Communications in Mathematical Physics - Volume 199

Commun. Math. Phys. 199, 1 – 24 (1998) Communications in Mathematical Physics © Springer-Verlag 1998 Grassmannian and...

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Commun. Math. Phys. 199, 1 – 24 (1998)

Communications in

Mathematical Physics © Springer-Verlag 1998

Grassmannian and String Theory Albert Schwarz Department of Mathematics, University of California, Davis, CA 95616, USA. E-mail: [email protected] Received: 19 December 1996 / Accepted: 27 March 1998

Abstract: The infinite-dimensional Grassmannian manifold contains moduli spaces of Riemann surfaces of all genera. This well known fact leads to a conjecture that nonperturbative string theory can be formulated in terms of the Grassmannian. We present new facts supporting this hypothesis. In particular, it is shown that Grassmannians can be considered as generalized moduli spaces; this statement permits us to define corresponding “string amplitudes” (at least formally). One can conjecture that it is possible to explain the relation between non-perturbative and perturbative string theory by means of localization theorems for equivariant cohomology; this conjecture is based on the characterization of moduli spaces, relevant to string theory, as sets consisting of points with large stabilizers in certain groups acting on the Grassmannian. We describe an involution on the Grassmannian that could be related to S-duality in string theory. 0. Introduction It is clear now that all versions of string theory are closely related.One should expect that all of them can be obtained from a unifying theory, where strings are not considered as fundamental objects (strings should be on equal footing with membranes). The present paper arose from attempts to understand the structure of the unifying theory. If we believe that the analysis of this hypothetical theory is related to calculation of integrals over infinite-dimensional supermanifolds then we can suggest the following picture. The integrand can have various odd symmetries. Under certain conditions the existence of odd symmetry leads to localization; in other words, one can replace the integration over the whole supermanifold with an integration over some part of it. (See [11, 13, 14, 15] for localization theorems in the framework of supergeometry and of equivariant cohomology.) Different odd symmetries lead to different localizations and therefore to theories that could look completely unrelated. A very natural candidate for the infinite-dimensional supermanifold arising in the universal theory is the infinite-dimensional (super)Grassmannian. I hope that appropri-

2

A. Schwarz

ate integrals could be localized to the Krichever locus; such a localization would lead immediately to relation to the string theory. I have only tentative results in this direction. Namely, I can prove that the Krichever locus consists of points having large stabilizers with respect to a group 0 acting on the Grassmannian. Well-known localization theorems for equivariant cohomology can be formulated as a statement that the “size” of the contribution of a point to this cohomology is related to the size of the corresponding stabilizer. These theorems are proved for compact groups therefore the use of them for the group 0 is not legitimate. However, formal application of localization theorems to 0-equivariant cohomology of the Grassmannian shows that this cohomology should be localized in some sense on the Krichever locus. It is essential to mention that there exists a multidimensional analog of Krichever construction studied in [20]. I hope that using this analog one can relate the Grassmannian not only to strings, but also to membranes. I was able to prove that Grassmannians can be considered as generalized moduli spaces, containing many other moduli spaces, and to describe a generalization of string theory that is defined in terms of the Grassmannian. Bosonic string theory is closely related to two-dimensional conformal field theory, other versions of string theory are related to corresponding generalizations of conformal field theory. We will use Segal’s approach to conformal field theory [9]. Let us consider the moduli space Pχ,n of (possibly, disconnected) compact conformal two-dimensional manifolds of Euler characterictic χ having a boundary consisting of n components (i.e. closed two-dimensional manifolds of Euler characteristic χ − n with n holes). We assume that boundaries of holes are parametrized and the holes are ordered. One can define natural maps νn,n0 : Pχ,n × Pχ0 ,n0 → Pχ+χ0 ,n+n0 and σ (n) : Pχ,n → Pχ,n−2 (the first map corresponds to the disjoint union of manifolds, the second one is constructed by means of pasting together of the boundary of the (n − 1)th hole and the boundary of nth hole). The symmetric group Sn acts naturally on Pχ,n (reordering the holes). Sometimes it is useful to take into account that pasting together boundaries of two holes we can rotate one of the boundaries. Then we obtain a map Pχ,n × S 1 → Pχ,n−2 . The situation when we have some spaces Pn with action of the symmetric group Sn and maps νn,n0 : Pn × Pn0 → Pn+n0 , σ (n) : Pn → Pn−2 appears quite often. If these data satisfy some compatibility conditions we will say that the spaces Pn constitute a MO (a modular operad). The spaces Pn = ∪χ Pχ,n form a MO; we reflect the fact Pn has such a decomposition, saying that we have a graded MO. If we include in our data also a map Pn × S 1 → Pn−2 , we will talk about an EMO (equivariant MO). The spaces Pχ,n constitute a graded EMO. If A is a linear space with inner product we can define a MO considering tensor 0 0 powers A⊗n and natural maps A⊗n × A⊗n → A⊗(n+n ) and A⊗n → A⊗(n−2) . An algebra over the MO Pn is defined as a collection of maps αn : Pn → A⊗n , that are compatible with the structures of MO in Pn and A⊗n . Conformal field theory can be defined as an algebra over the MO Pn = ∪Pχ,n , described above. (More precisely, in this case the maps αn are defined only up to a constant factor.) A superconformal field theory can be defined in a similar way. Moduli spaces of conformal manifolds (of complex curves) should be replaced in this case with moduli spaces of superconformal manifolds (super Riemann surfaces). We will be able to construct a lot of MOs starting with an infinite-dimensional Grassmannian. We will say that a semi-infinite structure in the Hilbert space H is specified by a decomposition H = H+ ⊕ H− and a unitary involution K interchanging H+ and H− .

Grassmannian and String Theory

3

The Segal–Wilson modification of the Sato Grassmannian Gr(H) can be defined as an infinite-dimensional manifold consisting of such subspaces V ⊂ H that V is close to H+ in some sense (one can require for example that the projection V → H+ is a Fredholm operator and the projection V → H− is a Hilbert–Schmidt operator). We will prove that the sequence of spaces Grn = Gr(H n ) can be considered as MO. Moreover, we will give conditions when Grn constitute an EMO. Using the well known fermionic construction of the Grassmannian we will obtain an algebra over this MO. Let us consider now an infinite-dimensional algebra G acting on H; then G n acts on H n and, under certain conditions, on Gr(H n ). We define the G-locus Pn (G) as a subspace of Gr(H n ) consisting of subspaces V ⊂ H n having “large stabilizers” in G. (More precisely, we introduce a semi-infinite structure in G and require that the stabilizer StabV contains a space W ∈ Gr(G n ).) We prove that the spaces Pn (G) also can be used to construct MO. For the appropriate choice of G we can consider conformal field theories, WZNW models, etc. as algebras over Pn (G). Replacing the Grassmannian with the super Grassmannian we can obtain also superconformal theories. It is well known that a conformal field theory can be considered as a background for a string theory and that one can construct corresponding string amplitudes as integrals over moduli spaces (at least in the case of critical central charge.) Generalizing these constructions we can define (at least formally) “string amplitudes” corresponding to an algebra over an EMO. It seems that the “string amplitudes” corresponding to Gr(H n ) could be universal in some sense. We describe an involution on Gr(H n ) that could be related to S-duality in string theory. The space Gr(H n ) can be represented as a union of connected components Gr(k) (H n ), where k stands for the index of the Fredholm operator V → H+ , corresponding to V ∈ Gr(H n ). It is natural to conjecture that in perturbation theory the contribution of Gr(k) (H n ) is proportional to g k , where g stands for the coupling constant. ( This conjecture is based on the remark that the contribution of a point to 0-equivariant cohomology is governed by the size of the stabilizer and this size is related to the index.) The involution we constructed transforms Gr(k) (H n ) into Gr(−k) (H n ). This means that g goes to g −1 .

1. Grassmannian Let us consider a (complex) Hilbert space H represented as a direct sum of subspaces H+ and H− . We will assume that there exists a unitary involution K on H transforming H+ into H− and H− into H+ . We will say that subspaces H+ , H− and the involution K specify a semi-infinite structure on H. As an example, we can take H = L2 (S 1 ), where S 1 is the unit circle in C with standard measure. Then we can define H− as a subspace spanned by z n = einϕ , n ≥ 0 and H+ as a subspace spanned by z n = einϕ , n < 0. The involution K can be chosen as a map transforming a function f (z) into a function z1 f ( z1 ). It is easy to check that every semi-infinite structure is isomorphic to the standard structure described above. (Notice that we include separability and infinite-dimensionality in the definition of Hilbert space. Subspaces are by definition closed linear submanifolds.) Using a semi-infinite structure in the Hilbert space H we can define a Grassmannian Gr(H) as a set of such subspaces V ⊂ H that the natural projection π+ of V into H+ is a Fredholm operator and the natural projection π− of V into H− is a compact operator. The Grassmannian Gr(H) can be represented as a union of connected components Gr(k) (H) where k stands for the index of the Fredholm operator π+ : V → H+ .

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A. Schwarz

Let us consider a map α : H+ → H = H+ ⊕ H− transforming h ∈ H+ into α(h) = (A + a)h + Bh, where A is an invertible operator acting on H+ and a : H+ → H+ , B : H+ → H− are compact operators. It is easy to check that the space α(H+ ) ⊂ H belongs to Gr(0) (H) and that every V ∈ Gr(0) (H) can be obtained by means of this construction. This gives us an alternative description of Gr(0) (H). One can give a similar description of Gr(k) (H): if F is a Fredholm operator of index k acting on H+ and B is a compact operator acting from H+ into H− , then the image of the operator α : H+ → H, where α(h) = F h + Bh, belongs to Gr(k) (H) and every element of Gr(k) (H) can be obtained this way. If H has the form H+ ⊕ H− a linear operator A acting in H can be represented in the form x˜ + = A++ x+ + A+− x− , x˜ − = A−+ x+ + A−− x− , where x+ , x˜ + ∈ H+ , x− , x˜ − ∈ H− . We will say that A ∈ GL(H) if A is an invertible operator, the operators A++ : H+ → H+ and A−− : H− → H− are Fredholm, the operators A+− : H− → H+ , A−+ : H+ → H− are compact. The Lie algebra gl(H) of GL(H) is defined as the set of operators A obeying exp(At) ∈ GL(H). It is easy to check that GL(H) acts on Gr(H) and that this action is transitive on every component Gr(k) (H). Notice that a semi-infinite structure on H induces naturally a semi-infinite structure on H n (on the direct sum of n copies of H). Namely, H n = H ⊕ · · · ⊕ H = (H+ ⊕ · · · ⊕ H+ ) ⊕ (H− ⊕ · · · ⊕ H− ); the involution K : H n → H n is a direct sum of n copies of K : H → H. Hence, we can speak about Gr(H n ) and about Gr(k) (H n ). One can modify the definition of Grassmannian and of GL(H) replacing compact operators with operators belonging to the trace class or with Hilbert–Schmidt operators. All considerations of this section remain valid after such a modification. The set of compact operators B can be considered as an ideal in the algebra L(H) of bounded operators; we can replace B with any other ideal consisting of compact operators and containing all finite-dimensional operators. One can define also a semi-infinite structure on a normed space H. Such a structure is specified by means of subspaces H+ and H− and an operator K, interchanging H+ and H− . (It is not necessary to assume that K is an involution.) We assume that K is an invertible operator on H and that H is equivalent to the direct sum H+ ⊕ H− (more precisely, there exists an invertible operator π = (π+ , π− ) : H → H+ ⊕ H− such that π± h = h for h ∈ H± ). Again we can define the Grassmannian Gr(H) and prove in this more general situation the results stated below. Let us fix a space H equipped with semi-infinite structure. We will use sometimes the notations Gr, Grn and Grn(k) instead of Gr(H), Gr(H n ) and Gr(k) (H n ) correspondingly. Let us consider linear subspaces V ⊂ H m and V 0 ⊂ H n . Then their direct sum V ⊕ V 0 belongs to Gr(H m+n ). We obtain a map νm,n : Gr(H m ) × Gr(H n ) → Gr(H m+n ). For every subspace V ⊂ H m we can construct a subspace σ(V ) ⊂ H m−2 consisting of points (f1 , . . . , fm−2 ) ∈ H m−2 , satisfying the condition that one can find u ∈ H in such a way that (f1 , . . . , fm−2 , Ku, u) ∈ V . Theorem 1. If V ∈ Gr(H m ), then σ(V ) ∈ Gr(H m−2 ).

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To prove this statement we represent V as the image of a map H+m → H m trans∈ H+m into a point (f1 + g1 , . . . , fm + gm ) ∈ H m , where forming P a point (h1 , . . . , hm )P Akj hj ∈ H+ , gk = Bkj hj ∈ H− , the operators Akk : H+ → H+ are Fredfk = holm, the operators Bkj : H+ → H− and operators Akj : H+ → H+ , k 6= j, are compact. To describe the space σ(V ) we impose the condition fm−1 + gm−1 = K(fm + gm ) on the points of V . Taking into account that fk ∈ H+ , gk ∈ H− we can rewrite this condition in the form X

Am−1,j hj = K

j

X

X

Bmj hj ,

j

Bm−1,j hj = K

X

j

Amj hj .

(1)

j

Now we can apply the following statement. Lemma 1. Let us consider an equation F x = Ly

(2)

where x, y are elements of infinite-dimensional Banach spaces E and E 0 . Assume that the operator F : E → E is Fredholm and the operator L : E 0 → E is compact. Then one can find a Fredholm operator G : E 0 → E 0 and a compact operator M : E 0 → E in such a way that for every element u ∈ E 0 the elements x = M u, y = Gu

(3)

obey (2). Moreover, one can find G and M in such a way that every solution to (2) can be represented in the form (3) and this representation is unique. (Then index G = − index F .) We apply Lemma 1 to Eq. (1) taking x = (hm−1 , hm ), y = (h1 , . . . , hm−2 ), E = H+ ⊕ H+ , E 0 = H+m−2 . The representation of σ(V ) obtained this way makes the statement of Theorem 1 obvious. Let us consider an infinite-dimensional Lie algebra G and a homomorphism α : G → gl(H), where H is provided with semi-infinite structure (H+ , H− , K). We assume that there exists an involution κ of the Lie algebra G obeying α(κ(γ)) = Kα(γ)K −1 and that one can find a semi-infinite structure on G specified by means of subspaces G+ and G− and involution κ. The Lie algebra G m (direct sum of m copies of G) acts on H m (an element (γ1 , . . . , γm ) ∈ G m transforms (f1 , . . . , fn ) ∈ H n into (α(γ1 )f1 , . . . , α(γm )fm ) ∈ H m ). It is easy to check that this action generates an action of G m on Gr(H m ). Let us denote by StabV a subalgebra of G n consisting of elements transforming an element V ∈ Gr(H m ) into itself (the stabilizer of V ). We will define a G-locus Pm (G) ∈ Gr(H m ) as a set of such points V ∈ Gr(H m ) that there exists an element W ∈ Gr(G m ) obeying StabV ⊃ W . In other words Pm (G) consists of elements V ∈ Gr(H m ) having “large stabilizers” in G m . Theorem 2. If V ∈ Pm (G), then σ(V ) ∈ Pm−2 (G).

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To prove this theorem we should check that σ(V ) has a large stabilizer. If (γ1 , . . . , γm ) ∈ StabV , then for (f1 , . . . , fm ) ∈ V we have (α(γ1 )f1 , . . . , α(γm )fm ) ∈ V . If we know that (γ1 , . . . , γm ) ∈ StabV , γm−1 = κ(γm ) we can check that (γ1 , . . . , γm−2 ) ∈ Stabσ(V ) . To verify this statement we take a point (f1 , . . . , fm−2 ) ∈ σ(V ) constructed by means of (f1 , . . . , fm ) ∈ V , where fm−1 = Kfm . Using the relation α(γm−1 )fm−1 = α(κ(γm ))Kfm = Kα(γm )fm we see that the point

(4)

(α(γ1 )f1 , . . . , α(γm−2 )fm−2 )

belongs to σ(V ) (one can construct it using the point (α(γ1 )f1 , . . . , α(γm )fm ) ∈ V ). This means that Stabσ(V ) ⊃ σ(StabV ). If StabV ⊃ W ∈ Gr(G m ) we can conclude from Theorem 1 that Stabσ(V ) ⊃ σ(W ) ∈ Gr(G m−2 ) and therefore σ(V ) ∈ Pm−2 (G). The following statement is almost evident: If V ∈ Pm (G), V 0 ∈ Pn (G), then V ⊕ V 0 ∈ Pm+n (G). We obtain a map νm,n : Pm (G) × Pn (G) → Pm+n (G). Let us consider a compact one-dimensional complex manifold (complex curve) 6 and s holomorphic maps of the standard disk D = {z ∈ C | |z| ≤ 1} into 6. We assume that the images D1 , . . . , Ds of these maps do not overlap. Let us fix a holomorphic line bundle ξ over 6 and trivializations of this bundle over the disks D1 , . . . , Ds . We denote by W = W (6, ξ) the space of holomorphic sections of the bundle over 6\(D1 ∪· · ·∪Ds ). Restricting a section s of ξ to the boundaries ∂D1 , . . . , ∂Ds and using the trivializations of ξ over the disks we obtain an element ρ(s) of H s , where H stands for the space L2 (S 1 ), S 1 denotes the standard circle |z| = 1. The image σ(W ) of the space W is a linear subspace of H s ; one can check [1] that σ(W ) ∈ Gr(H). (We equip H s with the standard semi-infinite structure.) One can generalize this construction allowing 6 to be a complete irreducible complex algebraic curve (possibly singular) and replacing the line bundle ξ with rank 1 torsion free coherent sheaf on 6. (If 6 is non-singular such a sheaf is a vector bundle.) Then again one can prove that σ(W ) ∈ Gr(H s ). The points σ(W ) obtained by means of the construction above constitute the so-called Krichever locus in Gr(H s ). Let us consider now the set 0 consisting of invertible twice differentiable functions on S 1 . Every element γ ∈ 0 generates an operator α(γ) : H → H transforming f ∈ H into γf ; it is easy to check that α(γ) ∈ GL(H) [1]. Considering 0 as a group with respect to multiplication we can say that α is a homomorphism of the group 0 into GL(H). Let us denote by A6 the space of invertible holomorphic functions on 6 \ (D1 ∪ · · · ∪ Ds ). Restricting these functions to the boundaries ∂D1 , . . . , ∂Ds we obtain an embedding of A6 into 0s ; the image of this embedding will be denoted by A06 . It is easy to see that A06 V ⊂ V for every V ∈ Gr(H s ) obtained by means of Krichever construction starting with 6, D1 , . . . , Ds (for any choice of the bundle or sheaf ξ and trivializations of it). The Lie algebra Lie 0 of the group 0 consists of all twice differentiable functions on the circle; the Lie algebra (Lie 0)s = Lie 0s can be represented by means of functions on a disjoint union of s circles. It is clear that for every V ∈ Gr(H s ) obtained by means of Krichever construction the stabilizer StabV of V in (Lie 0)s consists of all functions defined on the union of circles ∂D1 , . . . , ∂Ds and admitting a holomorphic continuation to 6 \ (D1 ∪ · · · ∪ Ds ). One can check that StabV belongs to Gr((Lie 0)s ) for an appropriate definition of semi-infinite structure in Lie 0. (We can consider Lie 0 as a subset of H = L2 (S 1 ); the standard semi-infinite structure on L2 (S 1 ) induces a semi-infinite structure on the pre-Hilbert space Lie 0.) We see that the locus Ps (Lie 0) contains all points of Gr(H s ) obtained by Krichever construction (Krichever locus).

Grassmannian and String Theory

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Moreover, one can prove that the Krichever locus coincides with Ps (Lie 0). (For s = 1 this follows from [2]; see also [1]. The case s > 1 can be treated in a similar way; see the paper [3], devoted to various generalizations of the above statement.) Let us describe an interesting duality transformation of Gr(H n ). We identify H with L2 (S 1 ) equipped with standard semi-infinite structure. Then we can introduce the bilinear inner product in H by means of the formula Z f (z)g(z)dz. (f, g) = |z|=1

This inner product induces the inner product in H m : if f = (f1 , . . . , fm ) ∈ H m , g = (g1 , . . . , gm ) ∈ H m , then XZ fk (z)gk (z)dz. (f, g) = k

|z|=1

Let us define a linear operator L : H → H by the formula (Lf )(z) = f (−z). The map Lm : H m → H m transforms (f1 , . . . , fm ) into (Lf1 , . . . , Lfm ). The symbol V ⊥ denotes the orthogonal complement of V ⊂ H m with respect to the bilinear inner product; the symbol λ(V ) stands for Lm V ⊥ . Theorem 3. If V ∈ Gr(k) (H m ), then V ⊥ ∈ Gr(−k) (H m ) and λ(V ) ∈ Gr(−k) (H m ). The maps λ : Gr(H m ) → Gr(H m ) commute with maps νm,n and σ (m) ; in particular λ ◦ σ (m) = σ (m) ◦ λ. To prove the first statement we represent the space V ⊂ H as an image of the operator α : H+ → H, where α(h) = (F h, Bh), F : H+ → H+ is Fredholm, and B : H+ → H− is compact. (We restrict ourselves to the case m = 1.) Then orthogonal complement V 0 R 2π to the space V with respect to the hermitian inner product < f, g >= 0 f (ϕ)g(ϕ)dϕ consists of pairs (h+ , h− ) obeying the equation F ∗ h+ = B ∗ h− = 0. (Here ∗ denotes Hermitian conjugation.) If V = H+ then V 0 = H− ; this means that we should expect that in the case when V is close to H+ the space V 0 should be close to H− . In other words, we should expect that for V ∈ Gr(H) the space V 0 belongs to Gr(H) defined by means of semi-infinite structure when the roles of H+ and H− are interchanged. This fact immediately follows from Lemma 1. Using that V ⊥ = z V¯ 0 , H¯ − = z −1 H+ , we obtain that V ⊥ belongs to the Gr(H) of original semi-infinite structure. It is easy to check that L ∈ GL(H), therefore LV ⊥ also belongs to Gr(H). Using Lemma 1 one can also calculate the index of V ⊥ and LV ⊥ . It is evident that λ commutes with νm,n ; let us prove that λ commutes with σ (m) . We start with a remark that σ (m) (V ) = π(V ∩ RK ), where RK denotes the subspace of H m , that consists of points (f1 , . . . , fm ) ∈ H m obeying f m−1 = Kfm , and π stands for natural projection of H m onto H m−2 (i.e. π(f1 , . . . , fm ) = (f1 , . . . , fm−2 .) It is easy to check that

8

A. Schwarz ⊥ σ (m) (V )⊥ = (π T )−1 ((V ∩ RK )⊥ ) = (π T )−1 (V ⊥ + RK ).

Here π T : H m−2 → H m is an operator, adjoint to π with respect to the bilinear inner product; it is easy to see that it transforms (f1 , . . . , fm−2 ) into (f1 , . . . , fm−2 , 0, 0). ⊥ consists of points (g1 , . . . , gm ) ∈ H m obeying gm−1 = −Kfm . UsThe space RK ing these facts we obtain that (f1 , . . . , fm−2 ) ∈ σ (m) (V )⊥ if there exists a point (f1 , . . . , fm−2 , fm−1 , fm ) ∈ V ⊥ satisfying fm−1 = −Kfm . In other words, (m) (V ⊥ ) σ (m) (V )⊥ = σ−K

(The subscript −K means that we replace the involution K in the definition of σ (m) with (m) we notice that the involution −K.) To check that λ commutes with σ (m) = σK (m) (m) (m) (V )) = Lm (σK (V )⊥ ) = Lm σ−K (V ⊥ ), λ(σK (m) (m) (m) ⊥ (λ(V )) = σK (Lm V ⊥ ) = Lm σL ). σK −1 KL (V

It remains to take into account that L−1 KL = −K. 2. Generalized Moduli Spaces (Modular Operads) One can reformulate the above results in the following way. Let us consider a sequence of Sm -spaces Pm . Here Sm denotes the symmetric group , i.e. a group of permutations of m elements {1, 2, . . . , m}. Sm -space is by definition a topological space with left action of the group Sm . The group Sk for k < m is embedded into the group Sm as a subgroup consisting of permutations leaving intact k + 1, . . . , m. The group Sm × Sn is naturally embedded in Sm+n ; namely, Sm permutes first m indices and Sn permutes last n indices. Let us fix maps νm,n : Pm × Pn → Pm+n and σ (m) : Pm → Pm−2 . We will say that these data specify a MO if the following conditions are satisfied: The maps νm,n determine an associative multiplication in ∪Pm . νm,n ◦ (ρ × τ ) = (ρ × τ ) ◦ νm,n for every ρ ∈ Sm , τ ∈ Sm . σ (m) ◦ ν = ν ◦ σ (m) for every ν ∈ Sm−2 . σ (m) ◦ λ = σ (m) if λ = (m, m − 1) (i.e. λ permutes two last indices). σ (m) commutes with σ (m) · µ, where µ ∈ Sm is a permutation obeying µ(m − 1) ≤ m − 2, µ(m) ≤ m − 2. 5. σ (n) ◦ νm,n = νm,n−2 ◦ σ (n) . 6. νn,m ◦ α = β ◦ νm,n , where α stands for the natural map Pm × Pn → Pn × Pm (transposition), β denotes a permutation (1, . . . , m + n) → (m + 1, . . . , m + n, 1, . . . , m).

0. 1. 2. 3. 4.

The notion of MO is almost equivalent to the notion of modular operad introduced in [4], as was pointed out to me by A. Voronov. The definition given in [4] is very complicated (in particular it is based on the notion of operad, which is not so simple by itself). I believe that the notion introduced above is simpler and more fundamental than the notion of operad, therefore it is unreasonable to use the term “operad” in its name. However, I did not find any good word for this notion. (m) : Pm → Pm−2 satisfying Notice that Conditions 2), 3) permit us to define maps σa,b (m) (m) (m) (m) (m) ◦ λ = σλ(a),λ(b) , σa,b = σb,a , σm−1,m = σ (m) . σa,b

Grassmannian and String Theory

9

(m) (Here a, b ∈ {1, . . . , m}, a 6= b, λ ∈ Sm .) Using the maps σa,b we can rewrite 4) in the (m) (m) (m) (m) 0 0 form: σa,b ◦ σa0 ,b0 = σa,b ◦ σa,b if a, b, a , b are distinct. The symmetric group Sm acts naturally on the spaces Grm = Gr(H m ) and Pm (G) described above. The maps Gr(H m ) → Gr(H m−2 ) and Pm (G) → Pm−2 (G) were constructed in Theorem 3. One can define νm,n as a map transforming a pair of the subspaces V ⊂ H m , V 0 ⊂ H n into subspace V ⊕ V 0 ⊂ H m+n = H m ⊕ H n ; it maps Gr(H m ) × Gr(H n ) into Gr(H m+n ) and Pm (G) × Pn (G) into Pm+n (G). It is easy to check that Conditions 0)–6) are satisfied. We obtain

Theorem 4. The Sm -spaces Gr(H m ) together with maps νm,n : Gr(H m )×Gr(H n ) → Gr(H m+n ) and σ (m) : Gr(H m ) → Gr(H m−2 ) constitute a MO. A similar statement is true for the Sm -spaces Pm (G). We mentioned already that it is not necessary to assume that the operator K in the definition of semi-infinite structure on H is an involution. It is possible to construct the maps νm,n : Gr(H m ) × Gr(H n ) → Gr(H m+n ) and σ (m) : Gr(H m ) → Gr(H m−2 ) without this assumption and to prove that all conditions in the definition of MO, except Condition 3), are satisfied. This statement remains correct if we replace Gr(H m ) with Pm (G). Let us assume that the Sm -spaces Pm have an Sm -invariant decomposition Pm = (k) and that the maps νm,n and σ (m) respect this decomposition (more precisely, ∪k Pm (k+l) (k) (k) (k) ×Pn(l) into Pm+n and σ (m) maps Pm into Pm−2 ). Then we will talk about νm,n maps Pm graded MOs. It is easy to see that the MO described in Theorem 4 can be considered as graded MOs with respect to decompositions Gr(H m ) = ∪Gr(k) (H m ), Pm (G) = (k) (G). ∪Pm Notice that for every MO (Pn , νm,n , σ (n) ) one can define maps Pm × Pn → Pm+n−2 taking composition σ (m+n) ◦ νm,n . In particular, for m = n = 2 we obtain a map P2 × P2 → P2 ; this map determines a structure of the semigroup on P2 . If Pn is a graded MO (Pn = ∪Pχ,n ) we obtain a structure of the semigroup on P0,2 . (For the MO constructed by means of moduli spaces of complex curves P0,2 is called the Neretin semigroup.) If we have an algebra (Pn , E, αn ) over Pn then under certain regularity conditions the map α2 determines a representation of the Lie algebra of the semigroup P0,2 (or P2 ) in the space E. Of course, not always we can speak about the Lie algebra of a semigroup, but such a Lie algebra exists in many interesting cases. In particular, the Lie algebra of the Neretin semigroup coincides with the complexified Lie algebra of the diffeomorphism group of a circle. MOs defined above should be called topological MOs. One can define linear MOs regarding Pm as Sm -modules. (Then we should consider νm,n as a bilinear map, i. e. as a linear map of the tensor product Pm ⊗ Pn into Pm+n , and σ (m) as a linear map Pm → Pm−2 . In the definition of a graded MO we should replace disjoint union with direct sum.) It is easy to see that homology groups of topological MO constitute a linear MO. A simple, but very important, example of linear MO (standard linear MO) can be defined if we have a linear space E equipped with a symmetric bilinear inner product. Then we can take the mth tensor degree E ⊗m as Pm . The definitions of maps νm,n : E ⊗m ⊗ E ⊗n → E ⊗(m+n) and σ (m) : E ⊗m → E ⊗(m−2) are obvious. If the bilinear inner product is not symmetric the maps νm,n and σ (m) obey all conditions in the definition of MO, except 3). 0 can be defined as a collection of maps A homomorphism of MO Pm into a MO Pm 0 αm : Pm → Pm , commuting with the operations νm,n , σ (m) . A homomorphism of MO into itself is called an automorphism. Theorem 3 can be interpreted as a statement that the

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maps λ : Gr(H m ) → Gr(H m ) constitute an automorphism of the MO Gr(H m ). Under certain conditions one can check that λ induces also an automorphism of MO Pm (G). (One should assume that for every γ ∈ α(G) we have γ T ∈ α(G) and LγL−1 ∈ α(G). Here α(G) stands for the image of G by the embedding α : G → gl(H), γ T denotes an operator adjoint to γ with respect to the bilinear inner product in H = L2 (S 1 ), (Lf )(z) = f (−z).) Let us define an algebra over (topological) MO as a homomorphism of it into the linear MO described above. In other words, an algebra over MO Pn is a collection of maps αm : Pm → E ⊗m such that σ (m) αm = αm−2 σ (m) , νm,n (αm × αn ) = αm+n νm,n . We will assume that αm (x), x ∈ Pm , is defined only up to a constant factor. (One can say that we consider projective algebras; if αm (x) is well defined we will talk about algebra in the strict sense.) Then for an appropriate choice of MO the notion of algebra over MO corresponds to the notion of conformal or superconformal field theory; if the maps αn are defined uniquely the central charge of corresponding conformal field theory vanishes. We will not exclude the case when the inner product in linear space E entering the definition of algebra over MO is determined only on a subset X of E 2 (then it is natural to assume that E is a topological linear space and X is dense in E 2 ). If it is necessary to emphasize that we are dealing with this case, we will use the term “generalized algebra”. If one uses Hilbert–Schmidt operators in the definition of Grassmannian, it is well known [1] that for every element V ∈ Gr(H) one can construct an element 9V of the fermionic Fock space F(H), defined up to a factor. We describe this construction below. (More general construction is given in Appendix.) We assume that the space H is equipped with antiunitary involution f → f¯, preserving H+ and H− . Then the Clifford algebra Cl(H) can be defined as an associative unital algebra with generators ψ(f ), ψ + (f ), depending linearly on f ∈ H and satisfying [ψ(f ), ψ + (f 0 )]+ = (f, f 0 ), [ψ + (f ), ψ + (f 0 )]+ = [ψ(f ), ψ(f 0 )]+ = 0. (Here ( , ) denotes the bilinear inner product related with the Hermitian inner product in H by the formula (f, g) =< f, g¯ >.) The Fock space F(H) can be defined as a space of representation of Cl(H), that contains a cyclic vector 8, obeying ψ(f )8 = 0 for f ∈ H+ , ψ + (f )8 = 0 for f ∈ H− . (We assume that ψ + (f¯) is Hermitian conjugate to ψ(f ).) If V ∈ Gr(H) then 9V can be defined as a vector from F(H) satisfying the conditions ψ(f )9V = 0 for f ∈ V, ψ + (f )9V = 0 for f ∈ V ⊥ . (Here V ⊥ stands for the orthogonal complement to V with respect to the bilinear inner product.) Notice that for every vector 9 ∈ F(H) one can define two orthogonal subspaces Ann 9 = {f ∈ H|ψ(f )9 = 0} and Ann+ 9 = {f ∈ H|ψ + (f )9 = 0}. One can prove the following Lemma 2. If Ann 9 ⊃ V, Ann+ 9 ⊃ KW, V ∈ Gr(k) (H), W ∈ Gr(−k) (H), then 9 = 9V , V = Ann 9, W = V ⊥ = K Ann+ 9.

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We will define also a bilinear inner product ( ,) in F = F(H), obeying the condition that −ψ(Kf ) is adjoint to the operator ψ(f ) and −ψ + (Kf ) is adjoint to ψ + (f ) with respect to this inner product. Using this bilinear inner product we define a linear MO F n. One can check the formula 9σ(n) V = σ (n) 9V for V ∈ Gr(H n ).

(5)

(We use the identification F(H n ) = F(H)⊗n in (5). This identification permits us to consider F(H n ) as a linear MO.) To prove (5) we represent 9V ∈ F ⊗n in the form X αi ⊗ Ai ⊗ Bi , 9V = where αi ∈ F ⊗(n−2) , Ai ∈ F, Bi ∈ F. By definition for every (f1 , . . . , fn ) ∈ V we have X X αi ⊗ (ψ(fn−1 )Ai ) ⊗ Bi (ψ(f )αi ) ⊗ Ai ⊗ Bi + (6) X + αi ⊗ Ai ⊗ (ψ(fn )Bi ) = 0, where ψ(f ) = ψ(f1 ) ⊗ 1 ⊗ · · · ⊗ 1 + 1 ⊗ ψ(f2 ) ⊗ 1 ⊗ · · · ⊗ 1 + 1 ⊗ · · · ⊗ 1 ⊗ ψ(fn−2 ). Let (f1 , . . . , fn−2 ) ∈ σ (n) V . Then it can be obtained from a point (f1 , . . . , fn−2 , fn−1 , fn ) ∈ V with fn−1 = Kfn . Applying the operator σ (n) to 9V we obtain X αi (Ai , Bi ). σ (n) 9V = Now we can apply σ (n) to (6). Using the relation X X αi (Ai , ψ(fn )Bi ) = 0 αi (ψ(fn−1 )Ai , Bi ) + for fn−1 = Kfn we see that

ψ(f )σ (n) 9V = 0

for every f = (f1 , . . . , fn−2 ) ∈ σ (n−2) V . In other words, σ (n) V ⊂ Ann σ (n) 9V . In a similar way we prove Kσ (n) (KV ⊥ ) ⊂ Ann+ σ (n) 9V . Taking into account that index σ (n) V + index σ (n) (KV ⊥ ) = 0 and using Lemma 2 we obtain (5). The following statement follows immediately from (5). Theorem 5. The sequence of maps V ∈ Gr(H m ) → 9V ∈ F(H m ) = F ⊗m determines an algebra over a MO Gr(H m ).

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It is important to emphasize that in all definitions and theorems above one can replace spaces with superspaces; only minor modifications are required. (The only exception is Theorem 5, that requires more essential modification; see Appendix.) In particular, one can consider the super Grassmannian. To give an example of a superspace with semiinfinite structure we can consider the space H = H m|n of functions on S 1 taking values in (m|n)-dimensional linear complex superspace Cm|n . The definitions of H+ , H− and K repeat definitions for H = L2 (S 1 ). Notice, that H 1|1 can be considered also as the space of functions on the supercircle, i.e. as the space of functions F (z, θ) = f (z) + ϕ(z)θ, |z| = 1, θ is an odd variable, f (z) is an even function on S 1 and θ(z) is an odd function on S 1 . The following construction will play later an important role. For every manifold M ˜ = 5T M as the space of a tangent bundle with reversed we define a supermanifold M ∂ i ˜ parity of fibers. One can define an odd vector field Qˆ = ξ i ∂x i on M (here x are i coordinates on M, ξ are odd coordinates in tangent spaces.) Functions on M can be identified with differential forms on M , then the operator Qˆ corresponds to the exterior differential. Notice that {Q, Q} = 0; this means that M is a Q-manifold in the ˜ also in the case when M is terminology of [6]. One can construct the supermanifold M ˜ can be considered as functions on a supermanifold. Then again differential forms on M ˜ correspond to differential forms. One can describe M , however not all functions on M ˜ M also as the space of maps of (0|1)-dimensional superspace R0|1 into M . It is easy to see that this construction is functorial: for every map f : M → M 0 one can define ˜ →M ˜ 0 . It follows from this remark that in the case when M is naturally a map f˜ : M ˜ is a (super) Lie group (the multiplication map M × M → M a (super) Lie group M ˜ ×M ˜ →M ˜ ). If M is a Lie algebra then M ˜ is a (super) Lie generates multiplication M i ˜ algebra. (If ln are generators of M , fjk are corresponding structure constants, then M has even generators ln and odd generators bn with commutation relations: [lm , ln ] = k k lk , [ln , bm ] = fmn bk , [bm , bn ]+ = 0.) fmn ˜ , transitive action on G induces transitive If a Lie group G acts on M then G˜ acts on M ˜ If M = G/G0 then M ˜ can be identified with G/ ˜ G˜ 0 . action of G. ˜ The supermanifold M = 5T M is equipped with the natural volume element dV = 5dxi dξ i (this volume element does not depend on the choice of coordinates on the ˜ over M ˜ or over L˜ (super)manifold M ). Therefore we can integrate a function on M where L is a submanifold of M . (Of course, we should make some assumptions about behavior of the function at infinity to guarantee the convergence of the integral.) If M is ˜ is equivalent to an ordinary manifold then the theory of integration of functions on M the theory of integration of differential forms on M . However, if M is a supermanifold ˜ , corresponding to differential forms on M , are not integrable. then the functions on M (By definition, such functions depend polynomially on ξ i . If some of coordinates xi are odd, corresponding ξ i are even and the function does not decrease at infinity.) It is necessary to mention that we can integrate also generalized functions (distributions) ˜. over M ˜ is Q-invariant. This means, that The volume element dV on M Z ˆ )dV = 0 (Qf ˜ L

˜ and every submanifold L ⊂ M . It is easy to check that for for every function f on M ˜ the integral the Q-invariant function ϕ on M Z ϕdV ˜ L

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does not change by continuous deformation of the submanifold L. (For the case when M is an ordinary manifold one can derive this fact from the remark that the Q-invariant ˜ can be considered as a closed differential form on M .) function ϕ on M 2 If H = L (S 1 ) then H˜ can be regarded as H 1|1 (as the space of functions on the ˜ supercircle). This remark permits us to embed Gr(H) into Gr(H 1|1 ). One can consider (k) ˜ ˜ (H) as a homogeneous space where GL(H) acts transitively and utilize for example Gr ˜ the fact that the GL(H) acts on H˜ = H 1|1 and therefore on Gr(H 1|1 ). However, it is ˜ useful to describe the embedding Gr(H) into Gr(H 1|1 ) explicitly. An arbitrary point 1|1 1|1 1|1 W ∈ Gr(H ) can be specified by means of a Fredholm operator A : H+ → H+ and 1|1 1|1 a compact operator B : H+ → H− . These operators can be written as 2 × 2-matrices A11 A12 B11 B12 , , A21 A22 B21 B22 where Aαβ : H+ → H+ , Bαβ : H+ → H− , diagonal entries are even, off-diagonal ˜ entries are odd. One can check that W belongs to Gr(H) ⊂ Gr(H 1|1 ) if it can be represented by means of operators A and B with matrices obeying A11 = A22 , A12 = 0, B11 = B22 , B12 = 0. 3. Generalized String Backgrounds To calculate string amplitudes corresponding to a conformal field theory with critical central charge we should “add ghosts” to obtain topological conformal field theory with vanishing central charge. Such a TCFT can be considered as “string background”. In other words one can define corresponding string amplitudes, even in the case when the TCFT does not correspond to any CFT (matter and ghosts are not separated). In this section we will describe an abstract analog of these constructions. We say that a MO (Pn , νm,n , σ (m) ) is a Q -MO if every space Pn is a Q-manifold (i. e. Pn is a supermanifold equipped with a vector field Q = Q(n) obeying {Q, Q} = 0) and maps νm,n , σ (m) are compatible with Q-structures on Pn (for example, σ∗(m) Q(m) = Q(m−2) ). A Q-homomorphism of Q-MO Pn into a Q-MO Pn0 is defined as a collection of maps ρn : Pn → Pn0 that determine a homomorphism of MOs and are compatible with Q-structures on Pn , Pn0 (i. e. ρn transforms the vector field Q on Pn into the corresponding vector field on Pn0 ). If we have a linear MO Pn we can introduce a notion of linear Q-structure on it, requiring that all vector fields Q(n) are linear, i. e. have the form (Q(n) )a = (Q(n) )ab z b , where (Q(n) )ab is a matrix of parity reversing linear operator, having zero square (such an operator is called a differential). In particular, if E is a linear superspace, equipped with a differential d and E is equipped with a d-invariant inner product (i. e. d = −d∗ ) then one can construct a linear Q-MO with spaces Pn = E ⊗n . (The differential d on E determines a differential d : E ⊗n → E ⊗n .) A linear Q-MO E ⊗n constructed this way will be called a standard linear Q-MO. We define a Q-algebra (Pn , E, αn ) over a Q-MO Pn as a Q-homomorphism of QMO Pn into standard linear Q-MO E ⊗n . Let us emphasize that although we assumed that the maps αn are defined only up to a factor when we considered an algebra over MO, in the definition of Q-algebra we assume that αn are well defined (i.e. the Q-algebra should be an algebra in the strict sense).

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Let us consider an arbitrary MO Pn . Then we can construct in a natural way a QMO P˜n . (Recall that P˜n = 5T Pn = {R0|1 → Pn }.) As it was explained above this construction is functorial; hence the maps νm,n , σ (m) , specifying the MO Pn induce maps ν˜ m,n , σ˜ (m) , that determine a MO P˜n . We obtain a Q-MO this way because every P˜n has a natural Q-structure. A Q-algebra (P˜n , F, βn ) over Q-MO P˜n is called a Qextension of an algebra (Pn , E, αn ) over a MO Pn if there exists such a linear map ρ : E → F for every x ∈ Pn ⊂ P˜n , e ∈ E, we have (ρ(e), βn (x)) = (e, αn (x)). If the algebra (Pn , E, αn ) corresponds to conformal field theory then we can obtain its Q-extension adding ghosts to the matter sector of the theory. Let us consider a MO (Pn , νm,n , σ (n) ). We say that this MO is an EMO (equivariant modular operad) if for every m the group (S 1 )m acts on Pm ; this action should be compatible the with Sm -action and with maps νm,n , σ (m) . More precisely, we assume that 1. s ◦ g ◦ s−1 = s(g) where g ∈ (S 1 )m , s ∈ Sm , g → s(g) denotes the natural action of Sm onto (S 1 )m . 2. νm,n ◦ (g × g 0 ) = (g × g 0 ) ◦ νm,n for g ∈ (S 1 )m , g 0 ∈ (S 1 )n . −1 3. If g = (g1 , . . . , gm ) ∈ (S 1 )m and gm−1 = gm then σ (m) ◦ g = g 0 ◦ σ (m) where 0 1 m−2 . g = (g1 , . . . , gm−2 ) ∈ (S ) We define ρ(m) : Pm ×S 1 → Pm−2 as a map transforming a point (x, γ) ∈ Pm ×S 1 into a point σ (m) (gx) where g = (1, . . . , 1, γ) ∈ (S 1 )m . It is possible to generalize a notion of EMO, taking as a starting point the maps ρ(m) ; we will not discuss this generalization here. The MO corresponding to conformal field theory can be considered as an EMO. Recall that the elements of Pm in this case can be considered as surfaces with m holes, having boundaries parametrized by the circle |z| = 1. The action of (S 1 )m changes the parametrization of boundary circles: (z1 , . . . , zm ) → (eiα1 z1 , . . . , eiαm zm ). Similarly, we can introduce a structure of EMO in the MO Gr(H k ), assuming that H has a semi-infinite structure (H+ , H− , K) and the group S 1 acts on H, in such a way that H+ and H− are invariant subspaces and K ◦ g ◦ K −1 = g −1 for every g ∈ S 1 . If the semi-infinite structure on H = L2 (S 1 ) is chosen in a standard way (H− spanned by z n , n ≥ 0, H+ by z n , n < 0, (Kf )(z) = z −1 f (z −1 )), then one can define an action of the group S 1 by the formula gα (z) = eiα g(e2iα z). As we know for every MO Pn the spaces P˜n constitute a Q-MO. We define a QEMO as a Q-MO with action of (S˜ 1 )n satisfying natural conditions. It is easy to see that if Pn constitute an EMO the spaces P˜n constitute a Q-EMO. Every linear space F with inner product and S 1 -action preserving this inner product determines a linear EMO. A linear superspace with inner product and linear operators Q, l, b, obeying Q2 = 0, b2 = 0, l = [Q, b]+ determines a linear Q-EMO. (Operators Q and b should be parity reversing. The inner product should be invariant with respect to Q, l, b. We assume that l generates an action of S 1 .) One can construct a BV-algebra corresponding to an EMO. (This construction is similar to the construction of [7].) Let us denote by Cnk the linear space of singular k-dimensional chains in Pn . Using the map ρ(n) : Pn × S 1 → Pn−2 we can define a k+1 1 1 by the formula 1(x) = ρ(n) map 1 : Cnk → Cn−2 ∗ (x × [S ]), where [S ] stands for P the fundamental cycle of S 1 . The space Cn = k Cnk can be considered as a graded Sn -module; the Sn -invariant part of Cn will be denoted by Ckinv . It is easy to check inv that 1 is a parity reversing operator acting from Cninv into Cn−2 and that 12 = 0 on inv k × Cnl → Cn . The map νm,n : Pm × Pn → Pn+m generates a map (νm,n )∗ : Cm

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k+l inv inv Cm+n . If x ∈ Cm , y ∈ Cninv we denote by x · y an element of Cm+n obtained by means ofP symmetrizatiom of (νm,n )∗ (x × y). Let us consider now a graded linear space Cninv (grading in C inv is induced by the grading in Cninv ). One can prove C inv = that 1 : C inv → C inv is a second order derivation with respect to multiplication (x, y) → x · y and therefore C inv can be considered as a BV-algebra.

4. String Amplitudes Let us consider a linear space E with inner product and with S 1 -action preserving this inner product. As we mentioned such a space determines a linear EMO. If Pn is an EMO we can consider an algebra (Pn , E, αn ) as a homomorphism of Pn into E ⊗n . (We require that maps αn : Pn → E ⊗n commute with (S 1 )n -action.) To define string amplitudes corresponding to the algebra (Pn , E, αn ) we should construct at first a Q-extension (P˜n , F, βn ) of (Pn , E, αn ) (we should “add the ghosts”). This means that we should construct a Q-algebra over Q-EMO P˜n , that can be considered as a Q-extension of our original algebra. More precisely, we should construct a linear space F with inner product and odd operators Q, b, respecting this inner product and obeying Q2 = b2 = 0. We assume that the operator l = [Q, b]+ generates an action of S 1 ; then l, b generate an action of S˜ 1 on F and an action of (S˜ 1 )n on F n . The Q-algebra (P˜n , F, βn ) is specified by the maps βn : P˜n → F ⊗n that are compatible with the action of Q and (S˜ 1 )n on P˜ and F ⊗n . Recall that we assumed that the maps αn are defined only up to a factor. However, we assume that the maps βn in (P˜n , F, βn ) are defined unambiguosly. As we mentioned the passage to Q-extension corresponds to adding ghosts to conformal field theory. Our assumption means that we consider critical theory when adding ghosts gives zero central charge. Now we define string amplitudes starting with the equivariant Q-algebra (P˜n , F, βn ). (Notice that we can forget about the original algebra (Pn , E, αn ) at this stage. This remark corresponds to the well known fact that string amplitudes can be defined also in the case when matter and ghosts are not separated; in other words we can consider a topological conformal field theory, that does not correspond to a conformal field theory, as a string background.) The space F can be interpreted as a space of states in the string theory. However, not all of them should be considered as physical states. The physical states A should satisfy the conditions lA = 0, bA = 0, QA = 0, where l and b are generators of the group S˜ 1 (l is even, b is odd, l = [b, Q]+ , hence lA = 0 follows from bA = 0, QA = 0). Two physical states A, A0 ∈ F should be considered equivalent if A0 − A = QB, where B ∈ F, lB = 0, bB = 0. In other words, the space of physical states can be identified with the homology of operator Q acting in the space F rel = {A ∈ F |lA = 0, bA = 0}. Now we can define the scattering amplitude for physical states A1 , . . . , An in the following way. Let us consider a function 8A1 ,...,An (x) = (A1 ⊗ · · · ⊗ An , βn (x))

(7)

on P˜n . We assumed that the map βn : P˜n → F ⊗n respects the action of (S˜ 1 )n and Q on P˜n and F ⊗n . It follows immediately from invariance of the inner product and from relation lAi = 0, bAi = 0, QAi = 0 that the function (7) is (S˜ 1 )n -invariant and Qinvariant. This means that 8A1 ,...,An (x) can be considered as a Q-invariant function on

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˜ n , where P˜n /(S˜ 1 )n . We define “string amplitude” as an integral of 8A1 ,...,An (x) over M 1 n Mn denotes the “fundamental cycle” of Pn /(S ) . One should emphasize that Pn /(S 1 )n is in general infinite-dimensional (super)manifold therefore the notion of “fundamental cycle” is ill-defined. However, say, in the case where Pn is a space of complex curves with n holes, the infinite-dimensional space Pn /(S 1 )n is homotopy equivalent to a disjoint union of finite-dimensional orbifolds and the fundamental cycle Mn of Pn /(S 1 )n can be defined as a sum of corresponding fundamental cycles. It follows from Q-invariance of 8A1 ,...,An (x) that the integral Z 8A1 ,...,An (x) ˜n M

depends only on the homology class of Mn . This integral gives the standard expression for the bosonic string amplitudes. Similar considerations can be applied to the case when Pn is a space of superconformal manifolds; we obtain an expression for the fermionic string amplitudes in this case. The definition of physical states given above is not completely general. One should define the space of physical states as the equivariant cohomology of F . More precisely, we should consider the subspace F inv [] of polynomials of indeterminate taking values in F inv = {x ∈ F |lx = 0}. One can define a differential d, acting on F inv , by the formula d = Q − b. Then equivariant cohomology of F can be identified with cohomology of d, acting on F inv []. It is clear that physical states in the old sense can be considered as physical states in the new sense. One can prove that both notions coincide if every element x of F obeying bx = 0 can be represented in the form x = by and F splits into a direct sum of eigenspaces of l [5]. If A1 , . . . , An are equivariant cocycles (i.e. A ∈ F inv , dA = 0) we define 8A1 ,...,An (x, 1 , . . . , n ) by means of the same formula (7). It is easy to see that the function 8 = 8A1 ,...,An (x, 1 , . . . , n ) is (S 1 )n -invariant (i.e. l1 8 = · · · = ln 8 = 0) and satisfies the condition X (Q − i bi )8 = 0. The function 8 on P˜n can be considered as an equivariant differential form on Pn ; the conditions above mean that this form is equivariantly closed (with respect to the action of the group (S 1 )n ); it determines therefore an equivariant cohomology class. In the case when (S 1 )n acts freely on Pn one can prove that equivariant cohomology of Pn is isomorphic to cohomology of Pn /(S 1 )n ; we obtain a cohomology class of Pn /(S 1 )n that can be used to define string amplitudes in the same way as above. Similar construction can be applied also in the case when Pn is a supermanifold. Notice that in the definition of EMO and in considerations of the present section one can replace the group S 1 with any other group G. (Condition 3 in the definition of −1 with the EMO should be slightly modified. Namely, we replace the relation gm−1 = gm relation gm−1 = ρ(gm ), where ρ is a map of G into itself.) Notice, that MO Gr(H m ), where H = L2 (S 1 ) can be considered as EMO with respect to the action of the group 0 of non-vanishing twice differentiable functions on S 1 . We will argue that this EMO and similar EMO’s should be related to non-perturbative string theory. Let us consider a collection of maps αn : Pn → E ⊗n , that determine an algebra over MO Pn . If 6 ∈ P1 has an automorphism group G, then this automorphism group acts naturally on E. In many interesting cases one can extend the action of the automorphism group G to an action of Gn onto Pn to obtain an EMO and an algebra over this EMO.

Grassmannian and String Theory

17

For example, if the algebra at hand corresponds to CFT, one can take as 6 a standard disk and consider G as a group S 1 of rotations of the disk. The space Pn for the MO related to CFT consists of complex curves with n embedded standard disks; the action of (S 1 )n onto Pn comes from automorphism groups of these disks. If Pn constitute an EMO with respect to the group G, then P˜n constitute an EMO with ˜ To define “string amplitudes” we can start with a G-equivariant respect to the group G. algebra (Pn , E, αn ) over Pn . (We say that an algebra (Pn , E, αn ) is G-equivariant, if G acts linearly on E, preserving the inner product, and the maps αn : Pn → E ⊗n ˜ commute with the action of Gn .) We should extend E to a G-module F (to a linear ˜ superspace F with linear action of the group G). At the level of Lie algebras such a module can be described by means of even operators Ln and odd operators bn obeying k k [Lm , Ln ] = fmn Lk , [Lm , bn ] = fmn bk , [bm , bn ]+ = 0,

(8)

k are structure constants of the Lie algebra G of G. One should assume also where fmn that F is equipped with an odd differential Q obeying

Lm = [Q, bm ]+

(9)

˜ and with an inner product which is Q-invariant and G-invariant. As we mentioned, the Lie algebra G˜ of the group G˜ has even generators Ln and odd generators bn obeying (8). Adding to Ln , bn an odd generator Q satisfying (9) and [Q, Q]+ = 0 we obtain a Lie algebra that will be denoted by G 0 . The corresponding extension of the group G˜ will be denoted by G0 . A Q-extension (P˜n , F, βn ) of an algebra (Pn , E, αn ) over EMO Pn should be considered as a Q-algebra, that respects the action of G˜ n in P˜n and F ⊗n . In other words P˜n should be considered as EMO with respect to the group G0 and (P˜n , F, βn ) should be a G0 -equivariant algebra. The space A of physical states must be identified with equivariant cohomology of F with respect to the action of G0 . ˜ where To define equivariant cohomology of F we begin with the space F ⊗ 8(5G), ˜ stands for a space of functions on 5G. ˜ One can consider various functional spaces 8(5G) and obtain different versions of the notion of equivariant cohomology. The standard notion of equivariant cohomology corresponds to the space of polynomial functions on ˜ If the Lie algebra G is m-dimensional polynomial functions on 5G˜ can be identified 5G. with polynomials of m odd variables ω1 , . . . , ωn and m even variables 1 , . . . n . The group G0 acts naturally on G 0 ; this action is linear and therefore determines a linear action of G0 on 5G 0 . One can identify functions on 5G˜ with homogeneous functions ˜ using this identification we obtain an action of G0 on 8(5G). ˜ (One can on 5G 0 \ 5G; ˜ using the embedding of 5G˜ into say also that we define the action of G0 on 8(5G) projective space corresponding to the linear space 5G 0 .) Combining the G0 -action on F ˜ we obtain G0 -action on F ⊗ 8(5G). ˜ In other words, we have and G0 -action on 8(5G) ˜ ˜ the G-action on F ⊗ 8(5G) and a differential Qtot on this space. ˜ inv of G-invariant ˜ elements of F ⊗ The differential Qtot acts on the set (F ⊗ 8(5G)) ˜ we define the equavariant cohomology HG (F ) of F as the cohomology of Qtot 8(5G); ˜ inv . If G is an m-dimensional connected Lie group we represent acting on (F ⊗ 8(5G)) ˜ as an F -valued function ϕ(ω1 , . . . , ωm , 1 , . . . , m ). The an element of F ⊗ 8(5G) ˜ condition of G-invariance means that k Li ϕ + (ωα fiα

∂ ∂ k + α fiα )ϕ = 0, ∂ωk ∂k

(10)

18

A. Schwarz

bi ϕ + (

∂ k ∂ + ωα fiα )ϕ = 0. ∂ωi ∂ωk

(11)

The second of these equations can be used to eliminate ω1 , . . . , ωm and to obtain a generalization of the definition of S 1 -equivariant cohomology that we gave above. More precisely, we assign to every solution ϕ(ω1 , . . . , ωm , 1 , . . . , m ) of (11) a function ϕ(0, . . . , 0, 1 , . . . , m ); we obtain one-to-one correspondence between solutions of (11) and functions depending on 1 , . . . , m . Using this correspondence we can define equivariant cohomology as cohomology of the operator Q − i bi acting on the space of G-invariant functions of 1 , . . . , m taking values in F . As we mentioned one can modify the definition of equivariant cohomology considering various spaces of functions ˜ If G (and therefore G) ˜ is an infinite-dimensional Lie algebra with semi-infinite on 5G. structure, one can define also the semi-infinite equivariant cohomology of F replacing ˜ with a Fock space F constructed by means of G˜ (see [12]). 8(5G) The definition of G-equivariant cohomology can be applied to every G0 -module. In ˜ ˜ other words, it can be applied to every differential G-module, i.e. to the G-module F ˜ equipped with an odd differential that is compatible with the structure of the G-module. (More precisely, if F is considered as a linear Q-manifold the map G˜ × F → F that determines an action of G˜ on F should be compatible with Q-structures on G˜ × F and ˜ F .) In particular, if a group G acts on a manifold M the group G˜ acts on the manifold M ˜ and therefore the space (M ) of differential forms of M (= the space of functions on M ) ˜ can be considered as a differential G-module. Equivariant cohomology of this module is called equivariant cohomology of the G-manifold M . (If M is a supermanifold there are various versions of this definition because we can consider different spaces of functions ˜ . A similar remark can be made in the case when G is a supergroup.) of M Let us come back to the definition of “string amplitudes”. We consider the Q-algebra ˜ F is a differential (P˜n , F, βn ), where P˜n constitute a Q-EMO with respect to the group G, ˜ G-module, βn are compatible with the action of G˜ and with Q. (It is not necessary to assume that this Q-algebra is obtained as a Q-extension of an algebra over Pn .) In other words one can say that P˜n constitute an EMO with respect to the group G0 , a superspace F is a G0 -module and the maps βn : P˜n → F ⊗n are compatible with the G0 -action. If A1 , . . . , An ∈ A are physical states (elements of equivariant cohomology of F ) we consider an expression

(12) Aˆ 1 ⊗ · · · ⊗ Aˆ n , βn (x) , where x ∈ P˜n and Aˆ k stands for a representative of cohomology class Ak ∈ A. It is easy to check that (12) determines an element of G-equivariant cohomology of Pn and that this element does not depend on the choice of representatives Aˆ k in the classes Ak . To get “string amplitudes” we should have a linear functional on G-equivariant cohomology of Pn (a kind of integration). It is important to emphasize that this construction can be applied also in the case when physical states are defined by means of semi-infinite equivariant cohomology. One can hope to obtain non-perturbative formulation of string theory applying the above consideration to the case when Pn = Gr(H n ) for an appropriate choice of (super)space H. This hope is based, in particular, on the relation between equivariant cohomology of the Grassmannian and cohomology of moduli spaces of conformal manifolds. To explain this relation we should recall some facts about equivariant cohomology. Let us assume that a connected compact abelian group T (a torus) acts on M . Then we can consider the equivariant cohomology HT (M ) as a module over the polynomial ring

Grassmannian and String Theory

19

C[1 , . . . , r ], where r =dimT . The ring C[1 , . . . , r ] can be considered as a ring 8(Lie T ) of polynomial functions on the Lie algebra of T . If T acts on M transitively then HT (M ) can be identified with cohomology of M/T with coefficients in the ring 8(Lie S) of polynomial functions on the subalgebra of Lie T consisting of elements t ∈ T obeying t(x) = 0 for the fixed point x ∈ M . (In other words Lie S is the stabilizer of x ∈ M .) We see that the “size” of HT (M ) is determined by the size of the stabilizer. If the action of T on M is not transitive it follows from the so-called localization theorems that the “contribution” of the point x ∈ M to the equivariant cohomology is governed by the stabilizer of x. In particular, the “free part” (the rank) of C[1 , . . . , r ]-module HT (M ) is determined by the fixed points of the action of T . We can try to apply formally the above statements to the action of the infinitedimensional abelian group 0m on Gr(H m ), where H = L2 (S 1 ). We will see that the 0m -equivariant cohomology of Gr(H m ) can be expressed in terms of the Krichever locus and that this cohomology is closely related to the cohomology of moduli spaces of conformal manifolds. (Recall that the Krichever locus consists of points having large stabilizers in 0m and that the space of orbits of 0m in the Krichever locus Pm (0) can be identified with the moduli space Pm used in the definition of CFT.) Of course, it is not clear that the statements proved for compact groups can be applied to a non-compact group 0. However, I was able to prove that many results of the theory of compact transformation groups can be transferred to the non-compact case if we understand equivariant cohomology as semi-infinite equivariant cohomology [12].

Appendix. Isotropic Grassmannian Let us consider a Hilbert space H provided with an antiunitary involution f → f ∗ . We will equip the direct sum H2 of two copies of H with semi-infinite structure considering the first copy as H+ , the second copy as H− and defining an operator K by the formula K(f, g) = (−g, f ). Notice that K is not an involution. However, as we mentioned, we can use K to define the spaces Gr(H2m ) and the maps νm,n : Gr(H2m ) × Gr(H2n ) → Gr(H2(m+n) ) and σ (m) : Gr(H2m ) → Gr(H2(m−2) ). We will exclude Condition 3) from the definition of MO; then these data constitute a MO. A linear subspace V ⊂ H2 is called isotropic if for every two points (f, g) ∈ V, (f 0 , g 0 ) ∈ V we have (f, g 0 ) + (g, f 0 ) = 0 (here ( ,) denotes bilinear inner product: (ϕ, ψ) =< ϕ, ψ ∗ >). The isotropic Grassmannian IGr(H) can be defined as a subset of Gr(H2 ) consisting of isotropic subspaces. Giving an obvious definition of isotropic subspace of H2m one can define also IGr(Hm ) as a subset of Gr(H2m ). It is easy to check that the direct sum of isotropic subspaces is again an isotropic subspace and that the map σ (m) : Gr(H2m ) → Gr(H2(m−2) ) transforms an isotropic subspace into isotropic subspace. In other words, the spaces IGr(Hm ) constitute a MO (a sub MO of the MO Gr(H2m )). The decomposition Gr(H2m ) = ∪Gr(k) (H2m ) induces decomposition IGr(Hm ) = ∪IGr(k) (Hm ), hence we consider the MO IGr(Hm ) as a graded MO. In particular, we can say that the spaces IGr(0) (Hm ) also constitute a MO. It is important to notice that, conversely, the usual Grassmannian can be embedded into the isotropic Grassmannian. Let us assume that H is equipped with a semi-infinite

20

A. Schwarz

structure (H = H+ ⊕ H− ) and that antiunitary involution f → f ∗ transforms H+ into H+ and H− into H− . Then for every linear subspace V ⊂ H we can construct an isotropic subspace ρ(V ) ⊂ H2 as a direct sum of subspaces π1 V and π2 V ⊥ , where π1 : H → H2 transforms f ∈ H into (π+ f, π− f ) and π2 : H → H2 transforms f ∈ H into (π− f, π+ f ). (Here π± : H → H± are orthogonal projections and the orthogonal complement V ⊥ is taken with respect to the bilinear inner product (ϕ, ψ) =< ϕ, ψ ∗ >.) It is easy to check that for V ∈ Gr(H) we have ρ(V ) ∈ IGr(H); hence we embedded Gr(H) into IGr(H). Moreover, one can verify that index ρ(V ) = 0, therefore we can say that we embedded Gr(H) into IGr(0) (H). Let us consider a Hilbert space H with antiunitary involution f → f ∗ . We define the space F as the space of antiholomorphic functionals 8(a∗ ) on 5H obeying the condition Z ∗ (13) 8(a∗ )8∗ (a)e−(a,a ) dada∗ < ∞. Here 5H stands for the superspace obtained from H by means of parity reversion; 8∗ (a) = (8(a∗ ))∗ . In other words, F consists of elements of the infinite-dimensional Grassmann algebra generated by H, i. e. of formal expressions X 1 Z 8n (x1 , . . . , xn )a∗ (x1 ) . . . a∗ (xn )dn x, 8(a∗ ) = 1/2 (n!) n where 8n (x1 , . . . , xn ) are antisymmetric and XZ |8n |2 dn x < ∞. (We consider H as the space of square integrable functions depending on x ∈ S, where S is a measure space; involution is realized as complex conjugation. This restriction is not essential, but it permits us to simplify notations.) It is well known that F can be considered as fermionic Fock space. (See [10]; we follow the notations of this book. A rigorous explanation of the meaning of infinite-dimensional integral in (13) also can be found in [10].) R ∗ + ROperatorδof multiplication on f (x)a (x)dx will be denoted by a (f ) and operator f (x) δa∗ (x) dx by a(f ). These operators obey canonical anticommutation relations (Clifford algebra relations) [a(f ), a(f 0 )]+ = [a+ (f ), a+ (f 0 )]+ = 0, [a(f ), a+ (f 0 )]+ = (f, f 0 ). The functional 80 = 1 can be considered as vacuum vector; it obeys a(f )80 = 0 for all f ∈ H. Operator a+ (f ∗ ) is adjoint to a(f ) with respect to the Hermitian inner product Z ∗ < 81 , 82 >= 81 (a∗ )8∗2 (a)e−(a,a ) dada∗ . For every vector 9 ∈ F we define a linear subspace Ann 9 ⊂ H2 consisting of such pairs (f, g) ∈ H2 that (a(f ) + a+ (g))9 = 0. It is easy to check that the subspace Ann 9 is isotropic. One can prove that for every V ∈ IGr(0) (H) there exists a vector 9 = 9V ∈ F obeying V = Ann 9; this vector is unique up to a factor. (We assume that the Grassmannian is defined by means of Hilbert–Schmidt operators; this is essential for the validity of the above statement.) To give a proof we represent V ∈ IGr(0) (H) as an image of a linear map H → H2

Grassmannian and String Theory

21

transforming ϕ ∈ H into (Aϕ, Bϕ). Here A is a Fredholm operator of index 0, B is a Hilbert–Schmidt operator. The condition V ⊂ Ann 9 means that for every ϕ ∈ H we have X X ∂ Bkl ϕl a∗k )9 = 0. (14) ( Akl ϕl ∗ + ∂ak (We have chosen a basis in H.) The condition that the space V is isotropic means that X (Akl Bkr + Bkl Akr ) = 0. (15) k

Without loss of generality we can assume that the matrix Akl is diagonal; moreover, non-zero entries can be taken equal to 1. We assume that Aii = 0 for i ≤ s, Aii = 1 for i > s. Then Eqs. (14) and the condition (15) take the form ! X ∗ (16) Bkl ak 9 = 0 for l ≤ s, k

∂9 X + Bkl a∗k ∂a∗l

! 9

for l > s,

(17)

k

Blr + Brl = 0 Blr = 0

if l > s, r > s,

if l > s, r ≤ s or l ≤ s, r > s.

The solution to Eqs. (16), (17) is unique (up to a factor) and can be written in the form P X 1 a∗ Bkl a∗ ∗ −2 k k,l≥s l (18) Bkl ak )e 9 = C · 5l≤s δ( k

or in the form 9 = C · 5l≤s (

X

− 21

Bkl a∗k )e

P k,l≥s

∗ a∗ l Bkl ak

.

(19)

k

These two forms are equivalent because δ(a∗k ) = a∗k . (In other words, we have R R δ(a∗k )ϕ(a∗ , a)dada∗ = a∗k ϕ(a∗ , a)dada∗ for every ϕ.) Notice that we proved a little bit more than claimed. Namely, it follows from the proof that for V ∈ IGr(0) (H) there exists a unique (up to a factor) vector 9 obeying Ann 9 ⊃ V and that we have Ann 9 = V for this vector. In the case when the operator A is invertible we can write the functional 9 = 9V in the form ∗ −1 ∗ 1 9V = Ce− 2 (a ,A Ba ) . (It is clear that 9V satisfies (14).) If the operator A has an even number of zero modes we can represent V ∈ IGr(0) (H) as a limit of Vn ∈ IGr(0) (H) in such a way that Vn is an image of a map H → H2 transforming ϕ ∈ H into (An ϕ, Bn ϕ), where An is an invertible operator, An − 1 belongs to the trace class and Bn is a Hilbert–Schmidt operator. Using this representation one can write 9V as a limit of functionals

22

A. Schwarz ∗

(det An )1/2 e− 2 (a 1

∗ ,A−1 n Bn a )

.

We constructed a map IGr(0) (H) → F (defined up to a factor). It is easy to check that applying this construction IGr(0) (Hm ) we obtain an algebra over MO IGr(0) (Hm ). The definition of the isotropic Grassmannian IGr(H) can be easily generalized to the case when H = H0 ⊕H1 is a complex Hilbert Z2 -graded space with antiunitary involution f → f ∗ . (The bilinear inner product (f, g) =< f, g ∗ > should be symmetric in the sense of the superalgebra: (g, f ) = (f, g) if f and g are even, (g, f ) = −(f, g) if f and g are odd.) Almost all considerations above can be repeated with some changes. In particular, one can relate IGr(H) to the Fock space F defined as a space of antiholomorphic functionals 8(a∗ ) on 5H satisfying the condition (13). Let us assume that the space H is realized as a space of functions on the measure space S taking values in Cp|q . (In other words elements of H are functions f (x, α) of x ∈ S and discrete index α; we assume that f (x, α) is even for 1 ≤ α ≤ p and f (x, α) is odd for p + 1 ≤ α ≤ p + q. We will say that α is a superindex, taking p even and q odd values.) One can represent an element of F as an expression on the form 8(a∗ ) =

X

1 (n!)1/2

X

8n (x1 , α1 , . . . , xn , αn )a∗ (x1 , α1 ) . . . a∗ (xn , αn )dn x

α1 ,...αn

obeying X

|8n (x1 , α1 , . . . , xn , αn )|2 dn x < ∞.

α1 ,...αn

(Here xi ∈ S, αi is a superindex taking p even values and q odd values, the function , αj )Rif αi and αj are even 8 is antisymmetric with respect to transposition (xi , αi ) and (xjP indices and symmetric in all other cases.) Multiplication by α f (x, α)a∗ (x, α)dx of f ∈ H. Operators determines an operator a+ (f ) acting onPF and R linearly depending δ dx generate a superanalog a+ (f ) together with operators a(f ) = α f (x, α) δa∗ (x,α) of the Clifford algebra. We can repeat the definition of the linear subspace Ann 9 ⊂ H2 and prove that Ann 9 is isotropic. (Here 9 ∈ F.) Representing V ∈ IGr(0) (H) as an image of the linear map ϕ → (Aϕ, Bϕ), where A is Fredholm operator of index 0, B is a Hilbert–Schmidt operator we obtain a condition of isotropicity and a condition that V ⊂ Ann 9. These conditions coincide with (15) and (14) correspondingly (up to irrelevant signs). For generic V we can assume that Akl is diagonal, Aii = 0 for i ≤ s, Aii = 1 for i > s. Then we can solve the analog of Eq. (16) and obtain the expression (18) for the functional 9 = 9V ; the solution is unique up to a factor. (Of course, (18) is not equivalent to (19) in the general case.) We see that the functional 9 can be considered as an element of F only in the case when the odd-odd block of the matrix A is invertible. A map V → 9V can be considered as a generalized function on IGr(0) (H) with values in F. (More precisely, this function takes values in an appropriate extension of the Fock space.) Analogously we obtain a generalized function IGr(0) (Hm ) → F ⊗m . It is easy to prove that this function can be considered as a (generalized) algebra over IGr(0) (Hm ). If V ∈ IGr(0) (H) is represented as an image of a map ϕ → (Aϕ, Bϕ), where A is an invertible operator, A − 1 belongs to trace class and B is a Hilbert–Schmidt operator, we can write ∗

9V = (det A)1/2 e− 2 (a 1

,A−1 Ba∗ )

.

(20)

Grassmannian and String Theory

23

Of course, det here and further stands for superdeterminant (Berezinian). For general A one can get 9V taking a limit in (20), as we explained above in the case when H is an ordinary Hilbert space. Of course all our consideration determine 9V only up to a factor. Using the embedding of the ordinary Grassmannian into isotropic Grassmannian we can construct a function V → 9V defined for V ∈ Gr(H) and taking values in Fock space (if H is a superspace then 9V takes values in an appropriate extension of Fock space). This function determines a (generalized) algebra over the Grassmannian MO. One can define 9V only up to a factor; however one can give an unambiguous definition of 9V ˜ ˜ (see below). We mentioned already that the on Gr(H) naturally embedded into Gr(H) maps 9V determine an algebra over MO Gr(H k ). One can reformulate the statement k ˜ ) ⊂ Gr(H˜ k ) specify a Q-extension of this above saying that the maps 9V on Gr(H k ˜ )). algebra (a Q-algebra over Q-MO Gr(H One should emphasize, that our consideration of the super Grassmannian and its connection with Fock space was neither rigorous, nor complete. More detailed treatment of related questions can be found in [17–19]. Let us discuss briefly a more general way to construct generalized algebras over Grassmannian MO’s. Every algebra over Gr(H n ), H = L2 (S 1 ) determines an algebra over MO Pn of surfaces with disks (a conformal field theory), because Pn is embedded into Gr(H n ) by means of Krichever construction. Conversely, a conformal field theory can be extended to an algebra over Gr(H n ) if this theory is defined by means of the quadratic Lagrangian (free fermions or bosons, bc-system, βγ-system or any combination of these theories). If a CFT (Pn , E, αn ) has non-vanishing central charge then αn should be considered not as a map Pn → E, but as a section of a bundle E ⊗ (det)k , where det stands for the so-called determinant bundle over Pn . It follows from well known results about determinant bundles over Pn and Gr(H n ) that an extension of a conformal field theory having vanishing central charge to the Grassmannian is an algebra in the strict sense (i.e. there exists an unambiguous definition of a vector corresponding to an element V ∈ Gr(H m )). Using the same idea one can check the above mentioned fact that 9V can be defined ˜ on Gr(H) unambiguously. Acknowledgement. I am indebted to L. Dickey, M. Duflo, M. Kontsevich, M. Mulase, M. Vergne, A. Voronov and B. Zwiebach for useful discussions.

References 1. Segal, G., Wilson, G.: Loup Groups and Equations of KdV type. IHES Publ Math. 61, 5–65 (1985) 2. Mulase, M.: Cohomological structure in soliton equations and Jacobian varieties. J. Diff. Geom. 19, 403–430 (1984) 3. Mulase, M.: Schwarz, A.: In preparation 4. Getzler, E.: Kapranov, M.: Modular operads. Preprint 5. Getzler, E.: Two-dimensional topological gravity and equivariant cohomology. Commun. Math. Phys. 163, 473–489 (1994) 6. Schwarz, A.: Geometry of Batalin-Vilkovisky quantization. Commun. Math. Phys. 155, 249–260 (1993); Semiclassical approximation in Batalin–Vilkovisky formalism. Commun. Math. Phys. 158, 373–396 (1993) 7. Sen, A., Zwiebach, B.: Quantum background independence of closed string field theory. Nucl. Phys. B423, 580 (1994) 8. Duflo, M., Kumar, S., Vergne, M.: Sur la cohomologie equivariante des varietes differentiables. Asterisque 215 (1993)

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9. Segal G.: Two-dimensional conformal field theory and modular functors. In: IXth International Conference on Mathematical Physics. B. Simon, A. Truman and I.M. Davies, eds. Bristol: Adam Hilger, 1989 10. Berezin, F.: The method of second quantization. New York–London: Academic Press, 1966 11. Hsiang, W.Y.: Cohomology theory of topological transformation groups. Berlin–Heidelberg–New York: Springer-Verlag, 1975 12. Schwarz, A.: Iin preparation 13. Atiyah, M., Bott, R.: The moment map and equivariant cohomology. Topology, 23, 1–28 (1993) 14. Berline, N., Getzler, E., Verline, M.: Heat kernels and Dirac operators. Berlin: Springer, 1991 15. Witten, E.: Two-dimensional gauge theories revisited. J. Geom. Phys. 9, 303 (1992) 16. Schwarz, A., Zaboronsky, O.: Supersymmetry and localization. hep-th 95 11112 to appear in CMP 17. Schwarz, A.: Fermionic string and universal moduli space. Nucl. Phys. B317, 323 (1989) 18. Alvarez-Gaume, L., Nelson, Ph., Gomez, C., Sierra, G., Vafa, C.: Fermionic strings in the operator formalism. Nucl. Phys. B311, 333 (1989) 19. Dolgikh, S., Schwarz, A.: Supergrassmannians, super tau-functions and strings. In: Physics and Mathematics of Strings, Editors: L. Brink, D. Friedan, A. Polyakov, Singapore: World Scientific, 1990, pp. 231–244 20. Friedlander, L., Schwarz, A.: Grassmannian and elliptic operators. funct-an 9704003 Communicated by R. H. Dijkgraaf

Commun. Math. Phys. 199, 25 – 69 (1998)

Communications in

Mathematical Physics © Springer-Verlag 1998

The Donaldson–Witten Function for Gauge Groups of Rank Larger Than One ˜ Gregory Moore Marcos Marino, Department of Physics, Yale University, New Haven, CT 06520, USA. E-mail: [email protected], [email protected] Received: 30 March 1998 / Accepted: 17 April 1998

Abstract: We study correlation functions in topologically twisted N = 2, d = 4 supersymmetric Yang–Mills theory for gauge groups of rank larger than one on compact four-manifolds X. We find that the topological invariance of the generator of correlation functions of BRST invariant observables is not spoiled by noncompactness of field space. We show how to express the correlators on simply connected manifolds of b2,+ (X) > 0 in terms of Seiberg–Witten invariants and the classical cohomology ring of X. For manifolds X of simple type and gauge group SU (N ) we give explicit expressions of the correlators as a sum over N = 1 vacua. We describe two applications of our expressions, one to superconformal field theory and one to large N expansions of SU (N ) N = 2, d = 4 supersymmetric Yang–Mills theory.

1. Introduction and Conclusion The Donaldson invariants of 4-manifolds have played an important role in the development of both mathematics and physics during the past fifteen years. Donaldson’s invariants are defined using nonabelian gauge theory for gauge group G = SU (2) or G = SO(3) on a compact oriented Riemannian 4-manifold X [1, 2]. They were interpreted by Witten in [3] as correlation functions in an N = 2 supersymmetric Yang–Mills theory (SYM) and as such are best presented as a function on H∗ (X) defined by a path integral: (1.1) ZDW (v · S + pP ) = exp[v · I(S) + pO(P )] , where P ∈ H0 (X; Z), S ∈ H2 (X; Z), I(S) and O(P ) are certain operators in the gauge theory, and the right hand side of (1.1) is an expectation value. We refer to [3] as the Donaldson–Witten function. Witten’s interpretation has lead to significant progress in the subject [4].

26

M. Mari˜no, G. Moore

1.1. Questions, and answers. Since (1.1) is a correlation function in an SU (2) or SO(3) gauge theory it is quite natural to ask about the generalization to compact simple gauge groups G of rank larger than one. The formal definitions, both mathematical and physical, proceed with little essential modification to the higher rank case so we may ask the following three basic questions: 1. Is ZDW an invariant of the diffeomorphism type of X? 2. Does ZDW define new 4-manifold invariants that go beyond the classical cohomology ring and the Seiberg–Witten invariants? 3. Can ZDW be evaluated explicitly? In this paper we answer these questions: 1. Yes, ZDW is a topological invariant for rank(G) ≡ r > 1. 2. No, ZDW does not contain any new topological information, at least for 4-manifolds of b2,+ > 0. 3. Yes, ZDW can be explicitly evaluated in terms of the classical cohomology ring and Seiberg–Witten invariants. These conclusions require further comment. The first question is not silly. From the mathematical point of view the instanton moduli spaces are quite singular and it is not obvious that there is a well-defined intersection theory on them. From the physical point of view, although the path integral is formally topologically invariant the expression for ZDW given below is, to say the least, intricate and delicate, and involves integrals over noncompact spaces with singular integrands. Given the phenomenon of wall-crossing [5–8] and the surprising discovery of [9] of continuous metric dependence in a superconformal SU (2) theory the topological invariance of ZDW is not obvious. Among other things one should worry about continuous metric dependence of ZDW arising from integration over the subvarieties in the moduli space of supersymmetric vacua with superconformal symmetry. The main technical work in this paper consists of carefully defining the integrals and checking their metric dependence. Our conclusion, as stated, is that there is no continuous variation. Somewhat surprisingly, in stark contrast to the rank one case, we find that there is no wall-crossing from the measure in the semiclassical regime. The answer to the second question is, of course, a disappointment. One of the main motivations for this work was the suggestion of E. Witten, made during the investigations of [9], that wall-crossing phenomena at superconformal points would lead to the discovery of new 4-manifold invariants. We would like to stress that we are not suggesting that N = 2 superconformal theories provide no new topological information (in fact we believe the opposite). However, if there are new invariants, they are inaccessible via the wall-crossing technique used in [9] and described below. Regarding the third question, the general formula is rather complicated and is only described in full detail for G = SU (3) in Eqs. (9.1)–(9.6) below. An important representative case is that of simply connected manifolds of b2,+ > 1 and of simple type. The resulting expression for G = SU (N ) is given in Eq. (9.13) below. It is a natural generalization of the expression found by Witten in [4] for the rank one case. 1.2. Method of derivation. Deriving the higher rank Donaldson invariants using the standard mathematical methods of [1, 2] does not work very well. Formal aspects of the problem, like the µ-map generalize straightforwardly but, because of the singularities of instanton moduli space, the intersection theory is difficult to define.

Donaldson–Witten Function for Rank Larger Than One

27

The physical approach to the problem turns out to be much more powerful. By the physical approach we mean the program proposed by Witten in [3], and brought to fruition in [4]. Some further technical developments described in [9] make the derivation of the main result of [4] and its higher rank generalizations conceptually straightforward, (although technically challenging in the higher rank case). The main insight we use from [9] is that one can derive the relation between Donaldson and Seiberg–Witten invariants from the phenomenon of wall-crossing. 1 This wall-crossing method proceeds as follows. One begins by considering the contribution to ZDW of the Coulomb branch of the moduli space of supersymmetric vacua on R4 . This contribution, denoted by ZCoulomb , is nonzero only for manifolds of b2,+ = 1. Nevertheless ZCoulomb turns out to contain the essential information for deriving the contributions of the SW branch to ZDW . In particular, cancellation of metric-dependence of ZCoulomb at strong coupling singularities in moduli space allows a complete derivation of the universal functions appearing in the Lagrangian of the magnetic dual theory with the light monopole or dyon hypermultiplet fields included in the theory. (See section seven of [9].) The wall-crossing method generalizes to simple gauge groups of rank r > 1. The Coulomb branch MCoulomb is now a quasi-affine variety of complex dimension r. In the weak coupling asymptotic region MCoulomb may be described as (t ⊗ C)/W , where t is a Cartan subalgebra for G and W is the Weyl group. More globally, the Coulomb branch has the form: (1.2) MCoulomb = Cr − D). In Seiberg–Witten theory the space of vacuum expectation values hTrφj i, for j ranging over the exponents of G, is identified with Cr . The low energy theory is characterized by a family of Abelian varieties over Cr , and D is the singular locus for this family. One can introduce local special coordinates aI on (1.2) , but these are never global coordinates and together with their duals aD,I transform in nontrivial ways under the quantum monodromy group 0, determined in principle from the explicit SW curve and differential. For the example of SU (N ), D is defined by the vanishing of the “quantum discriminant” of Eq. (2.10) below. Unfortunately a concise description of the discrete group 0 ⊂ Sp(2r; Z) does not appear to be available. The expression ZCoulomb turns out to have the general form: Z 2 [dad¯a]A(~u)χ B(~u)σ eU +S TV (~u) 9, (1.3) ZCoulomb = MCoulomb

where aI are local special coordinates, A, B are holomorphic automorphic forms for 0 described below, U, T are forms associated with operator insertions, χ, σ are the Euler character and signature of X, and 9 is a certain Narain-Siegel lattice theta function associated to H 2 (X; Z) ⊗ 3w (G), where 3w (G) is the weight lattice of G. The details of this expression are explained in Sects. 3 and 4 below. Some aspects in the derivation of the integrand were independently worked out in [12]. It is far from obvious that the integrand of (1.3) is single-valued on MCoulomb . We check this carefully for the case G = SU (3) and give a less-detailed general argument for single-valuedness for G = SU (N ) (although we prove the invariance under the semiclassical monodromies for any simply-laced group). We do not seriously doubt that 1

11].

Some mathematical papers dealing with the relation of Seiberg–Witten and Donaldson invariants are [10,

28

M. Mari˜no, G. Moore

the integrand of (1.3) is single-valued for all G of r > 1, but our arguments for this leave room for improvement. The integrand of (1.3) is singular on D and in the weak-coupling regime at infinity. Hence, some discussion is required to give rigorous meaning to the integral (1.3) . To do this we need to understand the structure of the divisor D more thoroughly. This divisor is a stratified space. The maximum dimension stratum is a union of several smooth complex codimension one components Di(1) corresponding physically to moduli for which a single u(1) ⊂ t becomes strong and for which a single monopole hypermultiplet becomes massless. The strata of D of higher codimension correspond to singularities in D where successively larger numbers of hypermultiplets become massless. We denote the smooth components of the codimension ` strata by Di(`) . In particular, Di(r) contains h(G) (the dual Coxeter number) points corresponding to the supersymmetric vacua of the N = 1 theory, as well as points corresponding to multicritical superconformal field theories. Following the discussion in [9] we define the integral by introducing a cutoff in the weak-coupling regime and by introducing tubular neighborhoods of Di(1) and doing a phase integral first over the relevant special coordinates. The definition of the integration near Di(`) for ` > 1 is more problematic and we only discuss it in full detail in the case G = SU (3). All this is described in Sects. 6 and 8. We expect our considerations to generalize to other gauge groups, but again our treatment leaves room for improvement. We then implement the wall-crossing argument of [9] by postulating that the metric variation of ZCoulomb from the singularities near Di(1) is cancelled by compensating metric-variation of a mixed Coulomb/monopole theory which describes the low-energy physics near Di(1) . A consequence of our postulate 2 is that ZDW must take the form: ZDW = ZCoulomb +

X i

ZD(1) + i

X i

ZD(2) + · · · + i

X i

ZD(r) . i

(1.4)

This is the generalization of Eq. (1.8) of [9]. The integrals ZD(1) along the codimension i one varieties are derived from the wall-crossing of ZCoulomb . This wall-crossing is described in Sect. 6, and the explicit formulae for G = SU (3) are derived in complete detail in Eqs. (9.1)–(9.6) below. The integrals ZD(1) themselves have wall-crossing behavior i which is compensated by metric dependence of ZD(2) . This allows a derivation of the i integrand of ZD(2) and so on. The procedure terminates at the codimension r singularities i of D. The central question of metric variation at a superconformal point is addressed in section eight. We analyze the behavior at the Argyres–Douglas points for G = SU (3) in detail and show that there cannot be any continuous metric dependence unless σ < −11. We also give a general argument that shows there cannot be any continuous metric dependence for any signature. This argument is based on the blow-up and wall-crossing formulae. The blow-up formula for the higher rank case is derived in section seven by a straightforward generalization of the derivation in [9]. Using this formula we can relate the invariants on X to invariants on a blowdown with sufficiently large signature that there can be no continuous metric variation. P In the case b2,+ > 1 only the last term i ZD(r) of (1.4) is nonvanishing, and indeed i only the N = 1 vacua contribute. This allows us to write the generalization of Witten’s formula [4] to G = SU (N ) in Eq. (9.13). 2

which may be justified physically from considerations of tunneling between vacua at finite volume

Donaldson–Witten Function for Rank Larger Than One

29

1.3. Applications. Ironically, our work, which was motivated by topology, might find its most interesting applications in physics. In Sects. 10 and 11 we describe two applications. In Sect. 10 we use the behavior of the integrand of (1.3) at superconformal points to deduce a selection rule for correlators of N = 2, d = 4 superconformal theories. In Sect. 11 we use the explicit result (9.13) to study some questions about the large N behavior of certain correlation functions in SU (N ) SYM theory. 1.4. Directions for future work. There are plenty of opportunities for future work. First, there are technical gaps in our discussion which have been indicated above. We are confident in our conclusions, but it should be possible to give a better treatment of the analysis of ZCoulomb . As in [9] the discussion is rather easily extended to theories with matter. For example, almost the same formulae hold for G = SU (Nc ) with Nf fundamental hypermultiplets, as long as the masses of the hypermultiplets are generic. For special values of the masses some very interesting things should happen and this remains an interesting avenue for future research. Another generalization worth studying is the case of SU (Nc ) with N = 4 supersymmetry perturbed to N = 2 by the addition of a mass perturbation for adjoint hypermultiplets. A discussion of ZCoulomb for these theories is technically challenging but might find interesting applications in string/M theory. The generalization of the u-plane integrals studied in [9] to the higher rank case is probably only the first of a series of interesting generalizations of similar integrals associated to special K¨ahler manifolds. 2. Higher Rank N = 2 Gauge Theories In this section, we review some properties of the low-energy structure of N = 2 gauge theories that we will need in this paper [13–15]. We then focus on the case of SU (N ) Yang–Mills theory, and in particular on the SU (3) moduli space, which has been explored in some detail 16, 14. We also work out some aspects of the solution near the superconformal, or Argyres–Douglas (AD) points that will be needed in the rest of the paper. 2.1. General structure. The classical moduli space of N = 2 SYM with a rank r gauge group is determined by the vacuum expectation value of the field φ, which can always be rotated into the Cartan subalgebra. Following [14], we will denote these expectation values by a vector ~a in the root lattice, and the components of ~a , aI with I = 1, · · · , r will correspond to a basis of simple roots. The charges will be specified by vectors ~q expanded in the Dynkin basis (i.e. the basis of fundamental weights). The central charges of electric BPS states are then written as Zq~ = ~q · ~a,

(2.1)

where the product is given by the usual bilinear form in the weight lattice. One can then introduce the Casimirs uk as Weyl-invariant coordinates in the classical moduli space. Singularities in this moduli space are associated semiclassically to massless gauge ~ is a positive root. They are located at the bosons, and they occur when Zα~ = 0, where α zeroes of the classical discriminant, Y Zα~2 . (2.2) 10 (~u) = α ~ >0

30

M. Mari˜no, G. Moore

The low energy effective action is determined by a prepotential F which depends on r N = 2 vector multiplets AI . The VEVs of the scalar components of these vector superfields are the aI . The dual variables and gauge couplings are defined as aD,I =

∂F , ∂aI

τIJ =

∂2F . ∂aI ∂aJ

(2.3)

The moduli space of vacua has a natural K¨ahler metric given by (ds)2 = ImτIJ daI d¯aJ ,

(2.4)

which is invariant under the group Sp(2r, Z) (the restriction to integer valued entries comes from the integrality requirement of the charges, as we will see in a moment). The inverse metric will be denoted by (Imτ )IJ . Matrices in Sp(2r, Z) have the structure AB γ= , (2.5) CD where the r × r matrices A, B, C, D satisfy: At D − C t B = 1,

At C = C t A,

B t D = Dt B.

The generators of the symplectic group Sp(2r, Z) are A 0 A= , A ∈ Gl(r, Z), 0 (At )−1 1θ Tθ = , θIJ ∈ Z, θt = θ, 01 0 1 = . −1 0

(2.6)

(2.7)

The symplectic group acts on the aI , aD,I variables as v → γv, where v t = (aD,I , aI ). In particular, we have the following transformation properties which will be useful later, ∂ ∂ → [(Cτ + D)−1 ]JI J , I ∂a ∂a τ → (Aτ + B)(Cτ + D)−1 ,

(2.8)

Imτ → [(Cτ + D)−1 ]t (Imτ )(Cτ + D)−1 .

2.2. SU (N ). In the SU (N ) case, the quantum theory is described by the hyperelliptic curve [17, 18]: y 2 = P (x)2 − 32N ,

P (x) = xN −

N X

uI xN −I ,

(2.9)

I=2

where the uI , I = 2, . . . , N , are the elementary symmetric polynomials in the roots of P (x). The quantum discriminant associated to this curve is given by 2

13 = 32N 10 (u2 , . . . , uN −1 , uN + 3N )10 (u2 , . . . , uN −1 , uN − 3N ).

(2.10)

Donaldson–Witten Function for Rank Larger Than One

31

The Coulomb branch of the quantum theory is then given by Cr − D, where D is the vanishing locus of (2.10) . To obtain the couplings τIJ and the masses of the BPS states, one chooses a symplectic homology basis for the genus r Riemann surface described by (2.9) , αI , β I , I = 1, · · · , r, and the basis of holomorphic differentials ωI = xI−1 dx/y. The periods of the curve are then I ∂aD,I ωJ = , AIJ = ∂uJ+1 αI (2.11) I ∂aI ωJ = , B IJ = ∂uJ+1 βI where I, J = 1, · · · , r. The gauge coupling is then given by τIJ = AIK (B −1 )KJ .

(2.12)

One then introduces a meromorphic one form on the hyperelliptic curve (usually known as Seiberg–Witten differential) λSW satisfying ∂λSW = ωI , ∂uI+1

(2.13)

which has the explicit expression [17]: λSW =

1 ∂P xdx . 2πi ∂x y

The BPS masses are then given by the periods of λSW : I I λSW , aI = λSW . aD,I = αI

(2.14)

(2.15)

βI

Quantum-mechanically, the singularities in the moduli space are associated to massless dyons. Their charges will be denoted by ~ν = (~g , ~q), where ~g , ~q are the r-component vectors of magnetic and electric charges, respectively. When one of these dyons becomes massless at a certain submanifold of the moduli space, one of the cycles of the hyperelliptic curve degenerates and there is an associated monodromy given by 1 + ~q ⊗ ~g ~q ⊗ ~q . (2.16) M~ν = −~g ⊗ ~g 1 − ~g ⊗ ~q 2.3. SU (3) and the AD points. In the case of N = 2 SYM theory with gauge group SU (3), the moduli space is parametrized by the Casimirs u = u2 , v = u3 . There is a discrete, anomaly-free subgroup Z6 of the R-symmetry which acts as u → e2πi/3 u, v → −v. The quantum discriminant is given by 13 = 318 [4u3 − 27(v + 33 )2 ][4u3 − 27(v − 33 )2 ].

(2.17)

There are two codimension one submanifolds given by 10 (u, v ± 33 ) = 0, which intersect in the three Z2 vacua 4u3 = (332 )3 , v = 0. At these points there are two mutually local dyons becoming massless, and when N = 2 is softly broken down to N = 1 with a superpotential Tr82 , they give the three vacua of N = 1 SYM [17]. The charges (n1m , n2m ; n1e , n2e ) of these states are the following [14]:

32

M. Mari˜no, G. Moore

~ν1 = (1, 0; 0, 0), ~ν2 = (0, 1; 0, 0), ~ν3 = (0, 1; −1, 2), ~ν4 = (−1, −1; 2, −1), ~ν5 = (−1, −1; 1, −2), ~ν6 = (1, 0; −2, 1).

(2.18)

The charges in the same row in (2.18) are mutually local. The first row corresponds to the Z2 vacuum at u1 = (27/4)1/3 32 , v = 0. The second and third rows correspond to the vacua located at u2 = e2πi/3 u1 , v = 0, and u3 = e4πi/3 u1 , v = 0, respectively. In fact one can find a matrix U [14] which implements the Z3 symmetry in moduli space,   −1 −1 1 2  1 0 −2 1  U = . (2.19) 0 0 0 −1 0 0 1 −1 One can check that, acting on the right on ~ν1 , ~ν2 , we obtain the other two pairs of massless states at the N = 1 points, i.e., ~ν1 U −1 = ~ν3 , ~ν2 U −1 = ~ν4 , etc. There are also singular points on each of the submanifolds (also called Z3 vacua) at the points u = 0, v = ±33 . These are the Argyres–Douglas (AD) points, where three mutually non-local hypermultiplets become massless [16]. We will be particularly interested in the behaviour of the theory near these points. Let’s focus on the point v = 33 , u = 0 (the behaviour near the other AD point can be obtained using the Z6 symmetry which sends v → −v). The states that become massless near this point are ~ν1 , ~ν3 and ~ν5 . The symplectic transformation Tθ −1 A, where −1 0 −1 −1 (2.20) A= , θ22 ∈ Z, , θ= 2 1 −1 θ22 gives a basis where all the states are charged only with respect to the first U (1) factor. Applying A−1 Tθ−1 on the right to the charge vectors νi , we find that the new charges ~ν · A−1 Tθ−1 are (in this new basis) (ne , nm ) = (−1, 0) for ~ν1 , (1, −1) for ~ν3 , and (0, 1) for ~ν5 , i.e. we have one electron, one dyon, and one monopole. In this basis, the hyperelliptic curve degenerates and at leading order in u, v − 33 , it splits into a “small” torus whose periods go to zero (and correspond to a1 , aD,1 ) and a “large” torus whose periods a2 , aD,2 are of order 3. We introduce now the useful parameters , ρ around the AD point, defined by u = 32 ρ,

v − 33 = 23 .

(2.21)

The variable ρ parametrizes the direction along which we approach the AD point in the u, v moduli space. The equation defining the small torus near the AD point is given by

with discriminant

w2 = z 3 − 3ρz − 2,

(2.22)

1ρ = 4 · 27(ρ3 − 1),

(2.23)

and the Seiberg–Witten differential on the curve (2.22) degenerates to λSW = p

5/2 33/2

wdz,

(2.24)

where p is some constant that depends on the normalization of λSW . The small torus theory gives us the dependence on ρ for the leading terms in of a1 , aD,1 . We can put

Donaldson–Witten Function for Rank Larger Than One

33

the curve (2.22) in Weierstrass form and compute a1 , aD,1 explicitly (at leading order in ) in terms of the periods of the curve ωρ , ωρ,D (with Im(ωρ,D /ωρ ) > 0) : a1 =

5/2

f (ρ), 33/2

aD,1 =

5/2 33/2

fD (ρ),

(2.25)

where

ωρ 48p ωρ,D 48p (ρη − ), fD (ρ) = (ρηD − ). (2.26) 5 8 5 8 In this equation, η = ζ(ωρ /2), ηD = ζ(ωρ,D /2) are the usual values of the Weierstrass zeta function at the half-periods. The curve (2.22) has singularities when ρ3 = 1 and also at infinity. At ρ3 = 1 we have A0 singularities and the behaviour of τ (ρ) is f (ρ) =

τ (ρ) =

1 log(ρ − ρk ), 2πi

(2.27)

where ρk = e2πik/3 , k = 0, 1, 2 are the corresponding singularities. At ρ → ∞, there is an H1 singularity (Kodaira’s type III) and the monodromy is given by S −1 . The behaviour of τ (ρ) is given by τ (ρ) = i +

C + ··· , ρ3/2

(2.28)

where C is a nonzero constant. The behaviour of a2 , aD,2 (the “long periods”) can be found [16, 19] to be a2 = b3 + c

v − 33 u +d + ··· , 3 32

aD,2 = bD 3 + cD

u v − 33 + dD + · · · , (2.29) 3 32

where b, c, d, bD , cD and dD are non-zero constants. From these explicit expressions we can compute the matrix of periods of the hyperelliptic curve, B IJ , at leading order: ! ! ∂a1 ∂a1 1/2 −1/2 0 f (ρ) H(ρ) ∂u ∂v 333/2 = 333/2c , (2.30) d ∂a2 ∂a2 ∂u

3

∂v

32

where the derivatives are with respect to ρ and f 0 (ρ) = 12pη,

H(ρ) =

5 3 f (ρ) − ρf 0 (ρ) = − pωρ . 4 2

(2.31)

The matrix AIJ has a similar expression in terms of cD , dD and fD (ρ). We then find that det

∂uI+1 235/2 1/2 = + O(3/2 ). ∂aJ pc ωρ

(2.32)

The gauge couplings can be also computed in a straightforward way and are given by 0 d fD (ρ) + O(2 ), c3 H(ρ) 4πip 1/2 =− + O(3/2 ), c31/2 ωρ dD f 0 (ρ) cD − + O(2 ), = c c3 H(ρ)

τ11 =τ (ρ) − τ12 τ22

(2.33)

34

M. Mari˜no, G. Moore

and cD /c = eπi/3 is the period of the large torus at the AD point (some aspects of the behaviour of the couplings at the AD point have been investigated in [20]). Finally, we will need the behaviour of the third derivatives of the prepotential near this point (and in particular their leading behaviour in ). These are given by: 33/2 −5/2 dτ (ρ) ρ + O(−3/2 ), H(ρ) dρ 53−2 f (ρ) dτ (ρ) + O(−1 ), = 12c H(ρ) dρ 31/2 −3/2 dD f 0 (ρ) 0 1 f 0 (ρ) + O(−1/2 ), = − cH(ρ) H(ρ) 2 H(ρ) −1 dD f 0 (ρ) 2 5 f (ρ) f 0 (ρ) 0 = − + O(1). 6c2 H(ρ) 2 H(ρ) H(ρ)

F111 = − F112 F122 F222

(2.34)

This behaviour is consistent with the R-charge assignment near the superconformal point, R(a1 ) = 1, R(a2 ) = R(u) = 4/5, R(F ) = 2. 3. The Twisted Effective Theory on the Coulomb Branch To study the twisted supersymmetric N = 2 SYM theory on a four-manifold X, we consider the low-energy description encoded in the solution presented in the last section. The procedure we will follow is a straightforward generalization of the one presented in [9]. The field content of the low-energy theory consists of r twisted abelian N = 2 vector multiplets. The Q-transformations are given by √ [Q, ψ I ] = 4 2daI , [Q, AI ] = ψ I , √ [Q, aI ] = 0, [Q, a¯ I ] = 2iη I , (3.1) [Q, χI ] = i(F+I − D+I ), [Q, η I ] = 0, [Q, DI ] = (dψ I )+ . We will also need the action of the one-form operator G, which gives a canonical solution to the descent equations (this operator was denoted by K in [9]). It is given by 1 [G, aI ] = √ ψ I , 4 2 [G, AI ] = −2iχI , √ i 2 I I d¯a , [G, η ] = − 2 √ 3i 2 ∗ d¯aI . [G, χI ] = − 4

[G, ψ I ] = −2(F−I + DI ), [G, a¯ I ] = 0, [G, DI ] = −

3i 3i ∗ dη I + dχ, 4 2

(3.2)

The twisted effective Lagrangian can be written in a manifestly topological way as: √ 1 i 2 i 4 G F(aI ) + {Q, F IJ χI (D + F+ )J } − {Q, F I d ∗ ψ I } 6π √ 16π 32π (3.3) 2i I νλJ µK {Q, F IJK χµν χ χλ }, − 3 · 25 π

Donaldson–Witten Function for Rank Larger Than One

35

which may be expanded out to give: 1 1 i τ IJ F+I ∧ F+J +τIJ F−I ∧ F−J + (ImτIJ )daI ∧ ∗d¯aJ − (ImτIJ )DI ∧ ∗DJ 16π 2π 8π 1 1 1 1 τIJ ψ I ∧ ∗dη J + − τ IJ η I ∧ d ∗ ψ J + τIJ ψ I ∧ dχJ − τ IJ χI ∧ (dψ J )+ 16π 16π 8π 8π √ √ i 2 i 2 + F IJK η I χJ ∧ (D+ + F+ )K − 7 FIJK (ψ I ∧ ψ J ) ∧ (F− + D+ )K 16π 2 π √ i 2i FIJKL ψ I ∧ ψ J ∧ ψ K ∧ ψ L − {Q, F IJK χIµν χνλJ χλµK }. + 11 3·2 π 3 · 25 π (3.4) It is important to notice that the part of the action involving the fourth descendant of the prepotential can be written as a Q-exact term plus terms which are topological (i.e. they do not involve the metric of the four-manifold X): i 4 G F(aI ) = 6π √ h √2i i 1 I 2 I J − J χ ∧ (F + D) + 7 FIJK ψ I ∧ ψ J ∧ χK Q, τIJ − 5 ψ ∧ ∗d¯a − 2 ·π 16π 2 ·π √ IJ i 2 i iτ F I ∧F J − 7 FIJK (ψ I ∧ψ J )∧F K + FIJKL ψ I ∧ψ J ∧ψ K ∧ψ L , + 16π 2 π 3 · 211 π (3.5) where integration over X is understood. 4. The Integrand in the Higher Rank Case The u-plane integral in the higher rank case is given by a general expression of the form: Z 2 [dad¯a]A(~u)χ B(~u)σ eU +S TV (~u) 9, (4.1) Zu (p, S; mi , τ0 ) = MCoulomb

where 9 is a certain lattice theta function. We will explain in some detail the structure of the different terms involved in (4.1) . The resulting expression, as we will see, holds for any simply-laced gauge group. 4.1. The observables. In (4.1) the 0-observable is a general invariant function U on the Lie algebra. This generalizes 2pu in the rank one case. We will restrict attention to expressions linear in the Casimirs of the group, U=

r+1 X

pI TrφI .

(4.2)

I=2

Here we are using the standard notation for the Casimirs of SU (N ). The VEVs of these operators can be related to the symmetric polynomials in (2.9) which parametrize the quantum moduli space by standard results on symmetric functions. The 2-observable is obtained by canonical descent from another general function V . When 2-observables are included one has to take into account contact terms, denoted by TV (~u). We will discuss them below. For simplicity, we will restrict attention to 2observables obtained from the quadratic Casimir, V = u2 . Other 2-observables involve

36

M. Mari˜no, G. Moore

new contact terms discussed in [12]. In general, the 2-observable is obtained as in [9] using the one-form operator G, which gives canonical solutions to the descent equations. In the rank r case we have √ 1 ∂ 2 uI 2 ∂uI 2 J K ψ ∧ψ − (F− + D+ )J . (4.3) G uI = 32 ∂aJ ∂aK 4 ∂aJ The two-observable associated with a surface S is given by Z i ˜ I(S) = √ G2 u I , π 2 S

(4.4)

where we use the normalization of [9]. The four-observables come again from a general function W . Using the canonical solution to the descent equations we see that they merely shift τIJ → τIJ + WIJ . This will involve further contact terms, which can be written by a process of covariantizing derivatives. 4.2. The measure factor. The A, B functions in (4.1) are the higher rank generalization of the gravitational factors considered in [21, 9]. They are given by: χ/2 ∂uI χ χ , (4.5) A = α det J ∂a σ/8

B σ = β σ 13 .

(4.6)

This may be proved by a modification of the argument of [21, 9]. The twisted theory with gauge group G has a gravitational contribution to the anomaly given by −(dimG)(χ + σ)/2. In the semiclassical regime the effective U (1)r theory gives the anomaly −r(χ+σ)/2. The remaining anomaly should be carried by the measure factor in the semiclassical region. On the other hand, near the divisor where a single hypermultiplet becomes massless, there is an accidental low-energy R-symmetry given by −σ/4 which should also show up in the measure factor in this region. We first check that the B σ factor gives the needed behaviour for the σ dependence. Near the divisor where a single hypermultiplet becomes massless, the quantum discriminant has the structure σ/8 ˜ 3, (4.7) 13 ∼ Z σ/8 1 ˜ 3 6= 0 along it. As Z has where Z is the transverse coordinate at the divisor and 1 R-charge two, we see that B σ gives the right behaviour. On the other hand, in the semiclassical region we have that 13 ∼ (10 )2 .

(4.8)

The Zα have R-charge 2. As there are (dimG − r)/2 positive roots, the R-charge of 13 in the semiclassical region is given by 4(dimG − r), and again we find the right charge. As 13 is a modular form of weight zero, B/13 has no zeroes and is a constant. This proves (4.6) . We now consider the Aχ factor. The R-charge at the semiclassical regime is easily computed to give χ(dimG − r)/2, again in agreement with the behaviour we need. On the other hand, we have to check that (in the appropriate local variables) this factor does not have zeros or singularities on the moduli space. Notice that, at a generic point in the moduli space, the Aχ factor can be written as

Donaldson–Witten Function for Rank Larger Than One

37

Aχ = αχ detB IJ

−χ/2

,

(4.9)

where detB IJ is the first minor of the period matrix of the hyperelliptic curve (2.9) , and is given in (2.11) . It follows from the Riemann bilinear relations that this minor is nonsingular. On a divisor where a hypermultiplet becomes massless, there are good coordinates aI around it in the sense that the Jacobian of the change of variables from uI to aJ is nonsingular, and again we see that in the appropriate variables this factor has no zeros or singularities. Since det ∂uI+1 /∂aJ is a modular form of weight (−1, 0), we have proved (4.5) . We will comment on the constant α below. 4.3. The lattice 0 and generalized Stiefel–Whitney classes. The function 9 in (4.1) , as we will see, involves the evaluation of the photon partition function for the effective U (1)r theory. Therefore, it includes a sum over electric line bundles [21]. We will consider theories with a non-abelian magnetic flux. This is possible, for instance, in the case of an SU (N ) theory, because the gauge group is actually SU (N )/ZN (provided all fields are in the adjoint representation of the group). A bundle E with this gauge group is characterized up to isomorphism by two topological invariants : the instanton number and the generalized Stiefel–Whitney class (or non-abelian magnetic flux) w ~ 2 (E) ∈ ~ 2 (E) takes values H 2 (X, ZN ). For a gauge group G, the non-abelian magnetic flux w in H 2 (X, π1 (G)). Equivalently [22], for any simply-laced gauge group, the magnetic fluxes are cohomology classes in H 2 (X, 3w /3r ), where 3w(r) are the weight and root lattices of the group, respectively. For every weight lattice, there is a set of weights called minimal weights which are in one-to-one correspondence with the cosets 3w /3r ([23] p. 72). There are in general c − 1 minimal weights, where c = det C is the “index of connection”, that is, the determinant of the Cartan matrix (notice that c is precisely the order of 3r in 3w ). The set of minimal weights is in general a subset of the set of fundamental weights. We will denote these weights by m ~ I , I = 1, · · · , c − 1. In the case of SU (N ), they are just the fundamental weights w ~ I , I = 1, · · · , N − 1. The electric line bundles are then classified by vectors of the form: ~λ = ~λZ + ~v ,

~λZ =

r X

λIZ α ~I,

~v =

I=1

c−1 X

πI m ~ I,

(4.10)

I=1

where α ~ I is a set of simple roots. In this expression, λIZ , π I are all integer classes in 2 ~ 2 (E) ∈ H 2 (X, 3w /3r ) lifted H (X; Z). The π I are fixed, and represent a choice of w 2 to H (X, 3r ). Notice that we can always expand the minimal weights in the basis of simple roots: m ~I =

r X

mIJ α ~J,

I = 1, · · · , c − 1,

J=1

mIJ ∈

1 Z, c

(4.11)

therefore we can write ~λ =

r X I=1

λI α ~I,

λI = λIZ +

c−1 X J=1

mJI π J ∈

1 2 H (X, Z). c

(4.12)

For SU (N ), one has mIJ = (C −1 )IJ , where CIJ is the Cartan matrix. Finally, later we will need the result that the instanton number of the original bundle E satisfies [22]

38

M. Mari˜no, G. Moore

c2 (E) = −

~v · ~v mod 1. 2

(4.13)

4.4. The lattice sum. The lattice sum 9 appearing in (4.1) is obtained after integrating over the zero modes of the fields, integrating out the auxiliary fields (after including the 2-observable (4.4) ) and taking into account the photon partition function. The procedure is entirely analogous to the one presented in [9] for SU (2). The only difference is that the photon partition function includes now a factor (det Imτ )−1/2 (for simplyconnected manifolds). We also have r zero modes for η I as well as for χI (when b+2 = 1). Because of the argument based on the scaling of the metric of [9], the contribution of the Coulomb branch vanishes if b+2 > 1. We also write F I = 4πλI , which is the appropriate normalization for the line bundles involved in the sum. The lattice 0 of λI has been already specified for the general case in which we have non-abelian magnetic fluxes. After taking all this into account, we finally obtain a formula for the factor 9 in (4.1) : 1 X VJ (Imτ )JK VK S+2 9 = (det Imτ )−1/2 exp 8π λ∈0 I J I J I ~ ~ exp −iπτ IJ (λ+ , λ+ ) − iπτIJ (λ− , λ− ) − iπ((λ − λ0 ) · ρ ~, w2 (X)) − iVI (S, λ− ) √ Z Y r i 2 KL dη I dχI exp − F IJK η I χJ [4π(λK VL (S, ω)] + , ω) + i(Imτ ) 16π I=1 1 + F KLI (Imτ )IJ F JP Q η K χL η P χQ . 64π (4.14) ∂V ~ ~ Here VI = ∂aI . The phase factor exp[−iπ((λ − λ0 ) · ρ ~, w2 (X))] can be derived by a generalization of Witten’s analysis in [21] (see [12] for a derivation along these lines). We found it (independently) from invariance of the Coulomb integral under the semiclassical monodromy. ~λ0 is an element in 0 such that ~λ − ~λ0 ∈ H 2 (X, 3r ), and corresponds to a choice of orientation of the higher rank instanton moduli spaces. Notice that its inclusion is necessary in order for the phase factor to be defined independently of the integral lift we choose for ~λ. One should also include in the lattice sum a global phase factor depending on the generalized Stiefel–Whitney class ~v , in order to obtain invariants that are real. 2 In the SU (2) case, this factor turns out to be eiπ~v·~v = eiπw2 (E) /2 [9]. We will find the appropriate factor for SU (N ) after computing the resulting invariants in Sect. 9. In the rank one case, this lattice sum is related to the sum 9r=1 introduced in [9] as follows: √ i du i 2 1 dτ X (λ−λ0 )·w2 (X) (S, ω) (−1) (λ, ω) + 9=− 4 y 1/2 da 4πy da λ∈0 du 2 2 (4.15) · exp −iπτ (λ+ ) − iπτ (λ− ) − i (S, λ− ) da √ 1 du 2 2 i 2 1 dτ exp − S 9r=1 , =− 1/2 4 y da 8πy da where τ = x + iy. We can explicitly evaluate the Grassmann integral in the rank two case, with the result:

Donaldson–Witten Function for Rank Larger Than One

39

√ i 2 KL dη dχ exp − F IJK η I χJ [4π(λK VL (S, ω)] + , ω) + i(Imτ ) 16π I=1,2 1 F KLI (ImτIJ )−1 F JP Q η K χL η P χQ + 64π 1 = − 2 7 F 11I F 22J − F 12I F 12J −4π(Imτ )IJ π 2 + [4π(λI+ , ω) + i(Imτ )IK VK (S, ω)][4π(λJ+ , ω) + i(Imτ )JL VL (S, ω)] .

Z Y

I

I

(4.16)

In general, the integration over the Grassmann variables will give a factor of the form det IJ (F IJK λK + ) + · · · , where the remaining terms should be regarded as contact terms. A more compact expression for (4.14) can be found if we introduce r bosonic auxiliary variables bI : 9=

r XZ Y I=1

λ∈0

Z I

dη dχ

I

r +∞ Y −∞ I=1

db exp −iπτ IJ (λI+ , λJ+ ) − iπτIJ (λI− , λJ− ) I

i 1 I b (Imτ )IJ bJ − iVI (S, λI− ) − VI (S, ω)bI 8π 4π √ i 2 ~λ − ~λ0 ) · ρ − F IJK η I χJ (bK + 4πλK ) − iπ(( ~ , w (X)) . 2 + 16π

+

(4.17)

We emphasize that the integral in (4.17) is finite-dimensional, and not a path integral. This expression can be formally considered as the partition function of a finite-dimensional topological “field” theory, obtained from the original one after restriction to the sector of harmonic forms. The topological invariance is obtained from (3.1) and reads: [Q, λ] = 0, √ I [Q, a¯ ] = 2iη I , I

[Q, η ] = 0,

[Q, aI ] = 0, [Q, bI ] = 0, I

[Q, χ ] =

i(4πλI+

(4.18) − b ). I

We can consider minus the exponent in (4.17) as the (euclidean) action SE of this topological field theory. It is Q-closed. 4.5. The contact term. As explained in [9], when 2-observables are taken into account there are possible contact terms in the low-energy description. As will become clear in the next section, the contact term TV must be such that 1 VJ (Imτ )JK VK TbV (~u) = TV (~u) + 8π

(4.19)

is duality invariant. We give its form for V = u2 and a general SU (N ) theory with Nf matter hypermultiplets, Nf ≤ 2N , following the approach of [9, 24]. Introduce 2N −N the parameter τ0 as 3N,Nf f = eiπτ0 for the asymptotically free theories, and as the microscopic gauge coupling for the theories with Nf = 2N . The prepotential verifies the relation [25–27]

40

M. Mari˜no, G. Moore

∂F 1 = u2 , ∂τ0 4 and under a symplectic transformation we have the following behaviour, ∂2F ∂2F ∂2F ∂2F → − [(Cτ + D)−1 ]IJ C JK . 2 2 I ∂τ0 ∂a ∂τ0 ∂aK ∂τ0 ∂τ0

(4.20)

(4.21)

If we take into account that VI = 4(∂ 2 F/∂aI ∂τ0 ), we see that the shift of (4.21) has the same structure as the shift of the second term in (4.19) under symplectic transformations. It follows that the contact term can be written as T (~u) =

4 ∂2F . πi ∂τ02

(4.22)

In some cases we can use the homogeneity properties of u2 to write more explicit expressions for T (~u). In the case of Nf < 2N massless hypermultiplets we have X ∂u2 1 2u2 − (4.23) aI I . T (~u) = 2N − Nf ∂a I

Notice from this expression that T (~u) vanishes in the semiclassical regime, as required by asymptotic freedom. This coincides with [9, 12] in the appropriate cases. Using the relation between higher rank SU (N ) Yang–Mills theory and the Toda– Whitham hierarchy [28–32], one can introduce a set of “times” in the prepotential which can be seen to be dual to the higher order Casimirs. This makes possible the computation of contact terms for the two-observables coming from these Casimirs using the same arguments we have given here, and generalizing the expression (4.22) to include the rest of the time variables [33]. These variables were also considered in [12] in the context of the twisted theory, and the contact terms for the higher Casimirs were derived using a blow-up argument. 4.6. Remark on the normalization. The overall normalization of the integral (4.1) has a meaning and can in principle be fixed by physical computations or by comparison to topological invariants. In particular the constants α, β in (4.5) , (4.6) are functions of the group and for G = SU (N ) are functions of N . Some constraints on these constants can be obtained from the factorization of the measure in certain regions of MCoulomb expected on physical grounds. Let us focus on G = SU (N ) and consider a semiclassical region of moduli space with scalar vevs: 0 φ1 0 N2 1N1 + , (4.24) φ=M 0 −N1 1N2 0 φ2 where φ1 , φ2 are traceless. Since it is important to keep track of quantum scales to understand the behavior of the measure we introduce the quantum scale 3N and require that ZDW be dimensionless. We work in the semiclassical region |M | |φa1 − φb1 |, |φi2 − φj2 | |3N |

(4.25)

for 1 ≤ a < b ≤ N1 , N1 + 1 ≤ i < j ≤ N1 + N2 . The physics of this region is that we have a hierarchy of symmetry breakings: SU (N )−→SU (N1 ) × SU (N2 ) × U (1) → U (1)N1 −1 × U (1)N2 −1 × U (1) M

(4.26)

Donaldson–Witten Function for Rank Larger Than One

41

with N = N1 + N2 . At the large scale M we integrate out N1 N2 vectormultiplets corresponding to the off-diagonal blocks. It is not difficult to show that, up to relative corrections of order O(φ/M ), the semiclassical prepotential reduces to: (φ1a − φ1b )2 i X (φ1a − φ1b )2 log F= 4π 32N1 a
(4.28)

One can then show that the SU (N ) 9-function with scale 3N , denoted 9SU (N ),3N factorizes in the region (4.25) as: (4.29) 9SU (N ),3N → 9SU (N1 ),3N1 9SU (N2 ),3N2 9U (1) 1 + O(φ/M ) . Moreover, in the semiclassical region we have: Y ~ (χ+σ)/2 α ~ ·φ Aχ B σ → α(N )χ β(N )σ 3N

(4.30)

α ~ >0

with the product over the positive roots of SU (N ). From this we easily find the factorization in the region (4.25) : Y Y Y ~ → (N M )N1 N2 ~1) ~ 2 ), (~ α · φ) (~ α1 · φ (~ α2 · φ (4.31) α ~ >0

α ~ 1 >0

α ~ 2 >0

~ 2 are positive roots of SU (N1 ), SU (N2 ), respectively. where α ~ 1, α Thus, the nontrivial functions in the measure factorize in the region (4.25) as expected dM on physical grounds. Factorization of the entire measure implies that the measure dM |3U (1) |2 picks up nontrivial dependence on |M |2 which we have not predicted on a priori grounds. However, the holomorphic part of the measure, (N M )N1 N2 can be expected on a priori grounds since it accounts for the R-charge anomaly of the vectormultiplets integrated out at scale M . Combining this insight with Seiberg’s trick of regarding constants in an effective Lagrangian as vev’s in some theory at a higher scale to determine holomorphic dependence, we can give an heuristic argument for the N -dependence of α(N ), β(N ). We regard the constants α(N ), β(N ) as well as 3N as carrying R-charge. Thus, as in (4.31) we expect the factorization α(N ) = α(N1 )α(N2 )(αU (1) )N1 N2

(4.32)

for some constant αU (1) . Consequently, there should be N -independent constants κ1 , κ2 such that 2 (4.33) α(N ) = eκ1 N +κ2 N . Similar formulae hold for β. Unfortunately we can only fix one linear combination using the known constants for the SU (2) case, which have been found in [4, 9] by comparing to explicit results for Donaldson invariants. As remarked in [34, 4], to compare the results of the physical theory to mathematical results, one has to multiply the Donaldson–Witten function by the order of the center of the gauge group.

42

M. Mari˜no, G. Moore

5. Single-Valuedness of the Integrand The generalized u-plane integral (4.1) derived in the previous section is not manifestly well-defined because of monodromy around divisors where the SW curve 6 (or the abelian variety J(6) in the integrable system) degenerates. In this section we perform a careful check of the monodromy invariance of the integral in the case of simplyconnected manifolds. The semiclassical analysis applies to any simply-laced gauge group. The strong-coupling analysis is only complete for G = SU (3). 5.1. Semiclassical monodromy. The classical monodromy group is isomorphic to the Weyl group of the gauge group, and it is generated by the Weyl reflections ri associated to the root basis, i = 1, · · · , r. Semiclassically this monodromy has a quantum correction due to the one-loop contribution to the prepotential. The general form of the semiclassical monodromy has been presented in [14, 15] for any gauge group. The action of the ri monodromy on ~a is given by the matrix ~i ⊗ w ~ i, ri = 1 − α

(5.1)

~ i are the fundamental where the simple roots α ~ i are expanded in the Dynkin basis, and w weights. The classical monodromy acting on (~aD , ~a) is given then by −1 (ri )t 0 (ri ) P = . (5.2) 0 ri The one-loop correction to the prepotential Fone−loop =

Z2 i X 2 Zα~ log α~2 , 4π 3 α ~ >0

(5.3)

(where the sum is over the positive roots) gives, in addition to the Weyl reflection, a theta-shift in the coupling constant of the form ~i ⊗ α ~ i ]ri−1 . τ → (ri−1 )t [τ − α The semiclassical monodromy matrix is then given by −1 αi ⊗ α ~ i) (ri )t −(ri−1 )t (~ (ri ) M = . 0 ri

(5.4)

(5.5)

The invariance of the Coulomb path integral under these monodromies is a non-trivial check of our expression. First we analyze the lattice sum, then the measure factor including the gravitational contributions. To analyze the lattice sum, it is convenient to redefine the variables λ, D, η and χ by performing the Weyl transformation ri−1 . Notice that the lattice 0 is invariant under this redefinition, as the fundamental weights are shifted by roots. The measure for the variables D, η, χ is invariant under this transformation, since it is an orthogonal transformation. Also, the two-observables VI are derivatives with respect to aI of duality-invariant quantities, so they transform as ∂/∂aI , therefore the terms involving the two-observables remain invariant after the Weyl transformation of λ and D. For the phase factor depending on w2 (X), we can use the properties of the CartanKilling form and the vector ρ ~ and see that it gives the additional term πi(w2 (X), α ~ i · ~λ), where the dot denotes the Cartan-Killing form on the weight space and (·, ·) denotes

Donaldson–Witten Function for Rank Larger Than One

43

the usual product in integer cohomology. We then have an additional phase factor in the lattice sum: ~ i ) + iπ(w2 (X), α ~ i · ~λ) . (5.6) exp iπ(~λ · α ~ i , ~λ · α The simple roots are expanded in the Dynkin basis. To see that this phase factor is one, we take into account the decomposition in (4.10) . The term (5.6) then reads c−1 X ~ i , ~λZ · α ~ i ) + (w2 (X), ~λZ · α ~ i) + (m ~I ·α ~ i )(m ~J ·α ~ i )(π I , π J ) iπ (~λZ · α I,J=1

+

c−1 X

(m ~I ·α ~ i )(w2 (X), π I ) + 2

I=1

c−1 X

(5.7)

(m ~I ·α ~ i )(π I , ~λZ · α ~ i) .

I=1

The last term is an even integer, and the other terms can be combined into even integers using the Wu formula (w2 (X), z) = (z, z) mod 2, z ∈ H 2 (X, Z).

(5.8)

Therefore, the lattice sum is invariant under the semiclassical monodromy. Next we examine the measure factor in the Coulomb path integral. The measure [dad¯a] is invariant under the monodromy, and for the gravitational factor involving χ we can use the symplectic transformation properties and the fact that det ri = −1 to derive χ/2 χ/2 ∂uI πiχ ∂uI ] det J → exp[ . (5.9) det J ∂a 2 ∂a Finally, we analyze the factor involving the discriminant. In the semiclassical regime we can use the expression (4.8) . The Weyl reflection acts as follows on the Zα , with α > 0: the basic root αi changes its sign, therefore Zαi → −Zαi ,

(5.10)

and the rest of the positive roots are permuted, so the product of the rest of the roots Zα in the classical discriminant is invariant. The only change in the discriminant comes from this minus sign, and we finally obtain iπσ σ/8 ]13 . (5.11) 2 For a four-manifold with b1 = 0 and b+2 = 1, χ + σ = 4, and the measure factor does not change under the monodromy. Therefore, the Coulomb integral is invariant under the semiclassical monodromies. σ/8

13

→ exp[

5.2. Duality transformations. To analyze the quantum monodromy, we have to consider the duality transformations of the Coulomb integral in the appropriate descriptions. To do this, we introduce the generalization of the lattice theta function of [9] to the higher rank case π I I IJ IJ 20 (τIJ , αI , β ; P, ξI ) = exp −iπ(αI , β )+ ξI,+ (Imτ ) ξJ,+ −ξI,− (Imτ ) ξJ,− 2 X I ˆJ I ˆJ I I ˆ ˆ ˆ ˆ × −iπτ IJ (λ+ , λ+ ) − iπτIJ (λ− , λ− ) − 2πi(λ , ξI ) + 2πi(λ , αI ) , λ∈0

(5.12)

44

M. Mari˜no, G. Moore

where λˆ I = λI + β I . Notice that in the rank one case we recover the complex conjugate of the theta function introduced in [9]. If we take √ 1 2 VI S − + F IJK η J χK ω, ξI = 2π 16π (5.13) c−1 X 1 I I J mJ π , αI = w2 (X), I = 1, · · · , r, β = 2 J=1

and consider λI as the integer class λIZ introduced in (4.10) , we see that the lattice sum (4.14) can be written as S2 VI (Imτ )IJ VJ exp[iπ(αI , β I )](det ImτIJ )−1/2 9 = exp 8π √ Z Y 2 × dηdχ exp F IJK η I χJ (Imτ )KL VL (S, ω) 20 (τIJ , αI , β I ; P, ξI ). 16π (5.14) The overall factor involving S 2 combines with T (~u) to give the duality-invariant quantity Tˆ (~u) introduced in (4.19) . We will now consider the transformation properties of this theta function under the group Sp(2r, Z). The generators of the symplectic group are given in (2.7) . The transformation properties are the following: Under we have: 20 (−(τ −1 )IJ , αI , β I ; P, −(τ −1 )IJ ξJ,+ − (τ −1 )IJ ξJ,− ) s |0| (det iτ IJ )b+ /2 (det −iτIJ )b− /2 200 (τIJ , β I , −αI ; P, ξI ), = |00 |

(5.15)

where 00 is the dual lattice. To derive the transformation law for ξI in (5.15) , one has to use that (5.16) (Imτ )IJ − 2i(τ −1 )IJ = (Imτ )IK τ KL (τ −1 )LJ . If there is a characteristic element w2 such that (λI , w2 ) = (λI , λI ) mod 2, the transformation law of (5.12) under Tθ is: 20 (τIJ + θIJ , αI , β I ; P, ξI ) πi X 1 (w2 , θII β I )]20 (τIJ , αI − θII w2 − θIJ β J , β I ; P, ξI ). = exp[− 2 2

(5.17)

I

Finally, under the transformation A we have 20 (Aτ At , αI , β I ; P, ξ I ) = 20 (τIJ , A−1 α, At β; P, A−1 ξ).

(5.18)

Using these transformations, it is easy to check that the lattice sum (5.14) (except for the exponential involving S 2 and the phase) is then a modular form of weights ((b− + 1)/2, (b+ − 3)/2). To derive this result, one formally considers the Grassmann variables η, χ as modular forms of weight (0, 1), and takes into account the change induced in the Grassmann measure. The modular factors then combine with the measure [dada] and the gravitational factors to give the Coulomb integral for the dual variables.

Donaldson–Witten Function for Rank Larger Than One

45

5.3. Explicit check of quantum monodromy invariance for SU (3). Using the above transformation properties, we can analyze the quantum monodromy in the SU (3) case, as we know the explicit strong coupling spectrum in this case [14]. To obtain the appropriate form of the integral, we will make a symplectic transformation for each pair of mutually local charges in such a way that in the resulting theory there are two electrically charged particles with charges qiI = δiI , i, I = 1, 2. For the two massless states ~ν1 , ~ν2 in (2.18) , we have to perform the transformation −1 . For the states ~ν3 , ~ν4 , the appropriate symplectic transformation is given by −1 A−1 T−θ , where −1 −1 1 1 A= , θ= . (5.19) 1 0 1 −2 Finally, for ~ν5 , ~ν6 , the symplectic transformation has again the structure −1 A−1 T−θ with 0 1 2 −1 A= , θ= . (5.20) −1 −1 −1 −1 Therefore, in this basis, the monodromies associated to the two mutually local massless states are given by: ii 1e , i = 1, 2, (5.21) Mi = 0 1 where (eii )IJ = δIi δJi , and this holds for each of the three pairs of mutually local dyons. An important aspect of these transformations is that in all cases we are left with dual theories where the shifts in the 0 lattice are given by βI =

1 w2 (X), 2

I = 1, · · · , r,

(5.22)

i.e. they are Spinc structures. This result is obtained using the higher rank theta function transformations (5.15) , (5.17) and (5.18) . It is important to notice that the shifts in the 0 lattice are defined modulo integer cohomology classes. Now the monodromy invariance of the integral can be easily checked. The strong coupling monodromies are just theta-angle shifts in the dual coupling constants, and they are given by (i) = δIi δJi . (5.23) θIJ We can now use (5.17) to see that the only change in the higher rank theta function is given by the phase exp[−πiw2 (X)2 /4]. There is also a change in the measure associated to the factor involving the discriminant. Near a singular locus this factor has the structure given in (4.7) , and the monodromy acts on Z as Z → e2πi Z. We then obtain a factor exp[iπσ/4]. But the second Stiefel–Whitney class verifies that w2 (X)2 = σ mod 8,

(5.24)

therefore both phases combine to 1 and the integral is invariant under the strong coupling monodromies. 5.4. General case. In the general case, the verification of quantum monodromy invariance requires a precise knowledge of the strong coupling spectrum (or, equivalently, of the monodromy subgroup of the symplectic group associated to the relevant hyperelliptic curve). On the other hand, one can always write the monodromy associated to a

46

M. Mari˜no, G. Moore

monopole divisor in the form (5.21) with a submatrix q 2 eii , where q = gcd(νk ), through an appropriate symplectic transformation [16]. If this symplectic transformation is such that the U (1)i factor has a shift like (5.22) , then the above argument goes through. Notice that, near the divisor where a hypermultiplet of charge q becomes massless, the discriminant of the curve has a zero of order q 2 . Thus a general proof of quantum monodromy invariance follows if the symplectic transformation taking the monodromy to (5.21) also makes the dual line bundle λi a Spinc structure. Unfortunately, the monodromy group 0 has not been studied in sufficient detail to make the explicit check, although we fully expect it to work.

6. Definition of the Integral and Wall-Crossing The generalized u-plane integral (4.1) above is a formal (albeit monodromy- invariant) expression. In order to give meaning to the integral and, ultimately, derive topological invariants for four-manifolds, we must define it carefully and examine its metric dependence properties. 6.1. Defining the integral. The integrand of (4.1) has bad behaviour at the singularities on the moduli space. Therefore, we should regularize it appropriately near the codimension one submanifolds where dyons become massless and also near infinity. The first step in the regularization is to choose appropriate coordinates along these submanifolds. A divisor where a hypermultiplet becomes massless can be given locally by the equation ai = 0, and this gives a preferred coordinate along this locus. The other coordinates should be chosen according to the region we are considering along the locus. At the N = 1 points, for instance, one should choose dyonic coordinates for all the dyonic U (1) factors, while at the region where a monopole locus goes to infinity, one should choose electric coordinates for the remaining variables. At a generic point at infinity, we choose electric variables for all the U (1) factors. Finally, near a point where mutually non-local dyons become massless, there are no truly appropriate coordinates, and the divergences of the quantities involved in the integral are quite different from the ones associated to the monopole loci and to infinity. In the case of the AD points of the SU (3) theory, we will check that the integral is well-behaved near these points by using the , ρ coordinates introduced in (2.21) . The second step in the regularization is to introduce appropriate cutoffs at the singularities. In the case of the monopole loci, we choose tubular neighbourhoods defined by |ai | > r, where r is some radius that we will take to zero at the end. 3 Near a generic point at infinity, the prepotential is naturally expressed in terms of the combinations ~ Zα~ , and the cutoff is given by |Zα~ | < R, where R → ∞, for some positive roots α (different choices of these roots give different directions at infinity). Near the AD points of the SU (3) theory, we will introduce an IR cutoff in the plane, || > r, and analyze the behaviour of the theory when we take r → 0 (recall that is a coordinate near this point). Notice that this regularization can be interpreted as the substitution of the original moduli space MCoulomb by a “regularized” moduli space Mreg Coulomb , which is a manifold with a non-connected boundary. Finally, we perform the integrals over the corresponding variables. The procedure is now similar to the one in [9]. We first perform the integral over the phase of the complex coordinates chosen for each region, and this procedure gives a projection of the terms of the form aν aµ onto terms with ν = µ. As we will see in the next sections, in the 3

We trust there will be no confusion with r as the rank of the gauge group, nor ri as a Weyl reflection.

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SU (3) case, the resulting integrals converge, although their metric dependence can be discontinuous, resulting in wall-crossing. 6.2. Metric dependence of the integral. To study the possible metric dependence of the integral, we follow the strategy in [9]. We consider the variation of the Coulomb integral with respect to a first order variation δω in the period point. All the dependence on ω in the Coulomb integral appears in the lattice sum 9. The variation is most easily expressed using the representation (4.17) , and reads δ9 =

r XZ Y λ∈0

Z I

dη dχ

I

r +∞ Y −∞ I=1

I=1

h i VI (S, δω)bI dbI e−SE −4π(ImτIJ )λI+ (λJ , δω) − 4π

√ i i 2 − F IJK η I χJ (λK , δω) + iVI (S, δω)λI+ + (λI , δω)S+ , 4

(6.1)

where 1 I b (Imτ )IJ bJ + iVI (S, λI− ) SE =iπτ IJ (λI+ , λJ+ ) + iπτIJ (λI− , λJ− ) − 8π √ i 2 i I ~ ~ VI (S, ω)b + F IJK η I χJ (bK + 4πλK ~, w2 (X)). + + ) + iπ((λ − λ0 ) · ρ 4π 16π (6.2) Topological field theory promises us that δ9 is the integral of a total derivative. In fact, the metric variation in (6.1) can be written as Z +∞ Y r r i h XZ Y ∂ I I dη dχ dbI {Q, 8e−SE } − 4π J e−SE (λJ , δω) , (6.3) δ9 = ∂b −∞ λ∈0

I=1

I=1

where 8 = i(Imτ )IJ χI (λJ , δω) +

1 (S, δω)VI χI . 4π

(6.4)

Now we can use the fact that, according to the transformations (4.18) the Q operator is given by √ ∂ ∂ Q = i 2η I I + i(4πλI+ − bI ) I (6.5) ∂ a¯ ∂χ and write the metric variation as a total derivative in field space with respect to the antiholomorphic coordinates: √ ∂ ¯ δ9 = i 2 I ϒI , ∂ a¯ where I¯

ϒ =

r XZ Y λ∈0

J=1

Z J

dη dχ

J

+∞

r Y

−∞ K=1

(6.6)

dbK η I 8e−SE .

(6.7)

We introduce now an (r, r − 1) form on the Coulomb moduli space as r √ X ¯ d¯ (−1)I+r−1 ϒI da1 ∧ · · · dar ∧ da1 ∧ · · · daI ∧ · · · ∧ dar , =i 2 I=1

(6.8)

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M. Mari˜no, G. Moore

which satisfies r √ X ∂ I¯ 1 ϒ da ∧ · · · dar ∧ da1 ∧ · · · ∧ dar . d = ∂ = i 2 I¯ ∂a I=1

(6.9)

Taking into account that the measure and the observables in the Coulomb integral are holomorphic, we can use Stokes theorem to write the metric dependence as an integral over the boundary of the regularized Coulomb branch: Z 2 reg Aχ B σ eU +S TV . (6.10) δZCoulomb = reg

∂MCoulomb

6.3. Wall-crossing formulae along the monopole loci. We will analyze now the generic wall-crossing at a monopole locus. For simplicity and concreteness, we will focus on the case of the pure SU (3) theory. The monopole locus is defined by the equation a2 = 0. As explained in Sect. 4.3, there is a symplectic transformation to a basis in which λ2 ∈ H 2 (X, Z) + 21 w2 (X). For the other U (1) factor, according to our remarks in Sect. 6.1, we should choose the appropriate coordinates depending on the region of the monopole locus, i.e. we are allowed to perform symplectic transformations that leave a2 fixed but change a1 . Along the monopole locus, then, the behaviour of the τ22 coupling is given by τ22 =

1 loga2 + · · · , 2πi

F 222 = −

1 1 + ··· , 2πi a2

(6.11)

while τ11 , τ12 and the other F IJK are smooth (except when we are at an N = 1 point or at infinity, where we will have “wall-crossing for wall-crossing”. This will be analyzed in a moment). Denote y ≡ Imτ22 . An analysis similar to the one performed in [9] shows that the possible discontinuities in the integral are associated to terms involving only 1/(y 1/2 a2 ), and they occur when (λ2 , λ2 ) < 0, λ2+ = 0. These are the usual conditions for SW wall-crossing for λ2 . Taking this into account, we can easily find the terms that contribute to wall-crossing in the integral (4.1) , using the explicit expression in (4.16) . First of all, the factor det ImτIJ appearing in the photon partition function has the structure det ImτIJ = y(Imτ11 + O(1/y)), and similarly (Imτ )

−1

=

(Imτ11 )−1 + O(1/y) O(1/y) . O(1/y) O(1/y)

(6.12)

(6.13)

Therefore, in the term written in (4.16) the only surviving contribution is given by −

1 2 λ+ F 111 F 222 4π(λ1 , ω) + i(Imτ11 )−1 V1 (S, ω) . 32π

(6.14)

The term in S+2 (involving (Imτ )−1 ) in (4.14) can be analyzed in the same way, with the result that the only contribution comes from V12 (Imτ11 )−1 (S+2 /8π). On the monopole locus, we also have the following expansion in powers of a2 : 2 (1) τ 11 (a1 , a2 ) = τ (0) 11 + a τ 11 + · · · ,

(6.15)

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49

1 2 2 where τ (0) 11 = τ 11 (a , a = 0) is the first term in an expansion in a and is different from 2 zero. As we have already noticed, the positive powers of a do not contribute to the discontinuity of the integral, therefore we can put τ 11 equal to τ (0) 11 in the wall-crossing formula and write it as the integral of a residue: √ Z 2 iφ(λ2 ) 2 1 1 e da da Resa2 =0 Aχ B σ exp pU + S 2 TV − iV2 (S, λ2 ) W C(λ ) = 8 D X −(λ2 ,λ2 )/2 −(λ1 ,λ2 ) 1 · q22 q12 9(λ ) . λ1 ∈01

(6.16) In this expression, D denotes the monopole divisor, φ(λ2 ) is a global phase which depends on λ2 and is obtained through the appropriate symplectic transformation to the monopole locus, qIJ = exp(2πiτIJ ), and 1 2 ∂τ (0) 1 2 11 V1 (Imτ11 )−1 exp (0) S+ 1 8π ∂a (Imτ11 )(0) 1 2 1 2 1 1 · exp −iπτ (0) (λ ) − iπτ (λ ) − iV (S, λ ) − iπ(λ , α ) 11 1 1 + − − 11

9(λ1 ) = p

(6.17)

· [4π(λ1 , ω) + i(Imτ11 )−1 (0) V1 (S, ω)], where we have denoted (Imτ11 )(0) = (1/(2i))(τ11 − τ (0) 11 ) and α1 is the phase we have for the a1 theory. This phase, as well as the shift in the lattice 01 , depends on our choice of symplectic basis (we will make definite choices when we consider the “wall-crossing for wall crossing,” because in this case there is also a preferred a1 variable). Notice that the wall-crossing formula involves an integral which is very similar to a rank one u-plane integral depending on a “background field” a2 , and where the antiholomorphic part of the theory involves a restriction to a2 = 0. 6.4. Wall-crossing for wall-crossing. An important aspect of the integral (6.16) is that it has wall-crossing by itself. Along the monopole locus, there are three distinguished points where the a1 theory has singularities. These are the N = 1 points where the divisors intersect, the region at infinity, and the AD points. The behaviour near the AD points will be analyzed later. In this section we will focus on the wall-crossing for wall-crossing near the N = 1 points and at infinity. Near an N = 1 point the appropriate variable for the a1 theory is also a “magnetic” one, therefore near this point we have a1 → 0 and the behaviour of τ11 is similar to (6.11) . For this choice of the variable, the shift in the lattice 01 is also given by w2 (X)/2, i.e. the λ1 are also Spinc structures. The wall-crossing behaviour of the integral for the a1 variable is very similar to the usual SW wall-crossing analyzed in [9]. Again, we have wall-crossing for (λ1 , λ1 ) < 0, λ1+ = 0. The discontinuity is now a double residue and is given by I

I

W C(W C(λ1 , λ2 )) = − 4π 2 e2πi(λ ,λ0 ) Resa1 ,a2 =0 2 Y − 1 (λI ,λJ ) · Aχ B σ exp pU + S 2 TV − iVI (S, λI ) qIJ2 . I,J=1

(6.18)

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M. Mari˜no, G. Moore

We have chosen the N = 1 point at u = (27/4)1/3 3. The phase factor involving ~λ0 is the generalization to the higher rank case of a similar factor considered in [9]. It gives the dependence of the SW contribution on the generalized Stiefel–Whitney class, and can be obtained from (5.15) . The wall-crossing for wall-crossing at the other N = 1 points can be obtained in a similar way (they will have different global phases, according to the Z3 symmetry). At infinity along the monopole locus, the physics is that of an SU (2) theory embedded in SU (3), i.e. we have the quantum-corrected gauge symmetry breaking pattern SU (3) → U (1) × SU (2), where the U (1) (corresponding to the a1 theory) is weakly coupled in electric variables, and the SU (2) → U (1) is weakly coupled in magnetic variables. There is a duality frame, therefore, where the behaviour of τ11 is given by τ11 =

i loga1 + · · · 2π

(6.19)

and corresponds to electric variables, i.e. the shift in the lattice 01 is given by β 1 = (C −1 )1J π J . The wall-crossing of the integral on a1 will then be a Donaldson wallcrossing, exactly like the one anlayzed in [9]. The expression we get is formally identical to the one in (6.18) , although the conditions for wall-crossing in λ1 are the ones for Donaldson wall-crossing, and one must use the appropriate duality frame. 6.5. Wall-crossing at infinity. The relevant information to analyze the wall-crossing at infinity is encoded in the semiclassical one-loop correction to the prepotential (5.3) . In the SU (3) case it is given by: Fone−loop =

3 i X 2 Z2 Zi log i2 , 4π 3 i=1

(6.20)

where we denote Zi = Zα~ i , corresponding to the three positive roots of SU (3), α ~ i, i = 1, 2, 3. The explicit expressions are Z1 = 2a1 − a2 , Z2 = −a1 + 2a2 , Z3 = Z1 + Z2 = a1 + a2 .To analyze the conditions for wall-crossing, we focus on the photon partition function of the lattice sum (4.14) : 2 ~ 2 ~ Y Z α~ −(λ+ ·~α) Zα~ (λ− ·~α) . (6.21) exp −iπτ IJ (λI+ , λJ+ ) − iπτIJ (λI− , λJ− ) ∼ 3 3 α ~ >0

We can approach the region at infinity in moduli space in many ways, keeping one of the Zi , i = 1, 2, 3 to be finite and the other two, Zj , j 6= i, going to infinity (notice that we cannot keep two of the Zi finite, as Z3 = Z1 + Z2 ). The conditions for a possible wallcrossing in ~λ are then given by ~λ+ · α ~ j = 0, j 6= i, as one can easily check from (6.21) . As any two positive roots are linearly independent, we find ~λ+ = 0. Therefore, there is no wall-crossing at infinity for SU (3) (across codimension one walls): the integral is not discontinuous when ~λ+ = 0. This is in contrast with the case of the non-simple rank two group SU (2) × SU (2), where there are only two positive roots and therefore there are directions at infinity where one finds wall-crossing (namely, the Donaldson wall-crossing associated to each of the SU (2) factors). One can also check this behaviour for SU (3) using the u, v variables, going to infinity along the u or the v planes, and using the explicit expressions for the behaviour of the prepotential given in [14]. Again, one finds that the condition for a possible wall-crossing along these directions is ~λ+ = 0 and there is no discontinuity in the integral.

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51

7. The Blowup Formula The blowup formula generalizing [35] can be easily derived following the method used in [9]. Since there are manifolds with vanishing SW contributions it suffices to derive the formula for ZCoulomb . The latter is easily derived by studying the change of the measure in (4.1) . One then applies a universality argument. Let X˜ = BlP (X) be the blowup at a smooth point. Then σ˜ = σ − 1, χ˜ = χ + 1. The change in the measure under X → X˜ is just: µX˜ =

1/2 ∂uJ α −1/8 det 13 µX . β ∂aI

(7.1)

Now let B denote the class of the exceptional divisor, with B 2 = −1. In the chamber B+ = 0 (or more properly, for a fixed correlation function, where B+ < for some sufficiently small ) the 9 function factorizes to a 9-function for X times a holomorphic 9-function involving a sum over the root lattice. Indeed we may write: λ˜ I+ = λI+ , λ˜ I− = λI− + nI B,

(7.2)

where nI is in Z + mJI eJ for integer eJ . The shift eJ depends on the generalized Steifel˜ In the chamber B+ = 0 the 9-function Whitney class of the gauge bundle E˜ → X. factorizes as: X iπτ nI nJ +itV nI −iπ P nI I ~ I 9X˜ = e IJ 9X ≡ 2mt~e,1 (7.3) ~ (tV |τ )9X , nI

~ = (1, . . . , 1). Thus, accounting for the where we have written S˜ = S + tB and 1 contact term, the integrand for the blown-up manifold X˜ is related to that for X by the replacement of zero-observables: e U → eU

1/2 2 ∂uJ α −1/8 ~ det 13 e−t TV 2mt~e,1 ~ (tV |τ ) I β ∂a

(7.4)

Note that the expression must be monodromy invariant. Indeed, it has modular weight zero. This observation can be used to derive the required contact terms TV for V other than the quadratic Casimir [12]. Moreover, since it is invariant, it is a function of t and the Casimirs u2 , . . . , ur . Physically, we expect the defect B creating the blown-up manifold can be represented by an infinite number of local observables. The ring of local BRST invariant observables is generated by the Casimirs u2 , . . . , ur+1 . Thus there must be polynomials B~e,k (u2 , . . . , ur+1 ) such that 1/2 X ∂uJ α −1/8 −t2 TV ~ |τ ) = det 1 e 2 (t V tk B~e,k (u2 , . . . , ur+1 ). t ~ 3 m ~ e,1 β ∂aI

(7.5)

k≥0

The fact that B~e,k (u2 , . . . , ur+1 ) are polynomials can be proven as follows: the blowup expression (7.4) is monodromy invariant, in particular of weight zero, so it must be a function of uI , I = 2, . . . , r + 1, and t. Using the R-symmetry, we see that t has to be of charge −2, hence the polynomial B~e,k has charge 2k. On the other hand, the expression

52

M. Mari˜no, G. Moore

(7.5) has no singularities in the moduli space. This is because the theta function involved in the expression never has singularities, and the only possible singularities come from −1/8 13 . But these must be cancelled by zeros of the theta function, as follows from monodromy invariance. In the case of the SU (2) theory, the explicit expression for these polynomials was obtained in [9] using the expansion of the theta functions in terms of Eisenstein series, but in the higher rank case these expansions are not available. However, these expressions can probably be obtained using the relation between Seiberg–Witten theory and integrable systems. For Ar , the integrable system relevant to the Seiberg–Witten solution is the periodic Toda lattice [28–30]. The solutions to both models are straightline motions in the Jacobian of a hyperelliptic curve. Indeed, we recognize that (7.5) is essentially the τ function for the Toda hierarchy. Solutions to the Toda equations can be obtained from the Baker-Akhiezer function, and comparing the t expansion of these solutions should determine the polynomials B~e,k (u2 , . . . , ur+1 ). We have not carried out the details of this procedure. In any case, the blowup formula at higher rank is: = exp I(S) + pO τ (t|O2 , . . . , Or+1 ) exp I(S) + tI(B) + pO ˜ X X (7.6) X = tk exp I(S)+pO B~e,k (O2 , . . . , Or+1 ) . k≥0

X

8. Behaviour at the Argyres–Douglas Points The Coulomb integral (4.1) depends on the metric of the four-manifold X. Its variation with respect to the metric can be written in terms of an integral over the boundary of the regularized Coulomb branch, as in (6.10) . In general, this integral over the boundary will vanish, due to the damping factors associated to the behaviour of the couplings near the singularities or in the semiclassical region. However, at the AD points of the SU (3) theory, there is an N = 2 superconformal field theory with a finite value of the gauge coupling. The situation is reminiscent of the behaviour of the Nf = 4 theory analyzed in [9], where it was found that generic correlation functions have a continuous dependence on the metric. Therefore, one should analyze the possible continuous metric dependence associated to these superconformal points. 8.1. A general argument. The blowup formula derived in Sect. 7 severely constrains the possibility of continuous metric dependence. This is because the blow-up formula relates the Donaldson–Witten function of manifolds with different signatures. As we will show below, for sufficiently large signature (e.g., σ > −11 for G = SU (3)) the measure near the superconformal points is sufficiently smooth that the metric variation vanishes. Now, the blow-up formula relates the invariants on X˜ to invariants on a manifold with ˜ + 1. If there is no continuous variation in the latter correlators there cannot σ(X) = σ(X) be any such variation in the former. Care should be taken with this argument since the blowup formula only applies for ω in certain chambers of the forward light cone in H 2 (X; R). For any given correlator, the formula applies in a chamber with B+ < for some sufficiently small , where B is the exceptional divisor of the blow-up. If there is no continuous metric variation in this chamber then, given metric-independent wallcrossing formulae, there cannot be any continuous variation in any other chamber. (In

Donaldson–Witten Function for Rank Larger Than One

53

fact, as we have seen there is no wall-crossing from infinity on codimension one walls, so there is really only one chamber.) One could ask why an argument like this doesn’t rule out continuous metric dependence in the Nf = 4 theory considered in [9]. The reason is that, in this case, the inequality involving the signature also includes the ghost number Q of the correlators, and the condition not to have any metric dependence has the form of an upper bound on 2σ + Q. The above argument does not apply in this case, and one can easily check that the blowup formula is perfectly compatible with continuous metric dependence for the Nf = 4 theory. The reason for the different behaviours (and for the different bounds on the signature) has to do with the fact that, in the Nf = 4 theory, the continuous metric dependence comes from the behaviour at infinity, while the superconformal points in SU (N ) super Yang–Mills theories are in an “interior” region of the moduli space. 8.2. An explicit check. The above argument is rather general and should be checked by explicit computation. We now give a detailed analysis of the behavior near the AD points for G = SU (3). In particular we explicitly show the absence of continuous variation for σ > −11. 8.2.1. Convergence near the AD points. First of all, we must analyze the convergence of the Coulomb integral itself, as the divergences of the integrand of (4.1) near the superconformal point are rather different from the ones we have considered. We have to introduce a cutoff r for the variable introduced in (2.21) , and study the behaviour of the integral as r goes to zero, as we have indicated in Sect. 6.1. To do that, we first consider the antiholomorphic terms with −n behaviour. These come from the terms in F IJK , in (4.16) , and whose structure near the AD point was presented in (2.34) . The most divergent term corresponds to F 111 F 221 ∼ −4 . We have to write the measure of the integral in terms of , ρ variables. The jacobian of the change of variables from aI to xJ = , ρ can be computed at leading order from (2.25) , (2.29) :

det

6c7/2 ∂aI = 5/2 H(ρ) + · · · . J ∂x 3

(8.1)

36c2 2 7 |H(ρ)| || + · · · dddρdρ. 35

(8.2)

The measure is then [dada] =

Because of the factor ||7 in the measure, we see that the leading behaviour of the integral is smooth, so it converges. Notice that the rest of the terms involved in the integrand (VI , τIJ , TV , u, v) are smooth as goes to zero, as one can see from (2.21) , (2.30) , and (2.33) . Thus we conclude that the integral is well-defined in the limit r → 0. 8.2.2. Explicit formulae for the metric variation. Now we want to study the possible metric dependence of the integral. The first step in doing that is to write explicit expressions for the ϒI quantities defined in (6.7) . After doing the Grassmann integrals in the rank two case, one obtains

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M. Mari˜no, G. Moore

√ 1 i 2 (det Imτ )−1/2 exp VJ (Imτ )JK VK S+2 ϒ =− 16π 8π X I J · exp −iπτ IJ (λ+ , λ+ ) − iπτIJ (λI− , λJ− ) 1

λ∈0

− iπ((~λ − ~λ0 ) · ρ ~, w2 (X)) − iVI (S, λI− ) KL VL (S, ω)] · [4π(λK + , ω) + i(Imτ ) 1 (S, ω)V ˙ 1 ˙ + · F 22K i(Imτ )1J (λJ , ω) 4π 1 − F 21K i(Imτ )2J (λJ , ω) (S, ω)V ˙ 2 , ˙ + 4π √ 1 i 2 (det Imτ )−1/2 exp VJ (Imτ )JK VK S+2 ϒ2 = − 16π 8π X · exp −iπτ IJ (λI+ , λJ+ ) λ∈0

(8.3)

−

iπτIJ (λI− , λJ− )

− iπ((~λ − ~λ0 ) · ρ ~, w2 (X)) − iVI (S, λI− )

KL VL (S, ω)] · [4π(λK + , ω) + i(Imτ ) 1 (S, ω)V ˙ 2 ˙ + · F 11K i(Imτ )2J (λJ , ω) 4π 1 J − F 12K i(Imτ )1J (λ , ω) (S, ω)V ˙ 1 . ˙ + 4π

To analyze the behaviour near the superconformal point, we use the duality frame specified by the symplectic transformation (2.20) , in order to use the “small torus” (2.22) and the explicit solutions in Sect. 2. The differential form of (6.8) can be written now in terms of the , ρ variables. The explicit expression follows from: √ ∂aI ∂a2 1 ∂a1 2 ϒ − ϒ d ∧ dρ ∧ d =i 2 det J ∂x ∂ ∂ (8.4) ∂a2 1 ∂a1 2 ϒ − ϒ d ∧ dρ ∧ dρ . + ∂ρ ∂ρ There are two terms in (8.4) which can lead to variation δZCoulomb . In the first term in (8.4) we take the integral over the ρ boundary, which will be a set of three tubular neighbourhoods of the monopole divisors ρ3 = 1. The contributions of these boundaries leads to discontinuous, wall-crossing type, metric dependence. This is just the monopole wall-crossing analyzed in Sect. 6.3. The second term in (8.4) is more interesting and it gives the possible metric dependence associated to the AD points. We regularize the integral by cutting a small disk of radius r around = 0. The boundary integral in will then be along the circle of radius r, Sr , with center at = 0. We want to know if there are surviving contributions as r → 0. To analyze the integral over Sr it is important to take into account monodromy invariance under → e2πi . This invariance can be verified explicitly using the fact,

Donaldson–Witten Function for Rank Larger Than One

55

crucial to the entire argument we are giving, that after the symplectic transformation (2.20) , the line bundles λ1 define Spinc structures. First, one can easily check, using the behaviour of F IJK in (2.34) , that all the powers of appearing in the expression are positive or zero. Actually the only contribution one can have when r → 0 comes from the terms with no powers of . These involve F 111 , F 112 . We can write the metric dependence then as the integral of a residue, in the same way that we have written the wall-crossing formulae: ∂aI U +S 2 TV d dρdρA B det J e ∂x X 1 1 2 J VJ (Imτ )JK (S, δω)V2 · (det Imτ(0) )−1/2 exp i(Imτ )(0) (0) VK S+ 2J (λ , δω) + 8π 4π λ∈0 2 2 I J I · exp −iπτ (ρ)(λ1+ )2 − iπτ (0) 22 (λ+ ) − iπτIJ (λ− , λ− ) − iVI (S, λ− )

1 δZCoulomb (ω) = − 8π

·

I

Z

χ

σ

dτ (ρ) [4π(λ1+ , ω) + i(Imτ )1L (0) VL (S, ω)], dρ

(8.5) where the (0) (sub)superscript means that in the antiholomorphic quantities we take = 0. In (8.5) we have omitted a global phase depending on the non-abelian magnetic fluxes. The expression (8.5) is not zero in general. We conclude that ZCoulomb has continuous metric dependence from the AD points. 8.2.3. The ρ-plane theory. We now examine the metric dependence we have discovered in more detail. One of the interesting things about (8.5) is that it involves, essentially, a rank one integral associated to the elliptic curve (2.22) . We will refer to this curve as the “ρ-curve.” To see this, let us study the leading behaviour for → 0 of the measure appearing in (8.5) . Up to a constant that can be computed from (2.17) , (2.32) , and (8.1) , together with (2.31) , we find the behaviour 3σ+χ+14 ∂aI 1− χ2 , Aχ B σ det J ∼ 4 1σ/8 ρ ωρ ∂x

(8.6)

where ωρ is the period of the curve (2.22) . Similarly, using (2.30) we also have for the 2-observable V1 ∼

1/2 , ωρ

(8.7)

which again behaves as the 2-observable of the rank one case (involving the period of the ρ-curve). Comparing the factors (8.6) (8.7) to the general expressions for the rank one u-plane integrals we see that the leading behavior for → 0 is governed by a family of effective supersymmetric theories described by the ρ-curve and which we will refer to as the “ρ-plane theory.” 4 4 One must excercise caution when expressing the behavior of the integral at the AD points in terms of the ρ-plane theory since the matrix (Imτ )IJ and (det Imτ )1/2 does lead to subleading terms in 1/(Imτ (ρ)).

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M. Mari˜no, G. Moore

8.2.4. Monopoles to the rescue. The nonvanishing continuous metric variation (8.5) of ZCoulomb appears to spell doom for the topological invariance of ZDW . Before jumping to this conclusion we must consider the possible continuous metric variation of the other terms in (1.4) . In particular, we must examine the continuous dependence of the mixed SW/Coulomb integrals along the monopole divisors ZD(1) . In the present case the relevant i

divisors are Di(1) , i = 1, 2, 3 defined by the roots of ρ3 = 1 . The integrals ZD(1) will be i analyzed in some detail in the next section, but their continuous metric dependence is easy to analyze here. The Seiberg–Witten contributions are obtained by cancellation of wall-crossing of ZCoulomb along the monopole divisors, and they are integrals along these subvarieties involving the Seiberg–Witten invariants at the singularities of the ρ-plane (corresponding to the dyons becoming massless at ρ3 = 1). They are obtained from the behaviour of ZCoulomb near ρ3 = 1 in such a way that wall-crossings cancel: W Cρi (ZCoulomb ) + W C(ZD(1) ) = 0, i

(8.8)

where ρi , i = 1, 2, 3 are the roots of ρ3 = 1 and label the three monopole divisors near the AD point. We want to know the continuous metric dependence of these Seiberg–Witten δ ZD(1) for generic ω, not just at walls. The contributions. That is we want to compute δω i continuous variation comes from the region = 0, and we will denote this variation by δ=0 ZD(1) . Since the continuous metric dependence and the discontinuous metric i dependence involve the behaviour with respect to different variables, we see that the wall-crossing of the integral over ρ, ρ in (8.5) near the ρ3 = 1 divisors has to match the Seiberg–Witten wall-crossing of δ=0 ZD(1) at these singularities. We then have i

I δ=0 ZD(1) =

d

i

3σ+χ+14 4

X

SW (λ1 )Resρ=ρi F (ρ, , λI , δω),

(8.9)

λ∈0

where F (ρ, , λI , δω) is a holomorphic function of , ρ which depends also on λI and δω. It can be obtained, as we have indicated, by computing the wall-crossing of the ρ, ρ integral in (8.5) and matching it to the wall-crossing of a Seiberg–Witten contribution with the appropriate insertion of observables, as in the following section. 5 Now we note that in the ρ-plane integral in (8.5) , all the terms that do not correspond to a rank one integral for the curve (2.22) do not contribute to wall-crossing, as they involve subleading powers in 1/(Imτ (ρ)). Thus, the continuous metric dependence of ZD(1) is expressed in i terms of the ρ-plane theory. Taking this into account, the metric dependence of ZDW near the AD point is a sum of two terms: one from the integral in (8.5) and one from the Seiberg–Witten contributions near the singularities at = 0, ρ3 = 1, and can therefore be written schematically as I δZDW =

d

3σ+χ+14 4

nZ

dρdρ [· · · ] +

3 XX

o SW (λ1 )Resρ=ρi F (ρ, , λI , δω) ,

λ∈0 i=1

(8.10) where [· · · ] denotes the integrand of (8.5) up to the global power of that we have factored out. 5 Notice once more that the consistency of this procedure requires that the λ1 bundles, which are the “line bundles” that couple to the ρ theory, define Spinc structures.

Donaldson–Witten Function for Rank Larger Than One

57

8.2.5. Vanishing of δZDW for G = SU (3), σ > −11. We are finally ready to justify the assertion that δZDW = 0 for sufficiently large signature. This is a simple consequence of (8.10) . From the scaling behaviour of the terms in (8.5) we see that all of the terms in the expansion of (8.5) have positive powers. Therefore, (8.10) will vanish if the power of in the measure is bigger than −1, i.e., if σ > −11,

(8.11)

where we have taken into account that χ + σ = 4. Notice that we can always make insertions of 2-observables which have no leading powers of (for example, V2 = 3/c + O()). Therefore, we can not write a general selection rule involving the ghost number of a given correlator, as in the Nf = 4 case analyzed in [9]. Rather we have a condition on the signature of the manifold, given by the bound (8.11) . This bound is particular to the gauge group SU (3). For other superconformal points associated to other gauge groups and/or matter content [36, 37], we expect other explicit bounds depending on the R-charge spectrum near these points. We can now complete the argument for topological invariance of ZDW by invoking the general argument at the beginning of this section since the blow-up formula holds for ZDW and the measure factor depending on is common to both contributions in (8.10) .

9. The Seiberg–Witten Contributions 9.1. The SW contribution along the monopole loci. As in [9], we expect that the higher rank Donaldson–Witten functional is given by the Coulomb integral (4.1) plus the contributions coming from the monopole divisors, as we have indicated in (1.4) . Generically, along these divisors a dyon becomes massless and the low-energy effective theory contains one hypermultiplet coupled to one of the U (1) factors. The twisted theory is now a “mixed” theory where one of the variables (the one that we have called a2 ) is a distinguished coordinate but we can still perform duality transformations which leave a2 fixed. We expect, however, that the twisted theory will localize to supersymmetric configurations for the A2 vector multiplet coupled to the hypermultiplet. These simply give the Seiberg–Witten monopole equations for the A2 variables. At the N = 1 points, there are distinguished coordinates for both a1 , a2 , the effective theory contains two mutually local hypermutiplets (each of them coupled to each of the vector multiplets), and the twisted theory will localize to supersymmetric configurations for both vector multiplets coupled to the hypermultiplets. At these points, the contribution will be given then by Seiberg–Witten invariants SW (λ1 ), SW (λ2 ). On general grounds, the Donaldson–Witten functional for SU (3) will be given by XZ XZ da1i da1i µiλ1 ,λ2 (a1i , a1i , a2 ) ZDW =ZCoulomb + i

+

3 X Z X i=1 λ1 ,λ2

Di(1)

λ1 ,λ2

MSW (λ2 )

(9.1)

8i (a , a ), 1

MSW (λ1 )×MSW (λ2 )

2

where we have included a sum over the components of the codimension one divisor Di(1) , and also the contribution of the three N = 1 points. MSW (λ) is the Seiberg–Witten moduli space for the Spinc structure λ. The structure of the functions µiλ1 ,λ2 (a1i , a1i , a2 ),

58

M. Mari˜no, G. Moore

8i (a1 , a2 ) can be obtained by cancellation of wall-crossing, as in [9], and comparing to the formulae derived in Sect. 6. We find that the Seiberg–Witten contribution along a monopole divisor is given by the function 2 µiλ1 ,λ2 (a1i , a1i , a2 ) = eφi (λ ) exp pU + S 2 TV − iV2 (S, λ2 ) · C22 (a1 , a2 )(λ

2 2

) /2

C12 (a1 , a2 )(λ

1

,λ2 )

(9.2)

· P (a1 , a2 )σ/8 L(a1 , a2 )χ/4 9(λ1 ), where 9(λ1 ) is given in (6.17) , φi (λ2 ) is the appropriate global phase depending on the divisor and the corresponding symplectic transformation, and the functions C22 , C12 , P (a1 , a2 ), L(a1 , a2 ) are given by a2 , q22 −1 C12 (a1 , a2 ) = q12 , √ ∂uI 2 2 2 α det J , L(a1 , a2 ) = − 8 ∂a 1 8 13 1 2 β 2 . P (a , a ) = 32 a C22 (a1 , a2 ) =

(9.3)

As we explained in Sect. 8, the Seiberg–Witten contributions along the monopole divisors have continuous metric dependence near the AD point, which can be obtained by matching the wall-crossing of the ρ, ρ integral in (8.5) near ρ3 = 1 to the wall-crossing coming from the Seiberg–Witten contributions at these singularities. This can be verified using the computations above, with the only difference that instead of having an integral over a1i , a¯ 1i (the coordinate which parametrizes the monopole divisors) we have a contour integral in . 9.2. Contributions from the N = 1 points. Now we follow the same approach to compute the functions involved at the N = 1 points. By comparison with wall-crossing, their structure is 8i (a1 , a2 ) =eiφi exp[2πi(λI , λI0 )] exp pU + S 2 TV − iVI (S, λI ) ·

2 Y

21 (λI ,λJ )

eIJ (a1 , a2 ) C

e 1 , a2 )χ/4 , Pe(a1 , a2 )σ/8 L(a

(9.4)

I,J=1

where φi is a global phase depending on the generalized Stiefel–Whitney class and on e 1 , a2 ) are given eIJ (a1 , a2 ), I, J = 1, 2, Pe(a1 , a2 ), L(a the N = 1 point. The functions C by I eII (a1 , a2 ) = a , I = 1, 2, C qII −1 1 2 e , I, J = 1, 2, I 6= J, CIJ (a , a ) = qIJ (9.5) e 1 , a2 ) = −4π 2 α2 det ∂uI 2 , L(a ∂aJ 1 3 Pe (a1 , a2 ) = 16π 4 β 8 1 2 . a a

Donaldson–Witten Function for Rank Larger Than One

59

We can thus write the SW contribution at the N = 1 points for SU (N ), which is a straightforward generalization of the above procedure, = αχ β σ eiφi e2πi(λ heU +I2 (S) i(i) λ1 ,··· ,λr · Resa1 =···ar =0 ·

1 Qr 3 I=1

Y r

σ/8

,λI0 )

r Y

(aI )

det

I 2 2χ+3σ − (λ2 ) 8

SW (λI )

I=1

I=1

aI

I

I 2 −1 −(λ ) /2 qeII

Y

−(λ qIJ

I

,λJ )

(9.6)

1≤I<J≤r

∂uI χ/2 2 I exp U + S T − iV (S, λ ) , V I ∂aJ

where qeII = qII /aI , and we have included in α, β the numerical factors that are obtained, as in (9.5) , from matching to wall-crossing. Qr It is important to notice that the quantities qeII as well as the factor involving 13 / I=1 aI in (9.6) are regular at aI = 0. 9.3. SU (N ) Donaldson invariants for manifolds of simple type. In this section we generalize the gauge group to G = SU (N ) for all N , but specialize the class of manifolds to those of simple type. In the simple type case, we can evaluate the contribution at the N = 1 points using the explicit expressions given in [38, 39] for the N = 1 point where N − 1 monopoles become massless, together with the discrete Z4N symmetry relating the N = 1 vacua. The eigenvalues for φ are [38] φn = 2 cos

π(n − 21 ) , n = 1, · · · , N, N

and from this expression we can easily compute the VEVs of the Casimirs, 2s 2s N, c2s+1 ≡ hTrφ2s+1 i = 0. c2s ≡ hTrφ i = s

(9.7)

(9.8)

One also finds a relation r X ∂aI I=1

∂φi

sin

πk(i − 21 ) πkI = i cos , N N

(9.9)

and from this we can obtain, πI ∂u2 . = −2i sin ∂aI N

(9.10)

This gives the value of VI . Finally, we need the value of the off-diagonal couplings at the N = 1 point. These can be obtained using the scaling trajectory for the eigenvalues φn obtained in [38]. This trajectory depends on a parameter s, and s = 0 corresponds to the N = 1 point. The value of the magnetic gauge coupling along this trajectory is given by: τIJ (s) =

N −1 πJK 2 X πIK sin , τK (s) sin N N N

(9.11)

K=1

and the eigenvalues of this matrix, τK (s), can be explicitly written in terms of some integrals. In particular, the leading behaviour of τK (s) as s → 0 is given by [38]

60

M. Mari˜no, G. Moore

τK (s) =

i 2π sin

Z πκ 2

cos(1 − κ)θ , dθ p −b e−2s − sin2 θ b

(9.12)

1 where κ = K/N and b = arcsin e−s . This integral has a divergent part 2πi log s as s → 0, and this produces the usual logarithmic divergence, diagonal in I and J, in τIJ . But the off-diagonal terms of τIJ are finite at s = 0 (i.e. at the N = 1 point), and they can be obtained from the finite part of the integral in (9.12) . For SU (3) we have for instance [14], i (9.13) τ12 (0) = log 2, π as one can check from (9.12) and (9.11) . Although we have not found an explicit expression for the finite part of (9.12) , there are two important properties of the couplings τIJ (0), I 6= J, that one can deduce from (9.12) and (9.11): they are imaginary, and they satisfy the following symmetry property:

τIJ (0) = τN −I

N −J (0).

(9.14)

We can already write the contribution from the N = 1 points to the SU (N ) invariants, using the fact that these points are related by the Z4N ⊂ U (1)R symmetry. We must take into account the R-charges of the different operators in the correlation function, as well as the gravitational contribution to the anomaly that appears on a curved four-manifold. This anomaly can be computed from the microscopic theory (as in [34]) or directly from the expression given in (9.6). The two computations must give the same result because the factors in the measure were actually determined from an R-charge argument. In fact, as the R-charge of qII is zero, we have σ/8 σ σ 13 = N (N − 1) − (N − 1), R Qr I 2 4 I=1 a ∂uI χ2 χ = N (N − 1), R det J (9.15) ∂a 2 r Y 2χ + 3σ −(λI )2 /2 , qeII =(N − 1) R 4 I=1

and the R-charges of these terms give the right R-charge coming from the underlying twisted SYM theory, namely N2 − 1 (χ + σ). (9.16) 2 We conclude that the contribution of the N = 1 vacua is χ eσ eN heU +I2 (S) iSU (N ) = α βN

N −1 X

ω k[(N

2

−1)δ+N ~ λ0 ·~ λ0 ]

·

1≤I<J≤r

−(λI ,λJ ) qIJ

exp

e2πi(λ

I

,λI0 )

−1 NY

λI

k=0

Y

X

N −1 [ X 2 ]

s=1

p2s ω

2ks

SW (λI )

I=1 2k

2

c2s + 2ω S + 2ω

k

πI , (S, λ ) sin N

N −1 X I=1

I

(9.17) where ω = exp[iπ/N ] and δ = (χ + σ)/4. In α e, βe we have reabsorbed numerical factors that come from the evaluation of L, P and qeII at the point where N − 1 monopoles become massless (the values at the other points are obtained using R-symmetry). The

Donaldson–Witten Function for Rank Larger Than One

61

qIJ , I < J, can be obtained from (9.11) and (9.12) when s = 0, as we have discussed. We have also included in the phase factor labeling the N = 1 vacua in (9.17) an additional term depending on the generalized Stiefel–Whitney class, which generalizes the SO(3) case considered in [34, 4]. This term can be obtained if we take into account that the instanton number of the bundle, once non-abelian fluxes are included, satisfies (4.13) . Equivalently, one can take into account the transformations (5.15) , (5.17) and (5.18) to find this extra factor when we go through the different N = 1 vacua and see that they generate precisely this extra phase. 2 We will now find a natural generalization of the phase factor eiπw2 (E)/2 obtained in [9] to guarantee that the resulting expression is real. Notice that this is also a consistency check of the above answer, as this factor must be a global phase depending on the generalized Stiefel–Whitney class. We then have to consider the properties of (9.17) χ eσ βN is real). First notice under complex conjugation (assuming that the overall factor α eN that the first term in the sum, k = 0, changes by conjugation of the global phase: exp[−2πi(λI , λI0 )] (the factors qIJ , I < J, are real). Now we take into account that, due to (9.14) and the expression (9.10) , the right hand side of (9.17) has the symmetry λI → λN −I , hence we can write the resulting phase as e2πi(λ

I

,λI0 ) −2πi[(λI ,λI0 )+(λN −I ,λI0 )]

e

.

(9.18)

(C −1 )JN −I

Using the fact that (C −1 )JI + is an integer, for all I, J = 1, . . . , N − 1, we can write the second factor as N −1 X [J(J + N − 2)(π J , w2 (X))] = exp[πiN~v · ~v ], (9.19) exp −πi J=1

where we have taken into account the Wu formula (5.8) and the explicit form of the inverse Cartan matrix for SU (N ). For the rest of the terms in the sum, k = 1, . . . , N − 1, we take into account that, under conjugation, ω k → −ω N −k , and we change λI → −λI . Using the transformation [4]: SW (−λ) = (−1)δ SW (λ),

(9.20)

one easily checks that the sum of the terms k = 1, . . . , N −1 changes by an overall sign of the form (−1)N~v·~v (notice that, for manifolds of simple type, δ is an integer). Comparing with (9.19) we see that, under conjugation, (9.17) picks a global sign depending on the generalized Stiefel–Whitney class ~v . Notice that N~v · ~v is always an integer. Moreover, for N odd it is an even integer, because in this case N C −1 is an even form. Therefore, for N odd, (9.17) is real. For N even, it is then natural to include a phase factor of the form eiπN~v·~v/2 to make the above expression real. This factor is independent of the lifting of ~v as long as N is even, and in the special case of SU (2) we recover the factor introduced in [9]. 10. Application 1: Twisted N = 2 Superconformal Field Theories At the AD points there is an N = 2 superconformal field theory, and as we are studying the twisted version of N = 2 super Yang–Mills theory, the relevant spacetime symmetry algebra describing the model there is the twisted version of the N = 2 extended superconformal algebra in four dimensions. Recall that this algebra includes extra bosonic generators Kµ and D, corresponding to the special conformal transformations and the

62

M. Mari˜no, G. Moore ˙

AI dilatations, respectively, as well as two Weyl spinors SAI , S , where A, A˙ are spinor indices of SU (2)− × SU (2)+ , and I is the SU (2)R index. There is also an R generator corresponding to the non-anomalous U (1)R -current. The topological twist changes the coupling to gravity of all the fields charged with respect to the internal symmetry group SU (2)R , in the usual way. This means that in the twisted theory we consider the diagonal subgroup SU (2)0 of SU (2)+ × SU (2)R , and the internal symmetry index I is promoted ˙ We then obtain two scalar supercharges, to a spinor index, I → A. ˙ ˙

Q = AB QA˙ B˙ ,

˙ ˙

S = AB S A˙ B˙ ,

(10.1)

where QAI ˙ is the usual supersymmetry charge. We can also define descent operators, Gµ =

i BA ˙ σ QAB˙ , 4 µ

Tµ =

i BA ˙ σ S ˙. 4 µ AB

(10.2)

The twisted N = 2 superconformal algebra includes the relations: 1 Q, 2 [Q, R] = −Q, 1 [Gµ , D] = Gµ , 2 [Gµ , R] = Gµ ,

1 [S, D] = − S, 2 [S, R] = S, 1 [Tµ , D] = − Tµ , 2 [Tµ , R] = −Tµ ,

{Q, Gµ } = iPµ ,

{S, Tµ } = iKµ ,

[Q, D] =

{Q, Q} = 0,

(10.3)

{S, S} = 0,

{Q, S} =2R − 4D. One can define in a natural way (topological) chiral primary fields as those fields satisfying (10.4) {Q, 8] = {S, 8] = 0. From the last relation in (10.3) , we recall the well-known fact that for such fields R(8) = 2D(8). The topological descendants of a chiral primary field are n-forms with the structure, (10.5) 8(n) µ1 ···µn = {Gµ1 , · · · , {Gµn , 8] · · · ], and we see from (10.3) that they are also annihilated by S. After integrating them on n-cycles, we find new chiral primary fields. Notice that the R-charge of the topological descendant 8(n) satisfies R(8(n) ) = R(8) − n. Now we can try to extract some information about the twisted superconformal field theory at the AD point from the results we have obtained above. An important new feature arising near the AD points is that the gravitational factors Aχ , B σ have the following behaviour: (10.6) Aχ B σ ∼ χ/4 3σ/4 . Recall that this factor measures the gravitational contribution to the anomaly of the Rcurrent. The factor involving σ is naturally interpreted as the U (1)R anomaly of the three mutually nonlocal hypermultiplets becoming massless at the AD point. We interpret the factor involving χ as a signal of the R charge of the superconformal vacuum, leading to an anomaly −χ(X)/4 in units where the anomaly of a single hypermultiplet is −σ(X)/4.

Donaldson–Witten Function for Rank Larger Than One

63

We conclude that in the twisted superconformal theory on a manifold X there is a selection rule for correlation functions: h81 · · · 8n iX 6= 0 only for

X

R[8i ] =

i

(10.7)

1 χ(X), 10

(10.8)

where we have taken into account that R() = 2/5. This has a striking ressemblance to the selection rule for correlators in a twisted d = 2, N = (2, 2) sigma model on a Riemann surface 6, where the R-charge of the vacuum is given by −ˆcχ(6)/2. The generalization of (10.7) , (10.8) to SU (N ) can be determined by examining ∂a . For simplicity, assume N is odd. Since uj ∼ j , and the order of vanishing of det ∂u 2 N +2 N −1 ∂a ∼ −(N −1) /8 and there are 2 vanishing β-periods, ai ∼ 2 , we expect that det ∂u hence the RHS of (10.8) becomes 18 (NN−1) +2 χ(X) for the ZN multicritical superconformal theories. 6 It is interesting to compare this result with some similar recent results for N = 1 theories [40]. We believe that further information about the behavior of superconformal theories can be extracted from the above results, in particular from the ρ-plane theory of Sect. 8 [41]. 2

11. Application 2: Large N Limits of SU (N ) N = 2 SYM As an application of (9.17) we now sketch the large N asymptotics of the Donaldson invariants for SU (N ). While this has no obvious interest for topology, explicit results for correlators in supersymmetric Yang–Mills theory are rarities. It is even rarer that one can explicitly study a large N limit using exact results. Since on hyperk¨ahler manifolds the correlators of Q-invariant operators are the same in the topologically twisted theory as the “physical correlators” [22] we may hope to understand something of the physics of large N N = 2 SU (N ) SYM theory. We know from [38] that this limit presents some unusual features. The following identities will be useful to evaluate the correlators from (9.17) : N −1 X

ω k` = N

` = 0 mod 2N,

k=0

` = 0 mod 2,

=0 =

2 1 − ω`

` 6= 0 mod 2N

(11.1)

` = 1 mod 2.

The last case, ` odd, leads to a nontrivial 1/N series: N −1 X k=1

∞

ω k` =

X B2t (−1)t π` 2t−1 2i N + 1 + 2i , π` (2t)! N

(11.2)

t=1

where the B2t are Bernoulli numbers. 6 In deriving this result we have simply counted factors of in the determinant det ∂a . In principle the ∂u O(1) factors in this determinant, which we have not computed, could lead to a cancellation of the coefficient 2 of the leading divergence −(N −1) /8 . We are assuming this does not happen.

64

M. Mari˜no, G. Moore

11.1. The torus X = T 4 . Although our considerations in this paper have been mostly on simply-connected manifolds, the extension to the non-simply connected case can be done along the same lines (some interesting issues arising in this case are addressed in [42]). For the four-torus, however, the situation is very simple because the only basic class is λ = 0, and the Donaldson invariants are still given by (9.17) (with λ = 0). The reason for this is the following: since T 4 is flat, the monopole field in the SW equations must vanish and the SW moduli space is simply the space of harmonic 1-forms on T 4 . However, this is a nongeneric situation and the complications of the bundle of antighost zeromodes are most easily handled, as is standard, by perturbing the equations. Then, since T 4 is hyperkahler the only basic class is λ = 0, with SW (λ) = 1. This is in accord with the physical argument using a nowhere-vanishing mass perturbation [34]. Let us consider the operators in the theory. Since the torus is not simply connected we could also introduce Q- closed 1-cycle and 3-cycle operators. These only contribute through their contact terms and do not change the following results in any essential way, so we omit these operators. It is convenient to rescale the 0-observables to Aj ≡

1 Trφ2j c2j

(11.3)

for j = 1, 2, . . . . Then we have, simply, N X −1 X X = exp tj ω 2kj + 2S 2 ω 2k . exp tj Aj + I(S) k=0

(11.4)

j≥1

Now we must decide how to define the large N limit. We consider the N → ∞ limit of finite polynomials in tj and S. By (11.1) we see that all correlators at fixed ghost number vanish identically for sufficiently large N . The large N limit exists, but it is utterly trivial. We can obtain some more interesting correlators in two ways. The first is to add SU (N )/ZN fluxes to the theory. These produce a factor ω −kf in the sum over N = 1 PN −1 I vacua, where f = −N~v 2 , ~v = ~ I with w ~ I the fundamental weights and I=1 π w π I integral classes. A second way to get interesting correlators is to introduce a new “conjugate” set of operators: A¯ j ≡

1 Trφ2L+2−2j , c2L+2−2j

(11.5)

where L ≡ [(N − 1)/2] and j = 1, 2, . . . . Note that these operators do not have welldefined ghost number in the N → ∞ limit. With these modifications (11.4) becomes a little more intricate: X X ¯ ¯ tj Aj + I(S) = exp tj Aj + f N −1 X

ω

−kf

exp

X

tj ω

2kj

+

X

t¯j ω

(2L+2−2j)k

2

+ 2S ω

2k

(11.6) .

k=0

Q ` `¯ Consider now a term ∼ tjj t¯jj (S 2 )r , where `j = `¯j = 0 for all but finitely many j. We now apply (11.1) with the exponent

Donaldson–Witten Function for Rank Larger Than One

−f + 2

X

j`j + 2

65

X

(L + 1 − j)`¯j + 2r.

(11.7)

We now must divide the problem into cases. First suppose N is odd so N = 2L + 1. In this case N (C −1 )IJ is an even integral quadratic form, and hence the squared-fluxes f are always even. Since f, `j , etc. are held fixed while N → ∞ we must have: X `¯j = 0 mod 2, (11.8) X X −f + 2 j`j − 2 (j − 1/2)`¯j + 2r = 0. Thus, the large N theory is summarized by: X X ¯ ¯ exp tj Aj + tj Aj + I(S) = N 2

I

X

dz −f /2 2S 2 z z e exp z

tj z

j

· exp

X

f

X −j+1/2 ¯tj z −j+1/2 + exp − ¯tj z .

(11.9) An interesting point is that the above 1/N “expansion” is exact. Now we consider N even. The integral form N (C −1 )IJ is odd, and hence there are two subcases depending on whether the flux-squared f is even or odd. If f is even then the evaluation proceeds as before and X X t¯j A¯ j + I(S) = exp tj Aj + N 2

I

f

X j X −j X −j dz −f /2 2S 2 z z e exp tj z · exp . t¯j z + exp − t¯j z z

(11.10) If f is odd then we evaluate the sums in (11.6) using (11.2) . Now the 1/N expansions become nontrivial and do not terminate. 11.2. X = K3. The situation for X = K3 is very similar. K3 is hyperkahler so the only 2 basic class is λ = 0. Now δ = 2 so ω (N −1)kδ = ω −2k . Thus the results (11.9) , (11.10) continue to hold with the simple modification: I I dz dz ˜ )−16 (· · · ) → α(N ˜ )24 β(N (· · · ). (11.11) z z2 11.3. Other 4-manifolds. For other four-manifolds with nonzero basic classes, the evaluation of the correlators is more complicated due to the off-diagonal couplings in (9.17) , which mix the different U (1) factors. We will give here some indications on the structure of these correlators in the simple case of minimal surfaces of general type, and sketch a possible strategy to perform a systematic large N expansion. In the case of minimal surfaces of general type, the only basic classes are ±K, where K is the canonical line bundle of the manifold, and moreover [4, 43, 44]: SW (−K) = 1, SW (+K) = (−1)δ .

(11.12)

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M. Mari˜no, G. Moore

We can then introduce the variables sI , I = 1, . . . , N − 1, taking the values ±1, and define 2λI = sI K. The sum over the basic classes for the k vacua now takes the form X

N −1 X πi 2 X πI I I J k s . (−s ) exp − K τIJ s s + ω (S, K) sin 2 N I δ

sI =±1

I<J

(11.13)

I=1

This is a correlation function for a one-dimensional spin chain with N − 1 sites, with long range interactions given by the off-diagonal couplings τIJ , and in the presence of a space dependent “magnetic field” proportional to sin(πI/N ). The large N limit of this expression corresponds then to the thermodynamic limit of the system. This suggests that one can study the large N limit using standard techniques in statistical mechanics. One possible strategy is to rewrite (11.13) by introducing auxiliary variables xI , I = 1, . . . , N − 1, and reexpress the quadratic term in the sI spin variables using a gaussian integral. To do this, it is useful to consider the invertible matrix of couplings τ˜IJ , I, J = 1, . . . , N − 1, where the diagonal couplings come from the regular part at the N = 1 point of the couplings defined in (9.11) (i.e. after subtracting the logarithmic divergence), and the off-diagonal terms are the ones in (11.13) . This is actually the matrix of couplings that naturally appears in the Seiberg–Witten contribution in (9.6) . The term involving the diagonal part of τ˜IJ is just an overall factor depending on N and K 2 . After introducing the auxiliary variables xI , the sum over the spin variables can be easily worked out, and the correlation function (11.13) becomes, for δ even: 2N −1 C(N )

Z

−1 +∞ N Y −∞ I=1

+

N −1 X I=1

i X −1 I J dxI exp − τ˜IJ x x πK 2

I,J

πI + xI log cosh ω k (S, K) sin N

(11.14)

,

where C(N ) is an overall constant depending on N and K 2 . An analogous expression involving sinh can be obtained for δ odd. Notice that, for minimal surfaces of general type one has K 2 > 0. One could evaluate this integral using a saddle-point approximation (which should be a good one in the large N limit) and then systematically computing the corrections to the saddle-point. For other four-manifolds, the set of basic classes is more complicated, but in the case of algebraic surfaces one has a complete description of this set and the Seiberg–Witten invariants have been explicitly computed [4, 43]. The structure of (9.17) indicates that the contribution of an N = 1 vacuum will be given again by a correlation function in some statistical mechanical system. As we have suggested, this analogy could prove useful in studying the properties of the Donaldson–Witten function in the large N limit, and one may likely find interesting phenomena (like phase transitions). Another reason to study the large N limit of Donaldson–Witten theory is a possible relation to topological strings, as was conjectured in [45] (a relation between the large N limit of Chern–Simons topological gauge theory and topological strings has been recently found in [46]). 11.4. Possible applications to D-branes and matrix theory. There are several ways in which N = 2 SYM is realized in the context of string theory, D-branes and M-theory. See [47] and [48] for recent reviews. We limit ourselves here to a few brief and superficial remarks.

Donaldson–Witten Function for Rank Larger Than One

67

Perhaps the most direct applications are to matrix theory. As noted above the physical theory and twisted theory correlators coincide for the hyperkahler 4-manifolds X = as a kind of finite temperature T 4 , K3. The correlators on the 4-torus can be interpreted P partition function: ZDW = Tre−βH (−1)F exp[ ti Ai ]. In [49] a matrix theory approach to studying Schwarzschild black holes was proposed. This approach requires the existence of a singularity in the equation of state, Q which should translate into a singularity in Tre−βH Oi at large N as a function of β. Although we are studying theories with 8 rather than 16 supercharges, one would expect the phenomenon required by [49] to be rather generic. Unfortunately we find no evidence of the discontinuity in N posited in [49]. This might be due to the insertion of (−1)F , and indeed that is consistent with the discussion in Sect. 3.2 of [50]. In the realization of Seiberg–Witten theory via M -theory 5-branes described in [51, 48] the correlators of Trφ2n carry information about the quantum distribution of positions of (D4)-branes. The tendency of large N correlators to vanish as described above would seem to suggest that if the 5-branes are wrapped in the x1,2,3 directions (to use the standard choice of coordinates in [51, 48]) at finite temperature then the D4-branes – or tubes between NS5-branes – are very uniformly distributed. When the 4-manifold X is not hyperkahler then the twisted and physical correlators can differ. However, topological correlators can very well be relevant in the theory of D-branes [52] so the above results might also find applications in the theory of branes in more complicated compactifications of string/M theory. Acknowledgement. We would like to thank E. Witten for several important discussions and remarks. We also thank M. Douglas, J. Harvey, and D. Freedman for some useful remarks. Many of these results were announced in [24] and in a talk at the CERN workshop on Non-perturbative Aspects of Strings, Branes and Fields, Dec. ´ 8–12. GM thanks the organizers Luis Alvarez-Gaum´ e, Costas Kounnas, and Boris Pioline for the opportunity to present these results. This work is supported by DOE grant DE-FG02-92ER40704.

References 1. Donaldson, S.K. and Kronheimer, P.B.: The Geometry of Four-Manifolds. Oxford: Clarendon Press, 1990 2. Friedman, R. and Morgan, J.W.: Smooth Four-Manifolds and Complex Surfaces. Berlin–Heidelberg– New York: Springer Verlag, 1991 3. Witten, E.: Topological Quantum Field Theory. Commun. Math. Phys. 117, 353 (1988) 4. Witten, E.: Monopoles and four-manifolds. hep-th/9411102; Math. Res. Letters 1, 769 (1994) 5. Donaldson, S.: Irrationality and the h-cobordism conjecture. J. Diff. Geom. 26, 141 (1987) 6. Ellingsrud, G. and G¨ottsche, L.: Wall-crossing formulas, Bott residue formula and the Donaldson invariants of rational surfaces. alg-geom/9506019 7. G¨ottsche, L.: Modular forms and Donaldson invariants for 4-manifolds with b+ = 1. alg-geom/9506018; J. Am. Math. Soc. 9, 827 (1996) 8. G¨ottsche, L. and Zagier, D.: Jacobi forms and the structure of Donaldson invariants for 4-manifolds with b+ = 1. alg-geom/9612020 9. Moore, G. and Witten, E.: Integration over the u-plane in Donaldson theory. hep-th/9709193 10. Pidstrigach, V. and Tyurin, A.: Localization of the Donaldson invariants along Seiberg–Witten classes. dg-ga/9507004 11. Feehan, P.M.N. and Leness, T.G.: P U (2) monopoles and relations between four-manifold invariants. dg-ga/9709022; P U (2) monopoles I: Regularity, Uhlenbeck compactness, and transversality. dgga/9710032; P U (2) monopoles II: Highest-level singularities and realtions between four-manifold invariants. dg-ga/9712005 12. Losev, A., Nekrasov, N. and Shatashvili, S.: Issues in topological gauge theory. hep-th/9711108; Testing Seiberg–Witten solution. hep-th/9801061

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13. Seiberg, N. and Witten, E.: Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang–Mills Theory. hep-th/9407087; Nucl. Phys. B426, 19 (1994) 14. Klemm, A., Lerche, W. and Theisen, S.: Nonperturbative effective actions of N = 2 supersymmetric gauge theories. hep-th/9505150; Int. J. Mod. Phys. A10, 1929 (1996) 15. Danielson, U.H. and Sundborg, B: The moduli space and monodromies of N = 2 supersmmetric gauge theories. hep-th/9504102; Phys. Lett. B38, 273 (1995) 16. Argyres, P.C. and Douglas, M.R.: New phenomena in SU (3) supersymmetric gauge theory. hepth/9505062; Nucl. Phys. B448, 93 (1995) 17. Argyres, P.C. and Faraggi, A.E.: Vacuum structure and spectrum of N = 2 supersymmetric SU (N ) gauge theory. hep-th/9411057; Phys. Rev. Lett. 74, 3931 (1995) 18. Klemm, A., Lerche, W., Theisen, S. and Yankielowicz, S.: Simple singularities and N = 2 supersymmetric Yang–Mills theory. hep-th/9411048; Phys. Lett. B344, 169 (1995) 19. Masuda, T. and Suzuki, H.: On explicit evaluations around the conformal point in N = 2 supersymmetric Yang–Mills theories. hep-th/9612240; Nucl. Phys. B495, 149 (1997) 20. Kubota, T. and Yokoi, N.: Renormalization group flow near the superconformal points in N = 2 supersymmetric gauge theories. hep-th/9712054 21. Witten, E.: On S-duality in abelian gauge theory. hep-th/9505186; Selecta Mathematica 1, 383 22. Vafa, C. and Witten, E.: A strong coupling test of S-duality. hep-th/9408074; Nucl. Phys. B431, 3 (1994) 23. Humphreys, J.E.: Introduction to Lie algebras and representation theory. Berlin–Heidelberg–New York: Springer-Verlag, 1972 24. Mari˜no, M. and Moore, G.: Integrating over the Coulomb branch in N = 2 gauge theory. hep-th/9712062 25. Matone, M.: Instantons and recursion relations in N = 2 supersymmetric gauge theory. hep-th/9506102; Phys. Lett. B357, 342 (1995) 26. Eguchi, T. and Yang, S.K.: Prepotentials of N = 2 supersymmetric gauge theories and soliton equations. hep-th/9510183; Mod. Phys. Lett. A11, 131 (1996) 27. Sonnenschein, J., Theisen, S. and Yankielowicz, S.: On the relation between the holomorphic prepotential and the quantum moduli in supersymmetric gauge theories. hep-th/9510129; Phys. Lett. B367, 145 (1996) 28. Gorsky, A., Krichever, I.M., Marshakov, A., Mironov, A. and Morozov, A.: Integrability and Seiberg– Witten exact solution. hep-th/9505035; Phys. Lett. B355, 466 (1995) 29. Martinec, E. and Warner, N.P.: Integrable systems and supersymmetric gauge theory. hep-th/9509161; Nucl. Phys. B459, 97 (1996) 30. Nakatsu, T. and Takasaki, K.: Whitham–Toda hierarchy and N = 2 supersymmetric Yang–Mills theory. hep-th/9509162; Mod. Phys. Lett. A11, 157 (1996) 31. Donagi, R. and Witten, E.: Supersymmetric Yang–Mills theory and integrable systems. hep-th/9510101; Nucl. Phys. B460, 299 (1996) 32. Donagi, R.: alg-geom/9705010 33. Gorsky, A., Marshakov, A., Mironov, A. and Morozov, A.: RG equations from Whitham hierarchy. hep-th/9802007 34. Witten, E.: Supersymmetric Yang–Mills theory on a four-manifold. hep-th/9403193; J. Math. Phys. 35, 5101 (1994) 35. Fintushel, R. and Stern, R.J.: The blowup formula for Donaldson invariants. alg-geom/9405002; Ann. Math. 143, 529 (1996) 36. Argyres, P.C., Plesser, M.R., Seiberg, N. and Witten, E.: New N = 2 superconformal field theories in four dimensions. hep-th/9511154; Nucl. Phys. B461, 71 (1996) 37. Eguchi, T., Hori, K., Ito, K. and Yang, S.K.: Study of N = 2 superconformal field theories in four dimensions. hep-th/9603002; Nucl. Phys. B471, 430 (1996) 38. Douglas, M.R. and Shenker, S.H.: Dynamics of SU (N ) supersymmetric gauge theory. hep-th/9503163; Nucl. Phys. B447, 271 (1995) 39. D’Hoker, E. and Phong, D.H.: Strong coupling expansions of SU (N ) Seiberg–Witten theory. hepth/9701055; Phys. Lett. B397, 94 (1997) 40. Anselmi, D., Erlich, J., Freedman, D.Z. and Johansen, A.A.: Nonperturbative formulas for central functions of supersymmetric gauge theories. hep-th/9708042; Positivity constraints on anomalies in supersymmetric gauge theories. hep-th/9711035 41. Mari˜no, M. and Moore, G.: Work in progress 42. Mari˜no, M. and Moore, G.: Donaldson invariants for nonsimply connected manifolds. hep-th/9804104 43. Friedman, R. and Morgan, J.W.: Algebraic surfaces and Seiberg–Witten invariants. alg-geom/9502026; J. Alg. Geom. 6, 445 (1997); Obstruction bundles, semiregularity, and Seiberg–Witten invariants. alggeom/9509007

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44. Morgan, J.W.: The Seiberg–Witten equations and applications to the topology of smooth four-manifolds. Princeton, NJ:Princeton University Press, 1996 45. Moore, G.: 2D Yang–Mills Theory and Topological Field Theory. hep-th/9409044; and in: Proceedings of the International Congress of Mathematicians 1994. Basel–Boston: Birkh¨auser, 1995 46. Gopakumar, R. and Vafa, C.: Topological gravity from large N gauge theory. hep-th/9802016 47. Klemm, A.: On the geometry behind N = 2 supersymmetric effective actions in four dimensions. hep-th/9705131 48. Giveon, A. and Kutasov, D.: Brane Dynamics and Gauge Theory. hep-th/9802067 49. Banks, T.,Fischler, W., Klebanov, I.R., Susskind, L.: Schwarzschild black holes from Matrix theory. hep-th/9709108 50. Witten, E.: Anti de Sitter space and holography. hep-th/9802150 51. Witten, E.: Solutions of four-dimensional field theories via M-theory. hep-th/9703066; Nucl. Phys. B500, 3 (1997) 52. Bershadsky, M., Sadov, V. and Vafa, C.: D-Branes and Topological Field Theories. Nucl. Phys. B463, 420 (1996); hep-th/9511222 Communicated by R. H. Dijkgraaf

Commun. Math. Phys. 199, 71 – 97 (1998)

Communications in

Mathematical Physics © Springer-Verlag 1998

Diffusive Mixing of Stable States in the Ginzburg–Landau Equation Thierry Gallay1 , Alexander Mielke2 1 Analyse Num´ erique et EDP, Universit´e de Paris XI, 91405 Orsay, France. E-mail: [email protected] 2 Institut f¨ ur Angewandte Mathematik, Universit¨at Hannover, 30167 Hannover, Germany. E-mail: [email protected]

Received: 22 January 1998 / Accepted: 19 April 1998

2 Abstract: The Ginzburg–Landau equation ∂t u = ∂x2 u p+ u − |u| u on the real line has spatially periodic steady states of the form Uη,β (x) = 1−η 2 ei(ηx+β) , with |η| ≤ 1 and β ∈ R. For η+ , η− ∈(− √13 , √13 ), β+ , β− ∈R, we construct solutions which converge for all t > 0 to the limiting pattern Uη± ,β± as x → ±∞. These solutions are stable with 2 and behave asymptotically in time like respect to√sufficiently small √ H perturbations, √ 2 1/2 e (x/ t )), where N e 0 = ηe ∈ C ∞ (R) is uniquely determined (1 − ηe(x/ t) ) exp(i t N by the boundary conditions ηe(±∞) = η± . This extends a previous result of [BrK92] by removing the assumption that η± should be close to zero. The existence of the limiting profile ηe is obtained as an application of the theory of monotone operators, and the longtime behavior of our solutions is controlled by rewriting the system in scaling variables and using energy estimates involving an exponentially growing damping term.

1. Introduction The dynamics of parabolic partial differential equations on unbounded spatial domains exhibit various interesting phenomena which are not present on bounded domains, since the semigroups associated to such extended systems are not compact. An important example is the development of so-called “side-band instabilities” for spatially periodic patterns bifurcating from a homogeneous steady state. Although this problem has been investigated at a formal level more than 30 years ago [Eck65], only recently a rigorous approach has been proposed and applied to various model problems, see [CoE90a], [Mi97b] and the references therein. In particular, nonlinear stability results have been obtained for the spatially periodic solutions of the one-dimensional Ginzburg–Landau [CEE92] and Swift–Hohenberg [Sch96] equations, as well as for the Taylor vortices in a cylindrical geometry [Sch98]. These periodic patterns appear to be stable with respect to sufficiently localized perturbations, and the convergence of the perturbed solutions towards the equilibrium state is only diffusive (typically O(t−1/2 )) as t → +∞. In higher

72

T. Gallay, A. Mielke

dimensions, we quote the recent results by [Mi97c] and [Uec98] which prove the linear and nonlinear stability of the roll solutions for the two-dimensional Swift–Hohenberg equation. Following up the work of Collet-Eckmann [CoE92] and Bricmont-Kupiainen [BrK92], we study here one particular aspect of the diffusive stability theory, namely the mixing of two spatially periodic steady states. Consider a partial differential equation on the real line having a continuous family of spatially periodic time-independent solutions, some of them being (diffusively) stable. For this system, we prescribe initial data which coincide with one stable pattern as x → −∞ and with another one as x → +∞. The question is to determine the long-time behavior of the solutions arising from such initial data. As in [CoE92, BrK92], we investigate this problem for the Ginzburg–Landau equation with real coefficients ∂t u = ∂x2 u + u − |u|2 u ,

t ≥ 0,

x ∈ R,

(1.1)

where u(t, x) ∈ C. This system has a family of periodic stationary states given by p Uη,β (x) = 1 − η 2 ei(ηx+β) , where |η| ≤ 1 and β ∈ R. These solutions are linearly stable if and only if η 2 ≤ 1/3 [Eck65], and nonlinearly stable with respect to H1 perturbations for all η 2 < 1/3 [CEE92, Ga95]. Thus, we shall study solutions of (1.1) satisfying u(t, x) − Uη± ,β± (x) → 0

as x → ±∞ ,

(1.2)

with η+ , η− ∈ (− √13 , √13 ) and β+ , β− ∈ R. If η+ = η− and β+ = β− , the problem reduces to the diffusive stability of the steady state Uη− ,β− . In the sequel, we shall concentrate on the most interesting case η+ 6= η− , and make only a few comments on the intermediate situation where η+ = η− and β+ 6= β− . Thus, we assume that η+ 6= η− and, without loss of generality, that β+ = β− = 0, see (2.1) below. If both η+ , η− are close to zero, it is shown in [BrK92] that, under technical assumptions on the initial data, the solutions of (1.1) satisfying (1.2) converge locally as t → +∞ to the circle of steady states S∗ = { Uη∗ ,β | β ∈ [0, 2π] }, where η∗ ∈ (− √13 , √13 ) is uniquely determined by the boundary values η+ , η− . The main

purpose of this paper is to generalize this result to all values η+ , η− ∈ (− √13 , √13 ). For our analysis to work, it will be necessary to make assumptions on the behavior of the solutions not only at infinity, but also in the transition region connecting the two limiting patterns. In particular, we shall assume that the solutions stay locally close to the family of steady states { Uη,β | |η| ≤ 1 , β ∈ R }. Writing u = r eiϕ , this implies that the amplitude r is slaved to the phase derivative ηb = ∂x ϕ in such a way that r2 + ηb2 ≈ 1 for all times. In fact, we shall construct a class of solutions for which r2 + ηb2 = 1+O(1/t) as t → +∞. The behavior of these solutions will be described in a good approximation by the phase diffusion equation ∂t ϕ = ∂x [A(∂x ϕ)] ,

where A(η) =

Rη 0

a(s) ds ,

a(s) =

1−3s2 1−s2

.

(1.3)

This equation is p obtained from (1.1) by rewriting it in polar coordinates (ϕ, r), and then substituting r = 1−(∂x ϕ)2 into the equation for ϕ.

Diffusive Mixing of Stable States in Ginzburg–Landau Equation

73

From the results in [BrK92], it is apparent that the long-time dynamics √ of Eq. (1.3) is most naturally described in the scaling coordinates τ = log t, ξ = x/ t. Thus, defining η(τ, ξ) = ∂x ϕ(eτ , eτ /2 ξ), Eq. (1.3) is transformed into ξ η˙ = [A(η)]00 + η 0 , 2

where (˙) = ∂τ ( ) and ( )0 = ∂ξ ( ),

(1.4)

and the boundary conditions (1.2) become lim η(τ, ξ) = η− ,

ξ→−∞

lim η(τ, ξ) = η+ .

ξ→+∞

(1.5)

In Sect. 3 below we show that, for all η+ , η− ∈(− √13 , √13 ), Eq. (1.4) has a unique steady state ηe(ξ) satisfying the boundary conditions (1.5). This profile is a monotone function of ξ and approaches its limits η± faster than any exponential. If |η± | 1, these results were already obtained in [BrK92] using a perturbation argument based on the implicit function theorem. In this case, one has the approximate expression Z ξ 2 η + − η− e−z /4 dz. ηe(ξ) ≈ η− + √ 4π −∞ Our construction is nonperturbative and relies on the monotonicity properties of the differential operator in the right-hand side of (1.4). At the same time, our monotonicity argument shows that ηe is a stable equilibrium point of (1.4), in the sense that any solution η(τ, ξ) which is sufficiently close to ηe in the H2 norm converges exponentially √ to ηe√as e (x/ t) τ → +∞. Returning to the original variables, this implies that ϕ(t, x) ≈ tN Rξ e (ξ) = η− ξ + as t → +∞, where N [e η (s) − η− ] ds. −∞ We are now able to state our main result in a precise way. Let η+ , η− ∈ (− √13 , √13 ), η+ 6= η− , and assume that β+ = β− = 0. For t > 0, we define the time-dependent profile q √ √ √ e (t, x) = 1 − ηe(x/ t )2 ei t Ne(x/ t ) , U (1.6) e are constructed above. We consider initial data u0 ∈ H2 (R) such that where ηe and N loc e (t0 , ·)kH2 ≤ ε for ε > 0 sufficiently small and t0 > 0 sufficiently large. ku0 (·) − U Then Theorem 4.2 below states that the solution u(t, x) of (1.1) with initial condition u(0, ·) = u0 satisfies e (t0 +t, ·)kL∞ = O(t−ν/4 ) ku(t, ·) − U

as t → +∞ ,

(1.7)

e (0) 6= 0, we have for all for any ν ∈ (0, 1). In particular, setting η∗ = ηe(0) and ϕ∗ = N x0 > 0: sup |u(t, x) − Uη∗ ,ϕ∗ √t (x)| = O(t−ν/4 )

|x|≤x0

as t → +∞ ,

(1.8)

see Corollary 4.3. This result shows that the solution u(t, ·) converges uniformly on compact sets to the circle of steady states S∗ = { Uη∗ ,β | β ∈ [0, 2π] }. Following [MiS95], we can also 1 and the uniformly local norm H1lu (see study the convergence in the weighted norm Hw (4.2) below for the definitions). We find

74

T. Gallay, A. Mielke

lim distHw1 (u(t, ·), S∗ ) = 0 ,

t→∞

and lim inf distH1lu (u(t, ·), S∗ ) > 0 . t→∞

1 topology the solution u(t, ·) approaches S∗ sliding along the circle Indeed, in the Hw with nonzero speed ϕ∗ t−1/2 , hence the ω-limit set of u(t, ·) is equal to S∗ . In the H1lu topology the ω-limit set of u(t, ·) is empty. The main ingredient in the proof of our convergence results is the use of the nonlinear change of variables defined by u = r eiϕ and r(eτ , eτ /2 ξ)2 −τ /2 τ τ /2 e ϕ(e , e ξ) − N (ξ) , s(τ, ξ) = log . ψ(τ, ξ) = e 1 − [∂x ϕ(eτ , eτ /2 ξ)]2 (1.9)

Remark that s = 0 is equivalent to r2 + (∂x ϕ)2 = 1, hence s measures the slaving of the amplitude to the phase derivative. Note also that s has the same regularity as ψ 0 = ∂ξ ψ. The new functions (ψ, s) are solutions of a quasilinear nonautonomous parabolic system given by (2.12) below. In Sect. 5, we show that this system is locally well-posed in the space H2 (R)×H1 (R). In Sect. 6, we prove using various energy estimates that the small solutions (ψ, s) of (2.12) satisfy kψ(τ )kH2 (R) = O(e−γτ ) ,

ks(τ )kH1 (R) = O(e−τ ) ,

τ → +∞ ,

(1.10)

for all γ ∈ (0, 43 ), see Theorem 4.1. Returning to the original variables, this implies the convergence result (1.7). We conclude this section with a few comments on possible generalizations. Without any additional difficulty, we can extend our results to cover the equation ∂t u = ∂x2 u + h(|u|2 )u , where h : [0, ∞) → R is a smooth, decreasing function such that h(0) > 0 and h(s) < 0 for sufficiently large s. This equation has stationary states Uη,β = H(η 2 )1/2 ei(ηx+β) , where |η| ≤ h(0) and H = h−1 . The Eckhaus stability criterion takes the form H(η 2 ) + 2η 2 H 0 (η 2 ) > 0, and thus is satisfied for all η in a symmetric interval (−η0 , η0 ). The phase diffusion equation (1.3) becomes Z η ∂t ϕ = ∂x A(∂x ϕ) , where A(η) = [1 + 2s2 H 0 (s2 )/H(s2 )] ds. 0

For all η+ , η− ∈ (−η0 , η0 ), the arguments of Sect. 3 show the existence and uniqueness of the steady state ηe satisfying (1.4), (1.5), and the stability of this profile is proved as in Sects. 5 and 6. Finally, an interesting open problem is the diffusive mixing in parabolic equations without phase invariance, such as the Swift–Hohenberg equation ∂t u = −(1 + ∂x2 )2 u + εu − u3 ,

ε > 0.

(1.11)

This equation also admits (if ε is sufficiently small) a continuous family of spatially periodic steady states, whose stability has been proved recently [Sch96, EWW97]. In addition, the solutions of (1.11) are well approximated for a long (but finite) interval of time by those of (1.1), see [CoE90b, KSM92]. Therefore, a natural question is whether the results of this paper can be extended to the higher order equation (1.11). The rest of the paper is organized as follows. In Sect. 2, we reformulate our problem in terms of the scaling variables τ, ξ and the new functions ψ, s. In Sect. 3, we prove

Diffusive Mixing of Stable States in Ginzburg–Landau Equation

75

the existence of the steady profile ηe, and give a few general results on the dynamics of the phase diffusion equation (1.4). Section 4 contains the precise statements of our convergence results in both the original and the rescaled variables. The local existence of solutions for the (ψ, s) system (2.12) is proved in Sect. 5, and Sect. 6 is devoted to the proof of our main result (Theorem 4.1) using energy estimates. 2. Reformulation of the Problem Let η+ , η− ∈ (− √13 , √13 ), β+ , β− ∈ R, and let u(t, x) be a solution of (1.1) satisfying u(t, x) 6= 0 for all t, x. Introducing the polar coordinates (ϕ, r) via u = reiϕ , we assume that the boundary conditions (1.2) hold, namely q 2 , as x → ±∞. ϕ(t, x) − η± x − β± → 0, r(t, x) → 1 − η± We suppose from now on that η+ 6= η− , and refer to the end of this section for a comment on the simpler case η+ = η− . We first observe that, due to the translation and phase symmetries of the Ginzburg–Landau equation (1.1), there is no loss of generality in assuming β+ = β− = 0. Indeed, this amounts to replacing u(t, x) with u(t, x − x0 )e−iϕ0 , where x0 =

β+ − β − , η+ − η−

ϕ0 =

η+ β − − η − β + . η+ − η−

(2.1)

Assuming thus β± = 0, we obtain for (ϕ, r) the system ∂t ϕ = ∂x2 ϕ + 2

∂x r ∂x ϕ, r

2 ∂t r = ∂x2 r + r(1 − r2 − ∂x ϕ ),

(2.2)

q 2 as together with the boundary conditions ϕ(t, x) − η± x → 0, r(t, x) → 1 − η± x → ±∞. For later use, we also introduce the local wave-vector ηb(t, x) = ∂x ϕ(t, x), which satisfies ∂x r ηb , ηb(t, x) → η± as x → ±∞. ∂t ηb = ∂x2 ηb + 2∂x r As √ explained in the introduction, we shall use the scaling coordinates τ = log t, ξ = x/ t to investigate the long-time dynamics of the system (2.2). Defining η(τ, ξ) = ηb(eτ , eτ /2 ξ), we obtain the rescaled system 0 0 ξ 0 ρη 00 η˙ = η + η + 2 , 2 ρ

ρ(τ, ξ) = r(eτ , eτ /2 ξ),

ξ ρ˙ = ρ00 + ρ0 + eτ ρ 1 − ρ2 − η 2 , 2

(2.3)

where (˙) = ∂τ ( ) and ( )0 = ∂ξ ( ). This nonautonomous system has an asymptotic equilibrium point (e η , ρe) (as τ → +∞) defined by ρe 2 = 1 − ηe 2 and Z η 00 ξ 0 1 − 3s2 ds. (2.4) A(e η ) + ηe = 0, where A(η) = 2 1 − s2 0

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In Theorem 3.1 below, we show that the differential equation (2.4) for ηe has a unique solution satisfying ηe(ξ) → η± as ξ → ±∞. This solution is monotone in ξ and approaches its limits η± faster than any exponential. In addition, we have ( Z if ξ > 0 η+ . (2.5) ηe(ξ) − η∞ (ξ) dξ = 0, where η∞ (ξ) = η if ξ < 0 − R The main purpose of this paper is to prove that, for suitable initial data, the solution (η, ρ) of (2.3) converges to (e η , ρe) as τ → +∞. For the phase ϕ(t, x), this will imply the long-time behavior ϕ(t, x) ≈ η− x +

√

Z t

√ x/ t −∞

√ ηe(ξ) − η− dξ ≡ η+ x − t

Z

+∞

√

x/ t

ηe(ξ) − η+ dξ, (2.6)

√ since ∂x ϕ(t, x) = ηb(t, x) ≈ ηe(x/ t) and ϕ(t, x) − η± x → 0 as x → ±∞ (note that the two expressions in the right-hand side of (2.6) are identical due to (2.5)). This motivates the following ansatz for ϕ(t, x). Let Z ξ Z +∞ e (ξ) = η− ξ + ηe(s) − η− ds ≡ η+ ξ − ηe(s) − η+ ds. (2.7) N −∞

ξ

We define e (ξ), ψ(τ, ξ) = e−τ /2 ϕ(eτ , eτ /2 ξ) − N

ρ(τ, ξ) = r(eτ , eτ /2 ξ),

(2.8)

or equivalently ϕ(t, x) =

√ √ √ e (x/ t) + ψ(log t, x/ t ) , t N

√ r(t, x) = ρ(log t, x/ t).

(2.9)

Then ψ(τ, ξ) → 0 as ξ → ±∞, and (2.2) is transformed into e + ψ)0 − 1 (N e + ψ) + 2 ρ0 η, e + ψ)00 + ξ (N ψ˙ = (N 2 2 ρ ρ˙ = ρ00 + ξ2 ρ0 + eτ ρ 1 − ρ2 − η 2 ,

(2.10)

e + ψ)0 = ηe + ψ 0 . where η = (N By construction, we expect that the solution (ψ, ρ) of (2.10) converges to (0, ρe) as τ → +∞, where ρe 2 = 1 − ηe 2 . Assuming this to hold, we deduce from the second equation in (2.10) that ρ(τ, ξ)2 = 1 − η(τ, ξ)2 + O(e−τ ). This is the so-called “slaving” of the amplitude ρ to the phase derivative η. To take this behavior into account, we parametrize the amplitude ρ with the new variable s defined by p p ρ = 1 − η 2 es/2 ≡ 1 − (e η + ψ 0 )2 es/2 . (2.11) After some elementary calculations, we obtain the following equations for ψ, s: 0 η ) + ξ2 ψ 0 − 21 ψ + ηs0 , ψ˙ = A(e η + ψ 0 ) − A(e 2

s˙ = a1 (η)s00 + a2 (η)η 00 + ξ2 s0 − 2eτ (1−η 2 )(es −1) + 21 s0 + a3 (η)η 0 , where η = ηe + ψ 0 . The coefficients a = A0 and a1 , a2 , a3 are given by

2

(2.12)

Diffusive Mixing of Stable States in Ginzburg–Landau Equation

a(η) =

1−3η 2 , 1−η 2

a1 (η) =

1+η 2 , 1−η 2

a2 (η) =

−4η 3 , (1−η 2 )2

77

a3 (η) =

−2−6η 2 . (1−η 2 )3 (2.13)

This is the final form of the problem which is the basis of our analysis in Sects. 5 and 6. In particular, our main result (Theorem 4.1) will show that sufficiently small solutions (ψ, s) of (2.12) satisfy the decay estimate (1.10). Remark that there is now an imbalance in the number of spatial derivatives of ψ and s in (2.12), since the s–equation contains η 00 = ηe00 + ψ 000 , whereas only s0 appears in the ψ–equation. This derives from the fact that ρ is slaved to η = ηe + ψ 0 . Due to this imbalance, we also have to consider the equation for δ = ψ 0 = η − ηe which reads 00 η +δ)s0 )0 . (2.14) δ˙ = A(e η +δ) − A(e η ) + ξ2 δ 0 + ((e Remark. When η+ = η− , the discussion above remains valid, except for an important difference: if β+ 6= β− , it is no longer possible to reduce the problem to β± = 0 using the symmetries of the Ginzburg–Landau equation. As a consequence, the function ψ(τ, ξ) R defined by (2.8) has non-trivial boundary values ψ(τ, ±∞) = β± , and δ(τ, ξ)dξ = β+ − β− 6= 0. It follows that δ(τ, ·) converges to zero like e−τ /2 as τ → +∞, and not faster as in the previous case. Indeed, since ηe(ξ) = η+ = η− is now constant, the linearized equation (2.14) becomes δ˙ = e aδ 00 + ξ2 δ 0 + ηes00 ,

where e a = a(e η ).

As is well-known, the largest eigenvalue of the operator e a∂ξ2 + ξ2 ∂ξ is −1/2, with eigenfunction ω(ξ) = (4πe a)−1/2 exp(−ξ 2 /4e a) and adjoint eigenfunction 1. Since s(τ, ·) = −τ O(e ) due to the slaving, we conclude that δ(τ, ξ) ≈ e−τ /2 m(ξ) as τ → +∞, where m(ξ) = (β+ −β− )ω(ξ). Therefore, a natural ansatz for ψ is ψ(τ, ξ) = e−τ /2 M (ξ)+χ(τ, ξ), where Z β+ − β− ξ −y2 /(4ea) e dy, M 0 (ξ) = m(ξ). M (ξ) = β− + (4πe a)1/2 −∞ With this definition, the equations for χ, s are 0 η + e−τ /2 m) + ξ2 χ0 − 21 χ + ηs0 + 3, χ˙ = A(e η + e−τ /2 m + χ0 ) − A(e

(2.15) 2 2 s˙ = a1 (η)s00 + a2 (η)η 00 + ξ2 s0 − 2eτ (1−η 2 )(es −1) + 21 s0 + a3 (η)η 0 , η ) m0 = O(e−τ ). η + e−τ /2 m) − a(e where η = ηe + e−τ /2 m + χ0 and 3(τ, ξ) = e−τ /2 a(e Then, proceeding as in Sect. 6, one can show that kχ(τ, ·)kH2 = O(e−γτ ) and ks(τ, ·)kH1 = O(e−τ ) as τ → +∞, for all γ ∈ (0, 3/4). Remark. Important features of the Ginzburg–Landau equation are its symmetry properties which are essential to our analysis in several respects. Equation (1.1) is invariant under the phase rotations Pα and the translations in time Tz and space Sy defined by Tz : (t, x, u) 7→ (t + z, x, u), z ∈ R, Sy : (t, x, u) 7→ (t, x + y, u), y ∈ R, Pα : (t, x, u) 7→ (t, x, eiα u), α ∈ R.

(2.16)

For the transformed system (2.3), the rotation Pα is trivial, and the translations Tz , Sy become

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T. Gallay, A. Mielke

√ Tz : (τ, ξ, η, ρ) 7→ (τ + log(1 + e−τ z), ξ/ 1 + e−τ z, η, ρ), Sy : (τ, ξ, η, ρ) 7→ (τ, ξ + e−τ /2 y, η, ρ).

(2.17)

The transformation rules for ψ(τ, ξ) and δ(τ, ξ) are more complicated and will not be used in the sequel. 3. The Phase Diffusion Equation In this section, we first prove the existence of a unique solution ηe(ξ) of (2.4) satisfying the boundary conditions ηe(ξ) → η± as ξ → ±∞. We next study the structure and dynamics of the phase diffusion equation (or slaved equation) 0 ξ 1 e η + ψ 0 ) − (N + ψ), ψ˙ = A(e η + ψ 0 ) + (e 2 2

(3.1)

which is obtained by setting s = 0 in the ψ–equation of (2.12). Since s(τ, ·) decays to zero like e−τ as τ → +∞ due to the slaving, it is reasonable to expect that the long-time dynamics of the full system (2.12) is well approximated by (3.1), even though s = 0 does not define an invariant submanifold. In addition, some structures of the full problem can be understood better on this simpler model. We begin with the existence of the steady state: Theorem 3.1. (a) For each η+ , η− ∈ (− √13 , √13 ) there exists a unique ηe ∈ C 2 (R, (− √13 , √13 )) solving

A(e η)

00

ξ + ηe0 = 0 on R, 2

and ηe(ξ) → η± as ξ → ±∞,

(3.2)

where A(η) is defined in (2.4). (b) The solution ηe is constant if η+ = η− and strictly monotone otherwise. In addition, 2 there exists C > 0 such that |e η 0 (ξ)| ≤ Ce−ξ /4 for all ξ ∈ R. (c) The function ηe satisfies Z Z ξ ηe(ξ)−η∞ (ξ) dξ = A(η− )−A(η+ ), ηe(ξ)−η∞ (ξ) dξ = 0 and R

R

R0 R +∞ η+ − ηe(ξ) dξ, where η∞ is defined in (2.5). Setting ϕ∗ = −∞ ηe(ξ) − η− dξ = 0 we have sign(ϕ∗ ) = sign(η+ −η− ), and hence ϕ∗ 6= 0 if η+ 6= η− . Proof. Let η+ , η− ∈ (− √13 , √13 ). The existence of the steady profile ηe will be obtained using the theory of (nonlinear) monotone operators. First, choosing ε > 0 sufficiently small, we modify a(η) = (1 − 3η 2 )/(1 − η 2 ) outside the interval containing η+ , η− in such a way that the resulting function (still denoted by a) is C ∞ and satisfies 1 ≥ a(η) ≥ Rη ε > 0 for all η ∈ R. Accordingly, the modified primitive A(η) = 0 a(y) dy satisfies (η1 − η2 )(A(η1 ) − A(η2 )) ≥ ε(η1 − η2 )2 for all η1 , η2 ∈ R. Next, we choose a smooth, monotone function n : R → R such that Z 0 −|ξ| −|ξ| |n (ξ)| ≤ Ce , |n(ξ) − η∞ (ξ)| ≤ Ce , n(ξ) − η∞ (ξ) dξ = 0, R

for some C > 0, where η∞ is given by (2.5). In analogy with (2.7), we set

Diffusive Mixing of Stable States in Ginzburg–Landau Equation

Z N (ξ) = η− ξ +

ξ −∞

n(s) − η− ds ≡ η+ ξ −

79

Z

+∞

n(s) − η+ ds,

ξ

and we observe that |N (ξ) − ξn(ξ)| ≤ Ce−|ξ| . Finally, we define the operators B1 : H1 (R) → H−1 (R) and B2 : D(B) ⊂ H1 (R) → H−1 (R) by D(B) = { ψ ∈ H1 (R) | ξψ 0 ∈ H−1 (R) }, 0 B1 (ψ) = − A(n+ψ 0 ) + 21 ψ, B2 ψ = − ξ2 ψ 0 .

(3.3)

The program is to show that the nonlinear equation 1 [ξn(ξ) − N (ξ)] ∈ H−1 (R) (3.4) 2 has a unique solution ψe ∈ D(B) ⊂ H1 (R). Then, ηe = n + ψe0 will be the desired solution of (3.2). To prove the existence and uniqueness of the solution of (3.4), we first observe that B1 , B2 are coercive (or strongly monotone) operators, with B1 (ψ) + B2 ψ =

hB1 (ψ1 )−B1 (ψ2 ), ψ1 −ψ2 iH−1 ×H1 ≥ εkψ10 −ψ20 k2L2 + 21 kψ1 −ψ2 k2L2 ∀ψ1 , ψ2 ∈ H1 , hB2 (ψ1 −ψ2 ), ψ1 −ψ2 iH−1 ×H1 = 41 kψ1 −ψ2 k2L2 ∀ψ1 , ψ2 ∈ D(B). Clearly the nonlinear operator B1 is maximal monotone since it is continuous (from H1 into its dual H−1 ) and monotone, see [Zei90] Proposition 32.7. On the other hand, the linear operator B2 is densely defined, and a direct calculation shows that the adjoint operator B2∗ satisfies D(B2∗ ) = D(B) and B2∗ = −B2 + 21 . Thus B2 is closed, B2∗ is monotone, hence B2 is maximal monotone by [Zei90] Theorem 32.L. Since intD(B1 ) ∩ D(B2 ) = H1 (R) ∩ D(B) = D(B) 6= ∅, it follows from the “sum theorem” (see [Zei90] Theorem 32.I) that B = B1 + B2 : D(B) → H−1 (R) is maximal monotone. As B is also coercive, we conclude that B is one-to-one from D(B) onto H−1 (R) ([Zei90] Theorem 32.G), hence there exists a unique ψe ∈ D(B) ⊂ H1 (R) such that (3.4) holds. It remains to show that ψe is smooth, and that ηe = n+ψe0 remains in the interval between η+ and η− where no modification of a was needed. Since a(η) = A0 (η) ≥ ε > 0, classical regularity theory for ordinary differential equations applied to (3.4) shows that ψe is as smooth as the coefficients in the equation, namely ψe ∈ C ∞ (R). Hence ηe = n + ψe0 solves 00 ξ ξ (a(e η )e η 0 ). η )e η 0 )0 = − A(e η ) + ηe0 = 0, or equivalently (a(e 2 2a(e η) Integrating this equation, we obtain for any ξ0 ∈ R, ! Z ξ a(e η (ξ0 ))e η 0 (ξ0 ) y 0 exp − dy , ηe (ξ) = a(e η (ξ)) η (y)) ξ0 2a(e

ξ ∈ R.

(3.5)

From this representation, it is obvious that ηe0 (ξ0 ) cannot vanish unless ηe is identically constant, hence ηe is a monotone function of ξ ∈ R. In addition, since 1 ≥ a(e η ) ≥ ε > 0, 2 it follows from (3.5) that |e η 0 (ξ)| ≤ Ce−ξ /4 for some C > 0 and all ξ ∈ R. Therefore, ηe converge to some limits as ξ → ±∞, and these limits must be the boundary value η± prescribed by n, since ηe − n = ψe0 ∈ L2 (R). Thus, parts (a) and (b) are established. The relations in (c) follow from (3.2) using integration by parts, taking care of the jump of η∞ at ξ = 0. Finally, sign(ϕ∗ ) = sign(η+ −η− ) since ηe is strictly monotone if η+ 6= η− .

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T. Gallay, A. Mielke

Strictly speaking, the remainder of this section is not needed for the proof of the main results as given in Sect. 4. However, the statements and proofs given here for the slaved problem (3.1) help to understand the analysis of the more difficult full problem (2.12). Moreover, the arguments here show that our decay results are in a certain sense optimal. In the above theorem we used the fact that the steady part of (3.1) has a monotone structure. This in fact implies that the dynamics of (3.1) is rather trivial: all solutions are contracted with prescribed rate towards the unique steady state ψ = 0. Let ψ1 , ψ2 be classical solutions of (3.1) in H2 (R) satisfying ηe(ξ) + ψj0 (τ, ξ) ∈ (− √13 , √13 ) for all τ ≥ 0 and ξ ∈ R. Then kψ1 (τ, ·) − ψ2 (τ, ·)kL2 (R) ≤ e−3τ /4 kψ1 (0, ·) − ψ2 (0, ·)kL2 (R) ,

(3.6)

for all τ ≥ 0. This follows from the energy estimate 1 d 2 dt

R R

0 0 (ψ1 −ψ2 ) A(e η +ψ20 )0 dξ ξη +ψ0 1 ) −A(e + RR (ψ1 −ψ2 ) 2 (ψ1 −ψ20 ) − 21 (ψ1 −ψ 2 ) dξ R η +ψ10 )−A(e η +ψ20 ) dξ − 43 R (ψ1 −ψ2 )2 dξ . = − R (ψ10 −ψ20 ) A(e

(ψ1 −ψ2 )2 dξ =

R

RR

The first integral in the last row is nonnegative since A is monotone, namely, A0 (η) = a(η) > 0 implying (η1 −η2 )(A(η1 )−A(η2 )) ≥ 0. Applying Gronwall’s inequality, we obtain (3.6). We now consider the linearization of the phase diffusion equation (3.1) at the origin, and we discuss the spectral properties of the corresponding differential operator L : ψ 7→ 0 a(e η )ψ 0 + ξ2 ψ 0 − 21 ψ. According to (3.6), the spectrum of L in L2 (R) is contained in the left half-plane 0 = { z ∈ C | Re(z) ≤ −3/4 }. In fact, our next result will show that σ(L) = 0 in L2 (R), so that the estimate (3.6) cannot be improved. However, the upper bound of the spectrum depends on the function space, and can be shifted to the left if we impose a faster decay at infinity. To see this, we introduce for ` ≥ 0 theR weighted space X ` = L2 (R, (1+ξ 2 )` dξ) equipped with the scalar product hψ, ϕi = R ψϕ(1+ξ 2 )` dξ. Then the following holds: Proposition 3.2. Let ` ≥ 0 and let L : D(L) ⊂ X ` → X ` be the linear operator defined by D(L) = { ψ ∈ X ` | ψ 00 ∈ X ` , ξψ 0 ∈ X ` } and Lψ = (a(e η )ψ 0 )0 + ξ2 ψ 0 − 21 ψ for all ψ ∈ D(L). Then there exists a sequence {λn }n≥1 ⊂ R− independent of ` such that the spectrum of L is σ(L) = { λn | n ≥ 1 } ∪ { z ∈ C | Re(z) ≤ −3/4 − `/2 } . In addition, λ1 = −1, λn+1 < λn for all n ≥ 1, λn → −∞ as n → +∞, and λn is a simple (isolated) eigenvalue of L for all n such that λn > −3/4 − `/2. Remarks. 1) In the case where η+ = η− = ηe, it is known that λn = −(n+1)/2 for all n ≥ 1, see for instance [GR97]. The remarkable fact that λ1 = −1 is the largest isolated eigenvalue of L for all η+ , η− originates in the translation invariance in time of the Ginzburg–Landau equation (1.1), see the discussion at the end of this section. 2) Proposition 3.2 gives the upper bound max{−1, −3/4 − `/2} on the real part of the spectrum of L in X ` . Thus, even for rapidly decreasing initial data, the solutions of (3.1) do not converge to zero faster than e−τ as τ → +∞.

