 2 (2z) ϑ32 (η) ϑ 2 (2z) ϑ 2 (2z) ϑ22 (η) ϑ12 (2z)ϑ42 (0) = 12 − 32 = 12 + 2 2 b ϑ1 (η) ϑ3 (0) ϑ3 (0) ϑ1 (η) ϑ22 (0) ϑ2 (0)ϑ32 (0) $ % ϑ22 (2z) ϑ12 (2z)ϑ42 (0) − + 2 ϑ22 (0) ϑ2 (0)ϑ32 (0) =

ϑ12 (2z) ϑ22 (η) ϑ12 (η) ϑ22 (0)

−

ϑ22 (2z) ϑ22 (0)

=

Z 02 − Z 12 . a

Let us check the second one using (9.29) to replace all occurrences of ϑ42 (2z) and ϑ42 (η) by the value given by (9.29). One gets ϑ 2 (2z) ϑ42 (η) Z 02 − Z 32 ϑ42 (2z) ϑ12 (2z) ϑ32 (η) ϑ12 (2z)ϑ22 (0) = 12 − = + 2 c ϑ1 (η) ϑ42 (0) ϑ42 (0) ϑ12 (η) ϑ32 (0) ϑ3 (0)ϑ42 (0) $ % ϑ32 (2z) ϑ12 (2z)ϑ22 (0) − + 2 ϑ32 (0) ϑ3 (0)ϑ42 (0) ×

ϑ12 (2z) ϑ32 (η) ϑ12 (η)

ϑ32 (0)

−

ϑ32 (2z) ϑ32 (0)

=

Z 02 − Z 22 . b

  The elements of S are obtained from the following values of z:      1 1 τ τ ψ(η) = p0 , ψ η + = p1 , ψ η + + = p2 , ψ η + = p3 2 2 2 2 and ψ(0) = q0 , ψ

    τ  1 1 τ = q1 , ψ + = q2 , ψ = q3 . 2 2 2 2

(9.30)

(9.31)

(We used M −1 .ψ to go back to the coordinates u µ , but note that M −1 = M and M(q j ) = q j .) Let H ∼ Z2 × Z2 be the Klein subgroup of the symmetric group S4 acting on P3 (C) by permutation of the coordinates (u 0 , u 1 , u 2 , u 3 ).

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79

For ρ in H one has  ◦ ρ = , so that ρ defines for each u an automorphism of Fu . For ρ in H the matrix Mρ M −1 is diagonal with ±1 on the diagonal and the quasiperiodicity of the ϑ-functions allows to check that these automorphisms are translations on Fu by the following 2-torsion elements of C/L,  ρ=  ρ=

0 1

1 0

2 3

3 2

0 2

1 3

2 0

3 1

 is translation by ω1 = 21 , (9.32)

 is translation by ω2 =

1 2

+ τ2 .

Let O ⊂ P3 (C) be the complement of the 4 hyperplanes {u µ = 0} with µ ∈ {0, 1, 2, 3}. −1 −1 −1 Then (u 0 , u 1 , u 2 , u 3 ) → (u −1 0 , u 1 , u 2 , u 3 ) defines an involutive automorphism I of O and since one has −1 −1 −1 −1 (u −1 0 + u k )(u  + u m ) = (u 0 u 1 u 2 u 3 ) (u 0 + u k )(u  + u m ),

(9.33)

it follows that  ◦ I = , so that I defines for each u ∈ O\{q0 , q1 , q2 , q3 } an involutive automorphism of Fu ∩ O which extends canonically to Fu . Note in fact that, as a birational map I continues to make sense on the complement of the 6 lines µ,ν = {u | u µ = 0, u ν = 0} for µ, ν ∈ {0, 1, 2, 3} using (u 0 , u 1 , u 2 , u 3 ) → (u 1 u 2 u 3 , u 0 u 2 u 3 , u 0 u 1 u 3 , u 0 u 1 u 2 ). Proposition 9.4. The restriction of I to Fu is the symmetry ψ(z) → ψ(−z) around any of the points qµ ∈ Fu in the elliptic curve Fu . This symmetry, as well as the above translations by two torsion elements does not refer to a choice of origin in the curve Fu . The proof follows from identities on theta functions but it can be seen directly using the isomorphism E 3  Fu of (9.16). Indeed the symmetry around q0 ∈ Fu corresponds to the transformation (X, Y ) → (X, −Y ) on E 3 and −1 −1 −1 the isomorphism (9.16) carries this back to (u 0 , u 1 , u 2 , u 3 ) → (u −1 0 , u 1 , u 2 , u 3 ) as one checks directly dividing each of Eqs. (9.16) (u k −u 0 )(X sk −1)− i (u 0 +u k ) Y = 0 by u 0 u k to get the same equation but with −Y instead of Y for the u −1 j . The torus T A of (9.7) gives a covering of the real moduli space M. For u ∈ T A , the point (u) is real with projective coordinates (u) = (s1 , s2 , s3 ), sk := 1 + t tm , tk := tan(ϕk − ϕ0 ).

(9.34)

The corresponding fiber Fu is stable under complex conjugation v → v and the intersection of Fu with the real moduli space is given by, FT (u) = Fu ∩ T A = {v ∈ Fu |I (v) = v}.

(9.35)

The curve Fu is defined over R and (9.35) determines its purely imaginary points. Note that FT (u) (9.35) is invariant under the Klein group H and thus has two connected components; we let FT (u)0 be the component containing q0 . The real points, {v ∈ Fu |v = v} = Fu ∩ P3 (R) of Fu do play a complementary role in the characteristic variety as we shall see below.

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Our aim now is to show that for u ∈ P3 (C) generic, there is an astute choice of generators of the quadratic algebra Au = Calg (C4 (u)) for which the characteristic variety E u actually coincides with the fiber variety Fu and to identify the corresponding automorphism σ . Since this coincidence no longer holds for non-generic values it is a quite remarkable fact which we first noticed by comparing the j-invariants of these two elliptic curves. Let u ∈ P3 (C) be generic; we perform the following change of generators: y0 = y1 = y2 = y3 =

√ √ √ √

u1 − u2 u0 + u2 u0 + u3 u0 + u1

√ √ √ √

u2 − u3

u2 − u3

√ √

u 3 − u 1 Y0 ,

u 0 + u 3 Y1 ,

u3 − u1

√ u 0 + u 1 Y2 ,

u1 − u2

√ u 0 + u 2 Y3 .

(9.36)

We let Jm be as before, given by (9.9), J12 =

a−b b−c c−a , J23 = , J31 = , c a b

(9.37)

with a, b, c given by (9.11). Finally let eν be the 4 points of P3 (C) whose homogeneous coordinates (Z µ ) all vanish but one. Theorem 9.5. 1) In terms of the Yµ , the relations of Au take the form [Y0 , Yk ]− = [Y , Ym ]+ , [Y , Ym ]− = −Jm [Y0 , Yk ]+

(9.38) (9.39)

for any k ∈ {1, 2, 3}, (k, , m) being a cyclic permutation of (1,2,3) 2) The characteristic variety E u is the union of Fu with the 4 points eν . 3) The automorphism σ of the characteristic variety E u is given by ψ(z) → ψ(z − η)

(9.40)

on Fu and σ = Id on the 4 points eν . 4) The automorphism σ is the restriction to Fu of a birational automorphism of P3 (C) independent of u and defined over Q. The similarity between the above presentation and the Sklyanin one (cf. (4.10), (4.12)) is misleading, indeed for the latter all the characteristic varieties are contained in the same quadric (cf. [29] § 2.4) 

xµ2 = 0

and can’t of course form a net of essentially disjoint curves.

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81

Proof. By construction E u = {Z | Rank N (Z ) < 4}, where ⎛

Z1

⎜ ⎜ Z2 ⎜ ⎜ ⎜ ⎜ Z3 ⎜ N (Z ) = ⎜ ⎜ ⎜ (b − c)Z 1 ⎜ ⎜ ⎜ ⎜ (c − a)Z 2 ⎝ (a − b)Z 3

−Z 0

Z3

Z2

Z3

−Z 0

Z1

Z2

Z1

−Z 0

(b − c)Z 0

−a Z 3

a Z2

bZ 3

(c − a)Z 0

−bZ 1

−cZ 2

cZ 1

(a − b)Z 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(9.41)

One checks that it is the union of the fiber Fu (in the generic case) with the above 4 points. In fact in terms of the original presentation (9.5), i.e. in terms of the y j the characteristic variety in the generic case is the intersection of the two quadrics,   y 2j = 0, u 2j y 2j = 0, (9.42) and after the change of variables (9.36) it just becomes Z 02 − Z 12 Z 2 − Z 22 Z 2 − Z 32 = 0 = 0 , a b c

(9.43)

as can easily be checked since only the squares Z 2j are involved (and linearly). Note that we already knew that the fiber Fu is abstractly isomorphic to the elliptic curve of the characteristic variety using Lemma 6.2, Proposition 9.2 and the isomorphism (9.16). But here we have shown that their respective embeddings in P3 (C) are the same (i.e. the corresponding line bundles are the same). The automorphism σ of the characteristic variety E u is given by definition by the equation, N (Z ) σ (Z ) = 0, (9.44) where σ (Z ) is the column vector σ (Z µ ) := M · σ (u) (in the variables Z λ ). One checks that σ (Z ) is already determined by the equations in (9.44) corresponding to the first three lines in N (Z ) which are independent of a, b, c (see below). Thus σ is in fact an automorphism of P3 (C) which is the identity on the above four points and which restricts as automorphism of Fu for each u generic. One checks that σ is the product of two involutions which both restrict to E u (for u generic) σ = I ◦ I0 ,

(9.45)

where I is the involution of Proposition 9.4 corresponding to u µ → u −1 µ and where I0 is given by (9.46) I0 (Z 0 ) = −Z 0 , I0 (Z k ) = Z k for k ∈ {1, 2, 3}, and which restricts obviously to E u in view of (9.22). Both I and I0 are the identity on the above four points and since I0 induces the symmetry ϕ(z) → ϕ(−z) around ϕ(0) = ψ(η/2) (Proposition 9.3) one gets the result using Proposition 9.4.

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The fact that σ does not depend on the parameters a, b, c plays an important role. Explicitly we get from the first 3 equations (9.44), ⎛ ⎞   Z ν2 − 2 Zλ⎠ (9.47) σ (Z )µ = ηµµ ⎝ Z µ3 − Z µ ν =µ

λ =µ

for µ ∈ {0, 1, 2, 3}, where η00 = 1 and ηnn = −1 for n ∈ {1, 2, 3}.

 

Remark. Notice that, since the u µ are homogeneous coordinates on P3 (C), the generators yµ as well as the generators Yµ are only defined modulo a non-zero multiplicative scalar. Indeed under a change u µ → ν 2 u µ of homogeneous coordinates (ν ∈ C\{0}), the yµ transform as yµ → ν −1 yµ while the Yµ transform as Yµ → ν −4 Yµ which leaves invariant (9.6) and (9.36) as well as the x µ ((x µ )2 = u µ (yµ )2 ). Later on, it will be convenient to choose a normalization for the generators, e.g. in (11.32). 10. The Map from T2η × [0, τ ] to Sϕ3 and the Pairing This section contains the main technical result of the paper, i.e. both the construction of the one-parameter family of ∗-homomorphisms from the algebra of Sϕ3 (with ϕ generic) to the algebra of the noncommutative torus T2η and the computation of the pairing of the image of the Hochschild 3-cycle with the natural “fundamental class” for the product T2η × [0, τ ]. 10.1. Central elements. We already saw in Sect. 2 (Lemma 2.1) that the algebra Calg (R4ϕ ) of R4ϕ contains in its center the following element:  (x µ )2 . (10.1) Q1 = C = To get another one, one first looks at the Sklyanin algebra defined by (4.10) and (4.12) whose center contains two natural “Casimir” elements C j ,   C1 = Sµ2 , C2 = jk Sk2 , (10.2) where the jk fulfill the relations −

j − jm = Jm . jk

(10.3)

Let us check that C1 commutes with S0 and S . The idea is to only use the relation (4.10) so that the Jm do not interfere with the computation. This means that one treats the commutators as follows: [S0 , Sk2 ] = (S0 Sk + Sk S0 ) Sk − Sk (S0 Sk + Sk S0 ) = i[Sk [S Sm ]]. Thus the sum over k gives 0 in view of the Jacobi identity: [S1 , S02 ] = (S1 S0 + S0 S1 ) S0 − S0 (S1 S0 + S0 S1 ) = − i(S2 S3 − S3 S2 ) S0 +i S0 (S2 S3 − S3 S2 ), [S1 , S22 ] = [S1 , S2 ] S2 + S2 [S1 , S2 ] = i(S0 S3 + S3 S0 ) S2 + i S2 (S0 S3 + S3 S0 ), [S1 , S32 ] = [S1 , S3 ] S3 + S3 [S1 , S3 ] = − i(S0 S2 + S2 S0 ) S3 − i S3 (S0 S2 + S2 S0 ).

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83

One checks that the sum of these terms gives 0. Using cyclic permutations the commutation with Sk easily follows. Remark. It is worth noticing here that the relation [C1 , Sν ] = 0 can be written in the form [18]  [Sµ , [Sµ , Sν ]+ ] = 0, µ

where it becomes apparent that it is a super Lie algebra version of the relation defining the Yang-Mills algebra studied in [16]. As pointed out in [18] the relation (4.10) is the corresponding super analog of the self-duality relation and the fact that it implies [C1 , Sν ] = 0 is the content of Lemma 1 in [18]. Using (4.9) we can then assert that in the generic case the following element is in the center, 

3   cos(ϕk − ϕ )cos(ϕk − ϕm )sinϕk (x k )2 . sin ϕk (x 0 )2 + 1

Subtracting (10.1) multiplied by



(10.4)

sinϕk and using

cos(ϕk − ϕ ) cos(ϕk − ϕm ) − sinϕ sinϕm = cosϕk cos(− ϕk + ϕ + ϕm ), we then get: Q2 =

3 1  sin 2ϕk cos(− ϕk + ϕ + ϕm ) (x k )2 2

(10.5)

1

as a central element. Note that to get that in (10.2), C1 is central, we did not use relation (4.12) and thus it holds irrespective of the finiteness of the Jm . Thus the case δ(ϕ) = 0 is entirely covered to show that (10.5) is central. Proposition 10.1. 1) Both Q j are in the center of Calg (R4ϕ ) for all values of ϕ. 2) Let Sµ = λµ x µ as in (4.9) and λ such that λ jk = (−sk + s + sm )/s sm , then C1 − λ C2 , Q 2 = λC2 . Q1 =  sin ϕk

(10.6)

Proof. 1) We just have to check for Q and this will be done replacing it by (10.4) and using instead of (4.10) the three relations (2.24), sinϕk [x 0 , x k ]+ = icos(ϕ − ϕm ) [x  , x m ].

(10.7)

We just have to repeat the same proof as for the Sµ ’s making sure that any [x 0 , x k ]+ has a sinϕk as coefficient and every [x  , x m ] a cos(ϕ − ϕm ). For the commutator with x 0 this follows from the term sinϕk (x k )2 in (10.4). For the commutator with x 1 this follows from the terms sinϕ1 (x 0 )2 and cos(ϕ1 − ϕk ) (x k )2 , k = 1. 2) Note that the existence of λ follows from Lemma 4.7, i.e. with s˜k = (−sk + s + sm )/s sm the equality −

s˜ − s˜m = Jm . s˜k

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 We have already shown that C1 = sin ϕk Q 1 + Q 2 . It remains to check that Q 2 = λ C2 . Since λ jk = s˜k this amounts to λ2k s˜k = 21 sin 2ϕk cos(− ϕk + ϕ + ϕm ), i.e. s˜k =

cos ϕk cos(− ϕk + ϕ + ϕm ) , cos(ϕk − ϕ ) cos(ϕk − ϕm )

which follows from the definition of s˜k = (−sk + s + sm )/s sm .

 

In terms of the presentation of Theorem 9.5 one gets: Proposition 10.2. Let Au = Calg (C4 (u)) at generic u, then the following three linearly dependent quadratic elements belong to the center of Au : Q m = Jk (Y02 + Ym2 ) + Yk2 − Y2 .

(10.8)

10.2. The Hochschild cycle ch3/2 (U ). With z k = eiϕk x k we have, with ϕ0 = 0,  τ0 = 1, τk = i σk . (10.9) U= τµ z µ , We need to compute ch3/2 (U ) = Ui0 i1 ⊗ Ui∗1 i2 ⊗ Ui2 i3 ⊗ Ui∗3 i3 − Ui∗0 i1 ⊗ Ui1 i2 ⊗ Ui∗2 i3 ⊗ Ui3 i0 .

(10.10)

The trace computation is given by: Lemma 10.3. One has 1 Trace (τα τβ∗ τγ τδ∗ ) = δαβ δγ δ + δβγ δδα − δαγ δβδ + εαβγ δ . 2 Proof. If the set {α, β, γ , δ} is {0, 1, 2, 3} we can assume α = 0 or β = 0 by symmetry in (α, β, γ , δ) → (γ , δ, α, β). For α = 0 we get 21 Trace (τβ τγ τδ ) since the two − signs from τ ∗ cancel. This is cyclic and antisymmetric and gives for (1, 2, 3) using σ1 σ2 = i σ3 the result i 4 × 21 × 2 = 1. For β = 0 we get 1 1 Trace (τα τγ τδ∗ ) = − Trace (τα τγ τδ ) = − ε0αγ δ = εαβγ δ . 2 2 If two of the elements α, β, γ , δ are equal and the two others are different we get 0. Thus there are 3 cases α = β, α = γ , α = δ. One has τα τα∗ = 1, thus if α = β we get 1. For α = γ we get 21 Trace (τα τβ∗ τα τβ∗ ). The two − signs cancel and give 1 1 2 2 Trace (τα τβ τα τβ ). We can assume α = β. If 0 ∈ {α, β} we get 2 Trace (τk ) = −1. If 0∈ / {α, β}, τβ τα = − τα τβ and we get again −1. The case α = δ, β = γ is as α = β, γ = δ. Finally if all indices are equal we get 1.   Proposition 10.4. The Hochschild cycle ch3/2 (U ) ∈ H Z 3 (C ∞ (Sϕ3 )) is given (using ϕ0 = 0 and up to a scalar factor) by  ch3/2 (U ) = εαβγ δ cos(ϕα − ϕβ + ϕγ − ϕδ ) x α ⊗ x β ⊗ x γ ⊗ x δ  −i sin2(ϕµ − ϕν ) x µ ⊗ x ν ⊗ x µ ⊗ x ν . µ,ν

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Proof. The coefficient of x α ⊗ x β ⊗ x γ ⊗ x δ in 21 ch3/2 (U ) is given by × 21 1 1 Trace (τα τβ∗ τγ τδ∗ ) ei(ϕα −ϕβ +ϕγ −ϕδ ) − Trace (τα∗ τβ τγ∗ τδ ) ei(−ϕα +ϕβ −ϕγ +ϕδ ) . 2 2 (10.11) It is non-zero only in the two cases (with cardinality denoted as #), # {α, β, γ , δ} = 4, # {α, β, γ , δ} ≤ 2. In the first case we get as coefficient of x α ⊗ x β ⊗ x γ ⊗ x δ the term εαβγ δ ei(ϕα −ϕβ +ϕγ −ϕδ ) − εβγ δα e−i(ϕα −ϕβ +ϕγ −ϕδ ) = 2εαβγ δ cos(ϕα − ϕβ + ϕγ − ϕδ ), since the cyclic permutation has signature −1. In the second case the terms δαβ δγ δ ei(ϕα −ϕβ +ϕγ −ϕδ ) and δβγ δδα ei(ϕα −ϕβ +ϕγ −ϕδ ) are just δαβ δγ δ and δβγ δδα and they cancel with the terms coming from the second part of 10.10, −δβγ δδα e−i(ϕα −ϕβ +ϕγ −ϕδ ) = −δβγ δδα and −δγ δ δαβ e−i(ϕα −ϕβ +ϕγ −ϕδ ) = −δγ δ δαβ . Thus the following terms remain: − δαγ δβδ ei(ϕα −ϕβ +ϕγ −ϕδ ) − (− δβδ δγ α e−i(ϕα −ϕβ +ϕγ −ϕδ ) ) = −2 i sin(2(ϕα − ϕβ )), which yield the second term in the proposition.

 

We now use the rescaling (4.9) in the case δ(ϕ) = 0 and rewrite ch3/2 in terms of the generators Sµ . We let
=

3 

(tan (ϕ j ) cos(ϕk − ϕ )),

(10.12)

1

s0 = 0, s j = 1 + tan ϕk tan ϕl , ∀ j ∈ {1, 2, 3},

(10.13)

n 0 = 0, n k = 1, ∀k ∈ {1, 2, 3}.

(10.14)

and

Corollary 10.5. One has 
ch3/2 = − εαβγ δ (n α − n β + n γ − n δ )(sα − sβ + sγ − sδ ) Sα ⊗ Sβ ⊗ Sγ ⊗ Sδ α,β,γ ,δ

+ 2i

 (−1)n µ −n ν (sµ − sν ) Sµ ⊗ Sν ⊗ Sµ ⊗ Sν . µ,ν

Proof. We write the above formula as
ch3/2 = −A + 2i B. We let the λµ be as in Lemma 4.4 so that 3  0

and λ20 =

3  1

λµ = −δ(ϕ) = −

3 

(sinϕk cos(ϕ − ϕm ))

(10.15)

1

sin ϕ j , λ2k = sin ϕk cos(ϕk − ϕ ) cos(ϕk − ϕm ).

(10.16)

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One has x µ =

Sµ λµ

and thus in
ch3/2 the first terms of Proposition 10.4 give


εαβγ δ  cos(ϕα − ϕβ + ϕγ − ϕδ )Sα ⊗ Sβ ⊗ Sγ ⊗ Sδ λµ

(ϕ0 = 0).

(10.17)

cos(ϕα − ϕβ + ϕγ − ϕδ )  = (n α − n β + n γ − n δ )(sα − sβ + sγ − sδ ) cosϕk

(10.18)

One has


1  = − . λµ cosϕk

Thus the presence of the term −A follows from the equality

which we now check. We let t j = tan ϕ j . One gets cos(ϕ1 − ϕ2 − ϕ3 )  = 1 − t2 t 3 + t 1 t 3 + t 1 t 2 , cosϕk and more generally for any permutation σ of 1, 2, 3 one has cos(ϕσ (1) − ϕσ (2) − ϕσ (3) )  = −sσ (1) + sσ (2) + sσ (3) . cosϕk

(10.19)

Let us prove (10.18). Since (α, β, γ , δ) is a permutation of (0, 1, 2, 3) one of the indices is 0. For α = 0 we get ϕα = 0 and cos(ϕα − ϕβ + ϕγ − ϕδ )  = −(s0 − sβ + sγ − sδ ), cosϕk since s0 = 0. But (n α − n β + n γ − n δ ) = −1 so that (10.18) holds. For β = 0 we get ϕβ = 0 and cos(ϕα − ϕβ + ϕγ − ϕδ )  = (sα − s0 + sγ − sδ ), cosϕk since s0 = 0. But (n α − n β + n γ − n δ ) = 1 so that (10.18) holds. For γ = 0 we get ϕγ = 0 and cos(ϕα − ϕβ + ϕγ − ϕδ )  = −(sα − sβ + s0 − sδ ), cosϕk since s0 = 0. But (n α − n β + n γ − n δ ) = −1 so that (10.18) holds. Finally for δ = 0, ϕδ = 0 and cos(ϕα − ϕβ + ϕγ − ϕδ )  = (sα − sβ + sγ − s0 ), cosϕk since s0 = 0. But (n α − n β + n γ − n δ ) = 1 so that (10.18) holds and we checked it in all cases. Let us now compute the contribution of the second terms of Proposition 10.4. For i, j ∈ {1, 2, 3} one has −


sin 2(ϕi − ϕ j ) = si − s j . 2λi2 λ2j

(10.20)

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87

Indeed say with i = 2, j = 3, one gets
sinϕ1 = . cosϕ1 cosϕ2 cosϕ3 cos(ϕ2 − ϕ3 ) λ22 λ23 Thus −


sinϕ1 sin(ϕ2 − ϕ3 ) sin 2(ϕ2 − ϕ3 ) = − cosϕ1 cosϕ2 cosϕ3 2λ22 λ23 = −t1 (t2 − t3 ) = (1 + t1 t3 ) − (1 + t1 t2 ) = s2 − s3 .

Next let us check that


sin 2ϕk = sk . (10.21) 2λ20 λ2k  Say with k = 1 one has λ20 = sinϕ j , λ21 = sinϕ1 cos(ϕ1 − ϕ2 )cos(ϕ1 − ϕ3 ) and
cos(ϕ2 − ϕ3 ) sin 2ϕ1 = = s1 . 2 2 cosϕ2 cosϕ3 2λ0 λ1 We thus get in general,
sin2(ϕµ − ϕν ) = −(−1)n µ −n ν 2(sµ − sν ). λ2µ λ2ν

(10.22)

Indeed, for µ, ν ∈ {1, 2, 3} this is (10.20). If both µ, ν = 0, both sides are 0. Now both sides are antisymmetric in µ, ν, thus one can take ν = 0, µ ∈ {1, 2, 3}. Then n µ − n ν = 1 and the result follows from (10.21).   10.3. Elliptic parameters. Let ϕ ∈ T A and using (9.26) let τ ∈ C, Im τ > 0 and η ∈ C and λ ∈ C such that   ϑ2 (0)2 ϑ3 (0)2 ϑ4 (0)2 . (10.23) λ (s1 , s2 , s3 ) = , , ϑ2 (η)2 ϑ3 (η)2 ϑ4 (η)2 We shall call the triplet (τ, η, λ) elliptic parameters for ϕ. They are not unique given ϕ but they determine uniquely the s j , and hence ϕ up to an overall sign. Lemma 10.6. With the above notations (9.27) is equivalent to   ϑ2 (0) ϑ2 (2η) ϑ3 (0) ϑ3 (2η) ϑ4 (0) ϑ4 (2η) (˜s1 , s˜2 , s˜3 ) = λ . , , ϑ2 (η)2 ϑ3 (η)2 ϑ4 (η)2 Proof. By Lemma 4.7 we just need to check that with   ϑ2 (0) ϑ2 (2η) ϑ3 (0) ϑ3 (2η) ϑ4 (0) ϑ4 (2η) ( j1 , j2 , j3 ) = , , ϑ2 (η)2 ϑ3 (η)2 ϑ4 (η)2 one has ( j˜1 , j˜2 , j˜3 ) =



 ϑ2 (0)2 ϑ3 (0)2 ϑ4 (0)2 . , , ϑ2 (η)2 ϑ3 (η)2 ϑ4 (η)2

(10.24)

(10.25)

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The three equalities jk j˜ + j˜k j = 2 follow from the following identities on ϑ-functions: ϑ3 (0)2 ϑ2 (0) ϑ2 (2η) = ϑ2 (η)2 ϑ3 (η)2 − ϑ1 (η)2 ϑ4 (η)2 , ϑ2 (0)2 ϑ3 (0) ϑ3 (2η) = ϑ2 (η)2 ϑ3 (η)2 + ϑ1 (η)2 ϑ4 (η)2 , and similarly ϑ4 (0)2 ϑ3 (0) ϑ3 (2η) = ϑ3 (η)2 ϑ4 (η)2 − ϑ1 (η)2 ϑ2 (η)2 , ϑ3 (0)2 ϑ4 (0) ϑ4 (2η) = ϑ3 (η)2 ϑ4 (η)2 + ϑ1 (η)2 ϑ2 (η)2 , and ϑ2 (0)2 ϑ4 (0) ϑ4 (2η) = ϑ2 (η)2 ϑ4 (η)2 + ϑ1 (η)2 ϑ3 (η)2 , ϑ4 (0)2 ϑ2 (0) ϑ2 (2η) = ϑ2 (η)2 ϑ4 (η)2 − ϑ1 (η)2 ϑ3 (η)2 .   This lemma allows to relate the above parameters with those used by Sklyanin and one has with the above notations ([28]), J12 =

ϑ1 (η)2 ϑ4 (η)2 , ϑ2 (η)2 ϑ3 (η)2

J23 =

ϑ1 (η)2 ϑ2 (η)2 , ϑ3 (η)2 ϑ4 (η)2

J31 = −

ϑ1 (η)2 ϑ3 (η)2 , ϑ2 (η)2 ϑ4 (η)2

(10.26)

which follows from the definition (9.27) of the elliptic parameters together with (9.19). 10.4. The sphere Sϕ3 and the noncommutative torus T2η . Let ϕ ∈ A◦ so that ϕ2 > ϕ3 > 0. We can then choose the elliptic parameters τ and η such that τ ∈ i R+ , η ∈ [0, 1].

π 2

> ϕ1 > (10.27)

Then the module q = eiτ ∈ ]0, 1[ and the ϑ functions ϑ j (z) are all real functions, i.e. fulfill ϑ j (¯z ) = ϑ j (z), ∀z ∈ C. (10.28) In particular the last elliptic parameter λ determined by (9.27) fulfills λ > 0. We shall explain in this section how to use the representations constructed by Sklyanin [28] to obtain ∗-homomorphisms from Calg (Sϕ3 ) to the algebra C ∞ (T2η ) = C ∞ (R/Z)
η Z,

(10.29)

obtained as the crossed product of the algebra C ∞ (R/Z) of smooth periodic functions by the translation η. Recall that a generic element of C ∞ (T2η ) is of the form f =



fn V n

Z

while the basic algebraic rule is given by V f V −1 (u) = f (u + η), ∀u ∈ R/Z, ∀ f ∈ C ∞ (R/Z).

(10.30)

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89

Moreover C ∞ (T2η ) is an involutive algebra with involution turning V into a unitary operator. Starting from the representations constructed in [28] and conjugating by the operator M(ξ )(u) = e−2πiu v/η ξ(u + τ/4) one performs a shift in the indices of the ϑ-functions based on   1 1 τ τ ϑ1 z + = q − 4 e−πi z iϑ4 (z), ϑ2 z + = q − 4 e−πi z ϑ3 (z), 2 2   1 1 τ τ ϑ3 z + = q − 4 e−πi z ϑ2 (z), ϑ4 z + = q − 4 e−πi z iϑ1 (z), 2 2 which allows to replace the singular denominator ϑ1 (2u) by ϑ4 (2u) which no longer vanishes for u ∈ R/Z. One obtains in this way a homomorphism from the Sklyanin algebra to C ∞ (T2η ), but it is not yet unitary and to make it so one needs to conjugate again by a multiplication operator of the form, ¯ ξ(u), N (ξ )(u) = d(u) where the function d ∈ C ∞ (R/Z) fulfills the following conditions, ¯ d(u)d(u) = ϑ4 (2u), ∀u ∈ R/Z. We use the identity ϑ3 (0)2 ϑ4 (0) ϑ4 (2u) = ϑ3 (u)2 ϑ4 (u)2 + ϑ1 (u)2 ϑ2 (u)2 , and thus take c d(u) = ϑ3 (u) ϑ4 (u) + i ϑ1 (u) ϑ2 (u), c2 = ϑ3 (0)2 ϑ4 (0).

(10.31)

Note that one has ¯ d(u) = d(−u), ∀u ∈ R/Z. The effect of the conjugacy N . N −1 on simple monomials is the following: f (u) f (u) V → V, ϑ4 (2 u) d(u) d(−u − η)

f (u) f (u) V∗ → V ∗. ϑ4 (2 u) d(u) d(−u + η)

The formulas which define the images ρ(Sα ) then become, with m ∈ [0, τ ], ϑ3 (2u + η + im) ϑ3 (2u − η − im) ∗ V + ϑ1 (η) V , (10.32) d(u) d(−u − η) d(u) d(−u + η) ϑ4 (2u + η + im) ϑ4 (2u − η − im) ∗ V + i ϑ2 (η) V , (10.33) ρ(S1 ) = −i ϑ2 (η) d(u) d(−u − η) d(u) d(−u + η) ϑ1 (2u + η + im) ϑ1 (2u − η − im) ∗ V + ϑ3 (η) V , (10.34) ρ(S2 ) = ϑ3 (η) d(u) d(−u − η) d(u) d(−u + η) ϑ2 (2u + η + im) ϑ2 (2u − η − im) ∗ V − ϑ4 (η) V , (10.35) ρ(S3 ) = −ϑ4 (η) d(u) d(−u − η) d(u) d(−u + η)

ρ(S0 ) = ϑ1 (η)

and one has

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Theorem 10.7. The formulas (10.32)...(10.35) define a ∗-homomorphism from the (C ∞ ([0, τ ]). Sklyanin algebra to C ∞ (T2η )⊗ Proof. Since by construction ρ is conjugate to an homomorphism, it is an homomorphism and we just need to check that the images ρ(Sµ ) of the generators are self-adjoint elements of C ∞ (T2η ) for each value of m ∈ [0, τ ]. One has ∗  f (u) f¯(u) = V∗ V , d(u) d(−u − η) d(−u) d(u + η) ¯ since η ∈ R and d(x) = d(−x) for x ∈ R. Thus using (10.30) one gets  ∗ f (u) f¯(u − η) V V ∗. = d(u) d(−u − η) d(−u + η) d(u) Since m is real one has ϑ¯ j (x + im) = ϑ j (x − im), ∀x ∈ R, ∀ j using (10.28). Thus one checks directly the required self-adjointness of the ρ(Sµ ).

 

To obtain a ∗-homomorphism from Calg (Sϕ3 ) to C ∞ (T2η ) ⊗ C ∞ ([0, τ ]) we need to normalize the above formulas so that the element Q 1 in the center of Calg (R4ϕ ) gets mapped to 1. By Proposition 10.1 this amounts to introduce an overall scaling factor given by 1/2  σ (m) = (C1 − λ C2 )−1/2 . (10.36) sin ϕ j where the explicit values of the Casimirs C j are given from [28] by C1 = 4 ϑ22 (im), C2 = 4 ϑ2 (η + im) ϑ2 (η − im).

(10.37)

We can now normalize the above homomorphism ρ as ρ(S ˜ j ) = σ (m) ρ(S j ).

(10.38)

we then get using the change of variables Sµ = λµ x µ . Corollary 10.8. The map ρ˜ defines a ∗-homomorphism (C ∞ ([0, τ ]). Calg (Sϕ3 ) → C ∞ (T2η )⊗ Proof. Since ϕ ∈ A one has λµ ∈ R and the above change of variables is a ∗-isomorphism of Calg (R4ϕ ) with the Sklyanin algebra. Thus we only need to check that with the above normalization the ∗-homomorphism ρ˜ maps the central element Q 1 which determines the sphere to the element (C ∞ ([0, τ ]). 1 ∈ C ∞ (T2η )⊗ This follows from Eq. (10.6).

 

In the following we shall use the notation C ∞ (T2η × [0, τ ]) to denote the completed (C ∞ ([0, τ ]). tensor product C ∞ (T2η )⊗

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91

10.5. Pairing with [T2η ]. In order to test the non-triviality of the ∗-homomorphism ρ˜ we shall compute what will be later interpreted as its Jacobian. In order to do this we shall pair the image ρ˜∗ (ch3/2 ) ∈ H Z 3 (C ∞ (T2η × [0, τ ])) (10.39) with the natural Hochschild three cocycle obtained using the fundamental class [T2η ] introduced in [10]. Since the variable m ∈ [0, τ ] labels the center we shall view the above pairing as defining a function of m. The basic Hochschild three cocycle on C ∞ (T2η × [0, τ ])) is given by  i jk τ0 (a0 δi (a1 ) δ j (a2 ) δk (a3 )), (10.40) τ (a0 , . . . , a3 ) = where τ0 is the trace obtained as the tensor product of the canonical trace χ on C ∞ (T2η ) by the trace on C ∞ ([0, τ ]) given by integration,  τ  1 τ0 (a)) = χ (a(m))dm, χ ( f ) = f (u)du, χ ( f V n ) = 0, ∀n = 0. 0

0

(10.41)

The three basic derivations δ j are given by δ1 = ∂/∂m, δ2 = ∂/∂u, δ3 = 2πi V ∂/∂ V,

(10.42)

where in the last term the differentiation V ∂/∂ V has the effect of multiplying by n any monomial f V n . As the product of τ by any function h(m) viewed as an element of the center of C ∞ (T2η × [0, τ ])) is still a Hochschild three cocycle, we obtain a differential one form on [0, τ ] as the pairing ω = ch3/2 , τ .

(10.43)

The basic lemma is then the following using the notation   ϑ2 (0)2 ϑ3 (0)2 ϑ4 (0)2 . (σ0 , σ1 , σ2 , σ3 ) = 0, , , ϑ2 (η)2 ϑ3 (η)2 ϑ4 (η)2

(10.44)

Lemma 10.9. With the above notations one has )  τ, εαβγ δ (n α −n β +n γ −n δ )(σα −σβ +σγ −σδ )ρ(Sα ) ⊗ ρ(Sβ ) ⊗ ρ(Sγ ) ⊗ ρ(Sδ ) α,β,γ ,δ

 −2 i (−1)n µ −n ν (σµ − σν ) ρ(Sµ ) ⊗ ρ(Sν ) ⊗ ρ(Sµ ) ⊗ ρ(Sν )

*

µ,ν

= 24 (2πi)3

ϑ1 (0)3 ϑ1 (η) ϑ1 (2im) . π 3 ϑ2 (η)ϑ3 (η)ϑ4 (η)

The precise meaning of the equality is that for any h ∈ C ∞ ([0, τ ])) the evaluation of the Hochschild cocycle τ on the product of the Hochschild cycle of the right-hand side by h gives the integral  τ h(m) g(m) dm, 0

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where6 g(m) = 24 (2πi)3

ϑ1 (0)3 ϑ1 (η) ϑ1 (2im) . π 3 ϑ2 (η)ϑ3 (η)ϑ4 (η)

(10.45)

The proof of this is a long computation based on the “a priori” properties of the pairing which allow to show that the dependence in the parameters η and m is of the expected form, while the dependence in the module q is that of a modular form. It then follows from the explicit knowledge of enough terms in the q-expansion that the above formula is valid. So far we have not been able to eliminate completely the use of the computer to check this validity and the lemma should thus be given the status of a conjecture rather than that of a mathematical result, due to the non-zero probability of a computational mistake. However what we shall see is that starting from this formula we shall, through the gradual simplifications below, come to a beautifully simple final result. We shall now show that the dependance in m of the normalization factor σ (m) in the definition (10.38) of the homomorphism ρ˜ can be ignored when one computes the pairing (10.43). Lemma 10.10. Let δ, δ  be derivations of the unital algebra A preserving a trace τ0 on A. Let φ j be the multilinear forms on A given by φ1 (a0 , a1 , a2 , a3 ) = τ0 (a0 a1 δ(a2 ) δ  (a3 )), φ2 (a0 , a1 , a2 , a3 ) = τ0 (a0 δ(a1 ) a2 δ  (a3 )). Then for any invertible U ∈ A one has φ j (U, U −1 , U, U −1 ) − φ j (U −1 , U, U −1 , U ) = 0. Proof. One has φ1 (U, U −1 , U, U −1 ) = τ0 (U U −1 δ(U ) δ  (U −1 )) = −τ0 (δ(U ) U −1 δ  (U ) U −1 ), φ1 (U −1 , U, U −1 , U ) = τ0 (U −1 U δ(U −1 ) δ  (U )) = −τ0 (U −1 δ(U ) U −1 δ  (U )), thus the cyclicity of the trace proves the statement for j = 1. Similarly one has φ2 (U, U −1 , U, U −1 ) = τ0 (U δ(U −1 ) U δ  (U −1 )) = τ0 (δ(U ) U −1 δ  (U ) U −1 ) φ2 (U −1 , U, U −1 , U ) = τ0 (U −1 δ(U ) U −1 δ  (U )), and the cyclicity of the trace proves the statement for j = 2.

 

We thus get the following result Corollary 10.11. The pairing of ρ˜∗ (ch3/2 ) with τ is given by the differential form ω= −

σ (m)4 g(m) dm, λ


with
given in (10.12), λ by (9.27), σ (m) by (10.36) and g(m) by (10.45).  6 The q-expansion of the fraction ϑ1 (0) has rational coefficients. π

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93

Proof. Using Lemma 10.9 and Corollary 10.5 one just needs to show that the terms of the form dσ (m) ˜ j )) − σ (m) δ1 (ρ(S j )) = δ1 (ρ(S ρ(S j ) dm do not contribute. But their total contribution is a sum of six terms each of which is of the form φ j (U, U −1 , U, U −1 ) − φ j (U −1 , U, U −1 , U ), where U ∈ M2 (Calg (Sϕ3 )) is the basic unitary element while φ j is as in Lemma 10.10  with δ, δ  ∈ {δ2 , δ3 }. Thus each of these terms vanishes by Lemma 10.10.  10.6. Simplifying the ∗-homomorphism ρ. ˜ We shall make several simplifications in the formulas involved in the construction of the ∗-homomorphism ρ˜ of Corollary 10.8 in order to gradually eliminate all ϑ-functions and express the result in purely algebraic terms. The denominators involved in the construction of the ∗-homomorphism ρ˜ are of the form d(u) d(−u ± η), (10.46) where by (10.31), c d(u) = ϑ3 (u) ϑ4 (u) + i ϑ1 (u) ϑ2 (u), c2 = ϑ3 (0)2 ϑ4 (0). Our first task will be to rewrite (10.46) as a linear form in terms of the projective coordinates ψ(u) of Proposition 9.3, i.e.   ϑ1 (2u − η) ϑ2 (2u − η) ϑ3 (2u − η) ϑ4 (2u − η) , , , = (Z 0 , Z 1 , Z 2 , Z 3 ). ψ(u) = ϑ1 (η) ϑ2 (η) ϑ3 (η) ϑ4 (η) Lemma 10.12. With the above notations one has ϑ3 (0) d(u) d(−u + η) = i ϑ1 (η)ϑ2 (η) Z 1 + ϑ3 (η)ϑ4 (η) Z 3 .

(10.47)

Proof. One has c2 d(u) d(−u + η) = (ϑ3 (u) ϑ4 (u) + i ϑ1 (u) ϑ2 (u))(ϑ3 (u − η) ϑ4 (u − η) − i ϑ1 (u − η) ϑ2 (u − η)) = ϑ3 (u) ϑ4 (u) ϑ3 (u − η) ϑ4 (u − η) + ϑ1 (u) ϑ2 (u) ϑ1 (u − η) ϑ2 (u − η) + i ϑ1 (u) ϑ2 (u) ϑ3 (u − η) ϑ4 (u − η) − i ϑ3 (u) ϑ4 (u) ϑ1 (u − η) ϑ2 (u − η). Thus using the basic addition formulas (obtained from (15.6) and (15.15)) ϑ3 (x)ϑ4 (x)ϑ3 (y)ϑ4 (y) − ϑ1 (x)ϑ2 (x)ϑ1 (y)ϑ2 (y)) = ϑ3 (0)ϑ4 (0)ϑ3 (x + y)ϑ4 (x − y), ϑ1 (x)ϑ2 (x)ϑ3 (y)ϑ4 (y) + ϑ3 (x)ϑ4 (x)ϑ1 (y)ϑ2 (y)) = ϑ3 (0)ϑ4 (0)ϑ1 (x + y)ϑ2 (x − y) for x = u, y = η − u, we get c2 d(u) d(−u + η) = ϑ3 (0) ϑ4 (0) (ϑ3 (η) ϑ4 (2u − η) + i ϑ1 (η) ϑ2 (2u − η)), which gives the required equality.

 

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To simplify the numerators involved in the construction of the ∗-homomorphism ρ˜ we pass from generators Sµ of the Sklyanin algebra to the generators Yµ of Theorem 9.5 by the following transformation: S0 = d Y2 , S1 = i Y3 , S2 = d Y0 , S3 = −Y1 ,

(10.48)

ϑ1 (η)ϑ3 (η) . One checks that the Yµ fulfill the presentation of Theorem 9.5 ϑ2 (η)ϑ4 (η) using the equality (10.26) d 2 = − J31 . We can then reformulate the construction of the homomorphism ρ in the following terms:

where d =

Lemma 10.13. With the above notations one has, up to an overall scalar factor γ , ρ(Yµ ) =

ψµ (u − i m/2) ∗ ψµ (u + i m/2) V + µ V , ¯ L(u) L(u)

(10.49)

where  = (1, 1, 1, −1) and ¯ = L(u). L(u) = i ϑ1 (η)ϑ2 (η) ψ1 (u) + ϑ3 (η)ϑ4 (η) ψ3 (u), L(u) Proof. One just needs to perform the transformation (10.48) on Eqs. (10.32)....(10.35). One gets an overall scalar factor γ = ϑ2 (η) ϑ4 (η) ϑ3 (0), multiplying the right-hand side of (10.49) (or equivalently dividing L(u)).

 

In order to understand (10.49) we let ¯ −1 . Z  =  ψ(u + i m/2), W = L(u)−1 V ∗ , W  = V L(u) (10.50) Note that one has Z  =  Z¯ and W  = W ∗ but we shall ignore that for a while and treat for instance Z and Z  as independent variables. The multiplicative terms such as L(u)−1 do not alter the cross product rules (10.30) but they alter the simplification rule V V ∗ = V ∗ V = 1. Our next task will thus be to give a simple expression for W W  in terms of (Z , Z  ). One has by construction Z = ψ(u − i m/2),

−1 ¯ , W W  = (L(u) L(u))

(10.51)

and we need to express the denominator in terms of Z and Z  . Note that we have the freedom to multiply by an arbitrary function of m since this only alters the normalization of ρ which is needed in any case to pass to ρ. ˜ Lemma 10.14. With the above notations one has, ¯ ν(m) L(u) L(u) = J23 (Z 0 Z 0 + Z 1 Z 1 ) + Z 2 Z 2 − Z 3 Z 3 , where ν(m) =

2 ϑ32 (im) ϑ32 (0)ϑ32 (η)ϑ42 (η)

.

(10.52)

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95

Proof. one has ¯ L(u) L(u) = ϑ12 (η)ϑ22 (2u − η) + ϑ32 (η)ϑ42 (2u − η), thus with a = 2u − η + im, b = 2u − η − im,

a+b = 2u − η, 2

a−b = im 2

we get     a−b a+b ¯ ϑ22 ϑ32 (im) L(u) L(u) = ϑ12 (η) ϑ32 2 2     a−b a+b ϑ42 . +ϑ32 (η) ϑ32 2 2

(10.53)

We now use the addition formulas     a−b a+b 2ϑ32 ϑ22 = ϑ22 (0)ϑ3 (a)ϑ3 (b) + ϑ32 (0)ϑ2 (a)ϑ2 (b) 2 2 −ϑ42 (0)ϑ1 (a)ϑ1 (b) (adding (15.9) and (15.10)) and     2 a−b 2 a+b 2 ϑ3 ϑ4 = ϑ22 (0) ϑ1 (a) ϑ1 (b) + ϑ32 (0) ϑ4 (a) ϑ4 (b) 2 2 + ϑ42 (0) ϑ3 (a) ϑ3 (b) (adding (15.5) and (15.6)) which allow to write (10.53) as a symmetric bilinear form in (Z , Z  ). One then uses (9.19) and (10.26) J23 = to obtain the required equality.

ϑ12 (η)ϑ22 (η) ϑ32 (η)ϑ42 (η)

,

 

Proposition 10.15. With the above notations one has, up to an overall scalar factor δ(m), (10.54) ρ(Yµ ) = Z µ W + W  Z µ with algebraic rules given by Z i W W  Z j =

Z i Z j Q(Z , Z  )

, W f (Z , Z  ) = f (σ (Z ), σ −1 (Z  )) W,

where σ is the translation by −η as in Theorem 9.5 and Q(Z , Z  ) = J23 (Z 0 Z 0 + Z 1 Z 1 ) + Z 2 Z 2 − Z 3 Z 3 .

(10.55)

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Proof. The first equality follows from Lemma 10.13 and Definition 10.50 of Z , Z  , W, W  . The first algebraic rule follows from (10.51) and Lemma 10.14. To obtain the second we need to understand the transformation  ψ(u + im/2) →  ψ(u − η + im/2), and to compare it with σ −1 , where σ is the translation by −η as in Theorem 9.5. By construction σ is the product (9.45) of two involutions σ = I ◦ I0 , where I0 just alters the sign of Z 0 (cf. (9.46)). Thus σ −1 = I0 ◦ I = I0 ◦ σ ◦ I0 and to show that the above tranformation is σ −1 it is enough to show that σ commutes with I0 ◦ I3 where I3 (Z ) =  Z . This follows from the commutation of translations on the elliptic curve and can be checked directly using (9.47).   11. Algebraic Geometry and C ∗-Algebras In this section we shall develop the basic relation between noncommutative differential geometry in the sense of [10] and noncommutative algebraic geometry. This will be obtained by abstracting the results of Proposition 10.15 of Subsect. 10.6 and giving a general construction, independent of ϑ-functions, of a homomorphism from a quadratic algebra to a crossed product algebra constructed from the geometric data. 11.1. Central quadratic forms and generalised cross-products. Let A = A(V, R) = T (V )/(R) be a quadratic algebra. Its geometric data {E, σ, L} is defined in such a way that A maps homomorphically to a cross-product algebra obtained from sections of powers of the line bundle L on powers of the correspondence σ ([4]). We shall begin by a purely algebraic result which considerably refines the above homomorphism and lands in a richer cross-product. We use the notations of Sect. 5 for general quadratic algebras. Definition 11.1. Let Q ∈ S 2 (V ) be a symmetric bilinear form on V ∗ and C a component of E × E. We shall say that Q is central on C iff for all (Z , Z  ) in C and ω ∈ R one has, ω(Z , Z  ) Q(σ (Z  ), σ −1 (Z )) + Q(Z , Z  ) ω(σ (Z  ), σ −1 (Z )) = 0.

(11.1)

By construction the space of symmetric bilinear form on V ∗ which are central on C is a linear subspace of S 2 (V ). Let C be a component of E × E globally invariant under the map σ˜ (Z , Z  ) := (σ (Z ), σ −1 (Z  )). (11.2) Given a quadratic form Q central and not identically zero on the component C, we define as follows an algebra C Q as a generalised cross-product of the ring R of meromorphic functions on C by the transformation σ˜ . Let L, L  ∈ V be such that L(Z ) L  (Z  ) does not vanish identically on C. We adjoin two generators W L and W L  which besides the usual cross-product rules, W L f = ( f ◦ σ˜ ) W L , W L  f = ( f ◦ σ˜ −1 ) W L  , ∀ f ∈ R

(11.3)

fulfill the following relations: W L W L  := π(Z , Z  ),

W L  W L := π(σ −1 (Z ), σ (Z  )),

(11.4)

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where the function π(Z , Z  ) is given by the ratio, π(Z , Z  ) :=

L(Z ) L  (Z  ) . Q(Z , Z  )

(11.5)

The a priori dependence on L, L  is eliminated by the rules, W L 2 :=

L 2 (Z ) WL 1 L 1 (Z )

W L  := W L  2

1

L 2 (Z  ) L 1 (Z  )

(11.6)

which allow to adjoin all W L and W L  for L and L  not identically zero on the projections of C, without changing the algebra and provides an intrinsic definition of C Q . Our first result is Lemma 11.2. Let Q be central and not identically zero on the component C. (i) The following equality defines a homomorphism ρ, A → C Q , √

2 ρ(Y ) :=

Y (Z  ) Y (Z ) W L + W L    , L(Z ) L (Z )

∀Y ∈ V.

(11.7)

(ii) If σ 4 = ?, then ρ(Q) = 1, where Q is viewed as an element of T (V )/(R). √ Proof. (i) Formula (11.7) is independent of L, L  using (11.6) and reduces (up to 2) to WY + WY when Y is non-trivial on the two projections of C. It is enough to check that the ρ(Y ) ∈ C Q fulfill the quadratic relations ω ∈ R. Let ω ∈ R,  ω(Z , Z  ) = ωi j Yi (Z ) Y j (Z  ), viewed as a bilinear form on V ∗ . One has    2 ωi j ρ(Yi )ρ(Y j ) = ωi j (WYi + WY i )(WY j + WY j ) = ωi j Yi (Z )Y j (σ (Z )W 2 +



ωi j

Yi (Z )Y j (Z  )  Yi (σ (Z  ))Y j (σ −1 (Z ))   + ω + ωi j W 2 Yi (σ −1 (Z  ))Y j (Z  ), i j  −1  Q(Z , Z ) Q(σ (Z ), σ (Z ))

where W2 =

1 1   W L2 , W 2 = W L2  −1    . L(Z )L(σ (Z )) L (σ (Z ))L (Z )

The vanishing of the terms in W 2 and in W correspondence σ , i.e. the equality

2

is automatic by construction of the

ω(Z , σ (Z )) = 0, ∀Z ∈ E. The sum of the middle terms is just ω(Z , Z  ) ω(σ (Z  ), σ −1 (Z )) + = 0,  Q(Z , Z ) Q(σ −1 (Z ), σ (Z  )) as follows from Definition 11.1 and the symmetry of Q. (ii) The above computation shows that ρ(Q) = 1, provided one can show that Q(Z , σ (Z )) = 0 and Q(σ −1 (Z  ), Z  ) = 0 for all Z , Z  in the projections E, E  of C. We

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assume that σ 4 (Z ) is not identically equal to Z on each connected component of E (resp. E  ) and use (11.1) with Z  = σ (Z ). The first term vanishes and we get ω(σ 2 (Z ), σ −1 (Z )) Q(Z , σ (Z )) = 0, ∀ω ∈ R. Thus if Q(Z , σ (Z )) does not vanish identically on a given connected component E 1 of E one gets that ω(σ 2 (Z ), σ −1 (Z )) = 0, ∀Z ∈ E 1 , ω ∈ R, so that σ −1 (Z ) = σ 3 (Z ) for all Z ∈ E 1 which contradicts the hypothesis.

 

Let Au = Calg (C4 (u)) at generic u, then by Proposition 10.2 the center of Au contains the three linearly dependent quadratic elements Q m := Jk (Y02 + Ym2 ) + Yk2 − Y2 .

(11.8)

Proposition 11.3. Let Au = Calg (C4 (u)) at generic u, then each Q m is central on Fu × Fu (⊂ E u × E u ).  

Proof. One uses (9.45) to check the algebraic identity.

Together with Lemma 11.2 this yields non-trivial homomorphisms of Au whose unitarity will be analysed in the next section. Note that for a general quadratic algebra A = A(V, R) = T (V )/(R) and a quadratic form Q ∈ S 2 (V ), such that Q ∈ Center(A), it does not automatically follow that Q is central on E × E. For instance Proposition 11.3 no longer holds on Fu × {eν }, where eν is any of the four points of E u not in Fu . In fact let us describe in some detail what happens in the case of the θ -deformations, i.e. C+ = {(ϕ, ϕ, 0)} (Case 7). We take the notations of Subsect. 5.9 to write the characteristic variety as the union of six lines  j . Proposition 11.4. Let Calg (R4ϕ ) for ϕ ∈ C+ , and Q k be defined by (10.1) and (10.5). (1) Each Q k is central on i × j provided i and j belong to the same subsets I = {1, 2} and J = {3, 4, 5, 6}. (2) The bilinear form Q 1 does not vanish identically on i ×  j iff i = j for i, j ∈ I and iff i = j for i, j ∈ J . (3) The bilinear form Q 2 vanishes identically on 2 ×  j and  j × 2 for all j. This is proved by direct computations. Note that since Q 1 fails to be central on 1 ×3 for instance, it was crucial to “localize” the notion of central quadratic form to components of the square E × E of the characteristic variety E. It is of course also crucial to check the non-vanishing of Q when applying Lemma 11.2, and the component 2 does not work for Q 2 in that respect. The precise table for the vanishing of the form Q 2 is the following, where = at (i, j) means that Q 2 does not vanish identically on i ×  j ;

= 0

=

=

=

=

0 0 0 0 0 0

= 0 0

=

= 0

= 0

= 0 0

=

= 0

= 0 0

=

= 0 0

=

= 0

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11.2. Positive central quadratic forms on quadratic ∗-algebras. The algebra Au , u ∈ T is by construction a quadratic ∗-algebra, i.e. a quadratic complex algebra A = A(V, R) which is also a ∗-algebra with involution x → x ∗ preserving the subspace V of generators. Equivalently one can take the generators of A (spanning V ) to be hermitian elements of A. In such a case the complex finite-dimensional vector space V has a real structure given by the antilinear involution v → j (v) obtained by restriction of x → x ∗ . Since one has (x y)∗ = y ∗ x ∗ for x, y ∈ A, it follows that the set R of relations satisfies ( j ⊗ j)(R) = t (R)

(11.9)

in V ⊗ V , where t : V ⊗ V → V ⊗ V is the transposition v ⊗ w → t (v ⊗ w) = w ⊗ v. This implies Lemma 11.5. The characteristic variety is stable under the involution Z → j (Z ) and one has σ ( j (Z )) = j (σ −1 (Z )). We let C be as above an invariant component of E × E; we say that C is j-real when it is globally invariant under the involution ˜ , Z  ) := ( j (Z  ), j (Z )). j(Z

(11.10)

By Lemma 11.5 this involution commutes with the automorphism σ˜ (11.2) and one has Proposition 11.6. Let C be a j-real invariant component of E × E and Q central on C be such that ˜ , Z  )) = Q(Z , Z  ), ∀(Z , Z  ) ∈ C. Q( j(Z (11.11) (1) The following turns the cross-product C Q into a ∗-algebra, ˜ , Z  )), f ∗ (Z , Z  ) := f ( j(Z

(W L )∗ = W j (L) ,

(W L  )∗ = W j (L  ) . (11.12)

(2) The homomorphism ρ of Lemma 11.2 is a ∗-homomorphism. Proof. We used the transpose of j to define j (L) in (11.12) by j (L)(Z ) = L( j (Z )),

∀Z ∈ V ∗ .

(11.13)

The compatibility of the involution with (11.3) follows from the commutation of j˜ with σ˜ . Its compatibility with (11.4) follows from π ∗ (Z , Z  ) :=



L( j (Z  )) L  ( j (Z )) ˜ , Z  )) Q( j(Z

−

=

j (L  )(Z ) j (L)(Z  ) . Q(Z , Z  )

To check 2) one writes for Y ∈ V , ρ(Y )∗ = (WY + WY )∗ = W j (Y ) + W j (Y ) = ρ( j (Y ).  

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We have treated so far Z and Z  as independent variables. We shall now restrict the above construction to the graph of j, i.e. to {(Z , Z  ) ∈ C | Z  = j (Z )}. Composing ρ with the restriction to the subset K = {Z | (Z , j (Z )) ∈ C}, one obtains in fact a ∗homomorphism θ of A = A(V, R) to a twisted cross-product C ∗ -algebra, C(K )×σ, L Z which involves the full geometric data (E, σ, L) and encodes the central quadratic form Q as a Hermitian metric on L provided Q fulfills the following positivity. Definition 11.7. Let C be a j-real invariant component of E × E and Q central on C. Then Q is positive on C iff it fullfills (11.11) and Q(Z , j (Z )) > 0, ∀Z ∈ K . One can then endow the line bundle L dual of the tautological line bundle on P(V ∗ ) with the Hermitian metric defined by f L , g L  Q (Z ) = f (Z ) g(Z )

L(Z ) L  (Z ) Q(Z , j (Z ))

L , L  ∈ V,

Z ∈ K , ∀ f, g ∈ C(K ).

(11.14) We view f L and g L  as sections of L and the right-hand side of the formula as a function on K which expresses their inner product f L , g L  . This defines a Hermitian metric on the restriction of L to K . Before we proceed we need to describe the general notion due to Pimsner [25] of twisted cross product. Given a compact space K , an homeomorphism σ of K and a hermitian line bundle L on K we define the C ∗ -algebra C(K ) ×σ, L Z as the twisted cross-product of C(K ) by the Hilbert C ∗ -bimodule associated to L and σ ([2,25]). We n let for each n ≥ 0, Lσ be the hermitian line bundle pullback of L by σ n and (cf. [4,29]) Ln := L ⊗ Lσ ⊗ · · · ⊗ Lσ

n−1

.

(11.15)

We first define a ∗-algebra as the linear span of the monomials ξ W n , W ∗n η∗ , ξ, η ∈ C(K , Ln )

(11.16)

with product given as in ([4,29]) for (ξ1 W n 1 ) (ξ2 W n 2 ) so that (ξ1 W n 1 ) (ξ2 W n 2 ) := (ξ1 ⊗ (ξ2 ◦ σ n 1 )) W n 1 +n 2 .

(11.17)

We use the hermitian structure of Ln to give meaning to the products η∗ ξ and ξ η∗ for ξ, η ∈ C(K , Ln ). The product then extends uniquely to an associative product of ∗-algebra fulfilling the following additional rules: (W ∗k η∗ ) (ξ W k ) := (η∗ ξ ) ◦ σ −k ,

(ξ W k ) (W ∗k η∗ ) := ξ η∗ .

(11.18)

The C ∗ -norm of C(K ) ×σ, L Z is defined as for ordinary cross-products and due to the amenability of the group Z there is no distinction between the reduced and maximal norms. The latter is obtained as the supremum of the norms in involutive representations in Hilbert space. The natural positive conditional expectation on the subalgebra C(K ) shows that the C ∗ -norm restricts to the usual sup norm on C(K ). To lighten notations in the next statement we abbreviate j (Z ) as Z¯ , but one should take care that in general the expression for j (Z ) can differ from Z¯ for instance with the notations of Subsect. 10.6 one gets j (Z ) =  Z¯ .

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Theorem 11.8. Let K ⊂ E be a compact σ -invariant subset and Q be central and strictly positive on {(Z , Z¯ ); Z ∈ K }. Let L be the restriction to K of the dual of the tautological line bundle on P(V ∗ ) endowed with the hermitian metric , Q . √ (i) The equality 2 θ (Y ) := Y W + W ∗ Y¯ ∗ yields a ∗-homomorphism θ : A = A(V, R) → C(K ) ×σ, L Z. (ii) For any Y ∈ V the C ∗ -norm of θ (Y ) fulfills √ Sup K !Y ! ≤ 2! θ (Y )! ≤ 2 Sup K !Y !. (iii) If σ 4 = ?, then θ (Q) = 1 where Q is viewed as an element of T (V )/(R). (i) The subset K˜ = {(Z , j (Z )); Z ∈ K } ⊂ C is globally invariant under σ˜ by Lemma 11.5. Moreover j˜ defined in (11.10) is the identity on K˜ . Each L ∈ V defines a section of L and hence an element L W ∈ C(K ) ×σ, L Z. The Definition (11.14) of the hermitian structure of L then shows that the elements L W and W ∗ j (L  )∗ of C(K ) ×σ, L Z fulfill the same algebraic rules (11.3), (11.4) as the W L and W L  while the involution of C(K ) ×σ, L Z is the restriction of the involution of Proposition 11.11. Thus the conclusion follows from Lemma 11.2. (ii) Since (Y W )(Y W )∗ = Y ∗ Y the C ∗ -norm of Y W is Sup K !Y !. It follows that √ 2! θ (Y )! ≤ 2 Sup K !Y !. For any complex number u of modulus one the map ξ W n → u n ξ W n extends to a ∗-automorphism of C(K ) ×σ, L Z. It √ follows taking u = i that !Y W − W ∗ Y¯ ∗ ! = !Y W + W ∗ Y¯ ∗ ! and Sup K !Y ! ≤ 2! θ (Y )!. (iii) follows from Lemma 11.2.  

Proof.

We shall now apply this general result to the algebras Calg (R4ϕ ). We take the quadratic form  X µ X µ (11.19) Q(X, X  ) := in the x-coordinates, so that Q is the canonical central element defining the sphere Sϕ3 by the equation Q = 1. Proposition 11.3 shows that in the generic case, i.e. for ϕ ∈ A ∪ B, the quadratic form Q is central on Fϕ × Fϕ with obvious notations. The positivity of Q is automatic since in the x-coordinates the involution jϕ coming from the involution of the quadratic ∗-algebra Calg (R4ϕ ) is simply complex conjugation jϕ (Z ) = Z¯ , so that Q(X, jϕ (X )) > 0 for X = 0. We thus get, Corollary 11.9. Let K ⊂ Fϕ be a compact σ -invariant subset. The homomorphism θ of Theorem 11.8 is a unital ∗-homomorphism from Calg (Sϕ3 ) to the cross-product C ∞ (K ) ×σ, L Z. This applies in particular to K = Fϕ . It follows that one obtains a non-trivial C ∗ -algebra C ∗ (Sϕ3 ) as the completion of Calg (Sϕ3 ) for the semi-norm, !P! := Sup! π(P)!,

(11.20)

where π varies through all unitary representations of Calg (Sϕ3 ). It was clear from the start that (11.20) defines a finite C ∗ -semi-norm on Calg (Sϕ3 ) since the equation of the sphere

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p3

q1

q2 p0 q0 p2 p1 q3 Fig. 7. The Elliptic Curve Fϕ ∩ P3 (R) (odd case)

 µ 2 (x ) = 1 together with the self-adjointness x µ = x µ∗ show that in any unitary representation one has ! π(x µ )! ≤ 1, ∀µ. What the above corollary gives is a lower bound for the C ∗ -norm such as that given by statement (ii) of Theorem 11.8 on the linear subspace V of generators. To analyse the compact σ -invariant subsets of Fϕ for generic ϕ, we distinguish the even case which corresponds to all sk having the same sign (cf. Fig. 4) (and holds for instance for ϕ ∈ A) from the odd case when all sk don’t have the same sign. First note that in all cases the real curve Fϕ ∩ P3 (R) is non-empty (it contains p0 ), and has two connected components since it is invariant under the Klein group H (9.32). In the even case σ preserves each of the two connected components of the real curve Fϕ ∩ P3 (R). In the odd case it permutes them (cf. Fig. 7). Proposition 11.10. Let ϕ be generic and even. (i) Each connected component of Fϕ ∩ P3 (R) is a minimal compact σ -invariant subset. (ii) Let K ⊂ Fϕ be a compact σ -invariant subset, then K is the sum in the elliptic curve Fϕ with origin p0 of K T = K ∩ FT (ϕ)0 (cf. 9.35) with the component Cϕ of Fϕ ∩ P3 (R) containing p0 . (iii) The cross-product C(Fϕ ) ×σ, L Z is isomorphic to the mapping torus of the automorphism β of the noncommutative torus T2η = Cϕ ×σ Z acting on the generators   1 4 by the matrix . 0 1 (i) This holds if we assume that ϕ is “generic” so that the elliptic parameter η fullfills η ∈ / Q. The diophantine approximations of η will play an important role later on. (ii) By construction the abelian compact group Fϕ is the product T1 × T2 of the one-dimensional tori T1 = Cϕ and T2 = FT (ϕ)0 , i.e. the component of FT (ϕ)

Proof.

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containing q0 (cf. 9.35). The translation σ is η × Id and the action of η is minimal on T1 = Cϕ . (iii) The isomorphism class of the cross-product C(Fϕ ) ×σ, L Z depends of L only through its class as a hermitian line bundle on the two torus Fϕ . This class is entirely specified by the first Chern class c1 (L). By construction one gets c1 (L) = 4 since the space of holomorphic sections of L is the 4-dimensional space V . Let U j be the generators of the algebra C(T2η ) where the presentation of the algebra is U1 U2 = e2πiη U2 U1 . For any integer k let βk be the automorphism of C(T2η ) acting on the generators U j by βk (U1 ) := U1 ,

βk (U2 ) := U1k U2 .

(11.21)

By construction the mapping torus T (βk ) of the automorphism βk is given by the algebra C(T (βk )) of continuous maps s ∈ R → x(s) ∈ C(T2η ) such that x(s + 1) = βk (x(s)), ∀s ∈ R. We just need to show that C(T (β4 )) is isomorphic to C(Fϕ ) ×σ, L Z and this follows from the general isomorphism C(T (βk ))  C(T1 × T2 ) ×η×Id, L Z,

(11.22)

(with T j = R/Z) for any hermitian line bundle L on T1 × T2 with c1 (L) = k. To check this one chooses L so that its continuous sections C(T1 × T2 , L) are scalar functions f (u, m) with u, m ∈ R such that f (u + 1, m) = f (u, m),

f (u, m + 1) = e2πik u f (u, m), ∀u, m ∈ R,

while its hermitian metric is given by f, g (u, m) = f (u, m) g(u, m), ∀u, m ∈ R. One defines a map α :

C(T (βk )) → C(T1 × T2 ) ×η×Id, L Z,

by writing for x ∈ C(T (βk )), x = (x(s)), x(s) ∈ C(T2η ) the Fourier expansion  x(s) = x(s, n) U2n , x(s, n) ∈ C(T1 ). Then the x(s, n) ∈ C(T1 ) define sections xn ∈ C(T1 × T2 , Ln ), and one just lets α(x) =



xn W n ∈ C(T1 × T2 ) ×η×Id, L Z, ∀x ∈ C(T (βk )).

One then checks that this gives the required isomorphism (11.22).

 

Corollary 11.11. Let ϕ be generic and even, then Fϕ ×σ, L Z is a noncommutative 3-manifold with an elliptic action of the three dimensional Heisenberg Lie algebra h3 and an invariant trace τ .

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Proof. This follows 7 from Proposition 11.10 (iii). One can construct directly the action of h3 on C ∞ (Fϕ ) ×σ, L Z by choosing a constant (translation invariant) curvature connection ∇, compatible with the metric, on the hermitian line bundle L on Fϕ (viewed in the C ∞ -category not in the holomorphic one). The two covariant differentials ∇ j corresponding to the two vector fields X j on Fϕ generating the translations of the elliptic curve, give a natural extension of X j as the unique derivations δ j of C ∞ (Fϕ ) ×σ, L Z fulfilling the rules, δ j ( f ) = X j ( f ), ∀ f ∈ C ∞ (Fϕ ), δ j (ξ W ) = ∇ j (ξ ) W, ∀ξ ∈ C ∞ (Fϕ , L).

(11.23)

We let δ be the unique derivation of C ∞ (Fϕ ) ×σ, L Z corresponding to the grading by powers of W . It vanishes on C ∞ (Fϕ ) and fulfills δ(ξ W k ) = i k ξ W k ,

δ(W ∗k η∗ ) = −i k W ∗k η∗ .

(11.24)

Let i κ be the constant curvature of the connection ∇; one gets [δ1 , δ2 ] = κ δ, [δ, δ j ] = 0, which provides the required action of the Lie algebra h3 on C ∞ (Fϕ ) ×σ, L Z.

(11.25)  

It follows that one is exactly in the framework developed in [10]. We refer to [26] and [1] where these noncommutative manifolds were analysed in terms of crossed products by Hilbert C ∗ -bimodules. Integration on the translation invariant volume form dv of Fϕ gives the h3 -invariant trace τ ,  τ( f ) = f dv, ∀ f ∈ C ∞ (Fϕ ), τ (ξ W k ) = τ (W ∗k η∗ ) = 0, ∀k = 0.

(11.26)

It follows in particular that the results of [10] apply to obtain the calculus. In particular the following gives the “fundamental class” as a 3-cyclic cocycle:  τ3 (a0 , a1 , a2 , a3 ) = i jk τ (a0 δi (a1 ) δ j (a2 ) δk (a3 )), (11.27) where the δ j are the above derivations with δ3 := δ. We shall in fact describe the same calculus in greater generality in the last section which will be devoted to the computation of the Jacobian of the homomorphism θ of Corollary 11.9. Similar results hold in the odd case. Then Fϕ ∩ P3 (R) is a minimal compact σ -invariant subset, any compact σ -invariant subset K ⊂ Fϕ is the sum in the elliptic curve Fϕ with origin p0 of Fϕ ∩ P3 (R) with K T = K ∩ FT (ϕ)0 but the latter is automatically invariant under the subgroup H0 ⊂ H of order 2 of the Klein group H (9.32), H0 := {h ∈ H | h(FT (ϕ)0 ) = FT (ϕ)0 }.

(11.28)

The group law in Fϕ is described geometrically as follows. It involves the point q0 . The sum z = x + y of two points x and y of Fϕ is z = I0 (w), where w is the 4th point of 7 It justifies the terminology “nilmanifold”.

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intersection of Fϕ with the plane determined by the three points {q0 , x, y}. It commutes by construction with complex conjugation so that x + y = x + y¯ , ∀x, y ∈ Fϕ . By Lemma 11.5 the translation σ is imaginary for the canonical involution jϕ . In terms of the coordinates Z µ this involution is described as follows, using (9.36) (multiplied by ei(π/4−ϕ1 −ϕ2 −ϕ3 ) 2−3/2 ) to change variables. Among the 3 real numbers λk = cosϕ cosϕm sin(ϕ − ϕm ),

k ∈ {1, 2, 3},

two have the same sign  and one, λk , k ∈ {1, 2, 3}, the opposite sign. Then jϕ =  Ik ◦ c,

(11.29)

where c is complex conjugation on the real elliptic curve Fϕ (Sect. 3) and Iµ the involution Iµ (Z µ ) = −Z µ , Iµ (Z ν ) = Z ν , ν = µ. (11.30) The index k and the sign  remain constant when ϕ varies in each of the four components of the complement of the four points qµ in FT (ϕ). The sign  matters for the action of jϕ on linear forms as in (11.13), but is irrelevant for the action on Fϕ . Each involution Iµ is a symmetry z → p − z in the elliptic curve Fϕ and the products Iµ ◦ Iν form the Klein subgroup H (9.32) acting by translations of order two on Fϕ . The quadratic form Q of (11.19) is given in the new coordinates by,   tk s k Q k (11.31) Q= cos2 ϕ with sk := 1 + t tm , tk := tan ϕk and Q k defined by (10.8). Let us assume that 0 < ϕ1 < ϕ2 < ϕ3 < π/2 for instance, then the relation between the x µ and the Yµ is given with the appropriate normalization of the Yµ by x µ = ρµ Yµ , where ρ02 = −sin(ϕ1 − ϕ2 ) sin(ϕ1 − ϕ3 ) sin(ϕ2 − ϕ3 ), ρ12 = −cosϕ2 cosϕ3 sin(ϕ2 − ϕ3 ), (11.32) 2 2 ρ2 = cosϕ1 cosϕ3 sin(ϕ1 − ϕ3 ), ρ3 = −cosϕ1 cosϕ2 sin(ϕ1 − ϕ2 )). All the ρµ are real except for ρ2 which is purely imaginary and the involution j is I2 ◦ c. One checks directly that    ρµ2 Yµ2 = tk s k Q k . cos2 ϕ We can now compare the ∗-homomorphism ρ˜ of Sect. 10.6 with the ∗-homomorphism obtained from a positive central quadratic form, one gets with the constants s and b given by,   s = −s2 cosϕ j cos(ϕk − ϕ ). sinϕ j , b2 = Proposition 11.12. Let 0 < ϕ1 < ϕ2 < ϕ3 < π/2. (i) The ∗-homomorphism ρ˜ is the ∗-homomorphism associated to the central quadratic form Q  , s Q  = Q 1 + Q 3 + s2 Q 2 , which is positive on E for the involution I3 ◦ c.

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(ii) Let β(Y0 ) = iY2 , β(Y1 ) =

+

+ + + J12 Y3 , β(Y2 ) = − J12 J23 Y0 , β(Y3 ) = i J23 Y1 ,

the map b β gives a ∗-isomorphism sending the form Q  into Q and the involution I3 ◦ c into I2 ◦ c.  Proof. (i) By Proposition 10.15 it is enough to show that Q  corresponds to (x µ )2 µ under the composition of the transformations Sµ = λµ x of Lemma 4.4 and (10.48), S0 = d Y2 , S1 = i Y3 , S2 = d Y0 , S3 = −Y1 , where d 2 = −J31 . (ii) The map b β is obtained as the composition of the isomorphisms (10.48), Sµ = λµ x µ of Lemma 4.4 and x µ = ρµ Yµ with ρµ given in (11.32). Thus the answer follows since each of these is a ∗-isomorphism and the image of Q  in the  maps µ µ 2 x variables is simply (x ) .   Let ϕ be generic and even and v ∈ FT (ϕ)0 . Let K (v) = v + Cϕ be the minimal compact σ -invariant subset containing v (Proposition 11.10 (ii)). By Corollary 11.9 we get a homomorphism, θv : Calg (Sϕ3 ) → C ∞ (T2η ), (11.33) whose non-triviality will be proved below in Corollary 12.8. We shall first show (Theorem 11.13) that it transits through the cross-product of the field K q of meromorphic functions on the elliptic curve by the subgroup of its Galois group AutC (K q ) generated by the translation σ . For Z = v + z , z ∈ Cϕ , one has using (11.29) and (9.35), jϕ (Z ) = Iµ (Z − v) − I (v),

(11.34)

which is a rational function r (v, Z ). Fixing ϕ, v and substituting Z and Z  = r (v, Z ) in the formulas (11.4) and (11.5) of Lemma 11.2 with L real such that 0 ∈ / L(K (v)), L  =  L ◦ Iµ and Q given by (11.31), we obtain rational formulas for a homomorphism θ˜v of Calg (Sϕ3 ) to the generalised cross-product of the field K q of meromorphic functions f (Z ) on the elliptic curve Fϕ by σ . The generalised cross-product rule (11.4) is given by W W  := γ (Z ), where γ is a rational function. Similarly W  W := γ (σ −1 (Z )). Using integration on the cycle K (v) to obtain a trace, together with Corollary 11.9, we get, Theorem 11.13. The homomorphism θv : Calg (Sϕ3 ) → C ∞ (T2η ) factorises with a homomorphism θ˜v : Calg (Sϕ3 ) → K q ×σ Z to the generalised cross-product of the field K q of meromorphic functions on the elliptic curve Fϕ by the subgroup of the Galois group AutC (K q ) generated by σ . Its image generates the hyperfinite factor of type II1 after weak closure relative to the trace given by integration on the cycle K (v). Elements of K q with poles on K (v) are unbounded and give elements of the regular ring of affiliated operators, but all elements of θv (Calg (Sϕ3 )) are regular on K (v). The above generalisation of the cross-product rules (11.4) with the rational formula for W W  := γ (Z ) is similar to the introduction of 2-cocycles in the standard Brauer theory of central simple algebras.

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12. The Jacobian of the Covering of Sϕ3 In this section we shall analyse the morphism of ∗-algebras θ : Calg (Sϕ3 ) → C ∞ (Fϕ ×σ, L Z)

(12.1)

of Corollary 11.9, by computing its Jacobian in the sense of noncommutative differential geometry ([11]). We postpone the analysis at the C ∗ -level, in particular the role of the discrete series, to another forthcoming publication. The usual Jacobian of a smooth map ϕ : M → N of manifolds is obtained as the ratio ϕ ∗ ( ω N )/ω M of the pullback of the volume form ω N of the target manifold N with the volume form ω M of the source manifold M. In noncommutative geometry, differential forms ω of degree d become Hochschild classes ω˜ ∈ H Hd (A), A = C ∞ (M). Moreover since one works with the dual formulation in terms of algebras, the pullback ϕ ∗ (ω N ) is replaced by the pushforward ϕ∗t (ω˜ N ) under the corresponding transposed morphism of algebras ϕ t ( f ) := f ◦ ϕ, ∀ f ∈ C ∞ (N ). The noncommutative sphere Sϕ3 admits a canonical “volume form” given by the Hochschild 3-cycle ch 3 (U ). Our goal is to compute the push-forward, 2

  θ∗ ch 3 (U ) ∈ H H3 (C ∞ (Fϕ ×σ, L Z)). 2

(12.2)

Let ϕ be generic and even. The noncommutative manifold Fϕ ×σ, L Z is, by Corollary 11.11, a noncommutative 3-dimensional nilmanifold isomorphic to the mapping torus of an automorphism of the noncommutative 2-torus Tη2 . Its Hochschild homology is easily computed using the corresponding result for the noncommutative torus ([11]). It admits in particular a canonical volume form V ∈ H H3 (C ∞ (Fϕ ×σ, L Z)) which corresponds to the natural class in H H2 (C ∞ (Tη2 )) ([11]). The volume form V is obtained in the cross-product Fϕ ×σ, L Z from the translation invariant 2-form dv on Fϕ . To compare θ∗ (ch 3 (U )) with V we shall pair it with the 3-dimensional Hochs2

child cocycle τh ∈ H H 3 (C ∞ (Fϕ ×σ, L Z)) given, for any element h of the center of C ∞ (Fϕ ×σ, L Z), by τh (a0 , a1 , a2 , a3 ) = τ3 (h a0 , a1 , a2 , a3 ),

(12.3)

H C 3 (C ∞ (Fϕ

where τ3 ∈ ×σ, L Z)) is the fundamental class in cyclic cohomology defined by (11.27). By Proposition 10.4 the component ch 3 (U ) of the Chern character 2 is given by,  ch 3 (U ) = αβγ δ cos(ϕα − ϕβ + ϕγ − ϕδ ) x α d x β d x γ d x δ 2  −i sin2(ϕµ − ϕν ) x µ d x ν d x µ d x ν , (12.4) where ϕ0 := 0. In terms of the Yµ one gets,  ch 3 (U ) = κ δαβγ δ (sα − sβ + sγ − sδ ) Yα dYβ dYγ dYδ 2  +κ αβγ δ (sα − sβ ) Yγ dYδ dYγ dYδ ,

(12.5)

where s0 := 0, sk := 1 + t tm , tk := tan ϕk and δαβγ δ = αβγ δ (n α − n β + n γ − n δ )

(12.6)

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with n 0 = 0 and n k = 1. The normalization factor is  κ= i cos2 (ϕk ) sin(ϕ − ϕm ).

(12.7)

Formula (12.5) shows that, up to normalization, ch 3 (U ) only depends on the fiber Fϕ 2 of ϕ. Let ϕ be generic and even; there is a similar formula in the odd case. In our case the involutions I and I0 are conjugate by a real translation κ of the elliptic curve Fϕ and we let Fϕ (0) be one of the two connected components of {Z ∈ Fϕ | I0 (Z ) = Z¯ }.

(12.8)

By Proposition 11.10 we can identify the center of C ∞ (Fϕ ×σ, L Z) with C ∞ (Fϕ (0)). We assume for simplicity that ϕ j ∈ [0, π2 ] are in cyclic order ϕk < ϕl < ϕm for some k ∈ {1, 2, 3}. Theorem 12.1. Let h ∈ Center (C ∞ (Fϕ ×σ, L Z)) ∼ C ∞ (Fϕ (0)). Then  h(Z ) d R(Z ), ch 3 (U ), τh = 6 π  Fϕ (0)

2

where  is the period given by the elliptic integral of the first kind,  Zk d Z0 − Z0d Zk , = Z Zm Cϕ and R the rational fraction, R(Z ) = tk

2 Zm 2 + c Z2 Zm k l

with ck = tg(ϕl ) cot(ϕk − ϕ ). We assume that 0 < ϕ1 < ϕ2 < ϕ3 < π/2, i.e. that k = 1. We start from Corollary 10.11 and express the result ω= −

σ (m)4 g(m) dm, λ


(12.9)

in terms of the trigonometric parameters ϕ j and the coordinates Z µ of Z . One has (cf. (10.36)) 2  σ (m)4 = (C1 − λ C2 )−2 , (12.10) sin ϕ j and we begin by giving a better formula for C1 − λ C2 . Lemma 12.2. One has C1 − λ C2 = b1 ϑ12 (im) + b2 ϑ22 (im), where b1 = 4

ϑ12 (η) s1 ϑ22 (η)

, b2 = 4

s1 − 1 . s1

(12.11)

(12.12)

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109

Proof. One uses (10.37), λ =

ϑ22 (0) s1 ϑ22 (η)

and the identity

ϑ22 (0) ϑ2 (η + im) ϑ2 (η − im) = ϑ22 (im) ϑ22 (η) − ϑ12 (im) ϑ12 (η).   The m-dependent terms are understood from the following lemma Lemma 12.3. Let b j be arbitrary constants, then ϑ32 (0)ϑ42 (0)

ϑ12 (u) ϑ1 (0)3 ϑ1 (2u) d . = b 2 2 2 2 2 du b1 ϑ1 (u) + b2 ϑ2 (u) π (b1 ϑ1 (u) + b2 ϑ22 (u))2 (12.13)

Proof. This follows from the classical identity d sn(u) = cn(u) dn(u), du

(12.14)

which implies ϑ  (0)3 d ϑ12 (u) ϑ1 (u)ϑ2 (u)ϑ3 (u)ϑ4 (u) ϑ1 (2u) = 2 π ϑ22 (0) = 2 21 , 2 4 du ϑ2 (u) ϑ2 (u) π ϑ3 (0)ϑ42 (0) ϑ24 (u) using the duplication formula ϑ2 (0)ϑ3 (0)ϑ4 (0)ϑ1 (2 u) = 2 ϑ1 (u)ϑ2 (u)ϑ3 (u)ϑ4 (u) and the Jacobi derivative formula ϑ1 (0) = ϑ2 (0)ϑ3 (0)ϑ4 (0). π   Taking u = im in (12.13) this allows to write ω as the differential of R(m) =

c ϑ12 (im) b1 ϑ12 (im) + b2 ϑ22 (im)

(12.15)

using (10.45)

ϑ1 (0)3 ϑ1 (η) ϑ1 (2im) . (12.16) π 3 ϑ2 (η)ϑ3 (η)ϑ4 (η) The constant c is uniquely determined and will be simplified later; we get so far,  sin ϕ j )2 ϑ1 (η) ϑ32 (0)ϑ42 (0) 3 ( . (12.17) c = 24 (2π ) π b2 λ
ϑ2 (η)ϑ3 (η)ϑ4 (η) g(m) = 24 (2πi)3

Lemma 12.4.

(i) The differential form χ :=

Zk d Z0 − Z0d Zk sk Z  Z m

(12.18)

is independent of k and is, up to scale, the only holomorphic form of type (1, 0) on Fϕ .

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(ii) One has 2π Proof.

ϑ42 (0)

ϑ1 (η) ϑ4 (η) = ϑ2 (η) ϑ3 (η)

 Cϕ

Z3d Z0 − Z0d Z3 . Z1 Z2

(12.19)

(i) Recall that the equations defining Fϕ are Z 02 − Z 12 Z 2 − Z 22 Z 2 − Z 32 = 0 = 0 . s1 s2 s3

(12.20)

One gets the required independence by differentiation. ϑ1 (η) ϑ4 (η) (ii) Let us check (12.19). The factor allows to replace the Z j by ϑ j+1 (2z) ϑ2 (η) ϑ3 (η) so that the right hand side gives, using (12.14), 

1

0

ϑ4 (2z)dϑ1 (2z) − ϑ1 (2z)dϑ4 (2z) = 2 π ϑ42 (0). ϑ2 (2z)ϑ3 (2z)

  In terms of Z = (Z j ) one has ϑ12 (η)

R(m) = c s1 where a1 =

Z 02

4 ϑ22 (η)

ϑ14 (η) ϑ24 (η)

a1 Z 02

+ a2 Z 12

,

(12.21)

, a2 = s1 − 1.

(12.22)

One can express the coefficient a1 in trigonometric terms using Lemma 12.5. a1 =

ϑ14 (η) ϑ24 (η)

=

(s1 − s2 )(s1 − s3 ) . s2 s3

(12.23)

Proof. By homogeneity of the right hand side one can replace the s j by the σ j of (10.44). One then gets (s1 − s2 )(s1 − s3 ) = − J12 J31 s2 s3 and the result follows from (10.26). We now simplify the product c s1 using (cf. (10.12))
=

3  1

  ϑ12 (η) 4 ϑ22 (η)

replacing ϑ42 (0) in (12.17) by (12.19) and

(tan (ϕ j ) cos(ϕk − ϕ )), λ =

ϑ32 (0) s2 ϑ32 (η)

.

(12.24)

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111

By elementary computations and using once more (12.23) one gets  ϑ 2 (η) s1 Z3d Z0 − Z0d Z3 c s1 1 2 . = 6 π t1 a 1 s3 C ϕ Z1 Z2 4 ϑ2 (η)

(12.25)

We thus obtain so far using the elementary equality a2 = cot(ϕ1 − ϕ2 ) cot(ϕ1 − ϕ3 ), a1 and Lemma 12.4 which allows to eliminate the term  the following formula for the rational fraction, R(Z ) = t1

s1 , using the definition of the period s3

Z 02 Z 02 + a Z 12

(12.26)

with a = cot(ϕ1 − ϕ2 ) cot(ϕ1 − ϕ3 ). What we have computed so far is the image of ch3/2 under the ∗-homomorphism ρ. ˜ By Proposition 11.12 (i) this amounts to the image of ch3/2 under the ∗-homomorphism associated to the central quadratic form Q  . By Proposition 11.12 (ii) we get the result for Q using the isomorphism β and this gives the following formula for R(Z ), R(Z ) = t1

Z 32 Z 32 + c1 Z 22

(12.27)

with c1 = tg(ϕ2 ) cot(ϕ1 − ϕ2 ). We shall explain below in Sect. 13 how to perform the transition from (12.26) to (12.27) in a conceptual manner. In fact the conceptual understanding of the simplicity of the final result of Theorem 12.1 is at the origin of many of the notions developed in the present paper and in particular of the “rational” formulation of the calculus which will be obtained in the last section. The geometric meaning of Theorem 12.1 is the computation of the Jacobian in the sense of noncommutative geometry of the morphism θ as explained above. The integral  is (up to a trivial normalization factor) a standard elliptic integral, it is given by an hypergeometric function in the variable m :=

sk (sl − sm ) sl (sk − sm )

(12.28)

or a modular form in terms of q. Lemma 12.6. The differential of R is given on Fϕ by d R = J (Z ) χ , where J (Z ) = 2 (sl − sm ) ck tk

Z0 Z1 Z2 Z3 2 + c Z 2 )2 (Z m k l

(12.29)

Proof. This can easily be checked using (12.14) but it is worthwhile to give a simple direct argument. One has indeed by definition of χ , d

Z 2j Z 02

= −2 s j

Z0 Z1 Z2 Z3 χ, Z 04

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which gives using 2 sl Z m − sm Z l2 = (sl − sm ) Z 02

the equality d and the required result.

Z l2 Z0 Z1 Z2 Z3 = 2 (sm − sl ) χ, 2 4 Zm Zm  

The period  does not vanish and J (Z ), Z ∈ Fϕ (0), only vanishes on the 4 “ramification points” necessarily present due to the symmetries. Corollary 12.7. The Jacobian of the map θ t is given by the equality θ∗ (ch 3 (U )) = 3  J V, 2

where J is the element of the center C ∞ (Fϕ (0)) of C ∞ (Fϕ ×σ, L Z) given by formula (12.29). This statement assumes that ϕ is generic in the measure theoretic sense so that η admits good diophantine approximation ([11]). It justifies in particular the terminology of “ramified covering” applied to θ t . The function J has only 4 zeros on Fϕ (0) which correspond to the ramification. As shown by Theorem 9.5 the algebra Aϕ is defined over R, i.e. admits a natural antilinear automorphism of period two, γ uniquely defined by γ (Yµ ) := Yµ , ∀µ.

(12.30)

Theorem 9.5 also shows that σ is defined over R and hence commutes with complex conjugation c(Z ) = Z¯ . This gives a natural real structure γ on the algebra C Q with C = Fϕ × Fϕ and Q as above,  γ ( f (Z , Z  )) := f (c(Z ), c(Z  )), γ (W L ) := Wc(L) , γ (W L  ) := Wc(L ).

One checks that the morphism ρ of Lemma 11.2 is “real”, i.e. that, γ ◦ ρ = ρ ◦ γ.

(12.31)

Since c(Z ) = Z¯ reverses the orientation of Fϕ , while γ preserves the orientation of Sϕ3 , it follows that J ( Z¯ ) = −J (Z ) and J necessarily vanishes on Fϕ (0) ∩ P3 (R). Note also that for general h one has ch 3 (U ), τh = 0 which shows that both 2

ch 3 (U ) ∈ H H3 and τh ∈ H H 3 are non trivial Hochschild classes. These results hold 2

in the smooth algebra C ∞ (Sϕ3 ) containing the closure of Calg (Sϕ3 ) under holomorphic functional calculus in the C ∗ algebra C ∗ (Sϕ3 ). We can also use Theorem 12.1 to show the non-triviality of the morphism θv : Calg (Sϕ3 ) → C ∞ (T2η ) of (11.33). Corollary 12.8. The pullback of the fundamental class [T2η ] of the noncommutative torus by the homomorphism θv : Calg (Sϕ3 ) → C ∞ (T2η ) of (11.33) is non zero, θv∗ ([T2η ]) = 0 ∈ H H 2 provided v is not a ramification point.

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We have shown above the non-triviality of the Hochschild homology and cohomology groups H H3 (C ∞ (Sϕ3 )) and H H 3 (C ∞ (Sϕ3 )) by exhibiting specific elements with nonzero pairing. Combining the ramified cover π = θ t with the natural spectral geometry (spectral triple) on the noncommutative 3-dimensional nilmanifold Fϕ ×σ, L Z yields a natural spectral triple on Sϕ3 in the generic case. 13. Calculus and Cyclic Cohomology Theorem 12.1 suggests the existence of a “rational” form of the calculus explaining the appearance of the elliptic period  and the rationality of R. We shall show in this last section that this is indeed the case. This will allow us to get a very simple conceptual form of the above computation of the Jacobian in Theorem 13.8 below. Let us first go back to the general framework of twisted cross products of the form A = C ∞ (M) ×σ, L Z,

(13.1)

where σ is a diffeomorphism of the manifold M. We shall follow [12] to construct cyclic cohomology classes from cocycles in the bicomplex of group cohomology (with group  = Z) with coefficients in de Rham currents on M. The twist by the line bundle L introduces a non-trivial interesting nuance. Let (M) be the algebra of smooth differential forms on M, endowed with the action of Z, α1,k (ω) := σ ∗k ω, k ∈ Z. (13.2) ˜ As in [14] p. 219 we let (M) be the graded algebra obtained as the (graded) tensor product of (M) by the exterior algebra ∧(C[Z] ) on the augmentation ideal C[Z] in the group ring C[Z]. With [n], n ∈ Z the canonical basis of C[Z], the augmentation  : C[Z] → C fulfills ([n]) = 1, ∀n , and δn := [n] − [0],

n ∈ Z, n = 0

(13.3)

is a linear basis of C[Z] . The left regular representation of Z on C[Z] restricts to C[Z] and is given on the above basis by α2,k (δn ) := δn+k − δk ,

k ∈ Z.

(13.4)

It extends to an action α2 of Z by automorphisms of ∧C[Z] . We let α = α1 ⊗ α2 be the tensor product action of Z on the graded tensor product ˜ (M) = (M) ⊗ ∧C[Z] .

(13.5)

We now use the hermitian line bundle L to form the twisted cross-product ˜ C := (M) ×α, L Z.

(13.6)

We let Ln be as in (11.15) for n > 0 and extend its definition for n < 0 so that L−n is the pullback by σ n of the dual Lˆ n of Ln for all n. The hermitian structure gives an antilinear isomorphism ∗ : Ln → Lˆ n . One has for all n, m ∈ Z a canonical isomorphism Ln ⊗ σ ∗m Lm  Ln+m , which is by construction compatible with the hermitian structures.

(13.7)

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The algebra C is the linear span of monomials ξ W n , where ˜ ξ ∈ C ∞ (M, Ln ) ⊗C ∞ (M) (M)

(13.8)

with the product rules (11.17), (11.18). Let ∇ be a hermitian connection on L. We shall turn C into a differential graded algebra. By functoriality ∇ gives a hermitian connection on the Lk and hence a graded derivation ∇n : C ∞ (M, Ln ) ⊗C ∞ (M) (M) → C ∞ (M, Ln ) ⊗C ∞ (M) (M)

(13.9)

whose square ∇n2 is multiplication by the curvature κn ∈ 2 (M) of Ln , κn+m = κn + σ ∗n (κm ), ∀n, m ∈ Z

(13.10)

2 (M)

the curvature of L. One has dκn = 0 and we extend the differwith κ1 = κ ∈ ˜ ential d to a graded derivation on (M) by dδn = κn .

(13.11)

We can then extend ∇n uniquely to the induced module ˜ En = C ∞ (M, Ln ) ⊗C ∞ (M) (M).

(13.12)

˜ ∇˜ n (ξ ω) = ∇n (ξ ) ω + (−1)deg(ξ ) ξ dω, ∀ω ∈ (M)

(13.13)

by the equality ˜ Proposition 13.1. a) The pair ((M), d) is a graded differential algebra. ˜ b) Let α be the tensor product action of Z, then α(k) ∈ Aut((M), d), ∀k ∈ Z. ˜ c) The following equality defines a flat connection on the induced module En on (M), ∇n (ξ ) = ∇˜ n (ξ ) − (−1)deg(ξ ) ξ δn .

(13.14)

˜ extends uniquely to a graded derivation of C such d) The graded derivation d of (M) that, d(ξ W n ) = ∇n (ξ ) W n , (13.15) which turns the pair (C, d) into a graded differential algebra. Proof. a) By construction d is the unique extension of the differential d of (M) to a graded derivation of the graded tensor product (13.5) such that (13.11) holds. One just needs to check that d 2 = 0 on simple tensors ω ⊗ δn , one gets d(ω ⊗ δn ) = dω ⊗ δn + (−1)deg(ω) ω κn ⊗ 1, d 2 (ω ⊗ δn ) = d 2 ω ⊗ δn + (−1)deg(ω)+1 dω κn ⊗ 1 + (−1)deg(ω) dω κn ⊗ 1 = 0. b) Let us check that α(k) commutes with the differentiation d on simple tensors ω ⊗δn . One has d(α(k)(ω ⊗ δn )) = d(σ ∗k (ω) ⊗ (δn+k − δk )) = σ ∗k (dω) ⊗ (δn+k − δk ) +(−1)deg(ω) σ ∗k (ω) (κn+k − κk ) ⊗ 1, ×α(k)(d(ω ⊗ δn )) = σ ∗k (dω) ⊗ (δn+k − δk ) +(−1)deg(ω) σ ∗k (ω κn ) ⊗ 1, and the equality follows from (13.10).

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c) Since δn2 = 0 one gets (∇n )2 (ξ ) = (∇˜ n )2 (ξ ) − ξ dδn = 0, since dδn = κn is the curvature. d) Since the algebra C is the linear span of monomials ξ W n the linear map d is well ˜ defined and coincides with the differential d on (M) since δ0 = 0. Let us show that the two terms of (13.14) separately define derivations of C. By construction the connections ∇n are compatible with the canonical isomorphisms (13.7) and the same holds for their extensions ∇˜ n which is enough to show that the first term of (13.14) separately defines a derivation of C. The proof for the second term 

d (ξ W n ) = (−1)deg(ξ ) ξ δn W n ,

(13.16)

follows from (13.4) and is identical to the proof of Lemma 12, Chapter III of [14]. The flatness (c) of the connections ∇n ensures that d 2 = 0 so that (C, d) is a graded differential algebra. To construct closed graded traces on this differential graded algebra we follow ([12]) and consider the double complex of group cochains (with group  = Z) with coefficients in de Rham currents on M. The cochains γ ∈ C n,m are totally antisymmetric maps from Zn+1 to the space −m (M) of de Rham currents of dimension −m, which fulfill γ (k0 + k, k1 + k, k2 + k, · · · , kn + k) = σ∗−k γ (k0 , k1 , k2 , · · · , kn ), ∀k, k j ∈ Z. (13.17) Besides the coboundary d1 of group cohomology, given by (d1 γ )(k0 , k1 , · · · , kn+1 ) =

n+1 

(−1) j+m γ (k0 , k1 , · · · , kˆj , · · · , kn+1 )

(13.18)

0

and the coboundary d2 of de Rham homology, (d2 γ )(k0 , k1 , · · · , kn ) = b(γ (k0 , k1 , · · · , kn )), the curvatures κn generate the further coboundary d3 defined on Ker d1 by, (d3 γ )(k0 , · · · , kn+1 ) =

n+1 

(−1) j+m+1 κ k j γ (k0 , · · · , kˆj , · · · , kn+1 ),

(13.19)

0

which maps Ker d1 ∩ C n,m to C n+1,m+2 . Translation invariance follows from (13.10) and ϕ∗ (ωC) = ϕ ∗−1 (ω)ϕ∗ (C) for C ∈ −m (M), ω ∈ ∗ (M). To each γ ∈ C n,m one associates the functional γ˜ on C given by, ˜ γ˜ ( ξ W n ) = 0, ∀ n = 0, ξ ∈ (M), γ˜ ( ω ⊗ δk1 · · · δkn ) = ω, γ (0, k1 · · · , kn ) , ∀k j ∈ Z. One then has

(13.20)

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Lemma 13.2. Let γ ∈ C n,m , then ˜ (i) One has for all ρ ∈ (M), γ˜ (ρ − α(−k)ρ) = −(d1˜γ )(δk ρ).

(13.21)

(ii) One has for all a, b ∈ C with d  defined in (13.16), γ˜ (a b − (−1)deg(a)deg(b) b a) = (−1)deg(a) (d1˜γ )(a d  b).

(13.22)

˜ (iii) One has for all ρ ∈ (M), γ˜ (dρ) = (d2˜γ )(ρ) + (d3˜γ )(ρ).

(13.23)

Proof. (i) We can assume that ρ is of the form ρ = ω ⊗ δk1 · · · δkn . The left side of (13.21) is by construction ω, γ (0, k1 · · · , kn ) − γ˜ (σ ∗−k (ω) ⊗ (δk1 −k − δ−k ) · · · (δkn −k − δ−k )). 2 = 0, When one expands the product one gets using δ−k

(δk1 −k − δ−k ) · · · (δkn −k − δ−k ) =



δk j −k + δ−k



(−1)i



δk j −k ,

j =i

and one uses the translation invariance (13.17) to write  γ˜ (σ ∗−k (ω) ⊗ δk j −k ) = ω, γ (k, k1 · · · , kn )

and



γ˜ (σ ∗−k (ω) ⊗ (−1)i δ−k

δk j −k ) = −(−1)i ω, γ (0, k, k1 · · · , kˆi , · · · , kn ) .

j =i

One thus obtains the same terms as in −(d1˜γ )(δk ρ) = − ω, (d1 γ )(0, k, k1 · · · , kn ) , using (13.18) and the graded commutation of δk with ω which yields a (−1)m overall sign. ˜ Eq. (11.13) holds (ii) It is enough to show that for any k ∈ Z and a  , b ∈ (M),  k  −k ˜ for a = a W and b = b W . The graded commutativity of (M) allows to write the ˜ graded commutator in (11.13) as ρ − α(−k)ρ, where ρ = a  α(k)(b ) ∈ (M). One has ρ = a b and (i) thus shows that γ˜ (a b − (−1)deg(a)deg(b) b a) = − (d1˜γ )(δk a b) = −(−1)deg(a)+deg(b) (d1˜γ )(a b δk ). One has 

a d (b) = a  α(k)((−1)deg(b) b δ−k ),

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thus the result follows from the equality α(k)(δ−k ) = − δk . (iii) We can assume that ρ is of the form ρ = ω ⊗ δk1 · · · δkn . One has dρ = dω ⊗ δk1 · · · δkn − (−1)deg(ω) which gives (13.23).



(−1) j ω κk j ⊗ δk1 · · · δkˆ j · · · δkn

 

To each γ ∈ C n,m one associates the (n − m + 1) linear form on A = C ∞ (M) ×σ, L Z given by, (γ )(a0 , a1 , · · · , an−m ) n−m  (−1) j (n−m− j) γ˜ (da j+1 · · · dan−m a0 da1 · · · da j−1 da j ), = λn,m

(13.24)

0

where λn,m :=

n! (n−m+1)! .

Lemma 13.3. (i) The Hochschild coboundary b(γ ) is equal to (d1 γ ). (ii) Let γ ∈ C n,m ∩ Ker d1 . Then (γ ) is a Hochschild cocycle and 1 B(γ ) = (d2 γ ) + n+1 (d3 γ ) Proof. (i) The proof is identical to that of Theorem 14 a) Chapter III of [14]. (ii) By (i) and the hypothesis d1 (γ ) = 0, (γ ) is a Hochschild cocycle. In fact by Lemma 13.2 the functional γ˜ is a graded trace and thus the formula for (γ ) simplifies to (γ )(a0 , a1 , · · · , an−m ) = (n − m + 1) λn,m γ˜ (a0 da1 · · · dan−m ).

(13.25)

It follows that B0 ((γ ))(a0 , a1 , · · · , an−m−1 ) = (n − m + 1) λn,m γ˜ (dρ), ρ = a0 da1 · · · dan−m−1 , and B0 ((γ )) is already cyclic so that B((γ )) = (n − m)B0 ((γ )). Since the coboundary d2 anticommutes with d1 , one has d2 γ ∈ Ker d1 and (d2 γ ) is also a Hochschild cocycle and is given by (13.25) for d2 γ . Let us check that d3 γ ∈ Ker d1 . One has up to an overall sign,  d1 d3 (γ ) = (−1)i d3 (γ )(k0 , · · · , kˆi · · · , kn+2 )  (i, j) κ k j γ (k0 , · · · , kˆi  , · · · , kˆj  , · · · , kn+1 ), = i, j

where (i  , j  ) is the permutation of (i, j) such that i  < j  and up to an overall sign (i, j) is the product of the signature of this permutation by (−1)i+ j . For each j the coefficient of κ k j is up to an overall sign given by d1 (γ )(k0 , · · · , kˆj · · · , kn+2 ) = 0, thus d1 d3 (γ ) = 0. The result thus follows from Lemma 13.2 (iii).  

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We shall now show how the above general framework allows to reformulate the calculus involved in Theorem 12.1 in rational terms. We let M be the elliptic curve Fϕ , where ϕ is generic and even. Let then ∇ be an  arbitrary hermitian connection on L and  κ its curvature. We first display a cocycle γ = γn,m ∈ C n,m which reproduces the cyclic cocycle τ3 . Lemma 13.4. There exists a two form α on M = Fϕ and a multiple λ dv of the translation invariant two form dv such that : (i) κn = n λ dv + (σ ∗n α − α), ∀n ∈ Z. (ii) d2 (γ j ) = 0, d1 (γ3 ) = 0, d1 (γ 1 ) + 21 d3 (γ3 ) = 0, B(γ1 ) = 0, where γ 1 ∈ C 1,0 and γ3 ∈ C 1,−2 are given by γ1 (k0 , k1 ) := 21 (k1 − k0 )(σ ∗k0 α + σ ∗k1 α), γ3 (k0 , k1 ) := k1 − k0 , ∀k j ∈ Z. (iii) The class of the cyclic cocycle (γ1 ) + (γ3 ) is equal to τ3 . We use the generic hypothesis in the measure theoretic sense to solve the “small denominator” problem in (i). In (ii) we identify differential forms ω ∈  d of degree d with the dual currents of dimension 2 − d. It is a general principle explained in [11] that a cyclic cocycle τ generates a calculus whose differential graded algebra is obtained as the quotient of the universal one by the radical of τ . We shall now explicitly describe the reduced calculus obtained from the cocycle of Lemma 13.4 (iii). We use as above the hermitian line bundle L to form the twisted cross-product B := (M) ×α, L Z (13.26) of the algebra (M) of differential forms on M by the diffeomorphism σ . Instead of having to adjoin the infinite number of odd elements δn we just adjoin two χ and X as follows. We let δ be the derivation of B such that δ(ξ W n ) := i n ξ W n , ∀ξ ∈ C ∞ (M, Ln ) ⊗C ∞ (M) (M).

(13.27)

We adjoin χ to B by tensoring B with the exterior algebra ∧{χ } generated by an element χ of degree 1, and extend the connection ∇ (13.9) to the unique graded derivation d  of  = B ⊗ ∧{χ } such that, d  ω = ∇ω + χ δ(ω), ∀ω ∈ B, d  χ = − λ dv

(13.28)

with λ dv as in Lemma 13.4. By construction, every element of  is of the form y = b0 + b1 χ , b j ∈ B. One does not yet have a graded differential algebra since d in Lemma 13.4 one has

(13.29) 2



= 0. However, with α as

d 2 (x) = [ x, α], ∀x ∈  = B ⊗ ∧{χ }

(13.30)

and one can apply Lemma 9, p. 229 of [14] to get a differential graded algebra by adjoining the degree 1 element X := “d1 fulfilling the rules X 2 = −α, x X y = 0, ∀x, y ∈  ,

(13.31)

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119

and defining the differential d by d x = d  x + [ X, x], ∀x ∈  , d X = 0,

(13.32)

where [ X, x] is the graded commutator. It follows from Lemma 9, p.229 of [14] that we obtain a differential graded algebra ∗ , generated by B, ξ and X . In fact using (13.31) every element of ∗ is of the form x = x1,1 + x1,2 X + X x2,1 + X x2,2 X, xi, j ∈  , , and we define the functional on ∗ by extending the ordinary integral,   ω := ω, ∀ω ∈ (M)

(13.33)

(13.34)

M

first to B := (M) ×α, L Z by

 ξ W n := 0, ∀n = 0

then to  by

(13.35)



 (b0 + b1 χ ) :=

b1 , ∀b j ∈ B

and finally to ∗ as in Lemma 9, p.229 of [14],    (x1,1 + x1,2 X + X x2,1 + X x2,2 X ) := x1,1 + (−1)deg( x2,2 ) x2,2 α.

(13.36)

(13.37)

∗ is a differential Theorem 13.5. Let M = Fϕ , ∇, α be as in Lemma 13.4. The algebra , ∞ graded algebra containing C (M) ×α, L Z. The functional is a closed graded trace on ∗ . The character of the corresponding cycle on C ∞ (M) ×α, L Z,  τ (a0 , · · · , a3 ) := a0 da1 · · · da3 , ∀a j ∈ C ∞ (M) ×α, L Z is cohomologous to the cyclic cocycle τ3 . It is worth noticing that the above calculus fits with [10,21 and 20]. Now in our case the line bundle L is holomorphic and we can apply Theorem 13.5 to its canonical hermitian connection ∇. We take the notations of Sect. 11, with C = Fϕ × Fϕ , and Q given by (11.31). This gives a particular “rational” form of the calculus which explains the rationality of the answer in Theorem 12.1. We first extend as follows the construction of C Q . We let (C, Q) be the generalised cross-product of the algebra (C) of meromorphic differential forms (in d Z and d Z  ) on C by the transformation σ˜ . The generators W L and W L  fulfill the cross-product rules, W L ω = σ˜ ∗ (ω) W L ,

W L  ω = (σ˜ −1 )∗ (ω) W L 

(13.38)

while (11.4) is unchanged. The connection ∇ is the restriction to the subspace {Z  = Z¯ } of the unique graded derivation ∇ on (C, Q) which induces the usual differential on (C) and satisfies, ∇W L = ( d Z log L(Z ) − d Z log Q(Z , Z  )) W L , ∇W L  = W L  ( d Z  log L  (Z  ) − d Z  log Q(Z , Z  )),

(13.39)

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where d Z and d Z  are the (partial) differentials relative to the variables Z and Z  . Note that one needs to check that the involved differential forms such as d Z log L(Z ) − d Z log Q(Z , Z  ) are not only invariant under the scaling transformations Z → λZ but are also basic, i.e. have zero restriction to the fibers of the map C4 → P3 (C), in both variables Z and Z  . By definition the derivation δκ = ∇ 2 of (C, Q) vanishes on (C) and fulfills δκ (W L ) = κ W L , δκ (W L  ) = − W L  κ, (13.40) where

κ = d Z d Z  log Q(Z , Z  )

(13.41) ¯ is a basic form which when restricted to the subspace = Z } is the curvature. We let as above δ be the derivation of (C, Q) which vanishes on (C) and is such that δW L = i W L and δW L  = −i W L  . We proceed exactly as above and get the graded algebras  = (C, Q) ⊗ ∧{χ } obtained by adjoining,χ and ∗ by adjoining X . We define d  , d as in (13.28) and (13.32) and the integral by integration (13.34) on the subspace {Z  = Z¯ } followed as above by steps (13.35), (13.36), (13.37). {Z 

Corollary 13.6. Let ρ: Calg (Sϕ3 ) → C Q be the morphism of Lemma 11.2. The equality    τalg (a0 , · · · , a3 ) := ρ(a0 ) d ρ(a1 ) · · · d ρ(a3 ) defines a 3-dimensional Hochschild cocycle τalg on Calg (Sϕ3 ). Let h ∈ Center (C ∞ (Fϕ ×σ, L Z)) ∼ C ∞ (Fϕ (0)). Then ch 3 (U ), τh = h ch 3 (U ), τalg . 2

2

The computation of d  only involves rational fractions in the variables Z , Z  (13.39), and the formula (12.5) for ch 3 (U ) is polynomial in the W L , W L  . 2 We are now ready for a better understanding and formulation of the result of Theorem 12.1. Indeed what the above shows is that the denominator that appears in the rational fraction R(Z ) of Theorem 12.1 should have to do with the central quadratic form Q(Z , Z  ) evaluated on the pairs (Z , Z¯ ). In fact the two dimensional space of central quadratic forms provides a natural space of functions of the form R(Z ) =

P(Z , Z¯ ) , Q(Z , Z¯ )

(13.42)

and this space is one dimensional when one mods out the constant functions. The following lemma shows that these functions are in fact invariant under the correspondence σ : Lemma 13.7. Let Q be central and not identically zero on the component C and P be central, then the function R(Z , Z  ) =

P(Z , Z  ) Q(Z , Z  )

is invariant under σ˜ . It is thus natural now to compare the differential form d R with the form that appears in Theorem 12.1.

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Theorem 13.8. Let ϕ be generic and even and let Q be the central quadratic form on R4ϕ defining the three sphere Sϕ3 . Let ρ Q be the associated ∗-homomorphism C ∞ (Sϕ3 ) → C ∞ (E ×σ, L Z). Then for any central quadratic form P not proportional to Q there exists a scalar µ such that  ch 3 (U ), τh = µ h(Z ) d R(Z ), ∀h ∈ Center C ∞ (E ×σ, L Z), 2

where R(Z ) =

P(Z , Z¯ ) . Q(Z , Z¯ )

Proof. Let us show that one can interpret (12.26) in the above terms. Thus with a = cot(ϕ1 − ϕ2 ) cot(ϕ1 − ϕ3 ) we need to show that R(Z ) = t1

Z 02 Z 02 + a Z 12

is in fact the restriction to the subset Fϕ (0) ⊂ E of elements Z with c(Z ) = I0 (Z ) of a ratio of the form P(Z , j (Z )) . Q(Z , j (Z )) One has j (Z ) = I3 (c(Z ) and thus j (Z ) = I3 ◦ I0 (Z ), ∀Z ∈ Fϕ (0).

(13.43)

We let Q  be the central quadratic form of Proposition 11.12, namely s Q  = Q 1 + Q 3 + s2 Q 2 , and we let P  be given by

s P  = Q1 + Q3.

(13.44)

A simple computation using (13.43) then shows that b

Z 02 P  (Z , j (Z )) 1 = − , ∀Z ∈ Fϕ (0), 2 2 Q  (Z , j (Z )) s Z0 + a Z1 1

(13.45)

where b=

sinϕ2 sinϕ3 . cos(ϕ2 − ϕ3 )

This gives the required result for the central quadratic form P  and since the space of central quadratic forms is two dimensional its quotient by multiples of Q  is one dimensional so that the result holds for all non-zero elements of this quotient.  

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With the above “invariant” formulation of the formulas of Theorem 12.1 we can now perform the change of variables required in the last part of its proof, i.e. explain how to pass from (12.26) to (12.27). We let P and Q be the central quadratic forms obtained from P  and Q  by the isomorphism β of Proposition 11.12. The form Q is given by (11.31), i.e. by    tk sk Q k = sin(ϕ − ϕm ) Z 02 (13.46) Q = ( cos2 ϕ )  − cosϕ cosϕm sin(ϕ − ϕm ) Z k2 . The form P is given by P =

3 

sinϕk sin(ϕ − ϕm ) cos(ϕk − ϕ − ϕm ) Z k2 .

(13.47)

1

The involution j2 is now given by j2 (Z ) = I2 (c(Z ) and one has j2 (Z ) = I2 ◦ I0 (Z ), ∀Z ∈ Fϕ (0).

(13.48)

A simple computation using (13.48) then shows that b

Z2 P(Z , j2 (Z )) 1 , ∀Z ∈ Fϕ (0), = 2 3 2− Q(Z , j2 (Z )) s Z 3 + c1 Z 2 1

(13.49)

with c1 = tg(ϕ2 ) cot(ϕ1 − ϕ2 ) and we thus obtain the formula required by Theorem 12.1. 14. Appendix 1: The List of Minors We give for convenience the list of the 15 minors of the matrix (5.3), which labels the missing lines, and in factorized form. By setting ⎧  ⎨ A = x02 + 3k=1 cos(2ϕk )xk2 (14.1)  ⎩ B = 3k=1 sin(2ϕk )xk2 one sees that these minors are combinations of the form Mi j = Pi j A + Q i j B: M12 = 2 (sin(ϕ1 − ϕ2 ) x1 x2 + i cos(ϕ3 ) x0 x3 ) (−cos(ϕ1 − ϕ2 ) (cos(ϕ1 − ϕ3 ) sin(ϕ1 ) x12 + cos(ϕ2 − ϕ3 ) sin(ϕ2 ) x22 ) +sin(ϕ3 ) (sin(ϕ1 ) sin(ϕ2 ) x02 − cos(ϕ1 − ϕ3 ) cos(ϕ2 − ϕ3 ) x32 )) = (sin(ϕ1 − ϕ2 ) x1 x2 + i cos(ϕ3 ) x0 x3 ) (2sin(ϕ1 ) sin(ϕ2 ) sin(ϕ3 ) A − (cos(ϕ1 − ϕ2 ) cos(ϕ3 ) + sin(ϕ1 + ϕ2 ) sin(ϕ3 ))B), (14.2) M13 = 2 i (cos(ϕ2 ) x0 x2 + i sin(ϕ1 − ϕ3 ) x1 x3 ) (−cos(ϕ1 − ϕ2 ) (cos(ϕ1 − ϕ3 ) sin(ϕ1 ) x12 + cos(ϕ2 − ϕ3 ) sin(ϕ2 ) x22 ) +sin(ϕ3 ) (sin(ϕ1 ) sin(ϕ2 ) x02 − cos(ϕ1 − ϕ3 ) cos(ϕ2 − ϕ3 ) x32 )) = i (cos(ϕ2 ) x0 x2 + i sin(ϕ1 − ϕ3 ) x1 x3 ) (2sin(ϕ1 ) sin(ϕ2 ) sin(ϕ3 ) A − (cos(ϕ1 − ϕ2 ) cos(ϕ3 ) + sin(ϕ1 + ϕ2 ) sin(ϕ3 ))B), (14.3)

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M14 = 2 (sin(ϕ2 ) x0 x2 + i cos(ϕ1 − ϕ3 ) x1 x3 ) (cos(ϕ2 ) (cos(ϕ3 ) sin(ϕ1 ) x02 + cos(ϕ2 − ϕ3 ) sin(ϕ1 − ϕ2 ) x22 ) +sin(ϕ1 − ϕ3 ) (−sin(ϕ1 ) sin(ϕ1 − ϕ2 ) x12 + cos(ϕ2 − ϕ3 ) cos(ϕ3 ) x32 )) = (sin(ϕ2 ) x0 x2 + i cos(ϕ1 − ϕ3 ) x1 x3 ) (2sin(ϕ1 ) cos(ϕ2 ) cos(ϕ3 ) A − (cos(ϕ1 ) cos(ϕ2 − ϕ3 ) − sin(ϕ1 ) sin(ϕ2 + ϕ3 )) B), (14.4) M15 = −i cos(ϕ1 − ϕ2 − ϕ3 ) (sin(2 (ϕ1 − ϕ2 )) x12 x22

+ sin(2 (ϕ1 − ϕ3 )) x12 x32 − x02 (sin(2 ϕ2 ) x22 + sin(2 ϕ3 ) x32 )) = i cos(ϕ1 − ϕ2 − ϕ3 ) ((sin(2ϕ2 ) x22 + sin(2ϕ3 ) x32 ) A − (cos(2ϕ2 ) x22 + cos(2ϕ3 ) x32 ) B), (14.5) M16 = −2 (−i cos(ϕ1 − ϕ2 ) x1 x2 + sin(ϕ3 ) x0 x3 ) (cos(ϕ2 ) (cos(ϕ3 ) sin(ϕ1 ) x02 + cos(ϕ2 − ϕ3 ) sin(ϕ1 − ϕ2 ) x22 )

+sin(ϕ1 − ϕ3 ) (−sin(ϕ1 ) sin(ϕ1 − ϕ2 ) x12 + cos(ϕ2 − ϕ3 ) cos(ϕ3 ) x32 )) = −(−i cos(ϕ1 − ϕ2 ) x1 x2 + sin(ϕ3 ) x0 x3 ) (2sin(ϕ1 ) cos(ϕ2 ) cos(ϕ3 ) A − (cos(ϕ1 ) cos(ϕ2 − ϕ3 ) − sin(ϕ1 ) sin(ϕ2 + ϕ3 )) B), (14.6) M23 = 2 (i cos(ϕ1 ) x0 x1 + sin(ϕ2 − ϕ3 ) x2 x3 ) (−cos(ϕ1 − ϕ2 ) (cos(ϕ1 − ϕ3 ) sin(ϕ1 ) x12 + cos(ϕ2 − ϕ3 ) sin(ϕ2 ) x22 ) +sin(ϕ3 ) (sin(ϕ1 ) sin(ϕ2 ) x02 − cos(ϕ1 − ϕ3 ) cos(ϕ2 − ϕ3 ) x32 )) = (i cos(ϕ1 ) x0 x1 + sin(ϕ2 − ϕ3 ) x2 x3 ) (2sin(ϕ1 ) sin(ϕ2 ) sin(ϕ3 ) A − (cos(ϕ1 − ϕ2 ) cos(ϕ3 ) + sin(ϕ1 + ϕ2 ) sin(ϕ3 ))B), (14.7) M24 = 2 (sin(ϕ1 ) x0 x1 − i cos(ϕ2 − ϕ3 ) x2 x3 ) (cos(ϕ1 ) (cos(ϕ3 ) sin(ϕ2 ) x02 − cos(ϕ1 − ϕ3 ) sin(ϕ1 − ϕ2 ) x12 ) +sin(ϕ2 − ϕ3 ) (sin(ϕ1 − ϕ2 ) sin(ϕ2 ) x22 + cos(ϕ1 − ϕ3 ) cos(ϕ3 ) x32 )) = (sin(ϕ1 ) x0 x1 − i cos(ϕ2 − ϕ3 ) x2 x3 ) (2cos(ϕ3 ) cos(ϕ1 ) sin(ϕ2 ) A − (cos(ϕ3 − ϕ1 ) cos(ϕ2 ) − sin(ϕ3 + ϕ1 ) sin(ϕ2 )) B), (14.8) M25 = −2 i (cos(ϕ1 − ϕ2 ) x1 x2 − i sin(ϕ3 ) x0 x3 ) (cos(ϕ1 ) (−cos(ϕ3 ) sin(ϕ2 ) x02 + cos(ϕ1 − ϕ3 ) sin(ϕ1 − ϕ2 ) x12 )− sin(ϕ2 − ϕ3 ) (sin(ϕ1 − ϕ2 ) sin(ϕ2 ) x22 + cos(ϕ1 − ϕ3 ) cos(ϕ3 ) x32 )) = i (cos(ϕ1 − ϕ2 ) x1 x2 − i sin(ϕ3 ) x0 x3 ) (2cos(ϕ3 ) cos(ϕ1 ) sin(ϕ2 ) A − (cos(ϕ3 − ϕ1 ) cos(ϕ2 ) − sin(ϕ3 + ϕ1 ) sin(ϕ2 )) B), (14.9) M26 = i cos(ϕ1 − ϕ2 + ϕ3 ) (sin(2 ϕ1 ) x02 x12

+ sin(2 (ϕ1 − ϕ2 )) x12 x22 + (sin(2 ϕ3 ) x02 − sin(2 (ϕ2 − ϕ3 )) x22 ) x32 ) = i cos(ϕ1 − ϕ2 + ϕ3 ) ((sin(2ϕ1 ) x12 + sin(2ϕ3 ) x32 ) A − (cos(2ϕ1 ) x12 + cos(2ϕ3 ) x32 ) B), (14.10)

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M34 = −i cos(ϕ1 + ϕ2 − ϕ3 ) (sin(2 ϕ1 ) x02 x12 + sin(2 ϕ2 ) x02 x22 + (sin(2 (ϕ1 − ϕ3 )) x12 + sin(2 (ϕ2 − ϕ3 )) x22 ) x32 ) = −i cos(ϕ1 + ϕ2 − ϕ3 ) ((sin(2ϕ1 ) x12 + sin(2ϕ2 ) x22 ) A − (cos(2ϕ1 ) x12 + cos(2ϕ2 ) x22 ) B), (14.11) M35 = 2 i (i sin(ϕ2 ) x0 x2 + cos(ϕ1 − ϕ3 ) x1 x3 ) (cos(ϕ1 ) (−cos(ϕ2 ) sin(ϕ3 ) x02 + cos(ϕ1 − ϕ2 ) sin(ϕ1 − ϕ3 ) x12 ) +sin(ϕ2 − ϕ3 ) (cos(ϕ1 − ϕ2 ) cos(ϕ2 ) x22 + sin(ϕ1 − ϕ3 ) sin(ϕ3 ) x32 )) = −i (i sin(ϕ2 ) x0 x2 + cos(ϕ1 − ϕ3 ) x1 x3 ) (2cos(ϕ1 ) cos(ϕ2 ) sin(ϕ3 ) A − (cos(ϕ1 − ϕ2 ) cos(ϕ3 ) − sin(ϕ1 + ϕ2 ) sin(ϕ3 )) B), (14.12) M36 = −2 (sin(ϕ1 ) x0 x1 + i cos(ϕ2 − ϕ3 ) x2 x3 ) (cos(ϕ1 ) (−cos(ϕ2 ) sin(ϕ3 ) x02 + cos(ϕ1 − ϕ2 ) sin(ϕ1 − ϕ3 ) x12 ) +sin(ϕ2 − ϕ3 ) (cos(ϕ1 − ϕ2 ) cos(ϕ2 ) x22 + sin(ϕ1 − ϕ3 ) sin(ϕ3 ) x32 )) = (sin(ϕ1 ) x0 x1 + i cos(ϕ2 − ϕ3 ) x2 x3 ) (2cos(ϕ1 ) cos(ϕ2 ) sin(ϕ3 ) A − (cos(ϕ1 − ϕ2 ) cos(ϕ3 ) − sin(ϕ1 + ϕ2 ) sin(ϕ3 )) B), (14.13) M45 = −2 i (cos(ϕ2 ) x0 x2 − i sin(ϕ1 − ϕ3 ) x1 x3 ) (cos(ϕ1 ) (cos(ϕ3 ) sin(ϕ2 ) x02 − cos(ϕ1 − ϕ3 ) sin(ϕ1 − ϕ2 ) x12 ) +sin(ϕ2 − ϕ3 ) (sin(ϕ1 − ϕ2 ) sin(ϕ2 ) x22 + cos(ϕ1 − ϕ3 ) cos(ϕ3 ) x32 )) = −i (cos(ϕ2 ) x0 x2 − i sin(ϕ1 − ϕ3 ) x1 x3 ) (2cos(ϕ3 ) cos(ϕ1 ) sin(ϕ2 ) A − (cos(ϕ3 − ϕ1 ) cos(ϕ2 ) − sin(ϕ3 + ϕ1 ) sin(ϕ2 )) B), (14.14) M46 = 2 (−i cos(ϕ1 ) x0 x1 + sin(ϕ2 − ϕ3 ) x2 x3 ) (cos(ϕ2 ) (cos(ϕ3 ) sin(ϕ1 ) x02 + cos(ϕ2 − ϕ3 ) sin(ϕ1 − ϕ2 ) x22 ) +sin(ϕ1 − ϕ3 ) (−sin(ϕ1 ) sin(ϕ1 − ϕ2 ) x12 + cos(ϕ2 − ϕ3 ) cos(ϕ3 ) x32 )) = (−i cos(ϕ1 ) x0 x1 + sin(ϕ2 − ϕ3 ) x2 x3 ) (2sin(ϕ1 ) cos(ϕ2 ) cos(ϕ3 ) A − (cos(ϕ1 ) cos(ϕ2 − ϕ3 ) − sin(ϕ1 ) sin(ϕ2 + ϕ3 )) B), (14.15) M56 = 2 (sin(ϕ1 − ϕ2 ) x1 x2 − i cos(ϕ3 ) x0 x3 ) (cos(ϕ1 ) (cos(ϕ2 ) sin(ϕ3 ) x02 − cos(ϕ1 − ϕ2 ) sin(ϕ1 − ϕ3 ) x12 ) − sin(ϕ2 − ϕ3 ) (cos(ϕ1 − ϕ2 ) cos(ϕ2 ) x22 + sin(ϕ1 − ϕ3 ) sin(ϕ3 ) x32 )) = (sin(ϕ1 − ϕ2 ) x1 x2 − i cos(ϕ3 ) x0 x3 ) (2cos(ϕ1 ) cos(ϕ2 ) sin(ϕ3 ) A − (cos(ϕ1 − ϕ2 ) cos(ϕ3 ) − sin(ϕ1 + ϕ2 ) sin(ϕ3 )) B). (14.16) 15. Appendix 2: The Sixteen Theta Relations The sixteen theta relations are the following, with (w, x, y, z) = M (a, b, c, d),

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where ⎡

1 1 1 ⎢1 1 M := ⎣ 2 1 −1 1 −1

1 −1 1 −1

⎤ 1 −1 ⎥ . −1 ⎦ 1

ϑ2 (a) ϑ2 (b) ϑ2 (c) ϑ2 (d) + ϑ3 (a) ϑ3 (b) ϑ3 (c) ϑ3 (d) = ϑ2 (x) ϑ2 (y) ϑ2 (z) ϑ2 (w) + ϑ3 (x) ϑ3 (y) ϑ3 (z) ϑ3 (w) ϑ3 (a) ϑ3 (b) ϑ3 (c) ϑ3 (d) − ϑ2 (a) ϑ2 (b) ϑ2 (c) ϑ2 (d) = ϑ1 (x) ϑ1 (y) ϑ1 (z) ϑ1 (w) + ϑ4 (x) ϑ4 (y) ϑ4 (z) ϑ4 (w) ϑ1 (a) ϑ1 (b) ϑ1 (c) ϑ1 (d) + ϑ4 (a) ϑ4 (b) ϑ4 (c) ϑ4 (d) = ϑ3 (w) ϑ3 (x) ϑ3 (y) ϑ3 (z) − ϑ2 (w) ϑ2 (x) ϑ2 (y) ϑ2 (z) ϑ4 (a) ϑ4 (b) ϑ4 (c) ϑ4 (d) − ϑ1 (a) ϑ1 (b) ϑ1 (c) ϑ1 (d) = ϑ4 (w) ϑ4 (x) ϑ4 (y) ϑ4 (z) − ϑ1 (w) ϑ1 (x) ϑ1 (y) ϑ1 (z) ϑ1 (a) ϑ1 (b) ϑ2 (c) ϑ2 (d) + ϑ3 (c) ϑ3 (d) ϑ4 (a) ϑ4 (b) = ϑ1 (x) ϑ1 (w) ϑ2 (y) ϑ2 (z) + ϑ3 (y) ϑ3 (z) ϑ4 (x) ϑ4 (w) ϑ4 (a) ϑ4 (b) ϑ3 (c) ϑ3 (d) − ϑ1 (a) ϑ1 (b) ϑ2 (c) ϑ2 (d) = ϑ1 (y) ϑ1 (z) ϑ2 (x) ϑ2 (w) + ϑ3 (x) ϑ3 (w) ϑ4 (y) ϑ4 (z) ϑ1 (a) ϑ1 (b) ϑ3 (c) ϑ3 (d) + ϑ2 (c) ϑ2 (d) ϑ4 (a) ϑ4 (b) = ϑ1 (x) ϑ1 (w) ϑ3 (y) ϑ3 (z) + ϑ2 (y) ϑ2 (z) ϑ4 (x) ϑ4 (w) ϑ4 (a) ϑ4 (b) ϑ2 (c) ϑ2 (d) − ϑ1 (a) ϑ1 (b) ϑ3 (c) ϑ3 (d) = ϑ1 (y) ϑ1 (z) ϑ3 (x) ϑ3 (w) + ϑ2 (x) ϑ2 (w) ϑ4 (y) ϑ4 (z) ϑ2 (c) ϑ2 (d) ϑ3 (a) ϑ3 (b) + ϑ2 (a) ϑ2 (b) ϑ3 (c) ϑ3 (d) = ϑ2 (x) ϑ2 (w) ϑ3 (y) ϑ3 (z) + ϑ2 (y) ϑ2 (z) ϑ3 (x) ϑ3 (w) ϑ3 (a) ϑ3 (b) ϑ2 (c) ϑ2 (d) − ϑ2 (a) ϑ2 (b) ϑ3 (c) ϑ3 (d) = ϑ1 (x) ϑ1 (w) ϑ4 (y) ϑ4 (z) + ϑ1 (y) ϑ1 (z) ϑ4 (x) ϑ4 (w) ϑ1 (c) ϑ1 (d) ϑ4 (a) ϑ4 (b) + ϑ1 (a) ϑ1 (b) ϑ4 (c) ϑ4 (d) = ϑ3 (w) ϑ3 (x) ϑ2 (y) ϑ2 (z) − ϑ2 (w) ϑ2 (x) ϑ3 (y) ϑ3 (z)

(15.1)

(15.2)

(15.3)

(15.4)

(15.5)

(15.6)

(15.7)

(15.8)

(15.9)

(15.10)

(15.11)

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ϑ4 (a) ϑ4 (b) ϑ1 (c) ϑ1 (d) − ϑ1 (a) ϑ1 (b) ϑ4 (c) ϑ4 (d) = ϑ4 (w) ϑ4 (x) ϑ1 (y) ϑ1 (z) − ϑ1 (w) ϑ1 (x) ϑ4 (y) ϑ4 (z) ϑ2 (c) ϑ2 (d) ϑ3 (a) ϑ3 (b) + ϑ1 (c) ϑ1 (d) ϑ4 (a) ϑ4 (b) = ϑ2 (y) ϑ2 (z) ϑ3 (x) ϑ3 (w) + ϑ1 (y) ϑ1 (z) ϑ4 (x) ϑ4 (w) ϑ3 (a) ϑ3 (b) ϑ2 (c) ϑ2 (d) − ϑ4 (a) ϑ4 (b) ϑ1 (c) ϑ1 (d) = ϑ2 (x) ϑ2 (w) ϑ3 (y) ϑ3 (z) + ϑ1 (x) ϑ1 (w) ϑ4 (y) ϑ4 (z) ϑ1 (d) ϑ2 (b) ϑ3 (a) ϑ4 (c) + ϑ1 (c) ϑ2 (a) ϑ3 (b) ϑ4 (d) = ϑ1 (w) ϑ4 (x) ϑ2 (y) ϑ3 (z) − ϑ4 (w) ϑ1 (x) ϑ3 (y) ϑ2 (z) ϑ3 (a) ϑ2 (b) ϑ4 (c) ϑ1 (d) − ϑ2 (a) ϑ3 (b) ϑ1 (c) ϑ4 (d) = ϑ3 (w) ϑ2 (x) ϑ4 (y) ϑ1 (z) − ϑ2 (w) ϑ3 (x) ϑ1 (y) ϑ4 (z)

(15.12)

(15.13)

(15.14)

(15.15)

(15.16)

References 1. Abadie, B., Exel, R.: Hilbert C ∗ -bimodules over commutative C ∗ -algebras and an isomorphism condition for quantum Heisenberg manifolds. Rev. Math. Phys. 9(4), 411–423 (1997) 2. Abadie, B., Eilers, S., Exel, R.: Morita equivalence for crossed products by Hilbert C ∗ -bimodules. Trans. Amer. Math. Soc. 350(8), 3043–3054 (1998) 3. Artin, M., Schelter, W.F.: Graded algebras of global dimension 3. Adv. Math. 66, 171–216 (1987) 4. Artin, M., Tate, J., Van den Bergh, M.: Some algebras associated to automorphisms of elliptic curves. The Grothendieck Festschrift. Vol. I. Prog. Math., 86, 33–85 (1990) 5. Artin, M., Tate, J., Vanden Bergh, M.: Modules over regular algebras of dimension 3. Invent. Math. 106, 335–388 (1991) 6. Berger, R.: Koszulity for nonquadratic algebras. J. Alg. 239, 705–734 (2001) 7. Berger, R., Dubois-Violette, M., Wambst, M.: Homogeneous algebras. J. Alg. 261, 172–185 (2003) 8. Bourbaki, N.: Groupes et algèbres de Lie. Chapitres 4,5 et 6. Parts: Hermann, 1968 9. Bourbaki, N.: Groupes et algèbres de Lie. Chapitre 9. Parts: Masson, 1982 10. Connes, A.: C ∗ algèbres et géométrie différentielle. C.R. Acad. Sci. Paris, Série A 290, 599–604 (1980) 11. Connes, A.: Noncommutative differential geometry. Part I: The Chern character in K -homology; Part II: de Rham homology and noncommutative algebra. Preprints IHES M/82/53; M83/19, 1982/83 12. Connes, A.: Cyclic cohomology and the transverse fundamental class of a foliation. In: Geometric methods in operator algebras, Pitman Res. Notes in Math. 123 (1986), Harlow: Longman 1986, pp. 52–144 13. Connes, A.: Non-commutative differential geometry. Publ. IHES 62, 257–360 (1986) 14. Connes, A.: Non-commutative geometry. London New York: Academic Press, 1994 15. Connes, A., Dubois-Violette, M.: Noncommutative finite-dimensional manifolds. I.Spherical manifolds and related examples. Commun. Math. Phys. 230, 539–579 (2002) 16. Connes, A., Dubois-Violette, M.: Yang-Mills algebra. Lett. Math. Phys. 61, 149–158 (2002) 17. Connes, A., Dubois-Violette, M.: Moduli space and structure of noncommutative 3-spheres. Lett. Math. Phys. 66, 91–121 (2003) 18. Connes, A., Dubois-Violette, M.: Yang-Mills and some related algebras. In: Rigorous Quantum Field Theory. J. Alg. 261, 172–185 (2003) 19. Connes, A., Landi, G.: Noncommutative manifolds, the instanton algebra and isospectral deformations. Commun. Math. Phys. 221, 141–159 (2001) 20. Dubois-Violette, M.: Lectures on graded differential algebras and noncommutative geometry. In: Maeda Y., Moriyoshi H. editors, Noncommutative Differential Geometry and Its Applications to Physics, (Shonan, Japan, 1999), Dordrecht: Kluwer Academic Publishers, 2001, pp. 245–306 21. Dubois-Violette, M., Michor, P.W.: Dérivations et calcul différentiel non commutatif, II. C.R. Acad. Sci. Paris 319, 927–931 (1994) 22. Helgason, S.: Differential geometry, Lie groups and symmetric spaces. London: Academic Press, 1978 23. Majid, S.: Foundations of quantum group theory. Cambridge: Cambridge University Press, 1995

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24. Odesskii, A.V., Feigin, B.L.: Sklyanin elliptic algebras. Func. Anal. Appl. 23, 207–214 (1989) 25. Pimsner, M.V.: A class of C*-algebra generalizing both Cuntz-Krieger algebras and crossed products by Z . In: Free probablility Theory, Fields Inst. Comm., Providence, RI: Amer. Math. Soc. 12, 1997, pp. 189–212 26. Rieffel, M.: Deformation quantization of Heisenberg Manifolds. Commun. Math. Phys. 122, 531–562 (1989) 27. Sklyanin, E.K.: Some algebraic structures connected with the Yang-Baxter equation. Func. Anal. Appl. 16, 263–270 (1982) 28. Sklyanin, E.K.: Some algebraic structures connected with the Yang-Baxter equation. Representation of Quantum Algebras. Func. Anal. Appl. 17, 273–284 (1983) 29. Smith, S.P., Stafford, J.T.: Regularity of the four dimensional Sklyanin algebra. Compos. Math. 83, 259–289 (1992) 30. Tate, J., Vanden Bergh, M.: Homological properties of Sklyanin algebras. Invent. Math. 124, 619–647 (1996) Communicated by N.A. Nekrasov

Commun. Math. Phys. 281, 129–177 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0481-x

Communications in

Mathematical Physics

A New Approach to the Modeling of Local Defects in Crystals: The Reduced Hartree-Fock Case Éric Cancès1 , Amélie Deleurence1 , Mathieu Lewin2 1 CERMICS, Ecole Nationale des Ponts et Chaussées (Paris Tech) & INRIA (Micmac Project),

6-8 Av. Pascal, 77455 Champs-sur-Marne, France. E-mail: [email protected]; [email protected] 2 CNRS & Laboratoire de Mathématiques UMR 8088, Université de Cergy-Pontoise, 2 Avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France. E-mail: [email protected] Received: 20 February 2007 / Accepted: 19 December 2007 Published online: 26 April 2008 – © Springer-Verlag 2008

Abstract: This article is concerned with the derivation and the mathematical study of a new mean-field model for the description of interacting electrons in crystals with local defects. We work with a reduced Hartree-Fock model, obtained from the usual Hartree-Fock model by neglecting the exchange term. First, we recall the definition of the self-consistent Fermi sea of the perfect crystal, which is obtained as a minimizer of some periodic problem, as was shown by Catto, Le Bris and Lions. We also prove some of its properties which were not mentioned before. Then, we define and study in detail a nonlinear model for the electrons of the crystal in the presence of a defect. We use formal analogies between the Fermi sea of a perturbed crystal and the Dirac sea in Quantum Electrodynamics in the presence of an external electrostatic field. The latter was recently studied by Hainzl, Lewin, Séré and Solovej, based on ideas from Chaix and Iracane. This enables us to define the ground state of the self-consistent Fermi sea in the presence of a defect. We end the paper by proving that our model is in fact the thermodynamic limit of the so-called supercell model, widely used in numerical simulations. Describing the electronic state of crystals with local defects is a major issue in solid-state physics, materials science and nano-electronics [17,25,33]. In this article, we develop a theory based on formal analogies between the Fermi sea of a perturbed crystal and the polarized Dirac sea in Quantum Electrodynamics in the presence of an external electrostatic field. Recently, the latter model was extensively studied by Hainzl, Lewin, Séré and Solovej in the Hartree-Fock approximation [10– 13], based on ideas from Chaix and Iracane [6] (see also [1,7]). This was summarized in the review [14]. Using and adapting these methods, we are able to propose a new mathematical approach for the self-consistent description of a crystal in the presence of local defects. We focus in this article on the reduced Hartree-Fock (rHF) model in which the so-called exchange term is neglected. To further simplify the mathematical formulas, we do not explicitly take the spin variable into account and we assume that the host

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crystal is cubic with a single atom of charge Z per unit cell. The arguments below can be easily extended to the general case. In the whole paper, the main object of interest will be the so-called density matrix of the electrons. This is a self-adjoint operator 0 ≤ γ ≤ 1 acting on the one-body space L 2 (R3 ). When γ has a finite rank, it models a finite number of electrons. In the periodic 0 has an infinite rank (it describes infinitely many case, the ground state density matrix γper electrons) and commutes with the translations of the lattice. We will see in the sequel that the ground state density matrix of a crystal with a local defect can be written as 0 + Q, where Q is a compact perturbation of the periodic density matrix γ 0 of γ = γper per the reference perfect crystal. In each of the above three cases (finite number of electrons, perfect crystal, defective crystal), the ground state density matrix can be obtained by minimizing some nonlinear energy functional depending on a set of admissible density matrices. In the case of a crystal with a local defect, the perturbation Q is a minimizer of some nonlinear minimization problem set in the whole space R3 , with a possible lack of compactness at infinity. The main unusual feature compared to standard variational problems is that Q is a self-adjoint operator of infinite rank. This was already the case in [10–13]. The paper is organized as follows. In Sect. 1, we recall the definition of the reduced Hartree-Fock model for a finite number of electrons, which serves as a basis for the theories of infinitely many electrons in a (possibly perturbed) periodic nuclear distribution. Sect. 2 is devoted to the definition of the model for the infinite periodic crystal, following mainly [4,5] (but we provide some additional material compared to what was done in [4,5]). In Sect. 3, we define a model for the crystal with local defects which takes the perfect crystal as reference. In Sect. 4, we prove that this model is the thermodynamic limit of the supercell model. For the convenience of the reader, we have gathered all the proofs in Sect. 5. Often, the proofs follow the same lines as those in [10–13] and we shall not detail identical arguments. But there are many difficulties associated with the particular model under study which do not appear in previous works and which are addressed in detail here.

1. The Reduced Hartree-Fock Model for N Electrons We start by recalling the definition of the reduced Hartree-Fock model [31] for a finite number of electrons. Note that the reduced Hartree-Fock model should not be confused with the restricted Hartree-Fock model commonly used in numerical simulations (see e.g. [8]). We consider a system containing N nonrelativistic quantum electrons and a set of nuclei having a density of charge ρnuc . If for instance there are K nuclei of charges z 1 , ..., z K ∈ N \ {0} located at R1 , ..., R K ∈ R3 , then ρnuc (x) :=

K


z k m k (x − Rk ),

k=1

where m 1 , ..., m K are positive measures on R3 of total mass one. Point-like nuclei would correspond to m k = δ (the Dirac measure) but for convenience we shall deal with smeared nuclei in the sequel,  i.e. we assume that for all k = 1...K , m k is a smooth nonnegative function such that R3 m k = 1. The technical difficulties arising with point-like nuclei will be dealt with elsewhere.

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The energy of the whole system in the reduced Hartree-Fock model reads [5,31]    1  1 EρrHF γ + D ργ − ρnuc , ργ − ρnuc . (1.1) (γ ) = Tr − nuc 2 2 We have chosen a system of units such that
= m = e = 4π10 = 1, where m and e are respectively the mass and the charge of an electron,
is the reduced Planck constant and 0 is the dielectric permittivity of the vacuum. The first term in the right-hand side of (1.1) is the kinetic energy of the electrons and D(·, ·) is the classical Coulomb interaction, which reads for f and g in L 6/5 (R3 ) as     f (x) g(y) f (k) g (k) d x d y = 4π D( f, g) = dk, (1.2) 2 3 3 3 |x − y| |k| R R R where  f denotes the Fourier transform of f . In this mean-field model, the state of the N electrons is described by the one-body density matrix γ , which is an element of the following class:

√  √ P N = γ ∈ S(L 2 (R3 )) | 0 ≤ γ ≤ 1, Tr(γ ) = N , Tr −γ − < ∞ . Here and below, S(H) denotes the space of bounded √ self-adjoint √ operators acting on the Hilbert space H. Also we define Tr((−)γ ) := Tr( −γ −) which makes sense when γ ∈ P N . The set P N is the closed convex hull of the set of orthogonal projectors of rank N acting on L 2 (R3 ) and having a finite kinetic energy. Each such projector

N |ϕi ϕi | is the density matrix of a Hartree-Fock state γ = i=1  = ϕ1 ∧ · · · ∧ ϕ N

(1.3)

in the usual N -body space of fermionic wavefunctions with finite kinetic energy N 1 3 i=1 H (R ). The function ργ appearing in (1.1) is the density associated with the operator γ , defined by ργ (x) = γ (x, x), where γ (x, y) is the kernel of the trace class operator γ . √ Notice that for all γ ∈ P N , one has ργ ≥ 0 and ργ ∈ H 1 (R3 ), hence the last term of 1 3 3 (1.1) is well-defined, since ργ ∈ L (R ) ∩ L (R3 ) ⊂ L 6/5 (R3 ).

M It can be proved (see the appendix of [31]) that if N ≤ k=1 z k (neutral or positively charged systems), the variational problem   N IrHF (ρnuc , N ) = inf EρrHF (1.4) (γ ), γ ∈ P nuc has a minimizer γ and that the corresponding minimizing density ργ is unique. The Hartree-Fock model [21] is the variational approximation of the time-independent Schrödinger equation obtained by restricting the set of fermionic wavefunctions under consideration to the subset of functions of the form (1.3). The HF functional reads  |γ (x, y)|2 1 HF rHF Eρnuc (γ ) = Eρnuc (γ ) − d x d y, (1.5) 2 R6 |x − y| the last term being called the exchange energy. As the Hartree-Fock energy functional is nonconvex, there is little hope to obtain rigorous thermodynamic limits in this setting, at least with current state-of-the-art techniques. For this reason, the exchange term is often neglected in mathematical studies.

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2. The Reduced Hartree-Fock Model for a Perfect Crystal In this article, we clamp the nuclei on a periodic lattice, optimizing only over the state of the electrons. More precisely we are interested in the change of the electronic state of the crystal when a local defect is introduced. To this end, we shall rely heavily on the rHF model for the infinite perfect crystal (with no defect) which was studied by Catto, Le Bris and Lions in [4,5]. The latter can be obtained as the thermodynamical limit of the rHF model for finite systems which was introduced in the previous section. This will be explained in Sect. 4 below. Let = [−1/2, 1/2)3 be the unit cell. We denote by ∗ = [−π, π )3 the first Brillouin zone of the lattice, and by τk the translation operator on L 2loc (R3 ) defined by τk u(x) = u(x − k). We then introduce Pper = γ ∈ S(L 2 (R3 )) | 0 ≤ γ ≤ 1, ∀k ∈ Z3 , τk γ = γ τk ,  Tr L 2 ( ) ((1 − ξ )1/2 γξ (1 − ξ )1/2 ) dξ < ∞ , ∗

ξ

where (γξ )ξ ∈ ∗ is the Bloch waves decomposition of γ , see [5,27]:  1 γ = γξ dξ, γξ ∈ S(L 2ξ ( )), (2π )3 ∗   L 2ξ ( ) = u ∈ L 2loc (R3 ) | τk u = e−ik·ξ u, ∀k ∈ Z3 ⊕ which corresponds to the decomposition in fibers L 2 (R3 ) = ∗ L 2ξ ( )dξ . For any γ ∈ Pper , we denote by γξ (x, y) the integral kernel of γξ . The density of γ is then the nonnegative Z3 -periodic function of L 1loc (R3 ) ∩ L 3loc (R3 ) defined as  1 ργ (x) := γξ (x, x) dξ. (2π )3 ∗ Notice that for any γ ∈ Pper ,  ργ (x)d x =

1 (2π )3

 ∗

Tr L 2 ( ) (γξ ) dξ, ξ

i.e. this gives the number of electrons per unit cell. Later we shall add the constraint that the system is neutral and restrict to states γ ∈ Pper satisfying  ργ (x)d x = Z ,

where Z is the total charge of the nuclei in each unit cell. We also introduce the Z3 -periodic Green kernel of the Poisson interaction [22], denoted by G 1 and uniquely defined by ⎧ ⎛ ⎞ ⎪
⎪ ⎪ ⎨ −G 1 = 4π ⎝ δk − 1⎠ k∈Z3 ⎪ ⎪ ⎪ ⎩ G 1 Z3 -periodic, min G 1 = 0.

R3

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The Fourier expansion of G 1 is


G 1 (x) = c +

k∈2π Z3 \{0}

4π ik·x e |k|2

 with c = G 1 > 0. The electrostatic potential associated with a Z3 -periodic density ρ ∈ L 1loc (R3 ) ∩ L 3loc (R3 ) is the Z3 -periodic function defined as  G 1 (x − y) ρ(y) dy. (ρ G 1 )(x) :=

We also set for any Z3 -periodic functions f and g,   DG 1 ( f, g) := G 1 (x − y) f (x) g(y)d x d y.



Throughout this article, we will denote by χ I the characteristic function of the set I ⊂ R and by χ I (A) the spectral projector on I of the self-adjoint operator A. The periodic density of the nuclei is given by
µper (x) = Z m(x − R). (2.1) R∈Z3

We assumefor simplicity that m is a nonnegative function of Cc∞ (R3 ) with support in , and that R3 m(x)d x = 1. Hence µper (x)d x = Z , the total charge of the nuclei in each unit cell. The periodic rHF energy is then defined for γ ∈ Pper as 0 Eper (γ ) =

1 (2π )3



    1 1 Tr L 2 ( ) − γξ dξ + DG 1 ργ − µper , ργ − µper . ξ 2 2 ∗

(2.2)

Introducing Z Pper

 := γ ∈ Pper | ργ = Z ,

(2.3)

the periodic rHF ground state energy (per unit cell) is given by   0 0 Z . Iper = inf Eper (γ ), γ ∈ Pper

(2.4)

0 ∈ PZ It was proved by Catto, Le Bris and Lions in [5] that there exists a minimizer γper per to the minimization problem (2.4), and that all the minimizers of (2.4) share the same density ργper 0 . We give in Appendix 5.10 the proof of the following

Theorem 1. (Definition of the periodic rHF minimizer) Let Z ∈ N \ {0}. The minimiza0 . Denoting by tion problem (2.4) admits a unique minimizer γper 0 Hper := −

 + (ργper 0 − µper ) G 1 , 2

(2.5)

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0 is the solution to the the corresponding periodic mean-field Hamiltonian, γper self-consistent equation 0 0 = χ(−∞, F ] (Hper ), γper

(2.6)

where  F is a Lagrange multiplier called Fermi level, which can be interpreted as a chemical potential. 0 is the unique minimizer on Additionally, for any  F ∈ R such that (2.6) holds, γper Pper of the energy functional  0 ργ . γ → Eper (γ ) −  F

0 is unique, Theorem 1 contains three main results that were not present in [5]: first γper second it is a projector, and third it satisfies Eq. (2.6). These three properties are crucial for a proper construction of the model for the crystal with a defect. 2 3 It can easily be seen that (ργper 0 − µper ) G 1 belongs to L loc (R ). By a result of 0 is purely absolutely continuous. This Thomas [34] this implies that the spectrum of Hper 0 . Let (λ (ξ )) is an essential property for the proof of the uniqueness of γper n n≥1 denote 0 the nondecreasing sequence of the eigenvalues of (Hper )ξ . Then 0 σ (Hper )

=



∗

λn ( ),

0 Hper

n≥1

1 = (2π )3

 ∗

0 (Hper )ξ dξ.

0 represents the state of the Fermi sea, i.e. of the infinite system of The projector γper all the electrons in the periodic crystal. Of course, it is an infinite rank projector, meaning that
0 γper = |ϕk ϕk | k

should be interpreted as the one-body matrix of a formal infinite Slater determinant  = ϕ1 ∧ ϕ2 ∧ · · · ∧ ϕk ∧ · · · . 0 is additionally a spectral projector associated with the continuous The fact that γper spectrum of an operator leads to the obvious analogy with the Dirac sea which is the projector on the negative spectral subspace of the Dirac operator [10–13]. Most of our results will hold true for insulators (or semi-conductors) only. When necessary, we shall take Z ∈ N \ {0} and make the following assumption: + (A1) There is a gap between the Z th and the (Z + 1)st bands, i.e.  +Z <  − Z +1 , where  Z − th st and  Z +1 are respectively the maximum and the minimum of the Z and the (Z + 1) 0 . bands of Hper 0 of the nonWe emphasize that Assumption (A1) is a condition on the solution γper 0 = χ 0 linear problem (2.4). Note that under (A1), one has γper (−∞, F ] (Hper ) for any − +  F ∈ ( Z ,  Z +1 ).

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3. The Reduced Hartree-Fock Model for a Crystal with a Defect In this section, we define the reduced Hartree-Fock model describing the behavior of the Fermi sea and possibly of a finite number of bound electrons (or holes) close to a local defect. Our model is an obvious transposition of the Bogoliubov-Dirac-Fock model which was proposed by Chaix and Iracane [6] to describe the polarized Dirac sea (and a finite number of relativistic electrons) in the presence of an external potential. Our mathematical definition of the reduced energy functional follows mainly ideas from [10,11]. We shall prove in Sect. 4 that this model can be obtained as the thermodynamic limit of the so-called supercell model. An analogous result was proved in [13] for the Bogoliubov-Dirac-Fock (BDF) model. Assume that the periodic nuclear density µper defined in (2.1) is replaced by a locally perturbed nuclear density µper +ν. The defect ν can model a vacancy, an interstitial atom, or an impurity, with possible local rearrangement of the neighboring atoms. The main idea underlying the model is to define a finite energy by subtracting the infinite energy 0 defined in the previous section, from the infinite energy of of the periodic Fermi sea γper the perturbed system under consideration. For the BDF model, this was proposed first in [13]. Formally, one obtains for a test state γ ,

 rHF 0 0 0 EµrHF (γ ) − E (γ ) “ = ” Tr H (γ − γ ) µ per per per +ν +ν per per   ν(x)ρ   ρ 0 ] (y) 0 ] (x)ρ[γ −γ 0 ] (y) [γ −γper [γ −γper 1 per dx dy + d x d y. (3.1) − 3 3 3 3 |x − y| 2 |x − y| R R R R Of course the two terms in the left-hand side of (3.1) are not well-defined because µper 0 have infinite ranks, but we shall be able to give a is periodic and because γ and γper 0 mathematical meaning to the right-hand side, exploiting the fact that Q := γ − γper 0 . The formal computation (3.1) induces a small perturbation of the reference state γper will be justified by means of thermodynamic limit arguments in Sect. 4. 3.1. Definition of the reduced Hartree-Fock energy of a defect. We now define properly the reduced Hartree-Fock energy of the Fermi sea in the presence of the defect ν. We denote by S p the Schatten class of operators Q acting on L 2 (R3 ) having a finite p trace, i.e. such that Tr(|Q| p ) < ∞. Note that S1 is the space of trace-class operators, and that S2 is the space of Hilbert-Schmidt operators. Let  be an orthogonal projector on L 2 (R3 ) such that both  and 1− have infinite ranks. A self-adjoint compact operator Q is said to be -trace class (Q ∈ S 1 ) when Q ∈ S2 and Q, (1−)Q(1−) ∈ S1 . Its -trace is then defined as Tr  (Q) = Tr(Q + (1 − )Q(1 − )). Notice that if Q ∈ S1 , then Q ∈ S 1 for any  and Tr  (Q) = Tr(Q). See [10, Sect. 2.1] for general properties related to this definition. In the following, we use the shorthand notation 0 0 Q −− := γper Qγper ,

0 0 Q + + := (1 − γper )Q(1 − γper ),

   γ0 S01 := S1per = Q ∈ S2  Q + + ∈ S1 , Q −− ∈ S1 and Tr 0 (Q) := Tr γper 0 (Q). We also introduce the Banach space    Q = Q ∈ S01  Q ∗ = Q, |∇|Q ∈ S2 , |∇|Q + + |∇| ∈ S1 , |∇|Q −− |∇| ∈ S1 ,

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endowed with its natural norm

    ||Q||Q := ||Q||S2 +  Q + + S +  Q −− S 1   1   + |||∇|Q||S2 + |∇|Q + + |∇|S + |∇|Q −− |∇|S . 1

1

The convex set on which the energy will be defined is   0 0 . K := Q ∈ Q | − γper ≤ Q ≤ 1 − γper

(3.2)

(3.3)

Notice that K is the closed convex hull of states Q ∈ Q of the special form Q = 0 , γ being an orthogonal projector on L 2 (R3 ). Besides, the number Tr (Q) can γ − γper 0 be interpreted as the charge of the system measured with respect to that of the unperturbed Fermi sea. It can be proved [10, Lemma 2] that Tr 0 (Q) is always an integer if Q is a 0 , with γ an orthogonal Hilbert-Schmidt operator of the special form Q = γ − γper projector. Additionally, in this case, Tr 0 (Q) = 0 when Q < 1. 0 ≤ Q ≤ 1 − γ 0 in (3.3) is equivalent [1,10] to the Note that the constraint −γper per inequality Q 2 ≤ Q + + − Q −−

(3.4)

and implies in particular that Q + + ≥ 0 and Q −− ≤ 0 for any Q ∈ K. In order to define properly the energy of Q, we need to associate a density ρ Q with any state Q ∈ K. We shall see that ρ Q can in fact be defined for any Q ∈ Q. This is not obvious a priori since Q does not only contain trace-class operators. Additionally we need to check that the last two terms of (3.1) are well-defined. For this purpose, we introduce the so-called Coulomb space C := {ρ ∈ S  (R3 ) | D(ρ, ρ) < ∞},  where D( f, g) = 4π R3 |k|−2  f (k) g (k)dk was already defined before in (1.2). The dual   space of C is the Beppo-Levi space C  := V ∈ L 6 (R3 ) | ∇V ∈ L 2 (R3 ) . We now use a duality argument to define ρ Q : that Q ∈ Q. Then Proposition 1. (Definition of the density  ρ Q for Q ∈ Q) Assume  QV ∈ S01 for any V = V1 + V2 ∈ C  + L 2 (R3 ) ∩ L ∞ (R3 ) and moreover there exists a constant C (independent of Q and V ) such that |Tr 0 (QV )| ≤ C ||Q||Q (||V1 ||C  + ||V2 || L 2 (R3 ) ).   Thus the linear form V ∈ C  + L 2 (R3 ) ∩ L ∞ (R3 ) → Tr 0 (QV ) can be continuously extended to C  + L 2 (R3 ) and there exists a uniquely defined function ρ Q ∈ C ∩ L 2 (R3 ) such that 

   ∀V = V1 + V2 ∈ C  + L 2 (R3 ) ∩ L ∞ (R3 ) , ρ Q , V1 C ,C  + ρ Q V2 = Tr 0 (QV ). R3

The linear map Q ∈ Q → ρ Q ∈ C ∩ L 2 (R3 ) is continuous:     ρ Q  + ρ Q  2 3 ≤ C ||Q||Q . C L (R ) Eventually when Q ∈ S1 ⊂ S01 , then ρ Q (x) = Q(x, x), where Q(x, y) is the integral kernel of Q.

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The proof of Proposition 1 is given in Sect. 5.2. Assuming that (A1) holds true, we are now in a position to give a rigorous sense to 0 = Q ∈ K. In the sequel, we use the following the right-hand side of (3.1) for γ − γper notation for any Q ∈ Q: 0 0 0 Tr 0 (Hper Q) := Tr(|Hper − κ|1/2 (Q + + − Q −− )|Hper − κ|1/2 ) + κTr 0 (Q), (3.5)

where κ is an arbitrary real number in the gap ( +Z ,  − Z +1 ) (this expression will be proved to be independent of κ, see Corollary 1 below). Then we define the energy of any state Q ∈ K as 0 E ν (Q) := Tr 0 (Hper Q) − D(ρ Q , ν) +

1 D(ρ Q , ρ Q ). 2

(3.6)

The function ν is an external density of charge representing the nuclear charge of the defect. For the sake of simplicity, we shall assume that ν ∈ L 1 (R3 ) ∩ L 2 (R3 ) ⊂ C throughout the paper, although some of our results are true with a weaker assumption. We shall need the following Lemma 1. Assume that (A1) holds true. For any fixed κ in the gap ( +Z ,  − Z +1 ), there exist two constants c1 , c2 > 0 such that 0 − κ| ≤ c2 (1 − ) c1 (1 − ) ≤ |Hper

(3.7)

as operators on L 2 (R3 ). In particular  √    √    0  0 − κ|−1/2 (1 − )1/2  ≤ 1/ c1 . |Hper − κ|1/2 (1 − )−1/2  ≤ c2 , |Hper 0 − κ)(1 − )−1 and its inverse are bounded operators. Similarly, (Hper

The proof of the above lemma is elementary; it will be given in Sect. 5.1.1. By the definition of Q and Lemma 1, it is clear that the right-hand side of (3.5) is a well-defined quantity for any Q ∈ Q and any κ ∈ ( +Z ,  − Z +1 ). By Proposition 1 which states that ρ Q ∈ C for any Q ∈ Q, we deduce that (3.6) is a well-defined functional. We shall need the following space of more regular operators Qr := {Q ∈ Q | (−)Q 2 (−) ∈ S1 , (−)(Q + + − Q −− )(−) ∈ S1 }

(3.8)

and the associated convex set Kr := K ∩ Qr . The following result will be useful (its proof will be given below in Sect. 5.3): Lemma 2. The space Qr (resp. the convex set Kr ) is dense in Q (resp. in K) for the topology of Q.

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0 Q ∈ S0 . For any Corollary 1. Assume that (A1) holds true. When Q ∈ Qr , then Hper 1 0 Q) does not depend on κ ∈ ( + ,  − ). If Q ∈ Q, the expression (3.5) for Tr 0 (Hper Z +1 Z Q ∈ K, then

0 ≤ c1 Tr((1 − )1/2 Q 2 (1 − )1/2 ) ≤ c1 Tr((1 − )1/2 (Q + + − Q −− )(1 − )1/2 ) 0 ≤ Tr 0 (Hper Q) − κTr 0 (Q)

(3.9)

≤ c2 Tr((1 − )1/2 (Q + + − Q −− )(1 − )1/2 ), where c1 and c2 are given by Lemma 1. 0 + + = |H 0 − κ|Q + + = Proof. Let Q ∈ Qr and κ ∈ ( +Z ,  − per Z +1 ). Then ((Hper − κ)Q) 0 −1 + + |Hper − κ|(1 − ) (1 − )Q ∈ S1 by Lemma 1 and the definition of Qr . A similar 0 − κ)Q)−− proves that H 0 Q ∈ S0 . Then for any Q ∈ Q , (3.9) is argument for ((Hper r per 1 a straightforward consequence of (3.7) and (3.4). We conclude using the density of Qr in Q and the density of Kr in K.  

The following is an adaptation of [10, Thm 1]: Corollary 2. Let ν ∈ L 1 (R3 ) ∩ L 2 (R3 ), Z ∈ N \ {0} and assume that (A1) holds. For any κ ∈ ( +Z ,  − Z +1 ), one has for some d1 , d2 > 0, 

    ∀Q ∈ K, E (Q) − κTr 0 (Q) ≥ d1  Q + + S +  Q −− S 1 1 

 1     + |∇|Q + + |∇|S + |∇|Q −− |∇|S + d2 ||Q||2S2 + |||∇|Q||2S2 − D(ν, ν). 1 1 2 (3.10) ν

Hence E ν − κTr 0 is bounded from below and coercive on K. Additionally, when ν ≡ 0, Q → E 0 (Q) − κTr 0 (Q) is nonnegative, 0 being its unique minimizer. Proof. Inequality (3.10) is a straightforward consequence of (3.9) and the fact that D(·, ·) defines a scalar product on C. The rest of the proof is obvious.   0 + Q with respect Remark 1. The energy E ν (Q) measures the energy of a state γ = γper 0 . Thus the last statement of Corollary 2 is another way of expressing the to that of γper 0 fact that γper is the state of lowest energy of the periodic system when there is no defect.

3.2. Existence of minimizers with a chemical potential. In view of Corollary 2, it is natural to introduce the following minimization problem E νF := inf{E ν (Q) −  F Tr 0 (Q), Q ∈ K} > −∞

(3.11)

for any Fermi level  F ∈ ( +Z ,  − Z +1 ). The following result is proved in Sect. 5.5, following ideas from [11]:

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Fig. 1. Decomposition Q¯ = Q pol + γe− for not too strong a positively charged nuclear defect (ν ≥ 0)

Theorem 2 (Existence of minimizers with a chemical potential). Let ν ∈ L 1 (R3 ) ∩ L 2 (R3 ), Z ∈ N \ {0} and assume that (A1) holds. Then for any  F ∈ ( +Z ,  − Z +1 ), there exists a minimizer Q¯ ∈ K for (3.11). Problem (3.11) may have several minimizers, but they all share the same density ρ¯ = ρ Q¯ . Any minimizer Q¯ of (3.11) satisfies the self-consistent equation 

0 + δ, Q¯ = χ(−∞, F ) (H Q¯ ) − γper 0 + (ρ − ν) ∗ | · |−1 , H Q¯ = Hper Q¯

(3.12)

where δ is a finite rank self-adjoint operator satisfying 0 ≤ δ ≤ 1 and Ran (δ) ⊆ ker(H Q¯ −  F ). 0 , Remark 2. It is easily seen that (ρ Q¯ − ν) ∗ | · |−1 is a compact perturbation of Hper 0 ) = D(−) = H 2 (R3 ) and that σ (H ) = implying that H Q¯ is self-adjoint on D(Hper ess Q¯ 0 σ (Hper ). Thus the discrete spectrum of H Q¯ is composed of isolated eigenvalues of finite multiplicity, possibly accumulating at the ends of the bands.

Recall that the charge of the minimizing state Q¯ obtained in Theorem 2 is defined ¯ Similarly to [10,11], it can be proved by perturbation theory that for any as Tr 0 ( Q). fixed  F , there exists a constant C( F ) such that when D(ν, ν) ≤ C( F ), one has ¯ = 0, i.e. the minimizer of the energy with chemical ker(H Q¯ −  F ) = {0} and Tr 0 ( Q) potential  F is a neutral perturbation of the periodic Fermi sea. For a fixed external density ν and an adequately chosen chemical potential  F , one ¯ = 0 meaning either that electron-hole pairs have been created from can have Tr 0 ( Q) the Fermi sea, and/or that the system of lowest energy contains a finite number of bound electrons or holes close to the defect. In the applications, one will usually have for a positively charged nuclear defect (ν ≥ 0) that the spectrum of H Q¯ contains a sequence of eigenvalues converging to the bottom  − Z +1 of the lowest unfilled band (conduction band), and that  F is chosen such that exactly q eigenvalues are filled, corresponding to q bound electrons: 

 0 + χ[, F ) (H Q¯ ) + δ := Q pol + γe− , Q¯ = χ(−∞,) (H Q¯ ) − γper where we have chosen as a reference the center of the gap  :=

 +Z +  − Z +1 . 2

(3.13)

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For not too strong a defect density ν, one has ker(H Q¯ − ) = {0} and Tr 0 (Q pol ) = 0. Hence Tr(γe− ) = q. Let us assume for simplicity that δ = 0 and that q ∈ N \ {0}. Then γe− = χ[, F ) (H Q¯ ) =

q


|ϕn ϕn |,

(3.14)

n=1

where (ϕn ) are eigenfunctions of H Q¯ corresponding to its q first eigenvalues in [,  F ): H Q¯ ϕn = λn ϕn .

(3.15)

H Q¯ = −/2 + (ργe− − ν) ∗ | · |−1 + Vpol ,

(3.16)

Notice that

where −1 Vpol = (ργper 0 − µper ) G 1 + ρ Q pol ∗ | · |

is the polarization potential created by the self-consistent Fermi sea and seen by the q electrons. Thus the q electrons solve a usual reduced Hartree-Fock equation (3.15) in which the mean-field operator (3.16) additionally contains the self-consistent polarization of the medium. The interpretation given in the previous paragraph is different if the positive density of charge ν of the defect is strong enough to create an electron-hole pair from the Fermi sea. We end this section by specifying the regularity of solutions of (3.12). The proof is given in Sect. 5.4. Proposition 2. Let ν ∈ L 1 (R3 ) ∩ L 2 (R3 ), Z ∈ N \ {0} and assume that (A1) holds. Any Q ∈ K solution of the self-consistent equation (3.12) belongs to Kr . Remark 3. Notice that it is natural to wonder whether Q ∈ S1 , which would in particular imply that ρ Q ∈ L 1 (R3 ). This is known to be false for the Bogoliubov-Dirac-Fock model studied in [10–13]. We do not answer this question for our model in the present paper. 3.3. Existence of minimizers under a charge constraint. In the previous section, we stated the existence of minimizers for any chemical potential in the gap of the periodic 0 , but of course the total charge Tr ( Q) operator Hper 0 ¯ of the obtained solution was unknown a priori. Here we tackle the more subtle problem of minimizing the energy while imposing a charge constraint. Mathematically this is more difficult because although the energy E ν (Q) is convex on K and weakly lower semi-continuous (wlsc) for the weak-∗ 0 -trace functional topology of Q (as will be shown in the proof of Theorem 2), the γper Q ∈ K → Tr 0 (Q) is continuous but not wlsc for the weak-∗ topology of Q: in principle it is possible that a (positive or negative) part of the charge of a minimizing sequence for the charge-constrained minimization problem escapes to infinity, leaving at the limit a state of a different (lower or higher) charge. In fact, we can prove that a minimizer exists under a charge constraint, if and only if some binding conditions hold, the role of which being to prevent the lack of compactness. As explained above, imposing Tr 0 (Q) = q should intuitively lead (for a sufficiently weak defect density ν) to a system of q electrons coupled to a polarized Fermi sea.

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 Notice that we do not impose that q = R3 ν, i.e. our model allows a priori to treat defects with non-zero total charge. As usual in reduced Hartree-Fock theories, we consider the case of a real charge constraint q ∈ R: E ν (q) := inf{E ν (Q), Q ∈ K, Tr 0 (Q) = q}.

(3.17)

When no defect is present, E 0 (q) can be computed explicitly: Proposition 3. (Defect-free charge-constrained energy) Let Z ∈ N \ {0} and assume that (A1) holds. Then one has  −  Z +1 q when q ≥ 0 0 E (q) = when q ≤ 0.  +Z q The minimization problem (3.17) with ν ≡ 0 has no solution except when q = 0. The proof of Proposition 3 is given in Sect. 5.6. We now state the main result of this section, which is directly inspired from [12]: Theorem 3. (Existence of minimizers under a charge constraint) Let ν ∈ L 1 (R3 ) ∩ L 2 (R3 ), Z ∈ N\{0} and assume that (A1) holds. The following assertions are equivalent: ¯ (a) Problem (3.17) admits a minimizer Q; (b) Every minimizing sequence for (3.17) is precompact in Q and converges towards a minimizer Q¯ of (3.17); (c) ∀q  ∈ R \ {0}, E ν (q) < E ν (q − q  ) + E 0 (q  ). Assume that the equivalent conditions (a), (b) and (c) above are fulfilled. In this case, the minimizer Q¯ is not necessarily unique, but all the minimizers share the same density ¯ ρ¯ = ρ Q¯ . Besides, there exists  F ∈ [ +Z ,  − Z +1 ] such that the obtained minimizer Q is a global minimizer for E νF defined in (3.11). It solves Eq. (3.12) for some 0 ≤ δ ≤ 1 with Ran(δ) ⊆ ker(H Q¯ −  F ). The operator δ is finite rank if  F ∈ ( +Z ,  − Z +1 ) and trace-class if  F ∈ { +Z ,  − }. Z +1 Additionally the set of q’s in R satisfying the above equivalent conditions is a nonempty closed interval I ⊆ R. This is the largest interval on which q → E ν (q) is strictly convex.   ¯ Q¯ min. of E ν ,  F ∈ [ + ,  − ] . Hence I = ∅ Remark 4. One has I = Tr 0 ( Q), Z +1 Z F by Theorem 2. Theorem 3 is proved in Sect. 5.8. Many of the above statements are very common in reduced Hartree-Fock theories and not all the details will be given (see, e.g. [31]). The difficult part is the proof that (b) is equivalent to (c), for which we use ideas from [12]. Conditions like (c) appear classically when analyzing the compactness properties of minimizing sequences, for instance by using the concentration-compactness principle of P.-L. Lions [24]. They are also very classical for linear models in which the bottom of the essential spectrum has the form of the minimum with respect to q  of the right hand side of (c), as expressed by the HVZ Theorem [16,35,36]. Assume for simplicity that q > 0 and that Q¯ can be written as in (3.13) and (3.14). When 0 < q  ≤ q, (c) means that it is not favorable to let q  electrons escape to infinity, while keeping q −q  electrons near the defect. When q  < 0, it means that it is not favorable to let |q  | holes escape to infinity, while keeping q + |q  | electrons near the defect. When q  > q, it means that it

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is not favorable to let q  electrons escape to infinity, while keeping q  − q holes near the defect. In this article we do not show when (c) holds true. Proving (c) usually requires some decay property of the density of charge ρ Q¯ for a solution Q¯ of the nonlinear equation (3.12). In particular, knowing that ρ Q¯ ∈ L 1 (R3 ) would be very useful (see Remark 3). We plan to investigate more closely the decay properties of ρ Q¯ and the validity of (c) in the near future. At present, the validity of a condition similar to (c) for the BogoliubovDirac-Fock model was only proved in the nonrelativistic limit or in the weak coupling limit, see [12].

4. Thermodynamic Limit of the Supercell Model As mentioned before, we shall now justify the model of the previous section by proving that it is the thermodynamic limit of the supercell model. Let us emphasize that there are several ways of performing thermodynamic limits. In [5], the authors consider a box of size L,  L := [−L/2, L/2)3 , and assume that the nuclei are located on Z3 ∩  L . Then they consider the rHF model of Sect. 1 for N electrons living in the whole space, with N = Z L 3 chosen to impose neutrality. Denoting by ρ L the ground state electronic density of the latter problem, it is proved in 0 , and that the following holds: [5, Thm 2.2] that the energy per unit cell converges to Iper L −3




ρ L (x − k) →

!

ργper 0

(4.1)

k∈Z3 ∩ L p

1 (R3 ), strongly in L (R3 ) for all 2 ≤ p < 6 and almost everywhere on weakly in Hloc loc 3 0 and γ 0 are defined in Sect. 2. R when L → ∞. Let us recall that Iper per Another way for performing thermodynamic limits is to confine the nuclei and the electrons in a domain  L with | L | → ∞, by means of Dirichlet boundary conditions for the electrons. The latter approach was chosen for the Schrödinger model with quantum nuclei in the canonical and grand canonical ensembles [28] by Lieb and Lebowitz in the seminal paper [19] (see also [18]), where the existence of a limit for the energy per unit volume is proved. The crystal case in the Schrödinger model was tackled by Fefferman [9] in the same spirit. We do not know whether Fefferman’s proof can be adapted to treat the Hartree-Fock case. Another possibility, perhaps less satisfactory from a physical viewpoint but more directly related to practical calculations (see e.g. [8]), is to take  L =  L and to impose periodic boundary conditions on the box  L . Usually the Coulomb interaction is also replaced by a (LZ3 )-periodic Coulomb potential, leading to the so-called supercell model which will be described in detail below. This approach has the advantage of respecting the symmetry of the system in the crystal case. It was used by Hainzl, Lewin and Solovej in [13] to justify the Hartree-Fock approximation of no-photon Quantum Electrodynamics. The supercell limit of a linear model for photonic crystals is studied in [32]. Of course the conjecture is that the final results (the energy per unit cell and the ground state density of the crystal) should not depend on the chosen thermodynamic limit procedure. This is actually the case for the reduced Hartree-Fock model of the crystal. See [15] for a result in this direction for a model with quantum nuclei.

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Let us now describe the supercell model. For L ∈ N \ {0}, we introduce the supercell  L = [−L/2, L/2)3 and the Hilbert space   L 2per ( L ) = ϕ ∈ L 2loc (R3 ) | ϕ (LZ3 )-periodic . We also introduce the LZ3 -periodic Coulomb potential G L defined as the unique solution to ⎧ ⎛ ⎞ ⎪
⎪ 1 ⎨ −G L = 4π ⎝ δk − 3 ⎠ L ⎪ k∈L Z3 ⎪ ⎩ G L L Z3 -periodic, minR3 G L = 0. It is easy to check that G L (x) = L −1 G 1 (x/L) and that G L (x) =

c + L


3 k∈ 2π L Z \{0}

4π 1 ik·x e . |k|2 L 3

For any LZ3 -periodic function g, we define    g  L G L (x) := G L (x − y) g(y) dy, L

 DG L ( f, g) :=

L

 L

G L (x − y) f (x) g(y) d x d y.

An admissible electronic state is then described by a one-body density matrix γ in Psc,L = γ ∈ S1 (L 2per ( L )) | γ ∗ = γ , 0 ≤ γ ≤ 1, Tr L 2per ( L ) (−γ ) < +∞ . Any γ ∈ Psc,L has a well-defined density of charge ργ (x) = γ (x, x), where γ (x, y) is the kernel of the operator γ . Notice that γ (x + Lz, y + Lz  ) = γ (x, y) for any z, z  ∈ Z3 , which implies that ργ is LZ3 -periodic. Throughout this section, we use the subscript ‘sc’ to indicate that we consider the thermodynamic limit of the supercell model.

4.1. Thermodynamic limit without defect. Because our model with defect uses the defect-free density matrix of the Fermi sea as a reference, we need to start with the study of the thermodynamic limit without defect. We are going to prove for the supercell model a result analogous to [5, Thm 2.2]. The reduced Hartree-Fock energy functional of the supercell model is defined for γ ∈ Psc,L as     1 1 0 Esc,L (γ ) = Tr L 2per ( L ) − γ + DG L ργ − µper , ργ − µper 2 2

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3 3 where we recall that µper (x) = R∈Z3 Z m(x − R) is a Z - (thus LZ -) periodic function. The reduced Hartree-Fock ground state energy for a neutral system in the box of size L is then given by  0 0 Isc,L = inf Esc,L (γ ), γ ∈ Psc,L ,

 L

ργ =

L

µper = Z L 3 .

(4.2)

0 , γ 0 and H 0 are defined in Sect. 2. In Sect. 5.9, we prove the Let us recall that Iper per per following

Theorem 4 (Thermodynamic limit of the defect-free supercell model). Let Z > 0. 0 i) For all L ∈ N \ {0}, the minimizing problem Isc,L has at least one minimizer, and all the minimizers share the same density. This density is Z3 -periodic. Besides, there is one 0 minimizer γsc,L of (4.2) which commutes with the translations τk , k ∈ Z3 . ii) The following thermodynamic limit properties hold true: • (Convergence of the energy per unit cell). lim

L→∞

0 Isc,L

L3

0 = Iper ;

• (Convergence of the density). ! ! 1 ργ 0  ργper weakly in Hloc (R3 ), 0 sc,L

ργ 0

sc,L

(4.3)

p

→ ργper strongly in L loc (R3 ) for 1 ≤ p < 3 and a.e.; 0

• (Convergence of the mean-field Hamiltonian and its spectrum). Let 0 Hsc,L =−

 + (ργ 0 − µper ) G 1 , sc,L 2

0 0 is a seen as an operator acting on L 2 (R3 ). Then, for all L ∈ N \ {0}, Hsc,L − Hper bounded operator and    0 0  lim  Hsc,L − Hper  = 0. L→∞

0 ) for Denoting by (λnL (ξ ))n∈N\{0} the nondecreasing sequence of eigenvalues of (Hsc,L ξ ∗ ξ ∈ , one has     (4.4) lim sup sup λnL (ξ ) − λn (ξ ) = 0, L→∞ n∈N\{0} ξ ∈ ∗

0 ) introduced in Theorem 1. where (λn (ξ ))n≥1 are the eigenvalues of (Hper ξ iii) Assume in addition that (A1) holds. Fix some  F ∈ ( +Z ,  − Z +1 ). Then for L large 0 0 enough, the minimizer γsc,L of Isc,L is unique. It is also the unique minimizer of the following problem:   0 0 (4.5) Isc,L , F := inf Esc,L (γ ) −  F Tr L 2per ( L ) (γ ), γ ∈ Psc,L .

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Remark 5. Notice that some of the above assertions are more precise for the supercell model than for the thermodynamic limit procedure considered in [5, Thm 2.2] (compare for instance (4.3) with (4.1)). This is because the supercell model respects the symmetry 0 in the box of size L 3 which of the system, allowing in particular to have a minimizer γsc,L 0 3 is periodic for the lattice Z . For an insulator, the uniqueness of γsc,L for large L and the convergence properties of iii) are also very interesting for computational purposes.

4.2. Thermodynamic limit with defect. We end this section by considering the thermodynamic limit of the supercell model with a defect. Recall that ν ∈ L 1 (R3 )∩ L 2 (R3 ) ⊂ C is the density of charge of the defect. First we need to periodize this function with respect to the large box  L , for instance by defining ν L (x) :=




(1 L ν)(· − Lz).

z∈Z3

The reduced Hartree-Fock energy functional of the supercell model with defect is then defined for γ ∈ Psc,L as ν Esc,L (γ )

    1 1 = Tr L 2per ( L ) − γ + DG L ργ − µper − ν L , ργ − µper − ν L . 2 2

For  F ∈ ( +Z ,  − Z +1 ), we consider the following minimization problem   ν ν Isc,L , F = inf Esc,L (γ ) −  F Tr L 2per ( L ) (γ ), γ ∈ Psc,L .

(4.6)

0 is defined in Sect. 2, that E ν and Q ¯ are defined in Sect. 3.2, and that We recall that γper κ 0 Isc,L , F is defined in Sect. 4.1. In Sect. 5.10, we prove the

Theorem 5 (Thermodynamic limit of the supercell model with defect). Let Z ∈ N \ {0}. Assume that (A1) holds and fix some  F ∈ ( +Z ,  − Z +1 ). Then one has

lim

L→∞

ν Isc,L , F

−

0 Isc,L , F



=

E νF

 −

 1

D(ν, ν). (4.7) ν (ργper 0 − µper ) G 1 + 2 R3

ν Additionally, if γsc,L denotes a minimizer for (4.6), then one has, up to extraction of a subsequence, ν 0 ¯ (γsc,L − γsc,L )(x, y) → Q(x, y) 1 (R3 × R3 ) and strongly in L 2 (R3 × R3 ), where Q ¯ is a minimizer of weakly in Hloc loc (3.11), as obtained in Theorem 2. Besides, ν ργsc,L − ργ 0

sc,L

→ ρ¯

weakly in L 2loc (R3 ), where ρ¯ is the common density of all the minimizers of (3.11).

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Remark 6. In numerical simulations, the right-hand side of (4.7) is approximated by ν 0 Isc,L , F − Isc,L , F for a given value of L. This approach has several drawbacks. First, the values of L that lead to tractable numerical simulations are in many cases much too small to obtain a correct estimation of the limit L → ∞. Second, it is not easy to extend this method for computing E νF , to the direct evaluation of E ν (q) for a given q (i.e. the energy of a defect with a prescribed total charge). The formalism introduced in the present article (Problems (3.11) and (3.17)) suggests an alternative way for computing energies of defects in crystalline materials. A work in this direction was already started [3]. 5. Proof of the Main Results Unless otherwise stated, the operators used in the following proofs are considered as operators on L 2 (R3 ). 5.1. Useful estimates. We gather in this section some results which we shall need throughout the proofs. We start with the 0 = −/2 + V 5.1.1. Proof of Lemma 1 Recall that Hper 0 − µper ) per with Vper := (ργper

0 −κ| ≥ H 0 −κ ≥ −/2−C for some large enough constant G 1 ∈ L ∞ (R3 ). Thus |Hper per    0  C. On the other hand, as κ ∈ ( +Z ,  − Z +1 ), there exists α > 0 such that Hper − κ  ≥ α. This implies that 0 |Hper − κ| ≥ max(−/2 − C, α) ≥ c1 (1 − )

for some constant c1 > 0. The proof of the upper bound in (3.7) is straightforward. 0 − κ + c) = 1 + (−/2 + c)−1 (V Then (−/2 + c)−1 (Hper per − κ) is a bounded invertible operator for c large enough, since   V  + |κ| per L ∞   −1 . (−/2 + c) (Vper − κ) ≤ c 0 − κ + c)−1 (−/2 + c) is bounded for a well-chosen c  1, which clearly Thus (Hper implies that 0 (Hper − κ)−1 (− + 1) =

0 −κ +c Hper 0 −κ Hper

is also bounded, together with its inverse.

0 (Hper − κ + c)−1 (−/2 + c)

− + 1 −/2 + c

 

5.1.2. Some commutator estimates Throughout this paper, we shall use Cauchy’s 0 : formula to express the projector γper  1 0 =− (H 0 − z)−1 dz, (5.1) γper 2iπ C per where C is a fixed regular bounded closed contour enclosing the lowest Z bands of the 0 . spectrum of Hper

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147

0 − z)−1 with The following result will be a useful tool to replace the resolvent (Hper the operator (− + 1)−1 which will be easier to manipulate. Its proof is the same as the one of Lemma 1. 0 − z)−1 (− + 1) and its inverse are bounded Lemma 3. The operator B(z) := (Hper uniformly in z ∈ C .

The next result provides some useful properties of commutators: 0 , ] and (1 − ) [γ 0 , |∇|] (1 − ) are bounded. Lemma 4. The operators [γper per 0 , ] follows from (5.1) and from the fact that [(H 0 − Proof. The boundedness of [γper per z)−1 , ] is bounded uniformly in z ∈ C by Lemma 3. 0 − z)−1 , |∇|](1 − ) is Using again (5.1), it suffices to prove that (1 − )[(Hper 0 bounded uniformly in z ∈ C to infer that (1 − ) [γper , |∇|] (1 − ) is bounded. In 0 − z)−1 , |∇|](1 − ), we use order to prove the uniform boundedness of (1 − )[(Hper the formal equality

[(A − z)−1 , B] = −(A − z)−1 [A, B](A − z)−1 .

(5.2)

" # 0 (1 − )[(Hper − z)−1 , |∇|](1 − ) = B(z)∗ |∇|, Vper B(z).

(5.3)

We thus obtain

Using (5.1) and Lemma 3, we obtain  "  # 0 (1 − )[γper , |∇|](1 − ) ≤ C  |∇|, Vper  ≤ C ∇Vper  L ∞ (R3 ) .   0 , V ] ∈ S and Lemma 5. Let V = V1 + V2 with V1 ∈ C  and V2 ∈ L 2 (R3 ). Then [γper 2 there exists a positive real constant C such that 0 [γper , V ]S2 ≤ C(V1 C  + ||V2 || L 2 (R3 ) ).

Proof. Formulas (5.1) and (5.2) lead to  1 0 [γper , V2 ] = − B(z)(− + 1)−1 [, V2 ](− + 1)−1 B(z)∗ dz 4iπ C  1 =− B(z)((− + 1)−1 )V2 (− + 1)−1 B(z)∗ dz 4iπ C  1 + B(z)(− + 1)−1 V2 ((− + 1)−1 )B(z)∗ dz. 4iπ C As (− + 1)−1 and (− + 1)−1  are bounded operators, we obtain, using Lemma 3,         0 [γper , V2 ] ≤ C (− + 1)−1 V2  , S2

S2

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for some constant C independent of V2 . Likewise,  1 0 , V1 ] = − B(z)(− + 1)−1 [, V1 ](− + 1)−1 B(z)∗ dz [γper 4iπ C  3  ∂V


1 1 B(z) (− + 1)−1 ∂xi (− + 1)−1 B(z)∗ dz =− 4iπ C ∂ xi +

i=1 3
i=1

which implies

  1 ∂ V1

∂xi (− + 1)−1 B(z)∗ dz, B(z)(− + 1)−1 4iπ C ∂ xi

   0  [γper , V1 ]

S2

    ≤ C (− + 1)−1 ∇V1 

S2

for some constant C independent of V1 . We then use the Kato-Seiler-Simon inequality (see [29] and [30, Thm 4.1]) ∀ p ≥ 2, to infer

|| f (−i∇)g(x)||Sp ≤ (2π )−3/ p ||g|| L p (R3 ) || f || L p (R3 )    0  [γper , V2 ]     0 [γper , V1 ]

S2

(5.4)

≤ C  ||V2 || L 2 (R3 ) ,

(5.5)

≤ C  ||∇V1 || L 2 (R3 ) = C  ||V1 ||C  .

(5.6)

S2

  0 − κ|, |∇| ] is bounded for any κ in the gap ( + ,  − ). Lemma 6. The operator [ |Hper Z +1 Z 0 − κ| = −(H 0 − κ)γ 0 + (H 0 − κ)(1 − γ 0 ) and thus Proof. We have |Hper per per per per 0 0 0 0 [ |Hper − κ|, |∇| ] = −2(Hper − κ)[γper , |∇|] + [|∇|, Vper ](2γper − 1) 0 0 = −2(B(κ)∗ )−1 (1 − )[γper , |∇|] + [|∇|, Vper ](2γper − 1)     0 , |∇|] is which gives the result since [|∇|, Vper ] ≤ ∇Vper  L ∞ (R3 ) and (1 − )[γper bounded by Lemma 4.  

5.2. Proof of Proposition 1. Let V = V1 +V2 where V1 ∈ C  and V2 ∈ L 2 (R3 )∩L ∞ (R3 ), and Q ∈ Q. Notice that 0 ⊥ 0 0 ⊥ (QV )+ + = Q + + V (γper ) + Q +− [γper , V ](γper ) , 0 0 0 (QV )−− = Q −− V γper − Q −+ [γper , V ]γper ,

(5.7)

0 )⊥ = 1 − γ 0 . We only treat the (QV )−− term, the argument being the same where (γper per for (QV )+ + .

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149

0 0 and notice that First we write Q −− V γper = Q −− (1 + |∇|)(1 + |∇|)−1 V γper (1 + |∇|)−1 V is bounded since V2 ∈ L ∞ (R3 ) by assumption and     (1 + |∇|)−1 V1  ≤ C ||V1 || L 6 ≤ C  ||∇V1 || L 2 = C  ||V1 ||C 

S6

by the Kato-Seiler-Simon inequality (5.4) and the critical Sobolev embedding of H 1 (R3 ) 0 is a trace-class operator. Thus the following is in L 6 (R3 ). This proves that Q −− V γper true: 0 |Tr(Q −− V γper )| = |Tr(Q −− V )|

= |Tr((1 + |∇|)Q −− (1 + |∇|)(1 + |∇|)−1 V (1 + |∇|)−1 )|     ≤ ||Q||Q (1 + |∇|)−1 V (1 + |∇|)−1  . S∞

Then

    (1 + |∇|)−1 V1 (1 + |∇|)−1 

S∞

    ≤ (1 + |∇|)−1 V1  

S6

    (1 + |∇|)−1 

≤ C ||V1 || L 6 ≤ C ||V1 ||C  , and

    (1 + |∇|)−1 V2 (1 + |∇|)−1 

    ≤ (1 + |∇|)−1 |V2 |(1 + |∇|)−1  S∞ S∞  2    ≤ (1 + |∇|)−1 |V2 |1/2  ≤ C ||V2 || L 2 . S4

Hence, 0 |Tr(Q −− V γper )| ≤ C ||Q||Q (||V1 ||C  + ||V2 || L 2 ). 0 , V ]γ 0 ∈ For the second term of (5.7), we just use Lemma 5 which tells us that Q −+ [γper per −+ 0 S1 since Q ∈ S2 and [γper , V ] ∈ S2 . Additionally      0 0 0 0  |Tr(Q −+ [γper , V ]γper )| ≤ Q −+ [γper , V ]γper  ≤ C  Q −+ S2 (||V1 ||C  + ||V2 || L 2 ).

S1

The end of the proof of Proposition 1 is then obvious.

 

5.3. Proof of Lemma 2. Let Q ∈ Q. For  > 0, we introduce the following regularization operator 0 R := (1 + |Hper − |)−1

(5.8)

and set Q  := R Q R . Notice first that Q  ∈ Qr . Indeed, using the same notation as in Lemma 3, we obtain 0 (1 − )R = (1 − )(Hper − )−1

0 − Hper 0 − | 1 + |Hper

= B()∗

0 − Hper 0 − | 1 + |Hper

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which shows that (1−)R  ≤  −1 B()∗ . Similarly, R (1−) ≤  −1 B(). 0 , we infer As R commutes with γper −− R (1 − ) ∈ S1 . (1 − )Q −−  (1 − ) = (1 − )R Q

Likewise, (1−)Q + + (1−) ∈ S1 . Then we show that Q  ∈ Kr ⊂ K when Q ∈ K. To 0 ≤ Q ≤ 1 − γ 0 is equivalent to Q 2 ≤ Q + + − Q −− prove this, we use the fact that −γper per (see Sect. 3.1). As R  ≤ 1, we obtain (Q  )2 = R Q(R )2 Q R ≤ R Q 2 R ≤ R (Q + + − Q −− )R = (Q  )+ + − (Q  )−− ,

(5.9)

0 commutes with R . Hence, it only where we have used that (R )2 ≤ 1 and that γper  remains to prove that Q  → Q for the Q-topology as  → 0, for any fixed Q ∈ Q. We shall need the

Lemma 7. For any 1 ≤ p < ∞ and any fixed Q ∈ S p , one has lim ||R Q − Q||Sp = 0.

(5.10)

→0

Proof. Notice that R − 1 = −

0 − | |Hper 0 − | 1 + |Hper

satisfies R − 1 ≤ 1. Hence ||(R − 1)Q||Sp ≤ ||Q||Sp . By linearity and density of “smooth” finite rank operators in S p for any 1 ≤ p < ∞, it suffices to prove (5.10) for Q = | f  f | with f ∈ H 2 (R3 ). Then    0  ||(R − 1)| f  f |||S1 ≤ ||(R − 1) f || L 2 || f || L 2 ≤  |Hper − | f  2 || f || L 2 L

which converges to 0 as  → 0 and controls ||(R − 1)| f  f |||Sp for 1 ≤ p < ∞.

 

We are now able to complete the proof of Lemma 2. First, by (5.2) $ % 0 − |, |∇| R , [R , |∇|] = − R |Hper and therefore by Lemma 6 there exists a constant C > 0 such that  [R , |∇|]  ≤ C.

(5.11)

Hence, lim→0  [R , |∇|]  = 0. Compute now for instance |∇|(R Q R − Q)−− |∇| = |∇|((R − 1)Q −− R + Q −− (R − 1))|∇| = [|∇|, R ]Q −− [R , |∇|] + [|∇|, R ]Q −− |∇|R +(R − 1)|∇|Q −− [R , |∇|] + |∇|Q −− [R , |∇|] +(R − 1)|∇|Q −− |∇|R + |∇|Q −− |∇|(R − 1). Applying either (5.11) or Lemma 7 to each term of the previous expression allows to conclude that   −− lim |∇|(Q −− )|∇|S = 0.  −Q →0

1

The proof is the same for the other terms in the definition of ||·||Q .

 

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151

5.4. Proof of Proposition 2: Regularity of solutions. Let Q ∈ Q be of the form 0 0 + V ) − γper + δ, Q = χ(−∞, F ) (Hper 0 + V −  ) and where 0 ≤ δ ≤ 1 is a finite rank operator with Ran(δ) ⊆ ker(Hper F V = ρ ∗ | · |−1 for some ρ ∈ L 1 (R3 ) ∩ L 2 (R3 ) (in our case ρ = ρ Q − ν). Note that V ∈ C  ∩ L ∞ (R3 ) since   | ρ (k)| (k)|dk = C ||V || L ∞ ≤ (2π )−3/2 |V dk 2 3 3 R R |k| 1/2  1/2  | ρ (k)|2 (1 + |k|2 ) dk ≤C dk < ∞. 2 2 |k|2 R3 R3 |k| (1 + |k| ) 0 + V −  ) ⊆ D(H 0 + V ) = D(H 0 ) = H 2 (R3 ), it is clear that the Since ker(Hper F per per finite rank operator δ satisfies (1 − )δ(1 − ) ∈ S1 . Thus, up to a change of  F , we 0 + V −  ) = {0} and that δ = 0: can assume that ker(Hper F 0 0 + V ) − γper . Q = χ(−∞, F ) (Hper

Then we remark that Q 2 = Q + + − Q −− , hence we only have to prove that (1 − )Q ∈ S2 . 0 + V below  . Since Let C be a smooth curve enclosing the whole spectrum of Hper F ∞ 3 0 V ∈ L (R ) and |Hper + V − z| ≥ c > 0 uniformly in z ∈ C , we can mimic the proof of Lemma 3 and find that 0 + V − z)−1  < ∞. sup (1 − )(Hper

z∈C

(5.12)

We then use Cauchy’s formula (5.1) and iterate the resolvent formula 0 0 0 0 + V − z)−1 = (Hper − z)−1 − (Hper + V − z)−1 V (Hper − z)−1 (Hper

to obtain

   1 0 0 (Hper + V − z)−1 − (Hper − z)−1 dz 2iπ C = Q1 + Q2 + Q3

Q=−

with

 1 0 Q1 = (H 0 − z)−1 V (Hper − z)−1 dz, 2iπ C per Q2 = −

Q3 =

 1 0 0 (H 0 − z)−1 V (Hper − z)−1 V (Hper − z)−1 dz, 2iπ C per

 1 0 0 0 (H 0 + V − z)−1 V (Hper − z)−1 V (Hper − z)−1 V (Hper − z)−1 dz. 2iπ C per

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Notice that (1 − )Q 3 ∈ S2 by (5.12) and the estimate           0 − z)−1  ≤ V (1 − )−1   B(z)∗  ≤ C ||V || L 6 , V (Hper S6

S6

where we have used (5.4) and Lemma 3. Let us now prove that (1 − )Q 1 ∈ S2 . First we notice that  0 0 0 0 (Hper − z)−1 γper V γper (Hper − z)−1 dz C  0 0 ⊥ 0 ⊥ 0 = (Hper − z)−1 (γper ) V (γper ) (Hper − z)−1 dz = 0 C

by the residuum formula. Then we have  0 0 0 ⊥ 0 (1 − ) (Hper − z)−1 γper V (γper ) (Hper − z)−1 dz C  0 0 ⊥ 0 = B(z)∗ [γper , V ](γper ) (Hper − z)−1 dz C

which belongs to S2 by Lemmas 3 and 5. The proof is the same for Q 2 .

 

5.5. Proof of Theorem 2: Existence of a minimizer with chemical potential. Let (Q n )n∈N be a minimizing sequence for (3.11). It follows from (3.10) that (Q n )n∈N is bounded in Q. By Proposition 1, (ρ Q n )n∈N is bounded in C ∩ L 2 (R3 ). Up to extraction, we can assume that there exists Q¯ in the convex set K such that i) Q n  Q¯ and |∇|Q n  |∇| Q¯ weakly in S2 ; 0 −  |1/2 Q + + |H 0 −  |1/2  |H 0 −  |1/2 Q 0 −  |1/2 , ¯ + + |Hper ii) |Hper F F F F n per per 0 1/2 −− 0 1/2 0 1/2 −− 0 −  |1/2 |Hper −  F | Q n |Hper −  F |  |Hper −  F | Q¯ |Hper F for the weak-∗ topology of S1 ;

iii) ρ Q n  ρ Q¯ in C ∩ L 2 (R3 ). Recall that S1 is the dual of the space of compact operators [26, Thm VI.26]. Thus here An  A for the weak-∗ topology of S1 means Tr(An K ) → Tr(AK ) for any compact operator K . Then, as D(·, ·) defines a scalar product on C, D(ρ Q¯ − ν, ρ Q¯ − ν) ≤ lim inf D(ρ Q n − ν, ρ Q n − ν). n→+∞

0 −  |1/2 Q + + |H 0 −  |1/2 is also a nonnegative operator Now since Q +n + ≥ 0, |Hper F F n per for any n. Thus Fatou’s Lemma [30] yields 0 0 −  F |1/2 Q¯ + + |Hper −  F |1/2 ) Tr(|Hper 0 0 ≤ lim inf Tr(|Hper −  F |1/2 Q +n + |Hper −  F |1/2 ). n→∞

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The same argument for the term involving − Q¯ −− ≥ 0 yields ¯ −  F Tr 0 ( Q) ¯ ≤ lim inf (E ν (Q n ) −  F Tr 0 (Q n )) = E ν , E ν ( Q) F n→∞

i.e. Q¯ ∈ K is a minimizer. The proof that Q¯ satisfies the self-consistent equation (3.12) is classical: writing that ν ¯ for any Q ∈ Kr and t ∈ [0, 1], one deduces that Q¯ E ((1 − t) Q¯ + t Q) ≥ E ν ( Q) minimizes the following linear functional: 0 0 Q ∈ K → F(Q) := Tr(|Hper −  F |1/2 (Q + + − Q −− )|Hper

− F |1/2 ) + D(ρ Q¯ − ν, ρ Q ).

(5.13)

Notice that when Q ∈ Kr ⊆ K, one has F(Q) = Tr 0 ((H Q¯ −  F )Q), where we have used the definition of ρ Q in Proposition 1 to infer

  D(ρ Q¯ − ν, ρ Q ) = Tr 0 (ρ Q¯ − ν) | · |−1 Q , since (ρ Q¯ − ν) | · |−1 ∈ C  when ρ Q¯ − ν ∈ C. Minimizers of the functional (5.13) are easily proved to be of the form (3.12). They belong to Kr by Proposition 2.   5.6. Proof of Proposition 3: The value of E 0 (q). Clearly 0 Q), Q ∈ K, Tr 0 (Q) = q} := E˜ 0 (q) E 0 (q) ≥ inf{Tr 0 (Hper

(5.14)

since D(ρ Q , ρ Q ) ≥ 0. It can be easily proved that  −  Z +1 q when q ≥ 0 0 E˜ (q) =  +Z q when q ≤ 0, see, e.g., the proof of Lemma 13 in [12]. Thus it remains to prove that E 0 (q) ≤ E˜ 0 (q) which we do by a kind of scaling argument. Let us deal with the case q ≥ 0, the other case being similar. We can assume ∗ that  − Z +1 = minξ ∈ λ Z +1 (ξ ) = λ Z +1 (ξ0 ) since each λn (ξ ) is known to be continuous on ∗ . For simplicity, we also assume that ξ0 is in the interior of ∗ (the proof can be easily adapted if this is not the case). Let us denote by u Z +1 (ξ, ·) ∈ L 2ξ an eigenvector associated with the eigenvalue λ Z +1 (ξ ) for any ξ ∈ ∗ . It will be convenient to extend it on R3 ×R3 0 u by u Z +1 (ξ, x) = 0 when ξ ∈ R3 \ ∗ . Since Hper Z +1 (ξ, x) = λ Z +1 (ξ )u Z +1 (ξ, x) for ∗ any ξ ∈ , it is clear that sup ||u Z +1 (ξ, ·)|| L 2 ( ) < ∞,

ξ ∈R3

ξ

2 (R3 )) and u ∞ 3 3 3 i.e. u Z +1 ∈ L ∞ (R3 , Hloc Z +1 ∈ L (R × R ) by the Z -periodicity (resp. 3 −iξ ·x the 2π Z -periodicity) of e u Z +1 (ξ, x) with respect to x (resp. to ξ ).

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Consider now a fixed space V of dimension d = [q] + 1, consisting of C0∞ functions f : R3 → C with support in the unit ball B(0, 1) of R3 . For any λ ≥ 1, we introduce the following subspace of L 2 (R3 ):  Wλ := gλ := λ3/2 f (λ(ξ − ξ0 ))u Z +1 (ξ, ·)dξ, f ∈ V R3

which has the same dimension as V by the properties of the Bloch decomposition, when λ is large enough such that the ball B(ξ0 , λ−1 ) is contained in ∗ . Noting that for any gλ ∈ Wλ arising from some f ∈ V ,  −3/2 | f (ξ  )u Z +1 (ξ0 + λ−1 ξ  , x)|dξ  |gλ (x)| ≤ λ R3  ≤ λ−3/2 ||u Z +1 || L ∞ (R3 ×R3 ) | f (ξ )|dξ, B(0,1)

≤ Cλ

−3/2

||u Z +1 || L ∞ (R3 ×R3 ) || f || L 2 (R3 ) ,

we deduce by interpolation that ∀ 2 < p ≤ ∞,

lim

λ→∞

sup g∈Wλ

||g|| L p (R3 ) = 0.

||g|| L 2 (R3 ) =1

By construction one also has for any fixed f ∈ V with associated gλ ∈ Wλ ,  & ' − 0 3 gλ , (Hper −  Z +1 )gλ = λ | f (λ(ξ − ξ0 ))|2 (λ Z +1 (ξ ) −  − Z +1 )dξ →λ→∞ 0. R3

λ Take now an orthonormal basis (ϕ1λ , ..., ϕ[q]+1 ) of Wλ and introduce the operator

Q λ :=

[q]


|ϕnλ ϕnλ | + (q − [q])|ϕdλ ϕdλ |.

n=1 0 ϕ λ = 0 for any n = 1, ..., [q] + 1 and Tr (Q λ ) = Tr(Q λ ) = q, thus By construction γper 0 n λ Q ∈ K satisfies the charge constraint. Then 0 Tr 0 (Hper Qλ) =

[q] &


' & ' 0 λ 0 λ ϕnλ , Hper → − ϕnλ + (q − [q]) ϕ[q]+1 , Hper ϕ[q]+1 Z +1 q

n=1 1/2

as λ → ∞. Besides, (ρ Q λ )λ≥1 is a bounded family in H 1 (R3 ) which converges to 0 in L p (R3 ) for any p > 2. By the Hardy-Littlewood-Sobolev inequality [20], one has  2 D(ρ Q λ , ρ Q λ ) ≤ C ρ Q λ  L 6/5

(5.15)

which implies D(ρ Q λ , ρ Q λ ) → 0 as λ → ∞. Eventually E 0 (Q λ ) → E˜ 0 (q) and Proposition 3 is proved.  

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5.7. Density of finite rank operators in K. This section is devoted to the generalization of results in [12, Appendix B] concerning the density of finite rank operators, which will be useful for proving Theorem 3. 0 ∈K Lemma 8. For any Q ∈ K there exists an orthogonal projector P such that P −γper and a trace-class operator δ ∈ Q such that 0 ≤ δ ≤ 1, [P, δ] = 0 and 0 Q = P − γper + δ.

 

Proof. This is an easy adaptation of [12, Lemma 19]. We denote for simplicity H+ =

0 )L 2 (R3 ) (1 − γper

0 L 2 (R3 ). and H− = γper

0 ∈ K. Proposition 4. Let P be an orthogonal projector on L 2 (R3 ) such that Q = P−γper 1 3 N 0 − Denote by ( f 1 , ..., f N ) ∈ (H+ ∩ H (R )) an orthonormal basis of E 1 = ker(P −γper 0 ) ∩ ker(1 − P) and by (g , ..., g ) ∈ (H ∩ H 1 (R3 )) M an orthonormal 1) = ker(γper 1 M − 0 + 1) = ker(1 − γ 0 ) ∩ ker(P). Then there exist an basis of E −1 = ker(P − γper per orthonormal basis (vi )i≥1 ⊂ H+ ∩ H 1 (R3 ) of (E 1 )⊥ in H+ , an orthonormal basis (u i )i≥1 ⊂ H− ∩ H 1 (R3 ) of (E −1 )⊥ in H− , and a sequence (λi )i≥1 ∈ 2 (R+ ) such that

P=

N


| f n  f n | +

n=1

1− P = Additionally

M


|gm gm | +

m=1

∞
|u i + λi vi u i + λi vi | , 1 + λi2 i=1

2 2 i≥1 λi (∇u i  L 2

∞
|vi − λi u i vi − λi u i | . 1 + λi2 i=1

(5.16)

(5.17)

+ ∇vi 2L 2 ) < ∞.

Proof. This is an obvious corollary of [12, Theorem 7].

 

Corollary 3. Let Q ∈ K. Then there exists a sequence {Q k }k≥1 of finite rank operators belonging to Kr such that ||Q k − Q||Q → 0 as k → ∞ and for any k ≥ 1, Tr 0 (Q k ) = Tr 0 (Q). Proof. Taking λi = 0 for i > k in the decomposition of Proposition 4, one can approxi0 → P − γ 0 in Q as k → ∞ and mate P by another projector Pk such that Pk − γper per 0 is finite rank: Pk − γper 0 Pk − γper =

N


| f n  f n | −

n=1

+

k
i=1

M
m=1

|gm gm | +

k
i=1

 λi  |u i vi | + |vi u i | . 2 1 + λi

 λi2  |vi vi | − |u i u i | 2 1 + λi (5.18)

It then suffices to approximate each function in (5.18) by a smoother one, for instance R u and orthonormalizing these new functions, by defining for   1, u˜ i := ||R u i ||−1 L2  i where R was defined previously in Eq. (5.8). 0 + δ of the form given by Lemma 8, it remains to Then for any Q = P − γper approximate δ by a finite rank operator δk such that [Pk , δk ] = 0, which is done in the same way.  

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5.8. Proof of Theorem 3: Existence of minimizers under a charge constraint. The proof of Theorem 3 follows ideas of [12]. The proof that any minimizer solves (3.12) is the same as before and will be omitted. Step 1. Large HVZ-type inequalities. Let us start by the following result, which indeed shows that (b)⇒(c): Lemma 9. (Large HVZ-type inequalities) Let Z ∈ N \ {0}, ν ∈ C and assume that (A1) holds. Then, for every q, q  ∈ R, one has E ν (q) ≤ E ν (q − q  ) + E 0 (q  ). If moreover there is a q  = 0 such that E ν (q) = E ν (q − q  ) + E 0 (q  ), then there is a minimizing sequence of E ν (q) which is not precompact. Proof. Thanks to Corollary 3, the proof is exactly the same as [12, Prop. 6].

 

Step 2. A necessary and sufficient condition for compactness. The following Proposition is the analogue of [12, Lemma 8]: Proposition 5. (Conservation of charge implies compactness) Let Z ∈ N \ {0}, ν ∈ C, q ∈ R and assume that (A1) holds. Assume that (Q n )n≥1 is a minimizing sequence in Kr for (3.11) such that Q n  Q ∈ K for the weak-∗ topology of Q. Then Q n → Q for the strong topology of Q if and only if Tr 0 (Q) = q. Proof. Let (Q n )n≥1 ⊆ Kr be as stated and assume that Tr 0 (Q) = q. We know from the proof of Theorem 2 that E ν (Q) ≤ lim E ν (Q n ) = E ν (q), n→∞

(5.19)

hence Q ∈ K is a minimizer of E ν (q). Therefore Q satisfies the equation 0 +δ Q = χ(−∞, F ) (H Q ) − γper

for some  F ∈ ( +Z ,  − Z +1 ) and where δ is a finite rank operator satisfying 0 ≤ δ ≤ 1 and Ran(δ) ⊆ ker(H Q −  F ). In particular Q ∈ Kr by Proposition 2. We now introduce P := χ(−∞, F ) (H Q ),

P  := χ( F ,∞) (H Q ) and π := χ{ F } (H Q ).

Let us write E ν (Q n ) = E ν (Q) + Tr 0 (H Q (Q n − Q)) +

1 D(ρ Q n − ρ Q , ρ Q n − ρ Q ). 2

Now using [10, Lemma 1] and the hypothesis Tr 0 (Q n ) = Tr 0 (Q), we obtain Tr 0 (H Q (Q n − Q)) = Tr 0 ((H Q −  F )(Q n − Q)) = Tr P ((H Q −  F )(Q n − Q)) = Tr(|H Q −  F |(P  (Q n − Q)P  − P(Q n − Q)P)), where we recall that by definition Tr P (A) = Tr(P A P + (1 − P)A(1 − P)). We have −P − δ ≤ Q n − Q ≤ 1 − P − δ which yields P(Q n − Q)2 P + P  (Q n − Q)2 P  ≤ P  (Q n − Q)P  − P(Q n − Q)P, (5.20)

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157

and in particular P  (Q n − Q)P  ≥ 0 and P(Q n − Q)P ≤ 0, i.e. Tr 0 (H Q (Q n − Q)) ≥ 0. Since we know that limn→∞ E ν (Q n ) = E ν (Q), we infer lim Tr(|H Q −  F |P  (Q n − Q)P  ) = lim Tr(|H Q −  F |P(Q n − Q)P) = 0

n→∞

n→∞

(5.21) and from (5.20) lim Tr(|H Q −  F |P  (Q n − Q)2 P  ) = lim Tr(|H Q −  F |P(Q n − Q)2 P) = 0.

n→∞

n→∞

On the one hand, it is easy to see that P|H Q −  F |P ≥ c P|H Q − κ|P and P  |H Q −  F |P  ≥ c P  |H Q − κ|P  for some small enough constant c > 0 and some κ ∈ / σ (H Q ) close enough to  F . On the other hand, the weak convergence of (Q n ) and the fact that π is a “smooth” finite rank operator imply that lim Tr(|H Q − κ|π(Q n − Q)2 π ) = lim Tr(|H Q − κ|π(Q n − Q)π ) = 0.

n→∞

n→∞

(5.22)

It is then clear that this yields lim Tr(|H Q − κ|(Q n − Q)2 ) = 0.

n→∞

(5.23)

As we have chosen κ ∈ / σ (H Q ), we can mimic the proof of Lemma 3 and obtain that c1 (1 − ) ≤ |H Q − κ| ≤ c2 (1 − ).

(5.24)

Hence (5.23) shows that (1 − )1/2 (Q n − Q) → 0 in S2 . Writing now 0 0 (Q n − Q)−− = P(Q n − Q)P + (γper − P)(Q n − Q)γper 0 0 0 0 −(P − γper )(Q n − Q)(P − γper ) + γper (Q n − Q)(γper − P) (5.25)

and using (5.24), (5.21) and (5.22), we easily see that |H Q − κ|1/2 (Q n − Q)−− |H Q − κ|1/2 → 0 and (1 − )1/2 (Q n − Q)−− (1 − )1/2 → 0 in S1 . The proof is the same for (Q n − Q)+ + .   Step 3: Proof that (c)⇒(b). We argue by contradiction. Let (Q n )n≥1 ⊆ K be a minimizing sequence for E ν (q) which is not precompact for the topology of Q. By the density of Kr in K, we can further assume that each Q n ∈ Kr . The bound (3.9) on the energy tells us that (Q n )n≥1 is bounded in Q. Then, up to extraction and by Proposition 5, we can assume that Q n  Q ∈ K, where Tr 0 (Q) = q, and that ρ Q n  ρ Q weakly in C. We write Tr 0 (Q) = q − q  with q  = 0. We now prove that E ν (q) ≥ E ν (q − q  ) + E 0 (q  )

(5.26)

which will contradict (c). To this end, we argue like in the proof of [12, Thm. 3]: consider a smooth radial function χ with support in B(0, 1) such that 0 ≤ χ ≤ 1 and χ ≡ 1

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É. Cancès, A. Deleurence, M. Lewin

in B(0, 1/2); define χ R (x) := χ (x/R). Then let η R := following localization operators: 0 0 0 ⊥ 0 ⊥ Y R := γper η R γper + (γper ) η R (γper ) ,

Lemma 10. We have for all 3 < p ≤ ∞,    0  ] = O(R −1+3/ p ), [η R , γper

!

1 − χ R2 . Let us introduce the

XR =

!

1 − Y R2 .

||Y R − η R ||Sp = O(R −1+3/ p ).

Sp

(5.27)

    Moreover  X 2R − χ R2 S = O R −1+3/ p . p

Proof. By (5.1) and (5.2)  1 0 =− (H 0 − z)−1 [, η R ] (Hper − z)−1 dz 4iπ C per      0 ] which yields [η R , γper ≤ C (1 − )−1 (∇η R )S ≤ C  ||∇η R || L p by the Kato p 0 , ηR ] [γper

Sp

Seiler-Simon inequality and following the proof of Lemma 5. Eventually we notice 0 0 , η ] and thus (5.27) is proved since γ 0 is bounded. The [γper Y R − η R = 1 − 2γper R per  last inequality is a consequence of ||Y R || ≤ 1, ||η R || ≤ 1.  Lemma 11. One has $  %   0 0 − |1/2 |Hper − |−1/2  = O(R −1 ),  Y R , |Hper

(5.28)

where we recall that  = ( +Z +  − Z +1 )/2 is the middle of the gap. Proof. We use the well-known integral representation of the square root [2] 0 |Hper − |1/2 =

1 π



0 − | |Hper

∞

0

s

0 + |Hper

ds √ . − | s

(5.29)

0 η γ 0 + (γ 0 )⊥ η (γ 0 )⊥ . For shortness, we only detail the Recall that Y R = γper R per R per per 0 )⊥ η (γ 0 )⊥ . Using that |H 0 − | commutes estimation of the term involving (γper R per per 0 )⊥ and that (γ 0 )⊥ is bounded, we see that it suffices to estimate with (γper per 0 ⊥ ) (γper



1 =− 2

∞

(

0



∞

0

−|−1/2

ηR ,

0 − | |Hper

s

0 + |Hper

ds 0 ⊥ 0 − |−1/2 √ (γper ) |Hper − | s

0 )⊥ (γper 0 − | s + |Hper 0 )⊥ (γper 0 − | s + |Hper

)

0 [η R , −] |Hper

√ sds.

(5.30)

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159

   0 − |−1/2  = O(R −1 ), Then [η R , −] = (η R ) + 2(∇η R ) · ∇, hence [η R , −] |Hper  0 − |−1/2 is a bounded operator by Lemma 1. Then we where we have used that ∇|Hper 0 use that |Hper − | ≥  for some  > 0 to estimate the right hand side of (5.30) in the operator norm by    ∞ √sds  0 −1/2  , −] |H − | = O(R −1 ). [η R  × per 2 ( + s) 0

  Notice now that Y R Q n Y R ∈ K for all R ≥ 1 (the same is true for X R Q n X R but we shall actually not need it). To see this, notice for instance that $ % 0 −1/2 0 (1 + |∇|)(Y R Q n Y R ) = −(1 + |∇|)|Hper − | Y R , |Hper − |1/2 ×  0 −1/2 0 1/2 0 1/2 (5.31) ×|Hper − | |Hper − | Q n Y R − Y R |Hper − | Q n Y R % $ 0 − |1/2 Q ∈ S and Y , |H 0 − |1/2 |H 0 − which belongs to S2 since |Hper n 2 R per per |−1/2 is bounded by Lemma 11. The proof that (1 + |∇|)(Y R Q n Y R )+ + (1 + |∇|) and (1 + |∇|)(Y R Q n Y R )−− (1 + |∇|) are trace-class is similar. Eventually, the proof that 0 ≤ Y QY ≤ 1 − γ 0 is easy, using the equivalent condition (3.4) and the fact −γper R R per 0 commutes with Y . that γper R We are now able to prove (5.26) as announced. We write, following [11], 0 0 1/2 E ν (Q n ) = Tr(X R |Hper − |1/2 (Q +n + − Q −− X R) n )|Hper − | 0 0 1/2 +Tr(Y R |Hper − |1/2 (Q +n + − Q −− Y R ) + Tr 0 (X R Q n X R ) n )|Hper − | 1 1 (5.32) +Tr 0 (Y R Q n Y R ) + D(ρ Q n − ν, ρ Q n − ν) − D(ν, ν), 2 2 0 , X ] = [γ 0 , Y ] = 0 to infer Tr (Q ) = Tr (X Q X )+ where we have used that [γper R R 0 n 0 R n R per Tr 0 (Y R Q n Y R ). Then, by Lemma 11 and using the fact that (Q n )n≥1 is a bounded sequence in Q, we deduce that 0 0 1/2 Tr(Y R |Hper − |1/2 (Q +n + − Q −− YR ) n )|Hper − | 0 0 1/2 ≥ Tr(|Hper − |1/2 Y R (Q +n + − Q −− ) − C/R n )Y R |Hper − |

(5.33)

for some constant C > 0. Arguing similarly for the other terms, we obtain 0 0 1/2 E ν (Q n ) ≥ E˜ 0 (Y R Q n Y R ) + Tr(χ R |Hper − |1/2 (Q +n + − Q −− χR ) n )|Hper − | 1 +Tr(χ R (Q +n + + Q −− n )χ R ) + D(ρ Q n − ν, ρ Q n − ν) 2 1  − D(ν, ν) − C /R (5.34) 2

for some constant C  , where 0 0 E˜ 0 (Q) := Tr(|Hper − |1/2 (Q + + − Q −− )|Hper − |1/2 ) + Tr 0 (Q).

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Recall (Proposition 3) E 0 (q) = inf{E˜ 0 (Q),

Q ∈ K, Tr 0 (Q) = q}.

Thus, using q = Tr 0 (Q n ) = Tr 0 (Y R Q n Y R ) + Tr 0 (X R Q n X R ), and the fact that q → E 0 (q) is Lipschitz by Proposition 3, (5.34) yields 0 0 1/2 E ν (Q n ) ≥ Tr(χ R |Hper − |1/2 (Q +n + − Q −− χR ) n )|Hper − |   ++ −− 0 ++ −− +Tr(χ R (Q n + Q n )χ R ) + E q − Tr(χ R (Q n + Q n )χ R ) 1 1 + D(ρ Q n − ν, ρ Q n − ν) − D(ν, ν) − C  /R. 2 2

(5.35)

Let us now pass to the limit n → ∞. First we notice 0 0 1/2 lim inf Tr(χ R |Hper − |1/2 (Q +n + − Q −− χR ) n )|Hper − | n→∞

0 0 ≥ Tr(χ R |Hper − |1/2 (Q + + − Q −− )|Hper − |1/2 χ R ),

lim inf D(ρ Q n − ν, ρ Q n − ν) ≥ D(ρ Q − ν, ρ Q − ν) n→∞

(5.36)

(5.37)

by Fatou’s Lemma and the weak convergence ρ Q n  ρ Q in C. Then −− χR ) lim Tr(χ R Q +n + χ R ) = Tr(χ R Q + + χ R ), lim Tr(χ R Q −− n χ R ) = Tr(χ R Q

n→∞

n→∞

which is obtained by writing for instance Tr(χ R Q +n + χ R ) = Tr(χ R (1 + |∇|)−1 (1 + |∇|)Q +n + (1 + |∇|)(1 + |∇|)−1 χ R ) and using that χ R (1 + |∇|)−1 is compact (it belongs to S p for p > 3 by the Kato-SeilerSimon inequality) and that (1 + |∇|)Q +n + (1 + |∇|)  (1 + |∇|)Q + + (1 + |∇|) for the weak-∗ topology of S1 . Thus, 0 0 E ν (q) = lim inf E ν (Q n ) ≥ Tr(χ R |Hper − |1/2 (Q + + − Q −− )|Hper − |1/2 χ R ) n→∞   +Tr(χ R (Q + + + Q −− )χ R ) + E 0 q − Tr(χ R (Q + + + Q −− )χ R ) 1 1 (5.38) + D(ρ Q − ν, ρ Q − ν) − D(ν, ν) − C  /R. 2 2

Passing now to the limit as R → ∞, we obtain (5.26). This contradicts (3) and shows that (b)⇔(c) in Theorem 3.

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161

Step 4. Characterization of the q’s such that (c) holds. Because q → E ν (q) is a convex function, it is classical that the set I = {q ∈ R, (c) holds} is a closed interval of R, see ¯ for any minimizer Q¯ of E ν obtained e.g. [31]. It is non empty since it contains Tr 0 ( Q) F − + in Theorem 2, for any  F in the gap ( Z ,  Z +1 ). Additionally, q → E ν (q) is linear on the connected components of R \ I and I is the largest interval on which this function is strictly convex. Let us now state and prove Lemma 12. Let Z ∈ N\{0}, ν ∈ L 1 (R3 )∩ L 2 (R3 ), and assume that (A1) holds. Assume that Q 1 and Q 2 are respectively two minimizers of E ν (q1 ) and E ν (q2 ) with q1 = q2 . Then ρ Q 1 = ρ Q 2 and therefore E ν (tq1 + (1 − t)q2 ) ≤ E ν (t Q 1 + (1 − t)Q 2 ) < t E ν (q1 ) + (1 − t)E ν (q2 ). Proof. Assume by contradiction that ρ Q 1 = ρ Q 2 . It is classical that the operators Q 1 and Q 2 satisfy the self-consistent equations 0 Q 1 = χ(−∞,1 ) (H Q 1 ) − γper + δ1 ,

0 Q 2 = χ(−∞,2 ) (H Q 2 ) − γper + δ2 ,

where 0 ≤ δk ≤ 1 and Ran(δk ) ⊆ ker(H Q k − k ) for k = 1, 2. Necessarily 1 and 2 are in [ +Z ,  − Z +1 ] otherwise Q 1 and Q 2 would not be compact, which is not possible since every operator of K is compact. Since H Q 1 = H Q 2 has only a point spectrum in the gap, − + we deduce that if k ∈ ( +Z ,  − Z +1 ), then necessarily δk is finite rank. If k ∈ { Z ,  Z +1 }, then it can easily be proved that at least δk ∈ S1 . Hence Q 1 and Q 2 differ by a trace-class operator: Q 2 = Q 1 +δ, Tr|δ| < ∞. Now 0 = q2 −q1 = Tr(δ) = ρδ which contradicts our assumption that ρδ = ρ Q 1 − ρ Q 2 = 0. The rest follows from the strict convexity of ρ → D(ρ, ρ).   / I , the interval on which (c) holds. Corollary 4. There is no minimizer for E ν (q) if q ∈ Thus (a) implies (c). / I , for instance q1 > max I := Proof. Assume that there is a minimizer Q 1 for some q1 ∈ q2 . Applying Lemma 12 to q1 and q2 shows that E ν (·) cannot be linear on [q2 , q1 ] which contradicts the definition of I .   5.9. Proof of Theorem 4: Thermodynamic limit of the supercell model for a perfect crystal. Step 1. Let us first prove that lim sup L→+∞

1 0 0 I ≤ Iper . Starting from the Bloch decomL 3 sc,L

position 0 γper =

1 (2π )3

 ∗

0 (γper )ξ dξ

0 it is possible to construct a convenient test function * γsc,L for (4.2) as follows: of γper

γsc,L (x, y) = *

1 (2π )3  ×


ξ∈

2π L

eiξ x

Z3 ∩ ∗

ξ +[− 2πL η , 2π(1−η) )3 L

e

−iξ  x

0 (γper )ξ  (x,

y) e

iξ  y

dξ





e−iξ y ,

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with η = 0 if L is even and η = 1/2 if L is odd. It is indeed easy to check that * γsc,L is in Psc,L and satisfies ρ* 0 . In particular, γsc,L = ργper 

 L

ρ* γsc,L =

L

µper = Z L 3 ,

3 and, since both ργper 0 and µper are Z -periodic,

   3 DG L ρ* 0 − µper , ργ 0 − µper . γsc,L − µper , ρ* γsc,L − µper = L DG 1 ργper per Besides,      1 1 1 1 0  − * −  γper dξ γsc,L = Tr 2 Tr 2 ξ L 3 L per ( L ) 2 (2π )3 ∗ L ξ ( ) 2 
1 0 − |ξ − ξ  |2 Tr L 2 ( ) ((γper )ξ  ) dξ  3 2π η 2π(1−η) 3 ξ 2(2π ) ξ +[− L , L ) 2π ξ∈

−

i (2π )3

L

Z3 ∩ ∗






ξ∈

2π L

Z3 ∩ ∗

ξ +[− 2πL η , 2π(1−η) )3 L

0 (ξ − ξ  ) · Tr L 2 ( ) (−i∇(γper )ξ  ) dξ  . ξ

 0 ) ) dξ and from the inequaIt follows from the boundedness of ∗ Tr L 2 ( ) ((1 − )(γper ξ ξ lity | − 2i∇| ≤ (1 − ) that the last two terms of the above expression go to zero, hence that      1 1 1 1 0 − = − dξ. * γ (γ lim Tr Tr ) 2 2 sc,L ξ L per ( L ) per L→+∞ L 3 2 (2π )3 ∗ L ξ ( ) 2 0 (* 0 (γ 0 ) = I 0 and consequently Therefore lim L→+∞ L −3 Esc,L γsc,L ) = Eper per per

lim sup L→+∞

1 0 1 0 0 Isc,L ≤ lim Esc,L (* γsc,L ) = Iper . 3 L→+∞ L 3 L

1 0 0 Isc,L ≥ Iper . First, the existence of a miniL→+∞ L 3 0 mizer γ L for (4.2) and the uniqueness of the corresponding density ρsc,L follows from 0 the same arguments as in the proof of [5, Thm 2.1]. Note that, by symmetry, ρsc,L is 0 3 Z -periodic. We now define the operator γsc,L as

Step 2. Let us now establish that lim inf

0 γsc,L =

1 L3




τk∗ γ L τk .

k∈Z3 ∩ L

By simple periodicity arguments, it is clear that τk∗ γ L τk is also a minimizer for (4.2) for 0 . Besides, γ 0 all k ∈ Z3 . By convexity, so is γsc,L sc,L commutes with the translations τk 0 3 for all k ∈ Z so that its kernel γsc,L (x, y) satisfies 0 0 ∀(x, y, z) ∈ R3 × R3 × Z3 , γsc,L (x + z, y + z) = γsc,L (x, y).

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163

0 The optimality conditions imply that γsc,L can be expanded as follows:

1 L3

0 (x, y) = γsc,L







L iξ ·x L L (y)e−iξ ·y , n k,ξ e vk,ξ (x) vk,ξ

3 ∗ k≥1 ξ ∈ 2π L Z ∩

L 3 ∗ 2 where for any ξ ∈ 2π L Z ∩ , (vk,ξ )k≥1 is a Hilbert basis of L per ( ) consisting of eigenfunctions of the self-adjoint operator on L 2per ( ) defined by

1 1 0 − µper ) G 1 + |ξ |2 −  − iξ · ∇ + (ρsc,L 2 2 L are in associated with eigenvalues λ1L (ξ ) ≤ λ2L (ξ ) ≤ · · · . The occupation numbers n k,ξ the range [0, 1] and such that

1 L n k,ξ = Z. 3 L 2π ξ∈

L

Z3 ∩ ∗ k≥1

L = 1 whenever λ L (ξ ) <  L and Lastly, there exists a Fermi level  FL ∈ R such that n k,ξ k F L L L n k,ξ = 0 whenever λk (ξ ) >  F . One has

1 0 1 1 0 0 I = 3 Esc,L (γsc,L )= 3 L 3 sc,L L L


ξ ∈ 2π L

L
n k,ξ

Z3 ∩ ∗ k≥1

2

L 2 (−i∇ + ξ )vk,ξ L 2

per ( )

1 DG L (ργ 0 − µper , ργ 0 − µper ). (5.39) sc,L sc,L 2 L3 We now introduce 
1 0 L L L (x, y) = n k,β eiξ ·x vk,β (x) vk,β (y)e−iξ ·y dξ, γsc,L * L (ξ ) L (ξ ) L (ξ ) (2π )3 ∗ +

k≥1

2π η 2π(1−η) 3 3 ∗ ) . where β L (ξ ) is the unique element of 2π L Z ∩ such that ξ ∈ β L (ξ )+[− L , L 0 0 Z Z It is easy to check that γ˜sc,L ∈ Pper . Thus γ˜sc,L ∈ Pper can be used as a test function for (2.4). Therefore 0 0 0 Eper (γ˜sc,L ) ≥ Iper .

As ργ˜ 0

sc,L

= ργ 0

is Z3 -periodic, one has

sc,L

DG 1 (ργ˜ 0

− µper , ργ˜ 0

sc,L

(5.40)

sc,L

− µper ) =

1 DG L (ργ 0 − µper , ργ 0 − µper ). sc,L sc,L L3

(5.41)

Besides, 1 (2π )3



 ∗

Tr L 2 ( ) ξ

1 0  −  γsc,L ξ 2



1 dξ = (2π )3



L
n k,β ∗ k≥1

L (ξ )

2

L ×(−i∇ + ξ )vk,β 2 L (ξ ) L 2

per ( )

=

1 L3


ξ ∈ 2π L

Z3 ∩ ∗


k≥1

L n k,ξ

2

L 2 (−i∇ + ξ )vk,ξ L 2

per ( )

+ RL

(5.42)

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É. Cancès, A. Deleurence, M. Lewin

with 1 RL = (2π )3



L
n k,β

∗ k≥1

L (ξ )

2

L (−i∇ + ξ )vk,β 2 L (ξ ) L 2

per ( )



L −(−i∇ + β L (ξ ))vk,β 2 L (ξ ) L 2

per ( )

.

0 0 . As + R L ≥ Iper Putting (5.39)–(5.42) together, we end up with L13 Isc,L + ,1/2   0 6Z Isc,L 2π 3Z 2π 2 + |R L | ≤ , L3 L 2 L

we finally obtain lim inf

L→+∞

1 0 0 I ≥ Iper . L 3 sc,L

0 ) Step 3. Convergence of the density. A byproduct of Steps 1 and 2 is that (γ˜sc,L L∈N\{0} 0 . The convergence results on the density ρ is a minimizing sequence for Iper = ργ˜ 0 0 γsc,L sc,L immediately follow from the proof of [5, Thm 2.1]. 0 Step 4. Convergence of the mean-field Hamiltonian and its spectrum. One has Hsc,L − 0 Hper =  L , where  L solves the Poisson equation − L = 4π(ργ 0 − ργper 0 ) on sc,L with periodic boundary conditions. As it follows from Step 3 that (ργ 0 − ργper 0 ) sc,L

2 ( ), hence in converges to zero in L 2per ( ), we obtain that  L converges to zero in Hper L ∞ (R3 ). Consequently,    0 0  (5.43)  ≤ || L || L ∞ → 0  Hsc,L − Hper

as L → ∞. This clearly implies, via the min-max principle, that     sup sup λnL (ξ ) − λn (ξ ) ≤  L  L ∞ −→ 0, n≥1 ξ ∈ ∗

L→∞

0 (resp. where (λnL (ξ ))n≥1, ξ ∈ ∗ (resp. (λn (ξ ))n≥1, ξ ∈ ∗ ) are the Bloch eigenvalues of Hsc,L 0 Hper ). 0 for large values of L. In the remainder of the proof, we Step 5. Uniqueness of γsc,L 0 has a gap. assume that (A1) holds, i.e. that Hper 0 considered as an operator on L 2per ( L ) is given by The spectrum of Hsc,L   0 σ L 2per ( L ) (Hsc,L )= λnL (ξ ). n∈N\{0} ξ ∈ 2π Z3 ∩ ∗ L

It follows from Step 4 that there exists some L 0 ∈ N \ {0} such that for all L ≥ L 0 , the 0 lowest Z L 3 eigenvalues of Hsc,L (including multiplicities) are   λnL (ξ ). 1≤n≤Z ξ ∈ 2π Z3 ∩ ∗ L

Local Defects in Periodic Crystals

165

and there is a gap between the (Z L 3 )th and the (Z L 3 +1)st eigenvalues. As a consequence, 0 γsc,L is uniquely defined: it is the spectral projector associated with the lowest Z L 3 0 , considered as an operator on L 2 ( ). eigenvalues of Hsc,L L per 0 0 Step 6. Let  F ∈ ( +Z ,  − Z +1 ). For L large enough, γsc,L = χ(−∞, F ] (Hsc,L ) = 0 0 χ(−∞,0] (Hsc,L −  F ) as operators acting on L 2per ( L ). This means that γsc,L satis0 fies the Euler-Lagrange equation associated with Isc,L , F , see (4.5). As the functional 0 (γ ) −  Tr 0 γ → Esc,L (γ ) is convex on P , γ F L 2per ( L ) sc,L sc,L is a minimizer of this functional. Its uniqueness follows as usual from the uniqueness of the minimizing density 0 and from the fact that 0 is not in the spectrum of Hsc,L − F .

5.10. Proof of Theorem 5: Thermodynamic limit of the supercell model for a crystal with local defects. We follow the method of [13]. As in the previous section, we denote 0 by γsc,L the minimizer of (4.2), which is unique for L large enough and is also the unique minimizer of (4.5). Let K L be the set of operators Q L on L 2per ( L ) such that 0 + Q L ∈ Psc,L . In fact γsc,L  K L = Q L ∈ S1 (L 2per ( L )) | Q ∗L = Q L , |∇|Q L |∇| ∈ S1 (L 2per ( L )),  0 0 −γsc,L . ≤ Q L ≤ 1 − γsc,L ν ν We introduce Esc,L , F := Esc,L −  F Tr L 2per ( L ) . Let Q L ∈ K L , one has ν 0 0 0 0 Esc,L , F (γsc,L + Q L ) − Esc,L , F (γsc,L ) = Tr L 2per ( L ) (Hsc,L Q L ) − DG L (ρ Q L , ν L )

1 1 + DG L (ρ Q L , ρ Q L ) −  F Tr L 2per ( L ) (Q L ) − DG L (ν L , ργ 0 − µper ) + DG L (ν L , ν L ). sc,L 2 2 0 is considered as an operator on L 2 ( ). Using Note that in the above expression, Hsc,L L per Theorem 4, this equality can be rewritten, for L large enough, as ++,L ν 0 0 0 0 − Q −−,L )) Esc,L , F (γsc,L + Q L ) − Esc,L , F (γsc,L ) = Tr L 2per ( L ) (|Hsc,L −  F |(Q L L

1 + DG L (ρ Q L − ν L , ρ Q L − ν L ) − DG L (ν L , ργ 0 − µper ), sc,L 2 where we have set

(5.44)

0 0 0 0 Q ++,L = (1 − γsc,L )Q L (1 − γsc,L ) and Q −−,L = γsc,L Q L γsc,L . L L

It follows from (5.44) that

  ν 0 ν Isc,L , F − Isc,L , F = inf E sc,L (Q L ) −  F Tr L 2per ( L ) (Q L ), Q L ∈ K L −DG L (ν L , ργ 0

sc,L

− µper ) +

1 DG L (ν L , ν L ), 2

where 1 ν (Q L ) −  F Tr L 2per ( L ) (Q L ) := −DG L (ρ Q L , ν L ) + DG L (ρ Q L , ρ Q L ) E sc,L 2

 ++,L −−,L 0 1/2 0 +Tr L 2per ( L ) |Hsc,L −  F | (Q L − Q L )|Hsc,L −  F |1/2 .

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É. Cancès, A. Deleurence, M. Lewin

First, using ν being in L 1 (R3 ) ∩ L 2 (R3 ) and the convergence of  L = (ργ 0 − sc,L ∞ (see Step 4 of the proof of Theorem 4 in Sect. 5.9), we obtain ργper 0 ) G 1 to zero in L 1 −DG L (ν L , ργ 0 − µper ) + DG L (ν L , ν L ) −→ sc,L 2   1  D(ν, ν). − ν (ργper 0 − µper ) G 1 + 2 R3 Our goal is to prove that lim E νF ,L = E νF ,

(5.45)

  ν E νF ,L = inf E sc,L (Q L ) −  F Tr L 2per ( L ) (Q L ), Q L ∈ K L .

(5.46)

L→∞

where

Step 1. Preliminaries. In the proof of (5.45), we shall need several times to compare states living in L 2per ( L ) with states living in L 2 (R3 ). To this end, we introduce the map i L : L 2 (R3 ) → L 2per ( L )
ϕ → (1 L ϕ)(· − Lz). z∈Z3

Notice that (i L )∗ : L 2per ( L ) → L 2 (R3 ) is the operator which to any periodic function ϕ ∈ L 2per ( L ) associates the function 1 L ϕ ∈ L 2 (R3 ). Remark that i L (i L )∗ = I d L 2per ( L ) whereas (i L )∗ i L = 1 L . Hence i L defines an isometry from L 2 ( L ) to

L 2per ( L ). The equality (i L )∗ i L ϕ = ϕ is only true when ϕ ∈ L 2 (R3 ) has its support in  L . When ϕ ∈ H 1 (R3 ) satisfies Supp(ϕ) ⊂  L , then one also has ∂xi i L (ϕ) = i L (∂xi ϕ). Notice in addition that if A ∈ S1 (L 2per ( L )), then (i L )∗ Ai L ∈ S1 (L 2 ( L )) ⊆ S1 (L 2 (R3 )) and   Tr L 2per ( L ) (A) = Tr L 2 (R3 ) (i L )∗ Ai L .

Similarly if A ∈ S p (L 2per ( L )),

  ||A||Sp (L 2per ( L )) = (i L )∗ Ai L S

p (L

2 (R3 ))

.

(5.47)

Finally, we shall use that for any Z3 -periodic bounded function f , i L f = f i L , where we use the same notation f to denote the multiplication operator by the function f acting 0 either on L 2per ( L ) or on L 2 (R3 ). Similarly, the operator Hsc,L = −/2 + (ργ 0 − sc,L

µper ) G 1 can be seen as acting on L 2per ( L ) or on L 2 (R3 ) and we use the same notation in the two cases. Then we have for any ϕ ∈ C0∞ (R3 ) satisfying Supp(ϕ) ⊆  L , 0 0 Hsc,L i L ϕ = i L Hsc,L ϕ.

(5.48)

Notice that one can also define −i∇ on L 2 (R3 ) or on L 2per ( L ) and we shall adopt the same notation for these two operators. We gather some useful limits in the following

Local Defects in Periodic Crystals

167

Lemma 13. Let be ψ ∈ L 2 (R3 ) and ϕ ∈ C0∞ (R3 ). Then we have as L → ∞, (1) (2) (3) (4) (5)

0 i ψ → γ 0 ψ in L 2 (R3 ); (i L )∗ γsc,L L per 0 γ 0 i ϕ → H 0 γ 0 ϕ in L 2 (R3 ); (i L )∗ Hsc,L per per sc,L L 0 i ϕ → γ 0 ϕ in L 2 (R3 ); (i L )∗ γsc,L L per (i L )∗ (1 + |∇|)i L ϕ → (1 + |∇|)ϕ in L 2 (R3 ); 0 0 −  |1/2 ϕ in L 2 (R3 ) for any fixed  in the gap (i L )∗ |Hsc,L −  F |1/2 i L ϕ → |Hper F F ). ( +Z ,  − Z +1

0 i being uniformly bounded with respect to L, it suffices Proof. The operator (i L )∗ γsc,L L to prove the first assertion for a dense subset of L 2 (R3 ) like C0∞ (R3 ). Hence we may assume that ψ = ϕ ∈ C0∞ (R3 ). 0 . We are going to prove that Let K be a compact set in the resolvent set of Hper 0 0 lim (i L )∗ (Hsc,L − z)−1 i L ϕ → L→∞ (Hper − z)−1 ϕ

(5.49)

L→∞

in L 2 (R3 ), uniformly for z ∈ K . To this end, we first notice that by Theorem 4, K is 0 contained in the resolvent set of Hsc,L for L large enough and thus      0   0 0 0  − z)−1  2 ≤ C(K ) Hsc,L − Hper →0  2 (Hsc,L − z)−1 − (Hper B(L per ( L ))

B(L per ( L ))

0 0 (seen as an opeby (5.43). Hence it suffices to prove (5.49) with Hsc,L replaced by Hper 2 rator acting on L per ( L )). Then we use its Bloch decomposition (detailed in Appendix, see (A.2)) and compute, assuming L large enough for Supp(ϕ) ⊂  L , 
L −3 
∗ 0 −1 (i L ) (Hper − z) i L ϕ(x) = en (k, ·)ϕ 1 L (x)en (k, x), λn (k) − z R3 2π k∈

L

Z3 ∩ ∗ n≥1

 2   0 − z)−1 i L ϕ  2 (i L )∗ (Hper

L (R 3 )




=




3 ∗ n≥1 k∈ 2π L Z ∩

 2   L −3   . e (k, ·)ϕ n   2 |λn (k) − z| R3

0 − z)−1 i ϕ  (H 0 − z)−1 ϕ weakly in L 2 (R3 ) It is then easy to see that (i L )∗ (Hper L per (one cantake the scalar productagainst a function ψ ∈ C0∞ (R3 ) to identify the limit)   0  0 − z)−1 i ϕ  −1  and that (i L )∗ (Hper L  2 3 → (Hper − z) ϕ  2 3 , yielding the strong L (R )

L (R )

convergence in L 2 (R3 ). For the proof of (1), it then suffices to choose a curve C around the first Z bands of 0 and use the above convergence of the resolvent in the Cauchy formula. Assertion Hper (2) is an easy consequence of (1) and (5.48). Indeed, by Theorem 4, we know that   0 0 i is bounded, this implies that 0  lim L→∞  Hsc,L − Hper  2 3 = 0. Since (i L )∗ γsc,L L B(L (R ))

for L large enough such that Supp(ϕ) ⊆  L ,         0 0 0 0 0 0 (Hsc,L i L − i L Hper )ϕ  2 3 = (i L )∗ γsc,L i L (Hsc,L − Hper )ϕ  (i L )∗ γsc,L L (R )

L 2 (R 3 )

→ 0,

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É. Cancès, A. Deleurence, M. Lewin

0 ϕ ∈ L 2 (R3 ). Hence (1) implies that where we have used (5.48). Then we notice that Hper    0 i − γ 0 )H 0 ϕ  lim L→∞ ((i L )∗ γsc,L L per per  2 3 = 0. The argument is exactly the same for L (R )

the third assertion (3). Assertion (5) can be proved in the same way, using (5.48) and the integral representation of the square root (5.29). Finally, it remains to prove that (4) is true, which is done by computing explicitly, for L large enough such that Supp(ϕ) ⊆  L ,
(2π )3/2 (1 + |k|) ϕ (k)eik·x 1 L (x), L3 2π

(i L )∗ (1 + |∇|)i L ϕ =

k∈

  (i L )∗ (1 + |∇|)i L ϕ 2 2

L (R 3 )

=

L

Z3


(2π )3 |(1 + |k|) ϕ (k)|2 → L→∞ ||(1 + |∇|)ϕ||2L 2 (R3 ) . 3 L 2π

k∈

L

Z3

The strong convergence is obtained as above.

 

Lemma 14. Let V ∈ C0∞ (R3 ). We have as L → ∞, (i L )∗ (1 − )−1 i L (V )i L → (1 − )−1 V,

(5.50)

(i L )∗ (1 + |∇|)−1 i L (V )(1 + |∇|)−1 i L → (1 + |∇|)−1 V (1 + |∇|)−1

(5.51)

strongly in S2 (L 2 (R3 )). Proof. For L large enough, we have     (i L )∗ (1 − )−1 i L (V )i L  2

S2 (L (R3 ))

    = (1 − )−1 i L (V ) ⎛ ≤

S2 (L 2per ( L ))

⎞1/2

||V || L 2 (R3 ) ⎜
⎟ ⎝ 2 )2 ⎠ (2π )3/2 (1 + |k| 2π (2π/L)3

k∈

L

which shows that (i L )∗ (1 − )−1 i L (V )i L is bounded in S2 (L 2 (R3 )) since 
(2π/L)3 = |g( p)|2 dp, g( p) = (1 + | p|2 )−1 . lim 2 )2 3 L→∞ (1 + |k| R 2π k∈

L

,

Z3

(5.52)

Z3

Arguing as in the proof of the fourth assertion of Lemma 13, we can prove that (5.50) holds in the strong sense, hence the convergence holds weakly in S2 (L 2 (R3 )) towards (1 − )−1 V . Now   ||V || L 2 (R3 ) ||g|| L 2 (R3 )   = lim (i L )∗ (1 − )−1 i L (V )i L  L→∞ S2 (L 2 (R3 )) (2π )3/2      = (1 − )−1 V  2 3 S2 (L (R ))

and the limit holds strongly in S2 (L 2 (R3 )).

Local Defects in Periodic Crystals

169

The argument is the same for (5.51), noticing that 2    (i L )∗ (1 + |∇|)−1 i L (V )(1 + |∇|)−1 i L  S2 (L 2 (R3 ))

 = Tr L 2per ( L ) (1 + |∇|)−2 i L (V )(1 + |∇|)−2 i L (V )  −3/2 = (2π ) |h L (x − y)|2 V (x)V (y)d x d y ( L )2

→ L→∞

 2   (1 + |∇|)−1 V (1 + |∇|)−1 

S2 (L 2 (R3 ))

,

where we have used that


h L (x) :=

3 k∈ 2π L Z

(2π )3/2 eik·x L 3 (1 + |k|)2

converges to the Fourier inverse F −1 (h) of h( p) = (1 + | p|)−2 , strongly in L 2loc (R3 ).   Step 2. Upper bound. We prove here that lim sup L→∞ E νF ,L ≤ E νF . Let  > 0. Using Lemma 8, Proposition 4, Corollary 3, and the notation therein, one can find a finite rank operator Q ∈ Kr such that E νF ≤ E ν (Q) −  F Tr 0 (Q) ≤ E νF + ,

(5.53)

of the form Q=

−1


|vm vm | −

m=−M

+

k
i=0

−1


|u n u n | +

n=−N

k
i=0

 λi2  |vi vi | − |u i u i | 1 + λi2


  λi   |u v | + |v u | + δ with δ = n j |w j w j |. (5.54) i i i i 1 + λi2 j=1 J

η

η

η

Let 0 < η << 1. It is possible to choose a family of orthonormal functions u n , vm , w j in C0∞ (R3 ) such that η

η u ηn − u n  H 2 ≤ η, vm − vm  H 2 ≤ η, w j − w j  H 2 ≤ η

(5.55)

for all n = −N . . . k, m = −M . . . k and j = 1 . . . J . Let us define the Gram matrices ' ' & & η η η η 0 η η 0 (S+ )i, j := (1 − γper (S− )i, j := γper ui , u j , )vi , v j η

η

which, by (5.55) satisfy S+ = I d M+k+1 + o(1)η→0 and S− = I d N +k+1 + o(1)η→0 . We 0 u η , (1 − γ 0 )v η } and define also introduce the orthogonal projector η on Span{γper n per m ' & η η η η (Sw )i, j := (1 − η )wi , w j . Clearly Sw = I d J + o(1)η→0 . Now we introduce a new orthonormal system in L 2per ( L ), η

u i,L :=

k
n=−N

−1/2

η

0 (S−,L )i,n γsc,L i L u ηn , vi,L :=

k
m=−M

−1/2

0 η (S+,L )i,m (1 − γsc,L )i L vm ,

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' & η η 0 (S−,L )i, j = γsc,L i L ui , i L u j

L 2per ( L )

' & η η 0 , (S+,L )i, j = (1 − γsc,L )i L vi , i L v j

L 2per ( L )

.

η

Notice that by the first assertion of Lemma 13, lim L→∞ S±,L = S± . Finally, we introη η duce the projector  L on Span(u n,L , vm,L ) and define η

w j,L :=

J ' &
−1/2 η η η (Sw,L ) j, (1 −  L )i L w , (Sw,L )i, j = (1 −  L )i L wi , i L w j . =1

We now define a state in K L by η

QL =

−1


η

η

|vm,L vm,L | −

m=−M

+

k
i=0

−1


η

η

|u n,L u n,L | +

n=−N

k
i=0

λi2  η η η η  |vi,L vi,L | − |u i,L u i,L | 2 1 + λi


λi  η η η η  η η |u i,L vi,L | + |vi,L u i,L | + n j |w j,L w j,L |. 2 1 + λi j=1 J

By Lemma 13, we have η

η

η

η

η (i L )∗ u n,L → L→∞ u˜ ηn , (i L )∗ vm,L → L→∞ v˜m and (i L )∗ w j,L → L→∞ w˜ j

(5.56)

in L 2 (R3 ) ∩ L ∞ (R3 ), where the limits are defined by η

u˜ i :=

k


η −1/2

η

0 η (S− )i,n γper u n , v˜i :=

n=−N

k


η −1/2

0 η (S+ )i,m, (1 − γper )vm ,

m=−M

η w˜ j

:=

J
=1

−1/2

η

(Swη ) j, (1 − η )w .