Diffusive Mixing of Stable States in Ginzburg–Landau Equation

81

Proof. Throughout the proof, we denote by ` the half-plane { z ∈ C | Re(z) ≤ −3/4 − `/2 }. Let b(ξ) = a(e η (ξ)), and let γ(ξ) be the solution of the differential equation 4b(ξ)γ 0 (ξ) + ξγ(ξ) = 0 with initial condition γ(0) = 1. By Theorem 3.1, b(ξ) converges (faster than any exponential) to positive limits b± as ξ → ±∞, hence γ(ξ) = O(exp(−ξ 2 /8b± )) as ξ → ±∞. For all λ ∈ C, it is easy to verify (using asymptotic expansions) that the linear ODE Lψ = λψ has two independent solutions ψ1+ , ψ2+ satisfying ψ1+ (ξ) = ξ 1+2λ 1 + O(ξ −2 ) , ψ2+ (ξ) = ξ −2(λ+1) γ(ξ)2 1 + O(ξ −2 ) , as ξ → +∞. Similarly, there exist independent solutions ψ1− , ψ2− such that ψ1− (ξ) = |ξ|1+2λ 1 + O(|ξ|−2 ) , ψ2− (ξ) = |ξ|−2(λ+1) γ(ξ)2 1 + O(|ξ|−2 ) , as ξ → −∞. Therefore, if Re(λ) < −3/4 − `/2, any solution ψ to Lψ = λψ belongs to X ` , so that λ is a double eigenvalue of L. This shows that σ(L) ⊃ ` . On the other hand, if λ is any eigenvalue of L with Re(λ) > −3/4 − `/2, then (by the same argument) λ is simple and the corresponding eigenvector ψ decays like |ξ|−2(λ+1) γ(ξ)2 as ξ → ±∞. Thus, setting ϕ(ξ) = γ(ξ)−1 ψ(ξ), we see that ϕ ∈ H2 (R) is a solution to 3ϕ = λϕ, where 3ϕ = (bϕ0 )0 − V ϕ ,

and V (ξ) =

ξ2 3 + . 4 16b(ξ)

Since b(ξ) → b± and V (ξ) → +∞ as ξ → ±∞, it is well-known (see for instance [CL55], chapter 9) that 3 is self-adjoint in L2 (R), and that its spectrum consists of a sequence {λn }n≥1 of real, simple eigenvalues satisfying λn+1 < λn and λn → −∞ as n → +∞. Moreover, the eigenfunction ϕn associated with λn has exactly n−1 zeros in R. This proves that, for all n ≥ 1, λn is an eigenvalue of L with eigenfunction ψn = γϕn , and that the sequence {λn }n≥1 exhausts the discrete spectrum of L in the half-plane c` . ˆ where ψ(ξ) ˆ Now, a direct calculation shows that Lψˆ = −ψ, = 2b(ξ)e η 0 (ξ) if η+ 6= η− and 2 ˆ ˆ ψ(ξ) = exp(−ξ /4b) if η+ = η− . Since ψ has no zero in R, it follows that λ1 = −1 and ˆ ψ0 = ψ. It remains to verify that L has no essential spectrum in c` (in other words, if λ ∈ σ(L) ∩ c` , then λ is an isolated eigenvalue of L with finite algebraic multiplicity.) Rξ To prove this, we use the change of variables defined by y = B(ξ) = 0 b(z)−1/2 dz, ψ(ξ) = χ(B(ξ)). Then (Lψ)(ξ) = (Lχ)(B(ξ)), where y 1 (Lχ)(y) = χ00 (y) + χ0 (y) − χ(y) + D(y)χ0 (y) ≡ (L0 χ)(y) + D(y)χ0 (y) , 2 2

and D(y) =

1 0 b (ξ) + ξb(ξ)−1/2 − B(ξ) . 2 ξ=B −1 (y)

From [GR97], we know that σess (L0 ) = ` in X ` . On the other hand, it follows from the Rellich criterion ([RS72], Theorem XIII.65) that the operator ∂y L0−1 is compact in X ` . Indeed, there exists a constant C > 0 such that, for all ψ ∈ X ` with kψkX ` ≤ 1, the function χ = ∂y (L0−1 ψ) satisfies kχkX ` ≤ C, kyχkX ` ≤ C, kχ0 kX ` ≤ C. Since the function D(y) is bounded, we conclude that L is a relatively compact perturbation of L0 . Now, it follows from a classical Theorem ([He81], Theorem A.1, p. 136) that either

82

T. Gallay, A. Mielke

σess (L) = σess (L0 ) = ` , or the half-plane c` is filled with eigenvalues of L. The second possibility is of course excluded, hence we have shown that σess (L) ≡ σess (L) = ` . Finally, we show that the eigenvalue λ1 = −1 of L is related to the symmetry properties (2.16) of the Ginzburg–Landau equation (1.1). It is advantageous to discuss these symmetries for the slaved equation for η given by 00 ξ (3.7) η˙ = A(η) + η 0 . 2 As was already mentioned, the phase rotation Pα leaves η invariant, but the symmetries of time translation Tz and space translation Sy as given in (2.17) are nontrivial. In addition, (3.7) is autonomous, a symmetry which was not present in the original system (2.12), due to the factor eτ in the right-hand side. Therefore, if η(τ, ξ) is a solution of (3.7), then for every τ0 , y, z ∈ R the function √ η τ + log(1+e−τ z) + τ0 , (ξ+e−τ /2 y)/ 1+e−τ z is also a solution. If η+ 6= η− we may choose for η the steady state ηe and obtain a two–dimensional invariant manifold √ M = { ηe((· + y)/ 1+z ) | y ∈ R, z > −1 } which is invariant under the flow of (3.7). Using y and z as coordinateson this manifold the reduced flow is y˙ = − 21 y and z˙ = −z. As a consequence, if K : δ 7→ a(e η (ξ))δ]00 + ξ2 δ 0 is the linearization of (3.7) at the steady state ηe, we have

since ∂y [Sy ηe]

1 Ke η 0 = − ηe0 , 2

y=0

and = e−τ /2 ηe0 and ∂z [Tz ηe]

z=0

K(ξe η 0 ) = −(ξe η0 ) , = e−τ (− 21 ξe η 0 ).

Thus, λ0 = −1/2 and λ1 = −1 are eigenvalues of K for all η+ , η− ∈ (− √13 , √13 ). 0 η = − 21 ηe and Since the operators K and L are related by Lψ = Kψ 0 , it follows that Le 0 0 Lψ0 = −ψ0 , where ψ0 (ξ) = 2a(e η (ξ))e η (ξ) is the primitive of −ξe η . This explains the eigenvalue λ1 = −1 of L in the general case η+ 6= η− (Note, however, that −1/2 is not an eigenvalue ofL, since 0 ηe does not decay at infinity.) In addition, using Proposition 3.2 and the relation Lψ = Kψ 0 , it is easy to show that the spectrum of K in the weighted space X ` is given by σ(K) = { λn | n ≥ 0 } ∪ { z ∈ C | Re(z) ≤ −1/4 − `/2 }. In fact, the global manifold M above is a special case of the local spectral manifolds as constructed in [Way97]. This recent theory for partial differential equations on unbounded domains uses rescaling and weighted norms to generate discrete spectra which then allows for a separation of the spectrum into a finite dimensional part and the remainder. Combining the arguments there and our Proposition 3.2 it should be possible to show that for all ` ≥ 0 and all initial data η0 such that distX` (η0 , M) is sufficiently small, there exist y, z ∈ R such that the solution η of (3.7) with η(0, ·) = η0 satisfies √ kη(τ, ·) − ηe((· + e−τ /2 y)/ 1+e−τ z )kX ` = O(e−ατ ) , as τ → +∞, for all α < min{−λ2 , 1/4 + `/2}. Clearly, only the case ` > 3/2 gives interesting results, since α > 1 is necessary to isolate the coordinate z properly.

Diffusive Mixing of Stable States in Ginzburg–Landau Equation

83

4. Convergence Results This section contains the convergence results for the diffusive mixing solutions. We first state our main result for the (ψ, s) system (2.12), and then we transform it back to the original Eq. (1.1). Theorem 4.1. Let η+ , η− ∈(− √13 , √13 ), η+ 6= η− , and let ηe be given by Theorem 3.1. There exist ε0 > 0 and τ0 > 0 such that, for all (ψ0 , s0 ) ∈ H2 (R)×H1 (R) satisfying kψ0 kH2 + ks0 kH1 ≤ ε0 , the system (2.12) has a unique global solution (ψ, s) ∈ C([τ0 , ∞), H2 (R)×H1 (R)) satisfying (ψ(τ0 ), s(τ0 )) = (ψ0 , s0 ). Moreover, for all γ ∈ (0, 3/4) we have kψ(τ )kH2 = O(e−γτ )

and

ks(τ )kH1 = O(e−τ )

as τ → +∞ .

(4.1)

Here and in the sequel, we simply write ψ(τ ), s(τ ) instead of ψ(τ, ·), s(τ, ·) when no confusion is possible. We shall prove Theorem 4.1 in several steps. In Sect. 5 we prove a local existence result for the solution (ψ, s) of the system (2.12) in the space H2 (R)×H1 (R). In Sect. 6 we introduce several energy functionals to control the behavior of the solution on the interval of existence. We derive differential inequalities for these functionals which imply, in particular, that for large τ0 every solution arising from sufficiently small data at time τ = τ0 exists and remains bounded for all subsequent times. Then, using the differential inequalities again, we show that the solution converges exponentially to zero as τ → +∞. These estimates are rather intricate for two reasons. Firstly, our system is nonautonomous through the factor eτ in the equation for s. This factor, which amounts to an exponentially growing damping, is very essential since it is responsible for the slaving of the amplitude ρ to the phase derivative η = ηe+ψ 0 . Note that the right-hand side of the equation for s does not vanish for (ψ, s) = 0, hence the origin is only an asymptotic fixed point of (2.12). Secondly, as was already mentioned in Sect. 2, there is an imbalance in the number of spatial derivatives in (2.12) which forces us to study this system in the rather unusual space H2 (R)×H1 (R). In this space, (2.12) behave like a quasilinear system, although the original equation (1.1) is semilinear. √ p η +ψ 0 )2 e t [Ne+ψ] . We now translate Theorem 4.1 back to the original variable u = 1−(e

Since the function u does not decay as |x| → ∞, it is convenient to use the uniformly local Sobolev spaces Hklu (R), see [MiS95, Mi97a]. These function spaces are equipped k norm) and a uniformly local norm (Hklu with two different norms: a weighted norm (Hw norm). Without loss of generality, we choose here the weight function w(x) = 1/(1+x2 ). For k ∈ N, we define Z kuk2Hwk = w(x)(|u(x)|2 + . . . +|∂xk u(x)|2 ) dx , kukHk = sup ku(·+y)kHwk . (4.2) R

lu

y∈R

In particular, Cb1 (R) ⊂ H1lu (R) ⊂ L∞ (R) and kukL∞ ≤ CkukH1lu (R) for some C > 0.

For all u0 ∈ Hklu (R), k ∈ N, the Ginzburg–Landau equation (1.1), has a unique global solution u ∈ C([0, ∞), Hklu (R)) with u(0, ·) = u0 (see [Col94, MiS95]). We now consider special initial conditions which are close to the mixing profiles (1.6) associated with ηe. Our result is

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e be given by (1.6). There exist Theorem 4.2. Let η+ , η− ∈(− √13 , √13 ), η+ 6= η− , and let U e (t0 , ·)kH2 ≤ ε, the t0 > 0 and ε > 0 such that, for all u0 ∈ H2lu (R) satisfying ku0 − U unique solution u of (1.1) in H2lu (R) with u(0, ·) = u0 satisfies, for all ν ∈ (0, 1), e (t0 +t, ·)kH1 = O(t−ν/4 ) , ku(t, ·) − U lu

e (t0 +t, ·)| kH1 = O(t−3ν/4 ) , k |u(t, ·)| − |U lu (4.3)

as t → +∞. Remarks. 1) The loss of regularity from H2 to H1 is generated by the imbalance in the number of derivatives of ψ and s in (2.12): we need an H2 condition on the data to ensure that ψ ∈ H2 , but since s ∈ H1 we can only recover an H1 regularity in our results. 2) The initial perturbations are assumed to be small in the H2 norm, but our convergence results are formulated in the uniformly local norm k · kH1lu only, because we want to keep the optimal decay rate in time. Indeed, since the estimates for (ψ, s) are obtained in Hk √ norms with respect to the diffusive variable ξ = x/ t (Theorem 4.1), we lose a factor t1/4 when transforming the results back to the corresponding Hk norms with respect to e (t0 +t, ·)| kH1 = O(t−ν/2 ), the original variable x. For instance, we find k |u(t, ·)| − |U √ since the amplitudes are just rescaled but do not involve a prefactor t like for the phase e (t0 +t, ·)kH1 ϕ. However, it is not possible to give a definite decay rate for ku(t, ·) − U unless we impose further spatial decay properties. Assuming the perturbation to lie in X ` (together with the first two derivatives) it is possible to improve the decay in (4.1) to O(e−ατ ) for all α < min{1, 3/4+`/2} (see [BrK92] and the arguments at the end of e (t0 +t, ·)kH1 = O(t3/4−α ) can be concluded, which gives Sect. 3 above). Then ku(t, ·)−U a positive decay rate whenever ` > 0. Proof. Let t0 = eτ0 , where τ0 is given by Theorem 4.1. If u0 ∈ H2lu (R) satisfies ku0 − e (t0 , ·)kH2 ≤ ε for some sufficiently small ε > 0, there exists a unique pair of functions U lu (ψ0 , s0 ) ∈ H2 (R)×H1 (R) such that q √ √ u0 ( t0 ξ) = 1−(e η (ξ)+ψ00 (ξ))2 es0 (ξ)/2 ei t0 [Ne(ξ)+ψ0 (ξ)] , ξ ∈ R . In addition, kψ0 kH2 + ks0 kH1 ≤ Cε for some C > 0. Thus, if Cε ≤ ε0 , Theorem 4.1 shows that the unique solution of (2.12) in H2 (R)×H1 (R) with initial data (ψ0 , s0 ) satisfies kψ(τ )kH2 + ks(τ )kH1 = O(e−γτ ) as τ → +∞, for all γ < 3/4. In particular, we have for all ν < 1: √ R 2 02 2 x/ t, log (ψ 2 +ψ 02 +s2 )(ξ, log t) = O(t−3ν/2 ) , t dx ≤√C sup R ξ∈R RR w(x)(ψ 002+ψ 02+s ) √ 002 w(x)(ψ +s ) x/ t, log t dx ≤ C t R (ψ + s02 )(ξ, log t)dξ = O(t−ν ) , R where w is the weight function 1/(1+x2 ) or any translate of it. Using these results, it is straightforward to verify that the unique solution u(t, x) of (1.1) in H2lu (R) with initial data u0 is given by u(t, x) = with τ = log(t+t0 ), ξ =

p

1−(e η (ξ)+ψ 0 (τ, ξ))2 es(τ,ξ)/2 ei

√x , t+t0

√

e(ξ)+ψ(τ,ξ)] t [N

,

and that the decay estimates (4.3) are satisfied.

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85

Theorem 4.2 implies that, up to a time-dependent phase, p the solution u(t, x) converges uniformly on compact sets to the stationary solution 1 − η∗2 eiη∗ x as t → +∞, R e (0) = 0 (e η (ξ)−η− )dξ 6= 0 as in Theorem 3.1, where η∗ = ηe(0). Indeed, setting ϕ∗ = N −∞ we have the following result (see also [BrK92]): Corollary 4.3. Under the assumptions of Theorem 4.2, we have for all x0 > 0 and all ν ∈ (0, 1) the estimate √ p (4.4) sup |u(t, x) − 1−η∗2 ei[ t ϕ∗ +η∗ x] | = O(t−ν/4 ) , t → +∞ , |x|≤x0

e (0). where η∗ = ηe(0) and ϕ∗ = N √ √ √ √ e (x/ t) = tϕ∗ + η∗ x + O(t−1/2 ) Proof. Since ηe(x/ t) = η∗ + O(t−1/2 ) and t N uniformly for |x| ≤ x0 , this result follows immediately from the first estimate in (4.3). Remark. In the case where both η+ , η− are close to zero, the estimate (4.4) is obtained in [BrK92] with the better decay rate O(t−ν/2 ) for the remainder. As was already mentioned in the previous remark, this is because we allow for initial perturbations with only L2 decay at infinity, while in [BrK92] stronger norms are used which imply in particular that the perturbations lie in X ` with ` ≥ 1. 5. Local Existence In this section, we prove a local existence result for the solutions (ψ, s) of the system (2.12) in the space H2 (R)×H1 (R). Although it originates from the simple Ginzburg– Landau equation (1.1), this system in nonautonomous due to the change of variables (2.8). In addition, the nonlinear transformation (2.11) forces us to study it in the unbalanced space H2 (R)×H1 (R), where it behaves like a quasilinear system. For these reasons, we have to take special care of the regularity of the solutions and the length of the local existence intervals. Throughout this section, we assume that η+ , η− ∈ (− √13 , √13 ) and that ηe is given by Theorem 3.1. Our result is: Proposition 5.1. There exist ε1 > 0, T1 > 0, and K1 ≥ 1 such that, for all τ1 ≥ 0 and all (ψ1 , s1 ) ∈ H2 (R)×H1 (R) with kψ1 kH2 + ks1 kH1 ≤ ε1 , the system (2.12) has a unique solution (ψ, s) ∈ C([τ1 , τ2 ], H2 (R)×H1 (R)) satisfying (ψ(τ1 ), s(τ1 )) = (ψ1 , s1 ), where τ2 = τ1 + log(1 + T1 e−τ1 ). This solution depends continuously on the initial data (ψ1 , s1 ) in H2 (R)×H1 (R), uniformly in τ ∈ [τ1 , τ2 ]. Moreover, the bound (5.1) kψ(τ )kH2 + ks(τ )kH1 ≤ K1 kψ1 kH2 + ks1 kH1 + eτ1 (τ − τ1 ) holds for all τ ∈ [τ1 , τ2 ]. Remarks. 1) In particular, Proposition 5.1 implies that, if (ψ, s) ∈ C([τ1 , τ ∗ ), H2 (R) ×H1 (R)) is a maximal solution of (2.12) which satisfies kψ(τ )kH2 + ks(τ )kH1 ≤ ε1 for all τ ∈ [τ1 , τ ∗ ), then actually τ ∗ = +∞, i.e. the solution can be continued to the whole interval [τ1 , +∞). 2) The proof shows that the solution ψ(τ, ξ), s(τ, ξ) of (2.12) is a C ∞ function of τ, ξ for all τ > τ1 . However, it is not true in general that (ψ, s) ∈ C k ((τ1 , τ2 ], H2 (R)×H1 (R)) for k > 0, unless we assume in addition that ψ1 (ξ), s1 (ξ) decay sufficiently fast as |ξ| → ∞.

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Proof. Instead of working directly on (2.12), we shall use the change of variables (2.9), (2.11) and solve the corresponding initial value problem for the simpler system (2.2). Let τ1 ≥ 0, and let (ψ1 , s1 ) ∈ H2 (R)×H1 (R) be initial data for (2.12) at time τ = τ1 satisfying kψ1 kH2 + ks1 kH1 = ε for some ε ≤ 1/4. The corresponding initial data for (2.2) are given by ϕ(t1 , x) = ϕL (x) + ϕ¯ 1 (x),

r(t1 , x) = rL (x) + r¯1 (x),

(5.2)

e (x/L), rL (x) = (1 − ηe(x/L)2 )1/2 , and where t1 = eτ1 , L = eτ1 /2 , ϕL (x) = LN ϕ¯ 1 (x) = Lψ1 (x/L),

r¯1 (x) = 1 − (e η (x/L)+ψ10 (x/L))2

1/2

1

e 2 s1 (x/L) − rL (x).

A direct calculation shows that there exists a constant C1 > 0 (independent of L) such 2 1 , HL are the Sobolev spaces that kϕ¯ 1 kHL2 + kr¯1 kHL1 ≤ C1 (kψ1 kH2 + ks1 kH1 ), where HL 2 1 H (R), H (R) equipped with the L−dependent norms ¯ 2L2 + L−1 kϕ¯ 0 k2L2 + Lkϕ¯ 00 k2L2 , kϕk ¯ 2H2 = L−3 kϕk L

krk ¯ 2H1 = L−1 krk ¯ 2L2 + Lkr¯0 k2L2 . L (5.3)

Motivated by (5.2), we look for a solution of (2.2) of the form ϕ(t, x) = ϕL (x) + ¯ x), where ϕ, ¯ r¯ satisfy the evolution system ϕ(t, ¯ x), r(t, x) = rL (x) + r(t, 0 ¯ −1 (rL +∂x r)(ϕ ¯ 0L +∂x ϕ), ¯ ∂t ϕ¯ = ∂x2 ϕ¯ + ϕ00L + 2(rL +r) 2 00 0 ∂t r¯ = ∂x r¯ + rL − (r(r ¯ L +2r) ¯ + (∂x ϕ)(2ϕ ¯ +∂ ϕ))(r ¯ ¯ x L +r), L

(5.4)

2 1 together with the initial condition (ϕ(t ¯ 1 ), r(t ¯ 1 )) = (ϕ¯ 1 , r¯1 ) ∈ HL ×HL . If C1 ε ≤ 1/4, Lemma 5.2 below shows that this initial value problem has a unique solution (ϕ, ¯ r) ¯ ∈ 2 1 C([t1 , t1 +T ], HL ×HL ), for some T > 0 (independent of L). This solution depends 2 1 continuously on the initial data (ϕ(t ¯ 1 ), r(t ¯ 1 )) in HL ×HL , uniformly in t ∈ [t1 , t1 +T ]. In addition, there exist C2 > 0, C3 ≥ 1 (independent of L) such that 2 + kr(t)k 1 ≤ C2 (t − t1 ) + C3 C1 ε, ¯ kϕ(t)k ¯ HL HL

(5.5)

¯ x) and r(t, ¯ x) are for all t ∈ [t1 , t1 +T ]. Finally, due to the parabolic regularization, ϕ(t, C ∞ functions of t, x for all t > t1 . Having constructed the solution (ϕ, ¯ r) ¯ of (5.4), we now return to the variables (ψ, s) defined by the relations (2.8), (2.11). Setting σ = τ − τ1 and using the definitions above of L, ϕL , rL , we arrive at the expressions σ/2 e (ξ) + e−σ/2 L−1 ϕ(ξLe e (ξeσ/2 ) − N ψ(τ1 +σ, ξ) = e−σ/2 N ¯ , L2 eσ ), −1 σ/2 σ/2 −1 σ/2 R(ξe )] + 2 log 1 + R(ξe ) r(ξLe ¯ , L2 e σ ) s(τ1 +σ, ξ) = 2 log[R(ξ) (5.6) η 0 (ξ)ψ 0 (ξ, τ )+ψ 0 (ξ, τ )2 ) , − log 1 − 2R(ξ)−2 (e p where 0 ≤ σ ≤ σ1 = log(1 + T e−τ1 ) and R(ξ) = 1 − ηe(ξ)2 . If we assume that C2 T +C1 C3 ε ≤ 1/4, then a direct calculation shows that (ψ, s) ∈ C([τ1 , τ1 +σ1 ], H2 ×H1 ) satisfies, for all τ ∈ [τ1 , τ1 +σ1 ],

¯ τ )kHL2 + kr(e ¯ τ )kHL1 ), kψ(τ )kH2 + ks(τ )kH1 ≤ C4 (τ − τ1 ) + C5 (kϕ(e

(5.7)

where C4 > 0 and C5 ≥ 1 are independent of L (or, equivalently, of τ1 ). The proof of (5.7) relies on the properties of ηe(ξ) listed in Theorem 3.1 and on the fact that the dilation λ 7→ ϕ(λ·) is a continuous operation in Hk (R), k ∈ N. The uniformity in L of

Diffusive Mixing of Stable States in Ginzburg–Landau Equation

87

the constants C4 , C5 follows from the definition of the scaled norms (5.3) and from the fact that 0 ≤ σ ≤ T uniformly in L. By construction, (ψ, s) ∈ C([τ1 , τ1 +σ1 ], H2 ×H1 ) is a solution of the system (2.12) satisfying (ψ(τ1 ), s(τ1 )) = (ψ1 , s1 ). The uniqueness of this solution and its continuous dependence on the data follow directly from the corresponding properties of the system (5.4), see Lemma 5.2. Finally, combining (5.5), (5.7), we obtain kψ(τ )kH2 + ks(τ )kH1 ≤ C1 C3 C5 ε + (C4 + C2 C5 eτ )(τ − τ1 ), for all τ ∈ [τ1 , τ1 +σ1 ]. Therefore, we see that (5.1) holds if we set, for example, 1 1 , K1 = max {C1 C3 C5 , C4 + C2 C5 (1+T1 )}. , ε1 = T1 = min T, 8C2 8C1 C3 This concludes the proof of Proposition 5.1.

Proposition 5.1 relies on a local existence result for the solutions of (5.4) which is the content of the next lemma. Since the system (5.4) is autonomous, we assume here 2 1 , HL are the (without loss of generality) that the initial time is t1 = 0. We recall that HL function spaces defined by the norms (5.3). Lemma 5.2. There exist T > 0, C2 > 0, C3 ≥ 1 such that, for all L ≥ 1 and for all 2 1 (ϕ¯ 1 , r¯1 ) ∈ HL ×HL such that kϕ¯ 1 kHL2 + kr¯1 kHL1 ≤ 1/4, the system (5.4) has a unique 2 1 ×HL ) satisfying (ϕ(0), ¯ r(0)) ¯ = (ϕ¯ 1 , r¯1 ). This solution solution (ϕ, ¯ r) ¯ ∈ C([0, T ], HL 2 1 depends continuously on the initial data (ϕ¯ 1 , r¯1 ) in HL ×HL , uniformly in t ∈ [0, T ]. Moreover, the bound 2 + kr(t)k 1 ≤ C2 t + C3 (kϕ kϕ(t)k ¯ ¯ ¯ 1 kHL2 + kr¯1 kHL1 ), HL HL

(5.8)

holds for all t ∈ [0, T ]. Proof. Lemma 5.2 is a standard local existence result, except that we have to control the dependence on the scaling parameter L ≥ 1, and that we need one more derivative for ϕ¯ than for r. ¯ Because of this imbalance, the apparently semilinear equation (5.4) behaves in fact like a quasilinear system. In particular, the constant C3 in (5.8) cannot be replaced by 1. Throughout the proof, we write (ϕ, r) instead of (ϕ, ¯ r) ¯ for simplicity. The system (5.4) becomes ∂t ϕ = ∂x2 ϕ + FL (ϕ, r), where

∂t r = ∂x2 r + GL (ϕ, r),

(5.9)

0 + ∂x r)(ϕ0L + ∂x ϕ), FL (ϕ, r) = ϕ00L + 2(rL + r)−1 (rL 00 − r(rL + 2r) + (∂x ϕ)(2ϕ0L + ∂x ϕ) (rL + r). GL (ϕ, r) = rL

To avoid the difficulty related to the imbalance of derivatives, we first prove the existence 2 1 ×HL in which the derivatives are of a unique local solution of (5.9) in a subspace of HL balanced. This will be done using a standard contraction mapping argument. 2 1 ×HL satisfying m1 ≡ kϕ¯ 1 kHL2 + kr¯1 kHL1 ≤ 1/4. Let L ≥ 1, and let (ϕ¯ 1 , r¯1 ) ∈ HL For any T > 0, we denote by X(T ) the Banach space o n 1 ), ∂x ϕ ∈ C([0, T ], L∞ ), ∂x2 ϕ ∈ L2 ([0, T ], L2 ) , X(T ) = (ϕ, r) ϕ, r ∈ C([0, T ], HL

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T. Gallay, A. Mielke

equipped with the norm k(ϕ, r)kX(T ) given by n max sup (kr(t)kHL1 + L−1 kϕ(t)kHL1 ), 0≤t≤T

Z sup k∂x ϕ(t)kL∞ ,

0≤t≤T 2

!1/2

T

k∂x2 ϕ(s)k2L2 ds

L 0

2

o .

2

˜ = et∂x r¯1 , where et∂x is the heat kernel. Then (ϕ, ˜ r) ˜ ∈ X(T ) We note ϕ(t) ˜ = et∂x ϕ¯ 1 , r(t) and k(ϕ, ˜ r)k ˜ X(T ) ≤ 1/4 for any T ≤ 1. Let B(T ) be the ball of radius 1/4 centered at (ϕ, ˜ r) ˜ in X(T ). For all (ϕ, r) ∈ B(T ), we define Z t Z t 2 2 (t−s)∂x e FL (ϕ(s), r(s)) ds, r(t) ˆ = e(t−s)∂x GL (ϕ(s), r(s)) ds. ϕ(t) ˆ = 0 0 (5.10) We shall show that (ϕ, ˆ r) ˆ ∈ X(T ) and k(ϕ, ˆ r)k ˆ X(T ) ≤ 1/4 if T is sufficiently small (uniformly inpL). Indeed, since (ϕ, r) ∈ B(T ), we have rL (x) + r(x, t) ≥ inf x rL (x) − kr(t)kL∞ ≥ 2/3 − 1/2 ≥ 1/4 for all x ∈ R, t ∈ [0, T ]. As a consequence, we have 0 +∂x rkL2 kϕ0L +∂x ϕkL∞ ≤ CL−1/2 , kFL (ϕ, r)kL2 ≤ kϕ00L kL2 + 8krL 00 kL2 + krkL2 krL +2rkL∞ + k∂x ϕkL2 k2ϕ0L +∂x ϕkL∞ krL +rkL∞ kGL (ϕ, r)kL2 ≤ krL (5.11) ≤ CL−3/2 + CL1/2 ≤ 2CL1/2 ,

for some C > 0 (independent of L). Therefore, using standard estimates for the heat kernel, we obtain for all t ∈ [0, T ]: √ −1/2 −1/2 −1/2 1/4 t, k∂x ϕ(t)k ˆ t, k∂x ϕ(t)k ˆ t , kϕ(t)k ˆ L∞ ≤ CL L2 ≤ CL L2 ≤ CL R t 1/2 2 2 −1 2 ≤ CL t, k∂ ϕ(s)k ˆ ds ≤ CL t. kr(t)k ˆ L x L2 0 (5.12) Moreover, differentiating GL with respect to x and proceeding as above, we find k∂x GL (ϕ, r)kL2 ≤ CL−1/2 + Ck∂x2 ϕkL2 , hence k∂x r(t)k ˆ L2

Z t 1/2 √ √ 2 ≤ t k∂x GL (ϕ, r)kL2 ds ≤ CL−1/2 t + CL−1/2 t ≤ CL−1/2 t , 0 (5.13)

for all t ∈ [0, T ]. From (5.12), (5.13), we conclude that k(ϕ, ˆ r)k ˆ X(T ) ≤ CT 1/4 , uniformly in L ≥ 1. This proves that the mapping M defined by M (ϕ, r) = (ϕ˜ + ϕ, ˆ r˜ + r) ˆ maps the ball B(T ) into itself, if T is sufficiently small. Using similar estimates, one shows that M is a contraction in B(T ), hence has a unique fixed point (ϕ, r) ∈ B(T ). By construction, this fixed point is the unique solution of (5.9) in X(T ) satisfying (ϕ(0), r(0)) = (ϕ¯ 1 , r¯1 ), and this solution depends continuously in X(T ) on the initial 2 1 ×HL . data (ϕ¯ 1 , r¯1 ) ∈ HL It remains to verify that ∂x2 ϕ ∈ C([0, T ], L2 ) and that (5.8) holds. Since (ϕ, r) = 2 + kr(t)k 1 ≤ m1 = ˜ (ϕ, ˜ r) ˜ + (ϕ, ˆ r), ˆ where (ϕ, ˆ r) ˆ is given by (5.10), and since kϕ(t)k ˜ HL HL 2 + ˆ kϕ¯ 1 kHL2 + kr¯1 kHL1 , it is sufficient to show that ∂x2 ϕˆ ∈ C([0, T ], L2 ) and kϕ(t)k HL 1 kr(t)k ˆ ≤ C(m1 + t) for t ∈ [0, T ]. Appropriate bounds on kϕk ˆ L2 and krk ˆ L2 are HL already contained in (5.12). To bound k∂x rk ˆ L2 , we observe that

Diffusive Mixing of Stable States in Ginzburg–Landau Equation

Z

t 0

Z k∂x2 ϕ(s)k2L2

t

ds ≤ 2 0

89

2 2 2 −1 ˜ ˆ k∂x2 ϕ(s)k L2 + k∂x ϕ(s)k L2 ds ≤ CL t,

(5.14)

by (5.12), hence the estimate (5.13) can be improved as follows: ˆ k∂x r(t)k L2

Z t 1/2 −1 2 2 ≤ t C(L + k∂x ϕ(s)kL2 ) ds ≤ CL−1/2 t.

(5.15)

0

To bound k∂x ϕk ˆ L2 and k∂x2 ϕk ˆ L2 , we need to estimate ∂x FL . It is convenient here to 0 0 ϕL /rL . Using (5.12), write FL (ϕ, r) = FL (0, 0) + FeL (ϕ, r), where FL (0, 0) = ϕ00L + 2rL (5.15), it is straightforward to verify that k∂x FL (0, 0)kL2 ≤ CL−3/2 , k∂x2 FL (0, 0)kL2 ≤ CL−5/2 , √ ˆ L2 ). k∂x FeL (ϕ, r)kL2 ≤ Ck∂x2 rk + C(L−1 + k∂x rkL∞ )(m1 + L−1/2 t + k∂x2 ϕk (5.16) On the other hand, using (5.12), (5.15) as well as various standard estimates for the heat kernel, it is not difficult to show that √ Rt Rt 2 k∂ r(s)k2L2 ds ≤ CL−1 (m21 + t2 ), k∂x r(s)k2L∞ ds ≤ CL−1 t(m21 + t2 ), 0 √ R0t x 2 −2 k∂x r(s)k2L∞ k∂x2 ϕ(s)k ˆ t(m21 + t2 ). L2 ds ≤ CL 0 (5.17) Note that the first estimate in (5.17) does not converge to zero as t → 0, since r(0) = r¯1 1 belongs to HL only. Combining (5.16), (5.17), we thus find Z t Z t k∂x FeL (ϕ, r)k2L2 ds ≤ CL−1 (m21 + t2 ) + CL−2 k∂x2 ϕk ˆ 2L2 ds . (5.18) 0

0

This result implies that ∂x FL ∈ L2 ([0, T ], L2 (R)) and Z t k∂x FL (ϕ, r)k2L2 ds ≤ CL−1 (m21 + t2 ) + CL−3 t 0

2 ) and by (5.12), (5.16). It follows that ϕˆ ∈ C([0, T ], HL

n R o1/2 √ t 2 2 ˆ ≤ t k∂ F (ϕ, r)k ds ≤ CL−1/2 t(m1 + t) + CL−3/2 t, k∂x ϕ(t)k 2 x L L L 0 nR o1/2 √ t 2 2 ˆ ≤ k∂ F (ϕ, r)k ds ≤ CL−1/2 (m1 + t) + CL−3/2 t. k∂x2 ϕ(t)k x L L L2 0 −1/2 In particular, k∂x ϕ(t)k ˆ (m1 +t), which is the desired estimate. The bound on L2 ≤ CL Rt 2 ˆ L2 is not sufficient yet, but replacing it into (5.18), we find 0 k∂x FeL (ϕ, r)k2L2 ds ≤ k∂x ϕk CL−1 (m21 + t2 ), which in turn implies

Z ˆ k∂x2 ϕ(t)k L2

t

≤ 0

Z k∂x2 FL (0, 0)kL2

t

ds+ 0

k∂x FeL (ϕ, r)k2L2 ds

1/2

≤ CL−1/2 (m1 +t) .

ˆ HL1 ≤ C(m1 + t) for all t ∈ [0, T ], and the conThus, we have shown that kϕk ˆ HL2 + krk tinuous dependence on the data follows by the same estimates. The proof of Lemma 5.2 is complete.

90

T. Gallay, A. Mielke

6. Energy Estimates This final section is devoted to the proof of Theorem 4.1 using energy estimates. Let η+ , η− ∈ (− √13 , √13 ), η+ 6= η− , and assume that (ψ, s) ∈ C([τ1 , τ2 ], H2 (R)×H1 (R)) is a solution of (2.12) defined on some time interval [τ1 , τ2 ] ⊂ R+ and satisfying kψ(τ )kH2 + ks(τ )kH1 ≤ ε for all τ ∈ [τ1 , τ2 ], for some sufficiently small ε > 0 (to be specified later). We set (ψ1 , s1 ) = (ψ(τ1 ), s(τ1 )). To control the evolution of the solutions (ψ, s), we introduce the energy functionals E0 (τ ) = Ek+1 (τ ) =

1 2 1 2

R RR R

ψ(τ, ξ)2 dξ, (3δ (k) (τ, ξ)2 + s(k) (τ, ξ)2 ) dξ,

k = 0, 1, 2 ,

(6.1)

where δ = ψ 0 . The use of the quadratic forms 3(δ (k) )2 + (s(k) )2 instead of (δ (k) )2 + (s(k) )2 in (6.1) can be understood as follows. For any β > 0, the system (2.12), (2.14) for δ, s can be written in the form 00 a(η) βη βδ β δ˙ + Rβ (ξ, τ ; δ, δ 0 , s, s0 ), (6.2) = s00 β −1 a2 (η) a1 (η) s˙ where a(η), a1 (η), a2 (η) are defined in (2.13) and the remainder Rβ contains only lower right-hand side of (6.2), order derivatives of δ, s. If Mβ (η) denotes the 2×2 matrix in the √ then TrMβ (η) = 2 and DetMβ (η) = 1 for all β > 0, |η| < 1/ 3, so that R 1 is always a double eigenvalue of Mβ . However, the evolution of the functionals (β 2 (δ (k) )2 + (s(k) )2 )dξ is determined (up to lower order terms) by the symmetrized matrix Mβs = 1 t E (τ ) in (6.1) 2 (Mβ + Mβ ), which is not necessarily positive definite. Our choice of √ k+1 s is motivated by the fact that Mβ is positive definite for all |η| < 1/ 3 if and only if √ β = 3. In this case, the eigenvalues are given by √

µ± (η) = 1 ±

3|η| (3 − 2η 2 + 3η 4 ). 6(1 − η 2 )2

(6.3)

√ √ ¯ In the sequel, we set η0 = max{|η+ |, |η− |} < 1/ 3, η¯ = 21 (η0 + 1/ 3), µ¯ = µ− (η), 2 2 a¯ = a(η) ¯ = (1 − 3η¯ )/(1 − η¯ ), and we assume that ε √ ≤ η¯ − η0 . This implies in particular that |η(τ, ξ)| = |e η (τ, ξ) + δ(τ, ξ)| ≤ η0 + ε ≤ η¯ < 1/ 3 for all ξ ∈ R, τ ∈ [τ1 , τ2 ]. Lemma 6.1. Under the assumptions above, we have E0 ∈ C 1 ([τ1 , τ2 ]) and, for all ν > 0, there exists a constant K2 > 0 such that Z Z 3 2 ˙ E0 (τ ) ≤ − − ν E0 (τ ) − a¯ δ 2 dξ + K2 s0 dξ, (6.4) 2 ¯ = (1 − 3η¯ 2 )/(1 − η¯ 2 ) > 0. for all τ ∈ [τ1 , τ2 ], where a¯ = a(η) Remark. Here and in the sequel, K2 , K3 , . . . denote positive constants depending on η+ , η− , but not on the interval [τ1 , τ2 ] ⊂ R+ nor on the solution (ψ, s), provided kψ(τ )kH2 + ks(τ )kH1 ≤ ε.

Diffusive Mixing of Stable States in Ginzburg–Landau Equation

91

Proof. By hypothesis on ψ, we have E0 ∈ C([τ1 , τ2 ]. To prove the differentiability, we first assume that the initial data (ψ1 , s1 ) belong to S(R), the space of rapidly decreasing C ∞ functions on R. Then ψ(τ ), s(τ ) ∈ S(R) for all τ ∈ [τ1 , τ2 ], E0 ∈ C ∞ ([τ1 , τ2 ]), and a direct calculation shows that Z Z Z 3 0 0 2 ˙ ψ dξ + ηs0 ψ dξ. η + ψ ) − A(e η )) dξ − (6.5) E0 (τ ) = − ψ (A(e 4 In the general case, we use the fact that the solution (ψ(τ ), s(τ )) depends continuously on the data (ψ1 , s1 ) ∈ H2 (R)×H1 (R), see Proposition 5.1. If F R(τ ) denotes the right-hand τ side of (6.5), we see that (for fixed τ ) both E0 (τ ) and E0 (τ1 ) + τ1 F (t) dt are continuous functions of (ψ1 , s1 ). Since these functions coincide on a dense subset (namely, S(R)2 ), they must be equal everywhere. This shows that E0 ∈ C 1 ([τ1 , τ2 ]) and satisfies (6.5). To prove (6.4), we first note that A(e η + ψ 0 ) − A(e η ) = a(η)ψ ˆ 0 for some ηˆ ∈ [e η , ηe + ψ 0 ]. 2 0 0 02 Since |η| ˆ ≤ η¯ by assumption, we have ψ (A(e η + ψ ) − A(e η )) = a(η)ψ ˆ ≥ a¯ ψ 0 . On the ν 2 1 02 0 0 other hand, for all ν > 0, |ηs ψ| ≤ |s ψ| ≤ 2 ψ + 2ν s . We thus find Z Z Z 3 ν 2 2 E˙ 0 (τ ) ≤ −¯a ψ 0 dξ − − ψ 2 dξ + K2 s0 dξ, 4 2 where K2 = (2ν)−1 . This concludes the proof of Lemma 6.1.

Lemma 6.2. Under the same assumptions, we have E1 ∈ C 1 ([τ1 , τ2 ]), and there exists a constant K3 > 0 such that Z Z 1 τ 2 −τ 2 2 ˙ s dξ + K3 e + (s + δ ) dξ , E1 (τ ) ≤ −µE ¯ 2 (τ ) − e 2 for all τ ∈ [τ1 , τ2 ], where µ¯ = µ− (η) ¯ is given by (6.3).

R R Proof. As in the proof of Lemma 6.1, the differentiability of δ 2 dξ and s2 dξ is easily verified using a density argument. From (2.14), we have R R R R δ δ˙ dξ = − δ 0 (A(e η + δ) − A(e η ))0 dξ − 41 δ 2 dξ − ηs0 δ 0 dξ R R R 2 η + δ) − a(e η )) dξ. = − (a(η)δ 0 + ηs0 δ 0 ) dξ − 41 δ 2 dξ − ηe0 δ 0 (a(e 0 Since |e η 0 δ 0 (a(e η + δ) − a(e η ))| ≤ |e η 0 ||a0 (η)||δδ ¯ | ≤ µ4¯ δ 0 + C1 δ 2 for some C1 > 0, we obtain Z Z Z Z µ¯ 2 2 δ 0 dξ + C1 δ 2 dξ. (6.6) δ δ˙ dξ ≤ − (a(η)δ 0 + ηs0 δ 0 ) dξ + 4 2

On the other hand, using (2.12) and integrating by parts, we find R R R R 2 ss˙ dξ = − (a1 (η)s0 + a2 (η)s0 δ 0 ) dξ − 41 s2 dξ − 2eτ (1 − η 2 )s(es− 1) dξ R 1 02 2 0 0 0 η 00 s − a02 (η)η 0 δ 0 s + a3 (η)sη 0 dξ. + 2 ss − a1 (η)η ss + a2 (η)e Since η 2 ≤ 1/3 and |s| ≤ ε ≤ 1/2, we have 2(1 − η 2 )s(es − 1) ≥ s2 . The other terms 2 2 are bounded as follows: |ss0 | ≤ εs0 , |a01 (η)η 0 ss0 | ≤ C|(e η 0 +δ 0 )ss0 | ≤ C(|ss0 |+ε|s0 δ 0 |), 2 2 00 −|ξ| 0 0 0 0 02 η s| ≤ C|s|e , |a2 (η)η δ s| ≤ C(|sδ |+εδ ), |a3 (η)sη 0 | ≤ C(|s|e−|ξ| +εδ 0 ). |a2 (η)e

92

We thus obtain R

T. Gallay, A. Mielke

R 2 R 2 s dξ ss˙ dξ ≤ − (a1 (η)s0 + a2 (η)s0 δ 0 ) dξ − 41 + eτ R −|ξ| 0 0 02 02 +C2 (|s|e + |ss | + |sδ | + ε(s + δ )) dξ,

0 0 ¯ for some C2 > 0. Now, assuming that ε ≤ µ/(4C 2 ), we have C2 ε(s +δ ) ≤ 1 2 τ 02 02 −|ξ| −τ −2|ξ| 0 µ(s ¯ +δ )/4. In addition, C2 |s|e ≤ 2 s e + Ce e and C2 (|ss | + |sδ 0 |) ≤ 02 02 2 + δ )/4 + Cs . Therefore, µ(s ¯ R 2 R R 2 s dξ ss˙ dξ = − (a1 (η)s0 + a2 (η)s0 δ 0 ) dξ + C3 − 21 eτ R (6.7) 2 2 + µ2¯ (s0 + δ 0 ) dξ + C3 e−τ , 2

for some C3 > 0. Finally, by definition of µ− (η) (see (6.3)), we have the inequality Z 2 2 ¯ 2 = 2µE ¯ 2. 3a(η)δ 0 + (3η + a2 (η))s0 δ 0 + a1 (η)s0 dξ ≥ 2µ− (η)E

2

(6.8)

Thus, multiplying (6.6) by 3 and adding the result to (6.7), we obtain Z Z 1 τ 2 ˙ s dξ + 3C1 δ 2 dξ + C3 e−τ , ¯ 2 (τ ) + C3 − e E1 (τ ) ≤ −µE 2 for all τ ∈ [τ1 , τ2 ]. This concludes the proof of Lemma 6.2.

Lemma 6.3. Under the same assumptions, (ψ, s) ∈ L2 ([τ1 , τ2 ], H3 (R)×H2 (R)), E2 ∈ W1,1 ([τ1 , τ2 ]), and there exists a constant K4 > 0 such that Z Z Z 1 2 2 2 ¯ 3 (τ ) − eτ s0 dξ + K4 e−τ + eτ s2 dξ + (s0 + δ 2 + δ 0 ) dξ . E˙ 2 (τ ) ≤ −µE 2 (6.9) Remark. Due to the regularization, one also has (ψ, s) ∈ C((τ1 , τ2 ], H3 (R)×H2 (R)), E2 ∈ C 1 ((τ1 , τ2 ]), and (6.9) holds for all τ ∈ (τ1 , τ2 ]. Proof. As in the proof of Lemma 6.1, we first assume that ψ(τ ), s(τ ) ∈ S(R) for all τ ∈ [τ1 , τ2 ]. Then, from (2.14) we have R R 0 0 δ δ˙ dξ = − δ 00 δ˙ dξ R R R 2 (6.10) = − δ 00 (A(e η + δ) − A(e η ))00 dξ + 41 δ 0 dξ − δ 00 (ηs0 )0 dξ. η + δ) − a(e η ))e η 00 + (a0 (e η + δ) − a0 (e η ))(e η 0 )2 + Since (A(e η + δ) − A(e η ))00 = a(η)δ 00 + (a(e 0 0 0 0 0 0 00 0 0 0 a (η)(2e η + δ )δ and (ηs ) = ηs + (e η + δ )s , we obtain using straightforward bounds R R 0 0 R 2 2 δ δ˙ dξ ≤ − (a(η)δ 00 + ηs00 δ 00 ) dξ + 41 δ 0 dξ R 2 +C4 |δ 00 |(|δ| + |δ 0 | + |s0 | + |δ 0 | + |s0 δ 0 |) dξ, for some C4 > 0. Now, C4 |δ 00 |(|δ| + |δ 0 | + |s0 |) ≤

µ¯ 00 2 8δ

+ C(δ 2 + δ 0 + s0 ) and 2

2

C4 kδ 00 δ 0 (|δ 0 | + |s0 |)kL1 ≤ C4 kδ 00 kL2 kδ 0 kL2 (kδ 0 kL2 + ks0 kL2 ) ≤ 2εC4 kδ 00 kL2 kδ 0 kL2 3/2

≤

µ¯ 00 2 8 kδ kL2

1/2

+ Cε4 kδ 0 k2L2 ,

3/2

1/2

Diffusive Mixing of Stable States in Ginzburg–Landau Equation

hence Z

δ 0 δ˙0 dξ ≤ −

Z

(a(η)δ 00 + ηs00 δ 00 ) dξ + 2

µ¯ 4

Z

93

δ 00 dξ + C5 2

Z

(δ 2 + δ 0 + s0 ) dξ, (6.11) 2

2

for some C5 > 0. On the other hand, assuming always ψ(τ ), s(τ ) ∈ S(R), we have from (2.12), R R R 0 0 2 s s˙ dξ = − s00 s˙ dξ = − (a1 (η)s00 + a2 (η)s00 η 00 ) dξ R R R 00 (6.12) 2 1 02 τ + 4 s dξ + 2e s (1 − η 2 )(es − 1) dξ − a3 (η)s00 η 0 dξ. Integrating by parts, we obtain Z Z Z η 000 + a02 (η)e η 00 (e η 0 + δ 0 )) dξ ≤ C |s0 |(e−|ξ| + |δ 0 |) dξ, − a2 (η)s00 ηe00 dξ = s0 (a2 (η)e R R 2 3 − a3 (η)s00 η 0 dξ = s0 (a03 (η)η 0 + 2a3 (η)η 0 η 00 ) dξ R 0 −|ξ| 3 ≤ C |s |(e + |δ 0 | + |δ 00 | + |δ 0 | + |δ 0 δ 00 |) dξ, R R R 2 2eτ s00 (1 − η 2 )(es − 1) dξ = −2eτ (1 − η 2 )es s0 dξ + 4eτ ηη 0 (es − 1)s0 dξ R 2 R ≤ −eτ s0 dξ + Ceτ |ss0 |(1 + |δ 0 |) dξ. Therefore, Z Z Z Z 1 0 0 00 2 00 00 τ 02 s s˙ dξ = − (a1 (η)s + a2 (η)s δ ) dξ + −e s dξ + R dξ, 4 where |R| ≤ C6 |s0 |(e−|ξ| + |δ 0 | + |δ 00 | + |δ 0 | + |δ 0 δ 00 |) + C6 eτ |ss0 |(1 + |δ 0 |) for some 2 C6 > 0. Using the bounds C6 |s0 |e−|ξ| ≤ 41 eτ s0 + Ce−τ e−2|ξ| and 3

C6 ks0 (|δ 0 | + |δ 00 | + |δ 0 | + |δ 0 δ 00 |)kL1 ≤ 3

C6 eτ kss0 (1 + |δ 0 |)kL1 ≤ we find

R

µ¯ 00 2 0 2 0 2 2 kδ kL2 + C(ks kL2 + kδ kL2 ), 1 τ 0 2 τ 2 4 e ks kL2 + Ce kskL2 ,

R 02 R 2 s dξ s0 s˙0 dξ ≤ − (a1 (η)s00 + a2 (η)s00 δ 00 ) dξ + C7 − 21 eτ R 00 2 R τ 2 −τ 02 +µ/2 ¯ δ dξ + C7 e + (e s + δ ) dξ ,

(6.13)

for some C7 > 0. Finally, combining (6.11), (6.13) and using the analogue of (6.8), we arrive at (6.9). This proves the claim when ψ(τ ), s(τ ) ∈ S(R). To infer the general case, we use again a density argument. Integrating (6.9), we obtain Z τ2 Z τ2 E3 (τ ) dτ ≤ E2 (τ1 ) + F (τ ) dτ, (6.14) E2 (τ2 ) + µ¯ τ1

ks(τ )k2H1 )

τ1

+ for all τ ∈ [τ1 , τ2 ]. Since S(R)2 is dense in where F (τ ) ≤ H2 (R)×H1 (R), the estimate (6.14) shows that, if (ψ, s) ∈ C([τ1 , τ2 ], H2 (R)×H1 (R)) is any solution satisfying the usual assumptions, then (ψ, s) ∈ L2 ([τ1 , τ2 ], H3 (R)×H2 (R)). Using this result, it follows immediately from (6.10), (6.12) that E˙ 2 ∈ L1 ([τ1 , τ2 ]), hence E2 ∈ W1,1 ([τ1 , τ2 ]), and the same calculations as above show that (6.9) holds in the general case. This concludes the proof of Lemma 6.3. C(kψ(τ )k2H2

94

T. Gallay, A. Mielke

As a consequence of Lemmas 6.1, 6.2, 6.3, we have the following estimate: Proposition 6.4. There exist ε2 > 0, K5 ≥ 1 such that, if (ψ, s) ∈ C([τ1 , τ2 ], H2 (R) ×H1 (R)) is any solution of (2.12) satisfying kψ(τ )kH2 +ks(τ )kH1 ≤ ε2 for all τ ∈ [τ1 , τ2 ], then kψ(τ )kH2 + ks(τ )kH1 ≤ K5 e−τ /2 1 + eτ1 /2 (kψ(τ1 )kH2 + ks(τ1 )kH1 ) , (6.15) for all τ ∈ [τ1 , τ2 ]. ¯ Proof. Fix γ ∈ (1/2, 3/4) and let ν = 3/2 − 2γ. We set ε2 = min{η¯ − η0 , µ/(4C 2 )}, where C2 is defined in the proof of Lemma 6.2. Then, for A, B > 0 large enough, we define G2 (τ ) = B(AE0 (τ ) + E1 (τ )) + E2 (τ ), τ ∈ [τ1 , τ2 ]. ¯ ≥ For instance, we may choose a¯ A = K3 + 11/4 and assume that B ≥ 2K4 + 1, 3µB 2K4 + 9/2. Then, using Lemmas 6.1, 6.2, 6.3, we find Z 1 2 G˙ 2 (τ ) ≤ −2γG2 (τ ) + (C8 − eτ ) (s2 + s0 ) dξ + C8 e−τ , τ ∈ [τ1 , τ2 ], 2 for some C8 > 0. Integrating this inequality, we easily obtain G2 (τ ) ≤ C9 G2 (τ1 )e−2γ(τ −τ1 ) + e−τ , τ ∈ [τ1 , τ2 ], for some C9 > 0. Since 2γ > 1 and G2 (τ ) is equivalent to kψ(τ )k2H2 + ks(τ )k2H1 , this proves (6.15). Combining Proposition 5.1 and Proposition 6.4, we are now able to complete the proof of Theorem 4.1. Proof of Theorem 4.1. Fix γ ∈ (1/2, 3/4), and let ε3 = min{ε1 , ε2 }, where ε1 is defined in Proposition 5.1 and ε2 in Proposition 6.4. We set ε3 ε3 , , τ0 = max 0, −2 log ε0 = 4K5 4K5 where K5 is defined in Proposition 6.4. Then, for all (ψ0 , s0 ) ∈ H2 (R)×H1 (R) satisfying kψ0 kH2 + ks0 kH1 ≤ ε0 , the system (2.12) has a unique global solution (ψ, s) ∈ C([τ0 , ∞), H2 (R)×H1 (R)) with initial data (ψ(τ0 ), s(τ0 )) = (ψ0 , s0 ). Indeed, according to our local existence result (Proposition 5.1), it suffices to show that the solution (ψ(τ ), s(τ )) satisfies kψ(τ )kH2 + ks(τ )kH1 < ε3 whenever it exists. Assume on the contrary that there exists a time τ3 > τ0 such that kψ(τ3 )kH2 + ks(τ3 )kH1 = ε3 and kψ(τ )kH2 + ks(τ )kH1 < ε3 for all τ ∈ [τ0 , τ3 ). Then, according to Proposition 6.4, we have ε 3 kψ(τ3 )kH2 + ks(τ3 )kH1 ≤ K5 e−τ3 /2 1 + ε0 eτ0 /2 ≤ K5 e−τ0 /2 + ε0 ≤ , 2 which is a contradiction. Therefore, (ψ(τ ), s(τ )) exists for all τ ≥ τ0 and satisfies kψ(τ )kH2 + ks(τ )kH1 ≤ ε3 /2. In particular, from (6.15) we have kψ(τ )kH2 + ks(τ )kH1 ≤ Ce−τ /2 ,

τ ≥ τ0 ,

(6.16)

for some C > 0. It remains to show that kψ(τ )kH2 + ks(τ )kH1 = O(e−γτ ) as τ → +∞. In fact, we shall prove that kψ(τ )kH2 = O(e−γτ ) and ks(τ )kH1 = O(e−τ ) as τ → +∞.

Diffusive Mixing of Stable States in Ginzburg–Landau Equation

95

We begin with ks(τ )kL2 . From (6.7) and (6.16), we have for all τ ≥ τ0 : Z Z Z Z d 2 τ 2 −τ 02 ¯ ¯ s dξ ≤ (C1 − e ) s dξ + C1 e + δ dξ ≤ −eτ s2 dξ + C¯ 2 e−τ , dτ for some C¯ 1 , C¯ 2 > 0. This differential inequality implies that ks(τ )kL2 = O(e−τ ) as τ → +∞. Indeed, let f (τ ) = e2τ ks(τ )k2L2 , and assume that eτ ≥ 4. Then f˙(τ ) ≤ 2f (τ ) + eτ (C¯ 2 − f (τ )) ≤ eτ (C¯ 2 − f (τ )/2), hence f (τ ) ≤ 2C¯ 2 + 1 if τ is sufficiently large. To prove that ks0 (τ )kL2 = O(e−τ ) as τ → +∞, we first have to show that E3 (τ ) = O(e−τ ), see (6.1). This estimate can be established using exactly the same techniques as above. Indeed, due to the parabolic regularization, E3 (τ ) is continuously differentiable for all τ > τ0 . Thus, proceeding as in Lemma 6.2 or 6.3, one verifies that there exists a K6 > 0 such that R 2 R 2 2 2 E˙ 3 (τ ) ≤ − 21 eτ s00 dξ + K6 (s00 + δ2 + δ 0 + δ 00 ) dξ R 2 +K6 e−τ + eτ (s2 + s0 ) dξ , for all τ ≥ τ0 + 1 (say). Then, defining G3 (τ ) = DG2 (τ ) + E3 (τ ) for some sufficiently large D > 0 and proceeding as in Proposition 6.4, we obtain Z 1 τ 2 2 ˙ (s2 + s0 + s00 ) dξ + K7 e−τ , G3 (τ ) ≤ −2γG3 (τ ) − e 2 for some K7 > 0. By Gronwall’s lemma, we have Z 1 τ −2γ(τ −t) t e e ks(t)k2H2 dt ≤ C¯ 3 e−τ , G3 (τ ) + 2 τ0 +1

τ ≥ τ0 + 1,

(6.17)

for some C¯ 3 > 0. In particular, E3 (τ ) = O(e−τ ) as τ → +∞. Using this result and (6.13), we deduce R 02 R R 02 d ¯ 4 − eτ ¯ 4 e−τ + (eτ s2 + δ 0 2 + δ 00 2 ) dξ dξ ≤ C dξ + C s s dτ R 02 ≤ C¯ 4 − eτ s dξ + C¯ 5 e−τ , hence ks0 (τ )kL2 = O(e−τ ) as τ → +∞. Finally, choosing A, B > 0 large enough, we define Z Z Z 2 H2 (τ ) = B A ψ 2 dξ + δ 2 dξ + δ 0 dξ. Using (6.4), (6.6), (6.11) and proceeding as in the proof of Proposition 6.4, we obtain H˙ 2 (τ ) ≤ −2γH2 (τ ) + C¯ 6 ks(τ )k2H2 ,

(6.18)

for some C¯ 6 > 0. To control the last term in the right-hand side, we recall that Z τ e(2γ+1)t ks(t)k2H2 dt ≤ 2C¯ 3 e(2γ−1)τ , τ ≥ τ0 + 1, τ0 +1

by (6.17). Since 2γ + 1 > 2, it follows that Therefore, we conclude from (6.18) that

R∞ τ0

eλτ ks(τ )k2H2 dτ < ∞ for all λ < 2.

96

T. Gallay, A. Mielke

H2 (τ ) ≤ e

−2γ(τ −τ0 )

Z H2 (τ0 ) + C¯ 6

τ τ0

e−2γ(τ −t) ks(t)k2H2 dt ≤ C¯ 7 e−2γτ ,

hence kψ(τ )kH2 = O(e−γτ ) as τ → +∞. The proof of Theorem 4.1 is now complete. Acknowledgement. The authors are grateful to G. Raugel and G. Schneider for helpful discussions. This work was begun when A.M. visited the University of Paris XI and Th.G. the University of Hannover. Both institutions are acknowledged for their hospitality. This research was partially supported by the DFGSchwerpunktprogramm “Dynamische Systeme” under the grant Mi 459/2-2.

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Diffusive Mixing of Stable States in Ginzburg–Landau Equation

[Uec98] [Way97] [Zei90]

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Communicated by A. Kupiainen

Commun. Math. Phys. 199, 99 – 115 (1998)

Communications in

Mathematical Physics © Springer-Verlag 1998

Upper Bounds for Regularized Determinants H. Gillet1,? , C. Soul´e2 1 Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607-7045, USA 2 CNRS, Institut des Hautes Etudes ´ Scientifiques, 35, Route de Chartres, 91440, Bures-sur-Yvette, France

Received: 7 July 1997 / Accepted: 20 April 1998

Abstract: We conjecture that the zeta-regularized determinant of the Laplace operator with coefficients in a holomorphic vector bundle on a compact K¨ahler manifold remains bounded when the metric on the bundle varies. This conjecture is shown to be true for certain classes of line bundles on Riemann surfaces. Introduction Let E be a holomorphic vector bundle on a compact K¨ahler manifold X. If we fix a metric h on E, we get a Laplace operator 1 acting upon smooth sections of E over X. Using the zeta function of 1, one defines its regularized determinant det 0 (1). In [5] §4.1.6, inspired by our arithmetic Riemann–Roch theorem, we were led to conjecture that, when h varies, this determinant det 0 (1) remains bounded from above. In this paper we prove this in two special cases. The first case is when X is a Riemann surface, E is a line bundle and dim H 0 (X, E) + dim H 1 (X, E) ≤ 2, and the second case is when X = P1 , E is a line bundle, and all metrics under consideration are invariant under rotation around a fixed axis. To get the desired upper bound in the first case we use an inequality of Moser and Trudinger, and its extension to arbitrary compact manifolds due to Fontana [3]. We prove the second case by direct estimates. Though our results deal with very few cases, we find it striking that inequalities as sharp as the theorem of Moser and Trudinger can be used to prove our conjecture. We hope the reader will get interested in the general question, and try to either prove or disprove our statement. In the first paragraph we phrase the conjecture in its most general form, and give a few facts about it. Next, in the case of a line bundle over a Riemann surface, we compute the anomaly A(ϕ) for the regularized determinant of the Laplace operator when a fixed metric h0 on the line bundle is replaced by h0 eϕ . To check our theorem, we then need to ?

Supported by N.S.F grant DMS-9501500

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bound from above the functional A(ϕ) when ϕ is any smooth function on the Riemann surface (resp. any function of the distance to the origin on the projective line). This is done in the next two paragraphs. At the end, we discuss the (much easier) case of the trivial line bundle on the circle, where the determinant is bounded from below. 1. Statement of the Results 1.1. Let X be a smooth, projective, equidimensional complex variety of dimension d and hX an hermitian metric on its tangent space. The associated (normalized) K¨ahler form µ is defined by the formula ∂ i X ∂ hX , dzα dz β , µ= 2π ∂z α ∂z β α,β

where (zα ) is any local holomorphic chart on X. Consider an holomorphic vector bundle E on X, equipped with a C ∞ hermitian metric h. Let A0q (X, E), q = 0, .., d, be the space of smooth forms of type (0, q) with values in E. The L2 -metric on A0q (X, E) is defined by the formula Z hs, tiL2 = hs(x), t(x)i µd /d! , X

where s, t ∈ A (X, E) and hs(x), t(x)i is the pointwise scalar product defined by h and hX ([5], §4.1.1). The Cauchy–Riemann operator 0q

∂ : A0q (X, E) → A0,q+1 (X, E) ∗

has an adjoint ∂ :

∗

h∂ s, tiL2 = hs, ∂ tiL2 . ∗

∗

We consider the Laplace operator 1q = ∂ ∂ + ∂ ∂ on A0q (X, E) and its zeta function Z ∞ 1 ζ1q (s) = tr (e−1q t ) ts−1 dt , Re (s) > 1. 0(s) 0 It is known that ζ1q (s) can be analytically continued to the whole complex plane and is regular at the origin. The regularized determinant of 1q is defined to be det 0 (1q ) = exp (−ζ10 q (0)), d where ζ10 q (0) is the value of ds ζ1 (s) at s = 0. Our goal is to find upper bounds for 0 det (1q ) when h varies. More precisely, for any q ≥ 0, consider the spaces

¯ 0,q−1 (X, E)) ⊂ A0q (X, E), B q = ∂(A

q ≥ 1,

B 0 = 0, and the zeta function q ζB q (s) = Tr(1−s q | B ),

Re(s) > d.

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By the Hodge decomposition theorem we have A0q (X, E) = B q ⊕ ∂¯ ∗ (A0,q+1 (X, E)) ⊕ Ker(1q ), and the Cauchy–Riemann operator induces an isomorphism ∼ ∂¯ : ∂¯ ∗ (A0q (X, E)) −→ B q

¯ It follows that such that ∂¯ 1q−1 = 1q ∂. ζ1q (s) = ζB q (s) + ζB q+1 (s), hence

ζB q+1 (s) = ζ1q (s) − ζ1q−1 (s) + ζ1q−2 (s) + · · · + (−1)q ζ10 (s).

This implies that ζB q (s) converges when Re(s) > d, has a meromorphic continuation to the whole complex plane, and is regular at the origin. Define 0 Dq (E, h) = exp(−ζB q (0)).

In [5], §4.1.6, we proposed the following Conjecture 1. There exists a constant Cq (E) such that, for any choice of a metric h on E, Dq (E, h) ≤ Cq (E). 1.2. Remarks on the conjecture. 1.2.1. In the conjecture above both Dq (E, h) and Cq (E) depend in general on the metric hX on X. Notice that, for any real constant t > 0, Dq (E, th) = Dq (E, h).

(1)

Indeed, when h is replaced by th, the L2 -metric on A0q (X, E) gets multiplied by the same factor t for all q ≥ 0, therefore 1q remains unchanged. Furthermore, if E ∨ = E ∗ ⊗ 3d (T X ∗ ) is Poincar´e dual of E and h∨ the metric on ∨ E induced by h and hX , the Poincar´e-Serre duality implies that Dq (E, h) = Dd+1−q (E ∨ , h∨ ).

(2)

Therefore the conjecture is stable under scaling and duality. 1.2.2. Our inspiration to make this conjecture was number theoretic. Assume that X is the set X (C) of complex points of a regular projective flat scheme X over Z and that E is the holomorphic vector bundle defined by an algebraic vector bundle E on X . In loc.cit. we defined arithmetic Betti numbers bq (E, h) ∈ R ,

0 ≤ q ≤ d + 1,

as follows . When M is a finitely generated abelian group, equipped with a norm k · k on its real span M ⊗ R, we let Z

h0 (M, k · k) = log # {m ∈ M/kmk ≤ 1}

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and h1 (M, k · k) = h0 (M ∗ , k · k∗ ). We consider the coherent cohomology groups H q (X , E), q ≥ 0, equipped with their L2 metric. Then we let bq (E, h) = h0 (H q (X , E)) + h1 (H q−1 (X , E)) +

1 0 ζ q (0). 2 B

(3)

In [5], loc.cit., we gave properties of these numbers (duality, Euler characteristic formula) which partially justified calling them Betti numbers. However, a basic property should be that each bq (E, h) is nonnegative, or at least bounded below. This led us to the conjecture in Sect. 1.1. 1.2.3. It would be of interest to find some interpretation of our conjecture in mathematical physics. In [6], the Moser–Trudinger inequality (see Sect. 2.3 below) is interpreted as the existence of a lower bound for a free energy functional, and it is derived in op.cit. Prop. 4 from the Gibbs variational principle. 1.2.4. A stronger version of the conjecture consists in requiring that Cq (E) depends only on the C ∞ bundle underlying E, and not on its holomorphic structure. Results like [10] Proposition 3 (due to Miyaoka and based on a result of Selberg, [10] Theorem 4) , which says that when d = 1 and when E is a flat unitary bundle, the following holds log D1 (E, h) ≤ constant · rank(E), points in this direction. 1.3. Results From now on we assume that X is a curve (d = 1) and that E is a line bundle L. We then take q = 1 and we write D(L, h) instead of D1 (E, h) = det0 (10 ). Theorem 1. i)

If dim H 0 (X, L) + dim H 1 (X, L) ≤ 2,

there exists a constant C(L) such that, for any metric h on L, D(L, h) ≤ C(L) ; ii) Assume that X = P1 (C) is the complex projective line, that hX is invariant under rotation, and that L = O(n), n ∈ Z. Then there is a constant C(n) such that, for any metric h on L invariant under rotation, D(L, h) ≤ C(n). To clarify statement ii) above, let us write z = r eiθ the standard coordinate on C ⊂ P1 (C). Given any α, we let rα (z) = eiα z be the rotation of angle α. A metric hX on X (resp. h on L) is said to be invariant under rotation when rα∗ (hX ) = hX (resp. rα∗ (h) = h) for all values of α.

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2. An Anomaly Formula 2.1. We fix X, hX and L as in Sect. 1.3. Let h0 and h = h0 exp (ϕ) be two hermitian metrics on L, with ϕ a smooth real valued function on X. We shall give a formula comparing the determinants D(L, h) and D(L, h0 ). Let b0 = dim H 0 (X, L) and b1 = dim H 1 (X, L). We endow H 0 (X, L) = ker (∂) ⊂ ∗ 00 A (X, L) and H 1 (X, L) = ker (∂ ) ⊂ A01 (X, L) with the L2 -metric coming from h0 . Let (αi ), i = 1, . . . , b0 , be an orthonormal basis of H 0 (X, L), and (βi ), i = 1, . . . , b1 , an orthonormal basis of H 1 (X, L). If 1 ≤ i, j ≤ b0 we let hαi , αj i be the pointwise scalar product of αi with αj . We define similarly hβi , βj i, 1 ≤ i, j ≤ b1 . ∂∂ c If d = ∂ + ∂ we let dc = ∂−∂ 4πi , so that dd = 2πi . Denote by c1 (TX , hX ) the first Chern form of the tangent bundle to X, and by c1 (L, h0 ) the first Chern form of (L, h0 ). Clearly, to prove Theorem 1 it will be enough to show that, under the given hypotheses, the quantity A(ϕ) = log D(L, h) − log D(L, h0 )

(4)

remains bounded from above when ϕ varies. Proposition 1. The following formula holds: Z Z 1 A(ϕ) = ϕ ddc (ϕ) − ϕ(c1 (TX , hX )/2 + c1 (L, h0 )) 2 X X Z eϕ hαi , αj i µ + log det X

Z + log det

1≤i,j≤b0

e−ϕ hβi , βj i µ

X

. 1≤i,j≤b1

2.2. To prove Proposition 1 we consider the Quillen metric [9] on the complex line λ(L) = 3b0 H 0 (X, L) ⊗ (3b1 H 0 (X, L))∗ . It is defined as the quotient of the L2 -metric by the determinant of the Laplace operator: k · k2Q,h = k · k2L2 ,h D(L, h)−1 . Therefore we get A(ϕ) = log

k · k2L2 ,h k · k2L2 ,h0

− log

k · k2Q,h k · k2Q,h0

,

(5)

and we are led to compute the variation of both the L2 -metric and the Quillen metric on λ(L). Concerning the Quillen metric, we can use the anomaly formula in [1], Theorem e (h, h0 ) be the Bott–Chern secondary characteristic class of L, which satisfies 0.3. Let ch e (h, h0 ) = ch (L, h) − ch (L, h0 ), ddc ch where ch (L, h) = exp (c1 (L, h)) is the Chern character form of (L, h). If Td (TX , hX ) is the form representing the Todd class of X defined using hX , the following holds (loc.cit.):

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− log

k · k2Q,h k · k2Q,h0

Z

e (h, h0 ) Td (TX , hX ). ch

= X

Since X has dimension one, we have Z Z Z e e e 1 (h, h0 ) c1 (TX , hX )/2, ch (h, h0 ) Td (TX , hX ) = ch2 (h, h0 ) + ch X

X

X

e p is the component of degree (p − 1, p − 1) of ch. e where ch ϕ Let (OX , e ) be the trivial line bundle equipped with the metric such that k1k2 = exp (ϕ). Since (L, h) = (L, h0 ) ⊗ (OX , eϕ ) we deduce from [4], Proposition 1.3.3 (and formula (1.3.5.2)) that e (eϕ , 1). e (h, h0 ) = ch (L, h0 ) ch ch e (eϕ , 1) comparing the metric such that k1k2 = To compute the Bott–Chern class ch exp (ϕ) with the trivial metric on OX , first notice that e 1 (eϕ , 1) = e ch c1 (eϕ , 1) = −ϕ by [4], (1.2.5.1). Furthermore, from [4], (1.3.1.2) we get 1 1 e 2 (eϕ , 1) = 1 e ch c21 (eϕ , 1) = c1 (eϕ ) e c1 (eϕ , 1) = ddc (ϕ) ϕ. 2 2 2 So we conclude that − log

k · k2Q,h k · k2Q,h0

Z = −

c1 (L, h0 ) ϕ + X

1 2

Z ϕ ddc (ϕ) − X

1 2

Z ϕ c1 (TX , hX ).

(6)

X

2.3. Now we have to compute the variation of the L2 -norm on λ(L). Since (αi ) is an orthonormal basis of Ker (∂) for the L2 -metric defined by h0 , the change of metric on 3b0 H 0 (X, L) is the determinant Z ϕ e hαi , αj i µ . (7) q0 = det (hαi , αj iL2 ,h ) = det X

For H 1 , as suggested by the referee, we use Serre’s duality to reduce to the previous ∗ case. Let βi be an orthonormal basis of ker (∂ ), and γi = ∗ βi , 1 ≤ i ≤ b1 , where ∗ is the star operator defined by hX and h0 . Then γi is a smooth section of the Serre dual ∗ 1X ⊗L∗ of L such that ∂ (γi ) = 0 (since ∂ h0 (βi ) = 0) and (γi ) is an orthonormal basis of H 0 (X, 1X ⊗ L∗ ), which is dual to H 1 (X, L). Note that, since γi = ∗ βi , the following pointwise equality holds: hβi , βj ih0 = hγj , γi ih0 ,

1 ≤ i, j ≤ b1 .

(8)

When the metric h0 is multiplied by eϕ , the metric on 1X ⊗ L∗ is multiplied by e−ϕ , hence, by (7), the L2 -metric on 3b1 H 1 (X, L) = 3b1 H 0 (X, 1X ⊗ L∗ )∗

Upper Bounds for Regularized Determinants

105

gets multiplied by the inverse of Z

e−ϕ hγi , γj ih0 µ ,

det X

i.e., using (8), by

Z

e−ϕ hβi , βj ih0 µ

q1 = det

−1 .

(9)

X

From (7) and (9) we get log

k · k2L2 ,h k · k2L2 ,h0

= log (q0 ) − log (q1 )

Z eϕ hαi , αj i µ + log det e−ϕ hβi , βj i µ .

Z = log det X

(10)

X

Proposition 1 follows from (5), (6) and (10). 2.4. Corollary 1. Under the assumption of Proposition 1, if b1 = 0 and if the conjecture holds for one choice of metric hX on X, it holds for any other choice of metric on X. Proof. Assume hX gets replaced by eρ hX , where ρ ∈ C ∞ (X). Then µ is replaced by µ0 = eρ µ. Let (αi0 ) be an orthonormal basis of H 0 (X, L) for the L2 metric defined by h0X and h0 . We may write b0 X αi0 = aij αj , j=1

1 ≤ i ≤ b0 , where the square matrix M = (aij ) is independent of ϕ. Therefore A(ϕ) is replaced by Z Z 1 ϕ ddc (ϕ) − ϕ(c1 (TX , hX )/2 − ddc ρ + c1 (L, h0 )) B(ϕ) = 2 X X Z ϕ+ρ e hαi , αj i µ + log det X

1≤i,j≤b0

+ 2 log | det(M )|. If we let ψ = ϕ + ρ, since Z

Z c

(ϕ ddc ρ + ρ ddc ϕ)

ϕ dd (ρ) =

2 X

X

we get 1 B(ϕ) = A(ψ) − 2

Z

Z ρ(c1 (TX , hX )/2 + c1 (L, h0 )) + 2 log | det(M )|.

c

ρ dd (ρ) + X

X

When ρ is chosen, if A(ψ) is bounded, so is B(ϕ).

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3. Proof of Theorem 1 in case i) First notice that, because of (1), we can impose the condition Z ϕ µ = 0.

(11)

X

On the other hand, we can choose the reference metric h0 in such a way that the form c1 (TX , hX )/2 + c1 (L, h0 ) is proportional to µ. Together with (11), this implies that the summand Z ϕ (c1 (TX , hX )/2 + c1 (L, h0 )) X

in Proposition 1 vanishes. Now let A be an upper bound on X for the C ∞ functions |hαi , αj i|, 1 ≤ i, j ≤ b0 , and |hβi , βj i|, 1 ≤ i, j ≤ b1 . We get Z Z eϕ hαi , αj i µ ≤ b0 log eϕ µ + b0 log (A) + log (b0 !) log det X

and

Z e

log det

X

−ϕ

hβi , βj i µ

Z

e−ϕ µ + b1 log (A) + log (b1 !).

≤ b1 log

X

X

So Proposition 1 implies Z Z Z 1 c ϕ ϕ dd ϕ + b0 log e µ + b1 log e−ϕ µ + c1 , A(ϕ) ≤ 2 X X X for some constant c1 ≥ 0 independent of ϕ. Let ∇ be the gradient defined by hX . A local computation shows that Z Z 1 ϕ ddc (ϕ) = − |∇ϕ|2 µ. 4π X X

(12)

(13)

We use now an inequality due to Fontana [3], Theorem 1.7, which extends to arbitrary compact manifolds a result of Moser and Trudinger for the sphere and open domains in Rn [7]. Namely, given any smooth real function f on X such that Z Z 2 |∇f | µ ≤ 1 and f µ = 0, X

X

Z

we have

exp(4π f 2 ) µ ≤ c2 ,

log

(14)

X

where c2 is a constant which does not depend on R f . From this inequality it follows that, for any smooth real function g on X such that X g µ = 0, Z Z 1 exp(g) µ ≤ c2 + |∇g|2 µ. (15) log 16π X X R Indeed, if we let B = X |∇g|2 µ and f = g B −1/2 , we have

Upper Bounds for Regularized Determinants

107

√ !2 B B = 4π f − 4π f − g + ≥ 0. 16π 8π 2

Therefore (14) gives Z Z Z B 1 log ≤ c2 + exp (g) µ ≤ log exp (4π f 2 ) µ + |∇g|2 µ. 16π 16π X X X If we apply the inequality (15) to ϕ and −ϕ we get, from (12) and (13), the inequality Z Z Z 1 b0 b1 A(ϕ) ≤ − |∇ϕ|2 µ + |∇ϕ|2 µ + |∇ϕ|2 µ + c1 + 2 c2 . 8π X 16π X 16π X When b0 + b1 ≤ 2 we conclude that A(ϕ) ≤ c1 + 2 c2 .

4. Proof of Theorem 1 in Case ii) 4.1. We assume that X = P1 is the complex projective line and that L = O(n), n ≥ 1. Then b1 = 0 and b0 = n + 1. Furthermore H 0 (X, O(n)) = S n H 0 (X, O(1)) is the space of homogeneous polynomials of degree n in two variables. Consider the canonical exact sequence of sheaves 0 → O(−1) → C2 → O(1) → 0, and denote by A, B ∈ H 0 (X, O(1)) the images of the vectors (1, 0), (0, 1) in C2 = H 0 (X, C2 ). Choose on O(1) the metric h0 induced by the standard metric on C2 . At a point P with homogeneous coordinates (u, v) in P1 the lift of A to C2 which is orthogonal to the vector (u, v) ∈ O(−1)P is given by A⊥ = (|v|2 , −u v) (|u|2 + |v|2 )−1 . Similarly B lifts to

B ⊥ = (−u v, |u|2 ) (|u|2 + |v|2 )−1 .

If P lies on the affine line A1 ⊂ P1 with affine coordinate z, these vectors become A⊥ = (1, −z) N −1 and

B ⊥ = (−z, |z|2 ) N −1 ,

where N = |z|2 + 1. The scalar products of the sections A and B of H 0 (X, O(1)) at the point P are thus given by hA, Ai = A⊥ · A⊥ = N −1 , ⊥

hA, Bi = A · B and

⊥

= −z N

−1

(16) ,

hB, Bi = B ⊥ · B ⊥ = |z|2 N −1 .

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H. Gillet, C. Soul´e

An orthogonal basis of H 0 (X, O(n)) is the set of monomials (Ai B j , i + j = n), where Ai B j is the symmetrization of the vector A⊗i ⊗ B ⊗j in H 0 (X, O(1))⊗n . Using (16) we see that the standard metric on S n H 0 (X, O(1)), which is (a constant multiple of) the L2 -metric on H 0 (X, O(n)), is such that n (−z)j (−z)` n i j k ` . (17) hA B , A B i = ` j Nn 4.2. To prove Theorem 1 ii), we may assume n > 0, because of (2). By the argument of Corollary 1, we may also assume that both h0 and hX are the standard metrics. In particular dz dz¯ µ= 2iπ(1 + |z|2 )2 and c1 (L, h0 ) is a multiple of µ. By 1.2 a) we may finally assume that Z ϕ µ = 0. X

From the previous section, an orthonormal basis (αi ) of H 0 (X, L) is given by the elements Ak B ` /kAk B ` kL2 , k + ` = n. From (17), since ϕ and µ are invariant under rotation, we conclude that Z eϕ hαi , αj i µ = 0 X

when i 6= j. From Proposition 1 we get Z 1 ϕ ddc (ϕ) A(ϕ) = 2 X Z n X log eϕ + i=0

X

|z|2i dz dz¯ + c1 , (1 + |z|2 )n+2

(18)

where c1 , c2 etc . . . will denote constants independent of ϕ. If we take polar coordinates z = r eiθ and if we make the change of coordinates r = et/2 , t ∈ R, we may write ϕ(x) = f (t), ρ(t) = (et/2 + e−t/2 )−2 , eit ρ(t), ρi (t) = (1 + et )n in which case

Z

Z ϕ ddc (ϕ) = − X

+∞

f˙(t)2 dt −∞

(where f˙(t) is the derivative of f (t)) and Z +∞ Z |z|2i eϕ dz d z ¯ = 2π ef (t) ρi (t) dt. 2 )n+2 (1 + |z| X −∞

(19)

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109

We conclude that Z 1 +∞ ˙ 2 f (t) dt 2 −∞ Z +∞ n X + log ef (t) ρi (t) dt + c2 .

A(ϕ) = −

−∞

i=0

Furthermore

(20)

Z

Z

+∞

ϕµ =

f (t) ρ (t) dt = 0.

(21)

−∞

X

4.3. Let A > 0 be such that

Z 2

∞

A =

f˙(t)2 dt.

(22)

−∞

We first deduce from (21) that there is a constant c3 such that |f (0)| ≤ c3 A

(23)

(compare [7] (8)). Indeed, the Cauchy-Schwarz inequality implies 2 Z t Z t Z t 2 ˙ ˙ f (t) dt ≤ f (t) dt dt, s

s

s

i.e. (f (t) − f (s))2 ≤ A2 |t − s| for all s and t, hence −A

p p |t − s| ≤ f (t) − f (s) ≤ A |t − s|.

(24)

We R ∞ multiply these inequalities by ρ(s) and we integrate with respect to s. Since ρ(s) ds = 1 we get from (21) that −∞ Z |f (t)| ≤ A

+∞

−∞

p |t − s| ρ (s) ds

and (23) follows when t = 0. 4.4. Lemma 1. There exists a function u(t), t ∈ R, t ≥ 0, which is C 1 and such that i) ii) iii) iv)

u(0) = f (0), u(+∞) = f (+∞); u(t) ˙ ≥ 0, u(t) ˙ is nonincreasing; u(t) ≥ f (t); R∞ R∞ u(t) ˙ 2 dt = 0 f˙(t)2 dt. 0

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Proof of Lemma 1. Let f˙∗ be the nonincreasing rearrangement of f˙ on [0, +∞[ (cf. e.g. [8]). In other words f˙∗ is the nonincreasing function on [0, +∞[ such that, for all y ≥ 0, (f˙∗ )−1 (y) is the Lebesgue measure of the set of numbers t ∈ [0, +∞[ such that f˙(t) ≥ y. Since f˙ is continuous, the same is true for f˙∗ and we may define Z t f˙∗ (s) ds. (25) u(t) = f (0) + 0

It is clear that u(0) = f (0), and the standard equalities ([8], Lemma 2.2) Z +∞ Z +∞ f˙k (s) ds, (f˙∗ )k (s) ds = 0

0

for k = 1, 2, imply that u(∞) = f (∞) and that iv) holds. Property ii) is a consequence of the definitions and iii) is equivalent to Z t Z t f˙∗ (s) ds ≥ f˙(s) ds , 0

0

a well-known property of rearrangements.

4.5. To bound the quantity A(ϕ) in (20) we may now assume, by Lemma 1, that f˙(t) ≥ 0, that f˙(t) is a nondecreasing function when t ≤ 0 and a nonincreasing function when t ≥ 0. Note that log(a + b) ≤ log(2) + log+ (a) + log+ (b), where log+ = M ax(log, 0). Therefore, Lemma 2 below, when applied to f (t) and f (−t), gives Z +∞ n X log ef (t) ρi (t) dt −∞

i=0

≤

n X

Z

ef (t) ρi (t) dt 0

i=0

+

n X

Z

≤

−∞

Z

+∞

+

log

n X

ef (t) ρi (t) dt + (n + 1) log(2)

ef (t) e−(i+1)t dt

0

i=0

+

0

log+

i=0 n X

+∞

log+

Z

0

log+

i=0

ef (t) e(n−i−1)t dt + (n + 1) log(2) −∞

≤ 2(n + 1) |f (0)| +

1 1 − 2 70n2

A2 + c 4 .

From (23) we conclude that 2(n + 1) |f (0)| ≤ 2(n + 1) c3 A ≤

1 A2 + c5 . 70n2

Upper Bounds for Regularized Determinants

Therefore we get

n X

Z

111

+∞

ef (t) ρi (t) ≤

log −∞

i=0

1 2 A + c6 , 2

i.e. (by (20) and (22)) A(ϕ) is bounded from above and Theorem 1 ii) is proved. 4.6. Lemma 2. Let M ≥ 1 be an integer and let u : R+ → R be a C 1 map such that u˙ is L2 and nonincreasing. Define X=

M X

Z

+∞

exp(u(t) − (j + 1)t) dt.

log

j=0

0

Then there exists a constant C ≥ 0 such that Z +∞ 1 1 − u(t) ˙ 2 dt + C. X ≤ (M + 1) |u(0)| + 2 70M 2 0

(26)

Proof of Lemma 2. For any integer k ≥ 1 we let λk = 1 +

1 5k 2

µk = 1 −

1 . 4k

and

Note that λk · k + µk > k and rk := k + 1 − λk · k − µk = as the smallest integer such that

1 4k

−

1 5k

> 0. Define N ≥ 0

u(0) ˙ ≤ λN +1 · (N + 1) + µN +1 . ∗ If j ≥ N + 1 and t ≥ 0 we have u(t) ˙ − (j + 1) ≤ u(0) ˙ − (j + 1) ≤ u(0) ˙ − (N + 2) ≤ −rN +1 . Therefore Z ∞ Z log exp(u(t) − (j + 1) t) dt ≤ u(0) + log 0

∞

exp(−rN +1 t) dt = u(0) + c. 0

∗ If N = 0 and 0 ≤ j < N + 1, i.e. j = 0, we know from (24) that √ u(t) ≤ u(0) + I · t, Z

where

∞

I=

u(t) ˙ 2 dt.

0

Therefore, by completing a square we get u(t) − t ≤ u(0) +

√

I · t − t ≤ u(0) +

t 3I − , 8 3

112

H. Gillet, C. Soul´e

from which it follows that Z ∞ Z ∞ 3I + log exp(u(t) − t) dt ≤ u(0) + exp(−t/3) dt log 8 0 0 hence, if N = 0, X ≤ (M + 1)u(0) +

3I +C . 8

∗ If N ≥ 1, we let λ = λN , µ = µN and we choose real numbers x0 ≥ x1 ≥ · · · ≥ xN > 0 such that (27) u(x ˙ j ) = λ · j + µ , 0 ≤ j < N. Then if 0 ≤ j < N , we have u(t) − (j + 1)t = u(t) − (λ · j + µ)t + (λ · j + µ − (j + 1))t, ≤ u(xj ) − (λ · j + µ)xj + (λ · N + µ − (N + 1))t Z

hence

∞

log

exp(u(t) − (j + 1) t) dt Z

0

∞

≤ u(xj ) − (λ · j + µ) xj + log

exp(−rN t) dt 0

= ϕ(xj ) + c0 , where

ϕ(x) = u(x) − u(x) ˙ x.

Therefore

X ≤ (M − N ) (u(0) + c) + N c0 + Y,

where Y =

N X

ϕ(xj ).

j=0

Using (27) we can write Y =

N X j=0

=

1 λ

1X ϕ(xj ) (u(x ˙ j ) − u(x ˙ j−1 )) + ϕ(x0 ) λ N

ϕ(xj ) =

N −1 X j=1

j=1

u(x ˙ j ) (ϕ(xj ) − ϕ(xj+1 )) −

1 ϕ(x1 ) u(x ˙ 0) λ

1 + u(x ˙ N ) ϕ(xN ) + ϕ(x0 ) λ N −1 1 1 X ˙ N ) (ϕ(xN ) − ϕ(0)) u(x ˙ j ) (ϕ(xj ) − ϕ(xj+1 )) + u(x = λ λ j=0 u(x ˙ 0) µ u(0) + ϕ(x0 ) 1 − + N+ λ λ Z x0 1 µ u(x ˙ 0) ≤ u(0) + ϕ(x0 ) 1 − . u˙ ϕ˙ dt + N + λ 0 λ λ

Upper Bounds for Regularized Determinants

But

Z

x0 0

Z x0 u˙ ϕ˙ dt = − tu˙ u¨ dt 0 x0 Z 1 x0 1 + u(t) ˙ 2 dt = − tu˙ u˙ 2 2 0 0 Z 1 x0 1 ˙ 0 )2 x 0 + u(t) ˙ 2 dt. = − u(x 2 2 0

So we get

1 µ I+ N+ u(0) + R, 2λ λ

Y ≤ where

113

(28)

1 u(x ˙ 0) u(x ˙ 0 )2 x0 + ϕ(x0 ) 1 − 2λ λ 2 µ µ = 1− u(x0 ) + − µ x0 . λ 2λ

R= −

Now, by (24), u(x0 ) ≤ u(0) +

p

x0 · I ,

and, by completing the square, α Therefore

√

x0 − β x0 ≤

α2 4β

for all α, β > 0.

(1 − µ/λ)2 µ I. u(0) + R≤ 1− µ2 λ 4 µ − 2λ

Using (28) we get Y ≤ (N + 1) u(0) + A · I, where

2 1 − µλ 1 . A= + 2λ 4 µ − µ2 2λ

From the values of λ = λN and µ = µN we compute A≤

1 1 1 1 − ≤ − . 2 2 70N 2 70M 2

Therefore X ≤ (M + 1) u(0) +

1 1 − 2 70M 2

Z

∞ 0

u(t) ˙ 2 dt + C.

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H. Gillet, C. Soul´e

5. Flat Bundles According to Bismut and Zhang [2] a flat C ∞ bundle (E, ∇), ∇2 = 0, together with a C ∞ metric h on E on a C ∞ manifold M is the analog in the differentiable category of a hermitian holomorphic bundle on a complex manifold. Inequalities similar to our conjecture might also hold in this case, but in some cases they must be lower bounds rather than upper bounds, as the following example suggests. Let M = S 1 be the circle, and let E = C be the trivial line bundle on M . We equip M with its standard metric and E with an arbitrary metric h. The connection ∇ = d has an adjoint d∗ (depending on h), and we consider the Laplace operator 1 = d∗ d on C ∞ (M ), and its regularized determinant det0 (1). Proposition 2. There is a constant C(E) such that, for any choice of a metric h on E = C, the following inequality holds: det0 (1) ≥ C(E). Proof. We have H 0 (S 1 , C) ' H 1 (S 1 , C) ' C. On λ(E) = H 0 (S 1 , C) ⊗ H 1 (S 1 , C)∗ we define the Quillen metric by k · k2Q = k · k2L2 det0 (1)−1 as in Sect. 1 above [2]. If ϕ ∈ C ∞ (M ) we let h be the metric on E = C such that h(1, 1) = exp(ϕ), and we denote by 1ϕ the corresponding Laplace operator on C ∞ (M, C). We define A(ϕ) = log(det 0 (1ϕ )) − log(det 0 (10 )) = log

k · k2L2 ,ϕ k·

k2L2 ,0

− log

k · k2Q,ϕ k · k2Q,0

.

According to [2] Theorem 0.1 we have k · kQ,ϕ = 0, k · kQ,0

log

and a computation similar to Sect. 2.3 gives log

k · k2L2 ,ϕ k · k2L2 ,0

Z

Z e

= log

ϕ(x)

e−ϕ(x) dx,

dx + log

S1

S1

where dx is the Haar measure of length one. The Cauchy-Schwarz inequality implies 2

Z dx S1

Z

Z

≤

e

ϕ(x)

dx

dx

S1

S1

from which we conclude that A(ϕ) ≥ 0.

e

−ϕ(x)

Acknowledgement. We thank W. Beckner, P. Chang and J. Lott for interesting discussions.

Upper Bounds for Regularized Determinants

115

References 1. Bismut, J.-M., Gillet H., Soul´e C.: Analytic torsion and holomorphic determinant bundles I. Bott–Chern forms and analytic torsion. Commun. Math. Phys. 115, 49–78 (1988) 2. Bismut, J.-M., Zhang, W.: An extension of the Cheeger–Muller theorem. Ast´erisque 205, (1992), Soc. Math. de France, Paris 3. Fontana, L.: Sharp borderline Sobolev inequalities on compact Riemannian manifolds. Commun. Math. Helv. 68, 415–454 (1993) 4. Gillet, H., Soul´e, C.: Characteristic classes for algebraic vector bundles with hermitian metrics I. Ann. Math. 131, 163–203 (1990) 5. Gillet, H., Soul´e, C.: An arithmetic Riemann–Roch theorem. Invent. Math. 110, 473–543 (1992) 6. Kiessling, M.K.H.: Statistical mechanics of classical particles with logarithmic interactions. Comm. Pure Appl. Math. 46, 27–56 (1993) 7. Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Math. J. 20, 1077–1092 (1971) 8. O’Neil, R.: Convolution operators and L(p, q) spaces. Duke Math. J. 30, 129–142 (1963) 9. Quillen, D.: Determinants of Cauchy–Riemann operators over a Riemann surface. Funct. Anal. Appl. 19, 31–34 (1985) 10. Soul´e, C.: A vanishing theorem on arithmetic surfaces. Invent. Math. 116, 577–599 (1994) Communicated by D. Brydges

Commun. Math. Phys. 199, 117 – 167 (1998)

Communications in

Mathematical Physics © Springer-Verlag 1998

Dobrushin–Koteck´y–Shlosman Theorem up to the Critical Temperature Dmitry Ioffe1,2,? , Roberto H. Schonmann3,?? 1

WIAS, Mohrenstrasse 39, D-10117 Berlin, Germany. E-mail: [email protected] Faculty of Industrial Engineering Technion, Haifa, 32000 Israel 3 Mathematics Department, University of California at Los Angeles, Los Angeles CA 90095, USA. E-mail: [email protected]

2

Received: 23 September 1997 / Accepted: 24 April 1998

Abstract: We develop a non-perturbative version of the Dobrushin–Koteck´y–Shlosman theory of phase separation in the canonical 2D Ising ensemble. The results are valid for all temperatures below critical. 1. Introduction Dobrushin–Koteck´y–Shlosman (DKS) Theorem [DKS, DS] gives a rigorous probabilistic content to the assertion that pure phases are separated on the macroscopic scale along the boundary of the equilibrium crystal shape. Their results were formulated and proved in the context of the 2D Ising model at very low temperatures. Despite this particular setting it would be appropriate to talk in terms of the DKS Theory, rather than in terms of only one theorem with a very long proof (as the authors of [DKS] modestly did). For many of their ideas and insights will certainly find a way into both more general lattice models and higher dimensions. During several years following the publication of [DKS], however, the main efforts have been invested into attempts to relax their proof [Pf] and, later on, to get rid of the “very low temperature” assumption. It has been commonly believed that one needs low temperature solely in order to have an additional technical tool of convergent cluster expansions readily available, whereas the results themselves should remain qualitatively the same in the whole of the phase transition region. And indeed, in a series of articles [I1, I2, SS1, SS3, CGMS] and culminating in [PV] and [V] some sort of the DKS theory has been developed in the non-perturbative regime and pushed all the way to the critical temperature. These results, however, have been based on the integral type limit theorems and are, in parts, closer in spirit to the non-perturbative treatment of the 2D Bernoulli percolation [ACC] than to the local limit setting of the exact canonical ensemble in the original monograph [DKS]. Subsequently, the phase ? ??

D. Ioffe was partially supported by the NSF grant DMS 9504513. R. H. Schonmann was partially supported by the NSF grants DMS 9400644 and DMS 9703814.

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separation geometry has been much less pronounced, and the accent has been generally shifted to the precise leading surface order of integral estimates. Moreover, it seems that the local limit part of the DKS theory is the one to be the most robust and amenable as opposed to the skeleton coarse graining techniques, which are probably too much oriented to the two-dimensional lattice and nearest neighbour interactions. In this paper we try to fill in this gap and to extend the theory up to the critical temperature in the original setting of [DKS], i.e., in the exact canonical ensemble. The results we obtain are comparable in strength and scope to those appearing in the corresponding chapters of [DKS] and [DS]. It should be noted, however, that we cover only part of their results - a delicate analysis of the phase separation line for all subcritical temperatures is currently beyond our reach, and we have to appeal to exact solution dependent facts about the Ornstein–Zernike behaviour of the two-point correlations [MW] and about analytic properties of the surface tension [AA], while deriving the lower bound of Sect. 3. But this is the only point at which our results fail to be self-contained. It also should be clear that the basic philosophy was already perceived in the original ground breaking monograph [DKS], our main contribution has been merely to understand how to implement it for all temperatures below critical, that is using only qualitative facts about ferromagnetic Ising measures, e.g., the Markov property, FKG and other correlation inequalities. The problem of defining the equilibrium crystal shape as the one which minimizes the interfacial surface energy was formulated at the turn of the century in [W]. In our setting of the 2D Ising model let β > βc be the inverse temperature, and let τβ : S1 7→ R+ be the corresponding anisotropic surface tension, see [DKS, Pf] and [A] for the definitions and properties. Then the equilibrium crystal shape (Wulff shape) of the area v is defined to be a solution of the following isoperimetric type problem: Z 1 τβ (ns )ds −→ min, Wβ (∂V ) = ∂V

Given: Area (V ) = v, where ds is the unit speed parametrization of the boundary ∂V and ns is the normal to ∂V at the point s. A nice feature of the variational problem above is its scale invariance: Wβ (∂(aV )) = aWβ (∂V ) . Consequently any dilatation of an optimal solution is itself optimal, and, therefore, one really should think in terms of shapes and, moreover, needs to specify solutions only at one prescribed value of the area v. A canonical way to pin down the solution is to define the unnormalized Wulff shape K=

\

{x : (x, n)R2 ≤ τβ (n) } ,

n∈S1

where (·, ·)R2 is the usual scalar product in R2 , and then to scale it down to the unit area 1 1 K1 = p K. |K| It is a known fact of the Brunn–Minkowski theory (cf., for example [Sc]), that all optimal

Dobrushin–Koteck´y–Shlosman Theorem up to Critical Temperature

119

shapes of the unit area are just shifts of K1 . We shall use the unit area as the reference scale and along with K1 also introduce a separate notation for the surface energy of the boundary ∂K1 respectively, 1

ω1 = Wβ (∂K1 ) . In the next subsection we introduce some basic notation. The results and the underlying heuristics are described in Subsect. 1.3. Finally, in the last subsection of the Introduction we try to outline the proof and, subsequently, to facilitate the orientation of the reader. 1.1. Notation. One of the main features of our approach is a uniform treatment of the appropriate family of subsets of Z2 . Lattice boxes. Z2 denotes the two-dimensional integer lattice. Given a point x = 1 (x1 , x2 ) ∈ Z2 , we use kxk1 = |x1 | + |x2 |. For any set A ⊆ Z2 we define its (outer) boundary ∂A via 1 ∂A = x ∈ Z2 \ A : min kx − yk1 = 1 . y∈A

Finally we define the closure of A as 1 b= A A ∪ ∂A = y ∈ Z2 : min ky − zk1 ≤ 1 . z∈A

Our results are asymptotic with the microscopic size of the system tending to infinity. In order to avoid irrelevant complications related to the finite box boundary effects, we choose our basic sequence of boxes 3N ⊂ Z2 as: 3 N = N K 1 ∩ Z2 , where, as before, K1 is the unit volume Wulff shape. More generally, we shall work with the following family DN of “boxes” in Z2 : A ∈ DN ⇐⇒ aN 2 ≤ |A| ≤ N 2 and |∂A| ≤ RN log N, where a and R are two respectively very small and very large numbers, which are fixed throughout the article. Measures. The results are obtained on different scales, which are quantified by the value of the large contour parameter s(N ). There are two typical scales for s(N ): 1. For K = K(β) sufficiently large we define the basic scale, s(N ) = K log N, which corresponds to the maximal probable size of contours inside 3N in the pure phase. 2. Intermediate scales: s(N ) = N b (log N )λ ; b ∈ (0, 1), λ ∈ R.

120

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Given a large contour parameter s(N ), we define an s(N )-restricted phase or, equivalently, the phase of s(N )-small contours as the corresponding Ising Gibbs measure, conditioned on the event that no configuration has a ± contour with the diameter exceeding s(N ). β β,s For a set A ∈ DN we use PβA,−,g ,Pβ,s A,−,g and h · iA,−,g , h · iA,−,g , to denote measures and expectations with minus boundary conditions, at the inverse temperature β and the magnetic field g in the unrestricted phase and in the phase of s(N )-small contours, respectively . We also use subindex N instead of 3N , whenever A is the 3N box itself, and we drop any finite box subindex while talking about infinite volume “−” state. Similarly, we drop the magnetic field subindex g, whenever g = 0. Finally, for each β > βc , we use m∗ to denote the spontaneous magnetization of the extremal Gibbs state, β m∗ = m∗ (β) = −hσ(0)i− > 0, and χ to denote its susceptibility, χ = χ(β) =

X

β

hσ(0); σ(x)i− < ∞.

x∈Z2 1

Magnetization. The space of spin configurations on A ⊂ Z2 is denoted by A = {−1, 1}A . Given a box A ⊂ Z2 we set X MA = σ(x), x∈A

to denote the total magnetization on A. In the case A = 3N , we use shortcuts N and MN respectively. One of the main objects of this article is to give precise asymptotics on the probabilities of the deviation of MN from −m∗ N 2 , PβN,− MN = −m∗ N 2 + aN , in the whole of the low temperature region β > βc , and, most importantly, to use results on such asymptotics in order to describe the phenomenon of the phase separation in the canonical ensemble PβN,− • | MN = −m∗ N 2 + aN . As in [DKS] our approach to this problem is built up upon uniform local limit type estimates on the deviations of MA ; A ∈ DN . Depending on the context it will be convenient to formulate such estimates either directly in terms of the deviation of MA from −m∗ |A|, or in terms of its deviations from the averages under current measures 1

β

MA = hMA iA,−

or

1

β,s

MsA = hMA iA,− .

We shall see (cf., Remark 2.2.2), however, that uniformly in A ⊂ Z2 , MA + m∗ |A| ≤ c1 (β)|∂A|, and MA − MsA ≤ c1 (β)|A|e−c2 (β)s(N ) .

(1.1.1)

Dobrushin–Koteck´y–Shlosman Theorem up to Critical Temperature

121

In particular, things are uniformly under control, once we restrict attention to the domains A ∈ DN . Of course, local limit estimates make sense only for the admissible values of the magnetization. Thus, for a deviation aN we write aN ∈ M A if, depending on the context, either −m∗ |A| + aN or MA + aN or MsA + aN lie in the range of MA , that is in between −|A| and |A| and equal to |A| mod(2). We, furthermore, concentrate on the positive values of aN , aN ∈ M+A , which point inside the phase transition region. Indeed, the probabilities PβA,− ( MA = bN ) exhibit, in the language of [DS], the classical limit behaviour, as soon as bN ≤ MA , and the corresponding asymptotics were thoroughly worked out in the latter article1 . Let us state a consequence of their results in the form we need it here: For any A ⊂ Z2 , non-positive magnetic field g ≤ 0 and a number a ≤ 0, the following Gaussian estimate is true: a2 β . (1.1.2) PβA,−,g MA = hMA iA,−,g + a ≤ exp −c3 (β) |A| Contours and skeletons. Our contours are always self avoiding objects, constructed according to one of the two possible splitting rules (see [DKS]). All contours lie on the 1 edges of the dual lattice Z2∗ = (1/2, 1/2) + Z2 . Given a ± contour γ and a large contour parameter s(N ), a set of dual vertices S = (u1 , . . . , un ) is called an s(N )-skeleton of γ if 1. All vertices of S lie on γ. 2. s(N )/2 ≤ kui −ui+1 k ≤ 2s(N ); ∀ i = 1, . . . , n, where we have identified un+1 ≡ u1 and k • k stands for the supremum norm; k(a, b)k = max{|a|, |b|}. 3. The Hausdorff distance dH between γ and the polygonal line P(S) through the vertices of S satisfies dH (γ, P(S)) ≤ s(N ). If S satisfies the above conditions, we say that γ and S are compatible and write γ ∼ S. Note that a contour might have many compatible skeletons. Note also that any s(N )-large contour, i.e., one whose diameter is at least s(N ), has a compatible s(N )-skeleton. We shall consider contours and their skeletons on different s(N )-scales. For each large contour parameter s(N ) fixed and for any given spin configuration σ we use a generic notation 0 = 0(σ) = (γ1 , . . . , γn ) for a set of all s(N )-large contours of σ. Each collection 0 of s(N )-large contours splits 3N into the disjoint union of its “−’ and “+” components, 3N = B ∪ C respectively. We use 1

Vol+ (0(σ)) = |C| for the cardinality of the “+” component |C|. 1

The first part of their Theorem 1.5.1 is true for any β 6= βc

122

D. Ioffe, R. H. Schonmann

A collection of s(N )-skeletons S = (S1 , . . . , Sn ) is then said to be compatible with σ (or with 0(σ)), which we denote as S ∼ 0(σ), if Si ∼ γi ; i = 1, . . . , n . The skeleton language is the main coarse graining tool - the surface tension is produced in the probabilistic estimates, once the events are started to be expressed in their skeleton approximation. The surface tension of a collection S = (S1 , . . . , Sn ) is defined in a standard way: n X Wβ (P(S)), Wβ (S) = i=1

where, as before, P(S) is the polygon through the vertices of the skeleton S. Though the contours are self avoiding objects, the polygonal lines through the vertices of their skeletons are, in general, not. Moreover, two polygons corresponding to two different skeletons of the same configurations might intersect as well. Thus, it is not immediately clear what should be the interior of a collection S. This difficulty was overcome in [DKS], and we stick to their definition of the phase volume. Moreover, we use S+ to denote the corresponding “+” phase component of S. As it was proved in [DKS], given any family of s(N )-large contours 0 and a compatible collection of s(N )-large skeletons S; S ∼ 0, | C1S+ | ≤ c(β)Wβ (S) s(N ),

(1.1.3)

where C is the “+” component of 3N under the 0-splitting. Note, by the way, that S+ is a disjoint union of connected (and simply connected if we restrict attention to exterior large contours) polygonal subsets of R2 , and, therefore, Wβ (∂S+ ) is well defined. Moreover, 1

Wβ (S+ ) = Wβ (∂S+ ) = Wβ (S).

(1.1.4)

1.2. Heuristics and results. The typical phase picture under the low temperature, β > βc , pure (minus) state could be loosely described as a sea of “−” spins with a homogeneous archipelago of “+” islands, some of which might contain “−” lakes, etc. The size of these islands inside 3N does not typically exceed K log N , and their density is such that the mean magnetization produced is close to −m∗ . One can think of two principal mechanisms behind a shift of the magnetization from its averaged value in the corresponding pure phase: 1. A homogeneous shift in the density of “+” islands, without, however, modifying their typical sizes. 2. A creation of abnormally huge islands of “+” phase. In particular, the aN shift of the magnetization should correspond to the excess area aN /2m∗ of those islands. The first scenario corresponds to the Gaussian fluctuations and its probabilistic price for the aN ∈ M+A shift of the magnetization should be close to a2 . exp − N 2χ|A| The phenomenon of the phase separation, of course, manifests itself in the second scenario. Its probabilistic price should be related to the surface tension of the optimal

Dobrushin–Koteck´y–Shlosman Theorem up to Critical Temperature

123

shape “huge islands” configuration, which, if there is no additional restriction on the size of those huge islands (e.g., if the estimates are performed in the unrestricted phase), leads to a value close to r aN ω exp − 1 . 2m∗ A comparison between the two expressions above indicates that the Gaussian contribution should win, whenever aN N 4/3 , whereas the creation of huge islands should become a dominant factor as soon as aN N 4/3 , which is the regime where the phase separation should be observed. Remark 1.2.1. The case of critical deviations aN ∼ N 4/3 requires both a more accurate heuristics and, respectively, more refined rigorous estimates. We relegate the corresponding discussion to a future publication. Another issue we do not work out in all the details here is the precise geometry of the phase picture in the canonical ensemble in the s(N )-restricted phase. Instead, we confine ourselves only to a derivation of imprecise bounds in this regime, which, nonetheless, capture the leading exponential order of decay of the corresponding probabilities and sharpen all the previous estimates [I2], [SS3] and [PV] of this sort. We attempt in this paper to develop a non-perturbative (∀ β > βc ) exact theory of the phase separation, which would be comparable in scope to the low temperature results obtained in [DKS] and [DS] using the method of cluster expansions. For any δ > 0 fixed we distinguish between small moderate deviation values of aN ∈ M+A , aN N 4/3−δ , and large moderate deviation values of aN ∈ M+A , aN N 4/3+δ . Theorems A,B and C were proved in [DKS] and [DS] only for sufficiently large values of the inverse temperature β. Our main result is their validity up to the critical temperature, i.e., for any value β > βc . Theorem A. Let δ ∈ (0, 2/3) and assume that aN ∈ M+N satisfies 4

aN ∼ N 3 +δ . Then, for every β > βc fixed, −1 r δ aN ω log PβN,− MN = −m∗ N 2 + aN = −1 + o N − 2 . (1.2.1) 1 ∗ 2m Moreover, if K is large enough, with the PβN,− ( · |MN = −N 2 m∗ + aN )-probability converging to 1 as N → ∞: 1. There is exactly one exterior K log N -large contour γ. 2. This γ satisfies ! r δ 2m∗ min dH γ, x + ∂K1 N − 4 . x aN

(1.2.2)

124

D. Ioffe, R. H. Schonmann

Theorem A deals with the case of large moderate deviations; aN N 2 , proper. The estimates on the right-hand sides of (1.2.1) and (1.2.2) can be improved at various particular values of δ chosen. Similarly, with additional care one is able to rule out all K log N -large non-Wulff contours and not only the external ones as we assert here. This would, however, give rise to rather messy formulas, and, as we feel, might obscure the common logic of the proof. This is also the reason why we decided not to address here the critical regime, described in Remark 1.2.1. We apply the theory in the full strength only in the traditionally interesting case of large deviations aN ∼ N 2 . Theorem B. Let the inverse temperature β > βc be fixed, and let the sequence {aN }; aN ∈ M+N be such that the limit lim

N →∞

aN ∈ (0, 2m∗ (β)) N2

exists. Then, r

aN ω1 2m∗

−1

log PβN,− MN = −m∗ N 2 + aN = −1 + O N −1/2 log N .

Moreover, if K is large enough, with the PβN,− ( · |MN = −N 2 m∗ + aN )-probability converging to 1 as N → ∞: 1. There is exactly one K log N -large contour γ. 2. This γ satisfies ! r p 2m∗ min dH γ, x + ∂K1 ≤ c1 (β)N −1/4 log N . x aN

(1.2.3)

Comparing Theorem B with the corresponding assertion (1.9.4) in Theorem 1.9 in [DKS], we find that our bound (1.2.3) on the typical Hausdorff distance to the dilatation of the Wulff shape even slightly improves their low temperature estimate. On the other hand, the best we can do in order to control the area of the symmetric difference between the interior of γ and the optimal deterministic shape is to apply (1.2.3), which yields r 2m∗ p int (γ) 1 (x + K1 ) ≤ c2 (β)N −1/4 log N . min x aN The above estimate falls short of the N −4/5 (log N )κ order in the corresponding result ([DKS], (1.9.3) ). This should not be too surprising, since their estimate is based on a more refined (though still not optimal) analysis of the fluctuations of phase separation lines and related fluctuations of the phase volume. Though such a refinement is apparently feasible in our setting as well, we decided not to perform it here. A much more challenging task would be to understand and develop a non-perturbative counterpart of the optimal results on the fluctuations of the phase separation line about Wulff shapes [DH], but this, as we have already mentioned, is beyond the scope of our work. Finally, in the case of small moderate deviations we obtain, uniformly in A ∈ DN , the following result:

Dobrushin–Koteck´y–Shlosman Theorem up to Critical Temperature

125

Theorem C. Let δ ∈ (0, 4/3) be fixed. Then, for every β > βc there exist, a value of K = K(β) < ∞, such that uniformly in A ∈ DN and aN ∈ M+A satisfying aN ≤ N 4/3−δ , the following two estimates hold:

a2 ( 1 + o(1)) , (1.2.4) exp − N 2χ|A| 2πχ|A|

PβA,− ( MA = MA + aN ) = p

1

and PβA,− ∃ K log N -large ± contour | MA = MA + aN

= o(1).

(1.2.5)

Remark 1.2.2. Given local limit asymptotics (1.2.4), one can derive various results on the statistical properties of configurations inside and outside the unique “Wulff” contour asserted in Theorems A and B, putting, in this way, more flesh on the corresponding notion of the phase separation. We refer to Subsect. 1.10 of [DKS] for the statement and to Subsects. 6.3 and 6.4 for respectively the proofs of such results. 1.3. Notes on the proofs. There are two main ingredients of the theory: 1. Coarse graining of contours. 2. Uniform local limit estimates on moderate deviations over domains A ∈ DN under various restricted phases Pβ,s A,− . Coarse graining is imperative for the production of the (macroscopic quantity) surface tension and it is performed in terms of skeletons: Important geometric events we deal with here are roughly of the following type: {There is a ± contour close to the boundary of a certain deterministic shape} . The point is that one-contour Peierls estimates never capture the precise order of decay of the probabilities of such events. In other words, the probability of having a contour close to some shape is substantially larger than the probability of each particular contour contributing to the event. Contrary to this the probability to observe a certain skeleton already integrates the entropy of the number of various contours compatible with this skeleton. Most of the coarse graining estimates we use were obtained before, and we refer to [Pf] for a detailed discussion. On every s(N )-scale, (1.3.1) PβN,− (S) ≤ exp −Wβ (S+ ) , and a scaling computation (cf., for example, [Pf]), to which we always refer as to the usual skeleton computation, gives rise to the following important “Energy” estimate on any s(N ) scale : There exists a constant c1 = c1 (β), such that X c1 log N β β . PN,− (S) ≤ exp −r 1 − PN,− Wβ (S) ≥ r ≤ s(N ) (1.3.2) Wβ (S)≥r Applying this bound on the basic K log N scale, and noting that if S is a K log N collection which is compatible with some family of K log N -large contours 0 = (γ1 , . . . , γn ), then

126

D. Ioffe, R. H. Schonmann 1

|0| =

n X

|γi | ≤ K log N

i=1

Wβ (S+ ) , minn τβ (n)

we infer the following simple but nonetheless useful consequence: There exists c2 = c2 (β), such that for any r > 0 and for an event C ∈ A with β PA,− (C) ≥ e−r , PβA,− |0| > c2 r log N | C = o(1). (1.3.3) In particular, since, as we shall see later, local limit estimates of Sect. 2 readily imply a lower bound of the form √ PβN,− MN = −m∗ N 2 + aN ≥ e−c3 (β) aN uniformly in large moderate deviations aN ∈ M+N , one can control the length of large ±-contours or, equivalently, the length of the boundaries of the “+” and “−” components in the corresponding splitting of 3N . This, by the way, explains our definition of the family DN . Local limit estimates are stated uniformly in domains from DN , which, as we just remarked, includes “−” components of admissible families 0 of large contours. In the phase separation regime or, equivalently, at large moderate values of deviations 4/3+δ , the only “stable” large contours are those of the maximal linear size a √N ∼ N aN . A mathematical interpretation and derivation of this fact comprises two steps: first we show that subcritical contours, i.e., those whose diameter is much smaller than N 2/3 do not appear in the unrestricted phase in any circumstances. This ought to be clear on the heuristic level: suppose that subcritical contours produce some a˜ N shift in the magnetization. Then for small moderate values of a˜ N , the surface tension price for this should be much higher than the corresponding quadratic term in Gaussian estimates in the basic K log N restricted phase. On the other hand, for large moderate values of a˜ N it would be √ much less expensive to produce such shifts by means of contours of the linear size ∼ a˜ N , and then to compensate a possible disbalance of magnetization in the restricted K log N phase inside and outside these contours. A rigorous implementation, therefore, strongly depends on accurate local limit bounds in the phase of K log N -small contours, which we subsequently derive in Sect. 2. Large unstable contours, in their turn, are ruled out by an isoperimetric type argument in Sect. 5. The efficiency of this argument, however, depends both on sharp coarse graining estimates, in particular on the lower bound of Sect. 3, and on sharp local estimates on various s(N ) scales. Many such estimates are derived according to the following pattern: In order to estimate the probability of a certain event C under PβA,− , we choose a certain large contour parameter s(N ) and make a skeleton decomposition of C, [ C ∩ {σ : σ ∼ S}. C= S

For each collection of skeletons S we control the length and the shape of any 0 ∼ S in terms of Wβ (S) through (1.3.3) and (1.1.3) respectively. Any S-compatible collection 0, on the other hand, decouples PβN,− into the product of the “+” and “−” states over the corresponding components in the induced decomposition of 3N . In the case C = {MA = −m∗ |A| + aN }, each admissible collection of large contours 0 leads to the following decomposition:

Dobrushin–Koteck´y–Shlosman Theorem up to Critical Temperature

127

C ∩ {σ : 0(σ) = 0} = [ { 0 ; MB = −m∗ |B| + bN ; MC = m∗ |C| + cN } , (1.3.4) bN +cN =aN −1(B,C)

where, as usual A = B ∪ C is the 0-induced decomposition of A into the respectively “−” and “+” components, while 1(B, C) = m∗ |A| − m∗ |B| + m∗ |C| + |∂+ 0| − |∂− 0| = 2m∗ |C| + (m∗ + 1)|∂+ 0| + (m∗ − 1)|∂− 0|, where ∂+ 0 and ∂− 0 are the set of sites in A where the presence of 0 forces the spins to be +1 and −1, respectively. Similar splitting is valid for the event C = {MA = MA + aN }, and we shall use in this case (1.1.1) and (1.3.3) in order to control bN +cN in terms of aN and |C|. We then use coarse graining estimates, e.g., (1.3.2), to control the probability of skeletons and local limit estimates to control the magnetization inside and outside compatible contours. The paper is organized in the following way: Sect. 2 is devoted to the local limit estimates over various domains A ⊂ Z2 . These results are the backbone of the theory. In particular, they imply a uniform (in A ∈ DN ) sharp lower bound a2 1 (1 + o(1)) exp − N (1.3.5) PβA,− ( MA = MA + aN ) ≥ p 2χ|A| 2πχ|A| in the case of small moderate deviations aN N 4/3−δ . Then, in Sect. 3, we use a combination of the skeleton coarse graining techniques and the local limit estimates of Sect. 2 in order to derive sharp lower bounds r √ aN 4 ω − c a log N , (1.3.6) PβN,− MN = −m∗ + aN ≥ exp − 1 4 N 2m∗ on the large moderate deviations aN N 4/3+δ . The central result of Sect. 4 asserts that no subcritical, i.e., with the diameter N 2/3 , K log N -large contours ever appear in the canonical ensemble PβN,− • | MN = −m∗ N 2 + aN

,

regardless of whether aN is in the regime of small or large moderate deviations. In order to prove this, the probabilities of events we want to rule out are tested against sharp lower bounds (1.3.5) and (1.3.6) of two previous sections via the skeleton type decompositions similar to (1.3.4). As a byproduct we derive the full statement of Theorem C. The proofs of Theorems A and B are concluded in Sect. 5. Combining the lower bound of Sect. 3 with the energy bound (1.3.2) and with the ubiquitous local limit estimates, we argue on the grounds of a simple isoperimetric stability statement, that with the exception of one large “Wulff” contour no other N 2/3−ν -large contours appear in the typical configuration in the canonical ensemble. Since contours, whose diameter lies in the interval [K log N, N 2/3−ν ], were already ruled out in Sect. 4, this implies the one-contour assertions in Theorems A and B, and, with some additional work, their full statements.

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Constants. The constants K = K(β) in the definition of the basic phase K log N and R, a in the definition of the family of domains DN are fixed throughout the article (regarding the choice of K see Theorem C and Remark 2.3.2). We use also finite positive constants c1 , c2 , . . . .Their values are updated with each subsection. Depending on the context the value of these constants might depend on the inverse temperature β and on the cutoff value δ in the large/small moderate deviation setting, but, unless mentioned explicitly, not on anything else. In particular they are always independent of N , of the current domain A ⊂ Z2 and the deviation aN ∈ MA under consideration. 2. Local Estimates for Moderate Deviations 2.1. Structure of local estimates. Let A ∈ DN and aN ∈ M+A . The main object of this section is to give precise estimates on the probability s Pβ,s A,− MA = MA + aN . The classical approach to such estimates is to find the value of magnetic field g = g(A, s(N ), aN ), such that the expected magnetization under the g-tilted state is precisely what we want, β,s

hMA iA,−,g = MsA + aN ,

(2.1.1)

β,s and, then, to rewrite the Pβ,s A,− -probability in terms of the PA,−,g one: s Pβ,s A,− MA = MA + aN n o β,s β,s = exp −(MsA + aN )g + log hegMA iA,− Pβ,s = hM M i A A A,−,g A,−,g   g g   Z Z β,s β,s hMA ; MA iA,−,h dhdr Pβ,s = exp − A,−,g MA = hMA iA,−,g .   (2.1.2) 0

r

Pushing the analogy with the classical case further, we encounter three types of problems : 1. Give a sufficiently precise estimate on g = g(A, s(N ), aN ) in (2.1.1). 2. Given such an estimate on g derive sufficiently sharp estimates on the semi-invariants of the family {Pβ,s A,−,r } for r ∈ (0, g). 3. Prove a local CLT under Pβ,s A,−,g . All three problems are, of course, inter-related, and in the classical case this procedure leads to Gaussian estimates in any moderate deviations regime. In our case, however, unless the values of aN and s are further qualified, this approach is in general doomed. Indeed, if , for example, there is no restriction on the size of large contours, i.e., the constraint s(N ) does not appear at all, then the mean magneβ tization hMA iA,−,g is extremely sensitive to the changes of g of order 1/N , and there is essentially a jump from the “−” to “+” phase within this range of the magnetic field. Moreover, as it was explained in the introduction, for aN N 4/3 the asymptotics of

Dobrushin–Koteck´y–Shlosman Theorem up to Critical Temperature

129

PβA,− (MA = MA + aN ) is not expected to be Gaussian at all. Therefore, the important thing for us is to understand how the magnetization β,s hMA iA,−,g and other semi-invariants of Pβ,s A,−,g change with g in the phase of s(N )small contours. Such questions were investigated in [SS2], and in the next subsection we state and explain a version of the corresponding results in the latter article. Then we proceed to derive precise estimates in what happens to be the most important regime of K log N -small contours. Finally, in the last subsection, we use these K log N -phase moderate deviation results to obtain useful upper bounds on various other s(N )-scales. 2.2. Estimates in cutoff ensembles. The breaking of the classical limit behaviour in the s(N )-restricted phase manifests itself by the jump of the magnetization and by the explosion of the susceptibility, which, in their turn, are related to the appearance of abnormally large ±-contours. On the heuristic level it is clear what should be the critical order of the magnetization g, at which those large contours should start to be favoured: for a ± contour of the linear size s(N ) one wins ∼ s2 g on the level of magnetization and loses ∼ s on the level of surface energy. These two terms start to be comparable when sg ∼ 1. Therefore no particular deviation from the classical behaviour should be expected as far as gs(N ) 1.

(2.2.1)

Lemma 2.2.1 below is a mathematical counterpart of these intuitive considerations. It also generalizes the corresponding results in [SS2] (see Lemmas 2.3.4 and 2.3.5 there). Let us first introduce some additional notation: For a finite subset B ⊆ Z2 and k ∈ R+ we define 3B (k) = {y ∈ Z2 : min ky − xk < k}. x∈B

If B = {x1 , x2 , . . . , xn } we use notations 3x1 x2 ...xn for 3B . Similarly for a local function φ we use 1

3φ = 3suppφ , 1

where suppφ is the support of φ. We use r(φ) = |suppφ| to denote the cardinality of the support of φ. Finally, for two local functions φ and ψ we let d(φ, ψ) denote the distance between the corresponding supports. We state results in the asymptotic form as N → ∞. Thus, the condition (2.2.1) should be understood in the sense that the sequence of nonnegative magnetic fields g = g(N ) satisfies lim g(N )s(N ) = 0.

N →∞

(2.2.2)

Finally the estimates we give deteriorate with the cardinality of supports of local functions. In order to give the estimates in the uniform way we fix a number M and impose the following restriction: r(φ) + r(ψ) ≤ M.

(2.2.3)

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D. Ioffe, R. H. Schonmann

Lemma 2.2.1. For any β > βc and M ∈ N there exists a constant c = c(M, β), such that for any sequence (s(N ), g(N )) satisfying (2.2.2) one can choose two constants c1 and c2 , such that |hφiA,−,h − hφi3φ (s)∩A,−,h | ≤ c1 kφke−cs(N ) ,

(2.2.4)

|hφ; ψiA,−,h | ≤ c2 kφkkψke−cs(N )∧d(φ,ψ) ,

(2.2.5)

β,s

β

and β,s

uniformly in N ∈ N, magnetic fields h ∈ [0, g(N )], domains A ⊂ Z2 and local functions φ, ψ : A 7→ R satisfying the support size constraint (2.2.3). Remark 2.2.2. For the value of M fixed, the lemma gives means for a uniform control of semi-invariants up to the order M under Pβ,s A,−,g . This proves to be a key to a classical treatment of both terms in the right-hand side of (2.1.2) already in the case M = 3. Note also that the control estimates (1.1.1) on the expected values MA and MsA at zero value of the magnetic field instantly follow from (2.2.4) and the fact (cf. [CCS] and Sect. 1.3 in [SS2]) that for any A, B ⊆ Z2 and any local function φ with suppφ ⊆ A ∩ B and the cardinality of the support |suppφ| = M , there exist c3 = c3 (M, β) and c4 = c4 (M, β), such that | hφiA,− − hφiB,− | ≤ c3 kφke−c4 d(suppφ,A1B) , β

β

(2.2.6)

where d(C, D) is the k · k-distance between the sets C and D. As in [SS2] in the heart of the proof lies an estimate on the exponential decay of certain connectivity functions (Proposition 2.2.3 below): Let us first of all recall the notions of + and +∗ connectedness: two sites x, y ∈ Z2 are called neighbours (∗-neighbours respectively), if kx−yk1 = 1 ( respectively kx−yk = 1), where, as before, k · k1 and k · k are respectively the l1 and the supremum lattice norms. A sequence of sites x1 , . . . , xn is called a connected (respectively ∗-connected) chain if each pair (xi , xi+1 ); i = 1, . . . , n − 1, is a pair of neighbours (∗-neighbours respectively). Finally, a set B ⊆ Z2 is + (+∗ respectively ) connected to a set C ⊆ Z2 , if there exists a connected (∗-connected) chain of sites x1 , . . . , xn , such that x1 ∈ B, + xn ∈ C and σ(xi ) = 1; i = 1, .., n. The corresponding event is denoted as {B −→ C} +∗ (respectively {B −→ C} ). The notions of − and −∗ connectedness are defined in a completely similar way. Proposition 2.2.3. Let the sequence (s(N ), g(N )) be as in the conditions of Lemma 2.2.1. Then, there exists a positive constant c5 = c5 (β), such that uniformly in k ∈ R+ , magnetic fields h ∈ [0, g(N )] and domains A ⊆ Z2 , + c Pβ,s ≤ e−c5 k , (2.2.7) A,−,h x −→ 3x (k) as soon as N is large enough. Proof. There is nothing to prove when k > s(N ). Indeed, in this case the very restriction of the s(N )-small phase rules out the possibility of x being + connected to the boundary ∂3x (k). Let us start by proving the assertion in the case k ∈ [s/2, s]. At this stage we almost literally follow the logic of the corresponding proof in [SS2] (Lemma 2.3.4 ): Let α = (γ1 , . . . , γn ) be the collection of all exterior s(N )-small contours, such that

Dobrushin–Koteck´y–Shlosman Theorem up to Critical Temperature

int(γi ) ∩ ∂3x (3s(N )) 6= ∅,

131

i = 1, . . . , n,

where ∂B is used to denote the outer boundary of a lattice domain B ⊂ Z2 . Define the random box 3α via [ [ 3α = 3x (3s(N )) \ int(γ), γ∈α

[ is the closure of int(γ). Note that due to the restriction on contours to be where int(γ) s(N )-small, 3x (k) ⊆ 3α for each k ≤ s(N ). Note also that the event corresponding to α does not depend on the spins inside the box 3α . Employing the decomposition with respect to all possible realizations of 3α , we obtain X + + c c pα Pβ,s = , (2.2.8) Pβ,s A,−,h x −→ 3x (k) 3α ∩A,−,h x → 3x (k) α

P where {pα } is a collection of probabilistic weights; α pα = 1, whose precise values are of no importance for us. Notice that the s(N )-restriction played the crucial role in the above reduction. Once, however, (2.2.8) is established, we do not need the phase of s(N )-small contours any more: Since the event {all contours are s(N ) − small} is non-increasing, one can take advantage of the FKG properties of Ising measures and develop the right-hand side of (2.2.8) further as, X + + β c c x ≤ x −→ 3 (k) p P → 3 (k) Pβ,s x α 3α ∩A,−,h x A,−,h α

+ ≤ Pβ3x (3s(N )),−,h x → 3x (k)c . On the other hand, |3x (3s(N )) | ≤ 36s(N )2 . Since, due to the condition (2.2.2), the magnetic field h ∈ [0, g] is uniformly under control, this means that the logarithm of the Radon–Nikodym derivative dPβ 3 (3s),−,h log β x dP3x (3s),−,0 is o(s(N )) uniformly in h ∈ [0, g]. Therefore, again uniformly in h ∈ [0, g], + + c o(s(N )) β c −→ 3 (k) P −→ 3 (k) x ≤ e x . Pβ,s x x 3x (3s),−,0 3x (3s),−,h By the results on the exponential decay of connectivities at zero value of the magnetic field [CCS] the latter quantity is bounded above by some e−c6 (β)k , and the claim of the proposition follows for k in the range k ∈ [s/2, s]. Let us now pick any value of k ∈ (0, s/2). The problem with smaller values of k is that from the first glance the exponential decay in the zero magnetic field phase might be overshot by the value of the Radon–Nikodym derivative over boxes of the s(N ) linear size. This, however, can be circumvented in the following way:

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D. Ioffe, R. H. Schonmann

We pick a finite sequence of numbers k1 , k2 , . . . , kn ; n = n(k) and construct the 1 corresponding sequence of boxes 3i = 3x (ki ), where k1 ≡ k

ki+1 = 2ki

and

n(k) =

max

2i k<3s(N )

i.

Now notice that, + (2.2.9) Pβ3x (3s),−,h x −→ 3x (k)c n oc + + + + Pβ3x (3s),−,h 3n−1 −→ 3cn . ≤ Pβ3x (3s),−,h x −→ 3x (k) ; 3n−1 −→ 3cn n

oc + 3n−1 −→ 3cn contains a −∗ connected loop inside 3n \ 3n−1 around 3n−1 . Thus, with a proper splitting argument we might hope to + reduce the computation of the probability of {x −→ 3x (k)c } in a first term on the righthand side of (2.2.9) to a box of the size smaller (roughly one half) than that of 3x (3s). This is very advantageous, since the impact of the magnetic field through the Radon– Nikodym derivative diminishes in this way. The second term on the right-hand side of (2.2.9) is already treatable by the first part of the proof. We then decompose probabilities in the smaller box exactly in the same fashion as in (2.2.9) and continue in this way until we reduce a computation to uniform estimates over boxes B; 31 ⊆ B ⊆ 32 . We now pass to the formal setting: Define a sequence of random domains 3αi via Each configuration σ ∈

3αn ≡ 3x (3s(N ))

1 d and 3αi = 3i+1 \ D αi ; i = 1, . . . , n − 1,

where random domains Dαi above are given by 1

+

Dαi = {y ∈ Di+1 : y −→ 3ci+1 }. Now observe that

n

+

3i −→ 3ci+1

oc

=⇒ 3i ⊆ 3αi ⊆ 3i+1 .

(2.2.10)

Also observe that, as before, for any B ⊆ 3i+1 the event {3αi = B} does not depend on the value of spins inside B. Unfolding (2.2.9) and splitting with respect to the different realizations of 3αi ; i = 1, . . . , n, we obtain: + Pβ3x (3s),−,h x −→ 3x (k)c (2.2.11) ( ) n X + + β β c c ≤ max P3α ,−,h 3i−1 −→ 3i P3α ,−,h x −→ 3x (k) + , α1 ,... ,αn−1

1

i=2

i

where the maximum is over all possible realizations of boxes 3α1 , . . . , 3αn−1 , which satisfy the inclusion on the right-hand side of (2.2.10). Estimating the Radon–Nikodym derivative with respect to the zero field measures we infer from (2.2.11), (2.2.10) and the FKG inequality, 2 + + Pβ3x (3s),−,h x −→ 3x (k)c ≤ e4k h Pβ− x −→ 3x (k)c +

n X i=2

2 + e4ki h Pβ− 3i−1 −→ 3ci .

Dobrushin–Koteck´y–Shlosman Theorem up to Critical Temperature

133

The result now follows by the exponential decay properties of Pβ− and the assumption (2.2.2). Remark 2.2.4. Notice that the only relevant feature of the box 3x (3s) we have used in the last part of the proof is the upper bound on its cardinality, |3x (3s)| ≤ 36s2 . A straightforward rerun of the arguments above leads to the following general estimate: Fix a number t ∈ R, and let s(N ) and g(N ) obey (2.2.2). Then, there exists c7 = c7 (β, t), such that once N is large enough, + PβB,− x → 3x (k)c ≤ exp (−c(β, t)k) , (2.2.12) uniformly in domains B ⊂ Z2 satisfying |B| ≤ ts2 , x ∈ B, magnetic fields h ∈ [0, g] and k ∈ N. Proof of Lemma 2.2.1. Given Proposition 2.2.3, we can now closely follow, for proving (2.2.4), the corresponding arguments in [SS2]. Since, however, the setting, notation and scaling of the latter article is different from ours, we present here the rest of the proof in full details for the sake of completeness. It is enough to consider only the case of monotone non-decreasing local φ and ψ. With A ⊂ Z2 being fixed, set c s(N ) s(N ) + Eφ = 3φ ⊆ A , −→ 3φ 8M 4M where M is the maximal support size in (2.2.3). By Proposition 2.2.3, −c9 s(N ) , Pβ,s A,−,h Eφ ≤ c8 e uniformly in A ⊆ Z2 and h ∈ [0, g(N )]. On the other hand every extended configuration b φ ( s(N ) ) surrounding σ∪{−1}∂A ; σ ∈ Eφc , contains a closed −∗ loop of spins inside A∩3 4M 1 s(N ) b b A ∩ 3φ ( 8M ), where, as before, A = A ∪ ∂A is the closure of A. Thus, splitting Eφc ; Eφc = ∪α, according to the realizations of the random set s(N ) s(N ) c 1 1 + cα and Dα = )\D ) , y : y −→ 3φ ( 3α = 3φ ( 4M 4M we obtain:

β,s

hφ; Eφc iA,−,h =

X

β,s

pα hφi3α ∩A,−,h ,

α

c (α) and, of course, pα = Pβ,s where pα = A,−,h Eφ . By construction, each realization of 3α satisfies s(N ) s(N ) ⊆ 3α ⊆ 3φ . 3φ 8M 4M Pβ,s A,−,h

P

(2.2.13)

Consequently the super-index s = s(N ) can be trivially omitted in Pβ,s 3α ,−,h ; indeed, the diameter of 3α is less than s for each α. But the unrestricted phase already enjoys FKG inequality. Subsequently, using the monotonicity assumption on φ, we infer:

134

D. Ioffe, R. H. Schonmann β

hφi3

β

s(N ) φ ( 8M )∩A,−,h

β,s

β

≤ hφi3α ∩A,−,h ≡ hφi3α ∩A,−,h ≤ hφi3φ (s(N ))∩A,−,h .

oc n + ) c ) with respect to the random As before, splitting the event supp(φ) −→ 3φ ( s(N 8M ) c + connected component of 3( s(N 8M ) , we obtain from the FKG inequality, β

β

0 ≤ hφi3φ (s)∩A,−,h − hφi3( s(N ) )∩A,−,h 8M

kφkPβ3φ (s)∩A,−,h

≤

+

supp(φ) −→ 3φ

s(N ) 8M

c

≤ kφke−c10 s(N ) , (2.2.14)

where the last inequality follows from (2.2.12) in Remark 2.2.4. This proves the first assertion (2.2.4) of Lemma 2.2.1. The covariance estimate (2.2.5) is a consequence of (2.2.4) and the decay property (2.2.7). Indeed, proceeding as in (2.2.14) above, we infer from the estimate (2.2.4), β,s

hφ; ψiA,−,h = β

hφψi3φ,ψ (

s M

β

)∩A,−,h

− hφi3φ,ψ (

s M

β s )∩A,−,h hψi3φ,ψ ( M )∩A,−,h

+ O kφkkψke−c11 s(N ) ,

1

where 3φ,ψ = 3φ ∪ 3ψ . Note that the diameter of each connected component of ) 3φ,ψ s(N does not exceed, by the condition (2.2.3), s(N ). Thus, the problem boils M down to the following: Prove that uniformly in connected domains B; diam(B) ≤ s(N ), monotone functions φ and ψ satisfying (2.2.3) and magnetic fields h ∈ [0, g] , hφ; ψiB,−,h = hφψiB,−,h − hφiB,−,h hψiB,−,h ≤ c12 kφkkψke−c13 d(φ,ψ) . (2.2.15) β

β

β

β

Define the “median” set M (φ, ψ) between the two supports as 1 M (φ, ψ) = z : | min kz − xk − min kz − yk| ≤ 1 , x∈suppφ y∈suppψ and consider the event n o[n o 1 + + suppψ −→ M (φ, ψ) . Eφ,ψ = suppφ −→ M (φ, ψ) Using the exponential decay of connectivities (2.2.12), one readily obtains that PβB,−,h Eφ,ψ ≤ c14 e−c15 d(φ,ψ) . On the complement of Eφ,ψ the supports of φ and ψ are necessarily separated by a −∗ connected chain, and, using the splitting decomposition with respect to the random domains n o 1 1 + cα and Dα = z : z −→ M (φ, ψ) , Bα = B \ D c render a non-positive contribution to the left-hand side of we readily obtain that Eφ,ψ (2.2.15). Lemma 2.2.1 is completely proven.

Dobrushin–Koteck´y–Shlosman Theorem up to Critical Temperature

135

2.3. Estimates in the K log N -Phase. We are going to give an estimate on g = g(A, s(N ), aN ) in (2.1.1) on the basic scale s(N ) = K log N and for the moderate values of deviations aN ∈ M+N satisfying aN

N2 . log N

(2.3.1)

Remark 2.3.1. The above condition on the range of aN is of a technical nature: The requirement aN N 2 / log N is related to the condition (2.2.2) and imposes no limiβ tations on the investigation of the unrestricted phase h · iN,− , which, in the end, is our primary object. Indeed, as we shall see later, large aN shifts of the empirical magnetization in the unrestricted phase lead to the creation of large contours on an appropriate scale and are, thereby, reduced to moderate deviations of the magnetization of much smaller order inside and outside those contours. On the other hand, the non-positive values of aN ; aN ≤ 0, correspond to the deviations outside the phase transition region. In particular the choice of g in (2.1.1) is, in this 1 case, also non-positive. Since the event As = { There are no s(N ) − large ± contours} is non-increasing, PβA,−,g (As ) ≥ PβA,− (As ) ≥ 1 − e−c1 s(N ) , where the first inequality follows from the FKG properties of Ising measures, whereas the second one is a consequence of the energy estimate (1.3.2). It is, then, easy to see that the aN ≤ 0 case essentially boils down to the corresponding classical regime of deviations in the unrestricted ensemble, which was summarized in (1.1.2). In order to have a convenient reference we restate the latter estimate here in the general case of s(N )-restricted phases: For any A ⊂ Z2 , large contour parameter s(N ), magnetic field g ≤ 0 and any a ≤ 0, the following Gaussian upper bound is valid: a2 β,s β,s . (2.3.2) PA,−,g MA = hMA iA,−,g + a ≤ exp −c2 (β) |A| We turn now to the moderate deviations in the phase transition range (2.3.1). On any scale s = s(N ), we have: Z g β,s β,s β,s hMA ; MA iA,−,r dr hMA iA,−,g = hMA iA,− + 0

β,s

Z

g

β,s

= hMA iA,− + ghMA ; MA iA,− +

0

Z

r 0

β,s

hMA ; MA ; MA iA,−,h dhdr. (2.3.3)

By (2.2.4) in Lemma 2.2.1, β,s

hMA ; MA iA,− = =

X x,y∈A

X

β,s

hσ(x); σ(y)iA,−

x,y∈A

β hσ(x); σ(y)iA∩3xy (s),− + O e−c3 (β)s(N ) |A|2 .

(2.3.4)

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D. Ioffe, R. H. Schonmann

Remark 2.3.2. The second summand in the bottom line above dictates the choice of the parameter K = K(β) in the definition of the basic s(N ) = K log N scale. Namely we pick up K to be so large that Kc3 (β) 1. As a result the second term on the right-hand side above is o(1). From (2.2.6) we obtain that the PβA∩3xy (s),− covariances are related to the corresponding infinite volume covariances via β β hσ(x); σ(y)i− − hσ(x); σ(y)iA∩3xy (s),− ≤ exp −c4 (β)d(∂A, {x, y}) ∧ s(N ) uniformly in domains A ⊂ Z2 . Consequently, β,s hMA ; MA iA,− − |A|χ ≤ c4 (β)|∂A|.

(2.3.5)

On the other hand, for each g satisfying condition (2.2.2), the covariance estimate (2.2.5) implies a similar bound on higher order semi-invariants (see Appendix B of [ML]). Therefore we have: Z

g 0

Z

r 0

β,s hMA ; MA ; MA iA,−,h dhdr ≤ c5 (β)g 2 |A|.

(2.3.6)

Since, by the definition , |∂A| ≤ RN log N for every A ∈ DN , the bounds (2.3.5) and (2.3.6) lead to the following estimate on the solution g = g(A, s, aN ) of (2.1.1): g=

a aN aN N +O ) . (log N ∨ χ|A| N3 N

(2.3.7)

Notice that in the s = K log N phase the range restriction aN N 2 / log N is, then, precisely translated into the smallness condition (2.2.2); lim g(N )s(N ) = 0, on the magnetic field g(N ) = g(A, s, aN ). Using the above results we can already estimate the crucial expression in the exponent on the right-hand side of (2.1.2): Z

g 0

Z

g r

β,s

hMA ; MA iA,−,h dhdr

Z gZ gZ h g2 β,s β,s = − hMA ; MA iA,− − hMA ; MA ; MA iA,−,q dqdhdr 2 0 r 0 g2 β,s = − hMA ; MA iA,− + O g 3 |A| 2 2 a2 aN aN = − N +O ) . (log N ∨ 2χ|A| N3 N

(2.3.8)

We switch now to the K log N scale proper (with some fixed choice of K = K(β) as it was specified in Remark 2.3.2).

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137

Lemma 2.3.3. Let s(N ) = K log N . Given a sequence of numbers aˆ N satisfying (2.3.1), the following asymptotic (in N ) estimate; s = Pβ,s A,− MA = MA + aN 2 1 aN a2 aN (1 + o(1)) , (2.3.9) = p ) exp − N + O (log N ∨ 2χ|A| N3 N 2πχ|A| holds uniformly in A ∈ DN and aN ∈ M+A ∩ [0, aˆ N ]. Proof. The prefactor near the exponent is due to the local CLT, which we prove in the next subsection. The expression in the exponent follows by the substitution of (2.3.8) into formula (2.1.2). Remark 2.3.4. Note in the rangeof small moderate deviations aN N 4/3−δ the that a2N correction term O N 3 (log N ∨ aNN ) is o(1) and, hence, can be omitted. 2.4. Local CLT estimate. We continue our investigation of the basic phase of s(N ) = K log N small contours and remain in framework of the previous subsection. Our main task here is to justify the form of the prefactor in (2.3.9). Let A ∈ DN and assume that aN ∈ M+A complies with the range restriction (2.3.1). We choose g = g(A, K log N, aN ) according to the formula (2.3.7), so that MsA +aN is produced as the mean magnetization under Pβ,s A,−,g , i.e., so that (2.1.1) is satisfied. As it was already noted in the lines following (2.3.7) such a choice of g is compatible with condition (2.2.2). It would be convenient to state our CLT result regardless of aN . Lemma 2.4.1. Assume that (the sequence of) magnetic fields g(N ¯ ) ≥ 0 satisfies (2.2.2). ¯ )], such that Then, uniformly in domains A ∈ DN and in magnetic fields g ∈ [0, g(N β,s

hMA iA,−,g ∈ M+A , the following asymptotic (N → ∞) estimate is valid: 1 β,s ( 1 + o(1)) . Pβ,s A,−,g MA = hMA iA,−,g = p 2πχ|A|

(2.4.1)

Lemma 2.4.1 is a local CLT type result. We are following here the idea of [DT] to prove it from an integral CLT estimate combined with decoupling properties of finite range Gibbs state. Set β,s MA − hMA iA,−,g d p . M A = χ|A| By the Fourier inversion formula, β,s Pβ,s A,−,g MA = hMA iA,−,q d = Pβ,s A,−,g MA = 0 =

π

1 p 2π χ|A|

√ Z χ|A|

cA i heitM A,−,g dt. β,s

−π

√

χ|A|

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D. Ioffe, R. H. Schonmann

√ R 2 Using the identity 1 = 1/ 2π e−t /2 dt, we obtain that for each choice of a large number r > 0, p 1 β,s | ≤ I1 + I2 + I3 , (2.4.2) − p 2π χ|A| | Pβ,s A,−,g MA = hMA iA,−,g 2πχ|A| where Zr t2 cA iβ,s e− 2 − heitM I1 = A,−,g dt ,

Z I2 =

−r

r<|t|<π

√

itM c β,s he A iA,−,g dt, χ|A|

R 2 t2 and finally, I3 = |t|>r e− 2 dt . Clearly, the last term satisfies I3 ≤ c1 e−r /2 . We are going to show that also for each r fixed, lim I1 = 0

N →∞

and

I2 ≤ e−c2 r

2

∧s(N )

(2.4.3)

¯ uniformly in A ∈ DN and g ∈ [0, g]. Let us start with the I1 part of (2.4.3). This is an integral CLT estimate. In order to prove it we shall follow the approach of [ML]. Namely, it happens to be a consequence of the following convergence result on the level on log-moment generating functions: cA i lim log hetM A,−,g = β,s

N →∞

t2 2

(2.4.4)

¯ Indeed, once (2.4.4) is uniformly on compact subsets of R, A ∈ DN and g ∈ [0, g]. established, the corresponding part of (2.4.3) is implied by the results of Appendix C in [ML] and an easy compactness argument. In its turn (2.4.4) is a consequence of the cA iβ,s up to the third estimates on semi-invariants of Lemma 2.2.1: Expanding hetM A,−,g order terms, cA i log hetM A,−,g = β,s

t2 d d β,s hMA ; MA iA,−,g 2 √ Zt Zh g+q/Z χ|A| d d d β,s + hM A ; MA ; MA iA,−,τ dτ dqdh 0 2

=

g

0

X

t β,s hσ(x); σ(y)iA,−,g (2.4.5) 2χ|A| x,y∈A   3 X t β,s max | hσ(x); σ(y); σ(z)iA,−,g0 |  . + O 3 N |g−g0 |≤t/√χ|A| x,y,z∈A

In order to estimate the second term in the right-hand side above, notice that the magnetic field g 0 also satisfies (2.2.2) in the whole range of its definition, more precisely, K log N max

max√

A∈DN |g 0 −g|≤t/

g 0 ≤ gK ¯ log N + χ|A

tK log N 1, √ N χ

Dobrushin–Koteck´y–Shlosman Theorem up to Critical Temperature

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on any compact interval t ∈ [0, T ], provided only that N is sufficiently large. Consequently, as with (2.3.6), it follows from (2.2.5), that X β,s max√ | hσ(x); σ(y); σ(z)iA,−,g0 | ≤ c3 (β)N 2 . max A∈DN |g−g 0 |≤t/

χ|A| x,y,z∈A

Similarly, the covariance term in (2.4.5) can be further expanded as Zg β,s hMA ; MA iA,−,g

=

β,s hMA ; MA iA,−

=

β,s hMA ; MA iA,−

β,s

hMA ; MA ; MA iA,−,h dh

+ 0

+ O gN 2 .

On the other hand, the zero-field covariance above was already estimated in (2.3.5). We, thus, infer: 3 t2 T cA iβ,s 2 = O + T − g(N ¯ ) , max max log hetM A,−,g 2 N |t|≤T A∈DN and (2.4.4) follows. Finally, the I2 part of (2.4.3) is implied by the following upper bound on characteristic cA iβ,s : functions heitM A,−,g For all A ∈ DN , g ∈ [0, g] ¯ and t ∈ R, cA i −c4 t ∨ e−c5 (β)s(N ) , | heitM A,−,g | ≤ e 2

β,s

(2.4.6)

as we prove next. The proof of (2.4.6) is inspired by the conditional argument of [DT]. With the basic large contour parameter s(N ) = K log N fixed, let us define a sub) 2 2 lattice Z˜2 = [ s(N 2 ]Z ⊂ Z and consider the covering of A by the family of boxes {3x (s(N )/4)}x∈Z˜2 . First of all, notice that, [

|3x (

) x∈Z˜2 : 3x ( s(N 4 )∩∂A6=∅

s2 s(N ) )| ≤ |∂A|, 4 4

which, since A was assumed to be in DN , is bounded above by R(K log N )2 /4. On the other hand, [ s(N ) )| ≥ |A| ≥ aN 2 . |3x ( 4 s(N ) ˜ x∈Z2 : 3x (

Thus,

4

)∩A6=∅

[

|3x (

) x∈Z˜2 : 3x ( s(N 4 )⊂A

|A| s(N ) )| ≥ . 4 2

Consequently, setting, Ax = A ∩ 3x

s(N ) 8

;

x ∈ Z˜2 ,

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D. Ioffe, R. H. Schonmann

we obtain A ∈ DN =⇒

X

| Ax | ≥

a 2 N . 8

(2.4.7)

Define now the event E ⊆ A as, c [ s(N ) s(N ) + E= 3x . −→ 3x 8 4 ˜ x∈A∩Z2

By virtue of Proposition 2.2.3 (and again, provided that K in the definition of s(N ) = K log N was picked large enough) , −c6 s(N ) . Pβ,s A,−,g (E) ≤ e

(2.4.8)

For each σ ∈ E c the extended configuration σ∪{−1}∂A necessarily contains a −∗ closed b ∩ 3x s(N ) for each x ∈ A ∩ Z˜2 . b ∩ 3x ( s(N ) ) surrounding A loop of spins inside A 4 8 We now introduce a domain splitting of the event E c , one of the type we have used so often in Subsect. 2.2: We set E c = ∪α, where α labels all possible realizations of random domains {Aα x }x∈A∩Z˜2 , c s(N ) s(N ) s(N ) 1 + α 1 α c Aα \ D ) : y = A∩3 and D . = y ∈ 3 ( −→ 3 x x x x x x 4 4 4 Performing the domain decomposition of E c , note that since diam(Aα x ) ≤ s/2, the super-index s can be dropped, and, as in the proof of Lemma 2.2.1, we obtain: X itM β,s c pα he A ; E c iA,−,g ≤ α

Y it √MAαx β he χ|A| iAαx ,−,g .

(2.4.9)

x∈A∩Z˜2

Since there are only characteristic functions of unrestricted ensembles on the righthand side above, we, at this stage, can simply apply the conditional variance argument of [DT] (see the derivation p of (1.17) in the latter paper) and, thereby, to conclude that uniformly in α, |t| ≤ χ|A| and x ∈ Z˜2 , it √MAαx |Aα | β −c t2 x he χ|A| iAαx ,−,g ≤ e 7 |A| . Since Ax ⊆ Aα x , the latter estimate , combined with (2.4.8), (2.4.7) and (2.4.9), implies the bound (2.4.6) and hence the claim of the lemma. 2.5. Upper bound in the s(N )-phase of small contours. The results of this subsection are used to derive super-surface order local estimates, which we shall need throughout the paper. In particular they are crucial for the isoperimetric stability argument of Sect. 5. Lemma 2.5.1. Let the large contour parameter s(N ) log N be fixed. There exists a constant c = c(β) > 0, such that for all N > 0, A ∈ DN and all aN ∈ M+A , aN a2N β,s s PA,− MA = MA + aN ≤ exp −c 2 ∧ . (2.5.1) N s(N )

Dobrushin–Koteck´y–Shlosman Theorem up to Critical Temperature

141

Remark 2.5.2. Lemma 2.5.1 provides an, in a certain sense, optimal generalization of various estimates in the phase of small contours (see [I2, SS3] and [PV]), which lie in the heart of all previous weak integral results on the phase separation up to the critical temperature Tc . Actually we need (2.5.1) only in the case of large moderate deviations aN N 4/3 . In the case of small moderate deviations aN N 4/3 a much more precise statement will be derived in Subsect. 4.1 independently of (2.5.1). Also the techniques of the latter subsection readily imply an asymptotic bound of the form (2.5.1) in the critical case aN ∼ N 4/3 as well. Thus, there is no loss to assume from the beginning that 4

aN N 3 .

(2.5.2)

Proof. The idea of the proof is simple: either a volume of order aN /2m∗ is exhausted by K log N -large contours, which, in the Pβ,s A,− -restricted phase, should have a surface tension price with the exponent of order aN /s(N ), or the K log N -large contours cover a volume much less than aN /2m∗ , and the remaining excess in magnetization ought to be compensated in the K log N restricted phase, where we can subsequently apply the estimates of Subsect. 2.3. So let 0(σ) be the collection of all K log N -large contours of σ . Recall, that Vol+ (0(σ)) was set to denote the area of the “+” component in the 0-induced decomposition of A. We write: aN s s ≤Pβ,s Pβ,s A,− MA = MA + aN A,− MA = MA + aN ; Vol+ (0(σ)) < 4m∗ aN + Pβ,s Vol+ (0(σ)) ≥ . (2.5.3) A,− 4m∗ By the estimates on the phase volume (1.1.3) on the K log N scale: Vol+ (0(σ)) ≥

aN aN =⇒ |S+ | ≥ − c1 K log N Wβ (S+ ) ∗ 4m 4m∗

(2.5.4)

for any collection of K log N skeletons S ∼ 0. We claim that in view of the s(N )restriction, (2.5.4) implies, in addition, that Wβ (S+ ) ≥ c2

aN , s(N )

(2.5.5)

as will be shown next. Indeed, the moment we assume, for example, that Wβ (S+ ) < aN /s(N ), we immediately infer from (2.5.4) and the choice s(N ) log N , that aN . 8m∗

|S+ | >

(2.5.6)

Since, 0 ∼ S is forced by Pβ,s A,− to comply with the s(N )-restriction on the size of its contours, i.e., 0 = (γ1 , . . . , γN ) ⇒ max diam(γi ) < s(N ), i

the diameter and the area of each connected component of S+ do not exceed 2s(N ) and 4s(N )2 respectively.

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D. Ioffe, R. H. Schonmann

In any case, (2.5.6) implies, by the isoperimetric inequality, that √ Wβ (S+ ) ≥ c3 aN . If aN /8m∗ ≤ 4s2 , then √

s aN

aN = s(N )

s2 aN . ≥ c4 aN s(N )

Otherwise, if aN ≥ 32m∗ s(N )2 , then it is easy to see that Wβ (S+ ) is always larger than or equal to the total surface energy of [aN /32m∗ s(N )2 ] Wulff droplets of the maximal permitted area 4s(N )2 (it is obvious that the surface energy is minimized on a collection of Wulff shaped droplets, and a simple computation shows that the transfer of “mass” from a smaller to a larger droplet decreases the surface energy). Since the surface tension of each of these droplets equals to 2s(N )ω1 , we obtain, aN aN Wβ (S+ ) ≥ 2ω1 , s(N ) ≥ c5 32m∗ s(N )2 s(N ) and (4.2.11) follows. From (2.5.5) and the energy inequality (1.3.2) on the K log N scale (see also (1.1.4)) the second term on the right-hand side of (2.5.3) is bounded above by exp −c6 aN /s(N ) . We proceed to investigate the first term on the right-hand side of (2.5.3): let 0 satisfy |C| = Vol+ (0(σ)) < aN /4m∗ , where C is the plus component of the corresponding 0-induced decomposition of A; A = B ∪ C. Since, by the definition, 0 is the collection of all K log N large contours, the following factorization of Pβ,s A,− is valid: β,K log N log N (•) Pβ,K (•) . Pβ,s A,− • | 0 = PB,− C,+ Using the corresponding analog of the decomposition (1.3.4), (2.5.7) { MA = MsA + aN ; 0 } n o [ log N K log N 0 ; MB = MK , + b ; M = −M + c = N C N B C bN +cN =aN −1(B,C)

where, of course, the compensator 1(B, C) is given by log N log N − MK − MsA + |∂+ 0| − |∂− 0|. 1(B, C) = MK B C

By Lemma 2.2.1 and (1.1.1), if K is large enough, 1(B, C) − 2m∗ |C| ≤ c7 |∂C| = c7 |0|.

(2.5.8)

In view of the energy estimate (1.3.2) there is no loss to assume that any 0-compatible collection of K log N skeletons S satisfies Wβ (S) ≤ aN /s(N ). Consequently, |0| ≤ c8 K log N Wβ (S) aN , which, by virtue of the assumption on the volume |C| ≤ aN /4m∗ and by the estimate (2.5.8), gives the following bound on 1(B, C): aN − 1 (B, C) ≥

aN . 3

Dobrushin–Koteck´y–Shlosman Theorem up to Critical Temperature

143

Therefore, the decomposition (2.5.7), in fact, implies: β,K log N K log N s = M + a | 0 ≤ max P = M + b M M Pβ,s A N B N A A,− B,− B bN ≥aN /6 _ log N log N − cN , max Pβ,K M C = MK C,− C cN ≥aN /6

where we have used the flip symmetry of Ising measures to rewrite the probability on C in the minus state setting. By the Gaussian bound (2.3.2), a2N log N K log N . M ≤ exp −c = M − c max Pβ,K C N 9 C,− C N2 cN ≥aN /6 As far as the “−” component B is concerned, we can, of course try to apply the local limit estimates of the previous subsection. Since, however, we are interested only in coarse upper bounds, the following usual large deviation type bound suffices, log N log N M B = MK + bN log Pβ,K B,− B n o log N gMB β,K log N + b i − log he . (2.5.9) ≤ − max g MK N B B,− g

For the values of magnetic field g satisfying g log N 1 one is entitled to use Lemma 2.2.1 and perform the corresponding expansion of the log-moment generating function, to estimate β,K log N

log hegMB iB,−

log N = gMK + B

g2 β,K log N hMB ; MB iB,− + O g3 N 2 2 log N + c10 g 2 N 2 , ≤ gMK B

where the O g 3 N 2 term above follows from (2.2.5) (as (2.3.6) did), whereas the last inequality is a consequence of (2.2.5) and the fact that |B| ≤ |A| ≤ N 2 . Substituting the above bound into (2.5.9) we obtain, ga N log N K log N 2 2 − c = M + b g N M ≤ − max . max log Pβ,K B N 10 B,− B 6 bN ≥aN /6 g1/ log N For the values of aN N 2 / log N the right-hand side above is of order a2N /N 2 . Otherwise an admissible choice of g = 1/s(N ) leads to the upper bound of order aN /s(N ), which is again compatible with the assertion of the lemma. The proof is, thereby, concluded. 3. Lower Bound 3.1. The estimate. With the exception of the references to the results of [CCS] this is the only part of the paper, where we use the FK representation. Likewise, this is the only place, where we digress from purely probabilistic considerations and use results obtained in the framework of exact solutions. Namely, we rely on two facts about the 2D Ising model: first of all it is proved in [AA], that the Wulff shape has everywhere positive radius of curvature, more precisely for any β > βc , the surface tension τβ enjoys the following positive stiffness property:

144

D. Ioffe, R. H. Schonmann

1 min τβ00 (n) + τβ (n) = R(β) > 0.

n∈S1

(3.1.1)

Secondly [MW], for any dual inverse temperature β ∗ < βc , the two point correlation functions of the (unique) infinite volume Gibbs distribution are subject to an Ornstein– Zernike type correction formula: u ∗ ) ≥ hσ(u)σ(0)iβ ∗ exp −kuk2 mβ ( kuk2 u ∗ ∗ ≥ exp −kuk2 mβ ( ) − c1 (β ) log kuk2 , (3.1.2) kuk2 where k · k2 is the Euclidean norm on R2 and mβ ∗ is the directionally dependent mass gap at β ∗ , which is related by the Krammer-Wannier duality (see [Pf] for more on this) to the surface tension of the direct model mβ ∗ ≡ τβ as soon as e2β = tanh β ∗ .

(3.1.3)

Both (3.1.1) and (3.1.2) play an important role in our approach to the lower bound. We would like to remark, however, that the technique of exact solutions itself does not seem to be indispensable. At least in the case of self-avoiding walks an alternative probabilistic treatment is developed in [I3], whereas various correction formulas similar to (3.1.2) were obtained in [A1] and in [A2] for two and three dimensional Bernoulli percolation and in the general framework of multidimensional subadditive arrays respectively. Moreover, (3.1.2) itself has recently been established in [A3] for a class of lattice models including the one we consider here. Theorem 3.1.1. For every β > βc there exists a number N0 = N0 (β) < ∞ and a constant c2 = c2 (β) > 0, such that for all N ≥ N0 and all aN ≥ N 4/3 in the range of MN ; aN ∈ M+N , the following estimate holds: r √ aN β 2 ∗ 4 ω1 − c2 (β) aN log N . PN,− MN = −N m + aN ≥ exp − 2m∗ (3.1.4) Remark 3.1.2. In fact an appropriate version of the above lower bound holds for the whole range of aN ∈ M+N . For small moderate values of aN , however, the leading asymptotics of PβN,− MN = −N 2 m∗ + aN stems from the Gaussian estimate (2.3.9) in the K log N phase and is, therefore, different from the one on the right-hand side of (3.1.4). Thus, the lower bound (3.1.1) becomes sharp only for large moderate values of aN . Notice also that a completely similar lower bound is, of course, valid for the deviations from the true average PβN,− ( MN = MN + aN ), whenever aN satisfies conditions of Theorem 3.1.1. 3.2. Outline of the proof. Note first of all that using results of Sect. 2, one can trivially derive a lower bound, which would capture the right surface √ order. Indeed, pick a ± contour γ, such that |int(γ)| = aN /2m∗ + O(1) and |γ| ∼ aN . Then, using local limit estimates of the previous section inside and outside of γ, one readily obtains: √ (3.2.1) PβN,− MN = −N 2 m∗ + aN ≥ exp −c(β) aN .

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As it was mentioned in the introduction, an important consequence of (3.2.1) is that by the observation (1.3.3), one can, from now on, restrict attention only√to those collections of large contours, which have admissible total length, i.e., of order aN log N at most. The logic behind the lower bound is transparent: in order to induce the aN -shift of the magnetization MN from its expected value under the PβN,− measure, one tries to “chop out” an island of area close to aN /2m∗ and of the optimal shape, and to enforce a typical “+”-phase configuration, i.e., of the mean magnetization close to m∗ , over this island. Thus, the proof of the lower bound boils down to the following procedure: we choose a certain parameter wN and let TN be the tube of the width wN around p the required dilatation aN /2m∗ K1 of the unit volume Wulff pshape. We shall let EN denote the event that there is a ± contour inside TN around aN /2m∗ K1 \ TN , and K EN be the subset of EN in which there is a unique such contour, all other contours being √ K log N -small, and the large contour has its length bounded above by K aN log N . Our estimate will be based on β K K PN,− MN = −N 2 m∗ + aN | EN . PβN,− MN = −N 2 m∗ + aN ≥ PβN,− EN (3.2.2) K This splits the problem into two: to an estimate on PβN,− EN and, provided √ that the area of the tube |TN | ∼ aN wN aN , to local limit estimates on K PβN,− MN = −N 2 m∗ + aN | EN . The latter problem happens to fall in the framework of the previous section. Clearly, the terms in the product on the right-hand side of (3.2.2) are of competing K should be a coarse grained one, and it is on nature. Indeed, an estimate on PβN,− EN this stage that the surface tension τβ enters the picture. In other words, the width wN of the tube TN is intimately related to the large skeleton scale s(N ), we choose to coarse grain. In fact, it happens that the optimal choice without going √ into detailed analysis of the fluctuations of the phase separation line is of order wN ∼ s(N ) log N . We shall, then, employ the Gaussian estimates of the previous section to conclude that K 2 ≥ exp −c3 (β)wN . (3.2.3) PβN,− MN = −N 2 m∗ + aN | EN Note that the estimate above deteriorates with s(N ). On the other hand, due to K becomes the Ornstein–Zernike correction formula (3.1.2), the estimate on PβN,− EN better at large values of the large contour parameter s(N ). Namely, we shall prove that on√ the s(N ) skeleton scale, i.e., with the corresponding choice of the tube width wN ∼ s(N ) log N , r log N aN β K . (3.2.4) ω1 1 − c3 (β) PN,− EN ≥ exp − 2m∗ s(N ) Consequently, the final assertion (3.1.4) is an outcome of an optimization in terms of the large skeleton scale s(N ). There are several possibilities to choose “scissors” for the implementation of the first part of the above program. The most straightforward one is to try to investigate directly the random line representation of contours in EN . Such an approach was pursued in [Pf, I1, PV] and [V]. The estimates obtained in the latter work are essentially equivalent to those we derive here. It is our opinion, however, that this direct approach leads to

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unnecessary complications, related to the fact that one is somehow compelled to move against the current and to reverse the natural direction of the inequalities related to the random line representation. Contrary to this, we shall follow [SS1] and use indirect “FKscissors” to chop out domains of the desirable shapes. As we shall see later the immediate advantage of this indirect approach is that the natural inequalities (FKG inequalities) become in this way friendly and point precisely in the direction we need. 3.3. FK representation. We refer to [ACCN, CCS, Pi] and [SS1] for the definition of the FK measures and the corresponding discussions of their properties. For any E-finite set of nearest neighbour edges of Z2 ; E ⊂ E 2 , we use GE = (VE , E) to denote the corresponding subgraph of (Z2 , E 2 ), where VE is the set of all the vertices of edges from E. Equally for any finite set of vertices B ⊂ Z2 we define EB to be the set of all the edges of E 2 with both endpoints in B, and write GB = (B, EB ) for the corresponding subgraph. Thus, given an inverse temperature β ∈ (0, ∞), the notation PβE,f , PβE,+ and PβE,− (respectively PβB,f , PβB,+ and PβB,− ) are reserved for finite volume Ising measures on GE (respectively on GB ) with free, plus and minus boundary conditions. Similarly, µβE,w and µβE,f (respectively µβB,w and µβB,f ) are used to denote the wired and free FK measures on GE (respectively on GB ) at the percolation parameter 1

p = p(β) = 1 − e−2β . It is convenient to use the above graph notation, once the dual quantities come into play: let Z2∗ = Z2 + (1/2, 1/2) be the dual lattice and E∗2 the corresponding set of dual edges. To each direct edge e ∈ E 2 there corresponds exactly one dual edge e∗ ; e ⊥ e∗ , which intersect e in the sense of geometric embedding of both lattices into R2 . Thus, given a finite set of direct edges E ⊂ E 2 , we define its dual E ∗ ⊂ E∗2 via E ∗ = e∗ ∈ E∗2 : e∗ ⊥ e for some e ∈ E . The notion of duality between subsets of Z2 and Z2∗ is defined, then, through the above notion of the edge duality: given B ∈ Z2 we define its dual B ∗ ⊂ Z2∗ via B ∗ = VEB∗ . We proceed by recalling the dual correspondence between the set of direct bond ∗ configurations E = {0, 1}E and the set of dual bond configurations E ∗ = {0, 1}E , n∗ (e∗ ) = 1 − n(e) for each pair e ⊥ e∗ . As a result any probability measure on E automatically induces and, in fact, can be identified with the corresponding measure on E ∗ . In particular [CCS], the direct ∗ wired measure µβB,w corresponds in this way to the dual free measure µβB ∗ ,f , where β ∗ is precisely the dual temperature given by the Krammer-Wannier relation (3.1.3). ∗ Similarly, the direct free measure µβB,f corresponds to the dual wired one µβB ∗ ,w . It should be stipulated, though, that the notion of duality is pronounced in the FK language much stronger than on the usual level of partition functions: not only both measures in duality are played on the same probability space, they, as we have already mentioned, are actually identified, and we shall frequently switch from the direct to the dual picture in the course of studying probabilities of occurrence of various geometric events related to the FK-percolation. So let a large skeleton parameter s(N ) be fixed, s(N ) log N,

(3.3.1)

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√ and set wN = M s(N ) log N , where M = M (β) is a large enough number to be specified later. Let also aN ∈ M+N be in the range of large moderate deviations; aN ∼ N 4/3+δ . The tube Tf N around the appropriate dilatation of the boundary of the Wulff shape is defined via: Tf N =

r

x : d x,

aN ∂K1 2m∗

1 ≤ wN 2

.

(3.3.2)

Below is our main probabilistic estimate on chopping out domains in terms of the “FK-scissors”: Lemma 3.3.1. Set r aN f f K1 \ TN . CN = ∃ loop of open dual bonds inside TN around 2m∗ Then, ∗

N

r

µβ3N ,w (CN ) = µβ3∗ ,f (CN ) ≥ exp

−

√ aN log N aN .(3.3.3) ω − c (β) 1 4 2m∗ s(N )

p around aN /2m∗ ∂K1 , i.e., assume Proof. Let S = (u1 , . . . , un ) be an s(N )-skeleton p ∗ that S is an s(N )-skeleton and that d(ui , aN /2m ∂K1 ) < 1; i = 1, . . . , n. As in [I1], using the positivep stiffness of τβ , it is not hard to see that the Hausdorff distance between the boundary aN /2m∗ ∂K1 and the polygonal line P(S) through the vertices of S is of order r s(N )2 aN ∂K , P(S) ∼ √ . dH 1 ∗ 2m aN Consequently, if s(N )3 aN log N,

(3.3.4)

one can think about the Tf N tube as being actually drawn around P(S) itself. In what follows we adopt a convenient construction in [V], based on the sharp triangle inequality of [I1]. For each pair of successive vertices {ui , ui+1 } in S, let us define the set Bi ⊂ Z2∗ via Bi =

z ∈ Z2∗ : kui − zk2 + kz − ui+1 k2 ≤ kui − ui+1 k2 + M 0 log N

, (3.3.5)

where k · k2 is the usual Euclidean norm on R2 and M 0 is some (big) number. Clearly, given M 0 > 0 one can always find M > 0, such that the following inclusion is true: Bi ⊂ Tf N ; i = 1, . . . , n .

(3.3.6)

On the other hand, once (3.3.6) is satisfied, the FKG property of random cluster measures implies:

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!

∗ µβ3N ∗ ,f

(CN ) ≥ ≥

∗ µβ3N ∗ ,f

Y i

≥

Y

\ FK {ui ←→ ui+1 inside Bi } i

∗ µβ3N ∗ ,f

FK

{ui ←→ ui+1 inside Bi }

(3.3.7)

Y ∗ FK β∗ µβBi ,f {ui ←→ ui+1 inside Bi } = hσ(ui )σ(ui+1 )iBi ,f .

i

i

By the results on the random line representation of the pair correlations [PV], for any set B ⊂ Z2∗ and for any two points u, v ∈ B one can compare the finite volume β∗ β∗ correlation hσ(u)σ(v)iB,f with the infinite volume one hσ(u)σ(v)if via β∗

β∗

β∗

β∗

hσ(u)σ(v)iB,f ≥ hσ(u)σ(v)if − c5 |∂B| max hσ(u)σ(z)if hσ(z)σ(v)if , z∈∂B

where c5 is a combinatorial constant, which depends only on the dimension and on the notion of the boundary ∂B chosen. In our case we define ∂B to be the outer boundary of B, (3.3.8) ∂B = z ∈ B c : min kz − xk = 1 . x∈B

The positive stiffness condition implies [I1], [V] the following form of the sharp triangle inequality: for any three distinct points u, v, z ∈ Z2∗ , ku − zk2 τβ (

u−z v−z u−v )+kv − zk2 τβ ( ) − ku − vk2 τβ ( ) ku − zk2 kv − zk2 ku − vk2 c6 ≥ ku − zk2 + kv − zk2 − ku − vk2 . (3.3.9) R(β)

By the Ornstein–Zernike correction formula (3.1.2) and by the very construction of the sets Bi , we thus infer from (3.3.9) that for i = 1, .., n, β∗

β∗

max hσ(ui )σ(z)if hσ(ui+1 )σ(z)if c6 β∗ M 0 log N + c7 log N . ≤ hσ(ui )σ(ui+1 )if exp − R(β)

z∈∂B

Consequently, choosing M 0 sufficiently large, which, as we have already remarked, amounts to choosing M in (3.3.2) large enough, we are able to conclude from (3.3.8) and (3.3.7): Y ∗ β∗ hσ(ui )σ(ui+1 )if . µβ3N ∗ ,f ( CN ) ≥ (1 + o(1)) i

Since the distance between any two neighbouring vertices of S lies in √ the interval [s(N ), 3s(N )], and the number of all such pairs of neighbours is of the order aN /s(N ), the claim of the lemma follows by another application of the Ornstein–Zernike lower bound (3.1.2) to each term in the product above. Notice that in this argument the error which results from approximating ω1 by a Riemann sum is negligible provided √ p s(N ) 4 aN log N . (3.3.10)

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3.4. Proof of the lower bound. We start by proving (3.2.4). Recall that the tube width wN was defined in the previous subsection as p wN = M s(N ) log N , and we now formally define the tube TN itself via r aN ∂K ) ≤ w TN = x : d(x, 1 N 2m∗ and r

EN =

There is a ± contour inside TN around

aN K 1 \ TN 2m∗

.

K , we are In order to prove the desired lower bound on the PβN,− -probability of EN going to construct a certain event Ef N stated in terms of the FK percolation geometry, such that r log N aN , 1 − c ≥ exp − ω (β) µβ3N ,w Ef N 1 7 2m∗ s(N ) (3.4.1) 1 PβN,− ( EN ) ≥ µβ3N ,w Ef (3.4.2) N 4

and K PβN,− EN

≥

1 β ( EN ) . P 2 N,−

(3.4.3)

The FK event Ef N we chose is n Ef ∃ two disjoint FK loops of occupied direct bonds inside N = r o aN . K \ T TN around 1 N 2m∗ We would like to recall at this stage a useful way [ES] to construct PβN,− from the wired FK measure µβ3N ,w . This comprises two steps: first play a bond configuration n ∈ {0, 1}E3N under µβ3N ,w , and second paint independently each maximal connected component of n into +1 or −1 with the probability 1/2 each, if this component is disconnected from ∂3N , while assigning identical −1 spin to the boundary cluster. Since the inner loop of Ef N is, clearly, disjoint from the boundary ∂3N , it will cost us exactly probability 1/2 to paint it into +1, which provides a + connected circuit of spins inside TN . The outer loop of Ef N may or may not be connected to ∂3N , in either case the probability to paint it into −1 (independently from the inner loop!) is at least 1/2. The presence of both + and − connected circuits of spins inside TN implies EN , and (3.4.2) follows. K into three events which we p define next. B1 To prove (3.4.3) we partition EN \ EN aN will denote the event that inside the outermost contour which surrounds 2m ∗ K1 \ TN

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there is some K log N -large B2 will denote the event that outside the innermost p contour. aN K \ contour which surrounds 2m ∗ 1 TN there is some K log N -large contour. √ And B3 will denote the event that there is some contour with length larger than K aN log N . Standard conditioning arguments show that when K is sufficiently large PβN,− B1 |EN = o(1), and PβN,− B2 |EN = o(1). Regarding B3 , one can just use (1.3.3) and a trivial lower bound on PβN,− (EN ) to show that PβN,− B3 |EN = o(1) as well. The inequality (3.4.3) is now immediate from K K | EN ≥ PβN,− (EN ) PβN,− EN PβN,− EN ≥ PβN,− (EN ) 1 − PβN,− B1 |EN − PβN,− B2 |EN − PβN,− B3 |EN . In order to prove (3.4.1) we use the estimate of Lemma 3.3.1 and the construction similar to the one already employed in [SS1]: for each bond configuration of CN (as in the statement of Lemma 3.3.1) let us define a (random) set r aN FK f . K ∪ T D = x : x ←→ 3N \ 1 N 2m∗ 1

Furthermore, let us split CN according to the realization of G = 3N \ D: [ CN = Gα , α

where, with a slight abuse of notations, we use Gα to denote both the subset of 3N \ D and the corresponding event {G = Gα }. Note that by the very virtue of CN , r r aN aN f K1 \ TN ⊆ Gα ⊆ K1 ∪ Tf (3.4.4) N. ∗ 2m 2m∗ Note also that the event Gα does not depend on the bond configuration on the edges of GGα . Let us define now the event Rin N via r aN in f K \ T RN = There is an occupied dual FK chain across TN ∩ . 1 N 2m∗ By the FKG inequality and the results of [CCS], for any realization of Gα , we have: p ∗ ∗ √ µβGα ,w Rin ≤ µβ√ aN Rin ≤ c8 aN exp −c9 (β)M s(N ) log N , N N f K 1 \T N ,w 2m∗ which is o(1) as soon as M is large enough, provided that (3.3.1) holds. Therefore, using the decoupling properties [K, Pi] of the FK measures and the duality, X β µ3N ,w Gα ; Rin = µβ3N ,w CN ; Rin N N =

X α

α ∗

µβ3N ,w ( Gα ) µβGα ,w Rin N

=o

µβ3N ,w ( CN )

.

(3.4.5)

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151

Exactly in the same fashion one might define r

Rout N

=

There is an occupied dual FK chain across TN \

aN K1 ∪ Tf N 2m∗

,

and, with the obvious modifications in the definitions of the random sets D and G, obtain: β ( ) µβ3N ,w CN ; Rout . (3.4.6) µ o = C N N 3N ,w Since

Ef N ⊃

out Rin N ∪ RN

c

,

the estimates (3.4.5),(3.4.6) and (3.3.3) readily imply (3.4.1), and the proof of the lower bound (3.2.4) on the occurrence of the ± contour close to the required dilatation of the Wulff shape is, thereby, complete. It remains, therefore, to perform the second step of the proof of Theorem 3.1.1, which is to give a local limit type estimate on K,γ , (3.4.7) min PβN,− MN = −N 2 m∗ + aN | EN γ

K,γ K K where EN partitions EN according to what the single K log N -large contour γ in EN is. And then to optimize the combined estimate in terms of the large skeleton scale s(N ) chosen. Each ± contour γ in (3.4.7) splits 3N into the disjoint union of the inner component C and the outer component B; 3N = B ∪ C, and the following decomposition, which we repeatedly use throughout the paper is valid: K,γ PβN,− MN = −m∗ N 2 + aN | EN X log N = Pβ,K MB = −m∗ |B| + bN B,− bN +cN =aN −1(B,C) log N × Pβ,K C,−

MC = m∗ |C| + cN

,

where 1(B, C) = 2m∗ |C| + (m∗ + 1)|∂+ γ| + (m∗ − 1)|∂− γ|. p √ However, by definition, γ lies in the wN = M s(N ) log N tube around aN /2m∗ ∂K1 . Consequently, any realization of the inner domain C satisfies: p √ | aN − 2m∗ |C| | ≤ c10 aN wN ≤ c11 aN s(N ) log N . √ K , |∂C| ≤ 2K aN log N . Combined with the above Moreover, by the definition of EN bound on the volume of C, this, in view of (3.3.10), implies that C ∈ D√aN . Furthermore, since |∂+ γ| ∨ |∂− γ| ≤ 2|∂C|, p aN − 1(B, C) ≤ c12 aN s(N ) log N as well. As a result Gaussian estimates of the previous section ((2.3.2) and Lemma 2.3.3 whose hypothesis are satisfied thanks to (3.3.10)) apply, and we, thereby, obtain:

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K,γ min PβN,− MN = −N 2 m∗ + aN |EN γ aN s(N ) log N ≥ exp (−c13 s(N ) log N ) . ≥ exp −c13 N2 Together with (3.2.2) and (3.2.4) this yields the following bound: PβN,− MN = −m∗ N 2 + aN r √ aN aN + s(N )) . ω − c log N ( ≥ exp − 1 14 2m∗ s(N )

(3.4.8)

In view of the right-hand side of (3.4.8) above we see that the optimal scale s(N ) corresponds to the minimal available order of √ aN + s(N ), s(N ) √ which is attained at s(N ) ∼ 4 aN . Note that such an optimal choice is always compatible with the restrictions (3.3.4), (3.3.1) and (3.3.10) which we were implicitly assuming in the course of the proof. √ Substituting s(N ) ∼ 4 aN into the estimate (3.4.8), we arrive at the conclusion of Theorem 3.1.1. Remark 3.4.1. The estimate on the optimal scale could be further refined if we minimize a more exact expression √ aN aN s(N ) + . s(N ) N2 4. Subcritical Contours In this section we show that contours γ with 2

K log N ≤ diam(γ) N 3 do not appear with the PβA,− ( · |MA = MA + aN )-probability tending to one essentially in the whole range of A ∈ DN and aN ∈ MA . In the small moderate deviation case this assertion is part of the claim of Theorem C, which we prove in the first subsection. The case of large moderate deviations aN N 4/3+δ is studied in Subsect. 4.2. 4.1. Proof of Theorem C. So assume that δ ∈ (0, 4/3) is fixed, and aN N 4/3−δ . We write, log N (MA = MA + aN ) (1 + o(1)) PβA,− (MA = MA + aN ) = Pβ,K A,−

+ PβA,− (MA = MA + aN ; ∃ K log N − large contour) . By the second of inequalities (1.1.1), log N |MA − MK | = o(1), A

(4.1.1)

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153

as soon as K is sufficiently large, which we always assume. Thus, by virtue of the results on the moderate deviations in the K log N restricted phase, i.e., by Lemma 2.3.3, the first term in (4.1.1) equals 1 a2N β,K log N (MA = MA + aN ) = p (1 + o(1)) , (4.1.2) exp − PA,− 2χ|A| 2πχ|A| uniformly in all A and aN satisfying the conditions of the theorem (see Remark 2.3.4 after the statement of Lemma 2.3.3). Thus, the claim of the theorem follows once we show that the second term in (4.1.1) is negligible with respect to the above expression. Notice that the existence of a K log N -large contour implies that the set of K log N large skeletons is not empty. Thus, X β PN,− (· ; S) , PβN,− (· ; ∃ K log N -large contour) ≤ S6=∅

and the main step of the proof will be to derive upper bounds on PβN,− (MA = MA + aN ; S) for different collections S of K log N -large skeletons. The expression (4.1.2) provides a lower bound for PβA,− (MA = MA + aN ) and we shall repeatedly use it in order to rule out various improbable events. Specifically, by the energy estimate (1.3.2) there exists c1 < ∞, such that the contribution to the right-hand side of (4.1.1) of all K log N -collections S, which do not comply with the energy bound Wβ (S+ ) ≤

c1 a2N 1+ 2χ|A| K

(4.1.3)

is negligible. Remark 4.1.1. In particular, the second term on the right-hand side of (4.1.1) is always negligible whenever a2N /N 2 = O(1) (provided that K is large enough). Consequently, it remains to study the case of δ ∈ (0, 1/3) only. By the isoperimetric inequality, we are entitled to disregard any collection of K log N -large skeletons S, unless 2 1 Wβ (S+ ) aN N −3δ . (4.1.4) |S+ | ≤ ω1 Let now S be a collection of K log N -large skeletons which satisfies both energy and (hence) volume constraints above. As usual, each 0 ∼ S splits A into the disjoint union of the “−” and “+” components; A = B ∪ C. Using the corresponding decomposition MA = MB + MC and the flip symmetry of the PC measures, we estimate; PβA,− (MA = MA + aN ; S) ≤e−Wβ (S+ ) × log N log N × max N 2 M B = MK max Pβ,K + bN × B,− B 0∼S

bN +cN =aN −1(B,C)

log N log N ×Pβ,K M C = MK − cN , C,− C

(4.1.5)

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D. Ioffe, R. H. Schonmann

where the compensator 1(B, C) is given this time by log N log N − MK − MA + |∂+ 0| − |∂− 0|. 1 (B, C) = MK B C

As usual, by the very notion of skeletons and by the phasevolume estimate (1.1.3), both the area |C| and the boundary |∂C| of the microscopic “+” phase component are controlled in terms the restrictions (4.1.3) and (4.1.4) on the K log N -large skeleton collections S. Specifically, |∂C| ≤ c1 K log N |∂S+ | N log N, which means that the “−” component B belongs to DN . Thus, by Lemma 2.3.3, log N Pβ,K B,−

MB =

log N MK B

+ bN

b2 ≤ exp − N 2χ|B|

.

(4.1.6)

On the other hand, the area of the “+” component C is bounded above by |C| ≤ |S+ | + c2 K log N Wβ (S+ ) aN N −3δ ,

(4.1.7)

as it readily follows from (4.1.3), (4.1.4) and the choice of δ in the range δ ∈ (0, 1/3) (see Remark 4.1.1 above). Let us now inspect the right-hand side of (4.1.5) more closely: If cN ≤ 0, then bN ≥ aN − 1(A, B) which, by (4.1.6), implies log N log N log N log N Pβ,K + bN Pβ,K − cN M B = MK M C = MK B,− B C,− C (aN − 1(B, C))2 . (4.1.8) ≤ exp − 2χ|B| For cN outside the phase transition region; cN > 0, we use the Gaussian estimate (2.3.2): c2N log N K log N . = M − c (β) M ≤ exp −c Pβ,K C N 3 C,− C |C| In this case we shall also need the fact that for large N , since |B| ≤ |A| and |C| |A|, (bN + cN )2 bN cN c2 b2N + c3 N ≥ −2 2χ|B| |C| 2χ|A| χ|A| 2 (aN − 1(B, C)) (aN − 1(B, C))|C| ≥ −2 2χ|A| χ|A| 2 (aN ) aN |C| ≥ − c4 , 2χ|A| |A| where we have used Lemma 2.2.1 and (2.2.6) to bound 1(B, C) ≤ c5 |C| in the last inequality.

Dobrushin–Koteck´y–Shlosman Theorem up to Critical Temperature

155

Combined with (4.1.8) this gives log N K log N β,K log N K log N max M P M Pβ,K = M +b = M −c B N C N B,− B C,− C

bN +cN =aN −1(B,C)

(aN )2 aN |C| ≤ exp − + c4 . 2χ|A| |A|

(4.1.9)

By the estimates (4.1.4), (4.1.7) and (4.1.3), |C| ≤ c6 Wβ (S+ ) + K log N Wβ (S+ ) ≤ c7 Wβ (S+ ) ≤ c8

a2N . N2

Thus, |C| aN a3 aN |C| = Wβ (S+ ) ≤ c9 Wβ (S+ ) N4 N −3δ Wβ (S+ ) . |A| Wβ (S+ ) |A| N Consequently, the right-hand side of (4.1.9) is bounded above by exp −a2N /2χ|A| + N −3δ Wβ (S+ ) . Back to the decomposition (4.1.5), we obtain: PβA,− (MA = MA + aN ; S) ≤ N 2 e−Wβ (S+ )(1−N

−3δ

)

a2 . exp − N 2χ|A|

Comparing the expression on the right-hand side above with (4.1.2), we see that it can be further bounded above by log N (MA = MA + aN ) . c10 N 3 e− 2 Wβ (S+ ) Pβ,K A,− 1

However, for large enough values of K, X 1 N3 e− 2 Wβ (S+ ) = o(1), S6=∅

and the claim of Theorem C follows. 4.2. Large moderate deviations. Lemma 4.2.1. Assume that the value of aN ∈ MN is in the domain of large or large 4 moderate deviations; aN ≥ N 3 +δ . Then, for large enough K, ∀ > 0, 2 PβN,− ∃ exterior contour γ : diam(γ) ∈ (K log N, N 3 − ) | MN = −N 2 m∗ + aN = o(1).

(4.2.1)

Remark 4.2.2. By (1.1.1) exactly the sameconclusion is valid under the conditions of the lemma for PβN,− · | MN = MN + aN .

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Proof. With > 0 fixed we choose a large contour parameter s(N ) of the form s(N ) = N 3 −η , 2

η ∈ (0, ).

(4.2.2)

We are going to prove that with PβN,− · | MN = −m∗ N 2 + aN -probability tending to one all K log N -large exterior contours of σ are s(N ) large as well. So let 0(σ) be the collection of all s(N )-large exterior contours of σ. As usual we consider the 0-induced decomposition 3N = B ∪ C of 3N into respectively “−” and “+” components. By the lower bound (3.1.4) and by the energy estimate (1.3.2) applied on the K log N -scale we routinely restrict attention only to the case of collections of large contours 0 of admissible length; |0| ≤ RN log N so that B ∈ DN . Pick now a ξ > 0, such that 4

N 1+ξ N 3 .

(4.2.3)

Assume that |C| = Vol+ (0(σ)) satisfies, |

aN − Vol+ (0(σ))| N 1+ξ . 2m∗

(4.2.4)

Remark 4.2.3. In the case of large deviations; aN ∼ N 2 , (4.2.4) immediately implies that the “+” component C ∈ DN as well. Consequently, a straightforward modification of the proof below, which is entirely built upon the uniform estimates of Theorem C, enables one to drop the adjective “exterior” in the statement of Lemma 4.2.1. We claim that under assumption (4.2.4) with PβN,− ( · |MN = −N 2 m∗ + aN )probability close to one there are no K log N -large exterior contours other than those belonging to 0. Indeed, such contours can appear only inside B, and, as in the proof of Theorem C, we can write for events from B , PβN,− · ; MN = −N 2 m∗ + aN |0 X β s = Pβ,s B,− · ; MB = MB + bN PC,− (MC = MC − cN ) , (4.2.5) bN +cN =aN −1(B,C) where the corresponding value of the compensator 1(B, C) is given by 1 (B, C) = N 2 m∗ + MsB − MC + (m∗ + 1)|∂+ 0| + (m∗ − 1)|∂− 0|, and

1 (B, C) − 2m∗ |C| ≤ c1 (β)|∂C| ≤ c1 RN log N,

(4.2.6)

where the last two inequalities follow respectively from (1.1.1) and the admissibility of contour lengths |0| under consideration. Furthermore, for each bN ≥ 0, we have by the upper bound of Lemma 2.5.1, bN b2N β,s s PB,− MB = MB + bN ≤ exp −c1 2 ∧ . N s(N ) In particular, there exists ν small enough, such that 2ξ+ν s Pβ,s ≤ e−c2 N , B,− MB = MB + bN

Dobrushin–Koteck´y–Shlosman Theorem up to Critical Temperature

157

for each bN satisfying bN > N 4/3−ν . On the other hand, by the Chebyshev inequality, PβC,− |MC − MC | > N 1+ξ = o(1). Consequently, it follows from the decomposition (4.2.5) and the local limit estimate of Lemma 2.3.3, that for each s(N )-collection of contours 0 satisfying (4.2.4), PβN,− MN = −N 2 m∗ + aN | 0 ≥

1 2N 1+ξ ≥

min

bN ∈MB |bN |≤N 1+ξ

s Pβ,s B,− MB = MB + bN

c3 exp −c4 N 2ξ . N 1+ξ

Thus we conclude that for any such 0, PβN,− MB ≥ MsB + N 4/3−ν | 0; MN = −N 2 m∗ + aN = o(1), provided that ν = ν(ξ, η) > 0 is sufficiently small. At this stage we can evoke Theorem C and assert that in the remaining range of values bN ≤ N 4/3−ν , s Pβ,s = o(1) B,− ∃ K log N -large contour | MB = MB + bN uniformly in all families 0 of s(N )-large contours satisfying the volume condition (4.2.4) (and having the admissible length |0| ≤ RN log N ). It therefore suffices to verify the following statement: Lemma 4.2.4. Suppose that aN ∈ M+N is in the range of large moderate deviations; aN ≥ N 4/3+δ , δ > 0. Suppose also that the large contour parameter s(N ); log N s(N ) N 2/3 , and ξ ∈ (0, 1/3) satisfy: √ aN 2 N2 √ log N ∨ ∨ 4 aN log N ∨ s(N ). N 3 N 2ξ s(N ) aN Let 0 = 0(σ) denote the collection of exterior s(N )-large contours. Then for each ν > 0, a N 1+ξ 2 ∗ = o(1). (4.2.7) − Vol (0(σ))| > νN | M = −N m + a PβN,− | + N N 2m∗ Remark 4.2.5. Note that by choosing η in (4.2.2) small enough, we can apply this lemma in our setting above, and so complete the proof of Lemma 4.2.1 . Proof of Lemma 4.2.4. As before we employ the skeleton decomposition of the event 1

C = {|

aN − Vol+ (0(σ))| > νN 1+ξ } 2m∗

on the s(N )-scale, PβN,− (C) ≤

X S

PβN,− (C ; S) .

158

D. Ioffe, R. H. Schonmann

√ It happens to be convenient to fix a parameter rN aN , the precise range pof values for which we specify later on, andpto distinguish between high; Wβ (S+ ) ≥ ( aN /2m∗ − rN )ω1 , and low; Wβ (S+ ) ≤ ( aN /2m∗ − rN )ω1 , energy collections S. Let us start with the high energy case: There are at most N 2 terms in the sum on the right-hand side of (4.2.5). Thus, applying for each term in this sum either Lemma 2.5.1 inside the phase transition region or Gaussian estimates (2.3.2) outside, we infer that 2ξ N 1+ξ β 2 ∗ 2ξ ≤ e−c6 N , PN,− MN = −N m + aN | 0 ≤ exp −c5 N ∧ s(N ) for each 0 violating (4.2.4). Therefore, by the energy estimate (1.3.2), X aN PβN,− MN = −N 2 m∗ + aN ; S ; | − Vol+ (0(σ))| 2m∗ √a Wβ (S+ )≥(

N 2m∗

−rN )ω1

> νN 1+ξ

X

≤ Wβ (S+ )≥

√ aN

2m∗

max 0∼S

e−Wβ (S+ )

(4.2.8)

−rN ω1

PβN,− MN = −N 2 m∗ + aN | 0

aN 1+ξ | 2m ∗ −Vol+ (0)|>νN r c7 log N aN 2ξ ≤ exp − ) − c − r (1 − N ω . N 1 8 2m∗ s(N )

Our next step amounts to a careful choice of the parameters rN and ξ in (4.2.8): By the lower bound (3.1.4) the right-hand side of (4.2.8) is negligible with respect to PβN,− (MN = −N 2 m∗ + aN ), as soon as √ aN log N √ 2ξ ∨ 4 aN log N. (4.2.9) N rN ∨ s(N ) and s(N ), satisfying (4.2.3) and In other words, for any choice of parameters ξ, rN p (4.2.9), the occurrence of high energy Wβ (S+) ≥ ω1 ( aN /2m∗ − rN ) collections is ruled out modulo {MN = −N 2 m∗ + aN } once (4.2.4) is violated. As a result, it remains to show that X PβN,− √a Wβ (S+ )<

N 2m∗

−rN ω1 ∗

MN = −N m + aN 2

aN 1+ξ ; S; | − Vol (0(σ))| > νN + 2m∗

(4.2.10)

is also negligible with respect to PβN,− (MN = −N 2 m∗ + aN ). To this end, note, first of all, that by the isoperimetric inequality and (4.2.9), for each S, such that r aN − rN ω 1 , Wβ (S+ ) ≤ 2m∗

Dobrushin–Koteck´y–Shlosman Theorem up to Critical Temperature

the following bound on |S+ | takes place: 2 √ aN 1 Wβ (S+ ) ≤ − c 9 rN a N . |S+ | ≤ ω1 2m∗

159

(4.2.11)

Since on the s(N )-scale, |Vol+ (0(σ)) − |S+ || ≤ c10 Wβ (S+ )s(N ), whenever 0 and S are compatible, we conclude from (4.2.11) that also Vol+ (0(σ)) ≤

√ aN − c11 rN aN , ∗ 2m

(4.2.12)

provided that rN s(N ).

(4.2.13)

Now, the low droplet energy expression in (4.2.10) is bounded above by X PβN,− MN = −N 2 m∗ + aN ; S √ aN Wβ (S+ )<(

≤

2m∗

−rN )ω1

X

Wβ (S+ )<(

√ aN

2m∗

e−Wβ (S+ ) max PβN,− MN = −N 2 m∗ + aN | 0 .

−rN )ω1

0∼S

By the decomposition (4.2.5) each 0-term in this sum is, in its turn, bounded above by β s max Pβ,s N2 B,− MB = MB + bN PC,− (MC = MC − cN ) . bN +cN =aN −1(B,C)

A careful look at (4.2.12) and (4.2.6) reveals that this max is attained for positive values of cN , such that MC − cN is outside the phase transition region. In such a case one always has a Gaussian bound on the PβC,− -probabilities, and, using Lemma 2.5.1 to bound Pβ,s B,− -term, we conclude that the above expression does not exceed c2N b2N bN + ) max exp −c12 ( 2 ∧ bN +cN =aN −1(B,C) N s(N ) |C| ! (aN − 1(B, C))2 (aN − 1(B, C)) ≤ exp −c13 . ∧ N2 s(N ) Since for each low energy collection S, aN s(N )Wβ (S+ ), and since, in addition, 2 |S+ | ≤ Wβ (S+ )/ω1 , a crude estimate , based on (4.2.6) and (1.1.3), yields, a N aN − 1 (B, C) ≥ aN − 2m∗ |S+ | − c14 Wβ (S+ )s(N ) ≥ c15 ω12 − (Wβ (S))2 . ∗ 2m In order to simplify notation let us introduce 1

Wβ (aN ) =

r

aN ω1 . 2m∗

Subsequently, each S term in (4.2.10) is bounded above by

160

D. Ioffe, R. H. Schonmann

( exp −Wβ (S+ ) − c16

Wβ (aN )2 − Wβ (S+ )2 N2

2

) Wβ (aN )2 − Wβ (S+ )2 . ∧ s(N ) (4.2.14)

We proceed to study both terms in the wedge product in (4.2.14): For each S satisfying Wβ (S+ ) < Wβ (aN ) − rN ω1 , we obtain: 2 Wβ (aN )2 − Wβ (S+ )2 N2 (Wβ (aN ) − Wβ (S+ ))2 (Wβ (aN ) + Wβ (S+ ))2 = Wβ (aN ) + c16 N2 − Wβ (aN ) − Wβ (S+ ) r a N N ≥ Wβ (aN ) + rN c16 −1 , 2 N

Wβ (S+ ) + c16

which is bounded below by r

2 aN rN aN ω + c , 1 16 ∗ 2m N2

(4.2.15)

provided that rN

N2 . aN

(4.2.16)

Similarly, Wβ (S+ ) + c16

c16 Wβ (aN ) − 1 s(N ) √ rN a N . (4.2.17) > Wβ (aN ) + c17 s(N )

Wβ (aN )2 − Wβ (S+ )2 ≥ Wβ (aN ) + rN s(N )

Substituting final estimates obtained (4.2.15) and (4.2.17) into the upper bound (4.2.14), we infer that each S term in the decomposition (4.2.10) can be bounded above by r aN ω − r N 2 # S : Wβ (S+ ) ≤ 1 N 2m∗ r √ 2 aN rN a N rN a N . ω1 − c18 ∧ exp − 2m∗ N2 s(N ) However, the number of s(N ) collections, whose surface tension does not exceed some value r, is , by the usual skeleton computation, bounded above by # {S : Wβ (S+ ) ≤ r} ≤ ec19 r log N/s(N ) . √ By (4.2.9) there is always room to choose rN aN log N/s(N ). On the other hand, such a choice of rN leads to

Dobrushin–Koteck´y–Shlosman Theorem up to Critical Temperature

161

√ √ √ 2 r N a N rN aN rN a N aN aN ∧ ∧ log N. rN = rN s(N ) N2 s(N ) N2 s(N ) √ Indeed, aN /s(N ) 1 by the very choice of aN N 4/3 and s(N ) N 2/3 , whereas the condition rN aN /N 2 1 was explicitly stipulated in (4.2.16). Therefore, X PβN,− MN = −N 2 m∗ + aN ; S √ aN Wβ (S+ )<( 2m ∗ −rN )ω1 (4.2.18) r aN ω1 − c20 rN , exp − 2m∗ which is, by the lower bound (3.1.4), negligible with respect to PβN,− (MN = −N 2 m∗ + aN ), and the claim of the lemma follows. Note that the requirements (4.2.3), (4.2.9), (4.2.13) and (4.2.16) are consistent, provided that the hypothesis of the lemma are satisfied.

5. Proof of the Main Result The proof of Theorems A and B comprises a certain simple isoperimetric estimate, which we establish in the first subsection, and an implementation of this inequality based on the results we have obtained earlier. 5.1. An Isoperimetric Estimate. Lemma 5.1.1. Assume that,

r Wβ (S+ ) ≤

aN ω 1 + tN 2m∗

(5.1.1)

and aN − kN . 2m∗ Let S1+ be the largest connected component of S+ and S2+ be the rest; |S+ | ≥

(5.1.2)

S+ = S1+ ∪ S2+ . Then, the following bounds on the volume of S2+ and on the energy Wβ (S2+ ) hold: 2 kN tN 2 + , (5.1.3) |S+ | ≤ c1 aN √ aN aN and Wβ (S2+ )

√

≤ c2 aN

kN tN + √ aN aN

.

Moreover, if S1+ is in addition simply connected, then s ! r kN 2m∗ tN min dH ∂S1+ , x + ∂K1 ≤ c3 √ + . x aN aN aN

(5.1.4)

(5.1.5)

162

D. Ioffe, R. H. Schonmann

Proof. The interesting case is, of course, kN tN + 1, √ aN aN

(5.1.6)

which, by virtue of the isoperimetric inequality, means that the ratio 1

x =

|S2+ | 1 |S+ |

as well. By the isoperimetric inequality and (5.1.2), q q 1 2 |S+ | + |S+ | Wβ (S+ ) ≥ ω1 q ≥ |S+ | r ≥

s

|S2 | 1− + + |S+ |

aN − kN 2m∗

√

s

|S2+ | |S+ |

1−x+

! ω1

√ x ω1 .

(5.1.7)

Since for small values of x, √ √ 1√ 1−x + x≥1+ x, 2 we infer from (5.1.1) and (5.1.7) that r r 2m∗ kN 1√ aN aN ω + t ≥ ω x . 1 − 1 + 1 N 1 2m∗ 2m∗ aN 2 As a result,

s

|S2+ | √ = x ≤ c4 |S+ |

kN tN + √ aN aN

,

and, since by (5.1.1) and (5.1.6) |S+ | < aN /m∗ , (5.1.3) follows. The second claim (5.1.4) is an easy consequence. Indeed, by the isoperimetric inequality, (5.1.2) and (5.1.3), s q |S2+ | 1 1 |S+ | 1 − Wβ S+ ≥ |S+ |ω1 = |S+ | v u 2 ! u aN kN tN t ≥ − kN + ω1 1 − c5 √ 2m∗ aN aN r ≥

aN ω1 2m∗

r

kN 1 − c5 aN

s 1 − c6

kN tN + √ aN aN

2 .

Dobrushin–Koteck´y–Shlosman Theorem up to Critical Temperature

163

Since Wβ (S+ ) = Wβ (S1+ ) + Wβ (S2+ ), (5.1.4) follows, using also (5.1.1). Finally connected and simply connected S1+ satisfy the following generalization of the Bonnensen inequality([DKS], Sect. 9 of Chapter 2): s ! r q c7 1 1 1 − ∂S , x + ∂K W |S1+ |ω1 . ≤ S min dH √ 1 β + + 4 a x |S1+ | N Using assumption (5.1.1) to bound Wβ (S1+ ) from above and assumption (5.1.2) together with the estimate (5.1.3) to bound |S1+ | from below, we conclude that the right-hand side above is bounded by s s kN kN tN c7 tN + √ = c7 √ + , √ 4 a a a a N N N N which implies the stability estimate (5.1.4).

5.2. Proof of Theorem A . Let us reformulate results of Sects. 3 and 4 in the following way: Fix δ ∈ (0, 2/3) and let aN ∼ N 4/3+δ ∈ M+N . Then with the PβN,− ( · |MN = −N 2 m∗ + aN )-probability tending to one: 1. By Theorem 4.2.1 for large K and for each > 0 fixed no exterior contour γ with K log N < diam(γ) ≤ N 3 − 2

appears. 2. By the lower bound (3.1.4) and the energy estimate (1.3.2) if, √ s(N ) 4 aN , only s(N )-collections S with

r

Wβ (S) ≤

√ aN ω1 + c1 4 aN log N 2m∗

(5.2.1)

(5.2.2)

appear. 3. By Lemma 4.2.4 for any choice of ξ < 1/3 and a large contour parameter s(N ), such that log N s(N ) N 2/3 and √ aN log N N 2 √ 2/3 2ξ ∨ N ∨ 4 aN log N ∨ s(N ), (5.2.3) N s(N ) aN the family 0 of exterior s(N )-large contours of σ satisfies aN N 1+ξ . Vol+ (0(σ)) − 2m∗

(5.2.4)

Therefore, by the phase volume estimate (1.1.3), any 0-compatible collections of skeletons on the s(N )-scale satisfies |S+ | ≥

√ aN − N 1+ξ − c2 s(N ) aN . ∗ 2m

(5.2.5)

164

D. Ioffe, R. H. Schonmann

Inequalities (5.2.2) and (5.2.5) set up the stage for the application of the isoperimetric estimate of Lemma 5.1.1 with √ tN = c1 4 aN log N

√ and kN = c3 N 1+ξ ∨ s(N ) aN .

(5.2.6)

We, therefore, conclude from (5.1.4) that any connected exterior component S of S+ , with the exception of the largest one, obeys the following bound: diam(S) ≤ c4

√ 4

N 1+ξ aN log N + √ ∨ s(N ) . aN

(5.2.7)

We shall suppose that for some η > 0, s(N ) N 3 −η . 2

(5.2.8)

Since aN ∼ N 4/3+δ with δ ∈ (0, 2/3), it follows from (5.2.7) then that for ν > 0 small enough, also diam(S) N 3 −ν . 2

Consequently all exterior s(N )-large contours γ, with the exception of the largest one, 2/3−ν satisfy diam(γ) N 2/3−ν . On the other hand, by virtue of Lemma 4.2.1, N β ∗ 2 small contours appear with PN,− · | MN = −m N + aN -probability tending to zero. Thus, in view of (5.2.4), there is exactly one K log N -large exterior contour γ. As a result we conclude that with the PβN,− • | MN = −m∗ N 2 + aN -probability tending to one, the collection Sext of all exterior s(N )-large skeletons actually consists exactly of one skeleton Sext = {S}; S ∼ γ, where γ is the unique exterior s(N )-large contour. Though γ is, by the definition, self-avoiding, the boundary ∂S+ of the “+” phase component of Sext in general may not be. However, thanks to (5.1.4), the sum of the diameters of all possible small connected components of S+ is under control. Consequently, the Hausdorff distance between γ and ∂S1+ is bounded above as, dH ∂S1+ , γ

√

≤ 2s(N ) + c5 aN

kN tN + √ aN aN

.

Specifying the values of tN and kN as in (5.2.6), we obtain: dH ∂S1+ , γ

≤ c6

√ 4

N 1+ξ aN log N + √ ∨ s(N ) aN

.

(5.2.9)

On the other hand, the distance r

min dH x

∂S1+ ,

x+

aN ∂K1 2m∗

is already controlled by the stability estimate (5.1.5). Thus, substituting the values of tN and kN from (5.2.5) , we obtain from the stability estimate (5.1.5), the bound (5.2.9) on the Hausdorff distance between γ and ∂S+ and the triangle inequality,

Dobrushin–Koteck´y–Shlosman Theorem up to Critical Temperature

r min dH x

2m∗ γ, x + ∂K1 aN s

+ c8

!

≤ c7

log N s(N ) N 1+ξ +√ ∨ √ 4 a aN aN N

log N s(N ) N 1+ξ +√ ∨ ≤ c9 √ 4 a aN aN N

s

165

+

log N s(N ) N 1+ξ +√ ∨ . √ 4 a aN aN N

(5.2.10)

It is not difficult to optimize in the right-hand side of the inequality (5.2.10) within the range of parameters ξ and s(N ) described in (5.2.3), (5.2.1) and (5.2.8). The answer, however, does not have a nice compact form, and we, therefore, simply observe that no matter what the value of δ ∈ (0, 2/3) is, it is always possible to choose admissible ξ and s(N ), such that δ log N s(N ) N 1+ξ +√ ∨ N− 2 . √ 4 a aN aN N This implies the stability statement (1.2.2). Finally, by the lower bound (5.2.5) on the volume of S+ , and the energy estimate (1.3.2), PβN,− MN = −m∗ N 2 + aN √ aN 1+ξ ≤ PβN,− |S+ | ≥ − c N ∨ s(N ) aN (1 + o(1)) 10 2m∗ r aN N 1+ξ β ω1 − c11 √ ∨ s(N ) ≤ PN,− Wβ (S+ ) ≥ 2m∗ aN r √ aN aN N 1+ξ log N . (5.2.11) ω1 + c12 √ ∨ s(N ) ∨ ≤ exp − 2m∗ aN s(N ) It is, again, possible to optimize in the right-hand side above within the admissible range of ξ and s(N ), specified in (5.2.3), (5.2.1) and (5.2.8). Note, however, that for any admissible choice of ξ, 2 δ N 1+ξ N 3− 2 . √ aN √ On the other hand, the s(N ) ∨ aN log N/s(N ) term is clearly minimized on the admissible scale p √ p s(N ) ∼ 4 aN log N ∼ N 1/3+δ/4 log N . Therefore, whatever δ ∈ (0, 2/3) is, one can always find admissible ξ and s(N ), such that √ 2 δ aN N 1+ξ log N = O N 3 − 4 . ∨ s(N ) ∨ √ aN s(N ) Consequently, PβN,−

∗

MN = −m N + aN 2

Since by the lower bound (3.1.4),

r aN − 3δ 4 ) . 1 + O(N ≤ exp − 2m∗

166

D. Ioffe, R. H. Schonmann

PβN,−

∗

MN = −m N + aN

and for each δ ∈ (0, 2/3),

2

r log N aN ) , ≥ exp − 1 + O( √ 4 a 2m∗ N

log N N −δ/2 , √ 4 a N

the estimate (1.2.1) follows, and the proof of Theorem A is, thereby concluded. 5.3. Proof of Theorem B. The proof essentially amounts to a more careful look on the results we have already obtained: Let aN satisfy the conditions of Theorem B. First of all, as it was already mentioned in Remark 4.2.3, the results and techniques of Sects. 4 and 5, can be easily adjusted to ruleout all interior K log N contours as well, i.e., with PβN,− • | MN = −m∗ N 2 + aN probability tending to one there is exactly one K log N -large contour γ. Furthermore, substituting aN ∼ N 2 into various formulas of the previous subsection, we infer that for any choice of ξ and large contour parameter s (see (5.2.3), (5.2.1) and (5.2.8)), such that, N 2/3 N 2ξ

N log N ∨ N 1/2 log N ∨ s(N ), s(N )

1

s(N ) N 2 ,

s(N ) N 3 −η , (5.3.1)

for some η > 0, this unique K log N -large contour γ satisfies (see (5.2.10)), ! r r 2m∗ log N s(N ) ∨ N ξ min dH , γ, x + ∂K1 ≤ c1 + x aN N N 1/2

2

(5.3.2)

√ which is bounded above by c2 N −1/4 log N for the admissible choice of s(N ) ∼ N 1/2 log N . Finally, by the lower bound (3.1.4), r aN β ∗ 2 1/2 ω − c N log N , PN,− MN = −m N + aN ≥ exp − 1 3 2m∗ and, by the upper bound (5.2.11) written at aN ∼ N 2 , r N log N aN ξ . ω + c N ∨ s(N ) ∨ PβN,− MN = −m∗ N 2 + aN ≤ exp − 1 4 2m∗ s(N ) √ The optimal choice of s(N ) for the upper bound above is given by s(N ) ∼ N 1/2 log N , which is again an admissible value. Theorem B is completely proven. References [A]

Abraham, D.B.: Surface Structures and Phase Transitions Exact Results. Phase Transitions and Critical Phenomena Vol 10, C. Domb and J.L. Leibowitz, eds., London: 1987, 1–74 [ACCN] Aizenman, M., Chayes J.T., Chayes, L. and Newman, C.M.: Discontinuity of the magnetization in one-dimensional 1/|x − y|2 Ising and Potts models. JSP 50, 1, 1–40 (1988) [AA] Akutsu, N. and Akutsu, Y.: Relationship between the anisotropic interface tension, the scaled interface width and the equilibrium shape in two dimensions. J. Phys. A: Math. Gen. 2813–2820 (1986) [A1] Alexander, K.S.: Lower bounds on the connectivity function in all directions for the Bernoulli percolation in two and three dimensions. Ann. Prob. 18, 1547–1562 (1990)

Dobrushin–Koteck´y–Shlosman Theorem up to Critical Temperature

[A2]

167

Alexander, K.S.: Approximation of subadditive functions and convergence rates in limiting-shape results. Ann. Prob. 25, 1, 30–55 (1997) [A3] Alexander, K.S.: Power-law corrections to exponential decay of connectivities and correlations in lattice models. Preprint (1997) [ACC] Alexander, K.S., Chayes, J.T. and Chayes, L.: The Wulff construction and asymptotics of the finite cluster distribution for two-dimensional Bernoulli percolation. Commun. Math. Phys. 131, 1–50 (1997) [CGMS] Cesi, F., Guadagni, G., Martinelli, F. and Schonmann, R.H.: On the 2D stochastic Ising model in the phase coexistence region near the critical point. J. Stat. Phys. 85, 55–102 (1996) [CCS] Chayes, J.T., Chayes, L. and Schonmann, R.M.: Exponential decay of connectivities in the twodimensional Ising model. J. Stat. Phys. 49, 433–445 (1987) [DH] Dobrushin, R.L and Hryniv, O.: Fluctuations of the phase boundary in the 2D Ising ferromagnet. Commun. Math. Phys. 189, 395–445 (1997) [DKS] Dobrushin, R.L., Koteck´y, R. and Shlosman, S.: Wulff Construction: a Global Shape from Local Interaction. AMS translations series, Vol. 104, Providence R.I.: AMS, (1992) [DS] Dobrushin, R.L. and Shlosman, S.: Large and moderate deviations in the Ising model. Adv. in Soviet Math. 20, 91–220 (1994) [DT] Dobrushin, R.L and Tirozzi, B.: The central limit theorem and the problem of equivalence of ensembles. Commun. Math. Phys. 54, 173–192 (1977) [ES] Edwards, R.G. and Sokal, A.D.: Generalization of the Fortuin–Kasteleyn–Swendsen–Wang representation and Monte Carlo algorithm. Phys. Rev. D 38, 2009–2012 (1988) [I1] Ioffe, D.: Large deviations for the 2D Ising model: A lower bound without cluster expansions. J. Stat. Phys. 74, 411–432 (1994) [I2] Ioffe, D.: Exact large deviations bounds up to Tc for the Ising model in two dimensions. Prob. Th. Rel. Fields 102, 313–330 (1995) [I3] Ioffe, D.: Ornstein–Zernike behaviour and analyticity of shapes for self-avoiding walks on Zd . Preprint (1998) [K] Kesten, H.: Asymptotics in high dimensions for the Fortuin–Kasteleyn random cluster model. In: Progress in Probability Vol. 19, Boston–Basel: Birkh¨auser, 1991 [ML] Martin-L¨of, A.: Mixing properties, differentiability of the free energy and the central limit theorem for a pure phase in the Ising model at low temperature. Commun. Math. Phys. 32, 75–92 (1973) [MW] McCoy, B.M. and Wu, T.T.: The Two-Dimensional Ising Model. Cambridge MA: Harvard Univ. Press, 1973 [Pi] Pisztora, A.: Surface order large deviations for Ising, Potts and percolation models. Prob. Th. Rel. Fields 104, 427–466 (1996) [Pf] Pfister, C-E.: Large deviations and phase separation in the two-dimensional Isingmodel. Helv. Phys. Acta 64, 953–1054 (1991) [PV] Pfister, C-E.: and Velenik, Y.: Large deviations and continuum limit in the 2D Ising model. Commun. Math. Phys. 194, 389–462 (1998) [Sc] Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory, Cambridge: Cambridge Univ.Press, 1993 [SS1] Schonmann, R.H and Shlosman, S.: Constrained variational problem with applications to the Ising model. J. Stat. Phys. 83, 867–905 (1996) [SS2] Schonmann, R.H and Shlosman, S.: Wulff droplets and the metastable relaxation of kinetic Ising models. Preprint (1997) [SS3] Schonmann, R.H and Shlosman, S.: Complete analyticity for the 2D Ising completed. Commun. Math. Phys. 170, 453–482 (1997) [V] Velenik, Y.: PhD Thesis EPF-L. (1997) [W] Wulff, G.: Zur Frage der Geschwindigkeit des Wachstums und der Aufl¨osung der Kristallfl¨achen. Zeitschrift f¨ur Kristallographie 34, 449–530 (1901) Communicated by J. L. Lebowitz

Commun. Math. Phys. 199, 169 – 202 (1998)

Communications in

Mathematical Physics © Springer-Verlag 1998

Spectral Form Factors of Rectangle Billiards Jens Marklof?,?? Division de Physique Th´eorique, Institut de Physique Nucl´eaire, F-91406 Orsay Cedex, France, and Institut ´ des Hautes Etudes Scientifique, 35, Route de Chartres, F-91440 Bures-sur-Yvette, France. E-mail: [email protected] Received: 16 February 1998 / Accepted: 24 April 1998

Abstract: The Berry–Tabor conjecture asserts that local statistical measures of the eigenvalues λj of a “generic” integrable quantum system coincide with those of a Poisson process. We prove P that, in the case of a rectangle billiard with random ratio of sides, the sum N −1/2 j≤N exp(2πi λj τ ) behaves for τ random and N large like a random walk in the complex plane with a non-Gaussian limit distribution. The expectation value of the distribution is zero; its variance, which is essentially the average pair correlation function, is one, in accordance with the Berry–Tabor conjecture, but all higher moments (≥ 4) diverge. The proof of the existence of the limit distribution uses the mixing property of a dynamical system defined on a product of hyperbolic surfaces. The Berry–Tabor conjecture and the existence of the limit distribution for a fixed generic rectangle are related to an equidistribution conjecture for long horocycles on this product space. Contents 1 1.1 1.2 1.3 2 3 4 5 6 6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Billiards in a rectangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 The main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Basic definitions and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Invariance Properties of the Form Factor . . . . . . . . . . . . . . . . . . . . . . . . . 177 Regime I: |τ | λ−1/2− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Regime II: λ−1/2− |τ | λ−1/2+ . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Ergodicity, Mixing and Equidistribution . . . . . . . . . . . . . . . . . . . . . . . . . 184 Regime III: τ ∼ const . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 The expectation value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

? Supported by the European Post-Doctoral Institute for the Mathematical Sciences, the Engineering and Physical Sciences Research Council (Grant #GR K99015), the European Commission (TMR Marie Curie Grant), and the Basic Research Institute in the Mathematical Sciences, Hewlett-Packard Laboratories, Bristol. ?? Permanent address: School of Mathematics, University of Bristol, Bristol BS8 1TW, UK. (E-mail: [email protected])

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6.2 The limit distribution – smooth cut-off functions . . . . . . . . . . . . . . . . . . 192 6.3 The limit distribution – general cut-off functions . . . . . . . . . . . . . . . . . . 196 6.4 Random walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 7 Rational α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 7.1 The expectation value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 7.2 The limit distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 1. Introduction Let λ1 ≤ λ2 ≤ λ3 ≤ . . . → ∞ be a sequence of numbers satisfying #{j : λj ≤ λ} ∼ λ,

λ → ∞,

(1)

which means that the average spacing between adjacent levels is asymptotically unity. One quantity measuring the “randomness” of the deterministic sequence {λj }j is the consecutive level spacing distribution, which is defined by N 1 X δ(s − λj+1 + λj ), P (s, N ) = N

(2)

j=1

where δ(x) is the Dirac mass. The limit distribution of P (s, N ) for N → ∞ (if it exists) shall be denoted by P (s). That is, for any sufficiently nice test function h, Z ∞ Z ∞ P (s, N ) h(s) ds = P (s) h(s) ds. (3) lim N →∞

0

0

Berry and Tabor conjectured [5] that, when the sequence {λj }j is a sequence of eigenvalues of a quantum Hamiltonian, whose classical dynamics is integrable, then the limit distribution P (s) should in general coincide with the one for a random sequence generated by a Poisson process, i.e., PPoisson (s) = exp(−s). This is particularly interesting, because the limit distribution for systems, which are not integrable but chaotic, is expected to be the Gaudin distribution for the eigenvalues of random matrices [18], which is approximately described by Wigner’s surmise. For the GUE ensemble, say, it reads PGUE (s) ≈

32 2 − 4 s2 s e π . π2

Obvious examples for “non-generic” integrable systems, which violate the Berry– Tabor conjecture, are two-dimensional harmonic oscillators, as was already noted by Berry and Tabor [5] and later studied in more detail by Pandey et al. [49], Bleher [7, 8] and Greenman [32].1 Further negative examples are Zoll surfaces, where, like on the sphere, all geodesics are closed and have the same length. In the case of the sphere the eigenvalues of the (negative) Laplacian −1 are (after rescaling) El,m = l(l + 1), m = −l, . . . , l, hence 1 The spacings of two-dimensional harmonic oscillators are directly related to the spacings between the fractional parts of the sequence nθ, which had been studied earlier, see [60] for a survey.

Spectral form factors of rectangle billiards

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with multiplicity 2l + 1. Label these numbers in increasing order by λ1 , λ2 , . . . ; due to the high multiplicity one thus has P (s) = δ(s).

(4)

The same result holds for all other Zoll surfaces where the eigenvalues are extremely clustered around the values l(l + 1), compare Duistermaat and Guillemin [26], Weinstein [65], Colin de Verdi`ere [23].2 Results in favor of the Berry–Tabor conjecture are rare and can so far be only proved for the pair correlation density3 , R2 (s, N ) =

N 1 X δ(s − λj + λk ), N

(5)

j,k=1

which measures the spacings between all elements of the sequence and is therefore not a probability distribution. In the case of a random sequence from a Poisson process, R2 (s, N ) converges to the limiting density4 R2 Poisson (s) = δ(s) + 1, which is consequently the expectation for integrable systems. In other words, Z ∞ Z ∞ R2 (s, N ) h(s) ds = h(0) + h(s) ds, lim N →∞

−∞

(6)

−∞

for a suitable class of test functions h. This variant of the Berry–Tabor conjecture with respect to R2 (s) was verified by Sarnak [54] for the eigenvalues of the Laplacian on almost every flat torus (almost every with respect to Lebesgue measure in the moduli space of two-dimensional flat tori), but he simultaneously disproved the conjecture for a set of second Baire category5 . His result was recently extended to four-dimensional tori by VanderKam [63, 64]. Similar studies in this direction are due to Rudnick and Sarnak [52], whose results can be related to the eigenvalues of boxed oscillators, and Zelditch [66], who considers the level spacings for quantum maps in genus zero. It should be pointed out that, although the above results hold almost everywhere in the corresponding parameter spaces, the Berry–Tabor conjecture could not be proved for a specific example.6 For a more detailed up-to-date review on these topics see Sarnak’s lectures [55, 56]. The Berry–Tabor conjecture can only be expected to hold for local statistics such as P (s) or R2 (s), i.e. statistics which only measure the independence of eigenvalues on 2 The local correlations of the eigenvalues of each individual cluster are studied by Uribe and Zelditch [62]. 3 Sinai [59] and Major [45] showed that the statistics of lattice points in certain generic strips follow Poisson statistics in all moments. The boundary of these domains is not twice differentiable and looks like a trajectory for Brownian motion. Spectra of integrable systems like integrable geodesic flows are, however, related via EBK quantization to lattice points in domains with piecewise smooth boundary [22]. 4 The delta mass δ(s) is a result of our definition which counts spacings between equal elements whose spacing is trivially zero. The interesting part is the “1”. 5 Sets of second Baire category are all sets which are not of first Baire category, and the latter are sets, which are countable unions of nowhere dense sets, so pretty sparse in the topological sense. 6 During the completion of this manuscript I have learned from A. Eskin that it is possible to prove relation √ (6) for rectangle billiards with ratio of sides α1/2 and α diophantine (e.g. α = 2), see [31]. This remarkable result is consistent with Conjecture 1.2 (Sect. 1).

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the scale of the mean level spacing (which by virtue of (1) is unity and thus independent of N ). Non-local statistics like the number variance 62 (L) are well known to violate the Poisson prediction due to non-universal long-range correlations, see Berry [6] and Bleher and Lebowitz [16, 17]. A different type of non-local statistics is connected with the fluctuations of the energy-level counting function (“the spectral staircase”) around its mean value. It was shown that for certain integrable systems7 these fluctuations are non-Gaussian, even though the local statistics follow in general the Poisson prediction.8

Fig. 1. A walk of N = 16 unit steps in the complex plane

In the present article, we shall study similar non-local statistical properties, which are, however much closer linked to the local level spacing statistics. The central object of our investigation will be the spectral form factor K2 (τ, N ) (also called pair correlation form factor), which is defined as the Fourier transform of the pair correlation density, Z ∞ R2 (s, N ) e(τ s) ds, K2 (τ, N ) = −∞

Z R2 (s, N ) =

∞

−∞

K2 (τ, N ) e(−s τ ) dτ

with e(z) ≡ e2πi z , hence, K2 (τ, N ) = |N −1/2

N X

2

e(λj τ )| .

(7)

j=1

The sum N X

e(λj τ )

(8)

j=1

may be viewed as a walk in the complex plane (Fig. 1), consisting of N steps of unit length, whose direction is determined by the phases ξj = 2πλj τ . In the case when the process, the probability of finding the end point after N steps λj come from a Poisson √ outside a disk of radius N R (R is a fixed constant) has – by virtue of the classical 7 E.g. billiards in a rectangle and tori [34, 9, 10, 14, 13], Liouville surfaces [41, 15], other surfaces of revolution [11], and Zoll surfaces [57]. For a survey see [12]. 8 In contrast, for chaotic systems the fluctuations of the spectral staircase are conjectured to be Gaussian [4, 3], based mainly on numerical evidence.

Spectral form factors of rectangle billiards

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central limit theorem – in the limit N → ∞ a Gaussian limit distribution in the complex plane. This means in particular that Z ∞ Poisson {K2 (τ, N ) > R} = e−r dr = e−R . (9) lim Prob N →∞

R

If, however, the λj are given by a deterministic sequence, we can test the “independence” of the λj by considering the distribution of the sum (8), i.e. the distribution of endpoints of the corresponding walk, for different values of τ . That is, we throw τ at random with probability density ρ, and ask if, as above, lim Probρ {K2 ( · , N ) > R} = e−R .

N →∞

(10)

Clearly, the answer to this question may now also depend on correlations between the λj , which do not only appear on the scale of the mean level spacing but as well on scales in units of N γ , for some power 0 ≤ γ ≤ 1, say. That is the reason why we have classified this statistic as non-local. For τ random as above, relation (6) can be reformulated as lim Eρ K2 ( · , N ) = 1 + ρ(0),

N →∞

(11)

where Eρ denotes the expectation, and h = ρˆ and ρ are related by Fourier transformation, Z ∞ ρ(s) ˆ = ρ(τ ) e(s τ ) ds. −∞

The statistical properties of the form factor K2 (τ, N ) have received great attention in the quantum chaos literature, see e.g. [43, 2, 44, 27, 50, 1] and references therein. It is generally believed that the normalized fluctuations of the form factor for a chaotic system are of Gaussian nature [50, 42], cf. also footnote 28 in [2]. The situation for integrable systems seems to be more subtle. We shall see that, even though a generic rectangle billiard is likely to follow Poisson statistics locally [5, 21, 17], the fluctuations of the form factor are not Gaussian but have a limit distribution with algebraic-logarithmic tail.9 On the other hand, numerical studies of the circle billiard exhibit Gaussian fluctuations of the form factor [61]. If these are truly Gaussian for the circle or other generic integrable cases (e.g. Liouville surfaces), remains an interesting open problem. 1.1. Billiards in a rectangle. The quantum energies of a rectangle billiard are given by the eigenvalues of the negative Laplacian −1 = −

∂2 ∂2 − 2 2 ∂q1 ∂q2

with Dirichlet conditions on the boundary of the rectangle. The eigenvalues scale trivially with the area of the rectangle, so the only parameter which will enter the theory is the side ratio α1/2 , 0 < α ≤ 1. (The reason why we work with α rather than the side ratio itself will become apparent later.) 9 Similar deviations from a Gaussian distribution have been observed by Casati et al. [21], who calculated (numerically) the fluctuations of the difference P (s, N ) − PPoisson (s). Other non-local statistics for rectangle billiards , such as the number variance and the fluctuations of the spectral staircase have been considered e.g. by Casati et al. [20, 21], Berry [6], Bleher et al. [10, 14, 13, 16, 17].

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Taking the area to be 4π we have eigenvalues (α) Em,n =

π 1/2 2 (α m + α−1/2 n2 ), 4

m, n ∈ N,

(α) (α) which we label in increasing order and with multiplicity by 0 < λ(α) 1 < λ2 ≤ λ3 ≤ . . . . This sequence clearly satisfies (1), since the number of lattice points in an ellipse of area λ is asymptotically λ. More precisely, we have −1/2 1/4 (α + α−1/4 ) λ1/2 + Oγ (λγ ), #{j : λ(α) j ≤ λ} = λ − π

(12)

where the relation is conjectured to hold for any γ > 41 ; the best bound so far, γ > is due to Huxley [36].