By Lemma 13, we know that for any fixed ϕ, ψ ∈ C0∞ (R3 ), & ' & ' 0 0 0 0 0 0 lim i L∗ (Hsc,L −  F )γsc,L i L ϕ, (i L )∗ γsc,L i L ψ = (Hper −  F )γper ϕ, γper ψ .

L→∞

η

Hence, inserting the definition of Q L in the kinetic energy and using the convergence of the Gram matrices, we obtain



 η 0 0 lim Tr L 2per ( L ) (Hsc,L −  F )Q L = Tr L 2 (R3 ) (Hper −  F ) Q˜ η , L→∞

η η η η where Q˜ η is defined similarly as Q L but with the functions (u˜ n , v˜m , w˜ j ) instead of η η η (u n,L , vm,L , w j,L ). Let us now prove that

lim DG L (ρ Q η , ρ Q η ) = D(ρ Q˜ η , ρ Q˜ η ).

L→∞

L

L

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The convergence (5.56) implies that 1 L ρ Q η converges to ρ Q˜ η in particular in L 1 (R3 ) ∩ L

L 2 (R3 ). Notice the definition of DG L (·, ·) implies that

DG L (ρ, ρ) ≤ C ||ρ||2L 1

∀ρ ∈ L 1per ( L ) ∩ L 2per ( L ),

per ( L

+ ||ρ||2L 2 )



per ( L )

(5.57)

for a constant C independent of L. Let us now write ρ Q η = ρ1,L + ρ2,L , where ρ1,L is L the periodic function which equals 1 B(0,L/4) ρ Q η on  L . The convergence of 1 L ρ Q η L

L

towards ρ Q˜ η in L 1 (R3 ) ∩ L 2 (R3 ) and (5.57) give that   lim ρ2,L  L 1

per ( L )

L→∞

  = lim ρ2,L  L 2

per ( L )

L→∞

= lim DG L (ρ2,L , ρ2,L ) = 0. L→∞

Hence, it remains to show that lim DG L (ρ1,L , ρ1,L ) = D(ρ Q˜ η , ρ Q˜ η ).

L→∞

(5.58)

To this end we use the estimate [22]    1   = O(L −1 ), sup G L (x) − |x|  x∈ L to obtain  DG L (ρ1,L , ρ1,L ) =

( L )2

G L (x − y)ρ1,L (x)ρ1,L (y)d x d y

= D(1 L ρ1,L , 1 L ρ1,L ) + O(L −1 ),   where we have used that ρ1,L  L 1

per ( L )

is uniformly bounded and that x − y ∈  L

for any x, y ∈ B(0, L/4), the support of ρ1,L . The convergence of 1 L ρ1,L towards ρ Q˜ η in L 1 (R3 ) ∩ L 2 (R3 ) then proves (5.58). Using the same argument for the term DG L (ρ Q η , ν L ) we obtain L



  1 1

lim −DG L (ρ Q η , ν L ) + DG L ρ Q η , ρ Q η = −D(ρ Q˜ η , ν) + D ρ Q˜ η , ρ Q˜ η . L L L L→∞ 2 2 Finally η η ν lim E sc,L (Q L ) −  F Tr L 2per ( L ) (Q L ) = E ν ( Q˜ η ) −  F Tr( Q˜ η ).

L→∞

(5.59)

Passing to the limit as η → 0 using (5.55) and the convergence of the Gram matrices η η S± and Sw , we eventually obtain lim sup E νF ,L ≤ E ν (Q) −  F Tr(Q) ≤ E νF + . L→∞

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É. Cancès, A. Deleurence, M. Lewin

Step 3. Lower bound. We end the proof by showing that lim inf L→∞ E νF ,L ≥ E νF . As  L is bounded for any fixed L, the existence of a minimizer Q L of   ν (Q L ) −  F Tr L 2per ( L ) (Q L ), Q L ∈ K L inf E sc,L 0 , considered as an operator on is straightforward. In addition, the spectrum of Hsc,L L 2per ( L ), being purely discrete and bounded below, Q L is finite rank. Using (4.4) and reasoning as in the proof of Lemma 1 (see Sect. 5.1.1), we prove that 0 there exists a constant c > 0 (independent of L) such that |Hsc,L −  F | ≥ c(1 − ) on 2 L per ( L ), for L large enough. The following uniform bounds follow from Step 2: 0 0 −  F |1/2 (Q ++,L − Q −−,L )|Hsc,L −  F |1/2 ) ≤ C, Tr L 2per ( L ) (|Hsc,L L L

(5.60)

− Q −−,L )(1 + |∇|)) ≤ C, Tr L 2per ( L ) ((1 + |∇|)(Q ++,L L L

(5.61)

Tr L 2per ( L ) ((1 + |∇|)Q 2L (1 + |∇|))

≤ C,

(5.62)

DG L (ρ Q L − ν L , ρ Q L − ν L ) ≤ C,

(5.63)

with C independent of L. Consider now the sequence of operators Q˜ L := (i L )∗ Q L i L acting on L 2 (R3 ). It is bounded in S2 (L 2 (R3 )) by (5.62) and since Tr L 2 (R3 ) ( Q˜ 2L ) = Tr L 2per ( L ) (Q 2L ) by (5.47). Hence Q˜ L weakly converges, up to extraction, to some Q ∈ S2 (L 2 (R3 )). Similarly, the Hilbert-Schmidt operator R L := (i L )∗ Q L (1 + |∇|)i L weakly converges up to extraction to some R in S2 (L 2 (R3 )). Let ϕ and ψ be in C0∞ (R3 ) and assume that Supp(ϕ) ∪ Supp(ψ) ⊂  L . Then & '   (i L )∗ Q L (1 + |∇|)i L ϕ, ψ L 2 (R3 ) = Q˜ L (i L )∗ (1 + |∇|)i L ϕ, ψ 2 3 L (R )

→ L→∞ Q(1 + |∇|)ϕ, ψ L 2 (R3 ) , where we have used that Q˜ L  Q weakly in S2 and that (i L )∗ (1+|∇|)i L ϕ → (1+|∇|)ϕ strongly in L 2 (R3 ) by the third assertion of Lemma 13. Hence Q(1 + |∇|) = R ∈ S2 (L 2 (R3 )). Similarly, define the operator SL := (i L )∗ Q −−,L i L which is nonpositive and yields L a bounded sequence in S1 (L 2 (R3 )) by (5.61). Up to extraction, we may assume that (SL ) converges for the weak-∗ topology to some S ∈ S1 (L 2 (R3 )). To identify the limit S, we compute as above for ϕ, ψ ∈ L 2 (R3 ), & ' 0 0 SL ϕ, ψ L 2 (R3 ) = (i L )∗ γsc,L Q L γsc,L i L ϕ, ψ 2 3 L (R ) & ' ∗ 0 ∗ 0 ˜ = Q L (i L ) γsc,L i L ϕ, (i L ) γsc,L i L ψ 2 3 . L (R )

Using now the first assertion of Lemma 13 we obtain lim L→∞ SL ϕ, ψ L 2 (R3 ) =   −− Q ϕ, ψ L 2 (R3 ) . Hence Q −− = S ∈ S1 . The same arguments allow to conclude that in fact, Q ∈ K. 0 0 Now, let TL := (i L )∗ |Hsc,L −  F |1/2 Q −−,L |Hsc,L −  F |1/2 i L which also defines a L 2 3 bounded sequence in S1 (L (R )). Up to extraction, we may assume that TL  T for

Local Defects in Periodic Crystals

173

the weak-∗ topology of S1 . Arguing as above and using Lemma 13, we deduce that 0 −  |1/2 Q −− |H 0 −  |1/2 . Now, Fatou’s Lemma yields T = |Hper F F per

 0 0 lim inf Tr L 2per ( L ) |Hsc,L −  F |1/2 (−Q −−,L )|Hsc,L −  F |1/2 L L→∞

 0 0 = lim inf Tr L 2 (R3 ) (−TL ) ≥ Tr L 2 (R3 ) |Hper −  F |1/2 (−Q −− )|Hper −  F |1/2 . L→∞

This proves that

 0 0 lim inf Tr L 2per ( L ) (Hsc,L −  F )Q L ≥ Tr 0 (Hper Q) −  F Tr 0 (Q). L→∞

We now study the term involving the density ρ Q L . First, following the proof of Proposition 1 and using the bounds (5.60)–(5.62), we can prove that there exists a   constant C such that for all L large enough ρ Q L  L 2 ( ) ≤ C. Hence, up to extraction, per

L

we have 1 L ρ Q L  ρ weakly in L 2 (R3 ) for some function ρ ∈ L 2 (R3 ). We now introduce an auxiliary function ρ L ∈ L 2 (R3 ) defined in Fourier space as follows:


ρ / L :=

3 k∈ 2π L Z \{0}

ck,L (ρ Q L ) c0,L (ρ Q L ) 1 Bk + 1 B0 , |Bk |1/2 |B0 |1/2

where for any k ∈ (2π/L)Z3 \ {0}, Bk := B k + 1/|k  |

k

k , 1  10L|k|1 10L 0, 10L .

 which is chosen to ensure

that ≤ 1/|k| for any ∈ Bk , and B0 := B Notice that ρ L is bounded in L 2 (R3 ) as we have by definition   
2 ρ L2 = |/ ρ L |2 = |ck,L (ρ Q L )|2 = ρQ . L R3

R3

3 k∈ 2π L Z

L

On the other hand (up to extraction) ρ L  ρ weakly in L 2 (R3 ), the same weak limit as 1 L ρ Q L . This is easily seen by considering a scalar product against a fixed function ϕ ∈ C0∞ (R3 ). Now by the choice of the balls Bk , we also have for L  1,  |/ ρ L (k  )|2  dk ≤ DG L (ρ Q L , ρ Q L ) ≤ C. D(ρ L , ρ L ) = 4π |k  |2 Hence, up to extraction we may assume that ρ L  ρ weakly in C. Using the regularity of  ν, we also deduce that   1 1 lim inf −DG L (ρ Q L , ν L ) + DG L (ρ Q L , ρ Q L ) ≥ −D(ρ, ν) + D(ρ, ρ). L→∞ 2 2 What remains to be proved is that ρ = ρ Q , where Q is the weak limit of (i L )∗ Q L i L obtained above. This will clearly show lim inf E νF ,L ≥ E ν (Q) −  F Tr 0 (Q) ≥ E νF L→∞

and end the proof of Theorem 5. We identify the limit of 1 L ρ Q L using its weak convergence to ρ in L 2 (R3 ).

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We start with ρ Q ++,L and write, fixing some V ∈ C0∞ (R3 ) and assuming L large L enough for Supp(V ) ⊂  L ,  ρ Q ++,L V = Tr L 2per ( L ) (Q ++,L i L (V )) = Tr L 2 (R3 ) (A L B L ) with L L

L

(1 + |∇|)i L , A L := (i L )∗ (1 + |∇|)Q ++,L L

B L := (i L )∗ (1 + |∇|)−1 i L (V )(1 + |∇|)−1 i L .

The sequence (A L ) is bounded in S1 (L 2 (R3 )), hence in S2 (L 2 (R3 )), by (5.61) and converges (up to extraction) towards (1 + |∇|)Q + + (1 + |∇|) weakly in S2 (L 2 (R3 )) (we proceed as above to identify the weak limit using the fourth assertion of Lemma 13). By Lemma 14, B L converges towards (1 + |∇|)−1 V (1 + |∇|)−1 strongly in S2 (L 2 (R3 )). We thus obtain   lim ρ Q ++,L V = Tr L 2 (R3 ) (Q + + V ) = ρ Q + + V. L→∞  L

R3

L

Likewise, it can be proved that the weak limit of ρ Q −−,L is ρ Q −− . L Let us now treat ρ Q +−,L (the other case ρ Q −+,L being similar). Following the proof of L L Proposition 1, we write 

 0 ρ Q +−,L V = Tr L 2per ( L ) Q +−,L [γ , i (V )] L sc,L L L L 

 1 0 −1 0 −1 . =− dzTr L 2per( L ) Q +−,L (H − z) (i (V ) − i (V ))(H − z) L L sc,L sc,L L 4iπ C We only detail the argument to pass to the limit in

 0 −1 0 −1 Tr L 2per ( L ) Q +−,L (H − z) i (V )(H − z) L sc,L sc,L L

 0 0 = Tr L 2per ( L ) (Hsc,L − z)−1 Q +−,L (Hsc,L − z)−1 (1 − )(1 − )−1 i L (V ) L

 = Tr L 2 (R3 ) C L (i L )∗ (1 − )−1 i L (V )i L 0 − z)−1 Q +−,L (H 0 − z)−1 (1 − )i . One has, up to extracwith C L := (i L )∗ (Hsc,L L L sc,L 0 − z)−1 Q +− (H 0 − z)−1 (1−) weakly in S (L 2 (R3 )). To see this, tion, C L  (Hper 2 per 2 3 one first remarks that C L is bounded in S2 (L (R )) and then identifies the weak limit by passing to the limit in C L ϕ, ψ for some fixed ϕ, ψ ∈ C0∞ (R3 ), using the uniform convergence of the resolvent for z ∈ C , as shown in the proof of Lemma 13. Then by Lemma 14 we know that (i L )∗ (1 − )−1 i L (V )i L converges towards (1 − )−1 V strongly in S2 (L 2 ( L )), hence we can pass to the limit in the above expression, uniformly in z ∈ C . We conclude that   lim 1 L ρ Q L V = ρQ V L→∞ R3

for any V ∈ C0∞ (R3 ), thus ρ = ρ Q .

R3

 

Local Defects in Periodic Crystals

175

Appendix A. Proof of Theorem 1 Our proof uses classical ideas for Hartree-Fock theories. See [23, Sect. 4] for a very 0 of I 0 (it is known to exist by [5, similar setting. Let us consider a minimizer γper per Thm 2.1]). First we note that the periodic potential Vper := (ργper 0 − µper ) G 1 is in L 2loc (R3 ). Thus Vper defines a -bounded operator on L 2 (R3 ) with relative bound 0 = −/2 + V zero (see [27, Thm XIII.96]) and therefore Hper per is self-adjoint on 2 3 1 3 0 is purely D(−) = H (R ) with form domain H (R ). Besides, the spectrum of Hper absolutely continuous, composed of bands as stated in [34, Thm 1-2] and [27, Thm XIII.100]. The Bloch eigenvalues λk (ξ ), k ≥ 1, ξ ∈ ∗ are known to be real analytic in each fixed direction and cannot be constant with respect to the variable ξ . Hence the function
  {ξ ∈ ∗ | λk (ξ ) ≤ κ} C : κ → k≥1 0 being bounded from below, we is continuous and nondecreasing on R. The operator Hper ∗ have C ≡ 0 on (−∞, inf λk ( )) and it is known [34, Lemma A-2] that limκ→∞ C(κ) = ∞. We can thus choose a Fermi level  F such that
  {ξ ∈ ∗ | λk (ξ ) ≤  F } . (A.1) Z = C( F ) = k≥1 0 + tγ for any γ ∈ P Z and t ∈ [0, 1], we deduce Considering a variation (1 − t)γper per 0 that γper minimizes the following linear functional: 

 1 Z 0 (H → Tr ) γ γ ∈ Pper 2 per ξ ξ dξ, (2π )3 ∗ L ξ ( ) 0 is the mean-field operator defined in (2.5). We subtract the chemical potential where Hper  F defined above and introduce the functional 

 1 0 (H dξ. γ ∈ Pper → F(γ ) := Tr −  ) γ 2 F ξ ξ per (2π )3 ∗ L ξ ( )    Z , then γ 0 also miniNotice that since (2π1 )3 ∗ Tr L 2 ( ) γξ dξ = Z for any γ ∈ Pper per ξ

Z . mizes F on Pper For any ξ ∈ ∗ , we can find orthonormal functions ek (ξ ) ∈ L 2ξ ( ) such that 0 (Hper )ξ =




λk (ξ )|ek (ξ )ek (ξ )|,

(A.2)

k≥1

each function ξ → ek (ξ ) ∈ L 2ξ ( ) being measurable on ∗ . Let us now define γ 0 ∈ Pper by
1 if λk (ξ ) ≤  F (γ 0 )ξ = δk (ξ )|ek (ξ )ek (ξ )|, δk (ξ ) = 0 if λk (ξ ) >  F . k≥1

0 ). Notice  was chosen to ensure γ 0 ∈ P Z . Stating this differently γ 0 = χ(−∞, F ] (Hper F per

176

É. Cancès, A. Deleurence, M. Lewin

We now prove that γ 0 is the unique minimizer of the function F defined above, on Z , this will prove that γ 0 = γ 0 the set Pper without a charge constraint. Since γ 0 ∈ Pper per 0 and that γper is the unique minimizer of F on Pper . We write 

 0 Tr L 2 ( ) (Hper −  F )ξ (γ − γ 0 )ξ dξ F(γ ) − F(γ 0 ) = (2π )−3 ξ ∗ 
   = (2π )−3 (λk (ξ ) −  F ) γξ ek (ξ ), ek (ξ ) ξ − δk (ξ ) dξ, ∗

k≥1

where ·, ·ξ is the usual scalar product of L 2ξ ( ). Since 0 ≤ γ ≤ 1 in L 2 (R3 ), we have   that 0 ≤ γξ ≤ 1 on L 2ξ ( ) and thus γξ ek (ξ ), ek (ξ ) ∈ [0, 1], for almost every ξ ∈ ∗ . Hence, using the definition of δk (ξ ), 
   0 −3 (2π ) |λk (ξ ) −  F | ×  γξ ek (ξ ), ek (ξ ) − δk (ξ )dξ ≥ 0. F(γ ) − F(γ ) = k≥1

∗

Z . If now F(γ ) = F(γ 0 ), then necessarily This shows that γ 0 minimizes F on Pper   γξ ek (ξ ), ek (ξ ) = δk (ξ ) for almost every ξ ∈ ∗ and any k ≥ 1, the set {ξ ∈ ∗ | ∃k, λk (ξ ) =  F } having a Lebesgue measure equal to zero by [34, Lemma 2]. Using now that the operators γξ and (1 − γ )ξ are nonnegative, we infer that γξ ek (ξ ) = δk (ξ )ek (ξ ) for all k ≥ 1 and almost all ξ ∈ ∗ . Hence γ = γ 0 and γ 0 is the unique 0 = γ 0 , i.e. γ 0 solves the self-consistent equation minimizer of F. In particular γper per (2.6). 0 on P Z , we recall that ρ = Consider now another minimizer γ of the energy Eper γ per 0 0 ργper 0 as was shown in [5]. Hence the operators Hper and γ defined above do not depend 0 , on the chosen minimizer. The above argument applied to γ shows that γ = γ 0 = γper 0 i.e. γper is unique.  

Acknowledgement. We are thankful to Éric Séré for useful advice on the model, to Claude Le Bris and Isabelle Catto for helpful comments on the manuscript. We all acknowledge support from the INRIA project MICMAC and from the ACI program SIMUMOL of the French Ministry of Research. M.L. acknowledges support from the ANR project “ACCQUAREL”. A.D. has been supported by a grant from Région Ile-de-France.

References 1. Bach, V., Barbaroux, J.-M., Helffer, B., Siedentop, H.: On the Stability of the Relativistic ElectronPositron Field. Commun. Math. Phys. 201, 445–460 (1999) 2. Bhatia, R.: Matrix analysis. Graduate Texts in Mathematics, 169. New York: Springer-Verlag, 1997 3. Cancès, É., Deleurence, A., Lewin, M.: Non-perturbative embedding of local defects in crystalline materials. http://arXiv.org/abs/:0706.0794, 2007 4. Catto, I., Le Bris, C., Lions, P.-L.: Sur la limite thermodynamique pour des modèles de type Hartree et Hartree-Fock. C.R. Acad. Sci. Paris, Série I 327, 259–266 (1998) 5. Catto, I., Le Bris, C., Lions, P.-L.: On the thermodynamic limit for Hartree-Fock type problems. Ann. I. H. Poincaré 18, 687–760 (2001) 6. Chaix, P., Iracane, D.: From quantum electrodynamics to mean field theory: I. The Bogoliubov-DiracFock Formalism. J. Phys. B. 22, 3791–3814 (1989) 7. Chaix, P., Iracane, D., Lions, P.L.: From quantum electrodynamics to mean field theory: II. Variational stability of the vacuum of quantum electrodynamics in the mean-field approximation. J. Phys. B. 22, 3815–3828 (1989)

Local Defects in Periodic Crystals

177

8. Dovesi, R., Orlando, R., Roetti, C., Pisani, C., Saunders, V.R.: periodic Hartree-Fock method and its implementation in the Crystal code. Phys. Stat. Sol. (B) 217, 63–88 (2000) 9. Fefferman, C.: The Thermodynamic Limit for a Crystal. Commun. Math. Phys. 98, 289–311 (1985) 10. Hainzl, Ch., Lewin, M., Séré, E.: Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation. Commun. Math. Phys. 257, 515–562 (2005) 11. Hainzl, Ch., Lewin, M., Séré, E.: Self-consistent solution for the polarized vacuum in a no-photon QED model. J. Phys. A: Math & Gen. 38(20), 4483–4499 (2005) 12. Hainzl, Ch., Lewin, M., Séré, E.: Existence of atoms and molecules in the mean-field approximation of no-photon quantum electrodynamics. Arch. Rat. Mech. Anal. (to appear) 13. Hainzl, Ch., Lewin, M., Solovej, J.P.: The mean-field approximation in quantum electrodynamics. The No-Photon Case. Comm. Pure Applied Math. 60(4), 546–596 (2007) 14. Hainzl, C., Lewin, M., Séré, É., Solovej, J.P.: A Minimization Method for Relativistic Electrons in a Mean-Field Approximation of Quantum Electrodynamics. Phys. Rev. A 76, 052104 (2007) 15. Hasler, D., Solovej, J.P.: The Independence on Boundary Conditions for the Thermodynamic Limit of Charged Systems. Commun. Math. Phys. 261(3), 549–568 (2006) 16. Hunziker, W.: On the Spectra of Schrödinger Multiparticle Hamiltonians. Helv. Phys. Acta 39, 451–462 (1996) 17. Kittel, Ch.: Quantum Theory of Solids. Second Edition. New York: Wiley, 1987 18. Lieb, E.H.: The Stability of Matter. Rev. Mod. Phys. 48, 553–569 (1976) 19. Lieb, E.H., Lebowitz, J.L.: The constitution of matter: existence of thermodynamics for systems composed of electrons and nuclei. Adv. Math. 9, 316–398 (1972) 20. Lieb, E.H., Loss, M.: Analysis, Second Edition. Graduate Studies in Mathematics, Vol. 14. Providence, RI: Amer. Math. Soc., 2001 21. Lieb, E.H., Simon, B.: The Hartree-Fock theory for Coulomb systems. Commun. Math. Phys. 53, 185–194 (1977) 22. Lieb, E.H., Simon, B.: The Thomas-Fermi theory of atoms, molecules and solids. Adv. Math. 23, 22–116 (1977) 23. Lieb, E.H., Solovej, J.P., Yngvason, J.: Asymptotics of heavy atoms in high magnetic fields. I. Lowest Landau band regions. Comm. Pure Appl. Math. 47(4), 513–591 (1994) 24. Lions, P.-L.: The concentration-compactness method in the Calculus of Variations. The locally compact case. Part. I: Anal. non linéaire, Ann. IHP 1 109–145, (1984); Part. II: Anal. non linéaire, Ann. IHP 1 223–283, (1984) 25. Pisani, C.: Quantum-mechanical treatment of the energetics of local defects in crystals: a few answers and many open questions. Phase Transitions 52, 123–136 (1994) 26. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol I, Functional Analysis, Second Ed. New York: Academic Press, 1980 27. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol IV, Analysis of Operators. New York: Academic Press, 1978 28. Ruelle, D.: Statistical Mechanics. Rigorous results. Singapore: Imperial College Press and World Scientific Publishing, 1999 29. Seiler, E., Simon, B.: Bounds in the Yukawa2 Quantum Field Theory: Upper Bound on the Pressure, Hamiltonian Bound and Linear Lower Bound. Commun. Math. Phys. 45, 99–114 (1975) 30. Simon, B.: Trace Ideals and their Applications. Vol 35 of London Mathematical Society Lecture Notes Series. Cambridge: Cambridge University Press, 1979 31. Solovej, J.P.: Proof of the ionization conjecture in a reduced Hartree-Fock model. Invent. Math. 104(2), 291–311 (1991) 32. Soussi, S.: Convergence of the supercell method for defect modes calculations in photonic crystals. SIAM J. Numer. Anal. 43(3), 1175–1201 (2005) 33. Stoneham, A.M.: Theory of Defects in Solids - Electronic Structure of Defects in Insulators and Semiconductors. Oxford: Oxford University Press, 2001 34. Thomas, L.E.: Time-Dependent Approach to Scattering from Impurities in a Crystal. Commun. Math. Phys. 33, 335–343 (1973) 35. Van Winter, C.: Theory of Finite Systems of Particles. I. The Green function. Mat.-Fys. Skr. Danske Vid. Selsk. 2(8), 1–60 (1964) 36. Zhislin, G.M.: A study of the spectrum of the Schrödinger operator for a system of several particles. (Russian) Trudy Moskov. Mat. Obšˇc. 9, 81–120 (1960) Communicated by H.-T. Yau

Commun. Math. Phys. 281, 179–202 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0480-y

Communications in

Mathematical Physics

Approach to Equilibrium for the Phonon Boltzmann Equation Jean Bricmont1,
, Antti Kupiainen2,

1 UCL, FYMA, chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium 2 Department of Mathematics, Helsinki University, P.O. Box 4, 00014 Helsinki, Finland.

E-mail: [email protected] Received: 21 February 2007 / Accepted: 3 December 2007 Published online: 26 April 2008 – © Springer-Verlag 2008

Abstract: We study the asymptotics of solutions of the Boltzmann equation describing the kinetic limit of a lattice of classical interacting anharmonic oscillators. We prove that, if the initial condition is a small perturbation of an equilibrium state, and vanishes at infinity, the dynamics tends diffusively to equilibrium. The solution is the sum of a local equilibrium state, associated to conserved quantities that diffuse to zero, and fast variables that are slaved to the slow ones. This slaving implies the Fourier law, which relates the induced currents to the gradients of the conserved quantities.

1. Introduction If a piece of solid is heated locally and left to cool, the initial temperature distribution will diffusively relax to a constant temperature. The process is accompanied by a heat flow that is proportional to the local temperature gradient, according to the Fourier law. A mathematical understanding of this phenomenon starting from a microscopic model of matter is a considerable challenge [2]. A simple classical mechanical system modeling heat transport in solids is given by a system of coupled oscillators organized on a d-dimensional cubic lattice Zd . The oscillators are indexed by lattice points x ∈ Zd , and carry momenta and coordinates ( px , qx ). In the simplest model px and qx take real values and their dynamics is generated by a Hamiltonian which is a perturbation of a harmonic system: H (q, p) =

1
2 1
λ
4 px + qx q y ω2 (x − y) + q , 2 x 2 xy 4 x x


Partially supported by the Belgian IAP program P6/02.

Partially supported by the Academy of Finland.

(1.1)

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J. Bricmont, A. Kupiainen

where ω2 denotes the Fourier transform of ω(k)2 , which will be given explicitly in (2.1, 2.2) below. In (1.1), the oscillators are coupled by harmonic forces generated by the coupling matrix ω2 which is taken short range. The parameter λ describing the strength of the anharmonicity is assumed to be small. The classical dynamics is given by Hamilton’s equations q˙ x = px , p˙ x = −

∂H . ∂qx

(1.2)

The dynamics (1.2) preserves the Gibbs measures in the phase space, formally given by Z −1 e−β H (q, p) dqdp

(1.3)

and describing an equilibrium state with constant inverse temperature β = 1/T . One expects initial states that agree with (1.3) “at infinity” to be attracted by (1.3) under the flow (1.2) (since one works here in infinite volume, one does not expect Poincaré’s recurrences to occur). While such a result is well beyond current techniques, one approach is to study this problem in an appropriate small coupling limit, the kinetic limit. In that limit, one takes λ2 = R, one rescales space and time by  and, in the limit as  → 0, one formally arrives at an evolution equation for the covariance of a Gaussian measure of the form [9]: 1   x W (x, k, t) = RC(W )(x, k, t), W˙ (x, k, t) + ∇ω(k) · ∇ 2π

(1.4)

where x ∈ Rd , k ∈ Td = Rd /2π Zd . The relation of W to the microscopic model is 
e−iky W (x, k, t)dk = lim ( ω(y − z)qx/+z qx/−z  + iqx/+y px/−y  ), (1.5) →0

z

where − is the state at time t/. W is real and positive. Since the function W (Rx, k, Rt) satisfies (1.4) with R = 1 we may, with no loss of generality, make that assumption. Equation (1.4) is a Boltzmann type equation where the collision term is given by  9π C(W )(x, k, t) = dk1 dk2 dk3 (ωω1 ω2 ω3 )−1 δ(ω + ω1 − ω2 − ω3 ) 4 T3d

δ(k + k1 − k2 − k3 )[W1 W2 W3 − W (W1 W2 + W1 W3 − W2 W3 )] (1.6) with ω = ω(k), W = W (x, k, t), ωi = ω(ki ), Wi = W (x, ki , t), i = 1, 2, 3. The sum k + k1 − k2 − k3 is mod (2π Z)d . We normalize Td dk = 1. Analogously to the collision term of the standard Boltzmann equation for gases, the collision term (1.6) describes a scattering process. The scattered “particles” are phonons, i.e. vibrational modes, and the scattering process in (1.6) involves four phonons. The phonons carry momenta, k, k1 , k2 , k3 and energies ω(k), ω(k1 ), ω(k2 ), ω(k3 ) and the scattering process conserves the total energy and momentum. We may pose the problem stated in the beginning, i.e. the question of relaxation to equilibrium, for Eq. (1.4). Thus, starting from some initial W that coincides “at infinity” with an equilibrium state we would like to prove that W (t) tends diffusively to equilibrium and the process is accompanied by heat fluxes governed by Fourier’s law. In this

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paper, we prove such a theorem (Theorem 1 in Sect. 3), for an initial condition that is close to equilibrium. Equation (1.4) becomes a coupled system for “fast” and “slow” variables (see (3.6), (3.7) below). The slow variables correspond to the temperature and the chemical potential of another conserved quantity, defined in (3.16). The slow variables flow under a nonlinear diffusion equation whereas the fast ones, that include the heat currents, are “slaved” to the slow ones by the Fourier law, which is discussed after stating Theorem 1 in Sect. 3. There is a rich literature on the derivation of hydrodynamics from the Boltzmann equation for gases. In particular in [1], it is proven that, in an appropriate limit, the Boltzmann equation gives rise to the Navier-Stokes equations (see also [6,5,7,11] for more results and background on the link between Boltzmann and Navier-Stokes). In Theorem 2 (Sect. 3), we consider the diffusive scaling limit, i.e. the limit where one scales the spatial variable by , and the time variable by  2 , and we show that a suitably rescaled solution of the Boltzmann equation (1.4) solves, in the limit  → 0, a nonlinear heat equation. Compared with the Boltzmann equation for gases, Eq. (1.4) has some nice features. In particular, its long time dynamics is described by hydrodynamic equations that are considerably simpler than the Navier-Stokes-Fourier system for the former (in the Euler scaling, where space and time are both scaled by , Eq. (1.4) reduces to the Fourier law). A mathematical theory of (1.4) however appears to be lacking. This paper should be considered as a step in that direction. For an attempt to prove the Fourier law for phonons, beyond the Boltzmann approximation, see [3,4]. 2. The Collision Term The collision term (1.6) involves integration over the subset determined by the constraints in the delta functions. Some assumptions on ω are needed for regularity of the resulting measure. The standard nearest neighbor coupling between the oscillators corresponds to ω = ω0 with ω0 (k)2 ≡ 2

d
(1 − cos k j ) + ν,

(2.1)

j=1

where ν ≥ 0 is the pinning parameter. The regularity properties stated in Proposition 2.1 should hold for this ω as long as ν > 0. The proof of this seems, however, quite tedious and, therefore, we take ω(k) = ω0 (k)2 ,

(2.2)

with ν > 0, which is simpler to analyze (see also the Remark after the proof of Proposition 2.1 below). Moreover, we need to take the spatial dimension d ≥ 2. From now on, we’ll make that assumption. For ν = 0 and ω(k) = ω0 (k) the regularity problems are more serious and we expect to need d ≥ 3. Moreover, the spectrum of the linearized collision operator has no gap and the nature of the flow of Eq. (1.4) is quite different since also the slaved modes are diffusive. Progress in that case is so far hampered by the complexity of the integrals appearing in (1.6). The basic properties of the collision term (1.6) are summarized in Proposition 2.1. Since the x variable plays no role in C(W ) we consider C as a map in B = C 0 (Td )

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equipped with the sup norm, denoted by · . We also need the Hilbert space H ≡ L 2 (Td , ω(k)2 dk),

(2.3)

whose norm will be denoted by · H . In this paper, we shall use C or c to denote constants that may vary from place to place. Proposition 2.1. For the map C defined in (1.6), with ω defined in (2.1, 2.2), we have: (a) C : B → B with C(W ) ≤ C W 3 , for some constant C < ∞. (b) The equation C(W ) = 0 has, for W > 0, exactly a two parameter family of solutions in B, WT,A =

T , ω(k) + A

where W > 0 for A > −ν. (c) For all W ∈ B,   dkC(W )(k) = dkω(k)C(W )(k) = 0.

(2.4)

(2.5)

(d) Let −L be the linearization of C(W ) around W0 = ω(k)−1 , i.e. L = −DC(W0 ).

(2.6)

Then, L is bounded in B and bounded and positive in H. It has two zero modes Lω−1 = Lω−2 = 0.

(2.7)

and the rest of its spectrum in B and H is contained in λ ≥ a > 0. Moreover L = M + K , where M is a multiplication operator by a strictly positive continuous function and K is an integral operator, compact in H and B, and satisfying  sup |K (k, k )|dk < ∞. (2.8) k

Proof. (b) We use an argument given in [9], for a similar kernel. Introduce the notation k0 = k, ω0 = ω(k0 ). Then, writing the [−] in (1.6) as 3 

  W (ki ) W (k0 )−1 − W (k3 )−1 − W (k2 )−1 + W (k1 )−1 ,

(2.9)

i=0

we have: 9π C(W )(k0 ) = 4

  3 T3d

i=1

dki

3 

(ωi−1 W (ki ))δ(ω0 + ω1 − ω2 − ω3 )

i=0

  ·δ(k0 + k1 − k2 − k3 ) W (k0 )−1 − W (k3 )−1 − W (k2 )−1 + W (k1 )−1 , (2.10)

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which vanishes, for W (ki ) > 0, if f (k) = W (k)−1 is a collisional invariant, i.e. a function satisfying f (k0 ) + f (k1 ) = f (k2 ) + f (k3 ),

(2.11)

on the set of vectors ki in the support of the delta functions in (2.10). In [10] it is proven that the only L 1 -solutions of (2.11) are of the form aω(k) + b. Hence, the functions in (2.4) satisfy C(W ) = 0. To show that these functions are the only solutions of C(W ) = 0, write, for g bounded,    3 9π (dki ωi−1 W (ki ))δ(ω0 + ω1 − ω2 − ω3 ) dkg(k)C(W )(k) = 4 T4d

i=0

·δ(k0 + k1 − k2 − k3 )g(k0 )   × W (k0 )−1 − W (k3 )−1 − W (k2 )−1 + W (k1 )−1 , (2.12) and use the symmetry 0 ↔ 1 of the rest of the integrand, to replace g(k0 ) in front of [−] in (2.12) by 21 (g(k0 ) + g(k1 )), and, using the antisymmetry of the rest of the integrand under the exchange 0 ↔ 3, 1 ↔ 2, we may replace 21 (g(k0 ) + g(k1 )) by − 12 (g(k3 ) + g(k2 )). This implies that (2.12) is proportional to   3 T4d

(dki ωi−1 W (ki ))δ(ω0 + ω1 − ω2 − ω3 )δ(k0 + k1 − k2 − k3 )

i=0

  · [g(k0 ) + g(k1 ) − g(k2 ) − g(k3 )] W (k0 )−1 − W (k3 )−1 − W (k2 )−1 + W (k1 )−1 . (2.13) W (k)−1 ,

Taking g(k) =   3 T4d

we obtain

(dki ωi−1 W (ki ))δ(ω0 + ω1 − ω2 − ω3 )δ(k0 + k1 − k2 − k3 )

i=0



· W (k0 )−1 − W (k3 )−1 − W (k2 )−1 + W (k1 )−1

2

,

(2.14)

which vanishes, for W (ki ) > 0, only if W (k)−1 is a collisional invariant, i.e., by [10], only if W is of the form (2.4). (c) Since dkg(k)C(W )(k) is proportional to (2.13), it vanishes, for any W , whenever g is a collisional invariant, in particular for g(k) = 1 and g(k) = ω(k). (d) Differentiating C(WT,A ) = 0 with respect to T and A at T = 1 and A = 0, we obtain Lω−1 = Lω−2 = 0.

(2.15)

Explicitly, we have (note the − sign in (2.6))   3 9π dki ωi−2 δ(ω0 + ω1 − ω2 − ω3 )δ(k0 + k1 − k2 − k3 ) L f (k0 ) = 4ω02 i=1 T3d   ω(k0 )2 f (k0 ) + ω(k1 )2 f (k1 ) − ω(k2 )2 f (k2 ) −ω(k3 )2 f (k3 ) . (2.16)

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J. Bricmont, A. Kupiainen

To obtain (2.16) from (1.6), write the [−] in (1.6) as in (2.9) and expand  −1 = ω(ki ) − ω(ki )2 f (ki ) + .... W (ki )−1 = ω(ki )−1 + f (ki )

(2.17)

If we take the first term in (2.17) in the [−], we get 0 because of the delta function in (1.6). So, the only linear terms in f correspond to taking the second term in (2.17) 3 3 W (ki ) by i=0 ωi−1 which implies (2.16). Using the inside the [−], and replacing i=0 symmetries that led to (2.13, 2.14), we see that the scalar product in H, ( f, L f ), defined in (2.3) (note the ω(k)2 there), is proportional to:   3 T4d

dki ωi−2 δ(ω0 + ω1 − ω2 − ω3 )δ(k0 + k1 − k2 − k3 )

i=0

 2 ω(k0 )2 f (k0 ) + ω(k1 )2 f (k1 ) − ω(k2 )2 f (k2 ) −ω(k3 )2 f (k3 ) ≥ 0.

(2.18)

Thus, the zero modes of L are of the form f = ω−2 g, where g are collisional invariants, i.e. necessarily of the form aω(k) + b (see part (b)) and therefore (2.7) gives the only zero modes in H and in B. We need to show that the zero modes are isolated from the rest of the spectrum of L. From (2.16), we get the decomposition L = M + K with 9π M(k) = 4ω02

  3 T3d

dki ωi−2 δ(ω0 + ω1 − ω2 − ω3 )δ(k0 + k1 − k2 − k3 )ω(k0 )2 (2.19)

i=1

(k = k0 ), and K is an integral operator  (K f )(k) = K (k, k ) f (k )ω(k )2 dk , Td

given by the last three terms in (2.16), i.e. K (k, k ) = −

9π (ω(k)ω(k ))−2 (I (1) (k, k ) + I (2) (k, k )), 4

(2.20)

where, with k = k0 , k = k2 or k = k3 , I (1) (k, k )  = 2 dk1 (ω(k1 )ω(k + k1 − k ))−2 δ(ω(k) + ω(k1 ) − ω(k ) − ω(k + k1 − k )), Td

(2.21) groups the last two terms in (2.16), and, with k = k0 ,

k

= k1 , k˜ = k2 ,

I (2) (k, k ) =  ˜ ˜ ˜ −2 δ(ω(k) + ω(k ) − ω(k) ˜ − ω(k + k − k)), ˜ − d k(ω( k)ω(k + k − k)) Td

is the remaining term.

(2.22)

Approach to Equilibrium for the Phonon Boltzmann Equation

185

We want to show that K defines a compact operator, both in B and in H, that the function M is strictly positive, and then use Weyl’s theorem on essential spectra to prove that the zero modes are isolated from the rest of the spectrum. To prove compactness, we need to see how singular the kernel of K is. The singularities of I (1) and I (2) are similar, and we will study I (1) first; let r = 2 (k − k ), 1

and change variables in (2.21) to k1 = q − r − ( π2 , . . . , π2 ), and write Recalling (2.1, 2.2), we get  (1)

I (k, k ) = dq δ( (q, r, k)) g(q, r )

π 2

for ( π2 , . . . , π2 ).

(2.23)

Td

with g = 2(ω(q − r − 21 π )ω(q + r − 21 π ))−2 , and

(q, r, k) = −ω(k − k + k1 ) + ω(k1 ) + ω(k) − ω(k ) =2

d
(sin(q j + r j ) − sin(q j − r j )) + ω(k) − ω(k − 2r ). (2.24) j=1

The summand equals 2 cos q j sin r j . We change variables to cos q j = s j (1 − y j ) with s j = 1 in the region q j ∈ [− π2 , π2 ], and s j = −1 in the region |q j | > π2 . In both cases, y j ∈ [0, 1]. Our integral becomes a sum of integrals of the form 
 −1 Is (k, k ) = δ(4 y j s j sin r j − m(k, r ))G(y, r ) (y j ) 2 h(y j )dy j , (2.25) j

[0,1]d

j

− 12

with h(y) = (2 − y)

, G smooth, bounded and strictly positive, and
s j sin r j . m(k, r ) = ω(k − 2r ) − ω(k) + 4

(2.26)

j

Let sin r j = 0 for all j, i.e. k j − k j = 0 modulo π and m(k, r ) = 0. Given k, the set of k such that these conditions hold is of full measure. Then, changing variables, y j → y j | sin r j |,  −1 Is (k, k ) = | sin r j | 2 Js (k, k ), (2.27) j

where Js (k, k ) =

j | sin r |

δ(4 0

j




y j s j − m(k, r ))G(yr , r )

j

with yr = y j /| sin r j | and s j = ±1.

 j

(y j )

− 21

j

h(yr )dy j ,

(2.28)

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Consider first d = 2. Note that G is bounded and strictly positive, and so is h(y), for y ∈ [0, 1]. So, integrating the delta function over the variable y 2 , we can bound from above and from below Js by the function 1 | sin r |

j (r, k, m) =

(sy + m ∈ [0, | sin r 2 |])dy , √ √ y sy + m

(2.29)

0

where (·) denotes the indicator function and where j is evaluated at m = s = −s 1 s 2 , s = s 2 , i.e. s, s = ±1. j is bounded by | j (r, k, m)| ≤ C(1 + | log |m||).

s m(k,r ) , 4

(2.30)

Since m(k, r ) is real analytic, it has at most polynomial singularities. Consider I (1) composed with itself:  (1) 2

(2.31) (I ) (k, k ) = dk I (1) (k, k )I (1) (k , k

). By Hölder’s inequality we can bound this by: ⎛ |(I (1) )2 (k, k

)| ≤ C ⎝



dk

 j

| sin r j |

− 2p



⎞1/ p | sin r j |

− 2p

⎠

,

(2.32)

j

with r as above and r = 21 (k − k

). This holds for any p > 1, because any power of p | log |m(k, r )| log |m(k , r )|| is integrable in k . Now j | sin r j |− 2 is in L q , as a function of r , for q < 2/ p (we choose p ∈ (1, 2)). The integral in (2.32) is a convolution of two functions in such an L q . Thus, as a function of k − k

, by Young’s inequality,

it is in L q , for q = q/(2 − q) which shows that (I (1) )2 (k, k

) is in L s (Td × Td ) for s = qp/(2 − q) < 2/(2 − 2/ p), for p arbitrarily close to 1, hence for s arbitrarily large. This means that the kernel (I (1) )2 defines a compact operator, and therefore so does I (1) , since I (1) (and I (2) ) are symmetric and thus define self-adjoint operators. We may repeat this analysis for I (2) , writing: I (2) (k, k + π )  ˜ ˜ − ω(k ) − ω(k˜ + k − k )), = − d k(ω( k˜ − π )ω(k˜ + k − k ))−2 δ(ω(k) + ω(k) Td

(2.33) where we changed variables k˜ → k˜ +(π, . . . , π ) in the integrand and used formula (2.1), (2.2), for ω(k) and cos(k j + π ) = − cos k j . Since the factor (ω(k˜ − π )ω(k˜ + k − k ))−2 is smooth, we can do exactly the same analysis as above and obtain the bound (2.30) for I (2) . So, K , given by (2.20), is compact in H. From (2.16), we see that L is self adjoint in H.

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To show compactness in B, we will use Hölder continuity properties of I (1) , I (2) . Consider first I (1) , i.e. Is (k, k ) given by (2.25). We want to bound  sup

dk

k,k˜

˜ k )| |Is (k, k ) − Is (k, , ˜α |k − k|

for some α > 0. We will analyze the dependence in k of each factor in (2.27), (2.28) separately. For u, v ∈ (0, 1] we have, for any n, |u

− 21

−v

− 12

1

1

| = |u 2n − v 2n ||

n


k

u − 2n v −

n+1−k 2n

1

n+1

n+1

| ≤ Cn |u − v| 2n max{u − 2n , v − 2n },

k=1

(2.34) 1

1

since u 2n ∈ C 2n ([0, 1]). We can use this, with n > 1, to show the Hölder continuity in −1 k of the factor j | sin r j | 2 in (2.27). For Js , let us, for simplicity, again study j. Let m˜ = m(k , r˜ ), with r˜ = 21 (k˜ − k ). 1 , Applying (2.34) to (2.29) we get, with α = 2n 1 | sin  r |

| 0

1 1 (sy + m ∈ [0, | sin r 2 |])( √ √ )dy −√  y sy + m y sy + m˜

˜ α (1 + max{|m|−α , |m| ˜ −α }) ≤ Cα |m − m| ˜ α (1 + max{|m|−α , |m| ≤ Cα |k − k| ˜ −α }),

(2.35)

where the last inequality holds because m is analytic in k.

−1 Since m is real analytic, its singularities are polynomial and j | sin r j | 2 |m(k, 21 (k− k ))|−α is integrable in k uniformly in k for α > 0 small enough (possibly smaller than 1 2n ). This fact and (2.35) prove the Hölder continuity of j, at least for its dependence on 1 k through √sy+m . Next, let | sin r˜ 1 | = | sin r 1 | + δ, assuming δ > 0. Then, using the fact that 1 | sin r |+δ

| sin r 1 |

δ (sy + m ∈ [0, | sin r 2 |])dy ≤ C min(1 + | log |m||, log(1 + )), √ √ y sy + m | sin r 1 |

we get, using | log(1 +

δ )| | sin r 1 |

1

1

≤ δ 2 if | sin r 1 | ≥ δ 2 , the following crude estimate: 1

1

| j (r 1 , r 2 , k, m) − j (˜r 1 , r 2 , k, m)| ≤ C(δ 2 + (| sin r 1 | ≤ δ 2 )(1 + | log |m||)). (2.36) A similar estimate holds for the dependence with respect to r 2 , and for the one with respect to m in (sy + m ∈ [0, | sin r 2 |]). 1 −1 Moreover, the integral of j | sin r j | 2 (| sin r 1 | ≤ δ 2 )(1 + | log |m(k, 21 (k − k )|) over k 1 is of order δ α , for some α small; this, combined with (2.36) proves the Hölder

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J. Bricmont, A. Kupiainen

continuity of j, for its dependence on k through | sin r 1 |, | sin r 2 |, and m in (sy + m ∈ [0, | sin r 2 |]). We get, by combining all these estimates, that for some α > 0 and all f ∈ B,  (2.37) Is (k, k ) f (k )dk α ≤ C f , where − α is the C α norm. We may repeat this analysis for I (2) , using (2.33), and obtain the bound (2.37) for I (2) . Then, the same bound holds for the kernel K , given by (2.20).This shows that K is a continuous map from B into the space C α of Hölder continuous functions. Since C α is compactly embedded in B (by Ascoli’s theorem), this proves that K is compact. Observe that, by symmetry, the integrals over k of the integrals in (2.21) and (2.22) are equal and, because of the factor 2 in (2.21) and the minus signs in (2.20) and (2.22), we get:  M(k) = − dk K (k, k )ω(k )2 , we also get M ∈ B (actually M is Hölder continuous) and that L is bounded (since K is compact). It remains to show that M > 0. Because of the symmetries mentioned above, it is enough to show that  dk I (1) (k, k )ω(k )2 > 0. Using the argument that led to (2.29), and integrating over r = 21 (k − k ) instead of k , we see that it is enough to show that  dr j (r, k, m) > 0, which in turn holds if we can satisfy the inequalities s m(k, r ) ≤ | sin r 2 |, 4 0 ≤ y ≤ | sin r1 |, 0 ≤ sy +

(2.38)

on a set of positive measure in the y, r variables, and for some choice of s, s = ±1. Consider first r1 = 0, y = 0. If we can show that the first inequalities in (2.38) hold in that situation with strict inequalities: 0<

s m(k, r ) < | sin r 2 |, 4

(2.39)

then, by continuity, (2.38) will hold on a set of positive measure in the y, r variables, with 0 ≤ y ≤ | sin r1 |. Since s = s 2 , we have, for r 1 = 0, using (2.26, 2.1, 2.2): s s m(k, r ) = (cos k 2 − cos(k 2 − 2r 2 )) + sin r 2 . 4 2

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If we take sin r 2 > 0 and s so that s (cos k 2 − cos(k 2 − 2r 2 )) < 0, the second inequality in (2.39) will be strict. To satisfy the first inequality, we must show that 1 | cos k 2 − cos(k 2 − 2r 2 )| < sin r 2 2 for some r 2 > 0. For k 2 = π/2 (mod π ), this holds for r 2 small enough, because then | sin k 2 | < 1. For k 2 = π/2, we need to show that | sin 2r 2 | < 2 sin r 2 , which again 3 holds for r 2 small enough and strictly positive, using sin x  x − x3! . So, the spectrum of the operator of multiplication by M is bounded away from zero, and, by Weyl’s theorem, adding a compact operator does not change the essential spectrum; hence the spectrum of L is bounded away from zero, apart from the two zero eigenvalues discussed above. Consider now d > 2. We start with the integration over the first two components of the vectors k, k etc., keeping the other components fixed; we proceed exactly as above and simply add the contributions of the other components to ω(k), ω(k ) . . . , to the function m in (2.26). This simply adds a constant to m and does not change the fact that m is real analytic, which is the only property of m that we used. Since we obtain the bounds (2.30, 2.37) after integrating over the first two components, for any fixed value of the other components, we obtain similar bounds after integrating over all the components. To prove strict positivity of M, it is enough to restrict the remaining components so that their contribution to m is small enough and that the arguments used for d = 2 still hold. (a) We proceed as in (d) with the delta functions and get  
 −1 3

|C(W )(k)| ≤ C W δ(4 y j s j sin r j−m(k, r )) (y j ) 2 h(y j )dy j dk j

[0,1]d

j

(2.40) which, using (2.27), (2.29), proves the claim.

 

Remark. If one takes ω(k) = ω0 (k), instead of ω0 (k)2 , see (2.1), (2.2), one expects to have similar singularities in d ≥ 2, but we cannot reduce the analysis to the one of the singularities of a simple integral, as we did here in the case of ω(k) = ω0 (k)2 . The identities (2.5) yield two conservation laws. To show this, define, for α = 1, 2, jα (x, t) = −

1 (ω−α , ∇ω W ), 2π

(2.41)

where the scalar product is in H, see (2.3); j1 is the thermal current and j2 can be called the phonon number current. Similarly, set Tα (x, t) = (ω−α , W ).

(2.42)

T1 is the temperature and T2 is related to the phonon chemical potential. Equations (1.4) and (2.5) give then the conservation laws T˙α = ∇ · jα .

(2.43)

Tα are the slow modes that will diffuse. The currents jα will be related to the gradients of Tα via the Fourier law, see (3.17) below.

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3. Results Let Wˆ ( p, k, t) denote the Fourier transform of W in the x variable. We shall look for solutions of (1.4) of the form: Wˆ ( p, k, t) ≡ W0 (k)δ( p) + w( p, k, t),

(3.1)

ω−1 .

where, we recall, W0 = In Theorems 1 and 2 below, we shall construct solutions with w( p, k, t) small, so that, due to the presence of the leading term W0 , W (x, k, t) will be everywhere strictly positive. Equation (1.4) becomes then, w˙ = −Dw + n(w),

(3.2)

where the linear operator is given by D=L+

i 2π

p · ∇ω(k),

(3.3)

and the nonlinear term is n(w) = C(W0 + w) + Lw = C(W0 + w) − DC(W0 )w. Written as an integral equation, (3.2) becomes w(t) = e

−t D

t w(0) +

dse−(t−s)D n(w(s)).

(3.4)

0

We need to decompose this in terms of the slow and the fast variables. Let P be the orthogonal projection in the Hilbert space H (defined by (2.3)) on E = span {ω−1 , ω−2 } and let Q be the one on the orthogonal complement of E. The identities (2.5) can then be written as: Pn = 0,

(3.5)

or n = Qn, since, by differentiation, (2.5) implies the same identities with C(W ) replaced by DC(W0 )w. Let Pw = T, Qw = v. Then, w(t) = T (t) + v(t), and Eq. (3.4) becomes T (t) = Pe

−t D

t w(0) +

ds Pe−(t−s)D Qn(w(s)),

(3.6)

ds Qe−(t−s)D Qn(w(s)).

(3.7)

0

v(t) = Qe−t D w(0) +

t 0

Define e( p, t) := (1 + (t + 1) p 2 )−n

(3.8)

with n > d/2 and let Et be the space of continuous functions f ( p, k) equipped with the norm f t ≡ sup e( p, t)−1 | f ( p, k)| = sup e( p, t)−1 f ( p, ·) , k, p

p

(3.9)

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191

and let E be the space of functions w(t) ∈ Et , with w E ≡ sup w(t) t .

(3.10)

t

Let κ : E → E be the linear operator κ = (2π )−2 Pω L −1 ω P,

(3.11)

∂ω . ∂k 1

κ is strictly positive by Proposition 2.1.d, since the set ω E forms a where ω = two dimensional subspace in E ⊥ (by symmetry: ω is odd in k, while the functions in E are even). Let T0 (t) = e−t p κ T (0) (| p| ≤ 1), 2

(3.12)

where T ( p, ·, 0) = Pw( p, ·, 0) ∈ E, and κ acts on E, and let v0 (t) =

−i −1 L p · ∇ωT0 (t), 2π

(3.13)

with v0 ∈ E ⊥ . Then, we have our main result: Theorem 1. There exists δ > 0 such that, for w(0) 0 ≤ δ, the Eq. (3.2) has a unique solution in E, w(t) = T (t) + v(t), satisfying, for t ≥ 1, −1

T (t) − T0 (t) t ≤ C(t)t 2 w(0) 0 , v(t) − v0 (t) t ≤ C(t)t −1 w(0) 0 ,

(3.14) (3.15)

where C(t) = C log(1 + t) for d = 2 and is constant for d > 2. Remark. The norm (3.9) and (3.14) imply that, in x space, T (x, t) is given by the d usual diffusive Gaussian term, which, for bounded x, decays like t − 2 , plus a correction d+1 bounded, uniformly in x, by O(t − 2 ). Equation (3.15) says that the fast mode v is slaved to the slow one T , i.e. it decays like an explicit term, the Fourier transform of d+2 v0 ( p, t), plus a correction uniformly bounded by O(C(t)t − 2 ). Equation (3.15) also implies the Fourier law for the leading terms of the solution. Write T in the basis
T ( p, k, t) = ω−β (k)T˜β ( p, t), (3.16) β=1,2

(since the basis is not orthogonal, T˜β does not coincide with the Fourier transform of Tβ in (2.42)). Then, in x-space, the currents (2.41) (where, by symmetry, only the v part of d+2 W contributes) become, up to terms O(C(t)t − 2 ), i.e. keeping only v0 ,
καβ ∇ T˜β , (3.17) jα = β

with the positive conductivity matrix καβ = (ω−α , κω−β ),

(3.18)

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with κ given by (3.11). To see this, observe that, by symmetry, (ω−α , (∂i ω)L −1 (∂ j ω)ω−β ) equals zero for i = j, and, for i = j, does not depend on i; so, one may choose i equal to 1, which shows that κ, given by (3.11), satisfies (3.18). Finally, we can also derive a nonlinear heat equation as the hydrodynamic scaling limit of the Boltzmann equation (1.4). We scale W˜ (x, k, t) = W (x, k,  2 t), where we define (with some abuse of notations) W˜ (x, k, t) as satisfying (1.4). Thus, we obtain, for W , the equation: W˙ (x, k, t) + (2π )−1 ∇ω(k) · ∇W (x, k, t) =  −2 C(W )(x, k, t).

(3.19)

We shall solve it with initial data W |t=0 = ω(k)−1 + w(x, k, 0) and w(x, k, 0) = T (x, k, 0) + v(x, k, 0),

(3.20)

with T (x, ·, 0) ∈ E, v0 (x, ·, 0) ∈ E ⊥ , ∀x. W˜ (x, k, 0) = W (x, k, 0) has spatial variations at scale  −1 . We have then Theorem 2. There exists δ > 0 such that, for all  ≤ 1 and T (0) 0 , v(0) 0 ≤ δ, Eq. (3.19) has a unique solution W = ω−1 + w  , with w  = T  + v  ∈ E. Moreover, T  (t) → T (t) and v  (t) → v(t) in Et , for all t > 0, as  → 0, where T and v are the unique solutions of DC(ω−1 + T )v = (2π )−1 ∇ω · ∇T, T˙ = −(2π )−1 P∇ω · ∇v.

(3.21) (3.22)

Remark. Since we have DC(ω−1 + T ) instead of DC(ω−1 ) = −L in (3.22), we get a nonlinear heat equation for T : T˙ = ∇ · (K(T )∇T )

(3.23)

K(T ) = −(2π )−2 Pω DC(ω−1 + T )−1 ω P,

(3.24)

with

(remembering that L is positive, we see that DC is negative and that K(T ) is a positive matrix). One can show that the long-time behavior of (3.23) is similar to the one in Theorem 1.

4. Proofs We start by proving some results on the spectrum of the operator D defined in (3.3). Recall that, by Proposition 2.1 d, (σ (L)\{0}) ≥ a.

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193

Proposition 4.1. There exist p0 > 0, and b > 0, such that, both in B and in H, the following holds: a) If | p| < p0 then σ (D) = {λ1 , λ2 } ∪ σ˜ with inf σ˜ > a/2 and λi = p 2 µi + O(| p|3 ).

(4.1)

where µi > 0 are the eigenvalues of the operator κ defined in (3.11). b) If | p| ≥ p0 then inf σ (D) ≥ b. Proof. a) Let L(β) = L + βρ, where ρ = (2π )−1 | p|−1 p · ∇ω(k) so that D = L(i| p|). ρ is a bounded operator in B and in H. Hence, for |β| < p0 small enough, the spectrum of L(β) consists of two eigenvalues close to the origin, with the rest of the spectrum having real part larger than a/2. Moreover, L(β) is self adjoint in H for β real and thus forms an analytic Kato family (see [8]). Therefore, there is a p0 such that, for |β| < p0 , the two eigenvalues of L(β), λ1 , λ2 , and the associated eigenfunctions ψi , are analytic in |β| < p0 . To compute their Taylor series, use the decomposition w = T + v and project the equation (L + βρ)w = λw onto QH and PH. Then, using PρT = 0 (since ρT is odd in k), we get a pair of equations: Lv + β Qρ(T + v) = λv, β Pρv = λT,

(4.2)

whereby, since (σ (L)\{0}) ≥ a, we get, for β small and λ = O(β), v = −β(L − λ + β Qρ)−1 QρT = −β L −1 QρT + O(β 2 )T,

(4.3)

where O(β 2 ) denotes a bound on the operator norm in B. Since QρT = ρT and T = P T , we get, inserting (4.3) in (4.2), −β 2 Pρ L −1 ρ P T + O(β 3 )T = λT. The 2×2 matrix Pρ L −1 ρ P is strictly positive, by Proposition 2.1 (d). The result follows by letting β = i| p|, and by observing that, since the functions in E are invariant under reflections and permutations of the coordinates of k, the matrix κ( p) = Pρ L −1 ρ P, where ρ depends on | p|−1 p, equals κ(e1 ), i.e. the one given in (3.11). b) Write D = M + i| p|ρ + K as a sum of a multiplication and a compact operator. Since σ (M + i| p|ρ) ≥ inf M > 0 we need only to show, by Weyl’s theorem on the essential spectrum, that all eigenvalues λ of D satisfy λ ≥ b, if | p| ≥ p0 . Let λ be an eigenvalue of D. Let us first show that, if λ < a2 , λ is real. We decompose H = He ⊕ Ho into even and odd subspaces. Then, since ρ is odd, and the operator L is parity-preserving, Lwe + i| p|ρwo = λwe , Lwo + i| p|ρwe = λwo .

(4.4)

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J. Bricmont, A. Kupiainen

Write λ = x + i y and suppose that x < a2 (where a = inf σ (L |Ho )). Then, since the zero eigenvectors are even, (L − λ) |Ho is invertible, and (4.4) becomes (L + p 2 ρ(L − λ)−1 ρ)we = λwe .

(4.5)

Let we H = 1. Then, taking the scalar product of (4.5) in H with we , and writing separately the real and imaginary parts, one gets: x = (we , (L + p 2 ρ(L − x)((L − x)2 + y 2 )−1 ρ)we ) a ≥ p 2 (we , ρ((L − x)2 + y 2 )−1 ρwe ), 2

(4.6)

since L ≥ 0 and, as operators in Ho (to which ρwe belongs, since ρ is odd), (L − x)((L − x)2 + y 2 )−1 ≥ a2 ((L − x)2 + y 2 )−1 . For the imaginary part, we get: y = yp 2 (we , ρ((L − x)2 + y 2 )−1 ρwe ).

(4.7)

If y = 0 (4.6) and (4.7) imply x ≥ a/2, against our assumption. Thus y = 0, i.e. all eigenvalues λ of D with λ < a2 are real, and, by (4.6), positive. Let us now define, motivated by (4.5), l(r, λ) = inf (we , (L + r 2 ρ(L − λ)−1 ρ)we ). we =1

Since, for λ ≤ a/2, (L − λ)−1 |Ho ≥ c > 0, and ρwe ∈ H0 , we get (we , (L + r 2 ρ(L − λ)−1 ρ)we ) ≥ (we , (L + cr 2 ρ 2 )we ). To bound this from below, observe that both terms in L + cr 2 ρ 2 are positive, that (we , Lwe ) ≥ c Qwe 2 , and that (Pwe , ρ 2 Pwe ) ≥ c1 Pwe 2 , for some c1 > 0, by inspection of the functions in PH and of ρ. So, if Qwe 2 ≥ c we 2 , we can use the first lower bound, while, if Qwe 2 ≤ c we 2 , i.e. Pwe 2 ≥ (1 − c ) we 2 , we can use the second one, if c is small enough (so that, if we expand we = Pwe + Qwe in (we , ρ 2 we ), the (Pwe , ρ 2 Pwe ) term dominates the sum), to get (L + cr 2 ρ 2 ) |He > c

r 2 for r < p0 , and p0 small enough. So, l( p, λ) ≥ c

p 2 , for | p| ≤ p0 , and λ ≤ a/2. Besides, l(r, λ) ≤ l(r , λ) if r < r . Since by (4.5) λ ≥ l(| p|, λ) we conclude λ ≥ c

p02 , for | p| ≥ p0 . Thus, we can take b = min( a2 , c

p02 ).   We are now ready to state the estimates for the heat kernels in Eqs. (3.6) and (3.7). Let R(z) = (z − D)−1 be the resolvent of the operator D and γ be a circle of radius a/4 around the origin. Then  dz ˜ P= R(z) 2πi , (4.8) γ

is, for | p| ≤ p0 , a projection onto the 2-dimensional eigenspace of D introduced in ˜ We have: Proposition 4.1 a. Let Q˜ = 1 − P.

Approach to Equilibrium for the Phonon Boltzmann Equation

195

Proposition 4.2. For all t ≥ 0 we have e−t D ≤ C(e−ct p + e−ct ), 2

(4.9)

where · is the operator norm in B. Moreover, there exists p0 > 0 so that for | p| ≤ p0 , Pe−t D Q + Qe−t D P ≤ C| p|(e−ct p + e−ct ), 2

Qe

−t D

e

−t D

2 −ct p 2

Q ≤ C( p e ˜ ≤ Ce−ct . Q

+e

−ct

),

(4.10) (4.11) (4.12)

Proof. Since D = L + 2πi p · ∇ω(k), we have, by the resolvent expansion:   i dz dz p · ∇ω(z − L)−1 2πi + O( p 2 ) P˜ = (z − L)−1 2πi + (z − L)−1 2π γ

γ

i i −1 L p · ∇ω P − P p · ∇ωL −1 Q + O( p 2 ) =P− 2π 2π = P + A P + P B Q + O( p 2 )

(4.13)

where, here and below, O( p 2 ) denotes a bound on the norm of operators in B, and we define A=−

i −1 L p · ∇ω, 2π

B=−

i p · ∇ωL −1 . 2π

(4.14)

(Note that the projectors in (4.13) and the oddness of ∇ω guarantee that L −1 acts on vectors on which it is well defined.) Hence, for | p| small, ˜ + Q˜ P + P˜ Q + Q P ˜ ≤ C| p|. P Q

(4.15)

˜ −t D P˜ + Qe ˜ −t D Q, ˜ e−t D = Pe

(4.16)

Now, write ˜ −t D P, ˜ ˜ and Q˜ project on invariant subspaces of D, we have e−t D P˜ = Pe (since P, −t D −t D ˜ ˜ and Q˜ = Qe Q), e  dz (4.17) e−t D Q˜ = e−t z R(z) 2πi , γ

where the curve γ goes around the part of the spectrum of D that lies in the complement of a ball of radius a/4, centered at the origin. Since L is bounded and | p| small, the length of γ is O(1), and we get (4.12). Then, the other claims follow, for | p| small, from (4.15), (4.16) and Proposition 4.1 a. To get (4.9) for | p| > p0 we note that by Proposition 4.1, e−t D is given by the right hand side of (4.17), where the curve γ can be chosen to be a rectangle in z > b, with vertical sides of length const | p|, and horizontal ones of length O(1) (since L is bounded). Hence, on the curve γ , we have |e−t z | ≤ e−bt , but the length of the contour of integration is not bounded as | p| → ∞. To control the integral, recall that D = M + i| p|ρ + K , and let R0 (z) = (z − M − i| p|ρ)−1 .

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J. Bricmont, A. Kupiainen

Then, by the resolvent formula, R = R0 + R0 K R0 + R0 K R0 K R.

(4.18)

The first term in (4.18), inserted in (4.17), gives the multiplication operator e−t (M+i| p|ρ) , which satisfies the bound (4.9) with c = inf M. For the second term, we use |e−t z | ≤ e−bt in the integral and write    1



, (4.19) | dk dz R0 (z, k)K (k, k )R0 (z, k )| ≤ dk |K (k, k )| dz |z − z 1 ||z − z 2 | γ

γ

where z 1 = (M + i| p|ρ)(k), z 2 = (M + i| p|ρ)(k ). Since |z − z i | ≥ c on γ , we have  1 dz ≤ C, (4.20) |z − z 1 ||z − z 2 | γ

 uniformly in k, k . Since, moreover, |K (k, k )|dk ≤ C, uniformly in k (which holds by (2.8)), we get the bound in (4.9) for the second term in (4.18). The third term in (4.18) is similar, since K R is bounded.   Proof of Theorem 1. Let us write Eq. (3.4) as w(t) = w (t) + N (w, t)

(4.21)

where w (t) is the solution of the linear problem: w ≡ e−t D w(0).

(4.22)

We will solve (4.21) in the space E. The linear term is bounded, using (4.9), and recalling the definition (3.8) of the weight function e, by w E ≤ C sup e( p, t)−1 e( p, 0)(e−ct p + e−ct ) w(0) 0 ≤ C w(0) 0 . 2

(4.23)

p,t

Consider then the nonlinear term in Eq. (3.4). By an easy extension of Proposition 2.1 a (see (2.40)), n(w(s)( p) ≤ C( w 2E e(s)∗2 ( p) + w 3E e(s)∗3 ( p)).

(4.24)

We need the simple estimate e(s)∗2 ( p) ≤ C(1 + s) which follows from ∗2

− d2

e( p, s),



e(s) ( p) ≤ 2

(4.25) 

dq e( p − q, s)e(q, s) ≤ Ce( p, s)

dq e(q, s),

(4.26)

| p−q|≥ 21 | p|

(the first inequality holds because either |q| ≥ 21 | p| or | p − q| ≥ 21 | p|). The last integral yields the power of 1 + s since n > 21 d in (3.8). The last term in (4.24) is smaller, for w E ≤ 1.

Approach to Equilibrium for the Phonon Boltzmann Equation

197

Thus, the second term in Eq. (3.6) and the second term in (3.7) are bounded by t R N (w, t, p) ≤ C w E 2

ds Re−(t−s)D Q (1 + s)

− d2

e( p, s),

(4.27)

0

where R equals P in (3.6) and Q in (3.7). Proposition 4.2 implies − 21 m − c (t−s) p 2 2

Re−(t−s)D Q ≤ C((1 + t − s)

e

c

with m = 1 if R = P and m = 2 if R = Q (using | p|m e− 2 p for t − s ≥ 1). Bounding c

e− 2 p

2 (t−s)

+ e−c(t−s) ), 2 (t−s)

(4.28)

≤ C(1 + t − s)

− 21 m

,

e( p, s) ≤ Ce( p, t)

and c

e− 2 (t−s) e( p, s) ≤ Ce( p, t), we may bound the remaining s-integral in (4.27), for m = 1, 2, i.e. for m ≤ d, by −1 m

C(t)(1 + t) 2 , where C(t) may be taken to be constant, except for d = 2, where C(t) = C log(1 + t). Hence we end up with R N (w, t) t ≤ C(t)(1 + t)

− 21 m

w 2E .

(4.29)

In particular, we have N (w) E ≤ C w 2E , so that N in (4.21) maps a ball of radius  in E into itself, for  small enough and is obviously a contraction there. The existence of a solution for Eq. (4.21) then follows from (4.23) and the Banach fixed point theorem and, since m = 1 if R = P, i.e. for (3.6), and m = 2 if R = Q, i.e. for (3.7), we obtain the bounds T (t) − T (t) t ≤ C(t)(1 + t)

− 21

v(t) − v (t) t ≤ C(t)(1 + t)

w(0) 20 ,

−1

w(0) 20 ,

(4.30) (4.31)

where we wrote w = Pw + Qw ≡ T + v .

(4.32)

To conclude the proof of Theorem 1 we need to relate T0 , v0 to T , v . For this we need to write the leading terms of w (t) more explicitly. Let us formulate this a little more generally, which will be useful also in the proof of Theorem 2. Let us denote K (t) = e−t p κ P, 2

(4.33)

where κ is given in Eq. (3.11). We have then the following lemma, proven at the end:

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Lemma 4.3. Let | p| ≤ p0 . Then the semigroup e−t D can be written with respect to the decomposition E ⊕ E ⊥ , as e−t D =



K (t) AK (t)

  2 K (t)B O(| p|e−ct p + p2 e−ct ) + 2 AK (t)B + R(t) O( p 2 e−ct p + | p|e−ct )

 2 O( p 2 e−ct p + | p|e−ct ) , 2 O(| p|3 e−ct p )

where O is with respect to the operator norm in B, ˜ −t D Q˜ Q, R(t) = Q Qe

(4.34)

and the operators A and B are defined in Eq. (4.14). Returning to the proof of Theorem 1 and recalling the definition (3.12) of T0 and (4.22), (4.32) of T , we get from Lemma 4.3, for | p| ≤ p0 , since B is of order | p|, see (4.14), T (t, p) − T0 (t, p) ≤ C| p|(e−ct p + e−ct )e( p, 0) w(0) 0 . 2

(4.35)

Since v0 , given in Eq. (3.13), equals v0 = AK (t)T (0) (see (4.14)), for | p| ≤ p0 ≤ 1, we have, from Lemma 4.3, v (t, p) − v0 (t, p) ≤ C( p 2 e−ct p + e−ct )e( p, 0) w(0) 0 , 2

(4.36)

where we used the bound, coming from (4.12), R(t) ≤ Ce−ct ,

(4.37)

and the fact that A and B are of order p. For | p| ≥ p0 , by (4.9), w0 (t, p) + w (t, p) ≤ Ce−ct e( p, 0) w(0) 0 . Using now the simple estimates e( p, t)−1 (e−ct p

2 /2

+ e−ct/2 ) ≤ Ce( p, 0)−1 ,

for t ≥ 1, e( p, t)−1 ≤ Ce( p, 0)−1 for t ≤ 1, and | p|m e−ct p

2 /2

+ e−ct/2 ≤ C(m)t

− 21 m

,

for t ≥ 1, m ≥ 0, we conclude that −1

T (t) − T0 (t) t ≤ C(1 + t) 2 w(0) 0 , v (t) − v0 (t) t ≤ C(1 + t)−1 w(0) 0 .

(4.38) (4.39)

The estimates (3.14) and (3.15) follow now from (4.30), (4.31), (4.38) and (4.39).

 

Approach to Equilibrium for the Phonon Boltzmann Equation

199

Proof of Theorem 2. The proof goes along the lines of the one of Theorem 1 and we will be brief. We expand 1 1 1 C(W ) = − 2 Lw + m(T, v) + n(T, v), 2    where m(T, v) is the term quadradic in T but linear in v, and n(T, v) collects the terms that are quadratic and cubic in v. Note that, since L T = 0, 12 Lw = 1 Lv and thus DC(ω−1 + T )v = −Lv + m(T, v).

(4.40)

We write (3.19) as w(t) = exp(−

t 1 D )w(0) + 2 

t ds exp(− 0

t −s D )(m(s) + n(s))ds, 2

where D = L + 

i p · ∇ω, 2π

(4.41)

i.e. it equals (3.3) evaluated at p. The decomposition w = T + v yields, as in (3.6), (3.7): T (t) = Pe−t D / P T (0) +  Pe−t D / Qv(0) t 1 2 + ds Pe−(t−s)D / Q(m(s) + n(s))  2

2

(4.42)

0

v(t) =

1 −t D / 2 2 Qe P T (0) + Qe−t D / Qv(0)  t 1 2 + 2 ds Qe−(t−s)D / Q(m(s) + n(s)). 

(4.43)

0

Consider first the case | p| ≤ p0 /. We use Lemma 4.3, with t replaced by t/ 2 and p by p. This leads to T = T ∗ + T1 , v = AT ∗ +

1 2

(4.44) t

˜ −(t−s)D / 2 Q˜ Qm(s) + v1 , ds Q Qe

(4.45)

0

where ∗

T =e

−t p 2 κ

t T (0) + 0

dse−(t−s) p κ P Bm(s), 2

(4.46)

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J. Bricmont, A. Kupiainen

and T1 =  Pe

−t D / 2

t Qv(0) +

ds Pe−(t−s)D / Qn(s) 2

0

t +

dsO(p 2 e−c(t−s) p + | p|e−c(t−s)/ )m(s), 2

2

0

v1 = Qe−t D / Qv(0) + 2

t +

1 

t

ds Qe−(t−s)D / Qn(s) 2

0

dsO(| p|3 e−c(t−s) p )m(s). 2

0

Then, following the proof of (4.38), (4.39), (4.30), (4.31), we get −1

T1 (t, p) ≤ C(1 + t) 2 e( p, t)( T (0) + v(0) ), v1 (t, p) ≤ C(t)(1 + t)−1 e( p, t)( T (0) + v(0) ),

(4.47) (4.48)

where C(t) = C max(t −1/2 , log(1 + t)) < ∞, for t > 0. We need to study the second term on the RHS of Eq. (4.45). Write m(s) = m(t) + (m(s) − m(t)) and consider the first term 1 2

t 0

˜ −(t−s)D / 2 Q˜ Qm(t) = 1 ds Q Qe 2

∞

˜ −s D / 2 Q˜ Qm(t) + O(e−ct/ 2 ) ds Q Qe

0

= Q Q˜ D−1 Q˜ Qm(t) + O(e−ct/ ) 2

= L −1 m(t) + O() + O(e−ct/ ), 2

(4.49)

where we used Q Q˜ D−1 Q˜ Q = Q L −1 Q + O() and where O(·) denotes a bound on the norm sup| p|≤ p0 / e( p, t)−1 · of the remainders, since sup| p|≤ p0 / e( p, t)−1 m(t) is bounded, uniformly in , because of the bounds on T , v, coming from (4.44)–(4.48). For the second term, we use Eqs. (4.42) and (4.43), to show that m(s) − m(t) t ≤ C|t − s|, for |t − s| small; whence 1 2

t

˜ −(t−s)D / 2 Q˜ Q(m(s) − m(t)) = O(). ds Q Qe

(4.50)

0

Altogether we obtain v(t) = AT ∗ (t) + L −1 m(t) + v2

(4.51)

sup e( p, t)−1 ( T1 + v2 ) → 0

(4.52)

and, for any t > 0, | p|≤ p0 /

Approach to Equilibrium for the Phonon Boltzmann Equation

201

as  → 0. For | p| > p0 / we have, from Proposition 4.2, e−t D / ≤ e−ct/ , 2

2

which implies that sup e( p, t)−1 ( T + v ) → 0,

(4.53)

| p|≥ p0 /

as  → 0. Hence, in the space Et , we have lim (T (t), v(t)) = (T ∗ (t), AT ∗ (t) + L −1 m(t)) ≡ (T ∗ (t), v ∗ (t)).

→0

(4.54)

Let us finally check that T ∗ and v ∗ satisfy Eqs. (3.21) and (3.22). From (4.40) and (4.54), we get AT ∗ = −L −1 DC(ω−1 + T ∗ )v ∗

(4.55)

which is (3.21) if we recall the definition (4.14) of A. From (4.46) and the definition (4.14) of B, we see that T ∗ satisfies i P p · ∇ωL −1 m. T˙ ∗ = − p 2 κ T ∗ − 2π

(4.56)

From (4.54) and the definition (4.14) of A, we have L −1 m = v ∗ − AT ∗ = v ∗ + i(2π )−1 L −1 p · ∇ωT ∗ . Substituting this into (4.56) and recalling the definition of κ in (3.11), which implies that − p 2 κ T ∗ + p 2 (2π )−2 P · ∇ωL −1 · ∇ωT ∗ = 0, we get Eq. (3.22).   Proof of Lemma 4.3. We need to calculate Re−t D R for R and R either P or Q. We use ˜ We need the following the representation (4.16) for e−t D and the formula (4.13) for P. expansions P P˜ = P + P B Q + O( p 2 ), P˜ Q = P B Q + O( p 2 ), ˜ = −P B Q + O( p 2 ), P Q˜ = P(1 − P) ˜ Q˜ Q = (1 − P)Q = Q − P B Q + O( p 2 ),

P˜ P = P + A P + O( p 2 ), Q P˜ = A P + O( p 2 ), Q˜ P = −Q A P + O( p 2 ), Q Q˜ = Q − A P + O( p 2 ),

where we used repeatedly P A P = 0 (A maps E into E ⊥ , because ∇ω is odd in k). Thus we get, using the identities on the left, ˜ −t D Q˜ Q + P Qe ˜ −t D Q˜ Q Pe−t D Q = P Pe = (P + P B Q)e−t D P B Q − P B Qe−t D (Q − P B Q) + O( p 2 e−ct p ) 2

= e−t p κ P B Q + O( p 2 e−ct p ) + O(| p|e−ct ), 2

2

(4.57)

where we use the bounds of Proposition 4.2, B = O(| p|), and λα = µα p 2 + O( p 3 ), with µα being the eigenvalues of κ, see (4.1). Similarly, we get ˜ −t D Q˜ Q + O(| p|3 e−ct p ) Qe−t D Q = Ae−t p κ P B Q + Q Qe 2

2

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and Pe−t D P = e−t p κ P + O(| p|e−ct p ) + O( p 2 e−ct ), 2

2

Qe−t D P = Ae−t p κ P + O( p 2 e−ct p ) + O(| p|e−ct ). 2

2

  Acknowledgement. We thank François Huveneers, Jani Lukkarinen and Herbert Spohn for useful discussions. J. B. thanks the Belgian Internuniversity Attraction Poles Program and A.K. thanks the Academy of Finland for financial support.

References 1. Bardos, C., Ukai, S.: The classical incompressible Navier-Stokes limit of the Boltzmann equation. Math. Models and Methods in Applied Sciences 1, 235–257 (1991) 2. Bonetto, F., Lebowitz, J.L., Rey-Bellet, L.: Fourier Law: A challenge to Theorists. In: Mathematical Physics 2000, London: Imp. Coll. Press, 2000, pp. 128–150 3. Bricmont, J., Kupiainen, A.: Towards a derivation of Fourier’s law for coupled anharmonic oscillators. Commun. Math. Phys. 274, 555–626 (2007) 4. Bricmont, J., Kupiainen, A.: Fourier’s law from closure equations. Phys. Rev. Lett. 98, 214301 (2007) 5. Esposito, R., Pulvirenti, M.: From particles to fluids. In: Handbook of Mathematical Fluid Dynamics, Vol. III, Friedlander, S., Serre, D. eds. Amsterdam: Elsevier Science, 2004 6. Golse, F., Saint-Raymond, L.: The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math. 155, 81–161 (2004) 7. Mouhot, C.: Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials. Commun. Math. Phys. 261, 629–672 (2006) 8. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Vol IV. Analysis of Operators, New York: Academic Press, 1978 9. Spohn, H.: The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics. J. Stat. Phys. 124, 1041–1104 (2006) 10. Spohn, H.: Collisional invariants for the phonon Boltzmann equation. J. Stat. Phys. 124, 1131–1135 (2006) 11. Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Handbook of Mathematical Fluid Dynamics, Vol. I, Friedlander, S., Serre, D. eds. Amsterdam: Elsevier Science, 2002 Communicated by H. Spohn

Commun. Math. Phys. 281, 203–229 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0484-7

Communications in

Mathematical Physics

Limit Correlation Functions for Fixed Trace Random Matrix Ensembles Friedrich Götze1,
, Mikhail Gordin2,

1 Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501, Bielefeld, Germany.

E-mail: [email protected]

2 V.A. Steklov Mathematical Institute at St. Petersburg, 27 Fontanka emb., St. Petersburg,

191023, Russia. E-mail: [email protected] Received: 14 March 2007 / Accepted: 18 December 2007 Published online: 9 May 2008 – © Springer-Verlag 2008

Abstract: Universal limits for the eigenvalue correlation functions in the bulk of the spectrum are shown for a class of nondeterminantal random matrices known as the fixed trace or the Hilbert-Schmidt ensemble. These universal limits have been proved before for determinantal Hermitian matrix ensembles and for some special classes of the Wigner random matrices. 1. Introduction and the Statement of the Result
N Let H N be the set of all N × N (complex) Hermitian matrices, and let tr A = i=1 aii denote the trace of a square matrix A = (ai j )i,N j=1 . H N is a real Hilbert space of dimension N 2 with respect to the symmetric bilinear form (A, B)
→ tr AB. Let l N denote the unique Lebesgue measure on H N which satisfies the relation l N (Q) = 1 for every cube Q ⊂ H N with edges of length 1. A Gaussian probability measure on H N invariant with respect to all orthogonal linear transformations of H N is uniquely defined up to a scaling transformation. Such measures form a one-parameter family (µsN )s>0 , where the measure µsN is specified by its density   1 1 s 2 (1.1) tr A exp − g N (A) = √ 2 2s ( s2π ) N with respect to l N . Thus, for a random matrix X distributed according to µsN we have E µsN tr X 2 = s N 2 .

(1.2)


Research supported by Sonderforschungsbereich 701 “Spektrale Strukturen und Topologische Methoden in der Mathematik”.

Research supported by Sonderforschungsbereich 701 “Spektrale Strukturen und Topologische Methoden in der Mathematik,” and grants RFBR-05-01-00911, DFG-RFBR-04-01-04000, and NS-638.2008.1.

204

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The set H N endowed with the measure µsN is called the Gaussian Unitary Ensemble (GUE). Let X be a random N × N Hermitian matrix (that is a random variable taking values in H N ). We consider the eigenvalues λ1 , λ2 , . . . , λ N of the random matrix X as a finite sequence of exchangeable random variables. By definition, this means that their joint distribution PNX does not change under any permutation of these variables. Let for X denote the joint distribution of some n of these N variables. each n, 1 ≤ n ≤ N , Pn,N X Obviously, Pn,N is a permutation invariant probability measure in Rn . In particular, the X describes the distribution of a single eigenvalue. By definition, the n-point measure P1,N X of a random matrix X is a non-normalized measure defined correlation measure ρn,N by the relation X ρn,N =

N! PX . (N − n)! n,N

(1.3)

X (A) can be interpreted as the average For a measurable set A ⊂ Rn the quantity ρn,N X is absolutely number of n−tuples of eigenvalues in the set A. If the measure ρn,N n continuous with respect to the Lebesgue measure on R , its Radon-Nikodym derivative X is called the n-point correlation function of the random matrix X . In particular, Rn,N X has total mass N . For a measurable set E ⊂ R1 , the quantity ρ X (E) the measure ρ1,N 1,N expresses the expected number of the eigenvalues belonging to E. The corresponding density with respect to the Lebesgue measure in R1 , if it exists, is called the eigenvalue density or the density of states (caution: under the same names the normalized versions of the same measures are considered in the literature as well). Let X N be a random matrix XN XN GUE,s GUE,s with the distribution µsN . For n = 1, . . . , N we set Pn,N = Pn,N and ρn,N = ρn,N . A classical result for the GUE says that we have GUE,1/N

P1,N

→ W,

N →∞

(1.4)

where the measures converge in the weak sense, and W is the standard Wigner measure on [−2, 2] defined by the density  w(x) = (2π )−1 (4 − x 2 )+ , x ∈ R. (1.5) In terms of the correlation measures the same relation reads 1 GUE,1/N ρ → W. N →∞ N 1,N

(1.6)

For the n−point correlation measures we have a similar relation 1 GUE,1/N ρ → W × ··· × W, N →∞ N n n,N n times

(1.7)

which means that the eigenvalues become independent in the limit. However, for n ≥ 2, the study of a finer asymptotics near a point from the principal diagonal in the cube (−2, 2)n shows [10,11]: for every u ∈ (−2, 2) and t1 , . . . , tn ∈ R1 ,   1  GUE, N  sin π(ti − t j ) n 1 tn t1 u + = det R , . . . , u + . lim N w(u) N w(u) N →∞ (N w(u))n n,N π(ti − t j ) i. j=1

Limit Correlation Functions for Fixed Trace Random Matrix Ensembles

205

This limit relation presents a pattern for many other results, in particular, for that of the present paper. The right hand side of this relation represents an example of the correlation function of a so-called determinantal (or fermionic) random point process [15]. In general the n−point correlation function Rn of such a process is given by the formula  n R(u 1 , . . . , u n ) = det  K (u i , u j )i, j=1 , where K is the kernel of an integral operator on the line, which is trace-class having been restricted to any finite interval in R, and subject to some further conditions (see [15] for a detailed exposition). Moreover, in the asymptotic Hermitian random matrix theory (K N ) N ≥0 are the reproducing kernels of the subspaces of polynomials of degree ≤ N −1 with respect to some weight on the line. In this case we call the corresponding matrix ensemble determinantal. The GUE gives an example of such an ensemble. To a large extent the asymptotic study of determinantal ensembles reduces to that of the respective kernels [3,5]. In particular, the same local limit as in the case of GUE is established in [12,3] for two broad classes of determinantal matrix ensembles. It is expected that the sin-kernel limit (first discovered by F. Dyson) is rather common. This is known as the universality conjecture. Outside the class of determinantal Hermitian random matrices only very few results on the asymptotics of the correlation functions are known (see, for instance, [10], where a mixture of determinantal measures is considered). In the present paper we investigate the following non-determinantal ensemble of Hermitian random matrices. Let SrN = {A ∈ H N : tr A2 = r 2 }

√

(1.8)

be√the sphere in H N of the radius r > 0 centered at the origin. Set r = s N . The sphere sN S N carries a unique probability measure ν Ns invariant with respect to all orthogonal linear transformations in the space H N . We call this measure the fixed Hilbert-Schmidt norm ensemble (or just HSE) to reserve the term “the fixed trace ensemble” for more general ones (see [1]). Let Y N be a random matrix distributed according to ν Ns . We set YN YN HSE,s HSE,s for n = 1, . . . , N Pn,N = Pn,N and ρn,N = ρn,N . It is a well known result (see [11,14]) that HSE,1/N

P1,N

→ W,

(1.9)

N →∞

HSE,1/N

of like in the case of GUE. In this paper we prove that the correlation functions Rn,N arbitrary order n (1 ≤ n ≤ N ) near every point u ∈ (−2, 2), have the same determinantal limit with the kernel sin π(t1 −t2 )/π(t1 −t2 ) as the GUE correlation functions (for n = 1 the limit equals 1). More precisely, we establish in this paper the following result. ν,1/N

Theorem 1.1. Let Rn,N

be the n−point correlation function of the eigenvalues for a √

random matrix uniformly distributed on the sphere S N N . Then for every u ∈ (−2, 2) and t1 , t2 , . . . , tn ∈ R1 ,     sin π(ti − t j ) n 1 ν,1/N tn t1 u + N w(u) , . . . , u + N w(u) → det R (1.10) (N w(u))n n,N π(ti − t j ) i, j=1 as N → ∞. For every α ∈ (0, 1) and A > 0 the relation (3.1) holds uniformly in all u ∈ [−2 + α, 2 − α], and t1 ∈ [−A, A], . . . , tn ∈ [−A, A].

206

F. Götze, M. Gordin

The class of fixed trace matrix ensembles was studied in several recent publications. The authors are indebted to G. Akemann for drawing their attention, after the first version of the present work has appeared as a preprint, to the paper [2], he published jointly with G. Vernizzi. In this paper the local universality is studied for a class of ensembles containing the HSE. The authors provide heuristic arguments for the universality near zero based on the complex inversion of the Laplace transform. Our proof is based on the complex Laplace inversion as well (more precisely, we arrive at the Fourier transform after a sequence of translations of the origin and rescalings depending on N ). However, establishing bounds for orthogonal functions and related kernels in the complex domain allows us to rigorously conclude convergence near every point of the interval (−2, 2). In the following we will sketch the main steps of the proof. The guiding principle is that results for the Hilbert-Schmidt ensembles are deducible from the corresponding results for the GUE using the ’equivalence of ensembles’ or the ’concentration phenomenon’. In our setup a primitive form of the concentration is given by the law of large numbers for the squares of the Hilbert-Schmidt norms of the GUE random matrices. Supplemented by some estimates of the probabilities of large deviations, this is the main tool in [9], where the universality is shown near zero for the correlation measures (rather than the correlation functions) of the Hilbert-Schmidt ensemble; convergence was shown there in the weak topology determined by the continuous compactly supported functions. Such simple arguments seem to be insufficient for proving local results for correlation functions of eigenvalues (maybe, with exception for eigenvalues near zero), particularly, for stronger topologies. This agrees with M.L.Mehta’s doubts ([11], Sect. 27.1, p.490) concerning the deducibility of the local results for correlation functions of the Hilbert-Schmidt ensembles from the corresponding results for GUE by using the equivalence of ensembles. Solving this open problem in the present paper, we use a local form of concentration given by the local central limit theorem for the densities of the squared Hilbert-Schmidt norms. First we represent the Hilbert-Schmidt measure as a conditional measure of the GUE, given the Hilbert-Schmidt norm of the GUE random matrix. Starting with the disintegration of the GUE according to the level sets of the Hilbert-Schmidt norm, we arrive at formula (2.9) which is the crucial ingredient of the proof. Here we have to extend the scaling parameter to the complex domain. The formal Fourier inversion applied to this formula gives an ‘heuristic proof’ of the result. For an outline of it see Sect. 2. To make this sketch rigorous we need asymptotic estimates in the complex domain for the kernels related to the Hermite functions. This is done in Sect. 3, based on the results in [3–5]. Unfortunately, some of the results we need are contained in these papers not explicitly enough and have to be extracted from the proofs rather than from the statements (see the proof of Lemma 3.2). The main convergence result we need is Lemma 3.2. All necessary bounds are summarized in Proposition 3.1. Based on these two facts, we complete the proof. Note that the analytic part of the present paper may be viewed as a form of the Tauberian theorem. 2. Disintegration, a Fourier Transform Formula and the Sketch of the Proof In this section we discuss a disintegration representation of the GUE in terms of the HSE, and derive a Fourier transform formula involving these matrix ensembles. We suppress in this section the “spectral” arguments of the correlation functions and related quantities assuming that these arguments vary inside the domain described in Sect. 1. For every r > 0 denote by SrN the sphere of radius r in H N centered at the origin. . Set Let for s > 0, X N be a random matrix in H N distributed according to µGUE,s N

Limit Correlation Functions for Fixed Trace Random Matrix Ensembles

207

 TN = tr(X 2N /s) and Y N = N X N / tr X 2N . The random variable TN can be represented as a sum of N 2 squares of independent standard Gaussian random variables, hence it follows the familiar χ N2 2 distribution. Moreover, TN and Y N are independent, and Y N is uniformly distributed on the sphere S NN in H N , that is, Y N is distributed according to ν N1 in the notation of the previous section. Then X N can be represented as YN  XN = sTN (2.1) N with TN and Y N as above. Let γ N 2 denote the probability density of TN . Then it follows from (2.1) that ∞ u/N 2 µsN = ν N γ N 2 (s −1 u)s −1 du. (2.2) 0

As a consequence of (2.2), the correlation functions of GUE and HSE for 1 ≤ n ≤ N −1 satisfy the relation ∞ HSE,u/N 2 GUE,s Rn,N = Rn,N γ N 2 (s −1 u)s −1 du. (2.3) 0

Note that E TN = N 2 , DTN2 = E(TN − E TN )2 = 2N 2 , and, for every m > 0,



γm (u) =

(2m/2 Γ (m/2))−1 x (m/2)−1 e−x/2 , if u ≥ 0, 0, if u < 0.

(2.4)

Set for s > 0 γm,s (·) = s −1 γm (s −1 ·), so that γm,1 = γm (note that the same set of densities with a different parametrization appears in Lemma 3.7 as ( f a, p )a, p>0 ). Observe now that the probability density of sTN is given by γ N 2 ,s (·). Then (2.3) can be rewritten as ∞ HSE,u/N 2 GUE,s = Rn,N γ N 2 ,s (u)du. (2.5) Rn,N 0

In particular, we have GUE,1/N

Rn,N

= 0

∞

HSE,u/N 2

Rn,N

γ N 2 ,1/N (u)du. HSE,1/N

Our goal is to investigate the limiting behavior of Rn,N

(2.6)

when n is fixed, N → ∞,

HSE,1/N Rn,N

and the “spectral” arguments of vary within an /N −neighborhood of a point from the main diagonal in the cube (−2, 2)n . However, we prefer to consider the function HSE,u/N 2

u
→ Rn,N

HSE,1/N

rather than its value Rn,N

at N . Moreover, we will perform the study HSE,u/N 2

γ N 2 ,1/N (u). of the latter function indirectly, first dealing with the product u
→ Rn,N √ After the change of variable (u − N )/ 2 = v in (2.6) we obtain the relation ∞ GUE,1/N Rn,N = q N 2 (v)dv, (2.7) −∞

208

F. Götze, M. Gordin

where

√ √ HSE,1/N +v 2/N 2 √ q N 2 (v) = Rn,N 2γ N 2 ,1/N (N + v 2). (2.8) √ √ Notice that v
→ 2 γ N 2 ,1/N (N + v 2) is the probability density of the centered and √ normalized random variable (TN 2 − N 2 )/ 2N 2 , and it tends to the standard normal √ density ϕ : v
→ (1/ √2π ) exp (−v 2 /2) as N → ∞. Thus, the limit behavior of the density γ N 2 ,1/N (N + · 2) is well understood, and we have to study q N 2 (·). For every fixed u, due to the relation γ N 2 ,s (·) = s −1 γ N 2 (s −1 ·), we can analytically extend γ N 2 ,s (u) to the domain s > 0. Moreover, in the formula (2.5) the integral in the right-hand side can be analytically continued in s in accordance with the continuation of γ N 2 ,s (u) GUE,s so that (2.5) mentioned above. This leads to the corresponding continuation of Rn,N holds for s from the right half-plane. Now we will evaluate the Fourier transform of the (nonprobabilistic) density q N 2 , keeping in mind that by (2.7) and (2.8) it is a nonnegative integrable function.

Lemma 2.1.



∞ −∞

√ GUE,1/((1−i p 2/N )N )

exp (i pv)q N 2 (v)dv = φ N 2 ( p)Rn,N

,

(2.9)

where

√ √ 2 φ N 2 ( p) = exp (−i pN / 2)(1 − i p 2/N )−(N /2) √ is the characteristic function of the random variable (TN 2 − N 2 )/ 2N 2 .

(2.10)

Proof. Denoting by Cm the normalizing constant in formula (2.4), we have ∞ exp (i pv)q N 2 (v)dv −∞ ∞ √ √ HSE,1/N +v 2/N 2 √ = 2γ N 2 ,1/N (N + v 2)dv exp (i pv)Rn,N −∞   ∞ i p(u − N ) HSE,u/N 2 Rn,N = exp γ N 2 ,1/N (u)du √ 2 −∞  ∞  √ HSE,u/N 2 = exp −i√pN exp (i pu 2)Rn,N γ N 2 ,1/N (u)du 2

= exp



−i√pN 2





−∞

HSE,u/N 2

Rn,N

C N 2 N (N u)

 = exp −i√pN 1 −

√ − N 2 +1 2 ip 2 N

∞



2

−∞

HSE,u/N 2 Rn,N CN2 N

=



Nu 1 −



−i√pN 2

 1−

 N2 2 −1 exp − N u 2



1−

√  ip 2 du N

×

√  N 2 −1  2 ip 2 exp − N2u N

√ − N 2 ∞ 2 ip 2 N −∞ √ GUE,1/((1−i p 2/N )N ) . φ N 2 ( p)Rn,N

= exp



−∞ ∞



HSE,u/N 2

Rn,N

 1−

√  ip 2 du N

γ N 2 ,1/((1−i p√2/N )N ) (u)du (2.11)

Limit Correlation Functions for Fixed Trace Random Matrix Ensembles

209

In the following we will outline our approach. Write ∞ √ 1 1 1 GUE,1/((1−i p 2/N )N ) q (0) = φ N 2 ( p) Rn,N dp. N2 n n (N w(u)) 2π −∞ (N w(u)) Passing to the limit in the integral on the right-hand side (uniformly with respect to the spectral variables), the right-hand side has the same limit as ∞ 1 GUE,1/N 1 R φ 2 ( p)dp (N w(u))n n,N 2π −∞ N or, in view of the local Central Limit Theorem (CLT) , as 1 GUE,1/N Rn,N . √ 2π (N w(u))n On the other hand, it follows from (2.8) that 1 1 HSE,1/N √ q N 2 (0) = Rn,N 2γ N 2 ,1/N (N ). n n (N w(u)) (N w(u)) √ √ Again, the local CLT implies that 2γ N 2 ,1/N (N ) → 1/ 2π . Therefore,

(2.12)

1 HSE,1/N R (N w(u))n n,N tends to the same limit as 1 GUE,1/N R , (N w(u))n n,N and the conclusion follows. In the next section these heuristic arguments will be made rigorous. 3. Proofs For every α ∈ R, through the rest of the paper, we will denote by (·)α the function (·)α : C \ (−∞, 0] → C : z
→ exp α log z,

(3.1) √

where log denotes the principal branch of the logarithm. We use · as a notation for √ (·)1/2 extended to 0 by 0 = 0. The same symbol (for example, C, L , M, R, α, . . . , δ with or without indices, N0 , and so on) may denote different constants in the bounds obtained in the rest of the paper (even in the same proof). However, we will indicate explicitly the dependence of such constants on parameters. First we are going to state some definitions and basic formulas related to the Hermite polynomials and Hermite functions. Then we will formulate some results on the asymptotics of the Hermite polynomials and Hermite functions in the complex plane. The asymptotic behavior of the Hermite polynomials at the scale we are interested in was first established in 1922 by Plancherel and Rotach. For a convenient form of these (and much more general) results we refer to the monograph [5] and the papers [3],[4] (in particular, Appendix B), and [6]. These results will be employed later in this section.

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Let s > 0, and let for every N , N ≥ 0, p˜ N (·, s) be a polynomial of degree N with a positive leading coefficient such that for ( p˜ N (·, s)) N ≥0 the relations p˜ M (x, s) p˜ N (x, s) exp (−x 2 /(2s)) d x = δ M,N , M, N = 0, 1, . . . , R

hold. The (monic) Hermite polynomials ( H˜ N (·, s)) N =0,1,... are defined by the expansion exp (γ x − γ 2 s/2) =

∞  N =0

γ , H˜ N (x, s) N! N

(3.2)

and for M, N = 0, 1, . . . satisfy the relations √ H˜ M (x, s) H˜ N (x, s) exp (−x 2 /(2s)) d x = δ M,N 2πs (M+N +1)/2 N ! R

so that p˜ N (·, s) =

1 (2π )1/4 s (2N +1)/4 (N !)1/2

H˜ N (·, s), N = 0, 1, . . . .

The polynomials p˜ N (·, s) satisfy the difference equations √ √ x p˜ N (x, s) = s 1/2 N + 1 p˜ N +1 (x, s) + s 1/2 N p˜ N −1 (x, s), N = 1, 2, . . . . (3.3) The Hermite functions (ϕ˜ N (·, s)) N =0,1,... defined by ϕ˜ N (x, s) = p˜ N (x, s) exp (−x 2 /(4s)), N = 0, 1, . . . , form an orthonormal sequence of functions in L 2 (R, λ), where λ is the Lebesgue measure in R. Let us introduce the standardized Hermite polynomials and Hermite functions (corresponding to the weight exp (−x 2 /2)) by the relations H N (·) = H˜ N (·, 1), p N (·) = p˜ N (·, 1), ϕ N (·) = ϕ˜ N (·, 1) so that H˜ N (·, s) = s N /2 H N (· s −1/2 ), p˜ N (·, s) = s −1/4 p N (· s −1/2 ), ϕ˜ N (·, s) = s −1/4 ϕ N (· s −1/2 ).

(3.4)

The above relations for the Hermite polynomials imply that the Hermite functions satisfy, for every k ≥ 1, the following system of differential equations: √ x ϕk (x) = − ϕk (x) + kϕk−1 (x), 2 √ x (3.5) ϕk−1 (x) = − kϕk (x) + ϕk−1 (x). 2 The reproducing kernel K˜ N (·, ·, s) of the orthogonal projection in L 2 (R, λ) onto the linear span of ϕ˜0 (·, s), . . . , ϕ˜ N −1 (·, s) is given by K˜ N (x, y, s) =

N −1  k=0

ϕ˜k (x, s)ϕ˜k (y, s).

(3.6)

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Note that the correlation functions of the GUE can be expressed in terms of the latter kernel as  n GUE,s Rn,N (x1 , . . . , xn ) = det K˜ N (xi , x j , s) (3.7) i, j=1

(see [11], (6.2.7)). Further, setting K N (x, y) = K˜ N (x, y, 1), we obtain K˜ N (x, y, s) = s −1/2 K N (xs −1/2 , ys −1/2 ).

(3.8)

The Christoffel - Darboux formula for the kernels K˜ N and K N reads ϕ˜ N (x, s)ϕ˜ N −1 (y, s) − ϕ˜ N −1 (x, s)ϕ˜ N (y, s) K˜ N (x, y, s) = x−y

(3.9)

and K N (x, y) =

ϕ N (x)ϕ N −1 (y) − ϕ N −1 (x)ϕ N (y) . x−y

(3.10)

Note that the kernels K˜ N , K N enjoy the property K˜ N (−z 1 , −z 2 , s) = K˜ N (z 1 , z 2 , s), K N (−z 1 , −z 2 ) = K N (z 1 , z 2 ).

(3.11)

As to K N , this is a consequence of (3.10) and the fact that the functions (ϕ N (·)) N ≥0 are even or odd depending on N being even or odd; for K˜ N it follows from (3.8). The following integral representation of the reproducing kernel is a version of formula (4.56) in [8]: K N (x, y) = N ∞ (ϕ N (x + τ )ϕ N −1 (y + τ ) + ϕ N −1 (x + τ )ϕ N (y + τ ))dτ 2 0

(3.12)

so that K˜ N (x, y, s) =          (3.13) N ∞ ϕ N √xs + τ ϕ N −1 √ys + τ + ϕ N −1 √xs + τ ϕ N √ys + τ dτ. 2s 0 The relations (3.4), (3.8), and our extension of the function (·)α to C \ (−∞, 0) allow us to continue H˜ N (x, ·), p˜ N (x, ·), ϕ˜ N (x, ·) and K˜ N (x, y, ·) to this domain analytically in the parameter s. The relations (3.2) - (3.11) remain valid under these continuations. So does (3.13) whenever the integrals there are well-defined. Denote by w the analytic continuation of the standard Wigner density (1.5) to the domain C \ ((−∞, 2] ∪ [2, ∞)) . Let us define for every α ∈ (0, 2) and β > 0 the sets Sα,β and Sα,β by the relations Sα,β = {z ∈ C : | z| < 2 − α, |u| < β}

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and Sα,β = {z ∈ C : | z| ≤ 2 − α, |u| ≤ β}. √ Set for H ∈ R d(H ) = 1 + i H and observe that |d(H )| = (1 + H 2 )1/4 ,

√ 1 + H2 + 1 , d(H ) = 2

√ 1 + H2 − 1 , d(H ) = sgn H 2

(3.14)

( d(H ))2 − (d(H ))2 = 1.

(3.17)

| (ud(H ))| ≥ |(ud(H ))|

(3.18)

(3.15)

(3.16)

and

It is clear from (3.17) that

for every u ∈ R. Note that for every b ≥ 0 the equation |d(H )| = b has a unique nonnegative solution Hb =



(1 + 2b2 )2 − 1.

(3.19)

Lemma 3.1. Let α, β, and u be real numbers such that α ∈ (0, 1), 0 < β < α, and u ∈ (−2 + 2α, 2 − 2α). Then the following assertions hold true: 1. For every b ∈ [−∞, ∞] the relations |H | < H|b| and |(d(H ))| < |b| are equivalent; the relations |H | ≤ H|b| and |(d(H ))| ≤ |b| are also equivalent (we set here H∞ = ∞, (d(∞)) = ∞, and (d(−∞)) = −∞). 2. For every real H we have ud(H ) ∈ Sα,β whenever |(ud(H ))| ≤ β; 3. If for a certain real H the relation |H | ≤ |Hβ/2 | (or, equivalently, | (d(H )) | ≤ β/2) holds, it follows that ud(H ) ∈ Sα,β . Proof. It is clear from (3.19) and the definition of Hb that for b ∈ [0, ∞] and H ∈ [0, ∞] the functions b
→ Hb and H
→ |d(H )| are strictly increasing mutually inverse mappings. This shows equivalence of the conditions in point 1 of the lemma for the functions in question restricted to [0, ∞]. The same is true for b, H ∈ [−∞, ∞] since both functions are even. This proves assertion 1. For every H such that | (ud(H ))| ≤ β

(3.20)

| (ud(H ))| < 2 − α

(3.21)

we have

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√ since it follows from the relation (3.17) and the assumptions β ≤ α, u ∈ (−2+2α, 2 − 2α) that    2 2 | (ud(H ))| = u + (u (d(H ))) ≤ (2 − 2α)2 + β 2 < (2 − 2α)2 + α 2 < 2 − α. In view of the definition of Sα,β , the second assertion is proved. The third assertion is an immediate consequence of the second one since |u| ≤ 2.

 Corollary 3.1. Let α, β, and u satisfy the assumptions of Lemma 3.1, and assume that the relation ud(H ) ∈ / Sα,β holds for some real H. Then we have |(ud(H ))| > β. Moreover, the relation |(ud(H ))| > β holds for every real H such that |H | ≥ |H |. Proof. The corollary follows immediately from the second and the first assertions of Lemma 3.1 combined with the fact that, by (3.16), |d(H )| is a strictly increasing function of |H |.

 In the next lemma the main result on convergence is formulated. Lemma 3.2. For every number α ∈ (0, 1) there exists a positive number H = H (α) such that for every A > 0 the relation      t1 t2 1 −1 ˜ u+ d(H ), u + d(H ), N KN N w(u) N w(u) N w(u) (3.22) sin (π(t1 − t2 )d(H )w(ud(H ))/w(u)) −→ N →∞ π(t1 − t2 )d(H ) holds uniformly for all real numbers u, t1 , t2 , and H provided that u ∈ [−2+2α, 2−2α], |t1 | ≤ A, |t2 | ≤ A, and |H | ≤ H . Proof. Note that for every α ∈ (0, 1) there exists β = β (α) > 0 such that for every A > 0 the relation   z1 z2 1 1 sin π(z 1 − z 2 ) ,v + , → (3.23) K˜ N v + N w(v) N w(v) N w(v) N N →∞ π(z 1 − z 2 ) holds uniformly for all complex numbers v ∈ Sα,β (α) , z 1 , and z 2 such that |z 1 | ≤ A, |z 2 | ≤ A. This assertion (and more general ones involving some class of weights) is implicitly contained in the papers [3] and [4] (see also the monograph [5]). Actually, Lemma 6.1 in [3] establishes the desired result for real u, z 1 and z 2 . However, the same reasoning applies to the complex u, z 1 , and z 2 satisfying the assumptions just made provided that β is chosen to be sufficiently small in accordance with α. The key ingredient of the proof in these references, the boundedness property of a certain derivative, is established in [3] and [4] for some complex neighborhood of an arbitrary real point u ∈ [−2 + 2α, 2 − 2α] (see relation (4.122) in [3]) which allows to bound this derivative in a rectangular strip Sα,β (α) . Lowering β if necessary, we can substitute β (α) by some β(α) ≤ α/4. Note that the function w(·) has no zeroes in C \ ((−∞, 2] ∪ [2, ∞)) ⊃ Sα,β(α) . Picking a point v ∈ Sα,β(α) and setting in (3.23) z i = z i w(v)/w(v ) (i = 1, 2), we obtain     sin π(z 1 − z 2 )w(v)/w(v ) z 1 z 2 1 1 ˜ ,v + , → . KN v + N w(v ) N w(v ) N w(v ) N N →∞ π(z 1 − z 2 ) (3.24)

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Note that for v ∈ S α,β(α) we have C −1 ≤ |w(v)| ≤ C with some C > 1. This fact and (3.23) imply that for every A > 0 (3.24) holds uniformly in v, v ∈ Sα,β(α) and |z 1 | ≤ A, |z 2 | ≤ A. According to assertion 3 of Lemma 3.1, for every u ∈ [−2 + 2α, 2 − 2α] and every H ∈ [−Hβ(α)/2 , Hβ(α)/2 ] we have ud(H ) ∈ Sα,β(α) . Moreover, it is clear from −1

< |d(H )| < C with some (3.14) that for H ∈ [−Hβ(α)/2 , Hβ(α)/2 ] we have C positive constant C > 1 depending on α. Then, setting v = u ∈ [−2 + 2α, 2 − 2α], v = ud(H ) ∈ Sα,β(α) , z i = ti d(H ) (i = 1, 2), we obtain (3.22).

 Remark 3.1. It is sometimes convenient to use the following reformulation of (3.23) and the related assumptions: For every number α ∈ (0, 1) there exists a real number β = β(α) ≤ α/4 such that for every A > 0 the relation     1 sin (N π(v1 − v2 )w(v))   K˜ N v1 , v2 , N −1 − 0 N  N−→ →∞ N π(v1 − v2 ) holds uniformly for all complex numbers v, v1 , v2 satisfying the constraints v ∈ Sα,β(α) , |v1 − v| ≤ A/N , |v2 − v| ≤ A/N .

 We will now derive upper bounds for the modulus of the kernel  1 ˜  K N ·, ·, N −1 N valid for various values of its arguments. We will use asymptotics of two types for the Hermite functions: one of them is valid in a narrow strip around the interval [−2 + α, 2 − α]; another one applies in the complement to some neighborhood of [−2, +2]. With this asymptotics, after obtaining certain intermediate bounds for the quantity  −1 −1 ˜ |N K N u + t1 /N )d(H ), u + t2 /N )d(H ), N | , we will finally determine for this quantity an upper estimate which does not depend on H ∈ R. This bound and analogous bounds for the correlation functions will be derived in Proposition 3.1. In the next two lemmas we present upper estimates for the Hermite functions. These estimates are immediate consequences of the Plancherel-Rotakh-type asymptotics. First we consider a bound valid in the strip S α,β(α) , where β(α) was defined in the proof of Lemma 3.2. Lemma 3.3. For any α ∈ (0, 1) there exist a constant C(α) > 0 such that we have   √    (3.25) ϕ N z N  ≤ (C(α)) N and

  √    ϕ N −1 z N  ≤ (C(α)) N −1

(3.26)

for every N ∈ N and z ∈ S α,β(α) . Proof. First we recall a known result about the Plancherel-Rotach asymptotics for the Hermite functions in a rectangular strip (Theorem 2.2, part (ii), in [4]; the particular case of the Hermite functions for the measure exp (−x 2 )d x is considered there in Appendix

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215

B). This strip in the complex plane contains, up to some neighborhoods of the ends, the interval where the zeroes of the Hermite functions asymptotically concentrate. Set ψ : C \ ((−∞, −1] ∪ [1, ∞)) → C : z
→

1 (1 − z)1/2 (1 + z)1/2 . 2π

The function arcsin is defined as the inverse function of  π → C \ ((−∞, −1] ∪ [1, ∞)). sin : z ∈ C : | (z)| < 2 In our notation, the statement in [4] reads: for every α ∈ (0, 1) uniformly in z ∈ {z ∈ C : | (z )| ≤ 1 − α, |(z )| ≤ α} we have √ ϕ˜ N ( 2N z, 1/2)

2 = √ (1 − z)−1/4 (1 + z)−1/4 π 2N      z 1 1 1+O × cos N π ψ(y)dy + arcsin z 2 N 1     z 1 1 . (3.27) + sin N π ψ(y)dy − arcsin z O 2 N 1 z The analytic functions z
→ 1 ψ(y)dy and arcsin are bounded on the set {z ∈ C : | (z )| ≤ 1 − α, |(z )| ≤ α}. Representing cos and sin as the sum and the difference of the exponentials, it follows from (3.27) (omitting the multiplier N −1/4 ) that the modulus of the left-hand side of (3.27) is bounded above by the N th power of a positive number depending on α. For every α ∈ (0, 2), due to the relations   z√  √ z √ −1/4 ϕ N (z N ) = 2 ϕ˜ N √ N , 1/2 = 2−1/4 ϕ˜ N 2N , 1/2 , 2 2 √ a similar bound by the N th power of some number depending on α holds for ϕ N (z N ) uniformly for z ∈ S α,α . For     √ ϕ N −1 (z N ) = ϕ N −1 z (N − 1) N /(N − 1) , √ we also have a bound of such type in S α,α/4 ⊂ S α/2,α/2 . Indeed, set ρ N = N /(N − 1), and let N0 = N0 (α) be defined by the equation ρ N0 = (2 − α/2)/(2 − α). Then we have ρ N ≤ 2 and ρ N S α,α/4 ⊂ S α/2,α/2 whenever N ∈ N satisfies N ≥ N0 . This means that √ for z ∈ S α,α/4 both z ∈ S α/2,α/2 and z N /(N − 1) ∈ S α/2,α/2 hold for every N ≥ N0 . Therefore, the conclusion of the lemma is valid for such values of N . Choosing a larger constant C(α) if necessary and recalling that β(α) ≤ α/4, the proof is completed.

 We will obtain now some estimates valid for the Hermite functions in the domain |z| ≥ δ > 0. Lemma 3.4. For every δ > 0 there exist constants C(δ) and M(δ) such that the inequalities  √   (3.28) ϕ N (z N ) ≤ C(δ)N −1/4 M N (δ) |z| N

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and

  √    ϕ N −1 z N  ≤ C(δ)(N )−1/4 M N −1 (δ) |z| N −1

(3.29)

hold for every N ∈ N and every z satisfying | z| ≥ |z| ≥ δ. Proof. Again we start with stating some known result about the Plancherel-Rotach asymptotics for the Hermite polynomials in the complex plane (Theorem 7.185 in [5], see also [18] and the references therein).√In our √ notation, uniformly in z from every compact set contained in (C ∪ {∞}) \ [− 2, 2], we have √    1 U (z) exp (N (z − z 2 − 2)2 /4) −1 ˜ , (3.30) 1+O H N (z, (2N ) ) = √ 2 N 2 N (z − z − 2) where

 U (z) :=

z − 2 1/4 z+2 ) +( z+2 z−2

1/4 .

From this relation we obtain for the monic orthogonal Hermite polynomials (H N ) N ≥0 with respect to the weight exp (−x 2 /2) that √ √ H N (z N ) = (2N ) N /2 H˜ N (z/ 2, (2N )−1 ) √    1 exp (N (z − z 2 − 4)2 /8) 1+O (3.31) = 2 N −1 N N /2 U (z) √ 2 N N (z − z − 4) uniformly for z from every compact set contained in (C ∪ {∞}) \ [−2, 2]. Passing to the normalized polynomials we see that √ p N (z N ) =

1 (2π )1/4 (N !)1/2

√ H N (z N )

√    1 2 N N N /2 U (z) exp (N (z − z 2 − 4)2 /8) 1 + O , (3.32) = √ 1/4 1/2 2(2π ) (N !) N (z − z 2 − 4) N √ 1 and, in view of the Stirling formula N ! = 2π N N + 2 e−N (1 + O(1/N )), we obtain √    √ U (z) exp (N ( 21 + log 2 + (z − z 2 − 4)2 /8)) 1 1+O p N (z N ) = √ 1/2 1/4 2 N N 2(2π ) (N ) (z − z − 4) (3.33) so that we have

√    U (z) exp ( N2 (1 + ((z − z 2 − 4)/2)2 )) 1 p N (z N ) =  N 1 + O √ N 2(2π )1/2 (N )1/4 (z − z 2 − 4)/2 √

(3.34)

uniformly in z from any compact subset of (C ∪ {∞}) \ [−2, 2]. For every complex number z ∈ C \ [−2, 2], let x = x(z) be the root of the equation x + x −1 = z satisfying |x| < 1. Then we have    x = x(z) = z − z 2 − 4 /2,

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√ where z
→ κ(z) = z 2 − 4 is, by definition, a univalent analytic function in C\[−2, 2] satisfying the relations κ 2 (z) = z 2 − 4 and κ(t) > 0 for t > 2. For every r ∈ (0, 1) the equation |x(z)| = r and the inequality |x(z)| ≤ r define the ellipse ( z)2 /(r −1 + r )2 + (z)2 /(r −1 − r )2 = 1

(3.35)

with focal points −2, 2 and the closed exterior domain Er of this ellipse, respectively. Thus for z ∈ Er we have       |x(z)| =  z − z 2 − 4 /2 ≤ r < 1, and  2  (z − z 2 − 4)/2 ≤ r 2 < 1.

(3.36)

Note that for every r ∈ (0, 1), min |z| = r −1 − r,

|x(z)|=r

which for z ∈ Er implies 2 1   = = |z − x(z)| ≤ |z| + 1 ≤ (1 + |z|−1 )|z| √   |x(z)| 2 − z − 4 z    1 |z|. ≤ 1 + −1 r −r

(3.37)

Further, for z ∈ Er (0 < r < 1) we can write      |z − 2| = (x(z) − 1) + x −1 (z) − 1  = 2|x(z) − 1||x −1 (z) − 1| = 2|x(z)|−1 |1 − x(z)|2 ≥ 2r −1 |1 − x(z)|2 ≥ 2r −1 |1 − r |2 = 2(r −1/2 − r 1/2 )2 , and analogously |z + 2| ≥ 2(r −1/2 − r 1/2 )2 . This implies that the function    z − 2 1/4  z + 2 1/4    |U (z)| =  +    z+2 z−2 is bounded above on Er by a constant depending on r ∈ (0, 1). Applying this estimate along with (3.36) and (3.37) to the relation (3.34) we arrive, for a fixed r ∈ (0, 1) and every z ∈ Er , at  √   (3.38)  p N (z N ) ≤ C1 (r )N −1/4 L N (r )|z| N .

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√ Now we set z n = z N /(N − 1) and note that, by convexity of C \ Er , z N ∈ Er if z ∈ Er . Hence, applying (3.38) to p N −1 , we see that for every z ∈ Er ,   √    √       p N −1 z N  =  p N −1 z N N − 1  ≤ C1 (r )(N − 1)−1/4 L N −1 (r )|z N | N −1 = C1 (r ) (1 + 1/(N − 1))(2N −3)/4 N −1/4 L N −1 (r )|z| N −1 . Since (1 + 1/(N − 1))(2N −3)/4 → e1/2 as N → ∞, we conclude that   √     p N −1 z N  ≤ C2 (r )(N )−1/4 L N −1 (r )|z| N −1 .

(3.39)

Observe that for every r ∈ (0, 1) the inequality z ≥ r −1 − r implies, by Eq. (3.35), that z ∈ Er . Finally set

δ δ2 r =− + +1 2 4 so that r is the solution of the equation r −1 − r = δ satisfying 0 < r < 1. We can now conclude from (3.38) and (3.39) that for every δ > 0 with some constants C(δ) and M(δ) the inequalities  √   (3.40)  p N (z N ) ≤ C(δ)N −1/4 M N (δ)|z| N and

  √     p N −1 z N  ≤ C(δ)(N )−1/4 M N −1 (δ)|z| N −1

(3.41)

hold for every natural N and every z with |z| ≥ δ. We pass now from Hermite polynomials to Hermite functions. Note that for every z satisfying | z| ≥ |z| we have   | exp (−z 2 /4)| ≤ exp (− 2 z + 2 z)/4 ≤ 1 (3.42) and, therefore, |ϕk (z)| ≤ | pk (z)| for every k ∈ N. Applying (3.40) and (3.41), the lemma follows.

(3.43) 

Corollary 3.2. For every real number δ > 0 and A > 0 there exist such constants C(δ, A) and M(δ) that the inequalities   √   (3.44) ϕ N (u + t/N ) d(H ) N  ≤ C(δ, A)M N (δ)|H | N and

  √   ϕ N −1 (u + t/N ) d(H ) N  ≤ C(δ, A)M N −1 (δ)|H | N −1

(3.45)

hold for every number u ∈ [−2, 2], t ∈ [−A, A], N ∈ N and H ∈ R such that | ((u + t/N )d(H ))| ≥ δ.

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Proof. Set v = (u + t/N ) d(H ). Since   √ |(d(H ))| = 1 + H 2 − 1 /2 ≤ |H |/2, we have that |(v)| ≤ (2 + (A/N ))|(d(H ))| ≤ (1 + (A/2N )) |H | and |(v)| N ≤ (1 + (A/2N )) N |H | N ≤ C(A)|H | N , since (1 + (A/2N )) N → exp (A/2). N →∞

Thus, we proved that |(v)| N ≤ C(A)|H | N .

(3.46)

In view of (3.18), | v| ≥ |v|, and we can apply Lemma 3.4. Omitting N −1/4 in the bound, the proof is completed.

 Now we will derive upper bounds for |N −1 K˜ N (·, ·, N −1 )| valid for various combinations of the values of its arguments. Lemma 3.5. For every real α ∈ (0, 1) and A > 0 there exists a constant R1 (α, A) > 0 such that for every real u ∈ [−2 + α, 2 − α], t1 , t2 ∈ [−A, A], H, and every N ∈ N we have   |N −1 K˜ N (u + t1 /N )d(H ), (u + t2 /N )d(H ), N −1 | ≤ (R1 (α, A))2N (3.47) whenever |(u + ti /N )d(H )| ≤ β(α/4)

(3.48)

holds for i = 1, 2. Proof. Set u i = u + ti /N , vi = (u + ti /N )d(H ) for i = 1, 2 and u = (u 1 + u 2 )/2, v = (v1 + v2 )/2. Let the number N0 = N0 (α, A) be large enough so that the inequality A/N ≤ α/2

(3.49)

holds for every N ≥ N0 . Observe that by (3.49) u i ∈ [−2 + α/2, 2 − α/2], i = 1, 2, for N ≥ N0 . Then it follows from (3.48) and Corollary 3.1 that v1 , v2 ∈ Sα/4,β(α/4) . Therefore, v ∈ Sα/4,β(α/4) , too.

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First assume now that we have |v1 − v2 | ≤ 1/N .

(3.50)

Then Remark 3.1 (with A = 1) applies, and it is clear that, for v1 , v2 , v = (v1 + v2 )/2 ∈ Sα/4,β(α/4) with v1 , v2 satisfying (3.50), the function | sin (N π(v1 − v2 )w(v)) / (N π(v1 − v2 )) | is bounded above by a constant depending on α and A. Therefore, by Remark 3.1, this applies to the left hand side of (3.47) as well. Now, instead of (3.50), we assume that the inequality |v1 − v2 | ≥ 1/N holds. Then, by the Christoffel-Darboux formula (3.9), we have   √  √ |N −1 K˜ N v1 , v2 , N −1 | = N 1/2 |K N N v1 , N v2 |  √  √ √ √ √ ≤ N |ϕ N ( N v1 )||ϕ N −1 ( N v2 )| + |ϕ N ( N v2 )||ϕ N −1 ( N v1 )| . Applying Lemma 3.3 and enlarging, if necessary, the constant in the bound, the lemma is proved.

 Lemma 3.6. For every α ∈ (0, 1) and A > 0 there exists a constant R2 (α, A) such that for every u ∈ [−2 + α, 2 − α], t1 , t2 ∈ [−A, A], H ∈ R and N ∈ N we have the relation   |N −1 K˜ N (u + t1 /N )d(H ), (u + t2 /N )d(H ), N −1 | ≤ Γ (N ) (R2 (α, A)) N H N (3.51) whenever none of the following pairs of inequalities | ((u + ti /N )d(H ))| ≤ β(α/4), i = 1, 2, | ((u + ti /N )d(H ))| ≥ β(α/4)/2, i = 1, 2,

(3.52)

holds. Proof. Set u i = (u + ti /N ), vi = u i d(H ), i = 1, 2. Since (3.52) does not hold we have |(vi )| < β(α/4)/2 and |(v j )| > β(α/4) for some choice of i, j ∈ {1, 2}, i = j. This implies |v2 − v1 | ≥ β(α/4)/2, and by the Christoffel-Darboux formula (3.9) we have 1 N

   √  √  √ ˜    N v1 , N v2   K N v1 , v2 , N −1  = N K N √  √ √ √ √ 2 N  ≤ |ϕ N ( N v1 )||ϕ N −1 ( N v2 )| + |ϕ N ( N v2 )||ϕ N −1 ( N v1 )| . β(α/4)

Assume now that, for example, |(v1 )| < β(α/4)/2 and |(v2 )| > β(α/4). In addition we have that u 1 ∈ [−2 + α/2, 2 − α/2] for N ≥ N (α, A). Then, by assertion 2 of Lemma 3.1, v1 ∈ S α/4,β(α/4)/2 ⊆ S α/4,β(α/4) , and we may apply Lemma 3.3 (with the √ √ parameter value α/4 instead of α) to bound ϕ N ( N v1 ) and ϕ N −1 ( N v1 ). Furthermore,

Limit Correlation Functions for Fixed Trace Random Matrix Ensembles

221

√ √ we apply Corollary 3.2 with δ = β(α/4) to bound ϕ N ( N v2 ) and ϕ N −1 ( N v2 ). Therefore, we arrive at the inequality 1 ˜ | K N ((u + t1 /N )d(H ), (u + t2 /N )d(H ), 1/N ) | N √ ≤ C(α, A) N Γ (N )M N (β(α/4))H N . Replacing the quantity M(β(α/4)) by an appropriate larger constant, we reduce this inequality to the form (3.51). Moreover, this way the bound can be made valid for all N ∈ N, which proves the lemma.

 Lemma 3.7. For every δ > 0 and certain constants C(δ), M(δ) the inequality 1 ˜  | K N z1, z2 , N

1 N



| ≤ C(δ)Γ (N )R 2N (δ) ((z 1 )) N ((z 2 )) N

(3.53)

holds for every complex number z 1 , z 2 such that |z i | ≥ δ and | z i | ≥ |z i | (i = 1, 2), and every natural number N ≥ δ −2 . Proof. First we consider the case when z 1 z 2 > 0. Let f a, p (x) =

1 a p x p−1 e−ax , p > 0, x > 0, Γ ( p)

be the gamma density [7] with the parameters a > 0, p > 0, and let P ≥ 0 and N ≥ 1 be some integers. Then for every complex number z satisfying z ≥ |z| > 0 we have ∞ ∞ N 2 2 |z + θ |2P e−N (z+θ) /2 dθ = |ξ + iz|2P e− 2 (ξ +iz) dξ 0 z ∞ ∞ N N 2 2 2 2P − 2 (ξ +iz) ≤ |ξ + iz| e dξ = (ξ 2 + (z)2 ) P e− 2 (ξ −(z) ) dξ |z|

=e

N (z)2



∞

|z|

2 P e N (z) N P+1 |z|

2

≤

|z|

(ξ 2 + (z)2 ) P e



∞ |z|



− N2

(ξ 2 +(z)2 )

dξ

  P N 2     2 N ξ 2 + (z)2 /2 e− 2 (ξ +(z) ) d N ξ 2 + (z)2 /2

∞ 2 2 P e N (z) Γ (P + 1) = P+1 η e dη = P+1 f 1,P (η)dη N |z| N (z)2 N |z| N (z)2   2 N (z)2 2 P e N (z) [N (z)2 ] P −N (z)2 1+ = P+1 Γ (P + 1)e + ··· + N |z| 1! P!   P 2 2 P N (z) [N (z) ] 2 Γ (P + 1) 1+ + ··· + , = P+1 N |z| 1! P! 2P

2 e N (z)



∞

P −η

(3.54) where we used a well known formula ([7], p. 11) while integrating f 1,P . If |z| ≥ δ, then N (z)2 ≥ 1 for N ≥ δ −2 , and the bound (3.54) yields ∞ 2 |z + θ |2P e−N (z+θ) /2 dθ ≤ e2 P N −1 Γ (P + 1)|(z)|2P−1 (3.55) 0

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whenever z ≥ |z| > 0. In particular, for P = N and P = N − 1 we have ∞ 2 |z + θ |2N e−N (z+θ) /2 dθ ≤ e2 N N −1 Γ (N + 1)|(z)|2N −1

(3.56)

0

and

∞

|z + θ |2(N −1) e−N (z+θ)

2 /2

dθ ≤ e2 N −1 N −1 Γ (N )|(z)|2N −3 .

(3.57)

0

Thus, in view of representation (3.12) and the bounds in Lemma 3.4, for z 1 , z 2 such that |z i | ≥ δ and z i ≥ |z i | (i = 1, 2) we have   √  √ N z1, N z2 | |N −1 K˜ N z 1 , z 2 , N −1 | = |N −1/2 K N  ∞  √ √  −1/2  = (2N ) ϕ N ( N z 1 + τ )ϕ N −1 ( N z 2 + τ )  0   √ √ + ϕ N −1 ( N z 1 + τ )ϕ N ( N z 2 + τ ) dτ   ∞  √ √  −1/2  =2 ϕ ( N (z + θ ))ϕ ( N (z 2 + θ )) N 1 N −1  0   √ √ + ϕ N −1 ( N (z 1 + θ ))ϕ N ( N (z 2 + θ )) dθ  ∞ 2 2 2 −1/2 2N −1 ≤ C (δ)N |z 1 + θ | N e−N (z 1 +θ) /4 |z 2 + θ | N −1 e−N (z 2 +θ) /4 M (δ) 0  2 N −1 −N (z 1 +θ)2 /4 + |z 1 + θ | e |z 2 + θ | N e−N (z 2 +θ) /4 dθ ≤ C 2 (δ)N −1/2 M 2N −1 (δ)  ∞ 2 |z 1 + θ |2N e−N (z 1 +θ) /2 dθ × 0



∞

+ 0

∞

|z 2 + θ |2(N −1) e−N (z 2 +θ)

2 /2

1/2 dθ

0

|z 1 + θ |

2(N −1) −N (z 1 +θ)2 /2

e

dθ

∞

(|z 2 + θ |

2N −N (z 2 +θ)2 /2

e

1/2  dθ

.

0

(3.58) Combining this bound with relations (3.56) and (3.57) we find that   | K˜ N z 1 , z 2 , N −1 | e21/2 C 2 (δ)M 2N −1 (δ)2 N Γ (N )|(z 1 )| N |(z 2 )| N ≤ . (3.59) N δ 2 N 3/2 The bound (3.59) applies as well when z 1 , z 2 satisfy |z i | ≥ δ and z i ≤ −|z| (i = 1, 2). Indeed, this follows from (3.11). Finally, we consider z 1 , z 2 such that | z i | ≥ |z| ≥ δ (i = 1, 2) and z 1 z 2 < 0. In this case we have 1 1 1 ≤ ≤ ≤ (2δ)−1 |z 2 − z 1 | | (z 1 )| + | (z 2 )| |(z 1 )| + |(z 2 )|

Limit Correlation Functions for Fixed Trace Random Matrix Ensembles

223

and, by (3.9),  √  √ 1 1 ˜  | K N z 1 , z 2 , N −1 | = √ |K N N z1, N z2 | N N  √ √ √ √ 2  ≤ √ |ϕ N ( N z 1 )||ϕ N −1 ( N z 2 )| + |ϕ N ( N z 2 )||ϕ N −1 ( N z 1 )| . δ N Applying Lemma 3.4 we obtain   |N −1 K˜ N z 1 , z 2 , N −1 | ≤ C(δ)N −1 M 2N −1 (δ)2 N Γ (N ) ((z 1 )) N ((z 2 )) N .

(3.60)

√ Setting R(δ) = M(δ) 2 and comparing 3.59 and 3.60 with the factor N −1 omitted, the proof is completed.

 Corollary 3.3. Let α ∈ (0, 1) and A be positive numbers. Then there exist positive constants C(α, A) and R2 (α) such that for every N ∈ N and real numbers u ∈ [−2, 2], t1 , t2 ∈ [−A, A], and H we have   1       K˜ N u + t1 d(H ), u + t2 d(H ), 1  ≤ Γ (N ) (R3 (α, A))2N H 2N (3.61) N N N N whenever the inequalities | ((u + ti /N )d(H ))| ≥ β(α/4)/2, i = 1, 2, hold. Proof. Set vi = (u + ti /N ) d(H ) for i = 1, 2. From (3.18) we conclude that | vi | ≥ |vi |, i = 1, 2. It follows from (3.46) that |(vi )| N ≤ C(A)|H | N , i = 1, 2. Then by Lemma 3.7 with δ = β(α/4)/2 we have   1       K˜ N u + t1 d(H ), u + t2 d(H ), 1  ≤ C(β(α/4)/2, A)Γ (N )R 2N (β(α/4)/2)H 2N N N N N for every N ≥ (β(α/4)/2)−2 . Substituting R in the latter inequality by a proper larger constant depending also on A, we obtain (3.61) which is valid for all N .

 Proposition 3.1. For every α ∈ (0, 1) and A > 0 there exists a constant R(α, A) such that for every u ∈ [−2 + α, 2 − α], t1 , . . . , tn ∈ [−A, A] and H ∈ R we have the relations    N 1       K˜ N u + t1 d(H ), u + t2 d(H ), 1  ≤ Γ (N ) R(α, A)(1 + H 2 ) (3.62) N N N N and 1 GUE, (1+i H )N

|Rn,N

(u + Nn

t1 N,...,u

+

tn N )|

≤ n! (Γ (N ))n (R(α, A))n N (1 + H 2 )

  1 n N+ 4 .

(3.63)

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F. Götze, M. Gordin

Proof. For every α ∈ (0, 1) and A > 0, every combination of t1 ∈ [−A, A], t2 ∈ [−A, A], H ∈ R, and N ∈ N satisfies the assumptions of at least one of the assertions in Lemma 3.5, Lemma 3.6 and Corollary 3.3. Hence, at least one of these   upper bounds is valid. Writing R(α, A) = max (R1 (α, A))2 , R2 (α, A), (R3 (α, A))2 , we obtain   1       K˜ N u + t1 d(H ), u + t2 d(H ), 1  N N N N    ≤ max (R1 (α, A))2N , Γ (N ) (R2 (α, A)) N H N , Γ (N ) (R3 (α, A))2N H 2N   ≤ (R(α, A)) N max 1, Γ (N )H N , Γ (N )H 2N   < (R(α, A)) N max 1, Γ (N )(1 + H 2 ) N /2 , Γ (N )(1 + H 2 ) N = Γ (N ) (R(α, A)) N (1 + H 2 ) N , and the bound (3.62) is proved. GUE,1/((1+i H )N ) Now we will derive a bound for the correlation function Rn,N . In view of formulas (3.7) and (3.8) we have GUE,σ s GUE,s −1/2 Rn,N (x1 , . . . , xn ) = σ −n/2 Rn,N (σ x1 , . . . , σ −1/2 xn ).

It follows from (2.5) that



GUE,σ s (x1 , . . . , xn ) = Rn,N

where

γm,s (u) =

∞ 0

HSE,u/N 2

Rn,N

γ N 2 ,σ s (u)du,

(2m/2 s −m/2 Γ (m/2))−1 u (m/2)−1 exp (−u/(2s)), if u ≥ 0, 0, if u < 0.

(3.64)

(3.65)

(3.66)

GUE,s (x1 , . . . , xn ) is an entire function, formula (3.64) can be Since (x1 , . . . , xn )
→ Rn,N GUE,σ s (x1 , . . . , xn ) in the parameter σ to used to analytically continue the function Rn,N the domain σ > 0. On the other hand, an analytic continuation can be obtained from formulas (3.65) and (3.66) (we omit the check of this fact requiring routine bounds for γm,s in the parameter s ∈ C). Since these continuations obviously agree, both (3.64) and (3.65) are valid for complex s with s > 0. In particular, we have the identity GUE,1/((1+i H )N )

Rn,N

GUE,1/N

(x1 , . . . , xn ) = d n (H )Rn,N

(x1 d(H ), . . . , xn d(H )). (3.67)

Along with the determinantal representation (3.7) (which also admits an analytic continuation) and inequality (3.62), this leads to GUE,1/((1+i H )N )

N −n |Rn,N = ≤

(u + t1 /N , . . . , u + tn /N )|

GUE,1/N ) N |d(H )| Rn,N ((u + t1 /N )d(H ), . . . , (u + t1 /N )d(H ))| n!|d(H )|n max0≤i< j≤n |N −1 K˜ ((u + ti /N )d(H ), (u + t j /N )d(H ))|n  n n N 2 N −n

≤ n!|d(H )|

n

Γ (N ) (R(α, A)) (1 + H )

= n! (Γ (N ))n (R(α, A))n N (1 + H 2 )n(N +1/4) , since |d(H )| = (1 + H 2 )1/4 by (3.14).



(3.68)

Limit Correlation Functions for Fixed Trace Random Matrix Ensembles

225

Proof of Theorem. As explained in the end of Sect. 2 we need to show that   ∞ √ 1 t1 tn GUE,1/((1−i h 2/N )N ) u+ ,...,u + dh φ 2 (h)Rn,N (N w(u))n −∞ N N w(u) N w(u) n  sin π(ti − t j ) 1 −→ √ det (3.69) N →∞ π(ti − t j ) i, j=1 2π uniformly in u, t1 , . . . , tn subject to the conditions formulated in the the statement of the theorem. Throughout the proof we will assume that these conditions are satisfied and GUE,· omit the arguments of Rn,N whenever that is possible. The existence of the integral in the left hand side of relation (3.69) will be a consequence of our estimates. GUE,s GUE,s (−u 1 , . . . , −u n ) = Rn,N (u 1 , . . . , u n ) since K N (−u 1 , −u 2 ) = Note that Rn,N K N (u 1 , u 2 ) by (3.11). In view of this property it suffices to prove (3.69) for u ≥ 0 only. Observe that for real h, φ N 2 (−h) = φ N 2 (h) and √ GUE,1/((1+i h 2/N )N )

Rn,N

√ GUE,1/((1−i h 2/N )N )

(u 1 , . . . , u n ) = Rn,N

(u 1 , . . . , u n )

for u 1 , . . . , u n ∈ R which shows that (3.69) is established if we prove ∞  √  1 GUE,1/((1+i h 2/N )N ) dh φ 2 (h)Rn,N N (N w(u))n 0 n  sin π(ti − t j ) 1 −→ √ det . N →∞ 2 2π π(ti − t j ) i, j=1

(3.70)

It follows from the central limit theorem for densities that ∞   1 φ N 2 (h) dh −→ √ . (3.71) N →∞ 2 2π 0 √ Here φ N 2 (·) is a prelimiting characteristic function and 1/2 2π is half the value of the limiting standard Gaussian density at 0. Thus, to prove (3.70) it suffices to establish the following relation: ∞  √  1 GUE,1/((1+i h 2/N )N ) dh φ (h)R 2 N n,N (N w(u))n 0 n  ∞   sin π(ti − t j ) (3.72) φ N 2 (h) dh − det π(ti − t j ) i, j=1 0 =: I (u, N ) − J (N ) −→ 0 N →∞

under the same uniformity constraints as above. Omitting the arguments t1 , . . . , tn , we set    n sin π(ti − t j )d(H )w(ud(H ))/w(u) Sn (u, H ) = det π(ti − t j )d(H ) i, j=1

226

F. Götze, M. Gordin

and

 Sn = Sn (u, 0) = det

sin π(ti − t j ) π(ti − t j )

n . i, j=1

For a given α ∈ (0, 1), let H (α) > 0 be a number whose existence is guaranteed by Lemma 3.2. Let > 0 be an arbitrary positive number, and H0 = H0 (α, A, ) ∈ (0, H (α)) be small enough to ensure that √ |Sn (u, H ) − Sn | ≤ 2 2/π (3.73) for all u ∈ [−2 + α, 2 − α], |H | ≤ H0 and t1 , . . . , tn satisfying |t1 | ≤ A, . . . , |tn | ≤ A. Such a number exists because of continuity of the sin-kernel √ and continuity near 0 of the function d(·). Performing the change of variable H = h 2/N , we may write ∞  √  1 GUE,1/((1+i h 2/N )N ) dh φ (h)R I (u, N ) = 2 N n,N n (N w(u)) 0 ∞   √ N GUE,1/((1+i H )N ) dH = √ φ N 2 (N H/ 2)Rn,N 2(N w(u))n 0 H0 ∞ N N = √ ... dH + √ ... dH 2(N w(u))n 0 2(N w(u))n H0 (3.74) =: I1 ( , u, N ) + I2 ( , u, N ). Quite analogously, we have ∞ √    N Sn ∞  φ N 2 (h) dh = √ φ N 2 (N H/ 2) d H J (N ) = Sn 2 0 0 √  √  N Sn ∞  N Sn H0  φ N 2 (N H/ 2) d H + √ φ N 2 (N H/ 2) d H = √ 2 0 2 H0 =: J1 ( , N ) + J2 ( , N ). (3.75) With this notation we need to show that I1 ( , u, N ) − J1 ( , N ) −→ 0,

(3.76)

I2 ( , u, N ) −→ 0,

(3.77)

N →∞

N →∞

and J2 ( , N ) −→ 0 N →∞

(3.78)

uniformly in u ∈ [−2 + α, 2 − α] and ti ∈ [−A, A], i = 1, . . . , n. By Proposition 3.1, for every , we obtain |I2 ( , u, N )| ∞   N   n! (Γ (N ))n (R(α, A))n N (1 + H 2 )n(N +1/4) φ N 2 ( N√H ) d H. ≤√ 2 2(w(u))n H0 (3.79)

Limit Correlation Functions for Fixed Trace Random Matrix Ensembles

227

Since √ 2 |φ N 2 (N H/ 2)| = (1 + H 2 )−(N /4) and

∞

(1 + H 2 )−K d H <

L

1 (1 + L 2 ) K −1



∞

(1 + H 2 )d H =

0

π (3.80) 2(1 + L 2 ) K −1

for every K > 1, we have, in particular, that

∞ L

√ |φ N 2 (N H/ 2)|d H <

π 2(1 + L 2 ) N

2 /4−1

.

(3.81)

 Note that N 2 /4 − n N − 1/4 > 1 whenever N > N0 = 2n + 4n 2 + 5/4. For every N > N0 we obtain from (3.80) with K = N 2 /4 − n N − 1/4 and L = H0 that 1 πN I2 ( , u, N ) ≤ √ n! (Γ (N ))n (R(α, A))n N  .  2 /4−n N −5/4 N n 2 2(w(u)) 1 + H02 (3.82) Since inf

u∈[−2+ ,2− ]

|w(u)| = w(2 − )

(3.83)

and, by the Stirling formula, (1 + H02 ) N /4 tends to infinity more rapidly than any fixed power of Γ (N ) (or, moreover, the N th power of any positive number), we conclude from (3.82) that 2

I2 ( , u, N ) −→ 0 N →∞

(3.84)

uniformly in u ∈ [−2 + α, 2 − α] and ti ∈ [−A, A], i = 1, . . . , n. The check of (3.78) is even more straightforward. Indeed, using (3.80) with K = N 2 /4 and L = H0 , we have N Sn |J2 ( , N )| = √ 2 N Sn = √ 2 and (3.78) follows.

 ∞   √  √    N Sn ∞    φ φ 2 (N H/ 2) d H ≤ √ 2 (N H/ 2) d H  N N   2 H0 H ∞0 π N Sn 2 (1 + H 2 )−(N /4) d H = √ , 2 2 2(1 + L 2 ) N /4−1 H0

228

F. Götze, M. Gordin

Now we complete the proof by establishing (3.76). Recall that we omit the arguments GUE,1/N of Rn,N . All bounds and limit transitions are uniform with respect to u ∈ [−2 + α, 2 − α] and t1 , t2 , . . . , tn satisfying |t1 | ≤ A, . . . , |tn | ≤ A. We have |I1 ( , u, N ) − J1 ( , N )|   H0  GUE,1/((1+i H )N ) √  N  Rn,N − Sn φ N 2 (N H/ 2) d H ≤√   (N w(u))n 2 0    H0  GUE,1/((1+i H )N ) √  N  Rn,N ≤√ − Sn (u, H ) φ N 2 (N H/ 2) d H   (N w(u))n 2 0  H0  √  N  +√ (Sn (u, H ) − Sn ) φ N 2 (N H/ 2) d H 2 0 (1)

(2)

=: D1 ( , u, N ) + D2 ( , u, N ).

(3.85)

It follows from the determinantal formula (3.7) and Lemma 3.2 that  GUE,1/((1+i H )N )  R   n,N  sup  − S (u, H )  −→ 0 n n   N →∞ (N w(u)) 0≤H ≤H0

(3.86)

uniformly in u ∈ [−2 + α, 2 − α], and |ti | ≤ A (i = 1, . . . , n). Therefore, using (3.81), we obtain (1)

D1 ( , u, N )  GUE,1/((1+i H )N )  R  ∞   n,N  φ N 2 (h) dh −→ 0 ≤ sup  − S (u, H )  n n  0 N →∞ (N w(u)) 0≤H ≤H0  since

 φ N 2 (h) dh −→

∞ 0

N →∞

1 √ . 2 2π

(3.87)

Finally, we see from (3.73) and (3.87) that D1(2) ( , u, N ) ≤ . Since is arbitrary small, this completes the proof.



Acknowledgements. The authors are indebted to the anonymous referees for careful reading of the manuscript. Moreover, the main theorem in the first version of the paper required excluding of a neighborhood of zero (this case has been considered in the note [9]). Stimulating questions of one of the referees and the editor led us to a revision of the proof which removes this restriction. Finally, we would like to thank Justine Swierkot for her assistance in the preparation of the text.

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References 1. Akemann, G., Cicuta, G.M., Molinari, L., Vernizzi, G.: Compact support probability distributions in random matrix theory. Phys. Rev. E (3) 59(2, part A), 1489–1497 (1999) 2. Akemann, G., Vernizzi, G.: Macroscopic and microscopic (non-)universality of compact support random matrix theory. Nucl. Phys. B 583(3), 739–757 (2000) 3. Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. J. Comm. Pure Appl. Math. 52(11), 1335–1425 (1999) 4. Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: Strong asymptotics for polynomials orthogonal with respect to exponential weights. J. Comm. Pure Appl. Math. 52(12), 1491– 1552 (1999) 5. Deift, P. A.: Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. Courant Lecture Notes in Mathematics, Vol. 3. N. Y.: New York University/Courant Institute of Mathematical Sciences, 1999 6. Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: A Riemann-Hilbert approach to asymptotic questions for orthogonal polynomials. J. Comput. Appl. Math. 133(1–2), 47–63 (2001) 7. Feller, W.: An introduction to probability theory and its applications. Vol. II. Second edition. New York: John Wiley & Sons Inc., 1971 8. Forrester, P.J.: Log-gases and random matrices. Book in progress. http://www.ms.unimelb.edu.au/ ~matpjf/matpjf.html, 2005 9. Götze, F., Gordin, M., Levina, A.: Limit correlation function at zero for fixed trace random matrix ensembles.(Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 341, 68–80 (2007); translation in J. Math. Sci. (N.Y.) 147(4), 6884–6890 (2007) 10. Johansson, K.: Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Commun. Math. Phys. 215(3), 683–705 (2001) 11. Mehta, M. L.: Random matrices. Third edition. Pure and Applied Mathematics (Amsterdam), 142. Amsterdam: Elsevier/Academic Press, 2004 12. Pastur, L., Shcherbina, M.: Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles. J. Stati. Phys. 86(1–2), 109–147 (1997) 13. Petrov, V.V.: Sums of independent random variables. N.Y.: Springer, 1975 14. Rosenzweig, N.: Statistical mechanics of equally likely quantum systems. In: Statistical physics (Brandeis Summer Institute, 1962, Vol. 3), N. Y.: W. A. Benjamin, (1963), pp. 91–158 15. Soshnikov, A.: Determinantal point random fields. Russ. Math. Surv. 55(5), 923–975 (2000) 16. Szego, G.: Orthogonal polynomials. Fourth edition. Providence, R.I.: Amer. Math. Soc., 1975 17. Tracy, C.A., Widom, H.: Correlation functions, cluster functions, and spacing distributions for random matrices. J. Stat. Phys. 92(5–6), 809–835 (1998) 18. Van Assche, W., Geronimo, J.S.: Asymptotics for orthogonal polynomials with regularly varying recurrence coefficients. Rocky Mountain J. Math. 19(1), 39–49 (1989) Communicated by H. Spohn

Commun. Math. Phys. 281, 231–249 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0487-4

Communications in

Mathematical Physics

Stability of Viscous Shocks in Isentropic Gas Dynamics
Blake Barker1 , Jeffrey Humpherys1 , Keith Rudd2 , Kevin Zumbrun3 1 Department of Mathematics, Brigham Young University, Provo, UT 84602, USA.

E-mail: [email protected]; [email protected]

2 Department of Eng. Sci. and App. Math., Northwestern University, Evanston,

IL 60208-3125, USA. E-mail: [email protected]

3 Department of Mathematics, Indiana University, Bloomington, IN 47402, USA.

E-mail: [email protected] Received: 25 March 2007 / Accepted: 21 December 2007 Published online: 6 May 2008 – © Springer-Verlag 2008

Abstract: In this paper, we examine the stability problem for viscous shock solutions of the isentropic compressible Navier–Stokes equations, or p-system with real viscosity. We first revisit the work of Matsumura and Nishihara, extending the known parameter regime for which small-amplitude viscous shocks are provably spectrally stable by an optimized version of their original argument. Next, using a novel spectral energy estimate, we show that there are no purely real unstable eigenvalues for any shock strength. By related estimates, we show that unstable eigenvalues are confined to a bounded region independent of shock strength. Then through an extensive numerical Evans function study, we show that there are no unstable spectra in the entire right-half plane, thus demonstrating numerically that large-amplitude shocks are spectrally stable up to Mach number M ≈ 3000 for 1 ≤ γ ≤ 3. This strongly suggests that shocks are stable independent of amplitude and the adiabatic constant γ . We complete our study by showing that finite-difference simulations of perturbed large-amplitude shocks converge to a translate of the original shock wave, as expected. 1. Introduction Consider the isentropic compressible Navier-Stokes equations in one spatial dimension expressed in Lagrangian coordinates, also known as the p-system with real viscosity: vt − u x = 0,
u  x , u t + p(v)x = v x

(1)

where, physically, v is the specific volume, u is the velocity, and p(v) is the pressure law. We assume that p(v) is adiabatic, that is, a γ -law gas satisfying p(v) = a0 v −γ for  This work was supported in part by the National Science Foundation award numbers DMS-0607721 and DMS-0300487.

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some constants a0 > 0 and γ ≥ 1. In the thermodynamical rarified gas approximation, γ > 1 is the average over constituent particles of γ = (n + 2)/n, where n is the number of internal degrees of freedom of an individual particle [3]: n = 3 (γ = 5/3) for monatomic, n = 5 (γ = 1.4) for diatomic gas. More generally, γ is usually taken within 1 ≤ γ ≤ 3 in physical approximations of gas-dynamical flow [27]. This system is an important and widely studied gas-dynamical model (see for example [27] and references within), and yet little is presently known about the stability of its large-amplitude viscous shock wave solutions. Over two decades ago, Matsumura and Nishihara [24] used a clever energy estimate to show that small-amplitude shocks are stable under zero-mass perturbations. The linear portion of their analysis is equivalent to proving spectral stability, which through the more recent work of Zumbrun and collaborators [32,21,23,22,31,17], implies asymptotic orbital stability, hereafter called nonlinear stability. We remark that the results of [23,22,31] hold for shocks of arbitrary amplitude, and thus nonlinear stability is implied by spectral stability. Hence, for largeamplitude shocks, spectral stability is the missing piece of the stability puzzle and the subject of our present focus. In this paper, we first improve upon the work in [24] slightly by extending the known parameter regime for which small-amplitude viscous shocks are provably spectrally stable, using an optimized version of the same method. We also show that this method cannot be extended any further to larger amplitudes. Using a novel spectral energy estimate, however, we are able to show that there are no purely real unstable eigenvalues for any shock strength. We note that this result is stronger than that which could be given by the Evans function stability index (sometimes called the orientation index), which only measures the parity of unstable real eigenvalues, see [4,14,15,31]. A consequence of this result (which follows also by the Evans function computations used to determine the stability index [31]) is that if an instability were to occur for large-amplitude viscous shocks, its onset, or indeed any change in the number of unstable eigenvalues, would be associated with a Hopf-like bifurcation in which one or more conjugate pairs of eigenvalues cross through the imaginary axis; see [28–30] for further discussion of this scenario. Continuing our investigations, we appeal to numerical Evans function computation to explore the large-amplitude regime through the use of winding number calculations via the argument principle. Before doing so, however, we rule out high-frequency instability through the combination of two spectral energy estimates, showing that unstable eigenvalues are confined to a bounded region independent of shock amplitude. This reduces the problem to investigation, feasible by numerics, of a compact set. Then by checking the low-frequency regime by repeating several Evans function computations, we determine whether or not a particular viscous shock is spectrally stable. As a final verification, we use a finite-difference method to simulate perturbed large-amplitude viscous shocks, and check whether they converge, as expected, to a translate of the original profile. Conclusions and results of numerical investigations. We carry out our numerical experiments far into the hypersonic shock regime, exploring up through Mach number M ≈ 3000 for 1 ≤ γ ≤ 3. Particular attention is given to the monatomic and diatomic cases, γ = 5/3 and γ = 7/5, respectively. In all cases, our results are consistent with spectral stability (hence also linear and nonlinear stability [22,23,31]). This strongly suggests that viscous shock profiles in an isentropic are spectrally stable independent of both amplitude and γ . Our bounds on the size of unstable eigenvalues, which are independent of shock strength, may be viewed as a first step in establishing such a result analytically.

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Extensions and open questions. The present study is not a numerical proof. However, as discussed in [6], it could be converted to numerical proof by the implementation of interval arithmetic and a posteriori error estimates for numerical solution of ODE. This would be an interesting direction for future investigation. A crucial step in carrying out numerical proof by interval arithmetic is by analytical estimates special to the problem at hand to sufficiently reduce the computational domain to make the computations feasible in realistic time. This we have carried out by the surprisingly strong estimates of Sect. 5 and Appendices B–C. A second very interesting direction would be to establish stability in the largeamplitude limit by a singular-perturbation analysis, an avenue we intend to follow in future work. Together, these two projects would give a complete, rigorous proof of stability for arbitrary-amplitude viscous shock solutions of (1) on the physical range γ ∈ [1, 3]. 2. Background By a viscous shock profile of (1), we mean a traveling wave solution v(x, t) = v(x ˆ − st), u(x, t) = u(x ˆ − st), moving with speed s and having asymptotically constant end-states (v± , u ± ). As an alternative, we can translate x → x − st, and consider instead stationary solutions of vt − svx − u x = 0,
u  x . u t − su x + (a0 v −γ )x = v x

(2)

Under the rescaling (x, t, v, u) → (−εsx, εs 2 t, v/ε, −u/(εs)), where ε is chosen so that 0 < v+ < v− = 1, our system takes the form vt + vx − u x = 0,
u  x , u t + u x + (av −γ )x = v x

(3)

where a = a0 ε−γ −1 s −2 . Thus, the shock profiles of (3) satisfy the ordinary differential equation v  − u  = 0,    u , u  + (av −γ ) = v subject to the boundary conditions (v(±∞), u(±∞)) = (v± , u ± ). This simplifies to    v  −γ  . v + (av ) = v By integrating from −∞ to x, we get the profile equation v  = v(v − 1 + a(v −γ − 1)),

(4)

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where a is found by setting x = +∞, thus yielding the Rankine-Hugoniot condition a=−

v+ − 1

−γ v+

−1

γ

= v+

1 − v+ γ. 1 − v+

(5) γ

Evidently, a → γ −1 in the weak shock limit v+ → 1, while a ∼ v+ in the strong shock limit v+ → 0. Remark 2.1. Since the profile equation (4) is first order scalar, it has a monotone solution. Since v+ < v− , we have that vˆ x < 0 for all x ∈ R. By linearizing (3) about the profile (v, ˆ u), ˆ we get the eigenvalue problem λv + v  − u  = 0,      h(v) ˆ u v = , λu + u  − vˆ γ +1 vˆ

(6)

h(v) ˆ = −vˆ γ +1 + a(γ − 1) + (a + 1)vˆ γ .

(7)

where

We say that a shock profile of (1) is spectrally stable if the linearized system (6) has no spectrum in the closed deleted right half-plane P = { e(λ) ≥ 0} \ {0}, that is, there are no growth or oscillatory modes for (6). We remark that a traveling wave profile always has a zero-eigenvalue associated with its translational invariance. This generally negates the possibility of good uniform bounds in energy estimates, and so we employ the standard technique (see [16,32]) of transforming into integrated coordinates. This goes as follows: Suppose that (v, u) is an eigenfunction of (6) with a non-zero eigenvalue λ. Then  u(x) ˜ =

x −∞

 u(z)dz, v(x) ˜ =

x

−∞

v(z)dz,

and their derivatives decay exponentially as x → ∞. Thus, by substituting and then integrating, (u, ˜ v) ˜ satisfies (suppressing the tilde) λv + v  − u  = 0, λu + u  −

(8a) u 

h(v) ˆ  . v = γ +1 vˆ vˆ

(8b)

This new eigenvalue problem differs spectrally from (6) only at λ = 0, hence spectral stability of (6) is implied by spectral stability of (8). Moreover, since (8) has no eigenvalue at λ = 0, one can expect to have a better chance of developing a successful spectral energy method to prove stability. We demonstrate this in the following section.

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3. Stability of Small-Amplitude Shocks In this section we further the work in [24] by extending slightly the known parameter regime for which small-amplitude viscous shocks are provably spectrally stable. We also show that this method cannot be extended any further for larger amplitudes. Theorem 3.1. ([24]) Viscous shocks of (1) are spectrally stable whenever 

γ +1

v+ aγ

2



γ +1

v + 2(γ − 1) + aγ

 − (γ − 1) ≥ 0.

(9)

In particular, as v+ → 1 (hence aγ → 1), the left-hand side of (9) approaches γ and so the inequality is satisfied. Therefore, small-amplitude viscous shocks of (1) are spectrally stable. Proof. We note that h(v) ˆ > 0. By multiplying (8b) by both the conjugate u¯ and vˆ γ +1 / h(v) ˆ and integrating along x from ∞ to −∞, we have   γ +1    γ λu u¯ vˆ γ +1 u u¯ vˆ u u¯ vˆ + − . v  u¯ = ˆ ˆ ˆ R h(v) R h(v) R R h(v)



Integrating the last three terms by parts and appropriately using (8a) to substitute for u  in the third term gives us   γ +1    γ   γ 2 vˆ λ|u|2 vˆ γ +1 u u¯ vˆ vˆ |u | + + =− v(λv + v  ) + u  u. ¯ h(v) ˆ ˆ ˆ ˆ R R h(v) R R h(v) R h(v)



We take the real part and appropriately integrate by parts to get   e(λ)

  vˆ γ +1 2 vˆ γ 2 |u| + |v|2 + |u  |2 = 0, g(v)|u| ˆ + h( v) ˆ h( v) ˆ R R R

where 1 g(v) ˆ =− 2



vˆ γ +1 h(v) ˆ



 +

vˆ γ h(v) ˆ

 

.

Thus, to prove stability it suffices to show that g(v) ˆ ≥ 0 on [v+ , 1]. By straightforward computation, we obtain identities: γ h(v) ˆ − vh ˆ  (v) ˆ = aγ (γ − 1) + vˆ γ +1 and γ −1 vˆ vˆ x = aγ − h(v). ˆ

(10) (11)

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(a)

(b)

Fig. 1. In (a), we have a graph of g(v) ˆ against vˆ for v+ = 1 × 10−4 and γ = 2.0. Note that g(v) ˆ dips down below zero on the left-hand size. Hence, the energy estimate will not generalize beyond the small-amplitude regime. In (b) we graph the stability boundaries (dark lines) given by (9) and (15), where γ and v+ are the horizontal and vertical axes, respectively. We see that in our scaling (15) is only a modest improvement over (9). The dotted lines correspond from top to bottom as the parameter regimes for Mach numbers 2, 5, and 10 (see the Appendix to see how to determine the Mach number). Hence, this energy estimate does not even hold for shocks at Mach 2 and γ > 1.084

Using (10) and (11), we abbreviate a few intermediate steps below:  (γ + 1)vˆ γ h(v) ˆ − vˆ γ +1 h  (v) ˆ ˆ − vˆ γ h  (v) ˆ d γ vˆ γ −1 h(v) + vˆ x h(v) ˆ 2 d vˆ h(v) ˆ 2



  ˆ − vh ˆ  (v) ˆ ˆ − vh ˆ  (v) ˆ d γ h(v) vˆ x vˆ γ (γ + 1)h(v) + (aγ − h(v)) ˆ =− 2 h(v) ˆ 2 d vˆ h(v) ˆ 2

vˆ x g(v) ˆ =− 2

=−



a vˆ x vˆ γ −1 × 2h(v) ˆ 3  γ 2 (γ + 1)vˆ γ +2 − 2(a + 1)γ (γ 2 − 1)vˆ γ +1 + (a + 1)2 γ 2 (γ − 1)vˆ γ  + aγ (γ + 2)(γ 2 − 1)vˆ − a(a + 1)γ 2 (γ 2 − 1)

=−



2 a vˆ x vˆ γ −1 [(γ + 1)vˆ γ +2 + vˆ γ (γ − 1) (γ + 1)vˆ − (a + 1)γ 3 2h(v) ˆ

(12)

+aγ (γ 2 − 1)(γ + 2)vˆ − a(a + 1)γ 2 (γ 2 − 1)] a vˆ x vˆ γ −1 [(γ + 1)vˆ γ +2 + aγ (γ 2 − 1)(γ + 2)vˆ − a(a + 1)γ 2 (γ 2 − 1)] 2h(v) ˆ 3 ⎡ ⎤  γ +1   γ +1 2 v+ γ 2 a 3 vˆ x (γ + 1) ⎣ v+ + 2(γ − 1) ≥− − (γ − 1)⎦ . (13) 2h(v) ˆ 3 v+ aγ aγ ≥−

Thus from (9), we have spectral stability.



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We note that the hypothesis in (9) is not sharp. Indeed, one can show from (12), that a stronger condition could be given as

2 (γ + 1)vˆ γ +2 + vˆ γ (γ − 1) (γ + 1)vˆ − (a + 1)γ (14) +aγ (γ 2 − 1)(γ + 2)vˆ − a(a + 1)γ 2 (γ 2 − 1) ≥ 0, which is sharp in the following sense: When this condition fails to be true, then g(v) ˆ is no longer nonnegative, and thus the energy method fails. In Fig. 1(a), we see that g(v) ˆ dips on the left-hand side when this inequality is compromised. We remark further that near v+ , the left-hand side of (14) is monotone increasing in v. ˆ Thus, (14) holds if and only if γ +2

γ

(γ + 1)v+ + v+ (γ − 1) ((γ + 1)v+ − (a + 1)γ )2 +aγ (γ 2 − 1)(γ + 2)v+ − a(a + 1)γ 2 (γ 2 − 1) ≥ 0.

(15)

We see from Fig. 1(b) that (15) is only a marginal improvement over (9). However, since (15) is sharp, we cannot hope to prove large-amplitude spectral stability using this approach. Instead, we proceed by a combined analytical and numerical approach as in [6–8], first showing that unstable eigenvalues can occur only in a bounded set, then searching for eigenvalues in this set by computing the Evans function numerically. Before doing so, however, we show in the following section that real unstable eigenvalues do not exist, even for large-amplitude viscous shocks. 4. No Real Unstable Eigenvalues In this section we use a novel spectral energy estimate to show that there are no purely real unstable eigenvalues for any shock strength. We note that this result is stronger than that which could be given by the Evans function stability index (sometimes called the orientation index), which only measures the parity of unstable real eigenvalues, see [4,15, 31]. The fact that this holds for all shock strengths is interesting because it is among the strongest statements about large-amplitude spectral stability that has been proven to date. Theorem 4.1. Viscous shocks of (1) have no unstable real spectra. Proof. We multiply (8b) by the conjugate v¯ and integrate along x from ∞ to −∞. This gives      h(v)v ˆ  v¯ u v¯  . λu v¯ + u v¯ − = γ +1 R R R vˆ R vˆ Notice that on the real line, λ¯ = λ. Using (8a) to substitute for λv in the first term and for u  in the last term, we get     h(v)v ˆ  v¯ (λv  + v  )v¯ . u(u¯  − v¯  ) + u  v¯ − = γ +1 vˆ R R R vˆ R Separating terms and simplifying gives        h(v)v ˆ  v¯ v v¯ v v¯ + . u u¯  + 2 u  v¯ − = λ γ +1 R R R vˆ R vˆ R vˆ

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We further simplify by substituting for u  in the second term and integrating the last terms by parts to give        h(v)v ˆ  v¯ v v¯ vˆ x  |v  |2   + . u u¯ + 2 (λv + v )v¯ − = λ v v ¯ − γ +1 2 R R R vˆ R vˆ R vˆ R vˆ Combining the third and fifth terms using (11) and simplifying, we have        v  v¯ |v  |2 v v¯  2  =λ . u u¯ + 2λ |v| + 2 v v¯ − aγ + γ +1 v ˆ v ˆ R R R R R R vˆ By taking the real part, we arrive at      λ vˆ x vˆ x |v  |2 aγ (γ + 1) 2 2 4 − 2 |v| − = 0. |v| + γ +2 2 R vˆ 2 R vˆ R vˆ This is a contradiction when λ ≥ 0.



We remark that the absence of positive real eigenvalues limits the admissible onset of instability to Hopf-like bifurcations where a pair of conjugate eigenvalues crosses the imaginary axis. In the following section, we give an upper bound on the spectral frequencies that are admissible. In other words, we show that if a pair of conjugate eigenvalues cross the imaginary axis, they must do so within these bounds. 5. High-Frequency Bounds In this section, we prove high-frequency spectral bounds. This is an important step because it gives a ceiling as to how far along both the imaginary and real axes that one must explore for spectrum when doing Evans function computations. Indeed if we want to check for roots of the Evans function in the unstable half-plane, say using the argument principle, then we need only compute within these bounds. If no roots are found therein, then we have strong numerical evidence that the given shock is spectrally stable. (Indeed, at the expense of further effort, such a calculation may be used as the basis of numerical proof, as described in [6,7].) In this section we show that the high-frequency bounds are quite strong, only allowing unstable eigenvalues to persist in a relatively small triangle adjoining the origin. Moreover, these bounds are independent of the shock amplitude. Lemma 5.1. The following holds for eλ ≥ 0:    1 2 2 ( e(λ) + | m(λ)|) v|u| ˆ − vˆ x |u| + |u  |2 2 R R R  √  h(v) ˆ  ≤ 2 |v ||u| + v|u ˆ  ||u|. γ v ˆ R R

(16)

Proof. We multiply (8b) by vˆ u¯ and integrate along x. This yields     h(v) ˆ  λ v|u| ˆ 2+ vu ˆ  u¯ + |u  |2 = v u. ¯ γ v ˆ R R R R We get (16) by taking the real √ and imaginary parts and adding them together, and noting  that | e(z)| + | m(z)| ≤ 2|z|.

Stability of Viscous Shocks in Isentropic Gas Dynamics

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Lemma 5.2. The following identity holds for eλ ≥ 0:      h(v) ˆ |v  |2 1 aγ  2 2 2 + |u | = 2 e(λ) |v| + e(λ) + |v  |2 , (17) 2 R vˆ γ +1 vˆ γ +1 R R R vˆ Proof. We multiply (8b) by v¯  and integrate along x. This yields      h(v) ˆ 2 1   1 u (λv  + v  )v¯  . u  v¯  − |v | = v ¯ = λ u v¯  + γ +1 R R R vˆ R vˆ R vˆ Using (8a) on the right-hand side, integrating by parts, and taking the real part gives       h(v) ˆ |v  |2 vˆ x     2 |v . e λ u v¯ + u v¯ = + | + e(λ) γ +1 2vˆ 2 R R R vˆ R vˆ The right hand side can be rewritten as       1 h(v) ˆ |v  |2 aγ     2 e λ u v¯ + |v . u v¯ = + | + e(λ) 2 R vˆ γ +1 vˆ γ +1 R R R vˆ

(18)

Now we manipulate the left-hand side. Note that         ¯ λ u v¯ + u v¯ = (λ + λ) u v¯ − u(λ¯ v¯  + v¯  ) R R R R   = −2 e(λ) u  v¯ − u u¯  R R    = −2 e(λ) (λv + v )v¯ + |u  |2 . R

R

Hence, by taking the real part we get      u  v¯  = |u  |2 − 2 e(λ)2 |v|2 . e λ u v¯  + R

R

R

R

This combines with (18) to give (17).

 Remark 5.3. Lemma 5.2 is a special case of the high-frequency bounds given in [22,31]. We also note that (17) follows from a “Kawashima-type” estimate as described in [22,31]. Lemma 5.4. For h(v) ˆ as in (7), we have    h(v) ˆ  1 − v+  sup  γ  = γ γ ≤ γ. vˆ 1 − v+ v∈[v ˆ + ,1]

(19)

Proof. Defining H (v) ˆ := h(v) ˆ vˆ −γ = −vˆ + a(γ − 1)vˆ −γ + (a + 1),

(20)

we have H  (v) ˆ = −1 − aγ (γ − 1)vˆ −γ −1 < 0 for 0 < v+ ≤ vˆ ≤ v− = 1, hence the maximum of H on vˆ ∈ [v+ , v− ] is achieved at vˆ = v+ . Substituting (5) into (20) and simplifying yields (19).

 We complete this section by proving our high-frequency bounds.

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B. Barker, J. Humpherys, K. Rudd, K. Zumbrun

Theorem 5.5. Any eigenvalue λ of (8) with nonnegative real part satisfies 1 √ e(λ) + | m(λ)| ≤ ( γ + )2 . 2

(21)

Proof. Using Young’s inequality twice on right-hand side of (16) together with (19), we get    1 ( e(λ) + | m(λ)|) v|u| ˆ 2− vˆ x |u|2 + |u  |2 2 R R R  √  h(v) ˆ  ≤ 2 |v ||u| + v|u ˆ  ||u| γ R vˆ R √     h(v) ˆ  2 ( 2)2 h(v) ˆ 1 2  2 ≤θ |v | + v|u| ˆ +  v|u ˆ | + v|u| ˆ 2 γ +1 γ 4θ 4 R R vˆ R vˆ R     1 h(v) ˆ 2 γ  2 + <θ |v | +  |u | + v|u| ˆ 2. γ +1 2θ 4 R R vˆ R Assuming that 0 <  < 1 and θ = (1 − )/2, this simplifies to   ( e(λ) + | m(λ)|) v|u| ˆ 2 + (1 − ) |u  |2 R R    1 h(v) ˆ 2 1− γ + < |v | + v|u| ˆ 2. γ +1 2 2θ 4 R R vˆ Applying (17) yields 



1 γ + ( e(λ) + | m(λ)|) v|u| ˆ < 1 −  4 R



2

R

v|u| ˆ 2,

or equivalently, ( e(λ) + | m(λ)|) < √ Setting  = 1/(2 γ + 1) gives (21).

(4γ − 1) − 1 . 4(1 − )



In the following section, we do an extensive Evans function study to explore numerically the rest of the right-half plane in an effort to locate the presence of unstable complex eigenvalues. 6. Evans Function Computation In this section, we numerically compute the Evans function to locate any unstable eigenvalues, if they exist, in our system. The Evans function D(λ) is analytic to the right of the essential spectrum and is defined as the Wronskian of decaying solutions of the eigenvalue equation for the linearized operator (8) (see [1,9–13]). In a spirit similar to the characteristic polynomial, we have that D(λ) = 0 if and only if λ is in the point spectrum of the linearized operator (8). While the Evans function is generally too complex to compute explicitly, it can readily be computed numerically, even for large systems [19].

Stability of Viscous Shocks in Isentropic Gas Dynamics

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Since the Evans function is analytic in the region of interest, we can numerically compute the winding number in the right-half plane. This allows us to systematically locate roots (and hence eigenvalues) within. As a result, spectral stability can be determined, and in the case of instability, one can produce bifurcation diagrams to illustrate and observe its onset. This approach was first used by Evans and Feroe [13] and has been advanced further in various directions (see for example [2,5–7,19,25,26]). We begin by writing (8) as a first-order system W  = A(x, λ)W , where ⎛ ⎞ ⎛ ⎞ 0 λ 1 u ⎠ , W = ⎝v ⎠ ,  = d , 1 A(x, λ) = ⎝ 0 0 (22) dx λvˆ λvˆ f (v) ˆ −λ v and f (v) ˆ = vˆ − vˆ −γ h(v), ˆ with h as in (7). Note that eigenvalues of (8) correspond to nontrivial solutions of W (x) for which the boundary conditions W (±∞) = 0 are satisfied. We remark that since vˆ is asymptotically constant in x, then so is A(x, λ). Thus at each end-state, we have the constant-coefficient system W  = A± (λ)W.

(23)

Hence solutions that satisfy the needed boundary condition must emerge from the unstable manifold W1− (x) at x = −∞ and the stable manifold W2+ (x) ∧ W3+ (x) at x = ∞. In other words, eigenvalues of (8) correspond to the values of λ for which these two manifolds intersect, or more precisely, when D(λ) := det(W1− W2+ W3+ )|x=0 is zero. As an alternative, we consider the adjoint formulation of the Evans function [4,26], where instead of computing the 2-dimensional stable manifold, we find the  + (x) coming from the unstable manifold of single trajectory W 1   = −A(x, λ)∗ W  W

(24)

 + (x) is orthogonal to both W + (x) and W + (x) since at x = ∞. Note that W 1 2 3 ∗    + is orthogonal to that of W + (W (x) W (x)) = 0 for all x and the initial data of W 1 2  + and W − are orthogonal, and and W3+ . Hence, the original manifolds intersect when W 1 1  + · W − )|x=0 . therefore, the adjoint Evans function takes the form D+ (λ) := (W 1 1 To further improve the numerical efficiency and accuracy of the shooting scheme,  to remove exponential growth/decay at infinity, and thus eliminate we rescale W and W − potential problems with stiffness. Specifically, we let W (x) = eµ x V (x), where µ− is the growth rate of the unstable manifold at x = −∞, and we solve instead V  (x) = (A(x, λ) − µ− I )V (x). We initialize V (x) at x = −∞ to be the eigenvector of A− (λ)  (x) at corresponding to µ− . Similarly, it is straightforward to rescale and initialize W x = ∞. Numerically, we use a finite domain for x, replacing the end states x = ±∞ with x = ±L, for sufficiently large L. Some care needs to be taken, however, to make sure that we go out far enough to produce good results. In Appendices B and C, we explore the decay rates of the profile vˆ and A(x, λ) and combine our analysis with numerical convergence experiments to conclude that our domain [−L , L] is sufficiently large. Our experiments, described below, were primarily conducted using L = 12, with relative error bounds ranging mostly between 10−3 and 10−4 . Larger choices of L were needed on the high end of the Mach scale, going up to L = 18 in some cases, to get the relative errors down to 10−4 . In Table 1, we provide a sample of relative errors in D(λ) for large-amplitude shocks.

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Table 1. Relative errors in D(λ) are computed by taking the maximum relative error for 60 contour points evaluated along the semicircle in Fig. 2(a). Samples were taken for varying L and γ , leaving v+ fixed at v+ = 10−4 (Mach M ≈ 1669). We used L = 8, 10, 12, 14, 16 and γ = 1.2, 1.4, 1.666, 1.8. Relative errors were computed using the next value of L as the baseline L

γ = 1.2

γ = 1.4

γ = 1.666

γ = 1.8

8 10 12 14

1.23(−1) 2.07(−2) 2.00(−3) 6.90(−4)

1.16(−1) 1.46(−2) 1.40(−3) 5.31(−4)

1.08(−1) 1.75(−2) 9.85(−4) 4.73(−4)

1.04(−1) 1.78(−2) 7.20(−4) 4.71(−4)

We remark also that in order to produce analytically varying Evans function output, (L), must be chosen analytically. The method of Kato [20, the initial data V (−L) and V p. 99], also described in [8], does this well by replacing the eigenvectors of (23) with analytically defined spectral projectors (see also [5,18]). Before we can compute the Evans function, we first need to compute the traveling wave profile. We use both Matlab’s ode45 routine, which is the adaptive fourthorder Runge-Kutta-Fehlberg method (RKF45), and Matlab’s bvp4c routine, which is an adaptive Lobatto quadrature scheme. Both methods work well and produce essentially equivalent profiles. Our experiments were carried out on the range (γ , M) ∈ [1, 3] × [1.6, 3000]. Recall, for γ ∈ [1, 3], that shocks are known to be stable for M ∈ [1, 1.6], by (15), Sect. 3, hence this completes the study of the range (γ , M) ∈ [1, 3] × [1, 3000] from minimum Mach number M = 1 far into the hypersonic shock regime, and encompassing the entire physically relevant range of γ . For each value of γ , the Mach number M was varied on logarithmic scale with regular mesh from M = 1.6 up to M = 3000. We used 50 mesh points for γ = 1.0 + 0.1k, where k = 1, 2, . . . , 20. For the special values γ = 1.4 and 1.666 (monatomic and diatomic cases), we did a refined study with 400 mesh points. For each value of (γ , M), we computed the Evans function along semi-circular contours that contain the triangular region found in the previous section via our high-frequency bounds; see Fig. 2(a). The ODE calculations for individual λ were carried out using Matlab’s ode45 routine, which is the adaptive 4th -order RungeKutta-Fehlberg method (RKF45), and after scaling out the exponential decay rate of the constant-coefficient solution at spatial infinity, as described in [5–8,19]. This method is known to have excellent accuracy [5,19]; in addition, the adaptive refinement gives automatic error control. Typical runs involved roughly 300 mesh points, with error tolerance set to AbsTol = 1e-6 and RelTol = 1e-8. Values of λ were varied on a contour with 60 points. As a check on winding number accuracy, it was tested a posteriori that the change in argument of D for each step was less than π/25. Recall, by Rouché’s Theorem, that accuracy is preserved so long as the argument varies by less than π along each mesh interval. In all the cases that we examined, the winding number was zero. This indicates that the shocks we considered are spectrally stable, and in view of [21–23], nonlinear stability follows. Moreover, in light of the large Mach numbers considered, this is highly suggestive of stability for all shock strengths.

Stability of Viscous Shocks in Isentropic Gas Dynamics

(a)

243

(b)

Fig. 2. The graph of the contour and its mapping via the Evans function. In (a), we have a contour, which is a large semi-circle aligned on the imaginary axis on the right-half plane (horizontal axis real, vertical axis imaginary). In (b) we have the image of the contour mapped by the Evans function. Note that the winding number of this graph is zero. Hence, there are no unstable eigenvalues in the semi-circle. Together with the high-frequency bounds, this implies spectral stability. Our computation was carried out for γ = 5/3 and v+ = 1 × 10−4 . This corresponds to a monatomic gas with Mach number M ≈ 1669 (see Appendix A)

7. Numerical Evolution of Strong Shocks In this section, we use a standard finite-difference method to simulate perturbed large-amplitude viscous shocks, and show that they converge, as expected, to a translate of the original profile. We do this with a nonlinear Crank-Nicholson scheme with a Newton solver to deal with the nonlinearity. This method provides second-order accuracy and will allow for larger time steps than a naive explicit scheme. By differentiating the viscosity term in (3), we have vt + vx − u x = 0, u x vx uxx − 2 . u t + u x − aγ v −γ −1 vx = v v By implementing the Crank-Nicholson averaging, we obtain − v nj v n+1 j

1 n n + (v n+1 − v n+1 j−1 + v j+1 − v j−1 ) t 4x j+1 1 n n (u n+1 − u n+1 − j−1 + u j+1 − u j−1 ) = 0 4x j+1

and u n+1 − u nj j

1 n n + (u n+1 − u n+1 j−1 + u j+1 − u j−1 ) t 4x j+1 a n+1 n n − γ (v nj )−γ −1 (v n+1 j+1 − v j−1 + v j+1 − v j−1 ) 4x 1 n n n − (u n+1 − 2u n+1 + u n+1 j j−1 + u j+1 − 2u j + u j−1 ) 2(x)2 v nj j+1

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Fig. 3. Snapshots of the evolution of a perturbed viscous shock wave solution generated by our extended Crank-Nicholson scheme (top curve: v against x; bottom curve: u against x, with time t fixed and increasing from figure to figure). The parameters used were γ = 1.4 and v+ = 9 × 10−6 . This corresponds to a diatomic gas with Mach number M ≈ 2877 (see Appendix A). As expected, the wave converges to a translate of the original shock

+

1 n n (u n+1 − u n+1 j−1 + u j+1 − u j−1 ) 16(x)2 (v nj )2 j+1

n+1 n n ×(v n+1 j+1 − v j−1 + v j+1 − v j−1 ) = 0,

where n and j are, respectively, the discretized temporal and spacial indices. To cope with the nonlinearities, we use the Newton solver −1   n+1   n+1    U U Du F Dv F F(U n+1 , V n+1 ) = − , , Du G Dv G k G(U n+1 , V n+1 ) k V n+1 k+1 V n+1 k where F(U n+1 , V n+1 ) and G(U n+1 , V n+1 ) are the above finite difference schemes, and Du F, Dv F, Du G, and Dv G are their corresponding partial derivatives. Hence, we use the previous time step as our initial guess in Newton’s method and then iterate until convergence. In Fig. 3, we see the evolution of a perturbed viscous shock of extremely high Mach number. We used a perturbation with a positive mass so that we could observe the convergence to a translate of the original profile, thus numerically demonstrating nonlinear stability. We remark that perturbed profiles as predicted by the nonlinear theory [22,23] exhibit also a “diffusion wave”, or approximate Gaussian signal, of constant

Stability of Viscous Shocks in Isentropic Gas Dynamics

245

mass, leaving the shock wave on the right and convecting with outgoing characteristic speed, which absorbs the portion of the mass not accounted for by shock shift. This is clearly visible for small- or intermediate-amplitude shocks, but for large-amplitude −1/2 shocks travels with characteristic speed ∼ v+ → +∞, leaving the viewing frame essentially instantaneously.

Appendix A. Mach Number for the p-System The Mach number is defined as M=

u+ − σ , c+

where u + is the downwind velocity, σ is the shock speed, and c+ is the downwind sound speed, all in Eulerian coordinates. By considering the conservation of mass equation, we have ρt + (ρu)x = 0. Hence, the jump condition is given by σ [ρ] = [ρu], which implies, in the original scaling for (1), that σ =

u + v− − u − v+ . v− − v+

Hence, M=

v+ [u] v+ v+ (u − − u + ) u+ − σ = = −s = c+ c+ (v− − v+ ) c+ [v] c+

or  M = 2

u+ − σ c+

2

γ +3

=

s 2 v+2 s 2 v+2 v = = + −γ −1  − p (v+ ) γ γ a0 v+

s2 . a0

To express this in our scaling, which is given in (3), we need to swap the pluses and minuses. Noting that 0 < v+ < v− = 1, we simplify to get γ

M2 =

1 1 − v+ 1 . = γ γa γ v+ 1 − v+

γ

Recalling that a ∼ v+ as v+ → 0, we find that v+ ∼ (γ M 2 )

− γ1

(25)

as M → ∞. In particular, | log v+ | ∼ γ −1 (log γ + 2 log M).

(26)

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Appendix B. Profile Bounds Denote profile equation (4) as v  = H (v, v+ ) := v(v − 1 + a(v −γ − 1)). Lemma B.1. For γ ≥ 1, 0 ≤ x < 1, 1≤

1 − xγ ≤ γ. 1−x

(27)

Proof. By convexity of f (x) = x γ , the difference quotient (27) is increasing in x, bounded above by f  (1) = γ , and below by its value at x = 0.

 Proposition B.2. For γ ≥ 1 and v+ ≤ v ≤ 16 , v+ ≤

1 12 ,

3 − γ (v − v+ ) ≤ H (v, v+ ) ≤ − (v − v+ ). 4 For γ ≥ 1, v+ ≤

1 4γ

, and

3 4

(28)

≤ v ≤ v− = 1,

1 (v − v− ) ≤ H (v, v+ ) ≤ (v − v− ). 2

(29)

Proof. By (5),
(v+ − 1)(v −γ − 1)  H (v, v+ ) = v (v − 1) − −γ v+ − 1
1 − v 
v 
+ + γ −1 = v (v − v+ ) + γ v 1 − v+

1 − v 
1 − v+ γ  + v . = (v − v+ ) v − γ 1 − vv+ 1 − v+ Applying (27) with x =

v+ v ,

we obtain (28) from

v−γ ≤

H (v, v+ ) ≤ v − (1 − v+ ). v − v+

Similarly, we may obtain (29) by the calculation

v γ
1 − v 
1 − v γ  H (v, v+ ) 3 1 + + =v 1− ≥ v − γ v+ ≥ − . γ v−1 v 1−v 4 4 1 − v+ 

Corollary B.3. For γ ≥ 1, 0 < v+ ≤ satisfies

1 12 ,

and v(0) ˆ := v+ +


1  3x e− 4 x ≥ 0, 12
1  x+12 |v(x) ˆ − v− | ≤ e 2 x ≤ 0. 4 |v(x) ˆ − v+ | ≤

1 12 ,

the solution vˆ of (4)

(30a) (30b)

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1 Proof. Observing that (vˆ − v+ )(0) = 12 , we obtain (30a) by Proposition B.2 and the comparison principle for first-order scalar ODE. Likewise, (30b) follows by (29) together with the observation that, by convexity of H , |H | is bounded below by estimates obtained 1 1 1 3 at vˆ = v+ + 12 and vˆ = 43 of 16 and 18 , respectively, so that vˆ traverses [v+ + 12 , 4 ] over 3/4  an x-interval of length ≤ 1/16 = 12.

Remark B.4. From Proposition B.2 and Corollary B.3, we obtain the remarkable fact that, in the scaling we have chosen, vˆ decays up to first derivative to endstates v± at a uniform exponential rate independent of shock strength, despite the apparent singularity at v = v+ . Appendix C. Initialization Error Lemma C.1. For A as in (22), | · | the Euclidean (2 ) matrix operator norm,
2|λ| + 1 + γ 2 (γ − 1)v −1  3x + e− 4 , x ≥ 0, 12
2|λ| + 1 + 2γ 3 (γ − 1)  x+12 e 2 , x ≤ 0. |A(x, λ) − A− (λ)| ≤ 4 |A(x, λ) − A+ (λ)| ≤

(31a) (31b)

ˆ − f (v± )|. As computed in the Proof. By (22), |A(x, λ) − A± (λ)| ≤ 2λ|vˆ − v± | + | f (v) proof of Lemma 5.4, f  (v) ˆ = −1 − aγ (γ − 1)vˆ −γ −1 . Applying (27) to the expression γ γ for a in (5), we obtain v+ ≤ a ≤ γ v+ , so that | f  (v)| ˆ ≤ 1 + γ 2 (γ − 1)vˆ −1 ≤ 1 + γ 2 (γ − 1)v+−1 , yielding (31a) by the Mean Value Theorem and (30a). Bound (31b) follows similarly.

 + and  µ+ be a left eigenvector Theorem C.2. (Simplified Gap Lemma [6,7,15]) Let V + and associated eigenvalue of −A (λ) and suppose that |e(−A

+ − µ+ )x

ˆ | ≤ C1 e−ηx x ≤ 0,

|(A − A+ )(x)| ≤ C2 e−ηx x ≥ 0,

(32)

µ+ x V (x, λ) of (24) with with 0 ≤ ηˆ < η. Then, there exists a solution W = e

(x, λ) − V + (λ)| C1 C2 e−ηx |V x≥L ≤ + (λ)| (η − η)(1 ˆ − ) |V

(33)

provided (η− η) ˆ −1 C1 C2 e−ηL ≤ . Similar estimates hold for solutions of (22) on x ≤ 0. (−A+ −  (−A + A+ ) and imposing the limiting behavior  = V µ+ ) + V Proof. Writing V +   V (+∞, λ) = V , we obtain by Duhamel’s Principle,  +∞ + µ+ )(x−y)   (y)e(−A+ − V (x) = V − (A − A+ )dy, V x

from which the result follows by a straightforward Contraction Mapping argument using (32). See [6,7,15] for details.



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Lemma C.3. For λ satisfying (21), γ ∈ [1, 3], and v+ ≤ η = 1/2γ , ηˆ = 1/4γ , and C1 = 104 , and similarly for x ≤ 0.

1 12 ,

(32)(a) holds with

Proof. Using the inverse Laplace transform representation  1 (−A+ − µ+ )x e = e zt (z + A+ +  µ+ )−1 dz, 2πi where is a contour enclosing the eigenvalues of (−A+ −  µ+ ) and distance ηˆ away, and + + −1 estimating the resolvent norm |(z + A +  µ ) | by Kramer’s rule, we obtain the stated crude bound, and similarly for x ≤ 0.

 For error tolerance θ := 10−k (| log θ | ∼ 2k), define
 L − (θ ) := 2 | log 10−4 | + | log(2γ + 7 + 2γ 3 (γ − 1))| + | log θ | + 12,  4
| log 10−4 | + | log(2γ + 7 + γ 2 (γ − 1)v+−1 )| + | log θ | . L + (θ, v+ ) := 3 Corollary C.4. For λ satisfying (21), we have relative error bounds −

|W1− (−L − , λ) − V − eµ x | , − |V − eµ x |

µ x| + e  + (L + , λ) − V |W 1 ≤ θ. µ+ x | + e |V

Proof. This follows by (33) with (31), Lemma C.3, and (21).

+

(34)



Remark C.5. Combining (34) with (26), we find that L+ ∼

4 (2 log M + 4 + | log 10−4 | + | log θ |) 3

suffices to obtain relative initialization error less than θ for γ ∈ [1, 3] and λ in the computational region (21). For θ = 10−3 , we thus obtain θ tolerance up to M = 3, 000 ∼ 103 for L + ∼ 40. This is conservative, as we have made little effort to optimize bounds, but still within the realm of our experiments. Our numerical convergence studies indicate that L ± = 18 in fact suffices for 10−3 accuracy.

References 1. Alexander, J., Gardner, R., Jonesm, C.: A topological invariant arising in the stability analysis of travelling waves. J. Reine Angew. Math. 410, 167–212 (1990) 2. Alexander, JC, Sachs, R.: Linear instability of solitary waves of a Boussinesq-type equation: a computer assisted computation. Nonlinear World 2(4), 471–507 (1995) 3. Batchelor, G.K.: An introduction to fluid dynamics. Cambridge Mathematical Library. Cambridge: Cambridge University Press, paperback edition, 1999 4. Benzoni-Gavage, S., Serre, D., Zumbrun, K.: Alternate Evans functions and viscous shock waves. SIAM J. Math. Anal. 32(5), 929–962 (electronic) (2001) 5. Bridges, T.J., Derks, G., Gottwald, G.: Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework. Phys. D 172(1–4), 190–216 (2002) 6. Brin, L.Q.: Numerical testing of the stability of viscous shock waves. PhD thesis, Indiana University, Bloomington, 1998 7. Brin, L.Q.: Numerical testing of the stability of viscous shock waves. Math. Comp. 70(235), 1071–1088 (2001)

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8. Brin, L.Q., Zumbrun, K.: Analytically varying eigenvectors and the stability of viscous shock waves. In: Seventh Workshop on Partial Differential Equations, Part I (Rio de Janeiro, 2001), P.L. Dias, D. Marchesin, A. Nelson, C. Tomei, eds. Mat. Contemp. 22, 19–32, (2002) 9. Evans, J.W.: Nerve axon equations. I. Linear approximations. Indiana Univ. Math. J. 21, 877–885 (1971) 10. Evans, J.W.: Nerve axon equations. II. Stability at rest. Indiana Univ. Math. J. 22, 75–90 (1972) 11. Evans, J.W.: Nerve axon equations. III. Stability of the Nerve Impulse. Indiana Univ. Math. J. 22, 577–593 (1972) 12. Evans, J.W.: Nerve axon equations. IV. The stable and the unstable impulse. Indiana Univ. Math. J. 24(12), 1169–1190 (1974) 13. Evans, J.W., Feroe, J.A.: Traveling waves of infinitely many pulses in nerve equations. Math. Biosci. 37, 23–50 (1977) 14. Gardner, R., Jones, C.K.R.T.: A stability index for steady state solutions of boundary value problems for parabolic systems. J. Differ. Eqs. 91(2), 181–203 (1991) 15. Gardner, R.A., Zumbrun, K.: The gap lemma and geometric criteria for instability of viscous shock profiles. Comm. Pure Appl. Math. 51(7), 797–855 (1998) 16. Goodman, J.: Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch. Rat. Mech. Anal. 95(4), 325–344 (1986) 17. Howard, P., Zumbrun, K.: Pointwise estimates and stability for dispersive-diffusive shock waves. Arch. Rat. Mech. Anal. 155(2), 85–169 (2000) 18. Humpherys, J., Sandstede, B., Zumbrun, K.: Efficient computation of analytic bases in Evans function analysis of large systems. Numer. Math. 103(4), 631–642 (2006) 19. Humpherys, J., Zumbrun, K.: An efficient shooting algorithm for evans function calculations in large systems. Physica D 220(2), 116–126 (2006) 20. Kato, T.: Perturbation theory for linear operators. Classics in Mathematics. Berlin: Springer-Verlag, 1995 (Reprint of the 1980 edition) 21. Mascia, C., Zumbrun, K.: Pointwise Green function bounds for shock profiles of systems with real viscosity. Arch. Rat. Mech. Anal. 169(3), 177–263 (2003) 22. Mascia, C., Zumbrun, K.: Stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems. Arch. Rat. Mech. Anal. 172(1), 93–131 (2004) 23. Mascia, C., Zumbrun, K.: Stability of small-amplitude shock profiles of symmetric hyperbolic-parabolic systems. Commun. Pure Appl. Math. 57(7), 841–876 (2004) 24. Matsumura, A., Nishihara, K.: On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas. Japan J. Appl. Math. 2(1), 17–25 (1985) 25. Pego, R.L., Smereka, P., Weinstein, M.I.: Oscillatory instability of traveling waves for a KdV-Burgers equation. Phys. D 67(1–3), 45–65 (1993) 26. Pego, R.L., Weinstein, M.I.: Eigenvalues, and instabilities of solitary waves. Philos. Trans. Roy. Soc. London Ser. A 340(1656), 47–94 (1992) 27. Smoller, J.: Shock waves and reaction-diffusion equations. New York: Springer-Verlag, Second edition, 1994 28. Texier, B., Zumbrun, K.: Relative Poincaré-Hopf bifurcation and galloping instability of traveling waves. Methods Appl. Anal. 12(4), 349–380 (2005) 29. Texier, B., Zumbrun, K.: Galloping instability of viscous shock waves. http://arxiv.org/list/math.Ap/ 0609331, 2006 30. Texier, B., Zumbrun, K.: Hopf bifurcation of viscous shock waves in compressible gas- and magnetohydrodynamics. Toappear Arch. Rat. Mech. Anal., doi:10.1007/s00205-008-0012-x 31. Zumbrun, K.: Multidimensional stability of planar viscous shock waves. Lecture notes, TMR summer school, 1999 32. Zumbrun, K., Howard, P.: Pointwise semigroup methods and stability of viscous shock waves. Indiana Univ. Math. J. 47(3), 741–871 (1998) Communicated by P. Constantin

Commun. Math. Phys. 281, 251–261 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0445-1

Communications in

Mathematical Physics

Nonlinear Dirac Operator and Quaternionic Analysis Andriy Haydys
Fakultät für mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany. E-mail: [email protected] Received: 28 March 2007 / Accepted: 11 September 2007 Published online: 28 March 2008 – © Springer-Verlag 2008

Abstract: Properties of the Cauchy–Riemann–Fueter equation for maps between quaternionic manifolds are studied. Spaces of solutions in case of maps from a K3–surface to the cotangent bundle of a complex projective space are computed. A relationship between harmonic spinors of a generalized nonlinear Dirac operator and solutions of the Cauchy–Riemann–Fueter equation are established. 1. Introduction Nonlinear generalizations of the Dirac operator were known to physicists long ago [3] and appeared in the framework of the σ –model. Much later they were considered by mathematicians [19,22] in the realm of the Seiberg–Witten theory. The basic idea of generalization is to replace the fibre of the spinor bundle C2 ∼ = H, i.e. the simplest hyperKähler manifold, by an arbitrary hyperKähler manifold with suitable symmetries. On the other hand, Anselmi and Fre [1] generalized quaternionic analysis in the form of Fueter [13,20] for maps between arbitrary hyperKähler manifolds. It is natural to expect that there is a connection between these two approaches, since in case of the flat source manifold harmonic spinors are exactly solutions to the Cauchy–Riemann–Fueter equation ∂u ∂u ∂u ∂u −i −j −k = 0, x ∈ H, u : H → H. ∂ x0 ∂ x1 ∂ x2 ∂ x3 One of the purposes of this paper is to establish a link between the generalized Dirac operator and quaternionic analysis. The paper is organized as follows. In Sect. 2 we study properties of the Cauchy– Riemann–Fueter equation for maps between quaternionic manifolds. In particular we show that its solutions are exactly those maps, whose differential has vanishing
This paper is based upon part of the author’s thesis;he was partially supported by the grant “Gauge theory and exceptional geometry” (Universität Bielefeld) while preparing the final form.

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quaternion–linear component. Therefore we call such maps aholomorphic. Usually a hyperKähler manifold comes equipped with some symmetries. It turns out that certain symmetries of the target manifold force aholomorphic maps to be (anti)holomorphic (in the usual complex sense). This in turn allows to apply a well–developed technique of algebraic geometry to compute certain spaces of aholomorphic maps. In the last section we show that the nonlinear Dirac operator can be regarded as an analogue of the ∂–operator in complex geometry. Harmonic spinors are shown to be twisted (in an appropriate sense) aholomorphic maps. 2. Aholomorphic Maps Algebraic preliminaries. Denote by H the R–algebra of quaternions and by Sp(1) the group of quaternions of unitary length. Let (U, J1 , J2 , J3 ) and (V, I1 , I2 , I3 ) be quaternionic vector spaces. One can regard U and V as natural real Sp(1)–representations. Further, denote by W the standard Sp(1)–representation given by the left multiplication on the space of quaternions H. As usual, sp(1) denotes the adjoint representation of Sp(1). Proposition 1. Let dimH U = m, dimH V = n. We have the following decomposition into irreducible components: H om R (U, V ) ∼ = 4mn sp(1) ⊕ R4mn ,

(1)

where R4mn denotes the trivial 4mn–dimensional representation. Proof. First observe, that a choice of quaternionic basis gives an isomorphism U∼ = W ⊗ Rn . Since W ∗ ∼ = W we get = W ⊗ Rm and similarly V ∼ H om R (U, V ) ∼ = W ⊗R W ⊗ Rmn . = U∗ ⊗ V ∼ ∼ WC ⊗C WC = ∼ W ⊗ W ⊗ C2 ∼ Further, since W¯ ∼ = (S 2 W ⊕ = W we have (W ⊗R W )C = 2 2 ∼ C) ⊗ C = (sp(1)C ⊕ C) ⊗ C and the statement follows. Our next aim is to find subspaces B± ⊂ H om R (U, V ) that give the decomposition (1). Consider a linear map C : H om R (U, V ) → H om R (U, V ), C(A) = I1 A J1 + I2 A J2 + I3 A J3 .

(2)

A direct computation shows that C satisfies the equation C 2 +2C −3 = 0. Consequently, C has two eigenvalues 1 and −3 and we have the decomposition H om R (U, V ) = B+ ⊕ B− , where B+ (resp. B− ) denotes the eigenspace corresponding to the eigenvalue 1 (resp. −3). Observe that since C is Sp(1)–invariant, the subspaces B± are also Sp(1)–invariant. The trivial Sp(1)–subrepresentation of H om R (U, V ) is by definition the space H om H (U, V ) of quaternion–linear maps. It is straightforward to check the inclusions H om H (U, V ) ⊂ B− , H om H (U, V ) ⊗ R3 ⊂ B+ , where the second one is given by A1 ⊗ e1 + A2 ⊗ e2 + A3 ⊗ e3 → I1 A1 + I2 A2 + I3 A3 .

Nonlinear Dirac Operator and Quaternionic Analysis

253

By dimension counting we conclude that these inclusions are in fact isomorphisms. Since B+ is Sp(1)–invariant and complementary to the trivial 4mn–dimensional representation, it must be isomorphic to 4mn sp(1). We summarize the above considerations in the following proposition. Proposition 2. The eigenspace B− of the linear map C (2) corresponding to the eigenvalue −3 consists of quaternion–linear maps and we have the following decomposition:

H om R (U, V ) ∼ = H om H (U, V ) ⊕ B+ . Definition 1. We say that a linear map A ∈ H om R (U, V ) between two quaternionic vector spaces (U, J1 , J2 , J3 ) and (V, I1 , I2 , I3 ) is aquaternionic if I1 A J1 + I2 A J2 + I3 A J3 = A. Corollary 1. An R–linear map is aquaternionic if and only if its quaternion–linear component vanishes.

Aholomorphic maps. Let (X, J1 , J2 , J3 ) and (M, I1 , I2 , I3 ) be almost hypercomplex manifolds and u : X → M be a smooth map. Then the differential u ∗ is pointwise an R–linear map between quaternionic vector spaces T.X and T.M. Definition 2. We say that a map u : X → M is aholomorphic, if it satisfies the Cauchy– Riemann–Fueter equation I1 u ∗ J1 + I2 u ∗ J2 + I3 u ∗ J3 = u ∗ .

(3)

Theorem 1. A map u : X → M is aholomorphic if and only if the quaternion–linear component of its differential vanishes at each point.

Aholomorphic maps were studied under a variety of different names. They naturally arise in the supersymmetric σ –model and appeared in physical literature [1,11] for the first time as “triholomorphic maps” or “hyperinstantons”. Such maps naturally appear in higher–dimensional gauge theory [9] and were also studied by Chen [7], Chen and Li [8] (“quaternionic maps”), Wang [23] (“triholomorphic maps”). Joyce [16] (“q–holomorphic functions”) considered the case of the flat target manifold H. Equation (3) was known long ago and was introduced in 1934 by Fueter [13](“regular functions”) for the simplest case of maps u : H → H in his attempts to construct a quaternionic analogue of the theory of complex holomorphic maps. An extensive exposition of this theory can be found in [20]. In the author’s opinion the proposed term aholomorphic map better reflects the properties of maps satisfying Eq. (3), namely the fact that the differential of solutions to (3) has vanishing quaternion–linear component. A reader can find examples of aholomorphic maps in the above mentioned sources. Other examples will appear below. Observe that the Cauchy–Riemann–Fuether equation (3) is elliptic only in the case when a source manifold X is four–dimensional. We will concentrate on this case below. We now consider aholomorphic maps between hyperKähler manifolds. A Riemannian 4n–dimensional manifold M is called hyperKähler if the holonomy group is a subgroup of Sp(n). In other words, a Riemannian manifold (M, g) is hyperKähler if it admits three covariantly constant complex structures I1 , I2 , I3 with quaternionic relations I1 I2 = −I2 I1 = I3 , I12 = I22 = I32 = −id, compatible with the Riemannian structure: g(Il ·, Il ·) = g(·, ·), l = 1, 2, 3. Let ωl denote the Kähler 2–form corresponding to Il .

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Proposition 3. Suppose X and M are both hyperKähler manifolds. If X is also compact and 4–dimensional (i.e. X is a torus or a K3 surface), then for any smooth map u : X → M the following identity holds:
1 1 u ∗ 2L 2 = u ∗ − C(u ∗ )2L 2 − 2 4 3



ωlX ∧ u ∗ ωlM .

(4)

l=1 X

This proposition was essentially proven by Chen and Li [8, Propo. 2.2]. Formula (4) immediately follows from the result of Chen and Li, once you observe that each Kähler form on X is self–dual and that the induced scalar product on Λ2 R4 is given by the sequence ∗ ⊗id · ∧· Λ2 R4 ⊗ Λ2 R4 −−−−→ Λ2 R4 ⊗ Λ2 R4 −−→ Λ4 R4 ∼ = R.

Corollary 2 (Vanishing theorem). Let X and M be as in Proposition 3. If the cohomology class of each Kähler form ωlM on M vanishes, then any aholomorphic map u : X → M is constant.

Let I = I M denote the trivial 3–dimensional subspace of Γ (End(T M)) spanned by I1 , I2 and I3 naturally identified with Im H = sp(1). Definition 3. An isometric action of the group Sp(1) (or S O(3)) on a hyperKähler manifold M is called permuting if the subspace I is preserved and the induced action on I is the adjoint one. The hypothesis of Corollary 2 is automatically satisfied for hyperKähler manifolds that admit a permuting action of Sp(1) or S O(3) [6]. Such actions will play a crucial role in Sect. 3. The class of hyperKähler manifolds that admit a permuting action is quite wide and includes a lot of interesting examples: Hn with the flat metric and its hyperKähler reductions with respect to the zero value of momentum map; different moduli spaces, obtained as infinite–dimensional hyperKähler reductions and in particular moduli spaces of framed instantons [18] over R4 and monopoles [2]. For any quaternionic Kähler manifold with positive scalar curvature Swann [21] constructed a hyperKähler manifold with permuting action of H∗ ⊃ Sp(1). Let us consider a lager class of target hyperKähler manifolds. Namely suppose that (M, I1 , I2 , I3 ) admits only an action of S 1 that fixes one complex structure, say I1 , and rotates the other two, i.e. (L z )∗ I1 = I1 (L z )∗ ,

(L z )∗ Iw (L z¯ )∗ = Izw ,

(5)

where L z : M → M denotes the left shift by z ∈ S 1, w = a + bi is a complex number of unitary length, Iw = a I2 + bI3 . Proposition 4. Let X be a compact hyperKähler 4–dimensional manifold. Assume that the target hyperKähler manifold M admits an isometric action of S 1 that fixes I1 while rotating I2 and I3 . Then a map u : X → M is aholomorphic if and only if it is (J1 , I1 )– antiholomorphic.

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Proof. First notice that existence of an isometric S 1 –action that fixes I1 and rotates the other two complex structures implies that ω2M and ω3M are exact [15]. Indeed, denote by K the Killing vector field of the S 1 –action and by L K the Lie derivative. It follows from (5) that L K ω2M = ω3M . Applying the Cartan formula, one gets ω3M = L K ω2M = d(ı K ω2M ). Therefore equality (4) takes the following form:  1 1 2 2 u ∗  L 2 = u ∗ − C(u ∗ ) L 2 − ω1X ∧ u ∗ ω1M . (6) 2 4 X

Further, since the action of E(u) =

S1

1 u ∗ 2L 2 2

is isometric, the energy functional  1 = u ∗ 2 dvol X , u:X→M 2

(7)

X

is S 1 –invariant. Suppose now that u is aholomorphic. In particular, u is an absolute minimum of the energy functional (7) within its homotopy class α = [u]. Then for each z ∈ S 1 the map u z = L z ◦ u lies in the same homotopy class α and is also an absolute minimum of the energy functional within α. From Eq. (6) we conclude that u z must be aholomorphic: I1 u ∗z J1 + I2 u ∗z J2 + I3 u ∗z J3 = u ∗z . The above equation can be rewritten as I1 u ∗ J1 + (L z¯ )∗ I2 (L z )∗ u ∗ J2 + (L z¯ )∗ I3 (L z )∗ u ∗ J3 = u ∗ . In particular, for z = −1 we get I1 u ∗ J1 − I2 u ∗ J2 − I3 u ∗ J3 = u ∗ . Since u satisfies the Cauchy–Riemann–Fueter equation (3), we obtain I1 u ∗ J1 = u ∗ , i.e. u is (J1 , I1 )– antiholomorphic. On the other hand, it easily follows from the definition that any antiholomorphic map is also aholomorphic. An example of the circle action preserving one complex structure and rotating the other two is the standard fiberwise action on the cotangent bundle T∗ Pn of the complex projective space Pn equipped with the Calabi metric. More generally, Kaledin [17] and independently Feix [10] constructed such metrics on a neighborhood of the zero section in T ∗ Z for real–analytic Kähler manifolds Z . Example 1 (Aholomorphic maps from a K3–surface into T∗ Pn ). It follows from Proposition 4 that any aholomorphic map u : X → T∗ Pn must be (J1 , I1 )–antiholomorphic. The standard antiholomorphic automorphism of Pn induces an I1 –antiholomorphic automorphism on T∗ Pn and therefore we have a natural bijection between the spaces of holomorphic and antiholomorphic maps into T∗ Pn . Further, each holomorphic map u : X → T∗ Pn naturally decomposes into the projection ϕ : X → Pn and a holomorphic section s ∈ Γ (ϕ ∗ T∗ Pn ). Each nonconstant holomorphic map ϕ into the projective space can be obtained as a morphism associated to a linear system of a positive base point free divisor D (see [14] for example). Then ϕ ∗ OPn (1) = O X (D) and the pull–back to X of the short exact sequence dual to the Euler one has the following form: 0 → ϕ ∗ T∗ Pn → (n + 1)O X (−D) → O X → 0. It follows that the lift s must be the zero section. Therefore our problem reduces to description of the space of holomorphic maps X → Pn .

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From now on we assume that X is a K3–surface. Let a map ϕ = ϕ D : X → Pn be given by the complete linear system |D| = P(H 0 (O X (D))) with empty base locus of a positive divisor D. Since D 2 is the degree of ϕ we may assume D 2 ≥ 0. It turns out that base point free divisors D on a K3–surface admit a purely numeric characterization at least if D is big (D 2 > 0). Indeed, for a big divisor D the associated morphism ϕ : X → Pn is generically finite. It follows that dim ϕ(X ) = 2 and by the Bertini’s theorem [4] the divisor D is linearly equivalent to a smooth irreducible curve C with pa (C) = 1 + C 2 /2 = 1 + D 2 /2 > 1. It follows that D is nef and by assumption D is also big (D 2 > 0). On the other hand, the complete linear system |D| of a nef and big divisor D has a non–empty base locus if and only if D = k E + R, where R is a smooth rational curve (R 2 = −2), E is a smooth elliptic curve (E 2 = 0), E · R = 1 and k ≥ 2 [12]. In case when D is big, nef and base point free the associated morphism is a map into the complex projective space of dimension N = 1 + D 2 /2. Now fix a positive integer n. Let ϕ : X → Pn be a holomorphic map and k be the dimension of the projective span of ϕ(X ), that is ϕ decomposes as i ◦ ψ, where i : Pk → Pn is a standard embedding. Then ψ is a morphism given by a projective basis of a linear k–dimensional system of D with k ≤ N = dim |D| = 1 + D 2 /2. One can describe ψ equivalently as the composition of a morphism ϕ D : X → P N , given by a projective basis of the complete linear system |D|, and a projection π : P N \ P N −k−1 → Pk with respect to a subspace P N −k−1 → P N that does not intersect the image of ϕ D . Denote pr by PV N −k (C N +1 ) the projective Stiefel manifold, PV N −k (C N +1 ) −−→ Gr N −k (C N +1 ). Then the space (D, k) of all holomorphic maps ψ : X → Pk that can be obtained by a choice of k + 1 linearly independent sections of O X (D) without base locus is the Zariski–open set pr −1 { V ∈ Gr N −k (C N +1 ) | [V ] ∩ ϕ D (X ) = ∅ } ⊂ PV N −k (C N +1 ).

(8)

In particular, (D, k) is connected. Observe also that P N \ P N −k−1 is the total space of a vector bundle over Pk → P N . Therefore π ◦ ϕ D is homotopic to ϕ D . It follows that the spaces (D, k) and (D  , k  ) lie in the same component of Map(X, Pn ) if and only if D = D  . Summing up the above considerations we get the following result. Proposition 5. Let X be a K3–surface and n be a positive integer. Then the space of all nonconstant aholomorphic maps u : X → T∗ Pn (equivalently, the space of all holomorphic maps ϕ : X → Pn ) is the stratified space  (D, k) × Grk+1 (Cn+1 ), D, k

where D denotes a divisor class on X such that the base locus of the complete linear system |D| is empty; k is an integer, 1 ≤ k ≤ n; (D, k) is a Zariski–open subset (8) of the projective Stiefel manifold PV N −k (C N +1 ), N = 1 + D 2 /2. The subsets (D, k) × Grk+1 (Cn+1 ) and (D  , k  ) × Grk+1 (Cn+1 ) lie in the same component of Map(X, T∗ Pn ) if and only if D = D  .

Observe that any holomorphic map X → Pn can be obtained from a morphism ϕ D , given by a primitive divisor D, composing it with holomorphic maps between complex projective spaces. In the examples below we will indicate only the primitive divisor classes.

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For a K3–surface X the Euclidean lattice (H 2 (X ; Z), ∪) is isomorphic to −2E 8 ⊕3H and is of rank 22. Notice that by the Torelli theorem a K3–surface can be specified by its period point (see [4] for details). The natural map c1 : Pic(X ) → H 2 (X, Z) is injective and the rank ρ of its image, the Neron–Severi group N S(X ), can take any integer value between 0 and 20. We shall consider some examples for small values of ρ (the examples are taken from [5]). 1) ρ = 1, N S(X ) ∼ = 2k 2 . We have just one primitive divisor class D with D 2 = 2k 2 . The complete linear system is base point free and we get a regular holomorphic map 2 ϕ D : X → P1+k . 2) ρ = 2, N S(X ) ∼ = H . In this case the two generators E 1 and E 2 are elliptic curves. The complete linear system |E i | is base point free [12] and defines a regular map ϕ Ei : X → P1 . Any other positive primitive divisor class has the form D = pE 1 + q E 2 , where p and q are positive coprime integers. The base locus of D is empty and therefore we get a map ϕ D : X → P2 pq .   2 1 3) ρ = 2, N S(X ) ∼ = 1 −2 . Let C and R be generators, C 2 = 2, R 2 = −2, C · R = 2. It is easy to check that there are no divisors with vanishing self–intersection number. A divisor D = pC + q R is nef iff p ≥ q. In this case D is also big. Consequently, for coprime p and q, p ≥ q the divisor D is primitive with empty 2 2 base locus and we get a holomorphic map ϕ D : X → P1+ p +2 pq−q . 3. Harmonic Spinors as Twisted Aholomorphic Maps In this section we establish connection between aholomorphic maps and (generalized) harmonic spinors. Before proceeding let us briefly outline the definition of the (generalized) Dirac operator in a form suitable for our purposes. Details can be found in the paper of Pidstrygach [19], where the Dirac operator is defined in a slightly more general context. Algebraic preliminaries. Recall that the group Spin(4) is isomorphic to the product of two copies of Sp(1): Spin(4) = Sp+ (1) × Sp− (1), where we use subscripts “±” to distinguish between different copies. The isomorphism Spin(4)/ ± 1 ∼ = S O(4) is given by [q+ , q− ] → Bq+ ,q− : H → H,

Bq+ ,q− h = q− h q¯+ .

(9)

Recall that W denotes the Sp(1)–representation on H by multiplication on the left. Further, R4 denotes the standard S O(4)–representation. The following homomorphism of Spin(4) = Sp+ (1) × Sp− (1)–representations R4 ⊗ W + → W − ,

h 1 ⊗ h 2 → h 1 h 2 ,

is called the Clifford multiplication, where W ± denote the representations induced by Sp± (1). Let (V, I1 , I2 , I3 ) be a quaternionic vector space. We identify the subspace span (I1 , I2 , I3 ) ⊂ EndR (V ) with Im H = sp(1).

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Lemma 1. Let V be a real representation of the group Sp(1) such that the induced action on EndR (V ) preserves span(I1 , I2 , I3 ). Assume that the induced representation on span(I1 , I2 , I3 ) coincides with the adjoint one. Then VC ∼ = W ⊗ E,

(10)

where E is an Sp(1)–representation. The proof is a straightforward calculation of the actions of i, j, k ∈ Sp(1) on C⊗R V ∼ = C2 ⊗C VI1 , where VI1 denotes the complex vector space (V, I1 ). From (10) one obtains a variant of the Clifford multiplication: R4C ⊗ VC ∼ = R4C ⊗ W + ⊗ E → W − ⊗ E.

(11)

Observe that the homomorphism (11) induces an homomorphism between real parts: R4 ⊗ V → [W − ⊗ E]r . The nonlinear Dirac operator. From now on X 4 is a smooth closed oriented Riemannian manifold. We also assume that X is a spin manifold with a Spin(4)–principal bundle π : P → X . The assumption that X is spin is not essential, since the bundle P can be replaced by a Spin c (4)–bundle, however the exposition becomes a bit clearer in the case of spin manifolds. Let (M, I1 , I2 , I3 ) be a hyperKähler manifold with permuting action of Sp(1) (see Definition 3). Then the group Spin(4) acts on M via the homomorphism Spin(4) → Sp+ (1) and we get the associated fibre bundle: M = P × Spin(4) M. Observe that in case M = H with the standard action of Sp(1) one gets the usual positive spinor bundle W + . Therefore sections of M are called (generalized) spinors. Notice that the space of spinors can be naturally identified with the space of equivariant maps: Γ (M) ∼ = Map Spin(4) (P, M). The Levi–Civita connection on P determines the covariant derivative: ∇v u = u ∗ (ˆv),

u ∈ Map Spin(4) (P, M),

where vˆ denotes the horizontal lift of v ∈ T X . In other words we get a map ∇ : Map Spin(4) (P, M) → Γ (T ∗ X ⊗ π! (u ∗ T M)), where π! (u ∗ T M) → X denotes the factor of u ∗ T M → P by the group action. Remark 1. Strictly speaking, the operator ∇ is not well–defined, since its range depends on the element of the domain. However one can define ∇ as a section of a certain vector bundle as follows. Consider the evaluation map ev : Map Spin(4) (P, M) × P → M, (u, p) → u( p). Then one gets the following diagram: ev ∗ T M ⏐ ⏐ 

−−−−→ T M ⏐ ⏐  ev

Map Spin(4) (P, M) × P −−−−→ M.

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Dividing the first column by the group action one gets a vector bundle E over the infinite dimensional space Map Spin(4) (P, M) × X ; the restriction of E to {u} × X coincides with π! (u ∗ T M). Then ∇ is well–defined as a section of E → Map Spin(4) (P, M) × X . In order to keep the exposition clear, we will not keep to the above formalism of vector bundles over infinite–dimensional spaces. The tangent space of M has a natural structure of quaternionic vector space at each point. Since we have a permuting action of Sp(1) on M, we get from Proposition 1 that ˜ where E˜ is a complex vector bundle over M with an action of Sp(1) TC M ∼ = W + ⊗ E, ( E˜ coicides with (T M, I1 ) as a complex vector bundle, however the action of Sp(1) is different from the induced one). Consequently, we get the Clifford multiplication Cl : T ∗ X ⊗ π! (u ∗ T M) → [W − ⊗ E]r ,

(12)

˜ and W − is the negative spinor bundle over X . where E = π! (u ∗ E) Definition 4 ([19]). The first order differential operator D defined by the sequence  Cl  ∇ D : Γ (M) −−→ Γ T ∗ X ⊗ π! (u ∗ T M) −−→ Γ [W − ⊗ E]r is called a (generalized) Dirac operator. Definition 5. A spinor u such that D u = 0 is called harmonic. Remark 2. The Dirac operator is well–defined as a section of a vector bundle over an infinite–dimensional space similarly to the covariant derivative (see Remark 1) and it is a Fredholm section [19]. In the case when the fibre M of the spinor bundle is a copy of quaternions with the standard action of Sp(1) one recovers the usual linear Dirac operator. Remark 3. Notice that if the target hyperKähler manifold admits a permuting action of S O(3) rather than Sp(1) one needs just the principal S O(4)–bundle of orthonormal frames rather than its Spin(4)–lifting to define the Dirac operator. A well–known example in classical theory is d + + d ∗ : 1 (X ) → 2+ (X ) ⊕ 0 (X ). One can also define a Dirac operator with the help of a Spin c (4)–structure. In this case the target manifold M is required to carry a triholomorphic action of S 1 commuting with the permuting action of Sp(1). The Levi–Civita connection splits T P into horizontal and vertical bundles: T P ∼ = H ⊕ V. Since we have a natural projection pr : P → PS O onto the principal bundle of orthonormal frames of X , the horizontal bundle H has a natural quaternionic structure (J1 , J2 , J3 ), which is defined as follows: a point p ∈ P determines an orthonormal basis v = pr ( p) = (v0 , v1 , v2 , v3 ) of Tπ( p) X and consequently a basis vˆ of H p ; then (J1 , J2 , J3 ) is defined as the unique quaternionic structure1 such that vˆ l = −Jl vˆ 0 for l = 1, 2, 3. Theorem 2. For a map u ∈ Map Spin(4) (P, M) denote by u ∗h the restriction of the differential u ∗ to the horizontal subbundle H ⊂ T P. Then the spinor u is harmonic if and only if the Cauchy–Riemann–Fueter–type equation holds: I1 u ∗h J1 + I2 u ∗h J2 + I3 u ∗h J3 = u ∗h . 1 The basis v provides an isomorphism T X ∼ H; then the quaternionic structure on H, compatible with x = the isomorphism (9), is the right one.

260

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Proof. Pick a point p ∈ P and denote by v = pr ( p) the basis of T X at the point x = π( p) as before. Let u ∈ Map Spin(4) (P, M) be a harmonic spinor, m = u( p) ∈ M. The pull–back of the Clifford multiplication (12) to P can be described by the following sequence(see (10), (11)):

 ∼ R4 ⊗ u ∗ T M → R4 ⊗ u ∗ T M = R4C ⊗ u ∗ TC M C (13) ∼ =(W − ⊗ W + ) ⊗ (W + ⊗ T M) → W − ⊗ T M. Notice that the image of the above map lies automatically in the real part of W − ⊗ T M. Further, R4 and W ± are isomorphic to H as vector spaces. Then the homomorphism (13) is the following map: h ⊗ w → h · 1 ⊗ w − h · j ⊗ I2 w,

h ∈ H, w ∈ T M.

Observe that h = 1 (resp. i, j, k) corresponds to the horizontal lift of v0 (resp. v1 , v2 , v3 ). According to the definition of the Dirac operator take h = 1, w = ∇vˆ 0 u = u ∗ (ˆv0 ) = u ∗h (ˆv0 ); h = i, w = u ∗h (ˆv1 ) . . . and sum up the result. It follows that u is harmonic at the point x if and only if the following equation holds:



 1 ⊗ u ∗h (ˆv0 ) − j ⊗ I2 u ∗h (ˆv0 ) + i ⊗ u ∗h (ˆv1 ) − k ⊗ I2 u ∗h (ˆv1 ) +



 + j ⊗ u ∗h (ˆv2 ) + 1 ⊗ I2 u ∗h (ˆv2 ) + k ⊗ u ∗h (ˆv3 ) + i ⊗ I2 u ∗h (ˆv3 ) = 0. After a simplification one gets

 1 ⊗ u ∗h (ˆv0 ) + I1 u ∗h (ˆv1 ) + I2 u ∗h (ˆv2 ) + I3 u ∗h (ˆv3 )

 + j ⊗ −I2 u ∗h (ˆv0 ) + I3 u ∗h (ˆv1 ) + u ∗h (ˆv2 ) − I1 u ∗h (ˆv3 ) = 0.

(14)

It is easy to see that (14) is equivalent to the single equation u ∗h (ˆv0 )+ I1 u ∗h (ˆv1 )+ I2 u ∗h (ˆv2 )+ I3 u ∗h (ˆv3 ) = 0. Recalling that vˆ l = −Jl vˆ 0 , l = 1, 2, 3 we get ˜ ∗h ) vˆ 0 = u ∗h vˆ 0 − I1 u ∗h J1 vˆ 0 − I2 u ∗h J2 vˆ 0 − I3 u ∗h J3 vˆ 0 = 0. C(u ˜ ∗h ) vˆ 0 = 0 and C(u ˜ ∗h ) Jl vˆ 0 = 0 are Observe that for any fixed l = 1, 2, 3 the equations C(u equivalent. It remains to note that vˆ 0 , J1 vˆ 0 , J2 vˆ 0 and J3 vˆ 0 span the horizontal subspace ˜ ∗h ) = 0. at the point p ∈ P and therefore C(u Corollary 3. A spinor u is harmonic if and only if the horizontal part u ∗h of its differential has no quaternion–linear component.

The above corollary reveals a deep analogy between the Dirac operator and the ∂–operator of complex geometry. Indeed, let (Y, IY ) and (Z , I Z ) be complex manifolds. Assume that a Lie group G acts holomorphically on Z and pick a G–principal bundle π G : PG → Y with a connection. Then the operator ∂ can be defined in the usual way, namely   ∇ ∂ : Γ (PG ×G Z ) −−→ 1 (Y ) ⊗ Γ π!G (u ∗ T Z ) → 1,0 (Y ) ⊗ Γ π!G (u ∗ T Z ) . Further, we also have a splitting of the tangent bundle of PG into vertical and horizontal parts: T PG = V ⊕ H. Observe that the bundle H inherits a complex structure from T Y . Then an equivariant map u satisfies the equation ∂u = 0 if and only if I Z u ∗h IY = u ∗h or, in other words, if and only if the horizontal component has no complex–linear component.

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Acknowledgement. I would like to thank my supervisor Prof. V. Pidstrygach for his constant encouragement and helpful discussions.

References 1. Anselmi, D., Fré, P.: Topological σ -models in four dimensions and triholomorphic maps. Nucl. Phys. B 416(1), 255–300 (1994) 2. Atiyah, M., Hitchin, N.: The geometry and dynamics of magnetic monopoles. M. B. Porter Lectures. Princeton, NJ: Princeton University Press, 1988 3. Bagger, J., Witten, E.: Matter couplings in N = 2 supergravity. Nucl. Phys. B 222(1), 1–10 (1983) 4. Barth, W.P., Hulek, K., Peters, C.A.M., Van de Ven, A.: Compact complex surfaces, Volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Berlin: Springer-Verlag, second edition, 2004 5. Belcastro, S.-M.: Picard lattices of families of K 3 surfaces. Comm. Alg. 30(1), 61–82 (2002) 6. Boyer, C.P., Galicki, K., Mann, B.M.: Quaternionic reduction and Einstein manifolds. Comm. Anal. Geom. 1(2), 229–279 (1993) 7. Chen, J.: Complex anti-self-dual connections on a product of Calabi-Yau surfaces and triholomorphic curves. Commun. Math. Phys. 201(1), 217–247 (1999) 8. Chen, J., Li, J.: Quaternionic maps between hyperkähler manifolds. J. Differ. Geom. 55(2), 355–384 (2000) 9. Donaldson, S.K., Thomas, R.P.: Gauge theory in higher dimensions. In: The geometric universe (Oxford, 1996), Oxford: Oxford Univ. Press 1998, pp. 31–47 10. Feix, B.: Hyperkahler metrics on cotangent bundles. J. Reine Angew. Math. 532, 33–46 (2001) 11. Figueroa-O’Farrill, J.M., Köhl, C., Spence, B.: Supersymmetric Yang-Mills, octonionic instantons and triholomorphic curves. Nucl. Phys. B 521(3), 419–443 (1998) 12. Friedman, R.: Algebraic surfaces and holomorphic vector bundles. Universitext. New York: SpringerVerlag, 1998 13. Fueter, R.: Die Funktionentheorie der Differentialgleichungen ∆u = 0 und ∆∆u = 0 mit vier reellen Variablen. Comment. Math. Helv. 7(1), 307–330 (1934) 14. Griffiths, P., Harris, J.: Principles of algebraic geometry. New York: Wiley-Interscience [John Wiley & Sons], 1978 15. Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc., III. Ser. 55, 59–126 (1987) 16. Joyce, D.: Hypercomplex algebraic geometry. Quart. J. Math. Oxford Ser. (2) 49(194), 129–162 (1998) 17. Kaledin, D.: Hyperkaehler structures on total spaces of holomorphic cotangent bundles. http://arxiv.org/ list/alg-geom/9710026, 1997 18. Maciocia, A.: Metrics on the moduli spaces of instantons over Euclidean 4-space. Commun. Math. Phys. 135(3), 467–482 (1991) 19. Pidstrygach, V.Ya.: Hyper-Kähler manifolds and the Seiberg-Witten equations. Tr. Mat. Inst. Steklova, 246(Algebr. Geom. Metody, Svyazi i Prilozh.), 263–276, (2004) 20. Sudbery, A.: Quaternionic analysis. Math. Proc. Camb. Philos. Soc. 85, 199–225 (1979) 21. Swann, A.: HyperKahler and quaternionic Kahler geometry. Math. Ann. 289(3), 421–450 (1991) 22. Taubes, C.H.: Nonlinear generalizations of a 3-manifold’s Dirac operator. In: Trends in mathematical physics (Knoxville, TN, 1998), Volume 13 of AMS/IP Stud. Adv. Math., Providence, RI: Amer. Math. Soc. pp. 475–486 1999 23. Wang, C.: Energy quantization for triholomorphic maps. Calc. Var. Partial Differ. Eqs. 18(2), 145–158 (2003) Communicated by G.W. Gibbons

Commun. Math. Phys. 281, 263–286 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0491-8

Communications in

Mathematical Physics

Mott Law as Upper Bound for a Random Walk in a Random Environment A. Faggionato1 , P. Mathieu2 1 Dipartimento di Matematica “G. Castelnuovo”, Università “La Sapienza”,

P.le Aldo Moro 2, 00185 Roma, Italy. E-mail: [email protected]

2 Centre de Mathématiques et d’Informatique, Université de Provence,

39 rue Joliot Curie, 13453 Marseille cedex 13, France. E-mail: [email protected] Received: 2 April 2007 / Accepted: 8 January 2008 Published online: 6 May 2008 – © Springer-Verlag 2008

Abstract: We consider a random walk on the support of an ergodic simple point process on Rd , d ≥ 2, furnished with independent energy marks. The jump rates of the random walk decay exponentially in the jump length and depend on the energy marks via a Boltzmann–type factor. This is an effective model for the phonon–induced hopping of electrons in disordered solids in the regime of strong Anderson localization. Under some technical assumption on the point process we prove an upper bound for the diffusion matrix of the random walk in agreement with Mott law. A lower bound for d ≥ 2 in agreement with Mott law was proved in [8]. 1. Introduction 1.1. Physical motivations. Phonon-assisted electron transport in disordered solids in which the Fermi level (set equal to 0 below) lies in a region of strong Anderson localization can be modeled by Mott variable–range hopping of the following interacting particle system in a random environment [17]. The environment is given by ξ := ({xi }, {E i }), where {xi } is an infinite and locally finite set of points in Rd such that each point xi is labeled by an energy mark E i belonging to some finite interval. Given ξ , particles can lie only at points of {xi } and perform random walks on {xi } with hard–core interaction. The probability rate for a jump of a particle at x to the vacant site y, with x = y in {xi }, is given by 
(1.1) r x,y (ξ ) = r0 exp −2|x − y|/∗ − β{E y − E x }+ , where E z is defined by E i for z = xi and {E y − E x }+ = max{E y − E x , 0}. Above | · | denotes the Euclidean norm in Rd , β = 1/(kT ), ∗ denotes the localization radius of wavefunctions and r0 is a constant depending on the material but which depends only weakly on β, |x − y| and the energies E x , E y . Without loss of generality, in what follows we set ∗ = 2, r0 = 1.

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The disorder of the solid is modeled by the randomness of the environment. The points {xi } correspond to the impurities of the disordered solid and the electron Hamiltonian has exponentially localized quantum eigenstates with localization centers xi if the corresponding energies E i are close to the Fermi level. The random set {xi } is a stationary simple point process with finite intensity (i.e. with finite mean number of points in finite boxes) and its stationarity reflects the homogeneity of the medium. Conditioned to {xi }, the energy marks {E i } are supposed to be i.i.d. random variables taking value in a finite interval (set in what follows equal to [−1, 1]) with common distribution ν, such that ν(d E) ∼ c |E|α d E, |E|  1, for a suitable nonnegative exponent α. The independence of the energy marks is compatible with Poisson level statistics, which is a general rough indicator for the localization regime and has been proven to hold for an Anderson model [13]. The exponent α allows to model a possible Coulomb pseudogap in the density of states [17]. In particular, the physically relevant possible values of α are 0 and d − 1, the latter corresponding to the Coulomb pseudogap. The DC conductivity matrix σ (β), measuring the linear response of the solid to a uniform external electric field, would vanish if it were not for the lattice vibrations (phonons) at nonzero temperature. For an isotropic solid at low temperature and with dimension d ≥ 2, Mott law [14,15] predicts that the conductivity decays exponentially as   α+1 σ (β) ∼ exp −c β α+1+d I, β  1, (1.2) where c is a β–independent positive constant and I denotes the identity matrix. The original derivation of (1.2) is based on an heuristic optimization argument, alternative and more robust derivations have been proposed after that (see [1,17] and references therein). Finally, we point out that due to the Einstein relation between σ (β) and the bulk diffusion matrix Dbulk (β) [18], (1.2) is equivalent to the asymptotic behavior   α+1 Dbulk (β) ∼ exp −c β α+1+d I, β  1. (1.3) A mean field version of Mott variable–range hopping at small temperature is given ξ by the continuous–time random walk X t on {xi } such that the jump from x to y, x = y in {xi }, has probability rate given by
 (1.4) cx,y (ξ ) = exp −|x − y| − β(|E x − E y | + |E x | + |E y |)/2 . ξ

We will call X t Mott variable–range random walk. The choice of the transition rates cx,y (ξ ) comes from the fact that, at small temperature, cx,y (ξ ) = r x,y (ξ ) µ(ηx = 1) µ(η y = 0) (1 + o(1)),

β  1,

(1.5)

where ηx is the particle number at site x in the above particle system,  while µ is the Gibbs measure of the particle system w.r.t. the Hamiltonian H = i E i ηxi with zero Fermi level, i.e. µ is the product measure on {0, 1}{xi } such that µ(ηxi = 1) = e−β Ei /(1 + e−β Ei ). In (1.5), the error term is negligible as β → ∞ uniformly in x, y belonging to a finite volume (as in realistic solids), for a typical environment ξ . Another derivation of the

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mean field version (1.4) from the original Mott variable range hopping (1.1) has been obtained in [12] (see also [1, Sect. IV]) by reduction to a random resistor network. The analogue of Mott law (1.3) for Mott variable–range random walk is given by   α+1 β  1, (1.6) D(β) ∼ exp −c β α+1+d I, where D(β) denotes the diffusion matrix of the random walk. For dimension d ≥ 2, a lower bound of D(β) in agreement with (1.6) has been recently proven in [8]. The present work addresses the problem of rigorously deriving an upper bound of D(β) in agreement with (1.6). The one dimensional case presents special features, and rigorous results on the corresponding Mott law have been obtained in [5].

1.2. Model and results. Let us give a precise definition of Mott variable–range random ξ walk X t , generalizing the choice of jump rates (1.4). The environment ξ = ({xi }, {E i }) is defined as follows. Let {xi } be a simple point process, i.e. a random locally finite subset of Rd , whose law is the Palm distribution Pˆ 0 associated to the law Pˆ of a stationary simple point process on Rd with finite intensity. Given {xi }, the energy marks {E i } are i.i.d. random variables having value in [−1, 1] and common law ν (restrictions on ν will be specified later). The law P0 of the environment is the so called ν–randomization of ˆ the Palm distribution Pˆ 0 associated to P. The reason for considering Palm distributions is the following: in order to make the random walk start at a fixed point, taken equal to the origin below, we condition to contain the origin the stationary point process of impurities introduced in the previous subsection. As discussed in Sect. 2, if the stationary process has finite intensity and law ˆ the law of the resulting process is given by the Palm distribution Pˆ 0 associated to P. ˆ P, d In what follows, we write ξ for a generic locally finite subset of R × [−1, 1] such that ξ has at most one point in each fiber {x} × [−1, 1], and we write ξˆ for a generic d ˆ locally finite subset of   R . Both ξ and ξ can be identified with the counting measures δ and δ respectively. Moreover, when there is no ambiguity, given x (x,E) ˆ (x,E)∈ξ x∈ξ d ξ its spatial projection on R will be denoted by ξˆ . We finally recall that the κ–moment ˆ κ > 0, is defined as ρκ of P,   ρκ := EPˆ |ξˆ ∩ [0, 1]d |κ . ˆ Then ρ := ρ1 is called the intensity of the process P. ξ Given a realization of the environment ξ , Mott variable–range random walk X t is defined as the continuous–time random walk on ξˆ = {xi } starting at the origin and jumping from x to y, x = y in {xi }, with probability rate
 cx,y (ξ ) = exp −|x − y| − βu(E x , E y ) , (1.7) where the function u satisfies κ1 (|E x | + |E y |) ≤ u(E x , E y ) ≤ κ2 (|E x | + |E y |)

(1.8)

for some positive constants κ1 ≤ κ2 . To simplify the notation, it is convenient to set cx,x (ξ ) ≡ 0 for all x ∈ {xi }.

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The law Pξ of X t is characterized by the following identities: ξ

Pξ (X 0 = 0) = 1, ξ

ξ

t ≥ 0, x = y, Pξ (X t+dt = y | X t = x) = cx,y (ξ )dt + o(dt),  ξ ξ cx,z (ξ )dt + o(dt), t ≥ 0. Pξ (X t+dt = x | X t = x) = 1 − z ξ

Equivalently, the dynamics of X t can be described as follows: after arriving at site x the particle waits an exponential time with parameter  λx (ξ ) = cx,z (ξ ), (1.9) z

and then jumps to site y, y = x, with probability cx,y (ξ ) . λx (ξ )

(1.10)

By standard methods (see e.g. [3,8, Appendix A]), one can check that the random walk ξ X t is well–defined for P0 –a.a. ξ as soon as Pˆ is ergodic w.r.t. spatial translations and EP0 (λ0 (ξ )) < ∞. As proven in [8] (see also Lemma 1 below) this last condition is equivalent to require that ρ2 < ∞. Assuming Pˆ to be ergodic, ρ12 to be finite and under some additional technical assumption on the law of the environment P0 , in [8] the authors prove that the diffusively rescaled random walk X ξ converges in P0 –probability to a Brownian motion whose covariance matrix coincides with D(β), where D(β) is the diffusion matrix of X ξ defined as the unique symmetric matrix such that    1 ξ EP0 EPξ (X t · a)2 , a ∈ Rd . (a · D(β)a) = lim (1.11) t→∞ t 0 Moreover, they prove the following variational characterization of D(β): P0 (ξ ) ξˆ (d x) c0,x (ξ ) (a · x − ∇x f (ξ ))2 , inf (a, D(β) a) = f ∈L ∞ (P0 )

a ∈ Rd , (1.12)

ˆ where the set ξˆ is defined as the spatial projection of ξ , i.e. ξ = {xi }, and is identified with the counting measure i δxi . Moreover, the gradient ∇x f (ξ ) is defined as ∇x f (ξ ) = f (Sx ξ ) − f (ξ ),

Sx ξ := {(xi − x, E i )} .

In addition, for d ≥ 2 and assuming that ν([−E, E]) ≥ c0 |E|α+1 ,

∀E ∈ [−1, 1],

(1.13)

for some positive constant c0 and some exponent α ≥ 0, the authors prove a lower bound on D(β): for β large enough it holds   α+1 (1.14) D(β) ≥ c1 β −c2 exp −C β α+1+d I,

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where c1 , c2 , C are positive constants independent of β. The above bound is in agreement with Mott law (1.6). We note that the requirement α > 0 in [8, Theorem 1] is due to a typing error. Moreover it is simple to check that, although in [8] the transition rates cx,y (ξ ) are defined as in (1.4), all the results in [8] remain true for transition rates cx,y (ξ ) defined as in (1.7) and assuming only that α > −1 in (1.13). Our main result consists in an upper bound for D(β) in agreement with Mott law. Roughly, we claim that if (1.13) holds with inverted sign, then also (1.14) remains valid with inverted sign. In order to precisely state our technical assumptions we fix some notation. ˆ its p–thinning is the simple Given p ∈ [0, 1] and a simple point process with law P, ˆ point process obtained as follows: for each realization ξ of the process with law Pˆ erase each point independently with probability 1 − p. We will write Pˆ ( p) for the law of the ˆ p–thinning of P. Finally, we recall that the Poisson point process on Rd with intensity ρ > 0 is a random locally finite subset ξˆ ⊂ Rd such that (i) for any A ⊂ Rd Borel and bounded, the cardinality ξˆ (A) = |ξˆ ∩ A| is a Poisson random variable with expectation ρ (A), where (A) is the Lebesgue measure of A; (ii) for any disjoint Borel subsets A1 , . . . , An ⊂ Rd , ξˆ (A1 ), . . . , ξˆ (An ) are independent random variables. We denote by Pˆ ρ the law of the Poisson point process with intensity ρ. The process Pˆ ρ is stationary. We can finally state our main result: Theorem 1. Let P0 be the ν–randomization of the Palm distribution Pˆ 0 associated to a stationary simple point process on Rd , d ≥ 2, with law Pˆ and finite intensity ρ, and let the following conditions be satisfied: • (i) For some constants c0 > 0 and α > −1, 0 < ν([−E, E]) ≤ c0 |E|α+1 ,

∀E ∈ (0, 1] ;

(1.15)

• (ii) There exist positive constants ρ  , K and there exists p ∈ (0, 1] such that, setting K (x) = x + [−K /2, K /2)d and defining the random field Y as Y = {Y (x) : x ∈ K Zd },

Y (x) := ξˆ ( K (x)) ,

(1.16)

ˆ is stochasthen the law of Y when ξˆ is chosen with law Pˆ ( p) (the p–thinning of P) ˆ ˆ tically dominated by the law of Y when ξ is chosen with law Pρ  (the Poisson point process with density ρ  ). Then the d × d symmetric matrix D(β) solving the variational problem (1.12) admits the following upper bound for β large enough:   α+1 (1.17) D(β) ≤ c1 β c2 exp −C β α+1+d I, for suitable β–independent positive constants c1 , c2 , C.

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We recall that due to the Strassen theorem the stochastic domination assumption in condition (ii) above is equivalent to the fact that one can construct on the same probability space processes Y1 = {Y1 (x) : x ∈ K Zd } and Y2 = {Y2 (x) : x ∈ K Zd } in such a way that Y1 (x) ≤ Y2 (x),

∀x ∈ K Zd ,

(1.18)

the law of Y1 equals the law of Y when ξˆ is chosen with law Pˆ ( p) and the law of Y2 equals the law of Y when ξˆ is chosen with law Pˆ ρ  . We have that D(β) ≥ e−Cβ I since |E| is assumed to be bounded from above. In case |E| is bounded from below by a positive constant (which violates the left condition in  (1.15)), then the reverted inequality holds: D(β) ≤ e−C β I. Indeed, due to (1.8) we can −|x−y|−2κ 2 β and, if ν([−E, E]) = 0 bound the transition rates cx,y (ξ ) from below by e for some E > 0, then we can bound cx,y (ξ ) from above by e−|x−y|−2κ1 Eβ . Using the variational characterization (1.12) and a similar formula for the diffusion matrix D˜ of the random walk on ξˆ with probability rate for a jump from x to y = x given by e−|x−y| , it is easy to derive from the above estimate that D(β) ≥ e−2κ2 β D˜ and, if ν([−E, E]) = 0 ˜ for some E > 0, D(β) ≤ e−2κ1 Eβ D. Theorem 1 applies to the case that Pˆ is or is dominated by a stationary Poisson point process Pˆ ρ  , i.e. when one can define random sets (ξ, ξ  ) such that ξ ⊂ ξ  almost surely and ξ, ξ  have marginal distributions Pˆ and Pˆ ρ  respectively. An example is given by Gibbsian random point fields with repulsive interactions (cf. [9] and [4, Sect. 5]). Moreover, the above theorem covers also the case of thinnings of point processes with uniform bounds on the local density, as for example diluted crystals. Note that in this case the point process Pˆ is not stochastically dominated by any stationary Poisson point process. We refer to Sect. 6 for a more detailed discussion. Finally we note that (1.14) and (1.17) may both hold. Assume for instance that points are distributed according to a Poisson point process and suppose that both conditions (1.13) and (1.15) are satisfied for the energy marks (the constants c0 may be different in (1.13) and (1.15)). Then the two bounds (1.14) and (1.17) both hold. In such a situation, one might wonder whether α+1

lim β − α+1+d log D(β)

β↑∞

exists. This question seems to be out of reach of the techniques used here and in [8], and seems to require more sophisticated arguments. 1.3. Overview. In Sect. 2 we recall some definitions and results about point processes (see [6,8] for more details) and state some technical results needed later on. The proof of Theorem 1 is based on the variational formula (1.12) since for each fixed function f ∈ L ∞ (P0 ) the r.h.s. in (1.12) gives an upper bound on (a, D(β)a). In Sect. 3 we derive an upper bound on D(β) by taking special test functions f in the r.h.s. of (1.12). The choice of such test functions is inspired by [16] and is related to the percolation approach of [1,8]. In Sect. 4 we first show that the above upper bound together with some scaling arguments leads to (1.17) if P0 is the ν-randomization of the Palm distribution associated to a stationary Poisson point process. In Sect. 5 we extend the proof to general point processes satisfying assumption (ii).

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As conjectured in the introduction of [8], the leading contribution to the conductivity of the medium as β ↑ ∞ comes from a subset of impurities that converges to a Poisson point process under suitable space rescaling. Due to the relevance of Poisson point processes in Mott’s law and since in the Poissonian case the proof of Theorem 1 is more transparent and simple, we have preferred to treat the Poissonian case and the general case separately. Finally, in Sect. 6 we show that Theorem 1 can be applied to thinnings of point processes with uniform bounds on the local density as diluted crystals. For a more detailed discussion about mathematical aspects and physical motivations of the Mott random walk we refer to [8,17] and references therein. 2. Simple Point Processes In this section we recall some basic definitions and results about simple point processes, referring to [6,8] for more details. In what follows, given a topological set Y we write B(Y ) for the σ –algebra of its Borel subsets. We denote Nˆ the space of simple counting measures ξˆ on Rd , i.e. integer– valued measures such that ξˆ (B) < ∞ for all bounded B ∈ B(Rd ), and ξˆ (x) ∈ {0, 1} for  all x ∈ Rd . One can show that ξˆ ∈ Nˆ if and only if ξˆ = j δx j , where {x j } ⊂ Rd is a locally finite set. Trivially, a simple counting measure ξˆ can be identified with its sup port. Given x ∈ Rd the translated counting measure Sx ξˆ is defined as Sx ξˆ = j δx j −x  if ξˆ = j δx j . The space Nˆ is endowed with the σ –algebra of measurable subsets generated by the maps Nˆ  ξˆ → ξˆ (B) ∈ N,

B ∈ B(Rd ) bounded.

A simple point process (on Rd ) is a measurable map from a probability space into Nˆ . With abuse of notation, we identify a simple point process with its distribution Pˆ on ˆ ˆ x A), for all x ∈ Rd and A ⊂ Nˆ Nˆ . Moreover, one calls it stationary if P(A) = P(S measurable. In this case, we define the κ–moment ρκ as   κ  , κ > 0. (2.1) ρκ := EPˆ ξˆ [0, 1]d Then ρ = ρ1 is the so–called intensity of the process. The Palm distribution Pˆ 0 associated to a stationary simple point process Pˆ on Rd with finite intensity ρ is the probability measure on the measurable subset Nˆ 0 ⊂ Nˆ ,   Nˆ 0 := ξˆ ∈ Nˆ : ξˆ (0) = 1 , characterized by the Campbell identity: 1 ˆ ˆ ˆ ξˆ (d x) χ A (Sx ξˆ ), P0 (A) = P(d ξ ) ρ K d Nˆ QK

∀A ⊂ Nˆ 0 measurable, (2.2)

where Q K = [−K /2, K /2]d , K > 0 (since Pˆ is stationary, the r.h.s. in Campbell identity does not depend on K ). As discussed in [6], the point process Pˆ 0 can be thought of

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as obtained from the point process Pˆ by conditioning the latter to give positive mass at the origin. ˆ Pˆ  one says that Pˆ is stochastically dominated Given two simple point processes P,   ˆ Pˆ  such that almost surely ξˆ ⊂ ξˆ  , by Pˆ , shortly Pˆ  Pˆ , if there exists a coupling of P, with (ξˆ , ξˆ  ) denoting the coupled random sets with marginal distributions given by Pˆ and Pˆ  respectively. We refer to [9] for more details on stochastic domination between point processes. We denote N the space of (marked) simple counting measures ξ on Rd ×[−1, 1] such that ξ(B × [−1, 1]) < ∞ for all B ∈ B(Rd ) bounded, and ξ({x} × [−1, 1]) ∈ {0, 1} for all x ∈ R. One can show that ξ ∈ N if and only if ξ = j δ(x j ,E j ) , where d {(x j , E j )} ⊂ R × [−1, 1] is a locally finite set such that for each x ∈ R there is at most one couple (x j , E j ) with x j = x. Trivially, a simple counting measure ξ can be identified with its support. The value E j is called the mark at x j . For physical reasons, we call it the energy mark. Given ξ ∈ N we write ξˆ for the simple counting measure on Rd defined as ξˆ (B) = ξ(B × [−1, 1]), for all B ∈ B(Rd ) bounded. Given x ∈ Rd the translated  simple counting measure Sx ξ is defined as Sx ξ = j δ(x j −x,E j ) if ξ = j δ(x j ,E j ) . The space N is endowed with the σ –algebra of measurable subsets generated by the maps N  ξ → ξ(B) ∈ N,

B ∈ B(Rd × [−1, 1]) bounded.

A marked simple point process on Rd is a measurable map from a probability space into N . Again, with abuse of notation, we will identify it with its distribution P on N . Moreover, it is called stationary if P(A) = P(Sx A), for all x ∈ Rd and A ⊂ N measurable. In this case, we define the κ–moment ρκ as   κ  ρκ := EP ξˆ [0, 1]d , κ > 0, and call ρ := ρ1 the intensity of the process. The Palm distribution P0 associated to a stationary marked simple point process P with finite intensity ρ is the probability measure on the measurable subset N0 ⊂ N ,   N0 := ξ ∈ N : ξˆ (0) = 1 , characterized by the Campbell identity 1 ξˆ (d x) χ A (Sx ξ ), P(dξ ) P0 (A) = ρKd N QK

∀A ⊂ N0 measurable, K > 0. (2.3)

A standard procedure for obtaining a marked simple point process from a given simple point process on Rd is the ν–randomization, where ν is a probability measure on [−1, 1]: given a realization of the simple point process on Rd , its points are marked by i.i.d. random variables with common law ν. It is simple to check that the ν-randomization of the Palm distribution associated to a given stationary simple point process coincides with the Palm distribution associated to the ν–randomization of the stationary simple point process. We conclude this section recalling some technical results derived in [8]. In particular, point (i) of the following lemma follows from [8, Lemma 1, (i)] and the Monotone Convergence Theorem, while the proof of point (ii) is similar to the proof of [8, Lemma 2]:

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Lemma 1. [8]. Let P0 be the Palm distribution associated to a stationary marked simple point process. function which is non negative or such (i) Let f : N0 × N0 → R be a measurable

ˆ ˆ that ξ (d x) | f (ξ, Sx ξ )| and ξ (d x) | f (Sx ξ, ξ )| are in L 1 (P0 ). Then P0 (dξ ) ξˆ (d x) f (ξ, Sx ξ ) = P0 (dξ ) ξˆ (d x) f (Sx ξ, ξ ) . (ii) Let n be a nonnegative integer such that ρn+1 < ∞. Then n  ξˆ (d x)e−γ |x| < ∞ P0 (dξ ) for any γ > 0. 3. Upper Bounds via Special Test Functions In this section we let P0 be the Palm distribution associated to the ν–randomization of a stationary simple point process with finite intensity and obtain upper bounds on D(β) by choosing special test functions f ∈ L ∞ (P0 ) in the r.h.s. of (1.12). We suppose that ν is not concentrated in a unique value, i.e. ν is not of the form ν = δ E . This implies that for P0 –a.a. ξ , Sx ξ = S y ξ if x, y ∈ ξˆ and x = y. Note that the above assumption is satisfied whenever (1.15) is fulfilled, moreover in the case ν = δ E the β–dependence of the diffusion matrix is trivial. Given ξ ∈ N0 , let E β (ξ ) be a family of non-oriented links in ξˆ , i.e.   (3.1) E β (ξ ) ⊂ {x, y} : x, y ∈ ξˆ and x = y , covariant w.r.t. space translations, i.e. E β (Sx ξ ) = E β (ξ ) − x,

∀ξ ∈ N0 , x ∈ ξˆ .