23 73 ,

1.2. The main results. Let us view τ as a random variable, which is distributed on the compact interval I ⊂ R with a piecewise continuous probability density ρ, and α as a random variable with piecewise continuous probability density σ on the compact interval A ⊂ R+ − {0}. The expectation Eρ,σ for the random variable f (τ, α) is then defined as Z Z f (τ, α) ρ(τ ) σ(α) dτ dα, Eρ,σ f = I

A

and its probability to be greater than R by Z Z DR (τ, α) ρ(τ ) σ(α) dτ dα, Probρ,σ {f > R} = I

A

with the distribution function ( DR (τ, α) =

if f (τ, α) > R otherwise.

1 0

Theorem 1.1. There exists a decreasing function 9 : R+ → R+ with Z ∞ 9(R) dR = 1, 9(0) = 1, 0

9(R) ∼ c R−2 log R

(for R → ∞),

discontinuous only for at most countably many R, such that, lim Probρ,σ {K2 ( · , N ) > R} = 9(R),

N →∞

except possibly at the discontinuities of 9(R). The constant c is given by the expression Z Z 2 4 c= 6 | e(t2 u1 + t22 u2 ) d2 t| d2 u. π R2 t21 +t22 ≤1 1

(13)

Spectral form factors of rectangle billiards

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R∞ Remarks. (A) The condition 0 9(R) dR = 1 is consistent with a Gaussian limit distribution and thus consistent with the fact that the pair correlation function is the one for Poisson random numbers. (B) The condition 9(R) ∼ c R−2 log R implies R ∞ that the limit distribution is nonGaussian. In particular, the higher moments 0 Rk d9(R) (k > 2) diverge. This divergence has the following origin. Just as the moment k = 1 is related to the pair correlation function R2 (s, N ) (i.e. counting spacings), the moment k = 2 is related to the density 1 N2

N X

δ(s − λj1 − λj2 + λk1 + λk2 ),

(14)

j1 ,j2 ,k1 ,k2 =1

and thus to (with the above test function h) 1 N2

N X

h(λj1 + λj2 − λk1 − λk2 ),

(15)

j1 ,j2 ,k1 ,k2 =1

which amounts to counting the number of quadruples of eigenvalues such that λj1 + λj2 − λk1 − λk2 is in the interval [−a, a], say. In the case of a rectangle billiard, this number grows like (N log N )2 , due to number-theoretic degeneracies: √ recall that in this case we count essentially the integers m1 , . . . , m4 , n1 , . . . , n4 ≤ N with (m21 + αn21 ) + (m22 + αn22 ) − (m23 + αn23 ) − (m24 + αn24 )

(16)

in some interval. The number of solutions of n21 + n22 = n23 + n24 m21 + m22 = m23 + m24 , √ with m1 , . . . , m4 , n1 , . . . , n4 ≤ N grows like (N log N )2 , due to Landau’s classical result on the number of ways of writing an integer as a sum of two squares. Hence the number of solutions of (16) in any arbitrarily small interval [−, ] grows like (N log N )2 , hence the divergence of the second moment, at least for test functions h with h(0) 6= 0. But since h is the Fourier transform of a probability density, we in fact have h(0) = 1. This explains the divergence of the moments k = 2 and higher. Consequently, we cannot employ the method of moments to prove the above theorem. We shall instead use another approach based on the transformation formulas of theta functions, relating the existence of the limit distribution to the mixing property of certain flows, see Sect. 5 for details. This is essentially the same idea as in the proofs of the limit theorems of theta sums N X e(n2 x) n=1

in [46, 47, 48], which is now generalized to theta sums with more variables. The methods presented here could be further generalized to Siegel theta sums of arbitrary quadratic forms Q(ξ) of d variables, X e(Q(ξ) τ ). ξ∈Zd ∩3N

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Here, 3N denotes some suitable domain 3 ⊂ Rd , which is magnified by a factor of N . It would be interesting to see, in which cases the values of the above theta sums have a limit distribution for τ random. The expectation value of this limit distribution is related to the quantitative version of the Oppenheim conjecture for Q(ξ), cf. Eskin et al. [29, 30] and Borel’s survey [19]. (C) Theorem 1.1 implies that there cannot be a set of α of non-zero Lebesgue measure, for which (for fixed α and random τ ) Probρ {K2 ( · , N ) > R} converges to e−R . We instead have good reasons to believe the following to be true for those α, which are diophantine, i.e., badly approximable by rationals.10 Conjecture 1.2. Theorem 1.1 even holds when α is not random but fixed, as long as α is diophantine. That means in particular lim Probρ {K2(α) ( · , N ) > R} = 9(R),

N →∞

with the same function 9(R) as before. The truth of this conjecture is related to the equidistribution of horocycles in the product space 0\ PSL(2, R) × 0\ PSL(2, R) (for details see Sect. 5, in particular Conjecture 5.6, and Sect. 6). The conjecture is obviously false for rational α = pq . The following theorem follows from the results in [39, 46], see Sect. 7 for details. Theorem 1.3. Let ψ be piecewise continuous and of compact support. Then there exists p a decreasing function 9( q ) : R+ → R+ with p

9( q ) (0) = 1,

p

p

9( q ) (R) ∼ c( q ) R−1

(for R → ∞),

discontinuous for at most countably many R, such that, (p)

p

lim Probρ {K2 q ( · , λ) > R} = 9( q ) (R),

λ→∞

p

except possibly at the discontinuities of 9( q ) (R). p

p q

= 1 the constant c( q ) reads c(1) = 1/π. R∞ Remark. The fact that now 0 9(R) dR = ∞ diverges is due to the well known degeneracy of values of rational quadratic forms at integers (cf. previous remark B). In particular, one has for the pair correlation function (Proposition 7.1) In the special case α =

(p)

p

Eρ K2 q ( · , N ) ∼ b( q ) log N

(N → ∞).

(17)

For α = pq = 1 we have b(1) = 1/π. The logarithmic divergence resembles the average number of ways to write an integer as a sum of two squares, which is a classical result by Landau, cf. the previous Remark (B). As a consequence, the consecutive level spacing distribution is P (s) = δ(s) for all rational α. For more information on two-level statistics for the square billiard, cf. also Connors and Keating [24]. 10 The precise definition of a diophantine number will be given later (Sect. 5). The set of diophantine numbers is of full Lebesgue measure in R.

Spectral form factors of rectangle billiards

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1.3. Basic definitions and notations. The expressions x a y and x = Oa (y) both mean there exists a constant Ca (which may depend on some additional parameter a) such that |x| ≤ Ca |y|. The notation x = O(y −∞ ) is an abbreviation for x = OM (y −M ) for every M ≥ M0 , for some suitably large constant M0 . A piecewise continuous function is discontinuous only on a set of measure zero, and bounded on all compact sets. We denote by S(Rn ) the Schwartz class on Rn , i.e. the space of smooth functions f (t1 , . . . , tn ) which decrease rapidly when t21 + . . . + t2n → ∞. The same must hold for all derivatives of f . 2. Invariance Properties of the Form Factor The pair correlation density R2 (s, λ) and the form factor K2 (τ, λ) measure the statistics of levels in an energy window [0, λ]. For technical reasons it is convenient to smooth this window, i.e., choose a smooth cut-off function ψ ∈ C∞ (R+ ), which is rapidly decreasing at ∞ and consider the smoothed (in λ) pair correlation density R2,ψ (s, λ) =

λk 1 X λj ψ( ) ψ( ) δ(s − λj + λk ), λ λ λ

(18)

λj ,λk

and the smoothed form factor K2,ψ (τ, λ) = |λ−1/2

X

ψ(

λj

λj 2 ) e(λj τ )| . λ

(19)

In the case of the rectangle billiard the form factor has the explicit expression (α) (τ, λ) = |λ−1/2 K2,ψ

X

ψ(

π 1/2 2 m 4 (α

m,n∈N

+ α−1/2 n2 ) )× λ π 2 × e( (α1/2 m2 + α−1/2 n2 ) τ )| . 4

For symmetry reasons, we can write the sum as X X X = − − 4 m,n∈N

m,n∈Z

m=0,n∈Z

X m∈Z,n=0

+

X

(20)

.

m=0,n=0

The form factor can now be expressed in terms of the theta functions X f (n y 1/2 ) e(n2 x), 2f (z) = y 1/4

(21)

n∈Z

and 1/4 1/4

2f (z1 ; z2 ) = y1 y2

X

1/2

1/2

f (m y1 , n y2 ) e(m2 x1 + n2 x2 ),

m,n∈Z

where zj = xj + i yj is a complex variable. Setting f (t) = ψ(t2 ), and

f (t1 , t2 ) = ψ(t21 + t22 ),

(22)

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J. Marklof

z1 =

π 1/2 α (τ + i λ−1 ), 4

z2 =

π −1/2 α (τ + i λ−1 ), 4

we have (α) K2,ψ (τ, λ) =

1 π −1/2 |( ) 2f (z1 , z2 ) 16 4 π π − λ−1/4 ( α1/2 )−1/4 2f (z1 ) + ( α−1/2 )−1/4 2f (z2 ) 4 4 2 + λ−1/2 ψ(0)| .

(23)

It is intuitively clear that in the limit λ → ∞ the most important contributions should come from the two-variable theta sum. However, for special values of α and τ this need not be the case, but the set of exceptions is fortunately of measure zero. The following lemma follows from standard estimates on theta sums [33]. Lemma 2.1. For almost all α and τ (with respect to Lebesgue measure) and any > 0 we have (α) ˆ (α) (τ, λ) + Oα,τ, (λ−1/4+ ), (τ, λ) = K K2,ψ 2,ψ with

ˆ (α) (τ, λ) = 1 |2f (z1 , z2 )|2 . K 2,ψ 4π

The transformation formulas of theta functions (21) in one complex variable were the starting point of the studies in [46, 47, 48], and can be readily generalized to the two-variable sum 2f (z1 ; z2 ), by considering each variable separately. Before stating the transformation formulas, we have to introduce some geometry. Every element g in the Lie group PSL(2, R) = SL(2, R)/{±1} has a unique Iwasawa decomposition 1/2 0 cos φ − sin φ 1x y , g= sin φ cos φ 01 0 y −1/2 where z = x + i y is a point in the upper half plane H = {z = x + i y : x, y ∈ R, y > 0}, and φ ∈ [0, π) parametrizes the circle S1 , so the underlying manifold of PSL(2, R) may be identified with the manifold H × S1 . The invariant volume element (Haar measure) reads in this choice of coordinates dµ(g) =

dx dy dφ . y2

By virtue of the relation 1/2 ab 0 cos φ − sin φ 1x y cd sin φ cos φ 01 0 y −1/2 ! 1/2 0 cos φ0 − sin φ0 y0 1 x0 , = −1/2 0 1 sin φ0 cos φ0 0 y0 where x0 + i y 0 =

a(x + i y) + b , c(x + i y) + d

φ0 = φ + arg[c(x + i y) + d],

(24)

Spectral form factors of rectangle billiards

179

the action of PSL(2, R) on H × S1 is canonically given by g (z, φ) = gz, φ + arg(cz + d) ,

g=

ab cd

,

(25)

where g acts on z ∈ H by fractional linear transformations, i.e., gz =

az + b . cz + d

The theta group 0θ , which is generated by the elements 11 0 −1/2 and , 01 2 0 corresponding to the transformations 11 (z, φ) = z + 1, φ), 01

0 −1/2 2 0

(z, φ) = −

1 , φ + arg z), 4z

is an example of a discrete subgroup of PSL(2, R), such that the quotient manifold Mθ = 0θ \ PSL(2, R) = {0θ h : h ∈ PSL(2, R)} has finite volume µ(Mθ ) = π 2 , but is not compact (with respect to the measure introduced above). A fundamental region of 0θ in H is (Fig. 2) Fθ = {z ∈ H : |x| < 1/2, |z| > 1/2}.

(26)

There are two cusps which are represented by the points at ∞ and 21 (− 21 is equivalent to 21 ). It is well known that every rational point on the boundary Im z = 0 of H is 0θ -equivalent to one of the cusp points, i.e. there is always an element ab ∈ 0θ cd

y

x -1/2

0

1/2

Fig. 2. The fundamental region Fθ = {z ∈ H : |x| < 1/2, |z| > 1/2} in H for the theta group 0θ . The cusps are represented by the points at ∞ and 21 . The thin lines indicate the symmetries of the surface

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J. Marklof

such that a pq + b c pq + d

1 = ∞ or . 2

(27)

We first review some properties of one-variable theta sums, cf. [46, 47, 48]. For f ∈ S(R), define the theta function 2f as a function on H × [0, 4π) by 2f (z, φ) = y 1/4

X

fφ (n y 1/2 ) e(n2 x),

(28)

n∈Z

where Z fφ (t) =

R

Gφ (t, t0 ) f (t0 ) dt0 ,

(29)

with the harmonic oscillator Green function # 2 (t2 + t0 ) cos φ − 2t t0 , e sin φ "

0

Gφ (t, t ) = 2

1/2

e(−σφ /8)| sin φ|

−1/2

where σφ = 2k + 1 when kπ < φ < (k + 1)π with k ∈ Z. Proposition 2.2. Let f ∈ S(R). Then 2f (z, φ) is infinitely differentiable and satisfies the following functional relations: 2f (z + 1, φ) = 2f (z, φ), 2f (−

1 , φ + arg z) = e− iπ/4 2f (z, φ). 4z

The function |2f |2 may thus be viewed as an infinitely differentiable function on the manifold Mθ = 0θ \ PSL(2, R). The second relation implies 2f (z, φ + π) = − i 2f (z, φ). Since our theta function is smooth, it is bounded except in the cusps. In order to state the asymptotic behaviour of the function in the cusp at 21 , we introduce a set of new coordinates, (w, θ) = (−(4z − 2)−1 , φ + arg(z − 1/2)), in which the cusp at 21 is represented as a cusp at infinity. That is, v = Im w is the coordinate pointing into the cusp, and u = Re w the one orthogonal to it. Proposition 2.3. Let f ∈ S(R). Then ( 2f (z, φ) =

y 1/4 fφ (0) + O(y −∞ ) O(v −∞ )

(y > (v >

1 100 ) 1 100 ).

Spectral form factors of rectangle billiards

181

The ranges for which the above relations hold, cover the entire fundamental region {z ∈ H : 0 < x < 1, |z| > 1/2, |z − 1| > 1/2}. (This new region is obtained from the old one Fθ by shifting the left half by x 7→ x + 1.) Within these ranges, the relations are uniform in (z, φ) (1st rel.), (w, θ) (2d rel.). For f ∈ S(R2 ), define the theta function 2f as a function on H×[0, 4π)×H×[0, 4π) by X 1/4 1/4 1/2 1/2 fφ1 ,φ2 (m y1 , n y2 ) e(m2 x1 + n2 x2 ), (30) 2f (z1 , φ1 ; z2 , φ2 ) = y1 y2 m,n∈Z

where ZZ fφ1 ,φ2 (t1 , t2 ) =

R2

Gφ1 (t1 , t1 0 ) Gφ2 (t2 , t2 0 ) f (t1 0 , t2 0 ) dt1 0 dt2 0 ,

(31)

with the same harmonic oscillator Green function Gφ (t, t0 ) as before. It can be readily verified that fφ1 ,φ2 (t1 , t2 ) is again ∈ S(R2 ) (use partial integration in t1 and t2 ), so the sum defining 2f is rapidly convergent for every (φ1 , φ2 ). Proposition 2.4. Let f ∈ S(R2 ). Then 2f (z1 , φ1 ; z2 , φ2 ) is infinitely differentiable and satisfies the following functional relations: 2f (z1 + 1, φ1 ; z2 , φ2 ) = 2f (z1 , φ1 ; z2 , φ2 ), 2f (−

1 , φ1 + arg z1 ; z2 , φ2 ) = e− iπ/4 2f (z1 , φ1 ; z2 , φ2 ), 4z1 2f (z2 , φ2 ; z1 , φ1 ) = 2f (z1 , φ1 ; z2 , φ2 ).

The function |2f |2 may thus be viewed as an infinitely differentiable function on the manifold M2θ = 0θ \ PSL(2, R) × 0θ \ PSL(2, R). The second relation implies 2f (z1 , φ1 + π; z2 , φ2 ) = − i 2f (z1 , φ1 ; z2 , φ2 ). As mentioned above, the proof of the proposition follows exactly the lines of the analogous proposition for one-variable theta functions, compare e.g. [47]. As a fundamental region of 0θ × 0θ we choose the set F = Fθ × [0, π) × Fθ × [0, π).

(32)

The four cusps of codimension two are represented by (∞, φ1 ; z2 , φ2 ),

1 ( , φ1 ; z2 , φ2 ), 2

(z1 , φ1 ; ∞, φ2 ),

1 (z1 , φ1 ; , φ2 ); 2

every point of the form p ( , φ1 ; z2 , φ2 ), q

p (z1 , φ1 ; , φ2 ), q

is equivalent to one of those four, compare (27).

p ∈Q q

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J. Marklof

Proposition 2.5. Let f ∈ S(R2 ). Then 2f (z1 , φ1 ; z2 , φ2 ) =  (y1 y2 )1/4 fφ1 ,φ2 (0, 0) + O((y1 y2 )−∞ )    O((v v )−∞ ) 1 2 = 1/4 −∞  ) O(y  1 v2   −∞ 1/4 O(v1 y2 )

(y1 (v1 (y1 (v1

> > > >

1 100 , 1 100 , 1 100 , 1 100 ,

y2 v2 v2 y2

> > > >

1 100 ) 1 100 ) 1 100 ) 1 100 ).

These relations are uniform in (z1 , φ1 ; z2 , φ2 ) (1st rel.), (w1 , θ1 ; w2 , θ2 ) (2d rel.), (z1 , φ1 ; w2 , θ2 ) (3d rel.), (w1 , θ1 ; z2 , φ2 ) (4th rel.). The proof follows directly from Proposition 2 in [46]. The inner products and norms of the L2 spaces in question are defined as usual by ZZ p (f, g)L2 (R2 ) = f g d2 x, kf kL2 (R2 ) = (f, f )L2 (R2 ) (33) R2

and

ZZ (F, G)L2 (M2 ) = θ

F G d2 µ, M2θ

kF kL2 (M2 ) = θ

p (F, F )L2 (M2 ) . θ

(34)

By virtue of the last proposition, it is clear that |2f | is in L2 (M2θ ). If f is even, i.e. f (t1 , t2 ) = f (−t1 , t2 ) = f (t1 , −t2 ), we have the relation 1 k2f kL2 (M2 ) = kf kL2 (R2 ) , θ 2π 2

(35)

compare [46], Proposition 4. 3. Regime I: |τ | λ−1/2− We begin with the regime where |τ | ≤ CI λ−γ with γ > 21 , and CI an arbitrary constant. The behaviour of the form factor is entirely determined by the values of the theta function in the cusp at ∞, since ˆ (α) (τ, λ) = 1 |2f (z1 , 0; z2 , 0)|2 = 1 |2f (− 1 , arg z1 ; − 1 , arg z2 )|2 K 2,ψ 4π 4π 4z1 4z2 with − −

1 −τ + i λ−1 1 = , 4z1 πα1/2 τ 2 + λ−2

1 1 −τ + i λ−1 = , 4z2 πα−1/2 τ 2 + λ−2

arg z1 = arg(τ + i λ−1 ), arg z2 = arg(τ + i λ−1 ).

For λ → ∞, the imaginary part of −1/4zj becomes infinitely large, so Proposition 2.5 is applicable, yielding −1 ˆ (α) (τ, λ) = 1 |farg(τ +i λ−1 ),arg(τ +i λ−1 ) (0, 0)|2 λ + O(λ−∞ ) K 2,ψ 4π 2 τ 2 + λ−2

which holds uniformly for |τ | ≤ CI λ−γ . It is easy to see that

(36)

Spectral form factors of rectangle billiards

183

|farg(τ +i λ−1 ),arg(τ +i λ−1 ) (0, 0)|2 2

2

= 4λ (τ + λ

−2

ZZ 2

)|

e[(t21 + t22 )τ λ] f (t1 , t2 ) dt1 dt2 | Z ∞ 2 = 4λ2 (τ 2 + λ−2 ) |π e(rτ λ) ψ(r) dr| .

(37)

0

Therefore

Z ˆ (α) (τ, λ) = | K 2,ψ

∞

e(rτ λ) ψ(r) dr| λ + O(λ−∞ ) 2

(38)

0

for λ → ∞ uniformly in |τ | ≤ CI λ−γ . In this regime, the form factor looks therefore asymptotically as a delta mass at zero: upon applying a test function ρ we have Z

CI λ−γ −CI

λ−γ

Z ˆ (α) (τ, λ) dτ ∼ ρ(τ )K 2,ψ

Z

CI λ1−γ

∞

ρ(τ /λ)| −CI

λ1−γ

2

e(rτ ) ψ(r) dr| dτ, 0

and with ρ(τ /λ) ∼ ρ(0) for τ ∈ [−CI λ1−γ , CI λ1−γ ] the above reduces to Z ∞ Z ∞ Z ∞ 2 ∼ ρ(0) | e(rτ ) ψ(r) dr| dτ = ρ(0) ψ(r)2 dr −∞

0

0

by Parseval’s equality. In summary, Z

Z

CI λ−γ −CI λ−γ

(α) ρ(τ )K2,ψ (τ, λ) dτ

∼

CI λ−γ

−CI λ−γ

ˆ (α) (τ, λ) dτ ρ(τ )K 2,ψ Z ∼ ρ(0)

∞

ψ(r)2 dr.

(39)

0

Regime I is usually referred to as the saturation regime, since the number variance 62 (L) saturates in this regime, in contrast to the prediction for a Poisson process [6, 16]. 4. Regime II: λ−1/2− |τ | λ−1/2+ Regime II is defined as the region where CI λ−1/2− ≤ |τ | ≤ CII λ−1/2+ , with CI and CII arbitrary positive constants. From the discussion of Regime I it is clear that the form factor in Regime II can at most grow like K2,ψ (τ, λ) = O (λ2 ),

(40)

which is obtained from (38) for τ = λ−1/2− . The other τ -values in Regime II are bounded farther away from the cusp, so that the above indeed gives an upper bound. The most interesting part of Regime II is when CI λ−1/2 ≤ |τ | ≤ CII λ−1/2 . Let us put ω = τ λ1/2 . Then we have for large λ, − and

1 1 1 −ωλ−1/2 + i λ−1 = ∼ (−ω −1 λ1/2 + i ω −2 ) ±1/2 4zj πα ω 2 λ−1 + λ−2 πα±1/2

184

J. Marklof

arg zj = arg(ωλ−1/2 + i λ−1 ) → 0+ . Since the series defining 2f (z1 , φ1 ; z2 , φ2 ) converges uniformly for y1 and y2 in compacta, we finally obtain (1/α)

(α) ˆ (α) (τ, λ) ∼ K ˆ (τ, λ) ∼ K K2,ψ 2,ψ (− 2,ψ

4 λ1/2 4 1 , ) π2 ω π2 ω2

(λ → ∞)

(41)

uniformly for |ω| = |τ |λ1/2 ∈ [CI , CII ]. Before turning to Regime III (τ ∼ const.), where we shall study the fluctuations of the form factor around its mean, we will have to prove some facts about the equidistribution of certain sets in M2θ . 5. Ergodicity, Mixing and Equidistribution In this section we discuss some ergodic properties of the geodesic flow on hyperbolic surfaces of finite area, whose unit tangent bundle is represented by the quotient M = 0\ PSL(2, R), where 0 is a discrete subgroup, such as the theta group 0θ . The surface should only have a finite number of cusps (≥ 0), and we assume that, if there is at least one cusp, the half-plane coordinates are chosen in such a way that one of the cusps appears as the standard cusp of unit width at ∞. Consider the following three flows on M = 0\ PSL(2, R), defined by right translation, t/2 e 0 t t 8 = , (42) g 7→ g8 , 0 e−t/2 and g 7→ g9t± ,

9t+ =

1t 01

,

9t− =

10 t1

.

(43)

These flows actually represent the geodesic and the (positive and negative) horocycle flow on the unit tangent bundle of the surface 0\H, which can be identified with 0\ PSL(2, R) (the angle φ is in fact − 21 times the orientation angle of the tangent vector). These flows are well known to be ergodic and mixing [25]. The mixing property can be stated as follows. Proposition 5.1. Let F, G ∈ L2 (M). Then Z F (g) G(g8t ) dµ(g) = lim t→±∞

M

1 µ(M)

Z

Z F dµ

M

G dµ. M

The mixing property has an interesting consequence for the asymptotic distribution of long arcs of horocycles, which is stated in the following corollary. In fact, the investigation of measures concentrated along unstable fibers (which in our case are the horocycles) is a central issue in the theory of dynamical systems. Our proof will follow an idea of Eskin and McMullen [28], Theorem 7.1.; for related methods, cf. Kleinbock and Margulis [40], Sect. 2.2.1. and references therein.

Spectral form factors of rectangle billiards

185

Corollary 5.2. Let F be bounded and piecewise continuous on M, and h be piecewise continuous and of compact support on R. Then, for any g0 ∈ M, Z Z Z 1 u t lim h(u) F (g0 9+ 8 ) du = F dµ h du. t→−∞ R µ(M) M R Proof. Every element g = ac db ∈ SL(2, R) with d 6= 0 can be written as a product g = ±9u+ 8a 9b− .

(44)

Let F be continuous on PSL(2, R), left-invariant under 0 and compactly supported when viewed as a function on M = 0\ PSL(2, R). Let furthermore H be the function on PSL(2, R) defined for some fixed > 0 by ( h(u) 1 χ( a ) χ(b) for g = 9u+ 8a 9b− H(g) = 0 for g = ac 0b , where χ is the characteristic function of the interval [− 21 , 21 ]. Consider the integral Z Z H(g) F (g0 g8t ) dµ(g) = H(9u+ 8a 9b− ) F (g0 9u+ 8a 9b− 8t ) dµ. PSL(2,R) PSL(2,R) (45) The bi-invariant Haar measure in these coordinates is given by dµ = e−a da db du.

(46)

t

Step A. Now using the relation 9b− 8t = 8t 9b−e the integral transforms to Z t H(9u+ 8a 9b− ) F (g0 9u+ 8a+t 9b−e ) dµ.

(47)

PSL(2,R)

The distance (with respect to the invariant metric on PSL(2, R)) between the points t g0 9u+ 8a+t 9b−e and g0 9u+ 8a+t is b et , hence t

F (g0 9u+ 8a+t 9b−e ) = F (g0 9u+ 8a+t ) + O(b et ),

(48)

where the implied constant does not depend on u, a, b or t, for F is uniformly continuous. For −t large we thus have Z t H(9u+ 8a 9b− ) F (g0 9u+ 8a+t 9b−e ) dµ PSL(2,R) Z 1 a h(u) χ( ) F (g0 9u+ 8a+t ) e−a da du + O(et ). (49) = 2 R Step B. We can rewrite (45) as Z Z H(g0−1 g) F (g8t ) dµ(g) = PSL(2,R)

M

X γ∈0

H(g0−1 γg) F (g8t ) dµ(g)

(50)

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J. Marklof

since F and dµ are 0-left-invariant. P The latter, by the mixing property, has the limit (notice that the function G(g) = γ∈0 H(g0−1 γg) is 0-left-invariant) Z Z X 1 F dµ H(g0−1 γg) dµ(g) µ(M) M M γ∈0 Z Z 1 = F dµ H dµ µ(M) M PSL(2,R) Z Z 1 a 1 F dµ h(u) χ( ) e−a da du. (51) = µ(M) M 2 R Step C. Combining Steps A and B we conclude that Z h(u) χ(a) F (g0 9u+ 8a+t ) e−a da du lim t→−∞ R2 Z Z 1 F dµ h(u) χ(a) e−a da du. = µ(M) M R2 By the uniform continuity of F , given any δ > 0, we find an > 0 such that Z h(u) χ(a) F (g0 9u+ 8a+t ) e−a da du − δ R2 Z h(u) χ(a) F (g0 9u+ 8t ) da du < R2 Z h(u) χ(a) F (g0 9u+ 8a+t ) e−a da du + δ. <

(52)

(53)

R2

Since the limits t → −∞ on the left and right hand side exist for every fixed , and differ by 2δ, which can be made arbitrarily small, the limit Z Z Z 1 h(u) F (g0 9u+ 8t ) du = F dµ h(u) du (54) lim t→−∞ R µ(M) M R must exist as well. In order to relax the condition of compact support for F notice that the assertion trivially holds for F ≡ const. One can then again use the inclusion principle to relax both the compact support hypothesis and the continuity hypothesis. Remarks. (A) It is crucial that the limit is t → −∞. The limit t → +∞ diverges, since in this case the support of the measure converges towards a single limit point on the boundary of H. (B) For g0 = 1 the assertion of the corollary can be stated in the (z, φ)-coordinates as Z Z Z 1 h(x)F (z, 0) dx = F dµ h dx. (55) lim y→0 R µ(M) M R In the case when the average is taken over a closed horocycle and h ≡ 1, this was proved by Sarnak [53]. Hejhal [35] extended the result to h of the above class, but still for closed horocycles and for F which are independent of φ, i.e. functions on 0\H. His proof involves estimates on Poincar´e series of weight zero.

Spectral form factors of rectangle billiards

187

The observation of the previous corollary shall now be extended from averages over translates of horocycles to averages over translates of more general curves, which are given by the equation (in suitable coordinates) y = ef (x) , where f is a bounded real function. The case f ≡ const corresponds to horocycles. Corollary 5.3. Let F , h be as before, and let f : R → R be continuous on the support of h. Then, for any g0 ∈ M, Z Z Z 1 h(u) F (g0 9u+ 8t+f (u) ) du = F dµ h du. lim t→−∞ R µ(M) M R That is, for g0 = 1, Z h(x)F (x + i y ef (x) , 0) dx = lim y→0

R

1 µ(M)

Z

Z F dµ

M

h dx. R

Proof. Repeat the proof of the previous corollary with ( h(u) 1 χ( a−f (u) ) χ(b) for g = 9u+ 8a 9b− H(g) = 0 for g = ac 0b , which still has compact support, since f is bounded on the support of h. The only main difference to the previous proof is in Step C where we have instead Z a − f (u) 1 ) F (g0 9u+ 8a+t ) da du h(u) χ( R2 Z h(u) χ(a) F (g0 9u+ 8a+t+f (u) ) da du, (56) = R2

and the conclusion is the same as before, for f is bounded on the domain of integration. The next step is to define flows on the product space M2 = 0\ PSL(2, R) × 0\ PSL(2, R) by the diagonal action 0\ PSL(2, R) × 0\ PSL(2, R) → 0\ PSL(2, R) × 0\ PSL(2, R),

(57)

(g1 , g2 ) 7→ (g1 , g2 )8t := (g1 8t , g2 8t ) and (g1 , g2 ) 7→ (g1 , g2 )9t± := (g1 9t± , g2 9t± ). We shall still call these flows geodesic or horocyclic, respectively. Each one is the direct product of two mixing dynamical systems, and thus mixing itself [25]: Proposition 5.4. Let F, G ∈ L2 (M2 ). Then Z lim F (g1 , g2 ) G((g1 , g2 )8t ) d2 µ(g1 , g2 ) = t→±∞

M2

1 µ(M2 )

Z

Z F d2 µ M2

G d2 µ. M2

188

J. Marklof

The analogue of Corollaries 5.2 and 5.5 is the following: Corollary 5.5. Let F be bounded and piecewise continuous on M2 , h be piecewise continuous and of compact support on R2 , and f = (f1 , f2 ) : R2 → R2 continuous on the support of h. Then, for any (g1 , g2 ) ∈ M2 , Z lim h(u1 , u2 ) F (g1 9u+ 1 8t+f1 (u1 ,u2 ) , g2 9u+ 2 8t+f2 (u1 ,u2 ) ) d2 u t→−∞ R2 Z Z 1 2 F d µ h d2 u. = µ(M2 ) M2 R2 The proof of Corollary 5.2 obviously generalizes to this case. For (g1 , g2 ) = 1 we have Z h(x1 , x2 )F (x1 + i y ef1 (x1 ,x2 ) , 0; x2 + i y ef2 (x1 ,x2 ) , 0) d2 x lim y→0 R2 Z Z 1 2 F d µ h d2 x. (58) = µ(M2 ) M2 R2 There are reasons to believe that the above equidistribution results do not only hold for two-dimensional averages over (x1 , x2 ), but even for averages over one-dimensional lines given by x2 = ηx1 . In the case when the lattice group 0 is a congruence subgroup of PSL(2, R) (such as the theta group 0θ ), it has to be assumed, however, that η is badly approximable by rationals, i.e. η − p ≥ C q qκ for all rationals pq and some κ ≥ 2. Numbers η satisfying this condition are called diophantine. The reason for the necessity of this condition will become clearer in Sect. 7. Conjecture 5.6. Suppose M = 0\ PSL(2, R) with 0 a congruence subgroup of PSL(2, R). Let F be bounded and piecewise continuous on M2 , h be piecewise continuous and of compact support on R. If η is diophantine, we have for all y1 , y2 > 0, Z Z Z 1 2 lim h(x)F (x + i y1 y, 0; ηx + i y2 y, 0) dx = F d µ h dx. y→0 R µ(M2 ) M2 R The above observations on the equidistribution of certain sets will now be applied to understand the value distribution of the form factor in the most subtle Regime III, where the order of magnitude of τ is independent of λ. 6. Regime III: τ ∼ const 6.1. The expectation value. The following proposition is the analogue of Sarnak’s Proposition 2.1(A) on the weak convergence of the pair correlation function in the case of a two-dimensional family of tori [54]. Here, the weak convergence is with respect to our one-dimensional family of rectangles, parametrized by α. (Recall the definition of ρ and σ in Sect. 1.2.)

Spectral form factors of rectangle billiards

189

Proposition 6.1. Let ψ be piecewise continuous and of compact support. Assume furthermore ρ is continuous at 0. Then Z ∞ ρ,σ ψ(r)2 dr. lim E K2,ψ ( · , λ) = 1 + ρ(0) λ→∞

0

Proof. We have to estimate the integral ZZ π X (α1/2 m2 + α−1/2 n2 ) |λ−1/2 )× ψ( 4 λ m,n∈N

π 2 × e( (α1/2 m2 + α−1/2 n2 ) τ )| ρ(τ ) σ(α) dτ dα. 4

(59)

Let us first assume all functions involved are infinitely differentiable. For later purposes it is furthermore convenient to assume that the functions ρ and σ are not necessarily probability densities, i.e., have averages also different from one. This freedom will be needed when we approximate piecewise continuous probability densities from above/below by smooth ρ and σ. To be able to use the inclusion principle for ψ, we have to replace (59) by the slightly more general expression ZZ π π X (α1/2 m21 + α−1/2 n21 ) (α1/2 m22 + α−1/2 n22 ) 1 ) ψ2 ( 4 )× ψ1 ( 4 λ λ λ m1 ,n1 ,m2 ,n2 ∈N π × e (α1/2 (m21 − m22 ) + α−1/2 (n21 − n22 )) τ ρ(τ ) σ(α) dτ dα. (60) 4 We get back to (59) by setting ψ1 = ψ2 = ψ. In the sequel, ψ1 and ψ2 are taken to be C∞ with compact support. Let us split the domain of integration over τ into the three parts corresponding to Regime I, Regime II and Regime III, that is, Z

Z

∞

Z

CI λ−1/2−

=

+

∞

+ CI λ−1/2−

0

0

Z

CII λ−1/2+

CII λ−1/2+

and similarly for the integral over the negative axis. The first integral is similar to the one calculated in Sect. 3 (where ψ1 = ψ2 = ψ), see (39), and we have for large λ, Z

CI λ−1/2− −CI λ−1/2−

Z ∼ ρ(0)

Z

∞

ψ1 (r) ψ2 (r)dr

σ(α) dα.

(61)

0

This result can be obtained most easily by replacing the modulus squared of the theta function, |2f (z1 , φ1 ; z2 , φ2 )|2 , by the product 2f1 (z1 , φ1 ; z2 , φ2 ) 2f2 (z1 , φ1 ; z2 , φ2 ),

(62)

with fν (t1 , t2 ) = ψν (t21 + t22 ). The function in (62) can still be viewed as a function on M2θ , cf. Proposition 2.4. The second integral vanishes to leading order, due to the bound analogous to bound (40) valid in Regime II (Sect. 4), so Z

Z

−CI λ−1/2−

CII λ−1/2+

+ −CII

λ−1/2+

CI

λ−1/2−

λ2 (CII λ−1/2+ − CI λ−1/2− ) λ−1/2+3 . (63)

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J. Marklof

We are left with the third integral, where |τ | ≥ CII λ−1/2+ , which we split into two parts, the diagonal part 1 λ

ZZ

X |τ |≥CII λ−1/2+ m,n∈N

π 1/2 2 m 4 (α

ψ1 (

× ψ2 (

+ α−1/2 n2 ) )× λ

π 1/2 2 m 4 (α

+ α−1/2 n2 ) ) ρ(τ ) σ(α) dτ dα, λ

(64)

which clearly converges for λ → ∞ to the desired expression11 Z

Z

∞

Z

ψ1 (r) ψ2 (r) dr

ρ(τ ) dτ

σ(α) dα,

0

and the off-diagonal part, ZZ X 1 × λ |τ |≥CII λ−1/2+ 2 (m1 ,n1 )6=(m2 ,n2 )∈N

× ψ1 (

π 1/2 2 m1 4 (α

π (α1/2 m22 + α−1/2 n22 ) + α−1/2 n21 ) ) ψ2 ( 4 )× λ λ

π × e( (α1/2 (m21 − m22 ) + α−1/2 (n21 − n22 )) τ ) ρ(τ ) σ(α) dτ dα. 4 Substituting x1 = 1 λ

π 1/2 τ, 4α

x2 =

π −1/2 τ 4α

(65)

the last expression equals

ZZ

X (m1 ,n1 )6=(m2 ,n2 )∈N2

| π2

q

π × ψ1 ( 4

√

x1 x2 |≥CII λ−1/2+

×

q q x1 2 x2 2 π x2 m2 + x1 n2 ) ψ2 ( )× λ 4 λ × e(x1 (m21 − m22 ) + x2 (n21 − n22 )) h(x1 , x2 ) dx1 dx2 ,

x1 2 x2 m1

q

+

x2 2 x 1 n1

(66)

where h is related to ρ, σ and the Jacobian of the substitution in the obvious way, h(x1 , x2 ) =

4 π

r

x1 x1 −1 4 √ x2 ρ( x1 x2 ) σ( ). x2 π x2

(67)

Let us divide (66) into the sums X (m1 ,n1 )6=(m2 ,n2 )∈N2

=

X m1 6=m2 ,n1 6=n2 ∈N

+

X m1 6=m2 ,n1 =n2 ∈N

+

X

.

m1 =m2 ,n1 6=n2 ∈N

Partial integration with respect to x1 and x2 shows the first sum diverges only logarithmically in λ, and, after some manipulations, this part of (66) turns out to be bounded by (use partial integration in x1 , x2 , and then re-substitute the old variables τ and α) 11

Notice that the sum over m, n converges to the corresponding Riemann integral.

Spectral form factors of rectangle billiards

1 λ

ZZ

X √

m1 6=m2 ,n1 6=n2 λ

|τ |≥CII λ

Z

1 λ1−0

191

|τ |≥λ−1/2+

−1/2+

ρ(τ ) σ(α) dτ dα τ 2 |m21 − m22 | |n21 − n22 |

0 ρ(τ ) dτ λ−1/2−+ τ2

Z |τ |≥1

ρ(τ λ−1/2+ ) dτ τ2 λ−1/2−+

0

(68)

for any 0 > 0. The remaining two sums can be shown to be of non-leading order by similar means. We have thus shown so far that for λ → ∞, 1 λ

Z

X

π 1/2 2 m1 4 (α

ψ1 (

(m1 ,n1 ),(m2 ,n2 )∈N2

π + α−1/2 n21 ) (α1/2 m22 + α−1/2 n22 ) ) ψ2 ( 4 )× λ λ

π × ρ( ˆ (α1/2 (m21 − m22 ) + α−1/2 (n21 − n22 ))) σ(α) dα 4 Z Z Z −→

ψ1 (r)ψ2 (r) dr

Taking finite linear combinations H(r1 , r2 ) = above type, we have 1 λ

Z

X

H(

π 1/2 2 m1 4 (α

(m1 ,n1 ),(m2 ,n2 )∈N2

ρ(0) + P

ρ(τ ) dτ

σ(α) dα.

(69)

ψj1 (r1 ) ψj2 (r2 ) of functions of the

+ α−1/2 n21 ) , λ

π 1/2 2 m2 4 (α

+ α−1/2 n22 ) )× λ

π × ρ( ˆ (α1/2 (m21 − m22 ) + α−1/2 (n21 − n22 ))) σ(α) dα 4 Z Z Z −→

H(r, r) dr

ρ(0) +

ρ(τ ) dτ

σ(α) dα.

(70)

Let us now see how the smoothness condition on ψ can be relaxed to piecewise continuous ψ with compact support. Following the lines of the proof of Theorem 3.2 in [51], ψ(r1 )ψ(r2 ) can now be approximated from above/below by smooth test functions H(r1 , r2 ), which are admissible for the previous derivation and for which Z (71) |H(r, r) − ψ(r)2 | dr < . By the inclusion principle the statement of the proposition holds thus for piecewise continuous ψ. The result can be further extended to piecewise continuous ρ and σ by approximating both sides of the relation Z ∞ ρ,σ ψ(r)2 dr lim E K2,ψ ( · , λ) = 1 + ρ(0) λ→∞

0

from above and below using smooth functions ρ and σ , which are -close12 to ρ and σ, respectively. 12 In the L1 sense but also in the sense that |ρ(0) − ρ (0)| < , which is where the continuity of ρ at 0 is required.

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J. Marklof

6.2. The limit distribution – smooth cut-off functions. Let us first consider the simpler case when ψ is a smooth cut-off function. The results will later be extended to more general cut-off functions. We denote by S(R+ ) the space of even functions in S(R), restricted to the positive half line R+ . Theorem 6.2. Let ψ ∈ S(R+ ). Then there exists a decreasing function 9 : R+ → R+ with Z ∞ Z ∞ 9ψ (0) = 1,

9ψ (R) dR =

0

ψ(r)2 dr,

0

discontinuous only for at most countably many R, such that, lim Probρ,σ {K2,ψ ( · , λ) > R} = 9ψ (R),

λ→∞

except possibly at the discontinuities of 9ψ (R). Theorem 6.3. For R → ∞, we have the asymptotic relation 9ψ (R) = cψ R−2 log R + dψ R−2 + Oψ (R−∞ ), where cψ =

2 π6

Z

Z R2

|

4

R2

e(t21 u1 + t22 u2 ) ψ(t21 + t22 ) d2 t| d2 u.

The constant dψ is given by a more complicated expression, see the proof of this theorem. The proofs of these theorems will be given below. The following conjecture is a direct consequence of the equidistribution hypothesis of Conjecture 5.6, compare the proof of Theorem 6.2. Conjecture 6.4. Theorem 6.2 even holds when α is not random but fixed, as long as α is diophantine. That means in particular (α) ( · , λ) > R} = 9ψ (R), lim Probρ {K2,ψ

λ→∞

with the same function 9ψ (R) as before. This conjecture does not hold when α is rational or well approximable by rationals, as we shall see in the next section. Proof of Theorem 6.2. Apply Corollary 5.5, i.e. relation (58), where χD is taken to be the characteristic function of the set D = {(z1 , φ1 ; z2 , φ2 ) ∈ M2θ :

1 |2f (z1 , φ1 ; z2 , φ2 )|2 > R}. 4π

The boundary of D can only have positive measure for a countable number of R, otherwise this would be a contradiction to the measurability of 2f . Hence χD is piecewise continuous except for countably many R. We chose the function h(x1 , x2 ) in a way that h(x1 , x2 ) χD (x1 + i y ef1 (x1 ,x2 ) , 0; x2 + i y ef2 (x1 ,x2 ) , 0) d2 x = DR (τ, α, λ) ρ(τ ) σ(α) dτ dα, where

(72)

Spectral form factors of rectangle billiards

x1 =

193

π 1/2 α τ, 4

x2 = r

π y = λ−1 , 4

f1 (x1 , x2 ) = log

and

( DR (τ, α, λ) =

1 0

x1 , x2

π −1/2 α τ, 4 r f2 (x1 , x2 ) = log

x2 , x1

(α) (τ, λ) > R if K2,ψ otherwise.

Since the compact interval A does not contain 0, the functions f1 and f2 are bounded on the domain of integration. Hence, except for at most countably many R, ˆ 2,ψ ( · , λ) > R} = 9ψ (R). lim Probρ,σ {K

λ→∞

(73)

To obtain the same result for K2,ψ , recall that by virtue of (23) we can express the ˆ 2,ψ as the sum difference between K2,ψ and K λ−1/4 F1 (z1 , φ1 ; z2 , φ2 ) + λ−1/2 F2 (z1 , φ1 ; z2 , φ2 ) + λ−3/4 F3 (z1 , φ1 ; z2 , φ2 ) + λ−1 ψ(0)2 , where the Fj are sums of products of theta functions and their modulus is therefore majorized by functions on M2θ ; it is therefore clear that ˆ 2,ψ ( · , λ)| > } = 0 lim Probρ,σ {|K2,ψ ( · , λ) − K

λ→∞

(74)

for arbitrarily small (but fixed) > 0. Hence for any given > 0, we have ˆ 2,ψ ( · , λ) > R + } Probρ,σ {K ≤ Probρ,σ {K2,ψ ( · , λ) > R} ˆ 2,ψ ( · , λ) > R − }, ≤ Probρ,σ {K

(75)

for every large enough λ. The limits 9ψ (R + ) and 9ψ (R − ) of the left- and right-hand side exist except for countably many R, . Since {(z1 , φ1 ; z2 , φ2 ) ∈ M2θ :

1 |2f (z1 , φ1 ; z2 , φ2 )|2 = R} 4π

has positive measure only for countably many R, the difference 9ψ (R − ) − 9ψ (R + ) can be made arbitrarily small for suitable small > 0, except for countably many R. Finally, the relation 9ψ (0) = 1 holds by definition,13 and the integral of 9ψ can be calculated as follows (notice that µ(M2θ ) = π 4 ): 13 It might happen that 9 () ≤ C < 1 for arbitrarily small , but since we allow 9 (R) to be discontinψ ψ uous at countably many R, it is most sensible to normalize 9ψ (0) = 1.

194

J. Marklof

Z

∞ 0

9ψ (R) dR Z ∞ 1 1 |2f (z1 , φ1 ; z2 , φ2 )|2 > R} dR µ{(z1 , φ1 ; z2 , φ2 ) ∈ F : = 4π µ(M2θ ) 0 Z Z Z ∞ 1 1 2 2 2 2 = |2 | d µ = |f (t , t )| d t = ψ(r)2 dr, (76) f 1 2 4π 5 M2θ π R2 0

by virtue of relation (35).

Proof of Theorem 6.3. It follows from the proof of Theorem 6.2 that for all but countably many R (recall the definition of the fundamental region F in (32)) 1 1 |2f (z1 , φ1 ; z2 , φ2 )|2 > R} µ{(z1 , φ1 ; z2 , φ2 ) ∈ F : 4 π 4π 1 2 |2f (z1 , φ1 ; z2 , φ2 )|2 > R}. = 4 µ{(z1 , φ1 ; z2 , φ2 ) ∈ F : y1 > y2 and π 4π

9ψ (R) =

(77)

The asymptotics of 9ψ (R) is clearly determined by the large values of the theta function 2f (z1 , φ1 ; z2 , φ2 ) in the cusps. By Proposition 2.5 and the refined asymptotic relation 2f (z1 , φ1 ; z2 , φ2 ) = y1 2f (φ1 ; z2 , φ2 ) + O(y1−∞ ) 1/4

with the theta function 1/4

2f (φ1 ; z2 , φ2 ) = y2

X

(78)

1/2

fφ1 ,φ2 (0, n y2 ) e(n2 x2 ),

n∈Z

we thus have 2 µ{(z1 , φ1 ; z2 , φ2 ) ∈ F : y1 > y2 , π4 1 1/2 y1 |2f (φ1 ; z2 , φ2 )|2 > R} + O(R−∞ ) y1 > 10 and 4π Z πZ Z π 1 1 2 dx2 dy2 dφ2 min { , , (4πR)−2 |2f (φ1 ; z2 , φ2 )|4 }dφ1 = 4 π 0 Fθ 0 y2 10 y22

9ψ (R) =

+ O(R−∞ ).

(79)

Let us first discuss the integral over the range 1 < (4πR)−2 |2f (φ1 ; z2 , φ2 )|4 , y2

i.e. (4πR)2 < y2 |2f (φ1 ; z2 , φ2 )|4 ,

which, for large R, requires y2 to be large as well. From the asymptotic relation ( 1/4 1 ) y2 fφ1 ,φ2 (0, 0) + O(y2−∞ ) (y2 > 100 2f (φ1 ; z2 , φ2 ) = −∞ 1 O(v2 ) (v2 > 100 ),

(80)

compare Proposition 2.5, it follows that the integral over the range in concern is bounded from above and below by the same integral over the ranges y2 |fφ1 ,φ2 (0, 0)|2 > 4πR ∓ CM R−M (for any M > 2 and a suitable constant CM ) which now can be worked out to give, for R large enough,

Spectral form factors of rectangle billiards

2 π4

Z

π 0

Z

π

Z

195

Z

∞

1

1 1 dx2 dy2 dφ2 , } dφ1 y2 10 y22 0 (4πR∓CM R−M )/|fφ1 ,φ2 (0,0)|2 0 Z πZ πZ ∞ Z 1 2 dx2 dy2 dφ2 = 4 dφ1 π 0 0 (4πR∓CM R−M )/|fφ1 ,φ2 (0,0)|2 0 y23 Z πZ π 1 = 4 (4πR)−2 |fφ1 ,φ2 (0, 0)|4 dφ1 dφ2 + OM (R−M ). π 0 0 min {

(81)

As to the remaining range (4πR)2 > y2 |2f (φ1 ; z2 , φ2 )|4 , the same reasoning as before permits to give the upper and lower bounds Z πZ πZ 2 × π 4 0 0 Fθ ((4πR±CM R−M )/|fφ1 ,φ2 (0,0)|2 ) × min {

1 dx2 dy2 dφ2 , (4πR)−2 |2f (φ1 ; z2 , φ2 )|4 } dφ1 10 y22

(82)

with the truncated fundamental region Fθ (T ) = {z ∈ Fθ : y < T }. 1 < (4πR)−2 |2f (φ1 ; z2 , φ2 )|4 , equals The above integral, restricted to the range 10 Z πZ πZ dy2 2 dφ2 dφ1 + O(R−∞ ) = O(R−∞ ), (83) 10π 4 0 0 (∗) y22

where the range (∗) of integration of the inner integral is 4πR (4πR)2 ≤ y2 ≤ . 10 |fφ1 ,φ2 (0, 0)|4 |fφ1 ,φ2 (0, 0)|2 1 > (4πR)−2 |2f (φ1 ; z2 , φ2 )|4 let us In order to work out the integral over the range 10 define the truncated function ( |2f (φ1 ; z2 , φ2 )|4 − y2 |fφ1 ,φ2 (0, 0)|4 for y2 > 1 (84) Hf (φ1 ; z2 , φ2 ) = |2f (φ1 ; z2 , φ2 )|4 otherwise,

which is rapidly decreasing in all cusps. The integral over the range under consideration can now be expressed as Z πZ πZ dx2 dy2 dφ2 1 Hf (φ1 ; z2 , φ2 ) dφ1 8π 6 R2 0 0 Fθ y22 Z π Z π Z 4πR/|fφ ,φ (0,0)|2 1 2 1 dy2 + 6 2 y2 |fφ1 ,φ2 (0, 0)|4 2 dφ2 dφ1 8π R 0 0 1 y2 + O(R−∞ ). The second integral yields Z πZ π 1 log R |fφ1 ,φ2 (0, 0)|4 dφ1 dφ2 8π 6 R2 0 0 Z πZ π 1 + 6 2 |fφ1 ,φ2 (0, 0)|4 log 4π/|fφ1 ,φ2 (0, 0)|2 ) dφ1 dφ2 . 8π R 0 0

(85)

(86)

196

J. Marklof

Collecting all leading-order terms in the above estimates, we obtain Z π Z π 1 4 |fφ1 ,φ2 (0, 0)| dφ1 dφ2 R−2 log R 9ψ (R) = 8π 6 0 0 Z π Z π 1 + 6 |fφ1 ,φ2 (0, 0)|4 log 4π/|fφ1 ,φ2 (0, 0)|2 ) dφ1 dφ2 8π 0 0 Z πZ πZ dx2 dy2 dφ2 + Hf (φ1 ; z2 , φ2 ) dφ1 y22 F 0 0 Z π θZ π 1 + |fφ1 ,φ2 (0, 0)|4 dφ1 dφ2 R−2 + O(R−∞ ). 2 0 0

(87)

The coefficients cψ and dψ are thus determined. By virtue of (31) we have, after some change of variables, Z Z Z πZ π 4 |fφ1 ,φ2 (0, 0)|4 dφ1 dφ2 = 24 | e(t21 u1 + t22 u2 ) f (t1 , t2 ) d2 t| d2 u, (88) 0

R2

0

R2

which concludes the proof of Theorem 6.3.

6.3. The limit distribution – general cut-off functions. We shall now assume only that f (and thus ψ) is piecewise continuous and of compact support. Theorem 6.5. Let ψ be piecewise continuous and of compact support. Then there exists a decreasing function 9 : R+ → R+ with Z ∞ Z ∞ 9ψ (R) dR = ψ(r)2 dr, 9ψ (0) = 1, 0

0

discontinuous only for at most countably many R, such that, lim Probρ,σ {K2,ψ ( · , λ) > R} = 9ψ (R),

λ→∞

except possibly at the discontinuities of 9ψ (R). Proof. The proof follows along the lines of the proof of Theorem 4 in [46]. For a given > 0 choose a function ψ ∈ C∞ (R+ ) of compact support, such that Z ∞ |ψ(r) − ψ (r)|2 dr < . (89) 0

The crucial observation is that, for λ large enough, we have Z Z (α) K2,ψ−ψ (τ, λ) dτ dα ≤ C, I

(90)

A

for some constant C independent of λ and . This fact is a consequence of Proposition 6.1 and Condition (89). Consider the set (α) (τ, λ) < 1/2 }. Sy = {τ ∈ I, α ∈ A : K2,ψ−ψ

Spectral form factors of rectangle billiards

197

The integral over the complement of this set must satisfy ZZ ZZ (α) K2,ψ−ψ (τ, λ) dτ dα ≥ C > (I×A)−Sy

1/2 dτ dα,

(I×A)−Sy

hence |Sy | > |I| |A| − C1/2 . Define the distribution function

( 1 0

DR,ψ (τ, α, λ) = and the probability

(α) (τ, λ) > R if K2,ψ otherwise,

(91)

(92)

Z Z

9ψ,y (R) =

DR,ψ (τ, α, λ) ρ(τ ) σ(α) dτ dα. I

(93)

A

By virtue of (91) we have the inclusions 9ψ ,y (R + 1/2 ) − C 0 1/2 ≤ 9ψ,y (R) ≤ 9ψ ,y (R − 1/2 ) + C 0 1/2 ,

(94)

where C 0 does not depend on , y, R. By Theorem 6.2, for y → 0, the left and right hand side have the limits lim 9ψ ,y (R ± 1/2 ) = 9ψ (R ± 1/2 )

y→0

(95)

except for countably many R, . Some analysis shows (for details compare [46]) that for every δ > 0 there is an > 0 such that 9ψ (R − 1/2 ) − 9ψ (R + 1/2 ) < δ

(96)

(except for countably many R). Hence there is a function 9ψ (R) such that lim 9ψ,y (R) = 9ψ (R),

y→0

which proves the claim.

(97)

6.4. Random walks. Proof of Theorem 1.1. The task is to show that the difference between 2 N 1 X >R e(λ(α) τ ) Probρ,σ √ j N j=1

and

2 X 1 >R Probρ,σ √ e(λ(α) τ ) j N λ(α) ≤N j

vanishes for N → ∞, since then Theorem 1.1 will follow from Theorem 6.5. First notice that

198

J. Marklof

2 2 N X 1 X 1 X 1 (α) (α) (α) √ √ √ e(λj τ ) − e(λj τ ) = e(λj τ ) , N N (α) N (α) j=1 λj ≤N

(98)

j∈XN

(α) is the set of j defined by where XN (α) (α) XN = {j ≤ N : λ(α) j > N } ∪ {j > N : λj ≤ N }.

From the asymptotic relation (12) it can be readily seen that the number of elements in (α) is bounded by XN √ (α) #XN N, (99) where the implied constant does not depend on α ∈ A (A fixed). It is therefore clear from the calculation done to prove Proposition 6.1 that 2 Z Z 1 X (α) √ e(λ τ ) (100) j dτ dα N I A (α) j∈XN

vanishes for large N . Hence we can use the inclusion principle in the same fashion as in the proof of Theorem 6.5 (the vanishing of (100) is the analogue of relation (90)) to prove the existence of the limit 9(R), which furthermore has to equal 9(R) = 9χ (R), with χ the characteristic function of the interval [0, 1].

7. Rational α The simplest case is α = 1, since z1 = z2 and so the form factor is related to the theta function 2f (z, φ; z, φ) by ˆ (1) (τ, λ) = 1 |2f (z, 0; z, 0)|2 . K 2,ψ 4π The function F (z, φ) = |2f (z, φ; z, φ)|2 can now be viewed as a function on the manifold Mθ , which is embedded as a three-dimensional submanifold in M2θ , and we can apply the theorems which were developed in the beginning of Sect. 5. This observation holds in a similar way for all rational α = pq ; however, the corresponding embedded threemanifold M pq becomes densely distributed in M2θ when the sequence of rationals pq approaches an irrational. Let us discuss this in more detail. For a given integer N we define the congruence subgroups 00 (N ) of SL(2, Z) by ab 00 (N ) = ∈ SL(2, Z) : c ≡ 0 mod N . (101) cd

With ηN =

N 0 0 1

we find [58] that −1 SL(2, Z)ηN ∩ SL(2, Z). 00 (N ) = ηN

(102)

Spectral form factors of rectangle billiards

199

The index of these subgroups in SL(2, Z) is finite, more precisely [58], Y [SL(2, Z) : 00 (N )] = N (1 + p−1 ).

(103)

p prime p|N

Consider now the form factor for rational α = pq , p ˆ ( q ) (τ, λ) = 1 |2f (z1 , 0; z2 , 0)|2 = 1 |2f (pz, 0; qz, 0)|2 , K 2,ψ 4π 4π

(104)

with

π z1 z2 = = √ (τ + i λ−1 ). p q 4 pq For γ˜ = ηp−1 γηp with γ = ac db ∈ 00 (4) we have the functional relations z=

˜ φ + arg(pc z + d); qz, φ)|2 |2f (pγz, = |2f (γpz, φ + arg(c pz + d); qz, φ)|2 = |2f (pz, φ; qz, φ)|2 ,

(105)

and similarly ˜ φ + arg(qc z + d))|2 = |2f (pz, φ; qz, φ)|2 |2f (pz, φ; q γz,

(106)

for γ˜ = ηq−1 γηq with γ ∈ 00 (4). Therefore the function (p)

|2fq (z, φ)|2 = |2f (pz, φ; qz, φ)|2

(107)

is invariant under the group p

0( q ) = ηp−1 00 (4)ηp ∩ ηq−1 00 (4)ηq , p

p

which, by virtue of (102), contains the congruence group 00 (4pq) ⊂ 0( q ) ; hence 0( q ) p is of finite index in SL(2, Z) and M pq = 0( q ) \ SL(2, R) has finite volume. Using the theory developed in [46, 48] and the equidistribution of horocycles (Corollary 5.2) we can now prove the analog statements of the previous sections, but now for rational α. 7.1. The expectation value. Proposition 7.1. Let ψ be piecewise continuous and of compact support. Then, for λ → ∞, (p)

(p)

q ( · , λ) ∼ bψq log λ Eρ K2,ψ

(p)

for some constant bψq . In the case b(1) ψ

p q

= 1 we have in particular

1 = π

Z

∞ 0

ψ(r)2 dr.

200

J. Marklof (p)

p

Sketch of the proof. The function |2fq (z, φ)|2 on M pq = 0( q ) \ SL(2, R) is not bounded so Corollary 5.2 is not directly applicable. However, there is a way of resolving this difficulty using a regularization with Eisenstein series, see [46, 48] for details. Compare alternatively Jurkat and van Horne [39] for a different approach. 7.2. The limit distribution. Theorem 7.2. Let ψ be piecewise continuous and of compact support. Then there exists (p)

(p)

a decreasing function 9ψq with 9ψq (0) = 1, discontinuous for at most countably many R, such that, (p (p) q) ( · , λ) > R} = 9ψq (R), lim Probρ {K2,ψ λ→∞

(p)

except possibly at the discontinuities of 9ψq (R). For large R, (p)

(p)

9ψq (R) ∼ cψq R−1 , (p)

with some constant cψq , which in the special case pq = 1 reads Z 1 ∞ (1) cψ = ψ(r)2 dr. π 0 Sketch of the proof. Simply use Corollary 5.2, and proceed as in the proof of Theorems 6.2 and 6.5. Compare also the corresponding theorems in [46, 47, 48] and Jurkat and van Horne’s results [37, 38, 39]. Acknowledgements. I thank G. Casati, A. Eskin, M.R. Haggerty, D.A. Hejhal, J.P. Keating, R.E. Prange, Z. Rudnick, U. Smilansky, G. Tanner and S. Zelditch for helpful discussions. The research was carried out at the Basic Research Institute in the Mathematical Sciences, Hewlett-Packard Laboratories, Bristol, and at the Programme on Disordered Systems and Quantum Chaos, Isaac Newton Instiute, Cambridge. I would like to thank both Institutes for their hospitality.

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39. Jurkat, W.B. and van Horne, J.W.: The uniform central limit theorem for theta sums. Duke Math. J. 50, 649–666 (1983) 40. Kleinbock, D.Y. and Margulis, G.A.: Bounded orbits of nonquasiunipotent flows on homogeneous spaces. In: L.A. Bunimovich et al. (eds.), Sinai’s Moscow Seminar on Dynamical Systems AMS Transl. (2) 171, Providence, RI: Am. Math. Soc., 1996, pp. 141–172 41. Kosygin, D.V., Minasov, A.A. and Sinai, Ya.G.: Statistical properties of the spectra of Laplace–Beltrami operators on iouville surfaces. Russ. Math. Surv. 48, 1–142 (1993) 42. Lebœuf, P. and Iacomelli, G.: Statistical properties of the time evolution of complex systems I. Preprint, Orsay 1997, cond-mat/970970 43. Leviandier, L., Lombardi, M., Jost, R. and Pique, R.: Fourier transform: A tool to measure statistical level properties in very complex spectra. Phys. Rev. Lett. 56, 2449–2452 (1986) 44. Lombardi, M. and Seligman, T.H.: Universal and nonuniversal statistical properties of levels and intensities for chaotic Rydberg molecules. Phys. Rev. A 47, 3571–3586 (1993) 45. Major, P.: Poisson law for the number of lattice points in a random strip with finite area. Probab. Theory Related Fields 92, 423–464 (1992) 46. Marklof, I.: Limit theorems for theta sums. Duke Math. J., to appear 47. Marklof, J.: Theta sums, Eisenstein series, and the semiclassical dynamics of a precessing spin. In: D. Hejhal et al. (eds.), Emerging Applications of Number Theory, IMA Volumes in Mathematics and its Applications, Vol. 109, New York: Springer, 1998, pp. 405–450 48. Marklof, J.: Limit Theorems for Theta Sums with Applications in Quantum Mechanics. Dissertation, Universit¨at Ulm, 1997, Aachen: Shaker Verlag, 1997 49. Pandey, A., Bohigas, O. and Giannoni, M.-J.: Level repulsion in the spectrum of two-dimensional harmonic oscillators. J. Phys. A 22, 4083–4088 (1989) 50. Prange, R.E.: The spectral form factor is not self-averaging. Phys. Rev. Lett. 78, 2280–2283 (1997) 51. Rudnick, Z. and Sarnak, P.: Zeros of principal L-functions and random matrix theory Duke Math. J. 81, 269–322. (1996) 52. Rudnick, Z. and Sarnak, P.: The pair correlation function for fractional parts of polynomials. Commun. Math. Phys. 194, 61–70 (1998) 53. Sarnak, P.: Asymptotic behavior of periodic orbits of the horocycle flow and Eisenstein series. Commun. Pure Appl. Math. 34, 719–739 (1981) 54. Sarnak, P.: Values at integers of binary quadratic forms. In: Harmonic analysis and number theory, Montreal, 1996, CMS Conf. Proc. 21, Providence, RI: Am. Math. Soc., 1997 pp. 181-203 55. Sarnak, P.: Quantum chaos, symmetry and zeta functions. Lecture I: Quantum chaos, Curr. Dev. Math. 84–101 (1997) 56. Sarnak, P.: Quantum chaos, symmetry and zeta functions. Lecture II: Zeta functions, Curr. Dev. Math. 102–115 (1997) 57. Schubert, R.: The trace formula and the distribution of eigenvalues of Schr¨odinger operators on manifolds all of whose geodesics are closed. DESY-Report 95–090 (1995) 58. Shimura, G.: Introduction to the Arithmetic Theory of Automorphic Functions. Princeton, NJ: Princeton Univ. Press, 1971 59. Sinai, Ya.G.: Poisson distribution in a geometric problem: In: Dynamical systems and statistical mechanics, (Moscow, 1991), Adv. Soviet Math. 3, Providence, RI: Am. Math. Soc., 1991, pp. 199–214 60. Slater, N.B.: Gaps and steps for the sequence nθ mod 1. Proc. Cambridge Philos. Soc. 63, 1115–1123 (1967) 61. Tanner, G.: private communication 62. Uribe, A. and Zelditch, S.: Spectral statistics on Zoll surfaces. Commun. Math. Phys. 154, 313–346 (1993) 63. VanderKam, J.M.: Values at integers of homogeneous polynomials. Duke Math. J., to appear 64. VanderKam, J.M.: Pair correlation of four-dimensional flat tori. Preprint, Princeton University, 1997 65. Weinstein, A.: Asymptotics of eigenvalue clusters for the Laplacian plus a potential. Duke Math. J. 44, 883–892 (1977) 66. Zelditch, S.: Level spacings for quantum maps in genus zero. Peprint, Isaac Newton Institute, Cambridge 1997 Communicated by Ya. G. Sinai

Commun. Math. Phys. 199, 203 – 242 (1998)

Communications in

Mathematical Physics © Springer-Verlag 1998

Hopf Algebras, Renormalization and Noncommutative Geometry Alain Connes1 , Dirk Kreimer2 1 2

IHES, 35, route de Chartres, F-91440 Bures-sur-Yvette, France Department of Physics, Universit¨at Mainz, Staudingerweg, 55099 Mainz, Germany

Received: 14 August 1998/ Accepted: 5 October 1998

Abstract: We explore the relation between the Hopf algebra associated to the renormalization of QFT and the Hopf algebra associated to the NCG computations of tranverse index theory for foliations.

Introduction In [1] it was shown that the combinatorics of the subtraction procedure inherent to perturbative renormalization gives rise to a Hopf algebra HR which provides a conceptual framework to understand the intricacies of the forest formula of Zimmermann. In [2], it was shown that the delicate computational problem which arises from the transverse hypoelliptic theory of foliations, as formulated in noncommutative geometry, can only be settled thanks to a Hopf algebra HT associated to each integer codimension. This Hopf algebra reduces transverse geometry to a universal geometry of affine nature. The aim of this paper is to establish a close relation between the above Hopf algebras HR and HT . We shall first recall the first results of [2] which describe in the simplest case of codimension 1 the presentation of the Hopf algebra HT . We shall then explain the origin of the Hopf algebra HR from the renormalization of the divergences of QFT and show following [1] how HR is used in concrete problems of renormalization theory. In the appendix we include the case of overlapping divergences. We then give the presentation of the simplest model of HR namely the Hopf algebra of rooted trees, and show that it is uniquely characterized as the solution of a universal problem in Hochschild cohomology. We then determine the formal Lie algebra G such that HR is obtained as the dual of the enveloping algebra of this Lie algebra. It turns out to be a refinement of the Lie algebra of formal vector fields in one dimension. We then show that many of the results of [2] actually extend to this refinement of formal vector fields. These results indicate that parallel to the ordinary differential calculus which underlies the transverse structure of foliations, the recipes of renormalization theory

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should be considered as a refined form of calculus and should be understandable on a conceptual ground. Concretely, in the first section we show in Theorem (8) that the algebraic rules of the Hopf algebra HT are the expression of the group law of composition of diffeomorphisms of R in terms of the coordinates δn given by the Taylor expansion of − log(ψ 0 (x)) at x = 0. In particular this shows that the antipode in HT is, modulo a change of variables, the same as the operation of inversion of a formal power series for the composition law. In the second section we begin by the simplest and most explicit examples of divergent integrals of the kind that are met in Quantum Field Theory computations. We describe in this toy case the explicit counterterm construction and the immediate problem which arises from divergent subintegrations and explain how the Hopf algebra HR finds the combinatorial solution of the subtraction problem from its antipode. We next explain why the same holds in QFT (the treatment of overlapping divergences is postponed to the appendix). In the third section we exhibit the precise relation and analogy between HR and HT to the point that the antipode in HR appears as a direct analogue of the antipode in HT which we understood above as the inversion of formal power series. The key nuance between the two Hopf algebras is that where HT uses integers to label the Taylor expansion, the Hopf algebra HR uses rooted trees for labels. 1. The Hopf Algebra HT The computation of the local index formula for transversally hypoelliptic operators ([2]) is governed by a very specific Hopf algebra HT , which only depends upon the codimension n of the foliation. The structure of this Hopf algebra, its relation with the Lie algebra of formal vector fields as well as the computation of its cyclic cohomology have been done in [2]. In order to pursue the analogy between this development and the discovery by D. K. ([1]) of the Hopf algebra underlying renormalization, we shall recall here in all details the presentation and first properties of the Hopf algebra HT in the simplest case of codimension one. Useless to say this does not dispense one from consulting [2], in particular in connection with the specific representation of HT on crossed product algebras and the corresponding analysis. We first define a bialgebra by generators and relations. As an algebra we view HT as the enveloping algebra of the Lie algebra which is the linear span of Y , X, δn , n ≥ 1 with the relations, [Y, X] = X, [Y, δn ] = n δn , [δn , δm ] = 0 ∀ n, m ≥ 1, [X, δn ] = δn+1 ∀ n ≥ 1. (1) The coproduct 1 in HT is defined by 1 Y = Y ⊗ 1 + 1 ⊗ Y , 1 X = X ⊗ 1 + 1 ⊗ X + δ1 ⊗ Y , 1 δ1 = δ1 ⊗ 1 + 1 ⊗ δ1 (2) with 1 δn defined by induction using (1) and the equality, 1(h1 h2 ) = 1h1 1h2

∀ hj ∈ HT .

Lemma 1. The above presentation defines a Hopf algebra HT .

(3)

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Proof. One checks that the Lie algebra relations (1) are fulfilled by the elements 1(Y ), 1(X), 1(δ1 ), so that, by the universal property of the enveloping algebra, 1 extends to an algebra homomorphism, 1 : HT → HT ⊗ HT

(4)

and using the uniqueness of the extension, one also checks the coassociativity. One needs to show the existence of the antipode S. It is characterized abstractly as the inverse of the element L(a) = a in the algebra of linear maps L from HT to HT endowed with the product X X L1 (a(1) ) L2 (a(2) ), 1 a = a(1) ⊗ a(2) , a ∈ HT . (5) (L1 ∗ L2 )(a) = A simple computation shows that S is the unique antiautomorphism of HT (n) such that, S(Y ) = −Y,

S(δ1 ) = −δ1 ,

Note that the square of S is not the identity.

S(X) = −X + δ1 Y.

(6)

In order to understand the Hopf algebra HT , we first analyse the commutative subalgebra generated by the δn . For each n we let Hn be the subalgebra generated by δ1 , . . . , δn , Hn = {P (δ1 , . . . , δn ) ; P polynomial in n variables}.

(7)

We let Hn,0 be the ideal, Hn,0 = {P ; P (0) = 0}.

(8)

By induction on n one proves the following Lemma 2. For each n there exists Rn−1 ∈ Hn−1,0 ⊗ Hn−1,0 such that 1 δn = δn ⊗ 1 + 1 ⊗ δn + Rn−1 . Proof. One has 1 δ1 = δ1 ⊗ 1 + 1 ⊗ δ1 , and a simple computation shows that 1 δ 2 = δ2 ⊗ 1 + 1 ⊗ δ2 + δ1 ⊗ δ 1 ,

(9)

1 δ3 = δ3 ⊗ 1 + 1 ⊗ δ3 + δ12 ⊗ δ1 + δ2 ⊗ δ1 + 3δ1 ⊗ δ2 .

(10)

and,

In general, one determines Rn by induction, using Rn = [X ⊗ 1 + 1 ⊗ X, Rn−1 ] + n δ1 ⊗ δn + [δ1 ⊗ Y, Rn−1 ].

(11)

Since [X, Hn−1,0 ] ⊂ Hn,0 and [Y, Hn−1,0 ] ⊂ Hn−1,0 ⊂ Hn,0 , one gets that Rn ∈ Hn,0 ⊗ Hn,0 . The equality (10) shows that Hn is not cocommutative for n ≥ 3. However, since it is commutative, we shall determine the corresponding Lie algebra, using the Milnor– Moore theorem. Let A1n be the Lie algebra of jets of order (n + 1) of vector fields on the line, f (x) ∂/∂x modulo xn+2 ∂/∂x.

,

f (0) = f 0 (0) = 0,

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Proposition 3. The Hopf algebra Hn is the dual of the enveloping algebra U(A1n ), Hn = U(A1n )∗ . Proof. For each k ≤ n we introduce a linear form Zk,n on Hn , ∂ P (0). hZk,n , P i = ∂ δk

(12)

One has by construction, hZk,n , P Qi = hZk,n , P i Q(0) + P (0) hZk,n , Qi.

(13)

Note that ε, hε, P i = P (0) is the counit of Hn , hL ⊗ ε, 1 P i = hε ⊗ L, 1 P i = hL, P i

∀ P ∈ Hn .

(14)

(Check both sides on a monomial P = δ1a1 . . . δnan .) Thus in the dual agebra Hn∗ one can write (13) as 1 Zk,n = Zk,n ⊗ 1 + 1 ⊗ Zk,n .

(15)

Moreover the Zk,n form a basis of the linear space of solutions of (15) and we just need to determine the Lie algebra structure determined by the bracket. Let, 0 = (k + 1)! Zk,n . Zk,n

(16)

0 0 , Z`,n ] = 0 if k + ` > n, and that, Let us show that [Zk,n 0 0 0 , Z`,n ] = (` − k) Zk+`,n , [Zk,n

(17)

if k + ` ≤ n. Let P = δ1a1 . . . δnan be a monomial. We need to compute h1 P, Zk,n ⊗ Z`,n − Z`,n ⊗ Zk,n i. One has 1 P = (δ1 ⊗ 1 + 1 ⊗ δ1 )a1 (δ2 ⊗ 1 + 1 ⊗ δ2 + R1 )a2 . . . (δn ⊗ 1 + 1 ⊗ δn + Rn−1 )an . We look for the terms in δk ⊗ δ` or δ` ⊗ δk and take the difference. The latter is non-zero only if all aj = 0 except aq = 1. Moreover since Rm is homogeneous of degree m + 1 0 0 , Z`,n ] = 0 if k + ` > n. One then computes by one gets q = k + ` and in particular [Zk,n induction using (11) the bilinear part of Rm . One has R1(1) = δ1 ⊗ δ1 , and from (11), (1) ] + n δ1 ⊗ δ n . Rn(1) = [(X ⊗ 1 + 1 ⊗ X), Rn−1

(18)

(1) = δn−1 ⊗ δ1 + Cn1 δn−2 ⊗ δ2 + . . . + Cnn−2 δ1 ⊗ δn−1 . Rn−1

(19)

This gives

`−1 and we get Thus the coefficient of δk ⊗ δ` is Ck+` `−1 k−1 − Ck+` ) Zk+`,n . [Zk,n , Z`,n ] = (Ck+`

One has

(k+1)! (`+1)! (k+`+1)!

`−1 k−1 (Ck+` − Ck+` ) =

xk+1 (k+1)!

`(`+1)−k(k+1) k+`+1

(20)

= ` − k, thus one gets (17). The

0 ∂/∂x of the Lie algebra A1n are related by (16) to Zk,n = elements Zk,n = k+1 x ∂/∂x which satisfy the Lie algebra relations (17). The result then follows from the Milnor–Moore theorem.

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The A1n form a projective system of Lie algebras, with limit the Lie algebra A1 of formal vector fields which vanish at order 2 at 0. Thus the inductive limit H1 of the Hopf algebras Hn is, H1 = U(A1 )∗ .

(21)

The Lie algebra A1 is a graded Lie algebra, with one parameter group of automorphisms, αt (Zn ) = ent Zn

(22)

which extends to U(A1 ) and transposes to U(A1 )∗ as ∂ αt (a)t=0 h[Y, P ], ai = P, ∀ P ∈ H1 , a ∈ U(A1 ). ∂t

(23)

Indeed (αt )t is a one parameter group of automorphisms of H1 such that αtt (δn ) = ent δn .

(24)

Now, using the Poincar´e–Birkhoff–Witt theorem, we take the basis of U(A1 ) given by the monomials, a

n−1 . . . Z2a2 Z1a1 , aj ≥ 0. Znan Zn−1

To each L ∈ U(A1 )∗ one associates the formal power series X L(Z an . . . Z a1 ) n 1 xa1 1 . . . xann , an ! . . . a1 !

(25)

(26)

in the commuting variables xj , j ∈ N. It follows from [3] 2.7.5 that we obtain in this way an isomorphism of the algebra of polynomials P (δ1 , . . . , δn ) on the algebra of polynomials in the xj ’s. To determine the formula for δn in terms of the xj ’s, we just need to compute hδn , Znan . . . Z1a1 i. P Note that, by homogeneity, (27) vanishes unless j aj = n. For n = 1, we get ρ (δ1 ) = x1 ,

(27) (28)

where ρ is the above isomorphism. We determine ρ (δn ) by induction, using the derivation X ∂ (P ) (29) D(P ) = δn+1 ∂ δn (which corresponds to P → [X, P ]). One has by construction, hδn , ai = hδn−1 , Dt (a)i

∀ a ∈ U(A1 ),

(30)

where Dt is the transpose of D. By definition of Zn as a linear form (12) one has, Dt Zn = Zn−1 , n ≥ 2 , Dt Z1 = 0.

(31)

Moreover the compatibility of Dt with the coproduct of H1 is given by Dt (ab) = Dt (a) b + a Dt (b) + (δ1 a) ∂t b

∀ a, b ∈ U(A1 ),

(32)

where a → δ1 a is the natural action of the algebra H1 on its dual hP, δ1 ai = hP δ1 , ai

∀ P ∈ H1 , a ∈ U(A1 ).

(33)

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Lemma 4. When restricted to U(A2 ), Dt is the unique derivation, with values in U(A1 ) satisfying (32), moreover Dt (Znan . . . Z2a2 Z1a1 ) = Dt (Znan . . . Z2a2 ) Z1a1 + Znan . . . Z2a2

a1 (a1 − 1) a1 −1 Z1 . 2

Proof. The equality 1 δ1 = δ1 ⊗ 1 + 1 ⊗ δ1 shows that a → δ1 a is a derivation of U(A1 ). One has δ1 Zn = 0 for n 6= 1 so that δ1 = 0 on U(A2 ) and the first statement follows from (31) and (32). The second statement follows from Dt (Z1m ) =

m(m − 1) m−1 Z1 , 2

which one proves by induction on m using (32).

(34)

Motivated by the first part of the lemma, we enlarge the Lie algebra A1 by adjoining an element Z−1 such that, [Z−1 , Zn ] = Zn−1

∀ n ≥ 2;

(35)

we then define Z0 by [Z−1 , Z1 ] = Z0 , [Z0 , Zk ] = k Zk .

(36)

The obtained Lie algebra A is the Lie algebra of formal vector fields with Z0 = x ∂∂x , xn+1 ∂ Z−1 = ∂∂x and as above Zn = (n+1)! ∂ x. Let L be the left ideal in U(A) generated by Z−1 , Z0 , Proposition 5. The linear map Dt : U(A1 ) → U(A1 ) is uniquely determined by the equality Dt (a) = [Z−1 , a] mod L. Proof. Let us compare Dt with the bracket with Z−1 . By Lemma 4, they agree on U(A2 ). Let us compute [Z−1 , Z1m ]. One has [Z−1 , Z1m ] =

m(m − 1) m−1 Z1 + m Z1m−1 Z0 . 2

(37)

For each monomial Znan . . . Z1a1 one has Dt (a) − [Z−1 , a] ∈ L. Thus this holds for any a ∈ U(A1 ). Moreover, using the basis of U(A) given by the a

Znan . . . Z1a1 Z0a0 Z−1−1 we see that U(A) is the direct sum L ⊕ U(A1 ).

We now define a linear form L0 on U(A) by a

L0 (Znan . . . Z1a1 Z0a0 Z−1−1 ) = 0 unless a0 = 1, aj = 0 and L0 (Z0 ) = 1. Lemma 6. For any n ≥ 1 one has . . . [Z−1 , a] . . . ]) hδn , ai = L0 ([ |{z} n times

∀ a ∈ U(A1 ).

∀ j,

(38)

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Proof. Let us first check it for n = 1. P We let a = Znan . . . Z1a1 . Then the degree of a is P j aj = 1 so that the only possibility is a1 = 1, j aj and L0 ([Z−1 , a]) 6= 0 requires aj = 0 ∀ j. In this case one gets L0 ([Z−1 , Z1 ]) = L0 (Z0 ) = 1. Thus by (28) we get the equality of Lemma 6 for n = 1. For the general case note first that L is stable under right multiplication by Z−1 and hence by the derivation [Z−1 , ·]. Thus one has (Dt )n (a) = [Z−1 , . . . [Z−1 , a] . . . ] mod L

∀ a ∈ U(A1 ).

(39)

Now for a ∈ L one has L0 ([Z−1 , a]) = 0. Indeed writing a

a = (Znan . . . Z1a1 )(Z0a0 Z−1−1 ) = bc a

with b ∈ U(A1 ), c = Z0a0 Z−1−1 , one has [Z−1 , a] = [Z−1 , b] c + b [Z−1 , c]. Since b ∈ U(A1 ) and [Z−1 , c] has strictly negative degree one has L0 (b [Z−1 , c]) = 0. Let Znbn . . . Z1b1 Z0b0 be a non zero component of [Z−1 , b], then unless all bi are 0 it contributes by 0 to L0 ([Z−1 , b] c). But [Z−1 , b] ∈ U(A0 )0 has no constant term. Thus one has L0 ([Z−1 , a]) = 0

a

∀ a = Znan . . . Z1a1 Z0a0 Z−1−1

(40)

except if all aj = 0, j 6= 1 and a1 = 1. L0 ([Z−1 , Z1 ]) = 1. Using (31) one has hδn , ai = hδ1 , (Dt )n−1 (a)i and the lemma follows. One can now easily compute the first values of ρ (δn ), ρ (δ1 ) = x1 , ρ (δ2 ) = x2 + x3

x21 2 ,

ρ (δ3 ) = x3 + x2 x1 + 21 , ρ (δ4 ) = x4 + x3 x1 + 2 x22 + 2 x2 x21 + 43 x41 . The affine structure provided by the δn has the following compatibility with left multiplication in U(A1 ). P k k Rn−1 ⊗ δk , Rn−1 ∈ Hn−1,0 . Lemma 7. a) One has Rn−1 = 1 k b) For fixed a0 ∈ U(A ) there are λn ∈ C such that X λkn hδk , ai. hδn , (a0 a)i = hδn , a0 i ε(a) + k , a0 i. Proof. a) By induction using (7). b) Follows, using λkn = hRn−1

The antipode S in U(A1 ) is the unique antiautomorphism such that S Zn = −Zn

∀ n.

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It is non-trivial to express in terms of the coordinates δn . In fact if we use the basis Zj of A1 but in reverse order to construct the map ρ we z2

obtain a map ρe whose first values are ρe (δ1 ) = z1 , ρe (δ2 ) = z2 + 21 , ρe (δ3 ) = z3 +3 z1 z2 + 21 z13 , ρe (δ4 ) = z4 + 2 z22 + 6 z1 z3 + 9 z12 z2 + 43 z14 . One has P am am . . . Z1a1 )i = (−1) aj hδn , Z1a1 . . . Zm i hδn , S (Zm so that

X am hδn , S (Zm . . . Z1a1 )i xa1 1 . . . xamm = ρ (S t δn ) = P X am i xa1 1 . . . xamm = ρe (δn ) = (−1) aj hδn , Z1a1 . . . Zm

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with zj = −xj in the latter expression.

x2

Thus ρ (S t δ1 ) = −x1 , ρ (S t δ2 ) = −x2 + 21 , ρ (S t δ3 ) = −x3 +3 x1 x2 − −x4 + 2 x22 + 6 x1 x3 − 9 x21 x2 + 43 x41 . We thus get S t δ1 = −δ1 ,

S t δ2 = −δ2 + δ12 ,

x31 t 2 , ρ (S δ4 )

S t δ3 = −δ3 + 4δ1 δ2 − 2δ13 ,

... .

=

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The meaning of all the above computations and their relation to the standard calculus of Taylor expansions is clarified by the following theorem ([2]). Theorem 8. Let G2 be the group of formal diffeomorphisms of R, of the form ψ(x) = x + o(x). For each n, let γn be the functional on G2 defined by, γn (ψ −1 ) = (∂xn log ψ 0 (x))x=0 . The equality 2(δn ) = γn determines a canonical isomorphism 2 of the Hopf algebra H1 with the Hopf algebra of coordinates on the group G2 . We refer to Theorem 8 of [2] for the proof, as well as for the more elaborate structure of the Hopf algebra HT . This theorem certainly shows that the antipode, i.e. the map ψ → ψ −1 is certainly non trivial to compute. Note also that the expression σ = δ2 − 21 δ12 is uniquely characterized by ρ (σ) = x2

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which suggests to define higher analogues of the Schwartzian as ρ−1 (xn ). 2. The Physics of Renormalization and the Hopf Algebra of Rooted Trees In this section we want to motivate the Hopf algebra structure behind the process of renormalization in Quantum Field Theories (QFTs) [1] and show how relations to the Hopf algebra of the previous section emerge. The renormalization procedure appears as the cure for the disease caused by the unavoidable presence of UV divergences in QFTs which describe the physics of local quantized fields. Such QFTs describe successfully all known particle physics phenomenology. The point of departure of the renormalization procedure is to alter the original Lagrangian by an infinite series of counterterms labelled by Feynman graphs, whose sole purpose is to cancel the UV-divergences coming from the presence of ill-defined integrals in the perturbative expansion of the theory. Recall that the perturbative expansion of the functional integral appears as a sum labelled by Feynman graphs 0. To each of these graphs corresponds an integral I0 which is in general ill-defined. To compensate for the resulting infinities one adds to the original Lagrangian LP 0 which appears as the argument of the exponential, an infinite series of counterterms 0 L0 , each term in the series corresponding to a Feynman graph P 0. The difficulty in finding the cut-off dependent counterterm Lagrangian L0 comes only from the presence of ill-defined subintegrations (usually dubbed subdivergences) in the integral I0 . Indeed in the special case of a diagram without subdivergences the counterterm is simply (in the MS scheme) just the pole part of I0 . As soon as subdivergences are present the extraction of L0 is much more complicated since we want to take into account the previous subtractions which is necessary to maintain locality in the theory.

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This obviously generates complicated combinatorial problems, which for the first time, acquire mathematical meaning thanks to the Hopf algebra HR . A Toy Model. It is possible to study the basic properties of the renormalization procedure with the help of toy models, to which we now turn. In the following we will consider integrals of the form Z ∞ 1 dy x(c) := y +c 0 for c > 0, which are to be regarded as functions of the parameter c. As it stands such an integral is ill-defined, due to its divergence at the upper boundary. Power counting reveals the presence of a logarithmic singularity, and in this respect the integral behaves no better or worse than a logarithmic divergent integral in QFT, which one typically confronts due to the presence of UV divergences in loop integrations. We will introduce a regularization, Z ∞ 1 y − dy, x(c) = y+c 0 where is a small positive parameter. We now easily evaluate the above integral x(c) = B(, 1 − )c− , where the presence of the pole term ∼ 0() = 0(1 + )/ indicates the UV divergence in the integral 1 . The process of renormalization demands the subtraction of this UV divergence, and at this level we can straightforwardly proceed by a simple subtraction Z ∞ (1 − c) dy = B(, 1 − )(c− − 1) y − x(c) − x(1) = (y + c)(y + 1) 0 which is evidently finite if we send → 0. Here, −x(1) acts as the counterterm for the ill-defined function x(c), and the difference x(c) − x(1) corresponds to the renormalized function associated to x(c). Physicists have good reason to demand that a counterterm like −x(1) above is independent of the external parameter c, as to maintain locality in the theory. Before we explain this in more detail we want to generalize this simple example to the presence of subdivergences. We consider Z ∞Z ∞ 1 1 y1− y2− dy1 dy2 x2 (c) := y + c y + y1 1 2 Z0 ∞ 0 1 = y1− x(y1 )dy1 . y 1+c 0 We say that x2 (c) has the function x(y1 ) as a subdivergence, but it still is overall divergent itself. Powercounting reveals that there is a divergent sector when the y2 integration variable tends to infinity for any fixed y1 , and when y1 , y2 tend to infinity jointly. There are no P∞ 1 B(, 1 − ) = 0(1 + )0(1 − )/, 0(1 + x) = exp(−γx) exp( ζ(j)xj /j), | x |< 1. j=2

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divergences when y2 is kept fixed and y1 tends to infinity, though. All the divergences are of logarithmic nature. Having successfully eliminated the divergence in the previous example by a naive subtraction procedure, it is interesting to see if we can eliminate the divergences in x2 (c) by subtracting x2 (1): Z ∞ 1 (1 − c) y1− y2− dy1 dy2 x2 (c) − x2 (1) = (y1 + c)(y1 + 1) y2 + y1 0 Z ∞ (1 − c) dy1 = B(, 1 − ) y1−2 (y1 + c)(y1 + 1) 0 = B(, 1 − ) B(2, 1 − 2)c−2 − B(2, 1 − 2) log(c) + finite terms. =− Unfortunately, this expression still suffers from a divergence in the y2 integration, and we were thus not successful with this naive attempt. Actually, we find that the divergence is ∼ log(c). The parameter c in our toy model is the remaining scale of the Green function. In realistic QFTs, this scale is furnished typically by an external momentum q, say, and divergences of the form log(q 2 )/ are non-local divergences: upon Fourier-transformation, they involve the logarithm of a differential operator, for example the logarithm of an external q 2 would translate as log(). Such terms can not be absorbed by local counterterms, and are strictly to be avoided if one wants to remain in the context of a local field theory. In the context of field theory, locality restricts counterterms to be polynomial in momenta. Correspondingly, in the context of our toy model, we thus look for counterterms which are at most polynomial in the parameter c. The failure above was twofold: the naive subtraction −x2 (1) not only failed to render x2 (c) finite, but also this failure could only be absorbed by a non-local counterterm ∼ log(c)/. To find a local counterterm, some more work is needed. Following the guidance of field theory we associate to x2 (c) (corresponding to a bare Green function) a function which has its subdivergences subtracted (a transition in field theory achieved by the R operation): Z ∞ 1 1 1 dy1 dy2 x2 (c) := x2 (c) − x(c)x(1) ≡ y1− y2− − y1 + c y2 + y1 y2 + 1 0 = B(, 1 − ) B(2, 1 − 2)c−2 − B(, 1 − )c− . Note that the subtraction term −x(c)x(1) involves the counterterm −x(1) times the analytic expression, x(c), which we obtain from x2 (c) when we set the subdivergence x(y1 ) in x2 (c) to one. We realize that xR 2 (c) = lim →0 [x2 (c) − x2 (1)] is a well-defined finite expression, the finite renormalized Green function xR 2 (c), −2 − B(, 1 − )c− xR 2 (c) = lim B(, 1 − ) B(2, 1 − 2)c →0

−B(, 1 − ) [B(2, 1 − 2) − B(, 1 − )]} 1 = log2 (c), 2 and thus identify

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−x2 (1) = −B(, 1 − )[B(2, 1 − 2) − B(, 1 − )] with the counterterm associated to x2 (c). R Note that xR 2 (1) = 0, by construction. The renormalized Green function x2 (c) becomes a power series in log(c) (without constant term). Note further that we can write an integral representation for it which eliminates the necessity to introduce a regularization at all: Z ∞Z ∞ 1 1 1 1 1 1 − − − dydx. (c) = xR 2 x+c y+x y+1 x+1 y+x y+1 0 0 This could be directly obtained following the BPHZ approach, and what we have just seen is the equivalence between on-shell renormalization (subtraction at the on-shell value c = 1) and the BPHZ renormalization in the toy model. The above example shows how to find a local counterterm for an ill-defined integral with ill-defined subintegrations. We first eliminated the ill-defined subintegration by a counterterm, and then proceeded to construct the counterterm for the integral as a whole. In QFT one proceeds in the same manner. A bare Green-function, given by an ill-defined integral, will suffer from a plethora of ill-defined subintegrations in general. These subintegrations can be disjoint, nested or overlapping [4]. We will see later that the overlapping case resolves into the other ones. This result was effectively already obtained in [1, 5, 6], and also known to others. An example how to resolve overlapping divergences in the case of φ3 theory in six dimensions will be given in an appendix. Thus, we introduce at this stage a generalization of the above toy model allowing only for arbitrary nested or disjoint subdivergences. This motivates to generalize the example to functions xt (c) of an external parameter c, indexed by a rooted tree t, due to the fact that any configurations of nested or disjoint subdivergences can be described by a rooted tree. The formal definition of a rooted tree is postponed to the next section, while here we continue to gain experience in the treatment of functions having nested and disjoint subdivergences. We define for a tree t with m vertices, enumerated such that the root has number 1, Z xt (c) := 0

∞

m

1 Y 1 y − dym . . . y1− dy1 , ∀c > 0, y1 + c yi + yj(i) m i=2

where j(i) is the number of the vertex to which the ith vertex is connected via its incoming edge. We can write this as Z xt (c) :=

r

1 Y xtj (y)y − dy, y+c j=1

if the root of t connects to r trees tj . Figure 1 defines some simple rooted trees. Note that each vertex vi of the rooted tree corresponds to an integration variable xi , and that an edge connecting vj to vi towards the root indicates that the xj integration is nested in the xi integration. Integration variables which correspond to vertices which are not connected by an edge correspond to disjoint integrations. For the rooted trees defined in Fig. 1 we find the following analytic expressions:

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Fig. 1. A toy model realizing rooted trees. We define the first couple of rooted trees t1 , t2 , t31 , t32 . The root is always drawn as the uppermost vertex. t2 gives rise to the function x2 (c)

Z xt1 (c) = 0

Z xt2 (c) =

0

Z xt31 (c) = xt32 (c) =

0

Z

0

∞

∞

∞

∞

y − dy, y+c y − xt1 (y) dy, y+c y − xt2 (y) dy, y+c y − xt1 (y) xt1 (y) dy. y+c

Note that x2 (c) ≡ xt2 (c). The Hopf algebra HR . The previous remarks motivate to introduce a Hopf algebra based on rooted trees. We still postpone all formal definitions to the next section and simply note that a rooted tree t is a connected and simply-connected set of oriented edges and vertices such that there is precisely one distinguished vertex with no incoming edge. This vertex is called the root of t. Further, every edge connects two vertices and the fertility f (v) of a vertex v is the number of edges outgoing from v. We consider the algebra of polynomials over Q in rooted trees. Note that for any rooted tree t with root r we have f (r) trees t1 , . . . , tf (r) which are the trees attached to r. Let B− be the operator which removes the root r from a tree t: B− : t → B− (t) = t1 t2 . . . tf (r) . Figure 2 gives an example.

Fig. 2. The action of B− on a rooted tree

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Let B+ be the operation which maps a monomial of n rooted trees to a new rooted tree t which has a root r with fertility f (r) = n which connects to the n roots of t1 , . . . , tn , B+ : t1 . . . tn → B+ (t1 . . . tn ) = t.

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This is clearly the inverse to the action of B− .

Fig. 3. The action of B+ on a monomial of trees

One has B+ (B− (t))) = B− (B+ (t))) = t

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for any rooted tree t. Figure 3 gives an example. We further set B− (t1 ) = 1, B+ (1) = t1 . We will introduce a Hopf algebra on such rooted trees by using the possibility to cut such trees in pieces. We start with the most elementary possibility. An elementary cut is a cut of a rooted tree at a single chosen edge, as indicated in Fig. 4. We will formalize all these notions in the next section. By such a cutting procedure, we will obtain the possibility to define a coproduct in a moment, as we can use the resulting pieces on either side of the coproduct.

Fig. 4. An elementary cut splits a rooted tree t into two components t1 , t2

But before doing so we finally introduce the notion of an admissible cut, also called a simple cut. It is any assignment of elementary cuts to a rooted tree t such that any path from any vertex of the tree to the root has at most one elementary cut. Figure 5 gives an example. An admissible cut C maps a tree to a monomial in trees. If the cut C contains n elementary cuts, it induces a map C : t → C(t) =

n+1 Y i=1

tji .

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Fig. 5. An admissible cut C acting on a tree t. It produces a monomial of trees. One of the factors, RC (t), contains the root of t

Note that precisely one of these trees tji will contain the root of t. Let us denote this distinguished tree by RC (t). The monomial which is delivered by the n − 1 other factors is denoted by P C (t). The definitions of C, P, R can be extended to monomials of trees in the obvious manner, by choosing a cut C i for every tree tji in the monomial:

C(tj1 . . . tjn ) := C 1 (tj1 ) . . . C n (tjn ), n

P C (tj1 . . . tjn ) := P C (tj1 ) . . . P C (tjn ), 1

n

RC (tj1 . . . tjn ) := RC (tj1 ) . . . RC (tjn ). 1

We have now collected a sufficient amount of structure to define a Hopf algebra on rooted trees. Our aim is to see the correspondence between the Hopf algebra formulated on rooted trees and the generation of a local counterterm for the functions xt (c) introduced above, and finally to see the correspondence between the Hopf algebra of rooted trees and the Hopf algebra of the previous section. Before we define the Hopf algebra of rooted trees, we leave it as an exercise to the reader to convince himself that any admissible cut in a rooted tree determines in the representation on functions xt (c) a divergent subintegration, and that vice versa any divergent subintegration corresponds to an admissible cut. For example, the single cut possible at x2 (c) ≡ xt2 (c) corresponds to the single divergent subintegration in this function. Let us now establish the Hopf algebra structure. Following [1] we define the counit and the coproduct. The counit : A → Q is simple: (X) = 0, for any X 6= e, (e) = 1. The coproduct 1 is defined by the equations

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1(e) = e ⊗ e, 1(t1 . . . tn ) = 1(t1 ) . . . 1(tn ), 1(t) = t ⊗ e + (id ⊗ B+ )[1(B− (t))],

(48) (49) (50)

which defines the coproduct on trees with n vertices iteratively through the coproduct on trees with a lesser number of vertices. The reader should work out the examples in Fig. 6 himself. One checks coassociativity of 1 [1]. Also, we will give a formal proof in the next section. ∆ ∆

∆

∆

1+1 1+1

+

1+1

+

+

+ 2

1+1

+

Ο Fig. 6. The coproduct. We work it out for the trees t1 , t2 , t31 , t32 . For the latter, the last line gives explicitly the simple admissible cuts which were used in the construction of the coproduct. The first two terms are generated by the full admissible and the empty cut, while the last three terms are generated by proper admissible cuts

The following statement follows directly from the results in the next section, but it is instructive to prove it here by elementary means to make contact with the previous section. We claim that the coproduct can be written as

1(t) = e ⊗ t + t ⊗ e +

X

P C (t) ⊗ RC (t).

(51)

adm. cuts C of t

Proof. The result is true for the tree t1 having only one vertex. The induction is on the number of vertices. We use that B− (t) has n vertices if t has n + 1. Thus,

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1(t) = t ⊗ e + (id ⊗ B+ )1(B− (t)) = t ⊗ id + (id ⊗ B+ ) e ⊗ B− (t) + B− (t) ⊗ e +

X

P C (B− (t)) ⊗ RC (B− (t))

adm. cuts C of B− (t)

= t ⊗ e + e ⊗ B+ (B− (t)) + B− (t) ⊗ δ1 X + P C (B− (t)) ⊗ B+ (RC (B− (t)))) adm. cuts C of B− (t)

= t⊗e+e⊗t X P C (t) ⊗ RC (t)). + adm. cuts C of t

We used the fact that B+ B− = id and that the only cut which distinguishes X P C (B− (t)) ⊗ B+ (RC (B− (t))) adm. cuts C of B− (t)

from

X

P C (t) ⊗ RC (t)

adm. cuts C of t

is the cut which generates B− (t) ⊗ δ1 .

Note that the above formula can be streamlined. X 0 C (t) ⊗ RC (t), 1(t) = adm. cuts C of t P where the primed sum indicates that we include the empty cut and the full admissible cut in the definition of admissible cuts in the manner indicated in Fig. 6. Any cut corresponds to a choice of a subset of edges on the set t(1) of all edges of a given rooted tree t. The empty cut C = ∅ corresponds to the empty set in this sense. Thus, P ∅ (t) = e, R∅ (t) = t. The full admissible cut Cf is defined by the complementary result: P Cf (t) = t, RCf (t) = e. It can be regarded as a cut c on the one new edge of B+ (t), defined as the intersection c = {t(1) ∩ B+ (t)(1) }. Note that P Cf (t) ⊗ RCf (t) = (id ⊗ B− )[P c (B+ (t)) ⊗ Rc (B+ (t))], with the cut c determined as above. In Fig. 6 we indicate the full cut Cf (t32 ) by a dashed box around the rooted tree P Cf (t32 ). The coproduct introduced here is linear in rooted trees in the right factor and polynomial in the left as one clearly sees in Eq. (51). This is a fundamental property shared with the coproduct of the previous section. We will explore this fact in some detail soon.

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Up to now we have established a bialgebra structure. It is actually a Hopf algebra. Following [1] we find the antipode S as S(e) = e, S(t) = −t −

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X

C

C

S[P (t)]R (t),

(53)

adm. cuts C of t

and one immediately checks that m[(S ⊗ id)1(t)] = t + S(t) +

X

S[P C (t)]RC (t),

(54)

adm. cuts C of t

= 0 = (t).

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To show that m[(id ⊗ S)1(t)] = 0 one uses induction on the number of vertices [1]. We mentioned already that a cut on a tree t is given by a subset of the set t(1) of the set of edges of t. So far, we allowed for a restricted class of subsets, corresponding to admissible cuts. We actually enlarged the set already and considered the set B+ (t)(1) of all edges of B+ (t), to construct the full admissible cut. We now consider all cuts corresponding to this set, that is all possible subsets of B+ (t)(1) , including the empty set. These subsets fall in two classes, one which contains the edge c(t) = t(1) ∩ B+ (t)(1) , one which does not contain this edge. Cuts corresponding to the first class we call full cuts, cuts not containing this distinguished edge we call normal cuts. Thus, the empty cut is a normal cut. Non-empty normal cuts are also called proper cuts. Note that for a given normal cut C ⊂ t(1) and the corresponding full cut {C, c(t)} = C ∩ c(t) we have P C (t) = P {C,c(t)} (t), RC (t) = R{C,c(t)} (t), while n{C,c(t)} = nC + 1, where nC is the cardinality of the set C. Let us give yet another formula to write the antipode, which one easily derives using induction on the number of vertices: X (−1)nC P C (t)RC (t). S(t) = all full cuts C of t

This time, we have a non-recursive expression, summing over all full cuts C, relaxing the restriction to admissible cuts. We introduced full cuts so that the overall sign agrees with the number of cuts employed. Note that we have for all t 6= e, X (−1)nC P C (t)RC (t) = 0 = (t), m[(S ⊗ id)1(t)] = all cuts C of t

as each cut appears twice, either as a full cut or as a normal cut, with opposite sign, due to the fact that the cardinality of {C, c(t)} extends the cardinality of C by one. By now we have established a Hopf algebra on rooted trees. It is instructive to calculate the coproduct and antipode for some simple trees. Figures 6, 7 give examples. Note that in Fig. 7 we represented cuts as boxes. This is possible in a unique way, as each cut on a simply connected rooted tree can be closed in the plane to a box without further intersecting the tree and so that the root is in the exterior of the box.

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At this time we can make contact to the previous section. We note that the sum of the two trees with three vertices behaves under the coproduct as the element δ3 in the first section. Defining δ1 := t1 , δ2 := t2 , δ3 := t31 + t32 , we find 1(δ1 ) = δ1 ⊗ e + e ⊗ δ1 , 1(δ2 ) = δ2 ⊗ e + e ⊗ δ2 + δ1 ⊗ δ1 , 1(δ3 ) = δ3 ⊗ e + e ⊗ δ3 + 3δ1 ⊗ δ2 + δ2 ⊗ δ1 + δ12 ⊗ δ1 , in accordance with Eq. (10) in the previous section. This is no accident, as we will soon see. The reader should also check the formulas for the coproduct and antipode on examples himself.

2

Fig. 7. The antipode for some simple trees. Introducing full cuts, we find a very convenient way to express it using boxes for cuts. The sign for each term can be easily memorized as (−1)nc , where nc is the number of full cuts

It is now not difficult to employ this Hopf algebra to regain the local counterterms for the toy model. By construction, subdivergent sectors correspond to admissible cuts. Also, forests in the sense of renormalization theory are in one to one correspondence with arbitrary cuts, with full cuts corresponding to full forests, and normal cuts to normal forests. This allows to recover local counterterms from the formula for the antipode in our Hopf algebra. The recursive and the non-recursive manner to write the antipode give rise to two equivalent formulas for the local counterterm, as it is standard in renormalization theory [4].

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To see all this, note that any cut defines a natural bracket structure on P c (t). In a moment we will see how this fact allows to introduce various different renormalization schemes R. It is instructive to come back to the toy model. Figure 8 summarizes how the standard notions of renormalization theory derive from the Hopf algebra of rooted trees.

Fig. 8. Cuts induce a bracket structure on trees. Exploring this fact, the toy model can be easily renormalized using the Hopf algebra structure of rooted trees

In Fig. 8 we see the tree t31 , corresponding to the function xt31 (c). The antipode S(t31 ) = −t31 + t1 t2 + t2 t1 − t1 t1 t1 derives from full cuts which induce the following bracket structure on toy model functions: −[xt31 (c)] + [[xt2 (c)]xt1 (c)] + [[xt1 (c)]xt2 (c)] − [[[xt1 (c)]xt1 (c)]xt1 (c)]. There is a certain freedom how to evaluate this bracket structure. Such a freedom is always there in renormalization. It corresponds to a choice of renormalization map R in the notation of [1], while in Collins textbook [4] it corresponds to the choice of the map T which extracts the divergent part of a given expression. As long as the evaluation of the bracket leaves the divergent part unchanged, it corresponds to a valid renormalization scheme, and we obtain the finite renormalized Green function from the consideration of m[(S ⊗ id)1(t31 )], which gives rise to the following expression by summing over brackets induced by full and normal cuts xR t3 (c) = xt31 (c) − [xt2 (c)]xt1 (c) − [xt1 (c)]xt2 (c) + [[xt1 (c)]xt1 (c)]xt1 (c) 1

− [xt31 (c)] + [[xt2 (c)]xt1 (c)] + [[xt1 (c)]xt2 (c)] − [[[xt1 (c)]xt1 (c)]xt1 (c)]. It is not to difficult to check the finiteness of this expression for the typical choices of renormalization schemes, on shell,

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[xt (c)] = xt (1), minimal subtraction, [xt (c)] = P oleP art (xt (1)), or BPHZ type schemes. Note that the only reason that the renormalized function xR t (c) does not vanish identically is that full cuts involve one more bracket evaluation than normal cuts. As the bracket evaluation respects the divergent part, it is clear that functions xR t (c) must be finite, due to the very fact that m[(S ⊗ id)1(t)] involves a sum over all cuts in pairs of normal cuts and associated full cuts. This gives rise to a sum of pairs of contributions, each pair being a difference X − [X] between an analytic contribution X and its bracket evaluation. Thus, as long as the bracket evaluation respects the divergent part of X, we will obtain a finite result for xR t (c). Let us now turn away from toy models and address QFTs. By its very definition, pQFT deals with the calculation of Feynman diagrams. As such, it confronts the problem of the presence of ultraviolet divergences in the diagrams. Hence, the diagrams refer to ill-defined analytic quantities. This is reflected by the fact that the analytic expressions provided by the diagrams become Laurent series in a regularization parameter. The presence of pole terms in this Laurent series then indicates the presence of UV-divergences in the first place. Equivalently, in the BPHZ spirit, we can Taylor expand the integrand in external momenta and would find that the first few terms are ill-defined quantities. The art of obtaining meaningful physical quantities from these Laurent series is known as renormalization. It is in this process that we will find the Hopf algebra structure realized. Let us recall a few basic properties of renormalization. Feynman diagrams consist of edges (propagators) and vertices of different types. To each such type we can assign a weight, and to a Feynman diagram we can assign an integral weight called degree of divergence which can be calculated from the dimension of spacetime, the numbers of closed cycles in the Feynman diagram, and the weights of its propagators and vertices. One finds that the analytic expression provided by a Feynman diagram under consideration provides UV-divergences if and only if its degree of divergence ω is ≥ 0. One speaks of logarithmic, linear, quadratic, . . . divergences for ω = 0, 1, 2, . . . . A Feynman diagram usually contains subdiagrams, which have their own degree of divergence, and thus might provide UV-divergent analytic expressions by themselves. All these divergences are to be compensated by local counterterms, which are to be calculated from a Feynman graph and its divergent subgraphs. Figure 9 shows how the previous discussion extends to a Feynman graph with one subdivergent graph. It is the main result of [1] that the transition from a bare Feynman diagram to its local counterterm, and to the renormalized Feynman graph, is described by a Hopf algebra structure. In this paper, we have changed the notation of [1] and formulate the Hopf algebra on rooted trees. A glance at Fig. 10 shows how to assign a rooted tree to a Feynman graph with subdivergences. This is possible in a unique manner as long as all subdivergences of the Feynman graph are either disjoint or nested [1]. In such circumstances, we can associate a unique rooted tree to the graph. If the subdivergences are overlapping, the renormalization will nevertheless follow the combinatorics dictated by the Hopf algebra of rooted trees. In such circumstances, a Feynman graph corresponds to a sum of rooted trees [5, 1]. We will comment on this fact in an appendix.

Hopf Algebras, Renormalization and Noncommutative Geometry

223

Γ

γ1

γ2

Fig. 9. This Feynman graph 0 behaves in the same manner as the toy function x2 (c). It contains a subdivergent graph γ2 ⊂ 0, contained in the smaller box. This subdivergence sits in a graph γ1 which we obtain if we shrink γ2 to a point in 0, γ1 = 0/γ2 . To get a local counterterm, we follow the same steps as before: we replace the divergent subgraph γ2 by its subtracted renormalized form γ2R = γ2 − [γ2 ], and can calculate the local counterterm for 0 from the resulting expression

Each Feynman diagram 0 furnishes a tree whose vertices are decorated by Feynman graphs γ ⊂ 0 which are free of subdivergences themselves. These decorations correspond to the letters in the parenthesized words of [1]. Note that the set of all admissible cuts is by construction in one to one correspondence with the set of all superficially divergent subgraphs of 0. Further the set of all cuts is in one to one correspondence with the set of all forests in the sense of renormalization theory. This is the main result of [1]. The antipode of the Hopf algebra AQF T delivers the Z-factor of a Feynman diagram.

Fig. 10. Rooted trees from Feynman diagrams. They are in one-to-one correspondence with the parenthesized words of [1]. In this example it is the rooted tree t31 which is associated to the configuration of subdivergences given by the graph

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At this stage, we can summarize the results in [1] using the language of the Hopf algebra of rooted trees. This is done in Fig. 11. Note that we even do not have to specify the renormalization scheme, but that the cuts used in the definition of the antipode on rooted trees extend to forests, so that we can apply any chosen renormalization prescription to evaluate the content of these forests. Thus, as before in the toy model, each cut corresponds to the operation T in Collins book [4], some operation which extracts the divergence of the expression on which it acts. As we mentioned already, this operation was called R in [1].

−>

−

+

+

−

Fig. 11. The steps involved in the process of renormalization are governed by the Hopf algebra on rooted trees, as this figure clearly exhibits. We indicate how the Hopf algebra of rooted trees acts on the Feynman diagrams. Subgraphs are indicated by grey rectangles and determine the tree structure, forests corresponding to cuts generated by the antipode of the associated rooted tree are given as dashed black rectangles

At this point, we succeeded in deriving the renormalization procedure from the Hopf algebra of rooted trees. The attentive reader will have noticed that we ignored overlapping divergences in our discussion. As promised, we will discuss them in an appendix, where it is shown how to assign a unique sum of rooted trees to any graph containing overlapping divergences, to which then our previous considerations apply. For us, these considerations of renormalization and the underlying Hopf algebra of rooted trees are sufficient motivation to get interested in this Hopf algebra. We will continue our exploration of this subject by showing how it relates to the Hopf algebra of the first section. We saw some of these relations already, and now continue to make this more precise.

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3. The Relation Between HR and HT Recall the relations 1(δ1 ) = δ1 ⊗ e + e ⊗ δ1 , 1(δ2 ) = δ2 ⊗ e + e ⊗ δ2 + δ1 ⊗ δ1 , 1(δ3 ) = δ3 ⊗ e + e ⊗ δ3 + 3δ1 ⊗ δ2 + δ2 ⊗ δ1 + δ12 ⊗ δ1 , which indicate an intimate connection to the Hopf algebra HT introduced in the first section. To find the general relation between the two Hopf algebras under consideration we first introduce naturally grown forests δk . To this end, we consider an operator N which maps a tree t with n vertices to a sum N (t) of n trees ti , each having n + 1 vertices, by attaching one more outgoing edge and vertex to each vertex of t, as in Fig. 12. The root remains the same in this operation.

+

+

Fig. 12. The operator N and the elements δk

Now we define δk := N k (e)

(56)

so that δk+1 = N (δk ). On products of trees N will act as a derivation, comparable to the derivation D introduced in Eq. (29).

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In Fig. 12 we see the first few elements δk . Note that there are non-trivial multiplicities as in δ4 . Let [X, δn ] = δn+1 , [Y, δn ] = nδn . The following result, which is a trivial consequence of the results in the next section, initiated this paper: i) ii)

With the coproduct of HR , the δk span a closed Hopf subalgebra of HR . 1(δn ) = e ⊗ δn + δn ⊗ e + Rn−1 , R0 = 0, R1 = δ 1 ⊗ δ1 , Rk = [X ⊗ e + e ⊗ X, Rk−1 ] + kδ1 ⊗ δk + [δ1 ⊗ Y, Rk−1 ].

(57) (58) (59) (60)

The proof follows from the results in the next section, but it is instructive to investigate directly the compatibility of the operation of natural P growth and the notion of an admissible cut: We note that δn is a sum of trees: δn = i ti , say. Thus, 1(δn ) = e ⊗ δn + δn ⊗ e +

X

X

i

all cuts C i of ti

i

i

P C (ti ) ⊗ RC (ti ).

(61)

Hence we can write, with the same ti as before, 1(δn+1 ) = e ⊗ δn+1 + δn+1 ⊗ e X X n i i N [P C (ti )] ⊗ RC (ti ) + i

all cuts C i of ti i

i

o

+(P C (ti ) ⊗ N [RC (ti )] +nδ1 ⊗ δn X X + i

i

i

l[RC (ti )]δ1 P C (ti ) ⊗ RCi (ti ),

(62)

all cuts C i of ti

where l(t) gives the number of vertices of a tree t. Thus, we decomposed the cuts at the components of δn+1 in four classes: either the edge to the new grown vertex is not cut, then we will have natural growth on either the former P C or RC part. Thus, the first two contributions deliver the operator N on either side of the tensorproduct. Or, for the remaining two cases, the edge to the new grown vertex is cut. These cases will always have a factor δ1 on the lhs of the tensorproduct. In these cases, it either was grown from the former RC part (admissibility of cuts forbid that it was grown from the P C part), or it was grown from the whole uncut former δn , which gives the term nδ1 ⊗ δn . Hence we have decomposed the cuts possible at the trees of δn+1 in terms of the cuts at the trees of δn . Figure 13 gives an instructive example. To finally prove the result, we note the following identities:

Hopf Algebras, Renormalization and Noncommutative Geometry

N (δi1 . . . δik ) = [X, δi1 . . . δik ], l(δk )δk = kδk = [Y, δk ],

227

(63) (64)

where we note that l[δk ] = k is well-defined, as δk is a homogeneous combination of trees with k vertices.

Fig. 13. The decomposition of the cuts at δn+1 in terms of the cuts at δn and the operator N . The first two terms of the bottom line indicate natural growth on the P C or the RC part. The third term gives the contribution for the case that the natural growth carries a cut itself. This can only happen at the RC part, due to admissibility of cuts. The last two terms are generated by the remaining possibility that the natural growth carries the sole cut

At this stage, we begin to see a fundamental connection between the process of renormalization and the results of [2]. Thus, we will now set out to define the Hopf algebra of rooted trees more formally and repeat the analysis of [2] for it. We shall formalize the simplest example from the last section as the Hopf algebra of rooted trees, and extend many of the results of the first section to this more involved case. By a rooted tree T we mean a finite, connected, simply connected, one dimensional simplicial complex with a base point ∗ ∈ T (0) = {set of vertices of T }. This base point is called the root. By the degree of the tree we mean deg(T ) = CardT (0) = # of vertices of T.

(65)

For each n we have a finite set of rooted trees T with deg(T ) = n, where we only consider isomorphism classes of trees and choose a representative in each isomorphism class. Thus for n = 1 we have one element t1 ≡ ∗, for n = 2 we also have only one, t2 , and for n = 3 we have two, t31 and t32 , all defined in Fig. 1. By a simple cut of a tree T we mean a subset c ⊂ T (1) of the set of edges of T such that, for any x ∈ T (0) the path (∗, x) only contains at most one element of c.

(66)

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Thus what is excluded is to have two cuts of the same path or branch. Given a cut c the new simplicial complex Tc with Tc(0) = T (0) and Tc(1) = T (1) \c,

(67)

is no longer connected, unless c = ∅. We let Rc (T ) be the connected component of ∗ with the same base point and call it the trunk, while for each other connected component, called a cut branch, we endow it with the base point which is the edge of the cut. We obtain in this way a set, with multiplicity, of finite rooted trees. For each n we let 6n be the set of trees of degree ≤ n, up to isomorphism, and let Hn be the polynomial commutative algebra generated by the symbols, δT , T ∈ 6n . One defines a coproduct on Hn by, 1 δ T = δ T ⊗ 1 + 1 ⊗ δT +

X

 

c

(68) 

Y

δTi  ⊗ δRc (T ) ,

(69)

Pc (T )

where the last sum is over all non trivial simple cuts (c 6= ∅) of T , while the product

Y Pc (T )

is over the cut branches, in accordance with Eq. (51). Equivalently, one can write (69) as,   X Y  δTi  ⊗ δRc (T ) , ) 1 δT = δ T ⊗ 1 + c

(70)

Pc (T )

where the last sum is over all simple cuts. This defines 1 on generators and it extends uniquely as an algebra homomorphism, 1 : Hn → Hn ⊗ Hn .

(71)

Lemma 1. The coproduct 1 is coassociative. Proof. It is enough to check the equality (1 ⊗ 1) 1 δT = (1 ⊗ 1) 1 δT

∀ T ∈ 6n ;

(72)

one can do it directly by introducing the notion of a double cut of T , but we shall use instead the following map from HR = ∪ Hn to HR , L(δT1 . . . δTm ) = δT , ∀ Tj ∈ 6 = ∪6n ,

(73)

where T is the pointed tree obtained by connecting a new base point ∗ to the base points of the pointed trees Tj . The map L is the unique linear map from HR to HR satisfying (73). It agrees with the map B+ introduced in the previous section. Let us show that, 1 ◦ L = L ⊗ 1 + (id ⊗ L) ◦ 1. Let a = δT1 . . . δTm and T be as in (73) so that L(a) = δT . From (70), one gets,

(74)

Hopf Algebras, Renormalization and Noncommutative Geometry

1(L(a)) − L(a) ⊗ 1 =

XY c

Pc

229

δTi0 ⊗ δRc ,

(75)

where all simple cuts of T , (including c = ∅) are allowed. Moreover, 1(a) =

n Y

 δTi ⊗ 1 +

XY ci Pci

i=1

 δTi00 ⊗ δRci  ,

(76)

j

where again all simple cuts ci of Ti are allowed. Let tn be the tree with base point ∗ and n other vertices vi labelled from i = 1 to i = n, all directly connected to the base point ∗. We view tn in an obvious way as a subgraph of the tree T , where the base points are the same and the vertex vi is the base point of Ti . Given a simple cut c of T one gets by restriction to the subgraph tn ⊂ T a cut of tn , it is characterized by the subset I ⊂ {1, ..., n}, I = {i ; (∗, vi ) ∈ c}. The simple cut c is uniquely determined by the subset I and for each i ∈ I c , i.e. each branch (∗, vi ) of tn which is not cut, by the simple cut ci of Ti given by the restriction of c to this subgraph. Thus the simple cuts c of T are Q in one to one correspondence with the various Q Q terms of the expression (76), namely the k∈I δTk ⊗ 1 i∈I c Pc δTi00 ⊗ δRci . i

j

The two sums match termwise and, applying id ⊗ L to (76) one gets, 1(L(a)) = L(a) ⊗ 1 + (id ⊗ L) 1(a).

(77)

This is Eq. (50) of the previous section. (Note that L(1) = δ∗ by definition.) One has, 1 δ ∗ = δ ∗ ⊗ 1 + 1 ⊗ δ∗

(78)

so that H1 is coassociative. Let us assume that Hn is coassociative and prove it for Hn+1 . It is enough to check (72) for the generators δT , with deg(T ) ≤ n + 1 one has δT = L(δT1 . . . δTm ) = L(a), where the degree of all Tj is ≤ n, i.e. a ∈ Hn . Using (77) we can replace 1 δT by L(a) ⊗ 1 + (id ⊗ L) 1(a),

(79)

where 1 is the coassociative coproduct in Hn . Thus we can use the notation which encodes the coassociativity of Hn , 1a = a(1) ⊗ a(2) , (id ⊗ 1) 1(a) = (1 ⊗ id) 1(a) = a(1) ⊗ a(2) ⊗ a(3) .

(80)

The first term of (72) is then: L(a) ⊗ 1 ⊗ 1 + a(1) ⊗ 1 ◦ L a(2) , which by (77) gives L(a) ⊗ 1 ⊗ 1 + a(1) ⊗ L(a(2) ) ⊗ 1 + a(1) ⊗ a(2) ⊗ L a(3) .

(81)

The second term of (72) is 1 ◦ L(a) ⊗ 1 + 1a(1) ⊗ La(2) , which by (77) gives, L(a) ⊗ 1 ⊗ 1 + a(1) ⊗ L a(2) ⊗ 1 + a(1) ⊗ a(2) ⊗ L a(3) . Thus we conclude that 1 is coassociative.

(82)

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We shall now characterize the Hopf algebra HR = ∪ Hn as the solution of a universal problem in Hochschild cohomology. First, given an algebra A with augmentation ε, let us consider the Hochschild cohomology of A with coefficients in the following bimodule M. As a vector space M = A, the left action of A on M is (a, ξ) → aξ, for all a ∈ A, ξ ∈ M. The right action of A on M is by (ξ, a) → ξ ε(a), ξ ∈ M, a ∈ A. Thus the right module structure is through the augmentation. Let us denote the corresponding cocycles by Zεn (A), the coboundaries by Bεn (A) and the cohomology as Hεn (A). D Thus for instance a 1-cocycle D ∈ Zε1 (A) is a linear map A → A such that D(ab) = D(a) ε(b) + a D(b) ∀ a, b ∈ A. Next, given a Hopf algebra H we use the unit of H and its coalgebra structure to transpose (as in the Harrison cohomology), the above complex. More precisely an n-cochain L is a linear map, L : H → H ⊗ ... ⊗ H {z } |

(83)

n times

and the coboundary b is given by, (bL)(a) = (id ⊗ L) 1(a) − 1(1) L(a) + 1(2) L(a) + . . . + (−1)j 1(j) L(a)

(84)

+ . . . + (−1)n 1(n) L(a) + (−1)n+1 L(a) ⊗ 1, where the lower index (j) in 1(j) indicates where the coproduct is applied. For n = 0, L is just a linear form on H and one has (bL)(a) = (id ⊗ L) 1(a) − L(a) 1.

(85)

For n = 1, L is a linear map from H to H and (bL)(a) = (id ⊗ L) 1(a) − 1 L(a) + L(a) ⊗ 1 ∈ H ⊗ H.

(86)

We shall use the notation Zεn (H∗ ), Hεn (H∗ ) . . . for the corresponding cocycles, cohomology classes, etc . . . . Theorem 2. There exists a pair (H, L), unique up to isomorphism, where H is a commutative Hopf algebra and L ∈ Zε1 (H∗ ) which is universal among all such pairs. In other words for any pair (H1 , L1 ) where H1 is a commutative Hopf algebra and L ∈ Zε1 (H1∗ ), ρ there exists a unique Hopf algebra morphism H → H1 such that L1 ◦ ρ = ρ ◦ L. Proof. Let HR be the Hopf algebra of rooted trees and L be the linear map defined by (73). The equality (74) shows that bL = 0. This shows that L is a 1-cocycle. It is clear that it is not a coboundary, indeed one has L(1) = δ∗ = 6 0,

(87)

where ∗ is the tree with only one vertex. Moreover, for any coboundary T = bZ one has T (1) = 0,

(88)

since T (1) = Z(1) 1 − (id ⊗ Z) 1(1) = 0. Next consider a pair (H1 , L1 ) where H1 is a commutative Hopf algebra and L1 ∈ Zε1 (H1∗ ) is a 1-cocycle. The equality L1 ◦ ρ = ρ ◦ L uniquely determines an algebra homomorphism ρ : HR → H1 . Indeed on the linear basis 5 δTi of HR one must have,

Hopf Algebras, Renormalization and Noncommutative Geometry

231

ρ (5 δTi ) = 5 ρ (δTi ),

(89)

by multiplicativity of ρ, while ρ (δT ) is determined by induction by ρ (δ∗ ) = L1 (1), and, ρ (L (5 δTi )) = L1 ρ (5 δTi ).

(90)

We need to check that it is a morphism of Hopf algebras, i.e. that it is compatible with the coproduct, (ρ ⊗ ρ) (1(a)) = 11 ρ(a) ∀ a ∈ HR .

(91)

It is enough to check (91) on generators of the form δT = L(5 δTi ). To do this one uses the cocycle property of L1 which allows to write, 11 L1 (ρ (5 δTi )) = L1 (ρ (5 δTi )) ⊗ 1 + (id ⊗ L1 ) 11 ρ (5 δTi ).

(92)

One uses an induction hypothesis on the validity of (91), to write, (id ⊗ L1 ) 11 ρ (5 δTi ) = (ρ ⊗ (ρ ◦ L)) 1(5 δTi ),

(93)

making use of the identity ρ ⊗ ρ ◦ L = (id ⊗ L1 ) (ρ ⊗ ρ). Thus one has, 11 L1 (ρ (5 δTi )) = L1 (ρ (5 δTi )) ⊗ 1 + (ρ ⊗ (ρ ◦ L)) 1(5 δTi ),

(94)

and the validity of (91) for a = δT follows from the equality (ρ ⊗ ρ) 1(L(5 δTi )) = (ρ ⊗ ρ) (L (5 δTi )) ⊗ 1 + (id ⊗ L) 1 (5 δTi )). We have thus shown the existence and uniqueness of the Hopf algebra morphism ρ. As the simplest example, let H1 be the Hopf algebra of polynomials P (δ1 ), as above, with, 1 δ 1 = δ 1 ⊗ 1 + 1 ⊗ δ1 .

(95)

The cohomology group Hε1 (H1∗ ) is one dimensional, and the natural generator is the cocycle, Z x P (a)da ∀P = P (δ1 ) ∈ H1 . (96) L1 (P )(x) = 0

The cocycle identity follows from the equality, Z x Z Z x+y P (a)da = P (a)da + 0

0

The coboundaries are of the form, Z L0 (P ) = (P (x + a) − P (a))φ(a)da

y

P (x + a)da.

(97)

0

∀P = P (δ1 ) ∈ H∞ ,

(98)

where φ is a distribution with support in the origin, and possibly infinite order. The tranpose ρt of the morphism of Hopf algebras given by Theorem 2 determines a Lie algebra homomorphism from the one dimensional Lie algebra (A11 with the notations of Sect. 1), to the Lie algebra L1 which corresponds, by the Milnor–Moore theorem to the commutative Hopf algebra HR .

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We proceed as in Sect. 1 to determine L1 . Let L∞ be the linear span of the elements ZT , indexed by rooted trees. We introduce an operation on L1 by the equality, X n(T1 , T2 ; T ) ZT , (99) ZT1 ∗ ZT2 = T

where the integer n(T1 , T2 ; T ) is determined as the number of simple cuts c of cardinality 1 such that the cut branch is T1 while the remaining trunk is T2 .2 Theorem 3. a) Let L1 be the linear span of the elements ZT , indexed by rooted trees. The following equality defines a structure of Lie algebra on L1 . The Lie bracket [ZT1 , ZT2 ] is ZT1 ∗ ZT2 − ZT2 ∗ ZT1 . b) The Hopf algebra HR is the dual of the enveloping algebra of the Lie algebra L1 . Define A (T1 , T2 , T3 ) = ZT1 ∗ (ZT2 ∗ ZT3 ) − (ZT1 ∗ ZT2 ) ∗ ZT3 .

(100)

We shall need the following lemma, Lemma 4. One has A(T1 , T2 , T3 ) = 6 n(T1 , T2 , T3 ; T ) ZT , where the integer n is the number of simple cuts c of T , |c| = 2 such that the two branches are T1 , T2 while Rc (T ) = T3 . Proof. When one evaluates (100) against ZT one gets the coefficient, X X n(T1 , T 0 ; T ) n(T2 , T3 ; T 0 ) − n(T1 , T2 ; T 00 ) n(T 00 , T3 ; T ), T0

(101)

T 00

the first sum corresponds to pairs of cuts, c, c0 of T with |c| = |c0 | = 1 and where c0 is a cut of Rc (T ). These pairs of cuts fall in two classes, either c ∪ c0 is a simple cut or it is not. The second sum corresponds to pairs of cuts c1 , c01 of T such that |c1 | = |c01 | = 1, Rc1 (T ) = T3 and c01 is a cut of Pc1 (T ). In such a case c1 ∪ c01 is never a simple cut so the difference (101) amounts to subtract from the first sum the pairs c, c0 such that c ∪ c0 is not a simple cut. This gives, X n(T1 , T2 , T3 ; T ) ZT , (102) A (T1 , T2 , T3 ) = T

where n(T1 , T2 , T3 ; T ) is the number of simple cuts c of T of cardinality 2 such that the two cut branches are T1 and T2 . It is thus clear that A (T1 , T2 , T3 ) = A (T2 , T1 , T3 ).

(103)

One then computes [[ZT1 , ZT2 ], ZT3 ] + [[ZT2 , ZT3 ], ZT1 ] + [[ZT3 , ZT1 ], ZT2 ]. One can write it, for short, as a sum of 24 terms, 2 The reader shall not confuse the operation which relates T and T to T with the transplantation used in 1 2 the theory of operads. Indeed, in the latter, the root of the tree T1 is restricted to be the end of a branch of T2 .

Hopf Algebras, Renormalization and Noncommutative Geometry

233

(T1 ∗ T2 ) ∗ T3 − (T2 ∗ T1 ) ∗ T3 − T3 ∗ (T1 ∗ T2 ) + T3 ∗ (T2 ∗ T1 ) + (T2 ∗ T3 ) ∗ T1 − (T3 ∗ T2 ) ∗ T1 − T1 ∗ (T2 ∗ T3 ) + T1 ∗ (T3 ∗ T2 ) + (T3 ∗ T1 ) ∗ T2 − (T1 ∗ T3 ) ∗ T2 − T2 ∗ (T3 ∗ T1 ) + T2 ∗ (T1 ∗ T3 ) = −A (T1 , T2 , T3 ) + A (T2 , T1 , T3 ) − A (T3 , T1 , T2 ) + A (T3 , T2 , T1 ) −A (T2 , T3 , T1 ) + A (T1 , T3 , T2 ) = 0. b) For each rooted tree T let us define a linear form ZT on HR by the equality, hZT , P (δTi )i = (∂/∂δT P )(0) .

(104)

Thus ZT vanishes when paired with any monomial δTn11 . . . δTnkk except when this monomial is δT while, hZT , δT i = 1.

(105)

Since P → P (0) is the counit of HR and since ZT satisfies hZT , P Qi = hZT , P i ε(Q) + ε(P ) hZT , Qi,

(106)

it follows that the coproduct of ZT is, 1 ZT = ZT ⊗ 1 + 1 ⊗ ZT ,

(107)

∗ is defined, when it makes sense, by dualizing the product where the coproduct on HR of HR . ∗ is defined by Similarly the product of two elements of HR

hZ1 Z2 , P i = hZ1 ⊗ Z2 , 1 P i.

(108)

∗ of elements Since the bracket of two derivations is still a derivation, the subspace of HR satisfying (102) is stable under bracket. What remains is to show that,

ZT1 ZT2 − ZT2 ZT1 = [ZT1 , ZT2 ],

(109)

where the r.h.s. is defined by the Lie algebra structure of Theorem 3. Let H0 be the augmentation ideal of HR , H0 = Ker ε. The formula defining the coproduct in HR shows that, 1 δT = δ T ⊗ 1 + 1 ⊗ δ T + RT ,

(110)

where RT ∈ H0 ⊗ H0 . In fact one can compute RT modulo higher powers of H0 , i.e. modulo H02 ⊗ H0 , it gives, X δTc0 ⊗ δTc , (111) RT(0) = c

where c varies among single cuts of the tree T , where Tc is the part of T that contains the base point, while Tc0 is the tree which remains. When one computes hZT1 ZT2 , 5 δTi i = hZT1 ⊗ ZT2 , 5 1 δTi i the part which is not symmetric in T1 , T2 is zero unless 5 δTi is equal to a single δT . When one computes hZT1 ZT2 , δT i = hZT1 ⊗ ZT2 , 1 δT i,

(112)

the only part which contributes comes from RT(0) and it counts the number of ways of obtaining T from T1 and T2 , which gives (109).

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Proposition 5. The equality degree ZT = # of vertices of T defines a grading of the Lie algebra L1 . Proof. The number of vertices of any tree obtained by gluing T1 to T2 is the sum of the number of vertices of T1 and T2 . We shall now show how to extend the Hopf algebra HR to include the generators X, Y of the Lie algebra of the affine group as in Sect. 1. The commutator of Y with δT will simply be given by, [Y, δT ] = deg(T )δT ,

(113)

i.e. by the above grading. The commutator with X will generate a derivation N of HR , uniquely determined by its value on the generators δT , by X δT 0 , (114) N δT = where the trees T 0 are obtained by adding one vertex and one edge to T in all possible ways without changing the base point. It is clear that the sum (114) contains deg(T ) terms. Using the derivation property of N , one has, ! n n Y X δTi = δT1 . . . N (δTi ) . . . δTn . (115) N 1

1

Our first task will be to get a formula for 1 N (δT ). Proposition 6. For any a ∈ HR one has 1 N a = (N ⊗ id) 1 a + (id ⊗ N ) 1 a + [δ1 ⊗ Y, 1 a]. Proof. First, it is enough to check the equality when a = δT . Indeed, both 1 ◦ N and (N ⊗ id + id ⊗ N + ad (δ1 ⊗ Y )) ◦ 1 are derivations from HR to the HR -bimodule HR ⊗ HR (using 1 to define the bimodule structure). Thus so is their difference ε0 which vanishes provided it does on the generators δT . Let thus T be a pointed tree and T 0 be obtained from T by adjoining an edge at v0 ∈ 10 (T ). One has X (5 δTj0 ⊗ δR0 0 ), (116) 1 δ T 0 = δ T 0 ⊗ 1 + 1 ⊗ δT 0 + c

c0

where c0 ∈ 11 (T 0 ) varies among the simple cuts of T 0 . One has 11 (T 0 ) = 11 (T ) ∪ {ε}, where ε is the new edge. Now the cuts c0 for T 0 are of two kinds, (A) The new edge is not cut, (A’) It is cut. There is also another dichotomy, (B) The vertex v0 belongs to the trunk, (B’) It belongs to one of the cut branches. If we sum (116) over all possible T 0 we get, XX 5 δTi0 ⊗ δRc0 . (117) 1 N δ T = N δ T ⊗ 1 + 1 ⊗ N δT + v

c0

Let us concentrate on the last term and consider first only the cuts c0 which satisfy (A). We also consider the term,

Hopf Algebras, Renormalization and Noncommutative Geometry

X

235

(5 δTc ) ⊗ δRc

(118)

c

over all the cuts c of the tree T . If we apply (id ⊗ N ) to (118), we obtain all possible cuts of a T 0 such that (A) (B) holds so that, X X = (id ⊗ N ) (5 δTc ) ⊗ δRc . (119) c

(A) (B)

It follows that, X

= (N ⊗ id)

X

(5 δTc ) ⊗ δRc .

(120)

c

(A) (B 0 )

We can thus summarize what we obtained so far by, 1 N δT = (N ⊗ id) 1 δT + (id ⊗ N ) 1 δT +

X

.

(121)

(A0 )

Now consider the sum

X

, the first case is when the only cut is the cut of the new edge.

(A0 )

The only cut branch gives us a δ1 and the number of ways of doing it is n = deg T , thus we get [δ1 ⊗ Y, δT ⊗ 1 + 1 ⊗ δT ] , [Y, δT ] = n δT .

(122)

The next case is when a non trivial cut c remains after we remove the new edge. For that cut c the new vertex necessarily belongs to the trunk (so that (A) (B) is excluded) as follows from the very definition of a cut. For such cuts, the result is to get an additional δ1 among the δTi , which comes from the cut new edge. The number of ways of doing it is exactly the degree of the trunk. Thus we get X (5 δTc ) ⊗ δRc ] . (123) [δ1 ⊗ Y, c

Combining (122) and (123) we get, X

= [δ1 ⊗ Y, 1 δT ].

(124)

(A0 )

This is enough to assert that for any tree T one has, 1 N δT = (N ⊗ id) 1 δT + (id ⊗ N ) 1 δT + [δ1 ⊗ Y, 1 δT ] which ends the proof of Proposition 6.

(125)

eR by adjoining the elements X, Y with In other words we can enlarge HR to H [X, a] = N (a), [Y, a] = (deg a) a ∀ a ∈ HR , [Y, X] = X, 1Y = Y ⊗ 1 + 1 ⊗ Y, 1X = X ⊗ 1 + 1 ⊗ X + δ1 ⊗ Y. ∗ . Let us translate Proposition 6 in terms of the transposed map N t acting on HR

(126)

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One has

hN t (AB), ai = hAB, N (a)i =

hA ⊗ B, 1 N (a)i = hA ⊗ B, (N ⊗ id + id ⊗ N + δ1 ⊗ deg) 1ai = hN t (A) ⊗ B + A ⊗ N t (B) + (δ1 )t ⊗ degt (A ⊗ B), 1ai = hN t (A) B + A N t (B) + δ1t (A) degt (B), ai ; thus, N t (AB) = N t (A) B + A N t (B) + δ1t (A) degt (B),

(127)

where δ1t (resp. degt ) is the transposed of the multiplication by δ1 (resp. deg) hδ1t A, ai = hA, δ1 ai.

(128)

One has hδ1t (AB), ai = hAB, δ1 ai = hA⊗B, 1 δ1 1 ai = hA⊗B, (δ1 ⊗1+1⊗δ1 ) 1 ai. Thus, δ1t (AB) = δ1t (A) B + A δ1t (B),

(129)

i.e. δ1t is a derivation. Moreover on the generator ZT , δ1t (ZT ) = 0

unless T = {∗}, δ1t (Z1 ) = 1.

(130)

Indeed, hZT , δ1 ai = 0 unless T = {∗}, while for T = {∗} one gets that hZ1 , δ1 ai = ε(a). Thus, δ1t =

∂ , ∂ Z1

(131)

wherePwe use the Poincar´e–Birkhoff–Witt theorem to write elements of U(L1 ) in the form 5 ZTi Z1a . Let us compute N t (ZT ), where T is a tree with more than one vertex. One has hN t ZT , δT1 δT2 . . . δTn i = hZT , N (δT1 . . . δTn )i, and this vanishes unless n = 1. Moreover for n = 1, hZT , N (δT1 )i = n(T ; T1 ),

(132)

where n(T ; T1 ) is the number of times the tree T is obtained by adjoining an edge and vertex to T1 . Thus one has, X n(T ; T1 )ZT1 , N t Z1 = 0. (133) N t ZT = We can now state the analogue of Lemma 4 of Sect. 1 as follows, where we let Lk be the Lie subalgebra of L1 generated by the ZT with deg(T ) ≥ k. Lemma 7. When restricted to U(L2 ), N t is the unique derivation, with values in U(L1 ) satisfying (133), moreover, for deg(Ti ) > 1 and A = 5ZTi one has N t (A Z1a1 ) = N t (A) Z1a1 + A

a1 (a1 − 1) a1 −1 Z1 . 2

Hopf Algebras, Renormalization and Noncommutative Geometry

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Proof. The first statement follows from (127) and (129). The second statement follows from, N t (Z1m ) = which one proves by induction on m.

m(m − 1) m−1 Z1 2

(134)

Motivated by Sect. 1 and the first part of the lemma, we enlarge the Lie algebra L1 by adjoining two elements Z0 and Z−1 such that, [Z−1 , Z1 ] = Z0 , [Z0 , ZT ] = deg(T ) ZT , X [Z−1 , ZT ] = n(T ; T1 )ZT1 ∀ T, deg(T ) > 1. The obtained Lie algebra L is an extension of the Lie algebra of formal vector fields xn+1 ∂ with Z0 = x ∂∂x , Z−1 = ∂∂x and as above Zn = (n+1)! ∂ x , as follows from, Theorem 8. The following equality defines a surjective Lie algebra homomorphism from L to A, 2(ZT ) = n(T )Zn , 2(Zi ) = Zi , i = 0, 1, where n(T ) is the number of times δT occurs in N deg(T )−1 (δ1 ). eR fulfill the presentation of Proof. The elements X, Y , and δ∗ of the Hopf algebra H eT , thus there exists a unique homomorphism of Hopf Sect. 1 for the Hopf algebra H eT such that, eR to H algebras h from H h(X) = X,

h(Y ) = Y,

h(δ1 ) = δ∗ .

By construction, h restricts to the subalgebra HR and defines a homomorphism to the Hopf algebra HT . Transposing this homomorphism to the Lie algebras, one obtains the restriction of 2 to the subalgebra L1 . At this stage we completed our understanding of the relation between the two Hopf algebras. It is best expressed by the Lie algebra homomorphism 2 from L1 to A1 . Its extension to the full L justifies the construction of the latter Lie algebra. By Theorem 3 the Hopf algebra HR should be thought of as the algebra of coordinates on a nilpotent formal group G whose Lie algebra is the graded Lie algebra L1 . Given a field K, elements of the group GK are obtained precisely as the characters of the algebra HR ⊗Q K. Indeed, such characters correspond to group-like elements u (i.e. elements u satisfying 1(u) = u ⊗ u) of a suitable completion of the enveloping algebra of L1 . Viewing u as a linear form on HR gives us the desired character. If we let K be the field of formal power series in a variable we thus obtain as points of GK the homomorphisms from HR to K. It is not difficult to check that the map which to every bare Feynman diagram 0 associates the corresponding Laurent expansion (in Dimensional Regularization, say, with regularized dimension D = 4 − 2, in four dimensions, say) is precisely such a character. This allows to reduce by the above conceptual mathematical structure of inversion in G the computation of renormalization in QFT to the primitive elements of the Hopf algebra, i.e. to Feynman diagrams without subdivergences.

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In order to better understand the extension of the group of diffeomorphisms provided by the group G, it would be desirable to find a non-commutative manifold X, whose diffeomorphism group is G. The coordinates δn = −(log(ψ 0 (x))(n) of a diffeomorphism ψ allow to reconstruct the latter by the formula Z x X δn un )du. exp(− ψ(x) = n! 0 This formula provides the clear meaning both for composition and inversion of diffeomorphisms. Of course, we would love to have a similar formula for the group G and it is tantalizing to consider the Feynman integral Z X L0 ) exp(−L0 + 0

as a direct analogue of the above expression. 4. Appendix 4.1. φ3 theory and overlapping divergences. A prominent problem in renormalization theory is the presence of overlapping divergences. We will soon see that to Green functions which suffer from such overlapping divergences we will have to associate a sum of trees, while so far our experience only lead to the identification of single trees with a given Green function. We will proceed by studying the example of φ3 theory in six dimensions. A full study will be given elsewhere [7], but we also mention that solutions to the problem of overlapping divergences were already found in [5], using combinatorical considerations concerning divergent sectors, in [1] and [6] using Schwinger Dyson equations, and were also known to others. In [7] we will show how overlapping divergences give rise to a slightly modified Hopf algebra, which eventually turns out to be identical to the Hopf algebra of rooted trees considered here. We sketch this more formal argument after the consideration of φ3 theory as an example. In whatever approach one takes, the final message is the same: Overlapping divergent functions can be resolved in sums of functions having only nested and disjoint divergences. To see how this comes about, we will here employ yet another approach, using differential equations on bare Green functions. Green functions in φ36 theory which are overall divergent are provided by two- and [3] three-point functions, to which we refer as G[2] n (q; m) and Gn (p, q; m). Here the subscript n refers to the number of loops in the Green-function, and m is the mass of the propagator, while p, q are external momenta. We first consider G[3] n (p, q; m): Z G[3] n (p, q; m) =

d6 l1 . . . d6 ln

3n Y 1 . Pi i:=1

For n ≥ 1, it is a product of 3n propagators Pi = 1/(ki2 − m2 + iη), where the ki are momentum vectors which are linear combinations of external momenta p, q and n internal momenta l1 , . . . , ln such that momentum conservation holds at each vertex.

Hopf Algebras, Renormalization and Noncommutative Geometry

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As each propagator 1/Pi contributes with weight two to the powercounting, we find that G[3] is overall logarithmic divergent, 3 × n × 2 − 6 × n = 0. For each Pi , let Pi := ki2 + iη, so that Pi = Pi − m2 . [3] Then, one immediately sees that G[3] n (p, q; m) − Gn (p, q; 0) is overall convergent. This follows directly from powercounting in the expression Q 3n Q 2 Y j Pj − j (Pj − m ) . Pi Pi i:=1 Thus, to determine the counterterm for a vertex function, it suffices to consider the massless case. 3 Hence all possible subdivergences of G[3] n (p, q; 0) are given by functions of the type [3] Gr (ki , kj ; 0) and G[2] s (ki ; 0), with s < n and r < n. In the context of φ3 theory in six dimensions, overlapping divergences can only be provided by two-point functions. The only circumstance which stops us to assign a unique tree to G[3] n (p, q; m) is the fact that there might be overlapping subdivergences provided by massless two-point functions G[2] s (ki ; 0), s < n. Before we handle these subdivergences, we turn to G[2] n (q; m) itself. At n loops, it consists of 3n − 1 propagators Z G[2] n (q; m) =

d6 l1 . . . d6 ln

3n−1 Y

1 . Pi

i:=1

Consider the difference Z G[2] n (q; m)

−

G[2] n (q; 0)

=m

2

d6 l1 . . . d6 ln

3n−1 Y i:=1

3n−1 1 X 1 Pi Pj j:=1

+overall finite terms which is of overall logarithmic degree of divergence. As far as the overall counterterm is concerned, we can even nullify masses in this difference and thus find that the divergences of G[2] n (q; m) can be separated as Z 2 G[2] n (q; m) = m

d6 l1 . . . d6 ln

3n−1 Y i:=1

+G[2] n (q; 0)

3n−1 1 X 1 Pi j:=1 Pj

+ U (q, m),

where U (q; m) collects all the overall finite terms. The first term on the rhs is overall logarithmic divergent. It only can provide overlapping divergences through massless functions G[2] s (q; 0) appearing as subgraphs in it, quite similar to the analysis of the vertex function, as the sum over j squares one propagator in turn. We have thus reduced all appearances of overlapping divergences to the presence of functions G[2] i (q; 0), i ≤ n. It remains to show how the overlapping divergences in G[2] n (q; 0) can be handled for all n. 3 Even better, again using powercounting, one immediately shows that it is sufficient to consider G[3] n (0, q; 0).

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δδ

δδ

=

=

Fig. 14. The resolution of overlapping divergences and the resulting sum of trees. A double derivative with respect to the external momentum resolves the graph in contributions each of which is free of overlapping divergences. We indicate by crosses on propagators the places where the derivative acts, for a chosen momentum flow

This is actually not that difficult. Necessarily, G[2] n (q; 0) has the form 2 1−n FGn (), G[2] n (q; 0) = (q )

where FGn () is a Laurent series in . Hence G[2] n (q; 0) fulfills the differential equation ∂ ∂ [2] 1 G (q; 0) = G[2] q2 n (q; 0). 2D(1 − n) ∂qµ ∂q µ n

Hopf Algebras, Renormalization and Noncommutative Geometry

241

This solves the problem. The remaining source of overlapping divergences, G[2] n (q; 0), ∂ 2 ∂ is expressed in terms of the overall logarithmic divergent function q ∂qµ ∂qµ G[2] n (q; 0) which is free of overlapping divergences. Such an approach is also very useful in practice [6]. Figure 14 gives two examples for the resolution of overlapping divergences. Crosses in the figure indicate where the derivatives with respect to q act for a chosen momentum flow through the graph. A general argument. So far, we decomposed graphs which have overlapping divergences into a sum of contributions each of which delivers a rooted tree. Thus, overlapping divergences correspond to a linear combination of rooted trees, while any Feynman diagram without overlapping divergences corresponds to a single rooted tree. One might suggest to enlarge the Hopf algebra HR of rooted trees to another Hopf algebra, HO say, so that HO directly contains elements which correspond to graphs with overlapping divergences [8]. Let us at this stage mention a general fact which shows that any such Hopf algebra HO is nothing else than the Hopf algebra of rooted trees. If we take into account the decorations of vertices by Feynman diagrams without subdivergences, any such Hopf algebra HO is a Hopf algebra HR for an appropriate set of decorations. Consider a Feynman graph 0 which has overlapping subdivergences, but in a way that any of its divergent subgraphs γ ⊂ 0 and any of the complementary graphs 0/γ is free of overlapping subdivergences. The first example in Fig. 14 is of this type. The cases we have excluded here will be handled later by a recursive argument. We want to construct a Hopf algebra HO which contains a single element t0 such that the antipode S(t0 ) delivers the counterterm without making recourse to the methods of the previous paragraph to disentangle t0 first as a sum of trees ti in some decorated algebra HR . The question is: Could such an algebra have a structure different from HR ? Now, as HO shall also be able to treat Feynman diagrams which only have nested or disjoint subdivergences, it will contain the Hopf algebra of rooted trees as a subalgebra. Let us actually construct HO by fairly general arguments. Let HR ⊂ HO be given, and let in particular all Feynman graphs without subdivergences be identified. Hence all possible decorations, and thus all primitive elements of HR are assumed to be determined. Note that the primitive elements of HO are identical with the primitive elements of HR as graphs with overlapping divergences necessarily contain subdivergences, and thus do not provide primitive elements per se. A Feynman graph 0 chosen as above has only subgraphs which can be described by proper rooted trees. Thus, its coproduct in HO will have the general form X t(γ) ⊗ t(0/γ), 1(t0 ) = t0 ⊗ e + e ⊗ t0 + γ

where the sum is over all subgraphs of 0, while t(γ) and t(0/γ) are the rooted trees assigned to the corresponding graphs. By the constraints which we imposed on 0 this is always possible. In HO we consider the above equation as the definition for the coproduct on elements t0 6∈ HR ⊂ HO . On the rhs of the above coproduct, the only part which is not in HR ⊗ HR is t 0 ⊗ e + e ⊗ t0 ,

242

and we write

A. Connes, D. Kreimer

1(t0 ) = t0 ⊗ e + e ⊗ t0 + R0 ,

with R0 ∈ HR ⊗ HR . Now, we know that there exists an element T0 ∈ HR such that 1(T0 ) = T0 ⊗ e + e ⊗ T0 + R0 . This element T is just the linear combination of rooted trees constructed in the previous section, but its existence can be established on general grounds from the consideration of maximal forests [5, 7]. Finally we set U := t0 − T0 and calculate 1(U ) = U ⊗ e + e ⊗ U. Now, if U is superficially divergent at all it is a primitive element. It thus can be described by the rooted tree t1 . To be able to do so we only have to enlarge the algebra HR to contain the decoration U . An easy recursion argument finally allows to drop the constraint on 0 [7]. One concludes that any Hopf algebra which contains HR but also elements t0 6∈ HR is isomorphic to the algebra of rooted trees HR with an enlarged set of primitive elements. In Fig. 14 we see some contributions which only generate the tree t1 . They correspond to such new primitive elements. A detailed version of this argument will be given elsewhere [7]. Acknowledgement. D.K. thanks the I.H.E.S., Bures-sur-Yvette, for hospitality during a stay Jan.–Feb. 1998 and the theory group at the CPT (Marseille) for interest and discussions. Also, support by a Heisenberg Fellowship of the DFG for D. K. is gratefully acknowledged. D. K. thanks R. Stora for motivating the investigation which is reported in [7].

References 1. Kreimer, D.: On the Hopf Algebra Structure of Perturbative Quantum Field Theories. Adv. Theor. Math. Phys.2.2, 303–334 (1998); q-alg/9707029 2. Connes, A., Moscovici, H.: Hopf Algebras, Cyclic Cohomology and the Transverse Index Theorem. IHES publication IHES/M/98/37 (1998); Commun. Math. Phys. 198, 198–246 (1998) 3. Dixmier, J.: Existence de traces non normales. C.R.Acad.Sci.Paris, Ser. A–B 262, A1107–A1108 (1966) 4. Collins, J.C.: Renormalization. Cambridge: Cambridge Univ. Press. 1984 5. Kreimer, D.: Renormalization and Knot Theory. J. Knot Th. Ram. 6, 479–581 (1997), q-alg/9607022 6. Broadhurst, D.J., Delbourgo, R., Kreimer, D.: Unknotting the polarized vacuum of quenched QED. Phys. Lett. B366, 421 (1996), hep-ph/9509296 7. Kreimer, D.: On Overlapping Divergences. hep-th/9810022 8. Krajewski, T., Wulkenhaar, R.: On Kreimer’s Hopf Algebra Structure of Feynman Graphs. CPT-98/P.3639; hep-th/9805098 Communicated by A. Jaffe

Commun. Math. Phys. 199, 243 – 256 (1998)

Communications in

Mathematical Physics © Springer-Verlag 1998

Analyse de Scattering d’un Op´erateur Cubique de Heun dans l’Espace de Bargmann Abdelkader Intissar Equipe d’Analyse Spectrale URA-CNRS, N◦ 2053, Universit´e de Corte, Quartier Grossetti, 20250 Corte, France Received: 20 November 1996 /Accepted: 18 February 1998

Abstract: The boundary conditions at infinity are used in a description of all maximal dissipative extensions in Bargmann space of the minimal Heun’s operator H = d2 2 d z dz 2 + z dz . The characteristic functions of the dissipative extensions are computed. Completeness theorems are obtained for the system of generalized eigenvectors.

1. Position du Probl`eme o n R 2 Soit E = ϕ : C −→ C analytique; C e−|z| |ϕ(z)|2 dx dy < ∞; ϕ(0) = 0 l’espace de Bargmann et ek (z) =

zk √ k!

la base canonique de cet espace. Le op´eraeurs d’annihilation

d de et de cr´eation A et A agissant sur l’espace de Bargmann sont respectivement A = dz 0 ∗ ∗ domaine D(A) = {ϕ ∈ E; ϕ ∈ E} et A = z de domaine D(A ) = {ϕ ∈ E; z.ϕ ∈ E}. On consid`ere l’´equation biconfluente de Heun dont l’expression g´en´erale [7, 17] s’´ecrit sous la forme: ∗

1 zu00 + z 2 + bz + a + 1 u0 + [(2 + a − c)z + d + (1 + a)b] u = 0 2 o`u a, b, et c sont des constantes complexes. Dans l´espace de Bargmann le hamiltonien attach´e a` cette derni`ere e´ quation n’est autre que: H = Ha.b.c =

1 (2 + a − c)A∗ + (1 + a)A + bA∗ A + A∗ (A + A∗ )A. 2

µ o`u µ est l’intercept de Pomeron, Pour a = −1, c = 1 et b imaginaire pur avec b = iλ 2 λ le triple coupling de Pomeron et i = −1, l’operateur H−1,b,1 not´e Hµ,λ caract´erise la th´eorie des champs de reggeons a` un seul site [9]. Dans ce cas, plusieurs propri´et´es spectrales de Hµλ sont e´ tablies dans [1, 2, 4] et [10–13]. On a en particulier:

244

1) 2) 3) 4) 5) 6) 7)

A. Intissar

Pour µ 6= 0, Hµ,λ est le fermeture de sa restriction aux polynˆomes. Pour µ 6= 0, Hµ,λ a un inverse de classe de Carleman d’ordre 1 + ε pour tout ε > 0. Pour µ 6= 0, les valeurs propres de Hµ,λ sont r´eelles. Pour µ > 0, Re hHµ,λ ϕ, ϕi ≥ µkϕk2 ∀ϕ ∈ D(Hµ,λ ). Pour µ > 0, Hµ,λ engendre un semi-groupe compact. Pour µ > 0, Hµ,λ admet au moins une valeur propre. Pour µ > 0, la plus petite valeur de Hµ,λ se prolonge en une fonction analytique positive par rapport a` µ. N´eanmoins la densit´e des vecteurs propres g´en´eralis´es de Hµ,λ dans l’espace de Bargmann est encore ouverte.

Pour a = −1, c = 1 et b = 0, Ando et Zerner ont commenc´e l’´etude math´ematique de l’op´erateur limite H−1,0,1 not´e H. On r´esume d’abord certains de leur r´esultats: a) Contrairement a` ce qui se passe pour µ 6= 0, l’op´erateur H et la fermeture de sa restriction aux plynˆomes ne coincident pas. b) H poss`ede un inverse a` droite. c) H est surjectif. d) H poss`ede une valeur propre. Dans ce travail, en utilisant l’arsenal de la th´eorie des op´erateurs aux diff´erences [6] chapitre VII, l’article [18] sur les extentions auto-adjointes des matrices de Jacobi de type cercle-limite et les techniques de Scattering de Pavlov sur la construction d’une dilatation auto-adjointe d’un op´erateur dissipatif de Schr¨odinger [16], on pr´esente quelques nouvelles popri´et´es spectrales de l’op´erateur H = H−1,0,1 . En particulier, on caract´erise son domaine minimal, puis on pr´esente une description de toute extension dissipative maximale d’un tel op´erateur. Enfin, on donne un th´eor`eme de compl´etude des vecteurs propres g´en´eralis´es d’une telle extension. 2. Caract´erisation du Domaine Minimal de l’Op´erateur H Soit ϕ un e´ l´ement de l’espace de Bargmann, on peut alors l’´ecrire sous la forme: ϕ(z) =

∞ X

zk o`u ek (z) = √ est la base canonique de E. k!

ϕk ek (z)

k=1

Une expression de H est donc donn´ee par: ( √ (Hϕ)1 = 2ϕ2 √ √ (Hϕ)k = (k − 1) k ϕk−1 + k k + 1 ϕk+1 On d´efinit:

( −l2 =

(ϕk )∞ k=1 ;

∞ X

pour k ≥ 2.

) |ϕk | < ∞ 2

k=1

et son produit scalaire par hϕ, ψi =

P∞

k=1

ϕk ψ k

√ −[ϕ, ψ]k = k k + 1 ϕk ψ k+1 − ϕk+1 ψ k ; On v´erifie facilement que:

k ≥ 1.

Analyse de Scattering d’un Op´erateur Cubique de Heun n X

245

(Hϕ)k ψ k − ϕk (Hψ)k = −[ϕ, ψ]n .

k=1

Soient Dmax = {ϕ ∈ l2 ; Hϕ ∈ l2 } et Hmax l’op´erateur agissant sur Dmax tel que Hmax ϕ = Hϕ. Alors on a: Lemme 1. Pour ϕ ∈ Dmax et ψ ∈ Dmax , la limite de [ϕ, ψ]k existe lorsque k tend vers l’infini. ∈ l2 et donc les suites D´emonstration. Pour ϕ ∈ Dmax et ψ ∈ Dmax on a Hϕ ∈ l2 et Hψ Pk (Hϕ)j ψ j et ϕj (Hψ)j sont dans l1 . Il en r´esulte que Lim j=1 (Hϕ)j ψ j −ϕj (Hψ)j existe lorsque k tend vers l’infini. Cons´equence. Si on pose Lim[ϕ, ψ]k = [ϕ, ψ]∞ lorsque k tend vers l’infini alors hHmax ϕ, ψi − hϕ, Hmax ψi = −[ϕ, ψ]∞ . Soit P0 l’ensemble des ϕ dans l2 ayant un nombre fini non nul de composantes, alors H|P0 est un op´erateur sym´etrique et admet une fermeture qu’on notera par: Hmin = H de domaine Dmin = {ϕ ∈ l2 tel que ∃pk ∈ P0 ; Lim pk = ϕ et ∃ψ ∈ l2 ; lim Hpk = ψ, k → ∞} . Lemme 2. 1) Dmin = {ϕ ∈ Dmax tel que [ϕ, ψ]∞ = 0 pour tout ψ ∈ Dmax } . 2) Hmin est ferm´e sym´etrique d’indices de d´efaut (1, 1). 3) Hmax est l’adjoint de Hmin . D´emonstration. Consid´erons la suite: u(σ) = {uk (σ)}∞ k=1 o`u   u1 (σ) = 1σ u2 (σ) = √2  √ √  (k − 1) kuk−1 (σ) + k k + 1uk+1 (σ) = σuk (σ)

pour k ≥ 2.

Cette suite r´ecurrente est toujours solvable et uk (σ) est un polynˆome de degr´e k − 1. En suivant la th´eorie de Yu. Berzanskii [6] sur les op´erateurs aux diff´erences, {uk (σ)} sera appel´ee une suite de polynˆomes du premier esp´ece associ´ee a` l’op´erateur H. Il est aussi clair que: i) Le coefficient de uk (σ) associ´e a` σ k−1 est positif. ii) Les indices de d´efaut de Hmin sont (1, 1) si et seulement si la s´erie de terme g´en´eral [uk (σ)]2 est convergente pour tout σ non r´eel. En utilisant le th´eor`eme 1.5, Ch. VII, √ [6], on en d´eduit que les indices de d´efaut de Hmin sont (1, 1). En effet, en posant ak = k k + 1, on v´erifie facilement que ak−1 ak+1 ≤ a2k et la s´erie de terme g´en´eral a1k est convergente. Les autres assertions du lemme ne posent aucune difficult´e. Consid´erons maintenant la suite:

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A. Intissar

v(σ) = {vk (σ)}∞ k=1 , o`u   v1 (σ) = 01 v2 (σ) = √2  √ √  (k − 1) k vk−1 (σ) + k k + 1 vk+1 (σ) = σvk

pour k ≥ 2.

ee une suite de polynˆomes du second esp`ece associ´ee Cette suite {vk (σ)}∞ k=1 sera appel´ a` l’op´erateur H. Remarque 1. 1) L’´equation Hϕ = σ · ϕ est e´ quivalente a` : ( ϕ0 = 0,√ √ (k − 1) kϕk−1 + k k + 1ϕk+1 = σ · ϕk

pour k ≥ 1.

(∗)

2) La suite u(σ) est une solution de Hu(σ) = σu(σ) mais Hv(σ) 6= σv(σ) et v(σ) n’est pas solution de (∗) ∞ ∞ 3) Pour k ≥ 1, le Wronskien W (ϕ, ψ) √ de deux solutions ϕ = {ϕk }k=1 et ψ = {ψk }k=1 de (*) est d´efini par Wk (ϕ, ψ) = k k + 1(ϕk ψk+1 − ϕk+1 ψk ) = [ϕ, ψ]k et ne d´epend pas de k. ∞ eduit que 4) De la d´efinition des suites u(σ) = {uk (σ)}∞ k=1 et v(σ) = {vk (σ)}k=1 , on d´ W1 (u(σ), v(σ)) = 1 et donc W (u(σ), v(σ)) = 1 et par cons´equent les suites u(σ) et v(σ) forment un syst`eme fondamental d’une e´ quation r´ecurrente de la forme: √ √ (∗∗) (k − 1) kwk−1 (σ) + k k + 1wk+1 (σ) = σwk (σ) pour k ≥ 2. Consid´erons les suites: ∞

∞

u = {uk (σ)}k=1 = {uk }k=1 o`u   u1 = 0 u2 = 0 √  (k − 1)√ku k−1 + k k + 1uk+1 = 0

pour k ≥ 2

et ∞

∞

v = {vk (σ)}k=1 = {vk }k=1 o`u   v1 = 0 v2 = √12 √  (k − 1)√kv k−1 + k k + 1vk+1 = 0

pour k ≥ 2.

Il en r´esulte que (Hu)k = 0 pour tout k ≥ 1, (Hv)1 = 1 et (Hv)k = 0 pour tout k ≥ 2. Et par cons´equent u et v sont des e´ l´ements de Dmax .

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Remarque 2. La suite {vk } est donn´ee par:  pour tout p ≥ 0 v2p+1 = 0 Q p−1

a2j+1

v2p = (−1)p−1 Qj=1 p−1 j=1

a2j

v2

√ pour tout p ≥ 2 avec aj = j j + 1.

Lemme 3. Pour tout couple de scalaires (a, b), il existe ϕ ∈ Dmax tel que [ϕ, u]∞ = a et [ϕ, v]∞ = b. D´emonstration. Soit f un e´ l´ement arbitraire de l2 v´erifant hf, ui = −a et hf, vi = −b, l’existence de f est assur´ee comme combinaison lin´eaire de u et v, c’est-`a-dire f = c1 u + c2 v o`u les constantes c1 , c2 v´erifient le syst`eme suivant: ( c1 hu, ui + c2 hv, ui = −a . c1 hu, vi + c2 hv, vi = −b Le d´eterminant principal est non nul comme d´eterminant de Gram du syst`eme libre {u, v}. Soit ϕ = {ϕk }∞ k=1 la solution de Hϕ = f telle que ϕ1 = 0. L’expression de cette solution est donn´ee par: ϕk =

k X

uj vk − uk vj fj

k = 1, 2, . . . .

j=1

En effet on a: ϕk−1 =

k−1 X

uj vk−1 − uk−1 vj fj

k = 1, 2, . . . ,

uj vk+1 − uk+1 vj fj

k = 1, 2, . . . ,

j=1

ϕk+1 =

k+1 X j=1

et donc k−1 X √ √ √ √ uj fj − (k − 1) kϕk−1 + k k + 1ϕk+1 = (k − 1) kvk+1 + k + 1 vk+1 j=1

√ √ (k − 1) kuk−1 + k k + 1uk+1

k−1 X

√ vj fj + k k + 1 (uk vk+1 − uk+1 vk ) fk .

j=1

√

Or comme k k + 1 (uk vk+1 − uk+1 vK ) = 1 et pour k ≥ 3 on a: √ √ √ √ (k − 1) kvk−1 + k k + 1vk+1 = (k − 1) kuk−1 + k k + 1uk+1 = 0, il en r´esulte que (Hϕ)k = fk pour k ≥ 3. Il nous reste a` v´erifier que pour ϕ1 = 0, ϕ2 = (u1 v2 − u2 v1 )f1 et ϕ3 = (u1 v3 − u3 v1 )f1 + (u2 v3 − u3 v2 )f2 on a: (Hϕ)1 = f1 et (Hϕ)2 = f2 . C’est-`a-dire: √ √ √ 2ϕ2 = f1 et 2 + 2 3ϕ3 = f2 .

248

A. Intissar

En utilisant ϕ1 = 0 cela revient a` v´erifier que √ √ 2ϕ2 = f1 et 2 3ϕ3 = f2 √ 2 (u1 v2 − u2 v1 ) f1 = f1

ou bien et

√ 2 3 ((u1 v3 − u3 v1 ) f1 + (u2 v3 − u3 v2 ) f2 ) = f2 . √ √ En rappellant que u1 = 1, u2 = 0, v1 = 0, v2 = √12 , 2(u1 v2 − u2 v1 ) = 1, 2 3(u2 v3 − √ u3 v2 ) = 1 et 2 3v3 = 0, on en d´eduit le r´esultat. Par cons´equent on a: hf, ui = hHφ, ui = hφ, Hui − [φ, u]∞ = −a, hf, vi = hHφ, vi = hϕ, Hvi − [ϕ, u]∞ = −b. Or d’une part, on a Hu = 0 donc hϕ, Hui = 0 et par cons´equent [ϕ, u]∞ = a et d’autre part, comme ϕ1 = 0 on a Hv = 0 donc hϕ, Hvi = 0 et par cons´equent [ϕ, v]∞ = b. ∞ Lemme 4. Pour des suites arbitraires ϕ = {ϕk }∞ k=1 et ψ = {ψk }k=1 on a:

[ϕ, ψ]k = −[ϕ, u]k [v, ψ]k + [ϕ, v]k [u, ψ]k . D´emonstration. Comme u et v sont des suites r´eelles on a: √ [ϕ, u]k = k k + 1 (ϕk uk+1 − ϕk+1 uk ) √ [ϕ, v]k = k k + 1 (ϕk vk+1 − ϕk+1 vk ) .

et D’o`u

√ √ [ϕ, u]k [v, ψ]k = k k + 1 (ϕk uk+1 − ϕk+1 uk ) k k + 1 vk ψ k+1 − vk+1 ψ k i2 h √ ϕk uk+1 vk ψ k+1 − ϕk uk+1 vk+1 ψ k = k k+1

− ϕk+1 uk vk ψ k+1 + ϕk+1 uk vk+1 ψ k , √ √ [ϕ, v]k [u, ψ]k = k k + 1 (ϕk vk+1 − ϕk+1 vk ) k k + 1 uk ψ k+1 − uk+1 ψ k i2 h √ ϕk vk+1 uk ψ k+1 − ϕk vk+1 uk+1 ψ k = k k+1 − ϕk+1 vk uk ψ k+1 + ϕk+1 vk uk+1 ψ k , et par cons´equent on a: h √ i2 ϕk vk+1 uk ψ k+1 − ϕk uk+1 vk ψ k+1 + [ϕ, v]k [u, ψ]k − [ϕ, u]k [v, ψ]k = k k + 1 √ ϕk+1 vk uk+1 ψ k − ϕk+1 uk vk+1 ψ k = k k + 1 ϕk ψ k+1 − ϕk+1 ψ k [u, v]k . Or [u, v]k = 1 d’o`u: [ϕ, ψ]k = [ϕ, v]k [u, ψ]k − [ϕ, u]k [v, ψ]k .

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Cons´equence. [ϕ, ψ]∞ = [ϕ, v]∞ [u, ψ]∞ − [ϕ, u]∞ [v, ψ]∞ . Th´eor`eme 1. A l’infini, le domaine minimal de Hmin est caract´eris´e par: Dmin = {ϕ ∈ Dmax

tel que [ϕ, u]∞ = [ϕ, v]∞ = 0} .

D´emonstration. D’apr`es le (1) du Lemme 2, on a: Dmin = {ϕ ∈ Dmax

tel que [ϕ, ψ]∞ = 0

pour tout ψ ∈ Dmax } .

Comme [ϕ, ψ]∞ = [ϕ, v]∞ [u, ψ]∞ − [ϕ, u]∞ [v, ψ]∞ , on en d´eduit que: [ϕ, v]∞ [u, ψ]∞ − [ϕ, u]∞ [v, ψ]∞ = 0. D’apr`es le Lemme 3, [u, ψ]∞ et [v, ψ]∞ sont arbitraires et donc [ϕ, v]∞ [u, ψ]∞ − [ϕ, u]∞ [v, ψ]∞ = 0 pour tout ψ ∈ Dmax si seulement si [ϕ, u]∞ = [ϕ, v]∞ = 0. 3. Sur les Extensions Dissipatives de l’Op´erateur Hmin D´efinition 1. Un op´erateur lin´eaire A agissant sur un espace de Hilbert E de domaine dense D(A) est dissipatif si Im hAf, f i ≥ 0 pour tout f ∈ D(A). Un op´erateur dissipatif est dit dissipatif-maximal s’il n’a d’extension dissipative que lui mˆeme. Remarque 3. Un op´erateur dissipatif est toujours fermable. La fermeture d’un op´erateur dissipatif est dissipative. Un op´erateur dissipatif-maximal est toujours ferm´e. Th´eor`eme 2 (V. I. Gorbachuk and M. L. Gorbachuk [8]). Le domaine d’une extension disspative d’un op´erateur sym´etrique A est inclus dans D(A∗ ). Soit A un op´erateur sym´etrique ferm´e agissant dans un espace de Hilbert E dont les indices d´efaut sont (n, n) (n ≤ ∞), un role important dans la th´eorie des extensions de ces op´erateurs est jou´e par le concept de l’espace des valeurs fronti´eres. D´efinition 2. Soit A un op´erateur sym´etrique ferm´e agissant dans un espace de Hilbert E. Le tripelt (E, 01 , 02 ) o`u E est un espace de Hilbert et 01 , 02 sont deux op´erateurs lin´eaires agissant de D(A∗ ) dans E, est appel´e espace valeurs fronti`eres si 1) hA∗ f, giE − hf, A∗ giE = h01 , f 02 giE − h02 f, 01 giE pour tout f , g dans D(A∗ ). 2) Pour tout F1 , F2 ∈ E il existe f ∈ D(A∗ ) tel que 01 f = F1 est 02 f = F2 . Il suit de cette d´efinition que f ∈ D(A) si et seulement si 01 f = 02 f = 0. Th´eor`eme 3. Pour tout op´erateur sym´etrique ferm´e d’indices de d´efaut (n, n) (n ≤ ∞), il existe un espace des valeurs fronti`eres (E, 01 , 02 ) avec dim E = n. Soit maintenant un espace des valeurs fronti`eres (E, 01 , 02 ) arbitraire d’un op´erateur sym´etrique ferm´e A agissant dans un espace de Hilbert E de domaine dense D(A). On donne une version simplifi´ee du Th´eor`eme 1.6, p. 156 [8] qui caract´erise une extension dissipative maximale de A:

250

A. Intissar

Th´eor`eme 4. Si K est une contraction sur E, alors la restriction de l’ op´erateur A∗ a` l’espace des e´ l´ements f ∈ D(A∗ ) v´erifiant la condition: (K − I)01 f + i(K + I)02 f = 0

ou (K − I)01 f − i(K + I)02 f = 0

est dissipative maximale. R´eciproquement, toute extension disspative maximale de A est la restriction de A∗ a` l’espace des e´ l´ements f ∈ D(A∗ ) v´erifiant l’une des conditions ci-dessus. Remarque 4. Pour l’op´erateur de Heun, on lui associe l’espace (E, 01 , 02 ) o`u E = C et 01 , 02 sont d´efinis sur Dmax a` valeurs dans C par 01 ϕ = [ϕ, v]∞ et 02 ϕ = [ϕ, u]∞ . Th´eor`eme 5. 1) Le triplet (C, 01 , 02 ) d´efini ci-dessus est un espace des valeurs fronti`eres de l’op´erateur Hmin . 2) Toute extension dissipative maximale Hα de Hmin est d´etermin´ee par Hα ϕ = Hϕ o`u le domaine de Hα est D(Hα ) = {ϕ ∈ Dmax tel que [ϕ, v]∞ − α[ϕ, u]∞ = 0; Im α > 0} . D´emonstration. 1) D’une part , comme cons´equence du Lemme 1 on a: hHmax ϕ, ψi − hϕ, Hmax ψi = −[ϕ, ψ]∞ . Et d’autre part comme cons´equence du Lemme 4 on a: −[ϕ, ψ]∞ = [ϕ, u]∞ [v, ψ]∞ − [ϕ, v]∞ [u, ψ]∞ = [ϕ, v]∞ [u, ψ]∞ − [ϕ, u]∞ [v, ψ]∞ = 01 ϕ02 ψ − 02 ϕ01 ψ. Il en r´esulte que: hHmax ϕ, ψi − hϕ, Hmax ψi = 01 ϕ02 ψ − 02 ϕ01 ψ = h01 ϕ, 02 ψiC − h02 ϕ, 01 ψiC . En fin appliquant le Lemme 3, on d´eduit que pour tout couple (a, b) ∈ C2 , il existe ϕ ∈ Dmax tel que [ϕ, u]∞ = a et [ϕ, v]∞ = b, c’est-`a-dire 01 ϕ = b et 02 ϕ = a. 2) La propri´et´e est une cons´equence imm´ediate du Th´eor`eme 1.6 [8] p. 159 rappel´e ci-dessus sous une forme simplifi´ee. Corollaire. Les extensions auto-adjointes de Hmin sont obtenues pr´ecis´ement par le th´eor`eme pr´ec´edent quand α est r´eel. Remarque 5. Pour notre exemple, on d´eduit du Th´eor`eme 5 les Th´eor`emes 3.3 et 3.5 dans [18] de Stephen T. Welstead qui caract´erisent les extensions auto-adjointes des matrices de Jacobi de type cercle-limite.

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251

4. Construction d’une Dilatation Auto-Adjointe de Hα En utilisant l’arsenal de la th´eorie de Scattering de Lax-Phillips [14] et les techniques de construction d’une dilatation auto-adjointe d’un op´erateur dissipatif de Schr¨odinger par Pavlov [16], on construit dans ce paragraphe une dilatation auto-adjointe de notre op´erateur Hα . Pour cela, on proc`ede de la fac¸on suivante: On pose H = L2 (−∞, 0) + l2 + L2 (0, ∞) l’espace de dilatation. Dans H, on d´efinit l’op´erateur Wα engendr´e par l’expression: de domaine : Pour U = (ϕ− , ϕ, ϕ+ ) ∈ H, Wα U = iϕ0− , Hϕ, iϕ0+ 1 D(Wα ) = (ϕ− , ϕ, ϕ+ ) ∈ H, ϕ− ∈ W2 (−∞, 0), ϕ ∈ Dmax , ϕ+ ∈ W21 (0, ∞), [ϕ, v]∞ − α[ϕ, u]∞ = γϕ− (0)

et [ϕ, v]∞ − α[ϕ, u]∞ = γϕ+ (0)}

avec γ 2 = 2Im α, γ > 0 et W21 (−∞, 0), W21 (0, ∞) sont les espaces classiques de Sobolev. Th´eor`eme 6. L’op´erateur Wα de domaine D(Wα ) est auto-adjoint sur H. D´emonstration. 1) Wα est un op´erateur sym´etrique. Pour U = ϕ− , ϕ, ϕ+ ∈ D(Wα ) et V = (ψ− , ψ, ψ+ ) ∈ D(Wα ), on a: Z 0 Z 0 iϕ0− ψ − dx − ϕ0− iψ − dx + hHϕ, ψi − hϕ, Hψi hWα U, V i − hU, Wα V i = −∞ −∞ Z ∞ Z ∞ 0 0 iϕ+ ψ + dx − ϕ+ iψ + dx. + 0

0

Or on peut remarquer que si I n’est pas born´e et f ∈ Wp1 (I) avec 1 ≤ p < ∞, alors lim f (x) = 0 lorsque |x| → ∞. Il en r´esulte la formule d’int´egration par parties dans Wp1 (I) et d’une cons´equence du Lemme 1 (hHmax ϕ, ψi − hϕ, Hmax ψi = −[ϕ, ψ]∞ ) qu’on a: hWα U, V i − hU, Wα V i = iϕ− (0)ψ − (0) − iϕ+ (0) − [ϕ, ψ]∞ . Comme U et V appartiennent a` D(Wα ) on a: γϕ− (0) = [ϕ, v]∞ − α[ϕ, u]∞ , γψ(0) = [ψ, v]∞ − α[ψ, u]∞ , γψ− (0) = [ψ, v]∞ − α[ψ, u]∞ , γϕ+ (0) = [ϕ, v]∞ − α[ϕ, u]∞ , et

γψ+ (0) = [ψ, v]∞ − α[ψu]∞ .

Il en r´esulte que: |γ|2 ϕ− (0)ψ− (0) − ϕ+ (0)ψ+ (0) = 2i Im α [ϕ, v]∞ [ψ, u]∞ − [ϕ, u]∞ [ψ, v]∞ . Comme cons´equence du Lemme 4 on a: −[ϕ, ψ]∞ = [ϕ, v]∞ [ψ, u]∞ − [ϕ, u]∞ [ψ, x]∞ . D’o`u:

i|γ|2 ϕ− (0)ψ− (0) − ϕ+ (0)ψ+ (0) − 2 Im α[ϕ, ψ]∞ = 0.

252

A. Intissar

On en d´eduit que pour |γ|2 = 2 Im α, l’op´erateur Wα est sym´etrique. 2) D(Wα∗ ) est inclus dans D(Wα ). Soit V = ψ− , ψ, ψ+ ∈ D(Wα∗ ) alors hWα U, V i = on a ψ− ∈ W21 (−∞, hU, Wα∗ V i pour tout U ∈ D(Wα ) et par cons´

equent 0), 1 0 ψ ∈ Dmax et ψ+ ∈ W2 (0, ∞) en particulier: iϕ− , Hϕ, iϕ0+ , ψ− , ψ, ψ+ H =

0 , Hψ, iψ+0 H pour tout ϕ− , ϕ, ϕ+ ∈ D(Wα ) ce qui entraine: ϕ− , ϕ, ϕ+ , iψ− i ϕ− (0)ψ− (0) − ϕ+ (0)ψ+ (0) − [ϕ, ψ]∞ = 0 pour tout ϕ− , ϕ, ϕ+ ∈ D(Wα ). (∗) Comme γϕ− (0) = [ϕ, v]∞ − α[ϕ, u]∞ et γϕ+ (0) = [ϕ, v]∞ − α[ϕ, u]∞ , on d´eduit que: 1 γ ϕ− (0) − ϕ+ (0) = ϕ− (0) − ϕ+ (0 ), 2i Im α iγ γ α = −αϕ+ (0) + αϕ− (0) = γϕ− (0) ϕ+ (0) − ϕ− (0 ), 2i Im α iγ

[ϕ, u]∞ = [ϕ, v]∞

et comme −[ϕ, ψ]∞ = [ϕ, v]∞ [ψ, u]∞ − [ϕ, u]∞ [ψ, v]∞ , on en d´eduit que: l’expression (∗) s’´ecrit comme suit: α ϕ+ (0) − ϕ− (0) [ψ, u]∞ + iϕ− (0)ψ− (0) − iϕ+ (0)ψ+ (0) = γϕ− (0) + iγ 1 ϕ− (0) − ϕ+ (0) [ψ, v]∞ , iγ c’est-`a-dire:

D’o`u on obtient: et

iγ 2 − α 1 [ψ, u]∞ − [ψ, v]∞ + ϕ− (0) iψ− (0) + iγ iγ α 1 ϕ+ (0) −iψ+ (0) + [ψ, u]∞ − [ψ, v]∞ = 0. iγ iγ

−γψ− (0) − α[ψ, u]∞ + [ψ, v]∞ = 0

γψ+ (0) + α[ψ, u]∞ − [ψ, v]∞ = 0.

Il en r´esulte que: [ψ, v]∞ − α[ψ, u]∞ = γψ− (0) et [ψ, v]∞ − α[ψ, u]∞ = γψ+ (0) avec γ 2 = 2 Im α, ce qui ach`eve la d´emonstration. L’op´erateur Wα engendre un groupe unitaire Exp (itWα ) sur H. Soient P : H → l2 et S : l2 → H, ϕ → (0, ϕ, 0). (ϕ− , ϕ, ϕ+ ) → ϕ Pour tout t ≥ 0 l’op´erateur Zt = P Exp (−itWα )S est un semi-groupe sur l2 dont le lorsque t tend vers 0+ . g´en´erateur infin´etisimal est d´efini par: Bα ϕ = lim Zt ϕ−ϕ it D´efinition 3. L’op´erateur Wα est appell´e dilatation auto-adjointe de Bα . Th´eor`eme 7. Wα est une dilatation auto-adjointe de Hα . D´emonstration. Soient ϕ ∈ l2 et σ ∈ C tel que Im σ < 0. On pose V = (ψ− , ψ, ψ+ ) = (Wα − σI)−1 Sϕ, alors on a:

Analyse de Scattering d’un Op´erateur Cubique de Heun

1) 2) 3) 4)

253

(Wα − σI)V = Sϕ, (H − σI)ψ = ϕ, ψ− (x) = ψ− (0)e−iσx , ψ+ (x) = ψ+ (0)e−iσx .

Comme ψ− ∈ L2 (−∞, 0) alors ψ− (0) = 0 et donc ψ v´erifie la condition: [ψ, v]∞ − α[ψ, u]∞ = 0 c’est-`a-dire ψ ∈ D(Hα ) le domaine de Hα , or ce dernier est disssipatif donc σ ne peut pas eˆ tre une valeur propre, il en r´esulte que ψ = (Hα − σI)−1 ϕ. On en d´eduit que: (Wα − σI)−1 Sϕ = 0, (Hα − σI)−1 ϕ, ψ+ (0) ψ et P (Wα − σI)−1 Sϕ = (Hα − σI]−1 ϕ pour tout ϕ ∈ l2 . R∞ Comme (P (Wα − σI)−1 S = −i 0 P e−iσt Exp (itWα )S dt = (Bα − σI)−1 . Alors (Hα − σI)−1 ϕ = (Bα − σI)−1 ϕ pour tout ϕ ∈ l2 et Hα = Bα . 5. Matrice de Scattering Associ´ee a` l’Op´erateur Hα Suivant la th´eorie de Lax–Phillips, on peut v´erifier que le groupe U = Exp(itWα ) poss`ede les propri`et´es survivantes: Soient D− = (L2 (−∞, 0), 0, 0) et D+ = (0, 0, L2 (0, ∞)) alors on a: i) Ut D− ⊂ D− , t ≤ 0 et Ut D+ ⊂ D+ , t ≥ 0, ii) ∩t≤0 Ut D− = ∩t≥0 Ut D+ = {0}, iii) ∪t≥0 Ut D− = ∪t≤0 Ut D+ = H, iv) D− ⊥ D+ . Soient H− = ∪t≥0 Ut D− et H+ = ∪t≤0 Ut D+ . Soient γ − −iσx −iσx , u(σ), sα (σ)e ζσ (x) = e ωα (σ) et

ζσ+ (x) =

sα (σ)e−iσx ,

γ u(σ), e−iσx ωα (σ)

o`u ωα (σ) = [u(σ), v]∞ − α [u(σ), u]∞ et s(σ) =

ωα (σ) ωα (σ)

D´efinition 4. Soit U = (ϕ− , ϕ, ϕ+ ) tel que ϕ− et ϕ+ sont des fonctions a` support el´ements non nuls. On d´efinit les compact et ϕ = {ϕk }∞ k une suite d’un nombre fini d’´ transformations suivivantes: 1

U, ζσ− (x) H F − U (σ) = √ 2π (Z ) Z ∞ ∞ 0 1 γ X iσx iσx √ ϕ− (x)e + ϕk uk (σ) + sα (σ) ϕ+ (x)e dx , ωα(σ) 2π −∞ 0 k=1

254

A. Intissar

1 F + U (σ) = √ hU, ζσ+ (x)iH 2π ( Z 0 1 =√ sα (σ) ϕ− (x)eiσx dx 2π ∞ ) Z ∞ ∞ γ X iσx ϕk uk (σ) + ϕ+ (x)e dx . + ωα (σ) 0 k=1

Th´eor`eme 8. 1) La transformation F − est une isom´etrie entre H− et L2 (−∞, ∞). 2) La transformation F + est une isom´etrie entre H+ et L2 (−∞, ∞). 3) Les identit´es de Parseval et les formules d’inversion respectives sont valides. D´emonstration. En utilisant des techniques similaires a` celles utilis´ees par Pavlov dans [16] qui rentre dans le cadre g´en´eral de la th´eorie de scattering de Lax–Phillips, on d´eduit sans difficult´e les r´esultats du th´eor`eme. Cons´equences. 1) H− = H+ . 2) ζσ− = sα (σ)ζσ+ , pour −∞ < σ < ∞. 3) F − est une repr´esentation spectrale entrante du groupe Ut . 4) F + est une repr´esentation spectrale sortante du groupe Ut . 5) Le passage d’un e´ l´ement de H dans la repr´esentation sortante a` la repr´esentation entrante est r´ealis´e par la multiplication de la fonction sα (σ). 6) s−1 α (σ) est la matrice de Scattering du groupe Ut . 7) La fonction caract´eristique de Hα est sα (σ). 6. Compl´etude des Vecteurs Propres G´en´eralises de l’Op´erateur Hα Soit Hα une extension dissipative asoci´ee a` l’op´erateur Hmin . Lemme 5. Pour tout ϕ ∈ D(Hα ), on a Im hHα ϕ, ϕi = (Im α)|[ϕ, u]∞ |2 D´emonstration. Pour chaque ϕ ∈ D(Hα ], on a: hHϕ, ψi − hϕ, Hψi = −[ϕ, ψ]∞ , [ϕ, v]∞ − α[ϕ, u]∞ = 0 et −[ϕ, ψ]∞ = [ϕ, u]∞ [v, ψ]∞ − [ϕ, v]∞ [u, ψ]∞ = [ϕ, v]∞ [ψ, u]∞ − [ϕ, u]∞ [ψ, v]∞ . Il en r´esulte que pour ψ = ϕ, on a: 2i Im hHϕ, ϕi = −[ϕ, ϕ]∞ , −[ϕ, ϕ]∞ = [ϕ, v]∞ [ϕ, u]∞ − [ϕ, v]∞ [ϕ, v]∞ , −[ϕ, ϕ]∞ = α[ϕ, u]∞ [ϕ, u)∞ − α[ϕ, u]∞ [ϕ, v]∞ = 2i Im α|[ϕ, u]∞ |2 . Par cons´equent on a: Im hHα ϕ, ϕi = (Im α)|[ϕ, u]∞ |2 , ce qui ach`eve la d´emonstration du lemme. u Consid´erons maintenant la suite: u(σ) = {uk (σ)}∞ k=1 o`   u1 (σ) = 1σ u2 (σ) = √2  √ √  (k − 1) kuk−1 (σ) + k k + 1uk+1 (σ) = σ · uk pour k ≥ 2.

Analyse de Scattering d’un Op´erateur Cubique de Heun

255

La suite u(σ) est la seule solution de Hu(σ) = σ · u(σ) appartenant a´ l2 pour tout scalaire complexe σ. Pour que u(σ) appartient au domaine de Hα , il faut qu’elle v`erifie: [u(σ), v]∞ − α[u(σ), u]∞ = 0

o´u Imα > 0.

On v´erifie facilement que: [u(σ), u]∞ = −σ

∞ X

uk uk (σ),

k=1 ∞ X

[u(σ), v]∞ = 1 − σ

vk uk (σ).

k=1

En utilisant le Th´eor`eme 2.4.3 de M. Riesz, p. 56 [3], on d´eduit que ces fonctions sont enti`eres d’ordre un et de type minimal. Elles sont r´eelles sur l’axe r´eel. On d´efinit la fonction caract´eristique de l’op´erateur de Heun par: s(σ) =

[u(σ), v]∞ − α[u(σ), u]∞ w(σ) − α = , [u(σ), v]∞ − α[u(σ), u]∞ w(σ) − α

[u(σ),v]∞ , cette d´erni`ere fonction est m´eromorphe sur le plan complexe, o`u w(σ) = [u(σ),u] ∞ elle est r´eelle sur l’axe r´eel et transforme le demi-plan (Imσ > 0) (respectivement le ku(σ)k2 demi-plan (Imσ < 0)) dans lui mˆeme, car Imw(σ) = Imσ |[u(σ),v] 2. ∞| 2 d´esigne Soient D− = (L2 (−∞, 0), 0, 0),D+ = (0, 0, L2 (0, ∞)), D = (0, l20 , 0) et E± l’espace de hardy dans L2 (−∞, ∞) des fonctions prolongeables analytiquement au demi-plan sup´erieur (respectivement au demi-plan inf´erieur). Alors on a:

1) 2) 3) 4)

F − (H) = L2 (−∞, ∞). 2 . F − (D− ) = E− − F (D+ ) = Sα E+2 . F − (D) = E+2 − Sα E+2 .

Soit Zt ϕ = P (eiλt ϕ) o`u P est la projection orthogonale de E+2 sur l’espace (E+2 − Sα E+2 ). Dans le livre de Lax–Phillips [14], le g´en´erateur infinit´esimal associ´e a` Zt est appel´e “op´erateur d’un model dissipatif” o`u s(λ) est sa fonction caract´eristique. Ces formules montrent que notre op´erateur Hα est unitairement e´ quivalent a` un “model dissipative operator” o`u s(σ) est sa fonction caract´eristique. Il en r´esulte d’une part que s(σ) est aussi fonction caract´eristique de Hα et d’autre part que les valeurs propres de Hα sont les z´eros de l’´equation [u(σ), u]∞ − α[u(σ), v]∞ = 0. Dans [15], il est bien connu que la fonction caract´eristique d’un op´erateur dissipatif caract´erise toutes les propri´et´es spectrales d’un tel op´erateur, en particulier si s(σ) est un produit de Blaschke alors le syst`eme des vecteurs propres g´en´eralis´es d’un tel op´erateur est complet. Th´eor`eme 9. 1) Pour tout α avec Imα > 0 sauf peut-ˆetre pour un point α0 = [u(iτ ),v]∞ Lim [u(iτ ),u]∞ lorsque τ tend vers l’infini, la fonction Sα (λ) est un produit de Blaschke de l’op´erateur dissipatif Hα . 2) Les spectre de Hα est purement ponctuel. 3) Le syst`eme des vecteurs propres g´en´eralis´es de Hα est complet dans l2 .

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A. Intissar

D´emonstration. 1) Comme Sα (σ) =

[u(σ), v]∞ − α[u(σ), u]∞ w(σ) − α , = [u(σ), v]∞ − α[u(σ), u]∞ w(σ) − α

[u(σ),v]∞ et les fonctions [u(σ), u]∞ , [u(σ), v]∞ sont enti`eres d’ordre un et o`u w(σ) = [u(σ),u] ∞ de type minimal, alors en utilisant le th´eor`eme de factorisation de Hadamard on d´eduit que Sα (σ) = eiσM (α) Bα (σ), M (α) > 0 et B(σ) est le produit de Blaschke. Il en r´esulte que |Sα (σ)| ≤ e−M (α)Imσ pour Imσ ≥ 0. Par cons´equent on a LimSα (iτ ) = 0 lorsque τ tend vers l’infini et Limw(iτ ) = α lorsque τ tend vers l’infini. [u(σ),v]∞ ),v]∞ ne d´epend pas de α alors Lim [u(iτ Comme w(σ) = [u(σ),u] [iτ ),u]∞ = α0 lorsque τ ∞ tend vers l’infini. Il en r´esulte que pour α 6= α0 , on a M (α) = 0 et par cons´equent Sα (σ) est un produit de Blaschke. Les propri´et´es 2) et 3) sont une cons´equence de 1).

References 1. Aimar, M.T., Intissar, A. et Paoli, J.M.: Quelques propri´et´es de r´egularit´e de l’op´erateur de Gribov. C.R. Acad. Sci. Paris, 320, S´erie I, (1995) 2. Aimar, M.T., Intissar, A. et Paoli, J.M.: Quelques nouvelles propri´et´es de r´egularit´e de l’op´erateur de Gribov. commun. Math. Phys. 172, (1995) 3. Akhiezer, N.I.: The classical moment problem and some related questions in analysis. Fizmatgiz, Moscow, 1961; English transl., London: Oliver and Boyd, and New York: Hafner, 1965 4. Ando, T. et Zerner, M.: Sur une valeur propre d’un op´erateur. Commun. Math. Phys. , (1984) 5. Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform I. Commun. Pure App. Math. 14, (1962) 6. Berezanskii, Yu.M.: Expansion in eigenfunctions of selfadjoint operators. Providence, RI: Am. Math Soc., 1968 7. Fedoriuk, M.: Asymptotic of the spectrum of Heun’s equation and Heun’s functions. Math. USSR Izvestiya 38, (1992) 8. Gorbachuk, V.I. and Gorbachuk, M.L.: Boundary value problems for operator differential equations. Dordrecht: Kluver Academic Publishers, 1991 9. Gribov, V.N.: A reggeon diagram technique. J.E.T.P., Sov. Phys. 26, 414–425 (1968) 10. Intissar, A.: Etude spectral d’une famille d’op´erateurs non-sym´etriques intervenant dans la th´eorie des champs de reggeons. Commun. Math. Phys. 113, 263–297 (1987) 11. Intissar, A.: Quelques nouvelles propri´et´es spectrales de l’hamiltonien de la th´eorie des champs de reggeons. C. R. Acad. Sci. Paris, S´erie I, 308, (1989) 12. Intissar, A.: The´eorie spectrale dans l’espace de Bargmann. Cours de D.E.A., Universit´e de Besanc¸on, 1989 13. Intissar, A.: Analyse fonctionnelle et Th´eorie spectrale pour les op´erateurs compacts non auto-adjoints. Editions-CEPADUES, 1997 14. Lax, P.D. and Phillips, R.S.: Scattering theory. New York: Academic Press, 1967 15. Nagy, B.Sz. and Foias, C.: Analyse harmonique des op´erateurs de l’espace de Hilbert. Paris: Masson et Cie , 1967 16. Pavlov, B.S.: Selfadjoint dilation of dissipative Schr¨odinger operator and its resolution in terms of eigenfunctions. Math. USSR. Sbornik, 31, n◦ 4, (1977) 17. Takazaki, K.: Analytic expression of Voros coefficients and its application to W-K-B connection problem. I-C.M-90. Satellite conference Proceeding Editeurs, M. Kashiwara, T. Miwa, Special-Functions, Berlin– Heidelberg–New York: Springer–Verlag 18. Welstead, S.T.: Selfadjoint extensions of Jacobi matrices of limit-circle type. Math. Anal. and Appl. 89, 315–326 (1982) Communicated by H. Araki

Commun. Math. Phys. 199, 257 – 279 (1998)

Communications in

Mathematical Physics © Springer-Verlag 1998

Gauge Theories on the Noncommutative Sphere C. Klimˇc´ık IHES, 91440 Bures-sur-Yvette, France, and Institut of Mathematics, University of Marseille II, 163, Avenue de Luminy, 13288 Marseille, France Received: 27 October 1997 / Accepted: 24 April 1998

Dedicated to my son Cedrik on the occasion of his birth

Abstract: Gauge theories are formulated on the noncommutative two-sphere. These theories have only finite number of degrees of freedom; nevertheless they exhibit both the gauge symmetry and the SU(2) symmetry of the sphere. In particular, the coupling of gauge fields to chiral fermions is naturally achieved.

1. Introduction There was a reappearing belief through this century that a possible understanding of a short distance behaviour of physical theories should stem from a theory that would incorporate a minimal length. At early times of quantum field theory, such an attitude was almost dictated by the pertinent problem of ultraviolet (UV) divergences. With the work of Wilson, we have learned better to get along with the divergences, nevertheless it is still expected that a fundamental theory should be in a sense finite. The divergences should appear in a controlled way in effective approaches to various aspects of the theory. String theory is often presented as a candidate for a fundamental description of our world. Its perturbation expansion is believed to be UV finite and the string tension sets a scale which is interpreted as the minimal length1 . Insights of the last two years even led to remarkable proposals of various matrix models [2, 3] which should constitute the nonperturbative formulation of string theory. We should like to stress, however, that the idea that theories containing large matrices naturally incorporate the concept of minimal length had appeared earlier in the field theoretical context [5, 6, 7, 8, 9, 10]. These works stem from the basic ideas of Connes’ noncommutative geometry [4] and argue that a nonperturbatively well defined path integral of various field theories can be formulated if the fields are replaced by big matrices in a way which correspond to the kinematical quantization of the spacetime on which the fields live. Because this is the crucial point 1

However, see recent progress in understanding substringy scales via D-branes [1].

C. Klimˇc´ık

258

of the whole matrix approach let me be more precise here (a reasoning quite similar in spirit now underlies the matrix approach to superstring theory [3]): Consider a Riemann sphere as a spacetime of an Euclidean field theory. This is a sufficiently general setting, since we can always effectively decompactify the spacetime by scaling the round metric (or radius) of the sphere and the Minkowski dynamics can be achieved by a suitable modification of the standard Osterwalder-Schrader procedure. An invariance of field theories with respect to the isometry group SO(3) of the sphere then in the limit of infinite scaling play the role of the Euclidean (or Poincar´e in the Minkowski case) invariance. Now the crucial observation is that the spacetime S 2 is naturally a symplectic manifold; the symplectic form ω is up to a normalization just the round volume form on the sphere. Using the standard complex coordinate z on the Riemann sphere, we have N dz¯ ∧ dz , (i) ω=− 2π (1 + zz) ¯ 2 with N a real parameter2 . If we consider a scalar field theory, then the scalar field φ is a function on the symplectic manifold or, in other words, a classical observable. The action of the massless (real) scalar field theory on S 2 is given by Z (ii) S = −i ωRi φRi φ, where Ri are the vector fields which generate the SO(3) rotations of S 2 and the Einstein summation convention is understood. The vector fields Ri are Hamiltonian; this means that there exists three concrete observables ri such that {ri , φ} = Ri φ.

(iii)

Here {., .} is the Poisson bracket which corresponds to the symplectic structure ω. The observables ri ∈ R3 are just the coordinates of the embedding of S 2 in R3 . Thus we can rewrite the action (ii) as Z S = −i

ω{ri , φ}{ri , φ}.

(iv)

Suppose we quantize the symplectic structure on S 2 (probably the first who has done it was Berezin [11]). Then the algebra of observables becomes the noncommutative algebra of all square matrices with entries in C; the quantization of S 2 can be only performed if N is an integer, the size of the scalar field matrices φ is then (N +1)×(N +1). This algebra of matrices defines the noncommutative (or fuzzy [6]) sphere. The integration over the phase space volume form iω is replaced by taking a properly normalized trace Tr over the matrices and the Poisson brackets are replaced by commutators (the Hamiltonians ri are also quantized, of course). Putting together, we can consider along with (iv) a noncommutative action S=−

1 Tr([ri , φ], [ri , φ]). N +1

(v)

The action (v) has a few nonstandard properties. First of all, the space of all “fields” (= matrices) is finite dimensional and the product of fields is noncommutative. The latter property may seem awkward but in all stages of analysis we shall never encounter 2 Note that we have chosen a normalization which makes the form ω purely imaginary. Under quantization, hence, the Poisson bracket is replaced by a commutator without any imaginary unit factor.

Gauge Theories on Noncommutative Sphere

259

a problem which this noncommutativity might create. The former property, however, is highly desirable, since all divergences of the usual field theories are automatically eliminated. We may interpret (v) as the regularized version of (iv); the fact that (v) goes to (iv) in the limit N → ∞ is just the statement that classical mechanics is the limit of the quantum one for the value of the Planck constant 1/N approaching zero. Remarkably, unlike lattice regularizations, (v) preserves the SO(3) isometry of the sphere (= “pacetime”). Indeed, under the variation δφ = [ri , φ] the action (v) remains invariant. A few more words are useful for understanding the nature of the commutative limit N → ∞ (see [8] for more details). Both classical commutative fields and their noncommutative counterparts (matrices) can be decomposed into spherical harmonics. This means that the algebra of observables is a representation of the group SO(3), which infinitesimally acts on φ via Poisson bracket or commutators (respectively) with the Hamiltonians ri . All spherical harmonics with the quantum number l form an irreducible representation of SO(3) (this is their definition in the noncommutative case). Recall that the Laplace operator on the sphere (the Casimir Ri Ri of SO(3)) has the spectrum l(l + 1). Now if a maximal l of φ in (v) is fixed, then the action (v) differs from (iv) by a factor which goes like 1/N for N → ∞. This means that, being at large scales (= small momenta l), the actions (iv) and (v) are equivalent. They differ only in the short distance limit. Hence we stress that there is no point in saying which one is better, or which action is an approximation of the other. Just from the historical reasons people were using (iv) earlier than (v) (they did not know about the noncommutative geometry). Both (iv) and (v) are extrapolations of the same long distance quantity also to the ultraviolet region. The symmetry principle cannot select one of them. Therefore we tend to believe that (v) might be a better choice, because it is from the outset manifestly regular. Such a statement may call for criticism; the most obvious would be: what about the other fields (spinor,vector etc.); can one construct classical field theories on noncommutative manifolds involving those fields? This paper constitutes a step of the program [8, 12, 13] which aims to show that the answer to this question is positive. Scalar fields, being observables, are naturally defined in the noncommutative context simply as elements of the algebra which defines the noncommutative manifold. However, no canonical procedure appears to exist of how to describe, say, spinor fields on a noncommutative manifold. The early approaches by Grosse and Madore [7] and later by Grosse and Preˇsnajder [14] considered spinors on S 2 as two component objects where both components are elements of the noncommutative algebra of matrices (N + 1) × (N + 1). Although this approach made it possible to construct field theories (in the case of [7] also the gauge fields with an inevitable addition of one propagating scalar), the noncommutative actions obtained in this way have somewhat lost their competitiveness with respect to the classical commutative actions. The reason was that the Dirac operator did not anticommute with the chiral grading for finite N . A shift in point of view was presented by Grosse, Klimˇc´ık and Preˇsnajder in [8], where it was argued that spinors should be understood as odd parts of scalar superfields. Then the chirality of the Dirac operator is automatically preserved. Thus the guiding principle for understanding the spinors consists in quantizing the supersymplectic manifolds. If they are compact (like a supersphere in two dimensions) then the quantized algebra of observables is just an algebra of supermatrices with a finite size. The odd (off-diagonal) part of the supermatrix then describes a spinor on the bosonic submanifold of the whole supermanifold. In this way, the scalar superfields on the supersphere were shown to describe both scalars and spinors on the sphere [8]. The moral of the story is that, by using supersymmetry,

260

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the construction of noncommutative spinors is as canonical as that of noncommutative scalars. The story of noncommutative gauge theories on S 2 , compatible with the chirality of the Dirac operator and with the already known description of the scalars and spinors, is so far missing. We are going to fill that gap in this article. Our strategy will consist in converting the standard derivatives into covariant ones in the noncommutative actions like (v). We shall do it in the supersymmetric framework which treats the scalars and spinors on the same footing and which uses only Hamiltonian vector fields as the derivatives appearing in the Lagrangian. This means that we shall covariantize these (odd) derivatives by adding suitable gauge fields. Then we identify a gauge invariant kinetic term for these gauge fields which, remarkably, can be also written solely in terms of the same (odd) Hamiltonian vector fields. At the commutative level, this means that we will be able to write actions for scalars and spinors interacting with gauge fields just (like in (iv)) in terms of 1) suitable Poisson brackets 2) integration over a suitable volume form on the spacetime. In this way, we may convert such actions into their noncommutative analogues (like (iv) to (v)). These noncommutative actions will be the same as their commutative counterparts at large distances, they will describe theories possessing the same set of symmetries as in the commutative case but containing only finite number of degrees of freedom! In Chapter 2, we describe a noncommutative complex of differential forms which underlies the notion of the gauge field in the noncommutative case. Then in Chapter 3 we subsequently construct scalar and spinor electrodynamics.

2. Hamiltonian de Rham Complex of S 2 In this chapter, we introduce a Hamiltonian de Rham complex. 0-forms of this complex will play the role of the scalar fields and 1-forms of the gauge fields, respectively. It should be emphasized that this new complex is something different from the ordinary de Rham complex, because an existing injective homomorphism of the latter in the former is not surjective. In some sense, the Hamiltonian complex is a “doubling” of the standard de Rham one. This doubling comes from the fact [8] that a scalar field on the fuzzy sphere in the presence of noncommutative spinors corresponds in the commutative limit to a pair of two ordinary scalar fields: one dynamical and one auxiliary. If we interpret that doubled scalar field as a 0-form of some larger complex, then it is clear that also 1-forms and 2-forms must be doubled. The Hamiltonian de Rham complex precisely plays the role of that larger complex and a simple counting of degrees of freedom will make it evident that it is a minimal complex which realizes the doubling. In the noncommutative version of our construction, there will be no noncommutative counterpart of the ordinary de Rham complex but the Hamiltonian complex will continue to make sense. We stress that in the noncommutative version of the field theoretical applications one can no longer distinguish between dynamical and auxiliary forms (= fields). In a sense, all fields are dynamical and some fields become irrelevant, or auxiliary, only in the commutative limit. 2.1. The commutative case. Consider an algebra A of functions in two conjugated bosonic variables z, z¯ and two conjugated fermionic ones b, b¯ with the standard graded commutative multiplication but linearly generated (over C) only by the functions of the form

Gauge Theories on Noncommutative Sphere ¯ ¯ z¯ k z k b¯ l bl ¯ m, (1 + zz ¯ + bb)

261

¯ k + l) ≤ m, max(k¯ + l,

¯ l, l, ¯ m ≥ 0. k, k,

(1)

The algebra is equipped with the graded involution ¯ z ‡ = z,

z¯ ‡ = z,

¯ b‡ = b,

(AB)‡ = (−1)AB B ‡ A‡ ,

b¯ ‡ = −b,

(2)

(A‡ )‡ = (−1)A A,

(3)

and it is known as the algebra of functions on the supersphere CP (1, 1). We can define an integral of an element f in the algebra as follows: Z dz¯ ∧ dz ∧ db¯ ∧ db i f. (4) I[f ] ≡ − ¯ 2π 1 + zz ¯ + bb (Note I[1] = 1.) Now an inner product on A is defined simply as (f, g) = I(f ‡ g).

(5)

The remaining structure to be defined is the graded symplectic structure, given by a non-degenerate super-Poisson bracket {., .} : A × A → A, ¯ {f, g} = (1 + zz)(1 ¯ + zz ¯ + bb)(∂ z f ∂z¯ g − ∂z¯ f ∂z g) f ¯ ¯ z(∂ ¯ b f ∂z¯ g − (−1)f ∂z¯ f ∂b g) +(1 + zz) ¯ bz((−1) ∂z f ∂b¯ g − ∂b¯ f ∂z g) + (1 + zz)b ¯ zz)(∂ ¯ − bb ¯ +(−1)(f +1) (1 + zz b f ∂b¯ g + ∂b¯ f ∂b g).

(6)

Now let us introduce four odd vector fields on A: ¯ b¯ − b∂z , T1 = z∂

T2 = ∂b¯ + zb∂z ,

¯ z¯ + z∂b , T¯1 = b∂

¯ z¯ . T¯2 = ∂b − z¯ b∂

(7)

It turns out that Ti , T¯i are Hamiltonian vector fields with respect to the super-Poisson bracket (6). They are generated by the Hamiltonians ti , t¯i : Ti f = {ti , f }, t1 ≡

zb ¯ ¯ , 1 + zz ¯ + bb

t2 ≡

T¯i f = {t¯i , f },

¯ bz b¯ b ¯1 ≡ ¯2 ≡ , t , t ¯ ¯ ¯ . 1 + zz ¯ + bb 1 + zz ¯ + bb 1 + zz ¯ + bb

(8)

(9)

Note also the properties of Ti , T¯i with respect to the graded involution: (Ti f )‡ = −T¯i f ‡ ,

(T¯i f )‡ = Ti f ‡ .

(10)

Define a complexified Hamiltonian de Rham complex over the standard sphere S 2 as the graded associative algebra with unit = 0 ⊕ 1 ⊕ 2 ,

(11)

0 = 2 = Ae

(12)

1 = Ab ⊕ Ab ⊕ Ab¯ ⊕ Ab¯ .

(13)

where

and

C. Klimˇc´ık

262

Here Ae is the even subalgebra of A linearly generated over C only by elements with l = l¯ (cf. Eq. (1)) and Ab (Ab¯ ) are (odd) bimodules over Ae linearly generated by the elements of the form (1) with l¯ = 0, l = 1 (l¯ = 1, l = 0). The multiplication in is entailed by one in A; the only non-obvious thing is to define the product of 1-forms. Here it is (A1 , A2 , A¯ 1 , A¯ 2 )(B1 , B2 , B¯ 1 , B¯ 2 ) ≡ A1 B¯ 1 + A2 B¯ 2 + A¯ 1 B1 + A¯ 2 B2 .

(14)

Of course, the r.h.s. is viewed as an element of 2 . The product of a 1-form and a 2-form is set to zero by definition. Now the coboundary operator d is given by df ≡ (T1 f, T2 f, T¯1 f, T¯2 f ), f ∈ 0 , d(A1 , A2 , A¯ 1 , A¯ 2 ) ≡ T1 A¯ 1 + T2 A¯ 2 + T¯1 A1 + T¯2 A2 ,

(15)

(A1 , A2 , A¯ 1 , A¯ 2 ) ∈ 1 , (16)

h ∈ 2 .

(17)

d(AB) = (dA)B + (−1)A A(dB).

(18)

dh = 0, It maps i to i+1 and it satisfies d2 = 0,

Now we show that the complex resembles very much the standard (complexified) de Rham complex dR of the commutative sphere S 2 (not of the supersphere!). The latter can be defined again as the graded associative algebra with unit given by dR = ω0 ⊕ ω1 ⊕ ω2 ,

(19)

where ω0,2 = B,

ω1 = B2 ⊕ B2¯ .

(20)

Here B is a unital algebra linearly generated (over C) by elements of the form z¯ k z k , (1 + zz) ¯ m ¯

¯ k) ≤ m, max(k,

¯ m ≥ 0, k, k,

(21)

that is, B is the algebra of complex functions on S 2 . B2 and B2¯ are B-bimodules linearly generated by the following elements of B: z¯ k z k , (1 + zz) ¯ m

¯ k + 2) ≤ m, max(k,

¯ m≥0 k, k,

(22)

z¯ k z k , (1 + zz) ¯ m

max(k¯ + 2, k) ≤ m,

¯ m ≥ 0, k, k,

(23)

¯

and ¯

respectively. One often writes the elements (V, V¯ ) of ω1 as V dz + V¯ dz¯

(24)

Gauge Theories on Noncommutative Sphere

263

and h of ω2 as h

2 dz¯ ∧ dz, i(zz ¯ + 1)2

h ∈ ω2 .

(25)

The multiplication in dR is now given by the standard wedge product in the representation (24,25) and the de Rham coboundary operator ddR is given in the standard way. One can easily verify the following Fact. The complex dR can be injected into the complex . The injection ν is a homomorphism which preserves all involved structures (i.e linear, multiplicative and differential ones). It is given explicitly as follows: ¯ ∈ 0 . f (z, z) ¯ ∈ ω0 → f (z, z)

(26)

¯ −V¯ (z, z) ¯ ∈ 1 , ¯ V (z, z)zb, ¯ +V¯ (z, z) ¯ b, ¯ z¯ b) (V (z, z), ¯ V¯ (z, z)) ¯ ∈ ω1 → (−V (z, z)b, (27) h(z, z) ¯ ∈ ω2 →

2i ¯ ∈ 2 . h(z, z) ¯ bb (1 + zz) ¯

(28)

The injection (26) of 0-forms requires, perhaps, some explanation. The elements of 0 ¯ However, were said to be linearly generated by the expressions of the form (1) with l = l. by noting the identity ¯ bb 1 1 + ≡ ¯ + 1 (zz ¯ + 1)2 zz ¯ +1 zz ¯ + bb ¯ + bb

(29)

one can easily see that any element of B ≡ ω0 (cf. (21)) can be written as a linear combination of the quantities (1). Thus we have injected the standard de Rham complex into the bigger Hamiltonian de Rham complex which has the virtue that all vector fields (i.e. Ti , T¯i ) needed for the definition of the exterior derivative d are now Hamiltonian. This means that we have a good chance to quantize the structure while maintaining all its properties (except graded commutativity of the multiplication in ). There remains to clarify the issues of reality, Hodge star and cohomology. As the name already suggests, the complexified Hamiltonian complex has a real subcomplex R given by all elements of real under an involution † defined as follows: f † = f ‡ , f ∈ 0 ,

h† = −h‡ , h ∈ 2 ,

(30)

(Ai , A¯ i )† = (A¯ ‡i , −A‡i ), (Ai , A¯ i ) ∈ 1 .

(31)

The involution † (†2 = 1) preserves the linear combinations with real coefficients and the multiplication, and commutes with the coboundary operator d: (af + bg)† = af † + bg † , a, b ∈ R, f, g ∈ ,

(32)

(f g)† = f † g † , f, g ∈ ,

(33)

C. Klimˇc´ık

264

(df )† = df † , f ∈ .

(34)

We see that R is indeed the real subcomplex which may be called the Hamiltonian de Rham complex of S 2 . We recall that the standard involution † on dR is given by f † = f ∗ , f ∈ ω0 ,

g † = g ∗ , g ∈ ω2 ,

(V, V¯ )† = (V¯ ∗ , V ∗ ), (V, V¯ ) ∈ ω1 ,

(35) (36)

where ∗ is the standard complex conjugation. It has also the property of preserving the real linear combinations and the multiplication in dR and it commutes with the standard de Rham coboundary operator. Thus the real elements of dR form the real de Rham complex of S 2 . It is now one easy to check that the homomorphism ν : dR → preserves the involution, i.e. (ν(f ))† = ν(f † ).

(37)

This is also the reason why we have chosen the same symbol for both involutions. It is instructive to compute the cohomology of both the real complexes: 1) The standard de Rham case: The only non-trivial classes occur in the 0th and in the 2nd cohomology and they are given by the elements 1 ∈ ω0 and 1 ∈ ω2 , respectively. 2) The Hamiltonian de Rham case: Using the homomorphism ν, the de Rham classes above can be injected into H ∗ (R ). It is not very difficult to verify, that they are the only non-trivial classes there. The “Hodge” star and the inner product. The Hodge star ∗H on the standard de Rham complex dR is given by ¯ ∈ ω0 → f (z, z) ¯ ∈ ω2 , ∗H : f (z, z)

h(z z) ¯ ∈ ω2 → h(z, z) ¯ ∈ ω0 ,

∗H : (V, V¯ ) ∈ ω1 → (iV, −iV¯ ) ∈ ω1 .

(38) (39)

Note that ∗H send real forms into real ones. The “Hodge” star ∗ on the Hamiltonian de Rham complex is given by ¯ b) ∈ 0 → ∗ : f (z, ¯ z, b,

¯ b)bb ¯ 2if (z, ¯ z, b, ∈ 2 , zz ¯ +1

(40)

∗ : (A1 , A2 , A¯ 1 , A¯ 2 ) ∈ 1 → (iA1 , iA2 , −iA¯ 1 , −iA¯ 2 ) ∈ 1 ,

(41)

¯ b) ∈ 0 . ¯ b) ∈ 2 → 1 (T¯i Ti − Ti T¯i − 2)h(z, ¯ z, b, ∗ : h(z, ¯ z, b, 4i

(42)

This “Hodge” star is also compatible with the involution † and has the property ∗ν(f ) = ν(∗H f ).

(43)

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265

This in turn means that the natural inner product in dR , Z 1 (∗H X † )Y, X, Y ∈ ω0 , ω1 , ω2 , (X, Y )dR ≡ 4π

(44)

does respect the natural inner product in , (X 0 , Y 0 ) ≡

i † I[(∗X 0 )Y 0 ], 2

X 0 , Y 0 ∈ 0 , 1 , 2 .

(45)

In other words: (ν(X), ν(Y )) = (X, Y )dR .

(46)

It should be perhaps noted, for clarity, that in (45) and in all the rest of the paper the integral I is applied always on an element of Ae . Though in (45) the argument of I is always a 2-form (to be understood as an element of Ae ), in subsequent applications we shall encounter also situations in which the argument will be a 0-form. The action of SU (2) . The standard action of the group SU (2) on S 2 induces the action of the same group on the Hamiltonian de Rham complex. The latter respects the grading of the complex; on the 0-forms (from 0 ≡ Ae ) and 2-forms (from 2 ≡ Ae ) it is given by the even Hamiltonian vector field R± , R3 obtained by taking suitable anticommutators of the odd vector fields Tj , T¯j :

R3 =

¯ b¯ , R+ = [T1 , T¯2 ]+ = −∂z − z¯ 2 ∂z¯ − z¯ b∂

(47)

R− = [T2 , T¯1 ]+ = ∂z¯ + z 2 ∂z + zb∂b ,

(48)

1 1¯ 1 ([T1 , T¯1 ]+ − [T2 , T¯2 ]+ ) = z∂ ¯ z¯ − z∂z + b∂ ¯ − b∂b. 2 2 b 2

(49)

The SU (2) Lie algebra commutation relations [R3 , R± ] = ±R± ,

[R+ , R− ] = 2R3

(50)

then directly follows. The Hamiltonians rj of the vector fields Rj are obtained by taking the corresponding Poisson brackets (6) of the Hamiltonians ti , t¯i given in (9). The vector fields Rj acting on the algebra A realize a (highly reducible) representation of SU (2). This representation is unitary with respect to the inner product (5) and the representation space A has several invariant subspaces which are of interest for us. They are Ae , Ab and Ab¯ ; all of them give rise to smaller unitary representations of SU (2) than A. In particular, since both 0 and 2 can be identified with Ae , we have an action of SU (2) on the 0-forms and 2-forms of the Hamiltonian de Rham complex . Now we realize that the space 1 of the Hamiltonian 1-forms can be written as 1 = Ab ⊗ C2 ⊕ Ab¯ ⊗ C2 .

(51)

The group SU (2) can be represented on the second copy of C2 in (51) by the standard spin 1/2 representation generated by the Pauli matrices 01 00 1 0 + − 3 , σ = , σ = , (52) σ = 00 10 0 −1

C. Klimˇc´ık

266

and on the first copy of C2 by its (equivalent) complex conjugated representation (σ ± → −σ ∓ , σ 3 → −σ 3 ). Now we can define the action of SU (2) on the 1-forms from 1 again by the formula (51) where ⊕ and ⊗ are understood to be the direct product and the direct sum of the SU (2) representations. Now it is easy to check that 1) the standard “de Rham” complex dR injected in by the homomorphism ν is also an invariant subspace of the just defined SU (2) action on ; 2) the SU (2) action on preserves the inner product (45) and restricted to the image of ν gives the standard (unitary) action of SU (2) on the de Rahm complex dR ; 3) the coboundary operator d of and the “Hodge” star ∗ are both SU (2) invariant; 4) the generators Ri of SU (2) act on as derivations with respect to the product on , i.e. Ri (89) = (Ri 8)9 + 8Ri 9. 2.2. The non-commutative case. In the previous section, we have described the Hamiltonian de Rham complex, by using substantially the structure of the algebra of functions on the supersphere (1). This algebra can be described in an alternative way, which makes the explicit form of the Poisson structure (6) much less cumbersome, though it makes more involved the relation between the standard de Rham complex and its Hamiltonian counterpart. Here are the details: Consider the algebra of functions on the complex C 2,1 superplane, i.e. algebra generated by bosonic variables χ¯ i , χi , i = 1, 2 and by fermionic ones a¯ , a. The algebra is equipped with the graded involution (χi )‡ = χ¯ i ,

, (χ¯ i )‡ = χi ,

, a‡ = a¯ ,

, a¯ ‡ = −a

(53)

and with the super-Poisson bracket {f, g} = ∂χi f ∂χ¯ i g − ∂χ¯ i f ∂χi g + (−1)f +1 [∂a f ∂a¯ g + ∂a¯ f ∂a g].

(54)

Here and in what follows, the Einstein summation convention applies. We can now apply the (super)symplectic reduction with respect to a moment map χ¯ i χi + a¯ a. The result is a smaller algebra A, that by definition consists of all functions f with the property {f, χ¯ i χi + a¯ a} = 0.

(55)

Moreover, two functions obeying (55) are considered to be equivalent if they differ just by a product of (χ¯ i χi + a¯ a − 1) with some other such function. The algebra A is just the same algebra (1) that we have considered in the previous section. The Poisson bracket (6) becomes the bracket (54) for the functions in A. The relation between the generators is as follows z=

χ1 , χ2

z¯ =

χ¯ 1 , χ¯ 2

b=

a , χ2

a¯ b¯ = 2 . χ¯

The integral (4) can be written as Z 1 dχ¯ 1 ∧ dχ1 ∧ dχ¯ 2 ∧ dχ2 ∧ d¯a ∧ da δ(χ¯ i χi + a¯ a − 1)f. I[f ] = − 2 4π

(56)

(57)

The vector fields Ti , T¯i turn out to be Ti = χ¯ i ∂a¯ − a∂χi ,

T¯i = a¯ ∂χ¯ i + χi ∂a .

(58)

Gauge Theories on Noncommutative Sphere

267

Of course, they annihilate the moment map (χ¯ i χi + a¯ a), otherwise they would not be well defined differential operators acting on A. Their Hamiltonians are ti = χ¯ i a,

t¯i = χi a¯ .

(59)

Now we are ready to quantize the infinitely dimensional algebra A with the goal of obtaining its (noncommutative) finite dimensional deformation. The quantization was actually performed in [8] using the representation theory of osp(2, 2) superalgebra. Here we adopt a different procedure, namely the quantum symplectic reduction (or, in other words, quantization with constraints). This method should be more transparent for anybody who knows the elements of quantum mechanics. We start with the well-known quantization of the complex plane C 2,1 . The generators χ¯ i , χi , a¯ , a become creation and annihilation operators on the Fock space whose commutation relations are given by the standard replacement {., .} →

1 [., .]. h

(60)

Here h is a real parameter (we have absorbed the imaginary unit into the definition of the Poisson bracket) referred to as the “Planck constant”. Explicitly, [χi , χ¯ j ]− = hδ ij ,

[a, a¯ ]+ = h,

(61)

and all remaining graded commutators vanish. The Fock space is built up as usual, applying the creation operators χ¯ i , a¯ on the vacuum |0i, which is in turn annihilated by the annihilation operators χ, a. The scalar product on the Fock space is fixed by the requirement that the barred generators are an adjoint of the unbarred ones. We hope that using the same symbols for the classical and quantum generators will not confuse the reader; it should be fairly obvious from the context which usage we have in mind. Now we perform the quantum symplectic reduction with the self-adjoint moment map (χ¯ i χi + a¯ a). First we restrict the Hilbert space only to the vectors ψ satisfying the constraint (χ¯ i χi + a¯ a − 1)ψ = 0.

(62)

Hence operators fˆ acting on this restricted space have to fulfill [fˆ, χ¯ i χi + a¯ a] = 0,

(63)

and they are to form our deformed version3 of A. The spectrum of the operator (χ¯ i χi + a¯ a−1) in the Fock space is given by a sequence mh − 1, where m’s are integers. In order to fulfil (62) for a non-vanishing ψ, we observe that the inverse Planck constant 1/h must be an integer N . The constraint (62) then selects only ψ’s living in the eigenspace HN of the operator (χ¯ i χi + a¯ a − 1) with the eigenvalue 0. This subspace of the Fock space has the dimension 2N + 1 and the algebra AN of operators fˆ acting on it is (2N + 1)2 -dimensional. When N → ∞ (the dimension (2N + 1)2 then also diverges) we have the Planck constant approaching 0 and, hence, the algebras AN tend to the classical limit A. The fact that the resulting finite-dimensional noncommutative algebras AN are deformations 3 Note that (63) is just a quantum version of (55) and it says that elements of the deformed algebra have to commute with the particle number operator.

C. Klimˇc´ık

268

of A is thus clear since the latter is just the classical limit of the former. The interested reader may find a rigorous proof of this fact in [8]. The Hilbert space HN is naturally graded. The even subspace HeN is created from the Fock vacuum by applying only the bosonic creation operators: (χ¯ 1 )n1 (χ¯ 2 )n2 |0i,

n1 + n2 = N,

(64)

while the odd one HoN by applying both bosonic and fermionic creation operators: (χ¯ 1 )n1 (χ¯ 2 )n2 a¯ |0i,

n1 + n2 = N − 1.

(65)

Correspondingly, the algebra of operators AN on HN consists of an even part AeN (operators respecting the grading) and an odd part (operators reversing the grading). The odd part can be itself written as a direct sum AaN ⊕ AaN ¯ . The two components in the sum are distinguished by their images: AaN HN = HeN while AaN ¯ HN = HoN . AaN is spanned by operators (χ¯ 1 )n1 (χ¯ 2 )n2 (χ1 )m1 (χ2 )m2 a,

n1 + n2 = m1 + m2 + 1 = N,

(66)

(χ¯ 1 )n1 (χ¯ 2 )n2 a¯ (χ1 )m1 (χ2 )m2 ,

n1 + n2 + 1 = m1 + m2 = N,

(67)

AaN by ¯

and AeN by (χ¯ 1 )n1 (χ¯ 2 )n2 (χ1 )m1 (χ2 )m2 (¯aa)k ,

n1 + n2 = m1 + m2 = N − k.

(68)

From this and (56), it is obvious that Ae from the previous section got deformed to AeN , while Ab and Ab¯ to AaN and AaN ¯ , respectively. The inner product (5) on A is given by the integral I (4) or (57). Its representation (57) is more convenient for finding its noncommutative deformation. At the level of supercomplex plane C 2,1 it is the textbook fact from quantum mechanics that the integral R i i dχ¯ dχ d¯ada (this is the Liouville integral over the superphase space) is replaced under the quantization procedure by the supertrace in the Fock space. (The supertrace is the trace over the indices of the zero-fermion states minus the trace over the one-fermion states.) The δ function of the operator (χ¯ i χi + a¯ a − 1) just restricts the supertrace to the trace over the indices of HeN minus the trace over the indices of HoN . Hence (fˆ, g) ˆ N ≡ STr[fˆ‡ g], ˆ

fˆ, gˆ ∈ AN .

(69)

Here the graded involution ‡ in the noncommutative algebra AN is defined exactly as in (53). It is now obvious that this inner product approaches in the limit N → ∞ the commutative one. The detailed proof of this fact was furnished in [8]. Define a non-commutative Hamiltonian de Rham complex N of the fuzzy sphere S 2 as the graded associative algebra with unit N = 0N ⊕ 1N ⊕ 2N ,

(70)

0N = 2N = AeN

(71)

⊕ AaN 1N = AaN ⊕ AaN ⊕ AaN ¯ ¯ .

(72)

where

and

Gauge Theories on Noncommutative Sphere

269

The multiplication in N with the standard properties with respect to the grading is entailed by one in AN . The product of 1-forms is given by the same formula as in the graded commutative case (14), (A1 , A2 , A¯ 1 , A¯ 2 )(B1 , B2 , B¯ 1 , B¯ 2 ) ≡ A1 B¯ 1 + A2 B¯ 2 + A¯ 1 B1 + A¯ 2 B2 .

(73)

Of course, by definition, the r.h.s. is viewed as an element of 2N . Here we note an important difference with the graded commutative case: the product AA of a 1-form A with itself automatically vanishes in the commutative case but may be a non-vanishing element of 2N in the deformed picture. The product of a 1-form and a 2-form is again set to zero by definition. Now the coboundary operator d is given by df ≡ (T1 f, T2 f, T¯1 f, T¯2 f ),

f ∈ 0N ,

d(A1 , A2 , A¯ 1 , A¯ 2 ) ≡ T1 A¯ 1 + T2 A¯ 2 + T¯1 A1 + T¯2 A2 ,

dh = 0,

(74)

(A1 , A2 , A¯ 1 , A¯ 2 ) ∈ 1N , (75)

h ∈ 2N ,

(76)

where the action of Ti , T¯i is given by the noncommutative version of (8): Ti X ≡ N (ti X − (−1)X Xti ),

T¯i X ≡ N (t¯i X − (−1)X X t¯i ),

X ∈ AN , (77)

where ti = χ¯ i a,

t¯i = χi a¯ .

(78)

d maps iN to i+1,N and it satisfies d2 = 0,

d(AB) = (dA)B + (−1)A A(dB).

(79)

Using the graded involution ‡, we define the standard involution †(†2 = 1) on the noncommutative complex N : f † = f ‡ , f ∈ 0N ,

g † = −g ‡ , g ∈ 2N ,

(Ai , A¯ i )† = (A¯ ‡i , −A‡i ), (Ai , A¯ i ) ∈ 1N .

(80) (81)

The coboundary map d is compatible with the involution, however, due to noncommutativity, it is no longer true that the product of two real elements of N gives a real element. Thus we cannot define the real noncommutative Hamiltonian de Rham complex. For field theoretical applications this is not a drawback, nevertheless, because for the formulation of the field theories we shall not need the structure of the real subcomplex, but only the involution on the complex Hamiltonian de Rham complex. The “Hodge” star ∗ in N is defined precisely as in the commutative case: ∗ : f ∈ 0N → 2i¯aaf ∈ 2N ,

(82)

∗ : (A1 , A2 , A¯ 1 , A¯ 2 ) ∈ 1N → (iA1 , iA2 , −iA¯ 1 , −iA¯ 2 ) ∈ 1N ,

(83)

C. Klimˇc´ık

270

∗ : h ∈ 2N →

1 ¯ (Ti Ti − Ti T¯i − 2)h ∈ 0N . 4i

(84)

This “Hodge” star is compatible with the involution †. Note that the definition of the “Hodge” star ∗ on the 0-forms does not involve any ordering problem, since the operator a¯ a commutes with the elements of 0N . The natural inner product on N , whose commutative limit is (45), is (X, Y )N ≡

i STr[(∗X † )Y ], 2

X, Y ∈ N 0 , N 1 , N 2 ,

(85)

where STr is the standard supertrace (cf. (69)). The action of SU (2). The study of the SU (2) action on the deformed Hamiltonian de Rham complex N is important in view of our field theoretical applications. We require that such a SU (2) action gives in the commutative limit the SU (2) action on the undeformed de Rham complex, described in the previous section. This can be easily arranged, however, because the action on the undeformed commutative complex is generated entirely in terms of the Hamiltonian vector fields Rj whose Hamiltonians are rj : r+ = χ†1 χ2 ,

r− = χ†2 χ1 ,

r3 =

1 † (χ χ1 − χ†2 χ2 ). 2 1

(86)

Hence the deformed action on AN will be generated by the same Hamiltonians (86) (now understood as operators on the Fock space) but the Poisson bracket will be replaced by the commutator: Rj X = N [rj , X],

X ∈ AN .

(87)

It is trivial to check that the commutation relations (50) are fulfilled for this definition and that the SU (2) representation so generated is unitary with respect to the deformed inner product (69) on AN . Since both 0N and 2N can be identified with AeN , which is (as in the nondeformed case) an invariant subspace of AN , we have just obtained the SU (2) action on the even forms of the deformed Hamiltonian complex N . Recall that the space 1N of the 1-forms can be written as ⊗ C2 . 1 = AaN ⊗ C2 ⊕ AaN ¯

(88)

are invariant subspaces of the As in the nondeformed case, the spaces AaN and AaN ¯ SU (2) action on AN , given by (87), thus the formula (88) makes sense at the level of representations of SU (2). In other words, ⊕ and ⊗ are operations on the SU (2) representations thus defining 1N as a SU (2) representation which we look for. (As in the nondeformed case (51), the SU (2) acts by (52) on the second copy of C2 in (88) and on the first copy it acts in the complex conjugated way.) The final three facts, we shall need, read: 1) the inner product (85) on N is invariant with respect to the just defined SU (2) action; 2) the coboundary operator d on N and the “Hodge” star ∗ are both SU (2) invariant; 3) the generators Ri of SU (2) act on N as derivations with respect to the deformed product on N . We have constructed the non-commutative deformation of the complexified commutative Hamiltonian de Rham complex. We have identified the noncommutative counterparts of all structures of the latter and shown that they have the correct commutative limit. In particular, we observed that the multiplication in N approaches for N → ∞ the standard commutative product in . We have described the involution, the inner

Gauge Theories on Noncommutative Sphere

271

product, the “Hodge” star and the SU (2) action on N which have also the correct commutative limits and we have established that the “d” in the noncommutative context has all the basic properties (79) in order to deserve to be called the coboundary operator. Thus, whatever commutative construction which we perform by using these structures can be rewritten in the deformed finite-dimensional case. In particular, we shall write the field theoretical actions in this way. Note that we did not inject any smaller deformed de Rham complex into the deformed Hamiltonian complex . If we could do this we would not have had to bother ourselves with the Hamiltonian case! The point of our construction is that we will be able to formulate the dynamics of the standard gauge theories using the commutative Hamiltonian complex; this means that we also can deform those theories directly at this level. The price to pay is relatively low: few auxiliary fields will appear on the top of the fields present in the more standard constructions. We are here in a similar position as people who introduce auxiliary fields in trying to achieve a closure of algebras of supersymmetry without imposing the equations of motion. It turns out that our auxiliary fields are just the same as those used for the closure of the algebra of supersymmetry! Thus the same auxiliary fields do the double job. We believe that this is not just a coincidence. It is of no immediate use to calculate the cohomology of the noncommutative complex for our discussion of the field theoretical actions in this article. It is interesting to remark, however, that the finite deformed complex N can be shown to preserve faithfully the cohomological content of . 3. Gauge Theories on Noncommutative S 2 3.1. Theories with a scalar matter. Consider a complex scalar field φ on S 2 (i.e. φ ∈ B; cf. (21)) and an U (1) gauge connection described on the complement of the north (south) ¯ z)dz + V ∗N (z, ¯ z)dz¯ (v S = V S (w, ¯ w)dw + pole by a real 1-form field v N = V N (z, ∗S ¯ w)dw), ¯ where z, w (z = 1/w) are the complex coordinates on the corresponding V (w, patches. In what follows, we understand φ to be always a section of the trivial line bundle on S 2 , i.e. the standard complex function. Thus we can describe the connection by one globally defined 1-form on S 2 . Here we shall work only with one patch — the complement of the north pole. We encode the global character of the 1-form v, v = V dz + V ∗ dz¯

(89)

on the patch by demanding that V (V ∗ ) is an element of a B-bimodule B2 (B2¯ ) (cf. (22,23)). This ensures that z 2 V does not diverge for z → ∞ and, thus, the form v is well defined globally over the whole sphere S 2 . The scalar electrodynamics is defined by an action Z 1 dz¯ ∧ dz{(∂z¯ + iV ∗ )φ∗ (∂z − iV )φ + (∂z¯ − iV ∗ )φ(∂z + iV )φ∗ S= 4πi 1 − 2 (1 + zz) ¯ 2 (∂z¯ V − ∂z V ∗ )2 }. (90) 2g Here g is a real coupling constant of the theory. As usual, the kinetic term of the scalar field in two dimensions does not “remember” the conformal factor of the round metric ¯ + zz) ¯ −2 ) but the kinetic term of the gauge fields ds2 on the sphere (i.e. ds2 = 4dzdz(1 must be multiplied by the inverse power of the conformal factor.

C. Klimˇc´ık

272

The action (90) can be written in the language of forms (elements of the standard de Rham complex) as follows: S = (dφ − ivφ, dφ − ivφ)dR +

1 (dv, dv)dR , g2

(91)

where the inner product (., .)dR was defined in (44). It is not difficult to rewrite the scalar electrodynamics (91) in terms of the fields taking values in the Hamiltonian de Rham complex. To do this, we have to note that, aside from applying the “Hodge” star *, there is another way to convert a 2-form C ∈ 2 into a 0-form in 0 . Indeed, since both 0 and 2 can be identified with Ae , the identity map does the job. In what follows, we shall understand by the symbol C0 the corresponding 0-form. The standard scalar electrodynamics is then described by the action S∞ = (d8 − iA8, d8 − iA8) −

1 i I[(dA)20 ] + I[∗(8† (dA)20 8)]. 2 4g 8

(92)

The multiplet of fields is given by a complex Hamiltonian 0-form 8 ∈ Ae (cf. (12)) and by a real Hamiltonian 1-form A ∈ 1 (A = (A1 , A2 , A¯ 1 , A¯ 2 ), A† = A). The inner product (., .) on was defined in (45). (The index ∞ refers to the fact that this action will be soon recovered as an N → ∞ limit of an action SN , defined on the deformed Hamiltonian complex.) In order to understand the content of this theory, let us parametrize the Hamiltonian forms 8 and A as 8=φ+

¯ F bb , zz ¯ +1

(93)

P z¯ P )b, A2 = (V z − )b, A1 = −(V + zz ¯ + 1 zz ¯ +∗ 1 ∗ P z ¯ P ¯ )b, A¯ 2 = −(V ∗ z¯ − )b, A¯ 1 = (V ∗ + zz ¯ +1 zz ¯ +1

(94)

where all the fields φ, F, V and P depend only on the variables z, ¯ z. It is easy to verify that if φ, F and P belong to the algebra B (cf. (21)) and V and V ∗ to the bimodules B2 and B2¯ respectively, then A given by (94) sweeps the space 1 . In other words, the multiplet consisting of the complex Hamiltonian 0-form 8 and the real Hamiltonian 1-form A contains three standard complex de Rham 0-forms (the scalar fields φ, P, F ) ¯ Moreover note that the 1-form v is and one real de Rham 1-form v = V dz + V ∗ dz. injected in A by using the homomorphism ν (cf.(27)). By using the ansatz (93) and (94), the action (92) gives S∞ = (dφ − ivφ, dφ − ivφ)dR + (F + iφReP, F + iφReP )dR 1 2 − 2 (ImP, ImP )dR + 2 (∗dv, ImP )dR . g g

(95)

Here the field ImP and the combination (F + iφReP ) play the role of auxiliary fields and therefore they can be integrated away. This gives the action (91). It is also interesting to study another action that “optically” resembles (91): S∞ = (d8 − iA8, d8 − iA8) +

1 (dA, dA). g2

(96)

Gauge Theories on Noncommutative Sphere

273

Inserting the same ansatz (93),(94) into (96), we obtain S∞ = (dφ − ivφ, dφ − ivφ)dR + +

1 (dv, dv)dR + g2

1 (dImP, dImP )dR + (φImP, φImP )dR . g2

(97)

We observe that the result (97) is not quite the action (91) of the scalar electrodynamics but, apart from φ and v, there is one more propagating interacting field present, namely the imaginary part of P . This new field is neutral and it couples to the field φ only. It is not difficult to recognize that the action (97) describes nothing but the bosonic sector of the supersymmetric extension of the Schwinger model [15]. Thus, we have a good chance that upon adding fermions to our framework we shall recover the whole supersymmetric theory! It is important to note that the actions (92) and (96) have a bigger gauge symmetry than the standard actions (91) and (97). Indeed, let Ge be a group of unitary elements of Ae , i.e. Ge = {U ∈ Ae ; U † U = U U † = 1}.

(98)

An element U of Ge acts on (8, A) as follows: 8 → U 8,

A → A − idU U −1 .

(99)

With the transformation law (99), the actions (92) and (96) are gauge invariant with respect to Ge . By inspecting (98), it is not difficult to find that the group Ge decomposes into the direct product of the standard U (1) gauge group consisting of all elements of the form eiλ , λ ∈ B and a R gauge group (gauged real line) which acts only on the auxiliary fields. This R-subgroup drops out from the formulation involving only the dynamical fields and we are left with the standard U (1) gauge transformations φ → eiλ φ,

V → V + ∂z λ,

V ∗ → V ∗ + ∂z¯ λ.

(100)

We have rewritten the standard action of the scalar electrodynamics in terms of the structures of the Hamiltonian de Rham complex. This in turn means that we can directly write down a finite dimensional deformation of the field theoretical model (92) by replacing all structures occurring in (92) by their noncommutative counterparts: SN = (d8 − iA8, d8 − iA8)N −

1 STr[(dA − iA2 )20 ] 4g 2

i + STr[∗(8† (dA − iA2 )20 8)]. 8

(101)

Here 8 ∈ 0N is a noncommutative Hamiltonian 0-form, A ∈ 1N a real noncommutative Hamiltonian 1-form and the inner product (., .)N is given by (85). Now we should examine the gauge invariance of this action. Consider a group GeN consisting of the unitary elements of AeN : GeN = {U ∈ AeN ; U † U = U U † = 1}.

(102)

An element U of GeN acts on (8, A) ∈ N as follows: 8 → U 8,

A → U AU −1 − idU U −1 .

(103)

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274

Note that due to noncommutativity the gauge transformation looks like a non-Abelian one (only in the limit N → ∞ it reduces to the standard Abelian one). In the action (101), the term A2 (which identically vanishes in the commutative limit) is crucial for the gauge invariance for only with it the field strength dA − iA2 transforms homogenously: dA − iA2 → U (dA − iA2 )U −1 .

(104)

It is important to keep in mind that the gauge group GeN is a deformation of the commutative gauge group Ge . The latter has its local U (1) subgroup acting as in (100). In the noncommutative case, we cannot say which part of the Hamiltonian connection A is auxiliary (i.e. P -part) and which dynamical (v-part). Thus neither can we identify a noncommutative “local” U (1) subgroup of GeN . This means that the full GeN group plays a role in the deformed theory. I believe that this fact will be crucial in getting a nonperturbative insight on the problem of chiral anomaly. The noncommutative deformation of the model (96) is slightly more involved than the one of the standard scalar electrodynamics (92) , for we have to add more terms like A2 which in the commutative limit trivially vanish but they are required for the gauge invariance of the deformed model. The easiest way to proceed consists first in rewriting the undeformed action in the form S∞ = (d8 − iA8, d8 − iA8) +

1 1 (d(dA)0 , d(dA)0 ) − 2 I[(dA)20 ]. (105) 2 4g 4g

In order to derive (105) from (96) we have used the explicit form (41,42) of the “Hodge” star * and the integration “per partes”. Now it is easy to write a deformation of the model (105), which is gauge invariant with respect to the transformation laws (103): 1 1 STr[(dA − iA2 )20 ] + 2 (d(dA − iA2 )0 4g 2 4g (106) − i[A, (dA − iA2 )0 ], d(dA − iA2 )0 − i[A, (dA − iA2 )0 ])N .

SN = (d8 − iA8, d8 − iA8)N −

As the term A2 , also i[A, (dA − iA2 )0 ] vanishes identically4 in the commutative limit, thus converting (106) into (105) for N → ∞. So far we did not mention a very important property of the classical scalar electrodynamics on the sphere. Namely, the non-deformed actions (92) and (96) are invariant with respect to the SU (2) group which rotates the sphere (this SU (2) symmetry is a compact euclidean version of the standard Poincar´e symmetry of field theories). This statement follows from the invariance of the inner product (45) on (recall that the action of the group SU (2) on Ae and on was defined in Sect. 2.1), the SU (2) invariance of the coboundary operator d and of the “Hodge” star * and the fact that the SU (2) generators act as derivations with respect to the product on (i.e. Rj (89) = (Rj 8)9 + 8(Rj 9)). But all these properties hold also in the deformed case, thus we conclude that also our deformed actions (101) and (106) are SU (2) invariant. We end up this section by noting that one can easily formulate theories with a nontrivial potential energy of the scalar field (like an Abelian Higgs model) by adding to the action (92) a term of the form (∗W (8† 8), 1) ≡ 2i I[∗W (8† 8)]. In the deformed case we have to add to (101) 2i STr[∗W (8† 8)]. Here the potential W is some real entire function. After eliminating the auxiliary fields in the commutative case, a standard 4 It may seem that the commutator gives a Poisson bracket in the commutative limit but, in fact, this is false. The truth is that only the commutator multiplied by N gives for N → ∞ the Poisson bracket.

Gauge Theories on Noncommutative Sphere

275

potential term (∗W (φ∗ φ), 1)dR turns out to be added to (91). In particular, the linear function W corresponds to assigning a mass to the charged field. 3.2. Spinor electrodynamics. The standard (chiral or Weyl) spinor bundle on S 2 can be identified with the complex line bundle with the winding number ±1 (this is to say that 1 the transition function on the overlap N ∩ S of the patches is ( zz¯ )± 2 = e±iϕ , where ϕ is the asimuthal angle on the sphere). The plus (minus) sign corresponds to the right (left) chirality of the spinor. We shall work only with the patch N (the complement of the north pole) parametrized by the complex coordinate z. A Dirac spinor is then a sum of right-and left-handed Weyl spinors: ψ D = ψR + ψL .

(107)

Here ψR and ψL have each one complex Grassmann valued component and they represent globally well defined sections of the corresponding chiral spinor bundles iff they are elements of B-bimodules B˜1 and B˜1¯ linearly generated by the elements of the form z¯ k z k ¯

(1 + zz) ¯ m+ 2

1

,

max(k¯ − 1, k) ≤ m,

¯ m≥0 k, k,

(108)

,

¯ k − 1) ≤ m, max(k,

¯ m ≥ 0, k, k,

(109)

and z¯ k z k ¯

(1 + zz) ¯ m+ 2

1

respectively. The fact that, say, a right handed spinor ψR should be an element of the bimodule (108) follows from the fact that a spinor bilinear composed of spinors of the same chirality must be in a line bundle with winding number 2. The elements of the latter, multiplied by the zweibein component euz =

2 1 + zz ¯

(110)

are therefore components of holomorphic 1-forms V dz and we already know that V must belong to B2 (cf. (22)). The action of the free euclidean massless Dirac field on S 2 is given by Z 1 i dz¯ ∧ dz eψ¯ D γ c eµc (∂µ + ωµ,ab [γ a , γ b ])ψD , (111) S= 8π 8 where e is the determinant of the zweibein eaµ (or square root of the determinant of the metric), γ c are flat euclidean Hermitian γ matrices satisfying {γ a , γ b } = 2δ ab ,

(112)

ψ¯ D = ψ¯ R + ψ¯ L

(113)

and

is the conjugated Dirac spinor (ψ¯ R(L) is now left(right)-handed). It is convenient to introduce the flat holomorphic index u by 02 u 0 1 . γ ≡ γ + iγ = 00

(114)

C. Klimˇc´ık

276

Then γ u¯ is defined as the Hermitian conjugate of γ u . The elements of the zweibeins are euz =

2 = euz¯¯ . 1 + zz ¯

(115)

The last thing to be explained in (111) is the notion of the spin connection ωµ,ab . It is defined by the requirement ∂µ eaν − 0λµν eaλ + ωµa b ebν = 0,

(116)

where 0λµν are the standard Christoffel symbols. For the round metric on the sphere one computes (ωz )uu¯ =

1 z¯ , 2 1 + zz ¯

(ωz¯ )uu ¯ =

1 z . 2 1 + zz ¯

(117)

It is important to note that the 1-form ωz dz + ωz¯ dz¯ (with values in the Lie algebra so(2) ≡ u(1)) given by (117) is not globally defined in the sense of (22, 23) since it is singular at the north pole N (it is easy to see the singularity by looking at ω in the coordinate patch w). This should be the case, however, since ω is the connection on the nontrivial spinor bundle. An arbitraty U (1) connection on the spinor bundle can be achieved by adding a globally well-defined form v = V dz + V ∗ dz¯ (cf. (22, 23)) to the spin connection ω. Thus, the interaction of the Dirac field on S 2 with an external U (1) field v is described by the action Z dz¯ ∧ dz ¯ i 1 z¯ 1 z {ψR (∂z − − iV )ψL + ψ¯ L (∂z¯ − + iV ∗ )ψR }. Sv = 2π zz ¯ +1 2 1 + zz ¯ 2 1 + zz ¯ (118) In the two-dimensional context, one often “gets rid” of the spin connection by renormalizing the spinors ψ¯ D , ψD : √ √ ¯ D , ψ¯ D = 1 + zz ¯ ξ¯D . (119) ψD = 1 + zzξ The action (118) then becomes Z i dz¯ ∧ dz{ξ¯R (∂z − iV )ξL + ξ¯L (∂z¯ + iV ∗ )ξR }. Sv = 2π

(120)

One should remember, however, that the spinors ξR ,ξ¯L now belong to a B-bimodule B1 linearly generated (over Grassmann numbers) by the elements of B of the form z¯ k z k , (1 + zz) ¯ m ¯

¯ k + 1) ≤ m, max(k,

¯ m ≥ 0, k, k,

(121)

¯ m ≥ 0. k, k,

(122)

and ξL ,ξ¯R to a bimodule B1¯ generated by elements z¯ k z k , (1 + zz) ¯ m ¯

max(k¯ + 1, k) ≤ m,

The action of the massless spinor electrodynamics (the Schwinger model) on S 2 can be then written as

Gauge Theories on Noncommutative Sphere

i S= 2π

Z

277

dz¯ ∧ dz ξ¯R (∂z − iV )ξL + ξ¯L (∂z¯ + iV ∗ )ξR +

1 2 ∗ 2 . (1 + zz) ¯ (∂ V − ∂ V ) z¯ z 4g 2

(123)

Our next task will be to rewrite the spinor electrodynamics (123) in the form which would use the Hamiltonian vector fields Ti , T¯i instead of ∂z , ∂z¯ and the (real) Hamiltonian 1-form A ∈ 1 instead of v. To do this we have first to encode the spinors ξ¯D , ξD as elements of the algebra A (cf.(1)). This is easy: define ¯ L, 8 ≡ bξ¯L + bξ

¯ ≡ bξR + b¯ ξ¯R , ; 8

(124)

¯ belong to A. We also the fact that ξR , ξ¯L ∈ B1 and ξL , ξ¯R ∈ B1¯ implies that 8 and 8 have to identify how the 1-form v enters into the real Hamiltonian 1-form A. This is given as before in (94). With the ansatz (94) and (124), it turns out that the standard Schwinger model on S 2 can be rewritten as S=−

1 1 ¯ T¯1 8 + T2 8T ¯ 1 8 + T¯2 8T¯1 8 ¯ + T2 8T1 8]. ¯ I[(dA)20 ] + I[T¯2 8 2 4g 2

(125)

The quantity P (coming from (94)) plays the role of a nondynamical auxiliary field in (125); it can be eliminated by its equation of motion to yield (123). The meaning of the symbols in (125) is the following: T¯j , Tj are covariant derivatives acting on 8 and ¯ as follows: 8 Tj 8 = Tj 8 − iAj 8,

T¯j 8 = T¯j 8 − iA¯ j 8,

(126)

¯ = Tj 8 ¯ + 8iA ¯ j, Tj 8

¯ = T¯j 8 ¯ + 8i ¯ A¯ j , T¯j 8

(127)

the integral I was defined in (5). It is easy to check that the action (125) has the gauge symmetry with the gauge group ¯ and A as follows: Ge (cf. (98)). An element U of Ge acts on 8, 8 8 → U 8,

¯ → 8U ¯ −1 , 8

A → A − idU U −1 .

(128)

Now it is straightforward to write the action of the deformed Schwinger model: SN = −

1 1 ¯ T¯1 8 + T2 8T ¯ 1 8 + T¯2 8T¯1 8 ¯ + T2 8T1 8]. ¯ STr[(dA − iA2 )20 ] + STr[T¯2 8 2 4g 2 (129)

As in the case of the scalar electrodynamics, here A ∈ 1N is a real noncommutative ¯ and 8 are elements of AaN ⊕AaN Hamiltonian 1-form and the deformed spinor fields 8 ¯ (cf. (66) and (67)) with Grassmann coefficients. The operators Tj , T¯j (by a little abuse of notation we denote them in the same way as the undeformed quantities appearing in (125)) act as Tj 8 = N [tj , 8]+ − iAj 8,

T¯j 8 = N [t¯j , 8]+ − iA¯ j 8,

(130)

¯ = N [tj , 8] ¯ + + 8iA ¯ j, Tj 8

¯ = N [t¯j , 8]+ + 8i ¯ A¯ j . T¯j 8

(131)

The quantities tj , t¯j were defined in (9).

C. Klimˇc´ık

278

It is obvious that for N → ∞ the deformed action (129) gives the undeformed one (125). The gauge symmetry group in the noncommutative case is GeN (cf. (102)) and the deformed fields transform as 8 → U 8,

¯ → 8U ¯ −1 , 8

A → U AU −1 − idU U −1 .

(132)

A proof of the SU (2) isometry of the undeformed action (125) and of the deformed one (129) is easy: 1) the invariance of the terms not containing the fermions was already proved in the case of the scalar electrodynamics ; 2) the invariance of the fermionic terms follows from the SU (2) invariance of the inner products (5) and (69) and from the following commutation relations: [R3 , T1 ] =

1 T1 , 2

1 [R3 , T2 ] = − T2 , 2

1 [R3 , T¯1 ] = − T¯1 , 2

[R+ , T1 ] = 0,

[R+ , T2 ] = T1 ,

[R+ , T¯1 ] = −T¯2 ,

[R− , T1 ] = T2 ,

[R− , T2 ] = 0,

[R− , T¯1 ] = 0,

[R3 , T¯2 ] =

1¯ T2 , 2 (133)

[R+ , T¯2 ] = 0,

(134)

[R− , T¯2 ] = −T¯1 .

(135)

It is interesting to remark that we can add to the Hamiltonian vector fields Ri , Ti , T¯i one ¯ bb more even vector field Z, generated by the Hamiltonian a¯ a = ( zz+1 ) and obeying ¯ [Z, Ri ] = 0,

[Z, Ti ] = −Ti ,

[Z, T¯i ] = T¯i .

(136)

Then the generators Ri , Ti , T¯i and Z fulfil the osp(2, 2) superalgebra commutation relations (47)–(50) and (133)–(136). Note that we can construct also the chiral electrodynamics by setting the fields ξL and ξ¯R to zero. In the noncommutative situation the latter case corresponds to saying that both ¯ are in AaN . The action will continue to be (125) in the commutative matrices 8 and 8 case and (129) in the noncommutative one and the gauge transformations will be (128) and (132), respectively. Thus we have achieved quite an interesting result: we have naturally coupled the gauge field to a chiral fermion while having only a finite number of degrees of freedom and no fermion doubling. Perhaps it would be somewhat premature to draw too optimistic conclusions from this two-dimensional story, nevertheless, there is a clear promise that the method might work also in higher dimensions. Acknowledgement. I am grateful to A. Connes for enlightening discussions.

References 1. 2. 3. 4. 5. 6. 7. 8. 9.

Douglas, M., Kabat, D., Pouliot, P. and Shenker, S.: Nucl. Phys. B485, 85 (1997) Banks, T., Fischler, W., Shenker, S. and Susskind,L.: Phys. Rev. D55, 5112 (1997) Ishibasi, N., Kawai, H., Kitazawa, Y. and Tsuchiya, A.: Nucl. Phys. B498, 467 (1997) Connes, A.: Noncommutative geometry. London: Academic Press, 1994 Hoppe, J.: MIT PhD thesis, 1982 and Elem. Part. Res. J. (Kyoto) 80, 145 (1989) Madore, J.: J. Math. Phys. 32, 332 (1991) and Class. Quant. Grav. 9, 69 (1992) Grosse, H. and Madore, J.: Phys. Lett. B283, 218 (1992) Grosse, H., Klimˇc´ık, C. and Preˇsnajder, P.: Commun. Math. Phys. 185, 155 (1997) Kempf, A.: Phys. Rev. D54, 5174 (1996)

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10. 11. 12. 13. 14. 15.

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Doplicher, S., Fredenhagen, K. and Roberts, J.: Commun. Math. Phys. 172, 187 (1995) Berezin, F.: Commun. Math. Phys. 40, 153 (1975) Hawkins, E.: Quantization of equivariant vector bundles. q-alg/9708030 Madore, J.: Gravity on fuzzy space-time. gr-qc/9709002; Fuzzy surfaces of genus zero. gr-qc/9706047 Grosse, H. and Preˇsnajder, P.: Lett. Math. Phys. 33, 171 (1995) Ferrara, S.: Lett. Nuov. Cim. 13, 629 (1975)

Communicated by A. Connes

Commun. Math. Phys. 199, 281 – 295 (1998)

Communications in

Mathematical Physics © Springer-Verlag 1998

Affine Weyl Groups, Discrete Dynamical Systems and Painlev´e Equations Masatoshi Noumi, Yasuhiko Yamada Department of Mathematics, Kobe University Rokko, Kobe 657-8501, Japan Received: 11 May 1998 / Accepted: 2 July 1998

Abstract: A new class of representations of affine Weyl groups on rational functions are constructed, in order to formulate discrete dynamical systems associated with affine root systems. As an application, some examples of difference and differential systems of Painlev´e type are discussed.

Introduction In this paper, we propose a class of discrete dynamical systems associated with affine root systems, by constructing new representations of affine Weyl groups. This class of difference systems covers certain types of discrete Painlev´e equations, and is expected also to provide a general framework to describe the structure of B¨acklund transformations of differential systems of Painlev´e type. By a series of works by K. Okamoto [14], it has been known since the 80’s that Painlev´e equations P II , P III , P IV , P V and P VI admit the affine Weyl groups of type (1) (1) (1) (1) acklund transformations. A(1) 1 , C2 , A2 , A3 and D4 , respectively, as groups of B¨ The relationship between the affine Weyl group symmetry and the structure of classical solutions has been clarified through the studies of irreducibility of Painlev´e equations in the modern sense of H. Umemura (see [14, 7, 16, 9], for instance). In a recent work [10], the authors introduced a new representation (5.12) of the fourth Painlev´e equation P IV from which the structures of B¨acklund transformations and of special solutions of P IV are understood naturally. This sort of “symmetric forms” can be formulated for other Painlev´e equations as well (see [13]). One important feature of symmetric forms is that the structure of B¨acklund transformations of these Painlev´e equations can be described in a unified manner, by introducing a class of representations of affine Weyl groups inside certain Cremona groups. Also, with the τ -functions appropriately defined, the dependent variables of the Painlev´e equations allow certain “multiplicative formulas” in terms of τ -functions. It is remarkable that, in our multi-

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M. Noumi, Y. Yamada

plicative formulas (2.2), the factors are completely determined by the Cartan matrix of the corresponding affine root system. Similar structures can be found commonly in various (discrete) integrable systems with Painlev´e (singularity confinement) property ([15, 5, 4]). The main purpose of this paper is to present a new class of representations of affine Weyl groups which provides a prototype of affine Weyl group symmetry in nonlinear differential and difference systems. In Sects. 1 and 2, we introduce a class of representations of the Coxeter groups of Kac–Moody type on certain fields of rational functions (on the levels of f -variables and τ -functions, respectively). This class of representations was found as a generalization of the structure of B¨acklund transformations in the symmetric forms of Painlev´e equations (1) (1) P IV , P V and P VI which are the cases of A(1) 2 , A3 and D4 respectively. Our representation in the case of an affine root system provides naturally a discrete dynamical system from the lattice part of the affine Weyl group. We introduce in Sect. 3 the discrete dynamical systems associated with affine root systems in this sense. The case of A(1) l is discussed in Sect. 4 in some detail as an example. One interesting aspect of our system is that continued fractions arise naturally in the discrete dynamical system, with variations depending on the affine root system. In the final section, we explain how one can apply our discrete dynamical systems to the problem of symmetry of nonlinear differential (or difference) systems. In particular, we present a series of nonlinear ordinary differential systems which have symmetry under the affine Weyl groups of type A(1) l . This series of nonlinear equations gives a generalization of the Painlev´e equations P IV and P V to higher orders. 1. A Representation of the Coxeter Group W (A) We fix a generalized Cartan matrix (or a root datum) A = (aij )i,j∈I with I being a finite indexing set. By definition, A is a square matrix with the properties (C1) (C2) (C3)

ajj = 2 for all j ∈ I, aij is a nonpositive integer if i 6= j, aij = 0 ⇔ aji = 0 (i, j ∈ I).

(See Kac [3] for the basic properties of generalized Cartan matrices. Although we assume that I is finite, a considerable part of the following argument can be formulated under the assumption that A is locally finite, namely, for each j ∈ I, aij = 0 except for a finite number of i’s. ) We define the root lattice Q = Q(A) and the coroot lattice Q∨ for A by M M Z αj and Q∨ = Z αj∨ (1.1) Q= j∈I

j∈I

respectively, together with the pairing h , i : Q∨ × Q → Z such that hαi∨ , αj i = aij for i, j ∈ I. We denote by W = W (A) the Coxeter group defined by the generators si (i ∈ I) and defining relations s2i = 1,

(si sj )mij = 1 (i, j ∈ I, i 6= j),

(1.2)

where mij = 2, 3, 4, 6 or ∞ according as aij aji = 0, 1, 2, 3 or ≥ 4. The generators si act naturally on Q by reflections

Affine Weyl Groups, Discrete Dynamical Systems and Painlev´e Equations

si (αj ) = αj − αi hαi∨ , αj i = αj − αi aij

283

(1.3)

for i, j ∈ I. Note that the action of each si on Q induces an automorphism of the field C(α) = C(αi ; i ∈ I) of rational functions in αi (i ∈ I) so that C(α) becomes a left W -module. Introducing a set of new “variables” fj (j ∈ I), we propose to extend the representation of W on C(α) to the field C(α; f ) = C(α)(fj ; j ∈ I) of rational functions in αj and fj (j ∈ I). In order to specify the action of si on fj , we fix a matrix U = (uij )i,j∈I with entries in C such that (0) uij = 0 if i = j or aij = 0, if (aij , aji ) = (−1, −1), (1) uij = −uji if (aij , aji ) = (−2, −1), (2) uij = −uji or −2uji (3) uij = −uji , − 23 uji , −2uji or −3uji if (aij , aji ) = (−3, −1). Theorem 1.1. Let A = (aij )i,j∈I be a generalized Cartan matrix and U = (uij )i,j∈I a matrix satisfying the conditions above. For each i ∈ I, we extend the action of si on C(α) to an automorphism of C(α; f ) such that si (fj ) = fj +

αi uij fi

(j ∈ I).

(1.4)

Then the actions of these si define a representation of the Coxeter group W = W (A) (i.e. a left W -module structure) on the field C(α; f ) of rational functions. We have only to check that the automorphisms si on C(α; f ) are involutions (s2i = 1 for all i ∈ I) and that they satisfy the Coxeter relations (si sj )mij = 1 when i 6= j and mij = 2, 3, 4, 6. This can be carried out by direct computations since, for any i ∈ I, the automorphism si stabilizes the subfield C(α)(fi , fk ) for each k ∈ I and, for any i, j ∈ I, both si and sj stabilize the subfield C(α)(fi , fj , fk ) for each k ∈ I. We remark that Theorem 1.1 provides a systematic method to realize the Coxeter groups of Kac–Moody type nontrivially inside Cremona groups (groups of the birational transformations of affine spaces). Remark 1.2. An important class of generalized Cartan matrices is that of symmetrizable ones, which includes the matrices of finite type and of affine type. Our condition on U = (uij )ij∈I described above requires that U should be “almost” skew-symmetrizable. The matrix U can be thought of as specifying a sort of orientation of the Coxeter graph of A. It is also related to Poisson structures of dynamical systems. f = W o Remark 1.3. Practically, it is sometimes necessary to consider the extension W of W = W (A) by a group of diagram automorphisms of A. Recall that a diagram automorphism ω is by definition a bijection on I such that aω(i)ω(j) = aij for all i, j ∈ I; the commutation relations of each ω ∈ with elements of W are given by ωsi = sω(i) ω for all i ∈ I. Suppose that the matrix U satisfies in addition the following compatibility condition with respect to : uω(i)ω(j) = uij for all i, j ∈ I, ω ∈ . Then, together with the automorphisms ω of C(α; f ) such that ω(αj ) = αω(j) , ω(fj ) = fω(j) (j ∈ I), the representation of W in Theorem 1.1 lifts to a representation of the extended Coxeter f = W o on C(α; f ). group W

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2. τ -Functions – A Further Extension of the Representation We now introduce another set of variables τj (j ∈ I), which we call the “τ -functions” for the f -variables fj (j ∈ I). Considering the field extension C(α; f ; τ ) = C(α; f )(τj ; j ∈ I), we propose a way to extend the representation of W of Theorem 1.1 to C(α; f ; τ ). Theorem 2.1. Let A be a generalized Cartan matrix and U = (uij )i,j∈I a matrix with entries in C satisfying the conditions (0) (1) (2)

ujj = 0 for all j ∈ I, uij = uji = 0 if aij = aji = 0, uij = −kuji if (aij , aji ) = (−k, −1) with k = 1, 2 or 3.

We extend the action of each generator si of W on C(α; f ) to an automorphism of C(α; f ; τ ) by the formulas Q |aki | Y −a k∈I\{i} τk ki τk = fi , (2.1) si (τj ) = τj (i 6= j), si (τi ) = fi τi τi k∈I

for all i, j ∈ I. Then these automorphisms define a representation of W on C(α; f ; τ ). The formulas (2.1) of Theorem 2.1 specify how the f -variables should be expressed in terms of the τ -functions: fj = Q

τj sj (τj ) |aij | i∈I\{j} τi

(2.2)

for all j ∈ I. We remark that this type of multiplicative formulas by τ -functions is of a universal nature as can be found in various discretized integrable systems such as T -systems, discrete Toda equations and discrete Painlev´e equations (see [5, 4, 15],. . . ). In that context, the existence of multiplicative formulas is thought of as a reflection of singularity confinement which is a discrete analogue of the Painlev´e property. Remark 2.2. If the matrix U is invariant with respect to a group of diagram automorf = W o on C(α; f ) extends phisms, then the action of the extended Coxeter group W naturally to C(α; f ; τ ) by ω.τj = τω(j) for all j ∈ I. Theorem 2.1 can be proved essentially by direct computation to verify the fundamental relations of the Coxeter group with respect to the action on the τ -functions τk (k ∈ I). Instead of giving the detail of such a proof, we will explain some of the ideas behind these multiplicative formulas. We consider that the τ -functions should correspond to the fundamental weights 3j , while the f -variables do to simple roots αi . Let ∨ us denote by L = HomZ (Q∨ , Z) the dual Z-module L of the coroot lattice Q , and take ∨ the dual basis {3j }j∈I of {αi }i∈I so that L = j∈I Z3j . Note that L, being the dual of Q, has a natural action of W and that there is a natural W -homomorphism Q → L such that X 3i aij (j ∈ I) (2.3) αj 7→ i∈I

through the pairing h , i. (The lattice L is in fact the weight lattice modulo the null roots.) The action of W on L is then described as

Affine Weyl Groups, Discrete Dynamical Systems and Painlev´e Equations

si (3j ) = 3j (i 6= j),

si (3i ) = 3i −

X

285

3k aki

(2.4)

k∈I

for i, j ∈ I. We remark that formulas (2.1) in Theorem 2.1 are a multiplicative analogue of (2.4) except for the factor fj . Let us introduce the notation of formal exponentials for τ -functions: Y X τiλi for each λ = λi 3i ∈ L, (2.5) τλ = i∈I

i∈I

where λi = hαi∨ , λi. In order to clarify the meaning of Theorem 2.1, we consider the action of each element w ∈ W on τ λ for λ ∈ L. Suppose now that the action of W on C(α; f ) can be extended to C(α; f ; τ ) as described in Theorem 2.1. Since formulas δ (2.1) read as si (τ 3j ) = fj ij τ si (3j ) for j ∈ I, we have by linearity si (τ λ ) = fiλi τ si (λ)

(2.6)

for each λ ∈ L. Hence, for each w ∈ W , we should have rational functions φw (λ) ∈ C(α; f ) indexed by λ ∈ L such that w(τ λ ) = φw (λ) τ w.λ

(w ∈ W, λ ∈ L).

(2.7)

Furthermore, these functions φw (λ) should satisfy the following cocycle condition: φw1 w2 (λ) = w1 (φw2 (λ)) φw1 (w2 .λ)

(2.8)

for all w1 , w2 ∈ W and λ ∈ L. Conversely, if one has a family (φw (λ))w∈W,λ∈L of rational functions satisfying the cocycle condition (2.8), one can define a representation of W on C(α; f ; τ ) by means of (2.7). Theorem 2.1 is thus equivalent to the following proposition. Proposition 2.3. Under the same assumption of Theorem 2.1, there exists a unique cocycle φ = (φw (λ))w∈W,λ∈L such that φ1 (λ) = 1,

hα∨ i ,λi

φsi (λ) = fi

(λ ∈ L)

(2.9)

for each i ∈ I. Remark 2.4. Any family {φw (λ)}w∈W,λ∈L of rational functions in C(α; f ), linear in λ, can be identified with a mapping φ : W → HomZ (L, C(α; f )× ) : w 7→ φw ,

(2.10)

where C(α; f )× stands for the multiplicative group of C(α; f ) regarded as a Z-module. The cocycle condition (2.8) is then equivalent to saying that the mapping φ of (2.10) is a Hochschild 1-cocycle of W with respect to the natural W -bimodule structure of HomZ (L, C(α; f )× ). Furthermore, formula (2.7) means that this cocycle φ becomes the coboundary of the 0-cochain τ ∈ HomZ (L, C(α; f ; τ )× ) : λ 7→ τ λ

(2.11)

after the extension of the W -module C(α; f ) to C(α; f ; τ ). Thus one could say that: The role of τ -functions is to trivialize the Hochschild 1-cocycle defined by the f -variables.

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From the cocycle condition, it follows that the cocycle φw : L → C(α; f )× of Proposition 2.3 can be expressed as φw (λ) =

p Y

∨

sj1 · · · sjr−1 (fjr )hαjr , sjr+1 ...sjp .λi

(λ ∈ L)

(2.12)

r=1

for any expression w = sj1 . . . sjp of w in terms of generators. The cocycle φ = (φw (λ))w∈W,λ∈L defined above plays a crucial role in application of our representation to discrete dynamical systems. One remarkable thing about this cocycle is that φ seems to have a very strong regularity as described in the following conjecture. Conjecture 2.5. In addition to conditions (1) and (2) of Theorem 2.1, suppose that the matrix U = (uij )i,j∈I satisfies the condition (30 )

uij aji + aij uji = 0

for all i, j ∈ I.

Then, for any k ∈ I, the rational functions φw (3k ) (w ∈ W ) of (2.7) are polynomials in αj , fj and uij (i, j ∈ I) with coefficients in Z. As for the root systems of type A(1) l and A∞ , this conjecture is known to be affirmative. Furthermore, the rational functions φw (3k ) in these cases have determinant formulas of Jacobi-Trudi type (see [18]). Remark 2.6. In Sects. 1 and 2, we presented a nontrivial class of representations of Coxeter groups W (A) over the fields of f -variables and τ -functions, with A being a generalized Cartan matrix. This class of representations appears in fact as B¨acklund transformations (or the Schlesinger transformations) of the Painlev´e equations P IV , P V and P VI , which correspond to the cases of the generalized Cartan matrices A of type A(1) 2 , (1) and D , respectively. As to these Painlev´ e equations, one can define appropriate A(1) 3 4 f -variables and τ -functions for which the B¨acklund transformations are described as in Theorems 1.1 and 2.1 (see [10, 13]). (In the cases of P II and P III , which have symmetries (1) of type A(1) 1 and C2 , the corresponding representations of the affine Weyl groups on f -variables must be modified appropriately, while the multiplicative formulas in terms of τ -functions keep the same structure.) In the context of B¨acklund transformations of Painlev´e equations, the functions φw (3k ), specialized to certain particular solutions, give rise to the special polynomials, called Umemura polynomials (see [17, 8]), which are defined to be the main factors of τ -functions for algebraic solutions of the Painlev´e equations. For this reason, we expect that the functions φw (3k ) (w ∈ W, k ∈ I) should supply an ample generalization of Umemura polynomials in terms of root systems.

3. Affine Weyl Groups and Discrete Dynamical Systems In what follows, we assume that the generalized Cartan matrix A is indecomposable and is of affine type. We use the standard notation of the indexing set I = {0, 1, . . . , l} so that α1 , . . . , αl form a basis for the corresponding finite root system. Recall that the null root δ is expressed as δ = a0 α0 + a1 α1 + · · · + al αl with certain positive integers a0 , a1 , . . . , al . The affine Weyl group W = W (A) is generated by the fundamental reflections s0 , s1 , . . . , sl with respect to the simple roots α0 , α1 , . . . , αl :

Affine Weyl Groups, Discrete Dynamical Systems and Painlev´e Equations

287

W = W (A) = hs0 , . . . , sl i.

(3.1)

One important aspect of the affine case is that the affine Weyl group W = W (A) has an alternative description as the semi-direct product of a free Z-submodule M of rank l of ◦ Ll hR = i=1 Rαi∨ , and the finite Weyl group W0 acting on M : ∼

W ← M o W0

with

W0 = hs1 , . . . , sl i.

(3.2)

For each element µ ∈ M , we denote by tµ the corresponding element of the affine Weyl group W , so that tµ+ν = tµ tν for all µ, ν ∈ M . Note that the structure of the lattice part M depends on the type of the affine root system and that, if A is nontwisted, ◦

i.e., of type Xl(1) , then M is identified with the coroot lattice Q∨ of the finite root system with basis {α1 , . . . , αl }. (There are descriptions analogous to (3.2) for certain f = W o as well.) As we already remarked, the field extended affine Weyl groups W C(α) = C(α0 , α1 , . . . , αl ) has a natural structure of W -module. The lattice part M in the decomposition (3.2) acts on C(α) by tµ (αj ) = αj − hµ, αj iδ

(j = 0, 1, . . . , l; µ ∈ M )

(3.3)

as shift operators with respect to the simple affine roots. (The null root δ is a W -invariant element of C(α). For this reason, it is sometimes more convenient to consider δ to be a nonzero constant which represents the scaling of the lattice M .) Suppose now that one has extended the action of W from C(α) to C(α; f ) = C(α)(f0 , f1 , . . . , fl ). At this moment, we can consider an arbitrary extension C(α; f ) as a W -module, assuming that each element of W acts on the function field as an automorphism; the representation of W presented in Sects. 1 and 2 provides a choice of such an extension. For each ν ∈ M , we define a family of rational functions Fνj (α; f ) ∈ C(α; f ) by tν (fj ) = Fνj (α; f )

(j = 0, 1, . . . , l).

(3.4)

Then these formulas can already be considered as a discrete dynamical system, defined by a set of commuting discrete time evolutions. In other words, we obtain a commuting family of rational mappings on the affine space where αj and fj play the role of coordinates of the discrete time variables and the dependent variables, respectively. To make clear the meaning of (3.4) as a difference system, we set αj [µ] = tµ (αj ) = αj − hµ, αj iδ,

fj [µ] = tµ (fj )

(j = 0, . . . , l)

(3.5)

for each µ ∈ M , and consider them as representing functions on M with initial values αj [0] = αj , fj [0] = fj (j = 0, . . . , l). Then formula (3.4) implies that fj [µ + ν] = Fνj (α[µ]; f [µ])

(j = 0, 1, . . . , l).

(3.6)

In this sense, the functions Fνj (α; f ) defined above provide a difference dynamical system on the lattice M . Since fj [µ] is a rational function in f0 , . . . , fl , for each µ ∈ M , the general solution of the difference system (3.6) a priori depends rationally on initial values f0 , f1 , . . . , fl . Note also that the action of the affine Weyl group W on fj [ν] is described as (w.fj )[µ] = w(fj [w−1 µ])

(j = 0, . . . , l; µ ∈ M )

(3.7)

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M. Noumi, Y. Yamada

for all w ∈ W . In this sense, our difference system admits the action of the affine Weyl group W (A). Note that, if one take the representation of Theorem 1.1, one has (si .fj )[µ] = fj [µ] +

αi [µ] uij fi [µ]

(3.8)

for i, j = 0, . . . , l. Suppose that one can extend the action of W further to the τ -functions as in Theorem 2.1 and set τi [µ] = tµ (τi ), regarding τi [0] = τi as initial values of the τ -functions. Then from (2.2) we obtain the multiplicative formulas τj [µ] sαj [µ] (τj [µ]) fj [µ] = Q |aij | i∈I\{j} τi [µ]

(j = 0, . . . , l)

(3.9)

for the f -variables in terms of τ -functions. In terms of the cocycle φ, these formulas are rewritten by (2.7) into φt (3j ) φtµ sj (3j ) fj [µ] = Q µ |aij | i∈I\{j} φtµ (3i )

(j = 0, . . . , l),

(3.10)

which give a complete description of the general solution of the difference system (3.4) in terms of the initial values f0 , . . . , fl . In this sense, the cocycle φ solves our difference system (3.4). It should be noted that all these properties of the difference system (3.4), or (3.6) equivalently, are already guaranteed when we take the representation of the affine Weyl group W (A) as in Theorem 2.1. Also, it is meaningful if one could find other types of representations of affine Weyl groups which have the properties of Theorem 2.1. We now take the representation of the affine Weyl group W (A) on C(α; f ; τ ) introduced in Theorem 2.1. One interesting feature of our representation is that continued fractions arise naturally in the description of discrete dynamical systems, and that the structure of continued fractions is determined by the affine root system. We assume for simplicity that the generalized Cartan matrix A is of type Xl(1) . For a given element w ∈ W (A), take a reduced decomposition w = si1 si2 · · · sip of w, and define the affine roots β1 , β2 , · · · , βp by β1 = α1 , β2 = si1 (αi2 ), . . . , βp = si1 · · · sip−1 (αip ).

(3.11)

Note that these βr (r = 1, . . . , p) give precisely the set of all positive real roots whose reflection hyperplanes separate the fundamental alcove C and its image w.C by w. Since the action of si on fj is given by si (fj ) = fj +

αi uij fi

(i, j = 0, . . . , l),

(3.12)

we have inductively w(fj ) = fj +

αip αi1 αi2 ui1 j + si1 ui2 j + · · · + si1 · · · sip−1 u ip j . fi1 fi2 fip

Each summand of this expression is given by the continued fraction

(3.13)

Affine Weyl Groups, Discrete Dynamical Systems and Painlev´e Equations

si1 · · · sir−1

αir = fir

289

βr fir + uir−1 ir

(3.14)

βr−1 fir−1 + . .

.

β2 fi2 + ui1 i2

β1 fi1

along the reduced decomposition w = si1 · · · sip . Note also that formula (3.13) for w(fj ) has an alternative expression φw (3j ) φwsj (3j ) w(fj ) = Q |aij | i∈I\{j} φw (3i )

(3.15)

in terms of the cocycle φ, which is implied by (2.2) and (2.7). If one take an element ν ∈ ◦

M = Q∨ of the dual root lattice, the rational functions tν (fj ) = Fνj (α; f ) (j = 0, . . . , l) for the time evolution with respect to ν are determined in the form Fνj (α; f ) = fj +

p X

si1 · · · sir−1

r=1

αir u ir j fir

(3.16)

as a sum of continued fractions along the reduced decomposition of tν , with positive real roots separating the fundamental alcove C and its translation C + ν. We remark that a similar description of the rational functions Fνj (α; f ) can be given also for the f = W o . A series of such discrete dynamical cases of extended affine Weyl groups W systems will be given in the next section. 4. Discrete Dynamical System of Type A(1) l As an example of our discrete dynamical systems associated with affine root systems, we will give an explicit description of the case of A(1) l with l ≥ 2. Consider the generalized Cartan matrix   2 −1 0 · · · 0 −1 −1 2 −1 · · · 0 0     0 −1 2 · · · 0 0   (4.1) A =  .. .. .. . . .. ..  . . .    . . .  0 0 0 · · · 2 −1 −1 0

0 · · · −1 2

(l ≥ 2), and identify the indexing set {0, 1, . . . , l} with Z/(l + 1)Z. We take of type the following matrix of “orientation” to specify our representation of W = W (A(1) l ):   0 1 0 · · · 0 −1 −1 0 1 · · · 0 0     0 −1 0 · · · 0 0   (4.2) U = . . . . .  . . . ... . .  . . .   . . .  0 0 0 ··· 0 1  A(1) l

1

0 0 · · · −1 0

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M. Noumi, Y. Yamada

Then the action of the affine Weyl group W = hs0 , . . . , sl i on the variables αj , fj and τj is given explicitly as follows: si (αj ) = αj + αi (j = i ± 1), si (αj ) = αj (j 6= i, i ± 1), si (αi ) = −αi , αi si (fj ) = fj ± (j = i ± 1), si (fj ) = fj (j = 6 i, i ± 1), si (fi ) = fi , fi τi−1 τi+1 (4.3) , si (τj ) = τj (j 6= i). si (τi ) = fi τi Note that U is invariant with respect to the diagram rotation π : i → i + 1. Hence this f = W o {1, π, . . . , π l } by action of W extends to the extended affine Weyl group W π(αj ) = αj+1 ,

π(fj ) = fj+1 , ◦

π(τj ) = τj+1 .

(4.4)

◦

f is now isomorphic to P o W0 , where P is the weight lattice of the finite The group W root system of type Al and W0 = hs1 , . . . , sl i ' Sl+1 . Taking the first fundamental weight $1 = (lα1 + (l − 1)α2 + · · · + αl )/(l + 1) of the finite root system, we set T 1 = t$ 1 ,

Ti = πTi−1 π −1

(i = 2, . . . , l + 1).

(4.5)

These shift operators are expressed as T1 = πsl sl−1 · · · s1 , T2 = s1 πsl . . . s2 , . . . , Tl+1 = sl · · · s1 π

(4.6)

f . Note that T1 · · · Tl+1 = 1 and that T1 , . . . , Tl form a in terms of the generators of W f. basis for the lattice part of W The simple affine roots α0 , . . . , αl are the dynamical variables for the shift operators T1 , . . . , Tl such that Ti (αi−1 ) = αi−1 + δ,

Ti (αi ) = αi − δ,

Ti (αj ) = αj (j 6= i − 1, i).

(4.7)

For each k ∈ Z/(l + 1)Z and r = 0, 1, . . . , l − 1, we define gk,r to be the continued fraction αk (4.8) gk,r = sk+r sk+r−1 · · · sk+1 fk αk+r | αk + . . . + αk+r | αk+1 + . . . + αk+r | − − ··· − . = | fk | fk+1 |fk+r Then the discrete time evolution by T1 is expressed as T1 (f0 ) = f1 − g2,l−1 + g0,0 , T1 (f1 ) = f2 − g3,l−2 , T1 (f2 ) = f3 − g4,l−3 + g2,l−1 , ··· T1 (fl−1 ) = fl − g0,0 + gl−1,2 , T1 (fl ) = f0 + gl,1 .

(4.9)

The corresponding formulas for T2 , . . . , Tl are obtained from these by applying the diagram rotation π.

Affine Weyl Groups, Discrete Dynamical Systems and Painlev´e Equations

291

5. Nonlinear Systems with Affine Weyl Group Symmetry As we already remarked in Sect. 2, the representation of W (A) introduced in Theo(1) (1) acklund transrems 1.1 and 2.1, for the cases A(1) 2 , A3 and D4 , arises in nature as B¨ formations of Painlev´e equations P IV , P V and P VI , respectively. Hence, the Painlev´e equations P IV , P V and P VI have the structure of discrete dynamical systems on the lattice as described in Sect. 3 with respect to B¨acklund transformations. As to PIV , this point has been discussed in detail in our previous paper [10]. “Symmetric forms” of all Painlev´e equations PII , . . . , P VI and their B¨acklund transformations will be discussed in our forthcoming paper [13]. From the viewpoint of nonlinear equations of Painlev´e type, an important problem would be the following: Problem 5.1. For each affine root system (or for each generalized Cartan matrix A, in general), find a system of differential (or difference)equations for which the Coxeter group W = W (A) acts as B¨acklund transformations. We believe that such differential (or difference) systems with affine Weyl group symmetry should provide an intriguing class of dynamical systems with rich mathematical structures, to be compared to Painlev´e equations. We also remark that, if one specifies the representation of W = W (A) in advance as in Theorem 2.1, then the problem mentioned above is equivalent to finding such derivations (or shift operators) on C(α; f ) and C(α; f ; τ ) that commute with the action of W (A). In this section, we will introduce some examples of type A(1) l of difference and differential systems with affine Weyl group symmetry, as well as remarks on the continuum limit from the difference to the differential systems. We first explain a general idea to construct difference systems with affine Weyl group symmetry by means of our discrete dynamical systems associated with affine root systems. Consider the discrete dynamical system defined by an affine root system as in Sect. 3. If we take a sublattice N ⊂ M of rank r, then the centralizer ZW (A) (N ) of N in W gives rise to a group of B¨acklund transformations of the discrete system tν (fj ) = Fνj (α; f )

(j = 0, . . . , l)

(5.1)

on the sublattice N of rank r, with αj (j = 0, . . . , l) regarded as functions on N such that tν (αj ) = αj −hν, αj i. The centralizer ZW (A) (N ) contains in fact subgroups generated by reflections acting on the quotient M/N . For instance, let WM/N be the group generated by the reflections sα with respect to the affine roots α that are perpendicular to the lattice N . Then WM/N is contained in the group of B¨acklund transformations of the discrete system (5.1). (The group of symmetry thus obtained may have a different structure from that of our representations of Sects. 1 and 2.) For example, the difference system (4.9) with respect to the shift operator T1 has symmetry under the affine Weyl group W (A(1) l−1 ) = hr, s2 , . . . , sl i, where r = s0 s1 s0 . The corresponding simple affine roots are given by α0 + α1 , α2 , . . . , αl . Note that the root α0 + α1 is invariant under T1 . The reflection r acts on the variables fj as follows: α0 + α1 α0 + α1 , r(f2 ) = f2 + , s1 (f0 ) s0 (f1 ) α0 + α1 α0 + α1 r(fl ) = fl − , r(f0 ) = f0 − , s1 (f0 ) s0 (f1 ) (j = 3, . . . , l − 1). r(fj ) = fj

r(f1 ) = f1 +

(5.2)

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We remark that the two elements s1 (f0 ) and s0 (f1 ) = T1 s1 (f0 ) are invariant under the action of r as well as f3 , . . . , fl−1 . Similarly, the difference system with respect to the commuting operators T1 , . . . , Tk has affine Weyl group symmetry under the subgroup W (A(1) l−k ) = hr, sk+1 , . . . , sl i, where r = s0 s1 · · · sk−1 sk sk−1 · · · s0 . If one can take an appropriate continuum limit of the sublattice N inside M , one would possibly obtain a differential system in r variables whose group of B¨acklund transformations contains a reasonable reflection group. We show an example in which the idea explained above works nicely, in some detail. Consider the discrete dynamical system of type A(1) 2 with extended affine Weyl group f which represents f = W o as in Sect. 4. We take the element T = T1 = πs2 s1 ∈ W W the translation t$1 with respect to the first fundamental weight $1 = (2α1 + α2 )/3 of the finite root system, so that T (α0 ) = α0 + δ,

T (α1 ) = α1 − δ,

T (α2 ) = α2 .

(5.3)

Note that T can be considered as a shift operator with respect to the variable α1 . Our discrete dynamical system for this case is described as follows: α0 α2 + α0 | α0 | − − , f0 | f2 |f0 α0 , T (f1 ) = f2 − f0 α2 + α0 | α0 | − T (f2 ) = f0 + | f2 |f0

T (f0 ) = f1 +

(5.4)

in terms of continued fractions. We remark that T −1 (f0 ) takes a simpler form than T (f0 ) above: α1 . (5.5) T −1 (f0 ) = f2 + f1 f , we set f0 +f1 +f2 = Noticing that the element f0 +f1 +f2 is invariant under the action of W c. Then from (5.4) and (5.5) we obtain the following equivalent form of our difference system: T −1 (f0 ) + f0 = c − f1 +

α1 , f1

f1 + T (f1 ) = c − f0 −

α0 . f0

(5.6)

With the notation fi [n] = T n (fi ) for n ∈ Z, this equation gives rise to a representation of the second discrete Painlev´e equation dP II (cf. [2]): α1 − nδ , f1 [n] α0 + nδ f1 [n] + f1 [n + 1] = c − f0 [n] − (n ∈ Z). f0 [n]

f0 [n − 1] + f0 [n] = c − f1 [n] +

(5.7)

Since the shift operator T = πs2 s1 commutes with the two reflections r0 = s0 s1 s0 and r1 = s2 , we see that the difference system (5.6) or (5.7) has symmetry of the affine Weyl group W (A(1) 1 ) = hr0 , r1 i. (The corresponding simple roots are β0 = α0 + α1 and β1 = α2 .)

Affine Weyl Groups, Discrete Dynamical Systems and Painlev´e Equations

293

The second Painlev´e equation P II arises as a continuum limit of the difference system (5.7), and that A(1) 1 -symmetry of (5.7) naturally passes to P II . Introduce a small parameter ε such that δ = ε3 , and set f0 [n] = 1 + εψ + ε2 ϕ0 ,

f1 [n] = 1 − εψ + ε2 ϕ1 ,

c = 2,

(5.8)

α0 + nδ = −1 + ε x + ε a0 , α1 − nδ = 1 − ε x + ε a1 , α2 = ε b1 . 2

3

2

3

3

Then in the limit as ε → 0, the difference equations (5.7) imply the following differential equation for ϕ0 , ϕ1 , ψ: 1 1 ϕ00 = 2ϕ1 ψ + a0 − , ϕ01 = 2ϕ0 ψ + a1 − , 2 2 ψ 0 = 2(ϕ0 + ϕ1 ) − ψ 2 + x.

(5.9)

From this we get the second Painlev´e equation for ψ, ψ 00 = 2ψ 3 − 2xψ − 2b1 + 1,

(5.10)

and the other dependent variables ϕ0 , ϕ1 are determined by quadrature from ψ. At the same time, we obtain the following B¨acklund transformations r0 and r1 for ψ: r0 (ψ) = ψ −

ψ0

2b0 , − ψ2 + x

r1 (ψ) = ψ −

ψ0

2b1 , + ψ2 − x

(5.11)

where b0 = a0 + a1 = 1 − b1 . The parameters b0 , b1 are the simple roots for the A(1) 1 symmetry of P II . Finally, we present a series of differential systems with A(1) l -symmetry (l ≥ 2), which give a generalization of the Painlev´e equations P IV and P V . In our previous paper [10], we introduced the symmetric form of the fourth Painlev´e equation: f00 = f0 (f1 − f2 ) + α0 , f10 f20

(5.12)

= f1 (f2 − f0 ) + α1 , = f2 (f0 − f1 ) + α2 .

This system defines in fact a derivation 0 of the field C(α; f ) which commute with the f of type A(1) as in (4.3) and (4.4). (Note that the action of the extended affine Weyl group W 2 convention of [10] corresponds to the transposition of U in (4.2).) We remark that the sum f , and satisfies the equation (f0 +f1 +f2 )0 = α0 +α1 +α2 = δ. f0 +f1 +f2 is invariant under W Introduce the independent variable x so that x0 = 1, and eliminate one of the three f variables, noting that f0 + f1 + f2 is a linear function of x. Then the differential system above is rewritten into a system of order 2, which is equivalent to the Painlev´e equation P IV . Differential system (5.12) has a generalization to higher orders. For example, when l = 4, the differential system f00 = f0 (f1 − f2 + f3 − f4 ) + α0 , f10 f20 f30 f40

= f1 (f2 − f3 + f4 − f0 ) + α1 , = f2 (f3 − f4 + f0 − f1 ) + α2 , = f3 (f4 − f0 + f1 − f2 ) + α3 , = f4 (f0 − f1 + f2 − f3 ) + α4

(5.13)

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has A(1) 4 -symmetry. Note that the sum f0 + f1 + f2 + f3 + f4 is a linear function of the independent variable x such that x0 = 1 and that the system above is essentially of order 4. In general, when l = 2n, the following differential system (essentially of order 2n) acklund transformations defined as in Sect. 4: turns out to have A(1) 2n -symmetry with the B¨ X (fj+2r−1 − fj+2r + αj (j = 0, 1, . . . , 2n). (5.14) fj0 = fj 1≤r≤n

We remark that this differential system is obtained as a continuum limit from the difference system with A(1) 2n -symmetry which arises from the discrete dynamical system of (1) type A2n+1 , in the manner as we explained above. We also found a series of differential systems with A(1) 2n+1 -symmetry (n = 1, 2, . . . ) which generalize the fifth Painlev´e equation P V : X X fj+2r−1 fj+2s − fj+2r fj+2s+1 (5.15) fj0 = fj 1≤r≤s≤n

1≤r≤s≤n

X X δ αj+2r )fj + αj ( fj+2r ) +( − 2 1≤r≤n

(j = 0, 1, . . . , 2n + 1),

1≤r≤n

where α0 + · · · + α2n+1 = δ. We remark Pn system (5.15) is also essentially Pn that differential of order 2n, since each of the sums r=0 f2r and r=0 f2r+1 is determined elementarily. The Painlev´e equation P V is covered as the case n = 1 (see [13]): δ f00 = f0 (f1 f2 − f2 f3 ) + ( − α2 )f0 + α0 f2 , 2 δ f10 = f1 (f2 f3 − f3 f0 ) + ( − α3 )f1 + α1 f3 , 2 δ 0 f2 = f2 (f3 f0 − f0 f1 ) + ( − α0 )f2 + α2 f0 , 2 δ 0 f3 = f3 (f0 f1 − f1 f2 ) + ( − α1 )f3 + α3 f1 , 2

(5.16)

where α0 + α1 + α2 + α3 = δ. These two series of differential systems with affine Weyl group symmetry can be considered as a variation of Lotka-Voltera equations and Bogoyavlensky lattices, including the parameters α0 , . . . , αl . Also, the structures of their B¨acklund transformations can be described completely in terms of the discrete dynamical systems we have introduced in this paper. (Details will be discussed in [12].) We expect that these systems of differential equations with affine Weyl group symmetry deserve to be studied individually from various aspects, since they already give a candidate for systematic generalization of Painlev´e equations to higher orders.

References ´ ements de Math´ematique, Paris: Masson, 1. Bourbaki, N.: Groupes et Alg`ebres de Lie, Chapitres 4,5 et 6. El´ 1981 2. Grammaticos, B., Nijhoff, F.W., Papageorgiou, V., Ramani, A. and Satsuma, J.: Linearization and solutions of the discrete Painlev´e III equation. Phys. Lett. A185, 446–452 (1994)

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3. Kac, V.G.: Infinite dimensional Lie algebras. Third edition, Cambridge: Cambridge University Press, 1990 4. Kuniba, A., Nakamura, S. and Hirota, R.: Pfaffian and determinant solutions to a discretized Toda equation for Br , Cr and Dr . J. Phys. A: Math. Gen. 29, 1759–1766 (1996) 5. Kuniba, A., Nakanishi, T. and Suzuki, J.: Functional relations in solvable lattice models I. Functional relations and representation theory. Int. J. Mod. Phys. A 9, 5215–5266 (1994) 6. Macdonald, I.G.: Affine root systems and Dedekind’s η-function. Inv. Math. 15, 91–143 (1972) 7. Murata, Y.: Rational solutions of the second and the fourth equations of Painlev´e. Funkcial. Ekvac. 28, 1–32 (1985) 8. Noumi, M., Okada, S., Okamoto, K. and Umemura, H.: Special polynomials associated with the Painlev´e equations II. To appear in the Proceedings of Taniguchi Symposium, 1997, “Integrable systems and Algebraic geometry”, M.H. Saito, et. al. ed. 9. Noumi, M. and Okamoto, K.: Irreducibility of the second and the fourth Painlev´e equations. Funkcial. Ekvac. 40, 139–163 (1997) 10. Noumi, M. and Yamada, Y.: Symmetries in the fourth Painlev´e equation and Okamoto polynomials. To appear in Nagoya Math. J. (q-alg/9708018) 11. Noumi, M. and Yamada, Y.: Umemura polynomials for Painlev´e V equation. To appear in Phys. Lett. A. . To appear in Funkcial. Ekvac. 12. Noumi, M. and Yamada, Y.: Higher order Painlev´e equations of type A(1) l (math.QA/9808003) 13. Noumi, M. and Yamada, Y.: Symmetric forms of the Painlev´e equations. In preparation 14. Okamoto, K.: Studies of the Painlev´e equations, I. Ann. Math. Pura Appl. 146, 337–381 (1987); II. Jap. J. Math. 13, 47–76 (1987); III. Math. Ann. 275, 221–255 (1986); IV. Funkcial. Ekvac. Ser. Int. 30, 305–332 (1987) 15. Ramani, A., Grammaticos, B. and Hietarinta, J.: Discrete versions of the Painlev´e equations. Phys. Rev. Lett. 67, 1829–1832 (1991) 16. Umemura, H.: On the irreducibility of the first differential equation of Painlev´e In: Algebraic Geometry and Commutative Algebra in honor of Masayoshi Nagata, Amsterdam: Kinokuniya-North-Holland, 1987, pp. 101–119 17. Umemura, H.: Special polynomials associated with the Painlev´e equations I. To appear in the Proceedings of the Workshop on “Painlev´e Transcendents”, CRM, Canada, 1996 18. Yamada, Y.: Determinant formulas for the τ -functions of the Painlev´e equations of type A. Preprint (math.QA/9808002) Communicated by T. Miwa

Commun. Math. Phys. 199, 297 – 325 (1998)

Communications in

Mathematical Physics © Springer-Verlag 1998

Asymptotic Metrics for SU (N )-Monopoles with Maximal Symmetry Breaking Roger Bielawski Max-Planck-Institut f¨ur Mathematik, Gottfried-Claren-Strasse 26, 53225 Bonn, Germany. E-mail: [email protected] Received: 19 March 1998 / Accepted: 4 May 1998

Abstract: We compute the asymptotic metrics for moduli spaces of SU (N ) monopoles with maximal symmetry breaking. These metrics are exponentially close to the exact monopole metric1 as soon as, for each simple root, the individual monopoles corresponding to that root are well separated. We also show that the estimates can be differentiated term by term in natural coordinates, which is a new result even for SU (2) monopoles.

1. Introduction It has been known since the work of Taubes [33] that an SU (2)-monopole of charge m with well-separated zeros of the Higgs field approximates a collection of m monopoles of charge 1. This fact is reflected in the asymptotic behaviour of the natural hyperk¨ahler metric on the moduli space Mm of charge m monopoles. Namely, in the asymptotic region of Mm , the monopole metric is exponentially close to another hyperk¨ahler metric, whose geodesics determine scattering of m particles with electric, magnetic and scalar charges. This metric was found by Gibbons and Manton in [15] and a proof that it differs from the exact monopole metric by an exponentially small amount was given in [3]. For m = 2, the Gibbons–Manton metric is just the product of a flat metric and the Taub-NUT metric with a negative mass parameter. Monopoles exist for any compact Lie group and Taubes’ estimates work equally well for SU (N )-monopoles with maximal symmetry breaking, that is monopoles whose Higgs field has distinct eigenvalues at infinity. This time a moduli space Mm1 ,...,mN −1 (µ1 , . . . , µN ), where mi are positive integers and µ1 < · · · < µN , is obtained by identifying gauge-equivalent framed monopoles whose Higgs field at infinity defines a map from the 2-sphere to the adjoint orbit O of diag(iµ1 , . . . , iµN ) and whose degree is (m1 , . . . , mN −1 ) ∈ H2 (O, Z). We should think of particles making up the monopole as coming in N − 1 distinguishable types, with mi being the number of particles of type i.

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In this paper we shall compute the asymptotic metric on Mm1 ,...,mN −1 (µ1 , . . . , µN ) 1 . This metric turns out to be a hybrid between the Gibbons–Manton metric and the metric on the moduli space of monopoles of charge (1, 1, . . . , 1) which was computed in [28, 29, 10]. Particles of the same type interact as in the Gibbons–Manton metric while the particles of different types as in the (1, 1, . . . , 1)-metric (i.e. neighboring types interact as in the Taub-NUT metric with a positive mass parameter and non-neighboring types do not interact). The precise formula for the asymptotic metric is given in the next section. Somewhat surprisingly, the monopole metric on Mm1 ,...,mN −1 (µ1 , . . . , µN ) is exponentially close to the asymptotic metric as soon as, for each i, particles of type i are far apart. Particles of different types can be as close to each other as we wish (in fact sometimes they can have the same position). Moreover, as we observe in Sect. 9, if particles of a single type, say i, are far apart, then the monopole metric is likely to be exponentially close to yet another hyperk¨ahler metric. This metric is simpler than the monopole metric, but still given by transcendental functions. It is only when the particles of each type are far apart, that the asymptotic metric becomes algebraic and we are able to compute it explicitly. The proof uses the idea of our previous work [3], i.e. replacing solutions to Nahm’s equations corresponding to monopoles with solutions defined on half-line. This gives a new moduli space whose metric is then computed by twistor methods. The main novelty is the way the metrics are compared. We prove (in Appendix B) a general theorem which allows us to deduce the estimates on all derivatives of components (in natural coordinates) of the difference of metric tensors from one-sided estimates on the metric tensors. Such a deduction is possible for two hyperk¨ahler metrics which are related via a complex-symplectic isomorphism providing that one of the metrics admits holomorphic coordinates in which the complex-symplectic form is standard and in which the components of the metric tensor are uniformly bounded. The paper is organized as follows. In the next section we collect some facts about T m -invariant hyperk¨ahler metrics in dimension 4m of which our asymptotic metric is an example. In Sect. 3 we recall the description, due to Nahm [30] and to Hurtubise and Murray [22], of SU (N )-monopoles in terms of Nahm’s equations. We also define there the moduli space of solutions to Nahm’s equations whose metric will be the asymptotic metric. This moduli space is a hyperk¨ahler quotient of the product of several simpler moduli spaces, and in the following four sections we compute the metrics on these. In Sect. 8 we discuss the topology of these moduli spaces. Finally, in Sect. 9, we put the results together to obtain an explicit formula for the asymptotic metric (Theorem 9.1). We also prove there that the rate of approximation is exponential (Theorem 9.2) and discuss the topology of the asymptotic moduli space. Appendix A deals with the question of identifying certain hyperk¨ahler quotients with corresponding complex-symplectic quotients which is needed in Sect. 8. In Appendix B we prove the above-mentioned comparison theorem for Ricci-flat K¨ahler manifolds. 2. T m -Invariant Hyperk¨ahler 4m-Dimensional Metrics The Gibbons–Manton metric [15, 16] is an example of a 4m-dimensional (pseudo)hyperk¨ahler metric admitting a tri-Hamiltonian (hence isometric) action of the m1 Strictly speaking, we only show that our asymptotic metric is close to the metric on the moduli space of solutions to Nahm’s equations. Thus we can conclude that the geodesics of the monopole metric and of the asymptotic metric are close to each other (see Remark 3.5). The metrics themselves are close if the Nahm transform for SU (N )-monopoles, N > 2, is an isometry.

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dimensional torus T m . Such metrics have particularly nice properties and were studied by several authors [27, 20, 32]. On the set where the action of T m is free such a metric can be locally written in the form: g = 8dx · dx + 8−1 (dt + A)2 ,

(2.1)

where x is the hyperk¨ahler moment map, dt is dual to the (m × 1)-matrix of Killing vector fields and the matrix 8 and the 1-form A depend only on the xi and satisfy certain linear PDE’s. In particular, 8 determines the metric up to a gauge equivalence. The set where the T m -action is free can be viewed as a T m -bundle over an open subset of R3m . For the Gibbons–Manton metric this open subset is the configuration space of C˜ m (R3 ) of m distinct points in R3 (i.e. R3m without the generalized diagonal) and ( P 1 if i = j − ν1 − k= 6 i kxi −xk k (2.2) 8ij = 1 if i 6 = j. kxi −xj k Here ν is the mass parameter. We can, in particular, take ν = ∞ and m = 2. Then the linearity of the equations for 8 and A implies that, for any mapping (i, j) 7→ sij of {1, . . . , m} × {1, . . . , m} such that sij = sji and sii = 0 for i, j = 1, . . . , m, and for any constants ci , i = 1, . . . , m, the following matrix 8 defines a T m -invariant (pseudo)-hyperk¨ahler metric: ( P sik ci + k= if i = j 6 i kxi −xk k (2.3) 8ij = sij if i 6 = j. − kxi −xj k The asymptotic metric on Mm1 ,...,mN −1 (µ1 , . . . , µN ) will turn out to be of this form. Namely m = Pm1 + · · · + mN −1 and, if we define the type t(i) of an i ≤ m by t(i) = min{k; i ≤ s≤k ms }, then ci = µk+1 − µk if t(i) = k and

  −2 sij = 1  0

if t(i) = t(j) if |t(i) − t(j)| = 1 otherwise.

(2.4)

(2.5)

3. Moduli Spaces of Solutions to Nahm’s Equations We shall define in this section several moduli spaces of solutions to Nahm’s equations. All of these spaces carry hyperk¨ahler metrics. In particular, we shall recall, after Nahm [30] and Hurtubise and Murray [22], the description of the moduli spaces of SU (N ) monopoles with maximal symmetry breaking in terms of solutions to Nahm’s equations. We shall also describe the asymptotic moduli spaces. We remark that from the point of view of hyperk¨ahler geometry many interesting metrics are obtained by replacing below the unitary group with an arbitrary compact Lie group. Such a generalization is straightforward, but, as our focus is on monopoles, we shall restrict ourselves to the unitary case.

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Nahm’s equations are the following ODE’s: 1 T˙i + [T0 , Ti ] + 2

X

ijk [Tj , Tk ] = 0 ,

i = 1, 2, 3.

(3.1)

j,k=1,2,3

The functions T0 , T1 , T2 , T3 are defined on some interval and are skew-hermitian and analytic. If the rank of the Ti is n, then the space of solutions is acted upon by the gauge group G of U (n)-valued functions g(t): 7 Ad(g)T0 − gg ˙ −1 , T0 → 7 Ad(g)Ti , i = 1, 2, 3. Ti →

(3.2)

We define fundamental moduli spaces of u(n)-valued solutions Fn (m; c) and F˜n (m; c). Here m is a nonnegative integer less than or equal to n and c is a positive real number (c can be negative or zero for F˜n (m; c)). The moduli spaces Fn (m; c) correspond to monopoles with minimal symmetry breaking and are the basic building blocks from which all moduli spaces of framed SU (N )-monopoles with maximal symmetry breaking can be obtained by means of the hyperk¨ahler quotient construction. The spaces F˜n (m; c) play a similar role for the asymptotic metrics. They are defined as follows: • Solutions in Fn (m; c) are defined on (0, c], while solutions in F˜n (m; c) are defined on (0, ∞]. • For a solution (T0 , T1 , T2 , T3 ) in either Fn (m; c) or F˜n (m; c), T0 and the m × m upper-diagonal blocks of T1 , T2 , T3 are analytic at t = 0, while the (n − m) × (n − m) lower-diagonal blocks have simple poles with residues defining the standard (n − m)dimensional irreducible representation of su(2). The off-diagonal blocks are of the form t(n−m−1)/2 × (analytic in t). • A solution in Fn (m; c) is analytic at t = c, while a solution in F˜n (m; c) approaches a diagonal limit at +∞ exponentially fast. Furthermore (T1 (+∞), T2 (+∞), T3 (+∞)) is a regular triple, i.e. its centralizer consists of diagonal matrices. • The gauge group for Fn (m; c) consists of gauge transformations g with g(0) = g(c) = 1, while the gauge group for F˜n (m; c) has the Lie algebra consisting of functions ρ : [0, +∞) → u(n) such that (i) ρ(0) = 0 and ρ˙ has a diagonal limit at +∞; (ii) (ρ˙ − ρ(+∞)) ˙ and [τ, ρ] decay exponentially fast for any regular diagonal matrix τ ∈ u(n); ˙ = 0. (iii) cρ(+∞) ˙ + limt→+∞ (ρ(t) − tρ(+∞)) Remark 3.1. Alternately, the space F˜n (m; c) can be viewed as the moduli space of solutions defined on [−c, +∞] with the gauge group given by the transformation which are exponentially close to exp(ht) for some diagonal h. The tangent space at a solution (T0 , T1 , T2 , T3 ) can be identified, for both Fn (m; c) and F˜n (m; c), with the space of solutions to the following system of linear equations: t˙0 + [T0 , t0 ] + [T1 , t1 ] + [T2 , t2 ] + [T3 , t3 ] = 0, t˙1 + [T0 , t1 ] − [T1 , t0 ] + [T2 , t3 ] − [T3 , t2 ] = 0, t˙2 + [T0 , t2 ] − [T1 , t3 ] − [T2 , t0 ] + [T3 , t1 ] = 0, t˙3 + [T0 , t3 ] + [T1 , t2 ] − [T2 , t1 ] − [T3 , t0 ] = 0. Fn (m; c) carries a hyperk¨ahler metric defined by

(3.3)

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Z k(t0 , t1 , t2 , t3 )k = 2

c

0

3 X

kti k2 ,

(3.4)

0

while F˜n (m; c) possesses an indefinite (and possibly degenerate) hyperk¨ahler metric given by: k(t0 , t1 , t2 , t3 )k2 = c

3 X 0

Z kti (+∞)k2 + 0

+∞

3 X

kti (s)k2 − kti (+∞)k2 ds. (3.5)

0

The moduli space Fn (m; c) has a tri-Hamiltonian action of U (n)×U (m) given by gauge transformations g with arbitrary values at t = c and with g(0) being block-diagonal with the off-diagonal blocks equal to 0 and the (n − m) × (n − m) lower-diagonal block being identity. Both U (n) and U (m) act freely. The hyperk¨ahler moment map for the action of U (n) is (−T1 (c), −T2 (c), −T3 (c)), while the one for the action of U (m) is π(T1 (0), T2 (0), T3 (0)), where π denotes the projection onto the m × m upper-diagonal block. The moduli space F˜n (m; c) has a similarly defined free tri-Hamiltonian action of U (m). In addition, it has a free tri-Hamiltonian action of the diagonal torus T n ≤ U (n) given by gauge transformations which are asymptotic to exp(−th + λh) for a diagonal h and real λ. The moment map for this action is (T1 (+∞), T2 (+∞), T3 (+∞)). We shall now consider hyperk¨ahler quotients of various products of Fn (m; c) and F˜n (m; c). We observe that the hyperk¨ahler quotient construction of say, Fn (m; c) × Fn (l; c0 ) matches solutions (T0 (t), T1 (t), T2 (t), T3 (t)) in Fn (m; c) with (−T0 (c + c0 − t), −T1 (c + c0 − t), −T2 (c + c0 − t), −T3 (c + c0 − t)) for a (T0 (t), T1 (t), T2 (t), T3 (t)) in F˜n (m; c). The resulting space can be identified with the moduli space of solutions to Nahm’s equations on [0, c + c0 ] having appropriate poles at t = 0 and at t = c + c0 . We recall that the triple backslash denotes hyperk¨ahler quotient (in all our constructions the moment map is canonical and we quotient its 0-set). We have basic hyperk¨ahler isomorphisms: Fn (m; c) × Fn (n; c0 ) ///U (n) ' Fn (m; c + c0 ), (3.6) Fn (m; c) × F˜n (n; c0 ) ///U (n) ' F˜n (m; c + c0 ). The group acts diagonally on the product. In particular all F˜n (m; c) can be obtained from the F˜n (n; c) and the Fn (m; c). 0 (c, c0 ), Fn,m˜ (c, c0 ), Fn, We now define auxiliary moduli spaces Fn,m (c, c0 ), Fn,m ˜ ˜ m ˜ (c, c ). Here n, m are arbitrary positive integers and c, c0 are arbitrary positive real numbers. The spaces are defined as follows: • if n < m, then Fn,m (c, c0 ) = Fn (n; c) × Fm (n; c0 ) ///U (n). The spaces with a tilde ˜ over n or m are obtained by replacing the corresponding F with F ; • if n > m, then Fn,m (c, c0 ) = Fn (m; c) × Fm (m; c0 ) ///U (m) and similarly for the other spaces; • if n = m, then Fn,n (c, c0 ) is the hyperk¨ahler quotient of Fn (n; c) × Fn (n; c0 ) × Hn by the diagonal action of U (n). Remark 3.2. Thus these moduli spaces consist of u(m)-valued solutions Ti− on [−x, 0) and of u(n)-valued solutions Ti+ on (0, y], where x = c0 or −∞ and y = c or y = +∞, with matching conditions at t = 0: if n > m (resp. n < m), then the limit of the m × m upper-diagonal block of Ti+ (resp. Ti− ) at t = 0 is equal to the limit of Ti− (resp. Ti+ );

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if n = m, then there exists a vector (V, W ) ∈ C2n such that (T2+ + iT3+ )(0+ ) − (T2− + iT3− )(0− ) = V W T and T1+ (0+ )−T1− (0− ) = (|V |2 −|W |2 )/2. The gauge transformations g(t) satisfy similar matching conditions: if n 6 = m, then the upper-diagonal m × m block is continuous, the lower-diagonal block is identity at t = 0 and the off-diagonal blocks vanish to order (n − m − 1)/2; if n = m, then g(t) is continuous at t = 0. Notice that Fn,m (c, c0 ) is isomorphic, as a hyperk¨ahler manifold, to Fm,n (c0 , c) and 0 similarly for Fn, ˜ m ˜ (c, c ). We can now define the moduli spaces we are really interested in. Let us fix an integer N and consider functions σ : {1, . . . , N − 1} → N t N is arbitrary and µ : {1, . . . , N } → R is increasing. We shall denote the second copy of ˜ and write its elements as 1, ˜ 2, ˜ 3, ˜ . . . . We define the moduli space Fσ (µ) as a N by N hyperk¨ahler quotient of Fσ1 (c1 ) × Fσ2 (c2 , c02 ) × . . . × FσN −1 (cN −1 , c0N −1 ) × FσN (c0N ). Here ci +c0i+1 = µ(i+1)−µ(i), σ1 , σN ∈ NtN and σi : {1, 2} → NtN for 2 ≤ i ≤ N −1. Furthermore σ1 = σ(1), σi (1) = σ(i), σi (2) = σ(i − 1) for 2 ≤ i ≤ N − 1, and σN = σ(N − 1). Finally Fn (c), Fn˜ (c) denote Fn (0; c) and F˜n (0; c). The group by which we quotient is a product of unitary groups and of tori acting on this product space: we take the diagonal action of U (n) on Fσi (ci , c0i ) × Fσi+1 (ci+1 , c0i+1 ) (or Fσ1 (c1 ) × Fσ2 (c2 , c02 ) for ˜ i = 1 and similarly for i = N − 1) if σ(i) = n and the diagonal action of T n if σ(i) = n. Remark 3.3. The moduli space Fσ (µ) should be viewed as consisting of solutions to Nahm’s equations on N − 1 “intervals” Ii , i = 1, . . . , N1 with matching conditions at ˜ then Ii = the boundary points. If σ(i) ∈ N, then Ii = [µ(i), µ(i + 1)], while if σ(i) ∈ N, [µ(i), +∞)∪(−∞, µ(i+1)] (see Remark 3.1). The solutions satisfy matching conditions of Remark 3.2 at each µi and are continuous at each infinity. The gauge transformations satisfy matching conditions of Remark 3.2 at each µi and are exponentially close to ˜ for some diagonal matrices hi , pi . exp(hi t + pi ) near ±∞ in Ii , σ(i) ∈ N, A theorem of Hurtubise and Murray [22], giving a full proof of the correspondence found by Nahm [30], and generalizing the SU (2) case due to Hitchin [19] can be phrased as follows (this formulation uses the connectivity of the moduli space of SU (N )-monopoles due to Jarvis [23]): Theorem 3.4. The moduli space Mm1 ,... ,mN −1 (µ1 , . . . , µN ) of framed SU (N ) monopoles of charge (m1 , . . . , mN −1 ) and the symmetry breaking at infinity equal to (µ1 , . . . , µN ), µi distinct, is diffeomorphic to the moduli space Fσ (µ) with σ(i) = mi and µ(i) = µi . Remark 3.5. It is expected, but at present not known (except for N = 2 [31]), that this diffeomorphism is an isometry. Nevertheless, the twistorial character of Hurtubise and Murray’s construction shows that this diffeomorphism preserves the three complex structures and, hence, the Levi–Civita connection. Thus, the geodesics are the same. Our aim is to show that the metric on Fσ (µ) with σ(i) = mi and µ(i) = µi is ˜ =m ˜ i and µ(i) = µi . We shall first compute asymptotic to the metric on Fσ˜ (µ) with σ(i) the metric on Fσ˜ (µ). 4. Complex Structures on Fn (m; c) and F˜n (n; c) All moduli spaces described in the previous section have an isometric action of SU (2) or SO(3) rotating the complex structures and therefore all complex structures are equivalent. We shall consider the complex structure I and describe the complex coordinates

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(and the complex symplectic form ω2 + iω3 ) on Fn (m; c) and F˜n (n; c). All other moduli spaces can be described as open subsets of complex-symplectic quotients of products of these. We set α = T0 + iT1 and β = T2 + iT3 . The Nahm equations can be then written as one complex and one real equation: dβ = [β, α], dt d (α + α∗ ) = [α∗ , α] + [β ∗ , β]. dt

(4.1) (4.2)

First, we consider Fn (m; c) (cf. [21, 3]). Let E1 , . . . , En denote the standard basis of Cn . There is a unique solution w1 of the equation dw = −αw dt with

(4.3)

lim t−(n−m−1)/2 w1 (t) − Em+1 = 0.

t→0

(4.4)

Setting wi (t) = β i−1 (t)w1 (t), we obtain a solution to (4.3) with lim ti−(n−m+1)/2 wi (t) − Em+i = 0. t→0

In addition there are solutions u1 , . . . , um to (4.3) whose last n − m components vanish to order (n − m + 1)/2, and which are linearly independent at t = 0. The complex gauge transformation g(t) with g −1 = (u1 , . . . , um , w1 , . . . , wn−m ) makes α identically zero and sends β(t) to the constant matrix (cf. [21])   h    f ... f 1 m B=    0 ... 0  . ..  . . . 0 ... 0

0 ... .. . 0 ... 0 ... . 1 .. ..

. 0 ...

 g1 ..  .   0 gm  0 e1  .   e2  .  .. . ..  1 en−m 0 .. .

(4.5)

The mapping (α, β) → (g(c), B) gives a biholomorphism between (Fn (m; c), I) and Gl(n, C)×gl(m, C)×Cn+m [5]. The action of Gl(n, C) is given by the right translations, and the action of Gl(m, C) is given by p · h, f, g, e, g(c) = (php−1 , f p−1 , pg, e, pg(c)), where for the last term we embedded Gl(m, C) in Gl(n, C) as the m×m upper-diagonal block. We can compute the complex symplectic form ω = ω2 + iω3 . We denote by b, bˆ vectors tangent to the space of B’s in (4.5) and by ρ, ρˆ right-invariant vector fields on Gl(n, C). We have [5]: ˆ = tr ρbˆ − ρb ω (ρ, b), (ρ, ˆ b) ˆ − B[ρ, ρ] ˆ . (4.6)

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Now we consider the complex structure of F˜n (n; c). Let n be a unipotent algebra corresponding to the Cartan algebra of diagonal matrices. We consider the open dense subset F˜ (n) of F˜n (n; c) defined as the set of all solutions (α, β) = (T0 + iT1 , T2 + iT3 ) such that the intersection of the sum of positive eigenvalues of ad(iT1 (+∞)) with the centralizer C(β(+∞)) is contained in n. We observe that, since (T1 (+∞), T2 (+∞), T3 (+∞)) is a regular triple, the projection of T1 (+∞) onto C(β(+∞)) is a regular element. Now, as in [3], we use results of Biquard [8] to deduce that F˜ (n) is biholomorphic to an open subset of Gl(n, C) ×N b, where N = exp n and b = d + n, d denoting the diagonal matrices. Briefly, the element g of Gl(n, C) is given by the value at t = 0 of the complex gauge transformation g(t) which makes (0, β(+∞) + n) into (α, β). The charts F˜ (n) are glued as follows: [g, d+n] ∼ [g 0 , d0 +n0 ] if and only if n ∈ n, n0 ∈ n0 , and either n0 ⊂ n and there exists an m ∈ N such that gm−1 = g 0 , Ad(m)(d + n) = d0 + n0 or vice versa (i.e. n ⊂ n0 etc.). We remark that F∅ is an open dense subset biholomorphic to an open subset of Gl(n, C) × d. We shall denote this subset by F˜nreg (n; c). If bd , bˆ d denote vectors tangent to the space of diagonal matrices and ρ, ρˆ denote this time left-invariant vector fields on Gl(n, C), then the form ω is given by [3]: ˆ d . (4.7) ω = − tr bd ρˆ − ρbˆ d − [ρ, ρ]β All other moduli spaces F˜n (m; c) and Fσ (µ) can be viewed as hyperk¨ahler quotients of products of Fn (m; c) and F˜n (n; c). Thus, as complex-symplectic manifolds, they are isomorphic to open subsets of complex-symplectic quotients of the corresponding complex-symplectic manifolds computed above. The description of the latter quotients is straightforward. Let us remark that Hurtubise showed in [21] that if σ(i) = mi , i = 1, . . . , N −1, then Fσ (µ) (i.e. the moduli space of SU (N )-monopoles) is biholomorphic to the space of based rational maps from CP 1 to SU (N )/T (maximal torus) of degree (m1 , . . . , mN −1 ). 0 5. Complex-Symplectic Structure of Fn, ˜ m ˜ (c, c ), n > m

˜ i . This space has dimension Our aim is to calculate the metric on Fσ (µ) where σ(i) = m 4p = 4(m1 + . . . + mN −1 ) and admits a tri-Hamiltonian action of T p . By the definition of Fσ (µ), it is a hyperk¨ahler quotient, by a torus, of the product spaces F˜n (0; c) and 0 ˜ Fn, ˜ m ˜ (c, c ). The metric on Fn (0; c) was calculated in [3] – it is the Gibbons–Manton met0 ric [15] with the mass parameter −1/c. It remains to calculate the metric on Fn, ˜ m ˜ (c, c ). For convenience we shall write 0 F˜n,m (c, c0 ) := Fn, ˜ m ˜ (c, c ).

The dimension of this space is 4(n + m) and it has a tri-Hamiltonian action of an (n + m)dimensional torus. The space F˜n,m (c, c0 ) should be thought of as consisting of solutions to Nahm’s equations on (−∞, 0) ∪ (0, ∞), which are u(m)-valued on (−∞, 0), u(n)-valued on (0, ∞), and satisfy appropriate matching conditions at zero. In what follows we shall usually say “F˜n,m (c, c0 ) is biholomorphic to . . . ” rather than “F˜n,m (c, c0 ) is biholomorphic to an open subset of . . . ”. This never leads to any problems (the former statement is actually correct, but this will be proved only in Sect. 8, as Corollary 8.2). We consider the space F˜n,m (c, c0 ) for n > m. From its description as a complexsymplectic quotient, F˜n,m (c, c0 ) is given by charts of the form {(b− , (g, b+ )} ∈ b ×

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305

Gl(n, C) ×N 0 b0 such that gb+ g −1 is of the form (4.5) with h = b− . Let us consider the chart on which b = dm and b0 = dn (dm and dn denote m × m and n × n diagonal κm ) and the elements of dn matrices). Let us write the elements Q of dm as diag(κ1 , . . . ,Q as diag(β1 , . . . , βn ). Let q+ (z) = (z − βi ) and q− (z) = (z − κi ). We assume that the roots of both these polynomials are distinct and we consider multiplication by z on C[z]/(q+ ). It is a linear operator which, in the basis Q 6 i (z − βj ) Q j= , i = 1, . . . , n, (5.1) j= 6 i (βi − βj ) is the diagonal matrix diag(β1 , . . . , βn ). On the other hand, in the basis Q 6 i (z − κj ) Q j= , q− (z), . . . , z n−m−1 q− (z), i = 1, . . . , m, (κ − κ ) i j j= 6 i

(5.2)

Q the multiplication by z is given by a matrix of the form (4.5) with fi = 1 j= 6 i (κi − κj ). Let Z be the matrix transforming the basis (5.1) into (5.2). Then any g which sends diag(β1 , . . . , βn ) to a matrix of the form (4.5) can be written as −1 , 1, . . . , 1)Z diag(u1 , . . . , un ). diag(v1−1 , . . . , vm

(5.3)

We shall now compute Z. We introduce one more basis of C[z]/(q+ ): 1, . . . , z n−1 .

(5.4)

The passage from (5.1) to (5.4) is given by V (β1 , . . . , βn )−1 , where V (β1 , . . . , βn ) is the Vandermonde matrix, i.e. its (i, j)th entry is (βi )j−1 . We then compute the passage from (5.4) to (5.2) as given by the matrix Vκ W , (5.5) L= 0 H where Vκ = V (κ1 , . . . , κm ), Wij = (κi )m+j−1 and H is upper-triangular with Hij = (−1)i−j Hi−j , where Hk denotes the k th complete symmetric polynomial in κ1 , . . . , κm , i.e. the sum of all monomials of degree k. Remark 5.1. The factorization Z = LV −1 is unique only if βi 6 = κj for all i, j. If, for instance, β1 = κ1 , then the above g sends diag(β1 , . . . , βn ) to a matrix of the form (4.5) with g1 = 0. However, there is then another g which makes f1 = 0. We calculate the complex symplectic form on F˜n,m (c, c0 ). The chart where βi 6 = βj , κr 6 = κs , βi 6 = κs for all i, j = 1, . . . , n, i 6 = j, r, s = 1, . . . , m, r 6 = s, can be described as consisting of pairs (g− , κd ), (g+ , βd ) , where κd = diag(κ1 , . . . , κm ), βd = diag(β1 , . . . , βn ), g− = Vκ−1 diag(v1 , . . . , vm ), g+ = Vκ−1 LVβ−1 diag(u1 , . . . , un ). According to the formula (4.7), the complex symplectic form in this chart is equal to: ω = − tr kd ρ˜− − ρ− k˜ d − κd [ρ− , ρ˜− ] + bd ρ˜+ − ρ+ b˜ d − βd [ρ+ , ρ˜+ ] . Here kd , ρ− , bd , ρ+ are dual to, respectively, κd , (g− )−1 dg− , dβd , (g+ )−1 dg+ . The first three terms can be computed as in [3] and give

306

R. Bielawski m X dvi i=1

vi

∧ dκi −

X dκi ∧ dκj i<j

κi − κj

.

(5.6)

To compute the remaining two terms let us write X = Vκ−1 L and Y = Vβ−1 diag(u1 , . . . , un ). Let us also write βc = Y βd Y −1 and bc , x, y for vector fields dual to βc , X −1 dX, Y −1 dY . Then ρ+ = Y −1 xY + y and bc = Y bd Y −1 + Y [y, βd ]Y −1 . Thus the last three terms in the above formula can be rewritten as ˜ − Y −1 xY b˜ d − βd [y, y] ˜ − tr bd ρ˜+ − ρ+ b˜ d − βd [ρ+ , ρ˜+ ] = − tr bd y˜ − y b˜ d + bd Y −1 xY ˜ ] − βd [Y −1 xY, y] ˜ − Y βd Y −1 [x, x] ˜ = ω+ − tr Y bd Y −1 x˜ − xY b˜ d Y −1 − βd [y, Y −1 xY − Y βd Y −1 [Y yY −1 , x] ˜ − Y βd Y −1 [x, Y yY ˜ −1 ] − Y βd Y −1 [x, x] ˜ ˜ . = ω+ − tr bc x˜ − xb˜ c − βc [x, x] ˜ , which, again as in [3], is equal to Here ω+ = − tr bd y˜ − y b˜ d − βd [y, y] n X dui i=1

ui

∧ dβi −

X dβi ∧ dβj i<j

βi − βj

.

(5.7)

For the remaining terms we observe that dβc is upper triangular and X −1 dX is strictly upper-triangular. Hence the remaining terms vanish and the complex-symplectic form on F˜n,m (c, c0 ) is given by: ω=

n X dui i=1

ui

∧ dβi −

X dβi ∧ dβj i<j

βi − βj

+

m X dvi i=1

vi

∧ dκi −

X dκi ∧ dκj i<j

κi − κj

. (5.8)

6. Twistor Space and the Metric of F˜n,m (c, c0 ), n > m First, we compute the twistor space of Fn (m; c). Let ζ be the affine coordinate on CP 1 . The twistor space of any hyperk¨ahler manifold admitting an action of SU (2) rotating the complex structures can be trivialized using just two charts ζ 6 = ∞ and ζ 6 = 0. For a moduli space of solutions to Nahm’s equations this is achieved by putting η = β + (α + α∗ )ζ − β ∗ ζ 2 , u = α − β ∗ ζ over ζ 6 = ∞ and η˜ = β/ζ 2 + (α + α∗ )/ζ − β ∗ , u˜ = −α∗ − β/ζ over ζ 6 = 0. Then, over ζ 6 = 0, ∞, we have η˜ = η/ζ 2 , u˜ = u − η/ζ. ¯ η 7→ −η ∗ /ζ¯2 , u 7→ −u∗ + η ∗ /ζ¯ (cf. [12, 9]). Moreover, the real structure is ζ 7→ −1/ζ, We consider the matrix (4.5). We have Lemma 6.1. The (i, j)th entry of exp Bt/ζ is of the form i > j > m.

1 i−j (i−j)! (t/ζ)

+ O(ti−j+1 ) for

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307

Proof. We write (Bt/ζ)k in the block form as P (k) Q(k) . R(k) S(k) One then checks by induction that: S(k)(r,s)

  if k < r − s 0 = (t/ζ)r−s if k = r − s  O(tr−s+1 ) if k > r − s

and similarly for the other blocks.

Therefore the (m + 1) column, denoted by p˜m+1 of g −1 (t) exp{Bt/ζ} is of the form p+O(tn−m ), for some vector p. This means that p belongs to the −(n−m−1)/2t eigenspace of Res u, ˜ and so is of the form aEn (En is the nth vector of the standard basis), for some constant a. Computing the tn−m−1 -term of the last entry of p˜m+1 gives −1 = ζ −(n−m−1) , and so a = ζ −(n−m−1) . (Res η)n−m−1 Em+1 ζ n−m−1 (n − m − 1)! d Thus, as in (4.4), w˜ 1 = ζ n−m−1 pm+1 is a solution w˜ 1 (t) to dt w˜ 1 = −u˜ w˜ 1 with lim t−(n−m−1)/2 w1 (t) − En = 0. th

n−m−1

t→0

In the same vein we see that w˜ i (t) = ζ n−m+1−2i p˜m+i , where p˜m+i is the (m + i)th column ˜ = d(ζ) exp{Bt/ζ}g(t), where of g −1 (t) exp{Bt/ζ}. In other words g(t) d(ζ) = diag{1, . . . , 1, ζ −(n−m−1) , . . . , ζ (n−m−1) }.

Similar computations show that the real structure sends B to −r(ζ) B ∗ /ζ¯2 r(ζ)−1 ¯ ∗ )−1 , where and g to r(ζ) exp{B ∗ /ζ}(g ( 0 if i + j 6 = n + m + 1 (6.1) rij (ζ) = (−1)j−1 ζ¯n+m+1−2j if i + j = n + m + 1. Now we consider the subset of F˜n (n; c), where the eigenvalues of β(+∞) are distinct. ˜ We have assigned to each element of this set the pair (β(+∞); g). We know that β(+∞) = 2 β(+∞)/ζ . The argument in Sect. 2 shows then that g˜ = g exp{−cβ(+∞)/ζ}. The real ¯ structure sends g to (g ∗ )−1 exp{cβ(+∞)∗ /ζ}. Now we wish to calculate the twistor space of F˜n,m (c, c0 ) for n > m. This space is a hyperk¨ahler quotient of Fn (m; 1) × F˜n (m; c − 1) × F˜m (m; c0 ). On the subset corresponding to the same chart as in Sect. 5 (i.e. b = dm and b0 = dn ), the coordinates are given by β(−∞), β(+∞) and g such that gβ(+∞)g −1 is of the form (4.5) with h = β(−∞). This g can be written as (g3 )−1 g1 g2 , where g1 (resp. g2 , resp. g3 ) is the g considered above for Fn (m; 1) (resp. F˜n (m; c − 1), resp. F˜m (m; c0 )). It follows that g˜ = exp{c0 β(−∞)/ζ}(g3 )−1 d(ζ) exp{B/ζ}g1 g2 exp{−(c − 1)β(+∞)/ζ} which can be rewritten as exp{c0 β(−∞)/ζ}d(ζ)(g3 )−1 g1 g2 exp{−cβ(+∞)/ζ}.

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Therefore

g˜ = exp{c0 β(−∞)/ζ}d(ζ)g exp{−cβ(+∞)/ζ}.

We now compute the twistor space in coordinates κ1 , . . . , κm ,v1 , . . . , vm ,β1 , . . . , βn , u1 , . . . , un of Sect. 5. We know that κ˜ i = κi /ζ 2 and β˜j = βj /ζ 2 . The matrix g is given by (5.3), where Z = LV −1 where L is described by (5.5) and V is the Vandermonde matrix for β1 , . . . , βn . We obtain equations (here vi = v˜ i = 1 and κi = κ˜ i = 0 for i > m): v˜ i−1 u˜ j Z˜ ij = exp{(c0 κi − cβj )/ζ}di (ζ)Zij vi−1 uj . In addition Z˜ ij = Zij if i ≤ m and Z˜ ij = ζ 2i−2 Zij if i > m. Hence ( exp{(c0 κi − cβj )/ζ}vi−1 uj if i ≤ m −1 v˜ i u˜ j = ζ −n−m+1 exp{−cβj /ζ}vi−1 uj if i > m. As vi = v˜ i = 1 for i > m, we finally obtain v˜ i = ζ −n−m+1 exp{−c0 κi /ζ}vi , u˜ j = ζ

−n−m+1

(6.2)

exp{−cβj /ζ}uj .

(6.3)

Finally, the real structure is computed as in [3]: βi 7→ −β¯i /ζ¯2 ,

¯ n+m−1 ecβi /ζ ui 7→ u¯ −1 i (1/ζ) ¯

¯

m Y Y (β¯i − β¯j ) (β¯i − κ¯ j ), (6.4) j= 6 i

κi 7→ −κ¯ i /ζ¯2 ,

¯ n+m−1 ec0 κ¯ i /ζ¯ vi 7→ v¯ i−1 (1/ζ)

Y (κ¯ i − κ¯ j ) j= 6 i

j=1 n Y

(κ¯ i − β¯j ). (6.5)

j=1

We now have to calculate the real sections. First of all we have βi (ζ) = zi + 2xi ζ − z¯i ζ 2 , κi (ζ) = zn+i + 2xn+i ζ − z¯n+i ζ , 2

for i = 1, . . . , n,

(6.6)

for i = 1, . . . , m,

(6.7)

where pi = (zi , xi ) ∈ C × R are such that pi 6 = pj if i 6 = j. These curves of genus 0 should be thought of as spectral curves of individual monopoles. Let Si denote either βi or κi . Two curves Si and Sj intersect in a pair of distinct points aij and aji , where q (xi − xj ) + rij , rij = (xi − xj )2 + |zi − zj |2 . (6.8) aij = z¯i − z¯j As in [3], if i, j ≤ n, then ui has a zero at aji and is nonzero at aij . Similarly, if i, j > n, then vi−n has a zero at aji and is nonzero at aij . Let us consider what happens when i ≤ n and j > n (and no other curves intersect Si at aji ). First of all, computing the characteristic polynomial of (4.5) gives [21]: det(η − B) = det(η − h)(η n−m − en−m η n−m−1 − . . . − e1 ) − f (η − h)adj g, (6.9) from which Q we conclude that fj−n gj−n is zero at both aij and aji . This implies, since fi = vi / s= 6 i (κi − κs ), that vj−n is zero precisely when fj−n is zero. Now, if the passage from diag(β1 , . . . , βn ) to (4.5) is given by the matrix G of the form (5.3) with

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309

−1 Z = LV −1 , then Gj−n,s = 0 if s 6 = i and Gj−n,i = vj−n ui . This has two implications: 1) ui is zero if and only if vj−n is, and 2) gj = 0 in (4.5). Hence, in this situation, vj−n 6 = 0. Thus ui and vj−n are zero at exactly one of the two points of intersection of Si and Sj . Furthermore, since κk (ζ) does not intersect βl (ζ) at aij or aji if k 6 = j − n or l 6 = i, we conclude that fk gk 6 = 0 at aij or aji if k 6 = j − n. Thus vk (aij ) 6 = 0 and vk (aji ) 6 = 0 for k 6 = j − n. Since G(aij ) and G(aji ) are invertible we also have ul (aij ) 6 = 0 and ul (aji ) 6 = 0 for l 6 = i. Summing up, ui (ζ), i ≤ n, and vi (ζ), i ≤ m, are of the form: Y Y (ζ − aji ) (ζ − cij )ec(xi −z¯i ζ) , (6.10) ui (ζ) = Ai j≤n j= 6 i

vi (ζ) = Bi

Y

j>n

(ζ − aj+n,i+n )

j≤m j= 6 i

Y

0

(ζ − ci+n,j )ec (xi −z¯i ζ) .

(6.11)

j≤n

Here cij can be either aij or aji and is at present undetermined. The reality condition implies that Y Y (xi − xj + rij ) ±(xi − xj ) + rij , Ai A¯ i = Bi B¯ i =

Y

j≤n j= 6 i

j>n

(xi+n − xj+n + ri+n,j+n )

j≤m j= 6 i

Y

±(xi+n − xj ) + ri+n,j ,

j≤n

where the undetermined signs are positive if cij = aji and negative if cij = aij . By continuity, these formulae extend to the case when more than two Si intersect at a point. We can now compute the metric on F˜n,m (c, c0 ) up to the above sign indeterminacy. This metric is of the form (2.1) and it is enough to compute the matrix 8. The complex symplectic form on the twistor space is given by the formula (5.8) and it follows from it that each factor in (6.10) (resp. (6.11)) together with the corresponding factor of |Ai | (resp. |Bi |) gives a separate contribution to 8 (the coefficient of ζ in the expansion of ω is the K¨ahler form ω1 ). The factors of ui , indexed by j ≤ n, together with the exponential term, describe exactly the twistor space of the Gibbons–Manton metric (2.2) with mass parameter −1/c, as computed in Proposition 6.2 of [3], providing that the complex symplectic form is taken to be the first two terms of (5.8). Thus these terms contribute X 1 c− rij j≤n j= 6 i

to 8ii , i ≤ n, and 1/rij to 8ij for i, j ≤ n, i 6 = j. An exactly parallel statement holds for the factors of vi indexed by j ≤ m plus the exponential term. The remaining factors dvi i contribute terms in du ui ∧ dβi and vi ∧ dκi . The calculation in the proof of Theorem 6.4 in [3] shows that the contribution of these factors to the matrix 8 are Gibbons– Manton-like terms with positive or negative sign. Thus we conclude that the matrix 8 for F˜n,m (c, c0 ) is given by (2.3) with ( c if i ≤ n (6.12) ci = c0 if i > n,

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R. Bielawski

( −1 sij = (−1)ij

if i, j ≤ n or i, j > n if i ≤ n, j > n or i > n, j ≤ n.

(6.13)

Here ij = 0 if cij = aij and ij = 1 if cij = aji . We shall eventually see (Lemma 9.6) that all ij are equal to zero. 7. The Metric on F˜n,n (c, c0 ) This space is the hyperk¨ahler quotient of F˜n (n; c) × F˜n (n; c0 ) × Hn by the diagonal action of U (n). According to Sect. 4 its complex charts can be described as b− , (g, b+ ), (V, W ) ∈ b × Gl(n, C) ×N 0 b0 × C2n with gb+ g −1 = b− + V W T . Our first step is to calculate the complex-symplectic form on the set where b = b0 = d. Let us write βd+ = diag(β1+ , . . . , βn+ ) for b+ and βd− = diag(β1− , . . . , βn− ) for b− on this set. The choice of our chart implies that βi+ 6 = βj+ and βi− 6 = βj− for i 6 = j. In addition we suppose that βi− 6 = βj+ for all i, j ≤ n. Then one of βd− , βd+ , say βd− , is invertible. Since all components of W must be nonzero (otherwise the spectra of βd− and βd+ are not disjoint), W is cyclic for βd− . Consider the basis given by columns of T (βd− )n−1 W, (βd− )n−2 W, . . . , W , in which β − is of form (4.5) (with m = 0) and W T = (0, 0, . . . , 1). Thus, β + is also of form (4.5). We can therefore describe this chart as consisting of pairs (g− , βd− ), (g+ , βd+ ) with g− βd− (g− )−1 and g+ βd+ (g+ )−1 both of ± form (4.5) (with m = 0). We have g± = V (β1± , . . . , βn± )−1 diag(u± 1 , . . . , un ). The ± form ω, via the complex-symplectic quotient, can be written as (bd , ρ± are dual to βd± , (g± )−1 dg± ): − ˜− − ρ− b˜ − ˜− ] , ω = − tr b+d ρ˜+ − ρ+ b˜ +d − βd+ [ρ+ , ρ˜+ ] + b− dρ d − βd [ρ− , ρ which can be computed as in [3] giving: ω=

n X du− i=1

i u− i

∧

dβi−

−

X dβi− ∧ dβj− i<j

βi−

−

βj−

+

n X du+ i=1

i u+i

∧ dβi+ −

X dβi+ ∧ dβj+ i<j

. βi+ − βj+ (7.1)

We now wish to compute the twistor space of F˜n,n (c, c0 ). We proceed as in the previous section. A calculation done there shows that the coordinates g, βd− , βd+ (here gβd+ g −1 = βd− + V W T ) change from ζ 6 = ∞ to ζ 6 = 0 as: β˜d± = βd± /ζ 2 ,

g˜ = exp{c0 βd− }g exp{−cβd+ }.

Moreover, since the twistor space of Hn is simply O(1) ⊗ C2n , we have V˜ = V /ζ,

˜ = W/ζ. W

We wish to pass to coordinates βi± , u± i , i = 1, . . . , n. The passage from the basis in which β ± are diagonal to the one in which they are of the form (4.5) is achieved by the matrix T H = (βd− )n−1 W, (βd− )n−2 W, . . . , W . Thus g− = Hd and g+ = Hdg for some diagonal d. We have d˜ = d exp{−c0 βd− } (d is g− in the chart in which b− is diagonal). On the other hand we have written

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311

± g± = V (β1± , . . . , βn± )−1 diag(u± 1 , . . . , un ). Comparing the two formulae in the two ± charts, we conclude that ui (ζ) changes from ζ 6 = ∞ to ζ 6 = 0 as:

u˜ +i = ζ 1−2n exp{−cβi+ /ζ}u+i , u˜ − i

=ζ

1−2n

exp{−c0 βi− /ζ}u− i ,

i = 1, . . . , n, i = 1, . . . , n.

The real structure is given by the formulae (6.4) and (6.5) with m = n, βi = βi+ , κi = βi− , ui = u+i , vi = u− i . ± We have to know what happens to the u± i (ζ) when two curves βi (ζ) intersect. As + 2 in the previous section we write Si (ζ) = βi (ζ) = zi + 2xi ζ − z¯i ζ , Sn+i (ζ) = βi− (ζ) = zn+i + 2xn+i ζ − z¯n+i ζ 2 , and we denote the intersection points of Si and Sj by aij , aji . These are given by the formula (6.8). Consider first the intersection of Si and Sj where both i and j are either less than or equal to n (and no other Sk intersect at aij , aji ). We can still assume generically that the spectra of βd− and βd+ are disjoint and, hence, W is cyclic for βd− . Then H is invertible at aij , aji and so are g± . We compute, as in [3], that each u± i has a zero at the intersection point aji and is nonzero at aij , and all other u± s , s 6 = i, j are nonzero at both aij and aji . The same argument works in the case when both i and j are greater than n. Now consider the intersection of Si and Sj where i ≤ n and j > n. In the chart in which β − is diagonal, we had gβd+ g −1 = βd− + V W T . Thus det(βd− − βi+ 1 + V W T ) = 0. Since V W T has rank one, and so all its k × k, k > 1, minors vanish, we have for any diagonal matrix d = diag(d1 , . . . , dn ) the formula   Y X Y Vk Wk dk + dl  . (7.2) det(d + V W T ) = k

k

l= 6 k

In our case dj = 0 and dk 6 = 0 for k 6 = j. We conclude that Vj Wj vanish at both aij and aji . However both Vj and Wj are sections of O(1) and so have exactly one zero. Thus Wj vanishes at either aij or aji (and only one of them). Furthermore, if we consider the diagonal matrix d = βd− −βs− 1, s 6 = j, then, by the above argument, the non-vanishing of det(d+V W T ) implies that Vs Ws does not vanish at either aij or aji if s 6 = j. In summary, it is precisely the j th column of H that vanishes at either aij or aji . Thus the same ± statement holds for both g− and g+ , and, as g± = V (β1± , . . . , βn± )−1 diag(u± 1 , . . . , un ), − + ± we conclude that both uj and uj vanish at either aij or aji and no other us vanishes at either aij or aji . This means that u+i is given by the formula (6.10) and u− i is given by the formula (6.11) (with n = m). Once more, the formulae extend to the non-generic case. The remainder of the previous section can be now repeated word by word, and we conclude that the metric on F˜n,n (c, c0 ) is of the form (2.1) with 8 given by (2.3) where the ci and sij are given by (6.12) and (6.13). 8. Topology of F˜n,m (c, c0 ) We shall discuss the topology of F˜n,m (c, c0 ). This space can be viewed as a moduli space of solutions to Nahm’s equations defined on (−∞, 0] ∪ (0, +∞) with the appropriate matching at 0. The tri-Hamiltonian action of T n+m = T n × T m gives us the moment map to R3 ⊗ Rn+m which is simply

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T1 (+∞), −T1 (−∞) , T2 (+∞), −T2 (−∞) , T3 (+∞), −T3 (−∞) .

Before stating the result let us recall that a basis of the second homology H2 C˜ p (R3 ), Z of a configuration space C˜ p (R3 ) is given by the p(p − 1)/2 2-spheres 2 = {(x1 , . . . , xp ) ∈ R3 ⊗ Rp ; |xi − xj | = const, xk = const if k 6 = i, j}, (8.1) Sij

where i < j. We have: Proposition 8.1. The above moment map induces a homeomorphism between the orbit space of F˜n,m (c, c0 ) and C˜ n (R3 ) × C˜ m (R3 ). The set of principal T n+m -orbits of F˜n,m (c, c0 ) maps to C˜ n+m (R3 ) and as a T n+m -bundle is determined by the element (h1 , . . . , hn+m ) of H 2 C˜ n+m (R3 ), Zn+m given by   if k = i sij 2 hk (Sij ) = −sij if k = j  0 otherwise, where the sij are given by (6.13). Proof. Let us fix an element (τ + , τ − ) of C˜ n (R3 ) × C˜ m (R3 ). Identify τ + with a regular triple (τ1+ , τ2+ , τ3+ ) of diagonal n × n matrices and similarly τ − with a regular triple (τ1− , τ2− , τ3− ) of diagonal m × m matrices. As in Proposition 5.2 of [3] the space of T n+m -orbits mapping to (τ + , τ − ) can be identified with the set of solutions to Nahm’s equations with T0 ≡ 0 and having values conjugate to (τ + , −τ − ) at +∞ and at −∞. If n > m, this space is diffeomorphic to the hyperk¨ahler quotient X of the product M (τ1+ , τ2+ , τ3+ ) × Fn (m; 1) × M (τ1− , τ2− , τ3− ) by U (n) × U (m), where the M ’s are Kronheimer’s hyperk¨ahler structures on Gl(n, C)/(T n )C and Gl(m, C)/(T m )C [26]. If n = m, this space is diffeomorphic to the hyperk¨ahler quotient X of the product M (τ1+ , τ2+ , τ3+ ) × Hn × M (τ1− , τ2− , τ3− ) by U (n). The first statement will be proved if we can show that these hyperk¨ahler quotients are single points. First we show that the corresponding complex-symplectic quotient, with respect to a generic complex structure I (i.e. one in which M (τ1± , τ2± , τ3± ) are biholomorphic to regular adjoint orbits), are single points. (1) n > m. Let M (τ1+ , τ2+ , τ3+ ) be complex-symplectic isomorphic to the adjoint orbit O+ of diag(β1+ , . . . , βn+ ) (βi+ distinct) and M (τ1− , τ2− , τ3− ) to the adjoint orbit O− − of diag(β1− , . . . , βm ) (βi− distinct). First of all, the complex symplectic quotient of + + + M (τ1 , τ2 , τ3 ) × Fn (m; 1) by Gl(n, C) can be identified with the set U of elements of O+ which are of the form (4.5). Then the zero-set of the complex moment map for the action of Gl(m, C) on U × O− can be identified with the set Y of matrices of the form (4.5) which belong to O+ and such that h belongs to O− . We have to show that Y is a single orbit of Gl(m, C). Since the βi− are distinct we can diagonalize h. Then Eq. (6.9) shows that the ei ’s and the products fi gi are determined. Thus we obtain a single (C∗ )m -orbit. (2) n = m. We make the same assumption about the complex-symplectic structure of the two M ’s. The zero set of the complex moment map for the action of Gl(n, C) on O+ × O− × C2n is the set {(a, b, V, W ) ∈ O+ × O− × Cn × Cn ; a = b + V W T }. Again we have to show that this set is a single orbit of Gl(n, C). Let us diagonalize b and use

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the formula (7.2) with d = b − η1. Substituting βi− for η shows that Vi Wi is determined, i = 1, . . . , n. We obtain a single (C∗ )n -orbit. We remark that the above proof shows that the action of GC , where GC is Gl(n, C) × Gl(m, C) in case (1) or Gl(n, C) in case (2), on the zero-set of the complex moment map has closed orbits of the form GC /T C for some subtorus T of G. Thus, to prove the first statement, we have to show that the complex-symplectic and the hyperk¨ahler quotient coincide. The proof of this requires a substantial detour from the main line of argument and will be given in Appendix A. Let us remark that Hurtubise’s argument [21] for matching solutions to Nahm’s equations on two (or more) intervals cannot be adapted to the case of two half-lines (in this case his Lemma 2.19 will not provide any information). It is clear from the description of the sections of the twistor space - formulae (6.10) and (6.11) - that the action is free precisely over C˜ n+m (R3 ). To determine the principal bundle, one merely has to repeat the calculation in the proof of Proposition 6.3 in [3]. Corollary 8.2. The action of T n+m on F˜n,m (c, c0 ) extends to the global action of (C∗ )n+m with respect to any complex structure. Proof. This is equivalent to showing that, if we fix ζ ∈ CP 1 , then the ui (ζ) and vj (ζ) of (6.10) and (6.11) can take arbitrary complex values (with appropriate degenerations at the intersection points of the βi (ζ)). If, for example ζ = 0, then the zi are fixed and one solves for the xi . One shows that a solution always exists and by the previous result the corresponding point lies in F˜n,m (c, c0 ). 9. Asymptotic Comparison of Metrics We consider the moduli space Mm1 ,... ,mN −1 (µ1 , . . . , µN ) of SU (N ) monopoles with maximal symmetry breaking. We wish to compare the metric on Fm1 ,... ,mN −1 (µ1 , . . . , µN ) (whose Levi–Civita connection coincides with that on Mm1 ,... ,mN −1 (µ1 , . . . , µN )) with the metric on Fm˜ 1 ,... ,m˜ N −1 (µ1 , . . . , µN ). As discussed in Remark 3.3, this space consists of solutions to Nahm’s equations on the union of Ik , where Ik = [µk , +∞) ∪ (−∞, µk+1 ] with matching conditions at the endpoints of each Ik . It will be convenient to write Ik = [[µk , µk+1 ]] and denote the “middle point” ±∞ by ∞k . We shall also use double brackets for any connected subset of [[µk , µk+1 ]]. The space Fm˜ 1 ,... ,m˜ N −1 (µ1 , . . . , µN ) should be thought of as consisting of m = m1 + k k k · · · + mN −1 particles with phases. The positions of particles are xik = (x i , Re zi , Im zi ), √ −1T1 (∞k ) and i ≤ mk , k = 1, . . . N − 1, where diag(xk1 , . . . , xkmk ) = √ k k diag(z1 , . . . , zmk ) = (T2 + −1T3 )(∞k ). We put Rk = min{|xik − xjk |; i 6 = j}.

(9.1)

Zk = min{|zik − zjk |; i 6 = j},

(9.2)

Let us also write

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R. Bielawski

reg and denote by Fm ˜ 1 ,... ,m ˜ N −1 (µ1 , . . . , µN ), where ˜ N −1 (µ1 , . . . , µN ) the subset of Fm ˜ 1 ,... ,m Zk > 0 for k = 1, . . . , N − 1. This subset depends on the chosen complex structure (which is I in the case at hand). If we write for this complex structure, as in Sect. 4, α reg for T0 + iT1 , β for T2 + iT3 , then we can define the subset Fm 1 ,... ,mN −1 (µ1 , . . . , µN ) of Fm1 ,... ,mN −1 (µ1 , . . . , µN ) as the set of (α, β) such that the eigenvalues of β restricted to the k th interval, k = 1, . . . , N − 1, are distinct. We define subset U (γ, δ, C) of Fm˜ 1 ,... ,m˜ N −1 (µ1 , . . . , µN ) as follows

U (γ, δ, C) = {x; min Zk (x) ≥ δ, min Rk (x) ≥ C, ζ T 8ζ ≥ γ|ζ|2 ∀ζ ∈ Rm }, k k (9.3) P where 8 is given by (2.3)–(2.5) and m = mk . reg We have canonical local complex coordinates on Fm ˜ N −1 (µ1 , . . . , µN ): ˜ 1 ,... ,m (w1 , . . . , w2m ) := {zik , uki ; i = 1, . . . , mk , k = 1, . . . , N − 1}, P mk . where the uki are given by the local Cm -action, m = Let g, ˜ g denote the metrics on reg Fm ˜ N −1 (µ1 , . . . , µN ) ˜ 1 ,... ,m

(9.4)

reg and Fm (µ1 , . . . , µN ) 1 ,... ,mN −1

respectively, and let 6 be the product of symmetric groups state the two main results of the paper.

QN −1 k=1

6mk . We can now

Theorem 9.1. The hyperk¨ahler metric on Fm˜ 1 ,...,m˜ N −1 (µ1 , . . . , µN ) is determined by the matrix 8 of the form (2.3) with the ci and sij given by (2.4) and (2.5). reg Theorem 9.2. There exists a complex-symplectic isomorphism φ from Fm ˜ N −1 ˜ 1 ,... ,m reg (µ1 , . . . , µN )/6 to Fm (µ , . . . , µ ) with the following property: ,... ,m 1 N 1 N −1 Let us write X φ∗ g − g˜ = Re Sij dwi ⊗ dw¯j

in coordinates (9.4). Then, for any positive γ, δ, there is a C = C(γ, δ) such that on the set U (γ, δ, C) defined by (9.3), we have |Dl Sij | ≤ Al e−λR ,

l = 0, 1, 2, . . . ,

(9.5)

where R = min{Rk ; k = 1, . . . , N − 1} and Al , λ > 0 are constants depending only on γ, δ. Remarks. 1. For a possible generalization see the discussion at the end of the section. 2. One can alternately use the coordinates given by positions and phases of particles and obtain a completely analogous statement. This follows at once from the explicit formulae for the metric and the twistor space of Fm˜ 1 ,... ,m˜ N −1 (µ1 , . . . , µN ). We see that the coordinate change map and its inverse have all derivatives uniformly bounded on U (γ, δ, C) (cf. [2], section 13.F for the case of Taub-NUT). The proof of Theorems 9.1 and 9.2 will be separated into several parts. We shall reg reg ˜ write M for Fm 1 ,... ,mN −1 (µ1 , . . . , µN ) and M for Fm ˜ N −1 (µ1 , . . . , µN ). As usual, ˜ 1 ,... ,m we use the same letter to denote constants varying from line to line. Part 1: Construction of φ. This is completely analogous to the SU (2) case. One goes via an intermediate moduli space MI consisting of solutions (α, β) to the complex

Asymptotic Metrics for SU (N )-Monopoles

315

Nahm equation which are constant and diagonal on each [[µk + c, µk+1 − c]] for some c < min{(µk+1 − µk )/2; k = 1, . . . , N − 1} and satisfy appropriate matching conditions at each µk , modulo gauge transformations g(t) which satisfy the matching conditions of Remark 3.3. In particular g(t) = exp{hk t − pk } near ∞k for some complex diagonal matrices for diagonal hk , pk . The passage from MI to M is given by restricting these solutions to the union of [[µk , (µk + µk+1 )/2]] ∪ [[(µk + µk+1 )/2, µk+1 ]], viewing them as solutions to the complex Nahm equation on [µ1 , µN −1 ] and solving the real equation as in [21]. The map from M to MI /6 is defined by using a complex gauge transformation to make an element (α, β) of M constant and diagonal on each [µk + c, µk+1 − c], cut off at the center of each interval and extend trivially onto [[µk , µk+1 ]]. ˜ to MI is given, as in Sect. 4, by making a solution constant The passage from M and diagonal on each [[µk + c, µk+1 − c]] by a complex gauge transformation with g(∞k ) = 1 and g(t) = 1 on each [µk , µk + c/2] ∪ [µk+1 − c/2, µk+1 ]. The inverse mapping is given by first solving the real Nahm equation by a complex transformation which is exponentially close to exp{hk t − pk } near ∞k for some diagonal hk , pk . To see that this can be done we argue as follows (cf. [3]). By the argument in the proof of Proposition 8.1, we can first solve the real equation on each (∞k−1 , µk ) ∪ (µk , ∞k ) by bounded gauge transformation gk (t) satisfying the matching condition at µk . Now, from ˜ as the hyperk¨ahler (and so complex-symplectic) Corollary 8.2 and the definition of M ˜. quotient of the F˜(n,m) (c, c0 ), we conclude that there is a global action of (C∗ )m on M Using this action allows us to replace the gk by a gauge transformation g(t) which is exponentially close to exp{hk t − pk } near each ∞k and which also solves the real equation. We still have to show that φ respects complex-symplectic forms. However, φ was constructed using only a) complex gauge transformations, and b) restriction or extension of constant solutions. Both of these operations respect the complex-symplectic forms involved. Part 2: Estimates on solutions. We first obtain estimates on solutions to Nahm’s equations. Recall that the biholomorphism φ was defined as the composition φ = φ2 φ1 with ˜ → MI and φ2 : MI → M . Let (α, β) be a solution to Nahm equations on a φ1 : M half-line [x, +∞), with min{|βii (+∞) − βjj (+∞)|; i 6 = j} ≥ δ > 0. For any > 0, we can assume that α and β are lower-triangular on [x + /2, +∞) (this is done as in [6]: one can conjugate β to be lower-triangular on [x + /2, +∞) by a unitary gauge transformation; (4.1) implies then that α is also lower-triangular). Then the apriori estimates from Sect. 1 in [6] show that |αij (t)| + |βij (t)| ≤ M e−λRij t

(9.6)

for i > j and t ≥ x + . Here Rij = | Re αii (+∞) − Re αjj (+∞)| + |βii (+∞) − βjj (+∞)| and M, λ > 0 are constants depending only on δ, (and the Lie-algebra to which α, β belong). For the diagonal part of α one has the following estimate ([6], end of Sect. 1): | Re αii (t) − Re αii (+∞)| ≤ K for all i and t ≥ x + , K = K(δ, ). Then the real Nahm equation (4.2) gives X d (Re αii ) ≤ M Rij e−λRij t , dt i>j from which we conclude, that for all i and t > x + ,

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R. Bielawski

| Re αii (t) − Re αii (+∞)| ≤ M e−λRt ,

R = min Rij .

(9.7)

Notice also that we can use the gauge freedom to make Im αii constant on [x + , +∞). Now, φ1 was defined by a complex gauge transformation p(t), with p(∞k ) = 1 and p(t) = 1 on each [µk , µk + c/2] ∪ [µk+1 − c/2, µk+1 ], making α and β constant and ˜ = φ1 (α, β) satisfies ˜ β) diagonal on [[µk + c, µk+1 − c]]. Thus we conclude that (α, ( 0 if t ∈ [µk , µk + c/2] ∪ [µk+1 − c/2, µk+1 ] |α(t) ˜ − α(t)| = O exp{−λRk t} if t ∈ [[µk + c, µk+1 − c]], (9.8) and similarly for β and for the derivative of α. We now consider φ2 . After cutting off the ˆ on [µ1 , µN ] to the complex Nahm equation which solutions, we obtain a solution (α, ˆ β) satisfies ˆ βˆ ∗ ] = O(e−λR ). ˆ := d αˆ + αˆ ∗ + [α, ˆ αˆ ∗ ] + [β, F (α, ˆ β) dt We know from the work of Hurtubise that there is a unique element of G C /G such that ˆ to an element of M . We have any element g(t) in this orbit takes (α, ˆ β) Lemma 9.3. The gauge transformation g satisfies |g ∗ g − 1| = O(e−λR ) uniformly on [µ1 , µN ]. Proof. Using Lemma 2.10 in [13] and a simple comparison theorem ([4], Lemma 2.8), one shows that the real equation can be solved on each [µk , µk+1 ] by a complex gauge transformation gk (t) with gk (µk ) = gk (µk+1 ) = 1 and gk∗ gk uniformly bounded by O(e−λR ). Furthermore, near µk , |gk∗ gk (t) − 1| ≤ (t − µk )ce−λR and similarly near µk+1 . Therefore the derivative of gk∗ gk at µk , µk+1 is bounded by ce−λR . This shows that, while the resulting αˇ does not satisfy the matching conditions at the µk , the jumps are of order O(e−λR ). Hurtubise shows in [21] that one can now match the solutions by ˇ and g 0 (α, ˇ satisfy the ˇ β) ˇ β) a unique (complex) gauge transformation g 0 . Since both (α, 0∗ 0 real equation, Lemma 2.10 in [13] implies that g g is bounded by its values at the points µk . Let φ (resp. −ψ) be the logarithm of maximum (resp. minimum) of eigenvalues of ˙ 1ψ˙ of g 0∗ g 0 . The proofs of Propositions 2.20 and 2.21 in [21] show that the jumps 1φ, derivatives φ˙ and ψ˙ are of order e−λR at each µk . We then conclude, by going through the proof of Lemma 2.19 in [21], that at each µk we have φ(µk ) ≤ c1φ˙ + O(e−λR ) and ψ(µk ) ≤ c1ψ˙ + O(e−λR ) for some c < 0 (depending only on the µj ). This shows that φ(µk ) and ψ(µk ) are both of order e−λR which finishes the proof. Part 3: Estimates for the tangent vectors. Recall that a tangent vector to a moduli space of solutions to Nahm’s equations is a quadruple t0 , . . . t3 satisfying Eqs. (3.3). We shall write a = t0 + it1 and b = t2 + it3 . Then Eqs. (3.3) can be written as a˙ = [α∗ , a] + [β ∗ , b], b˙ = [β, a] + [b, α].

(9.9) (9.10)

If the moduli space consists of solutions defined on several adjoining intervals, then a and b also satisfy appropriate matching conditions at the endpoints. 0 ˜ We shall need apriori estimates for solutions of the above equations in Fn,m (c, c ). − − + + ˜ Let us write (α , β ), (α , β ) for a representative of Fn,m (0, 0) (and so of any − + + − − + + F˜n,m (c, c0 )) and then x− i , zi (resp. xi , zi ) for the values of Re α , β (resp. Re α , β ) ± ± ± ± at −∞ (resp. at +∞)). Let us also write R± = min{|xi − xj | + |zi − zj |; i 6 = j}, − + + Z ± = min{|zi± − zj± |; i 6 = j} and S = min{|x− i − xj | + |zi − zj |}. We have

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317

Proposition 9.4. For any positive δ, , ν > 0 there exist constants M, C, λ > 0 depending only on m, n, , δ, ν with the following property: Let (α− , β − ), (α+ , β + ) be a representative of F˜n,m (0, 0) with Z ± ≥ δ > 0, S ≥ ν > 0 and R+ ,R− > C. If ((a− , b− ), (a+ , b+ )) is a tangent vector to F˜n,m (0, 0) at (α− , β − ), (α+ , β + ) and A2 = |a− (−∞)|2 + |b− (−∞)|2 + |a+ (+∞)|2 + |b+ (+∞)|2 ,

(9.11)

then for all t ≥ , −

|a− (−t) − a− (−∞)| + |b− (−t) − b− (−∞)| ≤ M e−λR t A, −λR+ t

|a (t) − a (+∞)| + |b (t) − b (+∞)| ≤ M e +

+

+

+

A.

(9.12) (9.13)

Proof. It is enough to prove the estimates for A = 1. We can assume, as in Part 2 of this proof, that α± (t), β ± (t) are lower-triangular for |t| ≥ /2. For the time being we consider only α+ , β + and we omit the superscript +. We choose C so that the right−1 and the right-hand side of (9.7) is small hand side of (9.6) is small compared to Rij −1 compared to R at t = . Then, if we write y for the diagonal components and x for the off-diagonal components of a and b, we obtain from Eqs. (9.9) and (9.10), y˙ = A(t)x,

|A(t)| ≤ M e−λRt .

(9.14)

On the other hand, if we differentiate Eqs. (9.9) and (9.10), we can write x¨ = D(t)x + B(t)y,

|B(t)| ≤ M e−λRt , ∃s>0 ∀z Re D(t)z, z) ≥ s2 R2 |z|2 . (9.15)

Let t0 ∈ [, +∞] be the first point for which |x(t0 )|2 + |y(t0 )|2 ≤ |a(+∞)|2 + |b(+∞)|2 ≤ 1. Let X = sup{x(t); t ∈ [t0 , +∞]}, Y = sup{y(t); t ∈ [t0 , +∞]}. Both X and Y are finite. Equation (9.14) implies that, for t ≥ t0 , |y(t) − y(t0 )| ≤

MX . R

(9.16)

Similarly, using (9.15) and a comparison theorem (the same argument as on p. 133 in [6]), one concludes that |x(t)| ≤ |x(t0 )|M Y e−λR(t−t0 ) .

(9.17)

From this, changing C if necessary (i.e. taking larger R), we conclude that there this a constant P such that X + Y ≤ P . Using again (9.14),(9.17) we obtain that the estimate (9.13) holds for t ≥ t0 . This implies that Z +∞ |a+ (t)|2 + |b+ (t)|2 − |a+ (+∞)|2 − |b+ (+∞)|2 dt ≤ ρ, (9.18) t0

where ρ = ρ(n, δ, ) and can be made arbitrarily small by changing C (recall that A = 1). We also have that for t ∈ [, t0 ] the expression under the integral sign in (9.18) is nonnegative. We can do exactly the same for α− , β − , a− , b− . Let s0 denote the negative number with the same properties as t0 . We now compute the length L of the vector (a− , b− ), (a+ , b+ ) in the metric of F˜(n,m) (, ). We can write (cf. (3.5)) (the fact that

318

R. Bielawski

below we have an inequality, rather than equality, stems from the fact that, for n = m, there are additional (positive) terms): Z 0 Z |a− (t)|2 + |b− (t)|2 dt + |a+ (t)|2 + |b+ (t)|2 dt L2 ≥ −

0

Z

−

|a− (t)|2 + |b− (t)|2 − |a− (−∞)|2 − |b− (−∞)|2 dt

+ s0

Z

+ Z

t0

s0

+ −∞

|a+ (t)|2 + |b+ (t)|2 − |a+ (+∞)|2 − |b+ (+∞)|2 dt |a− (t)|2 + |b− (t)|2 − |a− (−∞)|2 − |b− (−∞)|2 dt Z

+∞

+ t0

|a+ (t)|2 + |b+ (t)|2 − |a+ (+∞)|2 − |b+ (+∞)|2 dt.

(9.19)

Each of the first four terms is positive, while the last two have their absolute value bounded by 2ρ with ρ as small as we wish. Let us write T for the sum of the third and fourth term. It follows that T ≤ L2 + 2ρ. The explicit formula for the metric on F˜n,m (c, c0 ) found in Sects. 6 and 7 implies that L2 ≤ P , where P > 0 depends only on m, n, , ν, δ (notice that this bound is independent of the actual value of ij in (6.13)). Thus T ≤ P 0 . Now, if both t0 and −s0 are smaller than 2, then we are done (by replacing the original with 2). Suppose that t0 ≥ 2. Since the integrand in the second and third term is nonnegative, we conclude from T ≤ P 0 that there is a point t1 ∈ [, 2] with |a+ (t1 )|2 + |b+ (t1 )|2 ≤ P 0 /. We can now repeat the arguments after (9.15) and conclude that the estimate (9.13) holds for t ≥ 2. We can deal similarly with the case s0 ≤ −2. We shall also need the following strengthening of the last result: Lemma 9.5. With the same assumptions and notation as in Proposition 9.4, we can replace the estimates (9.12) and (9.13) with: −

−λRij t − A, |a− ij (−t)| + |bij (−t)| ≤ M e

|a+ij (t)| + |b+ij (t)| ≤ M e−λRij t A, +

± ± ± ± for all i 6 = j and t ≥ . Here Rij = |x± i − xj | + |zi − zj |.

Proof. We differentiate Eqs. (9.9) and (9.10) and proceed, as in Proposition 3.12 of [4] using (9.12) and (9.13). Part 4: Proof of Theorem 9.1. From its definition in Sect. 3, Fm˜ 1 ,...,m˜ N −1 (µ1 , . . . , µN ) is the hyperk¨ahler quotient, by a product of tori, of the product F˜m1 (c1 ) × F˜m2 ,m1 (c2 , c02 ) × · · · × F˜mN −1 ,mN −2 (cN −1 , c0N −1 ) × F˜mN (c0N ), where ci + c0i+1 = µi+1 − µi , i = 1, . . . , N − 1. The matrices 8 for each factor are of the form (2.3) with the sij given by (6.13) (for the first and last factor, the metric is the Gibbons–Manton metric given by (2.2)). On the hyperk¨ahler quotient these matrices are simply added together (after viewing each of them as a submatrix of an m × m matrix, m = m1 + · · · + mN −1 ). Thus the result is proved as soon as we show that all ij of (6.13) are zero. Let us show this. Lemma 9.6. In the formula (6.13), all ij are equal to zero.

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Proof. Suppose that this is not true. Let us write the norm of vector v tangent to F˜(n,m) (1, 1) as in formula (9.19) with = t0 = −s0 = 1. From the estimates (9.12) and (9.13), it follows that kvk2 ≥ −M A2 /R for a constant M depending only on m, n, δ and ν. Here R = min{R− , R+ }. Thus, for a sufficiently large R, kvk2 ≥ −ρA2 , A defined by (9.11), with ρ as small as we wish. However, if any ij = 1, then we can find a point x in F˜(n,m) (1, 1) with S = S(x) ≥ ν and R arbitrarily large such that there is a tangent vector v at x with kvk2 ≤ −cA2 /ν for some c = c(n, m). This contradicts kvk2 ≥ −ρA2 and so the lemma is proved. Part 5: Proof of Theorem 9.2. In Appendix B we prove a general theorem we allows us to reduce the estimates to one-sided estimates on the metric tensors. This is so because the asymptotic metric is quasi-isometric to the flat metric in coordinates (9.4). This last fact follows from the explicit formula for the metric and the twistor space (cf. Remark 2 after Theorem 9.2). Thus we only have to show that (9.20) φ∗ g ≤ 1 + M e−λR g˜ in the region U (γ, δ, C) for some C, M, λ > 0 depending only on γ, δ. Once we have this, we apply Theorem B.1 (and Remark B.2) to the region where R ≥ R0 and obtain that the estimate (9.5) with R = R0 , R0 arbitrary, holds in the region where R ≥ R0 + 1, in particular for all points with R = R0 + 1. Since R0 is arbitrary this will prove the theorem. Therefore we are going to show (9.20). We start with a vector (a, b) tangent to U (γ, δ, C), where C = C(γ, δ) is determined by the validity of estimates below. Since γ > 0, the metric is positive-definite and, furthermore, quasi-isometric to the flat metric in coordinates (9.4) or the coordinates given by positions Pand phases of particles.Let us assume that the norm of (a, b) is 1 in this metric. Then k |a(∞k )|2 + |b(∞k )|2 ≤ B, where B depends only on γ. We also have estimates of the form (9.12) and (9.13): |a(t) − a(∞k )| + |b(t) − b(∞k )| ≤ M e−λRk t B

if t ∈ [[µk + , µk+1 − ]], (9.21)

as well as the stronger estimates of Lemma 9.5. Also, by writing the metric as in (9.19) with = t0 = c and using (9.21), we get N Z X k=1

µk +c

µk −c

|a(t)|2 + |b(t)|2 dt ≤ M B.

(9.22)

The left-hand side includes the sum of Euclidean norms of pairs of vectors uk , vk which give us the matching conditions for (a, b) at µk in the case when mk−1 = mk . Recall that the map φ was a composition of a φ1 and φ2 . The map φ1 was given by a complex gauge transformation p(t) such that |p(t) − 1| = O(e−λRt ), by Part 2 of the proof. Therefore, after we conjugate a, b by p, they still satisfy (9.21). Moreover we have |k(pap−1 , pbp−1 )k − 1| = O(e−λR ). In order to obtain the vector dφ1 (a, b) , one has to make pap−1 and pbp−1 constant and diagonal on each [[µk + c, µk+1 ]] (c is defined in Part 1) by an infinitesimal complex gauge transformation ρ1 (with ρ1 (∞k ) = 0, etc.). From the estimates (9.21) and of Lemma 9.5 on pap−1 and pbp−1 , this changes the norm of p(a, b)p−1 by something of order e−λR . Furthermore the L2 -estimate (9.22) holds for dφ1 (a, b). At the next stage, we restrict dφ1 (a, b) to [µ1 , . . . , µN ]. Since dφ1 (a, b) is constant and diagonal on the union of [[µk + c, µk+1−c ]], its norm in the metric g˜ is the ˆ in the metric g. Now we conjugate (ˆa, b) ˆ by same as the norm of the restriction (ˆa, b)

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the complex gauge transformation g(t) of Lemma 9.3. Using the estimate of that lemma ˆ we conclude that and the estimate (9.22) for (ˆa, b) ˆ −1 )k − 1| ≤ M e−λR , |k(g aˆ g −1 , g bg

(9.23)

ˆ −1 ) solves for some M, λ > 0 depending only on γ and δ. The vector (g aˆ g −1 , g bg Eq. (9.10) but not (9.9). This is the final step: we obtain the vector dφ(a, b) by acting on ˆ −1 ) with a complex infinitesimal gauge transformation, so that the resulting (g aˆ g −1 , g bg vector solves (9.9). However, Eq. (9.9) is the condition of orthogonality to complex infinitesimal gauge transformations and, hence, the norm of the vector dφ(a, b) is not ˆ −1 ). This and (9.23) proves (9.20), and so, by the greater than the norm of (g aˆ g −1 , g bg discussion above, also Theorem 9.2. As remarked after the statement of the above theorem, there is a likely generalization of this result. Suppose that it is only particles of a given type, say k0 , that separate (recall that the type of particle i is the smallest k for which i ≤ mk ). Then the metric on Fσ (µ) = Fm1 ,... ,mN −1 (µ1 , . . . , µN )) should get close to the metric on Fσ˜ (µ), where ˜ 0) = m ˜ k0 . Similarly, if the particles of types k1 , . . . , ks σ(k) ˜ = mk if k 6 = i0 and σ(k separate, the metric should be close to the metric on Fσ˜ (µ), where σ(k) ˜ = mk if k 6 = ˜ j) = m ˜ kj , for j = 1, . . . , s. All of these moduli spaces have dimension k1 , . . . , ks and σ(k 4(m1 + . . . + mN −1 ). In general the metricPon Fσ˜ (µ) will be simpler than the one on s Fσ (µ) (it has a tri-Hamiltonian action of a ( i=1 mki )-dimensional torus), but it is only in the case when {k1 , . . . , ks } = {1, . . . , N − 1} that the metric is algebraic. Finally we shall discuss the topology of the asymptotic moduli space. Q First of all, from Proposition 8.1, the orbit space of Fm˜ 1 ,...,m˜ N −1 (µ1 , . . . , µN ) is C˜ mk R3 ), and so particles of different types can take the same position.P Now recall from Sect. 2 that the type t(i) of the particle i is defined as min{k; i ≤ s≤k ms }. It follows easily from Proposition 8.1 that the set of principal orbits of T m , m = m1 + · · · + mN −1 , on Fm˜ 1 ,...,m˜ N −1 (µ1 , . . . , µN ) is a bundle P over C = {(x1 , . . . , xm ) ∈ R3 ⊗ Rm ; |t(i) − t(j)| ≤ 1 =⇒ xi 6 = xj }. The basis of the second integer homology of C is given by the spheres Sij defined by (8.1), where now i, j run over the set {(i, j); i < j and |t(i) − t(j)| ≤ 1}. As in Lemma 7.1 in [3], we obtain that the bundle P is determined by the element (h1 , . . . , hm ) of H 2 C, Zm such that   if k = i sij hk (Sij ) = −sij if k = j  0 otherwise, where the sij are given by (2.5). A. Complex-Symplectic vs. Hyperk¨ahler Quotients In order to finish the proof of Proposition 8.1 we have to show that certain hyperk¨ahler and complex-symplectic quotients coincide. This question reduces, on the zero-set of the complex moment map, to identifying the K¨ahler quotient with the ordinary geometric quotient, i.e. to showing that all orbits of the complexified group are stable. The following useful criteria are given in [18], Lemma 3.3, and [17], Sects. A.1.3 and A.2.3.

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Proposition A.1. Let H be a connected closed subgroup of a compact semisimple Lie group G and suppose that M = GC /H C is equipped with a G-invariant K¨ahler form ω defined by a global G-invariant K¨ahler potential K for ω. Then the single GC -orbit M is stable if and only if K is proper. Proposition A.2. Suppose that the complex torus M = (C∗ )n is equipped with a T n invariant K¨ahler form ω. Then: The T n -action is Hamiltonian if and only if there exists a global T n -invariant K¨ahler potential K for ω; n (ii) If K has a quadratic growth at infinity, as a function on Rn where (C∗ )n = T n eR , then M is stable. (i)

In order to use these criteria we shall view the spaces involved in the proof of Proposition 8.1 as hyperk¨ahler quotients of simpler manifolds. Recall that, for a regular triple (τ1 , τ2 , τ3 ) of n × n diagonal matrices, M (τ1 , τ2 , τ3 ) denotes Kronheimer’s hyperk¨ahler structure on Gl(n, C)/T C (T is the diagonal torus in U (n)) with the cohomology class of ωi equal to τi , i = 1, 2, 3 (after identifying H 2 Gl(n, C)/T C , R with Lie(T )). We have Proposition A.3. M (τ1 , τ2 , τ3 ) is isomorphic to a hyperk¨ahler quotient of a flat quaternionic vector space by a product G of unitary groups. With respect to generic complex structure the action of GC on the zero-set of the complex moment map is free and its orbits are closed. Proof. Kobak and Swann [25] show how to construct nilpotent orbits as hyperk¨ahler quotients by a product G of unitary groups. Changing the level set of the moment map, from zero to appropriate values, gives a hyperk¨ahler manifold N which, with respect to a generic complex structure, is isomorphic, as a complex-symplectic manifold, to M (τ1 , τ2 , τ3 ) (if the levels of the complex moment map for the abelian factors of G are t1 , . . . , tn−1 , then the resulting orbit has eigenvalues 0, t1 , t1 + t2 , . . . , (t1 + · · · + tn−1 )). As a Riemannian manifold, N is complete and the uniqueness result of [7] shows that N ' M (τ1 , τ2 , τ3 ) as hyperk¨ahler manifolds. The flat space which we start with can be viewed as a space of matrices and the second statement follows by putting the products of matrices defining the zero set of the complex moment map into the Jordan normal form. We can finally finish the proof of Proposition 8.1. We considered there a hyperk¨ahler quotient X of the product of two M (τ1 , τ2 , τ3 )’s and of either Hn or of Fn (m; 1). By the above proposition, X can be viewed as a hyperk¨ahler quotient of either an Hp (when n = m), for some p, or of the product of Hp and of Fn (m; 1) (when n > m). Furthermore, the complexification H C of the group H by which we quotient acts freely and with closed orbits on the zero-set of the (generic) complex moment map. In addition the usual K¨ahler potential K1 , given by the square of the distance from the origin, on C2p ' Hp is proper and Sp(p)-invariant. Moreover it has quadratic (in fact exponential) growth on any closed orbit of a subtorus of Sp(p, C). Since the H C -orbits are closed on the zero set of the complex moment map, Propositions A.1 and A.2 show that all these H C -orbits are stable for n = m. The space Fn (m; 1) also has an U (n) × U (m) -invariant K¨ahler potential for any complex structure. This is a general phenomenon for hyperk¨ahler manifolds with an SU (2)-action rotating the complex structures, see [20]. For the complex structure I this K¨ahler potential is given by:

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Z K=

1

kT1 (t)k2 + kT2 (t)k2 dt.

0

Since the factor U (n) × U (m) of H is acts diagonally on Hp × Fn (m; 1), the K¨ahler potential K2 on each orbit of Gl(n, C)×Gl(m, C) is the sum of the restrictions of K1 and K. Since K1 is proper and has quadratic growth on each closed toral orbit and since K is positive, it follows that K2 is proper (on each orbit) and has quadratic growth on each closed toral orbit. Thus the complex-symplectic quotient coincides with the hyperk¨ahler quotient in this case as well. B. A Comparison Theorem for Ricci-Flat K¨ahler Metrics Our goal is the following theorem, which, under certain assumptions, reduces the comparison of Ricci-flat K¨ahler metrics and their derivatives to one-sided estimates on the metric tensors. Theorem B.1. Let V be an open subset of a Ricci-flat K¨ahler manifold (M, g) and suppose that there exists a biholomorphic map φ from a domain U in Cm onto V such that φ∗ g is bounded uniformly in the Euclidean metric on U by a constant C and that φ∗ ω m = ef ω0m with f bounded uniformly by a constant K (here ω, ω0 are the K¨ahler forms on X and on Cn ). Then, for any r, δ > 0, there exist constants Ak = Ak (m, C, K, r, δ), k = 0, 1, 2, . . . , with the following property: Let (M 0 , g 0 ) be another K¨ahler manifold and ψ : M → M 0 a volume-form preserving biholomorphism suchP that ψ ∗ g 0 ≤ (1 + )g, uniformly on V , where ≤ δ. Let us write φ∗ (ψ ∗ g 0 − g) = Re Sij dzi ⊗ dz¯j . Then, for any i, j ≤ m and any k ≥ 0, |Dk Sij | ≤ Ak

(B.1)

uniformly on the set {z ∈ U ; dist (z, ∂U ) ≥ r}. Remark B.2. In hyperk¨ahler geometry the conditions on the volume-form are very natural. Namely, if M is hyperk¨ahler with the complex-symplectic form = g(J , ) + ig(IJ , ) and φ∗ is the standard complex-symplectic form on C2n , then φ∗ ω 2n = ω02n , so that we can take K = 0 in the theorem. Similarly, if M 0 is hyperk¨ahler and ψ respects the complex-symplectic forms, then ψ respects the volume forms. We also remark that the assumption that φ∗ g and ln(ω m /ω0m ) are uniformly bounded is equivalent to φ being a quasi-isometry, i.e. to existence of a constant B such that B −1 g0 ≤ φ∗ g ≤ Bg0 , where g0 is the Euclidean metric on Cm . The fact that one-sided estimates on the metric plus estimates on the volume form give two-sided estimates on the metric is trivial but, given Remark B.2, worth stating separately: Proposition B.3. Let (M, g), (M 0 , g 0 ) be two oriented Riemannian manifolds of dimension n and let ψ : M → M 0 be a volume-form preserving diffeomorphism such that φ∗ g 0 ≤ Bg. Then B 1−n g ≤ φ∗ g 0 . P dxi ⊗ Proof. Choose a point m P∈ M and local coordinates x1 , . . . , xm so that gm = ∗ 0 dxi . Write (φ g )m = Xij dxi ⊗ dxj . Then the assumptions imply that det X = 1 and ζ T Xζ ≤ B for all ζ ∈ Rn . Since X is symmetric and positive-definite we can diagonalize X by an orthogonal matrix and conclude that ζ T Xζ ≥ B 1−n ζ T ζ for all ζ ∈ Rn .

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Proof of Theorem B.1. We write φ∗ g = Re

m X

Zij dzi ⊗ dz¯j ,

φ∗ ψ ∗ g 0 = Re

i,j=1

m X

0 Zij dzi ⊗ dz¯j

(B.2)

i,j=1

for hermitian Z, Z 0 . Let us fix an r, so that we have to estimate the derivatives at points z such that B(z, r) ⊂ U . We can assume, without loss of generality, that U = B(0, r) and estimate the derivatives at the origin. Since B(0, r) is strictly pseudo-convex, there are 0 . In general, smooth real-valued functions 8, 80 such that 8zi z¯j = Zij and 80zi z¯j = Zij we shall write L(u) for the complex Hessian (Levi form) uzi z¯j of a function u. Both 8 and 80 satisfy the complex Monge–Amp`ere equation det L(u) = ef .

(B.3)

Since M is Ricci-flat, f is a polyharmonic function. In particular all derivatives of g = ef have uniform bounds depending only on the bound K for f . We need the following Proposition B.4. Let u be a pluri-subharmonic solution to (B.3) in B(0, r). Then, for any k ≥ 0, kL(u)k∗k ≤ Bk , where Bk depends only on m, r, kL(u)k0 and kf kk+2 . Here k kk and k k∗k denote, respectively, the C k -norm and the interior C k -norm (see [14], formula (4.17), for the definition of the latter). Proof. Let B = kL(u)k0 . Since det L(u) = ef , we have (as in Proposition B.3) positive constants λ and 3 depending only on m, B and kf k0 such that λζ ∗ ζ ≤ ζ ∗ L(u)ζ ≤ 3ζ ∗ ζ for all ζ ∈ Cm . Thus Eq. (B.3) is uniformly elliptic with respect to u. Using the estimates ¯ on a solution of the ∂-problem (see, for example, [24]), we see that there is a smooth realvalued function u0 with L(u0 ) = L(u) and ku0 k0 ≤ CkL(u)k0 for a constant C = C(r). Therefore, we can assume from the beginning that u is bounded by some C0 depending only on m, r and kL(u)k0 . We now claim that there are constants C1 and α > 0, depending only on m, r, 3, kf k2 such that the following H¨older estimate holds kL(u)k∗α ≤ C1 .

(B.4)

The proof of this is essentially the same as the proof of (17.41) in [14] (applied to the equation log det L(u) = f ), but using the Hermitian analogue of Lemma 17.13 in [14]. Once we have this, we obtain inductively estimates on all derivatives of u by treating Eq. (B.3) and its successive linearizations in all directions as uniformly elliptic secondorder linear PDE’s (with the coefficients of second-order derivatives given by the adjoint matrix of L(u)). The desired estimates follow from standard Schauder interior estimates. To finish the proof of Theorem B.2, we write the equation det L(8) = det L(80 ) as (B.5) F L(8 − 80 ) = 0 for a linear function F . The coefficients of F depend only on L(8) and L(80 ). The function F is uniformly elliptic in B(0, 3r/4) with ellipticity constants depending only

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on m, r, C, K, δ. Moreover, by the previous proposition, the derivatives of coefficients also have bounds depending only on these constants. From Proposition B.3, v = 8 − 80 satisfies L(v) ≤ P for some constant P , and, as in the proof of Proposition B.4, we can ¯ use estimates (e.g. [24]) on the solution of the ∂-problem to conclude that there is a v 0 0 0 0 with L(v ) = L(v) and kv k0 ≤ P . We now obtain the estimates on all derivatives of v 0 from the Schauder interior estimates applied to Eq. (B.5) and its successive linearizations in all directions.

References 1. Atiyah, M.F. and Hitchin, N.J.: The geometry and dynamics of magnetic monopoles. Princeton, NJ: Princeton University Press, 1988 2. Besse, A.: Einstein manifolds. Berlin–Heidelberg–New York: Springer, 1987 3. Bielawski, R.: Monopoles and the Gibbons–Manton metric. Commun. Math. Phys. 194, 297–321 (1998) 4. Bielawski, R.: Asymptotic behaviour of SU (2) monopole metrics. J. reine angew. Math. 468, 139–165 (1995) 5. Bielawski, R.: Hyperk¨ahler structures and group actions. J. London Math. Soc. 55, 400–414 (1997) 6. Bielawski, R.: Monopoles, particles and rational functions. Ann. Glob. Anal. Geom. 14, 123–145 (1996) 7. Bielawski, R.: Invariant hyperk¨ahler metrics with a homogeneous complex structure. Math. Proc. Cambridge Phil. Soc. 122, 473–482 (1997) 8. Biquard, O.: Sur les e´ quations de Nahm et les orbites coadjointes des groupes de Lie semi-simples complexes. Math. Ann. 304, 253–276 (1996) 9. Biquard, O.: Twisteurs des orbites coadjointes. Ecole Polytechnique preprint (1997) 10. Chalmers, G.: Multi-monopole moduli spaces for SU (N ) gauge group. hep-th/9605182 11. Dancer, A.S.: Nahm’s equations and hyperk¨ahler geometry. Commun. Math. Phys. 158, 545–568 (1993) 12. Dancer, A.S.: A family of hyperk¨ahler manifolds. Quart. J. Math. Oxford 45, 463–478 (1994) 13. Donaldson, S.K.: Nahm’s equations and the classification of monopoles. Commun. Math. Phys. 96, 387–407 (1984) 14. Gilbarg, D. and Trudinger, N.S.: Elliptic partial differential equations of second order. 2nd ed., Berlin: Springer, 1983 15. Gibbons, G.W. and Manton, N.S.: The moduli space metric for well-separated BPS monopoles. Phys. Lett. B 356, 32–38 (1995) 16. Gibbons, G.W. and Rychenkova, P.: HyperK¨ahler quotient construction of BPS monopole moduli spaces. Commun. Math. Phys. 186, 581–599 (1997) 17. Guillemin, V.: Moment maps and combinatorial invariants of Hamiltonian T n -spaces. Boston: Birkh¨auser, 1994 18. Heinzner, P., Huckleberry, A.T. and Loose, F.: K¨ahlerian extensions of the symplectic reduction. J. reine angew. Math. 455, 123–140 (1994) 19. Hitchin, N.J.: On the construction of monopoles. Commun. Math. Phys. 89, 145–190 (1983) 20. Hitchin, N.J., Karlhede, A., Lindstr¨om, U. and Roˇcek, M.: Hyperk¨ahler metrics and supersymmetry. Commun. Math. Phys. 108, 535–586 (1985) 21. Hurtubise, J.C.: The classification of monopoles for the classical groups. Commun. Math. Phys. 120, 613–641 (1989) 22. Hurtubise, and Murray, M.K.: On the construction of monopoles for the classical groups. Commun. Math. Phys. 122, 35–89 (1989) 23. Jarvis, S.: A rational map for Euclidean monopoles via radial scattering. Oxford preprint (1996) ¯ = f in strongly pseudoconvex domains. Comm. 24. Kerzman, N.: H¨older and Lp estimates for solutions ∂u Pure Appl. Math. 24, 301–379 (1971) 25. Kobak, P.Z. and Swann, A.: Classical nilpotent orbits as hyper-K¨ahler quotients. Internat. J. Math. 7, 193–210 (1996) 26. Kronheimer, P.B.: A hyper-k¨ahlerian structure on coadjoint orbits of a semisimple complex group. J. London Math. Soc. 42, 193–208 (1990) 27. Lindstr¨om, U. and Roˇcek, M.: Scalar tensor duality and N = 1, 2 nonlinear σ-models. Nucl. Phys. 222B, 285–308 (1983)

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28. Lee, K., Weinberg, E.J. and Yi, P.: Moduli space of many BPS monopoles for arbitrary gauge groups. Phys. Rev. D 54, 1633–1643 (1996) 29. Murray, M.K.: A note on the (1, 1, . . . , 1) monopole metric. J. Geom. Phys. 23, 31–41 (1997) 30. Nahm, W.: The construction of all self-dual monopoles by the ADHM method. In: Monopoles in quantum field theory, Singapore: World Scientific, 1982 31. Nakajima, H.: Monopoles and Nahm’s equations. In: Einstein metrics and Yang–Mills connections, New York: Marcel Dekker, 1993 32. Pedersen, H. and Poon, Y.S.: Hyper-K¨ahler metrics and a generalization of the Bogomolny equations. Commun. Math. Phys. 117, 569–580 (1988) 33. Taubes, C.H.: Min-max theory for the Yang–Mills–Higgs equations. Commun. Math. Phys. 97, 473-540 (1985) Communicated by H. Nicolai

Commun. Math. Phys. 199, 327 – 349 (1998)

Communications in

Mathematical Physics © Springer-Verlag 1998

The First Eigenvalue of the Dirac Operator on Quaternionic K¨ahler Manifolds W. Kramer1,? , U. Semmelmann2,?? , G. Weingart1,? 1

Mathematisches Institut der Universit¨at Bonn, Beringstraße 1, 53115 Bonn, Germany. E-mail: [email protected]; [email protected] 2 Mathematisches Institut der Universit¨ at M¨unchen, Theresienstraße 39, 80333 M¨unchen, Germany. E-mail: [email protected] Received: 2 February 1998 / Accepted: 8 May 1998

Abstract: In [15] we proved a lower bound for the spectrum of the Dirac operator on quaternionic K¨ahler manifolds. In the present article we show that the only manifolds in the limit case, i. e. the only manifolds where the lower bound is attained as an eigenvalue, are the quaternionic projective spaces. We use the equivalent formulation in terms of the quaternionic Killing equation introduced in [16] and show that a nontrivial solution defines a parallel spinor on the associated hyperk¨ahler manifold. Contents 1 2 3 4 5 6 6.1 6.2 7 A B

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Semiquaternionic Vector Spaces and Representations . . . . . . . . . . . . . . 329 Principal Bundles on Quaternionic K¨ahler Manifolds . . . . . . . . . . . . . . . 332 c . . . . . . . . . . . . . . . . . . . . . . . . 335 The Levi–Civita Connection on S and M Quaternionic Killing Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 c and Application to Spinors . . . . . . . . . . . . . . . . . . . 342 The Geometry of M c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 The Hyperk¨ahler structure of M Reinterpretation of the quaternionic Killing equation . . . . . . . . . . . . . . . 344 Proof of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 Spinors on Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 The Curvature Tensor of M

1. Introduction The square of the first eigenvalue of the Dirac operator on the sphere S n of scalar n . In [6] Friedrich showed that this eigenvalue is a universal lower curvature κ is κ4 n−1 ? ??

Supported by the SFB 256 “Nichtlineare partielle Differentialgleichungen”. Supported by the Max-Planck-Institut f¨ur Mathematik.

328

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bound for all eigenvalues on an arbitrary compact Riemannian spin manifold (M n , g) with positive scalar curvature κ in the following sense: all eigenvalues of the Dirac operator satisfy minM κ n . λ2 ≥ 4 n−1 An eigenspinor realizing this lower bound is characterized by a special differential equation called the Killing equation. Conversely, on manifolds admitting Killing spinors, i. e. nontrivial solutions of this Killing equation, the lower bound is realized as an eigenvalue. These manifolds have been characterized by C. B¨ar [4] translating the Killing c = R+ ×t2 M . The equation on M into the equation of a parallel spinor on the cone M existence problem of solutions of the Killing equation was thus reduced to the description of manifolds with parallel spinors by M. Wang [24]. Despite the fact that Friedrich’s estimate is sharp, it is not optimal if M is assumed to have additional geometric structure, namely special holonomy. Due to a result of O. Hijazi [8], there is no solution of the Killing equation if M possesses a non-trivial parallel k-form, k 6= 0, n. There are two canonical classes of such manifolds which in addition have positive scalar curvature: K¨ahler manifolds and quaternionic K¨ahler manifolds. The eigenvalue estimate for K¨ahler manifolds has been improved by K.-D. Kirchberg in [12] and [13] (see also [17] and [9]). Again, this estimate is sharp: the lower bound is attained as first eigenvalue on the complex projective space CP m resp. on its product with a flat 2-torus in odd resp. even complex dimensions. On K¨ahler manifolds of odd complex dimension, a spinor with smallest possible eigenvalue is characterized by a suitable modification of the Killing equation, the K¨ahlerian Killing equation. A. Moroianu showed in [19] that a K¨ahlerian Killing spinor defines an ordinary Killing spinor on the canonical S 1 -bundle over M . Hence, the holonomy argument of B¨ar’s work can be used to study the limit case. In even complex dimensions, the problem is more difficult and could only recently be solved by A. Moroianu [20]. A quaternionic K¨ahler manifold is an oriented 4n-dimensional Riemannian manifold with n ≥ 2 whose holonomy group is contained in the subgroup Sp(1)·Sp(n) ⊂ SO(4n). Equivalently they are characterized by the existence of a certain parallel 4-form , the so-called fundamental or Kraines form (cf. [5, 14]). All quaternionic K¨ahler manifolds are Einstein with constant scalar curvature (cf. [2] or [11]) and possess a unique spin structure if n is even, whereas for odd n only the quaternionic projective spaces are spin (cf. [21]). In [15] we proved that on a compact quaternionic K¨ahler spin manifold (M 4n , g) of positive scalar curvature κ the eigenvalues λ of the Dirac operator satisfy λ2 ≥

κ n+3 . 4 n+2

As in the Riemannian or K¨ahler case, this estimate is sharp, and the lower bound is attained as first eigenvalue on the quaternionic projective space (cf. [18]). n+3 The natural task is to study the limit case and to find all manifolds which have κ4 n+2 2 in the spectrum of D . A first step in this direction was taken in [16], where we introduced the equation characterizing an eigenspinor with this particular eigenvalue. A new feature of this quaternionic Killing equation is that it involves not only the eigenspinor but also an auxiliary section of an additional bundle, which is not itself a spinor. We used it to show that no compact symmetric quaternionic K¨ahler manifold besides the quaternionic projective spaces carries quaternionic Killing spinors. In the present article we prove the following more general

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Theorem 1. Let M be a compact quaternionic K¨ahler manifold of quaternionic dimension n and positive scalar curvature κ. If there is an eigenspinor for the Dirac operator with eigenvalue λ satisfying κn+3 , λ2 = 4n+2 then M is isometric to the quaternionic projective space. For the proof we follow the approach of C. B¨ar and A. Moroianu. We consider the canonical SO(3)-bundle S associated with any quaternionic K¨ahler manifold. Introducing an c := R+ ×t2 S has a natural appropriate metric on the total space S the warped product M hyperk¨ahler metric. Reformulating the Killing equation in terms of equivariant functions on the Sp(1) · Sp(n)-frame bundle of the quaternionic K¨ahler manifold we show that a c. quaternionic Killing spinor induces a Killing spinor on S and a parallel spinor on M c The result of M. Wang then implies that the hyperk¨ahler manifold M has to be locally isometric to Hn+1 forcing M to be isometric to the quaternionic projective space. We would like to thank A. Swann for several hints and comments and W. Ballmann for encouragement and support.

2. Semiquaternionic Vector Spaces and Representations The tangent space of a quaternionic K¨ahler manifold is not a priori a quaternionic left vector space because in general the three local complex structures are not defined globally. This ambivalence gives rise to the weaker notion of a semiquaternionic structure on a real vector space: Definition. A semiquaternionic structure on a real vector space T is a subalgebra Q ⊂ EndT with idT ∈ Q and Q ∼ = H as R-algebras. It is said to be adapted to a euclidean scalar product h, i on T if h qt1 , t2 i = h t1 , qt2 i for all t1 , t2 ∈ T and q ∈ Q, where q denotes conjugation defined by Q = R ⊕ Im Q := R idT ⊕[Q, Q]. Thus, choosing an isomorphism from H to Q makes T a quaternionic left vector space, however no particular isomorphism is preferred. Accordingly, the notion of quaternionic linear map has to be refined: Definition. An R-linear map f : T → T 0 between vector spaces T , T 0 with semiquaternionic structures Q, Q0 is semilinear, if there exists an isomorphism of R-algebras f Q : Q → Q0 such that f (qt) = f Q (q)f (t) for all t ∈ T and q ∈ Q. If f is semilinear and not identically zero f Q is uniquely defined, because idT 0 ∈ Q0 and Q0 ∼ = H implies that every non-zero endomorphism in Q0 is invertible. If T is a euclidean vector space with an adapted semiquaternionic structure, then the group of all semilinear isometries of T is isomorphic to Sp(1) · Sp(n) := Sp(1) × Sp(n)/Z2 . Choosing a particular isomorphism makes T a true representation of Sp(1) · Sp(n) and any two representations T , T 0 defined this way are intertwined by a semilinear isometry, which is unique up to sign. For this reason we will call any such T the defining representation of Sp(1) · Sp(n) with a choice of isomorphism tacitly understood. It turns

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out that the defining representation T comes along with a preferred isomorphism H → Q given by the infinitesimal action of i, j, k ∈ sp(1) ∼ = Im H on T . Similarly, one may construct the defining representation of the group Sp(n) of unitary quaternionic n × n-matrices. If E is a complex vector space of dimension 2n endowed with a symplectic form σE and an adapted positive quaternionic structure J, i. e. σE ( Je1 , Je2 ) = σE ( e1 , e2 ) for all e1 , e2 ∈ E and σE ( e, Je ) > 0 for all 0 6= e ∈ E, then the group of all C-linear symplectic transformations of E commuting with J is isomorphic to Sp(n). Choosing a particular isomorphism makes E a true Sp(n)representation and any two representations E, E 0 defined this way are intertwined by a C-linear symplectic map preserving the quaternionic structure, which is unique up to sign. For this reason we will call any such E the defining representation of Sp(n). Note that the Lie algebra of all infinitesimal symplectic transformations of E is canonically isomorphic to Sym2 E with e1 e2 acting as the endomorphism σE (e1 , ·)e2 + σE (e2 , ·)e1 , and elements of Sym2 E commute with J if and only if they are real with respect to the real structure Sym2 J. Hence, the defining representation comes along with a canonical real Sp(n)-equivariant isomorphism C ⊗R sp(n) → Sym2 E of Lie algebras. We will denote the defining representation of Sp(1) by H. There are several possibilities to give explicit realizations of the representations introduced above. In calculations and proofs below we will use the following standard picture, differing somewhat from Salamon’s conventions (cf. [21]). Consider the space of row vectors Hn over the quaternions with complex and quaternionic structure given by multiplication with i and j from the left. The group of unitary quaternionic matrices Sp(n) := {A ∈ Mn,n H : AH A = 1} acts on Hn from the left by multiplying with AH from the right. Thus, it commutes with the complex and quaternionic structure and preserves the linear form σHn ( v1 , v2 ) := [ v1 v2H j ]C , where [q]C := 21 (q − iqi) ∈ C is the C-part of q ∈ H. The C-part is obviously C-bilinear, i. e. [xqy]C = xy[q]C for all x, y ∈ C ⊂ H, and satisfies [q]C = [q]C = [−jqj]C . Using these properties it is easily checked that σHn is indeed C-bilinear symplectic and that the quaternionic structure is adapted and positive. In this way Hn becomes the defining representation of Sp(n). With a slight modification of the construction above we can make Hn the defining representation of Sp(1) · Sp(n), too. The scalar multiplication with q ∈ H from the left on the row vectors in Hn determines a subspace Q ⊂ EndHn , which obviously is a semiquaternionic structure adapted to the standard scalar product on Hn given by h v1 , v2 i := Rev1 v2H = ReσHn ( v1 , jv2 ). The group Sp(1) · Sp(n) = {z · A : z ∈ Sp(1) and A ∈ Sp(n)} acts on Hn from the left through semilinear isometries by (z · A)v := zvAH . An important point is particularly obvious in this standard picture and in consequence true for every defining representation of Sp(1) · Sp(n). The infinitesimal action of i, j, k ∈ sp(1) ⊂ sp(1) ⊕ sp(n) on T defines a canonical Sp(1) · Sp(n)-equivariant isomorphism of algebras H → Q making T a quaternionic left vector space. Thus, the natural homomorphism Sp(1) · Sp(n) → AutQ, z · A 7→ (z · A)Q is trivial on the subgroup Sp(n) and descends to an isomorphism SO(3) := Sp(1)/Z2 ∼ = AutQ on the complementary subgroup Sp(1) sending z · 1 ∈ Sp(1) to z Q := (z · 1)Q .

First Eigenvalue of Dirac Operator on Quaternionic K¨ahler Manifolds

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This observation allows a construction which becomes fundamental for quaternionic K¨ahler geometry once it is “gauged”. Let T be the defining representation of Sp(1)·Sp(n) and T 0 an arbitrary euclidean vector space with an adapted semiquaternionic structure Q0 . The representations of Sp(1) · Sp(n) on T and SO(3) on Q define simply transitive right group actions on P = {f : T → T 0 semilinear isometry} and

S = {f Q : Q → Q0 isomorphism of algebras}

such that the natural projection f 7→ f Q is Sp(1)-equivariant. Fixing a base point in P to identify P with Sp(1) · Sp(n) we get the following diagram: z·A P A A A A A A A A U A AU 0 Q 1 z S {T } The quaternionic structure on T can be used to construct two important relations between the defining representations H, E and T of Sp(1), Sp(n) and Sp(1) · Sp(n): Lemma 2.1. Let T be the defining representation of Sp(1) · Sp(n). Define the complex and quaternionic structure on T by the infinitesimal action of i, j (and k) in sp(1) ∼ = ImH. With the C-bilinear symplectic form σT ( t1 , t2 ) = h jt1 , t2 i + ih kt1 , t2 i the vector space T becomes the defining representation of Sp(n). Conversely, the complex and quaternionic structure of the defining representation E of Sp(n) generate a subalgebra Q in the R-linear endomorphisms of E, which is a semiquaternionic structure adapted to the scalar product h ·, · i := Re σE ( ·, J· ) making E the defining representation of Sp(1) · Sp(n). In particular, there is up to sign a unique Sp(n)-equivariant, C-linear symplectic isomorphism 9 : T → E preserving the quaternionic structure. In the standard picture this isomorphism is simply the identity. This isomorphism has the disadvantage of spoiling the Sp(1)-action on T . Consequently, it is impossible to use it directly on quaternionic K¨ahler manifolds. Nevertheless, we may use it to define a family of Sp(n)-equivariant isomorphisms C ⊗R T → H ⊗C E depending on the choice of a canonical base p, q := Jp of H satisfying σH (p, q) = 1 by (2.1) 8 : x ⊗R t 7−→ √12 xp ⊗C 9(t) + xq ⊗C J9(t) . All these isomorphisms are real with respect to the real structure J ⊗C J on H ⊗C E and isometries from h, i to σH ⊗C σE : σH ⊗ σE (8(1 ⊗R t1 ) , 8(1 ⊗R t2 )) = 21 σE (9(t1 ), J9(t2 )) − σE (J9(t1 ), 9(t2 )) = 21 h jt1 , jt2 i + ih kt1 , jt2 i + h t1 , t2 i − ih it1 , t2 i = h t1 , t2 i, since J9(t) = 9(jt) and σE (9(t1 ), 9(t2 )) = h jt1 , t2 i + ih kt1 , t2 i by construction. It turns out that there are exactly two canonical bases p, q for which the isomorphism above is not only Sp(n)- but already Sp(1) · Sp(n)-equivariant, thus defining a more fundamental isomorphism better suited for globalization:

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Lemma 2.2. Up to sign there is a unique real Sp(1) · Sp(n)-equivariant isomorphism of complex vector spaces 8 : C ⊗R T ∼ = H ⊗C E, which is an isometry from the C-bilinear extension of h, i to σH ⊗ σE . Proof. It is sufficient to prove this lemma in the standard picture as it translates immediately to arbitrary realizations of the defining representations. In this picture the canonical base to choose is p = j and q = −1 (or p = −j and q = 1) leading to: 8 : C ⊗R Hn −→ x ⊗R v 7−→

H ⊗C Hn √1 (xj 2

⊗C v − x ⊗C jv)

8−1 : H ⊗C Hn −→ q ⊗C v 7−→

(2.2)

C ⊗R Hn √1 (1 2

⊗R qjv + i ⊗R qkv).

Note that 8−1 (iq ⊗C v) = 8−1 (q ⊗C iv) = i8−1 (q ⊗C v). Thus, 8−1 is well defined and C-linear. As Sp(1) · Sp(n)-equivariance of 8−1 is obvious and 8 is an isometry it remains to show that 8 and 8−1 are mutual inverses: (8−1 ◦ 8)(x ⊗R v) = 21 1 ⊗R (−jxjv + xv) + i ⊗R (−jxijv + xiv) = 21 1 ⊗R 2 Re(x)v + i ⊗R 2 Re(−xi)v = x ⊗R v. A direct calculation of 8 ◦ 8−1 = id is more tedious, but can be done.

3. Principal Bundles on Quaternionic K¨ahler Manifolds “Gauging” the pointwise constructions of the previous section leads to the definitions of the basic objects of quaternionic K¨ahler geometry. However, in strict analogy with K¨ahler geometry one has to impose an additional integrability condition: Definition. A quaternionic K¨ahler manifold is a Riemannian manifold M of dimension 4n, n ≥ 2 with an adapted semiquaternionic structure Qx M ⊂ EndTx M on every tangent space which is respected by the Levi–Civita connection of M : ∇0(Q) ⊂ 0(T ∗ M ⊗ Q). Thus, parallel transport of endomorphisms along arbitrary curves γ induces isomorphisms of R-algebras Qγ(0) M → Qγ(τ ) M , and a fortiori parallel transport of tangent vectors defines semilinear isometries Tγ(0) M → Tγ(τ ) M . In particular, the Levi–Civita connection is tangent to the reduction of the frame bundle to the principal Sp(1) · Sp(n)bundle of semilinear orthogonal frames P := {f : T → Tx M semilinear isometry}

(3.1)

with projection πM : P → M, f 7→ x. Additionally, P projects Sp(1)-equivariantly to S := {f Q : Q → Qx M isomorphism of algebras} Q

(3.2)

via πS : P → S, f 7→ f , which is in turn a principal SO(3)-bundle over M with projection π : S → M, f Q 7→ x. In this way P may be considered as a principal Sp(1) · Sp(n)-bundle over M or as a principal Sp(n)-bundle over S:

First Eigenvalue of Dirac Operator on Quaternionic K¨ahler Manifolds

333

f : T → Tx M A A πS A πM A A π -AUx f Q : Q → Qx M

P A

A πS A πM A AU π -M S

The tangent bundle T M of M is canonically isomorphic to the bundle associated to P by the defining representation T sending the class [f, t] ∈ P ×Sp(1)·Sp(n) T to f (t). Alternatively, this isomorphism can be expressed by the soldering form θM := f −1 ◦ (πM )∗ ∈ 0(T ∗ P ⊗ T ) of M . Considering the fundamental isomorphism 8 : C ⊗R T → H ⊗ E one might try to define vector bundles H and E from the defining representations H and E of Sp(1) and Sp(n). In this generality, this is only possible on the quaternionic projective spaces, because only for these manifolds the bundle P can be covered by a principal Sp(1) × Sp(n)-bundle. Nevertheless, representations of Sp(1) × Sp(n) contained in some H ⊗p ⊗ E ⊗q with p + q even descend to Sp(1) · Sp(n) defining vector bundles on M associated to P . In this way M carries a multitude of naturally defined vector bundles besides bundles constructed out of the tangent bundle. The usefulness of these vector bundles has been shown by Salamon [21] (see also [15]). In particular, the bundle H ⊗ E is globally defined and canonically isomorphic to T M C H⊗E as expressed e. g. by the soldering form θM := 8 ◦ θM ∈ 0(T ∗ P ⊗ (H ⊗ E)). By definition, the Levi–Civita connection is tangent to P and determined by a connection 1-form ωM with values in the direct sum sp(1) ⊕ sp(n); accordingly, ωM splits sp(1) sp(n) ⊕ ωM . Additionally, the Levi–Civita connection defines a connection into ωM on the principal bundle S. Recall that horizontal lifts X h ∈ Tf0 P of tangent vectors d |0 xτ ∈ Tx0 M can be represented by curves of semilinear orthogonal frames X = dτ ∇ |0 fτ = 0. Likewise the connection on S is defined fτ : T → Txτ M over xτ satisfying dτ h by representing horizontal lifts X ∈ Tf Q S by curves fτQ : Q → Qxτ M of algebra 0

∇ isomorphisms over xτ satisfying dτ |0 fτQ = 0. More succinctly, the connection 1-form d |0 fτQ is given by ω on S for arbitrary tangent vectors dτ

ω

d Q Q −1 ∇ Q fτ := f0 f . dτ 0 dτ 0 τ

Using the Leibniz rule for the covariant derivative along curves, it is immediately seen d d |0 fτ on P to horizontal vectors dτ |0 fτQ on S, that πS projects horizontal vectors dτ because for arbitrary q ∈ Q, t ∈ T , ∇ ∇ ∇ Q Q fτ (qt) = fτ (q) f0 (t) + f0 (q) fτ (t), dτ 0 dτ 0 dτ 0 ∇ ∇ |0 fτ = 0 implies dτ |0 fτQ = 0. Since the projection πS : P → S is and consequently dτ Sp(1)-equivariant, and vectors tangent to the Sp(n)-action on P are surely vertical with respect to the projection πS we conclude:

Lemma 3.1.

sp(1) . πS∗ ω = ωM

Remarkably, the curvature of this connection on S depends only on the scalar curvature κ of M . In fact, according to the classification of sp(1) ⊕ sp(n)-curvature tensors

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due to Alekseevskii (cf. [1, 21]) the curvature tensor of a quaternionic K¨ahler manifold can be expressed in terms of the scalar curvature κ and a section R of Sym4 E∗ . We have κ RH + RE + Rhyper , (3.3) R=− 8n(n + 2) where RH ,RE and Rhyper are Sym2 H- or Sym2 E-valued 2-forms defined on sections of H⊗E∼ = T M C: RhH1 ⊗e1 ,h2 ⊗e2 = σE (e1 , e2 )h1 · h2 ∈ Sym2 H, RhE1 ⊗e1 ,h2 ⊗e2 = σH (h1 , h2 )e1 · e2 ∈ Sym2 E,

(3.4)

= σH (h1 , h2 )R(e1 , e2 , ·, ·) ∈ Sym2 E∗ ∼ Rhhyper = Sym2 E, 1 ⊗e1 ,h2 ⊗e2 acting as endomorphisms on H⊗E. Analyzing these terms leads to the following formula for the pull-back of the curvature 2-form of the connection ω to P : Lemma 3.2. The curvature 2-form of the connection ω on S pulled back to P is given by κ h θM ∧ iθM ii + h θM ∧ jθM ij + h θM ∧ kθM ik , πS∗ = 16n(n + 2) where by definition h θM ∧ iθM i(X, Y ) := 2h θM (X), iθM (Y ) i. Proof. Instead of calculating the curvature on S directly we will calculate the curvature of the bundle Sym2 H, which can be associated to S or P inheriting the same connection sp(1) . Its curvature considered as an sp(1)-valued 2-form on P is thus due to πS∗ ω = ωM πS∗ . Obviously only RH acts non-trivially on Sym2 H and neglecting for a moment that it is defined for sections of vector bundles we may consider it as a real Sp(1) · Sp(n)equivariant morphism ∼ =

RH

∼ =

C ⊗R 32 T −→ 32 (H ⊗ E) −→ Sym2 H −→ C ⊗R sp(1), where the first isomorphism is the extension of C ⊗R T ∼ = H ⊗ E and the second is the canonical isomorphism C ⊗R sp(1) ∼ = Sym2 H, which makes H the defining representation of Sp(1). To make this isomorphism explicit in the standard picture we choose the canonical base j, −1 of H with σH (j, −1) = 1 and find for the infinitesimal action of i, j and k ∈ Im H: i :1 7→ −i = i(−1) j 7→ k = i(j)

j :1 7→ −j j 7→ 1

k :1 7→ −k = i(−j) j 7→ −i = i(−1).

Hence, the isomorphism C ⊗R sp(1) ∼ = Sym2 H maps i to i(1 j), j to 21 (j 2 + 12 ) and k to i 2 2 H 2 (j − 1 ). Accordingly, the morphism R reads in the standard picture H H H H H = 21 Rj⊗v − R1⊗jv − Rj⊗v + R1⊗jv R1⊗ 1 ,j⊗v2 1 ,j⊗v2 1 ,1⊗jv2 1 ,1⊗jv2 R v1 ,1⊗R v2 = 21 σHn (v1 , v2 )j 2 − σHn (jv1 , v2 )1 j − σHn (v1 , jv2 )1 j + σHn (jv1 , jv2 )12 = Re σHn (v1 , v2 ) · 21 (j 2 + 12 ) + Im σHn (v1 , v2 ) · 2i (j 2 − 12 ) − Im σHn (v1 , jv2 ) · i(1 j) = −h v1 , jv2 ij − h v1 , kv2 ik − h v1 , iv2 ii

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335

for v1 , v2 ∈ Hn . Being Sp(1)·Sp(n)-equivariant RH can be made a C⊗R (P ×Ad sp(1)) ∼ = Sym2 H-valued 2-form in a straightforward way and becomes the RH defined above. κ RH can be thought of as the horizontal Sp(1) · Sp(n)-equivariant Alternatively, − 8n(n+2) sp(1)-valued curvature form πS∗ on P and is given by the stated formula. The additional factor 21 comes from the definition of the wedge product. c 4. The Levi–Civita Connection on S and M Let π : S → M be the canonical SO(3)-bundle over M defined in (3.2), with connection form ω and Riemannian metric gS =

16n(n+2) κ

B(ω, ω) + π ∗ gM ,

where B is the standard metric on sp(1) ∼ = Im H, and κ denotes the scalar curvature of M . This metric is Einstein, and if we rescale the metrics of M and S so that κ = 16n(n + 2), then (S, gS ) has a natural Sasakian 3-structure. The structure group of S reduces to Sp(n), and we can embed the principal Sp(n)-bundle P into the frame bundle of S in such a way that the soldering form θS ∈ 0(T ∗ P ⊗ (sp(1) ⊕ T )) of S on P is given by θS =

q

16n(n+2) κ

sp(1) ωM ⊕ θM .

(4.1)

In this way the Riemannian metric of S is associated to the standard metric B ⊕ h, i on sp(1) ⊕ T . In terms of covariant derivatives the Levi–Civita connection of gS is easily computed, and we obtain: Lemma 4.1. Let U, V be vertical vector fields given as fundamental vector fields induced by elements of the Lie algebra sp(1). Further, let X h , Y h be the horizontal lifts of vector fields X, Y on M and {eν }ν=1,... ,4n a locally defined horizontal orthonormal frame on S. Then the only non-zero covariant derivatives are given by ∇SU V =

1 2

[U, V ],

∇SX h Y h = (∇X Y )h − 21 (X h , Y h ), P4n h ∇SX h U = ∇U X h = 21 ν=1 gS (X , eν ), U eν . Here and in the sequel we identify elements of sp(1) with their associated fundamental vector fields. The Levi–Civita connection of S is determined by an so(sp(1)⊕T )-valued 1-form ωS on the orthonormal frame bundle of S, but as the structure group reduces to Sp(n), it is sufficient to know its restriction to the Sp(n)-reduction P again denoted by ωS . Lemma 4.2. The connection form ωS on P can be written as sp(n) + ω S = ωM

1 2

q sp(1) κ ad ωM + 16n(n+2) i θM ∧ i + j θM ∧ j + k θM ∧ k .

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Proof. For the proof we identify vector fields on S with equivariant functions on P , i. e. 0(T S) ∼ = C ∞ (P, sp(1) ⊕ T )Sp(n) , b A 7→ A. With respect to this identification the covariant derivative translates as [ S B = dB( b A) e + ωS (A) e B, b ∇ A e denotes an arbitrary lift of A to a vector field on P . We will use this formula where A e B. b The definition of the soldering and Lemma 4.1 to compute all non-zero terms ωS (A) b is given by θS (A). e In particular, we form θS immediately implies that the function A have for a fundamental vector field U , e) = b = θS (U U

q

16n(n+2) κ

e) = πS∗ ω(U

q

16n(n+2) κ

ω(U ) =

q

16n(n+2) κ

U ∈ sp(1). (4.2)

Let X h , Y h denote the horizontal lifts of the vector fields X, Y on M . Because of fh = X. e Then, π ◦ πS = πM we can assume X S Y h = dY b (X) e + ωS (X) e Yb . ∇\ Xh

e as above, only the sp(n)-part ω sp(n) contributes. This Note that when ωM is applied to X M e projects onto a horizontal vector on S. Hence, using Lemma is due to the fact that X 4.1, Lemma 3.2 and Eq. (4.2) we find \ S Y )h − 1 (X \ h , Y h ) − dY e Yb = (∇ b (X) e ωS (X) X 2 q sp(n) e b κ e i θM (Ye )i i + hθM (X), e j θM (Ye )i j hθM (X), (X) Y − 16n(n+2) = ωM e k θM (Ye )i k + hθM (X), q sp(n) e b κ e Yb i i + hj θM (X), e Yb i j + hk θM (X), e Yb i k (X) Y + 16n(n+2) hi θM (X), = ωM q sp(n) e b κ e ∧ i + j θM (X) e ∧ j + k θM (X) e ∧ k Yb . i θM (X) (X) Y + 16n(n+2) = ωM e) = Let U, V be fundamental vector fields. Then Vb is a constant function and dVb (U e b U (V ) = 0. Hence, [ S V = ω (U e )Vb = ∇ S U

1 2

\ [U, V]=

q 1 2

16n(n+2) κ

b , Vb ] = [U

1 2

sp(1) e b ad ωM (U ) V .

Let U be fundamental, X h a horizontal lift of X and {eν }ν=1,... ,4n a horizontal orb (X) e = X( e U b ) = 0 and we thonormal frame on S, with Eν = θS (eeν ) = ebν . Then dU obtain

First Eigenvalue of Dirac Operator on Quaternionic K¨ahler Manifolds

\ S U = e U b=∇ ωS (X) Xh =

1 16n(n+2) 2 κ

P4n

1 2

P4n

337

gS (X h , eν ), U Eν sp(1) sp(1) e ) Eν B ωM (U (X h , eν ) , ωM

ν=1

P4n

ν=1

e i Eν i B(i, U ) Eν + hθM (X), e j Eν i B(j, U ) Eν hθM (X), e k Eν i B(k, U ) Eν + hθM (X), q κ e ∧ i + j θM (X) e ∧ j + k θM (X) e ∧k U b. i θM (X) = 16n(n+2) =

ν=1

Finally, we compute \ S X h − dX( e) X b =∇ b U e ) = 1 P4n gS (X h , eν ), U Eν − dX( b U e) ω S (U U ν=1 2 E P4n D b Eν Eν − dX( b U e) = − ν=1 i B(i, U ) + j B(j, U ) + k B(k, U ) X, sp(1) e b b U e ). (U ) X − dX( = − ωM

At this point one has to remember the equivariance of the soldering form. By identifying the fundamental vector field U with the corresponding element in the Lie algebra sp(1) we have d b d sp(1) e tU b b b U e )| = ) = − = −ωM (U ) · X(p), X(p · e dX( etU · X(p) p∈P dt t=0 dt t=0 e) X b = 0. Combining these three calculations we end up with the and we conclude ωS (U stated formula for ωS : q sp(n) sp(1) κ + 21 ad ωM + 16n(n+2) i θM ∧ i + j θM ∧ j + k θM ∧ k . ω S = ωM c over S, i. e. the warped Besides the manifold S we also need to consider the cone M + c := R ×t2 S with metric product M gb =

16n(n+2) κ

dt2 + t2 gS .

c reduces to Sp(n), and we can embed the principal Sp(n)The structure group of M + c bundle PM b := R × P into the bundle of orthonormal frames of M in such a way that the R ⊕ sp(1) ⊕ T -valued soldering form on PM b is given by θM b =

q

16n(n+2) κ

(−dt) ⊕ t θS =

q

16n(n+2) κ

sp(1) − dt ⊕ t ωM ⊕ t θM .

q κ This convention makes the inward pointing vector field 4 := − 16n(n+2)

∂ ∂t

correspond

to 1 := θM b (4) ∈ R. This may be surprising at first but it turns out that only this orientation c introduced later. With this choice of is compatible with a hyperk¨ahler structure on M soldering form the Riemannian metric gb is associated to the standard metric h, i⊕B⊕h, i on R ⊕ sp(1) ⊕ T .

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c Lemma 4.3. The restriction ωM b of the Levi–Civita connection of M to the reduction of the bundle of orthonormal frames reads: PM b ωM b = ωS +

q

κ 16n(n+2)

θS ∧ 1 sp(1) sp(1) ad ωM ∧1 + ωM

sp(n) + 21 = ωM q κ θM ∧ 1 + i θM ∧ i + j θM ∧ j + k θM ∧ k . + 16n(n+2)

In particular, the connection form ωM b is the pull-back of a well defined 1-form on P . b Proof. The proof is similar to the proof of the corresponding formula for ωS . Let ∇ denote the covariant derivative for the Levi–Civita connection of gb. The only nonvanishing terms are b X Y = ∇SX Y + ∇

q

κ 1 16n(n+2) t

gb(X, Y ) 4

and

q κ b X4 = ∇ b 4X = − ∇ 16n(n+2)

1 t

X.

Using the same notation as in the proof of Lemma 4.2 we obtain e b e b ωM b (X)Y = ωS (X)Y + e Yb + = ωS (X)

q q

1 κ 16n(n+2) t κ 16n(n+2)

e e b hθM b (X), θM b (Y )i 4

e Yb i 1. hθS (X),

Having this expression for horizontal vector fields X, Y we immediately derive ωM b = ωS +

q

κ 16n(n+2)

θ S ∧ 1 = ωS +

q

κ 16n(n+2)

sp(1) θ M ∧ 1 + ωM ∧ 1.

b of An interesting application of the formulas given above relates the curvature tensor R hyper c of the curvature tensor of (M, g). the manifold (M , gb) to the hyperk¨ahler part R The proof will be given in Appendix B. b is horizontal with respect to π c → M . Its Proposition 4.4. The curvature tensor R b:M only non-vanishing terms are b π∗ X, π b∗ Y ). R(X, Y ) = Rhyper (b c)⊥ of the vertical tangent The right-hand side acts on the orthogonal complement (T V M bundle, which is canonical isomorphic to π b∗ T M . c has been previously studied by A. Swann using the notation U(M ) The manifold M c as the Z2 -quotient of the total space of the locally defined bundle [22]. He constructs M H with zero section removed. The metric gb is a member of the family of hyperk¨ahler c introduced in [22]. In particular, the proposition above is implicit in his metrics on M work (see also [23]).

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5. Quaternionic Killing Spinors In this chapter we recall the quaternionic Killing equation introduced in [16]. It will provide us with an equivalent formulation of the limit case. First, we have to collect some facts for the spinor bundle and Clifford multiplication on a quaternionic K¨ahler manifold (cf. [15]). The spinor bundle of a 4n-dimensional quaternionic K¨ahler spin manifold decomposes into a sum of n + 1 subbundles, which can be expressed using the locally defined bundles E and H (cf. [3, 10] or [24]). For this we have to introduce the bundles 3s◦ E. They are associated to the irreducible Sp(n)-representations on the spaces 3s◦ E of primitive s-vectors, which are the kernels of the contraction σE y : 3s E −→ 3s−2 E with the symplectic form σE . With this notation the spinor bundle can be written S(M ) =

n M

Sr (M ) :=

r=0

n M

Symr H ⊗ 3n−r E. ◦

(5.1)

r=0

In order to define the Clifford multiplication we have to fix notations for modified contraction and multiplication on Symr H and 3s◦ E. Contraction preserves the primitive spaces, i. e. if η is in 3s◦ E then e] y η ∈ 3◦s−1 E, where e] := σE (e, ·) ∈ E ∗ denotes the dual of e ∈ E. However, this is not true for the wedge product and the projection e ∧◦ η of e ∧ η onto 3s+1 ◦ E is given by e ∧◦ η = e ∧ η −

1 n−s+1

LE ∧ (e] y η) ,

where LE is the canonical bivector associated to σE under the isomorphism 32 E ∼ = 32 E ∗ . Let h· denote the symmetric product with h ∈ H, and for h] := σH (h, ·) ∈ H ∗ we define h] y◦ : Symr H → Symr−1 H by h] y◦ := r1 h] y. Let h ⊗ e ∈ H ⊗ E = T M C be a tangent vector. Then, the Clifford multiplication µ(h ⊗ e) : S(M ) → S(M ) is given by √ 2(h · ⊗ e] y + h] y◦ ⊗ e ∧◦ ) . (5.2) µ(h ⊗ e) = In particular, it maps the subbundle Sr (M ) to the sum Sr−1 (M ) ⊕ Sr+1 (M ) and thus splits into two components T M ⊗ Sr (M ) −→ Sr+1 (M ) and µ− T M ⊗ Sr (M ) −→ Sr−1 (M ) , + : √ √ 2 (h] y◦ ⊗ e ∧◦ ). There are two with µ+− (e ⊗ h) = 2 (h · ⊗e] y) and µ− + (e ⊗ h) = operations defined similar to Clifford multiplication µ+− :

µ++ : T M ⊗ Symr H ⊗ 3s◦ E −→ Symr+1 H ⊗ 3s+1 ◦ E √ h⊗e⊗ψ 7−→ 2 (h · ⊗ e ∧◦ )ψ and

r r−1 s H ⊗ 3◦s−1 E µ− − : T M ⊗ Sym H ⊗ 3◦ E −→ Sym √ h⊗e⊗ψ 7−→ 2 (h] y◦ ⊗ e] y )ψ .

+ + D+− with Using these notations the Dirac operator D can be written D = D− + := µ+− ◦ ∇ : Sr (M ) −→ Sr+1 (M ) D−

D+− := µ− + ◦ ∇ : Sr (M ) −→ Sr−1 (M ) .

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The square of the Dirac operator respects the splitting of the spinor bundle, i. e. D2 : + + D− = 0 = D+− D+− . Sr (M ) −→ Sr (M ) and we have D− A quaternionic Killing spinor is by definition (cf. [16]) a section ψ = (ψ0 , ψ1 , ψ− ) of the Killing bundle E∼ E, SKilling (M ) := S0 (M ) ⊕ S1 (M ) ⊕ 3n−2 = 3n◦ E ⊕ (H ⊗ 3◦n−1 E) ⊕ 3n−2 ◦ ◦ satisfying the following quaternionic Killing equation for some parameter λ 6= 0 and all tangent vectors X: ∇ X ψ0 =

−

λ n+3

µ− + (X) ψ1

λ ∇X ψ1 = − 4n µ+− (X) ψ0

∇X ψ− =

+ −

λ 4n

3λ 2(n+3)

µ++ (X) ψ−

µ− − (X) ψ1 .

We remark that if (ψ0 , ψ1 , ψ− ) is a solution for parameter λ, then (ψ0 , −ψ1 , ψ− ) is a solution for parameter −λ. In [16] we showed that for any solution ψ 6= 0 the spinor ψ0 + ψ1 is an eigenspinor for the minimal eigenvalue q n+3 . λ = ± κ4 n+2 In particular, only for two values of the parameter λ 6= 0 there can possibly exist nontrivial solutions. Conversely, any eigenspinor for the minimal eigenvalue is of the form ψ0 + ψ1 ∈ 0(S0 (M ) ⊕ S1 (M )), and the augmented eigenspinor (ψ0 , ψ1 , ψ− ) with ψ− := − n−2 1 n+3 E) is a solution of the quaternionic Killing equation. 4λ n+4 (µ− ◦ ∇)ψ1 ∈ 0(3◦ Obviously, solutions are sections parallel with respect to a modified connection. Its curvature is precisely the hyperk¨ahler part Rhyper of the curvature of M . Since this part vanishes on the quaternionic projective space the Killing bundle of HP n is trivialized by augmented eigenspinors with minimal eigenvalue. Proposition 4.4 then motivates to c. lift a quaternionic Killing spinor to a parallel spinor on M For our purpose of characterizing the limit case it is more convenient to consider an equivalent version of the quaternionic Killing equation. Let ψ = (ψ0 , ψ1 , ψ− ) be a solution of the original equation, then the scaled section q q scal n+3 4n )= ψ , ψ , − ψ ψ scal := (ψ0scal , ψ1scal , ψ− 0 1 − 4n n+3 is a solution of the equation q κ AX ψ scal , ∇X ψ scal = − 16n(n+2) where ψ scal is considered as column vector with three entries and AX denotes the matrix   0 µ− 0 + (X)   + 3 +  0 AX =  2 µ+ (X)  .  µ− (X) 0

− µ− − (X)

0

For the remainder of this article, the index denoting the scaling will be omitted. The following lemma shows that AX can be interpreted as part of an sp(n + 1)-action.

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341

Lemma 5.1. Let F = H ⊕ E be the defining representation of Sp(n + 1) with symplectic form σF = σH + σE . Restricted to the subgroup Sp(1) × Sp(n) the Sp(n + 1)representation 3s◦ F decomposes into 3s◦ F ∼ = 3s◦ E ⊕ (H ⊗ 3◦s−1 E) ⊕ 3◦s−2 E, which descends to a well defined representation of Sp(1) · Sp(n) if s is even. Explicitly, the isomorphism ι : 3s◦ E ⊕ (H ⊗ 3◦s−1 E) ⊕ 3◦s−2 E −→ 3s◦ F is given by 1 LE ) ∧ φ− . ι φ0 ⊕ (h ⊗ φ1 ) ⊕ φ− = φ0 + (h ∧ φ1 ) + (LH − n−s+2 Similarly, Sym2 F ∼ = C ⊗R sp(n + 1) decomposes into Sym2 H ⊕ (H ⊗ E) ⊕ Sym2 E. For s = n, the subspace H ⊗ E ⊂ Sym2 F acts on 3n◦ F via (h ⊗ e) φ = √12 Ah⊗e φ. Proof. It is clear that ι is an injective map to 3s F . It remains to show that its image is already contained in 3s◦ F . Since σF = σH + σE the statement follows from σF y (LH −

1 n−s+2

LE ) ∧ φ− = [σH y, LH ∧] φ− −

1 n−s+2

[σE y, LE ∧] φ− = 0,

where we used the relations [σH y, LH ∧] φ− = φ− and [σE y, LE ∧] φ− = (n − s + 2) φ− . Comparing dimensions shows that ι is in addition surjective, hence it defines an isomorphism. The action of an element f1 f2 ∈ Sym2 F on F is given by (f1 f2 )(f ) = σF (f1 , f )f2 + σF (f2 , f ) f1 . It extends as derivation to 3∗◦ F and can be explicitly written as (f1 f2 )(ω) = (f2 ∧ f1] y + f1 ∧ f2] y)(ω). Hence, for h ∈ H and e ∈ E considered as elements of F , the element h ⊗ e ∈ H ⊗ E is identified with h e ∈ Sym2 F , and the action on ι(φ) = φ0 ⊕ (a ∧ φ1 ) ⊕ (LH − 21 LE ) ∧ φ− ∈ 3n◦ F is given by (h ⊗ e) ι(φ0 ) = h ∧ e] yφ0 =

√1 2

ι µ+− (h ⊗ e)φ0 ,

(h ⊗ e) ι(a ⊗ φ1 ) = (h ∧ e] y + e ∧ h] y)(a ∧ φ1 ) = −h ∧ a ∧ e] y φ1 + σH (h, a) e ∧ φ1 = −σH (h, a) LH − 21 LE ∧ e] y φ1 + σH (h, a) e ∧◦ φ1 − = √12 ι µ− (h ⊗ e) (a ⊗ φ ) − µ (h ⊗ e) (a ⊗ φ ) , 1 1 + − and

(h ⊗ e) ι(φ− ) = − 21 h ∧ e] y(LE ∧ φ− ) + e ∧ h] y(LH ∧ φ− ) =

1 2

h ∧ e ∧ φ− − 21 h ∧ LE ∧ (e] y φ− ) − e ∧ h ∧ φ−

=

3 2

h ∧ e ∧◦ φ−

=

3 √ 2 2

ι µ++ (h ⊗ e)φ− .

Hence, we see that operation of (h ⊗ e) on ι(φ) is just application of the matrix to the column vector φ.

√1 2

Ah⊗e

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W. Kramer, U. Semmelmann, G. Weingart

To stress the origin of the operation √12 AX from a group action, we introduce ? : H ⊗ E ⊗ 3n◦ F → 3n◦ F for the infinitesimal action of H ⊗ E on 3n◦ F . With this notation the quaternionic Killing equation reads ∇X ψ = −

q

κ 16n(n+2)

AX ψ = −

q

κ 8n(n+2)

8(X) ? ψ,

(5.3)

where 8 is the isomorphism defined in Lemma 2.2. c and Application to Spinors 6. The Geometry of M The aim of this section is to show that the quaternionic Killing equation, considered as a differential equation on equivariant functions on P can be interpreted in three different ways: first, of course, when we think of its solutions as sections of the Killing-bundle SKilling (M ) on M they are quaternionic Killing spinors. Interpreted as a section of the spinor bundle on S, the solutions are Killing spinors, and finally solutions pulled back + c to PM b = R × P are parallel sections of the spinor bundle of M . c. First we recall that the structure group of both 6.1. The Hyperk¨ahler structure of M c reduce to Sp(n), i. e. the tangent bundles may be associated to the principal S and M Sp(n)-bundles P and PM b through the Sp(n)-representations sp(1)⊕T and R⊕sp(1)⊕T respectively. However, as Sp(n)-representation T can be identified with E according to Lemma 2.1 reflecting the isomorphism π∗ T M ∼ = P ×Sp(n) T ∼ = P ×Sp(n) E of vector bundles on S. More important, this identification is respected by the Levi–Civita sp(1) sp(n) ⊕ωM is connection, because the infinitesimal sp(1)-action on T present in ωM = ωM canceled in the connection form ωS of S. For this reason we will consequently identify T with E and consider E as a euclidean vector space with scalar product h ·, · i = Re σE (·, J·). In the same vein we combine the obvious identification R ⊕ sp(1) ∼ = R ⊕ Im H = H with the isomorphism H → H of defining representations of Sp(1) unique up to sign to get an isometry from the standard metric on R ⊕ sp(1) to H with scalar product Re σH (·, J·) sending 1, i, j, k to 1, I, J, K ∈ H. In this way we get an isometry R ⊕ sp(1) ⊕ T −→ F with the defining representation F = H ⊕ E of Sp(n + 1) considered as a euclidean vector space with scalar product Re σF (·, J·). As this isometry is Sp(n)-equivariant by c and S are associated to the Sp(n)-representations construction the tangent bundles of M F and {1}⊥ =: (Im H) ⊕ E ⊂ F . c reduces to Sp(n), the holonomy of M c does not. Though the structure group of M c Nevertheless, we will show that it is a subgroup of Sp(n + 1), i. e. M is hyperk¨ahler. For c this purpose we group the summands of the connection form ωM b of M given in Lemma 4.3 as follows:

First Eigenvalue of Dirac Operator on Quaternionic K¨ahler Manifolds

343

sp(1) sp(1) ω sp(1) := 21 ad(ωM ) + ωM ∧ 1, b M sp(n) , ω sp(n) := ωM b M q n κ ω H := 16n(n+2) θM ∧ 1 + iθM ∧ I + jθM ∧ J + kθM ∧ K . b M sp(1) + ω sp(n) + ω H . Thus, the Levi–Civita connection can be written ωM b = ωM b b b M M n

sp(1) are the same, i. e. Lemma 6.1. The actions of ω sp(1) and ωM b M q sp(1) sp(n) κ = ω + ω + ωM M M 16n(n+2) θM ∧ 1 + iθM ∧ I + jθM ∧ J + kθM ∧ K . b

(6.1)

In particular, the connection form ωM b takes values in sp(n + 1). Proof. Note that for q ∈ H and imaginary z ∈ Im H we have 21 (zq + qz) = (Re q)z − h q, z i = −(z ∧ 1)q. Using this algebraic identity the infinitesimal sp(1)-action on H in the standard picture can be written as −qz =

1 2

ad(z)q + (z ∧ 1)q.

Hence, a particular merit of the identifications above is that the two summands sp(1) sp(1) 1 c acting on R ⊕ sp(1) ∧ 1 of the Levi–Civita connection of M 2 ad(ωM ) and ωM sp(1) combine into the infinitesimal action of ωM on H. Consequently, the summands ω sp(1) b M and ω sp(n) take values in sp(n + 1), i. e. in the infinitesimal quaternionic linear isometries b M n of F . The same is true for ω H because of its H-linearity. b M c is hyWith the Levi–Civita connection being sp(n + 1)-valued the manifold M perk¨ahler, and we may use the description of the spinor bundle for the more general quaternionic K¨ahler manifolds given in (5.1). Consider a complex vector space C2 endowed with a symplectic form σC2 and a positive quaternionic structure J. Choosing an isomorphism to the group of all symplectic transformations of C2 commuting with J would make C2 the defining representation of Sp(1). However, on a hyperk¨ahler manifold this Sp(1)-symmetry is not a local “gauged” symmetry, but a purely global one. c plays the role of H on M . Accordingly, the For this reason the trivial C2 -bundle on M c is associated to the Sp(n)-representation spinor bundle of M 6 =

n+1 M r=0

6r =

n+1 M

Symr C2 ⊗ 3n+1−r F, ◦

r=0

where Sp(n) operates trivially on H ⊂ F . The Clifford multiplication with complex tangent vectors in C2 ⊗ F is then given by the formula (5.2). To describe the Clifford multiplication with real tangent vectors however, we have to choose an isomorphism among the family of isometries 8 : C ⊗R F → C2 ⊗ F defined in Eq. (2.1). For quaternionic K¨ahler manifolds this isometry is uniquely fixed by the additional local Sp(1)-symmetry up to sign, but this is no longer true in the hyperk¨ahler case. In fact, we get a family of Clifford multiplications depending on the choice of 8, i. e. of a canonical base p, q of C2 satisfying Jp = q and σC2 ( p, q ) = 1. All these are intertwined by the

344

W. Kramer, U. Semmelmann, G. Weingart

global Sp(1)-symmetry acting on C2 . In this sense, to define the Clifford multiplication with real tangent vectors f ∈ F , we first have to apply the isomorphism (6.2) 8 : 1 ⊗R f 7−→ √12 p ⊗ f − q ⊗ Jf . Note that this isomorphism points out the ambivalence of the vector bundle associated to the defining representation F of Sp(n+1). Normally, this vector bundle is the real tangent bundle, but in the description of the spinor bundle it plays a role strictly analogous to the isotropic subspace T 0,1 M of the complexified tangent bundle on K¨ahler manifolds. To describe the spinor bundle on S we recall that the spinor module of Spin(4n + 4) decomposes into the two half-spin modules 6± . When restricted to Sp(n + 1) the representations 6± decompose further into a sum of certain 6r . Since Clifford multiplication maps 6r to 6r−1 ⊕ 6r+1 and interchanges 6+ with 6− , we conclude 6+ =

n+1 M r=0 r≡1 (2)

Symr C2 ⊗ 3n+1−r F, ◦

6− =

n+1 M

Symr C2 ⊗ 3n+1−r F, ◦

r=0 r≡0 (2)

at least if n is even. Due to Lemma 5.1 6+ and 6− are equivalent as Sp(n)-representations, and the spinor module of S is associated to the Sp(n)-principal bundle P through either one, e. g. 6+ . The Clifford multiplication with f ∈ {1}⊥ ⊂ F is then given by f · := (f ∧ 1)· = f · 1·. S

6.2. Reinterpretation of the quaternionic Killing equation. In this section we will translate the quaternionic Killing equation (5.3) on M into the equation for a parallel spinor c. Let ψ be a quaternionic Killing spinor on M , which we will consider as an on M equivariant function on P , i. e. ψ ∈ C ∞ (P, 3n◦ F )Sp(n) . Then the quaternionic Killing equation reads q H⊗E κ θM ? ψ = 0, (6.3) d + ωM + 8n(n+2) H⊗E c) of the spinor bundle of := 8 ◦ θM . The subbundle S1 (M where by definition θM n 2 c M is associated to PM b by the representation C ⊗ 3◦ F . To construct a section of this bundle, we proceed as follows: we lift ψ to PM b by extending it constantly along the R+ -direction and choose an arbitrary constant vector ξ ∈ C2 . Then, the function ξ ⊗ ψ c). As function on P it satisfies defines a section in S1 (M b M q H⊗E κ d + ωM + 8n(n+2) id ⊗θM ? (ξ ⊗ ψ) = 0,

where the connection and soldering form are considered to be pulled back to PM b. Obviously, they do not act on ξ but only on ψ. Using the crucial Lemma 6.2 below we q n H⊗E κ id ⊗θM ? by ω H , so that the resulting equation on PM can replace 8n(n+2) b reads b M n sp(1) sp(n) + ωM + ω Hb (ξ ⊗ ψ) = d + ωM d + ωM b (ξ ⊗ ψ) = 0. M c. Hence, ξ ⊗ ψ defines a parallel spinor on M We remark that the parallel spinor ξ ⊗ ψ also gives rise to a Killing spinor on S. This follows of course from the general equivalence between Killing spinors on a Riemannian

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345

manifold and parallel spinors on its cone, as described by C. B¨ar (cf. [4]) and briefly summarized in Appendix A. Nevertheless, we will include this construction since it is an immediate consequence of our approach. We have seen above that ξ ⊗ ψ, considered as function on PM b )(ξ ⊗ ψ) = 0 . If we substitute the expression ωM b b , satisfies (d + ωM given in Lemma 4.3 we obtain q κ d + ωS + 16n(n+2) θS ∧ 1 (ξ ⊗ ψ) = 0. Interpreted as equation on P this is just the Killing equation, i.e. q κ θS · (ξ ⊗ ψ) . (d + ωS ) (ξ ⊗ ψ) = − 21 16n(n+2) S

1 2

The additional factor is due to the action of the orthogonal Lie algebra on the spinor bundle. Before closing this section we formulate the lemma needed above. q

Lemma 6.2.

κ 8n(n+2)

n

H⊗E id ⊗θM ? = ω Hb . M

The proof needs an additional proposition which is analogous to Proposition 2.3 in [15]. Proposition 6.3. Let p, q be a base of C2 ∼ = H with σ( p, q ) = 1 and f1 , f2 ∈ F . Then we have the following identity of operators on the spinor bundle: (p ⊗ f1 ) ∧ (q ⊗ f2 ) − (q ⊗ f1 ) ∧ (p ⊗ f2 ) = id ⊗f1 · f2 . As an element of so(4n + 4), the left hand side acts on the spinor module via the isomorphism so(4n + 4) ∼ = spin(4n + 4) sending e1 ∧ e2 to 21 ( e1 e2 + he1 , e2 i ). We remark that the proof of these two technical propositions amounts to proving the decomposition (5.1) of the spinor bundle. With the help of this proposition it is easy to prove Lemma 6.2. Proof. Let p, q be the canonical base of C2 used to define Clifford multiplication. We can then extend the isomorphism C⊗R F ∼ = C2 ⊗F in Eq. (6.2) to C⊗R 32 F ∼ = 32 (C2 ⊗F ) to get (θM ∧ 1) = 21 (p ⊗ θM + q ⊗ JθM ) ∧ (p ⊗ 1 + q ⊗ J), (iθM ∧ I) = − 21 (p ⊗ θM − q ⊗ JθM ) ∧ (p ⊗ 1 − q ⊗ J), (jθM ∧ J) =

1 2

(p ⊗ JθM − q ⊗ θM ) ∧ (p ⊗ J − q ⊗ 1),

(kθM ∧ K) = − 21 (p ⊗ JθM + q ⊗ θM ) ∧ (p ⊗ J + q ⊗ 1). In the second and fourth line we use that J is conjugate linear. By summing up the four equations, some terms cancel, and we obtain q n κ (p ⊗ θM ) ∧ (q ⊗ J) − (q ⊗ θM ) ∧ (p ⊗ J) ω H = 16n(n+2) b M − (p ⊗ JθM ) ∧ (q ⊗ 1) + (q ⊗ JθM ) ∧ (p ⊗ 1) . Applying the proposition above yields the following equivalence q n κ (id ⊗θM · J − id ⊗JθM · 1) ω H = 16n(n+2) b M q q H⊗E κ κ id ⊗(J ⊗ θM − 1 ⊗ JθM )? = ?. = 16n(n+2) 8n(n+2) id ⊗θM

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7. Proof of the Theorem The fact that a quaternionic Killing spinor on M translates into a parallel spinor on the c is crucial to the proof of the main theorem. hyperk¨ahler manifold M Theorem 1. Let M be a compact quaternionic K¨ahler manifold of quaternionic dimension n and positive scalar curvature κ > 0. If there is an eigenspinor for the Dirac operator with eigenvalue λ satisfying λ2 =

κn+3 , 4n+2

then M is isometric to the quaternionic projective space. Proof. After the work done in the preceding sections the proof of this theorem reduces c is to a simple holonomy argument. The spinor bundle of the hyperk¨ahler manifold M associated to the Sp(n)-representation 6=

n+1 M r=0

6r =

n+1 M

Symr C2 ⊗ 3n+1−r F, ◦

r=0

c is contained in where F = H ⊕E, and Sp(n) operates trivially on H. The holonomy of M r 2 c c) associated Sp(n+1) and operates trivially on Sym C . The subbundle Sn+1 (M ) of S(M n+1 2 n+1 2 to 6n+1 = Sym C is consequently trivialized by n + 2 = dim (Sym C ) linearly independent parallel spinors. c, If M admits a quaternionic Killing spinor, there are additional parallel spinors on M n 2 c) associated to the representation C ⊗ 3 F . which are sections of the subbundle S1 (M ◦ Due to a result of Wang [24], on a manifold with holonomy equal to Sp(n + 1) there c). are exactly n + 2 linearly independent parallel spinors, just those trivializing Sn+1 (M The additional parallel spinor constructed out of a quaternionic Killing spinor reduces c is reducible the holonomy further. According to Berger’s list this can only happen if M c or locally symmetric. In the first case, as consequence of a theorem of Gallot [7], M has to be flat. But so it is in the second case, because it is hyperk¨ahler, hence Ricci-flat, c is flat which forces M to be isometric and in addition locally symmetric. Therefore M to the quaternionic projective space. A. Spinors on Cones In this appendix we will describe how to lift spinors on a Riemannian spin manifold N b := R+ × N . In particular, we will show that Killing spinors on to spinors on its cone N b . This construction is originally due to C. B¨ar [4]. N translate into parallel spinors on N b = R+ × N → N be Let (N, gN ) be a spin manifold of dimension n and let π : N the cone over N endowed with the warped product metric gNb = dt2 + e2λ π ∗ gN . The soldering and connection form on the principal bundle P := PSpin(n) N associated to the chosen spin structure are denoted by θN and ωN . Obviously, the cone is again spin and the spin structure reduces to the principal Spin(n)-bundle Pb := R+ × P . Forms on P give rise to forms on Pb by extending them constantly along the R+ -direction. It easy to b on Pb are given by see that the soldering resp. the connection form of N

First Eigenvalue of Dirac Operator on Quaternionic K¨ahler Manifolds

θNb = dt + eλ θN

resp.

347

∂λ θN ∧ 4, ωb = ωN − e λ N ∂t

∂ denotes the vertical unit vector. where 4 = ∂t A spinor ψ on N can be interpreted as a Spin(n)-equivariant function on P with values in the spinor module 6n with a fixed Clifford module structure. With the canonical isomorphism Cln ∼ = Cl0n+1 , ei 7→ ei · 4 in mind, we can consider 6n as an Cl0n+1 - and therefore as an Spin(n + 1)-representation. The values of ψ have now to be interpreted as lying in the Spin(n + 1)-representation. Let ψ be a Killing spinor on N . Interpreted as function on P , it satisfies:

dψ + (ωN − µθN ) ψ = 0, where µ is the Killing constant. Extending ψ constantly in R+ -direction and using the b . As a function on Pb, isomorphism of the Clifford algebras above defines a spinor on N it satisfies dψ + (ωN − µπ ∗ θN 4) ψ = 0. If the warping function is chosen such that µ = 21 eλ ∂λ ∂t , the expression in brackets is b . The other way round, it is also equal to ωNb , and therefore ψ has to be parallel on N b , then its associated function on Pb is constant in clear that, if ψ is a parallel spinor on N R+ -direction and it projects onto a Killing spinor on N . c B. The Curvature Tensor of M c to the In this appendix we will prove Proposition 4.4 relating the curvature tensor of M hyperk¨ahler part Rhyper of the curvature of M . We recall that the connection form of c restricted to P is pulled back from P and so is its curvature . We have seen in M b b M M Sect. 6 that the most convenient way to read the connection form ωM b given in Lemma 4.3 is n sp(1) + ω sp(n) + ω Hb , ωM b = ωM M b b M sp(1) sp(n) and ωM respectively, and where ω sp(1) and ω sp(n) are the pull-backs of ωM b b M M q n κ θM ∧ 1 + iθM ∧ I + jθM ∧ J + kθM ∧ K . ω Hb := 16n(n+2) M

Defining [α ∧ β]( X, Y ) := [α(X), β(Y )] − [α(Y ), β(X)] = [β ∧ α]( X, Y ) for Lie algebra valued 1-forms α, β, the curvature 2-form M of M on P can be written M = dωM + 21 ωM ∧ ωM sp(1) sp(n) sp(1) sp(1) sp(n) sp(n) + 21 ωM ∧ ωM + 21 ωM ∧ ωM + dωM . = dωM sp(1) sp(n) ∧ ωM ] = 0 since sp(1) and sp(n) centralize each other in sp(1) ⊕ Of course, [ωM sp(1) = πS∗ ω we conclude sp(n). Using the naturality of the exterior differential and ωM κ ∗ RH in the that the first summand is equal to πS . Thus, it corresponds to − 8n(n+2) sense of decomposition (3.3) as shown in the proof of Lemma 3.2. We conclude that the κ c RE + Rhyper . The curvature M second summand corresponds to − 8n(n+2) b of M can be

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calculated similarly. With sp(1) and sp(n) centralizing each other in sp(n + 1) we still have [ω sp(1) ∧ ω sp(n) ] = 0, and M b is the sum of the three terms: b b M M n n sp(1) dω sp(1) + 21 ω sp(1) ∧ ω sp(1) + 21 ω H ∧ ω H , b b b b b M M M M M n n n sp(1) sp(n) ω ∧ ωH + ω ∧ ωH , dω H + b b b b b M M M M M n n sp(n) , dω sp(n)+ 21 ω sp(n) ∧ ω sp(n) + 21 ω H ∧ ω H b b b b b M M M M M n

(B.1)

n

where [ω H ∧ ω H ] is projected onto its two components in sp(1) and sp(n) according b b M M to the Cartan decomposition sp(n + 1) = (sp(1) ⊕ sp(n)) ⊕ Hn . Using the formula [ a1 ∧ b1 , a2 ∧ b2 ] = ha1 , a2 ib1 ∧ b2 − hb1 , a2 ia1 ∧ b2 − ha1 , b2 ib1 ∧ a2 + hb1 , b2 ia1 ∧ a2 , we find Hn n ω b ∧ ω Hb M M κ θM ∧ θM + iθM ∧ iθM + jθM ∧ jθM + kθM ∧ kθM = 32n(n+2) 1 2

+2hθM ∧ iθM i1 ∧ I + 2hθM ∧ jθM i1 ∧ J + 2hθM ∧ kθM i1 ∧ K

+2hiθM ∧ jθM iI ∧ J + 2hiθM ∧ kθM iI ∧ K + 2hjθM ∧ kθM iJ ∧ K κ θM ∧ θM + iθM ∧ iθM + jθM ∧ jθM + kθM ∧ kθM (B.2) = 32n(n+2) κ (B.3) hθM ∧iθM i(1∧I−J∧K) + hθM ∧jθM i(1∧J−K∧I) + 16n(n+2) + hθM ∧kθM i(1∧K−I∧J) . We remark that by construction of the base 1, I, J and K the infinitesimal action of i, j and k ∈ sp(1) on H can be written i :1 7→ −I J 7→ K I 7→ 1 K → 7 −J

j :1 7→ −J K 7→ I J 7→ 1 I 7→ −K

k :1 7→ −K I 7→ J , K 7→ 1 J → 7 −I

i. e. i corresponds to −(1 ∧ I − J ∧ K). Thus, the summand (B.3) is equal to −πS∗ κ RH . Without proof we state that the sumaccording to Lemma 3.2, i. e. equal to 8n(n+2) κ E mand (B.2) is equal to 8n(n+2) R as this is certainly true on the quaternionic projective space. Consequently, in the decomposition (B.1) of M b the first term vanishes as does the second, because straightforward calculations show that it depends linearly on the torsion of the Levi–Civita connection ωM of M . Hence, the curvature M b reduces to sp(n) + M b = dωM b

1 2

sp(n) ∧ ω sp(n) + ω b b M M

E κ 8n(n+2) R

= Rhyper .

In this way M b ×Sp(n) E of the tangent bundle of b operates only on the subbundle PM c M . We remark that this subbundle is canonically isomorphic to PM b ×Sp(n) T , i. e. to π b∗ T M .