Consider the graph G β (ξ ) with vertices set V β (ξ ) and edges set E β (ξ ), where   V β (ξ ) = x ∈ ξˆ : ∃y ∈ ξˆ with {x, y} ∈ E β (ξ ) .

(3.2)

(3.3)

Given x, y in V β (ξ ) we say that they are connected if there exists a path in G β (ξ ) going β from x to y. Moreover, given x ∈ ξˆ , we denote by C x (ξ ) the connected component in β β β G (ξ ) containing x if x ∈ V (ξ ), while we set C x (ξ ) = ∅ if x ∈ ξˆ \V β (ξ ). It is clear β that C x (ξ ) depends on the edge set E β (ξ ). Proposition 1. Let P0 be the Palm distribution associated to the ν–randomization of a stationary simple point process with ρ2 < ∞ and ν = δ E for any  E ∈ [−1, 1]. Suppose that for each β > 0 a random graph G β (ξ ) = V β (ξ ), E β (ξ ) , satisfying (3.1), (3.2) β and (3.3), is assigned such that the function N0  ξ → C0 (ξ ) ∈ N is measurable. In addition, suppose that the following assumption (A1) is fulfilled: • (A1) There exists a positive function (β) such that |x − y| > (β) ⇒ {x, y} ∈ E β (ξ ),

∀x, y ∈ ξˆ , ∀ξ ∈ N0 .

(3.4)

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Then, for all i = 1, . . . , d and for all β,   2 β Di,i (β) ≤ 6 P0 (ξ ) ξˆ (d x)c0,x (ξ ) x (i) + (β)2 |C0 (ξ )|2 Ix∈C β (ξ ) , (3.5) 0

where x (i) denotes the i th coordinate of x and Ix∈C β (ξ ) is the characteristic function of 0

β

the event {x ∈ C0 (ξ )}. Moreover, suppose that the following additional assumptions (A2) and (A3) are fulfilled: • (A2) For some ε > 0,

  β lim sup EP0 |C0 (ξ )|2+ε < ∞,

(3.6)

ρ2(1+ε)/ε+1 < ∞

(3.7)

β↑∞

holds where a denotes the smallest integer larger than a. • (A3) There exists a positive function C(β) such that {x, y} ∈ E β (ξ ) ⇒ cx,y (ξ ) ≤ C(β),

∀x, y ∈ ξˆ , ∀ξ ∈ N0 .

(3.8)

Then, for all i = 1, . . . , d, κ ∈ (0, 1) and β large enough, Di,i (β) ≤ c(κ) C(β)κ (1 + (β)2 ),

(3.9)

for a suitable positive constant c(κ) depending on κ, but not on β. In particular, D(β) ≤ d c(κ) C(β)κ (1 + (β)2 ) I.

(3.10)

The proof of the above proposition is obtained by plugging suitable test functions f in the variational formula (1.12). Our test functions are similar to the ones used in [16, Proof of Theorem 3.12]. Remark 1. As one can easily deduce from the proof of the above proposition, the results (3.9) and (3.10) hold for all β > 0 if condition (3.6) is satisfied with lim supβ↑∞ replaced by supβ>0 . Proof. Assume (A1) to be satisfied and, given a positive integer N , consider the test β function f N : N0 → R≥0 defined as follows:    β β − min z (i) : z ∈ C0 (ξ ) if 1 ≤ |C0 (ξ )| ≤ N , β f N (ξ ) = (3.11) 0 otherwise, β

where z (i) denotes the i th coordinate of z. Due to (A2), f N is measurable, while due to (A1), β

β

0 ≤ f N (ξ ) ≤ |C0 (ξ )|(β),

∀ξ ∈ N0 .

(3.12)

In particular,    2 2 β β β x (i) − ∇x f N (ξ ) ≤ 3 x (i) + (β)2 |C0 (ξ )|2 + (β)2 |C0 (Sx ξ )|2 , ∀ξ ∈ N0 . (3.13)

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Due to (1.12),  2 β (1) (2) Di,i (β) ≤ P0 (ξ ) ξˆ (d x)c0,x (ξ ) x (i) − ∇x f N (ξ ) = I N (β) + I N (β), (3.14) where



(1)

I N (β) = (2) I N (β)

=



P0 (ξ ) P0 (ξ )



 2 β ξˆ (d x)c0,x (ξ ) x (i) − ∇x f N (ξ ) Ix∈C β (ξ ) , ξˆ (d x)c0,x (ξ )



0

2 β x (i) − ∇x f N (ξ ) Ix∈C β (ξ ) , 0

and I A denotes that characteristic function of the event A. (1)

β

β

β

• Estimate of I N (β). If x ∈ C0 (ξ ), then C0 (Sx ξ ) = C0 (ξ ) − x and in particular β β β β β |C0 (Sx ξ )| = |C0 (ξ )|. Hence, f N (Sx ξ ) = x (i) + f N (ξ ), i.e. x (i) − ∇x f N (ξ ) = 0, thus implying that (1)

I N (β) = 0.

(3.15)

• Estimate of I N(2) (β). Due to (3.13) we can bound   2 β β (2) I N (β) ≤ 3 P0 (ξ ) ξˆ (d x)c0,x (ξ ) x (i) + (β)2 |C0 (ξ )|2 + (β)2 |C0 (Sx ξ )|2 Ix∈C β (ξ ) .

(3.16)

0

We claim that, due to Lemma 1 (i), β P0 (ξ ) ξˆ (d x)c0,x (ξ )|C0 (ξ )|2 Ix∈C β (ξ ) 0 β = P0 (ξ ) ξˆ (d x)c0,x (ξ )|C0 (Sx ξ )|2 Ix∈C β (ξ ) holds.

(3.17)

0

In order to prove the above claim we define the function f on N0 × N0 as  β c0,x (ξ )|C0 (ξ )|2 Ix∈C β (ξ ) if ζ = Sx ξ, x ∈ ξˆ , 0 f (ξ, ζ ) = 0 otherwise. Note that the above definition is well posed P0 –a.s. since due to the choice ν = δ E we have that Sx ξ = S y ξ if x, y ∈ ξˆ , x = y, for P0 –a.a. ξ . Moreover, given x ∈ ξˆ , we can write β

f (ξ, Sx ξ ) = c0,x (ξ )|C0 (ξ )|2 Ix∈C β (ξ ) , 0

while, using the translation invariance, see (3.2), we get that β

f (Sx ξ, ξ ) = f (Sx ξ, S−x (Sx ξ )) = c0,−x (Sx ξ )|C0 (Sx ξ )|2 I−x∈C β (S 0

=

xξ)

β c0,x (ξ )|C0 (Sx ξ )|2 Ix∈C β (ξ ) . 0

At this point, (3.17) follows by applying Lemma 1 (i) to the above function f . Due to (3.17) we can conclude that   2 (2) (i) 2 β 2 ˆ I N (β) ≤ 6 P0 (ξ ) ξ (d x)c0,x (ξ ) x + (β) |C0 (ξ )| Ix∈C β (ξ ) . (3.18) 0

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• Conclusions. The bound (3.5) follows from (3.14), (3.15) and (3.18). Suppose now β that also assumptions (A2) and (A3) are valid. In particular, if x ∈ C0 (ξ ) then {0, x} ∈ E β (ξ ) and therefore c0,x (ξ ) ≤ C(β). In particular c0,x (ξ ) ≤ C(β)κ e−(1−κ)|x| , for all κ ∈ (0, 1). Due to (3.5) we get   β Di,i (β) ≤ 6 C(β)κ P0 (ξ ) ξˆ (d x)e−(1−κ)|x| |x|2 + (β)2 |C0 (ξ )|2 . (3.19)



Due to Lemma 1 (ii), the integral P0 (ξ ) ξˆ (d x)e−(1−κ)|x| |x|2 is finite. In order to bound the remaining contribution to the integral in the r.h.s. of (3.19), we apply the Hölder inequality: given p, q > 1 with 1/ p + 1/q = 1, β P0 (ξ ) ξˆ (d x)e−(1−κ)|x| (β)2 |C0 (ξ )|2   p 1/ p  1/q β ≤ (β)2 EP0 |C0 (ξ )|2q EP0 holds. ξˆ (d x)e−(1−κ)|x| Choosing 2q = 2+ε (hence p = (2+ε)/ε) we have that, due to (3.6), (3.7) and Lemma 1 (ii), both the expectations in the r.h.s. are uniformly bounded in β, for large enough β. The above observations, together with (3.19), imply (3.9). We claim that this last bound implies (3.10). Indeed, since D(β) is symmetric, there exists an orthonormal basis v1 , v2 , . . . , vd of Rd such that vi is an eigenvector of D(β), i.e. D(β)vi = λi vi . Since D(β) is a nonnegative matrix due to (1.12), all the eigenvalues λi must be nonnegative. Hence, given a ∈ Rd , we can estimate (a, D(β)a) =

d  i=1

 λi (a, vi ) ≤ 2

max λi

1≤i≤d

 d

(a, vi ) = (a, a) max λi ≤ (a, a) 2

1≤i≤d

i=1

d 

λi .

i=1

(3.20) The last term is the trace of the matrix U −1 D(β)U , where U is the d × d orthogonal matrix whose i th column coincides with vi . Note that the matrix U −1 D(β)U is a diagonal matrix with diagonal entries λ1 , λ2 , . . . , λd . Since T r (A B) = T r (B A) for generic matrixes A, B, d 

λi = T r (U −1 D(β)U ) = T r (D(β)) =

i=1

Finally, (3.10) follows from (3.9), (3.20) and (3.21).

d 

D(β)i,i holds.

(3.21)

i=1

 

4. Proof of Theorem 1 in the Poissonian Case In this section we prove Theorem 1 in the special case that P0 is the Palm distribution associated to the ν–randomization of the Poisson point process with intensity ρ > 0. In the next section, we extend the proof to the general case. We write Pˆ ρ for the Poisson point process with density ρ, Pρ for its ν–randomization, Pˆ 0,ρ for the Palm distribution associated to Pˆ ρ , and P0,ρ for the Palm distribution associated to Pρ . Equivalently, P0,ρ is the ν–randomization of Pˆ 0,ρ . In what follows, we write Q L for the cube [−L/2, L/2]d .

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The proof is based on Proposition 1, scaling arguments and continuum percolation. We recall some results of continuum percolation referring to [11] for a more detailed discussion. Given r > 0 we write Br (x) for the closed ball centered at x ∈ Rd with radius r . If x = 0 we simply write Br . We define the occupied region of the Boolean model with radius r driven by ξˆ ∈ Nˆ as Xr (ξˆ ) := ∪x∈ξˆ Br (x) = {y ∈ Rd : d(y, ξˆ ) ≤ r }, where d(·, ·) denotes the Euclidean distance. The connected components in the occupied region will be called occupied components. For A ⊂ Rd , we denote by Wr (A) = Wr (A)[ξˆ ] the union of all occupied components having non–empty intersection with r A. Given A ⊂ Rd and B ⊂ Rd we write A ←→ B if there exists a path inside Xr (ξˆ ) connecting A and B. The following results hold for stationary Poisson point processes [11]: there exists a positive density ρc such that for all bounded subsets A ⊂ Rd , A = ∅,   ρc = inf ρ > 0 : Pˆ ρ [ diam W1 (A) = ∞] > 0 ,   1 Pˆ ρ A ←→ ∂ Q L ≤ e−c(ρ,A) L , ∀L > 0, ∀ρ < ρc

(4.1) (4.2)

for a suitable positive constant c(ρ, A) depending on ρ, A. Above ∂ Q L denotes the border of the cube Q L . Moreover, one can prove that all occupied components in X1 are bounded Pˆ ρ –a.s. for ρ < ρc , while there exists a unique unbounded occupied component in X1 Pˆ ρ –a.s. for ρ > ρc . Note that the function Rd  x → x/r ∈ Rd maps Pˆ ρ in Pˆ ρ r d , namely if ξˆ has law Pˆ ρ then {x/r : x ∈ ξˆ } has law Pˆ ρ r d . This scaling property allows to restate the above results for a fixed density ρ and varying radius r : the positive constant rc (ρ) := (ρc /ρ)1/d = ρ −1/d rc (1)

(4.3)

satisfies   rc (ρ) = inf r > 0 : Pˆ ρ [ diam Wr (A) = ∞] > 0 ,   r Pˆ ρ A ←→ ∂ Q L ≤ e−c(r,ρ,A) L , ∀L > 0, ∀r < rc (ρ),

(4.4) (4.5)

for all bounded subsets A ⊂ Rd , A = ∅, and for a suitable positive constant c(r, ρ, A) depending on r, ρ, A. Moreover, all occupied components in Xr are bounded Pˆ ρ –a.s. for r < rc (ρ), while there exists a unique unbounded occupied component in Xr Pˆ ρ –a.s. for r > rc (ρ). We point out a simple consequence of (4.5): Lemma 2. If r < rc (ρ) then for all bounded sets A ⊂ Rd with A = ∅ and for all s > 0 EPˆ

ρ

  s  < ∞. ξˆ Wr (A)[ξˆ ]

(4.6)

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Proof. Due to the stationarity of Pˆ ρ , we can assume that 0 ∈ A without loss of generality. Fixed p, q > 1 with 1/ p + 1/q = 1, by the Hölder inequality and (4.5) we get ∞     s   ˆ ˆ r EPˆ ξ Wr (A)[ξ ] ≤ EPˆ ξˆ (Q L )s I{A←→∂ Q L−1 } ρ ρ L=1

≤

∞  L=1

 EPˆ

ρ

ξˆ (Q L )s p

1/ p

∞  1/q   1/ p r Pˆ ρ A ←→ ∂ Q L−1 ≤ EPˆ ξˆ (Q L )s p e−cL . L=1

ρ

(4.7) The thesis then follows by observing that, since ξˆ (Q L ) is a Poisson random variable with expectation ρ L d ,   EPˆ ξˆ (Q L )n ≤ c(ρ)L d n , ∀L > 0, n ∈ N, ρ

thus implying that the last member in (4.7) is summable.

 

We can now give the proof of Theorem 1 for Poisson point processes: Proof of Theorem 1. for P0 = P0,ρ . Given ξ ∈ N0 we set d

E(β) := β − α+1+d , ρ(β) := ρ ν ([−E(β), E(β)]) , α+1

(β) := λ β α+1+d ,   E β (ξ ) := {x, y} : x, y ∈ ξˆ , x = y, |E x | ≤ E(β), |E y | ≤ E(β) and |x − y| ≤ (β) , where λ is a β–independent positive constant that will be chosen at the end. Note that, due to assumption (1.15), d(α+1)

0 < ρ(β) ≤ c0 ρ β − α+1+d holds,

(4.8)

while, due to (1.8), {x, y} ∈ E β (ξ )

  α+1 ⇒ cx,y (ξ ) ≤ exp {−(β) ∧ [κ1 β E(β)]} = exp −(λ ∧ κ1 )β α+1+d =: C(β) holds. (4.9)

Then, due to (3.10), in order to conclude the proof of Theorem 1 it is enough

to check that the assumptions of Proposition 1 are fulfilled when the graph G β (ξ ) = V β (ξ ), E β (ξ ) is defined via (3.3). Conditions (3.1) and (3.2) are trivially satisfied. Assumption (A1) is obvious, (A3) has already been checked: it therefore only remains to consider (A2). Since Poisson point processes have finite moments, the only non-trivial condition to be checked is given by (3.6), which can be justified by means of scaling arguments and Lemma 2 as follows.

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As discussed in [6], the process ξˆ with law Pˆ 0,ρ can be constructed by setting ˆξ := ωˆ ∪ {0}, where ωˆ is a Poisson point process with law Pˆ ρ . Let ω be the ν–randomization of the process ωˆ and let E 0 be a random variable with law ν, independent from ω. Then ξ := ω ∪ {(0, E 0 )} has law P0,ρ . Setting
 ωˆ β = x ∈ ωˆ : |E x | ≤ E(β) , r (β) = (β)/2, we get

  β |C0 (ξ )| ≤ 1 + ωˆ β Wr (β) Br (β) [ωˆ β ] .

(4.10)

Above we have used the standard the set ωˆ β with the counting

convention

 to identify   measure x∈ωˆ β δx . Hence, ωˆ β Wr (β) Br (β) [ωˆ β ] is the cardinality of the set obtained

 as intersection of Wr (β) Br (β) [ωˆ β ] with ωˆ β . Moreover, we note that the process ωˆ β , obtained by thinning the Poisson process ωˆ with density ρ, has law Pˆ ρ(β) . We now consider the space rescaling Rd  x → x ρ(β)1/d ∈ Rd . Since the above function maps Pˆ ρ(β) onto Pˆ 1 and points at distance r (β) into points at distance r (β)ρ(β)1/d , the random variable

  (4.11) ωˆ β Wr (β) Br (β) [ωˆ β ] has the same law as

  ωˆ ∗ Wr∗ (β) Br∗ (β) [ωˆ ∗ ] ,

(4.12)

where ωˆ ∗ is a Poisson point process on Rd with intensity 1, i.e. ωˆ ∗ has law Pˆ 1 , and r∗ (β) := r (β)ρ(β)1/d . Since due to (4.8), r∗ (β) ≤ λ(c0 ρ)1/d /2, taking λ such that λ(c0 ρ)1/d /2 ≤ rc (1)/2, the random variable in (4.12) is bounded from above by

  ωˆ ∗ Wrc (1)/2 Brc (1)/2 [ωˆ ∗ ] . (4.13) This last random variable is β–independent and has finite moments due to Lemma 2. Hence all moments of (4.11) are β–independent and finite. Due to (4.10), (3.6) is satisfied and we can apply Proposition 1.   5. Proof of Theorem 1 in the General Case We now explain how one can derive Theorem 1 in the general (non Poissonian) case, under the domination assumption (ii). First we observe that due to assumption (ii) Pˆ has finite moments ρκ for all κ ≥ 0. In fact, assumption (ii) trivially implies that Pˆ ( p) has finite moments. Hence, denoting by X n a generic binomial variable with parameters n, p, we have  ∞ n      n j p (1 − p)n− j Pˆ ξˆ ([0, 1]d ) = n ∞ > EPˆ ( p) ξˆ ([0, 1]d )κ = jκ j n=1

=

∞  n=1

  Pˆ ξˆ ([0, 1]d ) = n E(X nκ ).

j=0

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Since for a suitable positive constant C it holds that C −1 n κ ≤ E(X nκ ) ≤ Cn κ for any n, κ ∈ N+ , this implies that ρκ =

∞ 

  Pˆ ξˆ ([0, 1]d ) = n n κ < ∞,

∀κ ∈ N.

n=1

Define d

E(β) := β − α+1+d .

(5.1)

Since E(β) ↓ 0 as β ↑ ∞, due to (1.15) we can find β∗ such that γ := ν ([−E(β∗ ), E(β∗ )]) ≤ p. Since the γ –thinning Pˆ (γ ) is stochastically dominated by the p–thinning Pˆ ( p) , assumption (ii) in Theorem 1 remains valid with p replaced by γ . Hence, without loss of generality, we can assume that γ = p in assumption (ii), i.e. p = ν ([−E(β∗ ), E(β∗ )])

(5.2)

for some β∗ . In what follows we take β ≥ β∗ and define α+1

(β) := λβ α+1+d , (5.3)   E β (ξ ) := {x, y} : x, y ∈ ξˆ , x = y, |E x | ≤ E(β), |E y | ≤ E(β) and |x − y| ≤ (β) , (5.4) where the positive β–independent constant λ will be fixed at the end.  We want to apply Proposition 1 where, given E β (ξ ), the graph G β (ξ ) = V β (ξ ), E β (ξ ) is defined by (3.3). Trivially, the set E β (ξ ) satisfies (3.1) and (3.2), and condition (A1) is fulfilled. Moreover, due to (1.8),   α+1 {x, y} ∈ E β (ξ ) ⇒ cx,y (ξ ) ≤ exp {−(β) ∧ [κ1 β E(β)]} ≤ exp −c β α+1+d =: C(β), for some β–independent positive constant c. Therefore, condition (A3) is satisfied. Moreover, as already observed, ρκ < ∞ for all κ > 0. Hence, in order to obtain the bound (3.10), which corresponds to (1.17), we only need to verify (3.6). We will prove that   β lim sup EP0 |C0 (ξ )|3 < ∞. (5.5) β↑∞

Due to the Campbell identity (2.3), which holds also with Q K replaced by K (0) [6], we can write    1 β β 3 ˆ EP0 |C0 (ξ )|3 = ξ (d x)|C , E (S ξ )| x P 0 ρKd K (0) ˆ where P denotes the ν–randomization of P. Let us define the conditioned measure ν∗ := ν (· | |E 0 | ≤ E(β∗ )) .

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Then, due to (5.2), we have the identity ν ([−E(β), E(β)]) = p ν∗ ([−E(β), E(β)]) .   Hence, for β ≥ β∗ , the random set x ∈ ξˆ : |E x | ≤ E(β) with ξ chosen with law P   and the random set x ∈ ξˆ : |E x | ≤ E(β) with ξ chosen with law P∗ , defined as the

β (ξ ) = V β (ξ ), ν∗ –randomization of P ( p) , have the same distribution. Since the graph G    E β (ξ ) is univocally determined by the set x ∈ ξˆ : |E x | ≤ E(β) , we conclude that   β EP0 |C0 (ξ )|3 =

1 EP ρKd ∗

 K (0)

β ξˆ (d x)|C0 (Sx ξ )|3 .

(5.6)

In order to bound the r.h.s. of (5.6) using the domination assumption (ii), we consider the partition of Rd in the cubes K (x), x ∈ K Zd , and to each A ⊂ Rd we associate the sets   VK (A) := x ∈ K Zd : K (x) ∩ A = ∅ ,  √  E K (A) := {x, y} : x, y ∈ VK (A), x = y, |x − y| ≤ (β) + d K . We define S K (A) as the connected cluster in the graph G K (A) = (VK (A), E K (A)) containing the origin if 0 ∈ VK (A), and as the empty set if 0 ∈ VK (A). Finally, we set C K (A) := ∪x∈SK (A) ( K (x) ∩ A) .   β If A = x ∈ ξˆ : |E x | ≤ E(β) we will simply write C K (ξ ) for C K (A).

(5.7)

Lemma 3. For all x ∈ K (0), β

β

C0 (Sx ξ ) + x ⊂ (ξ ∩ K (0)) ∪ C K (ξ ) holds. β

(5.8)

β

Proof. Due to the covariant property (3.2), C0 (Sx ξ ) + x = C x (ξ ). If this set is empty β the thesis is trivially true. Otherwise suppose that z ∈ C x (ξ ). If z ∈ K (0) then z belongs to the r.h.s. of (5.8). If z ∈ K (u) for some u ∈ K Zd \ {0}, by following the path connecting x to z in G β (ξ ) we can define a sequence of distinct points u 0 = 0, u 1 , u 2 , . . . , u n−1 , u n = u in K Zd such that for each i, 0 ≤ i ≤ n −1, there exist points ai ∈ K (u i ), bi ∈ K (u i+1 ) with {ai , bi } ∈ E β (ξ ). Hence {u i , u i+1 } ∈ E K (A) for all 0 ≤ i ≤ n − 1, where A = {x ∈ ξˆ : |E x | ≤ E(β)}. This proves that 0 and u are connected in the graph G K (A). In particular, u ∈ S K (A) and therefore β  z ∈ K (u) ∩ A ⊂ C K (ξ ).  Due to the above lemma we get the bound    c β 3 3 ˆ ˆ ξ (d x) ξ ( r.h.s. of (5.6) ≤ E (0)) + |C (ξ )| K P K ρ K d ∗ K (0)     β = c p EP∗ ξˆ ( K (0))3 + c p EP∗ |C K (ξ )|3

(5.9)

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(recall that P∗ is the ν∗ –randomization of P ( p) ). Since     EP∗ ξˆ ( K (0))3 = EP ( p) ξˆ ( K (0))3 < ∞, in order to conclude the proof we only need to prove that   β lim sup EP∗ |C K (ξ )|3 < ∞. β↑∞

Let us derive from the domination assumption (ii) that     β β EP∗ |C K (ξ )|3 ≤ EP∗,ρ  |C K (ξ )|3 ,

(5.10)

(5.11)

where P∗,ρ  is the ν∗ –randomization of the Poisson point process Pˆ ρ  . To this aim, we define   ξˆ ∈ N .

K (ξˆ ) = ξˆ ( K (x)) : x ∈ K Zd , We claim that, given a marked point process Q obtained as ν∗ –randomization of a stationary simple point process, the conditional expectation    β (5.12) EQ |C K (ξ )|3  K is an increasing function in K that does not depend on Q. In order to prove this statement, we write Bin(N , p) for a generic binomial variable with parameters N , p and recall that Bin(N , p) is stochastically dominated by Bin(N  , p) if N ≤ N  . Given 

d

K (ξˆ ), the random variables ax (ξ ), x ∈ K Z defined as   ax (ξ ) = {z ∈ ξˆ ∩ K (x) : |E z | ≤ E(β)} are independent binomial r.v.’s with parameters ξˆ( K (x)), ν∗ [−E(β), E(β)]. In par ticular, the conditional law of ax (ξ ), x ∈ K Zd given K does not depend on Q  β  and it increases with K . Besides, the cardinality C K (ξ ) is an increasing function

 of ax (ξ ), x ∈ K Zd . Since the composition of increasing functions is increasing, we conclude that the conditional expectation (5.12) is an increasing function of K , independent of Q. This proves our claim. Since by assumption (ii) the random field K (ξˆ ) with ξˆ chosen with law Pˆ ( p) is stochastically dominated by the random field K (ξˆ ) with ξˆ chosen with law Pˆ ρ  , we obtain that      β β EP∗ |C K (ξ )|3 = EP∗ EP∗ |C K (ξ )|3 | K (ξˆ )       β β = EPˆ ( p) EP∗ |C K (ξ )|3 | K (ξˆ ) ≤ EPˆ  EP∗ |C K (ξ )|3 | K (ξˆ ) ρ      β β 3 ˆ = EPˆ  EP∗,ρ  |C K (ξ )| | K (ξ ) = EP∗,ρ  |C K (ξ )|3 , ρ

thus concluding the proof of (5.11).

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Due to (5.11), in order to derive (5.10) and complete the proof of Theorem 1 we only need to show that   β lim sup EP∗,ρ  |C K (ξ )|3 < ∞. (5.13) β↑∞

To this aim we will use scaling and percolation arguments as in the previous section.   ˆ The random set A = x ∈ ξ : |E x | ≤ E(β) , where ξ is chosen with law P∗,ρ  , is a Poisson point process with intensity µ(β) = ρ  ν∗ ([−E(β), E(β)]) . β

Hence, by definition of C K (ξ ) (see (5.7)), we have     β EP∗,ρ  |C K (ξ )|3 = EPˆ |C K (ξˆ )|3 . µ(β)

Let

(5.14)

√ r (β) := 2 d(β)/3.

Recall that Bs denotes the closed ball of radius s centered at the origin. Using the same notation as in the previous section, for β large enough we can bound     3  3 ˆ ˆ ˆ . (5.15) |C K (ξ )| ≤ EPˆ EPˆ ξ Wr (β) (Br (β) )[ξ ] µ(β)

µ(β)

By the scaling invariance of the Poisson point process, for each γ > 0,     3  3  ˆ ˆ ˆ ˆ EPˆ = EPˆ γ d ξ Wr (β)/γ (Br (β)/γ )[ξ ] . ξ Wr (β) (Br (β) )[ξ ] µ(β)

µ(β)

Taking

−1/d

γ := µ(β)−1/d = ρ  ν∗ [−E(β), E(β)] , we get EPˆ

µ(β)

    3  3  ξˆ Wr (β) (Br (β) )[ξˆ ] = EPˆ ξˆ Wr (β)/γ (Br (β)/γ )[ξˆ ] . 1

(5.16)

Due to the definition of ν∗ , E(β) and assumption (1.15) we get that α+1

γ ≥ c β α+1+d for some positive constant c depending only on p, ρ  , d and on the constant c0 appearing α+1 in (1.15). Since (β) = λβ α+1+d , then √ √ √ α+1 r (β)/γ = 2 d(β)/(3γ ) ≤ 2 d(β)β − α+1+d /(3c) = 2 dλ/(3c). (5.17) It is enough to choose λ such that √ 2 dλ/(3c) ≤ rc (1)/2.

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With this choice (5.16) and (5.17) imply that     3  3  ξˆ Wr (β) (Br (β) )[ξˆ ] ≤ EPˆ ξˆ Wrc (1)/2 (Brc (1)/2 )[ξˆ ] . (5.18) sup EPˆ β>0

µ(β)

1

The r.h.s. in the above inequality is finite due to Lemma 2. This observation together with (5.14), (5.15) and (5.18) implies (5.13). This concludes the proof of Theorem 1. Remark 2. In the case that Pˆ is stochastically dominated by the Poisson point process Pˆ ρ  one can give a much simpler proof of Theorem 1, that we describe in what follows. Denote by P and Pρ  the ν–randomization of Pˆ and Pˆ ρ  , respectively. Due to the definition of stochastic domination given in Sect. 2, one can exhibit a coupling between P and Pρ  such that ξ ⊂ ξ  almost surely, with (ξ, ξ  ) denoting the random sets with marginal distributions given by Pˆ and Pˆ ρ  , respectively. Consider the graph G β (ξ ) introduced in Sect. 4. Due to the Campbell identity (2.3), we can write    

 1 β β ξˆ (d x) |C0 (Sx ξ )|3 . Fβ (ξ ) := EP0 |C0 (ξ )|3 = EP Fβ (ξ ) , ρ Q1 Since Fβ (ξ ) ≤ Fβ (ξ  ) if ξ ⊂ ξ  and due to the above coupling between P and Pρ  , we can conclude that

  EP Fβ (ξ ) ≤ EPρ  Fβ (ξ ) . Due to the Campbell identity (2.3), the r.h.s. in the above expression equals ρ  EP0,ρ    β |C0 (ξ )|3 which, as proven in Sect. 4, is bounded from above uniformly in β. Hence we have that   ρ   β β sup EP0 |C0 (ξ )|3 ≤ sup EP0,ρ  |C0 (ξ )|3 < ∞. (5.19) ρ β>0 β>0 At this point it is enough to apply Proposition 1: condition (3.6) is fulfilled due to (5.19), while all other conditions can be easily checked. 6. Point Processes with Uniform Bounds on the Local Density Let us prove that the conditions of Theorem 1 are fulfilled by stationary point processes with uniform bounds: Proposition 2. Let Pˆ be a stationary simple point process such that, for suitable positive constants K and N , ξˆ ( K (x)) ≤ N ,

∀x ∈ K Zd ,

Pˆ a.s.,

(6.1)

where K (x) = x + [−K /2, K /2)d . Then, for each q ∈ (0, 1], the q–thinning Pˆ (q) of Pˆ satisfies condition (ii) in Theorem 1. In order to prove the above statement, we use a standard result of stochastic domination (cf. Lemma 1.1 in [10]):

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Lemma 4. Let p ∈ [0, 1] and let σ = σx : x ∈ Zd be a random field s.t. σx ∈ {0, 1} for all x ∈ Zd . Suppose that   P σx = 1 | σ = ζ on Zd \{x} ≤ p, (6.2) for almost all ζ ∈ {0, 1}Z and for all x ∈ Zd . Then σ is stochastically dominated by the Bernoulli site percolation with parameter p, i.e. one can define random fields (σ  , ω) with d

σx ≤ ωx

∀x ∈ Zd ,

a.s., 

σ  having the same law of σ and ω being given by ω = ωx : x ∈ Zd , where ωx are i.i.d. random variables taking value 1 with probability p and value 0 with probability 1 − p. Proof of Proposition 2.

random fields σ = σx as  1 σx (ξˆ ) = 0

If q < 1 set p = q, otherwise fix some p ∈ (0, 1). Consider the 

 d : x ∈ Zd , τ = τx : x ∈ Zd having value in {0, 1}Z defined  if ξˆ ( K (x)) ≥ 1, 1 if ξˆ ( K (x)) ≥ N , τx (ξˆ ) = otherwise ; 0 otherwise.

Then, given ζ ∈ {0, 1}Z , we have   Pˆ ( p) σx = 1 | σ = ζ on Zd \{x} ≤ 1 − (1 − p) N =: p  . d

On the other hand, the random field τ with ξˆ chosen with Poisson law Pˆ ρ  is a Bernoulli site percolation with parameter p˜ := P(Z ≥ N ), where Z is a Poisson variable with mean ρ  K d . Since p  < 1 we can choose ρ  large enough so that p  ≤ p. ˜ Due to the ˆ previous lemma, we can conclude that the random field σ with ξ chosen with law Pˆ ( p) is stochastically dominated by the random field τ with ξˆ chosen with Poisson law Pˆ ρ  . Recall the definition of the random field Y from Theorem 1. Due to the transitivity of stochastic domination and since Y (x)[ξˆ ] ≤ N σx (ξˆ ), N τx (ξˆ ) ≤ Y (x)[ξˆ ],

∀x ∈ Zd , for Pˆ ( p) –a.a. ξˆ , ∀x ∈ Zd , ∀ξˆ ,

(the first inequality follows from (6.1)), we can conclude that the random field Y with ξˆ chosen with law Pˆ ( p) is stochastically dominated by the random field Y with ξˆ chosen with law Pˆ ρ  .   As a corollary of Proposition 2, we obtain that Theorem 1 can be applied to crystals or diluted crystals. In order to be more precise, let us start with a crystal, i.e. (cf. [2]) a locally finite set  ⊂ Rd such that for a suitable basis v1 , v2 , . . . , vd of Rd , −x =

∀x ∈ G := {z 1 v1 + z 2 v2 + · · · + z d vd : z i ∈ Z ∀i} holds.

Let  be the elementary cell  := {t1 v1 + t2 v2 + · · · + td vd : 0 ≤ ti < 1 ∀i} . Note that both the group G and the cell  depend on the basis v1 , v2 , . . . , vd .

(6.3)

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Let ω = (ωx : x ∈ ) be a site Bernoulli percolation on  with parameter p ∈ (0, 1] and let V be a random vector independent of ω, chosen in the elementary cell  with uniform distribution. Then consider the simple point process  ζˆ := ωx δV +x , x∈

obtained from the set  by a spatial randomization and a p–thinning, and call Pˆ its law. The following holds: Proposition 3. The simple point process ζˆ with law Pˆ is stationary and does not depend on the specific basis v1 , v2 , . . . , vd satisfying (6.3). Moreover, its Palm distribution Pˆ 0 is given by Pˆ 0 =

 1 Pu , | ∩ |

(6.4)

u∈∩

where Pu is the law of the simple point process  ζˆu := δ0 + ωx δx−u . x∈\{u}

ˆ the above result implies that Pˆ 0 Since the Palm distribution Pˆ 0 depends only on P, does not depend on the specific basis v1 , v2 , . . . , vd . If p = 1, the Palm distribution corresponds to choosing a point u of the crystal inside the elementary cell  with uniform probability and translating the crystal of −u. If p < 1, in addition to the previous step one erases points different from the origin independently with probability 1 − p. The resulting simple point process is what we called p–diluted crystal obtained from . Trivially, it has uniform bounds in the local density. Hence, due to Proposition 2, it fulfills the assumptions of Theorem 1. Proof. Due to the translation invariance (6.3) one easily proves that Pˆ is stationary. Let us prove that it does not depend on the specific basis v1 , v2 , . . . , vd . Given a Borel subset A ⊂ Rd with  finite positive Lebesgue measure, we define Pˆ A as the law of the simple point process x∈ ωx δW +x , where ω is a site Bernoulli percolation on  with parameter p and W is a vector independent from ω and chosen in A with uniform probability. Note that Pˆ = Pˆ  . By means of (6.3) one can easily check that Pˆ  = Pˆ A if A is a union of sets of the form  + x, x ∈ G. Now, consider the elementary cell  associated to another basis v1 , v2 , . . . , vd satisfying (6.3). By the previous observation, Pˆ  = Pˆ N  for each positive integer N . Define A N as the union of all the sets of the form  + x, x ∈ G, included in N  . Then Pˆ  = Pˆ A N . Finally observe that, given a bounded measurable function f : Nˆ → R,     EPˆ  ( f ) − EPˆ ( f ) ≤ C(,  , f )/N holds. N

AN

Hence the same estimate holds with Pˆ N  and Pˆ A N replaced by Pˆ  and Pˆ  , respectively. Taking N ↑ ∞, we conclude that Pˆ  = Pˆ  . Hence the law Pˆ does not depend on the specific basis v1 , v2 , . . . , vd satisfying (6.3).

Mott Law as Upper Bound for a Random Walk in a Random Environment

285

We prove the characterization (6.4) of the Palm distribution Pˆ 0 by means of the Campbell identity (2.2). Take a bounded measurable function f : Nˆ 0 → R and denote ˆ It is simple to check that by ρ the intensity of P. ρ = p| ∩ |/||,

(6.5)

where | ∩ | denotes the cardinality of  ∩  and || denotes the Lebesgue volume of . Since (2.2) holds also with Q K replaced by , we get that EPˆ ( f ) = 0

1 Eˆ ρ|| P

 

ξˆ (x) f (Sx ξˆ ) .

(6.6)

 Let us define ω as the support of the measure x∈ ωx δx . Denoting by Eω the expectation w.r.t. the site Bernoulli percolation ω we can write ⎛ ⎞⎤  1 ξˆ (x) f (Sx ξˆ ) = Eω ⎣ dy ⎝ f (ω + y − z)⎠⎦ ||   z∈∩(ω +y) ⎛ ⎞⎤ ⎡  1 = Eω ⎣ dy ⎝ f (ω − u)⎠⎦ ||  u∈(−y)∩ω ⎛ ⎞⎤ ⎡  1 = Eω ⎣ dy ⎝ ωu f ((ω − u) ∪ {0})⎠⎦ ||  u∈(−y)∩ ⎡ ⎛ ⎞⎤  1 = pEω ⎣ dy ⎝ f ((ω − u) ∪ {0})⎠⎦ ||  u∈(−y)∩  p = dy Eω [ f ((ω − u) ∪ {0})] . || 

 EPˆ



⎡

u∈(−y)∩

(6.7) Due to (6.3), it is simple to check that the last summation does not depend on y. Hence from the above identities (6.5), (6.6) and (6.7) we get that   p p Eω [ f ((ω − u) ∪ {0})] = E Pu ( f ) ρ|| ρ|| u∈∩ u∈∩  1 E Pu ( f ), = | ∩ |

EPˆ ( f ) = 0

(6.8)

u∈∩

thus concluding the proof of (6.4).

 

Acknowledgement. One of the authors, A.F., thanks the Centre de Mathématiques et d’Informatique, Université de Provence, for the kind hospitality and acknowledges the financial support of GREFI–MEFI.

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References 1. Ambegoakar, V., Halperin, B.I., Langer, J.S.: Hopping conductivity in disordered systems. Phys. Rev. B 4(8), 2612–2620 (1971) 2. Ashcroft, N.W., Mermin, N.D.: Solid state physics. Philadelphia PA: Saunders College, Publishing, 1976 3. Breiman, L.: Probability, Reading, MA: Addison–Wesley, 1953 4. Caputo, P., Faggionato, A.: Isoperimetric inequalities and mixing time for a random walk on a random point process. Ann. Appl. Probab. nos. 5/6, 1707–1744 (2007) 5. Caputo, P., Faggionato, A.: Diffusivity in one-dimensional generalized Mott variable-range hopping models. http://arxiv.org/list/math.PR/07101253, 2007 6. Daley, D.J., Vere–Jones, D.: An Introduction to the theory of point processes. New York: Springer, 1988 7. De Masi, A., Ferrari, P.A., Goldstein, S., Wick, W.D.: An invariance principle for reversible Markov processes. Applications to Random Motions in Random Environments. J. Stat. Phys. 55, 787–855 (1989) 8. Faggionato, A., Schulz–Baldes, H., Spehner, D.: Mott law as lower bound for a random walk in a random environment. Commun. Math. Phys. 263, 21–64 (2006) 9. Georgii, H.-O., Küneth, T.: Stochastic comparison of point random fields. J. Appl. Probab. 34, 868–881 (1997) 10. Liggett, T.M., Schonmann, R.H., Stacey, A.M.: Domination by product measures. The Annals of Probability 25, 71–95 (1997) 11. Meester, R., Roy, R.: Continuum Percolation. Cambridge: Cambridge University Press, 1996 12. Miller, A., Abrahams, E.: Impurity Conduction at Low Concentrations. Phys. Rev. 120, 745–755 (1960) 13. Minami, N.: Local fluctuation of the spectrum of a multidimensional Anderson tight binding model. Commun. Math. Phys. 177, 709–725 (1996) 14. Mott, N.F.: Conduction in non-crystalline materials. III. Localized states in a pseudogap and near extremities of conduction and valence bands. Phil. Mag. 19, 835–852 (1969) 15. Mott, N.F.: Charge transport in non-crystalline semiconductors. Festkörperprobleme 9, 22–45 (1969) 16. Piatnitski, A., Remy, E.: Homogenization of elliptic difference operators. SIAM J. Math. Anal. 33, 53–83 (2001) 17. Shklovskii, B.I., Efros, A.L.: Electronic Properties of Doped Semiconductors. Berlin: Springer, 1984 18. Spohn, H.: Large Scale Dynamics of Interacting Particles. Berlin: Springer, 1991 Communicated by M. Aizenman

Commun. Math. Phys. 281, 287–347 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0496-3

Communications in

Mathematical Physics

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation Shijin Deng1 , Weike Wang1 , Shih-Hsien Yu2 1 Department of Mathematics, Shanghai Jiaotong University, Shanghai, PRC.

E-mail: [email protected]; [email protected]

2 Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore,

Singapore. E-mail: [email protected] Received: 4 April 2007 / Accepted: 16 December 2007 Published online: 6 May 2008 – © Springer-Verlag 2008

Abstract: In this paper Green’s functions for the Boltzmann equation around a global Maxwellian are used to construct the non-characteristic nonlinear Knudsen layers as well as their time-asymptotic stability. Furthermore, the detailed pointwise structures, nonlinear wave couplings, and wave interactions with boundary are studied.

Contents 1. 2.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Tools: Green’s Functions for Initial Value Problem and Initial-Boundary Value Problem . . . . . . . . . . . . . . 2.1 Basic notions and the macro-micro decomposition . . . . 2.2 Upwind-Downwind decomposition and energy estimates 2.3 Grad’s lemma on collision operator . . . . . . . . . . . . 2.4 Green’s functions . . . . . . . . . . . . . . . . . . . . . 3. Existence of Knudsen’s Layers (Time-asymptotic Approach) 3.1 Green’s functions for the damped equation . . . . . . . . 3.2 Energy estimates on the boundary data . . . . . . . . . . 3.3 Nonlinear term . . . . . . . . . . . . . . . . . . . . . . 4. Coupling Waves . . . . . . . . . . . . . . . . . . . . . . . . 5. Stability of Knudsen’s Layers for M > 1 . . . . . . . . . . . 5.1 Energy estimates on the boundary data . . . . . . . . . . 5.2 Nonlinear term . . . . . . . . . . . . . . . . . . . . . . 6. Stability of Knudsen’s Layers for −1 < M < 1(M
= 0) . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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297 297 299 300 302 306 308 310 315 318 323 324 325 335 345 345

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1. Introduction There are two purposes of the present paper. The first one is to conclude the nonlinear time-asymptotic stability of the Knudsen layers constructed in [33] after the serial preparation works, the Green’s functions for the Boltzmann equation in [24–26]. The second one is to initiate the methodology: to use the Green’s function of a linearized equation around a constant state to solve the nonlinear problem around other non-uniform nonlinear wave patterns such as boundary layers, shock layers, etc. The Knudsen layers are in an important class of wave pattens for the Boltzmann equation:
∂t F(x, t, ξ ) + ξ 1 ∂x F(x, t, ξ ) = K1n Q(F), (x, t, ξ ) ∈ R+ × R+ × R3 , F(0, t, ξ )|ξ 1 >0 , F(x, 0, ξ ) : imposed. Here for simplicity, the collision operator is assumed to be the model for hard sphere molecules for which the collision operator is ⎧ ⎪ Q(F) ≡ B(F, F), ⎪  ⎪ ⎪ ⎪ ⎨B(g, h)(ξ ) ≡ 1 2 R3 × S2 (ξ −ξ∗ )·≥0 ⎪ ⎪ ⎪ (−g(ξ )h(ξ∗ ) − h(ξ )g(ξ∗ ) + g(ξ  )h(ξ∗ ) + h(ξ  )g(ξ∗ )) ⎪ ⎪ ⎩ · |(ξ − ξ∗ ) · | dξ∗ d,  ξ  = ξ − [(ξ − ξ∗ ) · ] , ξ∗ = ξ∗ + [(ξ − ξ∗ ) · ] , and the parameter K n > 0 is the Knudsen number which is the ratio of the mean free path to the physical dimension. The precise definition of a Knudsen layer  is a boundary layer solution of the Boltzmann equation for a half-space problem with an incoming data b+ (ξ )|ξ 1 >0 at boundary x = 0 and with a boundary value as a global Maxwellian state M at x = ∞, i.e. ⎧ 1 3 ⎪ ⎨ξ ∂x  = Q() for x > 0, ξ ∈ R , (0, ξ )|ξ 1 >0 = Fb (ξ )|ξ 1 >0 , (1.1) ⎪ ⎩ lim (x, ξ ) = M(ξ ), x→∞

where the global Maxwellian states M = M[ρ,u,θ] , (ρ, u, θ ) constant, ⎧ −u|2 ρ − |ξ2Rθ ⎪ ⎪ M = e , [ρ,u,θ] ⎪ ⎪ (2π Rθ )3/2 ⎨ ρ>0: bulk density, ⎪ 1 ⎪ u = (u , 0, 0) : bulk fluid velocity, ⎪ ⎪ ⎩ θ >0: bulk temperature, satisfy Q(M) = 0 and are constant solutions of the Boltzmann equation. The Knudsen layer is also an important subject in the study of condensation-vaporization problem for rarefied gas. See Sone [29] for a complete reference on this issue. Knudsen layers are the central components in the generalized Hilbert-Grad expansion. A Knudsen-layer correction to the Grad-Hilbert expansion was introduced by Y. Sone to fit the kinetic boundary data after the results on the linear Knudsen layers by [1,7,5,10].

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

289

This Knudsen layer correction opened an new chapter on the asymptotic expansion theories on the field of rarefied gas. Since the generalized Grad-Hilbert expansion, theories on thermal transpiration flow, condensation-vaporization, ghost-effect, etc. had been developed by Y. Sone, K. Aoki, and their coworkers on the base of the generalized Grad-Hilbert expansion. See [29] for comprehensive and detailed description of the asymptotic theories. The works of Y. Sone initiated the mathematical interest on the Knudsen layers. Here, the Grad-Hilbert expansion was proposed by H. Grad to develop a general asymptotic theory for the Boltzmann equation with small Knudsen number K n in [12], but the Grad-Hilbert expansion did not have sufficient freedom to fit kinetic boundary conditions imposed. The existence of nonlinear Knudsen layers with small magnitudes for the hard sphere collision model was obtained in [33] under the condition that the boundary is noncharacteristic , i.e. the speed of the boundary and the Maxwellian state M = M[ρ,u,θ] at x = ∞ satisfy that ⎧ 0
∈ {u 1 − c, u 1 , u 1 + c}, (speed of the boundary=0), ⎪ ⎪ ⎨ u = (u 1 , 0, 0) : bulk fluid velocity, ⎪ ⎪ ⎩c = 5θ : sound speed at rest. 3 This condition can be rephrased as |M |
= 0 or 1, where M ≡

u1 , (Mach Number). c

The nonlinear time-asymptotic stability of the Knudsen layers was obtained under the assumption that the boundary is negative supersonic, u 1 < −c, (M < −1) in [34]. However, since the boundary layer is a physical relevant issue, the condition M < −1 for the time-asymptotic stability is only a technical assumption. The methodologies in [33,34] are almost identical, though they solved different types of problems. (One is for the existence of the nonlinear Knudsen layers, and the other is for the time-asymptotic stability.) They are weighted energy estimates motivated by the Macro-Micro decomposition developed in [23]. Due to the constraint from the energy estimates one will need high order energy estimates and the Sobolev inequality together to close the a priori assumption on the smallness on the super norms of the lower terms which were used in the nonlinear terms for obtaining the high order energy estimates. However, it is not necessary for the solution of the Boltzmann equation with a kinetic boundary condition to be continuous at the boundary. This gives rise to a conflict with the possibility to perform the high order energy estimates. The condition M < −1 yields that an exponentially decaying energy estimate in time and this exponential decaying property compensates the discontinuity presence at the boundary. One still can obtain the nonlinear stability of the Knudsen layer. When M > −1, the exponential decaying property no longer exists. One needs to introduce a new methodology which does not require any regularity property in the solution to overcome the difficulty due to discontinuities at the boundary. Denote F and  to be the solution of the Boltzmann equation and the Knudsen layer respectively. Follow the custom way in [6] to rewrite the Boltzmann equation for the planar wave solution and the equation for the Knudsen layer as follows: 1

1

F = M 2 f + M,  = M 2 ψ + M, so that the Boltzmann equation becomes

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1

∂t f + ξ 1 ∂x f = Lf + (f), Lf ≡ 2M− 2 B(M, M 2 f), (f) ≡ M−1/2 Q(M1/2 f), while the equation for the Knudsen layer turns to be 1

1

ξ 1 ∂x ψ = Lψ + (ψ), Lψ ≡ 2M− 2 B(M, M 2 ψ), (ψ) ≡ M−1/2 Q(M1/2 ψ), where the definition the linearized collision operator L = −ν + K will be given later in Sect. 2. Set w = f − ψ. We consider the time stability problem of the Knudsen layer  in the sense of the following initial boundary value problem: ⎧ 1 ⎪ ⎨∂t w + ξ ∂x w − Lw = L δ w + (w), (1.2) w(0, t, ξ )|ξ 1 >0 = 0, ⎪ ⎩w(x, 0, ξ ) = w (x, ξ ), 0 where

⎧ −1/2 B(M, M1/2 w), ⎪ ⎨ Lw = 2M L δ w = 2M−1/2 B( − M, M1/2 w), ⎪ ⎩ (w) = M−1/2 Q(M1/2 w),

and w0 (x, ξ ) satisfies sup |w0 (x, ξ )|(1 + |ξ |)3 ≤ e−x

ξ ∈R3

with 1.

The stability of  is equivalent to the problem when in the initial data is sufficiently small, and the question whether the solution w(x, t, ξ ) tends to zero as t tends to infinity. Consider the backward Green’s function Gi (x, y, t, σ, ξ, ξ∗ ) for the initial value problem, ⎧ 1 3 ⎪ ⎨−∂s Gi − ξ∗ ∂ y Gi − LGi = 0 for 0 < s < t, x, y ∈ R ξ, ξ∗ ∈ R , 3 Gi (x, y, s, s, ξ, ξ∗ ) = δ(x − y)δ (ξ − ξ∗ ), ⎪ ⎩ lim Gi (x, y, t, s, ξ, ξ∗ ) = 0, y→∞

and the backward Green’s function Gb (x, y, t, σ, ξ, ξ∗ ) for the initial boundary value problem, ⎧ ⎪ −∂s Gb − ξ∗1 ∂ y Gb − LGb = 0 for 0 < s < t, x, y > 0, ξ, ξ∗ ∈ R3 , ⎪ ⎪ ⎪ ⎨Gb (x, y, s, s, ξ, ξ∗ ) = δ(x − y)δ 3 (ξ − ξ∗ ), Gb (x, 0, t, s, ξ, ξ∗ )|ξ∗1 <0 = 0, ⎪ ⎪ ⎪ ⎪ ⎩ lim Gb (x, y, t, s, ξ, ξ∗ ) = 0. y→∞

Then the solution w(x, t, ξ ) of (1.2) can be represented as follows:  ∞ Gi (x, y, t, 0, ξ, ξ∗ )w(y, 0, ξ∗ )dξ∗ dy w(x, t, ξ ) = R3 0  t + Gi (x, 0, t, s, ξ, ξ∗ )ξ∗1 f(0, s, ξ∗ d)dξ∗ ds 0

R3

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

 t + 

∞

0 0 ∞

w(x, t, ξ ) =

R3

0

R3

Gi (x, 0, t, s, ξ, ξ∗ ) (w)dξ∗ dyds,

291

(1.3)

Gb (x, y, t, 0, ξ, ξ∗ )w(y, 0, ξ∗ )dξ∗ dy

 t

+  + 

0

Gb (x, 0, t, s, ξ, ξ∗ )ξ∗1 f(0, s, ξ∗ d)dξ∗ ds R3 t  ∞

0 0 ∞

=

R3

Gb (x, 0, t, s, ξ, ξ∗ ) (w)dξ∗ dyds

Gb (x, y, t, 0, ξ, ξ∗ )w(y, 0, ξ∗ )dξ∗ dy R3 ∞

0

 t

+ 0

0

R3

Gb (x, 0, t, s, ξ, ξ∗ ) (w)dξ∗ dyds.

(1.4)

These representations do not require any regularity property to obtain the solution. It avoids the difficulty in obtaining the high order energy estimates, namely the discontinuities at the boundary. The main result of the paper is: Theorem 1.1 (Main Theorem for stability when M > −1 ∩ {|M |
= 1, 0}). Suppose that M > −1 and |M |
= 0, 1. Then, the Knudsen layer  is nonlinear stable, i.e. when is sufficiently small, the solution of (1.2), w(x, t, ξ ) satisfies that (x−λ j t)2

3 e− C(1+t) (1 + |ξ |)3 |w(x, t, ξ )| ≤ C + C e−(|x|+t)/C √ t + 1 j=1 ⎧ 2 ⎪ ⎪ 1 ⎪ ⎪ for x ∈ (λ1 t, λ3 t), √ ⎨ (|x − λl t| + 1)(|x − λl+1 t| + 1) +C l=1 (1.5) ⎪ ⎪ ⎪ ⎪ ⎩0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞),

where (λ1 , λ2 , λ3 ) = (u 1 − c, u 1 , u 1 + c), and C > 0 is an universal constant. The linearized equation around the Knudsen layer can be rewritten as a perturbation of the linearized problem around a global Maxwellian state. The perturbation term can be regarded as a small linear coupling term, but it takes some effort to justify that the effect from the small linear coupling remains uniformly small in all time. Then, we use the Green’s function of the linearized equation around a global Maxwellian state to rewrite it as an integral equation to generate the ansatz. As a result of the length calculation, the ansatz created is still bounded by the ansatz created by the nonlinear problem around a global Maxwellian state. This leads to an example to reduce a problem on non-uniform background to a problem around an uniform background. The proof of Theorem 1.1 is considered for the cases M > 1 and 0 < |M | < 1 respectively. The proof for the case M > 1 is also an interesting new methodology. In this case one can make a priori assumption on the smallness of the sup norm to obtain a priori estimate on the full boundary data. Then, the estimates on the boundary data combining with the Green’s function Gi and the representation (1.3) yields the ansatz. To combine the boundary a priori estimate with the Green’s function Gi is subtle. One

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needs to use the particlelike-wavelike decomposition to decompose the Green’s function Gi in order to resolve some singularities due to slow particles. One also needs to have a priori estimates on the small linear coupling terms. Finally, the ansatz will justify the smallness a priori assumption. Thus, the nonlinear problem is closed without a high order energy estimate. In contrast to this, for the case 0 < |M | < 1, one does not need any a priori estimates on the boundary data, since we will use the representation (1.4) and the Green’s function Gb for the initial-boundary value problem already takes care of the singularities which arise from the boundary. The second purpose of this paper is to reconsider some large-time stability problems by using the Green’s function. In fact, for the case M < −1, by using the Green’s function it’s more convenient to get the results stated in [33] and [34]. Consider the time-asymptotic stability property of an upwind-damped Boltzmann equation, ⎧ 1 + 1 ⎪ ⎨∂t f + ξ ∂x f − Lf = −γ P0 ξ f + (f), γ > 0, f(0, t, ξ )|ξ 1 >0 = b+ (ξ )|ξ 1 >0 , ⎪ ⎩f(x, 0, ξ ) = f (x, ξ ). 0 The stability of the solution of the damped equation together with the fact that limt→∞ f(x, t, ξ ) is the solution of  ξ 1 ∂x ψ − Lψ = −γ P+0 ξ 1 ψ + (ψ), γ > 0, ψ(0, t, ξ )|ξ 1 >0 = b+ (ξ )|ξ 1 >0 , yields the existence of the boundary layer: 1 Theorem 1.2 (Existence).

For1 given √ M = M[1,u,θ] , u = (u , 0, 0), with the condition on the Mach number M ≡ u / 5θ/3
∈ {−1, 0, 1}, there exist a map, 

3 2 3  : h(ξ )|ξ 1 >0 sup(1 + |ξ |) |h(ξ )| < ∞, h(ξ ) : Even function in ξ and ξ −→ Rd , ξ

  δ, γ , and C > 0 such that for any h ∈  −1 (0) ∩ supξ (1 + |ξ |)3 |h(ξ )| ≤ δ there is a Knudsen layer  connecting the kinetic boundary data Fb |ξ 1 >0 = (M + M1/2 h)|ξ 1 >0 to the Maxwellian state M, and the Knudsen layer  also satisfies that sup(1 + |ξ |)3 M−1/2 | − M| ≤ δe−γ x/C for x > 0, ξ

where     d = the number of elements in u 1 − 5θ/3, u 1 , u 1 + 5θ/3 ∩ R+ . When M < −1, the operator P+0 is a null operator. Thus stability for the case M < −1 follows: Theorem 1.3 (Stability for M < −1). When M < −1, the Knudsen layer  given in Theorem 1.2 is exponentially stable in time, i.e. the solution of (1.2), w(x, t, ξ ) converges to zero exponentially fast.

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293

The Green’s function G Di for the damped Boltzmann equation is used for the existence of the Knudsen layer and the stability for M < −1; the Green’s function Gi for the Cauchy problem is used for the stability for M > 1; and the Green’s function Gb for the initial boundary value problem is used for the stability when |M | < 1(M
= 0). The technology used depends on the Mach number M . However, it’s clear that we can deal with the existence and stability of the boundary layer by the Green’s function. In the above statement, we find that one needs more detailed information on the Green’s functions in order to obtain the pointwise estimate of the nonlinear problem. This seems a down side of this approach. Indeed, a full detailed understanding of the Green’s function would satisfy H. Grad’s requirement as a qualitative tool, “The main qualitative feature of all problem in kinetic theory is the transition from free flow to continuum flow“, Grad [15]. For the purpose, Grad’s project, to understand the detailed structure of the Green’s functions, it leads to the studies, [24–26], on the Boltzmann equation linearized around a global Maxwellian. The Green’s functions for an initial value problem for planar wave solutions are, [24]:  ∂t Gi (x, t, ξ, ξ0 )+ξ 1 ∂x Gi (x, t, ξ, ξ0 )−LGi (x, t, ξ, ξ0 ) = 0 for x ∈ R, t > 0, ξ, ξ0 ∈ R3, Gi (x, 0, ξ, ξ0 ) = δ(x)δ 3 (ξ − ξ0 ), for 3-D solutions are, [25]:  ∂t Gi (x, t, ξ, ξ0 ) + ξ · ∇x Gi (x, t, ξ, ξ0 )− LGi (x, t, ξ, ξ0 ) = 0 for t > 0, x, ξ, ξ0 ∈ R3, Gi (x, 0, ξ, ξ0 ) = δ 3 (x)δ 3 (ξ − ξ0 ), and for an initial-boundary value problem for 1-D case is, ⎧ 1 3 ⎪ ⎨∂t Gb + ξ ∂x Gb − LGb = 0 for 0 < s < t, x, y > 0, ξ, ξ∗ ∈ R , Gb (x, y, s, s, ξ, ξ∗ ) = δ(x − y)δ 3 (ξ − ξ∗ ), ⎪ ⎩G (x, 0, t, s, ξ, ξ )| 1 = 0. b ∗ ξ >0 In [9,20,28,31] the Green’s functions for an initial value problem had been studied in terms of the Fourier variable η:    1 Gi (x, t, ξ, ξ∗ )e−iη·x d x . e(−iη·ξ +L)t ≡ 3 π R3 This representation is also regarded as a semi-group. By using this semi-group representation in the Fourier variable η S. Ukai obtained the first global existence theory around a global Maxwellian state in [31]. The free flow feature is rather unclear in the semi-group approach due to the lack of a precise spectrum property of the operator iηξ 1 + L in the high wave number regime, |η|  1. Thus, the semi-group approach alone is insufficient to observe the transition from the free flow to the continuum flow, and it alone can not be a good qualitative tool for the Boltzmann equation. T.-P. Liu and S.-H. Yu started to analyze to the Green’s function instead of the semigroup and tried to understand the transition from the free flow to the continuum flow in [24]. They began from the following reduced problem:  ∂t f + ξ 1 ∂x f = Lf for (x, t, ξ ) ∈ R × R+ × R3 , f(x, 0, ξ ) = gin (x, ξ ), gin (x, ξ ) = 0 for |x| ≥ 1 and sup |gin (x, ξ )|(1 + |ξ |)3 ≤ 1. x,ξ

(1.6)

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In order to resolve the problem due to insufficient information on the high wave number spectrum, two complementary decompositions, namely the long wave-short wave decomposition and particlelike-wavelike decomposition, were introduced. Long wave-short wave decomposition. The long wave-short wave decomposition for the solution of (1.6) is: ⎧ ⎪ f = fL + fS ,  ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ei xη+(−iηξ +L)t gˆ in (η, ξ )dη, f L (x, t, ξ ) ≡ ⎪ ⎨  − 1 (x, t, ξ ) ≡ ei xη+(−iηξ +L)t gˆ in (η, ξ )dη, f ⎪ S ⎪ ⎪ |η|> ⎪  ⎪ ⎪ 1 ⎪ ⎪ e−i xη gin (x, ξ )d x, ⎩gˆ in (η, ξ ) ≡ π R where the parameter > 0 is an universal constant related to the spectrum gap of the operator −iξ η + L around the origin η = 0. The long wave component f L catches the continuum flow as the semi-group does. Furthermore, by inverse Fourier transformation and complex analysis one can obtain exponentially sharp pointwise estimates of the long wave component f L (x, t, ξ ) within a finite Mach number region |x| < αc(t +1) for some α > 2, 

R

(x−λ j t) 3 − C(1+t) e 2 f L (x, t, ξ ) dξ ≤ C √ 3 1+t

2

for |x| ≤ αc(1 + t),

j=1

where λ1 = u 1 − c, λ2 = u 1 , λ3 = u 1 + c, and the regularity in the x variable is assured for this long wave component in the following sense:   |∂xk f L (x, t, ξ )|2 dξ d x ≤ Ok , R R3

where Ok is independent of t. In the same time, the short wave component f S dissipates to zero exponentially in the L 2x (L 2ξ )-sense, i.e.   |f S (x, t, ξ )|2 dξ d x ≤ Ce− 0 t , R R3

where 0 > 0 is also an universal constant related to the spectrum gap of −iηξ 1 + L near η = 0. This energy estimate on the short-wave component is still not qualitative enough to reveal pointwise structure due to the integration over x. Here, to obtain the estimates on the long wave component f L in a finite Mach number region one needs to show the spectrum is an analytic function around η = 0. Such an analytic property on the spectrum is not necessarily true for a hard potential model, see [21]. Particlelike-wavelike decomposition (Wave Remainder decomposition in [24]). The particlelike-wavelike decomposition for the solution of (1.6) is based on the kinetic nature of (1.6). Rewrite (1.6) into the form to emphasize the feature of kinetic equation:  ∂t f + ξ 1 ∂x f + νf = Kf, f(x, 0, ξ ) = gin (x, ξ ).

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295

For any given i > 0 by the mixture lemma in [24] and by the solution operator e(−ξ ∂x +ν)t one can have two functions Wi (x, t, ξ ) (particlelike component) and Ri (x, t, ξ ) (wavelike component) such that ⎧ f(x, t, ξ ) = Wi (x, t, ξ ) + Ri (x, t, ξ ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Wi (x, 0, ξ ) = gin (x, ξ ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨Ri (x, 0, ξ ) ≡ 0, (1.7) |Wi (x, t, ξ )| ≤ Ce−(|x|+t)/C , ⎪ ⎪ ⎪    ⎪  2 ⎪ ⎪ i ⎪ 1 ⎪ ∂ ∂ + ξ ∂ + ν −K Wi dξ d x ≤ Ce−t/C (due to the mixture lemma), t x ⎪ x ⎪ 3 ⎪ ⎪ ⎪ R R 2 ⎩ 2 i R3 ∂x Ri + |Ri | dξ d x ≤ C, 1

where the constants C and C > 0 depend on i but not on t. Here, the mixture lemma is about the conversion of the microscopic regularity in ξ variable to the regularity property to the macroscopic variable x variable. It will be stated in Sect. 2 later. Eventually, by combining these two decompositions one has that f L − Ri = Wi − f S . This gives that ⎧  ⎪ |f S − Wi |2 dξ d x = Ce−t/C1 , ⎪ ⎪ ⎨ R R3   ⎪ ⎪ ⎪ ⎩

 

R R3

|∂x (f S − Wi )|2 dξ d x =

R R3

|∂x (f L − Ri )|2 dξ d x = C.

From the above and Sobolev inequality one can see that  sup |f S − Wi |2 dξ ≤ O(1)e−t/C2 x∈R R3

for some C2 > 0. Thus, from this and the information on Wi together one can obtain the structure of f S (x, t) without acquiring the whole spectrum of the operator −iηξ 1 + L . The most important fact is about the wavelike component Ri :  sup |f L − Ri |2 dξ ≤ Ce−t/C2 . x∈R R3

From this estimate one can conclude that the wavelike component Ri approaches the continuum structure exponentially fast within the region |x| ≤ αc(1 + t) : 

 R3

|Ri (x, t, ξ )|2 dξ ≤ C 0

t

e−τ/C dτ

 3 j=1

(x−λ j t)2

e− C(1+t) . √ 1+t

(1.8)

Thus, this and (1.7) give a clear picture about the transition from the free flow to the continuum flow in a finite Mach number region |x| ≤ αc(1 + t).

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Waves in the high Mach number region, (|x| > αc(1 + t)), are not necessarily related to the continuum flow. It is sufficient to use the particlelike-wavelike decomposition and energy estimates together to show all wave structures decay exponentially fast in x and t in the region |x| ≥ αc(1 + t) :  |Ri (x, t, ξ )|2 dξ ≤ Ce−(|x|+t)/C . (1.9) R3

The same framework for the planar wave solution can be applied to a 3-D wave, but one needs to apply a complicated S O(3)-symmetry to yield exponentially sharp estimates on the inverse Fourier transformation for the long-wave component, see [25]. When the Mach number is subsonic, (|M | < 1), one has to develop the Green’s function Gb (x, y, t, s, ξ, ξ∗ ) for the initial-boundary value problem, ⎧ 1 + 3 ⎪ ⎨∂t f + ξ ∂x f = Lf, (x, t, ξ ) ∈ R × R × R , f(0, t, ξ )|ξ 1 >0 = b+ (t, ξ ), (imposed boundary data,), ⎪ ⎩f(x, 0, ξ ) = f (x, ξ ), (imposed initial data), 0 since one can not have such a priori exponentially decaying estimate on the boundary data  ∞   ∞  εt 1 2 e |ξ |f(0, t, ξ ) dξ dt ≤ C eεt |ξ 1 |b+ (t, ξ )2 dξ dt for some ε > 0, 0

R3 ξ 1 <0

0

R3

ξ 1 >0

because u 1 − c, u 1 , and u 1 + c are not in the same signs in contrast to the case |M | > 1. To construct Gb the Fourier variable is no longer valid to approximate the continuum flow in a half space problem, but the Green’s function of the initial value problem remains valid in terms of the physical parameter (x, t, ξ ). The Green’s function for the initial value problem Gi (x, t, ξ, ξ∗ ) is used to approximate the continuum flow structure in Gb (x, y, t, s, ξ, ξ∗ ) in [26]. The basic tools in the construction of Gb are Gi and a upwind-damped Boltzmann equation, ⎧ 1 + 1 ⎪ ⎨∂t ˜f + ξ ∂x ˜f − L ˜f = −γ P0 ξ ˜f, γ > 0, ˜f(0, t, ξ )| 1 = b+ (t, ξ )| 1 , ξ >0 ξ >0 ⎪ ⎩˜f(x, 0, ξ ) = f (x, ξ ), 0 to approximate the boundary data f(0, t, ξ )|ξ 1 <0 by assuming γ 1. On the other hand with the extra upwind-damping −γ P+0 ξ 1˜f, one can have such an exponentially decaying a priori estimate on the boundary data:  ∞   ∞  eεt |ξ 1 |˜f(0, t, ξ )2 dξ dt ≤ C eεt |ξ 1 |b+ (t, ξ )2 dξ dt for some ε > 0. 0

R3 ξ 1 <0

0

R3

ξ 1 >0

(1.10) Using the boundary data ˜f(0, t, ξ ) and Gi one can obtain a solution of the Boltzmann equation, ˜f:

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

˜f(x, t, ξ ) =

297

 t

Gi (x, t − s, ξ, ξ∗ )ξ∗1˜f(0, s, ξ∗ )dξ∗ ds 0  ∞ Gi (x − y, t, ξ, ξ∗ )f0 (y, 0, ξ∗ )dξ∗ dy. + R3

0

R3

Then, one uses (1.7), (1.8), (1.9), and (1.10) to conclude the difference f(0, t, ξ )ξ 1 >0 − ˜f(0, t, ξ ) 1 is of the order γ 1/4 e−Cγ t , where C is a positive constant. By repeating the ξ >0 approximation procedure one can obtain the exponential shape estimates on the Green’s function Gb in [26]. With this Green’s function Gb the nonlinear stability of a global Maxwellian state M for a half space problem is proved. The estimates on the Green’s function Gb will be stated in Sect. 2. After the completion of the analysis on the Green’s functions Gi and Gb , the study of the Knudsen layer can be outlined as follows. In Sect. 2 mainly we will review the results on the Green’s functions for both the initial-value problems and the boundary-value problems as well as the necessary preliminaries for this paper. In Sect. 3 we obtain the existence of the Knudsen layer under the time-asymptotic approach. This approach also yields the stability of the boundary layer for the case M < −1 In Sect. 4 we prepare the technical integrals for estimating the nonlinear coupling wave in studying the case M > −1. Finally, we complete the nonlinear stability problems in Sects. 5 and 6 for the cases M > 1 and 0 < |M | < 1 respectively. 2. Tools: Green’s Functions for Initial Value Problem and Initial-Boundary Value Problem In this section we will summarize the Green’s functions developed for the initial value problem and the initial-boundary value problem developed in [24,26] for linearized Boltzmann equations around a global Maxwellian M = M[1,u,1] , u = (u 1 , 0, 0):  ∂t h + ξ 1 ∂x h = Lh for x ∈ R, t > 0, (2.1) h(x, 0) = h0 (x), ⎧ 1 ⎪ ⎨∂t j + ξ ∂x j = Lj for x, t > 0, (2.2) j(x, 0) = j0 (x), ⎪ ⎩j(0, t)| 1 = 0. ξ >0 2.1. Basic notions and the macro-micro decomposition. We denote L 2ξ the restricted Hilbert spaces with given inner products: L 2ξ ≡ {h ∈ L 2 (R3)|h(ξ ) : an even function in ξ 2 and ξ 3 .} ⎧ ⎨(h, g) ≡ h(ξ )g(ξ )dξ for h, g ∈ L 2ξ , R3 ⎩h 2 = (h, h). L ξ

Two auxiliary inner products (·, ·)− and (·, ·)+ on L 2ξ are introduced due to the presence of a physical boundary:

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(g, h)− ≡

g(ξ )h(ξ )dξ, (g, h)+ ≡ R3 ∩{ξ 1 ≤0}

g(ξ )h(ξ )dξ, R3 ∩{ξ 1 ≥0}

with the corresponding space L 2ξ,+ ≡ L 2 (R3+ ), R3+ ≡ {(ξ 1 , ξ 2 , ξ 3 ) : ξ 1 ≥ 0, ξ 2 , ξ 3 ∈ R}. We still use the same sets of notions in [26]: ⎧  1/2 ⎪ 2 ⎪ ⎪ g L 2 ≡ g(ξ ) dξ , ⎪ ⎪ ξ ⎪ R3 ⎪ ⎪ 2 ⎪ (L ξ ,  ·  L 2 ) : a Hilbert space with inner ⎪ ⎪ ξ ⎪ ⎪ ⎨ product  Measures locally in (x, t) : ⎪ h1 (ξ )h2 (ξ ) dξ, (h1 , h2 ) ≡ ⎪ ⎪ ⎪ R3 ⎪ ⎪ ⎪ ⎪ g L ∞ ≡ sup |g(ξ )|(1 + |ξ |)β , ⎪ ξ,β ⎪ ⎪ ξ ∈R3 ⎪ ⎪ ⎩ g ∞ ≡ sup 3 (1 + |ξ |β )|g(ξ )|, L ξ,β+ ξ ∈R+ ⎧ h L ∞ (L 2 ) ≡ supx∈R+ h(x, ·) L 2 , ⎪ ⎪ x ξ ξ ⎪   1/2 ⎪ ⎪ ⎪ ⎪ 2 ⎪ h ≡ h(x, ξ ) dξ d x , 2 ⎪ L 2x (L ξ ) ⎪ 3 ⎪ R R + ⎪ ⎛ ⎞1/2 ⎨   i Measures locally in t: j ⎪ h H i (L 2 ) ≡ ⎝ |∂x h(x, ξ )|2 dξ d x ⎠ , ⎪ ⎪ x ξ ⎪ R+ R3 j=0 ⎪ ⎪ ⎪ ⎪ h ∞ ∞ ≡ sup ⎪ , ⎪ L x (L ξ,β ) x∈R+ h(x, ·) L ∞ ⎪ ξ,β ⎪ ⎩ ∞ ∞ |||h||| ≡ h L x (L ξ,3 ) . The null space of L in the restricted Hilbert space L 2ξ is a three-dimensional vector space with orthogonal basis χi , i = 0, 1, 4: ker(L) ≡ span{χ0 , χ1 , χ4 }, ⎧ ⎪ ≡ M1/2 , χ 0 ⎨ χ1 ≡ (ξ 1 − u 1 )M1/2 , ⎪ ⎩χ4 ≡ √1 (|ξ − u|2 − 3)M1/2 . 6

The macro-micro decomposition (P0 , P1 ) on L 2ξ is given as follows: for any g ∈ L 2ξ , ⎧ ⎪ ⎨g ≡ P0 g + P1 g(≡ g0 + g1 ), P0 g ≡ (χ0 , g)χ0 + (χ1 , g)χ1 + (χ4 , g)χ4 , ⎪ ⎩P g ≡ g − P g. 1 0 The characteristic information of the Euler equations ∂t P0 f + ∂x P0 (ξ 1 P0 f) = 0

(2.3)

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

299

is computed from the flux operator P0 ξ 1 on P0 L 2ξ , [23]: dim(P0 L 2ξ ) = 3, P0 ξ 1 Ei = λi Ei for i = 1, 2, 3,   λ1 = −c + u 1 , λ2 = u 1 , λ3 = c + u 1 ,   ⎧  ⎪ 3 5 ⎪ E1 ≡ ⎪ 2 χ0 − 2 χ1 + χ 4 , ⎪ ⎪    ⎪ ⎪ ⎪ ⎨E ≡ − 2 χ + χ , 2 4 3 0    ⎪ ⎪ ⎪ 3 5 ⎪ E3 ≡ ⎪ 2 χ0 + 2 χ1 + χ 4 , ⎪ ⎪ ⎪ ⎩ i (Ei , E j ) = δ j (Kronecker’s delta function), (2.4) where c=



5/3

is the speed of sound. Lemma 2.1. There exist positive constants ν0 such that, for any h ∈ L 2ξ , ⎧ 3 ⎪ ⎪ ⎪ (P h, P h) = (E j , h)2 , ⎪ 0 0 ⎪ ⎪ ⎪ j=1 ⎨ 3 ⎪ (P0 h, ξ 1 P0 h) = λ j (E j , h)2 , ⎪ ⎪ ⎪ ⎪ ⎪ j=1 ⎪ ⎩ (P1 h, LP1 h) ≤ −ν0 (P1 h, (1 + |ξ |)P1 h).

2.2. Upwind-Downwind decomposition and energy estimates. When the fluid velocity u 1 > −c as well as not characteristic, i.e. u 1
∈ {−c, 0, c}, one has the eigendecomposition of the operator P0 ξ 1 | Range(P0 ) : ⎧ P0 | Range(P0 ) = P+0 | Range(P0 ) + P− ⎪ 0 | Range(P0 ) , ⎪ ⎪ − 1 ⎨P ξ 1 | + 1 0 Range(P0 ) = P0 ξ | Range(P0 ) ⊕ P0 ξ | Range(P0 ) , P+0 ξ 1 | Range(P+0 ) : positive-definite, ⎪ ⎪ ⎪ ⎩P− ξ 1 | − Range(P− ) : negative-definite or P0 ≡ 0. 0 0

The explicit expression of

and P− 0 is ⎧ +  P0 = Ej ⊗ Ej , ⎪ ⎪ ⎨ λ j >0  − ⎪ P = Ej ⊗ Ej . ⎪ 0 ⎩

P+0

λ j <0

Here, the operator g ⊗ h| is given by

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g ⊗ h| f ≡ (h, f)g for f ∈ L 2ξ . Lemma 2.2. If u 1 ∈ (−c, ∞) and it is not characteristic, then there exists δ0 > 0 such that whenever λ ∈ (0, δ0 ), 1 − γ (f, ξ 1 f) + γ (f, P+0 ξ 1 f) − (f, Lf) ≥ γ (P0 f, P0 f) + γ (P1 f, (1 + |ξ |)P1 f) holds, 2 (2.5) where ≡

1 min |λ j |. 8

2.3. Grad’s lemma on collision operator. For the hard sphere model we consider, the linearized collision operator L is of the following form, [16]: ⎧ Lg(ξ ) = −ν(ξ − u)g(ξ ) + Kg(ξ ), ⎪ ⎪  ⎪ ⎪ ⎪ Kg(ξ ) ≡ R3 K (ξ − u, ξ∗ − u)g(ξ∗ ) dξ∗ , ⎪ ⎪   ⎪ ⎪ 2 |ξ − ξ∗ |2 (|ξ |2 − |ξ∗ |2 )2 ⎪ ⎪ ) ≡ − exp − K (ξ, ξ √ ⎪ ∗ ⎪ 2 ⎨ 8|ξ − ξ 8  2π|ξ − ξ∗ | ∗| (2.6) (|ξ |2 + |ξ∗ |2 ) |ξ − ξ∗ | ⎪ exp − , − ⎪ ⎪ 4  ⎪ ⎪    |ξ |  2 ⎪ 2 2 ⎪ 1 1 ⎪ − |ξ2| − u2 ⎪ ν(ξ ) ≡ √ 2e + 2 |ξ | + e du , ⎪ ⎪ ⎪ |ξ | 0 2π ⎪ ⎩ ν(ξ ) ∼ 1 + |ξ |. This lemma follows from direct computations and Carleman’s theory, [4], on the negative definiteness of the operator L on Range(P1 ). Lemma 2.3 (Grad, [13]). For any given β ≥ 0 there exist positive constants C(β) and C1 such that  ≤ C(β)j L ∞ , K j L ∞ ξ,β ξ,β+1 ≤ C1 j L 2 , Kj L ∞ ξ,0 ξ

where

 K j (ξ ) ≡

R3

K (ξ, ξ∗ )j(ξ∗ )dξ∗ .

The proof of this lemma in [6] for β = 0 can easily be generalized for any β ∈ R. Lemma 2.4. For the hard sphere model, there exists a constant C > 0 such that for any α ≥ −1,        (1 + |ξ |)1+α f   (1 + |ξ |)1+α g   (1 + |ξ |)α B(f, g)        1 1 1   ∞ ≤C   ∞  ∞ , (2.7) M2 M2 M2 Lξ Lξ Lξ       1+α   1+α    (1 + |ξ |)α B(f, g)   ≤ C  (1 + |ξ |) f   (1 + |ξ |) g   (2.8) 1 1 1   2  2  2 .  M2 M2 M2 Lξ Lξ Lξ

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

301

The proof of (2.7) for α = −1 in [11] and that of (2.8) for α = − 21 quoted from [23] can be generalized for α ≥ −1. Proof. Since (ξ  , ξ∗ ) which satisfies 

ξ  = ξ − [(ξ − ξ∗ ) · ], ξ∗ = ξ∗ + [(ξ − ξ∗ ) · ],

(2.9)

conserve the kinetic energy and momentum after collision, |ξ |2 + |ξ∗ |2 = |ξ  |2 + |ξ∗ |2 . As a result, M(ξ )M(ξ∗ ) = M(ξ  )M(ξ∗ ).

(2.10)

Consider B(f, g) =

1 (B1 (f, g) + B1 (g, f) − B2 (f, g) − B2 (g, f)) , 2

(2.11)

where  B1 (f, g) =

R3 × S 2

 B2 (f, g) =

R3 × S 2

f(ξ  )g(ξ∗ ) |(ξ − ξ∗ ) · | dξ∗ d, f(ξ )g(ξ∗ ) |(ξ − ξ∗ ) · | dξ∗ d. 1

1

We consider B1 first. Without losing generality, we set f = M 2 u, g = M 2 v, (1+|ξ |)α B (f, g) (1+|ξ |)α  1 1 1 M(ξ ) 2 M(ξ∗ ) 2 |u(ξ  )||v(ξ∗ )| |(ξ −ξ∗ ) · | dξ∗ d ≤ 1 1 2 R3 × S M(ξ ) 2 M2 (1 + |ξ |)α+1 v L ∞ ≤ (1 + |ξ |)α (1 + |ξ |)α+1 u L ∞ ξ ξ 1  M(ξ∗ ) 2 |(ξ − ξ∗ ) · | × dξ∗ d.  α+1 (1 + |ξ  |)α+1 R3 × S2 (1 + |ξ |) ∗

(2.12)

From (2.9), we can get 1 + |ξ  | = 1 + |ξ − [(ξ − ξ∗ ) · ]| ≤ C(2 + |ξ | + |ξ∗ |), 1 + |ξ∗ | = 1 + |ξ∗ + [(ξ − ξ∗ ) · ]| ≤ C(2 + |ξ | + |ξ∗ |).

(2.13) (2.14)

Similarly, 1 + |ξ | ≤ C(2 + |ξ  | + |ξ∗ |), 1 + |ξ∗ | ≤ C(2 + |ξ  | + |ξ∗ |). This suggests 1 + |ξ | + |ξ∗ | ∼ 1 + |ξ  | + |ξ∗ |. As a result,

α+1 (1 + |ξ  |)α+1 (1 + |ξ∗ |)α+1 = (1 + |ξ  |)(1 + |ξ∗ |)

≥ (1 + |ξ  | + |ξ∗ |)α+1 ≥ C(1 + |ξ | + |ξ∗ |)α+1 ≥ C(1 + |ξ |)α+1 .

(2.15)

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 1 This together with the fact that R3 × S2 M(ξ∗ ) 2 |(ξ − ξ∗ ) · | dξ∗ d ≤ C(1 + |ξ |) yields that 

1

M(ξ∗ ) 2 |(ξ − ξ∗ ) · | 1 dξ∗ d ≤ C 2 (1 + |ξ  |)α+1 (1 + |ξ  |)α+1 3 (1 + |ξ |)α+1 R ×S ∗  1 1 × M(ξ∗ ) 2 |(ξ − ξ∗ ) · | dξ∗ d ≤ C . (1 + |ξ |)α R3 × S 2

Thus (2.16) and (2.12) result in (1 + |ξ |)α B (f, g) 1 (1 + |ξ |)α+1 v L ∞ . ≤ C(1 + |ξ |)α+1 u L ∞ 1 ξ ξ M(ξ ) 2

(2.16)

(2.17)

The proof for B2 is similar but much simpler. Now, we complete the proof of (2.7). By using the Hölder inequality and the analysis applied for (2.7), we can also get (2.8).   2.4. Green’s functions. Denote Gi (x, t, ξ, ξ∗ ) and Gb (x, y, t, s, ξ, ξ∗ ) is the Green’s function for the initial value problem (2.1) and for the initial-boundary value problem (2.2) respectively. The equations for Gi and Gb are
∂t Gi + ξ 1 ∂x Gi = LGi f or x ∈ R, t > 0ξ, ξ∗ ∈ R3 , (2.18) Gi (x, 0, ξ, ξ∗ ) = δ(x)δ 3 (ξ − ξ∗ ), ⎧ ⎨ ∂t Gb + ξ 1 ∂x Gb = LGb f or x, y > 0, t > s, ξ, ξ∗ ∈ R3 , (2.19) G (x, y, s, s, ξ, ξ∗ ) = δ(x − y)δ 3 (ξ − ξ∗ ), ⎩ b Gb (0, y, t, s, ξ, ξ∗ )|ξ 1 >0 = 0. The Green’s functions in the above also satisfy the backward equations:  (−∂s − ξ∗ ∂ y − L)Gi (x − y, t − s, ξ, ξ∗ ) = 0, Gi (x − y, 0, ξ, ξ∗ ) = δ 1 (y − x)δ 3 (ξ∗ − ξ ), x, y ∈ R, ξ, ξ∗ ∈ R3 , ⎧ ⎪ ⎨(−∂s − ξ∗ ∂ y − L)Gb (x, y, t, s, ξ, ξ∗ ) = 0, Gb (x, y, 0, ξ, ξ∗ ) = δ 1 (y − x)δ 3 (ξ∗ − ξ ), x, y > 0, ξ, ξ∗ ∈ R3 , ⎪ ⎩G (x, 0, t, s, ξ, ξ )| 1 = 0. b ∗ ξ∗ <0

(2.20)

(2.21)

We still keep the following abbreviations: For given (x, t) ∈ R × R+ , Gi(b) (x, t) : h ∈ L 2ξ −→ Gi(b) (x, t)h ∈ L 2ξ , Gi(b) (x, t)h(ξ ) ≡ Gi(b) (x, t, ξ, ξ∗ )h(ξ∗ )dξ∗ . R3

Let gin be a function satisfying  gin (x, ξ ) ≡ 0 for |x| ≥ 1, ∞ < ∞. |||gin ||| ≡ gin  L ∞ x (L ξ,3 )

(2.22)

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2.4.1. Green’s function for an initial value problem We recall the following theorems in Liu-Yu [24] on the Green’s function Gi for the initial value problem. Theorem 2.1. ([24], Theorem 1.1). Green’s function Gi (x, t, ξ ; ξ0 ) contains mainly the particlelike waves e−ν(ξ0 )t δ(x − ξ 1 t)δ 3 (ξ − ξ0 ) + j1 (x, t, ξ ; ξ0 ) + j2 (x, t, ξ ; ξ0 ) propagating with microscopic velocity, and fluidlike waves 3 k=1

−

(x−λ¯ k t)2

e 4 Ak (t+1) Ek (ξ )Ek (ξ0 ) √ 4 Ak π(t + 1)

propagating along Euler characteristics. Precisely, for some positive constant C1 , Gi (x, t, ξ ; ξ0 ) = G0i (x − u 1 t, t, ξ − u; ξ0 − u) satisfies   G0i (x, t, ξ ; ξ0 ) − e−ν(ξ0 )t δ(x − ξ 1 t)δ 3 (ξ − ξ0 ) − j1 (x, t, ξ ; ξ0 ) − j2 (x, t, ξ ; ξ0 )  ⎤ ⎡  (x−λ¯ k t)2 (x−λ¯ i t)2 − − 3 3  e 4 Ak (t+1)  ⎥ ⎢ e C0 (t+1) + e−(|x|+t)/C1 ⎦ , Ek (ξ )Ek (ξ0 ) = C ⎣ − √  (t + 1) 4 A π(t + 1) k k=1 k=1  2 Lξ

where {λ¯ 1 , λ¯ 2 , λ¯ 3 } = {−c, ⎧ 0, c}, 1 1 ⎪ ⎨0 if (x − ξ0 t)(ξ t − x) < 0, −ν(ξ )(t−t )−ν(ξ )t 1 1 0 1 K (ξ, ξ ) j (x, t, ξ ; ξ0 ) = e 0 ⎪ else if x − ξ0 t
= 0, ⎩ 1 (ξ 1 − ξ0 ) where t1 = t − j2 (x, t, ξ ; ξ

0)

(x−ξ01 t) , ξ 1 −ξ01

= O(1)e

−

ν0 (t+|x|) 3



|ξ − ξ0 |−1 (1 + |ξ |)−1 for |ξ − ξ0 | ≤ 1, (1 + |ξ |)−1 for |ξ − ξ0 | ≥ 1.

Here the parameters A j , j = 1, 2, 3, are given below, and the constant C0 satisfies 

A j ≡ −(P1 ξ 1 E j , L −1 P1 ξ 1 E j ) > 0, j = 1, 2, 3, C0 > 4 max(A1 , A2 , A3 ).

(2.23)

The dissipation parameters A j , j = 1, 2, 3, are the same as the viscosity and the heat conductivity coefficients for the Navier-Stokes equations obtained through the Chapman-Enskog expansion. Theorem 2.2. ([24], Theorem 5.6). For bounded initial data gin with compact support in x, (2.22), and for the constant C0 satisfying (2.23), there exists a positive constant C1 independent of gin such that

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for |x − u 1 t| ≤ 2ct, ⎞ ⎛ ⎧ |x−λi t|2 ⎪ − C (t+1) 3 ⎪ e 0 ⎪ ⎟ ⎪ Gt g (x) 2 = O(1) |||g ||| ⎜ ⎪ + e−(|x|+t)/C1 ⎠ , √ in ⎝ ⎪ i in Lξ ⎪ (t + 1) ⎪ ⎪ i=1 ⎪ ⎪ ⎪ ⎪ ⎪ Git P1 gin (x) L 2 , P1 Git gin (x) L 2 ⎪ ⎪ ξ ⎪ ⎞ ⎛ ξ ⎪ ⎪ |x−λi t|2 ⎪ − 3 ⎪ ⎪ ⎟ ⎜ e C0 (t+1) ⎪ ⎪ = O(1) |||gin ||| ⎝ + e−(|x|+t)/C1 ⎠ , ⎪ ⎪ ⎪ (t + 1) ⎪ i=1 ⎨ ⎞ ⎛ |x−λi t|2 − C (t+1) 3 ⎪ ⎪ (2.24) e 0 ⎟ ⎜ ⎪ t ⎪ + e−(|x|+t)/C1 ⎠ , ⎪ ⎪ P1 Gi P1 gin (x) L 2ξ = O(1) |||gin ||| ⎝ 3/2 ⎪ (t + 1) ⎪ i=1 ⎪ ⎪ ⎪ ⎪ t ⎪ ⎪ | E Gi Ek ⊗ k gin (x) L 2 ⎪ ξ ⎪ ⎞ ⎛ ⎪ ⎪ ⎪ |x−λk t|2 |x−λi t|2 ⎪ ⎪ e− C0 (t+1) ⎪ ⎟ ⎜e− C0 (t+1) ⎪ ⎪ ⎪ + e−(|x|+t)/C1⎟ + = O(1) |||gin ||| ⎜ √ ⎪ ⎠, ⎝ ⎪ (t + 1) 1≤i≤3 (t + 1) ⎪ ⎩ i
=k

for |x − u t| ≥ 2ct 1

Git gin (x) L 2 ≤ C |||gin ||| e−(|x|+t)/C1 , ξ

where

 Git gin (x) ≡ R Gi (x − y, t)gin (y)dy. Theorem 2.3 ([24], Theorem 7.2). For bounded, compact-supported initial data gin , (2.22), there exist positive constants C1 and C0 such that for all x ∈ R the main fluid part satisfies  [E j ⊗ E j Git Ek ⊗ Ek | gin ](x) L ∞ ξ,3 ⎞ ⎛ (x−λk t)2 (x−λi t)2 − C (t+1) − C (t+1) 3 0 0 e ⎟ ⎜ e + + e−(|x|+t)/C1 ⎠ , (2.25) = O(1)|||gin ||| ⎝δ kj √ (1 + t) (1 + t) i=1

and the non-fluid parts have higher rate of decay in time: ⎞ ⎛ (x−λi t)2 − C (t+1) 3 e 0 ⎟ ⎜ + e−(|x|+t)/C1 ⎠ , [Git P1 gin (x) L ∞ = O(1)|||gin ||| ⎝ ξ,3 (1 + t) i=1

P1 Git gin (x) L ∞ ξ,3

⎞ ⎛ (x−λi t)2 − 3 ⎟ ⎜ e C0 (t+1) + e−(|x|+t)/C1 ⎠ , = O(1)|||gin ||| ⎝ (1 + t) i=1

P1 Git P1 gin (x) L ∞ ξ,3

⎞ ⎛ (x−λi t)2 − C (t+1) 3 e 0 ⎟ ⎜ = O(1)|||gin ||| ⎝ + e−(|x|+t)/C1 ⎠ . (1 + t)3/2 i=1

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

305

2.4.2. Green’s function for the initial boundary value problem for the case −c < u 1 < c and u 1
= 0 Suppose −1 < M ≡ u 1 /c < 1(M
= 0). The following theorem which gives the properties of Green’s function Gb of the initial boundary value problem can be found in [26]: Theorem 2.4 ([26], Theorem 1.1). Suppose that the Euler characteristics may leave or enter the boundary x = 0. For definiteness, assume that λ1 < λ2 < 0 < λ3 . Then p Green’s function Gb (x, y, t, s, ξ, ξ∗ ), (2.19), consists of particlelike waves Gb , fluidlike f waves Gb and the remaining term Grb : p

f

Gb (x, y, t, s, ξ, ξ∗ ) = Gb + Gb + Grb

f or x, y, t − s > 0.

(2.26)

The particle like-waves consist of a δ-function propagating from the initial data and damped under the operator ν and non-smooth functions jkb , k = 1, 2 resulting from the action of the operator K in linear collision operator L, on the δ function: Gb = e−ν(ξ −u)(t−s) δ(x − y − ξ 1 (t − s))δ 3 (ξ − ξ∗ ) + j1b (x, y, t − s, ξ, ξ∗ ) p

+ j2b (x, y, t − s, ξ, ξ∗ ),

(2.27)

j1b (x, ⎧

y, t, ξ, ξ∗ ) 0 for (x − y − ξ∗1 t)(ξ 1 t − x + y) < 0, ⎪ ⎪ ⎪ ⎨0 for (x − y − ξ 1 t)(ξ 1 t − x + y) > 0, − y + x < t, ξ 1 > 0, −ξ 1 > 0, ∗ ∗ ξ∗1 ξ1 = (2.28) −ν(ξ −u)(t−t1 )−ν(ξ∗ −u)t1 K (ξ − u, ξ − u) ⎪ e ⎪ ∗ ⎪ otherwise, ⎩ (ξ 1 − ξ∗1 ) (x−y−ξ∗1 t) where t1 = t − ξ 1 −ξ 1 , ∗  ν (t+|x−y|) |ξ − ξ∗ |−1 (1 + |ξ |)−1 for |ξ − ξ∗ | ≤ 1, 2 − 0 3 |jb (x, y, t, ξ, ξ∗ ) = O(1)e (1 + |ξ |)−1 for |ξ − ξ∗ | ≥ 1. (2.29) The fluidlike waves propagate with Euler characteristic speeds and diffuses like the heat kernel, that is, for some positive constant C, ⎛

 (x−y−λ j t)2 − Ct

3 ⎜ e ⎜ f Gb (x, y, t, ξ, ξ∗ ) L 2 = O(1) ⎜ √ ξ ⎝ t +1 j=1

+

 λ y 2 x−λ3 t− λ3 j Ct

⎞

2 ⎟ e− ⎟ √ ⎟. ⎠ (t + 1)(y + 1) j=1

(2.30)

The remaining term is a continuous function and decays exponentially: Grb (x, y, t, ξ, ξ∗ ) L 2 = e−O(1)(|x|+|y|+t)/C . ξ

(2.31)

Furthermore, except for the explicitly given particlelike waves, the decay rates increase when the Green’s function acts on the non-fluid functions. Precisely, for any h in the

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S. Deng, W. Wang, S.-H. Yu

range of the non-fluid projection P1 = I − P0 and in L 2ξ ,      Gb (x, y, t, ξ, ξ∗ )h(ξ∗ )dξ∗    R3

L 2ξ

⎛



 λ y 2 x−λ3 t− λ3 j Ct

⎞

( ) 3 2 ⎜ ⎟ e− e− ⎜ ⎟ + = O(1)h L 2 ⎜ + e−O(1)(|x|+|y|+t)/C ⎟ . √ ξ ⎝ ⎠ t +1 (t + 1)(y + 1) j=1 j=1 x−y−λ j t Ct

2

(2.32) Remark 2.1. The construction of the Green’s function for the initial-boundary value is based on the Green’s function for the initial value problem through a complicated procedure to obtain. Thus, when the analysis can avoid to use the Green’s function for the initial-boundary value problem, one could consider it as a simple proof. 3. Existence of Knudsen’s Layers (Time-asymptotic Approach) The Knudsen’s layer  is a stationary solution of the Boltzmann equation with an imposed incoming boundary data: ⎧ 1 + 3 ⎪ ⎨ξ ∂x  = Q(), (x, ξ ) ∈ R × R , (0, ξ )|ξ 1 >0 = Fb (ξ ), (3.1) ⎪ ⎩ lim (x, ξ ) = M(ξ ). x→∞

We denote ψ(x, ξ ) as the quotient of  − M to ψ(x, ξ ) ≡

√

M:

(x, ξ ) − M . √ M

The equation for ψ(x, ξ ) is ⎧ 1 + 3 ⎪ ⎨ξ ψx − Lψ = (ψ), (x, ξ ) ∈ R × R , ψ(0, ξ )|ξ 1 >0 = b+ (ξ ), ⎪ ⎩ lim ψ(x, ξ ) = 0,

(3.2)

x→∞

where

√ √ Mψ) + B( Mψ, M) Q( Mψ) Lψ ≡ , (ψ) ≡ , √ √ M M (Fb − M) b+ = √ 1 . M ξ >0 B(M,

√

The existence of a solution of the boundary value problem (3.2) is subject to the solvability of the incoming boundary data b+ , [33]. This problem was solved by considering the following system which is (3.2) with an extra upwind damping −γ P+0 ξ 1 ψ, γ > 0: ⎧ 1 + 1 + 3 ⎪ ⎨ξ ψx − Lψ = (ψ) − γ P0 ξ ψ, (x, ξ ) ∈ R × R , ψ(0, ξ )|ξ 1 >0 = b+ (ξ ), (3.3) ⎪ ⎩ lim ψ(x, ξ ) = 0. x→∞

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

307

The methodology in solving (3.3) requires regularities on ψ(x, ξ ) for non-small |ξ 1 | due to energy estimates. Here, we implement the time-asymptotic approach to obtain the stationary solution, since this methodology will imply the time-asymptotic stability of Knudsen’s layer when M < −1. This is because P+0 ξ 1 = 0 when M < −1. We consider the time evolution equation: ⎧ ft + ξ 1 fx − Lf = (f) − γ P+0 ξ 1 f, (x, t, ξ ) ∈ R+ × R+ × R3 , ⎪ ⎪ ⎪ ⎨f(x, 0, ξ ) = f0 (x, ξ ), (3.4) f(0, t, ξ )|ξ 1 >0 = b+ (ξ ), ⎪ ⎪ ⎪ ⎩ lim f(x, 0, ξ ) = 0, x→∞

where the initial data f0 (x, ξ ) satisfies sup(1 + |ξ |)4 |f0 (x, ξ )| ≤ Cδe−|x|/C ,

(3.5)

sup(1 + |ξ |)4 |(f0 )x (x, ξ )| ≤ Cδe−|x|/C ,

(3.6)

ξ

and ξ

for some constant C > 0. Then sup(1 + |ξ |)3 |ft (x, t, ξ )|t=0 | ξ

= sup(1 + |ξ |)3 −ξ 1 fx (x, 0, ξ ) + Lf(x, 0, ξ ) + (f)(x, 0, ξ ) − γ P+0 ξ 1 f(x, 0, ξ ) ξ

≤ C sup(1 + |ξ |)4 (f0 (x, ξ ) + f0x (x, ξ )) ≤ Cδe−|x|/C . ξ

(3.7)

Lemma 3.1. Suppose that M
∈ {−1, 0, 1}. Then, there exist γ > 0 and δ > 0 such that whenever sup (1 + |ξ |)3 |b+ (ξ )| ≤ δ,

(3.8)

ξ 1 >0

the limit limt→∞ f(x, t, ξ ) exists and satisfies     e−γ x/C  lim f(x, t) ∞ ≤ Cb+  L ∞ ξ,3+ t→∞

(3.9)

L ξ,3

for some C > 0. Here, the constants C depend on M . Furthermore, ψ(x, ξ ) ≡ limt→∞ f(x, t, ξ ) solves (3.3). The proof of this lemma will be completed in the rest of this section. When P+0 ξ 1 ψ(x, ξ ) ≡ 0, the solution of (3.3) is also the solution of the original system (3.2). In [33], it has been proved that this condition can be reduced to P+0 ξ 1 ψ(0, ξ ) ≡ 0, where ψ(0, ξ ) is determined by the incoming boundary data b+ (ξ ) uniquely. That is, there exists a map,

3 −1/2 2 3  : b+ (ξ )|ξ 1 >0 sup(1+|ξ |) M |b+ (ξ )| < ∞, b+ (ξ ) : Even function in ξ and ξ  −→Rd , ξ 

such that when b+ (ξ ) ∈  −1 (0), the damped system (3.3) is just the original system (3.2). Thus, Theorem 1.2 is a direct consequence of Lemma 3.1.

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3.1. Green’s functions for the damped equation. We denote G Di (x, t, ξ, ξ∗ ) to be the Green’s function of  ft + ξ 1 fx − Lf + γ P+0 ξ 1 f = 0, (x, t, ξ ) ∈ R+ × R+ × R3 , (3.10) f(x, 0, ξ ) = f0 (x, ξ ). Then, the solution of (3.4) can be represented through the Green’s function G Di as follows:  ∞ f(x, t, ξ ) = G Di (x − y, t, ξ, ξ∗ )f0 (y, ξ∗ )dξ∗ dy R3 0  t + G Di (x, t − s, ξ, ξ∗ )ξ∗1 f(0, s, ξ∗ )dξ∗ ds 0 R3  t  ∞ + G Di (x − y, t − s, ξ, ξ∗ ) (f)(y, s, ξ∗ )dξ∗ dyds. (3.11) 0

R3

0

For convenience in writing, (3.11) can be written as follows without specifying the dependence on ξ :  ∞  t f(x, t) = G Di (x − y, t)f0 (y)dy + G Di (x, t − s)[ξ 1 f](0, s)ds 0 0  t ∞ + G Di (x − y, t − s) (f)(y, s)dyds. (3.12) 0

0

∞ In the representation (3.11), the only clear term is 0 R3 G Di (x − y, t, ξ, ξ∗ ) f0 (y, ξ∗ )dξ∗ dy which comes from the initial data. To get the estimate of this term, the properties of the Green’s function G Di are needed. In order to apply the framework in [24] to obtain the pointwise structure of the Green’s function G Di , we need the long wave information of the operator −iηξ 1 + L − γ P+0 ξ 1 . The following lemma can be proved by a straightforward calculation. Lemma 3.2. There exist κ0 > 0 and κ1 > 0 such that the discrete spectrum σ (η) of the operator −iηξ 1 + L − γ P+0 ξ 1 satisfies the following properties: for |η| > κ0 , σ (η) ⊂ {z ∈ C : Re(z) ≤ −κ1 };

(3.13)

for |η| ≤ κ0 , σ (η) = {σi (η), i = 1, 2, 3}, where the functions σi (η) are analytic functions satisfying ⎧ σ j (η) = −χ ( j)γ λ j − i(λ j + χ ( j)A j )η − B j |η|2 + O(1)|η|3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ 1, for λ j > 0 ⎪ ⎪ ⎨χ ( j) = 0, for λ < 0 , j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ A j = (E j , γ P+0 ξ 1 L−1 P1 ξ 1 E j ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ B j = −(P1 ξ 1 E j , L−1 P1 ξ 1 E j ) > 0, and λ j are the eigenvalues of the operator P0 ξ 1 P0 | Range(P0 ) .

(3.14)

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309

Following the method in [24] to get the properties of G Di , we only have to check ˜ = Kf − γ P+ ξ 1 f. Since the operator whether the mixture lemma still holds. Denote Kf 0 ˜ K˜ ξ and K˜ ξ are P+0 ξ 1 is an integration operator, it is straightforward to prove that K, ∗ 2 bounded L operators. Thus the mixture lemma still works. From the spectrum of the operator −iηξ 1 + L − γ P+0 ξ 1 stated in (3.14), we find that the damping term P+0 ξ 1 forces some branches to detach from the origin and lie in the interior of the left complex plane Re(z) ≤ −α0 γ for some positive constant α0 which is of the order O(1) minλ j >0 λ j . This fact results in the extra decaying factor e−α0 γ t before

)

−

(x−λ j t)2

e √C(t+1) λ j >0 t+1

.

Theorem 3.1. For the function gin specified in (2.22) and for the constant C0 satisfying (2.23), there exists a positive constant C1 independent of gin such that for |x − u 1 t| ≤ 2ct, ⎧ t ⎪ G Di gin (x) L 2 ⎪ ξ ⎛ ⎪ ⎞ ⎪ ⎪ |x−λi t|2 |x−λi t|2 ⎪ − − ⎪ C (t+1) C (t+1) ⎪ e 0 e 0 ⎪ ⎟ ⎜ ⎪ ⎪ = O(1) |||gin ||| ⎝ + e−α0 γ t + e−(|x|+t)/C1 ⎠ , √ √ ⎪ ⎪ (t + 1) (t + 1) ⎪ ⎪ λi <0 λi >0 ⎪ ⎪ ⎪ ⎪ ⎪ GtDi P1 gin (x) L 2 , P1 GtDi gin (x) L 2 ⎪ ⎪ ξ ξ ⎞ ⎛ ⎪ ⎨ |x−λi t|2 |x−λi t|2 − − C (t+1) C (t+1) 0 0 e e ⎟ ⎜ ⎪ + e−α0 γ t + e−(|x|+t)/C1 ⎠ , = O(1) |||gin ||| ⎝ ⎪ ⎪ (t + 1) (t + 1) ⎪ ⎪ λi <0 λi >0 ⎪ ⎪ ⎪ ⎪ t ⎪ P1 G Di P1 gin (x) L 2 ⎪ ⎪ ⎪ ⎞ ⎛ ξ ⎪ ⎪ |x−λi t|2 |x−λi t|2 ⎪ − − ⎪ C (t+1) C (t+1) ⎪ e 0 e 0 ⎪ ⎟ ⎜ ⎪ = O(1) |||gin ||| ⎝ + e−α0 γ t + e−(|x|+t)/C1 ⎠ , ⎪ ⎪ 3/2 3/2 ⎪ (t + 1) (t + 1) ⎩ λi <0

λi >0

for |x − u 1 t| ≥ 2ct GtDi gin (x) L 2 ≤ C |||gin ||| e−(|x|+t)/C1 , ξ

(3.15)

where  GtDi gin (x) ≡ R G Di (x − y, t)gin (y)dy. From Theorem 3.16 and (3.5),  ∞      G (x − y, t)f (y)dy Di 0   0

 ≤ 0

⎛ ∞

L 2ξ

⎞ |x−y−λi t|2 |x−y−λi t|2 − C (t+1) − C (t+1) 0 0 e e ⎜ ⎟ + e−α0 γ t + e−(|x−y|+t)/C1 ⎠ √ √ ⎝ (t + 1) (t + 1) λ <0 λ >0 i

i

×f0 (y) L 2 dy ≤ Ce−γ (|x|+t)/C . ξ

(3.16)

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S. Deng, W. Wang, S.-H. Yu

3.2. Energy estimates on the boundary data. In the representation (3.11), the full boundary data are needed while only f(0, t, ξ )|ξ 1 >0 are given. We shall get the information about f(0, t, ξ )|ξ 1 ≤0 by the energy method. The basic energy estimates for the solution f(x, t, ξ ) of (3.4) and its boundary data f(0, t, ξ ) follow from the standard method, cf. [1,33]. To control the nonlinear term (f), we make a priori smallness assumption: f L 2 1. ξ

Lemma 3.3. Suppose f L 2 < γ 2 .

(3.17)

ξ

Then there exists δ1 > 0 such that for any γ ∈ (0, δ1 ),  t  t e−γ (t−s) (f, |ξ 1 |f)− ds ≤ e−γ (t−s) (b+ , |ξ 1 |b+ )+ ds + e−γ t x=0 0 0  ∞ × eγ x (f0 , f0 )d x. (3.18) 0

Proof. From (3.4), we get  ∞   eγ x f · ∂t f + ξ 1 ∂x f − Lf − (f) + γ P+0 ξ 1 f dξ d x 0= R3 0  ∞  ∞  γ  1 γx d e (f, f)d x + eγ x − (f, ξ 1 f) + (f, γ P+0 ξ 1 f) − (f, Lf)−(f, (f)) d x = dt 0 2 2 0 1 . (3.19) − (f, ξ f) x=0

Lemma 2.4 and macro-micro decomposition yield the estimate of the nonlinear term 1

1

− (f, (f)) ≥ −(1 + |ξ |) 2 f L 2 (1 + |ξ |)− 2 (f) L 2 ξ

≥

ξ

−Cf L 2 (P1 f, (1 + |ξ |)P1 f) − Cf3L 2 . ξ ξ

(3.20)

A priori smallness assumption and (2.5) yield that γ − (f, ξ 1 f) + (f, γ P+0 ξ 1 f) − (f, Lf) − (f, (f)) 2 γ γ γ (P0 f, P0 f) + (P1 f, (1 + |ξ |)P1 f) > (f, f), ≥ 2 2 2 where  is the constant in Lemma 2.2. Equations (3.21) and (3.19) give that for any t > 0,  ∞  ∞ d eγ x (f, f)d x + γ eγ x (f, f)d x − (f, ξ 1 f) ≤ 0. x=0 dt 0 0 This inequality gives the estimate for any t > 0,  ∞  t eγ t eγ x (f, f)d x + eγ s (f, −ξ 1 f) 0

0

 x=0

∞

ds ≤

(3.22)

eγ x (f0 , f0 )d x. (3.23)

0

Therefore we get the estimate (3.18) for the full boundary data.

(3.21)

 

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

311

With the full boundary data f(0, t, ξ ), we can study the propagation of it into the interior through the integration 

t

I D (x, t) ≡

G Di (x, t − s)[ξ 1 f](0, s)ds.

0

Similar to the decomposition of Gi in Theorem 2.1,   G Di (x, t, ξ, ξ∗ ) ≡ δ 1 (x − ξ∗1 t)δ 3 (ξ − ξ∗ )e−ν(ξ∗ )t + j1D (x, t, ξ, ξ∗ ) + j2D (x, t, ξ, ξ∗ ) +r D (x, t, ξ, ξ∗ ),

(3.24)

where ⎧ 1 1 ⎪ ⎨0 if (x − ξ∗ t)(ξ t − x) < 0, ˜ j1D (x, t, ξ ; ξ∗ ) = e−ν(ξ )(t−t1 )−ν(ξ∗ )t1 K(ξ, ξ∗ ) ⎪ else if x − ξ∗ t
= 0, ⎩ 1 1 (ξ − ξ0 ) (x−ξ∗1 t) ˜ + 1 where t1 = t − 1 , K(ξ, ξ∗ ) = K(ξ, ξ∗ ) − γ P ξ , 0 ∗

ξ 1 −ξ0

j2D (x, t, ξ ; ξ∗ )

r D (x, t) L 2 ξ

= O(1)e

−

ν0 (t+|x|) 3

⎛ ) ≤ C ⎝ λ <0 i

−

 |ξ − ξ∗ |−1 (1 + |ξ |)−1 for |ξ − ξ∗ | ≤ 1, (1 + |ξ |)−1 for |ξ − ξ∗ | ≥ 1,

|x−λi t|2 C0 (t+1)

e√

(t+1)

+ e−α0

) γt

−

λi >0

|x−λi t|2 C0 (t+1)

e√

(t+1)

(3.25)

⎞ + e−(|x|+t)/C1 ⎠ .

This decomposition is obtained by Picard’s iteration:  (∂t + ξ 1 ∂x + ν(ξ ))h0D = 0, h0D (x, 0, ξ ) = δ 1 (x)δ 3 (ξ − ξ∗ ), 





˜ 0, (∂t + ξ 1 ∂x + ν(ξ ))j1D = Kh D j1D (x, 0, ξ ) = 0, ˜ 1, (∂t + ξ 1 ∂x + ν(ξ ))j2D = Kj D 2 j D (x, 0, ξ ) = 0, ˜ 2, (∂t + ξ 1 ∂x − L)r D = Kj D r D (x, 0, ξ ) = 0.

˜ we can get By solving the equations directly and from the properties of the operator K, (3.25). We omit the details. Thus the function I D (x, t) can be decomposed as follows: I D (x, t) = I D0 (x, t) + I D1 (x, t) + I D2 (x, t) + R D (x, t).

(3.26)

312

S. Deng, W. Wang, S.-H. Yu

For x, t > 0, I D0 (x, t, ξ ) ≡

I D1 (x, t, ξ ) ≡ I D2 (x, t, ξ ) ≡ R D (x, t, s, ξ ) ≡

⎧ 1 ⎪ ⎨0 if ξ x < 0, −ν(ξ ) 1 ξ b (ξ ) if {0 < x < ξ 1 t} ∩ {ξ 1 > 0}, e + ⎪ ⎩ 0 if {x > ξ 1 t} ∩ {ξ 1 > 0 }.  t j1D (x, t − s, ξ, ξ∗ )ξ∗1 f(0, s, ξ∗ )dξ∗ ds, 0 R3  t j2D (x, t − s, ξ, ξ∗ )ξ∗1 f(0, s, ξ∗ )dξ∗ ds, 0 R3  t r D (x, t − s, ξ, ξ∗ )ξ∗1 f(0, s, ξ∗ )dξ∗ ds. 0

R3

(3.27)

From (3.27), we have

 −ν(ξ ) x    ξ1 b  I D0 (x, t) L 2 ≤ e + ξ

L 2ξ

≤ Ce−ν0 |x| b+  L 2 ≤ Cδe−|x|/C . ξ

(3.28)

Also from (3.27), I D1 (x, t, ξ )  t = 0

(x−ξ 1 (t−s))(x−ξ∗1 (t−s))<0

˜ e−ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 K(ξ, ξ∗ ) 1 ξ∗ f(0, s, ξ∗ )dξ∗ ds, 1 1 ξ − ξ∗ (3.29)

where x − ξ∗1 (t − s) . ξ 1 − ξ∗1 In the special domain for integration, ξ 1 − ξ∗1 has a non-zero lower bound: x − ξ 1 (t − s) x 1 1 1 = − ξ . ξ − ξ∗ ≥ t −s t −s τ1 = t − s −

(3.30)

(3.31)

Note that from (3.30), x = ξ 1 (t − s − τ1 ) + ξ∗1 τ1 . Thus from ν(ξ ) ∼ 1 + |ξ |, we have ν(ξ )(t − s − τ1 ) + ν(ξ∗ )τ1 ≥ ν0 ((1 + |ξ |)(t − s − τ1 ) + (1 + |ξ∗ |)τ1 ) ≥ ν0 (t − s + |x|), (3.32) which implies e−ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 ≤ e−ν0 (t−s+|x|) .

(3.33)

We divide the domain {(ξ∗ , s) : s ∈ (0, t), (x − s))(x − s)) < 0} into two regions:
* 1 H1 = (ξ∗ , s) : s ∈ (0, t), (x − ξ 1 (t − s))(x − ξ∗1 (t − s)) < 0, |ξ 1 − ξ∗1 | ≥ |ξ∗1 | , 2 − ξ 1 (t

− ξ∗1 (t

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

313


* 1 H2 = (ξ∗ , s) : s ∈ (0, t), (x − ξ 1 (t − s))(x − ξ∗1 (t − s)) < 0, |ξ 1 − ξ∗1 | < |ξ∗1 | , 2 and denote   I1D1 =

˜ e−ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 K(ξ, ξ∗ ) 1 ξ∗ f(0, s, ξ∗ )dξ∗ ds, 1 1 ξ − ξ∗ ˜ e−ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 K(ξ, ξ∗ ) 1 ξ∗ f(0, s, ξ∗ )dξ∗ ds. 1 1 ξ − ξ∗

H1

  I2D1 =

H2

Then

In H1 ,

|I D1 (x, t, ξ )| ≤ I1D1 (x, t, ξ ) + I2D1 (x, t, ξ ) . |ξ∗1 | |ξ 1 −ξ∗1 |

(3.34)

≤ 2,  

I1D1 (x, t, ξ )2

˜ e−ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 K(ξ, ξ∗ ) 1 ξ∗ f(0, s, ξ∗ )dξ∗ ds 1 1 ξ − ξ∗

≤ H1

  ≤

2

e−ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 ˜ |K(ξ, ξ∗ )ξ∗1 |dξ∗ ds 3 1 1 H1 |ξ − ξ∗ | 2   −ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 |K(ξ, ˜ e ξ∗ )| 1 2 · |ξ∗ f (0, s, ξ∗ )|dξ∗ ds . 1 H1 |ξ 1 − ξ∗1 | 2 (3.35) ˜

∗) After a straightforward calculation, we can find that √K(ξ,ξ ∈ L 1ξ∗ . Thus 1 1

|ξ −ξ∗ |

 

e−ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 ˜ |K(ξ, ξ∗ )ξ∗1 |dξ∗ ds 3 1 1 2 H1 |ξ − ξ∗ |   ˜ |K(ξ, ξ∗ )| ≤ e−ν0 (t−s) dξ∗ ds ≤ C. 1 1 H1 |ξ − ξ∗1 | 2

(3.36)

˜

∗) The L 1 property of √K(ξ,ξ together with (3.35), (3.36) and (3.18) yields that 1 1

|ξ −ξ∗ |

 R3

I1D1 (x, t, ξ )2 dξ

≤ Ce

−ν0 |x|



t

e



t

≤C 0

2    e−ν0 (t−s+|x|)  |ξ 1 |f(0, s) 2 ds

−γ (t−s)

(b+ , |ξ |b+ )+ ds + e 1

−γ t

0 2 −ν0 |x|

≤ Cδ e and so



Lξ

∞

 e (f0 , f0 )d x γx

0

,

(3.37)    1 I D1 (x, t)

L 2ξ

≤ Cδe−|x|/C .

(3.38)

314

S. Deng, W. Wang, S.-H. Yu

In the domain H2 , |ξ∗1 | − |ξ 1 | ≤ |ξ 1 − ξ∗1 | ≤ 21 |ξ∗1 | ⇒ |ξ∗1 | ≤ 2|ξ 1 |. Since K˜ gains decaying rate in ξ variable,   e−ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 ˜ |K(ξ, ξ∗ )ξ∗1 |dξ∗ ds 3 H2 |ξ 1 − ξ∗1 | 2  2|ξ 1 | ≤ e−ν0 (t−s+|x|) ds 1 {s |s∈(0,t)∩|x−ξ 1 (t−s)|≤1 } |x − ξ 1 (t − s)| 2  1 + e−ν0 (t−s+|x|) ds 1 1 1 {s |s∈(0,t)∩|x−ξ (t−s)|≥1 } |x − ξ (t − s)| 2  2|ξ 1 | ≤ e−ν0 |x| ds 1 {s |s∈(0,t)∩|x−ξ 1 (t−s)|≤1 } |x − ξ 1 (t − s)| 2  + e−ν0 (t−s+|x|) ds ≤ Ce−ν0 |x| . (3.39) {s |s∈(0,t)∩|x−ξ 1 (t−s)|≤1 } ˜ K(ξ,ξ ∗)

The L 1 property of   2  I D1 (x, t)

L 2ξ

1

⎛  =⎝  ≤

|ξ 1 −ξ∗1 | 2

together with (3.39) yields that

 

R3

H2

 

R3

H2

 

˜ ξ∗ ) 1 e−ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 K(ξ, ξ∗ f(0, s, ξ∗ )dξ∗ ds ξ 1 − ξ∗1

1

H2

≤ Ce

−ν0 |x|

2

dξ ⎠

e−ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 ˜ |K(ξ, ξ∗ )ξ∗1 |dξ∗ ds 3 |ξ 1 − ξ∗1 | 2

˜ e−ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 |K(ξ, ξ∗ )|

·

⎞1

2

   R3

|ξ 1 − ξ∗1 | 2

1 2

|ξ∗1 f2 (0, s, ξ∗ )|dξ∗ ds

˜ e−ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 |K(ξ, ξ∗ )| 1

|ξ 1 − ξ∗1 | 2

H2

≤ Cδe−|x|/C .

dξ 1 2

|ξ∗1 f2 (0, s, ξ∗ )|dξ∗ dsdξ (3.40)

From (3.38) and (3.40),

    I D1 (x, t) L 2 ≤ I1D1 (x, t) ξ

L 2ξ

    + I2D1 (x, t)

L 2ξ

≤ Cδe−|x|/C .

(3.41)

1 Since j2D doesn’t contain the strong singularity |ξ 1 −ξ 1 | and only contains an integrable

singularity

1 |ξ −ξ∗ | ,

∗

it also gains decay in ξ variable. By direct computations, 

I D2 (x, t) L 2 ≤ C ξ

t

e

 21  2  1 ≤ Cδe−|x|/C .  |ξ |f(0, s) 2 ds

−ν0 (t−s+|x|) 

Lξ

0

(3.42)

r D doesn’t contain any singularity while contains an extra decaying rate in ξ . Thus from (3.18) and the Schwartz inequality, we get  t  (x−λ j (t−s))2  1   R D (x, t) L 2 ≤ C e− C(t−s+1)  |ξ 1 |f(0, s) 2 ds e−α0 γ (t−s) √ ξ Lξ t − s + 1 0 λ >0 j

≤ Cδe

−γ |x|/C

.

(3.43)

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

315

Equations (3.28), (3.41), (3.42) and (3.43) result in the estimate of the function I D (x, t): I D (x, t) L 2 ξ

 t    1  =  G Di (x, t − s)[ξ f](0, s)ds   0

L 2ξ

≤ Cδe−γ |x|/C .

(3.44)

3.3. Nonlinear term. From (3.16) and (3.44), we already have the estimates for the initial and boundary terms. There exists some constant A0 > 0 such that  ∞    A0 −γ |x|/C  G Di (x − y, t)f0 (y)dy  ,   2 ≤ 3 δe 0 Lξ  t     G Di (x, t − s)[ξ 1 f](0, s)ds  ≤ A0 δe−γ |x|/C .   2 3 0

(3.45) (3.46)

Lξ

Thus we make ansatz assumptions: f(x, t) L 2 ≤ C0 δe−γ |x|/C ,

(3.47)

f(x, t) L ∞ ≤ C1 δe−γ |x|/C , ξ,3

(3.48)

ξ

where C0 > A0 and C1 > 0 depends on C0 . Under such assumptions, the nonlinear term (f) can be controlled by  (f) L 2 ≤ Cδ 2 e−γ |x|/C ,

(3.49)

ξ

where C is independent of δ. The representation (3.12) of f(x, t, ξ ) and (3.45), (3.46) yield that   f(x, t) L 2 =   ξ

∞ 0

  G Di (x − y, t)f0 (y)dy  

 t   + 

L 2ξ

 t    1  + G (x, t − s)[ξ f](0, s)ds Di   0

 ∞  G Di (x − y, t − s) (f)(y, s)dyds   2 0 0 Lξ  t  ∞    2 A0 −γ |x|/C  δe ≤ + G Di (x − y, t − s) (f)(y, s)dyds   2. 3 0 0 L

L 2ξ

ξ

(3.50) To verify (3.47), we just have to proof  t     0

0

∞

  G Di (x − y, t − s) (f)(y, s)dyds  

L 2ξ



A0 −γ |x|/C . δe 3

(3.51)

316

S. Deng, W. Wang, S.-H. Yu

t ∞ Substitute (3.49) into 0 0 Gi (x − y, t − s) (f)(y, s)dyds,   t  ∞    G Di (x − y, t − s) (f)(y, s)dyds    0

≤C

L 2ξ

0

 t 0

∞

e−α0 γ (t−s)

0



t

≤ Cδ 2

e−α0 γ (t−s)

0

e

(x−y−λ j (t−s))2 − C(t−s+1)

√

λ j >0



t −s+1

δ 2 e−γ |y|/C dyds

e−γ |x−λ j (t−s)|/C ds ≤

λ j >0

C 2 −γ |x|/C A0 −γ |x|/C

, δ e δe γ 3 (3.52)

since δ is sufficiently small and γ is independent of δ. Thus we’ve proved (3.51) and justified the ansatz (3.47). To verify (3.48), we need another representation of f(x, t, ξ ),  t    ˜ + (f) x − ξ 1 (t − s), s, ξ ds, f(x, t, ξ ) = e−ν(ξ )t f0 (x − ξ 1 t, ξ )+ e−ν(ξ )(t−s) Kf 0

(3.53) ˜ = Kf − γ P+ ξ 1 f. Since K˜ gains decay in the ξ variable, where Kf 0  t  t ˜ 1 ˜ e−ν(ξ )(t−s) Kf(x ≤C (t − s), s)ds e−ν(ξ )(t−s) Kf(x − ξ 1 (t − s), s) ds − ξ 0 0  t −ν(ξ )(t−s) 1 ≤C e f(x − ξ (t − s), x) L 2 ds ξ 0  t δ −γ |x|/C 1 e ≤ CC0 δ e−ν(ξ )(t−s) e−γ |x−ξ (t−s)|/C ds ≤ C . (3.54) ν(ξ ) 0 Thus, consider sup eγ |x|/C f(x, t, ξ ) x,t

 t γ |x|/C−ν(ξ )t 1 γ |x|/C−ν(ξ )(t−s) ˜ 1 ≤ sup e f0 (x − ξ t, ξ ) + sup e Kf(x −ξ (t −s), s)ds x,t x,t 0  t + sup eγ |x|/C−ν(ξ )(t−s) (f)(x − ξ 1 (t − s), s)ds x,t 0  t Cδ Cδ −ν(ξ )(t−s)+γ |ξ 1 |(t−s) γ |x−ξ 1 (t−s)| 1 ≤ +sup + e δ|ξ |e f(x −ξ (t −s), s)ds (1+|ξ |)3 (1+|ξ |) x,t 0  t Cδ Cδ 1 γ |z|/C ≤ + δ sup + sup f(z, s, ξ ) e−ν(ξ )(t−s)+γ |ξ |(t−s) |ξ |ds. e 3 (1 + |ξ |) (1 + |ξ |) x,t z,s 0 (3.55) Since ν(ξ ) ∼ 1 + |ξ | and γ is a small parameter, we can get Cδ Cδ γ |x|/C e + f(x, t, ξ ) sup eγ |x|/C f(x, t, ξ ) ≤ + δ sup , (3.56) (1 + |ξ |)3 (1 + |ξ |) x,t x,t

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

317

which results in sup eγ |x|/C f(x, t, ξ ) ≤ x,t

2Cδ 2Cδ . + (1 + |ξ |)3 (1 + |ξ |)

(3.57)

Repeat the same argument for another two times. One will get sup eγ |x|/C f(x, t, ξ ) ≤ x,t

Cδ . (1 + |ξ |)3

(3.58)

Thus the ansatz assumption (3.48) holds. Until now, we’ve proved the a priori smallness assumption (3.17) and the fact that f(x, t, ξ ) is bounded under the norms || · || L 2 and || · || L ∞ . In the rest of this section, ξ,3 ξ we will check whether ft has a decaying rate fast enough to ensure the existence of limt→∞ f (x, t, ξ ). From (3.4),   0 = ft + ξ 1 fx − Lf − (f) + γ P+0 ξ 1 f = ftt + ξ 1 fxt − Lft − ( (f))t + γ P+0 ξ 1 ft . t

Set g = ft , then we get the equation about g, ⎧ gt + ξ 1 gx − Lg = ( (f))t − γ P+0 ξ 1 g, (x, t, ξ ) ∈ R+ × R+ × R3 , ⎪ ⎪ ⎪ ⎨g(x, 0, ξ ) = g0 (x, ξ ), g(0, t, ξ )|ξ 1 >0 = 0, ⎪ ⎪ ⎪ ⎩ lim g(x, t, ξ ) = 0,

(3.59)

x→∞

where  ( (f))t ≡

√ Q( Mf) √ M

= t

2B

√ √  Mf, Mg . √ M

From (3.7), the initial data g0 (x, ξ ) = ft (x, t, ξ )|t=0 satisfies sup(1 + |ξ |)3 |g0 | ≤ Cδe−|x|/C . ξ

Lemma 2.4 yields that the nonlinear term ( (f))t can be controlled by     ( (f))t  2 ≤ C f 2 (1 + |ξ |)g 2 + (1 + |ξ |)f 2 g 2 ≤ Cδe−γ |x|/C g L ∞ . L L L L L ξ,3 ξ

ξ

ξ

ξ

ξ

(3.60) Similar to what we have done with (3.4), then one can get that there exists some constant C which is independent of δ such that ||g(x, t)|| L 2 ≤ Cδe−γ (|x|+t)/C ,

(3.61)

||g(x, t)|| L ∞ ≤ Cδe−γ (|x|+t)/C . ξ,3

(3.62)

ξ

318

S. Deng, W. Wang, S.-H. Yu

t Therefore f(x, t, ξ ) = f0 (x, ξ ) + 0 g(x, s, ξ )ds gains exponential decaying rate in t. As a result, f(x, t, ξ ) does converge to ψ(x, ξ ) as t tends to infinity. In other words, the boundary layer solution exists and satisfies ||ψ(x)|| L 2 ≤ Cδe−γ |x|/C ,

(3.63)

≤ Cδe−γ |x|/C . ||ψ(x)|| L ∞ ξ,3

(3.64)

ξ

Therefore we have proved Lemma 3.1. When M < −1, the projection operator P+0 ξ 1 ≡ 0. Therefore, the damped system is just the original system. Then the fact that the solution f(x, t, ξ ) of (3.4) converges to the boundary layer solution ψ(x, ξ ) shows the stability of the boundary layer. 4. Coupling Waves We shall introduce some integrals which will be used in the following two chapters. Denote  A1 (x, t, ξ ; λ j ) ≡

t

√

0

1 s+1

e−

(x−ξ 1 (t−s)−λ j s)2 C(s+1)

e−ν(ξ )(t−s) ds,

A2 (x, t, ξ ; λk , λk+1 )  ≡ {s |s∈(0,t),x−ξ 1 (t−s)∈(λk s,λk+1 s) } e−ν(ξ )(t−s) ds, × (x − ξ 1 (t − s) − λk s + 1)(λk+1 s − x + ξ 1 (t − s) + 1)

(4.1)

(4.2)

for j = 1, 2, 3 and k = 1, 2. Lemma 4.1. For x, t > 0, 2

(x−λ j t) 1 1 e− C(t+1) , √ ν(ξ ) t + 1

A1 (x, t; λ j ) ≤ C

(4.3)

for j = 1, 2, 3. Proof. Consider A1 (x, t, ξ ; λ j ):  A1 (x, t, ξ ; λ j ) ≤

{s ||x−λ j s |>2|

ξ1



+

|(t−s) }

{s ||x−λ j s |<2|ξ 1 |(t−s) }

1

e−

(x−λ j s)2 4C(t+1)

e−ν(ξ )(t−s) ds s+1 |x−λ j s| ν(ξ )(t−s) 1 e− 4 e− 2 ds √ s+1

√

2

(x−λ j t) 1 1 ≤ e− C(t+1) . √ ν(ξ ) t + 1

Thus we get the estimate (4.3).

 

(4.4)

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

319

Lemma 4.2. For x, t > 0, A2 (x, t; λk , λk+1 ) ⎛ ⎞ (x−λ j t)2 3 − C(1+t) 1 ⎜ e ⎟ ≤C + e−(|x|+t)/C ⎠ √ ⎝ ν(ξ ) t +1 j=1 ⎧ 2 ⎪ 1 1 ⎨ for x ∈ (λ1 t, λ3 t), √ +C (|x − λl t| + 1)(|x − λl+1 t| + 1) l=1 ν(ξ ) ⎪ ⎩ 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞).

(4.5)

for k = 1, 2. Proof. The estimate of A2 (x, t, ξ ; λk , λk+1 ) should be considered in different regions of x. When λk t < x < λk+1 t,  t−5logt e−ν(ξ )(t−s) ds A2 (x, t, ξ ; λk , λk+1 ) ≤ 0  t 1 −ν(ξ )(t−s) + ds e √ (x − λ t + 1)(λ t−5logt k k+1 t − x + 1) 1 1 ≤ . √ ν(ξ ) (x − λk t + 1)(λk+1 t − x + 1)

(4.6)

When x > λk+1 t, in the special integral domain of A2 (x, t, ξ ; λk , λk+1 ), 0 > x − ξ 1 (t − s) − λk+1 s > λk+1 (t − s) − ξ 1 (t − s). Thus |x − ξ 1 (t − s) − λk+1 s| < C|ξ |(t − s) which implies  t (x−λk+1 t)2 ν(ξ )(t−s) 1 1 1 1 A2 (x, t, ξ ; λk , λk+1 ) ≤ √ e− 2 −|x−ξ (t−s)−λk+1 s|/C ds ≤ e− C(t+1) . √ ν(ξ ) t +1 0 t +1 When x < λk t, in the special integral domain of A2 (x, t, ξ ; λk , λk+1 ), 0 < x − ξ 1 (t − s) − λk s < λk (t − s) − ξ 1 (t − s). Thus |x − ξ 1 (t − s) − λk s| < C|ξ |(t − s) which implies  t (x−λk t)2 ν(ξ )(t−s) 1 1 1 1 A2 (x, t, ξ ; λk , λk+1 ) ≤ √ e− 2 −|x−ξ (t−s)−λk s|/C ds ≤ e− C(t+1) . √ ν(ξ ) t +1 t +1 0 Then we complete the proof.

 

320

S. Deng, W. Wang, S.-H. Yu

Denote J1 (x, t; λi , λ j ) ≡

 t 0

∞

0

 t

2

(x−y−λi (t−s))2 j s) 1 1 − (y−λ e− C(t−s+1) e C(s+1) dyds, (4.7) t −s+1 s+1

λk+1 s

2

(x−y−λ j (t−s)) 1 e− C(t−s+1) (t − s + 1) 0 λk s 1 × dyds, (4.8) (|y − λk s| + 1)(|y − λk+1 s| + 1)  t ∞ (y−λ j s)2 (x−y−λi (t−s))2 1 1 e−γ y/C e− C(s+1) dyds, e− C(t−s+1) √ J1 (x, t; λi , λ j ) ≡ s+1 0 0 t −s +1 (4.9)  t  λ3 s (x−y−λ j (t−s))2 1 J2 (x, t; λ j , λk , λk+1 ) ≡ e− C(t−s+1) (t − s + 1) 0 0 e−Cγ y ×√ dyds, (4.10) (|y − λk s| + 1)(|y − λk+1 s| + 1)

J2 (x, t; λ j , λk , λk+1 ) ≡

for i, j = 1, 2, 3 and k = 1, 2. The estimates of J1 (x, t; λi , λ j ) and J2 (x, t; λ j , λk , λk+1 ) can be gotten from [26]. Lemma 4.3. J1 (x, t; λi , λ j ), J2 (x, t; λ j , λk , λk+1 ) ⎛ ⎞ (x−λ j t)2 3 − C(1+t) e ⎜ ⎟ ≤ C ⎝ + e−(|x|+t)/C ⎠ √ t +1 j=1 ⎧ 2 ⎪ 1 ⎨ √ for x ∈ (λ1 t, λ3 t), +C (|x − λl t| + 1)(|x − λl+1 t| + 1) l=1 ⎪ ⎩ 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞).

(4.11)

for i, j = 1, 2, 3 and k = 1, 2. The following lemmas give explicit estimates for J1 (x, t; λi , λ j ) and J2 (x, t; λ j , λk , λk+1 ): Lemma 4.4. There exists some constant C > 0 which depends on γ such that ⎞ ⎛ (x−λ j t)2 3 − C(1+t) e ⎟ ⎜ J1 (x, t; λi , λ j ) ≤ C ⎝ + e−(|x|+t)/C ⎠ , √ t +1 j=1 for i, j = 1, 2, 3.

(4.12)

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

321

Proof. J1 (x, t; λi , λ j ) can be rewritten to be  t  λ j s/2 (x−y−λi (t−s))2 1 J1 (x, t; λi , λ j ) ≤ C e−Cγ y e− C(t−s+1) e−Cs dyds √ (t − s + 1) s + 1 0 0  t ∞ (x−y−λi (t−s))2 1 +C e−Cγ y/2+Cγ λ j s/4 e− C(t−s+1) dyds √ 0 λ j s/2 (t −s +1) s + 1  t ∞ (x−y−λi (t−s))2 1 ≤C e−Cγ |y|−Cγ s− C(t−s+1) dyds √ (t − s + 1) s + 1 0 0  t 1 e−Cγ |x−λi t| e−Cγ s−Cγ |x−λi (t−s)| ds ≤ C √ ≤C √ (t − s + 1)(s + 1) t +1 0 (x−λi t)2

e− C(t+1) . ≤C √ t +1

(4.13)

  Consider K α,β,σ (x, t; 0, t; λ, D) ≡

 t 0

∞ −∞

×e− Set α (t) ≡

 0

t

(t − s)(−β+σ )/2 (t − s + 1)−σ/2

(x−y−λ(t−s))2 D(t−s)

(s + 1)−α/2 e−

4γ |y| D

dyds.

(4.14)

⎧ ⎪ ⎨1 for α > 2, −α/2 (s + 1) ds = O(1) log(t + 1) for α = 2, ⎪ ⎩(t + 1)(2−α)/2 for α < 2. (4.15)

The following lemma is quoted from [22] and will be used to get the estimate of J2 (x, t; λ j , λk , λk+1 ). Lemma 4.5. Suppose that α ≥ 0, 3 > β −σ ≥ 0, σ ≥ 0 and λ a positive constant. Then there exists a positive constant C such that for any fixed positive constant E > D+O(1)γ and a bound O(1) independent of γ , K α,β,σ (x, t; 0, t; λ, D) = O(1) √ ⎧ (x + 1)(−β+2)/2 (λt − x)−α/2 (1 + γ x + 1)−1 + β−1 (γ −1 )(t + 1)−α/2 e−Cγ x ⎪ ⎪ √ ⎪ ⎪ ⎪ + α (γ −1 )(t + 1)(−β+1)/2 e−Cγ |λt−x| for 0 < x ≤ λ(t + 1) − t + 1, ⎪ √ √ ⎪ ⎪ ⎪ (t + 1)(−β+1)/2 [ α (t + 1)e−Cγ t + α ( t + 1)(1 + γ t)−1 ] ⎪ ⎪ √ ⎪ ⎪ ⎨+(t + 1)−α/2 β−1 (t + 1)e−Cγ t for |x − λ(t + 1)| ≤ t + 1, (x−λt)2 √ √ · ⎪ (t + 1)(−β+1)/2 [ α ( t + 1)(1 + γ t)−1 e− D(t+1) +(t +1)1/2 (x −λt)−α/2 e−Cγ (x−λt) ⎪ ⎪ ⎪ 4γ (x−λt) 2 ⎪ ⎪ + α (γ −1 )e−Cγ (x−λt) e−Cγ t + γ (α−2)/2 e− D ] ⎪ ⎪ ⎪ ⎪ ⎪ +(x − λt)(−α−2β+3)/2 e−Cγ (x−λt) + (t + 1)−α/2 ( β−1 (γ −1 ) + β−1 (t + 1)) ⎪ ⎪ √ 4γ (x−λt) ⎩ for x > λ(t + 1) + t + 1. e−Cγ t e− D (4.16)

322

S. Deng, W. Wang, S.-H. Yu

Lemma 4.6. There exists some constant C > 0 which depends on γ such that J2 (x, t; λ j , λk , λk+1 ) ⎛ ⎞ (x−λ j t)2 3 − ⎜ e C(1+t) ⎟ ≤C⎝ + e−(|x|+t)/C ⎠ √ t + 1 j=1 ⎧ 2 ⎪ 1 ⎨ √ for x ∈ (0, λ3 t), +C (|x − λ t| + 1)(|x − λl+1 t| + 1) l l=1 ⎪ ⎩ 0 for x ∈ (λ3 t, ∞).

(4.17)

for j = 1, 2, 3 and k = 1, 2. Proof. Similar to J1 (x, t; λi , λ j ), we’ll find J2 (x, t; λ j , λk , λk+1 )  t  λ3 s/2 (x−y−λ j (t−s))2 1 1 −Cγ y e− C(t−s+1) e ≤C dyds t − s + 1 s + 1 0 0  t  λ3 s (x−y−λ j (t−s))2 1 e− C(t−s+1) e−Cγ y/2−Cγ λ3 s/4 dyds +C 0 λ3 s/2 t − s + 1  t ∞ (x−y−λ j (t−s))2 1 e− C(t−s+1) e−Cγ y dyds. ≤C (t − s + 1)(s + 1) 0 0

(4.18)

When λ j > 0, J2 (x, t; λ j , λk , λk+1 ) ≤ O(1)K 2,2,2 (x, t; 0, t; λ j , C) ⎛ ⎞ (x−λ j t)2 3 − C(1+t) e ⎜ ⎟ ≤C⎝ + e−(|x|+t)/C ⎠ √ t +1 j=1 ⎧ 2 ⎪ 1 ⎨ √ for x ∈ (0, λ3 t), +C (|x − λl t| + 1)(|x − λl+1 t| + 1) l=1 ⎪ ⎩ 0 for x ∈ (λ3 t, ∞). When λ j < 0,

(4.19)



t e−Cγ |x−λ j (t−s)| e−Cγ |x| J2 (x, t; λ j , λk , λk+1 ) ≤ C ds ≤ C √ t +1 t − s + 1(s + 1) 0 ⎞ ⎛ (x−λ j t)2 3 − ⎟ ⎜ e C(1+t) ≤C⎝ + e−(|x|+t)/C ⎠ √ t + 1 j=1

+C  

⎧ 2 ⎪ ⎨ √

1 for x ∈ (0, λ3 t), (|x − λl t| + 1)(|x − λl+1 t| + 1)

⎪ ⎩ l=1 0 for x ∈ (λ3 t, ∞).

(4.20)

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

323

5. Stability of Knudsen’s Layers for M > 1 When M > −1, the damped system is no longer the original system. To study the stability problem, we treat the problem in the perturbation form. Set w(x, t, ξ ) = f(x, t, ξ ) − ψ(x, ξ ). Then w(x, t, ξ ) satisfies ⎧ wt + ξ 1 wx − Lw = L δ w + (w), (x, t, ξ ) ∈ R+ × R+ × R3 , ⎪ ⎪ ⎪ ⎨w(x, ξ ) = w0 (x, ξ ) = f0 (x, ξ ) − ψ(x, ξ ), w(0, t, ξ )|ξ 1 >0 = 0, ⎪ ⎪ ⎪ ⎩ lim w(x, t, ξ ) = 0,

(5.1)

x→∞

where

Lδw ≡

B

√

Mw,

√

 √  √ Mψ + B Mψ, Mw , √ M

and the initial data w0 (x, ξ ) satisfies sup(1 + |ξ |)3 |w0 (x, ξ )| ≤ C e−|x|/C , ξ

(5.2)

for some positive constant C independent of . As stated in Sect. 1, we’ll prove Theorem 1.1 under the assumption M > 1 in this section. Suppose M > 1. Then w(x, t, ξ ) can be represented directly by the Green’s function Gi (x, t) of the initial value problem:  w(x, t) =

 t Gi (x − y, t)w0 (y)dy + Gi (x, t − s)[ξ 1 w](0, s)ds 0 0  t ∞ + Gi (x − y, t − s)(L δ w + (w))(y, s)dyds. ∞

0

(5.3)

0

From Theorem 2.2,    

  Gi (x − y, t)w0 (y)dy   2 0 Lξ ⎛ ⎞ 2 (x−y−λ t)  ∞ 3 − C(t+1)j e ⎜ ⎟ ≤ C + e−(|x−y|+t)/C ⎠ e−|y|/C dy √ ⎝ t + 1 0 j=1 ⎞ ⎛ (x−λ j t)2 3 − C(t+1) e ⎟ ⎜ ≤ C ⎝ + e−(|x|+t)/C ⎠ . √ t +1 j=1 ∞

(5.4)

324

S. Deng, W. Wang, S.-H. Yu

5.1. Energy estimates on the boundary data. Similar to that in Sect. 3.2, we apply the weighted energy method to get the full boundary data. The weighted function is chosen to be e−βx , where β is a positive and sufficiently small constant. Lemma 5.1. Suppose the solution w(x, t, ξ ) of (5.1) satisfies w L 2 < β 2 ,

(5.5)

ξ

then there exists δ2 > 0 such that for any β ∈ (0, δ2 ),  t  ∞ eβt e−βx (w, w)d x + eβs (w, |ξ 1 |w)− ds x=0 0 0  ∞ e−βx (w0 , w0 )d x. ≤ 0

Proof. Following the process in Sect. 3.2, we only have to check whether L δ w)d x is small enough. Since     1 1     − (w, L δ w) ≥ −C (1 + |ξ ) 2 w 2 (1 + |ξ |)− 2 L δ w 2 Lξ

∞ 0

(5.6) e−βx (w,

Lξ

≥ −Cδ ((P1 w, (1 + |ξ |)P1 w) + (P0 w, P0 w)) and δ is independent of β, one can choose δ small enough and complete the process. Thus from the energy estimate, we have the full boundary data w(0, t, ξ ) and also the estimate of w(x, t, ξ ).   With the full boundary data, we now study how it propagates into the interior through the boundary integration  t I(x, t) ≡ Gi (x, t − s)[ξ 1 w](0, s)ds. 0

From Theorem 2.1,   Gi (x, t, ξ, ξ∗ ) ≡ δ 1 (x − ξ∗1 t)δ 3 (ξ − ξ∗ )e−ν(ξ∗ )t + j1 (x, t, ξ, ξ∗ ) + j2 (x, t, ξ, ξ∗ ) +r(x, t, ξ, ξ∗ ),

(5.7)

and the function I(x, t) can be decomposed as follows: I(x, t) = I0 (x, t) + I1 (x, t) + I2 (x, t) + R(x, t).

(5.8)

I0 (x, t, ξ ) ≡ 0,  t j1 (x, t − s, ξ, ξ∗ )ξ∗1 w(0, s, ξ∗ )dξ∗ ds, I1 (x, t, ξ ) ≡ 3 0 R  t I2 (x, t, ξ ) ≡ j2 (x, t − s, ξ, ξ∗ )ξ∗1 w(0, s, ξ∗ )dξ∗ ds, 3 0 R  t R(x, t, ξ ) ≡ r(x, t − s, ξ, ξ∗ )ξ∗1 w(0, s, ξ∗ )dξ∗ ds.

(5.9)

For x, t > 0,

0

R3

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

325

Similar to the analysis applied to I D1 (x, t, ξ ), I D2 (x, t, ξ ) and R D (x, t, ξ ) in (3.37), (3.40), (3.42) and (3.43), we get  Ii (x, t) L 2 ≤ C

t

e

ξ

 2  21  1 , for i = 1, 2, (5.10)  |ξ |w(0, s) 2

−ν0 (|x|+t−s) 

Lξ

0

R(x, t) L 2 ≤ C ξ



t

+C

e

3 

t

(x−λ j (t−s))2 C(t−s+1)

e− √

j=1 0

t −s+1

     |ξ 1 |w(0, s)

L 2ξ

 21  2  1  |ξ |w(0, s) 2 ds .

−ν0 (|x|+t−s) 

(5.11)

Lξ

0

ds

Equations (5.6), (5.10) and (5.11) together with the Schiwartz inequality yield that Ii (x, t) L 2 ≤ Ce

−ν0 |x|

ξ

⎜ ≤C⎝ 3

ξ

t

e

0 −βt−ν0 |x|

≤ C e ⎛ R(x, t) L 2





t

j=1 0

 ·

t

eβs

0

−β(t−s)

(w, |ξ |w)−

, for i = 1, 2, (x−λ j (t−s))2

 21 x=0

x=0

ds (5.12)

⎞1

e− C(t−s+1) −βs ⎟ e ds ⎠ t −s+1 (w, |ξ 1 |w)−

 21

1

ds

2

≤ C

3 

t

0

(x−λ j t)2 C(t+1)

t +1

j=1 0

where C depends on β but is independent of . Therefore we get the estimate of I(x, t, ξ ),  t    1  I(x, t) L 2 =  G (x, t − s)[ξ w](0, s)ds i   ξ

e

−

,

(5.13)

L 2ξ

⎞ ⎛ (x−λ j t)2 3 − C(t+1) e ⎟ ⎜ ≤ C ⎝ + e−(|x|+t)/C ⎠ , √ t +1 j=1

(5.14)

where the constant C is independent of . 5.2. Nonlinear term. From (5.4) and (5.14), one has that for some constant A0 > 0, ⎛ ⎞ (x−λ j t)2  ∞  3 − C(t+1)   e A0 ⎜ ⎟  (5.15) Gi (x − y, t)w0 (y)dy  + e−(|x|+t)/C ⎠ , √   2 ≤ 4 ⎝ t + 1 0 Lξ j=1 ⎞ ⎛ (x−λ j t)2  t  3 −   e C(t+1) ⎟  Gi (x, t − s)[ξ 1 w](0, s)ds  ≤ A0 ⎜ + e−(|x|+t)/C ⎠ . (5.16) √ ⎝   2 4 t +1 0

Lξ

j=1

326

S. Deng, W. Wang, S.-H. Yu

Take a Picard iteration several times: ⎧ ∞ t 1 ⎪ ⎨w1 (x, t) = 0 Gi (x − y, t)w0 (y)dy + 0 Gi (x, t − s)[ξ w](0, s)ds, ⎪ ⎩w (x, t) = w (x, t) +  t  ∞ G (x − y, t − s) (w )(y, s)dyds for k ≥ 2. k 1 i k−1 0 0 (5.17) Then we’ll get wk (x, t) L 2 ξ ⎞ ⎛ (x−λ j t)2 3 − C(1+t) e ⎟ ⎜ ≤ C ⎝ + e−(|x|+t)/C ⎠ √ t +1 j=1

+C

⎧ 2 ⎪ ⎨ √

1 for x ∈ (λ1 t, λ3 t), (|x − λl t| + 1)(|x − λl+1 t| + 1)

⎪ ⎩ l=1 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞),

(5.18)

for k ≥ 2. Thus we make ansatz assumptions: w(x, t) L 2 ξ ⎞ ⎛ (x−λ j t)2 3 − ⎟ ⎜ e C(1+t) ≤ C0 ⎝ + e−(|x|+t)/C ⎠ √ t + 1 j=1 ⎧ 2 ⎪ 1 ⎨ √ for x ∈ (λ1 t, λ3 t), +C0 (|x − λ t| + 1)(|x − λl+1 t| + 1) l l=1 ⎪ ⎩ 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞). w(x, t) L ∞ ξ,3 ⎞ ⎛ (x−λ j t)2 3 − ⎟ ⎜ e C(1+t) ≤ C1 ⎝ + e−(|x|+t)/C ⎠ √ t + 1 j=1 +C1

⎧ 2 ⎪ ⎨ √

1 for x ∈ (λ1 t, λ3 t), (|x − λl t| + 1)(|x −λl+1 t| + 1)

⎪ ⎩ l=1 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞).

(5.19)

(5.20)

for some C > 0, C0 > A0 and C1 > 0. Then Lemma 2.4 yields that (w) and L δ w can be controlled by

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

327

⎛

 (w) L 2 ≤ C(1 + |ξ |)w2L 2 ξ

ξ

+C 2

L δ w L 2 ξ

⎧ 2 ⎪ ⎨

⎞ (x−λ j t)2 3 − C(1+t) e ⎜ ⎟ + e−(|x|+t)/C ⎠ ≤ C 2 ⎝ t +1 j=1

1 for x ∈ (λ1 t, λ3 t), (|x − λl t| + 1)(|x − λl+1 t| + 1)

⎪ ⎩ l=1 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞),   ≤ C (1 + |ξ |)w L 2 (1 + |ξ |)ψ L 2 ξ ξ ⎛ ⎞ (x−λ j t)2 3 − ⎜ e C(1+t) ⎟ + e−(|x|+t)/C ⎠ ≤ C δe−γ |x|/C ⎝ √ t + 1 j=1

+C δe

⎧ 2 ⎪ ⎨ √

−γ |x|/C

(5.21)

1 for x ∈ (λ1 t, λ3 t), (|x −λl t|+1)(|x − λl+1 t|+1)

⎪ ⎩ l=1 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞),

(5.22) for some C > 0 independent of and δ. From the representation (5.3) of w(x, t) and (5.15), (5.16), we just have to prove the following fact to verify (5.19):   Gi (x − y, t − s)L δ w(y, s)dyds   2 0 0 Lξ ⎛ ⎞ 2 (x−λ j t) 3 − A0 −γ |x|/C ⎜ e C(1+t) ⎟ e

+ e−(|x|+t)/C ⎠ √ ⎝ 4 t + 1 j=1

 t    

∞

⎧ 2 ⎪ 1 A0 −γ |x|/C ⎨ for x ∈ (λ1 t, λ3 t), √ (5.23) + e (|x − λ t| + 1)(|x − λl+1 t| + 1) l l=1 ⎪ 4 ⎩ 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞),   t  ∞    Gi (x − y, t − s) (w)(y, s)dyds   2  0 0 Lξ ⎛ ⎞ 2 (x−λ j t) 3 − A0 −γ |x|/C ⎜ e C(1+t) ⎟ e

+ e−(|x|+t)/C ⎠ √ ⎝ 4 t + 1 j=1 ⎧ 2 ⎪ 1 A0 −γ |x|/C ⎨ for x ∈ (λ1 t, λ3 t), √ (5.24) + e (|x − λ t| + 1)(|x − λl+1 t| + 1) l l=1 ⎪ 4 ⎩ 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞).

328

S. Deng, W. Wang, S.-H. Yu

Linear coupling wave t ∞ Consider ˜I(x, t, ξ ) = 0 0 Gi (x − y, t − s)L δ w(y, s)dyds. To deal with the singularities, we decompose it into four parts like that in Sect. 5.1: for x, t > 0, ˜ ˜I(x, t) = ˜I0 (x, t) + ˜I1 (x, t) + ˜I2 (x, t) + R(x, t), (5.25)  t  ∞ ˜I0 (x, t, ξ ) ≡ δ 1 (x − y − ξ∗1 (t − s))δ 3 (ξ − ξ∗ )L δ w(y, s, ξ∗ )dξ∗ dyds, 0

˜I1 (x, t, ξ ) ≡

 t 0

˜I2 (x, t, ξ ) ≡

∞

(5.26)

∞

j (x − y, t − s, ξ, ξ∗ )L δ w(y, s, ξ∗ )dξ∗ dyds, 2

R3

0

 t 0

j1 (x − y, t − s, ξ, ξ∗ )L δ w(y, s, ξ∗ )dξ∗ dyds,

R3

0

 t 0

˜ R(x, t, ξ ) ≡

R3

0

∞

r(x − y, t − s, ξ, ξ∗ )L δ w(y, s, ξ∗ )dξ∗ dyds.

R3

0

To avoid complex computations, we treat the different parts of ˜I(x, t, ξ ) differently. Since r(x, t, ξ, ξ∗ ) doesn’t contain any singularity and gains are extra decaying rate in ˜ ξ , we substitute the result (5.6) of the energy estimate into R(x, t, ξ ),    ˜ R(x, t)

L 2ξ

≤C

 t 0

3 ∞ e−

√

0

j=1

⎛



t

≤ Cδ 0



⎜ ⎝



∞

×

(x−y−λ j (t−s))2 C(1+t−s)

e

0

t −s+1

3 ∞ e− j=1

−Cγ |y|

    (1 + |ξ |)−1 L δ w(y, s)

(x−y−λ j (t−s))2 C(1+t−s)

t −s+1

L 2ξ

dyds

⎞1 2

⎟ e−Cγ |y| dy ⎠

1 (w, w)dy

2

ds

0 2

2

(x−λ j (t−s)) (x−λ j t)  t 3 3 e− C(1+t−s) −Cγ s e− C(1+t) ≤ C δ e ds ≤ C δ , √ √ t −s+1 t +1 0 j=1 j=1

(5.27)

where C depends on γ but is independent of , δ. To deal with the singular parts ˜I0 (x, t, ξ ), ˜I1 (x, t, ξ ), ˜I2 (x, t, ξ ), we apply the ansatz assumptions directly. From (5.26),    ˜ I0 (x, t)

L 2ξ

 t    −ν(ξ )(t−s) 1  = e L δ w(x − ξ (t − s), s)ds   0

L 2ξ

.

(5.28)

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

329

For the convenience of representation, denote (x−λ j t)2

3 e− C(1+t) ϒ(x, t) = + e−(|x|+t)/C √ t + 1 j=1 ⎧ 2 ⎪ 1 ⎨ √ for x ∈ (λ1 t, λ3 t), + (|x − λ t| + 1)(|x − λl+1 t| + 1) l l=1 ⎪ ⎩ 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞).

Consider  t      1 1 ˜  −ν(ξ )(t−s) 1  L e w(x − ξ (t − s), s)ds I0 (x, t) 2 ≤ sup  δ   2 Lξ ϒ(x, t) x,t ϒ(x, t) x,t 0 Lξ   t 1   1  e−ν(ξ )(t−s) 1 ϒ(x − ξ (t − s), s) ds  . L δ w(x, t) L ∞ ≤ sup  2 ξ,1  ϒ(x, t) 1 + |ξ | ϒ(x, t) x,t 0 L sup

ξ

(5.29) From Lemma 4.1 and Lemma 4.2, 

t

e−ν(ξ )(t−s) ϒ(x − ξ 1 (t − s), s)ds =

0

+

3 

t

√

j=1 0

1 s+1

e−

(x−ξ 1 (t−s)−λ j s)2 C(s+1)

e−ν(ξ )(t−s) ds

2  k=1

{s |s∈(0,t),x−ξ 1 (t−s)∈(λk s,λk+1 s) }

e−ν(ξ )(t−s) ds, × (x − ξ 1 (t − s) − λk s + 1)(λk+1 s − x + ξ 1 (t − s) + 1) =

3 j=1

A1 (x, t; λ j ) +

2 k=1

A2 (x, t; λk , λk+1 ) ≤

C ϒ(x, t). ν(ξ )

(5.30)

Then    t  1     1  < C, (5.31)  e−ν(ξ )(t−s) 1 ϒ(x − ξ (t − s), s) ds  ≤     1 + |ξ | ϒ(x, t) ν(ξ )(1 + |ξ |)  L 2 0 L2 ξ

ξ

and so sup x,t

  1 1 ˜  L δ w(x, t) L ∞ I0 (x, t) 2 ≤ sup ξ,1 Lξ ϒ(x, t) x,t ϒ(x, t) 1 ψ(x, t) L ∞ w(x, t) L ∞ ≤ sup ≤ C δ. ξ,2 ξ,2 x,t ϒ(x, t) (5.32)

330

S. Deng, W. Wang, S.-H. Yu

Thus    ˜ I0 (x, t)

L 2ξ

⎞ ⎛ (x−λ j t)2 3 − C(1+t) e ⎟ ⎜ ≤ C δ ⎝ + e−(|x|+t)/C ⎠ √ t +1 j=1

⎧ 2 ⎪ 1 ⎨ √ for x ∈ (λ1 t, λ3 t), +C δ (|x − λl t| + 1)(|x − λl+1 t| + 1) l=1 ⎪ ⎩ 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞).

(5.33)

˜I1 (x, t, ξ ) is defined by ˜I1 (x, t, ξ ) =

 t 0

×

∞ (x−y−ξ 1 (t−s))(x−y−ξ∗1 (t−s))<0

0

K(ξ, ξ∗ ) −ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 ˜ e Lw(y, s, ξ∗ )dξ∗ dyds, |ξ 1 − ξ∗1 |

(5.34)

x−y−ξ 1 (t−s)

∗ where τ1 = t − s − . In the domain for integration, |ξ 1 − ξ∗1 | has a non-zero ξ 1 −ξ∗1 1 lower bound x−y t−s − ξ . From the formula of τ1 , we also have

ν(ξ )(t −s −τ1 )+ν(ξ∗ )τ1 ≥ ν0 ((1+|ξ |)(t − s − s1 )+(1+|ξ∗ |)s1 ) ≥ ν0 (t − s + |x − y|). Similar to the analysis in dealing with I D1 (x, t, ξ ), we divide the integration domain into two parts, ˜ 1 = {(ξ∗ , y, s) : s ∈ (0, t), y ∈ (0, ∞), (x − y − ξ 1 (t − s))(x − y − ξ 1 (t − s)) H ∗ < 0, |ξ 1 − ξ∗1 | ≥ 1},

˜ 2 = {(ξ∗ , y, s) : s ∈ (0, t), y ∈ (0, ∞), (x − y − ξ 1 (t − s))(x − y − ξ 1 (t − s)) H ∗ < 0, |ξ 1 − ξ∗1 | < 1}, and denote ˜I1 (x, t, ξ ) = 1 ˜I2 (x, t, ξ ) = 1

     

˜1 H ˜1 H

K(ξ, ξ∗ ) −ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 ˜ Lw(y, s, ξ∗ )dξ∗ dyds, e |ξ 1 − ξ∗1 | K(ξ, ξ∗ ) −ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 ˜ e Lw(y, s, ξ∗ )dξ∗ dyds. |ξ 1 − ξ∗1 |

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

331

By a straightforward calculation,  t ∞    ˜1 e−ν0 (t−s+|x−y|) Lδ w(y, s) L 2 dyds I1 (x, t) 2 ≤ C ξ Lξ 0 0 ⎛ (y−λ j s)2  t ∞ 3 − C(s+1) e ⎜ dyds e−ν0 (t−s+|x−y|) e−Cγ y ≤ C δ ⎝ √ s+1 0 0 j=1  t +

λ3 s

e−ν0 (t−s+|x−y|) e−Cγ y

λ1 s

0

2 k=1

⎛ ⎞ (x−λ j t)2 3 − C(1+t) e ⎜ ⎟ ≤ C δ ⎝ + e−(|x|+t)/C ⎠ √ t + 1 j=1

√

1 dyds (|y −λk s|+1)(|y −λk+1 s|+1)

⎧ 2 ⎪ ⎨ +C δ

1 for x ∈ (λ1 t, λ3 t), √ (|x − λ t| + 1)(|x − λl+1 t| + 1) l ⎪ l=1 ⎩ 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞).

(5.35)

˜ 2 , an analysis like (3.39) and (3.40) yields that In the domain H       K(ξ, ξ∗ ) −ν0 (t−s−|x−y|) ˜2  e dξ∗ dyds I1 (x, t) 2 ≤ C 3 3 Lξ ˜ H2 |ξ 1 − ξ∗1 | 2 R 1    2 K(ξ, ξ∗ ) −ν0 (t−s−|x−y|) 2 · e L δ w(y, s) dξ∗ dyds dξ . 1 ˜ 2 |ξ 1 − ξ 1 | 2 H ∗ (5.36) From

     ≤

K(ξ, ξ∗ )

˜2 H

|ξ 1

3

− ξ∗1 | 2

e−ν0 (t−s−|x−y|) dξ∗ dyds e−ν0 (t−s−|x−y|) 1

 ≤

{(y,s)|s∈(0,t),y∈(0,∞),|x−y−ξ 1 (t−s)|≤1 } |x − y − ξ 1 (t − s)| 2 t

e−ν0 (t−s) ds ≤ C

(5.37)

0 ∗) and the L 1 property of √K(ξ,ξ , we get 1 1

   ˜2 I1 (x, t)

L 2ξ

≤C

|ξ −ξ∗ |

 t 0

∞

0

e−ν0 (t−s+|x−y|) Lδ w(y, s) L 2 dyds

⎛  t ⎜ ≤ C δ ⎝ 0

 t + 0

λ3 s

λ1 s

dyds

ξ

(y−λ j s)2

∞

e

−ν0 (t−s+|x−y|) −Cγ y

e

0

e−ν0 (t−s+|x−y|) e−Cγ y

3 e− C(s+1) dyds √ s + 1 j=1

332

S. Deng, W. Wang, S.-H. Yu

×

2 k=1

1 dyds (|y − λk s| + 1)(|y − λk+1 s| + 1) ⎞ 2

⎛ (x−λ j t) 3 − ⎜ e C(1+t) ⎟ ≤ C δ ⎝ + e−(|x|+t)/C ⎠ √ t +1 j=1

⎧ 2 ⎪ 1 ⎨ √ for x ∈ (λ1 t, λ3 t), +C δ (|x − λl t| + 1)(|x − λl+1 t| + 1) l=1 ⎪ ⎩ 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞). (5.38) Thus    ˜ I1 (x, t)

L 2ξ

    ≤ ˜I11 (x, t)

L 2ξ

    + ˜I21 (x, t)

⎞ (x−λ j t)2 3 − C(1+t) e ⎟ ⎜ ≤ C δ ⎝ + e−(|x|+t)/C ⎠ √ t + 1 j=1 ⎛

L 2ξ

⎧ 2 ⎪ 1 ⎨ √ for x ∈ (λ1 t, λ3 t), +C δ (|x − λ t| + 1)(|x − λl+1 t| + 1) l l=1 ⎪ ⎩ 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞). ˜I2 (x, t, ξ ) only contains an integrable singularity in ξ . Therefore by a straight calculation, we get   ˜  I2 (x, t)

L 2ξ

1 |ξ −ξ∗ |

(5.39)

and gains extra decaying rate

⎞ ⎛ (x−λ j t)2 3 − C(1+t) e ⎟ ⎜ ≤ C δ ⎝ + e−(|x|+t)/C ⎠ √ t +1 j=1 ⎧ 2 ⎪ 1 ⎨ √ for x ∈ (λ1 t, λ3 t), +C δ (|x − λl t| + 1)(|x − λl+1 t| + 1) l=1 ⎪ ⎩ 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞). (5.40)

Since   ˜  I(x, t)

L 2ξ

    ≤ ˜I0 (x, t)

L 2ξ

we get the following conclusion.

    + ˜I1 (x, t)

L 2ξ

    + ˜I2 (x, t)

L 2ξ

  ˜  + R(x, t)

L 2ξ

,

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

333

Lemma 5.2.

 t  ∞     ˜I(x, t) L 2 =  G (x − y, t − s)L w(y, s)dyds i δ   2 ξ 0 0 Lξ ⎞ ⎛ (x−λ j t)2 3 − ⎟ ⎜ e C(1+t) ≤ C δ ⎝ + e−(|x|+t)/C ⎠ √ t +1 j=1

+C δ

⎧ 2 ⎪ ⎨ √

1 for x ∈ (λ1 t, λ3 t), (|x − λl t| + 1)(|x − λl+1 t| + 1)

⎪ ⎩ l=1 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞).

(5.41) where C is independent of δ and . Thus when δ is sufficiently small, (5.23) holds. t ∞ Nonlinear term Substitute (5.21) into 0 0 Gi (x − y, t − s) (w(y, s)dyds.  t  ∞      G (x − y, t − s) (w)(y, s)dyds i   2 0 0 Lξ ⎛   3 2 t ∞ (x−y−λi (t−s))2 j s) 1 1 − (y−λ e− C(t−s+1) e C(s+1) dyds ≤ C 2 ⎝ t −s+1 s+1 0 0 i, j=1

⎞ (x−y−λ j (t−s))2 1 1 − C(t−s+1) + e dyds⎠ (t −s + 1) (|y −λk s|+1)(|y −λk+1 s|+1) s 0 λ k j=1 k=1 ⎞ ⎛ 3 3 3 2 ≤ C 2 ⎝ J1 (x, t; λi , λ j ) + J2 (x, t; λ j , λk , λk+1 )⎠ . (5.42) 3 2  t

i=1 j=1

λk+1 s

j=1 k=1

Estimates

⎞ ⎛ (x−λ j t)2 3 − C(1+t) e ⎟ ⎜ |J1 |≤ C ⎝ + e−(|x|+t)/C ⎠ √ t + 1 j=1 ⎧ 2 ⎪ ⎪ ⎪ ⎪ ⎨

+C

√

⎪ l=1 ⎪ ⎪ ⎪ ⎩0 for

+C

√

⎪ l=1 ⎪ ⎪ ⎪ ⎩0 for

Lemma

4.3

1 for x ∈ (λ1 t, λ3 t), (|x − λl t| + 1)(|x − λl+1 t| + 1)

⎛ x ∈ (0, λ1 t)2 ∪ (λ3 t, ∞). ⎞ (x−λ j t) 3 − ⎟ ⎜ e C(1+t) |J2 |≤ C ⎝ + e−(|x|+t)/C ⎠ √ t + 1 j=1 ⎧ 2 ⎪ ⎪ ⎪ ⎪ ⎨

Reference

1 for x ∈ (λ1 t, λ3 t), (|x − λl t| + 1)(|x − λl+1 t| + 1) x ∈ (0, λ1 t) ∪ (λ3 t, ∞).

(5.43) Lemma

4.3

334

S. Deng, W. Wang, S.-H. Yu

Table (5.43) and (5.42) yield that  t  ∞      G (x − y, t − s) (w)(y, s)dyds i   0

0

⎛

⎞

(x−λ j t)2

L 2ξ

3 e− C(1+t) ⎟ ≤ C ⎝ + e−(|x|+t)/C ⎠ √ t + 1 j=1 2⎜

⎧ 2 ⎪ 1 ⎨ √ for x ∈ (λ1 t, λ3 t), 2 +C (|x − λl t| + 1)(|x − λl+1 t| + 1) l=1 ⎪ ⎩ 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞).

(5.44)

Thus we justify (5.24) and so the ansatz assumption (5.19) holds. To verify (5.20), we only have to consider w(x, t, ξ ) = e−ν(ξ )t w0 (x − ξ 1 t, ξ )  t e−ν(ξ )(t−s) (Kw + L δ w + (w)) (x − ξ 1 (t − s), s, ξ )ds. +

(5.45)

0

From Lemma 2.3 and the ansatz (5.19),  t  t    −ν(ξ )(t−s)  1 e−ν(ξ )(t−s) Kw(x −ξ 1 (t − s), s)ds ≤ C − ξ e (t − s), s) w(x  0

≤C

3  j=1 0

+C

0

t

(x−ξ 1 e− √ s+1

2 

1 (t−s)−λ s)2 j C(s+1)

L 2ξ

ds

e−ν(ξ )(t−s) ds



1 k=1 {s∈(0,t)}∩{s x−ξ (t−s)∈(λk s,λk+1 s)} e−ν(ξ )(t−s)

ds, × (x − ξ 1 (t − s) − λk s + 1)(λk+1 s − x + ξ 1 (t − s) + 1)

=

3

A1 (x, t; λ j ) +

j=1

2 k=1

A2 (x, t; λk , λk+1 ) ≤

C ϒ(x, t). ν(ξ )

(5.46)

Consider sup x,t

|w(x, t)| ϒ(x, t)

 t e−ν(ξ )t w0 (x −ξ 1 t, ξ ) 1 −ν(ξ )(t−s) 1 e Kw(x − ξ (t − s), s)ds ≤ sup +sup ϒ(x, t) x,t x,t ϒ(x, t) 0  t 1 −ν(ξ )(t−s) 1 + sup e (L δ w + (w)) (x − ξ (t − s), s)ds ϒ(x, t) x,t 0

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

≤

|(δ + )w(x, t)| C C + sup + (1 + |ξ |)3 1 + |ξ | x,t ϒ(x, t)



t

335

e−ν(ξ )(t−s) (1 + |ξ |)

0

ϒ(x − ξ 1 (t − s), s) × ds. ϒ(x, t)

(5.47)

From 

t

e−ν(ξ )(t−s) ϒ(x − ξ 1 (t − s), s)ds =

0

3 

t

j=1 0

+

√

1 s+1

e−

(x−ξ 1 (t−s)−λ j s)2 C(s+1)

e−ν(ξ )(t−s) ds

2  1 k=1 {s |s∈(0,t),x−ξ (t−s)∈(λk s,λk+1 s) } e−ν(ξ )(t−s)

ds, × (x − ξ 1 (t − s) − λk s + 1)(λk+1 s − x + ξ 1 (t − s) + 1) =

3

A1 (x, t; λ j ) +

j=1

2

A2 (x, t; λk , λk+1 ) ≤

k=1

C ϒ(x, t), ν(ξ )

(5.48)

one has 

ϒ(x − ξ 1 (t − s), s) ds e−ν(ξ )(t−s) (1 + |ξ |) ϒ(x, t) 0  t (1 + |ξ |) = e−ν(ξ )(t−s) ϒ(x − ξ 1 (t − s), s)ds ≤ C. ϒ(x, t) 0 t

(5.49)

Thus (5.47) and (5.49) yield that sup x,t

2C |w(x, t)| 2C ≤ , + ϒ(x, t) (1 + |ξ |)3 1 + |ξ |

(5.50)

when , δ 1. Repeat the process for another two times. Since the operator K gains the decaying rate in the ξ variable, we have sup x,t

C |w(x, t)| ≤ . ϒ(x, t) (1 + |ξ |)3

(5.51)

Thus the ansatz assumption (5.20) is justified. 6. Stability of Knudsen’s Layers for −1 < M < 1(M
= 0) When −1 < M < 1(M
= 0), we can no longer get full boundary data by a priori energy estimate. If we use the Green’s function Gi for the initial value problem, the incomplete boundary data can’t be complemented. We have to use the Green’s function Gb for the initial-boundary value problem to get the representation of w(x, t, ξ ):

336

S. Deng, W. Wang, S.-H. Yu



∞

w(x, t) =

Gb (x, y, t)w0 (y)dy +

0

 t 0

0

∞

Gb (x, y, t −s)(L δ w+ (w))(y, s)dyds. (6.1)

In the following part of this paper, we will assume −1 < M < 0 for definiteness. That is λ1 < λ2 < 0 < λ3 . The case of 0 < M < 1 is almost the same. From Theorem 2.4, one has ⎞ ⎛ (x−λ j t)2  ∞  3 − C(t+1)   e ⎟ ⎜  (6.2) Gb (x, y, t)w0 (y)dy  + e−(|x|+t)/C ⎠ , √   2 ≤ A0 ⎝ t + 1 0 Lξ j=1 for some constant A0 > 0. Make ansatz assumptions from the anastz of the perturbation around a global Maxwellian in the half-space problem in [26]: w(x, t) L 2 ξ ⎞ ⎛ (x−λ j t)2 3 − ⎟ ⎜ e C(1+t) ≤ C0 ⎝ + e−(|x|+t)/C ⎠ √ t + 1 j=1 +C0

⎧ 2 ⎪ ⎨ √

1 for x ∈ (0, λ3 t), (|x − λl t| + 1)(|x − λl+1 t| + 1)

(6.3)

1 for x ∈ (0, λ3 t), (|x − λl t| + 1)(|x − λl+1 t| + 1)

(6.4)

⎪ ⎩ l=1 0 for x ∈ (λ3 t, ∞), w(x, t) L ∞ ξ,3 ⎞ ⎛ (x−λ j t)2 3 − ⎟ ⎜ e C(1+t) ≤ C1 ⎝ + e−(|x|+t)/C ⎠ √ t + 1 j=1 +C1

⎧ 2 ⎪ ⎨ √

⎪ ⎩ l=1 0 for x ∈ (λ3 t, ∞),

for some positive constant C, C0 > 2 A0 and C1 > 0. Then Lemma 2.4 yields that (w) and L δ w can be controlled by  (w) L 2 ξ

+C 2

⎛ ⎞ (x−λ j t)2 3 − C(1+t) e ⎜ ⎟ + e−(|x|+t)/C ⎠ ≤ C 2 ⎝ t +1

⎧ 2 ⎪ ⎨

j=1

1 for x ∈ (0, λ3 t), (|x − λl t| + 1)(|x − λl+1 t| + 1)

⎪ ⎩ l=1 0 for x ∈ (λ3 t, ∞),

(6.5)

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

337

⎛

L δ w L 2 ξ

⎞ (x−λ j t)2 3 − C(1+t) e ⎜ ⎟ ≤ C δe−γ |x|/C ⎝ + e−(|x|+t)/C ⎠ √ t +1 j=1

⎧ 2 ⎪ 1 ⎨ √ for x ∈ (0, λ3 t), −γ |x|/C (6.6) +C δe (|x − λl t| + 1)(|x − λl+1 t| + 1) l=1 ⎪ ⎩ 0 for x ∈ (λ3 t, ∞).

1 of Gi P1 plays an important role in closing the nonlinThe algebraic decaying rate t+1 earity in the initial value problem. However, for Gb P1 one can not get such a global decaying rate. As a result, one needs rather detailed  t  ∞ spatial information on Gb P1 in order to close the nonlinearity. We shall consider 0 0 Gb (x, y, t−s) (L δ w + (w)) (y, s)dyds in the following different regions. Case 1. When 0 < x < λ3 t/2, we use the forward Green’s function. From (2.32) 1 1 in Theorem 2.4, the algebraic decaying rate of Gb P1 is t+1 + √(t+1)(y+1) . This suggests

that when y ≤ Ct, the dominant decaying rate is 1 t+1

get:

√

1 (t+1)(y+1)

while

1 t+1

+

√

1 (t+1)(y+1)

∼

when y ≥ Ct. Thus substitute (2.32) and (6.6) into the linear coupling wave to

 t     0

  Gb (x, y, t − s)L δ w(y, s)dyds  

∞ 0

⎛

 t

3

≤ C δ ⎝

i, j=1 0

+

0

3 2  t j=1 k=1 0

⎛

λ3 s 0

2

k=1 0

⎛ = C δ ⎝

0

0

3

⎜ +C δ ⎝ 2

J1 (x, t; λi , λ j ) +

k=1 0

3 2

λ1 (s−t) 0

L 2ξ

dyds

2 λ x− λ3 y−λ3 (t−s) 2 C(t−s+1)

e−Cγ y dyds √ (|y − λk s| + 1)(|y − λk+1 s| + 1) ⎞

e−Cγ y ⎟ dyds ⎠ √ (|y − λk s| + 1)(|y − λk+1 s| + 1) ⎞

J2 (x, t; λ j , λk , λk+1 )⎠

1 e− √ (t − s + 1)(y + 1) 

λ2 (s−t) 0

2 λ x− λ3 y−λ3 (t−s) 1 C(t−s+1)



 t

2  t

(y−λ j s)2 − C(s+1)

j=1 k=1

k=1 0

+

1 e− √ (t −s +1)(y +1)

1 e− √ (t − s + 1)(y + 1)

i, j=1

⎛

0

  Gb (x, y, t − s)P1 L δ w(y, s)dyds  

⎞ (x−y−λ j (t−s))2 1 e−Cγ y − C(t−s+1) e dyds ⎠ √ (t − s + 1) (|y − λk s| + 1)(|y − λk+1 s| + 1)

λ1 (s−t)

λ2 (s−t)

∞

(x−y−λi (t−s))2 1 e−Cγ y e− C(t−s+1) e √ (t − s + 1) s + 1



2  t k=1 0

0



 t

⎜ + C δ ⎝

+

∞

L 2ξ

 t   = 

√

1 e− (t −s +1)(y +1)

2 λ x− λ3 y−λ3 (t−s) 1 C(t−s+1)

2 λ x− λ3 y−λ3 (t−s) 2 C(t−s+1)

√

e−Cγ y dyds (|y −λk s|+1)(|y −λk+1 s|+1) ⎞ e−Cγ y

⎟ dyds ⎠ . √ (|y −λk s|+1)(|y −λk+1 s|+1)

(6.7)

338

S. Deng, W. Wang, S.-H. Yu Estimates |J1 |≤

⎛

⎜ C⎝

e

(x−λ j t)2 − C(1+t)

j=1

⎧ 2 ⎪ ⎪ ⎪ ⎪ ⎨ +C

3

√

⎪ l=1 ⎪ ⎪ ⎪ ⎩0 for

√ t +1

Reference

⎞ ⎟ + e−(|x|+t)/C ⎠

Lemma

4.4

1 for x ∈ (0, λ3 t), (|x − λl t| + 1)(|x − λl+1 t| + 1)

x ∈ (λ3 t, ∞). ⎞ (x−λ j t)2 3 e− C(1+t) ⎜ ⎟ + e−(|x|+t)/C ⎠ |J2 |≤ C ⎝ √ t + 1 j=1

(6.8)

⎛

⎧ 2 ⎪ ⎪ ⎪ ⎪ ⎨

+C

√

⎪ l=1 ⎪ ⎪ ⎪ ⎩0 for

Lemma

4.6

1 for x ∈ (0, λ3 t), (|x − λl t| + 1)(|x − λl+1 t| + 1) x ∈ (λ3 t, ∞).

From Table (6.8), we can conclude that ⎛ C δ ⎝

3

J1 (x, t; λi , λ j ) +

i, j=1

3 2

⎞ J2 (x, t; λ j , λk , λk+1 )⎠

j=1 k=1

⎛ ⎞ (x−λ j t)2 3 − C(1+t) e ⎜ ⎟ ≤ C δ ⎝ + e−(|x|+t)/C ⎠ √ t + 1 j=1 ⎧ 2 ⎪ 1 ⎨ √ for x ∈ (0, λ3 t). + C δ (|x − λ t| + 1)(|x − λl+1 t| + 1) l l=1 ⎪ ⎩ 0 for x ∈ (λ3 t, ∞).

(6.9)

Consider 2  t  λ1 (s−t)

1 − e √ (t − s + 1)(y + 1)

λ (x− λ3 y−λ3 (t−s))2 1 C(t−s+1)



e−Cγ y

(|y −λk s|+1)(|y −λk+1 s|+1)  t−x/λ3 1 1 −Cγ y e ds = O(1) √ √ λ1 y + 1 (|y − λ s| + 1)(|y − λ s| + 1) 0 1 2 y= λ (x−λ3 (t−s)) 3  t−x/λ3 1 1 −Cγ y e ds +O(1) √ √ λ1 y + 1 (|y − λ s| + 1)(|y − λ s| + 1) 0 2 3 y= λ (x−λ3 (t−s)) 3 ⎛ λ3  2 −λ1 1 1 ⎜ λ3 −λ1 (t−x/λ3 ) e−Cγ |x−λ3 t|  ds ≤C⎝ λ1 λ3 t − x 0 | λ (x − λ3 (t − s)) − λ3 s| 3 ⎞  t−λ3 x λ 1 1 −Cγ λ1 (x−λ3 (t−s)) ⎠ 3  + λ3 −λ e ds 1 2 λ t − x λ1 (x − λ (t − s)) + 1 λ −λ (t−x/λ3 ) 3 k=1 0

0

3

1

λ3

dyds

3

1 1 1 1 ≤C√ ≤C√ √ . γ λ3 t − x γ (|x − λ2 t| + 1)(|x − λ3 t| + 1)

(6.10)

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

339

Similarly, 2  t k=1 0

λ2 (s−t)

0

λ3

2

(x− λ y−λ3 (t−s)) 2 1 e−Cγ y C(t−s+1) e− dyds √ √ (t −s +1)(y +1) (|y −λk s|+1)(|y −λk+1 s|+1)

1 1 ≤ C√ √ . γ (|x − λ2 t| + 1)(|x − λ3 t| + 1)

(6.11)

Therefore (6.7), (6.9), (6.10) and (6.11) yield that the linear coupling wave satisfies  t     0

∞ 0

  Gb (x, y, t − s)L δ w(y, s)dyds  

L 2ξ

1 ≤ C δ √ (|x − λ2 t| + 1)(|x − λ3 t| + 1) (6.12)

when 0 < x < λ3 t/2. Then consider the nonlinear term:  t  ∞     Gb (x, y, t − s) (w)(y, s)dyds    2 0 0 Lξ  t  ∞     = Gb (x, y, t − s)P1 (w)(y, s)dyds   0

0

⎛

3  t

≤ C 2 ⎝

+

i, j=1 0

0

2  t 3

λ3 s

j=1 k=1 0

⎛

0

2  t ⎜ + C 2 ⎝ k=1 0

×

+

∞

2

(y−λ j s) (x−y−λi (t−s))2 1 e− C(t−s+1) e− C(s+1) dyds (t − s + 1)(s + 1)

1 e (t − s + 1)

(x−y−λ j (t−s))2 − C(t−s+1)



λ1 (s−t) 0

1 e− √ (t − s + 1)(y + 1)



λ2 (s−t) 0

1 e− √ (t − s + 1)(y + 1)

λ x− λ3 y−λ3 (t−s) 1 C(t−s+1)

λ x− λ3 y−λ3 (t−s) 2 C(t−s+1)

2

 1 dyds × (|y − λk s| + 1)(|y − λk+1 s| + 1) ⎞ ⎛ 3 3 2 = C 2 ⎝ J1 (x, t; λi , λ j ) + J2 (x, t; λ j , λk , λk+1 )⎠ i, j=1

j=1 k=1

⎞

1 dyds⎠ (|y −λk s| + 1)(|y −λk+1 s|+1)

1 dyds (|y − λk s| + 1)(|y − λk+1 s| + 1)

2  t k=1 0

L 2ξ

2

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S. Deng, W. Wang, S.-H. Yu

⎛ 2  t ⎜ + C 2 ⎝ k=1 0

0

1 e− √ (t − s + 1)(y + 1)

2 λ x− λ3 y−λ3 (t−s) 1 C(t−s+1)

1 dyds (|y − λk s| + 1)(|y − λk+1 s| + 1)

×

+



λ1 (s−t)



2  t k=1 0

λ2 (s−t)

0

1 e− √ (t − s + 1)(y + 1)

2 λ x− λ3 y−λ3 (t−s) 2 C(t−s+1)

 1 dyds . × (|y − λk s| + 1)(|y − λk+1 s| + 1)

Estimates

⎛

⎧ 2 ⎪ ⎪ ⎪ ⎪ ⎨ +C

Reference

⎞

(x−λ j t)2 − C(1+t)

⎟ ⎜ e C⎝ + e−(|x|+t)/C ⎠ √ t +1 j=1 3

|J1 |≤

Lemma

4.3

1 for x ∈ (0, λ3 t), √ (|x − λl t| + 1)(|x − λl+1 t| + 1)

⎪ l=1 ⎪ ⎪ ⎪ ⎩0 for

⎞ ⎛ x ∈ (λ3 t, ∞). (x−λ j t)2 3 − C(1+t) e ⎟ ⎜ |J2 |≤ C ⎝ + e−(|x|+t)/C ⎠ √ t +1 j=1 ⎧ 2 ⎪ ⎪ ⎪ ⎪ ⎨

+C

(6.13)

(6.14) Lemma

4.3

1 for x ∈ (0, λ3 t), √ (|x − λl t| + 1)(|x − λl+1 t| + 1)

⎪ l=1 ⎪ ⎪ ⎪ ⎩0 for

x ∈ (λ3 t, ∞).

From Table (6.14), we can conclude that ⎞ ⎛ 3 3 2 C 2 ⎝ J1 (x, t; λi , λ j ) + J2 (x, t; λ j , λk , λk+1 )⎠ i, j=1

j=1 k=1

⎛ ⎞ (x−λ j t)2 3 − C(1+t) e ⎜ ⎟ ≤ C 2 ⎝ + e−(|x|+t)/C ⎠ √ t +1 j=1 ⎧ 2 ⎪ 1 ⎨ √ for x ∈ (0, λ3 t), 2 + C (|x − λl t| + 1)(|x − λl+1 t| + 1) l=1 ⎪ ⎩ 0 for x ∈ (λ3 t, ∞).

(6.15)

Consider 2  t k=1 0

0

λ1 (s−t)

λ3

2

(x− λ y−λ3 (t−s)) 1 1 1 C(t−s+1) dyds e− √ (|y −λk s|+1)(|y −λk+1 s|+1) (t −s +1)(y +1)

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

341

1 1 = O(1) ds √ y + 1 (|y − λ1 s| + 1)(|y − λ2 s| + 1) y= λ1 (x−λ3 (t−s)) 0 λ3  t−x/λ3 1 1 +O(1) ds √ y + 1 (|y − λ2 s| + 1)(|y − λ3 s| + 1) y= λ1 (x−λ3 (t−s)) 0 λ3 ⎛ λ  23 −λ1 (t−x/λ3 ) 1 1 ⎜ λ3 −λ1 ds ≤C⎝ 3 λ1 0 (λ3 t − x) 2 | λ3 (x − λ3 (t − s)) − λ3 s| ⎞  t−λ3 x 1 1  + λ3 −λ ds ⎠ 1 2 (λ3 t − x)2 λ1 (x − λ (t − s)) + 1 λ −λ (t−x/λ3 ) 

3

≤C

t−x/λ3

λ3

1

log(λ3 t − x) (λ3 t − x)

3 2

≤ C√

3

1 , (|x − λ2 t| + 1)(|x − λ3 t| + 1)

(6.16)

and 2  t k=1 0

0

λ2 (s−t)

λ3

(x− λ y−λ3 (t−s)) 2 1 C(t−s+1) e− √ (t − s + 1)(y + 1)

2

1 dyds (|y − λk s| + 1)(|y − λk+1 s| + 1) 1 ≤ C√ . (|x − λ2 t| + 1)(|x − λ3 t| + 1) ×

(6.17)

Equations (6.13), (6.15), (6.16) and (6.17) result in  t     0

0

∞

  Gb (x, y, t − s) (w)(y, s)dyds  

L 2ξ

≤ C 2 √

1 (|x − λ2 t| + 1)(|x − λ3 t| + 1) (6.18)

when 0 < x < λ3 t/2. Thus, (6.12) and (6.18) together with (6.2) verify the ansatz assumption (6.3) for x ∈ (0, λ3 t/2). Case 2. When λ3 t/2 < x < λ3 t, we will fix (x, t) and use the backward Green’s function. By a symmetry consideration as stated in [26], one has ⎧ (x−y−λ j (t−s+1))2 ⎪ 3 − ⎪ C(t−s+1) ⎪ e ⎪ ⎪ + e−(|x−y|+t−s)/C ⎪ ⎪ ⎪ (t − s + 1) ⎪ ⎪ j=1 ⎪ ⎪ ⎨ for s ∈ (t − x/λ3 , t), λ x Gb (x, y, t, s)P1  L 2 = O(1) (6.19) (y−λ j (s−t)+ λi ))2 ξ ⎪ 3 ⎪ 2 − ⎪ C(t−s+1) ⎪ e ⎪ ⎪ + e−(|x−y|+t−s)/C √ ⎪ ⎪ (t − s + 1)x ⎪ ⎪ j=1 ⎪ ⎪ ⎩ for s ∈ (0, t − x/λ ). 3

342

S. Deng, W. Wang, S.-H. Yu

t ∞ We substitute (6.5), (6.6) and (6.19) into the representation 0 0 Gb (x, y, t − s) t ∞ L δ w(y, s)dyds and 0 0 Gb (x, y, t − s) (w)(y, s)dyds respectively:  t   

  t ∞       Gb (x, y, t − s)L δ w(y, s) =  Gb (x, y, t −s)P1 L δ w(y, s)dyds   2 0 0 0 0 L 2ξ Lξ ⎛ 3  t ∞ (y−λ j s)2 (x−y−λi (t−s))2 1 ≤ C δ ⎝ e−Cγ y e− C(t−s+1) e− C(s+1) dyds √ (t − s + 1) s + 1 i, j=1 0 0 +

∞

3 2  t j=1 k=1 0

λ3 s 0

2

(x−y−λ j (t−s)) 1 e− C(t−s+1) (t − s + 1) ⎞

e−Cγ y dyds ⎠ (|y − λk s| + 1)(|y − λk+1 s| + 1) ⎛ λ 2  t  λ1 (s−t) (x− λ3 y−λ3 (t−s))2 1 1 − C(t−s+1) ⎝ + C δ e √ (t − s + 1)(x + 1) k=1 0 0

× √

e−Cγ y dyds (|y − λk s| + 1)(|y − λk+1 s| + 1) λ 2  t  λ2 (s−t) (x− λ3 y−λ3 (t−s))2 2 1 − C(t−s+1) e + √ (t − s + 1)(x + 1) k=1 0 0  −Cγ y e × √ dyds (|y − λk s| + 1)(|y − λk+1 s| + 1) ⎞ ⎛ 3 3 2 J1 (x, t; λi , λ j ) + J2 (x, t; λ j , λk , λk+1 )⎠ = C δ ⎝ × √

i, j=1

⎛ + C δ ⎝

j=1 k=1

2  t k=1 0



λ1 (s−t)

0

λ3

(x− λ y−λ3 (t−s)) 1 1 C(t−s+1) e− √ (t − s + 1)(x + 1)

e−Cγ y dyds (|y − λk s| + 1)(|y − λk+1 s| + 1) λ 2  t  λ2 (s−t) (x− λ3 y−λ3 (t−s))2 2 1 − C(t−s+1) e + √ (t − s + 1)(x + 1) k=1 0 0  −Cγ y e × √ dyds . (|y − λk s| + 1)(|y − λk+1 s| + 1)

2

× √

The following estimates: 2  t k=1 0

0



λ1 (s−t)

1 e− √ (t − s + 1)(x + 1)

λ x− λ3 y−λ3 (t−s) 1 C(t−s+1)

2

(6.20)

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

343

e−Cγ y dyds (|y − λk s| + 1)(|y − λk+1 s| + 1)  t−x/λ3 1 1 −Cγ y = O(1) √ e ds √ (|y − λ1 s| + 1)(|y − λ2 s| + 1) y= λ1 (x−λ3 (t−s)) x +1 0 λ3  t−x/λ3 1 1 +O(1) √ e−Cγ y √ ds (|y − λ2 s| + 1)(|y − λ3 s| + 1) y= λ1 (x−λ3 (t−s)) x +1 0 λ3 ⎛ λ  23 −λ1 1 ⎜ λ3 −λ1 (t−x/λ3 ) −Cγ |x−λ3 t| 1  ≤ C√ e ds ⎝ √ x +1 0 λ3 t − x | λλ13 (x − λ3 (t − s)) − λ3 s| ⎞  t−λ3 x λ 1 1 −Cγ λ (x−λ3 (t−s)) ⎟ 3 e + λ3 −λ ds ⎠ 1 2 λ t − x 3 λ −λ (t−x/λ3 ) ×√

3

1

1 1 1 1 ≤C √ √ √ √ γ λ3 t − x x + 1 γ λ3 t − x t + 1 1 1 ≤C √ , γ (|x − λ2 t| + 1)(|x − λ3 t| + 1) λ 2  t  λ2 (s−t) (x− λ3 y−λ3 (t−s))2 2 1 − C(t−s+1) e √ (t − s + 1)(y + 1) k=1 0 0 ≤C

(6.21)

e−Cγ y ×√ dyds (|y − λk s| + 1)(|y − λk+1 s| + 1) 1 1 , ≤C √ γ (|x − λ2 t| + 1)(|x − λ3 t| + 1) together with Table (6.8) result in   t  ∞    Gb (x, y, t − s)L δ w(y, s)dyds    0

0

(6.22)

L 2ξ

⎛ ⎞ (x−λ j t)2 3 − C(1+t) e ⎜ ⎟ ≤ C δe−γ |x|/C ⎝ + e−(|x|+t)/C ⎠ √ t +1 j=1 ⎧ 2 ⎪ 1 ⎨ √ for x ∈ (0, λ3 t), −γ |x|/C (6.23) +C δe (|x − λl t| + 1)(|x − λl+1 t| + 1) l=1 ⎪ ⎩ 0 for x ∈ (λ3 t, ∞). The estimate for the nonlinear term is as follows:  t  ∞     Gb (x, y, t − s) (w)(y, s)dyds    2 0 0 Lξ  t  ∞     = Gb (x, y, t − s)P1 (w)(y, s)dyds   0

0

L 2ξ

344

S. Deng, W. Wang, S.-H. Yu

⎛ ≤ C

2⎝

3  t

i, j=1 0 3 2  t

∞

2

(y−λ j s) (x−y−λi (t−s))2 1 e− C(t−s+1) e− C(s+1) dyds (t − s + 1)(s + 1)

0 λ3 s

2

(x−y−λ j (t−s)) 1 e− C(t−s+1) (t − s + 1) j=1 k=1 0 0 ⎞ 1 dyds ⎠ × (|y − λk s| + 1)(|y − λk + 1s| + 1) ⎛ λ 2  t  λ1 (s−t) (x− λ3 y−λ3 (t−s))2 1 1 − 2⎝ C(t−s+1) + C e √ (t − s + 1)(x + 1) k=1 0 0

+

1 dyds (|y − λk s| + 1)(|y − λk+1 s| + 1) λ 2  t  λ2 (s−t) (x− λ3 y−λ3 (t−s))2 2 1 C(t−s+1) + e− √ (t − s + 1)(x + 1) k=1 0 0 ×

1 dyds (|y − λk s| + 1)(|y − λk+1 s| + 1) ⎛ ⎞ 3 3 2 = C 2 ⎝ J1 (x, t; λi , λ j ) + J2 (x, t; λ j , λk , λk+1 )⎠ ×

i, j=1

⎛

+ C

2⎝

j=1 k=1

2  t k=1 0

λ1 (s−t) 0

λ3

(x− λ y−λ3 (t−s)) 1 1 C(t−s+1) e− √ (t − s + 1)(x + 1)

1 dyds (|y − λk s| + 1)(|y − λk+1 s| + 1) λ 2  t  λ2 (s−t) (x− λ3 y−λ3 (t−s))2 2 1 − C(t−s+1) + e √ (t − s + 1)(x + 1) k=1 0 0 ⎞ 1 dyds ⎠ . × (|y − λk s| + 1)(|y − λk+1 s| + 1)

2

×

(6.24)

From Table (6.14) and 2  t



λ1 (s−t)

1 e− √ (t −s +1)(x +1)  t−x/λ3

λ x− λ3 y−λ3 (t−s) 1 C(t−s+1)

2

1 dyds (|y −λk s|+1)(|y −λk+1 s|+1) k=1 0 0 1 1 = O(1) √ ds (|y − λ1 s| + 1)(|y − λ2 s| + 1) y= λ1 (x−λ3 (t−s)) x +1 0 λ3  t−x/λ3 1 1 + O(1) √ ds (|y − λ2 s| + 1)(|y − λ3 s| + 1) y= λ1 (x−λ3 (t−s)) x +1 0 λ3

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

⎛ ≤ C√  +

1 x +1

⎜ ⎝



λ3 2 −λ1 λ3 −λ1 (t−x/λ3 )

0

t−λ3 x λ3 2 −λ1 λ3 −λ1 (t−x/λ3 )

≤C

345

1 1 ds (λ3 t − x) | λλ1 (x − λ3 (t − s)) − λ3 s| 3

1 ds (λ3 t − x)2

log(λ3 t − x) log(λ3 t − x) 1 ≤C ≤ C√ , √ √ (|x − λ2 t| + 1)(|x − λ3 t| + 1) (λ3 t − x) x + 1 (λ3 t − x) t + 1 (6.25) 

2  t k=1 0

λ2 (s−t) 0

√

1 e− (t − s + 1)(y + 1)

λ x− λ3 y−λ3 (t−s) 2 C(t−s+1)

1 dyds (|y − λk s| + 1)(|y − λk+1 s| + 1) 1 ≤ C√ , (|x − λ2 t| + 1)(|x − λ3 t| + 1)

2

×

(6.26)

we find that   t  ∞     G (x, y, t − s) (w)(y, s)dyds b  2  0 0 Lξ ⎛ ⎞ (x−λ j t)2 3 − C(1+t) e ⎜ ⎟ ≤ C 2 e−γ |x|/C ⎝ + e−(|x|+t)/C ⎠ √ t + 1 j=1 ⎧ 2 ⎪ 1 ⎨ √ for x ∈ (0, λ3 t), 2 −γ |x|/C + C e (|x − λl t| + 1)(|x − λl+1 t| + 1) l=1 ⎪ ⎩ 0 for x ∈ (λ3 t, ∞). (6.27) Thus (6.2), (6.23) and (6.27) verify the ansatz assumption (6.3) in the region x ∈ (λ3 t/2, λ3 t). The justification of the ansatz (6.4) is same to that in Sect. 5. Thus we complete the proof of Theorem 1.1. Acknowledgements. The research of the first and the second authors are supported by National Science Foundation of China 10531020. The research of the third author is supported by the National University of Singapore start-up grant R-146-000-108-133.

References 1. Bardos, C., Caflisch, R.E., Nicolaenko, B.: The Milne and Kramers problems for the Boltzmann equation of a hard sphere gas. Comm. Pure Appl. Math. 49, 323–352 (1986) 2. Boltzmann, L. (translated by Stephen G. Brush).: Lectures on Gas Theory. New York: Dover Publications, Inc. 1964

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3. Carleman, T.: Sur La Théorie de l’Équation Intégrodifférentielle de Boltzmann. Acta Mathematica 60, 91–142 (1933) 4. Carleman, T.: Problémes mathématiques dans la théorie cinétique des gaz. (French) Publ. Sci. Inst. Mittag-Leffler. 2, Uppsala, Almqvist & Wiksells Boktryckeri Ab, 1957 5. Cercignani, C.: Half-space problem in the kinetic theory of gases. In: Kroner, E., Kirchgassner, K. eds., Trends in Applications of Pure Mathematics to Mechanics, Berlin: Springer-Verlag, 1986, p. 3550 6. Cercignani, C., Illner, R., Pulvirenti, M.: The mathematical theory of dilute gases. Applied Mathematical Sciences. 106. New York: Springer-Verlag, 1994 7. Coron, F., Golse, F., Sulem, C.: A classification of well-posed kinetic layer problems. Commun. Pure Appl. Math. 41, 409–435 (1988) 8. DiPerna, R.J., Lions, P.-L.: On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. of Math. 130(2), 321–366 (1989) 9. Ellis, R., Pinsky, M.: The first and second fluid approximations to the linearized Boltzmann equation. J. Math. Pures Appl. 54(9), 125–156 (1975) 10. Golse, F., Poupaud, F.: Stationary solutions of the linearized Boltzmann equation in a half-space. Math. Meth. Appl. Sci. 11, 483–502 (1989) 11. Grad, H.: Asymptotic equivalence of the Navier-Stokes and nonlinear Boltzmann equations. Proc. Symp. Appl. Math. 17(ed. R.Finn), Providence, RI: Amer. Math. Soc., 1965, pp. 154–183 12. Grad, H.: Asymptotic theory of the Boltzmann equation. Phys. Fluids 6, 147–181 (1963a) 13. Grad, H.: Asymptotic theory of the Boltzmann equation. In: Rarefied Gas Dynamics, Laurmann, J. A. ed. 1, New York: Academic Press, 1963, pp. 26–59 14. Grad, H.: Singular and nonuniform limits of solutions of the Boltzmann equation. Transport Theory (Proc. Sympos. Appl. Math., New York, 1967), SIAM-AMS Proc., Vol. I Providence, RI: Amer. Math. Soc. 1969, pp. 269–308 15. Grad, H.: High Frequency sound according to the Boltzmann equation. J. SIAM Appl. Math. 14(4), 935–955 (1966) 16. Hilbert, D.: Grundzüge einer Allgemeinen Theorie der Linearen Integralgleichungen. Leipzig, Teubner Chap. 22, 1912 17. Kawashima, S., Matsumura, A., Nishida, T.: On the fluid-dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation. Commun. Math. Phys. 70(2), 97–124 (1979) 18. Kaniel, S., Shinbrot, M.: The Boltzmann equation. I. Uniqueness and local existence. Commun. Math. Phys. 58(1), 65–84 (1978) 19. Kato, T.: Perturbation theory for linear operators. Die Grundlehren der mathematischen Wissenschaften, Band 132, New York: Springer-Verlag, Inc., 1966 20. Kawashima, S., Matsumura, A., Nishida, T.: On the fluid-dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation. Commun. Math. Phys. 70(2), 97–124 (1979) 21. Lee, M.-Y., Liu, T.-P., Yu, S.-H.: Large-Time Behavior of Solutions for the Boltzmann Equation with Hard potentials. Commun. Math. Phys. 246(1), 133–179 (2006) 22. Liu, T.-P.: Pointwise Convergence to Shock Waves for Viscous Conservation Laws. Commun. Pure and Appl. Math. 50(11), 1113–1182 (1997) 23. Liu, T.-P., Yu, S.-H.: Boltzmann Equation: Micro-Macro Decompositions and Positivity of Shock Profiles. Commun. Math. Phys. 246(1), 133–179 (2004) 24. Liu, T.-P., Yu, S.-H.: Green’s function and large-Time Behavior of Solutions for One-Dimensional Boltzmann Equation. Commun. Pure Appl. Math. 57, 1543–1608 (2004) 25. Liu, T.-P., Yu, S.-H.: Green’s Function of Boltzmann Equation, 3-D waves. Bulletin, Inst. Math. Academia Sinica (N.S.) 1(1), 1–78 (2006) 26. Liu, T.-P., Yu, S.-H.: Initial-Boundary Value Problem for One-Dimensional Wave Solutions of the Boltzmann Equation. Comm. Pure Appl. Math. 60(3), 295–356 (2007) 27. Liu, T.-P., Zeng, Y.: Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws. Mem. Amer. Math. Soc. 125, 599 (1997) 28. Nishida, T.: Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation. Commun. Math. Phys. 61(2), 119–148 (1978) 29. Sone, Y.: Kinetic Theory and Fluid Dynamics. Basel-Boston: Birkhauser, 2002 30. Sone, Y., Ohwada, T., Aoki, K.: Evaporation and condensation on the plane condense phase: Numerical analysis of the linearized Boltzmann equation for hard-sphere molecules. Phys. Fluids A 1(8), 1398–1405 (1989) 31. Ukai, S.: On the existence of global solutions of mixed problem for non-linear Boltzmann equation. Proc. Japan Acad. 50, 179–184 (1974) 32. Ukai, S.: Les solutions globales de l’équation de Boltzmann dans l’espace tout entier et dans le demiespace. C.R.Acad. Sci. Paris Ser. A-B 282(6), A317–A320 (1976) 33. Ukai, S., Yang, T., Yu, S.-H.: Nonlinear Boundary Layers of the Boltzmann Equation: I. Existence. Commun. Math. Phys. 236, 373–393 (2003)

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

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34. Ukai, S., Yang, T., Yu, S.-H.: Nonlinear stability of boundary layers of the Boltzmann equation. I. The case M ∞ < −1. Commun. Math. Phys. 244, 99–109 (2004) 35. Zeng, Y.: L 1 asymptotic behavior of compressible, isentropic, viscous 1-D flow. Commun. Pure Appl. Math. 47(8), 1053–1082 (1994) Communicated by H.-T. Yau

Commun. Math. Phys. 281, 349–367 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0489-2

Communications in

Mathematical Physics

The BCS Functional for General Pair Interactions Christian Hainzl1 , Eman Hamza2 , Robert Seiringer3 , Jan Philip Solovej4 1 Departments of Mathematics and Physics, UAB, 1300 University Blvd,

Birmingham, AL 35294, USA. E-mail: [email protected]

2 Department of Mathematics, UAB, 1300 University Blvd, Birmingham, AL 35294, USA.

E-mail: [email protected]

3 Department of Physics, Princeton University, Princeton, NJ 08542-0708, USA.

E-mail: [email protected]

4 Department of Mathematics, University of Copenhagen, Universitetsparken 5,

2100 Copenhagen, Denmark. E-mail: [email protected] Received: 5 April 2007 / Accepted: 21 November 2007 Published online: 1 May 2008 – © The Authors 2008

Abstract: The Bardeen-Cooper-Schrieffer (BCS) functional has recently received renewed attention as a description of fermionic gases interacting with local pairwise interactions. We present here a rigorous analysis of the BCS functional for general pair interaction potentials. For both zero and positive temperature, we show that the existence of a non-trivial solution of the nonlinear BCS gap equation is equivalent to the existence of a negative eigenvalue of a certain linear operator. From this we conclude the existence of a critical temperature below which the BCS pairing wave function does not vanish identically. For attractive potentials, we prove that the critical temperature is non-zero and exponentially small in the strength of the potential.

1. Introduction Over the last few years, significant attention has been devoted to ultra-cold gases of fermionic atoms, both from the experimental and the theoretical point of view. Because of the ability to magnetically tune Feshbach resonances it has become possible to vary interatomic interactions via external magnetic fields, and hence to control the strength of the interactions among the atoms. This method was used in remarkable experiments to study the crossover from the regime of Bose-Einstein condensation (BEC) of tightly bound diatomic molecules to the Bardeen-Cooper-Schrieffer (BCS) regime of weakly bound Cooper pairs in ultra-cold gases of fermionic atoms. For a recent review on this topic we refer to [5,7]. The main purpose of this paper is to give a mathematically precise study of the BCS regime. This regime is usually described by the BCS functional, derived by Leggett [10] based on the original work of Bardeen-Cooper-Schrieffer [3]. We note that while in the original BCS model simple phonon-induced non-local interaction potentials were considered, the interaction potentials are local for atomic Fermi gases. In order to allow © 2008 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.

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for a wide range of applications, we will consider here the BCS functional for general pair interaction potentials V . One purpose of our paper is to obtain necessary and sufficient conditions on V for the system to display superfluid behavior. Recall that in the BCS model superfluidity (or, rather, superconductivity in the original BCS case, where electrons in a crystal structure have been considered) is related to the non-vanishing of the pairing wave function, describing particle pairs with opposite spin and zero total momentum. That is, superfluidity occurs if the paired state is energetically favored over the normal state described by the Fermi-Dirac distribution. More precisely, for arbitrary temperature T = 1/β ≥ 0 a system is in a superfluid state if there exists a non-trivial solution  to the BCS gap equation
E(q) 1 (q) tanh dq, (1.1) ( p) = − Vˆ ( p − q) 3/2 (2π ) E(q) 2T R3  with E( p) = ( p 2 − µ)2 + |( p)|2 . Here, µ denotes the chemical potential. Notice that the gap equation is highly nonlinear. In Theorem 1 below we will show that the existence of a non-trivial solution to the gap equation is equivalent to the existence of a negative eigenvalue of the corresponding linear operator K β,µ + V,

K β,µ = ( p 2 − µ)

eβ( p

2 −µ)

2 eβ( p −µ)

+1 −1

.

(1.2)

The operator K β,µ is understood as a multiplication operator in momentum space, i.e., p 2 equals − in configuration space. Observe that in the limit T → 0 this operator reduces to the Schrödinger type operator | p 2 − µ| + V.

(1.3)

This linear characterization of the existence of solutions to the nonlinear BCS equation represents a considerable simplification of the analysis. In particular, it proves the existence of solutions to (1.1) for a wide range of interaction potentials V , and hence generalizes previous results [4,13,18,19] valid only for non-local V ’s under suitable assumptions. Moreover, the monotonicity of K β,µ in β guarantees the existence of a critical temperature Tc , with 0 ≤ Tc < ∞, such that superfluidity occurs whenever T < Tc and normal behavior for T ≥ Tc . In particular, there is a phase transition at Tc . For positive values of µ, we shall show that Tc is strictly positive whenever the negative part of V , denoted by V− , is non-vanishing and the positive part V+ is small enough in a suitable sense. Moreover, we prove that Tc is exponentially small in the strength of the interaction potential. More precisely, if the potential is given by λV for some fixed V ∈ L 1 ∩ L 3/2 , then Tc
e−1/λ as λ → 0. This generalizes the corresponding result in [3, Eq. (3.29)] to a very large class of interaction potentials. For negative potentials, also a corresponding lower bound will be shown to hold. In a forthcoming work [9], the precise asymptotics of Tc for λ → 0 will be investigated. At zero temperature, the existence of a non-trivial solution of the BCS gap equation is also related to the non-vanishing of the energy gap between the ground state and the first excited state in the effective quadratic BCS Hamiltonian on Fock space. For very specific attractive interaction potentials, it has been argued that this gap is non-vanishing, see e.g. [10,14,15]. We note, however, that the existence of such an energy gap is not

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necessarily implied by the non-vanishing of the solution to the BCS equation. In fact, the presence of a gap is not necessary for our analysis. We consider general potentials where such an energy gap need not be present, a priori. We do prove that the gap is non-vanishing for a certain class of interaction potentials, however. We note that our results are relevant for actual experiments on cold atomic gases as long as the description via the BCS functional is applicable. There is an extensive literature on this subject, see e.g. [1,5–7,14–17]. According to Leggett [10] this is the case for weak interactions as long as the range of the potential V is much smaller than µ−1/2 (in appropriate units), i.e., for weakly coupled dilute gases. In particular, “weakly coupled” means that the corresponding Schrödinger operator p 2 + V typically does not allow for a negative energy bound state since otherwise the system is in the BEC regime of tightly bound bosonic molecules. We refer to [10] for a detailed discussion on this question. 2. Model and Main Results The precise definition of the BCS functional considered in this paper is given as follows. For the convenience of the reader, we describe in the Appendix the motivation and physical background; the discussion there follows Leggett’s derivation in [10]. See also [2] for a detailed study of the BCS approximation to the Hubbard model. We use the  standard convention α( ˆ p) = (2π )−3/2 R3 α(x)e−i px d x for the Fourier transform. Definition 1. Let D denote the set of pairs of functions (γ , α), with γ ∈ L 1 (R3 , (1 + p 2 )dp), 0 ≤ γ ( p) ≤ 1, and α ∈ H 1 (R3 , d x), satisfying |α( ˆ p)|2 ≤ γ ( p)(1 − γ ( p)). 3/2 3 Let V ∈ L (R , d x) be real-valued and µ ∈ R. For T = 1/β ≥ 0 and (γ , α) ∈ D, the energy functional Fβ is defined as

1 (2.1) Fβ (γ , α) = ( p 2 − µ)γ ( p)dp + |α(x)|2 V (x)d x − S(γ , α), β where


S(γ , α) = −

     s( p) ln s( p) + 1 − s( p) ln 1 − s( p) dp,

with s( p) determined by s(1 − s) = γ (1 − γ ) − |α| ˆ 2. As explained in the Appendix, (2.1) has the physical interpretation of − 21 (2π )3/2 times the pressure of a system of spin 1/2 fermions at temperature T interacting via a pair potential given by 2V . It is natural to require V ∈ L 3/2 (R3 ), as we do here, since this guarantees that V is relatively form-bounded with respect to p 2 (see, e.g., Sect. 11 in [11]). In fact, it would be enough to require the negative part of V to be in L 3/2 + L ∞  , and allow for more general positive parts of V . For simplicity of the presentation, we will not pursue this generalization here. In the absence of a potential V , it is easy to see that Fβ is minimized on D by the 2 choice α ≡ 0 and γ ( p) = γ0 ( p) = [eβ( p −µ) + 1]−1 . We refer to this state as the normal state of the system. We will be concerned with the question of whether a non-vanishing α can lower the energy for given potential V .

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Note that at T = 0 it is natural to formulate the model in terms of a functional that depends only on α and not on γ . Namely, for fixed |α( ˆ p)|2 , the optimal choice of γ ( p) is   1 (1 + 1 − 4|α( ˆ p)|2 ) for p 2 < µ γ ( p) = 21 (2.2) ˆ p)|2 ) for p 2 > µ 2 (1 − 1 − 4|α( in this case. Similarly, also at T > 0 it is enough to minimize Fβ only over pairs (γ , α) satisfying a certain relation independent of V , which is displayed in Eq. (3.4) below. According to the usual interpretation of the BCS theory, the normal state is unstable if the energy can be lowered by the formation of Cooper pairs, i.e., if Fβ can be lowered by choosing α = 0. The following theorem, which is the main result of this paper, shows that this property is equivalent to the existence of a negative eigenvalue of a certain linear operator. Theorem 1. Let V ∈ L 3/2 (R3 ), µ ∈ R, and ∞ > T = statements are equivalent:

1 β

≥ 0. Then the following

(i) The normal state (γ0 , 0) is unstable under pair formation, i.e., inf

(γ ,α)∈D

Fβ (γ , α) < Fβ (γ0 , 0).

(ii) There exists a pair (γ , α) ∈ D, with α = 0, such that ( p) = −

p2 − µ γ ( p) −

1 2

α( ˆ p)

(2.3)

satisfies the BCS gap equation

  βE  = −Vˆ ∗ tanh , with E( p) = ( p 2 − µ)2 + |( p)|2 . (2.4) E 2 (iii) The linear operator K β,µ + V, K β,µ = ( p − µ) 2

eβ( p

2 −µ)

2 eβ( p −µ)

+1 −1

,

(2.5)

has at least one negative eigenvalue. Here, ∗ denotes convolution in the BCS gap equation (2.4). In order to avoid having to write factors of 2π , we find it convenient to define convolution with a factor (2π )−3/2  in front, i.e., ( f ∗ g)( p) := (2π )−3/2 R3 f ( p − q)g(q)dq. Remark 1. The operator (2.5) is obtained by taking the second derivative of the functional Fβ with respect to α at the normal state (γ0 , 0) (see Eq. (3.10) below). Since the normal state is a critical point of Fβ , it is then easy to prove that (iii) implies (i). The opposite direction is not immediate, however. It says that if (γ0 , 0) is not a minimizer, it can also not be locally stable. Remark 2. As mentioned above, a minimizer of Fβ satisfies the V -independent relation (2.2) at T = 0, and (3.4) at T > 0. It is not necessary to assume that these equations hold in part (ii) of Theorem 1, however.

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Note that in the zero temperature case T = 0 the gap equation (2.4) takes the form   = −Vˆ ∗ , E (compare with [3,8,10,12]) and the operator in (2.5) is given by | p 2 − µ| + V . For any potential V we shall prove the existence of a critical temperature 0 ≤ Tc = 1 β c < ∞ at which a phase transition from a normal to superfluid state takes place. More precisely, we shall show that the gap equation (2.4) has a non-trivial solution for all T < Tc , whereas it has no non-trivial solution if T ≥ Tc . This follows from the next theorem, in combination with Theorem 1. Theorem 2. For any V ∈ L 3/2 (R3 ) there exists a critical temperature 0 ≤ Tc = β1 < ∞ c such that inf spec (K β,µ + V ) < 0 K β,µ + V ≥ 0

if T < Tc if T ≥ Tc .

With the aid of the Birman-Schwinger principle, it is possible to characterize the value of the critical temperature Tc , at least if Tc > 0. In order to do this, it is convenient to decompose V into its positive and negative parts, i.e., V = V+ − V− , with V+ ≥ 0, V− ≥ 0 and V− V+ = 0. We shall show in Proposition 4 that Tc > 0 satisfies the equation 1/2 1 1/2 (2.6) V V 1= − . − K βc ,µ + V+ Note that since K β,µ ≥ 2/β, this leads immediately to the general rough upper bound Tc ≤ 21 V− ∞ .

(2.7)

For negative potentials V and a positive chemical potential µ, we shall show that the critical temperature is always positive, i.e., that | p 2 − µ| + V has a negative energy bound state. We can even say more, namely we can guarantee superfluidity for small temperature whenever the positive part V+ is not too large in a suitable sense. Theorem 3. Let V ∈ L 3/2 (R3 ) be not identically zero, and let µ > 0. (i) If V ≤ 0, then the corresponding critical temperature is positive, Tc = β1c > 0. (ii) Let V = V+ − V− , and let β > βc (−V− ), where βc (−V− ) is the critical inverse temperature for the potential −V− . Then the system at temperature 1/β is still in 1/2 −1 1/2 a superfluid state if V+ K β,µ V+ is small enough. In particular, this theorem states that for any negative potential one can add a sufficiently small positive part such that the critical temperature Tc is still strictly positive. The next question concerns the magnitude of Tc . The rough upper bound given in (2.7) turns out to be much too rough, in general. In fact, the following theorem shows that Tc is always exponentially small in the strength of V . To be able to state the bound precisely, let a := inf t>0 t (et + 1)/((t + 2)(et − 1)). Numerically, a ≈ 0.654. Let further


∞ 1 1 1 2 . (2.8) f (t) := dp p − 2π 2 0 | p 2 − 1| + t p2 + 1 + t Note that f is a positive and monotone decreasing function, with f (t) ∼ ln(1/t) as t → 0. We obtain the following upper bound on the critical temperature Tc .

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Theorem 4. Let f −1 be the inverse of the function f in (2.8). Let µ > 0, and let  4/3 V = V+ −V− , with V ∈ L 3/2 , V− ∈ L 1 ∩L 3/2 , and with 13 π2

V− 3/2 < a ≈ 0.654. Then the critical temperature satisfies the upper bound

  4/3

V− 3/2 µ −1 a − 13 π2 Tc ≤ f . (2.9) 2 µ1/2 V− 1 Note that, in particular, f −1 (t) ∼ e−t for large t. Hence the critical temperature is exponentially small in the potential for V− ∈ L 1 ∩ L 3/2 . For negative potentials, one can show that the bound given in Theorem 4 is indeed optimal. More precisely, if the potential is given by λV for some fixed negative V ∈ L 3/2 , we shall show in Proposition 5 the lower bound Tc ≥ gµ,V (λ), where gµ,V is some positive function depending only on µ and V satisfying gµ,V (t) ∼ e−1/t for small t. Combining both bounds then implies that for negative potentials V the critical temperature is exponentially small in the coupling parameter. This generalizes well-known calculations in the physics literature, see e.g. [3, Eq. (3.29)], to a very large class of interaction potentials. Note that, in particular, this result implies that it is not possible to calculate the critical temperature via a perturbative expansion in V . Finally, we comment on the continuity properties of minimizers of Fβ . In fact, for positive temperature T > 0 the minimizing γ and αˆ can be shown to be continuous functions. (We show this in Prop. 3 below.) This may not be surprising, since already 2 the minimizer without potential, γ0 ( p) = 1/(eβ( p −µ) + 1), is continuous. But in the case of T = 0 the situation is not so clear, since the normal state is a step function with jump at p 2 = µ for µ > 0. The continuity of a minimizer γ at T = 0 is closely related to the existence of an energy gap, given by  := inf E( p), p

(2.10)

with E defined in (2.4). As explained in the Appendix, E( p) has the interpretation of the dispersion relation for quasi-particle excitations. The following shows that in case V has a strictly negative Fourier transform, γ is continuous even at T = 0 and the gap  is non-vanishing. Proposition 1. Assume that inf spec (| p 2 − µ| + V ) < 0, and that Vˆ is strictly negative. Let (γ , α) ∈ D be a minimizer of F∞ . Then  > 0 and γ is a continuous function. This result implies, in particular, that  > 0 for negative V with strictly negative Fourier transform. We note that this property need not necessarily be true for all potentials V , however. We conclude this section with two remarks. Remark 3. Our results can be generalized to non-local potentials given by an integral kernel V (x, y). Exactly such a non-local potential was used in the original paper of ˆ p) = 1 if BCS [3] who considered V of the form V (x, y) = −V0 φ(x)φ(y), with φ( 2 | p − µ| ≤ h and 0 otherwise. This models an effective potential arising from electronphonon interactions in crystals. In this case, the minimizer of Fβ can be evaluated explicitly (see, e.g., [12]).

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In particular, our analysis can be used to generalize previous results [4,13,18,19] on the existence of solutions of the gap equation, as already mentioned in the Introduction. None of this previous work allows for an interaction kernel Vˆ (k, k ) of the form Vˆ (k−k ), however, since such a kernel does not satisfy appropriate L p conditions. Such conditions were necessary for utilization of fixed point arguments in the previous investigations of this problem. Remark 4. In [6,15] the BCS-BEC crossover is studied numerically for certain potentials V that are negative and have negative Fourier transform. These potentials satisfy all the assumptions of our results stated above. In the following Sects. 3 and 4, the proofs of the claims of this section will be given. 3. Properties of the Minimizer of Fβ In this section we shall give the proof of Theorem 1 and Proposition 1 stated above. We start by showing the existence of a minimizer of Fβ . Proposition 2. There exists a minimizer of Fβ in D. Proof. We first show that Fβ dominates both the L 1 (R3 , (1 + p 2 )dp) norm of γ and the H 1 (R3 , d x) norm of α. Hence any minimizing sequence will be bounded in these norms. We have

3 2 Fβ (γ , α) ≥ C1 + ( p − µ)γ ( p)dp + |α(x)|2 V (x)d x, 4 where C1 =





β 1 1 1 2 ( p 2 − µ)γ ( p)dp − S(γ , α) = − ln(1 + e− 4 ( p −µ) )dp. β β (γ ,α)∈D 4 inf

Since V ∈ L 3/2 by assumption, it is relatively bounded with respect to − (in the sense of quadratic forms), and hence C2 = inf spec ( p 2 /4+V ) is finite. Using |α( ˆ p)|2 ≤ γ ( p), we thus have that


1 2 2 p γ ( p)dp + V (x)|α(x)| d x ≥ C2 γ ( p)dp. 4 Using again that |α( ˆ p)|2 ≤ γ ( p) ≤ 1, it follows that 1 1 Fβ (γ , α) ≥ −A + α 2H 1 (R3 ,d x) + γ L 1 (R3 ,(1+ p2 )d p) , 8 8 where A = −C1 −




p 2 /4 − 3µ/4 − 1/4 + C2

(3.1)

 −

dp,

with [ · ]− = min{ ·, 0} denoting the negative part. To show that a minimizer of Fβ exists in D, we pick a minimizing sequence (γn , αn ) ∈ D, with Fβ (γn , αn ) ≤ 0. From (3.1) we conclude that αn 2H 1 ≤ 8A,

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and hence we can find a subsequence that converges weakly to some α˜ ∈ H 1 . Since V ∈ L 3/2 (R3 ), this implies that

2 lim V (x)|αn (x)|2 d x = V (x)|α(x)| ˜ dx n→∞ R3

R3

[11, Thm. 11.4]. It remains to show that the remaining part of the functional, 
    1 1 0 2 Fβ (γ , α) = ( p − µ)γ ( p)dp+ s( p) ln s( p)+ 1 − s( p) ln 1 − s( p) dp, β β (3.2) is weakly lower semicontinuous. Note that Fβ0 is jointly convex in (γ , α), and that its domain D is a convex set. We already know that αn α˜ weakly in H 1 (R3 ). Moreover, since γn is uniformly bounded in L 1 (R3 ) ∩ L ∞ (R3 ), we can find a subsequence such that γn γ˜ weakly in L p (R3 ) for some 1 < p < ∞. We can then apply Mazur’s Theorem [11, Theorem 2.13] to construct a new sequence as convex combinations of the old one, which now converges strongly to (γ˜ , α) ˜ in L p (R3 ) × L 2 (R3 ). By going to a subsequence, we can also assume that γn → γ and αˆ n → αˆ˜ pointwise [11, Theorem 2.7]. Because of convexity of Fβ0 , this new sequence is again a minimizing sequence. Note that the integrand in (3.2) is bounded from below independently of γ and α 2 by −β −1 ln(1 + e−β( p −µ) ). Since this function is integrable, we can apply Fatou’s Lemma [11, Lemma 1.7], together with the pointwise convergence, to conclude that lim inf Fβ0 (γn , αn ) ≥ Fβ0 (γ˜ , α). ˜ We have thus shown that ˜ ≤ lim inf Fβ (γn , αn ). Fβ (γ˜ , α) n→∞

It is easy to see that (γ˜ , α) ˜ ∈ D, hence it is a minimizer. This proves the claim.

 

Next, we establish a few auxiliary results that will be needed below in the proof of Theorem 1. Lemma 1. Let (γ , α) ∈ D be a minimizer of Fβ for 0 < β < ∞. Then, for a.e. p ∈ R3 , α( ˆ p) , 2γ ( p) − 1 

2γ ( p) − 1 ( p 2 − µ) = − f 2 (γ ( p) − 21 )2 + |α( ˆ p)|2 , β

(Vˆ ∗ α)( ˆ p) = ( p 2 − µ)

where f (a) :=

1 a

ln

1+a 1−a

(3.3) (3.4)

for 0 ≤ a < 1.

ˆ p)|2 + (γ ( p) − 1/2)2 ≤ 1/4 − }. Let Proof. For  > 0, let A := { p ∈ R3 : |α( 1 3 1 3 g ∈ H (R ) ∩ L (R ), with Fourier transform supported in A . Then (γ , α + tg) ∈ D for small enough |t|. By the Lebesgue dominated convergence theorem, it follows that 
 d Fβ (γ , α + tg) = 2 Re V (x)g(x)α(x) d x dt t=0


2 ˆ p)α( + Re (3.5) g( ˆ p) f 2 (γ ( p) − 21 )2 + |α( ˆ p)|2 dp. β

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ˆ α) ∈ D for small |t|, and we get Similarly, for gˆ real, (γ + t g, 
 d ˆ α) ˆ p)dp Fβ (γ + t g, = ( p 2 − µ)g( (3.6) dt t=0 


1 ˆ p)(2γ ( p) − 1) f 2 (γ ( p) − 21 )2 + |α( ˆ p)|2 dp. + g( β Since (γ , α) minimize Fβ by assumption, the expressions in (3.5) and (3.6) vanish. It follows that for a.e. p ∈ A we have    1 ˆ p) f 2 (γ ( p) − 21 )2 + |α( ˆ p)|2 , (Vˆ ∗ α)( ˆ p) = − α( β    1 (3.7) ( p 2 − µ) = − (2γ ( p) − 1) f 2 (γ ( p) − 21 )2 + |α( ˆ p)|2 . β To conclude that (3.7) holds a.e. in R3 , it remains to show that B := R3 \ ∪>0 A has zero By definition, if p ∈ B we have γ ( p)(1 − γ ( p)) = |α( ˆ p)|2 , i.e.  measure.  s( p) s( p) − 1 = 0. It is then easy to see that if B has non-zero measure, Fβ (γ , α) can be lowered by modifying γ and αˆ on B (since t → t ln t has a diverging derivative at zero). This contradicts the fact that (γ , α) is a minimizer, and hence B has zero measure.   Remark 5. A similar strategy as in the proof of Lemma 1 leads to the conclusion that Eq. (3.3) is also valid at T = 0. The relation between γ and α, given in Eq. (3.4) for T > 0, has to be replaced by (2.2) at T = 0, however. We omit the details. Note that Eq. (3.4) and the positivity of f immediately imply that γ ( p) ≥ 1/2 for p 2 < µ, and γ ( p) ≤ 1/2 for p 2 > µ. Of greater importance below will be the following monotonicity property, which bounds γ in the opposite direction. Lemma 2. Assume that (γ , α) ∈ D satisfies Eq. (3.4), and let γ0 ( p) = [eβ( p −µ) + 1]−1 . For a.e. p ∈ R3 , we have γ ( p) ≤ γ0 ( p) if p 2 < µ, while for p 2 > µ we have γ ( p) ≥ γ0 ( p). Moreover, these inequalities are strict if α( ˆ p) = 0. In particular, p2 − µ p2 − µ < on the support of α. ˆ 1 − 2γ0 ( p) 1 − 2γ ( p) 2

Proof. Let p 2 > µ such that α( ˆ p) = 0 and assume by way of contradiction that γ ( p) ≤ γ0 ( p). We then have  1 − γ0 ( p) < |α( ˆ p)|2 + ( 21 − γ ( p))2 . 2 Since f is a strictly monotone increasing function, we obtain

   1 2 2 f 1 − 2γ0 ( p) < f 2 |α( ˆ p)| + ( 2 − γ ( p)) and hence   (1 − 2γ0 ( p)) f 1 − 2γ0 ( p) < (1 − 2γ ( p)) f

 1 2 2 2 |α( ˆ p)| + ( 2 − γ ( p)) .

(3.8)

This, however, contradicts the fact that both (γ0 , 0) and (γ , α) are solutions of (3.4), and hence both sides of (3.8) are equal to β( p 2 − µ). The case p 2 < µ is treated similarly.  

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We now have all the necessary prerequisites at our disposal to give the proof of Theorem 1. Proof of Theorem 1. [i] ⇒ [ii]: Consider first the case T > 0. According to Lemma 1 a minimizing pair (γ , α) ∈ D with α = 0 satisfies Eqs.  (3.3) and (3.4). Using the definitions of  and E in (2.3) and (2.4) we see that (γ ( p) − 1/2)2 + |α( ˆ p)|2 = 1/2−γ ( p) E( p). Plugging this into Eq. (3.4) leads to p 2 −µ 1 p2 − µ =− f 2γ ( p) − 1 β



1 − 2γ ( p) E( p) p2 − µ





2 1 − 2γ ( p) p2 − µ tanh−1 E( p) , β E( p) 2γ ( p) − 1 p2 − µ

= or

tanh

β E( p) 1 − 2γ ( p) = E( p). 2 p2 − µ

(3.9)

( p) β E( p) Therefore, we can rewrite the relation between αˆ and  as α( ˆ p) = 2E( p) tanh 2 . Together with Eq. (3.3) this gives the required form of the gap equation. The case T = 0 is treated similarly (cf. Remark 5). [ii] ⇒ [iii]: Given T ≥ 0 and a  satisfying (2.3) for some (γ , α) ∈ D, we first show that there exists a (γ˜ , α) ˜ ∈ D yielding the same  but satisfying in addition Eq. (2.2) at ˆ˜ p) = m( p)α( T = 0 or Eq. (3.4) at T > 0, respectively. Namely, we can let α( ˆ p) and γ˜ ( p) = 1/2 + m( p)(γ ( p) − 1/2) without changing . If we choose

m( p) = sgn

p2 − µ 1 − 2γ ( p)



tanh(β E( p)/2)  , 2 (γ ( p) − 21 )2 + |α( ˆ p)|2

with E( p) as in (2.4), it is easy to see that (3.9) is satisfied. Using in addition the gap equation, the same reasoning as in the previous part of the proof leads to the conclusion that Eqs. (3.3) and (2.2) (respectively (3.4)) are satisfied for this (γ˜ , α). ˜ Hence there exists a (γ , α) ∈ D satisfying (3.3) as well as (2.2) (at T = 0) or (3.4) (at T > 0). Using (3.3) and the definition of K β,µ , we have that  α, (K β,µ + V )α = α,

 1 1 − ( p 2 − µ)α . 1 − 2γ0 ( p) 1 − 2γ ( p)

It follows from Lemma 2 that this expression is negative whenever α is not identically zero. [iii] ⇒ [i]: Consider first the case T > 0. Showing that the existence of a negative eigenvalue of the operator (2.5) implies that the minimizing α is not identically zero is equivalent to proving that α ≡ 0 implies that inf spec (K β,µ + V ) ≥ 0. First, we note that, for any δ ∈ C0∞ , (tδ)2 + (γ0 − 21 )2 ∞ < 41 for all small |t|. With the aid of the 2 ˆ exists for Lebesgue dominated convergence theorem it is easy to see that d 2 Fβ (γ0 , t δ) dt

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small t, and is given by

(γ0 ( p) − 21 )2 d2 2 2 2 ˆ ˆ F (γ , t δ) = 2 V (x)| δ(x)| d x + |δ( p)| β 0 dt 2 β |tδ( p)|2 + (γ0 ( p) − 21 )2

 × f 2 |tδ( p)|2 + (γ0 ( p) − 21 )2 dp +

1 β




t 2 |δ( p)|4 [|tδ( p)|2 + (γ0 ( p) −

1 2 2 2 ) ][γ0 ( p)(1 − γ0 ( p)) − |tδ( p)| ]

dp.

Assuming that the minimizing α is identically zero, it follows that the corresponding d2 ˆ γ = γ0 and hence dt 2 Fβ (γ0 , t δ)|t=0 ≥ 0. But 

 d2 2  ˆ ˆ F (γ , t δ) = 2 V (x)| δ(x)| d x + 2 |δ( p)|2 K β,µ ( p) dp, β 0  dt 2 t=0

(3.10)

and hence δ, (K β,µ + V )δ ≥ 0 for all δ ∈ C0∞ . This proves the statement. In the case T = 0, it is sufficient to minimize Fβ over (γ , α) ∈ D with γ of the form (2.2), as remarked earlier. For simplicity, we abuse the notation slightly and denote the resulting functional by F∞ (α). Subtracting its value for α = 0, it is given by

 1 F∞ (α) − F∞ (0) = ˆ p)|2 )dp + V (x)|α(x)|2 d x. | p 2 − µ|(1 − 1 − 4|α( 2 (3.11) Assume that | p 2 − µ| + V has a negative eigenvalue. We can then find an α ∈ C0∞ (R3 ) such that α, (| p 2 − µ| + V )α < 0. For 0 <   1 small enough, we then have F∞ (α) − F∞ (0) =  2 α, (| p 2 − µ| + V )α + O( 4 ) < 0. This implies that a minimizer α does not vanish identically.

(3.12)

 

In the following, we investigate the continuity properties of a minimizing pair (γ , α) ∈ D. Proposition 3. If (γ , α) ∈ D is a minimizer of Fβ for 0 < β < ∞, then both γ and αˆ are continuous functions. Proof. Since V ∈ L 3/2 and α ∈ H 1 , we know that Vˆ ∈ L 3 and αˆ ∈ L 6/5+ for all  > 0. Hence Vˆ ∗ αˆ is continuous [11, Thm. 2.20]. From Eq. (3.3) and the definition of  and E given in (2.3) and (2.4), we observe that Vˆ ∗ αˆ = − 21  and hence both  and E are continuous functions. As in Eq. (3.9) in the proof of Thm. 1, Eq. (3.4) can be rewritten in the form p2 − µ eβ E( p) + 1 = − β E( p) E( p). 2γ ( p) − 1 e −1

(3.13)

The right side of this equation is continuous and does not vanish, not even when E( p) = 0 (which could happen, in principle, for p 2 = µ). Hence γ is continuous. Using again Eq. (3.3), this also implies continuity of α. ˆ  

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The situation is different at zero temperature, however. In the limit β → ∞, the right side of Eq. (3.13) vanishes in case E( p) = 0. Hence γ could possibly be discontinuous if p 2 = µ. In fact, this is what happens in the case of vanishing potential. On the other hand, a non-vanishing gap  = inf p E( p) > 0 implies continuity of γ . At T = 0, the gap equation (2.4) takes the form E( p)α( ˆ p) = −(Vˆ ∗ α)( ˆ p), with E( p) = 

(3.14)

| p 2 − µ|

. Hence the gap  is guaranteed to be non-zero if the right 1 − 4|α( ˆ p)|2 side of (3.14) vanishes nowhere on the sphere p 2 = µ. This can be easily shown in case Vˆ is strictly negative, which is the content of our Proposition 1. Proof of Proposition 1. Since Vˆ ≤ 0,

α( ˆ Vˆ ∗ α) ˆ ≥ |α|( ˆ Vˆ ∗ |α|). ˆ

(3.15)

Moreover, since Vˆ is assumed to be strictly negative, there can be equality in (3.15) only if α( ˆ p) = eiκ |α( ˆ p)| for some constant κ ∈ R. This implies that this is the case for a minimizer of F∞ , and hence Vˆ ∗ αˆ vanishes nowhere. The statement then follows from Eq. (3.14).   4. The Critical Temperature In this section we investigate the relation between properties of the interaction potential V and the critical temperature Tc at which the phase transition takes place. In particular, we shall prove Theorems 2–4. Proof of Theorem 2. Let βc = inf β {β > 0 : inf spec(K β,µ + V ) < 0}. It is easy to see that βc > 0, since V is relatively form-bounded with respect to | p 2 − µ| and K β,µ ≥ c(| p 2 − µ| + 1/β) for some c > 0 (cf. Lemma 3 below). From the definition of K β,µ in (2.5) one easily sees that K β1 ,µ ≤ K β2 ,µ

if β1 ≥ β2 .

(4.1)

It follows that inf spec(K β,µ + V ) < 0 for all β > βc . Moreover, since K β,µ depends continuously on β (in a norm resolvent sense), we have that K βc ,µ + V ≥ 0. Setting βc = ∞ in the case inf spec (K β,µ + V ) ≥ 0 for all β and letting Tc = 1/βc finishes the proof.   Next, we prove the first assertion of Theorem 3, namely that for any negative potential the critical temperature is strictly positive. Proof of Theorem 3(i). In light of Theorem 2 it suffices to prove that inf spec(| p 2 − µ| + V ) < 0. According to the Birman-Schwinger principle, this is equivalent to the fact that for some e > 0, the Birman-Schwinger operator Be := |V |1/2 (| p 2 − µ| + e)−1 |V |1/2

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has an eigenvalue 1. Notice that lime→∞ Be = 0 since V ∈ L 3/2 is relatively bounded with respect to p 2 (in the sense of quadratic forms). Because of continuity in e, it thus suffices to find an e > 0 such that Be > 1. Hence we have to show that there exists a φ ∈ L 2 (R3 ) with φ 2 = 1 such that φ, Be φ ≥ 1 for an appropriate e. For this purpose, we choose a normalized function
φ in such a way that φ|V |1/2 does not vanish in a neighborhood of some point p with
p 2 = µ. We can, for instance, choose φ to be rapidly decaying, in which case |φ|V |1/2 | is continuous and hence strictly positive on some small ball. This ball can then be made to overlap with the sphere p 2 = µ by simply multiplying φ by a factor eiq x for appropriate
q, which translates the function φ|V |1/2 by q. In this way,
1
φ, Be φ = |φ|V dp → ∞ as e → 0. |1/2 ( p)|2 2 | p − µ| + e Thus there exists e > 0 such that φ, Be φ > 1.

 

Before proceeding with the proof of the second part of Theorem 3, we introduce an alternative characterization for the critical temperature Tc in the case Tc > 0, which we already referred to in Eq. (2.6). Proposition 4. For all V ∈ L 3/2 (R3 ), such that V = V+ − V− , with V+ ≥ 0, V− ≥ 0 and V− V+ = 0, the critical temperature Tc = 1/βc is characterized by 1/2 1 1/2 V V − = 1, − K βc ,µ + V+ whenever Tc = 0. Recall that K β,µ ≥ 2/β, hence K β,µ + V+ has a bounded inverse. β

Proof. Let the operator Be be defined as Beβ = V−

1/2

1 1/2 V . K β,µ + V+ + e −

This operator is compact (in fact, Hilbert-Schmidt) and depends continuously on β and e. β Moreover, Be is strictly decreasing in e and 1/β. Since Tc > 0 by assumption, we know from Theorem 2 that K β,µ + V has a negative β eigenvalue for T < Tc . Hence the Birman-Schwinger principle implies that Be > 1 β for 1/β and e small enough. On the other hand, B0 → 0 as β → 0. Thus there exists β˜ a unique β˜ such that B0 = 1. ˜ Beβ > 1 for e small enough, and hence K β,µ + V has a negative energy For β > β, ˜ however, Beβ < 1 for all e ≥ 0, and hence K β,µ + V ≥ 0. bound state. For β < β, This shows that β˜ = βc .   From Prop. 4 it is then easy to conclude that the critical temperature stays strictly positive under small positive perturbations to negative potentials.

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Proof of Theorem 3(ii). According to Prop. 4, it suffices to prove that for β > βc (−V− ) 1/2 1/2 and for λ > 0 small enough, V− (K β,µ + λV+ )−1 V− > 1. In order to simplify the 1/2 1/2 notation let Bλ = V− (K β,µ + λV+ )−1 V− . It follows that Bλ = ≥

1/2 V−

1

1



−1/2 −1/2 1 + λK β,µ V+ K β,µ   −1/2 −1/2 −1 B0 1 + λ K β,µ V+ K β,µ

. 1/2 K β,µ

1 1/2 K β,µ

1/2

V−

Since V+ ∈ L 3/2 by assumption, it is relatively form-bounded with respect to K β,µ , and −1/2 −1/2 1/2 −1 1/2 hence K β,µ V+ K β,µ = V+ K β,µ V+ is finite. Therefore Bλ > 1 if B0 > 1 and λ is small enough.   Next, we shall derive upper and lower bounds on the critical temperature. These bounds are similar to the ones found in the literature, but generalize these results from simple rank 1 potentials as in [3] to very general pair interactions. We start with the upper bound given in Theorem 4. To this aim we first prove two auxiliary lemmas. Lemma 3. Let K β,µ ( p) = ( p 2 − µ)(eβ( p −µ) + 1)(eβ( p −µ) − 1)−1 . Then there is a constant a > 0 such that



2 2 ≤ K β,µ ( p) ≤ | p 2 − µ| + . (4.2) a | p 2 − µ| + β β 2

2

Proof. Let g(t) = t (et + 1)/((t + 2)(et − 1)) for t ≥ 0. It is easy to see that g(t) ≤ 1. Moreover, g(0) = 1 and limt→∞ g(t) = 1, hence a := inf t>0 g(t) > 0. In fact, a ≈ 0.654 numerically. Since K β,µ ( p) = (| p 2 − µ| + 2/β)g(β| p 2 − µ|), this implies the statement.   Lemma 4. Assume that V ∈ L 1 ∩ L 3/2 and that µ > 0. Then 1/2 |V |



1 1 2 4/3 1/2 1/2 ≤ µ |V |

V

f (e/µ) +

V 3/2 , 1 | p 2 − µ| + e 3 π

(4.3)

where f is defined in (2.8). Proof. Define ke,µ ( p) ≥ 0 by 1 1 = 2 + ke,µ ( p). | p 2 − µ| + e p +µ+e Then 1/2 |V |

(4.4)

1/2 1 1 1/2 1/2 1/2 1/2 |V | | ≤ + |V | |V | k ( p)|V | |V . e,µ | p 2 − µ| + e p2 + µ + e (4.5)

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363

The last term is bounded by 1/2 |V | ke,µ ( p)|V |1/2 = sup ψ|V |1/2 , kˆe,µ ∗ (ψ|V |1/2 )

ψ 2 =1

≤ (2π )−3/2 kˆe,µ ∞ V 1 ≤ (2π )−3 ke,µ 1 V 1 . Moreover, ke,µ 1 = (2π )3 µ1/2 f (e/µ), with f given in (2.8). For the first term on the right side of (4.5), the use of the Hardy-Littlewood-Sobolev inequality and Hölder’s inequality yields

  1/2 1 2 4/3 1 1/2 1/2 1 1/2 |V | |V | ≤ sup ψ|V | , 2 ψ|V | ≤

V 3/2 . p2 + µ + e p 3 π

ψ 2 =1   Equipped with these two lemmas we are now able to prove the upper bound on the critical temperature stated in Theorem 4. Proof of Theorem 4. Assume that Tc > 0, for otherwise there is nothing to prove. According to Proposition 4, the critical temperature Tc = 1/βc is determined by the equation 1/2 1 1/2 V V− − K = 1. βc ,µ + V+ Using Lemmas 3 and 4 as well as V+ ≥ 0, we thus have 1 1 1/2 1/2 V 1 ≤ V− a | p 2 − µ| + 2/βc −

 1 2 4/3 µ1/2 V− 1  ≤ f 2/(βc µ) +

V− 3/2 . a 3a π Since f is monotone decreasing, this implies the statement.

 

Finally, we turn to a lower bound for the critical temperature. We assume that the interaction potential is given by λV , with V ≤ 0 not identically zero, and λ > 0 a coupling parameter. Proposition 5. Let V ∈ L 3/2 , with V ≤ 0 not identically zero, and let µ > 0. Then there exists a monotone increasing strictly positive function gµ,V , with gµ,V (t) ∼ e−1/t as t → 0, such that the critical temperature for the potential λV satisfies the lower bound Tc ≥ gµ,V (λ)

(4.6)

for λ > 0. Proof. We have already shown in Thm. 3 that Tc > 0 for positive λ. By Prop. 4, the fact that V is non-positive and (4.2) we have that 1/2 1/2 1 1 1/2 1/2 |V | ≥ |V | |V | . 1/λ = |V | 2 K βc ,µ | p − µ| + 2/βc

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For any function ψ ∈ L 2 with ψ 2 = 1 we therefore obtain
 2 1  
|1/2 ( p) dp. 1/λ ≥ ψ|V | p 2 − µ| + 2/βc The latter integral is monotone increasing in βc and diverges logarithmically as βc → ∞
for appropriate ψ such that ψ|V |1/2 ( p) does not vanish identically if p 2 = µ. (See the proof of Thm. 3(i) for an argument concerning the existence of such a ψ.) This implies the statement.   Acknowledgements. The authors would like to thank George Bruun, Thomas Hoffmann-Ostenhof and Michael Loss for valuable suggestions and remarks. C.H., E.H., and R.S. thank the Erwin Schrödinger Institute in Vienna for the kind hospitality during part of this work. R.S. acknowledges partial support by U.S. National Science Foundation grant PHY-0353181 and by an Alfred P. Sloan Fellowship.

Appendix A. Motivation and Physical Background In this Appendix, we shall briefly explain the derivation of the BCS functional defined in Definition 1. Our presentation follows the one of Leggett in [10]. Similar derivations can be found in [8,12]. The BCS functional is obtained from several approximations and assumptions on the underlying many-body problem. We assume the reader to be familiar with the usual Fock space formalism in quantum statistical mechanics. Consider a system of spin 21 fermions confined to a cubic box  ⊂ R3 , with periodic boundary conditions, interacting via a two-body potential V . Assume, for simplicity, that V ∈ L 1 (R3 ) so that it has a bounded Fourier transform, denoted by Vˆ ( p) = (2π )−3/2 V (x)e−i px d x. The many-body Hamiltonian on Fock space is then given as H=

 σ,k

† k 2 ak,σ ak,σ +

(2π )3/2 2||



† Vˆ (k)a †p−k,σ aq+k,ν aq,ν a p,σ .

(A.1)

k, p,q,σ,ν

† and ak,σ denote the usual creation and annihilation operators for a particle Here, ak,σ of momentum k and spin σ , satisfying the canonical anticommutation relations. The momentum sums run over k ∈ (2π/L)Z3 , where L is the side length of the cube , whereas the spin sums are over σ ∈ {↑, ↓}. Units are chosen such that  = 2m = 1, m denoting the particle mass. Following [3,10] we restrict H to BCS type states. These are quasi-free states that do not have a fixed number of particles. For details on the formalism, we refer the reader to [2]. Quasi-free states are linear maps ρ from bounded operators on the Fock space F to the complex numbers, satisfying ρ(A† A) ≥ 0, ρ(I ) = 1, and

ρ(e1 e2 e3 e4 ) = ρ(e1 e2 )ρ(e3 e4 ) − ρ(e1 e3 )ρ(e2 e4 ) + ρ(e1 e4 )ρ(e2 e3 ), † . whenever ei is either ak,σ or ak,σ Assuming both translation invariance and SU (2) invariance for rotations of the spins, the state ρ is completely determined by the quantities † † ak,↑ ) = ρ(ak,↓ ak,↓ ), ϕ(k) = ρ(a−k,↑ ak,↓ ) = −ρ(a−k,↓ ak,↑ ). γ (k) = ρ(ak,↑

All other expectation values of quadratic expressions are zero under these symmetry assumptions. Note that 0 ≤ γ ( p) ≤ 1 and |ϕ( p)|2 ≤ γ ( p)(1 − γ (− p)) by Schwarz’s

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365

inequality and the canonical anticommutation relations. Since also ϕ( p) = ϕ(− p), this implies that |ϕ( p)|2 ≤ γ ( p)(1 − γ ( p)). The energy expectation value can be conveniently expressed in terms of these quantities: 

(2π )3/2  ˆ V (k)ϕ(k − p)ϕ( p) || k k, p

2  2(2π )3/2 ˆ (2π )3/2  ˆ V (0) + V (k)γ ( p − k)γ ( p). (A.2) γ ( p) − || || p

ρ(H) = 2

k 2 γ (k) +

k, p

The first term in the second line is just a constant times the square of the number of particles. The last term is the exchange terms. Both these terms will be dropped in the sequel. This truncations can be justified on physical grounds for interaction potentials V that are of short range such that s-wave scattering is dominant [10]. The von-Neumann entropy of the quasi-free state ρ under consideration here, S(ρ) = −Tr F ρ ln ρ, can be conveniently expressed in terms of the 2 × 2 matrix

γ ( p) ϕ( p) ( p) = . (A.3) ϕ( p) 1 − γ ( p) Note that 0 ≤ ( p) ≤ 1 as an operator on C2 . The entropy is then given by (compare with [2, Eq. (2.c.10)])  S(ρ) = −2 Tr C2 [( p) ln ( p)] . p

The factor 2 results from the 2 spin components. The thermodynamic pressure, P, of a state at a fixed temperature T = chemical potential µ is defined as P(ρ) =

1 β

≥ 0 and

1 [T S(ρ) − ρ(H − µN)] . ||

 † Here, N = σ,k ak,σ ak,σ denotes the number operator. Dropping the last two terms in (A.2) and taking a formal infinite volume limit, one thus obtains
  2 2 T Tr P(γ , α) := lim P(ρ) = − 2 ( p) ln ( p) + ( p − µ)γ ( p) dp C (2π )3 R3 →R3
1 |α(x)|2 V (x) d x. − (2π )3 R3  Here, α denotes the inverse Fourier transform of ϕ, α(x) = (2π )−3/2 ϕ( p)ei px dp. If we set Fβ = − 21 (2π )3 P and replace V by 2V to slightly simplify the notation, we are thus led to the definition of the BCS model given in Definition 1. Finally, we will briefly explain in which sense E( p) is an effective dispersion relation and how it is related to an energy gap between the ground state and the first excited state of the system.

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With ϕ = αˆ the Fourier transform of a maximizer of the pressure functional at T = 0, we introduce the following quadratic Hamiltonian as an approximation to the full Hamiltonian H: Hqu =

 σ,k



† k 2 ak,σ ak,σ +

(2π )3/2  ˆ V (k) || k, p

 † × ϕ(k − p)a− p,↑ a p,↓ + ϕ( p)a †p−k,↑ ak− p,↓ .