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References 1. Alekseevskii, D.V.: Riemannian spaces with exceptional holonomy groups. Funkt. Anal. Appl. 2, 97– 105(1968) 2. Alekseevskii, D.V.: Compact quaternion spaces. Funkt. Anal. Appl. 2, 106–114 (1968) 3. Barker, R. and Salamon, S.: Analysis on a generalized Heisenberg group. J. London Math Soc. 28, 184–192 (1983) 4. B¨ar, Ch.: Real Killing Spinors and Holonomy. Commun. Math. Phys. 154, 509–521 (1993) 5. Bonan, E.: Sur les G-structures de type quaternionien. Cahiers Top. et Geom. Diff. 9, 389–461 (1967) 6. Friedrich, Th.: Der erste Eigenwert des Dirac-Operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkr¨ummung: Math. Nachr. 97, 117–146 (1980) 7. Gallot, S.: Equations diff´erentielles charact´eristiques de la sph`ere: Ann. Sc. Ec. Norm. Sup. 12, 235–267 (1979) 8. Hijazi, O.: Op´erateurs de Dirac sur les vari´et´es riemanniennes: Minoration des valeurs propres. Th`ese ´ Polytechnique, 1984 de 3e cycle, Ecole 9. Hijazi, O.: Eigenvalues of the Dirac operator on compact K¨ahler manifolds. Commun. Math. Phys. 160, 563–579 (1994) 10. Hijazi, O., Milhorat, J.-L.: D´ecomposition du fibr´e des spineurs d’une vari´et´e spin K¨ahler-quaternionienne sous l’action de la 4-forme fondametale. J. Geom. Phys. 15, 320–332 (1995) 11. Ishihara, S.: Quaternionic K¨ahlerian manifolds, J. Diff. Geom. 9, 483–500 (1974) 12. Kirchberg, K.-D.: An estimation for the first eigenvalue of the Dirac operator on closed K¨ahler manifolds of positive scalar curvature. Ann. Global Anal. Geom. 4, 291–325 (1986) 13. Kirchberg, K.-D.: The first eigenvalue of the Dirac operator on K¨ahler manifolds. J. Geom. Phys. 7, 449–468 (1990) 14. Kraines, V.Y.: Topology of quaternionic manifolds. Trans. Am. Math. Soc. 122, 357–367 (1966) 15. Kramer, W., Semmelmann, U. and Weingart, G.: Eigenvalue Estimates for the Dirac Operator on Quaternionic K¨ahler Manifolds. SFB 256, preprint 507, Bonn 16. Kramer, W., Semmelmann, U. and Weingart, G.: Quaternionic Killing Spinors. Ann. Global Anal. Geom. 16, 63–87 (1998) 17. Lichnerowicz, A.: La premi`ere valeur propre de l’op´erateur de Dirac pour une vari´et´es K¨ahl´erienne et son cas limite. C. R. Acad. Sci. Paris 311, 717–722 (1990) 18. Milhorat, J.-L.: Spectre de l’op´erateur de Dirac sur les espaces projectifs quaternioniens. C. R. Acad. Sci. Paris 314, 69–72 (1992) 19. Moroianu, A.: La premi`ere valeur propre de l’op´erateur de Dirac sur les vari´et´es k¨ahl´eriennes compactes. Commun. Math. Phys. 169, 373–384 (1995) 20. Moroianu, A.: K¨ahler Manifolds with Small Eigenvalues of the Dirac Operator and a Conjecture of Lichnerowicz. Pr´epublications du Centre de Math. Ec. Polytechnique (1998) 98–4 21. Salamon, S. M.: Quaternionic K¨ahler manifolds. Invent. Math. 67, 143–171 (1982) 22. Swann, A.: HyperK¨ahler and quaternionic K¨ahler geometry. Math. Ann. 289, 421–450 (1991) 23. Pedersen, H., Poon, Y. S. and Swann, A. F.: Hypercomplex structures associated to quaternionic manifolds. Preprint 97/04, University of Bath, 1997 24. Wang, M.Y.: Parallel spinors and parallel forms. Anal. Global Anal. Geom. 7, 59–68 (1989) Communicated by A. Connes

Commun. Math. Phys. 199, 351 – 395 (1998)

Communications in

Mathematical Physics © Springer-Verlag 1998

Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs Jerrold E. Marsden1 , George W. Patrick2 , Steve Shkoller3 1

Control and Dynamical Systems, California Institute of Technology, 107-81, Pasadena, CA 91125, USA. E-mail: [email protected] 2 Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, SK S7N5E6, Canada. E-mail: [email protected] 3 Center for Nonlinear Studies, MS-B258, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. E-mail: [email protected] Received: 12 January 1998 / Accepted: 12 May 1998

Abstract: This paper presents a geometric-variational approach to continuous and discrete mechanics and field theories. Using multisymplectic geometry, we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle. In particular, we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function, and use this structure to prove covariant generalizations of conservation of symplecticity and Noether’s theorem. Natural discretization schemes for PDEs, which have these important preservation properties, then follow by choosing a discrete action functional. In the case of mechanics, we recover the variational symplectic integrators of Veselov type, while for PDEs we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries. We show that the usual notion of symplecticity along an infinite-dimensional space of fields can be naturally obtained by making a spacetime split. All of the aspects of our method are demonstrated with a nonlinear sine-Gordon equation, including computational results and a comparison with other discretization schemes. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 2 Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 3 Veselov Discretizations of Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 4 Variational Principles for Classical Field Theory . . . . . . . . . . . . . . . . . . 361 5 Veselov-type Discretizations of Multisymplectic Field Theory . . . . . . . 375 5.1 General theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 5.2 Numerical checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

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1. Introduction The purpose of this paper is to develop the geometric foundations for multisymplectic-momentum integrators for variational partial differential equations (PDEs). These integrators are the PDE generalizations of symplectic integrators that are popular for Hamiltonian ODEs (see, for example, the articles in Marsden, Patrick and Shadwick [1996], and especially the review article of McLachlan and Scovel [1996]) in that they are covariant spacetime integrators which preserve the geometric structures of the system. Because of the covariance of our method which we shall describe below, the resulting integrators are spacetime localizable in the context of hyperbolic PDEs, and generalize the notion of symplecticity and symmetry preservation in the context of elliptic problems. Herein, we shall primarily focus on spacetime integrators; however, we shall remark on the connection of our method with the finite element method for elliptic problems, as well as the Gregory and Lin [1991] method in optimal control. Historically, in the setting of ODEs, there have been many approaches devised for constructing symplectic integrators, beginning with the original derivations based on generating functions (see de Vogelaere [1956]) and proceeding to symplectic RungeKutta algorithms, the shake algorithm, and many others. In fact, in many areas of molecular dynamics, symplectic integrators such as the Verlet algorithm and variants thereof are quite popular, as are symplectic integrators for the integration of the solar system. In these domains, integrators that are either symplectic or which are adaptations of symplectic integrators, are amongst the most widely used. A fundamentally new approach to symplectic integration is that of Veselov [1988], [1991] who developed a discrete mechanics based on a discretization of Hamilton’s principle. This method leads in a natural way to symplectic-momentum integrators which include the shake and Verlet integrators as special cases (see Wendlandt and Marsden [1997]). In addition, Veselov integrators often have amazing properties with regard to preservation of integrable structures, as has been shown by Moser and Veselov [1991]. This aspect has yet to be exploited numerically, but it seems to be quite important. The approach we take in this paper is to develop a Veselov-type discretization for PDE’s in variational form. The relevant geometry for this situation is multisymplectic geometry (see Gotay, Isenberg, and Marsden [1997] and Marsden and Shkoller [1998]) and we develop it in a variational framework. As we have mentioned, this naturally leads to multisymplectic-momentum integrators. It is well-known that such integrators cannot in general preserve the Hamiltonian exactly (Ge and Marsden [1988]). However, these integrators have, under appropriate circumstances, very good energy performance in the sense of the conservation of a nearby Hamiltonian up to exponentially small errors, assuming small time steps, due to a result of Neishtadt [1984]. See also Dragt and Finn [1979], and Simo and Gonzales [1993]. This is related to backward error analysis; see Sanz-Serna and Calvo [1994], Calvo and Hairer [1995], and the recent work of Hyman, Newman and coworkers and references therein. It would be quite interesting to develop the links with Neishtadt’s analysis more thoroughly. An important part of our approach is to understand how the symplectic nature of the integrators is implied by the variational structure. In this way we are able to identify the symplectic and momentum conserving properties after discretizing the variational principle itself. Inspired by a paper of Wald [1993], we obtain a formal method for locating the symplectic or multisymplectic structures directly from the action function and its derivatives. We present the method in the context of ordinary Lagrangian mechanics, and apply it to discrete Lagrangian mechanics, and both continuous and discrete

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multisymplectic field theory. While in these contexts our variational method merely uncovers the well-known differential-geometric structures, our method forms an excellent pedagogical approach to those theories. Outline of paper. Section 2. In this section we sketch the three main aspects of our variational approach in the familiar context of particle mechanics. We show that the usual symplectic 2-form on the tangent bundle of the configuration manifold arises naturally as the boundary term in the first variational principle. We then show that application of d2 = 0 to the variational principle restricted to the space of solutions of the Euler–Lagrange equations produces the familiar concept of conservation of the symplectic form; this statement is obtained variationally in a non-dynamic context; that is, we do not require an evolutionary flow. We then show that if the action function is left invariant by a symmetry group, then Noether’s theorem follows directly and simply from the variational principle as well. Section 3. Here we use our variational approach to construct discretization schemes for mechanics which preserve the discrete symplectic form and the associated discrete momentum mappings. Section 4. This section defines the three aspects of our variational approach in the multisymplectic field-theoretic setting. Unlike the traditional approach of defining the canonical multisymplectic form on the dual of the first jet bundle and then pulling back to the Lagrangian side using the covariant Legendre transform, we obtain the geometric structure by staying entirely on the Lagrangian side. We prove the covariant analogue of the fact that the flow of conservative systems consists of symplectic maps; we call this result the multisymplectic form formula. After variationally proving a covariant version of Noether’s theorem, we show that one can use the multisymplectic form formula to recover the usual notion of symplecticity of the flow in an infinite-dimensional space of fields by making a spacetime split. We demonstrate this machinery using a nonlinear wave equation as an example. Section 5. In this section we develop discrete field theories from which the covariant integrators follow. We define discrete analogues of the first jet bundle of the configuration bundle whose sections are the fields of interest, and proceed to define the discrete action sum. We then apply our variational algorithm to this discrete action function to produce the discrete Euler–Lagrange equations and the discrete multisymplectic forms. As a consequence of our methodology, we show that the solutions of the discrete Euler– Lagrange equations satisfy the discrete version of the multisymplectic form formula as well as the discrete version of our generalized Noether’s theorem. Using our nonlinear wave equation example, we develop various multisymplectic-momentum integrators for the sine-Gordon equations, and compare our resulting numerical scheme with the energy-conserving methods of Li and Vu-Quoc [1995] and Guo, Pascual, Rodriguez, and Vazquez [1986]. Results are presented for long-time simulations of kink-antikink solutions for over 5000 soliton collisions. Section 6. This section contains some important remarks concerning the variational integrator methodology. For example, we discuss integrators for reduced systems, the role of grid uniformity, and the interesting connections with the finite-element methods for elliptic problems. We also make some comments on future work. 2. Lagrangian Mechanics Hamilton’s principle. We begin by recalling a problem going back to Euler, Lagrange and Hamilton in the period 1740–1830. Consider an n-dimensional configuration man-

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ifold Q with its tangent bundle T Q. We denote coordinates on Q by q i and those on T Q by (q i , q˙i ). Consider a Lagrangian L : T Q → R. Construct the corresponding action functional S on C 2 curves q(t) in Q by integration of L along the tangent to the curve. In coordinate notation, this reads Z b dq i i (t) dt. (2.1) L q (t), S q(t) ≡ dt a The action functional depends on a and b, but this is not explicit in the notation. Hamilton’s principle seeks the curves q(t) for which the functional S is stationary under variations of q(t) with fixed endpoints; namely, we seek curves q(t) which satisfy d S q (t) = 0 (2.2) dS q(t) · δq(t) ≡ d =0 for all δq(t) with δq(a) = δq(b) = 0, where q is a smooth family of curves with q0 = q and (d/d)|=0 q = δq. Using integration by parts, the calculation for this is simply Z b dqi d i (t) dt L q (t), dS q(t) · δq(t) = d =0 a dt b Z b d ∂L ∂L ∂L i δq i − δq . (2.3) = dt + ∂q i dt ∂ q˙i ∂ q˙i a a The last term in (2.3) vanishes since δq(a) = δq(b) = 0, so that the requirement (2.2) for S to be stationary yields the Euler–Lagrange equations d ∂L ∂L − = 0. ∂q i dt ∂ q˙i

(2.4)

Recall that L is called regular when the symmetric matrix [∂ 2 L/∂ q˙i ∂ q˙j ] is everywhere nonsingular. If L is regular, the Euler–Lagrange equations are second order ordinary differential equations for the required curves. The standard geometric setting. The action (2.1) is independent of the choice of coordinates, and thus the Euler–Lagrange equations are coordinate independent as well. Consequently, it is natural that the Euler–Lagrange equations may be intrinsically expressed using the language of differential geometry. This intrinsic development of mechanics is now standard, and can be seen, for example, in Arnold [1978], Abraham and Marsden [1978], and Marsden and Ratiu [1994]. The canonical 1-form θ0 on the 2n-dimensional cotangent bundle of Q, T ∗ Q is defined by θ0 (αq )wαq ≡ αq · T πQ wαq ,

αq ∈ Tq∗ Q, wαq ∈ Tαq T ∗ Q,

where πQ : T ∗ Q → Q is the canonical projection. The Lagrangian L intrinsically defines a fiber preserving bundle map FL : T Q → T ∗ Q, the Legendre transformation, by vertical differentiation: d L(vq + wq ). FL(vq )wq ≡ d =0

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We define the Lagrange 1-form on T Q, the Lagrangian side, by pull-back θL ≡ FL∗ θ0 , and the Lagrange 2-form by ωL = −dθL . We then seek a vector field XE (called the Lagrange vector field) on T Q such that XE ωL = dE, where the energy E is defined by E(vq ) ≡ FL(vq )vq − L(vq ). If FL is a local diffeomorphism then XE exists and is unique, and its integral curves solve the Euler–Lagrange equations (2.4). In addition, the flow Ft of XE preserves ωL ; that is, Ft∗ ωL = ωL . Such maps are symplectic, and the form ωL is called a symplectic 2form. This is an example of a symplectic manifold: a pair (M, ω) where M is a manifold and ω is closed nondegenerate 2-form. Despite the compactness and precision of this differential-geometric approach, it is difficult to motivate and, furthermore, is not entirely contained on the Lagrangian side. The canonical 1-form θ0 seems to appear from nowhere, as does the Legendre transform FL. Historically, after the Lagrangian picture on T Q was constructed, the canonical picture on T ∗ Q emerged through the work of Hamilton, but the modern approach described above treats the relation between the Hamiltonian and Lagrangian pictures of mechanics as a mathematical tautology, rather than what it is – a discovery of the highest order. The variational approach. More and more, one is finding that there are advantages to staying on the “Lagrangian side”. Many examples can be given, but the theory of Lagrangian reduction (the Euler–Poincar´e equations being an instance) is one example (see, for example, Marsden and Ratiu [1994] and Holm, Marsden and Ratiu [1998a,b]); another, of many, is the direct variational approach to questions in black hole dynamics given by Wald [1993]. In such studies, it is the variational principle that is the center of attention. We next show that one can derive in a natural way the fundamental differential geometric structures, including momentum mappings, directly from the variational approach. This development begins by removing the boundary condition δq(a) = δq(b) = 0 from (2.3). Eq. (2.3) becomes b Z b ∂L ∂L i d ∂L dt + δq i − δq , (2.5) dS q(t) · δq(t) = ∂q i dt ∂ q˙i ∂ q˙i a a where the left side now operates on more general δq (this generalization will be described in detail in Sect. 4), while the last term on the right side does not vanish. That last term of (2.5) is a linear pairing of the function ∂L/∂ q˙i , a function of q i and q˙i , with the tangent vector δq i . Thus, one may consider it to be a 1-form on T Q; namely the 1-form (∂L/∂ q˙i )dq i . This is exactly the Lagrange 1-form, and we can turn this into a formal theorem/definition: Theorem 2.1. Given a C k Lagrangian L, k ≥ 2, there exists a unique C k−2 mapping DEL L : Q¨ → T ∗ Q, defined on the second order submanifold 2 d q q a C 2 curve in Q ¨ (0) Q≡ dt2 of T T Q, and a unique C k−1 1-form θL on T Q, such that, for all C 2 variations q (t), b 2 Z b d q dq , ˆ · δq DEL L (2.6) · δq dt + θ dS q(t) · δq(t) = L dt2 dt a a where

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d δq(t) ≡ q (t), d =0

d d ˆ δq(t) ≡ q (t). d =0 dt t=0

The 1-form so defined is called the Lagrange 1-form. Indeed, uniqueness and local existence follow from the calculation (2.3), and the coordinate independence of the action, and then global existence is immediate. Here then, is the first aspect of our method: Using the variational principle, the Lagrange 1-form θL is the “boundary part” of the the functional derivative of the action when the boundary is varied. The analogue of the symplectic form is the (negative of) the exterior derivative of θL . For the mechanics example being discussed, we imagine a development wherein θL is so defined and we define ωL ≡ −dθL . Lagrangian flows are symplectic. One of Lagrange’s basic discoveries was that the solutions of the Euler–Lagrange equations give rise to a symplectic map. It is a curious twist of history that he did this without the machinery of either differential forms, of the Hamiltonian formalism or of Hamilton’s principle itself. (See Marsden and Ratiu [1994] for an account of some of this history.) Assuming that L is regular, the variational principle then gives coordinate independent second order ordinary differential equations, as we have noted. We temporarily denote the vector field on T Q so obtained by X, and its flow by Ft . Our further development relies on a change of viewpoint: we focus on the restriction of S to the subspace CL of solutions of the variational principle. The space CL may be identified with the initial conditions, elements of T Q, for the flow: to vq ∈ T Q, we associate the integral curve s 7→ Fs (vq ), s ∈ [0, t]. The value of S on that curve is denoted by St , and again called the action. Thus, we define the map St : T Q → R by Z t L(q(s), q(s)) ˙ ds, (2.7) St (vq ) = 0

where (q(s), q(s)) ˙ = Fs (vq ). The fundamental Eq. (2.6) becomes d Ft (vq ) − θL (vq ) · wvq , dSt (vq )wvq = θL Ft (vq ) · d =0 where 7→ vq is an arbitrary curve in T Q such that vq0 = vq and (d/d)|0 vq = wvq . We have thus derived the equation dSt = Ft∗ θL − θL .

(2.8)

Taking the exterior derivative of (2.8) yields the fundamental fact that the flow of X is symplectic: 0 = ddSt = d(Ft∗ θL − θL ) = −Ft∗ ωL + ωL , which is equivalent to

Ft∗ ωL = ωL .

This leads to the following: Using the variational principle, the fact that the evolution is symplectic is a consequence of the equation d2 = 0, applied to the action restricted to the space of solutions of the variational principle.

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In passing, we note that (2.8) also provides the differential-geometric equations for X. Indeed, one time derivative of (2.8) and using (2.7) gives dL = LX θL , so that X

ωL = −X

dθL = −LX θL + d(X

θL ) = d(X

θL − L) = dE,

if we define E ≡ X θL − L. Thus, we quite naturally find that X = XE . Of course, this set up also leads directly to Hamilton–Jacobi theory, which was one of the ways in which symplectic integrators were developed (see McLachlan and Scovel [1996] and references therein.) However, we shall not pursue the Hamilton–Jacobi aspect of the theory here. Momentum maps. Suppose that a Lie group G, with Lie algebra g, acts on Q, and hence on curves in Q, in such a way that the action S is invariant. Clearly, G leaves the set of solutions of the variational principle invariant, so the action of G restricts to CL , and the group action commutes with Ft . Denoting the infinitesimal generator of ξ ∈ g on T Q by ξT Q , we have by (2.8), 0 = ξT Q

dSt = ξT Q

(Ft∗ θL − θL ) = Ft∗ (ξT Q

θL ) − ξT Q

θL .

(2.9)

For ξ ∈ g, define Jξ : T Q → R by Jξ ≡ ξT Q θL . Then (2.9) says that Jξ is an integral of the flow of XE . We have arrived at a version of Noether’s theorem (rather close to the original derivation of Noether): Using the variational principle, Noether’s theorem results from the infinitesimal invariance of the action restricted to space of solutions of the variational principle. The conserved momentum associated to a Lie algebra element ξ is Jξ = ξT Q θL , where θL is the Lagrange one-form. Reformulation in terms of first variations. We have just seen that symplecticity of the flow and Noether’s theorem result from restricting the action to the space of solutions. One tacit assumption is that the space of solutions is a manifold in some appropriate sense. This is a potential problem, since solution spaces for field theories are known to have singularities (see, e.g., Arms, Marsden and Moncrief [1982]). More seriously there is the problem of finding a multisymplectic analogue of the statement that the Lagrangian flow map is symplectic, since for multisymplectic field theory one obtains an evolution picture only after splitting spacetime into space and time and adopting the “function space” point of view. Having the general formalism depend either on a spacetime split or an analysis of the associated Cauchy problem would be contrary to the general thrust of this article. We now give a formal argument, in the context of Lagrangian mechanics, which shows how both these problems can be simultaneously avoided. Given a solution q(t) ∈ CL , a first variation at q(t) is a vector field V on Q such that t 7→ FV ◦ q(t) is also a solution curve (i.e. a curve in CL ). We think of the solution space CL as being a (possibly) singular subset of the smooth space of all putative curves C in T Q, and the restriction of V to q(t) as being the derivative of some curve in CL at q(t). When CL is a manifold, a first variation is a vector at q(t) tangent to CL . Temporarily define α ≡ dS − θL , where by abuse of notation θL is the one form on C defined by θL q(t) δq(t) ≡ θL (b)δq(b) − θL (a)δq(a). Then CL is defined by α = 0 and we have the equation dS = α + θL , so if V and W are first variations at q(t), we obtain

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0=V

W

d2 S = V

W

dα + V

W

dθL .

(2.10)

We have the identity dα(V, W ) q(t) = V α(W ) − W α(V ) − α([V, W ]),

(2.11)

which we will use to evaluate (2.10) at the curve q(t). Let FV denote the flow of V , V V define q (t) ≡ F q(t) , and make similar definitions with W replacing V . For the first term of (2.11), we have d α(W )(qV ), V α(W ) q(t) = d =0 which vanishes, since α is zero along qV for every . Similarly the second term of (2.11) at q(t) also vanishes, while the third term vanishes since α q(t) = 0. Consequently, symplecticity of the Lagrangian flow Ft may be written V

W

dθL = 0,

for all first variations V and W . This formulation is valid whether or not the solution space is a manifold, and it does not explicitly refer to any temporal notion. Similarly, Noether’s theorem may be written in this way. Summarizing: Using the variational principle, the analogue of the evolution is symplectic is the equation d2 S = 0 restricted to first variations of the space of solutions of the variational principle. The analogue of Noether’s theorem is infinitesimal invariance of dS restricted to first variations of the space of solutions of the variational principle. The variational route to the differential-geometric formalism has obvious pedagogical advantages. More than that, however, it systematizes searching for the corresponding formalism in other contexts. We shall in the next sections show how this works in the context of discrete mechanics, classical field theory and multisymplectic geometry.

3. Veselov Discretizations of Mechanics The discrete Lagrangian formalism in Veselov [1988], [1991] fits nicely into our variational framework. Veselov uses Q × Q for the discrete version of the tangent bundle of a configuration space Q; heuristically, given some a priori choice of time interval 1t, a point (q1 , q0 ) ∈ Q × Q corresponds to the tangent vector (q1 − q0 )/1t. Define a discrete Lagrangian to be a smooth map L : Q × Q = {q1 , q0 } → R, and the corresponding action to be S≡

n X

L(qk , qk−1 ).

(3.1)

k=1

The variational principle is to extremize S for variations holding the endpoints q0 and qn fixed. This variational principle determines a “discrete flow” F : Q × Q → Q × Q by F (q1 , q0 ) = (q2 , q1 ), where q2 is found from the discrete Euler–Lagrange equations (DEL equations):

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∂L ∂L (q1 , q0 ) + (q2 , q1 ) = 0. ∂q1 ∂q0

(3.2)

In this section we work out the basic differential-geometric objects of this discrete mechanics directly from the variational point of view, consistent with our philosophy in the last section. A mathematically significant aspect of this theory is how it relates to integrable systems, a point taken up by Moser and Veselov [1991]. We will not explore this aspect in any detail in this paper, although later, we will briefly discuss the reduction process and we shall test an integrator for an integrable pde, the sine-Gordon equation. The Lagrange 1-form. We begin by calculating dS for variations that do not fix the endpoints: dS(q0 , · · · , qn ) · (δq0 , · · · , δqn ) n−1 X ∂L ∂L (qk+1 , qk )δqk+1 + (qk+1 , qk )δqk = ∂q1 ∂q0 k=0

n n−1 X X ∂L ∂L (qk , qk−1 )δqk + (qk+1 , qk )δqk = ∂q1 ∂q0 k=1 k=0 n−1 X ∂L ∂L = (qk , qk−1 ) + (qk+1 , qk ) δqk ∂q1 ∂q0 k=1

+

∂L ∂L (q1 , q0 )δq0 + (qn , qn−1 )δqn . ∂q0 ∂q1

(3.3)

It is the last two terms that arise from the boundary variations (i.e. these are the ones that are zero if the boundary is fixed), and so these are the terms amongst which we expect to find the discrete analogue of the Lagrange 1-form. Actually, interpretation of the boundary terms gives the two 1-forms on Q × Q − (q1 , q0 ) · (δq1 , δq0 ) ≡ θL

∂L (q1 , q0 )δq0 , ∂q0

(3.4)

+ (q1 , q0 ) · (δq1 , δq0 ) ≡ θL

∂L (q1 , q0 )δq1 , ∂q1

(3.5)

and

and we regard the pair (θ− , θ+ ) as being the analogue of the one form in this situation. Symplecticity of the flow. We parameterize the solutions of the variational principle by the initial conditions (q1 , q0 ), and restrict S to that solution space. Then Eq. (3.3) becomes − + + F ∗ θL . dS = θL

(3.6)

We should be able to obtain the symplecticity of F by determining what the equation ddS = 0 means for the right-hand-side of (3.6). At first, this does not appear to work, since ddS = 0 gives − + ) = −dθL , F ∗ (dθL

(3.7)

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which apparently says that F pulls a certain 2-form back to a different 2-form. The situation is aided by the observation that, from (3.4) and (3.5), − + + θL = dL, θL

(3.8)

− + + dθL = 0. dθL

(3.9)

and consequently,

Thus, there are two generally distinct 1-forms, but (up to sign) only one 2-form. If we make the definition − + = −dθL , ωL ≡ dθL

then (3.7) becomes F ∗ ωL = ωL . Eq. (3.4), in coordinates, gives ωL =

∂2L ∂q0i ∂q1j

dq0i ∧ dq1j ,

which agrees with the discrete symplectic form found in Veselov [1988], [1991]. Noether’s theorem. Suppose a Lie group G with Lie algebra g acts on Q, and hence diagonally on Q × Q, and that L is G-invariant. Clearly, S is also G-invariant and G sends critical points of S to themselves. Thus, the action of G restricts to the space of solutions, the map F is G-equivariant, and from (3.6), 0 = ξQ×Q

dS = ξQ×Q

− θL + ξQ×Q

+ (F ∗ θL ),

for ξ ∈ g, or equivalently, using the equivariance of F , ξQ×Q

− θL = −F ∗ (ξQ×Q

Since L is G-invariant, (3.8) gives ξQ×Q verts (3.10) to the conservation equation ξQ×Q

θ+ ).

(3.10)

− θL = −ξQ×Q

+ θL , which in turn con-

+ θL = F ∗ (ξQ×Q

θ+ ).

(3.11)

Defining the discrete momentum to be Jξ ≡ ξQ×Q

+ θL ,

we see that (3.11) becomes conservation of momentum. A virtually identical derivation of this discrete Noether theorem is found in Marsden and Wendlant [1997]. Reduction. As we mentioned above, this formalism lends itself to a discrete version of the theory of Lagrangian reduction (see Marsden and Scheurle [1993a,b], Holm, Marsden and Ratiu [1998a] and Cendra, Marsden and Ratiu [1998]). This theory is not the focus of this article, so we shall defer a brief discussion of it until the conclusions.

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4. Variational Principles for Classical Field Theory Multisymplectic geometry. We now review some aspects of multisymplectic geometry, following Gotay, Isenberg and Marsden [1997] and Marsden and Shkoller [1997]. We let πXY : Y → X be a fiber bundle over an oriented manifold X. Denote the first jet bundle over Y by J 1 (Y ) or J 1 Y and identify it with the affine bundle over Y −1 (x) consists of Aff(Tx X, Ty Y ), those linear mappings whose fiber over y ∈ Yx := πXY γ : Tx X → Ty Y satisfying T πXY ◦ γ = Identity on Tx X. We let dimX = n + 1 and the fiber dimension of Y be N . Coordinates on X are denoted xµ , µ = 1, 2, . . . , n, 0, and fiber coordinates on Y are denoted by y A , A = 1, . . . , N . These induce coordinates v A µ on the fibers of J 1 (Y ). If φ : X → Y is a section of πXY , its tangent map at x ∈ X, denoted Tx φ, is an element of J 1 (Y )φ(x) . Thus, the map x 7→ Tx φ defines a section of J 1 (Y ) regarded as a bundle over X. This section is denoted j 1 (φ) or j 1 φ and is called the first jet of φ. In coordinates, j 1 (φ) is given by xµ 7→ (xµ , φA (xµ ), ∂ν φA (xµ )),

(4.1)

where ∂ν = ∂/∂xν . Higher order jet bundles of Y , J m (Y ), then follow as J 1 (· · ·(J 1 (Y )). Analogous to the tangent map of the projection πY,J 1 (Y ) , T πY,J 1 (Y ) : T J 1 (Y ) → T Y , we may define the jet map of this projection which takes J 2 (Y ) onto J 1 (Y ) Definition 4.1. Let γ ∈ J 1 (Y ) so that πX,J 1 (Y ) (γ) = x. Then JπY,J 1 (Y ) : Aff(Tx X, Tγ J 1 (Y )) → Aff(Tx X, T πY,J 1 (Y ) · Tγ J 1 (Y )). We define the subbundle Y 00 of J 2 (Y ) over X which consists of second-order jets so that on each fiber Yx00 = {s ∈ J 2 (Y )γ | JπY,J 1 (Y ) (s) = γ}. In coordinates, if γ ∈ J 1 (Y ) is given by (xµ , y A , v A µ ), and s ∈ J 2 (Y )γ is given by (xµ , y A , v A µ , wA µ , κA µν ), then s is a second-order jet if v A µ = wA µ . Thus, the second jet of φ ∈ 0(πXY ), j 2 (φ), given in coordinates by the map xµ 7→ (xµ , φA , ∂ν φA , ∂µ ∂ν φA ), is an example of a second-order jet. Definition 4.2. The dual jet bundle J 1 (Y )? is the vector bundle over Y whose fiber at y ∈ Yx is the set of affine maps from J 1 (Y )y to 3n+1 (X)x , the bundle of (n + 1)-forms on X. A smooth section of J 1 (Y )? is therefore an affine bundle map of J 1 (Y ) to 3n+1 (X) covering πXY . Fiber coordinates on J 1 (Y )? are (p, pA µ ), which correspond to the affine map given in coordinates by v A µ 7→ (p + pA µ v A µ )dn+1 x, where dn+1 x = dx1 ∧ · · · ∧ dxn ∧ dx0 .

(4.2)

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Analogous to the canonical one- and two-forms on a cotangent bundle, there exist canonical (n + 1)- and (n + 2)-forms on the dual jet bundle J 1 (Y )? . In coordinates, with dn xµ := ∂µ dn+1 x, these forms are given by 2 = pA µ dy A ∧ dn xµ + pdn+1 x and = dy A ∧ dpA µ ∧ dn xµ − dp ∧ dn+1 x. (4.3) A Lagrangian density L : J 1 (Y ) → 3n+1 (X) is a smooth bundle map over X. In coordinates, we write L(γ) = L(xµ , y A , v A µ )dn+1 x.

(4.4)

The corresponding covariant Legendre transformation for L is a fiber preserving map over Y , FL : J 1 (Y ) → J 1 (Y )? , expressed intrinsically as the first order vertical Taylor approximation to L: d L(γ + ε(γ 0 − γ)), (4.5) FL(γ) · γ 0 = L(γ) + dε ε=0 where γ, γ 0 ∈ J 1 (Y )y . A straightforward calculation shows that the covariant Legendre transformation is given in coordinates by pA µ =

∂L , and ∂v A µ

p=L−

∂L A v µ. ∂v A µ

(4.6)

We can then define the Cartan form as the (n + 1)-form 2L on J 1 (Y ) given by 2L = (FL)∗ 2,

(4.7)

L = −d2L = (FL)∗ ,

(4.8)

and the (n + 2)-form L by

with local coordinate expressions

∂L ∂L A A n dy ∧ d xµ + L − A v µ dn+1 x, 2L = ∂v A µ ∂v µ ∂L A ∂L n n+1 x − d L − v x. ∧ d L = dy A ∧ d µ ∧d µ ∂v A µ ∂v A µ

(4.9)

This is the differential-geometric formulation of the multisymplectic structure. Subsequently, we shall show how we may obtain this structure directly from the variational principle, staying entirely on the Lagrangian side J 1 (Y ). The multisymplectic form formula. In this subsection we prove a formula that is the multisymplectic counterpart to the fact that in finite-dimensional mechanics, the flow of a mechanical system consists of symplectic maps. Again, we do this by studying the action function. Definition 4.3. Let U be a smooth manifold with (piecewise) smooth closed boundary. We define the set of smooth maps C ∞ = {φ : U → Y | πXY ◦ φ : U → X is an embedding}. For each φ ∈ C ∞ , we set φX := πXY ◦ φ and UX := πXY ◦ φ(U ) so that φX : U → UX is a diffeomorphism. .

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We may then define the infinite-dimensional manifold C to be the closure of C ∞ in either a Hilbert space or Banach space topology. For example, the manifold C may be given the topology of a Hilbert manifold of bundle mappings, H s (U, Y ), (U considered a bundle with fiber a point) for any integer s ≥ (n + 1)/2, so that the Hilbert sections φ ◦ φ−1 X in Y are those whose distributional derivatives up to order s are square-integrable in any chart. With our condition on s, the Sobolev embedding theorem makes such mappings well defined. Alternately, one may wish to consider the Banach manifold C as the closure of C ∞ in the usual C k -norm, or more generally, in a Holder space C k+α -norm. See Palais [1968] and Ebin and Marsden [1970] for a detailed account of manifolds of mappings. The choice of topology for C will not play a crucial role in this paper. Definition 4.4. Let G be the Lie group of πXY -bundle automorphisms ηY covering diffeomorphisms ηX , with Lie algebra g. We define the action 8 : G × C → C by 8(ηY , φ) = ηY ◦ φ.1 Furthermore, if φ ◦ φ−1 X ∈ 0(πUX ,Y ), then 8(ηY , φ) ∈ 0(πηX (UX ),Y ). The tangent space to the manifold C at a point φ is the set Tφ C defined by {V ∈ C ∞ (X, T Y ) | πY,T Y ◦ V = φ, &T πXY ◦ V = VX , a vector field on X} . (4.10) Of course, when these objects are topologized as we have described, the definition of the tangent space becomes a theorem, but as we have mentioned, this functional analytic aspect plays a minor role in what follows. Given vectors V, W ∈ Tφ C we may extend them to vector fields V, W on C by −1 fixing vector fields v, w ∈ T Y such that V = v ◦ (φ ◦ φ−1 X ) and W = w ◦ (φ ◦ φX ), and −1 letting Vρ = v ◦ (ρ ◦ ρ−1 X ) and Wρ = w ◦ (ρ ◦ ρX ). Thus, the flow of V on C is given by λ λ λ 8(ηY , ρ), where ηY covering ηX is the flow of v. The definition of the bracket of vector fields using their flows, then shows that [V, W](ρ) = [v, w] ◦ (ρ ◦ ρ−1 X ). Whenever it is contextually clear, we shall, for convenience, write V for v ◦ (φ ◦ φ−1 X ). Definition 4.5. The action function S on C is defined as follows: Z L(j 1 (φ ◦ φ−1 S(φ) = X )) for all φ ∈ C.

(4.11)

UX

Let λ 7→ ηYλ be an arbitrary smooth path in G such that ηY0 = e, and let V ∈ Tφ C be given by d d λ 8(ηY , φ), and VX = η λ ◦ φ. (4.12) V = dλ λ=0 dλ λ=0 X Definition 4.6. We say that φ is a stationary point, critical point, or extremum of S if d S(8(ηYλ , φ)) = 0. (4.13) dλ λ=0 1

−1 We shall also use the notation 8(ηY , φ) to denote the section ηY ◦ (φ ◦ φ−1 x ) ◦ ηx .

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Then,

Z d L(∂ 1 8(ηYλ , φ)) λ ◦φ (U ) dλ λ=0 ηX X Z d ∗ 1 λ ∗ j 1 (φ ◦ φ−1 = X ) j (ηY ) 2L , dλ

dSφ · V =

λ=0

(4.14)

φX (U )

where we have used the fact that L(z) = z ∗ 2L for all holonomic sections z of J 1 (Y ) (see Corollary 4.2 below), and that −1 −1 −1 1 1 j 1 (ηY ◦ φ ◦ φ−1 X ◦ ηX ) = j (ηY ) ◦ j (φ ◦ φX ) ◦ ηX .

Using the Cartan formula, we obtain that Z ∗ j 1 (φ ◦ φ−1 dSφ · V = X ) Lj 1 (V ) 2L UX Z ∗ 1 = −j 1 (φ ◦ φ−1 X ) [j (V ) L ] UX Z ∗ 1 j 1 (φ ◦ φ−1 + X ) [j (V ) 2L ].

(4.15)

∂UX

Hence, a necessary condition for φ ∈ C to be an extremum of S is that the first term in (4.15) vanishes. One may readily verify that the integrand of the first term in (4.15) is equal to zero whenever j 1 (V ) is replaced by W ∈ T J 1 (Y ) which is either πY,J 1 (Y ) -vertical or tangent to j 1 (φ ◦ φ−1 X ) (see Marsden and Shkoller [1998]), so that ∗ using a standard argument from the calculus of variations, j 1 (φ ◦ φ−1 L ] must X ) [W 1 vanish for all vectors W on J (Y ) in order for φ to be an extremum of the action. We shall call such elements φ ∈ C covering φX , solutions of the Euler–Lagrange equations. Definition 4.7. We let ∗ P = {φ ∈ C | j 1 (φ ◦ φ−1 X ) [W

L ] = 0 for all W ∈ T J 1 (Y )}.

(4.16)

A In coordinates, (φ ◦ φ−1 X ) is an element of P if ∂ ∂L 1 ∂L 1 −1 −1 (j (φ ◦ φX )) − µ (j (φ ◦ φX ) = 0 in UX . ∂y A ∂x ∂vµA

We are now ready to prove the multisymplectic form formula, a generalization of the symplectic flow theorem, but we first make the following remark. If P is a submanifold of C, then for any φ ∈ P, we may identify Tφ P with the set {V ∈ ∗ L ] = 0 for all W ∈ T J 1 (Y )} since such vectors Tφ C | j 1 (φ ◦ φ−1 X ) Lj 1 (V ) [W d arise by differentiating d |=0 j 1 (φ ◦ φX −1 )∗ [W L ] = 0, where φ is a smooth curve of solutions of the Euler–Lagrange equations in P (when such solutions exist). More generally, we do not require P to be a submanifold in order to define the first variation solution of the Euler–Lagrange equations. Definition 4.8. For any φ ∈ P ,we define the set ∗ F = {V ∈ Tφ C | j 1 (φ ◦ φ−1 X ) Lj 1 (V ) [W

L ] = 0 for all W ∈ T J 1 (Y )}. (4.17)

Elements of F solve the first variation equations of the Euler–Lagrange equations.

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Theorem 4.1 (Multisymplectic form formula). If φ ∈ P, then for all V and W in F, Z ∗ 1 1 j 1 (φ ◦ φ−1 (4.18) X ) [j (V ) j (W ) L ] = 0. ∂UX

Proof. We define the 1-forms α1 and α2 on C by Z ∗ 1 −j 1 (φ ◦ φ−1 α1 (φ) · V := X ) [j (V ) UX

Z

and α2 (φ) · V :=

∂UX

∗ 1 j 1 (φ ◦ φ−1 X ) [j (V )

L ]

2L ],

so that by (4.15), dSφ · V = α1 (φ) · V + α2 (φ) · V for all V ∈ Tφ C.

(4.19)

Recall that for any 1-form α on C and vector fields V, W on C, dα(V, W ) = V [α(W )] − W [α(V )] − α([V, W ]).

(4.20)

We let φ = ηY ◦ φ be a curve in C through φ, where ηY is a curve in G through the identity such that d W = |=0 ηY and W ∈ F, d and consider Eq. (4.19) restricted to all V ∈ F. Thus, d (α2 (V )(φ )) d(α2 (V ))(φ) · W = d =0 Z d ∗ 1 1 j 1 (φ ◦ φ−1 = X ) j (ηY )[j (V ) 2L ] d =0 ∂UX Z ∗ 1 j 1 (φ ◦ φ−1 = X ) Lj 1 (W ) [j (V ) 2L ] ∂UX Z ∗ 1 1 j 1 (φ ◦ φ−1 = X ) [j (W ) d(j (V ) 2L )] ∂UX Z ∗ 1 1 j 1 (φ ◦ φ−1 + X ) d[j (W ) j (V ) 2L ], ∂UX

where the last equality was obtained using Cartan’s formula. Using Stoke’s theorem, noting that ∂∂U is empty, and applying Cartan’s formula once again, we obtain that Z ∗ 1 1 − j 1 (φ ◦ φ−1 d(α2 (φ)(V )) · W = X ) [j (W ) j (V ) L ] ∂UX Z ∗ 1 j 1 (φ ◦ φ−1 + X ) [j (W ) Lj 1 (V ) 2L ], ∂UX

and

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Z d(α2 (φ)(W )) · V =

∗ 1 1 − j 1 (φ ◦ φ−1 X ) [j (V ) j (W ) L ] ∂UX Z ∗ 1 j 1 (φ ◦ φ−1 + X ) [j (V ) Lj 1 (W ) 2L ]. ∂UX

Also, since [j (V ), j (W )] = j ([V, W ]), we have Z ∗ 1 1 j 1 (φ ◦ φ−1 α2 (φ)([V, W ]) = X ) [j (V ), j (W )] 1

1

1

2L .

∂UX

Now [j 1 (V ), j 1 (W )] so that

2L = Lj 1 (V ) (j 1 (W )

Z

dα2 (φ)(V, W ) = 2

∂UX

Z +

∂UX

2L ) − j 1 (W )

Lj 1 (V ) 2L ,

∗ 1 j 1 (φ ◦ φ−1 X ) [j (V )

j 1 (W )

∗ 1 j 1 (φ ◦ φ−1 X ) [j (V )

Lj 1 (W ) 2L − Lj 1 (V ) (j 1 (W )

L ] 2L )].

But 2L ) = d(j 1 (V )

Lj 1 (V ) (j 1 (W )

j 1 (W )

2L ) + j 1 (V )

d(j 1 (W )

d(j 1 (W )

2L ) − j 1 (V )

j 1 (W )

2L )

and j 1 (V ) Hence,

Lj 1 (W ) 2L = j 1 (V ) Z ∂UX

Z

∗ 1 j 1 (φ ◦ φ−1 X ) [j (V )

= ∂UX

Z

−

Lj 1 (W ) 2L − Lj 1 (V ) (j 1 (W )

∗ 1 − j 1 (φ ◦ φ−1 X ) (j (V )

∂UX

∗ 1 j 1 (φ ◦ φ−1 X ) d(j (V )

j 1 (W ) j 1 (W )

L .

2L )]

L ) 2L ).

The last term once again vanishes by Stokes theorem together with the fact that ∂∂U is empty, and we obtain that Z ∗ 1 1 j 1 (φ ◦ φ−1 (4.21) dα2 (φ)(V, W ) = X ) (j (V ) j (W ) L ). ∂UX

We now use (4.20) on α1 . A similar computation as above yields Z ∗ 1 d(α1 (φ) · V ) · W = j 1 (φ ◦ φ−1 X ) Lj 1 (W ) [j (V ) L ] UX

which vanishes for all φ ∈ P and W ∈ F. Similarly, d(α1 (φ) · W ) · v = 0 for all φ ∈ P and V ∈ F. Finally, α1 (φ) = 0 for all φ ∈ P. Hence, since 0 = ddS(φ)(V, W ) = dα1 (φ)(V, W ) + dα2 (φ)(V, W ), we obtain the formula (4.18).

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Symplecticity revisited. Let 6 be a compact oriented connected boundaryless nmanifold which we think of as our reference Cauchy surface, and consider the space of embeddings of 6 into X, Emb(6, X); again, although it is unnecessary for this paper, we may topologize Emb(6, X) by completing the space in the appropriate C k or H s -norm. Let B be an m-dimensional manifold. For any fiber bundle πBK : K → B, we shall, in addition to 0(πBK ), use the corresponding script letter K to denote the space of sections of πBK . The space of sections of a fiber bundle is an infinite-dimensional manifold; in fact, it can be precisely defined and topologized as the manifold C of the previous section, where the diffeomorphisms on the base manifold are taken to be the identity map, so that the tangent space to K at σ is given simply by Tσ K = {W : B → V K |πK,T K ◦ W = σ}, where V K denotes the vertical tangent bundle of K. We let πK,L(V K,3m (B)) : L(V K, 3m (B)) → K be the vector bundle over K whose fiber at k ∈ Kx , x = πBK (k), is the set of linear mappings from Vk K to 3m (B)x . Then the cotangent space to K at σ is defined as Tσ∗ K = {π : B → L(V K, 3m (B)) | πK,L(V K,3n+1 (B)) ◦ π = σ}. Integration provides the natural pairing of Tσ∗ K with Tσ K: Z π · V. hπ, V i = B

In practice, the manifold B will either be X or some (n + 1)-dimensional subset of X, or the n-dimensional manifold 6τ , where for each τ ∈ Emb(6, X), 6τ := τ (6). We shall use the notation Yτ for the bundle π6τ ,Y , and Yτ for sections of this bundle. For the remainder of this section, we shall set the manifold C introduced earlier to Y. The infinite-dimensional manifold Yτ is called the τ -configuration space, its tangent bundle is called the τ -tangent space, and its cotangent bundle T ∗ Yτ is called the τ -phase space. Just as we described in Sect. 2, the cotangent bundle has a canonical 1-form θτ and a canonical 2-form ωτ . These differential forms are given by Z π(T πYτ ,T ∗ Yτ · V ) and ωτ = −dθτ , (4.22) θτ (ϕ, π) · V = 6τ

where (ϕ, π) ∈ Yτ , V ∈ T(ϕ,π) T ∗ Yτ , and πYτ ,T ∗ Yτ : T ∗ Yτ → Yτ is the cotangent bundle projection map. An infinitesimal slicing of the bundle πXY consists of Yτ together with a vector field ζ which is everywhere transverse to Yτ , and covers ζX which is everywhere transverse to 6τ . The existence of an infinitesimal slicing allows us to invariantly decompose the temporal from the spatial derivatives of the fields. Let φ ∈ Y, ϕ := φ|6τ , and let iτ : 6τ → X be the inclusion map. Then we may define the map βζ taking j 1 (Y)τ to j 1 (Yτ ) × 0(π6τ ,V Yτ ) over Yτ by ˙ where ϕ˙ := Lζ φ. βζ (j 1 (φ) ◦ iτ ) = (j 1 (ϕ), ϕ)

(4.23)

In our notation, j (Y)τ is the collection of restrictions of holonomic sections of J 1 (Y ) to 6τ , while j 1 (Yτ ) are the holonomic sections of π6τ ,J 1 (Y ) . It is easy to see that βζ is an isomorphism; it then follows that βζ is an isomorphism of j 1 (Y)τ with T Yτ , since j 1 (ϕ) is completely determined by ϕ. This bundle map is called the jet decomposition map, and its inverse is called the jet reconstruction map. Using this map, we can define the instantaneous Lagrangian. 1

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Definition 4.9. The instantaneous Lagrangian Lτ,ζ : T Yτ → R is given by Z ˙ = i∗τ [ζX L(βζ−1 (j 1 (ϕ), ϕ)] ˙ Lτ,ζ (ϕ, ϕ)

(4.24)

6τ

for all (ϕ, ϕ) ˙ ∈ T Yτ . The instantaneous Lagrangian Lτ,ζ has an instantaneous Legendre transform ˙ 7→ (ϕ, π) FLτ,ζ : T Yτ → T ∗ Yτ ; (ϕ, ϕ) which is defined in the usual way by vertical fiber differentiation of Lτ,ζ (see, for example, Abraham and Marsden [1978]). Using the instantaneous Legendre transformation, we can pull-back the canonical 1- and 2-forms on T ∗ Yτ . Definition 4.10. Denote, respectively, the instantaneous Lagrange 1- and 2-forms on T Yτ by θτL = FL∗τ,ζ θτ and ωτL = −dθτL .

(4.25)

Alternatively, we may define θτL using Theorem 2.1, in which case no reference to the cotangent bundle is necessary. We will show that our covariant multisymplectic form formula can be used to recover the fact that the flow of the Euler–Lagrange equations in the bundle πEmb(6,X),∪τ ∈Emb(6,X) T Yτ is symplectic with respect to ωτL . To do so, we must relate the multisymplectic Cartan (n + 2)-form L on J 1 (Y ) with the symplectic 2-form ωτL on T Yτ . 1 Theorem 4.2. Let 2L τ be the canonical 1-form on j (Y)τ given by Z 1 (j (φ) ◦ i ) · V = i∗τ j 1 (φ)∗ [V 2L ], 2L τ τ

(4.26)

6τ

where j 1 (φ) ◦ iτ ∈ j 1 (Y)τ , V ∈ Tj 1 (φ)◦iτ j 1 (Y)τ . L L 1 (a) If the 2-form L τ on j (Y)τ is defined by τ := −d2τ , then for V, W ∈ 1 Tj 1 (φ)◦iτ j (Y)τ , 1 L τ (j (φ) ◦ iτ )(V, W ) =

Z 6τ

i∗τ j 1 (φ)∗ [W

V

L ].

(4.27)

(b) Let the diffeomorphism sX : 6 × R → X be a slicing of X such that for λ ∈ R, 6λ := sX (6 × {λ}) and 6λ := τλ (6), where τλ ∈ Emb(6, X) is given by τλ (x) = sX (x, λ). For any φ ∈ P, let V, W ∈ Tφ Y ∩ F so that for each τ ∈ Emb(6, X), j 1 Vτ , j 1 Wτ ∈ Tj 1 (φ)◦iτ j 1 (Y)τ , and let τλ1 , τλ2 ∈ Emb(6, X). Then 1 1 L 1 1 L τλ (j Vτλ1 , j Wτλ1 ) = τλ (j Vτλ2 , j Wτλ2 ). 1

2

(4.28)

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Proof. Part (a) follows from the Cartan formula together with Stokes theorem using an argument like that in the proof of Theorem 4.1. For part (b), we recall that the multisymplectic form formula on Y states that for any subset UX ⊂ X with smooth closed boundary and vectors V, W ∈ Tφ Y ∩ F, φ ∈ Y, Z j 1 (φ)∗ [j 1 (V ) j 1 (W ) L ] = 0. (4.29) ∂UX

Let Then ∂UX

UX = ∪λ∈[λ1 ,λ2 ] 6λ . = 6λ1 − 6λ2 , so that (4.29) can be written as Z j 1 (φ ◦ iτλ2 )∗ [j 1 Vτλ2 j 1 Wτλ2 L ] 0= 6λ2

Z −

6λ1

j 1 (φ ◦ iτλ1 )∗ [j 1 Vτλ1

j 1 Wτλ1

L ]

1 1 L 1 1 = L τλ (j Vτλ1 , j Wτλ1 ) − τλ (j Vτλ2 , j Wτλ2 ), 1

2

which proves (4.28).

∗ L Theorem 4.3. The identity 2L τ = βζ θτ holds.

Proof. Let W ∈ Tj 1 (φ)◦iτ j 1 (Y)τ , which we identify with w ◦ φ ◦ iτ , where w is a πX,J 1 (Y ) -vertical vector. Choose a coordinate chart which is adapted to the slicing so that ∂0 |Yτ = ζ. With w = (0, W A , WµA ), we see that Z ∂L B B · W = (φ , φ ,µ )W A dn x0 . 2L τ A ∂v 0 6τ Now, from (4.24) we get ∂Lτ,ζ A dy ∂ y˙ A Z ∂ ∗ = i [∂ L(xµ , φA , φA ,µ )dn+1 x ⊗ dy A ] A τ 0 6τ ∂ y˙ Z ∂L B B (φ , φ ,µ )dy A ⊗ dn x0 , = A ∂v 0 6τ

˙ = θτL (ϕ, ϕ)

where we arrived at the last equality using the fact that y˙ A = v A 0 in this adapted chart. L Since (T βζ · W )A = W A , we see that 2L τ · W = θτ · (T βζ · W ), and this completes the proof. Let the instantaneous energy Eτ,ζ associated with Lτ,ζ be given by ˙ = FLτ,ζ (ϕ) ˙ · ϕ˙ − Lτ,ζ (ϕ, ϕ), ˙ Eτ,ζ (ϕ, ϕ)

(4.30)

and define the “time”-dependent Lagrangian vector field XEτ,ζ by XEτ,ζ

ωτL = dEτ,ζ .

Since ∪τ ∈Emb(6,X) T Yτ over Emb(6, X) is infinite-dimensional and wτL is only weakly nondegenerate, the second-order vector field XEτ,ζ does not, in general, exist. In the case that it does, we obtain the following result.

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Corollary 4.1. Assume XEτ,ζ exists and let Fτ be its semiflow, defined on some subset D of the bundle ∪τ ∈Emb(6,X) T Yτ over Emb(6, X). Fix τ¯ so that Fτ¯ (ϕ1 , ϕ˙1 ) = (ϕ2 , ϕ˙2 ), where (ϕ1 , ϕ˙1 ) ∈ T Yτ1 and (ϕ2 , ϕ˙2 ) ∈ T Yτ2 . Then Fτ¯∗ ωτL2 = ωτL1 . Proof. This follows immediately from Theorem 4.2(b) and Theorem 4.3 and the fact that βζ induces an isomorphism between j 1 (Y)τ and T Yτ . Example: Nonlinear wave equation. To illustrate the geometry that we have developed, let us consider the scalar nonlinear wave equation given by ∂2φ ∂x0

2

− 4φ − N 0 (φ) = 0, φ ∈ 0(πXY ),

(4.31)

where 4 is the Laplace-Beltrami operator and N is a real-valued C ∞ function of one variable. For concreteness, fix n=1 so that the spacetime manifold X := R2 , the configuration bundle Y := πR2 ,R , and the first jet bundle J 1 (Y ) := πR2 ,R3 . Equation (4.31) is governed by the Lagrangian density # ) ( " ∂φ 2 1 ∂φ 2 − 1 + N (φ) dx1 ∧ dx0 . (4.32) L= 2 ∂x0 ∂x Using coordinates (x0 , x1 , φ, φ,0 , φ,1 ) for J 1 (Y ), we write the multisymplectic 3-form for this nonlinear wave equation on R2 in coordinates as L = −dφ ∧ dφ,0 ∧ dx1 − dφ ∧ dφ,1 ∧ dx0 − N 0 (φ)dφ ∧ dx1 ∧ dx0 +φ,0 dφ,0 ∧ dx1 ∧ dx0 − φ,1 dφ,1 ∧ dx1 ∧ dx0 ;

(4.33)

a short computation verifies that solutions of (4.31) are elements of P, or that j 1 (φ ◦ ∗ L ] = 0 for all W ∈ T J 1 (Y ) (see Marsden and Shkoller [1998]). φ−1 X ) [W We will use this example to demonstrate that our multisymplectic form formula generalizes the notion of symplecticity given by Bridges [1997]. Since the Lagrangian (4.32) does not explicitly depend on time, it is convenient to identify sections of Y as mappings from R2 into R, and similarly, sections of J 1 (Y ) as mappings from R2 into R3 . Thus, for φ ∈ 0(πXY ), j 1 (φ)(xµ ) := (φ(xµ ), φ,0 (xµ ), φ,1 (xµ )) ∈ R3 , and if we set pµ := φ,µ , then (4.31) can be reformulated to J0 j 1 φ,0 + J1 j 1 φ,1 :=        0  φ N (φ) φ 0 10 0 0 −1  −1 0 0   p0  +  0 0 0   p0  =  −p0  . 0 00 10 0 p1 ,0 p1 ,1 p1

(4.34)

To each degenerate matrix Jµ , we associate the contact form ω µ on R3 given by ω µ (u1 , u2 ) = hJµ u1 , u2 i, where u1 , u2 ∈ R3 and h·, ·i is the standard inner product on R3 . Bridges obtains the following conservation of symplecticity: ∂ 1 1 ∂ 0 1 ω (j (φ,0 ), j 1 (φ,1 )) + ω (j (φ,0 ), j 1 (φ,1 )) = 0. ∂x0 ∂x1

(4.35)

This result is interesting, but has somewhat limited scope in that the vector fields in (4.35) upon which the contact forms act are not general solutions to the first variation equations; rather, they are the specific first variation solutions φ,µ . Bridges obtains this

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result by crucially relying on the multi-Hamiltonian structure of (4.31); in particular, the vector (N 0 (φ), −p0 , p1 ) on the right-hand-side of (4.34) is the gradient of a smooth multi-Hamiltonian function H(φ, p0 , p1 ) (although the multi-Hamiltonian formalism is not important for this article, we refer the reader to Marsden and Shkoller [1998] for the Hamiltonian version of our covariant framework, and to Bridges [1997]). Using Eq. (4.34), it is clear that H,0 = ω 0 (j 1 (φ,0 ), j 1 (φ,1 )) and H,1 = −ω 1 (j 1 (φ,0 ), j 1 (φ,1 )) so that (4.35) follows from the relation H,0,1 = H,1,0 . Proposition 4.1. The multisymplectic form formula is an intrinsic generalization of the conservation law (4.35); namely, for any V, W ∈ F that are πX,J 1 (Y ) -vertical, ∂ 1 1 ∂ 0 1 ω (j (V ), j 1 (W )) + ω (j (V ), j 1 (W )) = 0. 0 1 ∂x ∂x

(4.36)

Proof. Let j 1 (V ) and j 1 (W ) have the coordinate expressions (V, V 0 , V 1 ) and (W, W 0 , W 1 ), respectively. Using (4.33), we compute j 1 (W ) j 1 (V ) L = V W 0 − V 0 W dx + V W 1 − V 1 W dt, so that with Theorem 4.1 and the definition of ω µ , we have, for UX ⊂ X, Z ω 0 (j 1 (V ), j 1 (W ))dx − ω 1 (j 1 (V ), j 1 (W ))dt = 0, ∂UX

and hence by Green’s theorem, Z ∂ 0 1 ∂ 1 1 1 1 (j (V ), j (W )) + (j (V ), j (W )) dx1 ∧ dx0 = 0. ω ω ∂x0 ∂x1 UX

Since UX is arbitrary, we obtain the desired result.

In general, when V is πXY -vertical, j (V ) has the coordinate expression (V, V,µ + ∂V /∂φ · φ,µ ), but for the special case that V = φ,µ , j 1 (φµ ) = (j 1 φ),µ , and Proposition 4.1 gives ∂ ∂ φ0 φ,0,1 − φ1 φ,0,0 − 1 φ0 φ,1,1 − φ1 φ,0,1 = 0, 0 ∂x ∂x which simplifies to the trivial statement that 1

φ,0 N (φ),1 − φ,1 N (φ),0 = 0. The variational route to the Cartan form. We may alternatively define the Cartan form by beginning with Eq. (4.14). Using the infinitesimal generators defined in (4.12), we obtain that d S(8(ηYλ , φ)) dSφ · V = dλ λ=0 Z d L(j 1 (8(ηYλ , φ))) = λ (U ) dλ λ=0 ηX X Z Z d 1 λ L(j (8(η , φ))) + LVX L(j 1 (φ ◦ φ−1 (4.37) = Y X )) . dλ UX

λ=0

UX

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Using the natural splitting of T Y , any vector V ∈ Tφ C may decomposed as v h V = V h + V v , where V h = T (φ ◦ φ−1 X ) · VX and V = V − V ,

(4.38)

where we recall that VX = T πXY · V . Lemma 4.1. For any V ∈ Tφ C, dSφ · V

h

Z = ∂UX

and v

dSφ · V =

Z UX

[L(j 1 (φ ◦ φ−1 X ))],

(4.39)

d L(j 1 (8(ηYλ , φ))). dλ λ=0

(4.40)

VX

Proof. The equality (4.40) is obvious, since the second term in (4.37) clearly vanishes for all vertical vectors. For vectors V h , the first term in (4.37) vanishes; indeed, using the chain rule, we need only compute that d λ −1 η λ ◦ (φ0 φ−1 = V h − T (φ ◦ φ−1 X ) ◦ ηX X ) · VX , dλ λ=0 Y which is zero by (4.38). We then apply the Cartan formula to the second term in (4.37) and note that dL is an (n + 2)-form on the (n + 1)-dimensional manifold UX so that we obtain (4.39). Theorem 4.4. Given a smooth Lagrangian density L : J 1 (Y ) → 3n+1 (X), there exist a unique smooth section DEL L ∈ C ∞ (Y 00 , 3n+1 (X) ⊗ T ∗ Y )) and a unique differential form 2L ∈ 3n+1 (J 1 (Y )) such that for any V ∈ Tφ C, and any open subset UX such that U X ∩ ∂X = ∅, Z Z ∗ 1 DEL L(j 2 (φ ◦ φ−1 )) · V + j 1 (φ ◦ φ−1 dSφ · V = X X ) [j (V ) 2L ]. UX ∂UX (4.41) Furthermore, −1 ∗ 1 1 DEL L(j 2 (φ ◦ φ−1 X )) · V = j (φ ◦ φX ) [j (V )

L ] in UX .

(4.42)

00

In coordinates, the action of the Euler–Lagrange derivative DEL L on Y is given by ∂2L ∂L 1 −1 −1 2 (j (φ ◦ φ )) − (j 1 (φ ◦ φ−1 DEL L(j (φ ◦ φX )) = X X )) ∂y A ∂xµ ∂v A µ − −

∂2L −1 B (j 1 (φ ◦ φ−1 X )) · (φ ◦ φX ),µ ∂y B ∂v A µ

∂2L −1 −1 B 1 A n+1 (j (φ ◦ φ )) · (φ ◦ φ ) x, X X ,µν dy ∧ d ∂v B ν ∂v A µ

(4.43)

while the form 2L matches the definition of the Cartan form given in (4.9) and has the coordinate expression ∂L ∂L A A n n+1 dy ∧ d x + L − v x. (4.44) 2L = µ d µ ∂v A µ ∂v A µ

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Proof. Choose UX := φX (U ) small enough so that it is contained in a coordinate chart, say O1 . In these coordinates, let V = (V µ , V A ) so that along φ ◦ φ−1 X , our decomposition (4.38) may be written as ! −1 A ∂ µ ∂ v v A ∂ A µ ∂(φ ◦ φX ) and V = (V ) := V − V , VX = V ∂xµ ∂y A ∂xµ ∂y A and Eq. (4.40) gives Z ∂L 1 ∂L 1 ∂(V v )A n+1 −1 −1 v A d x, (j (φ ◦ φ )) · (V ) + (j (φ ◦ φ )) · dSφ · V v = X X A ∂v A µ ∂xµ UX ∂y (4.45) where we have used the fact that in coordinates along j 1 (φ ◦ φ−1 X ), {j 1 (V )}A µ = ∂µ [(V v )A (j 1 (φ ◦ φ−1 X ))]. Integrating (4.45) by parts, we obtain Z ∂L 1 ∂ ∂L 1 −1 −1 v A dn+1 x (j (φ ◦ φX )) − µ A (j (φ ◦ φX )) · V dSφ · V = ∂y A ∂x ∂v µ UX Z ∂L 1 A n (j (φ ◦ φ−1 + X )) · V d xµ ∂v A µ ∂UX ) −1 A ∂L 1 −1 ∂(φ ◦ φX ) ν n + (j (φ ◦ φX )) · V d xµ . (4.46) ∂v A µ ∂xν R Let α be the n-form integrand of the boundary integral in (4.46); then ∂UX α = R since α is invariant under this lift. Additionally, from Eq. (4.39), we ∂j 1 (φ◦φ−1 )(UX ) X obtain the horizontal contribution Z Z (V µ ∂µ ) (Ldn+1 x) = V µ Ldn xµ , (4.47) dSφ · V h = ∂j 1 (φ◦φ)X −1 )(UX )

∂UX

so combining Eqs. (4.46) and (4.47), a simple computation verifies that Z ∂ ∂L 1 ∂L 1 −1 −1 n+1 A (j (φ ◦ φ )) − (j (φ ◦ φ )) d x ⊗ dy ·V dSφ · V = X X ∂y A ∂xµ ∂v A µ UX Z ∂L 1 A n V (j (φ ◦ φ−1 + X ))dy ∧ d xµ A −1 ∂v 1 µ ∂j (φ◦φX )(UX ) # " ) −1 A ∂L 1 −1 ∂(φ ◦ φX ) n+1 d x . (4.48) + L − A (j (φ ◦ φX )) ∂v µ ∂xµ The vector V in the second term of (4.48) may be replaced by j 1 (V ) since πY,J 1 (Y ) vertical vectors are clearly in the kernel of the form that V is acting on. This shows that (4.43) and (4.44) hold, and hence that the boundary integral in (4.48) may be written as Z ∗ 1 j 1 (φ ◦ φ−1 X ) [j (V ) 2L ]. ∂UX

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Now, if we choose another coordinate chart O2 , the coordinate expressions of DEL L and 2L must agree on the overlap O1 ∩O2 since the left-hand-side of (4.41) is intrinsically defined. Thus, we have uniquely defined DEL L and 2L for any UX such that U X ∩∂X = ∅. Finally, (4.42) holds, since L = d2L is also intrinsically defined and both sides of the equation yield the same coordinate representation, the Euler–Lagrange equations in UX . Remark. To prove Theorem 4.4 for the case UX = X, we must modify the proof to take into account the boundary conditions which are prescribed on ∂X. Corollary 4.2. The (n + 1)-form 2L defined by the variational principle satisfies the relationship L(z) = z ∗ 2L for all holonomic sections z ∈ 0(πX,J 1 (Y ) ). Proof. This follows immediately by substituting (4.42) into (4.41) and integrating by parts using Cartan’s formula. Remark. We have thus far focused on holonomic sections of J 1 (Y ), those that are the first jets of sections of Y , and correspondingly, we have restricted the general splitting of T Y given by T Y = image γ ⊕ V Y for any γ ∈ 0(J 1 (Y )), to T Y = T φ ⊕ V Y , φ ∈ 0(Y ) as we specified in (4.38). For general sections γ ∈ 0(J 1 (Y )), the horizontal bundle is given by image γ, and the Frobenius theorem guarantees that γ is locally holonomic if the connection is flat, or equivalently if the curvature of the connection Rγ vanishes. Since this is a local statement, we may assume that Y = U × RN , where U ⊂ Rn+1 is open, and that πXY is simply the projection onto the first factor. For φ ∈ 0(Y ), and γ ∈ 0(J 1 (Y )), γ(x, φ(x)) : Rn+1 → RN is a linear operator which is holonomic if φ0 (x) = γ(x, φ(x)), where φ0 (x) is the differential of φ, and this is the case whenever the operator φ00 (x) is symmetric. Equivalently, the operator Sγ (x, y) · (v, w) := D1 γ(x, y) · (v, w) + D2 γ(x, y) · (γ(x, y) · v, w) is symmetric for all v, w ∈ Rn+1 . One may easily verify that the local curvature is given by Rγ (x, y) · (v, w) := Sγ (x, y) · (v, w) − Sγ (x, y) · (w, v) and that γ = j 1 (φ) locally for some φ ∈ 0(Y ), if and only if Rγ = 0. The variational route to Noether’s theorem. Suppose the Lie group G acts on C and leaves the action S invariant so that S(8(ηY , φ)) = S(φ) for all ηY ∈ G.

(4.49)

This implies that for each ηY ∈ G, 8(ηY , φ) ∈ P whenever φ ∈ P. We restrict the

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action of G to P, and let ξC be the corresponding infinitesimal generator on C restricted to points in P; then Z ∗ 1 j 1 (φ ◦ φ−1 0 = (ξC dS)φ = X ) [j (ξ) 2L ] ∂UX Z ∗ 1 j 1 (φ ◦ φ−1 = X ) [j (ξ) L ], UX

since Lj 1 (ξ) 2L = 0 by (4.49) and Corollary 4.2. We denote the covariant momentum map on J 1 (Y ) by J L ∈ L(g, 3n (J 1 (Y )) which we define as j 1 (ξ)

L = dJ L (ξ).

(4.50)

R ∗ L Using (4.50), we find that UX d[j 1 (φ ◦ φ−1 X ) J (ξ)] = 0, and since this must hold for all infinitesimal generators ξC at φ ∈ C, the integrand must also vanish so that ∗ L d[j 1 (φ ◦ φ−1 X ) J (ξ)] = 0,

(4.51)

which is precisely a restatement of the covariant Noether Theorem.

5. Veselov-type Discretizations of Multisymplectic Field Theory 5.1. General theory. We now generalize the Veselov discretization given in Sect. 3 to multisymplectic field theory, by discretizing the spacetime X. For simplicity we restrict to the discrete analogue of dim X = 2; i.e. n = 1. Thus, we take X = Z × Z = {(i, j)} and the fiber bundle Y to be X × F for some smooth manifold F. Notation. The development in this section is aided by a small amount of notation and terminology. Elements of Y over the base point (i, j) are written as yij and the projection πXY acts on Y by πXY (yij ) = (i, j). The fiber over (i, j) ∈ X is denoted Yij . A triangle 1 of X is an ordered triple of the form 1 = (i, j), (i, j + 1), (i + 1, j + 1) . The first component (i, j) of 1 is the first vertex of the triangle, denoted 11 , and similarly for the second and third vertices. The set of all triangles in X is denoted X 1 . By abuse of notation the same symbol is used for a triangle and the (unordered) set of its vertices. A point (i, j) ∈ X is touched by a triangle if it is a vertex of that triangle. If U ⊆ X, then (i, j) ∈ U is an interior point of U if U contains all three triangles of X that touch (i, j). The interior int U of U is the collection of the interior points of U . The closure cl U of U is the union of all triangles touching interior points of U . A boundary point of U is a point in U and cl U which is not an interior point. The boundary of U is the set of boundary points of U , so that ∂U ≡ (U ∩ cl U ) \ int U. Generally, U properly contains the union of its interior and boundary, and we call U regular if it is exactly that union. A section of Y is a map φ : U ⊆ X → Y such that πXY ◦ φ = idU .

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yi+1 j+1

yi,j

yi j+1

(i+1,j+1) x (i,j) (i,j+1) Fig. 5.1. Depiction of the heuristic interpretation of an element of J 1 Y when X is discrete

Multisymplectic phase space. We define the first jet bundle2 of Y to be J 1 Y ≡ {(yij , yi j+1 , yi+1 j+1 ) | (i, j) ∈ X, yij , yi j+1 , yi+1 j+1 ∈ F} ≡ X 1 × F 3. Heuristically (see Fig. 5.1), X corresponds to some grid of elements xij in continuous ˜ and yij , yi j+1 , yi+1 j+1 ∈ J 1 Y corresponds to j 1 φ(x), spacetime, say X, ¯ where x¯ is “inside” the triangle bounded by xij , xi j+1 , xi+1 j+1 , and φ is some smooth section of X˜ ×F interpolating the field values yij , yi j+1 , yi+1 j+1 . The first jet extension of a section φ of Y is the map j 1 φ : X 1 → J 1 Y defined by j 1 φ(1) ≡ 1, φ(11 ), φ(12 ), φ(13 ) . Given a vector field Z on Y , we denote its restriction to the fiber Yij by Zij , and similarly for vector fields on J 1 Y . The first jet extension of a vector field Z on Y is the vector field j 1 Z on J 1 Y defined by j 1 Z(y11 , y12 , y13 ) ≡ Z11 (y11 ), Z12 (y12 ), Z13 (y13 ) , for any triangle 1. The variational principle. Let us posit a discrete Lagrangian L : J 1 Y → R. Given a triangle 1, define the function L1 : F 3 → R by L1 (y1 , y2 , y3 ) ≡ L(1, y1 , y2 , y3 ), so that we may view the Lagrangian L as being a choice of a function L1 for each triangle 1 of X. The variables on the domain of L1 will be labeled y 1 , y 2 , y 3 , irrespective of the particular 1. Let U be regular and let CU be the set of sections of Y on U , so CU is the manifold F |U | . The action will assign real numbers to sections in CU by the rule X L ◦ j 1 φ(1). (5.1) S(φ) ≡ 1;1⊆U

Using three vertices is the simplest choice for approximating the two partial derivatives of the field φ, but may not lead to a good numerical scheme. Later, we shall also use four vertices together with averaging to define the partial derivatives of the fields. 2

Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs

377

(i + 1, j) (i + 1, j + 1)

(i, j − 1)

(i, j)

(i, j + 1)

(i − 1, j)

(i − 1, j − 1)

Fig. 5.2. The triangles which touch (i, j)

Given φ ∈ CU and a vector field V , there is the 1-parameter family of sections (FV φ)(i, j) ≡ FVij (φ(i, j)), where F Vij denotes the flow of Vij on F. The variational principle is to seek those φ for which d S(FV φ) = 0 d =0

for all vector fields V . The discrete Euler–Lagrange equations. The variational principle gives certain field equations, the discrete Euler–Lagrange field equations (DELF equations), as follows. Focus upon some (i, j) ∈ int U , and abuse notation by writing φ(i, j) ≡ yij . The action, written with its summands containing yij explicitly, is (see Fig. 5.2) S = · · · + L(yij , yi j+1 , yi+1 j+1 ) + L(yi j−1 , yij , yi+1 j ) + L(yi−1 j−1 , yi−1 j , yij ) + · · · so by differentiating in yij , the DELF equations are ∂L ∂L ∂L (yij , yi j+1 , yi+1 j+1 ) + 2 (yi j−1 , yij , yi+1 j ) + 3 (yi−1 j−1 , yi−1 j , yij ) = 0, ∂y 1 ∂y ∂y for all (i, j) ∈ int U . Equivalently, these equations may be written X l;1;(i,j)=1l

∂L1 (y 1 , y 2 , y 3 ) = 0, ∂y l 1 1 1

(5.2)

for all (i, j) ∈ int U . The discrete Cartan form. Now suppose we allow nonzero variations on the boundary ∂U , so we consider the effect on S of a vector field V which does not necessarily vanish on ∂U . For each (i, j) ∈ ∂U find the triangles in U touching (i, j). There is at least one such triangle since (i, j) ∈ cl U ; there are not three such triangles since (i, j) 6∈ int U . For each such triangle 1, (i, j) occurs as the lth vertex, for one or two of l = 1, 2, 3, and those lth expressions from the list

378

J. E. Marsden, G. W. Patrick, S. Shkoller

∂L (yij , yi j+1 , yi+1 j+1 )Vij (yij ), ∂y 1 ∂L (yi j−1 , yij , yi+1 j )Vij (yij ), ∂y 2 ∂L (yi−1 j−1 , yi−1 j , yij )Vij (yij ), ∂y 3 yielding one or two numbers. The contribution to dS from the boundary is the sum of all such numbers. To bring this into a recognizable format, we take our cue from discrete Lagrangian mechanics, which featured two 1-forms. Here the above list suggests the three 1-forms on J 1 Y , the first of which we define to be 21L (yij , yi j+1 , yi+1 j+1 ) · (vyij , vyi j+1 , vyi+1 j+1 ) ∂L (yij , yi j+1 , yi+1 j+1 ) · (vyij , 0, 0), ≡ ∂y 1 22L and 23L being defined analogously. With these notations, the contribution to dS from the boundary can be written θL (φ) · V , where θL is the 1-form on the space of sections CU defined by   X X  (5.3) (j 1 φ)∗ (j 1 V 2lL ) (1) . θL (φ) · V ≡ 1;1∩∂U 6=∅

l;1l ∈∂U

In comparing (5.3) with (4.41), the analogy with the multisymplectic formalism of Sect. 4 is immediate. The discrete multisymplectic form formula. Given a triangle 1 in X, we define the projection π1 : CU → J 1 Y by π1 (φ) ≡ (1, y11 , y12 , y13 ). In this notation, it is easily verified that (5.3) takes the convenient form   X X ∗ l   π1 2L . θL = 1;1∩∂U 6=∅

(5.4)

l;1l ∈∂U

A first-variation at a solution φ of the DELF equations is a vertical vector field V such that the associated flow F V maps φ to other solutions of the DELF equations. Set lL = −d2lL . Since 21L + 22L + 23L = dL,

(5.5)

one obtains 1L + 2L + 3L = 0, so that only two of the three 2-forms lL , l = 1, 2, 3 are essentially distinct. Exactly as in Sect. 2, the equation d2 S = 0, when specialized to two first-variations V and W now gives, by taking one exterior derivative of (5.4),

Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs

X

0 = dθL (φ)(V, W ) =



X



1;1∩∂U 6=∅

 V

W

∗ l  π1 L ,

l;1l ∈∂U

which in turn is equivalent to  X X  (j 1 φ)∗ (j 1 V 1;1∩∂U 6=∅

379

j1W



lL ) (1) = 0.

(5.6)

l;1l ∈∂U

Again, the analogy with the multisymplectic form formula for continuous spacetime (4.18) is immediate. The discrete Noether theorem. Suppose that a Lie group G with Lie algera g acts on F by vertical symmetries in such a way that the Lagrangian L is G-invariant. Then G acts on Y and J 1 Y in the obvious ways. Since there are three Lagrange 1-forms, there are three momentum maps J l , l = 1, 2, 3, each one a g∗ -valued function on triangles in X, and defined by Jξl ≡ ξJ 1 Y 2lL , for any ξ ∈ g. Invariance of L and (5.5) imply that J 1 + J 2 + J 3 = 0, so, as in the case of the 1-forms, only two of the three momenta are essentially distinct. For any ξ, the infinitesimal generator ξY is a first-variation, so invariance of S, namely ξY dS = 0 , becomes ξY θL = 0. By left insertion into (5.3), this becomes the discrete version of Noether’s theorem:   X X  J l (1) = 0. (5.7) 1;1∩∂U 6=∅

l;1l ∈∂U

Conservation in a space and time split. To understand the significance of (5.6) and (5.7) consider a discrete field theory with space a discrete version of the circle and time the real line, as depicted in Fig. 5.3, where space is split into space and time, with “constant time” being constant j and the “space index” 1 ≤ i ≤ N being cyclic. Applying (5.7) to the region {(i, j) | j = 0, 1, 2} shown in the figure, Noether’s theorem takes the conservation form N X

J 1 (yi0 , yi1 , yi+1 1 ) = −

i=1

N X

J 2 (yi1 , yi2 , yi+1 2 ) + J 3 (yi1 , yi2 , yi+1 2 )

i=1

=

N X

J 1 (yi1 , yi2 , yi+1 2 ).

i=1

Similarly, the discrete multisymplectic form formula also takes a conservation form. When there is spatial boundary, the discrete Noether theorem and the discrete multisymplectic form formulas automatically account for it, and thus form nontrivial generalizations of these conservation results.

380

J. E. Marsden, G. W. Patrick, S. Shkoller

i+1

Space i i- 1

j=1

j= 0

j=2

Time

Fig. 5.3. Symplectic flow and conservation of momentum from the discrete Noether theorem when the spatial boundary is empty and the temporal boundaries agree

Furthermore, as in the continuous case, we can achieve “evolution type” symplectic systems (i.e. discrete Moser–Veselov mechanical systems) if we define Q as the space of fields at constant j, so Q ≡ F N , and take as the discrete Lagrangian ˜ j0 ], [qj1 ]) ≡ L([q

N X

1 L(qi0 , qi1 , qi+1 ).

i=1

Then the Moser–Veselov DEL evolution-type equations (3.2) are equivalent to the DELF equations (5.2), the multisymplectic form formula implies symplecticity of the Moser– Veselov evolution map, and conservation of momentum gives identical results in both the “field” and “evolution” pictures. Example: Nonlinear wave equation. To illustrate the discretization method we have developed, let us consider the Lagrangian (4.32) of Sect. 4, which describes the nonlinear sine-Gordon wave equation. This is a completely integrable system with an extremely interesting hierarchy of soliton solutions, which we shall investigate by developing for it a variational multisymplectic-momentum integrator; see the recent article by Palais [1997] for a wonderful discussion on soliton theory. To discretize the continuous Lagrangian, we visualize each triangle 1 as having base length h and height k, and we think of the discrete jet (y11 , y12 , y13 ) as corresponding to the continuous jet yi j+1 − yij ∂φ , (y¯ij ) = ∂x0 h

∂φ yi+1 j+1 − yi j+1 , (y¯ij ) = ∂x1 k

where y¯ij is the center of the triangle 3 . This leads to the discrete Lagrangian L=

1 2

y2 − y1 h

2 −

1 2

y3 − y2 k

2 +N

y + y + y 1 2 3 , 3

with corresponding DELF equations 3 Other discretizations based on triangles are possible; for example, one could use the value y for insertion ij into the nonlinear term instead of y¯ij .

Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs

yi+1 j − 2 yij + yi−1 j yi j+1 − 2 yij + yi j−1 − k2 h2 yij + yi j+1 + yi+1 j+1 1 + N0 3 3 1 0 yi j−1 + yij + yi+1 j + N 3 3 1 0 yi−1 j−1 + yi−1 j + yij = 0. + N 3 3

381

(5.8)

When N = 0 (wave equation) this gives the explicit method yi j+1 =

h2 (yi+1 j − 2 yij + yi−1 j ) + 2 yij − yi j−1 , k2

which is stable whenever the Courant stability condition is satisfied. Extensions: Jets from rectangles and other polygons. Our choice of discrete jet bundle is obviously not restricted to triangles, and can be extended to rectangles or more general polygons (left of Fig. 5.4). A rectangle is a quadruple of the form, 1 = (i, j), (i, j + 1), (i + 1, j + 1), (i + 1, j) , a point is an interior point of a subset U of rectangles if U contains all four rectangles touching that point, the discrete Lagrangian depends on variables y1 , · · · , y4 , and the DELF equations become ∂L ∂L (yij , yi j+1 , yi+1 j+1 , yi+1 j ) + 2 (yi j−1 , yij , yi+1 j , yi+1 j−1 ) ∂y 1 ∂y ∂L ∂L + 3 (yi−1 j−1 , yi−1 j , yij , yi j−1 ) + 4 (yi−1 j , yi−1 j+1 , yi j+1 , yij ) = 0. ∂y ∂y The extension to polygons with even higher numbers of sides is straightforward; one example is illustrated on the right of Fig. 5.4. The motivation for consideration of these

(i +1, j −1)

(i +1, j) (i +1, j +1)

(i, j −1) (i, j +1) (i −1, j−1) (i−1, j)

(i−1, j +1)

Fig. 5.4. On the left, the method based on rectangles; on the right, a possible method based on hexagons

extensions is enhancing the stability of the triangle-based method in the nonlinear wave example just above.

382

J. E. Marsden, G. W. Patrick, S. Shkoller

Example: Nonlinear wave equation, rectangles. Think of each rectangle 1 as having length h and height k, and each discrete jet (y11 , y12 , y13 , y14 ) being associated to the continuous jet ∂φ yi j+1 − yij 1 yi+1 j − yi j yi+1 j+1 − yi j+1 ∂φ , + , (p) = (p) = ∂x0 h ∂x1 2 k k where p is a the center of the rectangle. This leads to the discrete Lagrangian 2 2 1 y4 − y1 y3 − y2 1 y2 − y1 − + L= 2 h 2 2k 2k y + y + y + y 1 2 3 4 . +N 4

(5.9)

If, for brevity, we set y¯ij ≡

yij + yi j+1 + yi+1 j+1 + yi+1 j , 4

then one verifies that the DELF equations become 1 yi+1 j − 2 yij + yi−1 j 1 yi+1 j+1 − 2 yi j+1 + yi−1 j+1 + 2 k2 4 k2 1 yi+1 j−1 − 2 yi j−1 + yi−1 j−1 yi j+1 − 2 yij + yi j−1 + − 4 k2 h2 1 0 0 0 0 + N (y¯ij ) + N (y¯i j−1 ) + N (y¯i−1 j−1 ) + N (y¯i−1 j ) = 0, 4 which, if we make the definitions ∂h2 yij ≡ yi j+1 − 2 yij + yi j−1 , ∂k2 yij ≡ yi+1 j − 2 yij + yi−1 j , 1 N 0 (y¯ij ) + N 0 (y¯i j−1 ) + N 0 (y¯i−1 j−1 ) + N 0 (y¯i−1 j ) , N¯ 0 (y¯ij ) ≡ 4 is (more compactly) 1 1 2 1 2 1 1 2 ∂ yi j+1 + ∂k yij + ∂k yi j−1 − 2 ∂h2 yij + N¯ 0 (y¯ij ) = 0. k2 4 k 2 4 h

(5.10)

These are implicit equations which must be solved for yi j+1 , 1 ≤ i ≤ N , given yi j , yi j−1 , 1 ≤ i ≤ N ; rearranging, an iterative form equivalent to (5.10) is h2 h2 yi+1 j+1 + yi j+1 − yi−1 j+1 − 2(h2 + 2k 2 ) 2(h2 + 2k 2 ) h2 1 = 2 − 2 y + y ) + − 2 y + y ) (y (y i+1 j ij i−1 j i+1 j−1 i j−1 i−1 j−1 h + 2k 2 2 2 2k 2 yij − yi j−1 + 2 h + 2k 2 h2 k 2 N 0 (y¯ij ) + N 0 (y¯i j−1 ) + N 0 (y¯i−1 j−1 ) + N 0 (y¯i−1 j ) . + 2(h2 + 2k 2 )

Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs

383

In the case of the sine-Gordon equation the values of the field ought to be considered as lying in S1 , by virtue of the vertical symmetry y 7→ y + 2π. Soliton solutions for example will have a jump of 2π and the method will fail unless field values at closetogether spacetime points are differenced modulo 2π. As a result it becomes important to calculate using integral multiples of small field-dependent quantities, so that it is clear when to discard multiples of 2π, and for this the above iterative form is inconvenient. But if we define ∂h1 yij ≡ yi j+1 − yij ,

∂k1 yij ≡ yi+1 j − yij ,

then there is the following iterative form, again equivalent to (5.10), yi j+1 = yij + ∂h1 yi j , −

h2 2 2(h + 2k 2 )

and

∂h1 yi+1 j + ∂h1 yi j −

h2 2 2(h + 2k 2 )

∂h1 yi−1 j

h2 2k 2 (3∂k2 yij + ∂k2 yi j−1 ) + 2 ∂ 1 yij 2 + 2k h + 2k 2 h h2 k 2 N¯ 0 (y¯ij ). + 2(h2 + 2k 2 )

=

h2

(5.11)

One can also modify (5.9) so as to treat space and time symmetrically, which leads to the discrete Lagrangian 1 L= 2

2 2 y2 − y 1 y 3 − y 4 1 y4 − y 1 y3 − y2 + + − 2h 2h 2 2k 2k y + y + y + y 1 2 3 4 , +N 4

and one verifies that the DELF equations become 1 2 1 2 1 1 2 ∂ ∂ ∂ y + y + y i j+1 ij i j−1 k2 4 k 2 k 4 k 1 1 1 1 − 2 ∂h2 yi+1 j + ∂h2 yij + ∂h2 yi−1 j + N¯ 0 (y¯ij ) = 0, h 4 2 4

(5.12)

an equivalent iterative form of which is yi j+1 = yij + ∂h1 yi j , and 2 2 h − k2 h − k2 1 1 ∂h yi+1 j + ∂h yi j − ∂h1 yi−1 j − 2(h2 + k 2 ) 2(h2 + k 2 ) =

h2 (3∂ 2 yij + ∂k2 yi j−1 ) 2(h2 + k 2 ) k h2 (2∂ 1 yij + ∂h1 yi+1 j + ∂h1 yi−1 j ) + 2(h2 + k 2 ) h h2 k 2 N¯ 0 (y¯ij ). + 2(h2 + k 2 )

(5.13)

384

J. E. Marsden, G. W. Patrick, S. Shkoller

0 6.28 0. -6.28 0

-.002 Time

40 40

Space

0

0

Time

40

6.28

6.28

0 0

Space

40

1

Space grid points

16

Fig. 5.5. Top left: the wave forms for a two soliton kink and antikink collision using (5.12). Top right: the energy error. Bottom left: the wave form at time t ≈ 11855. Bottom right: the portion of the bottom left graph for spatial grid points 1 . . . 16

5.2. Numerical checks. While the focus of this article is not the numerical implementation of the integrators which we have derived, we have, nevertheless, undertaken some preliminary numerical investigations of our multisymplectic methods in the context of the sine-Gordon equation with periodic boundary conditions. The rectangle-based multisymplectic method. The top half of Fig. 5.5 shows a simulation of the collision of “kink” and “antikink” solitons for the sine-Gordon equation, using the rectangle-based multisymplectic method (5.12). In the bottom half of that figure we show the result of running that simulation until the solitons have undergone about 460 collisions; shortly after this the simulation stops because the iteration (5.13) diverges. The anomalous spatial variations in the waveform of the bottom left of Fig. 5.5 have period 2 spatial grid divisions and are shown in finer scale on the bottom right of that figure. These variations are reminiscent of those found in Ablowitz, Herbst and Schober [1996] for the completely integrable discretization of Hirota, where the variations are attributed to independent evolution of waveforms supported on even vs. odd grid points. Observation of (5.12) indicates what is wrong: the nonlinear term N contributes to (5.12) in a way that will average out these variations, and consequently, once they have begun, (5.12) tends to continue such variations via the linear wave equation. In Ablowitz et. al., the situation is rectified when the number of spatial grid points is not even, and this is the case

Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs

385

for (5.12) as well. This is indicated on the left of Fig. 5.6, which shows the waveform after about 5000 soliton collisions when N = 255 rather than N = 256. Figure 5.7 summarizes the evolution of energy error4 for that simulation. 6.28

6.28

0

0 0

40

Space

0

40

Space

Fig. 5.6. On the left, the final wave form (after about 5000 soliton collisions at t ≈ 129133) obtained using the rectangle-based multisymplectic method (5.12). On the right, the final waveform (at t ≈ 129145) from the energy-conserving method (5.14) of Vu-Quoc and Li. In both simulations, temporal drift is occurring. For this reason the waveforms are inverted with respect to one another; moreover, the separate solitons are drifting at slightly different rates, as indicated by the off-center waveforms

0

0

Energy

Energy

-.002

-.002

0

120000

Time

129130

129155

Time

Fig. 5.7. On the left, the energy error corresponding to our multisymplectic method (5.13) for 5000 solition collisions; the three graphs correspond to the minimum, average, and maximum energy error over consecutive 5000 time step regions. On the right, the final energy error (i.e. the energy error after about 5000 soliton collisions), which can be compared with the initial energy error plot in the top left of Fig. 5.5

4

The discrete energy that we calculated was N X 1 yi j+1 − yij i=1

2

2h +

1 2

y

+

yi+1 j+1 − yi+1 j 2h

2

− yij yi+1 j+1 − yi j+1 + 2k 2k

i+1 j

2

− N y¯ij

.

386

J. E. Marsden, G. W. Patrick, S. Shkoller

Initial data. For the two-soliton-collision simulations, we used the following initial data: h = k/8 (except h = k/16 where noted), where k = 40/N and N = 255 spatial grid points (except Fig. 5.5 where N = 256). The circle that is space should be visualized as having circumference L = 40. Let κ = 1 − , where = 10−6 , L˜ = L/4 = 10, s Z 1/κ 1 L˜ 2 p p dy ≈ 15.90, c = 1 − 2 2 ≈ .7773, P =2 κ P 1 − y 2 1 − κ2 y 2 0 x ˜ √ ;κ . φ(x) ≡ 2 arcsin sn κ 1 − c2

and

˜ − ct) is a kink solution if space has a circumference of L. ˜ This kink and an Then φ(x oppositely moving antikink (but placed on the last quarter of space) made up the initial field, so that yi0 = φ(40(i − 1)/N ), i = 1, . . . , N , where   φ(x) 0 ≤ x < L/4 L/4 ≤ x < 3L/4 , φ(x) ≡ 2π  2π − φ(x − 3L/4) 3L/4 ≤ x < L ˙ − 1)/N )h, where while yi1 = yi0 + φ(40(i   (φ(x − hc) − φ(x))/h 0 ≤ x < L/4 ˙ L/4 ≤ x < 3L/4 . φ(x) ≡ 0  −(φ(x − hc) − φ(x))/h 3L/4 ≤ x < L Comparison with energy-conserving methods. As an example of how our method compares with an existing method, we considered the energy-conserving method of Vu-Quoc and Li [1993], p. 354: 1 2 1 2 1 1 1 2 ∂ ∂ ∂ y + y + y − 2 ∂h2 yij i j+1 ij i j−1 k k k 2 k 4 2 4 h 1 N (yi j+1 ) − N (yij ) N (yij ) − N (yi j−1 ) + + = 0. (5.14) 2 yi j+1 − yij yij − yi j−1 This has an iterative form similar to (5.13) and is quite comparable with (5.10) and (5.12) in terms of the computation required. Our method seems to preserve the soliton waveform better than (5.14), as is indicated by comparison of the left and right Fig. 5.6. In regards to the closely related papers Vu-Quoc and Li [1993] and Li and VuQuoc [1995], we could not verify in our simulations that their method conserves energy, nor could we verify their proof that their method conserves energy. So, as a further check, we implemented the following energy-conserving method of Guo, Pascual, Rodriguez, and Vazquez [1986]: ∂k2 yij − ∂h2 yij +

N (yi j+1 ) − N (yi j−1 ) , yi j+1 − yi j−1

which conserves the discrete energy N X j=1

1 (yi j+1 − yij )(yij − yi j−1 ) 1 + 2 h2 2

yi+1 j − yij k

2

(5.15) ! − N (yij ) .

Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs

387

.2

.00015

Energy

0 0

-.0001

0

Time

0

40

Time

120000

6.28 .18

Energy

.174

0 129130

Time

129155

0

Space

40

Fig. 5.8. A long-time simulation using the energy-conserving method (5.15) of Guo et al. Above left: the initial energy error. Above right: the average energy error over consecutive 5000 time step regions (the maximum and minimum closely parallel the average). Below left: the final energy error. Below right: the final waveform at t ≈ 129149

This method diverged after just 345 soliton collisions. As can be seen from (5.15), the nonlinear potential N enters as a difference over two grid spacings, which suggests that halving the time step might result in a fairer comparison with the methods (5.12) or (5.14). With this advantage, method (5.15) was able to simulate 5000 soliton collisions, with a waveform degradation similar to the energy-conserving method (5.14), as shown at the bottom right of Fig. 5.8. The same figure also shows that, although the energy behavior of (5.15) is excellent for short time simulations, it drifts significantly over long times, and the final energy error has a peculiar appearance. Figure 5.9 shows the time evolution of the waveform through the soliton collision that occurs just before the simulation stops. Apparently, at the soliton collisions, significant high frequency oscillations are present, and these are causing the jumps in the energy error in the bottom left plot of Fig. 5.8. This error then accumulates due to the energy-conserving property of the method. In these simulations, so as to guard against the possibility that this behavior of the energy was due to inadequately solving the implicit Eq. (5.15), we imposed a minimum limit of 3 iterations in the corresponding iterative loop, whereas this loop would otherwise have converged after just 1 iteration. Comparison with the triangle-based multisymplectic method. The discrete second derivatives in the method (5.15) are the same as in the triangle-based multisymplectic method (5.8); these derivatives are simpler than either our rectangle-based multisymplectic method (5.12) or the energy-conserving method of Vu-Quoc and Li (5.14). To explore this we implemented the triangle-based multisymplectic method (5.8). Even

388

J. E. Marsden, G. W. Patrick, S. Shkoller

3

2.3

1.2

0

0

0

.12

6.3

6.2

-.08

5.2

4.2

Fig. 5.9. The soliton collision at time t ≈ 129130, after the energy-conserving method (5.15) of Guo et al. has simulated about 5000 soliton collisions. The solitons collide beginning at the top left and proceed to the top right, then to the bottom left, and finally to the bottom right. The vertical scales are not constant and visually exaggerate the high frequency oscillations, which are small on the scale 0 to 2π

with the less complicated discrete second derivatives our triangle-based multisymplectic method simulated 5000 soliton collisions with comparable energy5 and waveform preservation properties as the rectangle-based multisymplectic method (5.12), as shown in Fig. 5.11. Figure 5.10 shows the time evolution of the waveform through the soliton collision just before the simulation stops, and may be compared to Fig. 5.9. As can be seen, the high frequency oscillations that are present during the soliton collisions are smaller and smoother for the triangle-based multisymplectic method than for the energy-conserving method (5.15). A similar statement is true irrespective which of the two multisymplectic or two energy conserving methods we tested, and is true all along the waveform, irrespective of whether or not a soliton collision is occurring. Summary. Our multisymplectic methods are finite difference methods that are computationally competitive with existing finite difference methods. Our methods show promise for long-time simulations of conservative partial differential equations, in that, for long-time simulations of the sine-Gordon equation, our method 1) had superior energy-conserving behavior, even when compared with energy-conserving methods; 2) better preserved the waveform than energy-conserving methods; and 3) exhibited superior stability, in that our methods excited smaller and more smooth high frequency oscillations than energy-conserving methods. However, further numerical investigation is certainly necessary to make any lasting conclusions about the long-time behavior of our integrator. 5

The discrete energy that we calculated was N X 1 yi j+1 − yij 2 i=1

2

h

+

1 2

y

− yij 2k

i+1 j

2

− N (y¯ij )

.

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3

2.3

1.2

0

0

0

.035

6.2

-.015

5.2

6.2

4

Fig. 5.10. Similar to the above plot but for our triangle-based multisymplectic method (5.8)

.1

.14

0 Energy

-.1 0

Time

40

-.14 0

.1

.1

0

0

-.1

-.1 0

Time

40

0

Time

Time

120000

40

Fig. 5.11. A simulation of 5000 soliton collisions using the triangle-based multisymplectic method (5.8). Above left: The initial energy error. Above right: The minimum, average and maximum energy as in the left of Fig. 5.7. Below left: the final waveform (at t ≈ 129130). Below right: the final energy error

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The programs. The programs that were used in the preceding simulations are “C” language implementations of the various methods. A simple tridiagonal LUD method was used to solve the linear equations (e.g. the left side of (5.13)), as in Vu-Quoc and Li [1993], p. 379. An 8th order extrapolator was used to provide a seed for the implicit step. All calculations were performed in double precision while the implicit step was terminated when the fields ceased to change to single precision; the program’s output was in single precision. The extrapolation usually provided a seed accurate enough so that the methods became practically explicit, in that for many of the time-steps the first or second run through the iterative loops solving the implicit equations solved those equations to single precision. However, in the absence of a regular spacetime grid the expenses of the extrapolation and solving the linear equation would grow. Our programs are freely available at URL http://www.cds.caltech.edu/shkoller/mps.

6. Concluding Remarks Here we make a few miscellaneous comments and remark on some work planned for the future. Lagrangian reduction. As mentioned in the text, it is useful to have a discrete counterpart to the Lagrangian reduction of Marsden and Scheurle [1993a,b], Holm, Marsden and Ratiu [1998a] and Cendra, Marsden and Ratiu [1998]. We sketch briefly how this theory might proceed. This reduction can be done for both the case of “particle mechanics” and for field theory. For particle mechanics, the simplest case to start with is an invariant (say left) Lagrangian on the tangent bundle of a Lie group: L : T G → R. The reduced Lagrangian is l : g → R and the corresponding Euler–Poincar´e equations have a variational principle of Lagrange d’Alembert type in that there are constraints on the allowed variations. This situation is described in Marsden and Ratiu [1994]. The discrete analogue of this would be to replace a discrete Lagrangian L : G×G → R by a reduced discrete Lagrangian ` : G → R related to L by `(g1 g2−1 ) = L(g1 , g2 ). In this situation, the algorithm from G × G to G × G reduces to one from G to G and it is generated by ` in a way that is similar to that for L. In addition, the discrete variational principle for L which states that one should find critical points of L(g1 , g2 ) + L(g2 , g3 ) with respect to g2 to implicitly define the map (g1 , g2 ) 7→ (g2 , g3 ), reduces naturally to the following principle: Find critical points of `(g) + `(h) with respect to variations of g and h of the form gξ := Lg ξ and ξh = Rh ξ, where Lg and Rh denote left and right translation and where ξ ∈ g. In other words, one sets to zero, the derivative of the sum `(gg−1 ) + `(g h) with respect to at = 0 for a curve g in G that passes through the identity at = 0. This defines (with caveats of regularity as before) a map of G to itself, which is the reduced algorithm. This algorithm can then be used to advance points in G × G itself, by advancing each component by the reduced trajectory, reproducing the algorithm on G × G. In addition, this can be used with the adjoint or coadjoint action to advance points in g∗ to approximate the Lie–Poisson dynamics.

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These equations for a discrete map, say φ` : G → G generated by ` on G are called the discrete Euler–Poincar´e equations as they are the discrete analogue of the Euler– Poincar´e equations on g. Notice that, at least in theory, computation can be done for this map first and then the dynamics on G × G is easily reconstructed by simply advancing ¯ 2 ), where h¯ = φ` (g −1 g2 ) = φ` (h). each pair as follows: (g1 , g2 ) 7→ (hg1 , hg 1 If one identifies the discrete Lagrangians with generating functions (as explained in Wendlandt and Marsden [1997]) then the reduced Lagrangian generates the reduced algorithm in the sense of Ge and Marsden [1988], and this in turn is closely related to the Lie–Poisson–Hamilton–Jacobi theory. Next, consider the more general case of T Q with its discretization Q × Q with a group action (assumed to be free and proper) by a Lie group G. The reduction of T Q by the action of G is T Q/G, which is a bundle over T (Q/G) with fiber isomorphic to g. The discrete analogue of this is (Q × Q)/G which is a bundle over (Q/G) × (Q/G) with fiber isomorphic to G itself. The projection map π : (Q × Q)/G → (Q/G) × (Q/G) is given by [(q1 , q2 )] 7→ ([q1 ], [q2 ]) where [ ] denotes the relevant equivalence class. Notice that in the case in which Q = G this bundle is “all fiber”. The reduced discrete Euler–Lagrange equations are similar to those in the continuous case, in which one has shape equations coupled with a version of the discrete Euler–Poincar´e equations. Of course all of the machinery in the continuous case can be contemplated here too, such as stability theory, geometric phases, etc. In addition, it would be useful to generalize this Lagrangian reduction theory to the multisymplectic case. All of these topics are planned for other papers. Role of uniformity of the grid. Consider an autonomous, continuous Lagrangian L : T Q → R where, for simplicity, Q is an open submanifold of Euclidean space. Imagine some not necessarily uniform temporal grid (t0 , t1 , · · · ) of R, so that t0 < t1 < t2 < · · · . In this situation, it is natural to consider the discrete action S=

n X k=1

Lk (qk , qk−1 ) ≡

n X k=1

L

qk + qk−1 qk − qk−1 , 2 tk − tk−1

(tk − tk−1 ).

(6.1)

This action principle deviates from the action principle (3.1) of Sect. 3 in that the discrete Lagrangian density depends explicitly on k. Of course nonautonomous continuous Lagrangians also yield k-dependent discrete Lagrangian densities, irrespective of uniformity of the grid. Thus, nonuniform temporal grids or nonautonomous Lagrangians give rise to discrete Lagrangian densities which are more general than those we have considered in Sect. 3. For field theories, the Lagrangian in the action (5.1) depends on the spacetime variables already, through its explicit dependence on the triangle 1. However, it is only in the context of a uniform grid that we have experimented numerically and only in that context that we have discussed the significance of the discrete multisymplectic form formula and the discrete Noether theorem. Using (6.1) as an example, will now indicate why the issue of grid uniformity may not be serious. The DEL equations corresponding to the action (6.1) are ∂Lk+1 ∂Lk (qk , qk−1 ) + (qk+1 , qk ) = 0, ∂q1 ∂q2

k = 1, 2, , · · · ,

and this gives evolution maps Fk+1,k : Q × Q → Q × Q defined so that Fk+1,k (qk , qk−1 ) = (qk+1 , qk ),

k = 1, 2, · · ·

(6.2)

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when (6.2) holds. For the canonical 1-forms corresponding to (3.4) and (3.5) we have the k-dependent one forms − (q1 , q0 ) · (δq1 , δq0 ) ≡ θL,k

∂Lk (q1 , q0 )δq0 , ∂q0

(6.3)

+ (q1 , q0 ) · (δq1 , δq0 ) ≡ θL,k

∂Lk (q1 , q0 )δq1 , ∂q1

(6.4)

and

and Eqs. (3.7) and (3.9) become − ∗ + (dθL,k ) = −dθL,k+1 , Fk+1,k

− + dθL,k + dθL,k =0

(6.5)

respectively. Together, these two equations give ∗ + + (dθL,k ) = dθL,k+1 , Fk+1,k

(6.6)

and if we set Fk ≡ Fk,k−1 ◦ Fk−1,k−2 ◦ · · · ◦ F2,1 + + then (6.6) chain together to imply Fk∗ (dθL,1 ) = dθL,k . This appears less than adequate since it merely says that the pull back by the evolution of a certain 2-form is, in general, a different 2-form. The significant point to note, however, is that this situation may be repaired at any k simply by choosing Lk = L1 . It is easily verified that the analogous statement is true with respect to momentum preservation via the discrete Noether theorem. Specifically, imagine integrating a symmetric autonomous mechanical system in a timestep adaptive way with Eqs. (6.2). As the integration proceeds, various timesteps are chosen, and if momentum is monitored it will show a dependence on those choices. A momentum-preserving symplectic simulation may be obtained by simply choosing the last timestep to be of equal duration to the first. This is the highly desirable situation which gives us some confidence that grid uniformity is a nonissue. There is one caveat: symplectic integration algorithms are evolutions which are high frequency perturbations of the actual system, the frequency being the inverse of the timestep, which is generally far smaller than the time scale of any process in the simulation. However, timestep adaptation schemes will make choices on a much larger time scale than the timestep itself, and then drift in the energy will appear on this larger time scale. A meaningful longtime simulation cannot be expected in the unfortunate case that the timestep adaptation makes repeated choices in a way that resonates with some process of the system being simulated.

The sphere. The sphere cannot be generally uniformly subdivided into spherical triangles; however, a good approximately uniform grid is obtained as follows: start from an inscribed icosahedron which produces a uniform subdivision into twenty spherical isosceles triangles; these are further subdivided by halving their sides and joining the resulting points by short geodesics. Elliptic PDEs. The variational approach we have developed allows us to examine the multisymplectic structure of elliptic boundary value problems as well. For a given Lagrangian, we form the associated action function, and by computing its first variation, we obtain the unique multisymplectic form of the elliptic operator. The multisymplectic

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form formula contains information on how symplecticity interacts with spatial boundaries. In the case of two spatial dimensions, X = R2 , Y = R3 , we see that Eq. (4.36) gives us the conservation law divX = 0, where the vector X = (ω 0 (j 1 V, j 1 W ), ω 1 (j 1 V, j 1 W )). Furthermore, using our generalized Noether theory, we may define momentummappings of the elliptic operator associated with its symmetries. It turns out that for important problems of spatial complexity arising in, for example, pattern formation systems, the covariant Noether current intrinsically contains the constrained toral variational principles whose solutions are the complex patterns (see Marsden and Shkoller [1998]). There is an interesting connection between our variational construction of multisymplectic-momentum integrators and the finite element method (FEM) for elliptic boundary value problems. FEM is also a variationally derived numerical scheme, fundamentally differing from our approach in the following way: whereas we form a discrete action sum and compute its first variation to obtain the discrete Euler–Lagrange equations, in FEM, it is the original continuum action function which is used together with a projection of the fields and their variations onto appropriately chosen finite-dimensional spaces. One varies the projected fields and integrates such variations over the spatial domain to recover the discrete equations. In general, the two discretization schemes do not agree, but for certain classes of finite element bases with particular integral approximations, the resulting discrete equations match the discrete Euler–Lagrange equations obtained by our method, and are hence naturally multisymplectic. To illustrate this concept, we consider the Gregory and Lin method of solving twopoint boundary value problems in optimal control. In this scheme, the discrete equations are obtained using a finite element method with a basis of linear interpolants. Over each one-dimensional element, let N1 and N2 be the two linear interpolating functions. RT ˙ Discretizing the As usual, we define the action function by S(q) = 0 L(q(t), q(t))dt. interval [0, T ] into N +1 uniform elements, we may write the action with fields projected onto the linear basis as N −1 Z k+1 X L({N1 φk + N2 φk+1 }, {N˙ 1 φk + N˙ 2 φk+1 })dt. S(q) = k=0

k

Since the Euler–Lagrange equations are obtained by linearizing the action and hence the Lagrangian, and as the functions Ni are linear, one may easily check that by evaluating the integrals in the linearized equations using a trapezoidal rule, the discrete Euler– Lagrange equations given in (3.3) are obtained. Thus, the Gregory and Lin method is actually a multisymplectic-momentum algorithm. Applicability to fluid problems. Fluid problems are not literally covered by the theory presented here because their symmetry groups (particle relabeling symmetries) are not vertical. A generalization is needed to cover this case and we propose to work out such a generalization in a future paper, along with numerical implementation, especially for geophysical fluid problems in which conservation laws such as conservation of enstrophy and Kelvin theorems more generally are quite important. Other types of integrators. It remains to link the approaches here with other types of integrators, such as volume preserving integrators (see, e.g., Kang and Shang [1995], Quispel [1995]) and reversible integrators (see, e.g., Stoffer [1995]). In particular since

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volume manifolds may be regarded as multisymplectic manifolds, it seems reasonable that there is an interesting link. Constraints. One of the very nice things about the Veselov construction is the way it handles constraints, both theoretically and numerically (see Wendlandt and Marsden [1997]). For field theories one would like to have a similar theory. For example, it is interesting that for fluids, the incompressibility constraint can be expressed as a pointwise constraint on the first jet of the particle placement field, namely that its Jacobian be unity. When viewed this way, it appears as a holonomic constraint and it should be amenable to the present approach. Under reduction by the particle relabeling group, such a constraint of course becomes the divergence free constraint and one would like to understand how these constraints behave under both reduction and discretization. Acknowledgement. We would like to extend our gratitude to Darryl Holm, Tudor Ratiu and Jeff Wendlandt for their time, encouragement and invaluable input. Work of J. Marsden was supported by the California Institute of Technology and NSF grant DMS 96–33161. Work by G. Patrick was partially supported by NSERC grant OGP0105716 and that of S. Shkoller was partially supported by the Cecil and Ida M. Green Foundation and DOE. We also thank the Control and Dynamical Systems Department at Caltech for providing a valuable setting for part of this work.

References Arms, J.M., Marsden, J.E. and Moncrief, V.: The structure of the space solutions of Einstein’s equations: II Several Killings fields and the Einstein–Yang–Mills equations. Ann. of Phys. 144, 81–106 (1982) Ablowitz, M.J., Herbst, B.M., and Schober, C.: On the numerical solution of the Sine-Gordon equation 1. Integrable discretizations and homoclinic manifolds. J. Comp. Phys. 126, 299–314 (1996) Abraham, R. and Marsden, J.E.: Foundations of Mechanics. Second Edition, Reading, MA: Addison-Wesley, 1978 Arms, J.M., Marsden, J.E. and Moncrief, V.: The structure of the space solutions of Einstein’s equations: II Several Killings fields and the Einstein–Yang–Mills equations. Ann. of Phys. 144, 81–106 (1982) Arnold, V.I.: Mathematical Methods of Classical Mechanics. Graduate Texts in Math. 60, Berlin–Heidelberg– New York: Springer Verlag. 1978 (Second Edition, 1989) Ben-Yu, G, Pascual, P.J., Rodriguez, M.J. and Vazquez, L.: Numerical solution of the sine-Gordon equation. Appl. Math. Comput. 18, 1–14 (1986) Bridges, T.J.: Multi-symplectic structures and wave propagation. Math. Proc. Camb. Phil. Soc., 121, 147–190 (1997) Calvo, M.P. and Hairer, E.: Accurate long-time integration of dynamical systems. Appl. Numer. Math., 18, 95 (1995) Cendra, H., Marsden, J.E. and Ratiu, T.S.: Lagrangian reduction by stages. Preprint (1998) de Vogela´ere, R.: Methods of integration which preserve the contact transformation property of the Hamiltonian equations. Department of Mathematics, University of Notre Dame, Report No. 4 (1956) Dragt, A.J. and Finn, J.M. [1979]: Normal form for mirror machine Hamiltonians. J. Math. Phys. 20, 2649– 2660 Ebin, D. and Marsden, J.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. 92, 102–163 (1970) Ge, Z. and Marsden, J.E.: Lie–Poisson integrators and Lie–Poisson Hamilton–Jacobi theory. Phys. Lett. A 133, 134–139 (1988) Gotay, M., Isenberg, J. and Marsden, J.E.: Momentum Maps and the Hamiltonian Structure of Classical Relativistic Field Theories, I. Preprint. (1997) Gregory, J and Lin, C.: The numerical solution of variable endpoint problems in the calculus of variations, Lecture Notes in Pure and Appl. Math., 127, 175–183 (1991) Guo, B. Y., Pascual, P.J., Rodriguez, M.J. and Vazquez, L.: Numerical solution of the sine-Gordon equation. Appl. Math. Comput. 18, 1–14 (1986)

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Holm, D.D., Marsden, J.E. and Ratiu, T.: The Euler–Poincar´e equations and semidirect products with applications to continuum theories. Adv. in Math. To appear (1998a) Holm, D. D., J.E. Marsden, J.E. and Ratiu, T.: The Euler–Poincar´e equations in geophysical fluid dynamics. In: Proceedings of the Isaac Newton Institute Programme on the Mathematics of Atmospheric and Ocean Dynamics, Cambridge: Cambridge University Press (to appear) (1998b) Li, S. and Vu-Quoc, L.: Finite-difference calculus invariant structure of a class of algorithms for the nonlinear Klein–Gordon equation. SIAM J. Num. Anal. 32, 1839–1875 (1995) Marsden, J.E., Patrick, G.W. and Shadwick, W.F. (Eds.): Integration Algorithms and Classical Mechanics. Fields Institute Communications 10, Providence, RI: Am. Math. Society, 1996 Marsden, J.E. and Ratiu, T.S.: Introduction to Mechanics and Symmetry. Texts in Applied Mathematics 17, Berlin–Heidelberg–New York: Springer-Verlag , 1994 Marsden, J.E. and Shkoller, S.M.: Multisymplectic geometry, covariant Hamiltonians and water waves. Math. Proc. Camb. Phil. Soc. 124 (1998) Marsden, J.E. and Wendlandt, J.M.: Mechanical systems with symmetry, variational principles and integration algorithms. Current and Future Directions in Applied Mathematics, Edited by M. Alber, B. Hu, and J. Rosenthal, Base–Boston: Birkh¨auser, pp. 219–261 (1997) McLachlan, R.I. and Scovel, C.: A survey of open problems in symplectic integration. Fields Institute Communications 10, 151–180 (1996) Moser, J. and Veselov, A.P.: Discrete versions of some classical integrable systems and factorization of matrix polynomials. Commun. Math. Phys. 139, 217–243 (1991( Neishtadt, A.: The separation of motions in systems with rapidly rotating phase. P.M.M. USSR 48, 133–139 (1984) Palais, R.S.: Foundations of global nonlinear analysis, New York: Benjamin, 1968 Palais, R.S.: The symmetries of solitons. Bull. Am. Math. Soc. 34, 339–403 (1997) Quispel, G.R.W.: Volume-preserving integrators. Phys. Lett. A. 206, 26 (1995) Sanz-Serna, J. M. and Calvo, M.: Numerical Hamiltonian Problems. London: Chapman and Hall, 1994 Simo, J.C.and Gonzalez., O.: Assessment of Energy-Momentum and Symplectic Schemes for Stiff Dynamical Systems. Proc. ASME Winter Annual Meeting, New Orleans, Dec. 1993 Stoffer, D.: Variable steps for reversible integration methods, Computing 55, 1 (1995) Veselov, A.P.: Integrable discrete-time systems and difference operators. Funkts. Anal. Prilozhen. 22, 1–13 (1988) Veselov, A.P.: Integrable Lagrangian correspondences and the factorization of matrix polynomials. Funkts. Anal. Prilozhen. 25, 38–49 (1991) Vu-Quoc, L. and Li, S.: Invariant-conserving finite difference algorithms for the nonlinear Klein–Gordon equation. Comput. Methods Appl. Mech. Engrg. 107, 341–391 (1993) Wald, R.M.: Variational principles, local symmetries and black hole entropy. Proc. Lanczos Centennary volume SIAM, 231–237 (1993) Wendlandt, J.M. and Marsden, J.E.: Mechanical integrators derived from a discrete variational principle. Physica D 106, 223–246 (1997) Communicated by A. Kupiainen

Commun. Math. Phys. 199, 397 – 416 (1998)

Communications in

Mathematical Physics © Springer-Verlag 1998

Wild Attractors of Polymodal Negative Schwarzian Maps? Alexander Blokh1,?? , Michał Misiurewicz2 1

Department of Mathematics, University of Alabama in Birmingham, University Station, Birmingham, AL 35294-2060 USA. E-mail: [email protected] 2 Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, IN 46202-3216, USA. E-mail: [email protected] Received: 15 December 1997 / Accepted: 13 May 1998

Abstract: We study “wild attractors” of polymodal negative Schwarzian interval maps and prove that they are super persistently recurrent (a polymodal version of persistent recurrence). We also prove that if a map has an attractor which is a cycle of intervals then at almost every point of this cycle the map has properties similar to the Markov property introduced by Martens. Thus, the lack of super persistent recurrence at a critical point c can be considered as a mild topological expanding property, and this expansion prevents ω(c) from being a wild attractor (in the previous paper we have shown that it also prevents the map from being C 2 -stable).

1. Introduction In his paper [Mi] Milnor suggested a new approach to the dynamics based on the notion of attractor. He showed that a smooth dynamical system has a unique so-called global attractor and posed a problem of decomposing it into minimal attractors, closely related to that of describing ω-limit sets of almost all points. Since then many papers have appeared dealing with the problem (see our list of references, which is of course far from being complete). We continue this study and consider piecewise monotone (polymodal) negative Schwarzian maps of an interval. The results can be extended to one-dimensional branched manifolds, but to avoid complications we restrict our attention to the interval case (and thus give definitions only in this case, although some of them are more general). Precise and full definitions of some notions, as well as a lot of standard definitions, are given later. ? The first author was partially supported by NSF grant DMS 9626303 and Humboldt Foundation and would also like to thank G. Keller for inviting him to Erlangen and for useful discussions of the results of the paper. The second author was partially supported by NSF grant DMS 9704422. ?? Current address: Mathematisches Institut, Universt¨ at Erlangen-N¨urnberg, 91054 Erlangen, Germany

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2 Mostly, we consider two spaces of maps. The space Cnf consists of all maps of [0, 1] into itself of class C 2 with finite number of critical points, all of them non-flat. The space 2 that have negative Schwarzian. We denote the ω-limit S consists of those maps from Cnf set of a point x by ω(x) and call it simply the limit set of x. 2 Let f ∈ Cnf . Then for a set A ⊂ [0, 1], the set rl(A) = {x : ω(x) ⊂ A} is called the realm of attraction of the set A. Also, the set RL(A) = {x : ω(x) = A} is called the realm of exact attraction of the set A. Denote the Lebesgue measure of a set X by |X|. A closed invariant set A is called an attractor if

(1) | rl(A)| > 0; (2) | rl(A) \ rl(A0 )| > 0 for any proper closed invariant set A0 ⊂ A. Clearly, if | RL(A)| > 0 then A is an attractor; such attractor is called primitive. An attractor A is called global if |[0, 1] \ rl(A)| = 0. In [Mi] it is shown that f has a unique global attractor. It is denoted by A(f ). The same holds for the restriction of f to a closed invariant set K. Let us describe types of primitive attractors which can be considered natural. The first is rather simple. A point x is called a (one-sided) periodic sink if there exists n > 0 and a (one-sided) neighborhood U of x such that f n (x) = x, f n (U ) ⊂ U and the diameter of f k (U ) tends to 0 as k → ∞. The orbit of a periodic sink gives an example (perhaps the easiest one) of a primitive attractor. To introduce the next type of primitive attractors we need more definitions. A closed interval I is called periodic (of period n) if the interiors of the intervals I, . . . , f n−1 (I) Sn−1 are disjoint, while f n (I) ⊂ I. Then the union i=0 f i (I) is called a cycle of intervals and denoted by cyc(I). This includes also the case of n = 1; then cyc(I) = I. Clearly, if J ⊂ I and both I, J are periodic then the period of J is a multiple of the period of I (yet sequence of periodic these periods may well coincide). Let I0 ⊃ I1 ⊃ . . . be a nested T ∞ intervals of periods m0 < m1 < . . . . Then the intersection X = i=0 cyc(Ii ) is called a solenoidal set and the cycles of intervals cyc(Ii ) are called X-generating. The dynamics on X are well known (see, e.g., [B]) even when f is just a continuous interval map. In the smooth case it can be specified even further because of the absence of wandering intervals (an interval J ⊂ [0, 1] is called wandering for f if its images f n (J), n ≥ 0, are pairwise disjoint and do not converge to a periodic orbit). The following theorem was proven in a series of papers ([G, L1, BL1, MMS]). 2 have no wandering intervals. Theorem 1.1. Maps from Cnf

Theorem 1.1 implies that the map on X is conjugate to a minimal translation in a compact infinite zero-dimensional Abelian group. In this case for every point x absorbed by all X-generating cycles of intervals we have ω(x) = X (a point x is absorbed by an invariant set D if f m (x) ∈ D for some m). Let us sketch the proof of the fact that any solenoidal set of f is indeed a primitive attractor. By a theorem of Martens, de Melo and van Strien [MMS] for any f there exists a number N such that all attracting or neutral periodic points of f have periods less than N . Now, if S is a solenoidal set of f then we can choose a generating cycle of intervals cyc(I) of period greater than N , so that there will be no attracting/neutral periodic orbits in cyc(I). Also, if C 0 is the set of all critical points of f belonging to S then we can also assume that C 0 is the set of all critical points of f belonging to cyc(I). Let us now apply a theorem of Ma˜ne´ [Man], according to which almost all points of cyc(I) contain a critical point from C 0 in their limit sets. Thus, ω(x) = S for a.e. x ∈ cyc(I), which proves our claim.

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It may also happen that there exist a cycle of intervals cyc(I) such that f |cyc(I) is transitive. This case plays an important role in one-dimensional dynamics. If the set of points x ∈ cyc(I) such that ω(x) = cyc(I) is of positive Lebesgue measure then cyc(I) is a primitive attractor. These three examples may be considered natural for the following reason: they all are also topological attractors in the sense that the set RL(A) for them is topologically big (of type Gδ , dense in some intervals). In fact it is proven in [B] that if a continuous interval map has no wandering intervals then for a dense Gδ set of points their limit set is either a periodic orbit, or a solenoidal set, or a cycle of intervals, on which the map is transitive. Hence, if there is a limit set D such that the set of points x attracted by D (i.e. such that ω(x) = D) is Gδ and dense in some interval then D is necessarily of one of these types. However, an amazing fact is that for Milnor attractors there is a fourth possibility. A primitive attractor which does not belong to any of the three classes described above is called a wild attractor. In other words, a wild attractor is an infinite nowhere dense and non-solenoidal primitive attractor. In [BKNS] an example of a wild attractor for a unimodal map was given. In the series of papers ([BL2, BL3] for polymodal negative Schwarzian maps, [BL4] for unimodal negative Schwarzian maps, and [L2] for polymodal C 2 -maps) the following theorem was proven. 2 is the union of all sinks of Theorem 1.2. The global attractor A(f ) of a map f ∈ Cnf f and finitely many infinite primitive attractors Ai which are either solenoidal sets, or cycles of intervals on which a map is transitive, conservative and ergodic (with respect to Lebesgue measure) or wild attractors (on which f is minimal). Each set Ai contains a critical point of f and intersections between two of them are possible only if they are cycles of intervals with a few common boundary points.

Unlike other primitive attractors, wild attractors are not well understood other than for the unimodal negative Schwarzian maps. Our work was motivated by this, and is an attempt to study wild attractors of polymodal negative Schwarzian maps. Hence, first we describe that case in more detail. Let f : [0, 1] → [0, 1] be a piecewise monotone map. For x ∈ [0, 1] let us denote by Hn (x) the maximal interval containing x on which f n is monotone and let f n (Hn (x)) = Mn (x). Let rn (x) be the minimal distance between f n (x) and the endpoints of Mn (x). If f n has a local extremum at x, there is an ambiguity in the choice of Hn (x) and Mn (x), but rn (x) = 0 independently of this choice. Moreover, in that case rm (x) = 0 for all m ≥ n. Also, if x = 0 or 1, then rn (x) = 0 for all n. Thus either for some m we have rm (x) = 0 (and then rn (x) = 0 for all n ≥ m) or rn (x) 6= 0 for any n, in which case x is neither a preimage of a turning point nor 0, 1. A recurrent critical point c ∈ [0, 1] of a unimodal map is called persistently recurrent if rn (f (c)) → 0. Now we summarize some information known about unimodal negative Schwarzian maps; by Theorem 1.2 in this case f has at most one infinite primitive attractor A = ω(c). We say that a map is purely dissipative if it is not conservative on any set of positive measure (we use the terms “conservative” and “dissipative” with respect to the Lebesgue measure). Theorem 1.3. Let f ∈ S be unimodal with the critical point c. Then the following holds. (1) ([BL4, GJ, Ma]) If A is a wild attractor of f then c is persistently recurrent. Moreover, there exists a cycle of intervals cyc(I) such that A ⊂ cyc(I), f |cyc(I) is purely dissipative and rn (x) → 0 as n → ∞ for a.e. x ∈ cyc(I).

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(2) ([Ma]) Let cyc(I) be an attractor. Then there is ε > 0 such that lim sup rn (x) > ε for a.e. x. Our main result generalizes Theorem 1.3 to the polymodal case. To state it we need more notions. To shorten the introduction we do this in brief, at least with respect to well-known notions (precise definitions will be given later). First we need the notion of a chain introduced in [L1] for polymodal negative Schwarzian maps. In that paper they helped to prove non-existence of wandering intervals for such maps. Later chains were used to prove an analogous result for smooth polymodal interval maps (see [BL1, MMS]) and became a popular tool in one-dimensional dynamics. A sequence (Gi )li=0 of closed intervals is called a chain if Gi is a maximal interval such that f (Gi ) ⊂ Gi+1 , i = 0, . . . , l − 1. Given a point x and an interval I 3 f n (x) we construct a chain of intervals (Gi )ni=0 whose last interval Gn is equal to I and whose first interval G0 contains x. If such a chain exists, it is unique. We call it the pull-back chain of I along x, . . . , f n (x). The number of intervals of the chain containing critical points of f is called the order of the chain. For a map f of class C 1 with finitely many critical points let Cr(f ) = Cr be its set of critical points. For every point x and ε > 0 we construct the pull-back chain of [f n (x) − ε, f n (x) + ε] along x, . . . , f n (x). We define rnk (x) as the supremum of all ε such that we get a chain of order k or less. Let Ek,ε (f ) be the set of all points y with lim supn→∞ rnk (y) > ε. We call a point x such that for every k we have rnk (x) → 0 as n → ∞ critically super persistent or Cr-super persistent. If x is additionally recurrent, we call it critically super persistently recurrent or Cr-super persistently recurrent (cf. [BM1]). An important property of limit sets of Cr-super persistently recurrent points is that they are minimal ([BM2], see also Theorem 2.5 below). Also, if rn (x) 6→ 0 then we call x critically reluctant or Cr-reluctant. Now we can state our main theorem. Theorem 5.3. For every f ∈ S and a primitive attractor A that is neither a periodic orbit nor a solenoidal set, one of the following holds. (1) The attractor A is wild. Then A = ω(c) for some Cr-super persistently recurrent critical point c. Furthermore, A is contained in a basic set B(cyc(I)) such that f |B is purely dissipative, |A| = 0, and almost all points of B are Cr-super persistent. (2) The attractor A equals B(cyc(I)) = cyc(I) and if I is of period m then f m |f j (I) is exact for 0 ≤ j ≤ m − 1, f |cyc(I) is ergodic, conservative and cyc(I) ⊂ E0,ε for some ε > 0 up to a set of measure zero (almost all points of A are Cr-reluctant). Moreover, there exists a finite set Y ⊂ I such that: (a) Cr(f ) ∩ I 6⊂ Y ; (b) for any two intervals U ⊂ int(V ) disjoint from Y and for almost every x ∈ I there is an arbitrarily large n and two intervals x ∈ W 0 ⊂ W 00 such that f n |W 00 has no critical points, f n (W 00 ) = V and f n (W 0 ) = U . In Sect. 4 prove Theorem 4.6 which is a version of Theorem 5.3 with a milder statement (2) and then in Sect. 5 we strengthen it in Proposition 5.2 by establishing mild expanding properties for polymodal negative Schwarzian maps on their attractors which are cycles of intervals. These properties are similar to the ones proved by [Ma] for the unimodal case. It is this expansion that prevents the attractor from being wild. Similarly, such mild expansion along the trajectory of a critical point causes C 2 instability of the map. Note that just like we define Cr-super persistent points, we can define Cr-persistent points. To avoid trivial cases we assume that x is not an eventual preimage of a critical

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point of f and call it Cr-persistent if rn (x) → 0 and Cr-persistently recurrent if x is also recurrent. While it could seem that for the polymodal maps the natural thing is to use Crpersistent recurrence (as for unimodal maps), it is not so. Sometimes it is crucial that the Cr-super persistent recurrence and related notions are used instead. One such place is the proof of Theorem 4.6, where we have to use the set E(f ), and not a similar set defined for Cr-persistent recurrence. Actually, the results of this paper hold if we replace in the definition of S the assumption that f is of class C 2 by the assumption that it is of class C 1 . The reason for this is that the assumption on negative Schwarzian is quite strong. However, with this weaker assumption (that f is of class C 1 ) we would not be able to invoke several results that we need and that are proved in the literature for functions of class C 2 . While we could reprove them for functions of class C 1 with negative Schwarzian (by repeating existing proofs with some estimates changed), this would make this paper considerably longer and the results only slightly stronger. Therefore we choose not to do it.

2. Topological Properties of Chains In this section we first summarize well-known facts about chains. Then we introduce some new notions and state new results, the main one establishing the minimality of the limit sets of super persistently recurrent points of f with arbitrarily small nice smart neighborhoods (see definitions below). This section contains almost no proofs. They can be found elsewhere, mainly in [BM2]. Throughout this section we assume that f ; [0, 1] → [0, 1] is a piecewise monotone continuous map, strictly monotone on any lap. We call the local extrema of f (except 0 and 1) turning points. Let Kf be the closure of the convex hull of the union of the trajectories of the turning points of f . Clearly, Kf is a closed invariant interval. This is where the important things from the dynamical point of view happen. We want to / Kf . We call such f loosely have some extra space around Kf , so we assume that 0, 1 ∈ packed. This assumption is not restrictive at all, since one can extend any f to a loosely packed map on a slightly larger interval, preserving smoothness and negative Schwarzian if necessary. This means that the properties of limit sets established with an additional assumption that f is loosely packed, hold also without this assumption. Thus, from now on we assume that f is a loosely packed map. Also, we fix a finite set of points C ⊂ Kf containing all turning points of f , call these points exceptional (cf. [BM2]) and assume that together with a map f there always comes the set C of exceptional points. In the smooth case C is usually chosen as the set Cr of all critical points of f . However, we would like to emphasize that the results of this section hold for any set C ⊂ Kf containing all turning points of f , mainly because the definitions and arguments are topological. If c is a turning point of f , let us take the largest interval [a, b] such that a < c < b, f (a) = f (b) and f is monotone on each of the intervals [a, c] and [c, b]. Then there is a unique continuous function τc : [a, b] → [a, b] such that f ◦ τ = f and f (x) 6= x for x 6= c. This function is an involution, that is τc2 is the identity. Although we are not dealing with smooth functions in this section, it is convenient to give some definitions related to them. We call an interval map f a C 2 -map with non-flat 2 ) if f is of class C 2 with critical critical points (and denote the class of such maps by Cnf βk points {ck } such that the inequalities A1 |x − ck | ≤ |f 0 (x)| ≤ A2 |x − ck |βk hold in

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neighborhoods of the points ck with positive A1 , A2 , βk (the inequalities are called nonflatness inequalities). Clearly, this implies that f has finitely many critical points and for some constant R > 0 we have 1/R > |τc0 | > R in the corresponding neighborhood of any turning point c. Now for given loosely packed f and C we choose a positive constant κ such that (A1) the distance between any two exceptional points of f is greater than κ, (A2) for any turning point c of f , the κ-neighborhood of c is contained in the domain of τc , (A3) for two exceptional points b, c either f (b) = c or |f (b) − c| > κ, (A4) Kf ⊂ (κ, 1 − κ), 2 then non-flatness inequalities hold in κ-neighborhoods of ck . (A5) if f ∈ Cnf Clearly, any sufficiently small κ satisfies the above conditions. Now we define a chain modifying traditional definitions (see [L1, BL1, MMS]) to serve our purposes (for instance, we add (B3) below). We call an interval smart if it does not contain any set of the form f k (V ), where k ≥ 0 and V is a one-sided κ-neighborhood of an exceptional point of f . Note that any subinterval of a smart interval is also smart. A sequence (Gi )li=0 of closed intervals is called a chain if (B1) Gi is a maximal interval such that f (Gi ) ⊂ Gi+1 , i = 0, . . . , l − 1, (B2) G0 ∩ Kf 6= ∅, (B3) Gl is smart. The number l is called the length of the chain, G0 is called the first interval of a chain, and Gl is called the last interval of the chain. The typical situation in which we deal with a chain of intervals is the following. Given a point x and an interval I 3 f n (x) we construct a chain of intervals (Gi )ni=0 whose last interval Gn is equal to I and whose first interval G0 contains x. If such chain exists, it is unique. We call it the pull-back chain of I along x, . . . , f n (x) or just the pull-back chain of I. Any Gi is called a pull-back of I. Construction of a pull-back is straightforward. Once we have Gi , we choose as Gi−1 the component of f −1 (Gi ) containing f i−1 (x). The only obstructions in the construction may be that (B2) or (B3) are not satisfied. However, condition (B2) is satisfied if x ∈ Kf and this will be always the case in the sequel. Condition (B3) says that I is smart. This is not a great problem, because of the following lemma. Lemma 2.1 ([BM2]). Assume that f has no wandering intervals. Then every nonperiodic point has a smart neighborhood. When we have a chain (Gi )li=0 , we cannot avoid the situation when Gi contains exceptional points of f . However, we have the following lemma. Lemma 2.2 (see, e.g., [BM2]). An interval Gi from a chain contains at most one exceptional point c of f , and if so then c is neither 0 nor 1. Moreover, if a turning point c of f belongs to Gi and i < l then τc (Gi ) = Gi . The intervals of a chain that contain elements of C play a special role. Their number in a chain is called the C-order (or just order) of the chain. The next lemma follows immediately from Lemma 2.2 and the definition of a chain. To state it we need the following definition. Let I be an interval, I 0 be a component of f −1 (I) such that either f |I 0 is monotone and f (I 0 ) = I or f |I 0 is unimodal, both

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endpoints of I 0 are mapped into one endpoint of I and (in the case of C 1 -map f ) nonflatness inequalities are satisfied in I 0 . Then we say that I 0 is a regular component of the preimage of I. Lemma 2.3 (see, e.g., [BM2]). The interval Gi is a regular component of the preimage of Gi+1 . Let us now repeat with more details the definition of super persistent recurrence. Let us fix the set C of exceptional points of f (recall that C must contain all turning points of f ) and consider the following construction. Fix a point x ∈ Kf . For every ε > 0 we try to construct the pull-back chain of [f n (x) − ε, f n (x) + ε] along x, . . . , f n (x) and denote by mx,n (ε) its order. Clearly, mx,n (ε) grows monotonically with ε. If there are no exceptional points among x, f (x), . . . , f n (x) then for sufficiently small ε we have mx,n (ε) = 0, otherwise even for arbitrarily small ε we have mx,n (ε) > 0. We define rnk (x) as the supremum of all ε such that mx,n (ε) exists and is smaller than or equal to k. In other words, ε is the biggest number such that for every ε0 < ε the ε0 -neighborhood of f n (x) can be pulled back along x, . . . , f n (x) with order at most k. Note that rnk (x) depends on f and C, yet for the sake of simplifying notation we avoid referring to them. We call a point x such that for every k we have rnk (x) → 0 as n → ∞ C-super persistent. If x is additionally recurrent, we call it C-super persistently recurrent (cf. [BM1]). If we only claim the existence of a set C of exceptional points for which x is C-super persistently recurrent, but do not fix it, we call x simply a super persistently recurrent point. Finally, if the map f is smooth then Cr(f ) = Cr denotes its set of critical points, so we get Cr-super persistent and Cr-super persistently recurrent points which will be the main focus S of our study. Let E(f ) = k,ε Ek,ε (f ); recall that Ek,ε (f ) is the set of all points y with lim sup rnk (y) > ε (where C = Cr). Thus, the set of Cr-super persistent points is [0, 1] \ E(f ). We will call an interval I ⊂ [0, 1] nice if for every n > 0 and an endpoint a of I the point f n (a) does not belong to the closure of I. In other words, the positive orbits of both endpoints of I miss the closure of I. A set A ⊂ [0, 1] is called minimal if f |A is minimal. Theorem 2.4 ([BM2]). Let x be a super persistently recurrent point of f having arbitrarily small smart nice neighborhoods. Then ω(x) is minimal. We also need some facts about so-called basic sets (see [B]) which will be used later on. Let M = cyc(I) be a cycle of intervals. Consider a set {x ∈ M : for any relative neighborhood U of x in M the orbit of U is dense in M }; it is easy to see that this is a closed invariant set. It is called a basic set and denoted by B(M ) (or simply by B) provided it is infinite. Let F : I → I and G : J → J be two interval maps, let ϕ : I → J be a (non-strict) monotone semiconjugacy between F and G and let B ⊂ I be an F -invariant closed set such that ϕ(B) = J and ϕ−1 (x) ∩ B = ∂ϕ−1 (x) for any x ∈ J. Then we say that ϕ almost conjugates F |B to G. Here ∂Z is the boundary of a set Z. Now we can list some of the properties of basic sets of interval maps. Theorem 2.5 ([B]). Let cyc(I) be a cycle of intervals of period n of a continuous interval map f and let B = B(cyc(I)) be the corresponding basic set. Then the following holds. (1) There exists a mixing map g : [0, 1] → [0, 1] and a monotone map ϕ : I → [0, 1] such that ϕ almost conjugates f n |(B∩I) to g. In particular, f maps complementary to B intervals one into another and also their boundaries one into another.

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The map f |B is transitive. The set B is perfect. The set B is contained in the closure of the set of periodic points of f . If K is a compact set contained in the interior of f j (I) and U is an open set intersecting B ∩ I then there exists a number l such that f mn+j (U ) ⊃ K for all m ≥ l. (6) Any limit set is contained in either a periodic orbit, or a solenoidal set, or a basic set. (2) (3) (4) (5)

The main properties here are (1) and (6). The rest can be deduced from it, yet in order to have convenient references we also include Properties (2)–(5). Theorem 2.5 implies the following corollary for maps without wandering intervals (here by (pre)periodic we mean points which are periodic or preperiodic). Corollary 2.6. Let f be a map without wandering intervals, and let B = B(cyc(I)) be its basic set. Then the following holds. (1) If J is a complementary to B interval then it is eventually mapped into a periodic complementary to B interval and its endpoints are (pre)periodic. (2) If ω(x) ⊂ B is infinite then f k (x) ∈ B for some k and, moreover, for no k is f k (x) an endpoint of a complementary to B interval. (3) If x is not an endpoint of a complementary to B interval and ε is less than the minimal length of an interval from cyc(I) then the maximal length of a neighborhood U of x such that |f N (U )| ≤ ε converges to 0 as N → ∞. Proof. (1) Let J = (a, b) be a complementary to B interval. Then we may assume that J ⊂ I. Suppose for the sake of definiteness that the period of I is m and consider the almost conjugacy ϕ between f m |I and a mixing map g : [0, 1] → [0, 1]. Then ϕ(J) = {x} is a point. If x is not (pre)periodic for g then J is wandering, a contradiction. So, x is (pre)periodic for g which implies that J is eventually mapped into a periodic complementary to B interval K. Since endpoints of complementary to B intervals are mapped into endpoints of complementary to B intervals by Theorem 2.5 (1) we see that endpoints of J are (pre)periodic. (2) Suppose that ω(x) ⊂ B(cyc(I)) is infinite. Then for some k we have f k (x) ∈ I. If k f (x) ∈ J where J is a complementary to B interval then by (1) x is eventually mapped into a periodic complementary to B interval. Since the limit set of x is contained in B then ω(x) is contained in the union of the endpoints of the intervals from cyc(K) which contradicts the fact that ω(x) is infinite. Also, by (1) all endpoints of complementary to B intervals are (pre)periodic, so f k (x) cannot be such an endpoint because ω(x) is infinite. (3) If the claim fails then there is a semi-neighborhood V of x and a sequence of integers Ni → ∞ such that |f Ni (V )| ≤ ε. Choose closed subintervals of f j (I) for all 0 ≤ j ≤ n so that the length of any such subinterval is greater than ε. By Theorem 2.5 (5) for some k and all m ≥ k the set f m (V ) contains at least one of these subintervals and therefore has length greater than ε; a contradiction. Theorem 2.7 (cf. [BM2]). If f has no wandering intervals then any super persistently recurrent point c of f has minimal limit set. Remark. In [BM2] we prove the same statement under the assumption that the point c is a turning point of f . The main reason for such restriction was that we only needed the result for turning points; in what follows we get rid of this restriction for maps without wandering intervals.

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Proof of Theorem 2.7. First of all notice that if c is a periodic point or belongs to a solenoidal set then ω(c) is indeed minimal. Thus we may assume from the very beginning that c has an infinite limit set which is not solenoidal. By Theorem 2.5 (6) then ω(c) ⊂ B(cyc(I)) for some basic set B(cyc(I)). Since c is recurrent we conclude that c ∈ B. Moreover, by Corollary 2.6 (2) no point of the orbit of c is an endpoint of a complementary to B interval. We claim that c can be approximated from both sides by the preimages of any point a from the interior of I. Indeed, since c ∈ B is not an endpoint of a complementary to B interval then any semi-neighborhood of c is non-disjoint from B. By Theorem 2.5 (5) images of this semi-neighborhood cover a which proves our claim. Our aim is to construct an arbitrarily small nice neighborhood of c. First observe that considering small neighborhoods U of c we may assume that there is a lot of periodic points with pairwise disjoint orbits which do not enter U . Indeed, by Theorem 2.5 (4) B is contained in the closure of the set of periodic points of f , so there are a lot of periodic points in B. We can choose several such points and then choose U to be disjoint from the union of their orbits. Now, take a periodic Sn point a ∈ I. By the previous paragraph for n sufficiently large there are points of i=0 f −i (a) in U at both sides of c very close to c. Choose the closest ones from both sides, x < c < y; we may assume that the orbit of a is disjoint from [x, y]. The neighborhood (x, y) of c is nice unless one of the points x, y is an image (under some iterate of f ) of the other one. In this case choose a periodic point b whose orbit is disjoint from both the orbit of a and the set [x, y]. Take the minimal k such that f −k (b) intersects (x, y). If f −k (b) intersects (x, c), replace x by the element of f −k (b) closest to c; similarly for (c, y) and y. The new neighborhood of c is nice. This is clear if we replaced only one of the points x, y. If we replaced both of them, the endpoints of the new neighborhood belong to G−k (b) with the same k. However, in this case none of them can be an image of the other one under any iterate of f because otherwise the orbit of b would not be disjoint from (x, y). On the other hand it is proven in [BM2] that c has arbitrarily small smart neighborhoods. We complete the proof by applying Theorem 2.4 to the point c. A set A will be called C-super persistently recurrent if for some C-super persistently recurrent point x ∈ A we have ω(x) = A. In view of the next proposition, it does not matter which point x ∈ A we choose. Proposition 2.8 ([BM2]). Let f have no wandering intervals and C be a set of exceptional points. Let X ⊂ [0, 1] be an infinite minimal set for f . Then either every point of X is C-super persistently recurrent or no point of X is C-super persistently recurrent. So far in Sect. 2 we have stated standard facts concerning chains or useful for us results from [BM2]. The next lemma establishes invariance of the sets E(f ) and Ek,ε (f ). By Cr(f ) we denote the union of all big orbits of the set of all exceptional points of f (that is of all their images and preimages). Lemma 2.9. The set Ek,ε (f ) \ Cr(f ) is invariant and the set E(f ) is fully invariant. Proof. This follows from the fact that if (Gi )li=0 is a chain of intervals of order m then the order of the chain (Gi )li=1 is either m or m − 1.

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3. Distortion Lemmas Normally, one defines the Schwarzian (or Schwarzian derivative) of a function f of class C 3 as Sf = f 000 /f 0 − (3/2)(f 00 /f 0 )2 . It is defined outside the set Cr(f ). Then negative Schwarzian means Sf < 0 at all non-critical points. As can be easily checked, this p 0 implies strict convexity of the function 1/ |f | on each component of the complement of the set Cr(f ), which requires only C 1 smoothness. Moreover, it is well known that distortion properties similar to those of maps f of class C 3 with Sf < 0 away from critical points hold for the maps of class C 1 described above as well. Thus we adopt this property as the definition of negative Schwarzian maps as was done in [BM2]: a p negative Schwarzian map is a map of class at least C 1 such that the function 1/ |f 0 | is strictly convex on each component of the complement of the set Cr(f ). The space of all 2 will be denoted by S. Note that we allow critical negative Schwarzian maps from Cnf inflection points. Some uniform estimates on distortion can be made for all negative Schwarzian diffeomorphisms. Also, iterates of negative Schwarzian maps have negative Schwarzian as well. This allows one to estimate distortion of an iterate of a negative Schwarzian map restricted to an interval where it has no critical points. Estimates of distortion for one iterate of a map in the presence of a critical point are necessary too. Since the “one-step” estimates can be made without negative Schwarzian assumptions, in Lemmas 3.1–3.3 2 . we consider general maps from the class Cnf We need some notions. The density of a set X in an interval I is ρ(X|I) =

|X ∩ I| . |I|

In the probability theory it is called conditional measure, but we prefer a more geometrical name density. As in [BM1], we introduce a function r : [0, 1]2 → R ∪ {∞} as follows: r(x, y) =

|f (x) − f (y)| |x − y| |f 0 (x)|

if x 6= y, and r(x, x)=1 and call the infimum of r(x, y) over the pairs of points x, y from the same lap the shrinkability of f . We denote it s (f ) (“s ” is a shrunken “s”). It is proven 2 have non-zero shrinkability. in [BM1] that maps from Cnf Let us now state our first distortion lemma. 2 , X is a measurable set and I is an interval such that Lemma 3.1. Assume that f ∈ Cnf f is monotone on I. Then

ρ(X|I) ≥ ρ(f (X)|f (I)) s (f ). Proof. If I = [a, b] and a < x < b then by the definition of s (f ) we have |f (b) − f (x)| ≥ s (f )|f 0 (x)| |b − x| and Hence,

|f (x) − f (a)| ≥ s (f )|f 0 (x)| |x − a|. |f (I)| |f (b) − f (a)| = ≥ s (f )|f 0 (x)|. |I| |b − a|

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This holds for every x ∈ (a, b), so taking into account that |f (X ∩ I)|/|X ∩ I| is the mean value of |f 0 (x)| over X ∩ I, we get |f (X ∩ I)| |f (I)| ≥ s (f ) . |I| |X ∩ I| This inequality is equivalent to the one we wanted to prove.

The next lemma relies upon Lemma 3.1. To state it we need the following definitions. A point x ∈ I is said to be η-centered in an interval I if the distance of x from the boundary of I is η|I| or larger. An interval K ⊃ I is said to be a δ-scaled neighborhood of I if the distance of both endpoints of K from I is at least δ|I|. 2 . Then there exists a positive constant ξ (depending only on f ) Lemma 3.2. Let f ∈ Cnf 0 such that if I is a regular component of the preimage of an interval I, x ∈ I 0 , and f (x) is η-centered in I then:

(1) the point x is ξη-centered in I 0 ; (2) if a set A has density at least α in both components of I \ {f (x)} then f −1 (A) has density at least ξηα in both components of I 0 \ {x}. Proof. If f is monotone on I 0 , by Lemma 3.1 both (1) and (2) hold with ξ = s (f ) (observe that η ≤ 1/2). Assume now that f is unimodal on I 0 . Let I = [a, b] and I 0 = [a0 , b0 ]. Without loss of generality we may assume that f (a0 ) = f (b0 ) = a and that there is c0 ∈ (a0 , b0 ) such that f is increasing on [a0 , c0 ] and decreasing on [b0 , c0 ]. Set c = f (c0 ). Recall that for τ = τc0 we have 1/R ≤ |τ 0 | ≤ R, where R = R(f ) depends only on f . Since τ ([a0 , c0 ]) = [c0 , b0 ] and τ ([c0 , b0 ]) = [a0 , c0 ], we get 1/R ≤ |c0 − a0 |/|b0 − c0 | ≤ R. Therefore |c0 − a0 |/|I 0 | ≥ 1/(R + 1) and similarly |b0 − c0 |/|I 0 | ≥ 1/(R + 1). Assume that x ∈ I 0 and f (x) is η-centered in I. Without loss of generality we may assume that x ∈ [a0 , c0 ]. We have |f (x) − a| ≥ η|I| ≥ η|c − a|, so by Lemma 3.1 and the preceding paragraph we get |x − a0 | ≥ s (f )η|c0 − a0 | ≥

s (f )

R+1

η|I 0 |.

On the other hand (since s (f ) ≤ 1 and η ≤ 1), |x − b0 | ≥ |b0 − c0 | ≥

s (f ) 1 |I 0 | ≥ η|I 0 |. R+1 R+1

This means that (1) holds in this case with ξ = s (f )/(R + 1). Assume now that a set A has density at least α in [a, f (x)]. Then by Lemma 3.1 f −1 (A) has density at least s (f )α in both [a0 , x] and [τ (x), b0 ]. Since f (τ (x)) = f (x), by the preceding paragraph the point τ (x) is η s (f )/(R + 1)-centered in I 0 , and hence η s (f ) |b0 − τ (x)| |b0 − τ (x)| ≥ ≥ . |b0 − x| |I 0 | R+1 Therefore f −1 (A) has density at least αη(s (f ))2 /(R + 1) in [x, b0 ]. This means that (2) holds in this case with ξ = (s (f ))2 /(R + 1). Thus, the whole lemma holds with ξ = (s (f ))2 /(R + 1).

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2 Lemma 3.3. Let f ∈ Cnf . Then there exists a positive constant ζ < 1 (depending only on f ) such that if I 0 is a regular component of the preimage of the interval I, J is an interval such that I is its δ-scaled neighborhood with δ ≤ 1, and J 0 is a component of (f |I 0 )−1 (J), then I 0 is a ζδ-scaled neighborhood of J 0 .

Proof. In the monotone case from Lemma 3.1 it follows that I 0 is a δ 0 -scaled neighborhood of J 0 , where δ δ0 . = s (f ) 1 + δ0 1+δ Since δ ≤ 1, we get δ 0 ≥ s (f )δ/2, so we can take ζ = s (f )/2 in this case. Assume now that f is unimodal on I 0 and use the same assumptions and notation as in the preceding proof. Suppose first that f (c0 ) does not belong to J 0 . Then we may assume that J 0 = [d0 , e0 ] ⊂ [a0 , c0 ] and then we get (as in the monotone case) s (f ) |d0 − a0 | ≥ δ. 0 |J | 2

On the other hand, |b0 − c0 | |τ ([c0 , a0 ])| 1 δ |b0 − e0 | ≥ = ≥ ≥ . 0 0 0 0 0 |J | |c − a | |[c , a ]| R R Hence, in this case we can take ζ = min(s (f )/2, 1/R). Suppose now that f (c0 ) belongs to J 0 = [d0 , e0 ]. Then f ([d0 , c0 ]) = f ([c0 , e0 ]) = J, so by the preceding case I 0 is a min(s (f )/2, 1/R)δ-scaled neighborhood of both [d0 , c0 ] and [c0 , e0 ]. However, |c0 − d0 | ≥ |J 0 |/(R + 1) and |e0 − c0 | ≥ |J 0 |/(R + 1), so I 0 is a min(s (f )/2, 1/R)δ/(R + 1)-scaled neighborhood of J 0 . Hence we can take in this case (and in all cases) ζ = min(s (f )/2, 1/R)/(R + 1). So far we have proven one step distortion lemmas which apply to f but not to its iterates. The famous Koebe Lemma fills this gap. Unlike Lemmas 3.1–3.3, it applies only to maps of negative Schwarzian and only if the interval contains no critical points in its interior. However, the estimates of the distortion do not depend on the map, which allows us to apply them to iterates of maps and makes the lemma very important. We state it in a form equivalent to the one from [BM2]. Koebe Lemma. If I is an interval, h : I → R is a negative Schwarzian map without critical points in the interior of I, and J ⊂ I is an interval such that h(I) is a δ-scaled neighborhood of h(J) then: (1) for every points x, y ∈ J we have h0 (x) ≤ h0 (y)

1+δ δ

2 ;

(2) the interval I is a δ 3 /(2(3δ + 2)2 )-scaled neighborhood of J. The next lemma shows what consequences similar to Lemma 3.2 can be drawn from the Koebe Lemma. Lemma 3.4. If I is an interval, h : I → R a negative Schwarzian map without critical points in the interior of I, and J ⊂ I an interval such that h(I) is a δ-scaled neighborhood of h(J) then:

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(1) if x ∈ J and h(x) is η-centered in h(J) then x is (δ/(1 + δ))2 η-centered in J; (2) for any measurable set A, ρ(f

−1

(A)|J) ≥

δ 1+δ

2 ρ(A|f (J)).

Proof. From the Koebe Lemma it follows that s (f |J ) ≥ (δ/(1 + δ))2 . Now (2) follows from this and Lemma 3.1 and (1) follows from (2). The next lemma is a distortion lemma for chains, which we will use in the next section. The Koebe Lemma shows that within segments of a chain which consist of intervals containing no critical points the distortion of the map remains bounded on smaller intervals. Therefore to estimate the distortion in the entire chain it is important to know how many intervals containing critical points it includes. Thus from now on we always consider maps f with the set of exceptional points C = Cr(f ), and estimate the distortion of a map along a chain provided that the order of the chain is known. Note that due to our definition of chain and Lemma 2.1 we do not have to include 0 and 1 in the set of exceptional points. If chains (Gi )li=0 and (Hi )li=0 are such that Gi ⊃ Hi for every i then we say that the chain (Gi )li=0 contains the chain (Hi )li=0 and denote this by (Gi )li=0 ⊃ (Hi )li=0 . Lemma 3.5. Assume that f ∈ S, ν is a natural number, δ ≤ 1, α and η are positive numbers. Then there exist positive numbers γ ≤ 1, ϑ and β (all of them depending on f , ν and δ; additionally ϑ depends on η, and β depends on η and α), such that whenever (Gi )li=0 ⊃ (Hi )li=0 are chains of order ν or smaller and Gl is a δ-scaled neighborhood of Hl , the following holds: (1) G0 is a γ-scaled neighborhood of H0 ; (2) if x ∈ H0 and f l (x) is η-centered in Hl then x is ϑ-centered in H0 ; (3) if x is as above and the density of a set A in both components of Hl \ {f l (x)} is at least α then the density of f −l (A) in both components of H0 \ {x} is at least β. Proof. We decompose our chain (Gi )li=0 into 2ν + 1 (or less) pieces. Each piece corresponds either to f restricted to some Gi that contains one exceptional point (Case 1) or to an iterate f j restricted to Gi such that there is no exceptional point of f j in Gi and f j |Gi has negative Schwarzian (Case 2). We go back along the chain piece by piece, using inductively appropriate statements of Lemmas 3.2 and 3.3 in Case 1, and of the Koebe Lemma and Lemma 3.4 in Case 2. In this way when we get to G0 and H0 , we obtain (1)–(3) for some constants γ, β and ϑ. In order to have these constants independent of the choice of chains (for given chains we may have a decomposition into less than 2ν + 1 pieces) we have to apply alternately Case 2 and Case 1, together 2ν + 1 times, starting with Case 2.

4. Super Persistent Recurrence of Wild Attractors In this section we consider maps from our class S with the set of exceptional points coinciding with the set of critical points Cr(f ). We study wild attractors and specify the dynamics on them. The main result of this section is that they are Cr-super persistently recurrent.

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We begin with a simple observation. By Theorem 2.5 (6) limit sets of points for interval maps are either periodic orbits, or are contained in solenoidal sets, or are contained in basic sets. The definitions imply that wild attractors are subsets of basic sets. Since by Theorem 1.2 a wild attractor A is the limit set of a recurrent critical point c we may assume that c ∈ A ⊂ B, where B is a basic set. Now we need a number of standard definitions from ergodic theory which we include here for the sake of completeness. Let T be a nonsingular map on (X, B, µ) (the measure µ here is finite but not assumed to be invariant). The set D ⊂ X is called fully invariant if T −1 (D) = D. The map T is said to be ergodic if all its fully invariant sets have either measure 0 or full measure. The map T is called conservative if for any set R of positive measure there exists n such that T n (R) ∩ R 6= ∅; the map T is conservative on its invariant set D if T |D is conservative in the above sense. The map T is said to be purely dissipative if there T are no invariant subsets on which it is conservative. The map T is said to be exact if n≥0 T −n (B) contains only sets of measure 0 or µ(X). Clearly, if a map is exact then it is ergodic. A useful tool in studying exactness is lim sup full sets and maps introduced by Julia Barnes in [Ba]. Let T be a nonsingular map on (X, B, µ). A set Y ⊂ X is called lim sup full provided lim sup T n (Y ) = µ(X). The map T is said to be lim sup full if every subset of positive measure is lim sup full. In her paper [Ba] Barnes proved the following theorem (the theorem is proven under the assumption that the map T is a non-singular d-to-1 map but the same argument goes through in a more general situation). Theorem 4.1 ([Ba]). Let T : X → X be a non-singular surjective map that is lim sup full with respect to a finite measure µ. Moreover, suppose that there exists a partition of X into finitely many subsets on each of which T is one-to-one. Then T is exact (and therefore ergodic) and conservative. Sm−1 Remark. If X = i=0 Xi , T (Xi ) = Xi+1 for all i and pairwise intersections of the sets {Xi } have zero µ-measure then one can consider the question of whether T m |Xi is lim sup full for all i. It is easy to see (we leave verification to the reader) that then if T m |Xi is lim sup full for some i then T m |Xi is lim sup full for all i and exact, while T : X → X is ergodic and conservative. In the sequel T will be our interval map f , B will be the σ-field of all Borel subsets of [0, 1], and µ will be the Lebesgue measure. In the following lemma we establish a sufficient condition for the restriction of a map f to its basic set to be lim sup full. A point y is said to be a point of semi-density of a set X if for components I − , I + of the set I \ {y}, where I is an interval centered at y we have max(ρ(X|I − ), ρ(X|I + )) → 1 as |I| → 0. Lemma 4.2. Let B = B(cyc(I)) be a basic set of a map f ∈ S, where I is of period m. Then the following holds. (1) Suppose that a point y ∈ E(f )∩B ∩I is a point of semi-density of some set X ⊂ B. Then X is lim sup full for f m |I (in particular if X is invariant then X = cyc(I) up to a set of measure zero). (2) Suppose that the set Y = E(f )∩B is of positive measure. Then B = cyc(I), f m |f j (I) is lim sup full for any j, and thus f m |f j (I) is exact and f |cyc(I) is ergodic and conservative. Moreover, then there exist k and ε > 0 such that Ek,ε (f ) ∩ cyc(I) = cyc(I) up to a set of measure zero.

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Proof. (1) We may assume that X ⊂ I. Since y ∈ E(f ), there exists a sequence of integers ni → ∞ and a number ε > 0 such that the pull back chain of the 4εneighborhood of f ni (y) along y, . . . , f ni (y) has order at most k. Choosing a subsequence we may assume that f ni (y) → z and |f ni (y)−z| < ε for all i. Then the pull back chains of the 3ε-neighborhood G of z along y, . . . , f ni (y) have order at most k. Together with these chains we consider the pull back chains of the 2ε-neighborhood H of z along y, . . . , f ni (y), whose first interval we denote by H0i . Observe that ε above may be chosen sufficiently small; in particular we can choose it smaller than the one half of the minimal length of the intervals from cyc(I). Then Corollary 2.6 (3) applies, so we see that |H0i | → 0 as ni → ∞. First we prove that for some semi-neighborhood U of z and a subsequence of iterates ni the measure of the sets U \ f ni (X) converges to 0. Let us check if appropriate claims of Lemma 3.5 can be applied to the two constructed pull back chains. Observe that G is a 1/4-scaled neighborhood of H and that the orders of the pull back chains of G along y, . . . , f ni (y) are at most k. Also, all points f ni (y) are 1/4-centered in H. Hence we can apply Lemma 3.5 with ν = k, δ = 1/4 and η = 1/4. We set Ai = [0, 1] \ f ni (X) and suppose that there is α > 0 such that for all i the set A on both components of H \ {f ni (y)} has density greater than α. By Lemma 3.5 there are positive constants ϑ and β such that for all i the point y is ϑ-centered in H0i and the density of f −ni (Ai ) in both components of H0i \ {y} is at least β. Let si ≤ ti be the lengths of the components of H0i \ {y}. Set Wi = [y − ti , y + ti ]. Since y is ϑ-centered in H0i , we have si /(si + ti ) ≥ ϑ, and hence si /ti ≥ ϑ/(1 − ϑ). The density of f −ni (Ai ) in the component of Wi which is also a component of H0i \ {y} is at least β; in the other component it is at least βsi /ti ≥ βϑ/(1 − ϑ). Since ϑ ≤ 1/2, we have ϑ/(1 − ϑ) ≤ 1, so the density of f −ni (Ai ) in each half of Wi is at least βϑ/(1 − ϑ). The set f −ni (Ai ) is disjoint from X. Since |H0i | → 0, we have ti → 0, and this contradicts the fact that y is the point of semi-density of X. Hence, after choosing a subsequence and without loss of generality we may assume that the density of f ni (X) on the left component V = [a, f ni (y)] of H \ {f ni (y)} approaches 1. Since f ni (y) → z we see that the density of f ni (X) on [a, z] approaches 1. This proves our claim for U = [a, z]. Now, by Theorem 2.5 (5) there exists j0 such that |I \ f mj (U )| is arbitrarily small for all j ≥ j0 . This means that the measure of f mj+ni (X) can be made arbitrarily close to that of I with the appropriate choice of j and i (choose j first and i next), and thus that X is lim sup full for f m |I . This completes the proof of (1). (2) Let us apply (1) to the set X = Y . Then Y = cyc(I) up to a set of measure zero, and since Y ⊂ B we have B = cyc(I). Moreover, E(f ) has full measure in I. We can now apply (1) to an arbitrary subset X of E(f ) ∩ I of positive measure. We see that X is lim sup full for f m |I . Since this holds for all positive measure sets X, we conclude that the map f m |I is lim sup full. Therefore by Theorem 4.1 and the remark after it f m |f j (I) is exact for 0 ≤ j ≤ m − 1 and f |cyc(I) is ergodic and conservative. The rest of the statement follows now easily from the invariance of sets (Ek,ε (f ) \ Cr(f )) ∩ cyc(I) (Lemma 2.9) and ergodicity of f |cyc(I) . It remains to consider a basic set B with |E(f ) ∩ B| = 0, which is exactly the situation where wild attractors appear. We do this in a sequence of lemmas. Lemma 4.3. Let A be a wild attractor contained in a basic set B = B(cyc(I)). Then for very ε > 0 there exists an invariant nowhere dense set X ⊂ B, contained in the ε-neighborhood of A, such that |X| > 0 and all points z with ω(z) = A are eventually mapped into X.

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Proof. We may assume that ε > 0 is so small that the compact ε-neighborhood U of A does not cover cyc(I). To see that, recall that wild attractors are always nowhere dense by the definition. Consider the set X = {x : x ∈ B, the orbit of x is contained in U and ω(x) = A}. For any point z such that ω(z) = A there is j0 such that f j (z) ∈ U for all j ≥ j0 . By Corollary 2.6 (2) there is j ≥ j0 such that f j (z) ∈ B. Therefore f j (z) ∈ X. Since the realm of exact attraction of A has positive measure, it follows that |X| > 0. Clearly, X is invariant. Let us show that X ⊂ B is nowhere dense in cyc(I). This is obvious if B itself is a nowhere dense subset of cyc(I). Otherwise B = cyc(I) and by Theorem 2.5 (2) f |cyc(I) is transitive. Hence, if x ∈ X then the orbit of every neighborhood of x intersects cyc(I) \ U . Therefore X is nowhere dense. We need an important estimate on the density of X. Lemma 4.4 ([L2]). Let x be a point of density of an invariant set X absorbed by a basic set B. Then any point of ω(x) is a point of semi-density of X. Now we are ready to prove the main result of this section. Theorem 4.5. Let A be a wild attractor of a map f ∈ S. Then A is Cr-super persistently recurrent. Proof. By Theorem 1.2 we may assume that A = ω(c), where c is a recurrent critical point. Let X be the set from Lemma 4.3. By Lemma 4.4 c is a point of semi-density of X. Thus, if c is not Cr-super persistently recurrent then by Lemma 4.2 (1) X = cyc(I) up to a set of measure zero, while on the other hand it is nowhere dense in cyc(I) by Lemma 4.3, a contradiction. The following theorem unites Lemma 4.2, Theorem 4.5 and some new arguments, thus giving a fuller description of Milnor attractors which are neither periodic orbits nor solenoidal sets. Note that if an attractor A is contained in a basic set B then since | rl(A)| > 0 and by Corollary 2.6 (2) the measure of B is positive. Theorem 4.6. For every f ∈ S and a primitive attractor A that is neither a periodic orbit nor a solenoidal set, one of the following holds. (1) The attractor A is wild. Then A = ω(c) for some Cr-super persistently recurrent critical point c. Furthermore, A is contained in a basic set B(cyc(I)) such that f |B is purely dissipative, |A| = 0, and almost all points of B are Cr-super persistent. (2) The attractor A is equal to B(cyc(I)) = cyc(I) and if I is of period m then f m |f j (I) is exact for 0 ≤ j ≤ m − 1 and f |cyc(I) is ergodic and conservative. Moreover, there exist k and ε > 0 such that Ek,ε ⊃ cyc(I) up to a set of measure zero. Proof. Let A be an attractor which is neither a periodic orbit nor a solenoidal set. By Theorem 2.5 (6) there is a basic set B = B(cyc(I)) such that A ⊂ B. If |E(f ) ∩ B| > 0 then by Lemma 4.2 (2) we have the case (2). Suppose that |E(f ) ∩ B| = 0 (that is, almost every point of B is Cr-super persistent) and show that this corresponds to the case (1) of the theorem. First we claim that in this case the set of recurrent points of f |B whose limit set is not minimal is of zero measure. Indeed, if the set of such recurrent points is of positive measure then almost all of them are not in E(f ) and thus are Cr-super persistently recurrent which by Theorem 2.7 implies that their limit sets are minimal, a contradiction. In particular, the set of

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413

points with the orbit dense in B is of zero measure (by Theorem 2.5 (1), B cannot be minimal). Therefore A cannot coincide with B and is a wild attractor. Let us prove the rest of the statements of claim (1) of the theorem. The fact that |A| = 0 follows from the results of [Va] (note that we actually need a weaker version of results of [Va], since we apply them only in the negative Schwarzian case for which a significant part of the arguments from [Va] can be omitted). It remains to show that f |B is purely dissipative, i.e. that there are no invariant subsets of B of positive measure on which f is conservative. To do this notice that by Theorem 1.2 there are finitely many primitive attractors Ai , 0 ≤ i ≤ k such that for almost every point x ∈ B the set ω(x) is one of them. Since |E(f ) ∩ B| = 0, all of them are wild. Now, suppose Sthat D ⊂ B is a set such that f |D is conservative and show that then it is contained in Ai modulo a set of measure zero. Indeed, otherwise there exist a number ε > 0 and a set D0 ⊂ D of positive measure disjoint from the ε-neighborhood S of Ai . By choosing a subset we may assume that D0 consists of points z such that ω(z) = A0 . Consider the set X constructed in Lemma 4.3 for this ε. By Lemma 4.3 all points of D0 are eventually mapped into X and therefore will only be mapped into D0 finitely many times. On the other hand by the Poincar´e Recurrence Theorem if a conservative then almost all points of D0 return to D0 infinitely many times. map f |D isS Thus D ⊂ Ai . However then by the results of [Va] quoted above we have |D| = 0, a contradiction. 5. Markov Maps Consider in more detail the case (2) of Theorem 4.6. We want to strengthen the last property of the attractor A = cyc(I) according to which cyc(I) ⊂ Ek,ε up to a set of measure zero. This means that for almost every point of cyc(I) we can find a large n and a neighborhood V of f n (x), “big” on both sides of f n (x), which can be pulled back with small order. Our stronger version of this property says that we can find V which is independent of x and n, and the order of the pull-back is 0. Moreover, we can choose V as a neighborhood of any given point outside some finite set. Since in the weaker version f n (x) is the midpoint of V , we mimic this property in the stronger version. Thus, we will not only specify V , but also its subinterval U and require that f n (x) ∈ U . This allows us to construct Markov maps introduced in the unimodal case by Martens in [Ma] and shows that the results of that paper related to Markov maps can be deduced from ours. To simplify notation, assume that the period of I is 1. The arguments can be repeated for any period almost literally. A map g : I → I is called topologically exact if for every interval J ⊂ I there is n with g n (J) = I. Theorem 2.5 (5) implies that f |I is topologically exact since the endpoints of I are images of some critical points from the interior of I (otherwise there would be an invariant proper subinterval of I). We start with a simple lemma. Lemma 5.1. Let f : I → I be topologically exact. Then for every ξ > 0 there exists N (ξ) such that every subinterval of I of length ξ is mapped by f N (ξ) onto I. Proof. If the lemma is false then we can find (by compactness of I) a subinterval of I that is not mapped onto I by arbitrarily large iterates of f , contrary to the exactness of f . To state the next proposition we need the following notation. Let CN (f ) be the set of all critical points of f N and the endpoints of the interval on which f is defined. Let YN (f ) = f N (CN (f )).

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Proposition 5.2. Let f ∈ S be topologically exact on an invariant interval I. Assume also that I ⊂ Ek,ε0 for some k and ε0 > 0 up to a set of measure zero. Then I ⊂ E0,ε for some ε > 0. Moreover, there is N such that for every pair of intervals U, V disjoint from the finite set Y = YN (f |I ) ⊂ I and such that U is contained in the interior of V , for almost every x ∈ I there exists an arbitrarily large n such that f n (x) ∈ U and the pull-back of V along x, . . . , f n (x) has order 0. Proof. By Lemma 5.1 there exists N = N (ε/2k+1 ) such that every subinterval I of length ε/2k+1 is mapped by f N onto I. Fix intervals U, V disjoint from Y and such that U is contained in the interior of V . If J is a subinterval of I of length ε/2k+1 then it is mapped by f N onto I, and thus there is an interval K ⊂ J such that f N (K) = V . Note that since V is disjoint from Y , we can choose K that is contained in the interior of J. Denote by Dn the set of all points x ∈ I for which there exists l > n such that f l (x) ∈ U and the pull-back of V along x, . . . , f l (x) has order 0. The set Dn is measurable. We show that it has full measure in I. Otherwise, there exists y ∈ Ek,ε which is a point of density of the complement of Dn . Thus, it is enough to show that there are arbitrarily small neighborhoods of every y ∈ Ek,ε in which the density of Dn is larger than some fixed β > 0. Fix y ∈ Ek,ε . There is an arbitrarily large t > n such that Gt = [f t (y) − ε, f t (y) + ε] can be pulled back along y, . . . , f t (y) with order at most k. Let the corresponding chain be (G0 , . . . , Gt ). Also, let G0t = [f t (y) − ε/2, f t (y) + ε/2]. The map f t has at most 2k − 1 critical points in G0 . Hence, there are intervals W1 ⊂ [f t (y) − ε/2, f t (y)] and W2 ⊂ [f t (y), f t (y) + ε/2] of length ε/2k+1 whose interiors are disjoint from Yt (f |G0 ). Thus, every component of f −t (Wi ) contained in G0 is mapped by f t onto Wi and there are no critical points of f t in its interior. Also, since the endpoints of G0 are mapped by f t to the endpoint(s) of Gt , each point of f −t (Wi ) ∩ G0 is contained in one of these components. Now we can choose intervals Ri contained in the interior of Wi (for i = 1, 2) which are mapped by f N onto V . Since V is disjoint from Y = f N (C), there are no critical points of f N in R1 or R2 . Let Qi be the subinterval of Ri that is mapped by f N onto U for i = 1, 2. The sets Q1 and Q2 are two of the finitely many components of f −N (U ). Thus, |Qi | ≥ δ, where δ is the minimum length of these components. Let (G00 , . . . , G0t ) be the pull-back chain of G0t along y, . . . , f t (y). Since G0t is 1/2 centered in Gt , the point f t (y) is 1/2 centered in G0t and the density of Q = Q1 ∪ Q2 in G0t is at least 2δ/ε, by Lemma 3.5 there exists β > 0, depending only on δ and ε, such that the density of f −t (Q) in G00 is at least β. By the construction, for every x ∈ f −t (Q) we have f t+N (x) ∈ U ⊂ V and the pull-back of V along x, . . . , f t+N (x) has order 0. Thus, f −t (Q) ⊂ Dn . By Lemma 5.1, |G00 | can be made arbitrarily small by taking sufficiently large t. This completes the proof that Dn has full measure. The intersection of all sets Dn , n ≥ 0 has also full measure and every point x from this intersection has the desired property. Note that f in Proposition 5.2 has no periodic critical points. Therefore there is at least one critical point which is not an eventual critical image. Denote this point by c. Then c ∈ / Y and therefore we can choose as intervals U, V small neighborhoods of c. By Proposition 5.2 for almost every point x ∈ U there exists n(x) and a pair of neighborhoods W 0 (x) ⊂ W 00 (x) such that f n(x) maps W 00 (x) onto V , has no critical points in the interior of W 00 (x) and f n(x) (W 0 (x)) = U . By the Koebe Lemma the quotient |(f n )0 (y)|/|(f n )0 (z)| is bounded from above for any y, z ∈ W 0 (x) by a constant which depends only on U and V . Choosing nice neighborhoods of c we see that the map T defined as f n(x) on intervals W 0 (x) is exactly the so-called Markov map defined in [Ma]

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in the unimodal case. Therefore the results of [Ma] related to Markov maps indeed follow from Theorem 4.6 and Proposition 5.2. Replacing the second part of Theorem 4.6 by Proposition 5.2 we finally get Theorem 5.3. Theorem 5.3. For every f ∈ S and a primitive attractor A that is neither a periodic orbit nor a solenoidal set, one of the following holds. (1) The attractor A is wild. Then A = ω(c) for some Cr-super persistently recurrent critical point c. Furthermore, A is contained in a basic set B(cyc(I)) such that f |B is purely dissipative, |A| = 0, and almost all points of B are Cr-super persistent. (2) The attractor A equals B(cyc(I)) = cyc(I) and if I is of period m then f m |f j (I) is exact for 0 ≤ j ≤ m − 1, f |cyc(I) is ergodic, conservative and cyc(I) ⊂ E0,ε for some ε > 0 up to a set of measure zero (almost all points of A are Cr-reluctant). Moreover, there exists a finite set Y ⊂ I such that: (a) Cr(f ) ∩ I 6⊂ Y ; (b) for any two intervals U ⊂ int(V ) disjoint from Y and for almost every x ∈ I there is an arbitrarily large n and two intervals x ∈ W 0 ⊂ W 00 such that f n |W 00 has no critical points, f n (W 00 ) = V and f n (W 0 ) = U .

References [Ba] [B]

Barnes, J.: Conservative exact rational maps of the sphere. Preprint (1997) Blokh, A.: The “spectral” decomposition for one-dimensional maps. Dynamics Reported 4, 1–59 (1995) [BL1] Blokh, A. and Lyubich, M.: Nonexistence of wandering intervals and structure of topological attractors of one-dimensional dynamical systems. II. The smooth case. Ergod. Th. & Dynam. Sys. 9, 751–758 (1989) [BL2] Blokh, A. and Lyubich, M.: Attractors of maps of the interval. In: Dynamical Systems and Ergodic Theory (Warsaw 1986). Banach Center Publ. 23, Warsaw: PWN, 1989, pp. 427–442 [BL3] Blokh, A. and Lyubich, M.: On decomposition of one-dimensional dynamical systems into ergodic components. The case of negative Schwarzian. Leningr. Math. J. 1, 137–155 (1990) [BL4] Blokh, A. and Lyubich, M.: Measurable dynamics of S-unimodal maps of the interval. Ann. Sci. Ecole Norm. Sup. (4) 24, 545–573 (1991) [BM1] Blokh, A. and Misiurewicz, M.: Collet–Eckmann maps are unstable. Commun. Math. Phys. 191, 61–70 (1998) [BM2] Blokh, A. and Misiurewicz, M.: Typical limit sets of critical points for smooth interval maps. Preprint (1997) [BKNS] Bruin, H., Keller, G., Nowicki, T. and van Strien, S.: Wild Cantor attractors exist. Ann. Math. 143, 97–130 (1996) [G] Guckenheimer, J.: Sensitive dependence to initial conditions for one-dimensional maps. Commun. Math. Phys. 70, 133–160 (1979) [GJ] Guckenheimer, J. and Johnson, S.: Distortion of S-unimodal maps. Ann. Math. 132, 71–130 (1990) [L1] Lyubich, M.: Nonexistence of wandering intervals and structure of topological attractors of onedimensional dynamical systems. I. The case of negative Schwarzian derivative Ergod. Th. & Dynam. Sys. 9, 737–749 (1989) [L2] Lyubich, M.: Ergodic theory for smooth one-dimensional dynamical systems. SUNY at Stony Brook, preprint #1991/11 [Man] Ma˜ne´ , R.: Hyperbolicity, sinks and measure in one-dimensional dynamics. Commun. Math. Phys. 100, 495–524 (1985;)(Erratum: vol. 112 , 721–724 (1987)) [Ma] Martens, M.: Distortion results and invariant Cantor sets of unimodal maps. Ergod. Th. & Dynam. Sys. 14, 331–349 (1994)

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[MMS] Martens, M., de Melo, W. and van Strien, S.: Julia–Fatou–Sullivan theory for real one-dimensional dynamics. Acta Math. 168, 273–318 (1992) [Mi] Milnor, J.: On the concept of attractor. Commun. Math. Phys. 99, 177–195 (1985)(Correction and remarks: vol. 102, 517–519 (1985)) [Va] Vargas, E.: Measure of minimal sets of polymodal maps. Ergod. Th. & Dynam. Sys. 16, 159–178 (1996) Communicated by Ya. G. Sinai

Commun. Math. Phys. 199, 417 – 439 (1998)

Communications in

Mathematical Physics © Springer-Verlag 1998

Rigorous Bounds on the Hausdorff Dimension of Siegel Disc Boundaries A. D. Burbanks? , A. H. Osbaldestin?? , A. Stirnemann Department of Mathematical Sciences, Loughborough University, Loughborough, Leics LE11 3TU, UK. E-mail: [email protected] Received: 30 March 1998 / Accepted: 13 May 1998

Abstract: We calculate rigorous bounds on the Hausdorff dimension of Siegel disc boundaries for maps that are attracted to the critical fixed point of the renormalization operator. This is done by expressing (a piece of) the universal invariant curve of the fixed-point maps as the limit set of an iterated function system. In particular, we prove (by computer-assisted means) that the Hausdorff dimension of these boundary curves is less than 1.08523 for maps that are close enough to the fixed point and attracted to it under renormalization. 1. Introduction 1.1. General background. For an integrable Hamiltonian system with N degrees of freedom, the motion in the 2N -dimensional phase space is confined to surfaces that are N -dimensional tori. The Kolmogorov–Arnold–Moser (KAM) theorem [1] states that sufficiently smooth and sufficiently small perturbations from integrability will preserve an arbitrarily large proportion of these surfaces (albeit in a distorted form). For systems far from integrable, there are generally no such invariant surfaces; an important question in dynamical systems theory is to understand how these surfaces are destroyed. A precursor of the KAM theorem was the proof by Siegel [23] of the existence of a domain of linearizability, known as a Siegel disc, around irrationally indifferent fixed points of certain complex maps. This problem forms a model for more complicated scenarios in nonlinear dynamics, in which the breakdown of quasiperiodic behaviour is found to have universal characteristics. One example is the breakdown of quasiperiodicity for families of maps of the circle [2]. The interest lies in what happens when a linear system (corresponding to rigid rotation) is perturbed by introducing a nonlinear component. An important question is ? ??

Current address: DPMMS, University of Cambridge, 16 Mill Lane, Cambridge, CB2 1SB, UK Corresponding author

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whether quasiperiodic motion persists, i.e. whether the motion remains conjugate to rigid rotation, as the magnitude of the nonlinear perturbation is increased. For the breakup of invariant tori in Hamiltonian systems, most attention has been focussed on the study of area-preserving twist maps [19]. The existence of a conjugacy to rigid rotation is plagued by small divisors; it is heavily dependent on the number-theoretic properties of the rotation number. Siegel’s work gave the first proof of convergence for a small divisor series. Quasiperiodic motion in these problems typically ceases as some parameter passes a critical value. The breakdown of quasiperiodicity, and the invariant structures present at critical parameter values, typically exhibit universal scaling properties. Recently, many of these properties have been analysed using renormalization group techniques. Perhaps the most straightforward application of renormalization methods to the study of universality in dynamical systems was the explanation offered by Feigenbaum ([12] and [13]) for the universal features of the period-doubling route to chaos observed in one-parameter families of unimodal maps of the interval. This explanation relies on the existence of a hyperbolic fixed point of a renormalization operator acting on a certain space of functions. Existence of the fixed point, and the nature of the spectrum of the derivative of the renormalization operator there, allows the deduction of universal features for a large class of maps in the space. A corresponding explanation for the universality of period doubling in area-preserving maps was given by Greene et al [15]. A renormalization explanation for the universality observed in families of circle maps was conjectured by Ostlund et al [22] and Feigenbaum et al [14], and for areapreserving twist maps by MacKay [19]. These explanations rely on the existence of both a simple fixed point of a renormalization operator (characterising the quasiperiodic motion present in the sub-critical case) and also a critical fixed point (controlling the universal scaling features associated with the critical parameter values in these problems). Manton and Nauenberg [18] made corresponding numerical observations of universality for Siegel discs, and a renormalization explanation of the type mentioned above was later offered by Widom [31]. There is currently a rich mixture of rigorous and numerical results in this area. The existence and hyperbolicity of a fixed point of the renormalization operator for period doubling in unimodal maps was established by Lanford ([16] and [17]) and Campanino et al ([5] and [6]), using rigorous computer-assisted proofs. More recently, existence of a fixed point has been established analytically by Epstein [10] and Sullivan [30]. The corresponding proof for period doubling in area-preserving twist maps was performed by Eckmann et al [8] by computer-assisted means. In the case of golden mean rotation number, a rigorous computer-assisted proof of the existence and hyperbolicity of the critical circle map fixed point, analogous to that of Lanford, was performed by Mestel [20]. Subsequently, an analytic proof has been given by Eckmann and Epstein [7]. For the breakup of invariant tori in area-preserving twist maps, significant steps have been made toward a computer-assisted proof of the existence of a critical renormalization fixed point by Stirnemann [29] in the case of golden mean rotation number. By assuming that such a fixed point exists, and that it satisfies certain hypotheses, it had earlier been proved that an invariant circle exists for all maps attracted to the fixed point [26]. For golden mean Siegel discs, existence of the corresponding critical fixed point was proved by computer-assisted means in [28]. Later, by verifying certain hypotheses concerning this fixed point, a proof was given of several conjectures concerning the boundaries of Siegel discs for maps attracted to the fixed point under renormalization [3]. One conjecture that remains outstanding is that the boundaries of these Siegel discs are fractals.

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1.2. Overview. The aim of this work is to find rigorous bounds for the Hausdorff dimension of the boundary curves for Siegel discs of maps attracted to the corresponding (critical) renormalization fixed point. It has been shown previously [3], that there is a (universal) invariant curve for (the maps of) the fixed-point itself, and that this curve characterises the universal features observed in Siegel disc boundaries. The first step is to examine the dimension of this curve. We then show that Hausdorff dimension is invariant under the renormalization scheme for maps that are both close enough to the fixed point and are attracted to it under renormalization, so that our results extend to the Siegel disc boundaries of these maps. In order to bound the Hausdorff dimension, we use essentially the techniques described by Falconer [11], in that we obtain upper bounds for the contractivities and lower bounds for the coercivities of the constituent maps of a suitable Iterated Function System (IFS), evaluated on a partition of the invariant curve. By solving the so-called “partition-function equations” we obtain upper and lower bounds for the dimension. In particular, we are able to prove that the Hausdorff dimension of the boundary curves is less than 1.08523. Unfortunately, the only lower bounds which our method has so far produced are less than 1. Organisation of the paper. In Sect. 2, we review the previous results concerning Siegel discs and their boundaries. In particular, we define the renormalization operator, discuss the proof that the critical fixed point exists, and describe briefly how the existence of the corresponding Siegel discs was deduced (along with certain properties of their boundary curves). In Sect. 3.1, we show how some preliminary bounds may be obtained analytically for the (universal) invariant curve of the fixed point maps. In Sect. 3.2, we demonstrate how (a piece of) this invariant curve may be written as the attractor of an IFS. This involves certain technical details which are presented in full in Appendix A. We then demonstrate, in Sect. 3.3, how upper bounds may be calculated on the dimension of this curve by a partition-function approach (Sect. 3.3.1). Section 3.4 gives a corresponding discussion for lower bounds. That the upper and lower bounds thus defined will converge to the Hausdorff dimension with successive steps of the this process (provided that certain bounds used in the definition are optimal) is stated in Sect. 3.5, with the full proof given in Appendix B. We then show, in Sect. 3.6, that the Hausdorff dimension is invariant under renormalization for maps that are both close enough to the fixed point and attracted to it. This means that the bounds we have calculated for the universal curve are also bounds on the dimension of Siegel boundaries for such maps. Finally, in Sect. 4, we conclude with some suggestions for the improvement of these results and directions for future research. 2. Renormalization for Siegel Discs 2.1. Siegel’s theorem. We consider analytic maps z → 7 f (z) of the form X an z n with λ = exp(2πiν), f (z) = λz + n≥2

where ν is an irrational real, i.e. the map has an irrationally indifferent fixed point at the origin.

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The question arises of whether there exists a domain of linearizability for such a map, i.e. whether there is some neighbourhood of the origin on which there is a conjugacy to the rigid rotation z 7→ λz, in the form of an invertible function ϕ such that f (z) = ϕ(λϕ−1 (z)). This question was answered by Siegel [23] who showed that the conjugacy ϕ exists provided that ν satisfies a Diophantine condition, i.e. if there exist constants c > 0, µ ≥ 0 such that ν − p > c for all integers p, q, q 6= 0. q |q|2+µ Equivalently, the condition is that sup

log qn+1 < ∞, log qn

where qn is the denominator of the nth rational convergent pn /qn of the continued fraction expansion for ν. Such a domain of linearizability for f is called a Siegel disc. Recently, Yoccoz [32] has shown that for quadratic maps the conjugacy exists if and only if ν satisfies the weaker Brjuno condition: X log qn+1 n

qn

< ∞.

2.2. Universal scaling. Manton and Nauenberg [18] made numerical studies of the behaviour both inside the Siegel disc and on the boundary (where the conjugacy ϕ breaks down) for a variety of maps. In particular, for those with golden mean rotation number, √ 5−1 , ν = ω := 2 they found results analogous to those observed for circle maps and area-preserving twist maps, namely the existence of sub-critical (interior of the Siegel disc) and critical (boundary of the Siegel disc) scaling scenarios and associated universal scaling constants. Figure 1 shows the boundary curve of a Siegel disc, along with some of the smooth interior curves. Figure 2 demonstrates the scaling scenarios for a golden mean Siegel disc; the Fibonacci-numbered iterates and preimages of an interior point zs , in the “simple” scaling regime, and a critical point zc of the map, are marked. Asymptotically, the Fibonacci iterates and preimages of an interior point approach that point along a straight line, at a geometric rate equal to the golden mean. In contrast, the Fibonacci iterates and preimages of the critical point zc are observed to accumulate to that point along two pairs of straight lines at a rate determined by a universal scaling exponent, with the angles of separation between the lines also tending to universal constants. In addition to identifying these scaling scenarios, it was conjectured that the boundary of the Siegel disc always passes through the critical point of the map and that the conjugacy function ϕ is not differentiable on the boundary. Further, it was conjectured that the boundary curves are fractals.

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ℑ

ℜ

Fig. 1. A Siegel disc boundary and some of the analytic interior curves

55

-34 144 Simple 377 Scaling zs 233

55

-89

89

144

-233 377

zc

Critical Scaling

θ+ 233

θ−

34

89 34

-144 -55

Fig. 2. Simple (interior) and critical (boundary) scaling scenarios for a golden mean Siegel disc

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A renormalization explanation for the scaling behaviour observed in golden mean Siegel discs was constructed by Widom [31], who considered a transformation on a space of pairs of functions. The scaling behaviour observed within the Siegel disc is characterised by a simple fixed point of the transformation, in the form of a pair of affine maps. Widom conjectured that the behaviour on the boundary is characterised by (the universal invariant curve of) a critical fixed point of the transformation. The existence of the critical fixed point was verified by Stirnemann [28], who considered the modified operator defined below. 2.3. The renormalization operator. Definition 1. Let DE and DF be non-empty connected open subsets of the complex plane, each containing the origin. Consider the space of pairs (E, F ) of analytic functions in one variable, defined on DE and DF respectively, such that αC(DE ) ⊂ DF , αC(DF ) ⊂ DE , E(αC(DF )) ⊂ DF , where C denotes complex conjugation, i.e. C : z 7→ z. ¯ The complex number α is defined by α = α(E, F ) := F (0), and is called the rescaling factor of the pair (E, F ). The renormalization operator is ˜ F˜ ), where then defined by (E, F ) 7→ (E, ˜ E(z) = C(α−1 F (αC(z))), F˜ (z) = C(α−1 F (E(αC(z)))). Thus α := F (0) enforces the normalization condition E(0) = 1. (This operator differs from that considered by Widom, in that we take α to be genuinely complex.) In what follows, we will omit parentheses wherever it makes the exposition clearer. Geometrically speaking, this operator encodes the process of successively magnifying the region around the critical point and reflecting it in the “symmetry axis” (indicated by the dotted line in Fig. 2) in such a way that, asymptotically, each Fibonacci-numbered iterate is successively mapped onto the previous ones. 2.4. Existence of the critical fixed point. Stirnemann established the existence of the critical renormalization fixed point, conjectured by Widom, in the form of a commuting pair (E, F ) of even maps, by means of a computer-assisted proof [28]. This was accomplished by writing E(z) = U (z 2 ) and F (z) = V (z 2 ). In terms of U and V the operator is now (U, V ) 7→ (U˜ , V˜ ), with U˜ (z) = Cα−1 V α2 C(z), V˜ (z) = Cα−1 V QU α2 C(z),

(1) (2)

where α := V (0). (Here, and in the following, Q denotes squaring, i.e. Q : z 7→ z 2 .) The basic strategy (cf. Lanford [16]) is to find a small ball around a good approximate fixed-point pair in a suitable space of (pairs of) functions and to demonstrate that a variant of Newton’s method for the fixed-point problem is a contraction on that ball. The contraction mapping theorem then yields the existence of a true (locally unique) fixed point of the operator within the ball. Thus Stirnemann’s proof yields the following:

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Theorem 1. There exists a pair (U, V ) of analytic functions, defined on suitable domains DU , respectively DV , satisfying the equations U (z) = Cα−1 V α2 C(z) on DU , V (z) = Cα−1 V QU α2 C(z) on DV .

(3) (4)

From these equations, two different pairs of fixed-point equations may be deduced by squaring: (1) for the pair (U Q, V Q) = (E, F ), we have √ (5) U Q(z) = Cα−1 V QαC(z) on DU , √ −1 (6) V Q(z) = Cα V QU QαC(z) on DV , which establishes the existence of a fixed point (E, F ) of the original renormalization operator, and (2) for the pair (QU, QV ):

(In the above,

√

QU (z) = Cα−2 QV α2 C(z) on DU , QV (z) = Cα−2 QV QU α2 C(z) on DV .

(7) (8)

D := {z : z 2 ∈ D}.)

2.5. The “generic” notation. Notice that the equations for both of the above fixed-point pairs are of the form (9) U = BT B −1 , (10) T = BT UB −1 , with (U, T , B) equal to either (U Q, V Q, Cα−1 ) or (QU, QV, Cα−2 ). In what follows, we will sometimes use this “generic” notation, with (U, T , B), for clarity. 2.6. The necklace construction. Using the ball of functions output by the existence proof (essentially a pair of high-degree polynomials, together with rigorous error bounds), we were then able to prove (in [3]) most of the conjectures made by Manton and Nauenberg in the case of golden mean rotation number. In particular, it was established that: Theorem 2. Any map attracted to the fixed point under iteration of the renormalization operator has a Siegel disc whose boundary is a non-smooth Jordan curve passing through a critical point of the map. (The boundary curve has a H¨older continuous parameterization; the conjugacy function ϕ is not differentiable at a dense set of points on the boundary.) Definition 2. We say that a single function f is “attracted to the fixed point” if the pair (I, f ) is attracted, where I is the identity function. This was accomplished by applying the necklace construction of [26], in which a sequence of sets, the domain pairs, is constructed iteratively by applying the maps of the renormalization fixed-point pair to suitable domains. Provided that certain hypotheses are satisfied (the necklace hypotheses) it follows that the sequence of domain pairs converges to a compact connected invariant set (the necklace) for the maps of the renormalization fixed point. By “piecing-together” copies of this necklace set, the (universal) invariant curve of the fixed-point pair is constructed. The necklace hypotheses had already been verified for the simple renormalization fixed point by Stirnemann [27], in order to prove a special case of Siegel’s theorem, yielding the existence of a smooth conjugacy to rigid rotation (within the Siegel disc)

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for maps attracted to the simple fixed point (corresponding results for circle mappings have been proved by Stark [25], and Sinai and Khanin [24]). The hypotheses were verified for the critical renormalization fixed point by means of a computer-assisted proof in [3]. The aim of the work presented here was to deal with the conjecture that the boundary curves of the Siegel discs are fractals for functions attracted to the critical fixed point. 3. Dimension of Siegel Disc Boundaries Numerical estimates for the dimension of golden mean Siegel disc boundaries have been made previously by Stirnemann (who used a non-rigorous numerical version of the method we present here) of 1.00119 to 1.07967, and by Osbaldestin [21], of 1.030. In this section, we first obtain an analytical upper bound on the Hausdorff dimension of the universal invariant curve by using the H¨older continuity of the parameterization function that was obtained from the necklace construction in [3]. We then cast (a piece of) the invariant curve as the attractor of an IFS and demonstrate how a partition-function approach may be used to yield improved bounds by means of rigorous numerics. (The invariant curve itself may be pieced-together from bi-Lipschitz copies of the IFS attractor, which means that the bounds calculated apply to the whole of this curve.) Finally, we demonstrate that Hausdorff dimension is invariant under the renormalization scheme for maps that are close enough to the fixed point and attracted under renormalization, so that the bounds also apply to the Siegel disc boundaries of such maps. 3.1. Analytical bounds. In [3], a H¨older continuous parameterization function φ(t) was constructed for a piece, L = φ([−ω, 1]), of the universal invariant curve of the renormalization fixed point. The H¨older continuity of φ(t) enables us to obtain analytical bounds on the Hausdorff dimension of the invariant curve. In fact, φ(t) is invariant for the pair (QU, QV ) rather than (U Q, V Q) = (E, F ). In [3] and [4], a parameterization χ(t) was then constructed for the invariant curve of (E, F ), essentially by “taking the square root” of φ. Since, apart from at the origin, squaring is a local similarity, it follows that the dimensions of these curves are the same (in fact, the relevant pieces of these curves are just different overlapping pieces of the same invariant curve, as demonstrated in Appendix A). From now on, we will work with φ(t). Since φ is H¨older continuous, there exist constants c > 0, µ > 0 such that |φ(s) − φ(t)| ≤ c|s − t|µ

for all s, t ∈ [−ω, 1].

It follows [11, Proposition 2.3] that µ1 is an upper bound for the Hausdorff dimension of L. Numerically, the upper bound thus found is about 1.6. A H¨older exponent µ is found by calculating (by computer) rigorous bounds on the contractivities of the maps used in the necklace construction; the full details are provided in [4]. It is also shown in [4] that the parameterization function φ is “H¨older coercive”, i.e. there exist constants c0 > 0, µ0 > 0 such that 0

c0 |s − t|µ ≤ |φ(s) − φ(t)| for all s, t ∈ [−ω, 1]. By using the “mass distribution principle” (see for instance Falconer [11]), it is then possible to obtain a lower bound of µ10 on the dimension. In practice, however, the lower bound thus found is less than 1. The following sections indicate how we calculate (improved) rigorous bounds.

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3.2. The IFS. 3.2.1. Composition order. In the generic notation of Sect. 2.5, the definition of the renormalization operator given above (and used in the existence proof of Stirnemann) contains the composition T U, corresponding to the fixed-point equations (9) and (10). However, the necklace construction was designed for the composition order UT , corresponding to the fixed-point equations U = BT B−1 , T = BUT B −1 .

(11) (12)

We will refer to the former (T U) as the inverse composition order, and the latter (UT ) as the accretive composition order. In order to verify the hypotheses for the necklace construction, it was necessary to obtain a fixed point for this second (accretive) order. This was done in [3], by taking an analytic continuation of the maps (U, V ), of the fixed point obtained from the existence proof, to new domains 1U , respectively 1V , such that the corresponding fixed-point equations are satisfied. It is more convenient here, computationally, to work with the original maps (U, V ) of the existence proof, on their domains DU and DV , rather than with the analytic continuations. By using the properties of the parameterization function φ, obtained from the necklace construction, the definition of the sequence of domain pairs (used to construct the invariant curve) may be converted into a form that works with the original “inverse” composition order. The details of this are presented in Appendix A. From now on, we will work with the domain pairs for the inverse order. 3.2.2. Inverse IFS. Converted to the inverse composition order (see Appendix A), the mapping defining the domain pairs in the necklace construction (cf. Stirnemann [26]) is (M, N ) 7→ (α2 C(N ), α2 C(M ) ∪ QU α2 C(N )). Consequently, the universal invariant curve (or, rather, a certain piece of it) may be written as the invariant set of an iterated function system by eliminating M from the fixed point equation of the above mapping, to obtain N 7→ |α|4 (N ) ∪ QU α2 C(N ).

(13)

This mapping defines an IFS, consisting of the two maps ψ0 : z 7→ |α|4 z and ψ1 : z 7→ QU α2 C(z) acting on some initial domain N0 . The advantage of expressing (a piece of) the universal curve in this way is that the dimension of the invariant set of an IFS can often be estimated in a relatively straightforward manner by examining the maps of the IFS themselves. By computer-assisted means, we verify that these maps are contractions on some carefully chosen initial domain N0 . It follows that the IFS has a unique attractor N . Note that one of the maps (ψ0 ) is a similarity, while the other (ψ1 ) is an analytic map of z¯ = C(z), i.e. an anti-analytic map. In particular, both maps are local similarities. This is a great advantage in that we can calculate the dimension of the IFS attractor by using the machinery described in [11], which mainly deals with similarities.

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3.2.3. Binary patches. Definition 3. The IFS defined by the above mapping (13) will be called a binary IFS; the j th generation, denoted Gj , consists of 2j binary patches. Each patch P ∈ Gj is obtained by applying a particular j-fold composition of the maps ψ0 and ψ1 to some initial domain N0 (chosen to contain the attractor N ), i.e. P := ψkj ψkj−1 · · · ψk1 (N0 ), with each k` ∈ {0, 1}. We denote the mapping defining P by 9P : 9P := ψkj · · · ψk1 . Since a binary patch is entirely determined by this mapping, we may identify P with 9P . 3.3. Upper bounds on the Hausdorff dimension. 3.3.1. The partition-function equation. In general, if an IFS consists of n contractive maps, with corresponding contraction factors κi , then it follows [11] that an upper bound on the Hausdorff dimension may be calculated by finding a solution s to the (upper) partition-function equation Zu (s) :=

n X

κsi = 1.

i=1

We apply this method to each generation Gj of the IFS by calculating the contraction factors of the 2j mappings defining the binary patches. In other words, for each generation Gj , we solve the equation X κ(P )sj = 1, (14) Zu(j) (sj ) := P ∈Gj

for sj , where κ(P ) is the contraction rate of the map 9P on N0 , i.e. |9P (x) − 9P (y)| ≤ κ(P )|x − y| for all x, y ∈ N0 . 3.3.2. Calculating the contraction factors. The calculation of the contraction rates for the first generation, i.e. κ(ψ0 ) and κ(ψ1 ), is performed as follows: For the similarity ψ0 : z 7→ |α|4 z, the contraction factor is simply |α|4 . We rewrite the anti-analytic map ψ1 : z 7→ QU α2 C(z) as an analytic map composed with complex conjugation: ψ1 (z) = QU α2 C(z) = F (C(z)). We then use rigorous methods to calculate bounds on |F 0 (C(z))| on a convex superset of N0 . By the mean value theorem, an upper bound on this quantity is a Lipschitz constant for F on C(N0 ), and is therefore also a Lipschitz constant for F C = ψ1 on N0 . (For this problem, these Lipschitz constants are less than 1 and are, therefore, contractivity constants.) The actual calculation involves covering the boundary of N0 by a large number of overlapping rectangles. An extension of the techniques of interval arithmetic then allows us to obtain rigorous bounds on |F 0 (C(z))| on this covering. (For details of the

Rigorous Bounds on Hausdorff Dimension of Siegel Disc Boundaries

427

rigorous computational framework, see [3] and [4]. It is based on a technique described by Eckmann, Koch, and Wittwer [8].) 3.3.3. Choosing the initial domain. The initial domain N0 is taken to be a convex region bounded by a polygon. It is chosen so that it contains its images ψ0 (N0 ) ∪ ψ1 (N0 ) under the constituent maps of the IFS and, therefore, contains the IFS attractor N . An additional constraint is that the domain N0 must be contained within the domains of definition of the corresponding maps. (If this were not the case, it might still be possible to use this method, by taking an analytic continuation of the maps to suitable domains.) 3.3.4. Calculation of bounds on the dimension. The contractivities κ(ψ0 ) and κ(ψ1 ) of the first generation give a crude upper bound s1 on the Hausdorff dimension of the IFS attractor by solving the partition-function equation (14). This bound is then refined by considering subsequent generations, which requires the calculation of contractivity constants for compositions of maps. For example, one of the patches in the second generation is ψ1 ψ0 (N0 ). A contractivity constant for the map ψ1 ψ0 on N0 is given by multiplying the contractivity of ψ1 on (a convex superset of) ψ0 (N0 ) by that of ψ0 on N0 . Notice that the latter is one of the contractivity constants that has already been calculated for the first generation, which suggests an efficient iterative algorithm. Solving the corresponding partition-function equation for the second generation yields a new upper bound on the Hausdorff dimension. The sets over which we obtain bounds on the contractivities of the two IFS mappings reduce in size with each subsequent generation, as the IFS converges toward the attractor, which entitles us to the hope that successive generations will give improved bounds on the dimension (this is proved for the case when the contractivity constants are optimal, i.e. the “best possible”, in Sect. 3.5 and Appendix B).

Fig. 3. The initial domain and 7th generation of the IFS construction

Figure 3 shows the initial domain and the 7th generation of the IFS (cf. the scaling picture for the Siegel disc boundary in Fig. 2). The upper bounds for the Hausdorff dimension obtained by this method are shown in Table 1 for generations 1 to 12 of the IFS. Recall that the necklace construction yields the (universal) invariant curve of the renormalization fixed-point functions. The results obtained above are bounds on the Hausdorff dimension of a piece of this curve. It follows from the details of the parameterization of the curve constructed in [3] (see also Appendix A) that the rest of the curve may be pieced together from bi-Lipschitz copies

428

A. D. Burbanks, A. H. Osbaldestin, A. Stirnemann Table 1. Upper bounds on the Hausdorff dimension for the inverse composition order j 1 2 3 4 5 6 7 8 9 10 11 12

Upper bound 1.21975 1.20298 1.17154 1.15548 1.13904 1.12748 1.11694 1.10864 1.10124 1.09515 1.08975 1.08523

of this piece. Since this process does not increase the Hausdorff dimension, it follows that we have calculated upper bounds on the dimension of the invariant curve itself. 3.4. Lower bounds on the Hausdorff dimension. It is, in principle, possible to obtain lower bounds on the dimension by solving the corresponding lower partition-function equations, X ρ(P )rj = 1, Z`(j) (rj ) := P ∈Gj

for rj , where ρ(P ) is a so-called coercivity constant for the map 9P on N0 , i.e. ρ(P )|x − y| ≤ |9P (x) − 9P (y)| ≤ κ(P )|x − y| for all x, y ∈ N0 .

(15)

Calculating these coercivity constants is more computationally expensive as it involves evaluating the derivative of maps on convex supersets of certain preimages of domains (for more details, see [4]). So far, the only lower bounds that we have been able to obtain by this method are less than 1. 3.4.1. The open set condition. In order to use this approach for the lower bounds, the IFS must satisfy an open set condition [11]: we must be able to find a non-empty bounded open set such that ⊆ ψ0 () ∪ ψ1 (), with the union on the right disjoint. Such a condition was proved in the case of the accretive composition order, by adapting the necklace construction approach (the full details are given in [4]). 3.5. Convergence of lower and upper bounds. Theorem 3. If the contractivities and coercivities, κ(P ) and ρ(P ), are optimal then the upper and lower bounds, tj and rj , on the Hausdorff dimension (defined by solutions of the upper, respectively lower, partition-function equations) converge to a common limit as j → ∞.

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429

By “optimal”, it is meant that ρ(P ) = κ(P ) =

sup

{ ρ : ρ|x − y| ≤ |9P (x) − 9P (y)| },

inf

{ κ : κ|x − y| ≥ |9P (x) − 9P (y)| }.

x,y∈N0 , x6=y x,y∈N0 , x6=y

The proof is given in Appendix B, and involves establishing that a “bounded variation principle” holds, namely that the quotient of (optimal) coercivity and contractivity, ρ(P )/κ(P ), is bounded uniformly away from zero. (In practice, however, the contractivity and coercivity constants that we calculate by using rigorous numerics and the mean value theorem will not generally be optimal.) 3.6. Renormalization invariance of Hausdorff dimension. We now show that the Hausdorff dimension is invariant under the renormalization scheme for maps that are both (1) close enough to the critical fixed point and (2) attracted to it under renormalization. It then follows that the bounds we have calculated for the (universal) invariant curve of the fixed-point functions also apply to the Siegel disc boundaries of these maps. (In what follows, we will use the “generic” notation of Sect. 2.5 for clarity.) 3.6.1. Asymptotic self-similarity. By analogy with [26], we consider “asymptotically self-similar” pairs of maps, i.e. we examine the iterates under renormalization of those maps that are attracted to the critical fixed point. Denote such a pair of maps by (U0 , T0 ) and denote the iterates under renormalization by (Uk , Tk ). The necklace construction defines sequences of domain pairs, (Mkj , Njk ), that converge to invariant sets for these iterates, by means of the following equations: −1 Njk+1 , Mkj+1 = Tk Bk+1 −1 k Nj+1 = Bk+1 (Mk+1 ∪ Njk+1 ). j

Letting j → ∞, gives equations for the invariant pairs of sets, (Mk , N k ): −1 N k+1 , Mk = Tk Bk+1 k

N =

−1 Bk+1 (Mk+1

∪N

(16) k+1

).

(17)

(Thus, Mk and N k are actually pieces of the invariant curve of the k th iterate (Uk , Tk ) under renormalization.) 3.6.2. Renormalization invariance. The map T B −1 (at the fixed point) is bi-Lipschitz. −1 are also bi-Lipschitz close enough to the By continuity it follows that the maps Tk Bk+1 −1 fixed point. The maps B −1 (at the fixed point) and Bk+1 are also bi-Lipschitz as they are simply rescalings. Lemma 1. Consider an asymptotically self-similar pair of maps (U0 , T0 ) that is close enough to the fixed point in the sense that the maps Tk are bi-Lipschitz for all k. Then the Hausdorff dimensions of the invariant sets (Mk , N k ), for the iterates of this pair, are all equal.

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Proof. Denote by dk , respectively ek , the Hausdorff dimensions of Mk , respectively N k: ek := dimH (N k ). dk := dimH (Mk ), Since the mappings in (16) and (17) are bi-Lipschitz, we obtain ek = max{dk+1 , ek+1 }, ek+1 = dk . If dk+1 ≤ ek+1 then we conclude that dk = ek . If not, i.e., if dk+1 > ek+1 , we obtain ek = dk+1 , which implies that dk < ek . This, according to the first remark, yields dk−1 = ek−1 . It follows that ek = dk =: d for all k, i.e., the Hausdorff dimension of the invariant sets of all the iterates are all the same; Hausdorff dimension is a renormalization invariant (close enough to the fixed point).

4. Conclusions We have given a rigorous computer-assisted proof that the dimension of the Siegel disc boundary for maps attracted (and close enough) to the critical renormalization fixed point lies below 1.08523. This was done by calculating bounds on the contractivities of the maps of an iterated function system, whose attractor is a piece of the universal invariant curve of the fixedpoint functions. By showing that the whole curve is pieced-together from bi-Lipschitz copies of this piece, and by proving that the Hausdorff dimension is invariant under renormalization for maps that are close enough to the fixed point and attracted to it, we deduce that the bound obtained also holds for the dimension of the Siegel disc boundaries of these maps. By considering modified partition-function equations, it should be possible to use the same techniques to obtain rigorous bounds on the multifractal spectrum [11] of the dynamically-invariant measure on the Siegel disc boundary. It seems possible to improve the bounds obtained here by using more computational power and by using the analytic continuations of the maps with the IFS for the accretive composition order (in particular, this might yield a useful lower bound). The nature of the spectrum of the derivative of the renormalization operator at the fixed point is still not rigorously known; in particular, it is not even known whether the fixed point has a stable manifold! Empirically, however, the situation is clear: the fixed point is hyperbolic in a suitable subspace (of commuting pairs of even maps), having a stable manifold of real codimension two. It may be possible to prove this by finding a suitable change of basis under which the linearisation of the derivative is sufficiently close to diagonal. It should be possible to generalise the necklace construction approach to handle other rotation numbers with eventually periodic continued fraction expansions. This extension is conceptually straightforward, but will entail computational difficulties. Acknowledgement. This work was carried out under EPSRC (UK) grant GR/H38386.

Rigorous Bounds on Hausdorff Dimension of Siegel Disc Boundaries

431

A. Inverse Composition-Order Domain Pairs A.1. Deriving the domain pairs. In the following, we will use the “generic” notation (U, T , B) of Sect. 2.5. The definition of the domain pairs (from which we obtain our IFS) in the necklace construction of [26] and [3] concerned the accretive composition order, corresponding to fixed-point equations (11) and (12). Here, we demonstrate how it is converted into a suitable form to use with the inverse composition order used in the bulk of this paper, corresponding to the fixed-point equations (9) and (10). The mapping used to construct the sequence of domain pairs for the accretive composition order is given by (Mj , Nj ) 7→ (Mj+1 , Nj+1 ) where Mj+1 = T B−1 Nj , Nj+1 = B

−1

(Mj ∪ Nj ).

(18) (19)

Let (M, N ) denote the limit pair of this mapping. Eliminating M from the fixed point equation for the above mapping yields an IFS for the accretive composition order, N 7→ B −1 T B−1 N ∪ B−1 N .

(20)

In [3], we used the necklace construction with (U, T , B) = (QU, QV, Cα−2 ) to show that the pair (QU, QV ) has a H¨older continuous invariant curve φ(t) which satisfies the following equations: Uφ(t) = φ(t − 1) for 1 − ω ≤ t ≤ 1, T φ(t) = φ(t + ω) for −ω ≤ t ≤ 1 − ω, φ(0) = 0, −1 B φ(t) = φ(−ωt) for −ω ≤ t ≤ 1,

(21) (22) (23) (24)

where ω is the golden mean. (It was then shown, essentially by “taking a square root” that this yields a curve χ(t) for the pair (U Q, V Q) = (E, F ) relevant to the Siegel disc problem.) It was also shown that the limit sets M, N of the domain pairs satisfy N = φ([−ω, 0]), M = φ([0, 1]), from which it follows that L := M ∪ N = φ([−ω, 1]) = T N ∪ UM. We use this to rewrite the mapping defining the domain pairs (18) and (19) as follows: Mj+1 = T B−1 Nj , Nj+1 = B −1 (T Nj ∪ UMj ).

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It follows that T UM = T UT B−1 N = B −1 (UT N ), UT N = UT B −1 T N ∪ UT B−1 UM = UB −1 (UT N ) ∪ B −1 (T UM). ˆ := T UM, Nˆ := UT N satisfy the fixed-point This, in turn, implies that the sets M equations ˆ = B −1 Nˆ , M ˆ ∪ UB −1 Nˆ . Nˆ = B −1 M

(25) (26)

Putting (U, T , B) = (QU, QV, Cα−2 ) then gives the definition of the domain pairs for the inverse composition order given in Sect. 3, Eq. (13). A.2. Continuous extension of the necklace curve. The above discussion suggests that we define ˆ := φ([0, ω]), M Nˆ := φ([−1, 0]). ˆ ∪ Nˆ = φ([−1, ω]). However, φ(t) is not defined, a priori, We would then have Lˆ := M for t < −ω (for the usual necklace construction, the necklace set L is φ([−ω, 1])). We therefore have to extend φ in some way. This is done quite simply by using (24) to show that φ(t) = Bφ(−ωt), for −ω ≤ t ≤ 1. The right hand side is defined for − ω1 ≤ t ≤ 1, and as a result we can use it to continuously extend φ to the interval [−1, 1]. ˆ and Nˆ are justified and may be With this extension, the above definitions of M written as ˆ := φ([0, ω]), M Nˆ := Bφ([0, ω]). ˆ and Nˆ actually satisfy the fixed-point It is now necessary to verify that the sets M equations (25) and (26). Firstly, ˆ B −1 Nˆ = φ([0, ω]) = M. For the second equation, observe that UB −1 Nˆ = Uφ([0, ω]) = BT B −1 φ([0, ω]) = BT φ([−ω 2 , 0]) = Bφ([−ω 2 + ω, ω]) = φ([−1, ω − 1]) = φ([−1, −ω 2 ]),

Rigorous Bounds on Hausdorff Dimension of Siegel Disc Boundaries

and that

433

ˆ = B −1 φ([0, ω]) = φ([−ω 2 , 0]). B−1 M

Finally, from (25) and (26), we obtain the fixed-point equation for the inverse binary IFS used in Sect. 3.2: Nˆ = B −2 Nˆ ∪ UB−1 Nˆ . B. Convergence of Upper and Lower Bounds Here, we prove Theorem 3, namely that the upper and lower bounds, tj and rj , defined by solutions of the corresponding partition-function equations, converge to a common limit as j → ∞ (provided that the contractivity and coercivity constants are optimal). In what follows, it is more convenient to reason in terms of the accretive composition order. Recall that (from the previous appendix) the invariant set of the domain pairs for the accretive composition order is L = φ([−ω, 1]) and that for the inverse composition order it is Lˆ = φ([−1, ω]). In other words, these two invariant sets are just different (overlapping) pieces of the same (invariant) curve φ(t). (Furthermore, one may be obtained from the other by applying a bi-Lipschitz mapping, which confirms that their Hausdorff dimensions are the same.) We will also use the “generic” notation to simplify things: here, (U, T , B) = (QU, QV, Cα−2 ) as used in [3]. B.1. Difference quotients. Definition 4 (Product of binary patches). The product P ∗ Q of two binary patches is the patch identified by the composition 9P ◦ 9Q . Recall that the union of all the binary patches with a fixed generation index j is called the j th generation and is denoted by Gj . (The above definition means that j(P ∗ Q) = j(P ) + j(Q), where j(P ) indicates the generation index for patch P .) Definition 5 (Product of generations). We define the product of two generations by Gj ∗ Gk := { P ∗ Q : P ∈ Gj , Q ∈ Gk }. (So we have Gj+1 = Gj ∗ G1 , and G2j = Gj ∗ Gj .) We are interested in the contractivity and coercivity of the maps 9P . To that end, we examine difference quotients: 1P (x, y) :=

9P (x) − 9P (y) , x−y

where x and y range over N (the initial domain of the relevant IFS) and x 6= y. Obviously, an upper bound of this difference quotient is a Lipschitz constant for 9P . Similarly, a lower bound is a coercivity constant (see Eq. (15)). (Because the maps ψ0 and ψ1 are both contractions, the Lipschitz constants are strictly smaller than 1 and are therefore contractivity constants.) Define ρ(P ) := κ(P ) :=

inf

|1P (x, y)|,

(27)

sup

|1P (x, y)|,

(28)

x,y∈N , x6=y x,y∈N , x6=y

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A. D. Burbanks, A. H. Osbaldestin, A. Stirnemann

i.e. let ρ(P ) and κ(P ) be optimal coercivity and contractivity constants, respectively. Notice that the diameter |P | of P admits the estimates ρ(P ) · |N | ≤ |P | ≤ κ(P ) · |N |.

(29)

Further, it follows that 9P (9Q (x)) − 9P (9Q (y)) x−y 9P (9Q (x)) − 9P (9Q (y)) 9Q (x) − 9Q (y) · . = 9Q (x) − 9Q (y) x−y

1P ∗Q (x, y) =

(Here, the right hand side is defined whenever x is different from y.) Therefore ρ(P ∗ Q) ≥ ρ(P )ρ(Q), κ(P ∗ Q) ≤ κ(P )κ(Q).

(30) (31)

Thus, ρ is super-multiplicative and κ is sub-multiplicative. B.2. Partition functions. Consider the upper and lower partition functions Zl(j) (s) and Zu(j) (s) defined by X ρ(P )s , (32) Zl(j) (s) := P ∈Gj

Zu(j) (s)

:=

X

κ(P )s .

(33)

P ∈Gj

(Notice that both functions are strictly decreasing in s.) Define real numbers rj and tj which solve the equations X ρ(P )rj = 1, (34) Zl(j) (rj ) := P ∈Gj

Zu(j) (tj )

:=

X

κ(P )tj = 1.

(35)

P ∈Gj

Lemma 2. If ρ(P ) is super-multiplicative and κ(P ) is sub-multiplicative, then the solutions, rj and tj , of the partition-function equations, satisfy rj ≤ r2j ≤ t2j ≤ tj .

(36)

Proof. Equation (30) (super-multiplicativity) implies that X X ρ(P )s = ρ(Q ∗ R)s Zl(2j) (s) = P ∈G2j

≥

X

Q,R∈Gj

Q,R∈Gj

ρ(Q)s ρ(R)s = (Zl(j) (s))2 .

(37)

In the same way, it follows by sub-multiplicativity (31) that Zu(2j) (s) ≤ (Zu(j) (s))2 .

(38)

Equations (37) and (38), together with the fact that the partition functions are strictly decreasing in s, imply Eq. (36).

Rigorous Bounds on Hausdorff Dimension of Siegel Disc Boundaries

435

If, in addition, it turns out that |rj − tj | → 0 as j → ∞, then it follows that the sequences rj and tj converge to a common limit. The next section proves that this is the case. The differences rj − tj are shown to form a zero sequence by using a bounded variation principle. B.3. The bounded-variation principle. In this section we prove that the quotient of optimal coercivity and contractivity, ρ(P )/κ(P ), remains uniformly bounded away from zero. B.3.1. Estimating difference quotients. Here, we will reason in terms of the accretive IFS, Eq. (20), as it makes the exposition simpler. Consider the map ψ1 = B −1 T B−1 . This map is analytic on an open neighbourhood of the set N . Let 0 be a contour surrounding N in , and let U be the interior of 0. The following estimates then follow: Proposition 1. 1. There exists a function γ(z, w) analytic on U × U such that ψ1 (z) − ψ1 (w) − ψ1 0 (z) = (z − w)γ(z, w). z−w

(39)

2. There exists a function δ(z, w) analytic on U × U such that ψ1 0 (z) − ψ1 0 (w) = (z − w)δ(z, w). Proof. By Cauchy’s theorem, we have I 1 2πi 0 I 1 ψ1 (w) = 2πi 0 ψ1 (z) =

ψ1 (ζ) dζ , ζ −z ψ1 (ζ) dζ , ζ −w

for any two points z and w of U . It follows that I ψ1 (ζ) dζ 1 ψ1 (z) − ψ1 (w) = . z−w 2πi 0 (ζ − z)(ζ − w) Moreover, 1 ψ1 (z) = 2πi 0

I 0

ψ1 (ζ) dζ . (ζ − z)2

Subtraction yields 1 ψ1 (z) − ψ1 (w) − ψ1 0 (z) = z−w 2πi

I 0

ψ1 (ζ) dζ · (w − z), (ζ − z)2 (ζ − w)

which verifies the first claim. Similarly, the equation I ψ1 0 (ζ)dζ 1 · (z − w) ψ1 0 (z) − ψ1 0 (w) = 2πi 0 (ζ − z)(ζ − w) proves the second claim.

(40)

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A. D. Burbanks, A. H. Osbaldestin, A. Stirnemann

B.3.2. Quotients of difference quotients. In order to discuss quotients of coercivity and contractivity, it is necessary to estimate the quotient of two difference quotients. Definition 6. Let x, y, x0 , y 0 ∈ N , with x 6= y, and x0 6= y 0 . Define Qψ (x, y, x0 , y 0 ) :=

ψ(x) − ψ(y) ψ(x0 ) − ψ(y 0 ) . x−y x0 − y 0

(41)

Proposition 2. There exists a constant c0 > 0 such that if x, y, x0 , y 0 are confined to a subset of N of diameter d, then log |Qψ0 (x, y, x0 , y 0 )| ≤ c0 d, log |Qψ1 (x, y, x0 , y 0 )| ≤ c0 d.

(42) (43)

Proof. ψ1 (x) − ψ1 (y) ψ1 (x0 ) − ψ1 (y 0 ) x−y x0 − y 0 ψ1 0 (x) + (x − y)γ(x, y) = 0 0 ψ1 (x ) + (x0 − y 0 )γ(x0 , y 0 ) ! 1 + (x − y) γ(x,y) ψ1 0 (x) ψ1 0 (x) = 0 0 0 ,y 0 ) ψ1 (x ) 1 + (x0 − y 0 ) γ(x ψ1 0 (x0 ) ! γ(x,y) 0 1 + (x − y) 0 (x) δ(x, x ) ψ 1 . = 1 + (x − x0 ) 0 0 0 ,y 0 ) ψ1 (x ) 1 + (x0 − y 0 ) γ(x ψ1 0 (x0 )

Qψ1 (x, y, x0 , y 0 ) =

Note that ψ1 0 , i.e. (B −1 T B−1 )0 , does not vanish on N . This follows from [3], where it was shown that the derivative of T B−2 is bounded away from zero on (a strict superset of) L during the verification of the necklace hypotheses. Notice that T B−2 on L is conjugate to B −1 T B−1 on N = B −1 L by the anti-similarity B−1 . Thus the right hand sides are well defined. N is compact (by the necklace construction [3]), hence there exists a constant c0 > 0 such that |Qψ1 (x, y, x0 , y 0 )| ≤ 1 + c0 max{|x − y|, |x0 − y 0 |, |x − x0 |}. In particular, if the points x, y, x0 , and y 0 are confined to a subset of N of diameter d, then the absolute value of Qψ1 is bounded by 1 + c0 d. Taking the logarithm yields the estimate log |Qψ1 | ≤ c0 d,

(44)

by the inequality log(1 + t) ≤ t valid for all t > −1. For the map ψ0 = B −1 , which is an anti-similarity, the difference quotients have constant absolute value. Therefore |Qψ0 | = 1, and the bound log |Qψ0 | ≤ c0 d follows immediately. B.3.3. Bounded variation for difference quotients. Proposition 3. The quotient of optimal coercivity and optimal contractivity, ρ(P )/κ(P ), is bounded uniformly away from zero.

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437

Proof. Note that, for any binary patch P and for any two different points x and y in N , we have x1 − y1 xj − yj 9P (x) − 9P (y) ··· = x−y xj−1 − yj−1 x0 − y0 ψ`j (xj−1 ) − ψ`j (yj−1 ) ψ` (x0 ) − ψ`1 (y0 ) = ··· 1 , xj−1 − yj−1 x0 − y0 (recall that j is the generation index of patch P ) where x0 = x, xi = ψ`i (xi−1 ),

y0 = y, yi = ψ`i (yi−1 ),

1 ≤ i ≤ j,

and each ψ`i is either ψ0 = B −1 or ψ1 = B −1 T B−1 . It follows that 9P (x) − 9P (y) 9P (x0 ) − 9P (y 0 ) 1P (x, y) = 1P (x0 , y 0 ) x−y x0 − y 0 0 0 = Qψ`j (xj−1 , yj−1 , xj−1 , yj−1 ) · · · Qψ`1 (x0 , y0 , x00 , y00 ).

(45)

Observe that, for any i, the points xi , yi , x0i , and yi0 are confined to one and the same binary patch of generation i (the sequence {ψ`i } being the same for each of them). The diameter of such a patch is bounded by c1 κi for some constant c1 , where κ is the quantity 1

κ := max(λ, µ 2 ), and λ and µ are bounds for the contractivities of the maps B −1 and T B−2 respectively. (This fact was established in [3] and [4].) Taking the logarithm of the absolute value of Eq. (45), and using Proposition 2 then yields | log 1P (x, y) − log 1P (x0 , y 0 )| ≤ c2 κj−1 + c2 κj−2 + · · · + c2 κ0 , where c2 = c0 c1 . The geometric series on the right hand side is bounded uniformly in j. Taking the exponential of both sides yields the estimate 1P (x, y) 1 ≤ c3 , ≤ c3 1P (x0 , y 0 ) where c3 > 1 is a constant. Since this holds for all pairs (x, y) and (x0 , y 0 ) in P × P , it continues to hold with the supremum of the numerator and the infimum of the denominator (optimality), giving 0<

κ(P ) 1 ≤ c3 . ≤ c3 ρ(P )

Since ρ(P ) ≤ κ(P ), we can restate this by saying that there exists a positive constant c > 1 such that 1 κ(P ) ≤ ρ(P ) ≤ κ(P ). c

(46)

This is the principle of bounded variation. (Notice that the bound depends neither on the patch P , nor on its generation.)

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A. D. Burbanks, A. H. Osbaldestin, A. Stirnemann

B.4. Convergence of upper and lower bounds. Lemma 3. The differences rj − tj converge to zero as j → ∞. Proof. In order to simplify notation, fix k := 2i (i ≥ 0) and put r := rk and t := tk . Equation (36) gives r ≤ t, i.e. t = r + δ, where δ ≥ 0. By definition, Eq. (34), we have X ρ(P )r = 1, P

X

κ(P )t = 1,

P

where the sums are taken over all patches of generation k. Using bounded variation, Eq. (46), it follows that X X κ(P )t ≤ ct ρ(P )t 1 = P t

= c

X

P r+δ

ρ(P )

P

≤ ct (ρmax )δ

X

ρ(P )r ,

for some ρmax

P

= ct (ρmax )δ . Observe that, since the sequence t2i is decreasing, the quantities ct = ct2i are bounded above as i → ∞. Moreover, ρmax → 0 as i → ∞ (see, for example, Eq. (29)). Therefore, the product ct (ρmax )δ can only stay bounded away from zero if δ → 0 as i → ∞, i.e. if |rk − tk | → 0. It then follows readily (using properties of sub-multiplicative and super-multiplicative sequences) that |rj − tj | → 0 as j → ∞. This, along with Lemma 2, completes the proof of Theorem 3, namely that rj and tj converge to a common limit as j → ∞. References 1. Arnold, V.I.: Mathematical Methods of Classical Mechanics. New York: Springer-Verlag, 1978 2. Arnold, V.I.: Geometrical methods in the theory of ordinary differential equations. New York: SpringerVerlag, 1983 3. Burbanks, A.D., Stirnemann, A.: H¨older continuous Siegel disc boundary curves. Nonlinearity 8, 901– 920 (1995) 4. Burbanks, A.D.: Renormalization for Siegel discs. PhD. thesis: Loughborough University, 1997 5. Campanino, M., Epstein, H.: On the existence of Feigenbaum’s fixed point. Commun. Math. Phys. 79, 261–302 (1981) 6. Campanino, M., Epstein, H., Ruelle, D.: On Feigenbaum’s functional equation g ◦ g(λx) + λg(x) = 0. Topology 21, 125–129 (1982) 7. Eckmann, J.-P., Epstein, H.: On the existence of fixed points of the composition operator for Circle Maps. Commun. Math. Phys. 107, 213–231 (1986) 8. Eckmann, J.-P., Koch, H., Wittwer, P.: A computer assisted proof of universality for area-preserving maps. Mem. Am. Math. Soc. 47, 1–122 (1984) 9. Eckmann, J.-P., Wittwer, P.: A complete proof of the Feigenbaum conjectures. J. Stat. Phys. 46, 455–477 (1987) 10. Epstein, H.: New proofs of the existence of the Feigenbaum functions. Commun. Math. Phys. 106, 395-426 (1986)

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11. Falconer, K.: Fractal Geometry, Mathematical Foundations and Applications. New York: Wiley, 1990 12. Feigenbaum, M.J.: Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 19, 25–52 (1978) 13. Feigenbaum, M.J.: The universal metric properties of nonlinear transformations. J. Stat. Phys. 21, 669– 706 (1979) 14. Feigenbaum, M.J., Kadanoff, L.P., Shenker, S.J.: Quasiperiodicity in dissipative systems: a renormalization group analysis. Physica 5D, 370–386 (1982) 15. Greene, J.M., MacKay, R.S., Vivaldi, F., Feigenbaum, M.J.: Universal behaviour in families of areapreserving maps. Physica 3D, 468–486 (1981) 16. Lanford, O.E.: A computer-assisted proof of the Feigenbaum conjectures. Bull. Am. Math. Soc. 6, 427–434 (1982) 17. Lanford, O.E.: Computer-assisted proofs in analysis. Proc. Int. Congr. of Mathematicians, Berkeley, California (1986) 18. Manton, N.S., Nauenberg, M.: Universal scaling behaviour for iterated maps of the complex plane. Commun. Math. Phys. 89, 555–570 (1983) 19. MacKay, R.S.: Renormalization in area-preserving maps. PhD thesis: Princeton, 1982 20. Mestel, B.D.: A computer-assisted proof of universality for cubic critical maps of the circle with golden mean rotation numbers. PhD thesis, Warwick University (1985) 21. Osbaldestin, A.H.: Siegel disc singularity spectra. J. Phys. A 25, 1169–1175 (1992) 22. Ostlund, S., Rand, D.A., Sethna, J., Siggia, E.D.: Universal properties of the transition from quasiperiodicity to chaos in dissipative systems. Physica 8D, 303–342 (1983) 23. Siegel, C.L.: Iteration of analytic functions. Ann. Math. 43, 607–612 (1942) 24. Sinai, Ya.G., Khanin, K.M.: Smoothness of conjugacies of diffeomorphisms of the circle with rotations. Russ. Math. Surv. 44, 69–99 (1989) 25. Stark, J.: Smooth conjugacy and renormalization for diffeomorphisms of the circle. Nonlinearity 1, 541–575 (1988) 26. Stirnemann, A.: Renormalization for golden circles. Dissertation ETH No. 9843. Swiss Federal Institute of Technology, Zurich (1992); Commun. Math. Phys. 152, 369–431 (1993) 27. Stirnemann, A.: A renormalization proof of Siegel’s theorem. Nonlinearity 7, 943–958 (1993) 28. Stirnemann, A.: Existence of the Siegel disc renormalization fixed point. Nonlinearity 7, 959–974 (1993) 29. Stirnemann, A.: Towards an existence proof of MacKay’s fixed point. Commun. Math. Phys. 188, 723– 735 (1997) 30. Sullivan, D.: Bounds, quadratic differentials and renormalization conjectures. In: Mathematics into the Twenty-first Century, Vol. 2. ed. F. Browder, Providence, RI: Am. Math. Soc. 1992, pp. 417–466 31. Widom, M.: Renormalization group analysis of quasi-periodicity in analytic maps. Commun. Math. Phys. 92, 121–136 (1983) 32. Yoccoz, J.-C.: Lin´earisation des germes de diff´eomorphisms holomorphes de (C, 0). C. R. Acad. Sci. Paris 306, 55–58 (1988) Communicated by Ya. G. Sinai

Commun. Math. Phys. 199, 441 – 470 (1998)

Communications in

Mathematical Physics © Springer-Verlag 1998

Coarsening by Ginzburg–Landau Dynamics J.-P. Eckmann1,2 , J. Rougemont1 1 2

D´epartement de Physique Th´eorique, Universit´e de Gen`eve, CH-1211 Gen`eve 4, Switzerland Section de Math´ematiques, Universit´e de Gen`eve, CH-1211 Gen`eve 4, Switzerland

Received: 26 June 1997 / Accepted: 18 May 1998

Abstract: We study slowly moving solutions of the real Ginzburg–Landau equation on the infinite line, by a method inspired by the work of J. Carr and R. L. Pego. These solutions are functions taking alternately positive or negative values on large intervals. We give a formula for the speed of the motion of the zeros of these functions and present a description of the collapse of two nearby zeros. Since we work on an infinite domain, we can find initial conditions which lead to infinite sequences of collapses of kinks. These results justify in part the approach to the coarsening problem by a simplified model proposed by A. J. Bray, B. Derrida, and C. Godr`eche. Their model starts from a random distribution of domain lengths and evolves by successive elimination of the smallest domain. Our model is on one hand more realistic, since it controls the original Ginzburg– Landau equation in continuous time. On the other hand, we fail to gain sufficient control over the vanishing of pairs of kinks “embedded” within pairs of kinks at a greater distance. 1. Introduction In a series of papers ([CP1,CP2]), Carr and Pego studied the evolution of multi-kink initial data of the real Ginzburg–Landau equation: ∂t v = ∂x2 v + v − v 3 ,

(1.1)

with v(x, t) : R ×R+ → R. These data are for most x very close to the stationary values v = ±1 with transitions from ±1 to ∓1 at certain points. We call these points “kinks”. √ Since an isolated kink moves to the stable stationary solution tanh((x − x0 )/ 2) for some x0 ∈ R, one expects the dynamics of multi-kink data to be slow when the inter-kink distance is very large. Carr and Pego showed that the dynamics of the position of the kinks can be approximated by a simple potential model with an exponentially decaying force between the kinks. They attract each other weakly and eventually two opposite

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kinks collide and “annihilate”. Carr and Pego proved their results for the equation on an interval subject to Neumann boundary conditions. In this paper, we extend their method to prove similar results for the equation on the infinite line. First we show that if the kinks are initially widely separated, then their annihilation goes through a well controlled sequence of shapes. One can then ask (and answer) some questions about the evolution of initial data with an infinity of kinks. They lead to a detailed relationship between this continuous dynamics and the stochastic model of coarsening studied by Bray et al. in [BDG, BD]. See also Bray [B] and references therein. There are several interesting technical points in this derivation, for example the problem to show that there can be no “conspiracy” between kinks which are far apart to change the basic potential evolution between neighboring kinks. Our analysis goes some way towards formulating a fully probabilistic set of “reasonable” initial conditions. However, we fall short of finding a set of positive measure on the full line of such conditions, but at least we can show that there are local conditions which guarantee the control of the evolution for all times. Our results hold for equations somewhat more general than the Ginzburg–Landau equation: if we introduce the notation VGL (z) =

z4 z2 − , 2 4

(1.2)

then the r.h.s. of Eq. (1.1) is 0 (v). LGL (v) = ∂x2 v + VGL

(1.3)

We can extend our results to all equations of the form

where

∂t v = L(v),

(1.4)

L(v) = ∂x2 v + V 0 (v),

(1.5)

for any function V ∈ C which satisfies: 3

V (+x) = V (−x), V 0 (±1) = V 0 (0) = 0, V 0 (x) 6 = 0 for x 6∈ {±1, 0}, V 00 (±1) < 0, V 00 (0) ≥ 1. Remark. The condition V (x) = V (−x) can be replaced by the simpler V (1) = V (−1), but then the notation gets somewhat more involved. We have fixed the scale by imposing V 0 (±1) = 0. Furthermore, to simplify the choice of cutoff functions, we have required V 00 (0) ≥ 1, and this could be generalized easily to V 00 (0) > 0. Note that our assumptions imply that any initial condition which is bounded by 1 in L∞ (R, dx) leads to a solution which is again bounded by 1, see [C,G]. We shall only consider such solutions. We start by listing all bounded stationary – i.e., time-independent – solutions of (1.4). They can be interpreted as trajectories of a free point particle moving in the potential V without friction, with x being the “time” variable (see Fig. 1). Note that this is an integrable Hamiltonian system. The stationary solutions are: – Three constant solutions, u± (x) = ±1, u0 (x) = 0.

Coarsening by Ginzburg–Landau Dynamics

443

V (v )

j ,1

,A

j

j

0

1

v

Fig. 1. The mechanical interpretation of the equation L(v) = 0

– Two heteroclinic solutions, which we will denote by ψ(x) and ψ(−x). We fix the notation by imposing xψ(x) ≤ 0. For V given by Eq. (1.2), such a solution is: x ψ(x) = − tanh √ . 2 – Periodic solutions, which are described in the following proposition (see Fig. 2).

' ( x) ^ D

D ( A)

0

> x

,A Fig. 2. A periodic solution ϕD (x)

Proposition 1.1. For every V as described above there is a D0 > 0 such that for every D ∈ (D0 , ∞), there exists a periodic stationary solution ϕD (x) of (1.4) with period 2D and amplitude A, and ϕD (x) has exactly one maximum and one minimum per period. Furthermore, ϕD ∈ C ∞ and there is a real analytic bijection between the amplitude A ∈ (0, 1) and D = D(A). There exist constants c1 > 0 and c2 > 0 such that the inverse function A(D) satisfies (1.6) 1 − A(D) ≤ c2 e−c1 D .

444

(

v x; t

J.-P. Eckmann, J. Rougemont

) t

x

Fig. 3. Numerical simulations. Top left: time 1 to 2.7, top right: time 25 to 29.2, middle left: time 29.2 to 31.3, middle right: time 3668 to 3675, bottom left: time 5043 to 5050 and bottom right: time 5266 to 5269

Remark. We shall choose the origin in such a way that ϕD (0) = ϕD (D) = 0, ϕD (x) < 0 for x ∈ (0, D). In fact, one has the formula D0 = V 00 (0)π. The result stated in this proposition is certainly not new, but in order to make the paper self-contained, we give a proof in Sect 9. Remark. The above list is exhaustive (up to translation). This can be seen by examining Fig. 1. The initial position −A of the particle (with zero initial speed) must satisfy A ∈ [0, 1] to keep the orbit bounded. But for any such initial condition, we have exhibited a solution of the equation L(v) = 0. Thus, there are no other bounded stationary solutions of Eq. (1.4). (See [G] for a detailed exposition of this result.) These periodic solutions will play an important role in the sequel. Our aim is to show the existence of “metastable” states, i.e., states that are unstable, but which creep for a very long time (see the numerical simulations in Fig. 3). It is common knowledge that the solutions u± are stable, that ψ is stable up to an eigenvalue 0 (corresponding to translations) and that all the other stationary solutions are unstable. We want to study the evolution of initial conditions v(t = 0) which are like “crenelations”: we define the set Z of zeros of v(t = 0) as Z = {zj ∈ R, j ∈ Z, zj < zj+1 and v(x = zj , t = 0) = 0 for all j}.

Coarsening by Ginzburg–Landau Dynamics

445

We assume v(x, t = 0) is negative for z2j < x < z2j+1 and positive for z2j−1 < x < z2j . We also introduce the interval lengths `j defined by: `j = zj − zj−1 . (The above definition is easily adapted to the case of a finite number of kinks.) Definition. A function which is of the form above will be called admissible. If we look only at the zeros of the solution of Eq. (1.4), then we have a reduced system of equations for the positions of the zeros. Thus Z becomes a function of time. One of the difficulties in the infinite domain is to show that there are “interesting” admissible initial data which remain admissible for all times when evolved with Eq. (1.4). The evolution of these initial data will look as follows. First, the positive (negative) part of u approaches√rapidly +1 (−1) and domain walls form in between, which (locally) look like ± tanh(x/ 2) (generally, the heteroclinic solutions). Intuitively, ±1 are stable fixed points, but the domain walls will move using the “only” degree of freedom they have, namely the translation mode associated with the eigenvalue zero of the heteroclinic solution. Since there is no reason for +1 to be preferred to −1 or vice-versa, the speed of the motion of a domain wall will, to first approximation, only depend on the sizes `i , `i+1 of the two domains adjacent to this wall. Carr and Pego showed, in the finite domain, that the speed of motion of the ith kink is roughly e−`i+1 − e−`i . We follow the method of Carr and Pego to prove similar results in the infinite domain: to a prescribed set Z, we associate a function u(0) Z having Z as the set of its zeros. In each (x) equal to a translate of ϕD with D = zi+1 − zi , so that interval (zi , zi+1 ), we set u(0) Z (0) uZ is a continuous function, alternately positive and negative between successive zeros. Then we slightly deform this non-differentiable function near each zero to get a smooth function uZ . (The idea of gluing near the zeros instead of gluing in the middle of the intervals was already present in Carr and Pego and is very fruitful.) This function uZ is, by construction, equal to a stationary solution of (1.4), except near the set Z. The next step is the study of the stability of these “almost stationary” functions. We show that the unstable directions are approximately tangent to M = {uZ : Z in some restricted set 0 } and the spectrum of the linearized operator corresponding to these unstable modes is contained in a ball of radius supi e−`i . Our main results are Theorem 2.6, Theorem 3.1, and Theorem 4.5. In Sect. 2, we analyze the behavior of initial conditions close to uZ , when all kinks are far apart, in particular, we show the existence of an invariant neighborhood of M. We also provide an explicit formula for the speed of the kinks. In Sect. 3, we discuss the annihilation of a pair of neighboring kinks. This analysis yields a version of the Bray–Derrida–Godr`eche dynamics of intervals which continuously eliminates the smallest interval, replacing it by the union of its neighbors ([BDG]). In Sect. 4, we construct a set C0 of initial conditions which never come to rest, i.e., that “coarsen” forever in the sense introduced in [BDG]. The remaining sections are devoted to the proofs. Our choice of initial configurations in Sect. 4 is as follows: if 4 consecutive zeros z0 , z1 , z2 , z3 have distances z2 −z1 z3 −z2 and z2 −z1 z1 −z0 , then we can predict with high precision the collision time and the annihilation of the pair z1 , z2 , independently of the position of the infinitely many other kinks. We show that this scenario can happen infinitely often in various places of the infinite line. It would be interesting to consider for example collections of 6 zeros, z0 , . . . , z5 with z3 − z2 min(z4 − z3 , z2 − z1 ), and max(z4 − z3 , z2 − z1 ) min(z5 − z4 , z1 − z0 ). One then expects a collapse of the pair

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z3 , z2 followed by a collapse of the pair z4 , z1 . The second collapse is expected to occur much later than the first, and the idea of the proof would be to show that the first collapse has completely “healed” (leading to a flat solution near z3 , z2 ) before the outer pair z4 , z1 makes any essential motion. Our bounds fall short of being sufficient for a complete control of this scenario: “flatness” would be reached when the solution in the center between z4 , z1 is no further than exp(−c1 |z4 − z1 |) from ±1, where c1 is some constant depending on the potential V . The time (after the collapse of z3 , z2 ) to reach such flatness is of order |z4 − z1 |. During this time the diffusive part of the problem will have changed the value of the solution near the kinks z4 and z1 by about exp(−c01 |z4 − z1 |). This is the same order of magnitude as the maximal perturbation we can allow in our bounds. If c01 were much larger than c1 (but independent of |z4 − z1 | and the position of the distant kinks) we could iterate our argument and control the collision of z4 , z1 in the same way as was done for z3 , z2 . However, our bounds only give c01 = O(c1 ). It is possible that a more careful control of the position of the kinks z4 and z1 during the healing process could resolve this problem. 2. Dynamics of N + 1 Kinks in a Dilute State The aim of our paper is to study the dynamics of an infinite number of kinks on the infinite line. We shall always assume that the kinks are well-separated, except perhaps for one pair which is going to collide. If we consider a (huge) interval 3, there will be a finite number of kinks (namely N + 1) in this interval. We shall consider the geometry where the inter-kink distance across the boundary of 3 is so big that the motion of the kinks inside 3 is essentially unaffected by what happens outside 3, for a time which is large enough for interesting dynamics to happen inside 3. In this section, we therefore restrict our attention to initial conditions with exactly N + 1 kinks, and show how their dynamics is controlled. Thus, we assume that outside 3, the initial data is close to the kink solution. In Sect. 4, we shall show that the control over the motion of the kinks inside 3 can be maintained even if the function outside 3 has kinks in addition to those in 3. Remark. As pointed out by the referee, the sequel of the paper only uses the study of the case of N = 1. Since there is not much work involved in doing an arbitrary number of kinks and the results will be used in [R], we stay with the general case. 2.1. Definitions. Let N be an integer which will count the number of kinks (in fact there are N + 1 kinks). Let Z = {zj }j=0,...,N ∈ RN +1 be a sequence of real numbers satisfying zj < zj+1 for j = 0, . . . , N . These numbers will be the approximate positions of the kinks. We next define the set N,0 of all sequences of “far apart” kinks: we let n o N,0 = Z = {z0 , . . . , zN } ∈ RN +1 : zj − zj−1 > 0, j = 1, . . . , N, . We next describe in detail the construction of uZ (as sketched in the introduction) which is defined for all Z ∈ N,0 when 0 > D0 . We first need some notation: we begin by choosing a convenient C ∞ cutoff function 1:  1  0 if x ≤ − 2 , 4x (2.1) if |x| < 21 , 1(x) = 21 1 + tanh 1−4x 2   1 1 if x ≥ 2 .

Coarsening by Ginzburg–Landau Dynamics

447

Given a Z ∈ N,0 , we next define several geometrical quantities. It is convenient to define z−1 = −∞ and zN +1 = +∞: `j = zj − zj−1 , |Z| = inf `j ,

j = 0, . . . , N + 1,

j=1,...,N

mj =

1 2 (zj

+ zj−1 ),

Ij = [mj , mj+1 ],

j = 0, . . . , N + 1, j = 0, . . . , N,

and χIj is the characteristic function of Ij . Now we can define the pieces which will make up uZ (we give details for odd N , and leave the case of even N to the reader): u(0) (x) = 1 − 1(x − z0 ) ψ(x − z0 ) + 1(x − z0 )ϕ`1 (x − z0 ), u(j) (x) = 1 − 1(x − zj ) ϕ`j (x − zj−1 ) + 1(x − zj )ϕ`j+1 (x − zj+1 ), for j = 1, 3, 5, . . . , N − 2, u (x) = 1 − 1(x − zj ) ϕ`j (x − zj ) + 1(x − zj )ϕ`j+1 (x − zj ), (j)

u

(N )

for j = 2, 4, 6, . . . , N − 1, (x) = 1 − 1(x − zN ) ϕ`N (x − zN −1 ) + 1(x − zN )ψ(zN − x).

Finally, uZ (x) is given by the formula uZ (x) =

N X

u(j) (x)χIj (x).

(2.2)

j=0

The reader can check that uZ ∈ C ∞ and that if |x − zj | > 21 then L(uZ (x)) = 0. Moreover, uZ (zj ) = 0 for j = 0, . . . , N , and (−1)j uZ (x) > 0 for x ∈ (zj−1 , zj ) (see Fig. 4).

uZ (x)

^

z0

z1

z2

z3

> x

Fig. 4. The function uZ . The zeros are labeled zi . Bold lines indicate the regions of gluing

With these tools, we can now define the set of admissible initial conditions for which we can control the flow. This set depends on three parameters, the number of kinks N +1, the minimal distance 0 between kinks, and an error-bound σ. Let

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J.-P. Eckmann, J. Rougemont

TN,0,σ =

n v ∈ L∞ (R) : kvk∞ ≤ 1, max inf k∂x (v − uZ )k2 , Z∈N,0

inf

Z∈N,0

o kv − uZ k2 < σ .

(2.3)

Important terminology. Throughout this paper, we shall use the condition “for sufficiently small TN,0,σ ” to mean: “For every N there is a 0N and a σN such that for all ∞ > 0 > 0N and all 0 < σ < σN ”. Note that if σ is small, then the shape of the function is essentially given by the positions of the zeros alone. We complete this section on notations and definitions with the following set of functions which “generate” the translation of the j th kink: τzj (x) = −1(x − mj − 21 )1(mj+1 − x − 21 )∂x uZ (x),

j = 0, . . . , N.

(2.4)

Note that τzj and all its derivatives are bounded uniformly in j and Z (for sufficiently large |Z|) and it has support in [mj , mj+1 ]. Definition. The scalar product in L2 (R, dx) is denoted by (·, ·). 2.2. Equation of motion for the kinks. We first fix a convenient way of decomposing functions v ∈ TN,0,σ as a sum of a function uZ given by Eq. (2.2) plus a (small) remainder w. Proposition 2.1. For sufficiently small TN,0,σ , there exists a differentiable function Z : TN,0,σ → N,0 such that v − uZ(v) , τzj (v) = 0, for j = 0, . . . , N . The proof of this proposition is an application of the Implicit Function Theorem and is detailed in Sect. 7. Using the function Z(v) ≡ {z1 (v), . . . , zN (v)}, we can write the equation of motion for the (approximate) kink positions. First we remark that X ∂zj uZ Z=Z(v) z˙j , ∂t uZ(v) = j=0,...,N

where z˙j = ∂t zj (v(·, t)). Differentiating the identities v − uZ(v) , τzj Z=Z(v) = 0,

for j = 0, . . . , N

with respect to t, we get: N X L(v), τzj (v) =

∂zk uZ , τzj (v) − v − uZ(v) , ∂zk τzj (v) z˙k (v)

k=0

≡

N X k=0

Sjk z˙k (v).

(2.5)

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449

We define the matrix S =

Sjk

0≤j,k≤N

.

It will be proved in Sect 5 that the matrix S is invertible. Hence Eqs.(2.5) can be written as N X −1 Sjk L(v), τzk . (2.6) z˙j (v) = k=0

We get a closed system of equations by also considering the derivative of w = v − uZ(v) : ∂t w = L(v) −

N X

∂zj uZ z˙j (v)

j=0

= L(w + uZ(v) ) −

N X j,k=0

−1 ∂zj uZ Sjk

(2.7)

L(v), τzk .

Expanding (1.5) around uZ by setting v = w + uZ gives: L(w + uZ ) = L(uZ ) − LZ w + w2 r(w, uZ ),

(2.8)

where the linear operator reads: LZ w = −∂x2 w − V 00 uZ w,

(2.9)

and the non-linear remainder is given by: Z 1 ds (1 − s) V 000 (sf + g). r(f, g) = 0

2.3. Properties of the linear operator. In this section, we present some properties of the evolution operator, linearized around the kink-function uZ . We assume throughout that Z ∈ N,0 with 0 > D0 . The differential expression (2.9) defines a self-adjoint operator with domain D(LZ ) dense in L2 (R, dx). One has the following: Proposition 2.2. There are a 0 = 0N < ∞, a c1 < ∞, and a M > 0 (both independent of N and 0) such that the following holds: For all Z ∈ N,0 , the spectrum of LZ ≡ −∂x2 − V 00 (uZ ) in (−∞, M ), consists of N + 1 eigenvalues λ0 , . . . , λN satisfying the bound |λj | < e−c1 |Z| , j = 0, . . . , N . Remark. The operator LZ also has spectrum beyond M , but this is unimportant for our analysis. The proof of this proposition is a classical application of perturbation theory and is provided in Sect. 7. We next show that a function w which is orthogonal to the set of vectors τzj (translation modes) is also almost orthogonal to the unstable modes of −LZ . Furthermore, the operator LZ acts like a positive operator on such functions. Let PN (Z) : L2 (R, dx) → L j=0,...,N Hλj denote the spectral projector associated with the eigenvalues λ0 , . . . , λN of LZ (and Hλj ⊂ L2 (R, dx) the corresponding spectral subspaces).

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Corollary 2.3. Let w ∈ D(LZ ). There exist constants c1 > 0 and c2 > 0 (independent of N ) such that for sufficiently large |Z|, if w satisfies w, τzj = 0 for j = 0, . . . , N , then kPN (Z)wk2 ≤ c2 e−c1 |Z|/2 kwk2 . Corollary 2.4. 2.4 Let w ∈ D(LZ ). Then, for sufficiently large |Z|, there exists a constant M1 > 0 (depending on N ) such that if w satisfies w, τzj = 0 for j = 0, . . . , N , then one has: w, LZ w ≥ M1 kwk22 , (2.10) kLZ wk22 ≥ M1 w, LZ w , w, LZ w ≥ M1 k∂x wk22 . 2.4. Geometric structure. Motivated by Corollary 2.4, we can introduce the following norm for perturbations w ∈ D(LZ ) which are orthogonal to τzj , j = 0, . . . , N , when Z ∈ N,0 : kwk2Z ≡ (w, LZ w). (2.11) We will use now the notation vt (·) = v(·, t). Using Corollary 2.4 and Proposition 2.1, we see that wt = vt − uZ(vt ) basically decays exponentially in time with rate M1 (it is orthogonal to the unstable manifold) and we only have to work out the evolution of Zt ≡ Z(vt ) (the motion on the unstable manifold). Theorem 2.5. There exists a B > 0 such that for sufficiently small TN,0,σ , the following holds: If v0 ∈ TN,0,σ then as long as vt ∈ TN,0,σ , (2.12) (∂t + 21 M1 ) kvt − uZt k2Zt − Bg12 (Zt ) ≤ 0, PN where M1 is like in Corollary 2.4, g12 (Z) = j=0 |(L(uZ ), τzj )|2 and k · k2Z is given by Eq. (2.11). Furthermore, g1 (Z) → 0 as |Z| → ∞. one has

This result can be converted into a contraction statement as follows: by Theorem 2.5, Corollary 2.4, and Gronwall’s Lemma, we have: (2.13) kwt k2Zt ≤ Bg12 (Zt ) + kw0 k2Z0 − Bg12 (Z0 ) e−M1 t/2 . Since g1 (Z) → 0 as |Z| → ∞, we have Bg12 (Z) < M1 σ 2 for sufficiently large |Z|. Thus, for sufficiently large 0, we can choose a positive s such that s2 is in the interval (Bg12 (Z), M1 σ 2 ). With this s, we can define A0 = {v = w + uZ ∈ TN,0,σ : kwkZ < s}, √ Z0 = {v ∈ A0 : kwkZ < Bg1 (Z)}.

(2.14)

By Eq. (2.13), we see that A0 is exponentially attracted towards Z0 and Corollary 2.4 shows that A0 ⊂ TN,0,σ . Denoting vt ≡ v(·, t) a solution of Eq. (1.4), we see that for v0 ∈ Z0 , there are only two possibilities: either the orbit of v0 stays forever in Z0 or there is a finite time t for which limt0 →t |Zt0 | = 0. This case is the one we will consider, and which leads ultimately to a “collapse” of the two kinks. The second theorem of this section provides an explicit formula for the speed of the domain walls located at x = zj , j = 0, . . . , N .

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Theorem 2.6. There exist a c1 > 0 and an E ∈ (0, ∞) such that for sufficiently small TN,0,σ , vt ∈ Z0 and Z = Z(vt ), one has: z˙j = E −1 e−c1 `j+1 − e−c1 `j + O(e−3c1 |Z|/2 ),

(2.15)

for j = 0, . . . , N . The proofs of both theorems are provided in Sect. 7. Remark. The parameters 0 and σ are now fixed by the requirement that Eq. (2.12) holds in TN,0,σ . They depend only on N .

3. Collapse of a Domain The discussion so far followed Carr and Pego quite closely. Now, we are going to use the freedom of working with an infinite line to get a more precise description of the collapsing mechanism. This is possible because any distribution of kinks which is sufficiently “dilute” and does not get stuck inside TN,0,σ for all times, leaves TN,0,σ through the “needle hole” at the ends |Z| = 0 of the tube. This means that we have some control over the shape of the solution in the interval [zj−1 , zj ] which has length 0, under the hypothesis that its neighbors are much larger. We conjecture that for initially widely separated kinks, the shapes just before collapse should be arbitrarily close to a universal shape. This is illustrated by numerical integration in Fig. 5. Once the critical distance of kinks (defined by 0) is reached the kinks will collapse in a time less than Tp < ∞, and the function will have constant sign in the interval [zj−1 , zj ].

t

= 153

t

= 2:9

Fig. 5. Two different initial conditions leading to the same shape just before the collapse

Suppose that vt ∈ Z0 for all t < T and that Z = Z(vT ) satisfies |Z| = 0. Let wt = vt − uZ(vt ) . Then, by Theorem 2.5 and Corollary 2.4, it is easily seen that for sufficiently small TN,0,σ ,

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kwT k∞ ≤ kwT k2 kwT0 k2

1/2

−1/2

≤ M1

−1/2

kwT kZ ≤ B 1/2 M1

g1 (Z),

(3.1)

which tends to 0 as 0 = |Z| goes to ∞. In the following theorem, proved in Sect. 8, we study the behavior of v0 = uZ + w0 where Z satisfies: there is a j ∈ {0, . . . , N } such that `j = 0, `j±1 > 00 . Theorem 3.1. For sufficiently large 00 > 0 and for vT satisfying Eq. (3.1), the following holds: suppose that for some j ∈ {0, . . . , N } one has zj (vT ) − zj−1 (vT ) = 0 and zk (vT ) − zk−1 (vT ) > 00 if k = j ± 1. Then there is a finite Tp > T such that inf x∈J |vTP (x)| > 0, where J = [zj−1 (vT ) − 1, zj (vT ) + 1]. Furthermore, the collision, i.e., the disappearance of 2 zeros takes place inside J. 4. Existence of the Coarsening Dynamics In this section, we describe a probabilistic point of view on the dynamics of the kinks. Since by the above discussion, the Ginzburg–Landau dynamics of many-kink states is essentially specified by the location of these kinks, we will treat a model which implements the dynamics of the (discrete) set of interval lengths. In the last section, we found an “effective” equation (Eq. (2.15)) for the coordinates {zj }j=0,...,N of the zeros of a solution of the Ginzburg–Landau equation (Eq. (1.4)). Assuming E = c1 = 1, neglecting higher order terms and extrapolating to infinitely many zeros, this equation is: z˙j = e−(zj+1 −zj ) − e−(zj −zj−1 ) ,

j ∈ Z.

˜ = e0 . Introducing the variables Throughout the section, we will use the notation 0 βj = ezj −zj−1 , yields the equations:

−1 −1 + βj−1 , β˙j = −2 + βj βj+1

(4.1) j ∈ Z.

(4.2)

Furthermore, we define a “boundary condition”: if there exists an index j ∈ Z and a ˜ (i.e., `j (t − 0) = 0) then βk (t), k ∈ Z is defined by time t > 0 such that βj (t − 0) = 0   if k < j − 1, βk (t − 0) ˜ j−1 (t − 0)βj+1 (t − 0) if k = j − 1, (4.3) βk (t) = 0β  β (t − 0) if k > j − 1. k+2 (This corresponds to the merging of the two intervals `k−1 and `k+1 when `k vanishes.) ˜ ∞)Z which we Eqs.(4.2) together with (4.3) define a dynamics on the space E0 = [0, baptize “coarsening dynamics” in reference to the Bray–Derrida–Godr`eche model. Definition 4.1. A collapse for β(t) satisfying the coarsening dynamics is a time τ such ˜ that β(t) is discontinuous at t = τ (i.e., there exists an integer j for which βj (τ −0) = 0). We will exhibit a set C0 of initial conditions in E0 such that the corresponding coarsening dynamics will have an infinite set of collapsing times spread over infinite time. In terms of the variables zj , this set can be viewed as a subset of R. Its restriction to any compact subset of R has positive measure with respect to the following probability measure π0 : we generalize the definition of N,0 , by letting n o 0 = Z = {zj , j ∈ Z} : zj − zj−1 > 0 for all j .

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Definition 4.2. We define π0 as the probability measure on 0 , which associates the weight ∞ Z Y d`j ρ0 (`j ) (4.4) j=−∞

Xj

to the set {Z : zj − zj−1 ∈ Xj }, Rwith Xj ⊂ [0, ∞), and ρ0 is a positive real function satisfying: supp(ρ0 ) = [0, ∞) and dxρ0 (x) = 1. This means that the `i are i.i.d. random variables. Let us define the set C0 : n C0 = β ∈ E0 : ∃{jn }n∈N ⊂ ZN s.t. o 2 ˜ (n + 1 )0 ˜ , βjn ±1 ∈ exp(en 0˜ ), ∞ , ∀n ∈ N, βjn ∈ n0, 2

(4.5)

and state the result: Proposition 4.3. Let t → β(t) ∈ E0 be the coarsening dynamics associated with an initial condition β(0) ∈ C0 . Then there exists an infinite sequence of numbers 0 < τ1 < τ2 < · · · < τn < . . . , such that limn→∞ τn = ∞ and such that for all n ∈ N, τn is a collapse for β(t). ˜ (n + 1 )0 ˜ and β±1 (0) > exp(en2 0˜ ). Then by (4.2), for Proof. Suppose β0 (0) ∈ n0, 2 sufficiently large n, −2 ≤ β˙0 (0) ≤ −1,

β˙±1 (0) ≥ 0.

˜ Note that the fact that These relations remain true for all times t < sup{t : β0 (t) > 0}. collapses may occur elsewhere in the meantime is irrelevant, since (apart from shifting the indices) it cannot modify β0 and it can only make β±1 even larger. It follows that ˜ (n − 1 )0)] ˜ such that β0 (sn − 0) = 0. ˜ Taking there is a time sn in the interval [ 21 (n − 1)0, 2 m the subsequence τm = sn where n = 3 proves the claim, since ˜ < τm ≤ (3m − 21 )0

1 m+1 2 (3

˜ ≤ τm+1 . − 1)0

Given a sequence {βj }j∈Z ∈ C0 , and a point z0 ∈ R, we can define a sequence {zj }j∈Z by using the correspondence Eq. (4.1). This defines a set C0,z0 . Given any finite interval B we then define the sequence {zj }j∈k,...,k0 of those zj which are in B. We associate with this sequence the set of all those sequences in 0 which coincide with {zj }j∈k,...,k0 , and which have no other point zj in B. Taking the union of these sets over all {βj }j∈Z ∈ C0 , we obtain a set C0,B,z0 of sequences which is a subset of 0 and which “looks like C0 in B” (when the origin is fixed by z0 ). Proposition 4.4. Let B be a finite interval in R and let |B| > 0 denote its length. Then there is a δ = δ(|B|) > 0 such that π0 C0,B,z0 ≥ δ, where π0 (·) is the probability measure defined in Definition 4.2.

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Remark. This proposition only shows that, when restricted to a (large) interval B, the measure of possible initial conditions is positive. This clearly does not mean that the measure on the infinite space is positive as well. Proof. Clearly, since each interval in C0 has at least length 0, we see that C0,B,z0 has at most N = |B|/0 + 2 intervals in B (the two intervals at the ends may have an intersection with B of length less than 0). Hence the product Eq. (4.4) contains only finitely many factors different from 1. It remains to show that none of them is 0. Among the configurations with N 0 ≤ N intervals, at least one must be “open” in the sense that the size of all intervals can be increased a little bit without changing the combinatorial type. (The intervals at the ends get a little shorter in that case (when considered in B).) This configuration has positive measure which yields a lower bound for π0 C0,B,z0 . Of course, in general, when |B| 0 the interesting configuration will contain a (large) number of intervals defined by the βj of Eq. (4.5). We next consider again the set C0,z0 which is C0 with a fixed “origin” z0 . Theorem 4.5. For sufficiently large 0 there exists a family {vγ }γ∈C0,z0 of functions in L∞ (R, dx) indexed by the elements of C0,z0 whose orbits (vγ )t under the evolution Eq. (1.4) satisfy: there exists a sequence of times {tn > 0}n∈N such that two zeros of vt annihilate at t = tn and limn→∞ tn = ∞. Furthermore, for any finite interval B, the restriction to B of this family is open in L∞ (R, dx). Proof. Let γ ∈ C0,z0 and let {jn }n∈N be the indices for which γjn − γjn −1 ≡ `jn ∈ 2˜ (log n + 0, log(n + 21 ) + 0) and γjn ±1 − γjn ±1−1 ≡ `jn ±1 > en 0 . We consider the 2˜ 2˜ intervals B (n) = [γjn −1 − 21 en 0 , γjn + 21 en 0 ], n ∈ Z. The set γ ∈ C0,z0 has exactly two points in B (n) , which we call for the moment z00 and z10 . Thus, we can define uZ 0 by Eq. (2.2), on the full line, where Z 0 = {z00 , z10 }. By Theorem 2.6, for sufficiently large 0, the dynamics of the kinks can be approximated arbitrarily well by Eq. (4.2) (through the change of variables Eq. (4.1) and the scaling c1 = 1) until times of order exp(|Z 0 |). Consequently, the evolution of any initial condition v sufficiently close to uZ 0 in the norm k · kZ 0 will lead to a collapse in a time TZ 0 . By Theorem 2.6, Proposition 4.3, and the geometric assumptions on C0,z0 , we get a bound C(n − 1)e0 ≤ TZ 0 ≤ Cne0 . This bound is valid for functions with exactly two kinks. We next show that the bound is changed very little if we assume that the initial condition is close to uZ 0 in B (n) in the L∞ (R, dx) norm, and bounded by 1 in modulus outside of B(n) . This is shown in the next Lemma 4.6. Let ε > 0. Let vt and vt∗ denote two orbits of Eq. (1.4) corresponding to initial conditions v0 and v0∗ with kv0 k∞ ≤ 1 and kv0∗ k∞ ≤ 1. For sufficiently large n, there is a δn > 0 such that the following holds: if sup |v0 (x) − v0∗ (x)| ≤ δn ,

x∈B(n)

then for all times t < TZ 0 one has the bound: sup |vt (x) − vt∗ (x)| ≤ ε,

x∈C (n)

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where C (n) = [γjn −1 − 41 en 0 , γjn + 41 en 0 ]. 2

˜

2

˜

We finish the proof of Theorem 4.5 before giving the proof of Lemma 4.6. By construction, the B (n) are disjoint, and the preceding calculation is valid for each (sufficiently large) n, provided the solution is bounded in modulus by 1 outside of B(n) . However, this is always the case for Eq. (1.4), provided it was true for the initial condition. Therefore, we have completed our construction. Since we have an open neighborhood for each interval B (n) and the initial condition is arbitrary (just bounded by 1) on the complement of the B (n) , the last part of the assertion follows. Proof of Lemma 4.6. The technique used in this proof is similar to what was done in [C,CE2]. By Duhamel’s principle, we have, for x ∈ C (n) : Z t ∗ 0 0 ∗ = Gt ? (v0 − v0 ) (x) + ds Gt−s ? (V (vs ) − V (vs )) (x) , |vt (x) − 0 (4.6) where Gt (x) = (4πt)−1/2 exp(−x2 /(4t)) and ? denotes convolution. The first term is bounded by ˜ Gt ? (v0 − v0∗ ) (x) ≤ C1 e−C2 exp(n2 0)/t + δn . vt∗ (x)|

The second term is more delicate, we perform preliminary calculations: 2 vt (y) − vt∗ (y) ∂t dy 1 + |x − y|4 R Z vt (y) − vt∗ (y) = 2 dy ∂y2 (vt (y) − vt∗ (y)) + V 0 (vt (y)) − V 0 (vt∗ (y)) 4 1 + |x − y| R 2 Z Z ∂y (vt (y) − vt∗ (y)) vt (y) − vt∗ (y) + C3 dy ≤ −2 dy ∂y (vt (y) − vt∗ (y)) 1 + |x − y|4 1 + |x − y|4 R R 2 Z vt (y) − vt∗ (y) + C4 dy 1 + |x − y|4 R 2 2 Z Z ∂y (vt (y) − vt∗ (y)) vt (y) − vt∗ (y) ≤ − dy + C5 dy 1 + |x − y|4 1 + |x − y|4 R R Z 2 vt (y) − vt∗ (y) ≤ C5 dy . 1 + |x − y|4 R Z

Using that the distance between C (n) and the complement of B(n) is larger than 41 en 0 , we get 2

Z R

2 ! 21 vt (y) − vt∗ ≤ 1 + |x − y|4

eC5 t

Z R

2 ! 21 v0 (y) − v0∗ 2˜ ≤ C6 eC5 t δn + e−C4 n 0 . 4 1 + |x − y|

˜

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We can now bound both terms of Eq. (4.6). Let t ≤ TZ 0 ≤ Cne0 , x ∈ C (n) , then 2˜ |vt (x) − vt∗ (x)| ≤ C6 e−C2 exp(n 0)/t + δn Z t Z (1 + |x − y|4 )1/2 0 ds dy Gt−s (x − y) |V (vs (y)) − V 0 (vs∗ (y))| + (1 + |x − y|4 )1/2 R 0 2˜ ≤ C6 e−C2 exp(n 0)/t + δn 2 ! 21 1 Z Z t Z ∗ 2 (y) − v (y) v s s ds dy G2t−s (x − y)(1 + |x − y|4 ) dy + C6 1 + |x − y|4 R R 0 Z t 2˜ 2˜ eC5 s δn + e−C4 n 0 . ≤ C6 e−C2 exp(n 0)/t + δn + C6 ds √ t − s 0 The claim follows immediately: sup |vt (x) − vt∗ (x)| ≤ C6 δn + e−C4 n

2

˜ C 5 n0 ˜ 0

x∈C (n)

e

,

which can be made as small as desired by taking δn sufficiently small and n sufficiently large.

5. Miscellaneous Bounds We first give estimates on the behavior of the function uZ given in Eq. (2.2). In particular we show that near the set Z, this function is so close to the heteroclinic solution ψ that it is almost stationary in an L∞ sense. Lemma 5.1. There exist positive K and c1 , such that for sufficiently large |Z| the following holds with the convention `0 = `N +1 = ∞: 1) |ψ((−1)j+1 (x − zj )) − uZ (x)| ≤ Ke−c1 min(`j ,`j+1 ) , for |x − zj | ≤ 21 , j = 0, . . . , N . 2) |L uZ (x) | ≤ Ke−c1 min(`j ,`j+1 ) , for |x − zj | ≤ 21 , j = 0, . . . , N . Proof. We first compare ϕD with ψ for fixed D. Let g(x) = ψ(x) − ϕD (x), α = V (ϕD (D/2)) = V (−A(D)), cf. Fig. 2 and Proposition 1.1, let x ∈ [−D/2, D/2]. If f is a stationary solution of Eq. (1.4) then f 00 + V 0 (f ) = 0 and thus f 00 f 0 + V 0 (f )f 0 = 0, i.e., 21 (f 0 )2 + V (f ) is constant. Taking x∗ such that f 0 (x∗ ) = 0, we get 0 2 1 2 (f ) (x)

+ V (f (x)) = V (f (x∗ )).

Therefore the derivative of g satisfies (note that for x ∈ [−D/2, D/2], ϕD (x) is monotone, and that |ψ(∞)| = 1): p √ p |g 0 (x)| = 2 V (1) − V (ψ(x)) − α − V (ϕD (x)) p √ p = 2 V (1) − V (ψ(x)) − −(V (1) − α) + V (1) − V (ϕD (x)) p p √ p ≤ 2 V (1) − V (ψ(x)) − V (1) − V (ϕD (x)) + V (1) − α ≤ C |ψ(x) − ϕD (x)| + e−c1 D = C |g(x)| + e−c1 D .

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√ √ √ In the third line, we have used the inequality − −a + b ≤ a − b, and in the last line, the first term comes from the differentiability of the function V while the second term is a consequence of Eq. (1.6) below. We apply Gronwall’s lemma (and |x| < 21 , g(0) = 0) to get |g(x)| ≤ Ke−c1 D and |g 0 (x)| ≤ Ke−c1 D .

Now, recalling the definition (2.2) of uZ , we have uZ (x) = 1 − 1(x − zj ) ϕ`j (x − zj−1 ) + 1(x − zj )ϕ`j+1 (x − zj+1 ) for |x − zj | < inf(`j , `j+1 )/2. Hence min(ϕ`j (x − zj−1 ), ϕ`j+1 (x − zj+1 )) ≤ uZ (x) ≤ max(ϕ`j (x − zj−1 ), ϕ`j+1 (x − zj+1 )). Consequently |ψ((−1)j+1 (x − zj )) − uZ (x)| = max ψ((−1)j+1 (x − zj )) − uZ (x), uZ (x) − ψ((−1)j+1 (x − zj )) h ≤ max ψ((−1)j+1 (x − zj )) − min ϕ`j (x − zj−1 ), ϕ`j+1 (x − zj+1 ) , i − ψ((−1)j+1 (x − zj )) + max ϕ`j (x − zj−1 ), ϕ`j+1 (x − zj+1 ) ≤ max Ke−c1 `j , Ke−c1 `j+1 , which proves claim 1). We write, for x ∈ Ij , L(uZ ) ≡ 100 (ϕ`j − ϕ`j+1 ) + 210 (ϕ0`j − ϕ0`j+1 ) − G. Using the fact that ϕD is a solution of L(u) = 0, we have G = (1 − 1)V 0 (ϕ`j ) + 1V 0 (ϕ`j+1 ) − V 0 (1 − 1)ϕ`j + 1ϕ`j+1 . We expand G near 0 and look at the coefficient of V 000 (0)/2, which is the first nonvanishing term: (1 − 1)ϕ2`j + 1ϕ2`j+1 − (1 − 1)ϕ`j + 1ϕ`j+1

2

= (1 − 1)ϕ2`j + 1ϕ2`j+1 − (1 − 1)2 ϕ2`j − 12 ϕ2`j+1 − 2(1 − 1)1ϕ`j ϕ`j+1 = (1 − 1)1 ϕ2`j + ϕ2`j+1 − 2ϕ`j ϕ`j+1 ≤ (ϕ`j+1 − ϕ`j )2 . Consequently, |G| ≤ κ3 |ϕ`j+1 − ϕ`j |2 and thus using 1), |L(uZ )| ≤ κ1 |g(x)| + κ2 |g 0 (x)| + κ3 |g(x)|2 ≤ Ke−c1 min(`j ,`j+1 ) , which completes the proof of claim 2).

We next give estimates related to the vectors τzj introduced in Eq. (2.4).

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Lemma 5.2. Let Z in N,0 , mj = 21 (zj + zj−1 ), −m0 = mN +1 = ∞, and τzj as PN defined in Eq. (2.4). Let g12 (Z) = j=0 | L(uZ ), τzj |2 and g2 (Z) = sup{ kLkτZkτ2k2 : τ ∈ span{τzj : j = 0, . . . , N }\{0}}. Then, there exist K > 0 and c1 > 0 such that for sufficiently large 0, one has: L(uZ ), τzj

= −V (uZ (mj+1 )) + V (uZ (mj )), −c1 |Z|

|g1 (Z)| ≤ Ke

for j = 0, . . . , N,

,

−c1 |Z|/2

|g2 (Z)| ≤ Ke

.

Proof. We compute directly, using τzj = −∂x uZ on the set supp L(uZ ) ∩ Ij : Z

dx L uZ (x) τzj (x) =

− R

=

2 1 2 (∂x uZ )

Z

1 zj + 2

1 zj − 2

dx ∂x2 uZ + V 0 (uZ ) ∂x uZ

+ V (uZ ) (zj + 21 ) −

2 1 2 (∂x uZ )

+ V (uZ ) (zj − 21 )

= V (uZ (mj+1 )) − V (uZ (mj )). This completes the proof of the first claim. The second one follows easily from Eq. (1.6). PN For the third claim, we make the following calculation: let τ ≡ j=0 tj τzj , then one has kLZ τ k22 ≤ kτ k22 ≥

sup kLZ τzj k22

N X

j=0,...,N

inf

j=0,...,N

|tj |2 ,

j=0

kτzj k22

N X

|tj |2 .

j=0

Recall that τzj (x) is strictly positive in [mj + 1, mj+1 − 1] with a lower bound uniform in j = 0, . . . , N . Hence the second norm above is uniformly bounded below. We complete the proof with a bound on kLZ τzj k22 : we use that τzj has compact support and is equal to the derivative of a stationary solution of Eq. (1.4) in the interval (mj + 1, mj+1 − 1): Z kLZ τzj k22 ≤ Z = ≤

mj+1 −1

mj +1 1 zj + 2

1 zj − 2

2 dx ∂x2 τzj (x) + V 00 (uZ (x))τzj (x)

2 dx ∂x ∂x2 uZ (x) + V 0 (uZ (x))

sup 1 |x−zj |< 2

L uZ (x) ·

and we use Lemma 5.1 to conclude.

sup 1 |x−zj |< 2

∂x L uZ (x) ,

Next we prove that certain matrices used in Sect. 7 have a bounded inverse.

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Lemma 5.3. For sufficiently small TN,0,σ , for all v ∈ TN,0,σ , and for N < ∞, the matrices S = Sjk 0≤j,k≤N = ∂zk uZ(v) , τzj (v) − (v − uZ(v) ), ∂zk (v) τzj (v) , 0≤j,k≤N = ∂zk uZ (v), τzj (v) , S1 = (S1 )jk 0≤j,k≤N

0≤j,k≤N

have inverses with an `∞ → `∞ norm uniformly bounded in Z and σ. Proof. By construction, τzj depends only on zk with |k−j| ≤ 1. Moreover, ∂zj uZ τzk ≡ 0 if |j − k| > 1. Therefore the matrix S is tridiagonal. To prove that S is invertible, we show that it is diagonally dominant, i.e., |Sii | >

X

|Sij |.

j= 6 i

1) The diagonal terms are Sii = ∂zi uZ , τzi − (v − uZ ), ∂zi τzi . The first term is uniformly bounded from below, by Proposition 2.3 of [CP1]. In fact, with 1i = 1(x − mi − 21 )1(mi+1 − x − 21 ) as in Eq. (2.4), it is a consequence of √ | ∂zi ϕ`i , 1i ∂x ϕ`i | ≈ | ∂zi ψ(· − zi ), 1i ∂x ψ(· − zi ) | = k 1i ∂x ψk22 > K. The second term in Sii is O(σ), thus for sufficiently small TN,0,σ , the whole expression is bounded below. 2) We next control the off-diagonal terms Sij = ∂zj uZ , τzi − (v − uZ ), ∂zj τzi . The first term is bounded by f (0) → 0 as 0 goes to infinity, see again Proposition 2.3 of [CP1] (recall that τzj has compact support, and the overlap between τzj and ∂zj±1 ϕ`j is very small). The second term is treated as before. 3) The proof for S1 is a special case of 1) and 2). We now estimate several expressions which will occur in the proof of Theorem 2.5 . Lemma 5.4. There exist constants c1 > 0, c2 > 0, and c3 > 0 such that for sufficiently small TN,0,σ , the following holds: S1 − E · 1 jk ≤ c2 e−c1 |Z| , j,k=0,...,N sup S −1 − S1−1 jk ≤ c3 kwk2 , sup

j,k=0,...,N

where E =

R1 −1

dy

√

2(V (1) − V (y)) =

R∞ −∞

2 dx ψ 0 (x) .

Proof. The first claim is proved with the following argument: the off-diagonal elements of S1 are ∂zj ±1 uZ (v), τzj (v) which is of order e−c1 |Z| (see Lemma 5.3) and the diagonal ones satisfy the bound:

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Z |E − (S1 )ii | ≤ |E +

mi+1 −1

mi +1

dx ∂zi uZ ∂x uZ |

≤ Ce−c1 |Z| + |E −

Z

mi+1 −1

mi +1

dx (∂x uZ )2 |

≤ c2 e−c1 |Z| , using Lemma 7.8 of [CP1] to compare ∂zj uZ with ∂x uZ . The second claim follows from |S −1 − S1−1 | = |S −1 (S1 − S)S1−1 | ≤ c3 kwk2 , because of Lemma 5.3 and using that (S1 − S)i,j = w, ∂zj τzi ≤ Ckwk2 .

We next introduce notations that will be used later for writing Eq. (2.7) and Eq. (2.6) in a simpler form: Q = L(uZ ) −

N X

∂zj uZ S1−1

j,k=0

Pj(1) =

N X

S1−1

k=0

Pj(2)

=

N X

jk

(S −1 −

jk

L(uZ ), τzk ,

L(uZ ), τzk ,

S1−1 )jk

(5.1)

L(uZ ), τzk ,

k=0

Pj(3) =

N X k=0

−1 Sjk −LZ w + w2 r(w, uZ ), τzk .

We have the following bounds: Lemma 5.5. There exist constants c1 > 0, c2 > 0, and c3 > 0 such that for sufficiently small TN,0,σ , the following holds: kQk22 ≤ c1 g12 (Z), sup |Pj(1) | ≤ c2 g1 (Z),

j=0,...,N

sup |Pj(2) + Pj(3) | ≤ c3 kwk2 g1 (Z) + g2 (Z) + kwk2 ,

j=0,...,N

where g1 and g2 are as in Lemma 5.2. Proof. One can bound |Q| by using Lemma 5.3: |Q| ≤ |L(uZ )| + Cg1 (Z), and |L(uZ )| ≤ C

N X j=0

L(uZ )τzj ,

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461

P since j τzj (x) > C −1 on the support of L(uZ ). The bound on kL(uZ )k22 follows immediately. The bound on P (1) is obvious from its definition, from the definition of g1 (Z) and from Lemma 5.4: 2 X 2 (1) −1 S1 )ji L(uZ ), τzi sup Pj = sup j=0,...,N j=0,...,N i=0,...,N X 2 L(uZ ), τz ≤ c2 i

i=0,...,N

= c2 g12 (Z). The bound on P (2) + P (3) follows from: | LZ w, τzj | = | w, LZ τzj | ≤ Ckwk2 g2 (Z), | w2 r(w, uZ ), τzj | ≤ Ckwk22 . The proof is completed by using Lemma 5.4.

6. Proofs of the Properties of the Linear Operator The first proof we provide in this section concerns the spectrum of LZ = −∂x2 −V 00 (uZ ). Proof of Proposition 2.2 . First we show that the operator LZ with N = 1, Z = {−z, z} has two eigenvalues satisfying the bounds of Proposition 2.2 . The function uZ is positive at |x| 1 and negative at x = 0. The operator LZ is a Sturm–Liouville operator with a potential U (x) = −V 00 (uZ (x)) which is a symmetric double well (see Fig. 6) bounded by Umin < U (x) < Umax . Its spectrum is made up of isolated eigenvalues in (Umin , Umax ) and absolutely continuous spectrum in (Umax , ∞). When |Z| → ∞, then the lowest eigenvalue is degenerate and it is (by translation) given as the lowest eigenvalue of L∗ = −∂x2 − V 00 (ψ). In this limit, ψ 0 (x) is an eigenfunction of L∗ with eigenvalue 0 (it corresponds to the invariance under translation). This is the ground state, because ψ 0 is a positive function. This double eigenvalue splits into λ− < λ+ when |Z| < ∞. It is a well-known consequence of the fact that e− (e+ ), the corresponding eigenfunctions, are the even (odd) extensions of the ground state of the same operator with Neumann (Dirichlet) boundary condition at x = 0, and the splitting is a consequence of the Dirichlet-Neumann bracketing. Furthermore, the splitting will be exponentially small as |Z| → ∞ (note that |ϕD (x) − ψ(x)| ≤ c2 e−c1 D , for |x| < D/2, see Lemma 5.1): one has λ+ − λ− ≤ Ke−c1 |Z| (for more details, see [RS4], p. 34, Example 6). By similar reasoning, using Dirichlet-Neumann bracketing, for any N < ∞, the spectrum of LZ has N + 1 eigenvalues satisfying the bound of Proposition 2.2. Proof of Corollary 2.3. We use the following notations: PZ (·) is the spectral measure associated with LZ , Z ∈ N,0 , and M > 0 is as in Proposition 2.2 . We show that the LN restriction of PN = PN (Z) ≡ j=0 PZ (λj ) to span{τzj : j = 0, . . . , N } has kernel {0}. Let τ ∈ span{τzj : j = 0, . . . , N }\{0}. Then

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J.-P. Eckmann, J. Rougemont

U (x)

^

uZ (x) > x

Fig. 6. The potential U (x) together with the function uZ (x). Note that the minima of U coincide with the zeros of uZ

Z ∞ M 2 k(1 − PN )τ k22 = M 2 PZ (dλ)τ, τ Z M λ2 PZ (dλ)τ, τ ≤ R

= kLZ τ k22 , hence k(1 − PN )τ k22 ≤ B 2 (Z)kτ k22 where B 2 (Z) ≡ g22 (Z)/M 2 . By Lemma 5.2, for sufficiently large |Z|, B(Z) < 1. Using kτ k2 ≤ kPN τ k2 + k(1 − PN )τ k2 , we get kPN τ k2 ≥ (1 − B(Z))kτ k2 . This proves that the map PN : span{τzj : j = 0, . . . , N } →

N M

Hλj

j=0

has trivial kernel. In addition, it is a map between N –dimensional spaces, hence it is a bijection. Now, let τ ∗ ∈ span{τzj : j = 0, . . . , N }\{0} be such that PN w = PN τ ∗ . Recall that by hypothesis, w, τ ∗ = 0, thus: kPN wk22 = | w, PN w | = | w, PN τ ∗ | = | w, (PN − 1)τ ∗ | ≤ kwk2 k(PN − 1)τ ∗ k2 ≤ B(Z)kwk2 kτ ∗ k2 B(Z) kwk22 , ≤ 1 − B(Z) using kwk2 ≥ kPN wk2 = kPN τ ∗ k2 ≥ (1 − B(Z))kτ ∗ k2 . The proof is completed by using the bound B(Z) ≤ c2 e−c1 |Z|/2 of Lemma 5.2.

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463

Proof of Corollary 2.4 . We use the following notations: PZ (·) is the spectral measure associated with LZ with Z ∈ N,0 , PN (Z) = PN is the restriction of PZ to the eigenspace of the first N + 1 eigenvalues, and M > 0 is as in Proposition 2.2 . We start by proving the first inequality of Eq. (2.10). By the Spectral Theorem, we have Z w, LZ w = λ PZ (dλ)w, w R

Z

PZ (dλ)w, w +

≥ M R

≥ M kwk22 +

N X

Z

M

−∞

λ−M

(λj − M ) w, ej

PZ (dλ)w, w

2

j=0

≥ M kwk22 + N (−e−c1 |Z| − M )kPN wk22 . By Corollary 2.3, for |Z| large, kPN wk2 ≤ c2 e−c1 |Z|/2 kwk2 . Thus w, LZ w 2 ≥ kwk22 M (1 − O(e−c1 |Z| )) ≡ M1 kwk22 .

(6.1)

If |Z| is sufficiently large, M1 is positive. To prove the second inequality of Eq. (2.10), we perform similar calculations: Z λ2 PZ (dλ)w, w LZ w, LZ w 2 = R

Z M λ PZ (dλ)w, w + λ − M λ PZ (dλ)w, w R −∞ ≥ M w, LZ w + N (−α − M )M kPN wk22 ≥ M (1 − O(e−c1 |Z| )) w, LZ w , Z

≥ M

where we have used Eq. (6.1). If |Z| is sufficiently large, then M2 ≡ M (1 − O(e−c1 |Z| )) is positive. The last inequality in Eq. (2.10) is next established: Z kw0 k22 ≤ w, LZ w + dx w2 |V 00 (uZ )| R ≤ w, LZ w + Kkwk22 ≤ (1 + K/M1 ) w, LZ w . Clearly 0 < (1 + K/M1 ) ≡ M3 < ∞. We denote again M1 the smallest of M1 , M2 , and M3 .

7. Proofs of the Geometric Structure We first prove the existence of an orthogonal coordinate system adapted to the problem. Proof of Proposition 2.1. We apply the Implicit Function Theorem to the function F : TN,0,σ × N,0 → RN defined by:

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J.-P. Eckmann, J. Rougemont

F(v, {z0 , . . . , zN }) = Fj (v, {z0 , . . . , zN }) =

j=0,...,N

(v − u{z0 ,...,zN } ), τzj

j=0,...,N

.

We can check that the hypotheses of the theorem are satisfied: 1) F(u{z0 ,...,zN } , {z0 , . . . , zN }) = 0, 2) (∂zk Fj (uZ , Z)) = −Sjk where S is as in Lemma 5.3, hence it has bounded inverse. The Implicit Function Theorem implies that for sufficiently small TN,0,σ , there exists a differentiable function Z : TN,0,σ → N,0 such that F(v, Z(v)) = 0. Before proceeding to the proof of Theorem 2.5 we put Eq. (2.6) in a more compact form. We have, using Eq. (2.8), Lemma 5.3, and Eq. (5.1): N X

z˙j (v) =

k=0

−1 Sjk L(uZ ) − LZ w + w2 r(w, uZ ) , τzk

N X

=

S1−1

k=0

+

N X k=0

jk

N X (S −1 − S1−1 )jk L(uZ ), τzk L(uZ ), τzk + k=0

−1 Sjk (−LZ w + w2 r(w, uZ )), τzk

(7.1)

≡ Pj(1) + Pj(2) + Pj(3) . Equation (2.7) takes the form:

∂t w = L(uZ ) − LZ w + w r(w, uZ ) − 2

N X j,k=0

= L(uZ ) −

N X

∂zj uZ S1−1

j,k=0

jk

−1 ∂zj uZ Sjk L(v), τzk

L(uZ ), τzk

− LZ w + w2 r(w, uZ ) −

N X j,k=0

∂zj uZ S −1 − S1−1

(7.2) jk

≡ Q − LZ w + w2 r(w, uZ ) −

L(uZ ), τzk N X

∂zj uZ Pj(2) + Pj(3) .

j=0

We are now prepared to give the proof of Theorem 2.5: Proof of Theorem 2.5 . We start by expanding the first term on the l.h.s. of Eq. (2.12). Denoting Z = Z(v) and w = v − uZ(v) , then by Eq. (2.7) and Eq. (2.8), we get

Coarsening by Ginzburg–Landau Dynamics

2 1 2 ∂t kwkZ

= w, ˙ LZ w −

465

Z 1 2

000

dx w V (uZ ) 2

R

N X

∂zj uZ z˙j

j=0

= Q, LZ w − kLZ wk22

N X + w2 r(w, uZ ), LZ w − ∂zj uZ (Pj(2) + Pj(3) ), LZ w j=0

Z −

1 2

dx w2 V 000 (uZ )

R

N X

∂zj uZ (Pj(1) + Pj(2) + Pj(3) ).

j=0

The next bound follows from Corollary 2.4 and Lemma 5.5 2 1 2 ∂t kwkZ

≤ − 21 kLZ wk22 + kQk22 ,

(7.3)

by taking TN,0,σ sufficiently small. We next expand and estimate the time derivative of g12 (Z): 2 1 2 ∂t g1 (Z)

X

=

L(uZ ), τzj

(1) −LZ ∂zk uZ , τzj + L(uZ ), ∂zk τzj Pk + Pk(2) + Pk(3) .

0≤j,k≤N

We have:

| LZ ∂zk uZ , τzj | = | ∂zk uZ , LZ τzj | ≤ Cg2 (Z), | L(uZ ), ∂zk τzj | ≤ Cg1 (Z).

Hence, using Lemma 5.5, we get 2 1 2 ∂t g1 (Z)

≤ Cg1 (Z) g1 (Z) + g2 (Z) g1 (Z) + kwk2 (g1 (Z) + g2 (Z) + kwk2 ) . (7.4)

Summing Eq. (7.3) and Eq. (7.4) and using kQk22 ≤ Cg12 (Z) (see Lemma 5.5), we have: 2 2 1 2 ∂t kwkZ − Bg1 (Z) ≤ − 21 kLZ wk22 + Cg12 (Z) + BCg12 (Z) g1 (Z) + g2 (Z) 2 + 41 kLZ wk22 + C 2 B 2 M1−4 g1 (Z) + g2 (Z) + kwk2 2 × g1 (Z) + g2 (Z) g12 (Z) (7.5) ≤ − 41 M1 kwk2Z − Bg12 (Z) M1 B + Cg12 (Z) 1 + B g1 (Z) + g2 (Z) − 4C 2 2 2 + B g1 (Z) + g2 (Z) g1 (Z) + g2 (Z) + kwk2 ≤ − 41 M1 kwk2Z − Bg12 (Z) + Cg12 (Z)F(B), where

F(B) = A1 B 2 + A2 B + 1 , 2 2 A1 = g1 (Z) + g2 (Z) g1 (Z) + g2 (Z) + kwk2 , M1 . A2 = g1 (Z) + g2 (Z) − 4C

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J.-P. Eckmann, J. Rougemont

We take TN,0,σ so small that g1 (Z) + g2 (Z) + kwk2 < 41 and g1 (Z) + g2 (Z) < 2 2 2 M1 /8C.√Thus, F(B) ≤ 1 − M1 B/(8C) √ + (M1 /(8C)) (1/4) B , hence it is negative for 1 − 3/2 < M1 B/(64C) < 1 + 3/2. We finally give the proof of Theorem 2.6. Note that in Theorem 2.6, we assume v = w + uZ(v) ∈ Z0 hence k∂x wk2 ≤ Cg1 (Z) and kwk2 ≤ Cg1 (Z) which is smaller than σ. Proof of Theorem 2.6. We start by recalling Eq. (7.1), with the shorthand Z = Z(v): z˙j (v) =

N X i,j=0

+ S

−1

S1−1

j,i

L(uZ ), τzi + S −1 − S1−1 j,i L(uZ ), τzi

j,i

−LZ w, τzi + S

−1

j,i

(7.6)

w r(w, uZ ), τzi . 2

Using Lemma 5.4, we find (S1−1 )ij = E −1 δij + O(e−c1 |Z| ). It follows that the first term of Eq. (7.6) is equal to E −1 L(uZ ), τzj + O(e−c1 |Z| )g1 (Z). Use here Lemma 5.2 to get the main contribution to the r.h.s. of Eq. (2.15). The remainder is next estimated: the second term of Eq. (7.6) is also estimated using Lemma 5.4 and again Lemma 5.2: |S −1 − S1−1 |jk ≤ c3 kwk2 ≤ c3 g1 (Z), thus, the second term in Eq. (7.6) is bounded by O(e−c1 |Z| )g1 (Z). The third term is O(g2 (Z)kwk2 ) = O(e−3c1 |Z|/2 ) and the last term is bounded by kwk22 ≤ c4 g12 (Z). This completes the proof of Theorem 2.6.

8. Proof of Theorem 3.1 First, we study the simpler case of the collapse of a function with two kinks separated by a distance 0 and then, we compare with the evolution of the many-kink solution. Lemma 8.1. Let Z = {0, 0}, and let v0 = uZ . Denote by vt the corresponding solution of Eq. (1.4). For sufficiently large 0 > D0 , and sufficiently large 00 > 0, there are a constant ε0 > 0 and a Tp < ∞ such that vTp (x) > ε0 for all x ∈ R. Proof. We use the parabolic maximum principle together with the existence of moving front solutions for Eq. (1.4). A front is a function fs (x − st), where s is a fixed number and fs (x) solves: (8.1) ∂x2 fs + s∂x fs + V 0 (fs ) = 0. In the mechanical interpretation shown in Fig. 1, fs (x) is an oscillating trajectory subject to a constant friction s. Let 0R > 0, let ϕ0R = ϕD with D = 0R , and let fs be the solution of Eq. (8.1) with initial values fs (0R ) = ϕ0R (0R ) = 0 and fs0 (0R ) = ϕ00R (0) < 0. Then we have 1) fs (x) < ϕ0R (x) for 0 < x < 0R and for all s > 0.

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467

2) For sufficiently small s > 0, there exist 0L 6 = 0C with 0L < −0R and 0C < 0 be such that fs (0L ) = fs (0C ) = 0. The first claim is a consequence of the following argument: integrating Eq. (8.1) from the initial values at x = 0R , we see that in a small neighborhood of 0R , we have fs (x) < ϕ0R (x). Let A be given by the equation 0R = D(A), cf. Proposition 1.1, and let x0 be such that ϕ0R (x0 ) = −A. Then, for x0 < x < 0R , √ p ϕ00R (x) = 2 V (A) − V (ϕ0R (x)), (8.2) √ p fs0 (x) > 2 V (A) − V (fs (x). If we suppose that there exists an x∗ , x0 < x∗ < 0R such that fs (x∗ ) = ϕ0R (x∗ ), since fs (x) < ϕ0R (x) near x = 0R , we conclude that fs0 (x∗ ) < ϕ00R (x∗ ) which is a contradiction with Eq. (8.2). In the interval (0, x0 ) the same argument applies with opposite signs for the square roots in Eq. (8.2). Hence fs does not intersect ϕ0R i.e., fs lies below ϕ0R in the interval (0, 0R ). The second claim follows from the observation that fs (x) → ϕ0R (x) when s → 0 hence, by continuity, there exist such zeros of fs for small s. Furthermore, we have that uZ (x) ≥ ψR (x) ≡ ψ(−(x − 0R )) and uZ (x) ≥ ψC (x) ≡ ψ(x − 0C ). Hence, by the maximum principle, vt (x) > max fs (x − st), ψL (x), ψC (x) , for all t > 0, and for fs satisfying 1) and 2). In particular, for Tp = (0R − 0C )/s, the function vTp is strictly positive (see Fig. 7, Fig. 8, and [CE1], p. 149, Example 4).

uZ (x)

^

j

,L

,C

j

,

s

jj

,R

> > x fs (x)

R(

x)

C(

x)

Fig. 7. The moving front fs , the stationary solutions ψR (x), ψC (x), and uZ with Z = {0, 0}

Proof of Theorem 3.1. We first remark that vT restricted to the interval I1 ≡ [zj − 00 /3, zj+1 + 00 /3] is close to the two-kinks function uZ˜ (see Eq. (3.1)), with Z˜ = {zj , zj + 0}. The evolution of uZ˜ has been described in Lemma 8.1, and is known to lead to a collapse. We now show that the evolutions of vT and uZ˜ remain close to each other for a time longer than the time Tp needed for collapse.

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J.-P. Eckmann, J. Rougemont

^ fs (x , sTp )

> x

R(

x)

C(

x)

Fig. 8. The position of the front fs at time Tp = (0R − 0C )/s, relative to ψR,C (x)

To perform the comparison, we consider the functions f0 = vT = w + uZ and f˜0 = w˜ + uZ˜ . By Duhamel’s Principle, with Gt (x) as in the proof of Lemma 4.6: ft (x) − f˜t (x) ≡ f (x, t) − f˜(x, t) Z t ds Gt−s ? V 0 (fs ) − V 0 (f˜s ) (x). = Gt ? (f0 − f˜0 ) (x) + 0

(8.3) Let I2 ≡ [zj − 1, zj + 0 + 1] ⊂ I1 . If 1t denotes the l.h.s of Eq. (8.3), we have that 10 (x) = 0 if x ∈ I1 and |V 0 (fs ) − V 0 (f˜s )| ≤ κ|1s | for some κ > 0 because V 0 is in C 2 and f, f˜ are bounded. By the same procedure as in the proof of Lemma 4.6, we find for x ∈ I2 , for sufficiently large 0 and 00 , 2 |1Tp (x)| ≤ C e−(00 /3−0) /(4Tp ) + e−c1 0 Z Tp ds (Tp − s)−1/2 eCs (00 /3 − 0)−2 + e−c1 0 +C 0

≤ ε0 /2, and since |fT∗p (x)| > ε0 by Lemma 8.1, we find |fTp (x)| > ε0 /2 for x ∈ I2 .

9. Proof of Proposition 1.1 Instead of the convention settled in the remark after Proposition 1.1, we shall choose the more symmetric definition: ϕD (x) has a minimum at x = 0, i.e., ϕD (0) = −A. We seek particular solutions of the equation L(u) = 0. In the mechanical interpretation of a free particle moving in the potential V without friction, u(x) is the position of the particle at time x (see Fig. 1). Intuitively, it is clear that if the particle starts at rest from a position u(0), with −1 < u(0) = −A < 0, its trajectory will oscillate around 0 with a certain period 2D. Looking for a relation between D and A, we show that if u solves the initial value problem

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469

u00 = −V 0 (u),

(9.1)

u(0) = −A, u0 (0) = 0,

then there exists a (minimal) D(A) such that u(D(A)/2) = 0. We can transform Eq. (9.1) into: 21 ((u0 )2 )0 = −(V (u))0 (supposing u0 6≡ 0), which, after integration, becomes (u0 )2 = −2(V (u) − V (−A)), where the integration constant was set to V (−A) in order to match the condition u0 (0) = 0. When −A ≤ u < 0, V (−A) ≥ V (u), hence we can take the square root: √ p u0 = 2 V (−A) − V (u). The r.h.s. is invertible if −A < u ≤ 0, yielding an equation for the inverse function x(u): 1 . x0 (u) = √ √ 2 V (−A) − V (u) There will be a solution satisfying the boundary condition u(0) = −A and the periodicity condition u(D(A)/2) = 0 if and only if the integral 1 √ 2

Z

0

−A

√

ds = x(0) − x(−A) = D(A)/2 − 0 V (−A) − V (s)

(9.2)

exists. This is an elliptic integral of the first kind, which is an analytic bijection from (0, 1) onto (D0 , ∞) (see Fig. 9 and, e.g., [A], pp. 322–324).

^ 2D(A)

2D0

, j

j> A 1

0 Fig. 9. The period 2D as a function of the amplitude A

We have described the solution u of Eq. (9.1) on the interval [0, D(A)/2], and we next check that it extends to a periodic function. The equation 21 (u0 )2 +V (u) = V (−A) = V (A) together with u00 = −V 0 (u) leads to the existence of a number D∗ (A) for which u D∗ (A)/2 = A, u0 D∗ (A)/2 = 0, u00 D∗ (A)/2 < 0, i.e., a maximum of u of height A. This number D∗ (A) is determined by (recall that V is even):

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J.-P. Eckmann, J. Rougemont

1 x(A) − x(0) = D∗ (A)/2 − D(A)/2 = √ 2 Z 0 ds 1 √ = D(A)/2. = √ 2 −A V (A) − V (s)

Z 0

A

√

ds V (A) − V (s)

The initial value problem (9.1) is invariant under x → −x, hence the solution is even. Altogether, we found the interval [−D(A), D(A)] and since the behavior of u on u −D(A) = u D(A) = A and u0 −D(A) = u0 D(A) = 0, periodic copies will match together. Equation (1.6) follows from Eq. (9.2), see [A]. Acknowledgement. We are grateful to P. Collet for suggesting this problem to us. We also thank C.-A. Pillet, L. Rey-Bellet, and P. Wittwer for useful comments. Notable improvements have been made possible by the critical reading of the referees. This work was supported by the Fonds National Suisse.

Note added in proof. In a subsequent paper by one of us [R], it will be shown that indeed the “healing process” can be rigorously controlled when the inside kinks are sufficiently far from the surrounding kinks. This is in marked difference with the problem on a finite interval, where it has been shown by an energy estimate that this healing is impossible [CP2]. References [A] [B] [BD]

Arfken, G.: Mathematical Methods for Physicists. 3d edition. Orlando: Academic Press, 1985 Bray, A.J.: Theory of phase-ordering kinetics. Adv. Phys. 43 (3), 357–459 (1994) Bray, A.J. and Derrida, B.: Exact Exponent λ of the Autocorrelation Function for a Soluble Model of Coarsening. Phys. Rev. E 51 (3), 1633–1636 (1995) [BDG] Bray, A.J., Derrida, B. and Godr`eche, C.: Non-trivial Algebraic Decay in a Soluble Model of Coarsening. Europhys. Lett. 27 (3), 175–180 (1994) [C] Collet, P.: Thermodynamic Limit of the Ginzburg–Landau Equations. Nonlinearity 7 (4), 1175–1190 (1994) [CE1] Collet, P. and Eckmann, J.-P.: Instabilities and Fronts in Extended Systems. Princeton, NJ: Princeton University Press, 1990 [CE2] Collet, P. and Eckmann, J.-P.: The time dependent amplitude equation for the Swift-Hohenberg problem. Commun. Math. Phys. 132, 139–153 (1990) [CP1] Carr, J. and Pego, R.L.: Metastable Patterns in Solutions of ut = ε2 uxx − f (u). Comm. Pure Appl. Math. XLII, 523–576 (1989) [CP2] Carr, J. and Pego, R.L.: Invariant Manifolds for Metastable Patterns in ut = ε2 uxx − f (u). Proc. Roy. Soc. Edinburgh 116A, 133–160 (1990) [G] Gallay, Th.: Existence et stabilit´e des fronts dans l’´equation de Ginzburg–Landau a` une dimension. PhD Thesis, University of Geneva, 1994 [R] Rougemont, J.: Dynamics of kinks in the Ginzburg–Landau equation: Approach to a metastable shape and collapse of embedded pairs of kinks. Preprint (1998) [RS4] Reed, M. and Simon, B.: Methods of Modern Mathematical Physics, IV: Analysis of Operators. San Diego: Academic Press, 1978 Communicated by A. Kupiainen

Commun. Math. Phys. 199, 471 – 491 (1998)

Communications in

Mathematical Physics © Springer-Verlag 1998

Inverse Scattering Problem in Anisotropic Media? Gregory Eskin Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA. E-mail: [email protected] Received: 9 January 1998 / Accepted: 19 May 1998

Abstract: A new family of scattering invariants is constructed that differs from the scattering invariants obtained by the geometric optics approximations. A local uniqueness (modulo a diffeomorphism) of the metric with a given hodograph function is proven under some restrictions on the curvature of the metric. 0. Introduction Consider a second order elliptic equation in Rn of the form ∂ ∂u aij (x) + k 2 ρ(x)u = 0, (0.1) 6nj,k=1 ∂xj ∂xk n where ρ(x), ajk (x) ∈ C ∞ (Rn ), n ≥ 3, ρ(x) ≥ C0 > 0, ajk (x) j,k=1 is a positively definite matrix, ajk (x) − δjk ∈ C0∞ (BR ), ρ(x) − 1 ∈ C0∞ (BR ), where BR is an open − 1 ball |x| < R. Introducing g jk (x) = det[ajk (x)] n−2 ajk we can rewrite (0.1) as a Laplace–Beltrami equation (cf. [U]): (0.2) L − k 2 c(x) u = 0, where Lu =

∂ g ∂xj

1 −6nj,k=1 √

√

∂u g g (x) ∂xk jk

,

(0.3)

n g = det gjk (x) , gjk (x) j,k=1 is the metric tensor and [g jk ]nj,k=1 is the inverse to

[gjk ]nj,k=1 , c = ?

ρ(x) √ . g

Vice versa Eq. (0.2) can be rewritten in the form (0.1) with ajk =

This research was supported by NSF grant DMS-9622310

472

G. Eskin

√ g g jk (x) and ρ = gc(x). We shall consider below Eq. (0.2) instead of Eq. (0.1). We look for the distorted plane wave solution u(x) of (L − k 2 c)u = 0 in the form √

u(x) = eikω·x + v(x, kω),

(0.4)

where |ω| = 1, k > 0 and v(x, kω) is obtained by the limiting absorption principle: v(x, kω) = lim vε (x, kω), ε→0

where

L − (k + iε)2 c vε = −q0 (x, kω) eikω·x ,

(0.5)

L − k 2 c eikω.·x .

(0.6)

q0 (x, kω) = e−kω·x ∞:

It is known that such v(x, kω) exists and has the following asymptotics when |x| → v(x, kω) =

eik|x| |x|

n−1 2

a(θ, ω, k) + O

1 |x|

,

(0.7)

x is fixed. Function a(θ, ω, k) is called the scattering amplitude. where θ = |x| 0 (x) = gij (x) − δij are small and a(θ, ω, k) is known for all Assuming that n ≥ 3, gij n k > 0, |θ| = 1, |ω| = 1 we shall recover matrix g ij (x) i,j=1 and c(x) modulo a change of variables y = ϕ(x), y = x for |x| ≥ R. Such a result was proved by Belyshev and Kurylev [BK] using the boundary control methods. Recently Stefanov and Uhlmann [SU1] gave a different proof in the case c = 1 using the time-dependent approach. In [SU2] Stefanov and Uhlmann gave a solution to the hodograph problem that leads to another method of the solution of the inverse problem in the case when c = 1. The hodograph problem is the recovery of the metric tensor (modulo diffeomorphism) from the hodograph function, i.e. the geodesic distance between any two points on ∂BR . Note earlier contributions to the solution of the hodograph problem by Michel [M], Gromov [G], Croke [C], Sharifutdinov [Sh] and others (see more details in [SU2] and [Sh]). Note that the two-dimensional case was solved earlier by Sylvester [Sy] and the analytic case (the inverse Dirichlet-to-Neumann problem) was done by Lee and Uhlmann [LU]. The content of the paper is the following: In Sect. 3 we prove a local uniqueness (modulo diffeomorphism) of the solution of the hodograph problem under the condition that the curvature of the metric is small (see Theorem 3.1 for details). The proof uses the estimates for the linearized problem obtained in [Sh]. In Sect. 1 we study a second order elliptic equation of the form (1.1) and we recover integrals (1.24) from the scattering amplitude. The recovering process is similar to the method of [ER1]. Note that there are not any smallness assumptions on the coefficients of (1.1). In Sect. 2 we combine results of Sect. 1 with known geometric optics constructions and the results of [ER2] to find a rich set of new scattering invariants (2.32) and (2.33) (we say that a quantity is a scattering invariant if it can be recovered from the scattering amplitude). Note that the hodograph function is also a scattering invariant but invariants (2.32) are different.

Inverse Scattering Problem in Anisotropic Media

473

Since the recovery of the metric from the hodograph is known only under severe restrictions on the curvature and only locally one may try to combine the hodograph function and the invariants (2.32) to recover the metric under milder restrictions. At the end of Sect. 2, using results of Sect. 3 to recover the metric tensor hij (x) = c(x)gij (x) from the hodograph, we put to work the invariants (2.33) to recover c(x). This leads to the following theorem: Theorem 0.1. Assume that n ≥ 3 and operators L1 − k 2 c1 (x) and L2 − k 2 c2 (x) have the same scattering amplitude a(θ, i ω,hk) for all |θ| i = |ω| = 1, k > 0. Assume h that i h (1) (1) = the curvature of metric hij (x) = c1 (x)gij (x) is small and the metric h(2) ij i h i h (2) (1) c2 (x)gij (x) is close to hij (x) . More exactly, introduce semigeodesics coordinates (see (3.1), (3.2) ) and let hˆ (1) and hˆ (2) be the metrics h(1) and h(2) in these coordinates. Assume that the Sobolev norm khˆ (2) − hˆ (1) k 3 n+10+ε , ε > 0, is small enough. Then there 2 ϕ of Rn , ϕ = I outside of BR such that g (2) = ϕ∗ g (1) and exists a C ∞ diffeomorfism c2 (x) = c1 ϕ−1 (x) , where ϕ∗ g (1) is the pullback of metric g (1) . 1. The Inverse Scattering for the Second Order Elliptic Operator in a Special Form Consider the second order elliptic equation of the form ∂ −1 + a0 x, −i 0 u − k 2 u = 0, ∂x where

(1.1)

n−1 a0 (x, ξ 0 ) = 6n−1 p,r=1 bpr (x)ξp ξr + 6p=1 bp (x)ξp + c(x),

n ) functions with x = (x0 , xn ), ξ 0 = (ξ1 , ..., ξn−1 ), bpr (x), bp (x), c(x) are C ∞ (R− compact support contained in BR when xn ≤ 0 and equal to zero for xn > 0. Note that bpr , bp , c may have jumps at xn = 0. As in (0.4), (0.5) u = eikω·x + v(x, kω),

where

∂ −1 + a0 x, −i 0 − k 2 v = −a0 (x, kω 0 )eikω·x , ∂x

(1.2)

ω = (ω 0 , ωn ), v is obtained by the limiting absorbtion principle and has the following form Z ˜ kω, k)eix·η dη h(η, 1 , (1.3) v(x, kω) = n (2π) Rn ηn2 + |η 0 |2 − k 2 − i0 ˜ kω, k) is the Fourier transform of h(x, kω, k). where h(x, kω, k) is the new unknown, h(η, Substituting (1.3) into (1.2) we obtain Z ˜ kω, k) a0 (x, η 0 )eix·η h(η, 1 dη (1.4) h(x, kω, k) + n 2 0 2 (2π) Rn ηn + |η | − k 2 − i0 = −a0 (x, kω 0 )eikω·x .

474

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Performing the Fourier transform in (1.4) we get Z ˜ kω, k) a˜ 0 (ξ − η, η 0 )h(η, 1 dη h˜ (ξ, kω, k) + n 2 0 2 (2π) Rn ηn + |η | − k 2 − i0 = −˜a0 (ξ − kω, kω 0 ).

(1.5)

It follows from (1.4) that supp h ⊂ BR since supp a0 (x, η 0 ) ⊂ BR for every η 0 . Note that the absence of the imbedded eigenvalues of (1.1) for k > 0 (assuming that (1.1) is formally self-adjoint) implies that (1.4) has a unique solution in L2 (BR ). Note ˜ also that the scattering amplitude a(θ, ω, k) is equal (modulo a constant) to h(kθ, kω, k) (cf. [ER1]). Let v(x, kω, k, z) be a tempered distribution solution of the equation ∂ ∂ 0 2 (1.6) + a (x, −i 0 ) − k v(x, kω, k, z) −1 − 2iz ∂xn ∂x = −a0 x, kω 0 eikω·x , where z = σ + iτ, τ ≥ 0. We look for the solution of (1.6) in the form Z ˜ ζ, k, z)eix·ξ h(ξ, 1 dξ, v(x, ζ, k, z) = n 2 (2π) Rn ξn + 2zξn + |ξ 0 |2 − k 2

(1.7)

where ζ = (ζ 0 , ζn ) = kω. Substituting (1.7) into (1.6) we get Z ˜ ζ, k, z)eix·η a0 (x, η 0 )h(η, 1 dη = −a0 (x, ζ 0 )eiζ·x . h(x, ζ, k, z) + (2π)n Rn ηn2 + 2zηn + |η 0 |2 − k 2

(1.8)

Performing the Fourier transform in x we obtain Z ˜ ζ, k, z) a˜ 0 (ξ − η, η 0 )h(η, ˜h(x, ζ, k, z) + 1 dη = −a0 (ξ − ζ, ζ 0 ). (2π)n Rn ηn2 + 2zηn + |η 0 |2 − k 2

(1.80 )

We shall study the solvability of (1.8). Note that supp h(x, ζ, k, z) ⊂ BR . Denote Z eix·η g(η) ˜ 1 dη, Ez g = n 2 (2π) Rn ηn + 2zηn + |η 0 |2 − k 2 where supp g ⊂ BR , g ∈ L2 (BR ). Let ψ(x) be a C0∞ (BR ) function such that ψ(x) = 1 in a neighborhood of supp a0 (x, η 0 ). Lemma 1.1. Operator ψEz is a bounded operator from L2 (BR ) to H2 (BR ) and kψEz gkH2 (BR ) ≤ CkgkL2 (BR ) , where

(1.9)

j+p

∂

(ψE g) , kψEz gk2H2 (BR ) = 62j,p=0 z

∂xj ∂xp L2 (BR )

C is independent of τ for τ ≥ 0, |σ| < C0 , z = σ + iτ . Moreover for every g ∈ L2 (BR ), kψEz gkH2 (BR ) → 0

when τ → ∞.

(1.10)

Inverse Scattering Problem in Anisotropic Media

475

Proof. Introducing polar coordinates η 0 = rω 0 , where r = |η 0 |, ω 0 =

η0 |η 0 |

we get

Fx (ψEz g) Z ∞ Z ∞Z 0 ˜ 0 − rω 0 , ξn − ηn )g(rω ψ(ξ ˜ , ηn )rn−2 1 drdω 0 dηn , (1.11) = (2π)n −∞ 0 ηn2 + 2σηn + r2 − k 2 + 2iτ ηn |ω 0 |=1 where Fx is the Fourier transform in x. Since g(η ˜ 0 , ηn ) is an entire function of exponential type we can deform for τ > 0 the contour of integration in (1.11) in the following way (cf. [ER1] ): (1.12) Fx (ψEz g) Z ∞Z Z 0 ˜ 0 − rω 0 , ξn − ηn )g(rω ψ(ξ ˜ , ηn )rn−2 1 drdω 0 dηn = (2π)n 0 ηn2 + 2σηn + r2 − k 2 + 2iτ ηn 0+ |ω 0 |=1 Z 0 Z Z ˜ 0 − rω 0 , ξn − ηn )g˜ rω 0 , ηn rn−2 ψ(ξ 1 + drdω 0 dηn , (2π)n −∞ 0− |ω0 |=1 ηn2 + 2σηn + r2 − k 2 + 2iτ ηn where r = t + iα(t), 0 ≤ t < +∞ is a parametric equation of 0+ and r = t − iα(t), 0 ≤ t < +∞ is a parametric equation of 0− , α(t) = t for 0 ≤ t ≤ 1, α(t) = 1 for t > 1. Parseval’s equality and the Cauchy theorem imply that for any b0 > 0 and M Z |g(η ˜ 0 + ib0 , ηn )|2 dη 0 dηn (1.13) Rn

+

max

|ηn |+|η 0 +ib0 |≤M

|g(η ˜ n , η 0 + ib0 )|2 ≤ CM,b0 kgk2L2 (BR ) ,

where |b0 | ≤ b0 . Divide the domain of integration in (1.12) in the integration over U1 where |r| < δ and |ηn | ≤ 2C0 + k and over the remaining domain that we denote by U2 . 2 +1 Denote I(τ ) = |η2 +2σηn|r| +r 2 −k2 +2iτ ηn | . n Using the Cauchy–Schwartz inequality for the integral over U2 and maximum norm estimate for the integral over U1 we get Z 2 0 I 2 (τ )|g(rω ˜ , ηn )|2 |rn−2 ||dr|dω 0 dηn (1.14) kψEz gkH2 (BR ) ≤ C U2

Z +C

U1

I(τ )|rn−2 ||dr|dωdηn

2

0 sup |g(rω ˜ , ηn )|2 . U1

Since I(τ ) ≤ C when (rω, ηn ) ∈ U2 and τ ≥ 0, |σ| ≤ C0 and since I(τ ) is integrable over U1 and the integral is bounded uniformly in τ ≥ 0, |σ| ≤ C0 we obtain (1.9). Estimate (1.10) follows from (1.13), (1.14) and the Lebesque dominant convergence theorem since I(τ ) → 0 for each rω, σ and ηn 6= 0. Denote by Az g, z = σ + iτ the operator Z 1 a0 (x, η 0 )eix·η g(η) ˜ dη. (1.15) Az g = n 2 0 (2π) ) Rn ηn + 2zηn + |η |2 − k 2 It follows from (1.9) that Az is a bounded operator in L2 (BR ). Taking the derivative of (1.15) in z one can see that Az depends analytically on z in the operator norm topology for τ > 0.

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Lemma 1.2. Equation (1.8) has a unique solution h(x, ζ, k, z) in L2 (BR ) for τ ≥ 0, σ ∈ (σ0 − ε, σ0 + ε) for some σ0 : kh(x, ζ, k, zkL2 (BR ) ≤ C,

(1.16)

where C is independent of z = σ + iτ, σ ∈ (σ0 − ε, σ0 + ε), τ ≥ 0. Moreover h(x, ζ, k, z) is analytic in z for τ > 0 and continuous for τ ≥ 0, σ ∈ (σ0 − ε, σ0 + ε). Proof. We shall show that Re ((I + Az )f, f ) ≥ Ckf k2L2 (BR )

(1.17)

for allf ∈ L2 (BR ), |σ| ≤ C0 , τ ≥ τ0 , where τ0 is large. Indeed the ellipticity of (1.1) implies that 1+

6n−1 p,r=1 bpr (x)ηp ηr ηn2 + |η 0 |2

≥ C.

(1.18)

We have that I(τ ) ≤ Cτ2 when |r| ≤ C1 , |σ| ≤ C0 , |ηn | ≥ δ. Perform the Fourier transform in (1.15) and change the contour of integration as in (1.12). Note that Z I(τ )|rn−2 ||dr|dω 0 |dηn < ε, (1.19) |ηn |<δ,|r|≤C1

where ε > 0 is small when δ is small. Therefore taking C1 large and using (1.18) when |r| > C1 , then choosing δ small and using (1.19) when |r| < C1 , |ηn | < δ (cf. (1.13) ) and finally choosing τ large and using that I(τ ) ≤ Cτ2 when |r| < C1 and |ηn | > δ we get (1.17). In fact (1.17) is a special case of the Garding inequality. Therefore (1.17) implies that I + Az is invertible for τ ≥ τ0 and the norm of (I + Az )−1 is bounded independently of τ for τ ≥ τ0 . Since Az is analytic in z for τ > 0 and continuous for τ ≥ 0 we get that (I + Az )−1 is meromorphic in z = σ + iτ and there exists a small σ0 such that (I + Az )−1 exists

and is bounded for all σ + iτ, 0 ≤ τ < +∞, |σ − σ0 | < ε. Since

a0 (x, ζ 0 )eiζ·x ≤ C we get (1.16). L2 (BR ) Taking the limit in (1.8’) as τ → +0 we will have Z ˜ ζ, k, σ + i0)dη a˜ 0 (ξ − η, η 0 )h(η, ˜h(ξ, ζ, k, σ + i0) + 1 (2π)n Rn (ηn + σ)2 + |η 0 |2 − k 2 − σ 2 + i0ηn 0 0 = −˜a (ξ − ζ, ζ ). (1.20) Changing variables ηn to ηn − σ, ξn to ξn − σ, ζn to ζn − σ we obtain ˜ 0 , ξn − σ, ζ 0 , ζn − σ, k, σ + i0) h(ξ Z a˜ 0 ξ − η, η 0 h˜ η 0 , ηn − σ, ζ 0 , ζn − σ, k, σ + i 1 dη + (2π)n Rn ηn2 + |η 0 |2 − k 2 − σ 2 + i0(ηn − σ) = −˜a0 (ξ − ζ, ζ 0 ). Therefore (cf. [ER1])

√ ˜ 0 , ξn − σ, ζ 0 , ζn − σ, k, σ + i0) = h˜ n,σ ξ 0 , ξn , ζ 0 , ζn , k 2 + σ 2 , h(ξ

(1.21)

Inverse Scattering Problem in Anisotropic Media

477

√ where h˜ n,σ ξ 0 , ξn , ζ 0 , ζn , k 2 + σ 2 is the solution of the integral equation analogous √ to (1.5) with k replaced by k 2 + σ 2 and the fundamental solution

E x,

√

k2

+

σ2

1 = (2π)n

Z Rn

eix·η dη |η|2 − k 2 − σ 2 − i0

replaced by the Faddeev fundamental solution √ En,σ x, k 2 + σ 2 =

1 (2π)n

Z Rn

|η|2

−

k2

eix·η dη. − σ 2 − i0(ηn − σ)

Therefore (cf. [ER1], (24)) there exists the following relation between √ √ 2 2 2 2 ˜ h(ξ, ζ, k + σ ) and hn,σ ξ, ζ, k + σ : √ Z √ iπ k 2 + σ 2 2 2 ˜ = h(ξ, ζ, k + σ ) − + (1.22) hn,σ ξ, ζ, √ (2π)n k2 +σ 2 ω>σ √ √ √ √ k 2 + σ 2 ω, ζ, k 2 + σ 2 dω ×h˜ ξ, k 2 + σ 2 ω, k 2 + σ 2 h˜ n,σ

√

k2

σ2

amFix ξ = (ξ 0 , 0), ζ = (ζ 0 , 0) where |ξ 0 | = |ζ 0 | = k. Knowing the scattering √ ˜ 0 , σ, ζ 0 , σ, k 2 + σ 2 ) for plitude a(θ, ω, k) for all |θ| = |ω| = 1, k > 0 we know h(ξ 0 2 2 0 2 2 2 2 all |σ| ≤ C0 since √ |ξ | + σ = |ζ | + σ = k + σ . Then from (1.22) we can find 0 0 hn,σ (ξ , σ, ζ , σ, k 2 + σ 2 ) for |σ − σ0 | < ε. ˜ 0 , 0, ζ 0 , 0, k, σ + i0) from (1.22) for all ξ 0 , ζ 0 such that |ξ 0 | = Therefore we know h(ξ |ζ 0 | = k and all σ ∈ (σ0 − ε, σ0 + ε). By the analytic continuation we can recover ˜ 0 , 0, ζ 0 , 0, k, z) for all z = σ + iτ, τ > 0. Now we take the limit in (1.8’) when h(ξ σ = σ0 , ξ 0 , ζ 0 are fixed and τ → ∞. It follows from (1.16) that there is a sequence hj = h(x, ζ, k, iτj ) weakly convergent in L2 (BR ) to some h∞ = h∞ (x, ζ, k) ∈ L2 (BR ) when τj → ∞. Therefore ˜ ζ, k, iτj ) = (hj , eix·ξ ) → (h∞ , eix·ξ ) when τj → ∞ and ξ = (ξ 0 , 0), ζ = (ζ 0 , 0), k h(ξ, R are fixed. Here (f, g) = BR f (x)g(x)dx is the inner product in L2 (BR ). We can rewrite (1.8’) in the following form: (hj , eix·ξ ) + (Aiτj hj , eix·ξ ) = −˜a0 (ξ − ζ, ζ 0 ).

(1.23)

∗ Note that (Aiτj hj , ψ(x)eix·ξ ) = (hj , A∗iτj ψeix·ξ ) since ψ = 1 on supp a0 (x, η 0 ). Since Eiτ has the same form as Eiτ with τ replaced by −τ we have using (1.10) that A∗iτj ψeix·ξ → 0 in L2 (BR ) norm. Therefore (hj , A∗iτj ψeik·ξ ) → (h∞ , 0) = 0 when τj → ∞. Thus taking the limit in (1.8’) when τj → ∞ we get

˜ 0 , 0, ζ 0 , 0, k, iτj ) = lim (hj , eix·ξ ) = −˜a0 (ξ 0 − ζ 0 , 0, ζ 0 ) lim h(ξ

τj →∞

τj →∞

(1.24)

˜ 0 , 0, ζ 0 , 0, k, iτj ) is determined by the for each (ξ 0 , ζ 0 ) such that |ξ 0 | = |ζ 0 | = k. Since h(ξ 0 0 0 scattering amplitude we can recover a˜ (ξ − ζ , 0, ζ 0 ) when |ξ 0 | = |ζ 0 |, ξn = ζn = 0.

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G. Eskin

2. Reduction to a Special Form Consider a change of variables yk = ϕk (x1 , ..., xn ), 1 ≤ k ≤ n,

(2.1)

and denote by x = ϕ−1 (y) the inverse change of variables. Then equation (L−k 2 c)u = 0 will have the following form in new coordinates: (Lˆ − k 2 )v(y) = 0,

(2.2)

where ˆ =− Lv

1 √

cˆ(y) g(y) ˆ

6np,r=1

∂ p ∂v pr ( g(y)b ˆ (y) ), ∂yp ∂yr ∂ϕp ∂ϕr , ∂xj ∂xk

(2.4)

∂ϕ −2 | , cˆ(y) = c(x). ∂x

(2.5)

bpr (y) = 6nj,k=1 g jk (x)

g(y) ˆ = g(x)|

(2.3)

We shall choose yk = ϕk (x), 1 ≤ k ≤ n, such that bnn (y) = 6nj,k=1 g jk (x)

∂ϕn ∂ϕn = c(x), ϕn = xn for xn < −R, ∂xj ∂xk

∂ϕp ∂ϕr = 0, ∂xj ∂xk 1 ≤ r ≤ n − 1, ϕr (x) = xr for xn < −R. bpr (y) = 6nj,k g jk (x)

(2.6)

(2.7)

Since supp (g jk (x) − δjk ) ⊂ BR , supp (c(x) − 1) ⊂ BR , we have y = x for xn < −R.

(2.8)

The smooth solution of (2.6) does not exist for all xn . Note that to solve (2.6) we consider the system of bicharacteristics dxk ∂H(x, p) , xk (0) = yk , 1 ≤ k ≤ n − 1, xn (0) = −R, = dt ∂pk ∂H(x, p) dpk , pj (0) = 0 for 1 ≤ j ≤ n − 1, pn (0) = 1, =− dt ∂xk

where H(x, p) =

q

∂H(x, p) dϕ = 6nk=1 pk , ϕ(0) = −R, dt ∂pk 6nj,k=1 hjk (x)pj pk , hjk = c−1 (x)g jk (x).

(2.9)

(2.10)

Let xk = xk (t, y 0 ), pk = pk (t, y 0 ) be the solution of (2.9) where y 0 = (y1 , ..., yn−1 ). Then

Inverse Scattering Problem in Anisotropic Media

Z

0

ϕ = ϕ(t, y ) = −R + 0

t

479

6nk=1 pk (τ, y 0 )

∂H(x(τ, y 0 ), p(τ, y 0 )) dτ ∂pk

(2.11)

∂(x1 ,...,xn ) 6= 0 for 0 ≤ t ≤ T , we is the solution of (2.10). Assuming that the Jacobian ∂(t,y 1 ,...,yn ) can solve t = t(x), y 0 = y 0 (x). Then ϕn (x) = ϕ(t(x), y 0 (x)) is the solution of (2.6). We assume that y = ϕ(x) exists and is one-to-one in a neighborhood of B¯ R . Denote ¯ˆ is contained in the Bˆ R = ϕ(BR ). We assume that there exists R1 > 0 such that B R half-space yn < R + R1 and that y = ϕ(x) is a diffeomorphism of some domain ¯ − ⊂ Rn onto the half-space yn ≤ R + R1 . These assumptions are certainly satisfied D if c−1 (x)gjk (x) − δjk and its derivatives are small. Denote 0 = ∂D− . Note that 0 is the hypersurface ϕn (x) = R + R1 , and 0 coincide with the plane xn = R + R1 for |x| large ¯ and |x| large. Obviously D− ⊃ B¯ R . since ϕ(x) = x for x ∈ D

Lemma 2.1. Let L1 and L2 be two Laplace–Beltrami operators, Li = −1 for |x| > R, i = 1, 2. If L1 − k 2 c1 and L2 − k 2 c2 have the same scattering amplitude for all |θ| = |ω| = 1, then 3L1 −k2 c1 = 3L2 −k2 c2 , where 3Li −k2 ci are the Dirichlet-to-Neumann 2 i operators in D− , i.e. 3Li −k2 ci v = ∂u ∂ν |0 , where (Li − k ci )ui = 0 in D− , ui = v on ∞ 0, ∀v ∈ C0 (0). Vice versa if 3L1 −k2 c1 = 3L2 −k2 c2 , then L1 − k 2 c1 and L2 − k 2 c2 have the same scattering amplitude in Rn . Lemma 2.1 is proven in [ER2]. Consider again two Laplace–Beltrami operators Lj such that Lj − k 2 cj have the same scattering amplitude. It is well-known that when L1 −k 2 c1 = L2 −k 2 c2 = −1−k 2 for |x| > R the equality of the scattering amplitudes imply the equality of the distorted wave solutions ψj of (Lj − k 2 cj )ψj = 0 for |x| > R, where ψj = eikω·x + vj (x),

(2.12)

vj satisfies radiation conditions (0.7) when |x| → ∞. Let y = ϕ(j) (x), j = 1, 2 be two changes of variables satisfying (2.6), (2.7) for L1 and L2 respectively. Using the solution ϕ(j) n (x) of the eiconal equation, one can construct the geometric optics solution wj (x, k) such that wj (x, k) = eikϕ

(j)

(x)

(a(j) 0 (x) +

1 (j) 1 (j) a (x) + ... + N aN (x)), k 1 k

(2.13)

where ap(j) (x) satisfies some transport equations, p = 0, ..., N and (Lj − k 2 cj )wj = O(

1 ), k → ∞, x ∈ D− . k N −1

(2.14)

It is proven in [V] (see also [GU, MR] and [SyU]) that ψj − wj = O kN1−2 on compact ¯ − . Here ψj have the form (2.12) with ω = (0, ..., 0, 1). Since ψ1 = ψ2 for sets in D ¯ − we obtain that R < |x|, x ∈ D (2) ¯ ¯ ϕ(1) n (x) = ϕn (x) on D− \ BR .

(2.15)

(2) ¯ ¯ ϕ(1) r (x) = ϕr (x) on D− \ BR , 1 ≤ r ≤ n − 1,

(2.16)

Now we shall show that

(p) where ϕ(p) r (x) are the solutions of (2.7) for p = 1, 2. Denote by ϕrε (x) the solution of equation

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6nj,k=1 gpjk (x) ϕ(p) rε = xn

p

(p) ∂ϕ(p) rε ∂ϕrε = cp , x ∈ D− , ∂xj ∂xk

1 − ε2 + εxr for xn = −R, 1 ≤ r ≤ n − 1.

d (p) (p) ϕrε (x)|ε=0 = ϕ(p) Then dε r (x), where ϕr (x) is the solution of (2.7) for p = 1, 2. Therefore if we construct the geometric optics solutions

wprε (x) = eikϕrε (x) (a(p) 0rε (x) + (p)

then wprε (x) − ψprε (x) = O

1 (p) 1 a (x) + ... + N a(p) (x)), k → ∞, k 1rε k N rε

¯ − , where on compact sets in D √ 2 ψprε (x) = eik 1−ε xn +ikεxr + vprε (x) 1 kN −2

2 are distorted plane waves √ solutions of (Lp −k cp )ψprε = 0, vprε have asymptotics (0.7) with ω = (0, ..., ε, ..., 1 − ε2 ), p = 1, 2, 1 ≤ r ≤ n−1. Since a1 (θ, ω, k) = a2 (θ, ω, k) (2) implies ψ1rε (x) = ψ2rε (x), ∀ε > 0, for |x| > R, we get that ϕ(1) rε (x) = ϕrε (x), 1 ≤ r ≤ ¯ ¯ n − 1, x ∈ D− \ BR . Taking the derivative in ε and putting ε = 0 we obtain that (2.16) holds. Therefore y (j) = ϕ(j) (x), j = 1, 2 map D− onto the half-space yn < R + R1 and (j) = ϕ(x) ϕ(1) (x) = ϕ(2) near 0 = ∂D− . Denote by Lˆ j operators c−1 j Lj in coordinates y (j) 2 and denote vj (y ) = uj (x), where uj is the solution to (Lj − k cj )uj = 0 in D− satisfying uj = g on 0. If 3L1 −k2 c1 = 3L2 −k2 c2 on 0, then 3Lˆ 1 −k2 = 3Lˆ 2 −k2 on yn = R + R1 since ϕ(1) (x) = ϕ(2) (x) near 0. Here 3Lˆ j −k2 are the Dirichlet-to-Neumann ∂v ˆ = j on yn = R + R1 , where (Lˆ j − k 2 )vj = 0 operators on yn = R + R1 i.e. 3 ˆ 2g Lj −k

∂yn

ˆ 0 ) ∈ C0∞ (Rn−1 ). for yn < R + R1 and vj = gˆ on yn = R + R1 , g(y We have for j = 1, 2, (Lˆ j − k 2 )vj = 0 for yn < R + R1 , where

p ∂vj gˆ j (y)ˆcj ∂yn p ∂vj ∂ 1 pr g ˆ (y)b (y) 6n−1 , − p j j ∂yn cˆj gˆ j (y) p,r=1 ∂yp

Lˆ j vj = −

1 p cˆj gˆ j (y)

∂ ∂yn

(2.17)

(2.18)

nr since bnn j = cˆj , bj = 0, j = 1, 2, 1 ≤ r ≤ n − 1. Take vj = mj wj , j = 1, 2.

Then ∂ 2 wj ∂ p ∂wj ∂mj ∂wj 1 p − 2 − g ˆ c ˆ j j mj ∂yn2 ∂yn ∂yn ∂yn cˆj gˆ j ∂yn p ∂mj 1 ∂ − p gˆ j cˆj wj ∂y ∂yn cˆj gˆ j n p pr ∂(mj wj ) 1 ∂ − p 6n−1 g ˆ b (y) − k 2 mj wj . (2.19) j ∂yr cˆj gˆ j p,r=1 ∂yp

(Lˆ j − k 2 )mj wj = −mj

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We choose mj such that 2

∂ p 1 ∂mj cˆj gˆ j mj = 0, mj = 1 for yn < −R. + p ∂yn cˆj gˆ j ∂yn

(2.20)

Then p − 21 , j = 1, 2. mj = cˆj gˆ j

(2.21)

Since cj = 1, gj (x) = 1 for |x| ≥ R and ϕ(1) (x) = ϕ(2) (x) near 0 we get that m1 (y) = m2 (y) near yn = R + R1 . Denote ˆˆ − k 2 )w = (L j j

1 ˆ (Lj − k 2 )(mwj ) for yn < R + R1 . mj

(2.22)

ˆˆ = −6n ∂ 2 for y ≥ R + R , j = 1, 2. ˆˆ to y ≥ R + R by taking L Extend L j n 1 j n 1 p=1 ∂yp2 ˆ Then operators Lˆ j − k 2 have the form (1.1) , j = 1, 2 in Rn . Denote by 3Lˆˆ −k2 the j Dirichlet-to-Neumann operator on yn = R+R1 . Since m1 (y) = m2 (y) near yn = R+R1 , we have that 3Lˆ 1 −k2 = 3Lˆ 2 −k2 implies that 3Lˆˆ −k2 = 3Lˆˆ −k2 . 1 2 Applying again Lemma 2.1 we get that 3Lˆˆ −k2 = 3Lˆˆ −k2 implies that the scattering 1 2 ˆˆ − k 2 to Rn are the same. ˆˆ − k 2 and L amplitudes of the extensions of operators L 1

2

Therefore we proved that if L1 − k 2 c1 and L2 − k 2 c2 have the same scattering ˆˆ − k 2 to ˆˆ − k 2 and L amplitudes for all k > 0, |θ| = |ω| = 1 then the extensions of L 1 2 n R have the same scattering amplitudes for all k > 0, |θ| = |ω| = 1. Therefore the integrals Z 0 0 0 a0 (y, ζ 0 )e−i(ξ −ζ )·y dy a˜ 0 (ξ 0 − ζ 0 , 0, ζ 0 ) = y
where |ξ 0 | = |ζ 0 | = k are the same for all Laplace–Beltrami operators L − k 2 c(x) having the same scattering amplitude for all k > 0, |θ| = |ω| = 1. The quantities which are the same for all operators L − k 2 c(x) having the same scattering amplitude will be called scattering invariants. Let η be any unit vector in Rn and let µk (η), 1 ≤ k ≤ n − 1, be the unit vectors such that η, µ1 (η), ..., µn−1 (η) is an orthonormal basis in Rn . Analogously to (2.6), (2.7) denote by yk = ϕk (x, η), 1 ≤ k ≤ n, the change of variables such that 6nj,k=1 hjk j (x)

6nj,k=1 hjk j (x)

∂ϕn (x, η) ϕn (x, η) = 1, ϕn = x · η for x · η < −R, ∂xj ∂xk

(2.24)

∂ϕn (x, η) ϕr (x, η) = 0, ϕr = x · µr (η) for x · η < −R, r = 1, ..., n − 1. ∂xj ∂xk (2.25)

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Equations (2.6), (2.7) correspond to the case η = (0, ..., 0, 1). Note that if ψp (x, η) is a solution of ∂ϕn (x, η) ψp (x, η) = 0, ∂xj ∂xk ψp = xp for x · η < −R, p = 1, ..., n,

6nj,k=1 hjk j (x)

(2.26)

then ϕr (x, η) = ψ(x, η) · µr (η), where ψ(x, η) = (ψ1 , ..., ψn ). Then as in the case η = (0, ..., 0, 1) let 0(η) be a smooth hypersurface such that 0(η) does not intersect B¯ R and 0(η) coincide with hyperplane x · η = R + R(η) for large |x|. Denote by D− (η) the domain in Rn bounded by 0(η) that contains B¯ R . We assume that y = ϕ(x, η) is a one-to-one map of D− (η) onto the half-space yn < R + R(η), R(η) > 0. Denote by L1 the differential operator obtained from p 1 ∂ ∂u g(x)g jk (x) c−1 (x)Lu = c−1 (x)g − 2 (x)6nj,k=1 ∂xj ∂xk by the change of variables y = ϕ(x, η). We have

p ∂v ∂ jk g(y) ˆ gˆ (y) L1 v(y) = cˆ (y)gˆ ∂yj ∂yk p ∂ ∂v −1 − 21 n jk ˆ gˆ cˆ(y)h (y) , = cˆ gˆ 6j,k=1 ∂yj ∂yk −1

− 21

(y)6ni,j=1

(2.27)

−2 jk = c(x)hjk (x), gˆ jk (y) = cˆ(y)hˆ jk (y), where cˆ(y) = c(x), g(y) ˆ = g(x)| ∂ϕ(x,η) ∂x | , g

∂ϕp (x, η) ∂ϕr (x, η) hˆ pr (y) = 6nj,k=1 hpr (x) . ∂xj ∂xk

(2.28)

Note that hˆ nn (y) = 1, hˆ nr (y) = 0 for 1 ≤ r ≤ n − 1. Denote n

m(y) ˆ = cˆ− 2 gˆ − 4 = cˆ− 2 cˆ 4 hˆ − 4 = cˆ 1

1

1

1

n−2 4

1 hˆ − 4 .

(2.29)

ˆˆ − k 2 )w = m ˆ −1 (L1 − k 2 )mw ˆ by −1 − k 2 to the half-space yn > Then extending (L ˆˆ − k 2 : R + R(η) we can recover the following integral from the scattering amplitude of L Z i h 0 iζ 0 ·y dy. (2.30) e−iξ ·y m ˆ −1 (y) (Lˆ 1 − k 2 ) m(y)e ˆ J(ξ 0 , ζ 0 , η) = yn
0

0

Note that |ξ | = |ζ | = k. Denote by BR (η) the image of BR by the map y = ϕ(x, η). Since ϕ(x, η) is a scattering invariant for |x| ≥ R and c = 1, gjk = δjk for |x| ≥ R we get that BR (η) and the integral in (2.30) over BR (η) are scattering invariants. Integrating the integral over BR (η) by parts and taking into account that all terms on ∂BR (η) are also scattering invariants we finally get the following scattering invariant: (2.31) J1 (ξ 0 , ζ 0 , η) Z p 0 0 1 ∂ ∂ 6nj,k=1 gˆ cˆhˆ jk (y) me ˆ iζ ·y (m ˆ −1 cˆ−1 gˆ − 2 e−iξ ·y )dy. = ∂y ∂y k j BR (η)

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n−2 1 1 1 1 Note that m ˆ 2 cˆgˆ 2 = m ˆ 2 cˆ− 2 hˆ 2 = 1, m ˆ −1 cˆ−1 gˆ − 2 = m, ˆ gˆ 2 cˆ = m ˆ −2 . We can rewrite 0 0 J1 (ξ , ζ , η) in the following form: Z 0 0 ∂m ˆ ∂m ˆ 6nj,k=1 m ˆ −2 hˆ jk iζk m ˆ + ˆ + −iξj m e−i(ξ −ζ )·y dy. J1 (ξ 0 , ζ 0 , η) = ∂yk ∂yj BR (η)

Since |ξ 0 | = |ζ 0 | is arbitrary we can for arbitrary l ∈ Rn−1 put ξ 0 = 21 l + sl⊥ , ζ 0 = − 21 l + sl⊥ , where l⊥ · l = 0, |l⊥ | = 1 and s ∈ R1 is arbitrary. Then we get J1 (ξ 0 , ζ 0 , η) = s2 J2 (l, l⊥ , η) + J3 (l, η), where

Z

⊥

J2 (l, l , η) = and

Z

1 J3 (l, η) = − 4

BR (η)

Z

BR (η)

⊥ ⊥ −il·y ˆ jk 6n−1 dy j,k=1 h (y)lj lk e

(2.32)

−il·y ˆ 6n−1 dy j,k=1 hjk (y)lj lk e

∂ ˆ 6n−1 (ln m)e ˆ −il·y dy j,k=1 hjk (y)lj ∂y k BR (η) ∂ ∂ ˆ 6n−1 (ln m) ˆ (ln m)e ˆ −il·y dy. j,k=1 hjk (y) ∂y ∂y j k BR (η)

−i Z +

(2.33)

Since s is arbitrary, each of J2 (l, η) and J3 (l, l⊥ , η) is a scattering invariant. Note that ln m ˆ =

1 pˆ n−2 ln cˆ − ln h. 4 2

For any x(1) , x(2) ∈ ∂BR denote by 0h (x(1) , x(2) ) the arclength of the geodesics in metric h that connects x(1) and x(2) . Our assumptions about existence of one-to-one maps y = ϕ(x, η) of BR onto BR (η) for each η ∈ S n−1 implies that h is a simple metric, i.e. there exists a unique geodesics in B¯ R connecting any two points of B¯ R . Function 0h (x(1) , x(2) ) is called the hodograph function. We shall show that 0h (x(1) , x(2) ) is a scattering invariant. Let h(1) = c1 g (1) and h(2) = c2 g (2) be two metrics such that operators L1 − k 2 c21 and L2 − k 2 c2 have the same scattering amplitude. Then y = ϕ(1) (x, η) and y = ϕ(2) (x, η), where ϕ(i) , i = 1, 2, satisfy (2.24), (2.25) coincide outside BR . Let γy(i)(1) ,η (t) be a geodesics in metric h(i) such that γy(i)(1) ,η (0) =

(i) (i) (2) d (i) dt γy (1) ,η (0) = η, γy (1) ,η (t) ∈ BR for 0 < t < 1 and γy (1) ,η (1) = xi ∈ ∂BR . (2) (1) (2) Since ϕ(1) = ϕ(2) on ∂BR we get that x(2) 1 = x2 , i.e. γy (1) ,η (t) and γy (2) ,h (t) have the same endpoints in BR . Denote γ˜ y(i)(1) ,η (t) = ϕ(i) ◦ γy(i)(1) ,η (t). Then γ˜ y(i)(1) ,η (t) are straight lines in BR (η) with yn being the arclength parameter in both metrics hˆ (i) = (ϕ(i) )∗ h(i) , i = 1, 2. Since the endpoints of γ˜ y(1)(1) ,η (t) and γ˜ y(2)(1) ,η (t) in BR (η) are the same, the arclengths of (1) (2) (1) (2)

y (1) ∈ ∂BR ,

γ˜ and γ˜ are the same and therefore the arclengths of γ and γ are the same. Thus the hodographs for metrics h(1) and h(2) are the same. It will be shown in the next section that if 0h(1) (x(1) , x(2) ) = 0h(2) (x(1) , x(2) ) for all x(1) , x(2) ∈ ∂BR , then hˆ (1) (y) = hˆ (2) (y) ˆ is a scattering invariant. for each η ∈ S n−1 . Therefore h(y)

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Therefore retaining in J3 (l, η) the terms containing cˆ and taking the inverse Fourier transform in l = (l1 , ..., ln−1 ) we get the following scattering invariant: Z ∂ n−2 ∂ 0 n jk ˆ 6j,k=1 (ln cˆ) dyn h (y) J4 (y , η) = − 4 ∂yj ∂yk γ˜ y0 ,η Z (n − 2)2 n ˆ jk ∂ ∂ 6j,k=1 h (y) + (ln cˆ) (ln cˆ)dyn 16 ∂y ∂y j k γ˜ y0 ,η Z n−2 n ∂ ∂ 1 pˆ jk ˆ )6j,k=1 2h (y) ln h dyn . (2.34) + (− (ln cˆ) 4 ∂yj ∂yk 2 γ˜ y0 ,η Here γ˜ y0 ,η is the intersection of the straight line y 0 = const with BR (η). Note that γ˜ y0 ,η is the geodesics in metric [hˆ jk (y)]nj,k=1 with initial data y 0 and initial direction η and dyn is the arclength element on this geodesics. Combining the first and the third integrals in (2.34) we get p Z 1 ∂ n−2 ∂ p 6nj,k=1 hˆ hˆ jk (y) (ln cˆ) dyn J4 (y 0 , η) = − 4 ∂yj ∂yk γ˜ y0 ,η hˆ 2 Z (n − 2) ∂ ∂ (ln cˆ) dyn . + 6nj,k=1 hˆ jk (y) (ln cˆ) 16 ∂yj ∂yk γ˜ y0 ,η Returning back to x-coordinates we obtain p Z n−2 1 ∂ ∂ √ (ln c(x)) − h(x)hjk (x) 6nj,k=1 J4 (y 0 , η) = 4 ∂xj ∂xk h(x) γy0 ,η 2 ∂ ∂ (n − 2) n jk (ln c(x)) (ln c(x)) ds, 6j,k=1 h (x) (2.35) + 16 ∂xj ∂xk where γy0 ,η is the geodesics in metric [hjk (k)]nj,k=1 , y 0 is the initial point of this geodesics, η is the initial direction and ds is the arclength of the geodesics in metric [hjk (x)]nj,k=1 . Since we know the integral (2.35) over all geodesics in BR we can recover (see [Sh], Theorem 4.3.3 ) the function R(x, c) = −

(n − 2)2 n ∂ ∂ n−2 Lh (ln c) + (ln c(x)) (ln c(x)) 6j,k=1 hjk (x) 4 16 ∂xj ∂xk

for all x ∈ B¯ R , where Lh is the Laplace–Beltrami operator for the metric [hjk ]nj,k=1 . Since we know that c = 1 outside BR we know the Cauchy data for ln c (in fact c = 1 ∂c ∂ = 0 on ∂BR , where ∂ν is the normal derivative on ∂BR ). on ∂BR and ∂ν Therefore the uniqueness of the Cauchy problem for the second order elliptic operator allows us to determine c(x) uniquely. 3. Solution of the Hodograph Problem (i) (x)] be two metric tensors on BR and h(1) = h(2) = I outside BR . Assume Let h(i) = [hpk that any two points x(1) , x(2) ∈ B¯ R can be uniquely connected by a geodesics in BR with respect to metric h(i) , i = 1, 2.

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Let 0h(i) (x(1) , x(2) ) be the arclength of this geodesics, i.e. 0h (x(1) , x(2) ) is the hodograph function. The hodograph problem consists in showing that if 0h(1) (x(1) , x(2) ) = 0h(2) (x(1) , x(2) ) for all (x(1) , x(2) ) ∈ ∂BR × ∂BR , then there exists a C ∞ map y = ϕ(x) such that ϕ(x) = x on ∂BR and h(2) = ϕ∗ h(1) , where ϕ∗ h(1) means the pullback of metric h(1) . Make the change of variables in Rn as in §2: yk = ψki (x), 1 ≤ k ≤ n, i = 1, 2, where ψki satisfies the equations 6np,r=1 hpr i (x) 6np,r=1 hpr i (x)

∂ψni ∂ψni = 1, ψni (x) = xn for xn < −R, ∂xp ∂xr

(3.1)

∂ψni ∂ψki = 0, ψki (x) = xk for xn < −R, 1 ≤ k ≤ n − 1. (3.2) ∂xp ∂xr

n (i) n Here [hpr i (x)]p,r=1 is the inverse to the metric tensor [hpr ]p,r=1 and we assume that these equations have a smooth solution for xn ≤ R + R2 , R2 > 0. −1 ˆ nn ˆ nr Denote hˆ (i) = ψi∗ h(i) , i = 1, 2. Then hˆ (i) (y) = [hˆ pr i (y)] , hi = 1, hi = 0, i = 1, 2, r = 1, ..., n − 1. Coordinates y = ψi (x) are called semigeodesics coordinates for metric h(i) . Note that y 0 = const, yn = t is a family of geodesics with respect to metrics hˆ (1) and ˆh(2) and t is the arclength parameter for both metrics. Denote Bˆ Ri = ψi (BR ), i = 1, 2. We shall show that if hodographs for h(1) and h(2) coincide then Bˆ R1 = Bˆ R2 , ψ1 (x) = ψ2 (x) on ∂BR and 0hˆ (1) (y (1) , y (2) ) = 0hˆ (2) (y (1) , y (2) ), where (y (1) , y (2) ) are any two points of ∂ Bˆ R . We denote Bˆ R = Bˆ R1 = Bˆ R2 . Let γ0 = γ0 (y 0 ) be any geodesics of the family y 0 = const, yn = t. Denote by ∂− Bˆ Ri the part of ∂ Bˆ Ri , where the geodesics y 0 = const, yn = t enter Bˆ Ri and by ∂+ BRi the part of ∂ Bˆ Ri where these geodesics leave Bˆ Ri . Denote γi = ψi−1 (γ0 ). Since ψ1 (x) = ψ2 (x) = x before γi enter BR we have that ∂− Bˆ R1 = ∂− Bˆ R2 = ∂− BR and ψ1−1 (y) = ψ2−1 (y) on ∂− Bˆ R1 . Note that the pullback preserves the arclength. Therefore (2) (2) (1) (2) 0h(i) (x(1) , x(2) ˆ (i) (y , yi ), where xi ∈ ∂+ BR is the endpoint of γi and yi ∈ i ) = 0h ˆ ˆ ∂+ BRi is the endpoint of γ0 in BRi , i = 1, 2. Since hodograph functions for h(1) and h(2) coincide and since γ1 and γ2 have the same initial point x(1) ∈ ∂BR and the same direction at x(1) , it follows (see [M]) that (2) (1) (2) (1) (2) the endpoints coincide, i.e. x(2) ˆ (1) (y , y1 ) = 0h ˆ (2) (y , y2 ). Since 1 = x2 . Therefore 0h (1) (2) ˆ ˆ yn is the arclength parameter in both metrics h and h we get that y1(2) = y2(2) . Therefore ∂+ Bˆ R1 = ∂+ Bˆ R2 and ψ1 (x) = ψ2 (x) on ∂+ BR . Thus ψ1 (x) = ψ2 (x) on ∂BR and ∂ Bˆ R1 = ∂ Bˆ R2 = ∂ Bˆ R . This implies that for any two points y (1) and y (2) on ∂ Bˆ R 0 ˆ (1) (y (1) , y (2) ) = 0 ˆ (2) (y (1) , y (2) ). Indeed if γ˜ 1 and γ˜ 2 are the geodesics of hˆ (1) and hˆ (2) h

h

connecting y (1) and y (2) and γk = ψk−1 (γ˜ k ), k = 1, 2, then γ1 and γ2 are geodesics of h(1) and h(2) having the same endpoints and therefore 0h(1) (γ1 ) = 0h(2) (γ2 ). Thus 0hˆ (1) (y (1) , y (2) ) = 0hˆ (2) (y (1) , y (2) ). We shall prove that this implies that hˆ (1) = hˆ (2) . Then ϕ(x) = ψ2−1 (ψ1 (x)) is a C ∞ map, ϕ(x) = x on ∂BR and h(2) = ϕ∗ h(1) . We have Z 1 dyp (t) dyr (t) 1 (1) (2) ) 2 dt, (6np,r=1 hˆ pr (y(t)) (3.3) 0hˆ (y , y ) = dt dt 0

486

G. Eskin

where y = y(t), 0 ≤ t ≤ 1, is the parametric equation of the geodesics in metric hˆ connecting points y (1) ∈ ∂ Bˆ R and y (2) ∈ ∂ Bˆ R , i.e. y(0) = y (1) , y(1) = y (2) . We assume that hˆ nn = 1 and hˆ nr = 0, 1 ≤ r ≤ n − 1, in Bˆ R . Note that 0hˆ (y (1) , y (2) ) is a twice continuously differentiable functional of hˆ assuming that metric hˆ is smooth enough and there exists a unique geodesics connecting any two points in Bˆ R . Expanding 0hˆ (y (1) , y (2) ) by the Taylor formula at hˆ = hˆ (1) we get 0hˆ (2) y (1) , y (2) = 0hˆ (1) y (1) , y (2) + Lhˆ (1) y (1) , y (2) hˆ (2) − hˆ (1) + R y (1) , y (2) , hˆ (1) , hˆ (2) , hˆ (2) − hˆ (1) ,

(3.4)

where Lhˆ (2) (y (1) , y (2) ) is the differential of 0hˆ at hˆ (1) (i.e. Lhˆ (1) (y (1) , y (2) ) is the linearization of 0hˆ at hˆ (1) ) and R(y (1) , y (2) , hˆ (1) , hˆ (2) , hˆ (2) − hˆ (1) ) is the remainder. To compute Lhˆ (1) (y (1) , y (2) ) (cf. [Sh]) we take hτ = hˆ (1) + τ (hˆ (2) − hˆ 1) ), 0 ≤ τ ≤ 1, take the derivative in τ and put τ = 0. We assume that any two points in B¯ R can be connected by unique geodesics in metric hτ , 0 ≤ τ ≤ 1. We have d (3.5) 0hτ y (1) , y (2) dτ Z 1 − 1 1 ∂ 6np,r=1 hτ pr y(t) y˙p y˙r 2 6np,r=1 ( hτ pr (y(t))y˙p y˙r = 2 ∂τ 0 ∂ y˙p ∂hτ pr (y(t)) ∂yk n n y˙p y˙r + 26p,r=1 hτ pr (y(t)) y˙r + 6k,p,r=1 dt, ∂yk ∂τ ∂τ where y˙p = dy dt . Taking τ = 0 and then choosing parameter t as the arclength parameter for the geodesics y = y(t, 0) in metric hˆ (1) we get that 6np,r=1 h0pr (y(t, 0)) y˙p (t, 0)y˙r (t, 0) = 1. Also the second and the third sum in (3.5) vanish when τ = 0. To see this one should integrate by parts in the third sum in (3.5) and use that y(t, 0) satisfies the equation for = ∂y(1,0) = 0. geodesics (see (3.8) below) and ∂y(0,0) ∂τ ∂τ Therefore we get Lhˆ (1) y , y (1)

(2)

Z h= 0

a1

6n−1 p,r=1 hpr (y(t)) y˙p y˙r dt,

(3.6)

where y = y(t) is the geodesics in metric hˆ (1) , t is the arclength parameter and a1 is the length of the geodesics: y(0) = y (1) , y(a1 ) = y (2) . (1) Here hpr = h(2) pr (y) − hpr (y). Note that hnr = 0 for 1 ≤ r ≤ n. Taking the second derivative in τ we have

Inverse Scattering Problem in Anisotropic Media

d2 0h (y (1) , y (2) ) = dτ 2 τ +2

Z 0

1

− 21 ∂ 2 hτ pr (y(t)) 1 hτ pr (y(t)) y˙p y˙r y˙p y˙r 2 ∂τ 2

487

(3.7)

∂hτ pr (y(t)) ∂ y˙p ∂ 2 hτ pr ∂yk ∂yj ∂ 2 hτ pr (y(t)) ∂yk y˙p y˙r + 4 y˙r + y˙p y˙r ∂τ ∂yk ∂τ ∂τ ∂τ ∂yk ∂yj ∂τ ∂τ

∂ y˙p ∂ y˙r ∂ 2 y˙p ∂hτ pr (y(t)) ∂ 2 yk ∂hτ pr ∂yk ∂ y˙p + 2h + 2h y ˙ y ˙ + 4 y ˙ p r τ pk τ pr r dt ∂yk ∂τ 2 ∂yk ∂τ ∂τ ∂τ ∂τ ∂τ 2 Z 1 − 3 1 hτ pr (y(t))y˙p y˙r 2 − + 4 0 2 ∂hτ pr (y(t)) ∂ y˙p ∂hτ pr ∂yk y˙p y˙r + y˙p y˙r + 2hτ pr y˙r dt. ∂τ ∂yk ∂τ ∂τ +

Here and below we assume the summation over repeating indices. Note that y(t, τ ) satisfies the equation of geodesics dyj dyk d 2 yi + 6nj,k=1 0ijk = 0, y(0, τ ) = y (1) , y(1, τ ) = y (2) , 2 dt dt dt where 0kij (y) =

(3.8)

1 kp ∂hτ jp ∂hτ ip ∂hτ ij h ( + − ), 2 τ ∂yi ∂yj ∂yp

[hkp τ ] is the inverse matrix to [hτ kp ]. Taking the derivative in τ and estimating the solution of ordinary differential equations we get ! ∂ ∂y ∂ y˙ n (2) (1) (2) (1) + ≤ C 6j=1 sup (hˆ − hˆ ) + sup hˆ − hˆ , (3.9) ∂τ ∂τ Bˆ R ∂yj Bˆ R ! 2 2 2 ∂ ∂ y ∂ y˙ n (2) (1) (2) (1) ˆ ˆ ˆ ˆ ∂τ 2 + ∂τ 2 ≤ C 6j=1 sup ∂yj (h − h ) + sup h − h , Bˆ R Bˆ R where C depends on C 2 (Bˆ R ) norms of hˆ (1) and hˆ (2) , C k (Bˆ R ) is the space of k times ¯ˆ with the sup norm.. continuous differentiable functions in B R Therefore taking into account that 0hτ (y (1) , y (2) ) ≤ C y (1) − y (2) we have R(y (1) , y (2) , hˆ (1) , hˆ (2) , hˆ (2) − hˆ (1) ) 2 k (3.10) ∂ (2) (1) 1 (2) (1) ˆ ˆ ≤ C1 y − y 6|k|=0 sup k (h − h ) , y∈Bˆ r ∂y

∂R where C1 depends on C 2 (Bˆ R ) norms of hˆ (1) and hˆ (2) . Analogously ∂y (r) , r = 1, 2 is bounded by the C 2 (Bˆ R ) norm of hˆ (2) − hˆ (1) with constant depending on C 3 (Bˆ R ) norms of hˆ (2) and hˆ (1) . Denote by ∂+ S Bˆ R the space of pairs (y, ξ), where y ∈ ∂ Bˆ R and ξ is a unit tangent vector at y in metric hˆ (1) such that < ν(y), ξ >= 6np,r=1 hˆ (1) pr (y)νp (y)ξr ≥ 0,

488

G. Eskin

where ν(y) is the inner normal vector to ∂ Bˆ R at y ∈ ∂ Bˆ R . We shall parametrize the d γy,ξ (0) = ξ. Let geodesics γy,ξ (t) in BˆR by (y, ξ) ∈ ∂+ S Bˆ R , where γy,ξ (0) = y, dt τ (y, ξ) be the length of geodesics γy,ξ (t). Any tensor [hjk (y)]nj,k=1 in Bˆ R admits a decomposition into solenoidal and potential parts (see [Sh], Theorem 3.3.2): h = hs + ds v, v|∂ Bˆ R = 0,

(3.11)

where hs is the solenoidal part, v = (v1 , ..., vn ) is a vector field in Bˆ R and (ds v)ij =

1 ∇i vj + ∇j vi , 1 ≤ i, j ≤ n, 2

(3.12)

where ∇i is the covariant derivative in metric hˆ (1) . Note that a tensor hs is called solenoidal if δ s (hs ) = 0, where δ s is dual operator to ds (see [Sh]). The following result is contained in [Sh], Theorem 4.3.3: Proposition 3.1. Assuming that the curvature tensor of metric hˆ (1) is small, the following estimate holds for any h ∈ H1 (Bˆ R ): (3.13) khs k20 ≤ C2 khk1 kLhˆ (1) (h)k0 + kLhˆ (1) (h)k21 , where hs is the solenoidal part of h and kLhˆ (1) (h)kk , k = 1, 2, is the Sobolev norm in ∂+ S Bˆ R , khk1 is the Sobolev norm in Bˆ R . ˆ (1) Since [hij (y)] is such that hnr = 0, 1 ≤ r ≤ n, in Bˆ R and since hˆ (1) nn = 1, hnr = ˆ 0, 1 ≤ r ≤ n − 1, in BR we can estimate kds vk−2 ≤ Ckhs k0 . Indeed we have ∇n vi + ∇i vn = −2hsin , i = 1, ..., n. In particular ∇n vn = −hsnn . We have

(1) ∂vj ∂vj 1 ∂ hˆ jr ˆ kr k h vk . − 0jn vk = − ∇n vj = ∂yn ∂yn 2 ∂y n 1

Therefore ∇n vn =

∂vn ∂yn

since

ˆ (1) ∂h nr ∂y n

= 0. Thus Z yn hsnn (y)dyn vn = − −R

and

kvn kk ≤ Ckhsnn kk , ∀k.

Then we can determine and estimate v1 , ..., vn−1 from the equation (1) 1 ∂ hˆ jr ˆ kr ∂vj h vk = −hsnj − ∇j vn . − ∂yn 2 ∂y n 1

Integrating in yn and applying Gronwall’s inequality we get n−1 s s s kvj kk ≤ C6n−1 p=1 khnp kk + Ckvn kk+1 ≤ C6p=1 khnp kk + khnn kk+1 .

Inverse Scattering Problem in Anisotropic Media

489

Therefore kds vkk ≤ Ck khs kk+2 .

(3.14)

Taking k = −2 we get khk−2 ≤ khs k−2 + kds vk−2 ≤ Ckhs k0 ≤ C(kLhˆ (1) hk21 + khk1 kLhˆ (1) hk0 ) 2 . (3.15) 1

There is one-to-one correspondence between ∂+ S Rˆ R and ∂ Bˆ R × ∂ Bˆ R . Indeed for any (y (1) , ξ) ∈ ∂+ S Rˆ R we assign a point (y (1) , y (2) ) ∈ ∂ B¯ R × ∂ Bˆ R , where y (2) = γy(1) ,ξ (τ (y (1) , ξ)) is the endpoint in Bˆ R of the geodesics starting at y (1) ∈ ∂ Bˆ R in direction ξ. The inverse map (y (1) , y (2) ) → (y (1) , ξ) is singular since ξ ≈ is small. Therefore we have the following estimate:

y (2) −y (1) |y (2) −y (1) |

when y (2) − y (1)

kLhˆ (1) (h)k1 ≤ CkLhˆ (1) (h)kw , where

Z kLhˆ (1) (h)k2w =

∂ Bˆ R ×∂ Bˆ R

(3.16)

h y (1) − y (2) 2−n |L ˆ (h)|2 h

(3.17)

!# ∂Lhˆ (1) (h) 2 ∂Lhˆ (1) (h) 2 + dsy(1) dsy(2) . + ∂y (1) ∂y (2)

Here dsy(1) is the area element of ∂ Bˆ R . Therefore we have the following estimate for the inverse of the linearized operator: khk2−2 ≤ C1 (kLhˆ (1) (h)k2w + khk1 kLhˆ (1) (h)kw ),

(3.18)

where C1 depends on hˆ (1) . Taking into account (3.10) and the subsequent remark, and the inequality k ∂ sup k h ≤ Cε khkk+ n2 +ε , ε > 0, y∈Bˆ R ∂y we obtain kRkw ≤ C2 khk2n1 , where n1 > n2 + 2, C2 depends on C 3 (Bˆ R ) norms of hˆ (1) and hˆ (2) . Therefore (3.18), (3.19) and (3.4) imply C1−1 khk2−2 ≤ kLhˆ (1) (h)kw khk1 + kLhˆ (1) (h)k2w ≤ khk1 k0hˆ (2) − 0hˆ (1) kw + kRkw + 2k0hˆ (2) − 0hˆ (1) k2w + 2kRk2w

(3.19)

(3.20)

≤ khk1 k0hˆ (2) − 0hˆ (1) kw + 2k0hˆ (2) − 0hˆ (1) k2w + C2 khk1 khk2n1 + 2C22 khk4n1 , where h = hˆ (2) − hˆ (1) . The following elementary interpolation inequality in Sobolev spaces Hk (Bˆ k ):

490

G. Eskin

khks ≤ khkθs1 khks1−θ , 2

(3.21)

s = θs1 + (1 − θ)s2 , 0 < θ < 1, holds for any real s1 and s2 . (The proof of (3.21) follows from the H¨older inequality.) Applying (3.21) first with s1 = −2, s = n1 and θ = 21 and second with s1 = −2, s = n1 and θ = 23 , we get khk4n1 ≤ khk2−2 khk22n1 +2 , khk3n1 ≤ khk2−2 khk23n1 +4 .

(3.22)

It follows from (3.20) and (3.22) that C1−1 khk2−2 ≤ khk1 k0hˆ (2) − 0hˆ (1) kw + 2k0hˆ (2) − 0hˆ (1) k2w + 2C22 khk2−2 khk22n1 +2 + C2 khk2−2 khk3n1 +4 , where h = hˆ (2) − hˆ (1) . We used that khk1 ≤ khkn1 since n1 > 1. Assuming that 2C1 C22 khˆ (2) − hˆ (1) k22n1 +2 + C1 C2 khˆ (2) − hˆ (1) k3n1 +4 < 1, we get khˆ (2) − hˆ (1) k2−2 ≤ C1 C3 2k0hˆ (2) − 0hˆ (1) k2w + C4 k0hˆ (2) − 0hˆ (1) kw , where C4 ≥ khˆ (2) − hˆ (1) k1 , C3 = (1 − 2C1 C22 khk22n1 +2 − C1 C2 khk3n1 +4 )−1 . This gives that hˆ (2) = hˆ (1) if 0hˆ (2) = 0hˆ (1) . Therefore we proved the following result: Theorem 3.1. Assume that metrics hτ = h(1) + τ (h(2) − h(1) ) are simple metrics for 0 ≤ τ ≤ 1 and that 0h(1) (x(1) , x(2) ) = 0h(2) (x(1) , x(2) ) for all x(1) , x(2) ∈ ∂BR . Assume that the curvature tensor of h(1) is small and that in semigeodesics coordinates the Sobolev norm khˆ (2) − hˆ (1) k 3 n+10+ε is small enough, ε > 0. Then there exists a diffeomorphism 2 y = ϕ(x) of Rn , ϕ(x) = x outside of BR such that h(2) = ϕ∗ h(1) , where ϕ∗ h(1) is the pullback of h(1) . Therefore if operators L1 − k 2 c1 (x) and L2 − k 2 c2 (x) have the same scattering (1) amplitude a(θ, ω, k) for all k > 0, |θ| = 1, |ω| = 1, then metrics h(1) ij (x) = c1 (x)gij and (2) (2) hij (x) = c2 (x)gij have the same hodograph functions and by Theorem 3.1 h(2) = ϕ∗ h(1) . Then using the scattering invariants (2.35) we get that c2 = ϕ∗ c1 . Therefore g (2) = ϕ∗ g (1) . This concludes the proof of Theorem 1.1. References [BK]

Belyshev, M. and Kurylev, Y.: To the reconstruction of a Riemannian manifold via its boundary spectral data (BC-methods). Comm. PDE 117, 767–804 (1992) [C] Croke, C.: Rigidity and the distance between boundary points. J. Differ. Geom. 33, no. 2, 445–464 (1991) [ER1] Eskin, G. and Ralston, J.: Inverse scattering problem for the Schr¨odinger equation with magnetic potential at a fixed energy. Commun. Math. Phys., 173, 199–224 (1995) [ER2] Eskin, G. and Ralston, J.: Inverse coefficient problems in perturbed halfspaces. Preprint [Gr] Gromov, M.: Filling Riemannian manifolds. J.Differ. Geom. 18 no.1, 1–148 (1983) [Gu] Guillemin, V.: Sojourn times and asymptotic properties of the scattering matrix. Publ. Res. Inst. Math. Sci. 12 , supplement, 69–88 (1976/77)

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[LU] [MR] [M] [Sh] [SU1] [SU2] [Sy] [SyU] [U] [V]

491

Lee, J. and Uhlmann, G.: Determining anisotropic real-analytic conductivities by boundary measurements. Comm. Pure Appl. Math. 45, 1097–1112 (1989) Majda, A. and Ralston, J.: An analogue of Weyl’s theorem for unbounded domains, II. Duke Math. J. 45, no. 1, 513–536 (1978) Michel, R.: Sur la ridigite imposee par la longuer des g´eod´esiques. Invent. Math. 65, 71–83 (1981) Sharafutdinov, V.A.: Integral geometry of tensor fields. VSP, Utrecht, the Netherlands (1994) Stefanov, P. and Uhlmann, G.: Stability estimates for the hyperbolic Dirichlet-to-Neumann map in anisotropic media. J. Funct. Anal., to appear Stefanov, P. and Uhlmann, G.: Rigidity for metrics with the same lengths of geodesics. Math. Res. Lett. 5, 83–96 (1998) Sylvester, J.: An anisotropic inverse boundary value problem. Comm. Pure Appl. Math. 46, 201–232 (1990) Sylvester, J. and Uhlmann, G.: Inverse problems in anisotropic media. Contemp. Math. 122, 105–117 (1991) Uhlmann, G.: Inverse boundary problems and applications. Ast´erisque 207, 153–221 (1992) Vainberg, B.R.: Asymptotic Methods in Equations of Mathematical Physics. New York: Gordon and Breach Sci. Publ., 1988

Communicated by B. Simon

Commun. Math. Phys. 199, 493 – 520 (1999)

Communications in

Mathematical Physics © Springer-Verlag 1999

Low Temperature Properties for Correlation Functions in Classical N-Vector Spin Models Tadeusz Balaban1 , Michael O’Carroll2 1 2

Mathematics Department, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA Departamento de F´ısica do ICEX, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brasil

Received: 13 October 1996 / Accepted: 12 January 1998

Abstract: We obtain convergent multi-scale expansions for the one-and two-point correlation functions of the low temperature lattice classical N – vector spin model in d ≥ 3 dimensions, N ≥ 2. The Gibbs factor is taken as exp[−β(1/2||∂φ||2 + λ/8|| |φ|2 − 1||2 + v/2||φ − h||2 )], where φ(x), h ∈ RN , x ∈ Z d , |h| = 1, β < ∞, λ ≥ ∞ are large and 0 < v ≤ 1. In the thermodynamic and v ↓ 0 limits, with h = e1 , and 1 ≡ ∂ ∗ ∂, the expansion gives hφ1 (x)i = 1 + 0(1/β 1/2 ) (spontaneous magnetization), hφ1 (x)φi (y)i = 0, hφi (x)φi (y)i = c0 1−1 (x, y) + R(x, y) (Goldstone Bosons), i = 2, 3, . . . , N , and hφ1 (x)φ1 (y)iT = R0 (x, y), where |R(x, y)|, |R0 (x, y)| < 0(1)(1 + |x − y|)d−2+ρ for some ρ > 0, and c0 is aprecisely determined constant. 1. Introduction and Results In [1–5] a block renormalization group analysis of the low temperature classical N vector spin model for lattice dimension d ≥ 3 is carried out. The generating function R of connected correlation functions is log Z(g) = log ρdφ, where the Gibbs factor ρ is taken as exp −β(1/2||∂φ||2 + λ/8|| |φ|2 − 1||2 + v/2||φ − h||2 ) + hg, φi − E . (1.1) E is chosen such that Z(0) = 1. In (1.1) h ∈ RN is a unit vector, φ(x) ∈ RN . the parameters β < ∞, λ ≤ ∞ are sufficiently large and v ∈ (0, 1], g is complex and ||g||l1 < 1. In this paper we extend the analysis of [1–5] to obtain multi-scale convergent expansions and properties for the one-and two-point functions of the model. In particular we obtain detailed information on the decay rate of the two-point function. Our main

494

T. Balaban, M. O’Carroll

interest and results are for v = 0 and we obtain them taking the limit v ↓ 0. Let us remark that for any fixed v > 0 there exists a β(v) such that the model can be treated by a single cluster expansion for any β ≥ β(v), but β(v) → +∞ if v → 0. We use a multi-scale analysis to abtain results which hold on some fixed interval β0 ≤ β < ∞ for any v ∈ (0, 1], including the limit v → 0. In [6, 7] it is shown that for large β the model has spontaneous magnetization and bounds are obtained on the decay rate of some correlation functions. Other results are obtained in [11, 12]. In the physics literature (see [8]) there is a well known “spin wave picture”, describing detailed properties of the correlation functions. We expect symmetry breaking, and in the v = 0 limit the transverse components of the two-point function have long-range decay like the inverse Laplacian, describing the so-called “Goldstone bosons”. This behavior is shown in a hierarchical version of the model in [9]. To get an understanding of the behavior of the correlation function (cf) we have set up a formal perturbation theory in β −1 and λ−1 for the generating function. This is done by making a non-linear change of coordinates and scaling as in the calculation of the first fluctuation integral in [1]. The results for λ = ∞ and to lowest non-vanishing order in 1/β are, with h = e1 and 1 ≡ −∂ ∗ ∂, N −1 (1 + v)−1 (0, 0), 2β N −1 (1 + v)−1 (x, y), i = 2, 3, . . . , N, hφi (x)φi (y)i = 2β 2 N −1 (1 + v)−1 (x, y) , hφ1 (x); φ1 (y)iT = 4β 2 hφ1 (x)i = 1 −

(1.2)

where T means truncated. From (1.2) for v = 0 we see the canonical massless behavior in the transverse 2-pt. cf and a massless non-canonical behavior in the longitudinal two-pt function. Also the parallel susceptibility is infinite for d ≤ 4, finite for d ≥ 5. These general features also hold for λ large and finite. Let us describe now the main results of the paper. We are working in the framework of the paper [5], where the finite volume densities are defined on tori T with periodic boundary conditions, and we denote the thermodynamic limit expectations by h·i = limh·iT . We denote the corresponding one-point function by M (x) = hφ(x)i, and the two-point truncated function by G(x, y) = hφ(x); φ(y)iT = hφ(x) ⊗ φ(y)i − hφ(x)i ⊗ hφ(y)i, where we suppress the vector indices. The scalar functions m, Gtr , Glg appearing in the theorem below are called the magnetization, transverse and longitudinal two-point truncated functions, respectively. Our main result is Theorem 1.1. There exist β0 < ∞, λ0 ≤ ∞ sufficiently large such that for β0 ≤ β < ∞ and λ0 ≤ λ ≤ ∞, M and G can be written as M (x) = m(β, v)h, G(x, y) = Gtr (x − y)(I − h ⊗ h) + Glg (x − y)(h ⊗ h). ∃ positive constants c, c0 , .ρ0 and small ρ1 such that a) for v > 0, |m(β, v) − 1| < cβ −1+ρ1 , |Gtr (x − y) − | , |Glg (x − y)| ≤

√ c exp −c0 v|x − y| . |x − y|d−2

Low Temperature Properties for Correlation Functions

495

b) For the v ↓ 0 limiting expectations, Gtr (x − y) =

1 m2 (β)1−1 (x − y) + R(x − y), βγ∞

where |γ∞ − 1| < cβ −1 and |R(x − y)| < c/(1 + |x − y|)d−2+ρ0 . Also

|Glg (x − y)| ≤ c/(1 + |x − y|)d−2+ρ0 .

Let us make some remarks on the above theorem. We actually obtain a much stronger result. We construct convergent multi-scale expansions of the correlation functions, which describe precisely contributions from the scales to the over-all behavior given in the teorem. In particular we can use this multi-scale expansion and elaborate on it to obtain sub-leading contributions to any desired accuracy. Another important methodological remark is connected with the analyticity properties of the effective actions formulated in the inductive hypotheses (H.2)–(H.4) [4], and their role in the proof of the theorem. The analyticity domains are determined by the constants εk , and they scale with a power lower than in the “canonical” d−2 1scaling, 1 more −γ, where 0 < γ < γ = min , . Actuprecisely εk+1 = (L−1 )α εk , α = d−2 0 2 2 2 = 2 ally a bit more careful analysis would give the best possible constant γ0 = min d−2 2 ,1 . Notice that for γ = 0 we obtain the “canonical” scaling. This means that the analyticity domains are larger than “canonical” ones. This property is connected with the fact that the effective action is a sum of “irrelevant” expressions with a positive “gap of irrelevancy”, which means that scaling powers of these expressions are greater than d plus a fixed positive number, the “gap”. It is a very important property, and it plays a crucial role in the proof given below, in particular it simplifies substantially the proof. It has not been stressed strongly enough in the previous papers [1–5], so we wanted to explain it here. Actually it should be put on a much more conceptual level and introduced as an important part of the inductive description, together with the concept of the “irrelevancy gap” mentioned above. We hope to do it on another occasion. We now describe the organization of the paper. In Sect. 2 we give an inductive description of the generating function for the 1- and 2-point cf after k RGT. We show in Sect. 3 that the inductive assumptions are satisfied applying the next RGT. A detailed description of the one- and two-point functions is obtained in Sect. 4. 2. Inductive Description of the Generating Function for the One- and Two-Point Correlation Functions The analysis of this section is based on the inductive description of the effective densities ρk given in the hypotheses (H.1)–(H.7) of [5]. Let us recall that ρk is the density obtained after k steps of the renormalization group procedure. The most important information about the correlation functions is contained in the function Fk described in the hypothesis (H.3) [5]. In the representation (1.1) of ρk in [5] we have the restricted functions Fk (Zkc ), but they are determined by the function Fk defined on the whole lattice T . We introduce here a more detailed description of this function, and it will give the corresponding description of the restricted functions. In this and the mext section it is enough to work with the function Fk defined on the whole lattice, only in the last step discussed in Sect. 4

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we have to take into account the representation (1.1) in [5] depending on the restricted functions Fk (Zkc ), and analyze its various contributions. We are interested in the precise behavior of the 1- and 2-pt. functions. This behavior will be obtained from a detailed description of the 1st and 2nd order in g contributions from the generating function Fk (ψ; h, g), where ψ denotes the new variable after k steps, defined on the unit lattice T1(k) and denoted previously by ψk . We expand the function Fk (ψ; h, g) up to the third order in g writing 1 Fk (ψ; h, g) = hg, Mk i1 + hg, Gk gi2 + 0(|g|3 ), 2

(2.1)

and we give detailed descriptions of the vector valued function Mk and the matrix valued kernel Gk . These descriptions are extensions of hypothesis (H.3) in [5]. We start with a hypothesis describing Mk . (H.1) The function Mk has the representation Mk (x; ψ, h) = mk (x)φk (x; ψ, h) +

k X j=1

− 21 βj−1 M (j) x; ψk(j) (ψ, h), h ,

x ∈ T, (2.2)

where the functions M (j) (x; ψ, h) are real analytic on 4j (1, εj ), ψj is a spin variable on the lattice T (j) , and satisfy the symmetry properties M (j) (x; Rψj , Rh) = RM (j) (x; ψj , h)

for R ∈ O(N ),

M (j) (τ x; τ ψj , h) = M (j) (x; ψj , h) for Euclidean transformations τ of the lattice T(j) . These functions have localization expansions X M (j) (x, X; ψj , h), Mk (x; ψj , h) =

(2.3) (2.4)

(2.5)

X∈Dj :x∈X

and the terms have analytic extensions onto the space 4cj (X; 1, εj ), satisfy the symmetry properties (2.3), (2.4) for transformations τ leaving invariant the partition πj and satisfy the bounds (j) 1 M (2.6) x, X; ψj , h < β − 2 +ρ1 ξ ρ0 exp(−κdj (X)), ξ = L−j , where ρ1 > 0 is a sufficiently small constant, ρ0 > 0 is a fixed constant. The constant κ is the same as in (H.3) [5]. By the symmetry property (2.3) we have M (j) (x; h, h) = uj (x)h, where uj (x) is a real-valued function, which is periodic with respect to translations of the lattice T (j) . We assume the renormalization condition uj (x) = 0,

or M (j) (x; h, h) = 0.

(2.7)

The function mk (x) in (2.2) has the representation mk (x) = 1 −

k X j=1

where m(j) 0 (x) satisfies the bounds

−1

2 βj−1 m(j) 0 (x),

(2.8)

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1 (j) m0 (x) < β − 2 +ρ1 ξ ρ0 .

(2.9)

Let us remark that by (2.8), (2.9) we have |mk (x) − 1| < 0(1)β −1+ρ1 , hence mk (x) is arbitrarily close to 1 for β large enough. Our next inductive hypothesis describes the kernel Gk . (H.2) Gk has the representation Gk = Ck + Rk ,

(2.10)

where Ck (x, x0 ; ψ, h) =

k X j=1

Rk (x, x0 ; ψ, h) =

−1 βj−1 mk (k)G(j) x, x0 ; ψk(j) (ψ, h), h mk (x0 ),

j k X X j=1 i=1

−1 ij βi−1 R x, x0 ; ψk(j) (ψ, h), h .

The kernels G(j) (x, x0 ; ψj , h) are explicitly defined by the formulas ∗ (x, x0 ), G(j) (x, x0 ) = Hj−1 C (j−1) Hj−1

(2.11) (2.12)

(2.13)

where C (j−1) (ψj , h) is the fluctuation covariance defined at the j th step in [4], and the linearized minimizer Hj−1 (ψj , h) is defined by Hj−1 (x, y; ψj , h) =

∂ φj−1 x; ψ (j−1) (ψj , h) + ψ, h |ψ=0 , ∂ψ(y)

x ∈ T,

y∈T

(j−1)

(2.14)

.

The kernels Rij (x, x0 ; ψj , h) are real analytic on 4j (1, εj ), and satisfy the symmetry properties Rij (x, x0 , Rψj , Rh) = RRij (x, x0 ; ψj , h)R−1

for R ∈ O(N ), (2.15)

ij

0

ij

0

R (τ x, τ x , τ ψj , h) = R (x, x ; ψj , h)R

−1

for τ as in (2.4) . (2.16)

They have the usual localization expansions X Rij (x, x0 , ψj , h) = X∈Dj

Rij (x, x0 ; X; ψj , h),

(2.17)

:x,x0 ∈X

which have the properties analogous to the ones formulated after (2.5), and satisfy the bounds ij R (x, x0 ; X; ψj , h) < β − 21 +ρ1 ξ ρ0 exp(−δ0 L−i |x − x0 |1 ) exp(−κi dj (X)) (2.18) with δ0 > 0 a fixed constant. Let us remark that the main contribution to the decay properties of the two-point cf comes from the kernel Ck , Rk gives a sub-dominant decay only, as follows from the lemma below.

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Lemma 1. If γ + δ ≥ d − 2 + ε0 > 0 then j k X X j=1 i=1

1 Lγi+δj

exp −δ0 L−i |x − x0 | ≤

c . (1 + |x − x0 |)d−2+ε0

(2.19)

The proof of the lemma follows from interchanging the sums and using the well-known bound ∞ X 1 c exp −δ0 L−n |x − x0 | ≤ . Lnσ (1 + |x − x0 |)σ n=0

Let us notice that the bounds (2.18) imply the global bound |Rij (x, x0 )| < K0 β − 2 +ρ1 ξ ρ0 exp(−δ0 |x − x0 |1 ). Recall also that βn−1 < 2β −1 L−n(d−2) . Applying the lemma shows that 1

cβ − 2 +ρ1 . (1 + |x − x0 |)d−2+ρ0 3

|Rk (x, x0 )| ≤

(2.20)

The renormalization condition (2.7) on M (j) plays an important role in our analysis. It allows us to improve bounds on the functions M (j) (x, X; ψj , h) restricted to some subspaces of 4cj (X; 1; εj ). The largest subspace is 9cj (X; σ1 , ε0 ), where σ1 = K1−1 α, ε0 = εj+1 , by the result of [1]. We consider this function on smaller subspaces −1 < σ1 , ε ≤ ε0 , and we prove better 9cj (X; σ, ε), where σ ≤ σ2 = 4dK1 LM α1 bounds depending on σ, ε. Such bounds will be used in the future to improve decay properties of new contributions added at the next step. We formulate the improved bounds in the lemma. Lemma 2. The function M (j) (x, X; ψj , h) considered on the subspace 9cj (X; σ, ε) with σ ≤ σ2 , ε ≤ ε0 satisfies the bound (j) M (x, X; ψj , h) − M (j) (x, X; h, h) 1 σε exp(−(κ − 1)dj (X)). (2.21) < 0(1)β − 2 +ρ1 ξ ρ0 εj Furthermore

(j) M (x; ψj , h) < 0(1)β − 21 +ρ1 ξ ρ0 σε . εj

(2.22)

Proof. We start with the decomposition M (j) (x, X; ψj , h) = M (j) (x, X; ψj (y), h) + δy M (j) (x, X; ψj , h), y ∈ T (j) ,

x ∈ Bj (y),

(2.23)

where δy M (j) (x, X; ψj , h) is defined by the above equality and ψj (y) is the constant configuration. We obtain the bound (2.21) considering separately the cases of small and large demains X. The small X is defined by the condition dj (X) ≤ d(ρ) =

1 α , 2dK1 M ρ

ρ=

For an explanation of this condition see Lemma 2.9 of [1].

σε . ε0

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499

In this case we write Z δy M (j) (x, X; ψj , h) =

1

dt 0

d (j) M (x, X; ψj (y) + tδψj , h). dt

By Lemmas 2.3, 2.7, 2.8 of [1] this function is analytic on 9cj (σ, ε) and by a Cauchy estimate it can be bounded by 1 1 − 1 +ρ1 ρ0 β 2 ξ exp(−κdj (X)), r0 = r0 24dLM 1 +

1 2L dj (X)

σ1 ε 0 . σε

This can be bounded further by 1 1 σε exp(−(κ − 1)dj (X)), 24dL1+α M K1 β − 2 +ρ1 ξ ρ0 α εj

which contributes to the required bound (2.21). In the case of large X, i.e. dj (X) > d(ρ), we have exp(−κdj (X)) ≤ exp(−(κ − 1)dj (X))2dLα M K1

1 σε . α εj

Using the inductive bound on M (j) and estimating the two terms in δy M (j) separately we obtain the bound 1 1 σε exp(−(κ − 1)dj (X)) |δy M (j) (x, X; ψj , h)| < 24dL1+α M K1 β − 2 +ρ1 ξ ρ0 α εj (2.24)

in both cases, thus for all X. Now we consider the first term in (2.23). It depends on the variables (ψj (y), h), so it is a function of one spin variable on a subdomain of RN × RN , not on space of configurations. Using the O(N) covariance from (2.3) we have M (j) (x, X; ψj (y), h) = R1−1 M (j) (x, X; |ψj (y)|e1 , R1 h), where we have taken R1 such that R1 ψj (y) = |ψj (y)|e1 . Writing R1 h = (R1 h)1 e 1 + (R1 h)⊥ we take the rotation R2 about the e1 axis such that R2 (R1 h)⊥ = (R1 h)⊥ e2 , R2 e1 = e1 . Thus M (j) (x, X; ψj (y), h) = R1−1 M (j) x, X; |ψ(y)|e1 , (R1 h)1 e1 + (R1 h)⊥ = R1−1 R2−1 M (j) x, X; |ψ(y)|e1 , (R1 h)1 e1 + (R1 h)⊥ e2 h (R1 h)⊥ 2 e1 + x, X; |ψ(y)|, = R1−1 R2−1 m(j) 1 2 i (j) + m2 x, X; |ψ(y)|, (R1 h)1 , (R1 h)⊥ vj (R1 h)⊥ e2 2 2 (h (y), h − · ψ (y)) x, X; |ψ(y)|, h · ψ ψ0 + = m(j) 0 0 1 x, X; |ψ(y)|, h · ψ0 (y), |h|2 − (h · ψ0 (y))2 × + m(j) 2 × vj (I − ψ0 (y) ⊗ ψ0 (y)) h,

(2.25)

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T. Balaban, M. O’Carroll

where we have used R1−1 R2−1 e1 = ψ0 (y) ≡ ψ(y)|ψ(y)|−1 , R1−1 (R1 h)⊥ = h− −1 (h · ψ0 (y))ψ0 (y) = (I − ψ0 (y) ⊗ ψ0 (y))h, and R2−1 e2 = (R1 h)⊥ (R1 h)⊥ . Also 1 (R1 h)⊥ = h2 − (h · ψ0 (y))2 2 and (R1 h)⊥ = R1 h − (R1 h · e1 )e1 . (j) The functions m(j) 1 , m2 have analytic extensions in variables u, v, w, introduced in [1], see (2.87), and they are defined by the formulas 2 (j) m(j) 1 (x, X; u, v, w ) = M (x, X; ue1 , ve1 + we2 )e1 , 1 2 M (j) (x, X; ue1 , ve1 + we2 )e2 . m(j) 2 (x, X; u, v, w ) = vj w

(2.26)

The analyticity domains are as in [1] (2.89), (2.91). As in [1] we have used the (j) symmetry to conclude that m(j) 1 , m2 are even functions in the variable vw. From the inductive assumptions we obtain the bounds 1 (j) m1 (x, X; u, v, w2 ) < β − 2 +ρ1 ξ ρ0 exp(−κdj (X)), √ 1 εj+1 (j) exp(−κdj (X)) m2 (x, X; u, v, w2 ) < β − 2 +ρ1 ξ ρ0 2 −1 (K1 αεj+1 )2 √ 1 1 1 = 2K12 Lα 2 β − 2 +ρ1 ξ ρ0 exp(−κdj (X)). α εj (2.27) By Lemma 2.4 of [1] we extend analytically the expressions on the right-hand sinde of (2.23) onto the domains 9cconst (σ, ε), ε ≤ ε0 = εj+1 , 7σε0 ≤ δ0 = σ1 εj+1 = K1−1 αεj+1 , so 7σ ≤ K1−1 α. These conditions are weaker than the conditions on σ, ε in the assumptions of the proposition. Consider the second term. From (2.99), (2.100) in [1] it follows that the analytically extended expression v(I − ψ0 ⊗ ψ0 )h can be bounded by 0(1)σ 2 ε, where 0(1) is an absolute constant. By the second inequality (2.27) we bound this term by 2 0(1)K12 Lα α12 σεjε exp(−κdj (X)), which is a better bound than required in (2.21). Consider the first term. We expand the function m1(j) (x, X; u, v, w2 ) up to first order around the point u = 1, v = 1, w2 = 0, so we have (j) (j) 2 2 m(j) 1 (x, X; u, v, w ) = m1 (x, X; 1, 1, 0) + m1.1 (x, X; u, v, w )(u − 1) (j) 2 2 2 2 + m(j) 1,2 (x, X; u, v, w )vj (v − 1) + m1.3 (x, X; u, v, w )vj w . (2.28)

By the maximum principle the functions m1j (j) satisfy the bounds 1 1 1 (j) exp(−κd(X)), m1.1 (x, X; u, v, w2 ) < 3β − 2 +ρ1 ξ ρ0 K1 Lα α εj 2 1 (j) 2 − 21 +ρ1 ρ0 α1 K1 L ξ exp(−κd(X)), m1.2 (x, X; u, v, w ) < β α ε2j 4 1 1 (j) 2 − 21 +ρ1 ρ0 (x, X; u, v, w ) < 2β ξ L2α 2 exp(−κdj (X)). K m1.3 1 α εj Using these bounds and the bounds (2.103) from [1] for the analytically extended expressions we bound the last three expressions in (2.28) by the required bound

Low Temperature Properties for Correlation Functions

501

(2.21) or even a better one with an additional factor σ. We are left with the terms (j) m(j) 1 (x, X; 1, 1, 0) = M (x, X; e1 , e1 )e1 , so we obtain the first inequality (2.21). To obtain the second we have to resum over domains X, and then we have X

M (j) (x, X; e1 , e1 ) · e1 = M (j) (x, e1 , e1 ) · e1 = 0

X∈Dj :x∈X

by the renormalization condition (2.7).

The above proposition will be used in the case where σ = C0 Lj η, ε = εk and then d εk σε = C0 Lj η = C0 (Lj η)1+α = C0 (Lj η) 2 −γ . εj εj

3. Inductive Asumptions Preserved by the RGT The k + 1st step gives three kinds of contributions to the generating functional. The first comes from the fluctuation integral, the second comes from the renormalization, the third from R(k+1) , which is described in hypothesis (H.4) in [5] and arises from the renormalization of the large field densities. We start with the analysis of the contribution coming from the fluctuation integral. We need to consider only the whole in lattice fluctuation tegral. This integral is determined by the measure Z (k)−1 exp −βk V (k) + δEk dµ0 (ψ), Q where dµ0 (ψ) = x∈T k exp −1/2|ψ(x)|2 dψ(x)/(2π)N/2 . This measure has been analysed and its properties have been discussed in [4]. We denote the expectation value by h i. Let us recall that by Theorem 2 of [4] the only effect of the renormalization transformation on the “old” effective action and generating functional is that the variable ψk is replaced by the background configuration ψ (k) (θ). The transformation adds the new contributions E0(k+1) (θ), F0(k+1) (θ, g) to the previous ones. We have to analyse only these. The 1st order in g added contribution is −1/2

βk

M0(k+1) ≡ hδMk i = mk hδφk i +

k X j=1

−1/2

βj−1 hδM (j) i,

(3.1)

where we have written −1/2 δφk = φk ψ (k) (θ) + βk ψ 0 − φk ψ (k) (θ) , −1/2 (j) (θ) , δM (j) = M (j) ψk(j) ψ (k) (θ) + βk ψ 0 − M (j) ψk+1

(3.2) ψ 0 = C (k)1/2 ψ,

where C (k) is the covariance given in [4]. ψ and θ satisfy the restrictions as in [4], namely |ψ| < p1 (βk ) and θ ∈ 4ck+1 (1, εk+1 ). The 2nd order in g added contribution is

502

T. Balaban, M. O’Carroll

hδGk i + hδMk ; δMk iT =

k X j=1

−1 βj−1 mk hδG(j) imk +

j k X X j=1 i=1

−1 βi−1 hδRij i +

  +T * k k X X −1/2 −1/2 βj−1 δM (j)  ; mk δφk + βj−1 δM (j)  + mk δφk + j=1

= βk−1 mk G(k+1) mk +

j=1

k+1 X i=1

−1 i,k+1 βi−1 R0 ,

(3.3)

where δG(j) , δRij are defined analogously to (3.2). The components Rij of Rk+1 with 1 ≤ i ≤ j ≤ k are equal to the previous ones discussed in (H.2), only with ψ replaced by ψ (k) (θ). We now identify the fluctuation integral contributions R0i,k+1 , i ≤ k + 1, as follows: R0i,k+1 ≡ R1i,k+1 + R2i,k+1 ,

i = 1, 2, . . . , k,

(3.4)

where R1i,k+1 = mk hδG(i) imk , R2i,k+1 =

k X

hδRij i,

(3.5)

(3.6)

j=1

and we write R0k+1,k+1 = R1k+1,k+1 + R2k+1,k+1 + R3k+1,k+1 ,

(3.7)

R1k+1,k+1 = −mk G(k+1) mk + βk hmk δφk ; δφk mk iT ,

(3.8)

where

R2k+1,k+1 =

k X j=1

−1/2

2βk hmk δφk ; βj−1 δM (j) iT

(3.9)

and R3k+1,k+1 =

k X i,j=1

−1/2 −1/2

βk βi−1 βj−1 hδM (i) ; δM (j) iT

(3.10)

We now bound the added contributions using the localization expansion [5]. Let us formulate a result on this expansion in a form we need here. Proposition 3.1. Consider a function F (X; ψj , h), X ∈ Dj , and assume it is analytic on a domain containing values of the function ψk(j) (ψ (k) (θ) + zψ 0 ) for (θ, h) ∈ 4ck+1 (1, ε), εk+1 ≤ ε ≤ εk+1 + 21 (εk − εk−1 ), the fluctuation field satisfying the bound |ψ| < p1 (βk ) = A1 (log βk )p1 and z in a disc |z| < r0 . Assume also that it satisfiesthe bound |F (X; ψj , h)| < F0 exp(−(κ − 1)dj (X)) on this domain and consider hF X; ψk(j) (ψ (k) (θ) + zψ 0 ), h iβ −1/2 σ , where h · iβ −1/2 σ means the expectation value k

k

Low Temperature Properties for Correlation Functions

503

−1/2

extended analytically in βk by introducing the complex parameter σ, as discussed 1/2 in [5] and where |σ| < r1 = cc1 βk /p1 (βk ), c1 is the constant in Prop. 1.1 of [2] and c > 0 is a small absolute constant. The expectation value of F has a localization expansion D X E −1/2 = F X, Y ; θ, h, β σ, z , F X; ψk(j) ψ (k) (θ) + zψ 0 , h k −1/2 βk

σ

Y ∈Dk+1 ;Y ⊃X

(3.11) where F (X, Y ; θ, h) are analytic functions of their arguments on the domains described above depending on θ, h localized to Y and satisfying the bounds −1/2 F (X, Y ; θ, h, βk σ, z) < CF0 exp(−κ0 dj (X)) exp(−2κdk+1 (Y )), (3.12) where C is an absolute constant depending on d only. If we have two functions F1 (X1 ; ψi , h), F2 (X2 ; ψj , h) satisfying the above assumptions for indices i, j, then the truncated expectation value of the product has the expansion D ET F1 X1 ; ψk(i) ψ (k) (θ) + z1 ψ 0 , h ; F2 X2 ; ψk(j) ψ (k) (θ) + z2 ψ 0 , h −1/2 σ βk X −1/2 = F X1 , X2 , Y ; θ, h, βk σ, z1 , z2 , (3.13) Y ∈Dk+1 ;Y ⊃X1 ∪X2

where

−1/2 F X1 , X2 , Y ; θ, h, βk σ, z1 , z2

< CF1 F2 exp (−κ0 di (X1 )) exp −κ0 dj (X2 ) exp (−2κdk+1 (Y )) .

Let us make some remarks on this proposition. The analyticity radius r0 has not been specified. There will be three cases. The first arises in connection with explicitly defined functions, like minimal functions φk and G(j) . Their analyticity domains are quite large, 1 . In the second determined by the constant c1 , and in this case we take r0 = cc1 p1 (β k) c case we consider functions analytic on 4j (1, εj ), and then, by the assumption on θ, we 1 1 , or rather its lower bound r0 = cαεj+1 p1 (β . In the third take r0 = c 21 εj − εk+1 p1 (β k) k) c 1 j . case we consider functions on 9j (C0 L η, εk ), and then we take r0 = cαεk+1 p1 (β k) The second remark concerns the expansion (3.13) of the truncated product of two functions. This can be generalized to truncated products of many functions, but we need here only the case of n = 2. The statement (3.13) of the above proposition has not been proved in [5]. It has been proved that if we take the expectation value of the product of the two functions F1 , F2 , then it has the representation (3.13), but with the sum over all domains Y containing X1 , X2 , which are unions of large cubes. These domains need not to be connected, and if Y = Y1 ∪ Y2 , Y1 ⊃ X1 , Y2 ⊃ X2 , Y1 , Y2 are connected components then −1/2 F X1 , X2 , Y ; θ, h, βk σ, z1 , z2 −1/2 −1/2 = F1 X1 , Y1 ; θ, h, βk σ, z1 , F2 X2 , Y2 ; θ, h, βk σ, z2 ,

504

T. Balaban, M. O’Carroll

−1/2 where Fi Xi , Yi ; θ, h, βk σ, zi , , i = 1, 2, are the terms of the expansions (3.11) for the expectation values of the functions Fi . From such an expansion it follows immediately that if we take the truncated expectation value, then the terms corresponding to disconnected domains Y are cancelled, and we obatain (3.13). We start with an analysis of (3.1). Our goal is to show that this function satisfies the inductive hypothesis (H.1), except for the renormalization condition (2.7). We have to −1/2 from the expectation values in (3.1). For the function in the first extract the factor βk one we write −1/2 −1/2 −1/2 φk x; ψ (k) (θ) + βk ψ 0 − φk x; ψ (k) (θ) = βk 0k x; βk ψ 0 ψ 0 , (3.14) where Z 0k (x; ψ)δψ =

1

dt 0

1 2πi

Z |τ |=r

dτ φk x; ψ (k) (θ) + tψ + τ δψ . 2 τ

cc1 . We have to construct a localization expansion of The radius r can be taken as r = |δψ| −1/2 0 h0k x; βk ψ ψ 0 i. To obtain proper bounds for terms of this expansion we have to −1/2

take an expansion in βk to the second order, and use the fact that the first order term is 0. We have E D −1/2 0k x; βk ψ 0 −1/2 βk

Z

E d D −1/2 0k (x; βk sψ 0 )ψ 0 −1/2 ds βk s 0 Z Z 1 D E dσ 1 −1/2 0k (x; βk σψ 0 )ψ 0 −1/2 = ds 2 2πi (σ − s) βk σ |σ|=r1 0 Z Z 1 Z 1 dσ 1 1 ds dt = 2 2πi (σ − s) 0 2πi 0 Z E dτ D −1/2 (k) 0 0 (θ) + tβ σψ + τ ψ , × φ x; ψ k k −1/2 τ2 βk σ Hk (x) = 0k (x; 0), = Hk (x)hψ 0 i0 +

1

ds

(3.15)

where we have used the equality hψ 0 i0 = C (k)1/2 hψi0 = 0. We apply Proposition 3.1 to the last expectation value above. It corresponds to j = 0, X = {x}, and the function F (F ; φ) = φ(x), which obviously satisfies all the assumptions. We take the complex −1/2 1 . We obtain the expansion (3.11) and parameter z = tβk σ + τ , and r = cc1 p1 (β k) applying the integrations in (3.15) we obtain an expansion of the hδφk (x)i. We write it in the form X −1/2 8k (x, Y ; θ, h), (3.16) mk (x)hδφk (x)i = βk Y ∈Dk+1 ;Y ∈x

where by (3.12), bounding the integrals in (3.15) in the obvious way and using the bound |φ| < 23 ,

Low Temperature Properties for Correlation Functions

505

1 3 r1 C exp(−2κdk+1 (Y )) (r1 − 1)2 r 2 C p21 (βk ) exp(−2κdk+1 (Y )) <3 (cc1 )2 β 1/2

|8k (x, Y ; θ, h)| <

k

−1/2+2ρ2

d−2

η 2 −2(d−2)ρ2 exp(−2κdk+1 (Y )) < 0(1)β d−2 1 < β −1/2+3ρ2 η 2 −2(d−2)ρ2 exp(−2κdk+1 (Y )), 2

(3.17)

small. We have used here the inductive assumpwhere ρ2 > 0 can be chosen arbitrarily tion on βk , i.e. βk = 0 βLk(d−2) , bound on the logarithmic function p1 (β) ≤ 0(1)β ρ2 , and β large enough so that β −ρ2 < 21 . Consider now a term hδM (j) (x)i in (3.1). We apply the localization expansion (2.5) to the function M (j), and we write the function δM (j) (x, X) in the form −1/2 −1/2 x, X; β ψ ψ0 , (3.18) δM (j) (x, X) = βk 0(j) k k where

Z

0(j) k (x, X; ψ)δψ

1

= 0

1 dt 2πi

Z

This yields the representation

−1/2 δM (j) (x) = βk

|τ |=r

dτ (j) (j) (k) x, X; ψ M (θ) + tψ + τ δψ . ψ k τ2

X X∈Dj :X∈x

D

E −1/2 0k(j) x, X; βk ψ 0 ψ 0 ,

(3.19)

and we obtain E D −1/2 0 x, X; β ψ0 ψ 0(j) k k Z Z 1 E dτ D (j) 1 −1/2 M x, X; ψk(j) ψ (k) (θ) + tβk ψ 0 + τ ψ 0 dt = 2 2πi |τ |=r τ 0 Z Z 1 X 1 dτ (j) −1/2 dt M + τ x, X, Y ; θ, h, tβ = k 2πi |τ |=r τ 2 0 (3.20) Y ∈Dk+1 :Y ⊃X

by a straightforward application of Propositon 3.1. We want to use the improved bounds (2.21) in Lemma 2 for the function M (j) (x, X), so we must take the analyticity domain c 1 . 9j C0 Lj η, εk , and the function ψk(j) inside has values in this space if r = cαεk+1 p1 (β k) A term of the expansion on the right-hand side of (3.20) can be bounded by d 1 C0(1)β −1/2+ρ1 ξ ρ0 C0 (Lj η) 2 −γ exp(−κ0 dj (X)) exp(−2κdk+1 (Y )) r d ≤ 0(1)β −1/2+ρ1 +ρ2 ξ ρ0 − 2 +γ η 1−(d−2)ρ2 exp(−κ0 dj (X)) exp(−2κdk+1 (Y )). (3.21)

For the sum over j in (3.1) we obtain the representation k X j=1

−1/2

βj−1

−1/2 δM (j) (x) = βk

X Y ∈D:Y 3x

M1(k+1) (x, Y ; θ, h),

506

T. Balaban, M. O’Carroll

where M1(k+1) (x, Y ; θ, h) =

k X j=1

−1/2 βj−1

Z

X X∈Dj :x∈X⊂Y

0

1

1 dt 2πi

Z |τ |=r

dτ (j) −1/2 M + τ . x, X, Y ; θ, h, tβ k τ2 (3.22)

From the bound (3.21) we obtain the following bound: k X (k+1) 0(1)K0 β −1+ρ1 +ρ2 ξ −1+ρ1 +γ η 1−(d−2)ρ2 exp(−2κdk+1 (Y )) M1 (x, Y ; θ, h) < j=1

≤

k X

0(1)β −1+ρ1 +ρ2 ξ γ−(d−2)ρ2 η ρ0 exp(−2κdk+1 (Y ))

j=1

≤ 0(1)β −1+ρ1 +ρ2 (L−1 η)ρ0 exp(−2κdk+1 (Y )),

(3.23)

where we have used the simple inequality η 1−ρ0 −(d−2)ρ2 ≤ ξ 1−ρ0 −(d−2)ρ2 , assuming 1 − ρ0 − (d − 2)ρ2 ≥ 0 and γ − (d − 2)ρ2 > 0. We combine the two expansions (3.16), (3.22), and we obtain a localization expansion of the function M0(k+1) (x; θ, h). A term of this expansion is equal to 8k (x, Y ; θ, h) + M1(k+1) (x, Y ; θ, h), and by (3.17), (3.23) it satisfies the bound required by the inductive assumption, if we take β large enough, so , 1 − 2(d − 2)ρ2 . This ends the that 0(1)ρ−1/2+ρ2 < 21 and if ρ1 ≥ 3ρ2 , ρ0 ≤ min d−2 2 analysis of the function (3.1). Now we analyse contributions to the second order terms. We start with R1i,k+1 given by (3.5), or more explicitly by

(3.24) R1i,k+1 (x, x0 ; θ, h) = mk (x) δG(i) (x, x0 ) mk (x0 ). Taking representation (2.13), the localization expansions of Hi−1 , C (i−1) constructed in [3], we obtain the localization expansion X G(i) (x, x0 , X; ψi , h), G(i) (x, x0 ; ψ 0 , h) = X∈Di :X⊃{x,x0 }

(i) G (x, x0 , X; ψi , h) < G0 exp −δ0 L−1 |x − x0 |1 ) exp(−κdi (X) . (3.25) The functions above are defined in connection with the basic variational problem, and the analyticity domain is quite large, independent of i and determined by the constant c1 . We apply the expansion (3.25) in the expectation value (3.24), and we have X

(i)

(i) δG (x, x0 , X) δG (x, x0 ) = X∈Di :X⊃{x,x0 }

X

=

X∈Di :X⊃{x,x0 }

D

Z 0

1

1 2πi

Z

|σ|=r1

dσ (σ − t)2 −1/2

× G(i) x, x0 , X; ψk(i) ψ (k) (θ) + βk

σψ 0

E

. (3.26)

Low Temperature Properties for Correlation Functions

By (3.11), (3.12) we have

(j) δG (x, x0 , X) =

X

507

G(i,k+1) (x, x0 , X, Y ; θ, h),

(3.27)

Y ∈Dk+i : Y ⊃X

where

(i,k+1) G (x, x0 , X, Y ; θ, h) <

r1 CG0 exp(−δ0 L−i |x − x0 |1 ) (r1 − 1)2 × exp(−κ0 di (X)) exp(−2κdk+1 (Y )) 1 1/2 β . and r1 = cc1 p1 (βk ) k

Resumming over X : {x, x0 } ⊂ X ⊂ Y gives X R1i,k+1 (x, x0 ; θ, h) =

R1i,k+1 (x, x0 , Y ; θ, h),

Y ∈Dk+1 :{x,x0 }⊂Y

where

i,k+1 R1 (x, x0 , Y ; θ, h) < 0(1)β −1/2−ρ2 η

d−2 2 −(d−2)ρ2

exp −δ0 L−i |x − x0 |1 exp (−2κdk+1 (Y )) , (3.28)

which is a better bound than required by the inductive assumption (2.18). The case of R2i,k+1 given by (3.6) is treated in a similar way. We have X

i,j δR (x, x0 ) = Z ×

1

1 2πi X

dt 0

=

X∈Dj :{x,x0 }⊂X

Z

|σ|=r

E dσ D ij −1/2 R x, x0 , X; ψk(j) ψ (k) (θ) + βk σψ 0 2 (σ − t) X Ri,j (x, x0 , X, Y ; θ), (3.29)

X∈Dj :{x,x0 }⊂X Y ∈Dk+1 :Y ⊃X

using the same representation as in (3.26), and Proposition 3.1. The only difference is that the functions Rij (x, x0 , X; ψj ) have the analyticity domain 4cj (1, εj ), so we take 1/2

1 β . A term of the last expansion has the bound the radius r = cαεj+1 p1 (β k) k i,j R (x, x0 , X, Y ; θ) r Cβ −1/2+ρ1 ξ ρ0 exp −δ0 L−i |x − x0 |1 exp −κ0 dj (X) exp (−2κdk+1 (Y )) < 2 (r − 1) d−2 d−2 < 0(1)β −1+ρ1 +ρ2 ξ ρ0 − 2 +γ η 2 −(d−2)ρ2 exp −δ0 L−i |x − x0 |1 exp (−2κdk+1 (Y )) . (3.30) d−2

d−2

2 −ρ0 −(d−2)ρ2 ≤ ξ 2 −ρ0 −(d−2)ρ2 Assuming that ρ0 ≤ d−2 2 − (d − 2)ρ2 we estimate η −1+ρ1 +ρ2 γ−(d−2)ρ2 ρ0 and we obtain the factor 0(1)β ξ η in front of the exponentials. Resumming over domains X ∈ Dj such that {x, x0 } ⊂ X ⊂ Y , and over j from i to k we obtain a term of a localization expansion of R2i,k+1 (x, x0 ). This term can be bounded by

508

T. Balaban, M. O’Carroll

0(1)β −1+ρ1 +ρ2 (L−1 η)ρ0 exp −δ0 L−i |x − x0 |1 exp (−2κdk+1 (Y )) , which for β sufficiently large is better than the bound required by the inductive assumptions. Now we consider the expectation values of the products of the two functions, included into the terms R1k+1,k+1 , R2k+1,k+1 , R3k+1,k+1 . We start with the term R1k+1,k+1 . We consider T

hδφk (x) ⊗ δφk (x0 )i ≡ hδφk (x) ⊗ δφk (x0 )i − hδφk (x)i ⊗ hδφk (x0 )i . (3.31) We used the representation (3.14) for δφk so we obtain T

hδφk (x) ⊗ δφk (x0 )i D E −1/2 −1/2 ≡ βk−1 0k x; βk ψ 0 ψ 0 ⊗ 0k x0 ; βk ψ 0 ψ 0 E D E D −1/2 −1/2 − 0k x; βk ψ 0 ψ 0 ⊗ 0k x0 ; βk ψ ψ D ET −1/2 −1/2 = βk−1 0k x; βk ψ ψ; 0k x0 ; βk ψ ψ 0 .

(3.32)

−1/2

The function in the parenthesis is an analytic function of βk , and we extend it to the −1/2 −1/2 function of βk σ as before. If we take βk σ = 0, we obtain T

h0k (x; 0)ψ 0 ; 0k (x0 ; 0)ψ 0 i0 = Hk (x)hψ 0 ; ψ 0 iT0 Hk∗ (x) = Hk (x)C (k) Hk∗ (x0 ), (3.33) which is equal to G(k+1) (x, x0 ), after multiplication by mk (x)mk (x0 ). Thus expanding −1/2 this function up to the first order in βk σ we obtain the formula for R1k+1,k+1 , R1k+1,k+1 (x, x0 ; θ)

Z

1

= 0

1 dt 2πi

Z |σ|=r1

D dσ −1/2 0 x; β ψ0 ; m (x) 0 σψ κ k κ (σ − t)2 (3.34)

ET −1/2 0k x0 ; βk σψ 0 ψ 0 −1/2 mk (x0 ) Z = 0

1

1 dt 2πi

Z

βk

|σ|=r1

σ

dσ (σ − t)2

Z

1

dt1 0

1 2πi

Z |τ1 |=r

dτ1 τ12

Z

1

dt2 0

1 2πi

Z |τ2 |=r

dτ2 . τ22

D −1/2 mk (x) φk x; ψ (k) (θ) + t1 βk σψ 0 + τ1 ψ 0 ; ET −1/2 mk (x0 ), φk x0 ; ψ (k) (θ) + t2 βk σψ 0 + τ2 ψ 0 −1/2 βk

r1 = cc

1 1/2 β , p1 (βk ) k

r = cc1

σ

1 . p1 (βk )

We have to construct a localization expansion for this function, and prove the proper bounds (2.18). This is again obtained by applying Proposition 3.1. We apply (3.13) to −1/2 the functions F1 (x, φ) = φ(x), F2 (x0 , φ) = φ(x0 ), where i = j = 0 and zl = tl βk σ + τl , l = 1, 2. We obtain

Low Temperature Properties for Correlation Functions

509

D

−1/2 φk x; ψ (k) (θ) + t1 βk σψ 0 + τ1 ψ 0 ; ET −1/2 φk x0 ; ψ (k) (θ) + t2 βk σψ 0 + τ2 ψ 0 −1/2 σ βk X −1/2 0 = 8k x, x , Y ; θ, βk σ, z1 , z2 ,

(3.35)

Y ∈Dk+1 :Y ⊃{x,x0 }

where

−1/2 8k x, x0 , Y ; θ, βk σ, z1 , z2 < C(3/2)2 exp(−2κdk+1 (Y )).

We substitute the above expansion into (3.34) and we obtain X R1k+1,k+1 (x, x0 , Y ; θ). R1k+1,k+1 (x, x0 ; θ) =

(3.36)

X∈Dκ+1 :X⊃{x,x0 }

A term in this expansion is obtained by applying the integrations in (3.34) to the corresponding term in (3.35), therefore it can be bounded by d−2 1 r1 9C exp(−2κdk+1 (Y )) < 0(1)β −1/2+3ρ2 η 2 −3(d−2)ρ2 exp(−2κdk+1 (Y )) 2 2 (r1 − 1) r d−2 −3(d−2)ρ2 exp(−δ0 L−(k+1) |x − x0 |1 ) exp(−κdk+1 (Y )) ≤ 0(1)β −1/2+3ρ2 L−1 η 2 (3.37)

We have used the definition of dk+1 (Y ) and the restriction {x, x0 } ⊂ Y , which give us the bound M dk+1 (Y ) + 2dLM ≥ |x − x0 |L−1 η = L−(k+1) |x − x0 |1 , hence e−κdk+1 (Y ) ≤ κ −(k+1) L |x − x0 |1 , and this yields the last inequality in (3.37) with δ0 = e2dLκ exp − M k+1,k+1 κ with the required M . Thus we have constructed the localization expansion of R1 d−2 bound, assuming ρ1 ≥ 3ρ2 , ρ0 ≤ 2 − 3(d − 2)ρ2 . Consider next the term R2k+1,k+1 given by (3.9). We use again the representations (3.14), (3.18) and we obtain R2k+1,k+1 (x, x0 ; θ) = 2

=2

k X

k X

Z

X

j=1

D ET −1/2 −1/2 −1/2 0 0 βj−1 mk (x) 0k x; βk ψ 0 ψ 0 ; 0(j) , β ψ x ψ0 k k

1

dt1

j=1 X∈Dj :X∈x

0

1 2πi

Z |τ1 |=r

dτ1 τ12

Z

1

dt2 0

1 2πi

Z |τ2

|=r 0

dτ2 −1/2 β . τ22 j−1

−1/2 mk (x)φk x; ψ (k) (θ) + t1 βk ψ 0 + τ1 ψ 0 ; ET −1/2 , M (j) x, X; ψk(j) ψ (k) (θ) + t2 βk ψ 0 + τ2 ψ 0

D

r = cc1

1 , p1 (βk )

r0 = cαεk+1

1 . p1 (βk )

(3.38)

510

T. Balaban, M. O’Carroll −1/2

We apply Proposition 3.1 with i = 0, F1 (x, φ) = mk (x)φ(x), F2 (X, ψj ) = βj−1 M (j) (x, X; ψj ), and we obtain a localization expansion of the expectation value in the last expression above, with terms bounded by −1/2

βj−1 0(1)β −1/2 ξ ρ0 Lj η

d2 −γ

exp −κ0 dj (X) exp (−2κdk+1 (Y )) .

To these terms we apply the integrations in (3.38), we resum them over X ∈ Dj , x0 ∈ X ⊂ Y , and finally over j from 1 to k. We obtain a localization expansion of the form (3.36), where |R2k+1,k+1 (x, x0 , Y ; θ)| <

k X

0(1)

j=1

<

k X

p21 (βk ) −1/2 −1/2+ρ1 ρ0 j d2 −γ β β ξ L η exp (−2κdk+1 (Y )) εκ+1 j−1

0(1)β −1+ρ1 +2ρ2 ξ ρ0 −1+γ η 1−2(d−2)ρ2 exp (−2κdk+1 (Y ))

j=1

< 0(1)β −1+ρ1 +2ρ2 L−1 η

ρ0

exp −δ0 L−(k+1) |x − x0 |1 exp (−κdk+1 (Y )) , (3.39)

where we have used the same bounds as in (3.23) and (3.37). This bound is smaller than the required bound (2.18) if we take 0(1)β −1/2+2ρ2 small enough. Consider briefly the term R3k+1,k+1 given by (3.10). It is handled in almost exactly the same way as the previous term. We write R3k+1,k+1 (x, x0 ; θ) = D

k X

X

X

i,j=1 X∈Di : X3x X 0 ∈Dj : X 0 3x0

−1/2

0(j) x, X; βk k

−1/2 −1/2

βi−1 βj−1

ET −1/2 0 0 0 ψ 0 ψ 0 ; 0(j) , X ; β ψ x ψ0 , k k

(3.40)

and we repeat the analysis used for (3.38), with the obvious modifications. We obtain a localization expansion of the form (3.36), where |R3k+1,k+1 (x, x0 , Y ; θ)| <

k X i,j=1

d

2 −1/2 −1/2 p1 (βk ) −1+2ρ1 β (L−i )ρ0 . ε2k+1

0(1)βi−1 βj−1

(3.41)

d

(Li η) 2 −γ (L−j )ρ0 (Lj η) 2 −γ exp(−2κdd+1 (Y )) 5

k X

0(1)β −2+2ρ1 +2ρ2 (L−i )ρ0 −1+γ (L−j )ρ0 −1+γ η 2−2(d−2)ρ2 exp(−2κdk+1 (Y ))

i,j=1

5 0(1)β −2+2ρ1 +2ρ2 (L−1 η)2ρ0 exp(−δ0 L−(k+1) |x − x0 |1 ) exp(−κdk+1 (Y )).

Low Temperature Properties for Correlation Functions

511

From the analysis of all the above cases we obtain the following restrictions on ρ0 , ρ1 , ρ2 , γ : ρ2 > 0, d−2 , 1 − 3(d − 2)ρ2 , 0 < ρ0 5 min 2 d−2 , 1 , γ − 2(d − 2)ρ2 > 0. 0 < γ < min 2 ρ1 = 3ρ2 ,

We may satisfy all of them by choosing ρ2 sufficiently small. This way we have finished the analysis of the new contributions coming from the fluctuation integral, and we have proved that they satisfy all the inductive hypotheses, except the renormalization condition (2.7) in (H.1). The remaining two contributions mentioned at the beginning of this section are considered together. In the inductive hypothesis (H.1) in [5] there is the term R(k+1) described in (H.4). After calculating the fluctuation integral we obtain R(k+1) which has . This term is roughly equal to an expectation value of R(k) an additional term R(k+1) 0 1 , and it has the same symmetry and other properties as E0(k+1) and F0(k+1) (g), which are the new contributions to the effective action and the generating functional. We expand (g) up to the first order in g writing R(k+1) 0 (k+1) (k+1) (g) = ER + FR (g), R(k+1) 0

(k+1) FR (g) = hg, M(k+1) R (g)i,

(3.42)

and we add the corresponding terms to E0(k+1) , F0(k+1) (g). Let us keep the same notations for the sums. The effective action Ek (ψ (k) (θ)) + E0(k+1) (θ) determines the renormalization procedure, in particular the renormaization “counterterms” and the renormalized coefficients βk+1 , αk+1 , λk+1 , νk+1 . The effect of the renormalization procedure on the generating function is that it changes the background configurations replacing them by the configuration determined by the renormalized coefficients. Let us denote the previous configurations by the subscript “u”, indicating the unrenormalized coefficients. We have the expansions φk+1,u = φk+1 + δφk+1 ,

αk+1,u = αk+1 + δαk+1 ,

(j) (j) (j) ψk+1,u = ψk+1 + δψk+1

(3.43)

(j) are determined by the renormalization condescribed in [4], where δφk+1 , δαk+1 , δψk+1 stants. The renormalization and the above expansions introduce new terms into the generating functional. Based on the formula (5.18) of [4] we obtain the following formulas for the first and second order contributions: 1/2

Mu(k+1) (ψk+1 , h) = M0(k+1) (zk+1 ψk+1 , h) + βk mk δφk+1 k h i X (j) (j) (j) −1 1/2 + (βk βj−1 ) + δψk+1 M (j) ψk+1 − M (j) ψk+1 j=1

1/2

(k+1) (zk+1 ψk+1 ) , + β k MR

(3.44)

512

T. Balaban, M. O’Carroll

(k+1) MR (θ) = M(k+1) R (θ, g = 0), k+1 X i=1

+

−1 i,k+1 βi−1 R0 (zk+1 ψk+1 ) + βk−1 mk G(k+1) (zk+1 ψk+1 )−G(k+1) (ψk+1 ) mk

k X j=1

+

(j) (j) (j) −1 βj−1 mk G(j) ψk+1 + δψk+1 − G(j) ψk+1 mk

j k X X j=1 i=1

(j) (j) (j) −1 Ri,j ψk+1 − Ri,j ψk+1 βi−1 + δψk+1

(k+1) (zk+1 ψk+1 ) , + βk−1 RR

(3.45)

(k+1) (θ) = βk RR

∂ M(k+1) (θ, g = 0), ∂g R

where βk−1 mk G(k+1) (ψk+1 )mk is the new addition to Ck . We remark that the sum (3.45) Pk+1 −1 i,k+1 R in Rk+1 . Now we have to see that the new contributes to the sum i=1 βi−1 contributions have the same properties as the contributions created by the fluctuation integral. This is in principle a simpler problem, we do not have the fluctuation integrals, just functions of the new variables which we have to localize and bound. This is mostly a repetition of the previous analysis, so we discuss a couple of cases as an illustration. −1 1 The function δφk+1 (x) satisfies the bound |δφk+1 | < 0(1)βk+1 εk+1 and it has a localization expansion X δ8k+1 (x, Y ), (3.46) δφk+1 (x) = Y ∈Dk+1 ;Y 3x

whose terms satisfy the bound |δ8k+1 (x, Y )| < 0(1) (βk+1 εk+1 )−1 exp (−2κdk+1 (Y )), 1/2 hence the term βk mk δφk+1 has a localization expansion whose terms are bounded by −1/2 −1 γ 0(1)β L η exp (−2κdk+1 (Y )), as required, assuming that ρ0 ≤ γ. The localization expansion is discussed in [5]. The localization expansions and bounds for the term of the sum in (3.44) are obtained in a similar way. We take the localization expansion of the function M (j) , and we consider the difference

−1 βk βj−1

1/2

(j) (j) (j) + δψk+1 M (j) x, X; ψk+1 − M (j) x, X; ψk+1 ,

which can be bounded by 0(1)β −1/2+ρ1 ξ ρ0 Lj η

d2 −γ− d−2 2

< 0(1)β −3/2+ρ1 ξ ρ0 Lj η

1−γ

−1 1 exp −(κ − 1)dj (X) βk+1 ε2k+1 2γ exp −(κ − 1)dj (X) . L−1 η

The difference has the localization expansion and terms of the expansion are bounded by 0(1)β − 2 +ρ1 ξ ρ0 L−1 η 3

2γ

exp −κ0 dj (X) exp (−2κdk+1 (Y )) .

Low Temperature Properties for Correlation Functions

513

Resumming over X : x ∈ X ⊂ Y , and over j = 1, . . . , k we obtain a localization expansion for the whole sum in (3.44). Terms of this expansion are bounded by 2γ 3 exp (−2κdk+1 (Y )) . 0(1)β − 2 +ρ1 L−1 η The last term in (3.44) is the simplest one. We obtain its localization expansion directly from the expansion of R(k+1) in (H.4) of [5], thus 0 ∂ (k+1) (k+1) R (Y ; θ, g = 0), MR (x, Y ; θ) = ∂g(x) 0 and we obtain immediately the bound 1/2

(k+1) (x, Y ; zx+1 ψk+1 )| |βk MR

< βk−1 exp(−2κdk+1 (Y )) < 2β −1 L−1 η

d−2

exp (−2κdk+1 (Y )) .

Thus the whole contribution (3.44) satisfies the inductive assumptions, if β is sufficiently large. −1 mk Consider now the second sum in (3.45). It is easy to see that βj−1 (j) (j) (j) −1 × G(j) ψk+1 + δψk+1 − G(j) ψk+1 mk contributes to the term βj−1 Rj,k+1 . It has the localization expansion (3.25), and the term (j) (j) (j) −1 mk (x) G(j) x, x0 , X; ψk+1 + δψk+1 − G(j) x, x0 , X; ψk+1 mk (x0 ) βj−1 (3.47) can be bounded by 1 −1 9 −1 G0 exp −δ0 L−j |x − x0 |1 exp −κdj (X) 0(1)βk−1 . βj−1 4 εk+1 εj+1 Taking the localization expansion of (3.47), and resumming over X, we get a localiza−1 Rj,k+1 . tion expansion satisfying bounds characterizing a term of an expansion of βj−1 Similarly for the terms of the last sum in (3.45) we take the localization expansion of Ri,j , and we get the sum over i, j, X, of the terms (j) (j) (j) −1 + δψk+1 Ri,j x, x0 , X; ψk+1 − Ri,j x, x0 , X; ψk+1 . (3.48) βi−1 Such a term can be bounded by 1 εk+1 εj+1 ρ0 +γ . L−1 η

−1 −1 0(1)β −1/2+ρ1 ξ ρ0 exp(−δ0 L−i |x − x0 |1 ) exp(−κdj (X))βk+1 βi−1 −1 < βi−1 0(1) exp(−δ0 L−i |x − x0 |1 ) exp(−κdj (X))β −3/2+ρ1 ξ γ

Taking the localization expansions of (3.48), and resumming over X ∈ Dj : {x, x0 } ⊂ X ⊂ Y , and over j = i, i+1, . . . , k, we get a localization expansion whose terms satisfy −1 i,k+1 R . bounds characteristic of the expansion for the term βi−1 The last term in (3.45) is treated similarly to the last term in (3.44), the difference is (k+1) is given by the second derivative with respect to that a term of the expansion of RR . Its bound is g of the corresponding term of the expansion of R(k+1) 0

514

T. Balaban, M. O’Carroll −1

(k+1) |RR (x, x0 , Y ; zk+1 ψk+1 )| < 4βk 2 exp(−2κdk+1 (Y ))

< 8L

d−2 2

e2dLκ β − 2 (L−1 η) 1

d−2 2

exp −δ0 L−(k+1) |x − x0 |1 exp(−κdk+1 (Y )).

We have finished the analysis of the sum (3.45), and we have shown that it contributes Pk+1 −1 i,k+1 R , in agreement with the inductive hypothesis (H.2). terms to the sum i=1 βi−1 Thus the added contributions satisfy (H.1), (H.2) except for the renormalization condition (2.7), to which we now turn. Consider Mu(k+1) (x; ψk+1 , h) with ψk+1 = h. By the rotational symmetry (2.3) there (x) defined by exists a scalar function m(k+1) 0 (x)h. Mu(k+1) (x; h, h) = m(k+1) 0 The renormalized M (k+1) is defined by (x)φk+1 (x; ψk+1 , h) M (k+1) (x; ψk+1 , h) = Mu(k+1) (x; ψk+1 , h) − m(k+1) 0 and satisfies the renormalization condition (2.7). The function mk+1 is now defined by the formula (2.8) with the summation extended to k + 1. This renormalized new contribution satisfies (H.1), assuming β sufficiently large. The second order in g contribution still does not have the correct form, because instead of Ck+1 we have k+1 X j=1

−1 βj−1 mk G(j) mk .

(3.49)

We arrive at Ck+1 by replacing (3.49) by Ck+1 plus the difference −

k+1 X j=1

h i −1/2 −1/2 −1 βj−1 G(j) mk + mk G(j) βk m(k+1) + βk−1 m(k+1) G(j) m(k+1) βk m(k+1) . 0 0 0 0

d−2 +ρ0 −1/2 can be bounded by 0(1)β −1+ρ1 L−1 η 2 and we incorThe function βk m(k+1) 0 −1 Rj,k+1 . This yields the second order porate the j th term in the above sum into βj−1 contributions satisfying exactly (H.2), again for β large enough. 4. Detailed Description of the One- and Two-Point Functions Up to now we have analyzed the detailed properties of the function Fk on the lattice T , but this function is not the complete generating function for correlation functions, which is log Z(g), where Z(g) is defined by (1.1). We have to explain the relation between the two functions, and in particular to understand when Fk becomes the main contribution to log Z(g). To clarify this connection we use the fundamental normalization property Z (4.1) Z(g) = Zk (g) = dψk ρk (ψk , g), where ρk is the density after k renormalization steps. This density is describred by the inductive hypotheses (H.1)–(H.7) in [5]. We write the general representation of this density following from the hypotheses (H.1), (H.5),

Low Temperature Properties for Correlation Functions

Z dψk ρk (ψk , g) =

XZ

515

dψk ρ0k (Zk ; ψk , g)χk (Zkc )

Zk

× exp Ak (Zkc ) + Fk (Zkc ) + R(k) (Zkc ) XXZ dψk Tk (Zk , Ak )χk (Zkc ) = Zk

Ak

× exp Ak (Zkc ) + Fk (Zkc ) + R(k) (Zkc ) + B (k) (zk , Ak ) .

(4.2)

Let us recall roughly the meaning of the representation appearing above. The domain Zk is the large field region and the partial density is divided into two parts, the large field part given by the integral operator Tk (Zk ) associated with unperformed integrations and the small field part given by the exponential. These two parts have almost disjoint supports in the basic variable ψk , contained in some “small” neighbourhoods of the domains Zk , Zkc correspondingly, except the term B(k) (Zk , Ak ), called a boundary term, which connects the two domains. The term Ak (Zkc ) is the effective action restricted to Zkc , Fk (Zkc ) is the whole lattice function Fk discussed in the previous sections and restricted to Zkc by the formulas in (H.3) [5], and R(k) (Zkc ) is the contribution coming from the renormalization transformation of the large field densities and restricted to Zkc also. The symbol Ak denotes a multiscale geometric structure, which is unimportant for this discussion, and χk (Zkc ) denotes an appropriate characteristic function imposing small field restrictions. Bounds and properties of all expressions in (4.2) are given in [5], and will be freely used to carry out our analysis. Let us explain now what index k we take in (4.1), (4.2). This has been discussed already in Sect. 6 of [4]. We start with some v > 0, actually it may be very small because we want to take the limit v ↓ 0, and by the renormalization procedure we generate the sequence of the “running” constants v1 , v2 , . . . , vk , where vj ≈ vL2j . There is the smallest index k(v) such that vk(v) > 78 L−2 , and we take k = k(v). In Sect. 6 of [4] we have shown that then the characteristic functions χk restrict the variables −1 ψk to a subdomain of {|ψk − h| < 0(1)βk 2 p0 (βk )}, the main action in Ak can be expanded in ψk − h, and the √ expansion starts with a quadratic form with a “massive” covariance, the “mass” ≈ vk ≈ L−1 . In this case the integral with respect to ψk , or rather the fluctuation variable ψk − h, has the same properties as the other fluctuation integrals in the procedure, so we can perform it and obtain a density satisfying all the inductive hypotheses of [4]. In the small field approximation of [4] it is reduced to the function Fk (h, g), because other functions are equal to 0 when taken at the background configuration ψk = h. In the complete model considered here we take the same k = k(v), and the small field part has basically the same properties, but there are additional contributions from the large field part. Let us discuss briefly the procedure we follow for the last step. For each term in (4.2) we perform at first the integral with respect to ψk c over Zkc , which is naturally carried out over a proper subdomain Zk+1 . For the reasons explained above this integral is like the previous fluctuation integrals, and the result of the integration can be written again in the form (4.2), but with the integral over the c of the exponential in (4.2) replaced by subdomain Zk+1 c c (4.3) ; h, g + R(k+1) Zk+1 ; h, g + B (k+1) (Zk+1 , Ak+1 ) , exp Fk+1 Zk+1 c of the corresponding whole lattice where the first two functions are restrictions to Zk+1 functions. Let us remark that the notation in (4.3) is kept in agreement with the structure

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these expressions have, they satisfy the inductive hypotheses for k + 1, but they have a different meaning. We have performed now all the integrations down to 0-momentum, so over all scales, not only over the k + 1 scales. Next we combine the exponential with the boundary term B (k+1) (Zk+1 , Ak+1 ) with the integral operators in (4.2), and construct a polymer expansion by expanding in the boundary terms1 . This polymer esxpansion has very small activities, bounded by arbitrary powers of βk−1 in addition to the usual exponential decay in size of domains, as follows from the bounds (1.26), (1.27) [5], for the integral operators. These bounds allow us to exponentiate the polymer expansion. Doing this we obtain the final representation of (4.2) in the form (4.4) exp Fk+1 (h, g) + R(k+1) (h, g) + P (k+1) (g) with the whole lattice functions. The new large field contribution P (k+1) (g) has the localization expansion X P (k+1) (Y ; g), (4.5) P (k+1) (g) = Y ∈Dk +1

and by the standard polymer expansion bounds, see [7], |P (k+1) (Y ; g)| < βk−2 exp(−2κdk+1 (Y )).

(4.6)

(k+1) (h, g). Furthermore it has the symmetry properties as the term R(k+1) m (h, g) of R From (4.4) we obtain the representation of the generating function

log Z(g) = Fk+1 (h, g) + R(k+1) (h, g) + P (k+1) (g);

(4.7)

for the one-point correlation function ∂ ∂ (k+1) (k+1) (h, g = 0) + (g = 0), R P hφ(x)iv = Mk+1 (h, h) + ∂g(x) ∂g(x) (4.8) and for the two-point truncated correlation function ∂2 0 T 0 (k+1) R (h, g = 0) hφ(x); φ(x )iv = Gk+1 (x, x , h, h) + ∂g(x)∂g(x0 ) ∂2 P (k+1) (g = 0). + ∂g(x)∂g(x0 )

(4.9)

Now it becames clear that the main contributions to the correlation fuctions are given by the first terms on the right-hand sides of (4.8), (4.9). The remaining terms are bounded (g) in Sect. 3. The sum of the second and third terms in as the terms defined by R(k+1) 0 (4.8) is bounded by m X n=1 1

K0 βk−n−1 + K0 βk−2 < 3K0 βk−2 ,

Such an expansion was already discussed twice in previous papers, once in Sect. 4 [6] and a second time in Sect. 3 [7]. In particular for details we refer the reader to Sect. 3 [7], where the expansion is of the same form as needed here.

Low Temperature Properties for Correlation Functions

517

which is obviously smaller than the contributions from the first term. Also it goes to 0 as k = k(v) → ∞, or as v → 0. Similarly the sum of the last two terms in (4.9) is bounded by ! m X −n−1 −2 K0 βk + K0 βk e2dLκ exp −δ0 L−(k+1) |x − x0 |1 n=1

< 0(1)β

−2

L

1

−(k+1) d−2 2

3

|x − x0 |12

(d−2)

1 −(k+1) 0 exp − δ0 L |x − x |1 . 2

This gives the exponential decay for a finite k, or v > 0, and a sub-dominant decay for k → ∞, where the constant in front converges to 0. Thus it remains to analyze the first terms on the right-hand sides. For simplicity we replace the index k + 1 by k. for the first term in (4.8) we have Mk (x, h, h) = mk (x)φk (x; h, h) +

k X j=1

−1/2

βj−1 M (j) (x; h, h) = mk (x)h,

(4.10)

by the inductive assumption (H.1) and the properties of φk , ψk(j) . By the symmetry properties Gk (x, x0 ; h, h) can be written in the form Gk (h, h) = Gtr [I − h ⊗ h] + Glg h ⊗ h, 1 tr {(Ck + Rk )(h, h)[I − h ⊗ h]} , Gtr = N −1 Glg = tr {(Ck + Rk )(h, h)[h ⊗ h]} ,

(4.11)

where Gtr , Glg are correspondingly the transverse and the longitudinal components of the two-point function. The contributions given by Rk have a sub-dominant long-range behavior, by applying Lemma 1, so we have to analyze the behavior of Ck . We will need explicit expressions for the minimizer Hj and the fluctuation covariance C (j) occurring −j in the representation (2.11), (2.13) of Ck . The operator (−1L + aj Q∗j Qj + vj )−1 will appear in these expressions and we will denote it by Gj as in the notation of [2]; it is not to be confused with Gk . For ψ = h from [2] we have −1 ∗ Hj = aj Gj Q∗j I − aj Gj (λ−1 j + Gj ) Gj Qj (h ⊗ h)

(4.12)

C (j) ≡ Ctr(j) (I − h ⊗ h) + Clg(j) (h ⊗ h),

(4.13)

and

where

and

−1 −2 , Ctr(j) = 1(j) tr + aL Q ∗ Q

−1 −2 ∗ + aL Q Q , Clg(j) = 1(j) lg

1jtr = aj − a2j Qj Gj Q∗j ,

(j) −1 2 −1 ∗ 1(j) lg = 1tr + aj Qj Gj (λj + Gj ) Gj Qj .

∗ the We first consider the transverse cf. In the expression for Gtr(j) = Hj−1 C (j) Hj−1 second terms of (4.12) and (4.13) do not appear. Thus the main contribution Ck tr is given by

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T. Balaban, M. O’Carroll

Ck tr

k−1 X 1 2 = mk a Gj Q∗j Ctr(j) Qj Gj mk . βj j

(4.14)

j=0

Using βj−1 ≤ 2L−j(d−2) β −1 the above is bounded by |Ck tr | ≤ 0(1)β

−1

k−1 X j=0

√ −δ( v+L−j )|x−y 0 |

1

e Lj(d−2)

√

≤ 0(1)β

−1

0

e−δ v|x−x | , (1 + |x − x0 |)d−2

which proves the first part of Theorem 1. Now we consider the v ↓ 0 limit so that k(v) ↑ ∞. We write the main contribution to Ck tr omitting the factors mk , as Ck tr =

k−1 X 1 (j) G (0) + S(k), βj tr

where S(k) =

j=0

k−1 X 1 (j) G (v) − G(j) (0) , βj (4.15) j=0

and show that the second term goes to zero as v ↓ 0. We break the sum into two parts S(k) = S(k/2) + S1 (k), |S1 (k)| has the bound |S1 (k)| ≤ 2

k−1 X

L−j(d−2) e−δL

−j

|x−x0 |

≤ 0(1)L−k/2 → 0

as

v ↓ 0.

j=k/2

Also the j th term in the sum representing S(k) is dominated by # " −j e−δL |x−y| (j) vj | ≤ 2 2 sup |G Lj(d−2) vj ∈(0,1] and the right side has a finite sum. For each j, lim G(j) (vj ) − G(j) (0) = 0 v↓0

so that by dominated convergence S(k/2) → 0 as v ↑ 0. By Eq. (4.15) we still have to analyze ∞ ∞ X X 1 (j) 1 2 G (0) = a Gj Q∗j Ctr(j) Qj Gj . βj βj j j=0

(4.16)

j=0

We separate a part that is a constant times 1−1 and show that the rest is subdominant. Recall from [10] that from applying RGT’s to the Gibbs factor exp −1/2(∂φ, ∂φ) we induce a decomposition of 1−1 given by 1−1 =

∞ X j=0

−2

a2j Lj(d−2)

Gj Q∗j Ctr(j) Qj Gj ,

(4.17)

1−L where aj = 1−L −2j a. In order to compare Eqs. (4.16) and (4.17) we need more precise properties of the sequence {βj }. We have

Low Temperature Properties for Correlation Functions

519

1 1 1 1 1 1 = = + 0(β −1 ) j(d−2+2γ) , βj βLj(d−2) γj βγ∞ Lj(d−2) L

(4.18)

and the contribution of the second term on the right-hand side above to (4.16) yields a subdominant long range behaviour, again by Lemma 1. The contribution of the first term is equal to βγ1∞ 1−1 , where |γ∞ − 1| < 0(1)β −1 , which is the required dominant behaviour. Finally we show that the v ↓ 0 limit of the longitudinal cf is subdominant.Denote spin indices by subindices, and assume that h = e1 . We analyze the term G(j) 11 = (j) −1 −1 −1 −1 −1 = Gj − Gj λ j Hj,11 C11 Hj,11 . Using the resolvent equation (Gj + λj ) −1 −1 we have Gj + λ j Hj,11 =

aj −1 ∗ (Gj + λ−1 j ) Gj Qj λj

and −1 (j) −1 ∗ −2 ∗ = 1(j) + a2j Qj Gj (λ−1 . C11 j + Gj ) Gj Qj + aL Q Q Furthermore from [2] we have the bounds i x 0(1) h x , y ≤ exp −δ − y , Hj,11 Lj λj Lj (j) (y, y 0 )| ≤ 0(1) exp −δ|y − y 0 | , |C11 −1 −2j so that we can bound Clg by for x, y on the unit lattice. Recall that λ−1 j =λ L

|Clg (x, x0 )| ≤

∞ X 1 Hj G(j) Hj∗ 11 (x, x0 ) βj j=0

≤ 0(1)

∞ X

βj λ2j

−1

e−δL

−j

|x−x0 |

≤ cβ −1 λ−2 /(1 + |x − x0 |)d+2 ,

j=0

i.e. it decays faster than 1−1 (x, x0 ). This ends the proof of the second part of Theorem 1. References 1. Balaban, T.: Low Temperature Expansions for Classical Vector Spin Models. Commun. Math. Phys. 167, 103–154 (1995) 2. Balaban, T.: The Variational Problem for Classical N-Vector Models. Commun. Math. Phys. 175, 607– 642 (1996) 3. Balaban, T.: Localization Expansions. I. Functions of the “Background Configurations”. Commun. Math. Phys. 182, 33–82 (1996) 4. Balaban, T.: Low Temperature Expansion for Classical N-Vector Models. II. Renormalization Group Equations. Commun. Math. Phys. 182, 675–721 (1997) 5. Balaban, T.: Low Temperature Expansion for classical N-Vector Models. III. A Complete Inductive Description, Fluctuation Integrals. Commun. Math. Phys. 196, 485–521 (1997) 6. Balaban, T.: Renormalization and Localization Expansions. II. Expectation Values of the “Fluctuation” Measures. Commun. Math. Phys., to appear 7. Balaban, T.: The Large Field Renormalization Operation in Classical N-Vector Models. Commun. Math. Phys., to appear

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8. Fr¨ohlich, J., Simon, B., Spencer, T.: Infrared bounds, phase transitions, and continuous symmetry breaking. Commun. Math. Phys. 50, 79–85 (1976) 9. Glimm, J., Jaffe, A.: Quantum physics. A Functional Integral Point of View. New York: Springer, 1989 10. Parisi, G.: Statistical Field Theory. New York: Addison-Wesley, 1987 11. Schor, R., O’Carroll, M.: Commun. Math. Phys. 138, 487–505 (1991); J. Stat. Phys. 64, 163–191 (1991) 12. Balaban, T.: Commun. Math. Phys. 89, 571–597 (1983) 13. Bricmont, J., Fontaine, J.R., Lebowitz, J.L., Spencer, T.: Lattice systems with a continuous symmetry. I, II. Commun. Math. Phys. 78, 281–302 and 363–372 (1980) 14. Bricmont, J., Fontaine, J.R., Lebowitz, J.L., Lieb, E., Spencer, T.: Commun. Math. Phys. 78, 566–595 (1981) Communicated by D. C. Brydges

Commun. Math. Phys. 305, 521 – 546 (1999)

Communications in

Mathematical Physics © Springer-Verlag 1999

Propagation of Smoothness and the Rate of Exponential Convergence to Equilibrium for a Spatially Homogeneous Maxwellian Gas E. A. Carlen1 , E. Gabetta2 , G. Toscani2 1 2

School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA Dipartimento di Matematica – Universit`a di Pavia, via Abbiategrasso 215, 27100 Pavia, Italy

Received: 8 January 1997 / Accepted: 12 May 1998

Abstract: We prove an inequality for the gain term in the Boltzmann equation for Maxwellian molecules that implies a uniform bound on Sobolev norms of the solution, provided the initial data has a finite norm in the corresponding Sobolev space. We then prove a sharp bound on the rate of exponential convergence to equilibrium in a weak norm. These results are then combined, using interpolation inequalities, to obtain the optimal rate of exponential convergence in the strong L1 norm, as well as various Sobolev norms. These results are the first showing that the spectral gap in the linearized collision operator actually does govern the rate of approach to equilibrium for the full non-linear Boltzmann equation, even for initial data that is far from equilibrium. 1. Introduction This paper concerns the large time behavior of solutions of the Boltzmann equation for Maxwellian molecules in the case of spatially homogeneous initial data: ∂ f (v, t) = Q(f, f )(v, t). ∂t

(1.1)

Here, f (v, t) is the probability density for the velocity space distribution of the molecules at time t, and Q, which represents the effects of binary collisions, has the form: Z B q, q · n/q f (v1 )g(w1 ) − f (v)g(w) dwdn. (1.2) Q(f, g)(v) R3 ×S 2

In expression (1.2), n is a unit vector, and dn denotes normalized surface measure on the unit sphere S 2 . Moreover q = v − w is the relative velocity, and in q · n, the dot denotes the usual inner product. The vector n parameterizes the set of all kinematicly possible (i.e., those conserving energy and momentum) post–collisional velocities (v1 and w1 ) by

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E. A. Carlen, E. Gabetta, G. Toscani

1 (v + w + qn), 2 (1.3) 1 w1 = (v + w − qn). 2 The relative likelihood of these kinematicly possible outcomes depends of course on the nature of the interaction between the molecules, and this is taken into account in the rate function B. Maxwell found that when this interaction is through an r−5 force law, B depends only on the scattering angle θ in cos θ = q · n/q, and not on q itself. By a Boltzmann equation for Maxwellian molecules, we mean throughout this paper one in which the rate function B has this simple form B(cos θ). We shall further suppose during most of our analysis that B is integrable: Z 1 B(u)du := b < ∞. (1.4) v1 =

−1

This condition is not satisfied for the actual rate function Maxwell considered, i.e., that one corresponding to an r−5 force law. In this case, the integral above diverges due to a singularity at u = 1, i.e., for small angle collisions. The standard strategy is to “cut off” these small angle collisions so that B becomes integrable, and then to seek estimates, that are independent of the cut-off. When (1.4) does hold, one can split Q(f, f ) into its “gain” and “loss” terms Q(f, f ) = Q+ (f, f )−Q− (f, f ). One easily sees (since f is a probability density) that Q− (f, f )(v) = bf (v) so that the Boltzmann equation can be rewritten ∂ f (v, t) + bf (v, t) = Q+ (f, f )(v, t). ∂t

(1.5)

This equation has been extensively investigated, and much is known. In particular, existence and uniqueness have been established, and moreover, it has been shown that, given enough moments for the initial density, the convergence is exponential in the strong L1 norm [Ar88, We93]. However, existing results provide little or no information on what the rate of this exponential convergence might be. This is significant for the following reasons. The unit time scale relevant for Eq. (1.1) is the mean time between collisions. This time scale is much, much shorter than the time scale governing macroscopic transport phenomena, so that it is commonly believed that (1.1) governs the rate of approach to local equilibrium even in non-homogeneous settings. There is a natural conjecture as to what this rate should be, which one obtains by linearizing the collision kernel Q(f, f ). That is, let Mf (v) be the Maxwellian density Mf (v) = (6πT )−3/2 exp(−|v − u|2 /6T ), Z vf (v)d3 v, u= R3 Z |v − u|2 f (v)d3 v. 3T =

(1.6)

R3

Then, Mf is the equilibrium solution of (1.1) towards which f (·, t) tends, and is, of course, independent of t: Mf (·,t) = Mf (·,0) for all t since the temperature T and bulk velocity u in (1.6) are conserved. Without loss of generality, we may suppose that our initial data is such that T = 1 and u = 0, and we shall simply write M for Mf in this case.

Smoothness and Convergence to Equilibrium for Maxwellian Gas

523

At this point losing generality (for the moment), suppose that the density f has the form f (v, t) = M (v)(1 + h(v, t)) (1.7) for some function h with

Z R3

|h(v, 0)|2 M (v)dv = 1

(1.8)

and some small number . Inserting this in (1.1), one obtains 1 ∂ h(v, t) = Lh(v, t) + Q(M h, M h)(v). ∂t M (v) Here, L is the linearized collision operator: 1 Q+ M, M h (v) + Q+ M h, M (v)) − Lh(v) = M (v)

(1.9)

Z R3

M h(v)dv − h(v). (1.10)

Observe the L is self-adjoint on the Hilbert space H with norm Z |h(v)|2 M (v)d3 v . khk2H =

(1.11)

R3

The natural conjecture is that the spectral properties of L govern the rate of approach to equilibrium in L1 for solutions of (1.1). Now, the spectrum of L has been computed [WU70], and the following facts are well known: L is negative semi–definite on H with a five dimensional null space due to the conservation of total probability, bulk momentum u, and temperature T . The remaining eigenvalues are discrete and strictly negative, and, in particular, let λ1 denote the absolute value of the first of these eigenvalues when they are arranged in order of increasing magnitudes. Thus λ1 is the “spectral gap” of the linearized collision operator. A concise statement of one of our main results is the following: Theorem 1.1. Let f0 (v) be initial data for (1.1) with Maxwellian collisions. Suppose that the bulk velocity u = 0, and the temperature T = 1. Let > 0 be given. Then there is a number n depending only on so that whenever Z Z 2n 3 |v| f0 (v)d v + |ξ|2n |fb0 (ξ)|2 d3 ξ < ∞ R3

then it holds that

R3

kf (·, t) − M kL1 ≤ C e−(1−)λ1 t .

Here, λ1 is the spectral gap of the linearized collision operator, fb0 denotes the Fourier transform of f0 , and C is computable in terms of the integral specified above. This result will be reformulated in more detail later in the paper, where in particular, we shall specify the relation between n, and C . Here in the introduction, we wish to focus on a few key points. First, transport coefficients for a rarefied gas, i.e., the bulk and thermal diffusivity, may be calculated in terms of the eigenvalues of L – assuming that this operator really does control the trend toward local equilibrium. This is not yet proved. In fact, until now, it had not even been proved that L governs the rate of approach to equilibrium in the spatially homogeneous case for initial data far from equilibrium.

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E. A. Carlen, E. Gabetta, G. Toscani

Cercignani, Lampis and Sgarra [CLS88] have proven an inequality for the non-linear term in (1.9) which shows this for initial data that is in a sufficiently small neighborhood of equilibrium, but for reasons we will now explain, the treatment of initial data that is far from equilibrium is delicate. This may be surprising to those who are encountering the problem for the first time. After all, we have said above that it is known that kf (·, t) − M kL1 tends to zero exponentially at some rate, so what can prevent it from eventually entering a small neighborhood of M in which L dominates the remaining evolution, with its spectrum governing the asymptotic speed of convergence? The answer lies with the meaning of “small neighborhood”. The operator L is selfadjoint on the Hilbert space H, and the requirement on f is that if we write f = M (1+h), then khkH < 0 for some sufficiently small number 0 . Stated in terms of f , this is a requirement that Z |f (v, t) − M (v)|2 3 (1.12) d v < 0 M (v) R3 should be sufficiently small. This requires more control on the tail of the distribution f (v, t), uniformly in t, than is available. If it were known that for some value r with 1/2 < r < 1, Z |f (v, t)|2 M −r (v)d3 v < C, (1.13) sup t>0

R3

then the eventual validity of (1.12) would follow from the decay of kf (·, t) − M kL1 . However, it remains an open problem to establish (1.13) for any reasonably general class of initial data – even, say, for initial data with compact support. This is true despite the fact that each individual moment of f will remain bounded, uniformly in time, in terms of the initial value of that moment. In short, the lack of sufficient control on the tails of the distribution f (v, t), uniformly in time, is a significant obstacle in the way of establishing the relevance of the spectrum of L in H to the rate of convergence to equilibrium for (1.1). We overcome this obstacle here by establishing a propagation of smoothness result for (1.1). This follows from an inequality on the gain term Q+ which is of independent interest, and indeed has already been applied in another problem in [CELMR96]. To state the result concisely, we introduce the Sobolev space norms k · kHk by Z |fb(ξ)|2 |ξ|2k d3 ξ kf k2H k = R3

for all k ≥ 0. Our convention for the Fourier transform is that Z f (v, t)e−iv·ξ d3 v. fb(ξ, t) = R3

We recall that the entropy of f , H(f ), is defined by Z f (v) ln f (v)d3 v. H(f ) = − R3

The key inequality enabling us to bound the Hm norm of solutions of (1.1) uniformly in time is the following:

Smoothness and Convergence to Equilibrium for Maxwellian Gas

525

Theorem 1.2. Let f be any probability density on R with unit variance, and kf kHm finite. Then, there are universal constants Cm < ∞ and Km > 0 so that for all such f , kQ+ (f )k2Hm ≤ (1/2)kf k2Hm + Cm whenever

(1.14)

H(Mf ) − H(f ) ≤ Km .

We shall later reformulate this inequality in more detail, and in fact a better form, with explicit determination of the constants. For now we observe that as soon as H(Mf ) − H(f ) ≤ Km holds at some time t0 , it holds at all succeeding times since H(f ) is strictly increasing for non-equilibrium solutions of (1.1). Then if we define φ(t) by φ(t) = kf (·, t)k2Hm it easily follows that 1 d φ(t) ≤ − φ(t) + K dt 2

for all t ≥ t0

with the consequence that for all t ≥ t0 .

φ(t) ≤ max{φ(t0 ), 2K}

Hence, once one has a bound on t0 , which can be obtained from entropy production bounds [CC94], it is a simple matter to bound φ(t0 ) in terms of φ(0). In this way, we obtain uniform bounds on the Hm norm of solutions of (1.1). We shall apply this by using an interpolation inequality to bound kf (·, t) − Mf (·)kL1 by the geometric mean of weak norm bound on f (·, t) − Mf (·), which decays at the required rate, and the Hm bound on this quantity which stays bounded above uniformly in t. Since, for n large enough, we shall be able to take arbitrarily little of the Hm norm in our geometric mean, this leads to Theorem 1.1. Clearly then, a crucial role is played by this weak norm convergence, which is obtained by further pushing the development of a recent method for obtaining exponential convergence for Maxwellian molecules in certain weak norms [GTW95]. To show that the convergence in these weak norms is taking place at the rate suggested by the spectral gap in the linearized collision operator L, we must work with a particular choice of these norms, outside the range originally considered. Namely, define the norm ||| · ||| by |||g||| = sup

ξ∈R3

|b g (ξ)| . |ξ|4

(1.15)

is well defined and R finite on the space of integrable functions g such that RThis norm |v|4 |g(v)|d3 v < ∞, and R3 P (v)|g(v)|d3 v = 0 whenever P (v) is a polynomial of R3 total degree three or less in the components of v. This space does not include f (·, t), or any probability density for that matter, but it does include f (·, t) − Mf (·) − S(·, t), where S is a subtraction term taking care of the first, second and third moments. Because of known results on the explicit exponential convergence of all of the moments of f to those of Mf , it will be easy to show that kS(t)kL1 converges to zero at faster than the required rate, and that kS(t)kHm remains bounded uniformly in time. Thus, as far as either the L1 norm or the Hm norm are concerned, both f (·, t) − Mf (·) − S(·, t) and f (·, t) − Mf (·) have the same decay and boundedness properties. Concerning the former, we have the following theorem:

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E. A. Carlen, E. Gabetta, G. Toscani

R Theorem 1.3. Let f0 be a probability density with R3 |v|4 f0 (v)d3 v < K, and let > 0 be given. Then there are constants B and C and a function S(·, t) such that |||f (·, t) − Mf (·) − S(·, t)||| ≤ Bte−t(1−)λ1 |||f (·, 0) − Mf (·) − S(·, 0)||| for all t ≥ 0, and with |||f (·, 0) − Mf (·) − S(·, 0)||| < ∞, such that for all m, etλ1 kS(·, t)kL1 + kS(·, t)kHm ≤ C, for all t ≥ 0. Here λ1 is the spectral gap of the linearized collision operator. Again, a more explicit version will be provided later. To combine the second and third theorem to prove the first, it is only necessary to use an interpolation inequality of the form kf − Mf − SkL1 ≤ C |||f − Mf − S|||1− kf − Mf − SkHm

(1.16)

which holds for any > 0 provided f0 , and hence f (·, t)−Mf (·)−S(·, t) has sufficiently many moments and belongs to Hm for m sufficiently large. Theorem 1.2 (and part of Theorem 1.3) says that the k · kHm terms stay bounded, and Theorem 1.3 says that the other norm is decaying at the desired rate. The methods will actually yield more: we can also prove convergence in Sobolev norms for sufficiently smooth and rapidly decaying initial data, again at the exponential rate given by the spectral gap in the linearized collision operator. We now briefly discuss related results in the literature. The result most closely related to Theorem 1.2 is the estimate of Lions [Li94]. In particular, in presence of smooth kernels B that vanish for small and large relative velocities, uniformly in the argument q · n/q, the gain term has been shown to possess a regularizing effect kQ+ (f, g)kH 1 ≤ Ckf kL1 kgkL2 .

(1.17)

The main application of the above result was to prove propagation of strong L1 compactness for renormalized solutions of the Boltzmann equation, and to prove that the weak solutions are strong, if any strong solution exists. For this purpose, the gain term is modified to have regular kernels, being the passage to the limit based on the averaging lemma. The paper by Wennberg, [We94], gives a simplified proof of this result, using a different representation of the gain term due to Carleman [Ca57], and Radon transform estimates. Furthermore, he was able to prove a similar inequality for smooth B that aren’t compactly supported, including the case of hard spheres, provided f and g posses sufficient additional Lp regularity and have sufficiently many moments, with norms on the right side reflecting these requirements. As application of this, Wennberg proves for the spatially homogeneous Boltzmann equation with hard sphere collisions that if the initial data f0 satisfies f0 (v)(1+|v|2 )1/2 ∈ L1 ∩ Lp with p > 6, and if f0 ∈ H 1 , then the same holds for the solutions, uniformly in time. The argument does not provide propagation of regularity in H k , for k > 1. There are few other results on propagation of smoothness for the Boltzmann equation, all of them obtained in recent years. These results are concerned with certain generalizations of the Fisher information, which, up to a constant is the square of H 1 norm of the square root of the density f . McKean [McK66] showed that this quantity was monotonically decreasing for solutions of the Kac equation. This monotonicity is

Smoothness and Convergence to Equilibrium for Maxwellian Gas

527

possible because of the uncertainty principle which says that among all densities with given variance, Maxwellians have the least Fisher information, i.e.: p p k∇ f kH1 ≥ k∇ Mf kH1 . This extremal property of Maxwellians does not hold for the Sobolev norms considered here; that is, one easily sees that there are densities f for which kf kH1 ≤ kMf kH1 and hence such a monotonicity property is impossible. Moreover it is not clear how far such monotonicity results can be extended from the Kac model to the Boltzmann equation. Carlen and Carvalho [CC92] showed that the Fisher information is decreasing for the Boltzmann equation in the case of constant B, Toscani [To92] showed this for Maxwellian molecules in two dimensions, and Bobylev and Toscani [BT92] also in three dimensions with certain symmetries effectively reducing the dimension to two. As far as the higher regularity of solutions is concerned, natural analogs of the Fisher information involving higher derivatives were recently studied by Gabetta [Ga95] and by Lions and Toscani [LT95]. They developed methods using these quantities to control the convergence towards the Gaussian density in the central limit theorem of probability theory as measured by these Sobolev norm like functionals. The methods of Lions and Toscani have been extended to the Kac equation by Gabetta and Pareschi [GP94] to prove propagation of regularity and convergence to equilibrium in various norms of the solution. Subsequently Toscani [To96] with the same tools obtained analogous results for the solution to the Boltzmann equation for Maxwell pseudomolecules, both in plane geometry and in the axially symmetric case. The key of his proof relies in the fact that, as already mentioned, in these cases Fisher information has been shown to be a nonincreasing Lyapunov functional [BT92]. Concerning Theorem 1.1, the rate at which the solution to the Boltzmann equation approaches equilibrium has been extensively studied starting from the fifties, when Ikenberry and Truesdell [IT56] proved that all moments of the solution to the spatially homogeneous Maxwell gas, that exist initially, converge exponentially to the corresponding ones of the equilibrium distribution. For intermolecular forces harder than Maxwellian ones, and in the presence of a cut-off, Arkeryd [Ar88] obtained stability results in L1 . These results were extended to pseudo–Maxwellian molecules by Wennberg [We93]. Here the method of proof is based on the spectral theory of the linearized collision operator, and gives exponential convergence to equilibrium, provided the initial data belong to an appropriately small neighborhood of the equilibrium itself. However, in these proofs, one uses the spectrum of the linearized operator not in its natural space, as discussed above, but in certain polynomially weighted L1 spaces. Here, it is not possible to explicitly compute the spectrum, and one must resort to compactness arguments to prove the existence of a spectral gap in the spaces considered. Hence, such an approach, while fully successful in establishing exponential convergence, gives no information as to what the exponential rate might be. The exponential convergence towards equilibrium has been obtained by Gabetta, Toscani and Wennberg [GBT95], for the Kac model and for Maxwellian molecules in a metric equivalent to the weak-? convergence of measures, closely related to the norm ||| · ||| considered here. In fact, they used a norm ||| · |||α which in definition differs from the norm ||| · ||| in that they divided by |ξ|2+α , α > 0 (but small) instead of |ξ|4 . The basic tool

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E. A. Carlen, E. Gabetta, G. Toscani

in [GBT95] is a Fourier transformed version of the Boltzmann equation for Maxwellian molecules, due to Bobylev [B88]. While this method gives exponential convergence in a very weak norm, it has the considerable advantage of doing so with an explicitly computable rate. By taking α = 2, we need to introduce extra subtraction terms (the function S in Theorem 1.3 becomes more complicated), but having done this, the approach can be extended to pick off the sharp behavior that we seek. We shall explain why this works in the course of proving Theorem 1.3. We shall begin the paper by first carrying out the program in the much simpler case of the Kac model. This not only adds considerable clarity, but the results for the Kac model are interesting in their own right. Indeed, McKean proved the strong L1 convergence to equilibrium at an exponential rate kf (·, t) − M kL1 ≤ Ce−λt with λ ≈ 0.016. He conjectured that the true rate should be given by λ = 1/4, which is the spectral gap in the linearized collision operator for the Kac model. We shall prove this conjecture here. Our result improves his bound on the rate of decay by more than an order of magnitude. The structure of the paper is as follows: In Sect. 2, we introduce the Kac model, and prove the analog of Theorem 1.2 in this context. In Sect. 3 we prove the analog of Theorem 1.3 in this context. At this point we need the interpolation inequalities. So we prove them in Sect. 4, in a general form suitable for both the Kac Model and the Boltzmann equation. Then, in Sect. 5, we prove the analog of Theorem 1.1 for the Kac model, and prove a conjecture of McKean. Sect. 6 then presents some geometric lemmas needed for our analysis of the Boltzmann equation. These are applied in Sect. 7 to prove Theorem 1.2. Next in Sect. 8 we prove Theorem 1.3, and finally, in Sect. 9, Theorem 1.1.

2. Propagation of Smoothness for the Kac Equation The Kac equation is a caricature of the Boltzmann equation introduced by Kac, and reduced to its essentials by McKean. It models a gas of one dimensional particles with collisions that conserve energy but not momentum (or else, in one dimension, the number of conserved quantities would equal the number of degrees of freedom). Thus, all of the kinematicly possible collisions (v, w) → (v 0 , w0 ) are given by v 0 = v cos θ + w sin θ

and

w0 = −v sin θ + w cos θ

(2.1)

for 0 ≤ θ < 2π. We could introduce a weight B(cos θ) favoring some collisions over others, as in [De94], but we shall follow McKean and simply take B to be constant. Then the gain term in the Kac model collision kernel is Z Z B 2π f (v 0 )f (w0 )dwdθ, (2.2) Q+ (f ) = 2π 0 R the loss term is B Q (f ) = 2π −

Z 0

2π

Z R

f (v)f (w)dwdθ = Bf (v),

Smoothness and Convergence to Equilibrium for Maxwellian Gas

529

and hence the equation itself is ∂f (v, t) + f (v, t) = Q+ (f ) ∂t

(2.3)

where, after a rescaling of time, we have taken B = 1. Further shifting the frame of reference and rescaling, we may freely suppose that 2 1 Mf (v) = √ e−v /2 . 2π

Then, as in McKean’s paper [Mk66], one linearizes about M = Mf by writing f = M (1 + h) and finds that

∂h(v, t) = Lh(v, t) + O(h2 ) ∂t where L, the linearized collision operator is given by Z 2π Z 1 M (v 0 )M (w0 ) h(v 0 ) + h(w0 ) − h(v) − h(w) dwdθ. Lh(v) = πM (v) 0 R

(2.4)

(2.5)

As McKean observed, the Hermite polynomials are a complete set of eigenfunctions for L (which is an average over Mehler kernels). All of the odd Hermite polynomials [Mk73] have eigenvalue -1. The null space of L consists of the span of the first two even such polynomials, h0 (v) = 1 and h1 (v) = 1 − v 2 . Let h2k (v) be the normalised Hermite polynomial of degree 2k. Since the leading coefficient is a multiple of v 2k , we need only apply L to v 2k to determine the corresponding eigenvalue. Doing so, one has repeated McKean’s calculation that Z 1 2π 2k sin (θ)dθ − 1 Lh2k (v) = π 0 for all k ≥ 1. The largest of these eigenvalues, λ1 , is given by λ1 = −1/4 and corresponds to h4 (v). Our goal in the next few sections is to show that for any > 0, there is a constant C so that for all sufficiently smooth and rapidly decaying initial data f0 (v), the corresponding solution f (v, t) of the Kac equation satisfies kf (·, t) − M (·)kL1 ≤ C e−(1−)λ1 t .