(A.4)

The Hamiltonian Hqu − µN can be diagonalized in a standard way (see, e.g., [12]) via a Bogoliubov transformation. Using the fact that ϕ  satisfies the gap equation E(k)ϕ(k) =  − 21 (2π )3/2 k Vˆ (k − k )ϕ(k ), where E(k) = (k 2 − µ)2 + |(k)|2 and  is determined from ϕ via (2.2)–(2.4), this leads to      (A.5) Hqu − µN − inf spec Hqu − µN = E(k) ck† ck + dk† dk . k

The operators ck , ck† , dk and dk† satisfy the canonical anticommutation relations, and both ck and dk annihilate the BCS ground state. Equation (A.5) thus explains why E(k) is interpreted as the quasi-particle dispersion relation and  = inf k E(k) as the energy gap of the system. References 1. Andrenacci, N., Perali, A., Pieri, P., Strinati, G.C.: Density-induced BCS to Bose-Einstein crossover. Phys. Rev. B 60, 12410 (1999) 2. Bach, V., Lieb, E., Solovej, J.: Generalized Hartree-Fock theory and the Hubbard model. J. Stat. Phys. 76, 3–89 (1994) 3. Bardeen, J., Cooper, L., Schrieffer, J.: Theory of Superconductivity. Phys. Rev. 108, 1175–1204 (1957) 4. Billard, P., Fano, G.: An existence proof for the gap equation in the superconductivity theory. Commun. Math. Phys. 10, 274–279 (1968) 5. Bloch, I., Dalibard, J., Zwerger, W.: Many-Body Physics with Ultracold Gases. http://arxiv.org/abs/:0704. 3011, 2007, to appear in Rev. Mod. Phys. 6. Carlson, J., Chang, S.-Y., Pandharipande, V.R., Schmidt, K.E.: Superfluid Fermi Gases with Large Scattering Length. Phys. Rev. Lett. 91, 0504011 (2003) 7. Chen, Q., Stajic, J., Tan, S., Levin, K.: BCS–BEC crossover: From high temperature superconductors to ultracold superfluids. Phys. Rep. 412, 1–88 (2005) 8. Fetter, A., Walecka, J.D.: Quantum theory of many-particle systems. New-York: McGraw-Hill, 1971 9. Frank, R.L., Hainzl, C., Naboko, S., Seiringer, R.: The critical temperature for the BCS equation at weak coupling. J. Geom. Anal. 17, 559–568 (2007) 10. Leggett, A.J.: Diatomic Molecules and Cooper Pairs. Modern trends in the theory of condensed matter, J. Phys. (Paris) Colloq, C7–19 Bertin-Heidelberg-New York: Springer, 1980 11. Lieb, E., Loss, M.: Analysis. Providence RI: Amer. Math. Soc., 2001 12. Martin, P.A., Rothen, F.: Many-body problems and Quantum Field Theory. Berlin-Heidelberg-New York: Springer, 2004 13. McLeod, J.B., Yang, Y.: The uniqueness and approximation of a positive solution of the Bardeen-CooperSchrieffer gap equation. J. Math. Phys. 41, 6007–6025 (2000) 14. Nozières, P., Schmitt-Rink, S.: Bose Condensation in an Attractive Fermion Gas: From Weak to Strong Coupling Superconductivity. J. Low Temp. Phys. 59, 195–211 (1985) 15. Parish, M., Mihaila, B., Timmermans, E., Blagoev, K., Littlewood, P.: BCS-BEC crossover with a finiterange interaction. Phys. Rev. B 71, 0645131–0645136 (2005) 16. Randeria, M.: In: Bose-Einstein Condensation, Griffin, A., Snoke, D.W., Stringari, S. eds., Cambridge: Cambridge University Press, 1995 17. Tiesinga, E., Verhaar, B.J., Stoof, H.T.C.: Threshold and resonance phenomena in ultracold ground-state collisions. Phys. Rev. A 47, 4114 (1993)

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18. Vansevenant, A.: The gap equation in superconductivity theory. Physica 17D, 339–344 (1985) 19. Yang, Y.: On the Bardeen-Cooper-Schrieffer integral equation in the theory of superconductivity. Lett. Math. Phys. 22, 27–37 (1991) Communicated by H. Spohn

Commun. Math. Phys. 281, 369–385 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0492-7

Communications in

Mathematical Physics

Electromagnetic Wormholes via Handlebody Constructions Allan Greenleaf1 , Yaroslav Kurylev2 , Matti Lassas3 , Gunther Uhlmann4 1 2 3 4

Department of Mathematics, University of Rochester, Rochester, NY 14627, USA Department of Mathematical Sciences, Loughborough Univ., Loughborough, LE11 3TU, UK Helsinki University of Technology, Institute of Mathematics, P.O.Box 1100, 02015 Helsinki, Finland Department of Mathematics, University of Washington, Seattle, WA 98195, USA. E-mail: [email protected]

Received: 10 April 2007 / Accepted: 24 December 2007 Published online: 16 May 2008 – © Springer-Verlag 2008

Abstract: Cloaking devices are prescriptions of electrostatic, optical or electromagnetic parameter fields (conductivity σ (x), index of refraction n(x), or electric permittivity (x) and magnetic permeability µ(x)) which are piecewise smooth on R3 and singular on a hypersurface , and such that objects in the region enclosed by  are not detectable to external observation by waves. Here, we give related constructions of invisible tunnels, which allow electromagnetic waves to pass between possibly distant points, but with only the ends of the tunnels visible to electromagnetic imaging. Effectively, these change the topology of space with respect to solutions of Maxwell’s equations, corresponding to attaching a handlebody to R3 . The resulting devices thus function as electromagnetic wormholes.

1. Introduction There has recently been considerable interest, both theoretical [3,13,16,18,19] and experimental [20], in invisibility (or “cloaking”) from observation by electromagnetic (EM) waves. (See also [17] for a treatment of cloaking in the context of elasticity.) Theoretically, cloaking devices are given by specifying the conductivity σ (x) (in the case of electrostatics), the index of refraction n(x) (for optics in the presence of polarization, where one uses the Helmholtz equation), or the electric permittivity (x) and magnetic permeability µ(x) (for the full system of Maxwell’s equations). In the constructions to date, the EM parameter fields ( σ ; n;  and µ ) have been piecewise smooth and anisotropic. (See, however, [5, Sect.4] for an example that can be interpreted as shielding with respect to Helmholtz by an isotropic negative index of refraction material). Furthermore, the EM parameters have singularities, with one or more eigenvalues of the tensors going to zero or infinity as one approaches, from one or both sides, the cloaking surface , which encloses the region within which objects may be hidden from external observation. Such constructions might have remained theoretical curiosities, but the advent of

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metamaterials [1] allows one, within the constraints of current technology, to construct media with fairly arbitrary (x) and µ(x). It thus becomes an interesting mathematical problem with practical significance to understand what other new phenomena of wave propagation can be produced by prescribing other arrangements of  and µ. Geometrically, cloaking can be viewed as arising from a singular transformation of R3 . Intuitively, for a spherical cloak [6,7,18], it is as if an infinitesimally small hole in space has been stretched to a ball D; an object can be inserted inside the hole so created and is then invisible to external observations. On the level of the EM parameters, homogeneous, isotropic parameters , µ are pushed forward to become inhomogeneous, anisotropic and singular as one approaches  = ∂ D from the exterior. There are then two ways, referred to as the single and double coating in [3], of continuing , µ to within D so as to rigorously obtain invisibility with respect to locally finite energy waves. We refer to either process as blowing up a point. As observed in [3], one can use the double coating to produce a manifold with a different topology, but with the change in topology invisible to external measurements. To define the solutions of Maxwell’s equations rigorously in the single coating case, one has to add boundary conditions on . Physically, this corresponds to the lining of the interior of the single coating material, e.g., in the case of blowing up a point, with a perfectly conducting layer, see [3]. We point out here that in the very interesting preprint [21], the single coating construction is supplemented with selfadjoint extensions of Maxwell operators in the interior of the cloaked regions; these implicitly impose interior boundary conditions on the boundary of the cloaked region, similar to the PEC boundary condition suggested in [3]. For the case of an infinite cylinder, the Soft-and-Hard surface (SHS) interior boundary condition is used in [3] to guarantee cloaking of active objects, and is needed even for passive ones. In this paper, we show how more elaborate geometric constructions, corresponding to blowing up a curve, enable the description of tunnels which allow the passage of waves between distant points, while only the ends of the tunnels are visible to external observation. These devices function as electromagnetic wormholes, essentially changing the topology of space with respect to solutions of Maxwell’s equations. Our proofs are valid of all positive frequencies; however current metamaterial designs which might be used to implement electromagnetic wormholes are fairly narrowband. Potential applications and further discussion can be found in [4]. We form the wormhole device around an obstacle K , a tubular neighborhood of a portion of a cylindrical surface in R3 as follows (see Fig. 1). First, one surrounds K with metamaterials, corresponding to a specification of EM parameters
ε and
µ. Secondly, one lines the surface of K with material implementing the SHS boundary condition from antenna theory [8,10,11]; this condition arose previously [3] in the context of cloaking an infinite cylinder. The EM parameters, which become singular as one approaches K , are given as the pushforwards of nonsingular parameters ε and µ on an abstract threemanifold M, described in Sect. 2. For a curve γ ⊂ M, we construct the diffeomorphism F from M\γ to the wormhole device in Sect. 3. For the resulting EM parameters
ε and
µ, we have singular coefficients of Maxwell’s equations at K , and so it is necessary to formulate an appropriate notion of locally finite energy solutions (see Def. 4.1). In Theorem 4.2, we then show that there is a perfect correspondence between the external measurements of EM waves propagating through the wormhole device and those propagating on the wormhole manifold, with the same boundary measurements. It was shown in [3] that the cloaking constructions are mathematically valid at all frequencies k. However, both cloaking and the wormhole effect studied here should be

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371

Fig. 1. The obstacle K without surrounding metamaterial

considered as essentially monochromatic, or at least very narrow-band, using current technology, since, from a practical point of view the metamaterials needed to implement the constructions have to be fabricated and assembled with a particular wavelength in mind, and theoretically are subject to significant dispersion [18]. Thus, as for cloaking in [3,16,18], here we describe the wormhole construction relative to electromagnetic waves at a fixed positive frequency k. We point out that the metamaterials used in the experimental verification of cloaking [20] should be readily adaptable to yield a physical implementation, at microwave frequencies, of the wormhole device described here. See Remark 1 in Sect. 4.2 for further discussion. The results proved here were announced in [4]. 2. The Wormhole Manifold M First we explain, somewhat informally, what we mean by a wormhole. The concept of a wormhole is familiar from general relativity [9,22], but here we define a wormhole as an object obtained by gluing together pieces of Euclidean space equipped with certain anisotropic EM parameter fields. We start by describing this process heuristically; later, we explain more precisely how this can be effectively realized vis-a-vis EM wave propagation using metamaterials. We first describe the wormhole as an abstract manifold M, see Fig. 2; in the next section we will show how to realize this concretely in R3 , as a wormhole device N . Start by making two holes in the Euclidean space R3 = {(x, y, z)|x, y, z ∈ R}, say by removing the open ball B1 = B(O, 1) with center at the origin O and of radius 1, and also the open ball B2 = B(P, 1), where P = (0, 0, L) is a point on the z-axis having the distance L > 3 to the origin. We denote by M1 the region so obtained, M1 = R3 \(B1 ∪ B2 ), which is the first component we need to construct a wormhole. Note that M1 is a 3dimensional manifold with boundary, the boundary of M1 being ∂ M1 = ∂ B1 ∪ ∂ B2 , the union of two 2-spheres. Thus, ∂ M1 can be considered as a disjoint union S2 ∪ S2 , where we will use S2 to denote various copies of the two-dimensional unit sphere. The second component needed is a 3−dimensional cylinder, M2 = S2 × [0, 1]. This cylinder can be constructed by taking the closed unit cube [0, 1]3 in R3 and, for each value of 0 < s < 1, gluing together, i.e., identifying, all of the points on the boundary of the cube with z = s. Note that we do not identify points at the top of the boundary,

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Fig. 2. Schematic figure: a wormhole manifold is glued from two components, the “handle” and space with two holes. Note that in the actual construction, the components are three dimensional

at z = 1, or at the bottom, at z = 0. We then glue together the boundary ∂ B(O, 1) ∼ S2 of the ball B(O, 1) with the lower end (boundary component) S2 × {0} of M2 , and the boundary ∂ B(P, 1) with the upper end, S2 × {1}. In doing so we identify the point (0, 0, 1) ∈ ∂ B(O, 1) with the point N P × {0} and the point (0, 0, L − 1) ∈ ∂ B(P, 1) with the point N P × {1}, where N P is the north pole on S2 . The resulting manifold M no longer lies in R3 , but rather is the connected sum of the components M1 and M2 and has the topology of R3 with a 3−dimensional handle attached. Note that adding this handle makes it possible to travel from one point in M1 to another point in M1 , not only along curves lying in M1 but also those in M2 . To consider Maxwell’s equations on M, let us start with Maxwell’s equations on R3 at frequency k ∈ R, given by ∇ × E = ik B, ∇ × H = −ik D,

D(x) = ε(x)E(x),

B(x) = µ(x)H (x).

Here E and H are the electric and magnetic fields, D and B are the electric displacement field and the magnetic flux density, ε and µ are matrices corresponding to permittivity and permeability. As the wormhole is topologically different from the Euclidean space R3 , we use a formulation of Maxwell’s equations on a manifold, and as in [3], do this in the setting of a general Riemannian manifold, (M, g). For our purposes, as in [3,14] it suffices to use ε, µ which are conformal, i.e., proportional by scalar fields, to the metric g. In this case, Maxwell’s equations can be written, in the coordinate invariant form, as d E = ik B, d H = −ik D,

D =  E, B = µH in M,

where E, H are 1-forms, D, B are 2-forms, d is the exterior derivative, and  and µ are scalar functions times the Hodge operator of (M, g), which maps 1-forms to the corresponding 2-forms [2]. In local coordinates these equations are written in the same form as Maxwell’s equations in Euclidean space with matrix valued ε and µ. Although not necessary, for simplicity one can choose a metric on the wormhole manifold M which is Euclidean on M1 , and on M2 is the product of a given metric g0 on S2 and the standard metric of [0, 1]. More generally, can also choose the metric on M2 to be a warped product. Even the simple choice of the product of the standard metric of S2 and the metric δ 2 ds 2 , where δ is the “length” of the wormhole, gives rise to interesting ray-tracing effects for rays passing through the wormhole tunnel. For δ << 1, the image through one end of the wormhole (of the region beyond the other end) would resemble the image in a a fisheye lens; for δ
1, multiple images and greater distortion occur. (See [4, Fig. 2]).

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The proof of the wormhole effect that we actually give is for yet another variation, where the balls that form the ends have their boundary spheres flattened; this may be useful for applications, since it allows for there to be a vacuum (or air) in a neighborhood of the axis of the wormhole, so that, e.g., instruments may be passed through the wormhole. We next show how to construct, using metamaterials, a device N in R3 that effectively realizes the geometry and topology of M, relative to solutions of Maxwell’s equations at frequency k, and hence functions as an electromagnetic wormhole. 3. The Wormhole Device N in R3 We now explain how to construct a “device” N in R3 , i.e., a specification of permittivity ε and permeability µ, which affects the propagation of electromagnetic waves in the same way as the presence of the handle M2 in the wormhole manifold M. What this means is that we prescribe a configuration of metamaterials which make the waves behave as if there were an invisible tube attached to R3 , analogous to the handle M2 in the wormhole manifold M. In other words, as far as external EM observations of the wormhole device are concerned, it appears as if the topology of space has been changed. We use cylindrical coordinates (θ, r, z) corresponding to a point (r cos θ, r sin θ, z) in R3 . The wormhole device is built around an obstacle K ⊂ R3 . To define K , let S be the two-dimensional finite cylinder {θ ∈ [0, 2π ], r = 2, 0 ≤ z ≤ L} ⊂ R3 . The open region K consists of all points in R3 that have distance less than one to S and has the shape of a long, thick-walled tube with smoothed corners. Let us first introduce a deformation map F from M to N = R3 \K or, more precisely, from M\γ to N \, where γ is a closed curve in M to be described shortly and  = ∂ K . We will define F separately on M1 and M2 denoting the corresponding parts by F1 and F2 . To describe F1 , let γ1 be the line segment on the z−axis connecting ∂ B(O, 1) and ∂ B(P, 1) in M1 , namely, γ1 = {r = 0, z ∈ [1, L −1]}. Let F1 (r, z) = (θ, R(r, z), Z (r, z)) be such that (R(r, z), Z (r, z)), shown in Fig. 3, transforms in the (r, z) coordinates the semicircles AB and C D in the left picture to the vertical line segments A B  = {r ∈ [0, 1], z = 0} and C  D  = {r ∈ [0, 1], z = L} in the right picture and the cut γ1 on the left picture to the curve B  C  on the right picture. This gives us a map F1 : M1 \γ1 → N1 \, where the closed region N1 in R3 is obtained by rotation of the region exterior to the curve A B  C  D  around the z−axis. We can choose F1 so that it is the identity map in the domain U = R3 \{−2 ≤ z ≤ L + 2, 0 ≤ r ≤ 4}. To describe F2 , consider the line segment, γ2 = {N P} × [0, 1] on M2 . The sphere without the north pole can be “flattened” and stretched to an open disc with radius one which, together with stretching [0, 1] to [0, L], gives us a map F2 from M2 \γ2 to N2 \. The region N2 is the 3−dimensional cylinder, N2 = {θ ∈ [0, 2π ], r ∈ [0, 1],

Fig. 3. The map (R(r, z), Z (r, z)) in cylindrical coordinates (z, r )

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z ∈ [0, L]}. When flattening S2 \N P, we do it in such a way that F1 on ∂ B(O, 1) and ∂ B(P, 1) coincides with F2 on (S2 \N P) × {0} and (S2 \N P) × {1}, respectively. Thus, F maps M\γ , where γ = γ1 ∪ γ2 is a closed curve in M, onto N \; in addition, F is the identity on the region U . Now we are ready to define the electromagnetic material parameter tensors on N . We define the permittivity to be  (D F)(x)· ε(x)· (D F(x))t 
ε = F∗ ε(y) = ,  det (D F) x=F −1 (y) where D F is the derivative matrix of F, and similarly the permeability to be
µ = F∗ µ. These deformation rules are based on the fact that permittivity and permeability are conductivity type tensors, see [14]. Maxwell’s equations are invariant under smooth changes of coordinates. This means that, by the chain rule, any solution to Maxwell’s equations in M\γ , endowed with material parameters ε, µ becomes, after transformation by F, a solution to Maxwell’s equations in N \ with material parameters
ε and
µ, and vice versa. However, when considering the fields on the entire spaces M and N , these observations are not enough, due to the singularities of
ε and
µ near ; the significance of this for cloaking was observed and analyzed in [3]. In the following, we will show that the physically relevant class of solutions to Maxwell’s equations, namely the (locally) finite energy solutions, remains the same, with respect to the transformation F, in (M; ε, µ) and (N ;
ε,
µ). One can analyze the rays in M and N endowed with the electromagnetic wave propagation √ √ metrics g = εµ and
g=
ε
µ, respectively. Then the rays on M are transformed by F into the rays in N . As almost all the rays on M do not intersect with γ , therefore, almost all the rays on N do not approach . This was the basis for [16,18] and was analyzed further in [19]; see also [17] for a similar analysis in the context of elasticity. Thus, heuristically one is led to conclude that the electromagnetic waves on (M; ε, µ) do not feel the presence of γ , while those on (N ;
ε,
µ) do not feel the presence of K , and these waves can be transformed into each other by the map F. Although the above considerations are mathematically rigorous, on the level both of the chain rule and of high frequency limits, i.e., ray tracing, in the exteriors M\γ and N \, they do not suffice to fully describe the behavior of physically meaningful solution fields on M and N . However, by carefully examining the class of the finite-energy waves in M and N and analyzing their behavior near γ and , respectively, we can give a complete analysis, justifying the conclusions above. Let us briefly explain the main steps of the analysis using methods developed for theory of invisibility (or cloaking) at frequency k > 0 [3] and at frequency k = 0 in [6,7]. The details will follow. First, to guarantee that the fields in N are finite energy solutions and do not blow up near , we have to impose at  the appropriate boundary condition, namely, the Soft-and-Hard surface condition, see [8,11], eθ · E| = 0, eθ · H | = 0, where eθ is the angular direction. Secondly, the map F can be considered as a smooth coordinate transformation on M\γ ; thus, the finite energy solutions on M\γ transform under F into the finite energy solutions on N \, and vice versa. Thirdly, the curve γ in M has Hausdorff dimension equal to one. This implies that the possible singularities of the finite energy electromagnetic fields near γ are removable [12], that is, the finite energy fields in M\γ are exactly the restriction to M\γ of the fields defined on all of M.

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Combining these steps we can see that measurements of the electromagnetic fields on (M; ε, µ) and on (R3 \K ;
ε,
µ) coincide in U . In other words, if we apply any current on U and measure the radiating electromagnetic fields it generates, then the fields on U in the wormhole manifold (M; ε, µ) coincide with the fields on U in (R3 \K ;
ε,
µ), 3-dimensional space equipped with the wormhole device construction. Summarizing our construction, the wormhole device consists of the metamaterial coating of the obstacle K . This coating should have the permittivity
ε and permeability
µ. In addition, we need to impose the SHS boundary condition on , which may be realized by fabricating the obstacle K from a perfectly conducting material with parallel corrugations on its surface [8,11]. In the next section, the permittivity
ε and and permeability
µ are described in a rather simple form. (As mentioned earlier, in order to allow for a tube around the axis of the wormhole to be a vacuum or air, we deal with a slightly different construction than was described above, starting with flattened spheres). It should be possible to physically implement an approximation to this mathematical idealization of the material parameters needed for the wormhole device, using concentric rings of split ring resonators as in the experimental verification of cloaking obtained in [20]. 4. Rigorous Construction of the Wormhole Here we present a rigorous model of a typical wormhole device and justify the claims above concerning the behavior of the electromagnetic fields in the wormhole device in R3 in terms of the fields on the wormhole manifold (M, g). 4.1. The wormhole manifold (M, g) and the wormhole device N . Here we prove the wormhole effect for a variant of the wormhole device described in the previous sections. Instead of using a round sphere S2 as before, we present a construction that uses a deformed sphere S2flat that is flat near the south and north poles, S P and N P. This makes it possible to have constant isotropic material parameters near the z-axis located inside the wormhole. For possible applications, see [4]. We use following notations. Let (θ, r, z) ∈ [0, 2π ] × R+ × R be the cylindrical coordinates of R3 , that is the map X : (θ, r, z) → (r cos θ, r sin θ, z) that maps X : [0, 2π ] × R+ × R → R3 . In the following, we identify [0, 2π ] and the unit circle S 1 . Let us start by removing from R3 two “deformed” balls which have flat portions near the south and north poles. More precisely, let M1 = R3 \(P1 ∪ P2 ), where in the cylindrical coordinates P1 = {X (θ, r, z) : −1 ≤ z ≤ 1, 0 ≤ r ≤ 1} ∪{X (θ, r, z) : (r − 1)2 + z 2 ≤ 1}, P2 = {X (θ, r, z) : 1 ≤ z − L ≤ 1, 0 ≤ r ≤ 1} ∪{X (θ, r, z) : (r − 1)2 + (z − L)2 ≤ 1}. We say that the boundary ∂ P1 of P1 is a deformed sphere with flat portions, and denote it by S2flat . We say that the intersection points of S2flat with the z-axis are the north pole, N P, and the south pole, S P.

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Fig. 4. A schematic figure on the map G 0 , considered in the (r, z) coordinates. Later, we impose the SHS boundary condition on the portion of the boundary, colored red in [4, Fig. 5]

Let g1 be the metric on M1 inherited from R3 , and let γ1 be the path γ1 = {X (0, 0, z) : 1 < z < L − 1} ⊂ M1 . Set A1 = M1 \V1/4 , Vt = {X (θ, r, z) : 0 ≤ r ≤ t, 1 < z < L − 1}, 0 < t < 1, and consider a map G 0 : M1 \γ1 → A1 ; see Fig. 4. G 0 defined as the identity map on M1 \V1/2 and, in cylindrical coordinates, as   1 r G 0 (X (θ, r, z)) = X θ, + , z , (θ, r, z) ∈ V1/2 . 4 2 Clearly, G 0 is C 0,1 −smooth. Let U (x) ∈ R3×3 , x = X (θ, r, z), be the orthogonal matrix that maps the standard unit vectors e1 , e2 , e3 of R3 to the Euclidean unit vectors corresponding to the θ , r , and z directions, that is, U (x)e1 = (− sin θ, cos θ, 0), U (x)e2 = (cos θ, sin θ, 0), U (x)e3 = (0, 0, 1). Then the differential of G 0 in the Euclidean coordinates at the point x ∈ V1/2 is the matrix ⎛1 1 r ⎞ r (4 + 2) 0 0 1 ⎠ U (x)−1 , x = X (θ, r, z), y = G (x). (1) DG 0 (x) = U (y) ⎝ 0 0 2 0 0 01 Later we impose on part of the boundary, 0 = ∂ A1 ∩ {1 < z < L − 1}, the SHS boundary condition. Next, let (θ, z, τ ) = (θ (x), z(x), τ (x)) be the Euclidean boundary normal coordinates associated to 0 , that is, τ (x) = distR3 (x, 0 ) and (θ (x), z(x)) are the θ and z-coordinates of the closest point of 0 to x. Denote by (G 0 )∗ g1 the push forward of the metric g1 in G 0 , that is, the metric obtained from g1 using the change of coordinates G 0 , see [2]. The metric (G 0 )∗ g1 coincides with g1 in A1 \V1/2 , and in the Euclidean boundary normal coordinates of 0 , on A1 ∩ V1/2 , the metric (G 0 )∗ g1 , has the length element ds 2 = 4τ 2 dθ 2 + dz 2 + 4dτ 2 .

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Fig. 5. Map G 1 in (r, z)-coordinates

Next, let

 q3 = conv {(r, z) : (r − 2)2 + (z − (−2))2 ≤ 1}  ∪{(r, z) : (r − 2) + (z − (L + 2)) ≤ 1} , 2

2

q4 = {(r, z) : 0 ≤ r ≤ 1, −1 ≤ z ≤ L + 1}, where conv(q) denotes the convex hull of the set q. Let N1 P3 P4 1

= = = =

R3 \(P3 ∪ P4 ), {X (θ, r, z) : (r, z) ∈ q3 }, {X (θ, r, z) : (r, z) ∈ q4 }, ∂ N1 \∂ P4 .

We can find a Lipschitz smooth map G 1 : A1 → N1 , see Fig. 5, of the form G 1 (X (θ, r, z)) = X (θ, R(r, z), Z (r, z)) such that it maps 0 to 1 , and in A1 near 0 it is given by G 1 (x + tν0 ) = G 1 (x) + tν1 .

(2)

Here, x ∈ 0 , ν0 is the Euclidean unit normal vector of 0 , ν1 is the Euclidean unit normal vector of 1 , and 0 < t < 41 . Moreover, we can find a G 1 so that it is the identity map near the z-axis, that is, G 1 (x) = x, x ∈ A1 ∩ {0 ≤ r <

1 } 4

(3)

and such that G 1 is also the identity map in the set of points with the Euclidean distance 4 or more from P1 ∪ P2 . Note that we can find such a G 1 such that both G 1 and its inverse G −1 1 are Lipschitz smooth up to the boundary. Thus the differential DG 1 of G 1 at x ∈ A1 in Euclidean coordinates is   a (r, z) 0 U (x)−1 , x = X (θ, r, z), y = G 1 (x), DG 1 (x) = U (y) 11 0 A(r, z) where c0 ≤ a11 (r, z) ≤ c1 and A(r, z) is a symmetric (2 × 2)-matrix satisfying c0 I ≤ A(r, z) ≤ c1 I with some c0 , c1 > 0.

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The map F1 (x) = G 1 (G 0 (x)) then maps F1 : M1 \γ1 → N1 . Let
g1 = (F1 )∗ g1 be metric on N1 . From the above considerations, we see that the differential D F1 of F1 at x ∈ M1 \γ1 near 0 , in Euclidean coordinates, is given by   b (θ, r, z) 0 U (x)−1 , D F1 (x) = U (y) 11 0 B(r, z) (4) c11 (r, z) x = X (θ, r, z), y = F1 (x), b11 (θ, r, z) = distR3 (X (θ, r, z), 0 ) where c0 ≤ c11 (r, z) ≤ c1 , and B(r, z) is a symmetric (2 × 2)-matrix satisfying c0 I ≤ B(r, z) ≤ c1 I, for some c0 , c1 > 0. Note that ∂ P4 ∩ {r < 1} consists of two two-dimensional discs, B2 (0, 1) × {−1} and B2 (0, 1) × {L + 1}. Below, we will use the map f 2 = F1 |∂ P1 \N P : ∂ P1 \N P → B2 (0, 1) × {−1} ⊂ ∂ N1 . The map f 2 can be considered as the deformation that “flattens” S2flat \N P to a two dimensional unit disc. To describe f 2 , consider S2flat as a surface in Euclidean space and define on it the θ coordinate corresponding to the θ coordinate of R3 \{z = 0}. Let then s(y) be the intrinsic distance of y ∈ S2flat to the south pole S P. Then (θ, s) define coordinates in S2flat \{S P, N P}. We denote by y(θ, s) ∈ S2flat \{S P, N P} the point corresponding to the coordinates (θ, s). By the above construction, the map f 2 has the form, with respect to the coordinates used above, f 2 (y(θ, s)) = X (θ, R(s), −1) ∈ B2 (0, 1) × {−1}, where 1 R(s) = s, for 0 < s < , 4 1 1 R(s) = 1 − [(π + 4) − s], for (π + 4) − < s < (π + 4), 2 4

(5)

cf. formulae (2) and (3). In the following we identify B2 (0, 1) × {−1} with the disc B2 (0, 1). Let h 1 be the metric on ∂ P1 \N P inherited from (M1 , g1 ). Let h 2 = ( f 2 )∗ h 1 be the metric on B2 (0, 1). We observe that the metric h 2 makes the disc B2 (0, 1) isometric to S2flat \N P, endowed with the metric inherited from R3 . Thus, let M2 = S2flat × [−1, L + 1]. On M2 , let the metric g2 be the product of the metric of S2flat inherited from R3 and the metric α2 (z)dz 2 , α2 > 0 on [−1, L + 1]. Let γ2 = {N P} × [−1, L + 1] be a path on M2 . Define N2 = P4 = {X (θ, r, z) : 0 ≤ r < 1, −1 ≤ z ≤ L + 1} ⊂ R3 , 2 = ∂ N2 ∩ {r = 1}, and let F2 : M2 \γ2 → N2 be the map of the form F2 (y, z) = ( f 2 (y), z) ∈ R3 , (y, z) ∈ (S2flat \N P) × [−1, L + 1]. Let
g2 = (F2 )∗ g2 be the resulting metric on N2 .

(6)

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Fig. 6. Left: The set N2 in the (r, z) coordinates. Right: The set N = N1 ∪ N2 ⊂ R3 having the complement K , presented in the (r, z) coordinates. Later, the SHS boundary condition is imposed on all of ∂ K

Denote by M 1 = M1 ∪ ∂ M1 the closure of M1 and let (M, g) = (M 1 , g1 )#(M2 , g2 ) be the connected sum of M 1 and M2 , that is, we glue the boundaries ∂ M1 and ∂ M2 . The set N = N1 ∪ N2 ⊂ R3 is open, and its boundary ∂ N is  = 1 ∪ 2 . Let F be the map F : M\γ → N defined by the maps F1 : M1 \γ1 → N1 and F2 : M2 \γ2 → N2 , and finally, let γ = γ1 ∪ γ2 and
g = F∗ g. t,
θ ), where Let K = R3 \N . On the surface  = ∂ K we can use local coordinates (

θ is the θ -coordinate of the ambient space R3 and
t is either the r or z -coordinate of the ambient space R3 restricted to . Denote also
τ =
τ (x) = distR3 (x, ∂ K ). Then by formula (2) we see that in N1 , in the Euclidean boundary normal coordinates (
θ ,
t,
τ ) associated to the surface 1 , the metric
g has the length element τ 2 + α1 (
t) d
t 2 + 4
τ 2 d
θ 2, 0 <
τ< ds 2 = 4d


1 , c0−1 ≤ α1 (
t) ≤ c0 , c0 ≥ 1. 4

The construction of F2 yields that in N2 , in the Euclidean boundary normal coordinates (
θ ,
t,
τ ) with
t = z, associated to the surface 2 = ∂ K ∩ ∂ N2 , the metric
g has the length element, near 2 , τ 2 + α2 (
t)d
t 2 + 4
τ 2 d
θ 2, 0 <
τ< ds 2 = 4d


1 . 4

Here, near ∂ N1 ∩ ∂ N2 , we use
t = z on 1 . Choosing the map G 1 in the construction of the map F1 appropriately, we have α2 (−1) = α1 (−1), α2 (L + 1) = α1 (L + 1), and the resulting map is Lipschitz. On M1 , N1 , and N2 that are subsets of R3 we have the well defined cylindrical coordinates (θ, r, z). Similarly, M2 = S2flat × [−1, L + 1] and we define the coordinates (θ, s, z), where (θ, s) are the above defined coordinates on S2flat \{S P, N P}. We can also consider on N ⊂ R3 also the Euclidean metric, denoted by g e . In Euclidean coordinates, (g e )i j = δ jk . Consider next the above defined Euclidean boundary normal coordinates (
θ ,
t,
τ ) associated to ∂ K . They are well defined in a neighborhood of ∂ K . We define the vector fields

ξ = ∂
η = ∂
τ,
t θ , ζ = ∂
on N near ∂ K . These vector fields are orthogonal with respect to the metric
g and to the metric g e . On M near γ , we use coordinates (θ, t, τ ). On M1 , near γ1 in the terms of the cylindrical coordinates they are (θ, t, τ ) = (θ, z, r ). On M2 , they are the coordinates

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(θ, t, τ ) = (θ, z, s), where s is the intrinsic distance to the north pole N P. We define also the vector fields ξ = ∂τ , η = ∂θ , ζ = ∂t on M\γ near γ . These vector fields are orthogonal with respect to the metric g. In the sequel, we consider the differential of F as the linear map D F : (Tx M, g) → (Ty N , g e ), y = F(x), x ∈ M\γ . Using formula (4) in M1 and formulas (5), (6) in M2 , we see that D F −1 (x) at x ∈ N near ∂ N is a bounded linear map that satisfies |(η, D F −1 (x)
η)g | ≤ C
τ (x), −1
(η, D F (x)ζ )g = 0,

(ζ, D F −1 (x)
η)g = 0, −1
|(ζ, D F (x)ζ )g | ≤ C,

(η, D F −1 (x)
ξ )g = 0,

|(ζ, D F −1 (x)
ξ )g | ≤ C,

(ξ, D F −1 (x)
η)g = 0, −1 |(ξ, D F (x)
ζ )g | ≤ C, (7) −1
|(ξ, D F (x)ξ )g | ≤ C,

where C > 0 and (· , · )g is the inner product defined by the metric g. Moreover, we obtain similar estimates for D F in terms of the Euclidean metric g e , (
η, D F(y)ζ )ge = 0,

(
ζ , D F(y)η)ge = 0,
|(ζ , D F(y)ζ )ge | ≤ C,

(
ξ , D F(y)η)ge = 0, |(
ξ , D F(y)ζ )ge | ≤ C,

(
η, D F(y)ξ )ge = 0,

|(
ζ , D F(y)ξ )ge | ≤ C,

|(
ξ , D F(y)ξ )ge | ≤ C

|(
η, D F(y)η)ge | ≤ C τ (y)−1 ,

(8)

for y ∈ M\γ near γ with C > 0. Next, consider D F(y) at y ∈ M\γ . Recall that the singular values s j (y), j = 1, 2, 3 of D F(y) are the square roots of the eigenvalues of (D F(y))t D F(y), where (D F)t is the transpose of D F. By (7), the singular values s j = s j (y), j = 1, 2, 3, of D F(y), numbered in increasing order, satisfy c1 ≤ s1 (y) ≤ c2 , c1 ≤ s2 (y) ≤ c2 , c1 c2 ≤ s3 (y) ≤ , τ (y) τ (y) where c1 , c2 > 0. The determinant of the matrix D F(y) can be computed in terms of its singular values by det (D F) = s1 s2 s3 . Later, we need the norm of the matrix det (D F(y))−1 D F(y). It satisfies by formula (8) 3 



−1 −1

det (D F(y)) D F(y) = sk sk−1 ≤ c1−2 . diag (s1 , s2 , s3 ) = max k=1

1≤ j≤3

k= j

(9) 4.2. Maxwell’s equations on the wormhole with the SHS lining. Let d V0 (x) denote the Euclidean volume element on N ⊂ R3 . Recall that N ⊂ R3 is an open set with boundary ∂ N = . Let d Vg be the Riemannian volume on (M, g). We consider below the map F : M\γ → N as a coordinate deformation. The map F induces for any differential
on N a form E = F ∗ E
in M\γ called the pull back of E
in F, see [2]. form E Next, we consider Maxwell equations with degenerate material parameters
ε and
µ on N with SH boundary conditions on . On M and N we define the permittivity and permeability by setting

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381

ε jk = µ jk = det (g)1/2 g jk , on M,

(10)

µ jk = det (
g )1/2
g jk , on N .
ε jk =


Here, and below, the matrix [g jk (x)] is the representation of the metric g in local coordinates, [g jk (x)] is the inverse of the matrix [g jk (x)], and det (g) is the determinant of [g jk (x)]. We note that the metric
g is degenerate near , and thus
ε and
µ, represented as matrices in the Euclidean coordinates, have elements that tend to infinity at , that is, the matrices
ε and
µ have a singularity near . Remark 1. Modifying the above construction by replacing M2 with M2 = S2flat × [l1 , l2 ] for appropriate l1 , l2 ∈ R and choosing F1 in an appropriate way, we can use local coordinates (
θ ,
t) on  such that the Euclidean distance along  of points (
θ ,
t1 ) and

(θ , t2 ) is proportional to |
t1 −
t2 |, and the metric
g in the Euclidean boundary normal coordinates (
θ ,
t,
τ ) associated to ∂ K has the form τ 2 + d
t 2 + 4
τ 2 d
θ 2, 0 <
τ< ds 2 = 4d


1 . 4

4.3. Finite energy solutions of Maxwell’s and the equivalence theorem. In the equations
=
=
x j and H
x j in the Euclidfollowing, we consider 1-forms E j E j d
j H j d
x 2,
x 3 ) of N ⊂ R3 . In the sequel, we use Einstein’s summation ean coordinates (
x 1,
convention and omit the sum signs. We use the Euclidean coordinates as we want to consider N with the differential structure inherited from the Euclidean space. We say
j and H
j are the (Euclidean) coefficients of the forms E
and H
, correspondingly. that E p We say that these coefficients are in L loc (N , d V0 ), 1 ≤ p < ∞, if

|E j (x)| p d V0 (x) < ∞, for all bounded measurable sets W ⊂ N . W


and H
are finite energy solutions of Maxwell’s Definition 1. We say that the 1-forms E equations in N with the soft-and-hard (SH) boundary conditions on  and the frequency k = 0,
= ik

, ∇ × H
= −ik

+ J
on N , ∇×E µ(x) H ε(x) E


η · E| = 0,
η · H | = 0,
and H
and 2-forms D
=

and

in N have coefficients in if 1-forms E εE B =
µH L 1loc (N , d V0 ) and satisfy

2

j E
k d V0 (x) < ∞,

E L 2 (W,|
=
ε jk E g |1/2 d V0 ) W


2 2
j H
k d V0 (x) < ∞ =
µ jk H

H 1/2 L (W,|
g|

d V0 ))

W

for all bounded measurable sets W ⊂ N , and finally,


− ik

) d V0 (x) = 0, ((∇ ×
h) · E h ·
µ(x) H N


+

− J
)) d V0 (x) = 0, ((∇ ×
e) · H e · (ik
ε(x) E N

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for all 1-forms
e and
h with coefficients in C0∞ (N ) that satisfy
η ·
e| = 0,
η ·
h| = 0,

(11)

where
η = ∂θ is the angular vector field that is tangential to . Below, we use for 1-forms E = E j d x j and H = H j d x j , given in local coordinates on M, the notations

(x 1 , x 2 , x 3 )

∇ × E = d H, ∇· (εE) = d ∗ E, ∇· (µH ) = d ∗ H, where d is the exterior derivative and ∗ is the Hodge operator on (M, g), cf. formula (10). We have the following “equivalent behavior of electromagnetic fields on N and M” result, analogous to the results of [3] for cloaking.
and H
be 1-forms with coefficients Theorem 1. Let E and H be 1-forms on M\γ and E 1 ∗ ∗


in L loc (N , d V0 ) such that E = F E, H = F H . Let J and J = F ∗ J
be 2-forms with smooth coefficients in N and M\γ that are supported away from  and γ . Then the following are equivalent:
and H
satisfy Maxwell’s equations with SHS boundary condi1. On N , the 1-forms E tions in the sense of Definition 1. 2. On M, the forms E and H can be extended on M so that they are classical solutions E and H of Maxwell’s equations, ∇ × E = ikµH, in M, ∇ × H = −ikεE + J, in M. Proof. Assume first that E and H satisfy Maxwell’s equations on M with source J supported away from γ . Then E and H are C ∞ smooth near γ .
H
and 2-form J
on N by Using F −1 : N → M\γ we define the 1-forms E,
= (F −1 )∗ E, H
= (F −1 )∗ H , and J
= (F −1 )∗ J. These fields satisfy Maxwell’s E equations in N ,
= ik

, ∇ × H
= −ik

+ J
in N . ∇×E µ(x) H ε(x) E

(12)

Now, writing E = E j (x)d x j on M near γ , we see using the transformation rule for
= (F −1 )∗ E is in local coordinates differential 1-forms that the form E
= E
j (
x )d
x j, E


j (
E x ) = (D F −1 )kj (
x ) E k (F −1 (
x )),
x ∈ N.

(13)


H
are Using the smoothness of E and H near γ on M and formulae (7), we see that E, forms on N with L 1loc (N , d V0 ) coefficients. Moreover,

ε(x) E(x) = det (D F(y))−1 D F(y)ε(y)D F(y)t (D F(y)t )−1 E(y) = det (D F(y))−1 D F(y)ε(y)E(y),
=

and

are ε E, B=
µH where x ∈ N , y = F −1 (x) ∈ M\γ . Formula (9) shows that D 2-forms on N with L 1loc (N , d V0 ) coefficients. Let (t) ⊂ N be the t-neighbourhood of  in the
g -metric. Note that for small t > 0 the set (t) is the Euclidean (t/2)-neighborhood of ∂ K . Denote by ν be the unit exterior
ge =

Euclidean normal vector of ∂(t) and the Euclidean inner product by (
η, E) η · E.

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Formulas (7) and (13) imply that the angular components satisfy
≤ Ct, x ∈ ∂(t), |
η · E| and
≤ C, x ∈ ∂(t) |
ζ · E| with some C > 0. Thus denoting by d S the Euclidean surface area on ∂(t), Stokes’ formula, formula (12), and the identity ν ×
ξ = ±
η yield


− ik

) d V0 (x) ((∇ ×
h) · E h ·
µH N


− ik

) d V0 (x) = lim ((∇ ×
h) · E h ·
µH t→0 N \(t)


·
= − lim (ν × E) h d S(x) t→0 ∂(t)


η + (


= − lim ν × ((
η · E)
ζ · E) ζ) ·
h d S(x) =0

t→0 ∂(t)

for a test function
h satisfying formula (11).
shows that 1-forms E
and H
satisfy Maxwell’s equations with Similar analysis for H SH boundary conditions in the sense of Definition 1.
and H
form a finite energy solution of Maxwell’s equations on Next, assume that E (N , g) with a source J
supported away from , implying in particular that
j E
k ∈ L 1 (W, d V0 ),

j H
k ∈ L 1 (W, d V0 )
ε jk E µ jk H where W = F(U \γ ) ⊂ N and U ⊂ M is a relatively compact open neighbourhood of
H = F∗ H
, and J = F ∗ J
on M\γ . Therefore γ , supp ( J
) ∩ W = ∅. Define E = F ∗ E, we conclude that ∇ × E = ikµ(x)H, ∇ × H = −ikε(x)E + J, in M\γ and ε jk E j E k ∈ L 1 (U \γ , d Vg ), µ jk H j Hk ∈ L 1 (U \γ , d Vg ). As representations of ε and µ, in local coordinates of M, are matrices that are bounded from above and below, these imply that ∇ × E ∈ L 2 (U \γ , d Vg ), ∇ × H ∈ L 2 (U \γ , d Vg ), ∇· (εE) = 0, ∇· (µH ) = 0, in U \γ . Let E e , H e ∈ L 2 (U, d Vg ) be measurable extensions of E and H to γ . Then ∇ × E e − ikµ(x)H e = 0, in U \γ , ∇ × E e − ikµ(x)H e ∈ H −1 (U, d Vg ), ∇ × H e + ikε(x)E e = 0, in U \γ , ∇ × H e + ikε(x)E e ∈ H −1 (U, d Vg ),

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where H −1 (U, d Vg ) is the Sobolev space with smoothness (−1) on (U, g). Since γ is a subset with (Hausdorff) dimension 1 in the 3-dimensional domain U , it has zero capacitance. Thus, the Lipschitz functions on U that vanish on γ are dense in H 1 (U ), see [12]. Therefore, there are no non-zero distributions in H −1 (U ) supported on γ . Hence we see that ∇ × E e − ikµ(x)H e = 0, ∇ × H e + ikε(x)E e = 0 in U. This also implies that ∇· (εE e ) = 0, ∇· (µH e ) = 0 in U, which, by elliptic regularity, imply that E e and H e are C ∞ smooth in U . In summary, E and H have unique continuous extensions to γ , and the extensions are classical solutions to Maxwell’s equations.   Acknowledgements. A.G. was supported by NSF, M.L. by CoE-program 213476 of Academy of Finland, G.U. was partially supported by the NSF and a Walker Family Endowed Professorship.

References 1. Eleftheriades, G., Balmain, K.: Negative-Refraction Metamaterials. New York IEEE Press (Wiley-Interscience), 2005 2. Frankel, T.: The geometry of physics. Cambridge University Press, Cambridge, 1997 3. Greenleaf, A., Kurylev, Y., Lassas, M., Uhlmann, G.: Full-wave invisibility of active devices at all frequencies. Commun. Math. Phys. 275(3), 749–789 (2007) 4. Greenleaf, A., Kurylev, Y., Lassas, M., Uhlmann, G.: Electromagnetic wormholes and virtual magnetic monopoles from metamaterials. Phys. Rev. Lett. 99, 183901 (2007) 5. Greenleaf, A., Lassas, M., Uhlmann, G.: The Calderón problem for conormal potentials, I: Global uniqueness and reconstruction. Comm. Pure Appl. Math 56(3), 328–352 (2003) 6. Greenleaf, A., Lassas, M., Uhlmann, G.: Anisotropic conductivities that cannot detected in EIT. Physiological Measurement (special issue on Impedance Tomography), 24, 413–420 (2003) 7. Greenleaf, A., Lassas, M., Uhlmann, G.: On nonuniqueness for Calderón’s inverse problem. Math. Res. Let. 10(5–6), 685–693 (2003) 8. Hänninen, I., Lindell, I., Sihvola, A.: Realization of generalized Soft-and-Hard Boundary. ProgrIn Electromag. Res. PIER 64, 317–333 (2006) 9. Hawking, S., Ellis, G.: The Large Scale Structure of Space-Time. Cambridge: Cambridge Univ. Press, 1973 10. Kildal, P.-S.: Definition of artificially soft and hard surfaces for electromagnetic waves. Electron. Lett. 24, 168–170 (1988) 11. Kildal, P.-S.: Artificially soft-and-hard surfaces in electromagnetics. IEEE Trans. Ant. and Propag. 10, 1537–1544 (1990) 12. Kilpeläinen, T., Kinnunen, J., Martio, O.: Sobolev spaces with zero boundary values on metric spaces. Potential Anal. 12(3), 233–247 (2000) 13. Kohn, R., Shen, H., Vogelius, M., Weinstein, M.: Cloaking via change of variables in electrical impedance tomography. Inverse Problems 24, 015016 (2008) 14. Kurylev, Y., Lassas, M., Somersalo, E.: Maxwell’s equations with a polarization independent wave velocity: Direct and inverse problems. J. Math. Pures Appl. 86, 237–270 (2006) 15. Lassas, M., Taylor, M., Uhlmann, G.: On determining a non-compact Riemannian manifold from the boundary values of harmonic functions. Comm. Geom. Anal. 11, 207–222 (2003) 16. Leonhardt, U.: Optical Conformal Mapping. Science 312, 1777–1780 (2006) 17. Milton, G., Briane, M., Willis, J.: On cloaking for elasticity and physical equations with a transformation invariant form. New J. Phys. 8, 248 (2006) 18. Pendry, J.B., Schurig, D., Smith, D.R.: Controlling Electromagnetic Fields. Science 312, 1780–1782 (2006) 19. Pendry, J.B., Schurig, D., Smith, D.R.: Calculation of material properties and ray tracing in transformation media. Optics Express 14, 9794 (2006)

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20. Schurig, D., Mock, J., Justice, B., Cummer, S., Pendry, J., Starr, A., Smith, D.: Metamaterial electromagnetic cloak at microwave frequencies. Science 314, 977–980 (2006) 21. Weder, R.: A rigorous time-domain analysis of full–wave electromagnetic cloaking (Invisibility). http://arxiv.org.abs/0704.0248, 2007 22. Visser, M.: Lorentzian Wormholes. New York: AIP Press, 1997 Communicated by P. Constantin

Commun. Math. Phys. 281, 387–400 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0475-8

Communications in

Mathematical Physics

Tangent Euler Top in General Relativity
José Natário Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal. E-mail: [email protected] Received: 11 April 2007 / Accepted: 9 October 2007 Published online: 25 April 2008 – © Springer-Verlag 2008

Abstract: We define the tangent Euler top in General Relativity through a constrained Lagrangian on the orthonormal frame bundle. The corresponding motions are studied to various degrees of approximation, the lowest of which is shown to yield the Mathisson-Papapetrou equations. 0. Introduction As is well known [Born09,Her10,Noe10,Ros47,ST54,NJ59,PW62,Boy65,WE66, WE67] there are two separate sets of problems preventing the consideration of generic rigid bodies in General Relativity. The first is the lack of a global notion of simultaneity, which renders the notion of distance between two particles undefined. Associated to this is the inexistence of arbitrarily fast signals, which would be necessary for a rigid body to keep its exact shape. The second problem is that even if a family of spacelike slices is somehow singled out to define a notion of simultaneity, it is in general impossible to keep the mutual distances of more than four particles constant, due to the varying curvature of the slices. Nevertheless, it is many times useful to consider approximate notions of rigidity in General Relativity [Dix70,ER77,TG83,Voi88,XTW04]. These are usually derived from multipole or post-Newtonian expansions, which can be quite formidable. A simpler method for obtaining the (approximate) motion of a small (approximately) rigid body from a finite-dimensional Lagrangian, mirroring what is done in Newtonian mechanics, would therefore be desirable. A possibility (suggested by Turakulov [Tur] in the context of Newtonian mechanics on a curved space) is to approximate the motion of a rigid body by the motion of
This work was partially supported by the Fundação para a Ciência e a Tecnologia through the Program POCI 2010/FEDER and by the grant POCI/MAT/58549/2004.

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an orthonormal frame (determined by a naturally defined Lagrangian). This avoids the curvature problem. The simultaneity problem can be overcome by adding the natural constraint that the frame’s timelike unit vector should be tangent to the motion of the base point (related ideas in flat spacetime can be found in [HR74,Ell82,Tak82]). The consequences of this model are explored in the present paper. Section 1 begins by briefly reviewing the Euler top, in order to set up the notation. The Newtonian tangent Euler top on a curved space, which is a toy model for its relativistic counterpart, is investigated in Sect. 2. This generalizes [Tur], where the discussion was restricted to spherical tops. The relativistic tangent Euler top is defined in Sect. 3, as well as a simpler Lagrangian determining the spinning motion only. An approximation scheme for these is devised in Sect. 4, where the quadratic and cubic approximations are examined. The first (but not the second) is seen to lead to the usual Euler top motion with respect to a Fermi-Walker transported frame. Section 5 studies the exact spinning motion when the base point is moving along a geodesic. The corresponding Lagrangian is shown to yield a (new) completely integrable system on S O(3). Finally, Sect. 6 analyzes the full Lagrangian in the quadratic approximation. It is found that the motion of the frame’s base point is determined by the Mathisson-Papapetrou equation of motion subject to the Mathisson-Pirani spin supplementary condition [Mat37,Pap51,Pir56,MTW73,Sem99,Ste04]. The present paper thus succeeds in deriving the Mathisson-Papapetrou equation from a Lagrangian, defined on the (finite-dimensional) orthonormal frame bundle, which is the natural generalization of the Lagrangian for a free rigid body in Newtonian mechanics. This is quite different from other approaches inspired on variational principles [K72,YB93,Por06], where the authors start with the Mathisson-Papapetrou equation and proceed to construct an action (generally with a much less clear physical meaning). Moreover, this Lagrangian formulation allows for the study of the rotational motion (which can be analyzed for arbitrary motions of the center of mass), instead of considering only the evolution of the angular momentum tensor. This leads to interesting new mechanical systems on S O(3). We use the conventions of [MTW73] (including the Einstein summation convention), except for the meaning of the indices, which is the following: latin indices will always be associated to an orthonormal frame; they will be spacetime indices (ranging from 0 to 3) if they are from the beginning of the alphabet (a, b, . . .), and space indices (ranging from 1 to 3) if the are from the middle of the alphabet (i, j, . . .). Similarly, greek indices will always be associated to a coordinate system; they will be spacetime indices if the are from the beginning of the alphabet (α, β, . . .), and space indices if the are from the middle of the alphabet (µ, ν, . . .)1 . As is usual, we will not worry about the vertical position of space indices on orthonormal frames. 1. Euler top We shall use this brief review of the Euler top to set up our notation. For simplicity we will consider only the case where the Euler top spins about its center of mass. Definition 1.1. The reference configuration of an Euler top is a positive finite Borel measure m on R3 , not supported on any 1-dimensional subspace, and such that

ξ dm = 0,
ξ
2 dm < +∞. R3

R3

1 This complication is unfortunate but inevitable, as we work in the orthonormal frame bundle and often distinguish between space and time.

Tangent Euler Top in General Relativity

389

The mass of the Euler top is M = m(R3 ), and its Euler tensor is the symmetric matrix
ξ i ξ j dm. Ii j = R3

A motion of the Euler top is an Euler-Lagrange curve for the Lagrangian L : T S O(3) → R given by2
1 ˙ ˙ , Sξ ˙  dm = 1 S˙i j I jk S˙ik = 1 tr( S˙ I S˙ t ). L(S, S) =  Sξ 2 R3 2 2 As is well known [Ar97,MR99], the Euler top is completely integrable: in addition to the total energy H = L, one can obtain commuting first integrals from the conservation of angular momentum. In fact, the Euler-Lagrange equations themselves are equivalent to the conservation of angular momentum. Theorem 1.2. If S : R → S O(3) is a motion of the Euler top, then the angular momentum matrix σ = S(I A + AI )S t ∈ so(3) is constant, where A : R → so(3) is such that S˙ = S A. Proof. S O(3) acts on itself by left multiplication, and L = − 21 tr(AI A) is clearly S O(3)-invariant. By Noether’s theorem, the function FL(X B ) is conserved along the motion for all B ∈ so(3), where FL is the fiber derivative of the Lagrangian and X B ∈ X(S O(3)) is the infinitesimal action of B ∈ so(3). Now  d  (X B ) S = exp(t B)S = B S = SS t B S, dt t=0 and hence 1 1 1 FL(X B ) = − tr(S t B S I A) − tr(AI S t B S) = − tr(S(I A + AI )S t B). 2 2 2 The formula C, D = − 21 tr(C D) defines an inner product on so(3). Since σ, B is conserved for all B ∈ so(3), so is σ .   Remark 1.3. (1) The matrix σ is related to the angular momentum vector σ through σξ = σ × ξ for all ξ ∈ R3 , i.e. σik = εi jk σ j . (2) We have σ = S S t with  = I A + AI ∈ so(3). Conservation of σ is equivalent to the Euler equation [Ar97,MR99,Oli02] ˙ =  A − A.  2 Here we use the usual convention that S˙ can either mean the time derivative of a curve S : R → S O(3) or a tangent vector on the tangent bundle T S O(3).

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2. Tangent Euler top on a curved space Let O M be the bundle of positive orthonormal frames on an oriented3 3-dimensional Riemannian manifold (M, ·, ·). A curve E : R → O M is given by E(t) = (c(t), E 1 (t), E 2 (t), E 3 (t)), where c : R → M is a curve on M and E 1 , E 2 , E 3 : R → T M are vector fields along c which form a positive orthonormal frame at c(t) for each t ∈ R. Definition 2.1. A motion of a tangent Euler top on (M, ·, ·) is an Euler-Lagrange curve for the Lagrangian L : T O M → R given by
  ˙ =1 L(E, E) c˙ + ξ i ∇c˙ E i , c˙ + ξ j ∇c˙ E j dm, 2 R3 where ∇ is the Levi-Civita connection and m is the reference configuration of an Euler top. Remark 2.2. (1) Notice that if (M, ·, ·) is R3 with the Euclidean metric then this Lagrangian is just the Lagrangian for a free rigid body. (2) Physically, one can expect a tangent Euler top to represent the motion of an approximately rigid extended body in the limit in which its size is much smaller than the local radius of curvature. Theorem 2.3. The motions of a tangent Euler top on (M, ·, ·) are as follows: the frame {E 1 , E 2 , E 3 } rotates exactly as an Euler top with respect to any frame { Eˆ 1 , Eˆ 2 , Eˆ 3 } which is parallel-transported along the motion of the base point; the base point moves according to the equation4 M

 D c˙ 1  ˆ i j σi j = 0, + ι(c) ˙

dt 2

(1)

ˆ i j are the curvature forms for the frame { Eˆ 1 , Eˆ 2 , Eˆ 3 },  : T ∗ M → T M is the where

isomorphism induced by the metric and σ ∈ so(3) is the angular momentum matrix. ∼ U × S O(3) by choosing a local orthoProof. We define a local trivialization O M|U = normal frame { Eˆ 1 , Eˆ 2 , Eˆ 3 } ⊂ X(U ) on a sufficiently small open set U ⊂ M. For this trivialization we have E i (t) = S ji (t)( Eˆ j )c(t) ⇒ ∇c˙ E i = S˙ ji Eˆ j + S ji ωˆ k j (c) ˙ Eˆ k , j

where ωˆ i are the connection forms associated to our local frame. The Lagrangian can therefore be written as L = T + K + C + F, 3 The hypothesis of orientability is not crucial, as one can always pass to the orientable double cover. 4 As is usual, D = ∇ is the covariant derivative along the curve c : R → M. c˙ dt

Tangent Euler Top in General Relativity

391

where


1 1 ˙ c; ˙ c, ˙ c ˙ dm = M c, 2 R3 2
1 1 K = ξ i S˙ki ξ j S˙l j  Eˆ k , Eˆ l  dm = tr( S˙ I S˙ t ); 2 R3 2
C= ξ i S˙ki ξ j Sl j ωˆ ml (c) ˙ Eˆ k , Eˆ m  dm = tr( S˙ I S t ωˆ t (c)); ˙ R3
1 1 F= ˆ c)S ˙ I S t ωˆ t (c)). ξ i Ski ωˆ l k (c)ξ ˙ j Sm j ωˆ nm (c) ˙ Eˆ l , Eˆ n  dm = tr(ω( ˙ 2 R3 2 T =

Our task is now to write the Euler-Lagrange equations. For the S O(3) part, we notice ˆ c) ˙ = 0, that if the frame { Eˆ 1 , Eˆ 2 , Eˆ 3 } happens to be parallel transported along c then ω( and hence only terms coming from the rotational kinetic energy K survive in the EulerLagrange equations. Therefore {E 1 , E 2 , E 3 } rotates with respect to { Eˆ 1 , Eˆ 2 , Eˆ 3 } exactly as an Euler top with reference configuration m. For the motion of the base point, there will be terms coming from both the translational kinetic energy T and the Coriolis term C (but not from the frame term F, which is quadratic in ω( ˆ c)). ˙ As is well known,   ∂T d ∂T D x˙ ν − µ = M gµν , µ dt ∂ x˙ ∂x dt where gµν is the matrix of the metric. Since the trace of the product of symmetric and anti-symmetric matrices is zero, we can write ˙ = C = tr(S AI S t ωˆ t (c)) Therefore d dt



∂C ∂ x˙ µ

 −

1 1 1 tr(S(AI − I At )S t ωˆ t (c)) ˙ = tr(σ ωˆ t (c)) ˙ = σi j ωˆ i jµ x˙ µ . 2 2 2

∂C 1 1 ˆ i jνµ x˙ ν , = σi j (∂ν ωˆ i jµ − ∂µ ωˆ i jν )x˙ ν = σi j

µ ∂x 2 2

ˆ i = d ωˆ i + ωˆ i ∧ ωˆ k are the curvature forms associated to { Eˆ 1 , Eˆ 2 , Eˆ 3 }. Conwhere

j j k j sequently, the equations of motion for the base point are M

D x˙ µ 1 ˆ µ x˙ ν = 0. + σi j

i jν dt 2

  Remark 2.4. (1) If (M, ·, ·) is R3 with the Euclidean metric then the curvature forms vanish and we obtain the familiar result that the center of mass of a free rigid body moves with constant velocity. (2) We can write Eq. (1) as M

D x˙ µ 1 µ = R νκλ x˙ ν s κλ , dt 2

where s = σi j Eˆ i ⊗ Eˆ j

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is the angular momentum tensor. As noticed by Turakulov [Tur], this equation is similar to the Mathisson-Papapetrou equation for a spinning particle in General Relativity [Mat37,Pap51]. The fact that the angular momentum matrix is constant on the parallel transported frame { Eˆ 1 , Eˆ 2 , Eˆ 3 } is translated into the fact that the angular momentum tensor satisfies the parallel transport equation Ds µν = 0. dt 3. Tangent Euler top in General Relativity Let O M be the bundle of positive future-pointing orthonormal frames on an oriented, time-oriented5 4-dimensional Lorentzian manifold (M, ·, ·). An admissible curve E : R → O M is given by E(t) = (c(t), E 0 (t), E 1 (t), E 2 (t), E 3 (t)), where c : R → M is a future-directed timelike curve on M parameterized by its proper time and E 0 , E 1 , E 2 , E 3 : R → T M are vector fields along c which form a positive orthonormal frame at c(t) satisfying E 0 (t) = c(t) ˙ for each t ∈ R. Definition 3.1. A motion of a tangent Euler top on (M, ·, ·) is a (constrained) EulerLagrange curve for the Lagrangian L : T O M → R given by
  1 2  ˙ = L(E, E)  c˙ + ξ i ∇c˙ E i , c˙ + ξ j ∇c˙ E j  dm, R3

where ∇ is the Levi-Civita connection and m is the reference configuration of an Euler top, chosen among all admissible curves, with unspecified final time. Remark 3.2. (1) This is the natural generalization of the Lagrangian for a (Newtonian) tangent Euler top on a curved space: one simply replaces the squared velocity by its relativistic length. (2) Restricting the action to admissible curves means considering bodies which are rigid as seen by an observer comoving with the base point. (3) We must leave the final time unspecified because we insisted that admissible curves should be parameterized by the proper time of the base point, which needs not be the same for all admissible curves with given endpoints. (4) A tangent Euler top in General Relativity can therefore be physically interpreted as the motion of an extended body which is approximately rigid as seen by an observer placed at its center of mass, in the limit in which the body’s size is much smaller than the local radius of curvature. A simpler problem is obtained by focussing on the spinning motion. To do this, we regard the motion c : R → M of the base point as given (which may or may not be the one arising from Definition 3.1) and look for the positive orthonormal frame {c(t), ˙ E 1 (t), E 2 (t), E 3 (t)} along c which maximizes the action. Choosing a positive orthonormal Fermi-Walker transported frame {c(t), ˙ Eˆ 1 (t), Eˆ 2 (t), Eˆ 3 (t)} along c, so that ∇c˙ c(t) ˙ = aˆ i (t) Eˆ i (t),

∇c˙ Eˆ i (t) = aˆ i (t)c(t), ˙

5 Again this is not a crucial hypothesis, as one can always pass to the time-orientable double cover.

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we have E i (t) = S ji (t)( Eˆ j )c(t) ⇒ ∇c˙ E i = S˙ ji Eˆ j + S ji a j c˙ for some curve S : R → S O(3). Therefore   c˙ + ξ i ∇c˙ E i , c˙ + ξ j ∇c˙ E j = −1 + S˙ki ξ i S˙k j ξ j − aˆ k Ski ξ i aˆ l Sl j ξ j − 2aˆ j S ji ξ i . Definition 3.3. A spinning motion of a tangent Euler top on (M, ·, ·) is an (unconstrained) Euler-Lagrange curve for the Lagrangian L : T S O(3) × R → R given by
 1  ˙ , Sξ ˙  + Sξ , aˆ (t)2 + 2Sξ , aˆ (t) 2 dm, ˙ t) = L(S, S, 1 −  Sξ R3

where m is the reference configuration of an Euler top and aˆ : R → R3 represents the proper acceleration on a Fermi-Walker transported frame. 4. Approximate Spinning Motions Expanding the integrand in the Lagrangian for a spinning motion in power series of ξ and integrating we can write L = M + L (2) + L (3) + . . . , where L (n) is the integral of the term of order n in ξ . Definition 4.1. An approximate spinning motion of order n of a tangent Euler top is an Euler-Lagrange curve of the truncated Lagrangian L = M + L (2) + . . . + L (n) . Theorem 4.2. The approximate spinning motions of order 2 of a tangent Euler top are exactly the motions of an Euler top with the same reference configuration. Proof. One just needs to check that L (2) = −  


1 ˙ , Sξ ˙  dm.  Sξ 2 R3

Remark 4.3. Notice that the proper acceleration aˆ does not appear in the expression of L (2) : in this approximation, the spinning motion completely decouples from the translational motion. This is similar to what happens in the Newtonian case (and is, of course, the principle underlying the operation of a gyroscope). Proposition 4.4. Generically, the approximate spinning motions of order 3 of a tangent Euler top are not the motions of an Euler top with the same reference configuration. Proof. One just needs to check that
1 ˙ , Sξ ˙ Sξ , aˆ  dm, L (3) =  Sξ 2 R3 which generically is not a total time derivative. L (3)

 

Remark 4.5. can be thought of as (minus) the potential energy corresponding to the weight of the rotational kinetic energy in the constant “gravitational field” −ˆa induced by the acceleration of the center of mass.

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5. Spinning Motions Along Geodesics If the spinning motion is along a timelike geodesic, we have aˆ = 0, and the Lagrangian becomes S O(3)-invariant. In this case, we can apply Noether’s Theorem. Theorem 5.1. If S : R → S O(3) is a spinning motion of a tangent Euler top along a timelike geodesic of (M, ·, ·), then σ = S S t ∈ so(3) is constant, where
=

(Aξ )ξ t − ξ (Aξ )t

R3

1

|1 − Aξ , Aξ | 2

dm ∈ so(3)

and A : R → so(3) is such that S˙ = S A. Proof. We can write the Lagrangian as



 1 t 2  ˙ ˙ ˙ 1 − tr ( Sξ )( Sξ ) dm = L(S, S) = R3

R3



 1 1 − tr Aξ ξ t At  2 dm.

By Noether’s theorem, FL(X B ) is conserved along the motion for all B ∈ so(3), where X B ∈ X(S O(3)) is the infinitesimal action of B. Since


tr S t B Sξ ξ t At + Aξ ξ t S t B t S 1 B dm FL(X ) = − 

 1 2 R3 1 − tr Aξ ξ t At  2  
1 Aξ ξ t − ξ ξ t At = tr B S dm S t , 1 3 2 R |1 − Aξ , Aξ | 2 we see that − 21 tr(Bσ ) = B, σ  is conserved for all B ∈ so(3), and so is σ .

 

Remark 5.2. (1) The angular momentum vector σ , related to the matrix σ through σξ = σ × ξ for all ξ ∈ R3 , can be seen to have the familiar expression
σ =

˙ ) (Sξ ) × ( Sξ dm. ˙ , Sξ ˙ | 21 R3 |1 −  Sξ

(2) This system is completely integrable: in addition to the total energy
1 dm, K = −H = −FL(A) + L = 1 3 R |1 − Aξ , Aξ | 2 one can obtain commuting first integrals from the conservation of angular momentum. In fact, the Euler-Lagrange equations themselves are equivalent to the conservation of angular momentum. The images of the motions A : R → so(3) can be obtained from the intersections of the level surfaces of the functions K (A) and (A), (A) = − 21 tr((A)2 ).

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(3) We have L(A) = M −

1 1 Ii j Aki Ak j − Ii jkl Ami Am j Ank Anl − . . . , 2 8

where


Ii jkl =

R3

ξ i ξ j ξ k ξ l dm, . . . .

Therefore the Euler tensor is not enough to characterize a relativistic tangent Euler top: the higher multipoles are also required (and become more important as the spinning motion becomes more relativistic). 6. Quadratically Approximate Motions We can extend Definition 4.1 to the general motion of a tangent Euler top. Definition 6.1. A quadratically approximated motion of a tangent Euler top on (M, ·, ·) is an Euler-Lagrange curve for the Lagrangian L : T O M → R given by ˙ =M− L(E, E)

1 Ii j ∇c˙ E i , E k ∇c˙ E j , E k , 2

chosen among all admissible curves, with unspecified final time. Theorem 6.2. Quadratically approximated motions of a tangent Euler top on (M, ·, ·) are as follows: {E 1 , E 2 , E 3 } rotates exactly as an Euler top with respect to { Eˆ 1 , Eˆ 2 , Eˆ 3 }, where { Eˆ 0 , Eˆ 1 , Eˆ 2 , Eˆ 3 } : R → O M is Fermi-Walker transported along the motion of the base point and satisfies Eˆ 0 = c; ˙ the base point moves according to the equation 

 d D c˙ ˆ D c˙ 1  ˆ i j σi j + Eˆ i σi j (2) + ι(c) ˙

, E j = 0, (M + K ) dt 2 dt dt ˆ ab are the curvature forms for the frame where K is the rotational kinetic energy,

 ∗ ˆ ˆ ˆ ˆ { E 0 , E 1 , E 2 , E 3 }, : T M → T M is the isomorphism induced by the metric and σ ∈ so(3) is the angular momentum matrix. Proof. The spinning motion with respect to a Fermi-Walker transported frame was already shown to be that of an Euler top in Sect. 4. To compute the motion of the base point we define a local trivialization O M|U ∼ = U × S O ↑ (3, 1) by choosing a local orthonormal frame { Eˆ 0 , Eˆ 1 , Eˆ 2 , Eˆ 3 } ⊂ X(U ) on a sufficiently small open set U ⊂ M. For this trivialization we have E i (t) = S ai (t)( Eˆ a )c(t) ⇒ ∇c˙ E i = S˙ ai Eˆ a + S ai ωˆ ba (c) ˙ Eˆ b , where ωˆ ba are the connection forms associated to our local frame, and hence the Lagrangian can be written as   1  L = M − Ii j ωˆ ab (c)S ˙ ak S bi + Aki ωˆ cd (c)S ˙ ck S d j + Ak j , 2

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where A = (Aab ) ∈ so(3, 1) is defined by S˙ = S A ⇔ S˙ ab = S ac Ac b . Notice that the conditions S ∈ O(3, 1) and A ∈ so(3, 1) are conveniently written by raising and lowering indices as (S −1 )ab = Sb a and Aab = −Aba . The best way to deal with the constraint while avoiding introducing coordinates on S O ↑ (3, 1) is to to set up our problem as an optimal control problem (see for instance [Law63,Pin93]). As state variables we choose the local coordinates x α of the base point, the components Sa b of the frame vectors and the action θ . As control variables we simply choose 6 independent components of the anti-symmetric matrix (Aab ). The state variables depend on the control variables through the solution of the Cauchy problem ⎧ α ˆα a ⎪ ⎨x˙ = E a S 0 a S˙ b = S ac Ac b    ⎪ ⎩θ˙ = M − 1 I ωˆ α S f Sd Se + A α Sc Sa Sb + A ˆ ˆ ω ˆ E E abα c ik deα f 0 j k jk . 0 i k 2 ij Our objective is to maximize θ1 = θ (t1 ), i.e. minimize  = −θ1 , where the final time t1 is not fixed. To do so, we introduce the multipliers λα , ab , λ (one for each differential equation) and set up the Hamiltonian function H = λα Eˆ aα S a0 + ab S ac Ac b    1  β f +λ M − Ii j ωˆ abα Eˆ cα S c0 S ai S bk + Aik ωˆ deβ Eˆ f S 0 S d j S ek + A jk . 2 According to the Pontryagin Maximum Principle, we must select the values of the control variables which maximize H . The minimizing curve will then be obtained by solving the Hamilton equations ⎧ ∂H ˙ ⎪ ⎨λ α = − ∂ x α ˙ ab = − ∂ Ha  ∂S b ⎪ ⎩˙ λ = −∂H ∂θ

together with the Cauchy problem. The multiplier λ is clearly constant; since θ1 is not specified, we must select λ(t1 ) = −

∂ =1 ∂θ1

(hence λ ≡ 1). Moreover, H is constant along the solution. Since t1 is not fixed, we must have H ≡ 0. Because we already know what the spinning motion will be, we do not have to write out all the minimum equations. Also, we shall assume that the trivializing frame { Eˆ 0 , Eˆ 1 , Eˆ 2 , Eˆ 3 } happens to be Fermi-Walker transported along the minimizing curve, ˙ As a consequence, we will have with Eˆ 0 = c. ωˆ i jα Eˆ 0α = 0,

ωˆ i0α Eˆ 0α = aˆ i ,

S a0 = δ a0

along this curve. In particular, S˙ a0 = 0 ⇔ S ac Ac 0 = 0 ⇔ Aa0 = 0 ⇔ Aa0 = −A0a = 0 along the minimizing curve.

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We now write the relevant minimum conditions. Maximizing H with respect to the control variables A0i yields ∂H = 0 ⇔ ai S a0 − a0 S ai = 0 ⇔ 0i − a0 S ai = 0. ∂ A0i

(3)

We have the Hamilton equations ˙l0 = − 

∂H ∂ Sl 0

= −λα Eˆ lα + l b A0b    β f −Ii j ωˆ abα Eˆ lα S ai S bk ωˆ deβ Eˆ f S 0 S d j S ek + A jk = −λα Eˆ lα

(4)

and ˙ 0l = − 

   ∂H β f d e b l α c b ˆ ˆ ω ˆ ω ˆ = − A − I S S S S S + A E E l j 0bα deβ jk c 0 0 k b f 0 j k ∂ S 0l    β f −Ii j ωˆ a0α Eˆ cα S c0 S ai ωˆ deβ Eˆ f S 0 S d j S el + A jl = 0i Ali + Il j aˆ m S mk A jk − Ii j aˆ m S mi A jl .

(5)

Differentiating (3) with respect to t and using (4), (5), and again (3) yields λα Eˆ αj S ji = (Ii j A jk + Ai j I jk )Smk aˆ m , from which one readily obtains λα Eˆ lα = σlm aˆ m .

(6)

We must also consider the Hamilton equation λ˙ µ = −

  ∂H ˆ 0α + Ii j ∂µ ωˆ abα Eˆ 0α S a S b A jk = −λ ∂ E α µ i k ∂xµ   +Ii j ωˆ abα ∂µ Eˆ 0α S a S b A jk . i

k

(7)

We now use our freedom in the choice of local coordinates to select Fermi normal coordinates (t, x 1 , x 2 , x 3 ), defined by the parameterization ϕ(t, x 1 , x 2 , x 3 ) = expc(t) (x 1 Eˆ 1 + x 2 Eˆ 2 + x 3 Eˆ 3 ), where exp p : T p M → M is the geodesic exponential map. To first order in x we can then choose6 ∂ Eˆ 0 = (1 − aˆ i x i ) , ∂t

∂ Eˆ i = i ∂x

6 Here we use latin indices i, j, . . . for the coordinates, as they are associated to the Fermi-Walker transported frame. By the same token, the coordinate t coincides with the proper time t along the minimizing curve.

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(see [Poi04]). In particular, ∇ Eˆ i Eˆ j = 0 for x = 0, from where we can deduce that ωˆ i j = 0 and ωˆ 0i ( Eˆ j ) = 0 along the minimizing curve. Equation (7) then becomes   λ˙ n = λt aˆ n + Ii j ∂n ωˆ lmα Eˆ 0α Sli Smk A jk . (8) The quantity λt in this equation can be obtained from H = 0 ⇔ λt + k j Ski Ai j + M −

1 Ii j Aik A jk = 0. 2

(9)

Maximizing H with respect to Ai j (with i < j, say) yields further relations: ∂H j = 0 ⇔ a S ai − ai S a j − Iik Ak j + I jk Aki = 0 ∂ Ai j ⇔ k j Ski − ki Sk j = Iik Ak j − I jk Aki . These can be used to rewrite the second term in (9), which is tr(S At ) =

1 1 1 tr(S At + At S t ) = tr(t S A − S t A) = tr((t S − S t )A). 2 2 2

After substitution we obtain λt = −M −

1 Ii j Aik A jk = −(M + K ), 2

where K is the rotational kinetic energy of the Euler top. Because ∂n ωˆ lmα Eˆ 0α is anti-symmetric in l, m, the quantity multiplying it in (8) can be replaced by 1 1 1 (Ii j Sli Smk A jk − Ii j Smi Slk A jk ) = Sli (Ii j A jk − Ik j A ji )Smk = σlm . 2 2 2 Consequently, Eq. (8) can be written as 1 λ˙ n = −(M + K )aˆ n + ∂n ωˆ lmα Eˆ 0α σlm . 2

(10)

The curvature forms are given by ˆ ba = d ωˆ ba + ωˆ bc ∧ ωˆ ca ⇒

ˆ i j = d ωˆ i j − ωˆ i0 ∧ ωˆ 0 j + ωˆ ik ∧ ωˆ k j .

In components, ˆ i jnt = ∂n ωˆ i jt − ∂t ωˆ i jn − ωˆ i0n ωˆ 0 jt + ωˆ i0t ωˆ 0 jn + ωˆ ikn ωˆ k jt − ωˆ ikt ωˆ k jn .

Recalling that with our choices ωˆ i j = 0 and ωˆ 0i ( Eˆ j ) = 0 along the minimizing curve, we have ˆ i jnt = ∂n ωˆ i jt

on this curve. Therefore (10) is equivalent to 1 ˆ lmnt σlm . λ˙ n = −(M + K )aˆ n +

2

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399

On the other hand, (6) can be written as λl = σlm aˆ m . From these two equations we finally deduce d aˆ 1 ˆ jkit σ jk + σi j j = 0, (M + K )aˆ i −

2 dt which are the nontrivial components of Equation (2).

 

Remark 6.3. (1) As one would expect, the remaining minimum conditions lead to the Euler equation. (2) We can write Eq. (2) as (M + K )

D x˙ α 1 D 2 x˙ β = R αβγ δ x˙ β s γ δ − s αβ , dt 2 dt 2

where s = σi j Eˆ i ⊗ Eˆ j is the angular momentum tensor7 . This equation is exactly the Mathisson-Papapetrou equation subject to the Mathisson-Pirani spin supplementary condition [Mat37, Pap51,Pir56,MTW73,Sem99,Ste04]. Notice that the total mass appearing in this equation is the sum of the Euler top’s rest mass M with its rotational kinetic energy K (which is constant along the motion). The fact that the angular momentum matrix is constant on the Fermi-Walker transported frame { Eˆ 1 , Eˆ 2 , Eˆ 3 } is translated into the fact that the angular momentum tensor satisfies the Fermi-Walker transport equation Ds αβ D 2 x˙ γ β D 2 x˙ γ α = x˙ α sγ + x˙ β s . 2 dt dt dt 2 γ References [Ar97] [Born09] [Boy65] [Dix70] [ER77] [Ell82] [K72] [HR74] [Her10]

Arnold, V.I.: Mathematical methods of classical mechanics. Berlin-Heidelberg-New York: Springer, 1997 Born, M.: Die theorie des starren elektrons in der kinematik des relativitätsprinzips. Ann. der Physik 30, 1–56 (1909) Boyer, R.: Rigid frames in general relativity. Proc. R. Soc. Lond. A 283, 343–355 (1965) Dixon, W.: Dynamics of extended bodies in general relativity i. Momentum and Angular Momentum. Proc. R. Soc. London A 314, 499 (1970) Ehlers, J., Rudolph, E.: Dynamics of extended bodies in general relativity – center-of-mass description and quasi-rigidity. Gen. Rel. Grav. 8, 197–217 (1977) Ellis, J.: Motion of a classical particle with spin. i. the canonical theory of multipliers. J. Phys. A: Math. Gen. 15, 409–427 (1982) Künzle, H.: Canonical dynamics of spinning particles in gravitational and electromagnetic fields. J. Math. Phys. 13, 739–744 (1972) Hanson, A., Regge, T.: The relativistic spherical top. Ann. Phys. 87, 498–566 (1974) Herglotz, G.: Über den vom standpunkt des relativitätsprinzips als “starr” zu bezeichnenden körper. Ann. der Physik 31, 393–415 (1910)

7 Notice however that many authors define the angular momentum tensor to be  s = −s, corresponding to the alternative relation  σi j = εi jk σ k between the angular momentum matrix and the angular momentum vector.

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[Law63] [MR99]

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Lawden, D.: Optimal trajectories for space navigation. Oxford: Butterworths, 1963 Marsden, J., Ratiu, T.: Introduction to mechanics and symmetry. Berlin-Heidelberg-New York: Springer, 1999 [Mat37] Mathisson, M.: Neue mechanik materieller systeme. Acta Phys. Polon. 6, 163–200 (1937) [MTW73] Misner, C., Thorne, K., Wheeler, J.A.: Gravitation. Newyark: Freeman, 1973 [NJ59] Newman, E., Janis, A.: Rigid frames in relativity. Phys. Rev. 116, 1610–1614 (1959) [Noe10] Noether, F.: Zur kinematik des starren körpers in der relativtheorie. Ann. der Physik 31, 919–944 (1910) [Oli02] Oliva, W.: Geometric mechanics. Berlin-Heidelberg-New York: Springer, 2002 [Pap51] Papapetrou, A.: Spinning test particles in general relativity i. Proc. R. Soc. London A 209, 248 (1951) [Pin93] Pinch, E.: Optimal control and the calculus of variations. Oxford, Oxford University Press, 1993 [Pir56] Pirani, F.: On the physical significance of the riemann tensor. Acta Phys. Polon. 15, 389– 405 (1956) [PW62] Pirani, F., Williams, G.: Rigid motion in a gravitational field. Semin. Mec. analytique Mec. celeste M. Janet 5(8-9), 1–16 (1961/62) [Poi04] Poisson, E.: The motion of point particles in curved spacetime. Living Rev. Relativity 7 (2004), available at http://www.livingreviews.org/lrr-2004-6 [Por06] Porto, R.: Post-newtonian corrections to the motion of spinning bodies in nonrelativistic general relativity. Phys. Rev. D 73, 104031 (2006) [Ros47] Rosen, N.: Notes on rotation and rigid bodies in relativity theory. Phys. Rev. 71, 54–58 (1947) [ST54] Salzman, G., Taub, A.: Born-type rigid motion in relativity. Phys. Rev. 95, 1659–1669 (1954) [Sem99] Semerák, O.: Spinning test particles in a kerr field - i. Mont. Not. Astron. Soc. 308, 863–875 (1999) [Ste04] Stephani, H.: Relativity. Cambridge: Cambridge University Press, 2004 [Tak82] Takabayasi, T.: Theory of relativistic top and electromagnetic interaction. Progress of Theoretical Physics 67, 357–374 (1982) [TG83] Thorne, K., Gürsel, Y.: The free precession of slowly rotating neutron stars: rigid-body motion in general relativity. Mon. Not. R. Astr. Soc. 205, 809–817 (1983) [Tur] Turakulov, Z.: Classical mechanics of spinning particle in a curved space, http://arxiv.org/list/ dg-ga/9703008, 1997 [Voi88] Voinov, A.: Motion and rotation of celestial bodies in the post-newtonian approximation. Celest. Mech. 42, 293–307 (1988) [WE66] Wahlquist, H., Estabrook, F.: Rigid motions in einstein spaces. J. Math. Phys. 7, 894–905 (1966) [WE67] Wahlquist, H., Estabrook, F.: Herglotz-noether theorem in conformal space-time. J. Math. Phys. 8, 919 (1967) [XTW04] Xu, C., Tao, J., Wu, X.: Post newtonian rigid body. Phys. Rev. D 69, 024003 (2004) [YB93] Yee, K., Bander, M.: Equations of motion for spinning particles in external electromagnetic and gravitational fields. Phys. Rev. D 48, 2797–2799 (1993)

Communicated by G.W. Gibbons

Commun. Math. Phys. 281, 401–444 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0495-4

Communications in

Mathematical Physics

Vanishing of Vacuum States and Blow-up Phenomena of the Compressible Navier-Stokes Equations Hai-Liang Li1,3 , Jing Li2,3 , Zhouping Xin1,3 1 Department of Mathematics, Capital Normal University, Beijing, P. R. China.

E-mail: [email protected]

2 Institute of Applied Mathematics, AMSS, Academia Sinica, Beijing 100080, P. R. China.

E-mail: [email protected]

3 Institute of Mathematical Science, The Chinese University of Hong Kong, Shatin, Hong Kong.

E-mail: [email protected] Received: 12 April 2007 / Accepted: 3 December 2007 Published online: 22 May 2008 – © Springer-Verlag 2008

Abstract: The Navier-Stokes systems for compressible fluids with density-dependent viscosities are considered in the present paper. These equations, in particular, include the ones which are rigorously derived recently as the Saint-Venant system for the motion of shallow water, from the Navier-Stokes system for incompressible flows with a moving free surface [14]. These compressible systems are degenerate when vacuum state appears. We study initial-boundary-value problems for such systems for both bounded spatial domains or periodic domains. The dynamics of weak solutions and vacuum states are investigated rigorously. First, it is proved that the entropy weak solutions for general large initial data satisfying finite initial entropy exist globally in time. Next, for more regular initial data, there is a global entropy weak solution which is unique and regular with well-defined velocity field for short time, and the interface of initial vacuum propagates along the particle path during this time period. Then, it is shown that for any global entropy weak solution, any (possibly existing) vacuum state must vanish within finite time. The velocity (even if regular enough and well-defined) blows up in finite time as the vacuum states vanish. Furthermore, after the vanishing of vacuum states, the global entropy weak solution becomes a strong solution and tends to the non-vacuum equilibrium state exponentially in time.

1. Introduction The compressible isentropic Navier-Stokes equations, which are the basic models describing the evolution of a viscous compressible fluid, read as follows:
ρt + div(ρu) = 0, (1.1) (ρu)t + div(ρu ⊗ u) − 2div(µD(u)) − ∇(ξ divu) + ∇ p(ρ) = 0,

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H.-L. Li, J. Li, Z. Xin

where x ∈  ⊂ R N , t ∈ (0, T ), D(u) = (∇u + (∇u)T )/2, and p(ρ) = aρ γ , a > 0, γ ≥ 1, the viscosity coefficients µ, ξ are assumed to satisfy µ ≥ 0 and ξ + 2µ/N ≥ 0. If µ and ξ are both constants, there is huge literature on the studies of the global existence and behavior of solutions to (1.1). For instance, the one-dimensional (1D) problems were addressed by Kazhikhov et al [24] for sufficiently smooth data, and by Serre [46,47] and Hoff [18] for discontinuous initial data, where the data are uniformly away from the vacuum; the multidimensional problems (1.1) were investigated by Matsumura et al [35–37], who proved global existence of smooth solutions for data close to a non-vacuum equilibrium, and later by Hoff for discontinuous initial data [19], and more recently, by Danchin [11], who obtained existence and uniqueness of global solutions in a functional space invariant by the natural scaling of the associated equations; and for the existence of solutions for arbitrary data (which may include vacuum states), Lions [29–31] (see also Feireisl et al [13]) obtained global existence of weak solutions - defined as solutions with finite energy - when the exponent γ is suitably large, where the only restriction on initial data is that the initial energy is finite, so that the density is allowed to vanish. Despite the important progress, the regularity, uniqueness and behavior of these weak solutions remain largely open. As emphasized in many papers related to compressible fluid dynamics [9,10,18,20,21,24,27,45,46,48,50,52], the possible appearance of vacuum is one of the major difficulties when trying to prove global existence and strong regularity results. Hoff and Smoller [21] proved that weak solutions of the compressible Navier-Stokes equations (1.1) in one space dimension do not exhibit vacuum states in a finite time provided that no vacuum is present initially under fairly general conditions on the data. Such a result was extended to the spherically symmetric case in [54] recently. On the other hand, the results of Xin [52] showed that there is no global smooth solution to Cauchy problem for (1.1) with a nontrivial compactly supported initial density, which gives results for finite time blow-up in the presence of vacuum. It is also proved in [27] that for bounded domain even one point initial vacuum shall cause the global strong solutions to 1D (1.1) to blow up as time goes to infinity provided the initial data satisfies some compatibility conditions. The independence of viscosities on density makes it possible to trace either the particle path or the trajectory of vacuum state. This, however, leads to the failure of continuous dependence of weak solutions containing vacuum state on initial data [20]. For the case that the density changes continuously across the interfaces separating the gas and vacuum, the global existence and uniqueness of weak solutions was obtained in [33], where the authors obtained that velocity is smooth enough up to the interfaces which are particle paths separating the gas from the vacuum and that the support of gas density expands outside as well as interface connecting gas and vacuum moves at an algebraic rate. Thus, viscous compressible fluids near vacuum should be better modeled by the compressible Navier-Stokes equations with density-dependent viscosities, as was derived in the fluid-dynamical approximation of Boltzmann equation for dilute gases. Further, as was first pointed out and investigated by Liu-Xin-Yang in [32], in the derivation of the compressible Navier-Stokes equations from the Boltzmann equation by the ChapmanEnskog expansions, the viscosity depends on the temperature,which is translated into the dependence of the viscosity on the density for isentropic flows. Moreover, it should be emphasized that a one-dimensional compressible flow model, called the viscous Saint-Venant system for laminar shallow water, derived rigorously from incompressible Navier-Stokes system with a moving free surface by Gerbeau-Perthame recently in [14],

Vanishing of Vacuum States and Blow-up Phenomena

has the form:




ρt + (ρu)x = 0, (ρu)t + (ρu 2 )x − a(ρu x )x + (ρ 2 )x = 0,

403

(1.2)

with the viscosity coefficients given by µ(ρ) = ξ(ρ) = aρ/3 for a given positive constant a. Indeed, such models appear naturally and often in geophysical flows [2,6,7]. In the case of the one-dimensional problem with µ = ξ = aρ α for some positive constants a and α, the well-posedness of the Cauchy problem has been studied by many authors for not only initially compact-supported density but initially non-vacuum density. Indeed, for the initially compact-supported density case, the local (in time) well-posedness of weak solutions to this problem was first established by Liu-XinYang in [32], where the initial density was assumed to be connected to vacuum with discontinuities. This property, as shown in [32], can be maintained for some finite time. And the global existence of weak solutions, together with α ∈ (0, 1) and the density function connecting to vacuum with discontinuities, was considered by many authors, see [23,43,55] and the references therein. It is noticed that the above analysis is based on the uniform positive lower bound of the density with respect to the construction of the approximate solutions. On the other hand, if the density function connects to vacuum continuously, there is no positive lower bound for the density and the viscosity coefficient vanishes at vacuum. This degeneracy in the viscosity coefficient gives rise to new difficulties in analysis because of the less regularizing effect on the solutions. Yang et al [56] first obtained a local existence result for this case under the free boundary condition with α > 1/2. The authors in [12,51,57] obtained the global existence of weak solutions with α ∈ (0, 1/2). However, almost all the above results concern mainly free boundary problems, and for the global existence results, the choices of viscosity do not fit the important physical model, the shallow water equation (1.2) with µ(ρ) = ρ (namely α = 1). For the constant viscosity case, one has known not only that the vacuum state will not develop later on time if there is no vacuum state initially [21], but also that the separate two initial vacuum states shall not meet together in a finite time [54], and that one point initial vacuum causes strong solutions to blow up [27] at infinity as well. However, little is known on the dynamics of the vacuum states of weak solutions to the compressible Navier-Stokes equations (1.1) with density-dependent viscosity on bounded domain. And in particular, it is not clear yet how the vacuum states evolve with respect to time and whether the initial vacuum states shall exist all the time or not for weak solutions. The study of these important dynamical problems about vacuum states is rather difficult because the nonlinear diffusion is degenerate as the vacuum appears, which is quite different from the case of constant viscosity. This causes the loss of information about the velocity and makes it difficult to trace the evolution of vacuum states in general. Across the interface (or vacuum boundary), it is usually difficult to obtain enough information about velocity even if considering specific cases such as point vacuum or continuous vacuum of one piece. It is important to get enough information about the velocity since the flow particles transport usually along particle path, and all interfaces of vacuum, such as free boundaries [23,33] which can be observed and dealt with, also move along the trajectories determined by velocity field. For the multidimensional case, Vaigant et al [49] first proved that for the 2D case and for the case µ is a constant and ξ(ρ) = aρ β , with a > 0, β > 3, (1.1) with periodic boundary condition has a unique strong and classical solution with density away from vacuum. More recently, Bresch and Desjardins [2,3,6,7] (see also [38]) have made

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important progress. Under the condition that ξ(ρ) = 2(µ (ρ)ρ − µ(ρ)), they establish a new Bresch-Desjardins (BD) entropy inequality which can not only be applied to the vacuum case but also used to get the compactness results for (1.1) which extended the compactness results due to Lions [29–31] to the case γ ≥ 1. On the other hand, the constructions of the approximation solutions do not seem routine in the general case of the appearance of vacuum. However, it should be noted that recently Bresch et al [4–6] have made significant progress on the construction of approximate solutions and existence of global weak solutions to the multi-dimensional compressible Navier-Stokes equations and the 2D shallow water model in the case that there is either a drag friction (for barotropic compressible flows) or a cold pressure (for viscous and heat conducting flows). In the case that there is neither drag friction nor cold pressure included, GuoJiu-Xin [16] recently have shown how to construct approximate smooth solutions and obtain the global existence of weak solutions via the BD entropy to the (2D and 3D) barotropic compressible Navier-Stokes for the spherical symmetric initial data. There are also other recent interesting applications of the BD entropy to one-dimensional compressible Navier-Stokes equations with degenerate viscosities, for instance, on the global existence and long time behavior of weak solutions for the free boundary problem [15,17], or the existence and uniqueness of the global strong solution away from vacuum in real line [39]. We study mainly the initial-boundary-value problem (IBVP) for (1.1), where µ = ρ α with α > 1/2, on spatial one-dimensional bounded spatial domains or periodic domains. This contains the physical important model for shallow water equations (1.2). The choice of α > 1/2 is necessary in order to consider the dynamics of vacuum states since it allows the existence of the initial vacuum in Eulerian coordinates as one can see later. We first establish the global existence of entropy weak solutions for the compressible Navier-Stokes equations (1.1), with pressure p = ρ γ and γ ≥ α/2, for general initial data with finite entropy and vacuum. The key in our analysis is the construction nonvacuum approximate solutions so that we can make use of the stability analysis in [38], where the Bresch-Desjardins (BD) entropy inequality (see [2,3,6,7]) was used to obtain the compactness results. Our construction of the approximate solutions is strongly motivated by the previous work of Jiang-Xin-Zhang [23] about the existence of global weak solutions to one-dimensional compressible Navier-Stokes with free boundary as vacuum interface. In general, it seems rather difficult to investigate the dynamics of vacuum states due to the degeneracy of nonlinear diffusion and the density function connecting to vacuum continuously. Therefore, we further consider the cases of more regular initial data containing point the vacuum or continuous vacuum of one piece, and we show that there is a global entropy weak solution which is unique and regular with well-defined velocity field at least for a short time, and the vacuum states remain for the short time. Then, we use some ideas due to [22,27,28] to prove that any possible vacuum state in such global weak solutions which satisfy the BD entropy must vanish within finite time. This shows that such short time structure and vacuum states of weak solutions can not be maintained all the time. And as the vacuum states vanish, the spatial derivative of velocity (if it exists) has to blow up even if the velocity is regular enough and welldefined before. After the vanishing of vacuum states, we can redefine the velocity field and recover the nonlinear diffusion term in terms of density and velocity. In addition, the global entropy weak solution is shown to become a strong solution and tends to the non-vacuum equilibrium state exponentially in time. This phenomena, applied to the compressible shallow water equations (1.2), seems to have never been observed for the compressible Navier-Stokes equations before.

Vanishing of Vacuum States and Blow-up Phenomena

405

The rest of the paper is as follows. In Sect. 2, the main results about the vanishing of vacuum states and blow-up phenomena of global entropy weak solutions for the compressible Navier-Stokes equations are stated. The global existence of entropy weak solutions for general large initial data with vacuum states allowed is proven in Sect. 3. The short time structure of global entropy weak solution with initial one point vacuum state or initial continuous vacuum states of one piece are investigated in Sect. 4. In Sect. 5, we show the vanishing of vacuum states and blow-up phenomena of any global entropy weak solution within finite time and analyze the regularity and large time asymptotic behavior of global entropy weak solutions after the vanishing of vacuum states. 2. Main Results We consider the initial-boundary-value-problem (IBVP) for the 1D compressible NavierStokes equations with density-dependent viscosity ρt + (ρu)x = 0,

(2.1)

(ρu)t + (ρu + p(ρ))x − (µ(ρ)u x )x = 0,

(2.2)

2

with ρ ≥ 0 the density, ρu the momentum. The pressure and viscosity are assumed to have the form: p(ρ) = a1 ρ γ , µ(ρ) = a2 ρ α , where γ ≥ 1, a1 > 0, a2 > 0, and α > 1/2 are constants, and for simplicity we set a1 = a2 = 1. The initial data is given for the density ρ and the momentum ρu, ρ(x, 0) = ρ0 (x) ≥ 0, ρu(x, 0) = m 0 (x), x ∈ ,

(2.3)

where the domain  is chosen as the unit interval denoting the spatial domain (0, 1) or periodic domain with period length one, and throughout the present paper the initial data is assumed to satisfy
α−1/2 ρ0 ≥ 0 a.e. in , ρ0 ∈ L 1 (), (ρ0 )x ∈ L 2 (), (2.4) |m 0 |2 m 0 = 0, a.e. on {x ∈  | ρ0 (x) = 0}, ρ0 ∈ L 1 (). Remark 2.1. Note here that the condition (2.4) implies ρ0 ∈ L ∞ (), ρ0 log+ ρ0 ∈ L 1 ().

(2.5)

It should be clear that a large class of initial data satisfy the conditions in (2.4). In particular, the assumptions (2.4) are satisfied for the following initial data: ρ0 (x) = (|x − x0 |2 )1/(2α−1) , m 0 (x) = 0, x ∈ . Without the loss of generality, the total initial mass is renormalized to be one throughout the present paper, i.e.,  ρ0 (x)d x = 1. 

The boundary conditions are one of the boundary conditions of Dirichlet type and periodic type for Eqs. (2.1)-(2.2) imposed as

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H.-L. Li, J. Li, Z. Xin

(1) Dirichlet case: ρu(0, t) = ρu(1, t) = 0, t ≥ 0,

(2.6)

ρ, u are periodic in x of period 1,

(2.7)

(2) Periodic case: where we consider Eqs. (2.1)–(2.2) on R × (0, ∞). Remark 2.2. For the case of the Dirichlet boundary (2.6), the boundary is given by the physical observable momentum instead of velocity. It is natural to employ such boundary condition as considering the dynamics of (global in time) weak solutions to the IBVP for the compressible Navier-Stokes equations with possible vacuum states included, since it is usually the case that, at vacuum states, the momentum is zero and is observed and controllable, but almost nothing is known yet for the velocity for weak solutions. Note here that for any (weak) solution away from the vacuum at the boundary, the boundary condition (2.6) reduces to u(0, t) = u(1, t) = 0, t ≥ 0. In order to define the weak solutions to the IBVP for the compressible Navier-Stokes equations (2.1)–(2.2) with initial data (2.3) and boundary condition (2.6) or (2.7), we define the set of test functions as follows:
C0∞ ( × [0, T )) for the Dirichlet case (2.6), 
∞ (R × [0, T )) for the periodic case (2.7), Cper and


C0∞ ( × [0, T )) 
∞ (R × [0, T )) Cper

for the Dirichlet case (2.6), for the periodic case (2.7),

∞ (R × [0, T )) defined by with Cper   ∞ Cper (R × [0, T )) = ϕ ∈ C ∞ (R × [0, T ))| ϕ is periodic in x of period 1 .

We define the weak solutions to the IBVP for the compressible Navier-Stokes Equations (2.1)–(2.2) as follows. Definition 2.3 (Global Weak Solutions). For any T > 0, (ρ, u) is said to be a weak solution to Eqs. (2.1)–(2.2) with initial data (2.3) and boundary value (2.6) or (2.7) in  × (0, T ), if ⎧ ∞ 1 γ α−1/2 ) ∈ L ∞ (0, T ; L 2 ()), ⎪ x ⎨0 ≤ ρ ∈ L (0, T ; L () ∩ L ()), (ρ (2.8) ⎪ ⎩√ρu ∈ L ∞ (0, T ; L 2 ()), ρ α u ∈ L 2 (0, T ; W −1,1 ()), x loc and (ρ, u) satisfies   ρ0 ψ(x, 0)d x + 

T 0



 

ρψt d xdt + 0

for any ψ ∈ , and   m 0 ϕ(x, 0)d x + 

0



T

+ 0

T

 

T

 

√ √ ρ ρuψx d xdt = 0

√ √ ρ( ρu)ϕt d xdt



√ ( ρu)2 + ρ γ ϕx d xdt − ρ α u x , ϕx = 0 

(2.9)

(2.10)

Vanishing of Vacuum States and Blow-up Phenomena

407

for all ϕ ∈ . The nonlinear diffusion term ρ α u x is defined as 

α

T

ρ u x , ϕ = −



√ ρ α−1/2 ρuϕx d xdt

0

2α − 2α − 1



T



√ (ρ α−1/2 )x ρuϕd xdt

(2.11)

0

for any ϕ ∈ , where √ ρ ∈ L ∞ ( × (0, T )) due to (2.8). Moreover, for the spatial periodic case (2.7), (ρ, ρu) is also periodic. Remark 2.4. For the Dirichlet case, (2.9), together with the fact ρ ∈ L ∞ ( × (0, T )) due to (2.8), implies that (ρ, u) satisfies Eq. (2.1) in the sense of distribution and justifies the boundary condition (2.6) in the sense that, for any time interval I ⊂ [0, T ], ε−1

  I

ε

√ √ ρ ρu(x, s)d xds → 0, ε−1

  I

0

ε

√ √ ρ ρu(x, s)d xds → 0

1−ε

√ √ as ε → 0+ . If further ρ ρu ∈ L p (0, T ; W 1,q ()) for some p ≥ 1, q ≥ 1, then (2.9) √ √ 1,q yields that ρ ρu ∈ L p (0, T ; W0 ()), that is, the Dirichlet boundary condition (2.6) is satisfied in the sense of trace. Definition 2.5 (Global Entropy Weak Solutions). Let (ρ, u) be a global weak solution (in the sense of Definition 2.3) to (2.1)–(2.2) with initial data (2.3) and boundary value (2.6) or (2.7) in  × (0, T ). Then, (ρ, u) is said to be a global entropy weak solution if there exists some function ∈ L 2 ( × (0, T )) satisfying (2.11), i.e., 

T





T

ϕd xdt = −

0



√ ρ α−1/2 ρuϕx d xdt

0

−

2α 2α − 1



T



√ (ρ α−1/2 )x ρuϕd xdt

(2.12)

0

for any ϕ ∈ , and the following uniform entropy inequality holds: 

√ | ρu|2 + |(ρ α−1/2 )x |2 + π(ρ) (x, t) d x sup 0≤t≤T  T

+ 

0

≤C0







(|(ρ (γ +α−1)/2 )x |2 + 2 )(x, t) d xdt 2



α−1/2

( |mρ00| + |(ρ0

)x |2 + π+ (ρ0 ))(x) d x

(2.13)

with C0 > 0 independent of T, and





ρ log ρ, if γ = 1, π(ρ)
1 γ γ −1 ρ , if γ > 1,

π+ (ρ)


ρ log+ ρ, if γ = 1, 1 γ if γ > 1. γ −1 ρ ,

(2.14)

We have the following result on the existence of global entropy weak solutions.

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H.-L. Li, J. Li, Z. Xin

Theorem 2.1. (Global Existence). Assume that α>

α 1 , γ > , 2 2

and that the initial data (ρ0 , m 0 ) satisfies (2.4) and

(2.15) |m 0 |2+ν ρ01+ν

∈ L 1 () for some positive

constant ν. Then for any T > 0, there exists a global entropy weak solution (ρ, u) to the IBVP for the compressible Navier-Stokes equations (2.1)–(2.3) with boundary condition (2.6) or (2.7) in  × (0, T ) in the sense of Definition 2.5. Moreover, for the case of Dirichlet boundary condition (2.6), if α ∈ (1/2, 3/2) √ and ν satisfies (2.17) (see Remark 2.6 below), then in addition to (2.8), the solution (ρ, ρu) satisfies √ √ 1,(4+2ν)/(4+ν) ρ ( ρu) ∈ L 2 (0, T ; W0 ()), i.e.,

√

(2.16)

√ ρ ( ρu) satisfies the Dirichlet boundary (2.6) in the sense of trace.

Remark 2.6. (1) Theorem 2.1 above holds for the compressible shallow water equation (1.2). (2) For the Dirichlet case (2.6), one of √ the √ available ways (available for the case α ∈ (1/2, 3/2) and γ ≥ 1) to obtain that ρ ρu ∈ L p (0, T ; W 1,q ()), for some p ≥ 1, q ≥ 1, is to choose the positive constant ν in Theorem 2.1 such that ⎧ ν ∈ (0, 2γ − α), for α ∈ ( 21 , 1], γ ≥ 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎨

1+α  −2) 2(2γ −α) 3 , ν ∈ 2(α+γ 3−α−γ 1+α−2γ , for α ∈ (1, 2 ), γ ∈ 1, 2 , ⎪ ⎪ ⎪ ⎪

 ⎪ ⎪ ⎩ ν ∈ 4(α−1) , ∞ , for α ∈ (1, 23 ), γ ≥ 1+α 3−2α 2 .

(2.17)

Next, we show that there is a global entropy weak solution (ρ, u) in the sense of Definition 2.5 for which the vacuum states and the structure of interface, if existing initially, can be preserved for a short time, so long as the initial data has additional regularity besides (2.4) and the fluids and the vacuum states in initial data are connected “smoothly”. In addition, the weak solution (ρ, u) is actually a unique regular solution for the short time. For simplicity, we consider the case of one point vacuum state contained at x = x0 ∈ (0, 1) in the initial data (ρ0 , m 0 ) = (ρ0 , ρ0 u 0 ) with additional regularity A0 |x − x0 |σ ≤ ρ0 (x) ≤ A1 |x − x0 |σ , for any x ∈ , ¯ u 0 ∈ C 1 (),

γ −1+1/2 j (ρ0 )x

∈ L 2 j (),

−1+1/2 j α ρ0 (ρ0 u 0x )x

(2.18)

∈ L 2 j (), j = 1, n, (2.19)

with n ≥ 2 an integer; and in the case of continuous vacuum state of one piece initially on 0 = [x0 , x1 ] ⊂ (0, 1) in the initial data, we require ⎧ σ σ ⎪ ⎨ A0 (x0 − x) ≤ ρ0 (x) ≤ A1 (x0 − x) , x ∈ [0, x0 ), (2.20) x ∈ [x0 , x1 ], ρ0 (x) = 0, m 0 (x) = ρ0 u 0 (x) = 0, ⎪ ⎩ B (x − x )σ ≤ ρ (x) ≤ B (x − x )σ , x ∈ (x , 1] 0 1 0 1 1 1

Vanishing of Vacuum States and Blow-up Phenomena

409

and ⎧ γ −1+1/2 j ¯ \ 0 ), ⎨ (ρ0 )x ∈ L 2 j (), j = 1, n, u 0 ∈ C 1 ( ⎩

−1+1/2 j

ρ0

(2.21)

(ρ0α u 0x )x ∈ L 2 j ( \ 0 ), j = 1, n,

with n ≥ 2 an integer. Here, σ , A0 , A1 , and B0 , B1 are positive constants, and the power σ ∈ (σ− , σ+ ) with positive constants σ± given in (2.33) later. We also require that the initial data (ρ0 , m 0 ) = (ρ0 , ρ0 u 0 ) given by (2.3) is consistent with boundary value for the Dirichlet boundary condition. We have the following results on short time structure of global entropy weak solutions. Theorem 2.2. (Short time structure of vacuum states). In addition to the assumptions of Theorem 2.1, assume further that α > 21 , γ > max{1, α}

(2.22)

and that there is either one point vacuum state in initial data (ρ0 , u 0 ) with (2.18)–(2.19) satisfied or a piece of continuous vacuum states in initial data (ρ0 , u 0 ) with (2.20)–(2.21) satisfied. Then, there exists a global entropy weak solution (ρ, u) to the IBVP for the compressible Navier-Stokes equations (2.1)–(2.2) with initial data (2.3) and boundary value (2.6) or (2.7) in the sense of Definition 2.5. Moreover, there is a short time T∗ > 0, so that the global entropy weak solution (ρ, u) is unique1 and regular on the domain  × [0, T∗ ], and the initial structure of vacuum states is maintained for t ∈ [0, T∗ ] in the following sense: For the case of the one point vacuum state initially, (2.18), the solution (ρ, u) is ¯ × [0, T∗ ], regular and unique on the domain  ¯ × [0, T∗ ]), u x ∈ L ∞ (0, T∗ ; C 0 ()), ¯ (ρ, u) ∈ C 0 ( u L ∞ (×[0,T + u x  L ∞ ([0,T∗ ];C 0 ()) ¯ ¯ ≤ C(T∗ ). ∗ ])

(2.23) (2.24)

The one point vacuum state propagates along the particle path, namely, there is one particle path x = X 0 (t) : [0, T∗ ] →  with X 0 (t) ∈ C([0, T∗ ]) defined by X˙ 0 (t) = u(X 0 (t), t), X 0 (0) = x0 ∈ (0, 1),

(2.25)

a− |x − X 0 (t)|σ ≤ ρ(x, t) ≤ a+ |x − X 0 (t)|σ

(2.26)

so that

for (x, t) ∈  × [0, T∗ ], where the two positive constants a± are independent of time T∗ . In the case of a piece of continuous vacuum states initially, (2.20), there are two particle paths x = X i (t) : [0, T∗ ] →  with X i (t) ∈ C([0, T∗ ]), i = 0, 1 defined by X˙ i (t) = u(X i (t), t), X i (0) = xi ∈ (0, 1), i = 0, 1, 1 Here the uniqueness is specified for density ρ and momentum ρu = states of one piece.

(2.27)

√ √ ρ ρu for continuous vacuum

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H.-L. Li, J. Li, Z. Xin

so that it holds for some positive constants a± , b± independent of the time T∗ , a− (X 0 (t) − x)σ ≤ ρ(x, t) ≤ a+ (X 0 (t) − x)σ ,

(2.28)

for (x, t) ∈ [0, X 0 (t)) × [0, T∗ ], and b− (x − X 1 (t))σ ≤ ρ(x, t) ≤ b+ (x − X 1 (t))σ

(2.29)

for (x, t) ∈ (X 1 (t), 1] × [0, T∗ ] respectively, and the interfaces separating the fluid and vacuum coincide with the particle paths ρ(x, t) = 0, ρu(x, t) = 0, (x, t) ∈ [X 0 (t), X 1 (t)] × [0, T∗ ].

(2.30)

The solution (ρ, u) is regular and unique up to the vacuum boundary ¯ × [0, T∗ ]), u ∈ C 0 ( ¯ × [0, T∗ ] \ 0T ), ρ ∈ C 0 ( ∗ u L ∞ (×[0,T + u x  L ∞ (×[0,T 0 0 ¯ ¯ ∗ ]\ ) ∗ ]\

T∗ )

T∗

≤ C(T∗ ),

(2.31) (2.32)

where 0T∗ = (X 0 (t), X 1 (t)) × [0, T∗ ]. Remark 2.7. (1) The constant exponents σ± are defined as σ± = β± /(1 − β± ) > 0 with β± determined by 1 β− = max{ 2α ,

1 1 γ (1 − 2n )},

β+ = min{1,

1 1 1 1 α (1 − 2n ), 1+3α (4 − n )},

(2.33)

while the positive constants a± are independent of the time T∗ . (2) The regularity assumptions (2.18)–(2.19) are satisfied for the following initial data ρ0 (x) = 21 (A0 + A1 )(|x − x0 |2 )(σ− +σ+ )/4 , u 0 (x) = 0, x ∈ , and the regularity assumptions (2.20)–(2.21) are satisfied for the initial data ⎧ 1 (σ +σ )/2 ⎪ ⎨ 2 (A0 + A1 )(x0 − x) − + , x ∈ [0, x0 ), ρ0 (x) = 0, x ∈ (x0 , x1 ), ⎪ ⎩ 1 (B + B )(x − x )(σ− +σ+ )/2 , x ∈ (x , 1]. 0 1 1 1 2

u 0 (x) = 0, x ∈ ,

Next, we prove that for any global entropy weak solution (ρ, u) to the IBVP (2.1)– (2.3) together with boundary condition (2.6) or (2.7) in the sense of Definition 2.5, even though in some cases the vacuum states may exist for some finite time, for instance, in the cases as shown by Theorem 2.2, any possible vacuum state has to vanish within finite time after which the density is always away from vacuum. Simultaneously, not only can the velocity field be defined in terms of the density and momentum, and the nonlinear diffusion is represented in terms of the density and velocity, but also the global entropy weak solution (ρ, u) is shown to be a unique and strong solution after the vanishing of vacuum states. We have the following result.

Vanishing of Vacuum States and Blow-up Phenomena

411

Theorem 2.3. (Vanishing of vacuum states). Assume that α>

1 , γ ≥ 1. 2

(2.34)

Let (ρ, u) be any global entropy weak solution to the IBVP (2.1)–(2.2) with initial data (2.3) and boundary value (2.6) or (2.7) in the sense of Definition 2.5. Then, there exist some time T0 > 0 (depending on initial data) and a constant ρ− so that inf ρ(x, t) ≥ ρ− > 0, t ≥ T0 ,

(2.35)

¯ x∈

and the global entropy weak solution (ρ, u) becomes a unique strong solution (ρ, u) for t ≥ T0 and satisfies
ρ ∈ L ∞ (T0 , t; H 1 ()), ρt ∈ L ∞ (T0 , t; L 2 ()), (2.36) u ∈ H 1 (T0 , t; L 2 ()) ∩ L 2 (T0 , t; H 2 ()), with velocity u and nonlinear diffusion term given by √ ρu u
√ , (ρ α u x )x = x , ρ respectively. In addition, for ⎧ ⎨0  us
1 ⎩ m0d x ρ¯0 

(2.37)

for the Dirichlet case, for the periodic case,

there exist two positive constants µ0 , c0 both depending on initial data (ρ0 , m 0 ) and ρ− , such that (ρ − ρ¯0 , u − u s )(·, t) L 2 () ≤ c0 e−µ0 (t−T0 ) , t > T0 ,

(2.38)

where (and in what follows) f denotes the average of f over the bounded domain , i.e.,   1 f = f (x)d x = f (x)d x. ||   Remark 2.8. (1) Theorem 2.3 shows that any possible vacuum states must vanish in finite time. This theory applies to the compressible shallow water equation (1.2). (2) It is easy to verify (see the proof of Proposition 5.1) that the phenomena of vacuum vanishing (2.35) in finite time actually happens for any global weak solution (ρ, u) to the IBVP (2.1)–(2.3) with boundary condition (2.6) or (2.7) in the sense of Definition 2.3 satisfying the following entropy inequality:  T 

√ 2 α−1/2 2 sup | ρu| + |(ρ )x | + π(ρ) (x, t) d x + (ρ (γ +α−1)/2 )x 2L 2 dt 0≤t≤T



≤ C0

 

0

|2

α−1/2

( |mρ00 + |(ρ0

with C0 independent of T.

)x |2 + π+ (ρ0 ))(x)d x

(2.39)

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Finally, for any global entropy weak solution (ρ, u) to the IBVP (2.1)-(2.3) together with boundary condition (2.6) or (2.7) in the sense of Definition 2.5, the density is continuous, i.e., ρ ∈ C( × [0, T ]) for any T > 0, due to (2.8) and (2.9). Thus, the continuity of ρ and Theorem 2.3 imply that if the density contains vacuum states at least at one point, then there exists some critical time T1 ∈ [0, T0 ) with T0 > 0 given by ¯ such that (2.35) and a nonempty subset 0 ⊂  ⎧ 0 ⎪ ⎨ρ(x, T1 ) = 0, ∀ x ∈  , ¯ \ 0 , (2.40) ρ(x, T1 ) > 0, ∀ x ∈  ⎪ ⎩ρ(x, t) > 0, ¯ × (T1 , T0 ]. ∀ (x, t) ∈  It follows from (2.36) easily that for any δ > 0, it holds 

T0

u x  L ∞ ds < ∞.

(2.41)

T1 +δ

Under the condition that vacuum states appear, we shall prove that the spatial derivative of velocity (if regular enough and definable) blows up in finite time as the vacuum states vanish, even if the solution is regular enough for short time so that the velocity field and its derivatives are bounded as shown by Theorem 2.2. Theorem 2.4. (Finite time blow-up). Let (ρ, u) be any global entropy weak solution, which contains vacuum states at least at one point for some finite time, to the IBVP for the compressible Navier-Stokes equations (2.1)–(2.3) with boundary condition (2.6) or (2.7) in the sense of Definition 2.5. Let T0 > 0 and T1 ∈ [0, T0 ) be the time such that (2.35) and (2.40) holds respectively. Then, the solution (ρ, u) blows up as vacuum states vanish. Namely, for T1 satisfying (2.40) and for any given fixed η > 0, it holds  lim+

t→T1

T1 +η

u x  L ∞ ds = ∞.

(2.42)

t

On the other hand, if there exists some T2 ∈ (0, T0 ) such that the weak solution (ρ, u) satisfies u L 1 (0,T2 ;W 1,∞ ()) < ∞, then, there is a time T3 ∈ [T2 , T0 ) so that the blowup phenomena happens for (ρ, u), i.e.,  lim

t→T3− 0

t

u x  L ∞ ds = ∞.

(2.43)

Remark 2.9. Theorem 2.4 implies that for any global entropy weak solution (ρ, u) to the IBVP for the compressible Navier-Stokes equations (2.1)–(2.2) with initial data (2.3) and boundary value (2.6) or (2.7) in the sense of Definition 2.5, which contains vacuum states at least at one point initially, the finite time blowup phenomena (2.42) happens for such a solution (ρ, u).

Vanishing of Vacuum States and Blow-up Phenomena

413

Remark 2.10. Theorems 2.1–2.4 provide a complete dynamical description on the vanishing of vacuum states and blow-up phenomena for the global entropy weak solutions to the compressible Navier-Stokes equations with density-dependent viscosity. That is, a global entropy weak solution exists for general large initial data with finite entropy. For short time, such weak solution is unique and regular with well-defined velocity field subject to additional initial regularity, and any existing vacuum state is maintained with the same interface structure as the initial. Then, within finite time the vacuum states vanish definitely and the velocity blows up (even if it is regular enough and definable along the interfaces). After the vanishing of vacuum states, the global entropy weak solution becomes a strong one and tends to the non-vacuum equilibrium state exponentially in time. This dynamical phenomena is quite similar to those well-known for the 3-D incompressible Navier-Stokes equations. However, before the time of vacuum-vanishing, the uniqueness of the global entropy weak solution to the compressible Navier-Stokes equations with density-dependent viscosity subject to the initial data is not known yet. Remark 2.11. All theories established in Theorems 2.1-2.4 fit the shallow water equation (1.2). We believe that such phenomena described by Theorems 2.1-2.4 are also observed for other compressible fluids with density-dependent viscosity, such as Navier-Stokes equations with capillarity and/or drag friction, Navier-Stokes–Poisson system, etc. Remark 2.12. It is interesting to investigate the (global) dynamics of (one-dimensional) interface connecting vacuum from initial time until the vanishing of vacuum in order to investigate the dynamics of interface and the vanishing of vacuum state and to verify the formation of singularity for the general case. Remark 2.13. (1) Another interesting problem is whether the phenomena of the vanishing of vacuum states and blow-up happens for a multi-dimensional compressible isentropic Navier-Stokes system with density-dependent viscosity, especially on spatial bounded domain with the Dirichlet boundary condition (2.6). (2) It is also interesting to study whether any vacuum states shall vanish within finite time and blow-up phenomena happen for the multi-dimensional full Navier-Stokes system with density-dependent viscosity. It is not obvious yet (although we expect) since it is not clearly understood yet how the dynamics of temperature and heat-conduction shall affect the global existence of weak solutions and the evolution of vacuum states, especially in the case of spherical symmetry under the Dirichlet boundary condition (2.6). This is under further investigation [26]. 3. Global Existence of Entropy Weak Solutions In this section, we will establish the existence of global entropy weak solutions to the IBVP for the compressible Navier-Stokes equations (2.1)–(2.3) together with (2.6) or (2.7). Since the compactness arguments are straightforward in the framework of MelletVasseur [38], we only need to construct a sequence of approximate solutions by using some ideas developed by Jiang-Xin-Zhang in [23] for one-dimensional compressible Navier-Stokes equations with free boundary, to establish some uniform a priori estimates, in particular, the lower and upper bounds of the density for the approximate solution sequence with the help of BD entropy, and to justify the boundary condition (2.16) for (limiting) weak solution for the Dirichlet case as follows. Outline of proof of Theorem 2.1. Step 1. Construction of smooth approximate solutions. Let us consider the following approximate compressible Navier-Stokes equations

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inspired by Jiang-Xin-Zhang in [23]: ρεt + (ρε u ε )x = 0, (ρε u ε )t + (ρε u 2ε

+ p(ρε ))x − (µε (ρε )u εx )x = 0, (ρε , ρε u ε )(x, 0) = (ρ0ε , m 0ε )(x),

(3.1) (3.2) (3.3)

with one of the following boundary conditions: u ε (0, t) = u ε (1, t) = 0,

(3.4)

ρε , u ε are periodic in x of period 1.

(3.5)

or The viscosity µε is given by µε (ρ) = ρ α + ερ θ , ε > 0, θ ∈ (0, 1/2),

(3.6)

where we remark (see Remark 3.1 below) that although the modified viscosities (3.6) are limited to the case θ ∈ (0, 1/2), the methods adapted here to construct approximate solutions also can be applied to the case θ ∈ (0, 1) and γ > θ after some modification, the interested reader can refer to [16,23] and references therein, we omit the details here. The initial data ρ0ε , m 0ε ∈ C ∞ () satisfies
α−1/2 α−1/2 → ρ0 in H 1 (), ρ0ε → ρ0 in L 1 (), ρ0ε (3.7) (m 0ε )2 (ρ0ε )−1 → m 20 ρ0−1 , |m 0ε |2+ν (ρ0ε )−1−ν → |m 0 |2+ν ρ0−1−ν in L 1 () as ε → 0+ , and ρ0ε ≥ c0 ε1/(2α−2θ)

(3.8)

for some c0 independent of ε. By (3.7) we can assume without the loss of generality that  ρ0ε d x = 1. (3.9) 

By the standard arguments (see [23] for reference), after applying the classical theory of parabolic and hyperbolic equations, one can obtain that there exists some T∗ > 0 such that the approximate problem (3.1)–(3.3) together with the boundary condition (3.4) or (3.5) has a unique smooth solution (ρε , u ε ) on [0, T∗ ] with the density away from vacuum, i.e., ρε (x, t) > 0, for all (x, t) ∈  × [0, T∗ ].

(3.10)

Step 2. The a-priori estimates. To extend the local solution globally in time, one needs to control the lower and upper bounds of the density and get some a-priori estimates. As mentioned in the introduction, this relies on the BD entropy inequality developed by Bresch and Desjardins (see [2,3,6,7] for instance). In the Eulerian coordinates, the BD entropy for the approximate solutions (ρε , u ε ) of the IBVP problem for the system (3.1)–(3.1) reads (see [2,3,6,7] for instance) as  θ √ √ ε )x |2 + | ε((ρ √ε ) )x |2 + π(ρε ))(x, t) d x (| ρε u ε |2 + | µ(ρ ρε ρε   t p  (ρε )µε (ρε ) + (µε (ρε )|u εx |2 + |ρεx |2 )(x, s) d xds ≤ C, t > 0 (3.11) ρε 0



Vanishing of Vacuum States and Blow-up Phenomena

415

with positive constant C independent of t and ε. Indeed, multiplying (3.2) by u ε , integrating by parts on [0, 1], and using (3.1), we get the classical entropy estimate  

√ (| ρε u ε |2 + π(ρε ))(x, t) d x +

 t 0



µε (ρε )|u εx |2 d xds ≤ C, t > 0 (3.12)

with positive constant C independent of t and ε. Multiplying the transport equation (3.1) by ϕε (ρε ) with ρϕε (ρ) = µε (ρ), i.e., ϕε (ρ) = α1 ρ α + θε ρ θ as in [8], we get ϕε (ρε )t + ϕε (ρε )x u ε + ϕε (ρε )ρε u εx = 0.

(3.13)

Differentiating the above equation with respect to x and denoting vε = ϕε (ρε )x /ρε , we have (ρε vε )t + (ρε vε u ε )x + (µε (ρε )u εx )x = 0.

(3.14)

Summing Eqs. (3.14) and (3.1) together, multiplying by u ε + vε and integrating by parts on [0, 1], we have after a straightforward computation that 

√ (| ρε u ε |2 + | ϕε√(ρρεε)x |2 + π(ρε ))(x, t) d x   t p  (ρε )ϕε (ρε ) + (µε (ρε )|u εx |2 + |ρεx |2 )(x, s) d xds ≤ C, t > 0 (3.15) ρε 0



with positive constant C independent of t and ε. This together with (3.12), (ρε2α−2 + ε2 ρε2θ−2 )(ρεx )2 ≤ (ϕε (ρε )x )2 = (ρε2α−2 + 2ερεα+θ−2 + ε2 ρε2θ−2 )(ρεx )2 ≤ 2(ρε2α−2 + ε2 ρε2θ−2 )(ρεx )2 , gives rise to the BD entropy (3.11). In order to derive the lower bound and upper bounds of density for the approximate solution (ρε , u ε ), it is convenient to make use of the Lagrangian coordinates (y, t) with 

x

y= 0

ρε (z, t)dz, t = t,

because the approximate solution (ρε , u ε ) is away from vacuum at least for short time due to (3.10) and fluid density transports along the particle path. In Lagrangian coordinates, we have the equivalent system for (ρε , u ε ) as follows: ρεt + ρε2 u εy = 0, u εt + ( p(ρε )) y − (ρε µε (ρε )u εy ) y = 0,

(3.16) (3.17)

−1 )(y), (ρε , u ε )(y, 0) = (ρ0ε , m 0ε ρ0ε

(3.18)

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together with the boundary conditions similar to (3.4) or (3.5), where (y, t) ∈  L × (0, T∗ ] and   L
(0, L ε ), with L ε = ρ0ε (x)d x = 1 

due to (3.9). Correspondingly, the BD entropy equality (3.11) becomes in Lagrangian coordinates as follows:  T∗  sup E ε (t) + ρε µε (ρε )(u εy )2 dydt 0≤t≤T∗



T∗

+ 0

L

0

 L



ρεγ +α−2 + ερεγ +θ−2 (ρεy )2 dydt ≤ C0 E ε (0)

(3.19)

with C0 independent of T∗ and ε > 0, and E ε (t) defined as 

u 2ε + ((ρεα ) y )2 + ε2 ((ρεθ ) y )2 + ρε−1 π(ρε ) dy. E ε (t)
L

Since

 L

ρε−1 dy

 =



d x = || = 1,

(3.20)

it follows from the continuity of ρε that there exists some y0 (t) ∈  L such that ρε (y0 (t), t) = | L |/|| = 1. Hence, it implies from (3.19) that ρεα (y, t) = ρεα (y0 (t), t) +



y y0 (t)

(ρεα ) y dy

≤ (| L |||−1 )α + C0 E ε (0) + | L |.

(3.21)

This yields the uniform upper bound (w.r.t. time t ∈ [0, T∗ ]) for the density. To obtain the lower bound for the density ρε , we employ the idea in [23]. That is, we consider the upper bound for vε = ρε−1 , which can be estimated by (3.19) and (3.20) as follows:   vε (y, t)dy + (vε )2 |ρεy |dy vε (y, t) ≤ L L  θ+1/2 ≤1 + C max (vε ) (ρεθ ) y  L 2 () ( vε (y, t)dy)1/2 L

y∈ L

1 ≤1 + max vε + C(ε−2 E ε (0))1/(1−2θ) . 2 y∈ L

(3.22)

This shows that for all (y, t) ∈  L × [0, T∗ ], ρε (y, t) ≥

ε2/(1−2θ) . 2ε2/(1−2θ) + C(E ε (0))1/(1−2θ)

(3.23)

Thus, we have obtained the uniform upper bound (3.21) and the lower bound (3.23) (w.r.t. time t ∈ [0, T∗ ]) for density ρε . Using these uniform bounds, one can extend

Vanishing of Vacuum States and Blow-up Phenomena

417

the local smooth solution globally in time by standard arguments (see [23] for details). Moreover, the total mass of the approximate solutions is conserved due to the boundary conditions (3.4),   ρε (x, t)d x = ρ0ε (x)d x, ∀ t > 0. (3.24) 



Remark 3.1. Although the estimates (3.22) and (3.23) are established for the approximate solutions in the case that the modified viscosities (3.6) are limited to the case θ ∈ (0, 1/2), the methods adapted here can be applied to establish the lower bounds like (3.23) in the case θ ∈ (0, 1) and γ > θ . Indeed, one can also establish the following estimates, similar to [23,16], for the approximate solutions (ρ, u ε ) as:  sup (ρ θ )2n x (x, t)d x ≤ C(T ) t∈[0,T ]

for any integer n ≥ 1 and C(T ) a constant independent of ε, θ . Thus, we can verify similar to (3.22) that   vε ≤ 1 + C vε1+θ |(ρ θ )x |d x ≤ 1 + C(T )( vεq(θ+1) dy)1/q ≤ C + C(T )vε(q(θ+1)−1)/q with q = 2n/(2n − 1). This together with Young’s inequality implies the positive lower bound of density for integer n sufficiently large but fixed so that θ < 2n−1 2n so long as θ ∈ (0, 1). η

In addition to (3.11), we show ρε u ε is actually bounded in L ∞ (0, T ; L 2+ν ()) for some η ≥ 1/(2 + ν) with ν > 0 given by
ν > 0 arbitrary if 2γ − α ≥ 1, (3.25) ν ∈ (0, 2(2γ − α)/(1 + α − 2γ )] if 2γ − α ∈ (0, 1). Note here that for α ∈ (1/2, 3/2) the constant ν defined by (2.17) actually satisfies (3.25). It follows from (3.11) that sup (ρεα−1/2 )x  L 2 () ≤ C,

(3.26)

t≥0

which together with (3.24) and the mean value theorem gives rise to α−1/2

sup ρε  L ∞ t≥0

≤ ρ0ε α−1/2 + (ρεα−1/2 )x  L 2 () ≤ C,

(3.27)

where C denotes some generic positive constant depending on E ε (0) but independent of T. With the help of (3.27), we can make use of the idea due to [38] to derive the following a-priori estimate:  sup ρε |u ε |2+ν d x ≤ C, (3.28) 0≤t≤T



which together with (3.27) implies for η(2 + ν) ≥ 1 that     η 2+ν η(2+ν) 2+ν ρε |u ε | dx = ρε |u ε | d x ≤ C sup ρε |u ε |2+ν d x ≤ C, 



0≤t≤T



(3.29)

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H.-L. Li, J. Li, Z. Xin

and, in particular, for η = 1/2 that  √ 2+ν ρε |u ε | dx ≤ C sup 0≤t≤T

(3.30)



with C one generic positive constant depending on both T and E ε (0). Thus, we can follow the compactness arguments in [38] to prove the convergence of the approximate solution (ρε , u ε ) to some expected weak solution (ρ, u) as ε → 0+. Namely, ρε −→ ρ

in C([0, T ] × ),

(3.31)

(ρεα−1/2 )x  (ρ α−1/2 )x weakly in L 2 ( × (0, T )), √ √ ρε u ε −→ ρu, ρεα u ε −→ ρ α u in L 2+ν/2 ( × (0, T )),

(3.32) (3.33)

holds and there exists some function ∈ L 2 ( × (0, T )) such that ρεα u εx 

weakly in L 2 ( × (0, T )), as ε → 0.

(3.34)

Step 3. Justification of the Dirichlet boundary condition (2.16) for α ∈ (1/2, 3/2). For α ∈ (1/2, 1], it follows from (3.11) and (3.29) that (ρε u ε )x  L 2 (0,T ;L (4+2ν)/(4+ν) ())    

   α−1/2 1/2  ρ ≤ C ρεα/2 u εx  2 + C ρ u  2  ε ε ε L (0,T ;L 2 ()) x L (0,T ;L (4+2ν)/(4+ν) ())   

  α−1/2   1/2  ≤ C + C  ρε · ρε u ε  ∞  ∞ 2+ν 2 ≤ C.

x L (0,T ;L ())

L (0,T ;L

())

(3.35)

For α ∈ (1, 3/2), some additional estimates are needed here. First, for γ ≥ (1+α)/2, it is easy to check that for ν defined by (2.17) that (3/2 − α)/(2 + ν) > 1. Hence, for β = (4 + 2ν)/(4 + ν) one deduces from (3.29) and (3.27) that   (ρε u ε )x  2 L (0,T ;L β ())    

 α/2   α−1/2 −α+3/2  ρ ≤ C ρε u εx  2 + C ρ u  2  ε ε ε L (0,T ;L 2 ()) x L (0,T ;L β ())   

  α−1/2   −α+3/2  ≤ C + C  ρε uε  ∞  ∞ ρε 2+ν 2 ≤ C.

x L (0,T ;L ())

L (0,T ;L

())

(3.36)

In the case that 1 ≤ γ < (1 + α)/2, (2.17) yields that (3 − α − γ )(2 + ν) ≥ 2. Hence, one derives from (3.29) and (3.11) that   (ρε u ε )x  2 L (0,T ;L β ())    

   (γ +α−1)/2 (3−α−γ )/2  ρ ≤ C ρεα/2 u εx  2 + C ρ u  2  ε ε ε L (0,T ;L 2 ()) x L (0,T ;L β ())   

   (3−α−γ )/2   ≤ C + C  ρε(γ +α−1)/2  uε  ∞ ρε 2+ν 2 2 ≤ C.

x L (0,T ;L ())

L (0,T ;L

())

(3.37)

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Thus, the estimates (3.4), (3.36) and (3.37) imply that (2.16) still holds for α ∈ (1, 3/2). The proof of Theorem 2.1 on the existence of global entropy weak solution (ρ, u) is completed. 4. Dynamics of Vacuum States for Short Time 4.1. Short time structure of vacuum states. We prove Theorem 2.2 in this subsection in order to study the short time structure of vacuum states for global entropy weak solutions. To this end, it is sufficient to show that there is a unique entropy weak solution in short time for the compressible Navier-Stokes equations (2.1)–(2.2) with initial data (2.3) and boundary value (2.6) or (2.7) under the assumptions of Theorem 2.2 as follows. Proposition 4.1. (Vacuum states for short time). Under the assumptions of Theorem 2.2, there is a short time T∗ > 0 so that the unique entropy weak solution (ρ, ˜ u) ˜ in the sense of Definition 2.5 of the IBVP problem for the compressible Navier-Stokes equations (2.1)–(2.2) with initial data (2.3) and boundary condition (2.6) or (2.7) exists on the domain  × [0, T∗ ]. The initial vacuum state (2.19) or (2.21) is also propagated for the short time t ∈ [0, T∗ ]; more precisely, the properties (2.23)–(2.26) or (2.27)–(2.32) hold respectively for (ρ, ˜ u). ˜ In addition, it holds that (ρ˜ α−1/2 )x  L ∞ (0,T∗ ;L 2 ()) + ((ρ˜ u) ˜ 2 /ρ, ˜ (ρ˜ u) ˜ 2+ν /ρ˜ 1+ν ) L ∞ (0,T∗ ;L 1 ()) ≤ C(T∗ ). (4.1) Proof. The proof of Proposition 4.1 will be completed in Subsects. 4.2–4.3 later. Proof of Theorem 2.2. This is a consequence of Proposition 4.1 and Theorem 2.1. In fact, Proposition 4.1 shows not only that there exists a unique entropy weak solution (ρ, ˜ u) ˜ on the domain ×(0, T∗ ) in the sense of Definition 2.5 to the IBVP problem for the compressible Navier-Stokes equations (2.1)–(2.2) with initial data (2.3) and boundary condition (2.6) or (2.7), but also that this short time entropy weak solution satisfies all the properties (2.23)–(2.32). Now, choose time T∗ = T∗ − δ with δ > 0 a constant small enough. One can verify that at time t = T∗ the density and momentum also satisfies the assumptions (2.4), and particularly ⎧ ⎪ ρ(x, ˜ T∗ ) ≥ 0 on , ρ˜ u(., ˜ T∗ ) = 0, on {x ∈  | ρ0 (x) = 0}, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ρ(., ˜ T∗ ) ∈ L 1 (), (ρ˜ α−1/2 (., T∗ ))x ∈ L 2 (), (4.2) ⎪ ⎪ ⎪ ⎪ ⎪ 2 2+ν ⎪ ˜ ˜ ∗ )| ∗ )| ⎩ |m(.,T + |ρm(.,T ∈ L 1 (). ρ(x,T ˜ ˜ 1+ν (x,T ) ∗) ∗

ˆ Thus, it follows from Theorem 2.1 that there is a global entropy weak solution (ρ, ˆ u) for time t ≥ T∗ in the sense of Definition 2.5 to the IBVP problem for the compressible Navier-Stokes equations (2.1)–(2.2) with initial data (ρ, ρu)(x, T∗ ) = (ρ, ˜ ρ˜ u)(x, ˜ T∗ ), x ∈  and boundary condition (2.6) or (2.7). Define (ρ, u) as
(ρ, ˜ u), ˜ for (x, t) ∈  × [0, T∗ ], (ρ, u) = ˆ (ρ, ˆ u), for (x, t) ∈  × [0, T∗ ].

(4.3)

(4.4)

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It is easy to verify that (ρ, u) is a global entropy weak solution in the sense of Definition 2.5 to the compressible Navier-Stokes equations (2.1)–(2.2) with initial data (2.3) and boundary condition (2.6) or (2.7) under the assumptions of Theorem 2.2. This global entropy weak solution is actually unique and regular for short time t ∈ [0, T∗ ] and satisfies all the properties (2.23)–(2.32). The proof of Theorem 2.2 is complete.   4.2. Compressible Navier-Stokes in Lagrangian coordinates. In order to prove Proposition 4.1, we present an equivalent proposition for the compressible Navier-Stokes equations in the Lagrangian coordinates (y, t), instead of the Eulerian coordinates (x, t), through the coordinate transformation  x y= ρ(z, t)dz, x ∈ (0, 1), t ≥ 0 (4.5) 0

for the Dirichlet boundary condition, or  X 0 (t)+x y= ρ(z, t)dz, x ∈ (0, 1), t ≥ 0 X 0 (t)

(4.6)

for the periodic boundary condition, where x = X 0 (t) is a particle path. In addition, both the case of initial one point vacuum state and the case of a piece of initial continuous vacuum states will be studied with some additional initial regularities. We first describe the equivalent proposition for the IBVP problem for the compressible Navier-Stokes equations in the Lagrangian coordinates in the case of initial one point vacuum state. Thus, consider the compressible Navier-Stokes equations ρt + ρ 2 u y = 0, u t + p(ρ) y − (ρµ(ρ)u y ) y = 0,

y ∈ , t > 0

(4.7) (4.8)

with initial data (ρ, u)(y, 0) = (ρ0 (y), u 0 (y)),

y ∈ ,

(4.9)

and the Dirichlet boundary condition u(0, t) = u(1, t) = 0, t ≥ 0

(4.10)

or the periodic boundary condition (ρ, u) is periodic w.r.t. x of period one.

(4.11)

For the case of Dirichlet boundary, the initial data (ρ0 , u 0 ) is assumed to be consistent with the boundary values. Moreover, we assume that there is a one point vacuum state at y = y0 for some fixed point y0 ∈ (0, 1) and that the initial data is of additional regularity ⎧ β β ⎪ ⎨ A0 |y − y0 | ≤ ρ0 (y) ≤ A1 |y − y0 | , y ∈ , (4.12) ⎪ γ ⎩u ∈ C 1 (), 1+α 2 2n ¯ (ρ0 ) y , (ρ0 u 0y ) y ∈ L () ∩ L () 0 with an integer n ≥ 2, where A0 , A1 are positive constants, and the constant β ∈ (β− , β+ ) with β± is determined by Remark 4.3. Then the following result holds in the case of the initial point vacuum state.

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Proposition 4.2. (Point vacuum state for short time). Assume that (2.22) and (4.12) hold. Then, there is a time T∗ > 0 so that the unique regular weak solution (ρ, u) with point vacuum of the IBVP (4.7)–(4.9) with boundary condition (4.10) or (4.11) exists on the domain  × [0, T∗ ] and satisfies ¯ × [0, T∗ ]) ∩ C 1 ([0, T∗ ]; L 2 ()), ρ ∈ C 0 ( ¯ × [0, T∗ ]) ∩ C 1 ([0, T∗ ]; L 2 ()), u ∈ C 0 (

(4.14)

ρ uy ∈ L (ρ α ) y  L ∞ ([0,T∗ ];L 2 ()) + ρu y  L ∞ (×[0,T∗ ]) ≤ C(T∗ ).

(4.15) (4.16)

1+α

∞

( × [0, T∗ ]) ∩ C 1/2 ([0, T∗ ]; L 2 ()),

(4.13)

In addition, the initial point vacuum state is maintained for the short time β

ρ(y0 , t) = 0, t ∈ [0, T∗ ], β

a− |y − y0 | ≤ ρ(y, t) ≤ a+ |y − y0 | , (y, t) ∈ [0, 1] × [0, T∗ ].

(4.17) (4.18)

Here, a± are positive constants independent of T∗ . Proof. The proof of Proposition 4.1 will be given in Subsect. 4.3. Remark 4.3. The choice of β± > 0 is such that 1 , β− = max{ 2α

1 1 γ (1 − 2n )},

β+ = min{1,

1 1 1 1 α (1 − 2n ), 1+3α (4 − n )}

(4.19)

for integer n ≥ 2. It should be emphasized here that all the assumptions required here are satisfied for the shallow water equations (1.2), i.e., γ = 2, α = 1. Next, we describe the equivalent proposition for the IBVP problem for the compressible Navier-Stokes equations in the Lagrangian coordinates for the case of initial continuous vacuum of one piece. In this case, we consider the IBVP problem for the compressible Navier-Stokes equation (4.7)–(4.9) with one of following boundary conditions: (1) Mixed boundary condition
u(0, t) = ρ(1, t) = 0, t ≥ 0, A0 (1 − y)β ≤ ρ0 (y) ≤ A1 (1 − y)β , y ∈ , or (2) Mixed free boundary condition
ρ(0, t) = u(1, t) = 0, t ≥ 0, B0 y β ≤ ρ0 (y) ≤ B1 y β ,

y ∈ ,

or (3) Free boundary condition
ρ(0, t) = ρ(1, t) = 0, t ≥ 0, A0 (y(1 − y))β ≤ ρ0 (y) ≤ A1 (y(1 − y))β , y ∈ ,

(4.20)

(4.21)

(4.22)

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where, β ∈ (β− , β+ ), and A0 , A1 , B0 , B1 , and A0 , A1 are some given positive constants. The initial data is also assumed to be regular γ

¯ (ρ0 ) y , (ρ01+α u 0y ) y ∈ L 2 () ∩ L 2n () u 0 ∈ C 1 (),

(4.23)

with an integer n ≥ 2. Then the following results for free boundary value problems for the compressible Navier-Stokes equations hold. Proposition 4.4. (Continuous vacuum state for short time). Assume that (2.22) and (4.23) hold. Then, there is a time T∗ > 0 so that the unique regular weak solution (ρ, u) of the IBVP problem for the compressible Navier-Stokes equations (4.7)–(4.8) with initial data (4.9) and free boundary condition (4.20), or (4.21), or (4.22) exists on the domain  × [0, T∗ ] and satisfies ¯ × [0, T∗ ]) ∩ C 1 ([0, T∗ ]; L 2 ()), ρ ∈ C 0 ( ¯ × [0, T∗ ]) ∩ C 1 ([0, T∗ ]; L 2 ()), u ∈ C 0 (

ρ 1+α u y ∈ L ∞ ( × [0, T∗ ]) ∩ C 1/2 ([0, T∗ ]; L 2 ()), and (ρ α ) y  L ∞ ([0,T∗ ];L 2 ()) + ρu y  L ∞ (×[0,T∗ ]) ≤ C(T∗ ). In addition, the initial vacuum state is also maintained for short time, namely, it holds that a− (1 − y)β ≤ ρ(y, t) ≤ a+ (1 − y)β , (y, t) ∈ (0, 1) × [0, T∗ ], corresponding to the mixed free boundary conditions (4.20), or b− y β ≤ ρ(y, t) ≤ b+ y β , (y, t) ∈ (0, 1) × [0, T∗ ] corresponding to the mixed free boundary conditions (4.21), or c− (y(1 − y))β ≤ ρ(y, t) ≤ c+ (y(1 − y))β , (y, t) ∈ (0, 1) × [0, T∗ ] corresponding to the free boundary (4.22). Here, a± , b± and c± are positive constants independent of T∗ . Proof. The short time existence, uniqueness and regularity of weak solutions of the free boundary problems for the compressible Navier-Stokes equations are well-investigated by many authors (see [23,32,33,56,57]). Proposition 4.4 can be proved in a similar way as [23,32,33,56], so we omit the details. Remark 4.5. Based on Proposition 4.4 for the compressible Navier-Stokes equations (4.7)–(4.9) with either mixed free boundary condition (4.20) and (4.21) or free boundary (4.22) and the coordinates transformation from the Lagrangian coordinates to the Eulerian coordinates, one can study easily the IBVP problem for compressible Navier-Stokes equations (2.1)–(2.2) in the Eulerian coordinates with either the Dirichlet boundary condition (2.6) or the periodic boundary condition (2.7) in the case of initial continuous vacuum state of one piece (2.20). In fact, for the compressible Navier-Stokes equations (2.1)–(2.2) with Dirichlet boundary condition (2.6) in the Eulerian coordinates, the short time existence of a unique

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solution subject to the case of a piece of initial continuous vacuum state (2.20)–(2.21) in initial data can be constructed, in the Lagrangian coordinates, by combining two mixed free boundary value problems (4.20) and (4.21) together with one continuous vacuum state in-between as follows. Denote two particle paths x = X i (t) (assumed to be definable for short time) starting from the initial vacuum boundary x = x0 and x = x1 respectively as X˙ i (t) = u(X i (t), t),

X i (0) = xi , i = 0, 1,

along which the vacuum boundary moves in the Eulerian coordinates so that ⎧ ⎪ x ∈ [X 0 (t), X 1 (t)], t ≥ 0, ⎨(ρ, ρu)(x, t) = 0, ⎪ ⎩ρu(x, t) > 0,

(4.24) x ∈ [0, X 0 (t)) ∪ (X 1 (t), 1], t ≥ 0.

We first choose the coordinate transformation from the Eulerian coordinates to the Lagrangian coordinates as ⎧ x ⎪ ⎨ y = 0 ρ(z, t)dz, x ∈ [0, X 0 (t)], (4.25) ⎪ ⎩ y =  X 0 (t) ρ(z, t)dz =  x0 ρ (z)dz < 1, conservation of mass, 0 0 0 0 which gives y ∈ [0, y0 ] and the mixed free boundary conditions (4.20). The application of Proposition 4.4  ywith  replaced by (0, y0 ) implies, via the inverse coordinate transformation x = 0 ρ −1 (z, t)dz, y ∈ [0, y0 ], the existence of unique solution (ρl , u l ) of the compressible Navier-Stokes equations (2.1)–(2.2) on [0, X 0 (t)] × [0, T∗ ] in the Eulerian coordinates with the initial data (ρl , u l )(x, 0) = (ρ0 , u 0 )(x), x ∈ (0, x0 ), ρ0 (x0 ) = 0, and mixed free boundary conditions u(0, t) = 0, ρ(X 0 (t), t) = 0, t ∈ [0, T∗ ]. Next, we choose the coordinate transformation from the Eulerian coordinates to the Lagrangian coordinates as
1 y = 1 − x ρ(z, t)dz, x ∈ [X 1 (t), 1], 1 1 (4.26) y1 = X 1 (t) ρ(z, t)dz = x1 ρ0 (z)dz < 1 conservation of mass, which gives y ∈ [1 − y1 , 1] and the mixed free boundary conditions (4.21). Then 1 Proposition 4.4 implies, via the inverse coordinate transformation x = 1− y ρ −1 (z, t)dz, y ∈ [1− y1 , 1], the existence of unique (ρr , u r ) of the compressible Navier-Stokes equations (2.1)–(2.2) on [X 1 (t), 1] × [0, T∗ ] in the Eulerian coordinates with initial data (ρr , u r )(x, 0) = (ρ0 , u 0 )(x), x ∈ (x1 , 1), ρ0 (x1 ) = 0, and mixed free boundary conditions u(1, t) = 0, ρ(X 1 (t), t) = 0, t ∈ [0, T∗ ].

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Consequently, we can construct the short time unique solution (ρ, u) to the IBVP problem for the compressible Navier-Stokes equations (2.1)–(2.3) with the Dirichlet boundary condition (2.6) and a piece of continuous vacuum state (2.20) in the initial data as ⎧ ⎪ ⎨(ρl , ρl u l ), on [0, X 0 (t)] × [0, T∗ ], (ρ, ρu) = (0, 0), on (X 0 (t), X 1 (t)) × [0, T∗ ], ⎪ ⎩(ρ , ρ u ), on [X (t), 1] × [0, T ]. r r r 1 ∗ Similarly, we can obtain the short time existence of unique solution to the IBVP problem for the compressible Navier-Stokes equations (2.1)–(2.2) with periodic boundary condition (2.7) which can be viewed as a free boundary problem after the choice of the spatial reference point. The details will be omitted.

4.3. Proof of Propositions 4.1–4.2. We first prove Proposition 4.2 in this subsection with the help of the a-priori estimates for (regularized) solutions and the construction of approximate solutions by a finite difference scheme, due to the modification of the ideas used in [23,33,43,56]. Without the loss of generality, we only prove Proposition 4.2 in the case of the Dirichlet boundary conditions with one point vacuum state in the initial data. First, we can easily derive some identities for (regularized) solutions as in [23,56]. Lemma 4.6. Let T > 0 and assume that the solution (ρ, u) of the IBVP problem (4.7)– (4.10) exists for t ∈ [0, T ] with ρ(y0 , t) = 0. Then, under the assumptions of Proposition 4.2, it holds that  y  y0 1+α γ ρ u y (y, t) = u t (z, t)dz + ρ (y, t) = − u t (z, t)dz + ρ γ (y, t), (4.27) y0

ρ α (y, t) + α

y



t 0

ρ γ (y, s)ds =ρ0α (y) + α

 t

=ρ0α (y) − α

0

y0

 t 0

u t (z, s)dzds

y y

(4.28) u t (z, s)dzds

y0

for all y ∈  with y = y0 . We also have the following useful a-priori estimates, whose proofs are similar to those for (3.11) and (3.19) and thus will be omitted. Lemma 4.7. Let T > 0 and assume that the solution (ρ, u) of the IBVP problem (4.7)– (4.10) exists for t ∈ [0, T ]. Then, under the assumptions of Proposition 4.2, it holds that  t γ −1 u(t)2L 2 () + ρ(t) L γ −1 () + |(ρ (γ +α)/2 ) y |2 d xds ≤ C0 , t ∈ [0, T ] , 0

ρ(t) L ∞ () + (ρ α (t)) y  L 2 () ≤ C0 , t ∈ [0, T ], with C0 > 0 a constant.

(4.29)

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Lemma 4.8. Let T > 0 and assume that the solution (ρ, u) of the IBVP problem (4.7)– (4.10) exists for t ∈ [0, T ] with ρ(y0 , t) = 0. Then, under the assumptions of Proposition 4.2, there is a time T∗ ∈ (0, T ] (depending on initial data) so that it holds that   t 2j 2 j−2 1+α 2 u t (y, t)dy + ρ u yt d xds ≤ C1 (4.30) ut 0

[0, T∗ ]

uniformly for t ∈ with j = 1, n and C1 > 0 a positive constant, and that ¯ is uniformly bounded for any t ∈ [0, T∗ ], ρu y (t) ∈ C 0 () ρu y  L ∞ (×[0,T∗ ]) ≤ C2

(4.31)

with C2 > 0 a constant. Moreover, the solution (ρ, u) is continuous ¯ × [0, T∗ ]), (ρ, u) ∈ C 0 (

(4.32)

and the initial one point vacuum state is maintained a− |y − y0 |β ≤ ρ(y, t) ≤ a+ |y − y0 |β , (y, t) ∈  × [0, T∗ ]

(4.33)

with a+ > a− > 0 two constants independent of time T∗ . Proof. Let us first assume that the weak solution is regular enough so that we can differentiate it through the equations and the interface as in [56], and the density is of the form a∗ |y − y0 |β ≤ ρ(y, t) ≤ a ∗ |y − y0 |β , (y, t) ∈  × [0, T ]

(4.34)

with a∗ , a ∗ two positive constants to be determined later. It will be assumed further that the following a-priori estimate holds: ρu y  L ∞ (×[0,T ]) ≤ M0

(4.35)

for some positive M0 to be determined later. We will prove (4.30) only for j = n below since the other case can be treated similarly. Taking the inner product between (4.8)t and 2n(u t )2n−1 over  leads to    d 2n γ 2n−1 dy − 2n (ρ 1+α u y ) yt (u t )2n−1 dy = 0 |u t | dy + 2n (ρ ) yt (u t ) dt from which, one deduces after integration by parts and using Eq. (4.7) that   d |u t |2n dy + 2n(2n − 1) ρ 1+α u 2yt (u t )2n−2 dy dt  =2n(2n − 1)γ ρ γ −1 ρt u yt (u t )2n−1 dy  + 2n(2n − 1)(1 + α) ρ α ρt u y u yt (u t )2n−2 dy  = − 2n(2n − 1)γ ρ γ +1 u y u yt (u t )2n−2 dy  − 2n(2n − 1)(1 + α) ρ 2+α u 2y u yt (u t )2n−2 dy

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 ≤ n(2n − 1) ρ 1+α u 2yt (u t )2n−2 dy   2γ +1−α 2 2n−2 +C ρ u y (u t ) dy + C ρ 3+α u 4y (u t )2n−2 dy  2γ −1−α ≤ n(2n − 1) ρ 1+α u 2yt (u t )2n−2 dy + Cρ L ∞ ρu y 2L ∞ u t (t)2n−2 L 2n−2  + C ρ 3+α u 4y (u t )2n−2 dy  2γ −1−α 2 ≤ n(2n − 1) ρ 1+α u 2yt (u t )2n−2 dy + CC0 M0 (1 + u t (t)2n+2 ) L 2n−2  (4.36) + C ρ 3+α u 4y (u t )2n−2 dy, where one has used (2.22), (4.29), the a-priori assumption (4.35), and Hölder’s inequality. The last term on the right hand side of (4.36) can be estimated as follows. For the case α ≥ 1, it follows from (4.29) and (4.35) that  ρ 3+α u 4y (u t )2n−2 dy ≤ρ 3+α u 4y  L ∞ u t (t)2n−2 ≤ C0α−1 M04 u t (t)2n−2 L 2n−2 L 2n−2 ≤ C M04 (1 + u t (t)2n+2 ). L 2n

(4.37)

For the case α ∈ ( 21 , 1), since it holds by (4.27) that  y |ρ 1+α u y | ≤ | u t (z, t)dz| + ρ γ ≤ u t  L 2n · |y − y0 |(2n−1)/2n + ρ γ ,

(4.38)

y0

which together (4.34) implies |ρ (3+α)n u 4n y |

=|ρ

−(1+3α)n

(ρ

1+α

uy) | = ρ 4n

−(1+3α)n



y

u t (z, t)dz + ρ

γ

4n

y0

≤ Cρ −(1+3α)n (u t 4n · |y − y0 |2(2n−1) + ρ 4nγ ) L 2n · |y − y0 |−(1+3α)nβ+2(2n−1) + ρ 3n(γ −α)+n(γ −1) , ≤ Cu t 4n L 2n

(4.39)

the last term in (4.36) is estimated by  1/n  (3+α)n 4n · u dy | ρ 3+α u 4y (u t )2n−2 dy| ≤ u t 2n−2 ρ y L 2n ≤ Cu t 2n−2 L 2n

 1/n   4n −(1+3α)nβ+2(2n−1) 3n(γ −α)+n(γ −1) · u t  L 2n |y − y0 | dy + ρ dy

≤ C(u t 2n+2 + 1), L 2n

(4.40)

4n−1 where one has used the fact β < n(1+3α) due to (4.19). Substituting (4.37) and (4.40) into (4.36) shows   d 2n |u t | dy + n(2n − 1) ρ 1+α u 2yt (u t )2n−2 dy dt ≤ C M02 (1 + M02 )(1 + u t (t)2n+2 ) ≤ C(1 + M04 )(u t (t)2n+2 + 1). (4.41) L 2n−2 L 2n

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Set u t (0)2n2n

2 −1 Ta = min{ n2n C(1+M 4 )C , n

0

L

C(1+M04 )

1

, T }.

(4.42)

One can apply Grönwall’s Lemma to obtain (4.30) for t ∈ [0, Ta ] with C1 given by C1 =: 2u t (0)2n ≥ u t (0)2n + C Ta (1 + M04 ). L 2n L 2n

(4.43)

To prove (4.31) and ensure the a-priori assumption (4.35), we use the equality (4.27) to get that near y = y0 , it holds that  y −α ρu y (y, t) = ρ (y, t) u t (z, t)dz + ρ γ −α (y, t), y = y0 , (4.44) y0

  1 which, together with (2.22), (4.29), and the fact β ≤ α1 1 − 2n due to (4.19), implies  y γ −α ≤C0 + |ρ −α (y, t) u t (z, t)dz| L ∞ ρu y  L ∞ ×[0,t] ×[0,t] γ −α ≤ C0 γ −α ≤ C0

y0 −α + a∗ |y − y0 |1−1/2n−αβ u t (t) L 2n γ −α 1/2n + a∗−α C1 ≤ C0 + 2a∗−α u t (0) L 2n

= : C 2 ≤ M0 so long as the constant M0 is chosen as

γ −α

M0 = 1 + C 0

+ 2a∗−α u t (0) L 2n .

(4.45)

Next, we verify the a-priori assumption (4.34). Set Tb = min{

Aα0

1/2n

3αC1

, Ta }.

It follows from Eq. (4.28) for y = y0 and t ∈ [0, Tb ] that  t  t ρ α (y, t) + α ρ γ (y, s)ds = ρ0α (y) + α 0

0

(4.46)

y

u t (z, s)dzds

y0

≥ Aα0 |y − y0 |αβ − αTb |y − y0 |1−1/2n sup u t (t) L 2n 0≤t≤Tb

≥

Aα0 |y

αβ

1/2n − C1 αTb |y

− y0 | − y0 | (4.47) 2 α 1 1/2n = A0 |y − y0 |αβ + |y − y0 |αβ ( Aα0 − C1 αTb |y − y0 |1−1/2n−αβ ) 3 3 2 ≥ Aα0 |y − y0 |αβ , 3   1 due to (4.19). On the where we have used the facts |y − y0 | ≤ 1 and β ≤ α1 1 − 2n other hand, it follows from (4.28) that  t  t y ρ α (y, t) + α ρ γ (y, s)ds = ρ0α (y) + α u t (z, s)dzds ≤ ≤

1−1/2n

0 0 y0 α αβ 1−1/2n A1 |y − y0 | + αt|y − y0 | u t (t) L 2n 1/2n α αβ (A1 + C1 αt)|y − y0 | .

(4.48)

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Define Tc ∈ (0, Tb ] by Tc = min{

Aα0 3αC(Aα1 + C1

1/2n

αTb )γ /α

, Ta , Tb }.

(4.49)

Set 

t

Z (t) =

ρ γ (y, s)ds.

0

It follows from (4.48) that (Z  (t))α/γ ≤ (Aα1 + C1

1/2n

αt)|y − y0 |αβ ,

which implies for t ∈ [0, Tc ] that 

t 0

ρ γ (y, s)ds ≤ Tc (Aα1 + C1

1/2n

αTc )γ /α |y − y0 |(γ −α)β |y − y0 |αβ

≤ C Tc (Aα1 + C1

1/2n

αTb )γ /α |y − y0 |αβ .

(4.50)

As a consequence of (4.48), (4.50), and (4.49), one gets 2 ρ (y, t) ≥ Aα0 |y − y0 |αβ − α 3 α



t

ρ γ (y, s)ds ≥

0

1 α A |y − y0 |αβ . 3 0

(4.51)

From (4.48) and (4.51), we can verify the a-priori assumption (4.34) and justify the property (4.33) by simply choosing a∗ = a− = (Aα0 /3)1/α , a ∗ = a+ = (Aα1 + C1

1/2n

αTc )1/α

(4.52)

for any t ∈ [0, T∗ ] with time T∗ = Tc determined by (4.49). One then derives from Eq. (4.7), (4.29), and (4.31) that ρ α ∈ L ∞ (0, T∗ , H 1 ()), (ρ α )t ∈ L ∞ (0, T∗ , L 2 ()),

(4.53)

while (4.31), (4.33), boundary condition (4.10), and (4.30) for j = 1 imply that u ∈ L ∞ (0, T∗ , W0 ()), u t ∈ L ∞ (0, T∗ , L 2 ()) 1, p

(4.54)

for any p ∈ (1, β −1 ), where one has used the fact β −1 > β+−1 ≥ 1 so that sup u y  L p () = sup ρ − p  L 1 () · ρu y  L ∞ (×[0,T  ]) ≤ C. p

t∈[0,T∗ ]

t∈[0,T∗ ]

∗

(4.55)

Equations (4.53)–(4.55) imply the continuity (4.32) of the solution (ρ, u), and the continuity of ρu y follows from Eq. (4.27) and that of (ρ, u). The proof of the lemma is completed.   Using Lemmas 4.6–4.8 and a direct computation, we can obtain the following result:

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Lemma 4.9. Let T∗ > 0 be given in Lemma 4.8 and (ρ, u) be the solution of the IBVP problem (4.7)–(4.10). Then, under the assumptions of Proposition 4.2, ρ 1+α u y (t) ∈ ¯ is uniformly bounded for any t ∈ [0, T∗ ], C 0 ()  lim ρ 1+α u y (y, t) = 0, ρ 1+α u y  L ∞ (0,T∗ ;C 0 ()) ¯ ≤ C(T∗ ),

y→y0

and (ρ, u) satisfies for 0 ≤ s < t ≤ T∗ that ρ(t) − ρ(s) L 2 () + u(t) − u(s) L 2 () ≤ C(T∗ )|t − s|, ρ 1+α u y (t) − ρ 1+α u y (s) L 2 () ≤ C(T∗ )|t − s|1/2 . ¯ and lim y→y0 Proof. The conclusions that for any t ∈ [0, T∗ ], ρ 1+α u y (t) ∈ C 0 () 1+α ρ u y (y, t) = 0 are due to the continuity of right hand side terms of (4.44) and (4.34). By (4.31) and (4.33), one can check easily that ρ 1+α u y  L ∞ (×[0,T∗ ]) ≤ ραL ∞ (×[0,T  ]) · ρu y  L ∞ (×[0,T∗ ]) ≤ C(T∗ ). ∗

Making use of Eq. (4.7), (4.30) and (4.31), one can obtain  t  t ρ(t) − ρ(s) L 2 () ≤ ρt (τ ) L 2 () dτ =  ρ 2 u y (τ ) L 2 () dτ s

s

≤ C(t − s)ρ L ∞ (×[0,T∗ ]) · ρu y  L ∞ (×[0,T∗ ]) ≤ C(T∗ )|t − s|,  u(t) − u(s) L 2 () ≤ 

t s

 u t (τ )dτ  L 2 () ≤ C

s

t

u t  L 2 () ≤ C(T∗ )|t − s|,

and  t ρ 1+α u y (t) − ρ 1+α u y (s) L 2 () ≤  (ρ 1+α u y (t))t dτ  L 2 () s  t  t ≤ C( ρt ρ α u y (τ ) L 2 () +  ρ 1+α u yt (τ )dτ  L 2 () ) s s  t  t ρ 2+α u 2y (τ ) L 2 () +  ρ 1+α u yt (τ )dτ  L 2 () ) ≤ C( s s  2+α 2 ∞ ≤ C(T∗ )(|t − s|ρ u y  L (×[0,T∗ ]) ≤C(T∗ )|t − s|1/2 .

The proof is completed.

+ |t − s|1/2 ρ (1+α)/2 u yt (τ ) L 2 (×[0,T∗ ]) )

 

Proof of the Proposition 4.2. With the help of Lemmas 4.6–4.8, we are ready to prove Proposition 4.2. (1) Existence of weak solution for short time. We only deal with the case for the Dirichlet boundary condition and one point vacuum state in the initial data, the case of periodic boundary and one point vacuum state in initial data can be done in a similar way. Once the a-priori estimates are established as in Lemmas 4.6–4.9, we are able to prove Proposition 4.2. First of all, we construct a sequence of approximate solutions by

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H.-L. Li, J. Li, Z. Xin

modifying the finite difference scheme used in [33,40,43]. Without the loss of generality, we assume y0 =

1 , namely, ρ0 ( 21 ) = 0. 2

(4.56)

For any given positive integer N = 2k + 1 with k ≥ 0 an integer, let h = 1/N . Consider the system of 2N ordinary differential equations ⎧ h h d h ⎪ h 2 u 2n+2 − u 2n ⎪ + (ρ ) ρ = 0, ⎪ 2n+1 2n+1 ⎪ ⎪ h ⎪ dt ⎨ h h (ρ2n+1 )γ − (ρ2n−1 )γ d h (4.57) u + ⎪ ⎪ dt 2n h ⎪ ⎪ ⎪ ⎪ ⎩ = 1 {(ρ h )1+α (u h − u h )/ h − (ρ h )1+α (u h − u h )/ h}, 2n+2 2n 2n−1 2n 2n−2 h 2n+1 with the boundary condition and point vacuum h u 0h (t) = u 2N (t) = 0, ρ2k+1 (t) = 0,

(4.58)

and initial data h ρ2n+1 (0) = ρ0 ((2n + 1) h2 ),

h u 2n (0) = u 0 ((2n) h2 ),

(4.59)

where n = 1, 2, 3, ..., N . Here, we also assume h h h h h h ρ2N +1 (t) = ρ2N −1 (t), ρ1 (t) = ρ−1 (t), u 2N +2 (t) = u −2 (t) = 0

which is consistent with the boundary condition and the fact that density is continuous and non-zero at the left boundary. By applying the idea as in [32,43] and the similar arguments mentioned above, we can obtain the following uniform (w.r.t. h) a-priori estimates about the solutions (ρ2n+1 , u 2n ) (here and below we omit the symbol h for simplicity) of (4.57)–(4.59) similar to Lemmas 4.6–4.9. Details will be omitted (the reader can refer to [32,43] for similar arguments in detail). Lemma 4.10. Let (ρ2n+1 , u 2n ) be the solution of (4.57)–(4.59). Then, it holds  t N N  1+α u 2n+2 (s)−u 2n (s) 2 1 2 ( 2 u 2n (t) + π(ρ2n+1 (t)))h + ρ2n+1 ( ) hds h 0 n=1

n=0

=

N  ( 21 u 22n (0) + π(ρ2n+1 (0)))h, n=0 N −1  n=0

−1 ρ2n+1 (t)h =

N −1 

−1 ρ2n+1 (0)h.

n=0

It follows from Lemma 4.10 and the standard theory of ordinary differential equations that there exists a global solution (ρ2n+1 , u 2n ) to (4.57)–(4.59) for any fixed positive N and h. Furthermore, the following properties hold:

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431

Lemma 4.11. Let (ρ2n+1 , u 2n ) with n = 0, 2, 3, ..., N − 1 with N = 2k + 1 be the solution of (4.57)–(4.59). Then, it holds for n ≥ k + 1 that  t   t n d γ α α ρ2n+1 (t) = ρ2n+1 (0) − α u 2 j (s)hds − α ρ2n+1 (s)ds, dt 0 0 j=k+1

1+α ρ2n+1 (t)

n  d u 2n+2 (t) − u 2n (t) γ = u 2 j (t)h + ρ2n+1 , h dt j=k+1

and for n ≤ k that α ρ2n−1 (t)

=

α ρ2n−1 (0) − α

 t  t k d γ u 2 j (s)hds − α ρ2n−1 (s)ds, dt 0 0 j=n

 d u 2n (t) − u 2n−2 (t) γ =− u 2 j (t)h + ρ2n−1 . h dt k

1+α ρ2n−1 (t)

j=n

Lemma 4.12. Under the assumptions of Proposition 4.2, there is a short time T∗ > 0 so that it holds for n = 0, 1, 2, ..., N that ρ2n+1 (t) +

N 

(

n=1 1 a− | 2 (2n

α α ρ2n+1 (t)−ρ2n−1 (t) 2 ) h h

2n (t) + |ρ2n+1 (t) u 2n+2 (t)−u | ≤ C(T∗ ), h

+ 1)h − 21 |β ≤ ρ2n (t) ≤ a+ | 21 (2n + 1)h − 21 |β ,

and for m = 1 or n that  t N N  d d 2m 2m−2 dt u 2n (s)− dt u 2n−2 (s) 2 d d ( dt u 2 j (t)) h + ρ21+α ( ) )h ≤ C(T∗ ). j+1 (s)( dt u 2 j (s)) h 0 j=0

j=0

Here C(T∗ ) > 0 and a± > 0 are constants. Lemma 4.13. Under the assumptions of Proposition 4.2 and Lemma 4.12, it holds for t ∈ [0, T∗ ] that 1+α 2n (t) |ρ2n+1 (t) u 2n+2 (t)−u | ≤ C(T∗ ), h

|u 2n (t)| +

N 

|u 2n (t) − u 2n−2 (t)| ≤ C(T∗ ),

n=1 N 

|(ρ2n+1 )1+α (u 2n+2 − u 2n )/ h − (ρ2n−1 )1+α (u 2n − u 2n−2 )/ h| ≤ C(T∗ ),

n=1 N  n=1 N  n=1

|ρ2n (t) − ρ2n (s)|2 h +

N 

|u 2n−1 (t) − u 2n−3 |2 h ≤ C(T∗ )|t − s|2 ,

n=1

2n−1 (t) 2n−3 (s) 2 |(ρ2n )1+α (t) u 2n+1 (t)−u − (ρ2n−2 )1+α (s) u 2n−1 (s)−u | h ≤ C(T∗ )|t − s|. h h

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With the help of Lemmas 4.10–4.13, we can define the sequence of approximate solutions (ρh , u h ) on the domain  × [0, T∗ ] as
ρh (y, t) = ρ2n+1 (t), u h (y, t) = h1 [(y − (n − 21 )h)u 2n+1 (t) + ((n + 21 )h − y)u 2n−1 (t)] for y ∈ ((2n) 21 h, (2n + 2) 21 h). It can be verified that the following properties hold for the approximate solutions ∂ y u h (y, t) = and

u 2n+1 (t) − u 2n−1 (t) h


a− |y − y0 |β ≤ ρh (y, t) ≤ a+ |y − y0 |β , |u h (y, t)| ≤ C(T∗ ), |ρh (y, t)∂x u h (y, t)| ≤ C(T∗ ),

for (y, t) ∈  × [0, T∗ ]. Then, the existence of a weak solution for short time t ∈ [0, t∗ ] follows from Helly’s theorem, the diagonal process together with Lebesgue’s theorem, and the a-priori estimates (see, for instance, [33]). The details are omitted. (2) Uniqueness of weak solution for short time. We prove the uniqueness of weak solutions for α ∈ ( 21 , γ ). Without loss of generality, only the case for the Dirichlet boundary condition (4.10) will be studied. Let (ρ1 , u 1 ) and (ρ2 , u 2 ) be two weak solutions of Eq. (4.7)–(4.10) satisfying Lemma 4.6–4.9. Denote n = ρ1 − ρ2 , ψ = u 1 − u 2 , (x, t) ∈ (0, 1) × [0, T ]. Obviously, the new unknown (n, ψ) with ψ(0, t) = ψ(1, t) = n(y0 , t) = 0 satisfies   n + ψ y = 0, y = y0 , (4.60) ρ1 ρ2 t γ

γ

ψt + (ρ1 − ρ2 ) y − (ρ11+α ψ y ) y − ((ρ11+α − ρ21+α )u 2y ) y = 0,

(4.61)

for (y, t) ∈ (0, 1) × (0, T∗ ] with zero initial data (n(y, 0), ψ(y, 0)) = (0, 0),

y ∈ (0, 1).

(4.62)

Take the inner product between ρ1α ρ2−1 n and (4.60) over [0, y0 ) ∪ (y0 , 1] to obtain    d ρ1−1+α ρ2−2 n 2 dy = −(1 + α) u 1y ρ1α ρ2−2 n 2 dy − 2 ρ1α ρ2−1 nψ y dy dt   1 ≤ (Cρ1 u 1y  L ∞ + 4) ρ1−1+α ρ2−2 n 2 dy + ρ11+α ψ y2 dy 4   1 ≤ C(T∗ ) ρ1−1+α ρ2−2 n 2 dy + (4.63) ρ11+α ψ y2 dy, 4 1  y where f d x =: 0 0 f d x + y0 f d x. Taking the inner product between ψ and (4.61) over , and noting that ψ(0, t) = ψ(1, t) = n(y0 , t) = 0 and ρi1+α u i (y0 , t) = 0, i = 1, 2, we have

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433

   1 d γ γ ψ(t)2 + ρ11+α ψ y2 dy = (ρ1 − ρ2 )ψ y dy − (ρ11+α − ρ21+α )u 2y ψ y dy 2 dt  γ  1+α γ ρ1 − ρ2 ρ1 − ρ21+α = nψ y dy − nu 2y ψ y dy ρ1 − ρ2 ρ1 − ρ2    γ  γ 2 ρ1 − ρ2 1 ρ1−1−α n 2 dy + ≤C ρ11+α ψ y2 dy ρ1 − ρ2 8 2   1+α  ρ1 − ρ21+α 1 ρ1−1−α n 2 u 22y dy + +4 ρ11+α ψ y2 dy ρ1 − ρ2 8   1 2(γ −1) 2(γ −1) −2α 2 1+α 2 + ρ2 )ρ1 ρ2 ρ1−1+α ρ2−2 n 2 dy ≤ ρ1 ψ y dy + C sup (ρ1 4 y= y0  + C sup (ρ12α + ρ22α )ρ1−2α ρ2 u 2y 2L ∞ ρ1−1+α ρ2−2 n 2 dy ≤

1 4



y= y0

ρ11+α ψ y2 dy + C sup |y − y0 |[2(γ −1)+2−2α]β y= y0

+ C sup |y − y0 | ≤

1 4



[2α−2α]β

y= y0

ρ11+α ψ y2 dy + C







ρ1−1+α ρ2−2 n 2 dy

ρ1−1+α ρ2−2 n 2 dy

ρ1−1+α ρ2−2 n 2 dy.

Summing up the two differential inequalities (4.63) and (4.64) leads to   d d −1+α −2 2 2 ψ(t) + ρ2 n (t)d x + ρ11+α ψ y2 dy ρ1 dt dt  ≤C(T∗ ) ρ1−1+α ρ2−2 n 2 dy

(4.64)

(4.65)

which, together with the initial data (4.62) and the Grönwall’s lemma, gives rise to  2 ψ(t) + ρ1−1+α ρ2−2 n 2 (x, t)d x ≡ 0, t ∈ [0, T ]. (4.66) This together with ρ1 (y0 , t) = ρ2 (y0 , t) = 0 implies the uniqueness of weak solution (ρ1 , u 1 ) ≡ (ρ2 , u 2 ).

(4.67)

The proof of Proposition 4.2 is completed.   Proof of Proposition 4.1. With the help of Propositions 4.2–4.4 and the inverse transformation from the Lagrangian coordinates to the Eulerian coordinates, we are able to prove Proposition 4.1. Since the case of periodic boundary conditions with one point vacuum state in the initial data can be dealt with in a similar framework, we only deal with the IBVP problem for the compressible Navier-Stokes equations (2.1)–(2.3) for the case of the Dirichlet boundary condition (2.6) with one point vacuum in the regular initial data (2.18)–(2.19). First, one can check easily that the IBVP problem (2.1)–(2.3), (2.6), and (2.18)– (2.19) in the Eulerian coordinates is equivalent to the corresponding IBVP problem in

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H.-L. Li, J. Li, Z. Xin

the Lagrangian coordinates for Eqs. (4.7)–(4.9) with the Dirichlet boundary conditions (4.10) and (4.12) through the coordinate transformation  x y= ρ(z, t)dz, x ∈ (0, 1), t ≥ 0. (4.68) 0

Note that y ∈ [0, 1] due to the conservation of mass. More importantly, the case of one point vacuum state in the initial  x data (2.18)–(2.19) isσ reformulated into the corresponding case (4.12) with y0 = 0 0 ρ0 (z)dz and β = 1+σ . It is easy to verify that all the assumptions of Proposition 4.2 are satisfied. Then Proposition 4.2 gives the existence and uniqueness of a weak solution (ρ, ˜ u) ˜ satisfying (4.13)–(4.18) in the Lagrangian coordinate to the compressible Navier-Stokes equations (4.7)–(4.9) with the Dirichlet boundary conditions (4.10) and one point vacuum state in the initial data. In particular, ¯ ¯ ρ, ˜ u˜ ∈ C 0 ([0, T∗ ]; C 0 ()), ρ˜ u˜ y (t) ∈ L ∞ (0, T∗ ; C 0 ()), ρ(y ˜ 0 , t) = 0, ρ(y, t) > 0, y = y0 .

(4.69) (4.70)

These in turn imply, in terms of the inverse coordinate transformation of (4.68), i.e.,  y ρ˜ −1 (z, t)dz, y ∈ [0, 1], x= 0

the existence and uniqueness of a weak solution (ρ, ˜ u) ˜ on the domain (x, t) ∈ [0, 1] × [0, T∗ ] to the compressible Navier-Stokes equations (2.1)–(2.3) with the Dirichlet boundary conditions (2.6) and initial one point vacuum state (2.18)–(2.19). Moreover, one can check that there exists one particle path x = X 0 (t) defined by ˜ 0 (t), t), t > 0, X˙ 0 (t) = u(X

X 0 (0) = x0

satisfying 

X 0 (t)

y0 ≡

 ρ(z, ˜ t)dz =

d d y(X 0 (t), t) = dt dt

ρ0 (z)dz

0

0

due to the fact

x0



X 0 (t)

ρ(z, ˜ t)dz = 0, t ≥ 0.

0

This, together with (4.70), gives rise to ρ(X ˜ 0 (t), t) = 0, and ρ(x, ˜ t) > 0, x = X 0 (t), t ≥ 0. Moreover, it is easy to verify that the solution (ρ, ˜ u) ˜ satisfies all the properties (2.23)– (2.26), and particularly ⎧ ˜ T∗ ) ≥ 0 on , m(., ˜ T∗ ) = ρ˜ u(., ˜ T∗ ) = 0, on {x ∈  | ρ0 (x) = 0}, ⎪ ⎨ρ(x, ρ(., ˜ T∗ ) ∈ L 1 () ∩ L γ (), (ρ˜ α−1/2 (., T∗ ))x ∈ L 2 (), 2 2+ν ⎪ ˜ ˜ ⎩ |m(.,T ∗ )| ∗ )| + |m(.,T ∈ L 1 (). ρ(x,T ˜ ∗)

ρ˜ 1+ν (x,T∗ )

The case of periodic boundary can be treated in a similar way; we omit the details. The proof of Proposition 4.1 is complete.  

Vanishing of Vacuum States and Blow-up Phenomena

435

5. Vanishing of Vacuum States and Blow-up Phenomena We shall prove that for any global entropy weak solution (ρ, u) to the IBVP for compressible Navier-Stokes equations (2.1)–(2.3) together with (2.6) or (2.7), any possible vacuum state vanishes in finite time and the velocity (if definable and regular enough) blows up in finite time if a vacuum state appears, for example, the density contains vacuum initially as in Theorem 2.2. The weak solution becomes a strong one after the vanishing of vacuum states and tends time-asymptotically to the non-vacuum equilibrium state exponentially. 5.1. Vanishing of vacuum states in finite time. Proposition 5.1. Let (2.34) hold. For any global entropy weak solution (ρ, u) to the IBVP for compressible Navier-Stokes equations (2.1)–(2.2) with initial data -(2.3) and boundary condition (2.6) or (2.7) in the sense of Definition 2.5, there exists a time T0 > 0 such that inf ρ(x, t) > 0, for all t ≥ T0 .

(5.1)

x∈

Proof. To prove (5.1), we will employ an idea which has been used in [28] (see also [22,27]) to show the blow up behavior of both the global strong solutions to the IBVP for (2.1)–(2.2) with the constant viscosity and the global strong solutions to the Stokes approximation equations, with initial data containing vacuum states. Let T ∈ (0, ∞) be fixed. In this subsection, C denotes some generic positive constant independent of T . First, it is noted that mass is conserved for any t ∈, (0, T ],   ρ(x, t)d x = ρ0 (x)d x. (5.2) 



Based on the entropy inequality (2.13), it can be deduced from (5.2) and (2.39) that for a constant b ≥ max{α + γ − 1, 2α + 1, 1},    T  2   b   + sup ρ L ∞ +  ρ b  (5.3)  ρ  dt ≤ C. x L2

0≤t≤T

It will be shown below that  4

  g(t)
 ρ b − ρ b (·, t) 4

L ()

where ρ b (t)

1 = ||

x L2

0

→ 0 as t → ∞,

(5.4)

 

ρ b (x, t)d x.

Now, we assume that (5.4) holds, and continue the proof of Proposition 5.1. In fact, the inequality (5.3) and the Poincaré-Sobolev inequality implies that  

  b  ρ − ρ b (·, t) C() 2/3 

  1/3 ≤ C  ρ b − ρ b (·, t) 4 (ρ b )x (·, t) L 2 L () 2/3 

  ≤ C  ρ b − ρ b (·, t) 4 → 0, as t → ∞. (5.5) L ()

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H.-L. Li, J. Li, Z. Xin

This suffices to finish the proof of Proposition 5.1 due to the following simple fact: ρ b (t) ≥ ρ b (t) ≡ ρ0 b = 1, for any t ≥ 0. It remains to prove (5.4). First, it follows directly from (5.3) and the Poincaré-Sobolev inequality that  T  2  T   g(t)dt ≤ C sup ρ b − ρ b  ∞ (ρ b )x 2L 2 dt 0

L

0≤t≤T

0

≤ C. Next, we prove that

(5.6) 

T

|g  (t)|dt ≤ C.

(5.7)

0

Note that (2.9) as well as the boundary condition (2.6) or (2.7) imply that  

3

3   b b−1 b g (t) = 4b ρ − ρ ρb − ρb d x ρ , ρt − 4 ρb t  H 1 ×H −1   

3

3  √ √ ρ b − ρ b ρ b−1 = −4b ρb − ρb d x ρ ρud x − 4 ρ b 

t

x




I1 + I2 .

(5.8)

It follows from (5.3) and (2.13) that   T  T 

2   b b b−1/2 √ b  ρ −ρ |I1 |dt ≤ C (ρ )x ρ ρud x  dt   0 0   T 

3   b b−1 √ √ b  +C ρ −ρ (ρ )x ρ ρud x  dt   0   T 

2   b α−1/2 b−α √ b   dt ≤C ρ − ρ (ρ ) ρ ρud x x    0  T   2 √  α−1/2    ≤C )x  2  ρu L 2 ρ b − ρ b  ∞ dt (ρ 

T

≤C ≤ C.

L

0

0

L

(ρ b )x 2L 2 dt (5.9)

The uniform entropy estimate (2.13), together with (2.9), gives that     d    sup  ρ b (t) = b sup  ρ b−1 , ρt  dt 0≤t≤T 0≤t≤T  

√ √   b−1  = b sup  ρ ρ ρud x  0≤t≤T



x

 

  α−1/2 b−α √  ≤ C sup  ρ ρ ρud x  x 0≤t≤T  



√  α−1/2  ≤ C sup  ρ   ρu L 2 ρb−α ∞ L 0≤t≤T

≤ C.

x L2

(5.10)

Vanishing of Vacuum States and Blow-up Phenomena

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This together with the Poincaré inequality and (5.3) yields 

T 0

 |I2 |dt ≤ C 0

T

  3  b  b ρ − ρ  ∞ dt ≤ C L

0

T

(ρ b )x 2L 2 dt ≤ C.

(5.11)

The estimate (5.7) thus follows directly from (5.8)–(5.11). Hence the desired estimate (5.4) follows from (5.6) and (5.7). The proof of Proposition 5.1 is complete.  

5.2. Regularity and asymptotics of weak solutions for large time. It is usually difficult √ to get information about the velocity field for the global entropy weak solution (ρ, ρu), in the sense of Definition 2.5 to the IBVP for the Compressible Navier-Stokes equations (2.1)–(2.2) with initial data (2.3) and boundary values (2.6) or (2.7) in the appearance of vacuum states. After vacuum states vanish, however, it will be shown that the velocity field u can be defined with enough regularity and the nonlinear diffusion term is represented in terms of the velocity u and the density ρ. The √ momentum equation becomes √ a uniform parabolic equation, and the weak solution (ρ, ρu) = (ρ, ρ · u) becomes a strong solution. Proposition 5.1 implies that there is a time T0 > 0 after which the density of the global entropy weak solution (ρ, u) to the IBVP problem for (2.1)–(2.3) together with (2.6) or (2.7) is strictly positive and (ρ, u) satisfies the finite entropy estimate (2.13). Consider the IBVP problem (2.1)–(2.2) again for time t ≥ T0 with data given at time t = T0 by √ ρu(x,t) √ , ρ(x,t) t→T0

ρ(x, T0 ) = lim ρ(x, t), u(x, T0 ) = lim t→T0

(5.1)

and note here that away from vacuum the Dirichlet boundary condition (2.6) reduces to u(0, t) = u(1, t) = 0, t ≥ T0 .

(5.2)

We then have the regularity property of the solution for the compressible Navier-Stokes equations (2.1)–(2.3) with the Dirichlet boundary condition (2.6) or the periodic boundary condition (2.7) for large time. √ Proposition 5.2. Under the assumptions of Theorem 2.3, let (ρ, ρu) be the global entropy weak solution to the IBVP for the compressible Navier-Stokes equations (2.1)– (2.2) with initial data (2.3) and boundary value (2.6) or (2.7) in the sense of Definition 2.5. Let T0 > 0 so that the global weak solution (ρ, u) satisfies for two positive constants ρ± that ¯ × [T0 , ∞). 0 < ρ− ≤ ρ(x, t) ≤ ρ+ , ∀ (x, t) ∈ 

(5.3)

√ √ Then, (ρ, ρu) = (ρ, ρ · u) is the unique strong solution to the IBVP for the compressible Navier–Stokes equations (2.1)–(2.2) and (5.1) with the boundary condition (2.6)2 or (2.7) for t ≥ T0 . Moreover, the regularity (2.36) and the long time behavior (2.38) hold. 2 Note here that away from vacuum the Dirichlet boundary condition (2.6) reduces to the usual one u(0, t) = u(1, t) = 0, t ≥ T0 .

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Proof. We only prove Proposition 5.2 for the Dirichlet case below, the periodic case can be treated similarly. Step 1. Regularity. It follows easily from Proposition 5.1 that there exist some T0 and a constant ρ− > 0 such that for all t ≥ T0 , inf ρ(x, t) ≥ ρ− > 0

(5.4)

x∈

which, together with (2.13), implies ρ ∈ L ∞ (T0 , T ; H 1 ())

(5.5)

¯ × [0, ∞)) there exists some σ > 0 small for any T > T0 . By the continuity of ρ ∈ C( enough such that for any t ≥ T0 − σ, inf ρ(x, t) ≥

x∈

ρ− > 0. 2

This implies that one can define the velocity u for any global entropy weak solution in the sense of Definition 2.5 after the vanishing of vacuum states by √ ρu u =: √ , for t ≥ T0 − σ. ρ It then follows from the definition and (2.8) that u ∈ L ∞ (T0 − σ, T ; L 2 ())

(5.6)

for any T > T0 . Noting that (2.12) implies that for any ϕ(x) ∈ C0∞ (), ψ(t) ∈ C0∞ (T0 − σ, T ), 



T T0 −σ

ψ(t) 

=−

T T0 −σ



ρ −α ϕd xdt 

ψ(t)



√ ρ α−1/2 ρu(ρ −α ϕ)x d xdt

 2α √ ψ(t) (ρ α−1/2 )x ρuρ −α ϕd xdt 2α − 1 T0 −σ    T √ ψ(t) ρuρ −1/2 ϕx d xdt =− 

−

 =−

T0 −σ T T0 −σ

T



 ψ(t)



uϕx d xdt,

we can define the spatial derivative of velocity and, together with (2.13), its regularity as ux =

∈ L 2 ( × (T0 − σ, T )). ρα

(5.7)

Vanishing of Vacuum States and Blow-up Phenomena

439

In terms of (5.6), (5.7) and (2.9) we are also able to justify the Dirichlet boundary condition (2.6) for the velocity u u(0, t) = u(1, t) = 0, for any t ≥ T0 − σ.

(5.8)

Thus, (5.6), (5.7) and (5.8) show u ∈ L 2 (T0 − σ, T ; H01 ()) ∩ L ∞ (T0 − σ, T ; L 2 ())

(5.9)

for the case of the Dirichlet boundary conditions. Note here that u ∈ L 2 (T0 − σ, T ; 1 ()) ∩ L ∞ (T − σ, T ; L 2 ()) in the case of periodic boundary conditions. We thus Hper 0 obtain from (2.9) and (2.10) that the solution (ρ, u) satisfies ρt + (ρu)x = 0

a.e. in

 × (T0 − σ, T ),

(5.10)

and 

T T0 −σ



 

ρuϕt d xdt +

T T0 −σ



ρu 2 − ρ α u x + ρ γ ϕx d xdt = 0 

(5.11)

for any ϕ(x, t) ∈ C0∞ ( × (T0 − σ, T )) for the Dirichlet case. Equation (5.11) can be re-written in terms of (5.9), (5.10) and (2.13) as follows:  T   T 

α−1 α−2 (uϕt − ρ u x ϕx + ρ ρx − u u x ϕ)d xdt = γρ γ −2 ρx ϕ d xdt T0 −σ



T0 −σ



(5.12) for any ϕ(x, t) ∈ C0∞ ( × (T0 − σ, T )) for the Dirichlet case. Noticing that ρ α−2 ρx − u ∈ L ∞ (T0 − σ, T ; L 2 ()) due to (2.8), (5.4) and (5.6), and using standard regularity results for linear parabolic equations (see [25]), we get that u ∈ L 2 (T0 , T ; H 2 ())) ∩ H 1 (T0 , T ; L 2 ())

(5.13)

2 ()) ∩ H 1 (T , T ; L 2 ()) for the Dirichlet case. It is noted here that u ∈ L 2 (T0 , T ; Hper 0 for the periodic case. Step 2. Uniqueness. We shall show that if there exists another solution (η, v) to the compressible Navier-Stokes equations (2.1)–(2.2) with the following initial data and Dirichlet boundary conditions
(η, v)(x, T0 ) = (ρ, u)(x, T0 ), (5.14) v(0, t) = v(1, t) = 0,

such that
ρ− ≤ η ∈ L ∞ (T0 , T ; H 1 ()), v ∈ L ∞ (T0 , T ; H 1 ()) ∩ L 2 (T0 , T ; H 2 ()) ∩ H 1 (T0 , T ; L 2 ()),

(5.15)

then ρ = η, u = v

a.e. in

 × (T0 , T ).

(5.16)

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H.-L. Li, J. Li, Z. Xin

In fact, it follows from (5.10), (5.11) and (5.13) that 1 2

 ρu d x + 2



 t 

T0

ρ α u 2x d xds

−

 t T0

1 ρ u x d xds = 2  γ

 

ρu 2 (x, T0 )d x

(5.17)

for all t ∈ (T0 , T ), while (5.11) and (5.15) imply that  

ρuvd x +

 t T0



=





 t

ρ α u x vx d xds −

ρuv(x, T0 )d x +



T0



T0

 t

ρ γ vx d xds

ρu(vt + uvx )d xds

(5.18)

for all t ∈ (T0 , T ). To estimate the second term on the right hand side of (5.18), we use the decomposition ρvt + ρuvx = (ρ − η)(vt + vvx ) + ρ(u − v)vx + (ηα vx )x − (ηγ )x .

(5.19)

Multiplying (5.19) by u and integrating by parts give  t T0

=



ρu(vt + uvx )d xds

 t 

T0

−

(ρ − η)u(vt + vvx )d xds +

 t 

T0

ηα vx u x d xds +



ρu(u − v)vx d xds



T0

 t T0

 t

ηγ u x d xds.

(5.20)

Substituting (5.20) into (5.18) gives, for a.e. t ∈ (T0 , T ), that  

ρuvd x +

 t T0



=





ρ α u x vx d xds −

ρuv(x, T0 )d x +

T0

 t

+



T0

 t 

 t 

T0

(ρ γ vx + ηγ u x )d xds

(ρ − η)u(vt + vvx )d xds

ρu(u − v)vx d xds −

 t 

T0

ηα vx u x d xds.

(5.21)

Multiplying (5.19) by v and integrating the result over  × (T0 , t) lead to 1 2

 

ρv 2 d x =

1 2

 ρv 2 (x, T0 )d x +

  t

+ T0  t

+ T0



 t T0







(ρ − η)v(vt + vvx )d xds

ρv(u − v)vx d xds −

 t T0

ηγ vx d xds.



ηα vx2 d xds (5.22)

Vanishing of Vacuum States and Blow-up Phenomena

441

We obtain after adding up (5.17) and (5.22) and subtracting (5.21) that for all t ∈ (T0 , T ),  t  1 2 ρ(u − v) d x + ρ α (u − v)2x d xds 2  T0   t  t α α (ρ − η )(v − u)x vx d xds + (ρ − η)(vt + vvx )(v − u)d xds = T0

−



 t 

T0

ρ(u − v)2 vx d xds −

 t T0



T0



 γ  ρ − ηγ (v − u)x d xds

 t

ρ α − ηα 2L 2 vx 2L ∞ + ρ − η2L 2 vt + vvx 2L 2 + ρ γ − ηγ 2L 2 ds ≤ Cε T0



t

+C



 vx 

T0

L∞

ρ(u − v) d xds + Cε 2



t T0

u − v2H 1 ds.

(5.23)

Next, we estimate the term g(t)
ρ − η2L 2 + ρ α − ηα 2L 2 + ρ γ − ηγ 2L 2 . For any β > 0, (5.15), (5.13) and (2.1) imply that (ρ β − ηβ )t + v(ρ β − ηβ )x + (u − v)(ρ β )x + βρ β (u − v)x + β(ρ β − ηβ )vx = 0. (5.24) One can derive from this that



ρ β − ηβ 2L 2 ≤ ρ β − ηβ 2L 2 Cvx  L ∞ + Cε ρx 2L 2 + Cε t

+εu − v2L ∞ + ε(u − v)x 2L 2 .

(5.25)

Thus, since g(T0 ) = 0, we obtain from (5.25) with β = 1, α, γ respectively that  t  t

2 g(s) Cvx  L ∞ + Cε ρx  L 2 + Cε ds + Cε u − v2H 1 ds. (5.26) g(t) ≤ T0

T0

Now (5.16) is a consequence from (5.23), (5.26), (5.15), (5.5) and (5.4). The proof of large time convergence (2.38) follows directly from the standard arguments (see [44] for instance) with the help of entropy inequality (2.13). The proof of Proposition 5.2 is completed.   5.3. Finite time blow-up. In this subsection, we shall prove Theorem 2.4 about the finite time blow-up phenomena as an immediate consequence of Theorem 2.3, Propositions 5.1 and Proposition 5.2. Proof of Theorem 2.4. We will prove (2.42) only. The proof of (2.43) is similar. If (2.42) fails, then there exists a fixed constant η > 0, such that  T1 +η u x  L ∞ ds < ∞. (5.27) T1

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For any (x, t) ∈  × (T1 , T1 + η], the particle path x(s) = X (s; t, x) through (x, t) is given by
∂ ∂s X (s; t, x) = u(X (s; t, x), s), T1 ≤ s < t ≤ T1 + η, (5.28) X (t; t, x) = x, T1 ≤ t ≤ T1 + η, x ∈ , which is well-defined due to (5.27) and (2.36). Consequently, one obtains via a standard argument from the transport equation (2.1) !  t u y (y, s)| y=X (s;t,x) ds (5.29) ρ(x, t) = ρ(X (T1 ; t, x), T1 ) exp − T1

for any (x, t) ∈  × (T1 , T1 + η]. On the other hand, it follows from (5.27) and (5.28) that for any x ∈  there exists a trajectory x = x(t) ∈  for t ∈ [T1 , T1 + η] so that X (T1 ; t, x(t)) = x. In particular, there exists a trajectory x = x1 (t) ∈  for t ∈ [T1 , T1 + η] so that X (T1 ; t, x(t)) = x1 with (x1 , T1 ) determined by (2.40), namely, ρ(x1 , T1 ) = 0. Thus, due to (5.29), we deduce from (5.27) that ρ(x1 (t), t) ≡ 0

for all

t ∈ (T1 , T1 + η],

which contradicts (2.40). Thus, the blowup phenomena (2.42) happens. The proof of Theorem 2.4 is complete.   Acknowledgements. The authors would like to thank the referee for informing them of the recently; published references [4,5,39] and the helpful comments on the manuscript. They also would like to thank Prof. Didier Bresch for his interest in this work and helpful discussions about the BD entropy estimates he and his collaborators introduced. The main part of this research was done when H.Li and J. Li were visiting the Institute of Mathematical Sciences (IMS) of The Chinese University of Hong Kong. Financial support from the IMS and the hospitality of the staff at the IMS are appreciated greatly. The research of H. Li is partially supported by Beijing Nova Program, NNSFC No.10431060, the NCET support of the Ministry of Education of China, the Institute of Mathematics and Interdisciplinary Science at CNU, the Huo Ying Dong Foundation 111003, and the Zheng Ge Ru Foundation. The research of J. Li is partially supported by the JSPS Research Fellowship for foreign researchers, NNSFC No.10601059. The research of J. Li and Xin are partially supported by Hong Kong RGC Earmarked Research Grants CUHK4028/04 and 4040/02, and the Zheng Ge Ru Foundation.

References 1. Balian, R.: From microphysics to macrophysics. Berlin: Springer, 1982 2. Bresch, D., Desjardins, B.: Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model. Commun. Math. Phys. 238, 211–223 (2003) 3. Bresch, D., Desjardins, B.: Some diffusive capillary models for Korteweg type. C. R. Mecanique 332(11), 881–886 (2004) 4. Bresch, D., Desjardins, B.: On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models. J. Math. Pures Appl. 86(4), 362–368 (2006) 5. Bresch, D., Desjardins, B., Gérard-Varet, D.: On compressible Navier-Stokes equations with density dependent viscosities in bounded domains. J. Math. Pures Appl. 87(2), 227–235 (2007) 6. Bresch, D., Desjardins, B.: On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids. J. Math. Pures Appl. 87(1), 57–90 (2007) 7. Bresch, D., Desjardins, B., Lin, C.-K.: On some compressible fluid models: Korteweg, lubrication, and shallow water systems. Comm. Partial Differ. Eqs. 28, 843–868 (2003) 8. Bresch, D., Desjardins, B., Métivier, G.: Recent Mathematical Results and Open Problems About Shallow Water Equations. Basel-Boston: Birkauser, 2007 9. Cho, Y., Choe, H.J., Kim, H.: Unique solvability of the initial boundary value problems for compressible viscous fluids. J. Math. Pures Appl. 83(2), 243–275 (2004)

Vanishing of Vacuum States and Blow-up Phenomena

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10. Choe, H.J., Kim, H.: Strong solutions of the Navier-Stokes equations for isentropic compressible fluids. J. Differ. Eqs. 190(2), 504–523 (2003) 11. Danchin, R.: Global existence in critical spaces for compressible Navier-Stokes equations. Invent. Math. 141, 579–614 (2000) 12. Fang, D., Zhang, T.: Compressible Navier-Stokes equations with vacuum state in one dimension. Comm. Pure Appl. Anal. 3, 675–694 (2004) 13. Feireisl, E., Novotny, A., Petzeltová, H.: On the existence of globally defined weak solutions to the Navier-Stokes equations. J. Math. Fluid Mech. 3(4), 358–392 (2001) 14. Gerbeau, J.-F., Perthame, B.: Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation. Discrete Contin. Dyn. Syst. Ser. B1, 89–102 (2001) 15. Guo, Z.-H., Jiang, S., Xie, F.: Global existence and asymptotic behavior of weak solutions to the 1D compressible Navier-Stokes equations with degenerate viscosity coefficient. Submitted for publication, 2007 16. Guo, Z.-H., Jiu, Q.-S., Xin, Z.: Spherically symmetric isentropic compressible flows with densitydependent viscosity coefficients. SIAM J. Math. Anal. 39(5), 1402–1427 (2008) 17. Guo, Z.-H., Zhu, C.-J.: Global weak solutions and asymptotic behavior to 1D compressible Navier-Stokes equations with density-dependent viscosity and vacuum. Submitted for publication, 2007 18. Hoff, D.: Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data. Trans. Amer. Math. Soc. 303(1), 169–181 (1987) 19. Hoff, D.: Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data. Arch. Rat. Mech. Anal. 132, 1–14 (1995) 20. Hoff, D., Serre, D.: The failure of continuous dependence on initial data for the Navier-Stokes equations of compressible flow. SIAM J. Appl. Math. 51(4), 887–898 (1991) 21. Hoff, D., Smoller, J.: Non-formation of vacuum states for compressible Navier-Stokes equations. Commun. Math. Phys. 216(2), 255–276 (2001) 22. Huang, F., Li, J., Xin, Z.: Convergence to equilibria and blowup behavior of global strong solutions to the Stokes approximation equations for two-dimensional compressible flows with large data. J. Math. Pures Appl. 86(6), 471–491 (2006) 23. Jiang, S., Xin, Z., Zhang, P.: Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity. Methods and Applications of Analysis 12, 239–251 (2005) 24. Kazhikhov, A.V., Shelukhin, V.V.: Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas. J. Appl. Math. Mech. 41(2), 273–282 (1977) 25. Ladyzenskaja, O.A., Solonnikov, V.A., Uraltseva, N.N.: Linear and quasilinear equations of parabolic type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. Providence, R.I.: American Mathematical Society, 1968 26. Li, H.-L., Li, J., Xin, Z.: Vanishing vacuum states and blow-up of global weak solutions for full NavierStokes equations. In preparation, 2006 27. Li, J.: Qualitative behavior of solutions to the compressible Navier-Stokes equations and its variants. PhD Thesis, Chinese University of Hong Kong, 2004 28. Li, J., Xin, Z.: Some uniform estimates and blowup behavior of global strong solutions to the Stokes approximation equations for two-dimensional compressible flows. J. Differ. Eqs. 221, 275–308 (2006) 29. Lions, P.L.: Existence globale de solutions pour les equations de Navier-Stokes compressibles isentropiques. C. R. Acad. Sci. Paris, Sér I Math. 316, 1335–1340 (1993) 30. Lions, P.L.: Limites incompressible et acoustique pour des fluides visqueux, compressibles et isentropiques. C. R. Acad. Sci. Paris Sér. I Math. 317, 1197–1202 (1993) 31. Lions, P.-L.: Mathematical topics in fluid mechanics. Vol. 2. Compressible models. New York: Oxford University Press, 1998 32. Liu, T.-P., Xin, Z., Yang, T.: Vacuum states for compressible flow. Discrete Contin. Dynam. Systems 4, 1–32 (1998) 33. Luo, T., Xin, Z., Yang, T.: Interface behavior of compressible Navier-Stokes equations with vacuum. SIAM J. Math. Anal. 31, 1175–1191 (2000) 34. Marche, F.: Derivation of a new two-dimensional shallow water model with varying topography, bottom friction and capillary effects. Eur. J. Mech. B Fluids 26(1), 49–63 (2007) 35. Matsumura, A., Nishida, T.: The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids. Proc. Japan Acad. Ser. A Math. Sci. 55(9), 337–342 (1979) 36. Matsumura, A., Nishida, T.: The initial value problem for the equations of motion of viscous and heatconductive gases. J. Math. Kyoto Univ. 20(1), 67–104 (1980) 37. Matsumura, A., Nishida, T.: The initial boundary value problems for the equations of motion of compressible and heat-conductive fluids. Commun. Math. Phys. 89, 445–464 (1983)

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38. Mellet, A., Vasseur, A.: On the barotropic compressible Navier-Stokes equations. Commun. Partial Differ. Eqs. 32(1–3), 431–452 (2007) 39. Mellet, A., Vasseur, A.: Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations. SIAM J. Math. Anal. 39(4), 1344–1365 (2008) 40. Nishida, T.: Motion of compressible fluids. In: “Patterns and waves. Qualitative analysis of nonlinear differential equations”, Ed. Nishida, Masayasu Mimura and Hiroshi Fujii, Tokyo: Kinokuniya Company Ltd., 1986, pp. 89–19 41. Okada, M.: Free boundary problem for the equation of one-dimensional motion of viscous gas. Japan J. Appl. Math. 6, 161–177 (1989) 42. Okada, M., Makino, T.: Free boundary problem for the equation of spherically symmetric motion of viscous gas. Japan J. Appl. Math. 10, 219–235 (1993) 43. Okada, M., Matsusu-Necasova, S., Makino, T.: Free boundary problem for the equation of onedimensional motion of compressible gas with density-dependent viscosity. Ann. Univ. Ferrara Sez. VII (N.S.) 48, 1–20 (2002) 44. Straškraba, I., Zlotnik, A.: Global properties of solutions to 1D-viscous compressible barotropic fluid equations with density dependent viscosity. Z. Angew. Math. Phys. 54(4), 593–607 (2003) 45. Salvi, R., Straškraba, I.: Global existence for viscous compressible fluids and their behavior as t → ∞. J. Fac. Sci. Univ. Tokyo Sect. IA, Math. 40, 17–51 (1993) 46. Serre, D.: Solutions faibles globales des quations de Navier-Stokes pour un fluide compressible. C. R. Acad. Sci. Paris Sér. I Math. 303(13), 639–642 (1986) 47. Serre, D.: On the one-dimensional equation of a viscous, compressible, heat-conducting fluid. C. R. Acad. Sci. Paris Sér. I Math. 303(14), 703–706 (1986) 48. Solonnikov, V.A.: On solvability of an initial boundary value problem for the equations of motion of viscous compressible fluid. Zap. Nauchn. Sem. LOMI 56, 128–142 (1976) 49. Vaigant, V.A., Kazhikhov, A.V.: On the existence of global solutions of two-dimensional Navier-Stokes equations of a compressible viscous fluid. (Russian). Sibirsk. Mat. Zh. 36(6), 1283–1316 (1995) 50. Valli, A., Zajaczkowski, W.M.: Navier-Stokes equations for compressible fluids: global existence and qualitative properties of the solutions in the general case. Commun. Math. Phys. 103(2), 259–296 (1986) 51. Vong, S.-W., Yang, T., Zhu, C.: Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. II. J. Differ. Eqs. 192(2), 475–501 (2003) 52. Xin, Z.: Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density. Comm. Pure Appl. Math. 51, 229–240 (1998) 53. Xin, Z.: On the behavior of solutions to the compressible Navier-Stokes equations. AMS/IP Stud. Adv. Math. 20, Providence, RI: Amen. Math. Soc., 2001, pp. 159–170 54. Xin, Z., Yuan, H.: Vacuum state for spherically symmetric solutions of the compressible Navier-Stokes equations. J Hyperbolic Differ. Eqs. 3, 403–442 (2006) 55. Yang, T., Yao, Z., Zhu, C.: Compressible Navier-Stokes equations with density-dependent viscosity and vacuum. Comm. Partial Differ Eqs. 26(5–6), 965–981 (2001) 56. Yang, T., Zhao, H.: A vacuum problem for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity. J. Differ. Eqs. 184, 163–184 (2002) 57. Yang, T., Zhu, C.: Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. Commun. Math. Phys. 230, 329–363 (2002) Communicated by P. Constantin

Commun. Math. Phys. 281, 445–468 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0488-3

Communications in

Mathematical Physics

Dimers on Surface Graphs and Spin Structures. II David Cimasoni1 , Nicolai Reshetikhin2 1 ETH Zurich, Zurich, Switzerland. E-mail: [email protected] 2 UC Berkeley, Berkeley, USA. E-mail: [email protected]

Received: 18 April 2007 / Accepted: 26 November 2007 Published online: 29 April 2008 – © Springer-Verlag 2008

Abstract: In a previous paper [3], we showed how certain orientations of the edges of a graph Γ embedded in a closed oriented surface Σ can be understood as discrete spin structures on Σ. We then used this correspondence to give a geometric proof of the Pfaffian formula for the partition function of the dimer model on Γ . In the present article, we generalize these results to the case of compact oriented surfaces with boundary. We also show how the operations of cutting and gluing act on discrete spin structures and how they change the partition function. These operations allow to reformulate the dimer model as a quantum field theory on surface graphs. 1. Introduction A dimer configuration on a graph Γ is a choice of a family of edges of Γ , called dimers, such that each vertex of Γ is adjacent to exactly one dimer. Assigning weights to the edges of Γ allows to define a probability measure on the set of dimer configurations. The study of this measure is called the dimer model on Γ . Dimer models on graphs have a long history in statistical mechanics [6,12], but also show interesting aspects involving combinatorics, probability theory [4,10], real algebraic geometry [8,9], etc... A remarkable fact about dimer models was discovered by P.W. Kasteleyn in the 60’s: the partition function of the dimer model can be written as a linear combination of 22g Pfaffians of N × N matrices, where N is the number of vertices in the graph and g the genus of a closed oriented surface Σ where the graph can be embedded. The matrices are signed-adjacency matrices, the sign being determined by an orientation of the edges of Γ called a Kasteleyn orientation. If the graph is embedded in a surface of genus g, there are exactly 22g equivalence classes of Kasteleyn orientations, defining the 22g matrices. This Pfaffian formula for the partition function was proved by Kasteleyn in [6] for the cases g = 0, 1, and only stated for the general case [7]. A combinatorial proof of this fact and the exact description of coefficients for all oriented surfaces first appeared much later [11,14].

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The number of equivalence classes of Kasteleyn orientations on a graph Γ embedded in Σ is also equal to the number of equivalence classes of spin structures on Σ. An explicit construction relating a spin structure on a surface with a Kasteleyn orientation on a graph with dimer configuration was suggested in [10]. In [3], we investigated further the relation between Kasteleyn orientations and spin structures. This allows to understand Kasteleyn orientations on a graph embedded in Σ as discrete spin structures on Σ. We also used this relation to give a geometric proof of the Pfaffian formula for closed surfaces. Our final formula can be expressed as follows: given a graph Γ embedded in a closed oriented surface Σ of genus g, the partition function of the dimer model on Γ is given by Z (Γ ) =

1
Arf(ξ )Pf(Aξ (Γ )), 2g ξ ∈S(Σ)

where S(Σ) denotes the set of equivalence classes of spin structures on Σ, Arf(ξ ) = ±1 is the Arf invariant of the spin structure ξ , and Aξ (Γ ) is the matrix given by the Kasteleyn orientation corresponding to ξ . The first part of the present paper is devoted to the extension of the results obtained in [3] to dimer models on graphs embedded in surfaces with boundary (Sects. 2 and 3). We then show how the operations of cutting and gluing act on discrete spin structures and how they change the partition function (Sect. 4). These operations define the structure of a functorial quantum field theory in the spirit of [2,13], as detailed in Sect. 5. We then give two equivalent reformulations of the dimer quantum field theory: the “Fermionic” version, which describes the partition function of the dimer model as a Grassman integral, and the “Bosonic” version, the equivalent description of dimer models on bipartite surface graphs in terms of height functions. This special case of bipartite graphs is the subject of Sect. 6. Throughout this paper, Σ is a compact surface, possibly disconnected and possibly with boundary, endowed with the counter-clockwise orientation. All results can be extended to the case of non-orientable surfaces, which will be done in a separate publication. We refer to [14] for a combinatorial treatment of dimer models on non-orientable surface graphs. 2. The Dimer Model on Graphs with Boundary 2.1. Dimers on graphs with boundary. In this paper, a graph with boundary is a finite graph Γ together with a set ∂Γ of one valent vertices called boundary vertices. A dimer configuration D on a graph with boundary (Γ, ∂Γ ) is a choice of edges of Γ , called dimers, such that each vertex that is not a boundary vertex is adjacent to exactly one dimer. Note that some of the boundary vertices may be adjacent to a dimer of D, and some may not. We shall denote by ∂ D this partition of boundary vertices into matched and nonmatched. Such a partition will be called a boundary condition for dimer configurations on Γ . A weight system on Γ is a positive real valued function w on the set of edges of Γ . It defines edge weights on the set D(Γ, ∂Γ ) of dimer configurations on (Γ, ∂Γ ) by  w(D) = w(e), e∈D

where the product is taken over all edges occupied by dimers of D.

Dimers on Surface Graphs and Spin Structures. II

447

Fix a boundary condition ∂ D0 . Then, the Gibbs measure for the dimer model on (Γ, ∂Γ ) with weight system w and boundary condition ∂ D0 is given by Prob(D | ∂ D0 ) = where

Z (Γ ; w | ∂ D0 ) =

w(D) , Z (Γ ; w | ∂ D0 )


w(D),

D:∂ D=∂ D0

the sum being on all D ∈ D(Γ, ∂Γ ) such that ∂ D = ∂ D0 . Let V (Γ ) denote the set of vertices of Γ . The group G(Γ ) = {s : V (Γ ) → R>0 } acts on the set of weight systems on Γ as follows: (sw)(e) = s(e+ )w(e)s(e − ), where e+ and e− are the two vertices adjacent to the edge e. Note that (sw)(D) = v s(v)w(D)  and Z (Γ ; sw | ∂ D0 ) = v s(v)Z (Γ ; w | ∂ D0 ), both products being on the set of vertices of Γ matched by D0 . Therefore, the Gibbs measure is invariant under the action of the group G(Γ ). Note that the dimer model on (Γ, ∂Γ ) with boundary condition ∂ D0 is equivalent to the dimer model on the graph obtained from Γ by removing all edges adjacent to non-matched boundary vertices. Given two dimer configurations D and D  on a graph with boundary (Γ, ∂Γ ), let us define the (D, D  )-composition cycles as the connected components of the symmetric difference C(D, D  ) = (D ∪ D  )\(D ∩ D  ). If ∂ D = ∂ D  , then C(D, D  ) is a 1-cycle in Γ with Z2 -coefficients. In general, it is only a 1-cycle (r el ∂Γ ). 2.2. Dimers on surface graphs with boundary. Let Σ be an oriented compact surface, not necessarily connected, with boundary ∂Σ. A surface graph with boundary Γ ⊂ Σ is a graph with boundary (Γ, ∂Γ ) embedded in Σ, so that Γ ∩ ∂Σ = ∂Γ and the complement of Γ \ ∂Γ in Σ \ ∂Σ consists of open 2-cells. These conditions imply that the graph Γ := Γ ∪ ∂Σ is the 1-skeleton of a cellular decomposition of Σ. Note that any graph with boundary can be realized as a surface graph with boundary. One way is to embed the graph in a closed surface of minimal genus, and then to remove one small open disc from this surface near each boundary vertex of the graph. A dimer configuration on a surface graph with boundary Γ ⊂ Σ is simply a dimer configuration on the underlying graph with boundary (Γ, ∂Γ ). Given two dimer configurations D and D  on a surface graph Γ ⊂ Σ, let ∆(D, D  ) denote the homology class of C(D, D  ) in H1 (Σ, ∂Σ; Z2 ). We shall say that two dimer configurations D and D  are equivalent if ∆(D, D  ) = 0 ∈ H1 (Σ, ∂Σ; Z2 ). Note that given any three dimer configurations D, D  , and D  on Γ ⊂ Σ, we have the identity ∆(D, D  ) + ∆(D  , D  ) = ∆(D, D  )

(1)

in H1 (Σ, ∂Σ; Z2 ). Fix a relative homology class β ∈ H1 (Σ, ∂Σ; Z2 ), a dimer configuration D1 ∈ D(Γ, ∂Γ ) and a boundary condition ∂ D0 . The associated partial partition function is defined by
Z β,D1 (Γ ; w | ∂ D0 ) = w(D), D:∂ D=∂ D0 ∆(D,D1 )=β

where the sum is taken over all D ∈ D(Γ, ∂Γ ) such that ∂ D = ∂ D0 and ∆(D, D1 ) = β.

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The equality (1) implies that Z β,D1 (Γ ; w | ∂ D0 ) = Z β+∆(D0 ,D1 ),D0 (Γ ; w | ∂ D0 ). Furthermore, the relative homology class β  = β + ∆(D0 , D1 ) lies in the image of the canonical homomorphism j : H1 (Σ; Z2 ) → H1 (Σ, ∂Σ; Z2 ). Hence,
Z α (Γ, w | ∂ D0 ), Z β  ,D0 (Γ ; w | ∂ D0 ) = α: j (α)=β 

where the sum is taken over all α ∈ H1 (Σ; Z2 ) such that j (α) = β  , and
Z α (Γ ; w | ∂ D0 ) = w(D). D:∂ D=∂ D0 ∆(D,D0 )=α

Therefore the computation of the partition function Z β,D1 (Γ ; w | ∂ D0 ) boils down to the computation of Z α (Γ ; w | ∂ D0 ) with α ∈ H1 (Σ; Z2 ). We shall give a Pfaffian formula for this latter partition function in the next section (see Theorem 2). 3. Kasteleyn Orientations on Surface Graphs with Boundary 3.1. Kasteleyn orientations. Let K be an orientation of the edges of a graph Γ , and let C be an oriented closed curve in Γ . We shall denote by n K (C) the number of times that, traveling once along C following its orientation, one runs along an edge in the direction opposite to the one given by K . A Kasteleyn orientation on a surface graph with boundary Γ ⊂ Σ is an orientation K of the edges of Γ = Γ ∪ ∂Σ which satisfies the following condition: for each face f of Σ, n K (∂ f ) is odd. Here ∂ f is oriented as the boundary of f , which inherits the orientation of Σ. Using the proof of [3, Theorem 1], one easily checks that if ∂Σ is non-empty, then there always exists a Kasteleyn orientation on Γ ⊂ Σ. More precisely, we have the following: Proposition 1. Let Γ ⊂ Σ be a connected surface graph, possibly with boundary, and let C1 , . . . , Cµ be the boundary components of Σ with the induced orientation. Finally, let n 1 , . . . , n µ be 0’s and 1’s. Then, there exists a Kasteleyn orientation on Γ ⊂ Σ such that 1 + n K (−Ci ) ≡ n i (mod 2) for all i if and only if n1 + · · · + nµ ≡ V

(mod 2),

where V is the number of vertices of Γ . Proof. First, let us assume that there is a Kasteleyn orientation K on Γ ⊂ Σ such that 1 + n K (−Ci ) ≡ n i for all i. Let Σ  be the closed surface obtained from Σ by pasting a 2-disc Di along each boundary component Ci . Let Γ  ⊂ Σ  be the surface graph obtained from Γ as follows: for each i such that n i = 1, add one vertex in the interior of Di and one edge (arbitrarily oriented) between this vertex and a vertex of Ci . The result is a Kasteleyn orientation on Γ  ⊂ Σ  , with Σ  closed. By [3, Theorem 1], the number V  of vertices of Γ  is even. Hence, 0 ≡ V  ≡ V + n1 + · · · + nµ

(mod 2).

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Conversely, assume Γ ⊂ Σ is a surface graph with n 1 + · · · + n µ ≡ V (mod 2). Paste 2-discs along the boundary components of Σ as before. This gives a surface graph Γ  ⊂ Σ  with Σ  closed and V  even. By [3, Theorem 1], there exists a Kasteleyn orientation K  on Γ  ⊂ Σ  . It restricts to a Kasteleyn orientation K on Γ ⊂ Σ with 1 + n K (−Ci ) ≡ n i for all i.  Recall that two Kasteleyn orientations are called equivalent if one can be obtained from the other by a sequence of moves reversing orientations of all edges adjacent to a vertex. The proof of [3, Theorem 2] goes through verbatim: if non-empty, the set of equivalence classes of Kasteleyn orientations on Γ ⊂ Σ is an affine H 1 (Σ; Z2 )-space. In particular, there are exactly 2b1 (Σ) equivalence classes of Kasteleyn orientations on Γ ⊂ Σ, where b1 (Σ) is the dimension of H 1 (Σ; Z2 ). 3.2. Discrete spin structures. As in the closed case, any dimer configuration D on a graph Γ allows to identify equivalence classes of Kasteleyn orientations on Γ ⊂ Σ with spin structures on Σ. Indeed, [3, Theorem 3] generalizes as follows. Given an oriented simple closed curve C in Γ , let  D (C) denote the number of vertices v in C whose adjacent dimer of D sticks out to the left of C in Σ. Also, let V∂ D (C) be the number of boundary vertices v in C not matched by D, and such that the interior of Σ lies to the right of C at v. Theorem 1. Fix a dimer configuration D on a surface graph with boundary Γ ⊂ Σ. Given a class α ∈ H1 (Σ; Z2 ), represent it by oriented simple closed curves C1 , . . . , Cm in Γ . If K is a Kasteleyn orientation on Γ ⊂ Σ, then the function q DK : H1 (Σ; Z2 ) → Z2 given by q DK (α) =


i< j

Ci · C j +

m
(1 + n K (Ci ) +  D (Ci ) + V∂ D (Ci )) (mod 2) i=1

is a well-defined quadratic form on H1 (Σ; Z2 ). Proof. Fix a dimer configuration D on (Γ, ∂Γ ) and a Kasteleyn orientation K on Γ ⊂ Σ. Let Σ  be the surface (homeomorphic to Σ) obtained from Σ by adding a small closed collar to its boundary. For every vertex v of ∂Γ that is not matched by a dimer of D, add a vertex v  near v in the interior of the collar and an edge between v and v  . Let us denote by Γ  the resulting graph in Σ  . Putting a dimer on each of these additional edges, and orienting them arbitrarily, we obtain a perfect matching D  and an orientation K  on Γ  . Although Γ  ⊂ Σ  is not strictly speaking a surface graph, all the methods of [3, Sect. 5] apply. Indeed, Kuperberg’s vector field defined near Γ  clearly extends continuously to the collar. As in the closed case, it also extends to the faces with even index singularities. Using the perfect matching D  on Γ  , we obtain a vector field f (K  , D  ) with even index singularities, which determines a spin structure ξ f (K  ,D  ) on Σ  . Johnson’s theorem [5] holds for surfaces with boundary, so this spin structure defines a quadratic form q on H1 (Σ  ; Z2 ) = H1 (Σ; Z2 ). If C is a simple closed curve  in Γ  ⊂ Σ  , then q([C]) + 1 = n K (C) +  D  (C) as in the closed case. The proof is  K completed using the equalities n (C) = n K (C) and  D  (C) =  D (C) + V∂ D (C).  Since Johnson’s theorem holds true for surfaces with boundary and [3, Prop. 1] easily extends, we have the following corollary.

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Corollary 1. Let Γ ⊂ Σ be a surface graph, non-necessarily connected, and possibly with boundary. Any dimer configuration D on Γ ⊂ Σ induces an isomorphism of affine H 1 (Σ; Z2 )-spaces ψ D : K(Γ ⊂ Σ) −→ S(Σ) from the set of equivalence classes of Kasteleyn orientations on Γ ⊂ Σ onto the set of spin structures on Σ. Furthermore, ψ D − ψ D  is given by the Poincaré dual of ∆(D, D  ) in H 1 (Σ; Z2 ). In particular, ψ D = ψ D  if and only if D and D  are equivalent dimer configurations. 3.3. The Pfaffian formula for the partition function. Let Γ be a graph, not necessarily connected, and possibly with boundary, endowed with a weight system w. Realize Γ as a surface graph Γ ⊂ Σ, and fix a Kasteleyn orientation K on it. The Kasteleyn coefficient associated to an ordered pair (v, v  ) of distinct vertices of Γ is the number K avv  =




K εvv  (e)w(e),

e

where the sum is on all edges e in Γ between the vertices v and v  , and  K εvv  (e)

=

1 if e is oriented by K from v to v  ; −1 otherwise.

K = 0. Let us fix a boundary condition ∂ D and enumerate the matched One also sets avv 0 vertices of Γ by 1, 2, . . . , 2n. Then, the corresponding coefficients form a 2n × 2n skew-symmetric matrix A K (Γ ; w | ∂ D0 ) = A K called the Kasteleyn matrix. Let D be a dimer configuration on (Γ, ∂Γ ) with ∂ D = ∂ D0 , given by edges e1 , . . . , en matching vertices i  and j for  = 1, . . . , n. Let σ be the permutation (1, . . . , 2n) → (i 1 , j1 , . . . , i n , jn ), and set

ε K (D) = (−1)σ

n  =1

εiK j (e ),

where (−1)σ denotes the sign of σ . Note that ε K (D) does not depend on the choice of σ , but only on the dimer configuration D. Finally, recall that the Arf invariant of a quadratic form q : H → Z2 , where (H, ·) is a Z2 -vector space with a symmetric bilinear form, can be defined by
1 Arf(q) = √ (−1)q(α) , |Ann ||H | α∈H where Ann = {α ∈ H | α · β = 0 for all β ∈ H }. If q(α) = 0 for some α ∈ Ann , then one easily checks that Arf(q) = 0. On the other hand, if q(α) = 0 for all α ∈ Ann , then Arf(q) = Arf(q  ), where q  denotes the induced non-degenerate quadratic form on H/Ann . In particular, Arf(q) then takes the values +1 or −1.

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Theorem 2. Let Γ ⊂ Σ be a surface graph, not necessarily connected, and possibly with boundary. Let b1 (Σ) denote the dimension of H1 (Σ; Z2 ), and let g denote the genus of Σ. Then, Z α (Γ ; w | ∂ D0 ) =

1 2b1 (Σ)


q K (α) (−1) D0 ε K (D0 )Pf(A K ) [K ]

for any α ∈ H1 (Σ; Z2 ), and Z (Γ ; w | ∂ D0 ) =

1
Arf(q DK0 )ε K (D0 )Pf(A K ), 2g [K ]

where both sums are over the 2b1 (Σ) equivalence classes of Kasteleyn orientations on Γ ⊂ Σ. Furthermore, Arf(q DK0 )ε K (D0 ) only depends on K and ∂ D0 . Proof. First note that if the theorem holds for two surface graphs, then it holds for their disjoint union. Therefore, it may be assumed that Σ is connected. The first formula follows from Theorem 1: the proof of Theorem 4 and the first half of the proof of Theorem 5 of [3] generalize verbatim to the case with (possible) boundary. The second formula is obtained from the first one by summing over all α ∈ H1 (Σ; Z2 ). Finally, note that Arf(q DK0 )ε K (D0 ) = 0 if and only if q DK0 (γ ) = 0 for some boundary component γ of Σ. But q DK0 (γ ) only depends on K and ∂ D0 . If q DK0 (γ ) = 0 for all boundary component γ , then Arf(q DK0 )ε K (D0 ) = ±1 only depends on K , as shown in [3, Theorem 5].  4. Cutting and Gluing 4.1. Cutting and gluing graphs with boundary. Let (Γ, ∂Γ ) be a graph with boundary, and let us fix an edge e of Γ . Let (Γ{e} , ∂Γ{e} ) denote the graph with boundary obtained from (Γ, ∂Γ ) as follows: cut the edge e in two, and set ∂Γ{e} = ∂Γ ∪ {v  , v  }, where v  and v  are the new one valent vertices. Iterating this procedure for some set of edges E leads to a graph with boundary (ΓE, ∂ΓE), which is said to be obtained by cutting (Γ, ∂Γ ) along E. Note that a dimer configuration D ∈ D(Γ, ∂Γ ) induces an obvious dimer configuration DE ∈ D(ΓE, ∂ΓE): cut in two the dimers of D that belong to E. t ) on Γ indexed A weight system w on Γ induces a family of weight systems (wE t E by t : E → R>0 , as follows: if e is an edge of Γ which does not belong to E, set t (e) = w(e); if e ∈ E is cut into two edges e , e of Γ , set w t (e ) = t (e)w(e)1/2 wE E E t (e ) = t (e)−1 w(e)1/2 . Note that this family of weight systems is an orbit under and wE the action of the subgroup of G(ΓE) consisting of elements s such that s(v) = 1 for all v ∈ V (Γ ) and s(v  ) = s(v  ) whenever v  , v  ∈ ∂ΓE come from the same edge of E. Let us now formulate how the cutting affects the partition function. The proof is straightforward. Proposition 2. Fix a boundary condition ∂ D0 on (Γ, ∂Γ ) and a set E of edges of Γ . Then, given any parameter t : E → R>0 ,
t Z (Γ ; w | ∂ D0 ) = Z (ΓE; wE | ∂ D0I ), I ⊂E

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where the sum is taken over all subsets I of E and ∂ D0I is the boundary condition on (ΓE, ∂ΓE) induced by ∂ D0 and I : a vertex of ∂ΓE is matched in ∂ D0I if and only if it is matched in ∂ D0 or it comes from an edge in I . The operation opposite to cutting is called gluing: pick a pair of boundary vertices of Γ , and glue the adjacent edges e , e along these vertices into a single edge e. In order for the result to be a graph, it should be assumed that e and e are different edges of Γ . We shall denote by (Γϕ , ∂Γϕ ) the graph obtained by gluing (Γ, ∂Γ ) according to a pairing ϕ of several vertices of ∂Γ . Note that a dimer configuration D ∈ D(Γ, ∂Γ ) induces a dimer configuration Dϕ ∈ D(Γϕ , ∂Γϕ ) if and only if the boundary condition ∂ D on ∂Γ is compatible with ϕ, i.e: ϕ relates matched vertices with matched vertices. Obviously, a dimer configuration DE is compatible with the pairing ϕ which glues back the edges of E, and (DE)ϕ = D on ((ΓE)ϕ , (∂ΓE)ϕ ) = (Γ, ∂Γ ). An edge weight system w on Γ induces an edge weight system wϕ on Γϕ as follows:  w(e) if e is an edge of Γ ; wϕ (e) = w(e )w(e ) if e is obtained by gluing the edges e and e of Γ . If E is a set of edges of Γ and ϕ is the pairing which glues back these edges, then t ) = w for any t : E → R . (wE ϕ >0 The effect of gluing on the partition function is best understood in the language of quantum field theory. We therefore postpone its study to Sect. 5. 4.2. Cutting and gluing surface graphs with boundary. Let Γ ⊂ Σ be a surface graph with boundary. Let C be a simple curve in Σ which is “in general position” with respect to Γ , in the following sense: i. it is disjoint from the set of vertices of Γ ; ii. it intersects the edges of Γ transversally; iii. its intersection with any given face of Σ is connected. Let ΣC be the surface with boundary obtained by cutting Σ open along C. Also, let ΓC := ΓE(C) be the graph with boundary obtained by cutting (Γ, ∂Γ ) along the set E(C) of edges of Γ which intersect C, as illustrated in Fig. 1. Obviously, ΓC ⊂ ΣC is a surface graph with boundary. We will say that it is obtained by cutting Γ ⊂ Σ along C. Abusing notation, we shall write wCt for the weight system t wE (C) on ΓC .

Fig. 1. Cutting a surface graph Γ ⊂ Σ along a curve C

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A class β ∈ H1 (Σ, ∂Σ; Z2 ) induces βC ∈ H1 (ΣC , ∂ΣC ; Z2 ) via H1 (Σ, ∂Σ; Z2 ) → H1 (Σ, ∂Σ ∪ N (C); Z2 ) H1 (ΣC , ∂ΣC ; Z2 ). Here N (C) denotes a neighborhood of C in Σ, the first homomorphism is induced by inclusion, and the second one is the excision isomorphism. Note that given any two dimers configurations D and D  on Γ ⊂ Σ, ∆(DC , DC ) = ∆(D, D  )C in H1 (ΣC , ∂ΣC ; Z2 ). This easily leads to the following refinement of Proposition 2. Proposition 3. Fix β ∈ H1 (Σ, ∂Σ; Z2 ), D  ∈ D(Γ, ∂Γ ), and a boundary condition ∂ D0 on (Γ, ∂Γ ). Then, given any parameter t : E(C) → R>0 ,
Z β,D  (Γ ; w | ∂ D0 ) = Z βC ,DC (ΓC ; wCt | ∂ D0I ), I ⊂E(C)

where the sum is taken over all subsets I of E(C) and ∂ D0I is the boundary condition on (ΓC , ∂ΓC ) induced by ∂ D0 and I . Let us now define the operation opposite to cutting a surface graph with boundary. Pick two closed connected subsets M1 , M2 of ∂Σ, which are not points, and satisfy the following properties: i. M1 ∩ M2 ⊂ ∂ M1 ∪ ∂ M2 and ∂ M1 ∪ ∂ M2 is disjoint from ∂Γ ; ii. the intersection of each given face of Σ with M1 ∪ M2 is connected; iii. there exists an orientation-reversing homeomorphism ϕ : M1 → M2 which induces a bijection M1 ∩ ∂Γ → M2 ∩ ∂Γ such that for all v in M1 ∩ ∂Γ , v and ϕ(v) are not adjacent to the same edge of Γ . Let Γϕ ⊂ Σϕ be obtained from the surface graph Γ ⊂ Σ by identifying M1 and M2 via ϕ and removing the corresponding vertices of Γ . This is illustrated in Fig. 2. By the conditions above, the pair Γϕ ⊂ Σϕ remains a surface graph. It is said to be obtained by gluing Γ ⊂ Σ along ϕ. Note that any surface graph ΓC ⊂ ΣC obtained by cutting Γ ⊂ Σ along some curve C in general position with respect to Γ satisfies the conditions listed above. Furthermore, (ΓC )ϕ ⊂ (ΣC )ϕ = Γ ⊂ Σ, where ϕ is the obvious homeomorphism identifying the two closed subsets of ∂ΣC coming from C. Conversely, if C denotes the curve in Σϕ given by the identification of M1 and M2 via ϕ, then it is in general position with respect to Γϕ , and (Γϕ )C ⊂ (Σϕ )C .

Fig. 2. Gluing a surface graph Γ ⊂ Σ along ϕ : M1 → M2

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4.3. Cutting and gluing discrete spin structures. Let Γ ⊂ Σ be a surface graph with boundary, and let C be a simple curve in Σ in general position with respect to Γ . As noted above, any dimer configuration D on (Γ, ∂Γ ) induces a dimer configuration DC on (ΓC , ∂ΓC ). If two dimer configurations D, D  ∈ D(Γ, ∂Γ ) are equivalent, then DC , DC ∈ D(ΓC , ∂ΓC ) are equivalent as well: ∆(DC , DC ) = ∆(D, D  )C = 0 ∈ H1 (ΣC , ∂ΣC ; Z2 ). A Kasteleyn orientation K on Γ ⊂ Σ induces a Kasteleyn orientation K C on ΓC ⊂ ΣC as follows. Let K C be equal to K on all edges of Γ C coming from edges of Γ . For all the new edges of Γ C , there is a unique orientation which satisfies the Kasteleyn condition, since each face of Σ is crossed at most once by C. One easily checks that if K and K  are equivalent Kasteleyn orientations, then K C and K C are also equivalent. Hence, there is a well-defined operation of cutting discrete spin structures on a surface with boundary. This is not a surprise. Indeed, the inclusion ΣC ⊂ ΣC ∪ N (C) = Σ induces a homomorphism i ∗ : H1 (ΣC ; Z2 ) → H1 (Σ; Z2 ). The assignment q → qC = q ◦ i ∗ defines a map from the quadratic forms on H1 (Σ; Z2 ) to the quadratic forms on H1 (ΣC ; Z2 ), which is affine over the restriction homomorphism i ∗ : H 1 (Σ; Z2 ) → H 1 (ΣC ; Z2 ). By Johnson’s theorem, it induces an affine map between the sets of spin structures S(Σ) → S(ΣC ). By Corollary 1, there is a unique map K(Γ ⊂ Σ) → K(ΓC ⊂ ΣC ) which makes the following diagram commute: K(Γ ⊂ Σ) ∼ = ψD

 S(Σ)

/ K(ΓC ⊂ ΣC )

(2)

∼ = ψ DC

 / S(ΣC ).

This map is nothing but [K ] → [K C ]. Now, let K be a Kasteleyn orientation on a surface graph Γ ⊂ Σ, and let ϕ : M1 → M2 be an orientation-reversing homeomorphism between two closed connected subsets in ∂Σ, as described above. We shall say that a Kasteleyn orientation K on Γ ⊂ Σ is compatible with ϕ if the following conditions hold: i. whenever two edges e , e of Γ are glued into a single edge e of Γ ϕ , the orientation K agrees on e and e , giving an orientation K ϕ on e; ii. the induced orientation K ϕ is a Kasteleyn orientation on Γϕ ⊂ Σϕ . The Kasteleyn orientation K ϕ on Γϕ ⊂ Σϕ is said to be obtained by gluing K along ϕ. Given any Kasteleyn orientation K on Γ ⊂ Σ, the induced orientation K C on ΓC ⊂ ΣC is compatible with the map ϕ such that (ΣC )ϕ = Σ; furthermore, (K C )ϕ is equal to K . Conversely, if K is a Kasteleyn orientation on Γ ⊂ Σ which is compatible with ϕ, and C denotes the curve in Σϕ given by the identification of M1 and M2 via ϕ, then (K ϕ )C is equal to K . With these notations, any dimer configuration D on Γ which is compatible with ϕ satisfies (Dϕ )C = D. Therefore, diagram (2) gives K(Γϕ ⊂ Σϕ ) ∼ = ψ Dϕ

 S(Σϕ )

/ K(Γ ⊂ Σ) ∼ = ψD

 / S(Σ),

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where both horizontal maps are affine over i ∗ : H 1 (Σϕ ; Z2 ) → H 1 (Σ; Z2 ). Understanding the gluing of Kasteleyn orientations (up to equivalence) now amounts to understanding the restriction homomorphism i ∗ . Using the exact sequence of the pair (Σϕ , Σ), one easily checks the following results: – The restriction homomorphism i ∗ is injective, unless M1 and M2 are disjoint and belong to the same connected component of Σ. In this case, the kernel of i ∗ has dimension 1. – The homomorphism i ∗ is onto unless M1 ∪ M2 is a 1-cycle and the corresponding connected component of Σϕ is not closed. In this case, the cokernel of i ∗ has dimension 1. This leads to the four following cases. Fix a Kasteleyn orientation K on Γ ⊂ Σ. 1. If i ∗ is an isomorphism, then there exist a Kasteleyn orientation K  equivalent to K which is compatible with ϕ. Furthermore, the assignment [K ] → [K ϕ ] gives a well-defined map between K(Γ ⊂ Σ) and K(Γϕ ⊂ Σϕ ). 2. If i ∗ is onto but not injective, then there exist K  , K  ∼ K which are compatible with ϕ, inducing two distinct well-defined maps [K ] → [K ϕ ] and [K ] → [K ϕ ] between K(Γ ⊂ Σ) and K(Γϕ ⊂ Σϕ ). 3. If i ∗ is injective but not onto, then M1 ∪ M2 is a 1-cycle, oriented as part of the boundary of Σ. There exist K  ∼ K which is compatible with ϕ if and only if the following condition holds: n K (M1 ) + n K (M2 ) ≡ 0

(mod 2) if M1 and M2 are disjoint;

n (M1 ∪ M2 ) ≡ 1 (mod 2) otherwise. K

(Note that this condition only depends on the equivalence class of K .) In this case, it induces a well-defined class [K ϕ ] in K(Γϕ ⊂ Σϕ ). 4. Finally, assume i ∗ is neither onto nor injective. If K satisfies the condition above, then there exist K  , K  ∼ K which are compatible with ϕ, inducing two welldefined maps [K ] → [K ϕ ] and [K ] → [K ϕ ]. On the other hand, if K does not satisfy the condition above, then it does not contain any representative which is compatible with ϕ. 4.4. Cutting Pfaffians. Let us conclude this section with one last observation. Let Γ ⊂ Σ be a surface graph with boundary, and let C be a simple curve in Σ. The equality
Z (Γ ; w | ∂ D0 ) = Z (ΓC ; wCt | ∂ D0I ) I ⊂E(C)

of Proposition 2 can be understood as the Taylor series expansion of the function Z (Γ ; w | ∂ D0 ) in the variables (w(e))e∈E(C) . Clearly, if E(C) = {ei1 , . . . , eik }, then k  =1

w(ei )

∂ k Z (Γ ; w | ∂ D0 ) (0) = Z (ΓC ; wCt | ∂ D0I ). ∂w(ei1 ) · · · ∂w(eik )

By Theorem 2, the partition function Z (Γ ; w | ∂ D0 ) can be expressed as a linear combination of Pfaffians of matrices A K (Γ ; w | ∂ D0 ) depending on Kasteleyn orientations

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K of Γ ⊂ Σ such that q DK0 (γ ) = 0 for all boundary component γ of Σ. Recall that any such orientation K extends to a Kasteleyn orientation K C on ΓC ⊂ ΣC . Furthermore, KC (γ ) = 0 for all bounall equivalence classes of Kasteleyn orientations such that q(D 0 )C dary component γ of ΣC are obtained in this way. (This follows from the fact that the map [K ] → [K C ] is affine over the restriction homomorphism.) Finally, the partition function Z (ΓC ; wCt | ∂ D0 ) can also be expressed as a linear combination of Pfaffians of matrices A K C (Γ C ; wCt | ∂ D0 ) via Theorem 2. Gathering all these equations, we obtain a relation between the Pfaffian of the matrix A K (Γ ; w | ∂ D0 ) and the Pfaffian of the matrix A K C (Γ C ; wCt | ∂ D0 ). This relation turns out to be exactly the equation below, a well-known property of Pfaffians. Proposition 4. Let A = (ai j ) be a skew-symmetric matrix of size 2n. Given an ordered subset I of the ordered set α = (1, . . . , 2n), let A I denote the matrix obtained from A by removing the i th row and the i th column for all i ∈ I . Then, for any ordered set of indices I = (i 1 , j1 , . . . , i k , jk ), ∂ k Pf(A) = (−1)σ (I ) Pf(A I ), ∂ai1 j1 · · · ∂aik jk where (−1)σ (I ) denote the signature of the permutation which sends α to the ordered set I (α\I ). 5. Quantum Field Theory for Dimers 5.1. Quantum field theory on graphs. Let (Γ, ∂Γ ) be a graph with boundary, and let us assume that each vertex v in ∂Γ is oriented, that is, endowed with some sign εv . In the spirit of the Atiyah-Segal axioms for a (0 + 1)-topological quantum field theory [2,13], let us define a quantum field theory on graphs as the following assignment: 1. Fix a finite dimensional complex vector space V . 2. To the oriented boundary ∂Γ , assign the vector space   Z (∂Γ ) = V⊗ V ∗, v∈∂Γ εv =+1

v∈∂Γ εv =−1

where V ∗ denotes the vector space dual to V . 3. To a finite graph Γ with oriented boundary ∂Γ and weight system w, assign some vector Z (Γ ; w) ∈ Z (∂Γ ), with Z (∅; w) = 1 ∈ C = Z (∅). Note that any orientation preserving bijection f : ∂Γ → ∂Γ  induces an isomorphism Z ( f ) : Z (∂Γ ) → Z (∂Γ  ) given by permutation of the factors. This assignment is functorial: if g : ∂Γ  → ∂Γ  is another orientation preserving bijection, then Z (g ◦ f ) = Z (g) ◦ Z ( f ). Finally, if f : ∂Γ → ∂Γ  extends to a simplicial bijection F : Γ → Γ  , then Z ( f ) maps Z (Γ ) to Z (Γ  ). Note also that Z (−∂Γ ) = Z (∂Γ )∗ , and that Z (∂Γ ∂Γ  ) = Z (∂Γ ) ⊗ Z (∂Γ  ). The main point is that we require the following gluing axiom. Let Γ be a graph with oriented boundary ∂Γ , such that there exists two disjoint subsets X 1 , X 2 of ∂Γ and an orientation reversing bijection ϕ : X 1 → X 2 (i.e. εϕ(v) = −εv for all v ∈ X 1 ). Obviously, ϕ induces a linear isomorphism Z (ϕ) : Z (X 1 ) → Z (X 2 )∗ . Let Γϕ denote the graph with boundary ∂Γϕ = ∂Γ \ (X 1 ∪ X 2 ) obtained by gluing Γ according to ϕ,

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and let wϕ be the corresponding weight system on Γϕ (recall Sect. 4.1). Let Bϕ denote the composition Z (∂Γ ) = Z (∂Γϕ ) ⊗ Z (X 1 ) ⊗ Z (X 2 ) → Z (∂Γϕ ) ⊗ Z (X 2 )∗ ⊗ Z (X 2 ) → Z (∂Γϕ ), where the first homomorphism is given by id ⊗ Z (ϕ) ⊗ id, and the second is induced by the natural pairing Z (X 2 )∗ ⊗ Z (X 2 ) → C. We require that Bϕ (Z (Γ ; w)) = Z (Γϕ ; wϕ ). Note that a bipartite structure on a graph with boundary (Γ, ∂Γ ) induces an orientation of ∂Γ . Furthermore, the graph Γϕ will remain bipartite for any orientation reversing bijection ϕ. Remark 1. In the same spirit, one can define a quantum field theory on surface graphs. Here, the vector Z (Γ ⊂ Σ; w) ∈ Z (∂Γ ) might depend on the realization of Γ as a surface graph Γ ⊂ Σ, and the gluing axiom concerns gluing of surface graphs, as defined in Sect. 4.2. 5.2. Quantum field theory for dimers on graphs. Let us now explain how the dimer model on weighted graphs with boundary defines a quantum field theory. As vector space V , choose the 2-dimensional complex vector space with fixed basis a0 , a1 . Let α0 , α1 denote the dual basis in V ∗ . To a finite graph Γ with oriented boundary ∂Γ and weight system w, assign
Z (Γ ; w) = Z (Γ ; w | ∂ D) a(∂ D) ∈ Z (∂Γ ), ∂D

where the sum is on all possible boundary conditions ∂ D on ∂Γ , and   a(∂ D) = aiv (∂ D) ⊗ αiv (∂ D) ∈ Z (∂Γ ). v∈∂Γ εv =+1

v∈∂Γ εv =−1

Here, i v (∂ D) = 1 if the vertex v is matched by ∂ D, and i v (∂ D) = 0 otherwise. Let us check the gluing axiom. First note that Bϕ (a(∂ D)) = 0 unless ∂ D is compatible with ϕ (i.e: unless ϕ(v) is matched in ∂ D if and only if v is matched in ∂ D). In such a case, Bϕ (a(∂ D)) = a(∂ D|∂Γϕ ), where ∂ D|∂Γϕ denotes the restriction of the boundary condition ∂ D to ∂Γϕ ⊂ ∂Γ . All the possible boundary conditions ∂ Dϕ on ∂Γϕ are given by such restrictions. Therefore, 

Bϕ (Z (Γ ; w)) = Z (Γ ; w | ∂ D) a(∂ Dϕ ), ∂ Dϕ

∂ D⊃∂ Dϕ

the interior sum being on all boundary conditions ∂ D on ∂Γ that are compatible with ϕ, and such that ∂ D|∂Γϕ = ∂ Dϕ . By definition,

Z (Γ ; w | ∂ D) = w(D) = Z (Γϕ ; wϕ | ∂ Dϕ ). ∂ D⊃∂ Dϕ

D:∂ D⊃∂ Dϕ

Therefore, the gluing axiom is satisfied.

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5.3. The dimer model as the theory of free Fermions. Let W be an n-dimensional vector space. The choice of an ordered basis in W induces an isomorphism between its exterior   algebra W = ⊕nk=0 k W and the algebra generated by elements φ1 , . . . , φn with defining relations φi φ j = −φ j φi . This is known as the Grassmann algebra generated by φ1 , . . . , φn . This choice of an ordered basis in W also defines a basis in the top exterior  power of W . The integral over the Grassmann algebra of W of an element a ∈ W is the coordinate of a in the top exterior power of W with respect to this basis. It is denoted by a dφ. There is a scalar product on the Grassmann algebra generated by φ1 , . . . , φn ; it is given by the Grassmann integral

< F, G >=

exp

n 


 φi ψi F(φ)G(ψ)dφdψ.