(2.7)

As indicated in the introduction, the first step will be to show that the smoothness of the initial data is propagated so that we have bounds on the smoothness uniform in time. To do this, we prove the analog of Theorem 1.2 for the Kac equation gain term. Theorem 2.1 (Smoothness bound on the gain term for Kac equation). Let f be any probability density on R with unit variance, and kf kHm finite. Then, whenever kf − Mf k1 ≤ (1/2)

m+1 2

,

kQ+ (f )k2Hm ≤ Cm Fm kf − Mf k1 kf k2Hm + Km , where

(2.8)

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E. A. Carlen, E. Gabetta, G. Toscani

4

Cm =

, (1 − 2m+1 2 8 (2m + 1)(m + 1) + Km = 2m + 1 e

1 −x2

1 −x2 /2

√ e

√ + 4 e 4

π

m

m ,

2π H H

(2.9)

1 2/(m+1) m+1/2 ) 2

and

x1/(m+1) + x2 , π

Fm (x) =

(2.10)

x ≥ 0.

(2.11)

Proof. We shall break the integral defining kQ+ (f )k2Hm into several pieces. We first consider those angles θ for which either cos θ or sin θ is small. Fix > 0, and define A = {θ | |θ − kπ/2| ≤ , k = 1, 2, 3, 4 and let

Ac

and

0 ≤ θ ≤ 2π}

be its complement in [0, 2π]. Then, by Jensen’s inequality, Z

kQ+ (f )k2Hm ≤

Z

1 |fb(ξ cos θ)|2 |fb(ξ sin θ)|2 |ξ|2m dξdθ + 2π A R Z Z 1 |fb(ξ cos θ)|2 |fb(ξ sin θ)|2 |ξ|2m dξdθ. + 2π Ac R

(2.12)

√ The integral over A has four parts. Consider the one with |θ| ≤ , on which cos θ ≥ 1 − 2 . Then with η = (cos θ)ξ, Z Z 1 |fb(ξ cos θ)|2 |fb(ξ sin θ)|2 |ξ|2m dξdθ ≤ 2π − R m+1/2 Z Z 1 1 |fb(ξ)|2 |ξ|2m dξdθ = 1 − 2 2π − R m+1/2 1 kf k2Hm , 2 1− π where we have used the fact that sup |fb(ξ sin θ)| ≤ 1. ξ

There are four contributions of this type, and hence m+1/2 Z Z 1 1 |fb(ξ cos θ)|2 |fb(ξ sin θ)|2 |ξ|2m dξdθ ≤ 4 kf k2Hm . (2.13) 2π A R 1 − 2 π Let us set c = kf − Mf k1 . Then cf (ξ) ≤ c + e−ξ cf (ξ)| + M |fb(ξ)| ≤ |fb(ξ) − M

2

/2

.

(2.14)

On Ac we split the integration into the two parts where |ξ| > R, and |ξ| ≤ R, for some R > 0 to be fixed later. On the latter region, using inequality (2.14) we obtain

Smoothness and Convergence to Equilibrium for Maxwellian Gas

Z Ac

Z

Z Ac

Z

|ξ|≤R

Z

Z

|ξ|≤R

Ac

|fb(ξ cos θ)|2 |fb(ξ sin θ)|2 |ξ|2m dξdθ ≤

2c2 + 2e−ξ

4 |ξ|≤R

531

2

c4 + c2 e−ξ

cos2 θ

2

sin2 θ

2c2 + 2e−ξ

+ c2 e−ξ

2

2

cos2 θ

sin2 θ

+ e−ξ

|ξ|2m dξdθ =

2

(2.15)

|ξ|2m dξdθ.

From now on, let us fix 2 ≤ 1/2. Then, on the set Ac , or sin2 θ ≥ 1/2, or cos2 θ ≥ 1/2. Hence, by (2.15) 1 2π Z 4 c4 + c2

Z

Z

Ac

|ξ|≤R

|fb(ξ cos θ)|2 |fb(ξ sin θ)|2 |ξ|2m dξdθ ≤

1 −x2

1 −x2 /2

√ e

√ |ξ|2m dξ + 4c2 + 4 e

π

2π

m

m≤ |ξ|≤R H

H

2 2 1 1 8 −x /2 −x

c2 R2m+1 + 4

√2π e

m + 4 √π e m . 2m + 1 H H

(2.16)

To handle the integration over the remaining region, we use the fact that

ξ2 exp − 2 2

≤c

if |ξ| ≥ − 1 log c2 . Thus, if we choose R = R(c, ) − 1 log c2 , by (2.14) we conclude that, if ξ ≥ R, or |fb(ξ sin θ)| ≤ 2c, or |fb(ξ cos θ)| ≤ 2c. So we obtain 1 2π

Z

Z

Ac

|ξ|>R

|fb(ξ cos θ)|2 |fb(ξ sin θ)|2 |ξ|2m dξdθ ≤ (2.17)

4c2 kf k2H m . (1 − 2 )m+1/2 Let us fix c = (m+1) . Then

sup c R(c, c 2

c≤1

1/(m+1)

2 )= (2m + 1)(m + 1) e

2m+1 ,

(2.18)

and, grouping inequalities (2.13), (2.16) and (2.17) we obtain c1/(m+1) 2 + c kf k2H m + π (1 − c2/(m+1) )m+1/2

2m+1

1 −x2 /2

1 −x2 1 8

+ 4 √ e (2m + 1)(m + 1)

m + 4 √π e 2m + 1 e 2π kQ+ (f )k2Hm ≤

4

H

Hm

.

(2.19) This proves the theorem, and gives at the same time the explicit form of the function Fm (·) and of the constants Cm and Km .

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E. A. Carlen, E. Gabetta, G. Toscani

Several useful variants of inequality (2.8) are easily established. In particular, this result becomes more useful if we replace the L1 norm, which is not monotonic, with the relative entropy, which is. This is easily done using the Kullback–Cszilar inequality which says that kf − Mf k2L1 ≤ 2 H(Mf ) − H(f ) and thus we can take c2 =

1 H(Mf ) − H(f ) 2

in inequality (2.14). We have Theorem 2.2. Smoothness bound on the gain term for Kac equation, entropic version. Let f be any density on R3 with kf kHm finite. Then, whenever kQ+ (f )k2Hm

H(Mf ) − H(f ) ≤ (1/2)m+2 , ≤ Cm Gm H(Mf ) − H(f ) kf k2Hm + Km ,

(2.20)

where Cm and Km are defined by (2.9) and (2.10) respectively, and 1 Gm (x) = π

x √ 2

1/2(m+1)

1 + x, 2

x ≥ 0.

(2.21)

The next version is given in terms of the Fisher information, and is different in that it does not require any “smallness” condition to apply. The main point is the determination of a bound on the decay of |fb(ξ)| in terms of the Fisher information I(f ). For the Kac equation, as well for Maxwell pseudomolecules and for certain rate functions b(cos θ), Fisher information is known to be non-increasing in time when evaluated along the solution [Mk66, BT92]. The result that follows is independent of the dimension, even regarding the constants, so we prove it on Rn . The Fisher information I(f ) is defined by Z Z p 2 |∇ f (v)| dv |∇ log f (v)|2 f (v) dv. (2.22) I(f ) = 4 Rn

Rn

We have Lemma 2.3. For any probability density f on Rn with I(f ) finite, √ I(f ) b . |f (ξ)| ≤ |ξ| Proof. As in the standard proof of the Riemann–Lebesgue Lemma, write Z 1 b f (v) − f (v + (π/|ξ|2 )ξ) eiξ·v dv, f (ξ) = 2 Rn so that |fb(ξ)| ≤ Next, write

1 2

Z Rn

|f (v) − f (v + (π/|ξ|2 )ξ)|dv.

(2.23)

Smoothness and Convergence to Equilibrium for Maxwellian Gas

533

| f (v) − f (v + (π/|ξ|2 )ξ) | = q q p p f (v) + f (v + (π/|ξ|2 )ξ) | f (v) − f (v + (π/|ξ|2 )ξ) | ≤ Z 1 q q p 2 f (v) + f (v + (π/|ξ| )ξ) (π/|ξ|) |∇ f (v + t(π/|ξ|2 )ξ)| dt. 0

Inserting this into the integral over v, and applying the Schwarz inequality, and then the Minkowski inequlity twice, one easily gets the stated result. Theorem 2.4. (Smoothness bound on the gain term for Kac equation, Fisher information version.) Let f be any density on R with kf kHm finite. Then there is a constant Km (I(f )) so that 1 (2.24) kQ+ (f )k2Hm ≤ kf k2Hm + Km (I(f )). 2 Proof. We proceed as in the proof of Theorem 2.1, except that we use I(f0 ) to control the size of |fb(ξ)| for large ξ. In consequence of Lemma 2.3, inequality (2.17) can now be substituted by Z Z 1 |fb(ξ cos θ)|2 |fb(ξ sin θ)|2 |ξ|2m dξdθ ≤ 2π Ac |ξ|>R (2.25) I(f ) 1 2 kf k . m H (1 − 2 )m+1/2 R2 2 Grouping inequalities (2.13), (2.16) and (2.25) we obtain 1 4 I 2 (f ) + 2 + 2 2 kf k2H m + kQ (f )kHm ≤ R (1 − 2 )m+1/2 π

1 −x2

2 1 8 2m+1 −x /2

R + 4 √ e

m + 4 √π e m . 2m + 1 2π H H √ Now, choose R = 2 I(f )/, and then = (I(f )) to satisfy 1 4 1 1 + = 2 m+1/2 π 4 2 (1 − ) and the result follows.

(2.26)

3. Optimal Exponential Convergence in the ||| · ||| Norm for the Kac Equation We present here, in the spirit of the proof of [GTW95], a new estimate for the rapid convergence towards equilibrium when sufficiently many moments exist initially. The new feature is that the best possible rate is obtained when the fourth moment exists. R Theorem 3.1. Let f0 be a probability density with R |v|4 |f0 (v)dv < K. Then there are constants B and C and a function S(·, t) such that |||f (·, t) − Mf (·) − S(·, t)||| ≤ Bte−tλ1 |||f (·, 0) − Mf (·) − S(·, 0)||| for all t ≥ 0 and with |||f (·, 0) − Mf (·) − S(·, 0)||| < ∞, such that for all m, etλ1 kS(·, t)kL1 + kS(·, t)kHm ≤ C, for all t ≥ 0. Here, λ1 = 1/4 is the spectral gap in the linearized collision operator for the Kac model.

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E. A. Carlen, E. Gabetta, G. Toscani

Proof. First, we need some preliminary bounds on the evolution of the moments to control the subtraction term S. These are simple analogs for the Kac model of certain bounds proved for the Boltzmann equation by Ikenberry and Truesdell [IT55]. Note in particular that the moments satisfy equations that are independent of the particular solution f . We will use this fact instead of direct calculation in the case of Maxwellian molecules. R For any natural number k, let us denote mk (t) = R v k f (v, t) dv, and let us put mk = mk (0). An easy computation shows that, Z v k Q+ (f, f )(v) dv R Z 1 (v cos θ + w sin θ)k f (v)f (w) dvdwdθ. R2 ×[0,2π] 2π Hence,

Z v 2k+1 Q+ (f, f )(v) dv = 0, R

which implies

m2k+1 (t) = m2k+1 e−t ,

and, if k = 4, owing to the conservation of the energy (m2 (t) = m2 ), we obtain that m4 (t) satisfies d 1 m4 (t) − 3m22 = − m4 (t) − 3m22 dt 4 so that m4 (t) = 3m22 + e−(1/4)t m4 − 3m22 . Let M denote the Maxwellian distribution with the same mass and temperature of f , and let us put c(ξ). φ(ξ, t) = fb(ξ, t) − M Now, we can’t divide by |ξ|4 and expect a finite supremum norm because of the possibly non-vanishing first and third moments of f . Thus we introduce a subtraction term to cancel these out. This is built by taking the third order Taylor polynomial of φ, and multiplying by a cut-off function as follows: Since f0 has four finite moments, taking a Taylor expansion of φ up to the fourth order, and using the above bounds, one gets ξ 3 −t ξ 4 e + m4 − 3m22 e−(1/4)t + o(ξ 4 ). φ(ξ, t) = −im1 ξ + im3 3! 4! Define X(ξ) = ξ if |ξ| ≤ 1, and X(ξ) = 0 otherwise, and let 3 4 b t) = −im1 X(ξ) + im3 X(ξ) e−t + X(ξ) m4 − 3m22 e−(1/4)t . S(ξ, 3! 4! b t). φ1 (ξ, t) = φ(ξ, t) − S(ξ, Consider that

4 ∂ b b t) 3 X(ξ) m4 − 3m22 e−(1/4)t . S(ξ, t) + S(ξ, ∂t 4 4!

Smoothness and Convergence to Equilibrium for Maxwellian Gas

Hence, φ1 satisfies ∂ b + (M c, φ1 ) + Q b + (S, b fb) + Q b + (M c, S) b − b + (φ1 , fb) + Q φ1 + φ1 Q ∂t 3 X(ξ)4 m4 − 3m22 e−(1/4)t . 4 4! Now, X(ξ cos θ)3 −t e + −im1 X(ξ cos θ) + im3 3! 0 X(ξ cos θ)4 m4 − 3m22 e−(1/4)t fb(ξ sin θ) dθ. 4! Z

b fb)(ξ, t) = 1 b + (S, Q 2π

2π

On the other hand, for all k ∈ N, Z 2π ξ k sink θX(ξ cos θ) dθ = 0, 0

Z

2π

ξ k sink θX(ξ cos θ)3 dθ = 0.

0

This implies that 1 2π

Z

2π

0

ξ2 −t dθ = 0 im1 X(ξ cos θ) 1 − im1 ξe − m2 2

and, since for a certain ξ¯ 2 3 fb(ξ, t) − 1 + im1 ξe−t + m2 ξ ≤ |ξ| fb000 (ξ, ¯ t) 2 3 and by H¨older inequality Z 3/4 b000 ¯ |v|3 f (v, t) dv ≤ m4 − 3m22 f (ξ, t) ≤ R

we obtain 1 2π

Z 2π im1 X(ξ cos θ)fb(ξ sin θ, t) dθ ≤ 0

3/4 |ξ|4 . |m1 | m4 − 3m22 3! Using the same method, and recalling that Z b0 ¯ 1/2 |v|f (v, t) dv ≤ m2 = 1, f (ξ, t) ≤ R

we obtain 1 2π

Z 2π X(ξ cos θ) b |ξ|4 f (ξ sin θ, t) dθ ≤ |m3 | . im3 0 3! 3!

535

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E. A. Carlen, E. Gabetta, G. Toscani

b + (M c, S). b Thus The exact same bound can be derived for Q b fb)(ξ, t)| + |Q b + (M c, S)| b ≤ b + (S, |Q h i |ξ|4 3/4 |ξ|4 + 2e−(1/4)t m4 − 3m22 . + |m3 | 2e−t |m1 | m4 − 3m22 3! 4! k/n

We can simplify this using the fact that |mk | ≤ mn becomes

for all even n > k. The bound

b fb)(ξ, t)| + |Q b + (M c, S)| b ≤ 4m4 e−t + [m4 − 3m2 ]e−t/4 b + (S, |Q 2 Hence

|ξ|4 . 3!

i h ∂ b + (M c, φ1 )| + ce−t + de−(1/4)t |ξ|4 b + (φ1 , fb) + Q φ1 + φ1 ≤ |Q ∂t

with constants c and d which are given explicitly just above. Define φ2 (ξ, t) = φ1 (ξ, t)/|ξ|4 . Then, φ2 satisfies h i ∂ b + (φ1 , fb) + Q b + (M c, φ1 )| + ce−t + de−(1/4)t . φ2 + φ2 ≤ 1 |Q |ξ|4 ∂t Now it remains to estimate 1 b+ b + (M c, φ1 )| , |Q (φ1 , fb) + Q |ξ|4 and to show how it is controlled by the spectral gap. c, φ1 )|/|ξ|4 . Clearly b + (M Consider first the term |Q

1 2π

Z

c, φ1 )|/|ξ|4 ≤ b + (M |Q 2π

c((cos θ)ξ)φ2 ((sin θ)ξ)|(sin θ)4 dθ ≤ |M

0

ckL∞ |||φ2 ||| kM

1 2π

Z

2π

(sin θ)4 dθ. 0

ckL∞ = 1, and since the same is true for fb, we get the same bound for the other Now, kM term. The value of the integral is 3/8, and so we have i h 1 ∂ |||φ2 ||| + |||φ2 ||| ≤ ce−t + de−(1/4)t ∂t 4 which is

i h ∂ (1/4)t e |||φ2 ||| ≤ ce−(3/4)t + d ∂t so that upon integration we have e(1/4)t |||φ2 (t)||| ≤ (4/3)c + dt. This proves the bound, and makes it a simple matter to reckon the constants.

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This proves the theorem, but it will be helpful for what follows to make the connection with L more explicit. To see the role of L explicitly, note that the bound we get for b + (M c, φ1 ), is, as we have pointed out, the same as the bound we would get b + (φ1 , fb) + Q Q + c b + (M c, φ1 ). But if we define h by φ1 = M h, then b for Q (φ1 , M ) + Q c) + Q b + (M c, φ1 ) − φ1 = M b + (φ1 , M \ Lh Q

R

since R M h(v)dv = 0. This is the reason that the constant we get is the spectral gap – which is naturally best possible. For the same reason we will see that the spectral gap controls the approach to equilibrium also for Maxwellian molecules. There, however, it will be more convenient to rely on a further development of the above explanation, instead of on direct calculation. Hence Sect. 8 will shed light on why this approach did yield the optimal bound. 4. Interpolation Inequalities This section contains the several interpolation inequalities that we shall use to extract strong convergence estimates from weak convergence estimates. The first result shows that ||| · |||α and arbitrarily little k · kHm control k · kHk for m sufficiently larger than k. Theorem 4.1. Let k ≥ 0 and α, β, r > 0, 0 < r < 1, be given. Then 2(1−r) 2r kf k2r kf kHk ≤ C(r, β)|||f |||α HM + kf kHN with

(1 − r)(n + β) k + (2 + α)(1 − r) , N =M+ , r 2r 1−r , C(r, β) = |B n |(1 + n/β) and where |B n | denotes the volume of the unit ball in Rn . M=

Proof. For any p > 0, and any r with 0 < r < 1, Z |fb(ξ)|2 |ξ|2k dn ξ = kf k2Hk = Z

Rn

|fb(ξ)|2−2r b 2r 2k+(2+α)(2−2r) |f (ξ)| |ξ| (1 + |ξ|p )r (1 + |ξ|p )−r dn ξ ≤ (2+α)(2−2r) |ξ| n R Z 2−2r |fb(ξ)|2r |ξ|2k+(2+α)(2−2r) (1 + |ξ|p )r (1 + |ξ|p )−r dn ξ ≤ |||f |||α Rn Z r 2−2r |||f |||α |fb(ξ)|2 |ξ|(2k+(2+α)(2−2r))/r (1 + |ξ|p )dn ξ × Rn

Z

p −r/(1−r) n

Rn

(1 + |ξ| )

d ξ

1−r .

For the last integral to converge, we require that p > r(1 − r)n, so let β > 0, and put p = r(1 − r)(n + β). Then the integral is Z (1 + |ξ|r(1−r)(n+β) )−1/r(1−r) dn ξ ≤ Rn Z ∞ n n r−(1+β) dr = |B n |(1 + n/β) . |B | + n|B | 0

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E. A. Carlen, E. Gabetta, G. Toscani

The next inequality shows that control of the sufficiently many moments and control on the L2 norm together control the L1 norm. Theorem 4.2. Let f be an integrable function on Rn . Then for all k > 0, Z

Z Rn

|f (v)|dn v ≤ C(n, k)

where

C(n, k) =

n 4k

Rn

|f (v)|2 dn v

4k/(n+4k)

+

2k Z (n+4k)

4k n

Rn

|v|2k f (v)dn v

n (n+4k)

,

n/(n+4k) |B n |2k/(n+4k)

and again, |B n | denotes the volume of the unit ball in Rn . Proof. We may assume that f is non-negative. Also, by translation invariance, we may assume that the infimum above is achieved at v = 0. Let R > 0 be chosen. Then Z Z Z n n f (v)d v = f (v)d v + f (v)dn v ≤ Rn |v|≤R |v|≥R Z n n 1/2 −2k kf kL2 + R |v|2k f (v)dn v. |B |R Rn

Optimizing in R now yields the result.

5. Optimal Exponential Convergence in the Strong L2 Norm and Sobolev Norms for the Kac Equation One now easily sees that the assertion in (1.16) is a consequence of Theorems 4.1 and 4.2, with a readily computed constant C . Since Theorem 2.3 gives us a uniform bound on kf (·, t)kHm provided it is finite initially, and provided the Fisher information I(f ) is initially finite, combining this with Theorem 3.1, we have: Theorem 5.1. Let f0 be any probability density on the real line with unit variance, finite fourth moment m4 and finite Fisher information I(f0 ). Then for any > 0 there is a fixed constant m, depending only on , so that if Z Z f (v)|v|2m dv + |fb(ξ)|2 |ξ|2m dξ ≤ K R

R

there is a universal, computable constant C depending only on , m4 , I(f0 ) and K so that the solution of the Kac equation f (·, t) with initial data f0 (·) satisfies kf (·, t) − Mf (·)kL1 ≤ Ce−(1−)λ1 t , where λ1 is the spectral gap in the linearized collision operator L for the Kac equation; i.e., λ1 = 1/4. Moreover, increasing m we obtain the same result if the L1 norm is replaced by any k H norm.

Smoothness and Convergence to Equilibrium for Maxwellian Gas

539

This result proves a conjecture of McKean [Mk66] concerning the optimal rate of convergence for the Kac equation, though the result requires moments and Sobolev space regularity of all orders to reach the optimal rate of convergence. In contrast, McKean’s proof used only a bound on the third moment and on the H1 norm of the square root of the density f , but yielded a bound on the rate that was smaller by more than an order of magnitude, and that did not improve in the presence of greater regularity. This concludes our treatment of the Kac model. We now turn to the Maxwellian case. 6. Geometric Lemmas for the Maxwellian Gain Term Fix a unit vector n in R3 , and consider the two maps ± : R3 → R3 defined by ± (ξ) = ξ± =

ξ ± |ξ|n . 2

(6.1)

Each of these maps “opens up” R3 into a half space with n as the bounding normal. More precisely, notice that (6.2) 2ξ+ · n = ξ · n + |ξ| which is always positive unless ξ is a negative multiple of n, in which case ξ+ = 0. Thus, if we delete the ray antiparallel to n from the domain of + , its range is H(n) = {η | η · n > 0}.

(6.3)

Moreover, this restricted map is one-to-one on this range, and one easily works out, using (6.4) |ξ+ |2 = |ξ|(ξ · n), that the inverse map is −1 + (η) = 2η −

|η|2 n. (η · n)

(6.5)

Moreover, let θ and 2± be defined by cos(θ) =

ξ·n |ξ|

and

ξ± · n . |ξ± |

(6.6)

|ξ+ ||ξ| cos(θ/2).

(6.7)

cos(2± )

Then one easily deduces from the above that 2+ = θ/2

and

It is now an easy matter to compute the Jacobian of the coordinate transformation given by + : 4 ∂ξ 4 = . (6.8) J = ∂ξ+ cos2 (2+ ) cos2 (θ/2) An exactly analogous analysis of − leads to: 4 4 ∂ξ . = J ∂ξ− cos2 (2− ) sin2 (θ/2)

(6.9)

The first Jacobian is singular only near θ = π, and the second only near θ = 0. In particular, away from the origin, at least one of the two is bounded by 8.

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E. A. Carlen, E. Gabetta, G. Toscani

7. Propagation of Smoothness for Maxwellian Molecules Once again, we use the Fourier transform. Since the Maxwellian collision kernel is less directly related to convolution than the Kac collision kernel, the computation is more involved. But it has been carried out by Bobylev [Bo88] who found that Z b+ (f, f )(ξ) = (7.1) fb(ξ+ )fb(ξ− )B(n · ξ/|ξ|)dn. Q S2

The notation is that introduced in the previous section. The next result contains Theorem 1.2, and provides some explicit information that we did not bring into the introduction. It is possible to present it in a form analogous to that of Theorem 2.1 for the Kac model, but this would require more work, and the present version suffices. Moreover, to keep the formulas readable, we explicitly treat R the case B = 1. It will be clear from the proof that all we really use is the fact that |B(ξ · n/|ξ|)dn < ∞. Thus Theorem 1.2 will be established in the stated generality. Theorem 7.1 (Smoothness bound on the gain term for Maxwellian molecules). Let f be any probability density with finite second moments on R3 such that kf kHm is finite. Then there is a constant C(m, T ) so that kQ+ (f )k2Hm ≤ (1/2)kf k2Hm + C(m, T )

(7.2)

kf − Mf k2L1 ≤ 2−(m+5) .

(7.3)

whenever

Proof. Fix any > 0. Then for j = 1, 2, define the sets A() j by j 2 2 A() j = {(ξ, n) | ((1 + (−1) cos θ) ≤ (2) }.

Z

Z

Then

S2

XZ Z

j=1,2

Z

S2

R3

R3

Z

S2

R3

(1 −

(7.4)

|fb(ξ+ )|2 |fb(ξ− )|2 |ξ|2m dξdn =

1A() (ξ, n)|fb(ξ+ )|2 |fb(ξ− )|2 |ξ|2m dξdn + j

X j=1,2

1A() (ξ, n))|fb(ξ+ )|2 |fb(ξ− )|2 |ξ|2m dξdn. j

Consider first the integral over A() 2 . On the set, ξ+ and ξ are close to one another, but we have little control over ξ− . Thus we use the Riemann-Lebesgue theorem to estimate: |fb(ξ− )|2 ≤ 1. Next we observe that 1 − cos θ = 1 − cos(22+ ) = 2 sin2 (2+ ) and that Hence,

sin2 (2+ ) ≥ (1 − cos 2+ ). () A() 2 (ξ, n) ⊂ A2 (ξ− , n).

(7.5)

Smoothness and Convergence to Equilibrium for Maxwellian Gas

541

2 Furthermore, on A() 2 , cos θ ≥ 1 − 2. Hence cos (θ/2) ≥ √ 1 − , and the Jacobian J(∂ξ/∂ξ± ) is bounded above 4/(1 − ). Finally, |ξ| ≤ |ξ− |/ 1 − . Hence, Z Z 1A() (ξ, n)|fb(ξ+ )|2 |fb(ξ− )|2 |ξ|2m dξdn ≤ 2 S 2 R3 Z Z 4 1 () (η, n)|fb(η)|2 ||η|2m dηdn ≤ (1 − )(m+1) S 2 H(n) A2 Z Z 4 1 () (η, n)|fb(η)|2 ||η|2m dηdn = (7.6) (1 − )(m+1) S 2 R3 A2 Z Z 4 b 2 2m dη = () (ξ, n)dn |f (η)| ||η| 1 (1 − )(m+1) R3 S 2 A2 4 4π kf k2Hm , (m+1) (1 − )

where H(n) is defined in (6.3). Clearly, we get the same bound for the integral over A() 1 . Now fix some R > 0 to be chosen later, and split the remaining integration into the two parts where |ξ| > R and |ξ| ≤ R. On the latter region, we again use the Riemann–Lebesgue estimate to obtain Z Z X (1 − 1A() (ξ, n))|fb(ξ+ )|2 |fb(ξ− )|2 |ξ|2m dξdn ≤ S2

|ξ|≤R

j=1,2

Z

S2

j

Z |ξ|2m dξdn ≤

(7.7)

|ξ|≤R 2m+1

π R . 2m + 1

To handle the integration over the remaining region, we first simplify notation by writing c = kf − Mf kL1 . Then by one more use of the Riemann–Lebesgue lemma, cf (ξ± )| = c + e−T |ξ± | |fb(ξ± )| ≤ c + |M

2

/2

.

(7.8)

Again with n fixed, consider those ξ in the final region of integration with ξ · n ≥ 0. Recall that |ξ+ | = |ξ| cos 2+ and |ξ− | = |ξ| sin 2+ . This is the region where cos 2 ≤ π/4, but sin2 2 ≥ . Hence in this region p

|fb(ξ− )| ≤ c + e−T R

2

/2

≤ 2cρ

when we choose R = −2 ln c/T . We now fix this choice of R. Making the same sort of estimates for ξ · n ≤ 0, we get Z Z X (1 − 1A() (ξ, n))|fb(ξ+ )|2 |fb(ξ− )|2 |ξ|2m dξdn ≤ S2

|ξ|>R

j=1,2

2c2 2m

4 1−

j

Z

Z

S 2 R3 m+2 2

|fb(η)|2 |η|2m dηdn ≤

c 2 kf k2Hm . 1− Putting together all of the pieces yields

(7.9)

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E. A. Carlen, E. Gabetta, G. Toscani

kQ+ (f )k2Hm ≤

8 2m+2 c2 + kf k2Hm + K(c, T, ), 1− (1 − )(m+1)

where K(c, T, ) =

1 (−2 ln c/T )(2m+1)/2 . 2m + 1

Finally, choose = c2 /2 to obtain the required multiple of kf k2Hm .

Several useful variants of the inequality are easily established. First, the Kullback– Cszilar inequality says that kf − Mf k2L1 ≤

1 H(Mf ) − H(f ) 2

and thus we immediately have Theorem 7.2. (Smoothness bound on the gain term for Maxwellian molecules, entropic version). Let f be any probability density on R3 with finite second moments such that kf kHm is finite. Then there is a constant C(m, T ) so that

whenever

kQ+ (f )k2Hm ≤ (1/2)kf k2Hm + C(m, T )

(7.10)

H(Mf ) − H(f ) ≤ 2−(m+4) .

(7.11)

The utility of this simple variant lies in the monotonicity of the entropy: once we arrive at a time for which the condition (7.11) is satisfied, it remains satisfied forever. The next version is given in terms of the Fisher information, and is different in that it does not require any “smallness” condition to apply. Theorem 7.3. (Smoothness bound on the gain term for Maxwellian molecules, Fisher information version). Let f be any probability density on R3 with finite second moments and such that kf kHm is finite. Then there is a constant C(m, T, I(f )) so that kQ+ (f )k2Hm ≤

1 kf k2Hm + C(m, T, I(f )). 2

(7.12)

Proof. We proceed as in the proof of the main theorem, except that we use I(f ), through Lemma 2.3, to control the size of |fb(ξ)| for large ξ. 8. Optimal Exponential Convergence in the ||| · ||| Norm for Maxwellian Molecules Here we prove Theorem 1.3, which is for Maxwellian molecules the analog of Theorem 3.1. In the course of the proof we obtain an interesting, somewhat indirect, explicit evaluation of λ1 . c(ξ), and we let P (ξ, t) Proof of Theorem 3.1. Once again, we put φ(ξ, t) = fb(ξ, t) − M denote the third degree Taylor polynomial of φ(ξ, t), and let R(ξ) denote the third degree c(ξ). This is a bit more complicated than in the one dimensional Taylor polynomial of M Kac case, in that now there can be second degree differences between P (ξ, t) and the c(ξ). third degree Taylor polynomial for M

Smoothness and Convergence to Equilibrium for Maxwellian Gas

543

However, we still proceed in the same way. First, let K(ξ), a cut-off function, be any smooth approximation to 1{|ξ|≤1} agreeing with it outside a thin neighborhood of the unit sphere. Then define b t) = K(ξ)(P (ξ, t) − R(ξ)). S(ξ,

(8.1)

Clearly, since the fourth moment of f is finite, so is |||φ(·, t) − S(·, t)|||. Moreover,[IT56], the coefficients of P are simply multiples of certain corresponding moments of f . And, as proved in this reference, these moments tend to their equilibrium values exponentially fast with a rate that is independent of the initial data, as in the Kac model case, provided only that the moment exists initially. Thus S(·, t) has the form K(ξ) times a polynomial of degree three whose coefficients are decaying to zero exponentially fast. Now any monomial in ξ multiplied by K(ξ) is the Fourier transform of a function that has a finite norm in both L1 and Hm for any n. Hence we have that (8.2) kS(·, t)kL1 + kS(·, t)kHm ≤ Ce−tλ for some C depending on f and n, and some universal λ. In our treatment of the Kac model, we computed the corresponding λ. Here we avoid this, and simply make the observation that we require: whatever the value of λ, it is the case that λ ≥ λ1 . Because of the universality of λ, this follows from the results of Cercignani, Lampis and Sgarra [CLS88] concerning the behavior of small perturbations of equilibrium. Specifically, consider the coefficient c1,2 (t) of K(ξ)ξ1 ξ2 in S(·, t). Pick δ small enough that δv1 v2 + |v|4 ≥ 0, and then pick small enough that f0 (v) = C(1 + (δv1 v2 + |v|4 ))M (v)

(8.3)

in which C is the normalizing constant, is close enough to equilibrium for the theorem of [CGS88] to apply. Then the corresponding solution of the Boltzmann equation approches equilibrium like e−tλ1 . But clearly it appraches equilibrium no faster than c1,2 (t) tends to zero. Hence c1,2 (t) must tend to zero at least as fast as e−tλ1 . The argument for the other coefficients is the same. Thus, (8.3) holds with λ = λ1 . Not only does S(·, t) decay at this rate, but since its coefficients in the Fourier transform respresentation satisfy the ordinary differential equations derived in [IT56], we have that the time derivative of S(·, t) does as well. At this point we have stepped around most of the direct computation in the proof of Theorem 3.1 – which was only possible there because we made the simplifying assumption of constant rate function – and can easily conclude the proof. Let φ1 = φ − S as before. Then ∂S ∂φ1 + φ1 = +S + ∂t ∂t + + b +Q b (S, b M c) + Q b+ (fb, φ1 ) + Q b+ (φ1 , M c). b (fb, S) Q Then, all the terms involving S are estimated as in the proof of Theorem 3.1, and we obtain that |

∂φ1 b+ (fb, φ1 ) + Q b+ (φ1 , M c) | ≤ C|ξ|4 e−tλ1 + φ1 − Q ∂t

544

E. A. Carlen, E. Gabetta, G. Toscani

for some constant C. Again as before, define φ2 (ξ, t) = φ1 (ξ, t)/|ξ|4 . Then the above estimate becomes ∂φ2 b+ (fb, φ1 ) + Q b+ (φ1 , M c) /|ξ|4 | ≤ Ce−tλ1 . + φ2 − Q | ∂t It remains to estimate b+ (φ1 , M c)|/|ξ|4 . b+ (fb, φ1 ) + Q |Q

(8.4)

Consider first c)|/|ξ|4 ≤ b+ (φ1 , M |Q

Z S2

c(ξ− )||φ2 (ξ+ )| cos4 (θ/2)B(cos(θ))dn, |M

where we have used the formulas (6.6) and (6.7). This in turn is dominated by Z ckL∞ kφ2 kL∞ cos4 (θ/2)B(cos(θ))dn = kM 2 S Z cos4 (θ/2)B(cos(θ))dn. |||φ1 ||| S2

b+ (fb, φ1 )|/|ξ|4 . Putting it all together, Naturally we obtain the same estimate for |Q we then have Z ∂|||φ1 ||| + 1−2 cos4 (θ/2)B(cos(θ))dn |||φ1 ||| ≤ Ce−tλ1 . ∂t 2 S Solving this differential inequality gives us exponential decay at rate λ with Z cos4 (θ/2)B(cos(θ))dn }. λ = min{λ1 , (1 − 2 S2

We now claim that the terms in the minimum are actually equal. To see this, let h(v) = |v|4 − b|v|2 − c, where b and c are chosen to make Z Z h(v)|v|2 M (v)d3 v = h(v)M (v)d3 v = 0. R3

R3

Of course, h is just the spherically symmetric fourth order Hermite polynomial. Now pick a small enough that 1 + ah(v) is positive, and consider f (v) = M (v)(1 + ah(v)), then φ1 (ξ) = aM (ξ)|ξ|4 , so that φ2 (ξ) = aM (ξ). In this case, we lose nothing in the estimate above: Z + 4 c b cos4 (θ/2)B(cos(θ))dn. |Q (φ1 , M )|/kξ| = aM (ξ) S2

Again, as we have observed in Sect. 3, c) + Q b + (M c, φ1 ) − φ1 = M b + (φ1 , M \ Lh, Q and hence \ M Lh = − 1 − 2

Z S2

cos4 (θ/2)B(cos(θ))dn h.

(8.5)

Smoothness and Convergence to Equilibrium for Maxwellian Gas

545

Thus, as is well known, h(v) is an eigenvector of L, and the corresponding eigenvalue is given by the above formula. This means that Z cos4 (θ/2)B(cos(θ))dn λ1 ≤ (1 − 2 S2

and while at first sight one might suppose that another Hermite polynomial of order higher than four produces the gap, the estimate that was applied to (8.4) can be equally well applied to (8.5) for these higher eigenfunctions to see that this is not the case. Thus we have computed λ1 , and proved that ∂|||φ2 ||| + λ1 |||φ2 ||| ≤ Ce−tλ1 . ∂t This together with our bounds on S yield the theorem.

9. Optimal Exponential Convergence in the Strong L1 Norm and Sobolev Norms for Maxwellian Molecules This section is very short; we only need collect results to provide a proof of Theorem 1.1, and to explain how to compute the constants involved in it. Proof of Theorem 1.1. Since we have established Theorem 1.3 in Sect. 8, and have established Theorem 1.2 in Sect. 7, Theorem 1.1 follows from the interpolation inequalities, Theorems 4.1 and 4.2 , just as in the case of the Kac model.

References Arkeryd, L.: Stability in L1 for the spatially homogeneous Boltzmann equation. Arch. Rat. Mech. Anal. 103, 151–167 (1988) [Bo88] Bobylev, A.V.: The theory of the nonlinear spatially uniform Boltzmann equation for Maxwellian molecules. Sov. Sci. Rev. c 7, 111–233 (1988) [BT92] Bobylev, A.V., Toscani, G.: On the generalization of the Boltzmann H-theorem for a spatially homogeneous Maxwell gas. J. Math. Phys. 33, 2578–2586 (1992) [Ca57] Carleman, T.: Probl`emes Matematiques dans la Th´eorie Cin´etique des Gaz. Uppsala: AlmqvistWiksells, 1957 [Ce88] Cercignani, C.: The Theory and Application of the Boltzmann Equation. New York: Springer, 1988 [CLS88] Cercignani, C., Lampis, M., Sgarra, C.: L2 -stability near equilibrium of the solution of the homogeneous Boltzmann equation in the case of Maxwellian molecules. Meccanica 23, 15–18 (1988) [CC92] Carlen, E.A., Carvalho, M.C.: Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation. J. Stat. Phys. 67, 575–608 (1992) [CC94] Carlen, E.A., Carvalho, M.C.: Entropy production estimates for Boltzmann equations with physically realistic collision kernels. [CELMR96] Carlen, E.A., Esposito, R., Lebowitz, J., Marra, R., Rokhlenko, A.: Kinetics of a Model Weakly Ionized Plasma in the Presence of Multiple Equilibria. Arch. Rat. Mech. Anal. 142, 193–218 (1998 [El83] Elmroth, T.: Global boundedness of moments of solutions of the Boltzmann equation for forces of infinite range. Arch. Rat. Mech. Anal. 82, 1–12 (1983) [De94] Desvillettes, L.: On the regularizing properties of the non cut-off Kac equation. Commun. Math. Phys. 168, 417–440 (1995) [Ar88]

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[GP96]

[GTW95] [IT56] [Ka57] [Mk66] [Li95] [LT95] [To92] [To95] [WU70]

[We93] [We94]

E. A. Carlen, E. Gabetta, G. Toscani

Gabetta, E.: On a conjecture of McKean with application to Kac’s model. Trans. Theo. Stat. Phys. 24, 305–317 (1995) Gabetta, E., Pareschi, L.: About the non cut-off Kac equation: Uniqueness and asymptotic behaviour. Pubbl. Istituto Analisi Numerica CNR n. 911, Pavia, 1994, Comm. Nonlinear Appl. Math. in press Gabetta, E., Pareschi, L.: Boundedness of moments and trend to equilibrium for the non “cutoff” Kac equation. Preprint n. 227, Dip. Matematica Universit`a di Ferrara, 1995, Rend. Sem. Math. Palermo, in press Gabetta, E., Toscani, G., Wennberg, B.: Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation. J. Stat. Phys. 81, 901–934 (1995) Ikenberry, E., Truesdell, C.: On the pressure and the flux of energy according to Maxwell’s kinetic energy I. J. Rat. Mech. Anal. 5, 1–54 (1956) Kac, M.: Probability and Related Topics in Physical Sciences. London, New York: Interscience Publ. LTD., 1957 McKean Jr., H.P.: Speed of approach to equilibrium for Kac’s caricature of a Maxwellian gas. Arch. Rational Mech. Anal. 21, 343–367 (1966) Lions, P.L.: Compactness in Boltzmann’s equation via Fourier integral operatorsand applications III. J. Math. Kyoto Univ. 34, 539–584 (1994) Lions, P.L., Toscani, G.: A sthrenghtened central limit theorem for smooth densities. J. Funct. Anal. 128, 148–167 (1995) Toscani, G.: New a priori estimates for the spatially homogeneous Boltzmann equation. Cont. Mech. Termodyn. 4, 81–93 (1992) Toscani, G.: On regularity and asymptotic behaviour of a spatially homogeneous Maxwell gas. Preprint (1995); Rend. Seminario Matem. Palermo, in press Wang Chang, C.S., Uhlenbeck, G.E.: The kinetic Theory of Gases. In: Studies in Statistical mechanics V, eds. J. de Boer and G.E. Uhlenbeck, Amsterdam: North Holland Publishing Co., 1970 (This is a collection of Michigan Research Institute Reports written by the authors during the period 1948 to 1956) Wennberg, B.: Stability and exponential convergence for the Boltzmann equation. Thesis, Chalmers University of Technology (1993) Wennberg, B.: Regularity in the Boltzmann equation and the Radon transform. Commun. Part. Diff. Eq. 19, 2057–2074 (1994)

Communicated by J. L. Lebowitz

Commun. Math. Phys. 199, 547 – 589 (1999)

Communications in

Mathematical Physics © Springer-Verlag 1999

Doubles of Quasi-Quantum Groups Frank Hausser?,?? , Florian Nill??? Freie Universit¨at Berlin, Institut f¨ur Theoretische Physik, Arnimallee 14, D-14195 Berlin, Germany. E-mail: [email protected]; [email protected] Received: 27 August 1997 / Accepted: 26 May 1998

Abstract: In [Dr1] Drinfeld showed that any finite dimensional Hopf algebra G extends to a quasitriangular Hopf algebra D(G), the quantum double of G. Based on the construction of a so-called diagonal crossed product developed by the authors in [HN], we generalize this result to the case of quasi-Hopf algebras G. As for ordinary Hopf algebras, as a vector space the “quasi-quantum double” D(G) is isomorphic to Gˆ ⊗ G, where Gˆ denotes the dual of G. We give explicit formulas for the product, the coproduct, the R-matrix and the antipode on D(G) and prove that they fulfill Drinfeld’s axioms of a quasitriangular quasi-Hopf algebra. In particular D(G) becomes an associative algebra containing G ≡ 1Gˆ ⊗ G as a quasi-Hopf subalgebra. On the other hand, Gˆ ≡ Gˆ ⊗ 1G is not a subalgebra of D(G) unless the coproduct on G is strictly coassociative. It is shown that the category Rep D(G) of finite dimensional representations of D(G) coincides with what has been called the double category of G-modules by S. Majid [M2]. Thus our construction gives a concrete realization of Majid’s abstract definition of quasi-quantum doubles in terms of a Tannaka–Krein-like reconstruction procedure. The whole construction is shown to generalize to weak quasi-Hopf algebras with D(G) now being linearly isomorphic to a subspace of Gˆ ⊗ G. Contents 1 2 2.1 2.2 2.3 3 ?

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548 Quasitriangular Quasi-Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 Basic definitions and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 Graphical calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554 The antipode image of the R-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 Doubles of Quasi-Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567

Supported by DFG, SFB 288 Differentialgeometrie und Quantenphysik. Current address: Dipartimento di Matematica, Universit´a “La Sapienza”, P. le Aldo Moro 2, 00185 Roma, Italy. E-mail: [email protected] ??? Supported by DFG, SFB 288 Differentialgeometrie und Quantenphysik. ??

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3.1 3.2 3.3 3.4 3.5 4 A B

D(G) as an associative algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567 Coherent 1-flip operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570 Left and right diagonal crossed products . . . . . . . . . . . . . . . . . . . . . . . . . 573 The quasitriangular quasi-Hopf structure . . . . . . . . . . . . . . . . . . . . . . . . 575 The category RepD(G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 Doubles of Weak Quasi-Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 582 The Twisted Double of a Finite Group . . . . . . . . . . . . . . . . . . . . . . . . . . 585 The Monodromy Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588

1. Introduction Given a finite dimensional Hopf algebra G and its dual Gˆ Drinfeld [Dr1] has introduced the quantum double D(G) ⊃ G as the universal Hopf algebra extension of G satisfying 1. There exists a unital algebra embedding D : Gˆ −→ D(G) such that D(G) is algeˆ braically generated by G and D(G). ˆ Then RD := eµ ⊗D(eµ ) ∈ D(G)⊗D(G) 2. Let eµ ∈ G be a basis with dual basis eµ ∈ G. is quasitriangular. It follows that as a coalgebra D(G) = Gˆ cop ⊗ G, where “cop” refers to the opposite coproduct. However, when realized on Gˆ ⊗ G, the algebraic structure of D(G) becomes more involved. It has been analyzed in detail by S. Majid as a particular example of his notion of double crossed products, see [M3, M4] and references therein. The dual version of the quantum double has been introduced for infinite dimensional compact quantum groups in [PW] as the mathematical structure underlying the quantum Lorentz group. During the 90’s the quantum double has become of increasing importance as a quantum symmetry in two-dimensional lattice and continuum QFT. In continuum theories the quantum double D(G) of a finite group G (i.e. G = CG) has first been applied (mostly in a twisted version, which we will come back to below) to describe the symmetry underlying the sector structure of orbifold models in [DPR]. Quite interestingly, the same structure appears as a residual generalized “dyon-symmetry” in spontanously broken (2+1)-dimensional Higgs models with a finite unbroken subgroup G [BaWi]. For the role of quantum doubles in integrable field theories see, e.g. [BL]. More recently, in the framework of algebraic QFT, M. M¨uger [M¨u] has also found the double of a finite group G acting as a global symmetry on a “disorder-field extension” Fˆ of a massive 2-dimensional field algebra F with global gauge symmetry G. As opposed to the above cases, in this type of models the “disorder-part” Gˆ of the double is also spontaneously broken, corresponding to a violation of Haag duality (for double cones) ˆ The Haag dual extension Aˆ ⊃ A is for the D(G)-invariant observable algebra A ⊂ F. then recovered as the invariant subalgebra of Fˆ under the unbroken symmetry G. On the lattice, related but prior to M¨uger’s work, the double of a finite group G has been realized by K. Szlach´anyi and P. Vecserny´es as a symmetry realized on the order×disorder field algebra of a G-spin quantum chain [SzV]. Since for G = ZN the double coincides with (the group algebra of) ZN × ZN , this generalizes the well known order×disorder symmetry of abelian G-spin models. This investigation has been substantially extended to arbitrary finite dimensional C ∗ -Hopf algebras G in [NSz], where the authors show that such “Hopf spin models” always have D(G) as a universal

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localized cosymmetry. This means that under the assumption of a Haag dual vacuum representation (i.e. absence of spontanous symmetry breaking) the full superselection structure of these models is precisely created by the irreducible representations of D(G). The formulation of [NSz] also allowed for a generalization of duality transformations to the non-commutative and non-cocommutative setting. As it has turned out meanwhile, very much related results have been obtained independently for lattice current algebras on finite periodic lattice chains by A. Alekseev et al. [AFFS]. For these models the authors have completely determined the representation category, showing that it is in one-to-one correspondence with Rep K1 , where K1 is the algebra living on a minimal loop consisting of one site and one link biting into its own tail. Using the braided-group theory of [M5] (see also [M4]), it has been realized by one of us [N1], that K1 is in fact again isomorphic to a quantum double D(G). Also, requiring G to be a modular Hopf algebra as in [AFFS], the Hopf spin model of [NSz] has been shown in [N1] to be isomorphic to the lattice current algebra of [AFFS] by a local transformation of the generators1 . As a common feature of all these models we emphasize that under the quantum physical requirement of positivity they only give rise to quantum symmetries with integer q-dimensions [N2]2 . Thus, to construct “rational” models with a finite sector theory and non-integer dimensions one is inevitably forced to depart from ordinary Hopf algebras G. Here, the most fashionable candidates are the truncated semisimple versions of the q-deformations Uq (g), g a simple Lie algebra, at roots of unity, q N = 1. Also, since lattice current algebras have been invented as regularized verions of WZNW-models [AFFS, AFSV, AFS, ByS, Fa, FG], they should eventually be studied at roots of unity. Following G. Mack and V. Schomerus [MS], truncated quantum groups at q N = 1 have to be described as weak quasi-Hopf algebras in the sense of Drinfeld [Dr2], with the additional feature 1(1) 6= 1 ⊗ 1, where 1 : G −→ G ⊗ G denotes the coproduct3 . To formulate lattice current algebras at roots of unity one may now combine the methods of [AFFS] with those developed by [AGS,AS] for lattice Chern–Simons theories. However, it remains unclear whether and how for q = root of unity the structural results of [AFFS] survive the truncation to the semi-simple (“physical”) quotients. Similarly, the generalizations of the model, the methods and the results of [NSz] to weak quasi quantum groups are by no means obvious. In particular one would like to know whether and in what sense in such models universal localized cosymmetries ρ : A −→ A ⊗ G still provide coactions and whether G would still be (an analogue of) a quantum double of a quasi-Hopf algebra. In fact, a definition of a quantum double D(G) for quasi-Hopf algebras G has recently been proposed by S. Majid [M2]. Unfortunately this has only been done in the form of an implicit Tannaka–Krein reconstruction procedure, which makes it hard to identify this algebra in terms of generators and relations in concrete models. In [HN] we have started a program where we generalize standard notions of Hopf algebra theory (like coactions and crossed products) to (weak) quasi-Hopf algebras and apply them to quantum chains based on weak quasi-quantum groups in the spirit of [NSz, AFFS]. As a central mathematical structure underlying these constructions we have developed the concept of a diagonal crossed product by the dual Gˆ of a (weak) 1 More generally, even without these assumptions on G, periodic Hopf spin chains are meanwhile known to be isomorphic to D(G) ⊗ Mat(NL ), where NL ∈ N depends on the length of the loop L [Sz], thus explaining the representation theory of [AFFS]. 2 This result is frequently ignored in the literature and relates to the finite dimensionality of G. By a Perron–Frobenius argument it also applies to the twisted double of [DPR]. 3 The twisted double of [DPR] is also a quasi-Hopf algebra but it still satisfies 1(1) = 1 ⊗ 1.

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quasi-quantum group G. In this way we have obtained as one of our main nontrivial examples an explicit algebraic definition of the double D(G). We have shown that, as for ordinary Hopf algebras, D(G) may be realized as a new quasi-bialgebra structure on Gˆ ⊗ G 4 (or, in the weak case, a certain subspace thereof) containing G ≡ 1Gˆ ⊗ G as a sub-bialgebra. Generalizing the results of [NSz, AFFS] we have also constructed the above lattice models for weak quasi-Hopf algebras G and established that they always admit localized coactions of D(G) in the sense of [NSz]. In this work we extend our analysis of D(G) by proving that it is always a quasitriangular quasi-Hopf algebra, which is weak if and only if 1(1G ) 6= 1G ⊗ 1G . Our main results are summarized by the following Theorem A. Let (G, 1, φ, S) be a finite dimensional quasi-Hopf algebra with coproduct 3 1 : G −→ G ⊗ G, reassociator φ ∈ G ⊗ and invertible antipode S. Assume D(G) to be a quasi-Hopf algebra extension D(G) ⊃ G satisfying There exists a linear map D : Gˆ −→ D(G) such that D(G) is algebraically generˆ ated byP G and D(G) (ii) RD := µ eµ ⊗ D(eµ ) ∈ D(G) ⊗ D(G) is quasitriangular. ˜ : Gˆ −→ D˜ have the same properties, then there exists a bialgebra (iii) If D˜ ⊃ G and D homomorphism f : D(G) −→ D˜ restricting to the identity on G and satisfying ˜ f ◦ D = D. (i)

Then D(G) exists uniquely up to equivalence and the map µ : Gˆ ⊗ G −→ D(G) given by5 µ(ϕ ⊗ a) = (id ⊗ ϕ(1) )(qρ ) D(ϕ(2) ) a,

where qρ = φ1 ⊗ S −1 (αφ3 )φ2 ∈ G ⊗ G (1.1)

provides a linear bijection. Theorem A will be proven at the end of Sect. 3.4. We will also have a generalization to weak quasi-Hopf algebras, which is stated as Theorem B in Sect. 4. The major achievement of Theorem A in comparison with [HN] consists in the construction of the antipode on D(G). To this end, as a central technical result we establish a formula for (S ⊗ S)(R) and the relations between R−1 , (S ⊗ id)(R) and (id ⊗ S −1 )(R) for a quasitriangular R ∈ G ⊗ G in any quasi-Hopf algebra G. Recall that in ordinary Hopf algebras the last three quantities coincide and therefore (S⊗S)(R) = R. To prove these results we combine the methods of [HN] with the very efficient graphical calculus developed by [RT, T, AC]. This will also allow to give nice intuitive interpretations of many of our almost untraceable identities derived in [HN]. In fact, without this graphical machinery we would have been lost in proving or even only trying to guess these formulas. In particular, a purely algebraic proof of the formulas for R−1 and (S ⊗ S)(R) in Theorem 2.1 would most likely be unreadable and therefore also untrustworthy6 . This is why we think it worthwhile to put more emphasis on this graphical technique in the present paper. We start in Sect. 2.1 with shortly reviewing Drinfeld’s theory of quasi-Hopf algebras and introduce our graphical conventions in Sect. 2.2. In Sect. 2.3 we derive our main 4

This has also been announced in [M2]. Here α ∈ G is one of the two structural elements appearing in Drinfeld’s antipode axioms, see Sect. 2.1. 6 Meanwhile we have been informed by A. Coste that the formula for R−1 has been obtained previously by [AC2]. 5

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formulas for R−1 and (S ⊗ S)(R) for any quasi-triangular R ∈ G ⊗ G. In Sect. 3.1 we review our construction [HN] of the double D(G) as an associative algebra on the vector space Gˆ ⊗ G. In Sect. 3.2 we reformulate this construction in the spirit of [N1] in terms of the universal 1-flip operator D ∈ G ⊗ D(G). Sect. 3.3 roughly sketches, how ˆ In Sect. 3.4 we establish the double may also be realized on the vector space G ⊗ G. the quasi-triangular quasi-Hopf structure of D(G) and prove Theorem A. Finally, in Sect. 3.5 we identify the category Rep D(G) as the double of the category Rep G in the sense of [M2], thus proving that our construction of D(G) provides a concrete realization of Majid’s Tannaka–Krein like reconstruction procedure. In Sect. 4 we generalize our results to weak quasi-Hopf algebras G. As an application we discuss the twisted double Dω (G) of [DPR] in Appendix A and generalize the results of [N1] on the relation with the monodromy algebras of [AGS, AS] in Appendix B. Throughout, all linear spaces are assumed finite dimensional over the field C. We will use standard Hopf algebra notations, see e.g. [A, Sw, K, M3]. By an extension B ⊃ A of algebras we always mean a unital injective algebra morphism A −→ B. Two extensions B1 ⊃ A and B2 ⊃ A are called equivalent, if there exists an isomorphism of algebras B1 ∼ = B2 restricting to the identity on A. 2. Quasitriangular Quasi-Hopf Algebras 2.1. Basic definitions and properties. In this subsection we review the basic definitions and properties of quasitriangular quasi-Hopf algebras as introduced by Drinfeld [Dr2], where the interested reader will find a more detailed discussion. A quasi-bialgebra (G, 1, , φ) is an associative algebra G with unity, algebra morphisms 1 : G −→ G ⊗ G and : G −→ C, and an invertible element φ ∈ G ⊗ G ⊗ G, such that (id ⊗ 1)(1(a))φ = φ(1 ⊗ id)(1(a)), a ∈ G, (id ⊗ id ⊗ 1)(φ)(1 ⊗ id ⊗ id)(φ) = (1 ⊗ φ)(id ⊗ 1 ⊗ id)(φ)(φ ⊗ 1), ( ⊗ id) ◦ 1 = id = (id ⊗ ) ◦ 1, (id ⊗ ⊗ id)(φ) = 1 ⊗ 1.

(2.1) (2.2) (2.3) (2.4)

The map 1 is called the coproduct and the counit. A coproduct with the above properties is called quasi-coassociative and the element φ will be called the reassociator. The identities (2.2) and (2.4) together imply ( ⊗ id ⊗ id)(φ) = (id ⊗ id ⊗ )(φ) = 1 ⊗ 1.

(2.5)

Let us briefly recall some of the main consequences of these definitions for the representation theory of G. Let Rep G be the category of finite dimensional representations of G, i.e. of pairs (πV , V ), where V is a finite dimensional vector space and πV : G −→ EndC (V ) is a unital algebra morphism. We will also use the equivalent notion of a G-module V with multiplication g · v ≡ πV (g)v. Given two pairs (πV , V ), (πU , U ), the coproduct allows for the definition of a tensor product (πV ⊗U , V ⊗U ) by setting πV ⊗U = (πV ⊗πU )◦1. The counit defines a one dimensional representation. Equation (2.3) says that this representation is a left and right unit with respect to the tensor product, and (2.1) says that given three representations (πU , πV , πW ), then π(U ⊗V )⊗W ∼ = πU ⊗(V ⊗W ) with intertwiner φU V W = (πU ⊗ πV ⊗ πW )(φ).

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The meaning of (2.2) is the commutativity of the pentagon - (U ⊗ V ) ⊗ (W ⊗ X)

((U ⊗ V ) ⊗ W ) ⊗ X

j (U ⊗ (V ⊗ W )) ⊗ X

- U ⊗ (V ⊗ (W ⊗ X)) *

- U ⊗ ((V ⊗ W ) ⊗ X) ,

(2.6)

where the arrows stand for the corresponding rebracketing intertwiners. For example the first one is given by (πU ⊗V ⊗ πW ⊗ πX )(φ) = (πU ⊗ πV ⊗ πW ⊗ πX ) (1 ⊗ id ⊗ id)(φ) . The diagram (2.6) explains the name pentagon identity for Eq. (2.2). The importance of axiom (2.2) lies in the fact that in any tensor product representation the intertwiner connecting two different bracket conventions is given by a suitable product of φ’s, as in (2.6). The pentagon identity then guarantees that this intertwiner is independent of the chosen sequence of intermediate rebracketings. This is known as Mac Lane’s coherence theorem [ML]. A quasi-bialgebra G is called quasi-Hopf algebra, if there is a linear antimorphism S : G → G and elements α, β ∈ G satisfying (for all a ∈ G) X X S(ai(1) )αai(2) = α(a), ai(1) βS(ai(2) ) = β(a), (2.7) i

X

i

j

j

j

X βS(Y )αZ = 1 =

X

j

(2.8)

j

Here and throughout we use the notation φ=

S(P j )αQj βS(Rj ).

X

P i

Xj ⊗ Y j ⊗ Zj;

ai(1) ⊗ ai(2) = 1(a) and

φ−1 =

j

X

P j ⊗ Qj ⊗ Rj .

(2.9)

j

To simplify the notation, we will in the following also frequently suppress the summation symbol and write φ = X i ⊗ Y i ⊗ Z i , 1(a) = a(1) ⊗ a(2) , etc. The map S is called an antipode. We will also always suppose that S is invertible. Note that as opposed to ordinary Hopf algebras, an antipode is not uniquely determined, provided it exists. The antipode allows to define the (left) dual representation (∗ π, ∗ V ) of (π, V ), where ∗ V is the dual space of V , by ∗ π(a) = π(S(a))t , the superscript t denoting the transposed map. Analogously one defines a right dual representation (π ∗ , V ∗ ), where V ∗ ≡ ∗ V and π ∗ (a) = π(S −1 (a))t . A quasi-Hopf algebra G is called quasitriangular, if there exists an invertible element R ∈ G ⊗ G, such that 1op (a)R = R1(a), (1 ⊗ id)(R) = φ

312

a ∈ G, −1 132

R (φ 13

−1 231

)

(2.10) R φ, 23

−1

(2.11)

R φ R φ , (2.12) P 1 m where we use the following notation: If ψ = i ψi ⊗ . . . ψin ∈ G ⊗ , then, for m ≤ n, n n ψ n1 n2 ...nm ∈ G ⊗ denotes the element of G ⊗ having ψik in the nk th slot and 1 in the remaining ones. The element R is called the R-matrix. The above relations imply the quasi-Yang-Baxter equation (id ⊗ 1)(R) = (φ

)

13

213

12

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R12 φ312 R13 (φ−1 )132 R23 φ = φ321 R23 (φ−1 )231 R13 φ213 R12

(2.13)

and the property ( ⊗ id)(R) = (id ⊗ )(R) = 1.

(2.14)

Equation (2.10) implies that for any pair πU , πV the two representations (πU ⊗V , U ⊗ V ) and (πV ⊗U , V ⊗ U ) are equivalent with intertwiner B U V := τ 12 ◦ (πU ⊗ πV )(R), where τ 12 denotes the permutation of tensor factors in U ⊗ V . Equations (2.11), (2.12) imply the commutativity of two hexagon diagrams obtained by taking πU ⊗ πV ⊗ πW on both sides. G being a quasitriangular quasi-Hopf algebra implies that Rep G is a rigid monoidal category with braiding, where the associativity and commutativity constraints for the tensor product functor ⊗ : Rep G × Rep G −→ Rep G are given by the natural families φU V W and τ 12 ◦ RU V and the (left) duality is defined with the help of the antipode S and the elements α, β, see (2.29–2.31) below. Together with a quasi-Hopf algebra G ≡ (G, 1, , φ, S, α, β) we also have Gop , G cop cop and Gop as quasi-Hopf algebras, where “op” means opposite multiplication and “cop” means opposite comultiplication. The quasi-Hopf structures are obtained by putting 321 cop −1 , Sop = S cop = (Sop ) := S −1 , αop := φop := φ−1 , φcop := (φ−1 )321 , φcop op := φ −1 −1 cop −1 cop −1 cop cop := S (α), β := S (β), αop := β and βop := α. S (β), βop := S (α), α −1 Also if R ∈ G ⊗ G is quasitriangular in G, then R is quasitriangular in Gop , R21 is cop . quasitriangular in G cop and (R−1 )21 is quasitriangular in Gop Next we recall that the definition of a quasitriangular quasi-Hopf algebra is “twist covariant” in the following sense: An element F ∈ G ⊗G which is invertible and satisfies ( ⊗ id)(F ) = (id ⊗ )(F ) = 1, induces a so-called twist transformation, 1F (a) : = F 1(a)F,

(2.15)

φF : = (1 ⊗ F ) (id ⊗ 1)(F ) φ (1 ⊗ id)(F

−1

) (F

−1

⊗ 1).

(2.16)

It has been noticed by Drinfeld [Dr2] that (G, 1F , , φF ) is again a quasi-bialgebra. Setting αF := S(hi )αk i ,

βF := f i βS(g i ),

where hi ⊗ k i = F −1 and f i ⊗ g i = F , (G, 1F , , φF , S, αF , βF ) is also a quasi-Hopf algebra. Moreover, if R is quasitriangular with respect to (1, φ), then RF := F 21 RF −1

(2.17)

is quasitriangular w.r.t. (1F , φF ). This means that a twist preserves the class of quasitriangular quasi-Hopf algebras [Dr2]. For Hopf algebras, one knows that the antipode is an anti coalgebra morphism, i.e. 1(a) = (S ⊗ S) 1op (S −1 (a)) . For quasi-Hopf algebras this is true only up to a twist: Following Drinfeld we define the elements γ, δ ∈ G ⊗ G by setting7 γ := (S(U i ) ⊗ S(T i )) · (α ⊗ α) · (V i ⊗ W i ), j

j

j

j

δ := (K ⊗ L ) · (β ⊗ β) · (S(N ) ⊗ S(M )), where 7

suppressing summation symbols

(2.18) (2.19)

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T i ⊗ U i ⊗ V i ⊗ W i = (1 ⊗ φ−1 ) · (id ⊗ id ⊗ 1)(φ), j

j

j

j

−1

K ⊗ L ⊗ M ⊗ N = (1 ⊗ id ⊗ id)(φ) · (φ

⊗ 1).

(2.20) (2.21)

With these definitions Drinfeld has shown in [Dr2], that f ∈ G ⊗ G given by f := (S ⊗ S)(1op (P i )) · γ · 1(Qi βRi )

(2.22)

defines a twist with inverse given by f −1 = 1(S(P j )αQj ) · δ · (S ⊗ S)(1op (Ri )), such that for all a ∈ G,

f 1(a)f −1 = (S ⊗ S) 1op (S −1 (a)) .

(2.23)

(2.24)

The elements γ, δ and the twist f fulfill the relation f 1(α) = γ,

1(β) f −1 = δ.

(2.25)

Furthermore, the corresponding twisted reassociator (2.16) is given by φf = (S ⊗ S ⊗ S)(φ321 ).

(2.26)

Setting h := (S −1 ⊗ S −1 )(f 21 ), the above relations imply

h1(a)h−1 = (S −1 ⊗ S −1 ) 1op (S(a)) , φh = (S

−1

⊗S

−1

⊗S

−1

321

)(φ

).

(2.27) (2.28)

The importance of the twist f for the representation theory of G is the existence of an intertwiner U ⊗ V −→ (∗ V ⊗ ∗ U )∗ given by τ 12 ◦ (πU ⊗ πV )(f ). Finally we introduce Gˆ as the dual space of G with its natural coassociative coalgebra ˆ ˆ) given by h1(ϕ) ˆ structure (1, | a ⊗ bi := hϕ | abi and ˆ(ϕ) := hϕ | 1G i, where ˆ ϕ ∈ G, a, b ∈ G and where h· | ·i : Gˆ ⊗ G −→ C denotes the dual pairing. On Gˆ we have the natural left and right G-actions a * ϕ := ϕ(1) hϕ(2) | ai,

ϕ ( a := ϕ(2) hϕ(1) | ai,

ˆ By transposing the coproduct on G we also get a multiplication where a ∈ G, ϕ ∈ G. ˆ ˆ ˆ G ⊗ G −→ G, which however is no longer associative hϕψ | ai := hϕ ⊗ ψ | 1(a)i,

h1Gˆ | ai := (a).

ˆ ˆ ˆ = 1(ϕ) 1(ψ), a * (ϕψ) = Yet, we have the identities 1Gˆ ϕ = ϕ1Gˆ = ϕ, 1(ϕψ) (a(1) * ϕ) (a(2) * ψ) and (ϕψ) ( a = (ϕ ( a(1) )(ψ ( a(2) ) for all ϕ, ψ ∈ Gˆ and a ∈ G. We also introduce Sˆ : Gˆ −→ Gˆ as the coalgebra anti-mophism dual to S, i.e. ˆ hS(ϕ) | ai := hϕ | S(a)i. 2.2. Graphical calculus. In the following it will be useful to have a graphical notation for the identities and definitions given so far. The graphical calculus introduced below has been developed and used in many papers, e.g. [RT, AC, T], mainly in the setting of ribbon–Hopf algebras. Formally speaking, it consists of a functor from the braided monoidal category Rep G into a category of colored graphs. For an introduction into the category terminology see [K, T]. We will use the graphical notation to have a pictorial

Doubles of Quasi-Quantum Groups

555

way to understand – and deduce – certain relations and identities between morphisms (intertwiners) in Rep G, which – written out algebraically – would look very complicated. By morphisms in Rep G we mean elements t ∈ HomG (U, V ), i.e. linear maps t : U → V satisfying t πU (a) = πV (a) t, ∀a ∈ G. As discussed in Sect. 2.1, the n-fold tensor product of G-modules is again a G-module (where one has to take care of the bracketing of the tensor factors). A morphism t from an n-fold to an m-fold tensor product of G-modules is represented by a graph consisting of a “coupon” with n lower legs and m upper legs “coloured” with the source and target modules respectively. The upper and lower legs are always equipped with a definite bracketing corresponding to the bracketing defining the associated tensor module. For example the picture U (

V

X )

t [( X

Y

) Z

] U

corresponds to the morphism

t : X ⊗ (Y ⊗ Z) ⊗ U −→ (U ⊗ V ) ⊗ X.

The tensor product of two morphisms corresponds to the juxtaposition of diagrams and the composition of morphisms is depicted by gluing the corresponding graphs together. Here one has to take care that the gluing t ◦ k is only admissible if source(t) = target(k), which in particular implies that the bracketing conventions of the associated tensor factors have to coincide. We also use the convention that the lower legs always represent the source, i.e. the graph t ◦ k is obtained by gluing t on top of k. Following the conventions of [AC] we now give a list of some special morphisms depicted by the following graphs: V := idV , (2.29) ∗

V V

∗

V

:=

bV

C

V ,

∗

V

V

∗

C W

V

V

W

V

, V

(2.30)

W := BV−1W ,

:= BV W , V

aV

:=

W

V

(2.31)

556

F. Hausser, F. Nill

where C stands for the one dimensional representation given by the counit and where bV : C −→ V ⊗ ∗ V, 1 7−→

X

βV vi ⊗ v i ,

i

aV : ∗ V ⊗ V −→ C, vˆ ⊗ w 7−→ hvˆ | αV wi, BV−1W = RV−1W ◦ τW V .

BV W = τ V W ◦ R V W ,

Here {vi } is a basis of V with dual basis {v i } and τV W denotes the permutation of tensor factors in V ⊗ W . We also use the shortcut notation αV ≡ πV (α), RV W ≡ (πV ⊗ πW )(R), etc. The properties of G being a quasitriangular quasi-Hopf algebra ensure that the above defined maps are in fact intertwiners (morphisms of G-modules). Note that within higher tensor products the graphs (2.30) and (2.31) are only admissible if their legs are “bracketed together”. In order to change the bracket convention one has to use rebracketing morphisms. These are given as products of the basic elements (

)

(

) = φ−1 UV W ,

= φU V W , ( U

) V

( V

U

W

) W

(2.32)

where each of the three individual legs in (2.32) may again represent a tensor product of G-modules. In this way we adopt the convention that any empty coupon with the same number of upper and lower legs – where the colouring only differs by the bracket convention – always represents the associated unique rebracketing morphism in Rep G given in terms of suitable products of φ’s. We have already remarked that the uniqueness of this rebracketing morphism (i.e. the independence of the chosen sequence of intermediate rebracketings) is guaranteed by Mac Lane’s coherence theorem and the “pentagon axiom” (2.2). This is why it is often not even necessary to spell out one of the possible formulas for such an intertwiner. Explicitly, the pentagon identity (2.2) may be expressed as

[

[(

)

(

)]

] [(

)

]

= [ [(

)

(

)

(

)]

(

)

] ( (

)

)

(2.33)

which is the graphical notation for (id ⊗ 1 ⊗ id)(φ) (φ ⊗ 1) (1 ⊗ id ⊗ id)(φ−1 ) = (1 ⊗ φ−1 ) (id ⊗ id ⊗ 1)(φ). In the same philosophy one may rewrite a simple rebracketing as a product of more complicated ones, as long as the overall source and target brackets coincide, for example

Doubles of Quasi-Quantum Groups

557

[ [

(

)]

(

)]

(

)

(

)

(

)

(

)

= [(

)

]

[(

)

]

.

(2.34)

As done in the above pictures we will frequently not specify the modules sitting at the source and target legs. Also note that by Eqs. (2.4) and (2.5) the rebracketing of the (invisible) “white” leg corresponding to the trivial G-module C is always given by the trivial identification. If G is finite dimensional, it may itself be viewed as a G-module under left multiplication and algebraic identities may directly be translated into identities of the corresponding graphs and vice versa. So e.g. Eq. (2.8) is equivalent to

(

)

( =

(

)

=

)

(

) (2.35)

and Eqs. (2.4) and (2.5) together with (2.7) imply [(

)

(

)]

] =

[

[

(

)] (2.36)

and (

)

(

)

(

)]

= [

[

(

)] (2.37)

as well as the upside–down and left–right mirror images of (2.36) and (2.37) and the graphs obtained by rotating by 180◦ in the drawing plane. In general, with every graphical rule, where the graph is built from elementary graphs of the above list, the rotated as well as the upside–down and left–right mirror images are also valid and are proven analogously. This induces a Z2 × Z2 - symmetry action on all graphical identities given below, which in fact is already apparent in the axioms of a quasi-triangular quasi-Hopf op instead of G. algebra given in Sect. 2.1 by taking G op , Gcop or Gcop Finally we point out the important “pull through” rule saying that morphisms built from representation matrices of special elements in G (like the braiding (2.31) or the reassociator (2.32)) always “commute” with all other intertwiners in the appropriate sense, i.e. by changing colours and orderings accordingly. For example one has

558

F. Hausser, F. Nill

U

V

U

h

V

,

= h

X

U

(

X

(

)

h0

[

U

=

[(

)

(

)]

(

) )

]

.

h0 [

(

)]

(2.38)

In the language of categories this means that the braidings and the reassociators provide natural transformations [ML]. 2.3. The antipode image of the R-matrix. In this subsection we exploit the full power of our graphical machinery by proving various important identities involving the action of the antipode on a quasitriangular R-matrix. We recall that for ordinary Hopf algebras (i.e. where φ, α and β are trivial) one has (S ⊗ id)(R) = R−1 = (id ⊗ S −1 )(R) and (S ⊗ S)(R) = R. To generalize these identities to the quasi-Hopf case we introduce the following four elements in G ⊗ G (using the notation (2.9)): pλ := Y i S −1 (X i β) ⊗ Z i , qλ := S(P i ) α Qi ⊗ Ri ,

pρ := P i ⊗ Qi β S(Ri ), qρ := X i ⊗ S −1 (α Z i ) Y i .

(2.39)

These elements have already been considered by [Dr2,S], see also Eqs. (9.20)–(9.23) of [HN]. They obey the commutation relations (for all a ∈ G) 1(a(2) ) pλ [S −1 (a(1) ) ⊗ 1] = pλ [1 ⊗ a], [S(a(1) ) ⊗ 1] qλ 1(a(2) ) = [1 ⊗ a] qλ ,

1(a(1) ) pρ [1 ⊗ S(a(2) )] = pρ [a ⊗ 1], [1 ⊗ S −1 (a(2) )] qρ 1(a(1) ) = [a ⊗ 1] qρ , (2.40)

see e.g. [HN,Lem. 9.1]. A graphical interpretation of these identities will be given in Eqs. (2.45) below. With these definitions we now have Theorem 2.1. Let (G, 1, φ, S, α, β) be a finite dimensional quasi-Hopf algebra, let γ ∈ G ⊗ G be as in (2.18) and let R ∈ G ⊗ G be quasitriangular. Then R−1 = [X j βS(P i Y j ) ⊗ 1] · [(S ⊗ id)(qρop R)] · [(Ri ⊗ Qi ) 1op (Z j )] = [1(Rj ) (Y i ⊗ Z i )] · [(id ⊗ S −1 )(R pρ )] · [1 ⊗ S −1 (αQj X i )P j ], (2.41) (S ⊗ S)(R) γ = γ R. 21

(2.42)

Doubles of Quasi-Quantum Groups

559

A direct implication of Eq. (2.42) is the following formula, which has already been stated without proof in [AC]. Corollary 2.2. Under the conditions of Theorem 2.1 let f ∈ G ⊗ G be the twist defined in (2.22), then f op R f −1 = (S ⊗ S)(R).

(2.43)

Proof of Corollary 2.2. Using the formula (2.22) for f and (2.42) one computes f op R = (S ⊗ S)(1(P i )) γ op 1op (Qi βRi ) R = (S ⊗ S)(1(P i )) (S ⊗ S)(R) γ 1(Qi βRi ) = (S ⊗ S)(R) (S ⊗ S)(1op (P i )) γ 1(Qi βRi ) = (S ⊗ S)(R) f.

To prepare the proof of Theorem 2.1 we need the following three lemmata. First we have Lemma 2.3. [(

)

[ (

)

]

(

)]

(

)

=

(2.44) and three mirror images. Proof. This is straightforward and left to the reader. (Use first (2.33), then (2.36) and (2.37) (or the suitable mirror image) and finally (2.35)). To give an algebraic formulation of the four identities of Lemma 2.3 let us introduce the notation V∗ PVλ W :=

V (

W )

(

)

(

)

QλV W :=

( ∗

V

V

) W

W ( ,

ρ PW V :=

,

QρW V :=

V

) (

)

(

)

( W

∗

V

.

) V

,

V∗

(2.45)

In a more general scenario these morphisms have already been introduced in [HN]. Algebraically, they are given by (using the module notation)

560

F. Hausser, F. Nill

PVλ W : w 7→ v i ⊗ pλ · (vi ⊗ w),

ρ i PW V : w 7→ pρ · (w ⊗ vi ) ⊗ v ,

QλV W : vˆ ⊗ v ⊗ w 7→ (vˆ ⊗ id) qλ · (v ⊗ w) , ˆ qρ · (w ⊗ v) , QρW V : w ⊗ v ⊗ vˆ 7→ (id ⊗ v)

(2.46)

see Eqs. (9.36), (9.37), (9.42) and (9.43) of [HN]. Note that the identities (2.40) precisely reflect the fact that these maps are morphisms in Rep G. In this way Lemma 2.3 is also contained in [HN, Lem 9.1], since it is equivalent to the four identities, respectively [1 ⊗ S −1 (p2ρ )] qρ 1(p1ρ ) = 1 ⊗ 1,

[S(p1λ ) ⊗ 1] qλ 1(p2λ ) = 1 ⊗ 1, 1(qλ2 ) pλ

[S

−1

(qλ1 )

1(qρ1 ) pρ

⊗ 1] = 1 ⊗ 1, ∗

[1 ⊗

S(qρ2 )]

(2.47)

= 1 ⊗ 1.

∗

Next we define intertwiners gV W : ( W ⊗ V ) ⊗ (V ⊗ W ) → C and dV W : C → (V ⊗ W ) ⊗ (∗ W ⊗ ∗ V ) by V ( [(

gV W := ( W

∗

∗

) ) V

]

( V

) W

W )

dV W :=

∗

W (

[(

,

)

∗

V ) ]

.

(2.48) One directly verifies that ˆ γV W (v ⊗ w)i, gV W (wˆ ⊗ vˆ ⊗ v ⊗ w) = hvˆ ⊗ w| X δV W (vi ⊗ wi ) ⊗ wi ⊗ v i , dV W (1) =

(2.49)

i ∗

∗

where vˆ ∈ V, wˆ ∈ W, v ∈ V, w ∈ W and where {vi ⊗ wi } is a basis of V ⊗ W with dual basis {v i ⊗ wi } ∈ ∗ V ⊗ ∗ W . Here γ,δ ∈ G ⊗ G are given in (2.18),(2.19) and γV W = (πV ⊗ πW )(γ), δV W = (πV ⊗ πW )(δ). We remark that in terms of these intertwiners the identities (2.25) may now be depicted as gV W ( = ∗

(V ⊗ W ) V ⊗ W

)

(

f −1 ∗

)

idV ⊗W

;

(V ⊗ W ) V ⊗ W

(V ⊗ W )∗ V ⊗ W (V ⊗ W )∗ V ⊗ W

f

= (

)

idV ⊗W . ( )

dV ∗ W ∗

(2.50)

Doubles of Quasi-Quantum Groups

561

Moreover, we have the following Lemma 2.4.

[

gV W =

(

( W

∗

∗

)] ) V

dV W =

( V

,

) W

V (

W )

[

(

∗

∗

W (

V ) .

)]

(2.51)

Proof. We prove the first identity:

[( [( (

) )

]

[

≡

]

(

)

)

(

)]

]

= [

) (

[

[(

(

)

(

)]

(

)

[(

)

(

)

] (

)

)] ≡

= [(

)

(

)

]

[ (

(

(

)] )

(

)

,

)

where in the second equality we have plugged in the pentagon identity (2.33) and in the third we have used (2.36). The second identity in (2.51) is the upside–down mirror image of the first one and is proved analogously. We invite the reader to check that algebraically Lemma 2.4 implies the following identities for γ and δ defined in (2.18) and (2.19): ˜ i ), γ = (S(U˜ i ) ⊗ S(T˜ i )) · (α ⊗ α) · (V˜ i ⊗ W ˜ j )), ˜ j ⊗ L˜ j ) · (β ⊗ β) · (S(N˜ j ) ⊗ S(M δ = (K ˜i

i

i

where

i

˜ = (φ ⊗ 1) · (1 ⊗ id ⊗ id)(φ−1 ), T ⊗ U˜ ⊗ V˜ ⊗ W ˜ j ⊗ N˜ j = (id ⊗ id ⊗ 1)(φ−1 ) · (1 ⊗ φ). ˜ j ⊗ L˜ j ⊗ M K These identities have already been obtained by Drinfeld [Dr2]. Finally we note the following linear isomorphisms of intertwiner spaces holding in fact in any rigid monoidal category.

562

F. Hausser, F. Nill

Lemma 2.5. Let X, V, W be finite dimensional G-modules. Then there exist linear bijections 9VX,W : HomG (X ⊗ W, V ) −→ HomG (X, V ⊗ ∗ W ),

(2.52)

8V,W X

(2.53)

: HomG (X, V ⊗ W ) −→ HomG (∗ V ⊗ X, W ),

given by ∗

V V 9VX,W :

h0 7−→

h0 X

W

(

;

)

W

(

)

X V ∗

V (9VX,W )−1 :

(

W (

7−→

h

) )

;

h

X

X

W

W V : 8V,W X

(

W h

) (

7−→

)

;

h

X ∗

V

X V

W

W −1 : (8V,W X )

h0 7−→

h0 ∗

V

X

( (

) ) X

.

Doubles of Quasi-Quantum Groups

563

Proof. We prove 9VX,W ◦ (9VX,W )−1 = id by determining its action on h ∈ HomG (X, V ⊗ ∗ W ) as follows: ∗

V ∗

V

(

(

W 7−→

h

)

) h (

X

W

) (

)

X ∗

V

)

(

W

∗

V [(

)

(

)

W

] h

=

= (

,

) X

h

X where in the first equality we have used a “pull through” rule for h, and in the second equality a left–right mirror image of (2.44). Analogously one shows that 9−1 ◦ 9 = id and 8 ◦ 8−1 = id = 8−1 ◦ 8. We are now in the position to prove Eqs. (2.41) and (2.42) of Theorem 2.1 by rewriting them as graphical identities as follows: Lemma 2.6. For all finite dimensional G-modules the inverse braiding BU−1V obeys U

U

V

(

)

(

[(

) )

( U

]

V

(

) )

( V

] ) U

(

[(

) )

] ,

=

= [(

)

V

V

[(

U ( V

) )

(

] )

U (2.54)

564

F. Hausser, F. Nill

and the (left) conjugate braiding B∗ U ∗ V obeys

[(

)

( ∗

) U

∗

]

(

)

] .

)

U

V

[(

= ( ∗

V

) ∗

U

(

)

U

V

V

(2.55)

Taking G itself as a G-module yields (2.54) ⇔ (2.41) and (2.55) ⇔ (2.42) and therefore the above identities prove Theorem 2.1. Proof. To prove the first equation in (2.54) note that (2.11) and (2.14) imply the identity V

(

)

(

)

(

)

( =

( ∗

(

)

U

U

)

∗

)

U

U

.

V

V

(2.56)

−1 of Lemma 2.5 to both sides of (2.56) to Now we apply the isomorphism (8U,V U ⊗V ) obtain

U U

[

V

(

)]

[

V

(

)]

[(

)

(

)

U

]

V [(

[(

)

] ≡

[( (

) )

]

(

[

) (

U

V

)

(

)

(

)]

( U

=

(

)

]

[

(

)]

[

(

)]

)

( U

) V

) V

(2.57)

where in the left identity we have used a “pull through” rule for the braiding. By the identity (2.37) the top of the graph in the middle of (2.57) may be replaced by

Doubles of Quasi-Quantum Groups

[

565

(

[(

)] )

(

=

)

(

) ,

]

[(

)

]

(2.58)

and using Lemma 2.3 for the r.h.s. of (2.57) we end up with U (

(

V

)

(

)

[(

)

]

[(

)

]

)

(

.

= U

)

U

V

V

(2.59)

Hence we get the first equality in (2.54)8 . Analogously one starts with (2.12),(2.14) to get (

)

U

(

)

(

)

(

(

=

(

U

V

,

)

) U

)

V

V∗

V∗ (2.60)

where we have used the trivial identification ∗ (V ∗ ) = V . Taking the mirror image of the above proof yields the second equality in (2.54). Equation (2.55) follows from Lemma 2.7 below by putting h0 = BU V and h = B∗ U ∗ V . The identifications (2.54) ⇔ (2.41) and (2.55) ⇔ (2.42) are straightforward and are left to the reader. This concludes the proof of Lemma 2.6 and therefore also of Theorem 2.1. We end this section with a lemma used in the above proof, which will also be used in the next section. 8

By finite dimensionality it suffices to prove the left inverse property.

566

F. Hausser, F. Nill

Lemma 2.7. Let V, W, X, Y be finite dimensional G-modules with intertwiners h : ∗ X ⊗ ∗ Y −→ ∗ W ⊗ ∗ V,

h0 : V ⊗ W −→ Y ⊗ X,

then the following two identities are equivalent

[( (

) )

(

]

[( (

)

(

) h0

h ∗

X

∗

∗

X

V

Y

∗

W

∗

X

)

(

)

(

)

V

Y

∗

X

W

(

] ,

=

)

)

(

W

(2.61)

W ) X

[(

)

]

[(

h

)

[(

)

]

H0

=:

[(

)

)

( ∗

)

( ∗

V

Y

) Y

.

] ∗

(

W

]

h0

=

∗

(

Y

V

)

V (2.62)

Proof. Using (2.48), (2.51), a “pull through” rule for h and h0 and rebracketing the source legs, Eq. (2.61) is equivalent to

[

(

(

)]

[(

) =

[(

)

[

∗

[ [

)] Y

V

W

) h0

]

(

X

]

(

h

∗

)

∗

(

(

X

∗

)]

.

)] Y

V

W

(2.63)

Note that the l.h.s. of (2.63) is of the form 9−1 (· · · ) (with a “white” target leg). Now we apply the isomorphism 9C [∗ X⊗(∗ Y ⊗V )],W of Lemma 2.5 to both sides of (2.63). Then using for the bottom part of the r.h.s. the identity

Doubles of Quasi-Quantum Groups

{

[

{[

(

567

)]}

( )]

{

} =

[

)] {

(

}

[

(

)]}

{[

(

)]

(

)

}

(

) (2.64)

and a pull through rule to raise the upper box of the r.h.s. of (2.64) to the top, we conclude that (2.63) is equivalent to { ∗

} {

W

(

(

{[ (

)

)

)

(

h [

)

∗

}

( (

H

)] ∗

∗

(

X

∗

)

∗

V

Y

(

X

∗

) Y

)

≡

0

( (

W

)

)

=

h

W }

]

(

∗

0

(

X

∗

.

) Y

V

)

V (2.65) ∗

W The proof is finished by applying the isomorphism (8−1 )X, ∗ Y ⊗V .

3. Doubles of Quasi-Hopf Algebras 3.1. D(G) as an associative algebra. In this section we review the definition of the double D(G) of a quasi-Hopf algebra G as a diagonal crossed product as introduced in [HN]. We also give a graphical description of this construction. Consider δ := (1 ⊗ id) ◦ 1 : G −→ G ⊗ G ⊗ G

(3.1)

and let 8 ∈ G ⊗ be given by 5

8 := [(id ⊗ 1 ⊗ id)(φ) ⊗ 1] [φ ⊗ 1 ⊗ 1] [(δ ⊗ id ⊗ id)(φ−1 )].

(3.2)

Then the pair (δ, 8) provides a two-sided coaction of G on itself as defined in [HN], i.e the following axioms are satisfied: (i)

The map δ is a unital algebra morphism satisfying ( ⊗ id ⊗ ) ◦ δ = id.

568

F. Hausser, F. Nill

(ii) The element 8 ∈ G ⊗ is invertible and fulfills 5

(id ⊗ δ ⊗ id)(δ(a)) 8 = 8 (1 ⊗ id ⊗ 1)(δ(a)),

∀a ∈ G, −1

(1 ⊗ 8 ⊗ 1) (id ⊗ 1 ⊗ id ⊗ 1 ⊗ id)(8) (φ ⊗ 1 ⊗ φ

,)

= (id⊗ ⊗ δ ⊗ id⊗ )(8) (1 ⊗ id⊗ ⊗ 1)(8), 2

2

3

(id ⊗ ⊗ id ⊗ ⊗ id)(8) = ( ⊗ id

⊗3

(3.3)

⊗ )(8) = 1 ⊗ 1 ⊗ 1.

(3.4) (3.5)

Next we define ≡ 1 ⊗ 2 ⊗ 3 ⊗ 4 ⊗ 5 ∈ G ⊗ by9 5

:= (id⊗ ⊗ S −1 ⊗ S −1 )(f 45 · 8−1 ) = (id⊗ ⊗ S −1 ⊗ S −1 )(8−1 ) · h54 , (3.6) 3

3

where f, h ∈ G ⊗ G are the twists defined in (2.22) and (2.27). Let now Gˆ be the coalgebra dual to G with its natural left and right G-action and the 3 nonassociative multiplication given by hϕψ | ai := hϕ ⊗ ψ | 1(a)i. With δ : G −→ G ⊗ being a two-sided coaction we then write ϕ . a / ψ := (ψ ⊗ id ⊗ ϕ)(δ(a)). Note that for the two-sided coaction δ in Eq. (3.1) we have the identity ϕ . a / ψ = (ϕ * a) ( ψ. Considered as an element of Gˆ we also write 1ˆ ≡ 1Gˆ ≡ . The following proposition has been proven in [HN]: Proposition 3.1. Let (δ, 8) be a two-sided coaction of G on G and define the diagonal crossed product Gˆ ./ G to be the vector space Gˆ ⊗ G with multiplication rule i h (ϕ ./ a)(ψ ./ b) := (1 * ϕ ( 5 )(2 * ψ(2) ( 4 ) i h (3.7) ./ 3 (Sˆ −1 (ψ(1) ) . a / ψ(3) ) b , where we write (ϕ ./ a) in place of (ϕ ⊗ a) to distinguish the new algebraic structure. Then Gˆ ./ G is an associative algebra with unit 1ˆ ./ 1, containing G ≡ 1ˆ ./ G as a unital subalgebra. The associativity of the above product follows from the axioms for two-sided coactions as has been proven in Theorem 10.2 of [HN]. Note that in general the subspace Gˆ ./ 1 is not a subalgebra of Gˆ ./ G. On the other hand if G is an ordinary Hopf algebra with φ ≡ 1 ⊗ 1 ⊗ 1, then Eq. (3.7) becomes (3.8) (ϕ ./ a)(ψ ./ b) = ϕψ(2) ./ (Sˆ −1 (ψ(1) ) . a / ψ(3) )b , which is the standard multiplication rule in the quantum double D(G) [Dr1,M3]. This motivates the Definition 3.2 ([HN]). The diagonal crossed product Gˆ ./ G defined in Proposition 3.1, with (δ, 8) given by (3.1),(3.2) is called the quantum double of G, denoted by D(G) ≡ Gˆ ./ G. We will now rewrite the multiplication (3.7) given in Proposition 3.1 using the “generating matrix” formalism of the St. Petersburg school. In this way we will be able to give a graphical (i.e. a categorical) description of the algebraic relations in Gˆ ./ G which should convince the reader that the multiplication given in (3.7) is indeed associative. The following corollary is a generalization of [N, Lemma 5.2] and coincides with [HN, Cor. 10.4] applied to the present scenario. 9

where we have again suppressed summation symbol and indices

Doubles of Quasi-Quantum Groups

569

Corollary 3.3 ([HN]). Let A be some unital algebra and γ : G −→ A a unital algebra map. Then the relation γL (ϕ ./ a) = (ϕ ⊗ id)(L) · γ(a)

(3.9)

provides a one to one correspondence between unital algebra morphisms γL : Gˆ ./ G −→ A extending γ and elements L ∈ G ⊗ A satisfying ( ⊗ id)(L) = 1A and [1G ⊗ γ(a)] L = [S −1 (a(1) ) ⊗ 1A ] L [a(−1) ⊗ γ(a(0) )], 13

L L

23

∀a ∈ G,

(3.10)

= [ ⊗ ⊗ 1A ][(1 ⊗ id)(L)][ ⊗ ⊗ γ( )], 5

4

1

2

3

(3.11) where has been defined in (3.6) and where δ(a) ≡ a(−1) ⊗ a(0) ⊗ a(1) . An element L ∈ G ⊗ A satisfying ( ⊗ id)(L) = 1A and (3.10/3.11) is called a normal coherent (left diagonal) δ-implementer (with respect to γ), see [HN]. Note that by choosing A = Gˆ ./ G and γ = id, Cor. 3.3 implies that the multiplication on Gˆ P ./ G may uniquely be described by the relations of one “generating matrix” L = eµ ⊗ (eµ ./ 1), where {eµ } is a basis in G with dual basis {eµ }. In fact this formulation is used in [HN] to prove the associativity of the multiplication (3.7). We now give a graphical interpretation of the identities (3.10),(3.11) by using that any unital algebra map γ : G −→ A defines a G-module structure on A via b · A ≡ πA (b)A := γ(b)A. Moreover, given two G-modules V, W then (V ⊗ A) ⊗ W becomes a G-module by setting πV ⊗A⊗W (a) = (πV ⊗ πA ⊗ πW )(δ(a)). Considering G as a G-module by left multiplication we now define the map ∗ (b ⊗ A ⊗ ϕ) 7→ (ϕ ⊗ id) L · (b ⊗ A) . LA (G⊗A)⊗G ∗ : (G ⊗ A) ⊗ G −→ A, being an intertwiner of G-modules, i.e. to Then Eq. (3.10) is equivalent to LA (G⊗A)⊗G ∗ ∗ LA ∈ Hom , G . We depict this intertwiner as (G ⊗ A) ⊗ G ∗ G (G⊗A)⊗G A LA (G⊗A)⊗G ∗ :=

,

L ( G

) A

G∗

and call this a d-fork (≡ down fork) graph. The “coherence condition” (3.11) is now equivalent to the graphical identity

(

A

A

L

L ( idG⊗G

) = L

{ G

[( G

) A

[( ]} G∗ G∗

{ G

) f )

[( G

] ) A

(

, )

]} G∗ G∗

(3.12)

570

F. Hausser, F. Nill

Note that the lowest box on the r.h.s. represents the rebracketing morphism 8−1 defined in (3.2). This explains why one has to choose the complicated multiplication rule (3.7) instead of (3.8) if φ and therefore f and 8 are non-trivial. 3.2. Coherent 1-flip operators. We are now going to provide another set of generators in D(G) which later will be more appropriate for defining the (quasitriangular) quasi-Hopf structure. Associated with any coherent (left diagonal) δ-implementer L we define the element T ∈ G ⊗ A by T := [S −1 (p2ρ ) ⊗ 1] · L · (id ⊗ γ)(1(p1ρ )),

(3.13)

where pρ ≡ p1ρ ⊗ p2ρ has been given in (2.39). Proposition 3.4 ([HN]). The relation (3.13) defines a one-to-one correspondence between elements L ∈ G ⊗ A satisfying (3.10) and (3.11) and elements T ∈ G ⊗ A satisfying (id ⊗ γ)(1op (a)) T = T(id ⊗ γ)(1(a)), φ312 A

13

T

(φ−1 )132 A

∀a ∈ G,

T φA = (1 ⊗ id)(T), 23

(3.14) (3.15)

where φA = (id ⊗ id ⊗ γ)(φ). L is recovered from T by L = (id ⊗ γ)(qρop ) T.

(3.16)

Moreover ( ⊗ id)(T) = 1A if and only if ( ⊗ id)(L) = 1A . Following [HN, Sect. 11] we call the elements T ∈ G ⊗ A satisfying (3.14) and (3.15) coherent 1-flip operators. They are special versions of coherent λρ-intertwiners [HN, Def. 10.8] associated with quasi-commuting pairs (λ, ρ) of left G-coactions λ and right Gcoactions ρ on an algebra M. Proposition 3.4 has been proven algebraically in [HN,Prop. 10.10]. Before giving an alternative proof below, using the graphical calculus developed in Sect. 2.2 and 2.3, let us state the following central consequence Theorem 3.5. Define the element D ∈ G ⊗ (Gˆ ⊗ G) by10 D :=

X µ

S −1 (p2ρ ) eµ p1ρ(1) ⊗ (eµ ⊗ p1ρ(2) ),

(3.17)

and denote iD : G ,→ Gˆ ⊗ G the canonical embedding iD (a) := 1Gˆ ⊗ a. Then there is a unique algebra structure on the vector space Gˆ ⊗ G satisfying ∀a, b ∈ G, iD (a)iD (b) = iD (ab) D · (id ⊗ iD )(1(a)) = (id ⊗ iD )(1op (a)) · D φ312 D13 (φ−1 )132 D23 φ = (1 ⊗ id)(D).

10

as before {eµ } denotes a basis of G with dual basis {eµ }

∀a ∈ G,

(3.18) (3.19) (3.20)

Doubles of Quasi-Quantum Groups

571

where we have identified φ ≡ (id ⊗ id ⊗ iD )(φ) ∈ G ⊗ G ⊗ D(G). This algebra is precisely the quantum double D(G) defined in Prop. 3.1 and we have ϕ ./ a = (iD ⊗ ϕ(1) )(qρ ) (ϕ(2) ⊗ id)(D) iD (a).

(3.21)

Proof. Follows from Proposition 3.4 and Cor. 3.3 by choosing A = D(G) which means that T = D. We call D the universal 1-flip operator in D(G). The description of the quantum double D(G) as given in Theorem 3.5 will be used in the next section to derive the quasi-Hopf structure of D(G). We now prove Proposition 3.4. Proof of Proposition 3.4. The equivalence ( ⊗ id)(T) = 1A ⇔ ( ⊗ id)(L) = 1A follows from property (2.5) of φ. To show that the relations (3.10) and (3.11) for L are equivalent to the relations (3.14) and (3.15) for T, respectively, we use the graphical calculus. First we use the isomorphism 9A (G⊗A),G ∗ of Lemma 2.5, to define the intertwiner TGA ∈ HomG (G ⊗ A, A ⊗ G) by G

A A

G

TGA ≡

L :=

G

[(

)

.

]

A ( G

) A

(

) (3.22)

Algebraically Definition (3.22) translate into TGA (b ⊗ A) := T21 · (A ⊗ b), where T ∈ G ⊗ A is expressed in terms of L by (3.13). Now note that the property of TGA being an intertwiner of G-modules is equivalent to T satisfying (3.14). Thus we have proven the equivalence (3.10) ⇔ (3.14) and since the map 9A (G⊗A),G ∗ is invertible also the invertibility of the transformation (3.13). In fact, (3.16) is equivalent to A

A =

L ( G

) A

G∗

( G

(

) .

) A

G∗

We are left to show that (3.11) ⇔ (3.15) (under the conditions (3.10) and (3.14), respectively). To this end we use that Eq. (3.15) is graphically expressed as

572

F. Hausser, F. Nill

G )

A (

G

A (

G

(

G

) idG⊗G

) .

= (

) idG⊗G

G

G

(

A

) ( G

G

) A

(3.23)

Thus the following lemma proves the equivalence (3.11) ⇔ (3.15) and therefore completes the proof of Proposition 3.4 Lemma 3.6. The graphical identities (3.12) for LA (A⊗G)⊗G ∗ and (3.23) for TGA are equivalent. Proof. Let us prove (3.12) ⇒ (3.23): Using the definition (3.22) we get for the l.h.s. of (3.23) (

)

L {[(

)

{(

)

}

]

{ L

)}

(

{[(

}

)

]

}

]

}

=

l.h.s = of (3.23)

(

L

) {[

L {[(

)

{(

)

] (

}

.

[(

) {( ) {[

] (

)]

}}

)}

Here we have used a pull through rule to push the lower d-fork up and then we have combined all rebracketing morphisms in one box. Now plugging in Eq. (3.12), splitting the rebracketing morphism at the bottom into four factors and pushing two of them up,

Doubles of Quasi-Quantum Groups

573

one obtains l.h.s. of (3.23) =

(

{ {[(

}

]

)

)

(

(

)

idG⊗G ] [

]

f

idG⊗G

}

)]

[(

L )

idG⊗G

( {(

) {[

(

)]

)

=

f

idG⊗G {[((

(

}

L )

)

}}

(

) (

)

[

(

(

) )]

)

Using the identity (2.50) the last picture equals the r.h.s. of (3.23). Hence we have shown (3.12) ⇒ (3.23). The implication (3.23) ⇒ (3.12) is shown similarly by bending the two upper G-legs in (3.23) down again. Thus we have proved Lemma 3.6 and therefore also Proposition 3.4. 3.3. Left and right diagonal crossed products. In this subsection we sketch how the quantum double D(G) may equivalently be modeled on G ⊗ Gˆ instead of Gˆ ⊗ G. (In fact this is true for any diagonal crossed product as has been shown in [HN,Thm. 10.2].) 5 With the notation as in Proposition 3.1 and with R ∈ G ⊗ given by R := (h−1 )21 · 3 ˆ (S −1 ⊗ S −1 ⊗ id⊗ G )(8) the right diagonal crossed product G ./ G is defined to be the ˆ vector space G ⊗ G with multiplication rule i h (a ./ ϕ)(b ./ ψ) := a ϕ(1) . b / Sˆ −1 (ϕ(3) ) 3R i h (3.24) ./ (2R * ϕ(2) ( 4R )(1R * ψ ( 5R ) . ˆ containing G ≡ G ./ 1ˆ as a This makes G ./ Gˆ an associative algebra with unit 1 ⊗ 1, ˆ unital subalgebra. To see that the two algebras G ./ G and G ./ Gˆ are isomorphic let us begin with stating the analogue of Lemma 3.3: Let γ : G −→ A be a unital algebra map into some target algebra A. Then the relation γR (a ./ ϕ) = γ(a) · (ϕ ⊗ id)(R)

(3.25)

provides a one-to-one correspondence between unital algebra morphisms γR : G ./ Gˆ −→ A extending γ and elements R ∈ G ⊗ A satisfying ( ⊗ id)(R) = 1A and R [1G ⊗ γ(a)] = [a(1) ⊗ γ(a(0) )] R [S −1 (a(−1) ) ⊗ 1A ], 13

23

R R

=

[4R

⊗

5R

⊗

γ(3R )] (1

⊗

id)(R) [2R

∀a ∈ G, ⊗

1R

(3.26)

⊗ 1A )]. (3.27)

We call such elements normal coherent right diagonal δ-implementers [HN]. With this definition one gets

574

F. Hausser, F. Nill

Lemma 3.7. Let γ : G −→ A be some unital algebra map. Then the relation R := [85 S −1 (84 β) ⊗ 1A ] L [82 S −1 (81 β) ⊗ γ(83 )]

(3.28)

defines a one-to-one correspondence between unital algebra maps γL : Gˆ ./ G −→ A and unital algebra maps γR : G ./ Gˆ −→ A extending γ, as defined in (3.9) and (3.25), respectively. Proof. We will sketch the proof, using graphical methods. For more details see [HN, (G ∗ ⊗A)⊗G : A −→ (G ∗ ⊗ A) ⊗ G by Prop. 10.5]. Defining the map RA X (G ∗ ⊗A)⊗G (A) := eµ ⊗ R21 · (A ⊗ eµ ) , A ∈ A, RA ∗

(G ⊗A)⊗G being an intertwiner of G-modules. property (3.26) of R is equivalent to RA Depicting this intertwiner as a u-fork (≡ up fork) graph

G∗ ( (G RA

∗

⊗A)⊗G

A )

G ,

R

:=

A the relation (3.28) may graphically be expressed as G (

∗

A )

G :=

R A

G∗

A

{

L )

[(

[(

)

] A

G

.

]} (

) (3.29)

Since the r.h.s. of (3.29) defines a G-module intertwiner if and only if L satisfies (3.10), the element R defined by (3.28),(3.29) satisfies (3.26) if and only if L satisfies (3.10). The equivalence of the coherence conditions (3.11) and (3.27) is shown by first expressing (3.27) as the graphical identity (3.12), then plugging in the definition (3.29) and using a pull through rule to collect all rebracketing morphisms at the bottom of the graph and finally using the identities (2.50). We leave it to the reader to draw the corresponding pictures. We are left to show that relation (3.28) may be inverted. This follows by a “two-sided version” of Lemma 2.5. The reader is invited to check that the inverse is given by ¯ )8 ¯ ⊗ γ(8 ¯ )] R [S −1 (α8 ¯ )8 ¯ ⊗ 1A ], L := [S −1 (α8 5

4

3

2

1

(3.30)

¯ ⊗8 ¯ ⊗8 ¯ ⊗8 ¯ ⊗8 ¯ . Equation (3.30) may be expressed graphically with 8−1 =: 8 as the up-side-down mirror image of picture (3.29) with L and R as well as G and G ∗ exchanged. 1

2

3

4

5

Doubles of Quasi-Quantum Groups

575

Putting A = Gˆ ./ G and γL = id or A = G ./ Gˆ and γR = id, respectively, Lemma 3.7 implies Corollary 3.8. Let (δ, 8) be the two-sided coaction of G on G given in (3.2) and define 3 V ≡ V 1 ⊗ V 2 ⊗ V 3 , W ≡ W 1 ⊗ W 2 ⊗ W 3 ∈ G ⊗ by ¯ ⊗8 ¯ ⊗ S −1 (α8 ¯ )8 ¯ , ¯ )α8 V := S(8 1

2

3

5

4

W := 82 S −1 (81 β) ⊗ 83 ⊗ 84 βS(85 ).

Then the map

ˆ ∈ G ./ Gˆ Gˆ ./ G 3 (ϕ ./ a) 7→ V 2 ./ S −1 (V 1 ) * ϕ ( V 3 · (a ./ 1)

provides an algebra isomorphism with inverse given by G ./ Gˆ 3 (a ./ ϕ) 7→ (1ˆ ./ a) · W 1 * ϕ ( S −1 (W 3 ) ./ W 2 ∈ Gˆ ./ G. Corollary 3.8 has been proven for general diagonal crossed products in [HN, Thm. 10.2.iii] using the notation V ≡ qδ and W ≡ pδ . We also remark that one may equally well use the two-sided G-coaction δ 0 := (id ⊗ 1) ◦ 1 with reassociator 80 := [1 ⊗ (id ⊗ 1 ⊗ id)(φ−1 )][1 ⊗ 1 ⊗ φ−1 ][(id ⊗ id ⊗ δ 0 )(φ)] to construct another to versions ˆ Since the two-sided coactions (δ, 8) and of quantum doubles Gˆ ./δ0 G and G ./δ0 G. (δ 0 , 80 ) are twist equivalent [HN, Prop.8.4], these constructions are also isomorphic to the previous ones, i.e. all four diagonal crossed products define equivalent extensions of G [HN, Prop. 10.6]. 3.4. The quasitriangular quasi-Hopf structure. In [HN] we have shown that D(G) is a quasi-bialgebra. As one might expect, D(G) is even a quasitriangular quasi-Hopf algebra. This is the content of the next theorem. Theorem 3.9. Let D(G) be the associative algebra defined in Theorem 3.5. Then (D(G), 1D , D , φD , SD , αD , βD , RD ) is a quasitriangular quasi-Hopf algebra, where φD := (iD ⊗ iD ⊗ iD )(φ), X iD (eµ ) ⊗ D(eµ ) RD := (iD ⊗ id)(D) =

(3.31) (3.32)

µ

and where the structural maps are given by 1D (iD (a)) := (iD ⊗ iD )(1(a)),

∀a ∈ G,

231 13 213 12 −1 (φ−1 D φD D φD , D )

(iD ⊗ 1D )(D) := D (iD (a)) := (a), ∀a ∈ G, (id ⊗ D )(D) := ( ⊗ id)(D) ≡ 1D(G) .

(3.33) (3.34) (3.35) (3.36)

Furthermore the antipode SD is defined by SD (iD (a)) := iD (S(a)),

∀a ∈ G, op

(S ⊗ SD )(D) := (id ⊗ iD )(f ) D(id ⊗ iD ) (f

(3.37) −1

),

(3.38)

where f ∈ G ⊗ G is the twist defined in (2.22). The elements αD , βD are given by αD := iD (α), βD := iD (β).

(3.39)

576

F. Hausser, F. Nill

Proof. To simplify the notation we will frequently suppress the embedding iD , if no confusion is possible, i.e. we write α ≡ iD (α) = αD , (id ⊗ id ⊗ iD )(φ) ≡ φ, etc. To show that (3.33),(3.34) define an algebra morphism 1D : D(G) −→ D(G) ⊗ D(G), it is sufficient to check the consistency with the defining relations (3.18)–(3.20). Consistency with (3.18) is obvious because of (3.33). Let us go on with (3.19): For the r.h.s. we get (id ⊗ 1D ) 1op (a) D = [(1 ⊗ id)(1(a))]231 · (φ−1 )231 D13 φ213 D12 φ−1 . The l.h.s. yields (id ⊗ 1D ) D 1(a) = (φ−1 )231 D13 φ213 D12 φ−1 [(id ⊗ 1)(1(a))], which, using (3.19) and the property φ(1 ⊗ id)(1) = (id ⊗ 1)(1)φ to shift the factor (id ⊗ 1)(1(a)) to the left, equals the r.h.s.. Consistency with the relation (3.20) may be checked in a longer but analogous calculation, where one also has to use the pentagon equation for φ several times, as in the proof of [HN, Lem. 11.2]. Hence 1D is an algebra map. To show that 1D is quasi-coassociative we compute by a similar calculation (1 ⊗ φD ) · (id ⊗ 1D ⊗ id) ◦ (id ⊗ 1D )(D) = (id ⊗ id ⊗ 1D ) ◦ (id ⊗ 1D )(D) · (1 ⊗ φD ), by using again the pentagon equation for φ and the covariance property (3.19). The property of D being a counit for 1D follows directly from the fact that (id ⊗ ⊗ id)(φ) = 1 ⊗ 1. Hence (D(G), 1D , D , φD ) is a quasi-bialgebra, see [HN, Thm. 11.3] for more details. To show quasitriangularity we first note that the element RD = (iD ⊗ id)(D) fulfills (2.11) and (2.12) so to say by definition because of (3.20) and (3.34). The invertibility of RD is equivalent to the invertibility of D which will be proved in Lemma 3.10 (i) below. We are left to show that RD intertwines 1D and 1op D , i.e. 1op D (iD (a)) · RD = RD · 1D (iD (a)), (id ⊗

1D op )(D))

·

23 RD

=

23 RD

∀a ∈ G,

· (id ⊗ 1D )(D).

(3.40) (3.41)

Now Eq. (3.40) follows from (3.19). Hence we also get in D(G)⊗ , 3

12 12 · (1D ⊗ id)(RD ) = (1op RD D ⊗ id)(RD ) · RD ,

(3.42)

which together with (2.11) implies the quasi-Yang Baxter equation 321 12 312 13 132 23 23 231 13 213 12 −1 RD φD RD (φ−1 R D = RD (φ−1 RD φD RD φD . (φ−1 D ) D ) D )

(3.43)

Using Definition (3.34), Eq. (3.43) is further equivalent to 23 23 (iD ⊗ 1op D )(D) · RD = RD · (iD ⊗ 1D )(D),

which also proves (3.41). Hence RD is quasitriangular. In order to prove that the definition of SD in (3.37),(3.38) may be extended antimultiplicatively to the entire algebra D(G), we have to show that this continuation is consistent with the defining relations (3.19),(3.20). This amounts to showing (S ⊗ SD )(D) · (S ⊗ SD )(1op (a)) = (S ⊗ SD )(1(a)) · (S ⊗ SD )(D),

(3.44)

Doubles of Quasi-Quantum Groups

577

and (S ⊗ S ⊗ SD ) (1 ⊗ id)(D) = (S ⊗ S ⊗ SD )(φ) · (S ⊗ S ⊗ SD )(D23 ) · (S ⊗ S ⊗ SD )((φ−1 )132 ) · (S ⊗ S ⊗ SD )(D13 ) · (S ⊗ S ⊗ SD )(φ312 ). (3.45) Since by definition (S ⊗ SD )(D) = f op Df −1 11 , Eq. 3.44) follows directly from (3.19) and the fact that by (2.24) f has the property f · 1(S(a)) = (S ⊗ S)(1op (a)) · f . For the proof of (3.45) let us recall that 1f := f 1(·)f −1 defines a twist equivalent quasicoassociative coproduct on G with twisted reassociator φf defined in (2.16) satisfying φf = (S ⊗ S ⊗ S)(φ321 ) (see (2.26)). Thus we get for the l.h.s. of (3.45) (with Df := f op Df −1 ) (S ⊗ S ⊗ SD ) (1 ⊗ id)(D) = (1fop ⊗ id) (S ⊗ SD )(D) = (1fop ⊗ id)(Df ) −1 231 13 213 23 Df φf , = φ321 f Df (φf )

where the last equality is exactly the transformation property of a quasitriangular Rmatrix under a twist [Dr2] and may be proven analogously using (3.19). By (2.26) this equals the r.h.s. of (3.45). Hence SD defines an anti-algebra morphism on D(G). We are left to show that the map SD fulfills the antipode axioms given in (2.7) and (2.8). Axiom (2.8) is clearly fulfilled since we have SD ◦ iD = iD ◦ S and αD = iD (α), βD = iD (β), φD = (iD ⊗iD ⊗iD )(φ). Noting that 1D (iD (a)) = (iD ⊗iD )(1(a)) , a ∈ G, the validity of axiom (2.7) follows from its validity in G and Lemma 3.10 (ii) below. Lemma 3.10. (i) The universal flip operator D ∈ G ⊗ D(G) is invertible where the inverse is given by D−1 = [X j βS(P i Y j ) ⊗ 1] · [(S ⊗ id)(qρop D)] · [(Ri ⊗ Qi ) 1op (Z j )], (3.46) where qρ ∈ G ⊗ G has been defined in (2.39). (ii) Let µD denote the multiplication map µD : D(G) ⊗ D(G) −→ D(G), then (id ⊗ µD ) ◦ (id ⊗ SD ⊗ id) (id ⊗ 1D )(D) · (1G ⊗ 1G ⊗ αD ) = 1G ⊗ αD , (3.47) (id ⊗ µD ) ◦ (id ⊗ id ⊗ SD ) (id ⊗ 1D )(D) · (1G ⊗ βD ⊗ 1G ) = 1G ⊗ βD . (3.48) Proof. We will use the graphical methods adopted in Sects. 2.2/2.3. To this end let us view G and D ≡ D(G) as left G-modules. Then, due to (3.19), B˜ GD := τGD ◦ D defines an intertwiner B˜ GD : G ⊗ D −→ D ⊗ G which will be depicted as D

G .

B˜ GD =: G 11

D

where we have again suppressed the embedding id ⊗ iD of f

578

F. Hausser, F. Nill

(In fact this is the intertwiner TGA defined in (3.22) for the special case A = D.) For the left modules ∗ G and ∗ D the corresponding intertwiners B˜ ∗ GD , B˜ G ∗ D , B˜ ∗ G ∗ D are defined with the help of the map S and/or SD . Graphically they are represented by the same picture, except that the colours of the legs are replaced by ∗ G and (or) ∗ D, respectively. The reason for distinguishing the D-line from the G-line lies in the fact that unlike in (2.31) B˜ GD is not given in terms of a quasitriangular R-matrix in G ⊗ G, which is why we write B˜ GD in place of BGD . Correspondingly, the identities derived in Sect. 2.3 are not automatically valid for B˜ GD . We now show, which of them still hold. First, since S is an antipode for 1, Eq. (3.20) together with ( ⊗ id)(D) = 1D implies the equality (compare with (2.56))

D

(

)

(

)

(

)

( =

( ∗

G

G

)

)

( ∗

)

G

G

D

D

(3.49)

and a step by step repetition of the proof of (2.59) yields

G

D

( G

D

(B˜ GD )−1 =:

)

(

[(

) )

] .

= D

[(

G (

) )

( D

] ) G

(3.50)

This means that algebraically we get the analogue of the first identity in Eq. (2.41) which yields (3.46). Thus we have proven part (i) To prove (ii) let us translate the two claims (3.47) and (3.48) into the graphical language as

Doubles of Quasi-Quantum Groups

579

(

G

) (

)

(

)

( (3.47) ⇔ (

=

(

)

)

G

G

∗

D

D

D

∗

D

G

)

∗

D

D (3.51)

and

( (

)

D

(3.48) ⇔

=

(

) (

G

D

(

)

(

∗

G )

)

.

) (3.52)

Note that as opposed to (3.49) the identities (3.51) and (3.52) are not automatically satisfied, since SD is not yet proved to be an antipode for 1D . To prove (3.51) and (3.52) we now proceed backwards along the proof of Lemma 2.6, i.e. we use Lemma 2.5 to show that either of these two identities is equivalent to G (

)

(

[(

∗

D )

)

] := (B˜ G ∗ D )−1 .

= [( ( ∗

) )

D

G

(

]

∗

D

G

) (3.53)

More precisely (3.51) is equivalent to (3.53) just as (2.60) is equivalent to the second equation in (2.54), and “rotating” this proof by 180◦ in the drawing plane we also get (3.52) ⇔ (3.53). Thus we are left with proving (3.53). To this end we remark that (3.49) equally holds if we replace D by ∗ D, and therefore (3.50) also holds with D replaced by ∗ D. Hence (3.53) follows from (3.50) provided we can show

580

F. Hausser, F. Nill ∗

G (

)

(

[(

G

D )

)

∗

D

(

]

)

(

[(

) )

] .

= [( ( ∗

) )

]

(

[(

)

(

) )

( ∗

G

D

] )

D

G

(3.54)

By Lemma 2.7 this further equivalent to

[(

)

]

[(

)

]

= ( ∗

) G

∗

D

( G

.

)

(

D

∗

) G

∗

D

( G

) D

(3.55)

Using (2.48) and (2.49), Eq. (3.55) is algebraically equivalent to γ op D = (S ⊗ SD )(D) γ, which finally holds by (3.38),(3.19) and (2.25). This concludes the proof of Lemma 3.10 (ii) and therefore of Theorem 3.9. Clearly, if G is a Hopf algebra and φ = 1 ⊗ 1 ⊗ 1, one recovers the well-known definitions of 1D , SD and RD in Drinfelds quantum double 1D (iD (a)) = (iD ⊗ iD )(1(a)), a ∈ G, ˆ ˆ op (ϕ)), ϕ ∈ G, 1D (D(ϕ)) = (D ⊗ D)(1 SD (iD (a)) = iD (S(a)), SD (D(ϕ)) = D(S −1 (ϕ)), RD = (1ˆ ⊗ eµ ) ⊗ (eµ ⊗ 1), ˆ As in the Hopf algebra case, one may take the where D(ϕ) := (ϕ ⊗ id)(D), ϕ ∈ G. construction of the quasitriangular R-Matrix in D(G) as the starting point and formulate Theorem 3.5 together with Theorem 3.9 differently: Corollary 3.11. Let G be a finite dimensional quasi-Hopf algebra with invertible antipode. Then there exists a unique quasi-Hopf algebra D(G) such that (i) D(G) = Gˆ ⊗ G as a vector space, (ii) the canonical embedding iD : G ,→ 1Gˆ ⊗ G ⊂ D(G) is a unital injective homomorphism of quasi-Hopf algebras,

Doubles of Quasi-Quantum Groups

581

(iii) Let D ∈ G ⊗ D(G) be given by Eq. (3.17), then the element RD := (iD ⊗ id)(D) ∈ D(G) ⊗ D(G) is quasitriangular. This quasi-Hopf algebra structure is given by the definitions in Theorem 3.5 and 3.9. Proof. The property (ii) implies (3.18), (3.33), (3.31), (3.35) and (3.39), yielding also fD = (iD ⊗ iD )(f ). The quasitriangularity of RD implies (3.19), (3.20), (3.34) and op −1 RD fD . Hence the antipode is (3.36) and according to (2.43) (SD ⊗ SD )(RD ) = fD uniquely fixed to be the one defined in Theorem 3.9. We are now in the position to prove Theorem A. Proof of Theorem A. First note that Corollary 3.11 already proves the existence parts (i), (ii) of Theorem A by putting D(ϕ) := (ϕ ⊗ id)(D). The fact that µ : Gˆ ⊗ G −→ D(G) provides a linear isomorphism follows from the last statement in Theorem 3.5. Moreover, ˜ : Gˆ −→ D˜ is a linear map such if D˜ ⊃ G is another Hopf algebra extension and if D ˜ µ ) ∈ D˜ ⊗ D˜ is ˜ ˜ ˆ that D is algebraically generated by G and D(G) and RD˜ := eµ ⊗ D(e quasitriangular, then ν : D(G) −→ D˜ ˜ (2) ) a ν(ϕ ⊗ a) := (id ⊗ ϕ(1) )(qρ ) D(ϕ

(3.56)

˜ by Prop. 3.4. In fact, is a uniquely and well defined algebra map satisfying ν ◦ D = D the quasitriangularity of RD˜ implies that ν is even a quasi-bialgebra homomorphism. Thus D(G) also solves the universality property (iii) of Theorem A. In particular the extension D(G) ⊃ G is unique up to equivalence. 3.5. The category RepD(G). We will now give a representation theoretical interpretation of the quantum double D(G) by describing its representation category in terms of the representation category of the underlying quasi-Hopf algebra G. In this way we will show that D(G) is a concrete realization of the quantum double as defined by Majid in [M2] with the help of a Tannaka–Krein-like reconstruction theorem. We denote the monoidal category of finite dimensional unital representations of D(G) and of G by Rep D(G) and Rep G, respectively. The next proposition states a necessary and sufficient condition, under which a representation of G extends to a representation of D(G): Proposition 3.12. 1) The objects of Rep D(G) are in one to one correspondence with pairs {(πV , V ), DV }, where (πV , V ) is a finite dimensional representation of G and where DV ∈ G ⊗ EndC (V ) is a normal coherent 1-flip, i.e. (i) ( ⊗ id)(DV ) = idV , (ii) DV · (id ⊗ πV )(1(a)) = (id ⊗ πV )(1op (a)) · DV , ∀a ∈ G, −1 132 23 13 DV φV = (1 ⊗ id)(DV ), where φV := (id ⊗ id ⊗ πV )(φ). (iii) φ312 V DV (φV ) 2) Let {(πV , V ), DV } and {(πW , W ), DW } be as above, then HomD(G) = {t ∈ HomG (V, W ) | (id ⊗ t)(DV ) = DW }. Proof. We define the extended representation πVD on the generators of D(G) by πVD (iD (g)) : = πV (g), πVD (D(ϕ))

g ∈ G,

: = (ϕ ⊗ idEndV )(DV ),

(3.57) ˆ ϕ ∈ G.

(3.58)

Condition (i) implies that πVD is unital whereas conditions (ii),(iii) just reflect the defining relations (3.19) and (3.20) of D(G), which ensures that πVD is a well defined algebra morphism. On the other hand, given a representation (πVD , V ) of D(G), we define

582

F. Hausser, F. Nill

DV := (idG ⊗ πVD )(D) which clearly satisfies conditions (i)–(iii). This proves Part 1. Part 2. follows trivially. To get the relation with Majid’s formalism [M2] we now write a · v := πV (a)v, a ∈ ¯ ¯ G, v ∈ V and define βV : V −→ G ⊗ V ; v 7→ v (1) ⊗ v (2) := DV 1G ⊗ v . With this notation we get the following corollary: Corollary 3.13. The conditions (i)-(iii) of Proposition 3.12 are equivalent to the following three conditions for βV (as before denoting P i ⊗ Qi ⊗ Ri = φ−1 ): (i’) ( ⊗ idV ) ◦ βV = idV , ¯ ¯ ¯ ¯ (ii’) (a(2) · v)(1) a(1) ⊗ (a(2) · v)(2) = a(2) v (1) ⊗ a(1) · v (2) , ∀v ∈ V , ¯ ¯ ¯ ¯ ¯ i (2) (1) i i (2) (2) (iii’) Ri v (1) ⊗ (Q h · v ) P ⊗ (Q · v ) = i ¯ ¯ ¯ −1 321 i (1) i i (φ ) · (R · v) (2) Q ⊗ (R · v)(1) (1) P i ⊗ (Ri · v)(2) , ∀v ∈ V . Proof. The equivalences (i) ⇔ (i’) and (ii) ⇔ (ii’) are obvious. The equivalence (iii) ⇔ 312 from the left and with φ−1 (iii’) follows by multiplying (iii) with (φ−1 V ) V from the right and permuting the first two tensor factors. The conditions stated in the above corollary agree with those formulated in [M2, Prop.2.2] by taking G cop ≡ (G, 1op , (φ−1 )321 ) instead of (G, 1, φ) as the underlying quasi-bialgebra. This means that we have identified the category Rep D(G) with what is called the double category of modules over G in [M2]. 4. Doubles of Weak Quasi-Hopf Algebras Allowing the coproduct 1 to be no-unital (i.e. 1(1) 6= 1 ⊗ 1) leads to the definition of weak quasi-Hopf algebras as introduced by G. Mack and V. Schomerus in [MS]. In this section we sketch how the construction of the quantum double D(G) generalizes to this case. As it will turn out, there are only minor adjustments to be made. The reason for this lies in the fact that we have used mostly graphical identities (i.e. identities in Rep G) to derive and describe our results. But since Rep G is a rigid monoidal category also in the case of weak quasi-Hopf algebras G, all graphical identities in Sects. 2.2 and 2.3 stay valid. Thus the only adjustments required refer to those points, where we have translated graphical identities into algebraic ones. Following [MS] we define a weak quasi-Hopf algebra (G, 1, 1, , φ) to be an associative algebra G with unit 1, a non-unital algebra map 1 : G −→ G ⊗ G, an algebra map : G −→ C and an element φ ∈ G ⊗ G ⊗ G satisfying (2.1)–(2.3), whereas (2.4) is replaced by (id ⊗ ⊗ id)(φ) = 1(1),

(4.1)

and where in place of invertibility φ is supposed to have a quasi-inverse φ¯ ≡ φ−1 with ¯ = φ, respect to the intertwining property (2.1). By this we mean that φ¯ satisfies φφφ ¯ φ¯ = φ¯ as well as φφ φ φ¯ = (id ⊗ 1)(1(1)), which implies the further identities

φ¯ φ = (1 ⊗ id)(1(1)),

(4.2)

Doubles of Quasi-Quantum Groups

583

(id ⊗ 1)(1(a)) = φ (1 ⊗ id)(1(a)) φ¯ , ∀a ∈ G, φ = φ (1 ⊗ id)(1(1)), φ¯ = φ¯ (id ⊗ 1)(1(1)), ¯ = 1(1). (id ⊗ ⊗ id)(φ)

(4.3) (4.4) (4.5)

More generally we call an element t ∈ A an intertwiner between two (possibly nonunital) algebra maps α, β : G −→ A, if t α(a) = β(a) t, ∀a ∈ G

and t α(1) ≡ β(1) t = t.

In this case by a quasi-inverse of t (with respect to this intertwiner property) we mean the unique (if existing) element t¯ ≡ t−1 ∈ A satisfying t¯t = α(1), tt¯ = β(1) and t¯tt¯ = t¯. Note that this implies t¯ β(a) = α(a) t¯,

t¯ β(1) ≡ α(1) t¯ = t¯,

and therefore t is also the quasi-inverse of t−1 . A weak quasi-bialgebra is called weak quasi-Hopf algebra, if there exists a unital algebra antimorphism S : G −→ G and elements α, β ∈ G satisfying (2.7) and (2.8). We will also always suppose that S is invertible. Furthermore, G is said to be quasitriangular if there exists an element R ∈ G ⊗ G satisfying (2.10)–(2.12) and possessing a quasi-inverse R¯ ≡ R−1 with respect to the intertwining property (2.10). With these substitutions Theorem A generalizes as follows Theorem B. Let (G, 1, φ) be a finite dimensional weak quasi-Hopf algebra with invertible antipode S. Assume D(G) ⊃ G to be a weak quasi-Hopf algebra extension satisfying (i)-(iii) of Theorem A. Then D(G) exists uniquely up to equivalence and the linear map µ : Gˆ ⊗ G −→ D(G) µ(ϕ ⊗ a) := (id ⊗ ϕ(1) )(qρ ) D(ϕ(2) ) a is surjective with Ker µ = Ker P , where P : Gˆ ⊗ G −→ Gˆ ⊗ G is the linear projection P (ϕ ⊗ a) := ϕ(2) ⊗ Sˆ −1 (ϕ(1) ) * 1G ( ϕ(3) a =: ϕ ./ a. To adapt our previous strategy to weak quasi-Hopf algebras we first recall that due to the coproduct being non-unital the definition of the tensor product functor in Rep G has to be slightly modified. First note that the element 1(1) (as well as higher coproducts of 1) is idempotent and commutes with all elements in 1(G). Thus, given two representations (V, πV ), (W, πW ), the operator (πV ⊗πW )(1(1)) is a projector, whose image is precisely the G-invariant subspace of V ⊗ W on which the tensor product representation operates non-trivially. Thus one is led to define the tensor product of two representations of G by setting V W := (πV ⊗ πW )(1(1)) (V ⊗ W ),

πV πW := (πV ⊗ πW ) ◦ 1|V W . (4.6)

One readily verifies that with these definitions φU V W – restricted to the subspace (U V ) W – furnish a natural family of isomorphisms defining an associativity constraint for the tensor product functor , where the tensor product of morphisms is defined by restricting the “usual” tensor product map to the truncated subspace.

584

F. Hausser, F. Nill

With these adjustments, the graphical calculus described in Sects. 2.2 and 2.3 carries over to the present case. The collection of colored upper (or lower) legs represent the (truncated) tensor product of G-modules associated with the individual legs. One just has to take care when translating the pictures into algebraic identities. For example the graph

≡ G

G

GG

is a pictorial representation of 1(1) and not of 1 ⊗ 1! Thus the graph (2.44) is equivalent to the algebraic identity [S(p1λ ) ⊗ 1] qλ 1(p2λ ) = 1(1) in place of the first equation of (2.47), etc. In this way all graphical identities of Sect. 2 stay valid as well as Theorem 2.1, where now R−1 is meant to be the quasi-inverse of R. The definition of the diagonal crossed product in Proposition 3.1 yields an associative algebra which in general is not unital, but (1Gˆ ⊗ 1G ) is still a right unit and in particular idempotent [HN, Thm. 14.2]. This may be cured by taking the right ideal generated by (1Gˆ ⊗ 1G ). Thus, let P : Gˆ ⊗ G −→ Gˆ ⊗ G be the linear projection given by left multiplication with 1Gˆ ⊗ 1G with respect to the algebra structure (3.7), i.e. P (ϕ ⊗ a) := ϕ(2) ⊗ Sˆ −1 (ϕ(1) ) . 1 / ϕ(3) a = 1(−1) * ϕ ( S −1 (1(1) ) ⊗ 1(0) a. (4.7) As in [HN, Sect. 14] we introduce the notation ϕ ./ a := P (ϕ ⊗ a) ∈ Gˆ ⊗ G

(4.8)

and define the quantum double D(G) as the subalgebra D(G) := Gˆ ./ G ≡ P (Gˆ ⊗ G).

(4.9)

Then 1Gˆ ./ 1G ≡ 1Gˆ ⊗ 1G is the unit of D(G) and in terms of the notation (4.8) the multiplication in D(G) is still given by (3.7). In particular iD : G 3 a 7→ 1Gˆ ./ a ≡ 1Gˆ ⊗ a ∈ D(G) still provides a unital algebra inclusion. Interpreting Eq. (3.9) also via (4.8), Corollary 3.3 likewise extends to the present scenario. However note that now the definition (3.10) for left diagonal δ-implementers L ∈ G ⊗ A also implies the nontrivial relation L ≡ [1G ⊗ 1A ] L = [S −1 (1(1) ) ⊗ 1A ] L [1(−1) ⊗ τ (1(0) )].

(4.10)

This leads to a slight modification of Proposition 3.4 where one has to add the requirement that 1-flip operators T fulfill also T (id ⊗ τ )(1(1)) ≡ (id ⊗ τ )(1op (1)) T = T which follows directly from (4.10) or by multiplicating both sides of (3.13) from the right with (id ⊗ τ )(1(1)). Taking this additional identity into account, Theorem 3.5 now reads

Doubles of Quasi-Quantum Groups

585

Theorem 4.1. Using the notation (4.8) we define the element D ∈ G ⊗ D(G) by X S −1 (p2ρ ) eµ p1ρ(1) ⊗ (eµ ./ p1ρ(2) ). D := µ

Then the multiplication (3.7) is the unique algebra structure on D(G) satisfying (3.18)– (3.20) together with D (id ⊗ iD )(1(1)) ≡ (id ⊗ iD )(1op (1)) D = D.

(4.11)

Moreover the identity (3.21) also stays valid. The quasitriangular quasi-Hopf structure is now defined precisely as in Theorem 3.9 and is proven analogously, where in (3.38) f −1 is the quasi-inverse of f . Correspondingly D−1 given by (3.46) becomes the quasi-inverse of D with respect to the 1-flip property (3.19). Thus we arrive at a Proof of Theorem B. The existence parts (i),(ii) of Theorem B follow by putting as before D(ϕ) = (ϕ ⊗ id)(D). The universality (and therefore uniqueness) property (iii) follows analogously as in the proof of Theorem A, Eq. (3.56). Here one just has to note that by Prop. 3.4, ˜ ∈ G ⊗ D˜ ˜ := qρop D L P ˜ µ ). Hence L ˜ satisfies (4.10) ˜ := is a normal coherent δ-implementer, where D eµ ⊗ D(e ˜ and therefore the map ν : Gˆ ⊗ G −→ D, ˜ a ν(ϕ ⊗ a) := (ϕ ⊗ id)(L) satisfies ν ◦P = ν, where P is the projection (4.7). Since as in Corollary 3.3 the relations (3.10),(3.11) guarantee that ν is an algebra map with respect to the multiplication (3.7) ˜ f (ϕ ./ a) := on Gˆ ⊗ G, it passes down to a well defined algebra map f : D(G) −→ D, ˜ ˜ ˜ ν(ϕ ⊗ a), thus proving (iii). Since D (as an element in D ⊗ D) is also required to be quasitriangular, f is even a quasi-bialgebra homomorphism. In the case D˜ = D(G) we have ν = µ and Ker P = Ker µ by definition, proving also the second part of Theorem B.

A. The Twisted Double of a Finite Group As an application we now use Theorem 3.5 and Theorem 3.9 to recover the “twisted” quantum double Dω (G) of [DPR] where G is a finite group and ω : G × G × G → U (1) is a normalized 3-cocycle. By definition this means ω(g, h, k) = 1 whenever at least one of the three arguments is equal to the unit e of G and ω(g, x, y)ω(gx, y, z)−1 ω(g, xy, z)ω(g, x, yz)−1 ω(x, y, z) = 1,

∀g, x, y, z ∈ G.

The Hopf algebra G := F un(G) of functions on G may then also be viewed as a quasiHopf algebra with its standard coproduct, counit and antipode but with reassociator given by X ω(g, h, k) · (δg ⊗ δh ⊗ δk ), (A.1) φ := g,h,k∈G

586

F. Hausser, F. Nill

where δg (x) := δg,x . The identities (2.2) and (2.4) for φ are equivalent to ω being a normalized 3-cocycle. Also note that choosing α = 1G the antipode axioms now require P β = g ω(g −1 , g, g −1 )δg . In this special example our quantum double D(G) ≡ Gˆ ./ G allows for another identification with the linear space Gˆ ⊗ G. Lemma A.1. Let G be as above and define σ : Gˆ ⊗ G −→ D(G) by σ(ϕ ⊗ a) := ˆ a ∈ G. Then σ is a linear bijection. D(ϕ) a, ϕ ∈ G, Proof. Since (G, 1, , S) is also an ordinary Hopf algebra, the relation (3.19) is equivalent to (suppressing the symbol iD ) ˆ a D(ϕ) = D a(1) * ϕ ( S −1 (a(3) ) a(2) , ∀a ∈ G, ϕ ∈ G. (A.2) Using (3.21) this implies

1 1 1 * ϕ ( (qρ2 S −1 (qρ(3) )) qρ(2) a, ϕ ./ a ≡ (id ⊗ ϕ(1) )(qρ ) D(ϕ(2) ) a = D qρ(1)

which lies in the image of σ. Hence, σ is surjective and therefore also injective.

We note that in general the map σ need not be surjective (nor injective). Due to Lemma A.1 we may now identify D(G) with the new algebraic structure on Gˆ ⊗ G induced by σ −1 . We call this algebra Gˆ ⊗D G. Putting a ≡ 1ˆ ⊗D a, a ∈ G and D := eµ ⊗ (eµ ⊗D 1) ∈ G ⊗ (Gˆ ⊗D G) it is described by the relations (A.2),(3.20) and the requirement of G ≡ 1ˆ ⊗D G being a unital subalgebra. To compute these multiplication rules we now use that the group elements g ∈ G provide a basis in Gˆ with dual basis δg ∈ G. Hence a basis of Gˆ ⊗D G is given by {h⊗D δg }h,g∈G . In this basis the generating matrix D is given by X X δk ⊗ (k ⊗D 1G ), 1G = δh . (A.3) D= k∈G

h∈G

Let us know compute the multiplication laws according to the definitions in Theorem 3.5. To begin with, we have (h ⊗ 1G )(e ⊗ δg ) = (h ⊗ δg )

(g ⊗ 1G )(h ⊗ 1G ) = (gh ⊗ 1G ). P Taking (x⊗id) of both sides of (3.19), where x ∈ G, and using 1(δg ) = k∈G δk ⊗δk−1 g we get and

(x ⊗ 1G )(e ⊗ δx−1 g ) = (e ⊗ δgx−1 )(x ⊗ 1G ), or equivalently (e ⊗ δg )(x ⊗ 1G ) = (x ⊗ δx−1 gx ).

(A.4)

Finally, pairing Eq. 3.20) with x ⊗ y ∈ Gˆ ⊗ Gˆ in the two auxiliary spaces, the l.h.s. yields X ω(s, x, y)(1 ⊗ δs )(x ⊗ 1G ) · ω(x, r, y)−1 (e ⊗ δr )(y ⊗ 1G ) · ω(x, y, t)(1 ⊗ δt ) s,r,t∈G

= (x ⊗ 1G )(y ⊗ 1G ) ·

X ω(s, x, y)ω(x, y, t) (e ⊗ δ(xy)−1 sxy δy−1 ry δt ) ω(x, r, y)

s,r,t∈G

=

X t∈G

(x ⊗ 1G )(y ⊗ 1G )(1 ⊗ δt )

ω(xyt(xy)−1 , x, y)ω(x, y, t) , ω(x, yty −1 , y)

Doubles of Quasi-Quantum Groups

587

where we have used (A.4) in the first equality. The right-hand side of (3.20) gives (xy ⊗ 1G ) so that we end up with (x ⊗ 1G )(y ⊗ 1G ) =

X t∈G

ω(x, yty −1 , y) (xy ⊗ δt ). ω(xyt(xy)−1 , x, y)ω(x, y, t)

Similarly the coproduct is computed as 1D (e ⊗ δg ) = 1D (x ⊗ 1G ) =

X r,s∈G

P

(A.5)

k∈G (e ⊗ δk ) ⊗ (e ⊗ δk−1 g )

and

ω(xrx−1 , x, s) (x ⊗ δr ) ⊗ (x ⊗ δs ) . (A.6) −1 −1 w(x, r, s)ω(xrx , xsx , x)

The above construction agrees with the definition of Dω (G) given in [DPR] up to the convention that they have built D(G) on G ⊗ Gˆ instead of Gˆ ⊗ G. B. The Monodromy Algebra The definition of monodromy algebras (see e.g. [AFFS]) associated with quasitriangular Hopf algebras may now easily be generalized to the case of quasi-Hopf algebras. This has already been done in [AGS]. We will give an explicit proof that the defining relations of [AGS] indeed define an associative algebra structure on Gˆ ⊗ G, which in fact is isomorphic to our quantum double D(G). For ordinary Hopf algebras this has recently been shown in [N1]. Let G be a finite dimensional quasi-Hopf algebra with quasitriangular R-matrix R ∈ G ⊗ G. Following [N1] we define the monodromy matrix M ∈ G ⊗ D(G) to be M := (id ⊗ iD )(Rop ) D. Defining also Rˆ ∈ G ⊗ G ⊗ D(G) by Rˆ := φ213 R12 φ−1 , we get the following lemma: Lemma B.1. The monodromy matrix M obeys the following three conditions (dropping the symbol iD ): ( ⊗ id)(M) = 1D(G) , 1(a) M = M 1(a), a ∈ G, 13 ˆ M R M23 = Rˆ φ (1 ⊗ id)(M) φ−1 .

(B.1) (B.2) (B.3)

Proof. We will freely suppress the embedding iD . Since the R-Matrix has the property (id ⊗ )(R) = 1, Eq. (B.1) follows from ( ⊗ id)(D) = 1D(G) . The identity (B.2) is implied by (3.19) and the intertwiner property of the R-matrix. Let us now compute the l.h.s. of (B.3): M13 Rˆ M23 = R31 D13 φ213 R12 φ−1 R32 D23 = R31 D13 [(1 ⊗ id)(R)φ−1 ]132 D23 312 13 −1 132 23 = (R ⊗ 1) · (1 ⊗ id)(R) D (φ ) D ,

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F. Hausser, F. Nill

where we have used the quasitriangularity of R in the second line and property (3.19) of D in the third line. The r.h.s. of (B.3) yields Rˆ φ (1 ⊗ id)(M) φ−1 = φ213 R12 (1 ⊗ id)(Rop ) φ312 D13 (φ−1 )132 D23 312 13 −1 132 23 = φ321 (1 ⊗ R) (id ⊗ 1)(R) φ D (φ ) D , where we have used the definitions of M and Rˆ and (3.20). Now, the quasitriangularity of R implies (R ⊗ 1) (1 ⊗ id)(R) = φ321 (1 ⊗ R) (id ⊗ 1)(R) φ which finally proves (B.3).

Note that the relations (B.1)–(B.3) are precisely the defining relations postulated by [AGS] to describe the algebra generated by the entries of a monodromy matrix around a closed loop together with the quantum group of gauge transformations sitting at the initial (≡ end) point of the loop. Thus we define similarly as in [N1] Definition B.2. The gauged monodromy algebra MR (G) ⊃ G is the algebra extension generated by G and elements M (ϕ), ϕ ∈ Gˆ with defining relations given by (B.1)–(B.3), where M (ϕ) ≡ (ϕ ⊗ id)(M). Lemma B.1 then implies the immediate Corollary B.3. Let (G, R) be a finite dimensional quasitriangular quasi-Hopf algebra. Then the monodromy algebra MR (G) and the quantum double D(G) are equivalent extensions of G, where the isomorphism is given on the generators by M (ϕ) ↔ (ϕ ⊗ id)(Rop D). References [A] [AC]

Abe, E.: Hopf Algebras, Cambridge University Press, Cambridge, 1980 Altschuler, D., Coste, A.: Quasi-Quantum Groups, Knots, Three-Manifolds and Topological Field Theory. Commun. Math. Phys. 150, 83–107 (1992) [AC2] Altschuler, D. and Coste, A.: Invariants de rubans et alg´ebres quasi Hopf. Rencontres de Strasbourg, (Strasbourg) (D. Bennequin and M. Rosso, eds.), 1992 [AFFS] Alekseev, A.Yu., Faddeev, L.D., Fr¨ohlich, J., Schomerus, V.: Representation Theory of Lattice Current Algebras. Commun. Math. Phys. 191, 31 (1998) [AFSV] Alekseev, A.Yu., Faddeev, L.D., Semenov-Tian-Shansky, L.D., Volkov, A.Yu.: The unraveling of the quantum group structure in WZNW theory. Preprint CERN-TH-5981/91 (1991) [AGS] Alekseev, A.Yu., Grosse, H., Schomerus, V.: Combinatorial Quantization of the Hamiltonian Chern Simons Theory, I, II. Commun. Math. Phys. 172, 317–358 (1995) and 174, 561–604 (1996) [AS] Alekseev, A.Yu., Schomerus, V.: Representation Theory of Chern Simons Observables. Duke Math. J. 85 No. 2, 447 (1996) [BaWi] Bais, F.A., de Wild Propitius, M.: Quantum groups in the Higgs phase. Theor. Math. Phys. 98, 425 (1994); Discrete Gauge Theories. hep-th/9511201 [BL] Bernard, D., LeClair, A.: The quantum double in integrable quantum field theory. Nucl. Phys. B399, 709 (1993) [ByS] Bytsko, A.G., Schomerus, V.: Vertex Operators – From a toy model to lattice algebras. Commun. Math. Phys. 191, 87 (1998) [DPR] Dijkgraaf, R., Pasquier, V., Roche, P.: Quasi Hopf algebras, group cohomology and orbifold models. Nucl. Phys. 18B (Proc. Suppl), 60 (1990) [Dr1] Drinfeld, V.G.: Quantum groups. In: Proc. Int. Cong. Math., Berkeley, 1986, p. 798

Doubles of Quasi-Quantum Groups

[Dr2] [Fa]

[FG] [HN] [K] [M1] [M2] [M3] [M4] [M5] [ML] [MS] [M¨u]

[N1] [N2] [NSz] [PW] [RT] [Sw] [Sz] [SzV] [T]

589

Drinfeld, V.G.: Quasi-Hopf Algebras. Leningrad Math. J. 1, No.6, 1419 (1990) Faddeev, L.D.: Quantum Symmetry in conformal field theory by Hamiltonian methods. In: New Symmetry Principles in Quantum Field Theory, (Proc. Carg`ese 1991) eds. J. Fr¨ohlich et al., New York: Plenum Press, 1992, pp. 159–175 Falceto, F., Gawedszki, K.: Lattice Wess–Zumino–Witten Model and Quantum Groups. J. Geom. Phys. 11, 251 (1993) Hausser, F., Nill, F.: Two-Sided Crossed Products by Duals of Quasi-Quantum Groups. qalg/9708004, to appear in Rev. Math. Phys. Kassel, C.: Quantum Groups. New York. Springer, 1995 Majid, S.: Tannaka–Krein Theorem for Quasi-Hopf algebras and other Results. Contemp. Math. Phys. 150, 219–232 (1992) Majid, S.: Quantum Double for Quasi-Hopf Algebras. q-alg/9701002 Majid, S.: Foundations of Quantum Group Theory. Cambridge: Cambridge University Press, 1995 Majid, S.: Quasitriangular Hopf algebras and Yang Baxter equations. Int. J. Mod. Phys. A5, 1 (1990) Majid, S.: Braided Groups. J. Pure Appl. Alg. 86, 187–221 (1993) Mac Lane, S.: Categories for the Working Mathematician. New York: Springer, 1971 G. Mack, V. Schomerus, Quasi Hopf Quantum Symmetry in Quantum Theory. Nucl. Phys. B370, 185 (1992) M¨uger, M.: Quantum Double Actions on Operator Algebras and Orbifold Quantum Field Theories. Commun. Math. Phys. 191, 31 (1998); Disorder Operators, Quantum Doubles and Haag Duality in 1 + 1 Dimensions. Preprint DESY 96-237; Superselection Structure of Quantum Field Theories in 1 + 1 Dimensions. PhD thesis, DESY 97–073 Nill, F.: On the Structure of Monodromy Algebras and Drinfeld Doubles. Rev. Math. Phys. 9, 371–397 (1997) Nill, F.: Fusion Structures from Quantum Groups II: Why truncation is necessary. Lett. Math. Phys. 29, 83–90 (1993) Nill, F., Szlach´anyi, K.: Quantum Chains of Hopf Algebras with Quantum Double Cosymmetry. Commun. Math. Phys. 187, 159 (1997) Podle´s, P., Woronowicz, S.L.: Quantum deformation of the Lorentz group. Commun. Math. Phys. 130, 381 (1990) Reshetikin, N.Yu. and Turaev, V.G.: Ribbon Graphs and their Invariants derived from Quantum Groups. Commun. Math. Phys. 127, 1–26 (1990) Sweedler, M.E.: Hopf algebras. New York: Benjamin, 1969 Szlach´anyi, K.: Unpublished Szlach´anyi, K., Vescerny´es, P.: Quantum symmetry and braid group statistics in G-spin models. Commun. Math. Phys. 156, 127 (1993) Turaev, V.G.: Quantum Invariants of Knots and 3-Manifolds. Berlin: Walter de Gruyter, 1994

Communicated by R. H. Dijkgraaf

Commun. Math. Phys. 199, 591 – 604 (1999)

Communications in

Mathematical Physics © Springer-Verlag 1999

Volume of Vortex Moduli Spaces N. S. Manton, S. M. Nasir? Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK. E-mail: [email protected] Received: 5 November 1997 / Accepted: 26 May 1998

Abstract: A gas of N Bogomol’nyi vortices in the Abelian Higgs model is studied on a compact Riemann surface of genus g and area A. The volume of the moduli space is computed and found to depend on N, g and A, but not on other details of the shape of the surface. The volume is then used to find the thermodynamic partition function and it is shown that the thermodynamical properties of such a gas do not depend on the genus of the Riemann surface. 1. Introduction Solitons are interesting objects to study and it is particularly interesting to study their dynamics. The moduli space approximation [6] gives an effective description of the dynamics of solitons at low energy when most of the degrees of freedom are frozen. The moduli space approximation works as follows: static multi-solitons are parametrized by the moduli space − the minima of the energy functional. At low energy, the actual field dynamics can be taken to be close to the moduli space, i.e. near the bottom of the valley of the energy functional. The dynamics projected onto the moduli space is then the geodesic motion on the moduli space [13]. For monopoles − solitons in three dimensions − the moduli space approximation has given important insight into the scattering and the bound states of the monopoles. It has found important application in proving various duality conjectures in supersymmetric field theories and in string theory [10]. On a plane, for Bogomol’nyi vortices in the Abelian Higgs model [1] − solitons in two dimensions − one can similarly describe their scattering [9]. For vortices on a compact Riemann surface, M , of genus g, one can do more − study their statistical mechanics [7]. Since the potential energy between the vortices is constant, in the moduli space approximation evaluation of the partition function of a gas of vortices effectively ? Current address: Mittag-Leffler Institute, Auravagan 17, S-18262 Djursholm, Sweden. E-mail: [email protected]

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reduces to the computation of the volume of the moduli space. As the moduli space is K¨ahler, in order to find the volume, one needs to know the K¨ahler form or more precisely, its cohomology class. For the genus g = 0, 1 cases, the K¨ahler forms have been computed in [7, 12], respectively. For the sphere (g = 0) the N -vortex moduli space is the complex projective space CPN . In this case symmetry arguments are enough to find the K¨ahler form. On the other hand, for the torus (g = 1), the K¨ahler form is found by exploiting the fibre bundle structure of the N -vortex moduli space. For genus g ≥ 1 and N ≥ 2g − 1, the N -vortex moduli space has a bundle structure, where the base is the Jacobian, J, of the Riemann surface, a torus of real dimension 2g, and the fibre is CPN −g . For N ≤ g, the N -vortex moduli space is homeomorphic to a 2N -dimensional analytic subvariety of the Jacobian. It would be interesting to find a general formula for the K¨ahler form and, hence, the volume of the moduli space for N vortices on an arbitrary Riemann surface, M , of genus g. Here, we will obtain such a formula. In the next section we will see that the K¨ahler form is the sum of two parts: one is related to the K¨ahler form of M , the other is determined by the vortex interactions. The cohomology classes of both of these can be determined. This then enables the required formula for the volume of the moduli space to be computed. It depends on N, g and the area of M . The various thermodynamical quantities for a gas of vortices can be deduced from this. It is found that the statistical mechanics of such a gas is independent of the genus of M . This is expected on physical grounds. This paper is organized as follows. In Sect. 2, we briefly describe the Bogomol’nyi vortices and the K¨ahler form on the moduli space. In Sect. 3, we present the cohomological formula for the volume. Then we compare it with the previously computed cases for vortices on the sphere and the torus. This serves as a check of the volume formula. Finally, in Sect. 4, we compute the various thermodynamical quantities. 2. Vortices and the K¨ahler Form on the Moduli Space (i) Bogomol’nyi vortices. Bogomol’nyi vortices are static, topologically stable, finite energy solutions of the critically coupled Abelian Higgs model in 2+1 dimensions [4]. We consider vortices on the space-time R × M , where M is a compact Riemann surface of genus g, and R parametrizes ordinary time x0 . The metric on R × M can be taken to be of the form (locally) ds2 = dx20 − (x1 , x2 )(dx21 + dx22 ),

(2.1)

where is positive. The Lagrangian density of the model is 1 1 1 L = − Fµν F µν + Dµ φDµ φ − (|φ|2 − 1)2 , 4 2 8

(2.2)

where φ is a complex Higgs scalar field, Aµ is a U (1) gauge potential, Dµ = ∂µ − iAµ and Fµν = ∂µ Aν − ∂ν Aµ (µ, ν = 0, 1, 2). The units are chosen such that both the gauge field coupling constant and the mass of the Higgs field are unity. Working in the gauge A0 = 0, the Lagrangian is L = T − V , where Z 1 ˙¯ d2 x (A˙ 1 A˙ 1 + A˙ 2 A˙ 2 + φ˙ φ), (2.3) T = 2 M 1 V = 2

Z M

−1

d x 2

2 F12

+ D1 φD1 φ + D2 φD2 φ + (|φ|2 − 1)2 4

(2.4)

Volume of Vortex Moduli Spaces

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are respectively the kinetic and the potential energies. Further, we need to impose Gauss’s law. This arises from the equation of motion of A0 , as the following constraint, ¯ = 0. ∂1 A˙ 1 + ∂2 A˙ 2 − Im(φ˙ φ)

(2.5)

In the static case the total energy, E = V , can be reexpressed as [1] Z 1 d2 x (D1 + iD2 )φ (D1 + iD2 )φ + −1 {F12 + (|φ|2 − 1)}2 + F12 . E= 2 M 2 (2.6) Here we have omitted a total derivative term, which vanishes as M has no boundary. Bogomol’nyi vortices, which minimize the above energy integral, satisfy the first order Bogomol’nyi equations (D1 + iD2 )φ = 0, F12 +

(2.7)

(|φ|2 − 1) = 0. 2

(2.8)

The solutions are classified into topologically stable sectors determined by the first Chern number [4, 14] Z 1 d2 xF12 = N, (2.9) 2π M where N is an integer. Note that, in general, there is an obstruction to the existence of N -vortex solutions on a compact surface. This is seen by integrating (2.8) over M . Since |φ|2 is non-negative, we deduce the bound, first obtained by Bradlow [2], 4πN ≤ A,

(2.10)

where A is the area of M . Assuming that this is satisfied, the solutions with the first Chern number N are uniquely determined by specifying N zeros of the Higgs field [2, 14]. Thus, N can also be interpreted as the vortex number. Since vortices are indistinguishable, the vortex moduli space, MN , is diffeomorphic to the symmetric product (M )N /SN , where SN is the permutation group of N elements. It should be noted that MN is a smooth manifold. In the sector with vortex number N , the potential energy of the vortices is E = πN . It is possible to eliminate the gauge potentials from Eq. (2.8), by solving (2.7), thereby obtaining an equation for |φ|2 , ∇2 log |φ|2 − (|φ|2 − 1) = 4π

N X

δ 2 (x − xi ),

(2.11)

i=1

where xi denotes the position of the zero of the Higgs field associated with the ith vortex and ∇2 = ∂12 + ∂22 . The kinetic energy (2.3) induces a natural Riemannian metric on the moduli space [6]. Let qα and gαβ (q)dq α dq β , where (α, β = 1, · · · , 2N ) denote real coordinates and the metric on MN . Then, in the moduli space approximation for vortex motion the Lagrangian can be written as

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L=

π gαβ (q)q˙α q˙β − N π, 2

(2.12)

where π is the mass of a single vortex. Below, we shall use the analogue of this expression using complex coordinates for the vortex positions. Although we cannot determine gαβ explicitly, we shall show that it is possible to compute the total volume of MN . (ii) The K¨ahler form on the moduli space. Samols [9] has obtained an expression for the metric gαβ and the associated K¨ahler form on the N -vortex moduli space by analyzing data around the N zeros of the Higgs field assuming these are distinct. Detailed computation shows that the metric has a smooth extension to the complete moduli space, where vortices may coincide. Let z be a local complex coordinate on M . Let the vortex positions be {zi = x1i + ix2i : i = 1, · · · , N }. Since zi is a simple zero of the Higgs field, log |φ|2 has the following series expansion obtained on using (2.11), 1 1 log |φ|2 = log |z−zi |2 + ai + bi (z − zi ) + b¯i (z¯ − z¯i ) + ci (z − zi )2 2 2 (zi ) (z − zi )(z¯ − z¯i ) + c¯i (z¯ − z¯i )2 + · · · . (2.13) − 4 From the expression for the kinetic energy Eq. (2.3), Samols shows, after some integrations, that the metric is ds2 =

N X

(zi )δij + 2

i,j=1

∂bi ∂ z¯j

dzi dz¯j .

(2.14)

Only the coefficients of the linear terms in (2.13) contribute to this formula. Notice that bi is a function of the positions of all N vortices. The reality property of the kinetic energy implies that ∂ b¯ i ∂bj = ∂zj ∂ z¯i

(2.15)

and from this follows the Hermiticity of the metric (2.14). One can then define the associated K¨ahler form as N ∂bi i X (2.16) (zi ) δij + 2 dzi ∧ dz¯j . ω= 2 ∂ z¯j i,j=1

Using (2.15) one can show that ω is a closed (1,1) form. The volume of the moduli space is Z 1 ωN . (2.17) VolN = N ! MN The K¨ahler form ω can be divided into two parts ω = ω1 + ω2 , where N

ω1 =

iX (zi )dzi ∧ dz¯i 2 i=1

is just N copies of the area form induced from M and

(2.18)

Volume of Vortex Moduli Spaces

595 N X ∂bi ω2 = i dzi ∧ dz¯j ∂ z¯j

(2.19)

i,j=1

contains information about the relative vortex positions. Our aim is to understand the topological nature of ω2 and its effect on VolN . If ω2 is ignored, VolN would simply be AN /N !. Notice that to obtain this result we have chosen a specific normalization of ω dictated by physics. In fact we can choose any normalization by multiplying the Lagrangian by an overall constant. ¯ where B is a one-form of degree (1,0), Notice that one can write ω2 = −i∂B, B=

N X

bi (z1 , z2 , · · · , zN , z¯1 , z¯2 , · · · , z¯N )dzi .

(2.20)

i=1

Since zi are natural coordinates on the Cartesian product (M )N , not on the moduli space MN , the symmetry of the one-form B is not manifest in the above equation. However, the indistinguishability of vortices implies that bi (· · · , zi , · · · , zj , · · · ) = bj (· · · , zj , · · · , zi , · · · ) .

(2.21)

Thus, the one-form B is symmetric and hence, defined on MN . Before proceeding further we would like to point out that the one-form B has poles whenever zi = zj for i 6= j. To see this let us consider the function ψ defined in a coordinate patch as follows: ψ = log |φ|2 −

N X

log |z − zi |2 .

(2.22)

i=1

Notice that ψ is a smooth function, since the singularities of log |φ|2 at the zeros of the PN Higgs field have been cancelled by the term i=1 log |z − zi |2 . Then, as zj approaches zi , one can see that bi =

2 + smooth part, zi − zj

(2.23)

hence, B has poles. It is useful to note that the residues of B are integers. This fact will be important later. One simple way to uncover the topological significance of B is to determine its transformation properties under change of coordinates. Let us assume that M is covered in such a way that all N vortices lie in one coordinate patch U whose local coordinate is denoted by z. The ith vortex in this patch has the coordinate zi . Under a holomorphic coordinate transformation, U goes into another coordinate patch U 0 . In terms of the local coordinate z → z 0 = ζ(z); and, also zi → zi0 = ζ(zi ). In the transformed coordinate, the expansion of log |φ|2 in (2.13) reads 1 1 log |φ|2 = log |z 0 − zi0 |2 + a0i+ b0i (z 0 − zi0 ) + b¯0i (z¯0 − z¯i0 ) + c0i (z 0 − zi0 )2 2 2 (zi0 ) 0 0 ¯0 (z − zi )(z − z¯i0 ) + c¯0i (z¯0 − z¯0 i )2 + · · · . (2.24) − 4 0 ), writing out the coordinate dependence explicitly. Here, b0i = b0i (z10 , · · · , zi0 , · · · , zN 2 Remember that |φ| is a globally well defined function on M . Thus, on the overlap

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region U ∩ U 0 , by comparing the coefficients of (z − zi ) on the right-hand sides of Eqs. (2.13) and (2.24) one finds bi = b0i

∂ζi ∂zi ∂ 2 ζi + , ∂zi ∂ζi ∂zi2

(2.25)

where ζi = ζ(zi ). Notice the striking similarity with the corresponding transformation of the Levi–Civita connection of M 0

0zzz = 0zz0 z0

∂z 0 ∂z ∂ 2 z 0 . + ∂z ∂z 0 ∂z 2

(2.26)

This heralds the topological nature of B. By looking at Eqs. (2.25) and (2.26), one concludes that B differs from the complex connection one-form on the co-tangent bundle of MN by a globally defined one-form. Generically, this one-form is not smooth as it ¯ would have been cohomologous contains poles. If the poles were absent then ω2 = −i∂B to the complex Ricci curvature two-form of the Levi–Civita connection on the co-tangent bundle of MN . This means that ω2 /2π would have been cohomologous to the first Chern class of the co-tangent bundle. In what follows we will need to evaluate the integrals of ω2 restricted to some special complex one-dimensional submanifolds. The integrals, as we will see shortly, receive two contributions: one is from the residues of B, and the other is due to the fact that B, restricted to these submanifolds, is related to the complex Levi–Civita connection. First, we consider the submanifold of N coincident vortices. The solutions with N coincident vortices are parametrized by a complex one-dimensional submanifold, Mco of the moduli space MN . Mco is diffeomorphic to M and lies inside the Jacobian J. Let Z be the position of the coincident vortices. |φ|2 now satisfies the equation 4

∂ 2 log |φ|2 − (|φ|2 − 1) = 4πN δ 2 (z − Z) ∂z∂ z¯

(2.27)

and log |φ|2 has the following series expansion around Z: 1¯ 1 ¯ + ··· . z¯ − Z) log |φ|2 = N log |z − Z|2 + a + b(z − Z) + b( 2 2

(2.28)

Then, the one-form B, restricted to Mco , simplifies to ¯ B = b(Z, Z)dZ.

(2.29)

By a similar analysis as in (2.25), one can determine the transformation properties of b under a holomorphic coordinate transformation z → ξ(z). One obtains b0 ∂ξ(Z) ∂Z ∂ 2 ξ(Z) b . = + N N ∂Z ∂ξ(Z) ∂Z 2

(2.30)

We remark that for N coincident vortices B does not contain any pole. By comparing (2.30) with (2.26), one finds that B/N restricted to the submanifold Mco differs from the complex Levi–Civita connection one-form of Mco by a smooth, globally defined oneform. Thus, ω2 /N restricted to Mco is cohomologous to the complex Ricci curvature two-form of the co-tangent bundle of Mco . Now, the volume of Mco is Z Z iN ∂b ω= + 2 ¯ dZ ∧ dZ¯ = N (A − 4πN (1 − g)) , Vco = 2 M ∂Z Mco (2.31)

Volume of Vortex Moduli Spaces

597

where use has been made of the Gauss–Bonnet formula for the integral of the curvature of the Levi–Civita connection, 0Z ZZ , on M Z Z ∂0ZZ −i dZ ∧ dZ¯ = 2(1 − g) (2.32) 2π M ∂ Z¯ which implies −i 2πN

Z M

∂b ∂ Z¯

dZ ∧ dZ¯ = 2(1 − g).

(2.33)

Z 1 ω2 = 2N 2 (g − 1). The volume Vco agrees with the volumes previ2π Mco ously computed for the sphere and the torus in [7, 12], respectively. Secondly, let us consider two clusters of vortices with m and (N − m) vortices, and let z1 and z2 be their positions on M , respectively. The solutions corresponding to such clusters are parametrized by a complex two-dimensional submanifold, Mc , of the moduli space MN . We can do a similar analysis as in the above to compute the integral of ω2 restricted to certain one-dimensional submanifolds of Mc . Restricted to Mc , B can be written as Notice that

B = b1 dz1 + b2 dz2 .

(2.34)

Notice that b1 has a pole at z1 = z2 and from the generalization of (2.23) to a pair of vortex clusters, one finds that Res(b1 ) = 2(N − m). Following (2.25) one can determine the transformation properties of b1 and b2 under holomorphic coordinate changes z1 → z10 and z2 → z20 . These are ∂z10 ∂z1 ∂ 2 z10 +m 0 , ∂z1 ∂z1 ∂z12

(2.35)

∂z20 ∂z2 ∂ 2 z20 + (N − m) 0 . ∂z2 ∂z2 ∂z22

(2.36)

b1 (z1 , z2 ) = b01 (z10 , z20 )

b2 (z1 , z2 ) = b02 (z10 , z20 )

We will be particularly interested in the case when the second cluster does not move, i.e. when z2 is a constant. The vortex motion is then restricted to a one-dimensional ˜ is diffeomorphic to M . ˜ , of Mc . M submanifold, M ˜ , differs from Now, comparing (2.35) with (2.26) one sees that B/m, restricted to M ˜ by a one-form which contains a the complex Levi–Civita connection one-form of M ˜ one can write pole at z1 = z2 . For the volume of M Z ω = Ir + I. (2.37) V˜ = ˜ M

Here, Ir is the contribution coming from the residues and I contains the rest of the contribution. Similarly as in the derivation of (2.31) we find I = m (A − 4πm(1 − g))

(2.38)

Ir = −2πm Res(b1 ) = −4πm(N − m).

(2.39)

and the residue contribution is

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N. S. Manton, S. M. Nasir

˜ is Thus, the total volume of M V˜ = m(A − 4πN + 4πmg).

(2.40)

As a consistency check, if m = N then we have one cluster of N coincident vortices. In this case we get back (2.31) Z by simply putting m = N in the above formula. 1 ω2 = 2m(mg − N ). Naturally, one would expect that the We remark that 2π M˜ (1,1) form ω2 belongs to H 2 (MN , R), since this is a part of the K¨ahler form of MN . However, because of the relationship between B and the complex Levi–Civita connection one-form, combined with the fact that the residues of B are integers, one sees that the integral of ω2 /2π over any complex one-dimensional submanifold is an integer. This means that ω2 /2π actually belongs to H 2 (MN , Z). This information will be used in the next section in obtaining a cohomological formula for ω2 . 3. Cohomology and the Volume of the Moduli Space (i) Cohomology ring of the symmetric products of a Riemann surface. Here, we quote several theorems without proof which will be used later. This also serves to fix the notation. The main reference is [5]. We have H 0 (M, Z) = Z, H 1 (M, Z) = Z2g and H 2 (M, Z) = Z. Let αi , i = 1, · · · , 2g be the generators of H 1 (M, Z) and β be the generator of H 2 (M, Z). It is useful to note that β is a normalized area form (i.e. its integral over M is unity) of type (1,1). The ring structure of H ∗ (M, Z) can be described as follows: αi αj = 0, i 6= j ± g ; αi αi+g = −αi+g αi = β, 1 ≤ i ≤ g.

(3.1)

Here, juxtaposition means cup product. Let αik = 1 ⊗ · · · ⊗ 1 ⊗ αi ⊗ 1 ⊗ · · · ⊗ 1 ∈ H 1 ((M )N , Z), βk = 1 ⊗ · · · ⊗ 1 ⊗ β ⊗ 1 ⊗ · · · ⊗ 1 ∈ H 2 ((M )N , Z),

(3.2)

the αi and β being in the k th place. Then, H ∗ ((M )N , Z) is generated by the αik and the βk (1 ≤ i ≤ 2g, 1 ≤ k ≤ N ) with the following relations being satisfied: αik αjk = 0, i 6= j ± g, αik αi+g,k = −αi+g,k αik = βk , 1 ≤ i ≤ g, αik αjl = −αjl αik , k 6= l.

(3.3)

Now, define the following symmetric linear combinations: ξi = αi1 + · · · + αiN , 1 ≤ i ≤ 2g, η = β1 + · · · + βN .

(3.4)

Further, define ξi0 = ξi+g (1 ≤ i ≤ g) and σi = ξi ξi0 . Then we have the following result [5] Theorem 1. Let M be a compact connected Riemann surface of genus g, MN its N th symmetric product. Then, the cohomology ring H ∗ (MN , Z) is generated by elements ξ1 , · · · , ξg , ξ10 , · · · , ξg0 of degree 1, and an element η of degree 2, subject to the following relations: (a) the ξ’s and ξ 0 ’s anti-commute with each other and commute with η;

Volume of Vortex Moduli Spaces

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(b) If i1 , · · · , ia , j1 , · · · , jb , k1 , · · · , kc are distinct integers from 1 to g inclusive, then ξi1 · · · ξia ξj0 1 · · · ξj0 b (σk1 − η) · · · (σkc − η)η q = 0

(3.5)

provided that a + b + 2c + q = N + 1. We will also need the following result on the cohomology of some particular submanifolds of MN . Let ν = (N1 · p1 + · · · + Nk · pk ) be a partition of N such that P Qk pi Ni . Then there exists a mapping from i=1 MNi p1 > p2 > · · · pk > 0 and N = onto a closed submanifold 4(ν) of MN , where 4(ν) has N1 clusters of p1 coincident vortices, N2 clusters of p2 coincident vortices, etc. This mapping is an isomorphism. For any submanifold Y , let us write [Y ] for its cohomology class in H ∗ (MN , Z). Then, one can show that [5] Theorem 2. [4(ν)] is the coefficient of τ1N1 · · · τkNk in P ρ−g η N −ρ−g

g Y (P η + Q(η − σi )),

(3.6)

i=1

where P = p 1 τ 1 + · · · + pk τ k , Q = (p21 − p1 )τ1 + · · · + (p2k − pk )τk , ρ = N 1 + · · · + Nk .

(3.7)

Now, if δs = [4(1 · s + (N − s) · 1)], s > 1, so that δs is the cohomology class of the submanifold of MN which consists of those points which have at least s vortices coinciding at one point, then one can show using Theorem 2 that δs = s(N + (g − 1)(s − 1))η s−1 − s(s − 1)η s−2 (σ1 + · · · + σg ).

(3.8)

In terms of the above notation, the submanifold Mco for N coincident vortices corresponds to 4(1 · N ) and its cohomology class is δN = N (N + (g − 1)(N − 1))η N −1 − N (N − 1)η N −2 (σ1 + · · · + σg ). (3.9) Qg Further, the total Chern class of the tangent bundle of MN is (1 + η)N −2g+1 i=1 (1 + η − σi ). So, the first Chern class of the tangent bundle is c1 (T MN ) = (N − g + 1)η − (σ1 + · · · + σg ).

(3.10)

(ii) Cohomological formula for the K¨ahler form and the volume of the moduli space1 . An expression for the cohomology class of the two-form ω2 can be obtained using the fact that ω2 /2π is a (1,1) form belonging to H 2 (MN , Z). Let us determine the generators of H 2 (MN , Z) which are of type (1,1). One can see that η is a generator of H 2 (MN , Z), and this is of type (1,1). The other type (1,1) generator of H 2 (MN , Z) comes from the pairing of the generators of H 1 (MN , Z). In Appendix (i) we show that it must be of the form D0 (σ1 + · · · + σg ),

(3.11)

1 An attempt to obtain a cohomological formula for the K¨ ahler form was first made by P. Shah [11]. His work has inspired us to look further into the problem from a cohomological point of view.

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where D0 is a non-zero integer. Thus, the general expression for ω2 reads ω2 = 2πC(g, N )η + 2πD(g, N )(σ1 + · · · + σg ),

(3.12)

where C(g, N ) and D(g, N ) are integers. The coefficients C(g, N ) and D(g, N ) can be determined by computing the volumes of the submanifolds describing different types of coincident vortices by cohomological means and then comparing them with the same volumes previously computed in Sect. 2. The volume of Mco − which describes the motion of N coincident vortices − is Z Z (ω1 + ω2 ) = (ω1 + ω2 ) ∧ δN . (3.13) Vco = Mco

MN

Using (3.9) and (3.5), one finds Vco = N (A + 2πC(g, N ) + 2πN gD(g, N )) ,

(3.14)

where we have used the fact that ω1 = Aη.

(3.15)

C(g, N ) + N gD(g, N ) = 2N (g − 1).

(3.16)

Equating this with (2.31), we require

˜ − which describes the In Appendix (ii) we show that the volume of the submanifold M motion of m coincident vortices with the remaining (N − m) vortices coincident and held fixed at a general position − is V˜ = m (A + 2πC(g, N ) + 2πmgD(g, N )) .

(3.17)

Comparing this with (2.40) gives C(g, N ) + mgD(g, N ) = −2N + 2mg.

(3.18)

From (3.16) and (3.18), we find C(g, N ) = −2N,

D(g, N ) = 2.

(3.19)

Thus, the K¨ahler form on MN is ω = ω1 + ω2 = (A − 4πN )η + 4π(σ1 + · · · + σg ).

(3.20)

ω2 /2π = −2c1 (T MN ) + 2(1 − g)η,

(3.21)

Notice that

where use has been made of (3.10). This shows that ω2 /2π is not just the first Chern class of the co-tangent bundle of MN . Now, putting all the ingredients together, and using (3.5), one finally obtains the following formula for the volume of the moduli space: Z g X ωN (4π)i (A − 4πN )g−i g! N −g . (3.22) = (A − 4πN ) VolN = (N − i)!(g − i)!i! MN N ! i=0

Volume of Vortex Moduli Spaces

601

In the formula above N ≥ g. An analogous formula can be written for N < g. The sum now runs from i = 0 to i = N , and the factors of A − 4πN are combined to give (A − 4πN )N −i in the sum. Notice that the volume is just a function of the area of M , its genus, and the number of vortices. It contains no information about the shape of M . (iii) Examples: The volume of the moduli space for the sphere and the torus. For the sphere (g = 0), (3.22) gives VolN =

(A − 4πN )N . N!

(3.23)

This is precisely the same as the formula obtained in [7]. On the other hand for the torus (g = 1) one gets VolN =

A(A − 4πN )N −1 . N!

(3.24)

Again this is the same as the formula obtained in [12]. Shah conjectured in [11] that the volume of the moduli space for any Riemann surface with genus g > 1 is given by (3.24). We, however, find this conjecture to be not true, e.g. for a Riemann surface of genus g = 2 and N ≥ 2, the volume is VolN =

(A2 − 16π 2 N )(A − 4πN )N −2 N!

(3.25)

which is different from (3.24). 4. Thermodynamics of the Vortices Following [7], the thermodynamics of N vortices at temperature T can be treated using the Gibbs distribution. The partition function is Z 1 [dp][dq]e−E(p,q)/T , (4.1) Z = 2N h MN where h is Planck’s constant, pα are the momenta conjugate to the coordinates qα and E is the energy. After doing the Gaussian momentum integrals, the partition function reduces to Z 2 2 N [dq](detgαβ )1/2 . (4.2) Z = (2π T /h ) MN

The second factor in this partition function is just the volume, VolN , of the moduli space MN . Using (4.2) and (3.22) one obtains the partition function for a gas of N vortices on M N (A − 4πN )N −g 2π 2 T R(g, A, N ), (4.3) Z= N! h2 where R(g, A, N ) =

g X (A − 4πN )g−i (4π)i g!N ! i=0

(N − i)!(g − i)!i!

.

(4.4)

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To obtain the thermodynamic limit, we let N → ∞, assuming that the density of the gas of vortices is a fixed constant given by N/A = n. Now, a short calculation shows that, at fixed n, (4.5) R(g, A, N ) = Ag 1 + O(1/N ) . Using Stirling’s formula for N !, when N is large, one obtains the free energy F = −T log Z, g g 2eπ 2 T log A + O(1/N ) . − log N + (1 − ) log(A − 4πN ) + F ' −N T log h2 N N (4.6) The pressure P = −∂F/∂A is P =

NT . A − 4πN

The entropy S = −∂F/∂T is 2 2 2e π T 1 − 4πn + log . S = N log n h2

(4.7)

(4.8)

These are precisely the same formulae as obtained in [7, 12]. Notice that the genus g appears nowhere in the formulae for the thermodynamical quantities. Thus, the thermodynamics of a gas of vortices is independent of the topology of the space on which the vortices are moving. 5. Conclusion Central to our study of the thermodynamics of a gas of vortices on an arbitrary Riemann surface is the computation of the volume of the vortex moduli space. The dependence of the volume on the area of the Riemann surface is quite noticeable. The area dependence disappears from the volume whenever A = 4πN − Bradlow’s limit. Then, for N ≤ g the volume of the moduli space is VolN = (4π)N g!/[N !(g − N )!], and for N > g the volume is zero. At A = 4πN the Higgs field vanishes everywhere and the problem of solving the Bogomol’nyi equations reduces to the problem of solving for a constant magnetic field on the Riemann surface M . It can be shown that for N = g the moduli space of this problem is related to the space of flat U (1) connections on M . Time-varying flat connections have non-trivial kinetic energy, and hence, following the argument of Sect. 2, there is a metric on this moduli space. The volume of this moduli space is a topological quantity. It is of interest to see that the volume of this moduli space is equal to Volg at Bradlow’s limit. This is shown in Ref. [8]. For N > g, it is also shown in [8], how VolN tends to zero as A approaches 4πN . Moduli spaces play an important role in diverse areas of physics and mathematics. In general it is desirable to know more about moduli spaces, e.g. their volume (compact cases), metric etc. Computation of the volume of a moduli space is not totally new. In [16], with a remarkable use of the Verlinde formula [15], Witten computed the volume of the moduli space of flat connections (for semi-simple gauge groups) on an arbitrary Riemann surface. In this case, however, the volume is a purely topological quantity. Thus, it is gratifying to see that in the case of the moduli space of Bogomol’nyi vortices on a compact Riemann surface one can also explicitly compute the volume. This is almost topological, but not exactly so, because the volume depends on the area of the Riemann surface, not on its shape.

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Appendix (i) A note on a (1,1) form belonging to H 2 (MN , Z). Let wρ , (ρ = 1, · · · , g) be a basis of holomorphic one-forms on M with the period matrix 3 = (λρi ), (i = 1, · · · , 2g). wρ P2g is related to the generators αi of H 1 (M, Z) as wρ = i=1 λρi αi . A basis of holomorphic one-forms on (M )N is given by wρk = 1 ⊗ · · · ⊗ 1 ⊗ wρ ⊗ 1 ⊗ · · · ⊗ 1, 1 ≤ ρ ≤ g, 1 ≤ k ≤ N

(A.1)

with wρ being in the k th place. Then, a basis ζρ of holomorphic one-forms on MN is given by the following symmetric linear combinations: ζρ = wρ1 + · · · + wρN , 1 ≤ ρ ≤ g.

(A.2)

One sees that ζρ =

2g X

λρi ξi .

(A.3)

i=1

Using the Riemann bilinear relations the period matrix can be written as 3t = (I 0), where I is the (g × g) unit matrix and 0 = (γjl ), (j, l = 1, · · · , g) is a symmetric matrix with Im(0) > 0. Notice that under the diffeomorphisms of M the elements (γjl ) can change. Let v ∈ H 2 (MN , Z) be expressed as v=

2g 1X qij ξi ξj , 2

(A.4)

i,j=1

where Q = (qij ) is an antisymmetric matrix with integer elements. Then, expressing v in terms of ζρ one can show that it is of type (1,1) if the following constraint is satisfied [3]: 3t Q−1 3 = 0.

(A.5)

This being a matrix constraint leaves one to freely choose g 2 elements of Q. However, for v to be invariant under diffeomorphisms of M , the above equation must be satisfied for arbitrary values of (γjl ) with Im(γjl ) > 0. This can be true only if Q has the following form: 0 I 0 , (A.6) Q=D −I 0 where I is the (g × g) unit matrix and D0 is a constant integer. Thus, on MN any integral (1,1) form v must be of the following type: v = D0 (σ1 + · · · + σg ).

(A.7)

(ii) Proof of (3.17). Consider the mapping j : M 0 × M 00 → MN , given by j(z1 , z2 ) = (z1 , · · · , z1 , z2 , · · · , z2 ), where z1 occurs m times and z2 occurs (N − m) times. M 0 and M 00 are two copies of M . For z2 fixed, the mapping j is an isomorphism onto the ˜ . One obtains submanifold M j ? (ξi ) = mαi0 ⊗ 1, j ? (η) = mβ 0 ⊗ 1.

(A.8)

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Here, αi0 and β 0 are, respectively, the generators of H 1 (M 0 , Z) and H 2 (M 0 , Z). Now, Z Z Z η= j ? (η) = m β 0 = m, (A.9) M0

˜ M

M0

and, similarly, since σi = ξi ξi+g , Z σi = m2 , 1 ≤ i ≤ g.

(A.10)

ω = m(A + 2πC(g, N ) + 2πmgD(g, N ))

(A.11)

˜ M

Thus,

Z ˜ M

as claimed. Acknowledgement. We are grateful to Professor G. Segal for very helpful discussions. Many thanks to Dr. C. Houghton and Dr. H. Merabet for their critical comments on this manuscript. The work of S.M.N. was supported by the Overseas Research Council, the Cambridge Commonwealth Trust and Wolfson College. This work was partly supported by EPSRC grant GR/K50641, part of the Applied Nonlinear Mathematics programme.

References 1. Bogomol’nyi, E.B.: The stability of classical solutions. Sov. J. Nucl. Phys. 24, 449–454 (1976) 2. Bradlow, S.: Vortices in holomorphic line bundles over closed K¨ahler manifolds. Commun. Math. Phys. 135, 1–17 (1990) 3. Griffiths, P. and Harris, J.: Principles of Algebraic Geometry. New York: John Wiley & sons, 1978 4. Jaffe, A. and Taubes, C.H.: Vortices and Monopoles. Structure of static gauge theories. Boston: Birkh¨auser, 1980 5. MacDonald, I.G.: Symmetric products of an algebraic curve. Topology 1, 319–343 (1962) 6. Manton, N.S.: A remark on the scattering of BPS monopoles. Phys. Lett. 110B, 54–56 (1982) 7. Manton, N.S: Statistical mechanics of vortices. Nucl. Phys. B400, 624–632 (1993) 8. Nasir, S.M.: Vortices and flat connections. Phys. Lett. 419B, 253–257 (1998) 9. Samols, T.M.: Vortex scattering. Commun. Math. Phys. 145, 149–180 (1992) 10. Sen, A.: Dyon-monopole bound states, self-dual harmonic forms on the multi-monopole moduli space, and SL(2,Z) invariance in string theory. Phys. Lett. 329B, 217–221 (1994) 11. Shah, P.A.: Ph.D Thesis. Cambridge University, 1995, (Unpublished) 12. Shah, P.A. and Manton, N.S.: Thermodynamics of vortices in the plane. J. Math. Phys. 35, 1171–1184 (1994) 13. Stuart, D.: Dynamics of Abelian Higgs vortices in the near Bogomolny regime. Commun. Math. Phys. 159, 51–91 (1994) 14. Taubes, C.H.: Arbitrary N-Vortex solutions to the first order Landau-Ginzburg equations. Commun. Math. Phys. 72, 277–292 (1980) 15. Verlinde, E.: Fusion rules and modular transformations in 2D conformal field theory. Nucl. Phys. B300, 360–376 (1988) 16. Witten, E.: On quantum gauge theories in two dimensions. Commun. Math. Phys. 141, 153–209 (1991) Communicated by R. H. Dijkgraaf

Commun. Math. Phys. 199, 605 – 647 (1999)

Communications in

Mathematical Physics © Springer-Verlag 1999

c Elliptic Algebra Uq,p (sl 2 ): Drinfeld Currents and Vertex Operators Michio Jimbo1 , Hitoshi Konno2 , Satoru Odake3 , Jun’ichi Shiraishi4 1

Division of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan. E-mail: [email protected] 2 Department of Mathematics, Faculty of Integrated Arts and Sciences, Hiroshima University, HigashiHiroshima 739-8521, Japan. E-mail: [email protected] 3 Department of Physics, Faculty of Science, Shinshu University, Matsumoto 390-8621, Japan. E-mail: [email protected] 4 Institute for Solid State Physics, University of Tokyo, Tokyo 106-0032, Japan. E-mail: [email protected] Received: 19 January 1998 / Accepted: 26 May 1998

b 2 ) introduced earlier Abstract: We investigate the structure of the elliptic algebra Uq,p (sl by one of the authors. Our construction is based on a new set of generating series in b 2 ), which are elliptic analogs of the Drinfeld currents. the quantum affine algebra Uq (sl b 2 ) with the tensor product of Uq (sl b 2 ) and a Heisenberg They enable us to identify Uq,p (sl algebra generated by P, Q with [Q, P ] = 1. In terms of these currents, we construct an L operator satisfying the dynamical RLL relation in the presence of the central element b 2 ) for c. The vertex operators of Lukyanov and Pugai arise as “intertwiners” of Uq,p (sl the level one representation, in the sense to be elaborated on in the text. We also present vertex operators with higher level/spin in the free field representation.

1. Introduction 1.1. Vertex operators in SOS models. The principle of infinite dimensional symmetry has seen an impressive success in conformal field theory (CFT). With the aim of understanding non-critical lattice models in the same spirit, the method of algebraic analysis [1, 2, 3] has been developed. In this approach, a central role is played by the notion of vertex operators (VO’s). There are two kinds of VO’s with distinct physical significance: the type I VO, which describes the operation of adding one lattice site, and the type II VO, which plays the role of particle creation/annihilation operators. In the most typical example of the XXZ spin chain, these VO’s have a clear mathematicalmeaning b2 . as intertwiners [4] of certain modules over the quantum affine algebra Uq sl An important class of CFT is the minimal unitary series [5]. Their lattice counterpart are the solvable models of Andrews–Baxter–Forrester (ABF) [6]. These are “solid-onsolid” (SOS, or “face”) models whose Boltzmann weights are expressed by elliptic functions. Their Lie theoretic generalizations have also been studied extensively [7, 8, 9]. The vertex operator approach to the ABF models and their fusion hierarchy was

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b 2 was used only as an formulated in [10] by a coset-type construction. In [10], Uq sl auxiliary tool to define the VO’s, and its role as a symmetry algebra was somewhat indirect. In [11], Lukyanov and Pugai constructed a free boson realization of type I VO’s for the ABF models. (The formulas for type II VO’s can be found in [12].) They have shown further that these VO’s commute with the action of the deformed Virasoro algebra (DVA) [13], making clear the parallelism with CFT. However, unlike the case of CFT, the VO’s did not allow for direct interpretation as intertwiners, because DVA lacks a coproduct1 . It has remained an open problem to understand the conceptual meaning of VO’s. b 2 and proposed it as In [14], one of the authors introduced an elliptic algebra Uq,p sl an algebra of screening currents of conjectural extended DVA associated with the fusion b 2 , and to SOS models. The aim of the present article is to continue the study of Uq,p sl show that it offers a characterization of the VO’s for SOS models in close analogy with the XXZ model. 1.2. Face type elliptic algebras. Through an attempt to understand integrable models based on elliptic Boltzmann weights, various versions of “elliptic quantum groups” [15– 19] have been introduced. According to Frønsdal [18, 19], elliptic quantum groups are nothing but quantum affine algebras Uq (g) equipped with a coproduct different from the original one. The resulting objects are quasi-Hopf algebras in the sense of Drinfeld [20]. Throughout this paper, we restrict our attention to the elliptic algebra of face type b 2 ) in [21]. b 2 , denoted as Bq,λ (sl associated with g = sl In Frønsdal’s approach, the quasi-Hopf structures are defined by twistors given as formal series in the deformation parameters. An explicit construction for the twistors was given in [21]. (A very similar construction was presented independently in [22].) The L-operators and the VO’s for the elliptic algebra can be obtained by “dressing” those b 2 with the twistor (up to some subtleties about the fractional powers which will of Uq sl be discussed shortly). From this point of view, the construction of the VO’s in bosonic representations is reduced to the determination of the image of the twistors. However, the solution of this issue is not known to us at this moment. For the bosonic realization of quantum affine algebras, the best suited presentation is in terms of the Drinfeld currents. In this paper we aim at an alternative construction of L operators and VO’s based on an elliptic analog of Drinfeld currents. These operators satisfy the same relations as those derived from the quasi-Hopf approach [21]. Though the precise relation is not known, we expect that these two methods give equivalent answers. Our construction is inspired by the work of Enriquez and Felder [17], who introduced Drinfeld-type currents defined on an elliptic curve and constructed the twistor b by a quantum factorization method. The algebra Uq,p sl2 in [14] and U~ g(τ ) in [17] are both central extensions of the same algebra, but there are significant differences. We shall discuss more about this in Sect. 6.2. 1.3. Outline of the results. Let us describe the content of this paper. Our starting point b 2 carrying a parameter r (the elliptic is to introduce a new set of currents of Uq sl modulus), obtained by modifying the usual Drinfeld currents. We shall refer to them as “elliptic currents”. They satisfy commutation relations with coefficients written in infinite products. The latter are essentially the Jacobi theta functions but not quite so, since the elliptic currents, and hence these coefficients, comprise only integral powers 1

The usual coproduct for the Virasoro algebra has no non-trivial deformation.

Elliptic Algebra Uq,p (slb2 ): Drinfeld Currents and Vertex Operators

607

in the Fourier mode expansions. In order to have relations written in theta functions alone, we need to supply fractional powers. For thispurpose we introduce ‘by hand” a b 2 and satisfy [Q, P ] = 1. Adjoining pair of generators P, Q which commute with Uq sl P, Q to the elliptic currents, we obtain “total currents” whose commutation relations b 2 ) [14] (see (3.34)–(3.43)). coincide with the defining relations of the algebra Uq,p (sl b b 2 and the In other words, we can identify Uq,p (sl2 ) with the tensor product of Uq sl b 2 ) mentioned above is the Heisenberg algebra generated by P, Q. The algebra Bq,λ (sl b b subalgebra of Uq,p (sl2 ) isomorphic to Uq (sl2 ), and is equipped with a coproduct defined via the twistor. However, this coproduct does not seem to extend naturally to the full b 2 ). That is, Bq,λ (sl b 2 ) is a quasi-Hopf algebra while (to our knowledge) algebra Uq,p (sl b 2 ) is not. We emphasize that the intertwining relations for VO’s will be based on Uq,p (sl the quasi-Hopf structure of the former. A characteristic feature of the elliptic algebras is that, in the presence of the central element c, we are forced to deal with two different elliptic moduli r and r∗ = r − c simultaneously [15]. From Frønsdal’s point of view, it is an effect of the quasi-Hopf twisting. The appearance of two different curves makes it difficult to apply the geometric method of [17]. Instead, we take a more pedestrian approach. Motivated by similar formulas in [17], we introduce “half currents” as certain contour integrals of the total currents. They have an advantage that the coefficients of the commutation relations can be written solely in terms of theta functions (as opposed to the delta functions appearing in the relations for the total currents). We then borrow the idea of the Gauß decomposition [23] to compose an L operator out of the half currents, and show that it satisfies the expected (dynamical) RLL relation (Proposition 4.4, 4.5). The construction of the L operator allows us to study the VO’s in the bosonic repreb 2 ). As is clear from the construction, the elliptic sentation. Let us first consider Bq,λ (sl currents can be realized in the same bosonic Fock spaces as with the Drinfeld currents of b 2 ) are a family b 2 . (We regard p = q 2r as a formal parameter.) The VO’s for Bq,λ (sl Uq sl ∗ of intertwiners 8(z, s), 9 (z, s) carrying a parameter s, and their intertwining relations b 2) involve a shift of s (see (2.9)–(2.10)). With the adjunction of P,L Q, the algebra Uq,p (sl has an enlarged Fock module. It has the decomposition F = s Fs into eigenspaces b 2 . Accordingly we modify of P , each eigenspace Fs being a Fock module for Uq sl further the VO’s with Q, b s) = z 2r1 ( 21 h(2) 8(u, ∗

2

+(s+h(1) )h(2) )

b (u, s) = 9 (z, s)z 9 ∗

− 2r1∗

8(q c z, s),

2 ( 21 h(1) +sh(1) )

Qh(1)

e

(1.1) ,

(1.2) b2 . where z = q 2u , h(1) = h ⊗ 1, h(2) = 1 ⊗ h, h being the “Cartan” generator of Uq sl Solving the intertwining relations for level one, we find that the VO’s of Lukyanov and Pugai arise in the form (1.1), (1.2), apart from certain signs in the intertwining relations for 8(z, s) and 9∗ (z, s). (For the discussion of the signs, see Subsects. 5.2 and 5.4.) We also calculate formulas for VO’s associated with higher spin representations. 1.4. Plan of the text. The text is organized as follows. In Sect. 2, we recall some results of [21] which are relevant to the following sections. b In Sect. 3, we introduce the elliptic currents of Uq sl2 , and discuss its relation to b 2 ). In Sect. 4, we introduce the “half currents” and derive their commutation Uq,p (sl relations. We then arrange them in the form of a Gauß decomposition to define the

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L operator. In Sect. 5, we describe the VO’s in the bosonic representation. In Sect. 6 we discuss the connection to other works and mention some open problems. The text is followed by four appendices. In Appendix A, we give the elliptic currents for the general non-twisted affine Lie algebra g. In Appendix B we discuss an elliptic analog of the Drinfeld coproduct, and show that it also arises as a quasi-Hopf twist from the usual Drinfeld coproduct. In Appendix C we study the evaluation modules and R matrix in the spin l/2 representation. Finally, in Appendix D we review the free field realization b 2 ). of the algebra Uq,p (sl While preparing this manuscript, we became aware of the paper by Hou et al. [24] which has some overlap with the content of the present paper. 2. RLL and Intertwining Relations The purpose of this section is to set up the form of the RLL and intertwining relations which we are going to study. 2.1. Previous results. In order to fix the notation, let us recall the results of [21] relevant b 2 with standard to the present paper. We consider the quantum affine algebra Uq = Uq sl generators ei , fi , hi (i = 0, 1) and d. The canonical central element is c = h0 + h1 . We retain the convention of [21] for the coproduct 1, though the details are not necessary here. Henceforth we shall write h = h1 . In [21], we have constructed a twistor F (λ) ∈ Uq⊗2 . Changing slightly the notation, let us set λ = (r∗ + 2)d + (s + 1) 21 h and write F (λ) as ∗ F (r∗ , s). Then F (r∗ , s) is a formal power series in q 2(r −s) and q 2s , satisfying the shifted cocycle condition F (12) (r∗ , s) (1 ⊗ id) F (r∗ , s) = F (23) (r∗ + c(1) , s + h(1) ) (id ⊗ 1) F (r∗ , s).

(2.1)

b 2 ) by twisting Uq via this F . Here We obtain the quasi-Hopf algebra Bq,λ = Bq,λ (sl and after, the superscripts refer to the tensor components; for instance, F (23) = 1 ⊗ F , h(1) = h ⊗ 1 ⊗ 1. Let R be the universal R matrix of Uq . The “dressed” R matrix R(r∗ , s) = F (21) (r∗ , s)RF (12) (r∗ , s)−1 of Bq,λ satisfies the dynamical YBE, R(12) (r∗ + c(3) , s + h(3) )R(13) (r∗ , s)R(23) (r∗ + c(1) , s + h(1) ) = R(23) (r∗ , s)R(13) (r∗ + c(2) , s + h(2) )R(12) (r∗ , s).

(2.2)

Let (πV , V ) be a finite dimensional Uq0 -module where Uq0 is the subalgebra generated by ei , fi , hi (i = 0, 1). Let (πV,z , Vz ) denote the evaluation module πV,z (a) = πV ◦ Ad(z d )(a)

(a ∈ Uq0 ),

πV,z (d) = z

d , dz

Vz = V [z, z −1 ].

Setting RV+ W (z1 /z2 , r∗ , s) = πV,z1 ⊗ πW,z2 R(r∗ , s), L+V (z, r∗ , s) = πV,z ⊗ id q c⊗d+d⊗c R(r∗ , s), we have the dynamical RLL relation for Bq,λ ,

(2.3) (2.4)

Elliptic Algebra Uq,p (slb2 ): Drinfeld Currents and Vertex Operators

609

+(1) +(2) ∗ ∗ ∗ (1) RV+(12) W (z1 /z2 , r + c, s + h)LV (z1 , r , s)LW (z2 , r , s + h ) +(1) +(12) ∗ ∗ (2) ∗ = L+(2) W (z2 , r , s)LV (z1 , r , s + h )RV W (z1 /z2 , r , s).

(2.5)

Hereafter we shall write r = r∗ + c,

RV+ W (z, s) = RV+ W (z, r, s),

RV∗+W (z, s) = RV+ W (z, r∗ , s), (2.6)

and normally suppress the r∗ -dependence. In this paper we will not consider the L− operator since it can be obtained from L+ , see Proposition 4.3 in [21]. Let now F, F 0 be highest weight Uq -modules on which c acts as a scalar k. Suppose we have intertwiners of Uq -modules 8V (z) : F −→ F 0 ⊗ Vz , 9∗V (z) : Vz ⊗ F −→ F 0 , which we refer to as vertex operators (VO’s) of type I and type II, respectively. Then the “dressed” VO’s for Bq,λ 8V (z, s) = id ⊗ πV,z F (r∗ , s) ◦ 8V (z), (2.7) ∗ ∗ ∗ −1 (2.8) 9V (z, s) = 9V (z) ◦ πV,z ⊗ id F (r , s) satisfy the following intertwining relations with the L operators: 8W (q k z2 , s)L+V (z1 , s) = RV+ W (z1 /z2 , s + h)L+V (z1 , s)8W (q k z2 , s + h(1) ), (2.9) L+V (z1 , s)9∗W (z2 , s + h(1) ) = 9∗W (z2 , s)L+V (z1 , s + h(2) )RV∗+W (z1 /z2 , s).

(2.10)

We note that all the operators (2.3), (2.4), (2.7), (2.8) are formal Laurent series comprising only integral powers of z. 0

−1

Remark. In the present paper, we shall adopt the universal R matrix R = R (21) , where R0 is given in (2.8) of [21]. This is purely a matter of convention. The properties (2.11)–(2.14) hold equally well for R and R0 , and hence the same construction applies. 2.2. Fractional powers. The RLL-relation (2.5) is unchanged under the transformation of the form 0

L+V (z, s) = µV (s, h(1) )µ(s + h(1) , h)L+V (z, s)µ∗V (s + h(2) , h(1) )−1 µ(s, h)−1 , (2.11) 0

RV+W (z1 /z2 , s) = µV (s, h(1) )µW (s + h(1) , h(2) ) ×RV+ W (z1 /z2 , s)µV (s + h(2) , h(1) )µW (s, h(2) ).

(2.12)

Here µV (s, h), µW (s, h), µ(s, h) are functions possibly depending on z and r, and µ∗V (s, h) means µV (s, h)|r→r∗ . This corresponds to the freedom of changing the twistor by “shifted coboundary”. Exploiting this freedom, we modify the R matrix by a fractional power of z so that it can be expressed in terms of the Jacobi theta functions. Consider the image of the R matrix in the evaluation modules (πl,z , Vl,z ) of spin l/2 (see Appendix C) + (z1 /z2 , s) = πl,z1 ⊗ πm,z2 R(r, s). Rlm + (z, s) = ρ+lm (z, p)Rlm (z, s), where In Appendix C, we show that it has the form Rlm + ρlm (z, p) is a scalar factor given in (C.10) and Rlm (z, s) has the transformation property (C.11). Write z = q 2u and set

610

M. Jimbo, H. Konno, S. Odake, J. Shiraishi 1 1 (1) + elm (u, s) = z 2r ( 2 h R

2

+(s+h(2) )h(1) )

(1) 2

+ Rlm (z, s)z − 2r ( 2 h 1

1

+sh(1) )

.

(2.13)

The R matrix (2.13) comprises fractional powers of z, but (up to a scalar factor) becomes completely periodic, 1 1 e+ (u + r, s) = e+ (u, s). R R ρ+lm (pz, p) lm ρ+lm (z, p) lm It turns out that the R matrix (2.13) is expressible in terms of the Jacobi theta functions. An explicit expression for the case l = m = 1 will be given in (4.18) below. In accordance with (2.13), we modify the L operator and VO’s as e +m (u, s) = z 2r1 ( 21 h(1) L e l (u, s) = z 8 e ∗l (u, s) 9

2

+(s+h)h(1) )

1 1 (2) 2 +(s+h(1) )h(2) ) 2r ( 2 h

= 9∗Vl (z, s)z

− 2r1∗

(1) 2

L+Vm (z, s)z − 2r∗ ( 2 h 1

1

+sh(1) )

c

8Vl (q z, s),

2 ( 21 h(1) +sh(1) )

,

(2.14) (2.15)

.

(2.16)

e + (u, s) associated with the spin e + (u, s) = L We shall focus attention to the L operators L 1 1/2 representation. The following are consequences of (2.5), (2.9), (2.10): e +(1) (u1 , s)L e +(2) (u2 , s + h(1) ) e+(12) (u1 − u2 , s + h)L R 11 e +(1) (u1 , s + h(2) )R e∗+(12) (u1 − u2 , s), e +(2) (u2 , s)L =L

(2.17) 11 + + + (1) e e e e e 8l (u2 , s)L (u1 , s) = R1l (u1 − u2 , s + h)L (u1 , s)8l (u2 , s + h ), (2.18) ∗ ∗ + (1) + (2) e ∗+ e e e e L (u1 , s)9l (u2 , s + h ) = 9l (u2 , s)L (u1 , s + h )R1l (u1 − u2 , s). (2.19) We shall study the relations (2.17)–(2.19) in the following sections. b2 3. Elliptic Currents and Uq,p sl

b 2 satisfying “elliptic” In this section, we introduce Drinfeld-type currents of Uq = Uq sl b 2 of [14] by commutation relations. We then relate them to the elliptic algebra Uq,p sl adjoining a pair of generators P, Q with [Q, P ] = 1. b 2 ). First let us recall the Drinfeld currents of Uq [25]. 3.1. Elliptic currents of Uq (sl Hereafter we fix a complex number q 6= 0, |q| < 1. We use the standard symbols [n] =

q n − q −n . q − q −1

Let x± n (n ∈ Z), an (n ∈ Z6=0 ), h, c, d denote the Drinfeld generators of Uq . In terms of the generating functions X −n x± , (3.1) x± (z) = nz n∈Z

ψ(q

c/2

h

z) = q exp (q − q

−1

)

X

! an z

−n

,

n>0

ϕ(q −c/2 z) = q −h exp −(q − q −1 )

X n>0

(3.2) !

a−n z n

,

(3.3)

Elliptic Algebra Uq,p (slb2 ): Drinfeld Currents and Vertex Operators

611

the defining relations read as follows: c : central, [h, d] = 0,

[d, an ] = nan ,

± [d, x± n ] = nxn ,

[h, x± (z)] = ±2x± (z), [2n][cn] −c|n| q δn+m,0 , [an , am ] = n [2n] −c|n| n + q z x (z), [an , x+ (z)] = n [2n] n − z x (z), [an , x− (z)] = − n (z − q ±2 w)x± (z)x± (w) = (q ±2 z − w)x± (w)x± (z), z z 1 δ q −c ψ(q c/2 w) − δ q c ϕ(q −c/2 w) . [x+ (z), x− (w)] = −1 q−q w w [h, an ] = 0,

We now introduce a new parameter p and modify (3.1)–(3.3) to define another set of currents. For notational convenience, we will frequently write p = q 2r ,

p∗ = pq −2c = q 2r

∗

(r∗ = r − c).

Let us introduce two currents u± (z, p) ∈ Uq depending on p by

u (z, p) = exp +

X n>0

! 1 a−n (q r z)n ∗ [r n]

,

X 1 an (q −r z)−n u− (z, p) = exp − [rn]

(3.4) ! .

(3.5)

n>0

Definition 3.1 (Elliptic currents). We define the currents e(z, p), f (z, p), ψ ± (z, p) by e(z, p) = u+ (z, p)x+ (z), −

−

(3.6)

f (z, p) = x (z)u (z, p),

(3.7)

ψ + (z, p) = u+ (q c/2 z, p)ψ(z)u− (q −c/2 z, p),

(3.8)

ψ − (z, p) = u+ (q −c/2 z, p)ϕ(z)u− (q c/2 z, p).

(3.9)

We will often drop p, and write e(z, p) as e(z) and so forth. The merit of these currents is that they obey the following “elliptic” commutation relations.

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M. Jimbo, H. Konno, S. Odake, J. Shiraishi

Proposition 3.2.

2p q −2 z/w 2p∗ q 2 z/w ψ ± (w)ψ ± (z), ψ (z)ψ (w) = 2p q 2 z/w 2p∗ q −2 z/w 2p pq −c−2 z/w 2p∗ p∗ q c+2 z/w − + − ψ (w)ψ + (z), ψ (z)ψ (w) = 2p pq −c+2 z/w 2p∗ p∗ q c−2 z/w ±

±

ψ ± (z)e(w)ψ ± (z)−1 = q −2

2p∗ (q ±c/2+2 z/w) e(w), 2p∗ (q ±c/2−2 z/w)

(3.10) (3.11) (3.12)

2p (q ∓c/2−2 z/w) f (w), (3.13) 2p (q ∓c/2+2 z/w) z z δ q −c ψ + (q c/2 w) − δ q c ψ − (q −c/2 w) . (3.14) w w

ψ ± (z)f (w)ψ ± (z)−1 = q 2 [e(z), f (w)] =

1 q − q −1

Here we have used the standard symbols 2p (z) = (z; p)∞ (pz −1 ; p)∞ (p; p)∞ , Y (1 − ztn1 1 · · · tnk k ). (z; t1 , · · · , tk )∞ = n1 ,··· ,nk ≥0

It will become convenient later to consider also the current ! ! X [n] X [n] a−n (q c z)n exp − an z −n . k(z) = exp [2n][r∗ n] [2n][rn] n>0

(3.15)

n>0

The ψ ± (z) are related to k(z) by the formula c

ψ ± (p∓(r− 2 ) z) = κq ±h k(qz)k(q −1 z), ξ(z; p∗ , q) , κ= ξ(z; p, q) z=q−2

(3.16) (3.17)

where the function ξ(z; p, q) =

(q 2 z; p, q 4 )∞ (pq 2 z; p, q 4 )∞ (q 4 z; p, q 4 )∞ (pz; p, q 4 )∞

(3.18)

is a solution of the difference equation ξ(z; p, q)ξ(q 2 z; p, q) =

(q 2 z; p)∞ . (pz; p)∞

We have the commutation relations supplementing (3.10)–(3.14), ξ(w/z; p, q) ξ(z/w; p∗ , q) k(w)k(z), ξ(w/z; p∗ , q) ξ(z/w; p, q) 2p∗ p∗1/2 qz/w e(w), k(z)e(w)k(z)−1 = 2p∗ p∗1/2 q −1 z/w 2p p1/2 q −1 z/w −1 f (w). k(z)f (w)k(z) = 2p p1/2 qz/w

k(z)k(w) =

(3.19) (3.20) (3.21)

Elliptic Algebra Uq,p (slb2 ): Drinfeld Currents and Vertex Operators

613

The commutation relations (3.10)–(3.14) have been proposed earlier in [26], but the direct connection with the usual Drinfeld currents was not known. In Appendix B we discuss also the Drinfeld type coproduct for the elliptic currents (3.6)–(3.9), (3.15). Remark. Strictly speaking, the currents (3.4)–(3.9), (3.15) are generating series whose b 2 ⊗ C[[p]]. At this level, p should be coefficients belong to a completion of Uq sl treated as an indeterminate. However, in the concrete representations we are going to discuss, such as evaluation modules and Fock modules, these currents have also analytical meaning. (For a formula for the currents in spin l/2 evaluation modules, see Appendix C.) We will not go into this point any further, and later treat p, p∗ as complex numbers satisfying |p|, |p∗ | < 1. b 2 . The elliptic algebra Uq,p sl b 2 in [14] is very similar 3.2. Elliptic algebra Uq,p sl to the algebra of the elliptic currents (3.10)–(3.14), (3.19)–(3.21). In the former, the coefficients of the relations are written in terms of the Jacobi elliptic theta function, which differs from 2p (z) used in the latter by a simple factor (see (3.22) below). Let us discuss the precise connection between the two algebras. For this purpose, it is more convenient to work with the “additive” notation. Following [14], we use the parameterization q = e−πi/rτ , p = e−2πi/τ ,

p∗ = e−2πi/τ

∗

(rτ = r∗ τ ∗ ),

z = q 2u = e−2πiu/rτ . We also use the Jacobi theta functions θ(u) = q

u2 r

−u 2p (q

2u

)

(p; p)3∞

,

u2

θ∗ (u) = q r∗ −u

2p∗ (q 2u ) . (p∗ ; p∗ )3∞

(3.22)

The function θ(u) has a zero at u = 0, enjoys the quasi-periodicity property θ(u + rτ ) = −e−πiτ −

θ(u + r) = −θ(u), and is so normalized that

I C0

2πiu r θ(u),

1 dz = 1, 2πiz θ(−u)

where C0 is a simple closed curve in the u-plane encircling u = 0 counterclockwise. The same holds for θ∗ (u), with r and τ replaced by r∗ and τ ∗ respectively. Now let us introduce new generators P, Q such that b2 . (3.23) [Q, P ] = 1, Q, P commute with Uq sl With the aid of them, we define the “total” currents obtained by modifying the elliptic currents of the previous Sect. 3.1. Below we shall use the notation for the conformal weight 1l,r =

l(l + 2) . 4r

(3.24)

614

M. Jimbo, H. Konno, S. Odake, J. Shiraishi

Definition 3.3 (Total currents). We define the currents K(u), E(u), F (u), H ± (u) by ∗

∗

K(u) = k(z)eQ z 1−P −h−1 −1−P −h −1−P −1 +1−P ,

(3.25)

∗ 2Q −1∗ −P −1 +1−P +1

E(u) = e(z)e z , 1−P −h−1 −1−P −h+1 , F (u) = f (z)z 1−P −h−1 −1−P −h+1 −1∗−P −1 +1∗−P +1 ± ± 2Q ±r¯ . q z H (u) = ψ (z)e

(3.26) (3.27) (3.28)

Here we have set z = q 2u , 1l = 1l,r , 1∗l = 1l,r∗ , and r¯ = r − c2 . The currents K(u) and H ± (u) are related by

r¯ 1 H (u) = κK u ± + 2 2 ±

r¯ 1 K u± − 2 2

,

¯ H − (u) = H + (u − r),

(3.29) (3.30)

with the same κ as in (3.17). We shall refer to (3.25)–(3.28) as total currents. From the commutation relations of the elliptic currents, we can derive those of the total currents. Let us introduce a function ρ(u) by ρ(u) =

ρ+∗ (u) , ρ+ (u)

(3.31)

{pq 2 z}2 {z −1 }{q 4 z −1 } , {pz}{pq 4 z} {q 2 z −1 }2

(3.32)

where 1

1

1

ρ+ (u) = z 2r ρ+11 (z, p) = z 2r q 2 {z} = (z; p, q 4 )∞ ,

(3.33)

ρ+lm (z, p) is given in (C.10), and ρ+∗ (u) = ρ+ (u)|r→r∗ . Proposition 3.4. The following commutation relations hold: K(u)K(v) = ρ(u − v)K(v)K(u), K(u)E(v) = K(u)F (v) =

∗ θ (u − v + 1−r 2 ) ∗ E(v)K(u), θ∗ (u − v − 1+r 2 ) θ(u − v − 1+r 2 ) F (v)K(u), θ(u − v + 1−r 2 ) ∗

θ (u − v + 1) E(v)E(u), θ∗ (u − v − 1) θ(u − v − 1) F (v)F (u), F (u)F (v) = θ(u − v + 1)

E(u)E(v) =

(3.34)

∗

(3.35) (3.36) (3.37) (3.38)

Elliptic Algebra Uq,p (slb2 ): Drinfeld Currents and Vertex Operators

615

c c 1 c c δ(u−v− )H + (u− )−δ(u−v + )H − (v − ) , (3.39) −1 q−q 2 4 2 4 c c ∗ θ − 1) (u − v + + 1) θ(u − v − 2 2 (3.40) H − (v)H + (u), H + (u)H − (v) = θ(u − v − c2 + 1) θ∗ (u − v + c2 − 1) θ(u − v − 1) θ∗ (u − v + 1) ± H (v)H ± (u), (3.41) H ± (u)H ± (v) = θ(u − v + 1) θ∗ (u − v − 1) θ∗ (u − v ± c4 + 1) E(v)H ± (u), (3.42) H ± (u)E(v) = ∗ θ (u − v ± c4 − 1) θ(u − v ∓ c4 − 1) F (v)H ± (u). (3.43) H ± (u)F (v) = θ(u − v ∓ c4 + 1) X z n (z = q 2u ). Here δ(u) means δ(z) = [E(u), F (v)] =

n∈Z

b 2 was presented in [14]. Thus we arrive at the It is in this form that the algebra Uq,p sl following interpretation: b 2 ). We define the algebra Uq,p sl b 2 to be the tensor Definition 3.5 (Algebra Uq,p sl b 2 and a Heisenberg algebra with generators Q, P (3.23). product of Uq sl We note that, when c = 0, the commutation relations (3.34)–(3.43) also coincide with those of Enriquez-Felder [17] with K = 0. In the bosonization of Appendix D, the elements P , h and Q are given as follows (c = k): r 2r(r − k) r−k b P0 − P2 , P −1=5= k k r b 0 = 2r(r − k) P0 − r P2 , P +h−1=5 k k s √ k iQ0 = − 2α0 iQ0 , Q=− 2r(r − k) 0

b 5 b are the notation of [14]. The physical meaning of the quantity 2α0 is the where 5, anomalous charge of the boson field φ0 . We note also that, using the notation (3.24) of the conformal weight, the L0 operator in [14] can be written as L0 = −d + 1−P +1,r−k − 1−P −h+1,r .

(3.44)

b 2 ) and the algebra Uq,p (sl b 2 ) are naturally extended to The elliptic currents of Uq (sl those associated with arbitrary non-twisted affine Lie algebras. In Appendix A, we give a summary of the results and discuss their significance. 4. The RLL Relations One of the goals of the present paper is to describe the vertex operators (VO’s) 8V (z, s), b 2 ) and Uq,p (sl b 2 ) given 9∗V (z, s) in the bosonic representation of the algebras Bq,λ (sl

616

M. Jimbo, H. Konno, S. Odake, J. Shiraishi

in Appendix D. The intertwining relations for the VO’s (2.9),(2.10) are based on the operator L+V (z, r∗ , s) defined in (2.4). In order to compute the VO’s, therefore, we need the image of the “dressed” universal R matrix R(r∗ , s) in the Fock space. The latter is b 2 ) [21], but we do not given as an infinite product of the universal R matrix for Uq (sl know how to calculate it at this moment. In this section, we take an alternative approach. Namely we utilize the elliptic currents to construct a 2×2 matrix operator L+ (u, P ) (see (4.17), (4.22)), and show that it satisfies the same RLL-relation (2.5) as for L+V (z, r∗ , s) with V being the spin 1/2 representation. Though we do not know a proof, from this construction we expect that (modulo perhaps b 2 ) (with s = P ). some base change) this L+ (u, P ) is the same as L+V (z, r∗ , s) of Bq,λ (sl 4.1. Half currents. The commutation relations of the total currents E(u) and F (u) involve delta functions. We are going to modify them so as to have commutation relations involving only “ordinary” functions. Motivated by a similar construction in [17], we b 2 as follows. define the half currents of Uq,p sl Definition 4.1 (Half currents). We set K + (u) = K(u + r+1 2 ), I ∗ u − u0 + c/2 − P + 1 θ∗ (1) dz 0 + ∗ 0 θ E(u ) , E (u) = a θ∗ (u − u0 + c/2) θ∗ (P − 1) 2πiz 0 C∗ I θ u − u0 + P + h − 1 θ(1) dz 0 F (u0 ) . F + (u) = a θ(P + h − 1) 2πiz 0 θ(u − u0 ) C

(4.1) (4.2) (4.3)

Here the contours are C ∗ : |p∗ q c z| < |z 0 | < |q c z|, C : |pz| < |z 0 | < |z|,

(4.4) (4.5)

and the constants a, a∗ are chosen to satisfy a∗ aθ∗ (1)κ = 1. q − q −1 We have to be careful about the ordering of P and E(u), F (u) in (4.2)–(4.3), since they do not commute. In fact we have the following commutation relations: [K(u), P ] = K(u), [E(u), P ] = 2E(u), [F (u), P ] = 0, [K(u), P + h] = K(u), [E(u), P + h] = 0, [F (u), P + h] = 2F (u). The specification of the contour (4.5) should be understood as an abbreviation of the prescription “C is a simple closed curve encircling the poles z 0 = pn z (n ≥ 1) of the integrand, but not containing z 0 = pn z (n ≤ 0) inside”. Similarly for (4.4). The half currents (4.2)–(4.3) can also be written in terms of the Fourier modes of the elliptic currents (3.6)–(3.7), X X en z −n , f (z, p) = fn z −n . e(z, p) = n∈Z

Substituting the Laurent expansion

n∈Z

Elliptic Algebra Uq,p (slb2 ): Drinfeld Currents and Vertex Operators

X s θ(u + s) 1 =− z −n+ r θ(u)θ(s) 1 − q −2s pn

617

(z = q 2u )

n∈Z

valid in the domain 1 < |z| < |p−1 |, we obtain E + (u) = e2Q a∗ θ∗ (1) F + (u) = −aθ(1)

X

X

en

n∈Z

fn

n∈Z

∗ ∗ 1 (q c z)−n−1−P −1 +1−P +1 , 1 − q 2(P −1) p∗n

1 z −n+1−P −h−1 −1−P −h+1 . 1 − q −2(P +h−1) pn

(4.6) (4.7)

Readers who prefer the formal series language may take (4.6), (4.7) as the definition of the half currents. We remark that a change of contours leads to different definitions. For instance, we can define another pair of currents E − (u), F − (u) by the formulas (4.2)-(4.3), with C ∗ , C ∗ : |q c z| < |z 0 | < |p∗−1 q c z| and C− : |z| < |z 0 | < |p−1 z|. changed respectively to C− Then we have −a∗ θ∗ (1)E(u) = E + (u) − E − (u), −aθ(1)F (u) = F + (u) − F − (u). This looks similar to the decomposition of the total currents to “positive” and “negative” parts in [17]. Notice however that in our case all the Fourier components en , fn appear in E + (u), F + (u), and hence the “half” currents already generate the full algebra. For this reason we will not consider E − (u), F − (u) and the analog of the L− -operator in [17]. From the commutation relations (3.34)–(3.43) for the total currents, we can obtain the relations for the half currents. Recall the function ρ(u) in (3.31)-(3.32), which satisfies ρ(0) = 1,

ρ(1) =

ρ(u)ρ(−u) = 1,

θ∗ (1) , θ(1) ρ(u)ρ(u + 1) =

θ∗ (u + 1) θ(u) . θ∗ (u) θ(u + 1)

Proposition 4.2. Set u = u1 − u2 . Then the following commutation relations hold: K + (u1 )K + (u2 ) = ρ(u)K + (u2 )K + (u1 ), K + (u1 )E + (u2 )K + (u1 )−1 = E + (u2 )

K + (u1 )−1 F + (u2 )K + (u1 ) =

(4.8)

θ∗ (1 + u) θ∗ (1) θ∗ (P + u) − E + (u1 ) ∗ , (4.9) ∗ θ (u) θ (P ) θ∗ (u)

θ(1 + u) + θ (1) θ (P + h − u) + F (u2 ) − F (u1 ), θ(u) θ (P + h) θ(u) (4.10)

θ∗ (1 + u) + θ∗ (1 − u) + + E E (u2 )E + (u1 ) (u )E (u ) + (4.11) 1 2 θ∗ (u) θ∗ (u) θ∗ (1) θ∗ (P − 2 + u) θ∗ (1) θ∗ (P − 2 − u) + 2 (u ) + E , = E + (u1 )2 ∗ 2 θ (P − 2) θ∗ (u) θ∗ (P − 2) θ∗ (u)

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M. Jimbo, H. Konno, S. Odake, J. Shiraishi

θ(1 + u) + θ(1 − u) + F (u1 )F + (u2 ) + F (u2 )F + (u1 ) (4.12) θ(u) θ(u) θ (P +h−2−u) θ (P + h − 2 + u) θ (1) θ (1) + F + (u2 )2 , = F + (u1 )2 θ (P +h−2) θ (u) θ (P +h−2) θ (u) θ∗ (P − 1 − u) θ∗ (1) (4.13) θ∗ (u) θ∗ (P − 1) θ(1) θ (P + h − 1 − u) − K + (u1 )K + (u1 − 1) . θ(u) θ(P + h − 1)

[E + (u1 ), F + (u2 )] = K + (u2 − 1)K + (u2 )

Proof. These relations can be proven by reducing them to identities of theta functions. Let us show (4.9). From the definition of the half currents (4.2) and the commutation relation (3.35), we have K + (u1 )E + (u2 )K + (u1 )−1 I 1−r ∗ ∗ 0 θ ∗ u 2 − u0 − P + 1 θ∗ (1) dz 0 ∗ 0 θ (u1 − u + 2 ) =a E(u ) ∗ . (4.14) ∗ θ∗ (u2 − u0 ) θ∗ (P − 1) 2πiz 0 θ (u1 − u0 − 1+r C∗ 2 ) Set ηs,t (u) =

θ(u + s)θ(t) . θ(u)θ(s)

Then the following identity holds: θ(u1 − u2 + t) θ(u1 + t) ηs,t (u2 ) = ηs+t,t (u2 ) + ηs,t (u2 − u1 )ηs+t,t (u1 ). (4.15) θ(u1 ) θ(u1 − u2 ) We obtain (4.9) by applying (4.15) to the integrand of (4.14) with the replacement θ(u) → θ∗ (u), a → 1, s → −P + 1, u → u0 and u1 → u1 − (1 + r∗ )/2. Likewise, (4.11) leads to an equality between two-fold integrals. It can be shown by symmetrizing the integration variables and applying the identity ψ(u1 , u2 ; u01 , u02 ) + ψ(u1 , u2 ; u02 , u01 ) ×

θ(u02 − u01 + t) = (u1 ←→ u2 ), θ(u02 − u01 − t)

where ψ(u1 , u2 ; u01 , u02 ) =

θ(u1 − u2 − t) ηs+t,t (u1 − u01 )ηs−t,t (u2 − u02 ) θ(u1 − u2 ) −ηs,t (u2 − u1 )ηs+t,t (u1 − u01 )ηs−t,t (u1 − u02 ).

(4.16)

The proofs of (4.10), (4.12) are similar. Finally let us show (4.13). Integrating the delta function in (3.39), we obtain (a∗ a)−1 (q − q −1 )[E + (u1 ), F + (u2 )] I c θ∗ (u1 − u0 − P + 1)θ∗ (1) θ(u2 − u0 + P + h − 1)θ(1) dz 0 H + (u0 + ) ∗ = 4 θ (u1 − u0 )θ∗ (P − 1) θ(u2 − u0 )θ(P + h − 1) 2πiz 0 C I1 ∗ 0 ∗ c θ (u1 − u + c − P + 1)θ (1) θ(u2 − u0 + P + h − 1)θ(1) dz 0 − H − (u0 − ) ∗ . 4 θ (u1 − u0 + c)θ∗ (P − 1) θ(u2 − u0 )θ(P + h − 1) 2πiz 0 C2

Elliptic Algebra Uq,p (slb2 ): Drinfeld Currents and Vertex Operators

619

Here the contours are C1 : |p∗ z1 |, |pz2 | < |z 0 | < |z1 |, |z2 |, C2 : |pz1 |, |pz2 | < |z 0 | < |q 2c z1 |, |z2 |. Change variables z 0 → pz 0 in the second term and use the periodicity of θ(u) along with the relation H − (u0 − c/4) = H + (u0 − r + c/4). We see that the integrand becomes the same as the first, whereas the contour becomes C20 : |z1 |, |z2 | < |z 0 | < |p−1 q 2c z1 |, |p−1 z2 |. Picking the residues at z 0 = z1 , z2 we find (4.13).

4.2. Gauß decomposition. Our next task is to rewrite the commutation relations (4.8)– (4.13) into an “RLL’-form. Following the idea of the Gauß decomposition of DingFrenkel [23], let us introduce the L-operator as follows. b 2 with b + (u) ∈ End(V ) ⊗ Uq,p sl Definition 4.3 (L-operator). We define the operator L V = C2 , by + + K1 (u) 0 1 0 b + (u) = 1 F (u) , (4.17) L 0 1 0 K2+ (u) E + (u) 1 where K2+ (u) = K + (u)−1 .

K1+ (u) = K + (u − 1), Note that

b + (u) = L b + (u)(P − h(1) ), PL b 2 , respectively. where h(1) and h mean h ⊗ 1 and 1 ⊗ h ∈ End(V ) ⊗ Uq,p sl We also need the formula for the R matrix (2.13) for l = m = 1. With a further e+ (u, s) takes the form transformation of the form (2.12), R+ (u, s) = R 11 b + (u), h(1) + h] = 0, [L



1



 b(u, s) c(u, s)  R+ (u, s) = ρ+ (u)  ¯ s)  . c¯(u, s) b(u, 1

(4.18)

Here ρ+ (u)2 is given in (3.32), and θ(1) θ(s + u) θ(s + 1)θ(s − 1) θ(u) , c(u, s) = , 2 θ(s) θ(1 + u) θ(s) θ(1 + u) θ(1) θ(s − u) ¯ s) = θ(u) . , b(u, c¯(u, s) = θ(s) θ(1 + u) θ(1 + u) b(u, s) =

Up to a scalar factor, this is the same R matrix as Eq. (93) in [17]. 2

This scalar factor differs from (3.16) in [21], see the remark at the end of Sect. 2.

(4.19) (4.20)

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M. Jimbo, H. Konno, S. Odake, J. Shiraishi

Proposition 4.4. The relations (4.8)–(4.13) are equivalent to the following RLL relation b +(1) (u1 )L b +(2) (u2 ) = L b +(2) (u2 )L b +(1) (u1 )R∗+(12) (u1 − u2 , P ). R+(12) (u1 − u2 , P + h)L (4.21) Proposition 4.4 can be shown by a direct computation. (A little care has to be taken since b + (u) do not commute with those of R+ (u, P ).) Since the calculation is the entries of L tedious but straightforward, we omit the details. The above RLL-relation is equivalent to the dynamical RLL relation (2.17). To see this, let us “strip off” the operator eQ from the half currents and define r−r ∗

k1+ (u, P ) = k(q r−1 z) × (q r−1 z)− 4rr∗

r−r ∗

1 (2P +1)+ 2r h

= K + (u − 1)e−Q ,

k2+ (u, P ) = k(q r+1 z)−1 × (q r+1 z) 4rr∗ (2P −1)− 2r h = K + (u)−1 eQ , X P 1 en (q c z)−n− r∗ = e−Q E + (u)e−Q , e+ (u, P ) = a∗ θ∗ (1) 2P ∗n 1−q p n∈Z X P +h−1 1 fn z −n+ r = F + (u). f + (u, P ) = −aθ(1) 1 − q −2(P +h−1) pn 1

n∈Z

b 2 having These currents all commute with P . We can regard them as currents in Uq sl P as a parameter (P plays the same role as λ used in [17]). We set −Q e 0 + + b (4.22) L (u, P ) = L (u) 0 eQ 1 0 k1+ (u, P ) 0 1 f + (u, P ) . = e+ (u, P ) 1 0 1 0 k2+ (u, P ) Then Proposition 4.4 is equivalently rephrased as follows. Proposition 4.5. The L-operator (4.22) satisfies the dynamical RLL relation R+(12) (u1 − u2 , P + h)L+(1) (u1 , P )L+(2) (u2 , P + h(1) ) = L+(2) (u2 , P )L+(1) (u1 , P + h(2) )R∗+(12) (u1 − u2 , P ).

(4.23)

5. Vertex Operators b 2 and compare them with In this section we shall study the VO’s for Uq,p = Uq,p sl those of Lukyanov and Pugai [11]. 5.1. Intertwining relations. As in Sect. 2, we start with VO’s in the sense of (2.7), (2.8) (or their modification by fractional powers (2.15), (2.16)) acting on some highest b weight modules FJ over Uq sl2 , where J is a label for the highest weight. Our main concern will be the Fock modules described in Appendix D. However, for the general considerations till the end of the next subsection, we do not need the details of FJ . For this purpose it is convenient to consider the VO’s as acting on the sum of the Fock spaces F = ⊕J FJ .

Elliptic Algebra Uq,p (slb2 ): Drinfeld Currents and Vertex Operators

621

We define the VO’s for Uq,p acting on the total Fock space Fˆ = ⊕µ F ⊗ eµQ by e l (v, P ) : Fˆ −→ Fˆ ⊗ Vl,v , b l (v) = 8 8 ∗ ˆ b l (v) = 9 e ∗l (v, P )eh⊗Q : Vl,v ⊗ Fˆ −→ F. 9

(5.1) (5.2)

∗

b + (u), the P in 8 ˆ For b l (v) and 9 b l (v) is regarded as an operator on F. Here, just as in L the notation about the spin l/2 representation Vl,v , see Appendix C.1. The basic relations we are going to investigate are the dynamical intertwining ree + (u, P ) e+ (u, P ) and L lations in (2.18) and (2.19). We shall solve them by replacing R 1l + (u, P ) in (C.17) and L+ (u, P ) in (4.22) respectively. Substituting (4.22), (5.1) with R1l and (5.2) into (2.18) and (2.19) and writing u1 = u, u2 = v, we get the following for b l (v) and 9 b ∗l (v) : 8 b + (u) = R+(13) (u − v, P + h)L b + (u)8 b l (v), b l (v)L 8 1l ∗ ∗ b + (u)R∗+(12) (u − v, P − h(1) − h(2) ). b + (u)9 b l (v) = 9 b l (v)L L 1l

(5.3) (5.4)

It should be noted that a natural coproduct for Uq,p is not known, and hence the meaning of intertwining relations for it is not clear. Equations (5.3), (5.4) should be b 2 ). regarded as a compact way of writing the family of intertwining relations for Bq,λ (sl With this understanding, we shall sometimes refer to (5.3),(5.4) (somewhat loosely) as “intertwining relations for Uq,p ”. Now using the explicit form of the R-matrix (C.17), let us write down the “interb + (u) as twining relations” (5.3) and (5.4). We shall write the entries of L b + (u) = L

b +++ (u) L b ++− (u) L b + (u) . b +−+ (u) L L −−

According to the Gauß decomposition (4.17), we have b ++− (u) =F + (u)K + (u)−1 , b +++ (u) = K + (u − 1) + F + (u)K + (u)−1 E + (u), L L b +−− (u) = K + (u)−1 . b +−+ (u) = K + (u)−1 E + (u), L L Define the components of VO’s by l

X l b l (v − 1 ) = 8l,m (v) ⊗ vm , 8 2 m=0

b ∗l (v 9

c+1 l − ⊗ · = 9∗l,m (v). ) vm 2

For brevity we set ϕ˜ l (u) = ϕl (u + 21 ), ϕ˜ ∗l (u) = ϕ∗l (u + c+1 2 ) and 3 = P + h. Here ϕl (u) is given in (C.16) and ϕ∗l (u) = ϕl (u)|r→r∗ . In these notations, (5.3) reads as follows:

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M. Jimbo, H. Konno, S. Odake, J. Shiraishi

b ++± (u) = ϕ˜ l (u − v)8l,m (v)L l 2

− m + 1)θ(3 + l − m + 1)θ(3 − m) b + L+± (u)8l,m (v) θ(3)θ(3 + l − 2m + 1) θ(u − v + 3 + 2l − m + 1)θ(m) b + (5.5) L−± (u)8l,m−1 (v), − θ(3 + l − 2m + 1) b + (u) = −θ(u − v − l + m + 1)L b + (u)8l,m (v) ϕ˜ l (u − v)8l,m (v)L −

θ(u − v +

−±

+

−±

2

θ(u − v − 3 −

l 2

+ m + 1)θ(l − m) b + L+± (u)8l,m+1 (v). θ(3)

(5.6)

Similarly (5.4) takes the form b +±+ (u)9∗l,m (v) = ϕ˜ ∗l (u − v)L

l

θ∗ (u − v +

l+c 2

− m + 1)θ∗ (P − l + m − 1)θ∗ (P + m) ∗ b +±+ (u) 9l,m (v)L θ∗ (P )θ∗ (P − l + 2m − 1) ∗ θ∗ (u − v − P + l+c 2 − m + 1)θ (l − m + 1) ∗ b +±− (u), 9l,m−1 (v)L (5.7) + θ∗ (P − l + 2m − 1) b + (u)9∗ (v) = −θ∗ (u − v − l−c + m + 1)9∗ (v)L b + (u) ϕ˜ ∗ (u − v)L −

±−

−

l,m

l,m

2

θ∗ (u − v + P −

l−c 2 +m θ∗ (P )

+ 1)θ∗ (m + 1)

±−

b +±+ (u). 9∗l,m+1 (v)L

(5.8)

Let us investigate the relations (5.5) and (5.6) in detail. For the highest component 8l,l (v), we can immediately obtain from (5.6) the relations ϕ˜ l (u − v)8l,l (v)K + (u)−1 = −θ(u − v +

l + 1)K + (u)−1 8l,l (v), 2

8l,l (v)E (u) = E (u)8l,l (v). +

+

(5.9) (5.10)

Notice that u = v − 2l − 1 is a zero of ϕ˜ l (u − v). Suppose that in the relation (5.5) the product of 8l,m (v) and F + (u) has no pole at this point. Then we obtain a relation which determines the components of VO’s recursively: l θ(3 + l − m) 8l,m (v) 8l,m−1 (v) = F + (v − ) 2 θ(3)

(m = 0, 1, .., l).