(3)

i=1

Note that the monomial basis is orthonormal with respect to this scalar product. One easily shows (see e.g. the Appendix to [3]) that the Pfaffian of an n × n skew symmetric matrix A = (ai j ) can be written as

Pf(A) =

exp

n 1


2

 φi ai j φ j dφ.

i, j=1

Let us now use this to reformulate the quantum field theory of dimers in terms of Grassmann integrals. Let Γ ⊂ Σ be a (possibly disconnected) surface graph, possibly with boundary. Let us fix a numbering of the vertices of Γ , a boundary condition ∂ D0 on ∂Γ and a Kasteleyn orientation K on Γ ⊂ Σ. Let aiKj be the Kasteleyn coefficient associated to K and the vertices i, j of Γ (recall Sect. 3). By Theorem 2 and the identity above,

 1
1
K K Z (Γ ; w | ∂ D0 ) = g Arf(q D0 )ε (D0 ) exp φi aiKj φ j dφ∂ D0 , 2 2 [K ]

i, j∈V (D0 )

where the sum is over all 2b1 (Σ) equivalence classes of Kasteleyn orientations on Γ ⊂ Σ, V (D0 ) denotes the set of vertices of Γ that are matched by D0 , and dφ∂ D0 = ∧i∈V (D0 ) dφi . This leads to the formula

 1
1
φi aiKj φ j D∂KD0 φ, exp Z (Γ ; w | ∂ D0 ) = g 2 2 [K ]

i, j∈V (D0 )

where D∂KD0 φ = Arf(q DK0 )ε K (D0 ) dφ∂ D0 . Let us point out that this measure does not depend on the choice of D0 , but only on K and on the induced boundary condition ∂ D0 (recall Theorem 2). Now, the numbering of the vertices of Γ gives a numbering of the vertices of ∂Γ . This induces a linear isomorphism between Z (∂Γ ) and the Grassmann algebra (∂Γ ) generated by (φi )i∈∂Γ . The image of the partition function under this isomorphism is the following element of the Grassmann algebra of boundary vertices:
  Z (Γ ; w) = Z (Γ ; w | ∂ D0 ) i∈V (∂ D0 ) φi ∈ (∂Γ ), ∂ D0

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where V (∂ D0 ) = V (D0 ) ∩ ∂Γ . This leads to

 1
 1

φi aiKj φ j D∂KD0 φ i∈V (∂ D0 ) φi Z (Γ ; w) = g exp 2 2 ∂ D0 [K ] i, j∈V (D0 )

 1
1
= g φi aiKj φ j D K φ, exp 2 2 [K ]

i, j∈V (Γ )

where D K φ = Arf(q DK0 )ε K (D0 ) ∧i ∈∂Γ dφi . This measure depends only on K , but not / on D0 . We can now formulate the dimer model as the theory of free (Gaussian) Fermions:  1. To the boundary of Γ ⊂ Σ, we assign (∂Γ ), the Grassmann algebra generated by the ordered set ∂Γ ; 2. To a surface graph Γ ⊂ Σ with ordered set  of vertices V (Γ ) and weight system w, we assign the element Z (Γ ⊂ Σ; w) of (∂Γ ) given by

 1
1
Z (Γ ⊂ Σ; w) = g φi aiKj φ j D K φ, exp 2 2 [K ]

i, j∈V (Γ )

where the sum is over all 2b1 (Σ) equivalence classes of Kasteleyn orientations on Γ ⊂ Σ, and D K φ = Arf(q DK0 )ε K (D0 ) ∧i ∈∂Γ dφi . / The gluing axiom now takes the following form. Let Γϕ ⊂ Σϕ denote the surface graph with boundary obtained by gluing Γ ⊂ Σ along some orientation-reversing homeomorphism ϕ : M1 → M2 (see Sect. 4.2). Recall that ϕ induces a bijection between the two disjoint sets  X 1 = ∂Γ  ∩ M1 and X 2 = ∂Γ ∩ M2 . Therefore, it induces an isomorphism Z (ϕ) : (X 1 ) → (X 2 ). Consider the map Bϕ given by the composition         (∂Γ ) = (∂Γϕ ) ⊗ (X 1 ) ⊗ (X 2 ) → (∂Γϕ ) ⊗ (X 2 )∗ ⊗ (X 2 ) → (∂Γϕ ).  Here, the first homomorphism is given by id ⊗ (h ◦ Z (ϕ)) ⊗ id, where h : (X 2 ) →  (X 2 )∗ is the isomorphism induced by the scalar product (3). Then, we require that Bϕ (Z (Γ ⊂ Σ; w)) = Z (Γϕ ⊂ Σϕ ; wϕ ). We already know that this equality holds. Indeed, Z (Γ ⊂ Σ; w) just depends on (Γ, w), and the formula above is nothing but the gluing axiom for Z (Γ ; w) translated in the formalism of Grassmann algebras. However, it can also be proved from scratch using the results of Sect. 4.3 together with well-known properties of Pfaffians. 6. Dimers on Bipartite Graphs and Height Functions 6.1. Composition cycles on bipartite graphs. Recall that a bipartite structure on a graph Γ is a partition of its set of vertices into two groups, say blacks and whites, such that no edge of Γ joins two vertices of the same group. Equivalently, a bipartite structure can be regarded as a 0-chain

β= v− v ∈ C0 (Γ ; Z). v black

v white

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A bipartite structure induces an orientation on the edges of Γ , called the bipartite orientation: simply orient all the edges from the white vertices to the black ones. Using this orientation, a dimer configuration D ∈ D(Γ, ∂Γ ) can now be regarded as a 1-chain with Z-coefficients
D= e ∈ C1 (Γ ; Z) e∈D

such that ∂ D = β in C0 (Γ, ∂Γ ; Z) = C0 (Γ ; Z)/C0 (∂Γ ; Z). Therefore, given two dimer configurations D, D  on Γ , their difference D − D  is a 1-cycle (r el ∂Γ ) with Z-coefficients, denoted by C(D, D  ). Its connected components are called (D, D  )composition cycles. In short, a bipartite structure on a graph allows to orient the composition cycles. 6.2. Height functions for planar bipartite graphs. Let us now assume that the bipartite graph Γ is planar without boundary, i.e. that it can be realized as a surface graph Γ ⊂ S 2 . Let X denote the induced cellular decomposition of the 2-sphere, which we endow with the counter-clockwise orientation. Since H1 (X ; Z) = H1 (S 2 ; Z) = 0, the 1-cycle C(D, D  ) is a 1-boundary, so there exists σ D,D  ∈ C2 (X ; Z) such that ∂σ D,D  = C(D, D  ). Let h D,D  ∈ C 2 (X ; Z) be given by the equality
σ D,D  = h D,D  ( f ) f ∈ C2 (X ; Z), f ∈F(X )

where the sum is over all faces of X . The cellular 2-cochain h D,D  is called a height function associated to D, D  . Since H2 (X ; Z) = H2 (S 2 ; Z) = Z, the 2-chain σ D,D  is uniquely defined by D, D  up to a constant, and the same holds for h D,D  . Hence, one can normalize all height functions by setting h D,D  ( f 0 ) = 0 for some fixed face f 0 . This is illustrated in Fig. 3. Alternatively, h D,D  can be defined as the only h ∈ C 2 (X ; Z) such that h( f 0 ) = 0 and h increases by 1 when a (D, D  )-composition cycle is crossed in the positive direction (left to right as we cross). It follows that for any height function h and any two 2-cells f 1 and f 2 , |h( f 1 ) − h( f 2 )| ≤ d( f 1 , f 2 ),

Fig. 3. An example of a bipartite planar graph with two dimer configurations D (solid) and D  (traced lines). The corresponding height function h D,D  (where f 0 is the outer face) and (D, D  )-composition cycles are pictured on the right hand side

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where d( f 1 , f 2 ) is the distance between f 1 and f 2 in the dual graph, i.e. the minimal number of edges crossed by a path connecting a point inside f 1 with a point inside f 2 . This can be regarded as a Lipschitz property of height functions. Note also that for any three dimer configurations D, D  and D  on Γ , the following cocycle equality holds: h D,D  + h D  ,D  = h D,D  . The Lipschitz condition stated above leads to the following definition. Given a fixed 2-cell f 0 of the cellular decomposition X induced by Γ ⊂ S 2 , set H(X, f 0 ) = {h ∈ C 2 (X ; Z) | h( f 0 ) = 0 and |h( f 1 ) − h( f 2 )| ≤ d( f 1 , f 2 ) ∀ f 1 , f 2 }. Given h ∈ H(X, f 0 ), let C(h) denote the oriented closed curves formed by the set of oriented edges e of Γ such that h increases its value by 1 when crossing e in the positive direction. (In other words, C(h) = ∂σ , where σ ∈ C2 (X ; Z) is dual to h ∈ C 2 (X ; Z).) Obviously, there is a well-defined map D(Γ ) × D(Γ ) → H(X, f 0 ), (D, D  ) → h D,D  with C(h D,D  ) = C(D, D  ). However, this map is neither injective nor surjective in general. Indeed, the number of preimages of a given h is equal to the number of dimer configurations on the graph obtained from Γ by removing the star of C(h). Depending on Γ ⊂ S 2 , this number can be zero, or arbitrarily large. To obtain a bijection, we proceed as follows. Fix a dimer configuration D0 on Γ . Let C(D0 ) denote the set of all C ⊂ Γ consisting of disjoint oriented simple 1-cycles, such that the following condition holds: for all e ∈ D0 , either e is contained in C or e is disjoint from C. Finally, set H D0 (X, f 0 ) = {h ∈ H(X, f 0 ) | C(h) ∈ C(D0 )}. Proposition 5. Given any h ∈ H D0 (X, f 0 ), there is unique dimer configuration D ∈ D(Γ ) such that h D,D0 = h. Furthermore, given any two dimer configurations D0 , D1 on Γ , we have a canonical bijection H D0 (X, f 0 ) → H D1 (X, f 0 ) given by h → h + h D0 ,D1 . Proof. One easily checks that the assignment D → C(D, D0 ) defines a bijection D(Γ ) → C(D0 ). Furthermore, there is an obvious bijection H D0 (X, f 0 ) → C(D0 ) given by h → C(h). This induces a bijection D(Γ ) → H D0 (X, f 0 ) and proves the first part of the proposition. The second part follows from the first one via the cocycle identity h D,D0 + h D0 ,D1 = h D,D1 .  Let us now consider an edge weight system w on the bipartite planar graph Γ . Recall that the Gibbs measure of D ∈ D(Γ ) is given by w(D) , Z (Γ ; w)   where w(D) = e∈D w(D) and Z (Γ ; w) = D∈D(Γ ) w(D). Let us now fix a dimer configuration D0 and a face f 0 of X , and use the bijection D(Γ ) → H D0 (X, f 0 ) given by D → h D,D0 to translate this measure into a probability measure on H D0 (X, f 0 ). Prob(D) =

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To do so, we shall need the following notations: given an oriented edge e of Γ , set  w(e) if the orientation on e is the bipartite orientation; wβ (e) = w(e)−1 otherwise. This defines a group homomorphism wβ : C1 (X ; Z) → R>0 . Finally, given any f ∈ F(X ), set q f = wβ (∂ f ), where ∂ f is oriented as the boundary of the counter-clockwise oriented face f . This number q f is called the volume weight of the face f . Proposition 6. The Gibbs measure on D(Γ ) given by the edge weight system w translates into the following probability measure on H D0 (X, f 0 ): Prob D0 (h) = where q(h) =



h( f )

qf

q(h) , Z D0 , f0 (X, q)


and Z D0 , f0 (X ; q) =

f ∈F(X )

q(h).

h∈HD0 (X, f 0 )

Furthermore, this measure is independent of the choice of f 0 . Finally, the bijection H D0 (X, f 0 ) → H D1 (X, f 0 ) given by h → h + h D0 ,D1 is invariant with respect to the measures Prob D0 and Prob D1 . Proof. For any D ∈ D(Γ ), we have   w(D)w(D0 )−1 = w(e) w(e)−1 = wβ (C(D, D0 )) e∈D

e∈D0

= wβ (∂σ D,D0 ) = wβ =






h D,D0 ( f )∂ f

f ∈F(X )

wβ (∂ f )

h D,D0 ( f )

=

f ∈F(X )

The proposition follows easily from this equality.





h D,D0 ( f )

qf

= q(h D,D0 ).

f ∈F(X )



Let V (Γ ) (resp. E(Γ )) denote the set of vertices (resp. of edges) of Γ . Recall that the group G(Γ ) = {s : V (Γ ) → R>0 } acts on the set of weight systems on Γ by (sw)(e) = s(e+ )w(e)s(e− ), where e+ and e− are the two vertices adjacent to the edge e. As observed in Sect. 2.1, the Gibbs measure on D(Γ ) is invariant under the action of the group G(Γ ). Note also that this action is free unless Γ is bipartite. In this later case, the 1-parameter family of elements sλ ∈ G(Γ ) given by sλ (v) = λ if v is black and sλ (v) = λ−1 if v is white act as the identity on the set of weight systems. Hence, if Γ is bipartite, the

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number of “essential” parameters is equal to |E(Γ )| − |V (Γ )| + 1. If this bipartite graph is planar, then |E(Γ )| − |V (Γ )| + 1 = |F(X )| − χ (S 2 ) + 1 = |F(X )| − 1. The |F(X )| volume weights q f are  invariant with respect to the action of G(Γ ). They can be normalized in such a way that f ∈F(X ) q f = 1, giving exactly |F(X )|−1 parameters. Thus, in the height function formulation of the Gibbs measure, only essential parameters appear.

6.3. Height functions for bipartite surface graphs. Let us now address the general case of a bipartite surface graph Γ ⊂ Σ, possibly disconnected, and possibly with boundary b1 ∂Γ ⊂ ∂Σ. Fix a family γ = {γi }i=1 of oriented simple curves in Γ representing a basis in H1 (Σ, ∂Σ; Z). Note that such a family of curves exists since Γ is the 1-squeletton of a cellular decomposition X of Σ. Given any D, D  ∈ D(Γ, ∂Γ ), the homology class of C(D, D  ) = D − D  can be written in a unique way [C(D, D  )] =

b1


γ

a D,D  (i)[γi ] ∈ H1 (Σ, ∂Σ; Z),

i=1

 b1 γ γ a D,D  (i)γi is a 1-boundary (r el ∂ X ), that with a D,D  (i) ∈ Z. Hence, C(D, D  ) − i=1 γ is, there exists σ D,D  ∈ C2 (X, ∂ X ; Z) = C2 (X ; Z) such that γ

C(D, D  ) − ∂σ D,D  −

b1


γ

a D,D  (i)γi ∈ C1 (∂ X ; Z).

(4)

i=1 γ

γ

The 2-cochain h D,D  ∈ C 2 (X ; Z) dual to σ D,D  is called a height function associated to D, D  with respect to γ . Since Z 2 (X, ∂ X ; Z) = H2 (X, ∂ X ; Z) = H2 (Σ, ∂Σ; Z) ∼ = γ H 0 (Σ; Z), the 2-chain σ D,D  is uniquely determined by D, D  and γ up to an element γ of H 0 (Σ; Z), and the same holds for h D,D  . In other words, the set of height functions associated to D, D  with respect to γ is an affine H 0 (Σ; Z)-space: it admits a freely transitive action of the abelian group H 0 (Σ; Z). One can normalize the height functions by choosing some family F0 of faces of X , one for each connected component of X , and γ by setting h D,D  ( f 0 ) = 0 for all f 0 ∈ F0 . Given h ∈ C 2 (X ; Z), set C(h) = ∂σ ∈ C1 (X ; Z), where σ ∈ C2 (X ; Z) is dual to h ∈ C 2 (X ; Z). Given a fixed D0 ∈ D(Γ, ∂Γ ), let C(D0 ) denote the set of all C ⊂ Γ consisting of disjoint oriented 1-cycles (r el ∂Γ ) such that the following condition holds: for all e ∈ D0 , either e is contained in C or e is disjoint from C. γ Finally, let H D0 (X, F0 ) denote the set of pairs (h, a) ∈ C 2 (X ; Z)×Zb1 which satisfy the following properties: – h( f 0 ) = 0 for all f 0 in F0 ; b1 – there exists C ∈ C(D0 ) such that C − C(h) − i=1 a(i)γi ∈ C1 (∂ X ; Z). We obtain the following generalization of Proposition 5. The proof is left to the reader.

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Proposition 7. Given any (h, a) ∈ H D0 (X, F0 ), there is a unique dimer configuration γ γ D ∈ D(Γ, ∂Γ ) such that h D,D0 = h and a D,D0 = a. Furthermore, given any two dimer configuration D0 , D1 ∈ D(Γ, ∂Γ ), there is a canonical bijection γ

γ

H D0 (X, F0 ) → H D1 (X, F0 ) γ

γ

given by (h, a) → (h + h D0 ,D1 , a + a D0 ,D1 ). Recall that the boundary conditions on dimer configurations induce a partition D(Γ, ∂Γ ) =



D(Γ, ∂Γ | ∂ D0 ),

∂ D0

where D(Γ, ∂Γ | ∂ D0 ) = {D ∈ D(Γ, ∂Γ ) | ∂ D = ∂ D0 }. This partition translates γ γ into a partition of H D0 (X, F0 ) via the bijection D(Γ, ∂Γ ) → H D0 (X, F0 ) given by γ γ D → (h D,D0 , a D,D0 ). Indeed, let F∂ (X ) denote the set of boundary faces of X , that is, the set of faces of X that are adjacent to ∂Σ. The choice of a boundary condition γ ∂ D0 (together with F0 ) determines h D,D0 ( f ) for all D such that ∂ D = ∂ D0 and all γ f ∈ F∂ (X ). The actual possible values of h D,D0 on the boundary faces depend on γ , D0 and F0 ; they can be determined explicitly. We shall denote by ∂h such a value of a height function on boundary faces, and call it a boundary condition for height functions. In short, we obtain a partition γ

H D0 (X, F0 ) =

∂h 0

γ

H D0 (X, F0 | ∂h 0 ) γ

indexed by all possible boundary conditions on height functions h D,D0 . Each boundary condition on dimer configurations corresponds to one boundary condition on height γ functions via D → h D,D0 . Let us now consider an edge weight system w on the bipartite graph Γ , and a fixed boundary condition ∂ D0 . Recall that the Gibbs measure for the dimer model on (Γ, ∂Γ ) with weight system w and boundary condition ∂ D0 is given by Prob(D | ∂ D0 ) =

w(D) , Z (Γ ; w | ∂ D0 )

where Z (Γ ; w | ∂ D0 ) =




w(D).

D∈D(Γ,∂Γ | ∂ D0 )

Let us realize Γ as a surface graph Γ ⊂ Σ, fix a dimer configuration D0 ∈ D(Γ, ∂Γ ), a family γ = {γi } of oriented simple curves in Γ representing a basis in H1 (Σ, ∂Σ; Z), and a collection F0 of faces of the induced cellular decomposition X of Σ, one face for each connected component of X . We can use the bijection D(Γ, ∂Γ | ∂ D0 ) → γ γ γ H D0 (X, F0 | ∂h 0 ) given by D → (h D,D0 , a D,D0 ) to translate the Gibbs measure into a γ probability measure on H D0 (X, F0 | ∂h 0 ).

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To do so, let us first extend the weight system w to all edges of X by setting w(e) = 1 for all boundary edges of X . As in the planar case, define wβ : C1 (X ; Z) → R>0 as the group homomorphism such that, for any oriented edge e of X ,  w(e) if the orientation on e is the bipartite orientation; wβ (e) = w(e)−1 otherwise. Note that this makes sense even for boundary edges where there is no bipartite orientation, as w(e) = 1 for such edges. Consider the parameters q f = wβ (∂ f ) for all f ∈ F(X ) \ F∂ (X ); qi = wβ (γi ) for all 1 ≤ i ≤ b1 . We obtain the following generalization of Proposition 6: γ

Proposition 8. Given an element (h, a) ∈ H D0 (X, F0 ), set 

q(h, a) =



h( f )

qf

f ∈F(X )\F∂ (X )

a(i)

qi

.

1≤i≤b1

Then, the Gibbs measure for the dimer model on (Γ, ∂Γ ) with weight system w and boundary condition ∂ D0 translates into the following probability measure on γ H D0 (X, F0 | ∂h 0 ): Prob D0 (h, a | ∂h 0 ) =

q(h, a) γ Z D0 ,F0 (X ; q

| ∂h 0 )

,

where


γ

Z D0 ,F0 (X ; q | ∂h 0 ) =

q(h, a).

γ

(h,a)∈HD (X,F0 | ∂h 0 ) 0

Furthermore, the measure is independent of the choice of F0 . Finally, the bijection γ γ γ γ H D0 (X, F0 | ∂h 0 ) → H D1 (X, F0 | ∂h 1 ) given by (h, a) → (h + h D0 ,D1 , a + a D0 ,D1 ) is invariant with respect to the measures Prob D0 and Prob D1 . Proof. For any D ∈ D(Γ, ∂Γ | ∂ D0 ), Eq. (4) leads to

 b1 γ γ wβ C(D, D0 ) − ∂σ D,D0 − i=1 a D,D0 (i)γi = 1. Computing the first term, we get   w(e) w(e)−1 = w(D)w(D0 )−1 . wβ (C(D, D0 )) = e∈D

e∈D0

As for the second one, γ

wβ (∂σ D,D0 ) = wβ


f ∈F(X )

h D,D0 ( f )∂ f



=

 f ∈F(X )

h D,D0 ( f )

qf

.

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Since wβ (γi ) = qi , these equations lead to w(D) = w(D0 )



γ

h D,D ( f )

qf

0

f ∈F(X )



γ

a D,D (i)

qi

0

γ

γ

= λ · q(h D,D0 , a D,D0 ),

1≤i≤b1

γ h D,D ( f )  where λ = w(D0 ) f ∈F∂ (X ) q f 0 depends only on D0 and ∂ D0 . The proposition follows easily from this equality. 

Let us count the number of essential parameters in the dimer model on (Γ, ∂Γ ) with some boundary condition partitioning ∂Γ into (∂Γ )nm (∂Γ )m , matched and non-matched vertices. We have |E(Γ )| − |(∂Γ )nm | edge weights, with an action of a (|V (Γ )| − |(∂Γ )nm |)-parameter group. Since Γ is bipartite, there is a b0 (Γ )-parameter subgroup acting as the identity. Therefore, the number of essential parameters is equal to |E(Γ )| − |V (Γ )| + b0 (Γ ) = |E(X )| − |∂Γ | − |V (X )| + b0 (X ) = |F(X )| − |∂Γ | − χ (X ) + b0 (X ) = |F(X ) \ F∂ (X )| + b1 (Σ) − b2 (Σ). to the parameters q f and qi . The numbers |F(X ) \ F∂ (X )| and b1 (Σ) correspond Furthermore, the parameters q f can be normalized by f q f = 1, the product being on all faces of a given closed component of Σ. Therefore, we obtain exactly the right number of parameters in this height function formulation of the dimer model. Remark 2. Note that all the results of the first part of the present section can be adapted to the general case of a non-necessarily bipartite surface graph: one simply needs to work with Z2 -coefficients. However, the height function formulation of the dimer model using volume weights does require a bipartite structure. It is unknown whether a reformulation of the dimer model with the right number of parameters is possible in the general case.

6.4. The dimer quantum field theory on bipartite surface graphs. Let us now use these results to reformulate the dimer quantum field theory on bipartite graphs. Let Γ ⊂ Σ be a bipartite surface graph, and let X denote the induced cellular decomposition of Σ. Fix a dimer configuration D0 ∈ D(Γ, ∂Γ ), a family γ = {γi } of oriented simple curves in Γ representing a basis in H1 (Σ, ∂Σ; Z), and a choice F0 of one face in each connected component of X . 1. To ∂ X , assign Z (∂ X ) =



W,

f ∈F∂ (X )

where W is the complex vector space with basis {αn }n∈Z , and F∂ (X ) denotes the set of faces of X adjacent to the boundary. 2. To X with weight system q = {q f } f ∈F(X ) ∪ {qi }1≤i≤b1 (Σ) , assign
γ γ Z D0 ,F0 (X ; q) = Z D0 ,F0 (X ; q | ∂h) α(∂h) ∈ Z (∂ X ), ∂h

Dimers on Surface Graphs and Spin Structures. II

where




γ

Z D0 ,F0 (X ; q | ∂h) =

467



h( f )

γ (h,a)∈HD (X,F0 | ∂h) f ∈F(X )\F∂ (X )

qf



a(i)

qi

1≤i≤b1 (Σ)

0

and α(∂h) =



h( f ) αh( f ) . f ∈F∂ (X ) q f

Recall the notation a(∂ D) ∈ Z (∂Γ ) of Sect. 5.2. The bijection D(Γ, ∂Γ ) → γ H D0 (X, F0 ) induces an inclusion j : Z (∂Γ ) → Z (∂ X ) such that j (a(∂ D)) =



f ∈F∂ (X )

αh γ

D,D0 ( f )

.

Therefore, using the proof of Proposition 8,
Z (Γ ; w | ∂ D) j (a(∂ D)) j (Z (Γ ; w)) = ∂D

=


∂h

γ

Z D0 ,F0 (X ; q | ∂h) w(D0 ) γ

 f ∈F∂ (X )

h( f )

qf



αh( f )

f ∈F∂ (X )

= w(D0 ) Z D0 ,F0 (X ; q), where the weight system q is obtained from w by q f = wβ (∂ f ) and qi = wβ (γi ). In this setting, the gluing axiom makes sense only when the data β, D0 and γ are compatible with the gluing map ϕ. In such a case, it holds by the equality above and the results of Sect. 5.2. The equivalence between the quantum field theories formulated in Sect. 5.3 and in the present section should be regarded as a discrete version of the boson-fermion correspondence on compact Riemann surfaces (see [1]). Acknowledgements. We are grateful to J. Andersen, M. Baillif, P. Teichner and A. Vershik for inspiring discussions. We also thankfully acknowledge the hospitality of the Department of Mathematics at the University of Aarhus. The work of D.C. was supported by the Swiss National Science Foundation. This work of N.R. was partially supported by the NSF grant DMS–0307599, by the CRDF grant RUM1–2622, by the Humboldt Foundation and by the Niels Bohr research grant.

References 1. Álvarez-Gaumé, L., Bost, J.-B., Moore, G., Nelson, P., Vafa, C.: Bosonization on higher genus Riemann surfaces. Commun. Math. Phys. 112, 503–552 (1987) 2. Atiyah, M.: Topological quantum field theories. Inst. Hautes Études Sci. Publ. Math. 68, 175–186 (1988) 3. Cimasoni, D., Reshetikhin, N.: Dimers on surface graphs and spin structures. I. Commun. Math. Phys. 275, 187–208 (2007) 4. Cohn, H., Kenyon, R., Propp, J.: A variational principle for domino tilings. J. Amer. Math. Soc. 14, 297–346 (2001) 5. Johnson, D.: Spin structures and quadratic forms on surfaces. J. London Math. Soc. (2) 22, 365–373 (1980) 6. Kasteleyn, W.: Dimer statistics and phase transitions. J. Math. Phys. 4, 287–293 (1963) 7. Kasteleyn, W.: Graph Theory and Theoretical Physics. London: Academic Press, 1967, pp. 43–110 8. Kenyon, R., Okounkov, A.: Planar dimers and Harnack curves. Duke Math. J. 131, 499–524 (2006) 9. Kenyon, R., Okounkov, A., Sheffield, S.: Dimers and amoebae. Ann. of Math. (2) 163, 1019–1056 (2006) 10. Kuperberg, G.: An exploration of the permanent-determinant method. Electron. J. Combin. 5, Research Paper 46, (1998) 34 pp. (electronic)

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11. Galluccio, A., Loebl, M.: On the theory of Pfaffian orientations. I. Perfect matchings and permanents. Electron. J. Combin. 6, Cambridge MA: Research Paper 6, (1999) 18 pp. (electronic) 12. McCoy, B., Wu, T.T.: The two-dimensional Ising model. Cambridge MA: Harvard University Press, 1973 13. Segal, G.: The definition of conformal field theory. In: Differential geometrical methods in theoretical physics (Como, 1987), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 250, Dordrecht: Kluwer Acad. Publ., 1988, pp. 165–171 14. Tesler, G.: Matchings in graphs on non-orientable surfaces, J. Combin. Theory Ser. B 78(2), 198–231 (2000) Communicated by L. Takhtajan

Commun. Math. Phys. 281, 469–497 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0482-9

Communications in

Mathematical Physics

The Equivariant Cohomology Theory of Twisted Generalized Complex Manifolds Yi Lin Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, M5S 3G3, Canada. E-mail: [email protected] Received: 20 April 2007 / Accepted: 23 October 2007 Published online: 17 May 2008 – © Springer-Verlag 2008

Abstract: It has been shown recently by Kapustin and Tomasiello that the mathematical notion of Hamiltonian actions on twisted generalized Kähler manifolds is in perfect agreement with the physical notion of general (2, 2) gauged sigma models with threeform fluxes. In this article, we study the twisted equivariant cohomology theory of Hamiltonian actions on H -twisted generalized complex manifolds. If the manifold satisfies the ∂∂-lemma, we establish the equivariant formality theorem. If in addition, the manifold satisfies the generalized Kähler condition, we prove the Kirwan injectivity in this setting. We then consider the Hamiltonian action of a torus on an H -twisted generalized Calabi-Yau manifold and extend to this case the Duistermaat-Heckman theorem for the push-forward measure. As a side result, we show in this paper that the generalized Kähler quotient of a generalized Kähler vector space can never have a (cohomologically) non-trivial twisting. This gives a negative answer to a question asked by physicists whether one can construct (2, 2) gauged linear sigma models with non-trivial fluxes. 1. Introduction Generalized complex geometry was initiated by Hitchin [H02] and further developed by Gualtieri in [Gua03]. It provides a unifying framework for both symplectic geometry and complex geometry, and turns out to be a useful geometric language understanding various aspects of string theory. Generalized Kähler geometry, the generalized complex version of Kähler geometry, was introduced by Gualtieri, who also shows that it is equivalent to the bi-Hermitian geometry which was first discovered by Gates, Hull and Rocek [GHR84] from the study of general (2, 2) supersymmetric sigma models. In [LT05], Tolman and the author proposed a definition of a Hamiltonian action and a moment map in both generalized complex geometry and generalized Kähler geometry. Recently, it has been shown by Kapustin and Tomasiello [KT06] that the mathematical notion of Hamiltonian actions on generalized Kähler manifolds introduced in [LT05]

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is in perfect agreement with the expectations from renormalization group flow, and corresponds exactly to the physical notion of general (2, 2) gauged sigma models with three-form fluxes. In this paper, we study the equivariant cohomology theory of Hamiltonian actions on twisted generalized complex and Kähler manifolds. To better explain the ideas involved in this work, let us first state an equivalent definition of Hamiltonian actions on generalized complex manifolds. Suppose a compact Lie group G with Lie algebra g acts on an H -twisted generalized complex manifold (M, J ), where H is a closed three form on M. The action is said to be Hamiltonian if there exists an equivariant smooth function µ : M → g∗ , called the generalized moment map, a g∗ -valued one form α ∈ 1 (M, g∗ ), called the moment one form, such that H + α is equivariantly closed (in the usual Cartan model) and such that the following diagram commutes: g

>

C ∞ (M) >

∨

C ∞ (T M ⊕ T ∗ M), where the horizontal map is given by ξ → f ξ , the vertical map is given by f → −J d f , and the diagonal map is given by ξ → ξ M + α ξ . Therefore, morally speaking, the Hamiltonian action of a compact Lie group G not only defines an action on the manifold M, but also defines an extended action on the exact Courant algebroid T M ⊕ T ∗ M.1 The equivariant cohomology theory has been an important topic in the study of group actions on manifolds. However, the usual equivariant cohomology theory encodes only the cohomological information of group actions on manifolds, and does not give us any information on the extended group actions on the exact Courant algebroid T M ⊕ T ∗ M. ¯ So when considering the consequence of the ∂∂-lemma for Hamiltonian actions on generalized complex manifolds, the author [Lin06] was forced to introduce two extensions of the usual equivariant de Rham cohomology theory, called the generalized equivariant cohomology and the generalized equivariant Dolbeault cohomology, to encode the information of the extended actions on T M ⊕ T ∗ M. In this paper, we explain that the generalized equivariant cohomology introduced in [Lin06] is actually the usual equivariant cohomology twisted by the equivariantly closed three form H + α. We note that the equivariant cohomology theory of general extended group actions on exact Courant algebroids has been developed by Hu and Uribe [HuUribe06]. 2 In particular, Hu and Uribe proved that the twisted equivariant cohomology has the usual expected properties of an equivariant cohomology theory such as functoriality, Mayer-Vietoris lemma, Thom isomorphism, localization theorem, etc. In this context, the equivariant ∂∂-lemma obtained in [Lin06] can be best understood in the framework of twisted equivariant cohomology theory. Let us consider the Hamiltonian action of a compact Lie group on an H -twisted generalized complex manifold satisfying the ∂∂-lemma. As an immediate consequence of the equivariant ∂∂-lemma, we obtain the equivariant formality of the twisted equivariant cohomology theory. Now assume the compact Lie group is a torus T , and that the manifold satisfies the generalized Kähler condition. Under these assumptions, we establish the Kirwan injectivity theorem for the twisted equivariant cohomology theory. 1 Indeed, the elements of extended group actions on exact courant algebroid have been developed in [Hu05], and, more generally, in [BCG05]. 2 We would also like to mention that another very useful version of equivariant cohomology theory in generalized complex geometry has already appeared in [BCG05, Sec. 5], where the equivariant Lie algebroid cohomology, as introduced in [BCRR05], was studied in the context of generalized complex geometry.

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It is an interesting question that if we can understand the cohomology of the generalized Kähler quotients from the cohomology of the Hamiltonian generalized Kähler manifold upstairs3 . We believe that the equivariant formality theorem and Kirwan injectivity theorem we established in this paper is useful for answering such a question. As a side result, we show that the generalized Kähler quotient of a generalized Kähler vector space can never have a (cohomologically) non-trivial twisting. This gives a negative answer to a question asked by physicists whether one can construct gauged linear sigma models with non-trivial fluxes. As an application of the twisted equivariant cohomology, we investigate the Hamiltonian action of a torus T on an H -twisted generalized Calabi-Yau manifold (M, ρ), where ρ is an H -twisted generalized Calabi-Yau structure, i.e., a nowhere vanishing section of the canonical line bundle of the generalized complex manifold M such that dρ = H ∧ ρ for the closed three form H . Indeed, the Hamiltonian action on generalized Calabi-Yau manifolds has been studied in the very recent works of Nitta, [NY06,NY07]. Under the assumption that both the twisting three form H and the moment one form α are zeroes, Nitta showed that the generalized Calabi-Yau structure descends to the generalized complex quotient. More interestingly, he observed that the Mukai pairing (ρ, ρ) gives rise to a volume form and so a measure on the generalized Calabi-Yau manifold; and if the type of the generalized Calabi-Yau structure is constant, then the push-forward measure via the generalized moment map is absolutely continuous, and its density function varies as a polynomial in the connected component of the regular values of the generalized moment map. This has thus generalized the usual Duistermaat-Heckman theorem to the setting of generalized Calabi-Yau manifolds. 4 However, the assumptions that both the twisting three form H and the moment one form α equal zero are not natural when one studies group actions on generalized complex manifolds. Indeed, even when the generalized complex manifold is untwisted, the moment one form may come up very naturally [LT05]. In this article, we show that neither one of these two assumptions made in [NY07] are necessary, establishing the Duistermaat-Heckman theorem in the setting of twisted generalized Calabi-Yau geometry in full generality. To achieve this we need to introduce the equivariant differential forms with coefficients in the ring of formal power series, which we discuss in the Appendix A. Along the same line, we explain that the notion of Hamiltonian action on generalized Calabi-Yau manifolds generalizes that of Hamiltonian action on symplectic manifolds. In addition, we give concrete examples of Hamiltonian action on (non-symplectic) generalized Calabi-Yau manifolds and calculate the density function of the push-forward measure via the generalized moment map explicitly. Finally we would like to mention that there are four other groups who have worked independently on the reduction and quotient construction in generalized geometry: Bursztyn, Cavalcanti and Gualtieri [BCG05], S. Hu[Hu05], P. Xu and M. Stiénon [SX05], Vaisman, I. [Va05]. In particular, the connection between the work of Bursztyn, Cavalcanti and Gualtieri and the gauged sigma model in physics is explained in [BCG05, Thm. 2.13]. This paper is organized as follows. Section 2 gives a quick review of some foundational aspects of twisted cohomology and twisted equivariant cohomology. Section 3 3 More recently, it has been shown by T. Baird and the author that Kirwan injectivity and surjectivity results hold for generalized complex quotients, c.f. [BL08]. 4 However, it seems that Nitta’s proof [NY07] has a small gap. Please see Remark 6.9 for an explanation.

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reviews elements of generalized complex geometry. Section 4 establishes the equivariant formality and Kirwan injectivity theorems for the twisted equivariant cohomology of Hamiltonian actions on twisted generalized Kähler manifolds. Section 5 proves that the generalized Kähler quotients of generalized Kähler vector spaces can never have a non-trivial twisting. Section 6 establishes the Duistermaat-Heckman theorem in the setting of twisted generalized Calabi-Yau manifolds in full generality. Section 7 presents concrete examples of Hamiltonian action on generalized Calabi-Yau manifolds which are not symplectic, and computes the density function of the push-forward measure explicitly. Appendix A discusses the equivariant differential forms with coefficients in the ring of formal power series. 2. Twisted Equivariant Cohomology 2.1. Twisted cohomology, non-equivariant case. In this subsection, we briefly review the elements of twisted cohomology theory. We refer to [AS05] for an excellent account of many fundamental aspects of the theory. An H -twisted manifold M is a smooth manifold in the usual sense together with a closed three form H . The closed three form gives rise to a twisted differential operator d H defined by d H γ = dγ − H ∧ γ , γ ∈ (M). 2 = 0. We define the H -twisted cohomology It is easy to check directly that d H

H (M, H ) = kerd H /imd H . Given a two form λ on M, then we can form the twisted differential d H +dλ and the group H (M, H + dλ). There is an isomorphism H (M, H ) → H (M, H + dλ), [α] → [exp(λ)α]. So any two closed differential three forms H and H , if representing the same cohomology class, determine isomorphic groups H (M, H ) and H (H, H ). However, as there are many two forms λ such that H = H + dλ, these two groups are not uniquely isomorphic. Note also there is a homomorphism of groups H (M, H ) ⊗ H (M, H ) → H (M, H + H ), [α] ⊗ [β] → [α ∧ β]. In particular, when H = 0, this homomorphism gives H (M, H ) a natural H (M)-module structure: if [α] ∈ H (M) and [β] ∈ H (M, H ), then [α∧β] ∈ H (M, H ).

2.2. Equivariant de Rham theory. Let G be a compact connected Lie group and let G (M) = (Sg∗ ⊗ (M))G be the Cartan complex of the G-manifold M. For brevity we will write  = (M) and G = G (M) when there is no confusion. By definition an element of G is an equivariant polynomial from g to  and is called an equivariant differential form on M. The bi-grading of the Cartan complex ij is defined by G = (S i g∗ ⊗  j−i )G . It is equipped with a vertical differential 1 ⊗ d, which is usually abbreviated to d, and the horizontal differential d , which is defined

Equivariant Cohomology Theory of Twisted Generalized Complex Manifolds

473

by d α(ξ ) = −ιξ M α(ξ ). Here ιξ M denotes inner product with the vector field ξ M on M induced by ξ ∈ g. As a total complex, G has the grading
ij kG = G (2.1) i+ j=k

and the total differential dG = d +d . The total cohomology kerdG /imdG is the de Rham equivariant cohomology HG (M). Now assume that the action of G on M is free. Choose a basis ξ1 , ξ2 , . . . , ξk of g. Then there exists θ 1 , θ 2 , . . . , θ k ∈ 1 (M), called the connection elements, such that ιξi,M θ j = δ ij , where δ ij is the Kronecker delta, and ξi,M is the vector field on M induced  by ξi . Any differential form γ ∈ (M) can be uniquely written as γ = γhor + I θ I γ I such that ιξi,M γhor = 0 and ιξi,M γ I = 0, 1 ≤ i ≤ k, where I is a multi-index, and γhor is said to be the horizontal component of the form γ . i be the coefficient constants of the Lie algebra g, let Let ck,l 1 i k l θ θ ci = dθ i + ck,l 2

(2.2)

be the curvature elements, and let (M)bas be the space of basic forms in (M), i.e., G-invariant forms which are annihilated by the vectors fields generated by the group action. The usual de Rham differential induces a differential d on bas and the cohomology of the differential complex (bas , d) is denoted by H (bas ). Then the Cartan map C : G (M) → (M)bas , which is defined by x I ⊗ γ → c I ∧ γhor ,

(2.3)

induces an isomorphism from HG (M) to H (bas ). More specifically, there is a chain ∂ homotopy operator Q : G → G such that dG Q − QdG = id − C. Define ∂r := r , ∂x 1 ≤ r ≤ k, where x 1 , x 2 , . . . , x k are indeterminates in the polynomial ring Sg∗ , and define the operators K and R on G (M) by K := −θ r ∂r , R := dθ r ∂r . Then the chain homotopy operator Q is given by Q = K F(I + R F + (R F)2 + · · · ),

(2.4)

where F is an operator on G (M) which preserves its bi-grading. We refer to [GS99, Chap. 5] for more details. Let π : M → M/G be the quotient map. It is straightforward to check that the pull-back map π ∗ : H (M/G) → H (bas ) is an isomorphism. In any event, we have the following result. Theorem 2.1. ([GS99]). Assume the action of G on M is free. Then the Cartan map G → (M)bas , x I ⊗ γ → c I ∧ γhor , induces a natural isomorphism from HG (M) to the ordinary cohomology H (M/G).

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2.3. Twisted equivariant cohomology. Throughout this subsection, we assume that η is an equivariantly closed three form in the Cartan model G (M). It gives rise to an 2 η-twisted differential dG,η := dG − η∧ on G (M) such that dG,η = 0. We define the (uncompleted) η-twisted equivariant cohomology HG (M, η) = kerdG,η /imdG,η . Let a = (x1 , x2 , . . . , xn ) be the ideal in Sg∗ = C[x1 , x2 , . . . , xn ] generated by x1 , x2 , . . . , xn . The sequence of modules     Sg∗ ⊗ (M) ⊇ a Sg∗ ⊗ (M) ⊇ · · · ⊇ a k Sg∗ ⊗ (M) ⊇ · · · defines an a-adic topology on Sg∗ ⊗ (M), and so defines an a-adic topology on its  G to be the completion of G with subspaces G = (Sg∗ ⊗ (M))G . We define  respect to this a-adic topology. It is easy to see that the η-twisted equivariant differential  G which squares to zero. For brevity, this on G extends naturally to a differential on   differential on G is also denoted by dG,η . The η-twisted equivariant cohomology is defined to be d

d

G,η G,η  G −−  G )
im(  G −−  G ). Hˆ G (M, η) = ker( → →

In the special case η = 0, the η-twisted equivariant cohomology is just Hˆ G (M), the equivariant cohomology with coefficients in the formal power series ring as we discussed in Appendix A. Similar to the non-equivariant case, we have the following results. Lemma 2.2. If λ is an equivariant differential two form, then there is an isomorphism Hˆ G (M, η) → Hˆ G (M, η + dG λ), [α] → [exp(λ)α]. Thus two equivariantly closed three forms η and η representing the same equivariant cohomology class determine the isomorphic groups HG (M, η) and HG (M, η ). Lemma 2.3. Let η and η be two equivariantly closed three forms. If γ is dG,η -closed, and γ is dG,η -closed, then γ ∧ γ is dG,η+η -closed. Consequently there is a homomorphism Hˆ G (M, η) ⊗ HG (M, η ) → HG (M, η + η ), [γ ] ⊗ [γ ] → [γ ∧ γ ]. In particular, when η = 0, the homomorphism Hˆ G (M) ⊗ Hˆ G (M, η) → Hˆ G (M, η), [γ ] ⊗ [γ ] → [γ ∧ γ ] defines an Hˆ G (M)-module structure on Hˆ G (M, η). ∗ )G -module, c.f. Appendix A, Hˆ G (M, η) carries a Note that since Hˆ G (M) is a ( Sg G ∗ ∗ )G , Hˆ G (M, η) is  ) -module structure. Moreover, since (Sg∗ )G is a subring of ( Sg ( Sg ∗ G also a (Sg ) -module. Now we describe the relationship between the twisted equivariant cohomologies dG,η ˆ HG (M, η) and HG (M, η). Observe that ker(G −−→ G ) inherits an a-adic topology from G , and that HG (M, η) obtains an a-adic topology from the quotient map dG,η

ker(G −−→ G ) → HG (M, η).

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It is straightforward to check by definition that the completion of HG (M, η) with respect ∗ )G -module structure. The following result is a simple to this topology also carries a ( Sg consequence of [AM99, Corollary 10.3]. Proposition 2.4. There is a natural ( Sg∗ )G -module isomorphism from the twisted equiˆ variant cohomology HG (M, η) to the completion of HG (M, η) with respect to the a-adic topology we described above. We refer to [HuUribe06] for a detailed treatment of the foundational aspects of the twisted equivariant cohomology. For our purpose, we will need the following result later on.5 Theorem 2.5. (Localization Theorem)([HuUribe06]). Suppose a compact connected torus T acts on a compact manifold M. Then the kernel and cokernel of the canonical map i ∗ : Hˆ T∗ (M, η) → Hˆ T∗ (M T , i ∗ η),  have support contained in H h, where η is an equivariantly closed three form, M T is the fixed points set of the torus T action, H runs over the stabilizers = T , and h is the Lie algebra of H . Remark 2.6. Let t∗ be the dual space of the Lie algebra t of T . Then St∗ = C[x1 , ∗ x2 , . . . , xn ] is a polynomial  ring in n variables. The support of an St -module A is defined to be supp(A) = { f ∈S t∗ | f ·A=0} V f , where V f = {x ∈ t | f (x) = 0}. The support of Hˆ ∗ (M, η) discussed in Theorem 2.5 is the support of it as an St∗ -module. T

3. Generalized Complex Geometry Let V be an n dimensional vector space. There is a natural bi-linear pairing of type (n, n) on V ⊕ V ∗ which is defined by X + α, Y + β =

1 (β(X ) + α(Y )). 2

A generalized complex structure on a vector space V is an orthogonal linear map √ J : V ⊕ V ∗ → V ⊕ V ∗ such that J 2 = −1. Let L ⊂ VC ⊕ VC∗ be the −1 eigenspace of the generalized complex structure J . Then L is maximal isotropic and L ∩ L = {0}. Conversely, given a maximal isotropic L ⊂ VC ⊕ VC∗ so that L ∩ L = {0}, there exists √ an unique generalized complex structure J whose −1 eigenspace is exactly L. Let π : VC ⊕ VC∗ → VC be the natural projection. The type of J is the codimension √ of π(L) in VC , where L is the −1 eigenspace of J . Let σ be the linear map on ∧V ∗ which acts on decomposables by σ (v1 ∧ v2 ∧ · · · ∧ vq ) = vq ∧ vq−1 ∧ · · · ∧ v1 ,

(3.1)

5 Strictly speaking, our definition of twisted equivariant cohomology differs slightly from the one used by Hu and Uribe. Nevertheless, the major results established in [HuUribe06] holds for our version of the twisted equivariant cohomology. Indeed, the argument used in [HuUribe06] applies verbatim to our situation.

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then we have the following bilinear pairing, called the Mukai pairing [Gua07], defined on ∧V ∗ : (ξ1 , ξ2 ) = (σ (ξ1 ) ∧ ξ2 )top , where (·, ·)top indicates taking the top degree component of the form. Let B ∈ ∧2 V ∗ be a two-form. For any ξ ∈ ∧V ∗ , throughout this paper we will denote by e B ξ the wedge product e B ∧ ξ . The following result was shown in [Gua07, Prop. 1.12]. Proposition 3.1. (e B ξ1 , e B ξ2 ) = (ξ1 , ξ2 ), for any ξ1 , ξ2 ∈ ∧V ∗ , where (·, ·) denotes the Mukai pairing. The Clifford algebra of VC ⊕ VC∗ acts on the space of forms ∧VC∗ via (X + ξ ) · ϕ = ι X ϕ + ξ ∧ ϕ. Since J is skew symmetric with respect to the natural pairing on V ⊕ V ∗ , J ∈ so(V ⊕ V ∗ ) ∼ = ∧2 (V ⊕ V ∗ ) ⊂ C L(V ⊕ V ∗ ). Therefore there is a Clifford action of J on the space of forms ∧V ∗ ( and so on its complexification ∧VC∗ ) which √ k U , where U k is the −k −1 eigenspace determines an alternative grading : ∧VC∗ = of the Clifford action of J . Note also that a classical result of Chevalley [Che97] asserts that {ϕ ∈ ∧VC∗ | (X + ξ ) · ϕ = 0, X + ξ ∈ L} is a one dimensional vector space in the space of exterior forms ∧VC∗ . Any form in this one dimensional subspace of ∧VC∗ is said to be a pure spinor form associated to the generalized complex structure J . On the other hand, given a form ϕ = 0 ∈ ∧VC∗ , L ϕ := {X + ξ ∈ VC ⊕ VC∗ | (X + ξ ) · ϕ = 0} is always an isotropic space. If in addition, L ϕ is maximal isotropic, and if (ϕ, ϕ) = 0, then √ L ϕ ∩ L ϕ = {0} and there exists an unique generalized complex structure J whose −1 eigenspace is exactly L ϕ . Let M be a manifold of dimension n. There is a natural pairing of type (n, n) which is defined on T M ⊕ T ∗ M by X + α, Y + β =

1 (β(Y ) + α(X )), 2

and which extends naturally to TC M ⊕ TC∗ M. For a closed three form H , the H -twisted Courant bracket of TC M ⊕ TC∗ M is defined by the identity 1 [X + ξ, Y + η] = [X, Y ] + L X η − L Y ξ − d (η(X ) − ξ(Y )) + ιY ι X H. 2 A generalized almost complex structure on a manifold M is an orthogonal bundle map J : T M ⊕ T ∗ M → T M ⊕ T ∗ M such that J 2 = −1.√Moreover, J is an H -twisted generalized complex structure if the sections of the −1 eigenbundle of J is closed under the H -twisted Courant bracket. The type of J at m ∈ M is the type of the restricted generalized complex structure on Tm M.

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The Clifford algebra of C ∞ (T M ⊕ T ∗ M) with the natural pairing acts on differential forms by (X + ξ ) · ϕ = ι X ϕ + ξ ∧ ϕ. √ If L ⊂ TC M ⊕ TC∗ M is the −1 eigenbundle of a generalized complex structure J , then the differential forms annihilated by the Clifford action of the sections of L span a line bundle sitting inside ∧TC∗ M, which is called the canonical line bundle of the generalized complex manifold (M, J ). Since J can be identified with a smooth section of the Clifford bundle C L(T M ⊕ k T ∗ M), √ there is a Clifford action of J on the space of differential forms. Let U be the −k −1 eigenbundle of J . [Gua07] shows that there is a grading of the differential forms: ∗ (M) = (U −n ) ⊕ · · · ⊕ (U 0 ) ⊕ · · · ⊕ (U n ).

(3.2)

It has been shown (See e.g. [Gua07] and [KL04]) that the integrability of an H -twisted generalized complex structure J implies that d H = d − H ∧ : (U k ) → (U k−1 ) ⊕ (U k+1 ), which gives rise to operators ∂ and ∂¯ via the projections ∂ : (U k ) → (U k−1 ), ∂¯ : (U k ) → (U k+1 ). ¯ In this context, a twisted generalized complex manifold is said to satisfy the ∂∂-lemma if and only if ¯ ker∂ ∩ im∂¯ = im∂ ∩ ker∂¯ = im∂∂. Let ∂ (M) = (M) ∩ ker∂. Since d H anti-commutes with ∂, (∂ , d H ) is a differential complex with the differential d H . The following result is a simple consequence of the ¯ ∂∂-lemma. Proposition 3.2. (c.f., [Ca06, Thm. 4.1]). If the generalized complex manifold (M, J ) ¯ satisfies the ∂∂-lemma, then the inclusion of the complex of ∂-closed differential forms ∂ (M) into the complex of differential forms (M) induces an isomorphism in cohomology: i : (∂ (M), d H ) → ((M), d H ), i ∗ : H (∂ (M)) ∼ = H (M, H ). An H -twisted generalized Calabi-Yau structure J is a generalized complex structure such that its canonical line bundle admits a nowhere vanishing d H -closed section ρ. On the other hand, given a nowhere vanishing d H -closed differential form ρ such that (ρ, ρ) = 0, there exists a generalized Calabi-Yau structure J such that ρ is the pure spinor associated to J . By the abuse of the notation, we will also call such a differential form ρ a generalized Calabi-Yau structure. Let B be a closed two-form on a manifold M, and consider the orthogonal bundle map defined by 

1 0 : T M ⊕ T ∗ M → T M ⊕ T ∗ M, eB = B1

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where B is regarded as a skew-symmetric map from T M to T ∗ M. This map preserves the canonical pairing < ·, · > and the H -twisted Courant bracket. As a consequence, if J is an H -twisted generalized complex structure on M, then J B := e B J e−B is another H -twisted generalized complex structure on M, called the B-transform of J . The effect of B-transforms on the canonical line bundle of a generalized complex structure has been studied in [Ca06]. Lemma 3.3. ([Ca06]). Let B be a closed two form and let J B be the B-transform of the generalized Calabi-Yau structure J with a nowhere vanishing closed pure spinor ρ. Then J B is a generalized Calabi-Yau structure with a nowhere vanishing closed pure spinor e−B ρ. Example 3.4. ([H02,Gua07]). • Let (V, ω) be a 2n dimensional symplectic vector space. Then the map J : V ⊕V ∗ → V ⊕ V ∗ defined by 

0 −ω−1 (3.3) J = ω 0 is a generalized complex structure on V . • Now let (M, ω) be a 2n dimensional symplectic manifold. Then (3.3) √ defines a gene√ ralized complex structure Jω with the −1 eigenbundle L = {X − −1ι X ω | X ∈ TC M}. It is easy to check that the canonical line bundle admits a nowhere vanishing closed section eiω . And so Jω is a generalized Calabi-Yau structure. On the other hand, the following lemma shows that any nowhere vanishing closed section of the canonical line bundle of Jω must be of the form aeiω for some non-zero constant a. Lemma 3.5. Let (M, ω) be a 2n dimensional connected symplectic manifold and Jω the generalized complex structure defined as in (3.3). Suppose ρ is a nowhere vanishing closed pure spinor associated to the generalized Calabi-Yau structure Jω . Then we have ρ = aeiω for some non-zero constant a. Proof. By assumption we have ρ = f eiω for some nowhere vanishing function f ∈ C ∞ (M). Since dρ = 0, we get d f ∧ eiω = 0. In particular, we have (iω)n−1 ∧ d f = 0. It is well known that the Lefschetz map : 1 (M) → 2n−1 (M), η → ωn−1 ∧ η L n−1 ω is an isomorphism, cf, [Yan96, Cor. 2.7]. It follows that d f = 0 and so f = a for some non-zero constant a. This finishes the proof.   It is important for our purpose to notice that given a 2n dimensional H -twisted generalized Calabi-Yau manifold M, the generalized Calabi-Yau structure ρ on M gives rise to a volume form (−1)n (ρ, ρ), (2i)n

(3.4)

where (·, ·) denotes the Mukai pairing. Note that if M is a symplectic manifold with a symplectic form ω and the generalized Calabi-Yau structure is given by eiω , then up to

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a normalization factor the volume form we defined above coincides with the symplectic 1 n volume n! ω . A manifold M is said to be an H -twisted generalized Kähler manifold if it has two commuting H -twisted generalized complex structures J1 , J2 such that −J1 J2 ξ, ξ  > 0 for any ξ = 0 ∈ C ∞ (TC M ⊕ TC∗ M), where ·, · is the canonical pairing on TC M ⊕ TC∗ M. The following remarkable result is due to Gualtieri. Theorem 3.6. ([Gua04]) Assume that (M, J1 , J2 ) is a compact H -twisted generalized ¯ Kähler manifold. Then it satisfies the ∂∂-lemma with respect to both J1 and J2 . 4. The Twisted Equivariant Cohomology of Hamiltonian Actions on Twisted Generalized Complex Manifolds First we recall the definition of Hamiltonian actions on H -twisted generalized complex manifolds given in [LT05]. Definition 4.1. 6 [LT05]). Let a compact Lie group G with Lie algebra g act on a manifold M, preserving an H -twisted generalized complex structure J , where H ∈ 3 (M)G is √ ∗ closed. Let L ⊂ TC M ⊕ TC M denote the −1 eigenbundle of J . The action of G is said to be Hamiltonian if there exists a smooth equivariant function µ : M → g∗ , called the generalized moment map, and a one form α ∈ 1 (M, g∗ ), called the moment one form, so that √ √ a) ξ M − −1 (dµξ + −1 α ξ ) lies in C ∞ (L) for all ξ ∈ g, where ξ M denotes the induced vector field. b) H + α is an equivariantly closed three form in the usual Cartan Model. Therefore given a Hamiltonian action on an H -twisted generalized complex manifold, there is an equivariantly closed three form η := H + α which is part of the data defining the Hamiltonian action. Next we recall the definition of the generalized equivariant cohomology as introduced in [Lin06] and compare it with the twisted equivariant cohomology HG (M, η). First observe that since H is an invariant three form, the operators d H = d − H ∧, ∂, and ∂ are G-equivariant and thus have natural extensions to the space of the usual equivariant differential forms, i.e., the Cartan model G . For brevity, we will also denote these extensions by d H , ∂ and ∂. To encode the information given by the moment one form, [Lin06] introduced the equivariant differential DG = d H +A in the Cartan model, where A is defined by √ √ (Aγ )(ξ ) = −ιξ M γ (ξ ) + −1(dµξ + −1α ξ ) ∧ γ (ξ ), ξ ∈ g, γ ∈ G . (4.1) 2 = 0. In particular, we have the following definition for the It is easy to check that DG generalized equivariant cohomology.

Definition 4.2. Let G = (Sg∗ ⊗ (M))G be Z 2 graded. Then DG = d H + A is a differential of degree 1. And the Z 2 graded generalized equivariant cohomology is defined to be 6 Indeed, Condition (b) was not imposed in [LT05, Def. A.2.]. However, in order to make the quotient construction work, Tolman and the author made it clear in [LT05, Prop. A.7, A.10] that H + α must be equivariantly closed in the usual Cartan model.

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DG odd/even ker even/odd − − →  G G .

H even/odd (G , DG ) = DG even/odd im odd/even − − →  G G Note that the differential operator DG on G extends naturally to a differential  G , which we will also denote by DG . Define operator on  

G  G −D G ker  −→   G , DG ) = 

H ( G  G −D G im  −→  to be the (completed) generalized equivariant cohomology. By the discussion in  G is nothing but the usual Sect. 2.3, it is clear that the differential operator DG on  √ equivariant differential dG twisted by the equivariantly closed three form η − −1 dG µ. (Note that dµ = dG µ is an exact equivariant three form.) The following result is a simple consequence of Lemma 2.2. Proposition 4.3. Consider the Hamiltonian action of a compact connected Lie group G on an H -twisted generalized complex manifold with generalized moment map µ : M → g∗ . Then the map √  G , DG ), [γ ] → [exp(− −1 µ)γ ] Hˆ G (M, η) → H ( defines an isomorphism between the η-twisted equivariant cohomology Hˆ G (M, η) and  G , DG ). the generalized equivariant cohomology H ( Recall that the following result is proved in [Lin06], which is a direct generalization of one of the major results in [LS03]. Theorem 4.4. If we assume that the H -twisted generalized complex manifold M satisfies ¯ the ∂∂-lemma, and assume the action of a compact connected Lie group G on M is Hamiltonian, then as (Sg∗ )G -modules, H (G , DG ) ∼ = (Sg∗ )G ⊗ H (M, H ). Using an argument similar to the one we give to prove Proposition 2.4, it is easy to ∗ )G ⊗ ∗ )G -module isomorphism from H (  G , DG ) to ( Sg see that there is a natural ( Sg H (M, H ) by Theorem 4.4. Combining this fact with Proposition 4.3, we have the following result. Theorem 4.5. (Equivariant Formality). Assume the H -twisted generalized complex ¯ manifold M satisfies the ∂∂-lemma, and assume the action of a compact connected Lie group G on M is Hamiltonian. Let α be the moment one form of the Hamiltonian action and let η = H + α. Then as ( Sg∗ )G -modules we have that Hˆ G (M, η) ∼ Sg∗ )G ⊗ H (M, H ), = ( where the isomorphism depends only on the generalized moment map.

(4.2)

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It is interesting to give a more concrete description of the isomorphism (4.2). Note that via the isomorphism (4.2) one gets a canonical map s : H (M, H ) → Hˆ G (M, η) such that the inverse of the canonical isomorphism (4.2) is given by 1 ⊗ s. Let us show how in principle one can compute the map s. Let ϕ be a d H -closed differential form representing a cohomology class in H (M, H ). Since the generalized complex manifold M satisfies the ∂∂-lemma, by Proposition 3.2 we may assume that ϕ is both ∂ and ∂ closed. We claim that we may assume that ϕ is G-invariant as well. Resorting to the usual averaging trick, to establish the claim one needs only to check that the induced action of G on H (M, H ) is trivial. Since G is connected, it suffices to show that all operators L ξ M act trivially on H (M, H ), where ξ M denotes the vector field on M induced by ξ ∈ g. However, since ιξ M H = dα ξ , we have L ξ M = dιξ M + ιξ M d = (d − H ∧)(ιξ M + α ξ ∧) + (ιξ M + α ξ ∧)(d − H ∧) = d H (ιξ M + α ξ ∧) + (ιξ M + α ξ ∧)d H . That is to say that L ξ M is chain homotopic to zero in H (M, H ) and our claim is esta blished. In view of the grading (3.2), write ϕ = i ϕ i , where ϕ i ∈ (U i ). Then each homogenous component ϕ i is G-invariant and is both ∂ and ∂ closed. Observe the grading (3.2) on the space of differential forms induces an alternative filtration on the Cartan model
F k G = (Sg∗ ⊗ (U j ))G . (4.3) j≤k

Now fix an i for the moment. It follows from [Lin06, Lemma 4.9] that Aϕ i is ∂-exact. Since ∂ anti-commutes with A, Aϕ i ∈ F i−1 G is both ∂-exact and ∂-closed. It then follows from the ∂∂-lemma that there exists γ i−1 ∈ F i−1 G such that Aϕ i = ∂∂γ i−1 . ( We refer to [Lin06, Lemma 4.9] for a detailed proof of this fact.) Also A∂γ i−1 = −∂Aγ i−1 is both ∂ exact and ∂-closed. Another application of the ∂∂-lemma implies that there exists γ i−3 ∈ F i−3 G such that A∂γ i−1 = ∂∂γ i−3 . Since the filtration (4.3) is finite, applying this argument recursively we get a finite chain γ k ∈ F i+1−2k G such that ϕgi := ϕ i + ∂γ k k

is ∂ G = (∂ + A)-closed. Note that by construction ϕgi is also ∂-closed. So ϕgi is DG = √  G , the completion (∂ G +∂)-closed. Thus by Proposition 4.3 e −1µ (ϕgi ) is dG,η -closed in  √  i −1µ  G , representing of G . Set ϕg := i ϕg . Then e ϕg is a dG,η -closed element in  a cohomology class in Hˆ G (M, η). A careful reading of the proof of Theorem 4.4 given in [Lin06] shows that s[ϕ] = [ϕg ]. Recall that an equivariant differential form η ∈ G (M) can be regarded as a polynomial mapping g → (M).

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Thus there is a natural map p : G (M) → (M), η → η(0), where η(0) is the value of the polynomial map η : g → (M) evaluated at the zero element in g. It is easy to see that p extends naturally to the completion of G and defines  G (M), dG,η ) and ((M), d H ). So a chain map between the differential complexes ( we have a natural map

p : Hˆ G (M, η) → H (M, H ); furthermore, by the above description of the map s, one checks easily that p s = id. In particular, we have proved the following result. Corollary 4.6. The canonical map s is a section of p . Thus every cohomology class in H (M, H ) has a canonical equivariant extension in Hˆ G (M, η). Recall from [LT05] that the action of a compact connected Lie group G on an H twisted generalized Kähler manifold (M, J1 , J2 ) is said to be Hamiltonian if the action preserves the generalized Kähler pair (J1 , J2 ) and if there exist a generalized moment map and a moment one-form for the twisted generalized complex structure J1 . By Theorem 3.6 an H -twisted generalized Kähler manifold always satisfies the ∂∂-lemma with respect to both J1 and J2 . The following result is a direct consequence of Theorem 4.5. Theorem 4.7. Suppose that a compact connected Lie group G acts on a compact H -twisted generalized Kähler manifold (M, J1 , J2 ) in a Hamiltonian fashion. Let α be the moment one form and η = H + α the equivariantly closed three form. Then we have Hˆ G (M, η) ∼ Sg∗ )G ⊗ H (M, H ), = ( where the isomorphism is an isomorphism of ( Sg∗ )G -modules and depends only on the generalized moment map. Next we extend the Kirwan injectivity theorem [Kir86] to the case of Hamiltonian actions on H -twisted generalized Kähler manifolds. We need the following intermediate result. Lemma 4.8. Suppose that a compact connected torus T acts on a generalized complex manifold (M, J ) in a Hamiltonian fashion. And suppose that W is a connected component of the fixed points set M T . Then W is a generalized complex manifold itself and the induced trivial T -action on W is Hamiltonian. √ Proof. Let L be the −1 eigenbundle of J : TC M ⊕ TC∗ M → TC M ⊕ TC∗ M. It is proved in [Lin06, Lemma 5.4] that W carries a generalized complex structure whose √ −1 eigenbudle is   L W,x = {X + (ξ |TC W ) | X + ξ ∈ L ∩ TC,x W ⊕ TC∗,x M }. Let µ : M → t∗ and α ∈ 1 (M, t∗ ) be the generalized moment map and the moment one form of the T action on M respectively. Let µW and αW be the restrictions of µ and √ √ ξ ξ α to W . Then it is easy to check that −1(dµW + −1αW ) ∈ C ∞ (L W ) for any ξ ∈ t. This proves that the trivial T action on W is Hamiltonian.  

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Theorem 4.9. (Kirwan injectivity). Suppose that a compact connected Lie group G acts on a compact H -twisted generalized Kähler manifold (M, J1 , J2 ) in a Hamiltonian fashion. Then the canonical map i ∗ : Hˆ T∗ (M, η) → Hˆ T∗ (M T , i ∗ η), induced by the inclusion i : M T → M is injective, where M T is the fixed points set of the torus T action, and η is the canonical equivariantly closed three form associated to the Hamiltonian T -action. Proof. It is proved in [Lin06, Prop. 5.5] any connected component of the fixed points set is a generalized Kähler manifold itself. Combining this fact with Lemma 4.8 and ∗ -module. Since St∗ Theorem 4.7, one sees immediately that Hˆ T (M T , i ∗ η) is a free St T ∗ ∗ ∗ ˆ  is a subring of St , HT (M , i η) must be a free St -module as well. So if [α] is a torsion element in HT (M, η), it gets mapped to zero in Hˆ T∗ (M T , i ∗ η). On the other hand, Theorem 2.5 implies that the kernel of the map i ∗ is a torsion module. So the kernel of i ∗ is the module of torsion elements in Hˆ T (M, η). However, by Theorem 4.7 ∗ -module. So it must be a free St∗ -module and it does not have Hˆ T (M, η) is a free St any non-trivial torsion elements. We conclude that the map i ∗ is injective.   5. Some Reflections on the Cohomology of Generalized Complex Quotients Let a compact Lie group G act on a twisted generalized complex manifold (M, J ) with generalized moment map µ. Let Oa be the co-adjoint orbit through a ∈ g∗ . If G acts freely on µ−1 (Oa ), then Oa consists of regular values and Ma = µ−1 (Oa )/G is a manifold, which is called the generalized complex quotient. The following two results were proved in [LT05]. Lemma 5.1. Let a compact Lie group G act freely on a manifold M. Let H be an invariant closed three form and let α be an equivariant mapping from g to 1 (M). Fix a connection θ ∈ (M, g∗ ). Then if H +α ∈ 3G (M) is equivariantly closed, there exists a natural form  ∈ 2 (M)G so that ιξ M  = α ξ . Thus H +α +dG  ∈ 3 (M)G ⊂ 3G (M)

] is the

∈ 3 (M/G) so that [ H is closed and basic and so descends to a closed form H image of [H + α] under the Kirwan map. Proposition 5.2. Assume there is a Hamiltonian action of a compact Lie group G on an H -twisted generalized complex manifold (M, J ) with generalized moment map µ : M → g∗ and moment one-form α ∈ 1 (M, g∗ ). Let Oa be a co-adjoint orbit through a ∈ g∗ so that G acts freely on µ−1 (Oa ). Given a connection on µ−1 (Oa ), the gene -twisted generalized complex structure J , ralized complex quotient Ma inherits an H

where H is defined as in Lemma 5.1. Up to B-transform, J is independent of the choice of connection. Finally, for all m ∈ µ−1 (Oa ), type(J )[m] = type(J )m . It is an interesting question if we can read off the cohomology of the generalized complex quotient Ma from the equivariant cohomology of the Hamiltonian generalized complex manifold M. The Kirwan map is the key to understanding this question.

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Assume that G acts freely on µ−1 (Oa ). Let i : µ−1 (Oa ) → M be the inclusion map and π : µ−1 (Oa ) → Ma the quotient map. Then there is a Kirwan map k : HG (M) → H (Ma ) such that the following diagram commutes: HG (M)

i∗ >

HG (µ−1 (Oa )) k >

∼ =

∨

H (Ma ). Here the vertical map is given by the inverse of the map π ∗ : H (Ma ) → HG (µ−1 (Oa )), [η] → [π ∗ η], which is an isomorphism due to Theorem 2.1. If the generalized complex structure J on M is induced by a symplectic structure, then it follows from the famous Kirwan surjectivity theorem [Kir86] that the Kirwan map is surjective. Let us give a very short explanation why the Kirwan surjectivity theorem holds in the symplectic case. For simplicity, let us only consider S 1 action here. In this case, the fixed points set of the action are exactly the critical points set of the moment map and the moment map is a Morse-Bott function. Besides, the norm square of the moment map induces a Morse stratification on the manifold M from which one derives the Kirwan surjectivity theorem. However, for Hamiltonian S 1 action on general complex manifolds, in general the generalized moment map is not a Morse-Bott function. Indeed, let µ : M → R be moment map and α the moment one form. Then the √ the generalized √ condition −X + −1(dµ + −1α) ∈ C ∞ (L) is equivalent to that J dµ = −X − α,

√ where X is the vector field generated by the S 1 action, and L is the −1 eigenbundle of the generalized complex structure. Let Crit(µ) be the set of critical points of the moment 1 map µ and M S the fixed points set of the S 1 action. It is easy to see that 1

Crit(µ) ⊂ M S . 1

However, in general Crit(µ) = M S . For instance, in [Hu05, Sect. 5] it was shown that there is a generalized Kähler structure (J1 , J2 ) on C2 \{0} and a Hamiltonian S 1 action on (C2 \{0}, J1 , J2 ) such that for any regular value a of the generalized moment map µ, the level set µ−1 (a) contains a fixed points submanifold. It is unclear to the author at the moment whether the Kirwan surjectivity still holds or not for Hamiltonian actions on twisted generalized complex manifolds in general. Nevertheless, the image of the Kirwan map forms a subgroup of the cohomology group of the generalized complex quotient which may contain many interesting cohomology classes. For instance, it is an interesting question when the generalized complex quotient has a non-trivial twisting. Observe that by Lemma 5.1 and Proposition 5.2 the

on Ma is exactly the image of the equicohomology class of the twisting three form H variant cohomology class [H + α] under the Kirwan map k : HG (M) → H (Ma ). This innocuous observation leads to the following result. Proposition 5.3. Assume that (M, J ) is an H -twisted generalized complex manifold such that H odd (M) = 0. And assume there is a Hamiltonian action of a compact connected Lie group G on M with generalized moment map µ and moment one form α. Let Oa be a co-adjoint orbit through a ∈ g∗ so that G acts freely on µ−1 (Oa ).

on Ma = µ−1 (Oa )/G, as defined in Lemma 5.1, is Then the twisted three form H cohomologically trivial.

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Proof. It is shown in [GS99, Thm. 6.5.3] that if H odd (M) = 0 then the spectral sequence of the Cartan complex of the G-manifold M degenerates at the E 1 stage. In particular, p,q we have that HGodd (M) ∼ = (Sg∗ )G ⊗ H odd (M) = 0. This shows = p+q=odd E 1 clearly that HG (M) contains no non-trivial odd dimensional equivariant cohomology

] = k([H + α]) = 0.  classes. We conclude that [H + α] = 0 and so [ H  Remark 5.4. Physicists are interested in producing explicit examples of gauged sigma models. When there is no flux, i.e., the target manifold does not have a cohomologically non-trivial twisting, a very useful tool producing such examples is to use the so called gauged linear model which corresponds to the Kähler quotient of a Kähler vector space. Naturally, physicists had hoped that one can produce explicit examples of generalized Kähler quotients of generalized Kähler vector spaces such that the quotient spaces have a cohomologically non-trivial twisting. The above result asserts that this is not possible since H odd (V ) = 0 for any vector space V . We finish this section by showing that in the framework of the twisted equivariant cohomology there is also a well-defined Kirwan map. This strongly suggests that the results we obtained in Sect. 4 do help us understand the cohomology of the generalized complex quotient. To describe the Kirwan map in this setting, we will need Proposition A.4 in the Appendix A. Proposition 5.5. Assume a compact connected Lie group G acts on an H -twisted generalized complex manifold M with generalized moment map µ : M → g∗ and moment one form α ∈ 1 (M, g∗ ). Assume that G acts freely on the level set µ−1 (0). Then there

) such that the following diagram is a Kirwan map k : Hˆ G (M, H + α) → H (M0 , H commutes: Hˆ G (M, H + α)

i∗ >

Hˆ G (µ−1 (0), i ∗ (H + α)) k >

∼ =∨

), H (M0 , H

where i ∗ : Hˆ G (M, H + α) → Hˆ G (µ−1 (0), i ∗ (H + α)) is induced by the inclusion map i : µ−1 (0) → M, M0 = µ−1 (0)/G is the generalized complex quotient taken at the zero

is the twisting three form on M0 obtained level of the generalized moment map, and H as in Lemma 5.1 and Proposition 5.2. Proof. By Lemma 5.1 and Proposition 5.2, we can find a closed three form η ∈

. Since η and i ∗ (H + α) 3 (µ−1 (0)) such that [η] = [i ∗ (H + α)] and such that η = π ∗ H represents the same cohomology class, we have Hˆ G (µ−1 (0), η) ∼ = Hˆ G (µ−1 (0), i ∗ (H + α)). Corollary 5.5 now follows easily from Proposition A.4.

 

6. The Duistermaat-Heckman Theorem for Generalized Calabi-Yau Manifolds Consider the Hamiltonian action of a compact connected Lie group G on an H -twisted generalized Calabi-Yau manifold M. Our first observation is that if the action is free at a level set of the generalized moment map, then the generalized complex quotient taken at the level set inherits a generalized Calabi-Yau structure. The same result, under the

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assumption that both the twisting three form H and the moment one form vanish, has already been proved by Nitta [NY06]. We will make use of the following fact established in the proof of [NY06, Thm. A]. Lemma 6.1. Suppose that there is a Hamiltonian action of a compact connected Lie group G on an H -twisted generalized Calabi-Yau manifold M with generalized moment map µ : M → g∗ and zero moment one form. Then the restriction of the generalized Calabi-Yau structure ρ to µ−1 (0) is nowhere vanishing. Remark 6.2. Although [NY06] assume the generalized Calabi-Yau structure is untwisted, the proof of the above fact given by Nitta uses only the linear algebra of generalized complex structures and so applies to the twisted case as well. Proposition 6.3. Assume that there is a Hamiltonian action of a compact connected Lie group G on an H -twisted generalized Calabi-Yau manifold (M, J ) with generalized moment map µ : M → g∗ and moment one form α ∈ 1 (M, g∗ ), preserving the generalized Calabi-Yau structure ρ. And assume Oa is a co-adjoint orbit through a ∈ g∗ so that G acts freely on µ−1 (Oa ). Then given a connection on µ−1 (Oa ), the generalized

-twisted generalized Calabi-Yau structure ρ complex quotient Ma inherits an H

, such

is defined that π ∗ ρ

= ρ |µ−1 (Oa ) , where π : µ−1 (Oa ) → Ma is the quotient map, H as in Lemma 5.1. Up to B-transform, ρ

is independent of the choice of connections. Finally, for all m ∈ µ−1 (Oa ), type(J )[m] = type(J )m , where J is the quotient generalized complex structure on Ma . Proof. It suffices to show that the restriction of ρ to µ−1 (Oa ) descends to a generalized Calabi-Yau structure ρ

on Ma such that π ∗ ρ

= ρ |µ−1 (Oa ) . Using the argument given in the proof of [LT05, Prop. 3.8, A.7], without the loss of generality, we can assume that the coadjoint orbit Oa = 0, and assume that in an open neighborhood of the level set µ−1 (0), the moment one form is zero and the twisting three form H is basic so that it descends

on the quotient M0 := µ−1 (0)/G. By the definition of the to the twisting three form H √ generalized moment map, we have that for any ξ ∈ g, ξ M − −1dµξ ∈ C ∞ (L), where √ L is the −1 eigenbundle of the generalized complex structure J . By the definition of a generalized Calabi-Yau structure, we have √ √ (ξ M − −1dµξ ) · ρ = ιξ M ρ − −1dµξ ∧ ρ = 0. √  It follows that ιξ M ρ |µ−1 (0) = −1dµξ ∧ ρ |µ−1 (0) = 0 for any ξ ∈ g. Therefore ρ | f −1 (0) is a basic form and so descends to a form ρ

such that π ∗ ρ

= ρ |µ−1 (0) . Since ρ is d H -closed, ρ

is d H -closed; moreover, since by Lemma 6.1 ρ|µ−1 (0) is nowhere vanishing, ρ

is nowhere vanishing as well. Thus L ρ (x) = {A ∈ TC,x M0 ⊕ TC∗,x M0 | A · ρ

= 0}

√ is an isotropic subspace of TC,x M0 ⊕ TC∗,x M0 , x ∈ M0 . Finally, let L be the −1 eigenbundle of the quotient generalized complex structure on M0 . Using the description of L in [LT05, Prop. 3.8] it is straightforward to check that ρ

is annihilated by the Clifford action of the sections of L. Hence we have L x ⊂ L ρ (x), x ∈ M0 . However, L x is a maximal isotropic subspace of TC,x M0 ⊕ TC∗,x M0 . So we must have L x = L ρ (x), x ∈ M. In other words, ρ

is the pure spinor associated to the quotient generalized

is a generalized Calabi-Yau complex structure J on M0 . This finishes the proof that ρ structure.  

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Example 6.4. Let G act on a symplectic manifold (M, ω) with moment map  : M → g∗ , that is,  is equivariant and ιξ M ω = −dξ for all ξ ∈ g. Then the action of G also preserves the generalized complex structure Jω , where Jω is defined as in Example 3.4, and it is straightforward to check that the action on the generalized complex manifold (M, Jω ) is Hamiltonian with generalized moment map  and trivial moment one form. By Example 3.4 ω defines a generalized Calabi-Yau structure eiω . Suppose that G acts freely on the level set µ−1 (0). Then the restriction of eiω to the level set µ−1 (0) descends to the quotient generalized Calabi-Yau structure eiω0 on M0 = µ−1 (0)/G, where ω0 is the reduced symplectic form on M0 . On the generalized complex quotient Ma , the quotient generalized Calabi-Yau structure ρ

a depends on the choice of a connection, cf, [LT05, Prop. A.7]. However, a different choice of the connection results in a new quotient generalized Calabi-Yau structure which is given by e−B ρ

a for some closed two form B ∈ 2 (Ma ). It follows immediately from Lemma 3.1 that, on the quotient Ma , ( ρa , ρ

a ) is well-defined, independent of the choice of connections, where (·, ·) denotes the Mukai pairing on the quotient space Ma . From now on we consider the Hamiltonian action of a compact connected torus T on an H -twisted generalized complex manifold M with a proper generalized moment map µ. Assume that a0 ∈ t∗ such that the action of T on the level set µ−1 (a0 ) is free. Since by Lemma 6.5 below the set on which T acts freely is open and since the generalized moment map is proper, there is an open subset U ⊂ t∗ such that for any a ∈ U the action of T on µ−1 (a) is free. We define the Duistermatt-Heckman function f on U ⊂ t∗ by the following formula. k(k+1)

f (a) =

(−1)n+ 2 (2π )k (2i)n−k

 Ma

( ρa , ρ

a ),

(6.1)

where ρ

a is the quotient generalized Calabi-Yau structure on Ma , and Ma is given the orientation induced by ( ρa , ρ

a ). Lemma 6.5. ([GS84]). Assume that a torus T acts on a connected manifold M effectively. Then the set M on which T acts freely is equal to the complement of a locally finite union of closed manifolds of codimension ≥ 2. In particular, M is open, connected, dense, and M \ M has measure zero. Theorem 6.6. Suppose that a k dimensional torus T acts on a 2n dimensional connected generalized Calabi-Yau manifold M freely such that the action is Hamiltonian with a proper generalized moment map µ : M → t∗ and a moment one form α ∈ 1 (M, t∗ ), and such that the action preserves the generalized Calabi-Yau structure ρ on M. Then the density function of the push-forward measure µ∗ (dm) coincides with the DuistermaatHeckman function f defined by (6.1), where dm is the measure on M defined by the volume form (3.4) on M. Proof. Since the torus action is free, after applying a B-transform, we may well assume that the moment one form α is trivial. Choose an integer lattice in t = Lie(T ), and identify T with S 1 × S 1 · · · × S1 = S 1 × T k−1 . Following the proof of [GGK97, Thm. k−1

5.8], we consider an open neighborhood of a free orbit in M. On such a neighborhood there exists a coordinate system θ1 , θ2 , . . . , θk , x1 , . . . , xk , y1 , . . . , y2d ,

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where 2d = 2n − 2k, θi ∈ R/2π Z, and xi are coordinates on t∗ , such that the T action is generated by the vector field ∂θ∂ 1 , ∂θ∂ 2 , . . . , ∂θ∂ k and the moment map µ is given by (x1 , x2 , . . . , xk ). We refer to the proof of [GGK97, Thm. 5.8] for a detailed explanation on the existence of such a coordinate system. Then we claim that for any a = (a1 , a2 , . . . , ak ) in an open subset of t∗ such that x1 , x2 , . . . , xk are well defined, we have (ρ(x), ρ(x)) M = (−2i)k (θ1 ∧ d x1 ∧ θ2 ∧ d x2 ∧ · · · ∧ θk ∧ d xk )   ∧ ρ

a (π(x)), ρ

a (π(x)) M , a

(6.2)

where π : µ−1 (a) → Ma is the quotient map, x is a point in the level set µ−1 (a), and (·, ·) M and (·, ·) Ma denote the Mukai pairing on M and Ma respectively. We will establish √our claim by induction on k. If√k = 1, then since the action is Hamiltonian, ( ∂θ∂ 1 − −1d x1 ) · ρ = 0, i.e., ι ∂ ρ = −1d x1 ∧ ρ. A simple calculation shows that ∂θ1

under the above coordinate system ρ must be of the form

d x1 ∧ α0 + iθ1 ∧ d x1 ∧ α1 + α1 , where αi is a differential form such that ι

∂ ∂θ1

αi = 0 and ι

∂ ∂ x1

αi = 0, i = 0, 1. It is easy

x1−1 (a)

to check directly that the restriction of α1 to the level set descends to a quotient Calabi-Yau structure α1 on the quotient x1−1 (a1 )/S 1 ; moreover, we have   α1 (π1 (x)), α 1 (π1 (x)) M , (ρ(x), ρ(x)) M = −2i(θ1 ∧ d x1 ) ∧ a

(6.3)

1

where π1 : x1−1 (a1 ) → Ma1 = x1−1 (a1 )/S 1 is the quotient map. If k = 1, then Ma1 = Ma and α1 is actually the quotient generalized Calabi-Yau structure ρ

a on it. The claim is proved for the case k = 1. Assume that the claim is true for k − 1 and consider the case k. Note that the action of T k−1 ⊂ T = S 1 × T k−1 on M commutes with that of S 1 and so descends to a Hamiltonian action on Ma1 with moment map (x2 , . . . , xk ). Here by abuse of notation, we have used the same xi to denote the functions on the quotient Ma1 whose pull-back under the quotient map π1 coincides with xi |x −1 (a1 ) , 2 ≤ i ≤ k. 1   −1 −1 k−1 Observe that Ma = x2 (a2 ) ∩ · · · xk (ak ) /T and the restriction of α1 to

a on x2−1 (a2 ) ∩ · · · xk−1 (ak ) descends to the quotient generalized Calabi-Yau structure ρ Ma . Using the induction assumption, we get that   α 1 (y) M

α1 (y),

a1

=(−2i)k−1 (θ2 ∧ d x2 ∧ · · · ∧ θk ∧ d xk )  

a (π2 (y)) M , ∧ ρ

a (π2 (y)), ρ

(6.4)

a

where π2 : x2−1 (a2 ) ∩ · · · xk−1 (ak ) → Ma =



 x2−1 (a2 ) ∩ · · · xk−1 (ak ) /T k−1 is the

quotient map, y is a point in the level set x2−1 (a2 ) ∩ · · · xk−1 (ak ). Combining (6.3) and (6.4), we conclude our claim holds for the case k. To finish the proof, we borrow the following argument from [GGK97]. Let λ j be an invariant partition of unit such that each λ j is supported in a neighborhood with coordinates described above. Then we have

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489

 (−1)n λ j (ρ(x), ρ(x)) M (2i)n M j    (−1)n−k =

a (π(x)), ρ

a (π(x)) M λ j (θ1 ∧ d x1 ∧ · · · ∧ θk ∧ d xk ) ∧ ρ n−k a (2i)

µ∗ (dm) =

j

=

(−1)

n+ k(k+1) 2

(2π )k

(2i)n−k

j



 λj

Ma

  ρ

a (π(x)), ρ

a (π(x)) M

 a

d1 d x2 · · · d xk

( Because of Fubini’s theorem) k(k+1)

     (−1)n+ 2 (2π )k d x1 d x2 · · · d xk ρ

λ (π(x)), ρ

(π(x)) = j a a Ma (2i)n−k Ma j  = f (a)d x1 d x2 · · · d xk . This proves Theorem 6.6.

 

Before we state the Duistermaat-Heckman theorem in the generalized Calabi-Yau setting, we first give two simple observations. Lemma 6.7. Let σ be the anti-automorphism defined on C ∞ (∧T ∗ M) as in (3.1), and let H be a closed three form on the manifold M. We have a) if a differential form ϕ is d H = (d − H ∧)-closed, then σ (ϕ) is d−H = (d + H ∧)closed; b) if X + γ ∈ C ∞ (T M ⊕ T ∗ M) and if (X + γ ) · ϕ = ι X γ + γ ∧ ϕ = 0, then we have (X − γ ) · σ (ϕ) = ι X σ (ϕ) − γ ∧ σ (ϕ) = 0. p Proof. Decompose ϕ = i=0 ϕi into homogeneous components by the usual grading of differential forms, where p = dimM. Then it follows from d H ϕ = 0 that dϕi = 0 for i ≤ 1 and dϕi = H ∧ ϕi−2 for i ≥ 2. It is straightforward to check that dσ (ϕi ) = 0 if i ≤ 1 and dσ (ϕi ) = −H ∧ σ (ϕi−2 ) if i ≥ 2. So we have that σ (ϕ) is d−H = (d + H ∧)closed. This proves Assertion (a) in Lemma 6.7. A similar argument proves Assertion (b).   Lemma 6.8. Assume that there is a Hamiltonian action of a compact connected Lie group G on an H -twisted generalized Calabi-Yau manifold M with proper generalized moment map µ : M → t∗ and moment one form α ∈ 1 (M, g∗ ). Then eiµ ρ is an η = (H + α)-twisted equivariantly closed form with coefficients in the ring of formal power series. Proof. For any ξ ∈ t, we have   √ ξ dG,η eiµ ρ (ξ ) = ( −1du ξ ∧ ρ − ιξ M ρ − α ξ ∧ ρ)eiµ + eiµ (d H ρ)   √ √ = −ξ M + −1(dµξ + −1α ξ ) · ρ = 0.

√ √ √ The last equality holds because −ξ M + −1(dµξ + −1α ξ ) is a section of the −1eigenbundle of the generalized Calabi-Yau structure so that its Clifford action annihilates ρ.  

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Remark 6.9. The same observation that eiµ ρ is an equivariant closed extension of ρ has been made in [HuUribe06], and in [NY07] for the case H = 0 and α = 0. However, eiµ ρ was treated as an equivariant differential form in the usual Cartan model in [NY07]. This is incorrect since eiµ ρ : g → (M) is not a polynomial map, where g is the Lie algebra of G. For instance, let G act on a symplectic manifold (M, ω) with moment map  : M → g∗ , and let ρ = eiω . Then ei eiω = ei(ω+) = eiωG , where ωG := ω+ is the equivariant symplectic form. It is well known that eiωG is not an equivariant differential form in the Cartan model, cf, [GS99, pp. 167]. Therefore, strictly speaking, the proof of the Duistermaat-Heckman theorem in the generalized Calabi-Yau setting given in [NY07] has a tiny gap. Theorem 6.10. Assume that there is a free Hamiltonian action of a k dimensional torus T on an H -twisted 2n dimensional generalized Calabi-Yau manifold M with proper generalized moment map µ : M → t∗ and moment one form α ∈ 1 (M, t∗ ), preserving the generalized Calabi-Yau structure ρ. Then the density function f for the push-forward measure on t∗ via the generalized moment map µ is a polynomial of degree at most n −k. Proof. As the action is free, after applying a B-transform, one can assume that the moment one form α is zero. Under this assumption, H = H +α is an equivariantly closed three form in the usual Cartan model. First, by Lemma 6.8 we have dG,H (eiµ ρ) = 0. Since dG,H is a real operator, we also have dG,H (e−iµ ρ) = 0. Next we observe that the canonical anti-automorphism σ : ∗ (M) → ∗ (M) extends naturally to the equivariant differential forms. Moreover, as an easy consequence of Lemma 6.7, we have that √ dG,−H (e−iµ σ (ρ)) = e−iµ (d + H ∧)σ (ρ) + e−iµ (− −1dµξ − ξ M ) · σ (ρ) = 0. Consequently by Lemma 2.3 e−2iµ σ (ρ) ∧ ρ is equivariantly closed under the usual (untwisted) equivariant differential dG . Now choose a T -invariant t∗ valued connection one form θ for the principal bundle M → M/T . Choose a basis of t so as to identify t∗ with Rk and suppose θ = (θ 1 , θ 2 , . . . , θ k ), µ = (µ1 , µ2 , . . . , µk ) under this identification. Let cl be the unique closed two form on M/T such that π ∗ cl = dθ l , 1 ≤ l ≤ k. As we explained in Appendix A, the usual Cartan map, as defined in (2.3), extends naturally to the equivariant differential forms with coefficients in the ring of formal power series. Moreover, it follows easily from Theorem A.3 that e−2iµ σ (ρ) ∧ ρ gets mapped to an ordinary closed differential form e−2iµ cl σ (τ ) ∧ τ , l

where τ is the unique differential form on M/T such that π ∗ τ = ρhor , and π : M → M/T is the quotient map. Let ja : Ma → M/T be the inclusion map and let πa : µ−1 (a) → Ma be the natural projection map. From the commutative diagram inclusion

µ−1 (a) −−−−−→ ⏐ ⏐ πa  Ma

M ⏐ ⏐ π

ja

−−−−→ M/T

on M/T we get that ja∗ τ = ρ

. The closed three form H descends to a closed three form H

= H and such that the quotient generalized Calabi-Yau structure ρa is such that π ∗ H

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491

d H -closed. By Lemma 6.7 σ (ρa ) is d− H -closed. Note that d H is a real operator and ρ

is d H -closed. Hence σ (ρa )∧ρ is an ordinary closed differential form, representing a cohol mology class in H (M/T, C). Now let ϕ be the cohomology class of [e−2iµ cl σ (τ ) ∧ τ ] in H (M/T, C). Then we have [σ ( ρa ) ∧ ρ

a ] = [ ja∗ (σ (τ ) ∧ τ )]   l l = [ ja∗ e2iµ cl ∧ (e−2iµ cl ∧ σ (τ ) ∧ τ ) ] = [e2ia cl ∧ ja∗ (e−2iµ cl ∧ σ (τ ) ∧ τ )] l

l

= [e2ia cl ] ∧ ja∗ ϕ, l

where ρa is the quotient generalized Calabi-Yau structure on Ma , and cl = ja∗ cl . Since Ma is compact and oriented, the embedding ja : Ma → M/T defines an integral homology class [Ma ] ∈ H2n−2k (M/T, Z). This homology class depends smoothly on a and is an integral class. So it is actually independent of a. Let us fix an a0 in the moment map image µ(M). Then we have [Ma0 ] = [Ma ]. Thus up to a normalization factor the integral (6.1) can be interpreted topologically as the pairing of the constant homology class [Ma0 ] with the cohomology class [e2ia cl ] ∧ ja∗ ϕ. l

(6.5)

Note that cl represents a degree two cohomology class, 1 ≤ l ≤ k. Therefore the density function (6.1) is a polynomial function of degree at most n − k.   Remark 6.11. If the generalized Calabi-Yau structure ρ is of constant type p, then by the generalized Darboux theorem locally it can always be written as e B+iω θ1 ∧ · · · ∧ θ p , where B + iω is a closed complex valued two form, and θ1 , . . . , θ p are complex valued one forms. (We refer to [Gua07, Thm. 4.35] for details.) Combining this with (6.5), we conclude that the Duistermaat-Heckman function is of degree at most n −k − p provided the generalized Calabi-Yau structure ρ is of constant type p, c.f., [NY07]. 7. Examples of Hamiltonian Actions on Generalized Calabi-Yau Manifolds The following theorem extends a useful construction of Hamiltonian actions in symplectic geometry [L04, Prop. 4.2] to generalized complex geometry, and it is an equivariant version of a construction proposed in [Ca05, Thm. 2.2]. It allows us to construct examples of Hamiltonian actions on compact twisted generalized Calabi-Yau manifolds which are not a direct product of a symplectic manifold and a complex manifold. Theorem 7.1. Let (N , J ) be a compact H -twisted generalized Calabi-Yau manifold. Then there exists a S 2 bundle π : M → N which satisfies a) M admits a π ∗ H -twisted generalized Calabi-Yau structure J ; b) there exists a Hamiltonian S 1 action on the generalized Calabi-Yau manifold (M, J ). Proof. Let S 2 be the set of unit vectors in R3 . In cylindrical coordinates (θ, h) away from the poles, 0 ≤ θ < 2π, −1 ≤ h ≤ 1, the standard symplectic form on S 2 is the area form σ = dθ ∧ dh. The circle S 1 acts on S 2 by rotations eit (θ, h) = (θ + t, h).

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This action is Hamiltonian with the moment map given by µ = h, i.e., the height function. Let π P : P → N be the principal S 1 -bundle with Euler class [c] ∈ H 2 (N , Z), let  be the connection 1-form such that π P∗ c = d, and let M be the associated bundle P × S 1 S 2 . Then π : M → N is a symplectic fibration over the compact H -twisted generalized complex manifold N . The standard symplectic form σ on S 2 gives rise to a symplectic form σx on each fibre π −1 (x) ∼ = S 2 , where x ∈ N , whereas the height 2 function h on S gives rise to a global function F on M whose restriction to each fiber is just h. Next we resort to minimal coupling construction to get a closed two form η which restricts to σx on each fiber π −1 (x). Let us give a sketch of this construction here and refer to [GS84] for technical details. Consider the closed two form −d(t) = −td−dt ∧ defined on P × R. It is easy to check that the S 1 action defined by eiθ ( p, t) = (eiθ p, t) is Hamiltonian with moment map t. Thus the diagonal action of S 1 on (P × R) × S 2 is also Hamiltonian, and M is just the reduced space of (P × R) × S 2 at the zero level. As a result, the closed two form (−d(t) + σ ) |zero level descends to a closed two form η on M which restricts to σx on each fibre π −1 (x) ∼ = S 2 , and the function (P × R) × S 2 → R, ( p, t, z) → h(z) descends to a function F on M which restricts to the height function h on each fibre S 2 . Moreover, there is another S 1 action on (P × R) × S 2 which is defined by letting S 1 act on (P × R) trivially and act on S 2 by rotation; this action commutes with the above diagonal S 1 action and so descends to a fibrewise S 1 action on M. Let X be the fundamental vector field generated by this fibrewise S 1 action. It is easy to check that the function F is invariant under the fibrewise S 1 action and ι X (η) = d F. Let ρ be the closed nowhere vanishing pure spinor associated to the generalized Calabi-Yau structure J on N , and let (·, ·) M be the Mukai pairing on M. We claim that for sufficiently small  > 0, (eiη ∧ π ∗ ρ, e−iη ∧ π ∗ ρ) M = 0. Since η is non-degenerate on the vertical tangent space kerπ∗ , at each point x ∈ M it determines a horizontal subspace Horx = {X ∈ Tx M : η(X, Y ) = 0, ∀Y ∈ Tx S 2 }. The subspace Horx is a complement to Tx S 2 and is isomorphic to Tπ(x) N via π∗ . So we have a spitting Tx M = Horx ⊕ Tx S 2 , which further induces a splitting of the space of differential forms of degree two ∧2 Tx∗ M = (∧2 Hor∗x ) ⊕ (∧2 Tx∗ S 2 ) ⊕ (Tx∗ M ⊗ Hor∗x ). Here Hor∗x denotes the dual space of Horx . By the definition of the horizontal subspaces, η splits into a direct sum η = η1 + η2 such that η1,x ∈ ∧2 Hor∗x , η2,x ∈ ∧2 Tx∗ S 2 . Let (·, ·)Hor be the Mukai pairing on Hor and (·, ·) N the Mukai pairing on N . Since

Equivariant Cohomology Theory of Twisted Generalized Complex Manifolds

493

Horx is isomorphic to Tπ(x) N via π∗ , (π ∗ ρ, π ∗ ρ)Hor = (ρ, ρ) N = 0. In view of the compactness of N , for sufficiently small  we have (eiη ∧ π ∗ ρ, e−iη ∧ π ∗ ρ) M = (2iη2 ) ∧ (eiη1 π ∗ ρ, e−iη1 π ∗ ρ)Hor = 0. It is straightforward to check that eiη ∧π ∗ ρ is a dπ ∗ H -closed form. Therefore eiη ∧π ∗ ρ determines a π ∗ H -twisted generalized Calabi-Yau structure J . Finally we note that (X − id F) · (eiη ∧ π ∗ ρ) = (iι X (η) − id F) ∧ (eiη ∧ π ∗ ρ) = 0. This proves that the X − id F ∈ C ∞ (L), where L is the i-eigenbundle of J . Thus the S 1 action is Hamiltonian with the generalized moment map  F and trivial moment one form.   Remark 7.2. a) Topologically, M = P × S 1 S 2 depends only on the choice of integral cohomology class [c] ∈ H 2 (N ). When [c] = 0, it is easy to see that the S 2 -bundle M is non-trivial, i.e., not a Cartesian product of N an S 2 . b) It is useful to have the following explicit description of η. Observe that dθ −  is a basic form on (P × R) × S 2 . Its restriction to the zero level of (P × R) × S 2 descends to a one form γ on M whose restriction to each fibre S 2 is just dθ . It is easy to see that on the associated bundle P × S 1 (S 2 − {two poles}) we actually have ∗ c + γ ∧ d F. Therefore, the generalized complex quotient taken at the level η = Fπ M  < t <  is M with the quotient generalized Calabi-Yau structure eitc ∧ ρ. Example 7.3. Let N = T 4 ∼ = C2 /Z4 be the four dimensional torus with periodic coordinates z i = xi + yi , 1 ≤ i ≤ 2, and let c = d x1 ∧ dy1 , ρ1 = e−ic dz 2 , ρ2 = dz 1 ∧ dz 2 . Then it is easy to check that both ρ1 and ρ2 are generalized Calabi-Yau structures on N which are of type 1 and 2 respectively. Let P be the principal S 1 -bundle P with Euler class [c], M = P × S 1 S 2 the 2 S -bundle associated to P, and ρi := e−ic π ∗ ρi the generalized Calabi-Yau structures on M as constructed in Theorem 7.1. By Remark 7.2 the generalized complex quotient taken at the level set − < t <  is M with the quotient generalized Calabi-Yau structure ρ

i = e−itc ∧ ρi , i = 1, 2. Note that N is endowed with the orientation ( ρi , ρ

i ), where (·, ·) stands for the Mukai pairing on N . For convenience, we will denote by N + the manifold N with the orientation d x1 ∧ d x2 ∧ d x3 ∧ d x4 , and N − the manifold N with the orientation −d x1 ∧ d x2 ∧ d x3 ∧ d x4 . A straightforward calculation shows that the Duistermaat-Heckman function for (M, ρ1 ) is  π f 1 (t) = − (e−i(t+1)c dz 2 , ei(t+1)c dz 2 ) 2 N+  π 4(t + 1)d x1 ∧ dy1 ∧ d x2 ∧ dy2 =− 2 N+ = −2π(t + 1); whereas the Dustermaat-Heckman function for (M, ρ2 ) is  π f 2 (t) = − (e−itc ∧ dz 1 ∧ dz 2 , eitc ∧ dz 1 ∧ dz 2 ) 2 N−  π −4d x1 ∧ dy1 ∧ d x2 ∧ dy2 =− 2 N− = −2π.

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Appendix A. The Equivariant Cohomology with Coefficients in the Ring of Formal Power Series In this Appendix, we introduce the equivariant differential forms with coefficients in the ring of formal power series. Let an n dimensional compact connected Lie group G act on an m dimensional manifold M, let g be the Lie algebra of G. Choose a basis ξ1 , ξ2 , . . . , ξn of g and let x1 , x2 , . . . , xn denote the corresponding dual basis in g∗ which generates the polynomial ring Sg∗ . Let R = C[[x1 , x2 , . . . , xn ]] be the ring of formal power series over C. Let a = (x1 , x2 , . . . , xn ) be the ideal in R generated by x1 , x2 , . . . , xn . Then the sequence of ideals Sg∗ = a 0 ⊇ a ⊇ · · · ⊇ a k ⊇ · · · defines an a-adic topology on the polynomial ring Sg∗ , c.f. [AM99, Chapter 10], and the sequence of modules     Sg∗ ⊗ (M) ⊇ a Sg∗ ⊗ (M) ⊇ · · · ⊇ a k Sg∗ ⊗ (M) ⊇ · · · defines an a-adic topology on Sg∗ ⊗ (M). So (Sg∗ ⊗ (M))G inherits a topology from Sg∗ ⊗ (M) as a subspace.  G (M), the space of equivariant differential forms with coeffiDefinition A.1. Define  cients in R, to be the completion of G with respect to the a-adic topology we described above, and define the equivariant differential    G (M), dG ( x I ⊗ dα I + (x j x I ) ⊗ ιξ j,M α I , x I ⊗ αI ) = x I ⊗ αI ∈  I

I

I

(A.1) where I is a multi-index, and ξ j,M denotes the vector field induced by ξ j ∈ g. It is easy to check that dG2 = 0. Define the cohomology G (M) = kerdG /imdG H to be the equivariant cohomology with coefficients in R. Lemma A.2. Assume that G acts freely on the manifold M. Then the usual equivariant differential dG : G → G , the chain homotopy operator Q : G → G as defined in (2.4), and the Cartan map C : G → bas as defined in (2.3) are continuous map from G to itself. Proof. We divide our proofs into three steps. 1) dG : G → G is continuous. Note that as an operator dG is also defined on Sg∗ ⊗ (M). Furthermore, we have   dG a k (Sg∗ ⊗ (M)) ⊂ a k (Sg∗ ⊗ (M)), for any k ≥ 0. So dG : Sg∗ ⊗ (M) → Sg∗ ⊗ (M) is continuous and so its restriction to G is also continuous.

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2) The chain homotopy map Q is continuous. Let K , R and F be the operators we introduced in Sect. 2.2. Note that K , R F and Q are all defined on Sg∗ ⊗ (M) as well, and we have   K a k (Sg∗ ⊗ (M)) ⊂ a k−1 (Sg∗ ⊗ (M)),   (R F)i a k (Sg∗ ⊗ (M)) ⊂ a k−i (Sg∗ ⊗ (M)), where 0 ≤ i ≤ k. Thus K and R F are both continuous operators and so their restrictions to G are also continuous. Note that given any γ ∈ Sg∗ ⊗ (M), R = dθ r ∂r increases the form degree of γ by two, whereas F preserves both the form degree and polynomial degree of γ . So for i > 21 dimM we have (R F)i = 0   for dimension reasons. Therefore we conclude that Q = K F R F + (R F)2 + · · · is also continuous on Sg∗ ⊗ (M) and so the restriction of Q is continuous on G . 3) The Cartan map C : G → bas is continuous. This follows from the simple   observation that, for dimension reasons, by (2.3) we have C a k (Sg∗ ⊗ (M) = 0  if k > 21 dimM.   G (M) is the completion of G (M). Thus dG , Q, and C have an By definition,   G (M). One checks easily that the extension of dG to   G is the one unique extension to   G → bas is as we defined in Definition A.1, and the extension of the Cartan map C :  given by x I ⊗ αI → c I ∧ αI , (A.2) I

I

where is the curvature elements  as defined in (2.2), and I = (i 1 , i 2 , . . . , i n ) is a multi-index. Note that if | I | = nk=1 i k > 21 dim M, then c I ∧ α I = 0 for dimension reasons. So the right hand side of (A.2) is actually a finite sum. Since the equality G . dG Q + QdG = I −C holds on G , it also holds on the completion of G , i.e.,  In summary, we have proved the following result. ci

Theorem A.3. Assume the action of G on M is free. Then the Cartan map (A.2) induces a natural isomorphism from Hˆ G (M) to the ordinary cohomology H (M/G). Indeed, we can prove a slightly more general result. Suppose that H ∈ 3 (M) is a closed basic three form. Then the map H ∧ : G → G , α  → H ∧ α is obviously continuous. Since H is a basic form, a straightforward check shows that Q(H ∧ α) + H ∧ (Qα) = 0 for any α ∈ G . As a result, dG,H Q + QdG,H = I − C  G . This proves the following result. holds on G , and so it also holds on  Theorem A.4. Assume the action of G on M is free. Let π : M → M/G be the quotient map. And assume that H ∈ 3 (X ) is a basic form so that there is a three

∈ 3 (X/G) which satisfies π ∗ H

= H . Then the Cartan map (A.2) induces a form H natural isomorphism from the twisted equivariant cohomology Hˆ G (M, H ) to the twisted

). cohomology H (M/G, H Acknowledgement. I would like to thank Tom Baird, Kentaro Hori, Lisa Jeffrey, Anton kapustin, Yael Karshon, Eckhard Meinrenken, Reyer Sjamaar, and Allessandro Tomasiello for useful conversations. And I would like to thank Marco Gualtieri for useful comments on an earlier version of this article. A special thank goes to Lisa Jeffrey. The present work would not have been possible without her kind support.

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References [AS05]

Atiyah, M., Segal, G.: Twisted K-theory and cohomology. http://arxiv.org/list/math.KT/ 0510674, 2005 ref. [AS05] [AM99] Atiyah, M., Macdonald, I.G.: Introduction to commutative algebra. Addison-Wesley Publishing Co, Reading, Mass-London-Don Mills, Ont: 1969 [BCRR05] Bruzzo, U., Cirio, L., Rossi, P., Rubtsov, V.: Equivariant cohomology and locolization for Lie algebroids. http://arxiv.org/list/mah.DG/0506392, 2005 [BCG05] Bursztyn, H., Cavalcanti, G., Gualtieri, M.: Reduction of courant algebroids and generalized complex structures. Adv. in Math. 211(2), 726–765 (2007) [BL08] Tom Baird, Yi Lin: Cohomology of generalized complex quotients I, http://arxiv.org/list/arXiv: math.DG/0802.1341 [Ca05] Cavalcanti, G.: New aspects of dd c lemma. Oxford D. Phil. thesis, http://arxiv.org/list/math. DG/0501406, 2005 [Ca06] Cavalcanti, G.: The decomposition of forms and cohomology of generalized complex manifolds. J. Geom. Phyi. 57, 121–132 (2006) [Che97] Chevalley, C.: The algebraic theory of spinors and Clifford algebras. In: Collected works, Vol. 2., Edited and with a forward by Pierre Cartier and Catherine Chevalley. With a postface by J.-P. Bourguignon. Berlin: Springer-Verlag, 1997 [GHR84] Gates, S. Jr., Hull, C., Rocek, M.: Twisted multiplets and new supersymmetric non-linear σ -model. Nucl. Phys. B. 248(1), 157–186 (1984) [Gua03] Gualtieri, M.: Generalized complex geometry. Oxford D. Phil. thesis, http://arxiv.org/list/math. DG/0401221, 2004 [Gua07] Gualtieri, M.: Generalized complex geometry. http://arxiv.org/list/math.DG/0703298, 2007 [Gua04] Gualtieri, M.: Generalized geometry and the Hodge decomposition, http://arxiv.org/list/math. DG/0409093, 2004 [GS84] Guillemin, V., Sternberg, S.: Symplectic Techniques in Physics. Cambridge: Cambridge University Press, 1984 [GS99] Guillemin, V., Sternberg, S.: Super-symmetry and equivariant de Rham theory, with an appendix containing two reprints by Henri Cartan. In: Mathematics Past and Present. Berlin: Springer-Verlag, 1999 [GGK97] Guillemin, V., Ginzburg, V., Karshon, Y.: Moment maps, Cobordisms, and Hamiltonian actions, American Mathematical Society, Mathematical surveys and Monographys, Vol. 98, Providence, RI: Amer. Math. Soc., 2002 [H02] Hitchin, N.: Generalized calabi-yau manifolds. Q.J. Math. 54(3), 281–308 (2003) [Hu05] Hu, S.: Hamiltonian symmetries and reduction in generalized geometry, http://arxiv.org/list/ math.DG/0509060, 2005 [HuUribe06] Hu, S., Uribe, B.: Extended manifolds and extended equivariant cohomology. http://arxiv.org/ list/math.DG/0608319, 2006 [Kir86] Kirwan, F.: Cohomology of quotients in symplectic and algebraic geometry. Mathematical Notes, 31, Princeton, NJ: Princeton Univ. Press, 1984 [KL04] Kapustin, A., Li, Y.: Topological sigma-models with H-flux and twisted generalized complex manifolds, http://arxiv.org/list/hep-th/0407249, 2004 [KT06] Kapustin, A., Tomasiello, A.: The general (2, 2) gauged sigma model with three-form flux. http://arxiv.org/list/hep-th/0610210, 2006 [LS03] Lin, Y., Sjamaar, R.: Equivariant symplectic hodge theory and the dg δ-lemma. J. Sympl. Geom. 2(2), 267–278 (2004) [L04] Lin, Y.: Examples of non-Kähler Hamiltonian circle manifolds with the strong Lefschetz property. Adv. in Math., 208, issue 2, 699–709 (207) [LT05] Lin, Y., Tolman, S.: Symmetries in generalized kähler geometry. Commun. Math. Phys. 208, 199–222 (2006) [Lin06] Lin, Y.: Generalized geometry, equivariant ∂∂-lemma and torus actions. J. Geom. Phys. 57, 1842–1860 (2007) [Mc01] McCleary, J.: A user’s Guide to Spectral Sequences, Second edition. Cambridge Studies in Advanced Mathematics, 58, Cambridge: Cambridge University press, 2001 [NY06] Nitta, Y.: Reduction of generalized calabi-yau structures. J. Math. Soc. Japan 59(4), 1179–1198 (2007) [NY07] Nitta, Y.: Duistermaat-Heckman formula for a Torus action on a generalized Calabi-Yau manifold and localization formula. http://arxiv.org/list/math.DG/0702264, 2007 [SX05] Stiénon, M., Ping, X.: Reduction of generalized complex structure. J. Geom. Phys. 58, 105– 121 (2008)

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[Va05] [Yan96]

497

Vaisman, I.: Reduction and submanifolds of generalized complex manifolds. http://arxiv.org/ list/math.DG/0511013, 2005 Yan, D.: Hodge structure on symplectic manifolds. Adv. in Math. 120(1), 143–154 (1996)

Communicated by N.A. Nekrasov

Commun. Math. Phys. 281, 499–528 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0483-8

Communications in

Mathematical Physics

Symmetry Transitions in Random Matrix Theory & L -functions J. P. Keating
, B. E. Odgers

School of Mathematics, University of Bristol, Bristol BS8 1TW, UK. E-mail: [email protected] Received: 28 April 2007 / Accepted: 23 October 2007 Published online: 6 May 2008 – © Springer-Verlag 2008

Abstract: We compute the moments of the characteristic polynomials of random orthogonal and symplectic matrices, defined by averages with respect to Haar measure on S O(2N ) and U Sp(2N ), to leading order as N → ∞, on the unit circle as functions of the angle θ measured from one of the two symmetry points in the eigenvalue spectrum {exp(±iθn )}1≤n≤N . Our results extend previous formulae that relate just to the symmetry points, i.e. to θ = 0. Local spectral statistics are expected to converge to those of random unitary matrices in the limit as N → ∞ when θ is fixed, and to show a transition from the orthogonal or symplectic to the unitary forms on the scale of the mean eigenvalue spacing: if θ = π y/N they become functions of y in the limit when N → ∞. We verify that this is true for the spectral two-point correlation function, but show that it is not true for the moments of the characteristic polynomials, for which the leading order asymptotic approximation is a function of θ rather than y. Symmetry points therefore influence the moments asymptotically far away on the scale of the mean eigenvalue spacing. We also investigate the moments of the logarithms of the characteristic polynomials in the same context. The moments of the characteristic polynomials of random matrices are conjectured to be related to the moments of families of L-functions. Previously, moments at the symmetry point θ = 0 have been related to the moments of families of L-functions evaluated at the centre of the critical strip. Our results motivate general conjectures for the moments of orthogonal and symplectic families of L-functions evaluated at a fixed height t up the critical line. These conjectures suggest that the symmetry of the non-trivial zeros of the L-functions influences the moments asymptotically far, on the scale of the mean zero spacing, from the centre of the critical strip. We verify that the second moments of real quadratic Dirichlet L-functions and a family of automorphic L-functions are consistent with our conjectures.
JPK is supported by an EPSRC Senior Research Fellowship.

BEO was supported by an Overseas Research Scholarship and a University of Bristol Research

Scholarship.

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1. Introduction It is believed that deep connections exist between the statistical distributions of the eigenvalues of certain ensembles of random matrices and those of the non-trivial zeros of L-functions, for example the Riemann zeta-function ζ (s). Such connections, if established, would have profound implications in number theory. They have their origins in a conjecture due to Montgomery, which asserts that the pair correlation of the non-trivial zeros of the zeta function close to a height T up the critical line s = 1/2 + it, on the scale of the mean zero spacing 2π/ log(|T |/2π ), coincides, in the limit T → ∞, with the N → ∞ limit of the pair correlation of the eigenvalues exp(iθn ) of N × N unitary matrices, on the scale of the mean eigenphase spacing 2π/N , defined by averaging uniformly with respect to Haar measure on U (N ) [27]. For a particular class of test functions – those whose Fourier transform have support in (−1, 1) – this conjecture was proved by Montgomery [27]. Its truth in general, and its extension to all n-point correlations, is supported by extensive numerical computations [28] and heuristic arguments [3,4]. These results and conjectures generalize to all principal L-functions, in the sense that for each such L-function the statistical distribution of the zeros high up on the critical line is believed to match the corresponding distribution of the eigenvalues of large random unitary matrices [30]. They also extend to lower order terms in the asymptotic expansion of the correlation functions, viewed as functions of the mean spacing between the zeros [5,2]. It was suggested by Katz and Sarnak [21] that statistical properties of the zeros defined with respect to averages over families of L-functions, at a fixed height on the critical line, should be related to averages over one of the compact classical groups, the particular group being determined by symmetries of the family in question. For example, L-functions associated with quadratic twists of elliptic curves are expected to be described by averages over the orthogonal group, and real quadratic Dirichlet L-functions by averages over the symplectic group. The differences show up most clearly in the statistical distribution of the zeros closest to the centre of the critical strip, where the critical line cuts the real s axis, since these are most sensitive to any symmetry of the zeros with respect to reflection through this point. The corresponding random-matrix distributions relate to the eigenvalues closest to the symmetry points at θ = 0 and θ = π in the spectrum of orthogonal and symplectic matrices, associated with the fact that these eigenvalues come in complex conjugate pairs. For example, these differences show up in the one-level density near to the respective symmetry points, and in the distribution of the zeros/eigenvalues closest to these points [29], when both are evaluated asymptotically on the scale of the mean zero/eigenvalue spacing. In the case of the one-level density, the situation is thus that in the limit as one looks infinitely high up the critical line all L-functions behave in the same way: their one-level densities coincide with that of random unitary matrices. By contrast, near to the centre of the critical strip, the various families show different behaviour, corresponding to that found in either unitary, orthogonal or symplectic matrices close to θ = 0 and θ = π (i.e. the symmetry points in the latter two cases). This exactly mirrors the form of the one level densities of the eigenvalues themselves: close to θ = 0 and θ = π these are different for unitary, orthogonal or symplectic matrices, but far from these points (on the scale of the mean eigenvalue spacing) they coincide. The reason for this is that the one-level density is a local statistic: its value at any point is determined by zeros/eigenvalues close, on the scale of the mean zero/eigenvalue spacing, to that point. Thus asymptotically far from θ = 0 and θ = π it is insensitive to any symmetry and so takes the unitary form.

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In recent years there has been considerable interest in the connections between the moments of L-functions and those of the characteristic polynomials of random matrices. It was first conjectured by Keating and Snaith that the leading order asymptotics of the Riemann zeta function, computed with respect to an average high up along the critical line, are related to the moments of the characteristic polynomials of random unitary matrices, defined with respect to an average over the unitary group U (N ) [22]. The moments of other individual principal L-functions computed in the same way (i.e. with respect to an average high up the critical line) are also expected to be related to the moments of the characteristic polynomials of random unitary matrices. In accordance with the Katz-Sarnak philosophy, the moments of L-functions computed at the centre of the critical strip with respect to family averages are likewise expected to be related to the corresponding moments of the characteristic polynomials of random matrices drawn from any one of the classical compact groups, the group in question being determined by symmetries of the family [8,23]. This connection extends to shifted moments (or correlation functions), and hence to lower order terms in the moment asymptotics [9–11], and to ratios of shifted moments [12], allowing, for example, the formulae for the lower order terms in the asymptotics of the correlation functions of the zeros derived in [5] to be recovered. For a review of results in this general direction, see [24,26]. We again draw attention to the fact that the asymptotics of the moments at the centre of the critical strip, when the average runs over a family, apparently differs from that high up the critical line for a fixed L-function: there is a transition between family-dependent behaviour at the centre, governed by various possible symmetries, and universally unitary behaviour high up on the critical line. One of our primary aims here is to investigate the nature of the transition in the statistical properties of the eigenvalues and characteristic polynomials of random orthogonal and symplectic matrices as one moves away from the symmetry points, at θ = 0 for example. First, we consider the spectral two-point correlation function. For unitary matrices this is independent of θ . For orthogonal and symplectic matrices at θ = 0 it takes different forms. When θ is fixed and takes a non-zero value, it tends to the unitary expression as N → ∞. The transition from the orthogonal/symplectic forms to the unitary form takes place on the scale of the mean eigenvalue spacing; that is when θ = π y/N the transition is described in each case by a function of y when N → ∞, and we determine this function explicitly. These results are entirely consistent with the fact that the two-point correlation function is a local statistic. The behaviour of the moments of the characteristic polynomials of random orthogonal and symplectic matrices is much more interesting in this context. The moments at θ = 0 were calculated in [23] and differ significantly from the moments of random unitary matrices (which are θ -independent). We here calculate the asymptotics of the moments as N → ∞ when θ = 0. In this case the results do not coincide with the unitary expression, even though one is infinitely far from the symmetry point on the scale of the mean eigenvalue spacing. We derive explicit formulae for the second moments as functions of θ . These illustrate the nature of the transition in this case. We also investigate the corresponding transition in the moments of the logarithms of the characteristic polynomials. Our second main aim is to investigate the asymptotics of the moments of L-functions, at a fixed height t up the critical line, with respect to an average over orthogonal and symplectic families. Using the formalism developed in [10] we develop general conjectures which again suggest that, even though one is infinitely far from the centre of the critical strip on the scale of the mean zero spacing, the moments do not coincide with

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the unitary forms that describe a fixed L-function high up on the critical line. We give two examples for which the second moments can be calculated rigorously, the results being consistent with the conjectured formulae. The fact that the transition in the L-function moments is rather subtle is due to the fact that they are not sufficiently ‘local’. The influence of this ‘nonlocality’ was already observed in the context of determining the arithmetical contributions to the moments in [18,6,7]. Our main point here is that even in random matrix theory this property plays a central role in determining the nature of the transition as one moves away from a symmetry point. We list our results explicitly in the next section. The proofs of our theorems, and the calculations supporting our conjectures are then described in the later sections. 2. Results 2.1. Two-point correlations. Let A be an N × N unitary matrix. We denote its eigenvalues by exp(iθn ), n = 1, . . . , N , and the unfolded eigenangles by N . 2π

φn = θn

(1)

The two-point correlation function for the matrix A is conventionally defined as R2 (A; x) =

N ∞ N 1


δ(x + N k − φn + φm ). N

(2)

n=1 m=1 k=−∞

We further define the local two-point correlation function by R2 (A, φ; x) :=

N
N
∞


∞


δ(φ −

x 2

− φm + N j)δ(φ +

x 2

− φn + N j). (3)

m=1 n=1 j=−∞ k=−∞

Note that R2 (A, φ; x) is N -periodic in both φ and x and that  1 N /2 R2 (A, φ; x)dφ = R2 (A; x). N −N /2 For matrices in U Sp(2N ) and S O(2N ) the eigenvalues form conjugate pairs exp (±iθn ), and so, explicitly, R2 (A, φ; x) :=

N
∞ N



∞


[δ(φ −

x 2

− φm + 2N j) + δ(φ −

m=1 n=1 j=−∞ k=−∞ ×[δ(φ + x2 − φn + 2N k) + δ(φ

+

x 2

+ φn + 2N k)].

x 2

+ φm + 2N j)] (4)

Here φ is the distance from the symmetry point at θ = 0, measured in units of the mean eigenvalue spacing. Throughout this paper, we denote averages with respect to Haar measure by  f (A)d A. EG(N ) [ f (A)] = G(N )

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503

Calculations over U Sp(2N ) or S O(2N ) will often be combined, with G U (N ) denoting either group and  1 for G U (N ) = U Sp(2N ), a= 0 for G U (N ) = S O(2N ). With this in mind, in Sect. 3 the following expectations for R2 (A, φ; x) are found. Theorem 1. For φ, x ∈ R, ∞


EU (N ) [R2 (A, φ; x)] =

δ (x − j N ) + 1 −

j=−∞

sin2 (π x) . N 2 sin2 πNx

This formula was to be expected, since for the unitary group statistics are independent of θ , and the right-hand side coincides with the expression for EU (N ) [R2 (A; x)] due to Dyson [17]. Theorem 2. For φ, x ∈ R, EG U (N ) [R2 (A, φ; x)] ∞


=



δ(x − 2N j) 1 + (−1)

j=−∞ ∞


 δ(2φ − 2N j) 1 + (−1)

+

j=−∞

− 1 + 2a)2φπ/2N ) (−1)a − 2N sin(2φπ/2N ) 2N

a sin((2N



− 1 + 2a)xπ/2N ) (−1)a − 2N sin(xπ/2N ) 2N

a sin((2N



sin2 (π x) sin2 (2π φ)   πx  − 2 φ (2N )2 sin 2N (2N )2 sin2 2π 2N   1 sin((2N − 1 + 2a)(2φ − x)π/2N ) − + (−1)a + 2N sin((2φ − x)π/2N ) 2N   1 sin((2N − 1 + 2a)(2φ + x)π/2N ) − × (−1)a + 2N sin((2φ + x)π/2N ) 2N    sin((2N − 1 + 2a)xπ/2N ) sin((2N − 1 + 2a)2φπ/2N ) −2(−1)a 2N sin(xπ/2N ) 2N sin(2φπ/2N )   (−1)a sin((4N − 1 + 2a)2φπ/2N ) sin((4N − 1 + 2a)xπ/2N ) + . + 2N 2N sin(2φπ/2N ) 2N sin(xπ/2N ) −

As N → ∞ we then have the following corollaries. Corollary 1. For φ, x ∈ R and fixed, lim EU (N ) [R2 (A, φ; x)] = δ(x) + 1 −

N →∞

sin2 (π x) , (π x)2

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J. P. Keating, B. E. Odgers

and lim EG U (N ) [R2 (A, φ; x)]     sin(2φπ ) sin(π x) = δ(x) 1 + (−1)a + δ(2φ) 1 + (−1)a 2π φ πx 2  sin(2π φ) sin(π x) − + (−1)a πx 2π φ    sin(π(2φ − x)) a a sin(π(2φ + x)) 1 + (−1) . + 1 + (−1) π(2φ − x) π(2φ + x)

N →∞

Thus, at a fixed unfolded distance (i.e. at a fixed distance on the scale of the mean eigenvalue spacing) from the critical point in the limit of large matrix size we see distinct behaviours. If we further let φ → ∞, we then see a smooth transition to the following universal form: Corollary 2. For x ∈ R, lim

lim EG(N ) [R2 (A, φ; x)] = δ(x) + 1 −

φ→∞ N →∞

sin2 (π x) (π x)2

for G(N ) = U (N ), U Sp(2N ) and S O(2N ). This limit is consistent with the fact that the two-point correlation function is a local statistic, and therefore only influenced by any symmetry in the spectrum close, on the scale of the mean eigenvalue separation, to the symmetry point. 2.2. Log-moments of characteristic polynomials. For an N × N unitary matrix A with Hermitian conjugate A∗ such that A∗ A = I , the characteristic polynomial of A is given by N

Λ A (s) = det(I − A∗ s) = (1 − se−iθn ). n=1

Keating and Snaith showed in [22] that for any θ , EU (N ) [log Λ A (eiθ )] = 0;

(5)

and in [23] that EG U (N ) [log Λ A (e0 )] =

N −1+a (−1)1+a (−1)1+a
1 ∼ log N . 2 j 2 j=1

Specifically, the log-moments for each ensemble are distinct at the critical point. In Sect. 4 we prove the following theorem: Theorem 3. For θ ∈ R, EG U (N ) [log Λ A (eiθ )] =

N −1+a (−1)1+a
ei2 jθ . 2 j j=1

(6)

Symmetry Transitions in Random Matrix Theory & L-functions

505

Letting θ = 0 clearly gives the result of Keating and Snaith above; whilst away from the symmetry point, letting N → ∞ we have the following corollary: Corollary 3. For fixed θ = nπ where n ∈ Z, lim EG U (N ) [log Λ A (eiθ )] =

N →∞

(−1)a log(1 − ei2θ ). 2

Hence at a fixed angle θ we see a dependence on θ retained as N → ∞. Given that θ = πNφ , fixing θ as N → ∞ corresponds to letting φ → ∞. Therefore Corollary 3 implies that even at an infinite unfolded distance (i.e. measured on the scale of the mean eigenvalue spacing) from the symmetry point, symmetry still has an effect on the 1st log-moment. 2.3. Moments of characteristic polynomials. In [22], Keating and Snaith showed that   k−1 k

j!

iθ 2k |Λ A (e )| d A = (N + i + j) (k + j)! U (N ) j=0 i=1 ⎛ ⎞ k−1

j! ⎠ N k2 . ∼⎝ (k + j)! j=0

The corresponding formulae for the 2k th moments at the symmetry points of U Sp(2N ) and S O(2N ), given in [23], are  |Λ A (e0 )|2k d A G U (N )

⎛

⎞ 2k



j! ⎠ = 22k(2k+1)/2 ⎝ (N + j)a (N − 1 + a + (2 j)! j=1 j=1 1≤i< j≤2k ⎛ ⎞ 2k−1+

a j! ⎠ N 2k(2k−1+2a)/2 . ∼ 22k(2k+1)/2 ⎝ (2 j)! 2k−1+

a

i+ j 2 )

(7)

j=1

We again see distinct behaviours at the symmetry point for all three ensembles. For later comparison we note that for k = 1,  |Λ A (eiθ )|2 d A ∼2N U (N )

and



 1 a 1+2a |Λ A (e )| d A ∼4 N . 12 G U (N ) 

0 2

Moving away from the symmetry point, in Sect. 5.1 the following formula for the 2nd moments of the characteristic polynomial over U Sp(2N ) and S O(2N ) is proved:

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J. P. Keating, B. E. Odgers

Theorem 4. For θ ∈ R,  |Λ A (eiθ )|2 d A G U (N )

⎡ ⎤ 2N
+1+2a 1 ⎣(2N + 1 + 2a) + (−1)a = e(N +a− j)2iθ ⎦. |1 − e−2iθ |2a j=0

This implies Corollary 4. For fixed θ = nπ , where n ∈ Z, as N → ∞,  1 |Λ A (eiθ )|2 d A ∼2N . −2iθ |2a |1 − e G U (N ) As for the 1st log-moment, the dependence on θ (in the symplectic case) indicates that symmetry has an effect on the 2nd moment even at an infinite unfolded distance from the symmetry point. This is extended to all 2k th moments in the following theorem, proved in Sect. 5.2. Theorem 5. For fixed θ = nπ , as N → ∞,  2 |Λ A (eiθ )|2k d A ∼ gU (2k)N k G U (N )

where gU (2k) := lim N −k N →∞

2

1 , |1 − e−2iθ |k(k−1+2a)

 U (2N )

|Λ A (e0 )|2k d A = 2k

2

k−1

j=0

j! . (k + j)!

2

The gU (2k) and N k terms combine to give the large N asymptotic behaviour for U (N ), and so the effect of symmetry is purely in the term |1 − e−2iθ |−k(k−1+2a) . 2.4. Moments of an L-function family with symplectic symmetry. We have seen that at a fixed angle θ from the symmetry point, the asymptotic behaviour of the even integral moments for U Sp(2N ) and S O(2N ) as N → ∞ retains a dependence on θ . The L-function analogue is now considered; that is, the large conductor asymptotics of the even integral moments over a family of L-functions at a fixed height t up the critical line. Consider the L-functions with Dirichlet series and Euler product  ∞


 χd ( p) −1 χd (n) L(s, χd ) = 1 − and L(s, χ ) = d ns ps p n=1

respectively, where χd (n) is the Kronecker Symbol, ( dn ). χd (n) are real quadratic characters. The Dirichlet series is absolutely convergent for (s) > 1, and χd (n) is a primitive

Symmetry Transitions in Random Matrix Theory & L-functions

507

character for fundamental discriminants d. Such L-functions have a functional equation and can be analytically continued to the rest of the complex plane. This family of L-functions has symmetries which correspond to those of U Sp(2N ). (See for example [29].) For ease of calculation, we work with a subset of this family which retains such symmetries, namely fundamental discriminants d such that d < 0 and d ≡ 0 (mod 8). For this sub-family the functional equation is given by 1

L(s, χd ) = |d| 2 −s X (s)L(1 − s, χd ),

(8)

where 1

X (s) = π −( 2 −s)

Γ (1 − 2s ) Γ ( 21 + 2s )

.

2.4.1. Fixed height t, large conductor asymptotics In [10], Conrey, Farmer, Keating, Rubinstein and Snaith provide a method for conjecturing the even integral moments over such a family of L-functions. In Sect. 6.2, their conjecture for this family is combined with a lemma underlying the earlier observations on the moments to establish the following conjecture for the large conductor asymptotics of the 2k th moments. Define



ζ (1 + αi + α j ) R2k,8, p ( 21 , α1 , . . . , α2k ) R2k,8 ( 21 , α1 , . . . , α2k ) = 1≤i≤ j≤2k

×

1≤i≤ j≤2k

 1−

p

1



p 1+αi +α j

,

(9)

where R2k,8,2 ( 21 , α1 , . . . , α2k ) = 1, and for p ≥ 3, R2k,8, p ( 21 , α1 , . . . , α2k ) ⎛ ⎛  2k  ⎞⎞ −1  2k



1 1 1 ⎝1 + 1 ⎝ ⎠⎠. = 1+ + − 21 −α j − 21 −α j p p 2 1 − p 1 + p j=1 j=1 Conjecture 1. Let g(x) be a test function compactly supported in the range [1, α], where α > 1 is fixed relative to D. We conjecture that as D → ∞: 1. At the symmetry point: t = 0, 
∗ |d| 1 2k d<0 |L( 2 , χd )| g D d≡0(8) 
∗ |d| g d<0 D d≡0(8) ⎛ ⎞ 2k

j! ⎠ ∼ ( 21 ln D)2k(2k+1)/2 ⎝22k(2k+1)/2 (2 j)! ×

p

 R2k,8, p ( 21 , 0, . . . , 0)

j=1

1 1− p

2k(2k+1)/2 .

508

J. P. Keating, B. E. Odgers

2. Away from the symmetry point: t = 0, 
∗ |d| 1 2k d<0 |L( 2 + it, χd )| g D d≡0(8) 
∗ |d| d<0 g D d≡0(8)

∼

( 21

k2

ln D) gU (2k)



R2k,8, p ( 21 ; it, . . . , it, −it, . . . , −it)

p

 k(k+1)   k(k+1)  2 2 2 1 k 1 1 1− 1 − 1−2it × 1 − 1+2it |ζ (1 + 2it)|k(k+1) , p p p 

where


∗

indicates that the sum is taken over fundamental discriminants, d.

That is, we see a dependence on the height t (and hence the symmetry type of the family), with the ζ (1 + 2it) term in the above conjecture corresponding to the (1 − e−2iθ )−1 term in the RMT results. 2.4.2. Theorem: 2nd Moment In [13], Conrey and Soundararajan calculate a mollified 2nd moment for this family for any point in a region near the critical point s = 21 . Their calculation is based on earlier work of Soundararajan [31,32], within which he explains that the method can be used to calculate the moment at any point in the critical strip. In Sect. 6.3 we outline the relevant results in [13] which lead to confirmation of Conjecture 1 1 for this family in the case where k = 1 and t is in the range [0, 100 ). Theorem 6 (following Soundararajan [31,32] and Conrey-Soundararajan [13]). Let Ψ be a sufficiently smooth function supported in the interval [1, 2]. Then as D → ∞, 1. For t = 0, 
∗ 1 2 Ψ |d| |L( , χ )|  d d<0 2 D

 p − 1  2 2 1 d≡0(8) 3 1 1  1+ − 2 + 3 ; ∼ ( 2 log D)
∗ |d| 4! p+1 p p p p≥3 d<0 Ψ D d≡0(8)

2. For 0 < t <
∗

1 100 ,

1 2 d<0 |L( 2 + it, χd )| Ψ d≡0(8) 
∗ |d| d<0 Ψ D d≡0(8)

×



|d| D

∼ ( 21 log D)(1 − 2−1−2it )(1 − 2−1+2it )



 p − 1  2 p 2it + p −2it 1 |ζ (1 + 2it)|2 . 1+ − + p+1 p p2 p3

p≥3

Letting

 Ψ(v) = max

0≤ j≤v 0

∞

|Ψ ( j) (t)|dt,

3 = O(1). Ψ being “sufficiently smooth” means that Ψ(2) Ψ(3) 1 The restriction to t ∈ [0, 100 ) was made by Conrey and Soundararajan in [13] for ease of exposition and has been retained here for similar reasons.

Symmetry Transitions in Random Matrix Theory & L-functions

509

2.5. Moments of an L-function family with orthogonal symmetry. The same steps are now followed for a family with orthogonal symmetry. For q squarefree and even weight w, let Sw (q) denote the linear space of holomorphic cusp forms of weight w and level q. Further, let Hw (q) denote the set of newforms in Sw (q). Any f (z) ∈ Hw (q) admits a Fourier expansion of the form f (z) :=

∞


n (w−1)/2 λ f (n)e(nz),

n=1

where we normalise so that λ f (1) = 1. The coefficients λ f (n) are real. To f is attached an L-function with Dirichlet series
L( f, s) := λ f (n)n −s . n≥1

This series is absolutely convergent when (s) is large and admits analytic continuation to all of C. The L-function satisfies the functional equation L( f, s) = ε f Xw,q (s)L( f, 1 − s), where

 q 1 −s Γ  1 − s + √ 2 ε f = i µ(q)λ f (q) q = ±1 and Xw,q (s) = 2 4π 2 Γ s − 21 + w



w 2 . w 2

± (q) the collections of newforms with  = ±1 respectively. We denote by Hw f We will consider moments using the harmonic average over Hw (q), defined as
h αf Γ (k − 1)
, α f := k−1 (4π ) f, f f ∈Hw (q)

f

where f, f is the Petersson norm. Such L-function moments have been studied previously by Duke [16], Duke, Friedlander and Iwaniec [14,15], Iwaniec and Sarnak [20], and Kowalski, Michel and VanderKam [25]. 2.5.1. Fixed t, large conductor asymptotics In Sect. 7.2, we establish the following conjectures as a consequence of the formulae conjectured in [10]. Firstly, for the newforms split into collections of even and odd functions we have Conjecture 2. For level q squarefree and even weight w fixed, as q → ∞, 1. At the symmetry point: t = 0, ⎛ 2k−1
h

L f ( 21 )2k ∼ 21 ⎝22k(2k+1)/2 + (q) f ∈Hw

and

j=1


h − f ∈Hw (q)

⎞ j! ⎠ 1 ( 2 log q)2k(2k−1)/2 A(0, . . . , 0) (2 j)!

L f ( 21 )2k = 0;

510

J. P. Keating, B. E. Odgers

2. Away from the symmetry point: t = 0,
h

2

|L f ( 21 + it)|2k ∼ 21 ( 21 log q)k gU (2k)A(it, . . . , it, −it, . . . , −it)|ζ (1+2it)|k(k−1),

f ∈Hw± (q)

where A2k (z 1 , . . . , z 2k ) =





 1−

p
q 1≤i< j≤2k

×

2 π



π

sin2 θ

0

2k



1 p 1+zi +z j

 1−

eiθ

−1

eiθ 1 +z

p2

j

eiθ

j=1

− e−iθ

 1−

−1

e−iθ 1 +z

p2

− e−iθ

j

dθ.

Then, combining the odd and even newforms together we obtain Conjecture 3. For level q squarefree and even weight w fixed, as q → ∞, 1. At the symmetry point: t = 0, ⎛ 2k−1
h

L f ( 21 )2k ∼ 21 ⎝22k(2k+1)/2 f ∈Hw (q)

j=1

⎞ j! ⎠ 1 ( 2 log q)2k(2k−1)/2 A2k (0, . . . , 0); (2 j)!

2. Away from the symmetry point: t = 0,
h f ∈Hw (q)

|L f ( 21 + it)|2k 2

∼ ( 21 log q)k gU (2k)A2k (it, . . . , it, −it, . . . , −it)|ζ (1 + 2it)|k(k−1) . 2.5.2. Theorem: 2nd Moment The following theorem is proved in Sect. 7.3 with the main term coming from the “diagonal” term (of the Petersson Trace Formula). It confirms Conjecture 3 in the case when k = 1, w = 2 and q is prime. Theorem 7. For q prime, for all real t as q → ∞,
h f ∈H2 (q)

|L f

1 2

 + it |2 ∼ log q.

This theorem sees the 2nd moment not having any t-dependence away from the critical point. This is as expected, since in the orthogonal case the t-dependence is not conjectured until the 4th moment. It should also be noted that when making direct comparisons with the RMT conjec+ (q) which are directly comparable with the moments tures, it is the moments over Hw over S O(2N ).

Symmetry Transitions in Random Matrix Theory & L-functions

511

3. Local Two-point Correlations 3.1. U (N ). Beginning with Eq. (3) and using Poisson summation we can write EU (N ) [R2 (A, φ; x)] ⎡ N N ∞ 1


= EU (N ) ⎣ 2 N

∞


⎤ e

i j2π(φ− x2 −φm )/N

e

ik2π(φ+ x2 −φn )/N

⎦

m=1 n=1 j=−∞ k=−∞

=

1 N2

∞


∞


e−i j (2φ−x)π/N e−ik(2φ+x)π/N EU (N ) [Tr(A j ) Tr(Ak )].

(10)

j=−∞ k=−∞

To evaluate this expectation of traces we call on the following lemma of Dyson. Lemma 1 (Dyson [17]). For j, k ∈ Z,  EU (N ) [Tr(A )] = k

N k = 0; 0 otherwise

and ⎧ 2 ⎪ ⎨N k = 0 j −k EU (N ) [Tr(A ) Tr(A )] = δ( j, k) |k| 1 ≤ |k| ≤ N ⎪ ⎩ N |k| > N .

(This lemma is reproved at the end of this section using results of Hughes and Rudnick [19].) Equation (10) and Lemma 1 then combine to give

EU (N ) [R2 (A, φ; x)] =

1 N2

∞
k=−∞

∞


=

ei2kxπ/N

⎧ 2 ⎪ ⎨N k = 0 |k| 1 ≤ |k| ≤ N ⎪ ⎩ N |k| > N

δ (x − j N ) + 1 −

j=−∞

sin2 (π x) , N 2 sin2 πNx

(11)

which is the result for U (N ) stated in Theorem 1. 3.2. U Sp(2N ) and S O(2N ). Beginning with Eq. (4), Poisson summation again allows us to write R2 (A, φ; x) =

∞
1 (2N )2

∞


j=−∞ k=−∞

e−i j (2φ−x)π/2N e−ik(2φ+x)π/2N Tr(A j ) Tr(Ak ).

512

J. P. Keating, B. E. Odgers

Noting that for U Sp(2N ) and S O(2N ), Tr(U k ) = Tr(U −k ) and Tr(A0 ) = 2N , EG U (N ) [R2 (A, φ; x)] =1+ +e +

k=1 ik(2φ+x)π/2N

+ e−ik(2φ+x)π/2N ]EG U (N ) [Tr(Ak )]

∞ 1
ik(2φ−x)π/2N [e + e−ik(2φ−x)π/2N ] (2N )2

× [e +

∞ 1
ik(2φ−x)π/2N [e + e−ik(2φ−x)π/2N 2N

k=1 ik(2φ+x)π/2N

+ e−ik(2φ+x)π/2N ]EG U (N ) [Tr(Ak )2 ]


1 [ei j (2φ−x)π/2N + e−i j (2φ−x)π/2N ] 2 (2N ) 1≤ j
× [e +

ik(2φ+x)π/2N

+ e−ik(2φ+x)π/2N ]EG U (N ) [Tr(A j ) Tr(Ak )]


1 [ei j (2φ+x)π/2N + e−i j (2φ+x)π/2N ] 2 (2N ) 1≤ j
× [e

ik(2φ−x)π/2N

+ e−ik(2φ−x)π/2N ]EG U (N ) [Tr(A j ) Tr(Ak )].

(12)

We then call on the following lemma detailing the expected value of the traces, based on the work of Hughes and Rudnick [19] and proved at the end of this section. Lemma 2. Letting

 1 for k even ηk = 0 for k odd

and



1 for G U (N ) = U Sp(2N ) 0 for G U (N ) = S O(2N ), ⎧ ⎪ k = 0, ⎨2N k a EG U (N ) [Tr(A )] = (−1) ηk 1 ≤ |k| ≤ 2N − 2 + 2a, ⎪ ⎩0 otherwise, a=

and for 0 < j ≤ k, EG U (N ) [Tr(A j ) Tr(Ak )] = k ( for 1 ≤ j = k ≤ 2N ) + 2N ( for j = k > 2N ) + (−1)a ( for N + a ≤ j = k ≤ 2N − 1 + a) + η j ηk ( for 1 ≤ j ≤ k ≤ 2N − 2 + 2a) + (−1)a ( for †), where † denotes that the pair j, k must be such that we can write j = n − m and k = n + m with 1 ≤ m ≤ N − 1 + a and n ≥ N + a. Applying Lemma 2 to (12) then gives Theorem 2, after some straightforward simplification.

Symmetry Transitions in Random Matrix Theory & L-functions

513

3.3. Proof of Lemmas 1 and 2. We base our proof on the results of Hughes and Rudnick [19]. Let A be an N × N unitary matrix with eigenvalues exp(iθn ), 1 ≤ n ≤ N , and let g be a test function which we assume is 2π -periodic. Consider the linear statistic Tr g(A) :=

N


g(θn ) =

G(N )

gn Tr(An ).

n=−∞

n=1

The cumulants Cl

∞


(g) of Tr g(A), are defined by the expansion ∞


log EG(N ) [et Tr g(A) ] =

l=1

tl ClG(N ) (g) , l!

where the expectation is taken over G(N ) = U (N ), U Sp(2N ) or S O(2N ). Differentiating this expression twice gives the relations G(N )

C1

(g) = EG(N ) [Tr g(A)] =

∞


gn EG(N ) [Tr(An )],

(13)

n=−∞

and G(N )

C2

G(N )

(g) + C1

(g)2 = EG(N ) [Tr g(A)2 ] ∞


=

gm gn EG(N ) [Tr(Am ) Tr(An )].

(14)

m,n=−∞ U (N )

be the l th cumulant of Theorem 8 (Soshnikov, Theorem 7 from [19]). Let Cl Tr g(A), averaged over all N × N unitary matrices with Haar measure. Then U (N ) C1

= N g0 and

U (N ) C2

=

∞


min(|n|, N )gn g−n .

n=−∞ n=0

Substituting the values from Theorem 8 into Eqs. (13) and (14) and equating coefficients gives Lemma 1. For U Sp(2N ) and S O(2N ) we restrict our attention to g even (that is gn = g−n ) since the corresponding matrix eigenvalues occur in conjugate pairs, ensuring Tr(Ak ) is even. Further since Tr(A0 ) = 2N trivially, we take g0 = 0. This sees (13) and (14) become G (N )

C 1 U

(g) = 2

∞


gn EG U (N ) [Tr(An )],

(15)

n=1

and G (N )

C 2 U

G (N )

(g) + C1 U

(g)2 = 4

∞


gm gn EG U (N ) [Tr(Am ) Tr(An )].

m,n=1

Theorems 4 and 5 from [19] are then combined using our notation to be

(16)

514

J. P. Keating, B. E. Odgers

Theorem 9 (Hughes & Rudnick). G (N )

C 1 U

N
−1+a

(g) = 2N g0 + (−1)a2

g2n ,

n=1 G (N )

C 2 U

(g) = 4

∞


∞


min(n, 2N − 1 + 2a)gn2 + (−1)a4

gn2

n=N +a

n=1

+ (−1)a8

N
−1+a

∞


gn+m gn−m .

m=1 n=N +a

Substituting the values from Theorem 9 into (15) and (16) and recalling that g0 = 0 gives (−1)a

N
−1+a

g2n =

n=1

∞


gn EG U (N ) [Tr(An )],

n=1

and ∞


gn2 + (−1)a2

N
−1+a

n=N +a

n=1

+

∞


min(n, 2N − 1 + 2a)gn2 + (−1)a −1+a N
−1+a N
m=1

g2m g2n =

n=1

∞


∞


gn+m gn−m

m=1 n=N +a

gm gn EG U (N ) [Tr(Am ) Tr(An )].

m,n=1

Equating coefficients then gives Lemma 2. 4. Log-moments Consider the expectation EG(N ) [log Λ A (e )] = EG(N ) iθ

 N


 log(1 − e

i(θ−θn )

) .

n=1

The Mercator Series expansion for the logarithm can then be used to give ⎡ ⎤ N
∞ i jθ ∞ i j (θ−θn )  

e e ⎦=− EG(N ) [log Λ A (eiθ )] = −EG(N ) ⎣ EG(N ) Tr(A− j ) . j j n=1 j=1

j=1

(Since we are taking a continuous expectation, the finite number of points which will cause problems for convergence of the Mercator series can be ignored.) Lemmas 1 and 2 give the expectations for the trace above, and so we have EU (N ) [log Λ A (eiθ )] = 0 and EG U (N ) [log Λ A (eiθ )] = (−1)1+a

N
−1+a i2 jθ e j=1

as required.

2j

,

Symmetry Transitions in Random Matrix Theory & L-functions

515

5. Moments 5.1. 2nd moments. Combining Eqs. (3.6) and (4.19) from [9], Conrey, Farmer, Keating, Rubinstein and Snaith give the following permutation sum for the autocorrelation of a characteristic polynomial:  G U (N )

Λ A (es1 )Λ A (es2 )d A


= e N (s1 +s2 )

e N (ε1 s1 +ε2 s2 )

ε j ∈{−1,1}

(1 − e−(ε1 s1 +ε2 s2 ) )(1 − e−2ε1 s1 )a(1 − e−2ε2 s2 )a

.

(17)

Setting s1 = iθ and s2 = −iθ gives the 2nd moment. This leads to poles in the individual terms of the permutation sum, but these cancel on summation. (Poles also arise in the above expression if θ = 0; however since an expression for the moments at the symmetry point is already known [23], this will not concern us.) Noting that εaj ea(ε j −1)s j = , (1 − e−2ε j s j )a (1 − e−2s j )a 1

rearrangement of (17) gives  G U (N )

 ×

Λ A (es1 )Λ A (es2 )d A =

e(N −a)(s1 +s2 ) (1 − e−2s1 )a(1 − e−2s2 )a

e(N +a)(s1 +s2 ) − e−(N +1+a)(s1 +s2 ) (1 − e−(s1 +s2 ) )

+(−1)a

 e(N +a)(s1 −s2 ) − e−(N +1+a)(s1 −s2 ) , (1 − e−(s1 −s2 ) )

which can be rewritten as  G U (N )

×

Λ A (es1 )Λ A (es2 )d A =

e(N −a)(s1 +s2 ) (1 − e−2s1 )a(1 − e−2s2 )a

2N
+1+2a 

 e(N +a− j)(s1 +s2 ) + (−1)ae(N +a− j)(s1 −s2 ) .

j=0

Letting s1 = iθ and s2 = −iθ then gives Theorem 4 which, in the limit as θ → 0, matches the results obtained in [23] at the symmetry point.

5.2. 2k th moments. We now extend our analysis from the 2nd moments to look at the fixed θ , large N asymptotics of all 2k th moments.

516

J. P. Keating, B. E. Odgers

Generalizing (17), [9] gives the following autocorrelation for 2k characteristic polynomials: ⎛ ⎞  2k 2k




⎝ Λ A (es1 ) . . . Λ A (es2k )d A = eNs j eε j N s j ⎠ G U (N )

×

ε j ∈{−1,1}

j=1



(1 − e−(εi si +ε j s j ) )−1

1≤i< j≤2k

2k

j=1

(1 − e−2ε j s j )−a.

(18)

j=1

An expression for the 2k th moment is then obtained by setting s j = iθ for 1 ≤ j ≤ k, and s j = −iθ for k + 1 ≤ j ≤ 2k. The large N asymptotic behaviour of such a moment can be found using the following lemma. (This lemma is more general than is required for finding the asymptotic behaviour of the 2k th moments of the characteristic polynomial, but will be called on again later when we deal with the moments of L-function families.) Lemma 3. Suppose F is a symmetric function of 2k variables, regular near (0, . . . , 0), and that f (s) has a simple pole of residue 1 at s = 0 and is otherwise analytic in a neighbourhood of s = 0. For a, b ∈ {0, 1}, let Ha,b be the permutation sum ⎛ ⎞ 2k 2k


⎝ Ha,b(X ; ω1 , . . . , ω2k ) :=  bj ⎠ e X j=1  j ω j F(1 ω1 , . . . , 2k ω2k )  j =±1

j=1



×

f (i ωi +  j ω j )

2k

1≤i< j≤2k

f (2 j ω j )a.

j=1

Noting that Ha,b is symmetric with respect to the ω j ’s, we define Ia,b(X, k; iβ) to be the value of Ha,b when ω1 , . . . ωk = iβ, and ωk+1 , . . . ω2k = −iβ. We then have that: 1. For β = 0 and b = 0,

⎛

Ia,0 (X, k; 0) ∼ X 2k(2k−1+2a)/2 F(0, . . . , 0) ⎝22k(2k+1)/2

2k−1+

a j=1

⎞ j! ⎠ , (2 j)!

while if b = 1, Ia,1 (X, k; 0) = 0. 2. For fixed real β = 0, 2

Ia,b(X, k; iβ) ∼ X k gU (2k)F(iβ, . . . , iβ, −iβ, . . . , −iβ) × f (2iβ)k(k−1+2a)/2 f (−2iβ)k(k−1+2a)/2 .

Here gU (2k) denotes the coefficient of the large N asymptotic of the 2k th moment over U (2N ), namely gU (2k) := lim N −k N →∞

2

 U (2N )

|Λ A (e0 )|2k d A = 2k

2

k−1

j=0

j! . (k + j)!

Symmetry Transitions in Random Matrix Theory & L-functions

517

Setting – a = 1 for U Sp(2N ) or 0 for S O(2N ), – b = 0, – F(s1 , . . . , s2k ) = 1, and – f (s) = (1 − e−s )−1 , we can apply this lemma to (18), giving Theorem 5. 5.3. Proof of Lemma 3. We begin the proof by making the following definition related to Ha,b: G a(X ; ω1 , . . . , ω2k ) := e X

2k

j=1 ω j



F(ω1 , . . . , ω2k )

f (ωi + ω j )

1≤i< j≤2k

2k

f (2ω j )a.

j=1

(19) Translated into the notation of Lemma 3 and Eq. (19), Lemma 2.5.2 from [10] states that Lemma 4 (Conrey-Farmer-Keating-Rubinstein-Snaith). If 2iβ and −2iβ are contained in the region of analyticity of f (s), then (−1)2k(2k−1)/2 22k . . . G a(X ; z 1 , . . . , z 2k ) (2πi)2k (2k)! !2k 1−b b 2 )2 ∆(z 12 , . . . , z 2k j=1 z j β × !2k dz 1 . . . dz 2k , 2k 2k j=1 (z j − iβ) (z j + iβ)

Ia,b(X, k; iβ) =

where the path of integration encloses iβ and −iβ. We analyze this integral separately for β = 0 and β = 0. Letting β → 0, Lemma 4 gives Ia,b(X, k; 0) =

(−1)2k(2k−1)/2 22k . . . G a(X ; z 1 , . . . , z 2k ) (2πi)2k (2k)! !2k 1−b b 2 )2 ∆(z 12 , . . . , z 2k j=1 z j 0 × dz 1 . . . dz 2k ; !2k 4k j=1 z j

so clearly Ia,1 (X, k; 0) = 0. For b = 0, expanding G a and letting z j = v j / X gives Ia,0 (X, k; 0) = X 2k(2k−1+2a)/2 22k(1−a) × ×

(−1)2k(2k−1)/2 1 (2πi)2k (2k)!

"

...

e

2k

j=1 v j

f ((vi + v j )/ X )(vi + v j )/ X

1≤i< j≤2k

×



1≤i< j≤2k

F(v1 / X, . . . , v2k / X )

2k " #

f (2v j / X )2v j / X

j=1

(vi + v j )(vi − v j )2 !2k

1

4k−1+a j=1 v j

dv1 . . . dv2k .

#a

518

J. P. Keating, B. E. Odgers

As X → ∞, we have Ia,0 (X, k; 0) ∼ X 2k(2k−1+2a)/2 22k(1−a) F(0, . . . , 0) 2k (−1)2k(2k−1)/2 1 j=1 v j . . . e (2πi)2k (2k)! 2 )2

1 ∆(v12 , . . . , v2k × !2k 4k−1+a dv1 . . . dv2k . vi + v j j=1 v j

×

1≤i< j≤2k

Equations (3.36) and (4.43) of [9] give the following multiple contour integral form of the autocorrelation function  Λ A (es1 ) . . . Λ A (esk )d A G U (N )

=

(−1)k(k−1)/2 2k (1−2a)N kj=1 s j e (2πi)k k!

×

...



(1 − e−z m −zl )−1

1≤l<m≤k

! k ∆(z 12 , . . . , z k2 )2 kj=1 z j −2z j −a e N j=1 z j dz 1 . . . dz k . (1 − e ) !k !k i=1 j=1 (z j + si )(z j − si ) j=1

k

Now recalling the multiple contour integral in Lemma 4; the results from [23], given in (7), for the moments at the symmetry point, combined with the above equation, imply that for F(0, . . . , 0) = 1, ⎛ ⎞  2k−1+

a j! ⎠. Ia,0 (N , k; 0) = |Λ A (e0 )|2k d A ∼ N 2k(2k−1+2a)/2 ⎝22k(2k+1)/2 (2 j)! G U (N ) j=1

Hence, noting that the integral in our expression for the large X behaviour of Ia,0 (X, k; 0) is independent of F, f and X , we conclude it must have the same form as above. That is ⎛ ⎞ 2k−1+

a j! ⎠, Ia,0 (X, k; 0) ∼ X 2k(2k−1+2a)/2 F(0, . . . , 0) ⎝22k(2k+1)/2 (2 j)! j=1

thus completing the proof of the β = 0 case. For β = 0, in the integral for Ia,b(X, k; iβ) each contour can be continuously deformed to two small circular contours centred respectively at the poles ±iβ, connected by two straight line paths (whose contributions will cancel). We can therefore consider the multiple contour integral as a sum of 22k integrals in which each z j runs over one of the smaller circular paths. (There are 2k variables, each following one of 2 possible paths; hence the sum of 22k integrals.) With  j = ±1, let Γ j iβ be a circle with centre  j iβ, and radius less than |β|. We then let Ja,b(X, k; iβ; Γ1 iβ , . . . , Γ2k iβ ) be the value of the above multiple integral, but with the z j contour changed to Γ j iβ . Hence Ja,b(X, k; iβ; Γ1 iβ , . . . , Γ2k iβ ) (−1)2k(2k−1)/2 22k ... G a(X ; z 1 , . . . , z 2k ) (2πi)2k (2k)! Γ1 iβ Γ2k iβ !2k 1−b b 2 )2 ∆(z 12 , . . . , z 2k j=1 z j β × !2k dz 1 . . . dz 2k , 2k 2k j=1 (z j − iβ) (z j + iβ)

=

Symmetry Transitions in Random Matrix Theory & L-functions

519

and


Ia,b(X, k; iβ) =

Ja,b(X, k; iβ; Γ1 iβ , . . . , Γ2k iβ ).

 j =±1

Focusing our attention on Ja,b(X, k; iβ; Γ1 iβ , . . . , Γ2k iβ ), we now change variables to v j = X (z j −  j iβ). This gives Ja,b(X, k; iβ; Γ1 iβ , . . . , Γ2k iβ ) = X 2k(2k−1) ×

Γ0

...

(−1)2k(2k−1)/2 22k (2πi)2k (2k)! Γ0

G a(X ; v1 / X + 1 iβ, . . . , v2k / X + 2k iβ)

× ∆((v1 / X + 1 iβ)2 , . . . , (v2k / X + 2k iβ)2 )2 !2k 1−bβ b j=1 (v j / X +  j iβ) × !2k 2k dv1 . . . dv2k , 2k j=1 v j (v j / X + 2 j iβ) where Γ0 is a circle centred at the origin with radius less than |β|. Hence, expanding G a and ∆, Ja,b(X, k; iβ; Γ1 iβ , . . . , Γ2k iβ ) 2k

(−1)2k(2k−1)/2 22k (2πi)2k (2k)! × F(v1 / X + 1 iβ, . . . , v2k / X + 2k iβ)

f ((v j + vi )/ X + ( j + i )iβ) ×

= eX

j=1  j iβ

X 2k(2k−1)

Γ0

...

Γ0

e

2k

j=1 v j

1≤i< j≤2k

× ((v j + vi )/ X + ( j + i )iβ)2 ((v j − vi )/ X + ( j − i )iβ)2 !2k 2k 1−bβ b

j=1 (v j / X +  j iβ) a × f (2v j / X + 2 j iβ) !2k 2k dv1 . . . dv2k . 2k j=1 v j (v j / X + 2 j iβ) j=1 As X → ∞, with β fixed away from 0, we therefore have Ja,b(X, k; iβ; Γ1 iβ , . . . , Γ2k iβ ) ∼ (−1)k be X

2k

j=1  j iβ

X 2k(2k−1) F(1 iβ, . . . , 2k iβ)

2k

b  1− j

j=1

×

1≤i< j≤2k  j =i

×



1≤i< j≤2k  j =i

f (2 j iβ)

2k

f (2 j iβ)a

j=1

(v j − vi )2 / X 2

1≤i< j≤2k  j =−i

1 1 (2πi)2k (2k)!

Γ0

(v j + vi )/ X !2k

...

1

2k j=1 v j

Γ0

e

2k

j=1 v j

dv1 . . . dv2k .

520

J. P. Keating, B. E. Odgers

The dominant contribution arises when we maximize the number of pairs {i ,  j } with    j = −i , which will occur when exactly k of the  j ’s are 1. There are 2k k ways of achieving this, but the symmetry of the expression means that all such alternatives are identical. Hence   2k 2 Ia,b(X, k; iβ) ∼ (−1)k X k F(iβ, . . . , iβ, −iβ, . . . , −iβ) k × f (2iβ)k(k−1+2a)/2 f (−2iβ)k(k−1+2a)/2 2k 1 1 j=1 v j × . . . e (2πi)2k (2k)! Γ0 Γ0



× (v j − vi )2 (v j − vi )2 1≤i< j≤k

×

k

2k

k+1≤i< j≤2k

(v j + vi ) !2k

1

2k j=1 v j

i=1 j=k+1

dv1 . . . dv2k .

Making the change of variables v j → −v j for j = k + 1, . . . , 2k and rearranging we obtain Ia,b(X, k; iβ) ∼ X k F(iβ, . . . , iβ, −iβ, . . . , −iβ) f (2iβ)k(k−1+2a)/2 f (−2iβ)k(k−1+2a)/2 k 1 1 j=1 v j −vk+ j × 2 . . . e k! (2πi)2k Γ0 Γ0 2

×

k

k

i=1 j=k

1 ∆(v1 , . . . , v2k )2 dv1 . . . dv2k . !2k 2k vk+ j − vi j=1 v j

We next use the following lemma. Lemma 5. 1 1 2 k! (2πi)2k ×

k

k

i=1 j=1

...

e

k

j=1 v j −vk+ j

1 ∆(v1 , . . . , v2k )2 dv1 . . . dv2k = gU (2k), !2k 2k vk+ j − vi j=1 v j

where gU (2k) := lim N −k

2



N →∞

U (2N )

|Λ A (e0 )|2k d A = 2k

2

k−1

j=0

j! . (k + j)!

This allows us to write 2

Ia,b(X, k; iβ) ∼ X k gU (2k)F(iβ, . . . , iβ, −iβ, . . . , −iβ) × f (2iβ)k(k−1+2a)/2 f (−2iβ)k(k−1+2a)/2 ,

as required.

Symmetry Transitions in Random Matrix Theory & L-functions

521

5.4. Proof of Lemma 5. Let 1

Z A (s) = (e−iφ )− 2 s −N Λ A (s), where eiφ = det A and Λ A (s) is the characteristic polynomial of A. Equation 1.2.7 of [10] tells us that Z A (s) has functional equation Z A (s) = Z A (1/s). Hence in the notation of Eq. (1.4.3) from [10], we have 

 |Λ A (e )| d A = 0 2k

U (2N )

Z A (e ) Z A 0 k

U (2N )

(e0 )

k

 dA =

U (2N )

Z A (e0 )2k d A

=Jk (U (2N ), (0, . . . , 0)). Equation (1.5.9) of [10] gives Jk (U (2N ), (0, . . . , 0)) =

1 (−1)k k!2 (2πi)2k ×

k

k

eN

...

(1 − e z k+ j −zi )−1

i=1 j=1

k j=1

z j −z k+ j

∆(z 1 , . . . , z 2k )2 dz 1 . . . dz 2k . !2k 2k j=1 z j

Making the transformation z j = v j /N , we have Jk (U (2N ), (0, . . . , 0)) =

1 (−1)k 2 k! (2πi)2k ×

k

k

...

e

k

(1 − e(vk+ j−vi )/N )−1

i=1 j=1

For large N , (1 − e(vk+ j −vi )/N )−1 ∼ asymptotic form: Jk (U (2N ), (0, . . . , 0)) ∼ N k ×

2

j=1 v j −vk+ j

−N vk+ j −vi

, and so the integral has the following

1 1 2 k! (2πi)2k

k

k

i=1 j=1

∆(v1 , . . . , v2k )2 dv1 . . . dv2k . !2k 2k j=1 v j

...

e

k

j=1 v j −vk+ j

1 ∆(v1 , . . . , v2k )2 dv1 . . . dv2k . !2k 2k vk+ j − vi j=1 v j

2

Dividing by N k and taking the limit as N → ∞ gives the lemma. 6. A Family of L-functions with Symplectic Symmetry For the family of L-functions with symplectic symmetry described in Sect. 2.4, we now write down the moment conjecture according to the recipe detailed in [10], use Lemma 3 to determine the large conductor asymptotics of these moments, and then verify the resulting conjecture rigorously for the 2nd moment.

522

J. P. Keating, B. E. Odgers

6.1. Conjecture for the integral moments. Recalling the functional equation given in (8), let 1 1 Z d (s, χd ) = (|d| 2 −s X (s))− 2 L(s, χd ). Further recalling the definition of R2k,8 ( 21 , α1 , . . . , α2k ) given in (9); for g(d) a suitable weight function, Eq. (4.4.22) of [10] gives Conjecture 4 (Conrey-Farmer-Keating-Rubinstein-Snaith). For |(α j )| < 21 ,
∗

2k
∗


Z d ( 21 + α1 , χd ) . . . Z d ( 21 + α2k , χd )g(|d|) =

d<0 d≡0(8) 1

×R2k,8 ( 21 , ε1 α1 , . . . , ε2k α2k )|d| 2

d<0 εi =±1 j=1 d≡0(8)

2k

j=1 ε j α j

1

X ( 21 + ε j α j )− 2

1

(1 + O(|d|− 2 +ε ))g(|d|).

If we let α1 = . . . αk = it and αk+1 = . . . α2k = −it, re-labelling the εi ’s gives the following expression for the 2k th moment:
∗

|L( 21 + it, χd )|2k g(|d|) =

2k
∗


1

d<0 εi =±1 j=1 d≡0(8) 1 2k j=1 ε j it 2

d<0 d≡0(8)

X ( 21 + ε j it)− 2 1

(1 + O(|d|− 2 +ε ))g(|d|).

×R2k,8 ( 21 , ε1 it, . . . , ε2k it)|d|

(20)

6.2. Fixed t, large conductor asymptotics. Consider the inner permutation sum S=

2k


εi =±1 j=1

=


εi =±1

×

1

1

X ( 21 + ε j it)− 2 R2k,8 ( 21 , ε1 it, . . . , ε2k it)|d| 2

1

2k

2k

 1−

j=1



e( 2 ln |d|)

j=1 ε j it

1 p 1+εi it+ε j it

1≤i≤ j≤2k

1

X ( 21 + ε j it)− 2 



2k

j=1 ε j it

R2k,8, p ( 21 , ε1 it, . . . , ε2k it)

p



ζ (1 + εi it + ε j it).

1≤i≤ j≤2k

We use Lemma 3 to obtain the large |d| asymptotics for S. In the notation of Lemma 3, if we let a = 1, X = 21 ln |d|, f (s) = ζ (1 + s) and F(s1 , . . . , s2k ) =

2k

1

X ( 21 + s j )− 2

p

j=1

then as |d| → ∞: when t = 0,



R2k,8, p ( 21 , s1 , . . . , s2k )

p

1−

1≤i≤ j≤2k

⎛

S ∼ ( 21 ln |d|)2k(2k+1)/2 ⎝22k(2k+1)/2 ×



 R2k,8, p ( 21 , 0, . . . , 0) 1 −

2k

j=1

1 p

⎞ j! ⎠ (2 j)!

2k(2k+1)/2 ;

1 p 1+si +s j

 ;

Symmetry Transitions in Random Matrix Theory & L-functions

523

and when t = 0, S∼

k 2 k k ln |d| gU (2k)X ( 21 + it)− 2 X ( 21 − it)− 2

× R2k,8, p ( 21 , it, . . . , it, −it, . . . , −it)

1 2

p

 × 1−

k(k+1)/2  1− 1+2it 1

p

k(k+1)/2  2 1 k 1− |ζ (1 + 2it)|k(k+1) . p 1−2it p 1

Substituting this equation into (20) we obtain Conjecture 1.

6.3. Theorem: 2nd moment. We now give an example for which Conjecture 1 holds: when k = 1, in a small t-range on the critical line. The tools for such a calculation were put together by Soundararajan in [31] and [32], and were expanded upon by Conrey and Soundararajan in [13] where a mollified 2nd moment is evaluated for this family. 1 we In the notation of [13], for Ψ a smooth function supported in (1, 2) and |t| < 100 have 
∗ 1 2 Ψ |d| |L( + it, χ )| d d<0 2 D S(Ait,0 (e); Ψ ) d≡0(8)  . (21) =
∗ |d| S(1; Ψ ) d<0 Ψ D d≡0(8)

Let ηw (s) =



η p;w (s);

p

where η2;w (s) = (1 − 2−s−2w )(1 − 2−s )(1 − 2−s+2w ), and for p ≥ 3,  η p;w (s) =

p p+1

Next let Ψˇ (s) =



   1 1 1 p 2w + p −2w 1 1+ + s − . 1− s + p p p p s+1 p 2s+1

2 1

 Ψ (t)t s dt and Ψ(ν) = max

0≤ j≤ν 1

2

|Ψ ( j) (t)|dt.

Splitting the numerator in (21) into main and remainder components, S(Aδ,τ (e); Ψ ) = S M (Aδ,τ (e); Ψ ) + O(S R (Aδ,τ (e); Ψ )), with the value of the main term given in the following proposition.

524

J. P. Keating, B. E. Odgers

Proposition 1 (Proposition 2.3 from [13]). With notation as above, and |δ|, |τ | <

1 100 ,

   8D µτ 2
ˇ S M (Aδ,τ (e); Ψ ) ∼ Γδ (µτ )Ψ (µτ )ηδ (1 + 2µτ ) 3ζ (2) µ=± π × ζ (1 + 2µτ − 2δ)ζ (1 + 2µτ )ζ (1 + 2µτ + 2δ)   8D µδ + Γτ (µδ)Ψˇ (µδ)ητ (1 + 2µδ) π  × ζ (1 + 2µδ − 2τ )ζ (1 + 2µδ)ζ (1 + 2µδ + 2τ ) .

The remainder term can then be dealt with by removing the mollification from the proof of Proposition 2.2 from [13]. Proposition 2 (Proposition 2.2 from [13] (with mollification removed)). Let Ψ be a smooth function supported on [1, 2], with Ψ (t)  1. With notation as above S R (Aδ,τ (e); Ψ ) 

D 1/100+ . Y

We are now left to deal with the denominator, which upon use of Theorem 3.1.23 of [10] is found to be 2 Ψˇ (0) . S (1; Ψ ) ∼ 3ζ (2) Combining these results into (21) gives an expression for a shifted moment. If we further δ1 +δ2 2 let δ = δ1 −δ 2 and τ = 2 , this expression can be written as a permutation sum, leading to the following theorem. Theorem 10. For Ψ a sufficiently smooth function supported in [1, 2], |δ1 | < 1 |δ2 | < 100 , and Ψˇ and η as defined earlier, as D → ∞,
∗

∼

d<0 e d≡0(8)

2

(δ1 +δ2 )( 21 log

Γ ( 43 +

|d| π )


∗

L( 21 + δ1 , χd )L( 21 + δ2 , χd )Ψ ( |d| D)

|d| d<0 Ψ ( D ) d≡0(8)

δ j −1 ˇ −1 2 ) Ψ (0)

j=1

×Ψˇ ( 21 (1 δ1




1

e(1 δ1 +2 δ2 )( 2 log

8D π )

 j =±1

+ 2 δ2 ))η 1 (1 δ1 −2 δ2 ) (1 + 1 δ1 + 2 δ2 )

×ζ (1 + 1 δ1 + 2 δ2 )

2

2

j=1

ζ (1 + 2 j δ j ).

2

j=1

Γ ( 43 + 21  j δ j )

1 100 ,

Symmetry Transitions in Random Matrix Theory & L-functions

525

We are now left to let δ1 = −δ2 = it and determine the large D behaviour of this expression. This can be done by calling on Lemma 3. Letting a = 1, X = ( 21 log 8D π ), f (s) = ζ (1 + s) and F(s1 , s2 ) =

2

Γ ( 43 + 21 s j )Ψˇ ( 21 (s1 + s2 ))η 1 (s1 −s2 ) (1 + s1 + s2 ), 2

j=1

we obtain Theorem 6. 7. A Family of L-functions with Orthogonal Symmetry 7.1. Integral Moments Conjecture. Recalling the definition of the family given in Sect. 2.5, we wish to obtain a conjecture for the large q asymptotics of
h |L f ( 21 + it)|2k . ± f ∈Hw (q)

Conjecture 4.5.2 of [10] gives the corresponding shifted moments: Conjecture 5 (Conrey-Farmer-Keating-Rubinstein-Snaith). For level q squarefree and even weight w, for |(α j )| < 21 ,
h L f ( 21 + α1 ) . . . L f ( 21 + α2k ) + (q) f ∈Hw

2k 2k 1


1 1

Xw,q ( 21 − α j )− 2 Xw,q ( 21 +  j α j )− 2 A(1 α1 , . . . , 2k α2k ) 2  j =±1 j=1 j=1

1 × ζ (1 + i αi +  j α j )(1 + O(wq)− 2 + )

=

1≤i< j≤2k

and
h − f ∈Hw (q)

L f ( 21 + α1 ) . . . L f ( 21 + α2k )

⎛ ⎞ 2k 2k 2k



1
1 1

⎝ = Xw,q ( 21 −α j )− 2  j⎠ Xw,q ( 21 +  j α j )− 2 A(1 α1 , . . . , 2k α2k ) 2  j =±1 j=1 j=1 j=1

1 × ζ (1 + i αi +  j α j )(1 + O(wq)− 2 + ), 1≤i< j≤2k

where A2k (z 1 , . . . , z 2k ) =





 1−

p
q 1≤i< j≤2k

×

2 π



π 0

sin2 θ

2k

j=1

1



p 1+zi +z j eiθ

 1−

−1

eiθ 1 +z

p2

j

eiθ

− e−iθ

 1−

− e−iθ

The asymptotics for these expressions are now dealt with individually.

−1

e−iθ 1 +z

p2

j

dθ.

526

J. P. Keating, B. E. Odgers

7.2. Fixed t, large conductor asymptotics. Observing that Xw,q ( 21

+ s)

− 21

=e

( 12 log q)s

(2π )

−s



Γ ( w2 − s) Γ ( w2 + s)

using Lemma 3 and letting a = 0, f (s) = ζ (1 + s), X = F(s1 , . . . , s2k ) =

2k

(2π )

−s j

j=1



Γ ( w2 − s j ) Γ ( w2 + s j )

1 2

− 21

− 21

,

log q, and A(s1 , . . . , s2k ),

we obtain Conjecture 2 and further Conjecture 3. 7.3. Theorem: 2nd moment. It can easily be shown that A2 (it, −it) = 1 for all t. When k = 1, we therefore conjecture that as q → ∞, for all t,
h |L f ( 21 + it)|2 ∼ log q. f ∈Hw (q)

We now prove this conjecture in the case where q is prime and w = 2. Let G(s) = G t (s) be an even function such that G(0) = 1 and G(s)Γ (1+s +it)Γ (1+ s − it) is holomorphic for (s) ≥ −A for some large fixed constant A. The functional equation, Stirling’s approximation and a contour shift then imply the “approximate functional equation” |L( f,

1 2

+ it)| = 2|Γ (1 + it)| 2

−2

  2 ∞
λ f (m)λ f (n)( mn )it 4π mn , W √ q mn

m,n=1

where W (y) =

1 2πi

 (c)

G(s)Γ (1 + s + it)Γ (1 + s − it)y −s

ds . s

To evaluate the sum over f , we introduce the following lemma based on the Petersson Trace Formula. Lemma 6 (Lemma 2.2 from [25]). For q prime and w = 2 we have 
h 1 1 3 λ f (m)λ f (n) = δ(m, n) + O (m, n, q) 2 (mn) 2 q − 2 . f ∈H2 (q)

The diagonal term then gives the main contribution to the moment, and we write
h |L( f, 21 + it)|2 = M(t) + Ot (R(t)), f ∈H2 (q)

where M(t) = 2|Γ (1 + it)|−2

 2 2 ∞
4π n 1 W , n q n=1

Symmetry Transitions in Random Matrix Theory & L-functions

and R(t) = q

∞


− 32

m,n=1

(m, n, q)

1 2

 m it n

527

 W

4π 2 mn q

 .

From the behaviour of the integral for W (y) at (A) and (−A), W (y) satisfies the bounds  |Γ (1 + it)|2 + O A (y A ) for 0 < y < 1 W (y) = for y > 1. O A (y −A ) This leads to M(t) = 2


√ n< q/2π

1 + O A,t (1), and R(t) = O A,t (q −1/2 ), n

and so as q → ∞ we therefore have
h |L( f, 21 + it)|2 ∼ log q, f ∈H2 (q)

as required. References 1. Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions. New York: Dover, 1965 2. Berry, M.V., Keating, J.P.: The Riemann zeros and Eigenvalue Asymptotics. SIAM Rev. 41, 236–266 (1999) 3. Bogomolny, E.B., Keating, J.P.: Random matrix theory and the Riemann zeros I: three- and four-point correlations. Nonlinearity 8, 1115–1131 (1995) 4. Bogomolny, E.B., Keating, J.P.: Random matrix theory and the Riemann zeros II: n-point correlations. Nonlinearity 9, 911–935 (1996) 5. Bogomolny, E.B., Keating, J.P.: Gutzwiller’s trace formula and spectral statistics: beyond the diagonal approximation. Phys. Rev. Lett. 77, 1472–1475 (1996) 6. Bui, H.M., Keating, J.P.: On the mean values of Dirichlet L-functions. Proc. Lond. Math. Soc. 95, 273–298 (2007) 7. Bui, H.M., Keating, J.P.: On the mean values of L-functions in orthogonal and symplectic families. Proc. Lond. Math. Soc. 96(2), 335–366 (2008) 8. Conrey, J.B., Farmer, D.W.: Mean values of L-functions and symmetry. Int. Math. Res. Notices 17, 883–908 (2000) 9. Conrey, J.B., Farmer, D.W., Keating, J.P., Rubinstein, M.O., Snaith, N.C.: Autocorrelation of random matrix polynomials. Commun. Math. Phys. 237, 365–395 (2003) 10. Conrey, J.B., Farmer, D.W., Keating, J.P., Rubinstein, M.O., Snaith, N.C.: Integral moments of L-functions. P. Lond. Math. Soc. 91, 33–104 (2005) 11. Conrey, J.B., Farmer, D.W., Keating, J.P., Rubinstein, M.O., Snaith, N.C.: Lower order terms in the full moment conjecture for the Riemann zeta function. J. Number. Th. (2007), doi:10.1016/j.int.2007.05.013 12. Conrey, J.B., Snaith, N.C.: Applications of the L-functions ratios conjectures. Proc. Lond. Math. Soc. 94(3), 594–646 (2007) 13. Conrey, J.B., Soundararajan, K.: Real zeros of quadratic Dirichlet L-functions. Invent. Math. 150, 1–44 (2002) 14. Duke, W., Friedlander, J.B., Iwaniec, H.: Bounds for automorphic L-functions. Invent. Math. 112, 1–8 (1993) 15. Duke, W., Friedlander, J.B., Iwaniec, H.: Bounds for automorphic L-functions II. Invent. Math. 115, 219–239 (1994) 16. Duke, W.: The critical order of vanishing of automorphic L-functions with large level. Invent. Math. 119, 165–174 (1995) 17. Dyson, F.J.: Statistical theory of the energy levels of complex systems, I, II and III. J. Math. Phys. 3, 140–175 (1962)

528

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18. Gonek, S.M., Hughes, C.P., Keating, J.P.: A hybrid Euler-Hadamard product for the Riemann zeta function. Duke Math. J. 136, 507–549 (2007) 19. Hughes, C.P., Rudnick, Z.: Mock-Gaussian behaviour for linear statistics of classical compact groups. J. Phys. A: Math. Gen. 36, 2919–2932 (2003) 20. Iwaniec, H., Sarnak, P.: The non-vanishing of central values of automorphic L-functions and SiegelLandau zeros. Israel J. Math. 120, 155–177 (2000) 21. Katz, N.M., Sarnak, P.: Random matrices, Frobenius eigenvalues, and monodromy: colloq. Pub. 45. Providence, RI: Amer. Math. Soc., 1999 22. Keating, J.P., Snaith, N.C.: Random matrix theory and ζ (1/2 + it). Commun. Math. Phys. 214, 57–89 (2000) 23. Keating, J.P., Snaith, N.C.: Random matrix theory and L-functions at s = 1/2. Commun. Math. Phys. 214, 91–110 (2000) 24. Keating, J.P., Snaith, N.C.: Random matrices and L-functions. J. Phys. A 36, 2859–2881 (2003) 25. Kowalski, E., Michel, P., VanderKam, J.: Mollification of the fourth moment of automorphic L-functions and arithmetic applications. Invent. Math. 142, 95–151 (2000) 26. Mezzadri, F., Snaith, N.C.: Recent perspectives in random matrix theory and number theory: London Mathematic Society Lecture Note Series 322, London: London Math. Soc., 2005 27. Montgomery, H.L.: The pair correlation of zeros of the Riemann zeta-function. Proc. Symp. Pure Math. 24, 181–193 (1973) 28. Odlyzko, A.M.: The 1020 th zero of the Riemann zeta function and 70 million of its neighbours. Preprint, 1989 29. Rubinstein, M.O.: PhD Thesis: Evidence for a spectral interpretation of the zeros of L-functions. Princeton University, June 1998 30. Rudnick, Z., Sarnak, P.: Zeros of principal L-functions and random matrix theory. Duke Math. J. 81, 269–322 (1996) 31. Soundararajan, K.: PhD Thesis: Quadratic twists of Dirichlet L-functions,. Princeton University, 1998 32. Soundararajan, K.: Nonvanishing of quadratic Dirichlet L-functions at s = 0.5. Ann. Math. 152, 447–488 (2000) Communicated by P. Sarnak

Commun. Math. Phys. 281, 529–571 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0447-z

Communications in

Mathematical Physics

Generalized CCR Flows
Masaki Izumi1 , R. Srinivasan2 1 Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto, Japan.

E-mail: [email protected]

2 Chennai Mathematical Institute, Siruseri 603103, India. E-mail: [email protected]

Received: 24 May 2007 / Accepted: 23 July 2007 Published online: 6 March 2008 – © Springer-Verlag 2008

Abstract: We introduce a new construction of E 0 -semigroups, called generalized CCR flows, with two kinds of descriptions: those arising from sum systems and those arising from pairs of C0 -semigroups. We get a new necessary and sufficient condition for them to be of type III, when the associated sum system is of finite index. Using this criterion, we construct examples of type III E 0 -semigroups, which can not be distinguished from E 0 -semigroups of type I by the invariants introduced by Boris Tsirelson. Finally, by considering the local von Neumann algebras, and by associating a type III factor to a given type III E 0 -semigroup, we show that there exist uncountably many type III E 0 -semigroups in this family, which are mutually non-cocycle conjugate.

1. Introduction An E 0 -semigroup is a weakly continuous semigroup of unital ∗-endomorphisms on B(H ), the algebra of all bounded operators on a separable Hilbert space H . E 0 -semigroups are classified into three broad categories, namely type I, II, and III, depending upon the existence of intertwining semigroups called units. William Arveson completely classified the E 0 -semigroups of type I, by showing that the CCR flows exhaust all the type I E 0 -semigroups, up to the identification of cocycle conjugacy ([5]). But the theory of E 0 -semigroups belonging to type II and type III remained mysterious for quite some time. There is no hope of completely classifying the whole class of E 0 -semigroups even now, basically due to the presence of type II and type III examples. For quite some time there were essentially only one example for each type II and type III E 0 -semigroups, due to R. T. Powers ([14,15]). In this context Boris Tsirelson produced uncountable families of both type II and type III E 0 -semigroups by using measure type spaces arising from several models in probability theory ([18]). It is equivalent to


Work supported by JSPS.

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study the product systems of Hilbert spaces, a complete invariant introduced by Arveson, in order to understand the associated E 0 -semigroups. Tsirelson basically produced uncountable families of both type II and type III product systems of Hilbert spaces. Tsirelson’s construction of type III product systems uses off white noises, which are Gaussian generalized (i.e. distribution valued) processes with a slight correlation between “past and future”. After discussing Tsirelson’s results, Arveson concludes his book ([5]) by saying, ‘It is clear that we have not achieved a satisfactory understanding of the existence of ‘logarithms’ in the category of product systems.’ This was clarified for Tsirelson’s type III examples in [6] from the viewpoint of operator algebras. Inspired by the results of Tsirelson, a purely operator algebraic construction of Tsirelson’s type III examples was provided in [6]. They were called ‘product systems arising from sum systems’. In particular, a dichotomy result about types was proved in [6], namely, it was proved that the product system arising from a divisible sum system is either of type I or of type III. A sum system is said to be divisible if it has sufficiently many real and imaginary addits (called additive units in [6]). On the other hand, motivated by Tsirelson’s construction of type III examples, a class of C0 -semigroups acting on L 2 (0, ∞), which are Hilbert-Schmidt perturbations of the unilateral shift semigroup, was investigated in [10]. Description of such semigroups in terms of analytic functions on the right-half plane was given and several examples were constructed. In this paper we discuss the consequences of these developments. First we describe the E 0 -semigroups associated with the above mentioned product systems arising from sum systems. This would generalize the simplest kind of E 0 -semigroups called CCR flows. These generalized CCR flows are given by a pair of C0 -semigroups. By studying the product system we get a new necessary and sufficient condition for the E 0 -semigroup to be of type III, when the associated sum system is of finite index. This criterion is much more powerful than the sufficient condition already proved in [6]. Using the results proved in [10], we compute the additive cocycles for the pairs of the shift semigroup and its perturbations, and show that the sum systems associated with them are always divisible. This class of sum systems include those coming from off white noises of Tsirelson. Then we concentrate on a special subclass of C0 -semigroups, which give rise to new type III E 0 -semigroups. These new examples cannot be distinguished from type I E 0 -semigroups by the invariants introduced by Tsirelson, and later discussed in [6]. Let us give an intuitive explanation of this phenomenon in terms of Tsirelson’s off white noise picture here. Although our examples also come from Tsirelson’s off white noises, spectral density functions for them tend to 1 at infinity. This means that our off white noises are so close to white noise that Tsirelson’s invariant can not work for them. Finally, we associate a type III factor as an invariant to each of these type III E 0 -semigroups, and using that we prove that there are uncountably many examples in this family which are not cocycle conjugate to each other. Toeplitz operator plays an essential role throughout these discussions. We end this section by reviewing some of the very basic definitions about E 0 -semigroups. For the definitions of notions related to E 0 -semigroups, such as cocycle conjugacy, index etc., we refer to [5]. A unit for an E 0 -semigroup {αt } acting on B(H ) is a strongly continuous semigroup of bounded operators {Tt }, which intertwines α and the identity, that is αt (X )Tt = Tt X,

∀ A ∈ B(H ), t ≥ 0.

Product systems, a complete invariant for cocycle conjugacy introduced by Arveson, are demonstrated to be a very useful tool in studying E 0 -semigroups. For an E 0 -semi-

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group {αt ; t ≥ 0} these are Hilbert spaces of the intertwining operators {Ht ; t ≥ 0}, defined by Ht = {T ∈ B(H ); αt (X )T = T X, ∀ X ∈ B(H )} with inner product T, S1 H = S ∗ T (see [5]). Every notion related to E 0 -semigroups can be translated into the framework of the associated product systems. We provide here a slightly different (but equivalent) definition than originally defined in [4] and [5]. The difference is in the measurability axiom (and we do not need a separate nontriviality axiom), which is due to Volkmar Liebcher. He has also proved in [12] that any two measurable structures on a given algebraic product system give rise to isomorphic product systems, and as a consequence we get that two product systems are isomorphic if they are algebraically isomorphic. So we need not consider measurable structures while dealing with isomorphism of product systems. Definition 1.1. A product system of Hilbert spaces is an one parameter family of separable complex Hilbert spaces {Ht ; t ≥ 0}, together with unitary operators Us,t : Hs ⊗ Ht → Hs+t for s, t ∈ (0, ∞), satisfying the following two axioms of associativity and measurability. (i) Associativity. For any s1 , s2 , s3 ∈ (0, ∞), Us1 ,s2 +s3 (1 Hs1 ⊗ Us2 ,s3 ) = Us1 +s2 ,s3 (Us1 ,s2 ⊗ 1 Hs3 ). (ii) Measurability. There exists a countable set H 0 of sections R  t → h t ∈ Ht such that t → h t , h t  is measurable for any two h, h ∈ H 0 , and the set {h t ; h ∈ H 0 } is total in Ht , for each t ∈ (0, ∞). Further it is also assumed that the map (s, t) → Us,t (h s ⊗ h t ), h s+t  is measurable for any two h, h ∈ H 0 . }) are said to be Definition 1.2. Two product systems ({Ht }, {Us,t }) and ({Ht }, {Us,t isomorphic if there exists a unitary operator Vt : Ht → Ht , for each t ∈ (0, ∞) satisfying Vs+t Us,t = Us,t (Vs ⊗ Vt ).

Definition 1.3. A unit for a product system is a non-zero section {u t ; t ≥ 0}, such that the map t → u t , h t  is measurable for any h ∈ H 0 and Us,t (u s ⊗ u t ) = u s+t , ∀s, t ∈ (0, ∞). A product system (E 0 -semigroup) is said to be of type I, if units exist for the product system and they generate the product system, i.e. for any fixed t ∈ (0, ∞), the set {u 1t1 u 2t2 · · · u ntn ;

n


ti = t, u i ∈ U},

i=1

is a total set in Ht , where U is the set of all units and the product is defined as the image of u 1t1 ⊗ u 2t2 · · · ⊗ u ntn in Ht , under the canonical unitary given by the associativity axiom. It is of type II if units exist but they do not generate the product system. We say a product system is of type III or unitless if there does not exist any unit for the product system.

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2. Preliminaries and Notation In this section we fix the notation used in this paper, recall some of the results proved earlier and make a few definitions. For an operator A, we denote the range of A by Ran (A) and the kernel of A by Ker (A). We denote the identity operator on a Hilbert space H by 1 H (or simply by 1 if no confusion arises). For a complex Hilbert space K , we denote by (K ) the symmetric Fock space associated with K , and by  the vacuum vector in (K ) (see [13]). For any x ∈ K , the exponential vector of x is defined by x⊗ e(x) = ⊕∞ n=0 √ , n! n

where x 0 = . Then the set of all exponential vectors {e(x) : x ∈ K } is a linearly independent total set in (K ). For a unitary U ∈ B(K ), we denote by Exp (U ) the unitary in B((K )) given by Exp (U )e(x) = e(U x). The Weyl operator, corresponding to an element x ∈ K , is defined by W (x)(e(y)) = e−

x2 2 −y,x

e(y + x),

and W (x) extends to a unitary operator on (K ). The ∗-algebra generated by {W (x)}x∈K is called the Weyl algebra for K . For a real Hilbert space G, we denote the complexification of G by G C . For two Hilbert spaces G 1 , G 2 , define 1

S(G 1 , G 2 ) = {A ∈ B(G 1 , G 2 ); A invertible and I − (A∗ A) 2 Hilbert-Schmidt}. In the above definition and elsewhere, by invertibility we mean that the inverse is also bounded. The set S(., .) is well behaved with respect to taking inverses, adjoints, products, and restrictions (see [6]). For two real Hilbert spaces G 1 , G 2 and A ∈ S(G 1 , G 2 ), C −1 ∗ define a real linear operator S A : G C 1 → G 2 by S A (u + iv) = Au + i(A ) v for C C u, v ∈ G 1 . Then S A is a symplectic isomorphism between G 1 and G 2 (i.e. S A is a real linear, bounded, invertible map satisfying Im (S A x, S A y) = Im x, y for all x, y ∈ G C 1 , see [13, p. 162]). In general S A is not complex linear, unless A is unitary. The following theorem, a generalization of Shales theorem, is used to construct the product system from a given sum system in [6]. Theorem 2.1. (i) Let G 1 , G 2 be real Hilbert spaces and A ∈ S(G 1 , G 2 ) , then there C exists a unique unitary operator (A) : (G C 1 ) → (G 2 ) such that (A)W (u)(A)∗ = W (S A u) ∀ u ∈ G C 1, (A)1 , 2  ∈ R+ ,

(2.1) (2.2)

C where 1 and 2 are the vacuum vectors in (G C 1 ) and (G 2 ) respectively. Conversely, for a given bounded operator A : G 1 → G 2 , if there exists a uni∗ C tary operator (A) : (G C 1 ) → (G 2 ), satisfying, (A)W (u)(A) = W (S A u), ∀ u ∈ GC 1 , then A ∈ S(G 1 , G 2 ). (ii) Suppose G 1 , G 2 , G 3 are three real Hilbert spaces, and A ∈ S(G 1 , G 2 ), B ∈ S(G 2 , G 3 ), then

(A−1 ) = (A)∗ , (B A) = (B)(A).

(2.3) (2.4)

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Proof. We only have to prove the converse part of (i). Remaining parts are already proved in [6]. For A ∈ B(G 1 , G 2 ), consider the polar decomposition of A = U A0 , where A0 ∈ B(G 1 ), and U ∈ B(G 1 , G 2 ), a unitary operator. Suppose there exists a unitary ∗ C operator (A) : (G C 1 ) → (G 2 ), satisfying (A)W (u)(A) = W (S A u), then ∗ ∗ (A0 ) = Exp (U )(A) will satisfy, (A0 )W (u)(A0 ) = W (S A0 u). The original version of Shales theorem will imply that A0 ∈ S(G 1 , G 1 ) (see [13, p. 169]). So we can conclude that A ∈ S(G 1 , G 2 ).   Next we define the notion of a sum system. Definition 2.2. A sum system is a two parameter family of real Hilbert spaces {G s,t } for 0 < s < t ≤ ∞, satisfying G s,t ⊂ G s ,t if the interval (s, t) is contained in the interval (s , t ), together with a one parameter semigroup {St }, of bounded linear operators on G (0,∞) for t ∈ (0, ∞) such that (i) Ss |G 0,t ∈ S(G 0,t , G s,s+t ) ∀ t ∈ (0, ∞], s ∈ [0, ∞). (ii) If As,t : G 0,s ⊕G s,s+t → G 0,s+t , is the map As,t (x ⊕ y) = x + y, for x ∈ G 0,s , y ∈ G s,s+t , then As,t ∈ S(G 0,s ⊕ G s,s+t , G 0,s+t ), ∀ s ∈ (0, ∞), ∀ t ∈ (0, ∞]. (iii) The semigroup {St } is strongly continuous. }, {S }) are isomorphic if there We say that two sum systems ({G a,b }, {St }) and ({G a,b t exists a family of operator Ut ∈ S(G 0,t , G 0,t ) preserving every structure of the sum systems, (i.e) {Ut } satisfies

A s,t (Us ⊕ Ss |G 0,t Ut ) = Us+t As,t (1G 0,s ⊕ Ss |G 0,t ), }, {S }) similar to A where A s,t is defined for ({G a,b s,t above. t

The above definition is slightly stronger than the one given in [6]. Namely, we require here that {St } is a C0 -semigroup acting on the global Hilbert space G 0,∞ and axiom (ii) holds for t = ∞ also, though they were not assumed in [6]. Given a sum system ({G s,t }, {St }), we define Hilbert spaces Ht = (G C 0,t ), and unitary operators Us,t : Hs ⊗ Ht → Hs+t , by Us,t = (As,t )(1 Hs ⊗ (Ss |G 0,t )). It is proved in [6] that ({Ht }, {Us,t }) forms a product system. Isomorphic sum systems give rise to isomorphic product systems. Let K be a complex Hilbert space and let {St } be the shift semigroup of L 2 ((0, ∞), K ) defined by (St f )(s) = 0, s < t, = f (s − t), s ≥ t, for f ∈ L 2 ((0, ∞), K ). The CCR flow of index dim K is the E 0 -semigroup α acting on B((L 2 ((0, ∞), K ))) defined by αt (W ( f )) = W (St f ). To generalize the CCR flows, we ask the following question. Let G be a real Hilbert space and H = (G C ). Suppose we have two semigroups of linear operators, St , Tt : G → G for t ≥ 0. Consider the association αt (W (x)) → W (St x), αt (W (i y)) → W (i Tt y), x, y ∈ G. When can we extend this map to an E 0 -semigroup on B(H )? The continuity and the semigroup property of {αt } will immediately imply that both {St } and {Tt } have to be C0 -semigroups. Also, αt being an endomorphism satisfies αt (W (u)W (v)) = αt (W (u))αt (W (v)), u, v ∈ G C .

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Comparing both sides, using the canonical commutation relation, we get St x, Tt y = x, y ∀ x, y ∈ G which is the same as saying Tt∗ St = 1. Assume that these conditions are satisfied. Then for αt to extend as an endomorphism of B(H ), it is necessary and sufficient that αt , as a representation of the Weyl algebra, is quasi-equivalent to the defining (vacuum) representation. The following lemma, probably well-known among specialists, gives a complete answer to the above question. For convenience of the reader, we include a proof here. Lemma 2.3. Let G be a real Hilbert space and let S and T be operators in B(G) satisfying T ∗ S = 1. Then the representation π of the Weyl algebra for G C given by π(W (x + i y)) = W (Sx + i T y), x, y ∈ G is quasi-equivalent to the defining representation if and only if S −T is a Hilbert-Schmidt class operator. Proof. Thanks to the relation T ∗ S = 1, the above π actually gives a representation of the Weyl algebra. Since T ∗ (resp. S ∗ ) is a left inverse of S (resp. T ), the range of S (resp. T ) is closed and |S| (resp. |T |) is invertible. We first claim that π is a factor representation. Let K 1 = Ran (S) and K 2 = Ran (T ), which are closed subspaces of G. Then the relation T ∗ S = 1 implies that we have K 1 ∩ K 2⊥ = K 2 ∩ K 1⊥ = {0}, and so M := {π(W (z)); z ∈ G C } = {W (x + i y); x ∈ K 1 , y ∈ K 2 } is a factor thanks to [1, Theorem 1]. The claim implies that π is quasi-equivalent to the restriction of π to M, which is unitarily equivalent to the GNS representation of the quasi-free state ω (x) = π(x), . Therefore to prove the statement, it suffices to show that the GNS representations of the vacuum state ω and the quasi-free state ω are quasi-equivalent if and only if S − T is a Hilbert-Schmidt operator. For that we can apply well-known criteria in [3,7]. Note that ω and ω are given by ω(W (x + i y)) = e−(x ω (W (x

+ i y)) =

2 +y2 )/2

, x, y ∈ G,

2 2 e−(|S| x,x+|T | y,y)/2 ,

x, y ∈ G.

Since we deal with Weyl operators rather than creation and annihilation operators, we use the criterion in [7], and we first recall the notation there. Let σ (z 1 , z 2 ) = −Im z 1 , z 2 , which is a symplectic form of G C as a real vector space. Then we have the Weyl relation W (z 1 )W (z 2 ) = e−iσ (z 1 ,z 2 ) W (z 1 + z 2 ). Let A(x + i y) = y − i x and B(x + i y) = |T |2 y − i|S|2 x and let s A (z 1 , z 2 ) = σ (Az 1 , z 2 ), s B (z 1 , z 2 ) = σ (Bz 1 , z 2 ). Then the two states are given by ω(W (z)) = e−s A (z,z)/2 and ω (W (z)) = e−s B (z,z)/2 . To see that ω is a quasi-free state, we have to verify that B ∗ B − 1 is positive as an operator acting on the real Hilbert space G C equipped with the inner product s B . Since B ∗ = −B, all we have to show is |S|2 |T |2 |S|2 ≥ |S|2 and |T |2 |S|2 |T |2 ≥ |T |2 , or equivalently just |S||T |2 |S| ≥ 1. Indeed, let S = U |S| and T = V |T | be the polar decomposition of S and T respectively. Then thanks to T ∗ S = 1, we have V ∗ U = |T |−1 |S|−1 , and so we get 1 ≥ V ∗ UU ∗ V = |T |−1 |S|−2 |T |−1 , which shows |S||T |2 |S| ≥ 1.

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√ √ √ Let Q A = A + A 1 + A−2 = A and Q B = B + B 1 + B −2 , where 1 + B −2 √ 2 is a positive operator acting on (G C , s B ) satisfying 1 + B −2 = 1 + B −2 . Then the two GNS representations are quasi-equivalent if and only if Q −1 A Q B − 1 is a HilbertSchmidt operator [7, p. 190]. In the rest of the proof, the symbol ≡ means equality up to a Hilbert-Schmidt class operator. Straightforward computation yields  Q −1 Q (x + i y) = |S|(1 + 1 − |S|−1 |T |−2 |S|−1 )|S|x B A  + i|T |(1 + 1 − |T |−1 |S|−2 |T |−1 )|T |y, and so Q −1 A Q B −1 is a Hilbert-Schmidt operator if and only if the following two relations hold:  1 − |S|−1 |T |−2 |S|−1 ≡ |S|−2 − 1, (2.5)  −2 −1 −2 −1 (2.6) 1 − |T | |S| |T | ≡ |T | − 1. Assume first that Eqs. (2.5) and (2.6) hold. Then (2.5) implies 1 − |S|−1 |T |−2 |S|−1 ≡ 1 − 2|S|−2 + |S|−4 , which is equivalent to |S|−2 + |T |−2 ≡ 2. Therefore (2.5), (2.6), and this imply   1 − |S|−1 |T |−2 |S|−1 + 1 − |T |−1 |S|−2 |T |−1 ≡ 0. Since the left-hand side is a positive operator, each term must be a Hilbert-Schmidt operator. This means that both sides of (2.5) and (2.6) are Hilbert-Schmidt operators. In particular, we have |S|−2 ≡ |T |−2 ≡ 1 and the two operators 1 − |S|−1 |T |−2 |S|−1 and 1 − |T |−1 |S|−2 |T |−1 are trace class operators. Since this shows that |S|2 − |T |−2 is a trace class operator, the operator (S − T )∗ (S − T ) = |S 2 | + |T |2 − 2 = |S|2 − |T |−2 + (|T |−2 − 1)2 |T |2 is a trace class operator. This shows that S − T is a Hilbert-Schmidt operator. Assume conversely that S − T is a Hilbert-Schmidt operator now. Then 0 ≡ S ∗ (S − T ) = |S|2 − 1, 0 ≡ T ∗ (T − S) = |T |2 − 1 and (S − T )∗ (S − T ) = |S|2 + |T |2 − 2 is a trace class operator. Therefore a similar computation as above shows that (2.5) and (2.6) hold.   The above lemma allows us to introduce the notion of a generalized CCR flow. Definition 2.4. Let {St } and {Tt } be C0 -semigroups acting on a real Hilbert space G. We say that {Tt } is a perturbation of {St }, if they satisfy, (i) Tt ∗ St = 1. (ii) St − Tt is a Hilbert-Schmidt operator. Given a perturbation {Tt } of {St }, we say that the E 0 -semigroup {αt } acting on B((G C )) given by αt (W (x + i y)) = W (St x + i Tt y), x, y ∈ G is a generalized CCR flow associated with the pair {St } and {Tt }.

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In fact we can also show from Proposition 3.6 (see Sect. 3) that a pair of ({St }, {Tt }) gives rise to a generalized CCR flow if Tt is a perturbation of St . From Sect. 5 onwards we will follow the notations used in [10]. We will denote by {St } the shift semigroup of L 2 (0, ∞). For f, g ∈ L 2 (0, ∞), we denote  ∞ ( f, g) = f (x)g(x)d x, 0

while  f, g denotes the usual complex inner product. Let Hr be the right-half plane {z ∈ C : Re x > 0}. For z ∈ Hr we set ez (x) = e−zx . We denote by L 1loc [0, ∞) the set of all measurable functions on [0, ∞) which are integrable on every compact subsets of [0, ∞). For f ∈ L 1loc [0, ∞) and a > 1, such that the function e−ax f (x) ∈ L 1 (0, ∞), we denote by L[ f ](z) the Laplace transform  ∞ L[ f ](z) = f (x)e−zx d x, Re z > a. 0

Let HD be the set of holomorphic functions M(z) on the right-half plane Hr such that M(z)/(1 + z) belongs to the Hardy space H 2 (Hr ) and M(z) does not belong to H 2 (Hr ). Then we can associate a differential operator A M through the procedure described in [10, Sect. 2 and 3]. Namely, let q ∈ L 2 (0, ∞) such that L[q](z) = M(z)/(1 + z). Then A M is the differential operator A M f (x) = − f (x) whose domain D(A M ) is the set of locally absolutely continuous functions f ∈ L 2 (0, ∞) such that f ∈ L 2 (0, ∞) and ( f − f , q) = 0. We denote by HDb the set of M ∈ HD such that A M generates a C0 -semigroup, and by HD2 the set of M ∈ HDb such that et A M − St is a Hilbert-Schmidt operator for all t ≥ 0, where {St } is the shift semigroup. The semigroup {Tt = et A M } will satisfy the relation Tt∗ St = 1 for all t ≥ 0. Conversely for any given C0 -semigroup Tt satisfying Tt∗ St = 1, its generator A can be described by a M(z) ∈ HDb as above and M(z) is unique up to a non-zero scalar multiple. On the other hand suppose {Tt = St + K t } be a C0 -semigroup on L 2 (0, ∞) such that {Tt } is a perturbation of the shift semigroup {St }. Then it is proved in [10, Sect. 4] that there exists a measurable function k(x, y) defined on (0, ∞)2 and a > 0, satisfying  t k(x, s)k(t − s, y)ds, ∀ t ≥ 0, a.e (x, y) ∈ (0, ∞)2 , k(x + t, y) = k(x, y + t) + 0



∞ ∞

0

such that

|e−ax k(x, y)|2 d xd y < ∞,

(2.8)

0

 K t f (x) = 1(0,t) (x)

(2.7)

∞

k(t − x, y) f (y)dy,

f ∈ L 2 (0, ∞).

0

Conversely if K t is given by a measurable function k(x, y) as above, satisfying (2.7) and (2.8), then the C0 -semigroup {Tt = St + K t } satisfies the above conditions (i) and (ii). As described in [10, Sect. 6], Tsirelson’s type III E 0 -semigroups arising from off white noises (see [18,19]) are isomorphic to generalized CCR flows associated with {St } and {Tt } as above with the spectral density functions |M(i y)|2 .

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Now we describe a particular subclass of C0 -semigroups which are perturbations of the shift. We denote x ∧ y = min{x, y}. It is proved in [10, Lemma 5.1] that for any ϕ ∈ L 1loc [0, ∞) ∩ L 2 ((0, ∞), x ∧ 1d x), there exists a unique ψ and a positive real number a satisfying ea ψ ∈ L 1 (0, ∞) and L[ψ] =

L[ϕ] . 1 − L[ϕ]

The following theorem is also proved in [10, Theorem 5.2]. Theorem 2.5. Let ϕ ∈ L 1loc [0, ∞) ∩ L 2 ((0, ∞), x ∧ 1d x) and ψ be defined as above. Then k(x, y) defined by  x ϕ(x + y − s)ψ(s)ds, k(x, y) = ϕ(x + y) + 0

satisfies the conditions (2.7), (2.8), and consequently gives rise to a C0 -semigroup {Tt } satisfying conditions (i) and (ii) of Definition 2.4. We will often be using the following theorem [10, Theorem 5.3]. Theorem 2.6. Let ϕ ∈ L 1loc [0, ∞) ∩ L 2 ((0, ∞), x ∧ 1d x) and {Tt } be the C0 -semigroup constructed from ϕ as in the above theorem, which is a perturbation of the shift. Let A M , with M ∈ HD2 , be the generator of {Tt } and let q ∈ L 2 (0, ∞)\D(A) be the function satisfying (1 + z)L[q](z) = M(z). Then q is continuous at 0 and M(z) = q(0)(1 − L[ϕ](z)). When ϕ is a real function in L 1loc [0, ∞) ∩ L 2 ((0, ∞), x ∧ 1d x), let M = 1 − L[ϕ], then we get a C0 -semigroup {Tt } as described above, which preserves the real functions in L 2 (0, ∞). We will show in Sect. 7 that the generalized CCR flows α ϕ associated with such pairs ({St }, {Tt }) is type III if and only if ϕ ∈ / L 2 (0, ∞). In the last section we associate von Neumann algebras to bounded open sets in (0, ∞), for each E 0 -semigroup in this family. These local von Neumann algebras form an invariant for the E 0 -semigroup. For a subfamily of the above constructed E 0 -semigroups, we provide an open set for two different given E 0 -semigroups, whose associated von Neumann algebra is a type III factor for the first E 0 -semigroup and type I factor for the second E 0 -semigroup. This way we show that there exists an uncountable family of type III E 0 -semigroups which are mutually non-cocycle conjugate. 3. Sum Systems and Generalized CCR Flows In this section we study the E 0 -semigroup associated with the product system constructed out of a given sum system. Fix a sum system ({G a,b }, {St }) and let ({Ht }, {Us,t }) be the product system constructed out of it. Denote G = G 0,∞ , At = At,∞ . For x ∈ G, define xs ∈ G 0,s , xs,t ∈ G s,t , and xt,∞ ∈ G t,∞ by the unique decomposition, x = xs + xs,t + xt,∞ . We may consider St as a bounded linear invertible map from G onto G t,∞ . Hence (St∗ )−1 is a well-defined bounded operator from G onto G t,∞ . When there is no confusion, by misusing the notation, we consider (St∗ )−1 as an element of B(G) itself. Define Tt ∈ B(G), by ∗ −1 Tt = (A∗t )−1 A−1 ∀ t ∈ [0, ∞). t (St )

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Lemma 3.1. For any y ∈ G 0,∞ , the family {St−1 yt,∞ }t>0 converges to y, as t → 0. Proof. We have St−1 yt,∞ − y = St−1 (yt,∞ − St y) ≤ St−1 (yt,∞ − y + St y − y). We claim that there exist positive numbers c and ε such that for all x ∈ G and t ∈ (0, ε) we have ||St x|| ≥ cx. Indeed, if it were not the case, we would have a decreasing ∞ sequence {tn }∞ n=1 of positive numbers converging to zero and a sequence {x n }n=1 in G ∞ satisfying xn  = 1 such that {Stn xn }n=1 converges to zero. Note that since {St } is a C0 -semigroup, there exist positive constants a and b such that St  ≤ aebt holds for all t > 0 (see [20, p. 232]). We choose s > t1 . Then Ss xn  = Ss−tn Stn xn  ≤ Ss−tn Stn xn  ≤ aebs Stn xn  → 0, (n → ∞), which is a contradiction because Ss is invertible as a map from G onto G s,∞ . Therefore the claim is proved, and so St−1  is bounded when t tends to 0. On the other hand St y converges to y by the strong continuity of the semigroup {St } and yt,∞ converges to y (see [6, Prop. 24]), which finishes the proof.   Lemma 3.2. {Tt } forms a C0 -semigroup on G. Proof. Note that Tt x = (A∗t )−1 (0 ⊕ (St∗ )−1 x), and therefore −1 Tt x, y = (0 ⊕ (St∗ )−1 x), A−1 t y = x, St yt,∞ , ∀ x, y ∈ G.

(3.1)

Hence for any x, y ∈ G, Ts Tt x, y = Tt x, Ss−1 (ys,∞ ) = (St∗ )−1 x, (Ss−1 ys,∞ )t,∞  ∗ −1 ) x, Ss (Ss−1 ys,∞ )t,∞ . = (Ss+t Suppose ys,∞ = Ss y for some y ∈ G, then Ss (Ss−1 ys,∞ )t,∞ = Ss yt,∞ = ys+t,∞ . Hence we get ∗ −1 ) x, ys+t,∞  Ts Tt x, y = (Ss+t = Ts+t x, y.

So {Tt } forms a semigroup. Now Eq. (3.1), thanks to Lemma 3.1, will imply that the semigroup {Tt } is weak and hence strongly continuous (see [20, p. 233]). So {Tt } forms a C0 -semigroup.   We say that the pair ({St }, {Tt }) is associated with the sum system (G a,b , St ). The E 0 -semigroup associated with the product system (Ht , Us,t ) can be described in terms of these two semigroups, {St } and {Tt } as follows. Let H = (G C ). Proposition 3.3. Let the notation be as above. Then there is a unique E 0 -semigroup αt on B(H ) satisfying αt (W (x)) = W (St x), αt (W (i y)) = W (i Tt y), x, y ∈ G. Moreover the product system associated with this E 0 -semigroup is the one constructed out of the sum system.

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Proof. Fix t ≥ 0. Set Ht,∞ = (G C t,∞ ) and Ut = (At ) : Ht ⊗ Ht,∞ → H . The map T → (St )T (St )∗ is an isomorphism between B(H ) and B(Ht,∞ ). So we get a ∗-endomorphism αt0 of B(H ) defined by T → Ut (1 H0,t ⊗ (St )T (St )∗ )Ut∗ .

(3.2)

We claim that the above endomorphism is αt . Indeed, for x, y ∈ G we have αt0 (W (x + i y)) = (At )W (0 ⊕ (St x + i St−1∗ y))(At )∗ = W (St x + i A−1∗ (0 ⊕ St−1∗ y)) t = W (St x + i Tt y), where we identify 1 Ht ⊗ W (z) with W (0 ⊕ z) for z ∈ G C t,∞ . Hence αt defined in the proposition extends to a ∗-endomorphism on B(H ), and we get a generalized CCR flow. Since any endomorphism on B(H ) is determined uniquely by its restriction to the Weyl algebra, αt is the unique extension. To determine the product system of this E 0 -semigroup, we want to determine the family of Hilbert spaces E t = {T ∈ B(H ) : αt (X )T = T X ∀ X ∈ B(H )}. Given a ξt ∈ Ht , we define a bounded operator Tξt on H by the following prescription, Tξt ξ = Ut (ξt ⊗ (St )ξ ), ∀ ξ ∈ H. It is easy to verify that Tξt defines a bounded operator, and that Tξ∗ Tξt ξ, η = ξt , ξt  Ht ξ, η, ∀ ξt , ξt ∈ Ht , ∀ξ, η ∈ H. t

We claim that Tξt ∈ E t . In fact by the definition of αt (see Eq. (3.2)) αt (X )Tξt ξ = Ut (1 H0,t ⊗ (St )X (St )∗ )Ut∗ Ut (ξt ⊗ (St )ξ ) = Ut (ξt ⊗ (St )X ξ ) = Tξt X ξ. So the association ξt → Tξt provides an isometry from Ht into E t . We only need to prove that this map is surjective. Suppose T ∈ E t is such that T ∗ Tξt = 0 ∀ ξt ∈ Ht , that is T ∗ vanishes on all vectors of the form Ut (ξt ⊗ (St )ξ ), ξt ∈ Ht , ξ ∈ H . But such vectors form a total subset of H , and hence we conclude that T = 0. It is also easy to verify that the product structure is also preserved.   The above proposition together with Lemma 2.3 imply Corollary 3.4. Let the notation be as above. Then {Tt } is a perturbation of {St }. Now we investigate the reverse question. Let G be a real Hilbert space and H = (G C ). We assume that a C0 -semigroup {Tt } is a perturbation of another C0 -semigroup {St } acting on G. Our task is to determine the product system for the generalized CCR flow associated with the pair ({St }, {Tt }). Define  G 0,t = Ker (Tt∗ ), G 0,∞ = G 0,t , G a,b = Sa (G 0,b−a ). t>0

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We verify that G a,b ⊆ G c,d whenever (a, b) ⊆ (c, d). This is same as saying Sc Sa−c (G 0,b−a ) ⊆ Sc (G 0,d−c ). So we only need to verify that Sa−c x ∈ G 0,d−c , for any x ∈ G 0,b−a . First we assume that d = ∞. We verify that Sa−c x ⊥ Ran (Td−c ) as follows. For z ∈ G, ∗ Ta−c Td−a z Sa−c x, Td−c z = x, Sa−c = x, Td−a z = x, Tb−a Td−b z = 0.

Now let d = ∞. If b = ∞ then the above verification itself implies that, for any finite d ≥ b, we have Sc Sa−c (G 0,b−a ) ⊆ Sc (G 0,d −c ) ⊆ Sc (G 0,∞ ). So we have to consider only the case when b = ∞, and equivalently we only have to show that St (G 0,∞ ) ⊆ G 0,∞ for any t ≥ 0. But this follows immediately as we have already shown that for any fixed s ≥ 0, Ss (G 0,t ) ⊂ G 0,∞ for all s, t ≥ 0. Note that since St has a left inverse Tt∗ , the range of St is closed and the operator St from G onto Ran (St ) has a bounded inverse. Lemma 3.5. Let the notation be as above and 0 < t < s ≤ ∞. Then (i) We have Tt∗ G 0,s ⊂ G 0,s−t . (ii) The two operators St Tt∗ and 1 − St Tt∗ are idempotents such that Ran (St Tt∗ ) = Ker (1− St Tt∗ ) = Ran (St ) and Ran (1− St Tt∗ ) = Ker (St Tt∗ ) = G 0,t . In particular, the Hilbert space G is a topological direct sum of G 0,t and Ran (St ). (iii) The Hilbert space G 0,s is a topological direct sum of G 0,t and St G 0,s−t . Proof. (i) It is easy to verify the statement for finite s. Assume x ∈ G 0,∞ . Then there exists a sequence {xn } converging to x such that xn ∈ G 0,n . Thus for n larger than t, we get Tt∗ xn ∈ G 0,n−t . Since {Tt∗ xn } converges to Tt∗ x, the statement holds. (ii) follows from direct computation. (iii) follows from (i) and (ii).   Let P : G → G 0,∞ be the orthogonal projection. We define St0 and Tt0 by St0 = P St P, Tt0 = P Tt P. Then {St0 } and {Tt0 } are C0 -semigroups and one is a perturbation of the other. Indeed, it follows from the fact that the inclusion relation St (G 0,∞ ) ⊆ G 0,∞ implies (1 − P) St P = 0 and Lemma 3.5,(i) implies P Tt (1 − P) = 0 for all t > 0. Proposition 3.6. Let G be a real Hilbert space and let {St } and {Tt } be C0 -semigroups acting on G such that {Tt } is a perturbation of {St }. Let {G s,t }, {St0 }, and {Tt0 } be as above. Then (a) The system ({G a,b }, {St0 }) forms a sum system. (b) The pair of C0 -semigroups ({St0 }, {Tt0 }) is associated with ({G a,b }, {St0 }). In consequence, the product system for the generalized CCR flow arising from ({St0 }, {Tt0 }) is isomorphic to the one arising from ({G a,b }, {St0 }).

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541

(c) The product system for the generalized CCR flow arising from ({St }, {Tt }) is isomorphic to the product system arising from ({G a,b }, {St0 }). In consequence, the generalized CCR flow arising from the pair ({St }, {Tt }) is cocycle conjugate to that arising from ({St0 }, {Tt0 }). Proof. (a) We have already shown the axiom (iii) of Definition 2.2 and G a,b ⊆ G c,d for (a, b) ⊆ (c, d). Since {Tt0 } is a perturbation of {St0 }, the operator St0 has a left inverse ∗ Tt0 . Therefore the restriction of St0 to G 0,s is an invertible operator from G 0,s onto St G 0,s with the bounded inverse. Since St0∗ (Tt0 − St0 ) = P − St0∗ St0 is a Hilbert Schmidt operator, the axiom (i) is satisfied. To prove the axiom (ii), it is enough if we prove that the operator At,∞ : G 0,t ⊕ G t,∞  x ⊕ y → x + y ∈ G 0,∞ , is in the class S(G 0,t ⊕ G t,∞ , G 0,∞ ), for all t ∈ (0, ∞). Let G t = Ran (St ). We will prove a stronger statement that the operator, A t : G 0,t ⊕ G t  x ⊕ y → x + y ∈ G, is in the class S(G 0,t ⊕ G t , G), for all t ∈ (0, ∞). Thanks to Lemma 3.5, the operator A t has a bounded inverse. Let x1 , x2 ∈ G 0,t and y1 , y2 ∈ G. Then ∗

A t A t (x1 ⊕ St y1 ), x2 ⊕ St y2  = x1 + St y1 , x2 + St y2  = x1 , x2  + St y1 , St y2  + x1 , St y2  + St y1 , x2  = x1 ⊕ St y1 , x2 ⊕ St y2  + Tt (St∗ − Tt∗ )x1 , St y2  + St y1 , Tt (St∗ − Tt∗ )x2 , where we use Tt∗ x1 = Tt∗ x2 = 0. This shows that A t ∈ S(G 0,t ⊕ G t , G) and in consequence As,t ∈ S(G 0,s ⊕ G s,s+t , G 0,s+t ). Therefore the axiom (ii) holds and ({G a,b }, {St }) is a sum system. (b) It suffices to verify 0∗ −1 Tt0 = (A∗t,∞ )−1 (A−1 t,∞ )(St ) ,

where St0 is regarded as an element of B(G 0,∞ , G t,∞ ). For y ∈ G, let y = yt + yt,∞ be the unique decomposition such that yt ∈ G 0,t , yt,∞ ∈ G t,∞ . By a calculation we have already done in Lemma 3.2, for x, y ∈ G, we have, ∗ −1 −1 (A∗t,∞ )−1 (A−1 t,∞ )(St ) x, y = x, St yt,∞ .

On the other hand Tt0 x, y = x, Tt ∗ (yt + St St−1 yt,∞ ) = x, St−1 yt,∞ . Therefore ({St0 }, {Tt0 }) is associated with ({G s,t }, {St0 }). (c) We use the notation of the proof of (a). Proceeding exactly in the same way as in the proof of (b), by replacing G t,∞ with G t , we can verify that ∗

Tt = (A t )−1 A t

−1

(St∗ )−1 ,

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C with (St∗ )−1 regarded as an element in S(G, G t ). By replacing (G C t,∞ ) with (G t ), and by exactly imitating the proof of Proposition 3.3, we conclude that

R → (A t )(1(G C ) ⊗ (St )R(St )∗ )(A t )∗ , 0,t

is the generalized CCR flow given by the pair ({St }, {Tt }). If we again imitate the proof of Proposition 3.3, the part where the product system is computed, we will be able to see that the product system associated with the generalized CCR flow given by the pair ({St }, {Tt }) also coincides with the product system constructed out of the sum system ({G a,b }, {St }). Since the product systems determine E 0 -semigroups up to cocycle conjugacy, we conclude that the generalized CCR flow given by ({St }, {Tt }) is cocycle conjugate to the generalized CCR flow given by ({St0 }, {Tt0 }).   Remark 3.7. It is practically impossible to classify pairs ({St }, {Tt }) of C0 -semigroups acting on G as above without posing any condition; let us consider the case where {St } is a semigroup of isometries. Then up to unitary equivalence we may assume that G = L 2 ((0, ∞), K ) ⊕ L and St = St ⊕ Ut , where {St } is the shift semigroup and {Ut } is a 1-parameter unitary group. In this case, we have  Tt =

Tt Bt 0 Ut

 ,

where {Tt } is a perturbation of {St }. It is routine work to show that the two sum systems for ({St }, {Tt }) and ({St }, {Tt }) are isomorphic (though not identical in general), and so Proposition 3.6 implies that the two generalized CCR flows arising from them are cocycle conjugate. Therefore it is worth investigating perturbations of the shift semigroup. This has been already done in [10] in the case where dim K = 1 and we will analyze the structure of the resulting generalized CCR flows in Sect. 5–7.

4. Type III Criterion In this section we derive a necessary and sufficient condition, which would determine the type of the product system (in other words the type of the associated E 0 -semigroup), arising from a divisible sum system with finite index. To begin with we define the notion of divisibility for sum systems and recall some results from [6]. We first define addits for a sum system, which were called additive units in [6]. Definition 4.1. Let ({G a,b }, {St }) be a sum system. A real addit for the sum system ({G (a,b) }, {St }) is a family {xt }t∈(0,∞) such that xt ∈ G 0,t , ∀ t ∈ (0, ∞), satisfying the following conditions. (i) The map t → xt , x is measurable for any x ∈ G 0,∞ . (ii) xs + Ss xt = xs+t , ∀s, t, ∈ (0, ∞), (i. e.) As,t (xs ⊕ Ss xt ) = xs+t . An imaginary addit for the sum system ({G a,b }, {St }) is a family {yt }t∈(0,∞) such that yt ∈ G 0,t , ∀ t ∈ (0, ∞), satisfying the following conditions. (i) The map t → yt , y is measurable for any y ∈ G 0,∞ . (ii) {yt } satisfies (A∗s,t )−1 (ys ⊕ (Ss∗ )−1 yt ) = ys+t , ∀s, t, ∈ (0, ∞).

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We denote by RAU and I AU the set of all real and imaginary addits respectively, which are real linear spaces. For a given real addit {xt }, define xs,t = Ss (xt−s ) ∈ G s,t . Similarly for a given imaginary addit {yt } define ys,t = (Ss∗ )−1 (yt−s ) ∈ G s,t . We also define for an imaginary addit {yt }, G 0,s  s ys 1 ,s2 = (A∗ )−1 (0 ⊕ ys1 ,s2 ⊕ 0), for any (s1 , s2 ) ⊂ (0, s), where A : G 0,s1 ⊕ G s1 ,s2 ⊕ G s2 ,s → G 0,s is defined by x ⊕ y ⊕ z → x + y + z. It is easy ⊥   to check that s ys 1 ,s2 ∈ G 0,s1 G s2 ,s ∩ G 0,s . When s = 1, we just denote t ys 1 ,s2 by by just y . Finally note that ys 1 ,s2 , and y0,t t = ys+t . xs + xs,s+t = xs+t , ys + ys,s+t

Definition 4.2. A sum system ({G a,b }, {St }) is called a divisible sum system if the addits exist and generate the sum system, (i. e.) G 0,s = span R [xs1 ,s2 ; (s1 , s2 ) ⊆ (0, s), {xt } ∈ RAU] and G 0,s = span R [s ys 1 ,s2 ; (s1 , s2 ) ⊆ (0, s), {yt } ∈ I AU]. The following proposition has been already proved in [6], except that {St } need not be a semigroup of isometries. But in the proof of [6, Prop. 37 (ii)], the verification of measurability of the function t → xt , yt  does not need this assumption, and also the relation xs+t , ys+t  = xs , ys  + xt , yt , can also be verified without this assumption. Proposition 4.3. Let ({G (a,b) }, {St }) be a divisible sum system. If {xt } ∈ RAU and {yt } ∈ I AU, then xt , yt  = x1 , y1 t ∀ t ∈ (0, ∞). In general for any two intervals (s1 , s2 ), (t1 , t2 ) ⊂ (0, s), it is true that xs1 ,s2 , s yt 1 ,t2  = x1 , y1 |(s1 , s2 ) ∩ (t1 , t2 )|,

(4.1)

where |.| is the Lebesgue measure on R. The next lemma immediately follows from the definition of a divisible sum system and Proposition 4.3, which allows us to introduce the notion of the index of a divisible sum system. Lemma 4.4. Let ({G a,b }, {St }) be a divisible sum system. For x = {xt } ∈ RAU and y = {yt } ∈ I AU , we set bG (x, y) = x1 , y1 . Then bG is non-degenerate as a bilinear form bG : RAU × I AU → R. Definition 4.5. For a divisible sum system ({G a,b }, {St }), the index ind G is the number dim RAU = dim I AU ∈ N ∪ {∞}.

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From now on we assume that ({G a,b }, {St }) is a divisible sum system. We further assume that ind G = n is finite. In that case, both RAU and I AU carry unique linear topologies. Denote G 00,t = span R [xs1 ,s2 ; (s1 , s2 ) ⊆ (0, t), {xt } ∈ RAU] ⊆ G 0,t ,

G 00,t = span R [t ys 1 ,s2 ; (s1 , s2 ) ⊆ (0, t), {yt } ∈ I AU] ⊆ G 0,t . For a given linear map J : RAU → I AU, we set Jt,0 to be the linear map Jt,0 : G 00,t →

G 00,t determined by Jt,0 (xs1 ,s2 ) = t J (x) s1 ,s2 , for (s1 , s2 ) ⊆ (0, t) and x ∈ RAU. When Jt,0 has a bounded extension to G 0,t we denote it by Jt . We need the following lemma. Lemma 4.6. Let G be a real Hilbert space and let R0 be a real linear operator on G C with dense domain D(R0 ), which preserves imaginary parts of the inner product. Suppose there exists a unitary operator U ∈ B((G C )) satisfying U W (x)U ∗ = W (R0 x), ∀ x ∈ D(R0 ),

(4.2)

then R0 extends to a bounded invertible operator R on G C . Further it is true that R ∈ S(G ⊕ G, G ⊕ G), where we identify G ⊕ G with G C equipped with the real inner product ·, ·R = Re ·, ·. Proof. First let us prove that R0 is bounded. Suppose {xn } ⊆ D(R0 ) is any sequence which converges to 0. Then, by the strong continuity of the Weyl representation, we conclude that W (xn ) converges strongly to 1. Therefore, by our assumption (4.2), it follows that W (R0 xn ) also converges strongly to 1. Consequently W (R0 xn ),  = 2 e−R0 xn  /2 converges to 1 and hence we conclude that R0 xn also converges to 0. We claim that the range of R0 is dense in G C . Let K = Ran (R0 )⊥ with respect to the real inner product. Then [1, Theorem 1,(5)] shows {W (R0 z); z ∈ G C } = {W (z); z ∈ i K } . Since the left-hand side is C thanks to W (R0 z) = U W (z)U ∗ , we get K = {0}, which shows the claim. Note that R0 is automatically injective and the inverse of R0 is well defined as a densely defined operator. A similar argument as above shows that the inverse is also bounded. The remaining part follows from the converse statement in the original Shales theorem ([13]).   We will also be using the following lemmas. Lemma 4.7. Let G be a second countable locally compact abelian group and let (w, K ) be a continuous unitary representation of G on a complex Hilbert space K without containing the trivial representation. Let c : G → K be a continuous 1-cocycle, that is, the map c satisfies the cocycle relation c(r + s) = c(r ) + w(r )c(s), ∀ r, s ∈ G. Then the following equation holds for all r, s ∈ G: c(r ), w(r )c(s) = c(s), w(s)c(r ).

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Proof. Thanks to [17, Theorem 4.2.1], there exists a sequence {z n } in K such that {z n − w(g)z n } uniformly converges to c(g) on every compact subset of G. Then we have c(r ), w(r )c(s) = lim z n − w(r )z n , w(r )z n − w(r )w(s)z n  n→∞  = lim z n , w(r )z n  + z n , w(s)z n  − z n , z n  − z n , w(r )w(s)z n  . n→∞

On the other hand, c(s), w(s)c(r ) = lim z n − w(s)z n , w(s)z n − w(s)w(r )z n  n→∞  = lim z n , w(s)z n  + z n , w(r )z n  − z n , z n  − z n , w(s)w(r )z n  , n→∞

which shows the statement.

 

Let K be a complex Hilbert space. Recall that the automorphism group G K of the exponential product system of index dim K is described as follows (see [5]): Let U (K ) be the unitary group of K . Then G K is homeomorphic to R × K × U (K ) with the group operation (a, ξ, u) · (b, η, v) = (a + b + Im ξ, uη, ξ + uη, uv). For (a, ξ, u) ∈ G K , the corresponding automorphism is realized by the family of unitary operators eiat W (1(0,t] ξ )Exp (1(0,t] u), t > 0. Direct computation shows the following: Lemma 4.8. Let G be an abelian group and let ρ : G  r → (a(r ), ξ(r ), u(r )) ∈ G K be a map. Then ρ is a homomorphism if and only if the following relation holds for every r, s ∈ G: a(r + s) = a(r ) + a(s) + Im ξ(r ), u(r )ξ(s), ξ(r + s) = ξ(r ) + u(r )ξ(s), u(r + s) = u(r )u(s).

(4.3) (4.4) (4.5)

In particular, when u(r ) = 1 for all r ∈ G, then ρ is a homomorphism if and only if a(r + s) = a(r ) + a(s), ξ(r + s) = ξ(r ) + ξ(s), Im ξ(r ), ξ(s) = 0.

(4.6) (4.7) (4.8)

Proof. The first statement is obvious. Assume u(r ) = 1 now. Then Eq. (4.3) implies a(r + s) − a(r ) − a(s) = Im ξ(r ), ξ(s). Note that the left-hand side is symmetric in r and s while the right-hand side is antisymmetric. Thus the second statement holds.   Theorem 4.9. Let ({G a,b }, {St }) be a divisible sum system of finite index and let ({Ht }, {Us,t }) be the product system constructed out of the above sum system. Then the following statements are equivalent. (i) The product system (Ht , Us,t ) is of type I .

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(ii) There exists a linear isomorphism J : RAU → I AU satisfying the following property: the bilinear form bG (·, J ·) is an inner product of RAU, and for each t > 0, the operator Jt,0 extends to a bounded operator Jt on G 0,t such that Jt ∈ S(G 0,t , G 0,t ). (iii) There exists a linear isomorphism J : RAU → I AU satisfying the following property: the bilinear form bG (·, J ·) is an inner product of RAU and the operator J1,0 extends to a bounded operator J1 on G 0,1 such that J1 ∈ S(G 0,1 , G 0,1 ). Proof. We set n = ind G. (i) ⇒ (ii) We assume that the product system (Ht , Us,t ) is of type I . Then it is isomorphic to an exponential product system ([5]). We first claim that this exponential product system is also of index n. For t > 0, let Vt : (G 0,t ) → (L 2 ((0, t), K )) be a family of unitary operators, implementing the isomorphism between the above product systems. Here K is some separable complex Hilbert space. We want to show that the dimension of K is n. For each x ∈ RAU and y ∈ I AU, the families {W (xt )}t>0 and {W (i yt )}t>0 form automorphisms for the product system (Ht , Us,t ) ([6, Theorem 26]) satisfying the relations: (1)

(2)

(1)

W (xt )W (xt ) = W (xt

(2)

+ xt ), ∀ x (1) , x (2) ∈ RAU,

(1) (2) (1) (2) W (i yt )W (i yt ) = W (i yt + i yt ), W (xt )W (i yt ) = e−2itbG (x,y) W (i yt )W (xt ),

∀

y (1) , y (2)

∈ I AU,

(4.9) (4.10)

∀ x ∈ RAU, ∀ y ∈ I AU. (4.11)

Therefore there exist two continuous homomorphisms ρ : RAU  x → (a(x), ξ(x), u(x)) ∈ G K, σ : I AU  y → (b(y), η(y), v(y)) ∈ G K, satisfying Vt W (xt )Vt∗ = eita(x) W (1(0,t] ξ(x))Exp (1(0,t] u(x)),

(4.12)

Vt W (i yt )Vt∗

(4.13)

=

eitb(y) W (1(0,t] η(y))Exp (1(0,t] v(y)),

where 1(0,t] denotes the characteristic function of the interval (0, t]. Equation (4.11) implies that in addition to the relations in Lemma 4.8, we have 2bG (x, y) = Im η(y), v(y)ξ(x) − Im ξ(x), u(x)η(y), ξ(x) + u(x)η(y) = η(y) + v(y)ξ(x), u(x)v(y) = v(y)u(x).

(4.14) (4.15) (4.16)

Let w((x, y)) = u(x)v(y) and c(x, y) = ξ(x) + u(x)η(y). Then (4.5) and (4.16) imply that (K , w) is a continuous unitary representation of RAU × I AU, and (4.4) and (4.15) imply that c is a continuous 1-cocycle. Let K 0 = {z ∈ K ; w(g)z = z, ∀g ∈ RAU × I AU}, and let K 1 = K 0⊥ . Let ξi (x) be the projection of ξ(x) to K i and let ηi (y) be the projection of η(y) to K i . Then Lemma 4.7 implies ξ1 (x), u(x)η1 (y) = η1 (y), v(y)ξ1 (x),

(4.17)

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and Eq. (4.14) is equivalent to bG (x, y) = Im η0 (y), ξ0 (x).

(4.18)

Assume that K 1 is not trivial. Let 0 < p < q < t. Then it is routine work to show Vt W (x p,q )Vt∗ = ei(q− p)a(x) W (1( p,q] ξ(x))Exp (1(0, p] + 1( p,q] u(x) + 1(q,t] ), Vt W (i t y p,q )Vt∗ = ei(q− p)b(y) W (1( p,q] η(y))Exp (1(0, p] + 1( p,q] v(y) + 1(q,t] ). By definition of K 1 , either u(x 0 ) or v(y 0 ) is not trivial for some x 0 ∈ RAU and y 0 ∈ I AU. Thus we assume that u(x 0 ) = 1 (the case with non-trivial v(y 0 ) can be treated in the same way). Direct computation using Lemma 4.7 and (4.3) shows that the operator W (1(0,t] ξ1 (x 0 ))Exp (1(0,t] u(x 0 )) commutes with Vt W (x p,q )Vt∗ and Vt W (i t y p,q )Vt∗ for all x ∈ RAU, y ∈ I AU, and 0 < p < q < t. However, this contradicts the irreducibility of the vacuum representation of the Weyl algebra, since the sets {x p,q ; ( p, q) ⊆ (0, t), x ∈ RAU} and {t y p,q ; ( p, q) ⊆ (0, t), y ∈ I AU} are total in G 0,t due to the divisibility of the sum system. Hence K = K 0 . Now assume that dim K is strictly larger than n. Then there exists non-zero ζ ∈ K orthogonal to ξ(RAU) and η(I AU) with respect to the real inner product Re ·, ·. Again we can show that W (i1(0,t] ζ ) would commute with Vt W (x p,q )Vt∗ and Vt W (i t y p,q )Vt∗ , for all x ∈ RAU, y ∈ I AU, and 0 < p < q < t, which is a contradiction. Therefore we conclude that dim K ≤ n. In the above argument, we have shown the following: there exist continuous homomorphisms ξ : RAU → K , η : I AU → K , a : RAU → R, and b : I AU → R satisfying Vt W (xt )Vt∗ = eita(x) W (1(0,t] ξ(x)),

(4.19)

Vt W (i yt )Vt∗ = eitb(y) W (1(0,t] η(y)), Im ξ(x (1) ), ξ(x (2) ) = 0, ∀ x (1) , x (2) ∈ RAU,

(4.20)

Im η(y (1) ), η(y (2) )

y (1) , y (2)

= 0, ∀ ∈ I AU, bG (x, y) = Im η(y), ξ(x), ∀ x ∈ RAU, ∀ y ∈ I AU.

(4.21) (4.22) (4.23)

Since bG is non-degenerate, Eq. (4.23) shows that ξ and η are injective, and in particular, the real dimension of the image of ξ is n. Thus Eq. (4.21) shows that there exists an orthonormal basis {e j }nj=1 of K consisting of elements in ξ(RAU), and so dim K = n. Let x j = ξ −1 (e j ). Then {x j }nj=1 is a basis of RAU. Let {y j }nj=1 be the dual basis of {x j }nj=1 in I AU with respect to the bilinear form bG . Then there exists a real matrix (λ jk ) such that η(y j ) = ie j +

n


λ jk ek .

k=1

Moreover, Eq. (4.22) implies that the matrix (λ jk ) is symmetric. Thus by changing the basis {e j }nj=1 if necessary, we may and do assume that (λ jk ) is diagonal and there exist real numbers λ j such that η(y j ) = (λ j + i)e j .

(4.24)

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Next we want to get rid of a and b in the above. Let 1
1
b(y j )ξ(x j ) − a(x j )η(y j ). 2 2 n

ζ =

j=1

j=1

Then direct computation yields Im ζ, ξ(x k ) = −a(x k )/2 and Im ζ, η(y k ) = −b(y k )/2. Now by replacing Vt with W (1(0,t] ζ )Vt , we may and do assume a and b are trivial. Let λ j + i = r j eiθ j with r j > 0 and θ j ∈ (0, π ). Let g ∈ U (K ) be the unitary operator determined by ge j = eiθ j e j for all j. We denote Ut = Vt∗ Exp (1(0,t] g)Vt . Then for 0 < p < q < t, we have     1 1 j 1( p,q] η(y j ) Vt = W i t y j p,q , Ut W (x p,q )Ut∗ = Vt∗ W rj rj

Ut W (i t y j p,q )Ut∗ = Vt∗ W (1( p,q] r j ei2θ j e j )Vt = Vt∗ W (1( p,q] (2 cos θ j r j eiθ j − r j )e j )Vt = Vt∗ W (1( p,q] (ξ(−r j x j ) + η(2 cos θ j y j )))Vt = e−i2(q− p)r j cos θ j Vt∗ W (1( p,q] ξ(−r j x j ))Vt Vt∗ W (1( p,q] η(2 cos θ j y j ))Vt



= e−i2(q− p)r j cos θ j W (−r j x p,q )W (i2 cos θ j t y j p,q ) j



= W (−r j x p,q + i2 cos θ j t y j p,q ).

We introduce a real linear operator R0 with domain D(R0 ) = G 00,t + i G 00,t by setting j

R0 x p,q =

i t j y p,q , rj





R0 i t y j p,q = −r j x p,q + i2 cos θ j t y j p,q ,

which preserves the imaginary part of the inner product. Then we have the relation Ut W (x)Ut∗ = W (R0 x) for every x ∈ D(R0 ). Let J : RAU → I AU be the linear j operator determined by J x j = r −1 j y . Lemma 4.6 implies that R0 extends to an operator R ∈ S(G 0,t ⊕ G 0,t , G 0,t ⊕ G 0,t ), and in consequence Jt exists. Moreover, the operator Jt is invertible and R can be expressed in a matrix form as,   0 −Jt−1 R= , Jt B



where B is the bounded linear extension of the map i t y j p,q → 2 cos θ j i t y j p,q . Thus   Jt∗ B Jt∗ Jt . R∗ R = B Jt Jt−1∗ Jt−1 + B 2 Therefore Jt ∈ S(G 0,t , G 0,t ) and B is a Hilbert-Schmidt operator, and in consequence, we have cos θ j = 0 for all j. This shows J x j = y j and bG (x j , J x k ) = δ j,k , and so bG (·, J ·) is an inner product (with an orthonormal basis {x j }nj=1 ). (ii) ⇒ (iii) Clear.

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(iii) ⇒ (i) Assume that there exists a linear isomorphism J : RAU → I AU such that bG (·, J ·) gives an inner product of RAU and that the bounded extension J1 of J1,0 exists and J1 ∈ S(G 0,1 ). We first claim Jt ∈ S(G 0,t ) for each t ∈ (0, 1). Denote ∗ G 0,t = G ⊥ t,1 ∩ G 0,1 . Notice that J1 |G 0,t maps G 0,t to G 0,t and At,1−t maps G 0,t to G 0,t . Now our claim follows immediately from the observation that Jt,0 = (A∗t,1−t |G )(J1,0 |G 0,t ). 0,t

With the claim, it is routine work to show that Jt exists and Jt ∈ S(G 0,1 ) for all t > 0. Since bG (·, J ·) gives an inner product of RAU, we choose an orthonormal basis {x j }nj=1 with respect to this inner product. We set y j = J x j . Let K be an n-dimensional real Hilbert space with an orthonormal basis {e j }nj=1 . Denote L (0) ((0, t), K) = span R {1( p,q] ξ : ( p, q) ⊆ (0, t), ξ ∈ K} which is a dense subspace of the real Hilbert space L 2 ((0, t), K). Define Bt : G 00,t → L 2 ((0, t), K), Bt : L (0) ((0, t), K) → G 0,t by

Bt (x p,q ) = 1( p,q] e j , Bt (1( p,q] e j ) = t y j p,q , j

and extend by linearity. Notice that Jt,0 = Bt Bt . Using Proposition 4.3, we get Bt x p,q , 1(r,s] ek  = δ j,k |[ p, q] ∩ [r, s]| = x p,q , Bt 1(r,s] ek . j

j

By taking linear sums, we observe that Bt and Bt satisfy the adjoint condition on a dense subspace. So for any x ∈ G 00,t we have Bt x2 = Bt Bt x, x ≤ Jt x2 , which shows that Bt extends to an element in B(G 0,t , L 2 ((0, t), K)). The operator Bt also extends to a bounded operator because it is restriction of Bt∗ . We use the same symbols Bt and Bt for their bounded extension. Then we have Bt∗ Bt = Jt . Now from our assumption Jt ∈ S(G 0,t , G 0,t ) we have Bt ∈ S(G 0,t , L 2 ((0, t), K)). It is routine verification to see that Bt satisfies Bs ⊕ Ss Bt = Bs+t As,t (1G 0,s ⊕ Ss |G 0,t ) ∀ s, t ∈ (0, 1), where we have identified L 2 ((0, s), K) ⊕ L 2 ((s, s + t), K) with L 2 ((0, s + t), K). Therefore ({G a,b }, {St }) is isomorphic, as a sum system, to ({L 2 ((a, b), K)}), {St }) where {St } is the unilateral shift. So ({Ht }, {Us,t }) is isomorphic to the exponential product system, and hence type I.  

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Remark 4.10. It has been proved in [6, Theorem 39] that only type I and type III product systems can be constructed from divisible sum systems. So thanks to the above theorem, violating the condition J1 ∈ S(G 0,1 , G 0,1 ) is necessary and sufficient for the associated product system to be of type III. This criterion is much more powerful than the necessary condition for type I already proved in [6, Theorem 40]. In fact we can arrive at that condition just by assuming that J1 is bounded. Suppose that E n ⊆ (0, 1) is a sequence of elementary sets satisfying lim inf G E n = G 0,1 . That is, given any x ∈ G 0,1 there exists a sequence {xn } satisfying xn ∈ G E n and xn → x. If we set yn = J1 xn , then yn ∈ G ⊥ E nc . Also if we assume that J1 is bounded, then yn converges to J1 x. Thanks to [6, Lemma 28] this would imply that lim sup G E nc = {0}. In Sect. 6, we will see that there are examples of divisible sum systems of finite index with bounded J1 , which give rise to type III product systems. 5. Additive Cocycles In this section we compute the addits when G = L 2 (0, ∞)R and {St } is the shift semigroup. We use the notation L 2 (0, t)R for the set of all real functions in L 2 (0, t). We follow the notations in [10] and use many results from there. The reader may refer to the summary of notations and results reviewed at the end of Sect. 2. We denote by HD2,R the set of functions M ∈ HD2 satisfying M(z) = M(z). We fix M(z) in HD2,R and set Tt = et A M , and K t = Tt − St . With the above condition, the real part L 2 (0, ∞)R is preserved by Tt . As defined in Sect. 3, for t > 0, we set M = { f ∈ L 2 (0, ∞)R ; Tt∗ f = 0}, G 0,t

M to be M M and set G 0,∞ t>0 G 0,t . When M is an outer function, one can see that G 0,∞ = 2 L (0, ∞)R from [10, Theorem 6.4]. Definition 5.1. A measurable family {ct }t>0 of elements in L 2 (0, ∞)R is said to be M for all t > 0, and the a real additive cocycle for the pair ({St }, {Tt }), if ct ∈ G 0,t cocycle relation cs+t = cs + Ss ct holds for all s, t > 0. A measurable family {dt }t>0 of elements in L 2 (0, ∞)R is said to be an imaginary additive cocycle for the pair ({St }, {Tt }), if dt ∈ L 2 (0, t)R for all t > 0, and the cocycle relation ds+t = ds + Ts dt holds. Remark 5.2. Clearly the real additive cocycles are the same as the real addits for the sum M }, {S }), as defined in Sect. 3. system ({G a,b t While proving part (a) in Proposition 3.6, it has been proved, for any t ≥ 0, that M ⊕ L 2 (t, ∞) 2 the map A t : G 0,t R → L (0, ∞)R , given by x ⊕ y → x + y, is in M M is an imaginary addit for the sum 2 2 S(G 0,t ⊕ L (t, ∞)R , L (0, ∞)R ). If {yt } ∈ G 0,t M }, {S }), then we have system ({G a,b t ∗

(A t )−1 yt , St x = yt ⊕ 0, 0 ⊕ St x = 0, ∀ x ∈ L 2 (0, ∞), and hence St∗ (A t ∗ )−1 yt = 0. Moreover ∗

∗

∗

∗

(A s )−1 ys + Tt (A t )−1 yt = (A s )−1 (ys ⊕ Ss (A t )−1 yt ) ∗

= (A s+t )−1 ((A∗s,t )−1 (ys ⊕ Ss yt )) ∗

= (A s+t )−1 ys+t .

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In the above verification (and in the verification below) we have used our earlier observation that Tt = (A t ∗ )−1 (A t )−1 St (see the proof of Proposition 3.6,(c)) and the M }, {S }). Conversely if {d } is an imaginary cocycle associativity of the sum system ({G a,b t t for ({St }, {Tt }) then we have ∗

A t dt , 0 ⊕ St x = dt , St x = 0, ∀ x ∈ L 2 (0, ∞), M . Moreover and hence A t ∗ dt ∈ G 0,t ∗

∗

∗

∗

∗

∗

(A∗s,t )−1 (A s ds ⊕ Ss A t dt ) = A s+t (A s+t )−1 (A∗s,t )−1 (A s ds ⊕ Ss A t dt )  ∗ ∗ ∗ = A s+t (A s )−1 A s ds ⊕ Ss dt ∗

= A s+t (ds + Ts dt ) ∗

= A s+t ds+t . Hence an imaginary additive cocycle {dt }t>0 for the pair ({St }, {Tt }) is given by an M }, {S }) through the imaginary addit {yt }t>0 of the corresponding sum system ({G a,b t ∗ −1 bijective correspondence dt = (At ) yt and vice versa. (The above verification can as well be done without assuming {St } being a semigroup of isometries, by replacing St by (St∗ )−1 .) The following lemma is probably well-known. We include a proof here for the reader’s convenience. Lemma 5.3. Let {ct } be a real additive cocycle. Then t → ct is continuous. Proof. We regard L 2 (0, ∞)R as a subspace of L 2 (R) in a natural way. Let {Ut } be the shift of L 2 (R). By setting ct = −Ut c−t for negative t, we can extend c to an R-cocycle. Let V (t) = W (ct )Exp (U (t)) ∈ B((L 2 (R))). Then {V (t)} is a measurable unitary representation of R. It is well-known that such a representation is in fact continuous and so t → W (ct ) is continuous in the strong operator topology. Thus t → W (ct ),  = e−ct /2 is continuous, where  is the vacuum vector. Let P be the projection from (L 2 (R)) onto the one particle subspace of (L 2 (R)). Then we have ect /2 P W (ct ) = ct and ct is continuous.   Let {ct }t>0 be a real additive cocycle. Then Arveson’s Theorem [5, Theorem 5.3.2] shows that there exists c ∈ L 2loc [0, ∞) such that ct = c − St c, where we extend the shift St to L 2loc [0, ∞) in an obvious way. Lemma 5.3 and the cocycle relation imply that there exist positive constants a, b such that ||ct || ≤ a + bt, and so  t |c(x)|2 d x ≤ ||ct ||2 ≤ (a + bt)2 . 0

Note that for every ε > 0,  ∞  |c(x)|2 e−εx d x = 0

∞

0



=ε

e−εx

∞

e 0

d dx 



x

 |c(y)|2 dy d x

0 x

−εx

0

|c(y)|2 d yd x < ∞.

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Thus for w ∈ Hr , we have



0 = (ct , Tt ew ) = (c − St c, St ew + K t ew ) =

∞ 0

c(x + t)e−xw d x −L[c](w) + (c, K t ew ).

Performing the Laplace transformation in the variable t, we get the following from [10, Lemma 3.1] for z ∈ Hr with sufficiently large Re z: L[c](w) − L[c](z) L[c](w) − + (c, ez )(ξ M,z , ew ) = 0, z−w z where ξ M,z is as defined in the beginning of [10, Sect. 3]. Now [10, Lemma 3.3] implies wL[c](w) zL[c](z) = , M(z) M(w) and so L[c](z) is proportional to M(z)/z. Thanks to the fact that M(z)/(1 + z) ∈ H 2 (Hr ), indeed there exists a function c M such that c M ea ∈ L 2 (0, ∞) for all a > 0, and L[c M ](z) = M(z)/z. Note that c M is a real function. We set ctM = c M − St c M . Then L[ctM ] =

(1 − e−t z )M(z) , z

which belongs to H 2 (Hr ). Thus ctM ∈ L 2 (0, ∞)R . Tracing back the above argument, we can actually show the following lemma. Lemma 5.4. Let c M be a function in L 2loc [0, ∞) such that L[c M ](z) = M(z)/z and let ctM = c M − St c M . Then {ctM }t>0 is a real additive cocycle. Every real additive cocycle is a scalar multiple of {ctM }t>0 . Let {dt } be an imaginary additive cocycle now. The condition St∗ dt = 0 is equivalent to that the support of dt is in (0, t]. The cocycle relation ds+t (x) = ds (x) + (K s dt )(x) + (Ss dt )(x) shows that for every s < x ≤ t, we have ds+t (x) = dt (x − s) or equivalently, for every 0 < x ≤ t, we have ds+t (s + t − x) = dt (t − x). This means that there exists a function d ∈ L 2loc [0, ∞) such that dt (x) = 1(0,t] (x)d(t − x). The cocycle relation is now equivalent to  t d(t + x) = d(x) + k(x, y)d(t − y)dy, t, x > 0. 0

Note that t → dt  is a non-decreasing function. Thanks to the cocycle relation, we can see that  t t → dt 2 = |d(s)|2 ds 0

grows at most exponentially and the Laplace transformation L[d](z) of d exists for z with sufficiently large Re z. Performing the Laplace transformation of the above equation for the two variables t and x we get L[d](w) − L[d](z) L[d](z) = + L[ξ M,z ](w)L[d](w), z−w w

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553

which is equivalent to z M(z)L[d](z) = wM(w)L[d](w). Thus L[d](z) should be proportional to 1/(z M(z)). Indeed, [10, Cor. 4.3] shows that there exists d M ∈ L 2loc [0, ∞) such that L[d M ](z) = 1/(z M(z)). Lemma 5.5. Let d M ∈ L 2loc [0, ∞) such that L[d M ](z) = 1/(z M(z)) and we set dtM (x) = 1(0,t] (x)d M (t − x). Then {dtM }t>0 is an imaginary additive cocycle. Every imaginary additive cocycle is a scalar multiple of {dtM }t>0 . M = S c M and d M = T d M for 0 < s < t. Then due to Remark 5.6. We set cs,t s t−s s t−s s,t M , d M  = c M , d M  + c M , d M  can easily Lemma 4.3 (also for instance the relation cs+t s+t s s t t M , d M  = C|[q, r ] ∩ [s, t]|, where be verified), there exists a constant C such that the cq,r s,t |.| is the Lebesgue measure. As we have

 ctM , dtM  =

t

c M (x)d M (t − x)d x = c M ∗ d M (t),

0

and L[c M ∗ d M ](z) = L[c M ](z)L[d M ](z) = 1/z 2 , we actually get C = 1 in our case. A similar argument implies that the linear span of {dtM }t>0 is dense in L 2 (0, ∞)R . Indeed, if f ∈ L 2 (0, ∞)R is orthogonal to dtM for every t ≥ 0,  0 = dtM , f  =

t

d M (t − x) f (x)d x = d M ∗ f (t),

0

and so 0 = L[d M ∗ f ](z) = L[d M ](z)L[ f ](z) =

L[ f ](z) , z M(z)

for z ∈ Hr with sufficiently large Re z. This implies L[ f ](z) = 0 and f = 0. M M} We will show that the linear span of {cr,s 0
Lemma 5.7. Let M I (z) be the inner component of M(z) (see [10, Theorem 4.5]). Then M} the closed linear span of {cs,t 0<s0 and M L[cs,t ](z) =

(e−sz − e−t z )M(z) . z

Therefore, the statement follows from the Beurling-Lax Theorem [9, p. 107].

 

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Recall that there exists a measurable function k(x, y) such that  ∞ K t f (x) = 1(0,t) (x) k(t − x, y) f (y)dy, f ∈ L 2 (0, ∞), 0

and there exists a > 0 with  ∞ 0

∞

|e−ax k(x, y)|2 d xd y < ∞,

0

[10, Eq. 3.1, Lemma 4.1] shows  ∞ ∞ M(z) − M(w) , k(x, y)e−x z−yw d xd y = M(z)(z − w) 0 0 for z, w ∈ Hr with Re z > a/2. ⊥

M Lemma 5.8. A function f ∈ L 2 (0, ∞)R belongs to G 0,∞ if and only if there exists a positive number a such that  ∞ L[ f ](−iλ)M(iλ) dλ = 0 −∞ (z + iλ)M(w + iλ)

holds for all z, w ∈ Hr with Re w > a. Proof. To make sense of the statement, first we claim that for z, w ∈ Hr with Re w > a, the function M(iλ) iλ → (z + iλ)M(w + iλ)  x −sw 2 belongs to H (Hr ). Let h(x) = 0 e k(s, x − s)ds. Then since  x  x |h(x)|2 ≤ |e−as k(s, x − s)|2 ds × e−2Re (w−a)u du 0 0  x 1 −as ≤ |e k(s, x − s)|2 ds, 2Re (w − a) 0 



∞

dx 0

x

ds|e−as k(s, x − s)|2 =

0





∞

∞

ds 0

 =

s



∞

ds 0

∞

d x|e−as k(s, x − s)|2 du|e−as k(s, u)|2 < ∞,

0

the function h belongs to L 2 (0, ∞). In consequence, we have ez ∗ h ∈ L 2 (0, ∞). On the other hand, for ζ ∈ Hr we have  x  ∞ 1 −xζ d xe dse−sw k(s, x − s) L[ez ∗ h](ζ ) = z+ζ 0 0  ∞  ∞ 1 dse−sw d xe−xζ k(s, x − s) = z+ζ 0 s  ∞  ∞ 1 −s(w+ζ ) = dse dye−yζ k(s, y) z+ζ 0 0   M(ζ ) 1 1− . = w(z + ζ ) M(w + ζ )

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As the function iλ → 1/(z + iλ) belongs H 2 (Hr ), we get the claim. If we interchange St and Tt in Lemma 3.5, (ii), we get that the orthogonal complement M , which is Ran (T ), is same as the range of the idempotent T S ∗ . Hence we of G 0,t t t t M conclude that a function f ∈ L 2 (0, ∞) belongs to the orthogonal complement of G 0,t if and only if f = Tt St∗ f = (K t St∗ + St St∗ ) f, t > 0. This is equivalent to  f (x) =

∞

k(s, y) f (s + x + y)dy, x, s > 0.

(5.1)

0

The Plancherel theorem implies that (5.1) is equivalent to  ∞ 1 f (x) = L[k(s, ·)](iλ)L[ f ](−iλ)e−iλ(s+x) dλ. 2π −∞

(5.2)

Let z, w ∈ Hr with Re w > a. Then via the Laplace transformation, (5.2) is equivalent to    ∞ 1 M(iλ) L[ f ](−iλ) L[ f ](z) = 1− dλ. (5.3) 2π −∞ z + iλ M(w + iλ) Note that since L[ f ] ∈ H 2 (Hr ), we have  ∞ L[ f ](iλ) 1 dλ. L[ f ](z) = 2π −∞ z − iλ Thus (5.3) is equivalent to  ∞ −∞

L[ f ](−iλ) M(iλ) dλ = 0. z + iλ M(w + iλ)

  Theorem 5.9. Let the notation be as above. Then M M} (1) The linear span of {cr,s 0
iλ →

1 (z + iλ)M(w + iλ)

is an outer function in H 2 (Hr ). Thus Lemma 5.7 and Lemma 5.8 imply that ⊥

⊥

M,c M ⊂ G 0,∞ , G 0,∞ M = G M,c . which shows that G 0,∞ 0,∞

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M. Izumi, R. Srinivasan

M,c,0 M,c,0 M} Let G 0,∞ be the linear span of {cr,s 0r >0 is dense in L 2 (0, ∞)R . A similar argument works using the fact that

L 2 (0, t)R + Tt L 2 (0, ∞)R = L 2 (0, ∞)R is a topological direct sum.

 

M }, {S }) is divisible and of index 1. Corollary 5.10. The sum system ({G a,b t

Remark 5.11. Note that we have  ∞ −iqλ

1 (e − e−ir λ )(eisλ − eitλ )|M(iλ)|2 M M cq,r cs,t = dλ, 2π −∞ λ2 M } which depends only on the outer component of M(z). Since {cq,r 00 is determined by the sum system for M, and so we conclude that the cocycle conjugacy class is determined by the outer component of M. Note that this is consistent M = L 2 (0, ∞) when M is an outer function and with part (c) in with the fact that G 0,∞ R Proposition 3.6.

6. Application of the Type III Criterion In what follows, we assume that M ∈ HD2,R is an outer function unless otherwise M and the sum system {G M } stated. Then we have L 2 (0, ∞)R = G 0,∞ 0,t t>0 coincides with that discussed in [10, Sect. 6] except that complex Hilbert spaces are considered in [10]. The function |M(iλ)|2 corresponds to the spectral density function considered in [19] (see [10, Sect. 6]). M be a linear operator whose domain is the linear span of {c M } Let Jt,0 r,s 0
M M Jt,0 = (A t )−1 |G M Jt,0 and A t ∈ S(G 0,t ⊕ L 2 (t, ∞), L 2 (0, ∞)), 0,t

where A t is as defined in Remark 5.2.

Generalized CCR Flows

557

Theorem 6.1. Let M ∈ HD2,R be an outer function. Then the generalized CCR flow arising from {et A M }t>0 is of type I if and only if the bounded extension JtM exists and it is invertible such that for some positive number a, the operator (JtM )∗ JtM − a I is in the Hilbert-Schmidt class for all (some) t > 0. For a measurable (not necessarily bounded) function F on the imaginary axis, the Toeplitz operator T F on L 2 (0, ∞) with domain D(T F ) is defined as follows: ∞ D(T F ) = { f ∈ L 2 (0, ∞); −∞ |F(iλ)|2 |L[ f ](iλ)|2 dλ < ∞}, ∞ 1 i xλ dλ, (T F f )(x) = 1(0,∞) (x) 2π −∞ F(iλ)L[ f ](iλ)e where the above integral should be appropriately interpreted as usual. Assume that there exists a bounded operator J M ∈ B(L 2 (0, ∞)) whose restriction M is J M . Then by construction, we have J M S = T J M and so S ∗ J M S = J M . to G 0,t t t t t t Such an operator must be a Toeplitz operator (see, for example, [5, Chap. 13.3]). Indeed, we can determine the symbol even in the case where the global extension J M does not exist. M is in the domain of T Lemma 6.2. If 1/{(1 + z)M(z)} ∈ H 2 (Hr ), then cs,t 1/|M|2 for 0 < s < t and

T

1 |M|2

M M cs,t = ds,t .

Proof. The first statement is obvious. To prove the second statement, it suffices to show M , c M  = d M , c M  for all 0 < q < r . Indeed, T1/|M|2 cs,t q,r s,t q,r 

 T

M M 1 cs,t , cq,r 2

|M|

1 = 2π



∞

−∞

1 (e−isλ − e−itλ )M(iλ) (e−iqλ − e−ir λ )M(iλ) dλ |M(iλ)|2 iλ iλ

 ∞ −isλ 1 (e − e−itλ )(eiqλ − eir λ ) = dλ 2π −∞ λ2 = 1(s,t] , 1(q,r ] 

M M , cq,r . = ds,t

  Remark 6.3. The condition 1/{(1 + z)M(z)} ∈ H 2 (Hr ) in the above lemma is automatically satisfied for the class of {et A M } coming from Tsirelson’s off white noises. An off white noise is a generalized Gaussian process with the spectral density functions eρ(λ) (denoted by W (λ) in [19]) satisfying the following two conditions (in fact, the first one automatically follows from the second one as we will see below):  ∞ ρ(λ) e < ∞, (6.1) 2 1 −∞ + λ 

∞



∞

−∞ −∞

|ρ(λ1 ) − ρ(λ2 )|2 dλ1 dλ2 < ∞. |λ1 − λ2 |2

(6.2)

As described in [10, Sect. 6], Tsirelson’s E 0 -semigroup arising from the spectral density function eρ(λ) is conjugate to the generalized CCR flow given by an outer function

558

M. Izumi, R. Srinivasan

M(z) with the relation |M(iλ)|2 = eρ(λ) . Indeed, given a real function ρ(λ) satisfying Eq. (6.2), the function M(z) ∈ HD2,R is obtained by the integral    ∞ 1 λz + i ρ(λ)dλ M(z) = exp . 2π −∞ λ + i z 1 + λ2 Via the change of variable ζ = (z − 1)/(z + 1), the condition (6.1) is equivalent to that N (ζ ) = M(z) belongs to the Hardy space H 2 (D), where D is the unit disc, while the condition (6.2) is equivalent to that log N (ζ ) belongs to the analytic Besov space B2 (D) (see [19] and [21, Chap. 5]). Since we have the inclusion relation B2 (D) ⊂ VMOA(D) (see [21, Lemma 9.4.2]), condition (6.2) solely implies both (6.1) and 1/N (ζ ) ∈ H 2 (D) thanks to [11, p. 100], and so 1/{(1 + z)M(z)} ∈ H 2 (Hr ) holds. Tsirelson [18, Lemma 10.2] shows that if lim M(iλ) = 0,

λ→±∞

the E 0 -semigroup arising from {Tt }t>0 is of type III. With Theorem 6.1, we can easily treat the other extreme case. Theorem 6.4. Let M ∈ HD2,R be an outer function such that 1/{(1 + z)M(z)} ∈ H 2 (Hr ). If lim |M(iλ)| = ∞,

λ→±∞

then the generalized CCR flow arising from {Tt }t>0 is of type III. Proof. We use the same function as in [18, Lemma 10.1]. For a natural number n, we set  n−1 
M M cM ∈ G 0,1 . fn = 2k 2k+1 − c 2k+1 2k+2 , , 2n

k=0

2n

2n

2n

Then L[ f n ](z) = Fn (z)M(z) with 2k 2k+1 z n−1 − 2k+2 z
(1 − e−z )(1 − e− 2n )2 e 2n + e− 2n z − 2e− 2n z = Fn (z) = z z (1 − e− n )z

k=0

z

=

(1 − e−z )(1 − e− 2n ) z

(1 + e− 2n )z

.

The Plancherel theorem applied to the case with M = 1 implies  ∞ |Fn (iλ)|2 dλ = 2π. −∞

Suppose that J1M exists and it is invertible. Then || f n ||2 ≤ ||(J1M )−1 ||2 ||J1M f n ||2 = ||(J1M )−1 ||2 ||T ≤

||(J1M )−1 ||2 2π



∞ −∞

|Fn (iλ)|2 dλ, |M(iλ)|2

1 |M|2

f n ||

Generalized CCR Flows

559

which would imply  ∞  |Fn (iλ)|2 |M(iλ)|2 dλ ≤ ||(J1M )−1 ||2 −∞

∞ −∞

|Fn (iλ)|2 dλ. |M(iλ)|2

However, since the sequence {|Fn (z)|2 }∞ n=1 uniformly converges to 0 on every compact subset of the imaginary axis, we get a contradiction.   One can also reproduce Tsirelson’s result [18, Lemma 10.2] using Theorem 6.1. Indeed, assume that limλ→±∞ M(iλ) = 0 and 1/{(1+ z)M(z)} ∈ H 2 (Hr ). If {Tt }t≥0 gave a type I E 0 -semigroup, a similar computation as above would show that |(J1M f n , f n )| ≤ ||J1M |||| f n ||2 and  ∞  ∞ 2π = |Fn (iλ)|2 dλ ≤ ||J1M || |Fn (iλ)|2 |M(iλ)|2 dλ, −∞

−∞

which is a contradiction. Among the spectral density functions treated in [19], the ones to which the above Theorem 6.4 applies are as follows: strictly positive smooth functions such that for large |λ|, (1) |M(iλ)|2 = logβ |λ| for β > 0, or (2) |M(iλ)|2 = exp(a logβ |λ|) with a > 0 and 0 < β < 1/2. It is quite likely that these families of spectral density functions give rise to mutually non-isomorphic product systems. Problem 6.5. Show that the product systems arising from the above spectral density functions are mutually non-isomorphic. Now we treat the case where |M(iλ)| converges to a non-zero constant at infinity. Let L tM be the restriction of T M to D(T M ) ∩ L 2 (0, t)R . We claim that for f ∈ D(T M ), the function M(z)L[ f ](z) belongs to H 2 (Hr ). Recall that a function F(z) belongs to H 2 (Hr ) if and only if (1 + z)F(z) belongs to the Hardy space H 2 (D) of the unit disc via the change of variable ζ = (z − 1)/(z + 1). Since M(z)/(1 + z) and L[ f ](z) are in H 2 (Hr ) and M(iλ)L[ f ](iλ) is square integrable, the function (1 + z)M(z)L[ f ](z) belongs to H 1 (D) ∩ L 2 (T) ⊂ H 2 (D) and we get the claim. As a consequence, we have M , and the operator L M is densely defined and its image is dense in G M . T M (1(r,s] ) = cr,s t 0,t We denote by Pt the orthogonal projection from L 2 (0, ∞)R onto L 2 (0, t)R . Theorem 6.6. Let M1 , M2 be (not necessarily outer) functions in HD2,R . We assume Mi that L tMi is a bounded invertible operator from L 2 (0, t)R onto G 0,t for i = 1, 2. Then the sum system for M1 and M2 are isomorphic if and only if there exists a positive constant a such that ∗

∗

L tM1 L tM1 − a L tM2 L tM2 is a Hilbert-Schmidt operator for all (some) 0 < t < ∞. ∗

Proof. Since the sum system for M and L tM L tM depend only on the outer component of M, we may and do assume that M1 and M2 are outer. Since L tM1 (L tM2 )−1 ctM2 = ctM1 , the two sum systems are isomorphic if and only if there exists a positive constant a such that (L tM2 )−1∗ (L tM1 )∗ L tM1 (L tM2 )−1 − a1

560

M. Izumi, R. Srinivasan

M2 is a Hilbert-Schmidt operator in B(G 0,t ) for all (some) 0 < t < ∞. Since L tM2 is

invertible, this is equivalent to that (L tM1 )∗ L tM1 − a(L tM2 )∗ L tM2 is a Hilbert-Schmidt operator, which shows the statement.  

Theorem 6.7. Let M ∈ HD2,R be an outer function. We assume that there exist two positive constants b1 , b2 such that b1 ≤ |M(iλ)| ≤ b2 for almost every λ. Then the semigroup {et A M }t>0 gives rise to a type I E 0 -semigroup if and only if there exists a positive constant a such that ∗ Pt T|M|2 Pt − a Pt T1/M T1/M Pt

is a Hilbert-Schmidt operator for all (some) 0 < t < ∞. If moreover (I − Pt )T|M|2 Pt is a Hilbert-Schmidt operator for all 0 < t < ∞, then this condition is equivalent to the statement that Pt T|M|2 −√a Pt is a Hilbert-Schmidt for all 0 < t < ∞. Proof. By assumption, M and 1/M are in H ∞ (Hr ) and T M and T1/M are bounded. We ∗ T M also have T|M|2 = T M∗ T M and T1/|M|2 = T1/M 1/M . Lemma 6.2 implies that Jt is the ∗ M restriction of T1/M T1/M to G 0,t . Thus a similar argument as above using Theorem 6.1 implies the first statement. Now assume that (I − Pt )T|M|2 Pt is a Hilbert-Schmidt operator for all 0 < t < ∞. Since ∗ ∗ (Pt T1/M T1/M Pt )(Pt T|M|2 Pt ) = Pt − Pt T1/M T1/M (I − Pt )T|M|2 Pt , ∗ P modulo the Hilbert-Schmidt the operator Pt T|M|2 Pt is the inverse of Pt T1/M T1/M t operators. Thus thanks to the first statement, the semigroup {et A M }t>0 gives rise to a type I E 0 -semigroup if and only if there exists a positive constant a such that

(Pt T|M|2 Pt )2 − a Pt is a Hilbert-Schmidt operator for all 0 < t < ∞, which is equivalent that Pt T|M|2 −√a I Pt is a Hilbert-Schmidt operator for all 0 < t < ∞.   Note that the assumptions of the above two theorems can be checked by using the Fourier transformation of the symbols in distribution sense. We now collect a few useful criteria for local boundedness of Toeplitz operators with unbounded symbols. For f ∈ L 2 (R) we denote by fˆ(ξ ) its Fourier transformation with normalization  ∞ f (x)e−iξ x d x, f ∈ L 1 (R) ∩ L 2 (R). fˆ(ξ ) = −∞

The next lemma may be regarded as a version of the well-known uncertainty principle. Lemma 6.8. Let E and F be measurable subsets of R with finite Lebesgue measures |E| and |F| respectively. For f ∈ L 2 (R) whose support is in E, we have  | fˆ(ξ )|2 dξ ≤ |E||F||| f ||2 . F

Generalized CCR Flows

561

Proof. First we assume f ∈ L 1 (R) ∩ L 2 (R). We take g ∈ L 2 (R) such that gˆ is the characteristic function of F. Then the Plancherel theorem implies   ∞  ∞ 2 2 ˆ ˆ ˆ )| dξ = 2π | f (ξ )| dξ = | f (ξ )g(ξ | f ∗ g(x)|2 d x. −∞

F

On the other hand,

−∞





| f ∗ g(x)|2 = |

f (t)g(x − t)dt|2 ≤ || f ||2 E

and so

|g(x − t)|2 dt, E



 | fˆ(ξ )|2 dξ ≤ 2π || f ||2 F



∞

|g(x − t)|2 dtd x = 2π || f ||2 ||g||2 |E|

−∞

E

= || f ||2 |E||F|. The statement in the general case follows from an easy approximation argument.

 

Lemma 6.9. Let M ∈ HD2,R . If there exists ε > 0 such that the Lebesgue measure of {λ ∈ R; |M(iλ)| ≤ ε} is finite, then there exists a positive constant Ct for each 0 < t < ∞ such that ||L tM f || ≥ Ct || f ||, ∀ f ∈ L 2 (0, t)R ∩ D(T M ). Proof. Note that we have ||T M

1 f || = 2π 2



∞

−∞

|M(iλ)|2 |L[ f ](iλ)|2 dλ,

for f ∈ D(T M ) as we have M(z)L[ f ](z) ∈ H 2 (Hr ). Let 1 }. n By assumption, |E n | is finite for large n. Note that M(iλ) = 0 almost everywhere as M(z)/(1 + z) ∈ H 2 (Hr ), which implies that {|E n |}∞ n=1 converges to 0. For f ∈ L 2 (0, t)R ∩ D(T M ), we have   ∞ |M(iλ)|2 |L[ f ](iλ)|2 dλ ≥ |M(iλ)|2 |L[ f ](iλ)|2 dλ −∞ R\E n  1 ≥ |L[ f ](iλ)|2 dλ n R\E n  2π || f ||2 − E n |L[ f ](iλ)|2 dλ = n (2π − t|E n |)|| f ||2 , ≥ n where we use Lemma 6.8. Thus for sufficiently large n with 2π > t|E n |, we get E n = {λ ∈ R; |M(iλ)| ≤

||T M f ||2 ≥ which finishes the proof.

 

2π − t|E n | || f ||2 , 2π n

562

M. Izumi, R. Srinivasan

Lemma 6.10. Let ϕ ∈ L 2 (R) and let (iλ) = ϕ(λ). ˆ Then L 2 (0, t)R ⊂ D(T ) and T Pt is a Hilbert-Schmidt operator for all 0 < t < ∞. Proof. Let f ∈ L 2 (0, t)R . Since L 2 (0, t)R ⊂ L 1 (0, ∞), the convolution ϕ ∗ f makes sense as an element in L 2 (R) and we get T f (x) = 1(0,∞) (x)ϕ ∗ f (x). In particular, T Pt is the integral operator with the kernel 1(0,∞) (x)ϕ(x − y)1(0,t] (y) and so  t ∞  ∞  ||T Pt ||2H.S = |ϕ(x − y)|2 d xd y = |ϕ(u)|2 t ∧ (t + u) du ≤ t||ϕ||2 . 0

−t

0

  In a similar way as above, we can show the following: Lemma 6.11. Let ϕ ∈ L 1 (R) and let (iλ) = ϕ(λ). ˆ (1) T is bounded. (2) Pt T Pt is a Hilbert-Schmidt operator for all 0 < t < ∞ if and only if ϕ ∈ L 2loc (R). (3) (1 − Pt )T Pt is a Hilbert-Schmidt operator for all 0 < t < ∞ if and only if  ∞ |ϕ(x)|2 (1 ∧ x)d x < ∞. 0

7. Examples Let ϕ be a real function in L 1loc [0, ∞) ∩ L 2 ((0, ∞), x ∧ 1d x) and let M = 1 − L[ϕ]. Then M belongs to HD2,R . We determine the type of the generalized CCR flow arising from such M. Lemma 7.1. Let the notation be as above. Then L tM is a bounded invertible operator M for all 0 < t < ∞. from L 2 (0, t)R onto G 0,t Proof. Let ϕ1 (x) = 1(0,1] (x)ϕ(x) and ϕ2 (x) = 1(1,∞) (x)ϕ(x). Then ϕ1 ∈ L 1 (0, ∞) and ϕ2 ∈ L 2 (0, ∞), and so Lemma 6.9, Lemma 6.10, and Lemma 6.11 imply the statement.   Lemma 7.2. Let the notation be as above. For a natural number n, we set gn (x) =

n−1
 1( 2k , 2k+1 ] (x) − 1( 2k+1 , 2k+2 ] (x) . k=0

2n

2n

2n

2n

Then {T M gn − gn }∞ n=1 converges to 0. Proof. Note that L[gn ](z) = Fn (z) holds, where Fn is as in the proof of Theorem 6.4. Let i (z) = L[ϕi ](z) for i = 1, 2, where ϕ = ϕ1 + ϕ2 is the decomposition as in the proof of Lemma 7.1. Then T M gn − gn = −T1 gn − T2 gn . Since {gn } converges to 0 weakly, the second term converges to 0 as T2 P1 is a compact operator thanks to Lemma 6.10. Since 1 (iλ) is a continuous function vanishing at infinity, we get  ∞ 1 |1 (iλ)|2 |Fn (iλ)|2 dλ → 0, (n → ∞). ||T1 gn ||2 = 2π −∞  

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563

Lemma 7.3. Let f, g, h ∈ L 1 (0, 1)∩ L 2 ((0, 1), xd x). We regard L 1 (0, 1) as a subspace of L 1 (R) naturally and set g(x) ˜ = g(−x). If f, h ∈ / L 2 (0, 1) such that ||h|| L 2 (t,1)

lim sup

|| f || L 2 (t,1)

t→+0

= C < ∞,

then ||h ∗ g|| ˜ L 2 (t,1)

lim

|| f || L 2 (t,1)

t→+0

= 0.

Proof. Let 0 < t, ε < 1 with t + ε < 1. Then  1  1  1 ||h ∗ g|| ˜ 2L 2 (t,1) ≤ dx dr ds|g(r )||g(s)||h(x + r )||h(x + s)| t



≤

0



1

dx t

0



1

1

dr 0 1



= ||g||1

0 1

 dx



t



0

ds|g(r )||g(s)|

dr |g(r )||h(x + r )|2

0 1−t

= ||g||1 ≤ ||g||1 0

ε

|g(r )|||h||2L 2 (t+r,1) dr

|g(r )|dr ||h||2L 2 (t,1) + ||g||21 ||h||2L 2 (ε,1) .

Thus lim sup

||h ∗ g|| ˜ L 2 (t,1)

t→+0

and the statement holds.

|h(x + r )|2 + |h(x + s)|2 2

|| f || L 2 (t,1)



ε

≤ C||g||1

|g(r )|dr,

0

 

Theorem 7.4. Let ϕ1 , ϕ2 ∈ L 1loc [0, ∞) ∩ L 2 ((0, ∞), x ∧ 1d x)R and let Mi = 1 − L[ϕi ] for i = 1, 2. (1) If ϕ1 − ϕ2 ∈ L 2 (0, ∞), the two sum systems for M1 and M2 are isomorphic. (2) Assume ϕ1 ∈ / L 2 (0, ∞). If the two sum systems for M1 and M2 are isomorphic, then the following two statements hold: lim

t→+0

lim

t→+0

||ϕ2 || L 2 (t,∞) ||ϕ1 || L 2 (t,∞)

= 1,

||ϕ1 − ϕ2 || L 2 (t,∞) ||ϕ1 || L 2 (t,∞)

= 0.

(7.1)

(7.2)

Proof. Lemma 7.1 shows that L tMi is a bounded invertible operator from L 2 (0, t)R onto Mi . Therefore we can apply Theorem 6.6 to prove the theorem. G 0,t (1) Assume that ϕ1 − ϕ2 ∈ L 2 (0, ∞). Lemma 6.10 implies that L tM1 − L tM2 is a Hilbert-Schmidt operator for all 0 < t < ∞ and Theorem 6.6 shows that the sum systems for M1 and M2 are isomorphic.

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(2) Assume now that the sum systems for M1 and M2 are isomorphic. By the result just proved above, we may add a function in L 2 (0, ∞) to ϕi without changing the assumption. Therefore we may and do assume that ϕi has support in (0, 1) and ||ϕi ||1 < 1 by truncating ϕi . As a consequence, T Mi is a bounded operator and we have (L tMi )∗ L tMi = / L 2 (0, 1). Pt T|Mi |2 Pt now. Assume that ϕ1 ∈ Theorem 6.6 implies that there exists a positive constant a such that Pt T|M1 |2 −a|M2 |2 Pt is a Hilbert-Schmidt operator for all 0 < t < ∞. This is possible only if a = 1 thanks to Lemma 7.2. Let i = L[ϕi ]. Then |M1 |2 − |M2 |2 = 2 − 1 + 2 − 1 + |1 |2 − |2 |2 , and we have T|M1 |2 −|M2 |2 f = 1(0,∞) (x)g ∗ f (x) for f ∈ L 2 (0, ∞) with g = ϕ2 − ϕ1 + ϕ˜2 − ϕ˜1 + ϕ˜1 ∗ ϕ1 − ϕ˜2 ∗ ϕ2 ∈ L 1 (R), where the notation in Lemma 7.3 is used. Lemma 6.11 implies that g ∈ L 2loc (R) or equivalently g ∈ L 2 ([−1, 1]) as g is supported by [−1, 1]. Setting h = ϕ2 − ϕ1 , we have g = h + h˜ − h ∗ h˜ − ϕ1 ∗ h˜ − h ∗ ϕ˜1 . First suppose that (7.1) does not hold. Exchanging the roles of ϕ1 and ϕ2 if necessary, we may assume that there exist a number 0 < δ < 1 and a decreasing sequence of positive numbers {tn }∞ n=1 converging to 0 such that ||ϕ2 || L 2 (tn ,1) ≤ (1 − δ)||ϕ1 || L 2 (tn ,1) holds for all n ∈ N. Then ||h|| L 2 (tn ,1) ≥ δ||ϕ1 || L 2 (tn ,1) . On the other hand we have 1=

||g + h ∗ h˜ + ϕ1 ∗ h˜ + h ∗ ϕ˜1 || L 2 (tn ,1) ||h|| L 2 (tn ,1)

,

whose right-hand side would converge to 0 due to Lemma 7.3, which is a contradiction. Thus Eq. (7.1) holds. Equation (7.2) can be shown in a similar way in the presence of (7.1).   Theorem 7.5. Let ϕ ∈ L 1loc [0, ∞) ∩ L 2 ((0, ∞), x ∧ 1d x)R and let M = 1 − L[ϕ]. Then the generalized CCR flow arising from {et A M }t>0 is of type I if and only if ϕ ∈ L 2 (0, ∞). In consequence, if ϕ ∈ / L 2 (0, ∞), then the corresponding generalized CCR flow is of type III. Proof. Thanks to Theorem 7.4, (1), to prove the theorem we may assume that ϕ is supported by (0, 1) and ||ϕ||1 < 1. In this case, the function M(z) is continuous and 1 − ||ϕ||1 ≤ |M(z)| ≤ 1 + ||ϕ||1 holds on the closure of Hr . In particular M is an outer function. Therefore we can apply Theorem 6.7 to M(z). First we claim that (1 − Pt )T|M|2 Pt is a Hilbert-Schmidt operator for all 0 < t < ∞. Let  = L[ϕ]. Then (1 − Pt )T|M|2 Pt = −(1 − Pt )T Pt − (1 − Pt )T Pt + (1 − Pt )T||2 Pt .

Generalized CCR Flows

565

Thanks to Lemma 6.11, the first two terms above are Hilbert-Schmidt operators. Note that we have T||2 f = 1(0,∞) ϕ ∗ ϕ˜ ∗ f for f ∈ L 2 (0, ∞). Thus to prove the claim, it suffices to show that 

1

t|ϕ ∗ ϕ(t)| ˜ 2 dt < ∞.

0

Indeed, we have 

1

 2 x|ϕ ∗ ϕ(x)| ˜ dx ≤

0



1



1

xd x 0

 ≤



1

0



1



1

= ||ϕ||1

0 1

xd x 

0

0

ds|ϕ(r )||ϕ(s)|

1

ds|ϕ(s)| 0

 ||ϕ||21

|ϕ(x + r )|2 + |ϕ(x + s)|2 2

ds|ϕ(s)||ϕ(x + s)|2 

1

= ||ϕ||1 ≤

1

dr 0



ds|ϕ(r )||ϕ(s)||ϕ(x + r )||ϕ(x + s)|

0

xd x 0

1

dr

dy(y − s)|ϕ(y)|2

s 1

x|ϕ(x)|2 d x < ∞.

0

Theorem 6.7 together with the above claim shows that the generalized CCR flow arising from {et A M }t>0 is of type I if and only if Pt T|M|2 −1 Pt is a Hilbert-Schmidt operator for all t, which is further equivalent to that g = ϕ ∗ ϕ˜ − ϕ − ϕ, ˜ is in L 2 (−1, 1) thanks to Lemma 6.11,(2). Lemma 7.3 shows that this is equivalent to the condition ϕ ∈ L 2 (0, 1).   Remark 7.6. Let ϕ be as above and ||ϕ||1 < 1. Then as shown in [10, Sect. 6], the sum M can be realized as follows: G M = L 2 (0, t) as a topological vector space system G 0,t R 0,t with a new inner product  f, gG =

1 2π



∞

−∞

2 ˆ fˆ(λ)g(λ)|M(iλ)| dλ.

M is equivalent to the usual L 2 -norm as we have Note that the norm of G 0,t

1 − ||ϕ||1 ≤ |M(iλ)| ≤ 1 + ||ϕ||1 . This means that the invariant for a product system (through the associated sum system) discussed in [18] and [6] can not distinguish the product system corresponding to this M from the exponential product system of index 1.

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8. Type III factors and type III E0 -semigroups 8.1. Local algebras. For a product system E = (E t )t>0 , following [12,18 and 6], we introduce an analogue of the observable algebra for a finite interval I = (s, t) ⊂ (0, a) by a a ∗ A E (I ) = Us,t (C1 E s ⊗ B(E t−s ) ⊗ C1 Ea−t )Us,t , a is the unique unitary operator between the Hilbert spaces E ⊗ E where Us,t s t−s ⊗ E a−t and Ha , determined uniquely by the associativity of the product system.  For a family {Aλ }λ∈ of von Neumann algebras, we use the notation λ∈ Aλ for the von Neumann algebra generated by {Aλ }λ∈ . For an open subset U of (0, a), we set  A E (I ), A E (U ) = I ⊂U

where I runs over all intervals contained in U . Then the isomorphism class of A E (U ) does not depend on the choice of a. Note that we can always ignore a finite subset of (0, ∞) whenever we deal with A E (U ) as we have A E ((r, s) ∪ (s, t)) = A E ((r, t)) (see for instance Corollary 25 in [6]). When U is an elementary set, that is a finite union of intervals, then A E (U ) is a type I factor because it is generated by finitely many mutually commuting type I factors. However, when U has infinitely many components, it is not clear whether A E (U ) is of type I or even it is not clear whether A E (U ) is a factor. Indeed, Murray von Neumann’s notion of the type of A E (U ) gives a computable invariant of the product system (E t )t>0 . Lemma 8.1. Let E = (E t )t>0 be a product system and let U be a bounded non-empty open subset of (0, ∞). Then (1) If E has a unit, then A E (U ) has a direct summand that is a type I∞ factor. (2) If E is of type I, then A E (U ) is a type I∞ factor. Proof. (1) We choose a > 0 such that U ⊂ (0, a) and assume that A E (U ) acts

on E a . Since U is open, there exist mutually disjoint open intervals In such that U = ∞ n=1 In . Let v = (vt )t>0 be a unit. Then we may and do assume that ||vt || = 1 for all t > 0. Let L = A E (U )va and PL be the projection from E a onto L, which belongs to A E (U ) . We introduce a state ω of A E (U ) by ω(x) = xv1 , v1 . Then for xi ∈ A E (Ii ), we have ω(x1 x2 · · · xn ) = ω(x1 )ω(x2 ) · · · ω(xn ), n A(Ii ) ⊂ A E (U ) for all n. Therefore which shows that ω is a product pure state of i=1 E A (U )PL is a type I∞ factor. (2) We may assume that E is the exponential product system with the test function space L 2 ((0, ∞), K ), where K is the multiplicity space. Then A E (U ) is nothing but B((L 2 (U, K ))).  

Remark 8.2. If E has a unit and U = ∞ n=1 (an , bn ) with bn < an+1 , then it is easy to show that A E (U ) is a type I factor. Indeed, using the notation of the proof of (1) above, we have A E (U ) L = E(0, a) in this case. Therefore the defining representation of A E (U ) is quasi-equivalent to its restriction to L.

Generalized CCR Flows

567

8.2. Type of Aϕ (U ). We fix ϕ ∈ L 1loc [0, ∞) ∩ L 2 ((0, ∞), 1 ∧ xd x)R \ L 2 (0, ∞) and set  = L[ϕ], M = 1 − . Let H = (L 2 (0, ∞)) and α ϕ be the E 0 -semigroup acting on B(H ) defined by ϕ

αt (W ( f + ig)) = W (St + i Tt g),

f, g ∈ L 2 (0, ∞)R .

Then α ϕ is of type III as we saw in Theorem 6.7. Since the cocycle conjugacy class of α ϕ does not change under a L 2 -perturbation of ϕ, we assume that ϕ is supported by [0, 1) and ||ϕ||1 < 1/3. Note that the Toeplitz operator T M is a bounded invertible operator with T M−1 = T1/M in this case. ϕ Let E ϕ be the product system for α ϕ and Aϕ (U ) := A E (U ). Then we may and do ϕ ϕ identify Aϕ ((0, t)) with αt (B(H )) and Aϕ ((s, t)) with αs (Aϕ ((0, t − s))). Lemma 8.3. Let the notation be as above. Then Aϕ (U ) = {W (T M f + iT M∗−1 g); f, g ∈ L 2 (U )R } . In consequence, the von Neumann algebra Aϕ (U ) is either a type I factor or a type III factor. Proof. Let I = (s, t) be an open interval. When s = 0, using the duality theorem [1, Theorem 1’,(5)], we have Aϕ (0, t) = {W (St f + i Tt g); f, g ∈ L 2 (0, ∞) R } M = {W ( f + ig); f ∈ G 0,t , g ∈ L 2 (0, t)R } . Therefore M Aϕ ((s, t)) = {W (Ss f + i Ts g); f ∈ G 0,t−s , g ∈ L 2 (0, t − s)R } ,

and thanks to Theorem 5.9, Aϕ ((s, t)) = {W (ξ cx,y + iηdu,v ); s < x < y < t, s < u < v < t, ξ, η ∈ R} . Note that we have cx,y = T M 1(x,y) and ∗ du,v = T1/|M|2 cu,v = T1/M T1/M cu,v = T M∗−1 1(s,t) ,

and so we get Aϕ (I ) = {W (T M f + iT M∗−1 g); f, g ∈ L 2 (I )R } . For an open subset U of (0, 1), we take mutually disjoint open intervals In such that U= ∞ n=1 In . Then Aϕ (U ) =

∞ 

{W (T M f + iT M∗−1 g); f, g ∈ L 2 (In )R }

n=1

= {W (T M f + iT M∗−1 g); f, g ∈ L 2 (U )R } . Let K 1 = T M L 2 (U )R and K 2 = T M−1∗ L 2 (U )R , which are closed spaces of L 2 (0, ∞)R . To show that Aϕ (U ) is a factor, it suffices to show that K 1 ∩ K 2⊥ = K 1⊥ ∩ K 2 = {0} (see [1, Theorem 1’,(4)]). This immediately follows from T M f, T M−1∗ g =  f, g. The rest of the statement follows from [2].  

568

M. Izumi, R. Srinivasan

Let K 1 and K 2 be as above. It is shown in [1, Lemma 4.1, 2] that there exists an unique closed operator V : K 1 → K 1⊥ such that K 2 is the graph of V . Moreover, the factor Aϕ (U ) is of type I (resp. type III) if and only if V is (resp. is not) a Hilbert-Schmidt class operator. (Though K 1 ∩ K 2 = K 1⊥ ∩ K 2⊥ = {0} is also assumed in [1], it is not really necessary.) Lemma 8.4. Let the notation be as above and let PU be the projection onto L 2 (U )R . Then V is the restriction of −T M−1∗ (I − PU )T M∗ to K 1 . In consequence, the factor Aϕ (U ) is of type I if and only if (I − PU )T+−||2 PU is a Hilbert-Schmidt class operator. Proof. Let V0 := −T M−1∗ (I − PU )T M∗ | K 1 . For f, g ∈ L 2 (U )R , we have V0 T M f, T M g = −(I − PU )T M∗ T M f, g = 0, which shows that V0 is a bounded operator from K 1 to K 1⊥ . Moreover, we have T M f + V0 T M f = T M f − T M−1∗ (I − PU )T M∗ T M f = T M−1∗ PU T M∗ T M f ∈ K 2 . Therefore to prove that V = V0 , it suffices to show that PU T M∗ T M PU is invertible in B(L 2 (U )). Since T M∗ T M = I − T − T∗ + T∗ T , this follows from ||T || ≤ ||ϕ||1 < 1/3. Since T M PU is an invertible operator from L 2 (U )R onto K 1 , we conclude that V is a Hilbert-Schmidt class operator if and only if T M∗ V T M PU = −(I − PU )T|M|2 PU = (I − PU )T+−||2 PU is a Hilbert-Schmidt class operator.

 

For two subsets E, F of R, we denote by E  F the symmetric difference of E and F. Theorem 8.5. Let ϕ ∈ L 1loc [0, ∞) ∩ L 2 ((0, ∞), 1 ∧ xd x)R , and let U be a bounded open subset of (0, ∞). We regard ϕ as an element of L 1 (R) and set ϕ(x) ˜ = ϕ(−x). Then Aϕ (U ) is a type I factor (resp. type III factor) if and only if the integral  I1 =

1

2 |ϕ(x) − ϕ ∗ ϕ(x)| ˜ |(U + x)  U |d x

0

converges (resp. diverges). When there exists a > 0 such that ϕ(x) is monotone on (0, a), the above integral converges if and only if the integral 

1

I2 = 0

converges.

|ϕ(x)|2 |(U + x)  U |d x

Generalized CCR Flows

569

Proof. Note that the isomorphism class of Aϕ (U ) and whether I1 is finite or not are stable under L 2 -perturbation of ϕ. Therefore we may and do assume that ϕ is supported by [0, 1] and ||ϕ||1 < 1/3. Then thanks to Lemma 8.4, the factor A E (U ) is of type I if and only if   I3 := dx dy|ϕ(x − y) + ϕ(x ˜ − y) − ϕ ∗ ϕ(x ˜ − y)|2 < ∞, (0,∞)\U

U

because (I − PU )T+−||2 PU is the integral operator with the kernel (ϕ(x − y) + ϕ(x ˜ − y) − ϕ ∗ ϕ(x ˜ − y))1(0,∞)\U (x)1U (y). On the other hand, we have  1 2 I3 = |ϕ(s) + ϕ(s) ˜ − ϕ ∗ ϕ(s)| ˜ |([0, ∞) \ U ) ∩ (U + s)|ds  = 

−1 1

 2 |([0, ∞) \ U ) ∩ (U + s)| + |([0, ∞) \ U ) ∩ (U − s)| ds |ϕ(s) − ϕ ∗ ϕ(s)| ˜

0 1

=

2 |ϕ(s) − ϕ ∗ ϕ(s)| ˜ (|(U + s)  U | − |[0, s] ∩ U |)ds.

0

Since



1

 2 |ϕ(s) − ϕ ∗ ϕ(s)| ˜ |[0, s] ∩ U |ds ≤

0

1

2 |ϕ(s) − ϕ ∗ ϕ(s)| ˜ sds < ∞,

0

we get the first statement. Assume now that ϕ is monotone on (0, a). When lim x→+0 ϕ(x) is finite, the function ϕ is in L 1 (0, ∞) ∩ L 2 (0, ∞) and I1 and I2 are finite. Assume that the limit at 0 does not exist. Then by truncating ϕ if necessary, we may assume that either ϕ(x) is non-negative and decreasing or ϕ(x) is non-positive and increasing. Assume that ϕ(x) is non-negative and decreasing. (The other case can be treated in a similar way.) Then for 0 < x < 1 we have  1 ϕ(x) ≥ ϕ(x) − ϕ ∗ ϕ(x) ˜ = ϕ(x) − ϕ(x + t)ϕ(t)dt 0



1

≥ ϕ(x) − ϕ(x)

ϕ(t)dt = (1 − ||ϕ||1 )ϕ(x).

0

Therefore I1 is finite if and only if I2 is finite.

 

When U is an elementary set, we have |(U + x)  U | = O(x), (x → 0) and I1 is always finite, which recovers the fact that Aϕ (U ) is a type I factor. To construct U such that Aϕ (U ) is of type III, we have to control the speed of convergence of lim x→+0 |(U + x)  U | = 0. Lemma 8.6. Let f (x) be a non-negative strictly decreasing continuous function on (0, ∞) satisfying lim x→+0 f (x) = ∞, lim x→∞ f (x) = 0, and  1 f (x)d x < ∞. 0

570

M. Izumi, R. Srinivasan

n We set a0 = 0, a2n−1 = a2n = f −1 (n) for n ∈ N, and bn = k=0 ak . Let In =

∞ (b2n , b2n+1 ) and U = I . Then U is a bounded open set and the following n n=0 estimate holds for every 0 < x < f −1 (1):  x (2 f (x) − 1)x ≤ |(U + x)  U | ≤ (2 f (x) + 1)x + 2 f (s)ds. 0

Proof. Note that we have ∞


f

−1

 (k) ≤

f −1 (n+1)

f (s)ds − n f −1 (n + 1)

0

k= n+1

and in particular, the set U is bounded. For 0 < x ≤ f −1 (1), we choose n so that f −1 (n + 1) < x ≤ f −1 (n), which is equivalent to n ≤ f (x) < n + 1. Then we have nx ≤ |(U + x) \ U | ≤ nx +

∞


|Ik |,

k= n+1

(n + 1)x ≤ |(U − x) \ U | ≤ (n + 1)x +

∞


|Ik |,

k= n+2

and so (2n + 1)x ≤ |(U + x)  U | ≤ (2n + 1)x + 2

∞


f −1 (k).

k= n+1

Since f (x) − 1 < n, we get the lower estimate. Since  f −1 (n+1)  ∞
−1 −1 f (k) ≤ f (s)ds − n f (n + 1) ≤ k= n+1

0

x

f (s)ds

0

and n ≤ f (x), we get the upper estimate.

 

For two real valued functions h(x) and g(x) on (0, ∞), we denote h(x)  g(x) if there exist positive numbers a, b, c > 0 such that bh(x) ≤ g(x) ≤ ch(x), ∀x ∈ (0, a). Corollary 8.7. Assume that f (x) = x γ −1 L(x) satisfies the assumption of Lemma 8.6 such that 0 < γ ≤ 1 and L(x) is a slowly varying function (see [8] for the definition). If U is a bounded open subset of (0, ∞) constructed from f in Lemma 8.6, then |(U + x)  U |  x γ L(x). For 0 < β ≤ 1/2, we set ϕβ (x) = x β−1 e−x . Then ϕβ satisfies the assumption of Theorem 8.5. Corollary 8.7 shows that for any 0 < β1 < β2 ≤ 1/2, there exists a bounded open set U ⊂ (0, ∞) such that Aϕβ1 (U ) is a type III factor and Aϕβ2 (U ) is a type I factor. In particular α ϕβ1 and α ϕβ2 are not cocycle conjugate. Theorem 8.8. There exist uncountably many mutually non-cocycle conjugate type III E 0 -semigroups of the form α ϕ with ϕ ∈ L 1loc [0, ∞) ∩ L 2 ((0, ∞), 1 ∧ xd x)R .

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571

References 1. Araki, H.: A lattice of von Neumann algebras associated with the quantum theory of a free Bose field. J. Math. Phys. 4, 1343–1362 (1963) 2. Araki, H.: Type of von Neumann algebra associated with free Bose field. Progr. Theoret. Phys. 32, 956–965 (1964) 3. Araki, H.: On quasifree states of the canonical commutation relations. II. Publ. Res. Inst. Math. Sci. 7, 121–152 (1971) 4. Arveson, W.: Continuous analogues of Fock spaces. Mem. Americ. Math. Soc. 80(409), 1–66 (1989) 5. Arveson, W.: Non-commutative dynamics and E-semigroups. Springer Monograph in Math, BerlinHeidelberg-New York: Springer, 2003 6. Rajarama Bhat, B.V., Srinivasan, R.: On product systems arising from sum systems. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8, 1–31 (2005) 7. Daele, A.van : Quasi-equivalence of quasi-free states on the Weyl algebra. Commun. Math. Phys. 21, 171–191 (1971) 8. Feller, W.: An Introduction to Probability Theory and its Applications. Vol. II. Second edition, New York-London-Sydney: John Wiley & Sons, Inc., 1971 9. Hoffman, K.: Banach Spaces of Analytic Functions. Englewood Cliffs, N J: Prentice-Hall, Inc., 1962 10. Izumi, M.: A perturbation problem for the shift semigroup. J. Funct. Anal 251(2), 498–595 (2007) 11. Koosis, P.: Introduction to H p Spaces. Second edition. With two appendices by V. P. Havin. Cambridge Tracts in Mathematics, 115. Cambridge: Cambridge University Press, 1998 12. Liebscher, V.: Random sets and invariants for (type I I ) continuous product systems of Hilbert spaces, http://arxiv.org/list/math.PR/0306365, 2003 13. Parthasarathy, K.R.: An Introduction to Quantum Stochastic Calculus. Basel-Boston-Berlin: Birkauser, 1992 14. Powers, R.T.: A nonspatial continuous semigroup of ∗-endomorphisms of B(H ). Publ. Res. Inst. Math. Sci. 23, 1053–1069 (1987) 15. Powers, R.T.: New examples of continuous spatial semigroups of endomorphisms of B(H ). Int. J. Math. 10(2), 215–288 (1999) 16. Price, G.L., Baker, B.M., Jorgensen, P.E.T., Muhly, P.S. (eds.),: Advances in Quantum Dynamics, (South Hadley, MA, 2002) Contemp. Math. 335, Providence, RI: Amer. Math. Soc., 2003 17. Shalom, Y.: Harmonic analysis, cohomology, and the large-scale geometry of amenable groups. Acta Math. 192, 119–185 (2004) 18. Tsirelson, B.: Non-isomorphic product systems. Advances in Quantum Dynamics (South Hadley, MA, 2002), Contemp. Math., 335, Providence, RI: Amer. Math. Soc., 2003, pp. 273–328 19. Tsirelson, B.: Spectral densities describing off-white noises. Ann. Inst. H. Poincaré Probab. Statist. 38, 1059–1069 (2002) 20. Yosida, K.: Functional Analysis. Sixth edition. Berlin-New York: Springer-Verlag, 1980 21. Zhu, K.H.: Operator Theory in Function Spaces. Monographs and Textbooks in Pure and Applied Mathematics 139. New York: Marcel Dekker, Inc., 1990 Communicated by Y. Kawahigashi

Commun. Math. Phys. 281, 573–596 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0523-4

Communications in

Mathematical Physics

Asymptotic Analysis for a Vlasov-Fokker-Planck/ Compressible Navier-Stokes System of Equations A. Mellet
, A. Vasseur Department of Mathematics, University of Texas at Austin, Austin, TX 78712, USA. E-mail: [email protected]; [email protected]; [email protected] Received: 16 May 2006 / Accepted: 12 February 2008 Published online: 6 June 2008 – © Springer-Verlag 2008

Abstract: This article is devoted to the asymptotic analysis of a system of coupled kinetic and fluid equations, namely the Vlasov-Fokker-Planck equation and a compressible Navier-Stokes equation. Such a system is used, for example, to model fluid-particle interactions arising in sprays, aerosols or sedimentation problems. The asymptotic regime corresponding to a strong drag force and a strong Brownian motion is studied and the convergence toward a two phase macroscopic model is proved. The proof relies on a relative entropy method.

1. Introduction 1.1. The model. This paper is devoted to the asymptotic analysis of a system of equations modeling the evolution of dispersed particles in a compressible fluid. This kind of system arises in a lot of industrial applications. One example is the analysis of sedimentation phenomenon, with applications in medicine, chemical engineering or waste water treatment (see Berres, Bürger, Karlsen, and Tory [5], Gidaspow [13], Sartory [21], Spannenberg and Galvin [22]). Such systems are also used in the modeling of aerosols and sprays with applications, for instance, in the study of Diesel engines (see Williams [23,24]). At the microscopic scale, the cloud of particles is described by its distribution function f (x, v, t), solution to a Vlasov-Fokker-Planck equation: ∂t f + v · ∇x f + divv (Fd f − ∇v f ) = 0.

(1)

The fluid, on the other hand, is modeled by macroscopic quantities, namely its density ρ(x, t) ≥ 0 and its velocity field u(x, t) ∈ R N . We assume that the fluid is compressible
A. Mellet was partially supported by NSF grant DMS-0456647.

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A. Mellet, A. Vasseur

and isentropic, so that (ρ, u) solves the compressible Euler or Navier-Stokes system of equations:
∂t ρ + divx (ρu) = 0 (2) ∂t (ρu) + divx (ρu ⊗ u) + ∇x p − νu = F f (with ν = 0 in the inviscid case and ν > 0 in the viscous case). The fluid-particle interactions are modeled by a friction (or drag) force exerted by the fluid onto the particles. This force is assumed to be proportional to the relative velocity of the fluid and the particles: Fd = F0 (u(x, t) − v), where F0 is supposed to be constant equal to 1 (see Remark 1.1 below). The right handside in the Euler equation takes into account the action of the cloud of particles on the fluid:   F f = − Fd f dv = F0 (v − u(x, t)) f (x, v, t) dv. For the sake of simplicity, we assume that the pressure term is given by p = ργ , though more general pressure terms could be taken into consideration. This particular Vlasov-Navier-Stokes system of equations is used, for instance, in the modeling of reaction flows of sprays (see Williams [23,24]) and is at the basis of the code KIVA-II of the Los Alamos National Laboratory (see O’Rourke et al. [1] and Amsden [2]). More recently, some properties of this model were investigated by J. Carrillo and T. Goudon in [11]. In particular the stability properties of the coupled system (1)–(2) were studied, and the asymptotic analysis, which is at stake in the present paper, was formally discussed for the first time. A first issue, when dealing with such a kinetic/fluid system of equations is the existence of solutions. Global existence results for the coupling of kinetic equations with incompressible Navier-Stokes equations was proved by Hamdache in [17]. The existence of solutions for short time in the case of the hyperbolic system (i.e. no viscosity in the Navier-Stokes equation (ν = 0) and no Brownian effect in the kinetic equation) is proved by Baranger and Desvillettes in [3]. In [20], we study the coupled system (1)–(2) and prove the existence of weak solutions ( f, ρ, u) for any γ > 3/2 under minimal assumption (finite mass and energy) on the initial data. Our result is recalled in Sect. 1.5. In this paper we address the question on the asymptotic regime corresponding to a strong drag force and strong brownian motion. More precisely, we consider the following system of singular equations: 1 ∂t f ε + v · ∇x f ε + divv ((u − v) f ε − ∇v f ε ) = 0, (3) ε (4) ∂t ρε + divx (ρε u ε ) = 0, 1 ∂t (ρε u ε ) + divx (ρε u ε ⊗ u ε ) + ∇x ρεγ − νu = ( jε − n ε u ε ), (5) ε   where n ε = f ε (x, v, t) dv and jε = v f ε (x, v, t) dv. Note that large time behavior in the case of incompressible fluids has been studied previously by Hamdache [17] and Goudon, Jabin and Vasseur [15,16].

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The main result stated in the present paper is the convergence of weak solutions ( f ε , ρε , u ε ) of (3)–(5) to (Mn,u , ρ, u) (Mn,u denotes the Maxwellian distribution with density n and velocity u) where (n, ρ, u) is solution of the following system of hydrodynamic equations: ⎧ ⎨ ∂t n + divx (nu) = 0, ∂t ρ + divx (ρu) = 0, (6) ⎩ ∂ ((ρ + n)u) + div ((ρ + n)u ⊗ u) + ∇ (n + ρ γ ) − νu = 0. t x x This kind of multi-fluid system is also widely used in the modeling of particle/fluid interaction (see for instance Laurent, Massot, and Villedieu [18], or Berthonnaud [7]). Notice that the Brownian effect in the kinetic equation ends up as an additional pressure term in the velocity equation. The derivation of (6) from (3)–(5) was first addressed, formally, by Carrillo and Goudon in [11]. In this paper, we rigorously justify this asymptotic analysis. Remark 1.1. The choice of drag force Fd = F0 (u − v) may not be the most relevant one from a physical point of view. In particular, a quadratic dependence in the velocity may be more appropriate in certain regimes. It could also seem more relevant from a physical point of view to assume that |Fd | depends on the density of the fluid ρ, such as Fd = ρ(u − v).

(7)

However, the asymptotic analysis is not valid in those situations, unless we can establish a priori lower bounds on the density. Indeed, we cannot expect to have relaxation of the kinetic regime toward an hydrodynamic regime with velocity u on the vacuum {ρ = 0}. The main tool in the proof is a relative entropy method. It relies on the “weakstrong” uniqueness principle established by Dafermos for multidimensional systems of hyperbolic conservation laws admitting a convex entropy functional [12]. It has been frequently used for a system of particles and rarefied gas dynamics, see Yau [25]. It is the main tool also for the asymptotic limit of the Boltzmann equation to the incompressible Navier-Stokes equation, see Bardos, Golse, Levermore [4], Golse, Saint-Raymond [14], Lions, Masmoudi [19]. See also Berthelin, Vasseur [6] and Goudon, Jabin, Vasseur [16] for other kind of Hydrodynamical limit. For different asymptotic problems it is called the “modulated energy” method (Brenier [9,10]).

1.2. Boundary conditions and notion of weak solutions. As usual, the kinetic variable v (the velocity) lies in R N , while the space variable x lies in a subset  of R N . We will be considering two situations: The case of a  bounded subset with periodic boundary condition ( = T N ), and the case of a  bounded subset of R3 with smooth boundary ∂. In the later case, the system (1)–(2) has to be supplemented with boundary condition along ∂. The natural assumption for Navier-Stokes equations is homogeneous Dirichlet condition for the velocity: u(x, t) = 0

∀x ∈ ∂.

To write the boundary condition for the kinetic equation, we introduce γ f , the trace of f on ∂, and we denote γ ± f (x, v, t) = f | ± , where ± = {(x, v) ∈ ∂ × R N | ± v · r (x) > 0}

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(r (x) denotes the outward normal unit vector). In this paper, our only requirement is that the boundary condition is such that the following conditions are satisfied:  (v · r (x)) γ f (x, v, t) dv = 0 ∀x ∈ ∂, (8) RN

and 



 |v|2 (v · r (x)) + log(γ f ) γ f dv ≥ 0 ∀x ∈ ∂. 2 RN

(9)

Those conditions are very classical in kinetic theory and they are satisfied if we consider local or diffusive reflection conditions. Typically, we write such conditions in the form γ − f (x, v, t) = B(γ + f )

∀(x, v) ∈ − ,

where the operator B is given by B(g) = α J (g) + (1 − α)D(g)

α ∈ [0, 1]

with a specular reflection operator J defined by J (g)(x, v) = g(x, Rx v)

Rx v = v − 2(v · r (x)) r (x),

and a diffusive operator given by (for example)  g(x, v)v · r dv , D(g)(x, v) = M(v) v·r >0

 where M(v) is a Maxwellian distribution satisfying v·r >0 M(v) |v · r | dv = 1 for all r ∈ S2. Other boundary conditions can be considered (such as elastic reflection conditions), we refer the reader to [20] for further considerations on those boundary conditions. Finally, we recall that ( f, ρ, u) is a weak solution of (3)–(5) on [0, T ] if f (x, v, t) ≥ 0 ∀(x, v, t) ∈  × R3 × (0, T ), f ∈ C([0, T ]; L 1 ( × R3 )) ∩ L ∞ (0, T ; L 1 ∩ L ∞ ( × R3 )), |v|2 f ∈ L ∞ (0, T ; L 1 ( × R3 )), and ρ(x, t) ≥ 0 ∀(x, t) ∈  × (0, T ), ρ ∈ L ∞ (0, T ; L γ ()) ∩ C([0, T ]; L 1 ()), u ∈ L 2 (0, T ; H01 ()), ρ |u|2 ∈ L ∞ (0, T ; L 1 ()), ρ u ∈ C([0, T ]; L 2γ /(γ +1) () − w). Moreover, we ask that (4)–(5) holds in the sense of distribution (Note that the conditions on f yield n(x, t) ∈ L ∞ (0, T ; L 6/5 ()) which is enough to give a meaning to the

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product n u in L 1 ((0, T ) × )), and in the case of reflection boundary conditions, we ask that the kinetic equation (3) holds in the following sense:  T f [∂t ϕ + v · ∇x ϕ + (u − v) · ∇v ϕ + v ϕ] d x dv dt ×R N 0  f 0 ϕ(x, v, 0) d x dv = 0 (10) + ×R N

for any ϕ ∈

C ∞ ( × R3

× [0, T ]) such that ϕ(·, T ) = 0 and γ + ϕ = B∗ γ − ϕ

on + × [0, T ]

(11)

(which holds, in particular, if ϕ is independent of v). 1.3. Formal derivation of the asymptotic model. The formal derivation of the asymptotic model was first investigated by J. Carrillo and T. Goudon in [11]. We recall here the main steps for the sake of completeness. First of all, assuming that limε→0 f ε = f and limε→0 u ε = u, (3) formally yields (u − v) f − ∇v f = 0, which implies f (x, v, t) = n(x, t)

2 1 − |u(x,t)−v| 2 e , (2π )3/2

 where n(x, t) = f (x, v, t) dv denotes the density of particles. Furthermore, integrating (3) with respect to v yields ∂t n + div j = 0, where

 j (x, t) =

v f (x, v, t) dv = n(x, t)u(x, t),

is the current of particles. Finally, multiplying (3) by v and integrating with respect to v, we are led to:  1 ∂t (nu) + divx (n u ⊗ u) + ∇x n = − (u − v) f dv, (12) ε in which we used the equality  v ⊗ v f dv = u ⊗ u n + n I (where I denotes the identity matrix). Adding (12) and (5), it is readily seen that the right-hand sides cancel, so that (n, ρ, u) is solution to the following system of equations: ⎧ ⎨ ∂t n + divx (nu) = 0, ∂t ρ + divx (ρu) = 0, ⎩ ∂ ((ρ + n)u) + div ((ρ + n)u ⊗ u) + ∇ (n + ρ γ ) − νu = 0, t x x with boundary condition u(x, t) = 0

∀x ∈ ∂.

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1.4. Entropies. The rigorous proof of the convergence towards the asymptotic equation (6) relies heavily on the use of energy inequalities and relative entropy methods. Before stating our main result, we need to review the classical inequalities satisfied by smooth solutions of (3) and (4)–(5). The inequalities presented in this section can also be found in [11]. First of all, setting    2 |v| E1 ( f ) = f + f log f dv, 2 RN it is readily seen (multipying (3) by |v|2 + log f ε + 1 and integrating with respect to x, v) that smooth solutions of (3) satisfy:    d 1 1 E1 ( f ε ) d x + |(u ε − v) f ε − ∇v f ε |2 dv d x N dt  ε  R fε  2   |v| + log γ f ε + 1 γ f ε dσ (x) dv (v · r (x)) + 2 ∂×R N   1 u ε (u ε − v) f ε dv d x, = ε  RN 2

where the boundary term is either 0 (periodic boundary condition) or non-negative (using (8) and (9)). In either case, we deduce    d 1 1 E1 ( f ε ) d x + |(u ε − v) f ε − ∇v f ε |2 dv d x dt  ε  RN fε   1 u ε (u ε − v) f ε dv d x. ≤ ε  RN Moreover, defining E2 (ρ, u) = ρ

1 |u|2 + ργ , 2 γ −1

it is well-known that smooth solutions of (4)–(5) satisfy:     d 1 E2 (ρε , u ε ) d x + ν |∇x u ε |2 d x = − u ε (u ε − v) f ε dv d x. dt  ε  RN  We deduce the following proposition: Proposition 1.1. Let ( f ε , ρε , u ε ) be a smooth solution of (3)–(5), then the following energy equality holds:  d [E1 ( f ε ) + E2 (ρε , u ε )] d x dt     1 2 1 + |(u ε − v) f ε − ∇v f ε | dv d x + ν |∇x u ε |2 d x ≤ 0. (13) ε  RN fε  In particular  E ( f, ρ, u) =

RN



|v|2 f + f log f 2

is an entropy for the system (3)–(5).

dv + ρ

|u|2 1 + ργ 2 γ −1

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From now on, we denote by   D( f, u) =



RN

|(u − v) f − ∇v f |2

1 dv d x f

the kinetic dissipation. Next, for a given density distribution n and velocity field u we introduce the Maxwellian distribution 2 n − |u−v| 2 e . (2π )3/2  We know that, for any ( f, ρ, u) satisfying f dv + E ( f, ρ, u) < ∞, the following minimization principle holds (see Bouchut [8]):

M(n,u) =

E (M(n,u) , ρ, u) ≤ E ( f, ρ, u), where

 n(x, t) =

f (x, v, t) dv.

In particular, introducing H (n, ρ, u) = E (M(n,u) , ρ, u) = (n + ρ)

1 u2 2 + ρ γ + n log n − log(2π )n, 2 γ −1 3

we have

 H (n, ρ, u) ≤ E ( f, ρ, u),

if

n=

f dv.

(14)

Moreover, H is an entropy for the asymptotic system (6). More precisely, we have: Proposition 1.2. Let (n, ρ, u) be a solution of (6), and let U = (n, ρ, P) = (n, ρ, (ρ + n)u). We define H (U ) =

P2 1 2 + ρ γ + n log n − log(2π )n. 2(n + ρ) γ − 1 3

Then we have  

 P 2 d P P H (U ) + divx F(U ) − ν ∇ + ν ∇ = 0, dt n+ρ n+ρ n +ρ    P 2 where ∇ n+ρ  = i, j |∂ j u i |2 and with P F(U ) = n+ρ



P2 γ 2 γ + ρ + n log n − log(2π )n . 2(n + ρ) γ − 1 3

In other words, H is a convex entropy for the system (6).

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Finally, we recall that given an entropy H (U ), we can define the relative entropy by H (V |U ) = H (V ) − H (U ) − DH (U )(V − U ), where D stand for the derivation with respect to the conservative variables (n, ρ, P). A simple computation shows that the relative entropy associated with (6) is H (U |U ∗ ) = (n + ρ)

|u − u ∗ |2 1 + p1 (ρ|ρ ∗ ) + p2 (n|n ∗ ) 2 γ −1

with p1 (ρ|ρ ∗ ) = ρ γ − ρ ∗ γ − γ ρ ∗ γ −1 (ρ − ρ ∗ ), p2 (n|n ∗ ) = n log n − n ∗ log n ∗ − (log n ∗ + 1)(n − n ∗ ) n = n log ∗ + (n ∗ − n). n (Note that the pi (·|·) are the relative entropies associated to p1 (ρ) = ρ γ and p2 (n) = n log n.) 1.5. Main results. We now have all the notations necessary to state our main results. Throughout the paper, we will assume that the initial data have finite mass and energy. More precisely, we assume that f 0 (x, v), ρ0 (x) and u 0 (x) are such that  ⎧ ⎪ ⎪ ρ0 d x < ∞, f 0 d x dv < ∞ ⎨   (15)  ⎪ ⎪ ⎩ E ( f 0 , ρ0 , u 0 ) d x < +∞. 

We recall that under those hypotheses, we proved in [20] the following result: Theorem 1.1 ([20]). Let f 0 (x, v), ρ0 (x) and u 0 (x) satisfy (15). Assume moreover that f 0 ∈ L ∞ ( × R N ) and that ν>0

and γ > 3/2.

Then, for any ε > 0 there exists a weak solution ( f ε , ρε , u ε ) of (3)–(5) defined globally in time. Moreover, this solution satisfies the usual entropy inequality:  t   1 t E ( f ε (t), ρε (t), u ε (t)) d x + D( f ε , u ε ) ds + ν |∇u ε |2 d x ds ε 0  0   ≤ E ( f 0 , ρ0 , u 0 ) d x. (16) 

In this paper, we are concerned with the asymptotic behavior of weak solutions as ε goes to zero. To simplify the analysis, we will assume that the initial data are well prepared, which amounts to assuming that f 0 (x, v) = M(n 0 ,u 0 ) =

n 0 (x) − |u 0 (x)−v|2 2 e . (2π )3/2

(17)

Analysis of Vlasov-Fokker-Planck/Compressible Navier-Stokes Equations

Note that (15) can then be rewritten as ⎧  ⎪ ⎪ n0 d x < ∞ ⎨ ρ0 d x < ∞,   2 1 |u 0 | ⎪ γ ⎪ + ρ + n 0 log n 0 d x < ∞. ⎩ (n 0 + ρ0 ) 2 γ −1 0 

581

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Remark 1.2. Instead of (17), it would actually be enough to assume that the initial data f 0ε converges to a maxwellian distribution in the following sense:   E ( f 0ε , ρ0 , u 0 ) d x −→ H (n 0 , ρ0 , u 0 ) d x. 



Finally, we need to assume that solutions of the asymptotic system of equations associated to initial value (ρ0 , n 0 , u 0 ) stays away from vacuum. This will be satisfied (at least for small time) if the following condition holds: λ0 ≤ n 0 (x) + ρ0 (x) ≤ 0

∀x ∈ ,

(19)

with λ0 and 0 positive constants. Indeed, under this assumption, we have the following proposition (see Dafermos [12]): Proposition 1.3. Under hypothesis (18) and (19), there exists T ∗ > 0 and the n ∗ , ρ ∗ , u ∗ solution of (6) on [0, T ∗ ) such that 0 < λ ≤ n ∗ (x, t) + ρ ∗ (x, t) ≤ 

for all x ∈ , t ∈ [0, T ∗ )

for some positive constants λ and . The main result of this paper says that any solutions of (3)–(5) satisfying the entropy inequality (and thus in particular the solution constructed in Theorem 1.1), converges, as ε goes to zero to (n ∗ , ρ ∗ , u ∗ ): Theorem 1.2. Assume that n 0 (x), ρ0 (x) and u 0 (x) satisfy (15)–(19), and let ( f ε , ρε , u ε ) be a weak solution of (3)–(5) with initial conditions f ε (x, v, 0) = f 0 (x, v), ρε (x, 0) = ρ0 (x), u ε (x, 0) = u 0 (x), and satisfying the entropy inequality (16). Assume moreover that ν≥0

and γ ∈ (1, 2).

Then there exists a constant C such that  T     √ ν T ∗ ∇(u ε − u ∗ )2 d x dt ≤ C ε H (Uε |U ) dt + (20) 2 0  0  for all T < T ∗ where Uε = (n ε , ρε , Pε ), n ε = f ε (x, v, t) dv and Pε = (n ε + ρε )u ε . Moreover, any sequence of functions ( f ε , ρε , u ε ) satisfying Inequality (20) satisfies f ε −→ Mn ∗ ,u ∗ n ε −→ n ∗

1 (0, T ∗ ; L 1 ( × R3 ))-strong , a.e. and L loc 1 (0, T ∗ ; L 1 ())-strong , a.e. and L loc

ρε −→ ρ ∗ a.e. and L loc (0, T ∗ ; L p ())-strong ∀ p < γ , √ √ p ρε u ε −→ ρ ∗ u ∗ L loc (0, T ∗ ; L 2 ())-strong. p

When ν > 0, we also have u ε −→ u ∗

2 (0, T ∗ ; L 2 ())-strong. L loc

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We stress the fact that it is not necessary to have a positive viscosity coefficient ν to carry out the proof of this theorem. We now turn to the proof of Theorem 1.2. The next section is devoted to the central argument of the proof, namely the relative entropy inequality and its consequences. The (more technical) results needed to complete the proof (control of the relative flux and of the kinetics approximation) are then detailed in the following two sections. 2. Relative Entropy: Proof of Theorem 1.2 In this section we derive a relative entropy inequality for the asymptotic system (6) which is the cornerstone of this paper. This inequality is much more general than the system under consideration and is valid for a general system of conservation laws of the form      ∂t Ui + ∂xk Aik (U ) = ∂xk Bi j (U )∂xk D j H (U ) , (21) k

k

where B(U ) is a positive symmetric matrix and DH denotes the derivative (with respect to U ) of the entropy H (U ) associated with the flux A(U ) (the system (6) can be written in this form, see below). The existence of such an entropy is equivalent to the existence of an entropy flux function F such that  D j Fk (U ) = Di H (U )D j Aik (U ) (22) i

for all U . Then we have   ∂tH (U ) + ∂xk [Fk (U ) − D j H (U )Bi j (U )∂xk Di H (U )] k

+



i, j

   Bi j (U ) ∂xk DiH (U ) ∂xk D j H (U ) = 0,

i, j,k

which in particular gives d dt

 

H (U ) d x ≤ 0.

The asymptotic system (6) can be written in this form, with U = (n, ρ, P) (we recall that P = (n + ρ)u) and ⎞ ⎛ ⎞ ⎛ 00000 n P1 n P2 n P3 ρ P1 ρ P2 ρ P3 ⎟ ⎜0 0 0 0 0⎟ 1 ⎜ ⎟ ⎜ ⎟ ⎜ A(U ) = ⎜ C11 C12 C13 ⎟ , B(U ) = ν ⎜ 0 0 1 0 0 ⎟ , ⎝0 0 0 1 0⎠ n+ρ ⎝ C C C ⎠ 21 22 23 00001 C31 C32 C33 where Ci j = Pi P j + (n + ρ γ )(n + ρ)δi j . Note that we have

  2 2 u2 γ u ρ γ −1 , u . DH (U ) = − + log n + 1 − log(2π ), − + 2 3 2 γ −1

The cornerstone of the method is the following very general proposition:

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583

Proposition 2.1. Consider a system of conservation laws (21) and assume that there exists a smooth convex entropy H and a corresponding entropy flux F defined by (22). Then, for any smooth solution U ∗ of (21) and for any smooth function V , the following inequality holds: d dt



+





H (V |U ∗ ) d x

   Bi j (V ) ∇ Di H (V ) − ∇ Di H (U ∗ ) ∇ D j H (V ) − ∇ D j H (U ∗ ) d x    d H (V )d x + Bi j (V )∂xk DiH (V )∂xk D j H (V ) d x ≤ dt       − DH (U ∗ )[∂t V + divx A(V ) − ∂xk B j (V )∂xk D j H (V ) ] d x   − ∂xk [∂ j H (U ∗ )]A jk (V |U ∗ ) d x  + −

 j,k

div (B(U ∗ )∇ DH (U ∗ ))DH (V |U ∗ ) d x







   B(V ) − B(U ∗ ) ∇ DH (U ∗ ) ∇ DH (V ) − ∇ DH (U ∗ ) d x,

(23)

where A(V |U ) = A(V ) − A(U ) − D A(U ) · (V − U ), DH (V |U ) = DH (V ) − DH (U ) − D 2 H (U ) · (V − U ). A similar proposition was first established by Dafermos [12] for general system of hyperbolic conservation laws, without viscosity (see also [6]). This inequality has been extensively used ever since, and proved to be an important tool in stability/asymptotic analysis of the system of conservation laws. However, it is the first time, to our knowledge, that such an inequality is derived when a viscosity term arises in the equation. We believe that the main interest of this proposition is to show that as long as the viscosity term is of the form div [B(U )∇ (DH (U ))] , the natural relative entropy inequality (and its consequences) is preserved. Moreover, it is not very hard to see that a viscosity term of the form ∇ [E(U )div (DH (U ))] =



  ∂xi E i j ∂x j (D j H (U )) ,

j

could also be added in the system. In particular, the usual viscosity term for the NavierStokes equation div(ν∇u) + ∇(λdivu) can be used, but more general viscosity terms are allowed (in particular with diffusion terms involving the density). Finally, we observe that in our model, the viscosity matrix B does not depend on U , so the last term in (23) vanishes.

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Proof of Proposition 2.1. First, for any V, U ∈ [C 1 (Rn )] p , we have ∂t H (V |U ) = ∂t H (V ) − ∂t H (U ) − D 2 H (U )∂t U · (V − U ) − DH (U )∂t V + DH (U )∂t U. Next, we observe that, using (22), we have, for all U ∈ R N :  Di j H (U )∂xk (A jk (U ))(Vi − Ui ) − i, j,k

+



∂xk [Di Fk (U )(Vi − Ui )]

i, j,k

−



  D j H (U )∂xk Di A jk (U )(Vi − Ui ) = 0.

i, j,k

Using this equality, a careful computation yields: ∂t H (V |U ) = ∂t H (V ) − ∂t H (U ) −div(B(V )∇ DH (V ))DH (V ) + div(B(U )∇ DH (U ))DH (U ) −D 2 H (U )[∂t U + divx A(U ) − div (B(U )∇ (DH (U )))](V − U ) −DH (U )[∂t V + divx A(V ) − div (B(V )∇ (DH (V )))] +DH (U )[∂t U + divx A(U ) − div (B(U )∇ (DH (U )))]  ∂xk [∂i Fk (U )(Vi − Ui )] + i,k

+



∂ j H (U )∂xk [A(V |U )]

j,k

+div [B(V )∇ DH (V ) − B(U )∇ DH (U )] [DH (V ) − DH (U )] +div(B(U )∇ DH (U ))DH (V |U ). Thus, if U = U ∗ is a smooth solution of (21), we get ∂t H (V |U ∗ ) = ∂t H (V ) − div(B(V )∇ DH (V ))DH (V ) + divx F(U ∗ ) −DH (U ∗ ) · [∂t V + divx A(V ) − div (B(V )∇ (DH (V )))]  ∂xk [∂i Fk (U ∗ )(Vi − Ui∗ )] + i,k

+



∂ j H (U ∗ )∂xk [A(V |U ∗ )]

j,k

   +div B(V )(∇ DH (V ) − ∇ DH (U ∗ )) DH (V ) − DH (U ∗ )    +div (B(V ) − B(U ∗ ))∇ DH (U ∗ ) DH (V ) − DH (U ∗ ) +div(B(U ∗ )∇ DH (U ∗ ))DH (V |U ∗ ). Integrating with respect to x (using the boundary condition u = 0 on ∂ when  is a

bounded subset of R N ), it is relatively easy to deduce (23).

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585

We can now write the main proposition of this section: Proposition 2.2. Let ( f ε , ρ ε , u ε ) be a weak solution of (3)–(5) with initial data satisfying (15)–(19) and verifying the entropy decay inequality (13), and let U ∗ = (n ∗ , ρ ∗ , (n ∗ + ρ ∗ )u ∗ ) be a smooth solution of (6). Denote by U ε the macroscopic quantities corresponding to ( f ε , ρ ε , u ε ):  U ε = (n ε , ρ ε , (n ε + ρ ε )u ε ) with n ε = f ε dv. Then U ε satisfies the following inequality:   t   ∇(u ε − u ∗ )2 d x ds H (U ε |U ∗ )(t) d x + ν  0   t  ≤− ∂xk [∂ j H (U ∗ )]A jk (U ε |U ∗ ) d x ds  j,k

0

 t + −

div(B∇ DH (U ∗ ))DH (U ε |U ∗ ) d x ds

0



0



 t

  DH (U ∗ ) ∂t U ε + divx (A(U ε )) − divx (B∇ DH (U ε )) d x ds

(where the last integral has to be understood in the distributional sense). Proof of Proposition 2.2. First, we note that since U ∗ is a smooth solution of (6) and using the expression of B(U ), Inequality (23) gives:    ε 2 d ∇u − ∇u ∗ d x H (U ε |U ∗ ) d x + ν dt    d H (U ε ) d x + ν |∇u ε | d x = dt     − ∂xk [∂ j H (U ∗ )]A jk (U ε |U ∗ ) d x 

 j,k

− DH (U ∗ )(∂t U ε + divx (A(U ε )) − div(B∇ DH (U ε ))) d x.   + div (B∇ DH (U ∗ )) · DH (U ε |U ∗ ) d x. 

Integrating this equality with respect to t, we get:   t  ε 2 H (U ε |U ∗ )(t) d x + ν ∇u − ∇u ∗ d x ds 0      ε ∗ ≤ H (U |U )(0) d x + H (U ε )(t) − H (U0 ) d x    t  − ∂xk [∂ j H (U ∗ )]A jk (U ε |U ∗ ) d x ds 0

 t

 j,k

DH (U ∗ )(∂t U ε + divx (A(U ε )) − divx (B∇ DH (U ε ))) d x ds − 0   t + div (B∇ DH (U ∗ )) · DH (U ε |U ∗ ) d x ds. 0



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Next, we recall that E denotes the entropy associated to the initial system (3–5) (see Proposition 1.1) and we write: H (U ε )(t) − H (U0 ) = H (U ε )(t) − E ( f ε , ρ ε , u ε )(t) +E ( f ε , ρ ε , u ε )(t) − E ( f 0 , ρ0 , u 0 ) +E ( f 0 , ρ0 , u 0 ) − H (U0 ). The well-preparedness of the initial data (17) yields E ( f 0 , ρ0 , u 0 ) − H (U0 ) = 0, and (14) gives: H (U ε )(t) − E ( f ε , ρ ε , u ε )(t) ≤ 0. Moreover, the entropy inequality (13) implies that   E ( f ε , ρ ε , u ε )(t) d x − E ( f ε , ρ ε , u ε )(0) d x ≤ 0, 



and so







 H (U ε )(t) − H (U0 ) d x ≤ 0.

Finally, hypothesis (17) yields H (U ε |U ∗ )(0) = 0. We deduce  t   ε 2 ε ∗ ∇u − ∇u ∗ d x ds H (U |U )(t) d x + ν 0   t  ≤− ∂xk [∂ j H (U ∗ )]A jk (U ε |U ∗ ) d x ds 0

 t

 j,k

DH (U ∗ )(∂t U ε + divx (A(U ε )) − div(B∇ DH (U ε ))) d x ds − 0   t + div (B(U ∗ )∇ DH (U ∗ )) · DH (U ε |U ∗ ) d x ds, 

0

which gives Proposition 2.2.



It is readily seen that Inequality (20) and Theorem 1.2 follow from Proposition 2.2 if we can prove that the following facts hold (under the hypotheses of Theorem 1.2): (i) The relative flux is controlled by the relative entropy:   |A j,k (U |U ∗ )| d x ≤ C H (U |U ∗ ) d x for all function U , 



where the constant C depends on λ = inf(n ∗ + ρ ∗ ) and  = sup(n ∗ + ρ ∗ ) (see Lemma 3.1).

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(ii) The term due to the viscosity can be controlled by the relative entropy and the viscosity:  t 0

div(B j (U ∗ )∇∂ j H (U ∗ ))DH (U |U ∗ ) d x ds      u ∗  ν ∗   H (U |U ) d x + |∇(u − u ∗ )|2 d x ds ≤C ∗ n + ρ ∗ L ∞  2  

for all U = (n, ρ, (n + ρ)u) (see Lemma 3.3). (iii) The asymptotic system (6) is consistent with the original system, in the sense that     t    ε ε ε    ∂ U + div(A(U )) − div(B∇H (U )) d x dt t   0 

√ ≤ C(T ) ε

L 1 (0,T )

for any smooth function (x, t) (see Proposition 4.1). Note that the first two points are concerned with the stability of the asymptotic system and will result from an algebraic computation (see Sect. 3), while the last point says that (6) is indeed the correct asymptotic system for (3)–(5) and relies on the dissipative properties of the entropy (see Sect. 4). When those three conditions are fullfilled, we can deduce (20). More precisely, we have (using Proposition 2.2, and Lemma 3.1, 3.3 and Proposition 4.1): Proposition 2.3. Let ( f ε , ρ ε , u ε ) be a weak solution of (3)–(5) with initial data satisfying (15)–(19), and verifying the entropy decay inequality (13). Let U ∗ = (n ∗ , ρ ∗ , (n ∗ + ρ ∗ )u ∗ ) be a smooth solution of (6) such that λ ≤ ρ ∗ + n ∗ ≤ . Then, there exists a constant C depending on λ−1 , , ||∇u|| L ∞ , ||∂t u|| L ∞ , ||∇u 2 || L ∞ , ||u|| L ∞ and ||∇ log n|| L ∞ such that     ν t ∇(u ε − u ∗ )2 d x ds H (U ε |U ∗ )(t) d x + 2 0    t ≤C H (U ε |U ∗ ) d x ds + R(t)



0



√ with ||R|| L 1 (0,T ∗ ) ≤ C ε. A Gronwall argument then leads to 

T 0

 

H (U ε |U )(t) d x dt +

ν 2



T



0

which concludes the proof of Theorem 1.2.



  √ ∇(u ε − u ∗ )2 d x dt ≤ C(T ) ε,

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3. Relative Flux In this section, we study the structure of the asymptotic system (6). We denote respectively by A(·|·) and H (·|·) the relative flux and the relative entropy associated with the system (6) (see Sect. 1.4 for details). We establish the following lemma: Lemma 3.1. Assume 1 < γ < 2, then for any positive constants λ and , there exists C(λ−1 , ) such that   |A(U |U ∗ )| d x ≤ C H (U |U ∗ ) d x 



for any U = (n, ρ, (n + ρ)u),

U∗

=

(n ∗ , ρ ∗ , (n ∗

+ ρ ∗ )u ∗ ) smooth functions satisfying

λ ≤ ρ ∗ + n ∗ ≤ . Proof of Lemma 3.1. First, we check that the relative flux is given by ⎛ ⎞ (α − α ∗ )(ρ + n)(u i − u i∗ ) ⎠ (β − β ∗ )(ρ + n)(u i − u i∗ ) A(U |U ∗ ) = ⎝ (ρ + n)(u i − u i∗ )(u j − u ∗j ) + p1 (ρ|ρ ∗ )δi j n with α = n+ρ , α∗ = entropy satisfies

n∗ n ∗ +ρ ∗ ,

β =

H (U |U ∗ ) = (n + ρ)

ρ n+ρ

and β ∗ =

ρ∗ n ∗ +ρ ∗ ,

and we recall that the relative

1 |u − u ∗ |2 + p1 (ρ|ρ ∗ ) + p2 (n|n ∗ ) 2 γ −1

with γ γ −2 ξ (ρ − ρ ∗ )2 , 2 1 1 1 p2 (n|n ∗ ) = n log n − n ∗ log n ∗ − (log n ∗ + 1)(n − n ∗ ) = (n − n ∗ )2 , 2 ξ2

p1 (ρ|ρ ∗ ) = ρ γ − ρ ∗ γ − γ ρ ∗ γ −1 (ρ − ρ ∗ ) =

where ξ1 between ρ and ρ ∗ and ξ2 between n and n ∗ . Those computations are very close to the ones developed in [6], and we refer the reader to [6] for more details. The L 1 norm of A(U |U ∗ ) involves the following terms:   |(β − β ∗ )(ρ + n)(u i − u i∗ )| d x (24) |(α − α ∗ )(ρ + n)(u i − u i∗ )| d x , and 

(ρ + n)(u i − u i∗ )(u j − u ∗j ) d x ,



p1 (ρ|ρ ∗ ) d x,

(25)

 and it is readily seen that the last two terms (25) are bounded above by H (U |U ∗ ) d x. Moreover, the terms in (24) are alike, so we only need to treat in detail one of them (the first one).

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We note that the Cauchy-Schwartz inequality gives  |(α − α ∗ )(ρ + n)(u i − u i∗ )| d x  ≤

1/2 

∗ 2

(α − α ) (ρ + n) d x

where the second term is bounded above by the task of showing that the quantity

(ρ 

+ n)|(u i − u i∗ )|2 d x

H (U |U ∗ )d x

1/2

1/2 ,

. So we are left with

I = (α − α ∗ )2 (n + ρ) is bounded above by H (U |U ∗ ). To that purpose, we need to distinguish the case where n + ρ is larger than  and the case where n + ρ is smaller than . In each case, we will use one of the following expressions for α − α ∗ : ρ(n − n ∗ ) + n(ρ ∗ − ρ) (n + ρ)(n ∗ + ρ ∗ )

(26)

ρ ∗ (n − n ∗ ) + n ∗ (ρ ∗ − ρ) . (n + ρ)(n ∗ + ρ ∗ )

(27)

α − α∗ = or α − α∗ =

1) When n + ρ < , using the fact that ρ <  and ρ ∗ ≤ , we get ξ1 < . Since γ − 2 < 0 we deduce p1 (ρ|ρ ∗ ) ≥ C()(ρ − ρ ∗ )2 . Similarly, using the fact that n <  and n ∗ ≤ , we have ξ2 <  which yields p2 (n|n ∗ ) ≥ C()(n − n ∗ )2 . Finally, using (26) together with the fact that n/(n + ρ) ≤ 1 and ρ/(n + ρ) ≤ 1, we get   |ρ ∗ − ρ| 2 |n − n ∗ | + I ≤ (n + ρ) (n ∗ + ρ ∗ ) (n ∗ + ρ ∗ )  ≤ ((n − n ∗ )2 + (ρ ∗ − ρ)2 ), λ and therefore   I ≤ C(, λ) p1 (ρ|ρ ∗ ) + p2 (n|n ∗ ) . 2) When n + ρ > , we first note that using (27) and the fact that n ∗ /(n ∗ + ρ ∗ ) ≤ 1, we have I ≤

(|ρ ∗ − ρ| + |n − n ∗ |)2 (n − n ∗ )2 (ρ − ρ ∗ )2 ≤ + . (n + ρ) n+ρ n+ρ

In order to control the first term, we again distinguish two situations:

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− When n ≥ , then n ∗ < n, and so ξ2 < n. Therefore p2 (n|n ∗ ) >

1 (n − n ∗ )2 (n − n ∗ )2 > . n n+ρ

− When n < , then 1/ξ2 > 1/ max(n, n ∗ ) > 1/, and since n ∗ )2 , we get p2 (n|n ∗ ) > C()(n − n ∗ )2 > C()

(n−n ∗ )2 ρ+n

<

1  (n

−

(n − n ∗ )2 , ρ+n

where we used the fact that n + ρ ≥ . In either case, we have (n − n ∗ )2 ≤ C() p2 (n|n ∗ ). n+ρ ∗ 2

) ∗ Finally, we proceed similarly to show that the term (ρ−ρ n+ρ is controlled by p1 (ρ|ρ ): γ −2

− When ρ > , then ξ1 ≤ ρ and so ξ1 γ > 1). Since

> ρ γ −2 > C()/ρ (using the fact that

(ρ ∗ − ρ)2 (ρ − ρ ∗ )2 ≤ , n+ρ ρ we deduce (ρ ∗ − ρ)2 γ −2 < Cξ1 (ρ ∗ − ρ)2 ≤ p1 (ρ|ρ ∗ ). n+ρ γ −2

− When ρ < , then ξ1 1  (ρ

− ρ ∗ )2

> (max(n, n ∗ ))γ −2 > γ −2 , and since

(we recall that we still have n + ρ > ), we get p1 (n|n ∗ ) > C()(ρ − ρ ∗ )2 > C()

The proof of Lemma 3.1 is now complete.

(ρ−ρ ∗ )2 ρ+n

<

(ρ − ρ ∗ )2 . ρ+n



Note that we actually proved the following fact: Lemma 3.2. Assume 1 < γ < 2, then for any positive constants λ and , there exists C(λ, ) such that for any non-negative functions (n, ρ) and (n ∗ , ρ ∗ ) satisfying λ ≤ ρ ∗ + n ∗ ≤ , we have

 1 1 |ρ − ρ ∗ |2 , , n+ρ    1 1 , |n − n ∗ |2 . p2 (n|n ∗ ) ≥ C min n+ρ 

p1 (ρ|ρ ∗ ) ≥ C min

We deduce the following result:



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Lemma 3.3. Assume 1 < γ < 2, then for any positive constants λ and , there exists C(λ, ) such that for every t > 0:  t div(B∇ DH (U ∗ ))DH (U |U ∗ ) d x ds 0     t    u ∗  ν t ∗   ≤C ∗ H (U |U ) d x ds + |∇(u − u ∗ )|2 d x ds n + ρ ∗ L ∞ 0  2 0  for any U = (n, ρ, (n + ρ)u), U ∗ = (n ∗ , ρ ∗ , (n ∗ + ρ ∗ )u ∗ ) smooth functions satisfying λ ≤ ρ ∗ + n ∗ ≤ , (and u = u ∗ = 0 on ∂ when  is a bounded domain). Proof. We first check that we have B DH (U |U ∗ ) = (B DH )(U |U ∗ ), where B DH (U ) =

P n+ρ .

Thus we have

B DH (U |U ∗ ) =

n∗

1 (n − n ∗ + ρ − ρ ∗ )(u ∗ − u). + ρ∗

It follows that div(B∇ DH (U ∗ ))DH (U |U ∗ ) = and so  t 0



 u ∗  n − n ∗ + ρ − ρ ∗ (u ∗ − u) ∗ ∗ n +ρ

div(B∇ DH (U ∗ ))DH (U |U ∗ ) d x ds

   u ∗    ≤ ∗ n + ρ∗ 

   u ∗    ≤ ∗ n + ρ∗ 

 t 

 1/2  1  1 ∗ 2 ∗ 2 , |ρ − ρ | + |n − n | d x ds min n+ρ  0   t  1/2 max(ρ + n, )(u − u ∗ )2 d x ds

L∞

0

 t  L∞

0

 0





∗

1/2

∗

p1 (n|n ) + p2 (ρ|ρ ) d x ds

 t



∗ 2

max(ρ + n, )(u − u ) d x ds

1/2 .

Moreover, we can write  t max(ρ + n, )(u − u ∗ )2 d x ds 0   t  t ≤ (ρ + n)(u − u ∗ )2 d x ds +  (u − u ∗ )2 d x ds 0  0   t  t ≤ H (U |U ∗ ) d x ds + C ∇(u − u ∗ )2 d x ds 0



0



thanks to the Poincaré inequality (and the fact that u = u ∗ on ∂). The lemma follows easily using Young’s inequality.

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4. Control of the Kinetic Approximation In this section, we prove the consistency of the asymptotic system with the kinetic model. More precisely, we prove: Proposition 4.1. Let Uε be a weak solution of (3)–(5) satisfying the entropy inequality (13), and let U = (n, ρ, (n + ρ)u) be a smooth function. Then, there exists a constant C depending only on ||∇u|| L ∞ , ||∂t u|| L ∞ , ||∇u 2 || L ∞ and ||∇ log n|| L ∞ such that     t  √   ≤ C ε. DH (U ) U + div(A(U )) − div(B∇(DH (U )))] d xds [∂t ε ε ε   0 

L 1 (0,T )

The proof relies on inequality (13). However in order to control the dissipation term, we first need to show that the negative part of the entropy can be controlled by its positive part. More precisely, we need to show: Lemma 4.1. There exists a constant C such that for every T > 0 and ε:    |v|2 1 T D( f ε ) dt ≤ C. f ε + | f ε log f ε | dv d x + ε 0  RN 4 Proof of Lemma 4.1. This is a fairly classical result, and its proof can be found in particular in [15]. We recall it here for the sake of completeness: We write |s ln s| = s ln s − 2s ln sχ0≤s≤1 , and for ω > 0, we note that √ −s ln sχ0≤s≤1 ≤ sω + C sχe−w ≥s ≤ sω + Ce−ω/2 . Using these relations with s = f ε and ω = v 2 /8, we deduce:   | f ε ln( f ε )| d x dv ≤ f ε ln( f ε ) d x dv   1 2 + e−v /16 d x dv. v 2 f ε d x dv + 2C 4 Since  is bounded, it follows that   1 f ε (1 + | ln( f ε )|) d x dv + v 2 f ε dv d x 4

  1 t 1 u 2ε γ ρε + ρ dx + D( f ε ) ds + 2 γ −1 ε 0 ≤ E ( f ε , ρε , u ε )(t) + C ≤ E ( f ε , ρε , u ε )(0) + C.

Proof of Proposition 4.1. Formally, we have, integrating (3) with respect to v:  ∂t n ε = −divx v f ε dv, and thus

 ∂t n ε + divx (n ε u ε ) = divx

(u ε − v) f ε dv.

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593

Moreover, multiplying (3) by v and integrating with respect to v, we get (still formally):    1 ∂t v f ε dv = −divx v ⊗ v f ε dv + (u ε − v) f ε dv, ε and thus ∂t ((n ε + ρε )u ε ) + divx (u ε ⊗ u ε (n ε + ρε )) + ∇x (ρεγ + n ε ) − νu ε    = ∂t (u ε − v) f ε dv + divx (u ε ⊗ u ε − v ⊗ v + I ) f ε dv . Hence, we can write  t   DH (U ) ∂t U ε + divx (A(U ε )) − divx (B∇ DH (U ε )) d x ds 0     t =− [∇x D1 H (U ) + ∂t D3 H (U )] · (u ε − v) f ε dv d x ds 0     D3 H (U (t)) + (u ε − v) f ε (t) dv d x    t   − (u ε ⊗ u ε − v ⊗ v + I ) f ε dv : ∇x D3 H (U ) d x ds, 

0

where we recall that D1 H (U ) = −

2 u2 + log n + 1 − log(2π ), 2 3

and

D3 H (U ) = u.

This equality can be made rigorous by taking D1 H (U ) + v · D3 H (U ) as a test function in (10) (note that this function satisfies the compatibilty condition (11) along ∂ in the case of reflection boundary conditions) and D3 H (U ) as a test function in (5). We deduce that there exists a constant C depending only on ||∇u|| L ∞ , ||∂t u|| L ∞ , ||∇u 2 || L ∞ and ||∇ log n|| L ∞ such that  t DH (U )(∂t U ε + divx (A(U ε )) − divx (B∇ DH (U ε ))) d x ds 0    t       (28) ≤C  N (u ε − v) f ε dv  d x ds 0  R        +C  N (u ε − v) f ε (t) dv  d x  R   t      +C (29) (u ε ⊗ u ε − v ⊗ v + I ) f ε dv  d x ds.  0



RN

We are thus left with the task of showing that              (u ε − v) f ε dv  d x and (u ε ⊗ u ε − v ⊗ v + I ) f ε dv  d x   

RN

can be controlled by the dissipation.



RN

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1) We have      (u ε − v) f ε − ∇v f ε dv     1/2  1/2 1 ≤ f ε dv , |(u ε − v) f ε − ∇v f ε |2 dv fε

     (u ε − v) f ε dv  =  

and therefore    t      (u ε − v) f ε dv  d x ds ≤ C   0



0

t

√ D( f ε ) ds ≤ C ε,

which gives a bound for the first two terms in (29) (note that only the L 1 (0, t)-norm of the second one is bounded). 2) Next, we write  (u ε ⊗ u ε − v ⊗ v + I ) f ε dv  = [u ε ⊗ (u ε − v) + (u ε − v) ⊗ v + I ] f ε dv         = u ε f ε ⊗ (u ε − v) f ε − 2∇v f ε + u ε ⊗ 2 f ε ∇v f ε dv         + (u ε − v) f ε − 2∇v f ε ⊗ v f ε + 2 f ε ∇v f ε ⊗ v + I f ε dv       = u ε f ε ⊗ (u ε − v) f ε − 2∇v f ε + u ε ⊗ ∇v f ε dv       + (u ε − v) f ε − 2∇v f ε ⊗ v f ε + ∇v f ε ⊗ v + I f ε dv       = u ε f ε ⊗ (u ε − v) f ε − 2∇v f ε dv       + (u ε − v) f ε − 2∇v f ε ⊗ v f ε dv, and so   t      (u ε ⊗ u ε − v ⊗ v + I ) f ε dv  d x ds   0



≤

 t   0



(|u ε |2 + |v|2 ) f ε dv d x ds

1/2  0

t

D( f ε ) ds.

 So it only remains to see that (|u ε |2 + |v|2 ) f ε dv d x is bounded  2 uniformly by a constant. Using the entropy inequality, we already know that |v| f ε dv d x is boun ded, so it is enough to check that (u ε − v)2 f ε dv d x is bounded. To that purpose, we write

Analysis of Vlasov-Fokker-Planck/Compressible Navier-Stokes Equations

595

 

 (u ε − v)2 f ε dv d x =

     (u ε − v) f ε (u ε − v) f ε − 2∇v f ε dv d x   + (u ε − v)∇v f ε dv d x

 ≤ which gives  t 0

1/2 (u ε − v) f ε dv d x 2

 (u ε − v) f ε dv d x ds ≤ 2

0

t



D( f ε ) ds + 2

 D( f ε ) +

f ε dv d x

 t 0

f ε dv d x ds,

and yields the result. Putting all the pieces together, (29) gives     t    ε ε ε  DH (U ) ∂t U + divx (A(U )) − divx (B∇ DH (U )) d x ds    0





≤ (1 + T ) sup x

L 1 (0,T )

 T √ |∇x D1 H (U )| + |∂t D3 H (U )| + |∇x D3 H (U )|ds ε

0

which is the desired result.



References 1. Amsden, A.A.: Kiva-3V Release 2, improvements to Kiva-3V. Tech. Rep., Los Alamos National Laboratory, 1999 2. Amsden, A.A., O’Rourke, P.J., Butler, T.D.: Kiva-2, a computer program for chemical reactive flows with sprays. Tech. Rep., Los Alamos National Laboratory, 1989 3. Baranger, C., Desvillettes, L.: Coupling Euler and Vlasov equations in the context of sprays: the local-in-time, classical solutions. J. Hyperbolic Differ. Eq. 3(1), 1–26 (2006) 4. Bardos, C., Golse, F., Levermore, C.D.: The acoustic limit for the Boltzmann equation. Arch. Ration. Mech. Anal. 153(3), 177–204 (2000) 5. Berres, S., Bürger, R., Karlsen, K.H., Tory, E.M.: Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression. SIAM J. Appl. Math. 64(1), 41–80 (2003) (electronic) 6. Berthelin, F., Vasseur, A.: From kinetic equations to multidimensional isentropic gas dynamics before shocks. SIAM J. Math. Anal. 36(6), 1807–1835 (2005) (electronic) 7. Berthonnaud, P.: Limites fluides pour des modèles cinétiques de brouillards de gouttes monodispersés. C. R. Acad. Sci. Paris Sér. I Math. 331(8), 651–654 (2000) 8. Bouchut, F.: Construction of BGK models with a family of kinetic entropies for a given system of conservation laws. J. Stat. Phys. 95(1–2), 113–170 (1999) 9. Brenier, Y.: Convergence of the Vlasov-Poisson system to the incompressible Euler equations. Comm. Partial Differ. Eqs. 25(3–4), 737–754 (2000) 10. Brenier, Y.: Hydrodynamic structure of the augmented Born-Infeld equations. Arch. Ration. Mech. Anal. 172(1), 65–91 (2004) 11. Carrillo, J.A., Goudon, T.: Stability and asymptotic analysis of a fluid-particle interaction model. Comm. Partial Differ. Eqs. 31(7–9), 1349–1379 (2006) 12. Dafermos, C.M.: The second law of thermodynamics and stability. Arch. Rational Mech. Anal. 70(2), 167–179 (1979) 13. Gidaspow, D.: Multiphase flow and fluidization. Boston, MA: Academic Press Inc., 1994 14. Golse, F., Saint-Raymond, L.: The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math. 155(1), 81–161 (2004) 15. Goudon, T., Jabin, P.-E., Vasseur, A.: Hydrodynamic limit for the Vlasov-Navier-Stokes equations. I. Light particles regime. Indiana Univ. Math. J. 53(6), 1495–1515 (2004)

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16. Goudon, T., Jabin, P.-E., Vasseur, A.: Hydrodynamic limit for the Vlasov-Navier-Stokes equations. II. Fine particles regime. Indiana Univ. Math. J. 53(6), 1517–1536 (2004) 17. Hamdache, K.: Global existence and large time behaviour of solutions for the Vlasov-Stokes equations. Japan J. Indust. Appl. Math. 15(1), 51–74 (1998) 18. Laurent, F., Massot, M., Villedieu, P.: Eulerian multi-fluid modeling for the numerical simulation of coalescence in polydisperse dense liquid sprays. J. Comput. Phys. 194(2), 505–543 (2004) 19. Lions, P.-L., Masmoudi, N.: From the Boltzmann equations to the equations of incompressible fluid mechanics. I, II. Arch. Ration. Mech. Anal. 158(3), 173–193, 195–211 (2001) 20. Mellet, A., Vasseur, A.: Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system of equations. Math. Models Methods Appl. Sci. 17(7), 1039–1063 (2007) 21. Sartory, W.K.: Three-component analysis of blood sedimentation by the method of characteristics. Math. Biosci. 33, 145–165 (1977) 22. Spannenberg, A., Galvin, K.P.: Continuous differential sedimentation of a binary suspension. Chem. Eng. in Australia 21, 7–11 (1996) 23. Williams, F.A.: Spray combustion and atomization. Phys. of Fluids 1(6), 541–545 (1958) 24. Williams, F.A.: Combustion theory. Reading, MA: Benjamin Cummings Publ., 2nd ed., 1985 25. Yau, H.-T.: Relative entropy and hydrodynamics of Ginzburg-Landau models. Lett. Math. Phys. 22(1), 63–80 (1991) Communicated by J.L. Lebowitz

Commun. Math. Phys. 281, 597–619 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0500-y

Communications in

Mathematical Physics

Generic Dynamics of 4-Dimensional C 2 Hamiltonian Systems Mário Bessa1 , João Lopes Dias2 1 Centro de Matemática da Universidade do Porto, Rua do Campo Alegre, 687,

4169-007 Porto, Portugal. E-mail: [email protected]

2 Departamento de Matemática, ISEG, Universidade Técnica de Lisboa, Rua do Quelhas 6,

1200-781 Lisboa, Portugal. E-mail: [email protected] Received: 20 April 2007 / Accepted: 2 January 2008 Published online: 9 May 2008 – © Springer-Verlag 2008

Abstract: We study the dynamical behaviour of Hamiltonian flows defined on 4-dimensional compact symplectic manifolds. We find the existence of a C 2 -residual set of Hamiltonians for which there is an open mod 0 dense set of regular energy surfaces each either being Anosov or having zero Lyapunov exponents almost everywhere. This is in the spirit of the Bochi-Mañé dichotomy for area-preserving diffeomorphisms on compact surfaces [2] and its continuous-time version for 3-dimensional volume-preserving flows [1]. 1. Introduction and statement of the results The computation of Lyapunov exponents is one of the main problems in the modern theory of dynamical systems. They give us fundamental information on the asymptotic exponential behaviour of the linearized system. It is therefore important to understand these objects in order to study the time evolution of orbits. In particular, Pesin’s theory deals with non-vanishing Lyapunov exponents systems (non-uniformly hyperbolic). This setting jointly with a C α regularity, α > 0, of the tangent map allows us to derive a very complete geometric picture of the dynamics (stable/unstable invariant manifolds). On the other hand, if we aim at understanding both local and global dynamics, the presence of zero Lyapunov exponents creates lots of obstacles. An example is the case of conservative systems: using enough differentiability, the celebrated KAM theory guarantees persistence of invariant quasiperiodic motion on tori yielding zero Lyapunov exponents. In this paper we study the dependence of the Lyapunov exponents on the dynamics of Hamiltonian flows. Despite the fact that the theory of Hamiltonian systems ask, in general, for more refined topologies, here we work in the framework of the C 1 topology of the Hamiltonian vector field. Our motivation comes from a recent result of Bochi [2] for area-preserving diffeomorphisms on compact surfaces, followed by its continuous time counterpart [1] for volume-preserving flows on compact 3-manifolds. We point out that these results are based on the outlined approach of Mañé [10,11]. Furthermore,

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Bochi and Viana (see [4]) generalized the result in [2] and proved also a version for linear cocycles and symplectomorphisms in any finite dimension. For a survey of the theory see [5] and references therein. Here we prove that zero Lyapunov exponents for 4-dimensional Hamiltonian systems are very common, at least for a C 2 -residual subset. This picture changes radically for the C ∞ topology, the setting of most Hamiltonian systems coming from applications. In this case Markus and Meyer showed that there exists a residual of C ∞ Hamiltonians neither integrable nor ergodic [12]. Let (M, ω) be a compact symplectic manifold. We will be interested in the Hamiltonian dynamics of real-valued C s , 2 ≤ s ≤ +∞, functions on M that are constant on each connected component of the boundary ∂ M. These functions are referred to as Hamiltonians on M and their set will be denoted by C s (M, R) which we endow with the C 2 -topology. Moreover, we include in this definition the case of M without boundary ∂ M = ∅. We assume M and ∂ M (when it exists) to both be smooth. Given a Hamiltonian H , any scalar e ∈ H (M) ⊂ R is called an energy of H and H −1 (e) = {x ∈ M : H (x) = e} the corresponding invariant energy level set. It is regular if it does not contain critical points. An energy surface E is a connected component of H −1 (e). Notice that connected components of ∂ M = ∅ correspond to energy surfaces. The volume form ωd gives a measure µ on M that is preserved by the Hamiltonian flow. Recall that for a C 2 -generic Hamiltonian all but finitely many points are regular (hence a full µ-measure set), since Morse functions are C 2 -open and dense. On each regular energy surface E ⊂ M there is a natural finite invariant volume measure µE (see Sect. 2.1). Theorem 1. Let (M, ω) be a 4-dim compact symplectic manifold. For a C 2 -generic Hamiltonian H ∈ C 2 (M, R), the union of the regular energy surfaces E that are either Anosov or have zero Lyapunov exponents µE -a.e. for the Hamiltonian flow, forms an open µ-mod 0 and dense subset of M. A regular energy surface is Anosov if it is uniformly hyperbolic for the Hamiltonian flow (cf. Sect. 2.5). Geodesic flows on negative curvature surfaces are well-known systems yielding Anosov energy levels. An example of a mechanical system which is Anosov on each positive energy level was obtained by Hunt and MacKay [9]. We prove another dichotomy result for the transversal linear Poincaré flow on the tangent bundle (see Sect. 2.3). This projected tangent flow can present a weaker form of hyperbolicity, a dominated splitting (see Sect. 2.6). Theorem 2. Let (M, ω) be a 4-dimensional compact symplectic manifold. There exists a C 2 -dense subset D of C 2 (M, R) such that, if H ∈ D, there exists an invariant decomposition M = D ∪ Z (mod 0) satisfying:
• D = n∈N Dm n , where Dm n is a set with m n -dominated splitting for the transversal linear Poincaré flow of H , and • the Hamiltonian flow of H has zero Lyapunov exponents for x ∈ Z . The results above follow closely the strategy applied in [1] for volume-preserving flows. Besides the decomposition of the manifold into invariant sets for each energy, the main novelty here is the construction of Hamiltonian perturbations. Once those are built, we use abstract arguments developed in [2] and [1] to conclude the proofs. Nevertheless, for completeness, we will present all the ingredients in the Hamiltonian framework. At the end of Sect. 3.4 we discuss why in Theorem 2 we are only able to prove the existence of a dense subset instead of residual.

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At this point it is interesting to recall a related C 2 -generic dichotomy by Newhouse [14]. That states the existence of a C 2 -residual set of all Hamiltonians on a compact symplectic 2d-manifold, for which an energy surface through any p ∈ M is Anosov or is in the closure of 1-elliptical periodic orbits. For another related result, in the topological point of view, we mention a recent theorem by Vivier [17]: any 4-dimensional Hamiltonian vector field admitting a robustly transitive regular energy surface is Anosov on that surface. In Sect. 2 we introduce the main tools for the proofs of the above theorems (Sect. 3). These are based on Proposition 3.1 to which we devote the rest of the paper. The fundamental point is the construction of the perturbations of the Hamiltonian in Sect. 4. Finally, we conclude the proof in Sect. 5 by an abstract construction already contained in [1], which works equally in the present setting.

2. Preliminaries 2.1. Basic notions. Let M be a 2d-dimensional smooth manifold endowed with a symplectic structure, i.e. a closed and nondegenerate 2-form ω. The pair (M, ω) is called a symplectic manifold which is also a volume manifold by Liouville’s theorem. Let µ be the so-called Lebesgue measure associated to the volume form ωd = ω ∧ · · · ∧ ω. A diffeomorphism g : (M, ω) → (N , ω ) between two symplectic manifolds is called a symplectomorphism if g ∗ ω = ω. The action of a diffeomorphism on a 2-form is given by the pull-back (g ∗ ω )(X, Y ) = ω (g∗ X, g∗ Y ). Here X and Y are vector fields on M and the push-forward g∗ X = Dg X is a vector field on N . Notice that a symplectomorphism g : M → M preserves the Lebesgue measure µ since g ∗ ωd = ωd . For any smooth Hamiltonian function H : M → R there is a corresponding Hamiltonian vector field X H : M → T M determined by ι X H ω = d H being exact, where ιv ω = ω(v, ·) is a 1-form. Notice that H is C s iff X H is C s−1 . The Hamiltonian vector field generates the Hamiltonian flow, a smooth 1-parameter group of d t symplectomorphisms ϕ tH on M satisfying dt ϕ H = X H ◦ ϕ tH and ϕ 0H = id. Since d H (X H ) = ω(X H , X H ) = 0, X H is tangent to the energy level sets H −1 (e). In addition, the Hamiltonian flow is globally defined with respect to time because H |∂ M is constant or, equivalently, X H is tangent to ∂ M. If v ∈ Tx H −1 (e), i.e. d H (v)(x) = ω(X H , v)(x) = 0, then its push-forward by ϕ tH is again tangent to H −1 (e) on ϕ tH (x) since ∗

d H (Dϕ tH v)(ϕ tH (x)) = ω(X H , Dϕ tH v)(ϕ tH (x)) = ϕ tH ω(X H , v)(x) = 0. We consider also the tangent flow Dϕ tH : T M → T M that satisfies the linear variational equation (the linearized differential equation) d Dϕ tH = D X H (ϕ tH ) Dϕ tH dt with D X H : M → T T M. We say that x is a regular point if d H (x) = 0 (x is not critical). We denote the set of regular points by R(H ) and the set of critical points by Crit(H ). We call H −1 (e) a regular energy level of H if H −1 (e) ∩ Crit(H ) = ∅. A regular energy surface is a connected component of a regular energy level.

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Given any regular energy level or surface E, we induce a volume form ωE on the (2d − 1)-dimensional manifold E in the following way. For each x ∈ E, ωE (x) = ιY ωd (x) on Tx E defines a (2d − 1) non-degenerate form if Y ∈ Tx M satisfies d H (Y )(x) = 1. Notice that this definition does not depend on Y (up to normalization) as long as it is transversal to E at x. Moreover, d H (Dϕ tH Y )(ϕ tH (x)) = d(H ◦ ϕ tH )(Y )(x) = 1. Thus, ωE is ϕ tH -invariant, and the measure µE induced by ωE is again invariant. In order to obtain finite measures, we need to consider compact energy levels. On the manifold M we also fix any Riemannian structure which induces a norm  ·  on the fibers Tx M. We will use the standard norm of a bounded linear map A given by A = supv=1 A v. The symplectic structure guarantees by Darboux theorem the existence of an atlas {h j : U j → R2d } satisfying h ∗j ω0 = ω with ω0 =

d 

dyi ∧ dyd+i .

(2.1)

i=1

On the other hand, when dealing with volume manifolds (N , ) of dimension p, Moser’s theorem [13] gives an atlas {h j : U j → R p } such that h ∗j (dy1 ∧ · · · ∧ dy p ) = . 2.2. Oseledets’ theorem for 4-dim Hamiltonian systems. Unless indicated, for the rest of this paper we fix a 4-dimensional compact symplectic manifold (M, ω). Take H ∈ C 2 (M, R). Since the time-1 map of any tangent flow derived from a Hamiltonian vector field is measure preserving, we obtain a version of Oseledets’ theorem [15] for Hamiltonian systems. Given µ-a.e. point x ∈ M we have two possible splittings: (1) Tx M = E x with E x 4-dimensional and lim

1

t→±∞ t

log Dϕ tH (x) v = 0,

v ∈ Ex .

(2) Tx M = E x+ ⊕ E x− ⊕ E x0 ⊕ RX H (x), where RX H (x) denotes the vector field direction, each one of these subspaces being 1-dimensional and • lim 1t log Dϕ tH (x)| E x0 ⊕R X H (x)  = 0; t→±∞

• λ+ (H, x) = lim • λ− (H, x)

1 log Dϕ tH (x)| E x+  > 0; t→±∞ t = lim 1t log Dϕ tH (x)| E x−  = −λ+ (H, x). t→±∞

Moreover,  1 log det Dϕ tH (x) = λi (H, x) dim(E xi ) = 0 t→±∞ t lim

(2.2)

i∈{+,−}

and lim

t→±∞

1 log sin αt = 0, t

where αt is the angle at time t between any subspaces of the splitting.

(2.3)

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The splitting of the tangent bundle is called Oseledets splitting and the real numbers λ± (H, x) are called the Lyapunov exponents. In the case (1) we say that the Oseledets splitting is trivial. The full measure set of the Oseledets points is denoted by O(H ). The vector field direction RX H (x) is trivially an Oseledets’s direction with zero Lyapunov exponent.

2.3. The transversal linear Poincaré flow of a Hamiltonian. For each x ∈ R (we omit H when there is no ambiguity) take the orthogonal splitting Tx M = RX H (x) ⊕ N x , where N x = (RX H (x))⊥ is the normal fiber at x. Consider the automorphism of vector bundles Dϕ tH : TR M → TR M (x, v) → (ϕ tH (x), Dϕ tH (x) v).

(2.4)

Of course, in general, the subbundle NR is not Dϕ tH -invariant. So we relate to the Dϕ tH R = TR M/RX H (R) with an isomorphism φ1 : NR → N R invariant quotient space N (which is also an isometry). The unique map PHt : NR → NR such that φ1 ◦ PHt = Dϕ tH ◦ φ1 is called the linear Poincaré flow for H . Denoting by x : Tx M → N x the canonical orthogonal projection, the linear map PHt (x) : N x → Nϕ t (x) is H

PHt (x) v = ϕ t

H (x)

◦ Dϕ tH (x) v.

We now consider Nx = N x ∩ Tx H −1 (e), where Tx H −1 (e) = ker d H (x) is the tangent space to the energy level set with e = H (x). Thus, NR is invariant under PHt . So we define the map

tH : NR → NR ,

tH = PHt |NR ,

called the transversal linear Poincaré flow for H such that

tH (x) : Nx → Nϕ t

H (x)

, tH (x) v = ϕ t

H (x)

◦ Dϕ tH (x) v

is a linear symplectomorphism for the symplectic form induced on NR by ω. If x ∈ R ∩ O and λ+ (x) > 0, the Oseledets splitting on Tx M induces a

tH (x)-invariant splitting Nx = Nx+ ⊕ Nx− , where Nx± = x (E x± ).

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2.4. Lyapunov exponents. Our next lemma makes explicit that the dynamics of Dϕ tH and tH are coherent so that the Lyapunov exponents for both cases are related. Lemma 2.1. Given x ∈ R ∩ O, the Lyapunov exponents of the tH -invariant decomposition are equal to the ones of the Dϕ tH -invariant decomposition. Proof. If Oseledets’ splitting is trivial there is nothing to prove. Otherwise, let n + = α X H (x) + v + ∈ Nx+ with v + ∈ E x+ and α ∈ R. We want to study the asymptotic behavior of  tH (x) n + . From the following two equalities • ϕ t (x) Dϕ tH (x) X H (x) = ϕ t (x) X H ◦ ϕ tH (x) = 0, H H •  ϕ t (x) Dϕ tH (x) v +  = sin(θt )Dϕ tH (x) v + , H

we get lim

t→±∞

  1 1 log  tH (x) n +  = lim log sin(θt )Dϕ tH (x) v +  , t→±∞ t t

where θt is the angle between X H ◦ ϕ tH (x) and E ϕ+t

H (x)

. By (2.3), we obtain

  1 1 log sin(θt )Dϕ tH (x) v +  = lim log Dϕ tH (x) v +  t→±∞ t t→±∞ t = λ+ (H, x). lim

We proceed analogously for Nx− .

 

Below we state the Oseledets theorem for the transversal linear Poincaré flow. Theorem 2.2. Let H ∈ C 2 (M, R). For µ-a.e. x ∈ M there exists the upper Lyapunov exponent, 1 log  tH (x) ≥ 0, t→+∞ t

λ+ (H, x) = lim

and x → λ+ (H, x) is measurable. For µ-a.e. x with λ+ (H, x) > 0, there is a splitting Nx = Nx+ ⊕ Nx− which varies measurably with x such that: ⎧ + ⎪ v ∈ Nx+ \ {0} ⎨λ (H, x), 1 lim log  tH (x) v = −λ+ (H, x), v ∈ Nx− \ {0} t→±∞ t ⎪ ⎩±λ+ (H, x), v ∈ / Nx+ ∪ Nx− .

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2.5. Hyperbolic structure. Let H ∈ C 2 (M, R). Given any compact and ϕ tH -invariant set ⊂ H −1 (e), we say that is a hyperbolic set for ϕ tH if there exist m ∈ N and a Dϕ tH -invariant splitting T H −1 (e) = E + ⊕ E − ⊕ E such that for all x ∈ we have: 1 • Dϕ m H (x)| E x−  ≤ 2 (uniform contraction), −m • Dϕ H (x)| E x+  ≤ 21 (uniform expansion) • and E includes the directions of the vector field and of the gradient of H .

If is a regular energy surface, then ϕ tH | is said to be Anosov (for simplicity, we often say that is Anosov). Notice that there are no minimal hyperbolic sets larger than energy level sets. Similarly, we can define a hyperbolic structure for the transversal linear Poincaré flow

tH . We say that is hyperbolic for tH on if tH | is a hyperbolic vector bundle automorphism. The next lemma relates the hyperbolicity for tH with the hyperbolicity for ϕ tH . It is an immediate consequence of a result by Doering [8] for the linear Poincaré flow extended to our Hamiltonian setting and the transversal linear Poincaré flow. Lemma 2.3. Let be an ϕ tH -invariant and compact set. Then is hyperbolic for ϕ tH iff is hyperbolic for tH . We end this section with a well-known result about the measure of hyperbolic sets for C 2 (or more general C 1+ ) dynamical systems, proved by Bowen [7], Bochi-Viana [5] and Bessa [1] in several contexts. Here, following [1], it is stated for Hamiltonian functions, meaning a higher differentiability degree. Lemma 2.4. Let H ∈ C 3 (M, R) and a regular energy surface E. If ⊂ E is hyperbolic, then µE ( ) = 0 or = E (i.e. Anosov). 2.6. Dominated splitting. We now study a weaker form of hyperbolicity. Definition 2.5. Let ⊂ M be an ϕ tH -invariant set and m ∈ N. A splitting of the bundle N = N − ⊕ N + is an m-dominated splitting for the transversal linear Poincaré flow if it is tH -invariant and continuous such that  mH (x)|Nx−  1 ≤ ,  mH (x)|Nx+  2

x ∈ .

(2.5)

We shall call N = N − ⊕ N + a dominated splitting if it is m-dominated for some m ∈ N. If has a dominated splitting, then we may extend the splitting to its closure, except to critical points. Moreover, the angle between N − and N + is bounded away from zero on . Due to our low dimensional assumption, the decomposition is unique. For more details about dominated splitting see [6]. The above definition of dominated splitting is equivalent to the existence of C > 0 and 0 < θ < 1 so that  tH (x)|Nx−  ≤ Cθ t ,  tH (x)|Nx+ 

x ∈ , t ≥ 0.

(2.6)

The proof of the next lemma hints to the fact that the 4-dimensional setting is crucial in obtaining hyperbolicity from the dominated splitting structure.

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Lemma 2.6. Let H ∈ C 2 (M, R) and a regular energy surface E. If ⊂ E has a dominated splitting for tH , then is hyperbolic. Proof. Since E is compact it is at a fixed distance away from critical points, hence there is K > 1 such that 1 x ∈ E. ≤ X H (x) ≤ K , K On the other hand, because X H is volume-preserving on the 3-dimensional submanifold E, we get sin(γ0 ) X H (x) = sin(γt ) X H ◦ ϕ tH (x)  tH (x)|Nx+   tH (x)|Nx− .

(2.7)

Here γt is the angle between the subspaces N − and N + at ϕ tH (x), which is bounded from below by some β > 0 for any x ∈ . We can now rewrite (2.7) as t sin(γ0 ) X H (x)  H (x)|Nx−  sin(γt ) X H ◦ ϕ tH (x)  tH (x)|Nx+  sin(γ0 ) t ≤ K2 Cθ , sin(β)

 tH (x)|Nx− 2 =

where we also have used (2.6). Thus we have uniform contraction on Nx− . The above procedure can be adapted for Nx+ to find uniform expansion, hence is hyperbolic for tH . Lemma 2.3 concludes the proof.   Combining Lemmas 2.4 and 2.6 we get the following: Proposition 2.7. Let H ∈ C 3 (M, R) and a regular energy surface E. If ⊂ E has a dominated splitting for tH , then µE ( ) = 0 or E is Anosov. In particular, there is a C 2 -dense set of C 2 -Hamiltonians for which the above holds. Remark 2.8. It is an open problem to decide whether for every H ∈ C 3 (M, R) the following holds: an invariant set containing critical points of H and admitting a dominated splitting can only be of zero measure or Anosov. 3. Proof of the Main Theorems 3.1. Integrated Lyapunov exponent. Let H ∈ C 2 (M, R). We take any measurable ϕ tH -invariant subset  of M and we define the integrated upper Lyapunov exponent over  by

λ+ (H, x) dµ(x). (3.1) LE(H, ) = 

The sequence

an (H ) =



log  nH (x) dµ(x)

) ) = inf an (H is subaditive (an+m ≤ an + am ), hence lim an (H n n . That is,

1 LE(H, ) = inf log  nH (x) dµ(x). (3.2) n≥1 n   Since H → n1  log  nH (x)dµ(x) is continuous for each n, we conclude that LE(·, ) is upper semicontinuous among C 2 Hamiltonians having a common invariant set .

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3.2. Decay of Lyapunov exponent. For a given Hamiltonian H ∈ C 2 (M, R) and m ∈ N, we define the open set m (H ) = M \ Dm (H ), where Dm (H ) is the invariant set with m-dominated splitting for tH . This means that m (H ) is the set of points without m-dominated splitting. Furthermore, there exists m˜ ∈ N such that for all m ≥ m˜ we have m (H ) ⊂ m (H ). On the other hand, if H = H on Dm (H ), then m (H ) ⊂ m (H ). The equivalent relations for Dm (H ) are immediate. The next proposition is fundamental because it allows us to decay the integrated Lyapunov exponent over a full measure subset of m (H ). Proposition 3.1. Let H ∈ C s+1 (M, R) with s ≥ 2 or s = ∞, and , δ > 0. Then there  ∈ C s (M, R), -C 2 -close to H , such that H  = H on Dm (H ) and exists m ∈ N and H , m (H )) < δ. LE( H

(3.3)

We assume that LE(H, m (H )) > 0, otherwise the claim holds trivially. We postpone the proof of this proposition to Sect. 5 and complete the ones of our main results. 3.3. Proof of Theorem 1. Here we look at the product set M = M × C 2 (M, R) endowed with the standard product topology. Given a point p on the manifold M, we denote by E p (H ) the energy surface in H −1 (H ( p)) passing through p. The subset A = {( p, H ) ∈ M : E p (H ) is an Anosov regular energy surface} is open by structural stability of Anosov systems. Moreover, for each ( p, H ) ∈ A there is a tubular neighbourhood of E p (H ) in M, consisting of regular energy surfaces supporting Anosov flows. On the complement of the closure of A, denoted by B = M \ A, there is a continuous positive function η : B → R+ guaranteeing for ( p, H ) ∈ B that V p,H is a connected component of {x ∈ M : |H (x) − H ( p)| < η( p, H )} containing p and made entirely of non-Anosov energy surfaces. Now, for each k ∈ N write

1 . Ak = ( p, H ) ∈ B : LE(H, V p,H ) < k This is an open set because the function in its definition is upper semicontinuous. Lemma 3.2. Ak is dense in B.

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Proof. Let ( p, H ) ∈ B. We want to find an arbitrarily close pair ( p, H ) in Ak . Notice that we will not need to approximate on the first component, the point on the manifold, but only on the Hamiltonian. Denote the set of C 3 Morse functions on M by K . Since Morse functions are C 2 -dense and have a finite number of critical points, it is sufficient to prove the claim by restricting to (M × K ) ∩ B. Moreover, small perturbations of a Hamiltonian in K will have regular energy surfaces through p. Therefore, we have a dense subset D ⊂ M × K in B such that E p (H ) is regular for ( p, H ) ∈ D and away from Anosov. This means that in fact we only need to show the claim for D. ) ∈ D, and ε > 0 such that ( p, H ) ∈ B for any H that is ε-C 2 -close to Let ( p, H . Proposition 3.1 guarantees that for all δ > 0 we can find H ∈ C 2 (M, R) which is H  and satisfies H = H  on Dm ( H ) (hence m (H ) ⊂ m ( H )) and ε-C 2 -close to H )) < δ. LE(H, m ( H Notice that µ(m (H ) ∩ V p,H ) = µ(V p,H ) for all m ∈ N. Otherwise, if there was an energy surface E ⊂ V p,H and m ∈ N such that µE (Dm (H )∩E) > 0, by Proposition 2.7, it would be Anosov, thus contradicting that ( p, H ) ∈ B. Therefore, since the upper Lyapunov exponent is non-negative, LE(H, V p,H ) = LE(H, m (H ) ∩ V p,H ) )) < δ. ≤ LE(H, m ( H

(3.4)

 

The choice δ = 1/k yields ( p, H ) ∈ Ak .

From the above, A ∪ Ak is open and dense. Finally,   (A ∪ Ak ) = A ∪ Ak A= k∈N

k∈N







= A ∪ ( p, H ) ∈ B :

λ (H, x) dµ(x) = 0 +

V p,H

is residual. By [3] Prop. A.7, we can thus write  M H × {H }, A= H ∈R

where R is C 2 -residual in C 2 (M, R) and, for each H ∈ R, M H is a residual subset of M, having the following property: if H ∈ R and p ∈ M H , then E p (H ) is Anosov or



λ+ dµE d H = 0. The latter implies that d H -a.e. the Lyapunov exponents on each energy surface E in V are µE -a.e. equal to zero. Recall that we can split the measure µ into µE on the energy surfaces and d H corresponding to the 1-form transversal to E. Therefore, for a C 2 -generic H , in a neighbourhood of a generic point in M we have the above dichotomy, thus being valid everywhere in the manifold. That completes the proof of Theorem 1.

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3.4. Proof of Theorem 2. It is enough to show that we can arbitrarly C 2 -approximate any H ∈ C ∞ (M, R) by H ∈ C 2 (M, R) satisfying L E(H , Z ) = 0 for some Z to be determined, without domination and whose mod 0-complement is dominated. We use an inductive scheme built on (3.3) and the fact that LE(·, ) is an upper semicontinuous function among Hamiltonians having a common invariant set , to define a convenient sequence Hn ∈ C ∞ (M, R) with C 2 -limit H . Choose a sequence n ≤ 0 2−n (to be further specified later) for some 0 > 0. By Proposition 3.1 we construct the sequence of Hamiltonians Hn in the following way: (1) H0 = H , (2) Hn and Hn−1 are n -C 2 -close, (3) Hn = Hn−1 on Dm n (Hn−1 ), (4) LE(Hn , m n (Hn−1 )) ≤ 2−n . That is, each term Hn of the sequence is the perturbation of the previous one Hn−1 as given by Proposition 3.1. Then, the C 2 -limit H exists and is n -C 2 -close to any Hn . For each n and an invariant set  for Hn , because LE(·, ) is upper semicontinuous, for any θ > 0 we can find ηn > 0 such that LE(H∗ , ) ≤ (1 + θ ) LE(Hn , ) as long as Hn and H∗ are ηn -C 2 -close and have the common invariant set . Impose now additionally that n < ηn . So, for any n, LE(H , ∩i m i (Hi−1 )) ≤ LE(H , m n (Hn−1 )) ≤ (1 + θ ) LE(Hn , m n (Hn−1 )) ≤ (1 + θ )2−n . Therefore, LE(H , ∩i m i (Hi−1 )) = 0 and the Lyapunov exponents vanish on  m i (Hi−1 ) (mod 0). Z= i∈N

is

Consider an increasing subsequence m n k . The complementary set of ∩i m ni (Hn i −1 ) D=

 i∈N

Di , where Di = Dm ni (Hn i −1 ).

By the inductive scheme above, Di ⊂ Di+1 and H = Hn i on Di . So, H has an m n i -dominated splitting on Di . Finally, we would like to explain why, unfortunately, the strategy in [4] to obtain residual instead of dense in the hypothesis of Theorem 2, does not apply in our case. We start with a C 2 Hamiltonian which is a continuity point of the upper semicontinuous function H → LE(H, M) (it is well-known that the set of points of continuity is residual) and define the jump (see [4], p. 1467) by LE(H, ∞ (H )), where ∞ (H ) = ∩m m (H ). A continuity point means a zero jump, so that λ+ (H, x) = 0 for a.e. x ∈ ∞ (H ) or else ∞ (H ) has zero measure. Now, in order to estimate a lower bound for the jump, we will need to perturb the original Hamiltonian H as done in Sect. 4. But Theorem 4.1 becomes useless if H is C 2 , because the conjugacy symplectomorphism will only be C 1 . Finally, we should note that C 3 (M, R) equipped with the C 2 -topology is not a Baire space, thus residual sets can be meaningless.

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4. Perturbing the Hamiltonian 4.1. A symplectic straightening-out lemma. Here we present an improved version of a lemma by Robinson [16] that provides us with symplectic flowbox coordinates useful to perform local perturbations to our original Hamiltonian. Consider the canonical symplectic form on R2d given by ω0 as in (2.1). The Hamiltonian vector field of any smooth H : R2d → R is then   0 I ∇ H, XH = −I 0 where I is the d × d identity matrix. Let the Hamiltonian function H0 : R2d → R be given by y → yd+1 , so that X H0 =

∂ . ∂ y1

Theorem 4.1 (Symplectic flowbox coordinates). Let (M 2d , ω) be a C s symplectic manifold, a Hamiltonian H ∈ C s (M, R), s ≥ 2 or s = ∞, and x ∈ M. If x ∈ R(H ), there exists a neighborhood U ⊂ M of x and a local C s−1 -symplectomorphism g : (U, ω) → (R2d , ω0 ) such that H = H0 ◦ g on U . Proof. Fix e = H (x). Choose any C s function G : M → R such that G(x) = 0 and ω(X H , X G )(x) = 0.

(4.1)

This defines a transversal  to X H at x in the following way. If U ⊂ M is a small enough neighborhood of x in M (U will always be allowed to remain as small as needed), then  = G −1 (0) ∩ U is a C s regular connected submanifold of dimension 2d − 1. Notice also that (4.1) holds in U . Locally there is a C s regular (2d − 2)-dimensional hypersurface of H −1 (e) where H and G are both constant: e =  ∩ H −1 (e). Notice that for m ∈ e , Tm e = {v ∈ Tm M : d H (v)(m) = dG(v)(m) = 0} = ker(ι X H ω(m)) ∩ ker(ι X G ω(m)).

(4.2)

Since ω(X H , X G ) = 0, we have X G (m), X H (m) ∈ Tm e and Tm M = Tm e ⊕ RX H (m) ⊕ RX G (m). Now, consider the closed 2-form ωe = ω|e defined on T e × T e . To show that (e , ωe ) is a C s symplectic manifold it is enough to check that ωe is non-degenerate. So, suppose there is v ∈ Tm e such that ωe (w, v) = 0 for any w ∈ Tm e . As in addition ω(X H , v)(m) = ω(X G , v)(m) = 0, m ∈ e , due to the fact that ω is non-degenerate we have to have v = 0. Thus, ωe is non-degenerate. So, Darboux’s theorem assures us the existence of a local diffeomorphism h : e → R2d−2 such that h ∗ ω0 = ωe where ω0 =

d  i=2

dyi ∧ dyd+i .

(4.3)

Generic Dynamics of 4-Dim Hamiltonians

Σ

609

XG (x)

2d−1

Σe

2d−2

m

τ H

ϕ

g

X (x)

x

H

φ

e−H

h

Fig. 1. The symplectic flowbox

The next step is to extend the above symplectic coordinates from e to U . For this purpose we use the parametrization by the flows ϕ tH and φ t generated by X H and Y := ω(X H , X G )−1 X G , respectively. The time reparametrization in the definition of Y is necessary to normalize the pull-back of the form as it will become clear later. τ (m) The tranversality condition (4.1) is again used in solving the equation G ◦ϕ H (m) = 0, m ∈ U , with respect to a function τ : U → R. That is, we want to find τ and U such that ϕ τH(m) (m) ∈  for each m ∈ U . By the implicit function theorem, since G ◦ ϕ 0H (m) = 0 and d G ◦ ϕ tH (m)|t=0 = dG(X H )(m) = ω(X G , X H )(m) = 0, dt there exists U and a unique τ ∈ C s−1 (U, R) as required. Moreover, φ t preserves the level sets of G as LY G = ω(X G , Y ) = 0, and LY H =

d H ◦ φ t (m) = ω(X H , Y ) ◦ φ t (m) = 1. dt

Thus, H ◦ φ t (m) = H (m) + t and in particular H ◦ φ e−H (m) (m) = e meaning that φ e−H (m) (m) ∈ H −1 (e) for m ∈ U . So, we define the map g : U → R2d given by τ (m)

g(m) = (−τ (m), h 1 ◦ φ e−H (m) ◦ ϕ H

τ (m)

(m), H (m), h 2 ◦ φ e−H (m) ◦ ϕ H

(m)),

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where h = (h 1 , h 2 ) as in (4.3) and h i : e → Rd−1 . In particular, H0 ◦ g = H . It remains to prove that g is a C s−1 -symplectomorphism. It follows that g is C s−1 and it has a C s−1 inverse g −1 : g(U ) → U given by g −1 (y) = ϕ H1 ◦ φ yd+1 −e ◦ h −1 ( y), y

where  y = (y2 , . . . , yd , yd+2 , . . . , y2d ). In addition, for y ∈ g(U ), g∗−1 X H0 (y) = ϕ˙ H1 ◦ φ yd+1 −e ◦ h −1 ( y) y

= X H ◦ ϕ H1 ◦ φ yd+1 −e ◦ h −1 ( y) y

= X H ◦ g −1 (y).

(4.4)

Hence, g∗ X H = X H0 . Similarly, we can show that g∗ Y = Notice that on g(e ) we have taking in addition k ∈ {1, d + 1},

g∗−1 ∂∂y j

=

∂ h −1 ∗ ∂yj

∂ ∂ yd+1

when restricting to .

for j ∈ {1, d + 1}. Furthermore,

      ∂ ∂ ∂ ∂ ∂ ∂ ∗ ∗ (g −1 ω) = (h −1 ω) = ω0 , , , , ∂ y j ∂ yk ∂ y j ∂ yk ∂ y j ∂ yk   ∂ ∂ ∗ (g −1 ω) = ω(X H , Y ) ◦ g −1 = 1. , ∂ y1 ∂ yd+1 Since Dh −1 ∂∂y j ∈ T e , and H and G are constant on e , (g

−1 ∗



∂ ∂ ω) , ∂ y1 ∂ y j



 = ω X H , Dh

−1

∂ ∂yj



 = dH

Dh

−1

∂ ∂yj

 =0

  ∗ ∗ and analogously (g −1 ω) ∂ y∂d+1 , ∂∂y j = 0. Therefore g −1 ω has to be the canonical 2-form, i.e. g ∗ ω0 = ω on e . Now, we show that g ∗ ω0 = ω also holds on . Using Cartan’s formula for the Lie derivative Lv = ιv d + dιv with respect to a vector field v and the identities d f ∗ = f ∗ d and f ∗ ιv ω = ι f∗−1 v f ∗ ω, then LY g ∗ ω0 = g ∗ dι∂/∂ yd+1 ω0 = g ∗ d 2 (−y1 ) = 0. As we also have L X G ω = 0 and LY ω = 0, the forms g ∗ ω0 and ω are constant and coincide along the flow of Y passing through e , i.e. on . In order to see that we can have g ∗ ω0 = ω on all of U , we compute L X H g ∗ ω0 = dι X H0 ω0 = d(d H0 ) = 0. Recall that L X H ω = 0. So, g ∗ ω0 = ω along the flow of X H through , thus on all U . This concludes the proof that g is a symplectomorphism.  

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4.2. Hamiltonian local perturbation. In the next lemma we introduce the main tool to perturb 2d = 4-dimensional Hamiltonians. We will then be able to perturb the transversal linear Poincaré flow in order to rotate its action by a small angle. As we shall see later, that is all we need to interchange N + with N − using the lack of dominance. For functions on R4 consider the C k -norm, with k ≥ 0 integer,    ∂ |σ | f (y)   ,  f C k = sup max  σ  σ y 0≤|σ |≤k ∂ 1 y1 . . . ∂ 4 y4  where σ = (σ1 , . . . , σ4 ) ∈ N40 with |σ | = i σi . Define the “tube”  Va,b,c = {(y1 , y2 , y3 , y4 ) ∈ R4 : a < y1 < b, y22 + y42 < c, |y3 | < c}. Moreover, take the 2-dim plane 0 = {(0, y2 , 0, y4 ) ∈ R4 } and the orthogonal projection π0 : R4 → 0 . Notice that the transversal linear Poincaré flow of H0 (y) = y3 on 0 is given by tH0 (y2 , y4 ) = π0 . In the following we fix a universal 0 <  < 1. Lemma 4.2. Given 0 < ν < 1 and  > 0, there exists α0 > 0 such that, for every 0 < r < 1 and 0 < α ≤ α0 , we can find H ∈ C ∞ (R4 , R) satisfying • H = H0 outside V0,,r , • H − H0 C 2 < , • D X H (y) = 0 for y ∈ {0, } × R3 and • 1H (0, y2 , 0, y4 ) := (π0 Dϕ 1H )(0, y2 , 0, y4 ) = Rα on 0 with ⎡

0 ⎢0 Rα = ⎣ 0 0

0 cos α 0 sin α

0 0 0 0



y22 + y42 < r ν, where

⎤ 0 − sin α ⎥ . 0 ⎦ cos α

Proof. Consider the Hamiltonian flow ϕ tH0 (y) = (y1 + t, y2 , y3 , y4 ). We want to -C 2 -perturb H0 to get a Hamiltonian flow that rotates on the (y2 , y4 )-plane while the orbit is inside Vξ,ξ ,r ν for some fixed universal constants 0 < ξ < ξ <  < 1. Outside the slightly larger tube V0,,r we impose no perturbation. In order to construct a C ∞ perturbation on those terms, we need to consider three bump functions. It is possible to find C ∞ maps  : R → R along the time direction and   : R → R on the y3 -direction satisfying   0 , y1 ∈ [ξ, ξ ] 1, |y3 | ≤ r ν  (y3 ) = (y1 ) = 0, y1 ∈ (0, ), 0, |y3 | ≥ r, 1 C 0 all bounded from above 0 > 0, 0  = 1, the norms C 0 ,  C 0 , 

C 0 ,  2

by a constant (recall that ξ , ξ and  are seen as universal),   C 0 ≤ (1−ν)r and 4  

C 0 ≤ [(1−ν)r ]2 . Similarly, get a C ∞ map φ : R+0 → R for the plane (y2 , y4 ) such that  2 ρ , ρ ≤ rν φ(ρ) = 2 0, ρ ≥ r,

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2  ν2 2ν

 φC 0 ≤ (r ν)2 , φ C 0 ≤ 2r and φ ≤ . 0 C 1−ν 1−ν Now, we construct the perturbed Hamiltonian

where ρ =



(y3 ) φ(ρ), H (y) = H0 (y) − α(y1 ) 

(4.5)

y22 + y42 . Clearly, it is equal to H0 outside V0,,r . Hence, for y ∈ V0,1,r ν , " # ∇ H (y) = −α (y1 )φ(ρ), −α y2  (y1 ), 1, −α y4  (y1 ) . (4.6)

So, on this domain, X H generates the flow   

t t (y1 + s)ds , ϕ H (y) = y1 + t, ρ cos θ + α 0

y3 + α φ(ρ) [(y1 + t) − (y1 )],  

t ρ sin θ + α (y1 + s)ds ,

(4.7)

0 d 2 dt ρ

= 0 so that ρ is ϕ tH -invariant. That is, where θ = arctan(y4 /y2 ). Notice that on the (y2 , y4 )-plane the motion consists of a rotation. Furthermore, by fixing y3 = 0, 2 |y3 (t)| ≤ α | ρ2 | |(t)| ≤ r ν if α ≤ 2(C 0 r ν)−1 . Now, if ρ < r ν, ϕ 1H (0, y2 , 0, y4 ) = (1, ρ cos(θ + α), 0, ρ sin(θ + α)) and (π0 Dϕ 1H )(0, y2 , 0, y4 ) v = Rα v, v ∈ 0 . Finally, we need to estimate the C 2 -norm of the perturbation. From (4.5) and (4.6) we get H − H0 C 1  αr ν(1 − ν)−1 ,

(4.8)

where we are using the notation A  B to mean that there is a constant C > 0 such that A ≤ C B. The second order derivatives are ∂2 H = −α

(y1 ) (y3 )φ(ρ), ∂ y12 ∂2 H = −α (y1 )  (y3 )φ(ρ), ∂ y1 ∂ y3   ∂2 H y2 y4 = −α 2 (y1 ) (y3 ) φ

(ρ) − φ (ρ)ρ −1 , ∂ y2 ∂ y4 ρ 2 ∂ H = −αy j ρ −1  (y1 ) (y3 )φ (ρ), ∂ y1 ∂ y j % $ ∂2 H = −α(y1 ) (y3 ) φ

(ρ)y 2j ρ −2 + φ (ρ)ρ −1 − φ (ρ)y 2j ρ −3 , 2 ∂yj ∂2 H = −α(y1 )  (y3 )φ (ρ)y j ρ −1 , ∂ y3 ∂ y j ∂2 H = −α(y1 ) 

(y3 )φ(ρ), ∂ y32

(4.9)

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613

where j = 2, 4. So, D X H = 0 if y1 ≤ 0 or y1 ≥ , and D 2 (H − H0 )C 0  α(1 − ν)−2 .

(4.10)

Hence, there is α0  (1 − ν)2 such that H − H0 C 2 <  for all 0 < α ≤ α0 .

 

Remark 4.3. It is not possible to find α as above if we require C 3 -closeness. This can 3 easily be seen in the proof by computing the third order derivatives. e.g. ∂∂y 3 H contains 2

the term α(y1 ) (y3 )y23 ρ −3 φ

(ρ) that can not be controlled by a bound of smaller order −1 than αr .

4.3. Realizing Hamiltonian systems. In this section we define the central objects for the proof of Proposition 3.1, the achievable or realizable linear flows. These will be constructed by perturbations of tH . We start with a point x ∈ O(H ) with lack of hyperbolic behavior and mix the directions Nx+ and Nx− to cause the decay of the upper Lyapunov exponent. In fact we are interested in “a lot” of points (related to the Lebesgue measure on transversal sections). Therefore, we perturb the Hamiltonian to make sure that “many” points y near x have tH (y) close to tH (x). For this reason we must be very careful in our procedure. Consider a Darboux atlas {h j : U j → R4 } j∈{1,...,} . For each x ∈ R(H ) choose j such that x ∈ U j , and take the 3-dimensional normal section Nx to the flow. In the sequel we abuse notation to write Nx for h j (Nx ∩ U j ), so √ that we work in R4 instead of M. Furthermore, denote by B(x, r ) = {(u, v, w) ∈ R3 : u 2 + v 2 < r, |w| < r } the open ball in Nx about x with small enough radius r . We estimate the distance between linear maps on tangent fibers at different base points by using the atlas and translating the objects to the origin in R4 . That is, At1 − At2  for linear flows Ait : Txi M → Tϕ t (xi ) M, H is given by Dh j1,t (ϕ tH (x1 ))At1 (Dh j1,0 (x1 ))−1 − Dh j2,t (ϕ tH (x2 ))At2 (Dh j2,0 (x2 ))−1 , where ji,t is the indice of the chart corresponding to ϕ tH (xi ). Consider the standard Poincaré map t (x) : U → Nϕ t PH

H (x)

,

where U ⊂ Nx is chosen sufficiently small. Given T > 0, the self-disjoint set & t ' (x) y ∈ M : y ∈ U, t ∈ [0, T ] , F HT (x, U ) = P H is called a T -length flowbox at x associated to the Hamiltonian H . There is a natural way to define a measure µ in the transversal sections by considering the invariant volume form ι X H ωd . We easily obtain an estimate on the time evolution of the measure of transversal sets: for ν, t > 0 there is r > 0 such that for any measurable A ⊂ B(x, r ) we have   µ(A) − α(t) µ(P t (x) A) < ν, (4.11) H where α(t) =

X H (ϕ tH (x)) . X H (x)

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Definition 4.4. Fix a Hamiltonian H ∈ C s+1 (M, R), s ≥ 2 or s = ∞, T,  > 0, 0 < κ < 1 and a non-periodic point x ∈ M (or with period larger than T ). The flow L of symplectic linear maps: L t (x) : Nx → Nϕ t

H (x)

, 0 ≤ t ≤ T,

is (, κ)-realizable of length T at x if the following holds: For γ > 0 there is r > 0 such that for any open set U ⊂ B(x, r ) ⊂ Nx we can find (1) K ⊂ U with µ(U \ K ) ≤ κ µ(U ), and  ∈ C s (M, R) -C 2 -close to H , verifying (2) H  outside F T (x, U ), (a) H = H H T (x) U , and (b) D X H (y) = D X H(y) for y ∈ U ∪ P H T T (c)  H(y) − L (x) < γ for all y ∈ K . Let us add a few words about this definition: (2a) and (2b) guarantee that the support of the perturbation is restricted to the flowbox and it C 1 “glues” to its complement; (2c) says that a large percentage of points (given numerically by (1)) have the transversal  (as in (2)) very close to the abstract linear action of the central linear Poincaré flow of H point x along the orbit. Notice that the realizability is with respect to the C 2 topology. Remark 4.5. Using Vitali covering arguments we may replace any open set U of Definition 4.4 by open balls. That turns out to be very useful because the basic perturbation Lemma 4.2 works for balls. It is an immediate consequence of the definition that the transversal linear Poincaré flow of H is itself a realizable linear flow. In addition, the concatenation of two realizable linear flows is still a realizable linear flow as it is shown in the following lemma. Lemma 4.6. Let H ∈ C s+1 (M, R), s ≥ 2 or s = ∞, and x ∈ M non-periodic. If L 1 is T1 (x) so (, κ1 )-realizable of length T1 at x and L 2 is (, κ2 )-realizable of length T2 at ϕ H that κ = κ1 + κ2 < 1, then the concatenated linear flow  L t1 (x), 0 ≤ t ≤ T1 L t (x) = T1 T1 1 L t−T (ϕ (x)) L (x), T1 < t ≤ T1 + T2 2 1 H is (, κ)-realizable of length T1 + T2 at x. Remark 4.7. Notice that concatenation of realizable flows worsens κ. 1 , H 2 the obvious variables in the definition Proof. For γ > 0, take r1 , r2 , K 1 , K 2 , H  for L satisfying the for L 1 and L 2 . We want to find the corresponding ones r, K , H T1 properties of realizable flows. Let x2 = ϕ H (x). • First, choose r ≤ r1 such that T1 U2 := P H (x) U ⊂ B(x2 , r2 )

with U = B(x, r ).

Generic Dynamics of 4-Dim Hamiltonians

 as • Now, we construct H

615

⎧ 1 on F T1 (x, U ) ⎪ ⎨H H = H 2 on F T2 (x2 , U2 ) H H ⎪ ⎩ H otherwise.

Notice that F HT1 +T2 (x, U ) = F HT1 (x, U ) ∪ F HT2 (x2 , U2 ). −T1 • Consider K = K 1 ∩ P H (x) (K 2 ∩ U2 ). Hence, −T1 µ(U \ K ) ≤ µ(U \ K 1 ) + µ(U \ P H (x) (K 2 ∩ U2 )) −T1 (x) (K 2 ∩ U2 )). ≤ (κ1 + 1) µ(U ) − µ(P H −T1 (x) (K 2 ∩ U2 ) we know that Now, by (4.11) applied to A = P H −T1 µ(P H (x) (K 2 ∩ U2 )) ≥ α(T1 ) µ(K 2 ∩ U2 ) = α(T1 ) [µ(U2 ) − µ(U2 \ K 2 )] ≥ α(T1 )(1 − κ2 ) µ(U2 ).

On the other hand, using (4.11) for A = U , µ(U2 ) ≥ α(T1 )−1 µ(U ). Combining all the above estimates we get µ(U \ K ) ≤ (κ1 + κ2 ) µ(U ).  yields that D X H = D X  on U because that is true for H 1 . The • The choice of H H T1 +T2 2 . same holds on P H (x) U related to H  is C s it is enough to look at U2 . That follows from the same • In order to check that H reason as the previous item. T1 • Finally, there is C > 0 verifying for y ∈ K and writing y2 = P H (x) y,  TH1 +T2 (y) − L T1 +T2 (x) ≤ TH2 (y2 )[ TH1 (y) − L T1 (x)] + [ TH2 (y2 ) − L T2 (x2 )]L T1 (x) 0 and 0 < κ < 1. Then, there exists α0 = α0 (H, , κ) > 0 such that for any non-periodic point x ∈ M (or with period larger than 1) and 0 < α ≤ α0 , the linear flow tH (x) ◦ Rα : Nx → Nϕ t (x) is H (, κ)-realizable of length 1 at x. Proof. Let γ > 0. We start by choosing r > 0 sufficiently small such that:

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• B(x, r ) is inside the neighbourhood given by Lemma 4.1. Notice that by taking the transversal section  = B(x, r ) in the proof of the lemma, such neighbourhood can be extended along its orbit to an open set A containing FH1 (x, r ), where
FHτ (x, r ) = 0≤t≤τ ϕ tH (B(x, r )). So, a C s -symplectomorphism g : A → R4 exists satisfying: g(B(x, r )) is an orthogonal section to X H0 at g(x) = 0, H = H0 ◦ g, and all norms of the derivatives are bounded. Moreover, the derivatives of g and g −1 are of order r -close to the identity tangent map I on local coordinates; 1 (x, B(x, r )) is not self-intersecting; • FH  1 (x, B(x, r )) (recall that 0 <  < 1 is a fixed constant introduced • FH (x, r ) ⊂ F H before Lemma 4.2).  > 0 and 0 <  κ < 1. Define gx = g − g(x ) and Let U = B(x , r ) ⊂ B(x, r ),    ⊂ B(0, r2 ) and U = gx (U ). For r small we find r1 , r2 > 0 such that B(0, r1 ) ⊂ U r2 /r1 = [(1 −  κ )−1/3 + 1]/2 > 1. Setting ν = [1 + (1 −  κ )1/3 ]/2 < 1, Lemma 4.2 gives us that there is α0 = α0 (H, , κ ) > 0 such that for any 0 < α ≤ α0 and using the radius r1 we have that tH0 (0) ◦ Rα is ( , κ )-realizable of length-1 at the origin. Take   and H  such that K  = B(0, r1 ν) with the obvious variables in the definition K ⊂ U ) = π(r1 ν)3 and  H  − H0 C 2 <  ) ≥ (1 −  ). µ( K . Then, µ( K κ )µ(U ) ⊂ U and H = H  ◦ gx . If  Define K = gx ( K  and  κ are small enough (depending on , κ and the norms of the derivatives of g), we get that Definition 4.4 (1) is satisfied and  − H C 2 = ( H  − H0 ) ◦ gx C 2   H  < . We use the same notation  as in the proof of Lemma 4.2. By construction it is simple to check that Definition 4.4 (2a) and (2b) also hold. As discussed before, Lemma 4.1 determines the existence of a neighborhood at each regular point of M and a C s -symplectomorphism straightening the flow. By compacity of M the derivatives of the symplectomorphism up to the order s are uniformly bounded on small length 1 flowboxes. For this reason α0 given above (depending on   and  κ ) was chosen to be independent of x ∈ M. It remains to check (c) in Definition 4.4. This will require further restrictions on r , depending on γ . By definition, the time-1 transversal linear Poincaré flow on N y ⊂ Ty H −1 (H (y)) is y) Dϕ 1H(g(y)) Dg(y)

1H(y) = ϕ 1 (y) Dg −1 (  H

for y ∈ K , and in x yields

1H (x) ◦ Rα = ϕ 1 (x) Dg −1 ( x ) Dg(x) Rα , H

where  x = ϕ 1H0 ◦ g(x) = (1, 0, 0, 0) and  y = ϕ 1H ◦ g(y) are of order r -close. Notice that  ϕ 1 (y) − ϕ 1 (x)   r and  H

H

Dg −1 ( y) − Dg −1 ( x )  r. Therefore,  1H(y) − 1H (x) ◦ Rα   r + ϒ, where

% $ x ) Dϕ 1H(g(y)) Dg(y) − Dg(x) Rα . ϒ = ϕ 1 (x) Dg −1 ( H

Generic Dynamics of 4-Dim Hamiltonians

617

Moreover, Dg −1 ( x ) − I   r . So, ϒ  r +  ϕ 1 (x) (Rα Dg(y) − Dg(x) Rα ) , H

where we have also used  ϕ 1 (x) − π0   r and π0 Dϕ 1H(0, y2 , 0, y4 ) = Rα . Finally, H since Rα Dg(y) − Dg(x) Rα = Rα (Dg(y) − I ) + (I − Dg(x))Rα , we obtain the bound  1H(y) − 1H (x) ◦ Rα   r < γ for r  γ small enough.

 

Remark 4.9. A similar result holds true also for Rα ◦ tH (x) using essentially the same proof. 5. Proof of Proposition 3.1 We present here a sketch of how to complete the proof of Proposition 3.1; see [1] for full details. We would like to highlight the fact that our result does not hold for a C 2  has to be one degree of differentiability less. Hamiltonian H , since the perturbed one H The differentiability loss comes from the symplectomorphism obtained in Theorem 4.1 that rectifies the flow. 5.1. Local. The lemma below states that the absence of dominated splitting is sufficient to interchange the two directions of non-zero Lyapunov exponents along an orbit segment by the means of a realizable flow. Lemma 5.1. Let H ∈ C s+1 (M, R), s ≥ 2 or s = ∞,  > 0 and 0 < κ < 1. There exists m ∈ N, such that for every x ∈ R(H ) ∩ O(H ) with a positive Lyapunov exponent and satisfying  mH (x)|Nx−  1 ≥ ,  mH (x)|Nx+  2 there exists a (, κ)-realizable linear flow L of length m at x such that L m (x) Nx+ = Nϕ−m (x) . H

Proof. The proof is the same as for Lemma 3.15 of [1] in which the constructions of Lemma 4.8 are used, namely the concatenation of rotated Poincaré linear maps.   Now we aim at locally decaying the upper Lyapunov exponent. Lemma 5.2. Let H ∈ C s+1 (M, R), s ≥ 2 or s = ∞, and , δ > 0, 0 < κ < 1. There is T : m (H ) → R measurable, such that for µ-a.e. x ∈ m (H ) and t ≥ T (x), we can find a (, κ)-realizable linear flow L at x with length t satisfying 1 log L t (x) < δ. t

(5.1)

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M. Bessa, J. Lopes Dias

Proof. We follow Lemma 3.18 of [1]. Notice that for µ-a.e. x ∈ m (H ) with λ = λ+ (H, x) > 0 and due to the nice recurrence properties of the function T (see Lemma 3.12 of [2]) we obtain for every (very large) t ≥ T (x) that  mH (y)|N y−   mH (y)|N y+ 

≥

1 2

for y = ϕ sH (x) with s ≈ t/2. + Now, by Lemma 5.1 we obtain a (, κ)-realizable linear flow L t2 such that L m 2 Ny = − t t Nϕ m (y) . We consider also the realizable linear flows L 1 : Nx → N y and L 3 : Nϕ mH (y) → H

Nϕ t (x) given by tH for 0 ≤ t ≤ s and t ≥ m, respectively. Then we use Lemma 4.6 H and concatenate L 1 → L 2 → L 3 as L t , which is a (, κ)-realizable linear flow at x with length t. The choice of t  m and the exchange of the directions will cause a decay on the norm of L t . Roughly that is: • in Nx+ the action of L 1 is approximately eλt/2 , • in Nϕ−m (y) the action of L 3 is approximately e−λt/2 , and H • L 2 exchange these two rates. Therefore, L t (x) < etδ .

 

5.2. Global. Notice that, in Lemma 5.2, we obtained L t (x) < etδ . However, we still need to get an upper estimate of the upper Lyapunov exponent. Due to (3.2) this can be done without taking limits, say in finite time computations. In other words, we will be using the inequality



1 + ˜ log  tH˜ (x)dµ(x), λ ( H , x)dµ(x) ≤ (5.2) m (H ) m (H ) t which is true for all t ∈ R. Therefore, δ is larger than the upper Lyapunov exponent of at least most of the points near x. To prove Proposition 3.1 we turn Lemma 5.2 global. This is done by a recurrence argument based in the Kakutani towers techniques entirely described in [1], Sect. 3.6. In broad terms the construction goes as follows: • Take a very large m ∈ N from Lemma 5.1. Then Lemma 5.2 gives us a measurable function T : m (H ) → R depending on κ and δ. Let δ 2 = κ. • For x1 ∈ m (H ), the realizability of the flow L t (x1 ) guarantees that we have a 1 . t-length flowbox at x1 (a tower T1 ) associated to the perturbed Hamiltonian H If we take a point in the measurable set K 1 (cf. (1) of Definition 4.4) contained in the base of the tower, then by (2c) of Definition 4.4 and Lemma 5.2, we have  tH (y) < e2δt for all y ∈ K 1 . 1 • Now, for x2 , . . . , x j ∈ m (H ), where j ∈ N is large enough, we define self-disjoint towers Ti , i = 1, . . . , j, which (almost) cover the set m (H ) in the measure theoretical sense. We take these towers such that their heights are approximately the same, say h.  is defined by glueing together all perturbations H i , • The C s Hamiltonian H i = 1, . . . , j.

Generic Dynamics of 4-Dim Hamiltonians

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• Consider T = ∪i Ti , U = ∪i Ui and K = ∪i K i . Clearly K ⊂ U . Note that for points in U \ K we may not have  tH (·) < e2δt . 1

• Denote by T K the subtowers of T with base K instead of U . By (1) of Definition 4.4 we obtain that µ(U \ K ) ≤ κµ(U ), hence µ(T \ T K ) < µ(T ) ≤ δ 2 .

We claim that it is sufficient to take t = hδ −1 in (5.2). It follows from (5.1) that we only control the iterates that enter the base of T K . Since the height of each tower is approximately h the orbits leave T K at most δ −1 times. For each of those times the chance of not re-entering again is less than δ 2 . So, the probability of leaving T K along t iterates is less than δ. In conclusion, most of the points in m (H ) satisfy the inequality (5.1) and Proposition 3.1 is proved. Acknowledgements. We would like to thank Gonzalo Contreras and the anonymous referee for useful comments. MB was supported by Fundação para a Ciência e a Tecnologia, SFRH/BPD/20890/2004. JLD was partially supported by Fundação para a Ciência e a Tecnologia through the Program FEDER/POCI 2010.

References 1. Bessa, M.: The Lyapunov exponents of generic zero divergence 3-dimensional vector fields. Erg. Theor. Dyn. Syst. 27, 1445–1472 (2007) 2. Bochi, J.: Genericity of zero Lyapunov exponents. Erg. Theor. Dyn. Syst. 22, 1667–1696 (2002) 3. Bochi, J., Fayad, B.: Dichotomies between uniform hyperbolicity and zero Lyapunov exponents for S L(2, R) cocycles. Bull. Braz. Math. Soc. 37, 307–349 (2006) 4. Bochi, J., Viana, M.: The Lyapunov exponents of generic volume preserving and symplectic maps. Ann. Math. 161, 1423–1485 (2005) 5. Bochi, J., Viana, M.: Lyapunov exponents: How frequently are dynamical systems hyperbolic? In: Advances in Dynamical Systems. Cambridge: Cambridge Univ. Press, 2004 6. Bonatti, C., Díaz, L., Viana, M.: Dynamics beyond uniform hyperbolicity. A global geometric and probabilistic perspective. Encycl. of Math. Sc. 102. Math. Phys. 3. Berlin: Springer-Verlag, 2005 7. Bowen, R.: Equilibrium states and ergodic theory of Anosov diffeomorphisms. Lect. Notes in Math. 470, Berlin-Heidelberg New York: Springer-Verlag, 1975 8. Doering, C.: Persistently transitive vector fields on three-dimensional manifolds. In: Proceedings on Dynamical Systems and Bifurcation Theory, Pitman Res. Notes in Math 160, 1987, pp. 59–89 9. Hunt, T., MacKay, R.S.: Anosov parameter values for the triple linkage and a physical system with a uniformly chaotic attractor. Nonlinearity 16, 1499–1510 (2003) 10. Mañé, R.: Oseledec’s theorem from generic viewpoint. In: Proceedings of the international Congress of Mathematicians, Warszawa, Vol. 2, Warszawa: PWN,1983, pp. 1259–1276 11. Mañé, R.: The Lyapunov exponents of generic area preserving diffeomorphisms. International Conference on Dynamical Systems (Montevideo, 1995), Res. Notes Math. Ser. 362, London: Pitman, pp. 1996, 110–119 12. Markus, L., Meyer, K.R.: Generic Hamiltonian Dynamical Systems are neither Integrable nor Ergodic. Memoirs AMS Providence, RI: Amer. Math. Soc. 144, 1974 13. Moser, J.: On the volume elements on a manifold. Trans. Amer. Math. Soc. 120, 286–294 (1965) 14. Newhouse, S.: Quasi-elliptic periodic points in conservative dynamical systems. Am. J. Math. 99, 1061–1087 (1977) 15. Oseledets, V.I.: A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19, 197–231 (1968) 16. Robinson, C.: Lectures on Hamiltonian Systems. Monograf. Mat. Rio de Janeiro IMPA, 1971 17. Vivier, T.: Robustly transitive 3-dimensional regular energy surfaces are Anosov. Preprint Dijon, 2005, available at http://math.u-bourgogne.fr/topolog/prepub/HamHyp11.pdf, 2005 Communicated by G. Gallavotti

Commun. Math. Phys. 281, 621–653 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0490-9

Communications in

Mathematical Physics

Enumerative Geometry of Calabi-Yau 4-Folds A. Klemm1 , R. Pandharipande2 1 Department of Physics, Univ. of Wisconsin, Madison, WI 53706, USA.

E-mail: [email protected]

2 Department of Mathematics, Princeton University, Princeton, NJ 08544, USA.

E-mail: [email protected] Received: 11 May 2007 / Accepted: 23 November 2007 Published online: 30 May 2008 – © Springer-Verlag 2008

Abstract: Gromov-Witten theory is used to define an enumerative geometry of curves in Calabi-Yau 4-folds. The main technique is to find exact solutions to moving multiple cover integrals. The resulting invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi-Yau 3-folds. We conjecture the 4-fold invariants to be integers and expect a sheaf theoretic explanation. Several local Calabi-Yau 4-folds are solved exactly. Compact cases, including the sextic Calabi-Yau in P5 , are also studied. A complete solution of the Gromov-Witten theory of the sextic is conjecturally obtained by the holomorphic anomaly equation. 0. Introduction 0.1. Gromov-Witten theory. Let X be a nonsingular projective variety over C. Let M g,n (X, β) be the moduli space of genus g, n-pointed stable maps to X representing the class β ∈ H2 (X, Z). The Gromov-Witten theory of primary fields1 concerns the integrals
n  X (γ1 , . . . , γn ) = evi∗ (γn ), (1) N g,β [M g,n (X,β)]vir i=1

where evi : M g,n (X, β) → X is the ith evaluation map and γi ∈ H ∗ (X, Z). The notation
X N g, = 1 β [M g (X,β)]vir

is used in case there are no insertions. 1 We consider only primary Gromov-Witten theory in the paper.

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Since the moduli space M g,n (X, β) is a Deligne-Mumford stack, the Gromov-Witten invariants (1) are Q-valued. 0.2. Enumerative geometry. The relationship between Gromov-Witten theory and the enumerative geometry of curves in X is straightforward in three cases: (i) X is convex (in genus 0), (ii) X is a curve, (iii) X is P2 or P1 × P1 . For (i–iii), the Gromov-Witten theory with primary insertions equals the classical enumerative geometry of curves. A discussion of convex varieties (i) in genus 0 can be found in [11]. Examples (ii) and (iii) hold for all genera and recover the étale Hurwitz numbers and the classical Severi degrees respectively. Case (iii) certainly extends in some form to all rational surfaces viewed as generic blow-ups. The genus 0 case is treated in [16]. The first nontrivial cases occur for irrational surfaces. When X is a minimal surface of general type, Taubes’ results exactly determine the primary Gromov-Witten invariant for the adjunction genus in the canonical class, N gXX ,K X = (−1)χ (X,O X ) , see [36–39]. While much is known about surfaces of general type [23,28], surfaces in between are more mysterious. For example, many questions about the relationship of Gromov-Witten theory to the enumerative geometry of the K 3 and Enriques surfaces remain open [5,20,22,28]. The enumerative significance of Gromov-Witten theory in dimension 3 has been studied since the beginning of the subject. For Calabi-Yau 3-folds, essentially all GromovWitten invariants, even in genus 0, have large denominators. The Aspinwall-Morrison formula [1] was conjectured to produce integer invariants in genus 0. A full integrality conjecture for the Gromov-Witten theory of Calabi-Yau 3-folds in terms of BPS states was formulated by Gopakumar and Vafa [14,15]. Later, integral invariants for all 3-folds were conjectured in [32,33]. Various mathematical attempts to capture the BPS counts in terms of the cohomologies of associated moduli of sheaves on X were put forward without a definitive treatment. However, the integral invariants of [14,15,33] can be conjecturally interpreted in terms of the sheaf enumeration of Donaldson-Thomas theory [25,26,40]. Our main point here is to show the integrality of Gromov-Witten theory persists in higher dimensions as well. We speculate there exist universal transformations in every dimension which express Gromov-Witten theory in terms of Z-valued invariants. We conjecture the exact form of the transformation for Calabi-Yau 4-folds. A sheaf theoretic interpretation of the resulting invariants remains to be found.

0.3. Calabi-Yau 4-folds. Let X be a nonsingular, projective, Calabi-Yau 4-fold, and let β ∈ H2 (X, Z) be a curve class. Since
vir dim M g (X, β) = c1 (X ) + (dim X − 3)(1 − g) = 1 − g, β

Gromov-Witten theory vanishes for g ≥ 2. We need only consider genus 0 and 1.

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We measure the degree of β with respect to a fixed ample polarization L on X ,
deg(β) = c1 (L). β

All effective curve classes2 satisfy deg(β) > 0. We abbreviate the latter condition by β > 0. We are only interested here in Gromov-Witten invariants for classes satisfying β > 0. Integrality in genus 0 is expressed by the following generalization of the AspinwallMorrison formula. Invariants n 0,β (γ1 , . . . , γn ) virtually enumerating rational curves of class β incident to cycles dual to the classes γi are uniquely defined by 

N0,β (γ1 , . . . , γn )q β =

β>0



n 0,β (γ1 , . . . , γn )

β>0

∞ 

d −3+n q dβ .

(2)

d=1

A justification for the definition via multiple coverings is given in Sect. 1.1. Conjecture 0. The invariants n 0,β (γ1 , . . . , γn ) are integers. Let S1 , . . . , Ss be a basis of H 4 (X, Z) mod torsion. Let
Si ∪ S j gi j = X

be the intersection form, and let  g i j [Si ⊗ S j ] ∈ H 8 (X × X, Z) i, j

be the H 4 (X, Z) ⊗ H 4 (X, Z) part of the Künneth decomposition of the diagonal (mod torsion). For β1 , β2 ∈ H2 (X, Z), we define invariants m β1 ,β2 virtually enumerating rational curves of class β1 meeting rational curves of class β2 . The meeting invariants are uniquely determined by the following rules: (i) The invariants are symmetric, m β1 ,β2 = m β2 ,β1 . (ii) If either deg(β1 ) ≤ 0 or deg(β2 ) ≤ 0, then m β1 ,β2 = 0. (iii) If β1 = β2 , then,  m β1 ,β2 = n 0,β1 (Si ) g i j n 0,β2 (S j ) + m β1 ,β2 −β1 + m β1 −β2 ,β2 . i, j 2 Integrality constraints for Gromov-Witten theory always exclude constant maps. The constant contributions are easily determined in terms of the classical cohomology of X . For D ∈ H 2 (X, Z), the genus 1 invariant
1 c3 (X ) ∪ D N1,0 (D) = − 24 X

has denominator bounded by 24.

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(iv) In case of equality, m β,β = n 0,β (c2 (TX )) +



n 0,β (Si ) g i j n 0,β (S j ) −



m β1 ,β2 .

β1 +β2 =β

i, j

A geometric derivation of the rules (i-iv) is presented in Sect. 1.2. The conjectural integrality of the invariants n 0,β (γ ) implies the integrality of the meeting invariants m β1 ,β2 . In genus 1, we need only consider Gromov-Witten invariants N1,β of X with no insertions since the virtual dimension is 0. The invariants n 1,β virtually enumerating elliptic curves are uniquely defined by 

N1,β q β =

β>0



n 1,β

β>0

+

∞  σ (d) d=1

d

q dβ

(3)

1  n 0,β (c2 (TX )) log(1 − q β ) 24 β>0

1  m β1 ,β2 log(1 − q β1 +β2 ). − 24 β1 ,β2

The function σ is defined by σ (d) =



i.

i|d

The number of automorphism-weighted, connected, degree d, étale covers of an elliptic curve is σ (d)/d. Conjecture 1. The invariants n 1,β are integers. The explicit form of (3) is derived from studying a particular solvable local Calabi-Yau 4-fold in Sect. 2. 0.4. Examples. The last four sections of the paper are devoted to the calculation of basic examples of Calabi-Yau 4-folds. The two local cases, OP2 (−1) ⊕ OP2 (−2) → P2 , OP1 ×P1 (−1, −1) ⊕ OP1 ×P1 (−1, −1) → P1 × P1 , are solved in closed form by virtual localization in Sect. 3. The local case OP3 (−4) → P3 and the compact Calabi-Yau 4-fold hypersurfaces X 6 ⊂ P5 , X 10 ⊂ P5 (1, 1, 2, 2, 2, 2),

X 2,5 ⊂ P1 × P4 ,

X 24 ⊂ P5 (1, 1, 1, 1, 8, 12) are solved by the conjectural holomorphic anomaly equation3 in Sects. 4–6. The compact cases are much more interesting than the local toric examples. In all calculations, the integralities of Conjectures 0 and 1 are verified. 3 While mathematical approaches to the genus 1 invariants in the compact case are available [12,24,27], the methods are much less effective than the anomaly equation.

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0.5. Physical interpretation. Type IIA string compactifications on Calabi-Yau 4-folds give rise to massive theories with (2, 2) supergravity in 2 dimensions. Such theories and their BPS states were extensively studied in general [6,7] and in particular for type IIA 4-folds in [13,18]. The effective action, worked out in [13], contains an on Calabi-Yau d2 z R (2) term, and the topological string at genus 1 calculates a 1-loop correction to this term. The latter comes from the famous 1-loop term in 10 dimensional type IIA theory that was discovered in the context of heterotic type II duality in [41] and gives the following contribution to the 10 dimensional effective action:
δS = − d 10 x B Y8 (R). (4) Here, B is the NS − NS 2-form of type IIA coupling to the string and Y8 (R) is an 8-form constructed as a quartic polynomial in the curvature. In 10 dimensions, the term can be directly calculated from the 1-loop amplitude with 4 gravitons and the antisymmetric B-field as external legs. If the latter is in the 2 non-compact dimensions, in the absence (X ) of further flux terms, the tadpole condition that − χ24 vanishes is obtained. The topo 2 (2) logical string computes the correction to the d z R term calculated from a loop with 1 external graviton, 3 internal gravitons, and the B-field.4 As in [14,15], the loop integral receives only contributions from BPS states. The behavior of the topological string amplitude in the large volume limit appears as the zero mass contribution and supports the claim that the amplitude computes the reduction of (4). BPS states with a D2-brane charge β contribute ∼ log(1−q β ) to the integral. The integer expansion (3) can be alternatively written as   N1,β q β = − n˜ 1,β log(1 − q β ) β>0

β>0

1  n 0,β (c2 (TX )) log(1 − q β ) + 24 β>0

−

1  m β1 ,β2 log(1 − q β1 +β2 ). 24 β1 ,β2

The integrality condition for the invariants n˜ 1,β is equivalent to the conjectured integrality for n 1,β . We intepret the n˜ 1,β as counting BPS states. Futhermore, the structure of the m β1 ,β2 log(1 − q β1 +β2 ) term suggests that the invariants m β1 ,β2 count bound states at the threshold of BPS states with D2-brane charge β1 and β2 respectively. 0.6. Outlook. The meeting invariants make integrality in genus 1 for Calabi-Yau 4-folds considerably more subtle than the corresponding integrality for Calabi-Yau 3-folds. The integrality transformations in the higher dimensional Calabi-Yau cases should include all genus 0 meeting configurations. In the non-Calabi-Yau cases, higher genus meeting configurations should occur as well. Finding the correct coefficients for such a universal transformation is an interesting problem. 4 Work in progress.

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1. Genus 0 1.1. Aspinwall-Morrison. Let π and ι denote the universal curve and map over the moduli space, π : C → M 0,0 (P1 , d), ι : C → P1 . The Aspinwall-Morrison formula is
  1 ctop R 1 π∗ ι∗ (OP1 (−1) ⊕ OP1 (−1)) = 3 . 1 d M 0,0 (P ,d) By the divisor equation, we obtain
n     ctop R 1 π∗ f ∗ OP1 (−1) ⊕ OP1 (−1) ∪ evi∗ ([P]) = d −3+n , M 0,n (P1 ,d)

(5)

i=1

where [P] ∈ H 2 (P1 , Z) is the class of a point. Let X be a Calabi-Yau 4-fold, and let V1 , . . . , Vn ⊂ X be cycles imposing a 1-dimensional incidence constraint for curves. Let C⊂X be a nonsingular rational curve transversely incident to the cycles Vi . If the rational curve has generic normal bundle splitting, ∼

N X/C = OP1 (−1) ⊕ OP1 (−1) ⊕ OP1 , the contribution of C to the genus 0 Gromov-Witten theory of X is ∞ 

d −3+n q d[C]

d=1

by (5). The constraints kill the trivial normal direction. The justification for definition (2) for the virtually enumerative invariants n 0,n (γ1 , . . . , γn ) is complete. Of course, since transversality and genericity were assumed in the justification, we do not have a proof of Conjecture 0. 1.2. Meeting invariants. 1.2.1. Rules (i) and (ii). Let X be a Calabi-Yau 4-fold. The meeting invariant m β1 ,β2 virtually enumerates rational curves of class β1 meeting rational curves of class β2 . Rules (i) and (ii) have clear geometric motivation. In fact, rule (i) is consequence of rules (ii–iv). Rule (ii) may be viewed as a boundary condition. Ultimately, m β1 ,β2 is defined by rules (i–iv). Rules (iii) and (iv) are derived by assuming the best possible behavior for rational curves. However, the ideal assumptions are typically false. As in Sect. 1.1, our derivation can be viewed, rather, as a justification for the definitions.

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1.2.2. Boundary divisor. For nonzero classes β1 , β2 ∈ H2 (X, Z), let β1 ,β2 denote the virtual boundary divisor 

β1 ,β2 → M 0,0 (X, β1 + β2 ) corresponding to reducible nodal curves with degree splitting of type (β1 , β2 ). In the balanced case β1 = β2 , an ordering is taken in β1 ,β2 , and  is of degree 2. The virtual dimension of β1 ,β2 is 0. Let
Mβ1 ,β2 = 1 ∈Q [ β1 ,β2 ]vir

be the associated Gromov-Witten invariant. By the splitting axiom of Gromov-Witten theory,  Mβ1 ,β2 = N0,β1 (Si ) g i j N0,β2 (S j ), i, j

following the notation of Sect. 0.3. The meeting invariants m β1 ,β2 , defined by rules (i-iv), may be viewed as an integral version of Mβ1 ,β2 . 1.2.3. Rule (iii). Ideally, the embedded rational curves in X of class βi occur in complete, nonsingular, 1-dimensional families Fi ⊂ M 0,0 (X, βi ). Let πi and ιi denote the universal curve and map over Fi , πi : Si → Fi , ιi : Si → X. Since β1 = β2 , the families F1 and F2 are distinct. Ideally, the surfaces Si are nonsingular and the morphisms πi are smooth except for finitely many 1-nodal fibers. The meeting number m β1 ,β2 is related to the intersection ι1∗ (S1 ) ∩ ι2∗ (S2 ) ⊂ X.

(6)

However, the intersection (6) is not transverse (even ideally). A fiber of π1 may be a component of a reducible fiber of π2 or vice versa. The meeting number m β1 ,β2 is defined to count the ideal number of isolated points of the intersection (6). Hence,
ι1∗ [S1 ] ∩ ι2∗ [S2 ], m β1 ,β2 + δ = X

where the correction δ is determined by the non-transversal intersection loci. The number of times a fiber of π1 occurs as a component of a reducible fiber of π2 is simply m β1 ,β2 −β1 . Similarly, the opposite event occurs m β1 −β2 ,β2 times. The contribution to δ of each non-transversal is easily determined. Let C⊂X

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A. Klemm, R. Pandharipande

be a fiber of π1 and a component of a reducible fiber of π2 . Then
δ(C) = c1 (E), C

where 0 → N S1 /C ⊕ N S2 /C → N X/C → E → 0 is the normal bundle sequence. Certainly N S1 /C is trivial and N S2 /C has degree −1. By the Calabi-Yau condition, N X/C is of degree −2. Hence, δ(C) = −1. Rule (iii) is obtained by expanding the intersection (6) via the Künneth decomposition of the diagonal. We have
ι∗ [S1 ] ∩ ι∗ [S2 ] − δ m β1 ,β2 = X  n 0,β1 (Si ) g i j n 0,β2 (S j ) + m β1 ,β2 −β1 + m β1 −β2 ,β2 . = i, j

1.2.4. Rule (iv). In case of equality, the meeting number is more subtle. While the surface Sβ is ideally nonsingular and ι : Sβ → X is ideally an immersion, ι is not (even ideally) an embedding. The correct interpretation of m β,β is twice the number of ideal double points of ι. The factor of 2 arises from ordering. The double point formula [10] yields a calculation of m β,β as a correction to the self-intersection,

c(TX ) , m β,β = ι(Sβ ) ∩ ι(Sβ ) − X Sβ c(TSβ ) where c(TX ) and c(TSβ ) denote the total Chern classes of the respective bundles. Expanding the correction term (and using the Calabi-Yau condition) we find

c(TX ) = c2 (TX ) + c1 (TSβ )2 − c2 (TSβ ). Sβ c(TSβ ) S Certainly,
n 0,β (c2 (TX )) =

c2 (TX ). Sβ

There is a decomposition c1 (TSβ ) = −ψ + c1 (Fβ ),

Enumerative Geometry of Calabi-Yau 4-Folds

629

where ψ is the cotangent line on Sβ viewed as the 1-pointed space. Hence

c1 (TSβ )2 = ψ 2 + 4χ (Fβ ). Sβ

Sβ

An elementary geometric argument shows
1  ψ2 = − m β1 ,β2 , 2 Sβ β+β2 =β

where the right side is the number of reducible fibers of πβ . Since
1  c2 (TSβ ) = χ (Sβ ) = 2χ (Fβ ) + m β1 ,β2 , 2 Sβ β+β2 =β

only a calculation of the topological Euler characteristic χ (Fβ ) remains. The formula  χ (Fβ ) = −n 0,β (c2 (TX )) + m β1 ,β2 β+β2 =β

is obtained via a Grothendieck-Riemann-Roch calculation applied to the deformation theoretic characterization of TFβ , 0 → R 0 π∗ (ωπ∨ ) → R 0 π∗ ι∗ (TX ) → TFβ → O Fβ (D) → 0, where D ⊂ Fβ is the divisor corresponding to nodal fibers of π : S → Fβ , see [31] for a similar discussion. Rule (iv) is obtained by expanding the self-intersection of ι∗ [Sβ ] via the Künneth decomposition of the diagonal and putting together the above surface calculations. We have

c(TX ) ι∗ [Sβ ] ∩ ι∗ [Sβ ] − m β,β = X S c(TSβ )   ij n 0,β (Si ) g n 0,β (S j ) + n 0,β (c2 (TX )) − m β1 ,β2 . = β1 +β2 =β

i, j

2. Genus 1 2.1. Step I. The equation defining the virtually enumerative invariants n 1,β ,  β>0

N1,β q β =



n 1,β

β>0

∞  σ (d) d=1

d

q dβ

1  + n 0,β (c2 (TX )) log(1 − q β ) 24 β>0

1  m β1 ,β2 log(1 − q β1 +β2 ), − 24 β1 ,β2

is justified in three steps—one for each term on the right.

(7)

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The first term is the easiest since the contribution of an embedded, super-rigid, elliptic curve E ⊂ X of class β to the genus 1 Gromov-Witten theory of X is ∞  σ (d) d=1

d

q d[E] .

See [32] for a discussion of super-rigidity and multiple covers of elliptic curves. 2.2. Step II. The second term of (7) is obtained from the contributions of families of rational curves to the genus 1 Gromov-Witten invariants of X . Let F ⊂ M 0,0 (X, β) be the ideal nonsingular family of embedded rational curves (as considered in Sect. 1.2.3). We make two further hypotheses. Condition 1. The family F contains no nodal elements. Hence, the morphism π:S→F is a P1 -bundle. In fact, Condition 1 is rarely valid, even ideally, and will be corrected in Step III. The contribution of F to N1,dβ is expressed as an excess integral over the moduli π and space of maps M 1 (S, d) to the P1 -bundle representing d times the fiber class. Let ι denote the universal curve and map over the moduli space, π : C → M 1 (S, d), ι : C → S. Condition 2. R 0 π∗ ι∗ (N X/S ) vanishes. With the vanishing of Condition 2,
Cont F (N1,β ) =

[M 1 (S,d)]vir

  ctop R 1 π∗ ι∗ N X/S .

Lemma 1. Under the above hypotheses, Cont F (N1,β ) = −

1 24d


c2 (TX ). S

Proof. Consider the relative moduli space of maps to the fibers of π, M 1 ( π , d) → F.

(8)

We will use the isomorphism ∼

M 1 (S, d) = M 1 ( π , d). The two virtual classes are easily compared, [M 1 (S, d)]vir = −λ ∩ [M 1 ( π , d)]vir + χ (F) · [M 1 (P1 , d)]vir ∈ H∗ (M 1 (S, d), Q). (9)

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On the right, λ is the Chern class of the Hodge bundle, χ (F) is the topological Euler characteristic, and M 1 (P1 , d) a fiber of (8). Consider first the integral
  − (10) λ · ctop R 1 π∗ ι∗ N X/S . [M 1 ( π ,d)]vir

Using the basic boundary relation5 λ=

1

0 ∈ H 2 (M 1,1 , Q), 12

and the normalization sequence, we can rewrite (10) as
  1 − (ev1 × ev2 )∗ ([ Diag ]) · ev∗1 (c2 (N X/S )) · ctop R 1 π∗ ι∗ N X/S . 24 [M 0,2 ( π ,d)]vir Finally, using the Aspinwall-Morrison formula,
−

[M 1 ( π ,d)]vir


  d −3+2 λ · ctop R 1 π∗ ι∗ N X/S = − c2 (N X/S ). 24 S

For the second integral, we use the formula from [17] for genus 1 contributions,
  χ (F) ctop R 1 π∗ ι∗ N X/S = [M 1 (P1 ,d)]vir




χ (F)

  χ (F) ctop R 1 π∗ ι∗ (OP1 (−1) ⊕ OP1 (−1)) = . 12d M 1 (P1 ,d)

Summing the first and second integrals, we obtain, by (9),


 1 − c2 (N X/S ) + 2χ (F) . Cont F (N1,β ) = 24d S Finally, using the Calabi-Yau condition and the geometry of P1 -bundles, c2 (N X/S ) = c2 (TX ) + c1 (TS )2 − c2 (TS ) = c2 (TX ) + c2 (TS ) = c2 (TX ) + 2χ (F), concluding the lemma.

 

Modulo the corrections from nodal elements of F to be discussed in Step III, the derivation of the second term of (7) is complete since
n 0,β (c2 (TX )) = c2 (TX ). S 5 The required marked point can be added and removed by the divisor equation.

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2.3. Step III. We now relax Condition 1 of Sect. 2.2, but keep Condition 2 in the following stronger form. Let π:S→F

(11)

) be the moduli be the ideal family of embedded rational curves of class β. Let M 1 (S, β space of maps to S representing a π -vertical curve class ∈ H2 (S, Z). β The morphism (11) is the blow-up of a P1 -bundle over finitely many points corresponding to the 1  m β1 ,β2 2 β1 +β2 =β

need not be a multiple of the fiber class. As nodal fibers. Since π is not a P1 -bundle, β ). before let π and ι denote the universal curve and map over the moduli space M 1 (S, β satisfying Condition 2
. R 0 π∗ ι∗ (N X/S ) vanishes for every class β − [Fiber(π )] > 0. β The inequality is required in Condition 2 . Ideally, the inequality is violated for con equals a multiple of a single component of a reducible nodal nected curves only if β fiber of π . Then, R 0 π∗ ι∗ (N X/S ) = 0. We view F as not contributing at all to the Gromov-Witten invariants in classes violating the inequality (as these curves deform away from F). With the vanishing of Condition 2 ,
  Cont F (N1,β ) = ctop R 1 π∗ ι∗ N X/S )]vir [M 1 (S,β

satisfying the inequality. for classes β Since the family F may contain nodal elements, Lemma 1 must be modified. We have ⎛ ⎞  1 n 0,β (c2 (TX )) log(1 − q β ) Cont F ⎝ N1,β q β ⎠ = (12) 24 β−[Fiber ( π )]>0

∞ ∞ 1   + 2



cd1 ,d2 m β1 ,β2 q d1 β1 +d2 β2

d1 =1 d2 =1 β1 +β2 =β

for universal constants cd1 ,d2 . The first term on the right is the uncorrected answer of Lemma 1. The second term is the correction. The factor of 2 is included for the double counting induced by the ordering. The universal form of the correction terms follows from the canonical local analytic geometry near the nodal fibers . Let π −1 ( p) = E 1 ∪ E 2

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633

be a nodal fiber. The local geometry of S near π −1 ( p) is the total space of the node smoothing deformation. The restriction of N X/S splits in the form O S (E 1 ) ⊕ O S (E 2 ). The universality of the correction terms then follows. Lemma 2. We have ∞ ∞ 1   1 log(1 − q1 q2 ). cd1 ,d2 q1d1 q2d2 = − 2 24 d1 =1 d2 =1

Lemma 2 concludes Step III and completes the justification of definition (7) of the invariants n 1,β . 2.3.1. Proof of Lemma 2 By universality, we can prove Lemma 2 by considering any exactly solved geometry that is sufficiently rich to yield all the constants cd1 ,d2 . The simplest is the following local geometry. Let S be the blow-up of P1 × P1 at the point (∞, ∞), ν

S = BL(∞,∞) (P1 × P1 ) → P1 × P1 . Let L 1 and L 2 be line bundles on S,     L 1 = ν ∗ OP1 ×P1 (−1, −1) , L 2 = ν ∗ OP1 ×P1 (−1, −1) (E), where E is the exceptional divisor. Let X be the Calabi-Yau total space X = L 1 ⊕ L 2 → S. Of course, X is non-compact. The homology H2 (X, Z) is freely spanned by H2 (P1 × P1 , Z) = Z[C1 ] ⊕ Z[C2 ] and [E]. Let β = [C1 ] ∈ H2 (X, Z). ∼

Certainly, Fβ = P1 and the associated universal family is π : S → P1 , obtained by composing ν with the projection onto the second factor. The morphism π has a unique nodal fiber over ∞ ∈ F which splits as β1 = [C1 ] − [E], β2 = [E]. Hence, the only nonzero meeting numbers for X are m β1 ,β2 = m β2 ,β1 = 1. Condition 2 is easily verified for the family F.

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Proposition 1. We have Cont F (N1,d1 β1 +d2 β2 ) =

δd1 ,d2 12d1

for d1 , d2 > 0. Proof. Let T2 = C∗ × C∗ act on P1 × P1 by (ξ1 , ξ2 ) · ([x0 , x1 ], [y0 , y1 ]) = ([ξ1 x0 , x1 ], [ξ2 y0 , y1 ]) with fixed points (0, 0), (0, ∞), (∞, 0), (∞, ∞). The action of T2 lifts canonically to S. We calculate
  ctop R 1 π∗ ι∗ (L 1 ⊕ L 2 ) Cont F (N1,d1 β1 +d2 β2 ) = [M 1 (S,d1 β1 +d2 β2 )]vir

(13)

(14)

by T2 -localization. With the correct T2 -equivariant linearizations of L 1 and L 2 , the integral is possible to evaluate explicitly. Let s1 and s2 denote the weights of the two torus factors of T2 . The tangent weights of the T-action on P1 × P1 are (−s1 , −s2 ), (−s1 , s2 ), (s1 , −s2 ), (s1 , s2 ) at the respective fixed points (13). (i) Let T2 act on OP1 ×P1 (−1, −1) with weights s1 + s2 , s1 , s2 , 0 at the respective fixed points (13). The choice induces a canonical T2 -linearization on L 1 . (ii) Let T2 act on OP1 ×P1 (−1, −1) with weights s1 , s1 − s2 , 0, −s2 at the fixed points (13). Together with the canonical linearization on O S (E), the choice induces a canonical T2 -linearization on L 2 . The T2 -localization contributions of the integral (14) over 0 ∈ F must first be calculated. The contribution over 0 ∈ F certainly vanishes unless d1 = d2 . An unravelling of the formulas shows Cont0∈F (N1,dβ1 +dβ2 ) =


   −s1 − λ ctop −s1 ⊗ R 1 π∗ ι ∗ (OP1 (−1) ⊕ OP1 (−1)) . s1 M 1 (P1 ,d) Then, by a straightforward expansion similar to the proof of Lemma 1, we obtain the vanishing Cont0∈F (N1,dβ1 +dβ2 ) = 0.

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635

Fig. 1. A double comb

By the vanishing over 0 ∈ F, the contribution over ∞ ∈ F must be a constant,6 Cont∞∈F (N1,d1 β1 +d2 β2 ) ∈ Q. The T2 -action on S has 3 fixed points  p0 , p∞ , p∞

over ∞ ∈ F. Here, p0 is the fixed point lying over (0, ∞), p∞ is the node of π −1 (∞),  is the remaining fixed point. With the linearizations (i) and (ii), L has weight and p∞ 1  , and L has weight 0 over p , p . 0 over p∞ , p∞ 2 0 ∞  is 0, each node of the fixed map over Since the T2 -weight of L 1 at p∞ and p∞ 2 1 ∗ these produces a T -trivial factor of R π∗ ι (L 1 ) by the normalization sequence. Each T2 -fixed component mapping to β2 produces a cancelling T2 -trivial factor of R 1 π∗ ι∗ (L 1 ). Similarly for L 2 . The only localization graphs7 which survive the T2 -trivial factors from the 0 weights of L 1 and L 2 are double combs. A double comb is a connected graph with a single vertex  over p  , and a single path v0 over p0 , a single vertex v∞ ∞  v0 −v∞ −v∞ ,  through p . connecting p0 to p∞ ∞ Since a double comb has no loops, one of the vertices must have genus 1. The localization contribution of the double comb is understood to include all possible genus 1 vertex assignments. The final part of the analysis requires taking the nonequivariant limit

Lims1 →0 Cont∞∈F (N1,d1 β1 +d2 β2 ).

(15)

Since the contribution on the right is a constant, no information is lost. Nonequivariant limits are often difficult to study, but for double combs the analysis is simple. A factor of s1 in the denominator of the localization contribution of double  have equal comb can only occur if the two edges of the unique path connecting p0 to p∞ degrees e

e

 v0 − v∞ − v∞ . 6 By definition, the contribution over ∞ is a rational function in s and s . 1 2 7 We follow [17] for the graphical terminology for the virtual localization formula.

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A. Klemm, R. Pandharipande

The s1 factor occurs here from the node smoothing deformation at v∞ . Even then, the s1 factor in the denominator is cancelled by s1 factors in the numerator if either v0 or  has valence greater than 1. v∞ In case d1 = d2 , the latter valence condition must be satisfied, and the nonequivariant limit (15) can be taken for each double comb. In fact, each nonequivariant limit is easily seen to vanish, proving the proposition in the unequal case. If d1 = d2 , there is a unique double comb which does not satisfy the valence condition, d1

d2

 v0 − v∞ − v∞ .

(16)

However, since the nonequivariant limit Lims1 →0 exists for all other double combs, the limit must exist as well for (16). As in the unequal case, the nonequivariant limit vanishes for all double combs except (16). The limit for (16) is explicit calculated to equal 1 12d1 in the equal case.

 

To complete the proof of Lemma 2, we expand (12) for the local Calabi-Yau X . Since c2 (TX ) = 0, ∞ ∞   1   Cont F N1,d1 β1 +d2 β2 = 2



cd1 ,d2 m β1 ,β2 q d1 β1 +d2 β2

d1 =1 d2 =1 β1 +β2 =β

= cd1 ,d2 . Hence,

cd1 ,d2 =

δd1 .d2 12d

by Lemma 1.   The justification for definition (7) of the invariants n 1,β is based on ideal geometry. Since the ideal hypotheses are typically false in algebraic geometry, Conjectures 0 and 1 are not proven. In fact, one may be suspicious of their validity. In the remaining sections, we will compute many examples and find the conjectures to always be valid. 3. Local Examples I 3.1. Solutions. Proposition 1 already describes an exactly solved local Calabi-Yau 4-fold geometry. However, a complete solution is not given by Proposition 1 since only special curve classes of BL(∞,∞) (P1 × P1 ) are considered. The two simplest nontrivial local Calabi-Yau 4-folds are studied here. The examples may be viewed as the analogues of the OP1 (−1) ⊕ OP1 (−1) → P1

(17)

local Calabi-Yau 3-fold. As in (17), we find closed form solutions for all curve classes.

Enumerative Geometry of Calabi-Yau 4-Folds

637

3.2. Local P2 . Let Y be the local Calabi-Yau determined by the total space of the rank 2 bundle OP2 (−1) ⊕ OP2 (−2) → P2 . Let P denote the point class on P2 . Proposition 2. We have

 (−1)d 2d , d 2d 2 ⎛ ⎞

   2d 1 d d d q ⎠ log ⎝ (−1) N1,d q = d 12 N0,d (P) =

d>0

d≥0

1 = − log(1 + 4q). 24 The proposition is proven by localization. Let T2 act on P2 with fixed points [1, 0, 0], [0, 1, 0], [0, 0, 1]

(18)

and respective tangent weights (s1 , s2 ), (−s2 , s1 − s2 ), (s2 − s1 , −s1). Let P be the equivariant class of the fixed point [1, 0, 0]. Let T2 act on OP1 (−1) with weights 0, s2 , s1 at the respective fixed points (18). Similarly, let T2 act on OP1 (−2) with weights −s1 − s2 , s2 − s1 , s1 − s2 . The above choices kill the localization contributions to N0,d (P) and N1,d of all graphs with either a node over [1, 0, 0] or an edge connecting [0, 1, 0] and [0, 0, 1]. The sum over remaining comb graphs is not difficult and left to the reader. The integral invariants n 0,d (P) and n 1,d can be easily calculated from the GromovWitten invariants by the defining formulas (2) and (3). n 0,d (P) n 1,d

1 −1 0

2 1 0

3 −1 −1

4 2 2

5 −5 −8

6 13 27

7 −35 −90

8 100 314

9 −300 −1140

10 925 4158

The underlying moduli space of maps (with the point condition imposed) for the invariants n 0,1 (P) and n 0,2 (P) are projective spaces of dimension 1 and 4 respectively. It appears when the underlying moduli space is Pk , the invariant is (−1)k reminiscent of Seiberg-Witten theory for surfaces. The genus 1 invariants vanish in case there are no embedded genus 1 curves. The underlying moduli space for n 1,3 is P9 .

638

A. Klemm, R. Pandharipande

3.3. Local P1 × P1 . Let Z be the local Calabi-Yau determined by the total space of the rank 2 bundle OP1 ×P1 (−1, −1) ⊕ OP1 ×P1 (−1, −1) → P1 × P1 . Appropriate localization formulas8 for Y in genus 0 and 1 yield 

N0,(d1 ,d2 ) (P) =

 (−1)d1 +d2 · z(m)z(n)

m∈P(d1 ) n∈P(d2 )


M 0,(m)+(n)+1

(m) i=1

(1 + m i ψi )

1 (n)

j=1 (1 − n j ψ(m)+ j )

and 

N1,(d1 ,d2 ) =

 (−1)d1 +d2 · z(m)z(n)

m∈P(d1 ) n∈P(d2 )


M 1,(m)+(n)

(m) i=1

(1 + m i ψi )

1 (n)

j=1 (1 − n j ψ(m)+ j )

.

Here, P is the point class on P1 × P1 , and P(d) denotes the set of partitions of d. For p ∈ P(d), the length is denoted by ( p). The function z( p) = |Aut( p)| ·

( p) 

pi

i=1

is the usual factor. By evaluating the above localization sums, we obtain the following exact solutions. Proposition 3. We have,

 (d1 ,d2 ) =(0,0)

 d1 + d2 2 1 N0,(d1 ,d2 ) (P) = , (d1 + d2 )2 d1 ⎛ ⎞

2   d1 + d2 1 log ⎝ N1,(d1 ,d2 ) q1d1 q2d2 = q1d1 q2d2 ⎠. 12 d1 d1 ≥0 d2 ≥0

The integral invariants n 0,(d1 ,d2 ) (P) and n 1,(d1 ,d2 ) can be easily calculated from the Gromov-Witten invariants by the defining formulas (2) and (3). n 0,(d1 ,d2 ) (P) 0 1 2 3 4 5 6

0 * 1 0 0 0 0 0

1 1 1 1 1 1 1 1

2 0 1 2 4 6 9 12

3 0 1 4 11 25 49 87

4 0 1 6 25 76 196 440

8 We now leave the optimal weight choice for the reader to discover.

5 0 1 9 49 196 635 1764

6 0 1 12 87 440 1764 5926

Enumerative Geometry of Calabi-Yau 4-Folds

n 1,(d1 ,d2 ) 0 1 2 3 4 5 6

0 * 0 0 0 0 0 0

1 0 0 0 0 0 0 0

639

2 0 0 1 2 5 8 14

3 0 0 2 10 28 68 144

4 0 0 5 28 112 350 922

5 0 0 8 68 350 1370 4426

6 0 0 14 144 922 4426 17220

For n 0,(1,d) (P) the underlying moduli space is P2d . The elliptic invariants vanish in classes in which there are no embedded elliptic curves. For n 1,(2,2) the moduli space is P8 . 4. 1-Loop Amplitude and Ray-Singer Torsion Let X be a nonsingular Calabi-Yau n-fold. The string amplitude which contains information about the genus 1 Gromov-Witten theory of X is the twisted 1-loop amplitude
  d2 τ 1 Tr H (−1) F FL FR Q HL Q¯ H R . (19) F1 = 2 F Imτ Here, the integral is over the fundamental domain F of the mapping class group of the world-sheet torus with respect to the SL(2, Z)-invariant measure. The trace is over the Ramond sector H of the twisted non-linear σ -model on X . The operators FL and FR are the left and right fermion number operators, F = FL + FR , and HL and H R are the left and right moving Hamilton operators. The parameter τ is the complex modulus of the world-sheet torus, and Q = exp(2πiτ ). The object F1 is an index which depends either only on the complexified Kähler structure moduli of X in the A-model or only on the complex structure moduli of Xˆ in the B-model. The dependence on the moduli is via the spectra of HL and H R . We will use the B-model analysis to evaluate F1 on the mirror Xˆ of X . Predictions for the genus 1 invariants of X are then made by the mirror map. By the world-sheet analysis of [7,2], F1 satisfies the holomorphic anomaly equation   1 1 ∂i ∂¯j¯ F1 = Tr H (−) F Ci C¯ j¯ − Tr H (−) F G i j¯ . (20) 2 24 Here G i j¯ is the Zamolodchikov metric and the derivatives are with respect to the N = 2 moduli. For N = 2σ -models on Calabi-Yau n-folds, we can specialize to the complex moduli on Xˆ . Then, Tr H (−) F becomes the Euler number χ of the Xˆ and G i j¯ becomes the Weil-Peterson metric on the complex structure moduli space of Xˆ . The Ci are genus 0, 3-point functions in the A-model. In the B-model on Xˆ , the Ci can be calculated from the Picard-Fuchs equation for periods of the holomorphic (n, 0) form on Xˆ . The indices i, j¯ run from 1 to h n−1,1 ( Xˆ ).

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There are two methods to integrate Eq. (20). One can use the integrability conditions of special geometry for Calabi-Yau n-folds or, somewhat more generally, the tt ∗ -equations. The latter apply to any N = 2 conformal world-sheet theory. If the central charge satisfies 3c = n ∈ Z, then the tt ∗ -equations imply the special geometry relations for Calabi-Yau n-folds. The tt ∗ equations are used in [7,2] to obtain p+q 1 χ Tr p,q [log(g)] − K + log | f |2 . F1 = (−1) p+q (21) 2 p,q 2 24 The sum here is over the Ramond-Ramond degenerate lowest energy states labeled by p, q which range for the σ -model case in the left and the right moving sector as follows: n n n − , − + 1, . . . , . 2 2 2 By the usual argument [42], the states are identified in the A-model with harmonic (k, l) = ( p +

n n ,q + ) 2 2

forms. For 4-folds the (2, 2) forms correspond to ( p, q) = (0, 0) and decouple from the sum in (21). Finally, g is the tt ∗ metric, K is the Kähler potential for Weil-Peterson metric, and f is the holomorphic ambiguity. The metric g is related to the Weil-Peterson metric by G i j¯ =

gi j¯ = ∂i ∂¯j¯ K , g00¯

The Kähler potential K is given by e−K =


Xˆ

g00¯ = e K .

¯ , ∧

(22)

(23)

where  is the holomorphic (n, 0)-form on Xˆ — e−K can be calculated from the periods on Xˆ . In summary, specializing to 4-folds9 with h 21 = 0, we evaluate (21) to 

χ (X ) K − log det G + log | f |2 . (24) F1 = 2 + h 11 (X ) − 24 For 3-folds, in our normalization,10

 1 1 χ (X ) (3) 3 + h 11 (X ) − K − log det G + log | f (3) |2 . F1 = 2 12 2

(25)

The Gromov-Witten invariants are extracted in the holomorphic limit of (24) in the large volume of X corresponding to the point P of maximal unipotent monodromy of 9 All of our compact examples will satisfy h = 0. 21 10 This is, up to the normalization factor 1 , the result in [2]. 2

Enumerative Geometry of Calabi-Yau 4-Folds

641

Xˆ . Taking the holomorphic limit is very similar for all dimensions. We introduce the flat coordinates near P ti =

X i (z) , X 0 (z)

(26)

which are identified with the complexified Kähler parameters of X . As the coordinates z are the complex structure moduli of Xˆ , Eq. (26) defines the mirror map between the complex structure on Xˆ and the complexified Kähler structure on X . The function X 0 is the unique holomorphic period at P, which we chose to lie at z i = 0. The functions  1  0 X log(z) + holomorphic Xi = 2πi are the h 11 (X ) = h n−1,1 ( Xˆ ) single logarithmic periods. The existence of X 0 and X i satisfying the above conditions is part of the defining property of P. Using further the structure of the periods in an integer symplectic basis at P, we conclude limt¯→i∞ K = − log(X 0 ), limt¯→i∞ G i j¯ =

(27)

∂ti j δ . ∂z j j¯

After substitution in (24), we obtain the holomorphic limit of F1 at P,

 χ  ∂z 0 + log | f |2 . F1 = − h 11 − 2 log(X ) + log det 24 ∂t

(28)

Here, ∂z/∂t is the Jacobian of the inverse mirror map, and f (z) is the holomorphic ambiguity at genus 1. The latter is restricted by the space time modular invariance of F1 (t, t¯). The first two terms in (24) can be shown to be modular invariant. Therefore f (z) must be modular invariant as well. The modular constraints together with the large volume behavior, which in physical terms comes from a zero mode analysis, F1 →

h n−1,1
(−1)n+1  ti cn−1 (T ) ∧ Hi , 24 X

t → ∞,

(29)

i=1

and the expected universal local behavior at other singular limits in the complex structure moduli space fix the holomorphic ambiguity f (z). As explained in [3], the the genus 1 free energy F1 is related to the Ray-Singer torsion [35]. The latter describes aspects of the spectrum of the Laplacians of V,q = ∂¯ V ∂¯ V† + ∂¯ V† ∂¯ V of a del-bar operator ∂¯ V : ∧q T¯ ∗ ⊗ V → ∧q+1 T¯ ∗ ⊗ V coupled to a holomorphic vector bundle V over M. More precisely, with a regularized determinant over the non-zero mode spectrum of V,q , one defines11 [35] I R S (V ) =

n     q (−1)q+1 det V,q 2 .

(30)

q=0 11 [34] reviews these facts and relates the Ray-Singer torsion to Hitchin’s generalized 3-form action at one loop.

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A. Klemm, R. Pandharipande

The case V = ∧ p T ∗ with  p,q = ∧ p T ∗ ,q leads to the definition of a family index F1 =

n n     (−1) p+q pq 1 det  pq log 2

(31)

p=0 q=0

depending only on the complex structure of Xˆ . 5. Local Examples II We now consider the local Calabi-Yau geometry O(−n) → Pn−1 . Since the space is toric, Batyrev’s reflexive cone construction produces the mirror geometry: a compact Calabi-Yau (n − 1)-fold together with a meromorphic (n − 1, 0)-form λ. The latter can be obtained as a reduction of the holomorphic (n, 0)-form to the Calabi-Yau (n − 1)-fold and has a non-vanishing residuum.12 The n periods of λ fulfill the Picard-Fuchs equation   n−1  n−1 n LX = θ − (−1) nz (nθ + k) θ X = 0, (32) k=1 d where θ = z dz . The discriminant of the Picard-Fuchs equation is

˜ = (1 − (−n)n z).  Equation (32) has a constant solution corresponding to the residuum of λ, a general property of non-compact Calabi-Yau manifolds. We normalize constant period to X 0 = 1. The system (32) has 3 regular singular points: (i) the point z = 0 of maximal unipotent monodromy, ˜ = 0 corresponding to a nodal singularity (called the conifold point), (ii) the point  (iii) the point z → ∞ (a Zn orbifold point). Because of (iii), a single cover variable ψ is sometimes more convenient. It is customary to introduce the latter as z=

(−1)n , (nψ)n

(33)

so the conifold is at ψ n = 1. More precisely we define the conifold divisor as  = (1 − ψ n ) . Solutions to (32) can be obtained as  k

 ∂  Xk = X 0 (z, ρ)  2πi∂ρ

(34)

,

(35)

ρ=0

12 One can also consider the elliptic fibration over Pn given by the hypersurface of degree 6n in the weighted projective space Pn+1 (1n , 2n, 3n), apply Batyrev’s reflexive polyhederal mirror construction, and take the large fiber limit on both sides.

Enumerative Geometry of Calabi-Yau 4-Folds

643

where we define X (z, ρ) := 0

∞  k=0

z k+ρ . (−n(k + ρ) + 1)(k + ρ + 1)n

(36)

Specializing to n = 4, we find the compact part of the mirror geometry is related to the K3 given by the quartic in P3 obtained by setting n = 4 in the above equations. The meromorphic differential is given by
d 1 , λ= 2πi γ0 p where the contour γ0 is around p = 0 and d is the canonical measure on P3 . The single logarithmic solution is  1  log(z) + 24 z + 1260 z 2 + 123200 z 3 + O(z 4 ) . X1 = (37) 2πi We define q = exp(2πit) and with t = X 1 / X 0 = X 1 we obtain by inverting the mirror map the series z = q − 24 q 2 − 396 q 3 − 39104 q 4 + O(q 5 ) .

(38)

The first term of (28) vanishes in the local case as X 0 = 1. The holomorphic limit of the Kähler potential term is trivial. We must determine the holomorphic ambiguity. As f is a modular invariant, f can be expressed in terms of ψ 4 . As there is a non-degenerate conformal field theory description at ψ = 0 given by the σ -model on the orbifold Cn /Zn , F1 cannot be singular at this point. On the other hand the CFT degenerates at ψ n = 1 and at ψ = ∞, and F1 is expected to be logarithmically divergent at the conifold and at the point of maximal unipotent monodromy. The former behavior can be argued by comparison with the 3-fold case while the latter follows directly from (29) and the leading behavior of (37). Therefore, we are left with the ansatz f = x log(), where x is unknown. We obtain 

1 ∂ψ F1 = log − log(). (39) ∂t 24 The first term comes from the holomorphic limit of the Weil-Peterson metric. The x coefficient of the last term is matched to the first term in the localization calculation that can be done in the local case. The leading behavior at boundary divisors in the moduli space will only depend on the type of the singularities. We expect therefore that the 1 leading − 24 log() behavior will be universal at every conifold in 4-folds. The series F1 generates the genus 1 Gromov-Witten invariants of the local Calabi-Yau O(−4) → P4 ,  N1,d q d , F1 = C + d>0

where C is an integration constant. Our result for F1 agrees up to degree 6 with the calculation of Mayr [30]. Mayr uses the localization and Hodge integral formulas of [9,17] to calculate up to degree 6. The associated invariants n 0,d (L) and n 1,d are given in Table 1. We have checked integrality of the invariants up to degree d = 100.

644

A. Klemm, R. Pandharipande Table 1. Integer invariants n 0,d (L) and n 1,d for O(−4) → P3 g=0

g=1

−20 −820 −68060 −7486440 −965038900 −137569841980 −21025364147340 −3381701440136400 −565563880222390140 −97547208266548098900 −17249904137787210605980 −3113965536138337597215480

0 0 11200 3747900 963762432 225851278400 50819375678400 11209456846594400 2447078892879536000 531302247998293196352 115033243754049262028000 24874518281284024213236000

d 1 2 3 4 5 6 7 8 9 10 11 12

6. Compact Calabi-Yau 4-Folds The holomorphic anomaly equation will now be used to verify the integrality conjectures for several compact Calabi-Yau 4-folds. The compact cases have much more interesting geometry than the local models previously considered.

6.1. The sextic 4-fold. Batyrev’s mirror construction gives the 1-parameter complex mirror family for degree n hypersurfaces in Pn−1 as p=

n 

xkn − nψ

k=1

n 

xk = 0

(40)

k=1

in13 Pn−1/Zn−2 n . The holomorphic (n, 0)-form can be written as =

1 2πi


γ0

ψd , p

where the contour  γ0 is around p = 0. We obtain the Picard-Fuchs operator for the period integrals  (4)  in this family parameterized by the variable z = (−nψ)−n as L = θ n−1 − n z

n−1 

(n θ + k) .

(41)

k=1

˜ = 1 − n n z, and z = 0 is the point of maximal unipotent monodrThe discriminant is  omy. Solutions as in (35) are obtained from X (z, ρ) := 0

∞  (n(k + ρ) + 1) k=0

(k + ρ + 1)n

z k+ρ .

(42)

13 The orbifold is essentially irrelevant for the B-model period calculation. It only changes the normalization 1 . of the periods by a factor n−2 n

Enumerative Geometry of Calabi-Yau 4-Folds

645

Here, there is a non-trivial holomorphic solution X0 =

∞  (nk)! k=0

(k!)n

zk .

(43)

Let X 6 ⊂ P5 be the sextic 4-fold. For X 6 , the first few terms of the inverse mirror map are z = q − 6264 q 2 − 8627796 q 3 − 237290958144 q 4 + O(q 5 ) .

(44)

The nonvanishing Hodge numbers of X 6 up to the symmetries of the Hodge diamond are h 00 = h 4,0 = 1, h 11 = 1, h 31 = 426, h 22 = 1752. Further one has

χ= c4 = 2610, c2 = 15H 2 , c3 = −70H 3 , H 4 = 6, X

(45)

X

where H is the hyperplane class. The holomorphic ambiguity can be fixed as follows. A simple analytic continuation argument shows that X 0 ∼ ψ at the orbifold point ψ = 0. As there is no singularity 0 in F1 at this point, only the combination Xψ can appear in F1 . Furthermore, we use the universal behavior at the conifold  = (1 − ψ 6 ) = 0 and obtain, using (28), 423 log F1 = 4



X0 ψ





∂ψ + log ∂t

 −

1 log(). 24

(46)

As a consistency check, Eq. (29) is fulfilled. A further consistency check is obtained from classical methods for degrees d ≤ 4. The vanishing of the integer invariants n 1,1 = n 1,2 = 0 is expected from geometrical considerations. The invariants n 1,3 and n 1,4 are enumerative. The number of elliptic cubics on a general sextic X 6 ⊂ P5 can easily be computed by Schubert calculus14 to be 2734099200. The number of elliptic quartics on X 6 was computed to be 387176346729900 by Ellingsrud and Stromme in [8]. Finally, the integrality of n 1,d , which we have checked to d = 100, is highly non-trivial. The values for the first few n 1,d are listed in Table 2. We also report a few of the meeting invariants as they have an interesting interpretation as BPS bound states at threshold in Table 3. 14 We used the Maple package Schubert written by S. Katz and S. Stromme.

646

A. Klemm, R. Pandharipande Table 2. Integer invariants n 0,d (H 2 ) and n 1,d for X 6

d

g=0

g=1

1 2 3 4 5 6 7 8

60480 440884080 6255156277440 117715791990353760 2591176156368821985600 63022367592536650014764880 1642558496795158117310144372160 45038918271966862868230872208340160

0 0 2734099200 387176346729900 26873294164654597632 1418722120880095142462400 65673027816957718149246220800 2828627118403192025358734275898400

Table 3. Meeting invariants for X 6 m d1 ,d2

d2 = 1

d1 = 1 2 3

15245496000 0 0

2

3

111118033656000 809911567810170000 0

1576410499948536000 11490828530432030136000 16302908356356789337413600

6.2. Quintic fibrations over P1 . Genus 0 Gromov-Witten invariants for multiparameter Calabi-Yau 4-folds have been calculated in [21,29]. We determine the genus 1 GromowWitten invariants and test the integer expansion of F1 for two such cases. We first consider the quintic fibration over P1 realized as the resolution of the degree 10 orbifold hypersurface X 10 ⊂ P5 (1, 1, 2, 2, 2, 2). The non-vanishing Hodge numbers are h 0,0 = h 4,0 = 1, h 11 = 2, h 31 = 1452 up to symmetries of the Hodge diamond. We introduce the divisor F associated to the linear system generated by monomials of degree 2. For example, a representative would be x3 = 0 yielding a degree 10 hypersurface in P5 (1, 1, 2, 2, 2). The dual curve to F lies as a degree 1 curve in the quintic fiber with size t1 . Another divisor B is associated to the linear system generated by monomials of degree 1. Since B lies in a linear pencil of quintic fibers, B 2 = 0. The dual curve is the base P1 with size t2 . We calculate the classical intersection data by toric geometry as follows
F4




= 10,

F3 B


= 5,

c2 ∧ F = 110, c2 ∧ B F = 50,
M c3 ∧ F = −200, c3 ∧ B = −410 . M

c4 = 2160, M


2

M

(47)

M

By Batyrev’s construction the mirror is given also as a degree 10 hypersurface in P(1, 1, 2, 2, 2, 2)/(Z10 Z45 ), 2  k=1

2(n+1)

xk

+

n+2  k=3

(n+1)

xk

− 2φ

2  i=1

xin+1 − ψ

n+2  k=1

xi

(48)

Enumerative Geometry of Calabi-Yau 4-Folds

647

with15 n = 4. We derive the Picard-Fuchs operators as   n+1 L1 = θ1n−1 (2θ2 − θ1 ) − (n + 2) k=1 ((n + 2)θ1 − k) z 1 , L2 = θ2 − (2θ2 − θ1 − 1)(2θ2 − θ1 − 2)z 2 ,

(49)

φ 1 where θi = z i dzd i with z 1 = (−(n+2)ψ) n+2 and z 2 = (2φ)2 . The system has one conifold discriminant con and a ‘strong coupling’ discriminant s at

con = 1 − (ψ n+2 − φ)2 ,

s = 1 − φ 2 .

(50) (1)

Let us now turn to the calculation of n 0,β (Si ) and n 0,β (c2 ). We denote by A1 = J F (1) and A2 = J B the harmonic (1, 1)-forms dual to F and B. We chose further a basis (2) 2 2,2 (M). Toric geometry A(2) 1 = J F and A2 = J B J F for the vertical subspace of H implies that the latter can be obtained from the leading θ -polynomials Lˆ i of the PicardFuchs operators. More precisely the subspace is spanned by the degree two elements of the graded multiplicative ring (1) (1) R = C[A1 , . . . , Ah 11 ]/Id(Lˆ i |θ →A(1) ) , i

(51)

i

where the Lˆ i are the formal limits Lˆ i = lim zi →0 Li of the Picard-Fuchs operators. Following [19,21,29], we calculate the genus 0 quantum cohomology intersection
(1) (1) (1) Ci jα = Ai ∧ A j ∧ A(2) (52) α + instanton corrections, M

in the B model as follows. Using the flat coordinates ti =

Xi 1 log(z i ) + O(z) = X0 2πi

for the identification of the B-model structure at the point of maximal unipotent monodromy with the A-model structure [19], we find Ci(1) jα = ∂ti ∂t j

(2) α , X0

(53)

(2) where given an A(2) α the dual period α is specified by the leading quadratic behavior 16 in the logarithms as
(2) log(z i ) log(z j ) α 1 (1) (1) = Ai ∧ A j ∧ A(2) + O(z) . (54) α × 0 X 2 (2πi)2 M ij

For example, using (47), the period c2 , whose expansion in qi yields the n 0,β (c2 ), is specified by the leading logarithmic behavior c2 1 = (55 log(z 1 )2 + 50 log(z 1 ) log(z 2 )). X0 (2πi)2 15 These formulas apply to n-dimensional degree 2(n + 1) hypersurfaces in P(1, 1, 2n ). 16 Note that admixtures of periods with lower leading logarithmic behavior does not affect C (1) due to the i jα

derivatives in (52).

648

A. Klemm, R. Pandharipande Table 4. Integer invariants for the resolution of X 10 ⊂ P5 (1, 1, 2, 2, 2)

n 10,β d1 = 0 1 2 3 4

d2 = 0

n 20,β d1 = 0 1 2 3 4

d2 = 0 * 2875 1218500 951619125 969870120000

2

3

4

1

2

3

4

5 0 0 0 9375 0 0 0 17669375 5243750 0 0 34150175000 50975575000 4766665625 0 66623314796875 253824223203750 125716582171875 5379339875000

d2 = 0

1

2

3

4

* 0 0 −2768250 −17325370250

0 0 0 7297250 90447173500

0 0 0 7297250 699252105750

0 0 0 −2768250 90447173500

0 0 0 0 −17325370250

n 1,β d1 = 0 1 2 3 4

1

* 0 0 0 0 12250 12250 0 0 0 6462250 35338750 6462250 0 0 5718284750 85125750000 85125750000 5718284750 0 6349209995000 192339896968750 507648446407500 192339896968750 6349209995000

Table 5. Meeting invariants for the resolution of X 10 ⊂ P5 (1, 1, 2, 2, 2) m β1 ,β2 (0, 1) (0, 2) (1, 0) (1, 1) (1, 2) (2, 0) (2, 1)

(0, 1)

(0, 2)

(1, 0)

(1, 1)

(1, 2)

(2, 0)

(2, 1)

−10

−10 10

6500 0 10781250

0 0 19237750 10768250

0 0 0 0 0

4025250 0 5310625000 10532668750 0 2555792968750

4025250 0 43309206500 43309206500 0 22836787744000 124882678630250

With this information, we calculate the invariants (2)

n i0,β = n 0,β (Ai ) as well as n 0,β (c2 ). Finally, for the genus 1 Gromov-Witten invariants, we obtain

F1 = 86 log

X0 ψ



+ log det

∂(ψ, φ) ∂(t1 , t2 )

 −

1 7 log(con ) − log(s ). 24 24

(55)

The only difference from the calculation for the sextic is that the behavior at s must be  determined. The latter determination is made by imposing (29) with M c3 ∧ B = −410. The second condition in (29) is a check. At s = 0 we have divisor collapsing, which is an P1 fibration over the degree 5 hypersurface in P3 . Integer as well as meeting invariants are listed in Tables 4 and 5. It is interesting to compare the above quintic fibration with a different quintic fibration given by the hypersurface of bidegree (2, 5), X 2,5 ∈ P1 × P4 .

Enumerative Geometry of Calabi-Yau 4-Folds

649

The Hodge diamond of the second fibration is the same as the previous case. The divisors B and F correspond to the pull-backs of the hyperplane classes on P1 and P4 respec2 tively. We use the same basis for the vertical subspace of H 2,2 (M) as before A(2) 1 = JF (2) and A2 = J B J F . Due to the different fibration structure, the topological data differ from the previous case:

c2 ∧ F 2 = 70, c2 ∧ BF = 50, F 4 = 2, F 3 B = 5,

M M
(56) c4 = 2160, c3 ∧ F = −200, c3 ∧ B = −330 . M

M

M

We derive the Picard-Fuchs equations in the standard large volume variables z 1 and z 2 [19] as L1 = θ13 (5θ1 − 2θ2 ) − 55 z 1 L2 = θ2 − z 2

2 

4 

(5θ1 + 2θ2 + k) + 4z 2 (5θ1 + 2θ2 + 1),

k=1

(57)

(5θ2 + 2θ1 + k) .

k=1

The system has only one conifold discriminant  = (1 − x12 ) − 5x2 (1 + 4x1 ) + 10x22 (1 − x1 ) − x23 (10 − 5x2 + x22 ) .

(58)

We have introduced rescaled variables x1 = 55 z 1 and x2 = 22 z 2 . Here, we know no further regularity conditions in the interior of the moduli space. Therefore, we simply impose (29) with

c3 ∧ B = −330 and c3 ∧ B = −200. M

M

That fixes the coefficients of the log(z 1 ) and log(z 2 ) terms in the most general ansatz of the ambiguity 

  1 51 ∂(z 1 , z 2 ) 22 − log() + log(z 1 ) + log(z 2 ). F1 = 86 log X 0 + log det ∂(t1 , t2 ) 24 4 3 (59) The integer invariants listed in Table 6 are compatible with the previous quintic fibration — we get the same invariants in the fiber direction, as expected. 6.3. Elliptically fibered Calabi-Yau 4-folds. A simple elliptic fibration over P3 compactifies the local model in Sect. 5. Consider the resolution of the degree 24 orbifold hypersurface X 24 ⊂ P5 (1, 1, 1, 1, 8, 12) in weighted projective space. The genus 0 invariants have be calculated in [21]. The resolution has the following non-vanishing Hodge numbers: h 0,0 = h 4,0 = 1, h 11 = 2, h 31 = 3878, h 22 = 15564 up to symmetries.

650

A. Klemm, R. Pandharipande Table 6. Integer invariants for X 2,5 ⊂ P1 × P4

n 10,β d1 = 0 1 2 3 4

d2 = 0

1

2

3

* 9950 5487450 4956989450 5573313899000

0 171750 533197250 1342522028500 3120681190272750

0 609500 9651689750 64483365881000 301443864603401500

0 609500 63917722000 1152361680367750 10812807897775185750

n 20,β d1 = 0 1 2 3 4

d2 = 0

1

2

3

* 2875 1218500 951619125 969870120000

125 195875 369229625 713334157250 1390949237651750

0 1248250 10980854250 53873269172000 205222409245164750

0 1799250 101591346500 1308427978728875 9819953566670512000

d2 = 0

1

2

3

* 0 0 −2768250 −17325370250

0 0 0 218986250 2510820252500

0 0 0 82508848750 1468762788741875

0 0 0 2759605738750 94873159058300000

n 1,β d1 = 0 1 2 3 4

We introduce the linear system B generated by linear polynomials in the four degree 1 variables. The linear system maps X 24 to P3 with fibers given by elliptic curves. The second linear system E is generated by polynomials of degree 4. The curve dual to E is a curve extending over the fiber E with size denoted by t1 . The curve dual to B is a degree one curve in P3 with size denoted by t2 . The intersections of the divisors are E 4 = 64,

E 3 B = 16,

E 2 B 2 = 4,

E B 3 = 1,

B4 = 0 .

Further topological data are


c4 = 23328, c3 ∧ B = −960, c3 ∧ E = −3860, M

M
M c2 ∧ B 2 = 48, c2 ∧ B E = 182, c2 ∧ E 2 = 728. M

M

(60)

(61)

M

The mirror family is likewise given by an hypersurface of degree 24 in P(1, 1, 1, 1, 8, 12)/(Z324 ), n 

2 3 x16n + xn+1 + xn+2 − nφ

n 

xi2n − 6nψ

n+2 

xi

(62)

L1 = θ1 (θ1  − nθ2 ) − 12(6 θ1 − 5)(6 θ1 − 1)z 1 , L2 = θ2n − nk=1 (n θ2 − θ1 − k)z 2 ,

(63)

k=1

k=1

i=1

with17 n = 4. We derive the Picard-Fuchs operators as

nφ (−1) where θi = z i dzd i with z 1 = (nψ) 6 and z 2 = (nφ)n . The system has two conifold discriminants n

1 = 1 − φ n ,

2 = 1 − φ˜ n ,

17 These formulas apply to n-dimensional degree 6n hypersurfaces in P(1n , 2n, 3n).

(64)

Enumerative Geometry of Calabi-Yau 4-Folds

651

Table 7. Integer invariants for the resolution of X 24 n 10,β d1 = 0 1 2 3 4 n 20,β d1 = 0 1 2 3 4 n 1,β d1 = 0 1 2 3 4

d2 = 0

1

2

3

4

0 960 1920 2880 3840

0 5760 −1817280 421685760 2555202430080

0 181440 −98640000 29972448000 −6353500619520

0 13791360 −10715760000 4447212981120 −1273702762398720

0 1458000000 −1476352644480 783432258136320 −285239128072550400

d2 = 0

1

2

3

4

0 0 0 0 0

−20 7680 −1800000 278394880 623056099920

−820 491520 −159801600 35703398400 −6039828417600

−68060 56256000 −24602371200 7380433205760 −1683081588149760

−7486440 7943424000 −4394584496640 1662353371955200 −478655396625235200

d2 = 0

1

2

3

4

0 −20 0 0 0

0 −120 45720 −10662240 1638152760

0 −3780 2245680 −719326800 160844654520

11200 −7852120 2858334000 −719497580160 140278855296640

3747900 −3536410200 1724679193440 −573686979645680 145314212874711600

where we defined φ˜ = ψ 6 − φ. The solutions to the Picard-Fuchs equations can be obtained similarly as in Sect. 6.1 using the methods outlined in [21]. For example the holomorphic solution at the point of maximal unipotent monodromy is given by ∞ 

X = 0

k1 =0,k2 =0

(6k1 )!(nk2 )! z k1 z k2 . (2k1 )!(3k1 )!k1 !(k2 !)n 1 2

(65)

The considerations, which lead to the expression of F1 in the holomorphic limit, are very similar to those of Sect. 6.1,

F1 = 928 log

X0 ψ





∂(ψ, φ) + log det ∂(t1 , t2 )

A new feature here is

lim log det

ψ→0



∂(ψ, φ) ∂(t1 , t2 )

1  log(i ) . 24 2

+ 3 log(ψ) −

(66)

i=1



∼ ψ −3 ,

as is shown by simple analytic continuation of X 0 and the two logarithmic solutions to ψ = 0. To maintain the expected regularity at ψ = 0, we have to add the explicit 3 log(ψ) term to the holomorphic ambiguity. As a check of the result (66), we note again that (29) with (61) is fulfilled. (2) (2) 1 We chose further a basis A1 = 17 (4J E2 + J E J B ) and A2 = J B2 and calculate as before the genus 0 and genus 1 invariants. As a consistency check we note that scaling the size of the elliptic fiber t1 to infinity leaves us precisely with the O(−4) → P3 geometry. The corresponding invariants are listed in Table 7. Acknowledgements. The paper began in a conversation with I. Coskun about the classical enumerative geometry of canonical curves.

652

A. Klemm, R. Pandharipande

We thank M. Aganagic, J. Bryan, D. Maulik, S. Theisen, and especially C. Vafa for useful discussions. A. K. was partially supported by the DOE grant DE-FG02-95ER40986. R. P. was partially supported by the Packard foundation and the NSF grant DMS-0500187. The project was started during a visit of R. P. to MIT in the fall of 2006.

References 1. Aspinwall, P., Morrison, D.: Topological field theory and rational curves. Comm. Math. Phys. 151, 245–262 (1993) 2. Bershadski, M., Cecotti, S., Ooguri, H., Vafa, C.: Holomorphic anomalies in topological field theories. Nucl. Phys. B 405, 279 (1993) 3. Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes. Commun. Math. Phys. 165, 311 (1994) 4. Bismut, J.-M., Gillet, H., Soulé, C.: Analytic Torsion and Holomorphic Determinant Bundles I,II and III. Commun. Math. Phys. 115, 49 (1988), Commun. Math. Phys. 115, 79 (1988) and Commun. Math. Phys. 165, 301 (1988) 5. Bryan, J., Leung, N.C.: The enumerative geometry of K 3 surfaces and modular forms. J. AMS 13, 371–410 (2000) 6. Cecotti, S., Fendley, P., Intriligator, K.A., Vafa, C.: A New supersymmetric index. Nucl. Phys. B 386, 405 (1992) 7. Cecotti, S., Vafa, C.: Ising Model and N = 2 Supersymmetric Theories. Commun. Math. Phys. 157, 139 (1993) 8. Ellingsrud, G., Stromme, S.: Bott’s formula and enumerative geometry. JAMS 9, 175–193 (1996) 9. Faber, C., Pandharipande, R.: Hodge integrals and Gromov-Witten theory. Invent. Math. 139, 173–199 (2000) 10. Fulton, W.: Intersection theory. Berlin: Springer-Verlag 1998 11. Fulton, W., Pandharipande, R.: Notes on stable maps and quantum cohomology. In: Algebraic Geometry (Santa Cruz 1995) Kollár, J., Lazarsfeld, R., Morrison, D. eds., Volume 62, Part 2, Providence, RI: Amer. Math. Soc., 1997, pp. 45–96 12. Gathmann, A.: Gromov-Witten invariants of hypersurfaces. Habilitation thesis, Univ. of Kaiserslautern, 2003 13. Gates, S.J.J., Gukov, S., Witten, E.: Two two-dimensional supergravity theories from Calabi-Yau four-folds. Nucl. Phys. B 584, 109 (2000) 14. Gopakumar, R., Vafa, C.: M-theory and topological strings I. http://arixiv.org/list/hepth/9809187, 1998 15. Gopakumar, R., Vafa, C.: M-theory and topological strings II. http://arixiv.org/list/hepth/9812127, 1998 16. Göttsche, L., Pandharipande, R.: The quantum cohomology of blow-ups of P2 and enumerative geometry. J. Diff. Geom. 48, 61–90 (1998) 17. Graber, T., Pandharipande, R.: Localization of virtual classes. Invent. Math. 135, 487–518 (1999) 18. Gukov, S., Vafa, C., Witten, E.: CFT’s from Calabi-Yau four-folds, Nucl. Phys. B 584, 69 (2000), Erratum-ibid. B 608, 477 (2001) 19. Hosono, S., Klemm, A., Theisen, S., Yau, S.T.: Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces. Nucl. Phys. B 433, 501 (1995) 20. Katz, A., Klemm, A., Vafa, C.: M-theory, topological strings, and spinning blackholes. Adv. Theor. Math. Phys. 3, 1445–1537 (1999) 21. Klemm, A., Lian, B., Roan, S.S., Yau, S.T.: Calabi-Yau fourfolds for M- and F-theory compactifications, Nucl. Phys. B 518 515–574 (1998) 22. Klemm, A., Mariño, M.: Counting BPS states on the Enriques Calabi-Yau. http://arixiv.org/list/hepth/ 0512227, 2005 23. Lee, J.L., Parker, T.: A structure theorem for the Gromov-Witten invariants of Kähler surfaces. http://arixiv.org/list/math.SG/0610570, 2006 24. Li, J., Zinger, A.: On the genus 1 Gromov-Witten invariants of complete intersection threefolds. http://arixiv.org/list/math.AG/0507104, 2005 25. Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov-Witten theory and DonaldsonThomas theory I. Comp. Math. 142, 1263–1285 (2006) 26. Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov-Witten theory and DonaldsonThomas theory II. Comp. Math. 142, 1286–1304 (2006) 27. Maulik, D., Pandharipande, R.: A topological view of Gromov-Witten theory. Topology 45, 887–918 (2006) 28. Maulik, D., Pandharipande, R.: New calculations in Gromov-Witten theory. http://arixiv.org/list/math. AG/0601395, 2006

Enumerative Geometry of Calabi-Yau 4-Folds

653

29. Mayr, P.: Mirror symmetry, N = 1 superpotentials and tensionless strings on Calabi-Yau four-folds, Nucl. Phys. B 494, 489 (1997) 30. Mayr, P.: Summing up open string instantons and N=1 string amplitudes. http://arixiv.org/list/hepth/ 0203237, 2002 31. Pandharipande, R.: The canonical class of M 0,n (Pr , d) and enumerative geometry. IMRN 173–186 (1997) 32. Pandharipande, R.: Hodge integrals and degenerate contributions. Comm. Math. Phys. 208, 489–506 (1999) 33. Pandharipande, R.: Three questions in Gromov-Witten theory, Proceedings of the ICM (Beijing 2002), Vol II., Beijing: Higher Ed. Press, 2002, pp. 503–512 34. Pestun, V., Witten, E.: The Hitchin functionals and the topological B-model at one loop. Lett. Math. Phys. 74, 21–51 (2005) 35. Ray, D.B., Singer, I.M.: Analytic torsion for complex manifolds, Ann. of. Math. 98, 154 (1973) 36. Taubes, C.: SW ⇒ Gr: from the Seiberg-Witten equations to pseudo-holomorphic curves. J. AMS 9, 845–918 (1996) 37. Taubes, C.: Counting pseudo-holomorphic submanifolds in dimension 4. J. Diff. Geom. 44, 818–893 (1996) 38. Taubes, C.: Gr ⇒ SW: from pseudo-holomorphic curves to Seiberg-Witten solutions. J. Diff. Geom. 51, 203–334 (1999) 39. Taubes, C.: GR = SW: counting curves and connections. J. Diff. Geom. 52, 453–609 (1999) 40. Thomas, R.: A holomorphic Casson invariant for Calabi-Yau 3-folds and bundles on K3 fibrations. JDG 54, 367–438 (2000) 41. Vafa, C., Witten, E.: A one loop test of string duality, Nucl. Phys. B 447, 261 (1995) 42. Witten, E.: Mirror manifolds and topological field theory. http://arixiv.org/list/hepth/9112056, 1991 Communicated by N.A. Nekrasov

Commun. Math. Phys. 281, 655–673 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0454-0

Communications in

Mathematical Physics

Quasi-Linear Dynamics in Nonlinear Schrödinger Equation with Periodic Boundary Conditions
M. Burak Erdo˘gan, Vadim Zharnitsky Department of Mathematics, University of Illinois, Urbana, IL 61801, USA. E-mail: [email protected]; [email protected] Received: 27 May 2007 / Accepted: 7 August 2007 Published online: 7 March 2008 – © Springer-Verlag 2008

Abstract: It is shown that a large subset of initial data with finite energy (L 2 norm) evolves nearly linearly in nonlinear Schrödinger equation with periodic boundary conditions. These new solutions are not perturbations of the known ones such as solitons, semiclassical or weakly linear solutions.

1. Introduction The nonlinear Schrödinger (NLS) equation iqt +
q + |q|2 q = 0,

(1)

where q : Rt × Mx → C, frequently appears as the leading approximation of the envelope dynamics of a quasi-monochromatic plane wave propagating in a weakly nonlinear dispersive medium. It arises in a number of physical models in the description of nonlinear waves such as the propagation of a laser beam in a medium whose index of reflection is sensitive to the wave amplitude. NLS has been considered on various domains such as M = Rn , Tn , with periodic or Dirichlet boundary conditions. One dimensional cubic NLS is integrable [17] and the explicit (or approximately explicit) solutions can be obtained as solitons, cnoidal waves, and their perturbations. There have been also many interesting results on the long time asymptotics of solutions of integrable NLS in the limit of small dispersion, see e.g. the recent monograph [10,6,16,3] and references therein. Recent results in optical communication literature (see, e.g. [2,7,13], and the Appendix) suggest that for some initial data (highly localized pulses) the evolution is nearly linear. Based on these studies, we introduce a large class of solutions, which we call quasi-linear, for one dimensional cubic NLS with periodic boundary conditions. These  The authors were partially supported by NSF grants DMS-0505216 (V. Z.) and DMS-0600101 (B. E.).

656

M. B. Erdo˘gan, V. Zharnitsky

solutions can be characterized by the magnitude of Fourier coefficients of the initial data. We prove that these solutions evolve nearly linearly using a normal form reduction and estimates on Fourier sums. Although we do not explicitly use integrability, we do rely on the integrability of the quartic normal form which is partially responsible for quasi-linear behavior. Therefore, similar results can be obtained for some nonlinear PDEs, such as iqt + qx x x x + |q|2 q = 0, for which there are no integrability results. We do not study long time asymptotics but rather the finite time dynamics in the limit of spectral broadening of initial data. This broadening forces q(x, 0) H s to grow to infinity, making the analysis rather nontrivial even for the finite time interval. While, we consider the focusing case, our result holds for the defocusing case as well. The reader will be able to see that our proof can be immediately adapted for the defocusing case, since nowhere our arguments rely on the nonlinearity sign. In many engineering and physics applications, nonlinearity is unavoidable while modeling and optimizing a linear behavior is much easier than a nonlinear one. Therefore, it is an important question whether a nonlinear system can be made to behave linearly. In applied mathematics and physics literature, such a behavior has been observed in e.g. [1,7,8,14,15]. We believe that our result gives a systematic way to analyze this behavior in nonlinear systems when the energy is distributed over many Fourier harmonics. 2. Main Results We consider the nonlinear Schrödinger equation with periodic boundary conditions, iqt + qx x + 2|q|2 q = 0, with initial data in q(0) ∈ L 2 (−π, π ). In [4], Bourgain proved the L 2 global wellposedness of this equation. The numerical simulations of quasi-linear regime for light wave communication systems suggest that the following statement should hold (see, e.g., [7,14]) Observation 1. Assume that initial data is a localized Gaussian 1 − x2 q(x, 0) = √ e ε2 h(x), ε where h(x) is a smooth cutoff near x = ±π/2. Then the initial data evolves quasi-linearly, q(x, t) − eit (
+4P) q(x, 0)2 → 0,

(2)

as ε → 0 and for t ≤ T , where T is a fixed positive number, and P = q(·, 0)22 /2π . We will prove (2) for a large class of initial data (including the ones above) characterized by the magnitude of Fourier coefficients. We will use Fourier transform in the form
q(x, t) = u(n, t)einx , n∈Z

u(m, t) =

1 2π



π

−π

q(x, t)e−imx d x,

Quasi-Linear Dynamics in NLS

657

so that the NLS equation takes the form i

du(m) − m 2 u(m) + 2 dt m




u(m 1 )u(m 2 )u(m ¯ 3 ) = 0.

(3)

1 +m 2 −m 3 =m

Our main result is the following theorem. Theorem 2.1. Let P > 0 and C > 0 be fixed. Assume that the Fourier sequence of the
initial data u(n, 0) = q(·, 0)(n) satisfies 1

1

u(·, 0)1 ≤ Cε− 2 ,

u(·, 0)∞ ≤ Cε 2 ,

for sufficiently small ε ∈ (0, 1). Then, for each t > 0, q(·, t) − eit (
+4P) q(·, 0) L 2
t ε1− , (4) √ where P = q(·, 0)22 /2π , t = 1 + t 2 and the implicit constant depends only on C. Remark 2.1. The initial data in the observation above satisfies the hypothesis of the theorem. In fact, if f is an H s function for some s > 1/2 with compact support on (−π, π ), then 1 f ε (x) = √ f (x/ε) ε satisfies the hypothesis of the theorem. By continuous dependence on initial data in L 2 , it suffices to prove (4) for any δ > 0 and for any initial data in the following subset of L 2 : ⎧ ⎫  ∞ 1/ p ⎨ ⎬
1 1 − δ = f ∈ L 2 :  fˆ p,δ := | fˆ(n)| p eδ|n| p ≤ Cε 2 p , p ∈ [1, ∞] . Bε,C ⎩ ⎭ n=−∞

δ ⊂ H 1 , we can introduce the Hamiltonian [11] Since Bε,C

H (u) = i n 2 |u(n)|2 − i u(n 1 )u(n 2 )u(n ¯ 3 )u(n ¯ 4 ), n

l(n)=0

with conjugated variables {u(n), u(n)} ¯ n∈Z , where l(n) = n 1 + n 2 − n 3 − n 4 . The Hamiltonian flow is then given by u(n) ˙ =

∂H · ∂ u(n) ¯

Theorem 2.1 follows from the following by continuous dependence on initial data in L 2. Theorem 2.2. Let P > 0 and C > 0 be fixed. Assume that q(0)22 = 2π P, and δ q(·, 0) ∈ Bε,C for some δ > 0, and for sufficiently small ε ∈ (0, 1). Then, for each t > 0, q(·, t) − eit (
+4P) q(·, 0)2
t ε1− , (5) where the implicit constant depends only on C.

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The proof of Theorem 2.2 is based on the normal form transformations, see, e.g., [11,12] and [5]. In Sect. 3, we introduce a canonical transformation u = u(v) in the Fourier space which brings the equation into the form1 , see (15) and (16) below, v(n) ˙ = i(n 2 + 4P)v(n) + E(v)(n).

(6)

We prove that the transformation u = u(v) is near-identical in the following sense. δ or v ∈ B δ , then Proposition 2.1. If u ∈ Bε,C ε,C

u2 = v2 , and

3

u − v p,δ
ε 2

− 1p −

for 1 ≤ p ≤ ∞, where the implicit constant depends on C and p. In particular, if ε is δ implies v ∈ B δ sufficiently small, then u ∈ Bε,C ε,2C and vice versa. Then, we estimate the error term E(v) as follows δ , then the error term E(v) in the transformed equation (6) Proposition 2.2. If v ∈ Bε,C satisfies 3

E(v) p,δ
ε 2

− 1p −

,

for 1 ≤ p ≤ ∞, where the implicit constant depends on C and p. δ for some Propositions 2.1 and 2.2 imply Theorem 2.2. Indeed, assume that q(·, 0) ∈ Bε,C

δ > 0, C > 0, and for sufficiently small ε ∈ (0, 1). Multiplying (6) with e−i(n and integrating over t, we obtain  t 2 −i(n 2 +4P)t − v(n, 0) = e−i(n +4P)τ E(v)dτ. v(n, t)e

2 +4P)t

0

This and Propositions 2.1 and 2.2 imply, for each p ∈ [1, ∞], that 3

v(t) − ei Lt v(0) p,δ = v(t)e−i Lt − v(0) p,δ
t ε 2

− 1p −

,

(7)

where L(v)(n) = (n 2 + 4P)v(n). Finally, Proposition 2.1 and (7) imply, for p ∈ [1, ∞], that u(t) − ei Lt u(0) p,δ ≤ u(t) − v(t) p,δ + v(t) − ei Lt v(0) p,δ + ei Lt v(0) − ei Lt u(0) p,δ 3


t ε 2

− 1p −

,

where the implicit constant depends on C. In particular, this yields the assertion of Theorem 2.2 as follows q(t) − eit (
+2P) q(0)2 = u(t) − ei Lt u(0)2 ≤ u(t) − ei Lt u(0)2,δ
t ε1− .

Notation. We will frequently use convolution with 1/|n|, which √ will be denoted by ρ(n) =

1 |n| χZ\{0} (n).,

and we will also use the notation n =

1 + n2.

1 Similar quasi-linear behavior can be obtained for the nonintegrable NLS iq + q 2 t x x x x + |q| q = 0 with the leading behavior given by v(n) ˙ = i(n 4 + 4P)v(n).

Quasi-Linear Dynamics in NLS

659

We always assume by default that the summation index avoids the terms with vanishing denominators. To avoid using unimportant constants, we will use
sign: A
B means there is an absolute constant K such that A ≤ K B. In some cases the constant will depend on parameters such as p. A
B(η−) means that for any γ > 0, A ≤ Cγ B(η − γ ). A
B(η+) is defined similarly. 3. Normal Form Calculations Consider the change of variables u n → vn , generated by the time 1 flow of a purely imaginary Hamiltonian F. Namely, solve dw ∂F = , w|s=0 = v, ds ∂ w¯ thus producing a symplectic transformation u = u(v) := w|s=1 . Let X sF be the time s map of the flow of F. Using Taylor expansion [11,12], we have H ◦ X 1F (v) = H (v) + {H, F}(v) + · · · + 

(8)

k 1

+ 0

where

1 {· · · {{H, F}, F}, . . . , F }(v)   k!

(1 − s)k {. . . {{H, F}, F}, · · · , F } ◦ X sF (v) ds,   k! k+1


 ∂A ∂B ∂A ∂B {A, B} = − ∂u(n) ∂ u(n) ¯ ∂ u(n) ¯ ∂u(n) n

(9)

is the Poisson bracket. Recall that H has a quadratic and a quartic part H = 2 + H4 , where

2 = i




m 2 |u(m)|2 .

(10)

(11)

We write H4 = H4nr + H4r , where the superscripts “nr” and “r” denotes the non-resonant and resonant terms:
H4nr = i v(m 1 )v(m 2 )v(m ¯ 3 )v(m ¯ 4 ), l(m)=0, q(m)=0

H4r

=i




v(m 1 )v(m 2 )v(m ¯ 3 )v(m ¯ 4 ),

l(m)=0, q(m)=0

where l(m) = m 1 + m 2 − m 3 − m 4 and q(m) = m 21 + m 22 − m 23 − m 24 . As usual H4r is the part of the Hamiltonian that commutes with 2 . Note that we can further decompose H4r as

H4r = −i |v(m)|4 + 2i |v(m 1 )|2 |v(m 2 )|2 := H4r1 + H4r2 . m

m 1 ,m 2

660

M. B. Erdo˘gan, V. Zharnitsky

We sequentially apply two normal form transformations generated by F1 and F2 . We choose F1 so that the following cancellation property holds: { 2 , F1 } = −H4nr . We will prove that F1 commutes with with k = 2, we obtain

H4r2 .

(12)

Using these cancellation properties in (8)

1 1 H ◦ X 1F1 = 2 + H4r1 + H4r2 + {H4r1 , F1 } + {H4nr , F1 } + g 2F1 (H4 ) 2 2  1 (1 − s)2 3 + g F1 (H ) ◦ X sF1 ds, 2 0 where we used the notation g 0F (H ) = H,

k g k+1 F (H ) = {g F , F}, k = 0, 1, 2, . . . .

Now, we apply the second transformation2 generated by F2 to eliminate the non-resonant terms in 21 {H4nr , F1 }, i.e., 1 (13) { 2 , F2 } = − {H4nr , F1 }nr . 2 We will also prove that F2 commutes with H4r2 . Using these cancellation properties as above in (8) (with k = 1), we obtain H ◦ X 1F1 ◦ X 1F2 = 2 + H4r2 + R, where 1 R = H4r1 + {H4r1 , F1 } + {H4nr , F1 }r + {H4r1 , F2 } + {{H4r1 , F1 }, F2 } + K 2  1 1 + {{H4nr , F1 }, F2 } + {K , F2 } + (1 − s)g 2F2 (H ◦ X 1F1 ) ◦ X sF2 ds, 2 0 where

 1 1 2 (1 − s)2 3 g F1 (H4 ) + g F1 (H ) ◦ X sF1 ds. 2 2 0 The transformed evolution equation is given by K =

∂(H ◦ X 1F1 ◦ X 1F2 )

. ∂ v¯ Note that the contribution of the “leading” terms, 2 + H4r2 , is given by  

∂ |v(m 1 )|2 |v(m 2 )|2 = i(n 2 + 4P)v(n). i m 2 |v(m)|2 + 2i ∂ v(n) ¯ m ,m v(n) ˙ =

1

(14)

2

Therefore, we can rewrite (14) as v(n) ˙ = i(n 2 + 4P)v(n) + E(v)(n),

(15)

where E(v)(n) =

∂R . ∂ v(n) ¯

(16)

2 It turns out that the transform generated by F is not enough since the term {H nr , F } is present in the 1 1 4 Hamiltonian. The direct estimate of this term produces a finite order nonlinear effect (see Subsect. 4.4).

Quasi-Linear Dynamics in NLS

661

3.1. Calculation of F1 and F2 . To obtain (12), we take F1 of the form
F1 = f (m 1 , m 2 , m 3 , m 4 )v(m 1 )v(m 2 )v(m ¯ 3 )v(m ¯ 4 ). l(m)=0

We have

  ∂ F1 ∂ F1 ¯ − v(m) m 2 v(m) ∂ v(m) ¯ ∂v(m) m
=i (m 21 + m 22 − m 23 − m 24 ) f (m 1 , m 2 , m 3 , m 4 )v(m 1 )v(m 2 )v(m ¯ 3 )v(m ¯ 4 ).

{ 2 , F1 } = i




l(m)=0

Therefore, we let F1 =


v(m 1 )v(m 2 )v(m
v(m 1 )v(m 2 )v(m ¯ 3 )v(m ¯ 4) ¯ 3 )v(m ¯ 4) . = 2 2 2 2 2(m 1 − m 3 )(m 2 − m 3 ) m1 + m2 − m3 − m4 l(m)=0 l(m)=0

(17)

Now, we calculate F2 . Using the Hamiltonian structure3 ∂H ∂H =− , ∂ v(n) ¯ ∂v(n)

∂ F2 ∂ F2 =− ∂ v(n) ¯ ∂v(n)

we obtain {H4nr , F1 }nr = 2i




v(m 1 )v(m 2 )v(m 3 )v(m ¯ 4 )v(m ¯ 5 )v(m ¯ 6) − c.c. (m 2 − m 6 )(m 3 − m 6 )

m 4 ,m 5  =m 1 , m 2 ,m 3  =m 6

l(m)=0, q(m)=0

Therefore, a calculation similar to the one for F1 yields F2 =


m 4 ,m 5  =m 1 , m 2 ,m 3  =m 6

v(m 1 )v(m 2 )v(m 3 )v(m ¯ 4 )v(m ¯ 5 )v(m ¯ 6) − c.c. q(m)(m 2 − m 6 )(m 6 − m 3 )

(18)

l(m)=0, q(m)=0

Here, l(m) = m 1 +m 2 +m 3 −m 4 −m 5 −m 6 , and q(m) = m 21 +m 22 +m 23 −m 24 −m 25 −m 26 . 3.2. Proof of Proposition 2.1. First we state a simple corollary of Young’s inequality. Recall that ρ(n) = 1/|n| for n = 0 and ρ(0) = 0. Lemma 3.1. For any p > 1, for any choices of ± signs,    
  
w p− .  w(±n ± j)ρ(± j)    p  j n

With some abuse of notation, we denote each sum of the above form by w ∗ ρ. 3 These identities follow from the following easily checked ones: ¯ = 0 and ∂v H¯ (v, v) ¯ = ∂v¯ H (v, v). ¯ + ∂v H¯ (v, v) ¯

(H ) = 0, ∂v H (v, v)

662

M. B. Erdo˘gan, V. Zharnitsky

Proof. Recall that by Young’s inequality, w∗ρ p
wq ρr , where 1+ 1p = q1 + r1 . The lemma follows since ρ ∈ q for any q > 1.   Proof of Proposition 2.1. First note that the equality of the 2 norms follows from Hamiltonian formalism. Indeed, it is straightforward to verify that {F, Q} = 0 (where Q(u) = u22 ), which implies 2 norm conservation. To prove the second statement, we should estimate the time 1 map of the flow of F1 and of F2 . We start with F1 , dw(n) ∂ F1 = = ds ∂ w(n) ¯


m 1 +m 2 −m 3 −n=0

w(m 1 )w(m 2 )w(m ¯ 3) . (m 1 − n)(m 2 − n)

(19)

δ ) Multiplying with eδ|n| , we estimate (assuming that w ∈ Bε,C

   δ|n| dw(n)   e  ds 
≤ m 1 +m 2 −m 3 −n=0

≤ w∞,δ

e−δ(|m 1 |+|m 2 |+|m 3 |−|n|) |w(m 1 )eδ|m 1 | w(m 2 )eδ|m 2 | w(m 3 )eδ|m 3 | | |m 1 − n||m 2 − n|


|w(m 1 )|eδ|m 1 | |w(m 2 )|eδ|m 2 | ≤ w∞,δ [|w|eδ|·| ∗ ρ]2 (n). |m − n||m − n| 1 2 m ,m 1

2

In the second line, we used the fact that |m 1 | + |m 2 | + |m 3 | − |n| ≥ 0. Therefore, by Lemma 3.1, we obtain    dw  δ|·| 2 2    ds  ∞,δ ≤ w∞,δ |w| e ∗ ρ∞ ≤ w∞,δ wq,δ  for any 1 ≤ q < ∞. Similarly, using Lemma 3.1, we obtain    dw  δ|·| 2 2    ds  1,δ ≤ w∞,δ |w| e ∗ ρ2
w∞,δ w2−,δ .  δ (or w(1) ∈ B δ ) then The last two inequalities imply that if w(0) ∈ Bε,C ε,C 3

1

w(s) − w(0)∞,δ
ε 2 − ,

w(s) − w(0)1,δ
ε 2 − .

This completes the proof for F1 . In the proof for F2 , we omit some of the details, in particular the multiplication with eδ|n| argument above, since it works exactly the same way. To estimate the  p norm of the right-hand side of dw(n) ∂ F2 = , ds ∂ w(n) ¯ we use duality:    ∂ F2     ∂ w(n)  ¯

p

=

sup

h

p

 =1

 
 ∂ F2   h(n) .  ∂ w(n) ¯

(20)

Quasi-Linear Dynamics in NLS

663

Note that the right-hand side of (20) can be estimated by the sum of six terms of the form


F˜2 (w1 , . . . , w6 ) :=

m 4 ,m 5  =m 1 , m 2 ,m 3  =m 6

w1 (m 1 ) · · · w6 (m 6 ) , |q(m)||m 2 − m 6 ||m 6 − m 3 |

(21)

l(m)=0, q(m)=0

where in the j th term w j = |h| and the others are |v|. The required estimates for these terms follow by applying Lemma 3.2 below with arbitrarily small η and with i = j if p  = 1 and with k = j if p  = ∞.   Lemma 3.2. For any η > 0 and for any distinct i, k ∈ {1, 2, 3, 4, 5, 6}, there is a permutation (i 1 , i 2 , i 3 , i 4 ) of the remaining indices such that F˜2 (w1 , . . . , w6 )
wi 1 wk ∞ wi1 1

4 

1

η

1+η wil 1+η ∞ wil  1 . 

l=2

Proof. Fix η > 0, i, and k. By Holder’s inequality we have ⎤

⎡ ⎢ F˜2 ≤ ⎢ ⎣


m 4 ,m 5  =m 1 , m 2 ,m 3  =m 6

⎡ ×⎣

⎥ w1 (m 1 ) · · · w6 (m 6 ) ⎥ |q(m)|1+η |m 2 − m 6 |1+η |m 6 − m 3 |1+η ⎦

l(m)=0, q(m)=0




⎤

1 1+η

(22)

η 1+η

w1 (m 1 ) · · · w6 (m 6 )⎦

.

l(m)=0

The second line is bounded by η

wk 1+η ∞

6  l=1,l=k

η

wl 1+η 1 .

The required estimate for the sum in the first line follows from the following claim: For any permutation ( j1 , j2 , j3 ) of {1, 4, 5}, and for any permutation (n 1 , n 2 , n 3 ) of {2, 3, 6}, we have
m 4 ,m 5  =m 1 , m 2 ,m 3  =m 6

w1 (m 1 ) · · · w6 (m 6 ) |q(m)|1+η |m 2 − m 6 |1+η |m 6 − m 3 |1+η

(23)

l(m)=0, q(m)=0


w j1 ∞ w j2 ∞ w j3 1 wn 1 ∞ wn 2 ∞ wn 3 1 . To prove this inequality, replace m j1 in the sum with a linear combination of other indices using the identity l(m) = 0. We claim that |q(m)|−1−η 1m

j2


1,

664

M. B. Erdo˘gan, V. Zharnitsky

where the implicit constant is independent of the remaining indices. Indeed, it suffices to consider the cases j1 = 1, j2 = 4 and j1 = 4, j2 = 5 since m 4 and m 5 enter symmetrically. In the former case q(m) = (m 4 + m 5 + m 6 − m 2 − m 3 )2 + m 22 + m 23 − m 24 − m 25 − m 26 = C1 m 4 + C2 , where the integers C1 , C2 depend on m 2 , m 3 , m 5 , m 6 . Moreover, C1 = 0 since m 1 = m 4 . Therefore, sup

m 2 ,m 3 ,m 5 ,m 6

|q(m)|−1−η 1m
1. 4

In the latter case q(m) = C1 + C2 m 5 − 2m 25 , where the integers C1 , C2 depend on m 1 , m 2 , m 3 , m 6 . Since for any integers n, C1 , C2 , the equation n = C1 + C2 m 5 − 2m 25 has at most two solutions, we have sup

m 1 ,m 2 ,m 3 ,m 6

|q(m)|−1−η 1m
1. 5

Using this claim, we obtain


w2 (m 2 )w3 (m 3 )w6 (m 6 ) |m 2 − m 6 |1+η |m 6 − m 3 |1+η
wn 3 (m n 3 ) ≤ w j1 ∞ w j2 ∞ w j3 1 wn 1 ∞ wn 2 ∞ |m 2 − m 6 |1+η |m 6 − m 3 |1+η
w j1 ∞ w j2 ∞ w j3 1 wn 1 ∞ wn 2 ∞ wn 3 1 .

(23)
w j1 ∞ w j2 ∞ w j3 1

  3.3. Cancellation property of H4r2 . We claim that {H4r2 , F j } = 0, j = 1, 2. Indeed, by (17) and (18), both F1 and F2 have the phase invariant property F j (v) = F j (veiφ ), but the evolution induced by H4r2 is just uniform phase rotation, v(n, t) = ei2Pt v(n, 0). Thus, {H4r2 , F j } :=

d ) = 0, F j (X t=0 H4r2 dt

j = 1, 2.

Quasi-Linear Dynamics in NLS

665

4. Proof of Proposition 2.2 δ , we should prove that the  p,δ norm of each of the summands Assuming that v ∈ Bε,C 3/2−1/ p− in (16) is
ε for p = 1 and p = ∞. To simplify the exposition, we will do this only in the case δ = 0. The proof for the case δ > 0 is similar by using the simple multiplication by eδ|·| argument we used in the proof of Proposition 2.1. Note that it suffices to consider the ∂v(k) derivatives of the following terms ¯

H4r1 , {H4r1 , F1 }, {H4r1 , F2 }, {H4nr , F1 }r , g bF2 g aF1 (H4 ), a + b ≥ 2, and the terms involving integrals. We define


f 1 (v1 , v2 , v3 )(k) :=

m 1 ,m 2 =k

v1 (m 1 )v2 (m 2 )v3 (m 1 + m 2 − k) (m 1 − k)(m 2 − k)

¯ = ∂v(k) so that f 1 (v, v, v)(k) ¯ F1 . Similarly we define f 2 (v1 , v2 , v3 , v4 , v5 )(k) so that f 2 (v, v, v, v, ¯ v)(k) ¯ = ∂v(k) ¯ F2 . The following lemma will be used repeatedly: Lemma 4.1. I) For any q ∈ [1, ∞] and any permutation (i 1 , i 2 , i 3 ) of (1, 2, 3), we have  f 1 (v1 , v2 , v3 )q
vi1 q vi2 ∞− vi3 ∞− . II) For any q ∈ [1, ∞], for any η > 0, and for any i ∈ {1, 2, 3, 4, 5} there is a permutation (i 1 , i 2 , i 3 , i 4 ) of the set {1, 2, 3, 4, 5}\{i} such that  f 2 (v1 , v2 , v3 , v4 , v5 )q
vi q vi1 1

4 

1

η

1+η vil 1+η ∞ vil  1 . 

l=2

Proof. Part I can easily be verified following the proof of Proposition 2.1 with δ = 0. Part II follows from Lemma 3.2 and interpolation.   r1 4.1. Estimate of ∂v(k) ¯ H4 . Recall that

H4r1 = i




|v(m)|4 ,

m

and hence ∂ H4r1 = 2i|v(k)|2 v(k). ∂ v(k) ¯ We estimate the contribution of this term as    ∂ H r1   4   
v 3 ∞
ε3/2 ,  ∂ v(·) ¯ ∞ 

and

   ∂ H r1   4     ∂ v(·) ¯ 

3

1


v 3 1 = v33
ε 2 −1 .

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M. B. Erdo˘gan, V. Zharnitsky

r1 r1 4.2. Estimates for ∂v(k) ¯ {H4 , F1 } and ∂v(k) ¯ {H4 , F2 }. Let

H˜ 4r1 (v1 , v2 , v3 , v4 ) :=




v1 (n)v2 (n)v3 (n)v4 (n).

n

We use duality as in (20). Note that the following two terms:

 k

r1 |∂v(k) ¯ {H4 , F1 }||h(k)| is bounded by the sum of

H˜ 4r1 (| f 1 (|v|, |v|, |v|)|, |h|, |v|, |v|),

H˜ 4r1 (| f 1 (|h|, |v|, |v|)|, |v|, |v|, |v|),

and similar terms obtained by permuting the arguments. The following estimates (with p = 1 and p = ∞), which follow from the definition of H˜ 4r1 and Lemma 4.1, completes r1 the analysis of ∂v(k) ¯ {H4 , F1 }: H˜ 4r1 (| f 1 (|v|, |v|, |v|)|, |h|, |v|, |v|)
h p v p v∞  f 1 (|v|, |v|, |v|)∞
h p v p v2∞ v2∞−
ε 2

5

− 1p −

,

5

− 1p −

.

H˜ 4r1 (| f 1 (|h|, |v|, |v|)|, |v|, |v|, |v|)
 f 1 (|h|, |v|, |v|) p v p v2∞
h p v p v2∞ v2∞−
ε 2

r1 We estimate ∂v(k) ¯ {H4 , F2 } similarly. The estimates below imply the required bound

H˜ 4r1 (| f 2 (|v|, . . . , |v|)|, |h|, |v|, |v|)
h p v p v∞  f 2 (|v|, . . . , |v|)∞ 3

3η

1+η
h p v p v2∞ v1 v1+η ∞ v 1  5


ε2

− 1p −

, r1 ˜ H4 (| f 2 (|h|, |v|, |v|, |v|, |v|)|, |v|, |v|, |v|)
 f 2 (|h|, . . . , |v|) p v p v2∞ 3

3η

1+η
h p v1 v p v2∞ v1+η ∞ v 1  5


ε2

− 1p −

.

In both estimates, the last inequality is obtained by taking η sufficiently small. nr r 4.3. Estimate of ∂v(k) ¯ {H4 , F1 } . Based on the calculations in Sect. 3.1, we have




{H4nr , F1 }r = 2i

m 4 ,m 5  =m 1 , m 2 ,m 3  =m 6

v(m 1 )v(m 2 )v(m 3 )v(m ¯ 4 )v(m ¯ 5 )v(m ¯ 6) − c.c. (m 2 − m 6 )(m 3 − m 6 )

l(m)=0, q(m)=0

Using duality as above we need to estimate 6 terms of the form
m 4 ,m 5  =m 1 , m 2 ,m 3  =m 6

l(m)=0, q(m)=0

v1 (m 1 )v2 (m 2 )v3 (m 3 )v4 (m 4 )v5 (m 5 )v6 (m 6 ) , |m 2 − m 6 ||m 3 − m 6 |

(24)

Quasi-Linear Dynamics in NLS

667

where v j = |h| and others are |v|. The required estimates follow from the following claim: For any η > 0, for any permutation ( j1 , j2 , j3 ) of {1, 4, 5}, and for any permutation (n 1 , n 2 , n 3 ) of {2, 3, 6}, we have   1   η (24)
v j1 ∞ v j3 1 vn 1 ∞ vn 3 1 v j2 ∞ vn 2 ∞ 1+η v j2 1 vn 2 1 1+η . As in the proof of Lemma 3.2, see (22), the claim follows from an estimate for the following sum:
m 4 ,m 5  =m 1 , m 2 ,m 3  =m 6

v1 (m 1 )2 v(m 2 )v3 (m 3 )v4 (m 4 )v5 (m 5 )v6 (m 6 ) . |m 2 − m 6 |1+η |m 3 − m 6 |1+η

(25)

l(m)=0, q(m)=0

First we replace j1 in the equation q(m) = 0 using l(m) = 0. By symmetry it suffices to consider two cases j1 = 1, j1 = 4. In the former case we have 0 = (m 2 + m 3 − j2 − j3 − m 6 )2 + m 22 + m 23 − j22 − j32 − m 26 = −2 j2 (m 2 + m 3 − j3 − m 6 ) + (m 2 + m 3 − j3 − m 6 )2 + m 22 + m 23 − j32 − m 26 . Moreover, m 2 + m 3 − j3 − m 6 = 0 since m 1 = m 4 , m 5 . Therefore, both j1 and j2 are determined by the remaining indices. This implies that (25)
v j1 ∞ v j2 ∞ v j3 1


m 2 ,m 3 =m 6

v(m 2 )v3 (m 3 )v6 (m 6 ) |m 2 − m 6 |1+η |m 3 − m 6 |1+η


v j1 ∞ v j2 ∞ v j3 1 vn 1 ∞ vn 2 ∞ vn 3 1 , which leads to the desired estimate as in the previous sections. The case j1 = 4 is similar, the only difference is that j2 is determined as roots of a quadratic polynomial instead of a linear one. 4.4. Estimate of ∂v(k) g bF2 g aF1 (H4 ).. The bounds for ∂v(k) g bF2 g aF1 (H4 ) will be obtained ¯ ¯ inductively. Although we only need to consider the cases when a + b ≥ 2, we start with the case a = 1, b = 0 for clarity. Note that g 1F1 (H4 ) is a sum of terms of the form
H4 (v1 , v2 , v3 , v4 ) = v1 (n 1 )v2 (n 2 )v3 (n 3 )v4 (n 4 ), n 1 −n 2 +n 3 −n 4 =0

    1 ¯ To estimate ∂v(k) where one of vi ’s is f 1 or f¯1 and the others are v or v. ¯ g F1 (H4 ) p ,  we use duality as before:   
 ∂  ∂   1 1    (26) g F1 (H4 ) ≤ sup g F1 (H4 ) |h(k)|.  ∂ v(k)  ¯ ∂ v(k) ¯ h p =1 p k

Note that the sum in the right-hand side of (26) is bounded by the sum of the following two terms: H4 (| f 1 (|v|, |v|, |v|)|, |h|, |v|, |v|),

H4 (| f 1 (|h|, |v|, |v|)|, |v|, |v|, |v|),

and similar terms obtained by permuting the arguments. The following lemma will be used to estimate these terms and the ones appearing in the higher order commutators.

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M. B. Erdo˘gan, V. Zharnitsky

Lemma 4.2. For any q ∈ [1, ∞] and any permutation (i 1 , i 2 , i 3 , i 4 ) of (1, 2, 3, 4), we have |H4 (v1 , v2 , v3 , v4 )| ≤ vi1 q vi2 q  vi3 1 vi4 1 . Proof. Note that for any permutation we can write
H4 (v1 , v2 , v3 , v4 ) = vi1 ( j) vi2 ∗ vi3 ∗ vi4 ( j). j

The statement follows from Hölder’s and Young’s inequalities.   Using Lemma 4.2 and Lemma 4.1, we obtain H4 (| f 1 (|v|, |v|, |v|)|, |h|, |v|, |v|)
h p v p v1  f 1 (|v|, |v|, |v|)1
h p v p v1 v1 v2∞− 1


h p ε 2

− 1p −

.

Similarly, we have H4 (| f 1 (|h|, |v|, |v|)|, |v|, |v|, |v|)
 f 1 (|h|, |v|, |v|) p v p v21
h p v2∞− v p v21 1


h p ε 2

− 1p −

.

Similar bounds follow for the terms obtained by permuting the arguments. Therefore we have    ∂  1 1 1  2 − p −. g F1 (H4 )  ∂ v(k)  p
ε ¯ 

Note that this gives an error of order 1 when p = 2. This explains why we consider higher order commutators and a second normal form transform (see footnote 2). This proof motivates the following generalization: Lemma 4.3. Consider H4 (|v|, |v|, |v|, |v|). Repeatedly (a times) replace one of the v’s with f 1 (|v|, |v|, |v|). Then repeatedly (b times) replace one of the v’s with f 2 (|v|, |v|, |v|, |v|, |v|). Finally, replace one of the v’s with h. We denote any such function by H4,a,b ( f 1 , f 2 , h, v). Then, for p = 1 and p = ∞, we have |H4,a,b ( f 1 , f 2 , h, v)|
h p ε

a+b− 21 − 1p −

.

Proof. First by using Lemma 4.1 repeatedly (with sufficiently small η, we see that any composition of f 1 ’s and f 2 ’s satisfy  b 3− 1+ , (27)  · q
vq v2a ∞− v1 v∞ where a is the number of f 1 ’s and b is the number of f 2 ’s appearing in the composition. Now, note that H4 has four arguments. Let a j (resp. b j ) be the number of f 1 ’s (resp. f 2 ’s) appearing in the j th argument. Only one of the arguments contains h, say the first one. Using Lemma 4.2, we have |H4 (v1 , v2 , v3 , v4 )|
v1  p v2  p v3 1 v4 1 .

Quasi-Linear Dynamics in NLS

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Using (27), we have 2(a +a3 +a4 )

v2  p v3 1 v4 1
v p v1 v1 v∞−2

 v1+ v3− ∞ 1

b2 +b3 +b4

.

Next, note that v1 is either |h| (in which case we stop) or f 1 (v1,1 , v1,2 , v1,3 ) or f 2 (v1,1 , . . . , v1,5 ). In the latter cases, without loss of generality, v1,1 contains |h|. We estimate, using (27) and a simple induction,  b1 1 v1+ v1  p
h p v2a v3− . ∞ ∞− 1 Combining these estimates we obtain  3− 1+ |H4,a,b ( f 1 , f 2 , h, v)|
h p v p v1 v1 v2a ∞− v1 v∞
h p ε

a+b− 21 − 1p −

b

.

  b a Using duality as above we see that the right-hand side of (26) for ∂v(k) ¯ g F2 g F1 (H4 ) can be bounded by a finite sum of functions H4,a,b ( f 1 , f 2 , h, v). Therefore, Lemma 4.3 implies that    ∂g b g a (H4 )  3 1  F2 F1  a+b− 21 − 1p − − −
ε 2 p , if a + b ≥ 2.  
ε   p ∂ v(k) ¯ 

4.5. Remainder Estimates. It remains to estimate the terms involving integrals. Note that it suffices to prove the inequalities    ∂  3 1 − − 3 s   sup  g F1 (H ) ◦ X F1 
ε 2 p , ∂ v(k) ¯ s∈[0,1] p    ∂  3 1 − − 3 s  sup  {g F1 (H ) ◦ X F1 , F2 }
ε2 p ,  ¯ p s∈[0,1] ∂ v(k)     ∂  3 1 − − 2 1 s sup  g F2 (H ◦ X F1 ) ◦ X F2 
ε2 p  ∂ v(k)  ¯ p s∈[0,1]  1

−1

for p = 1, ∞ assuming that v p
ε 2 p , p ∈ [1, ∞]. Since we have to estimate the composite function derivative, we first study the bounds on the derivatives of X sF j (v), j = 1, 2, s ∈ [0, 1], more precisely, let w(m) = [X sF j (v)](m), which is the solution at t = s of the system ∂ Fj dw(m) = , dt ∂ w(m) ¯

w|t=0 = v.

Differentiating this equation with  d respect to initial condition w(n)|t=0 = v(n) and using the notation Dn , we see that  dt Dn w(m) is bounded by a sum of terms of the form f 1 (v1 , v2 , v3 )(m), for j = 1, f 2 (v1 , v2 , v3 , v4 , v5 )(m), for j = 2,

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M. B. Erdo˘gan, V. Zharnitsky

where one of the vk ’s is |Dn w| and the others are |w|. Without loss of generality we can d Dn w. ¯ Note that at s = 0, we assume that v1 = |Dn w|. We have a similar formula for ds have Dn w(m)∞ 1 = Dn w(m)∞ 1 = 1. m n n m 1 We will prove that both of these norms remain bounded for s ∈ [0, 1]. Taking the ∞ m n norm of f j we obtain

   d  Dn w(m)   dt

1 ∞ m n

 
 f j (|Dn w|, . . . , |w|)(m)∞ 1 m n

    ≤  f j (Dn w1n , . . . , |w|)(m)

∞ m

1−
Dn w(m)∞ . 1ε m n

In the last line, we used Lemma 4.1 (for sufficiently small η). This implies that (with w(m) = [X sF j (v)](m), j = 1, 2) sup Dn w(m)∞ 1
1. m n

(28)

sup Dn w(m)∞ 1
1. n m

(29)

0≤s≤1

Similarly, we obtain 0≤s≤1

We also need the following estimates for the higher order derivatives of w = X sF1 (v) with respect to the initial conditions: 1

D j Dn w(k)∞ 1
ε 2 − , j,n k

D j Dm Dn w(k)∞

1 j,m,n k


1,

1

D j Dn w(k)∞ 1
ε 2 − , k,n j

D j Dm Dn w(k)∞


1,

1 k,m,n  j

which can be obtained using Lemma 4.1 as in the proof of (28), (29). Remark 4.1. For δ > 0, a similar argument implies      δ|n−m|   δ|n−m|  Dn w(m) ∞ 1
1, Dn w(m) e e

1 ∞ n m

m n

and for higher order derivatives of w = X sF1 (v) we have     δ| j1 +···+ jk −m| D j1 . . . D jk w(m) ∞
1, e  j ,..., j 1m 1 k     δ| j1 +···+ jk −m| D j1 . . . D jk w(m) ∞
1. e 1 m, j

 2 ,..., jk j1

The rest of the argument follows as in other sections.


1,

Quasi-Linear Dynamics in NLS

671

s 3 nr 4.5.1. Estimation of ∂v(k) ¯ g F1 (H ) ◦ X F1 (v). Since g F1 ( 2 ) = −H4 , it suffices to estimate    ∂  a s  sup  g F1 (H4 ) ◦ X F1 (v)  p , a = 2, 3. ¯ s∈[0,1] ∂ v(k) 

When a = 2, we estimate this expression rather than the one containing H4nr (as we should have) because it simplifies the notation and still implies the estimate for the required expression. Note that     
∂g a (H4 ) ∂ w(i)  ∂  ¯    F1 a s   g F1 (H4 ) ◦ X F1 (v) ≤   + c.c.1  ∂ v(k) k  ¯ ∂ w(i) ¯ ∂ v(k) ¯ 1 1 k

k

i

  a  ∂g F1 (H4 )   Dk w(i)  ≤ ¯ i∞ 1k + c.c.1k 1 ∂ w(i) ¯  i


ε

1 2−

,

for a ≥ 2 by (28) and the estimates we obtained in Subsect. 4.4. Similarly, using (29), we obtain    ∂  3 a s  g F1 (H4 ) ◦ X F1 (v)
ε 2 − , for a ≥ 2.  ∂ v(k)  ¯ ∞ k

s s 3 2 1 The estimates for ∂v(k) ¯ {g F1 (H ) ◦ X F1 , F2 } and ∂v(k) ¯ g F2 (H ◦ X F1 ) ◦ X F2 are similar. The only difference is that we also require the higher order derivative estimates of w = X sF1 (v) listed above. We omit the details.

Appendix A. Nonlinear Fiber Optics Application One of the most important applications of NLS concerns light-wave communication systems, where optical pulses in a retarded time frame evolve according to the one dimensional NLS i A z + Sd(z)Aτ τ + g(z)|A|2 A = 0.

(30)

Here z is the rescaled distance, τ is the rescaled retarded time, A is the amplitude of the optical wave envelope, d(z) is the group velocity dispersion, which is usually piecewise constant, and S is the dispersion strength parameter. Finally, g(z) > 0 is the nonlinear coefficient which accounts for the losses and amplifications. For the derivation of NLS from Maxwell’s equations, one can consult many references, e.g. [9]. It is a standard assumption that d(z) and g(z) are periodic. In general, in light-wave communication, the information is transmitted with localized pulses (with Gaussian or exponential tails) in allocated time slots. The presence of pulse corresponds to “1” and the absence of pulse corresponds to “0” in binary format. Naturally, it is preferable that the incoming waveform would appear undistorted at the end of the transmission line. It can be achieved by optimizing an individual pulse, so it would propagate without distortion, and sending such pulses together, keeping them sufficiently far apart (i.e. taking time slots sufficiently large), so they would not interact. Such regime is usually called “soliton regime” in the optical communication literature,

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where the word “soliton” does not usually mean that the equation is integrable. The pulses could be, for example, dispersion managed solitons, which are approximately periodic localized solutions of the above equation. In other words, the main feature of the soliton regime is that the pulses do not interact (or rather pulse to pulse interaction is weak compared to the pulse self-interaction) during the propagation through the transmission line. An alternative regime (often called the quasi-linear regime) has been found when the pulses strongly overlap during the transmission, see e.g. the survey paper [7]. Surprisingly, it was observed that up to a linear transformation of the transmitted waveform, the pulses appeared undistorted. Note that even though the pulses spread over many time slots, the average optical energy (L 2 norm square) per bit does not change and therefore nonlinear effects remain strong. It is usually implicitly assumed in the engineering literature that “nonlinearity gets averaged out” due to the high frequency of the initial data. In this article, we rigorously explain the quasi-linear phenomenon for a model problem when d(z) and g(z) are constant and all bits are occupied by 1 s, in the limit of vanishing pulse width. This case (of all identical 1’s) leads to the formulation with periodic boundary conditions. Although, this is a special case, we hope that our proof can be extended to the more general case: pseudo-random sequence of 1 s and 0 s. Note that constant d(z) and g(z) assumption is not restrictive since if the evolution is quasi-linear on each interval where d(z), g(z) are constant, then the evolution is quasi-linear on their union. There has been previous work on the quasi-linear regime. In [15], the limit of the short pulse width for dispersion managed NLS on the real line is considered. An effective evolution equation was derived which turned out to be integrable and weakly nonlinear. The equation was later improved in [1]. On the real line the energy disperses to infinity and therefore nonlinearity becomes small. This leaves an open question: what will happen if the energy does not disperse to infinity or in other words, there is an infinite bit stream. The problem considered in this paper models this situation: nonlinearity remains strong which is due to the periodic boundary conditions. Finally, we note that on R, the dispersion strength S and pulse width ε can be combined into a single effective parameter S/ε2 by scaling τ . This implies that the limits S → ∞ and ε → 0 are equivalent. This is not the case in our model since the characteristic τ -scale, bit size, is already present. Therefore, the two parameter problem in S, ε should be considered. However, the limit S → ∞ is insufficient to achieve quasi-linear evolution and must be supplemented with ε → 0. On the other hand, the limit ε → 0 does produce quasi-linear evolution with S being fixed but arbitrary. This motivated us to consider only this case. We put S = 1 in order not to obscure the exposition. Acknowledgement. The authors would like to thank Eugene Wayne for many helpful discussions.

References 1. Ablowitz, M., Hirooka, T., Biondini, G.: Quasi-linear optical pulses in strongly dispersion-managed transmission systems. Optics Lett. 26, 459–462 (2000) 2. Bergano, N.S. et al.: Dig. Optical Fiber Communications Conf., 1998, Postdeadline Paper 12, Washington, DC: Opt. Soc of Amer., 1998 3. Biondini, G., Kodama, Y.: On the Whitham equations for the defocusing nonlinear Schrödinger equation with step initial data. J. Nonlinear Sci. 16(5), 435–481 (2006)

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4. Bourgain, J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part I: Schrödinger equations. GAFA 3(2), 107–156 (1993) 5. Bourgain, J.: A remark on normal forms and the “I -method” for periodic NLS. J. Anal. Math. 94, 125–157 (2004) 6. Deift, P., Zhou, X.: Perturbation theory for infinite-dimensional integrable systems on the line. A Case Study. Acta Math. 188(2), 1871–2509 (2002) 7. Essiambre, R.-J., Raybon, G., Mikkelsen, B.: Pseudo-linear transmission of high-speed TDM signals: 40 and 160 Gb/s. Optical Fiber Communications IV, I. Kaminow, T. Li, ed. San Diego, CA: Academic, 2002, pp. 232–304 8. Gershgorin, B., Lvov, Y., Cai, D.: Renormalized waves and discrete breathers in β-Fermi-Pasta-Ulam chains. Phys. Rev. Lett. 95, 264302 (2005) 9. Hasegawa, A., Kodama, Y.: Solitons in Optical Communication. New York: Oxford University Press, 1995 10. Kamvissis, S., McLaughlin, K., Miller, P.: Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation. Annals of Mathematical Studies 154, Princeton, NJ: Princeton University Press, 2003 11. Kuksin, S.: Analysis of Hamiltonian PDEs. New York: Oxford University Press, 2000 12. Kuksin, S., Poschel, J.: Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation. Ann. Math. 142, 149–179 (1995) 13. Mamyshev, P.V., Mamysheva, N.A.: Pulse-overlapped dispersion-managed data transmission and inrachannel four-wave mixing. Opt. Lett. 24, 1454–1456 (1999) 14. Mikkelsen, B. et al.: 320-Gb/s Single-Channel pseudolinear transmission over 200 km of nonzero-dispersion fiber. IEEE Photon. Technol. Lett. 12, 1400–1402 (2000) 15. Manakov, S.V., Zakharov, V.E.: On propagation of short pulses in strong dispersion managed optical lines. Sov. Phys. JETP Lett. 70, 578–582 (1999) 16. Tovbis, A., Venakides, S., Zhou, X.: On the long-time limit of semiclassical (zero dispersion limit) solutions of the focusing nonlinear Schrödinger equation: pure radiation case. Comm. Pure Appl. Math. 59, 1379–1432 (2006) 17. Zakharov, V.E., Shabat, A.B.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP Lett. 37, 823–828 (1973) Communicated by P. Constantin

Commun. Math. Phys. 281, 675–709 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0511-8

Communications in

Mathematical Physics

Height Fluctuations in the Honeycomb Dimer Model Richard Kenyon Department of Mathematics, Brown University, Providence, RI, USA. E-mail: [email protected] Received: 29 May 2007 / Accepted: 18 January 2008 Published online: 29 May 2008 – © Springer-Verlag 2008

Abstract: We study a model of random surfaces arising in the dimer model on the honeycomb lattice. For a fixed “wire frame” boundary condition, as the lattice spacing
→ 0, Cohn, Kenyon and Propp [3] showed the almost sure convergence of a random surface to a non-random limit shape 0 . In [12], Okounkov and the author showed how to parametrize the limit shapes in terms of analytic functions, in particular constructing a natural conformal structure on them. We show here that when 0 has no facets, for a family of boundary conditions approximating the wire frame, the large-scale surface fluctuations (height fluctuations) about 0 converge as
→ 0 to a Gaussian free field for the above conformal structure. We also show that the local statistics of the fluctuations near a given point x are, as conjectured in [3], given by the unique ergodic Gibbs measure (on plane configurations) whose slope is the slope of the tangent plane of 0 at x. Contents 1.

2.

Introduction . . . . . . . . 1.1 Dimers and surfaces . . 1.2 Results . . . . . . . . . 1.2.1 Limit shape. . . . . 1.2.2 Local statistics. . . 1.2.3 Fluctuations. . . . 1.3 The Gaussian free field 1.4 Beltrami coefficient . . 1.5 Examples . . . . . . . 1.6 Proof outline . . . . . . Definitions . . . . . . . . . 2.1 Graphs . . . . . . . . . 2.1.1 Dual graph. . . . .

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676 676 677 677 678 678 680 680 681 682 682 682 682

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2.1.2 Forms. . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Heights and asymptotics . . . . . . . . . . . . . . . . 2.3 Measures and gauge equivalence . . . . . . . . . . . . 2.4 Kasteleyn matrices . . . . . . . . . . . . . . . . . . . 2.5 Measures in infinite volume . . . . . . . . . . . . . . . 2.6 T -graphs . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Definition. . . . . . . . . . . . . . . . . . . . . . 2.6.2 Associated dimer graph and Kasteleyn matrix. . . . 2.6.3 Harmonic functions and discrete analytic functions. 2.6.4 Green’s function and K G−1 . . . . . . . . . . . . . . D Constant-Slope Case . . . . . . . . . . . . . . . . . . . . . 3.1 T -graph construction . . . . . . . . . . . . . . . . . . 3.2 Boundary behavior . . . . . . . . . . . . . . . . . . . 3.2.1 Flows and dimer configurations . . . . . . . . . . 3.2.2 Canonical flow. . . . . . . . . . . . . . . . . . . . 3.2.3 Boundary height. . . . . . . . . . . . . . . . . . . 3.3 Continuous and discrete harmonic functions . . . . . . 3.3.1 Discrete and continuous Green’s functions. . . . . 3.3.2 Smooth functions. . . . . . . . . . . . . . . . . . 3.4 K G−1 in constant-slope case . . . . . . . . . . . . . . . D 3.4.1 Values near the diagonal. . . . . . . . . . . . . . . General Boundary Conditions . . . . . . . . . . . . . . . . 4.1 The complex height function . . . . . . . . . . . . . . 4.2 Gauge transformation . . . . . . . . . . . . . . . . . . 4.3 Embedding . . . . . . . . . . . . . . . . . . . . . . . 4.4 Boundary . . . . . . . . . . . . . . . . . . . . . . . . Continuity of K −1 . . . . . . . . . . . . . . . . . . . . . . Asymptotic Coupling Function . . . . . . . . . . . . . . . Free Field Moments . . . . . . . . . . . . . . . . . . . . . Boxed Plane Partition Example . . . . . . . . . . . . . . .

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683 683 684 684 685 686 686 688 688 689 690 691 693 693 693 694 695 695 696 698 699 699 699 700 701 703 703 704 704 707

1. Introduction 1.1. Dimers and surfaces. A dimer covering, or perfect matching, of a finite graph is a set of edges covering all the vertices exactly once. The dimer model is the study of random dimer coverings of a graph. Here we shall for the most part deal with the uniform measure on dimer coverings. A good reference is [11]. In this paper we study the dimer model on the honeycomb lattice (the periodic planar graph whose faces are regular hexagons), or rather, on large pieces of it. This model, and more generally dimer models on other periodic bipartite planar graphs, are statistical mechanical models for discrete random interfaces. Part of their interest lies in the conformal invariance properties of their scaling limits [8,9]. Dimer coverings of the honeycomb graph are dual to tilings with 60◦ rhombi, also known as lozenges, see Fig. 1. Lozenge tilings can in turn be viewed as orthogonal projections onto the plane P111 = {x + y + z = 0} of stepped surfaces which are polygonal surfaces in R3 whose faces are squares in the 2-skeleton of Z3 (the stepped surfaces are monotone in the sense that the projection is injective), see Figs. 1,3. Each stepped surface is the graph of a function, the normalized height function, on the √ underlying tiling, which is linear on each tile. This function is defined simply as 3 times the

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Fig. 1. Honeycomb dimers (solid) and the corresponding “lozenge” tiling (green)

√ distance from the surface to the plane P111 . (The scaling factor 3 is just to make the function integer-valued on Z3 . The value of the height function at a vertex is the sum of its coordinates.) 1.2. Results. We are interested in studying the scaling limit of the honeycomb dimer model, that is, the limiting behavior of a uniform random dimer covering of a fixed plane region U when the lattice spacing
goes to zero. Equivalently, we take stepped surfaces in
Z3 and let
→ 0. As boundary conditions we are interested in stepped surfaces spanning a “wire frame” which is a simple closed polygonal path γ
in
Z3 . We take γ
converging as
→ 0 to a smooth path γ which projects to ∂U . 1.2.1. Limit shape. Let U be a domain in P111 , and γ be a smooth closed curve in R3 , projecting orthogonally to ∂U . For each
> 0 let γ
be a nearest-neighbor path in
Z3 approximating γ (in the Hausdorff metric) and which can be spanned by a monotone stepped surface 
, monotone in the sense that it projects injectively to P111 , or in other words it is the graph of a function on P111 . See for example Fig. 3 (although there the boundary is only piecewise smooth). The existence of such an approximating sequence imposes constraints on γ , see [3,5], as follows: the curve γ can be spanned by a surface , which is the graph of a continuous function on U , and such that the normal ν to the surface points into the positive orthant R3≥0 . Conversely, any such curve γ can be approximated by γ
as above, see [3]. The condition of positivity of the normal to  can be stated in terms of the gradient of the function h whose graph is : this gradient must lie in a certain triangle. The formulation in terms of the normal is more symmetric, however. For a given γ
there are, typically, many spanning surfaces 
and we study the limiting properties of the uniform measure on the set of 
as
→ 0. For a surface 
spanning γ
, let h
: P111 → R be the normalized height function, defined on the region enclosed by U
:= π111 (γ
), whose graph is 
. Under the above hypotheses Cohn, Kenyon and Propp proved the existence of a limit shape: Theorem 1.1 (Cohn, Kenyon, Propp [3]). The distribution of h
converges as
→ 0 a.s. to a nonrandom function h¯ : U → R. The function h¯ is the unique function h which minimizes the “surface tension” functional

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σ (∇h) d x d y,

min h

U

where, in terms of the normal vector ( pa , pb , pc ) ∈ R3 to the graph of h scaled so that pa + pb + pc = 1, we have σ ( pa , pb , pc ) = − π1 (L(π pa ) + L(π pb ) + L(π pc )) and x L(x) = − 0 log(2 sin t) dt is the Lobachevsky function. Here the minimum is over Lipschitz functions whose graph has normal with nonnegative coordinates. Equivalently, these are functions whose gradient lies in a certain triangle. In the above formula the surface tension σ is the negative of the exponential growth rate of the number of discrete surfaces of average slope ∇h. The function h¯ is called the asymptotic height function. Its graph is a surface 0 spanning γ . 1.2.2. Local statistics. Suppose that the gradient of h¯ is not maximal at any point in U¯ , i.e. the normal to 0 has nonzero coordinates at every point of U¯ . In this paper we show that, if the precise local behavior of the approximating curves γ
is chosen in a particular way, then both the local statistics and the global height fluctuations of 
can be determined. Here is the result on the local statistics. Theorem 1.2. Suppose that the gradient of h¯ is not maximal at any point in U¯ , that is, the normal vector to the surface has nonzero coordinates at every point. Under appropriate hypotheses on the local structure of the approximating curves γ
, the local statistics of 
near a given point are given by the unique Gibbs measure1 of slope equal to the slope of h¯ at that point. For the precise statement see Theorem 6.1. In particular the hypotheses on γ
are explained in Sects. 2.6 and 3.2. Suppose that ( pa , pb , pc ) is a normal vector to the surface at a point. Recall that pa , pb , pc > 0. If we rescale so that pa + pb + pc = 1, then one consequence of Theorem 6.1 is that the quantities pa , pb , pc are the densities of the three orientations of lozenges near the corresponding point on the surface. 1.2.3. Fluctuations. The fluctuations are the image of the Gaussian free field under a certain diffeomorphism from the unit disk D to U . To describe the fluctuations, we first describe the relevant conformal structure on U . It is a function of the normal to the ¯ and is defined as follows. Let ( pa , pb , pc ) be the normal to h, ¯ scaled so graph of h, that pa + pb + pc = 1. Let θa = π pa , θb = π pb , θc = π pc . Let a, b, c be the edges of a Euclidean triangle with angles θa , θb , θc . Let z = −e−iθc and w = −eiθb so that a + bz + cw = 0. See Fig. 2; here the triangle on the left has edges a, bz, cw when these edges are oriented counterclockwise. We define = −cw/a. All of these quantities are functions on U , although a, b, c are only defined up to scale. Let x, ˆ yˆ , zˆ be the unit vectors in P111 in the directions of the projections of the standard basis vectors in R3 . 1 Sheffield [19] showed that there is a 2-parameter family of Gibbs measures on dimer covers of the honeycomb lattice, one for each possible average slope of the height function; see Sect. 2.5 below.

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679

Φ θa

θa

b

c

θc

θb a

θb

θc

0

1

Fig. 2. The triangle and a scaled copy with vertices 0, 1,

Theorem 1.3 ([12]). The function satisfies the complex Burgers equation xˆ + yˆ = 0,

(1)

where xˆ , yˆ are directional derivatives of in directions x, ˆ yˆ respectively. The function : U → C can be used to define a conformal structure on U , as follows. A function g : U → C is defined to be analytic in this conformal structure on U if it satisfies gxˆ + g yˆ = 0, where gxˆ , g yˆ are the directional derivatives of g in the directions x, ˆ yˆ respectively. By the Alhfors-Bers theorem there is a diffeomorphism f : U → D satisfying f xˆ + f yˆ = 0; the conformal structure on U is the pull-back of the standard conformal structure on D under f . The conformal structure on U can be described by a Beltrami coefficient ξ (see below) which in the current case is ξ = ( − eiπ/3 )/( − e−iπ/3 ). In the special cases that is constant (which correspond to the cases where γ is contained in a plane), this means that the conformal structure is just a linear image of the standard conformal structure. For example, note that in the standard conformal structure on P111 , a function g is analytic if gxˆ + eiπ/3 g yˆ = 0. So the case = eiπ/3 , which corresponds to the case a = b = c, gives the standard conformal structure (recall that the vectors xˆ and yˆ are 120◦ apart). Theorem 1.4. Suppose that the gradient of h¯ is not maximal at any point in U¯ . Under the same hypotheses on γ
as in Theorem 1.2, the fluctuations of the unnormalized height ¯ have a weak limit as
→ 0 which is the Gaussian free field in the function, 1
(h
− h), complex structure defined by , that is, the pull-back under the map f : U → D above, of the Gaussian free field on the unit disk D. For the definition of the Gaussian free field see below. As mentioned above, we require that the normal to the graph of h¯ be strictly inside the positive orthant, so that we have a positive lower bound on the values of pa , pb , pc , whereas the results of [3] and [12] do not require this restriction. Indeed, in many of the simplest cases the surface 0 will have facets, which are regions on which ( pa , pb , pc ) = (1, 0, 0), (0, 1, 0) or (0, 0, 1). Our results do not apply to these situations. This work builds on work of [9,13,12]. Previously Theorem 1.4 was proved for the dimer model on Z2 in a special case which in our context corresponds to the wire frame γ lying in the plane P111 , see [9]. In that case the conformal structure is the standard conformal structure on U . In the present case we still require special boundary conditions, which generalize the “Temperleyan” boundary conditions of [8,9]. It remains an open question whether the result holds for all boundary conditions. The fluctuations in the presence of facets are also unknown, and the current techniques do not seem to immediately extend to this more general setting.

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1.3. The Gaussian free field. The Gaussian free field X on D [18] is a random object in the space of distributions on D, defined on smooth test functions as follows. For any smooth test function ψ on D, D ψ(x)X (x)|d x|2 is a real Gaussian random variable of mean zero and variance given by

D D

ψ(x1 )ψ(x2 )G(x1 , x2 )|d x1 |2 |d x2 |2 ,

where the kernel G is the Dirichlet Green’s function on D:    x1 − x2  1  . log  G(x1 , x2 ) = − 2π 1 − x¯1 x2  A similar definition holds (for the standard conformal structure) on any bounded domain in C, only the expression for the Green’s function is different. An alternative description of the Gaussian free field is that it is the unique Gaussian process which satisfies E[X (x1 )X (x2 )] = G(x1 , x2 ). Higher moments of Gaussian processes can always be written in terms of the moments of order 2; for the Gaussian free field we have E[X (x1 ) . . . X (xn )] = 0 if n is odd, and E[X (x1 ) . . . X (x2k )] =



G(xσ (1) , xσ (2) ) . . . G(xσ (2k−1) , xσ (2k) ),

(2)

pairings

where the sum is over all (2k − 1)!! pairings of the indices. Any process whose moments satisfy (2) is the Gaussian free field [9].

1.4. Beltrami coefficient. A conformal structure on U can be defined as an equivalence class of diffeomorphisms φ : U → D, where mappings φ1 , φ2 are equivalent if the composition φ1 ◦ φ2−1 is a conformal self-map of D. The Beltrami differential ξ(z) ddzz¯ of φ is defined by the formula ξ(z)

φz¯ d z¯ d z¯ = . dz φz dz

The Beltrami differential is invariant under post-composition of φ with a conformal map, so it is a function only of the conformal structure (and in fact defines the conformal structure as well). It is not hard to show that |ξ(z)| < 1; note that ξ(z) = 0 if and only if the map is conformal. The Ahlfors-Bers uniformization theorem [1] says that any smooth function (even any measurable function) ξ(z) satisfying |ξ(z)| < 1 defines a conformal structure.

Height Fluctuations in the Honeycomb Dimer Model

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Fig. 3. Boxed plane partition

1.5. Examples. The simplest case is when the wire frame γ is contained in a plane {(x, y, z) ∈ R3 | pa x + pb y + pc z = const}. In this case the limit surface 0 is linear. The normal ( pa , pb , pc ) is constant, and the conformal structure is a linear image of the standard conformal structure. That is, the map f : U → D is a linear map L composed with a conformal map. For a more interesting case, consider the boxed plane partition (BPP) shown in Fig. 3, which is a random lozenge tiling of a regular hexagon. In [4] it was shown that, for a random tiling of the hexagon, the asymptotic height function h¯ is linear outside of the inscribed circle and analytic inside (with an explicit but somewhat complicated formula). Although our theorem does not apply to this case because of the facets outside the inscribed circle, if we choose boundary conditions inside the inscribed circle, and boundary values equal to the graph of the function there, our results apply.√ Suppose that the hexagon has sides of √ length 1, so that the inscribed circle has radius 3/2. Let U be a disk of radius r < 3/2 concentric with it. Suppose that the normalized height function on the boundary of U is chosen to agree with the asymptotic height function of the corresponding region in the BPP, so that the asymptotic height function of U equals the asymptotic height function of the BPP restricted to U . Then the fluctuations on U can be computed using Theorem 7.1. In fact in this setting, the conformal structure can be explicitly computed: take the standard conformal structure on a hemisphere in R3 , and project it orthogonally onto the plane containing its equator. Identifying the equator with the inscribed circle in the BPP gives the relevant conformal structure in U . Remarkably, in this example the Beltrami coefficient is rotationally invariant, even though h¯ itself is not. See Sect. 8. √ We conjecture that the fluctuations for the BPP are given by the limit r → 3/2 of this construction (it is known [4] that fluctuations in the “frozen” regions outside the circle are exponentially small in 1/
).

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Fig. 4. The honeycomb graph

1.6. Proof outline. The fundamental tool in the study of the dimer model is the Kasteleyn matrix (defined below). Minors of the inverse Kasteleyn matrix compute (multiple-) edge probabilities in the model. The main goal of the paper is to obtain an asymptotic expansion of the inverse Kasteleyn matrix. This is complicated by the fact that it grows exponentially in the distance between vertices (except in the special case when the boundary height function is horizontal). However by pre- and post-composition with an appropriate diagonal matrix, we can remove the exponential growth and relate K −1 to the standard Green’s function with Dirichlet boundary conditions on a related graph GT . Here is a sketch of the main ideas. 1. We construct a discrete version of the map f of Theorem 1.4. For each
we define a directed graph GT embedded in the upper half plane H, and a geometric map φ from U
to GT , such that the Laplacian on GT is related (via the construction of [14,15]) with the Kasteleyn matrix on U
. The existence of such a graph GT follows from [15]. This is done in Sect. 3.1 in the “constant slope” case and Sect. 4.3 in the general case. 2. Standard techniques for discrete harmonic functions yield an asymptotic expansion for the Green’s function on GT . This is done in Sect. 2.6.4. 3. The asymptotic expansion of the inverse Kasteleyn matrix on U
is obtained from the derivative of the Green’s function on GT , pulled back under the mapping φ. See Sects. 2.6.4 and 6. 4. Asymptotic expansions of the moments of the height fluctuations are computed via integrals of the asymptotic inverse Kasteleyn matrix. These moments are the moments of the Gaussian free field on H pulled back under φ. 2. Definitions 2.1. Graphs. Let π111 be the orthogonal projection of R3 onto P111 . Let x, ˆ yˆ , zˆ be the π111 -projections of the unit basis vectors. Define e1 = 13 (xˆ − zˆ ), e2 = 13 ( yˆ − x), ˆ 1 e3 = 3 (ˆz − yˆ ), so that xˆ = e1 − e2 , yˆ = e2 − e3 , zˆ = e3 − e1 . Let H be the honeycomb lattice in P111 : vertices of H are L ∪ (L + e1 ), where L is the lattice L = Z(e1 − e2 ) + Z(e2 − e3 ) = Zxˆ + Z yˆ , and edges connect nearest neighbors. Vertices in L are colored white, those in L + e1 are black. See Fig. 4. 2.1.1. Dual graph. Let U be a Jordan domain in P111 with smooth boundary. In
H, take a simple closed polygonal path with approximates ∂U in a reasonable way, for example the polygonal curve is locally monotone in the same direction as the curve ∂U .

Height Fluctuations in the Honeycomb Dimer Model

683

Fig. 5. The “dual” graph G ∗ (solid lines) of the graph G of Fig. 4

Let G be the subgraph of
H bounded by this polygonal path. We define a special kind of dual graph G ∗ as follows. Let
H∗ be the usual planar dual of
H. For each white vertex of G take the corresponding triangular face of
H∗ ; the union of the edges forming these triangles, along with the corresponding vertices, forms G ∗ . In other words, G ∗ has a face for each white vertex of G, as well as for black vertices which have all three neighbors in G. See Fig. 5. Throughout the paper the graph G and its related graphs will be scaled by a factor
over the corresponding graphs H, and so the G graphs have edge lengths of order
, and the graph H and its related graphs have edge lengths of order 1. 2.1.2. Forms. For an edge in G joining vertices b and w, we denote by (bw)∗ the dual edge in G ∗ , which we orient at +90◦ from the edge bw (when this edge is oriented from b to w). A 1-form ω on a graph is a function on directed edges which is antisymmetric with respect to reversing the orientation: ω(v1 v2 ) = −ω(v2 v1 ). A 1-form is also called a flow. If the graph is planar one can similarly define a 1-form on the dual graph. If ω is a 1-form, ω∗ is the dual 1-form, defined by ω∗ ((v1 v2 )∗ ) := ω(v1 v2 ). On a planar  graph with a 1-form ω, dω is a function on oriented faces defined by dω( f ) = e ω(e), where the sum is over the edges on a path around the face going counterclockwise. This is also known as the curl of the flow ω.  The form dω∗ is a function on vertices (faces of the dual graph), defined by dω∗ (v) = v ∼v ω(v, v ). In other words it is the divergence of the flow ω. A 1-form ω is closed if dω = 0, that is, the sum of ω along any cycle is zero (in the language of flows, the flow has zero curl). It is exact if ω = d f for some function f on the vertices, that is ω(v1 v2 ) = f (v1 ) − f (v2 ). A 1-form is co-closed if its dual form is closed. The corresponding flow is divergence-free. If dω∗ = 0, the integral of ω∗ between two faces of G (i.e. on a path in the dual graph) is the flux, or total flow, between those faces.

2.2. Heights and asymptotics. The unnormalized height function, or just height function, of a tiling is the integer-valued function on the vertices of the lozenges (faces of G) which is the sum of the coordinates of the corresponding point in Z3 . It changes by

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±1 along each edge of a tile. When we scale the lattice by
, so that we are discussing surfaces in
Z3 , the height function is defined as 1/
times the coordinate sum, so that it is still integer-valued. The normalized height √ function is
times the height function, and is the function which, when scaled by 3, has graph which is the surface in
Z3 . Let u : ∂U → R be a continuous function with the property that u can be extended to a Lipschitz function u˜ on the interior of U having the property that the normal to the graph of u˜ has nonnegative coordinates, that is, the normal points into the positive orthant R3≥0 . In other words, the graph of u is a wire frame γ of the type discussed before. Let h¯ : U → R be the asymptotic height function with boundary values u, from Theorem 1.1. It is smooth assuming the hypothesis of Theorem 1.4. ¯ scaled so that pa + pb + Let ν = ( pa , pb , pc ) be the normal vector to the graph of h, pc = 1. The directional derivatives of h¯ in the directions x, ˆ yˆ , zˆ are respectively 3 pc − 1, 3 pa − 1, 3 pb − 1.

(3)

2.3. Measures and gauge equivalence. We let µ = µ(G) be the uniform measure on dimer configurations on a finite graph G. If edges of G are given positive real weights, we can define a new probability measure, the Boltzmann measure, giving a configuration a probability proportional to the product of its edge weights. Certain edge-weight functions lead to the same Boltzmann measure: in particular if we multiply by a constant the weights of all the edges in G having a fixed vertex, the Boltzmann measure does not change, since exactly one of these weights is used in every configuration. More generally, two weight functions ν1 , ν2 are said to be gauge equivalent if ν1 /ν2 is a product of such operations, that is, if there are functions F1 on white vertices and F2 on black vertices so that for each edge wb, ν1 (wb)/ν2 (wb) = F1 (w)F2 (b). Gauge equivalent weights define the same Boltzmann measure. It is not hard to show that for planar graphs, two edge-weight functions are gauge equivalent if and only if they have the same face weights, where the weight of a face is defined to be the alternating product of the edge weights around the face (that is, the first, divided by the second, times the third, and so on), see e.g. [13]. In this paper we will only consider weights which are gauge equivalent to constant weights (or nearly so), so the Boltzmann measure will always be (nearly) the uniform measure.

2.4. Kasteleyn matrices. Kasteleyn showed that one can count dimer configurations on planar graph with the determinant of the certain matrix, the “Kasteleyn matrix” [6]. In the current case, when the underlying graph is part of the honeycomb graph, the Kasteleyn matrix K is just the adjacency matrix from white vertices to black vertices. For more general bipartite planar graphs, and when the edges have weights, the matrix is a signed, weighted version of the adjacency matrix [17], whose determinant is the sum of the weights of dimer coverings. Each entry K (w, b) is a complex number with modulus given by the corresponding edge weight (or zero if the vertices are not adjacent), and an argument which must be chosen in such a way that around each face the alternating product of the entries (the first, divided by the second, times the third, and so on) is positive if the face has 2 mod 4 edges and negative if the face has 0 mod 4

Height Fluctuations in the Honeycomb Dimer Model

685

edges (since we are assuming the graph is bipartite, each face has an even number of edges. For nonbipartite graphs, a more complicated condition is necessary). The Kasteleyn matrix is unique up to gauge transformations, which consist of preand post-multiplication by diagonal matrices (with, in general, complex entries). If the weights are real then we can choose a gauge in which K is real, although in certain cases it is convenient to allow complex numbers (we will below). Probabilities of individual edges occurring in a random tiling can likewise be computed using the minors of the inverse Kasteleyn matrix: Theorem 2.1 ([7]). The probability of edges {(b1 , w1 ), . . . , (bk , wk )} occurring in a random dimer covering is  k 

 K (wi , bi ) det{K −1 (bi , w j )}1≤i, j≤k .

i=1

On an infinite graph K is defined similarly but K −1 is not unique in general. This is related to the fact that there are potentially many different measures which could be obtained as limits of Boltzmann measures on sequences of finite graphs filling out the infinite graph. The edge probabilities for these measures can all be described as in the theorem above, but where the matrix “K −1 ” now depends on the measure; see the next section for examples.

2.5. Measures in infinite volume. On the infinite honeycomb graph H there is a twoparameter family of natural translation-invariant and ergodic probability measures on dimer configurations, which restrict to the uniform measure on finite regions (i.e. when conditioned on the complement of the finite region: we say they are conditionally uniform). Such measures are also known as ergodic Gibbs measures. They are classified in the following theorem due to Sheffield. Theorem 2.2 ([19]). For each ν = ( pa , pb , pc ) with pa , pb , pc ≥ 0 and scaled so that pa + pb + pc = 1 there is a unique translation-invariant ergodic Gibbs measure µν on the set of dimer coverings of H, for which the height function has average normal ν. This measure can be obtained as the limit as n → ∞ of the uniform measure on the set of those dimer coverings of Hn = H/n L whose proportion of dimers in the three orientations is ( pa : pb : pc ), up to errors tending to zero as n → ∞. Moreover every ergodic Gibbs measure on H is of the above type for some ν. The unicity in the above statement is a deep and important result. Associated to µν is an infinite matrix, the inverse Kasteleyn matrix of µν , K ν−1 = −1 (K ν (b, w)) whose rows index the black vertices and columns index the white vertices, and whose minors give local statistics for µν , just as in Theorem 2.1. From [13] there is an explicit formula for K ν−1 : let w = m 1 xˆ +n 1 yˆ and b = e1 +m 2 xˆ +n 2 yˆ = w +e1 +m xˆ +n yˆ , where m = m 2 − m 1 , n = n 2 − n 1 . Then a b −1 K ν−1 (b, w) = a( )m ( )n K abc (b, w), b c

(4)

686

where a, b, c, z, w are as defined in Sect. (1.2.3) and
z 1−m+n w1−n dz 1 dw1 1 −1 K abc (b, w) = (2πi)2 |z|=|w|=1 a + bz 1 + cw1 z 1 w1

−m+n −n

1 1 w z . +O = Im π cwm + an m 2 + n2

R. Kenyon

(5) (6)

−1 This formula for K abc and its asymptotics were derived in [13]: they are obtained from the limit n → ∞ of the inverse Kasteleyn matrix on the torus H/n L with edge weights a, b, c according to direction. It is not hard to check from (5) that K K −1 = Id. −1 , that is, obtained by From (4), the matrix K ν−1 is just a gauge transformation of K abc pre- and post-composing with diagonal matrices. Defining F(w) = (bz/a)m 1 (cw/bz)n 1 and F(b) = a(bz/a)−m 2 (cw/bz)−n 2 we can write, using (6),

F(w)F(b) 1 1 + |F(b)F(w)|O( 2 K ν−1 (b, w) = Im ). (7) π cwm + an m + n2 We’ll use this function F below. As a sample calculation, the µν -probability of a single horizontal edge, from w = 0 to b = e1 , being present in a random dimer covering is (see Theorem 2.1)
1 dz 1 dw1 a θa −1 K ν−1 (b, w) = a K abc (b, w) = = , (2πi)2 T2 a + bz 1 + cw1 z 1 w1 π where θa is, as before, the angle opposite side a in a triangle with sides a, b, c. This is consistent with (3).

2.6. T -graphs. 2.6.1. Definition. T -graphs were defined and studied in [15]. A pairwise disjoint coln L is lection L 1 , L 2 , . . . , L n of open line segments in R2 forms a T-graph in R2 if ∪i=1 i connected and contains all of its limit points except for some finite set R = {r1 , . . . , rm }, where each ri lies on the boundary of the infinite component of R2 minus the closure of n L . See Fig. 6 for an example where the outer boundary is a polygon. Elements in R ∪i=1 i are called root vertices and are labeled in cyclic order; the L i are called complete edges. We only consider the case that the outer boundary of the T -graph is a simple polygon, and the root vertices are the convex corners of this polygon. (An example where the outer boundary is not a polygon is a “T” formed from two edges, one ending in the interior of the other.) Associated to a T -graph is a Markov chain GT , whose vertices are the points which are endpoints of some L i . Each non-root vertex is in the interior of a unique L j (because the L j are disjoint); there is a transition from that vertex to its adjacent vertices along L j , and the transition probabilities are proportional to the inverses of the Euclidean distances. Root vertices are sinks of the Markov chain. See Fig. 7. Note that the coordinate functions on GT are harmonic functions on GT \ R. More generally, any function f on GT which is harmonic on GT \ R (we refer to such functions as harmonic functions on GT ) has the property that it is linear along edges, that is, if v1 , v2 , v3 are vertices on the same complete edge then f (v2 ) − f (v3 ) f (v1 ) − f (v2 ) = . (8) v1 − v2 v2 − v3

Height Fluctuations in the Honeycomb Dimer Model

r

687

4

r

r

3

r2

5

r1

r

6

Fig. 6. A T-graph (solid) and associated graph G D (dotted)

Fig. 7. The Markov chain associated to the T -graph of Fig. 6. Note that root vertices are sinks of the Markov chain

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Fig. 8. A face of G D (dotted)

2.6.2. Associated dimer graph and Kasteleyn matrix. Associated to a T -graph is a weighted bipartite planar graph G D constructed as follows, see Fig. 6. Black vertices of G D are the L j . White vertices are the bounded complementary regions, as well as one white vertex for each boundary path joining consecutive root vertices r j and r j+1 (but not for the path from rm to r1 ). The complementary regions are called faces; the paths between adjacent root vertices are called outer faces. Edges connect the L i to each face it borders along a positive-length subsegment. The edge weights are equal to the Euclidean length of the bounding segment. To G D there is a canonically associated Kasteleyn matrix of G D : this is the n × n matrix K G D = (K G D (w, b)) with rows indexing the white vertices and columns indexing the black vertices of G D . We have K G D (w, b) = 0 if w and b are not adjacent, and otherwise K G D (w, b) is the complex number equal to the edge vector corresponding to the edge of the region w along complete edge b (taken in the counterclockwise direction around w). In particular |K G D (w, b)| is the length of the corresponding edge of w. Lemma 2.3. K G D is a Kasteleyn matrix for G D , that is, the alternating product of the matrix entries for edges around a bounded face is positive real or negative real according to whether the face has 2 mod 4 or 0 mod 4 edges, respectively. By alternating product we mean the first, divided by the second, times the third, etc. Proof. Let f be a bounded face of G D (we mean not one of the outer faces). It corresponds to a meeting point of two or more complete edges; this meeting point is in the interior of exactly one of these complete edges, L. See Fig. 8. In G D , for each other black vertex on that face the two edges of the T -graph to neighboring white vertices have opposite orientations. The two edges parallel to L (horizontal in the figure) have the same orientation, so their ratio is positive. This implies the result.  Although we won’t need this fact, in [15] it is shown that the set of in-directed spanning forests of GT (rooted at the root vertices and weighted by the product of the transition probabilities) is in measure-preserving (up to a global constant) bijection with the set of dimer coverings of G D . 2.6.3. Harmonic functions and discrete analytic functions. To a harmonic function f on a T -graph GT we associate a derivative d f which is a function on black vertices of

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G D as follows. Let v1 and v2 be two distinct points on complete edge b, considered as complex numbers. We define d f (b) =

f (v2 ) − f (v1 ) . v2 − v1

(9)

Since f is linear along any complete edge (Eq. (8)), d f is independent of the choice of v1 and v2 . Lemma 2.4. If f  is harmonic on a T -graph GT and K = K G D is the associated Kasteleny matrix, then b∈B K (w, b)d f (b) = 0 for any interior white vertex w. Proof. Let b1 , . . . , bk be the neighbors of w in cyclic order. To each neighbor bi is associated a segment of a complete edge L i . Let vi and vi+1 be the endpoints of that segment, and wi and wi be the endpoints of L i . Then f (wi ) − f (wi ) f (vi+1 ) − f (vi ) = vi+1 − vi wi − wi since the harmonic function is linear along L i . In particular K G D (w, bi )d f (bi ) = (vi+1 − vi )

f (wi ) − f (wi ) = f (vi+1 ) − f (vi ). wi − wi

Summing over i (with cyclic indices) yields the result.   Note that at a boundary white vertex w, B K G D (w, b)d f (b) is the difference in f -values at the adjacent root vertices of GT .  We will refer to a function g on black vertices of G D satisfying b∈B K G D (w, b)g(b) = 0 for all interior white vertices w as a discrete analytic function. The construction in the above lemma can be reversed, starting from a discrete analytic function d f (on black vertices of G D ) and integrating to get a harmonic function f on GT : define f arbitrarily at a vertex of GT and then extend to neighboring vertices (on a same complete edge) using (9). The extension is well-defined by discrete analyticity. . We can relate K G−1 to the conjugate Green’s function 2.6.4. Green’s function and K G−1 D D on GT using the construction of the previous section, as follows. Let w be an interior face of GT , and  a path from a point in w to the outer boundary of GT which misses all the vertices of GT . For vertices v of GT , define the conjugate Green’s function G ∗ (w, v) to be the expected algebraic number of crossings of  by the random walk started at v and stopped at the boundary. This is the unique function with zero boundary values which is harmonic everywhere except for a jump discontinuity of −1 across  when going counterclockwise around w. (If there were two such functions, their difference would be harmonic everywhere with zero boundary values.) See Fig. 9 for an example. Let K G−1 (b, w) be the discrete analytic function of b defined from G ∗ (w, v) as in the D (b, w) using two points on b previous section; on an edge which crosses , define K G−1 D = I by Lemma 2.4, and on the same side of . This function clearly satisfies K G D K G−1 D therefore is independent of the choice of .

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Fig. 9. Conjugate Green’s function example. Here the sides of the triangle are bisected in ratio 2 : 3 and interior complete edges 1 : 1

Note that K G−1 also has the following probabilistic interpretation: take two particles, D started simultaneously at two different points v1 , v2 of the same complete edge, and couple their random walks so that they start independently, take simultaneous steps, and when they meet they stick together for all future times. Then the difference in their winding numbers around w is determined by their crossings of  before they meet. That is, K G−1 (b, w)(v1 − v2 ) is the expected difference in crossings before the particles meet D or until they hit the boundary, whichever comes first. 3. Constant-Slope Case (Theorem 3.7) in the speIn this section we compute the asymptotic expansion of K G−1 D cial case is when γ is planar. In this case the normalized asymptotic height function h¯ is linear, and its normal ν is constant. This case already contains most of the complexity of the general case, which is treated in Sect. 4. ¯ scaled as usual so that pa + pb + pc = 1. Let ν = ( pa , pb , pc ) be the normal to h, The angles θa , θb , θc are constant, and we choose a, b, c as before to be constant as well. Define a function F(w) = (bz/a)m (cw/bz)n at a white vertex w = m xˆ + n yˆ and F(b) = a(bz/a)−m (cw/bz)−n at a black vertex b = e1 + m xˆ + n yˆ . These functions are defined on all of the honeycomb graph H. Let K H be the adjacency matrix of H (which, as we mentioned earlier, is a Kasteleyn matrix for H). Lemma 3.1. We have  b

K H (w, b)F(b) = 0 =

 w

F(w)K H (w, b),

(10)

Height Fluctuations in the Honeycomb Dimer Model

691

where K H is the adjacency matrix of H. Proof. This follows from the equation a + bz + cw = 0.



3.1. T -graph construction. Define a 1-form  on edges of H by (wb) = −(bw) = 2Re(F(w))F(b). ∗

(11)

∗ ((wb)∗ )

By (10) the dual form (defined by = (wb)) is closed (the integral around any closed cycle is zero) and therefore ∗ = d for a complex-valued function  on H∗ . Here H∗ , the dual of the honeycomb, is the graph of the equilateral triangulation of the plane. Extend  linearly over the edges of H∗ . This defines a mapping from H∗ to C with the property that the images of the white faces are triangles similar to the a, b, c-triangle (via orientation-preserving similarities), and the images of black faces are segments. This follows immediately from the definitions: if b1 , b2 , b3 are the three neighbors of a white vertex w of H, the edges wbi have values 2Re(F(w))F(bi ) which are proportional to F(b1 ) : F(b2 ) : F(b3 ), which in turn are proportional to a : bz : cw by (10). If w1 , w2 , w3 are the three neighbors of a black vertex then the corresponding edge values are 2Re(F(wi ))F(b) which are proportional to Re(F(w1 )) : Re(F(w2 )) : Re(F(w3 )), that is, they all have the same slope and sum to zero. It is not hard to see that the images of the white triangles are in fact non-overlapping (see [15], Sect. 5 for the proof, or look at Fig. 10). It may be that ReF(w) = 0 for some w; in this case choose a generic modulus-1 complex number λ and replace F(w) by λF(w) and F(b) by λF(b). So we can assume that each white triangle is similar to the a, b, c-triangle. In fact, this operation will be important later; note that by varying λ the size of an individual triangle varies; by an appropriate choice we can make any particular triangle have maximal size (side lengths a, b, c). Lemma 3.2. The mapping  is almost linear, that is, it is a linear map φ(m, n) = cwm + an plus a bounded function. Proof. Consider for example a vertical column of horizontal edges {w1 b1 , . . . , wk bk } of H connecting a face f 1 to face f k+1 . We have k 

∗ ((wi bi )∗ ) =

i=1

k 

(wi bi )

i=1

=

k  (F(wi ) + F(wi ))F(bi ) i=1

= ka +

k 

F(wi )F(bi )

i=1

= ka + a

k 

z 2i w −2i

i=1

= φ( f k+1 ) − φ( f 1 ) + osc,

(12)

where osc is oscillating and in fact O(1) independently of k (by hypothesis z/w ∈ R). In the other two lattice directions the linear part of  is again φ, so that  is almost linear in all directions. 

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Fig. 10. The -image of H∗ is a T -graph covering R2

This lemma shows that the image of H∗ under  is an infinite T -graph HT covering all of R2 . The images of the black triangles are the complete edges and have lengths O(1). If we insert a λ as above and let λ vary over the unit circle, one sees all possible local structures of the T -graph, that is, the geometry of the T -graph HT = HT (λ) in a neighborhood of a triangle (w) only depends up to homothety on the argument of λF(w). Recall that G ∗ is a subgraph of
H∗ approximating U . We can restrict  to 1
G ∗ thought of as a subgraph of 1
H, and then multiply its image by
. Thus we get a finite sub-T -graph GT of
HT . Let 
=
◦ ◦ 1
so that 
acts on G ∗ . The union of the 
-images of the white triangles in G ∗ forms a polygon P. Define the “dimer” graphs H D associated to HT , and G D associated to GT as in Sect. 2.6.2. See Fig. 6 for the T -graph arising from the graph G ∗ of Fig. 5. Note that G D contains G (defined in Sect. 2.1.1) but has extra white vertices along the boundaries. These extra vertices make the height function on G D approximate the desired linear function (whose graph has normal ν), see the next section. From (11) we have Lemma 3.3. Edge weights of H D and G D are gauge equivalent to constant edge weights. Indeed, for w not on the boundary of G D we have K G D (w, b) = 2
Re(F(w))F(b)K G (w, b)

(13)

and similarly for all w, K H D = 2Re(F(w))F(b)K H (w, b).

(14)

Height Fluctuations in the Honeycomb Dimer Model

693

Asymptotics of K G−1 are described below in Sect. 3.4. For K H D , from (7) and (14) D we have

F(w)F(b) 1 1 −1 Im + O( (b, w) = ). (15) KH D 2π Re(F(w))F(b) φ(b) − φ(w) |φ(b) − φ(w)|2 −1 Here K H is the inverse constructed from the conjugate Green’s function on the T -graph D

−1 (b, w) coincides with K ν−1 of (7) since HT . We use the fact that 2Re(F(w))F(b)K H D both satisfy the equation that d K −1 equals the conjugate Green’s function.

3.2. Boundary behavior. Recall that from our region U we constructed a graph G (Sect. 2.1.1). From the normalized height function u on ∂U (which is the restriction of a linear function to ∂U ) we constructed the T -graph GT and then the dimer graph G D . In this section we show that the normalized boundary height function of G D when G D ¯ This is proved in a roundabout way: we first show is chosen as above approximates h. that the boundary height does not depend (up to local fluctuations) on the exact choice of boundary conditions, as long as we construct GT from G ∗ as in Sect. 2.1. Then we compute the height change for “simple” boundary conditions. 3.2.1. Flows and dimer configurations Any dimer configuration m defines a flow (or 1-form) [m] with divergence 1 at each white vertex and divergence −1 at each black vertex: just flow by 1 along each edge in m and 0 on the other edges. The set 1 ⊂ [0, 1] E of unit white-to-black flows (i.e. flows with divergence 1 at each white vertex and −1 at each black vertex) and with capacity 1 on each edge is a convex polytope whose vertices are the dimer configurations [16]. On the graph H define the flow ω1/3 ∈ 1 to be the flow with value 1/3 on each edge wb from w to b. Up to a factor 1/3, this flow can be used to define the height function, in the sense that for any dimer configuration m, [m] − ω1/3 is a divergence-free flow and the integral of its dual (which is closed) is 1/3 times the height function of m. That is, the height difference between two points is three times the flux of [m] − ω1/3 between those points. This is easy to see: across an edge which contains a dimer the height changes by ±2 (depending on the orientation); if the edge does not contain a dimer the height changes by ∓1, for so that the height change can be written ±(3χ − 1) where χ is the indicator function of a dimer on that edge. For a finite subgraph of H the boundary height function can be obtained by integrating around the boundary (3 times) the dual of [m] − ω1/3 for some m. 3.2.2. Canonical flow. On H D there is a canonical flow ω ∈ 1 defined as illustrated in Fig. 11: let v1 v2 be the two vertices of GT along b adjacent to w. The flow from w to b is 1/(2π ) times the sum of the two angles that the complete edges through v1 and v2 make with b. One or both of these angles may be zero. Note that the total flow out of w is 1. For an edge b, the total flow into b is also 1 as illustrated on the right in Fig. 11. Now G D is a subgraph of H D with extra white vertices around its boundary. The canonical flow on H D restricts to a flow on G D except on edges connecting to these extra white vertices (we define the canonical flow there to be zero). This flow has divergence 1, −1 at white/black vertices, except at the black vertices of G D connected to the boundary white vertices, and the boundary white vertices themselves. If m is a dimer

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θ5

θ4

θ 3+ θ4

θ4 θ1

θ5+ θ 6 θ6

θ3

θ1 + θ 2 θ1

θ1

θ2 θ3 θ3 + θ 4

θ2

θ2

Fig. 11. Defining the canonical flow (divide the angles by 2π ).

covering of G D , the flow [m] − ω is now a divergence-free flow on G D except at these black and white boundary vertices. Lemma 3.4. Along the boundary of G D the divergence of [m] − ω for any dimer configuration m is the turning angle of the boundary of P. Proof. Consider a complete edge L corresponding to black vertex b. The canonical flow into b has a contribution from the two endpoints of L. The flow [m] can be considered to contribute −1/2 for each endpoint. Recall that each endpoint of L ends in the interior of another complete edge or at a convex vertex of the polygon P. If an endpoint of L ends in the interior of another complete edge, and there are white faces adjacent to the two sides of this endpoint (that is, the endpoint is not a concave vertex of P) then the contribution of the canonical flow is also −1/2, so the contribution of [m] − ω is zero. Suppose the endpoint is at a concave vertex v of P with exterior angle θ < π . The θ contribution from v to the flow of [m] − ω into L is −1/2 + 2π . The other complete edge at v does not end at v and so has no contribution. This quantity is 1/2π times the turning angle of the boundary. Suppose the endpoint is at a convex vertex v of P of interior angle θ < π . There is some complete edge L of HT , containing v in its interior, which is not in GT . The sum of the contributions of [m] − ω for the endpoints of the two complete edges L 1 , L 2 of GT meeting at v is −1/2 − θ/2π . This is the 1/2π times the turning angle at the convex vertex, minus 1. The contribution from [m] from the boundary white vertices is 1 per white boundary vertex, that is, one per convex vertex of P.  This lemma proves in particular that the divergence of [m] − ω is bounded for any [m]. 3.2.3. Boundary height. Recall that the height function of a dimer covering m can be defined as the flux of ω1/3 − [m]. In particular, the flux of ω1/3 − ω defines the boundary height function up to O(1) (the turning of the boundary), since the flux of [m] − ω is the boundary turning angle. The flux of ω1/3 − ω between two faces can be computed along any path, and in fact because both ω1/3 and ω are locally defined from H D , we see that the flux does not depend on the choice of the nearby boundary. Let us compute this flux and show that it is linear along lattice directions, and therefore linear everywhere in H D . Take a vertical column of horizontal edges in G, and let us compute ω on this set of edges. The  image of the dual of this column is a polygonal curve η whose j th edge is

Height Fluctuations in the Honeycomb Dimer Model

695

Fig. 12. A column of triangles from GT

(using (12)) a constant times 1 + ( wz )2 j . The image of the triangles in the vertical column of G ∗ is as shown in Fig. 12. The j th edge is part of a complete edge corresponding to the j th black vertex in the column. By the argument of the previous section, the flux is equal to the number of convex corners of this polygonal curve η, that is, corners where the curve turns left. The curve η has a convex corner when

1 + (z/w)2 j+2 ≥ 0, Im 1 + (z/w)2 j that is, using z/w = −eiθa , when 2 jθa ∈ [π, π + 2θa ]. Assuming that θa is irrational, a this happens with frequency 2θ 2π , so the flux of ω1/3 − ω along a column of length n θa 1 is n( π − 3 ), and the average flux per edge is pa − 1/3. Therefore the average height change per horizontal edge is 3 pa − 1. If θa is rational, a continuity argument shows that 3 pa − 1 is still the average height change per horizontal edge. A similar result holds in the other directions, and (3) shows that the average height has normal ( pa , pb , pc ) as desired. 3.3. Continuous and discrete harmonic functions. To understand the asymptotic expansion of G ∗ , the conjugate Green’s function on GT , from which we can get K −1 , we need two ingredients. We need to understand the conjugate Green’s function on HT , and also the relation between continuous harmonic functions on domains in R2 and discrete harmonic functions on (domains in)
HT . 3.3.1. Discrete and continuous Green’s functions. On HT , the discrete conjugate Green’s function G ∗ (w, v), for w ∈ H and v ∈ HT , can be obtained from integrating the exact formula for K ν−1 given in (4) as discussed in Sect. 2.6.4. As we shall see, this formula differs even in its leading term from the continuous conjugate Green’s function, due to the singularity at the diagonal. Basically, because our Markov chain is directed, a long random walk can have a nonzero expected winding number around the origin. This causes the conjugate Green’s function, which measures this winding, to have a component of Re log v as well as a part Im log v. The continuous conjugate Green’s function on the whole plane is g ∗ (v1 , v2 ) =

1 Im log(v2 − v1 ). 2π

(We use G ∗ to denote the discrete conjugate Green’s function and g ∗ the continuous version.) To compute the discrete conjugate Green’s function G ∗ , if we simply restrict g ∗ to the vertices of
HT , it will be very nearly harmonic as a function of v2 for large |v2 −v1 |, but the discrete Laplacian of g ∗ at vertices v2 near v1 (within O(
) of v1 ) will be of constant order in general. We can correct for the non-harmonicity at a vertex v by adding an

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appropriate multiple of the actual (non-conjugate) discrete Green’s function G(v, v2 ). The large scale behavior of this correction term is a constant times Re log(v2 − v ). So we can expect the long-range behavior of the discrete conjugate Green’s function G ∗ (w, v), for |w − v| large, to be equal to g ∗ (w, v) plus a sum of terms involving the real part Re log(v − v ) for v s within O(
) of φ(w). These extra terms sum to a function of the form c log |v − φ(w)| +
s(v) + O(
2 ), where s is a smooth function and c is a real constant, both c and s depending on the local structure of HT near φ(w). This form of G ∗ can be seen explicitly, of course, if we integrate the exact formula (15). We have Lemma 3.5. The discrete conjugate Green’s function G ∗ on the plane
HT is asymptotically

Im(F(w)) 1 Im log(v − φ(w)) + Re log(v − φ(w)) G ∗ (w, v) = 2π Re(F(w)) 1 +
s(w, v) + O(
2 ) + O( ) (16) |v − φ(w)| 1 Im (F(w) log(v − φ(w))) +
s(w, v) = 2π Re(F(w)) 1 ), (17) +O(
2 ) + O( |v − φ(w)| where s(w, v) is a smooth function of v. Proof. From the argument of the previous paragraph, it suffices to compute the constant in front of the Re log(v − φ(w)) term. This can be computed by differentiating the above formula with respect to v and comparing with formula (15) for K −1 . The differential of
s(w, v) is O(
). Let v1 , v2 be two vertices of complete edge b, coming from adjacent faces of H, adjacent across an edge bw1 (so that v1 − v2 = ∗ (bw1 ) = 2
Re(F(w1 ))F(b)). We have for w far from w1 , −1 (b, w) = KH D

=

G ∗ (w, v1 ) − G ∗ (w, v2 ) v − v2  1 F(w) v1 −v2 1 2π Im Re(F(w)) φ(b)−φ(w)

+ O(
) 2
Re(F(w1 ))F(b)

1 F(w)F(b) = Im + O(
). 2π Re(F(w))F(b) φ(b) − φ(w)



Note that in fact for any complex number λ of modulus 1, we get a discrete conjugate Green’s function G ∗λ on the graph
HT (λ) (from Sect. 3.1) with similar asymptotics. 3.3.2. Smooth functions. Constant and linear functions on R2 , when restricted to
HT , are exactly harmonic. More generally, if we take a continuous harmonic function f on R2 and evaluate it on the vertices of
HT , the result will be close to a discrete harmonic function f
, in the sense that the discrete Laplacian will be O(
2 ): if v is a vertex of
HT

Height Fluctuations in the Honeycomb Dimer Model

697

and v1 , v2 are its (forward) neighbors located at v1 = v −
d1 eiθ and v2 = v +
d2 eiθ then the Taylor expansion of f about v yields  f (v) = f (v) −

d1 d2 f (v −
d1 eiθ ) − f (v +
d2 eiθ ) = O(
2 ). d1 + d2 d1 + d2

This situation is not as good as in the (more standard) case of a graph like
Z2 , where if we evaluate a continuous harmonic function on the vertices, the Laplacian of the resulting discrete function is O(
4 ): f (v) −

1 ( f (v +
) + f (v +
i) + f (v −
) + f (v −
i)) = O(
4 ). 4

In the present case the principal error is due to the second derivatives of f . To get an error smaller than O(
2 ), we need to add to f a term which cancels out the
2 error. We can add to f a function which is
2 times a bounded function f 2 whose value at a point v depends only on the local structure of
HT near v and on the second derivatives of f at v. Lemma 3.6. For a smooth harmonic function f on R2 whose second partial derivatives don’t all vanish at any point, there is a bounded function f
on HT such that f
(z) = f (z) +
2 f 2 (z) has discrete Laplacian of order O(
3 ), where f 2 (z) depends only the second derivatives of f at z and on the local structure of the graph HT at z. Proof. We have exact formulas for one discrete harmonic function, the conjugate Green’s function on
HT , and we know its asymptotics (Lemma 3.5), Eq. (17), which are G ∗ (w, z 2 ) ≈ η(z 1 , z 2 ) =

1 Im(c log(z 2 − z 1 )) 2π

for a constant c depending on the local structure of the graph near w, and where z 1 is a point in φ(w). We’ll let w be the origin in
HT and z 1 = 0; η(0, z 2 ) is a continuous harmonic function of z 2 . For z ∈ R2 consider the second derivatives of the function f , which by hypothesis are not all zero. There is a point z 2 = β(z) ∈ R2 at which η(0, z 2 ) has the same second derivatives. Indeed, f has three second partial derivatives, f x x , f x y , and f yy , but because f is harmonic f yy = − f x x . We have ∂z∂ 2 η(0, z 2 ) = 21pi Im zc2 , and the second partial derivatives of η are the real and imaginary parts of const/z 22 , which is surjective, in fact 2 to 1, as a mapping of R2 − {0} to itself. In particular there are two choices of z 2 for which f (z) and η(β(z)) have the same second derivatives. Since f and η are smooth, by taking a consistent choice, β can be chosen to be a smooth function as well. Consider the function f
(z) = f (z) + G ∗λ (0, β(z)) − η(0, β(z)), where G ∗λ is the discrete conjugate Green’s function and λ is chosen so that HT (λ) (see Sect. 3.1) has local structure at β(z) identical to that of HT at z. We claim that the discrete Laplacian of f
is O(
3 ). This is because G ∗λ (0, β(z)) is discrete harmonic, and f (z) − η(0, β(z)) has vanishing second derivatives. We also have that G ∗ (0, β(z)) − η(0, β(z)) = O(
2 ), see Lemma 3.5 above. 

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3.4. K G−1 in constant-slope case. D Theorem 3.7. In the case of constant slope ν and a bounded domain U , let ξ be a conformal diffeomorphism from φ(U ) to H. When b and w are converging to different points as
→ 0 we have K G−1 (b, w) = D

1 2π Re(F(w))F(b)   ξ (φ(b))F(w)F(b) ξ (φ(b))F(w)F(b) + + O(
). (18) ×Im ξ(φ(b)) − ξ(φ(w)) ξ(φ(b)) − ξ(φ(w))

Proof. The function G ∗GT (w, v) is equal to the function (16) for the whole plane, plus a harmonic function on GT whose boundary values are the negative of the values of (16) on the boundary of GT . Since discrete harmonic functions on GT are close to continuous harmonic functions on U , we can work with the corresponding continuous functions. From (17) we have G ∗HT (w, v) =



F(w) 1 Im log(v − z 1 ) +
s(z 1 , v) + O(
)2 , 2π Re(F(w))

(19)

where z 1 is a point in face φ(w). The continuous harmonic function of v on U whose values on ∂U are the negative of the values of (19) on ∂U is

F(w) 1 Im log(v − z 1 ) 2π ReF(w)   F(w) F(w) 1 Im log(ξ(v) − ξ(z 1 ))+ log(ξ(v)−ξ(z 1 )) +
s2 (z 1 , v)+ O(
)2, + 2π ReF(w) ReF(w)

−

(20) where s2 is smooth. The discrete Green’s function G ∗GT must be the sum of (19) and (20): 1 Im G ∗GT (w, v) = 2π



F(w) F(w) log(ξ(v) − ξ(z 1 )) + log(ξ(v) − ξ(z 1 )) ReF(w) ReF(w)



+
s3 (z 1 , v) + O(
)2 . Differentiating gives the result (as in Lemma 3.5).



It is instructive to compare the discrete conjugate Green’s function in the above proof with the continuous conjugate Green’s function on U which is g ∗ (z 1 , z 2 ) =

1 1 Im log(ξ(z 1 ) − ξ(z 2 )) + Im log(ξ(z 1 ) − ξ(z 2 )). 2π 2π

Height Fluctuations in the Honeycomb Dimer Model

699

3.4.1. Values near the diagonal. Note that when b is within O(
) of w, and neither is close to the boundary, the discrete Green’s function G ∗ (w, v) for v on b is equal to the discrete Green’s function on the plane G ∗HT (w, v) plus an error which is O(1) coming from the corrective term due to the boundary. The error is smooth plus oscillations of order O(
2 ), so that within O(
) of w the error is a linear function plus O(
). Therefore when we take derivatives −1 K G−1 (b, w) = K H (b, w) + O(1), D D −1 is of order O(
−1 ) when |b − w| = O(
), implies that the local which, since K H D statistics are given by µν .

Theorem 3.8. In the case of constant slope ν, the local statistics at any point in the interior of U are given in the limit
→ 0 by µν , the ergodic Gibbs measure on tilings of the plane of slope ν. 4. General Boundary Conditions ¯ the normalized Here we consider the general setting: U is a smooth Jordan domain and h, asymptotic height function on U , is not necessarily linear.

4.1. The complex height function. The equation (1) implies that the form log( − 1)d xˆ − log(

1 − 1)d yˆ

is closed. Since U is simply connected it is d H for a function H : U → C which we call the complex height function. ¯ we have arg( − 1) = π − θc (Fig. 2) and The imaginary part of H is related to h: arg( 1 − 1) = θa − π , which gives Im d H = (π − θc )d xˆ + (π − θa )d yˆ . From (3) we have d h¯ = (3 pc − 1)d xˆ + (3 pa − 1)d yˆ , so 3 ¯ Im d H = 2(d xˆ + d yˆ ) − d h. π The real part of H is the logarithm of a special gauge function which we describe below. We have Hxˆ = log( − 1), 1 Hyˆ = − log( − 1), − yˆ xˆ = , Hxˆ xˆ = −1 −1 1 yˆ . Hyˆ yˆ = − −1

(21) (22) (23) (24)

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4.2. Gauge transformation. The mapping is a real analytic mapping from U to the upper half plane. It is an open mapping since Im( xˆ / yˆ ) = −Im = 0, but may have isolated critical points. The Ahlfors-Bers theorem gives us a diffeomorphism φ from U onto the upper half plane satisfying the Beltrami equation dφ d z¯ dφ dz

=

d d z¯ d dz

=

− eiπ/3 , − e−iπ/3

that is φxˆ = − φ yˆ . Such a φ exists by the Ahlfors-Bers theorem [1]. It follows that is of the form f (φ) for some holomorphic function f from H into H. Since ∂U is smooth, φ is smooth up to and including the boundary, and φxˆ , φ yˆ are both nonzero. For white vertices of G define

1 F(w) = e
H (w) φ yˆ (w)(1 +
M(w)), (25) where M(w) is any function which satisfies 

 φx2ˆ yˆ φxˆ yˆ yˆ H φ H x ˆ x ˆ x ˆ y ˆ x ˆ x ˆ x ˆ e−Hxˆ Mxˆ − e Hyˆ M yˆ = e−Hxˆ − − − 2+ 8 6 4φ yˆ 4φ yˆ 8φ yˆ  2  Hyˆ yˆ Hyˆ yˆ yˆ Hyˆ yˆ φ yˆ yˆ φ y2ˆ yˆ φ yˆ yˆ yˆ H yˆ + + +e − 2+ . 8 6 4φ yˆ 4φ yˆ 8φ yˆ Hxˆ2xˆ

(26)

The existence of such an M follows from the fact that the ratio of the coefficients of Mxˆ and M yˆ is −e Hxˆ +Hyˆ = , so (26) is of the form Mxˆ + M yˆ = J (x, y) for some smooth function J . This is the ∂¯ equation in coordinate φ. We don’t need to know M explicitly; the final result is independent of M. We just need its existence to get better estimates on the error terms in Lemma 4.1 below. For black vertices b define

1 F(b) = e−
H (w) φ yˆ (w)(1 −
M(w)), (27) where w is the vertex adjacent to and left of b and M(w) is as above. Lemma 4.1. For each black vertex b with three neighbors in G we have 

F(w)K (w, b) = O(
3 )

w

and for each white vertex w we have  b

K (w, b)F(b) = O(
3 ).

Height Fluctuations in the Honeycomb Dimer Model

701

Proof. This is a calculation. Let w, w −
x, ˆ w +
yˆ be the three neighbors of b. Then, setting H = H (w), φ yˆ = φ yˆ (w), and M = M(w) we have  1
2
3
2 4 F(w −
x) ˆ = e
(H −
Hxˆ + 2 Hxˆ xˆ − 6 Hxˆ xˆ xˆ +O(
)) φ yˆ −
φ yˆ xˆ + φ yx x + O(
3 ) 2 ×(1 +
M −
2 Mxˆ + O(
3 )), F(w +
yˆ ) = e

1
2
(H +
H yˆ + 2

3

H yˆ yˆ +
6 H yˆ yˆ yˆ +O(
4 ))

 φ yˆ +
φ yˆ yˆ +


2 φ yˆ yˆ yˆ + O(
3 ) 2

×(1 +
M +
2 M yˆ + O(
3 )). The sum of the leading order terms in F(w) + F(w −
x) ˆ + F(w +
yˆ ) is

1 1  1 + ) = 0. e
H φ yˆ (1 + e−Hxˆ + e Hyˆ ) = e
H φ(1 + −1 1− 1  The sum of the terms of order
is 2
e
H φ yˆ times φ yˆ xˆ φ yˆ yˆ + Hxˆ xˆ ) + e Hyˆ ( + Hyˆ yˆ ) φ yˆ φ yˆ



− yˆ yˆ φ yˆ yˆ φ yˆ yˆ 1 + = 0, + yˆ + − = −1 φ yˆ −1 1 − φ yˆ ( − 1)

e−Hxˆ (−

1  and the sum of the order-
2 terms is
2 e
H φ yˆ times   2 φx2ˆ yˆ φxˆ yˆ yˆ H H φ H x ˆ x ˆ x ˆ y ˆ x ˆ x ˆ x ˆ xˆ xˆ − − − 2+ −e−Hxˆ Mxˆ + e Hyˆ M yˆ + e−Hxˆ 8 6 4φ yˆ 4φ yˆ 8φ yˆ  2  Hyˆ yˆ Hyˆ yˆ yˆ Hyˆ yˆ φ yˆ yˆ φ y2ˆ yˆ φ yˆ yˆ yˆ H yˆ + + +e − 2+ = 0. (28) 8 6 4φ yˆ 4φ yˆ 8φ yˆ

A similar calculation holds at a white vertex, and we get the same expression for the
2 contribution (changing the signs of M, H, d/d xˆ and d/d yˆ gives the same expression). 

It is clear from this proof that the error in the statement can be improved to any order O(
3+k ) by replacing M(w) with M0 (w) +
M1 (w) + · · · +
k Mk (w), where each M j satisfies an equation of the form (26) except with a different right hand side—the right-hand side will depend on derivatives of H, φ and the Mi for i < j. For our proof below we need an error O(
4 ) and therefore the M0 and M1 terms, even though the final result will depend on neither M0 nor M1 . 4.3. Embedding. Define a 1-form (wb) = 2Re(F(w))F(b)K (w, b) on edges of G, where F is defined in (25,27) (since K (w, b) =
is a constant, this is an unimportant factor for now, but in a moment we will perturb K ). By the comments

702

R. Kenyon

after the proof of Lemma 4.1, the dual form ∗ on G ∗ can be chosen to be closed up to O(
4 ) and so there is a function φ˜ on G ∗ , defined up to an additive constant, satisfying d φ˜ = ∗ + O(
3 ). In fact up to the choice of the additive constant, φ˜ is equal to φ plus an oscillating function. This can be seen as follows. For a horizontal edge wb we have F(w)F(b) = φ yˆ (w) + O(
2 ). Thus on a vertical column {w1 b1 , . . . , wk bk } of horizontal edges we have k 

∗ (wi bi ) =


i=1

k 

(F(wi ) + F(wi ))F(bi )

i=1

=




φ yˆ (wi ) +


k 

F(wi )F(bi ) + O(
3 ).

i=1

The first sum gives the change in φ from one endpoint of the column to the other, and the second sum is oscillating (F(wi ) and F(bi ) have the same argument which is in (0, π ) and which is a continuous function of the position) and so contributes O(
). Similarly, in the other two lattice directions the sum is given by the change in φ plus an oscillating term. Therefore by choosing the additive constant appropriately, φ˜ maps G ∗ to a small ˆ which shrinks to H as
→ 0. neighborhood of H in the spherical metric on C, The map φ˜ has the following additional properties. The image of the three edges of a black face of G ∗ are nearly collinear, and the image of a white triangle is a triangle nearly similar to the a, b, c-triangle, and of the same orientation. Thus it is nearly a mapping onto a T -graph. In fact near a point where the relative weights are a, b, c (weights which are slowly varying on the scale of the lattice) the map is up to small errors the map  of Sect. 3. We can adjust the mapping φ˜ by O(
3 ) so that the image of each black triangle is an exact line segment: this can be arranged by choosing for each black face a line such that ˜ the φ-image of the corresponding black face is within O(
3 ) of that line; the intersections of these lines can then be used to define a new mapping  : G ∗ → C which is an exact T -graph mapping. The mapping  will then correspond to the above 1-form  but for a matrix K˜ with slightly different edge weights. Let us check how much the weights differ from the original weights (which are
). As long as the triangular faces are of size of order
(which they are typically), the adjustment will change the edge lengths locally by O(
3 ) and therefore their relative lengths by 1 + O(
2 ). There will be some isolated triangular faces, however, which will be smaller—of order O(
2 ) because of the possibility that F(w) might be nearly pure imaginary. We can deal with these as follows. Once we have readjusted the “large” triangular faces we have an exact T -graph mapping on most of the graph. We can then multiply F(w) by λ = i and F(b) by λ = −i: the readjusted weights give (for most of the graph) a new exact T -graph mapping (because we now have K˜ F = F K˜ = 0 exactly for these weights), but now all faces which were too small before become O(
) in size and we can readjust their dimensions locally by a factor 1 + O(
2 ). In the end we have an exact mapping  of G ∗ onto a T -graph GT and it distorts the edge weights of K by at most 1 + O(
2 ). We shall see in Sect. 5 that this is close enough to get a good approximation to K −1 .

Height Fluctuations in the Honeycomb Dimer Model

703

In conclusion the Kasteleyn matrix for G D is equal to 2Re(F(w))F(b) K˜ (w, b) where K˜ has edge weights
+ O(
3 ).

4.4. Boundary. Along the boundary we claim that the normalized height function of G D follows u. Since G D arises from a T -graph, we can use its canonical flow (Sect. 3.2.2). Near any given point the canonical flow looks like the canonical flow in the constantweight case—since the weights vary continuously, they vary slowly at the scale
, the scale of the graph. Since the canonical flow defines the slope of the normalized height function, we have pointwise convergence of the derivative of h¯ along the boundary to the derivative of u. Thus h¯ converges to u. In fact this argument shows that the normalized height function of the canonical flow converges to the asymptotic height function in the interior of U as well. 5. Continuity of K −1 In this section we show how K −1 changes under a small change in edge weights. Lemma 5.1. Suppose 0 < δ 
. If G is a graph identical to G D but with edge weights which differ by a factor 1 + O(δ), then K G−1 = K G−1 (1 + O(δ/
)). D In particular since δ =
2 in our case this will be sufficient to approximate K −1 to within 1 + O(
). Proof. For any matrix A we have ⎛

(K + δ
A)−1

⎞ ∞  = K −1 ⎝1 + (−1) j (δ
) j (AK −1 ) j ⎠ , j=1

as long as this sum converges. From Theorem 6.1 below we have that K G−1 (b, w) = O(1/|b − w|). When A represents a bounded, weighted adjacency matrix of G D , the matrix norm of AK −1 is then AK −1  ≤ max w

   | A(w , b)K −1 (b, w)| ≤ 3a max w

b

w

w=w

|w

1 = O(
−2 ), − w|

where a is the maximum entry of A. In particular when δ 
we have (K + δ
A)−1 = K −1 (1 + O(δ/
)). 

704

R. Kenyon

6. Asymptotic Coupling Function We define K G D as in (13) using the values (25,27) for F (and K G is the adjacency matrix of G). Theorem 6.1. If b, w are not within o(1) of the boundary of U , we have   F(w)F(b) F(w)F(b) 1 −1 K G D (b, w) = + O(
). (29) Im + 2π Re(F(w))F(b) φ(b) − φ(w) φ(b) − φ(w) When one or both of b, w are near the boundary, but they are not within o(1) of each other, then K G−1 (b, w) = O(1). D Proof. The proof is identical to the proof in the constant-slope case, see Theorem 3.7, except that ξ φ there is φ here. The ξ factors from (18) are here absorbed in the definition of the functions φ and F.  As in the case of constant slope, when b and w are close to each other (within O(
)) and not within o(1) of the boundary, Theorem 3.8 applies to show that the local statistics are given by µν . 7. Free Field Moments As before let φ be a diffeomorphism from U to H satisfying φxˆ = − φ yˆ , where is defined from h¯ as in Sect. 4.1. Theorem 7.1. Let h¯ be the asymptotic √ height function on G D , h
the normalized height ¯ Then hˆ converges weakly as
→ 0 function of a random tiling, and hˆ = 23
π (h
− h). to φ ∗ F, the pull-back under φ of F, the Gaussian free field on H. Here weak convergence means that for any smooth test function ψ on U , zero on the boundary, we have
 ˆ f) →
2 ψ( f )h( ψ(x)F(φ(x))|d x|2 , GD

U

where the sum on the left is over faces f of G D . Proof. We compute the moments of F. Let ψ1 , . . . , ψk be smooth functions on U , each zero on the boundary. We have ⎡⎛ ⎞ ⎛ ⎞⎤   ˆ f 1 )⎠ · · · ⎝
2 ˆ f k )⎠ ⎦ E ⎣⎝
2 ψ1 ( f 1 )h( ψk ( f k )h( f 1 ∈G D

=




2k

f k ∈G D

ˆ f 1 ) . . . h( ˆ f k )]. ψ1 ( f 1 ) · · · ψk ( f k )E[h(

f 1 ,..., f k

From Theorem 7.2 below the sum becomes

 = ··· E[F(φ(x1 )) . . . F(φ(xk ))] ψi (xi )|d xi |2 + o(1), U

U

that is, the moments of hˆ converge to the moments of the free field φ ∗ (F). Since the free field is a Gaussian process, it is determined by its moments. This completes the proof. 

Height Fluctuations in the Honeycomb Dimer Model

705

Theorem 7.2. Let s1 , . . . , sk ∈ U be distinct points in the interior of U . For each
let f 1 , f 2 , . . . , f k be faces of G D , with f i converging to si as
→ 0. If k is odd we have ˆ f 1 ) . . . h( ˆ f k )] = 0, lim E[h(


→0

and if k is even we have ˆ f 1 ) . . . h( ˆ f k )] = lim E[h(


→0



k/2 

G(φ(sσ (2 j−1) ), φ(sσ (2 j) )),

pairings σ j=1

where    z − z  1   G(z, z ) = − log  2π z − z¯ 

is the Dirichlet Green’s function on H and the sum is over all pairings of the indices. If two or more of the si are equal, we have ˆ f 1 ) . . . h( ˆ f k )] = O(
− ), E[h( where  is the number of coincidences (i.e. k −  is the number of distinct si ). Proof. We first deal with the case that the si are distinct. Let γ1 , . . . , γk be pairwise disjoint paths of faces from points si on the boundary to the f i . We assume that these paths are far apart from each other (that is, as
→ 0 they converge to disjoint paths). The height h
( f i ) can be measured as a sum along γi . We suppose without loss of generality that each γi is a polygonal path consisting of a bounded number of straight segments which are parallel to the lattice directions x, ˆ yˆ , zˆ . In this case, by additivity of the height change along γi and linearity of the moment in each index, we may as well assume that γi is in a single lattice direction. Now the change in hˆ along γi is given by the sum of ai j − E(ai j ) where ai j is the indicator function of the jth edge crossing γi (with a sign according to the direction of γi ). So the moment is ˆ f 1 ) . . . h( ˆ f k )] = E[h(



E[(a1 j1 − E[a1 j1 ]) . . . (ak jk − E[ak jk ])].

j1 ∈γ1 ,..., jk ∈γk

If wi ji , bi ji are the vertices of edge ai ji , this moment becomes (see [8])    0 K −1 (b2 j , w1 j ) ... K −1 (bk j , w1 j )   1  2 1 k ⎞  −1   K (b1 j , w2 j ) 0 k     1 2 . ⎝ K (wi j , bi j )⎠  . .  . i i . . .   j1 ,..., jk i=1 .   . .   −1 −1   K (b1 j , wk j ) 0 , wk j ) ... K (bk−1 j 1 k k k−1 ⎛

In particular, the effect of subtracting off the mean values of the ai ji is equivalent to cancelling the diagonal terms K −1 (bi ji , wi ji ) in the matrix.

706

R. Kenyon

Expand the determinant as a sum over the symmetric group. For a given permutation σ , which must be fixed-point free or else the term is zero, we expand out the corresponding product, and sum along the paths. For example if σ is the k-cycle σ = (12 . . . k), the corresponding term is   k   sgn(σ ) K (wi ji , bi ji ) K −1 (b1 j1 , w2 j2 ) j1 ,..., jk

×K

= −(

−1 k ) 4πi

⎛



⎝

j1 ,..., jk

−1

i=1

(b2 j2 , w3 j3 ) · · · K −1 (bk jk , w1 j1 ) =

⎞ F(b1 j )F(w2 j ) F(b1 j )F(w2 j ) F(b1 j )F(w2 j ) F(b1 j )F(w2 j ) 2 − 1 2 − 1 2 ⎠... 1 1 2 + φ(b1 j ) − φ(w2 j ) φ(b1 j ) − φ(w2 j ) φ(b1 j ) − φ(w2 j ) φ(b1 j ) − φ(w2 j ) 1 2 1 2 1 2 1 2 ⎞ ⎛ F(bk j )F(w1 j ) F(bk j )F(w1 j ) F(bk j )F(w1 j ) F(bk j )F(w1 j ) 1 1 − 1 ⎠ 1 − k k k k ...⎝ + φ(b1 j ) − φ(w2 j ) φ(bk j ) − φ(w1 j ) φ(bk j ) − φ(w1 j ) φ(bk j ) − φ(w1 j ) 1 2 1 k 1 1 k k

(30)

plus lower-order terms. Multiplying out this product, all 4k terms have an oscillating coefficient (as some ji varies) except for the terms in which the pairs F(bi ji ) and F(wi ji ) in the numerator are either both conjugated or both unconjugated for each i. That is, any term involving F(bi ji )F(wi ji ) or its conjugate will oscillate as ji varies and so contribute negligibly to the sum. There are only 2k terms which survive. Let z i denote the point z i = φ(bi ji ) ≈ φ(wi ji ). For a term with F(bi ji ) and F(wi ji ) both conjugated, the coefficient F(bi ji )F(wi ji ) is equal to dz i , otherwise it is equal to dz i , where dz i is the amount that φ changes when moving by one step along path γi , that is, when ji increases by 1. So the above term for the k-cycle σ becomes ⎛ ⎞   −1 k ⎝ εj⎠ −( ) 4πi
×

ε1 ,...,εk =±1

j




φ(γ1 )

...

(ε )

(ε )

dz 1 1 . . . dz k k

φ(γk )

(z 1(ε1 ) − z 2(ε2 ) )(z 2(ε2 ) − z 3(ε3 ) ) . . . (z k(εk ) − z 1(ε1 ) ) (1)

(−1)

plus an error of lower order, where z j = z j and z j = z¯ j , with similar expressions for other σ . When we now sum over all permutations σ , only the fixed-point free involutions do not cancel: Lemma 7.3 ([2]). For n > 2 let Cn be the set of n-cycles in the symmetric group Sn . Then n   σ ∈Cn i=1

1 = 0, z σ (i) − z σ (i+1)

where the indices are taken cyclically.

Height Fluctuations in the Honeycomb Dimer Model

707

Proof. This is true for n odd by antisymmetry (pair each cycle with its inverse). For n even, the left-hand side is a symmetric rational function whose denominator is the Van dermonde i< j (z i − z j ) and whose numerator is of lower degree than the denominator. Since the denominator is antisymmetric, the numerator must be as well. But the only antisymmetric polynomial of lower degree than the Vandermonde is 0.  By the lemma, in the big determinant all terms cancel except those for which σ is a fixed-point free involution. It remains to evaluate what happens for a single transposition, since a general fixed-point free involution σ will be a disjoint product of these: 1 (4πi)2



φ(s )
φ(s )
φ(s )
φ(s )
φ(s )
φ(s ) φ(s1 )
φ(s2 ) dz dz d z¯ 1 dz 2 dz 1 d z¯ 2 d z¯ 1 d z¯ 2 1 2 1 2 1 2 1 2 − − + 2 2 2 2 φ(s1 ) φ(s2 ) (z 1 − z 2 ) φ(s1 ) φ(s2 ) (¯z 1 − z 2 ) φ(s1 ) φ(s2 ) (z 1 − z¯ 2 ) φ(s1 ) φ(s2 ) (¯z 1 − z¯ 2 )

1 = (4πi)2 =




φ(s1 )
φ(s2 )

φ(s1 )

φ(s2 )

dz 1 dz 2 (z 1 − z 2 )2

1 G(φ(s1 ), φ(s2 )), 2π

where we used φ(si ) ∈ R. Now suppose that some of the si coincide. We choose paths γi as before but √ suppose that the γi are close only at those endpoints where the si coincide. Let δ =
. For pairs of edges ai ji , ai , ji on different paths, both within δ of such an endpoint we use 1 −1 the  bound K (b, w) = O(
), so that the big determinant, multiplied by the prefactor i K (wi ji , bi ji ), is O(1). In the sum over paths the net contribution for each coincidence is then O(δ/
)2 = O(
−1 ), since this is the number of terms in which both edges of two different paths are near the endpoint.  8. Boxed Plane Partition Example For the boxed plane partition, whose hexagon has vertices {(1, 0), (1, 1), (0, 1), (−1, 0), (−1, −1), (0, −1)} in (x, ˆ yˆ ) coordinates, see Fig. 3, it is shown in [12] that satisfies (− x + y)2 = 1 − + 2 , or  1 − 2x y + 4(x 2 − x y + y 2 ) − 3 (x, y) = 2(1 − x 2 ) for (x, y) inside the circle x 2 − x y + y 2 ≤ 3/4. maps the region inside the inscribed circle with degree 2 onto the upper half-plane, with critical point (0, 0) mapping to z−eπi/3 eπi/3 . If we map the half plane to the unit disk with the mapping z → z−e −πi/3 , the composition is √ (x, y) − eπi/3 2r 2 − 3 + 9 − 12r 2 = (31) (x, y) − e−iπ/3 2(y − ωx) ¯ 2 (where r 2 = x 2 − x y + y 2 ) which maps circles concentric about the origin to circles concentric about the origin. To see this, note that z = i(y − ωx) defines the standard

708

R. Kenyon

conformal structure, and |z|2 = x 2 − x y + y 2 = r 2 , so that the right-hand side of Eq. (31) is f (r )/¯z 2 . Note also that the Beltrami differential of , which is µ(x, y) =

z¯ − eπi/3 = , z − e−πi/3

satisfies µ=

µz¯ , µz

that is, it is its own Beltrami differential! This is simply a restatement of the PDE (1) in terms of µ. The diffeomorphism φ from√the region inside the inscribed circle to the unit disk is also very simple, it is just φ = µ, or √ √ 3 − 3 − 4r 2 iθ iθ e . φ(r e ) = 2r The inverse of φ which maps D to U is even simpler: it is √ z 3 −1 φ (z) = . 1 + |z|2 This map can be viewed as the orthogonal projection of a hemisphere onto the plane through its equator, if we identify D conformally with the upper hemisphere sending 0 to the north pole. In conclusion if the domain U is the disk {(x, y) | x 2 −x y + y 2 ≤ r 2 }, where r 2 < 3/4, and the height function on the boundary of U is given by the height function h¯ of the BPP on ∂U , then the h¯ on U will equal the h¯ on BPP restricted to U , and the fluctuations of the height function are the pull-back of the Gaussian free field on the disk of radius √ √ 3− 3−4r 2 under φ. 2r Acknowledgements. Many ideas in this paper were inspired by conversations with Henry Cohn, Jim Propp, Jean-René Geoffroy, Scott Sheffield, Béatrice deTilière, Cédric Boutillier, and Andrei Okounkov. We thank the referees for useful comments. This paper was partially completed while the author was visiting Princeton University and IPAM, and the bulk of the paper was prepared while the author was at the University of British Columbia.

References 1. 2. 3. 4.

Ahlfors, L., Bers, L.: Riemann’s mapping theorem for variable metrics. Ann. Math 72, 385–404 (1960) Boutillier, C.: Thesis, Université Paris-Sud, 2004 Cohn, H., Kenyon, R., Propp, J.: A variational principle for domino tilings. J. AMS 14, 297–346 (2001) Cohn, H., Larsen, M., Propp, J.: The shape of a typical boxed plane partition. New York J. Math 4, 137–165 (1998) 5. Fournier, J.C.: Pavage des figures planes sans trous par des dominos: fondement graphique de l’algorithme de Thurston, parallélisation, unicité et décomposition. C. R. Acad. Sci. Paris Sér. I Math. 320(1), 107–112 (1995) 6. Kasteleyn, P.: Graph theory and crystal physics. In: Proc. 1967 Graph Theory and Theoretical Physics, London: Academic Press, pp. 43–110

Height Fluctuations in the Honeycomb Dimer Model

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7. Kenyon, R.: Local statistics of lattice dimers. Ann. Inst. H. Poincar Probab. Statist. 33(5), 591–618 (1997) 8. Kenyon, R.: Conformal invariance of domino tiling. Ann. Probab. 28, 759–795 (2000) 9. Kenyon, R.: Dominos and the Gaussian free field. Ann. Probab. 29, 1128–1137 (2001) 10. Kenyon, R.: The Laplacian and Dirac operators on critical planar graphs. Invent. Math. 150, 409–439 (2002) 11. Kenyon, R.: Lectures on dimers. PCMI Lecture notes, to appear 12. Kenyon, R., Okounkov, A.: Limit shapes and the complex Burgers equation. Acta. Math. 199(2), 263–302 (2007) 13. Kenyon, R., Okounkov, A., Sheffield, S.: Dimers and amoebae. Ann. of Math. (2) 163(3), 1019–1056 (2006) 14. Kenyon, R., Propp, J., Wilson, D.: Trees and matchings. Electr. J. Combin. 7 (2000), research paper 25 15. Kenyon, R., Sheffield, S.: Dimers, tilings and trees. J. Combin. Theory Ser. B 92(2), 295–317 (2004) 16. Lovasz, L., Plummer, M.: Matching Theory, North-Holland Mathematics Studies, 121 Annals of Discrete Mathematics 29, Amsterdam: North-Holland Publishing Co., 1986 17. Percus, J.: One more technique for the dimer problem. J. Math. Phys. 10, 1881–1888 (1969) 18. Sheffield, S.: Gaussian free fields for mathematicians. Probab. Theory Related Fields 139(3–4), 521–541 (2007) 19. Sheffield, S.: Random surfaces. Astérisque No. 304 (2005) Communicated by H. Spohn

Commun. Math. Phys. 281, 711–751 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0498-1

Communications in

Mathematical Physics

Ergodic Theory of Parabolic Horseshoes Mariusz Urbanski ´ 1,
, Christian Wolf2,

1 Department of Mathematics, University of North Texas, P.O. 311430, Denton,

TX 76203-1430, USA. E-mail: [email protected]

2 Department of Mathematics, Wichita State University, Wichita, KS 67260, USA.

E-mail: [email protected] Received: 10 June 2007 / Accepted: 18 November 2007 Published online: 15 May 2008 – © Springer-Verlag 2008

Abstract: In this paper we develop the ergodic theory for a horseshoe map f which is uniformly hyperbolic, except at one parabolic fixed point ω and possibly also on W s (ω). We call f a parabolic horseshoe map. In order to analyze dynamical and geometric properties of such horseshoes, by making use of induced maps, we establish, in the context of σ -finite measures, an appropriate version of the variational principle for continuous potentials with mild distortion defined on subshifts of finite type. Staying in this setting, we propose a concept of σ -finite equilibrium states (each classical probability equilibrium state is a σ -finite equilibrium state). We then study the unstable pressure function t
→ P(−t log |D f |E u |), the corresponding finite and σ -finite equilibrium states and their associated conditional measures. The main idea is to relate the pressure function to the pressure of an embedded parabolic iterated function system and to apply the developed theory of the symbolic σ -finite thermodynamic formalism. We prove, in particular, an appropriate form of the Bowen-Ruelle-Manning-McCluskey formula, the existence of exactly two σ -finite ergodic conservative equilibrium states for the potential −t u log |D f |E u | (where t u denotes the unstable dimension), one of which is the Dirac δ-measure supported at the parabolic fixed point and the other being nonatomic. We also show that the conditional measures of this non-atomic equilibrium state on unstable manifolds, are equivalent to (finite and positive) packing measures, whereas the Hausdorff measures vanish. As an application of our results we obtain a classification for the existence of a generalized physical measure, as well as a criteria implying the non-existence of an ergodic measure of maximal dimension.


The research of the first author was supported in part by the NSF Grant DMS 0400481.

The research of the second author was supported in part by the National Science Foundation under Grant

No. EPS-0236913 and matching support from the State of Kansas through Kansas Technology Enterprise Corporation.

712

M. Urba´nski, C. Wolf

Contents 1.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Statement of the main results . . . . . . . . . . . . . . . . . . . . . . . 2. Symbol Space and the Shift Map . . . . . . . . . . . . . . . . . . . . . . . 3. Variational Principle and σ -Finite Equilibrium States . . . . . . . . . . . . . 4. One-Dimensional Parabolic Iterated Function Systems . . . . . . . . . . . . 5. Parabolic Smale’s Horseshoes . . . . . . . . . . . . . . . . . . . . . . . . . 6. Horseshoe and the Associated Parabolic Iterated Function System . . . . . . 7. Topological Pressures, Unstable Dimension, Hausdorff and Packing Measures 8. Equilibrium States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Conditional Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Dimension of the Horseshoe . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Generalized Physical Measures . . . . . . . . . . . . . . . . . . . . . . . . 12. Measures of Maximal Dimension . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction 1.1. Motivation. Even mild parabolic features in one dimensional non-invertible dynamics have a profound impact on the dynamical and geometric character of the reference dynamical systems. Our goal in this paper is to understand whether in the case of higher dimensional systems the presence of, even weakly, parabolic points deeply affects the dynamics and, especially the geometry of the corresponding invariant set. We test this issue on one of the simplest, at least in our opinion, examples of parabolic higher dimensional systems. Namely, we introduce the concept of a parabolic horseshoe, and investigate it in detail throughout the paper. Even though parabolic horseshoes can be derived by slightly perturbing a hyperbolic horseshoe at one of its fixed points (see Example 1 in Sect. 5), the phenomena we discover differentiate promptly our parabolic system from hyperbolic ones. The main results are stated below in Subsect. 1.2. In order to perform our analysis of a parabolic horseshoe, we develop an appropriate form of the thermodynamic formalism of σ -finite measures, we borrow from the theory of parabolic iterated function systems and develop the “parabolic” approach to study generalized physical measures and measures of maximal dimension, existing up to our knowledge, so far only in hyperbolic contexts. 1.2. Statement of the main results. Let S ⊂ R2 be a closed topological disk whose boundary is smooth except at finitely many (possibly none) points. Let f : S → R2 be a parabolic horseshoe map of smooth type and let ω ∈ S be its parabolic fixed point (see Sect. 5 for the definition and details). We call the set  = {x ∈ S : f n (x) ∈ S for all n ∈ Z} the parabolic horseshoe of f . Consider the potential φu :  → R defined by φu (x) = log |D f (x)| E xu |. We define the unstable pressure function by P u (t) = P( f | , −tφu ) : R → R, where P( f | , .) denotes the topological pressure with respect to the dynamical system f |. Our first result is the following (Proposition 7.3 and Theorem 7.4) version of a Bowen-Ruelle-Manning-McCluskey type of formula for the unstable dimension of .

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Theorem 1.1. Let f : S → R2 be a parabolic horseshoe map of smooth type. Then the unstable dimension t u = dim H W u (x) ∩  is independent of x ∈ . Moreover, t u is the smallest zero of the unstable pressure function t
→ P u (t) and 0 < t u < 1. Let Ht (A) and Pt (A) denote the t-dimensional Hausdorff respectively packing measure of a set A. In the case of hyperbolic horseshoes and more generally for uniformly u (x) ∩  has positive and finite hyperbolic sets on surfaces it is well-known that Wloc u t -dimensional Hausdorff measure. The next result (see Theorem 7.5 in the text) shows that alone the occurrence of one parabolic fixed point can cause a drastic change on this phenomenon. Theorem 1.2. Let f : S → R2 be a parabolic horseshoe map of smooth type and let u (x) ∩ ) = 0 and 0 < P u (W u (x) ∩ ) < ∞. x ∈ . Then Ht u (Wloc t loc Next, we discuss results concerning the equilibrium states of the potential −t u φu . In particular, we consider finite as well as σ -finite equilibrium states. Let µω denote the Dirac δ-measure supported on the parabolic fixed point ω which is clearly an equilibrium state of the potential −t u φu . We show in Theorem 8.3 that there exists a unique (up to a multiplicative constant) ergodic conservative σ -finite equilibrium state µt u of the potential −t u φu which is distinct from µω . It turns out that the question whether µt u is finite is closely related to the behavior of f near ω. Let β be defined as in Eq. (5.1). Roughly speaking, the exponent β determines the rate at which orbits starting close to ω escape from ω. The following theorem compiles results from Theorems 8.1 and 8.3 in the text. Theorem 1.3. Let f : S → R2 be a parabolic horseshoe map of smooth type. Then the following are equivalent: (i) µt u is finite, in which case P u is not differentiable at t u ; (ii) t u > 2β/(β + 1). Since t u < 1, the measure µt u being finite implies that β < 1. On the other hand, by Eq. (5.1), 1 + β is an upper bound for the maximal possible regularity of f , i.e., f is at most of class C 1+β . Therefore, if f is a C 2 -diffeomorphism then µt u is always an infinite measure. In order to reasonably speak about σ -finite equilibrium states, we develop in Sect. 3 an appropriate form of a thermodynamic formalism: variational principle (Theorem 3.7) and equilibrium states (Definition 3.8) for σ -finite measures on subshifts of finite type (keep in mind that our parabolic horseshoe is topologically conjugate to the full shift on two elements). We now discuss two applications of our results concerning the existence of certain natural invariant measures of f . Recall that an ergodic invariant probability measure µ is called a generalized physical measure if its basin B(µ) has the same Hausdorff dimension as the stable set of  (see Sect. 11 and [Wo] for more details). Applying Theorem 1.3 we are able to prove that the finiteness of the equilibrium state µt u is equivalent to the existence of a generalized physical measure (see Theorem 11.2). In particular, there are parabolic horseshoes having no generalized physical measure (see Corollary 11.3). This contrasts the case of hyperbolic surface diffeomorphisms which always have a unique generalized physical measure (see [Wo]). Another application concerns the existence of ergodic measures of maximal dimension. Given an invariant probability measure µ, we denote by dim H µ the Hausdorff

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dimension of µ (see (12.1) for the definition). Assume now that µ is ergodic. Following [BW1] we say that µ is an ergodic measure of maximal dimension if dim H µ = sup dim H ν, ν

where the supremum is taken over all ergodic invariant probability measures ν. These measures have been intensively studied in the context of hyperbolic diffeomorphism in [BW1]. It turns out that for hyperbolic sets on surfaces there always exists an ergodic measure of maximal dimension. Moreover, this measure is, in general, not unique (see the example in [Ra]). In contrast to these results, we show in Theorem 12.4 that for certain parabolic horseshoe maps there exists no ergodic measure of maximal dimension. 2. Symbol Space and the Shift Map In this section we recall some notions from symbolic dynamics. We will discuss simultaneously one-sided and two-sided shift maps. We denote by Z the set of all integers and by N = {0, 1, 2, 3, . . .} the set of all non-negative integers. Given a countable, either finite or infinite set E and a function A : E × E → {0, 1}, called and incidence matrix, we define
 E +A = (ωn )+∞ ∈ E N : Aωn ωn+1 = 1 ∀ n ∈ N and 0
 +∞ Z E +− = (ω ) ∈ E : A = 1 ∀ n ∈ Z . n ω ω n n+1 −∞ A We refer to either of these sets as a shift or a symbol space. In the case when we do not want to specify the shift space or also when it is clear from the context which shift space is meant, we write E A instead of E +A or E +− A . Given any s ∈ (0, 1), the space E A can be endowed with the metric ρ = ρs defined by   ρ (ωn )n , (τn )n = s min{n≥0: ωn =τn or ω−n =τ−n } . Here we use the common convention that s +∞ = 0. All the metrics ρs , s ∈ (0, 1) are Hölder continuously equivalent and induce (the same) Tychonoff topology on E A . It is well-known that E A endowed with the product (Tychonov) topology is compact in the case when E is a finite set. The (left) shift map σ : E A → E A is defined by the formula   σ (xn )n ) = (xn+1 )n . Occasionally, in order to avoid confusion, we write σ+ for σ : E +A → E +A and σ+− for +− +− σ : E +− A → E A . Note that in the case of E A , the shift map is injective and in the case of E +A , the shift map performs cutting off the zeroth coordinate. Let m ≤ n and let ω = (ωm , ωm+1 , · · · , ωn ) ∈ E n−m+1 . We call [ω] = {τ ∈ E A : τ j = ω j for all m ≤ j ≤ n}, the cylinder generated by ω. If ω ∈ E A and m ≤ n, we define ω|nm = (ωm , ωm+1 , . . . , ωn ), and, if m = 0, we frequently write ω|n instead of ω|n0 . For every n ≥ 1 an n-tuple τ of elements in E is said to be A-admissible provided that Aab = 1 for all pairs

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of consecutive elements ab in τ . The number n is then called the length of τ and is denoted by |τ |. We denote by E nA the set of A-admissible tuples of length n. We also put  +− +− n + + E ∗A = ∞ n=1 E A . Denote by  : E A → E A the projection from E A to E A defined by   +∞  (ωn )+∞ −∞ = (ωn )0 . If D ⊂ E then we write D A for D A| D×D . Let g : E A → R be a function. Given an integer n ∈ N we define the n th partition function Z n (D, g) by      Z n (D, g) = exp sup Sn g(τ ) , ω∈D nA

τ ∈[ω]∩D A

where Sn g :=

n−1 

g ◦ σ j.

j=0

In the case when D = E, then we simply write Z n (g) instead of Z n (E, g). A straightforward argument shows that the sequence (log Z n (D, f ))n∈N is subadditive. Therefore, we can define the topological pressure of g with respect to the shift map σ : D A → D A by

1 1 log Z n (D, g) : n ∈ N . (2.1) PD (g) = lim log Z n (D, g) = inf n→∞ n n If D = E, then we simply write P(g) instead of PE (g). Recall from [MU2] that a function g : E +A → R is said to be acceptable if it is uniformly continuous and     sup sup g|[e] − inf g|[e] ) : e ∈ E < ∞. The following fact, which will be needed in the next section, was proven in [MU2]. Theorem 2.1. If g : E +A → R is acceptable, then



P(g) = sup hµ (σ ) + gdµ = sup{PD (g)}, where the first supremum is taken over all Borel probability (ergodic) σ -invariant measures µ on E such that the function −g is µ-integrable and the second supremum is taken over all finite subsets D of E. In the case when E = {0, 1, 2, . . . , d − 1}, where d ≥ 2, and the incidence matrix A consists of 1s only, we rather use the notation d+ := {0, 1, 2, . . . , d − 1}+A = {0, 1, 2, . . . , d − 1}N and Z d+− := {0, 1, 2, . . . , d − 1}+− A = {0, 1, 2, . . . , d − 1} .

Similarly as above, in the case when we do not want to specify the shift space or also when it is clear from the context which shift space is meant, we write d instead of d+ or d+− .

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3. Variational Principle and σ -Finite Equilibrium States Let (X, A, µ) be a measure space, where µ is a σ -finite measure. Moreover, let T : X → X be a measurable map which is µ-invariant (that is, µ ◦ T −1 = µ), ergodic and conservative (see [A] for the definitions). Consider a fixed set F ∈ A with µ(F) > 0. Then the first return time τ := τ F : F → {1, 2, . . .} ∪ {∞} given by the formula τ (x) = min{k ≥ 1 : T k (x) ∈ F} is finite in the complement (in F) of a set of measure zero. Therefore, the first return map TF : F → F given by the formula TF (x) = T τ F (x) (x) is well-defined µ-a.e. on F. If φ : X → R is a measurable function, we set φ F (x) =

τ (x)−1 

φ ◦ T j (x), x ∈ F.

j=0

The function φ F is A F -measurable, where A F is the σ -algebra of all subsets of F that belong to A. If in addition µ(F) < ∞, then it is well-known (see [A] for example) that the conditional measure µ F on F defined by µ F (B) = µ(B)/µ(F), B ∈ A F , is TF -invariant. Also if φ is µ-integrable, that is |φ|dµ < +∞, then φ F is µ F -integrable and 

φdµ . (3.1) φ F dµ F = µ(F) F However, the converse is not in general true, i.e. µ F integrability of φ F does not imply µ-integrability of φ, even if the measure µ is finite. We now shall provide a short proof of the following, essentially immediate, consequence of Abramov’s formula, which is due to Krengel ([K]). Theorem 3.1. Let µ be a σ -finite measure on X , and let T : X → X be an ergodic, conservative and µ-invariant map. If E and F are measurable sets with 0 < µ(E), µ(F) < ∞, then µ(E)hµ E (TE ) = µ(F)hµ F (TF ). Proof. Put D = E ∪ F. Obviously 0 < µ(D) < +∞. Let TED and TFD be the first return maps respectively to E and F induced by the map TD : D → D. It follows from Abramov’s formula that µ D (E)h(µ D ) E (TED ) = hµ D (TD ) = µ D (F)h(µ D ) F (TFD ).

(3.2)

Since µ D (E) = µ(E)/µ(D), µ D (F) = µ(F)/µ(D), TE = TED , TF = TFD (as E, F ⊂ D), and (µ D ) E = µ E , (µ D ) F = µ F , we may conclude that µ(E)hµ E (TE ) = µ(F)hµ F (TF ).   Following Krengel we denote this common value µ(F)hµ F (TF ) by hµ (T ) and call it the entropy of the transformation T with respect to the invariant measure µ. Assume now that T : X → X is a continuous self-map of a compact metric space X . Given a Borel

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measure µ on X let X µ (∞) be the set of all points x ∈ X such that µ(U ) = +∞ for every open set U containing x. Following [U3] we call X µ (∞) the set of points of infinite condensation of µ. Notice that X µ (∞) is a closed subset of X and X µ (∞) = ∅ if and only if the measure µ is finite. Denote by M∞ T the family of all Borel ergodic conservative T invariant measures µ for which µ(X \ X µ (∞)) > 0. Note that if µ ∈ M∞ T , then X µ (∞) is forward invariant, i.e. T (X µ (∞)) ⊂ X µ (∞). The following simple proposition shows that X µ (∞) is measurably negligable and, in the transitive case, it is also topologically negligable. Proposition 3.2. If T : X → X is a continuous self-map of a compact metric space X and µ ∈ M∞ T , then µ(X µ (∞)) = 0. If, in addition, T : X → X is topologically transitive, then X µ (∞) is a nowhere dense subset of X . Proof. Since T(X µ (∞))  since, by ergodicity and conservativity of µ,  ⊂ nX µ (∞) and the measure µ X \ ∞ T (X (∞)) = 0 whenever µ(X µ (∞)) = 0, the condiµ n=0 tion µ(X \ X µ (∞)) > 0 implies that µ(X µ (∞)) = 0. Suppose now that T is topologically transitive and assume X µ (∞) is not nowhere dense. Then it follows that ∞ ∞ n n n=0 T (X µ (∞)) is a dense subset of X . Since however n=0 T (X µ (∞)) ⊂ X µ (∞) and since X \ X µ (∞) = ∅ (as µ(X \ X µ (∞)) > 0), we get a contradiction which completes the proof.   Proposition 3.3. If T : X → X is a continuous self-map of a compact metric space X and µ ∈ M∞ T , then µ is σ -finite. Proof. By the definition of X µ (∞), for every point x ∈ X \ X µ (∞) there exists an open ball Bx ⊂ X \ X µ (∞) such that x ∈ Bx and µ(Bx ) < +∞. Since  X \ X µ (∞) is a separable metric space, there exists a countable set Y ⊂ X such that y∈Y B y = X \ X µ (∞). Thus {X µ (∞)} ∪ {B y } y∈Y forms a countable cover of X by Borel sets of finite measure (µ(X µ (∞)) = 0) by Proposition 3.2.   Note that if F is a closed (compact) subset of X \ X µ (∞), then µ(F) < +∞. Motivated by the remark after formula (3.1), we say that a Borel function  φ : X → R is dynamically integrable with respect to µ ∈ M∞ T on X provided that F |φ F |dµ F < +∞ for every closed set F ⊂ X \ X µ (∞) with µ(F) > 0 (closed sets can be equivalently repla ced by those whose closure is disjoint from X µ (∞)). Then the number µ(F) F φ F dµ F is independent of F, its common value is denoted by

ˆ

φdµ,

and it is referred to as the dynamical integral of φ against the measure µ. Thus, we have

ˆ

φ F dµ F

φdµ = µ(F)

(3.3)

F

whenever F is a closed subset of X \ X µ (∞) with positive measure µ. Of course, each integrable function is dynamically integrable, but, as has been said, not conversly. Keep T : X → X as a continuous self-map of a compact metric space X . A closed forward invariant subset Q of X is called ergodically finite provided T | Q : Q → Q admits no conservative ergodic infinite σ -finite invariant measure. Fix a continuous

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function φ : X → R. Then it is a consequence of the well-known variational principle that



(3.4) sup hµ (T ) + (φ − P(φ))dµ = 0, where P(φ) denotes the topological pressure of the potential φ, and the supremum is taken over all ergodic T -invariant Borel probability measures on X . Denote by M∞ φ the family of all measures µ ∈ M∞ for which the function φ − P(φ) is dynamically T integrable with respect to the measure µ. The potential φ : X → R is called VPA (Variational Principle Admissible) provided that  

ˆ ∞ sup hµ (T ) + (φ − P(φ))dµ : µ ∈ Mφ = 0. (3.5) One of the most common ergodic theory constructions is the concept of Rokhlin’s natural extension. The next lemma and the corollaries following establish close canonical relations between the classes M∞ and M∞ as well as M∞ and M∞ , where T˜ : X˜ → X˜ T

φ

T˜

φ◦

is Rokhlin’s natural extension of T : X → X and  : X˜ → X is the projection onto the 0th coordinate. It will be the basic tool in Sect. 7 to establish links between parabolic horseshoes and parabolic iterated function systems. Lemma 3.4. Suppose that T : X → X is a continuous map of a compact metric space X . Let T˜ : X˜ → X˜ be Rokhlin’s natural extension of T . Then the map µ
→ µ ◦ −1 is ∞ a bijection between M∞ ˜ and MT . In addition, T

hµ (T˜ ) = hµ◦−1 (T ),

(3.6)

, the function φ and, if φ : X → R is a Borel function, then for every measure µ ∈ M∞ T˜ is dynamically integrable against µ ◦ −1 if and only if φ ◦  is dynamically integrable against the measure µ. Furthermore, if one of these integrabilities holds, then

ˆ

ˆ φ ◦ dµ = φdµ ◦ −1 . X˜

X

Proof. Recall that X˜ = {(xn )+∞ n=0 : T (x n+1 ) = x n for all n ≥ 0}, and +∞ T˜ ((xn )+∞ n=0 ) = (T (x n ))n=0 .

Recall that T˜ : X˜ → X˜ is a homeomorphism and its inverse is the left shift map (cutting off the 0th coordinate). For every n ≥ 0 let n : X˜ → X be the projection onto the n th coordinate. We put  := 0 . Fix µ ∈ M∞ . Since T˜   lim dist T˜ n (−1 (( X˜ µ (∞)))), X˜ µ (∞) = 0, n→∞

  it follows from conservativity of the measure µ that µ −1 (( X˜ µ (∞))) \ X µ (∞) = 0. Therefore,

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µ ◦ −1 (( X˜ µ (∞))) = µ(−1 (( X˜ µ (∞)))) = µ(( X˜ µ (∞) ∪ (−1 (( X˜ µ (∞))) \ X˜ µ (∞))) = µ(( X˜ µ (∞)) + µ(−1 (( X˜ µ (∞))) \ X˜ µ (∞)) = 0 + 0 = 0. (3.7) Now, if ω ∈ X \ ( X˜ µ (∞)), then there exists an open neighbourhood G ⊂ X of ω such that G ∩ ( X˜ µ (∞)) = ∅. Then −1 (G) ∩ X˜ µ (∞) = ∅, and, as −1 (G) is compact, µ(−1 (G)) < +∞. Hence, µ ◦ −1 (G) < ∞, so ω ∈ / X µ◦−1 (∞) and X µ◦−1 (∞) ⊂ ( X˜ µ (∞)), and, by virtue of (3.7), µ ◦ −1 (X µ◦−1 (∞)) = 0. On the other hand, if µ is T˜ -invariant and µ ◦ −1 ∈ M∞ T , then X˜ µ (∞) ⊂ −1 (X µ◦−1 (∞)),

(3.8)

µ( X˜ µ (∞)) ≤ µ ◦ −1 (X µ◦−1 (∞)) = 0.

(3.9)

( X˜ µ (∞)) = X µ◦−1 (∞).

(3.10)

and (therefore),

We also claim that

Indeed, the inclusion ( X˜ µ (∞)) ⊂ X µ◦−1 (∞) follows directly from (3.8). Suppose now for a contradiction that there exists a point x ∈ X µ◦−1 (∞)\( X˜ µ (∞)). Then there exists a closed neighborhood U of x such that U ∩ ( X˜ µ (∞)) = ∅. But then −1 (U ) is a compact set disjoint from X˜ µ (∞)). Hence µ ◦ −1 (U ) = µ(−1 (U )) < +∞, contrary to the assumption that x ∈ X µ◦−1 (∞). Thus, identity (3.10) is proven. Now, it is obvious that if µ ∈ M∞ (it suffices to know that µ is shift-invariant and T˜ ergodic conservative), then µ ◦ −1 is ergodic conservative. On the other hand, suppose that µ is a Borel T˜ -invariant measure on X˜ such that µ ◦ −1 ∈ M∞ T . We want to show that µ is ergodic conservative. Indeed, if µ has atoms, then µ ◦ −1 has them as well (images of atoms of µ under ), and since µ◦−1 is ergodic, the measure µ◦−1 must be finite and supported on a single periodic orbit of T . Therefore, µ must be also finite and supported on a single periodic orbit of T˜ . Thus, µ is ergodic conservative. If however µ has no atoms, then in virtue of Proposition 1.2.1 from [A] (saying that for invertible nonsingular transformations with respect to a non-atomic measure, ergodicity implies conservativity), it suffices to show that µ is ergodic. Since µ ◦ −1 ∈ M∞ T , there exists a compact set F ⊂ X \ X µ◦−1 (∞) such that µ◦−1 (F) > 0 (then also µ(F) < +∞). By Proposition 1.5.2(1) in [A], the first return map TF : F → F is ergodic with respect to the conditional measure µ◦−1 | F on F. Using the uniqueness property of Rokhlin’s natural extension, it is easy to see that the authomorphism T˜−1 (F) : −1 (F) → −1 (F) with the invariant measure µ−1 (F) is canonically isomorphic to Rokhlin’s natural extension T˜F : F˜ → F˜ of TF : F → F. But, by ergodicity of the latter, the former map (T˜F : F˜ →

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˜ is also ergodic (see [KFS], Theorem 1 in Sect. 10.4). Hence, T˜−1 (F) : −1 (F) → F) −1 is ergodic conservative for T : X → X , −1 (F) is ergodic. Since the  measure µ ◦    ∞ −1 −n ˜ −n (π −1 (F))) = 0. At this point X \ n=0 T (F) = 0. Hence, µ X˜ \ ∞ µ◦ n=0 T Proposition 1.5.2(2) in [A] applies, to yield that the map T˜ : X˜ → X˜ is ergodic with respect to the measure µ. Since µ is non-atomic, it now follows from Proposition 1.2.1 in [A] that µ is conservative. So, µ is ergodic conservative, and by using (3.7), we may conclude that µ ∈ M∞ . The above considerations with the set F provide also the T˜ −1 = ν ◦ −1 , injectivity of the map µ → µ ◦ −1 . Indeed, if µ, ν ∈ M∞ ˜ and µ ◦  T

−1 then µ◦−1 F = ν ◦ F . So, from injectivity of Rokhlin’s natural extension construction on probability measures, and because of the canonical isomorphism established above, we conclude that µ−1 (F) = ν−1 (F) . Thus, using also the fact that µ(−1 (F)) = ν(−1 (F)), we may conclude that µ = ν. Summarizing what we have done so far, we have shown that the map µ → µ ◦ −1 is an injection from M∞ to M∞ T , and that in T˜ order to prove this map to be bijective, it suffices to demonstrate that if m ∈ M∞ T , then −1 ˜ ˜ there exists a Borel T -invariant measure µ on X such that m = µ ◦  . And indeed, fix again a compact set F ⊂ X \ X µ◦−1 (∞) such that µ ◦ −1 (F) > 0 (then also µ(F) < +∞). Since the homeomorphism T˜−1 (F) : −1 (F) → −1 (F) is canonically isomorphic (topologically conjugate) with (topological) natural extension T˜F : F˜ → F˜ of TF : F → F, there exists a finite Borel T˜−1 (F) -invariant measure m˜F on −1 (F) such that m F = m˜F ◦ −1 . Then there exists a finite Borel T˜ -invariant measure µ on X˜ such that µ−1 (F) = m˜F . Hence, (µ◦−1 ) F = µ−1 (F) ◦−1 = m˜F ◦−1 = m F . So, m = µ ◦ −1 , and bijectivity of the map ν → ν ◦ −1 from M∞ to M∞ T is established. T˜ In order to prove the entropy formula (3.6), we first notice that Rokhlin’s natural extension of probability invariant measures preserves entropies. Fix now µ ∈ M∞ and T˜ −1 a Borel set F ⊂ X such that 0 < µ ◦  (F) < +∞. Then

hµ◦−1 (T ) = µ ◦ −1 (F)h(µ◦−1 ) F (TF ) = µ(−1 (F))hµ−1 (F) (T˜−1 (F) ) = hµ (T˜ ). Passing to the proof of the last claim of our lemma, fix µ ∈ M∞ and a Borel function T˜ φ : X → R. Suppose that φ and φ ◦  are dynamically integrable respectively against measures µ ◦ −1 and µ. Fix a compact set F ⊂ X \ X µ (∞) such that µ ◦ −1 (F) > 0. Then

ˆ

φdµ ◦ −1 = µ ◦ −1 (F) φ F d(µ ◦ −1 ) F X

F = µ(−1 (F)) φ F dµ−1 (F) ◦ −1

F −1 = µ( (F)) φ F ◦ dµ−1 (F) −1 (F)

= µ(−1 (F)) (φ F ◦ )−1 (F) dµ−1 (F) =

−1 (F)

ˆ X˜

φ ◦ dµ.

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Since dynamical integrability of φ ◦  against µ clearly implies dynamical integrability of φ against µ ◦ −1 , we are therefore left to show that dynamical integrability of φ against µ ◦ −1 implies dynamical integrability of φ ◦  against µ. In order to do this, fix a compact set F ⊂ X˜ \ X˜ µ (∞) such that µ(F) > 0. Since the sets of the form −1 n (G), where G is an open subset of X and n is an integer ≥ 0, form a basis of topology for X˜ , there exist finitely many integers 0 ≤ n 1 < n 2 < . . . < n k and open sets G 1 , G 2 , . . . , G k ⊂ X such that F⊂

k 

−1 −1 ˜ ˜ −1 n j (G j ) and n 1 (G 1 ) ∪ . . . ∪ n k (G k ) ⊂ X \ X µ (∞).

(3.11)

j=1

Since n j = T n k −n j ◦ n k for all j = 1, 2, . . . , k, we get that F ⊂ −1 n k (), where  =

k j=1

(3.12)

T −(n k −n j ) (G j ). Set l = n k . By (3.12), we get T˜ −l (F) ⊂ (l ◦ T˜ l )−1 () = −1 ()

(3.13)

and, l−1 () =

k 

k k     l−1 T −(n k −n j ) (G j ) = (T (n k −n j ) ◦ l )−1 (G j ) = −1 n j (G j ).

j=1

j=1

j=1

Hence, −1 () ∩ X˜ µ = (T l ◦ l )−1 () ∩ T˜ −l ( X˜ µ ) = (l ◦ T˜ l )−1 () ∩ T˜ −l ( X˜ µ ) = T˜ −l (−1 ()) ∩ T˜ −l ( X˜ µ ) = T˜ −l (−1 ()) ∩ X˜ µ ) l

l

⊂ T˜ −l (∅) = ∅. Thus, using also (3.10), we get  ∩ X µ◦−1 =  ◦ ( X˜ µ ) = ∅. Since  is compact and φ is dynamically integrable against µ ◦ −1 , we conclude that φ is integrable against (µ ◦ −1 ) , or equivalently, φ ◦ −1 () is integrable against µ−1 () . We are now going to show that φT˜ l (−1 ()) is integrable against µT˜ l (−1 ()) . Indeed, since T˜ ˜ l −1 ◦ T˜ l = T˜ l ◦ T˜−1 () and since the measure µ is T˜ l T (

())

invariant, the map T˜ l : −1 () → T˜ l (−1 iso   ()) establishes a measure-preserving morphism between the dynamical systems T˜−1 () ,−1 (),µ−1 () and T˜T˜ l (−1 ()) ,

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 T˜ l (−1 ()), µT˜ l (−1 ()) . Therefore,

T˜ l (−1 ())

|φ ◦ T˜ l (−1 ()) |dµT˜ l (−1 ())

= |φ ◦ T˜ l (−1 ()) |dµ−1 () ◦ T˜ −l l −1 ˜ T ( ())

= |φ ◦ T˜ l (−1 ()) ◦ T˜ l |dµ−1 () −1  ()

= |(φ ◦  ◦ T˜ l )−1 () |dµ−1 () −1 ()

=

−1 ()

−

l−1 

τ−1 () (x)+l−1  τ−1 () (x)−1    ˜ j (x)) + φ ◦ ( T φ ◦ (T˜ j (x))  j=0

  j ˜ φ ◦ (T (x))dµ−1 ()

j=τ−1 () (x)

j=0

  |φ ◦ )−1 () | + 2l||φ||∞ dµ−1 ()

= 2l||φ||∞ + |φ ◦ )−1 () |dµ−1 () < +∞. ≤

−1 ()

−1 ()

Thus φ ◦ T˜ l (−1 ()) is integrable against µT˜ l (−1 ()) . By (3.13), F ⊂ T˜ l (−1 ()),   and therefore, (φ ◦ ) F = (φ ◦ )T˜ l (−1 ()) F is integrable against µ F . We are done.   We will need the following. Lemma 3.5. Suppose that T : X → X is a continuous self-map of a compact metric space X . Suppose φ, ψ : X → R are two continuous functions cohomologous in the class of continuous functions. Then for every µ ∈ M∞ T , the function ψ is dynamically ˆ  integrable against µ if and only if φ is, and ψdµ = ˆ φdµ. Furthermore, ψ satisfies the generalized variational principle if and only if φ does, and E S ∞ (ψ) = E S ∞ (φ), the sets of all (σ -finite) equilibrium states of ψ and φ respectively. Proof. By our hypothesis there exists a continuous function u : X → R such that ψ − φ = u − u ◦ T . Fix F, a compact subset of X \ X µ (∞). Then ψ F − φ F = u − u ◦ TF . Since  u is bounded, ψ F is thus integrable against µ F if and only if φ F is, and ψ dµ = φ F dµ f . Therefore ψ is dynamically integrable against µ if and only F f F ˆ F  if φ is, and ψdµ = ˆ φdµ. The rest of the lemma follows now from the observation that P(ψ) = P(φ) and (ψ − P(ψ)) − (φ − P(φ)) = u − u ◦ T .   We remark that this lemma fails if dynamical integrability against µ is replaced by (sheer) integrability against µ. In order to prove the main results of this section, a sufficient condition for a potential to be VPA, our function φ : X → R needs to satisfy some further conditions. Let d be

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723

a metric on X compatible with the topology of X . Given n ≥ 1 the metric dn on X is defined as follows: dn (x, y) = max{d(T j (x), T j (y)) : 0 ≤ j ≤ n − 1}. Each metric dn is equivalent to d (they induce the same topology) and the set Bn (x, δ) = Bdn (x, δ), an open ball with respect to the metric dn , is called the Bowen ball centered at x with radius δ. Considering again a function φ : X → R, we need the following definition. Definition 3.6. A continuous function φ : X → R is said to be of mild distortion provided that there exists an ergodically finite set Q ⊂ X such that ∀ε > 0 ∃δ > 0 s.th. ∀x ∈ X ∀n ≥ 1 with T n (x) ∈ / B(Q, ε) we have |Sn φ(y) − Sn φ(x)| ≤ ε

(3.14)

whenever y ∈ Bn (x, δ). If we need to be more specific about the set Q, we say that the function φ is of mild distortion modulo Q. In the context of subshifts of finite type and potentials of mild distortion we obtain the following stronger version of the variational principle (3.4) which now also takes σ -finite measures into account. Theorem 3.7. Suppose that E = ∅ is a finite set and let A : E × E → {0, 1} be an irreducible incidence matrix. If φ : E +A → R is a continuous function with mild distortion, then φ is VPA, i.e.  

ˆ ∞ sup hµ (σ ) + (φ − P(φ))dµ : µ ∈ Mφ = 0. Proof. In view of (3.3), (3.4) and Theorem 3.1 it is sufficient to demonstrate that

hµ F (σ F ) + (φ − P(φ)) F dµ F ≤ 0 (3.15) F

+ + for every infinite measure µ ∈ M∞ φ and some set F ⊂ F ⊂ E A \ E A,µ (∞) (depending on µ) with µ(F) > 0. Consider such an infinite measure µ ∈ M∞ φ . It follows from + Proposition 3.2 and the ergodic finiteness of Q that µ(E A,µ (∞) ∪ Q) = 0. Hence, there exists an A-admissible finite word ω such that [ω] ⊂ E A \ (E +A,µ (∞) ∪ Q) and µ([ω]) ∈ (0, ∞). Let F be the set of all those ρ ∈ E A that return to [ω] infinitely often under the forward iteration of the shift map σ : E +A → E +A . Since µ is conservative it follows that µ(F) = µ([ω]) ∈ (0, ∞). Let

E ω = {ρ|τ F (ρ)−1 : ρ ∈ F}. Obviously E ω is a countable set (as a subset of E ∗A ). Define an incidence matrix Aω : E ω × E ω → {0, 1} by declaring that Aω (α, β) = 1 if and only if αβ ∈ E ∗A . Clearly Aω is irreducible and F coincides with the set E ω∞ of all infinite concatenations of elements from E ω admissible by Aω . Furthermore, the map I : F → E ω∞ which ascribes to each element ρ ∈ F its unique representation as an infinite word over the alphabet E ω , is a

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homeomorphism, and I establishes topological conjugacy between σ F : F → F and the full shift map σ : E ω∞ → E ω∞ . That is, the following diagram commutes, F ⏐ ⏐ I E ω∞

σF −→ σ −→

F ⏐ ⏐ I E ω∞

or equivalently I ◦ σ F = σ ◦ I . For every n ≥ 1 let Dn be the (finite) set of all those elements in E ω whose length (when treated as elements in E ∞ A ) is bounded above by n. Then I −1 (Dn∞ ) is a compact subset of E ∞ , the time of the first return from I −1 (Dn∞ ) to A −1 ∞ −1 ∞ I (Dn ) of every element x ∈ I (Dn ) is equal to τ F (x) ≤ n, and the first return map σn : I −1 (Dn∞ ) → I −1 (Dn∞ ) is continuous. In what follows, subtracting P(φ) from φ, we may assume without loss of generality that P(φ) = 0 and we may treat, via the conjugacy I , the function ψ = φ F as defined on the shift space E ω∞ . Since [ω] ∩ Q = ∅ and both Q and [ω] are compact sets, we have ε = dist(Q, [ω]) > 0. Let δ > 0 be associated to ε according to Definition 3.6. Replacing [ω] by its sufficiently long subcylinder, we may assume without loss of generality that diam([ω]) < δ. Then σ |ω| ([ρ|τ F (ρ)−1 ω]) ⊂ Bτ F (ρ)−|ω| (σ |ω| (ρ), δ), and it therefore follows from Definition 3.6 (note also that σ τ F (ρ)−|ω| (σ |ω| (ρ)) = σ τ F (ρ) (ρ) ∈ / Q) that the function φ F is acceptable. Thus, in view of Theorem 2.1 and the classical variational principle, for every θ > 0 there exists n ≥ 1 and a Borel probability σ F -invariant measure m supported on I −1 (Dn ) such that

hµ F (σ F ) + (φ) F dµ F ≤ hm (σ F ) + φ F dm + θ. (3.16) F

F

Since m is supported on I −1 (Dn ) (τ F ≤ n!), there exists a Borel  probability σ -invariant measure mˆ on E +A such that mˆ F = m. Hence, by (3.4), hmˆ (σ )+ E + φd mˆ ≤ 0. Therefore, A  by (3.1) and Theorem 3.1, we get that hm (σ F ) + F φdm ≤ 0. Inserting this-into (3.16)  gives hµ F (σ F )+ F (φ) F dµ F ≤ θ . Letting finally θ  0 yields hµ F (σ F )+ F (φ) F dµ F ≤ 0. We are done.   It is clear that every Hölder continuous potential has mild distortion with respect to Q = ∅. In the forthcoming sections we will provide further classes of potentials with mild distortion. Passing to σ -finite equilibrium states, we start with the following. Definition 3.8. Let T : X → X be a continuous self-map of a compact metric space and let φ : X → R be a continuous VPA potential. A σ -finite measure µ ∈ M∞ φ is said to be an equilibrium state of φ provided that

ˆ (φ − P(φ))dµ = 0. hµ (T ) + X

The space of all (σ -finite) equilibrium states of φ is denoted by E S ∞ (φ).

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725

+− Since the two-sided shift map σ+− : E +− A → E A is canonically topologically conjugate to Rokhlin’s natural extension of the one-side shift σ+ : E +A → E +A , we now can easily prove the following.

Proposition 3.9. If ψ : E +A → R has mild distortion and φ : E +− A → R is a continuous function such that ψ ◦  and φ are cohomologous in the class of continuous functions, −1 is a bijection between the sets of then φ : E +− A → R is VPA, the map µ → µ ◦  equilibrium states of φ and ψ respectively, P(φ) = P(ψ), and

ˆ

ˆ hµ (σ+− ) = hµ◦−1 (σ+ ) and φdµ = ψdµ ◦ −1 . E +− A

E +A

Proof. It follows from Lemma 3.4 and the classical Variational principle that P(φ) = P(ψ ◦ ) = P(ψ). Our proposition is then an immediate consequence of Theorem 3.7, Lemma 3.4 and Lemma 3.5.   4. One-Dimensional Parabolic Iterated Function Systems The concept of a parabolic Cantor set was introduced in [U2]. The concept of a parabolic iterated function system was formally introduced in [MU1]. Both concepts largely overlap and the object dealt with in this and subsequent sections belongs to this overlap. We briefly summarize here the definition of a parabolic iterated function system following [MU1] and partially adopting it to the much more concrete setting we will need in the sequel for our applications. Let  be a compact line segment. Suppose that we have at least two and at most finitely many C 1+ε maps φi :  → , i ∈ I , satisfying the following conditions: (1) Open Set Condition: φi (int()) ∩ φ j (int()) = ∅ for all i = j. (2) |φi (x)| < 1 everywhere except for finitely many pairs (i, xi ), i ∈ I , for which xi is the unique fixed point of φi and |φi (xi )| = 1. Such pairs and indices i will be called parabolic and the set of parabolic indices will be denoted by . All other indices will be called hyperbolic. (3) ∀n ≥ 1 ∀ω = (ω1 , ω2 , . . . , ωn ) ∈ I n if ωn is a hyperbolic index or ωn−1 = ωn , then φω = φω1 ◦ φω2 . . . φωn extends in a C 1+ε manner to an  open line segment V and maps V into itself. (4) If i is a parabolic index, then n≥0 φi n () = {xi } and so the diameters of the sets φi n () converge to 0. Here i n denotes the n-tuple whose entries are all i. (5) ∃s < 1 ∀n ≥ 1 ∀ω ∈ I n if ωn is a hyperbolic index or ωn−1 = ωn , then ||φω || ≤ s. (6) Bounded Distortion Property: ∃K ≥ 1 ∀n ≥ 1 ∀ω = (ω1 , . . . , ωn ) ∈ I n ∀x, y ∈ V if ωn is a hyperbolic index or ωn−1 = ωn , then |φω (y)| ≤ K. |φω (x)| We call such a system of maps F = {φi : i ∈ I } a 1-dimensional subparabolic iterated function system. If  = ∅, we call the system F = {φi : i ∈ I } a 1-dimensional parabolic iterated function system. From now on throughout the entire section we assume the system F to be parabolic.

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The Bounded Distortion Property (6) is not obvious in the 1-dimensional case. But there is a natural easily verifiable sufficient condition for this property to hold. Indeed, it was proved in [U1] and [U2] that condition (6) (Bounded Distortion Property) follows from all conditions (1)-(5) enlarged by the requirement that if i is a parabolic element, then the map φi has the following representation in a neighborhood of the the parabolic point xi : φi (x) = x ± a|x − xi |βi +1 + o(|x − xi |βi +1 )

(4.1)

with some βi > 0 and with the sign “−” if x − xi ≥ 0 and the sign “+” if x − xi ≤ 0. It was also proven in [U1] and [U2] that for every parabolic element i and every n ≥ 0, |φin (x)|  (n + 1)

−

βi +1 βi

and |φin (x) − xi |  (n + 1)

− β1

i

outside every fixed open neighborhood of xi . In particular |φin j (x)|  (n + 1)

−

βi +1 βi

and |φin j (x) − xi |  (n + 1)

− β1

i

for every parabolic element i and every j ∈ I \{i} and all n ≥ 0. Recall that the elements of the set I \  are called hyperbolic (see Condition 2). We extend this name to all the words appearing in Conditions (5) and (6). By I ∗ we denote the set of all finite words with alphabet I and by I ∞ all infinite sequences with elements in I . It follows from (3) that for every hyperbolic word ω, φω (V ) ⊂ V . Note that our conditions insure that φi (x) = 0, for all i and  x ∈ V . It has been proven in [MU1] that for all ω = (ωn )n≥0 ∈ I ∞ the intersection n≥0 φωn () is a singleton. Furthermore, lim sup{diam(φω ()) : ω ∈ I ∗ , |ω| = n)} = 0.

n→∞

Thus we can define the coding map π : I ∞ → , defining π(ω) to be the only element of the intersection n≥0 φωn (), and this map is uniformly continuous. The limit set J = JF of the system F = {φi }i∈I is defined to be π(I ∞ ). It turns out that J is compact and satisfies the following invariance property: J = ∪i∈I φi (J ). Consider now the system F ∗ generated by I ∗ defined by F ∗ = {φi n j : n ≥ 1, i ∈ , i = j} ∪ {φk : k ∈ I \ }. It immediately follows from our assumptions that the following holds. Theorem 4.1. The system F ∗ is a hyperbolic (though with the infinite alphabet I ∗ ) conformal iterated function system, i.e. F ∗ has no parabolic elements. F ∗ is called the hyperbolic iterated function system associated to the parabolic system F. The limit set generated by the system F ∗ is denoted by J ∗ . A proof of the following lemma can be found in [MU1]. Lemma 4.2. The limit sets J and J ∗ of the systems F and F ∗ respectively differ only by a countable set. More precisely, J ∗ ⊂ J and J \ J ∗ is countable. In this paper we will only be interested in the special case when I = {0, 1}, φ0 () ∩ φ1 () = ∅, and 0 is the only parabolic index with the parabolic point x0 . The following result is an immediate consequence of the Bounded Distortion Property (6).

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Proposition 4.3. The function ω
→ − log |φω 1 (π(σ (ω))|, ω ∈ 2+ , has mild distortion modulo Q = {0∞ }. Note that now the projection π : 2+ → JF is a homeomorphism, there is a welldefined map F : JF → JF , where F(x) = φi−1 (x) if x ∈ φi (), which has a C 1+ε extension φi−1 to each interval φi (V ), and the projection π : 2+ → JF establishes a canonical conjugacy between the shift map σ : 2+ → 2+ and the map F : JF → JF . Consequently, the entire discussion from Sect. 2 about symbol spaces and shift maps applies to the map F : JF → JF . We will frequently invoke the results proven in [MU1] and [U2] concerning the dynamics of the map F and the geometry of the limit set JF called a parabolic Cantor set in [U2]. 5. Parabolic Smale’s Horseshoes In this section we describe the main object of interest in this paper, a parabolic horseshoe of smooth type. Let S ⊂ R2 be a closed topological disk whose boundary is smooth except at finitely many (possibly none) points. We consider a C 1+ε -diffeomorphism f : S → R2 onto its image having the following properties: (a) The intersection f (S) ∩ S consists of two disjoint closed topological disks R0 and R1 . (b) The closure f (S) \ S consists of three mutually disjoint topological disks R2 , R3 , R4 . Furthermore, it is required that there exist two 1-dimensional mutually transversal foliations W u and W s of S ∪ f (S) consisting of smooth connected leaves with the following properties: (c) If x ∈ Ri , i = 0, 1, 2, 3, 4, then the sets W u (x) ∩ Ri and W s (x) ∩ Ri are connected. (d) For all points x, y ∈ Ri , i = 0, 1, 2, 3, 4, W u (x) ∩ W s (y) ∩ Ri is a singleton denoted by [x, y]. (e) For every point x ∈ Ri , i = 0, 1, 2, 3, 4, the map [·, ·] : W s (x)∩Ri ×W u (x)∩Ri → Ri , (y, z)
→ [y, z], is a homeomorphism.   (f) For every point x ∈ Ri , i = 0, 1, f W u (x) ∩ Ri = W u ( f (x)). If f (x) ∈ R j ,   j = 0, 1, 2, 3, 4, then f −1 W s ( f (x)) ∩ R j = W s (x). u/s

For every point x ∈ S we denote by E x the tangent space of W u/s (x) at x. We use the notation Du/s f (x) for the derivative of the map f : W u/s (x) ∩ S → W u/s ( f (x)); u/s hence Du/s f (x) = D f (x)|E x for all x ∈ S. Let ω be the fixed point of f lying in R0 . We require that the following conditions hold: (g) (h) (i) (j)

There exists 0 < γ < 1 such that |Ds f (x)| ≤ γ for all x ∈ S. If x ∈ S then |Du f (x)| ≥ 1. Moreover, if x ∈ S \ W s (ω), then |Du f (x)| > 1. Du f (ω) = 1. In the oriented arc-length parametrization of W u (ω) starting at ω, we have f −1 (x) = x ± a|x|β+1 + o(|x|β+1 )

(5.1)

in a sufficiently small neighborhood of 0 with some positive constants a and β and with the sign “−” if x ≥ 0 and the sign “+” if x ≤ 0. We call a map f : S → R2 satisfying properties (a) - (j) a parabolic horseshoe map.

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Remark. We note that property (j) restricts the regularity of f , namely by (5.1) the map f is at most of class C 1+β . In particular, if β < 1, then f is not twice differentiable. It is easy to see that for every τ ∈ 2+− , the intersection +∞ 

f −n (Rτn )

n=−∞ +− is a singleton, which  +− we denote by (τ ). The map  : 2 → S is a homeomorphism on its image  2 , which we denote by . Hence,

 = {x ∈ S : f ±n (x) ∈ S for all n ∈ N}.

(5.2)

Obviously,  is a compact f -invariant set, i.e. f () =  = f −1 (). Moreover, the map  : 2+− →  establishes a topological conjugacy between the shift map σ : 2+− → 2+− and f :  → , i.e., the following diagram commutes: 2+− ⏐ ⏐  

σ −→ f −→

2+− ⏐ ⏐  

In the sequel we will frequently identify the cylinders on the symbol space 2+− and their images under the homeomorphism . Given a point x ∈ Ri , i = 0, 1, 2, 3, 4, we put s Wloc (x) = W s (x) ∩ Ri .

These sets are called local stable manifolds. Given a point x ∈ Ri , i = 0, 1, 2, 3, 4, we define Wiu (x) = W u (x) ∩ Ri . For i ∈ {0, 1} we define u (x) = W u (x) ∩ Ri⊕1 (x), Wi⊕1

where the symbol “⊕” denotes the addition mod(2) in the group {0, 1}. Given two points x, y ∈ f (S) we denote by u Hx,y : W u (x) → W u (y)

the holonomy map along local stable manifolds. Moreover, we denote by H u : f (S) → W u (ω) the holonomy map from f (S) to W u (ω) along local stable manifolds, that is, u . H u (x) = Hx,ω

Ergodic Theory of Parabolic Horseshoes

729

We define ω0 = ω and pick ω1 ∈  arbitrary having the property that f (W1u (ω)) = W u (ω1 ). We say that the parabolic horseshoe f : S → R2 is of smooth type if the u : W u (x) → holonomy map H u : f (S) → W u (ω) and all the holonomy maps Hx,y W u (y) are C 1+ε , and in addition, the derivative along unstable manifolds of the map f −1 ◦ Hωu, ω1 : W u (ω) → W u (ω) has norm less than 1 at every point of W u (ω) except possibly at ω. Given two points x, y ∈ f (S) we denote by s s s : Wloc (x) → Wloc (y) Hx,y

the holonomy map along unstable manifolds. We say that the parabolic horseshoe f : s : S → R2 has a Lipschitz continuous unstable foliation if all the holonomy maps Hx,y s s Wloc (x) → Wloc (y) are Lipschitz continuous with a uniform Lipschitz constant. The following example provides a parabolic horseshoe map f : S → R2 of smooth type having a Lipschitz continuous unstable foliation. In particular, the unstable and stable manifolds in R0 and R1 are vertical and horizontal lines respectively. Example 1. An Almost Linear Parabolic Horseshoe Map. Let S ⊂ R2 be a unit square and let h : S → R2 be a linear horseshoe map with constant contraction rate λs < 21 and constant expansion rate λu > 2, see Fig. 1. Let ω = (ω1 , ω2 ) be the fixed point of h contained in R0 . Let δ > 0 such that (ω1 , ω2 + δ), (ω1 , ω2 − δ) ∈ intS. Fix constants β > 0, a > 0 and η > 1. Let ϕ : [−δ, δ] → [−δ, δ] be a C 1+ -diffeomorphism satisfying the following properties: (i) ϕ −1 (t) = λu t ± λu a|t|β+1 for t sufficiently close to 0, and with the sign "−" if t ≥ 0 and the sign "+" if t ≤ 0;  (ii) λ−1 u < ϕ (t) ≤ η for all t ∈ [−δ, δ] \ {0}; (iii) ϕ  (−δ) = ϕ  (δ) = 1. Let Aδ ⊂ h(S) be defined as in Fig. 1, that is, Aδ = [a1 , a2 ] × [b1 , b2 ], where b1 = ω2 − δ, b2 = ω2 + δ and a2 − a1 = λs . We define the map g : h(S) → h(S) by ⎧ ⎨ (x1 , ω2 + ϕ(x2 − ω2 )) if x ∈ Aδ g(x) = (5.3) ⎩x if x ∈ h(S) \ Aδ for all x = (x1 , x2 ) ∈ h(S). Clearly, g is a C 1+ -diffeomorphism which preserves vertical and horizontal lines in R0 ∪ R1 = S ∩ h(S). We now define the map f = g ◦ h. It follows immediately from the construction that f is a parabolic horseshoe map. In particular, property (5.1) holds. Indeed (5.1) is a consequence of the facts that the foliations W u |R0 ∪ R1 and W s |R0 ∪ R1 of f are given by vertical and horizontal lines −1 respectively and that pr 2 (h −1 (ω1 , x2 )) = ω2 + λ−1 u (x 2 − ω2 ) and pr 2 (g (ω1 , x 2 )) = ω2 + ϕ −1 (x2 − ω2 ) if x2 is sufficiently close to ω2 . Here pr 2 denotes the projection in R2 on the 2nd coordinate. Moreover, it is easy to see that f is of smooth type and has a Lipschitz continuous unstable foliation. A simple calculation shows that |Ds f (x)| = λs and

1 ≤ |Du f (x)| ≤ ηλu for all x ∈ .

(5.4)

Remark. (i) We note that the diffeomorphism f is uniquely determined on S ∩ h(S) once we have chosen the constants a, β, η, δ, λu , λs , the map ϕ and h. In particular, the contraction rate λs ∈ (0, 1/2) can be chosen independently of a, β, η, δ, λu , and ϕ. (ii) The method of construction of an almost linear parabolic horseshoe map in Example 1

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M. Urba´nski, C. Wolf

Fig. 1. An Almost Linear Parabolic Horseshoe Map f = g ◦ h

can be generalized. Namely, let h be a hyperbolic (not necessarily linear) C 1+ -horseshoe map which has the property that h as well as h −1 is of smooth type. Then we can construct analogously to Example 1 a C 1+ -diffeomorphism g : h(S) → h(S) such that f = g ◦ h is a parabolic horseshoe map of smooth type with a Lipschitz (even C 1+α ) unstable foliation. 6. Horseshoe and the Associated Parabolic Iterated Function System In this section we introduce for a parabolic horseshoe map an associated parabolic iterated function system on the unstable manifold of the parabolic fixed point. The goal is to do this in such a way that the parabolic horseshoe and the parabolic iterated function system share many ergodic-theoretical features. Let f be a parabolic horseshoe map of smooth type. We define ω0 = ω and pick ω1 ∈  arbitrary having the property that f (W1u (ω)) = W u (ω1 ). We introduce the 1-dimensional iterated function system  on W u (ω) defined by the following two maps: u φi = f −1 ◦ Hω,ω : W u (ω) → Wiu (ω) ⊂ W u (ω), i = 0, 1. i

(6.1)

It is easy to check that the iterated function system  = {φ0 , φ1 } satisfies all the requirements (in particular (4.1)) of a 1-dimensional parabolic iterated function system introduced in Sect. 4 except possibly item (2) (it may happen that |φ1 (ω)| = 1). We shall now demonstrate that after a C ∞ change of the Riemannian metric on W u (ω), Condition (2) is also satisfied and  = {φ0 , φ1 } becomes a parabolic iterated function system. The following formula immediately follows from the definition of a smooth parabolic horseshoe: |φi (x)| < 1 for all x ∈ W u (ω) \ {ω} and |φi (ω)| ≤ 1, i = 0, 1.

(6.2)

Since φ1 (φ1 (ω)), φ0 (φ1 (ω)) = φ1 (ω), there exists a closed segment T ⊂ W u (ω) with the following properties: (a) T ⊂ W1u (ω), (b) φ1 (ω) ∈ intW1u (ω) T , (c) φ1 (T ) ∩ T = ∅. Define M = sup{max{|φ0 (x)|, |φ1 (x)| : x ∈ T }.

(6.3)

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In view of (6.2), item (a) above, and the continuity of the functions x
→ |φ0 (x)|, |φ1 (x)|, we conclude that M < 1.

(6.4)

Therefore there exists a C ∞ function ρ : W u (ω) → (0, 1] such that M < ρ(x) < 1

for all x ∈ intT

(6.5)

and ρ(x) = 1

for all x ∈ W u (ω) \ T.

(6.6)

Consider the Riemannian metric ρ(x)d x on W u (ω) and let |g  (x)|ρ = ρ(g(x))|g  (x)|/ ρ(x) be the norm of the derivative of a differentiable function g : W u (ω) → W u (ω) calculated with respect to the Riemannian metric ρ(x)d x. We shall prove the following. Lemma 6.1. We have |φi (x)|ρ < 1 for all x ∈ W u (ω) \ {ω}, i = 0, 1, |φ0 (ω)|ρ = 1 and |φ1 (ω)|ρ < 1. Proof. The equality |φ0 (ω)|ρ = 1 is immediate. Consider an arbitrary point x ∈ intT . Then both φ0 (x), φ1 (x) ∈ / T by (a) and (c) respectively. Thus, using (6.2)-(6.3), we obtain for i = 0, 1 that |φi (x)|ρ = ρ(φi (x))|φi (x)|/ρ(x) = |φi (x)|/ρ(x) < M −1 |φi (x)| ≤ 1. Next we assume that x ∈ W u (ω) \ intT . Then |φi (x)|ρ = ρ(φi (x))|φi (x)| ≤ |φi (x)|.

(6.7)

So, if x = ω, then it follows from (6.2) that |φi (x)|ρ ≤ |φi (x)| < 1. Hence, we are left to consider the case when x = ω and i = 1. It then follows from (b), (6.5) and (6.2) that |φ1 (ω)|ρ = ρ(φ1 (ω))|φ1 (ω)|/ρ(φ1 (ω)) < 1.   Therefore, as long as we are dealing with the iterated function system  itself, we may assume without loss of generality that  is a parabolic iterated function system and, in particular, all the considerations from Sect. 4 apply. Similarly as in the case of the horseshoe  we will frequently identify in the sequel the cylinders on the symbol space 2+ and their images under the homeomorphism π : 2+ → J . Note that u φi−1 = (Hω,ω )−1 ◦ f = Hωui ,ω ◦ f : Wiu (ω) → W u (ω), i = 0, 1 i

(6.8)

Hωui ,ω = H u |W u (ωi ) .

(6.9)

and

We first prove a preliminary result. Lemma 6.2. For all x ∈ S and all n ≥ 0 we have that (H u ◦ f )k (x) ∈ S for all k = 0, 1, . . . , n if and only if f k (x) ∈ S for all k = 0, 1, . . . , n. In this case we have  u  s s (H ◦ f )n (x) = Wloc ( f n (x)). Wloc

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Proof. We shall prove this lemma by induction with respect to n ≥ 0. For n = 0 there is nothing to prove. So, take n ≥ 1 and suppose that our lemma is true for all non-negative integers less than n. Assume that f k (x) ∈ S for all k = 0, 1, . . . , n. Then    s  ( f n−1 (x)) (H u ◦ f )n (x) = H u ◦ f (H u ◦ f )n−1 (x) ∈ H u ◦ f Wloc  s s ( f n (x)) ⊂ Wloc ( f n (x)), ⊂ H u (Wloc and, in particular, (H u ◦ f )n (x) ∈ S. We now assume that (H u ◦ f )k (x) ∈ S for all k = 0, 1, . . . , n. Thus,  s  u  (H ◦ f )n−1 (x) f n (x) = f ( f n−1 (x)) ∈ f Wloc   u   s s ⊂ Wloc f ◦ (H u ◦ f )n−1 (x) = Wloc H ◦ f ) ◦ (H u ◦ f )n−1 (x) s ((H u ◦ f )n (x)). = Wloc

This implies that f n (x) ∈ S. The desired result now follows from the induction assumption.   For all integers n ≥ 0 and also for n = +∞ we define Sn =

n 

f −k (S)

k=0

and S−n =

n 

f k (S).

k=0

Next, we prove the following. s (x) ⊂ S . Lemma 6.3. If n ≥ 0 and x ∈ Sn , then Wloc n s (z) ⊂ S = S for all z ∈ S. So, suppose Proof. For n = 0 this is obvious because Wloc 0 that our lemma is true for some n ≥ 0 and fix a point x ∈ Sn+1  . Then x ∈ Sn , and in s (x) ⊂ S . Hence, f n+1 W s (x) is well-defined view of our inductive hypothesis, Wloc n  s  n+1 loc   s and, as f n+1 (x) ∈ S, we get that f n+1 Wloc f (x) ⊂ S. This completes (x) ⊂ Wloc the proof.  

We recall that J is the limit set of the iterated function system . The relation between this limit set and the horseshoe  is given by the following. Lemma 6.4. J =  ∩ W u (ω). Proof. We shall show first by induction that J ⊂ Sn for all n ≥ 0. Indeed, if z ∈ J , then z ∈ φ0 (W u (ω)) ∪ φ1 (W u (ω)) = W0u (ω) ∪ W1u (ω) ⊂ S = S0 and our inclusion is proved for n = 0. So, suppose that n ≥ 1 and that J ⊂ Sk for all k = 0, 1, . . . , n − 1. Let us consider an arbitrary point z ∈ J . We write z = π(τ ), where τ ∈ I ∞ . Then z ∈ J ⊂ Sn−1 and z = φτ0 (π(σ (τ ))). Hence,  u  u (π(σ (τ ))) = f n−1 Hω,ω (π(σ (τ ))) . f n (z) = f n ◦ φτ0 (π(σ (τ ))) = f n ◦ f −1 ◦ Hω,ω τ τ 0

0

Ergodic Theory of Parabolic Horseshoes

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s (π(σ (τ ))) and since π(σ (τ )) ∈ J ⊂ S u Since Hω,ω (π(σ (τ ))) ∈ Wloc  n−1 , we may τ0 u n conclude from Lemma 6.3 that Hω,ωτ (π(σ (τ ))) ∈ Sn−1 . Thus, f (z) ∈ S, which 0 implies that z ∈ Sn . Therefore, the inductive proof is complete, and we obtain that J ⊂ S+∞ . Since also J ⊂ W0u (ω) ∪ W1u (ω) ⊂ S−∞ , we therefore obtain that

J ⊂ W u (ω) ∩ S+∞ ∩ S−∞ = W u (ω) ∩ .

(6.10)

In order to prove the opposite inclusion we consider an arbitrary point z ∈  ∩ W u (ω). We shall prove by induction that there exists an infinite word τ = ∅τ1 τ2 . . . ∈ {∅} × I ∞ , such that for every n ≥ 0, z = φτ |n (x) for some x ∈ W u (ω). Indeed, for n = 0 we have z = φ∅ (z) and z ∈ W u (ω). So, suppose that for some n ≥ 1, the word τ |n = ∅τ1 τ2 . . . τn has been constructed. This means that z = φτ |n (x) with some x ∈ W u (ω). As z ∈ , we have f n (z) ∈ S, and, in view of Lemma 6.2, (H ◦ f )n (z) ∈ S. Using (6.8) and (6.9), u n we therefore obtain that x = φτ−1 |n (z) = (H ◦ f ) (z) ∈ S. And applying the last part of Lemma 6.2 along with the fact that f n+1 (z) ∈ S, we get that  s  s ( f n (z)) ⊂ Wloc ( f n+1 (z)) ⊂ S. f (x) = f ◦ (H u ◦ f )n (z) ∈ f Wloc Moreover,       f (x) ∈ f (S ∩ W u (ω)) = f W0u (ω) ∪ W1u (ω) = f W0u (ω) ∪ f W1u (ω) = W u (ω) ∪ W u (ω1 ), u (y) with some i ∈ {0, 1} and y ∈ W u (ω). Consequently, and thus f (x) = Hω,ω i u (y) = φ (y). Thus z = φ (φ (y)) = φ (y), and the inductive proof x = f −1 ◦ Hω,ω i τ i τi i is complete by putting τn+1 = i. Note that z = π(τ ) ∈ J . This gives the inclusion W u (ω) ∩  ⊂ J . Combining this with (6.10) completes the proof.  

7. Topological Pressures, Unstable Dimension, Hausdorff and Packing Measures Let f be a parabolic horseshoe map of smooth type. Since the holonomy maps along local stable manifolds are smooth, it follows that def

t u = dim H W u (ω) ∩  = dim H W u (x) ∩  is independent of x ∈ . We call the quantity t u the unstable dimension of the set . The main goal of this section is to establish a Bowen-Ruelle-Manning-McCluskey type of formula for t u . In order to do this we will make use of Lemma 6.4, will introduce the topological pressure of several dynamical systems and potentials, and will apply results from the thermodynamic formalism of parabolic iterated function systems derived in ˆ [MU1] and [U2]. First we consider different pressure functions. For all t ≥ 0 let P(t) denote the pressure function associated with the iterated function system  as defined in Sect. 4 (see [MU1,U2] for details). Moreover, we define P u (t) = P( f, −t log |Du f |), where P( f, −t log |Du f |) is the ordinary topological pressure of the potential −t log |Du f | :  → R with respect to the dynamical system f |. We claim that the following diagram commutes:

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M. Urba´nski, C. Wolf

 ⏐ ⏐ Hu J

f −→ ◦ f −→

Hu

 ⏐ ⏐ u H

(7.1)

J

that is, H u ◦ f = (H u ◦ f ) ◦ H u .

(7.2)

s (x) = W s (H u (x)). This implies that W s ( f (x)) = Indeed, if x ∈ , then Wloc loc loc s u Wloc ( f (H (x))), and therefore H u ( f (x)) = H u ( f (H u (x))), which proves the claim. We define     ˜ = P H u ◦ f, −t log |D(H u ◦ f )| and P(t) = P f, −t log |D(H u ◦ f )| ◦ H u . P(t)

We also set, φu = log |Du f | :  → R, and φ˜ u = log |D(H u ◦ f )| : J → R. Since H u ◦ f : J → J is the dynamical system generated by the inverses φ0−1 and φ1−1 , we immediately obtain that ˜ ˆ P(t) = P(t), t ≥ 0.

(7.3)

Differentiating both sides of Eq. (7.2) along the unstable manifolds, we get (Du H u ◦ f ) · Du f = D(H u ◦ f ) ◦ H u · Du H u . This implies the following. Lemma 7.1. log |D(H u ◦ f ) ◦ H u | − log |Du f | = log |Du H u ◦ f | − log |Du H u |, and consequently the potentials −t log |D(H u ◦ f ) ◦ H u | and −t log |Du f | are, for all t ≥ 0, cohomologous in the class of continuous functions. As an immediate consequence of this lemma along with Proposition 3.9 and Proposition 4.3, we get the following. Lemma 7.2. For every t ≥ 0, we have that ¯ ˆ P u (t) = P(t) = P(t). In addition, the potentials −tφu :  → R and −t φ˜ u : J → R, are VPA and the map µ
→ µ ◦ (H u )−1 is a bijection between E S ∞ (−tφu ) and E S ∞ (−t φ˜ u ), the sets of (σ -finite) equilibrium states for −tφu and −t φ˜ u respectively. Combining Lemma 7.2 and Bowen’s formula proven in [U2] (compare [MU1]) provides the following.

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Proposition 7.3. The unstable dimension t u of  is the smallest zero of the pressure ˆ function t
→ P(t) = P(t), t ≥ 0. Now, let us take more fruits of the results proven in [U2]. First, Theorem 6.5 from [U2] and Lemma 6.4 give the following. Theorem 7.4. The unstable dimension t u ∈ (0, 1). Remark. It follows from work in [DV] that if f is a C 2 -diffeomorphism (which in particular implies that β ≥ 1), then t u ≥ 21 . Let us denote by Ht (A) respectively Pt (A) the t-dimensional Hausdorff respectively packing measure of the set A. Combining Theorem 7.4, Lemma 6.4, and Theorem 6.4 of [U2] provides the following. u (z) ∩ ) = 0 and 0 < P u (W u (z) ∩ ) < ∞ for Theorem 7.5. We have that Ht u (Wloc t loc all z ∈ .

We end this section with the following result. Theorem 7.6. The unstable pressure function t
→ P u (t) is real-analytic on (0, t u ). Proof. Consider the hyperbolic iterated function system ∗ = {φ0n 1 }∞ n=0 associated to the system  as in Sect. 4. Consider the two-parameter family G t,s of the functions 0 1 (z) = t log |φ0 n 1 (z)| − s(n + 1) : t, s ∈ R, n ≥ 0, z ∈ W u (ω). gt,s n

With the terminology of Sect. 3 of [MU2] (see also [U4] and [HMU], where these were introduced) we shall prove the following. Lemma 7.7. For all t, s ∈ R the family G t,s is Hölder continuous. For all (t, s) ∈ R × (0, +∞) the family G t,s is summable. Proof. The fact that the family G t,s is Hölder follows immediately from the sentence located just beneath the proof of Theorem 8.4.2 in [MU2]. Since ||φ0 n 1 ||  (n + 1) (see Lemma 2.3 in [U2]), we see that if t ∈ R and s > 0, then    0n 1      = exp sup gt,s exp sup t log |φ0 n 1 (z)| − s(n + 1) : z ∈ W u (ω) n≥0

n≥0

=



e−s(n+1) ||φ0 n 1 ||t 

n≥0



(n + 1)

− β+1 β

− β+1 β t −s(n+1)

e

n≥0

< +∞. This precisely means that our family G t,s is summable, and we are done.   Let gt,s : {0n 1 : n ≥ 0}N → R be the amalgamated function (see [MU2], comp. [U4] 0n 1 }∞ . This function is given by the formula and [HMU]) of the family {gt,s n=0 τ0 gt,s (τ ) = gt,s (π∗ (σ∗ (τ ))) = t log |φτ 0 | − s|τ0 |,

(7.4)

where σ∗ : {0n 1 : n ≥ 0}N → {0n 1 : n ≥ 0}N is the shift map associated with the iterated function system ∗ and π∗ : {0n 1 : n ≥ 0}N → J∗ is the corresponding canonical projection. Summability of the family G t,s proven in Lemma 7.7 precisely means

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summability of the amalgamated function gt,s . Hölder continuity of this amalgamated function follows from Lemma 7.7 and Lemma 3.1.3 from [MU2]. The following lemma is now an immediate consequence of Theorem 2.6.12 and Proposition 3.1.4, both from [MU2]. Lemma 7.8. The function (t, s)
→ P(gt,s ), (t, s) ∈ R × (0, +∞) is real-analytic in both variables t and s, where the topological pressure P(gt,s ) is defined in Sect. 3.1 of [MU2]. We now prove the conclusion of Theorem 7.6. Since the dynamical system H u ◦ f : J → J is expansive, it follows from Theorem 3.12 in [DU] that for every t ≥ 0 there exists a Borel probability measure m t supported on J and such that

˜ m t (H u ◦ f (A)) = e P(t) |(H u ◦ f ) |t dm t (7.5) A

for all Borel sets A ⊂ J having the property H u ◦ f | A is one-to-one. Hence,

˜ m t (φi (E)) = e− P(t) |φi |t dm t E

  for i = 0, 1 and E, any Borel subset of J . In addition, m t φ0 (W u (ω))∩φ1 (W u (ω)) = 0 as these sets φ0 (W u (ω)) and φ1 (W u (ω)) are disjoint. We therefore get by a straightforward induction that

   0n 1  m t φ0n 1 (E) = dm t exp gt,P(t) E

and   m t φ0n 1 (W u (ω)) ∩ φ0k 1 (W u (ω)) = 0 ˜ by P u (t) due to for all t ≥ 0 and all n, k ≥ 0 with n = k. We have replaced here P(t) u u Lemma 7.2 and (7.3). If now t ∈ (0, t ), then P (t) > 0 and the family G t P(t) is Hölder and summable due to Lemma 7.7. So, looking at the definition of G t P(t) -conformal measures, i.e. formulas (3.5) and (3.6) from [MU2], we see that m t is a unique G t P(t) conformal measure and that     P gt,s = P G t P(t) = 0, t ∈ (0, t u ). (7.6) Looking at Theorem 3.2.3, Corollary 2.7.5 and Proposition 2.6.13 from [MU2] and at the formula (7.4), we conclude that for all t ∈ (0, t u ),  

∂ P gt,s |(t,P(t)) = −|τ0 |d µ˜ t (τ ) = 0, (7.7) ∂s where µ˜ t = µ˜ gt,P(t) is the σ∗ -invariant Gibbs state proved to exist by Corollary 2.7.5 of [MU2]. Hence, applying formula (7.7) (also using (7.6)), the proof follows by applying Lemma 7.8 and the Implicit Function Theorem.  

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8. Equilibrium States In this section we provide a complete description of all ergodic σ -finite equilibrium states of the potential −t u φu , where, we recall, φu = log |Du f | :  → R, with respect to the dynamical system f :  → . We start our analysis with the potential −t u φ˜ u , where, we also recall, φ˜ u = log |D(H u ◦ f )| : J → R with respect to the dynamical system H u ◦ f : J → J . Let µ˜ ω be the Dirac δ-measure supported on ω. Let m˜ t u be the t u -conformal measure established in (7.5) with t = t u . ˜ u ) = 0. It follows from Theorem 7.2 in [U2] that there exists a unique We note that P(t (up to a multiplicative constant) ergodic σ -finite H u ◦ f -invariant measure µ˜ t u on J equivalent to m˜ t u . The measure µ˜ t u is ergodic and conservative. The following necessary and sufficient condition for the measure µ˜ t u to be finite was established in [U2]. Theorem 8.1. The measure µ˜ t u is finite if and only if tu > 2

β . β +1

We shall prove the following. Theorem 8.2. The measures µ˜ t u and µ˜ ω are the only ergodic equilibrium states of the potential −t u φ˜ u with respect to the dynamical system H u ◦ f : J → J . u˜ ˜ Proof.  u Since by Lemma 3.3 and Theorem 4.3 in [U2], P(−t φu ) = 0 and since −t φ˜ u d µ˜ ω = 0, it follows that the Dirac δ-measure µ˜ ω is an equilibrium state of the potential −t u φ˜ u . Next, we demonstrate that µ˜ t u is also an equilibrium state. We define

ρ=

d µ˜ t u . d m˜ t u

(8.1)

  It follows from Theorem 7.2 in [U2] that ρ|[0n 1|n0 ]  n+1. Since in addition m˜ t u [0n 1|n0 ]  (n + 1)

u − β+1 β t

and since |(H u ◦ f ) (z) − 1|  (β + 1)z β  (n + 1)−1

(8.2)

for all z ∈ [0n 1|n0 ] (so log |(H u ◦ f ) (z)|  (n + 1)−1 ), we may conclude that

0 ≤ −t u φ˜ u d µ˜ t u < ∞.

(8.3)

Now notice that 0 < µ˜ t u ([1|00 ]) < ∞. In order to see that µ˜ t u is an equilibrium state for −t u φ˜ u let us induce the map H u ◦ f on the cylinder [1|00 ]. Denote µ˜ t u [1|0 ] by µ˜ 1 . Consider 0

the partition α of [1|00 ] generated by the first return time. Obviously α is a generating partition for the measure µ˜ 1 and the first return map on [1|00 ]. Since for µ˜ 1 -a.e. point τ ∈ [1|00 ] there is an infinite sequence {kn }∞ n=1 of positive integers such that τkn = 1, and

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M. Urba´nski, C. Wolf

since µ˜ 1 is equivalent to m˜ t u |[1|0 ] with the Radon-Nikodym derivative bounded away 0 from zero and infinity, we get that       µ˜ 1 [τ |k0n ]  m˜ t u [τ |k0n ]  exp −t u Skn φ˜ u (τ ) .     But [τ |k0n ] = α n (τ ) and Skn −t u φ˜ u (τ ) = Sn1 −t u φ˜ u1 (τ ), where α n is the n th refined partition with respect to the first return map on [1|00 ], S 1j is the j th ergodic sum with   respect to the first return map on [1|00 ], and −t u φ˜ u1 = −t u φ˜ u |[1|0 ] . So, µ˜ 1 (β n (τ )) = 0   exp −t u Sn1 φ˜ u1 (τ ) . Applying now the Shannon-McMillan-Breiman Theorem and Biru khoff’s Ergodic Theorem  (along with the observation that µ˜ 1 is ergodic because m˜ t is), we get that hµ˜ 1 + (−t u φ˜ u1 )d µ˜ 1 = 0, the equality that demonstrates that µ˜ t u is an equilibrium state for −t u φ˜ u . In order to prove that µ˜ ω and µ˜ t u are the only ergodic conservative equilibrium states for −t u φ˜ u , suppose that µ˜ is an ergodic conservative equilibrium state for −t u φ˜ u different from µ˜ ω . Suppose that µ˜ has an atom. Because of ergodicity and conservativity of µ, ˜ this measure must be supported on a periodic orbit of σ containing this  atom. But then µ˜ is (up to a multiplicative constant) a probability measure, hµ˜ = 0 and φ˜ u d µ˜ > 0  since µ˜ = µ˜ ω . Consequently, hµ˜ + (−t u φ˜ u )d µ˜ < 0 contrary to the fact that µ˜ is an ergodic conservative equilibrium state for −t u φ˜ u . So, µ˜ is atomless, and similarly as in the proof of Theorem 3.7, there exists an initial cylinder F, with the first coordinate equal to 1, such that µ(F) ˜ ∈ (0, ∞). Let α be the countable partition of F induced by the first return time. We shall prove the following. Claim 1. hµ˜ F (α) < ∞. Proof. Suppose on the contrary that hµ˜ F (α) = ∞. It then immediately follows from Shannon-McMillan-Breiman Theorem that   − log µ˜ F (α n (ω)) =∞ (8.4) lim n→∞ n for µ˜ F -a.e. ω ∈ F, say ω ∈ F1 . Since µ˜ ∈ M−t u φ˜u , the function |φ˜ u | = −φ˜ u is  µ-integrable, ˜ and therefore 0 < χ := |φ˜ u |d µ˜ F < ∞. Thus, by Birkhoff’s Ergodic Theorem, lim

n→∞

1 F F S φ˜ (ω) = χ n n u

(8.5)

for µ-a.e. ˜ ω ∈ F, say ω ∈ F2 . Put F3 = F1 ∩ F2 . Then µ˜ F (F3 ) = 1. Since φ1 is a hyperbolic element of our parabolic iterated function system , it follows from item (b) of Sect. 4 and the definition of measure µ˜ t u that for every ω ∈ F and every n ≥ 1, µ˜ t u F (α n (ω))  exp −t u SnF φ˜ u (ω) . Therefore, using (8.5), we get for all ω ∈ F3 and all n ≥ 1 large enough, that   µ˜ t u F (α n (ω)) ≥ exp −t u (χ + 1)n . Combining this and (8.4), we see that for all ω ∈ F3 , µ˜ F (α n (ω)) = 0. n→∞ µ ˜ t u F (α n (ω)) lim

Thus µ˜ F (F3 ) = 0 and this contradiction finishes the proof of Claim 1.

 

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Let Jµ−1 ˜ F with ˜ F : F → [0, 1] be the inverse of the (weak) Jacobian of the measure µ respect to the first return map σ F : F → F, i.e

−1 µ˜ F (B) = Jµ−1 ˜F ˜ F ◦ (σ F | B ) d µ σ F (B)

for every Borel set B ⊂ F such that σ F | B is injective. Let Lµ˜ F : L 1 (µ˜ F ) → L 1 (µ˜ F ) be the Perron-Frobenius operator associated to the measure µ˜ F . The operator Lµ˜ F is given by the formula  Jµ−1 Lµ˜ F g(z) = ˜ F (x)g(x). x∈σ F−1 (z)

Notice that for the measure µ˜ t u we have Jµ−1 ˜ u (z) = t F

ρ(z) u |(H u ◦ f )F (z)|−t ρ(σ F (z))

(8.6)

and this Jacobian is a continuous function. Furthermore, Lµ˜ t u F g(z) =

 x∈σ F−1 (z)

ρ(x) u |(H u ◦ f )F (z)|−t g(x) ρ(z)

(8.7)

and Lµ˜ t u F g : F → R is continuous for every continuous function g : F → R. In particular, Lµ˜ t u F ρ = ρ, and this equality holds throughout the whole set F. We now shall prove the following: −1 ˜ F -a.e. z ∈ F. Claim 2. Jµ−1 ˜ F (z) = Jµ˜ u (z) for µ t F

Proof. Since Lµ˜ F (11) = 11 and since Lµ˜ t u F (11) = 11, applying (8.7), we get

  −1 −1 1 = 11d µ˜ F = Lµ˜ t u F (11)d µ˜ F Jµ−1 Jµ˜ u (x)Jµ−1 ˜ F (z) ˜ F (x) ˜ F (x)d µ

x∈σ F−1 (z)

−1 −1  Lµ˜ F = Lµ˜ F Jµ−1 Jµ˜ u d µ˜ F ˜ F (x) t F

 −1 −1 −1  ≥ 1 + log Jµ˜ F (x) Jµ˜ u d µ˜ F ≥ 1 t F

 u + log Jµ˜ F ρ · (ρ ◦ σ F )−1 |(H u ◦ f )F |−t d µ˜ F

= 1 + log(Jµ˜ F )d µ˜ F + log ρd µ˜ F − t u log(ρ ◦ σ F )

u −t log |(H u ◦ f )F |d µ˜ F

= 1 + log(Jµ˜ F )d µ˜ F − t u φ˜ uF d µ˜ F . =



−1 −1  Jµ˜ u d µ˜ F Jµ−1 ˜ F (x) t F

t F

(8.8)

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 Applying Claim 1 yields log(Jµ˜ F )d µ˜ F = hµ˜ F (σ F ). Therefore, since P(−t u φ˜ u ) = 0 and since µ˜ is an equilibrium state of −t u φ˜ u , we conclude that the most right-hand sided formula in (8.8) is equal to 1. Hence, the signs "≥" appearing in (8.8) must be all equal −1 ˜ F a.e. The proof of Claim 2 is to the "=" signs. As a consequence, Jµ−1 ˜ F = Jµ˜ t u F -µ complete.   Since σ Fn (A) = F for all n ≥ 1 and all A ∈ α n , it immediately follows from Claim 2, (8.6), (8.1), and the Bounded Distortion Property (item (6) in Sect. 4), that if µ˜ F (A) > 0, then µ˜ F (A)  µ˜ t u F (A). Thus, the measure µ˜ F is absolutely continuous with respect to the measure µ˜ t u F (A), and since the latter is ergodic, µ˜ F = µ˜ t u F . Hence, µ˜ = µ˜ t u , and we are done.   Let µω be the Dirac δ-measure on  supported on ω. The following, main result of this section, provides a complete description of the structure of equilibrium states of the potential −t u φu :  → R, where φu is given by the formula φu (x) = log |Du f (x)|,

(8.9)

with respect to the dynamical system f :  → . As an immediate consequence of Theorem 8.2 and Lemma 7.2, we get the following. Theorem 8.3. There are exactly two (up to a multiplicative constant) ergodic equilibrium states for the potential −t u φu . Namely, µω and µt u , the latter uniquely determined by the condition that µt u ◦ (H u )−1 = µ˜ t u . 9. Conditional Measures Suppose that (X, A, ν) is a σ -finite measure space. Suppose also that A is a sub-σ algebra of A. It easily follows from the probabilistic case that for every ν-integrable function g : X → R there exists E(g|A) : X → R, a unique expected value of g with respect to the σ -algebra A, i.e. an A-measurable function for which

E(g|A)dν = gdν A

A

for every set A ∈ A. Let P = PA be the measurable partition generated by the σ -algebra A. The canonical system {ν x }x∈X of ν conditional measures with respect to the σ -algebra A (and partition P) is given by the following formula: ν x (B ∩ P(x)) = E(11 B |A)(x)

(9.1)

for every set B ∈ A. We note that for ν-a.e. x ∈ X the value ν x (B ∩P(x)) is independent of the choice of a set B  ∈ A with the property that B  ∩ P(x) = B ∩ P(x). Since ν is σ -finite, it easily follows from Martingale’s Convergence Theorem that if {An }∞ n=0 is an ascending sequence of sub-σ -algebras of A, generating A, then   (9.2) E(11 B |A)(x) = lim E 11 B |An (x) n→∞

for ν-a.e. x ∈ X . We now consider the σ -finite measure space (, B, µt u ), the partition P u of  into unstable manifolds Wiu (x), x ∈ , i = 0, 1, and the σ -algebra B u

Ergodic Theory of Parabolic Horseshoes

741

generated by the partition P u . In the language of the symbol space 2+− the unstable manifolds take on the form W0u (γ ) = [γ |0−∞ ]. Without confusion we will frequently use either of the two languages: symbolic or "differentiable". For every n ≥ 0 let Bnu be the finite σ -algebra generated by the cylinders [γ |0−n ], γ ∈ 2+− . Obviously, {Bnu }∞ n=0 is an ascending sequence of sub-σ -algebras generating B u . Applying (9.1) and (9.2), we see that for every γ ∈ 2+− and every τ ∈ {0, 1}q , we have     γ  γ  q q µt u [γ |0−∞ τ |−∞ ] = µt u [τ |1 ] ∩ [γ |0−∞ ] = E 11|[τ |q ] |B u (γ ) 1   u = lim E 11|[τ |q ] |Bn (γ ) 1 n→∞  q   q  µt u [τ |1 ] ∩ [γ |0−n ] µt u [γ |0−n τ |−n ] = lim = lim n→∞ n→∞ µt u ([γ |0−n ]) µt u ([γ |0−n ])   0  n+q n+q  µt u [γ |−n τ |0 ] µ˜ t u [γ |0−n τ |0 ]+ = lim = lim , n→∞ µt u ([γ |0 |n ]) n→∞ µ ˜ t u ([γ |0−n |n0 ]+ ) −n 0 where we could have written the second last equality sign since the measure µt u is shiftinvariant and we wrote the second equality sign because of Theorem 8.3. We recall that µ˜ t u u ˜ u ρ = ddm ˜ t u . Using the fact that P(t ) = P(t ) = 0, we further can write by making use of (7.5) that   u u |φγ |0 τ |t ρ ◦ φγ |0 τ d µ˜ t u |φγ |0 τ |t d µ˜ t u J J     −n γ q −n −n  lim  , µt u [γ |0−∞ τ |−∞ ] = lim    tu ρ ◦ φ 0 dµ tu ˜ u n→∞ n→∞ u |φ | ˜ |φ t t 0 γ | J γ | J γ |0 | d µ −n

−n

−n

where we wrote the comparability sign using (8.3). We now assume that τ = 0q and γ |0−∞ = 0∞ |0−∞ . Let i ≥ 0 be the least integer such that γ−i = 1. Then the distortion property allows us to continue as follows:  γ  q µt u [γ |0−∞ τ |−∞ ]



K i,(3)j

||φγ |0

−n τ

||t

u

u ||φγ |0 ||t −n

 K i, j ||φρ 0 τ ||h  (3)

||φγ |0 ||t ||φτ ||t u



K i,(2)j

−n

u ||φγ |0 ||t −n  q  K i, j m˜ h [ρ0 τ |0 ] ,

u

= K i,(2)j ||φτ ||t

u

(2)

where the constants K i, j , K i, j , K i, j and K i, j depend on i and j = q − l, where l is the last position of the letter 1 in the word τ = τ1 τ2 . . . τq . Since the conformal measure µ˜ t u is a constant multiple of packing measure Pt u | J (see Theorem 6.4 in [U2]), since the holonomy maps between unstable manifolds are uniformly Lipschitz, and since  q q H [γ |0−∞ τ |−∞ ] = [γ0 τ |0 ], we obtain that    γ  q q µt u [γ |0−∞ τ |−∞ ]  K i, j Pt u [γ |0−∞ τ |−∞ ] . We now may conclude that γ

µt u |[γ |0

n −∞ ]∩[1|n ]

 K i,0 Pt u |[γ |0

n −∞ ]∩[1|n ]

   γ  for all n ≥ 0. Since in addition µt u [γ |0−∞ 0∞ ] = Pt u [γ |0−∞ 0∞ ] = 0 (also using that  0  n µt u is non-atomic), and since [γ |−∞ ] = [γ |0−∞ 0∞ ] ∪ ∞ n=0 [1|n ], we therefore have proven the following main result of this section.

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Theorem 9.1. For every i = 0, 1 and all x ∈  ∩ Ri the conditional measure µtxu on the unstable manifold Wiu (x) is equivalent to the packing measure Pt u restricted to the manifold Wiu (x). 10. Dimension of the Horseshoe In this section we establish formulas for the stable dimension and the dimension of the parabolic horseshoe. In this and in the subsequent sections we consider probability measures rather than σ -finite measures. In particular, the notion of equilibrium states will be from this point on exclusively used in the context of probability measures. Let f be a parabolic horseshoe map. We denote by M the space of all Borel f -invariant probability measures on  endowed with weak* topology. This makes M a compact convex space. Moreover, we denote by M E ⊂ M the subset of ergodic measures. Let µ ∈ M. We define

log |Du f | dµ and λs (µ) = log |Ds f | dµ. (10.1) λu (µ) = 



Note that λu (µ) and λs (µ) coincide with the µ-average of the pointwise Lyapunov exponents of f . It follows from properties (g), (h) and (i) of the parabolic horseshoe (see Sect. 5) that λu (µ) ≥ 0 and λs (µ) ≤ log γ < 0

(10.2)

for all µ ∈ M, where γ < 1 is the constant in property (g) of the parabolic horseshoe. We say that µ ∈ M is a hyperbolic measure if λu (µ) > 0. In this case we refer to λu/s (µ) as the positive/negative Lyapunov exponent of µ. Recall that µω denotes the Dirac-δ measure supported on the parabolic fixed point ω. We begin with a preliminary result. Lemma 10.1. Let µ ∈ M. Then µ is a hyperbolic measure if and only if µ = µω . Proof. We first consider the case when µ is ergodic. Obviously, if µ = µω then λu (µ) = 0, and µ is not hyperbolic. Assume now that µ = µω . Therefore, Birkhoff’s Ergodic Theorem implies that µ(W s (ω)) < 1, and since µ is ergodic we obtain µ(W s (ω)) = 0. Note that |Du f (x)| > 1 for all x ∈  \ W s (ω). Therefore, it follows from the definition of the Lyapunov exponent and the fact that x
→ Du f (x) is continuous that λu (µ) > 0. By using that λs (ν) < 0 for all ν ∈ M we obtain that µ is hyperbolic. Finally, the case when µ is not ergodic follows from the ergodic case by using an ergodic decomposition of µ.   We define the stable pressure function P s : R → R by P s (t) = P( f |, tφs ), where P s ( f |, .) is the ordinary topological pressure of the dynamical system f | and φs = log |Ds f | :  → R. Next we establish a Bowen-Ruelle-Manning-McCluskey type of formula for the stable dimension of . Theorem 10.2. Let f be a parabolic horseshoe map having a Lipschitz continuous def s (x) ∩  does not depend on x ∈ . Moreover, unstable foliation. Then t s = dim H Wloc s t is given by the unique solution of P s (t) = 0, and 0 <

ts

< 1.

(10.3)

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Proof. The proof is analogous to the case of uniformly hyperbolic sets on surfaces, see [MM]. Therefore, we provide only a sketch. First, note that since the unstable foliation s (x) ∩  is independent of is Lipschitz continuous, it follows immediately that dim H Wloc x ∈ . Observe that t
→ tφs is strictly decreasing. Therefore, P s is also strictly decreasing. Hence t s is well-defined. Note that h µω ( f ) = 0 and λs (µω ) < 0. In particular, µω is not an equilibrium state of t s φs . On the other hand, f | is expansive, which implies that the entropy map ν
→ h ν ( f ) is upper semi-continuous on M. Hence, for every ϕ ∈ C(, R) there exists at least one equilibrium state. Let µ be an ergodic equilibrium state of t s φs . It now follows from the definition of t s that t s = hµ ( f )/λs (µ). Applying s (x) ∩ . a result of Mendosa (see [Me, Theorem 1]) we deduce that t s ≤ dim H Wloc s s Finally, the proofs of the inequalities dim H Wloc (x) ∩  ≤ t < 1 and t s > 0 are analogous to the case of uniformly hyperbolic sets on surfaces (see [MM]).   Remark. (i) Similar to the case of uniformly hyperbolic surface diffeomorphisms one can show that the potential tφs has a unique equilibrium state for all t ≥ 0. (ii) We note that Theorem 10.2 in particular applies to almost linear parabolic horseshoe maps (see Example 1) as well as to the more general case of small perturbations of hyperbolic horseshoes of smooth type which were discussed in part (ii) of the remark after Example 1. Theorem 10.3. Let f be a parabolic horseshoe map of smooth type having a Lipschitz continuous unstable foliation. Then dim H  = t u + t s . Proof. Let i = 0, 1. We consider the set i =  ∩ Ri . Given x ∈ i it follows from property (e) of the parabolic horseshoe (see Sect. 5) that [·, ·] : W u (x) ∩ i × W s (x) ∩ i → i is a homeomorphism. Moreover, since f is of smooth type and since the unstable foliation of f is Lipschitz continuous, it follows that [·, ·] as well as [·, ·]−1 are Lipschitz continuous and therefore preserve Hausdorff dimension. Recall that dim B A denotes the upper box dimension of the set A. Similar, as in the case of hyperbolic sets on surfaces one can show that dim H W s (x) ∩ i = dim B W s (x) ∩ i . Using the formula for the Hausdorff dimension of products (see for example [F]) we obtain dim H W u (x)∩i ×W s (x)∩i = dim H W s (x) ∩ i + dim H W u (x) ∩ i . This completes the proof.   We define the stable set of  by
 W s () = x ∈ S : f n (x) ∈ S for all n ∈ N, lim dist ( f n (x), ) = 0 . n→∞

(10.4)

Similarly we define the unstable set W u () of  as the stable set of  with respect to f −1 . It follows immediately from the properties of the parabolic horseshoe that  W s/u () = W s/u (x). (10.5) x∈

Note that (10.5) is also a consequence of the Shadowing Lemma. Theorem 10.4. Let f be a parabolic horseshoe map of smooth type having a Lipschitz continuous unstable foliation. Then dim H W s/u () = t u/s + 1 < 2.

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Proof. We only prove the formula for the dimension of the stable set. The proof for the unstable set is analogous. Note that by Theorem 7.4, t u < 1, which gives the right-hand side inequality. Combining that  has a local product structure with (10.5) implies that it suffices to prove that ⎞ ⎛  s dim H ⎝ Wloc (y)⎠ = t u + 1 (10.6) u (x)∩ y∈Wloc

u ( p) ∩ . Since  has a local product structure, for all x ∈ . Let x ∈ . Set A x = Wloc there exists a homeomorphism  s Wloc (y), (10.7) H : A x × (−1, 1) → y∈A x s (y) for all y ∈ A . Moreover, since f with the property that H (y × (−1, 1)) = Wloc x is of smooth type and since f has a Lipschitz continuous unstable foliation it follows that H as well as H −1 are Lipschitz continuous. Applying Theorem 10.2 completes the proof.  

11. Generalized Physical Measures Let f be a parabolic horseshoe map of smooth type having a Lipschitz continuous unstable foliation. In this section provide a classification for f having a generalized physical measure. Given µ ∈ M we define the basin of µ by 

 n−1 1 B(µ) = x ∈ S : f (x) ∈ S for all n ∈ N, lim δ f i (x) = µ . n→∞ n n

(11.1)

i=0

Here δ f i (x) denotes the Dirac-δ measure on f i (x). The basin of µ is sometimes also called the set of future generic points of µ, see [DGS] and [Ma]. A measure µ ∈ M E is called a physical measure if B(µ) has positive Lebesgue measure. Obviously, B(µ) ⊂ W s ().

(11.2)

Therefore, by Theorem 10.4 the map f can not have a physical measure. Following [Wo] we say that µ ∈ M E is a generalized physical measure if it is non-atomic and B(µ) is as large as possible in the sense that dim H B(µ) = dim H W s ().

(11.3)

We now prove a formula for the Hausdorff dimension of the basin of a hyperbolic measure. Proposition 11.1. Let µ ∈ M E \ {µω }. Then dim H B(µ) =

hµ( f ) + 1. λu (µ)

(11.4)

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745

Proof. Note that by Lemma 10.1, λu (µ) > 0; hence the right-hand side of (11.4) is well-defined. It is easy to see that if y ∈ f (S) then s y ∈ B(µ) if and only if ∃x ∈  ∩ B(µ) with y ∈ Wloc (x).

(11.5)

Combining (11.5) with property (e) of the parabolic horseshoe (see Sect. 5) it follows that it is sufficient to show that for each x ∈ , ⎛ ⎞  hν ( f ) s Wloc (y)⎠ = dim H ⎝ + 1, (11.6) λu (ν) y∈A x

u (x) ∩ B(µ). Pick x ∈ . The same methods used by Manning [Ma] where A x = Wloc in the context of hyperbolic surface diffeomorphisms (see [Me] for the analogous result in the non-uniformly hyperbolic setting) can be used to show that

dim H A x =

hµ( f ) . λu (µ)

(11.7)

Therefore, (11.6) can be shown analogously as Theorem 10.4 by using the bi-Lipschitz continuous homeomorphism H .   Recall that φu = log |Du f | :  → R. We now present our main result about generalized physical measures. Theorem 11.2. Let f be a parabolic horseshoe of smooth type having a Lipschitz continuous unstable foliation. Then the following are equivalent: (i) (ii) (iii) (iv) (v)

f admits a generalized physical measure; The potential −t u φu has more than one (finite) equilibrium state; The potential −t u φu has precisely two ergodic (finite) equilibrium states; P u is not differentiable at t u ; β . t u > 2 β+1

Proof. (i)⇒(ii): Let µ be a generalized physical measure of f . It follows from Theorem 10.4 and Proposition 11.1 that t u = h µ ( f )/λu (µ). Thus, µ is an ergodic equilibrium state of the potential −t u φu . Using that µ = µω implies (ii). The implication (ii)⇒(iii) is a consequence of Theorem 8.3. Moreover, (iii)⇒(iv) follows from [J, Cor. 1]. The implication (iv)⇒(v) follows from [J, Cor. 1] and Theorem 8.1. Finally, if (v) holds, then by Theorem 8.1 there exists a probability ergodic equilibrium state µ of the potential −t u φu with µ = µω . Hence, t u = h µ ( f )/λu (µ), and we may conclude from Theorem 10.4 and Proposition 11.1 that µ is a generalized physical measure for f .   As an application of Theorem 11.2 we construct parabolic horseshoe maps with as well as without generalized physical measures. Corollary 11.3. There exists a parabolic horseshoe map having a generalized physical measure as well as one having no generalized physical measure. Proof. We first construct an example of a parabolic horseshoe map having a generalized physical measure. Pick 0 < β < 1, η > 1 and λu > 2 such that β log 2 . >2 log η + log λu β +1

(11.8)

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Let f be a parabolic horseshoe map as defined in Example 1 with the corresponding constants β, η and λu . In particular, f is of smooth type and has a Lipschitz continuous unstable foliation. It follows from (5.4) that λu (ν) ≤ log λu + log η

(11.9)

for all ν ∈ M. Let µ denote the measure of maximal entropy of f , i.e. the unique measure satisfying h µ ( f ) = log 2. Therefore, (11.9) and Theorem 1 in [Me] imply that β t u > 2 β+1 . We now may conclude from Theorem 11.2 that f has a generalized physical measure. The existence of a parabolic horseshoe map having no generalized physical measure can be easily seen. Just pick any parabolic horseshoe map f of smooth type having a Lipschitz continuous unstable foliation (for instance a map as in Example 1) with β ≥ 1. Recall that t u < 1 (see Theorem 7.4); hence t u < 2β/(β + 1), and therefore, Theorem 11.2 implies that f does not have a generalized physical measure.   Corollary 11.4. The existence of a generalized physical measure is not a (topological) conjugacy invariant. Proof. By Corollary 11.3 there exist parabolic horseshoe maps f k , k = 1, 2 such that f 1 has a generalized physical measure and f 2 does not have a generalized physical measure. Since f 1 |1 and f |2 are both topological conjugate to the shift map σ : 2+− → 2+− , the result follows.   12. Measures of Maximal Dimension In this section we discuss the existence of ergodic measures of maximal dimension for a particular subclass of parabolic horseshoe maps. In particular, we provide a criterion which guarantees that no ergodic measure of maximal dimension exists. Recall that in this section the notion of equilibrium states is meant in the space of probability measures. Let f : S → R2 be a parabolic horseshoe map. We say that f has constant contraction rate if there is 0 < c < 1 such that |Ds f (x)| = c for all x ∈ . For example the almost linear parabolic horseshoe maps in Example 1 have constant contraction rate c < 1/2. Given µ ∈ M we define the Hausdorff dimension of µ by dim H µ = inf{dim H A : µ(A) = 1}.

(12.1)

Following [BW1] we say that µ ∈ M E is an ergodic measure of maximal dimension if def

dim H µ = sup{dim H ν : ν ∈ M E } = δ( f ).

(12.2)

It follows from work in [BW2] that the definition of δ( f ) in (12.2) is the same when the supremum is taken over all (not necessarily ergodic) measures in M. Let µ ∈ M \ {µω }. It follows from Young’s formula [Y] (also using Lemma 10.1) that if µ is ergodic, then dim H µ = d(µ), where def

d(µ) = h µ ( f )

! 1 1 − . λu (µ) λs (µ)

(12.3)

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747

We now define a one-parameter family of measures (νt )t∈[0,t u ] which will be crucial for the analysis of measures of maximal dimension. For t ∈ [0, t u ) we define νt to be the unique equilibrium state of the potential −tφu . Note that νt is well-defined. Indeed, P u is differentiable in [0, t u ). This is a consequence of Theorem 7.6 and the fact that f | has a unique measure of maximal entropy. Thus, by [J, Cor. 1] the potential −tφu has a unique equilibrium state. Next, we define νt u . In the case when the potential −t u φu has more than one equilibrium state we define νt u to be the unique hyperbolic ergodic equilibrium state of −t u φu . Otherwise, we define νt u = µω . The results of this section will be based on a careful analysis of the dimension of the measures νt . For simplicity we write h(t) = h νt ( f ) and λu (t) = λu (νt ) for all t ∈ [0, t u ]. Thus, P u (t) = h(t) − tλu (t).

(12.4)

It follows from standard properties of the topological pressure (see for example [J]) that if P u is differentiable at t0 then d P u (t0 ) = −λu (t0 ). dt

(12.5)

We first prove a preliminary result. Lemma 12.1. We have the following: (i) If νt u = µω then P u ∈ C 1 ([0, t u ]); (ii) If νt u = µω then P u ∈ C 1 ([0, t u )). Proof. We already know that P u is real-analytic on (0, t u ) (see Theorem 7.6). Therefore, we only have to consider t = 0 and t = t u . Since ν0 is the unique measure of maximal entropy it follows from [J] that P u is differentiable at 0. Similarly, if νt u = µω then P u is differentiable at t u . We claim that if νt u = µω then P u is C 1 in a left neighborhood of t u . Let tn ≤ t u with tn → t u for n → ∞. By (12.5) is suffices to show that λu (tn ) → λu (t u ) for n → ∞. By convexity of P u and (12.5), λu is decreasing. Thus, a = limn→∞ λu (tn ) exists. By compactness of M there exists µ ∈ M such that µ is a weak* cluster point of the measures νtn . Since λu is continuous on M we may conclude that λu (µ) = a. Using that the entropy map ν
→ h ν ( f ) is upper semi-continuous on M (also using (12.4)) we obtain that µ is an equilibrium state of the potential −t u φu ; hence µ = νt u which proves the claim. The proof of the statement that P u is C 1 in a right neighborhood of 0 is entirely analogous.   Since P u is real-analytic in (0, t u ), (12.5) implies that λu is also real-analytic in (0, t u ). Hence, by (12.4), h is also real-analytic in (0, t u ). Moreover, t0 dλu (t0 ) dh(t0 ) = dt dt

(12.6)

for all t0 ∈ (0, t u ). We now prove another preliminary result. Lemma 12.2. The functions λu and h are continuous in [0, t u ]. Moreover, {λu (t) : t ∈ [0, t u ]} = [λu (t u ), λu (0)].

(12.7)

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Proof. We first consider the case νt u = µω . In this case (12.4), (12.5) and Lemma 12.1 imply that the functions λu and h are continuous in [0, t u ]. Moreover, by (12.5), {λu (t) : t ∈ [0, t u ]} is a compact interval; thus, (12.7) follows from the fact that t
→ λu (t) is decreasing in [0, t u ]. Next, we consider the case νt u = µω . Similarly as above, we can show that λu is continuous in [0, t u ). We now prove the continuity of λu at t u . Let E S(t u ) ⊂ M be the set of all equilibrium states of the potential −t u φu . It is well-known that E S(t u ) is a compact convex set whose extreme points are the ergodic measures in E S(t u ), see e.g. [J]. Therefore, Theorem 8.3 implies that E S(t u ) = {sµω + (1 − s)νt u : s ∈ [0, 1]}.

(12.8)

Since λu is decreasing in [0, t u ) it follows that limt→t u − λu (t) exists. Since M is compact, there exist a sequence (tn )n∈N converging to t u from the left and a measure µ ∈ M such that µ is a weak* limit of the sequence of measures (νtn )n∈N . Since the entropy map is upper semi-continuous, we may conclude from the continuity of λu that µ is an equilibrium state of the potential −t u φu . Thus, µ = sµω +(1−s)νt u for some s ∈ [0, 1]. We claim that s = 0. Indeed, otherwise there would exist  > 0 and δ > 0 such that λu (t) < λu (t u )− for all t < t u with t u −t < δ. But this contradicts the convexity of P u and the fact that νt u is an equilibrium state of −t u φu . We conclude that λu is continuous at t0 . Finally, the continuity of h and identity (12.7) can be shown analogously as in the case νt u = µω (see above).   The following result shows that each ergodic measure of maximal dimension must be contained in the family of measures (νt )t∈[0,t u ] . Theorem 12.3. Let f be a parabolic horseshoe map of smooth type with constant contraction rate. Then δ( f ) = sup{dim H νt : t ∈ [0, t u ]}.

(12.9)

Moreover, if µmax is an ergodic measure of maximal dimension for f then there is t ∈ [0, t u ] such that µ = νt . Proof. Since f has constant contraction rate, there exists 0 < c < 1 such that λs (ν) = log c

(12.10)

for all ν ∈ M. Let µ ∈ M E . We claim that there exists t ∈ [0, t u ] such that dim H µ ≤ dim H νt . Note that the claim clearly implies (12.9). To prove the claim we first consider the case λu (µ) > λu (ν0 ). Using that ν0 is the unique measure of maximal entropy, we conclude from (12.3) and (12.10) that d(µ) < d(ν0 ). Next we assume that λu (µ) ∈ [λu (ν0 ), λu (νt u )]. It follows from (12.7) that there exists t ∈ [0, t u ] with λu (νt ) = λu (µ). We now may conclude from the definition of νt and (12.4) that h νt ( f ) ≥ h µ ( f ). As before, (12.3) and (12.10) gives d(µ) ≤ d(νt ). Finally, we consider the case λu (µ) < λu (νt u ). Clearly, this is only possible if νt u = µω . Hence, νtu is the unique ergodic hyperbolic equilibrium state of the potential −t u φu . We now may conclude from (12.3), (12.10) and the definition of t u that dim H νt u = t u −

h νt u ( f ) . log c

(12.11)

Ergodic Theory of Parabolic Horseshoes

749

On the other hand, by (12.4), h νt u ( f ) ≥ h µ ( f ). We conclude that d(νt u ) > d(µ). This completes the proof of the claim. Assume now that µmax is an ergodic measure of maximal dimension for f . Applying the same arguments as before to µmax instead of µ implies that there exists tmax ∈ [0, t u ] such that λu (µmax ) = λu (νtmax ) and h νtmax ( f ) ≥ h µmax ( f ). On the other hand, the fact that µmax is an ergodic measure of maximal dimension and (12.3) imply h µmax ( f ) ≥ h νtmax ( f ); hence h µmax ( f ) = h νtmax ( f ). We conclude that µmax is an ergodic equilibrium state of the potential −tmax φu . Using that µmax = µω implies µmax = νtmax . This completes the proof.   We now present the main result of this section. Theorem 12.4. Let f be a parabolic horseshoe map of smooth type with constant contraction rate c. Suppose that µω is the unique equilibrium state of the potential −t u φu and that u

P (t) u η = inf : t ∈ [0, t ) > 0. (12.12) λu (νt )2 Then there exists c0 = c0 (η) > 0 such that if 0 < c < c0 , then f has no ergodic measure of maximal dimension. Proof. Note that in order to show that f has no ergodic measure of maximal dimension it suffices to prove that t
→ d(νt )

(12.13)

is strictly increasing on [0, t u ). This follows from Theorem 12.3 and the fact that dim H νt u = 0. We claim that P u is not affine on [0, t u ). Indeed, because otherwise (12.5) would imply that the measure of maximal entropy ν0 is an equilibrium state of the potential −t u φu , in contradiction to the hypothesis. Since P u a real-analytic convex function on (0, t u ), we may conclude that d 2 P u (t0 ) ≥ 0, dt 2

(12.14)

where all the zeros of d 2 P u /d 2 t in (0, t u ) are isolated. Moreover, νt0 = µω for all t0 ∈ [0, t u ). Thus νt0 is a hyperbolic measure (see Lemma 10.1). Let now t0 ∈ (0, t u ). It follows from (12.3), (12.5), (12.6) and an elementary calculation that ! h(t0 ) h(t0 ) − λu (t0 ) log c u (t ) P t0 λu (t0 ) 0 − = −λu (t0 ) 2 λu (t0 ) log c ! 2 u u t0 d P (t0 ) P (t0 ) + = dt 2 λu (t0 )2 log c

d d d(νt0 ) = dt dt

def

=

d 2 P u (t0 ) A(t0 ). dt 2

(12.15)

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M. Urba´nski, C. Wolf

We define c0 = exp(−t u /η). It follows from an easy computation that if 0 < c < c0 then A(t0 ) > 0 for all t0 ∈ (0, t u ). Therefore, (12.15) implies that d d(νt0 ) ≥ 0, dt

(12.16)

d d(νt0 ) in (0, t u ) are isolated. We conclude that d(νt ) is strictly where all the zeros of dt u increasing in [0, t ). This completes the proof.  

Remark. (i) We note that by L’Hospital’s rule the hypotheses (12.12) holds for instance in the case when P u has uniformly bounded second derivatives on [0, t u ). (ii) A strategy to construct a parabolic horseshoe map having no ergodic measure of maximal dimension is as follows: Consider an almost linear parabolic horseshoe map f as given in Example 1 such that −t u log |Du f | has only one ergodic equilibrium state (for example one can choose β ≥ 1) and that (12.12) holds. In particular, f is of smooth type and has constant contraction rate; hence Theorem 12.4 applies. As mentioned in Remark (i) after Example 1, we may decrease the contraction rate c without changing Du f on W u (ω), insuring that c < c0 . Therefore, Lemma 7.2 guarantees that P u (t), t ∈ [0, t u ) is not affected by this procedure, and therefore, (12.12) still holds. Finally, applying Theorem 12.4 shows that f does not have an ergodic measure of maximal dimension.

References [A] [BW1] [BW2] [DU] [DV] [DGS] [F] [HMU] [J] [KFS] [K] [Ma] [MM] [MU1] [MU2] [Me] [PU]

Aaronson, J.: An introduction to infinite ergodic theory. Mathematical Surveys and Monographs, 50. Providence, RI: Amer. Math. Soc., 1997 Barreira, L, Wolf, C.: Measures of maximal dimension for hyperbolic diffeomorphisms. Commun. Math. Phys. 239, 93–113 (2003) Barreira, L., Wolf, C.: Pointwise dimension and ergodic decompositions. Erg. Theory Dyn. Syst. 26(3), 653–671 (2006) Denker, M., Urba´nski, M.: On the existence of conformal measures. Trans. A.M.S. 328, 563–587 (1991) Diaz, L., Viana, M.: Discontinuity of hausdorff dimension and limit capacity on arcs of diffeomorphisms. Erg. Theory Dyn. Syst. 9(3), 403–425 (1989) Denker, M., Grillenberger, C., Sigmund, K.: Ergodic theory on compact spaces, Lecture Notes in Math. 527. Berlin: Springer-Verlag, 1976 Falconer, K.: Fractal Geometry: Mathematical Foundations and applications. New York: Wiley, 2003 Hanus, P., Mauldin, D., Urba´nski, M.: Thermodynamic formalism and multifractal analysis of conformal infinite iterated function systems. Acta Math. Hungarica 96, 27–98 (2002) Jenkinson, O.: Rotation, Entropy, and Equilibrium States. Trans. Amer. Math. Soc. 353, 3713–3739 (2001) Kornfeld, P., Fomin, S.V., Sinai, Y.G.: Ergodic Theory. Berlin-Heidelberg-New York: Springer, 1982 Krengel, U.: Entropy of conservative transformations. Z. Wahr. Verw. Geb. 7, 161–181 (1967) Manning, A.: A relation between Lyapunov exponents, Hausdorff dimension and entropy. Ergodic Theory Dy. Syst. 1(4), 451–459 (1982) Manning, A., McCluskey, H.: Hausdorff dimension for Horseshoes. Erg. Theory Dyn. Syst. 3, 251–260 (1983) Mauldin, D., Urba´nski, M.: Parabolic iterated function systems. Erg. Th. & Dyna. Syst. 20, 1423–1447 (2000) Mauldin, D., Urba´nski, M.: Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets. Cambridge: Cambridge Univ. Press, 2003 Mendoza, L.: The entropy of C 2 surface diffeomorphisms in terms of Hausdorff dimension and a Lyapunov exponent. Erg. Theory Dyn. Syst. 5(2), 273–283 (1985) Przytycki, F., Urba´nski, M.: Fractals in the Plane - Ergodic Theory Methods. To appear Cambridge Univ. Press, available on Urba´nski’s webpage, http://www.math.unt.edu/~urbanski/

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[Ra] [Ru] [U1] [U2] [U3] [U4] [Wa] [Wo] [Y]

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Rams, M.: Measures of maximal dimension for linear horseshoes. Real Analysis Exchange 31, 55–62 (2006) Ruelle, D.: Thermodynamic formalism. Reading, MA: Addison-Wesley, 1978 Urba´nski, M.: Parabolic Cantor sets. Preprint 1995, available on Urba´nski’s webpage, http://www. math.unt.edu/~urbanski/ Urba´nski, M.: Parabolic cantor sets. Fund. Math. 151, 241–277 (1996) Urba´nski, M.: Geometry and ergodic theory of conformal nonrecurrent dynamics. Erg. Th. Dyn. Syst. 17, 1449–1476 (1997) Urba´nski, M.: Hausdorff measures versus equilibrium states of conformal infinite iterated function systems. Periodica Math. Hung. 37, 153–205 (1998) Walters, P.: An introduction to ergodic theory. Graduate Texts in Mathematics 79, Berlin-Heidelberg-New York: Springer, 1981 Wolf, C.: Generalized physical and SRB measures for hyperbolic diffeomorphisms. J. Stat. Phys. 122(6), 1111–1138 (2006) Young, L.-S.: Dimension, entropy and Lyapunov exponents. Erg. Theory Dyn. Syst. 2, 109–124 (1982)

Communicated by G. Gallavotti

Commun. Math. Phys. 281, 753–774 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0493-6

Communications in

Mathematical Physics

Entanglement of Positive Definite Functions on Compact Groups J. K. Korbicz1,2 , J. Wehr2,3 , M. Lewenstein2 1 Dept. d’Estructura i Constituents de la Matèria, Universitat de Barcelona, 647 Diagonal,

08028 Barcelona, Spain. E-mail: [email protected]

2 ICREA and ICFO–Institut de Ciències Fotòniques, Mediterranean Technology Park,

08860 Castelldefels (Barcelona), Spain

3 Department of Mathematics, University of Arizona, 617 N. Santa Rita Ave.,

Tucson, AZ 85721-0089, USA Received: 13 June 2007 / Accepted: 17 October 2007 Published online: 15 May 2008 – © Springer-Verlag 2008

Abstract: We define and study entanglement of continuous positive definite functions on products of compact groups. We formulate and prove an infinite-dimensional analog of the Horodecki Theorem, giving a necessary and sufficient criterion for separability of such functions. The resulting characterisation is given in terms of mappings of the space of continuous functions, preserving positive definiteness. A relation between the developed group-theoretical formalism and the conventional one, given in terms of density matrices, is established through the non-commutative Fourier analysis. It shows that the presented method plays the role of a “generating function” formalism for the theory of entanglement. 1. Introduction Entanglement is a property of states of composite quantum mechanical systems. This concept lies at the very heart of quantum mechanics, and it concerns all of the important aspects of quantum theory: from philosophical aspects [1,2], through physical [3] and mathematical [4,5] fundamentals1 , to applications in quantum information and metrology [7]. The importance of entangled states for the understanding of quantum theory was recognized quite early, mainly thanks to Einstein (e.g. in the famous EPR paper by Einstein, Podolsky, and Rosen [8]). Only with the advent of new experimental techniques in recent years, it became clear that entanglement may in fact also be used as a resource for transmission and processing of (quantum) information, e.g. for quantum cryptography or quantum computing (for a recent review, see Ref. [4] ; see also Ref. [7]). In the present work we develop a novel framework for studying quantum entanglement, based on analysis of continuous functions on compact groups. With respect to the standard formalism of entanglement theory, our approach plays a role analogous 1 For a description of positive maps from a physical point of view, see e.g. Ref. [6].

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J. K. Korbicz, J. Wehr, M. Lewenstein

to that of a “generating function” method—various group-theoretical objects serve as “generating functions” for the corresponding families of operator-algebraic objects (like density matrices, positive maps, etc), operating in different dimensions. This allows one to formulate and address the questions of entanglement theory in a unified, dimensionwise, way. Before we proceed with the group-theoretical formalism, let us first recall some basic facts and define the notion of entanglement precisely. A quantum system is associated with a Hilbert space H, which we will assume to have a countable basis. A state of the system is then represented by a positive, trace-class operator
(a density matrix), satisfying normalization condition tr
= 1. If the system under consideration is composite, i.e. it can be thought of being composed of two subsystems A and B, each of which is treated as an independent individual, then, according to the postulates of quantum theory, the Hilbert space of the system is H = HA ⊗ HB . The following definition thus makes sense [9]: Definition 1.1. A state
on HA ⊗ HB is called separable if it can be approximated in the trace norm by convex combinations of the form: K


pm |xm xm | ⊗ |ym ym |, where xm ∈ HA , ym ∈ HB ,

m=1

pm
0,

K


pm = 1. (1)

m=1

Otherwise
is called entangled. This definition can be easily generalized to multipartite systems with more than two parties involved. In the light of Definition 1.1 a natural question arises, known as the separability problem: Given a state
decide if it is separable or not. The problem turns out to be computationally very hard: although efficient algorithms employing positive definite programming methods exist in lower dimensions [10] (for a specific formulation of semi-definite approach for 2 ⊗ N systems see Ref. [11]), it has been proven that the problem belongs to the N P complexity class as dimensions of the Hilbert spaces involved grow [12]. In terms of operational entanglement criteria up to date there are only partial answers known, in both finite and infinite dimensions. We briefly quote below few basic results, referring the reader to Ref. [4] for a complete overview. One astonishingly powerful, given its simplicity, necessary criterion for separability follows immediately from the definition of separable states [13,14]: Theorem 1.1 (Positivity of Partial Transpose (PPT)). If a state
on HA ⊗ HB is separable then the partially transposed operator
TB := (1A ⊗ T )
is positive, where T is a transposition map and 1A is the identity operator on HA . In the lowest non-trivial dimensions dimHA = dimHB = 2 and dimHA = 2, dimHA = 3 PPT criterion provides both necessary and sufficient conditions for separability (see Ref. [14], Theorem 3). However, in higher dimensions there exist states, called PPT or bound entangled, which satisfy the PPT criterion, but are nevertheless entangled. The first examples of such states were constructed in Ref. [15] (although in a different context of, so called, indecomposable maps) and in Ref. [16]. In infinite dimension, a complete solution to the separability problem exists only for a special family of states—so called Gaussian states [17]. As mentioned above the separability problem is connected to other open mathematical problems. In their fundamental work [14] Horodecki et al. established an important link

Entanglement of Positive Definite Functions on Compact Groups

755

between this problem and the problem of characterization of positive maps on finitedimensional matrix algebras (cf. Ref. [14], Theorem 2; see also Refs. [5,15,18,19] ): Theorem 1.2 (M., P., and R. Horodecki). Let L(H) denote the space of linear operators on H and let
∈ L(HA ⊗HB ) be a density matrix on a finite dimensional Hilbert space HA ⊗HB . Matrix
is separable if and only if for all linear maps  : L(HB ) → L(HA ) preserving positive operators (such maps are called positive), (1A ⊗ )

0 as an operator on HA ⊗ HA . The starting point for the present work is the non-commutative Fourier analysis on a compact group, which we proposed to employ for studying entanglement in Ref. [20] (cf. Ref. [21] where the same method was used for a rigorous derivation of the classical limit of quantum state space). Namely, one can pass from operators A ∈ L(HA ⊗ HB ) to their non-commutative Fourier transforms2 in two steps: i) identify the spaces HA , HB with representation spaces of unitary, irreducible representations πα , τβ of some compact groups G 1 and G 2 respectively (there are no a priori restrictions on G 1 , G 2 apart from possessing representations in suitable dimensions); ii) pass from A to a function ϕ A : G 1 × G 2 → C, the non-commutative Fourier transform of A, through:   A → ϕ A (g1 , g2 ) := tr Aπα (g1 ) ⊗ τβ (g2 ) .

(2)

The above transform is called non-commutative, since apart from the trivial case dimHA = dimHB = 1, groups G 1 and G 2 are necessarily non-Abelian. In case A =
is a quantum state, the corresponding function ϕ
is called a non-commutative characteristic function of
. The transformation (2) is invertible—one can recover A from ϕ A . Hence, one expects that for density matrices their non-commutative characteristic functions should encode entanglement in some way [20]. This is indeed the case and in what follows we define and study the notion of separability for suitably generalized noncommutative characteristic functions (general continuous positive definite functions on G 1 × G 2 ; cf. Definition 2.1). We then prove an analog of the Horodecki Theorem 1.2 for such functions, which constitutes the main result of the paper. Since the framework we work in is countably infinite-dimensional (unless both G 1 , G 2 are finite) our result can be viewed as a generalization of the Horodecki Theorem to an infinite-dimensional setting. The usual quantum-mechanical formalism, given by density matrices, and the presented group-theoretical one are then shown to be related through non-commutative harmonic analysis. In particular, by employing the non-commutative Fourier transform we demonstrate how our approach turns out to be a “generating function” method for the theory of entanglement. Let us finally remark that the formalism of non-commutative Fourier transform (2) is closely related to that of generalized coherent states [23]. The difference is that in the coherent state formalism one assigns to an operator A a function (called P-representation of A), which is defined not on the whole group G, but on a homogeneous space G/H , where H is an isotropy subgroup of a fixed vector. However, unlike the non-commutative Fourier transform ϕ A , P-representation is generally non-unique (e.g. in the SU (2) case) and does not encode positivity of a density matrix in a simple manner. For some applications of generalized coherent states to the study of entanglement see e.g. Refs. [24]. 2 We note that in Ref. [22] the term “noncommutative Fourier transform” is used in a slightly different— though very closely related—sense.

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2. Preliminary Notions In the main part of the work G 1 , G 2 will be compact groups. The principal object of our study are continuous positive definite functions on the product group G 1 × G 2 . But first we recall some basic definitions and facts, valid for any locally compact G (see e.g. Refs. [25–27] for a complete exposition). Definition 2.1. A continuous complex function ϕ on a group G with the Haar measure dg is called positive definite if it is bounded and satisfies:  dgdh f (g)ϕ(g −1 h) f (h)
0 (3) (bar denotes complex conjugation) for any continuous function f with compact support. We will denote by P(G) the set of positive definite functions on G and by P1 (G) its subset consisting of the functions which satisfy the normalization ϕ(e) = 1, where e is the neutral element of G. P(G) is a closed convex cone in C(G) — the space of continuous complex-valued functions on G equipped with the topology of uniform convergence on compact sets, called compact convergence in the sequel. The structure of P(G) is described by the following deep, fundamental result of representation theory, often referred to as the GNS construction (see e.g. Ref. [25], Theorem 3.20; Ref. [26], Theorem 13.4.5): Theorem 2.1 (Gel’fand, Naimark, Segal). With every ϕ ∈ P(G) we can associate a Hilbert space Hϕ , a unitary representation πϕ of G in Hϕ and a vector vϕ , cyclic for πϕ , such that: ϕ(g) = vϕ |πϕ (g)vϕ .

(4)

The representation πϕ is unique up to a unitary equivalence. The above result provides a tool for a systematic study of P(G) in terms of representations of G (and conversely). In the sequel we will need some basic properties of positive-definite functions. While they all follow from the definition by standard, elementary arguments, we find the proofs based on the GNS representation particularly transparent. A function ϕ will be called pure if πϕ is irreducible. Pure normalized functions are the extreme points of P1 (G) (cf. Ref. [25], Theorem 3.25); we denote their set by E1 (G). Every ϕ ∈ P1 (G) is a limit, in the topology of compact convergence, of convex combinations of extreme points of P1 (G) (cf. Ref. [26], Theorem 13.6.4): g →

N


pm εm (g), where εm ∈ E1 (G), pm
0,

m=1

N


pm = 1.

(5)

m=1

There is also an integral representation (provided G is separable as a topological space), sometimes called the generalized Bochner Theorem. Namely, for any ϕ ∈ P1 (G) there exists a probability measure µϕ concentrated on E1 (G) such that (cf. Ref. [26], Propo. 13.6.8):  ϕ(g) = dµϕ (ε) ε(g) for any g ∈ G. (6) E1 (G)

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From this point on, we assume that G 1 , G 2 are compact and consider positive definite functions on G 1 × G 2 . Let us introduce the algebraic tensor product C(G 1 ) ⊗ C(G 2 ) as the space of finite (complex) linear combinations of product functions f ⊗ξ : (g1 , g2 ) → f (g1 )ξ(g2 ). Then C(G 1 )⊗C(G 2 ) is uniformly dense in C(G 1 × G 2 ). This standard fact follows, for example, from the Stone-Weierstrass Theorem (see e.g. Ref. [28]). Every product φ ⊗ ψ, where φ ∈ P1 (G 1 ), ψ ∈ P1 (G 2 ), is positive definite on G 1 × G 2 , since, by the GNS Theorem 2.1, φ(g1 )ψ(g2 ) = vφ ⊗ vψ |πφ (g1 ) ⊗ πψ (g2 )vφ ⊗ vψ  which is of the form (4) on G 1 × G 2 . It follows that convex combinations of such products are positive definite and hence so are uniform limits of such convex combinations. The resulting class of positive definite functions, introduced formally in the next definition, is our fundamental object of study (compare Definition 1.1). Definition 2.2. We define Sep0 as the set of all functions ϕ ∈ P1 (G 1 × G 2 ) which can be represented as finite convex combinations ϕ(g1 , g2 ) =

K


pm εm (g1 )ηm (g2 ), where εm ∈ E1 (G 1 ), ηm ∈ E1 (G 2 ).

(7)

m=1

A function ϕ ∈ P1 (G 1 × G 2 ) is called separable if it is a uniform limit of elements of Sep0 . The set of separable functions is denoted by Sep. Functions which are not separable are called entangled. The definitions of separable and entangled functions generalize without any change to arbitrary (i.e. not necessarily normalized) positive definite functions. This includes our main result, Theorem 3.2, together with its proof (since for a nonzero positive definite function ϕ(e1 , e2 ) = ||ϕ||∞ > 0, we can replace ϕ by ϕ/ϕ(e1 , e2 ) and reduce the proof to the normalized case). The normalization is, however, natural from the physical point of view. Geometrically E1 (G 1 ) × E1 (G 2 ) is embedded into E1 (G 1 × G 2 ) through the map (ε, η) → ε ⊗ η. Then Sep is a closed convex hull of E1 (G 1 ) × E1 (G 2 ). We note that every ϕ ∈ Sep admits an integral representation:  ϕ(g1 , g2 ) = dµϕ (ε, η) ε(g1 )η(g2 ) for any (g1 , g2 ) ∈ G 1 × G 2 , (8) E1 (G 1 )×E1 (G 2 )

but we will not use this fact.

3. Necessary and Sufficient Criterion for Separability of Positive Definite Functions The main problem we would like to address is that of finding an intrinsic characterization of separable functions ϕ ∈ Sep. This is known as the generalized separability problem [20]. By Eq. (4) for every positive definite function φ, φ(g −1 ) = φ(g) and this function is again positive definite. It now follows immediately from Definition 2.2 and from uniform closedness of P(G 1 × G 2 ) in C(G 1 × G 2 ) that: Theorem 3.1. If ϕ ∈ Sep then the function (g1 , g2 ) → ϕ(g1 , g2−1 ) is positive definite.

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The above simple criterion is only a necessary condition—there are functions satisfying it which are nevertheless entangled. This can be seen by noting that Theorem 3.1 is a group-theoretical analog of the PPT criterion, given by Theorem 1.1, (we will show it in Sect. 5; see also Ref. [20], Theorem 2) and, as we mentioned in the Introduction, there exist PPT entangled (or, equivalently, bound entangled) quantum states [16]. A natural question arises whether one obtains a complete characterization of separable functions when in place of the inverse g → g −1 one considers all possible linear maps of functions, preserving positive definiteness. The affirmative answer is the main result of our work: Theorem 3.2. A function ϕ ∈ P1 (G 1 × G 2 ) is separable if and only if for every bounded linear map : C(G 2 ) → C(G 1 ), such that P(G 2 ) ⊂ P(G 1 ), function (id ⊗ )ϕ is positive definite on G 1 × G 1 . Tensor product id ⊗ : C(G 1 × G 2 ) → C(G 1 × G 1 ) is defined in the natural way: we first define it on the algebraic product C(G 1 ) ⊗ C(G 2 ) and then extend by continuity to all of C(G 1 × G 2 ). The above theorem is a group-theoretical analog of the Horodecki Theorem 1.2. In fact, we derive a version of the Horodecki result as a corollary in Sect. 4 (cf. Theorem 4.2). We will adopt the standard terminology of entanglement theory and say that an entangled function ϕ is detected by a map if the function (id ⊗ )ϕ is not positive definite. As mentioned earlier, the above theorem, as well as the following proof, hold for arbitrary positive definite functions, but in order to use more natural concepts from the physical point of view we state and prove it for normalized ones. The proof in one K direction is immediate and follows directly from Definition 2.2: (id⊗ ) m=1 pm εm ⊗ K ηm = m=1 pm εm ⊗ ηm ∈ P(G 1 × G 1 ) and since P(G 1 × G 1 ) is uniformly closed in C(G 1 × G 1 ) this holds on all of Sep. For the proof in the other direction, let C(G 1 × G 2 ) denote the space of continuous linear functionals on C(G 1 × G 2 ) (the space dual to C(G 1 × G 2 )). Since Sep is a closed convex set, it follows from the Hahn-Banach Theorem (see e.g. Ref. [29], Theorem V.4) that for every ϕ ∈ / Sep there exists a functional l ∈ C(G 1 × G 2 ) and a real number γ , such that: Rel(ϕ) < γ  Rel(σ ) for any σ ∈ Sep,

(9)

where Rel denotes the real part of the functional l: Rel(ϕ) := Re[l(ϕ)]. From the Riesz Representation Theorem (see e.g Ref. [29], Theorem IV.17) we know that each linear functional l on C(G 1 × G 2 ) can be uniquely represented by a complex measure µl with finite total variation |µl | on G 1 × G 2 . Denoting the space of such measures by M(G 1 × G 2 ) we have: C(G 1 × G 2 ) = M(G 1 × G 2 ). We will interchangeably treat elements of C(G 1 × G 2 ) as either linear functionals or as the corresponding measures. To work with the normalized functions ϕ which we are interested in, it is convenient to introduce a modification of the functional l in (9): L := l − γ δ(e1 ,e2 ) , where δ(e1 ,e2 ) is the Dirac delta (point mass), concentrated at the neutral element (e1 , e2 ) of G 1 × G 2 . Hence, for every entangled ϕ ∈ P1 (G 1 × G 2 ) there exists L ∈ C(G 1 × G 2 ) such that: ReL(ϕ) < 0  ReL(σ ) for any σ ∈ Sep.

(10)

As an easy consequence of the above condition we obtain the following lemma, crucial for the rest of the proof:

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Lemma 3.1. A function ϕ is separable if and only if for every functional L ∈ C(G 1 × G 2 ) , satisfying ReL(ψ1 ⊗ ψ2 )
0 for every ψ1 ∈ P1 (G 1 ), ψ2 ∈ P1 (G 2 ), we have ReL(ϕ)
0. K K Indeed, for every Sep0 ϕ = m=1 pm εm ⊗ηm , L(ϕ) = m=1 pm L(εm ⊗ηm )
0 and by continuity of L this extends to all of Sep. Conversely, assume that for every L satisfying the condition in the statement of the lemma, ReL(ϕ)
0 but ϕ ∈ / Sep. Then from the Hahn-Banach Theorem (see (10)) we know that there exists a functional L 0 such that ReL 0 (σ )
0 for every separable σ —in particular for every function of the form ψ1 ⊗ ψ2 —and ReL 0 (ϕ) < 0, which contradicts our assumption.  In order to pass from linear functionals on C(G 1 × G 2 ) to linear maps from C(G 2 ) to C(G 1 ), we first employ an algebraic isomorphism between functionals from C(G 1 ×G 2 )  : C(G 2 ) → C(G 1 ) . For each L ∈ C(G 1 × G 2 ) we define and bounded linear maps  the corresponding map L by:  L ξ( f ) := L( f ⊗ ξ );

(11)

 L is bounded:

 L || = || =

 L ξ || ∞ = sup ||

||ξ ||∞ =1

sup

sup

sup |L( f ⊗ ξ )|,

||ξ ||∞ =1 || f ||∞ =1

 L ξ( f )| sup |

||ξ ||∞ =1 || f ||∞ =1

(12)

where || · || ∞ is the norm on C(G 1 ) induced by the supremum norm || · ||∞ . Conversely,  : C(G 2 ) → C(G 1 ) , Eq. (11) defines a functional L for any bounded map  on C(G 1 ) ⊗ C(G 2 ), which is bounded by Eq. (12), and uniquely defined, since if L ( f ⊗  ≡ 0 (note that L ξ ) = 0 for all f ∈ C(G 1 ), ξ ∈ C(G 2 ), then by Eq. (11)  depends  linearly on ) . As C(G 1 ) ⊗ C(G 2 ) is uniformly dense in C(G 1 × G 2 ), L  can be uniquely extended to a continuous functional on all of C(G 1 × G 2 ). This establishes the L . claimed isomorphism L ↔  L , analogous to the one given by Next, we establish a positivity criterion Jamiołkowski in Ref. [30] for operators on finite dimensional Hilbert spaces: Lemma 3.2. A functional L ∈ C(G 1 × G 2 ) satisfies ReL(ψ1 ⊗ ψ2 )
0 for all  L maps positive definite functions from ψ1 ∈ P1 (G 1 ), ψ2 ∈ P1 (G 2 ) if and only if Re P(G 2 ) to positive definite measures from M(G 1 ). Positive definite measure on G is a measure µ satisfying a generalization of the condition (3)3 :  (13) dµ ( f ∗ ∗ f )
0 for any f ∈ C(G),  where f ∗ (g) := f (g −1 ) is the involution and ( f ∗ ξ )(h) := dg f (g)ξ(g −1 h) is the  L on f is defined as the real part of the functional convolution. The action of the map Re  L f . The condition ReL(ψ1 ⊗ ψ2 )
0 for all normalized ψ1 ∈ P1 (G 1 ), ψ2 ∈ P1 (G 2 )

3 On an arbitrary locally compact G positive definite measures are defined by requiring that condition (13) hold for all continuous functions with compact support (cf. Ref. [26], Def. 13.7.1).

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is equivalent to ReL(ψ1 ⊗ ψ2 )
0 for all ψ1 ∈ P(G 1 ), ψ2 ∈ P(G 2 ), since we can replace ψ1 , ψ2 = 0 by ψ1 /ψ1 (e) and ψ2 /ψ2 (e). From definition (11) it follows that:  L ψ2 (ψ1 )] = Re  L ψ2 (ψ1 ). ReL(ψ1 ⊗ ψ2 ) = Re[

(14)

A theorem by Godement (cf. Ref. [31], Theorem 17; Ref. [26], Theorem 13.8.6) states that every positive definite function can be uniformly approximated by functions of the form f ∗ ∗ f , where f is continuous with compact support. Hence, since G 1 , G 2 are compact, ReL(ψ1 ⊗ ψ2 )
0 for every ψ1 ∈ P(G 1 ), ψ2 ∈ P(G 2 ) if and only if   L ψ2 ( f ∗ ∗ f )
0 for every f ∈ C(G 1 ) and ψ2 ∈ P(G 2 ). ReL ( f ∗ ∗ f ) ⊗ ψ2 = Re  L ψ2 ∈ M(G 1 ) being But by Eqs. (13) and (14) this is equivalent to the measure Re positive definite for every ψ2 ∈ P(G 2 ).  . Note that for any µ ∈ M(G 1 ) and As the next step we will regularize maps f ∈ C(G 1 ) the convolution:  µ ∗ f (h) = dµ(g) f (g −1 h) (15) is a continuous function on G 1 . Let {ψU }, where U ⊂ G 1 runs through a neighbourhood base of the neutral element e1 ∈ G 1 , be an approximate identity in C(G 1 ). That is, for every U we have: o) ψU ∈ C(G 1 ), i) suppψU is a compact subset of U,  ψU = 1. ii) ψU
0, iii) ψU (g −1 ) = ψU (g), iv)

(16) (17)

Using definition (15), let us define for every f ∈ C(G 1 ) function,  f ∗ ψU .

U f :=

(18)

 f as U → {e1 }. To see this, let us Then U f converges in weak-∗ topology to calculate dg U f (g)ξ(g) for an arbitrary ξ ∈ C(G 1 ):     f ∗ ψU )(g)ξ(g) = dg d(  f )(h)ψU (h −1 g)ξ(g) dg (    = d( f )(h) dgψU (g −1 h)ξ(g)  f (ξ ∗ ψU ), =

(19)

where in the second step we used the symmetry of ψU : ψU (g −1 ) = ψU (g). But from the properties (16), (17) of {ψU } it follows that ξ ∗ ψU → ξ uniformly as U → {e1 } (cf. Ref. [25], Theorem 2.42), which proves the desired weak-∗ convergence. Thus any  : C(G 2 ) → M(G 1 ) can be weakly-∗ approximated by bounded maps bounded map

U : C(G 2 ) → C(G 1 ) (boundedness of U for every U ⊂ G 1 follows immediately from the definitions (15) and (18)). ’s, introduced in Lemma 3.2, we In order to preserve the positivity property of the choose regularizing functions ψU in a special way. Namely, for every neighbourhood U of e1 ∈ G 1 we can find such open V e1 that: (cf. Ref. [25], Lemma 5.24): o) V ⊂ U, i) V −1 = V, ii) gV g −1 = V for every g ∈ G 1 .

(20)

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Let us define the functions: 1 χV , |V | ψU := κV ∗ κV , κV :=

(21) (22)

 where χV is the indicator function of V and |V | = dgχV (g) is the Haar measure of V . Then one easily shows that {ψU } form an approximate identity in C(G 1 ). Moreover, κV , and hence ψU , are central functions, i.e. for every g and h, κV (gh) = κV (hg), which follows from the property (ii) of the sets V . Using functions (22) to regularize , we find that: an arbitrary map  f ∗ ψU = κV ∗  f ∗ κV = κV∗ ∗  f ∗ κV ,

U f =

(23)

where in the second step we used the fact that κV are central and hence µ ∗ κV = κV ∗ µ for any µ. In the last step we used the symmetry condition (i) and χV = χV . Now, from the special form of the regularization (23) we obtain:  : C(G 2 ) → M(G 1 ) maps positive definite functions into positive Lemma 3.3. If definite measures, then for every neighbourhood U of e1 ∈ G 1 the regularized maps

U , defined by Eqs. (21–23), map P(G 2 ) into P(G 1 ). To prove Lemma 3.3, note that for an arbitrary φ ∈ P(G 2 ) and f ∈ C(G 1 ) one has: 

 φ ∗ κV (g −1 h) f (h) dgdh f (g) κV∗ ∗   φ)(b) κV (b−1 a −1 g −1 h) f (h) = dgdhda f (g) κV (a −1 ) d(      = d( φ)(b) da dg f (g) κV (ag) dh f (h)κV (b−1 ah)    



φ)(b) da f ∗ κˇ V ∗ (a) f ∗ κˇ V (a −1 b) = d(

 φ ( f ∗ κV ∗ ∗ ( f ∗ κV ) , =

(24)

φ is a positive where κˇ V (g) := κV (g −1 ) = κV (g) by property (i) in Eq. (20). Hence, if definite measure from M(G 1 ) then for every U , U φ is a positive definite function on G 1 (note that f ∗ κV is continuous since f is).  Let us introduce some terminology, analogous to that used in the theory of linear mappings of operators on finite-dimensional Hilbert spaces (see e.g. Refs. [32,19,18]): Definition 3.1. Let : C(G 2 ) → C(G 1 ) be a bounded linear map. Then is called: • positive definite (PD) if it preserves positive definite functions, i.e. if P(G 2 ) ⊂ P(G 1 ); • H -positive definite (H -PD), where H is a compact group, if id ⊗ : C(H × G 2 ) →  C(H × G 1 ) is positive definite, i.e. if id ⊗ P(H × G 2 ) ⊂ P(H × G 1 ); • completely positive definite (CPD) if it is H -positive definite for any compact H .

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J. K. Korbicz, J. Wehr, M. Lewenstein

Thus, rephrased in the terms introduced above, Lemma 3.3 states that every bounded  : C(G 2 ) → M(G 1 ), mapping P(G 2 ) into positive definite measures, can linear map be weakly-∗ approximated by positive definite maps from C(G 2 ) to C(G 1 ). After we have established almost all the necessary facts, we return to the main Lemma 3.1. First, rewrite Eq. (11) using the regularization (3.3) in order to be able to write down explicitly the right-hand side of Eq. (11) for arbitrary functions, not only product ones. For product functions we have:  L ξ( f ). ReL( f ⊗ ξ ) = Re

(25)

 L , denoting the regularized operators by RU , We apply the regularization (23) to Re

 RU : C(G 2 ) → C(G 1 ), rather than by Re L U , to obtain: 

 dg1 RU ξ (g1 ) f (g1 ) ReL( f ⊗ ξ ) = lim RU ξ( f ) = lim U →{e1 }

 = lim

U →{e1 } G1

U →{e1 } G1



 dg1 id ⊗ RU f ⊗ ξ (g1 , g1 ).

(26)

Formula (26) immediately extends to all of C(G 1 × G 2 ), so in particular for any ϕ ∈ P1 (G 1 × G 2 ) we have: 

 ReL(ϕ) = lim dg1 id ⊗ RU ϕ (g1 , g1 ). (27) U →{e1 } G1

Summarizing, by application of Lemmas 3.2 and 3.3 we obtain that for an arbitrary functional L ∈ C(G 1 × G 2 ) , such that ReL(ψ1 ⊗ ψ2 )
0 for every ψ1 , ψ2 ∈ P1 (G), ReL(ϕ) is given by Eq. (27), where for every neighbourhood U ⊂ G 1 the maps RU : C(G 2 ) → C(G 1 ) are bounded and positive definite. We need two more simple facts. First, we note that if ϕ is a positive definite function on the product of two copies  of G 1 , i.e. ϕ ∈ P(G 1 × G 1 ), then its restriction to the diagonal ϕ  (g) := ϕ(g, g) is a positive definite function on G 1 . A particularly direct proof of this fact follows from the GNS construction (cf. Theorem 2.1):    −1  dgdh f (g)vϕ |πϕ (g −1 , g −1 )πϕ (h, h) vϕ  f (h) dgdh f (g)ϕ  (g h) f (h) =     (28) = dg f (g)πϕ (g, g)vϕ  dh f (h)πϕ (h, h)vϕ
0.  Second, we note that since G 1 is compact, dg φ(g)
0 for any φ ∈ P(G 1 ) (cf. Ref. [27], Theorem 34.8). Indeed, if G 1 is compact then the constant function 1 is a function support  with compact  and we can use it in the condition (3), which then implies that dgdh φ(g −1 h) = dh φ(h)
0, where we changed variables h → gh and used the normalization of dg. To finish the proof of the main Theorem 3.2, let us assume that for every bounded and positive definite map : C(G 2 ) → C(G 1 ), the function (id ⊗ )ϕ is positive definite. It then follows from the above discussion and from Eq. (27) that ReL(ϕ)
0 for every L, such that ReL(ψ1 ⊗ ψ2 )
0 for every ψ1 ∈ P1 (G 1 ), ψ2 ∈ P1 (G 2 ). But then Lemma 3.1 implies that ϕ ∈ Sep. 

Entanglement of Positive Definite Functions on Compact Groups

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4. Fourier Transforms and “Generating Function” Formalism In this section we establish a connection between the formalism of positive definite functions and standard notions of entanglement theory, thus ascribing a “physical meaning” to the former. Namely, as advertised in the Introduction, we show that various group-theoretical objects studied in the previous sections turn out to be “generating functions” for the corresponding operator-algebraic objects. For example a positive definite function generates a family of (subnormalized) density matrices, a positive definite map (cf. Definition 3.1) generates a family of positive maps, etc. We also derive here a weaker version of the Horodecki Theorem (cf. Theorem 1.2) from Theorem 3.2 and prove a number of other useful results. Our main tool will be non-commutative Fourier analysis of continuous functions on compact groups. Below we recall some basic notions and methods, which we will need (see e.g. Refs. [25,26] for more). The goal which we have in mind is to construct uniformly convergent Fourier series for continuous functions. By (a part of) the fundamental theorem of the theory—the Peter-Weyl Theorem (see e.g. Ref. [25], Theorem 5.12), any continuous function on a compact group can be uniformly approximated by linear combinations of matrix elements of irreducible representations, taken in some fixed orthonormal bases of the corresponding representation spaces. Here, we are primarily interested in product groups G 1 × G 2 , so we first recall 1 (G 2 ) the set of equivalence classes their representation structure. Let us denote by G of irreducible, strongly continuous, unitary representations (irreps) of G 1 (G 2 ). Since 1 and G 2 are discrete and (the classes of equivalent) irreducible G 1 , G 2 are compact, G representations can be labelled by discrete indices. We will denote irreps of G 1 and G 2 β . It can then by πα and τβ respectively and the spaces where they act by Hα and H be shown that for a large family of groups, including compact ones, every irrep of the product G 1 × G 2 can be chosen in the form πα ⊗ τβ , where: πα ⊗ τβ (g1 , g2 ) := πα (g1 ) ⊗ τβ (g2 )

(29)

β (cf. Ref. [25], Theorem 7.25; Ref. [26], Prop. 13.1.8). In acts in the space Hα ⊗ H 1 × G 2 .
other words, G 1 × G 2 can be identified with G Next, we need matrix elements of the irreps πα ⊗ τβ . It is natural here to take them β . Thus, for each pair with respect to product bases of the representation spaces Hα ⊗ H of the representation indices α and β we fix an orthonormal base {ei }i=1,...,dimHα of β (we do not indicate explicitly the Hα and an orthonormal base {e˜k }k=1,...,dimH β of H dependence of {ei }, {e˜k } on the representation indices α, β in order not to complicate the notation). The corresponding matrix elements of πα ⊗ τβ are then simply given by products of the matrix elements of πα and τβ —that is, they are given by the functions β πiαj ⊗ τkl , where: β

πiαj (g1 ) := ei |πα (g1 )e j , τkl (g2 ) := e˜k |τβ (g2 )e˜l .

(30)

Now, for a given f ∈ C(G 1 × G 2 ) we can formally write the Fourier series: 

i jkl i jkl β β f = f αβ πiαj ⊗ τkl , f αβ := n α m β dg1 dg2 πiαj (g1 ) τkl (g2 ) f (g1 , g2 ), α,β i,...,l

G1 G2

(31)

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J. K. Korbicz, J. Wehr, M. Lewenstein

where n α := dimHα , m β := dimHβ . However, for a generic function f ∈ C(G 1 × G 2 ) the Fourier series (31) converges only in the L 2 norm (since G 1 × G 2 is compact C(G 1 × G 2 ) ⊂ L 2 (G 1 × G 2 )) and not uniformly. The standard way around this difficulty is the following: i) regularize f so that the Fourier series of the regularized function converges uniformly; ii) perform the desired manipulations with the series; iii) at the end uniformly remove the regularization. For the regularization the same technique as in Sect. 3 (cf. Eqs. (16–18) and Eqs. (20–23)) is used. Thus, for a given f we consider a function f U := f ∗ ψU , where the regularizing functions ψU ∈ C(G 1 × G 2 ) are defined in an analogous way as in Eqs. (20–22), but this time on the product G 1 × G 2 . The sets U ⊂ G 1 × G 2 now run through a neighborhood base of the neutral element {e1 , e2 } ∈ G 1 × G 2 . Since, by construction, the ψU ’s are central on G 1 × G 2 (cf. property (ii) in Eq. (20)), a simple calculation shows that the Fourier series of f ∗ ψU takes the following form:

αβ i jkl β cU f αβ πiαj ⊗ τkl , f ∗ ψU = αβ



α,β i,...,l

cU :=

dg1 dg2 ψU (g1 , g2 )χα (g1 ) χβ (g2 ),

(32)

G1 G2

where χα (g) := trπα (g) is the character of the representation πα and, analogously, χβ is the character of τβ . It can then be shown that the series (32) converges uniformly αβ for every U (cf. Ref. [25], p. 137). Thus, the role of the constants cU is to enhance ∗ convergence of the Fourier series (31). Note that since f ∗ ψU = κV ∗ f ∗ κV , where the sets V are defined as in Eq. (20) but on G 1 × G 2 , the regularization preserves positive definiteness (cf. Eq. (24) where we proved it for measures on a single group G 1 ). In fact, it preserves separability as well, as we will show later (see Lemma 4.1). Finally, the initial function f can be recovered from f ∗ ψU by letting U → {e1 , e2 } as then f ∗ ψU → f uniformly (cf. Ref. [25], Theorem 2.42). β ) by: Let us define operators fˆαβ ∈ L(Hα ⊗ H
jilk fˆαβ := f αβ |ei e j | ⊗ |e˜k e˜l | i,...,l



= nα m β

dg1 dg2 f (g1 , g2 )πα (g1 )† ⊗ τβ (g2 )† ,

(33)

G1 G2

(note the change of the order of indices). Operators fˆαβ are inverse Fourier transforms of f : fˆαβ ≡ fˆ(πα ⊗ τβ ) [25,20] and Fourier series (31) and (32) can be rewritten as:
αβ 
   tr fˆαβ πα ⊗ τβ , f ∗ ψU = cU tr fˆαβ πα ⊗ τβ . f = (34) α,β

α,β

The last equation again explicitly shows the role of regularization in enhancing convergence of the Fourier series (31). Having recalled the technicalities of the Fourier analysis, we proceed to relate the group-theoretical formalism to the standard one. We begin by quoting a standard fact, which we will extensively use in what follows (see e.g. Ref. [27], Theorem 34.10): 1 and [τβ ] ∈ G 2 . Theorem 4.1. ϕ ∈ P(G 1 × G 2 ) if and only if ϕˆαβ
0 for all [πα ] ∈ G

Entanglement of Positive Definite Functions on Compact Groups

765

This is a non-commutative analog of the fact that positive definiteness corresponds under (usual) Fourier transform to positivity. We present a proof of the above theorem just for the sake of completeness. Let us first introduce an abbreviation g := (g1 , g2 ) ∈ G 1 × G 2 . From Eq. (33) we then obtain β : for any v ∈ Hα ⊗ H     v|ϕˆαβ v = n α m β dg1 dg2 ϕ(g1 , g2 ) v πα (g1 )† ⊗ τβ (g2 )† v     = nα m β d hdgϕ(h−1 g) πα (h 1 )† ⊗ τβ (h 2 )† v πα (g1 )† ⊗ τβ (g2 )† v 

n α ,m β

=




nα m β

αβ

αβ

d hdg vik (h) ϕ(h−1 g)vik (g)
0,

(35)

i,k=1

   αβ where vik (h) := ei ⊗ e˜k πα (h 1 )† ⊗ τβ (h 2 )† v . In the second step above we inserted  1 = G 1 ×G 2 dh 1 dh 2 and then changed the variables g → h−1 g. Then we inserted the β unit matrix 1α ⊗ 1β , decomposed with respect to the fixed bases {ei }, {e˜k } of Hα , H and used positive definiteness of ϕ. Note that we do not need uniform convergence of the Fourier series here, and hence we used the L 2 -convergent series (31) of ϕ. The same applies to the proof in the other direction. Let us now assume that ϕˆαβ
0 for all α, β. Then from Eq. (31) we obtain:  dgd h f (g)ϕ(g −1 h) f (h)

i jkl   β = ϕαβ dgd h f (g1 , g2 )πiαj (g1−1 h 1 )τkl (g2−1 h 2 ) f (h 1 , h 2 ) α,β i,...,l

=



α,β i,...,l,r,s

i jkl ϕαβ



β

dg1 dg2 f (g1 , g2 ) πriα (g1 ) τsk (g2 )

 β × dh 1 dh 2 f (h 1 , h 2 )πrαj (h 1 )τsl (h 2 )

= cr s |ϕˆ αβ cr s 
0

(36)

α,β r,s

  β dg1 dg2 f (g1 , g2 )πriα (g1 )τsk (g2 ) ei ⊗ e˜k . for any f ∈ C(G 1 × G 2 ), where cr s := ik  Thus, by the above theorem, a positive definite function generates a family of subnormalized states ϕˆαβ . The operators ϕˆαβ are not normalized (except in the trivial case when sum (34) consists of one term only) even if ϕ is, since from Eq. (34) we obtain that  α,β tr(ϕˆ αβ ) = ϕ(e1 , e2 ) = 1, so tr(ϕˆ αβ )  1. However, we can still speak of separability of the operators ϕˆαβ in the sense that they are decomposable into convex combinations of products of positive operators (cf. the corresponding remark after Definition 2.2). The following result holds (cf. Ref. [20] where a weaker version was proven): Lemma 4.1. A function ϕ ∈ P1 (G 1 × G 2 ) is separable if and only if the operators β ) are separable for all [πα ] ∈ G 1 and [τβ ] ∈ G 2 . ϕˆαβ ∈ L(Hα ⊗ H

766

J. K. Korbicz, J. Wehr, M. Lewenstein

K Indeed, from Eq. (33) it follows that if Sep0 ϕ = m=1 pm εm ⊗ ηm , then K (m) (m) (m) (m) ϕˆ αβ = m=1 pm εˆ α ⊗ηˆ β , where all the operators εˆ α , ηˆ β are positive by Lemma 4.1, since εm ∈ E1 (G 1 ), ηm ∈ E1 (G 2 ) for all m. This extends to all of Sep by the continuity for all α, β of the inverse Fourier transform (33) ϕ → ϕˆαβ and the fact that positive β ) [16]. separable matrices σ with trσ  1 form a compact convex subset of L(Hα ⊗ H The latter follows from the facts that i) the set of extreme points of the latter subset can be identified with CP n α × CP m β ∪ {0} and ii) the convex hull of a compact subset of R N is compact.  K αβ αβ α α β β Conversely, assume that for every α, β, ϕˆαβ = m=1 pm |x m x m | ⊗ |ym ym |, β β . We will prove that ϕ is then a uniform limit of separawhere xmα ∈ Hα , ym ∈ H ble functions and hence is itself separable. First, we pass to the regularized function ϕU := ϕ ∗ ψU , according to the procedure we described above (cf. Eq. (32) and the surrounding paragraph). As we mentioned, ϕU is positive definite and hence ϕU (e1 , e2 ) = ||ϕU ||∞ > 0 (except in the trivial case ϕ ≡ 0), which allows us to pass to the normalized function ϕU /||ϕU ||∞ . Then Eq. (34) implies that: K αβ

1 1 αβ αβ ϕU (g1 , g2 ) = cU pm xmα |πα (g1 )xmα ymβ |τβ (g2 )ymβ  ||ϕU ||∞ ||ϕU ||∞ α,β m=1

=

K αβ

α,β m=1

×

1 αβ c p αβ ||x α ||2 ||ymβ ||2 ||ϕU ||∞ U m m

β β  xα  xmα  ym  ym  m  π τ , (g ) (g )   α 1 β 2 β ||xmα || ||xmα || ||ymβ || ||ym ||

(37)

and the series converges uniformly. The functions given by scalar products belong to E1 (G 1 ) and E1 (G 2 ) respectively, since πα and τβ are irreducible. From their definition in αβ Eq. (32) and the definition of ψU (22) it also follows that the factors cU are non-negative:   αβ −1 cU = dgd hκV (h)κV (h g)χαβ (g) = dgd hκV (h−1 )κV (g)χαβ (hg)    1 = dgd hκV (h)κV (g)χαβ (h−1 g) = 2 2 tr κ (38) κV αβ )†
0, V αβ ( nα m β where we used definition (33) and the fact that κV is symmetric [cf. property (i) in Eq. (20)] and real. Evaluating ϕU /||ϕU ||∞ at the neutral element we see that the sum in Eq. (37) is in fact a convex combination of pure product functions (cf. the definition of a pure function in Sect. 2), since: K αβ

α,β m=1

and

1 ϕU (e1 , e2 ) αβ αβ cU pm ||xmα ||2 ||ymβ ||2 = =1 ||ϕU ||∞ ||ϕU ||∞

1 αβ c p αβ ||x α ||2 ||ymβ ||2
0. ||ϕU ||∞ U m m

(39)

Thus, ϕU /||ϕU ||∞ is separable, as a uniform limit of separable functions. Since ϕU −−−−−−→ ϕ uniformly, ϕ ∈ Sep.  U →{e1 ,e2 }

Entanglement of Positive Definite Functions on Compact Groups

767

From Lemma 4.1 it follows that the problem of describing separable functions on G 1 × G 2 generates a family of separability problems in all pairs of dimensions where G 1 and G 2 have irreducible representations. In other words, it plays a role of a “generating function” for this family. Conversely, from the form of the Fourier transformation (2) β are in one-to-one and its inverse (33) it follows that density matrices on Hα ⊗ H correspondence with those functions ϕ from P1 (G 1 × G 2 ), which belong to the (finiteβ dimensional) linear span of πiαj ⊗ τkl , where πα , τβ are fixed. Moreover, since for an β ) its Fourier transform ϕ
(cf. Eq. (2)) satisfies: arbitrary density matrix
∈ L(Hα ⊗ H (ϕ
)γ ν = δαγ δβν
,

(40)

Lemma 4.1 implies that (cf. Ref. [20], Theorem 1): Corollary 4.1. A state
is separable if and only if ϕ
∈ Sep . Next we examine bounded linear maps : C(G 2 ) → C(G 1 ). For an arbitrary function f ∈ C(G 2 ), we consider a function fU ∈ C(G 1 ), where fU := f ∗ ψU is the regularization of f . Now, the regularizing functions ψU ∈ C(G 2 ) are the single-group functions defined in Eq. (22), but now on the group G 2 , and the sets U run through a neighborhood base of e2 ∈ G 2 . Note, however, that unlike in Sect. 3 here we are regularizing the argument of and not its value. Calculating the Fourier transform of

fU from the single-group version of the definition (33) we obtain:   





fU α = n α dg1 f ∗ ψU (g1 )πα (g1 )† G1



= nα

dg1 G1

=


β

β cU




β

cU




β




f βkl

β f βkl τkl (g1 )πα (g1 )†

k,l



nα

k,l

β dg1 τkl (g1 )πα (g1 )† ,

(41)

G1

where we used the uniform convergence of Fourier series for fU and the fact that is β continuous in the uniform norm. The regularizing constants cU are defined analogously as in Eq. (32), i.e.  β (42) cU := dg2 ψU (g2 )χβ (g2 ). G2

β ) → L(Hα ) through an analog of Eq. ˆ βα : L(H We will find it useful to define maps (33): 
jiβ 

kl |, i jβ := n α dg1 π α (g1 ) τ β (g1 ) ˆ βα :=

αlk |E i j  H S  E (43)

ij kl αkl i,...,l

G1

kl := |e˜k e˜l |, E i j := |ei e j | are the (note the change of the order of indices), where E  bases of L(Hβ ) and L(Hα ) respectively, and A|B H S = tr(A† B). We can then rewrite Eq. (41) as follows:
β

  ˆ βα fˆβ .

fU α = cU (44) β

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J. K. Korbicz, J. Wehr, M. Lewenstein

ˆ βα fˆβ ∈ L(Hα ) are finite-dimensional and Note that in the above series all operators hence the convergence can be understood in any of the equivalent norms on L(Hα ). β ) → L(Hα ) we can Fourier transform Conversely, given an arbitrary map  : L(H it and assign to it a map  : C(G 2 ) → C(G 1 ) through the following formula (compare with Eq. (2) where Fourier transform of operators was defined):



 f (g1 ) := tr  fˆβ πα (g1 ) for every f ∈ C(G 2 ). (45) It is obvious that  f ∈ C(G 1 ), because we consider only continuous representations. Moreover from the definition of fˆβ we have: 

   †  ||  f ||∞ = m β sup  dg2 f (g2 )tr τβ (g2 ) πα (g1 )  g1 ∈G 1

G2



 m β || f ||∞ sup

g1 ∈G 1



  †   dg2 tr τβ (g2 ) πα (g1 ) .

(46)

G2

The last supremum is finite, as the integrand is continuous and G is compact, and independent of f , so that  is bounded. The transformation (45) is an inverse of the ˆ βα given by Eq. (43), since by an easy direct calculation one finds that mapping → (compare Eq. (40)):   )µ ( ν = δνα δµβ .

(47)

By analogy with Theorem 4.1, which characterizes positive definite functions in terms of their inverse Fourier transforms, one would expect a corresponding characterization of positive definite (PD) maps (cf. Definition 3.1) in terms of their inverse Fourier ˆ βα . The next lemma provides such a characterization: transforms Lemma 4.2. A bounded linear map : C(G 2 ) → C(G 1 ) is positive definite, that is β ) → L(Hα ) are positive for all ˆ βα : L(H

P(G 2 ) ⊂ P(G 1 ), if and only if the maps   [πα ] ∈ G 1 , [τβ ] ∈ G 2 . To prove it, let us take an arbitrary φ ∈ P(G 2 ), which by Theorem 4.1 is equivalent to ˆβ ˆ ˆ βα are positive, then cβ φˆ β
0 for all β. We employ Eq. (44). If all maps U α φβ
0 for β all β, since cU
0 (cf. Eq. (38) where we proved it for G 1 ×G 2 ). Since positive operators  β β ˆ α φˆ β converges to a positive operator form a closed cone in L(Hα ), the series β cU

  and hence φU α
0 for all α. Theorem 4.1 implies then that (φ ∗ ψU ) ∈ P(G 1 ). Taking the limit U → {e2 }, φ ∗ ψU converges to φ uniformly. Thus, using continuity of and uniform closedness of P(G 1 ) ⊂ C(G 1 ), we obtain that φ ∈ P(G 1 ) for any φ ∈ P(G 2 ). 2 and take an Conversely, assume to be positive definite. Let us fix [τβ ] ∈ G  arbitrary positive operator
∈ L(Hβ ). Applying Fourier transform (2) to
we obtain its characteristic function:  
kl β φ
(g2 ) := tr
τβ (g2 ) =
τlk (g2 ), (48) k,l

Entanglement of Positive Definite Functions on Compact Groups

769

which by Theorem 4.1 is positive definite, since (φ
)γ = δγβ

0. Hence, φ
is positive definite too. Then from Eq. (44), where we can neglect the regularization and β put cU = 1 since the Fourier series of φ
contains only one term, and from Theorem 4.1

 ˆ βα

0 for all α. Since τβ and
were arbitrary, the result we obtain that 

φ
α = follows.  Thus, from the above lemma and Eq. (47) it follows that (compare Corollary 4.1): β ) → L(Hα ) is positive if and only if the map Corollary 4.2. A map  : L(H

 : C(G 2 ) → C(G 1 ), defined in Eq. (45), is positive definite. Next we present a characterization of completely positive definite maps (cf. Definition 3.1) in terms of their Fourier transforms. We first prove the following fact: Lemma 4.3. A bounded linear map : C(G 2 ) → C(G 1 ) is G 2 -positive definite, i.e. β ) → L(Hα ) are ˆ βα : L(H (id ⊗ )P(G 2 × G 2 ) ⊂ P(G 2 × G 1 ), if and only if maps 1 , [τβ ] ∈ G 2 . completely positive for all [πα ] ∈ G 2 . Then the Let ϕ ∈ P(G 2 × G 2 ), so by Theorem 4.1 ϕˆβγ
0 for all [τβ ], [τγ ] ∈ G obvious generalization of Eq. (44) to G 2 × G 2 , implies that:


δγ


βγ δγ ˆ γα )ϕˆβγ ,
⊗ βα ϕˆδγ = = cU id cU (1β ⊗ (49) (id
⊗ )ϕU βα

γ

δ,γ

where now U ⊂ G 2 × G 2 runs through a neighborhood base of {e2 , e2 } ∈ G 2 × G 2 . ˆ γα are completely positive, then cβγ (1β ⊗ ˆ γα )ϕˆ βγ
0 as operators from If all maps U α ), since cβγ
0 for all β, γ (cf. Eq. (38)). From closedness of the cone β ⊗ H L(H U   of positive operators in L( Hβ ⊗ Hα ), the series in Eq. (49) converges to a positive operator as well. Hence (id
⊗ )ϕU
0 for all α, β and from Theorem 4.1 it βα

follows that (id ⊗ )(ϕ ∗ ψU ) ∈ P(G 2 × G 1 ). We remove the regularization by letting U → {e2 , e2 } so that ϕ ∗ ψU → ϕ uniformly. Then from the continuity of id ⊗ and uniform closedness of P(G 2 × G 1 ) it follows that (id ⊗ )ϕ ∈ P(G 2 × G 1 ) for any ϕ ∈ P(G 2 × G 2 ). For the proof in the other direction, we proceed along the same lines as in the proof of β ⊗ H γ ), 2 and arbitrary 0 
∈ L(H the previous lemma—for arbitrary [τβ ], [τγ ] ∈ G we consider the Fourier transform of
, ϕ
, defined in Eq. (2). Then positive definiteness µν of id⊗ , Theorem 4.1, and Eq.(44) generalized to G 2 ×G 2 (with cU = 1 as the Fourier

ˆ γα )

0 series (31) of ϕ
contains only one term) imply that (id
⊗ )ϕ
= (1β ⊗ βα

for all γ . Since [τβ ], [τγ ], and
are arbitrary, G 2 -positive definiteness of implies that 2 . Thus, in particular, every ˆ γα ˆ γα is (dimτ )-positive for all possible [τ ] ∈ G every map  is m γ -positive, m γ = dimHγ . But then by the Choi Theorem (cf. Ref. [18], Theorem 2) ˆ γα being completely positive.  this is equivalent to As a by-product we obtain an analog of the Choi Theorem (cf. Ref. [18], Theorem 2) for PD maps: Corollary 4.3. A map : C(G 2 ) → C(G 1 ) is completely positive definite if and only if it is G 2 -positive definite.

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J. K. Korbicz, J. Wehr, M. Lewenstein

The proof in one direction follows immediately from Definition 3.1. For the opposite implication, let us assume that is G 2 -PD. Let H be an arbitrary compact group and let Latin indices a, b, . . . enumerate irreps of H . From the complete positivity of Fourier ˆ βα , guaranteed by Lemma 4.3, Theorem 4.1, Eq. (38) applied to H × G 2 , transforms and closedness of the cone of positive operators, we obtain that:


bβ ˆ βα )ϕˆbβ
0, (id
⊗ )ϕU = cU (1b ⊗ (50) bα

β

for every b and α and every ϕ ∈ P(H × G 2 ). Thus from Theorem 4.1, (id ⊗ )ϕU ∈ P(H × G 1 ). Letting U → {e H , e2 } ∈ H × G 2 , so that ϕU converges to ϕ uniformly, and using continuity of id ⊗ and uniform closedness of P(H × G 1 ), we obtain that (id ⊗ )ϕ ∈ P(H × G 1 ) for every compact H and every ϕ ∈ P(H × G 2 ).  From Lemma 4.3 and Corollary 4.3 we finally obtain: Lemma 4.4. A bounded linear map : C(G 2 ) → C(G 1 ) is completely positive defβ ) → L(Hα ) are completely positive for all ˆ βα : L(H inite if and only if the maps 1 , [τβ ] ∈ G 2 . [πα ] ∈ G Comparison of Theorem 3.2 with the Horodecki Theorem 1.2 shows that positive definite mappings of continuous functions on compact groups play an analogous role to that of positive mappings of density matrices in the standard theory of entanglement [14]. The harmonic-analytical formalism described above makes this observation, as well as the “generating function” analogy, formal. Namely, Lemma 4.2 implies that every PD ˆ βα , acting between algebras of operators map generates a family of positive maps on representation spaces of G 1 and G 2 . Conversely, Fourier transform (45), together with property (47) and Lemma 4.2 allows one to assign a unique PD map to every suitable (cf. definition (45)) positive map. Analogously, Lemma 4.4 shows that each CPD map gives rise to a family of completely positive maps, and by Fourier transform (45) every (suitable) completely positive map defines a CPD map. Thus, PD and CPD maps between groups play a role of “generating functions” of families of positive and completely positive maps respectively. In order to compare Theorem 3.2 with the Horodecki Theorem 1.2, we first choose G 1 and G 2 so that they possess irreps in dimensions dimHA and dimHB , so that β for some [πα ] ∈ G 1 and [τβ ] ∈ G 2 we may identify HA ∼ = Hα and HB ∼ = H [20]. Apart from that there are no further restrictions on G 1 , G 2 . Finding such a group for a given finite-dimensional HA , HB is always possible—for example we can take G 1 = G 2 = SU (2), which possesses irreps in all finite dimensions. Having made the above identification, we obtain from Theorem 3.2 that: β ) is separable if and only if for all Theorem 4.2. A density matrix
∈ L(Hα ⊗ H β ) → L(Hγ ), (1α ⊗ γ )

0 as an 1 and all positive maps γ : L(H [πγ ] ∈ G operator on Hα ⊗ Hγ . To prove it, we first Fourier transform
, passing to its characteristic function ϕ
= tr(
πα ⊗ τβ ) ∈ P1 (G 1 × G 2 ). From Lemma 4.1
is separable if and only if ϕ ∈ Sep. Applying Theorem 3.2 to ϕ
we obtain that
is separable if and only if for all positive definite maps : C(G 2 ) → C(G 1 ), (id ⊗ )ϕ
∈ P(G 1 × G 1 ). From the δγ generalization of Eq. (44) to G 1 × G 2 with all cU = 1 (no regularization of ϕ
is needed

Entanglement of Positive Definite Functions on Compact Groups

771

because the Fourier series of ϕ
contains only one non-zero term; cf. Eq. (40)) we obtain that:

ˆ βγ )
. = (1α ⊗ (51) (id
⊗ )ϕ
αγ

Then from Theorem 4.1 and Lemma 4.2 it follows that (id ⊗ )ϕ
is a positive definite β ) → ˆ βγ )

0 for every γ , where every map ˆ βγ : L(H function if and only if (1α ⊗ L(Hγ ) is positive.  Thus, when applied to finite dimension, Theorem 3.2 turns out to be weaker than Theorem 1.2, since generically one has to check positive maps operating between the β ) and the whole family of spaces L(Hγ ), [πγ ] ∈ G 1 , and not only fixed space L(H β ) and L(Hα ) as in Theorem 1.2. Note, however, that the number of between L(H spaces Hγ to check need not be infinite, since it may be possible to find discrete G 1 [20]. This is particularly easy for low-dimensional initial spaces HA , HB . 5. Examples of Positive Definite Maps for G 1 = G 2 From the point of view of classification of separable functions using Theorem 3.2 only those positive definite maps which are not completely positive definite (cf. Definition 3.1) are interesting: if is a CPD map then all the functions (id⊗ )ϕ, ϕ ∈ P1 (G 1 ×G 2 ), are positive definite, whether ϕ is separable or not. Hence we encounter a similar problem as in the finite-dimensional linear algebra [19]: classify all positive definite but not completely positive definite maps from C(G 2 ) to C(G 1 ). In this section we give some examples of PD and CPD maps for the case G 1 = G 2 ≡ G. The first example of PD but not CPD map was already encountered in Theorem 3.1— the inversion map θ : θ f (g) := f (g −1 ).

(52)

To show that θ is not CPD (positive definiteness will be proven below in a more general setting), observe that θ corresponds through the Fourier transform to the transposition map T , T
:=
T , acting on each representation space Hα of G:



f (g −1 ) = f αi j πiαj (g −1 ) = f αji πiαj (g), and hence α

i, j

 θf α¯ = T fˆα = fˆαT .

α

i, j

(53)

Here index α¯ denotes the complex conjugate π α of representation πα : π α (g) := πα (g). Equation (53) establishes the connection between the PPT criterion (Theorem 1.1) and Theorem 3.1 (see Ref. [20] for more details). Now, let
∈ L(Hα ⊗ Hβ ) be any positive operator such that its partial transpose (1α ⊗ T )
is not positive. Then, by Theorem 4.1, the Fourier transform ϕ
of
is positive definite, but (id ⊗ θ )ϕ
is not. We propose to use the same terminology as in quantum information theory and call entangled functions ϕ not detected by θ (see the remark after Theorem 3.2) bound entangled. Let us consider more general substitutions of the argument in the tested function. Let α be an arbitrary automorphism of G and β an arbitrary anti-automorphism of G, i.e.: α(gh) = α(g)α(h), β(gh) = β(h)β(g).

(54) (55)

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J. K. Korbicz, J. Wehr, M. Lewenstein

We define the corresponding maps from C(G) to C(G):

α f (g) := f (α(g)), β f (g) := f (β(g)).

(56)

Both α and β are positive definite, which follows most directly from the GNS construction (cf. Theorem 2.1): 

 dgdh f (g) α φ (g −1 h) f (h)  

 

  = dg f (g)πφ α(g) vφ  dh f (h)πφ α(h) vφ
0, 

 dgdh f (g) β φ (g −1 h) f (h)  

† 

†  = dh f (h)πφ β(h) vφ  dg f (g)πφ β(h) vφ
0, where φ ∈ P(G) and we used the fact that α(g −1 ) = α(g)−1 and β(g −1 ) = β(g)−1 . Moreover, maps arising from automorphisms are completely positive definite. Indeed, from Corollary 4.3 it is enough to check the extension of to C(G × G). But then we obtain that (with the boldface characters denoting elements of G × G): 

 dgd h f (g) id ⊗ α ϕ (g −1 h) f ( h)   



    dh 1 dh 2 f (h 1 , h 2 )πϕ h 1 , α (h 2 ) vϕ
0, = dg1 dg2 f (g1 , g2 )πϕ g1 , α (g2 ) vϕ  (57)

for an arbitrary f ∈ C(G × G). The maps arising from anti-automorphisms are not necessarily CPD—the above calculation leading to the inequality (57) cannot be repeated.  −1 However, since every anti-automorphism can be written in the form β(g) = α(g) , where α is an automorphism, every map β arising from an anti-automorphism is of the form:

β = C P D ◦ θ,

(58)

where C P D is some CPD map. Hence, the use of a general anti-homomorphism β in Theorem 3.1 gives no improvement, since the function (id ⊗ β )ϕ is positive definite if (id ⊗ θ )ϕ is. In other words, PD maps of the type (58) cannot detect bound entangled functions. This is in close analogy to what one encounters in the study of standard separability problems. Indeed, from Lemma 4.3, Corollary 4.3, and Eq. (53) PD maps of the form (58) generate positive maps of the type C P ◦ T , where C P is a completely positive map. Clearly, by Theorem 1.2, such maps cannot detect bound entangled states
, for which (1 ⊗ T )

0. We can give a more general example of a CPD map, motivated by the Kraus decomposition of a completely positive map (cf. Ref. [32] and Ref. [18], Theorem 1). For an arbitrary measure µ from M(G) we define a map: 



µ f (g) := µ∗ ∗ f ∗ µ (g) = dµ(a)dµ(b) f (agb−1 ), (59) G G

Entanglement of Positive Definite Functions on Compact Groups

773

where the adjoint µ∗ is defined as µ∗ () := µ(−1 ) for any Borel set  ⊂ G (cf. the corresponding definition for functions after Eq. (13)) and the convolution is defined through Eq. (15). Obviously, µ maps C(G) to C(G) and is a generalization of the regularization formula (23). The map µ is bounded on C(G), since      −1   dµ(a)dµ(b) f (agb ) = |µ|2 (G). sup sup  g∈G || f ||∞ =1

It is also completely positive definite, which can be easily proven using the GNS Theorem 2.1):     dµ(a)dµ(b) vϕ πϕ (g1 , ag2 b−1 )vϕ (id ⊗ µ )ϕ(g1 , g2 ) =     = dµ(a)dµ(b) πϕ (e, a)† vϕ πϕ (g1 , g2 )πϕ (e, b)† vϕ    = πϕ (e, µ)† vϕ πϕ (g1 , g2 )πϕ (e, µ)† vϕ , (60)  where πϕ (e, µ) := dµ(g)πϕ (e, g). Thus, (id ⊗ µ )ϕ is positive definite. Map (59) can be further generalized: 



M f (g) := dM(µ) µ∗ ∗ f ∗ µ (g), (61) M(G)

 where M is a positive measure on M(G) with a finite total variation, i.e. M ∈ M M(G) . We conjecture that any CPD map from C(G) to C(G) is of this form for some measure M, so that Eq. (61) is an analog of the Kraus decomposition of a completely positive map. 6. Conclusions The main conclusions are twofold. On one hand, this paper is directed to the audience of mathematicians working in the area of harmonic analysis. We have formulated here the separability problem in purely abstract terms of positive definite functions on compact groups. To our knowledge this is a new problem in harmonic analysis. One may hope that some well established methods of harmonic analysis will help to get more insight into the problem, solving it, at least partially. Several generalizations call for immediate attention: applications to quantum groups being perhaps one of the most fascinating ones. We hope that studies of entanglement within the harmonic analysis framework will open new avenues here. One the other hand, we expect that the harmonic analysis methods will help to study the physics of separability and entanglement. We expect to find new entanglement criteria, and get a better understanding of the whole problem, by looking at concrete examples and applications of our theoretical results. In particular, finite groups (which have finitely many irreducible representations) of various types (nilpotent, solvable) seem a rich source of interesting examples. As seen above, passing from linear-algebraic entanglement criteria for quantum states to the theory of positive-definite functions on compact groups is somewhat involved and requires technical work. We expect, on the other hand, that, going in the opposite direction, the above general results, when specialized to concrete groups (e.g. finite groups or SU (2)) will directly yield physically interesting results.

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Acknowledgements. We would like to thank M. Bo˙zejko, P. Horodecki, M. Marciniak, P. Sołtan, and S. L. Woronowicz for discussions. We gratefully acknowledge the financial support of EU IP Programme “SCALA”, ESF PESC Programme “QUDEDIS”, Spanish MEC grants (FIS 2005-04627, Conslider Ingenio 2010 “QOIT”), and Trup Cualitat Generalitat de Catalunya. JW was partially supported by the NSF grant DMS 9706915.

References 1. Bell, J.S.: Speakable and Unspeakable in Quantum Mechamics. Cambridge: Cambridge University Press, 2004 2. Norsen, T.: Found. Phys. Lett. 19, 633 (2006); Found. Phys. 37, 311 (2007) 3. Peres, A.: Quantum Theory: Concepts and Mehtods. Dordrecht: Kluwer Academic Publishers, 1993 4. Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Rev. Mod. Phys., in press, available at http://arxiv.org/list/quant-ph/0702225v2, 2007 5. Strømer, E.: Acta Math. 110, 233 (1963) 6. Terhal, B.M.: Lin. Alg. Appl. 323, 61 (2001); Lewenstein, M., Kraus, B., Horodecki, P., Cirac, J.I.: Phys. Rev. A 63, 044304 (2001); Breuer, H.-P.: Phys. Rev. Lett. 97, 080501 (2006) 7. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge: Cambridge University Press, 2000 8. Einstein, A., Podolsky, B., Rosen, N.: Phys. Rev. 47, 777 (1935) 9. Werner, R.F.: Phys. Rev. A 40, 4277 (1989) 10. Doherty, A.C., Parrilo, P.A., Spedalieri, F.M.: Phys. Rev. Lett. 88, 187904 (2002) 11. Woerdmann, H.J.: Phys. Rev. A 67, 010303 (2003) 12. Gurvits, L.: J. Comp. Sys. Sci. 69, 448 (2004) 13. Peres, A.: Phys. Rev. Lett. 77, 1413 (1996) 14. Horodecki, M., Horodecki, P., Horodecki, R.: Phys. Lett. A 223, 1 (1996) 15. Choi, M.-D.: Linear Algebra Appl. 12, 95 (1975) 16. Horodecki, P.: Phys. Lett. A 232, 333 (1997) 17. Giedke, G., Kraus, B., Lewenstein, M., Cirac, J.I.: Phys. Rev. Lett. 87, 167904 (2001) 18. Choi, M.-D.: Linear Algebra Appl. 10, 285 (1975) 19. Woronowicz, S.L.: Rep. Math. Phys. 10, 165 (1976); Commun. Math. Phys. 51, 243 (1976); Kruszy´nski, P., Woronowicz, S.L.: Lett. Math. Phys. 3, 317 (1979) 20. Korbicz, J.K., Lewenstein, M.: Phys. Rev. A 74, 022318 (2006) 21. Korbicz, J.K., Lewenstein, M.: Found. Phys. 37, 879 (2007) 22. Holevo, A.S.: Probabilistic and statistical aspects of quantum theory. Amsterdam, North Holland, 1982 23. Perelomov, A.: Generalized Coherent States and Their Applications. Berlin, Springer, 1986 24. Barnum, H., Knill, E., Ortiz, G., Viola, L.: Phys. Rev. A 68, 032308 (2003); Barnum, H., Knill, E., Ortiz, G., Somma, R., Viola, L.: Phys. Rev. Lett. 92, 107902 (2004); Braunstein, S.L., Caves, C.M., Jozsa, R., ˙ Linden, N., Popescu, S., Schack, R.: Phys. Rev. Lett 83, 1054 (1999); Mintert, F., Zyczkowski, K.: Phys. Rev. A 69, 022317 (2004) 25. Folland, G.: A Course in Abstract Harmonic Analysis. Boca Raton: CRC Press, 1995 26. Dixmier, J.: C ∗ -Algebras. Amsterdam: North Holland, 1977 27. Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis. Vol. II, Berlin: Springer-Verlag, 1963 28. Folland, G.: Real Analysis. Modern Techniques and Their Applications. New York: Wiley, 1999 29. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Vol. I, San Diego: Academic Press, 1980 30. Jamiołkowski, A.: Rep. Math. Phys. 3, 275 (1972) 31. Godement, R.: Trans. Amer. Math. Soc. 63, 1 (1948) 32. Kraus, K.: States, Effects, and Operators: Fundamental Notions of Quantum Theory. Berlin: Springer, 1983 Communicated by M.B. Ruskai

Commun. Math. Phys. 281, 775–791 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0499-0

Communications in

Mathematical Physics

Strange Attractors in Periodically-Kicked Degenerate Hopf Bifurcations William Ott Courant Institute of Mathematical Sciences, New York, New York 10012, USA. E-mail: [email protected] Received: 18 June 2007 / Accepted: 18 November 2007 Published online: 15 May 2008 – © Springer-Verlag 2008

Abstract: We prove that spiral sinks (stable foci of vector fields) can be transformed into strange attractors exhibiting sustained, observable chaos if subjected to periodic pulsatile forcing. We show that this phenomenon occurs in the context of periodicallykicked degenerate supercritical Hopf bifurcations. The results and their proofs make use of a k-parameter version of the theory of rank one maps. 1. Introduction This paper aims to support the idea that shear and twist are natural mechanisms for the production of sustained, observable chaos in forced dynamical systems. Consider a weakly stable dynamical structure such as an equilibrium point or a limit cycle. If shear or twist is present, then forcing of various types can transform the weakly stable structure into a strange attractor. The nature of the forcing is not essential. Admissible types of forcing include periodic pulsatile drives, deterministic continuous-time signals, and random signals generated by stochastic processes. The strange attractors possess many of the dynamical, statistical, and geometrical properties commonly associated with chaotic dynamics. We study the simplest weakly stable dynamical structure. This is the spiral sink (or stable focus), an equilibrium point of a vector field with the property that the linearization of the field at the equilibrium point has a pair of complex conjugate eigenvalues α ± iβ satisfying α < 0 and β
= 0. We consider the degenerate supercritical Hopf bifurcation in two dimensions. When a generic supercritical Hopf bifurcation occurs, the spiral sink becomes unstable and a limit cycle is born. In the degenerate case, the spiral sink loses its stability but no limit cycle is born. We prove that in the case of the degenerate supercritical Hopf bifurcation, periodic pulsatile drives transform the spiral sink into a strange attractor. The analysis is certainly not limited to Hopf bifurcations. We work in this context because the origin of the shear is transparent in the defining differential equations.

776

W. Ott

The analysis is based on the beautiful dynamical theory of rank one maps formulated by Wang and Young (8,7). Speaking impressionistically, rank one maps are strongly dissipative maps exhibiting a single direction of instability. Rank one theory provides checkable conditions that imply the existence of strange attractors for a positive-measure set of parameters within a given parametrized family of rank one maps. The conditions appear within the following scheme. (1) Let dissipation go to infinity. This procedure produces the singular limit, a parametrized family of one-dimensional maps. (2) Check that the singular limit includes a map with strong expanding properties (a map of Misiurewicz type). (3) Verify a parameter transversality condition. (4) Verify a nondegeneracy condition. This allows information about the singular limit to be passed to the maps with finite dissipation. Steps (3) and (4) cumulatively require verifying that only finitely many quantities do not vanish. For good parameters, parameters corresponding to maps admitting strange attractors, rank one theory provides a reasonably complete dynamical description of the map. The attractor supports a positive, finite number of ergodic SRB measures. The orbit of Lebesgue almost-every point in the basin of attraction has a positive Lyapunov exponent and is asymptotically distributed according to one of the ergodic SRB measures. Each SRB measure satisfies the central limit theorem and exhibits exponential decay of correlations. A symbolic coding exists for orbits on the attractor. This symbolic coding implies the existence of equilibrium states and a measure of maximal entropy. Summarizing, the map has a nonuniformly hyperbolic character and exhibits sustained, observable chaos. Of the four steps in the rank one scheme, step (2) is the most fundamental and typically requires the most work. A Misiurewicz map has the property that the positive orbit of every critical point remains bounded away from the critical set. Existing papers on rank one theory view the singular limit as a one-parameter family { f a } of one-dimensional maps. This view makes locating Misiurewicz parameters difficult if the maps have multiple critical points. If each map f a has exactly one critical point c(a), then locating Misiurevicz parameters is relatively easy. Assuming that f a (c(a)) moves reasonably quickly as one varies a, simply locate an invariant set (a) that is disjoint from the critical set and then choose a ∗ such that f a ∗ (c(a ∗ )) ∈ (a ∗ ). The set (a) could be a periodic orbit or a Cantor set. If the singular limit consists of maps with multiple critical points, then one must locate a parameter a ∗ for which all of the critical orbits of f a ∗ are contained in good invariant sets. This is a serious challenge because good invariant sets such as periodic orbits and Cantor sets typically have Lebesgue measure zero. Wang and Young (9) overcome this challenge. However, their results assume that the maps in the singular limit possess an extremely large amount of expansion. We prove that significantly less expansion is needed if the singular limit is viewed as an m-parameter family for m sufficiently large. Assume that the singular limit consists of maps with k critical points. We prove that if the singular limit is viewed as a k-parameter family, then it contains Misiurewicz points assuming the maps are mildly expanding and assuming the parameters are independent in a sense to be made precise. This result widens the scope of rank one theory. We view this work as an element of a growing list of applications of rank one theory. The theory has been rigorously applied to simple mechanical systems (9), periodicallykicked limit cycles and Hopf bifurcations (10), and the Chua circuit (6). Guckenheimer,

Strange Attractors in Periodically-Kicked Degenerate Hopf Bifurcations

777

Wechselberger, and Young (1) connect rank one theory and geometric singular perturbation theory by formulating a general technique for proving the existence of chaotic attractors for three-dimensional vector fields with two time scales. Lin (2) demonstrates how rank one theory can be combined with sophisticated computational techniques to analyze the response of concrete nonlinear oscillators of interest in biological applications to periodic pulsatile drives. Lin and Young (3) study shear-induced chaos numerically in situations beyond the reach of current analytical tools. In particular, they consider stochastic forcing. This work supports the belief that shear-induced chaos is both widespread and robust. We organize the presentation of ideas as follows. In Sect. 2, we present the main results for periodically-kicked degenerate Hopf bifurcations. In Sect. 3, we prove the result concerning the existence of Misiurewicz points in k-parameter families of onedimensional maps and we present a two-parameter example. Section 4 presents rank one theory viewing the singular limit as a k-parameter family. Finally, in Sect. 5 we prove the results presented in Sect. 2.

2. Periodically-Kicked Degenerate Hopf Bifurcations The normal form for the supercritical Hopf bifurcation in two spatial dimensions is given in polar coordinates by ⎧ ⎨ r˙ = (µ − αµr 2 )r + r 5 gµ (r, θ ) ⎩˙ θ = ω + γµ µ + βµr 2 + r 4 h µ (r, θ ). Here µ is the bifurcation parameter and ω is a constant. The multipliers αµ , γµ , and βµ depend smoothly on µ. The functions gµ and h µ depend smoothly on µ and they are of class C 4 with respect to r and θ . The normal form for the degenerate Hopf bifurcation in two spatial dimensions is obtained by setting αµ = 0 for all µ and replacing µ with −µ, yielding ⎧ ⎨ r˙ = −µr + r 5 gµ (r, θ ) (2.1) ⎩˙ θ = ω + γµ µ + βµr 2 + r 4 h µ (r, θ ). For µ > 0, the origin is an asymptotically stable equilibrium point (a sink). We study (2.1) t denote the flow generated by (2.1). We perturb the flow F t with in this µ-range. Let F a ‘kick’ map κ defined as follows. Let L > 0 and let ρ2 > 0. The map κ = κµ,L ,ρ2 is given in rectangular coordinates by κ

  r cos(θ ) r sin(θ )

 =

r cos(θ )

r sin(θ ) + Lµρ2

 .

t ◦ κ may be thought of as a perturbation followed by a period of The composition F t ◦ κ. Let A denote the annulus relaxation. We define an annulus map associated with F defined by A = {(r, θ ) : K 4−1 µρ1
r
K 4 µρ1 },

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where K 4 > 1 and 0 < ρ2 < ρ1 . Let r˜ denote the distance from κ(A) to the origin. We have r˜ = Lµρ2 − K 4 µρ1 . Define the relaxation time τ (µ) by r˜ e−µτ (µ) = µρ1 . τ (µ) ◦ κ maps A into A. For µ sufficiently large, F τ (µ) ◦κ The following theorem states that under certain conditions, the annulus map F admits a strange attractor for a positive-measure set of values of µ. We make the crucial assumption that the twist factor β0 is nonzero. A nonzero twist factor implies the existence of an angular-velocity gradient in the radial direction for values of µ in a neighborhood of the bifurcation parameter µ = 0. This angular-velocity gradient allows the flow to stretch and fold the phase space, thereby producing chaos. The chaos in this setting is sustained in time and observable. The strange attractors possess many of the geometric and dynamical properties normally associated with chaotic systems. These properties include the existence of a positive Lyapunov exponent (SA1), the existence of SRB measures and basin property (SA2), and statistical properties such as exponential decay of correlations and the central limit theorem for dynamical observations (SA3). In addition, if |βL0 | is sufficiently large, then the annulus map admits a unique SRB measure (SA4). Properties (SA1)–(SA4) are described in detail in Sect. 4. Theorem 2.1. Assume β0
= 0. Let ρ1 and ρ2 satisfy ρ2 ∈ ( 38 , 21 ) and ρ1 + ρ2 = 1. M0 , then there exist L ∗ ∈ [L , L + |βπ0 | ] (1) There exists M0 > 0 such that if L  |β 0| and µ0 > 0 satisfying the following. The parameter interval (0, µ0 ] contains a set τ (µ) ◦ κ admits  = (L ∗ ) of positive measure such that for µ ∈ , the map F a strange attractor with properties (SA1), (SA2), and (SA3). The set  intersects every interval of the form (0, µ] ˜ in a set of positive measure. M1 (2) There exists M1  M0 such that for all L  |β , there exists a set  = (L) with 0| the properties described in (1). (3) If L is sufficiently large and µ ∈ (L), then (SA4) holds as well.

3. Locating Misiurewicz Points Let I denote an interval or the circle S 1 . Let F : I × [a1 , a2 ] × [b1 , b2 ] → I be a C 2 map. The map F defines a two-parameter family F = { f a,b : a ∈ [a1 , a2 ], b ∈ [b1 , b2 ]} via f a,b (x) = F(x, a, b). Set A = [a1 , a2 ] and B = [b1 , b2 ]. We assume that for each (a, b) ∈ A × B, f a,b has two critical points. We label these critical points c(1) (a, b) and c(2) (a, b). Let C = C(a, b) = {c(1) (a, b), c(2) (a, b)}. For δ > 0, let Cδ denote the δ-neighborhood of C in I . We seek to identify conditions under which F contains strongly expanding (Misiurewicz) maps. We now introduce this class. Definition 3.1. We say that f ∈ C 2 (I, I ) is a Misiurewicz map and we write f ∈ M if the following hold for some neighborhood V of C. (A) (Outside of V ) There exist λ0 > 0, M0 ∈ Z+ , and 0 < d0
1 such that (1) for all n  M0 , if f k (x) ∈ / V for 0
k
n − 1, then |( f n ) (x)|  eλ0 n , + k (2) for any n ∈ Z , if f (x) ∈ / V for 0
k
n − 1 and f n (x) ∈ V , then n  λ n 0 |( f ) (x)|  d0 e .

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(B) (Critical orbits) For all c ∈ C and n > 0, f n (c) ∈ / V. (C) (Inside V ) (1) We have f  (x)
= 0 for all x ∈ V , and (2) for all x ∈ V \C, there exists p0 (x) > 0 such that f j (x) ∈ / V for all j < p0 (x) 1

and |( f p0 (x) ) (x)|  d0−1 e 3 λ0 p0 (x) . We first formulate hypotheses that imply the existence of maps in F that satisfy Definition 3.1(B). 3.1. The general result. We formulate the result for two-parameter families consisting of maps with two critical points. The result generalizes in a natural way for k-parameter families consisting of maps with k critical points. The first hypothesis is formulated in terms of the evolutions n (a, b) → γn(i) (a, b) where γn(i) (a, b) = f a,b (c(i) (a, b)). (i)

The evolutions {γn : n ∈ N} generate critical curve dynamics. Define n : A × B → (1) (2) I × I by n = (γn , γn ). We now present the general hypotheses. For J ⊂ I and ε > 0, let J ε denote the ε-neighborhood of J . Suppose there exist subintervals I1 and I2 of I , subintervals 1 ⊂ A and 2 ⊂ B, δ1 > 0, and ε1 > 0 such that the following hold. (H1) (Finite Misiurevicz condition) There exists n 0 ∈ Z+ such that n 0 (1 × 2 ) ⊃ I1 × I2 and for i ∈ {1, 2}, (a, b) ∈ 1 × 2 , and n < n 0 , we have (i) γn (a, b) ∈ I \Cδ1 (a, b). (H2) There exist fixed parameters aˆ ∈ 1 and bˆ ∈ 2 satisfying f ˆ (I1 ) × a, ˆ b

ε1 ε1 f a, ˆ bˆ (I2 ) ⊃ I1 × I2 . (H3) For all (a, b) ∈ 1 × 2 , we have I1 × I2 ⊂ I \Cδ1 (a, b) × I \Cδ1 (a, b).

Proposition 3.2. Suppose F satisfies (H1)–(H3). If √ 2 2 max{ ∂a F C 0 , ∂b F C 0 } · max{|1 |, |2 |} < ε1 ,

(3.1)

(a ∗ , b∗ )

then there exists ∈ 1 × 2 such that for i ∈ {1, 2} and for every n ∈ N, (i) γn (a ∗ , b∗ ) ∈ I \ Cδ1 (a ∗ , b∗ ). Proof of Proposition 3.2. Define G = ( f a, ˆ bˆ , f a, ˆ bˆ ). Applying (H1) and (H2), we have ε1 ε1 G(n 0 (1 × 2 )) ⊃ I1 × I2 . For every (a, b) ∈ 1 × 2 , we have n 0 +1 (a, b) − G(n 0 (a, b)) < ε1 , since ε1 satisfies (3.1). Therefore, n 0 +1 (1 ×2 ) ⊃ I1 ×I2 . Inductively, n (1 ×2 ) ⊃ I1 × I2 for all n  n 0 . Define n 0 = (1 × 2 ) ∩ n−1 (I1 × I2 ). 0 For n > n 0 , define

−1 n+1 = n ∩ n+1 (I1 × I2 ).

Let = and choose (a ∗ , b∗ ) ∈ .  

∞  k=n 0

k

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3.2. Verifying (H1). We present a two-step procedure for the verification of hypothesis (H1). First, we assume that 1 is a diffeomorphism on 1 × 2 . This implies that the image of 1 ×2 contains a rectangle in I × I . Second, if we assume that each map f a,b is expanding on I \Cδ1 , then the evolutions γ (1) and γ (2) will enlarge the rectangle to macroscopic size. The required time for this enlargement depends upon the magnitude of the expansion. Therefore, greater expansion results in a smaller value of n 0 . We now make these ideas precise. Suppose that 1 is a diffeomorphism on 1 × 2 such that for i ∈ {1, 2} and for (i) every (a, b) ∈ 1 × 2 , we have γ1 (a, b) ∈ I \Cδ1 . Define   (1) ∂ γ (a, b) J (a, b) =  a 1(2) ∂a γ1 (a, b)

  ∂b γ1(1) (a, b) . (2) ∂b γ1 (a, b)

Assume that there exists k0 > 0 such that |J |  k0 on 1 × 2 . This implies that 1 (1 × 2 ) contains a box with side length bounded below by k02 min{|1 |, |2 |}, λM where λM =

sup

(a,b)∈1 ×2

sup{|λ| : λ is an eigenvalue of D1∗ D1 }.

 |  K > 1 on I\C . Lower Now suppose that for every (a, b) ∈ 1 × 2 we have | f a,b δ1 bounds on K will be given as the discussion proceeds. (1) (2) We choose K based on the magnitudes of the partial derivatives of γ1 and γ1 . Assume there exists ρ > 0 such that on 1 × 2 we have

(1) |∂a γ1(1) |  ρ or |∂b γ1(1) |  ρ, and (2) (2) (2) |∂a γ1 |  ρ or |∂b γ1 |  ρ. Suppose for the sake of definiteness that (1) and (2) hold with respect to the operator ∂a . We now relate spatial and parametric derivatives. The equation ∂ (i) ∂ (i) ∂ F (i)  γn+1 (a, b) = f a,b γn (a, b) + (γ (a, b), a, b) (γn(i) (a, b)) · ∂a ∂a ∂a n

(3.2)

implies that parametric derivatives grow exponentially provided that spatial derivatives grow exponentially. If K satisfies 3 K , and 4 ∞ 1 ∂a F C 0 K−j
, 4

Kρ − ∂a F C 0 

j=2

then (3.2) implies that |∂a γn(i) (a, b)| 

1 n K 2

(3.3)

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(i)

provided γ j (a, b) ∈ I \Cδ1 for j < n. Hypothesis (H1) may be verified as follows. (i)

Look for a time n 0 such that for i ∈ {1, 2}, γn 0 (1 × {b}) ⊃ Ii for every b ∈ 2 and γk(i) (a, b) ∈ I \Cδ1 for all k < n 0 and (a, b) ∈ 1 × 2 . By (3.3), we have K n0 ≈

2λ M max{|I1 |, |I2 |} . k02 min{|1 |, |2 |}

3.3. A two-parameter example. Let S 1 = R/2π Z. Let  : S 1 → R be a C 3 function with two nondegenerate critical points c(1) and c(2) . We assume that (c(1) )
= (c(2) ). Fix ζ ∈ S 1 . Consider the two-parameter family of circle maps F = { f a,L : a ∈ S 1 , L ∈ R+ } defined by f a,L (θ ) = ζ + L(θ ) + a. Small perturbations of this family frequently arise as singular limits of rank one families. Definition 3.3. We say that (a, L) is a Misiurewicz pair if f a,L ∈ M. The goal of this subsection is to prove the following result. Theorem 3.4. There exists L 0 > 0 such that if L  L 0 , then there exists a Misiurewicz pair (a ∗ , L ∗ ) with L ∗ ∈ [L , L + 2π/|(c(2) ) − (c(1) )|] and a ∗ ∈ [0, 2π ). Remark 3.5. Misiurewicz points occur with greater frequency as L increases. Wang and Young (9) prove that there exists L 1  L 0 such that if L  L 1 , then f a,L ∈ M for a O(1/L)-dense subset of parameters a ∈ [0, 2π ). Proof of Theorem 3.4. We prove Theorem 3.4 in two steps. We first show that for L sufficiently large, if f a,L satisfies Definition 3.1(B), then f a,L ∈ M. We then prove the existence of parameters for which f a,L satisfies Definition 3.1(B). Set f = f a,L for the sake of simplicity. Let k1 = 21 min{ (c(1) ),  (c(2) )}. There exists δ2 = δ2 () such that |c(2) −c(1) | > 2δ2 and | | > k1 on Cδ2 . Notice that | f  | > k1 L on Cδ2 . At this point we introduce the auxiliary constant K . This constant will be used to bound the derivative of f from below away from the critical set. Lower bounds on K will be given as the proof develops. We choose K before we choose L. Let σ = 2k1−1 L −1 K 3 and assume σ2 < δ2 . For x ∈ Cδ2 \C 1 σ we have | f  (x)|  K 3 . Choose L sufficiently large so that | f  |  K 3 2 outside C 1 σ . Summarizing, the map f has the following properties. 2

(P1) | f  | > k1 L on Cδ2 , (P2) | f  |  K 3 outside C 1 σ . 2

The following recovery lemma asserts that if an orbit visits a small neighborhood of a critical point, then the derivative along this orbit regains a definite amount of exponential growth as this orbit tracks the orbit of the critical point for a period of time. Set K 2 =  C 2 . Let V = {x ∈ S 1 : | f  (x)|
K } and note that V ⊂ C 1 σ . Together 2 with (P1) and (P2), Lemma 3.6 implies that if f satisfies Definition 3.1(B), then f ∈ M.

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Lemma 3.6 (Recovery estimate). Let c ∈ C be such that f n (c) ∈ / Cσ for all n ∈ N. For x ∈ V , let n(x) be the smallest value of n such that | f n (x) − f n (c)| > 4K1 2 K 3 L −1 . We have n(x) > 1 and |( f n(x) ) (x)|  k3 K n(x) for some k3 = k3 (k1 , K 2 ). The proof of Lemma 3.6 uses the following distortion estimate. Sublemma 3.7 (Local distortion estimate). Let x, y ∈ S 1 . For i ∈ Z+ , let ωi denote the segment between f i (x) and f i (y). If n ∈ Z+ is such that |ωi |
4K1 2 K 3 L −1 and d(ωi , C)  21 σ for all 0
i < n, then

( f n ) (x) ( f n ) (y)


2.

Proof of Sublemma 3.7. We have

n   

 i n−1 ( f ) (x) f ( f (x)) log = log ( f n ) (y) f  ( f i (y)) i=0

n−1 | f  ( f i (x)) − f  ( f i (y))|
| f  ( f i (y))| i=0

n−1 L K 2 | f i (x) − f i (y)|
K3 i=0 n−1  L K2 1
| f n−1 (x) − f n−1 (y)| < log(2) K3 K 3i i=0

provided K is sufficiently large.   Proof of Lemma 3.6. We first show that n(x) > 1. Since x ∈ V , we have K  | f  (x)| = | f  (γ1 )| · |x − c| and therefore | f (x) − f (c)| =

| f  (γ2 )| 2  C 0 2 1  | f (γ2 )| · |x − c|2
K . K
 2 2 2| f (γ1 )| 2k1 L

We may assume the final quantity is less than ma 3.7 implies

1 4K 2

K 3 L −1 . If n(x) = 2, then Sublem-

1 K 3 L −1 < | f 2 (x) − f 2 (c)| = |( f 2 ) (γ3 )| · |x − c|
2|( f 2 ) (x)| · |x − c|. 4K 2 This inequality coupled with the estimate |x − c|
|( f 2 ) (x)| >

K L  C 0

implies

 C 0 2 K . 8K 2

Now assume n = n(x)  3. Applying Sublemma 3.7 to estimate | f n−1 (x) − f n−1 (c)| and | f n (x) − f n (c)|, we have 1  1 1 | f (γ2 )| · |x − c|2 · |( f n−2 ) ( f (c))|
K 3 L −1 , 2 2 4K 2 1  1 K 3 L −1 . | f (γ2 )| · |x − c|2 · 2|( f n−1 ) ( f (c))| > 2 4K 2

(3.4) (3.5)

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The recovery estimate follows from the lower bound |( f n ) (x)| 

1  | f (γ1 )| · |x − c| · |( f n−1 ) ( f (c))|. 2

Replacing |( f n−1 ) ( f (c))| with the lower bound provided by (3.5) and then replacing |x − c|−1 with the lower bound provided by (3.4) yields |( f n ) (x)| 

3 3 k1 k1 n K 2 n− 2  K . 8K 2 8K 2

  We have shown that for L sufficiently large, if f satisfies Definition 3.1(B), then f ∈ M with V = {x ∈ S 1 : | f  (x)|
K }. We now find a ∗ ∈ [0, 2π ) and L ∗ ∈ [L , L + 2π/|(c(2) ) − (c(1) )|) such that f a ∗ ,L ∗ satisfies Definition 3.1(B) by applying Proposition 3.2. Additional lower bounds on K will be given as the need arises. Hypotheses (H1)–(H3) are verified as follows. Let z ∈ S 1 be such that d(z, C)  d(y, C) for all y ∈ S 1 . We have d(z, C 1 σ )  π2 − 21 σ . There exists a˜ ∈ [0, 2π ) and 2

(1) (2) ˜ L) = γ1 (a, ˜ L) = z. This is so L ∈ [L , L + 2π/|(c(2) ) − (c(1) )|) such that γ1 (a, because (2) (1) |∂ L γ1 (a, L) − ∂ L γ1 (a, L)| = |(c(2) ) − (c(1) )|.

Referring to the setting of Subsects. 3.1 and 3.2, we have J (a, L) = (c(2) ) − (c(1) ) and we therefore set k0 = |(c(2) ) − (c(1) )|. Let 1 be a parameter interval in a-space of length λkM2 K −3 centered at a˜ and let 2 be a parameter interval in L-space 0

of the same length centered at L. We assume K is sufficiently large so that 2 ⊂ [L , L + 2π/|(c(2) ) − (c(1) )|). The image 1 (1 × 2 ) contains a box such that the length of each of the sides is equal to K −3 . Let I1 and I2 be the vertical and horizontal projections of this box onto I , respectively. Since |γ1(i) (a, L) − z|
max{1, |(c(1) )|, |(c(2) )|} ·

1 λ M −3 π K < − σ 2 2 2k02

for i ∈ {1, 2} and for all (a, L) ∈ 1 × 2 provided K is sufficiently large, we have I1 ⊂ S 1 \C 1 σ and I2 ⊂ S 1 \C 1 σ . 2 2 By construction, (H1) is satisfied with n 0 = 1 and the intervals I1 and I2 satisfy (H3).  |  K 3 on S 1\C Setting ε1 = 1, (H2) is satisfied because | f a,L 1 for all (a, L) ∈ 1 ×2 . 2σ If K is large enough so that (3.1) holds, then the application of Proposition 3.2 with δ1 = 21 σ produces a Misiurewicz pair (a ∗ , L ∗ ) ∈ 1 × 2 .   4. Theory of Rank One Attractors Let D denote the closed unit disk in Rn−1 and let M = S 1 × D. We consider a family of maps Ta,b : M → M, where a = (a1 , . . . , ak ) ⊂  is a vector of parameters and b ∈ B0 is a scalar parameter. Here  = 1 × · · · × k ⊂ Rk is a product of intervals and B0 ⊂ R\{0} is a subset of R with an accumulation point at 0. Points in M are denoted by (x, y) with x ∈ S 1 and y ∈ D. Rank one theory postulates the following.

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(G1) Regularity conditions. (a) For each b ∈ B0 , the function (x, y, a) → Ta,b (x, y) is C 3 . (b) Each map Ta,b is an embedding of M into itself. (c) There exists K D > 0 independent of a and b such that for all a ∈ , b ∈ B0 , and z, z  ∈ M, we have | det DTa,b (z)|
K D. | det DTa,b (z  )| (G2) Existence of a singular limit. For a ∈ , there exists a map Ta,0 : M → S 1 ×{0} such that the following holds. We select a special index j ∈ {1, . . . , k}. For every fixed set {ai ∈ i : i
= j}, the maps (x, y, a j ) → Ta,b (x, y) converge in the C 3 topology to (x, y, a j ) → Ta,0 (x, y). Identifying S 1 × {0} with S 1 , we refer to Ta,0 and the restriction f a : S 1 → S 1 defined by f a (x) = Ta,0 (x, 0) as the singular limit of Ta,b . (G3) Existence of a sufficiently expanding map within the singular limit. There exists a∗ = (a1∗ , . . . , ak∗ ) ∈  such that f a∗ ∈ M. (G4) Parameter transversality. Let Ca∗ denote the critical set of f a∗ . Define a˜ j = (a1∗ , . . . , a ∗j−1 , a j , a ∗j+1 , . . . , ak∗ ). We say that the family { f a } satisfies the parameter transversality condition with respect to parameter a j if the following holds. For each x ∈ Ca∗ , let p = f (x) and let x(˜a j ) and p(˜a j ) denote the continuations of x and p, respectively, as the parameter a j varies around a ∗j . The point p(˜a j ) is the unique point such that p(˜a j ) and p have identical itineraries under f a˜ j and f a∗ , respectively. We have     d d f a˜ j (x(˜a j ))
= p(˜a j ) . da j da j a j =a ∗ a j =a ∗ j

j

(G5) Nondegeneracy at ‘turns’. For each x ∈ Ca∗ , there exists 1

n − 1 such that ∂ Ta∗ ,0 (x, 0)
= 0. ∂ y (G6) Conditions for mixing. 1

(a) We have e 3 λ0 > 2, where λ0 is defined within Definition 3.1. (b) Let J1 , . . . , Jr be the intervals of monotonicity of f a∗ . Let Q = (qi j ) be the matrix defined by

1, if f a∗ (Ji ) ⊃ J j , qi j = 0, otherwise. There exists N > 0 such that Q N > 0. The following lemma often facilitates the verification of (G4). Lemma 4.1 ((5,4)). Let f = f a∗ . Suppose that for all x ∈ Ca∗ , we have ∞

1

k=0

|( f k ) ( f (x))|

< ∞.

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Then for each x ∈ Ca∗ , ∞ [(∂ f )( f k (x))] a j a˜ j a j =a ∗j k=0

( f k ) ( f (x))

 =

d d f a˜ (x(˜a j )) − p(˜a j ) da j j da j

 a j =a ∗j

.

(4.1)

Rank one theory states that given a family {Ta,b } satisfying (G1)–(G5), a measuretheoretically significant subset of this family consists of maps admitting attractors with strong chaotic and stochastic properties. We formulate the precise results and we then describe the properties that the attractors possess. Theorem 4.2 ((8,7)). Suppose the family {Ta,b } satisfies (G1)–(G3) and (G5). For all 1
j
k such that the parameter a j satisfies (G4) and for all sufficiently small b ∈ B0 , there exists a subset A j ⊂  j of positive Lebesgue measure such that for a j ∈ A j , Ta˜ j ,b admits a strange attractor  with properties (SA1), (SA2), and (SA3). Theorem 4.3 ((8,9,7)). In the sense of Theorem 4.2, (G1)–(G6) =⇒ (SA1)–(SA4). (SA1) Positive Lyapunov exponent. Let U denote the basin of attraction of the attractor . For almost every (x, y) ∈ U with respect to Lebesgue measure, the orbit of (x, y) has a positive Lyapunov exponent. That is, lim

n→∞

1 log DT n (x, y) > 0. n

(SA2) Existence of SRB measures and basin property. (a) The map T admits at least one and at most finitely many ergodic SRB measures all of which have no zero Lyapunov exponents. Let ν1 , · · · , νr denote these measures. (b) For Lebesgue-a.e. (x, y) ∈ U , there exists j (x) ∈ {1, . . . , r } such that for every continuous function ϕ : U → R,  n−1 1 ϕ(T i (x, y)) → ϕ dν j (x) . n i=0

(SA3) Statistical properties of dynamical observations. (a) For every ergodic SRB measure ν and every Hölder continuous function i + ϕ :  → R,  the sequence {ϕ ◦ T : i ∈ Z } obeys a central limit theorem. That is, if ϕ dν = 0, then the sequence n−1 1 ϕ ◦ Ti √ n i=0

converges in distribution to the normal distribution. The variance of the limiting normal distribution is strictly positive unless ϕ ◦ T = ψ ◦ T − ψ for some ψ.

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(b) Suppose that for some N  1, T N has an SRB measure ν that is mixing. Then given a Hölder exponent η, there exists τ = τ (η) < 1 such that for all Hölder ϕ, ψ :  → R with Hölder exponent η, there exists L = L(ϕ, ψ) such that for all n ∈ N,        (ϕ ◦ T n N )ψ dν − ϕ dν ψ dν 
L(ϕ, ψ)τ n .   (SA4) Uniqueness of SRB measures and ergodic properties. (a) The map T admits a unique (and therefore ergodic) SRB measure ν, and (b) the dynamical system (T, ν) is mixing, or, equivalently, isomorphic to a Bernoulli shift. 5. Proof of Theorem 2.1 5.1. Degenerate Hopf bifurcation: the reduced equations. We study the two-dimensional system

r˙ = −µr (5.1) θ˙ = ω + γµ µ + βµr 2 . System (5.1) is obtained from (2.1) by setting gµ = h µ = 0. Let Ft denote the flow of (5.1). For µ sufficiently large, Fτ (µ) ◦ κ maps A into A. The study of this annulus map is the central goal of this subsection. We introduce a new coordinate system in order to standardize the position and size of A. Let r = µρ1 z. Written in terms of z and θ , system (5.1) becomes

z˙ = −µz (5.2) θ˙ = ω + γµ µ + µ2ρ1 βµ z 2 . Let G t denote the flow associated with (5.2). The kick map κ is now given in rectangular coordinates by 



z cos(θ ) z cos(θ ) = . κ z sin(θ ) z sin(θ ) + Lµρ2 −ρ1 We have A = {(z, θ ) : K 4−1
z
K 4 }. The relaxation time τ (µ) is given by z˜ e−µτ (µ) = 1,

(5.3)

where z˜ = Lµρ2 −ρ1 − K 4 . Let µ = G τ (µ) ◦ κ. For µ sufficiently large, µ maps A into A. We now derive µ : A → A explicitly. Let (z 0 , θ0 ) ∈ A. Writing κ(z 0 , θ0 ) = (z 1 , θ1 ), we have z 12 = z 02 + 2Lµρ2 −ρ1 z 0 sin(θ0 ) + L 2 µ2(ρ2 −ρ1 ) ,

 z 0 cos(θ0 ) π −1 . θ1 = − tan 2 z 0 sin(θ0 ) + Lµρ2 −ρ1 Integrating (5.2) and writing G t ◦ κ = (z(t), θ (t)), we have z(t) = z 1 e−µt , θ (t) = θ1 + t (ω + γµ µ) +

βµ 2ρ1 −1 2 µ z 1 (1 − e−2µt ). 2

(5.4)

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Evaluating θ (τ (µ)) using (5.3), we have

  βµ 2ρ1 −1 2 z 12 µ θ (τ (µ)) = θ1 + (ω + γµ µ)τ (µ) + z1 − 2 . 2 z˜

(5.5)

Replacing the first occurrence of z 12 in (5.5) with the right side of (5.4), we obtain   2 z βµ θ (τ (µ)) = θ1 + ξ(µ) + µ2ρ1 −1 z 02 + 2Lz 0 sin(θ0 )µρ1 +ρ2 −1 − µ2ρ1 −1 12 , 2 z˜ where ξ(µ) = (ω + γµ µ)τ (µ) +

βµ 2 2ρ2 −1 L µ . 2

The second component of µ is given by z(τ (µ)) =

z1 . z˜

We wish to show that the family {µ } converges to a singular limit as µ → 0. This cannot be accomplished directly because ξ(µ) diverges as µ → 0, preventing the convergence of θ (τ (µ)). We overcome this difficulty by taking advantage of the fact that θ (t) is computed modulo 2π . Assume that ω > 0. For µ sufficiently small, ξ(µ) is monotone. In addition, ξ(µ) → ∞ as µ → 0. Let (µn ) be a sequence such that µn → 0 monotonically, ξ is monotone on (0, µ1 ], and ξ(µn ) ∈ 2π Z for all n ∈ N. We introduce the parameter a ∈ [0, 2π ) and write µ in terms of a. For n ∈ N and a ∈ [0, 2π ), let µ(a, n) = ξ −1 (ξ(µn ) + a). When referring to µ(a, n), we will henceforth suppress the dependence on n and simply write µ(a). The problematic term ξ(µ) becomes ξ(µ(a)) = a. Writing µ(a) = Ta,L ,µn , we have 1

Ta,L ,µn (z 0 , θ0 ) =

z1 , z˜

2

Ta,L ,µn (z 0 , θ0 ) = θ1 + a +

βµ(a) µ(a)2ρ1 −1 z 02 2

+ 2Lµ(a)ρ1 +ρ2 −1 z 0 sin(θ0 ) − µ(a)2ρ1 −1

 z 12 , z˜ 2

where T 1 and T 2 are the components of T . Let ρ1 and ρ2 satisfy 21 < ρ1 < 1 and ρ1 + ρ2 = 1. Then as n → ∞, Ta,L ,µn converges in the C 0 topology to the map Ta,L ,0 defined by 1

Ta,L ,0 = 1, π 2 Ta,L ,0 = + β0 Lz 0 sin(θ0 ) + a. 2 The following lemma asserts that the convergence is strong enough for the application of rank one theory. Lemma 5.1. Fix L > 0. The maps (z 0 , θ0 , a) → Ta,L ,µn (z 0 , θ0 ) converge in the C 3 topology to the map (z 0 , θ0 , a) → Ta,L ,0 (z 0 , θ0 ) as n → ∞ on the domain A × [0, 2π ).

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Proof of Lemma 5.1. Holding a fixed, the derivatives of θ1 , zz˜1 , and z˜ 12 of orders 1, 2, and 3 with respect to z 0 and θ0 are O(µρ1 −ρ2 ). When differentiating with respect to a, use the fact that for i = 1, 2, 3,   µi+1 n (i) ∂a µ(a) = O  i . log(µ−1 n ) We finish this subsection with a distortion estimate.   Lemma 5.2 (Distortion estimate). Let 0 < L 2 < L 3 . There exists K D > 0 such that for all n ∈ N, a ∈ [0, 2π ), L ∈ [L 2 , L 3 ], and (z 0 , θ0 ), (z 0 , θ0 ) ∈ A, we have | det DTa,L ,µn (z 0 , θ0 )|
K D. | det DTa,L ,µn (z 0 , θ0 )| Proof of Lemma 5.2. Recall that Ta,L ,µn = G τ (µ(a)) ◦ κ. We bound the distortion by analyzing G and κ independently. Let (z 0 , θ0 ), (z 0 , θ0 ) ∈ A. Writing µ = µ(a), det Dκ(z 0 , θ0 ) is given by z 03 + µρ2 −ρ1 · Lz 0 (1 + z 0 ) sin(θ0 ) + µ2(ρ2 −ρ1 ) · L 2 (1 + (z 0 − 1) cos2 (θ0 ))   . z 1 (z 0 sin(θ0 ) + Lµρ2 −ρ1 )2 + z 02 cos2 (θ0 ) This explicit formula implies the estimate det Dκ(z 0 , θ0 ) = O(1). det Dκ(z 0 , θ0 ) Now set G = G τ (µ(a)) . For any point in κ(A), the determinant of the derivative of G is precisely z˜ −1 . Therefore, det DG(z 1 , θ1 ) = 1. det DG(z 1 , θ1 ) 5.2. Inclusion of the higher-order terms in the normal form. We show that the inclusion of the higher-order terms in the differential equations defining the flow does not affect ˆ Written the form of the singular limit derived in Subsect. 5.1. Set r = µρ1 zˆ and θ = θ. ˆ in terms of zˆ and θ , the normal form (2.1) becomes

z˙ˆ = −µˆz + µ4ρ1 zˆ 5 gµ (µρ1 zˆ , θˆ ) (5.6) θ˙ˆ = ω + γ µ + β µ2ρ1 zˆ 2 + µ4ρ1 zˆ 4 h (µρ1 zˆ , θˆ ) µ

µ

µ

t denote the flow generated by (5.6). We define the family {T } on A by first applyLet G t -flow to return κ(A) to A. Set T a,L ,µn = ing the kick map κ and then allowing the G  G τ (µ(a)) ◦ κ. a,L ,µn (z 0 , θ0 ) Lemma 5.3. Fix L > 0. If ρ2 > 13 , then the maps (z 0 , θ0 , a) → T 3 converge in the C topology to the map (z 0 , θ0 , a) → Ta,L ,0 (z 0 , θ0 ) as n → ∞ on the domain A × [0, 2π ).

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789

t ◦ κ, we have Proof of Lemma 5.3. Computing the first component of G

  t −1 µs −µt 4ρ1 5 ρ1 ˆ 1+µ z 1 e zˆ (s) gµ (µ zˆ (s), θ (s)) ds zˆ (t) = z 1 e 0

= z(t) + ζ (t), where the perturbative term ζ (t) is defined by  t eµs zˆ (s)5 gµ (µρ1 zˆ (s), θˆ (s)) ds. ζ (t) = µ4ρ1 e−µt

(5.7)

0

t ◦ κ, we have θˆ (t) = θ (t) + θ(t), ˜ Computing the second component of G where  t  v 2µ4ρ1 z 1 e−2µv eµs zˆ (s)5 gµ (µρ1 zˆ (s), θˆ (s)) ds θ˜ (t) =βµ µ2ρ1 0

8ρ1 −2µv

+µ

e 

0



v 0

t

+ µ4ρ1 0

e zˆ (s) gµ (µ zˆ (s), θˆ (s)) ds µs

5

ρ1

2  dv

(5.8)

zˆ (s)4 h µ (µρ1 zˆ (s), θˆ (s)) ds.

In order to establish C 0 convergence, it suffices to show that the perturbative terms ζ (τ (µ(a))) and θ˜ (τ (µ(a))) converge to 0 in the C 0 topology as n → ∞. Estimating the integrals in (5.7) and (5.8), we obtain ζ (τ (µ)) = O(µ5ρ2 −ρ1 −1 log(µ−1 )),     θ˜ (τ (µ)) = O µ6ρ2 −2 (log(µ−1 ))2 + O µ10ρ2 −3 (log(µ−1 ))3 . Since ρ2 ∈ ( 13 , 21 ) and ρ1 ∈ ( 21 , 23 ), we have ζ (τ (µ)) C 0 → 0 and θ˜ (τ (µ)) C 0 → 0 as µ → 0. We complete the proof of Lemma 5.3 by showing that τ (µ(a)) ◦ κ − D i G τ (µ(a)) ◦ κ C 0 → 0 D i G for 1
i
3. In light of Lemma 5.1, this establishes the asserted C 3 convergence. τ (µ(a)) − Since D i κ C 0 is bounded for 1
i
3, it is sufficient to show that D i G D i G τ (µ(a)) C 0 → 0 for 1
i
3. We use the following elementary Gronwall-type lemma.  be C 1 vector Lemma 5.4 ((10)). Let  ⊂ R N be a convex open domain. Let W and W fields on . Suppose that for t ∈ [0, t0 ], ϕˆ and ϕ solve the equations dϕ d ϕˆ  (ϕ) =W ˆ and = W (ϕ) dt dt with ϕ(0) ˆ = ϕ(0). Then for all t ∈ [0, t0 ], we have ϕ(t) ˆ − ϕ(t)


A1 A2 t (e − 1), A2

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where  (x) − W (x) and A2 = A1 = sup W x∈

N

sup DW  j (x) .

j=1 x∈

We rescale time in (5.2) and (5.6) by setting t = t  τ (µ(a)). Let η and ηˆ denote the rescaled vector fields. We have

η1 = τ (µ(a))(−µz)

η2 = τ (µ(a))(ω + γµ µ + µ2ρ1 βµ z 2 ), ηˆ 1 = τ (µ(a))(−µˆz + µ4ρ1 zˆ 5 gµ (µρ1 zˆ , θˆ )) ηˆ 2 = τ (µ(a))(ω + γµ µ + βµ µ2ρ1 zˆ 2 + µ4ρ1 zˆ 4 h µ (µρ1 zˆ , θˆ ))

We explicitly treat the case i = 1. The cases i = 2 and i = 3 are handled using the same  ϕ = DG, W  = D η, technique. Apply Lemma 5.4 with ϕˆ = D G, ˆ W = Dη, and t = 1. The quantity A2 is bounded. Therefore, the estimate A1 = O(µ5ρ2 −ρ1 −1 log(µ−1 )) implies that 1 − DG 1 C 0 = O(µ5ρ2 −ρ1 −1 log(µ−1 )). (5.9) D G   5.3. Verification of (G1)-(G6). Theorem 2.1 follows from an application of Theorems 4.2 and 4.3. Statements (1) and (2) of Theorem 2.1 require the verification of (G1)-(G5) for a,L ,µn }. Statement (3) of Theorem 2.1 requires the additional verification the family {T of (G6). We proceed with the verification of statement (1) of Theorem 2.1. Properties (G1)(a) and (G1)(b) follow from the general theory of ordinary differential equations. For (G1)(c), τ (µ(a)) is bounded because the distortion of κ it suffices to show that the distortion of G is bounded. Using (5.9), we have   ε2 z˜−1 + ε1  τ (µ(a)) (z 1 , θ1 ) = βµ 2ρ −1 DG , 1 2z 1 − 2zz˜ 21 + ε3 1 + ε4 2 µ where ε j = O(µ5ρ2 −ρ1 −1 log(µ−1 )) for 1
j
4. Since ρ2 > 38 , for 1
j
4. Therefore, we have

εj z˜ −1

→ 0 as µ → 0

τ (µ(a)) (z 1 , θ1 )) = z˜ −1 + O(µ5ρ2 −ρ1 −1 log(µ−1 )). det(D G τ (µ(a)) is bounded. This estimate implies that the distortion of G 2 Lemma 5.3 establishes (G2). Let f a,L denote the restriction of Ta,L ,0 to the circle S 1 = {(z 0 , θ0 ) : z 0 = 1}. We have f a,L (θ ) =

π + β0 L sin(θ ) + a. 2

Applying Theorem 3.4 with (θ ) = sin(θ ), c(1) = π2 , and c(2) = 3π 2 , if L is sufficiently large then there exist L ∗ ∈ [L , L + |βπ0 | ] and a ∗ ∈ [0, 2π ) such that f a ∗ ,L ∗ ∈ M. This is (G3). We establish parameter transversality (G4) by applying Lemma 4.1. Write f = f a ∗ ,L ∗ and f a = f a,L ∗ . We have ∂a f a (·) = 1 and |( f k ) ( f (x))|  K k . Therefore,

Strange Attractors in Periodically-Kicked Degenerate Hopf Bifurcations

the absolute value of the left side of (4.1) is bounded below by 1 − quantity is positive if K > 2. For (G5), observe that 2

791

∞

k=1

K −k . This

2

∂z 0 Ta,L ,0 (1, c(1) ) = β0 L
= 0 and ∂z 0 Ta,L ,0 (1, c(2) ) = −β0 L
= 0. This completes the verification of statement (1) of Theorem 2.1. Statement (2) of Theorem 2.1 follows from the fact that for all L sufficiently large, f a,L ∈ M for a O(L −1 )-dense set of values of a. Wang and Young (9) prove this result in a slightly different context. The proof for the family { f a,L } is essentially the same. Statement (3) of Theorem 2.1 requires the verification of the conditions for mixing (G6). Property (G6)(a) holds provided eλ0 = K > 8. Property (G6)(b) is satisfied with N = 1 provided L is sufficiently large. References 1. Guckenheimer, J., Wechselberger, M., Young, L.-S.: Chaotic attractors of relaxation oscillators. Nonlinearity 19(3), 701–720 (2006) 2. Lin, K.K.: Entrainment and chaos in a pulse-driven Hodgkin-Huxley oscillator, SIAM J. Appl. Dyn. Syst. 5(2), 179–204 (2006) (electronic) 3. Lin, K.K., Young, L.-S.: Shear-induced chaos. http://arxiv.org/pdf/0705.3294, 2007 4. Thieullen, Ph., Tresser, C., Young, L.-S.: Positive Lyapunov exponent for generic one-parameter families of unimodal maps. J. Anal. Math. 64, 121–172 (1994) 5. Thieullen, Ph., Tresser, C., Young, L.-S.: Exposant de Lyapunov positif dans des familles à un paramètre d’applications unimodales. C. R. Acad. Sci. Paris Sér. I Math. 315(1), 69–72 (1992) 6. Wang, Q., Oksasoglu, A.: Strange attractors in periodically kicked Chua’s circuit. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 15(1), 83–98 (2005) 7. Wang, Q., Young, L.-S.: Toward a theory of rank one attractors. Ann. Math. 16(2), 349–480 (2008) 8. Wang, Q., Young, L.-S.: Strange attractors with one direction of instability. Comm. Math. Phys. 218(1), 1–97 (2001) 9. Wang, Q., Young, L.-S.: From invariant curves to strange attractors. Comm. Math. Phys. 225(2), 275–304 (2002) 10. Wang, Q., Young, L.-S.: Strange attractors in periodically-kicked limit cycles and Hopf bifurcations. Comm. Math. Phys. 240(3), 509–529 (2003) Communicated by G. Gallavotti

Commun. Math. Phys. 281, 793–803 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0494-5

Communications in

Mathematical Physics

Deligne Products of Line Bundles over Moduli Spaces of Curves L. Weng1,
, D. Zagier2,3 1 Graduate School of Mathematics, Kyushu University, Fukuoka, Japan.

E-mail: [email protected]

2 Max-Planck-Institut für Mathematik, Bonn, Germany 3 Collège de France, Paris, France

Received: 22 June 2007 / Accepted: 9 October 2007 Published online: 20 May 2008 – © The Author(s) 2008

Abstract: We study Deligne products for forgetful maps between moduli spaces of marked curves by offering a closed formula for tautological line bundles associated to marked points. In particular, we show that the Deligne products for line bundles on the total spaces corresponding to “forgotten” marked points are positive integral multiples of the Weil-Petersson bundles on the base moduli spaces. 1. Introduction Let Mg,N denote the moduli space of curves C of genus g with N ordered marked points P1 , . . . , PN , and π = π N : Cg,N → Mg,N the universal curve over Mg,N . (We are using the language of stacks here [3].) The marked points give sections Pi : Mg,N → Cg,N , i = 1, . . . , N of π . The Picard group of Mg,N is known to be free of rank N + 1 [4] and has a Z-basis given by the Mumford class λ (the line bundle whose fiber at C is detH 0 (C, K C ) ⊗ detH 1 (C, K C )−1 ) and the “tautological line bundles” i := Pi∗ (K N ), where K N is the relative canonical line bundle (relative dualizing sheaf) of π [1,5]. The i carry metrics in such a way that their first Chern forms give the Kähler metrics on Mg,N defined by Takhtajan-Zograf [9,10] in terms of Eisenstein series associated to punctured Riemann surfaces [11,12]. There is a further interesting element  ∈ Pic(Mg,N ), whose associated first Chern form (for a certain natural metric) gives the Kähler form for the Weil-Petersson metric on Mg,N [11,12]; it is given in terms of λ and the i
N by the Riemann-Roch formula  = 12λ + i=1 i . We  by the for  can also  define  mula  := K N (P1 + · · · + P N ), K N (P1 + · · · + P N ) π , where ·, · π : Pic(Cg,N )2 → Pic(M  g,N ) denotes the Deligne pairing. We recall that the Deligne pairing is a bilinear map ·, · p : Pic(Y )2 → Pic(X ) which is defined for any flat morphism p : Y → X
Partially supported by the Japan Society for the Promotion of Science.

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  of relative dimension 1 and that it can be generalized to a multilinear map ·, . . . , · p : Pic(Y )n+1 → Pic(X ), the Deligne product, defined for any flat morphism p : Y → X of relative dimension n. (The precise definitions will be given in §3.) We are interested in computing the Deligne product explicitly for the forgetful map π N ,m : Mg,N +m → Mg,N ,

(C; P1 , . . . , PN +m ) → (C; P1 , . . . , PN ).

˜ , ˜ ˜ j to denote the Mumford, Weil-Petersson and tautological In other words, if we use λ,   line bundles on Mg,N +m , respectively, then we would like to compute L 1 , . . . , L m+1 π N ,m as a linear combination of 1 , . . . ,  N and λ (or ) on Mg,N , where each L ν is one of ˜ We have not solved this problem in general (though it is inter˜1 , . . . , ˜N +m and λ˜ (or ). ˜ esting and perhaps not intractable), but only in the case where each L ν is one of the  i , ˜ does not appear. The formula we find expresses L 1 , . . . , L m+1 i.e., where λ˜ (or ) π N ,m in this case as a positive linear combination of  and those i (i = 1, . . . , N ) for which if eachof L 1 , . . . , L m+1 is one of the last m line ˜i appear among the L ν . In particular,  is simply a positive integer multiple bundles ˜N +1 , . . . , ˜N +m , then L 1 , . . . , L m+1 π N ,m of the Weil-Petersson class , giving an interesting relation between the Weil-Petersson and the tautological line bundles.

2. Statement of the Theorem   As just explained, we want to compute the Deligne product L 1 , . . . , L m+1 π , where N ,m each L ν belongs to the set {˜1 , . . . , ˜N +m }. It turns out to be more convenient to use the multiplicities where the ˜i occur in {L 1 , . . . , L m+1 } as coordinates. We therefore introduce the notation   TN ,m (a1 , . . . , a N +m ) := ˜1 , . . . , ˜1 , . . . , ˜N +m , . . . , ˜N +m π       N ,m a1

a N +m

∈ Pic(Mg,N ),

(1)

where a1 , . . . , a N +m ∈ Z≥0 with a1 + · · · + a N +m = m + 1. We will sometimes denote this element by TN ,m (a1 , . . . , a N ; a N +1 , . . . , a N +m ) or even, setting a N +i =: di , by TN ,m (a1 , . . . , a N ; d1 , . . . , dm ), to emphasize the different roles played by the indices corresponding to the points which are also marked in Mg,N and to those which are “forgotten” by the projection map π N ,m . Our main result is then: Theorem. Let a1 , . . . , a N , d1 , . . . , dm ≥ 0 be non-negative integers with sum m + 1. Then the line bundle TN ,m (a1 , . . . , a N ; d1 , . . . , dm ) defined in (1) is given in terms of the elements i ,  ∈ Pic(Mg,N ) by the formula  N

ai !

 m

i=1

= C1 (d)

d j ! TN ,m (a1 , . . . , a N ; d1 , . . . , dm )

j=1 N  i=1

  + C2 (d) , ai i − N

(2)

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795

= N + 2g − 3 and the coefficients Cν (d) = Cν ( N , d1 , . . . , dm ) (ν = 1, 2) where N and are given explicitly by depend only on the di and on N C1 (d) = C2 (d) =

m  + n − 1)! (m − n)! ( N n=0 m  n=0

− 1)! (N

σn ,

+ n − 2)! (m − n + 1)! ( N σn N ( N − 2)!

(3)

with σn = σn (d1 , . . . , dm ) the n th elementary symmetric polynomial in the di . = 0 or 1, then the factors 1/( N − 1)! and 1/ N ( N − 2)! occurring in Remark. 1. If N − 1)/ N !, the formulas for C1 (d) and C2 (d) are to be interpreted as N / N ! and ( N respectively. 2. The proof (or rather, the recursive description of the TN ,m on which it is based) will show that in the formula for TN ,m (a1 , . . . , a N ; d1 , . . . , dm ) in terms of i and , all the coefficients are non-negative and integral (even though 1 , . . . ,  N ,  is not a Z-basis of Pic(Mg,N )). Neither property is obvious from the formulas (2) and , (3), though one can see easily that both C1 (d) and C2 (d) − (m + 1 − σ1 )C1 (d)/ N the coefficient of  on the right-hand side of (2), are polynomials in N . 3. In Sect. 6 we will give an alternative explicit formula for the coefficients C1 (d) and C2 (d). 4. Notice that, as already mentioned in the Introduction, formula (2) in the special case when all the ai are 0 says that TN ,m (0, . . . , 0; d1 , . . . , dm ) is a multiple of  alone. In other words, all Deligne products of line bundles corresponding to points which are “forgotten” by π N ,m are positive integral multiples of the Weil-Petersson bundle . 3. The Deligne product We start with some basic facts about Deligne products [2]. Let π : X → S be a flat family of algebraic varieties of relative dimension n. Then for any n + 1 invertible sheaves L 0 , . . . , L n over X , following Deligne [2], we    may introduce  prod the Deligne uct, denoted by L 0 , . . . , L n (X/S) or L 0 , . . . , L n π or simply L 0 , . . . , L n , defined uniquely axioms:  by the following  (DP1) L 0 , . . . , L n (X/S) is an invertible sheaf on S, and is symmetric and multi-linear in the L i ’s.    (DP2) L 0 , . . . , L n (X/S) is locally generated by the symbols t0 , . . . , tn , where the ti are sections of L i whose divisors have no common intersection, and these symbols satisfy the following

property: if one
multiplies one of the sections ti by a rational function f on X , where j=i div(t j ) = n k Yk is finite over S and div( f ) has no intersection with any Yk , then       NormYk /S ( f )n k · t0 , . . . , tn . t0 , . . . , f ti , . . . , tn = k

(Here NormYk /S ( f ) is defined as follows: Since Yk is finite over S, the function field of Yk is a finite extension of that of S, and hence can be viewed as a finite dimensional vector space. Since f is in the function field of X , via restriction we may view f as an

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L. Weng, D. Zagier

element of the function field of Yk . Then multiplication by f defines a linear map of the vector space of the function field of Yk over the function field of S. By definition, the determinant of this linear map is called the norm of f with respect to Yk /S.) (DP3) If tn is a section of L n such that all components Dα of the divisor div(tn ) =
α n α Dα are flat (of relative dimension n − 1) over S, then we have a canonical isomorphism 

     L 0 , . . . , L n (X/S)  L 0 , . . . , L n−1 (div(tn )/S) := ⊗α L 0 , . . . , L n−1 (Dα /S)⊗n α .

Roughly speaking, the Deligne product may be built up as follows using the above axioms: one first uses (DP3) to make an induction on the relative dimension so as to reduce to special cases, say, n = 1, by using a certain choice of sections, and then shows with the help of axiom (DP2) that this construction does not depend on the choice of sections of line bundles and is symmetric by virtue of the Weil reciprocity law. As a consequence of these axioms and the uniqueness, we know that the Deligne product formalism is compatible with any base change, and that the products satisfy the following compatibility relations with respect to compositions of flat morphisms: Proposition 1. ([2]). Let f : X → Y and g : Y → Z be flat morphisms of relative dimension n and m, respectively. Then: (a) For invertible sheaves L 0 , . . . , L n on X and H1 , . . . , Hm on Y , we have 

    L 0 , . . . , L n f , H1 , . . . , Hm g  L 0 , . . . , L n , f ∗ H1 , . . . , f ∗ Hm g◦ f . (4a)

(b) For invertible sheaves L 1 , . . . , L n on X and H0 , . . . , Hm on Y , we have 

   f (c (L )···c (L ))  f ∗ H0 , L 1 . . . , L n f , H1 , . . . , Hm g  H0 , H1 , . . . , Hm g∗ 1 1 1 n .

(4b)

A special case of Proposition 1 which will be needed later is the formula 

L , f ∗ H0 , . . . , f ∗ Hm

 g◦ f

   deg f (L) H0 , . . . , Hm g

(5)

for f , g as in the proposition with n = 1 and for any bundles L on X and H0 , . . . , Hm on  Y . ∗To get this, we use part (a) of the proposition  to write the left-hand side as L , f H0 f , H1 , . . . , Hm g and then part (b) to write L , f ∗ H0 f as deg f (L)H0 . Remark. Recall that there is a map c1 from Pic(X ) to the codimension 1 part CH1 (X ) of the Chow group of X . If f : X → Y is flat of relative dimension n and L 0 , . . . , L n belong to Pic(X ), then the image of L 0 , . . . , L n under c1 is equal to the image of the product c1 (L 0 ) · · · c1 (L n ) under the push-forward map f ∗ : CHn+1 (X ) → CH1 (Y ). At this level, formulas (4a), (4b) and (5) are just specializations of the general projection formula g∗ ( f ∗ (A) · B) = (g f )∗ (A · f ∗ (B)), valid for any flat morphisms f : X → Y and g : Y → Z and elements A ∈ CH(X ), B ∈ CH(Y ).

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4. Geometric Preparations We now apply the Deligne product to universal curves over moduli spaces. As in §1, we denote by K N the relative canonical line bundle of π = π N : Cg,N → Mg,N , by Pi (1 ≤ i ≤ N ) the N sections of π and by i = Pi∗ (K N ) the i th tautological line bundle on Mg,N . We write L i (1 ≤ i ≤ N + 1) for the line bundles on Cg,N defined in the same way, where Cg,N is identified with Mg,N +1 . Also for convenience, we denote the bundle OCg,N (Pi ) (1 ≤ i ≤ N ) simply by Pi . The following properties can be found in [6,7]. (Deligne products are not used in Knudsen’s original papers, but the verbatim change is rather trivial. See e.g., [11,12].) Proposition 2. ([6]) With the above notations, we have   (a) Pi , P j π  O (i, j = 1, . . . , N , i = j);   (b) K N (Pi ), Pi π  O (i = 1, . . . , N ); (c) L i  π ∗ i + Pi (i = 1, . . . , N ); (d) L N +1  K N (P1 + · · · + P N ). The next proposition, which is a slight extension of Proposition 2, contains all the geometric information which we will need to compute the Deligne products in (1). We use the same notations as above, but also denote by π  = π N −m,m the forgetful map from Mg,N to Mg,N −m for some m ≥ 0 and use ξ1 , . . . , ξm to denote m general elements of Pic(Cg,N ). Proposition 3. With the above notations, we have   (a) Pi , P j , ξ1 , . . . , ξm π  ◦π  O (i, j = 1, . . . , N , i = j);   (b) Pi , L i , ξ1 , . . . , ξm π  ◦π  O (i = 1, . . . , N );   (c) Pi , L N +1 π  O (i = 1, . . . , N );   (d) degπ L N +1 = 2g − 2 + N . Proof. Since the sections Pi and P j are disjoint for i = j, the pull-back of O(Pi ) to P j is trivial (Prop. 2(a)). Therefore axiom (DP3) from §3 implies (a). Next, we use Prop. 2(c) to write       L i , Pi , ξ1 , . . . , ξm π  ◦π  π ∗ i , Pi , ξ1 , . . . , ξm π  ◦π + Pi , Pi , ξ1 , . . . , ξm π  ◦π .     By (DP3), the first term equals i , ξ1 P , . . . , ξm P π  (actually multiplied by degπ (Pi ), i i     but the relative degree of a section is 1), while the second term is − i , ξ1  , . . . , ξm   Pi

Pi π

by Prop. formula). This proves (b). Part (c) follows from  2(b) (adjunction  Prop. 2(d),   since Pi , K N (Pi ) π vanishes by the adjunction formula and all Pi , P j π with j = i vanish by (a). Part (d) also follows from Prop. 2(d) by  taking the  relative degree of both sides. We also mention the stronger statement that L N +1 , L i π  (2g − 2 + N ) i for i = 1, . . . , N . The proof of this is similar to the other parts of the proposition, but we omit it since this result will not be used in the sequel.

5. The Recursion Formula for TN,m In this section, we use the results of §4 to give a recursion formula and initial data for the line bundles (1) which determine them completely in Pic(Mg,N ). These recursions will be solved in §6. The recursion formula which we will prove for the TN ,m is as follows.

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Proposition 4. (String Equation) For m ≥ 0 and any integers a1 , . . . , a N +m ≥ 0, we have TN ,m+1 (a1 , . . . , a N +m , 0) =

N +m 

TN ,m (a1 , . . . , ai − 1, . . . , a N +m ),

i=1

with the convention that TN ,m+1 (a1 , . . . , a N +m ) = 0 if any ai < 0. Recall that the indices ai with i > N in TN ,m (a1 , . . . , a N +m ) play a different role than the ai with i ≤ N and that we also use the notations d j for a N + j (1 ≤ j ≤ m) and TN ,m (a1 , . . . , a N ; d1 , . . . , dm ) for TN ,m (a1 , . . . , a N +m ). Proposition 4 lets us reduce the calculation of these bundles by induction to the case when every di is strictly positive. (If any d j is zero, we can put it in the last position, because TN ,m is symmetric in the d’s.)
N
But since i=1 ai + mj=1 d j = m +1, this can only happen if (d1 , . . . , dm ) = (1, . . . , 1) or (2, 1, . . . , 1). There are therefore only two initial cases which have to be considered. The values of TN ,m in these two cases are given by the following: Proposition 5. The line bundles TN ,m (a1 , . . . , a N ; d1 , . . . , dm ) in the two cases when all the di are strictly positive are given by the formulas TN ,m (1, 0, . . . , 0; 1, . . . , 1) =       N −1

m

+ m)! (N 1 ! N

(m ≥ 0)

and + m)! (N TN ,m (0, . . . , 0; 2, 1, . . . , 1) =        + 1)! (N

(m ≥ 1),

m−1

N

= N + 2g − 3. where N Proposition 5 in turn can be deduced by induction over m from the special cases TN ,0 (1, 0, . . . , 0; ) = 1 ,    N −1

TN ,1 (0, . . . , 0; 2) =     N

(the first of which is trivial because the Deligne product is simply the identity map, and the second by the very definition of ) and from the following companion result to Proposition 4. Proposition 6. (Dilaton Equation) For m ≥ 0 and any integers a1 , . . . , a N +m ≥ 0, we have TN ,m+1 (a1 , . . . , a N +m , 1) = (N + m + 2g − 2) TN ,m (a1 , . . . , a N +m ). The proofs of Propositions 4 and 6 are similar to one another and will be given together.   For convenience, we use the abbreviated notation S1◦k1 , S2◦k2 , . . . , Sn◦kn f to denote the Deligne product   S , . . . , S , S , . . . , S , . . . , Sn , . . . , Sn f  1  1  2  2    k1

k2

kn

Deligne Products of Line Bundles over Moduli Spaces of Curves

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for any line bundles Si and any integers ki ≥ 0. We also replace the “N ” of §4 by “N + m” and use the same conventions as there, i.e., i (1 ≤ i ≤ N + m) denotes the i th tautological line bundle on Mg,N +m and L i (1 ≤ i ≤ N + m + 1) the i th tautological line bundle on Mg,N +m+1 , while π and π  denote the projections from Mg,N +m+1 to Mg,N +m and from Mg,N +m to Mg,N , respectively. Finally, we set M = N + m. With these notations, the two formulas to be proved become 

◦a M 1 L ◦a 1 ,..., LM

 π  ◦π

=

M  

◦(ai −1)

1 ◦a 1 , . . . , i

M , . . . , ◦a M

 π

(6)

i=1

  ◦(a −1) (with the usual convention that · · · , i i , · · · = 0 if ai = 0) and  ◦a1   ◦a1 ◦a M  M L 1 , . . . , L ◦a M , L M+1 π  ◦π = (2g − 2 + M) 1 , . . . ,  M π  .

(7)

To prove these equations, we proceed as follows. Using Prop. 2(c), Prop. 3(b) and then Prop. 2(c) again (all of them with N replaced by M), we obtain  ◦a1  ◦(a −1)    L 1 , ξ1 , . . . , ξr π  ◦π = L 1 , π ∗ 1 + P1 1 , ξ1 , . . . , ξr π  ◦π ◦(a −1)    = L 1 , π ∗ 1 1 , ξ1 , . . . , ξr π  ◦π ◦a   = π ∗ 1 1 , ξ1 , . . . , ξr π  ◦π ◦(a −1)    + P1 , π ∗ 1 1 , ξ1 , . . . , ξr π  ◦π for any line bundles ξ1 , . . . , ξr in Pic(Cg,M ), where a1 + r = m + 2. Now if there are a2 indices i with ξi = L 2 , then we can do the same with L 2 as we did with L 1 . This gives  ◦(a −1) ◦(a2 −1)    four terms a priori, but one of them is P1 , π ∗ 1 1 , P2 , π ∗ 2 , . . . π  ◦π and this vanishes by Prop. 3(a). Continuing, we find  ◦a1  ◦a M L 1 , . . . , L ◦a M , L M+1 π  ◦π ◦a ◦a    = π ∗ 1 1 , . . . , π ∗  M M , L ◦a M+1 π  ◦π +

M  

   ◦a ◦(ai −1) ◦a  , . . . , π ∗  M M , L ◦a Pi , π ∗ 1 1 , . . . , π ∗ i M+1 π  ◦π

(8)

i=1

for any ai , a ≥ 0. We have to evaluate this in two cases, when a = 0 and when a = 1. If a = 0, then the first term in (8) vanishes, because each of the arguments of the Deligne product is a pull-back under π . For the second term, we note that     Pi , π ∗ ξ1 , . . . , π ∗ ξm+1 π  ◦π = ξ1 , . . . , ξm+1 π  for any ξ1 , . . . , ξm+1 ∈ Pic(Mg,M ). (This follows from (DP3), because Pi is a section of π .) This gives Eq. (6) and hence Proposition 4 (string equation). If a = 1, then the second term of (8) vanishes, because      Pi , π ∗ ξ1 , . . . , π ∗ ξm , L M+1 π  ◦π = Pi , L M+1 π , ξ1 , . . . , ξm π  = 0 for any ξ1 , . . . , ξm ∈ Pic(Mg,M ), by Prop. 1 (a) and Prop. 3 (c). The first term in (8) is equal to the right-hand side of (7) by Eq. (5) and Prop. 3 (d). This proves Eq. (7) and hence Proposition 6 (dilaton equation).

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6. Proof of the Main Theorem Since the recursion formula and initial values given in Propositions 4 and 5 determine the elements TN (a1 , . . . , a N +m ) ∈ Pic(Mg,N ) uniquely, we can prove Theorem 1 by showing that the elements TN (a1 , . . . , a N +m ) defined by (2) and (3) satisfy these three equations. A direct proof of this is possible, but rather complicated, involving a series of lemmas about elementary symmetric polynomials and multinomial coefficients. This proof can be simplified considerably by a judicious use of generating functions, but remains quite complicated. A much simpler proof is obtained using the following trick. By the substitution t = x/(1 + x) and Euler’s formula for the beta integral (or simply by integration by parts and induction on α and β) we see that  ∞  1 xα dx α! β! = t α (1 − t)β dt = α+β+2 (x + 1) (α + β + 1)! 0 0 for any integers α, β ≥ 0. Hence, if we define a polynomial F(x) = Fd1 ,...,dm (x) by F(x) = then we have  ∞ (x

0

m 

m   x + dj = σn x m−n ,

j=1

n=0

F(x) d x + 1) N +m+1

=

m 

σn

n=0

σn = σn (d1 , . . . , dm ),

+ n − 1)! − 1)! (m − n)! ( N (N = C (d) + m)! + m)! 1 (N (N

(9a)

and 

∞ 0

x F(x) d x (x

+ 1) N +m+1

=

m 

σn

n=0

+ n − 2)! (N − 2)! (m − n + 1)! ( N N = C (d) + m)! + m)! 2 (N (N (9b)

; d1 , . . . , dm ) as in Eq. (3) (or the first remark after Theorem 1 if N with Cν (d) = Cν ( N is 0 or 1). In particular, we have + m)!  ∞ + m)! dx (N (N = C1 ( N ; 1, . . . , 1) = ,    − 1)! 0 (x + 1) N +1 ! (N N m  ∞ + m)! x dx (N ; 1, . . . , 1) = ( N + m)! = C2 ( N ,    ( N − 2)! 0 (x + 1) N +1 · N ! N N m  ∞ + m)! x (x + 2) d x 2 (N ; 2, 1, . . . , 1) = ( N + m)! = C2 ( N ,    + 1)! ( N − 2)! 0 (N N (x + 1) N +2 m−1

so Eq. (2) in the two cases when all di are strictly positive reduces to TN ,m (1, 0, . . . , 0; 1, . . . , 1)       N −1

m

  ; 1, . . . , 1) 1 −  + C2 ( N ; 1, . . . , 1)  = ( N + m)! 1 = C1 ( N ! N N

Deligne Products of Line Bundles over Moduli Spaces of Curves

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and TN ,m (0, . . . , 0; 2, 1, . . . , 1)       m−1

N

1 ; 2, 1, . . . , 1)  = ( N + m)! , C2 ( N + 1)! 2! (N in accordance with the initial values in Proposition 5. To prove the recursion formula of Proposition 4 (string equation), we will show that it is equivalent to a pair of recurrences for the coefficients Cν (d) (Eq. (11) below) and then prove these recurrences using the integral representation (9). Denote the right-hand side of (2) by t N ,m (a1 , . . . , a N +m ) or t N ,m (a1 , . . . , a N ; d1 , . . . , dm ), with d j = a N + j for  N +m 1 ≤ j ≤ m. Since we want TN ,m (a1 , . . . , a N +m ) = t N ,m (a1 , . . . , a N +m )/ i=1 ai !, we have to prove the recursion =

t N ,m+1 (a1 , . . . , a N +m , 0) =

N +m 

ai t N ,m (a1 , . . . , ai − 1, . . . , a N +m ).

i=1

(The extra factor ai in front of t N ,m (a1 , . . . , ai − 1, . . . , a N +m ) comes from the change  N +m in i=1 ai ! when ai is decreased by 1.) Now again separating the ai (1 ≤ i ≤ N ) and the d j = a N + j (1 ≤ j ≤ m), we can write this out more explicitly as t N ,m+1 (a1 , . . . , a N ; d1 , . . . , dm , 0) =

N 

ai t N ,m (a1 , . . . , ai − 1, . . . , a N ; d1 , . . . , dm )

i=1 m 

+

d j t N ,m (a1 , . . . , a N ; d1 , . . . , d j − 1, . . . , dm ).

(10)

j=1

We can write the definition of t N ,m (a1 , . . . , a N ; d1 , . . . , dm ) in an abbreviated notation as t N ,m (a1 , . . . , a N ; d1 , . . . , dm ) = C1 (d)

N 

ak  k + C2 (d) ,

k=1

; d1 , . . . , dm ) as before ( N does not change when we change m where Cν (d) = Cν ( N of by 1, so can be omitted from the notation) and where  k is the element k − / N Pic(Mg,N ) ⊗ Q. Then the left-hand side of (10) equals C1 (d, 0)

N 

ak  k + C2 (d, 0) ,

i=1

while the right-hand side equals   N N      ak − δki k + C2 (d)  ai C1 (d) i=1

k=1

+

m  j=1

  N   d j C1 (d1 , . . . , d j − 1, . . . , dm ) ak k + C2 (d1 , . . . , d j −1, . . . , dm ) k=1

802

=

L. Weng, D. Zagier



  N  N m  ai − 1 + d j C1 (d1 , . . . , d j − 1, . . . , dm ) ak  k C1 (d) k=1

i=1

j=1

    N m + C2 (d) ai + d j C2 (d1 , . . . , d j − 1, . . . , dm ) . i=1

j=1


N ai = m + 2 − σ1 (d) (because Comparing these two expressions, and recalling that i=1 the sum of all the indices a1 , . . . , a N , d1 , . . . , dm , 0 in Eq. (10) must equal m + 2), we find that the theorem will follow from the two identities: m    d j C1 (d1 , . . . , d j − 1, . . . , dm ), C1 (d, 0) = m + 1 − σ1 C1 (d) +

(11a)

j=1 m    C2 (d, 0) = m + 2 − σ1 C2 (d) + d j C2 (d1 , . . . , d j − 1, . . . , dm ).

(11b)

j=1

To prove the first of these, we use Eq. (9a). Replacing d = (d1 , . . . , dm ) by (d, 0) = (d1 , . . . , dm , 0) increases m by 1 and replaces the polynomial F(x) by x F(x), whereas replacing d by (d1 , . . . , d j − 1, . . . , dm ) leaves m unchanged and replaces F(x) by F(x)(x + d j − 1)/(x + d j ). Therefore substituting (9a) into (11a) and dividing both sides + m)!/( N − 1)! gives by ( N  ∞ x F(x) d x + m + 1) (N (x + 1) N +m+2 0  

  ∞ m  x + dj − 1 F(x) d x m+1+ = d j −1 + x + dj (x + 1) N +m+1 0 j=1 as the identity to be proved. But this is immediate by integration by parts, since the expres
 (x) x sion in square brackets equals 1 + mj=1 x+d = 1 + x FF(x) . The proof of Eq. (11b) is j exactly the same, using (9b) instead of (9a), with F(x) replaced by x F(x). This completes the proof of the theorem. 7. Final Remark A formula very similar to Eq. (2) (in the case when all ai = 0) appears in §4.6 of [8], but in a somewhat different situation: the formula there is for the moduli space of curves of genus 1 with N marked points and deals with the Gromov-Witten invariants, which are integers, whereas our formula is for arbitary genus (although in the final result the ) and gives the Deligne products, genus does not appear except in the shift from N to N which take values in Pic(Mg,N ). Both proofs are based on the string and dilaton equations, which are valid in both contexts. This suggests a possible common generalization. Our situation concerns codimension one cycles, while Gromov-Witten invariants have to do with zero-dimensional cycles. It therefore seems reasonable to ask whether (2) and the equation in [8] are special cases of a more general result valid for intermediate dimensions, for which the string and dilaton equations still hold.

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References 1. Arbarello, E., Cornalba, M.: The Picard groups of the moduli spaces of curves. Topology 26(2), 153–171 (1987) 2. Deligne, P.: Le déterminant de la cohomologie. In: Current trends in arithmetic algebraic geometry, Contemporary Math. Vol. 67, Providence, RI: Amer. Math. Soc., pp. 93–178, 1987 3. Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. IHES Publ. Math. 36, 75–109 (1969) 4. Harer, J.: The second homology group of the mapping class group of an orientable surface. Invent. Math. 72(2), 221–239 (1983) 5. Kaufmann, R., Manin, Yu., Zagier, D.: Higher Weil-Petersson volumes of moduli spaces of stable n-pointed curves. Commun. Math. Phys. 181(3), 763–787 (1996) 6. Knudsen, F.: The projectivity of the moduli space of stable curves, II. Math. Scand. 52, 161–199 (1983) 7. Knudsen, F., Mumford, D.: The projectivity of the moduli space of stable curves. I. Math. Scand. 39, 19–55 (1976) 8. Lando, S.K., Zvonkin, A.K.: Graphs on Surfaces and Their Applications, In: Encyclopedia of Mathematical Sciences, Berlin–Heidelberg: Springer-Verlag, 2004 9. Takhtajan, L., Zograf, P.: The Selberg zeta function and a new Kähler metric on the moduli space of punctured Riemann surfaces. J. Geo. Phys. 5, 551–570 (1988) ¯ 10. Takhtajan, L., Zograf, P.: A local index theorem for families of ∂-operators on punctured Riemann surfaces and a new Kähler metric on their moduli spaces. Commun. Math. Phys. 137, 399–426 (1991) 11. Weng, L.: Hyperbolic Metrics, Selberg Zeta Functions and Arakelov Theory for Punctured Riemann Surfaces. Lecture Note Series in Mathematics 6, Osaka: Osaka University, 1998 12. Weng, L.: -admissible theory, II.. Math. Ann. 320, 239–283 (2001) Communicated by L. Takhtajan

Commun. Math. Phys. 281, 805–826 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0503-8

Communications in

Mathematical Physics

The Paraboson Fock Space and Unitary Irreducible Representations of the Lie Superalgebra osp(1|2n) S. Lievens, N. I. Stoilova
,

, J. Van der Jeugt Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281-S9, B-9000 Gent, Belgium. E-mail: [email protected]; [email protected]; [email protected] Received: 2 July 2007 / Accepted: 30 November 2007 Published online: 15 May 2008 – © Springer-Verlag 2008

Abstract: It is known that the defining relations of the orthosymplectic Lie superalgebra osp(1|2n) are equivalent to the defining (triple) relations of n pairs of paraboson operators bi± . In particular, with the usual star conditions, this implies that the “parabosons of order p” correspond to a unitary irreducible (infinite-dimensional) lowest weight representation V ( p) of osp(1|2n). Apart from the simple cases p = 1 or n = 1, these representations had never been constructed due to computational difficulties, despite their importance. In the present paper we give an explicit and elegant construction of these representations V ( p), and we present explicit actions or matrix elements of the osp(1|2n) generators. The orthogonal basis vectors of V ( p) are written in terms of Gelfand-Zetlin patterns, where the subalgebra u(n) of osp(1|2n) plays a crucial role. Our results also lead to character formulas for these infinite-dimensional osp(1|2n) representations. Furthermore, by considering the branching osp(1|2n) ⊃ sp(2n) ⊃ u(n), we find explicit infinite-dimensional unitary irreducible lowest weight representations of sp(2n) and their characters. 1. Introduction The classical notion of Bose operators or bosons has been generalized a long time ago to parabose operators or parabosons [8]. These parabosons are of interest in many applications, in particular in quantum field theory [6,24,25], generalizations of quantum statistics (para-statistics) [8,10–12], and in Wigner quantum systems [17,19,21,31]. The generalization of the usual boson Fock space is characterized by a parameter p, referred to as the order. For a single paraboson, n = 1, the structure of the paraboson Fock space is well known [26]. Surprisingly, for a system of n parabosons with n > 1, the structure
Permanent address: Institute for Nuclear Research and Nuclear Energy, Boul. Tsarigradsko Chaussee 72, 1784 Sofia, Bulgaria.

NIS was supported by a project from the Fund for Scientific Research – Flanders (Belgium) and by project P6/02 of the Interuniversity Attraction Poles Programme (Belgian State – Belgian Science Policy).

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S. Lievens, N. I. Stoilova, J. Van der Jeugt

of the paraboson Fock space is not known, even though it can in principle be constructed by means of the so-called Green ansatz [8]. The computational difficulties of the Green ansatz are related to finding a proper basis of an irreducible constituent of a p-fold tensor product [10]. An important result was given by Ganchev and Palev [7], who observed that the triple relations for n pairs of parabosons are in fact defining relations for the orthosymplectic Lie superalgebra osp(1|2n) [15]. This implies also that the paraboson Fock space of order p is a certain infinite-dimensional unitary irreducible representation (unirrep) of osp(1|2n). However, the construction of this representation, for n > 1 and arbitrary p, also turned out to be difficult. In the present paper we present a solution to this problem. Our solution is based upon group theoretical techniques, in particular related to the branching osp(1|2n) ⊃ sp(2n) ⊃ u(n). This allows us to construct a proper GelfandZetlin (GZ) basis for some induced representation [16], from which the basis for the irreducible representation follows. Our result is thus a complete solution to the problem: for the paraboson Fock space we give not only an orthogonal GZ-basis but also the action (matrix elements) of the paraboson operators in this basis. The structure of the paper is as follows. In Sect. 2, we recall the definition of parabosons and of the paraboson Fock space V ( p). In Sect. 3, we discuss the important relation between parabosons and the Lie superalgebra osp(1|2n), and give a description of V ( p) in terms of representations of osp(1|2n). The following section is devoted to the first non-trivial example of osp(1|4). The analysis of the representations V ( p) for osp(1|4) is performed in detail, and the techniques used here can be lifted to the general case osp(1|2n). This general case is investigated in Sect. 5, where the main computational result (matrix elements) is given in Proposition 6 and the main structural result (characters) in Theorem 7. Section 6 is devoted to the branching to the even subalgebra sp(2n) = sp(2n, R), and gives some sp(2n) characters. We end the paper with some final remarks. 2. The Paraboson Fock Space V ( p) Before introducing the paraboson Fock space, let us recall some aspects of the usual boson Fock space. For a single pair (n = 1) of boson operators B + , B − , the defining relation is given by [B − , B + ] = 1. (2.1) The boson Fock space is defined as a Hilbert space with vacuum vector |0, in which the action of the operators B + , B − is defined and satisfies 0|0 = 1,

B − |0 = 0,

(B ± )† = B ∓ .

Moreover, under the action of the algebra spanned by {B + ,

(2.2)

B − , 1} (subject to (2.1)), the

Hilbert space is irreducible. A set of basis vectors of this space, denoted by V (1), is given by (B + )k k ∈ Z+ = {0, 1, 2 . . .}. (2.3) |k = √ |0, k! These vectors are orthogonal and normalized. The space V (1) is a unitary irreducible representation (unirrep) of the Lie superalgebra osp(1|2) [26] (see the next paragraph). It is also the direct sum of two unirreps of the Lie algebra sp(2) = sp(2, R) (one with even k’s and one with odd k’s), known as the metaplectic representations (certain positive discrete series representations of sp(2)) [3,14,29].

Paraboson Fock Space and Unitary Irreducible Representations

807

For a single pair (n = 1) of paraboson operators b+ , b− [8], the defining relation is a triple relation (with anticommutator {., .} and commutator [., .]) given by [{b− , b+ }, b± ] = ±2b± .

(2.4)

The paraboson Fock space [26] is again a Hilbert space with vacuum vector |0, defined by means of 0|0 = 1, b− |0 = 0, {b− , b+ }|0 = p |0,

(b± )† = b∓ , (2.5)

and by irreducibility under the action of b+ , b− . Herein, p is a parameter, known as the order of the paraboson. In order to have a genuine inner product, p should be positive and real: p > 0. A set of basis vectors for this space, denoted by V ( p), is given by |2k =

(b+ )2k |0, √ 2k k!( p/2)k

|2k + 1 =

(b+ )2k+1 |0. √ 2k k!2( p/2)k+1

(2.6)

This basis is orthogonal and normalized; the symbol (a)k = a(a + 1) · · · (a + k − 1) is the common Pochhammer symbol. If one considers b+ and b− as odd generators of a Lie superalgebra, then the elements + {b , b+ }, {b+ , b− } and {b− , b− } form a basis for the even part of this superalgebra. Using the relations (2.4) it is easy to see that this superalgebra is the orthosymplectic Lie superalgebra osp(1|2), with even part sp(2). The paraboson Fock space V ( p) is then a unirrep of osp(1|2). It splits as the direct sum of two positive discrete series representations of sp(2): one with lowest weight vector |0 (lowest weight p/2) and basis vectors |2k, and one with lowest weight vector |1 (lowest weight 1 + p/2) and basis vectors |2k + 1. For p = 1 the paraboson Fock space coincides with the ordinary boson Fock space. This also follows from the general action (b− b+ − b+ b− )|2k = p |2k,

(b− b+ − b+ b− )|2k + 1 = (2 − p) |2k + 1.

(2.7)

Let us now consider the case of n pairs of boson operators Bi± (i = 1, 2, . . . , n), satisfying the standard commutation relations [Bi− , B +j ] = δi j .

(2.8)

The n-boson Fock space is again defined as a Hilbert space with vacuum vector |0, with 0|0 = 1,

Bi− |0 = 0,

(Bi± )† = Bi∓

(i = 1, . . . , n).

(2.9)

The Hilbert space is irreducible under the action of the algebra spanned by the elements 1, Bi+ , Bi− (i = 1, . . . , n), subject to (2.8). A set of (orthogonal and normalized) basis vectors of this space is given by |k1 , . . . , kn  =

(B1+ )k1 · · · (Bn+ )kn |0, √ k1 ! · · · kn !

k1 , . . . , kn ∈ Z+ .

(2.10)

We shall see that this Fock space is a certain unirrep of the Lie superalgebra osp(1|2n), with lowest weight ( 21 , . . . , 21 ).

808

S. Lievens, N. I. Stoilova, J. Van der Jeugt

We are primarily interested in a system of n pairs of paraboson operators b±j ( j = 1, . . . , n). The defining triple relations for such a system are given by [8] ξ

η

η

ξ

[{b j , bk }, bl ] = ( − ξ )δ jl bk + ( − η)δkl b j ,

(2.11)

where j, k, l ∈ {1, 2, . . . , n} and η, , ξ ∈ {+, −} (to be interpreted as +1 and −1 in the algebraic expressions  − ξ and  − η). The paraboson Fock space V ( p) is the Hilbert space with vacuum vector |0, defined by means of ( j, k = 1, 2, . . . , n) 0|0 = 1,

b−j |0 = 0,

(b±j )† = b∓j ,

{b−j , bk+ }|0 = pδ jk |0,

(2.12)

and by irreducibility under the action of the algebra spanned by the elements b+j , b−j ( j = 1, . . . , n), subject to (2.11). The parameter p is referred to as the order of the paraboson system. In general p is thought of as a positive integer, and for p = 1 the paraboson Fock space V ( p) coincides with the boson Fock space V (1). We shall see that also certain non-integer p-values are allowed. Constructing a basis for the Fock space V ( p) turns out to be a difficult problem, unsolved so far. Even the simpler question of finding the structure of V ( p) (weight structure) is not solved. In the present paper we shall unravel the structure of V ( p), determine for which p-values V ( p) is actually a Hilbert space, construct an orthogonal (normalized) basis for V ( p), and give the actions of the generators b±j on the basis vectors. 3. The Lie Superalgebra osp(1|2n) The Lie superalgebra osp(1|2n) [15] consists of matrices of the form ⎛ ⎞ 0 a a1 ⎝ at b c ⎠ , 1 −a t d −bt

(3.1)

where a and a1 are (1 × n)-matrices, b is any (n × n)-matrix, and c and d are symmetric (n × n)-matrices. The even elements have a = a1 = 0 and the odd elements are those with b = c = d = 0. It will be convenient to have the row and column indices running from 0 to 2n (instead of 1 to 2n+1), and to denote by ei j the matrix with zeros everywhere except a 1 on position (i, j). Then the Cartan subalgebra h of osp(1|2n) is spanned by the diagonal elements h j = e j j − en+ j,n+ j

( j = 1, . . . , n).

(3.2)

In terms of the dual basis δ j of h∗ , the odd root vectors and corresponding roots of osp(1|2n) are given by: e0,k − en+k,0 ↔ −δk , k = 1, . . . , n, e0,n+k + ek,0 ↔ δk , k = 1, . . . , n. The even roots and root vectors are e j,k − en+k,n+ j ↔ δ j − δk , j = k = 1, . . . , n, e j,n+k + ek,n+ j ↔ δ j + δk , j ≤ k = 1, . . . , n, en+ j,k + en+k, j ↔ −δ j − δk , j ≤ k = 1, . . . , n.

Paraboson Fock Space and Unitary Irreducible Representations

809

If we introduce the following multiples of the odd root vectors bk+ =

√ 2(e0,n+k + ek,0 ),

bk− =

√

2(e0,k − en+k,0 )

(k = 1, . . . , n)

(3.3)

then it is easy to verify that these operators satisfy the triple relations (2.11). Since all η ξ even root vectors can be obtained by anticommutators {b j , bk }, the following holds [7]: Theorem 1 (Ganchev and Palev). As a Lie superalgebra defined by generators and relations, osp(1|2n) is generated by 2n odd elements bk± subject to the following (paraboson) relations: ξ

η

η

ξ

[{b j , bk }, bl ] = ( − ξ )δ jl bk + ( − η)δkl b j .

(3.4)

The paraboson operators b+j are the positive odd root vectors, and the b−j are the negative odd root vectors. Recall that the paraboson Fock space V ( p) is characterized by ( j, k = 1, . . . , n) (b±j )† = b∓j ,

b−j |0 = 0,

{b−j , bk+ }|0 = p δ jk |0.

(3.5)

Furthermore, it is easy to verify that {b−j , b+j } = 2h j

( j = 1, . . . , n).

(3.6)

Hence we have the following: Corollary 2. The paraboson Fock space V ( p) is the unitary irreducible representation of osp(1|2n) with lowest weight ( 2p , 2p , . . . , 2p ). In order to construct the representation V ( p) [27] one can use an induced module construction. The relevant subalgebras of osp(1|2n) are easy to describe by means of the odd generators b±j . Proposition 3. A basis for the even subalgebra sp(2n) of osp(1|2n) is given by the elements {b±j , bk± } (1 ≤ j ≤ k ≤ n), {b+j , bk− } (1 ≤ j, k ≤ n).

(3.7)

The n 2 elements {b+j , bk− }

( j, k = 1, . . . , n)

(3.8)

are a basis for the sp(2n) subalgebra u(n). Note that with {b+j , bk− } = 2E jk , the triple relations (3.4) imply the relations [E i j , E kl ] = δ jk E il − δli E k j . In other words, the elements {b+j , bk− } form, up to a factor 2, the standard u(n) or gl(n) basis elements. So the odd generators b±j clearly reveal the subalgebra chain osp(1|2n) ⊃ sp(2n) ⊃ u(n). Note that u(n) is, algebraically, the same as the general linear Lie algebra gl(n). But the condition (b±j )† = b∓j implies that we are dealing here with the “compact form” u(n).

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The subalgebra u(n) can be extended to a parabolic subalgebra P of osp(1|2n) [27]: P = span{{b+j , bk− }, b−j , {b−j , bk− } | j, k = 1, . . . , n}.

(3.9)

Recall that {b−j , bk+ }|0 = p δ jk |0, with {b−j , b+j } = 2h j . This means that the space spanned by |0 is a trivial one-dimensional u(n) module C|0 of weight ( 2p , . . . , 2p ). Since b−j |0 = 0, the module C|0 can be extended to a one-dimensional P module. Now we are in a position to define the induced osp(1|2n) module V ( p): osp(1|2n)

V ( p) = IndP

C|0.

(3.10)

This is an osp(1|2n) representation with lowest weight ( 2p , . . . , 2p ). By the PoincaréBirkhoff-Witt theorem [16,27], it is easy to give a basis for V ( p): + , b+ })kn−1,n |0, (b1+ )k1 · · · (bn+ )kn ({b1+ , b2+ })k12 ({b1+ , b3+ })k13 · · · ({bn−1 n k1 , . . . , kn , k12 , k13 . . . , kn−1,n ∈ Z+ .

(3.11)

This is all rather formal, and it sounds easy. The difficulty however comes from the fact that in general V ( p) is not a simple module (i.e. not an irreducible representation) of osp(1|2n). Let M( p) be the maximal nontrivial submodule of V ( p). Then the simple module (irreducible module), corresponding to the paraboson Fock space, is V ( p) = V ( p)/M( p).

(3.12)

The purpose is now to determine the vectors belonging to M( p), and hence to find the structure of V ( p). Furthermore, we want to find explicit matrix elements of the osp(1|2n) generators in an appropriate basis of V ( p). As an illustrative example, we shall first treat the case of osp(1|4). 4. Paraboson Fock Representations of osp(1|4) We shall examine the induced module V ( p) in the case n = 2, with basis vectors |k, l, m ≡ (b1+ )k (b2+ )l ({b1+ , b2+ })m |0

(k, l, m ∈ Z+ ).

Note that the weight of this vector is  p p , + kδ1 + lδ2 + m(δ1 + δ2 ). 2 2

(4.1)

(4.2)

The level of this vector is defined as k + l + 2m. The basis vectors |k, l, m are easy to define, and the actions of the generators b1± , b2± on these vectors can be computed, using the triple relations. For the positive root vectors this is easy: b1+ |k, l, m = |k + 1, l, m, b2+ |2k, l, m = |2k, l + 1, m, b2+ |2k

+ 1, l, m = |2k, l, m + 1 − |2k + 1, l + 1, m.

(4.3)

Paraboson Fock Space and Unitary Irreducible Representations

811

For the negative root vectors this requires some tough computations, yielding: b1− |2k, l, m = 2k|2k − 1, l, m + 2m|2k, l + 1, m − 1,

b1− |2k + 1, l, m = ( p + 2m + 2k)|2k, l, m − 2m|2k + 1, l + 1, m − 1,

b2− |2k, 2l, m = 2l|2k, 2l − 1, m + 2m|2k + 1, 2l, m − 1,

b2− |2k, 2l + 1, m = ( p + 2l)|2k, 2l, m + 2m|2k + 1, 2l + 1, m − 1,

(4.4)

b2− |2k + 1, 2l, m = 2l|2k, 2l − 2, m + 1 − 2l|2k + 1, 2l − 1, m + 2m|2k + 2, 2l, m − 1,

b2− |2k + 1, 2l + 1, m = 2l|2k, 2l − 1, m + 1 − ( p + 2l − 2)|2k + 1, 2l, m + 2m|2k + 2, 2l + 1, m − 1. It is now possible to compute “inner products” of vectors |k, l, m, using 0|0 = 1 and (bi± )† = bi∓ . Clearly, vectors of different weight have inner product zero. Let us compute a number of the nonzero inner products. At weight ( 2p , 2p ) there is one vector only, |0, 0, 0 = |0, with 0, 0, 0|0, 0, 0 = 1. (4.5) At level 1 there is one vector of weight ( 2p + 1, 2p ) and one of weight ( 2p , inner products respectively: 1, 0, 0|1, 0, 0 = p,

p 2

+ 1), with

0, 1, 0|0, 1, 0 = p.

At level 2 there is one vector of weight ( 2p + 2, 2p ), one of weight ( 2p , vectors of weight ( 2p + 1, 2p + 1). The inner products are given by:

(4.6) p 2

+ 2), and two

2, 0, 0|2, 0, 0 = 0, 2, 0|0, 2, 0 = 2 p, 1, 1, 0|1, 1, 0 = p 2 , 1, 1, 0|0, 0, 1 = 2 p, 0, 0, 1|0, 0, 1 = 4 p.

(4.7)

From (4.6) it follows already that p should be a positive number, otherwise the inner product (bilinear form) is not positive definite. The matrix of inner products of the vectors of weight ( 2p + 1, 2p + 1) has determinant  det

p2 2 p 2p 4p

 = 4 p 2 ( p − 1).

(4.8)

So this matrix is positive definite only if p > 1. Thus, for p > 1 both vectors of weight ( 2p + 1, 2p + 1) belong to V ( p); but for p = 1 one vector (2|1, 1, 0 − |0, 0, 1) belongs to M( p) and the subspace of V ( p) of weight ( 2p + 1, 2p + 1) is one-dimensional. One could continue this analysis level by level, but the computations become rather complicated and in order to find a technique that works for arbitrary n one should find a better way of analysing V ( p). For this purpose, we shall construct a different basis for V ( p). This new basis is indicated by the character of V ( p): this is a formal infij j nite series of terms µx11 x22 , with ( j1 , j2 ) a weight of V ( p) and µ the dimension of this weight space. So the vacuum vector |0 of V ( p), of weight ( 2p , 2p ), yields a term p

p

x12 x22 = (x1 x2 ) p/2 in the character char V ( p). Since the basis vectors are given by (b1+ )k (b2+ )l ({b1+ , b2+ })m |0, where k, l, m ∈ Z+ , it follows that

812

S. Lievens, N. I. Stoilova, J. Van der Jeugt

char V ( p) =

(x1 x2 ) p/2 . (1 − x1 )(1 − x2 )(1 − x1 x2 )

(4.9)

Such expressions have an interesting expansion in terms of Schur functions, valid for general n. Proposition 4 (Cauchy, Littlewood). Let x1 , . . . , xn be a set of n variables. Then [22] n

i=1 (1 − x i )



1 1≤ j
x j xk )

=



sλ (x1 , . . . , xn ) =

λ



sλ (x).

(4.10)

λ

In the right-hand side, the sum is over all partitions λ and sλ (x) is the Schur symmetric function [23]. For n variables, sλ (x) = 0 if the length (λ) is greater than n, so in practice the sum is over all partitions of length less than or equal to n. For example, for n = 2 one has char V ( p) = (x1 x2 ) p/2 (1 + s1 (x) + s2 (x) + s1,1 (x) + s3 (x) + s2,1 (x) + · · · ). The characters of finite dimensional u(n) representations (here u(2)) are given by such Schur functions sλ (x). Hence such expansions are useful since they yield the branching to u(n) of the osp(1|2n) representation V ( p). But for such finite dimensional u(n) representations labelled by a partition λ, there is a known basis: the Gelfand-Zetlin basis (GZ) [9,1]. We shall use the u(n) GZ basis vectors as our new basis for V ( p). For our current example, n = 2, the new basis vectors are thus given by:     m ,m  m ,m |m) =  p; 12 22 ≡  12 22 , m 11 m 11

(4.11)

where (m 12 , m 22 ) is the partition λ of length 2 (that is m 12 and m 22 are integers with m 12 ≥ m 22 ≥ 0), and the GZ pattern m satisfies the betweenness condition m 12 ≥ m 11 ≥ m 22 . As p is supposed to be fixed, it will usually be dropped from the notation of the vectors |m), as in (4.11). For such vectors, (m 12 , m 22 ) is the u(2) representation label, and m 11 is the u(2) internal label of the vector |m). We assume that the action of the u(2) generators is as usual, thus the weight of the vector |m) is given by p p ( , ) + (m 11 , m 12 + m 22 − m 11 ). 2 2 In this new basis, we need to compute the action of b1± and b2± on the basis vectors |m). In fact, it will be sufficient to compute only those of b1+ and b2+ (as b−j = (b+j )† ).    0, 0  . Then, the relation with the old basis |k, l, m follows implicitly from |0, 0, 0 =  0 From the triple relations (3.4), it follows that under the u(2) basis (3.8), the set (b1+ , b2+ ) forms a tensor of rank (1,0), with weights δ1 and δ2 , and GZ patterns

Paraboson Fock Space and Unitary Irreducible Representations

 b1+ ∼

 10 , 1

813

 b2+ ∼

 10 . 0

(4.12)

Therefore, (m



 |b1+ |m)

=

  m 12 m 22 1 0  m 12 m 22 × (m 12 , m 22 ||b+ ||m 12 , m 22 ). ; 1  m 11 m 11

(4.13)

The first factor in the right-hand side is a classical u(2) Clebsch-Gordan coefficient (CGC) [30,2], and the second factor is a reduced matrix element [30]. The possible values of the patterns m  are determined by the u(2) tensor product (1, 0) ⊗ (m 12 , m 22 ) = (m 12 + 1, m 22 ) ⊕ (m 12 , m 22 + 1), and by the additivity property of the internal labels (m 11 = m 11 + 1 in the above expression). So the only u(2) CGCs of relevance are given below, their values taken from [30, p. 385]:   m 12 , m 22 1, 0  m 12 + 1, m 22 m 11 −m 22 +1 = m ; , 12 −m 22 +1 1  m 11 + 1 m 11    m 12 , m 22 1, 0  m 12 , m 22 + 1 −m 11 = − mm1212−m ; ,  22 +1 1 m 11 + 1 m 11    m 12 , m 22 1, 0  m 12 + 1, m 22 m 12 −m 11 +1 = m ; , 12 −m 22 +1 0  m 11 m 11    m 12 , m 22 1, 0  m 12 , m 22 + 1 −m 22 = mm1211−m ; . 22 +1 0  m 11 m 11 

(4.14) (4.15) (4.16) (4.17)

The problem is thus reduced to finding explicit expressions for two functions F1 and F2 , where F1 (m) = (m 12 + 1, m 22 ||b+ ||m 12 , m 22 ),

F2 (m) = (m 12 , m 22 + 1||b+ ||m 12 , m 22 ). (4.18)

We can write: b1+ |m) =

b2+ |m) =





 

m 11 −m 22 +1  m 12 m 12 −m 22 +1 F1 (m)  m 11

+ 1, m 22 +1

 −



 

m 12 −m 11  m 12 , m 22 m 12 −m 22 +1 F2 (m)  m 11 + 1

+1

 ,

(4.19)       m 12 + 1, m 22 m 12 , m 22 + 1 m 12 −m 11 +1 m 11 −m 22   + , F (m) F (m)  m 11  m 11 m 12 −m 22 +1 1 m 12 −m 22 +1 2 (4.20)

and the action of b−j follows from (m  |b−j |m) = (m|b+j |m  ). Now it remains to determine the functions F1 and F2 . From the action {b2− , b2+ }|m) = 2h 2 |m) = ( p + 2(m 12 + m 22 − m 11 ))|m),

(4.21)

814

S. Lievens, N. I. Stoilova, J. Van der Jeugt

one deduces the following recurrence relations for F1 and F2 : F1 (m 12 , m 22 )F2 (m 12 + 1, m 22 − 1) √ √ m 12 − m 22 + 1 m 12 − m 22 + 3 F1 (m 12 , m 22 − 1)F2 (m 12 , m 22 − 1) = 0, (4.22) + m 12 − m 22 + 2 m 12 − m 11 + 1 m 22 − m 11 F1 (m 12 , m 22 )2 − F2 (m 12 , m 22 )2 m 12 − m 22 + 1 m 12 − m 22 + 1 m 12 − m 11 + F1 (m 12 − 1, m 22 )2 m 12 − m 22 m 22 − m 11 − 1 F2 (m 12 , m 22 − 1)2 = p + 2m 12 + 2m 22 − 2m 11 . (4.23) − m 12 − m 22 + 2 The action b2− |m) leads to the boundary condition: F2 (m 12 , m 22 − 1) = 0 if m 22 = 0. The boundary condition together with the recurrence relations (4.22)–(4.23) lead to the following solution for the unknown functions F1 and F2 : (m 12 − m 22 + 1)1/2 , (m 12 − m 22 + 1 + Om 12 −m 22 )1/2 (4.24) 1/2 (m − m + 1) 12 22 F2 (m 12 , m 22 ) = (m 22 + 1 + Em 22 ( p − 2))1/2 , (4.25) (m 12 − m 22 + 1 − Om 12 −m 22 )1/2 F1 (m 12 , m 22 ) = (−1)m 22 (m 12 + 2 + Em 12 ( p − 2))1/2

where the even and odd functions E j and O j are defined by E j = 1 if j is even and 0 otherwise, O j = 1 if j is odd and 0 otherwise.

(4.26)

Note that O j = 1 − E j , but it is convenient to use both notations. The solution for F1 and F2 is unique up to a choice of the sign factor. At this moment, only the action of {b2− , b2+ } has been used in the process. Now it remains to verify whether the actions of b1± and b2± thus determined do indeed yield a solution, i.e. one should verify that all triple relations (3.4) are satisfied. This is a straightforward but tedious computation; the only result provided by this calculation is that the sign factors are restricted. The choice of sign factors presented in (4.24)–(4.25) is the simplest solution. So we finally present the complete solution in the case of osp(1|4):  

 m + 1, m 22 b1+ |m) = m 11 − m 22 + 1 f 1 (m 12 , m 22 )  12 m 11 + 1    m 12 , m 22 + 1 √ , (4.27) − m 12 − m 11 f 2 (m 12 , m 22 )  m 11 + 1  

 m + 1, m 22 b2+ |m) = m 12 − m 11 + 1 f 1 (m 12 , m 22 )  12 m 11   m ,m + 1 √ , (4.28) + m 11 − m 22 f 2 (m 12 , m 22 )  12 22 m 11

Paraboson Fock Space and Unitary Irreducible Representations

815

Fig. 1. Relevant factors of the reduced matrix elements between two partitions (case of osp(1|4))

   m − 1, m 22 √ = m 11 − m 22 f 1 (m 12 − 1, m 22 )  12 m 11 − 1  

m ,m − 1 − m 12 − m 11 + 1 f 2 (m 12 , m 22 − 1)  12 22 , m 11 − 1    m − 1, m 22 √ b2− |m) = m 12 − m 11 f 1 (m 12 − 1, m 22 )  12 m 11  

m ,m − 1 , + m 11 − m 22 + 1 f 2 (m 12 , m 22 − 1)  12 22 m 11

b1− |m)

(4.29)

(4.30)

where f 1 (m 12 , m 22 ) = (−1)m 22 f 2 (m 12 , m 22 ) =

(m 12 + 2 + Em 12 ( p − 2))1/2 , (m 12 − m 22 + 1 + Om 12 −m 22 )1/2

(m 22 + 1 + Em 22 ( p − 2))1/2 . (m 12 − m 22 + 1 − Om 12 −m 22 )1/2

(4.31) (4.32)

In this whole analysis we have assumed that the parameter p is sufficiently large such that all factors appearing under square root symbols are positive, in other words such that V ( p) itself is irreducible. The general expressions thus obtained now allow us to examine the structure in more detail. In the expressions (4.24)–(4.25), the first factor in the right-hand side determines the essential cases when the reduced matrix element is zero or not. Let us depict these factors (or rather their squares) in a scheme as given in Fig. 1. In this diagram, there is an edge between two partitions (m) and (m  ) if the reduced matrix element (m  ||b+ ||m) is nonzero, and the number on top of the edge is the crucial factor of (m  ||b+ ||m)2 . Clearly, these factors tell us when these partitions “are part of” the irreducible representation V ( p) or not (to be more precise, whether the vectors of the u(2) module labelled by (m) belong to M( p) or not). Obviously, for p = 1 only the top line of the scheme survives, and forms the u(2) content of V (1). For p > 1, all u(2) representations (m) survive, and V ( p) = V ( p). Theorem 5. The osp(1|4) representation V ( p) with lowest weight ( 2p , 2p ) is a unirrep if and only if p ≥ 1. For p > 1, V ( p) = V ( p) and char V ( p) = (x1 x2 ) p/2 /((1 − x1 )(1 − x2 )(1 − x1 x2 )). For p = 1, V ( p) = V ( p)/M( p) with M( p) = 0. V (1) is the boson Fock space, and char V (1) = (x1 x2 )1/2 /((1 − x1 )(1 − x2 )).

816

S. Lievens, N. I. Stoilova, J. Van der Jeugt

The explicit action of the osp(1|4) generators in V ( p) is given by (4.27)-(4.30). The basis is orthogonal and normalized. Note that also for p = 1 this action remains valid, provided one keeps in mind that all vectors with m 22 = 0 must vanish. We end this section mentioning that more general osp(1|4) irreducible representations with no unique “vacuum” were investigated in [4,13]. However their matrix elements were not determined. 5. Paraboson Fock Representations of osp(1|2n) Similarly as in the previous section, we start our analysis by considering the induced module V ( p). First, a new basis for V ( p) will be introduced. In this basis, matrix elements are computed, and from these expressions it will be clear which vectors belong to M( p). A basis for V ( p) was already given in (3.11). From this expression, one finds (x1 · · · xn ) p/2 . i=1 (1 − x i ) 1≤ j
char V ( p) = n

(5.1)

The denominator can be expanded as in (4.10). The relation between Schur functions and u(n) characters [32] makes it again natural to consider a basis consisting of all GZpatterns [1,9] for all possible partitions of length at most n. Thus the new basis of V ( p) consists of vectors of the form  ⎞ · · · · · · m n−1,n m nn  m 1n     m 1,n−1 · · · · · · m n−1,n−1 ⎟  [m]n n ⎟ = |m) ≡ |m) ≡  .. (5.2) . ⎠  |m)n−1 . .. . m 11

Just as for osp(1|4), the label p is dropped in the notation of the vectors |m). Herein, the top line of the pattern, also denoted by the n-tuple [m]n , is any partition (consisting of non-increasing nonnegative numbers). The remaining n − 1 lines of the pattern will sometimes be denoted by |m)n−1 . So all m i j in the above GZ-pattern are nonnegative integers, satisfying the betweenness conditions m i, j+1 ≥ m i j ≥ m i+1, j+1

(1 ≤ i ≤ j ≤ n − 1).

(5.3)

Note that, since the weight of |0 is ( 2p , . . . , 2p ), the weight of the above vector is determined by ⎛ ⎞ h k |m) = ⎝

p

+ m jk − m j,k−1 ⎠ |m). 2 k

k−1

j=1

j=1

(5.4)

Now we use a technique similar as in the previous section. By the triple relations, one obtains [{bi+ , b−j }, bk+ ] = 2δ jk bi+ . With the identification {bi+ , b−j } = 2E i j in the standard u(n) basis, this is equivalent to the action E i j ·ek = δ jk ei . In other words, the triple relations imply that (b1+ , b2+ , . . . , bn+ )

Paraboson Fock Space and Unitary Irreducible Representations

817

is a standard u(n) tensor of rank (1, 0, . . . , 0). This means that one can attach a unique GZ-pattern with top line 10 · · · 0 to every b+j , corresponding to the weight +δ j . Explicitly: 10 · · · 000 10 · · · 00 , b+j ∼ ·0···· · 0 ··· 0

(5.5)

where the pattern consists of j − 1 zero rows at the bottom, and the first n − j + 1 rows are of the form 10 · · · 0. The tensor product rule in u(n) reads ([m]n ) ⊗ (10 · · · 0) = ([m]n+1 ) ⊕ ([m]n+2 ) ⊕ · · · ⊕ ([m]n+n ),

(5.6)

where ([m]n ) = (m 1n , m 2n , . . . , m nn ) and a subscript ±k indicates an increment of the k th label by ±1: ([m]n±k ) = (m 1n , . . . , m kn ± 1, . . . , m nn ). (5.7) In the right-hand side of (5.6), only those components which are still partitions (i.e. consisting of nondecreasing integers) survive. A general matrix element of b+j can now be written as follows: (m



   [m]n+k  +  [m]n   b = |m  )n−1  j  |m)n−1  10 · · · 00  [m]n  [m]n +k · · · 0  = ; 10 × ([m]n+k ||b+ ||[m]n ). · · · n−1  n−1  |m) |m ) 0 

|b+j |m)

(5.8)

The first factor in the right hand side is a u(n) Clebsch-Gordan coefficient [30], the second factor is a reduced matrix element. By the tensor product rule, the first line of |m  ) has to be of the form (5.7), i.e. [m  ]n = [m]n+k for some k-value. The special u(n) CGCs appearing here are well known, and have fairly simple expressions. They can be found, e.g. in [30]. They can be expressed by means of u(n)-u(n − 1) isoscalar factors and u(n−1) CGC’s, which on their turn are written by means of u(n−1)u(n − 2) isoscalar factors and u(n − 2) CGC’s, etc. The explicit form of the special u(n) CGCs appearing here is given in Appendix A. The actual problem is now converted into finding expressions for the reduced matrix elements, i.e. for the functions Fk ([m]n ), for arbitrary n-tuples of non-increasing nonnegative integers [m]n = (m 1n , m 2n , . . . , m nn ): Fk ([m]n ) = Fk (m 1n , m 2n , . . . , m nn ) = ([m]n+k ||b+ ||[m]n ).

(5.9)

So one can write: b+j |m) b−j |m)

=

=





 k,m 

|m)n−1 

k,m 

[m]n

[m]n−k |m  )n−1

     [m]n  [m]n+k +k  n)  ([m] F   n−1 k  |m  )n−1 , (5.10)  |m )   10 · · · 00  [m]n   [m]n−k  10 · · · 0 n  ; ··· Fk ([m]−k )   n−1 . (5.11)   |m)n−1 |m ) 0

10 · · · 00 ···0 ; 10 ··· 0

818

S. Lievens, N. I. Stoilova, J. Van der Jeugt

In order to determine the unknown functions Fk , one can again start from the following action: n n−1



{bn− , bn+ }|m) = 2h n |m) = ( p + 2( m jn − m j,n−1 ))|m). (5.12) j=1

j=1

Expressing the left hand side by means of (5.10)-(5.11), using the explicit form of the CGCs and isoscalar factors given in Appendix A (which are simple here since j = n), one finds a system of coupled recurrence relations for the functions Fk . Together with the appropriate boundary conditions, we have been able to solve this, in particular using Maple. So our main computational result is: Proposition 6. The reduced matrix elements Fk appearing in the actions of b±j on vectors |m) of V ( p) are given by: Fk (m 1n , m 2n , . . . , m nn ) = (−1)m k+1,n +···+m nn (m kn + n + 1 − k + Em kn ( p − n))1/2  1/2 n  m jn − m kn − j + k × , (5.13) m jn − m kn − j + k − Om jn −m kn j =k=1

where E and O are the even and odd functions defined in (4.26). It would be unfeasible to present all the details of this computational result. Essentially, the proof consists of verifying that all triple relations (3.4) hold when acting on any vector |m). Each such verification leads to an algebraic identity in n variables m 1n , . . . , m nn . In these computations, there are some intermediate verifications: e.g. the action {b+j , bk− }|m) should leave the top row of the GZ-patter |m) invariant (since {b+j , bk− } belongs to u(n)). Furthermore, it must yield (up to a factor 2) the known action of the standard u(n) matrix elements E jk in the classical GZ-basis. The purpose is now to deduce the structure of V ( p) from the general expression of the matrix elements in V ( p). Just as for osp(1|4), this will be governed by the factor (m kn + n + 1 − k + Em kn ( p − n)) in the expression of Fk ([m]n ), since this is the only factor in the right hand side of (5.13) that may become zero. If this factor is zero or negative, the assigned vector |m) belongs to M( p). The integers m jn satisfy m 1n ≥ m 2n ≥ · · · ≥ m nn ≥ 0. If m kn = 0 (its smallest possible value), then this factor in Fk takes the value ( p − k + 1). So the p-values 1, 2, . . . , n − 1 will play a special role. Let us again depict these factors in a scheme, shown here in Fig. 2 for n = 3. In this diagram, there is an edge between two partitions ([m]n ) and ([m  ]n ) = ([m]n+k ) if the reduced matrix element ([m]n+k ||b+ ||[m]n ) = Fk ([m]n ) is in general nonzero. For the boldface lines, the relevant factor (m kn + n + 1 − k + Em kn ( p − n)) is equal to p − 1; for the dotted lines, this factor is equal to p − 2. As a consequence, for p = 1 the irreducible module V ( p) corresponds to the first line of the scheme only, i.e. only partitions ([m]n ) of length 1 appear (the length of a partition is the number of nonzero parts). For p = 2, the irreducible module V ( p) is composed of partitions ([m]n ) of length 1 or 2 only. This observation holds in general, due to the factor ( p − k + 1) for Fk . This finally leads to the following result: Theorem 7. The osp(1|2n) representation V ( p) with lowest weight ( 2p , . . . , 2p ) is a unirrep if and only if p ∈ {1, 2, . . . , n − 1} or p > n − 1.

Paraboson Fock Space and Unitary Irreducible Representations

819

Fig. 2. Structure of the osp(1|2n) representations V ( p) and V ( p), here illustrated for n = 3 (so only partitions of length at most 3 appear). The boldface line indicates that the corresponding reduced matrix element has a factor ( p − 1); the dotted line indicates that it has a factor ( p − 2)

For p > n − 1, V ( p) = V ( p) and (x1 · · · xn ) p/2 i (1 − x i ) j
= (x1 · · · xn ) p/2 sλ (x).

char V ( p) =

(5.14) (5.15)

λ

For p ∈ {1, 2, . . . , n − 1}, V ( p) = V ( p)/M( p) with M( p) = 0. The structure of V ( p) is determined by

char V ( p) = (x1 · · · xn ) p/2 sλ (x), (5.16) λ, (λ)≤ p

where (λ) is the length of the partition λ. The explicit action of the osp(1|2n) generators in V ( p) is given by (5.10)–(5.11) or (A.11)–(A.12), and the basis is orthogonal and normalized. For p ∈ {1, 2, . . . , n − 1} this action remains valid, provided one keeps in mind that all vectors with m p+1,n = 0 must vanish.

820

S. Lievens, N. I. Stoilova, J. Van der Jeugt

Now starting from  0  0 |0 =  .. . 0

⎞ ··· ··· 0 0 ··· ··· 0 ⎟ ⎟ . ⎠ ..

and using the explicit actions (A.11)–(A.12) of the osp(1|2n) generators on the GZ-basis vectors one can obtain the connection between the GZ-basis (5.2) and the relevant elements of the induced basis (3.11). This relation is however not simple and it is not possible to give it in an unified form. For example, the coefficients of the linear combinations expressing the GZ-basis vectors in terms of the basis (3.11) consist in general of algebraic expressions containing non-factorizable polynomials in p. Note that the first line of Theorem 7 can also be deduced from [5], where all lowest weight unirreps of osp(1|2n) are classified by means of their lowest weight. There is an interesting question related to the character of V ( p). For p > n − 1, the character is written in the form (5.14), i.e. with a genuine osp(1|2n) denominator (each factor in this denominator corresponds to a positive root α of osp(1|2n) for which 2α is not a root). Can the characters of V ( p) for p ∈ {1, 2, . . . , n − 1} be written in a similar form? In other words, what is E p in the expression char V ( p) = x p/2



sλ (x) = x p/2

i (1 − x i )

λ, (λ)≤ p

Ep

j
x j xk )

,

(5.17)

where we have used the short hand notation x = x1 · · · xn . Some initial computations lead us to the following conjecture:

Ep = (−1)cη sη (x), (5.18) η

where the sum is over all partitions η of the form   a1 a2 ··· ar η= a1 + p a2 + p · · · ar + p

(5.19)

in Frobenius notation, and cη = a1 + a2 + · · · + ar + r.

(5.20)

The notation is a special way of denoting partitions, see [23], related to the lengths of rows and columns in the Young diagram of the partition, counted from the diagonal. In the current case, the partitions η are all those with a Young diagram of the shape depicted in Fig. 3. Since the number of variables x1 , . . . , xn is finite, the expression E p is also finite. Some special cases are:  E1 = (1 − x j xk ), (5.21) 1≤ j
E n−1 = 1 − x1 x2 · · · xn .

(5.22)

Paraboson Fock Space and Unitary Irreducible Representations

821

Fig. 3. Typical shape of the Young diagram for the partition η, given by the Frobenius notation (5.19) (illustrated here for r = 3)

The first expression is a known S-function series, also due to Littlewood [22]. The first special case leads to x 1/2 char V (1) = , (5.23) i (1 − x i ) as it should, since this corresponds to the ordinary boson Fock space. The second special case yields (1 − x1 x2 · · · xn ) char V (n − 1) = x p/2 . (5.24) (1 − xi ) j


η (−1)

sλ (x) =

i (1 − x i )

λ, (λ)≤ p



cη s

η (x)

j
(5.25)

x j xk )

was proved by R.C. King [20]. In fact, he also obtained an alternative expression in terms of determinants, which will be presented below. It leads to the following alternative forms of the character, for p = 1, 2, . . . , n:

char V ( p) = x p/2 sλ (x) λ, (λ)≤ p



=x

η (−1)

p/2

i (1 − x i )



= x p/2



cη s

j
n− j

det xi

η (x)

x j xk )

 n− p+ j−1 n

− (−1) p χ ( j > p)xi   n− j n+ j−1 n det xi − xi

i, j=1

where χ (w) = 1 if w is true and 0 otherwise.

i, j=1

,

822

S. Lievens, N. I. Stoilova, J. Van der Jeugt

6. Branching to sp(2n) and sp(2n) Characters The osp(1|2n) representations V ( p) have been completely determined, including the action of the osp(1|2n) generators b±j . A basis for the even subalgebra sp(2n) has been given in (3.7). As a consequence, the explicit action of the sp(2n) basis elements {b±j , bk± } can also be determined. Under the action of sp(2n), however, V ( p) is in general not irreducible. In this section we shall determine the decomposition of V ( p) in irreducible sp(2n) components. We shall first consider the generic case p > n − 1, where V ( p) = V ( p). Since V ( p) is a lowest weight representation, all irreducible sp(2n) components will also be lowest weight representations. Each such component is thus characterized by a lowest weight vector v. Such a vector should satisfy: {b−j , bk− } · v = 0 for 1 ≤ j, k ≤ n,

{b−j , bk+ } · v = 0 for 1 ≤ j < k ≤ n.

(6.1)

From the explicit action of b±j it is possible to deduce the following Proposition 8. The n + 1 vectors vk (k = 0, 1, . . . , n) of the form  ⎞  11 · · · 110 · · · 0 row 1 : k ones   11 · · · 10 · · · 0 ⎟ row 2 : k − 1 ones ⎟  ··· ⎟ ··· ⎟  row k : 1 one vk =  10 · · · 0 ⎟ ⎟ 0···0 row k + 1 : all zeros ⎟  ··· ⎠ ···  0 row n : zero

(6.2)

satisfy (6.1). These are the only vectors of V ( p) satisfying (6.1). The weight of vk is given by p p (6.3) ( , · · · , ) + (0, . . . , 0, 1, . . . , 1) (k ones). 2 2 As a consequence, the decomposition of V ( p) with respect to sp(2n) can be written as V ( p) →

n 

V ( p, k),

(6.4)

k=0

where V ( p, k) is a unirrep of sp(2n) with lowest weight vector vk and lowest weight given by (6.3). The structure of these unirreps V ( p, k) can be deduced from the character formula of V ( p). Since n  (1 + xi ) = 1 + s1 (x) + s1,1 (x) + · · · + s1,1,...,1 (x), i=1

it follows that char V ( p) = =

x p/2

i (1 − x i ) j
Using the knowledge of the sp(2n) lowest weight vectors, one finds

(6.5)

Paraboson Fock Space and Unitary Irreducible Representations

823

Proposition 9. The characters of the sp(2n) unirreps V ( p, k) with lowest weight (6.3) and with p > n − 1 are given by char V ( p, k) = x p/2

i (1 −

s1,...,1 (x) 2 xi ) j
x j xk )

,

(6.6)

where (1, . . . , 1) = 1k . For the non-generic case, V ( p) with p ∈ {1, 2, . . . , n − 1}, the analysis is similar and we will not give all the details. In fact, the vectors vk with k = 0, 1, . . . , p are again sp(2n) lowest weight vectors (this time k ≤ p because otherwise vk would belong to M( p)). This yields: p  V ( p) → V ( p, k). (6.7) k=0

The characters are essentially the same as in the generic case, except that one has to exclude u(n) components with partitions of length greater than p (as the corresponding reduced matrix elements vanish). So we can write   s1,...,1 (x) p/2 char V ( p, k) = x L p , (6.8) 2 i (1 − x i ) j
824

S. Lievens, N. I. Stoilova, J. Van der Jeugt

Note that in Theorem 7 in the second case, p ∈ {1, 2, . . . , n − 1}, the induced module V ( p) is not irreducible. So it must have certain primitive vectors with respect to osp(1|2n). In fact, the weights of these primitive vectors follow from (5.17) and (5.18). In this paper we have constructed osp(1|2n) unirreps with a particular lowest weight, namely of the form ( 2p , . . . , 2p ). It looks as if our techniques could also be used to construct explicitly osp(1|2n) unirreps with a more general lowest weight of the form ( p1 , . . . , pn ), though the computational difficulties might be extremely hard. A. Appendix In this Appendix we shall give the special u(n) CGCs [2,30] needed in the action (5.10)–(5.11) of the osp(1|2n) generators. For this purpose, let us write [m] j for the j th row in a GZ-pattern, and we shall denote the increment of element i in such a j-tuple j by [m]±i : [m] j = [m 1 j , m 2 j , . . . , m j j ],

(A.1)

j [m]±i

(A.2)

= [m 1 j , . . . , m i j ± 1, . . . , m j j ].

The u(n) CGCs can now be written as (a 0˙ stands for a sequence of zeros of appropriate length)  ⎞ ⎛ n [m]n 10˙  [m]+k ⎜ [m]n−1 10˙  [m  ]n−1 ⎟ ⎟ ⎜ ⎟ · · ·  · · · ⎜··· ⎟ ⎜ j  j  ⎟ ; 10˙  [m ] ⎜ [m] ⎟ ⎜ j−1 j−1  ⎟  ⎜ [m] 0˙  [m ] ⎟ ⎝···  ⎠ ··· ···   m 11 0 m 11  ⎞ ⎛ [m]n−1 10˙  [m  ]n−1 ⎟ · · ·  · · · ⎜··· ⎟  

 [m]n  10˙  [m]n  ⎜ j  j  ⎟ ⎜ [m] 10˙  [m ] +k  ⎟.  = (A.3) × ; ⎜ n−1 n−1  j−1 j−1   ⎟   ˙ [m]  0 [m ] ⎜ [m] 0˙  [m ] ⎟  m ⎝ ⎠ · · ·  · · · ···   0 m 11 m 11 In the right-hand side, the first factor is an isoscalar factor, and the second factor is a CGC of u(n − 1). The middle pattern in the u(n − 1) CGC is that of the u(n) CGC with the first row deleted. The middle pattern in the isoscalar factor consists of the first two rows of the middle pattern in the left hand side, so  is 0 or 1. If  = 0, then [m  ]n−1 = [m]n−1 . If  = 1, then [m  ]n−1 = [m 1,n−1 , . . . , m r,n−1 + 1, . . . , m n−1,n−1 ] = [m]n−1 +r for some r -value. The explicit forms of these isoscalar factors is given in [30, p. 385]. We have: 1/2      n−1 [m]n  10˙  [m]n+k j=1 (l j,n−1 − lkn − 1) n = , (A.4) [m]n−1  00˙  [m]n−1 j =k=1 (l jn − lkn ) where, as usual in this context,

li j = m i j − i;

(A.5)

Paraboson Fock Space and Unitary Irreducible Representations

825

and 

1/2  n−1    (l j,n−1 − lkn − 1) nj =k=1 (l jn − lr,n−1 )  10˙  [m]n [m]n j = r =1 +k   , = S(k, r ) n [m]n−1  10˙  [m]n−1 (l jn − lkn ) n−1 (l j,n−1 − lr,n−1 − 1) +r j =k=1

j =r =1

(A.6)



where

1 for k ≤ r (A.7) −1 for k > r. We are now in a position to write down the explicit actions of the osp(1|2n) generators on the basis vectors |m). First, note that for the diagonal elements one has ⎞ ⎛ k k−1



p h k |m) = ⎝ + (1 ≤ k ≤ n). (A.8) m jk − m j,k−1 ⎠ |m), 2 S(k, r ) =

j=1

j=1

Due to the simplicity of the CGCs, the actions of bn± are the simplest to describe:  1/2 n n−1

k=1 (lk,n−1 − lin − 1) + n bn |m) = Fi (m 1n , m 2n , . . . , m nn ) |m)+in ; (A.9) k =i=1 (lkn − lin ) i=1  1/2 n n−1

k=1 (lk,n−1 − lin ) − n Fi (m 1n , . . . , m in − 1, . . . , m nn ) |m)−in . bn |m) = k =i=1 (lkn − lin + 1) i=1

(A.10) Herein, Fi are the functions given in (5.13), and |m)±in indicates the replacement m in → m in ± 1. Finally, the actions of the remaining generators b±j ( j = 1, 2, . . . , n − 1) are somewhat more involved, due to the various isoscalar factors. They can be written in the following form: b+j |m)

=

n−1 n



i n =1 i n−1 =1

×

n− j r =1

...

 n−r

j

⎞1/2 j−1 k=1 (lk, j−1 − li j , j − 1) ⎠ j k =i j =1 (lk j − li j , j )

⎛ S(i n , i n−1 )S(i n−1 , i n−2 ) . . . S(i j+1 , i j ) ⎝

i j =1

1/2 n−r +1 k =i n−r =1 (lk,n−r − li n−r +1 ,n−r +1 − 1) k =i n−r +1 =1 (lk,n−r +1 − li n−r ,n−r ) n−r +1 n−r k =i n−r +1 =1 (lk,n−r +1 − li n−r +1 ,n−r +1 ) k =i n−r =1 (lk,n−r − li n−r ,n−r − 1)

× Fin (m 1n , m 2n , . . . , m nn ) |m)+in ,n;+in−1 ,n−1;...;+i j , j ;

(A.11)

⎞1/2 ⎛ j−1 (l − l ) k, j−1 i , j j k=1 ⎠ ... S(i n , i n−1 )S(i n−1 , i n−2 ) . . . S(i j+1 , i j ) ⎝ j b−j |m) = i n =1 i n−1 =1 i j =1 k =i j =1 (lk j − li j , j + 1)  n−r 1/2 n−r +1 n− j k =i n−r =1 (lk,n−r − li n−r +1 ,n−r +1 ) k =i n−r +1 =1 (lk,n−r +1 − li n−r ,n−r + 1) × n−r +1 n−r k =i n−r +1 =1 (lk,n−r +1 − li n−r +1 ,n−r +1 + 1) k =i n−r =1 (lk,n−r − li n−r ,n−r ) r =1 n n−1



j

× Fin (m 1n , . . . , m in ,n − 1, . . . , m nn ) |m)−in ,n;−in−1 ,n−1;...;−i j , j .

(A.12)

Herein, each symbol ±i k , k attached as a subscript to |m) indicates a replacement m ik ,k → m ik ,k ± 1. Acknowledgements. The authors would like to thank Professor T.D. Palev for his interest.

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S. Lievens, N. I. Stoilova, J. Van der Jeugt

References 1. Baird, G.E., Biedenharn, L.C.: On the Representations of the Semisimple Lie Groups. II. J. Math. Phys. 4, 1449–1466 (1963) 2. Baird, G.E., Biedenharn, L.C.: On the Representations of the Semisimple Lie Groups. V. Some Explicit Wigner Operators for SU3 . J. Math. Phys. 6, 1847–1854 (1965) 3. Bargmann, V.: Irreducible Unitary Representations of the Lorentz Group. Ann. of Math. 48, 568–640 (1947) 4. Blank, J., Havlicˇek, M.: Irreducible ∗-Representations of the Lie Superalgebras B(0, n) with FiniteDegenerated Vacuum. II. J Math. Phys. 29, 546–559 (1988) 5. Dobrev, V.K., Zhang, R.B.: Positive Energy Unitary Irreducible Representations of the Superalgebras osp(1| 2n; R). Phys. Atom. Nuclei 68, 1660–1669 (2005) 6. Drühl, K., Haag, R., Roberts, J.E.: On parastatistics. Commun. Math. Phys. 18, 204–226 (1970) 7. Ganchev, A.Ch., Palev, T.D.: A Lie Superalgebraic Interpretation of the Para-Bose Statistics. J. Math. Phys. 21, 797–799 (1980) 8. Green, H.S.: A Generalized Method of Field Quantization. Phys. Rev. 90, 270–273 (1953) 9. Gel’fand, I.M., Zetlin, M.L.: Finite-Dimensional Representations of the Group of Unimodular Matrices. Dokl. Akad. Nauk SSSR 71, 825–828 (1950) 10. Greenberg, O.W., Messiah, A.M.L.: Selection Rules for Parafields and the Absence of Para particles in Nature. Phys. Rev. B 138, 1155–1167 (1965) 11. Greenberg, O.W., Nelson, C.A.: Color Models of Hadrons. Phys. Rep. C 32, 69–121 (1977) 12. Greenberg, O.W., Macrae, K.I.: Locally Gauge-Invariant Formulation of Parastatistics. Nucl. Phys. B 219, 358–366 (1983) 13. Heidenreich, W.: All Linear Unitary Representations of de Sitter Supersymmetry with Positive Energy. Phys. Lett. B 110, 461–464 (1982) 14. Itzykson, C.: Remarks on Boson Commutation Rules. Commun. Math. Phys. 4, 92–122 (1967) 15. Kac, V.G.: Lie Superalgebras. Adv. Math. 26, 8–96 (1977) 16. Kac, V.G.: Representations of Classical Lie Superalgebras. Lect. Notes in Math. 626, Berlin: Springer Verlag, 1978, pp. 597–626 17. Kamupingene, A.H., Palev, T.D., Tsavena, S.P.: Wigner Quantum Systems Two Particles Interacting via a Harmonic Potential. I. Two-Dimensional Space. J. Math. Phys. 27, 2067–2075 (1986) 18. King, R.C., Wybourne, B.G.: Holomorphic Discrete Series and Harmonic Series Unitary Irreducible Representations of Non-Compact Lie Groups: Sp(2n, R), U(p, q) and SO∗ (2n). J. Phys. A: Math. Gen. 18, 3113–3139 (1985) 19. King, R.C., Palev, T.D., Stoilova, N.I., Vander Jeugt, J.: A Non-Commutative n-Particle 3D Wigner Quantum Oscillator. J. Phys. A 36, 11999–12019 (2003) 20. King, R.C.: Private communication 21. Lievens, S., Stoilova, N.I., Van der Jeugt, J.: Harmonic Oscillators Coupled by Springs: Discrete Solutions as a Wigner Quantum System. J. Math. Phys. 47, 113504, 23 pp (2006) 22. Littlewood, D.E.: The theory of Group Characters and Matrix Representations of Groups. Oxford: Oxford University Press, 1950 23. Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford: Oxford University Press, 2nd edition, 1995 24. Mansouri, F., Wu, Xi Zeng: Parastatistics and Conformal Field Theories in Two Dimensions. J. Math. Phys. 30, 892-901 (1989) 25. Ohnuki, Y., Kamefuchi, S.: Quantum Field Theory and Parastatistics. Berlin: Springer, 1982 26. Palev, T.D.: Wigner Approach to Quantization. Noncanonical Quantization of Two Particles Interacting Via a Harmonic Potential. J. Math. Phys. 23, 1778–1784 (1982) 27. Palev, T.D.: Lie Superalgebras, Infinite-Dimensional Algebras and Quantum Statistics. Rep. Math. Phys. 31, 241–262 (1992) 28. Rowe, D.J., Wybourne, B.G., Butler, P.H.: Unitary Representations, Branching Rules and Matrix Elements for the Noncompact Symplectic Groups. J. Phys. A 18, 939–953 (1985) 29. Sternberg, S., Wolf, J.A.: Hermitian Lie Algebras and Metaplectic representations. Trans. Amer. Math. Soc. 238, 1–43 (1978) 30. Vilenkin, N.Ja., Klimyk, A.U.: Representation of Lie Groups and Special Functions, Vol. 3: Classical and Quantum Groups and Special Functions. Dordrecht: Kluwer Academic Publishers, 1992 31. Wigner, E.P.: Do the Equations of Motion Determine the Quantum Mechanical Commutation Relations? Phys. Rev. 77, 711–712 (1950) 32. Weyl, H.: The Classical Groups. Princeton, NJ: Princeton University Press, 1946 Communicated by Y. Kawahigashi

Commun. Math. Phys. 281, 827–857 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0501-x

Communications in

Mathematical Physics

Equivariant Volumes of Non-Compact Quotients and Instanton Counting Johan Martens Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada. E-mail: [email protected] Received: 12 July 2007 / Accepted: 14 October 2007 Published online: 9 May 2008 – © Springer-Verlag 2008

Abstract: Motivated by Nekrasov’s instanton counting, we discuss a method for calculating equivariant volumes of non-compact quotients in symplectic and hyper-Kähler geometry by means of the Jeffrey-Kirwan residue formula of non-abelian localization. In order to overcome the non-compactness, we use varying symplectic cuts to reduce the problem to a compact setting, and study what happens in the limit that recovers the original problem. We implement this method for the ADHM construction of the moduli spaces of framed Yang-Mills instantons on R4 and rederive the formulas for the equivariant volumes obtained earlier by Nekrasov-Shadchin, expressing these volumes as iterated residues of a single rational function.

1. Introduction In this paper we develop a method for calculating equivariant volumes of non-compact symplectic spaces (where they are also known as regularized volumes) equipped with a Hamiltonian action of a torus T , in the case when these spaces are finite-dimensional symplectic reductions or hyper-Kähler quotients. The Hamiltonian action gives a moment map µ : M → t∗ , and we are interested in the improper integrals


eµ,t M

ωn n!

(1)

of the function eµ,t with respect to the Liouville measure ωn! . Under certain conditions this integral converges for t in an open subset of the Lie algebra tC . Formally we may write it as
eω+µ,t , n

M

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where eω+µ,t is a class in a completion of the equivariant cohomology ring HT∗ M. Because M is not compact, however, the integral of this class cannot be directly defined cohomologically. The method described here is essentially a combination of the symplectic cut of Lerman [Ler95] and the Jeffrey-Kirwan residue formula of non-abelian localization [JK95b]. The symplectic cut allows us to reduce the integral to an integral over a compact space Mλ , such that the original integral is recovered in the limit as λ goes to infinity. Once in the compact setting, the residue formulas (which require compactness) allow the integral over the quotient to be expressed in terms of equivariant data on the space that one takes the quotient of. The key observation is that the cutting operation introduces new fixed points ‘at infinity’, whose contribution is crucial. In the particular case of symplectic reductions or hyper-Kähler quotients of vector spaces by linear actions, we are able to describe the outcome of this procedure in greater detail, and express the corresponding volumes as iterated residues of a single rational function. For hyper-Kähler quotients for instance we have
 2 e(µC ) K˜ Res+X 1 ,...,X n  eω+µ = . |WG | V ///G j ρj In the rational function

2 e(µ )  C j ρj

in question,  is the product of the positive roots of G,

and both e(µC ) and the denominator are products of linear factors entirely determined by the action of G on V . The iterated residue Res+X 1 ,...,X n extracts the relevant information out of the rational function; its definition is given in Sect. 4. Our motivating examples come from physics: the equivariant volumes of moduli spaces of framed instantons, which are the main ingredient in the recent instanton counting of Nekrasov [Nek03]. The famous ADHM construction [AHDM78] realizes these spaces as hyper-Kähler quotients, and using this we completely carry out the calculational strategy described below for them. We would like to emphasize that we interpret the integrals we study as regularized  volumes eω+µ , i.e. bona fide improper integrals, firmly rooted in symplectic geometry, which is true to the initial formulation of the problem of instanton counting in physics. This is somewhat in contrast to other authors [NY05a,NO06], who work with a formal  integral 1 in equivariant cohomology of the class 1, essentially defined through a localization formula. From our more naive point of view, all localization formulas are theorems rather than definitions, and more importantly for us, this perspective allows for our calculational method, as the moment map gives the exact recipe for handling the residues of the new fixed points introduced by the cut. 1.1. Organization. The rest of this paper is organized as follows. In Sect. 2 we discuss the basic object of our study, the regularized volumes of non-compact symplectic spaces with Hamiltonian torus action, and review the Prato-Wu theorem that calculates their volume. In Sect. 3 we discuss the Jeffrey-Kirwan residue theorem, and develop our main conceptual idea, combining the residue theorem with symplectic cutting to calculate the equivariant volume of quotients. In Sect. 4 we specialize to linear symplectic reductions and hyper-Kähler quotients, and discuss the specific implementation of our calculational method in these cases. As examples of this calculational method we implement it in Sect. 5 for the ADHM spaces for the classical gauge groups, and compare the results with the work of Nekrasov-Shadchin [NS04] whose formulas we rederive.

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1.2. Notation. We will use the following notation throughout: ., .

natural pairing between t∗ and t or their complexifications t∗C and tC

M//λ G

symplectic reduction of M by G at a central value λ of the moment map

Mλ , Vλ

symplectic cut of M, V with respect to a circle action at value λ ∈ u(1)∗

T n  (e1 , . . . , en ) elements of a split torus T = U (1)n , and = (e1 , . . . , en ) corresponding coordinates on Lie algebra t; we will use corresponding Greek and Roman lower-case letters for elements of t and T respectively 2. The Duistermaat-Heckman Theorem for Non-Compact Spaces The famous formula of Duistermaat-Heckman [DH82] establishes an expression for the integral of the equivariant volume of a compact symplectic manifold (M, ω) equipped with a Hamiltonian action of a torus T with moment map µ:

eω+µ ω+µ , e = e (ν F ) M T F T F⊂M

where the sum in the right-hand side is over the connected components of the fixed point set for the action of T , ν F is the normal bundle of F in M, and eT (ν F ) is its equivariant Euler class. In [AB84,BV82] this formula was established as a particular instance of localization in equivariant cohomology, using the Cartan model. In [PW94], this theorem is discussed in the case where M is no longer compact: Theorem 1 (Prato-Wu). Let (M, ω) be a (non-compact) symplectic manifold equipped with a Hamiltonian action of a torus T . If the fixed-point set M T is compact and there exists a t0 ∈ t = Lie(T ) such that the corresponding component of the moment map µt0 = µ, t0  is proper and bounded on one side (assume bounded above) then the Duistermaat-Heckman formula


eµ,t M


eω+µ,t ωn = n! eT (ν F ) T F

(2)

F⊂M

still holds true, provided one interprets the right-hand side as the integral of the function n eµ,t with respect to the measure induced by the Liouville volume ωn! , and one restricts oneself to t inside of an certain open cone C ⊂ tC . The open cone occurring in the theorem is exactly the set of all t for which the integral in the left hand side of (2) converges. The right-hand side can now be interpreted as a function on tC with poles on an arrangement of hyperplanes containing the origin, and the cone used is contained in a component of the complement of the hyperplanes.

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One can prove this theorem by means of the symplectic cut construction introduced by Lerman [Ler95]: the conditions stated above guarantee the existence of a circle U (1) in T whose moment map µc is proper and bounded above, hence the symplectic cut Mλ = (M × C) //λ U (1) = µ−1 c (−∞, λ) M//λ U (1) with respect to this circle is a compact symplectic space, still equipped with a T -action. As the open dense subset inherits its symplectic structure from the inclusion of µ−1 c (−∞, λ) in M, one can moreover recover the integral over M we are interested in as the limit of integrals over the symplectic cuts:

eω+µ = lim eω+µ . (3) M

λ→∞ Mλ

For the compact spaces Mλ the original Duistermaat-Heckman formula holds, and as M T is assumed to be compact, one can remark that the fixed point set (Mλ )T , for λ large enough, consists of the ‘original’ fixed point components of M, together with ‘new’ ones introduced by the cut:  T MλT = µ−1 (−∞, λ) (M//λ U (1))T = M T (M//λ U (1))T . c As the components of M T don’t depend on λ, (3) becomes

eω+µ  eω+µ = + lim e (ν F ) λ→∞

M T F T F⊂M

F

⊂M//U (1)T


F

eω+µ . eT (ν F )

(4)

Moreover, in the limit as λ → ∞, one can easily show that the contributions of the new fixed points (i.e. the T -fixed points of M//λ U (1)) will vanish,
 eω+µ,t lim = 0, λ→∞

eT (ν F )

T F F ⊂M//U (1)

provided that one exactly restricts to values of t in the open cone C, which establishes (2). Indeed, the values of any new fixed-point component F under the moment map µ will vary with the parameter λ, and the cone C is such that the exponent of eµ(F),t is negative. It was pointed out to the author by M. Vergne that the original proof of [PW94] essentially follows this line, although the technology of symplectic cuts had not been introduced at the time. In [Par00], Paradan discusses the above formula in the much broader setting of equivariant cohomology with generalized coefficients. For our purposes the naive interpretation given above suffices however, and we will restrict ourselves to this framework. 3. Equivariant Volumes of Non-Compact Quotients In this section we describe the main idea of this paper: in order to calculate the equivariant volume of a non-compact quotient M//G, compactify both M and M//G by a symplectic cut, use a residue formula to calculate the volume of the cut of M//G, and recover the result in the limit.

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3.1. Residue Theorem. Kirwan showed in [Kir84] that for the symplectic reduction of a compact symplectic space (M, ω) by a compact connected Lie group G there is a surjective map, the Kirwan map, κ : HG∗ (M, Q) → H ∗ (M//G, Q). In [JK95b] Jeffrey and Kirwan established a method for calculating integrals over M//G of images of this map, by means of the residue formula:1 Theorem 2.




ω

M//G

κ(α)e =

 F⊂M TG

K JKResΛ  2 |WG |


F

i F∗ αeω+µTG , eTG (ν F )

(5)

(−1) where TG is a maximal torus of G, K is the constant vol(T , n + is the number of positive G) roots, the volume vol(TG ) is taken with respect to an invariant metric on TG , WG is the Weyl group of G,  is the product of the positive roots of G, i F is the inclusion i F : F → M, and JKResΛ is the Jeffrey-Kirwan residue operation. n+

The residue JKResΛ is a linear operation defined on a class of meromorphic forms on a vector space, with poles on an arrangement of hyperplanes (i.e. the denominator has to split as a product of linear factors). The residue depends on the choice of a cone Λ ⊂ t; for different choices of Λ the residue will affect the terms in the right hand side differently, but the total result is independent of the choice for Λ. The original definition of the residue was given in terms of formal Fourier transforms, and a characterizing list of axioms was determined in [JK95b, Prop. 8.11]. Another intrinsic formulation of the residue was given in [BV99]. We will however restrict ourselves to using another definition of the residue, given in [JK97], as iterated one dimensional residues. Begin by looking at the one-dimensional case. Let f be a function on tC ∼ = C of the form  g j eλ j x , f (x) = j

where g j are rational functions on C, and λ j ∈ R. Then define the residue of the 1-form f (x)d x to be  jkres+ ( f (x)d x) = resb (g j (x)eλ j x ). (6) λ j ≥0 b∈C

This one-dimensional residue will be the building block for defining higher-dimensional versions. A very useful little fact that we shall use often later on is: if p(x), q(x) are polynomials on tC such that deg( p) + 1 = deg(q), then jkres+

p(x) d x = 0, q(x)

(7)

1 Various appearances of this formula in the literature [JK95a,JK97,JK98] differ in the constants used because of alternative notational conventions, e.g. the use of ω + iµ instead of ω + µ as the equivariant symplectic form. We will follow the version given in [JK98]. Other approaches to the residue formula were given in [Ver96,GK96].

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as can easily be seen by interpreting this rational form as a rational form on CP1 , calculating jkres+ by using the Cauchy theorem, and looking at the residue at ∞. Now the higher-dimensional version that we want to use is in practice obtained by iterating such one-dimensional residues jkres+ . Suppose (X 1 , . . . , X n ) is a linear coordinate system on t. Then for any function given as a sum of terms of the form p(x)eλ(x) f (z) =  ρi (x)

(8)

with q(x) a polynomial on tC , and λ, ρi ∈ t∗ , we can still define one-dimensional residues jkres+X i for any of the given coordinates, by treating all other coordinates as generic constants. The result of this is a new function of the form(8), but this time only in the remaining variables X 1 , . . . , Xˆ i , . . . , X n . We then follow [JK97,JK05] in defining the residue operation JKResΛ inductively as an iterated residue: JKResΛ (h[d x]) = jkres+X 1 . . . jkres+X n h(X 1 , . . . , X n )[d X ]n1

(9)

for any generic coordinate system (X 1 , . . . , X n ) such that (0, . . . , 0, 1) ∈ Λ, where  is the determinant of any n × n matrix whose columns form an orthonormal basis of t defining the same orientation as (X 1 , . . . , X n ) (a metric on t is tacitly understood). Different choices for Λ will give different results when applying JKResΛ to the summands in the RHS of (5), but the total result will be unaffected. In general JKResΛ will only give a non-zero contribution for the summands corresponding to components F ⊂ M T , where µ(F) ∈ Λ∗ , the dual cone of Λ. In [JK95b,JK97] it was shown that this definition of JKResΛ coincides with the earlier one given in terms of a Fourier transform. As indicated above the residue operation acts on meromorphic forms rather than functions, but for notational convenience we will suppress the relevant form part [d X ]n1 throughout.

3.2. Equivariant volumes of symplectic reductions. We want to apply Theorem 2 for the calculation of equivariant volumes of non-compact spaces that arise as symplectic reductions, in the sense described in Sect. 2. We will take the same approach as the one outlined above for the theorem of Prato-Wu in (3): one can relate the integral on the non-compact space to a limit of integrals on compact spaces by means of varying symplectic cuts, and then use the available residue theorems for the compact spaces. In sharp contrast however with the theorem of Prato-Wu is that the contributions of the ‘new’ fixed points introduced by the cuts will not vanish in the limit. In fact, in the cases that we study later on, these ‘fixed points at infinity’ carry all the information. In order to implement this we need the following straightforward equivariant generalization of the Jeffrey-Kirwan theorem: Theorem 3. Let G and H be two (compact, connected) Lie groups with commuting Hamiltonian actions on a compact symplectic manifold (M, ω). Then the action of H descends to the symplectic reduction M//G, and with the equivariant Kirwan map ∗ κ : HG×H M → H H∗ (M//G) and the notation as in Theorem 2 we have
M//G

κ(α)eω+µ H =

 F⊂M TG

K JKResΛ  2 |WG |


F

i F∗ αeω+µTG +µ H . eTG (ν F )

(10)

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As our proof of this is a fairly straightforward reduction of the equivariant to the non-equivariant H = 1 case and mainly consists of technical remarks, we postpone it to Appendix A. Now suppose that we have two commuting Hamiltonian group actions on a symplectic manifold (M, ω), of a compact connected Lie group G and a torus T with moment maps µG and µT , and suppose that µT has a component that is proper and bounded from below. Then, as before, we can find a circle in T such that the symplectic cuts of both M and M//G with respect to this circle are compact. Moreover, since all group actions commute and a symplectic cut is a symplectic reduction at heart, we have (Mλ )//G = (M//G)λ . If we express the equivariant volume of the reduced space M//G as the limit of the equivariant volumes of the cut spaces (M//G)λ , the equivariant residue formula allows us to calculate the equivariant volumes of M//G as

ω+µT e = lim eω+µT λ→∞ (M//G)λ M//G
= lim eω+µT λ→∞ (Mλ )//G

= lim

λ→∞



F⊂(Mλ )TG

K JKResΛ  2 |W |


F

eω+µT +µTG . eT ×TG (ν F )

(11)

One now has to make the basic but important observation that the symplectic cut introduces new fixed points—under the conditions stated above we have, for large λ, MλTG = M TG (M//λ U (1))TG . Furthermore, when considering the limit as λ → ∞ we restrict ourselves to the open cone in tC as in Sect. 2. On this open cone the limit of most — but not all — of the terms given by the residue formula vanishes. This is somewhat reminiscent of the physical concept of renormalization for quantum field theories: we have introduced a ‘scale’ (the momentum cut λ), but it turns out that there are quantities independent of the scale, and those are the ones that survive the limit as λ goes to infinity. We further remark that for our purposes, we shall use Theorem 3 in the case where the second group H = T is a torus. In this situation, the integrals over the fixed point components are integrals in T -equivariant cohomology that we can calculate via the usual localization. Therefore we can actually replace the sum on the right-hand side with the sum over all the TG × T -fixed point components F ⊂ M TG ×T . 3.3. Basic Example. Let us illustrate this principle in the simplest possible case: the equivariant integral on N = C with respect to a circle H = T 1 acting linearly with weight two. This is nothing but a Gaussian integral, which can be calculated using the Prato-Wu theorem

1 2 i , eω+µT (τ ) = e−2||z|| τ dz ∧ dz = 2 2τ N C where the right-hand side is the contribution of the unique fixed point to the localization formula. This result needs to be interpreted as holding only on the open cone τ ∈ (0, ∞)

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Fig. 1. Moment maps for the groups G and H

∼ R, or if preferred on the cone (0, ∞) × iR in the complexification in Lie(T ) = tC . Indeed, if we would follow the reasoning of Eq. (4) and compactify N through a symplectic cut to obtain Nλ = CP1 , then this cut would introduce a single extra fixed −λτ point, with contribution e−2τ . In the limit λ → ∞, the contribution of this fixed point ‘at infinity’ would vanish for τ ∈ (0, ∞) or (0, ∞) × iR. Alternatively, we can construct N as well as the reduction of M = C2 by the action of a circle G = T 1 acting with weights 1 and −1: s(z 1 , z 2 ) = (sz 1 , s −1 z 2 ). The action of H lifts to M as well, by t (z 1 , z 2 ) = (t z 1 , t z 2 ). We can depict the images under the moment maps of M and Mλ as in Fig. 1. After making the cut Mλ ∼ = C2 CP1 and using affine coordinates for C2 and projective coordinates for CP1 , it is easy to see that there are three fixed points in Mλ : the original (0, 0) in M, and the two new fixed points, ‘at infinity’, [1 : 0] and [0 : 1]. For the calculation of the residue, we only need to take [1 : 0] into account, as the residues of the contributions of the other fixed points give zero. The weights of the isotropy representation of T on its tangent space are directly read off from figure 1:

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σ − τ and 2σ . Hence we get as an integral:

eω+µT = lim M//H

eω+µ   e−λ(−σ +τ ) Λ = lim JKRes λ→∞ (σ − τ )(2σ ) −λτ

e 1 = lim + λ→∞ −2τ 2τ 1 . = 2τ λ→∞ Mλ //H

The reason only one of the two terms given by the residue survives the limit as λ → ∞ is exactly that the cone on which we consider the outcome (namely τ ∈ (0, ∞) or (0, ∞) × iR) is such that every exponential that still occurs vanishes. 4. Equivariant Volumes of Quotients by Linear Actions In order to put the above idea to use, one must have a way of handling the limit of the residues as λ goes to infinity. Therefore we restrict ourselves from now on to symplectic reductions and hyper-Kähler quotients of vector spaces by linear group actions, where this becomes feasible. We first need some remarks about how we interpret the symplectic cut for hyper-Kähler quotients. 4.1. Complex Moment Maps and hyper-Kähler quotients. We will be interested in applying this philosophy to calculate equivariant volumes of hyper-Kähler quotients of a vector space V by a linear group action, −1 −1 V ///(λ1 ,λ2 ,λ3 ) G = µ−1 1 (λ1 ) ∩ µ2 (λ2 ) ∩ µ3 (λ3 )/G.

However, we will not be interested in the full hyper-Kähler structure, but rather in the symplectic structure induced by one of the complex structures. With a choice of a preferred complex structure we can single out the moment map µ1 associated with the corresponding Kähler form, and pack the ‘other’ moment maps together in a complex valued µC = µ2 + iµ3 : V → g∗ ⊗ C. We will therefore mainly think of the hyperKähler quotients as symplectic reductions of a level set of this complex moment map: V ///(λ1 ,λ2 ,λ3 ) G = µ−1 C (λ2 + iλ3 )//λ1 G. The torus actions of T on V that we will consider do not preserve the hyper-Kähler structure, but as µC is (quadratic) homogeneous it will preserve the variety µ−1 C (0) ⊂ V −1 and therefore it will descend to an action on V ///(λ1 ,0,0) G = µC (0)//λ1 G. Now in order to apply the  calculational scheme outlined above, we need to study the symplectic cut µ−1 (0) . Our calculations become much simpler, however, if we C λ remark here that this symplectic cut, a subspace of Vλ , can still be interpreted as a zero-set of a section of a vector bundle. Indeed, in the cases studied later on, the symplectic cut happens with respect to a circle acting with global weight, and hence the cut space Vλ is a projective space. As µC is quadratic, we can therefore associate with it an equivariant

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section of the T -equivariant bundle O(2) ⊗ gC over the projective space Vλ , which we will also denote as µC : t∗C ⊗ O(2) B

(12)

µC

 Vλ By further abuse of notation we will say that (Vλ )///G = µ−1 (0)//G. This allows us now to reduce integration over (V ///G)λ to integration over (V //G)λ , by using the (equivariant) Euler class of the bundle as equivariant Poincaré dual to this zero-set. Strictly speaking there is no Poincaré duality in equivariant cohomology, but  one still has the property that M eG (E) ∧ α = s −1 (0) i s∗ α for a sufficiently general2 section s of an equivariant bundle E over a compact M, see e.g. [Mei06]. Hence we can adapt the calculational method given above in (11) to the case of hyper-Kähler quotients of vector spaces: 

V ///G

eω+µT = lim

λ→∞




= lim

λ→∞

(Vλ )///G

eω+µT

e




= lim

λ→∞

(V ///G)λ

⎛

(Vλ )//G

K = lim ⎝ λ→∞ |W |

e

ω+µT

ω+µT



eT (gC ⊗ O(2))

 F⊂(Vλ )TG ×T

JKResΛ  2

⎞




e

F

ω+µT +µT G eT × T (g C ⊗O(2)) G eT × T (ν F ) G

⎠. (13)

We have been conspicuously silent about an important aspect of this method: the possibility that the quotient M//G or V ///G is singular. (The simple example discussed in Sect. 3.3 above was a particular case where a reduction at a singular value of the moment map was nevertheless smooth.) In essence, everything still goes through, and we postpone the discussion about this to Sect. 5. 4.2. Calculational strategies. Since in practice most, if not all, finite-dimensional hyper-Kähler quotients are quotients of vector spaces by linear group actions, we will now restrict ourself to those cases. For these, as outlined above, we need a circle inside T that acts with a single global weight, and hence the cut space Vλ will be a projective 2 For the spaces that we are interested in this is indeed the case: it was shown by Crawley-Boevey [CB01, Theorem 1.1 and remarks to §1] that the zero-level of the complex moment map for ADHM spaces is a complete intersection.

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space Vλ = V PV . As all the fixed point data in this projective space are determined by the representation of TG × T on V , we can expect a more direct approach that bypasses the procedure of making the cut to compactify, examining the fixed points on the cut locus PV and their residues, and then taking the limit λ → ∞. Indeed this is the case, and below we give a description for the outcome that no longer mentions the cut. We will take a shamelessly pragmatic approach, explicitly using coordinates and working with a residue operation defined through iterated one-dimensional residues as in (9). For simplicity we will assume that all relevant (i.e. whose residue is not manifestly zero) T × TG fixed points on Vλ are isolated—which is the case for all the examples we will discuss below. We will use the following notation: for a meromorphic function pρ expressed in i linear coordinates (X 1 , . . . , X n ) and with all ρi linear, denote by p res Xj i  ρi the meromorphic function in the variables (X 1 , . . . , X i−1 , X i+1 , . . . , X n ) obtained by taking the residue in the variable X i of the pole determined by the j th factor in the denominator (and meanwhile considering all other variables as generic constants). The new form lacks a j th factor in the denominator, but for convenience we keep the same name and labelling of factors (i.e. the index skips j). As an example we have res3X 2

X 1 − 4X 2 (−X 1 + X 2 + X 3 )(X 1 − X 2 + X 3 )(X 1 + X 2 − X 3 ) X 1 − 4(X 3 − X 1 ) . = (−X 1 + (X 3 − X 1 ) + X 3 )(X 1 − (X 3 − X 1 ) + X 3 )

4.2.1. Fixed points in Vλ . Now, let us examine how to implement the above calculational method (13) when we have a (compact connected) Lie group G with maximal torus TG which acts tri-Hamiltonianly on a hyper-Kähler vector space V , with complex moment map µC . (Everything below still goes through if V is just a complex vector space of which we want to take a symplectic reduction, by just dropping the extra factors coming from µC throughout.) We assume that V is further equipped with the action of a torus T , commuting with G, such that T contains a circle acting with global weight one. Let ρi be the weights (repeated in the case of multiplicities) occurring in the weight decomposition of the torus action of T × TG on V . Clearly the ‘new’ fixed point components of the cut space in V //λ U (1) ⊂ Vλ will correspond exactly to weight spaces Vρi for some i. With the assumption of isolated fixed points made above, the relevant such weights will have multiplicity one, and the image under the moment map µT ×TG of such an isolated fixed point of Vλ will be λρi . One can also easily see that the weights of the isotropy representation at the fixed point corresponding to the weight ρi are given by i i ρ1i = ρ1 − ρi , . . . , ρi−1 = ρi−1 − ρi , ρii = −ρi , ρi+1 = ρi+1 − ρi , . . . . We will further abuse notation by writing the equivariant Euler class of the bundle (12), of which the complex moment map is a section, at the fixed point 0 as e(µC ) as well (this is also a polynomial, a product of linear factors), and at the fixed point determined by ρi as e(µiC ). 4.2.2. Admissible paths Let us examine the contributions of such a fixed-point component to the formula given above. Clearly the iterated residue JKResΛ (9) gives rise

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Fig. 2. Example of a path P, taking successive residues at various poles

to a sum of terms, each of which is obtained by taking the residue at one pole for each variable X i on t∗ : JKResΛ

 X  2 e(µiC )eλρi  2 e(µiC )eλρi = res j11 . . . res Xjnn .  i  i ρj j ρj

(14)

Calculating (14) in practice comes down to determining exactly which iterated residues need to be included in the right-hand side. For each term in the right-hand side of (14) we can indicate which poles were chosen for which variables by a path P as in Fig. 2, and denote this by Res P . As taking the residue at a pole eliminates that pole, each path can only contain every pole at most once, but certainly this is not a sufficient condition. Let us examine more closely which paths can occur in the sum (14). Recall from (6) the definition of the one-dimensional residue  jkres+ ( f (x)d x) = resb (g j (x)eλ j x ). λ j ≥0 b∈C

Iterating this gives the condition that just before taking the residue for the X j variable determined by the path, the coefficient of the X j variable of the exponent has to be positive, and this holds for all j. Let us call this Condition A. Furthermore, we are only interested in the terms that survive the limit λ → ∞ (as always, only considering the answer as a function on an open cone in t∗ ). In order for there to be any term that survives the limit, the last pole that is evaluated has to be the i th one, so our path has to end at the i th vertex in the right-most column. Call this Condition B. Indeed, if the i th pole occurs sooner in the path, the next step in the full iterated residue JKResΛ clearly yields zero. Likewise, if the i th pole is never used in a residue coming from a path, the term corresponding to that path would still have a nonzero coefficient in the exponent, and the cone in the Lie algebra on which we evaluate the final result is such that in the limit λ → ∞ the term would vanish.

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Let us call a path satisfying both these Conditions A and B an admissible path. Hence we have lim JKResΛ

λ→∞

 2 e(µiC )eλρi =  i j ρj



Res P

 2 e(µiC )eλρi  i j ρj

admissible paths P ending at ρi

4.2.3. Yoga of admissible paths From the above it is clear that in practice calculating the residue JKResΛ boils down to determining which paths P are admissible, and calculating the corresponding contribution Res P

 2 e(µiC )eλρi  i j ρj

for each admissible path P ending at

ρii .

Let us use the following notation: given a linear form ρ and a choice of (linear) coordinates (X 1 , . . . , X n ) on a vector space, we say that ρ > j 0 if the X j coefficient of ρ is greater than 0. With this notation, we can state the conditions for a path to be admissible as follows: first, we need to verify that ρi >n 0. After taking the residue in the X n variable at the pole determined by ρ ij1 , we then want ρi >n−1 0, and this keeps repeating until variable X n−1 . In particular, when testing if a path is admissible, this requires at each step calculating res Xjl l before we can determine if the vertex (l, jl ) can be part of the path, which is quite inefficient. Using some elementary linear algebra, however, we can reshuffle the calculations for contributions of admissible paths in such a way that we gain two distinct advantages. Firstly, the conditions for a path to be admissible become more straightforward. Secondly, we can apply the iterated residues corresponding to all admissible paths to the same single function, irrespective of the vertex at which the paths end (i.e. the fixed point to whose contribution the term belongs). Hence we eliminate referring to the actual ‘fixed points at infinity’, as we directly extract their contributions from one single function. Lemma 1. A path P = {(X n , jn ), . . . (X 2 , j2 ), (X 1 , i)} giving rise to the term Res P

 2 e(µiC )eλρi  2 e(µiC )eλρi = resiX 1 res Xj22 . . . res Xjnn  i  i j ρi j ρj

in the contribution of the fixed point corresponding to the weight ρi is admissible if and only if we have at the beginning ρi >n 0 and then at each step ρ jl >l−1 0. Furthermore, its contribution can also be computed as Res P

 2 e(µiC )eλρi  2 e(µC ) X = (−1)n res Xj21 . . . res jnn−1 resiX n   i j ρj j ρi  2 e(µC ) = (−1)n Res P  , j ρj

 is the path {(X n , i), (X n−1 , jn ), . . . , (X 1 , j2 )}. where P Proof. For the first statement it suffices to remark the following. We each time want to test the function ρi in the exponent after consecutive residues: ρi >n 0, ρi >n−1 0, . . . . However the poles at which we take the consecutive residues are exactly

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ρ ijn = ρ jn −ρi , ρ ijn−1 = ρ jn−1 − ρi , . . . . As taking a residue at such a pole essentially sets that term to zero, we might as well test for ρi >n 0, ρ jn >n−1 0, ρ jn−1 >n−2 0, . . . . For the second statement, notice that calculating one-dimensional residues res Xil can ρj

essentially be understood in terms of elementary linear algebra as Gaussian elimination for a matrix. Indeed, if you organize all the occurring linear functions (expressed in coordinates) as rows in a matrix as follows: variables used for residues

   ⎡X n . . . X 1

factors used for poles

all other factors

⎧ ρ ijn ⎪ ⎪ ⎪ ⎪. ⎢ ⎨ .. ⎢ ⎢ ⎢ ⎪ ⎪ρ ij2 ⎢ ⎪ ⎪ ⎩ i ⎢ ⎢  ρi ⎢ ⎢ ⎣

remaining variables

   Y1 . . . Ym⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

then taking a residue in a variable can be interpreted as doing the column-elimination for the corresponding column (given by the variable used for the residue) and row (given by the factor), and replacing all but the used row in the original function with the corresponding new rows. The factor of the pivotal entry is all that remains of the row used. From this one immediately sees that taking the consecutive one-dimensional residues comes down to doing row-reduction for the first n rows, replacing the remaining factors in the original function by the corresponding new rows, and dividing the function by the determinant of the upper n × n submatrix in the above matrix. With this in mind it is now clear that one can in fact change the ordering of the poles used for the consecutive residues, as well as the ordering of the variables used, with the sign change that occurs through the determinant as the only penalty. The single condition that needs to be satisfied is that for the new ordering, all the pivotal elements need to be non-zero. Now, given our ordering of the poles (i.e. ρ ijn , . . . , ρ ij2 , ρii ), change the ordering by making the last pole used first (hence obtaining ρii , ρ ijn , . . . , ρ ij2 ). As we have used ρii = −ρi for the first pole, this immediately implies that we might as well change all the other factors (i.e. ρ ijn , . . . , ρ ij2 as well as e(µiC )) to the corresponding factors (i.e. C) ρ jn , . . . , ρ j2 and e(µC )) for the ‘central function’ e(µ . This ordering also has the j ρj advantage that it exactly corresponds to the ordering of the testing explained above, and furthermore the consecutive conditions that the pivotal elements are non-zero is indeed implied through the consecutive tests ρ jl >l+1 0.  2

With this it makes sense to define the following: a complex vector Definition 1. Let X 1 , . . . , X n be a choice of linear coordinates on  space V . Then for any rational function f = qp on V , where q = j ρ j , and where each ρ j comes with a preferred polarization, we define Res+X 1 ,...X n inductively as Res+X 1 ,...X n = res+X 1 . . . res+X n

p , q

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where in each step the other variables are treated as generic constants, and where the single-variable residues are defined as res+X

 p p = res Xj , q q ρ j >X 0

ρ j in the sum on the right hand side being the factors of q. With this definition we now can summarize the discussion of this section: Theorem 4. Let a compact connected Lie group G act tri-Hamiltonianly on a hyperKähler vector space V , such that the group action commutes with the action of a torus T . Assume that T acts Hamiltonianly with respect to one of the Kähler structures on V . Let ω and µT denote the corresponding Kähler form and moment map on the hyper-Kähler quotient V ///G. Then with the notation as above we have
V ///(0,0,0) G

where K˜ is the constant

eω+µT =

 2 e(µC ) K˜ Res+X 1 ,...,X n  , |WG | j ρj

(−1)n + +n vol(TG ) .

Here X 1 , . . . , X n is a linear coordinate system on the Lie algebra of G, suitable in the sense of (9). For the volume of a symplectic reduction of a vector space with a linear action, the formula is identical except that e(µC ) is missing. We remark here the difference in the definitions of the various residue operations: JKResΛ and the one-dimensional jkres+X on the one side, and Res+X 1 ,...X n on the other. The former, originally coming from a Fourier-transform, involve a positivity test for the coefficient in the exponent of the numerator (hence the choice of the position of Λ and + in the notation). The latter involves a test of positivity on the factors of the denominator. The advantage of the reordering of the poles is that in practice it offers an easier way to go through the calculations of residues: at each step (i.e. when taking the residue for each successive variable) just take the sum of the residues at the poles corresponding to factors where the variable has a positive coefficient. We remark that the way we have written the residue above does not simply apply to rational functions—in particular each factor in the denominator has to come with a preferred sign, which is similar to the choice of polarization used in the Guillemin-Lerman-Sternberg approach to the DuistermaatHeckman formula [GLS96]. In the next section we will use this result further to compare our calculational method with the one given by Nekrasov and Shadchin [NS04] in the physics literature. 5. Instanton Counting We are now ready to apply the calculational techniques described above to our guiding examples: the calculation of the equivariant volumes of the ADHM spaces as they occur in instanton counting. We discuss the corresponding residue formulas, both for their own sake and as examples of how to implement the calculational heuristic given above in various settings, and rederive the results of Nekrasov and Shadchin [NS04]. The most important aspect still requiring settlement is the matter of the singularities of the quotients.

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5.1. Physical background. The recent work regarding instanton counting [Nek03] is a direct answer to an open problem in physics. In particular, it is concerned with N = 2 supersymmetric quantum Yang-Mills theory. One now wants to describe the low-energy behavior of this theory, and using the ideas of the Wilson renormalization group, one finds that this low-energy limit is described by means of another action, the effective action. In general it is extremely difficult to explicitly write down the effective Lagrangian. However, in the case of N = 2 quantum Yang-Mills theory it was shown by Seiberg and Witten in the seminal [SW94] that the entire Lagrangian for the low energy effective theory is determined by a single (multivalued) holomorphic function F SW , the SeibergWitten prepotential. Furthermore, using an assumption that a form of electric-magnetic duality (called Montonen-Olive duality) holds for the quantum theory, Seiberg and Witten managed to describe this prepotential analytically, in terms of period integrals of a family of curves. Their work was a major breakthrough in physics, but led to spectacular advances in mathematics as well, in particular in the framework of the Seiberg-Witten invariants in differential geometry. The original work of Seiberg and Witten was based on an assumption of duality, and though this duality is widely believed to hold, it is conjectural even for physicists as no derivation from first principles is known. In the last 10 years attempts were made to derive and verify their results directly. It was shown that this reduces to calculating certain integrals over moduli spaces of framed instantons on R4 (see e.g. [DHKM02] for background), but the actual calculations of these integrals were very difficult in general. The solution to this outstanding problem was finally accomplished by Nekrasov in 2002 [Nek03]. The strategy Nekrasov employs is to use maximal symmetry on the moduli spaces, induced by change of framing of instantons and rotations in R4 , and then compute equivariant volumes with respect to this group action by means of localization techniques in equivariant cohomology. In particular Nekrasov considers the following generating function3
∞  Z inst (q, τ, 1 , 2 ) = qk eω+µT ,(1 ,2 ,τ ) , (15) k=0

Mon,c

where Mon,c is the instanton-moduli space of rank n, charge c framed instantons on R4 . The torus T = T 2 × TG is the product of T 2 , the maximal torus of SU (2) that acts diagonally on R4 after the identification R4 = C2 and hence has an induced action on Mon,c , and TG , the maximal torus of the gauge group that acts on Mon,c by changing the framing of the instantons at infinity. Here i and τ are coordinates on the Lie algebras of T 2 and TG respectively. With the interpretation given in Sect. 2 above this is a mathematically well-defined object, a function on an open subset of the Lie algebra of T2 × TG , which by analytic continuation gives a meromorphic (even rational) function on the whole of the complexified Lie algebra of T2 × TG . Nekrasov argues that one can write Z inst (q, τ, 1 , 2 ) as

inst F (q, τ, 1 , 2 ) Z inst (q, τ, 1 , 2 ) = exp , 1 2 where the function F inst (q, τ, 1 , 2 ) is analytic and regular near 1 , 2 = 0. Furthermore, he claims that F inst |1 ,2 =0 corresponds to the instanton part of the prepotential 3 The full generating function Nekrasov studies Z = Z pert Z inst also has an extra factor, Z pert , the perturbative one-loop contribution, which is of no concern to us.

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of Seiberg-Witten. As the latter can be defined rigorously in terms of periods of certain families of curves, this correspondence gives rise to a remarkable conjecture in geometry. In 2003, this conjecture was proven independently by Nakajima and Yoshioka [NY05a,NY04] and Nekrasov and Okounkov [NO06] for the gauge group SU (n), using very different techniques. In [NS04] Nekrasov and Shadchin indicated how the proof of [NO06] could be adapted to the other classical gauge groups. More recently, a noncomputational proof for all simple gauge groups was given by Braverman and Etingof in [Bra04,BE06]. 5.2. Equivariant volumes of ADHM spaces for SU (n) instantons. 5.2.1. ADHM spaces The space Mon,c we are considering here is the moduli-space of framed SU (n) instantons with instanton number c (cf. [AHDM78,NY04]), constructed as a hyper-Kähler quotient of the vector space Mn,c of ADHM data Mn,c = {(A, B, i, j)}, where A, B, i, j are linear maps between complex hermitian vector spaces V and W of dimension c and n respectively as follows: 8 VT f

A

j

B

i

 W and the quotient is taken by the group action of U (V ) which acts on A and B by conjugation and on i and j by left and right multiplication. Strictly speaking, the moduli space of framed instantons (or, equivalently, framed rank n holomorphic bundles on CP2 with second Chern number c) is the non-singular locus of this quotient. This non-singular locus doesn’t satisfy the conditions to apply either the Prato-Wu theorem (2) or the method described above—in particular the moment map for the torus actions we are considering is not proper, due to the fact that the space is not metrically complete in the Kähler metric. One can however extend (‘partially compactify’) these spaces in various ways: first of all, one can allow for so-called ideal instantons, which are interpreted in differential geometry as being (framed) connections whose curvature is concentrated at certain points. This leads to the Uhlenbeck space, which is the full hyper-Kähler quotient: −1 −1 Mun,c = Mn,c ///(0,0) U (V ) = µC (0)//0 U (V ) = µ−1 R (0) ∩ µC (0)/U (V ).

The Uhlenbeck space is highly singular, however. A better option is to extend the nonsingular locus in another way, which provides a desingularization of the Uhlenbeck space. As the group U (V ) occurring in the hyper-Kähler quotient has characters, we can vary the symplectic reduction or GIT quotient (cf. [Tha96,DH98]) by changing the value of the real moment map. This gives the Gieseker space −1 −1 Mn,c = Mn,c ///(+,0) U (V ) = µ−1 C (0)//+ U (V ) = µR (+) ∩ µC (0)//U (V ). g

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The Gieseker space has a modular interpretation in algebraic geometry as the modulispace of framed torsion-free sheaves on CP2 , i.e. torsion-free sheaves that come with a fixed trivialization on a line ∞ ⊂ CP2 (in physics this space is also thought of as the moduli space of instantons on a non-commutative R4 [NS98]). Of particular relevance g for us is that Mn,c is smooth. 5.2.2. Torus actions on ADHM spaces In instanton counting one is interested in the group action given by T 2 × TU (n) , where TU (n) changes the framing at infinity by acting by the maximal torus of the gauge group (for notational convenience it is easier to allow this to be U (n) rather than SU (n)), and T 2 acts by rescaling C2 ⊂ CP2 . In order to lift this action to the space of ADHM data, we must examine the monad constructed out of ADHM data (see e.g. [Nak99, Chap. 2 Don84]). The monad is a sequence of vector bundle maps over CP2 :



x A−x V ⊗O a = x00 B−x12 ⊕ x0 j b =(−x0 B+x2 x0 A−x1 x0 i ) V ⊗ O(−1) −−−−−−−−→ V ⊗ O −−−−−−−−−−−−−−→ V ⊗ O(1). ⊕ W ⊗O

If a set of ADHM data satisfies the vanishing of the complex moment map, this sequence of bundles on CP2 is actually a complex, and the cohomology E = ker b/im a is a bundle (or sheaf) on CP2 , which indeed satisfies the required triviality on the line at infinity. By examining the effect of T 2 on the monad constructed out of the ADHM data, and by interpreting W ∼ = H 0 ( ∞ , E| ∞ ) as the trivialization on the line at infinity ∞ by means of the Beilinson spectral sequence (see e.g.[Nak99,OSS80]), one can lift the action of TU (n) × T 2 to Mn,c as was done in [NY05a]: for (e1 , e2 ) ∈ T 2 , t ∈ TUn this gives (e1 , e2 , t).(A, B, i, j) = (e1 A, e2 B, it −1 , e1 e2 t j).

(16)

5.2.3. Volumes For this torus action one can now calculate the regularized or equivariant volume of the moduli space. We remark that the original question asks for the equivariant volume of the moduli space with respect to the Kähler form it inherits from being included in the Uhlenbeck space (which is an affine variety). Nevertheless, we can work with the Gieseker space by pulling back the symplectic form from the Uhlenbeck space to the Gieseker space—so we desingularize in algebraic geometry but not in symplectic g geometry. This gives a closed 2-form on Mn,c which is degenerate on the exceptional g g set of Mn,c → Mun,c ; on Mon,c ⊂ Mn,c it is exactly the form we are concerned with. Alternatively, one could think of the degenerate symplectic form as a limit of proper symplectic (even Kähler) forms that degenerate in the limit, and the regularized volume that we are interested in as the limit of the corresponding equivariant volumes for the Gieseker space. Despite the degeneracy, one can speak of moment maps with respect to this form. As usual, the sum of the 2-form and the moment map determines a cohomology class in the Cartan model of equivariant cohomology, and one can look at the localization formula for the formal exponential of this class, as was already remarked, even with the degeneracy, in [AB84]. From our viewpoint, we look at the integral of a function with g respect to a volume form, and as Mn,c \Moc,n has measure zero we have

ω+µ e = eω+µ . g Mon,c

Mn,c

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For the torus action described above, one can now give explicit descriptions of the fixed points (which are all isolated), and the isotropy representations on their tangent g spaces, as was done for Mn,c in e.g. [NY05a]. In particular, they  can be described very nicely by n-tuples of partitions Y = (Y1 , . . . , Yn ) such that i |Yi | = k. As all these fixed points lie above the same unique fixed point in Mun,c under the desingularization g Mn,c → Mun,c , they all take the same value (i.e. 0) under the moment map for the g degenerate symplectic form on Mn,c , and hence we have by the Prato-Wu theorem
g

Mn,c

eω+µ =


 1  eµ(Y ) = = 1, g eT (νY ) eT (νY ) Mn,c Y Y

(17)

where the right-hand side has to be interpreted as a formal integral (of the class 1) defined by a localization formula. This is the viewpoint taken by several authors: the integrals that form the coefficients of the Nekrasov partition function are the formal equivariant g integrals of 1 over the Gieseker space Mn,c . On the other hand, we could apply the technique described in the sections above. In order to do this, however, it is very crucial that one thinks of the integral as an equivariant volume rather than the formal integral of 1, as one now has to use the new fixed points ‘at infinity’ introduced by the cut. The value of these new fixed points under the moment map will not be zero, and through the residue the geometry of the moment map gives the recipe for obtaining their contribution. From (16) we can see that the necessary condition for our cutting construction to work is clearly satisfied: there is a subgroup in T 2 × TU (n) acting with global weight 1 on Mn,c . With this lift we can now implement the calculational method described above. The resulting formula is
Mon,c

eω+µ =

 2 eT (µC ) 1 Resσ+i , c! eT (Mn,c )

where ! " σe − σ f ,

2 = e = f

! " 1 + 2 + σ g − σ h ,

eT (µC ) = 1≤g,h≤c

and eT (Mn,c ) =

! " 1 +σi −σ j

1≤i, j≤c

(2 +σk −σl )

(σm −τo )

! " 1 +2 −σ p +τq ,

1≤k,l≤c 1≤m≤c 1≤o≤n

1≤ p≤c 1≤q≤n

where we use a maximal torus of the form diag(s1 , . . . , sc ) = (eσ1 , . . . , eσc ) for U (c).

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5.3. Equivariant volumes for the other classical gauge groups. We can now try to implement the same method to compute the regularized volumes of the moduli spaces of S O(n) and Sp(n) instantons. We remark that for these other classical groups no equivalent of the Gieseker space is known; hence a direct localization calculation as (17) is not available. The Uhlenbeck spaces do exist here, but as before they are highly singular. We repeat that the integrals we are interested in are the indefinite integrals of n a function, eµT , depending on a parameter in tC with respect to a volume form ωn! on the non-singular locus of these Uhlenbeck spaces. When implementing our method, two minor issues need to be addressed: we need to treat the singularities differently because of this lack of a Gieseker space, and the lifting of the torus action to the space of ADHM data is not automatic. The ADHM construction for gauge groups Sp(n) and S O(n) was discussed in [Don84] and described in greater detail in e.g. [BS00]. In both of these cases the groups occurring by which one has to quotient are simple, and hence one cannot hope to obtain a desingularization of the hyper-Kähler quotient by means of a variation of GIT quotient as was the case for the gauge group SU (n). However, in [Kir85], Kirwan describes a method for constructing desingularizations of singular quotients M//G by blowing up certain  and then constructing the desingularizasubvarieties in M to obtain a new space M,  tion of M//G as M//G. One could therefore use this approach to obtain an equivariant desingularization of the Uhlenbeck spaces for gauge groups Sp(n) and S O(n), and use these to calculate the equivariant volumes. Nevertheless, since an interpretation of these desingularizations as moduli spaces is at least a priori lacking (see however [Fre05] for related discussions), determining the fixed point data in the hope of applying a direct localization formula is a non-trivial matter. In [JKKW03] the Kirwan desingularization construction was used to develop a residue formula as for intersection pairings in the intersection cohomology of a singular GIT quotient. While we are not directly interested in the intersection cohomology of the Uhlenbeck spaces, we can take a similar approach for calculating the equivariant volumes of the ADHM spaces for symplectic and special orthogonal gauge groups, to obtain the equivalent formula for these volumes to the one derived in the previous section for SU (n), by considering a degenerate form on the Kirwan desingularization.

5.3.1. Sp(n) Following [BS00], we can describe the ADHM construction of the Uhlenbeck space for gauge group Sp(n) as a hyper-Kähler quotient as follows: look at the diagram of linear maps

A

j

&

x V D B DD DD D Φ DD !

WC CC CC C J CC! W∗

j∗

(18)

VJ ∗

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where Φ is a (fixed) real structure on V , i.e. an isomorphism V → V ∗ such that J ∼ =

Φ ∗ = Φ, and J is a (fixed) symplectic structure on W (W → W ∗ , J ∗ = −J ). The Sp

space of ADHM data Mn,c here consists of {(A, B, j)}, with the extra conditions that Φ A, Φ B ∈ S 2 V ∗ , and the group divided by O(V ), is determined by Φ. We can write the vanishing of the complex moment map as Φ[A, B] − j ∗ J j = 0, and we can again put together a monad:



x A−x V ⊗O a = x00 B−x12 ⊕ x0 j b =( x2 Φ−x0 B ∗ Φ −x1 Φ+x0 A∗ Φ −x0 j ∗ J ) V ⊗ O(−1) −−−−−−−−→ V ⊗ O −−−−−−−−−−−−−−−−−−−−−−→ V ⊗ O(1). ⊕ W ⊗O (19)

Composing this with Φ on the right gives a self-dual monad, whose cohomology is an Sp(n) instanton. As in the case of SU (n), we are again interested in a torus action given by changing the framing at infinity and rescaling C2 ⊂ CP2 . The former action is readily lifted to Sp Mn,c : t ∈ TSp(n) ⇒ t.(A, B, j) = (A, B, t j). As for the scaling, things become a bit more cumbersome—a lift of the torus action to the space of ADHM data does not seem to be available. However, we can still proceed as before if we temper our ambition and only try to lift the action with weight two—that is to say the action induced by the scaling of C2 ⊂ CP2 given by (e1 , e2 ).(x0 , x1 , x2 ) = (x0 , e12 x1 , e22 x2 ). Indeed, if we now introduce the bundle isomorphism ⎛e ⎞ V ⊗O V ⊗O   1 e v1 v1 ⊕ ⊕ ⎜ e22 ⎟ φ : V ⊗ O −→ V ⊗ O : v2 → ⎝ e v2 ⎠, 1 ⊕ ⊕ w w W ⊗O W ⊗O then by using this isomorphism we obtain



x0 A−e1−2 x1 φ 1 x0 (e12 A)−x1 im x0 B−e2−2 x2 ∼ = im x0 (e22 B)−x2 e1 e2 x0 (e1 e2 j) x0 j ker ker

!

e2−2 x2 Φ−x0 B ∗ Φ −e1−2 x1 Φ+x0 A∗ Φ x0 j

∼ =

and

φ

" ,

1 x Φ−x0 (e22 B)Φ −x1 Φ+xo (e22 B)∗ Φ x0 (e1 e2 j)). e1 e2 ( 2

Hence we can lift the scaling to (e1 , e2 ).(A, B, j) = (e12 A, e22 B, e1 e2 j). As we are just interested in calculating the equivariant volumes there is no problem in lifting a ‘higher weight’—as the equivariant volumes for the different weights are related

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by a scaling of the Lie algebra. Again we can remark that there is a U (1) in T whose action is given by a constant global weight (two, in this case): U (1) → T = T 2 × TSp(n) : s → (s, s, 1). If we now apply the symplectic cut with respect to this circle action — using a weight 2 action however on the copy of C used in the symplectic cut — to compactify we again get a projective space,   Sp Sp Sp Mn,c = Mn,c PMn,c , λ

··

·

and we can implement the method as before. Two remarks here: there is an extra factor 1 2 needed to account for the fact that the moduli space is the hyper-Kähler quotient by O(V ) rather than just S O(V ) (the residue formula assumes that the group is connected), and secondly, because we only lifted the ‘weight 2’ action the space of ADHM data the corresponding variables 1 , 2 need to be rescaled. For the calculations below we will fix isomorphisms V ∼ = Cc , W ∼ = C2n , and

  0 1 0 I represent Φ by the off-diagonal matrix , and J by −I 0n . Using maximal n 1 0 −1 tori of the form diag(s1 , . . . , sm , sm , . . . , s1 ) for even c = 2m, diag(s1 , . . . , sm , 1, sm−1 , . . . , s1−1 ) for odd c = 2m + 1 for S O(c) and diag(t1 , . . . , tn , t1−1 , . . . , tn−1 ) for Sp(n), we can easily write down all the weights involved. Implementing the calculational method we get
2 1 1 σi  eT (µC ) ω+µ Res e = , + Sp Sp 2 |W | Mn,c eT (Mn,c ) where for c = 2m even |W | = 2m−1 m!, 2 =

(σi2 − σ j2 )2 , i< j

 "2   "2  ! ! , (1 + 2 )2 − σi + σ j (1 + 2 )2 − σi − σ j

eT (µC ) = (1 + 2 )m i< j

and

⎛ ⎝

Sp

eT (Mn,c ) = k=1,2



(k + σi − σ j ) i, j

i

⎞

(k )2 − (σi + σ j )2 ⎠

i≤ j

l







2 2 1 + 2 1 +  2 2 2 + τl − σi − τl − σi , 2 2

and for c = 2m + 1 odd |W | = 2m m!,

Equivariant Volumes of Quotients & Instanton Counting

2 =

849

(σi2 − σ j2 )2

σi2 ,

i< j

i

  (1 + 2 )2 − σi2

eT (µC ) = (1 + 2 )m i

 "2   "2  ! ! , (1 + 2 )2 − σi + σ j (1 + 2 )2 − σi − σ j

i< j

and

⎝k k=1,2

i

l

⎞   (k )2 − (σ1 + σ2 )2 ⎠

⎛

  Sp eT Mn,c =

(k2 − σi2 ) i

(k + σi − σ j ) i, j

i≤ j







2 2 1 +  2 1 +  2 1 + 2 2 2 2 2 + τl − σi − τl − σi − τl . 2 2 2 l

5.3.2. SO(n) For the gauge group S O(n) the ADHM construction goes similarly: again using the diagram (18), where in this case Φ : V → V ∗ is a symplectic structure, and S O consists of the triples J : W → W ∗ is a real structure, the space of ADHM data Mc,n % of linear maps (A, B, j), this time with the extra condition that Φ A, Φ B ∈ 2 V ∗ . With this understood, (and the vanishing of the complex moment map again being Φ[A, B] − j ∗ J j = 0) nominally the same monad (19) again gives the desired bundle, from which we can directly see that the same lift of the torus action works (again with ‘double weight’ for the 2-torus that scales): (e1 , e2 , t).(A, B, j) = (e12 A, e22 B, e1 e2 t j) for (e1 , e2 ) ∈ T 2 and j ∈ TS O(n) . With maximal tori written as before it is again just a matter of identifying the weights S O and the complex moment map. We obtain again for all tori involved on the space Mn,c
2 1 σi  eT (µC ) , eω+µ = Res + SO ) SO c!2c eT (Mn,c Mn,c where 2 =

(σi2 − σ j2 )2 i< j

i

  (1 + 2 )2 − (σi − σ j )2

n

eT (µC ) = (1 + 2 ) 2 i< j

for even n = 2m 

(2σi )2 ,

i≤ j

⎛



k=1,2

i, j

i

⎞   k2 − (σi + σ j )2 ⎠

! " k + σi − σ j

⎝

SO eT Mn,c =

  (1 + 2 )2 − (σi + σ j )2 ,

i< j

l





2 2 1 + 2 1 +  2 + τl − σi2 − τl − σi2 , 2 2

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and for odd n = 2m + 1 ⎛   SO ⎝ = eT Mn,c k=1,2

i

⎞

    +  2 1 2 2 2⎠ 2 k − (σi + σ j ) − σi 2 i, j i< j i 



2 2 1 + 2 1 +  2 + τl − σi2 − τl − σi2 . 2 2 ! " k +σi −σ j

l

The above formulas exactly correspond to those found in [NS04, §5.3]. 5.4. Comparison. We will now compare the calculational strategy outlined above with the work of Nekrasov-Shadchin [NS04]. In [NY05a, §4] a K -theoretic interpretation of the coefficients of Nekrasov’s partition g function (15) is given as follows. Let E be a T -equivariant coherent sheaf on Mn,c and look at 2nr  g (−1)i chH i (Mn,c , E),

(20)

i=0

where the character ch denotes the trace of the representation, & defined as a Hilbert series for a representation M with weight decomposition M = µ Mµ as chM =



dim Mµ eµ ,

µ

where the sum is over all characters. For the case of the structure sheaf O, (20) reduces to chH 0 (Muc,n , O), and the following K -theoretic localization formula is explained [NY05a, Prop. 4.1]: 



g

(−1)i chH i (Mn,c , E) =

i

! g "T F⊂ Mn,c

i∗ E %F ∗ , −1 ν F

% where as usual the sum in the right-hand side is over all fixed points, and −1 stands for the alternating sum of exterior powers as an element in the equivariant K -group K T (for all details see [NY05a]). For the case of the structure sheaf O this reduces to chH 0 (Mun,c , O) =

 ! g "T F⊂ Mn,c

%

1 ∗ −1 ν F

.

In the (co)-homological setting, in the mathematics literature on  instanton counting [NY05a,NY05b,Bra04] most authors use the equivariant integral 1 as the coefficients for the generating  function, where the integral is defined in practice through a localization theorem 1 = F e(ν1F ) . In [NY05a] the link between this and the K -theoretic approach is made by
 β 2nc  1 % = lim lim β 2nc chH 0 . 1 = ∗ = β→0 g e(ν F) β→0 Mn,c −1 (ν F )

Equivariant Volumes of Quotients & Instanton Counting

851

One can think of the limit β → 0 as formally inducing a multiplication by the inverse Todd class prescribed by a Riemann-Roch theorem; this has the effect of changing the denominators of the localization theorem for K -theory to those of (co)homology. In [NS04], Nekrasov–Shadchin use the same philosophy to calculate the partition functions for all classical groups. They physically interpret the limit β → 0 as arising from considering the 4-dimensional theory as the limit of a 5-dimensional theory compactified on a circle of radius β. Then they calculate chH 0 (Mun,c ) by means of the ADHM construction as follows. Let ρ1 , . . . , ρ. be the characters of the T × TG action on Mc,n . Then the T × TG -equivariant character is 1 . chH 0 (Mn,c , O) =  (1 − ρi ) Furthermore, the T × TG action on µC is homogeneous, say with weights ν1 , . . . , ν L for L = dim G. Hence we have  (1 − νi ) 0 −1 . chH (µC (0), O) =  (1 − ρi ) −1 In order to get the character over µC (0)//G, we need to get the G-invariant part of this. The projection onto the G-invariant part is given by averaging over the whole of G,
1 0 −1 chH (µC (0)//G, O) = chH 0 (µ−1 C (0), O), volG G

but by the Weyl integral formula this can be reduced to an integral over the maximal torus,
1 chH 0 (µ−1 (0)/ /G, O) = chH 0 (µ−1 C C (0), O). |WG |volTG TG  Now factor the torus as T = U (1) and break up the integral into circle integrals. The integrand in each lives on C∗ , and one can think of these as contour integrals in the plane. These contour integrals can be calculated by Cauchy’s theorem, for each integral keeping the other variables as generic constants. Finally, take the limit β → 0+ to reduce everything to cohomology. Nekrasov-Shadchin interpret this limit as a contour integral of a meromorphic top degree form over the complexified Lie algebra. In either case the actual evaluation happens by means of iterated residues, for each variable choosing a half-space in which to consider the poles. It is thus that the formula gives exactly the same result as our method outlined above. It is interesting to remark that the non-compactness manifests itself in different ways in the two methods. Let us illustrate this by calculating the regularized volume of the simplest hyperKähler quotient, C4 ///C∗ , where s ∈ C∗ acts as s.(x1 , x2 , x3 , x4 ) = (sx1 , sx2 , s −1 x3 , s −1 x4 ). Furthermore we consider a T 2 action on C4 by (t1 , t2 )(x1 , x2 , x3 , x4 ) = (t1 x1 , t2 x2 , t2 x3 , t1 x4 ). As the complex moment map µC for the C∗ action is x1 x3 + x2 x4 , the T 2 action indeed 2 4 preserves µ−1 C (0), and furthermore the moment map for the T action on C clearly has a component that is proper and bounded below, hence all the conditions are satisfied.

852

J. Martens

For the calculation according to Nekrasov-Shadchin we need to find the Hilbert-series of O on µ−1 C (0), which is 1 − t1 t2 . (1 − st1 )(1 − st2 )(1 − t2 s −1 )(1 − t1 s −1 ) ∗ In order to calculate the Hilbert series over the quotient µ−1 C (0)//C we then have to take the contour integral ' 1 − t1 t 2 1 ds ch = 2πi s (1 − st1 )(1 − st2 )(s − ts2 )(1 − ts1 ) 1 − t1 t2 1 − t1 t 2 . = t1 + 2 2 (1 − t1 t2 )(1 − t2 )(1 − t2 ) (1 − t1 )(1 − t1 t2 )(1 − tt21 )

Finally set t1 = e−βτ1 , t1 = e−βτ2 and s = e−βσ and take limβ→0 β 2 ch, which gives τ1 + τ2 τ1 + τ2 1 + = . (τ2 + τ1 )(2τ2 )(τ1 − τ2 ) 2τ1 (τ1 + τ2 )(τ2 − τ1 ) 2τ1 τ2

(21)

On the other hand, in the method we have outlined above, we can use the symplectic cut on C4 with respect to the diagonal T 1 ⊂ T 2 , which gives (C4 )λ = CP3 C4 . For clarity we shall not take the calculational shortcut discussed in Sect. 4.2, but actually go through the procedure of making the symplectic cut and taking the limit as λ → ∞. There are five fixed points for the C∗ × T 2 action on this space, of which we only need to consider [1 : 0 : 0 : 0] and [0 : 1 : 0 : 0] for the Jeffrey-Kirwan residue formula. Applying this gives eλ(σ +τ1 ) (−2σ + τ2 − τ1 ) (−σ − τ1 )(τ2 − τ1 )(τ2 − τ1 − 2σ )(−2σ ) eλ(σ +τ2 ) (−2σ + τ1 − τ2 ) . + jkres+σ (τ1 − τ2 )(−σ − τ2 )(−2σ )(τ1 − τ2 − 2σ )

jkres+σ

As usual, we are only interested in the terms that survive the λ → ∞ limit on the open cone of Lie(T 2 ); hence for each of the two terms above we only need to consider the residue at the pole that cancels the exponent, which gives the exact same expression as (21). A. Proof of Theorem 3 One could essentially try to adapt any proof of Theorem 3 to the equivariant setting. The approach we take here is almost completely based on [JK05], and we restrict ourselves to commenting on how to adjust it. Proof. We can break up the proof in two steps: first reduce the case where G is a general compact connected Lie group to the corresponding statement for a maximal torus TG of G, and then prove the theorem just for the case when G is a torus. The first step is achieved by the abelianization theorem:

Equivariant Volumes of Quotients & Instanton Counting

853

Theorem 5. Let M as above be equipped with commuting Hamiltonian group actions of compact connected Lie groups G, H , let TG be a maximal torus of G, and let a be an ∗ element of HG×H M. Then in H H∗ (∗),

1 κG (a) = κTG ( 2 a). |W | G M//G M//TG This theorem is a corollary of the Jeffrey-Kirwan residue theorem [JK95b], but was also proven directly by Martin [Mar]. The original theorem of Martin was not formulated in an equivariant setting (i.e. no H was present), but it suffices to remark here that the proof given in [Mar] is valid without modifications in the equivariant case as well. Now, to prove Theorem 3 for a torus G = T , we will first outline the strategy of the proof given in [JK05], and then comment on how it can be used for our equivariant purposes. Begin by choosing a cone Λ, as follows. Look at the weights of the torus T at all the components of M T . Each of these determines a hyperplane in t. Choose a connected component of the complement of their union; call this Λ. Then look at a polyhedral cone containing the dual cone, say Λ∗ ⊂ Σ. The next step is to take the symplectic cut MΣ of M with respect to Σ—having chosen Σ in such a way that this is smooth. For technical purposes we here first alter our moment map for the T -action a little by choosing a point p ∈ Σ, and replacing µT by µ = µT − p. As we inherit T. a commuting action of T and H on this, we can investigate the fixed-point set MΣ The fixed-point components come in three flavors: ‘old ones,’ i.e. components F with ˜ ∈ ∂(Σ), and the component µ(F) ∈ Σ 0 , ‘new ones,’ i.e. components F˜ with 0 = µ( F) at the vertex of Σ, which we can identify with (M//p T ). Applying the localization theorem to MΣ gives


eω˜ η = MΣ


eω˜ i ∗ η 
eω˜ i ∗˜ ηΣ
eω˜ i ∗ η F F + + . F˜ eT1 (νG ) F eT (ν F ) M//p T eT1 (ν M//T ) F

(22)

F˜

As a function on t⊕h this is ‘holomorphic’ (i.e. the analytic continuation is holomorphic on the complexification tC ⊕ hC ), and for such functions it is proven by Jeffrey and Kogan [JK05, Lemma 3.3] that the residue with respect to Λ and −Λ give the same results. This leads to the equality ⎛ ⎞ ω˜ i ∗ η 
eω˜ i ∗ η 
eω˜ i ∗˜ ηΣ
e Λ⎝ F F ⎠ [d x] JKRes + + eT (ν F ) eT1 (ν M//T ) ˜ eT1 (νG ) F M/ / T F p F F˜ ⎛ ⎞


eω˜ i ∗ η ∗ ω ˜ ω ˜ ∗   e iF η e i η ⎠ F˜ Σ + + = JKRes−Λ ⎝ [d x]. e (ν ) e (ν ) e ˜ F T F M//p T T1 (ν M//T ) F T1 G F

F˜

(23) The modification of the moment map from µT to µ is done exactly to ensure that the residue operation is valid for all the terms appearing above. Next it is shown in [JK05, §5.2] that in fact, for small enough  > 0 and with the choices of Λ, Σ and µ as above, 4 terms in this equality are zero, leading to

eω˜ i ∗ η eω˜ i ∗ η F JKRes−Λ [d x] = JKResΛ [d x]. M//p T eT1 (ν M//T ) F eT (ν F ) F

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J. Martens

Finally, take the limit as  → 0. By identifying the weights of T on ν M//T (which is a bundle of rank dim t), one obtains lim+ JKRes−Λ

→0


M//T

eω˜ i ∗ η [d x] = eT1 (ν M//T )


M//T

eω κη,

(24)

and finally this gives
M//T



κ(η)eω =

JKResΛ

F⊂M T


F

eω˜ i F∗ η [d x]. eT (ν F )

This is how Theorem 2 is proved in the non-equivariant case in [JK05]. Now in order to make this valid in the presence of the extra group action of H , we remark that because the actions of T and H on M commute, the action of H descends to MΣ . Now we can use our approximation of the Borel model by smooth finite-dimensional spaces M × E i H/H , determined by finite-dimensional approximations E i H → Bi H to the classifying space E H → B H . These spaces are not symplectic, but they still are Poisson manifolds, with the fibers of M × H E i H → Bi H as the symplectic leaves. Furthermore, as the moment map µG : M → g∗ -action is invariant under H , we get moment maps M ×G E i H → g∗ in the sense of Poisson geometry; see e.g. [OR04]. Also, as the proof of the localization theorem used in (22) essentially only uses the functoriality of the integral as a push-forward (see [AB84]), we can apply it to other (proper) push-forwards as well. In particular, consider (MΣ × E i H )/H → Bi H.  Denote the corresponding push-forward by i , so that
i

: HT∗G (((MΣ × E i H )/H ) → HT∗G (Bi H ) = HT∗G (∗) ⊗ H ∗ (Bi H )

and
i

: HT∗G ((F × E i H )/H ) → HT∗G (∗) ⊗ H ∗ (Bi H ).

The equivalent of the localization formula is then

 i F∗ α . α= i i eT (ν F×Ei H ) T F×E i H H

⊂

M ×E i H H

H

We can now apply the residue operation JKResΛ to both sides (or rather the obvious extension of the residue from k(t) to the various k(t) ⊗ H ∗ (Bi H )), and we observe that for the same reasons as in [JK05], the equivalences of (23) and the cancellations of the four terms are still valid—this time interpreted as happening in H ∗ (Bi H ). The same is the case for the equivalent statement of (24): essentially the only thing we need to remark for this is that the weights for the action of T occurring on the normal bundles of M//T iH . in MΣ are the same as the weights for the normal bundle of M//TH×Ei H in MΣ ×E H

Equivariant Volumes of Quotients & Instanton Counting

855

Furthermore, as all the spaces are finite-dimensional, the inverses of the Euler classes exist in the usual way. So we have

η [d x]. κ(η) = −JKResΛ e (ν F×E i,M iH ) T i T F⊂M

H

Now we want to take the limit i → ∞ to obtain the desired H -equivariant result. We need to observe in which ring we allow all of the manipulations to take place. Begin by remarking that in the proof of the abelian localization theorem [AB84], which was the basis of all of our localization results, one had to kill some of the torsion in the equivariant cohomology ring, but tensoring with the full fraction field k(t) isa bit excessive. In particular, we can do with just inverting all products of linear factors i βi , with βi ∈ t∗ . Doing this has the advantage that we still have a grading on the ring we ∗ ∗
∗ ∗

obtain (say H T (∗)). This gives a bigrading on HT ×H (F) = HT (∗) ⊗ H H F for all T F ⊂ M . Now complete these rings by means of the filtration · · · ⊂ Rk+1 ⊂ Rk ⊂ Rk−1 ⊂ . . . , where Rk consists of finite sums of elements of bidegree (i, j) such that i + 2 j ≥ k. This completion has all the desired properties (though it is of course not unique as such): it contains the formal cohomology classes eω˜ , and it allows us to invert the equivariant Euler classes — which because  of the splitting principle we can write without loss of generality as eT ×H (ν F ) = i (βi + c1 νi,F ) with βi ∈ t, c1 νi,F ∈ H H2 F—as 1 = eT ×H (ν F )

i

∞ c1 (νi,F ) j 1  − . βi βi j=0

The residue JKResΛ of an integral of such a class will lie in the usual completion of H H∗ (∗) by degree. This completes the proof of Theorem 3.  Acknowledgements. I would like to thank Lisa Jeffrey, Mikhail Kogan, Nikita Nekrasov, Michèle Vergne and Peter Woit for useful conversations. The bulk of the work presented here was done while I was a graduate student at Columbia University, and I would like to thank my advisor Michael Thaddeus for continuous guidance and encouragement, as well as a careful reading of the final version of this article. Finally I would like to add that this work was largely inspired by reading the early paper [MNS00]. This article was partially written during a stay at the Max-Planck-Institut für Mathematik, Bonn, whose hospitality and support is gratefully acknowledged.

References [AB84] [AHDM78] [BE06] [Bra04]

Atiyah, M.F., Bott, R.: The moment map and equivariant cohomology. Topology 23(1), 1–28 (1984) Atiyah, M.F., Hitchin, N.J., Drinfel’d, V.G., Manin, Yu.I.: Construction of instantons. Phys. Lett. A 65(3), 185–187 (1978) Braverman, A., Etingof, P.: Instanton counting via affine Lie algebras. II. From Whittaker vectors to the Seiberg-Witten prepotential. In Studies in Lie theory, Volume 243 of Progr. Math., Boston, MA: Birkhäuser Boston, 2006, pp. 61–78 Braverman, A.: Instanton counting via affine Lie algebras. I. Equivariant J -functions of (affine) flag manifolds and Whittaker vectors. In Algebraic structures and moduli spaces, Volume 38 of CRM Proc. Lecture Notes, Providence, RI: Amer. Math. Soc., 2004, pp. 113– 132

856

[BS00] [BV82] [BV99] [CB01] [DH82] [DH98] [DHKM02] [Don84] [Fre05] [GK96] [GLS96] [JK95a] [JK95b] [JK97] [JK98] [JK05] [JKKW03] [Kir84] [Kir85] [Ler95] [Mar] [Mei06] [MNS00] [Nak99] [Nek03] [NO06] [NS98] [NS04] [NY04]

J. Martens Bryan, J., Sanders, M.: Instantons on S 4 and CP2 , rank stabilization, and Bott periodicity. Topology 39(2), 331–352 (2000) Berline, N., Vergne, M.: Classes caractéristiques équivariantes. formule de localisation en cohomologie équivariante. C. R. Acad. Sci. Paris Sér. I Math. 295(9), 539–541 (1982) Brion, M., Vergne, M.: Arrangement of hyperplanes. i. rational functions and Jeffrey-Kirwan residue. Ann. Sci. École Norm. Sup. (4) 32(5), 715–741 (1999) Crawley-Boevey, W.: Geometry of the moment map for representations of quivers. Comp. Math. 126(3), 257–293 (2001) Duistermaat, J.J., Heckman, G.J.: On the variation in the cohomology of the symplectic form of the reduced phase space. Invent. Math. 69(2), 259–268 (1982) Dolgachev, I.V., Hu, Yi.: Variation of geometric invariant theory quotients. Inst. Hautes Études Sci. Publ. Math. (87):5–56, (1998). with an appendix by N. Ressayre Dorey, N., Hollowood, T.J., Khoze, V.V., Mattis, M.P.: The calculus of many instantons. Phys. Rep. 371(4–5), 231–459 (2002) Donaldson, S.K.: Instantons and geometric invariant theory. Commun. Math. Phys. 93(4), 453–460 (1984) Freiermuth, H.-G.: On partial compactifications of the space of framed vector bundles on the projective plane. PhD thesis, Columbia University, 2005 Guillemin, V., Kalkman, J.: The Jeffrey-Kirwan localization theorem and residue operations in equivariant cohomology. J. Reine Angew. Math. 470, 123–142 (1996) Guillemin, V., Lerman, E., Sternberg, S.: Symplectic fibrations and multiplicity diagrams. Cambridge: Cambridge University Press, 1996 Jeffrey, L.C., Kirwan, F.C.: Intersection pairings in moduli spaces of holomorphic bundles on a Riemann surface. Electron. Res. Announc. Amer. Math. Soc. 1(2), 57–71 (electronic), (1995) Jeffrey, L.C., Kirwan, F.C.: Localization for nonabelian group actions. Topology 34(2), 291–327 (1995) Jeffrey, L.C., Kirwan, F.C.: Localization and the quantization conjecture. Topology 36(3), 647–693 (1997) Jeffrey, L.C., Kirwan, F.C.: Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface. Ann. of Math. (2) 148(1), 109–196 (1998) Jeffrey, L., Kogan, M.: Localization theorems by symplectic cuts. In: The breadth of symplectic and Poisson geometry, Volume 232 of Progr. Math., Boston, MA: Birkhäuser Boston, 2005, pp. 303–326 Jeffrey, L.C., Kiem, Y.-H., Kirwan, F., Woolf, J.: Cohomology pairings on singular quotients in geometric invariant theory. Transform. Groups 8(3), 217–259 (2003) Kirwan, F.C.: Cohomology of quotients in symplectic and algebraic geometry, Volume 31 of Mathematical Notes. Princeton, NJ: Princeton University Press, 1984 Kirwan, F.C.: Partial desingularisations of quotients of nonsingular varieties and their Betti numbers. Ann. of Math. (2) 122(1), 41–85 (1985) Lerman, E.: Symplectic cuts. Math. Res. Lett. 2(3), 247–258 (1995) Martin, S.K.: Symplectic quotients by a nonabelian group and by its maximal torus. To appear in Annals of Math., http://arXiv.org/list/math.SG/0001001, 2000 Meinrenken, E.: Equivariant cohomology and the Cartan model. In: Françoise, J.-P., Naber, G.L., Tsun, T.S. editors, Encyclopedia of mathematical physics. Oxford: Academic Press/Elsevier Science, 2006 Moore, G., Nekrasov, N., Shatashvili, S.: Integrating over Higgs branches. Commun. Math. Phys. 209(1), 97–121 (2000) Nakajima, H.: Lectures on Hilbert schemes of points on surfaces, Volume 18 of University Lecture Series. Providence, RI: Amer. Math. Soc. 1999 Nekrasov, N.A.: Seiberg-Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7(5), 831–864 (2003) Nekrasov, N.A., Okounkov, A.: Seiberg-Witten theory and random partitions. In: The unity of mathematics, Volume 244 of Progr. Math., Boston, MA: Birkhäuser Boston, 2006, pp. 525– 596 Nekrasov, N., Schwarz, A.: Instantons on noncommutative R4 , and (2, 0) superconformal six-dimensional theory. Commun. Math. Phys. 198(3), 689–703 (1998) Nekrasov, N., Shadchin S.: ABCD of instantons. Commun. Math. Phys. 252(1–3), 359–391 (2004) Nakajima, H., Yoshioka, K.: Lectures on instanton counting. In: Algebraic structures and moduli spaces, Volume 38 of CRM Proc. Lecture Notes, Providence, RI: Amer. Math. Soc. 2004, pp. 31–101

Equivariant Volumes of Quotients & Instanton Counting

[NY05a] [NY05b] [OR04] [OSS80] [Par00] [PW94] [SW94] [Tha96] [Ver96]

857

Nakajima, H., Yoshioka, K.: Instanton counting on blowup, i. 4-dimensional pure gauge theory. Invent. Math. 162(2), 313–355 (2005) Nakajima, H., Yoshioka, K.: Instanton counting on blowup. ii. K -theoretic partition function. Transform. Groups 10(3–4), 489–519 (2005) Ortega, J.-P., Ratiu, T.S.: Momentum maps and Hamiltonian reduction, Volume 222 of Progress in Mathematics. Boston, MA: Birkhäuser Boston Inc., 2004 Okonek, C., Schneider, M., Spindler, H.: Vector bundles on complex projective spaces, Volume 3 of Progress in Mathematics. Boston, MA: Birkhäuser Boston, 1980 Paradan, P.-E.: The moment map and equivariant cohomology with generalized coefficients. Topology 39(2), 401–444 (2000) Prato, E., Wu, S.: Duistermaat-Heckman measures in a non-compact setting. Comp. Math. 94(2), 113–128 (1994) Seiberg, N., Witten, E.: Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory. Nucl. Phys. B 426(1), 19–52 (1994) Thaddeus, M.: Geometric invariant theory and flips. J. Amer. Math. Soc. 9(3), 691–723 (1996) Vergne, M.: A note on the Jeffrey-Kirwan-Witten localisation formula. Topology 35(1), 243–266 (1996)

Communicated by N.A. Nekrasov