(5.11)

We will show below (Proposition 5.1) that this assumption is satisfied in the free field realization of Uq,p . Substituting (5.9), (5.11) into (5.5) and using Riemann’s theta identity, we find the following relation as the sufficient condition for (5.5) with the choice of the lower sign: 8l,l (v)F (u) =

θ(u − v − 2l ) θ(u − v + 2l )

F (u)8l,l (v).

(5.12)

By (5.9)–(5.12), the remaining relations in (5.5)–(5.6) are reduced to those involving the highest component 8l,l (v) and K + (u), E + (u), F + (u). In order to ensure the existence of VO, we need to verify them. For level one (c = 1), we have verified that they are consequences of Proposition 4.2. In general, such a direct check seems complicated, and it would be better to invoke the fusion procedure. We do not go into this issue

Elliptic Algebra Uq,p (slb2 ): Drinfeld Currents and Vertex Operators

623

any further. Note however that, had we known the equivalence with the quasi-Hopf construction, there would be no need for the check because the existence of VO is clear in the latter context. Similarly, the intertwining relations (5.7) and (5.8) for the type II vertex operator lead to the following relations as the sufficient condition for the highest component: ϕ˜ ∗l (u − v)K + (u)−1 9∗l,l (v) = −9∗l,l (v)K + (u)−1 θ∗ (u − v + 9∗l,l (v)E(u) = E(u)9∗l,l (v)

θ∗ (u − v + 2l )

θ∗ (u − v − 2l )

l+c + 1), (5.13) 2

,

(5.14)

9∗l,l (v)F + (u) = F + (u)9∗l,l (v),

(5.15)

and the relation for the lower component 9∗l,m−1 (v) = 9∗l,m (v)E + (v −

l+c θ∗ (m)θ∗ (P − l + m − 2) − r∗ ) ∗ 2 θ (l − m + 1)θ∗ (P − 2)

(m = 0, 1, .., l). (5.16)

∗ We remark that in the derivation of (5.16), we took u = v − l+c 2 − 1 − r as a zero of ∗ ∗ ϕ˜ l (u − v). If we chose a zero without the shift by r , we would have an extra term in the RHS of (5.14). The shift of u by r∗ in E + (u) yields a change of the contour in (4.2). For example, we have I θ∗ v − u0 − l/2 − P + 1 θ∗ (1) dz 0 l+c E(u0 ) − r∗ ) = a∗ E + (v − ∗ 0 ∗ 2 θ (u − u − l/2) θ (P − 1) 2πiz 0 C˜ ∗

with the contour being C˜ ∗ : |q −l z| < |z 0 | < |p∗−1 q −l z|.

(5.17)

5.2. “Twisted” intertwining relations. In order to compare these results with those in [11, 12] and [14], we need a further modification by signs. In [27], the modified VO’s are called “twisted” intertwiners 3 . b 2 ), the twisted type I 80 (v) and the type II 9∗ 0 (v) VO’s are In the case of Uq (sl V V the intertwiners of the same type as 8V (v) and 9∗V (v), but satisfying the following intertwining relations twisted with signs: 80V (v)ι(a) = 1(a)80V (v),

9∗ 0V

(v)1(a) =

ι(a)9∗ 0V

(v),

(5.18) b 2 ), ∀a ∈ Uq (sl

(5.19)

b 2 ), where ι denotes the involution of Uq (sl ± ι(x± n ) = −xn ,

ι(an ) = an .

Analogously to the procedures (2.7), (2.8), (5.1) and (5.2), we can define the 0 b ∗l (v) for Uq,p as the operators of the same type as (5.1) b 0l (v) and 9 “twisted” VO’s 8 and (5.2) but satisfying the following “twisted” intertwining relations. 3

This terminology is not to be confused with the “twisting” in the sense of quasi-Hopf algebras.

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M. Jimbo, H. Konno, S. Odake, J. Shiraishi

b +0 (u) = R+(13) (u − v, P + h)L b + (u)8 b 0l (v)L b 0l (v), 8 b +0

L

0 b ∗l (v) (u)9

=

0 b + (u)R∗+(12) (u b ∗l (v)L 9

(5.20)

− v, P − h

(1)

− h ). (2)

(5.21)

b + (u) has the following components: Here the “twisted” L-operator L 0

0 b + (u), b +±± (u) = L L ±±

b +0 (u) = −L b + (u). L ±∓ ±∓

Defining the components of the “twisted” VO’s as l

X l b 0l (v − 1 ) = 80l,m (v) ⊗ vm , 8 2 m=0

0 b ∗l (v 9

c+1 l ) vm − ⊗ · = 9∗ 0l,m (v), 2

we obtain the “twisted” counterpart of (5.9)–(5.16) as follows: ϕ˜ l (u − v)80l,l (v)K + (u)−1 = −θ(u − v + 80l,l (v)E + (u) = −E + (u)80l,l (v), 80l,l (v)F (u) = −

θ(u − v −

l + 1)K + (u)−1 80l,l (v), 2

(5.22) (5.23)

l 2) F (u)80l,l (v), l ) 2

(5.24)

θ(u − v + l θ(3 + l − m) 0 8l,m (v), 80l,m−1 (v) = F + (v − ) 2 θ(3)

(5.25)

for type I, and ϕ˜ ∗l (u − v)K + (u)−1 9∗ 0l,l (v) = −9∗ 0l,l (v)K + (u)−1 θ∗ (u − v + 9∗ 0l,l (v)E(u) = −E(u)9∗ 0l,l (v)

θ∗ (u − v + 2l )

θ∗ (u − v − 2l )

l+c + 1), (5.26) 2

,

9∗ 0l,l (v)F + (u) = −F + (u)9∗ 0l,l (v), l+c θ∗ (m)θ∗ (P − l + m − 2) − r∗ ) ∗ , 9∗ 0l,m−1 (v) = 9∗ 0l,m (v)E + (v − 2 θ (l − m + 1)θ∗ (P − 2)

(5.27) (5.28) (5.29)

for type II. (See the remark below (5.16).) b 2 admits a free field representation for arbitrary level 5.3. Free field realization. Uq,p sl c = k(6= 0, −2) [14](see Appendix D). Using this, we obtain a realization of the VO’s. We consider below the “twisted” VO’s. The non-twisted ones are obtained in a similar way. Proposition 5.1. Using the notation in Appendix D, the intertwining relation (5.20) has the following solution: (l = 0, 1, .., k), (5.30) 80l,l (v) = φk−l,−(k−l) (w) : exp −φ00 (l; 2, k|w) :

Elliptic Algebra Uq,p (slb2 ): Drinfeld Currents and Vertex Operators



l−m Y

80l,m (v) = 

I a

j=1

×

l−m Y

Cj

 dzj  2πizj

θ(v − uj −

×

l−m Y

l 2

+ 3 − 1 + 2j)θ(1)θ(3 + l − m − 1 + 2j)

θ(v − uj − 2l )θ(3 − 1 + 2j)θ(3 + 2j) 

j=1



625

F (uj ) 80l,l (v)

(m = 0, 1, .., l),

(5.31)

j=1

where w = q 2v , zj = q 2uj , 3 = P + h and k k + 2 − φ1 l; 2, k + 2|w; ± :. φl,±l (w) = : exp −φ2 ±l; 2, k|w; ± 2 2 (5.32) Similarly (5.21) has the solution = φk−l,k−l (w) : exp φ0 (l; 2, k|z) : (l = 0, 1, 2, .., k),     I l−m l−m Y Y dzj  ∗ 0 a∗ E(uj ) 9 l,l (v)  9∗ 0l,m (v) =  C˜ j∗ 2πizj

9∗ 0l,l (v)

j=1

×

l−m Y j=1

θ∗ (v − uj − θ∗ (v − uj −

(5.33)

j=1

l ∗ ∗ ∗ 2 − P + 1 + 2(l − m − j))θ (1)θ (P − j − 1)θ (l − j + 1) l ∗ ∗ ∗ 2 )θ (P − 1 − 2(l − m − j))θ (P − 2 − 2(l − m − j))θ (j)

(m = 0, 1, .., l).

(5.34)

The contours Cj and C˜ j∗ (j = 1, 2, .., l − m) are taken as Cl−m : |q l w|, |pq −l w| < |zl−m | < |q −l w|,

(5.35)

Cj : |q l w|, |pq −l w|, |q −2 zj+1 | < |zj | < |q −l w| (j = 1, 2, .., l − m − 1), (5.36) ∗ : |q −l w| < |zl−m | < |p∗−1 q −l w|, |q l w|, C˜ l−m

(5.37)

C˜ j∗ : |q −l w|, |q 2 zj+1 | < |zj | < |p∗−1 q −l w|, |q l w| (j = 1, 2, .., l − m − 1). (5.38)

Sketch of proof. First of all, we should note that the relations (5.22)–(5.24) and (5.26)– (5.28) coincide with those in Proposition 4.4 in [14] if we identify 80l,l (u) and 9∗ 0l,l (u) ˜ (l) ˜ (l)∗ with 8 l (u) and 9l (u) in [14], respectively. Using the bosons a0,m , a1,m , a2,m defined in Appendix D, the expressions (5.30) and (5.33) are the unique solutions of them up to a scalar.

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M. Jimbo, H. Konno, S. Odake, J. Shiraishi

Next, the expressions (5.31) and (5.34) are the direct consequence of (5.25) and (5.29). The contours are determined from (4.5) with the replacement z → q −l w, (5.17) and the following OPE’s derived from (5.30), (5.33) and Proposition D.3: (pq −l z/w; p)∞ : 80l,l (v)F (u) :, (q l z/w; p)∞

(5.39)

(pq −l w/z; p)∞ : 80l,l (v)F (u) :, (q l w/z; p)∞

(5.40)

l

80l,l (v)F (u) = w r −1 l

F (u)80l,l (v) = z r −1 0

l

9∗l,l (v)E(u) = w− r−k −1 0

l

E(u)9∗l,l (v) = z − r−k −1

0 (p∗ q l z/w; p∗ )∞ : E(u)9∗l,l (v) :, (q −l z/w; p∗ )∞

(5.41)

0 (p∗ q l w/z; p∗ )∞ : E(u)9∗l,l (v) :, −l ∗ (q w/z; p )∞

(5.42)

(p∗ q −2 z 0 /z; p∗ )∞ −φ0 (k|z) −φ0 (k|z0 ) :e e : (q 2 z 0 /z; p∗ )∞ z0 1 − 0 − − 0 )(: 9− q(1 − × I (z)9I (z ) : + : 9II (z)9II (z ) : (q − q −1 )2 z z0 − − 0 0 −1 : 9− − (1 − q −2 )(q : 9− I (z)9II (z ) : +q II (z)9I (z ) :) , (5.43) z

E(u)E(u0 ) = z r−k 2

0 0 0 (pq 2 z 0 /z; p)∞ : eφ0 (k|z) eφ0 (k|z ) : −2 0 (q z /z; p)∞ z0 1 q −1 (1 − )(: 9+I (z)9+I (z 0 ) : + : 9+II (z)9+II (z 0 ) : × −1 2 (q − q ) z 0 z (5.44) −(1 − q 2 )(q −1 : 9+I (z)9+II (z 0 ) : +q : 9+II (z)9+I (z 0 ) :) , z

F (u)F (u0 ) = z − r 2

0

2u 0 2u , w = q 2v . where 9± I,II (z) are given in (D.12) and (D.28), and z = q , z = q In the derivation of (5.25) and (5.29), we made an assumption that no counter poles to the zeros of ϕ˜ l (u − v) and ϕ˜ ∗l (u − v) appear from the OPE’s 80l,m (v)F + (u) and 0 E + (u)9∗l,m (v). The verification of this assumption is not so hard. Substitute the OPE’s 0 (5.39)- (5.44) into the products 80l,m (v)F + (u) and E + (u)9∗l,m (v), we can show that such counter poles do not appear. We have also checked at level one (k = 1) that, upon elimination of 80 1,m (v) and 0 9∗1,m (v), the rest of the intertwining relations are consequences of the commutation relations for the half currents in Proposition 4.2. As we mentioned already, such a direct check is difficult for higher levels. However, modulo the assumption about the equivalence with the quasi-Hopf construction (where the existence of the VO’s is known), the expressions (5.30)-(5.34) are unique up to a scalar factor and a choice of an equivalent set of three bosons.

5.4. Level one case (k = 1). In this case, writing 8− (v) = 801,1 (v), 8+ (v) = 0 0 801,0 (v), 9∗− (v) = 9∗1,1 (v) and 9∗+ (v) = 9∗1,0 (v), we have from Proposition 5.1,

Elliptic Algebra Uq,p (slb2 ): Drinfeld Currents and Vertex Operators

0 8− (v) = : exp −φ0 (2|w) :, I 8+ (v) = a

C

(5.45)

dz 0 0 − r−1 (pq −1 w/z 0 ; p)∞ z r 0 2πiz (qw/z 0 ; p)∞

θ(v − u0 − 21 + 3 + 1)θ(1) : F (u0 )8− (v) :, θ(v − u0 − 21 )θ(3 + 1) 9∗ − (v) = : exp φ0 (2|z) :, 9∗ + (v) = a∗

I C˜ ∗

627

(5.46)

(5.47)

∗ 0 ∗ r (p qz /w; p )∞ dz 0 : 9∗− (v)E(u0 ) : w− r−1 −1 0 0 2πiz (q z /w; p∗ )∞

θ∗ (v − u0 − 21 − P + 1)θ∗ (1) , θ∗ (v − u0 − 21 )θ∗ (P − 1)

(5.48)

where the contours C and C˜ ∗ are given by (5.35) and (5.37) letting l = 1, m = 0. Since the level one parafermion theory is a trivial theory, one can neglect the parafermion currents in E(u) and F (u). Then, the expressions (5.45)–(5.48) agree with the results in [11], [12]. Remark. Our notation here is related to those of [12] as follows: x = −q, Q = Q0 ,

1 a0,m , [2m] P = P0 ,

αm =

βn =

ˆ + 1, L − 1 = P (in (5.48)) = 5 A(z) = F (u),

1 0 a , [2m] 0,m

ˆ 0 + 1, K −1=3=5

B(z) = E(u).

6. Discussions 6.1. Classical limit. We have not been able so far to identify the L-operator constructed in Sect. 4 with the one obtained by a quasi-Hopftwist [21]. In this subsection, we study b 2 in the RLL formulation, and compare the classical limit of the elliptic algebra Bq,λ sl it with the half currents (4.6), (4.7) in the quantum case. Let a = sl2 , with the standard generators e, f, h. Let ( , ) be the invariant inner product normalized as (h, h) = 2, (e, f ) = (f, e) = 1. We consider the homogeneous c2 , realization of the affine Lie algebra g = sl g = span{en , fn , hn (n ∈ Z), d, c}, [xm , yn ] = [x, y]m+n + m(x, y)δm+n,0 c, [d, xm ] = mxm , c : central . In what follows, we identify h0 ∈ g with h ∈ a.

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Let us recall the notion of a quasi-Lie bialgebra [20] which is the classical counterpart of a quasi-Hopf algebra. By definition, it is a triple (g, δ, ϕ) consisting of a Lie algebra g, a 1-cocycle (cobracket) δ : g → ∧2 g and a tensor ϕ ∈ ∧3 g, satisfying 1 Alt(δ ⊗ 1)δ(x) = [x(1) + x(2) + x(3) , ϕ], 2 Alt(δ ⊗ 1 ⊗ 1)ϕ = 0. Here the symbol Alt stands for skew-symmetrization. In the case of Bq,λ corresponding quasi-Lie bialgebra structure on g is given as follows [19]: δ(x) = [x(1) + x(2) , r], ϕ = −2 D(1) r(23) − D(2) r(13) + D(3) r(12) .

(6.1) (6.2) b 2 , the sl (6.3) (6.4)

Here r denotes the classical r matrix X X 1 1 1 hn ⊗ h−n + 2 en ⊗ f−n r = h⊗h+ n 2 1−p 1 − wpn +2

X n∈Z

n6=0

n∈Z

1 fn ⊗ e−n + c ⊗ d + d ⊗ c, 1 − w−1 pn

(6.5)

with p, w being parameters (having the same meaning as in the body of the text), and we have set ∂ ∂ + h(i) w r(jk) . D(i) r(jk) = c(i) p ∂p ∂w Let ρz : g0 → a ⊗ C[z, z −1 ] (g0 = [g, g]) be the evaluation morphism given by ρz (xn ) = xz n (x = e, f, h), ρz (c) = 0. Setting r0 = r − c ⊗ d − d ⊗ c, we define L+ (z) = (ρz ⊗ id) r0 ,

r(z) = (id ⊗ ρ1 ) L+ (z).

These are formal series with values in a ⊗ g and a ⊗ a , respectively. Explicitly we have   X 1 1 z n h−n  L+ (z) = h ⊗  h + 2 1 − pn n6=0 ! ! X X 2 2 n n z f−n + f ⊗ z e−n , (6.6) +e ⊗ 1 − wpn 1 − w−1 pn n∈Z n∈Z   X 1 1 zn r(z) = h ⊗ h  + 2 1 − pn n6=0 ! ! X X 2 2 n n +f ⊗e . (6.7) z z +e ⊗ f 1 − wpn 1 − w−1 pn n∈Z

n∈Z

Up to a change of “gauge” and the extra zero-mode operators P, Q, (6.6) agrees with the classical limit of the L operator based on the half currents (4.6), (4.7). According to Drinfeld [20], there is a bijective correspondence between quasi-Lie bialgebras and Manin pairs. This means the following. Let Dg = g ⊕ g∗ be the direct

Elliptic Algebra Uq,p (slb2 ): Drinfeld Currents and Vertex Operators

629

sum with the dual vector space g∗ . Equip it with the inner product ( , ) requiring that it vanishes on g × g, g∗ × g∗ and coincides with the canonical paring on g × g∗ . Then, for (g, δ, ϕ) a quasi-Lie bialgebra, Dg is endowed with a unique Lie algebra structure (the classical double) such that ( , ) is an invariant inner product for Dg. Moreover the correspondence (g, δ, ϕ) ↔ Dg is one-to-one. In the present case, let us take the dual basis e∗n , fn∗ , h∗n (n ∈ Z), d∗ , c∗ of g∗ , with the dual pairing given by hxm , yn∗ i = hx, yiδm+n,0 ,

hd, c∗ i = 1,

hc, d∗ i = 1,

others = 0.

Set4 1 h ⊗ h− (z) + e ⊗ f − (z) + f ⊗ e− (z), 2 X x∗n z −n (x = e, f, h). x− (z) =

L− (z) =

(6.8)

n∈Z

Then the dual pairing takes the form hL+(1) (z1 ), L−(2) (z2 )i = r(12) (z1 /z2 ). With the above notation, the Lie algebra structure of Dg can be described as follows: [L±(1) (z1 ), L±(2) (z2 )] = −[r(12) (z1 /z2 ), L±(1) (z1 ) + L±(2) (z2 )] ∂ ∂ h(1) L+(2) (z2 ) − h(2) L+(1) (z1 ) + hr(12) (z1 /z2 ) ± 2c p r(12) (z1 /z2 ), ±2w ∂w ∂p [L+(1) (z1 ), L−(2) (z2 )] = −[r(12) (z1 /z2 ), L+(1) (z1 ) + L−(2) (z2 )] ∂ h(1) L+(2) (z2 ) − h(2) L+(1) (z1 ) + hr(12) (z1 /z2 ) +2w ∂w ∂ ∂ +2cp r(12) (z1 /z2 ) − c∗ z r(12) (z1 /z2 ), ∂p ∂z c, c∗ : central , [d, d∗ ] = 0, ∂ [d, L± (z)] = −z L± (z), ∂z ∂ ∂ ∗ ± [d , L (z)] = ±2p L+ (z) − z Le− (z). ∂p ∂z Here we have set



1 Le− (z) = h ⊗  h + 2 +e ⊗

X



1 z n h∗−n  1 − pn n6=0 ! X 2 n ∗ z f−n + f ⊗ 1 − wpn

n∈Z

X n∈Z

! 2 z n e∗−n 1 − w−1 pn

.

Notice that, because of the quasi-Lie nature of (g, δ, ϕ), g is a Lie subalgebra of the double Dg whereas g∗ is only a linear subspace (the Lie bracket does not close inside g∗ ). 4 The L− (z) is a generating series in g∗ and is independent of L+ (z). It should not be confused with the classical limit of the L− (z) operator in [21] which has a simple relation with L+ (z).

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M. Jimbo, H. Konno, S. Odake, J. Shiraishi

b 2 , but we omit the A similar description is possible for the classical limit of Aq,p sl details. 6.2. Comparison with Enriquez-Felder. In [17], Enriquez and Felder studied the quasiHopf structure of an elliptic algebra U~ g(τ ) associated with the face-type R matrix. This b 2) algebra U~ g(τ ) contains a central element K. Roughly speaking, U~ g(τ ) and Uq,p (sl are central extensions of the same algebra, as we already mentioned in Sect. 3.2. Let us examine the main differences between the two algebras. The formulation of [17] starts with an elliptic curve with modulus τ and a coordinate u on it. The latter plays the role of an “additive” spectral parameter, to be compared with our “multiplicative” spectral parameter z = q 2u . In the classical case, the relevant Manin pair in [17] is defined by assigning “positive” and “negative” parts in powers of u. Accordingly, in the construction of the half currents, the integration contours are chosen around a point in the u-plane. In our case, the integrations are taken along a circle around the origin on the z-plane. A more serious (perhaps related) difference arises in the quantum case. In [17], the curve is fixed throughout quantization. In contrast, in the presence of the central term c, we have to deal simultaneously with two different elliptic curves with nomes p and p∗ = pq −2c . This makes it difficult to adapt the geometric construction of [17]. b 2 ) are quite different. The nature of the central element K ∈ U~ g(τ ) and c ∈ Uq,p (sl Being dual to the grading element d ∈ g, K seems similar to the element c∗ ∈ g∗ in the double Dg discussed in the previous subsection. For infinite dimensional representations and bosonization, we feel that it is more natural to consider extension by c. A similar distinction has been discussed in the context of the double Yangian [28]. In [17], U~ g(τ ) is initially endowed with a simple Hopf-algebra structure given by a Drinfeld-type coproduct. The quasi-Hopf structure related to the dynamical RLL relation is then obtained by constructing a suitable twist from the initial Hopf structure. Such a Drinfeld-type coproduct persists in the presence of c as well, but it provides only a quasi-Hopf structure (Appendix B). Let us also mention a “physical” reason why we prefer z to u. Recall Baxter’s corner transfer matrix (CTM) method [29]. A CTM is composed of a product of infinitely many Boltzmann weights. For the elliptic models, the individual weights (with appropriate normalization) are doubly periodic functions in u. The most important property of CTM is that, in the infinite lattice limit, the eigenvalues are all simple integral powers of z. This means that, in passing to the infinite lattice limit, one of the periods is lost. It is because the infinite lattice limit makes sense only inside the physical region, whereas shifting by another period takes us out of that region. Intuitively the L-operators generating the elliptic algebra are also products of Boltzmann weights on a single row of the lattice. The above argument indicates that in infinite dimensional representations the currents of the algebra possess only one period, and that z is the natural variable to use. 6.3. Space of states. Let us discuss how we view the space of states of the k-fusion unreb 2 ), Uq,p (sl b 2 ). The parameter stricted ABF model in connection with the algebras Bq,λ (sl 1 ∗ ∗ r in λ = (r + 2)d + (s + 1) 2 h corresponds to the elliptic nome. We shall argue below that the parameter s corresponds to the boundary height degrees of freedom. First consider the “low temperature” limit p, q → 0. Let us recall the “paths” of the spin k/2-XXZ model [30]. A vertex-path v is a semi-infinite sequence v = (· · · , v(2), v(1)), where v(l) ∈ {0, 1, · · · , k}. We have k + 1 different ground state vertex-paths v¯ m (m = 0, 1, · · · , k) given by

Elliptic Algebra Uq,p (slb2 ): Drinfeld Currents and Vertex Operators

v¯ m (l) =

m for l ≡ 0 mod 2 k − m for l ≡ 1 mod 2.

We say v is an m-vertex-path if it satisfies the boundary condition v(l) = v¯ m (l) 1. We assign a weight to an m-vertex-path by the formula X (v¯ m (l) − v(l))α1 − h(v)(α0 + α1 ), wt(v) = m(31 − 30 ) +

631

(6.9) for l

l>0

where h(v) is the “energy” of the path v (see [30] for definition). The collection of all m-vertex-paths can be regarded as (the low temperature P limit of) the space on which the CTM of the fusion vertex model acts. Its character v q h(v) z (wt(v),h) (sum over the m-vertex-paths) is known to be the same as the character of the integrable highest weight b 2 )-module V (µm ) of highest weight Uq (sl µm = (k − m)30 + m31 .

(6.10)

Next we consider the paths for the unrestricted k-fusion ABF model. A face-path s is a semi-infinite sequence s = (· · · , s(1), s(0)) of integers s(l) ∈ Z, subject to the admissibility condition s(l) − s(l − 1) ∈ {k, k − 2, · · · , −k} for l ≥ 1. A face-path s is called an (m, n)-face-path (m ∈ {0, 1, · · · , k}, n ∈ Z) if it satisfies the boundary condition s(l) = s¯m,n (l) for l 1, where s¯m,n signifies the ground state face-path s¯m,n (0) = n + m, s¯m,n (l) − s¯m,n (l − 1) = 2v¯ m (l) − k. From a face-path s, we can construct a vertex-path as (· · · , (s(2) − s(1) + k)/2, (s(1) − s(0) + k)/2), and conversely we obtain a unique face-path from a vertex-path up to a uniform shift s(l) → s(l)+a (for all l). Thus an (m, n)-face-path s is uniquely represented by an m-vertex-path v as X 2(v¯ m (l) − v(l)), s(0) = n + m + l>0

s(l) − s(l − 1) = 2v(l) − k. The parameter n determines the “boundary height at infinity’, and the “bulk configuration” is described by some m-vertex-path v. Returning to the finite temperature situation 0 < |p| < |q| < 1, let us consider the space Hm,n for the face model CTM under the boundary condition determined by m, n. As we have seen, for each fixed n, the character of Hm,n is the same as that of the b 2 )-module V (µm ). Since Bq,λ (sl b 2 ) has the same representations as the underlying Uq (sl b algebra Uq (sl2 ), Hm,n can also be viewed as the level k irreducible highest weight b 2 ) with λ = (r∗ + 2)d + (s + 1) 1 h. Let us consider module V (µm ; r∗ , s) over Bq,λ (sl 2 the relation between n and the parameter s. As we have discussed in the main text, the b 2 ) consists of two sectors, Uq (sl b 2 ) and a Heisenberg algebra generated algebra Uq,p (sl by P, Q with [P, Q] = −1. A representation of this Heisenberg algebra is given by the zero mode lattice spanned by e−nQ |0i (n ∈ Z), where P |0i = 0. The operator P takes a fixed value n on each state e−nQ |0i. Thus we can regard the direct sum M V (µm ; r∗ , n) ⊗ e−nQ (6.11) n∈Z m=0,1,··· ,k

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M. Jimbo, H. Konno, S. Odake, J. Shiraishi

b 2 )-module. This suggests that it is natural to identify s with n. The above as a Uq,p (sl picture is consistent with the manner in which the dynamical shift appears in the RLL relation for L+ (u, P ). b 2 )-module (6.11) with the space H, In summary, we identify the Uq,p (sl M Hm,n , (6.12) H= n∈Z m=0,1,··· ,k

on which the CTM of the k-fusion unrestricted ABF model acts under all possible boundary conditions. 6.4. Further issues. Finally let us mention some related works and open problems. b 2 ) was found by deforming the free field realization of (i) In [14], the algebra Uq,p (sl b 2 )k /U (sl b 2 )l+k . As has been pointed out in [11], in the b the coset CFT U (sl2 )l ⊗ U (sl b case k = 1, Uq,p (sl2 ) appears as the algebra of screening currents for the deformed Virasoro algebra (DVA). (For the coset-type description of the RSOS model and DVA, see [10, 31].) We wish to understand the conceptual meaning of DVA from the quasi-Hopf point of view. (ii) We note that the screening operators for the deformed Wn+1 - algebra coincide b 2 ) at level one up to some cocycle factors which adjust with E(u), F (u) of Uq,p (sl signs in the commutation relations (Appendix A). A “higher rank extension” of DVA (the deformed Wn+1 -algebra) and its screening currents have been studied in [32, 33]. The VO’s for the A(1) n face models is constructed in [34] by the use of the screening operator for the deformed Wn+1 -algebra. The cohomological structure of the Fock module of the deformed Wn+1 -algebra is studied in [35]. Even though not everything has yet been made clear, these works indicate that the deformed Wn+1 -algebras play the role of the dynamical symmetry of the A(1) n face models. (iii) In [36], the deformed W -algebra Wq,t (¯g) associated to an arbitrary simple Lie algebra g¯ has been proposed. It can be regarded as a quantization of the deformed Poisson W -algebra obtained from a difference analogue of the Drinfeld-Sokolov reduction [37, 38]. On the other hand, we have obtained the algebra Uq,p (g) for b 2) an arbitrary non-twisted affine Lie algebra g, extending the results for Uq,p (sl as shown in Appendix A. For non-simply laced g, there is a considerable difference between the Drinfeld-type currents for Uq,p (g) at level one and the screening currents for Wq,t (¯g). It seems natural to have such a difference since these two have different CFT limits; the former originates from the coset construction U (g)k ⊗ U (g)l /U (g)k+l and the latter from the Drinfeld-Sokolov reduction of the loop group G((z)). We thus expect the existence of another type of deformed W algebra for non-simply laced g corresponding to Uq,p (g). (iv) A superalgebra version of the W -algebra was proposed recently in [39]. We hope this class can be treated through the Lie superalgebra version of the quasi-Hopf structure [22]. We note also that a construction of some extended version of DVA and b n+1 ) is discussed deformed W -algebras from the vertex-type elliptic algebra Aq,p (sl in the works [40, 41, 42, 43]. (v) It is also tempting to guess that there is a “higher level extension” of the deformed b 2) Virasoro algebra whose screening currents are given by E(u), F (u) of Uq,p (sl at level k. Such an algebra would be a deformation of (fractional) super-Virasoro algebra in CFT (see [14] for detailed discussions). It is an open problem to find such an extended algebraic structure.

Elliptic Algebra Uq,p (slb2 ): Drinfeld Currents and Vertex Operators

633

(vi) In this paper as well as in [21], we focused attention to the case where the parameters p, q are generic. In terms of lattice models, we considered only the unrestricted SOS case. The RSOS case corresponds to non-generic p and needs special treatment. We wish to understand in particular the mechanism of obtaining possible extra singular vectors.5 The structure of the states of the RSOS model was also approached by b 2 ) [10], and the DVA current was constructed using the quantum affine algebra Uq (sl within this language [10, 31]. It is desirable to study the relationship between this description and the one based on the quasi-Hopf algebra. (vii) An algebraic approach to the fusion ABF models has been presented on the basis b 2 ) and the elliptic algebra Uq,p (sl b 2 ) in this work. of the quasi-Hopf algebra Bq,λ (sl Another interesting direction is to study Baxter’s eight vertex model and Belavin’s generalization. Recently, a remarkable bosonization formula of the type I VO for the eight vertex model was proposed by Lashkevich and Pugai [46]. They succeeded in reducing the problem to the already known bosonization for the ABF model through the use of intertwining vectors and Lukyanov’s screening operators. To understand their bosonization scheme, it seems necessary to clarify the relationship between b n) the intertwining vectors and the two twistors F (λ) and E(r), which define Bq,λ (sl b n ) respectively. It is also interesting to seek a more direct bosonization, and Aq,p (sl b 2 ) and does which is intrinsically connected with the quasi-Hopf structure of Aq,p (sl not rely on the bosonization of the ABF model. Acknowledgement. We thank Hidetoshi Awata, Jintai Ding, Benjamin Enriquez, Boris Feigin, Ian Grojnowski, Koji Hasegawa, Hiroaki Kanno, Harunobu Kubo, Michael Lashkevich, Tetsuji Miwa , Yaroslav Pugai, Takashi Takebe and Jun Uchiyama for discussions and interest.

A. Elliptic Currents for General g In this appendix we give the elliptic currents and the algebra Uq,p (g) for non-twisted affine Lie algebras g. A.1. Uq (g). Let g be an affine Lie algebra of non-twisted type associated with a generalized Cartan matrix A = (aij ) and let g be a corresponding simple finite dimensional Lie algebra. Fixing an invariant bilinear form ( , ) on the Cartan subalgebra h, we identify h∗ with h via ( , ). Denoting the simple roots by αj , we set bij = di aij = bji with di = (αi , αi )/2. Hence B = (bij ) is the symmetrized Cartan matrix. We also set qi = q di and q n − qi−n , [n]i = i qi − qi−1

q n − q −n [n)i = , qi − qi−1

[m]i ! m . = n i [n]i ![m − n]i !

(A.1)

5 In this connection we refer to the works [44, 45] on DVA and deformed W -algebras, where a detailed study is made on the Kac-determinant and properties of extra singular vectors at roots of unity.

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Consider the quantum affine algebra Uq (g) of non-twisted type, and let x± i (z) =

X

−n x± , i,n z

n∈Z

ψi (q

c/2

z) =

qihi

exp (qi −

qi−1 )

X

! −n

,

ai,−n z

n

ai,n z

n>0

ϕi (q

−c/2

z) =

qi−hi

exp −(qi −

qi−1 )

X

!

n>0

be the Drinfeld currents (i = 1, 2, .., rank g). The defining relations for Uq (g) read as follows: c : central , [hi , d] = 0, [d, ai,n ] = nai,n ,

± [d, x± i,n ] = nxi,n ,

± [hi , x± j (z)] = ±aij xj (z), [aij n]i [cn)j −c|n| q δn+m,0 , [ai,n , aj,m ] = n [aij n]i −c|n| n + z xj (z), [ai,n , x+j (z)] = q n [aij n]i n − z xj (z), [ai,n , x− j (z)] = − n ± ± ±bij z − w)x± (z − q ±bij w)x± i (z)xj (w) = (q i (w)xj (z), δi,j −c z c/2 c z −c/2 ψ ϕ δ q (w)] = (q w) − δ q (q w) , [x+i (z), x− i j j w w qi − qi−1 a X X ± ± ± l a (−) x± · · · x± i,mσ(l) xj,m xi,mσ(l+1) · · · xi,mσ(a) = 0, l i i,mσ(1)

[hi , aj,n ] = 0,

σ∈Sa l=0

(i 6= j, a = 1 − aij , m1 , · · · , ma ∈ Z). In the last line, Sa denotes the symmetric group on a letters. 2r by A.2. Elliptic currents. Let us introduce the currents u± i (z, p) ∈ Uq (g) with p = q

u+i (z, p)

= exp

X n>0

u− i (z, p)

= exp −

! 1 ai,−n (q r z)n [r∗ n)i

X n>0

,

1 ai,n (q −r z)−n [rn)i

Then the following commutation relations hold.

! .

Elliptic Algebra Uq,p (slb2 ): Drinfeld Currents and Vertex Operators

635

Lemma A.1. (p∗ q bij z/w; p∗ )∞ + x (w)u+i (z, p), (A.2) (p∗ q −bij z/w; p∗ )∞ j (p∗ q −bij +c z/w; p∗ )∞ − x (w)u+i (z, p), (A.3) u+i (z, p)x− j (w) = (p∗ q bij +c z/w; p∗ )∞ j (pq −bij −c z/w; p)∞ + + (z, p)x (w) = x (w)u− (A.4) u− j i i (z, p), (pq bij −c z/w; p)∞ j (pq bij z/w; p)∞ − − (z, p)x (w) = x (w)u− (A.5) u− i j i (z, p), (pq −bij z/w; p)∞ j (pq −c−bij z/w; p)∞ (p∗ q c+bij z/w; p∗ )∞ − u (w, p)u+i (z, p). (A.6) u+i (z, p)u− j (w, p) = (pq −c+bij z/w; p)∞ (p∗ q c−bij z/w; p∗ )∞ j u+i (z, p)x+j (w) =

± Define the “dressed” currents x± i (z, p), ψi (z, p) in Uq (g) by

x+i (z, p) = u+i (z, p)x+i (z), − − x− i (z, p) = xi (z)ui (z, p),

(A.7) (A.8)

−c/2 z, p), ψi+ (z, p) = u+i (q c/2 z, p)ψi (z)u− i (q

(A.9)

ψi− (z, p)

=

c/2 u+i (q −c/2 z, p)ϕi (z)u− z, p). i (q

(A.10)

Set ei (z) = x+i (z, p) and fi (z) = x− i (z, p). From Lemma A.1, we obtain Proposition A.2. [hi , aj,n ] = 0, [hi , ej (z)] = aij ej (z), [hi , fj (z)] = −aij fj (z), [d, hi ] = 0, [d, ai,n ] = nai,n , ∂ ∂ [d, ei (z)] = −z ei (z), [d, fi (z)] = −z fi (z), ∂z ∂z [aij n]i [cn)j −c|n| q δn+m,0 , [ai,n , aj,m ] = n [aij n]i −c|n| n q z ej (z), [ai,n , ej (z)] = n [aij n]i n z fj (z), [ai,n , fj (z)] = − n z2p∗ q bij w/z ei (z)ej (w) = −w2p∗ q bij z/w ej (w)ei (z), z2p q −bij w/z fi (z)fj (w) = −w2p q −bij z/w fj (w)fi (z), δi,j [ei (z), fj (w)] = qi − qi−1 z z δ q −c ψi+ (q c/2 w, p) − δ q c ψi− (q −c/2 w, p) , w w

(A.11) (A.12) (A.13) (A.14) (A.15) (A.16) (A.17) (A.18)

(A.19)

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M. Jimbo, H. Konno, S. Odake, J. Shiraishi

X

Y

σ∈Sa 1≤k<m≤a

×

a X l=0

(p∗ q 2 zσ(m) /zσ(k) ; p∗ )∞ (p∗ q −2 zσ(m) /zσ(k) ; p∗ )∞

Y l (p∗ q bij z/zσ(k) ; p∗ )∞ (p∗ q −bij zσ(k) /z; p∗ )∞ a (−) l i (p∗ q −bij z/zσ(k) ; p∗ )∞ (p∗ q bij zσ(k) /z; p∗ )∞ l

×ei (zσ(1) ) · · · ei (zσ(l) )ej (z)ei (zσ(l+1) ) · · · ei (zσ(a) ) = 0 X Y (pq −2 zσ(k) /zσ(m) ; p)∞ σ∈Sa 1≤k<m≤a

×

a X l=0

(A.20)

k=1

(i 6= j, a = 1 − aij ),

(pq 2 zσ(k) /zσ(m) ; p)∞

Y l (pq bij z/zσ(k) ; p)∞ (pq −bij zσ(k) /z; p)∞ a (−) l i (pq −bij z/zσ(k) ; p)∞ (pq bij zσ(k) /z; p)∞ l

(A.21)

k=1

×fi (zσ(1) ) · · · fi (zσ(l) )fj (z)fi (zσ(l+1) ) · · · fi (zσ(a) ) = 0

(i 6= j, a = 1 − aij ).

A.3. Uq,p (g). Let us introduce further a set of generators of the Heisenberg algebra {Pi , Qi } (i = 1, 2, .., rank g) which commute with Uq (g) and satisfy [Pi , eQj ] = −

aij Qj e . 2

(A.22)

Setting P˜i = di Pi , h˜ i = di hi and Ei (u) = ei (z)e2Qi z − Fi (u) = fi (z)z Hi± (z)

P˜ i −1 r∗

˜ −1 P˜ i +h i r

,

,

r−r ∗ ˜ c 1 ˜ ψi± (z)e2Qi (q ±(r− 2 ) z)− rr∗ Pi + r hi , ∗

= ˆ d = d − 1 + 1, 1 X −1 (B )ij (P˜i − 1)(P˜j − 3), 1∗ = ∗ 2r i,j 1=

1 X −1 (B )ij (P˜i + h˜ i − 1)(P˜j + h˜ j − 3), 2r i,j

with z = q 2u , we have c : central, (A.23) [hi , aj,n ] = 0, [hi , Ej (u)] = aij Ej (u), [hi , Fj (u)] = −aij Fj (u), (A.24) ˆ ai,n ] = nai,n , ˆ hi ] = 0, [d, (A.25) [d, ˆ Fi (u)] = −z ∂ + 1 Fi (u), (A.26) ˆ Ei (u)] = −z ∂ + 1 Ei (u), [d, [d, ∂z r∗ ∂z r [aij n]i [cn)j −c|n| δn+m,0 , (A.27) q [ai,n , aj,m ] = n [aij n]i −c|n| n q z Ej (u), (A.28) [ai,n , Ej (u)] = n [aij n]i n z Fj (u), (A.29) [ai,n , Fj (u)] = − n

Elliptic Algebra Uq,p (slb2 ): Drinfeld Currents and Vertex Operators

637

bij bij ∗ Ei (u)Ej (v) = θ u − v + Ej (v)Ei (u), θ u−v− (A.30) 2 2 bij bij Fi (u)Fj (v) = θ u − v − Fj (v)Fi (u), (A.31) θ u−v+ 2 2 z δi,j −c z + c/2 Hi (q w) − δ q c Hi− (q −c/2 w) , (A.32) δ q [Ei (u), Fj (v)] = −1 w w qi − qi ∗ 2 ∗ Y X − r2∗ (p q zσ(m) /zσ(k) ; p )∞ zσ(k) (p∗ q −2 zσ(m) /zσ(k) ; p∗ )∞ ∗

σ∈Sa 1≤k<m≤a

×

a X l=0

Y brij∗ ∗ bij l z (p q z/zσ(k) ; p∗ )∞ (p∗ q −bij zσ(k) /z; p∗ )∞ a (−) (A.33) l i zσ(k) (p∗ q −bij z/zσ(k) ; p∗ )∞ (p∗ q bij zσ(k) /z; p∗ )∞ l

k=1

×Ei (uσ(1) ) · · · Ei (uσ(l) )Ej (u)Ei (uσ(l+1) ) · · · Ei (uσ(a) ) = 0 Y X 2 (pq −2 zσ(k) /zσ(m) ; p)∞ r zσ(k) (pq 2 zσ(k) /zσ(m) ; p)∞

(i 6= j, a = 1 − aij ),

σ∈Sa 1≤k<m≤a

×

a X l=0

Y brij l (pq bij z/zσ(k) ; p)∞ (pq −bij zσ(k) /z; p)∞ z a (−)l l i zσ(k) (pq −bij z/zσ(k) ; p)∞ (pq bij zσ(k) /z; p)∞

(A.34)

k=1

×Fi (uσ(1) ) · · · Fi (uσ(l) )Fj (u)Fi (uσ(l+1) ) · · · Fi (uσ(a) ) = 0

(i 6= j, a = 1 − aij ).

b 2 ). Free field realThese are the generalizations of the defining relations of Uq,p (sl (1) (1) (1) izations of Uq,p (g) for g = An , Bn and Dn are easily obtained. We will report on this subject in a future publication. b n ) at level one with the relations among the screening curRemark. Comparing Uq,p (sl rents of the deformed Wn -algebra [32], we have a difference by a sign factor (−)aij . However, such discrepancy always occurs in the free field realization of (quantum) affine Lie algebras and is known to be adjusted by using cocycle factors [47] or by using a procedure of central extension of the group algebra of the weight lattice. After such b n ) at level one as the algebra of the an adjustment, one can regard the algebra Uq,p (sl screening currents of the deformed Wn -algebra. B. Drinfeld Coproduct b 2 is also Besides the standard Hopf algebra structure, the quantum affine algebra Uq sl endowed with the so-called Drinfeld coproduct: 1∞ x = x ⊗ 1 + 1 ⊗ x

(x = h, c, d),

1∞ an = an ⊗ 1 + q −c

(1)

1∞ x (z) = x (q +

−

+

−

−c

(2)

|n|

(B.1)

⊗ an ,

(B.2)

+

−c(2) /2

z) + 1 ⊗ x (z),

−

−c(1) /2

−

z) ⊗ ψ (q

1∞ x (z) = x (z) ⊗ 1 + ψ (q

+

z) ⊗ x (q

−c(1)

z).

(B.3) (B.4)

The universal R matrix R∞ associated with this coproduct is given in [48]. We have also an elliptic analog of the Drinfeld coproduct given as follows:

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M. Jimbo, H. Konno, S. Odake, J. Shiraishi

1p,∞ x = x ⊗ 1 + 1 ⊗ x (x = h, c, d),  [rn]  ⊗ an an ⊗ 1 + [(r − c(1) )n] 1p,∞ an = [(r − c(1) − c(2) )n] (2) (1)   an ⊗ q c n + q c n ⊗ an [(r − c(1) )n] 1p,∞ e(z, p) = e(q −c z, p) ⊗ ψ + (q −c

/2

1p,∞ f (z, p) = f (z, p) ⊗ 1 + ψ − (q −c

/2

(2)

(2)

(1)

(B.5) (n>0), (B.6) (n<0),

z, pq −2c ) + 1 ⊗ e(z, pq −2c ), (B.7) (1)

(1)

z, p) ⊗ f (q −c z, pq −2c ). (1)

(1)

(B.8)

±

In terms of ψ (z, p) we have 1p,∞ ψ ± (z, p) = ψ ± (q ∓c

(2)

/2

z, p) ⊗ ψ ± (q ±c

(1)

/2

z, pq −2c ). (1)

As it turns out, this coproduct is obtained by a twist of (B.1)-(B.4). Set ! X n q rn a−n ⊗ an . F∞ (p) = exp − [(r − c(1) )n][2n]

(B.9)

(B.10)

n>0

Then a simple computation shows the following. Proposition B.1. The twistor (B.10) enjoys the shifted cocycle property (12) (23) (p) (1∞ ⊗ id) F∞ (p) = F∞ (pq −2c ) (id ⊗ 1∞ ) F∞ (p), F∞ (1)

and satisfies 1p,∞ (a) = F∞ (p) · 1∞ (a) · F∞ (p)−1

b2 . ∀a ∈ Uq sl

The universal R matrix associated with this coproduct 1p,∞ is given by (21) (p)R∞ F∞ (p)−1 . R∞ (p) = F∞

(B.11)

The classical limit of (B.11) reads X pn X 1 h ∧ h + 2 en ⊗ f−n + c ⊗ d + d ⊗ c. r∞ (p) = h ⊗ h + 2 n −n 2 1 − pn n n>0

Upon skew-symmetrization, it gives the limiting case w → 0 of the classical r matrix (6.5). Note that the elliptic parameter p enters only via the “Cartan” part. A similar classical r-matrix appeared also in the Drinfeld-Sokolov reduction [37]. C. Evaluation Modules C.1. Spin l/2 modules. Let l be a non-negative integer. We recall here the evaluation b 2 based on the spin l/2 representation. module of Uq sl l , Vl,z = Vl [z, z −1 ]. Define operators h, S ± on Vl by Let Vl = ⊕lm=0 Cvm l l = (l − 2m)vm , hvm

l l S ± vm = vm∓1 ,

(C.1)

l = 0 for m < 0 or m > l. In terms of the Drinfeld where by convention we set vm generators, the evaluation module (πl,z , Vl,z ) is defined by the following formulas:

Elliptic Algebra Uq,p (slb2 ): Drinfeld Currents and Vertex Operators

πl,z (c) = 0,

πl,z (d) = z

639

d , dz

(C.2)

zn 1 (q n + q −n )q nh − (q (l+1)n + q −(l+1)n ) , −1 n q−q h ±h + l + 2 i z δ q h±1 0 . πl,z (x± (z 0 )) = S ± 2 z πl,z (an ) =

(C.3) (C.4)

In (C.4), [x] means (q x − q −x )/(q − q −1 ). The images of the elliptic currents (3.6)-(3.9), (3.15) are then given as follows: 0

0

{q r−l z0 }{q r+l+2 zz0 } {q r−l zz }{q r+l+2 zz } (p; p)∞ , πl,z (k(z )) = r−l+2z z r+l+4 z } r−l+2 z 0 }{q r+l+4 z 0 } 2 (q r−h z 0 ) {q }{q {q p z0 z0 z z z 2p q −l−1 zz0 2p q l+1 zz0 , πl,z (ψ + (z 0 , p)) = q h 2p q −1+h zz0 2p q 1+h zz0 0 0 2p q −l−1 zz 2p q l+1 zz πl,z (ψ − (z 0 , p)) = q −h 0 0 , 2p q 1−h zz 2p q −1−h zz q (−h−l)/2 (q h+l+2 ; p)∞ (pq h−l ; p)∞ h+1 z , δ q πl,z (e(z 0 , p)) = S + 1 − q2 (p; p)∞ (pq −2 ; p)∞ z0 q (h−l)/2 (q −h+l+2 ; p)∞ (pq −h−l ; p)∞ h−1 z , δ q πl,z (f (z 0 , p)) = S − 1 − q2 (p; p)∞ (pq 2 ; p)∞ z0 0

(C.5) (C.6)

(C.7) (C.8) (C.9)

where {z} = (z; p, q 4 )∞ . In the text, we shall also write Vl,z as Vl,u with z = q 2u . −1

C.2. R matrix for spin l/2 representation. Let R+ = R, R− = R(21) be the universal b 2 , and let R+ (r, s) = R(r, s), R− (r, s) = R(21) (r, s)−1 be the R matrices6 of Uq sl elliptic counterparts. We consider their images in Vl,z1 ⊗ Vm,z2 , ± ± (z1 /z2 ) = πl,z1 ⊗ πm,z2 R± , Rlm (z1 /z2 , s) = πl,z1 ⊗ πm,z2 R± (r, s). Rlm The former has the form ± (z) = ρ± Rlm lm (z)Rlm (z), ±1 ±1 ) is a scalar factor and Rlm (z) is normalized as Rlm (z)v0l ⊗ where ρ± lm (z) = ρlm (z m l m v0 = v0 ⊗ v0 . From the formula (3.10) of [30] for ρlm (z), we find that ± l m (z, s)v0l ⊗ v0m = ρ± Rlm lm (z, p)v0 ⊗ v0 ,

ρ+lm (z, p) = q lm/2

{pq l−m+2 z}{pq −l+m+2 z} {q l+m+2 z −1 }{q −l−m+2 z −1 } . (C.10) {pq l+m+2 z}{pq −l−m+2 z} {q l−m+2 z −1 }{q −l+m+2 z −1 }

± (z, s) = ρ± Set further Rlm lm (z, p)Rlm (z, s). Noting that Rlm (z) is a rational function in z, we find the following relation from (4.8) of [21]: (1) 2

Rlm (pz, s) = q − 2 h 1

6

Our R here is R(21)

−1

−(s+h(2) )h(1)

(1) 2

· Rlm (z, s) · q 2 h

in [21], see the remark at the end of Sect. 2.1.

1

+sh(1)

.

(C.11)

640

M. Jimbo, H. Konno, S. Odake, J. Shiraishi

Let us consider the image of the L+ operator (4.22) in the spin l/2 representation l l → g(h)vm , we find the follow(C.5)–(C.9). With a suitable base change of the form vm ing expression: 2 l+h+2 h+1 3 + 4r + + θ( 2 )θ(u − v − 2 − s) h+1 − rs − r1 l+h+2 2 , (C.12) b(q w) q πv (e (u, s)) = −S θ(u − v − h+1 )θ(s) 2 θ l−h+2 θ u − v + h−1 2 2 +s πv (f + (u, s)) = S − θ(u − v − h−1 2 )θ(s + h − 1) 2 s+h−1 1 − r1 l−h+2 + 4r −1 h−1 2 r w) q , (C.13) ×b (q ϕl (u − v − 1) h h2 −l(l+2) + (C.14) w 2r q 4r , πv (k1 (u, s)) = − θ(u − v − h+1 2 ) πv (k2+ (u, s)) = −

h2 −l(l+2) θ(u − v − h−1 h 2 ) − 2r q − 4r . w ϕl (u − v)

(C.15)

Here b is a constant, w = q 2v , and l

ϕl (u) = −z − 2r θ u +

l + 1 + ρ1l (z, p)−1 . 2

(C.16)

We note the relation l+1 l+1 )θ(u + ). 2 2 We can get rid of the powers of w and q appearing in (C.12)-(C.15) by the transformation (2.11). Choosing ϕl (u)ϕl (u − 1) = θ(u −

h

µ(s, h) = w 2r h(s+ 2 ) q 1

3 2 2 h2 −l(l+2) s+ h +(l+1)h8r−(l+1) h 4r

and b = 1 we set 0

+ (u − v, s) = (id ⊗ πv )L+ (u, s). R1l

The result is as follows: + (u, s) = R1l

1 ϕl (u)

R++ (u) R+− (u) R−+ (u) R−− (u)

.

(C.17)

Here ϕl (u) is given in (C.16), and the entries Rε,ε0 (u) ∈ End(Vl,v ) are given by + s)θ( h+l 2 + s + 1) , (C.18) θ(s + h + 1)θ(s) h−1 θ( l−h+2 2 )θ(u + 2 + s) , (C.19) R+− (u) = −S − θ(s + h − 1) h+1 θ( l+h+2 2 )θ(u − 2 − s) , (C.20) R−+ (u) = S + θ(s) h−1 ). (C.21) R−− (u) = −θ(u − 2 In the simplest case l = 1, this R-matrix coincides with (4.18) constructed from the image of twistors. R++ (u) = −

θ(u +

h+1 h−l 2 )θ( 2

Elliptic Algebra Uq,p (slb2 ): Drinfeld Currents and Vertex Operators

641

b 2) D. Free Field Representation of Uq,p (sl b 2 . We then construct a In this section we review the free field representation of Uq sl b 2 ) following the prescription of Sect. 3. free field representation of Uq,p (sl D.1. Bosons. Let an be the bosons in Sect. 3.1. In addition to them, we introduce two more kinds of bosons aj,m (m ∈ Z6=0 j = 1, 2) satisfying the commutation relations [2m][(c + 2)m] δm+n,0 , m [2m][cm] δm+n,0 . [a2,m , a2,n ] = − m [a1,m , a1,n ] =

(D.1) (D.2)

We need also the zero-mode operators Qj and Pj (j = 0, 1, 2) satisfying [P0 , Q0 ] = −i,

[P1 , Q1 ] = 2(c + 2),

It is also convenient to introduce the notation r r r r∗ , , α− = − α+ = ∗ cr cr

[P2 , Q2 ] = −2c. r

2α0 = α+ + α− =

c . rr∗

(D.3)

(D.4)

Let us set h = −P2 ,

α=

√ β = − 2α− iQ0 .

1 Q2 , c

(D.5)

Then [h, α] = 2. We define the Fock space FJ,M˜ by M FJ,M˜ ;m,m0 , FJ,M˜ =

(D.6)

m,m0 ∈Z

FJ,M˜ ;m,m0 = C[a−1 , a−2 , ..; aj,−1 , aj,−2 , ..(j = 1, 2)] J

⊗Ce 2(c+2) Q1 ⊗ Cemβ ⊗ Ce

˜ M 2

α+m0 α

.

(D.7)

D.2. Bosonization of total currents. Let us introduce the generating functions of bosons (boson fields), A (Qj + Pj log z) + φ˜ j (A; B, C|z; D), φj (A; B, C|z; D) = − BC X [Am] aj,m z −m q D|m| , φ˜ j (A; B, C|z; D) = [Bm][Cm]

(D.8) (D.9)

m6=0

and φj(±) (A; B|z; C) =

X [Am] Pj log q + (q − q −1 ) aj,±m z ∓m q Cm 2 [Bm] m>0

(j = 1, 2). (D.10)

We sometimes use the abridgment φj (C|z; D) = φj (A; A, C|z; D),

φj (C|z) = φj (C|z; 0).

(D.11)

642

M. Jimbo, H. Konno, S. Odake, J. Shiraishi

˜ ˜ † (z) by 9(z) ˜ ˜ − (z), 9 ˜ † (z) = Now let us define the “parafermion fields” 9(z) and 9 =9 ˜ (z), with 9 1 ˜± ˜± ˜ ± (z) = ∓ 9 (D.12) 9 I (z) − 9II (z) , −1 (q − q ) c c + 2 c (+) (+) ˜ ˜± ) exp −φ ± φ :, (z) =: exp ± φ (c|z; ± 1; 2|z; ∓ 1; 2|z; ∓ 9 2 I 2 1 2 2 2 c c + 2 c (−) (−) ˜ ˜± ) exp φ ∓ φ :. (z) =: exp ± φ (c|z; ± 9 1; 2|z; ∓ 1; 2|z; ∓ 2 II 2 1 2 2 2 +

Then we have ± Proposition D.1. The following currents x (z) and operator d with h, c give a repreb 2 on FJ = FJ,J : sentation of Uq sl X 1 + −n ˜ (D.13) an z : eβ eα , x (z) = 9(z) : exp − [cn] n6=0 X c|n| q ˜ † (z) : exp (D.14) an z −n : e−β e−α , x− (z) = 9 [cn] n6=0

d = d1,2 + da ,

(D.15)

where d1,2 = −

X m>0

X

+

m>0

da = −

m2 a1,−m a1,m [2m][(c + 2)m] m2 P1 (P1 + 2) a2,−m a2,m − , [2m][cm] 4(c + 2)

X m2 q cm a−m am . [2m][cm]

(D.16) (D.17)

m>0

Note that this representation is slightly different from the one obtained by Matsuo [49]. The main difference is in the identification of the Cartan operator h. See the discussion in Subsect. D.3. b 2 ) for c = k Note also that FJ gives a level k highest weight representation of Uq (sl with the highest weight state J

J

|Ji = 1 ⊗ e 2(k+2) Q1 ⊗ 1 ⊗ e 2 α . With a substitution of (D.13) and (D.14) into (3.6) and (3.7), the boson an (n ∈ Z6=0 ), the currents e(z, p), f (z, p) and h, c, d give a representation of the elliptic currents of b 2 on FJ . Explicitly, we have Uq sl Proposition D.2.

X 1 −n ˜ : eβ eα , a0,n z e(z, p) = 9(z) : exp − [cn] n6=0 X c|n| q ˜ † (z) : exp a00,n z −n : e−β e−α . f (z, p) = 9 [cn] n6=0

(D.18)

(D.19)

Elliptic Algebra Uq,p (slb2 ): Drinfeld Currents and Vertex Operators

643

Here we introduced “dressed” bosons a0,n and a00,n by  an for n > 0 [rn] c|n| a0,n =  ∗ q an for n < 0, [r n] ∗ n] [r a0,n a00,n = [rn] satisfying [a0,m , a0,n ] =

[2m][cm] [rm] m [r ∗ m] δm+n,0

and [a00,m , a00,n ] =

(D.20) (D.21) [2m][cm] [r ∗ m] m [rm] δm+n,0 .

Let us next introduce the Heisenberg algebra generated by P and Q. We realize them as

r P −1=

2rr∗ r∗ P0 + h, c c

√ Q = − 2α0 iQ0 .

b2 . It is easy to check that [Q, P ] = 1 and that P and Q commute with Uq sl Accordingly, we modify the Fock space FJ by eQ to Fˆ J , M Fˆ J,µ , Fˆ J,µ = FJ ⊗ eµQ . Fˆ J =

(D.22)

(D.23)

µ∈Z

Now we define the currents K(z), E(z) and F (z) by (3.25)–(3.27) replacing e(z) and f (z) with (D.18) and (D.19), respectively. Let us define also dˆ = d − 1−P +1,r∗ + 1−P −h+1,r

(D.24)

Then we have [14] Proposition D.3. The currents K(z), E(z) and F (z) and h, c, dˆ give a representation b 2 ) on Fˆ J . Explicitly, these currents are given by of Uq,p (sl (D.25) K(z) = : exp −φ0 (1; 2, r|z) :, (D.26) E(z) = 9(z) : exp −φ0 (c|z) :, (D.27) F (z) = 9† (z) : exp φ00 (c|z) :, where

1 9† (z) ± ± 9 =∓ (z) − 9 (z) , II 9(z) (q − q −1 ) I

(D.28)

∓α ± c1 h ˜± z , 9± I,II (z) = 9I,II (z) e

and

r A 2cr (iQ0 + P0 log z) φ0 (A; B, C|z; D) = BC r∗ X [Am] + a0,m z −m q D|m| , [Bm][Cm]

(D.29)

m6=0

φ00 (A; B, C|z; D) = φ0 (A; B, C|z; D)

with r ↔ r∗ , a0,m → a00,m . (D.30)

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M. Jimbo, H. Konno, S. Odake, J. Shiraishi

Using the field φ2 (A|z; D), the boson expression for the parafermion current 9(z) ˜ ˜ † (z)) by replacing the field (resp. 9 (resp. 9† (z)) is obtained from the one for 9(z) ˜ ˜ φ2 (c|z; −c/2) with φ2 (c|z; −c/2) (resp. φ2 (c|z; c/2) with φ2 (c|z; c/2)). Remark. The parameterization of the vacuum charges of the Fock space Fˆ J,M˜ ;m,m0 ,µ = FJ,M˜ ;m,m0 ⊗ eµQ is related to those of FJ,M ;n0 n in [14] as follows. Let us set αn0 ,n = 1−n0 1−n 2 α− + 2 α+ . Then √ ˜ + 2m0 = M, mβ + µQ = − 2αn0 n iQ0 (D.31) M with 1 − n0 = 2m + µ and 1 − n = µ. b 2 ) from Uq sl b2 D.3. An alternative form. There is another way of constructing Uq,p (sl in terms of free bosons. Let us set r √ ¯h = − 2cP0 , α¯ = − 2 iQ0 . (D.32) c ¯ α] Then [h, ¯ = 2. Define the Fock space F¯ J,M¯ by F¯ J,M¯ =

M

F¯ J,M¯ ;m,m¯ 0 ,

(D.33)

m,m ¯ 0 ∈Z

F¯ J,M¯ ;m,m¯ 0 = C[a−1 , a−2 , ..; aj,−1 , aj,−2 , ..(j = 1, 2)] 0

J

⊗Ce 2(c+2) Q1 ⊗ Cem¯ α ⊗ Ce

¯ M 2

α+m ¯ α ¯

.

(D.34)

± ¯ ¯ Proposition D.4. The following currents x (z) and operator d with h, c give a repreb 2 on F¯ J = F¯ J,J [49]: sentation of Uq sl X 1 ¯ 1 (D.35) an z −n : eα¯ z c h , x+ (z) = 9(z) : exp − [cn] n6=0 X c|n| 1 ¯ q − † −n an z : e−α¯ z − c h , (D.36) x (z) = 9 (z) : exp [cn] n6=0

d¯ = d¯1,2 + d¯a ,

(D.37)

where X

d¯1,2 = −

m>0

X m2 m2 a1,−m a1,m + a2,−m a2,m (D.38) [2m][(c + 2)m] [2m][cm] m>0

P22

P1 (P1 + 2) + , 4(c + 2) 4c X m2 q cm h¯ 2 a−m am − . d¯a = − [2m][cm] 4c −

m>0

Then we have

(D.39)

Elliptic Algebra Uq,p (slb2 ): Drinfeld Currents and Vertex Operators

645

Proposition D.5. Dressing x± (z) by the procedure (3.6) and (3.7), we have the following currents e(z, p), f (z, p) with which the boson an (n ∈ Z6=0 ) and h, c, d give a b 2 on F¯ J . representation of the elliptic currents of Uq sl X 1 ¯ 1 −n a0,n z (D.40) : eα¯ z c h , e(z, p) = 9(z) : exp − [cn] n6=0 X c|n| 1 ¯ q a00,n z −n : e−α¯ z − c h . (D.41) f (z, p) = 9† (z) : exp [cn] n6=0

In this case, we can obtain Uq,p by dressing the elliptic currents via α¯ and h¯ instead of adjoining P and Q. This is a procedure of turning on the anomalous background charge 2α0 in φ0 . In conformal field theory, this corresponds to the twist of the energymomentum tensor by the Cartan operator. Then, the zero-mode lattice associated with α¯ gains one additional dimension and becomes 2-dimensional. Hence the Fock space F¯ J,M¯ is changed to F¯ J0 =

M

F¯ J;m¯ 0 ,n, ¯ n ¯ 0,

(D.42)

m ¯ 0 ,n, ¯ n ¯ 0 ∈Z 0 F¯ J; ¯ n ¯ 0 = C[a−1 , a−2 , ..; aj,−1 , aj,−2 , ..(j = 1, 2)] m ¯ 0 ,n, √ r n¯ 0 √ r∗ n ¯ J + 2 α ¯ Q1 m ¯ 0α 2 r r∗ 2(c+2) ⊗ Ce ⊗ Ce . ⊗Ce

(D.43)

¯ ¯ Proposition D.6. The following currents K(z), E(z), F¯ (z) and d˜ with h, c give a 0 b ¯ representation of Uq,p (sl2 ) on FJ : √ cα0 α ¯

¯ K(z) =e

√

−(1−

¯ E(z) =e

k(z) z 2 r r∗

√ r∗

F¯ (z) = e(1− r α0 ¯ d˜ = d¯ − √ h. c

)α ¯

)α ¯

√1 rr ∗

¯ h

,

e(z, p) z

f (z, p) z

√

− c1 (1−

√ r∗

1 c (1−

r

(D.44) r r∗

¯ )h

¯ )h

,

,

(D.45) (D.46) (D.47)

¯ ¯ Expressing P , h¯ by Pj (j = 0, 1, 2) and α, ¯ γ by Q0 , Q2 , the resultant K(z), E(z), F¯ (z) and d˜ coincide with K(z), E(z), F (z) and dˆ in Proposition D.3, respectively. The 0 ˆ ˜ ;m,m0 ,µ by Fock space F¯ J; ¯ n ¯ 0 is isomorphic to FJ,M m ¯ 0 ,n, m ¯0=

˜ M + m0 , 2

n¯ = µ,

n¯ 0 = −2m − µ.

References 1. Davies, B., Foda, O., Jimbo, M., Miwa, T. and Nakayashiki, A.: Diagonalization of the XXZ Hamiltonian by vertex operators. Commun. Math. Phys. 151, 89–153 (1993) 2. Jimbo, M., Miki, K., Miwa, T. and Nakayashiki, A.: Correlation functions of the XXZ model for 1 < −1. Phys. Lett. A 168, 256–263 (1992) 3. Jimbo, M. and Miwa, T.: Algebraic Analysis of Solvable Lattice Models. CBMS Regional Conference Series in Mathematics, Vol. 85, Providence, RI: AMS, (1994)

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4. Frenkel, I.B. and Reshetikhin, N.Yu.: Quantum affine algebras and holonomic difference equations. Commun. Math. Phys. 146, 1–60 (1992) 5. Belavin, A.A., Polyakov, A.M. and Zamolodchikov, A.B.: Infinite conformal symmetry in twodimensional quantum field theory. Nucl. Phys. B241, 333–380 (1984) 6. Andrews, G.E., Baxter, R.J. and Forrester, P.J.: Eight-vertex SOS model and generalized RogersRamanujan-type identities. J. Stat. Phys. 35, 193–266 (1984) 7. Jimbo, M., Miwa, T. and Okado, M.: Local state probabilities of solvable lattice models: an A(1) n−1 family. Nucl. Phys. B300 [FS22], 74–108 (1988) 8. Date, E., Jimbo, M., Kuniba, A., Miwa, T. and Okado, M.: Exactly solvable SOS models II: Proof of the star-triangle relation and combinatorial identities. Adv. Stud. Pure Math. 16, 17–122 (1988) 9. Jimbo, M., Miwa, T. and Okado, M.: Solvable lattice models related to the vector representation of classical simple Lie algebras. Commun. Math. Phys. 116, 507–525 (1988) 10. Jimbo, M., Miwa, T. and Ohta, Y.: Structure of the space of states in RSOS models. Int. J. Mod. Phys. A8, 1457–1477 (1993) 11. Lukyanov, S. and Pugai, Y.: Multi-point local height probabilities in the integrable RSOS model. Nucl. Phys. B 473 [FS], 631–658 (1996) 12. Miwa, T. and Weston, R.: Boundary ABF models. Nucl. Phys. B 486[PM], 517–545 (1997) 13. Shiraishi, J., Kubo, H., Awata, H. and Odake, S.: A quantum deformation of the Virasoro algebra and the Macdonald symmetric functions. Lett. Math. Phys. 38, 33–57 (1996) 14. Konno, H.: An elliptic algebra Uq,p slb2 and the fusion RSOS model. Commun. Math. Phys. 195, 373–403 (1998) 15. Foda, O., Iohara, K., Jimbo, M., Kedem, R., Miwa, T. and Yan, H.: An elliptic quantum algebra for slb2 . Lett. Math. Phys. 32, 259–268 (1994) 16. Felder, G.: Elliptic quantum groups. Proc. ICMP Paris 1994, Cambridge–Hong Kong: International Press: 1995, pp. 211–218 17. Enriquez, B. and Felder, G.: Elliptic quantum groups Eτ,η (slb2 ) and quasi-Hopf algebras, 1997. qalg/9703018 18. Frønsdal, C.: Generalization and exact deformations of quantum groups. Publ.RIMS, Kyoto Univ. 33, 91–149 (1997) 19. Frønsdal, C.: Quasi-Hopf deformation of quantum groups. Lett. Math. Phys. 40, 117–134 (1997) 20. Drinfeld, V.G.: Quasi-Hopf algebras. Leningrad Math. J. 1, 1419–1457 (1990) 21. Jimbo, M., Konno, H., Odake, S. and Shiraishi, J.: Quasi-Hopf twistors for elliptic quantum groups. 1997, q-alg/9712029, to appear in Transformation Groups 22. Arnaudon, D.,Buffenoir, E., Ragoucy, E. and Roche, Ph.: Universal solutions of quantum dynamical Yang-Baxter equations. 1997, q-alg/9712037 23. Ding, J. and Frenkel, I.: Isomorphism of two realizations of quantum affine algebra Uq gbln . Commun. Math. Phys. 156, 277–300 (1993) 24. Hou, B. and Yang, W.: Dynamically twisted algebra Aq,p;πˆ slb2 as current algebra generalizing screening currents of q-deformed Virasoro algebra 1997, q-alg/9709024 25. Drinfeld, V.G.: A new realization of Yangians and quantized affine algebras. Soviet Math. Dokl. 36, 212–216 (1988) 26. Ding, J. and Iohara, K.: Generalization of Drinfeld quantum affine algebras. Lett. Math. Phys. 41, 183–193 (1997) 27. Khoroshkin, S., Lebedev, D. and Pakuliak, S.: Elliptic algebra Aq,p (slb2 ) in the scaling limit. Commun. Math. Phys. 190, 597–627 (1998) 28. Khoroshkin, S., Lebedev, D. and Pakuliak, S.: Yangian algebras and classical Riemann problems. 1997, q-alg/9712057 29. Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. London: Academic, 1982 30. Idzumi, M., Iohara, K., Jimbo, M., Miwa, T., Nakashima, T. and Tokihiro, T.: Quantum affine symmetry in vertex models. Int. J. Mod. Phys. A8, 1479–1511 (1993) 31. Jimbo, M. and Shiraishi, J.: A coset-type construction for the deformed Virasoro algebra. Lett. Math. Phys. 43, 173–185 (1998) 32. Feigin, B.L. and Frenkel, E.V.: Quantum W-algebras and elliptic algebras. Commun. Math. Phys. 178, 653–678 (1996) 33. Awata, H., Kubo, H., Odake, S. and Shiraishi, J.: Quantum WN algebras and Macdonald polynomials. Commun. Math. Phys. 179, 401–416 (1996)

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34. Asai, Y., Jimbo, M., Miwa, T. and Pugai, Y.: Bosonization of vertex operators for the A(1) n−1 face model. J. Phys. A29, 6595–6616 (1996) 35. Feigin, B., Jimbo, M., Miwa, T., Odesskii, A. and Pugai, Y.: Algebra of screening operators for the deformed Wn algebra. Commun. Math. Phys. 191, 501–541 (1998) 36. Frenkel, E. and Reshetikhin, N.: Deformations of W-algebras associated to simple Lie algebras. 1997, q-alg/9708006 37. Frenkel, E., Reshetikhin, N., Semenov-Tian-Shansky, M.A.: Drinfeld-Sokolov reduction for difference operators and deformations of W-algebras I. The case of Virasoro algebra. Commun. Math. Phys. 192, 605–629 (1998) 38. Semenov-Tian-Shansky, M.A. and Sevostyanov, A.V.: Drinfeld-Sokolov reduction for difference operators and deformations of W-algebras. II. General Semisimple Case. Commun. Math. Phys. 192, 631–647 (1998) 39. Ding, J. and Feigin, B.: Quantized W-algebra of sl(2, 1): A construction from the quantization of screening operators. 1998, math.QA/9801084 40. Avan, J., Frappat, L., Rossi, M. and Sorba, P.: Poisson structures on the center of the elliptic algebra ˆ Aq,p (sl(2) c ). Phys. Lett. A235, 323–334 (1997) 41. Avan, J., Frappat, L., Rossi, M. and Sorba, P.: New Wq,p sl(2) algebras from the elliptic algebra

42. 43. 44. 45. 46. 47. 48. 49.

Aq,p sl(2)c . Phys. Lett. A239, 27–35 (1998) Avan, J., Frappat, L., Rossi, M. and Sorba, P.: From quantum to elliptic algebras. 1997, q-alg/9707034 Avan, J., Frappat, L., Rossi, M. and Sorba, P.: Deformed WN algebras from elliptic sl(N ) algebras. 1998, math.QA/9801105 Bouwknegt, P. and Pilch, K.: The deformed Virasoro algebra at roots of unity. 1997, q-alg/9710026 Bouwknegt, P. and Pilch, K.: On deformed W-algebras and quantum affine algebras. 1998, math.QA/9801112 Lashkevich, M. and Pugai, Y.: Free field construction for correlation functions of the eight-vertex model. Nucl. Phys. B516, 623–651 (1998) Frenkel, I. and Kac, V.G.: Basic representations of affine Lie algebras and dual resonance models. Invent.Math. 62, 23–66 (1980) Ding, J. and Khoroshkin, S.: Weyl group extension of quantized current algebras. math.QA/9804139 Matsuo, A.: A q-deformation of Wakimoto modules, primary fields and screening operators. Commun. Math. Phys. 161, 33–48 (1994)

Communicated by G. Felder

Commun. Math. Phys. 199, 649 – 681 (1999)

Communications in

Mathematical Physics © Springer-Verlag 1999

Transport by Time Dependent Stationary Flow Leonid Koralov? School of Mathematics, Institute for Advanced Study, Olden Lane, Princeton, NJ 08540, USA. E-mail: [email protected] Received: 29 December 1997 / Accepted: 29 May 1998

Abstract: We consider transport properties for Gaussian, stationary, divergence free, random velocity fields in R2 , which are Markov in time. We prove the existence of effective diffusivity. We also obtain its full asymptotics in the case of short time correlations, on a fully rigorous level. The main regularity assumption is that almost every realization of the random velocity field should be continuous in space and time, and Lipschitz continuous in space. 1. Introduction Consider the motion of a particle in the random velocity field V (x, t), x ∈ R2 , which is described by the system of random ordinary differential equations, X˙ t = V (Xt , t), X0 = x0 .

(1)

The matrix of effective diffusivity is defined as Dab =

E (Xta − X0a )(Xtb − X0b ) 1 lim , a, b = 1, 2, 2 t→∞ t

(2)

where a and b are coordinate directions. The problem of expressing the effective diffusivity, which is a Lagrangian characteristic of the flow, in terms of the correlation function of the flow vector field itself, which is Eulerian data, is an important question which has been discussed extensively in physical and mathematical publications. We refer to [15] for an introduction to this literature. Most of the results have been obtained in two cases. Either the time correlation scale of the random vector field is infinite, that is the field is time-independent, or ? Supported in part by Applied Mathematics Subprogram of the US Department of Energy DE-FG0290ER25084.

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on the contrary, the field has short time correlations. Some of the important results in the case of time independent vector fields are due to Fannjiang and Papanicolaou [6], Papanicolaou and Varadhan [18], and Kozlov [14]. For the short time correlated vector fields the first results date back to Taylor [19]. Taylor’s method gives the answer (on a physical level) for the main term of effective diffusivity in the short time correlated limit. Recently Molchanov [16], and Carmona, Grishin and Molchanov [2] considered random vector fields with a finite number of spatial modes. It was shown by Molchanov [16] that for a class of vector fields with a finite number of spatial modes the effective diffusivity can be expressed in terms of the solution of a certain hypoelliptic partial differential equation, provided the solution of the PDE exists. This equation couples velocity mode space to physical space through (1). Through the coupling to physical space, it gives the influence of the velocity modes upon diffusive transport. The existence and uniqueness of solutions to the PDE, which imply the existence of effective diffusivity for a finite number of velocity modes, were shown in [13]. The fully rigorous asymptotics of the effective diffusivity in the short time correlated limit was also obtained there. In the present paper, while utilizing some of the ideas and technical results developed in [16, 13], we avoid the non-physical assumption of a finite number of modes. As a result we obtain the existence of effective diffusivity under a mild regularity hypothesis on the random velocity field. We also obtain its full asymptotics as the time correlation scale of the vector field tends to zero. The main term of the asymptotic expansion coincides with Taylor’s answer, as is required from physical considerations. Among the related results we would like to note a recent paper by Komorowski and Papanicolaou [12]. The existence of effective diffusivity is proved there for a Gaussian, stationary, incompressible velocity field under the assumption that the correlation function of the field has finite support in time. The main regularity assumption is that almost every realization of the velocity field is continuous in t and C 1 smooth in x. While the regularity assumptions in our paper are essentially the same, we study the fields which are Markov in time, rather than the ones with finite time correlations. Markov assumption implies that the correlation function is exponentially decreasing in time, and thus excludes the case considered in [12]. A different derivation of the existence of effective diffusivity for Markov in time vector fields was obtained independently by Fannjiang and Komorowski [5]. The technique developed in our paper allows us to prove the following two consequences. The first is a complete, and rigorous asymptotic analysis of diffusivity in the short time correlated limit. The second is a rigorous analysis of the divergence of diffusivity for generalized random fields with pure Kolmogorov spectrum. The first topic is presented here, and the second will be contained in a following paper. We assume a physical model of turbulence described by a Gaussian random field, which is stationary in time and space and Markov correlated in time. Following [16, 13] and also using the ideas which succeeded in constructive quantum field theory [8], we use the discretization of the spectrum of the random field V (x, t) in order to approximate the system of random ordinary differential equations (1) by a finite dimensional system of stochastic differential equations. Two types of cutoffs are needed to obtain a finite dimensional system. A finite volume (periodic) cutoff gives a discrete structure to mode space, and a truncation with a finite number of periodic modes gives a finite dimensional velocity space. Thus, together with (1) we consider an auxiliary system X˙ tn = V n (Xtn , t), X0n = x0 .

(3)

Transport by Time Dependent Stationary Flow

651

From the Markov assumption governing the time correlations of the random velocity statistics, each Fourier mode in the random velocity field V n (x, t) is represented by an Ornstein–Uhlenbeck process. Thus (3) can be also viewed as a system of stochastic differential equations dXtn,a =

n X

Ytn,i via (Xtn )dt,

a = 1, 2,

(4)

i = 1, ..., n,

(5)

i=1

dYtn,i =

p 2i dWti − i Ytn,i dt,

where vi are periodic with common period p, and Ytn,i are independent Ornstein– Uhlenbeck processes. The Markov process (Xtn , Ytn ) is ergodic on T2p × Rn . However, it appears to be impossible to prove directly sufficient mixing properties, which would allow one to apply the functional Central Limit Theorem to (Xtn , Ytn ). Instead we use the harmonic coordinates method [17, 7] to approximate the process E(Mtn,a Mtn,b ) 2t E(Xtn,a Xtn,b ) . limt→∞ 2t a

Xtn,a by a stochastic integral Mtn,a . The mean square expectation

can

The be calculated explicitly, and its limit as t → ∞ is equal to harmonic coordinates ua + xa , a = 1, 2 are defined by the solutions u of hypoelliptic equations M (ua + xa ) = 0 on T2p × Rn , where M is the infinitesimal generator of the system (4),(5). The existence and uniqueness of solutions to this PDE is one of the main technical results of [13]. Hormander’s hypoellipticity principle [9] applied to the differential operator and its adjoint is a key element in the proof of existence and regularity. The effective diffusivity can be then expressed in terms of the harmonic coordinates. In the present paper we obtain a priori estimates for the operator M which are uniform in the number of modes n in the spectrum of the velocity field. These estimates allow us to perform the removal of the cutoffs and prove the existence of the effective diffusivity for the field V (x, t). In Sect. 2 of this paper we introduce assumptions on the random field V (x, t) and formulate the theorems on existence of effective diffusivity and on its asymptotic expansion in the case of short time correlations. In Sect. 3 we describe the discretization of the spectrum of the velocity field. We state the theorem that for finite fixed time the mean square displacement of a particle in the field V n (x, t) tends to the mean square displacement in the field V (x, t) as the discretization gets finer, that is as n → ∞. The proof of this theorem, being of a purely technical nature, is given in the end of the paper in Sect. 8. In Sect. 4 we relate the diffusivity in the field V n (x, t) to the solution of a hypoelliptic PDE. This PDE couples the infinitesimal generator of the Ornstein–Uhlenbeck process, in each of the mode variables, to the transport operator v∇x u in physical space. Section 5 contains the proof of the main technical result, namely a priori estimates for the hypoelliptic PDE, which are uniform in the number of modes in the spectrum of the velocity field. Section 6 is devoted to the full asymptotic expansion of the solution of the hypoelliptic PDE. First we construct the series which satisfies the equation at a formal level. Then we use the estimates obtained in Sect. 5 to show that this series is the true asymptotic series for the solution. The asymptotic expansion for the solution of the hypoelliptic

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PDE provides in turn the asymptotic expansion for the effective diffusivity of the field V n (x, t). We then justify the removal of the cut-offs in the expansion in order to get the asymptotics of the effective diffusivity in the field V (x, t). In Sect. 7 we calculate explicitly the first two terms of the expansion for the effective diffusivity. 2. Definitions, Assumptions, and Results Let F (x, t) = E(H(x, t)H(0, 0)) be the correlation function of a zero mean Gaussian field, which is stationary in x and t and Markov in time. We shall consider the motion of a particle in the random vector field 2 V (x, t) = ∇⊥ x H(x, t), x ∈ R ,

ab = E V a (x, t)V b (0, 0) with the stream function H, where ∇⊥ x = (∂x2 , −∂x1 ). Let G be the correlation matrix of the field V (x, t). Note that a ⊥ b Gab = −(∇⊥ x ) (∇x ) F.

(6)

Let Hε be the field given by the formula t 1 Hε (x, t) = √ H(x, ). ε ε

(7)

Then Hε has the same stationarity and Markov properties as H. We shall also consider the 2 motion of a particle in the random field Vε (x, t) = ∇⊥ x Hε (x, t), x ∈ R . The meaning of assumption (7) is that as ε → 0 the stream function Hε becomes short time correlated. The multiplicative factor √1ε in front of H ensures that the main term of effective diffusivity is of order one as ε → 0. The properties of H(x, t) listed above imply aRparticular form for the correlation F (x, t) and the spectral measure Fb(k, t) = (2π)−2 e−ikx F (x, t)dx of H. Recall that since F is a positive definite function Fb is a positive measure, which is finite on bounded sets in k-space. Here we follow [16]. By stationarity, Gaussian and Markov properties there exists a function K(t, x) such that for t0 ≤ t1 ≤ t2 , E[H(x, t1 )|σ(H(·, s), s ≤ t0 )] = [K(t1 − t0 ) ∗ H(t0 )](x), where the symbol ∗ denotes convolution in the space variables. From the Markov property F (·, t2 − t0 ) = K(t2 − t1 ) ∗ K(t1 − t0 ) ∗ F (·, 0) = K(t2 − t0 ) ∗ F (·, 0). A Fourier transform in the space variables shows that b 2 − t1 , k)K(t b 1 − t0 , k)Fb(k, 0) = K(t b 2 − t0 , k)Fb(k, 0). Fb(k, t2 − t0 ) = K(t Thus Fb(k, t) = exp(−|t|(k))Fb(k, 0),

(8)

Transport by Time Dependent Stationary Flow

653

and also

cε (k, t) = 1 exp(− |t| (k))Fb(k, 0). F ε ε Necessarily (k) ≥ 0 on suppFb(k, 0) ([13]). We shall assume that (k) > C > 0 in order to exclude the time independent modes. The classical solution of the equation of motion (1) exists whenever the function V (x, t) on the RHS is continuous in (x, t) and Lipschitz continuous in x uniformly on any compact. This will hold for almost every realization of the random vector field V (x, t) under certain smoothness conditions on its correlation function (we are allowed to take a modification of the random field V (x, t)). It is preferable to deal with the correlation F (x, t) of the stream function. Thus we introduce the following assumptions: Assumption A. H(x, t) is a zero mean Gaussian field, stationary in x and t and Markov in time. Assumption B. For the measure Fb(k, 0) and the function (k) determined by (8): span{suppFb(k, 0)} = R2 , and Z

(9)

(1 + |k|4+δ )Fb(dk, 0) < ∞

(10)

for some δ > 0. Moreover, (k) > C > 0; (k) is Lipschitz continuous uniformly on any compact, and grows not faster than some power of |k| as k → ∞. It will be shown that under the above assumptions there exists a modification of the vector field V (x, t), such that the solutions to Eq. (1) exist for all t for almost all realizations of the vector field V (x, t). Let the initial data x0 be a random variable. With t ≥ 0 fixed, Xt − X0 is the displacement in time t under the action of the random field V (x, t). Since V (x, t) is stationary, the distribution of the vector Xt − X0 does not depend on the initial data x0 , provided that x0 is independent of the vector field V (x, t). Theorem 2.1. Suppose Assumptions A and B hold. Then the effective diffusivity, which is defined by (2), exists and is finite. We denote by Dεab the effective diffusivity in the vector field Vε . In order to prove the asymptotic expansion of Dεab through order εm we need stronger local regularity assumptions on the correlation function. Assumption Cm . The Fourier transform of the correlation function F (x, t) satisfies the condition Z |k|4+4m Fb(dk, 0) < ∞. We can now formulate the theorem on asymptotic expansion of effective diffusivity. Theorem 2.2. Suppose Assumptions A, B, and Cm hold. Then there exist constant maab trices dab 0 , ..., dm , such that ab ab m m Dεab = dab 0 + d1 ε + ... + dm ε + o(ε )

when

ε → 0.

(11)

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3. Preliminary Considerations The proofs of Theorems 2.1 and 2.2 are based on approximation of the stream funcn tion Hε (x, t) by the stream functions Hεn (x, t), whose spectral measures Fc ε (k, t) are supported on finite sets in k-space. Let us describe the construction of the field Hεn (x, t). Consider the partition of the square Sm = {|k a | ≤ 2m−1 , a = 1, 2} into n = 24m squares 1i , i = 1, ..., n of the size 21m . Let ki be the center of 1i . We shall use the following notation: if k = (k 1 , k 2 ), then k ⊥ = (k 2 , −k 1 ). Let αi be the interior of 1i , βi be the boundary of 1i excluding the corners, and γi be the set which consists of four corners of the square 1i . Define Z Z Z 1 1 (12) i = (ki ), and σi = (2 Fb(dk, 0) + Fb(dk, 0) + Fb(dk, 0)) 2 . 2 αi

βi

γi

It is important to note that 1i , ki , i , αi , βi , γi , and σi depend on n. The transition from Fb(k, 0) to Fcn (k, 0) consists of integrating Fb(k, 0) over each square, and placing all the mass in the center. The three different integrals enter (12) with their specified factors because each side belongs to two different squares, and each corner to four different squares. Define Fcn (k, 0) =

n X

δ(ki )

i=1

σi2 , and Fcn (k, t) = exp(−|t|(k))Fcn (k, 0). 2

(13)

Then H n (x, t) is defined to be the real valued Gaussian random field whose spectral measure is given by (13). Hεn (x, t) is defined to be the real valued Gaussian random field whose spectral measure is given by |t| 1 n exp(− (k))Fcn (k, 0). Fc ε (k, t) = ε ε From (10) it follows that there exists a constant C1 > 0, such that for all n, Z |k|q Fcn (dk, 0) < C1 , if 0 ≤ q ≤ 4 + δ,

(14)

and the integrals converge at infinity uniformly in n. Define the 2 × 2 matrix M with entries n X σj2 (kj⊥ )a (kj⊥ )b a, b = 1, 2. Mab = 2j j=1

Then for any x ∈ R , 2

2 X a,b=1

Mab xa xb =

n X σj2 ⊥ 2 (k , x) . 2j j

(15)

j=1

R ⊥ ,x)2 dk From the definition of σj it follows that the RHS of (15) tends to φ(x) = Fb(k,0)(k (k) uniformly on the set {x ∈ R2 , |x| = 1} as n → ∞. By (9) φ(x) ≥ C2 > 0 on the set {x ∈ R2 , |x| = 1}, and therefore there exist constants n0 and C3 > 0 such that for n ≥ n0 and any x ∈ R2 ,

Transport by Time Dependent Stationary Flow 2 X

655

Mab xa xb ≥ C3 |x|2 .

(16)

a,b=1

Throughout the rest of the paper we shall only consider n ≥ n0 . The Fourier representation of the field Hεn (x, t) in space variables is Z 1 t eikx Z(dk, ). Hεn (x, t) = √ ε ε For t fixed Z(k, t) is an orthogonal Gaussian measure, which depends on n. Since suppFcn (k, 0) ⊂ {ki }, n 1 X iki x t e z(ki , ), Hεn (x, t) = √ ε ε

(17)

i=1

where z(ki , t) are complex-valued Gaussian stationary processes. The normalization of z(ki , t) is fixed by (13) so that σ2 E z(ki , t)z(kj , 0) = δij i exp(−|t|i ). 2 The fact that H(x, t) is a real valued field implies that together with the mode ki = (ki1 , ki2 ) the set {ki } also contains (−ki1 , −ki2 ) with the same σi2 and i . We shall write {ki , i = 1, ..., n} = {ki , −ki , i = 1, ..., n/2}. Therefore we can write (17) as n/2 1 X t t σi A1i ( ) cos(ki x) + A2i ( ) sin(ki x) . Hεn (x, t) = √ ε ε ε

(18)

i=1

Here Ali (t), i = 1, ..., n/2; l = 1, 2 are independent real-valued stationary Gaussian processes and 0 E Ali (t)Ali0 (0) = δii0 δll0 exp(−|t|i ). This implies that the Ali (t) are independent Ornstein–Uhlenbeck processes with correlation scales i and variances 1. We shall write {Y i , i = 1, ..., n} = {Ali , i = 1, ..., n/2; l = 1, 2}. Applying ∇⊥ x to (18) and using the indices i = 1, ..., n again, we see that n 1 X i t n √ H (x, t) = Y ( )vi (x), Vεn (x, t) = ∇⊥ x ε ε ε

(19)

i=1

where the vectors vi (x), i = 1, ..., n are of the following form ⊥ {vi , i = 1, ..., n} = {σi ∇⊥ x cos ki x, σi ∇x sin ki x, i = 1, ..., n/2}.

(20)

Therefore the vector fields vi are divergence free and infinitely smooth. By the definition of ki the vector fields are periodic with common period p = 2m+2 π.

656

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The fact that d d d d ( )α1 cos x( )α2 cos x + ( )α1 sin x( )α2 sin x = 0, if α1 + α2 is odd, (21) dx dx dx dx the particular form (20) of the vector fields vi , and the fact that (kj ) = (−kj ) imply that n X 1 vj ∇x vja = 0 . (22) j j=1

The equation of motion for the particle in the vector field Vεn (x, t) has the form X˙ tn = Vεn (Xtn , t), X0n = xn0 . (23) As before we assume xn0 to be independent of the random field Vεn (x, t). Whenever the subscript ε is omitted from Vεn we shall imply that ε = 1 is being considered. Provided that the following expectations exist we define the finite time displacement tensors: 1 Dn,ab (t) = E (Xtn,a − X0n,a )(Xtn,b − X0n,b ) , 2 1 ab D (t) = E (Xta − X0a )(Xtb − X0b ) , 2 where Xtn , and Xt are the solutions of (23) with ε = 1, and (1) respectively. The proof of Theorem 2.1 is based on the following two theorems. Theorem 3.1. Suppose Assumptions A and B hold. Then for arbitrary t ≥ 0 the displacement tensors Dn,ab (t) and Dab (t) exist for some modification of the field V (x, t), and for t fixed, lim Dn,ab (t) = Dab (t).

n→∞

(24)

Theorem 3.2. Suppose Assumptions A and B hold. Then the following limit Dn,ab (t) , a, b = 1, 2 t→∞ t lim

exists and is uniform in n.

(25)

Theorem 3.1 and Theorem 3.2 will be proved in Sects. 8 and 4 respectively. Notice that the limit in (25) is the effective diffusivity for the vector field V n (x, t). We shall denote it by Dn,ab . The effective diffusivity for the vector field Vεn (x, t) will be denoted by Dεn,ab . Proof of Theorem 2.1. By Theorem 3.2, Dn,ab (t) = Dn,ab t→∞ t uniformly in n. By Theorem 3.1, lim

Dn,ab (t) Dab (t) = . n→∞ t t Therefore by the theorem on uniform convergence the following limits exist: lim

Dab (t) = lim Dn,ab . t→∞ n→∞ t This completes the proof of Theorem 2.1. Dab = lim

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657

4. Formulation of the Theorem on the Hypoelliptic Estimate and the Proof of Theorem 3.2 In this section we reduce the assertion of Theorem 3.2 to a hypoelliptic estimate for a PDE. The estimate itself is proved in Sect. 5. Since the same estimate will be used in the proof of Theorem 2.2 we preserve the ε-dependence throughout this section. By (19), (23) the equation of motion in the vector field Vεn (x, t) has the form n

1 X n,i a n Yt/ε vi (Xt )dt, dXtn,a = √ ε

a = 1, 2,

(26)

i=1

where the Ytn,i are independent Ornstein–Uhlenbeck processes dYtn,i =

p 2i dWti − i Ytn,i dt,

i = 1, ..., n.

(27)

We rescale the time variable in (26) so that we can consider (26)–(27) as a system n,a etn,a = Xεt . In the new variables (26) of stochastic differential equations. Thus define X takes the form etn,a = dX

n √ X e n )dt, ε Ytn,i via (X t

a = 1, 2.

(28)

i=1

The operator n n X √ X i ∂y2i − yi ∂yi + ε yi vi (x)∇x Mε = i=1

(29)

i=1

is the infinitesimal generator of the system (27)–(28). Recall that p is the common period of the velocity modes defined in Sect. 3. Let n

y2 1Y (2π)−1/4 exp(− i ). η(x, y) = p 4 i=1

As initial conditions for the system we take the distribution η 2 , which is invariant for etn , Ytn ) on T2p × Rn . We shall repeatedly use the following elementary the process (X integrals: Z Z 1 yi yj η 2 dy = 2 δij . (30) yi η 2 dy = 0, p Consider the equation √ Mε ( εun,a + xa ) = 0 for a function un,a (x, y) defined on T2p × Rn . The function analog of the harmonic coordinate of [17, 7].

(31) √

εun,a + xa of (31) is the

658

L. Koralov

Theorem 4.1. Suppose Assumptions A and B hold. Then Eq. (31) has a unique solution in the class of C ∞ functions which satisfy the relations n X

ZZ i

(∂yi un,a )2 + yi2 (un,a )2 + (un,a )2 η 2 dxdy < ∞,

ZZ

ua η 2 dxdy = 0.

i=1

Moreover, there exists a constant C independent of n, ε, such that the solution of (31) satisfies ZZ (u

n,a 2

) +

n X

! i (∂yi u

n,a 2

)

η 2 dxdy < C.

(32)

i=1

The proof of Theorem 4.1 is given as a consequence of the more general Theorems 5.1 and 5.2 in Sect. 5 below. For the proof of Theorem 3.2 we shall need the following simple lemma Lemma 4.2. Let ftn,a and gtn,a be random variables, which depend on parameters n and t. Suppose E (ftn,a + gtn,a )(ftn,b + gtn,b ) = φab (n).

(33)

Suppose there are constants C1 and C2 , which do not depend on n and t, such that tE(gtn,a )2 < C1 , and φab (n) < C2 .

(34)

Then lim E(ftn,a ftn,b ) = φab (n),

t→∞

and the limit is uniform in n. Proof. From (33) with a = b it follows that E(ftn,a )2 = φaa (n) − E(gtn,a )2 − 2E(ftn,a gtn,a ).

(35)

From (35) and (34) we conclude that there exists a constant C3 such that E(ftn,a )2 < C3 for all n and t ≥ 1.

(36)

E(ftn,a ftn,b ) − φab (n) = −E(gtn,a gtn,b ) − E(ftn,a gtn,b ) − E(ftn,b gtn,a ).

(37)

From (33)

By the Schwartz inequality, (34), and (36) the RHS of (37) tends to zero uniformly in n as t → ∞. This completes the proof of Lemma 4.2. √ Proof of Theorem 3.2. We are here using the result of Theorem 4.1. Since εun,a + xa √ n,a e n , Y n ). By (27) and (28) we is smooth we can apply Ito’s formula to ( εu + xa )(X t t obtain

Transport by Time Dependent Stationary Flow

659

√ √ etn , Ytn ) = ( εun,a + xa )(X e0n , Y0n ) ( εun,a + xa )(X Zt √ esn , Ysn )ds + + Mε ( εun,a + xa )(X 0

Z n √ X √ √ e n , Y n )dW i . (38) i ∂yi ( εun,a + xa )(X + 2 s s s t

i=1

0

√

Since Mε ( εun,a + xa ) = 0 the expression above can be rewritten as √ √ etn , Ytn ) − ( εun,a + xa )(X e0n , Y0n ) = ( εun,a + xa )(X Zt n √ X √ esn , Ysn )dWsi . 2ε i ∂yi un,a (X i=1

(39)

0

Similarly √ √ e n , Y n ) − ( εun,b + xb )(X e n, Y n) ( εun,b + xb )(X t t 0 0 t Z n √ X √ e n , Y n )dW i . i ∂yi un,b (X = 2ε s s s i=1

(40)

0

e n , Y n ), the expectation of the product Since the measure η 2 is invariant for the process (X t t of the right sides of (39) and (40) is equal to 2ε

n X

Zt i

i=1

2εt

e n , Y n )∂y un,b (X e n , Y n )]ds = E[∂yi un,a (X s s s s i

0 n X

ZZ i

i=1

(∂yi un,a )(∂yi un,b )η 2 dxdy,

T2 ×Rn

which is finite by Theorem 4.1. Thus multiplying (39) by (40), dividing both sides by 2tε, and taking expectations of both sides of the resulting equality we obtain √ √ 1 etn,a − X e n,a + εun,a (X etn , Ytn ) − εun,a (X e0n , Y0n ) E[ X 0 2tε √ √ e n,b + εun,b (X e n , Y n ) − εun,b (X e n, Y n) ] = etn,b − X X t t 0 0 0 ZZ n X i (∂yi un,a )(∂yi un,b )η 2 dxdy. (41) i=1

T2p ×Rn

Set ftn,a =

e n,a etn,a − X X 0 √ , 2tε

gtn,a =

e n , Y n ) − un,a (X e n, Y n) un,a (X t t 0 0 √ , 2t

and

660

L. Koralov

ZZ n X φab (n) = i i=1

(∂yi un,a )(∂yi un,b )η 2 dxdy.

T2p ×Rn

With this choice of ftn,a , gtn,a and φab (n) Eq. (41) takes the form (33). Since the measure e n , Y n ), η 2 is invariant for the process (X t t ZZ e n , Y n ))2 = E(un,a (X e n , Y n ))2 = (un,a )2 η 2 dxdy, (42) E(un,a (X t t 0 0 T2p ×Rn

which does not depend on t. By Theorem 4.1 the RHS of (42) is bounded uniformly in n. Therefore tE(gtn,a )2 < C1 . From Theorem 4.1 it also follows that φab (n) < C2 . Therefore we can apply Lemma 4.2 to conclude that the limit etn,a − X e n,a )(X etn,b − X e n,b ) 1 E (X 0 0 lim E (ftn,a )(ftn,b ) = lim t→∞ 2 t→∞ tε E (Xtn,a − X0n,a )(Xtn,b − X0n,b ) 1 = lim 2 t→∞ t ZZ n X i (∂yi un,a )(∂yi un,b )η 2 dxdy = i=1

T2p ×Rn

exists uniformly in n. Put ε = 1 to obtain the statement of Theorem 3.2.

Corollary 4.3. Suppose Assumptions A and B hold. Then the effective diffusivity Dεn,ab is expressed in terms of the solution un,a of Eq. (31) by the formula ZZ n X i (∂yi un,a )(∂yi un,b )η 2 dxdy. (43) Dεn,ab = i=1

T2p ×Rn

We make a change of variables so that Mε becomes the sum of a formally self adjoint operator in the y variables (a simple harmonic oscillator) and a formally antisymmetric operator in the x variables. Thus let n,a η, un,a new = u

(44)

n,a

and rename the unknown function by u again. Then Eq. (31) becomes n n n X X √ X yi2 1 2 n,a n,a u + ε + i ∂yi − yi vi ∇x u = − yi via (x)η(y). (45) 4 2 i=1

i=1

i=1

Note that the first term on the LHS of (45) is the simple harmonic oscillator and η(y) is its eigenfunction with zero eigenvalue. We transform (43) by the above change of variables. Corollary 4.4 ([13]). Suppose Assumptions A and B hold. Then the effective diffusivity Dεn,ab is expressed in terms of the solution un,a of Eq. (45) by the formula n ZZ 1 X (un,a vib + un,b via )yi ηdxdy. Dεn,ab = 2 i=1

T2p ×Rn

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661

5. The Hypoelliptic Estimate Let us introduce notation needed for the statement of Theorem 5.1. S ⊥ is the space of functions on T2p × Rn which are infinitely smooth, orthogonal to η, and decay faster than any polynomial together with all their derivatives. That is f ∈ S ⊥ if ZZ f (x, y)η(y)dxdy = 0, and sup Q(y)P1 (Dy )P2 (Dx )f < ∞ x,y

⊥ 2 n for any polynomials P1 , P2 , and Q. L⊥ 2 is the completion of S in L2 (Tp × R ). Clearly 2 n L⊥ 2 ⊕ {const · η(y)} = L2 (Tp × R ).

|| · || is the usual norm of L2 (T2p × Rn ). H⊥ is the completion of S ⊥ in the harmonic oscillator inner product ZZ n X (∂yi f )(∂yi g) + yi2 f g + f g dxdy. i (f, g)H⊥ = i=1

H−1 (T2p ), L2 (T2p ), H1 (T2p ), and H2 (T2p ) are the usual Sobolev spaces of scalar or vector valued functions on the torus. Write n n X X √ y2 1 i (∂y2i u − i u + u), Au = yi vi (x)∇x u, Mε = L + εA. Lu = 4 2 i=1

i=1

Here Mε is the transform by (44) of the expression (29), with change of notation to the new variables. For the proof of the next theorem we refer the reader to [13]. Theorem 5.1 ([13]). Suppose Assumptions A and B hold, and f ∈ L⊥ 2 . Then the equation Mε u = f

(46)

has a unique weak solution u ∈ H⊥ . There is a constant C(n, ε), such that ||u||H⊥ ≤ C(n, ε)||f ||.

(47)

Moreover, if f ∈ C ∞ , then u ∈ C ∞ also. Theorem 5.1 proves existence, uniqueness, and regularity for Eq. (45), and consequently for Eq. (31). However the estimate (47) does not imply (32) since the constant in the RHS of (47) depends on n and ε. The following theorem provides the estimate of the L2 norm of the solution, which is uniform in n and ε. This estimate will allow us to prove (32). Theorem 5.2. Suppose Assumptions A and B hold, and f ∈ S ⊥ satisfies Z n Z X 1 vj ∇x f yj ηdy = 0 f or all x. f ηdy = 0 f or all x, and j

(48)

j=1

Then there exists a constant C, which does not depend on n or ε, such that the solution u of (46), given by Theorem 5.1 satisfies ||u|| ≤ C||f ||.

(49)

662

L. Koralov

Proof. We represent u ∈ H⊥ uniquely as a sum of two functions which are orthogonal in L2 (Rny ) for all x, that is u(x, y) = w(x, y) + u0 (x)η(y),

(50)

R R where u0 (x)dx = 0, and w(x, y)η(y)dy = 0 as an element of L2 (T2p ). To prove (49) it is sufficient to estimate ||w|| and ||u0 ||L2 (T2p ) separately. ⊥ Consider Mε as an (unbounded) operator from H⊥ to L⊥ 2 with domain S . We need the following lemma, for the proof of which we refer the reader to [13]: Lemma 5.3 ([13]). Under the assumptions of Theorem 5.1, the closure of Mε S ⊥ coincides with L⊥ 2 . By Lemma 5.3 there exists a sequence uk ∈ S ⊥ , such that Mε uk = f k → f in L⊥ 2 .

(51)

Then {f k } is a Cauchy sequence in L⊥ 2 , and by (47) uk → u in H⊥ . Let

(52)

uk = wk + uk0 η,

f k = g k + f0k η R as in (50). Note that by (51), (52), and since f ηdy = 0, wk → w in H⊥ ; uk0 → u0 in L2 (T2p ), g k → f in H⊥ ; f0k → 0 in L2 (T2p ). The equation Mε uk = f k can be written as √ √ Lwk + εAuk0 (x)η(y) + εAwk = g k + f0k η.

(53)

In order to estimate the norms of wk and uk0 we first derive three integral and differential relations satisfied by wk and uk0 . In order to do so we multiply (53) successively by η, wk , and yj η and integrate in y. We could not perform this integration with w and u0 replacing wk and uk0 in (53), since w may not belong to S ⊥ , and thus the integral over y may not converge. Then we let k → ∞ in order to obtain three corresponding relations on w and u0 . The first two of these are used to bound w, while the third gives the bound on u0 . Multiplying (53) by η and integrating in y yields Z Z Z Z Z √ √ k k 2 k k ηLw dy + ε Au0 (x)η (y)dy + ε Aw η(y)dy = g ηdy + f0k η 2 dy. (54) R R The RHS of (54) is equal to p12 f0k (x). Note that also ηLwk dy = wk Lηdy = 0 since R R Lη = 0, and Auk0 (x)η 2 (y)dy = 0 since yi η 2 (y)dy = 0. Thus our first relation has the form Z √ 1 ε Awk ηdy = 2 f0k (x). (55) p

Transport by Time Dependent Stationary Flow

663

Multiplying (53) by wk and integrating in y, we obtain Z Z Z √ √ k k k k w Lw dy + ε w Au0 (x)η(y)dy + ε wk Awk dy Z Z = g k wk dy + f0k (x)ηwk dy.

(56)

Note that Z Z Z n n X 1X yi wk vi (x)∇x wk dx = yi div(vi (wk )2 )dx = 0, wk Awk dx = 2 i=1

i=1

and thus the last term of the LHS of (56)R vanishes after integration over x. The second term on the RHS of (56) vanishes since ηwk dy = 0. By (55), since A∗ = −A ZZ ZZ Z √ √ 1 k k k k ε w Au0 (x)η(y)dxdy = − ε u0 Aw η(y)dxdy = − 2 f0k (x)uk0 (x)dx . p Therefore, by (56), we obtain our second relation, ZZ Z ZZ 1 g k wk dxdy + 2 f0k (x)uk0 (x)dx. wk Lwk dxdy = p Multiplying (53) by yj η and integrating in y, we obtain Z Z Z Z √ √ yj ηLwk dy + ε Auk0 yj η 2 dy + ε Awk yj ηdy = g k yj ηdy. The first term on the LHS is equal to n P yi vi ∇x . We evaluate A=

R

wk L(yj η)dy = j

R

(57)

(58)

wk yj ηdy. Recall that

i=1

Z X n

vi ∇x uk0 (x)yi yj η 2 dy =

i=1

1 vj ∇x uk0 (x) p2

with only the i = j term contributing. By carrying the first and the third terms from the left to the right side of (58), we have the identity Z Z Z √ √ k 2 k k εvj ∇x u0 (x) = p ( g yj ηdy − j w yj ηdy − ε yj Awk ηdy). (59) n P Applying the operator 1j vj ∇ to both sides of (59), taking the sum , and dividing by j=1 √ ε, we obtain due to (22), n 2 X X 1 a b k v v u = j j j 0 xa xb

(60)

a,b=1 j=1 n

p2 X 1 √ vj ∇( j ε j=1

Z

g k yj ηdy − j

Z

wk yj ηdy −

√

Z ε

yj Awk ηdy).

664

L. Koralov

Formulas (55), (57) and (60) are the desired relations on wk and uk0 . In order to obtain the integral relations of the type (55) and (57) for the limits w and u0 , n R √ P we consider k → ∞ in those formulas. In (55) the LHS tends to ε vi ∇x wyi ηdy i=1

in H−1 (T2p ), while the RHS tends to zero in H−1 (T2p ) since f0k → 0. We conclude that n X

Z vi ∇

wyi ηdy = 0.

(61)

i=1

From (57) we conclude that ZZ

ZZ wLwdxdy =

f wdxdy.

(62)

Formulas (61) and (62) are the first two relations for w. Using (62), we bound ||w||. By the elementary properties of the simple harmonic oscillator there exists a constant C1 > 0, such that for w ∈ H⊥ , ZZ 2 wLwdxdy. ||w|| ≤ −C1 By Schwartz inequality we conclude from (62) that ||w|| ≤ C3 ||f ||.

(63)

Next we estimate u0 . We start by deriving our third relation, an elliptic equation which n 2 P P 1 a b u0 satisfies. In (60) the LHS tends in H−2 (T2p ) to j vj vj u0 xa xb as k → ∞. The first term on the RHS tends in H

−2

2 (T2p ) to √p ε

n P j=1

a,b=1 j=1

1 j vj ∇(

R

f yj ηdy). The latter quantity

vanishes by the assumptions of the theorem. The second term on the RHS of (60) tends in H−2 (T2p ) to n

p2 X √ vj ∇ ε

Z wyj ηdy,

j=1

which is equal to zero by (61). The last term on the RHS of (60) tends in H−2 (T2p ) to Z n X 1 vj ∇(vi ∇( yi yj wηdy)). p j 2

i,j=1

Thus (60) yields Z n n 2 X X X 1 a b 1 vj vj u0 xa xb = p2 vj ∇(vi ∇( yi yj wηdy)). j j

a,b=1 j=1

(64)

i,j=1

For the following lemma it is important to note that T2p is a torus (a compact manifold), but not a square with a boundary.

Transport by Time Dependent Stationary Flow

665

Lemma 5.4. Suppose u ∈ L2 (T2p ), f ∈ H −2 (T2p ), and

R T2p

udx = 0. Let c be a constant,

and Mab a constant matrix, such that for any x ∈ R2 , 2 X

Mab xa xb ≥ c||x||2 .

a,b=1

Then there exists a constant c1 , which depends only on c, such that the equality 2 X

Mab uxa xb (x) = f on T2p

a,b=1

implies the estimate ||u||L2 (T2p ) ≤ c1 ||f ||H −2 (T2p ) .

(65)

Proof. In the case when p = 1 the statement of the lemma is a standard a priori estimate from general elliptic theory ([10]). If p 6= 1, then the change of variables u e(x) = u(px), fe(x) = p2 f (px), x ∈ T2 reduces the statement of the lemma to the case when p = 1. This completes the proof of Lemma 5.4. Note that by (20) n n X 1 a b X σj2 (kj⊥ )a (kj⊥ )b v v = , j j j 2j j=1

(66)

j=1

and thus by (16) we can apply the result of Lemma 5.4 to (64). We need to estimate the H−2 (T2p ) norm of the RHS of (64). Since vi is divergence free the RHS of (64) can be written as p2

Z n X 1 vj ∇(div(vi yi yj wηdy)). j

i,j=1

We have the following inequality: Z n X 1 vj ∇div(vi yi yj wηdy)||H−2 (T2p ) j i,j=1 Z n X supx |vj | supx |vi | ≤ C4 || yi yj wηdy||L2 (T2p ) . j

||

(67)

i,j=1

By (20) supx |vi | = σi |ki |. Using the Schwartz inequality and the fact that the j are uniformly bounded from below, we estimate the RHS of (67) as follows:

666

L. Koralov

Z n X supx |vj | supx |vi | || yi yj wηdy||L2 (T2p ) ≤ j

i,j=1 n

n

X C5 X 2 2 2 ( σi |ki | σj |kj |2 )1/2 ( || p i,j=1

By (14) the factor

n P i,j=1

Z pyi yj ηwdy||2L2 (T2p ) )1/2 .

i,j=1

(68)

σi2 |ki |2 σj2 |kj |2 is bounded uniformly in n. To estimate the second

factor on the RHS of (68) we note that Z Z pyi yj ηwdy = p(yi yj − δij )ηwdy, R since wηdy = 0. The functions p(yi yj − δij )η form an orthogonal system in L2 (Rn ). Moreover, ||p(yi yj − δij )η||2L2 (Rn ) = 1 + δij . Therefore Z Z X n n Z X || pyi yj wηdy||2L2 (T2p ) = ( p(yi yj − δij )wηdy)2 dx

i,j=1

i,j=1

(69)

ZZ ≤2

w dxdy ≤ C6 ||f || . 2

2

The last inequality in (69) is due to (63). From the chain of inequalities (67) - (69) we conclude that the H−2 (T2p ) norm of the RHS of (64) is estimated from above by C7 p||f ||. In the view of (66) and (16) we can apply Lemma 5.4 to Eq. (64) and obtain ||u0 ||L2 (T2p ) ≤ C8 p||f ||. Therefore ||u0 η|| ≤ C9 ||f ||.

(70)

Combining (70) with (63) we obtain (49), which completes the proof of Theorem 5.2. Corollary 5.5. Theorem 4.1 is a consequence of Theorems 5.1 and 5.2. Proof. With the change of variables (44) Eq. (31) takes the form (45). The existence and uniqueness result of Theorem 4.1 follows immediately from Theorem 5.1 applied to Eq. (45). We transform the LHS of (32) by the change of variables (44) ([13]). Thus the LHS of (32) is equal to ! ZZ n X n,a 2 n,a a (u ) + u vi yi η dxdy, (71) T2p ×Rn

i=1

Transport by Time Dependent Stationary Flow

667

where un,a is the solution of Eq. (45). We estimate the norm of the RHS of (45) as follows ZZ X n n X yi via η||2 = ( yi via η)2 dxdy || i=1

i=1

n n n Z X X 1 X (via )2 dx ≤ sup(via )2 ≤ σi2 |ki |2 ≤ C1 . = 2 p x i=1

i=1

i=1

(72)

The last inequality in (72) is due to (14). From (72) and Schwartz inequality it follows that the expression in (71) is estimated from above by ZZ (un,a )2 dxdy. C2 T2p ×Rn

To complete the proof of Theorem 4.1 it remains to apply Theorem 5.2 to Eq. (45). n P yi via η satisfies (48). The first part, In order to do that we need to show that f = R

f ηdy = 0, is trivial since

write

n Z X j=1

R

i=1

yi η dy = 0. To show that 2

n R P j=1

1 j vj ∇x f yj ηdy

= 0 we

n n Z X 1 1 1 X 1 vj ∇x f yj ηdy = vj ∇x via yi yj η 2 dy = 2 vj ∇x vja = 0 . j j p j (73) i,j=1 j=1

The second equality in (73) is due to (30), and the last one is due to (22). This completes the proof of Corollary 5.5. 6. Effective Diffusivity in Vector Fields with Short Time Correlation This section is devoted to the proof of Theorem 2.2. Let P be the space of functions on T2p × Rn which have the form u(x, y) = p(x, y)η(y), where p(x, y) is a polynomial in y whose coefficients are infinitely smooth functions of x. Let P ⊥ be the subspace of P of functions which satisfy Z u(x, y)η(y)dy = 0 for all x. We shall say that u(x, y) is an odd (even) function if u ∈ P and the corresponding polynomial p(x, y) contains only odd (even) powers of y. The next lemma follows from the general properties of the simple harmonic oscillator [8]. Lemma 6.1. Suppose that f (x, y) ∈ P ⊥ . Then the equation Lu = f has a unique solution u ∈ P ⊥ . If f is an odd (even) function, then u is also odd (even).

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L. Koralov

In the next theorem we state the asymptotic expansion for the solution of Eq. (45), which in turn provides the asymptotics for the effective diffusivity Dεn,ab by Corollary 4.4. Theorem 6.2. Suppose Assumptions A, B, and Cm hold. Let un,a be the solution of ⊥ Eq. (45) given by Theorem 5.1. Then there exist functions un,a k (x, y) ∈ H with k = 0, 1, ..., 2m, and constants c(k) independent of n and ε, such that k

+ ε 2 un,a + ... + ε 2 un,a ||un,a − (un,a 0 1 k )|| ≤ c(k)ε 1

k+1 2

.

(74)

n,a Moreover, un,a 2k is an odd function and u2k−1 is an even function for all k ≤ 2m.

Let us substitute the series

∞ P k=0

ε 2 un,a k formally into the equation Mε u = f . Equating k

the terms with the same powers of ε, we obtain = f, Lun,a 0

(75)

n,a Lun,a k+1 = −Auk .

(76)

By Lemma 6.1 in order for the solutions of (75), (76) to exist in P ⊥ it is enough to show that the right sides are in P ⊥ . Unfortunately, the fact that u ∈ P ⊥ does not guarantee that −Au ∈ P ⊥ . We use the particular form of the vector fields vi to describe the subspaces of P ⊥ appropriate for solving (75), (76). By staying within these subspaces we verify ∈ P ⊥ . Then we use (75), (76) as an inductive definition of the sequence {un,a −Aun,a k }. k k n,a n,a n,a 2 From (75), (76) it follows that wk = u − (u0 + ... + ε uk ) satisfies Mε wk = −ε

k+1 2

Aun,a k .

(77)

We then estimate the RHS of (77) uniformly in n and employ Theorem 5.2 to obtain the desired estimate of wk . These arguments will be made rigorous below. Let us introduce notation needed for the proof of Theorem 6.2. To simplify the notation we consider a = 1 and drop the superscripts on the function u and the terms of the asymptotic expansion. Recall that the set {vi , i = 1, ..., n} can be written as follows: {vi , i = 1, ..., n} = {σi ∇⊥ cos ki x, σi ∇⊥ sin ki x, i = 1, ..., n/2}.

(78)

On the set {i = 1, ..., n} we introduce a reflection operation which interchanges sin and cos terms of the same wave length as follows: if vi = σi ∇⊥ cos ki x, then i0 is the index for which vi0 = σi ∇⊥ sin ki x. Similarly, if vi = σi ∇⊥ sin ki x, then vi0 = σi ∇⊥ cos ki x. Recall that σi = σi0 and i = i0 .

(79)

In what follows the constants c(i1 , ..., ik ) are assumed to be invariant under this reflection operation: c(i1 , ..., il , ..., ik ) = c(i1 , ..., i0l , ..., ik ) for any 1 ≤ l ≤ k.

(80)

We next describe the subspaces Rk of P ⊥ appropriate for solving (75), (76). Rk will be defined as a linear space of functions of (n, x, y). Note that we include the

Transport by Time Dependent Stationary Flow

669

dependence on n in the definition of the space Rk . Thus we shall be solving (75), (76) for all n simultaneously. First we define the set of functions which span Rk . We shall say that a function u(n; x, y) belongs to T k if for some t and s, such that t + 2s = k, u=

n X i1 ,...,ik =1

vik ∇(...∇(vi2 ∇vi11 )...)yir1 ...yirt δil1 im1 ...δils ims cn (i1 , ..., ik )η, (81)

where (r1 , ..., rt , l1 , ..., ls , m1 , ..., ms ) is some permutation of (1, ..., k), the constants cn (i1 , ..., ik ) satisfy (80) and |cn | ≤ c,

(82)

with a constant c which may depend on u, but does not depend on n. Note that T k is not a linear space since it is a set theoretic union over choices t, s, and the permutations, and thus is not closed under addition. We shall say that a function u(n, x, y) belongs to Rk if u = u1 + ... + ud , where u1 , ..., ud ∈ T k . Thus Rk = span{T k }. Note that the number d of functions of T k which comprise an element of Rk can not depend on n, since n is already an argument in each of the functions u, u1 , ..., ud . An element of Rk can combine terms of the form (81) with different t and s. If k is even, then all elements of Rk are even, if k is odd, then all elements of Rk are odd. We prove several lemmas which are needed for the proof of Theorem 6.2. Lemmas 6.3, 6.4, and 6.5 imply that (75), (76) can be used as an inductive definition for the sequence un,a k . Lemma 6.6 provides the estimate of the RHS of (77), which is uniform in n. Lemma 6.3. Suppose Assumptions A and B hold. If u ∈ Rk , then Au ∈ Rk+1 , and n P 1 k+1 . j yj vj ∇x u ∈ R j=1

Proof. The statement follows from the fact that the same is true when Rk and Rk+1 are replaced by T k and T k+1 . Lemma 6.4. Suppose Assumptions A and B hold. If u ∈ Rk , then u ∈ P ⊥ . Proof. Without loss of generality weRcan consider u ∈ T k given by formula (81). If k ⊥ is odd, then R u is an odd function and uηdy = 0, that is u ∈ P . Assuming now that k is even, uηdy is equal to a sum of the terms, each of which has the following form: n X i1 ,...,ik =1

vik ∇(...∇(vi2 ∇vi11 )...)δil1 im1 ...δilk/2 imk/2 cn (i1 , ..., ik ),

(83)

where (l1 , ..., lk/2 , m1 , ..., mk/2 ) is a permutation of (1, ..., k). From (21) we derive the following: if D1α1 and D2α2 are (tensor valued) homogeneous partial differential operators of orders α1 and α2 respectively, then D1α1 vi ⊗ D2α2 vi + D1α1 vi0 ⊗ D2α2 vi0 = 0,

(84)

provided α1 +α2 is odd. Here product ⊗ denotes either a tensor product, or a convolution in one or more indices. From (84) we conclude that

670

L. Koralov n X

D1α1 vi ⊗ D2α2 vi c(i) = 0,

(85)

i=1

if α1 + α2 is odd and c(i) = c(i0 ). Distributing the derivatives in the expression vik ∇(...∇(vi2 ∇vi11 )...) we see that (83) is equal to a sum of the terms, each of which has the form: n X i1 ,...,ik =1

Note that

k P j=1

Dkαk vik ⊗ ... ⊗ D1α1 vi1 δil1 im1 ...δilk/2 imk/2 cn (i1 , ..., ik ).

(86)

αj = k − 1 is odd. Therefore there exist lp and mp such that αlp + αmp is

odd. By (85) n X ilp ,imp =1

Dkαk vik ⊗ ... ⊗ D1α1 vi1 δil1 im1 ...δilp imp ...δilk/2 imk/2 cn (i1 , ..., ik ) = 0.

Therefore, the expression in (86) is equal to zero. Therefore, the expression in (83) is equal to zero. This completes the proof of Lemma 6.4. Lemmas 6.1 and 6.4 imply that if u ∈ Rk , then the equation Lw = u

(87)

has the unique solution w ∈ P ⊥ . Lemma 6.5. Suppose Assumptions A and B hold. If u ∈ Rk , then the solution w of (87) belongs to Rk . Proof. Without loss of generality we may consider u ∈ T k given by formula (81). The proof will be by induction with step 2 on the number t of the y-factors in (81). Thus we introduce the following induction hypothesis: to Rk . (Hj ) If u ∈ T k is given by (81) and t = j, then the solution w of (87) belongs R If t = j = 0, then u has the form u = u0 (x)η. Since, by Lemma 6.4, uηdy = 0 for all x, we conclude that u = 0, and therefore w = 0. Thus H0 holds. If t = j = 1, then n X

u=

i1 ,...,ik =1

vik ∇(...∇(vi2 ∇vi11 )...)yir1 δil1 im1 ...δils ims cn (i1 , ..., ik )η, yi

where (r1 , l1 , ..., ls , m1 , ..., ms ) is a permutation of (1, ..., k). Since L( ir1 η) = −yir1 η, r1 the solution of (87) is w=−

n X i1 ,...,ik =1

vik ∇(...∇(vi2 ∇vi11 )...)

y i r1 δi i ...δils ims cn (i1 , ..., ik )η, ir1 l1 m1

which belongs to T k since i are positive and uniformly bounded away from zero. Thus H1 holds.

Transport by Time Dependent Stationary Flow

671

Assume a value of j for which Hj 0 holds for any j 0 ≤ j. We shall verify that Hj+2 holds. Let u ∈ T k be given by (81) with t = j + 2. Define n X

w0 = −

vik ∇ ...∇(vi2 ∇vi11 )...

i1 ,...,ik =1

yir1 ...yirt δi i ...δils ims cn (i1 , ..., ik )η. (ir1 + ... + irt ) l1 m1

Since i are positive and uniformly bounded away from zero w0 belongs to T k . Let u0 = u − Lw0 . Then u0 ∈ Rk and it can be represented as a sum of elements of T k , each of which has the form (81) with t ≤ j. Since L(w − w0 ) = u0 , 0

and Hj 0 holds for any j ≤ j, we conclude that w − w0 ∈ Rk , and therefore w ∈ Rk . Thus Hj+2 holds. This completes the proof of Lemma 6.5. Lemma 6.6. Suppose Assumptions A, B, and Cm hold. Then for any function u(n; x, y) ∈ R2m+2 the L2 norms ||u(n; x, y)|| are bounded uniformly in n. Proof. Without loss of generality we can consider u ∈ T 2m+2 given by formula (81) with k = 2m + 2. We square both sides of (81) and integrate the resulting equality in x and y. On the RHS we can perform the integration in y explicitly. Note that Z 1 X δiα1 iα2 ...δiα2t−1 iα2t , yir1 ...yirt yiq1 ...yiqt η 2 dy = 2 p 0 σ

where σ 0 = (α1 , ..., α2t ) is a permutation of (r1 , ..., rt , q1 , ..., qt ), and

P

is the sum over

σ0

the set of all such permutations. Note that the number of such permutations depends only on t, but not on n. Thus, using (82), we see that ||u||2 is estimated from above by a sum of terms, the number of terms being independent of n, each of which has the following form: Z n X c(σ) |vi4m+4 ∇(...∇(vi2m+4 ∇vi12m+3 )...) p2 T2p

i1 ,...,i4m+4 =1

vi2m+2 ∇(...∇(vi2 ∇vi11 )...)δil1 ik1 ...δil2m+2 ik2m+2 |dx ≤ n X

c(σ) sup x

(88)

|vi4m+4 ∇(...∇(vi2m+4 ∇vi12m+3 )...)

i1 ,...,i4m+4 =1

vi2m+2 ∇(...∇(vi2 ∇vi11 )...)δil1 ik1 ...δil2m+2 ik2m+2 |, where σ = (l1 , ..., l2m+2 , k1 , ..., k2m+2 ) is a permutation of (1, ..., 4m + 4). Distributing the derivatives in the RHS of (88) we see that it can be estimated from above by a sum of the terms, the number of terms being independent of n, each of which has the following form: c(σ) sup x

n X i1 ,...,i4m+4 =1

α4m+4 |D4m+4 vi4m+4 ⊗ ... ⊗ D1α1 vi1 δil1 ik1 ...δil2m+2 ik2m+2 |.

(89)

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L. Koralov

Note that maxi αi ≤ 2m + 1. For an arbitrary tensor T let |T | be the supremum of its component’s absolute values. Then sup |Dα cos(ki x)| = sup |Dα sin(ki x)| ≤ c(D)|ki |α . x

x

Thus due to the particular form (78) of the vector fields vi the expression (89) is estimated from above by a constant independent of n times the product of 2m + 2 factors, each of n n P P which has the form σi2 |ki |q with q ≤ 4m+4. Each of the factors σi2 |ki |q is bounded i=1

i=1

uniformly in n by assumption Cm . Therefore the absolute value of the expression in (89) is bounded uniformly in n. Thus the RHS of (88) is bounded uniformly in n. This completes the proof of Lemma 6.6. Proof of Theorem 6.2. The right side of (75) belongs to R1 . From Lemmas 6.3, 6.4 and 6.5 it follows that (75), (76) can be used as the inductive definition of the sequence {uk }. Moreover, uk ∈ Rk+1 . Thus u2k is an odd function and u2k+1 is an even function for all k. Since uk ∈ Rk+1 the function Auk belongs to Rk+2 . Thus, by Lemma 6.6 the norm of the RHS of (77) is estimated as follows: || − ε

k+1 2

Auk || ≤ C1 (k)ε

k+1 2

.

By Lemmas 6.3 and 6.4 the RHS of (77) satisfies (48). Therefore we can employ Thek+1 orem 5.2 to conclude that ||wk || ≤ C2 (k)ε 2 . Thus (74) is proved. This completes the proof of Theorem 6.2. We next employ Theorem 6.2 and Corollary 4.4 to obtain the asymptotic expansion for Dεn,ab . While the asymptotic expansion for the solution of Eq. (45) is in the powers 1 of ε 2 , only the terms with integer powers of ε contribute to the asymptotic expansion of Dεn,ab . The fractional powers in ε vanish because the integrals in y of the odd terms vanish. Theorem 6.7. Suppose Assumptions A, B, and Cm hold. Then the effective diffusivity Dεn,ab has the asymptotic expansion m m n + dn,ab ε + ... + dn,ab Dεn,ab = dn,ab m ε + ε φ (ε, m), where 0 1

= dn,ab k

n

1X 2

ZZ

(90)

n,b a b (un,a 2k vi + u2k vi )yi ηdxdy,

i=1

n

and limε→0 φ (ε, m) = 0 uniformly in n. Proof. As in the proof of Theorem 6.2 we consider the case a = b = 1. By Corollary 4.4 Dεn,11 =

2m X k=0

k

ε2

ZZ

+

n ZZ X

unk vi1 yi ηdxdy

i=1

[un − (un0 + ε 2 un1 + · · · + εm un2m )]vi1 yi ηdxdy. 1

(91)

Transport by Time Dependent Stationary Flow

Note that ||

n P i=1

673

vi1 yi η|| ≤ C1 by (72). By Theorem 6.2 ||un − (un0 + ε 2 un1 + ... + εm un2m )|| ≤ C2 (m)εm+ 2 , 1

1

therefore the last term on the RHS of (91) does not exceed C3 (m)εm+ 2 . It remains to show that the terms with non integer powers of ε vanish. By Theorem 6.2 the integrand in the first term on the RHS of (91) is a product of η times an odd function if k is odd, and therefore the integral is equal to zero. This completes the proof of Theorem 6.7. 1

Proof of Theorem 2.2. We introduce the induction hypothesis: n,ab exists. (Ik ) The limit dab k = lim n→∞ dk From the proof of Theorem 2.1 it follows that limn→∞ Dεn,ab = Dεab . By Theorem 6.7 uniformly in n. Therefore by the theorem on uniform convergence limε→0 Dεn,ab = dn,ab 0 the following limits exist: n,ab = lim Dεab . dab 0 = lim d0 n→∞

ε→0

Thus I0 holds. Assume a value of k, k < m, for which Ik holds. We shall verify that Ik+1 holds. Consider ψkab (n, ε) =

Dεn,ab − dn,ab − εdn,ab − ... − εk dn,ab 0 1 k . εk+1

By Ik

ab k ab Dεab − dab 0 − εd1 − ... − ε dk . n→∞ εk+1 By Theorem 6.7 limε→0 ψkab (n, ε) = dn,ab k+1 uniformly in n. Therefore by the theorem on uniform convergence the following limits exist:

lim ψkab (n, ε) =

ab k ab Dεab − dab 0 − εd1 − ... − ε dk . (92) n→∞ ε→0 εk+1 Therefore Ik+1 holds for all k < m. From (92) with k = m − 1 we obtain (11). This completes the proof of Theorem 2.2. n,ab dab k+1 = lim dk+1 = lim

7. Explicit Calculations Now we calculate explicitly the first two terms of the expansion (11) in terms of the correlation matrix G of the velocity field. Theorem 7.1. Suppose Assumptions A, B, and C1 hold. Then the effective diffusivity Dεab has the asymptotic expansion Dεab

Z∞ =

Gab (0, t)dt

0 2 Z∞ Z∞ Z∞ X ∂ ∂ Gab (x, t1 )|x=0 Glm (0, t3 )dt1 dt2 dt3 + o(ε). +ε ∂xl ∂xm l,m=1 0 t t 3 2

(93)

674

L. Koralov

Proof. Solving (75) we obtain n X 1 = yi ηvia (x). un,a 0 i i=1

From (76) with k = 0 and k = 1 we obtain consecutively = un,a 1

n X

i,j=1

= un,a 2

n X i,j,k=1 n X i,j,k=1 n X i,k,j=1 n X

1 yi yj ηvi ∇vja , (i + j )j

1 yi yj yk ηvi ∇(vj ∇vka ) + (i + j + k )(j + k )k δij

2 i yk ηvi ∇(vj ∇vka ) + k (i + j + k )(j + k )k

δik

2 i yj ηvi ∇(vj ∇vka ) + j (i + j + k )(j + k )k

δjk

2 k yi ηvi ∇(vj ∇vka ). i (i + j + k )(j + k )k

i,j,k=1

(94)

we see that the first coefUsing Theorem 6.7 together with the expression for un,a 0 ficient (with ε0 ) in the expansion (90) is equal to Z n ZZ n X 1 a b p−2 1 X n,ab b a 2 yi (vi vi + vi vi )σi yi η dxdy = via vib dx. (95) d0 = 2 i i i=1

i=1

The coefficient at the first power of ε is equal to n ZZ 1 X n,ab (ua2 vib + via ub2 )yi ηdxdy. d1 = 2 i=1

Using (94) and (85) this is seen to be equal to Z n X p−2 (vj ∇via )(vj ∇vib )dx. − 2 ( + ) i j i i,j=1

(96)

Using (78) and (79) we evaluate the integrals in (95) and (96). Thus we obtain = dn,ab 0

n 1 X σi2 ⊥ a ⊥ b (k ) (ki ) , 2 i i

n X

σi2 σj2

i,j=1

2i (i + j )

i=1

=− dn,ab 1

1 4

(ki kj⊥ )(ki kj⊥ )(ki⊥ )a (ki⊥ )b .

The expression (8) for the Fourier transform of the correlation function F (x, t) and the relation (6) imply that

Transport by Time Dependent Stationary Flow

dab 0

=

lim dn,ab n→∞ 0

675

Z∞ =

Gab (0, t)dt,

0

n,ab = dab 1 = lim d1 n→∞

2 Z∞ Z∞ Z∞ X ∂ ∂ Gab (x, t1 )|x=0 Glm (0, t3 )dt1 dt2 dt3 . ∂xl ∂xm

l,m=1 0 t t 3 2

Therefore (93) is valid. This completes the proof of Theorem 7.1.

8. Proof of Theorem 3.1 In order to prove Theorem 3.1 we need a number of results from [11, 3, 1]. First we show that the RHS of (1) is smooth enough for some modification of the field V (x, t), so that we can solve (1) for almost every realization of V . In Lemmas 8.1, 8.5, and Corollary 8.2 we state the general results on the regularity and on the behavior at infinity of a typical realization of a Gaussian field, whose correlation function is sufficiently regular. It follows from Lemma 8.6 that our vector field V (x) indeed satisfies the assumptions of Corollary 8.2 and Lemma 8.5, and thus we can solve (1) for almost every realization of V . Note that the Gaussian fields V n (x) are defined in Sect. 3 through their correlation matrices, and the underlining probability space for V n may be different from the one for V . The argument which proves the convergence of the displacement tensors (24) is based on the Skorohod Theorem, which allows us to realize the random fields V and V n on the same probability space. Let f (x, t) be a scalar, vector, or tensor valued function on R3 . Let K ⊂ R3 be a compact set, and let α > 0. We shall say that f (x, t) ∈ Hα (K) if there exist positive constants C1 and C2 such that for (x1 , t1 ), (x2 , t2 ) ∈ K, |f (x1 , t1 ) − f (s2 , t2 )| ≤ C1 |(x1 − x2 , t1 − t2 )|α whenever |(x1 − x2 , t1 − t2 )| ≤ C2 . We shall say that f (x, t) ∈ Hα if f (x, t) ∈ Hα (K) for every compact set K. Lemma 8.1 ([11]). Let φ be a Gaussian stationary random field (possibly vector or matrix valued), whose correlation tensor belongs to Hα . Then there is a modification of φ, whose almost every realization belongs to Hα/2−ε for every ε > 0. Corollary 8.2. Let g ab (x, t) be the correlation matrix of a Gaussian stationary vector field v(x, t). Suppose that for every partial differential operator Dx of order not greater than two with constant coefficients the following holds: Dx g ab (x, t) ∈ Hα . Then there exists a modification of v, such that almost every realization of the vector field belongs to Hα/2−ε together with the first order partial derivatives in x. For a compact set K ⊂ R3 we define C(K) to be the set of all continuous R2 valued functions on K. C(K) is endowed with the usual Borel σ-algebra. It is assumed that 0 ∈ K. We state several lemmas, which will be used in the proof of Theorem 3.1. Lemmas 8.3 and 8.4 are contained in Chapter 2 of Billingsley’s book [1], while Lemma 8.5 follows from Lemma 4.5 and Proposition 2.5 of Collela and Lanford [3].

676

L. Koralov

Lemma 8.3 ([1]). Let v n (x, t) and v(x, t) be continuous random fields on a compact set K ⊂ R3 . If finite dimensional distributions of v n converge weakly to those of v and if the sequence v n is tight on C(K), then v n converge weakly to v as measures on C(K). Lemma 8.4 ([1]). The sequence v n is tight on C(K) if and only if these two conditions hold: (a) For each positive η, there exists an a such that P {v n : |v n (0, 0)| > a} ≤ η, n ≥ 1.

(97)

(b) For each positive ε and η, there exist a δ, with 0 < δ < 1, and an integer n0 such that P {v n :

sup |(x1 −x2 ,t1 −t2 )|<δ

|v n (x1 , t1 ) − v n (x2 , t2 )| ≥ ε} ≤ η, n ≥ n0 .

(98)

Lemma 8.5 ([3]). Suppose that v(x, t) is a Gaussian stationary random vector field on R3(x,t) with continuous realizations, such that its correlation matrix g ab (x, t) is continuous. Suppose there exist constants C1 , C2 , α > 0 such that (a) |g ab (0, 0)| < C1 . (b) |g ab (x, t) − g ab (0, 0)| ≤ C2 |(x, t)|α f or |(x, t)| ≤ 21 . Then for every T0 , γ > 0 there exist constants k1 and k2 , which depend only on C1 , C2 , α, T0 and γ such that P {v :

sup x>ε−γ k1 ;t≤T0

|v(x, t)| p > k2 } < ε f or all ε ≤ 1. log |x|

(99)

Moreover, for every compact set K ⊂ R3 and for each positive ε and η there exists a δ, which depends only on C1 , C2 , α, K, ε and η such that P {v :

sup |(x1 −x2 ,t1 −t2 )|<δ

|v(x1 , t1 ) − v(x2 , t2 )| ≥ ε} ≤ η.

We next show that the vector field V (x, t) satisfies the assumptions of Corollary 8.2, and V n (x, t) satisfy the assumptions of Lemmas 8.4 and 8.5. Note that due to (19) almost all the realizations of V n (x, t) belong to Hα together with the first order partial derivatives in x for some α > 0. Let Gn,ab (x, t) = E V n,a (x, t)V n,b (0, 0) be the correlation matrix of the vector field V n (x, t). Lemma 8.6. Suppose Assumptions A and B hold. Then (a) The correlation matrix Gab (x, t) of the vector field V (x, t) satisfies the assumptions of Corollary 8.2 with some α > 0. (b) For every compact set K the vector fields V n (x, t) satisfy the assumptions of Lemma 8.4. (c) The correlation matrices Gab and Gn,ab satisfy the assumptions of Lemma 8.5 with C1 , C2 and α independent of n. (d) For every compact set K Gn,ab → Gab uniformly on K.

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Proof. Note the following: |e−|t1 |(k) − e−|t2 |(k) | ≤ |t1 − t2 |α (k)α , for all t1 , t2 , k, and 0 ≤ α ≤ 1, (100) |eikx1 − eikx2 | ≤ 2|x1 − x2 |α k α , for all x1 , x2 , k, and 0 ≤ α ≤ 1.

(101)

We shall use (100) and (101) without reference. Note that if Dx is of order not greater than two, then Z Dx Gab (x, t) = p(k)e−|t|(k) eikx Fb(dk, 0), where p(k) is a polynomial of degree not greater than four. Thus, Z |Dx Gab (x1 , t) − Dx Gab (x2 , t)| = | p(k)e−|t|(k) (eikx1 − eikx2 )Fb(dk, 0)| ≤ 2|x1 − x2 |δ

Z

|p(k)|e−|t|(k) |k|δ Fb(dk, 0).

(102)

The integral on the RHS of (102) converges by Assumption B, and thus |Dx Gab (x1 , t) − Dx Gab (x2 , t)| ≤ c1 |x1 − x2 |δ . Similarly |Dx Gab (x, t1 ) − Dx Gab (x, t2 )| = | |t1 − t2 |γ

Z

Z

(103)

p(k)(e−|t1 |(k) − e−|t2 |(k) )eikx Fb(dk, 0)| ≤

|p(k)|((k))γ Fb(dk, 0).

(104)

For γ small enough the integral on the RHS of (104) converges by Assumption B, and thus |Dx Gab (x, t1 ) − Dx Gab (x, t2 )| ≤ c2 |t1 − t2 |γ .

(105)

By (103) and (105) |Dx Gab (x1 , t1 ) − Dx Gab (x2 , t2 )| ≤ c3 |(x1 , t1 ) − (x2 , t2 )|α with α = min{δ, γ}. (106) Therefore the function Dx Gab belongs to Hα with constants C1 and C2 independent of the compact set K. This completes the proof of part (a) of Lemma 8.6. The arguments leading to (106) can be applied to Gn,ab instead of Gab . The only difference is that for Gn,ab one needs to use (14) instead of (10). Thus |Gn,ab (x1 , t1 ) − Gn,ab (x2 , t2 )| ≤ c4 |(x1 , t1 ) − (x2 , t2 )|α , where c4 does not depend on n. Recall that Z n,ab (x, t) = e−|t|(k) eikx (k ⊥ )a (k ⊥ )b Fcn (dk, 0). G

(107)

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L. Koralov

Therefore, by (14) |Gn,ab (0, 0)| ≤

Z

|(k ⊥ )a (k ⊥ )b |Fcn (dk, 0) ≤ c5 ,

(108)

where c5 does not depend on n. The assumptions of Lemma 8.5 are satisfied by (107) and (108). This completes the proof of part (c) of Lemma 8.6. The relation (97) for the fields V n follows from (108) by Chebyshev inequality. The relation (98) is a consequence of Lemma 8.5 applied to V n . This completes the proof of part (b) of Lemma 8.6. It remains to prove part (d). Let a compact K ⊂ R3 be given. We also fix a compact set MN = {|k a | < N, a = 1, 2} ⊂ R2 in Fourier space. Recall from Sect. 3 that αi , βi and γi are the interior, the sides and the corners of the square 1i , respectively. We introduce the following notation, for any function φ(k) and measure µ: Z Z Z Z 0 1 1 φ(k)µ(dk) = φ(k)µ(dk) + φ(k)µ(dk) + φ(k)µ(dk). 2 4 αi

1i

γi

βi

Then, ab

|G (x, t) − G

n,ab

Z

X

(x, t)| ≤ |

{i,ki ∈MN }

0

1i

− Fcn (dk, 0))| + Z +

e−|t|(k) (k ⊥ )a (k ⊥ )b eikx (Fb(dk, 0) Z

|e−|t|(k) (k ⊥ )a (k ⊥ )b eikx |Fb(dk, 0)

k∈M / N

(109)

|e−|t|(k) (k ⊥ )a (k ⊥ )b eikx |Fcn (dk, 0).

k∈M / N

By (10) and (14) the last two terms on the RHS of (109) can be made arbitrarily small uniformly in n, t and x by selecting N large enough. In order to estimate the first term we write Z 0 e−|t|(k) (k ⊥ )a (k ⊥ )b eikx (Fb(dk, 0) − Fcn (dk, 0))| | 1i

Z =|

0 1i

Z

(e−|t|(k) − e−|t|(ki ) )(k ⊥ )a (k ⊥ )b eikx Fb(dk, 0) 0

σ2 e−|t|(ki ) (k ⊥ )a (k ⊥ )b eikx (Fb(dk, 0) − i δ(ki )dk)| 2 1i Z 0 |((k) − (ki ))(k ⊥ )a (k ⊥ )b |Fb(dk, 0) ≤ c1 (K)

+

1i

⊥ a

⊥ b ikx

+ sup {|(k ) (k ) e k∈1i

Z

−

(ki⊥ )a (ki⊥ )b eiki x |}

1i

0

Fb(dk, 0).

(110)

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Since (k) is Lipschitz continuous on MN the RHS of (110) is estimated from above by Z 0 1 + |(k ⊥ )a (k ⊥ )b | Fb(dk, 0), c2 (K, N )|1i | 1i

where |1i | is the length of the diagonal of 1i . Thus the first term on the RHS of (109) is estimated from above by Z 1 + |(k ⊥ )a (k ⊥ )b | Fb(dk, 0). (111) c3 (K, N ) max |1i | i

Due to (10), since maxi |1i | tends to zero as n → ∞, the quantity in (111) tends to zero as n → ∞. Therefore lim

sup |Gn,ab (x, t) − Gab (x, t)| = 0.

n→∞ (x,t)∈K

This completes the proof of Lemma 8.6.

Proof of Theorem 3.1. Without loss of generality we may consider X0 = X0n = 0. Let t = T0 be fixed. By Lemma 8.5 there exist constants k1 > 1 and k2 such that (99) holds 1 for v = V and v = Vn for every n. Let Rε = 2ε− 4 k1 ek2 T0 and Kε = {|x| ≤ Rε }×[0, T0 ]. n The finite dimensional distributions of V converge weakly to those of V , since V n and V are Gaussian, and the correlation functions converge pointwise (Lemma 8.6, part (d)). By Lemma 8.4 the sequence V n is tight on C(Kε ), and therefore, by Lemma 8.3 V n converges weakly to V on C(Kε ). By the Skorohod Theorem ([4]) there exist vector fields Wεn and Wε , which induce the same measures on C(Kε ) as V n and V , and which are defined on the same probability space ε with Wεn → Wε in C(Kε ) for almost all ω ∈ ε . By Corollary 8.2 the realizations of the vector fields Wεn and Wε belong to Hα (Kε ) together with their first order partial derivatives in x. We do not need to take the modifications since we know a priori that almost all the realizations belong to C(Kε ). n Define Yε,s and Yε,s to be the solutions of the equations Y˙ε,s = Wε (Yε,s , s), Yε,0 = 0, n n n = Wεn (Yε,s , s), Yε,0 = 0. Y˙ε,s

(112) (113)

Since (99) holds for v = Wε , on a subset of ε of measure not less than 1 − ε the norm of Yε,s is estimated as follows: |Yε,s | ≤ ys , s ≤ T0 on 0ε , µ(0ε ) ≥ 1 − ε, where ys is the solution of the equation p 1 y˙s = k2 log ys , y0 = ε− 4 k1 . Clearly sup ys ≤

s≤T0

Rε , 2

and therefore sup |Yε,s | ≤

s≤T0

Rε . 2

(114)

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L. Koralov

This shows that for ω ∈ 0ε Eq. (112) can be solved for 0 ≤ s ≤ T0 , and that P { sup |Yε,s |2 > s≤T0

Rε2 } ≤ ε. 4

(115)

Since Yε,T0 and XT0 have the same distributions when they are restricted to the events {sups≤T0 |Yε,s | < R2ε } and {sups≤T0 |Xs | < R2ε } respectively, we conclude from (115) that P { sup |Xs |2 > s≤T0

Rε2 } ≤ ε. 4

(116)

Therefore |XT0 |2 is integrable. Since (99) holds for v = Wεn we can repeat the arguments leading to (116) to conclude that P { sup |Xsn |2 > s≤T0

Rε2 } ≤ ε, 4

which shows that |XTn0 |2 are uniformly integrable. Since X0 = X0n = 0, in order to complete the proof

of Theorem 3.1 we need to show

that the difference E(Xtn,a Xtn,b ) − E(Xta Xtb )

(117)

tends to zero as n → ∞. Let an arbitrary γ > 0 be given. Since |XTn0 |2 and |XT0 |2 are uniformly integrable, we can select ε > 0 such that for every measurable set A, with P (A) < 2ε, E|XTn0 |2 χA < γ ; E|XT0 |2 χA < γ.

(118)

n − Yε,s | tends to zero almost surely on 0ε . By (112) and We next show that sups≤T0 |Yε,s

(113),

n n − Y˙ε,s | ≤ |Yε,s − Yε,s | |Y˙ε,s

+

sup

(x,s0 )∈Kε

sup

|Dx Wε (x, s0 )|

(x,s0 )∈Kε |Wε (x, s0 ) −

Wεn (x, s0 )|,

(119)

n whenever sups0 ≤s |Yε,s 0 | < Rε . Since the second term on the RHS of (119) tends to zero n |< as n → ∞, and sups≤T0 |Yε,s | ≤ R2ε on 0ε by (114), we conclude that sups≤T0 |Yε,s 0 0 n Rε on ε for n > n(ω), and sups≤T0 |Yε,s − Yε,s | → 0 almost surely on ε . Therefore there exists a set 1ε ⊂ 0ε with µ(1ε ) ≥ 1 − 2ε, such that n − Yε,s | → 0 uniformly on 1ε . sup |Yε,s

s≤T0

n −Yε,s and Xsn −Xs have the same distributions when Since for each n the variables Yε,s n |, |Yε,s | ≤ Rε } and {sups≤T0 |Xsn |, |Xs | ≤ Rε } restricted to the events {sups≤T0 |Yε,s respectively, and since the constant γ in (118) was arbitrary, we conclude that (117) tends to zero as n → ∞. This completes the proof of Theorem 3.1.

Acknowledgement. I would like to thank Professors J. Glimm and S. Molchanov for introducing me to this problem and for numerous useful discussions. I would also like to thank the Center for Nonlinear Studies at Los Alamos National Laboratory for its hospitality during the completion of this paper.

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References [1] [2]

[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

[18]

[19]

Billingsley, P.: Convergence of Probability Measures, New York: Wiley, 1968 Carmona, R., Grishin, S.A. and Molchanov, S.A.: Massively Parallel Simulations of Motions in a Gaussian Velocity Field. In: Stochastic Modeling in Physical Oceanography. Progress in Probability, Basel– Boston: Birkhauser, Vol 39, 1996, pp. 47–69 Collela, P., Lanford, O.: Sample Field Behavior for the Free Markov Random Field. Lecture Notes in Physics, 25, Berlin–Heidelberg–New York: Springer-Verlag, 1973 Dudley, R.: Real Analysis and Probability. London: Chapman and Hall, 1989 Fannjiang, A., Komorowski, T.: Turbulent Diffusion in Markovian Flows. Preprint Fannjiang, A., and Papanicolaou, G.C.:. Diffusion in Turbulence. Probability Theory and Related Fields 105, 279–334 (1996) Freidlin, M.I.: Dirichlet Problem for Equations with Periodic Coefficients. Probability Theory and Appl. 9, 133–139 (1964) Glimm, J., and Jaffe, A.: Quantum Physics Berlin–Heidelberg–New York: Springer-Verlag, 1987 Hormander, L.: Hypoelliptic Second Order Differential Equations. Acta Math. 119, 147–171 (1967) Hormander, L.: The Analysis of Linear Partial Differential Operators. Vol 3, Berlin–Heidelberg–New York: Springer Verlag, 1985 Ibragimov, I.A., Rosanov Y.A.: Gaussian Random Processes. New York: Springer-Verlag, 1978 Komorowski, T., Papanicolaou, G.: Motion in a Gaussian, Incompressible Flow. The Annals of Applied Probability 7, 229–264 (1997) Koralov, L.: Effective Diffusivity of Stationary Vector Fields with Short Time Correlations. Random Ops and Stochastic Eqs. 5, no 4, 303–324 (1997) Kozlov, S.: Averaging of Random Operators. Matem. Sbornik 151, 188–202 (1979) McComb, W. D.: The Physics of Fluid Turbulence. Oxford: Clarendon Press, 1990 Molchanov, S.A.: Topics in Statistical Oceanography. In: Stochastic Modeling in Physical Oceanography. Progress in Probability, Basel–Boston: Birkhauser, 39, 1996, pp. 343–381 Molchanov, S.A.: Lectures on Random Media in Lectures on Probability Theory. In: Ecole d’Ete de Probabilites de Saint-Flour XXII-1992. Editor P. Bernard. Berlin–Heidelberg–New York:: SpringerVerlag Papanicolaou, G., Varadhan, S.R.S.: Boundary Value Problem with Rapidly Oscillating Random Coefficients. Coll Math Soc. Janos Bolyai, 27, Random Fields, Vol. 2, Amsterdam: North-Holland, 1981, pp. 835–873 Taylor, G.I.: Diffusion by Continuous Movements. Proc. R. Soc., Second Series. 20, 196–211 (1922)

Communicated by Ya. G. Sinai

Commun. Math. Phys. 199, 683 – 695 (1999)

Communications in

Mathematical Physics

A Note on the Eigenvalue Density of Random Matrices? Michael K.-H. Kiessling1 , Herbert Spohn1,?? Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway, N.J. 08854, USA. E-mail: [email protected] Received: 1 April 1998 / Accepted: 5 June 1998

Abstract: The distribution of eigenvalues of N × N random matrices in the limit N → ∞ is the solution to a variational principle that determines the ground state energy of a confined fluid of classical unit charges. This fact is a consequence of a more general theorem, proven here, in the statistical mechanics of unstable interactions. Our result establishes the eigenvalue density of some ensembles of random matrices which were not covered by previous theorems. 1. Introduction Since the pioneering work of Wigner [37, 38, 39], there has been a considerable effort to understand the statistics of eigenvalues of N × N random matrices. The problem has three scales: (i) the density of eigenvalues which converges to a deterministic limit as N → ∞; (ii) the fluctuations of order one around this deterministic density; (iii) the local statistics on the scale of the typical distance between eigenvalues. Item (i) depends on the particular matrix ensemble while (ii) and (iii) are “universal”, in the sense that they depend only on some overall matrix characteristics (e.g. the matrices being real and symmetric). The “classical” results are reviewed in [28]; see also [31] for a collection of early work. Recently, in the context of the double scaling limit of 2D quantum gravity [10], (ii) and (iii) have been studied at the edge of the support of the density of states where novel universality classes occur. Among recent work on (ii) we also mention [29, 34], and [30, 4, 9] regarding (iii). In our paper we will consider only the largest scale (i). For various N × N random matrix ensembles of the form m(N ) (dM ) = Q(N )−1 e−κN Tr V (M ) dM , ?

(1.1)

c (1998) The authors. Reproduction of this article for non-commercial purposes by any means is

permitted. ?? On leave from: Theoretische Physik, Ludwig-Maximilians-Universit¨ at, Theresienstr. 37, D-80333 M¨unchen, Germany. E-mail: [email protected]

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M. K.-H. Kiessling, H. Spohn

the joint probability distribution for the N eigenvalues λ1 , ..., λN (which may be real or complex) is identical to the (configurational) canonical ensemble at inverse temperature β of N unit point charges at positions λ1 , ..., λN ∈ 3 ⊂ R2 . The region 3 can be all of R2 , the unit disk B1 ⊂ R2 , the unit circle S1 , the entire real line R, or some other set, depending on the type of random matrices. This joint probability distribution has the general form (1.2) dµ(N ) = Q(N ) (β)−1 exp −βH (N ) dλ1 · · · dλN on 3N , where dλk is the uniform measure on 3 and Q(N ) (β) the normalizing partition function. The classical Hamiltonian, H (N ) , is of the form X X G(λj , λk ) + F (λk ) + N V (λk ), (1.3) H (N ) (λ1 , ..., λN ) = 1≤j
1≤k≤N

where G(λj , λk ) = G(λk , λj ) is (2π×) a Green’s function for −1 in two dimensions, F (λ) = limη→λ (G(λ, η) + ln |λ − η|) is the regular part of G, and V (λ) given in (1.1). Let us list a few examples. The joint eigenvalue distribution, µ(N ) , of the Gaussian ensembles (i.e., V (M ) = M † M in (1.1)) for the real symmetric [22], the complex Hermitian [14], the general complex [20], and the Hermitian self-dual quaternionic [14] N × N random matrices takes the form Y Y 2 1 |λi − λj |β e−βN |λk | dλk ; (1.4) dµ(N ) = (N ) Q (β) 1≤i<j≤N

1≤k≤N

see also [28]. Clearly, this corresponds to (1.2), (1.3), where G(λ1 , λ2 ) = − ln |λ1 − λ2 | is the free space Green’s function (whence F ≡ 0) and V (λ) = |λ|2 a quadratic potential. The eigenvalues for real symmetric, complex Hermitian, and Hermitian selfdual quaternionic matrices are real. Therefore, the charges are confined to R, i.e. 3 = R. The parameters have the values β = 1, 2, 4 and κ = 1, 2, 2, respectively. For each of the associated ensembles of unitary matrices the charges are confined to the unit circle, i.e. 3 = S1 . Then β is unmodified but κ = 0 [13]. For general complex random matrices, the eigenvalues are complex, corresponding to unconfined charges, i.e. 3 = R2 = C, and β = 2, κ = 2 [20]. The joint eigenvalue distribution for the general real quaternionic N × N matrices takes the more complicated form [20] dµ(N ) =

1

Y

Q(N ) (β)

1≤i<j≤N

|λi −λj |β |λi −λ∗j |β

Y

|λk −λ∗k |β e−βN |λk | dλk (1.5) 2

1≤k≤N

with β = 2, κ = 1, 3 = R2 = C. Although apparently not noticed previously, (1.5) can be interpreted as a configurational canonical Coulomb ensemble, with a Hamiltonian (1.3) in which now G(λ1 , λ2 ) = − ln |λ1 − λ2 | − ln |λ1 − λ∗2 | is the Green’s function of −1 for the upper half space R × R+ = C+ equipped with a perfectly dia-electric condition at its boundary ∂C+ (the real axis) and asymptotic free conditions at infinity, extended symmetrically to R2 , having a regular part given by F (λ) = − ln(2|Im(λ)|). Moreover, V (λ) = |λ|2 . This Hamiltonian describes N Coulomb point charges interacting via the free space Green’s function −ln |λ − λ0 | amongst each other and also with N identical image charges with respect to the line Im(λ) = 0. Since the interaction of a charge with its own image contributes only an amount F/2 to the Hamiltonian, F/2 + N V is now to be counted as the external potential. For symmetric random matrices, properties of the eigenvalue statistics have been computed in great detail for arbitrary N , using explicit expansion techniques [28, 27,

Eigenvalue Density of Random Matrices

685

20], group theoretical methods [13], the method of orthogonal polynomials [28, 30, 4, 9], as well as some more recent developments in soliton theory and two-dimensional quantum gravity [1, 2]. The beautiful connection to two-dimensional Coulomb systems suggests to use the general methods of statistical mechanics when the above algebraic methods fail. Even in the exactly solvable situations the statistical mechanics approach may provide us rather readily with certain relevant asymptotic (N → ∞) results which to extract from the exact finite N solutions would require quite tedious and lengthy computations. Furthermore, as Dyson has pointed out [13], in the framework of statistical mechanics the limit N → ∞ makes sense for arbitrary β and not only for the discrete values of β = 1, 2, 4. Thereby one achieves a “thermodynamic view point” which yields valuable new insights into random matrices. Early results in this direction are in [13, 40], and more recent ones in [8, 17, 18]. The prime example of the exploitation of the Coulomb analogy is Wigner’s [39] electrostatic derivation of his semi-circle law ( (2/π)(1 − λ2 )1/2 ; |λ| < 1 (1.6) ρ(λ) = 0; |λ| ≥ 1 for the eigenvalue density ρ(λ) in the limit N = ∞ of the Gaussian ensemble (1.1) in the case of real symmetric random matrices. Wigner [39] argued, heuristically, that when N → ∞, the eigenvalue density for (1.4) can be obtained from a variational principle for a continuum charge density ρ(λ) of total charge 1, restricted to the real line, that satisfies the requirement of mechanical force balance between its own electrostatic force field and the applied force field −∂λ |λ|2 = −2λ. Previously [37] he had proved, by the method of moments, that (1.6) holds true for a Bernoulli ensemble of bordered random sign matrices, which suggested that (1.6) was the limiting law under more general circumstances [39]. In [38, 39] he announced that (1.6) can indeed be proved to hold for a wider class of ensembles under a mild set of conditions, including the Gaussian real symmetric ensembles. Interestingly in itself, entropy plays no role in Wigner’s variational principle, which is concerned only with the ground state energy of the classical continuum Coulomb fluid. To get an intuitive idea how this can arise from the canonical measure (1.4) with fixed β (< ∞), we rewrite (1.4) as Y N X X 1 1 (N ) 2 = (N ) ln |λi −λj |− |λk | dλk . (1.7) exp βN dµ Q (β) N 1≤i<j≤N

k=1

1≤k≤N

Apparently, the limit N → ∞ for (1.7) is a simultaneous thermodynamic and zerotemperature limit for an unstable Hamiltonian with mean-field scaling. If in (1.7) we replace βN by βN0 , with N0 fixed, and then let N → ∞, we obtain the variational principle of [6,23] for a continuum free energy functional at inverse mean-field temperature βM F = βN0 , see also [26]. Letting N0 → ∞ subsequently, the entropy contribution to the free energy drops out, giving formally Wigner’s variational principle. With a leap of faith one may thus expect that the limit N → ∞ in (1.7) will give the same result directly. Recently, Boutet de Monvel, Pastur, and Shcherbina [5] have studied the limit N → ∞ of the joint eigenvalue distribution of real symmetric random matrix ensembles (1.1) for a large class of V (M ), satisfying certain regularity conditions. They prove that in the limit N → ∞ the nth marginal measure of (1.4), with λ2 replaced by V (λ), factors into an n-fold tensor product of identical one-particle measures whose density is precisely

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M. K.-H. Kiessling, H. Spohn

that of the two-dimensional Coulomb fluid restricted to a line, satisfying the equations of electrostatic equilibrium in the applied potential field V . The proof in [5] is based on the classical Stieltjes transform. It is carried out explicitly for β = 1, but the method covers all β > 0 (in particular, including β = 2, 4), as well as certain H¨older continuous many-body interactions. We here generalize this result in [5] to a wider class of interactions and an arbitrary (finite) number of space dimension. Our result covers also the limit N → ∞ of the eigenvalue distribution of Ginibre’s complex and real quaternionic Gaussian random matrix ensembles [20]. However, our method is very different from that in [5]. Instead of using the Stieltjes transform, we adapt the strategy of Messer and Spohn [26], Kiessling [23] and Caglioti et al. [6] to the combined mean-field and zero temperature limit. Interestingly enough, we can work without the detailed control of [5] on the finiteN marginal densities, and in this sense our proof of the variational principle is also considerably shorter and simpler than the proof in [5]. 2. Main Result We now prepare the statements of our main result. Let 3 ⊂ Rd be closed and connected, and let dx denote uniform measure on 3. Note that 3 may be all of Rd , or it may be a lower dimensional manifold, e.g. the sphere Sd−1 . We denote by P (3) the probability measures on 3, and by P s (3N ) the permutation symmetric probability measures on the infinite Cartesian product 3N . We recall the decomposition theorem of de Finetti [16] and Dynkin [12] (see also [21, 15]), which states that µ ∈ P s (3N ) is uniquely presentable as a linear convex superposition of product measures, i.e., for each µ ∈ P s (3N ) there exists a unique probability measure ν(d%|µ) on P (3), such that for each n ∈ N, Z n ν(d%|µ)%⊗n (dn x), (2.1) µn (d x) = P (3)

where µn denotes the nth marginal measure of µ. This is also the extremal decomposition for the convex set P s (3N ), see [21]. To establish the limit N → ∞ for measures of the form (1.2), (1.3) only some general properties of the interactions enter. Also, the particular value of β plays no role, and it is convenient to absorb it into the Hamiltonian. Therefore, we study the sequence of probability measures µ(N ) dN x =

Y 1 exp −H (N ) (x1 , ..., xN ) dxk (N ) Q

(2.2)

1≤k≤N

for Hamiltonians of the form H (N ) (x1 , ..., xN ) =

X

w(xi , xj ) +

1≤i<j≤N

X

u(xi ) + N v(xi ).

(2.3)

1≤i≤N

The pair interaction w and one-particle potentials u, v satisfy the following conditions. Condition on w(x, y): (C1)

Symmetry : w(x, y) = w(y, x).

Eigenvalue Density of Random Matrices

687

Condition on U (x) = u(x) + v(x): (C2)

Integrability : e−U (x) ∈ L1 (3, dx).

Conditions on W (x, y) = w(x, y) + v(x) + v(y): (C3)

Lower semicontinuity : W is l.s.c. on 3 × 3.

(C4)

Integrability : W (x, y) ∈ L1 32 , e−U (x) dx ⊗ e−U (y) dy .

In case of an unbounded 3 we also need a growth condition at infinity. Let W− (x) ≡ miny W (x, y). (C5)

Confinement : lim W− (x) = ∞, uniformly in x. |x|→∞

Theorem. Let 3 ⊂ Rd be closed and connected, and let w, v and u satisfy the conditions (C1) − (C5). Consider (2.2) as extended to a probability on 3N . Then there exists a 0 µ ∈ P s (3N ) such that, after extraction of a subsequence µ(N ) , 0

lim µ(N ) = µ.

N 0 →∞

(2.4)

For each limit point µ, the decomposition measure ν(d%|µ) is concentrated on the subset of P (3) which consists of the probability measures % that minimize the functional E(%) =

1 ⊗2 % (W ) 2

(2.5)

over P (3). In general we have little information on the decomposition measure ν(d%|µ). More is known for regular mean-field Hamiltonians, see [25]. However, if it can be shown, as is the case for many random matrix ensembles, that (2.5) has a unique minimizer, say %0 , then in (2.4) we have in fact convergence, and the limit is of the form µn = %⊗n 0 .

(2.6)

As discussed in [35], the factorization property (2.6) is equivalent to a law of large numbers. Consider the eigenvalues averaged over some continuous test function f , hf iN ≡

N 1 X f (λj ). N

Then (2.6) implies that, for all such f , lim hf iN =

N →∞

(2.7)

j=1

Z 3

%0 (dλ)f (λ)

(2.8)

in probability. We summarize as Corollary. If (2.5) has a unique minimizer, %0 , then the weak law of large numbers (2.8) holds for all f ∈ C 0 (3). Applications of our theorem to random matrix ensembles are presented in the concluding section.

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3. Proof of the Theorem The proof looks somewhat technical because we work under fairly minimal assumptions on W . But, in essence, we only have to show, through sharp upper and lower bounds, that the pair-specific free energy converges to the minimum continuum energy. The remainder of the theorem follows from the permutation invariance of the measures. We define the absolutely continuous (w.r.t. dx) a-priori probability measure on 3, µ0 (dx) = Z0−1 e−U (x) dx.

(3.1)

is defined by For each %(N ) ∈ P (3N ) its entropy w.r.t. µ⊗N 0 ! Z (N ) d% %(N ) (dN x) ln S (N ) %(N ) = − dµ⊗N 3N 0

(3.2)

⊗N if %(N ) is absolutely continuous w.r.t. a-priori measure µ0 , and provided the integral (N ) (N ) % = −∞. in (3.2) exists. In all other cases, S

Lemma 1. The relative entropy (3.2) is non-positive, S (N ) %(N ) ≤ 0.

(3.3)

Proof of Lemma 1. A standard convexity argument, see [15].

We introduce the symmetric Hamiltonian K (N ) (x1 , ..., xN ) =

1 2

X

W (xj , xk ).

(3.4)

1≤j6=k≤N

For %(N ) ∈ P (3N ), (β×) its Helmholtz free energy (or just free energy) is now defined by (3.5) F (N ) %(N ) = %(N ) K (N ) − S (N ) %(N ) if the right side exists, and by F (N ) %(N ) = +∞ in all other cases. Lemma 2. The free energy (3.5) takes its unique minimum at the probability measure µ(N ) (dN x) =

1 Z (N )

N exp −K (N ) (x1 , ..., xN ) µ⊗N 0 (d x),

(3.6)

where Z (N ) = Q(N ) /Z0N . Thus, min

%(N ) ∈P (3N )

F (N ) %(N ) = F (N ) µ(N ) = − ln Z (N ) .

(3.7)

Proof of Lemma 2. The variational principle is verified by a standard convexity estimate [15,32], which shows that F (N ) %(N ) − F (N ) µ(N ) ≥ 0, with equality holding if and only if %(N ) = µ(N ) . The second identity in (3.7) is verified by explicit calculation. Notice that the canonical probability measure (3.6) is just (2.2) rewritten in terms of K (N ) and µ0 . Lemma 3. The functional E(%) has a finite minimum over P (3).

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Proof of Lemma 3. For bounded 3 the lower semicontinuity (C3) establishes that E(%) is a lower semicontinuous functional for the topology of measures in P (3). For an unbounded region, the same conclusion holds because of (C3) and (C5). Now the claim follows from standard facts about lower semicontinuous functionals [32]. In the following, let %0 denote a minimizing probability measure for (2.5), and let E0 = E(%0 ). Lemma 4. The pair specific free energy is bounded above by lim sup −N −2 ln Z (N ) ≤ E0 . N →∞

(3.8)

We will prove Lemma 4 in two blocks, distinguishing minimizers %0 with finite entropy from those with negative infinite entropy. For instance, Wigner’s semicircle density (1.6) has finite entropy, but other ensembles can have a singular measure as eigenvalue “density”. Proof of Lemma 4. Assume first that the entropy of %0 is finite, i.e. (recalling Lemma 1), S (1) (%0 ) = S0 ∈ (−∞, 0].

(3.9)

Notice that the non-positive constant S0 is N -independent. We then can use Lemma 2 to estimate for all N that −1 (3.10) E(%0 ) − N −1 S (1) (%0 ) . −N −2 ln Z (N ) ≤ N −2 F (N ) (%⊗N 0 )= 1−N The identity in (3.10) follows from (C1). By (3.9), S (1) (%0 ) = S0 , and S0 is finite and N -independent, and by Lemma 3, E(%0 ) = E0 is finite and N -independent. Thus (3.8) in the finite entropy case follows by taking N → ∞ in (3.10). Assume now that (3.9) is false, so that S (1) (%0 ) = −∞. In that case, (3.10) becomes useless. Now, since the C0∞ functions are dense in P (3), the obvious way out is to modify the above argument and to work with a regular approximation to %0 . However, since E(%) is only lower semicontinuous, we have to employ also a continuous approximation to E(%). By (C3) for bounded 3, and by (C3), (C5) in case of an unbounded 3, W is the pointwise upper limit of a continuous increasing map γ 7→ Wγ ∈ C 0 (3 × 3) that is uniformly bounded below, see [32]. By (C1), we can assume that Wγ (x, y) = Wγ (y, x). ) dN x In the following, let Kγ(N ) be defined by (3.4) with W replaced by Wγ , and let µ(N γ and Zγ(N ) , respectively Fγ(N ) %(N ) , be defined by (3.6), respectively (3.5), with K (N ) replaced by Kγ(N ) . Since W satisfies (C3), and Wγ is of class C 0 , and Wγ % W pointwise, for each positive 1 we can find a γ such that, simultaneously,

and

1 1 E0 − %⊗2 0 (Wγ ) < 2 3

(3.11)

1 lim sup −N −2 ln Z (N ) ≤ lim sup −N −2 ln Zγ(N ) + 3 N →∞ N →∞

(3.12)

whenever γ ≥ γ . Moreover, since Wγ is bounded below on 3 × 3 uniformly in γ, and Wγ ∈ C 0 (3 × 3), for each γ we can find a measure % ∈ P (3) that is equivalent to a positive function of class C0∞ (3), such that

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M. K.-H. Kiessling, H. Spohn

1 ⊗2 % (Wγ ) − 1 %⊗2 (Wγ ) < 1 . 0 3 2 2

(3.13)

On the other hand, given any γ and any %δ ∈ P (3) of class C0∞ , we have the estimate ) Fγ(N ) %(N ) − ln Zγ(N ) = Fγ(N ) µ(N = min γ ⊗N

≤ Fγ(N ) %δ

%(N ) ∈P (3N )

1 (Wγ ) + N S (1) (%δ ), = N (N − 1) %⊗2 2 δ

(3.14)

where the first line is the analog of Lemma 2, the inequality obvious, and the last identity an explicit computation, using the symmetry of the Wγ . In particular, for any given we can choose γ = γ and %δ = % in (3.14), then multiply (3.14) by N −2 and take the limsup. Since % is of class C0∞ , we have |S (1) (% )| = C() < ∞, independent of N , whence N −1 S (1) (% ) → 0. Next we use (3.11), (3.12), (3.13), and the triangle inequality, and conclude 1 1 lim sup −N −2 ln Z (N ) ≤ %⊗2 (Wγ ) + ≤ E0 + , 2 3 N →∞

(3.15)

for arbitrarily small . This proves (3.8) for the infinite entropy case. The proof of Lemma 4 is complete. Lemma 5. The sequence N 7→ µ(N ) given by (3.6) is compact for bounded 3, and tight for unbounded 3. Proof of Lemma 5. If 3 is bounded then it is also compact, for 3 ⊂ Rd is closed, and in that case P (3) is compact for the topology of measures. By Tychonov’s theorem, P s (3N ) is now compact in the product topology. Hence, for bounded 3, the sequence (3.6) is compact. If 3 is unbounded, we have to estimate the contribution from outside of BRn to ) of (3.6), when N → ∞. Recall that a sequence the mass of the nth marginal µ(N n (N ) of probability measures µn is tight if for each 1 there exists a R = R() such ) n that µ(N n (BR ) > 1 − , independent of N , see [11]. Since our marginal measures m−n ) n (N ) n ) for m > n) and permutation are compatible (i.e., µ(N n (d x) = µm (d x ⊗ 3 symmetric by (C1), it suffices to prove tightness for any particular n. We pick n = 2. We notice that by (C3) and (C5), min

(x,y)∈3×3

W (x, y) = W0 > −∞.

(3.16)

Since (3.6) is invariant under the transformation W → W + C, we can even assume, without loss of generality, that W0 > 0. We then have the following sandwich bounds, independent of N , ) ⊗2 (3.17) 0 < µ(N 2 (W ) ≤ µ0 (W ), with µ⊗2 0 (W ) < ∞, by (C4). The lower bound in (3.17) is obvious, for W0 > 0. To prove the upper bound in (3.17) we use the strategy of [23]. We can replace K (N ) by αK (N ) in (3.6), with 0 ≤ α < ∞, so that −2N −2 (1 − N −1 ) ln Z (N ) = 0N (α) is now a function of α. Clearly, 0N (0) = 0, and W ≥ 0 implies 0N (α) ≥ 0 as well as 00N (α) ≥ 0, while the Cauchy–Schwarz inequality implies 000N (α) ≤ 0. Moreover, Jensen’s inequality, applied w.r.t. µ⊗N 0 , and (C4) imply

Eigenvalue Density of Random Matrices

691

0N (α) ≤ αµ⊗2 0 (W ).

(3.18)

Obviously µ⊗2 0 (W ) is N -independent. Thus, 0N (α) is a nonnegative, increasing, concave real function, bounded above by (3.18), and mapping zero into itself. A simple geometrical argument now reveals that the slope of any tangent to 0N (α) never exceeds (N ) 0 the slope of the ray on the r.h.s. of (3.18), i.e., 00N (α) ≤ µ⊗2 0 (W ). But 0N (1) = µ2 (W ), which proves the right inequality in (3.17). Now pick 1 arbitrary. By (C5) we can find a R = R() such that inf

(x,y) 6∈ B2R

W (x, y) ≥

1 ⊗2 µ (W ). 0

(3.19)

Let χ denote the characteristic function of the complement of BR in 3. We then have the chain of estimates (N ) (N ) ) W χ⊗2 ≥ inf W (x, y) µ(N χ⊗2 µ⊗2 0 (W ) ≥ µ2 (W ) ≥ µ2 2

1 ) (W ) 1 − µ(N BR2 ≥ µ⊗2 2 0

(x,y) 6∈ B2R

(3.20)

.

Division of (3.20) by −1 µ⊗2 0 (W ) and a simple rewriting reveals that, independent of N, ) BR2 ≥ 1 − , (3.21) µ(N 2

which was to be shown. The proof is complete.

Lemma 6. The pair specific free energy is bounded below by lim inf −N −2 ln Z (N ) ≥ E0 .

(3.22)

Proof of Lemma 6. By Lemma 1, S (N ) (µ(N ) ) ≤ 0. Therefore, − ln Z (N ) ≥ µ(N ) K (N ) .

(3.23)

N →∞

By (C1),

1 (N ) (N ) µ K = N2

1 1− N

1 (N ) µ (W ). 2 2

(3.24)

0

Now pick a converging subsequence of (3.6), µ(N ) * µ ∈ P s (3N ). Such a converging subsequence exists by Lemma 5 and the Bolzano–Weierstrass theorem. Then, by (C3), we have 0 (3.25) lim inf µ2(N ) (W ) ≥ µ2 (W ), 0 N →∞

while 1 − N

0 −1

→ 1 trivially. Thus, lim inf −N −2 ln Z (N ) N →∞

≥

1 µ2 (W ). 2

Finally, using the representation (2.1), we see that Z 1 ν(d%|µ) E(%) ≥ E(%0 ), µ2 (W ) = 2 P (3) and the proof of Lemma 6 is complete.

(3.26)

(3.27)

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M. K.-H. Kiessling, H. Spohn

Proof of the Theorem. By Lemma 4 and Lemma 6, lim −N −2 ln Z (N ) = E0 .

(3.28)

Recalling (3.26) and (3.27), we see that (3.28) implies Z ν(d%|µ) E(%) = E(%0 )

(3.29)

N →∞

P (3)

for every limit point µ of µ(N ) . Equation (3.29) in turn implies that the decomposition measure ν(d%|µ) is concentrated on the minimizers of E(%), for assume not, then Z ν(d%|µ) E(%) > E(%0 ), P (3)

which contradicts (3.29). The proof of the theorem is complete.

We are now also in the position to vindicate our remark on the existence of the limit in (2.4) in case the minimizer %0 is unique. Indeed, in that case the set of limit points of {µ(N ) , N = 1, 2, . . . } consists of the single measure. 4. Applications With the specifications in (2.2), (2.3) that x = λ ∈ 3 ⊂ R2 , v(λ) = βV (λ), u(λ) = βF (λ), and w(λ, η) = βG(λ, η), where G is a Green’s function for −1 in 2D and F its regular part, our theorem characterizes the limit N = ∞ of (1.2), (1.3), which for β = 1, 2, 4 is the joint eigenvalue distribution of various random matrix ensembles of the form (1.1). The decomposition measure of the limit is concentrated on the ground state(s) of the electrostatic energy functional ε(%) = β −1 E(%) of a charged continuum fluid with “charge density” d%/dλ (which may be a singular measure) of total charge 1, subject to an external potential V . Explicitly, the energy functional reads ε(%) =

1 ⊗2 % (G) + %(V ). 2

(4.1)

The regular part, F , of G does not contribute to the limit. We list a few examples. 4.1. Real symmetric, complex Hermitian, and quaternionic self-dual Hermitian matrices. As mentioned in the introduction we have 3 = R, G(λ, η) = − ln |λ − η|, β = 1, 2, 4, and κ = 1, 2, 2, respectively. Our electrostatic variational principle (VP) for (4.1) then becomes the VP of Boutet de Monvel et al. [5], but here with a slightly wider class of potentials V . In particular, for β = 1 we can allow continuous V with V (λ) ∼ (1+) ln |λ| asymptotically, as compared to H¨older continuous V with V (λ) ∼ (2 + ) ln |λ| in [5]. The VP has been studied extensively in [33]. A unique minimizer is known to exist under certain regularity conditions on V . Of course, for V (λ) = |λ|2 , the quadratic potential of the Gaussian ensembles, the minimizer of (4.1) is given by Wigner’s semicircle law (1.6). 4.2. General complex matrices. We have 3 = R2 , G(λ, η) = − ln |λ − η|. In this case our variational principle for (4.1) generalizes the VP of [5] to two-dimensional domains.

Eigenvalue Density of Random Matrices

693

Under mild conditions on V , and in particular for all our examples, it can be shown that the minimizer is unique. We consider only the Gaussian ensemble with κV (M ) = M † M in (1.1), whence V (λ) = |λ|2 /2 in (1.3), and β = 2 in (1.2). The minimizer of (4.1) is given by d%0 = π −1 χB1 (λ) dλ,

(4.2)

where χB1 (λ) is the indicator function of the unit disk B1 in R2 . This result can also be obtained from Ginibre’s exact finite N formula, see [20]. 4.3. Normal complex matrices. We have 3 = R2 , G(λ, η) = − ln |λ − η|, and β = 2. Consider first (1.1) with κV (M ) = ln(1 + M † M )1+1/N . Then in (1.3) we have V (λ) = − ln[πρC |λ|]1/2 , where ρC (ξ) = π −1 (1 + ξ 2 )−1 is the density of the Cauchy distribution. With these identifications (C5) is violated, but (1.2) is well defined for all β > 1 and the minimizer of the electrostatic energy functional is found to be d%0 = π −1 (1 + |λ|2 )−2 dλ.

(4.3)

The measure (4.3) has geometrical significance. Recall that |J|2 (λ) = 4/(1 + |λ|2 )2 is the Jacobian of the stereographic projection map S2 → R2 , arranged such that the equator of S2 coincides with the unit circle in R2 . Therefore, (4.3) is the stereographic projection onto the Euclidean plane of the uniform probability measure on S2 . Also the finite-N measure (1.2), with (1.3) specified as above, is itself a stereographic projection onto Euclidean space of a canonical ensemble measure of N point charges in the twosphere S2 . For β = 2, this spherical ensemble is given by (1.2) with Hamiltonian X ln |λj − λk |, (4.4) H (N ) (λ1 , ..., λN ) = − 1≤j
where λj ∈ S2 , |λj − λk | is the chordal distance on S2 , and dλk in (1.2) now means uniform measure on S2 . (If β 6= 2, a non-constant one-particle potential has to be added to H (N ) .) Our theorem applies directly to this ensemble on (S2 )×N . We arrive at (4.3) by taking the limit N → ∞ on the sphere, arguing that the minimizer is a constant, and projecting the result onto the Euclidean plane. In a sense, the spherical ensemble is the counterpart on S2 to Dyson’s circular ensembles on S1 , although these ensembles are related to random matrices in a different manner. The statistical mechanics of the spherical ensemble has been studied in some detail in [7, 19], using exact algebraic techniques. The spherical ensemble is also related to J.J. Thomson’s celebrated problem of determining the minimum energy configuration of N point charges on S2 [36], which recently has received much attention in physics [3], topology and Knot theory [24]. As a second example, consider entries restricted by kM k∞ ≤ 1 for every M , with uniform distribution otherwise. This corresponds to V (M ) = (1 + tanh[t (I − M M † )])−1 as t → ∞. Alternatively, we can choose 3 = B1 and V = 0 in (1.2). The unique minimizer of the corresponding variational principle is the electrostatic charge distribution on a circular perfect conductor. It is well known that any surplus charge accumulates at the “surface”, i.e., the minimizer is given by a Dirac mass concentrated uniformly on the boundary of the unit disk, d%0 = δS1 (dx) . With the help of our corollary this gives us the following.

(4.5)

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Proposition. Let hf iN be defined as in (2.7), the summation running over the eigenvalues, restricted to B1 ⊂ C, of a complex normal N × N random matrix whose free entries R 2π are uniformly distributed otherwise. Then, in probability, hfiN → (2π)−1 0 f eiϕ dϕ as N → ∞. This result is worth rephrasing in less technical terms. As far as averages over the spectrum are concerned, an infinite normal random matrix with eigenvalues in the unit disk and independent entries uniformly distributed otherwise, is almost surely equivalent to some infinite unitary matrix. 4.4. Real quaternion matrices. We consider only the Gaussian ensemble. The joint probability density is given by (1.5) with 3 = R2 , β = 2, κ = 1, [20,28]. The limiting eigenvalue distribution now minimizes the slightly more complicated electrostatic energy functional ε(%) =

1 ⊗2 − ln |λ − η| − ln |λ − η ∗ | + |λ|2 + |η|2 , % 2

(4.6)

where η ∗ is the mirror image of η with respect to the real axis. The minimizer of (4.6) is, once again, unique and given by the measure (4.2). Acknowledgement. We thank the referee for a careful reading of the manuscript, and L. Pastur and P. ZinnJustin for valuable comments. The work of M.K. was supported by NSF Grant # DMS 96-23220.

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14. Dyson, F.J.: A Brownian motion model for the eigenvalues of a random matrix. J. Math. Phys. 3, 1191– 1198 (1962) 15. Ellis, R.S.: Entropy, large deviations, and statistical mechanics. New York: Springer-Verlag, 1985 16. de Finetti, B.: Funzione caratteristica di un fenomeno aleatorio. Atti della R. Accad. Naz. dei Lincei, Ser. 6, Memorie, Classe di Scienze Fisiche, Matematiche e Naturali 4, 251–300 (1931) 17. Fogler, M.M., and Shklovskii, B.I.: Probability of an eigenvalue fluctuation in an interval of a random matrix spectrum. Phys. Rev. Lett. 74, 3312–3315 (1995) 18. Forrester, P.J.: Fluctuation formula for complex random matrices. Preprint, Univ. Melbourne (1998) 19. Forrester, P.J., and Jancovici, B.: The two-dimensional two-component plasma plus background on a sphere: exact results. J. Stat. Phys. 84, 337–357 (1996) 20. Ginibre, J.: Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6, 440–449 (1964) 21. Hewitt, E., and Savage, L. J.: Symmetric measures on Cartesian products. Trans. Am. Math. Soc. 80, 470–501 (1955) 22. Hsu, P.L.: On the distribution of roots of certain determinantal equations. Ann. Eug. 9, 240–258 (1939) 23. Kiessling, M. K.-H.: Statistical mechanics of classical particles with logarithmic interactions. Commun. Pure Appl. Math. 47, 27–56 (1993) 24. Kusner, R.B., and Sullivan, J.M.: M¨obius energies for knots and links, surfaces and submanifolds. Preprint (1995) 25. Kusuoka, S., and Tamura, Y.: Gibbs measures for mean field potentials. J. Fac. Sci. Univ. Tokyo, Sec. IA, Math. 31, 223–245 (1984) 26. Messer, J., and Spohn, H.: Statistical mechanics of the isothermal Lane-Emden equation. J. Stat. Phys. 29, 561–578 (1982) 27. Mehta, M.L., and Srivastava, P.K.: Correlation functions for eigenvalues of real quaternian matrices. J. Math. Phys. 7, 341–344 (1966) 28. Mehta, M.L.: Random Matrices. 2nd ed., New York: Acad. Press, 1991 29. K. Johansson: On the fluctuations of eigenvalues of random hermitian matrices. Duke Math. J., 91, 151–204 (1998) 30. Pastur, L., and Shcherbina, M.: Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles. J. Stat. Phys. 86, 109–147 (1997) 31. Porter, C.E. (Ed.): Statistical theories of spectra: fluctuations. New York: Academic Press, 1965 32. Ruelle, D.: Statistical Mechanics: Rigorous Results. Reading, MA: Addison Wesley (1989) 33. Saff, E.B., and Totik, V.: Logarithmic potentials with external fields. Grundl. d. Math. Wiss. vol. 316, Berlin–Heidelberg–New York: Springer, 1997 34. Sinai, Ya., and Soshnikov, A.: Central limit theorem for traces of large random symmetric matrices with independent matrix elements. Preprint, Princeton Univ. (1997) 35. Spohn, H.: Large Scale Dynamics of Interacting Particles. Texts and Monographs in Physics, Berlin– Heidelberg–New York: Springer, 1991 36. Thomson, J.J.: Philos. Mag. 7, 237 (1904) 37. Wigner, E.: Characteristic vectors of bordered matrices with infinite dimension. Part I, Ann. Math. 62, 548–564 (1955]; Part II, ibid. 65, 203–207 (1957) 38. Wigner, E.: On the distribution of the roots of certain symmetric matrices. Ann. Math. 67, 325–327 (1958) 39. Wigner, E.: Statistical properties of real symmetric matrices with many dimensions. 4th Can. Math. Congress (Banff 1957), Proc., Toronto: Univ. Toronto Press, 1959, pp. 174–184 40. Wilson, K.G.: Proof of a conjecture by Dyson. J. Math. Phys. 3, 1040–1043 (1962) Communicated by Ya. G. Sinai

Commun. Math. Phys. 199, 697 – 728 (1999)

Communications in

Mathematical Physics © Springer-Verlag 1999

Deformed WN Algebras from Elliptic sl(N ) Algebras J. Avan1 , L. Frappat2 , M. Rossi2 , P. Sorba2 1 2

LPTHE, CNRS-URA 280, Universit´es Paris VI/VII, France Laboratoire de Physique Th´eorique LAPTH, URA 1436, CNRS and Universit´e de Savoie, France

Received: 23 January 1998 / Accepted: 8 June 1998

Abstract: We extend to the sl(N ) case the results that we previously obtained on the b c ). The elliptic algebra construction of Wq,p algebras from the elliptic algebra Aq,p (sl(2) b Aq,p (sl(N )c ) at the critical level c = −N has an extended center containing trace-like operators t(z). Families of Poisson structures indexed by N (N − 1)/2 integers, defining q-deformations of the WN algebra, are constructed. The operators t(z) also close an 1

exchange algebra when (−p 2 )N M = q −c−N for M ∈ Z. It becomes Abelian when in addition p = q N h , where h is a non-zero integer. The Poisson structures obtained in these classical limits contain different q-deformed WN algebras depending on the parity of h, characterizing the exchange structures at p 6= q N h as new Wq,p (sl(N )) algebras. R´esum´e: Nous e´ tendons au cas sl(N ) les r´esultats que nous avons obtenus pr´ec´edemment concernant la construction des alg`ebres Wq,p a` partir de l’alg`ebre elliptique b c ). L’alg`ebre elliptique Aq,p (sl(N b )c ) au niveau critique c = −N poss`ede un Aq,p (sl(2) centre e´ tendu contenant des op´erateurs de trace t(z). On construit sur ce centre des familles de structures de Poisson indic´ees par N (N − 1)/2 entiers, d´efinissant des qd´eformations de l’alg`ebre WN . Les op´erateurs t(z) engendrent une alg`ebre d’´echange 1

lorsque (−p 2 )N M = q −c−N o`u M ∈ Z. Cette alg`ebre devient ab´elienne si de plus p = q N h avec h entier non nul. Les structures de Poisson obtenues dans ces limites classiques contiennent diff´erentes alg`ebres WN q-d´eform´ees d´ependant de la parit´e de l’entier h, caract´erisant les structures d’´echange a` p 6= q N h comme de nouvelles alg`ebres Wq,p (sl(N )). 1. Introduction b c ) was introduced in [1] and further studied in [2, 3, 4] The elliptic algebra Aq,p (sl(2) where several trigonometric limits were derived and shown to be relevant as symmetries

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b c ) was proposed as basic symmetry algebra for the for the XXZ model, whereas Aq,p (sl(2) b c) XYZ model [5, 6]. More recently another, possibly related elliptic algebra Uq,p (sl(2) was introduced [7] in connection with the k-fusion RSOS model [8, 9]. A dynamical b c ), making use of the dynamical “eight-vertex” elliptic version of the algebra Aq,p (sl(2) R-matrix [10, 11], was also recently proposed in [12]. The domain was unified recently b c ) algebra (and its extension to sl(N )) and Felder’s el[13, 14] where both the Aq,p (sl(2) liptic dynamical algebra [11, 15] were shown to be obtained from application of universal b c ). As a consequence twisting operators (see also [16]) to the quantum algebra Uq (sl(2) the construction [1] was validated and extended to the sl(N ) case. The q-Virasoro algebra was introduced in [17] as an extension to the Ruijsenaar– Schneider model of the Virasoro algebra arising in the collective theory formulation of the Calogero–Moser model [18, 19, 20]. It arose simultaneously as quantization of a classical quadratic Poisson structure on the center at c = −2 of the quantum affine algebra Uq (sl(2)c ) [21]. It was shown [22, 23] to be the symmetry algebra of the restricted ABF b c ) algebra. On the model, itself connected as previously said to the elliptic Uq,p (sl(2) other hand, the limit of the q-Virasoro current coincides [22, 2] with the field obtained b c ), hinting by concatenation of vertex operators of the degeneracy limit of Aq,p (sl(2) indeed at a deep connection between the two structures. This identification was recently extended to the full q-deformed Virasoro case [24]. A direct connection was indeed established in two recent papers [25, 26]: we derived b c ) algebra, exchange algebras and limit Poisson structures from the original Aq,p (sl(2) m c+2 by establishing the existence of a center at p = q for any integer m. The Poisson structures all boil down to the q-Virasoro (classical) algebra defined in [17, 21]. The exchange algebras are however quite distinct from the quantization of q-Virasoro constructed in [17, 27]. Their eventual central (and linear ?) extensions also exhibit a much richer structure than was originally derived in [27] and are currently under investigation [28]. It is known that q-Virasoro structures admit extensions to the sl(N ) case as classical and quantum q-WN algebras [21, 27, 29]. These have been the object of many investigations recently [30, 31]. In particular, they were also shown to be in general symmetry algebras of RSOS models [32, 23]. It is a natural question in view of our previous results to investigate whether classical and quantized q-WN algebras arise as structures b )c ). embedded in the generalized elliptic algebra Aq,p (sl(N The plan of the paper is as follows. We first recall in Sect.2 useful properties of the elliptic functions and the elliptic sl(N ) R-matrix, and introduce the definition of b )c ) [14]. We then prove in Sect. 3 the existence the quantum elliptic algebra Aq,p (sl(N of an extended center at c = −N and compute the Poisson structures on this center, using the notion of the sector-depending mode Poisson bracket already introduced in [25]. These structures are identified as q-deformed WN algebras. They are obtained from analytic continuations of a classical algebra identical to the initial version of qWN given in [21]. In Sect. 4 we show the existence of closed (quadratic) exchange 1

algebras whenever (−p 2 )N M = q −c−N for any integer M ∈ Z. These algebras differ b )) structures introduced in [27]. They admit a classical from the quantum Wq,p (sl(N limit (commuting algebras) at p = q N h with h ∈ Z\{0}. In Sect. 5 we compute the related Poisson structures. They include for h even the Poisson structures in [21]. The exchange algebras therefore realize new quantizations of these Poisson structures. When h is odd, by contrast, this classical limit takes a form different from the initial q-WN

Deformed WN Algebras from Elliptic sl(N ) Algebras

699

b )c ) structures. This emphasizes the key role of the initial 3-parameter structure Aq,p (sl(N in allowing for an intermediate quantum q-deformed WN algebra. Finally in Sect. 6, we compute the mode expansion of the quantum exchange algebra structures. As in the classical case, a “sector-type” labeling is needed due to the singularity behaviour of the structure function viewed as an analytic continuation and therefore exhibiting different formal series expansions corresponding to different convergent series expansions in distinct domains of the complex plane. We give an explicit example of this treatment applied to the spin one field for the sake of simplicity, and we describe the essential features of the extension of our computation to higher spin fields. 2. Notations and Basic Definitions 2.1. Definition of the N -elliptic R-matrix. The N -elliptic R-matrix in End(CN ) ⊗ End(CN ), associated to the ZN -vertex model, is defined as follows [33, 34]: 1 ϑ 21 (ζ, τ ) e q, p) = z 2/N −2 1 1 2 R(z, κ(z 2 ) ϑ 21 (ξ + ζ, τ ) (2.1) 2 X −1 W(α1 ,α2 ) (ξ, ζ, τ ) I(α1 ,α2 ) ⊗ I(α , × 1 ,α2 ) (α1 ,α2 )∈ZN ×ZN

where the variables z, q, p are related to the variables ξ, ζ, τ by z = eiπξ ,

q = eiπζ ,

p = e2iπτ . (2.2) γ The Jacobi theta functions with rational characteristics ϑ 1 (ξ, τ ) are defined in γ2 Appendix A. The normalization factor is chosen as follows: 1 = κ(z 2 ) (q 2N z −2 ; p, q 2N )∞ (q 2 z 2 ; p, q 2N )∞ (pz −2 ; p, q 2N )∞ (pq 2N −2 z 2 ; p, q 2N )∞ . (q 2N z 2 ; p, q 2N )∞ (q 2 z −2 ; p, q 2N )∞ (pz 2 ; p, q 2N )∞ (pq 2N −2 z −2 ; p, q 2N )∞ The functions W(α1 ,α2 ) are given by

W(α1 ,α2 ) (ξ, ζ, τ ) =

1

(2.3)

+ α1 /N ϑ 21 (ξ + ζ/N, τ ) + 2 α2 /N . 1 + α1 /N N ϑ 21 (ζ/N, τ ) 2 + α2 /N

(2.4)

The matrices I(α1 ,α2 ) are defined as follows: I(α1 ,α2 ) = g α2 hα1 ,

(2.5)

where the N × N matrices g and h are given by gij = ω i δij and hij = δi+1,j , the addition of indices being understood modulo N .

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Let us set

X

S(ξ, ζ, τ ) =

(α1 ,α2 )∈ZN ×ZN

−1 W(α1 ,α2 ) (ξ, ζ, τ ) I(α1 ,α2 ) ⊗ I(α . 1 ,α2 )

(2.6)

The matrix S is ZN -symmetric, that is a+s, b+s a, b Sc+s, d+s = Sc, d

(2.7)

for any indices a, b, c, d, s ∈ ZN (the addition of indices being understood modulo N ) a, c+b and the non-vanishing elements of the matrix S are Sc, a+b . One finds explicitly: a, c+b Sc, a+b =

X

W(a−c,s) ω −bs .

(2.8)

s∈ZN

0, a+b The matrix S being ZN -symmetric, it is sufficient to examine the terms S ab ≡ Sa, b . One finds: N −1 Y k/N + 21 (b − a)/N + 21 ϑ (ξ, N τ ) 1 ϑ (ξ + ζ, N τ )

S ab (ξ, ζ, τ ) =

ϑ

1 2

−a/N +

1 2

1 2

k=0,k6=b

N −1 Y

(ζ, N τ )

ϑ

k=1

2

k/N + 1 2

1 2

(0, N τ )

. (2.9)

It satisfies the following shift properties: S ab (ξ, ζ + λτ, τ ) = exp(−2iπλξ/N ) S a−λ,b (ξ, ζ, τ ), ab

S (ξ + λτ, ζ, τ ) = exp(−iπλ τ − 2iπλ(ξ + ζ/N + 2

(2.10a)

a,b+λ 1 (ξ, ζ, τ ). 2 )) S

(2.10b)

Using the following “gluing” formula: N −1 Y k=0

ϑ

k/N + 1 2

1 2

(ξ, N τ ) = p

2 1 24 (N −1)

QN −1

(pN −k ; pN )∞ ϑ (pN ; pN )N ∞

k=0

1 2 1 2

(ξ, τ ), (2.11)

and inserting Eq. (2.9) into Eqs. (2.6) and (2.1), one then finds the following expression eab elements of the matrix (2.1): in terms of the Jacobi 2 functions for the R eab (z, q, p) = R

ab 2b 2 1 p− N q N z − N (N +a−1) 2 κ(z ) 2pN (pN +b−a q 2 z 2 ) 2pN (pN ) 2p (pq 2 ) 2p (pz 2 ) . × 2pN (pN +b z 2 ) 2pN (pN −a q 2 ) 2p (pq 2 z 2 ) 2p (p)

(2.12)

2.2. Gauge-transformed R-matrix. In order to make the comparison with our previous results easier [25, 26], one needs to introduce the following “gauge-transformed” matrix: 1 1 e q, p)(g − 21 ⊗ g − 21 ). R(z, q, p) = (g 2 ⊗ g 2 )R(z,

Theorem 1. The matrix R(z, q, p) satisfies the following properties:

(2.13)

Deformed WN Algebras from Elliptic sl(N ) Algebras

701

– Yang–Baxter equation: R12 (z) R13 (w) R23 (w/z) = R23 (w/z) R13 (w) R12 (z),

(2.14)

R12 (z) R21 (z −1 ) = 1,

(2.15)

R12 (z)t2 R21 (z −1 q −N )t2 = 1,

(2.16)

R12 (−z) = ω (g −1 ⊗ I) R12 (z) (g ⊗ I),

(2.17)

b21 (z −1 )−1 (g 21 hg 21 ⊗ I), b12 (−zp 21 ) = (g 21 hg 21 ⊗ I)−1 R R

(2.18)

b12 (z, q, p) = τN (q 21 z −1 ) R12 (z, q, p), b12 (z) ≡ R R

(2.19)

– Unitarity:

– Crossing-symmetry:

– Antisymmetry:

– Quasi-periodicity:

where

the function τN (z) being defined by τN (z) = z N −2 2

2q2N (qz 2 ) . 2q2N (qz −2 )

(2.20)

The function τN (z) is periodic with period q N : τN (q N z) = τN (z) and satisfies τN (z −1 ) = τN (z)−1 . Proof. The proof of the Yang–Baxter equation has been given in [35]. The proof of the unitarity and the crossing-symmetry is done by a direct calculation. One has to use the following two identities (the first one for the unitarity and the second one for the crossing-symmetry, see [36] for a proof of Eqs. (2.21) and (2.22)): X S −i−k,i−k (ξ, ζ, τ )S −j−k,k−j (−ξ, ζ, τ ) k∈ZN

ϑ

1 2 1 2

=

(2.21)

2

X

S

k∈ZN

ϑ =

1 (ξ + ζ, τ ) ϑ 21 (−ξ + ζ, τ ) 1 2 δij , 2 2 ϑ 1 (ζ, τ )

i−k,0

1 2 1 2

(z, ζ, τ )S j−k,0 (−ξ − N ζ, ζ, τ )

1 (z, τ ) ϑ 21 (−ξ − N ζ, τ ) 1 2 δij . 2 2 ϑ 1 (ζ, τ )

(2.22)

2

Finally, the antisymmetry and the quasi-periodicity are explicitly checked from the expressions of the matrix elements of R.

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Remark. The crossing-symmetry and the unitarity properties of R12 allow exchange of inversion and transposition for the matrix R12 as (the same property also holds for the b12 ): matrix R

R12 (x)t2

−1

t2 = R12 (q N x)−1 .

(2.23)

b )c ). We now define the elliptic quantum 2.3. The quantum elliptic algebra Aq,p (sl(N P b algebra Aq,p (sl(N )c ) [1, 14] as an algebra of operators Lij (z) ≡ n∈Z Lij (n) z n , where i, j ∈ ZN , encapsulated into a N × N matrix   L11 (z) · · · L1N (z)   .. .. (2.24) L(z) =  . . . LN 1 (z) · · · LN N (z)

b )c ) by imposing the following constraints on the Lij (z) (with the One defines Aq,p (gl(N b matrix R12 given by Eq. (2.19)): ∗ b12 b12 (z/w) L1 (z) L2 (w) = L2 (w) L1 (z) R (z/w), R

(2.25)

b∗ is defined by R b∗ (z, q, p) ≡ where L1 (z) ≡ L(z) ⊗ I, L2 (z) ≡ I ⊗ L(z) and R 12 12 b12 (z, q, p∗ = pq −2c ). This definition is the most immediate generalization to N of the R definition adopted in [1] for N = 2. b∗ obeys also the unitarity, crossing-symmetry, antisymmetry and quasiThe matrix R 12 periodicity conditions of Theorem 1 (note that the quasi-periodicity condition (2.18) for b∗ has to be understood with the modified elliptic nome p∗ ). R 12 The q-determinant q-det L(z) given by q- det L(z) ≡

X

ε(σ)

σ∈SN

N Y

Li,σ(i) (zq i−N −1 )

(2.26)

i=1

b )c ). It can (ε(σ) being the signature of the permutation σ) is in the center of Aq,p (gl(N c 2 be “factored out”, and set to the value q so as to get b )c ) = Aq,p (gl(N b )c )/hq- det L − q c2 i. Aq,p (sl(N

(2.27)

It is useful to introduce the following two matrices: c

L+ (z) ≡ L(q 2 z),

(2.28a)

L− (z) ≡ (g hg ) L(−p z) (g hg )−1 . 1 2

1 2

1 2

1 2

1 2

(2.28b)

They obey coupled exchange relations following from (2.25) and periodicity/unitarity b∗ : b12 and R properties of the matrices R 12 ∗ b12 b12 (z/w) L± (z) L± (w) = L± (w) L± (z) R (z/w), R 1 2 2 1 c c − − + + ∗ b b 2 R12 (q z/w) L1 (z) L (w) = L (w) L1 (z) R12 (q − 2 z/w). 2

2

(2.29a) (2.29b)

The parameters c, p, q in our definition are related to the corresponding parameters 2/N 1/N c0 , p0 , q 0 of [14] by c = 2c0 , p = (p0 ) , q = (q 0 ) .

Deformed WN Algebras from Elliptic sl(N ) Algebras

703

b 3. The Center of Aq,p (sl(N )c ) at the Critical Level c = −N b )c ). 3.1. Center of Aq,p (sl(N Theorem 2. At the critical level c = −N , the operators generated by c t(z) ≡ Tr(L(z)) ≡ Tr L+ (q 2 z)L− (z)−1

(3.1)

b )c ). lie in the center of the algebra Aq,p (sl(N e ± (z) ≡ (L± (z)−1 )t , one can derive from Eqs. (2.29) further exchange Proof. Defining L e− : relations between the operators L+ and L −1 t2 −1 ∗ e ± (w) = L e ± (w) L± (z) R b12 b12 (z/w)t2 L± (z) L (z/w) , (3.2a) R 1 2 2 1 −1 −1 e − (w) = L e − (w) L+ (z) R b∗ (q − c2 z/w)t2 b12 (q c2 z/w)t2 L+1 (z) L . R 1 12 2 2 (3.2b) i h Let us now compute t(z), L+ (w) . One rewrites: c e − (z)t1 L+2 (w) = Tr 1 L+1 (q c2 z)t1 L e − (z)L+2 (w) , (3.3) t(z) L+ (w) = Tr 1 L+1 (q 2 z)L 1 1 since one is allowed to exchange transposition under a trace procedure. Commuting e − (z) using Eq. (3.2b), one gets: L+2 (w) through L 1 c c ∗ b21 (q c2 w/z)t1 )−1 L+2 (w)L e − (z)R b21 (q − 2 w/z)t1 . t(z) L+2 (w) = Tr 1 L+1 (q 2 z)t1 (R 1 (3.4) b12 , one has: Using the unitarity property of R c b21 (q − c2 w/z)−1 )t1 L+ (w) = L+ (w) (R b∗ (q − c2 w/z)−1 )t1 L+ (q c2 z)t1 . L+1 (q 2 z)t1 (R 2 2 21 1 (3.5)

Then applying Eq. (2.23) to (3.5) gives: c c c ∗ b21 (wq − c2 −N /z)t1 )−1 L+2 (w) = L+2 (w) (R b21 (q − 2 w/z)−1 )t1 L+1 (q 2 z)t1 , L+1 (q 2 z)t1 (R (3.6)

which leads at the critical value c = −N to c b21 (q c2 w/z)t1 )−1 L+ (w) = L+ (w) (R b∗ (q − c2 w/z)−1 )t1 L+ (q c2 z)t1 . (3.7) L+1 (q 2 z)t1 (R 2 2 21 1

Now, inserting (3.7) into (3.4), the first three terms in (3.4) can be rearranged and one obtains: c c ∗ b∗ (q − c2 w/z)t1 . e − (z)R b21 (q − 2 w/z)−1 )t1 L+1 (q 2 z)t1 L t(z) L+2 (w) = L+2 (w) Tr 1 (R 21 1 (3.8) Using the fact that under a trace over the space 1 one has Tr1 R21 Q1 R0 21 = t2 0 t2 R21 t2 , one gets Tr 1 Q1 R21

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J. Avan, L. Frappat, M. Rossi, P. Sorba

t(z) L+2 (w)

t2 (3.9) c b∗ (q − c2 w/z)t1 t2 (R b∗ (q − c2 w/z)−1 )t1 t2 . e − (z)R = L+2 (w) Tr 1 L+1 (q 2 z)t1 L 21 21 1

The last two termsin the right hand side cancel each other, leaving a trivial dependence c t e− (z) ≡ t(z) in space 1. This shows the commutation of in space 2 and Tr 1 L+1 (q 2 z) 1 L 1 c

t(z) with L+ (w) and therefore with L− (w) = (g 2 hg 2 )L+ (−p 2 q − 2 w)(g 2 hg 2 )−1 , hence b )c ) at c = −N . with the full algebra Aq,p (sl(N 1

1

1

1

1

This demonstration reproduces the proof for N = 2 given in [25]; note that only in the sl(N ) crossing-symmetry relation (2.16) does N appear explicitly. The form of the operator (3.1) is identical to the form of the commuting operator derived by [37] in b )c ). As in [25] the center of Aq,p (sl(N b )c ) at the case of the quantum algebra Uq (sl(N c = −N may contain other generators which we have not yet derived. However, t(z) do close on their own a Poisson algebra as we are going to show. 3.2. Exchange algebra. In order to get the Poisson structure on t(z), we need to compute the exchange algebra between the operators t(z) and t(w) when c 6= −N . From the definition of the element t(z), one has e − (z)j1 L+ (q 2 w)j2 L e − (w)j2 . t(z)t(w) = L(z)ii11 L(w)ii22 = L+ (q 2 z)ji11 L i1 i2 i2 c

c

(3.10)

Suitable rewritings of the relations (2.29b) lead to the following exchange relations e− : between the operators L+ and L −1 c ∗ b21 (q c2 w/z)t1 e − (z) R b21 e − (z) L+2 (w) = R L+2 (w) L (q − 2 w/z)t1 , L 1 1 (3.11a) −1 c c e − (w) = R b12 (q 2 z/w)t2 L b∗ (q − 2 z/w)t2 e − (w) L+1 (z) R , L+1 (z) L 12 2 2 (3.11b) −1 e − (w) = R b12 (z/w)t1 t2 e − (z) R b∗ (z/w)t1 t2 . e − (w) L e − (z) L L L 12 1 2 2 1 (3.11c) The exchange relations (2.29b), (3.11c) and the properties of the matrix R12 given in e − (w) to the left of the matrices Theorem 1 then allow to move the matrices L+ (w), L + − e L (z), L (z). One obtains t(z)t(w) = Y(w/z)ij11ij22 L(w)ji22 L(z)ji11 ,

(3.12)

where Y(w/z) = T (w/z)M(w/z) with t2 t −1 −1 2 t R12 (q c z/w) 2 M(w/z) = R21 (w/z) R21 (q c+N w/z) R12 (z/w) (3.13) and τN (q 2 z/w)τN (q 2 −c w/z) 1

T (w/z) =

1

τN (q 2 w/z)τN (q 2 −c z/w) 1

1

.

(3.14)

Deformed WN Algebras from Elliptic sl(N ) Algebras

705

b )c ) at c = −N . One enounces: 3.3. Poisson structures on the center of Aq,p (sl(N Theorem 3. The elements t(z) form a closed algebra under the natural Poisson bracket b )c ) given by (with x = w/z) on the center of Aq,p (sl(N 1 1 d d {t(z), t(w)} = − ln q x ln τN (q 2 x) − x−1 −1 ln τN (q 2 x−1 ) t(z) t(w). dx dx (3.15) Proof. At the critical level c = −N , it is easy to show by direct calculation from (3.13) and (3.14) that T (x)cr = 1

and M(x)cr = I2 ⊗ I2 .

(3.16)

One recovers immediately that t(z)t(w) = t(w)t(z) at the critical level c = −N . Hence a natural Poisson structure can be defined by i1 i2 n o dY L(w)ji22 L(z)ji11 . (3.17) (w/z) t(z), t(w) = dc j1 j2 cr From Eq. (3.16), one has dT dM dY (x) = (x) I2 ⊗ I2 + (x) . dc dc dc cr cr cr

(3.18)

Now, one has  !t2  d dM = − ln q y R R12 (q −N z/w)t2  dy 21 y=w/z dc cr !t2   d − (R12 (z/w)−1 )t2 y R12 .  dy y=q −N z/w

(3.19)

Taking now the derivative of the relation (2.23), one obtains !t2 !t2 d −1 d t2 −1 t2 −1 = −y R12 , (3.20) R12 (x) y R12 R12 (x) dy dy y=x y=xq N which can be rewritten by virtue of Eq. (2.23) as: !t2 t2 d N −1 t2 R12 (xq N )−1 = y R12 R12 (xq ) dy y=x

!t2 d −1 −y R12 . dy y=xq N (3.21)

Using then Eq. (3.21) for the value x = q −N z/w and inserting it into Eq. (3.19), one finds  !t2 !t2    d d dM −1 = − ln q y + y R12 (q −N z/w)t2 . R R 21 12  dy  dc cr dy y=w/z y=z/w (3.22)

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J. Avan, L. Frappat, M. Rossi, P. Sorba

dM dY dT From the unitarity property, it follows that = 0. Hence = I2 ⊗ I2 . dc cr dc cr dc cr This guarantees that the Poisson bracket of t(z) closes on t(z), a property not obvious b )c ). since we have indicated that t(z) may not exhaust the center of Aq,p (sl(N Finally, using the q N -periodicity property of the function τN and Eq. (3.16), the derivative of T (x) is given by: 1 1 d d ln T d dT (x) = (x) = − ln q x ln τN (q 2 x) − x−1 −1 ln τN (q 2 x−1 ) . dc dc dx dx cr cr (3.23) 3.4. Explicit Poisson structures at c = −N . The structure function in Eq. (3.15) is easily computed. One obtains: {t(z), t(w)} = f (w/z) t(z) t(w), where

(3.24)

 X 2x2 q 2N ` x2 q 2N `+2 x2 q 2N `−2  − − f (x) = −2 ln q 1 − x2 q 2N ` 1 − x2 q 2N `+2 1 − x2 q 2N `−2 `≥0 x2 q 2 x2 q −2 x2 −1 1 1 + + − (x ↔ x ) . (3.25) − 1 − x2 2 1 − x2 q 2 2 1 − x2 q −2

We now define the Poisson structure for modes of the generating function derived from (3.24, 3.25). The modes of t(z) are defined in the sense of generating functions (or formal series expansions): I dz −n z t(z), (3.26) tn = C 2πiz where C is a contour encircling the origin. Mode expansions when the structure function f (x) has an infinite set of poles at x = q P (k) (where P (k) is integer) require a specific definition using the notion of “sectors”. This was done at the classical level in [25]. The procedure runs as follows: The Poisson bracket between the modes is given by a double contour integral I I dz dw −n −m f (w/z) t(w) t(z). (3.27) z w {tn , tm } = 2πiz 2πiw C1 C2 The function f (x) has here simple poles at x = ±q ±N ` and x = ±q ±N `±1 . Hence the relative position of the contours C1 and C2 must be specified in order to have an unambiguous result for (3.27). In addition, the antisymmetry of the Poisson brackets is only guaranteed at the mode level by an explicit symmetrization of (3.27) with respect to the position of the contours C1 and C2 . We shall comment on this fact when discussing the quantum mode-exchange structure. The mode Poisson bracket is thus defined as: I I I I dz dw dz dw 1 + {tn , tm } = 2 C1 2πiz C2 2πiw C2 2πiz C1 2πiw (3.28) z −n w−m f (w/z) t(z) t(w),

Deformed WN Algebras from Elliptic sl(N ) Algebras

707

where C1 and C2 are circles of radii R1 and R2 and one chooses R1 > R2 . Explicit evaluation of (3.28) now requires expressing f (w/z) as a convergent Laurent series in the appropriate domains for |w/z|. h R1 i −P (k) −P (k+1) , P (k) being the ordered ∈ |q| , |q| Let us define the sector (k) by R2 set of powers of q −1 , where the poles of f are located (here {P (k), k ∈ N} = {0, 1; N − 1, N, N + 1; . . . ; N ` − 1, N `, N ` + 1; . . . } respectively). For every sector (k), Eq. (3.28) defines a distinct mode Poisson bracket. As in the sl(2) case, one observes the difference between the analytic continuation formula (3.15), which is unique, and the formal series formula (3.28), where every klabeled convergent Laurent series expansion for f may be taken as the formal series expansion of f . This fact is also mentioned in [27], considering the quantum problem. h R1 i −1 ), one finds ∈ 1, |q| Proposition 1. In the case k = 0 (i.e. R2 {tn , tm }k=0 = −2 ln q(q − q −1 )

X [(N − 1)r]q [r]q tn−2r tm+2r , [N r]q

(3.29)

r∈Z

where the q-numbers [r]q are defined as usual: [r]q ≡

q r − q −r . q − q −1

(3.30)

The proof is immediate. When k 6= 0, one must add to (3.29) contributions arising from the poles at q −P (j) with j = 1, . . . , k. Proposition 2. The convergent series expansions in any sector (k) are obtained by adding to the coefficients of the convergent series at k = 0, coefficients obtained from the w/z)+δ(−q −P (j) w/z)− canonical formal series expansion of the distributions δ(q −P (j)P δ(q P (j) w/z) − δ(−q P (j) w/z) for j = 1, . . . , k, where δ(x) ≡ n∈Z xn for x ∈ C. Proof. Moving from a sector (k) to a sector (k + 1) requires to rewrite the only term from x2 q 2P (k) , by a convergent (3.25) whose series expansion becomes divergent, namely 1 − x2 q 2P (k) P −P (k) . This substitutes the series − r≥0 x−2r q −2P (k)r , series expansion for |x| > |q| P −P (k) convergent for |x| > |q| , to the series r>0 x2r q 2P (k)r , convergent when |x| < −P (k) . The overall result, in the |q| P full series expansion, is to “add” the difference (order by order in x2r ), namely − r∈Z x2r q 2P (k)r = −δ(x2 q 2P (k) ). A similar reasoning generates the term δ(x2 q −2P (k) ) from the x−1 terms in (3.25). ±N `

get an overall 2 factor while the terms Beware that the terms obtained at |x| = |q| ±N `±1 get an overall −1 factor. at |x| = |q| Specification of a Poisson structure in the context of a multiple-singularities structure function therefore does require going to an explicit mode expansion. 3.5. Realization of the higher spin generators. To realize deformed WN Poisson structures, we need to introduce generating functions for the higher spin objects. Having at our disposal only one commuting generating function t(z), we are led by comparison

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J. Avan, L. Frappat, M. Rossi, P. Sorba

with [21] to define shifted products, although with the same generator. Notice that such ordered shifted products were used a long time ago, to construct trigonometric and elliptic R-matrices from rational ones by the so-called “mean procedure” [38, 39]. We define accordingly: Y

(i−1)/2

si (z) =

t(q u z),

(3.31)

u=−(i−1)/2

where i = 1, . . . , N − 1. The generators s1 (z), . . . , sN −1 (z) then close a Poisson algebra with the following Poisson brackets (i, j = 1, . . . , N − 1): X

(i−1)/2

{si (z), sj (w)} =

X

(j−1)/2

u=−(i−1)/2 v=−(j−1)/2

w si (z) sj (w). f q v−u z

(3.32)

Although the generating functions (3.31) are here all constructed from one single object t(z), we shall see that the Poisson structure deduced from (3.32) do not reflect this dependence and give rise indeed to genuine WN -type structures, in particular recovering the q-WN algebra in [21], as a consequence of the sector structure of the q-WN algebra in terms of modes. Notice also from (3.32) that the limitation of the index i to values smaller than N is justifed by the fact that a product of N functions t(q u z) in (3.31) in fact Poisson commute with all si (w) owing to: N X

f (q u x) = 0,

∀ x ∈ C.

(3.33)

u=1

We now study the mode expansion of (3.32). The singularities of the structure function lie at x = ±q ±N ` q u and ±q ±N `±1 q u with ` a positive integer and u an integer (resp. a half-integer) from 1 − 21 (i + j) to 21 (i + j) − 1 for (i + j) even (resp. (i + j) odd). They fall into sets of (i + j + 1) poles symmetrically arranged around q ±N ` , separated by one power of q. This setting defines a labeling of sectors for the Poisson brackets as follows: h R1 i −Pij (k) −P (k+1) ∈ |q| , |q| ij , Pij (k) being for fixed i and j, we define a sector (k) by R2 the ordered set of powers of q −1 , positive and negative, where the poles of the structure function of (3.32) are located, such that Pij (0) = 0. The rules which define the sectors in which the Poisson bracket of the modes of a given couple (si , sj ) is computed, are the following: • Poisson brackets between modes of the same field {si (z), si (w)} are required by antisymmetry to be computed on contours C1 and C2 symmetrized as in formula (3.28). Hence they are labeled by positive numbers (k) only, corresponding to the h R1 i −Pii (k) −P (k+1) ∈ |q| , |q| ii choice . R2 • Poisson brackets between modes of different fields {si (z), sj (w)} can be computed h i R1 P (k) P (k+1) where ∈ |q| ij , |q| ij on a single set of contours C1 and C2 such that R2 Pij (k) may be positive or negative. Symmetrization over C1 and C2 is not required. Antisymmetry of the Poisson bracket is imposed by computing {si , sj }(k) for i < j and setting {sj , si }(k) ≡ −{si , sj }(k) . Hence these Poisson brackets are labeled by positive and negative numbers, one for each couple (i, j).

Deformed WN Algebras from Elliptic sl(N ) Algebras

709

• The choice of sectors k(i, j) on which Poisson brackets of different couples are computed is arbitrary. In fact, quadratic Poisson bracket structures obey the Jacobi identity as soon as they are antisymmetric, hence any antisymmetric Poisson structure is consistent. To summarize, a complete Poisson structure for {si , sj } is characterized by the choice of N − 1 positive integer labels and 21 (N − 1)(N − 2) integer labels. Proposition 3. When one chooses k = 0 for all sectors, one obtains a compact generic expression (with n, m ∈ Z): {si (n), sj (m)}

(3.34) X [(N − max(i, j))r]q [min(i, j)r]q si (n − 2r)sj (m + 2r). = −2 ln q(q − q −1 ) [N r]q r∈Z

The proof of this formula is given in Appendix B. This Poisson bracket structure is identical to the Poisson bracket structure obtained in [21] from a bosonization construction, excluding the extra δ-type terms in si−p sj+p . We shall comment on the possibility of occurrence for such terms in the conclusion. This realizes, as in the sl(2) case, a non-trivial connection between the q-WN algebra and the sl(N ) elliptic algebra. Proposition 4. Any Poisson structure in a given sector k(i, j) can be obtained from (3.34) by adding to the r-dependent structure coefficient contributions from the relevant singularities of the structure function. They are given by formal power series expansions of terms δ(±q ±Pij (s) w/z)si (z)sj (w) for s = 1, . . . , k(i, j). One should now emphasize that the Poisson structures (3.34) in any sector k(i, j) are not identified to the structure which P would beQobtained from application by the Leibniz rule to the mode expansion a1 +···+ai =m tai q ki obtained from (3.31) by a single contour integral on a contour C1 for z of any particular k-sector Poisson structure for the generators tm derived in (3.29). Indeed, this structure would simply be given by [(N − 1)r]q [r]q the corresponding structure function plus its δ-contributions. [N r]q By contrast, it can be seen in (3.32)-(3.34) and in the derivation (Appendix B) that (forgetting for the time being the further symmetrization requirements over the double contour integral) the Poisson structure (3.34) follows in fact from application of the Leibniz rule to expansions of the form above, but where each individual Poisson bracket {tn , tm } must be computed in distinct relative sectors since they stem from Poisson brackets between generating functions t(q u1 z) and t(q u2 w) given by f (q u1 −u2 z/w) as a contribution to the structure where u1 andu2 respectively live in two in function, tervals − 21 (i − 1), 21 (i − 1) and − 21 (j − 1), 21 (j − 1) : hence the relative position of integration contours depends on the difference between the indices u1 and u2 which lives in − 21 (i + j) − 1, 21 (i + j) − 1 and no individual Poisson bracket {tn , tm } can be factored out. This is in particular true when considering identical fields {si , si }. The symmetrization procedure required for the tn Poisson brackets adds a further obstacle to attempts at factoring out symmetrized Poisson brackets for the modes tn . To summarize, once the Poisson bracket of composite fields si (z), sj (w) are computed for the modes defined by contour integrals in specified relative positions for z and w, giving (3.32)-(3.34), the nature of composite fields si (z) as products of the initial t(z) generators (3.31) is obliterated from the new mode Poisson structure thus obtained. The composite fields then assume the nature of independent objects with the Poisson structure (3.34), thereby validating the seemingly redundant definition (3.31).

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J. Avan, L. Frappat, M. Rossi, P. Sorba

Remark. Reciprocally, all supplementary terms denoted (δ(±q u w/z)si (z)sj (w))n,m from the mode expansion in k 6= 0 sectors are defined in the sense of formal series expansions as: X q up si (n − p)sj (m + p). (δ(±q u w/z)si (z)sj (w))n,m ≡ p∈Z

Their Poisson brackets must therefore be computed consistently by the Leibniz rule applied to their mode expansion, and not by using si (z)sj (q u z) as a generating functional. In particular, the extra terms δ(q −1 z/w)t(z)t(w) in the sl(2) case must not be understood as a central extension although the generating function t(z)t(zq) Poisson commutes with t(w). Central and lower-spin terms do not occur in our derivation. We shall comment on their absence here, and their possible reconstruction, in the conclusion. b )c ) 4. Quadratic Algebras in Aq,p (sl(N Theorem 4. In the three-dimensional parameter space generated by p, q, c, one defines a two-dimensional surface 6N,M for any integer M ∈ Z by the set of triplets (p, q, c) 1 connected by the relation (−p 2 )N M = q −c−N . On the surface 6N,M , the generators b )c ): t(z) realize an exchange algebra with the generators L(w) of Aq,p (sl(N w L(w) t(z), (4.1) t(z) L(w) = F M, z where F (M, x) = q 2M (N −1)

NY M −1 k=0

2q2N (x−2 p−k ) 2q2N (x2 pk ) 2q2N (x−2 q 2 p−k ) 2q2N (x2 q 2 pk )

for M > 0, (4.2a)

N |M |

Y 2q2N (x−2 q 2 pk ) 2q2N (x2 q 2 p−k ) 2q2N (x−2 pk ) 2q2N (x2 p−k )

F (M, x) = q −2|M |(N −1)

k=1

for M < 0. (4.2b)

Proof. The proof runs along similar lines to the commutativity proof of Theorem 2. From Eqs. (3.1) to (3.4), one gets c c ∗ b21 (q c2 w/z)t1 )−1 L+2 (w)L e − (z)R b21 (q − 2 w/z)t1 . t(z) L+2 (w) = Tr 1 L+1 (q 2 z)t1 (R 1 (4.3) One also has, from (3.6): c c c ∗ b21 (q c2 −c−N w/z)t1 )−1 L+2 (w) = L+2 (w) (R b21 (q − 2 w/z)−1 )t1 L+1 (q 2 z)t1 . L+1 (q 2 z)t1 (R (4.4)

One realizes that the only obvious condition that allows a substitution of Eq. (4.4) into b12 is the following: Eq. (4.3) using the quasi-periodicity of the matrix R (−p 2 )N M = q −c−N 1

with M ∈ Z.

b12 , one has: Actually, from the quasi-periodicity property of R

(4.5)

Deformed WN Algebras from Elliptic sl(N ) Algebras

711

b21 (x), b21 ((−p 21 )N M x) = F (M, x)R R

(4.6)

where

F (M, x) =

 N M −1 Y  1   F (−p 2 )k x     k=0

for M > 0, (4.7)

 N |M |  −1  Y 1    F (−p 2 )−k x for M < 0,  k=1

the function F (x) being given by −1 2 −1 2 −1 (q x)τN (q x ) = q 2−2/N F (x) = τN 1

1

2q2N (x2 )2q2N (x−2 ) . 2q2N (q 2 x2 )2q2N (q 2 x−2 )

(4.8)

Then on the two-dimensional surface defined by (4.5), Eq. (4.4) becomes b21 (q 2 w/z)t1 )−1 L+2 (w) L+1 (q 2 z)t1 (R c w b∗ (q − c2 w/z)−1 )t1 L+ (q c2 z)t1 . L+2 (w) (R = F M, q 2 21 1 z c

c

(4.9)

It follows that c w L+2 (w)Tr 1 (4.10) t(z) L+2 (w) = F M, q 2 z c c −1 t1 ∗ ∗ − c2 b b21 (q − 2 w/z)−1 )t1 L+1 (q 2 z)t1 (L− w/z)t1 . × (R 1 (z) ) R21 (q t2 0 t2 R21 t2 . The two R matrices cancel due to the relation Tr 1 R21 Q1 R0 21 = Tr 1 Q1 R21 c

Hence recalling that L+ (w) = L(q 2 w), one gets the desired result. Finally, one needs to compute the factor F (M, x). Using Eq. (4.8), one obtains the expressions of Theorem 4. Theorem 5. On the surface 6N,M , t(z) closes a quadratic subalgebra: t(z)t(w) = YN,p,q,M

w z

t(w)t(z) ,

(4.11)

where

YN,p,q,M (x) =

 NM 2 2 −k 2 2 k 2 −2 k   Y 2q2N (x p ) 2q2N (x q p ) 2q2N (x q p )   2  2 k 2 2 −k 2 −2 −k   k=1 2q2N (x p ) 2q2N (x q p ) 2q2N (x q p )

for M > 0,

 N |M |−1  (4.12)  Y 22q2N (x2 p−k ) 2q2N (x2 q 2 pk ) 2q2N (x2 q −2 pk )    for M < 0.  22q2N (x2 pk ) 2q2N (x2 q 2 p−k ) 2q2N (x2 q −2 p−k ) k=1

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J. Avan, L. Frappat, M. Rossi, P. Sorba

Proof. From Theorem 4, one has c w L+ (w) t(z), t(z) L+ (w) = F M, q 2 z 1 w (L− (w))−1 t(z), t(z) (L− (w))−1 = F −1 M, −p 2 z Hence, the definition (3.1) of t(z) immediately implies: w F M, q c z t(w) t(z). t(z) t(w) = 1 w 2 F M, −p z The explicit expression (4.2b) for F (M, x) gives the result as stated above.

(4.13a) (4.13b)

(4.14)

We shall discuss the mode expansion of (4.11) in Sect. 6. b )c ) 5. Poisson Structures Associated to Commuting Subalgebras in Aq,p (sl(N Theorem 6. On the surface 6(N, M ), when p = q N h with h ∈ Z\{0}, the function b )c ). YN,p,q,M is equal to 1. Hence t(z) realizes an Abelian subalgebra in Aq,p (sl(N Proof. Theorem 6 is easily proved using the explicit expression for F (M, x) and the periodicity properties of the 2q2N functions. Remark. Except in the case N = 2 (see [26]), no value of h here allows for t(z) to be b )c ). an element of a possibly extended center of Aq,p (sl(N The result of Theorem 6 nevertheless allows us to define Poisson structures on the corresponding Abelian algebras. They are obtained as limits of the exchange algebra (4.11) when p = q N h with h ∈ Z\{0}. Conversely it follows that (4.14) realizes a natural quantization of the Poisson structures obtained by this limit, since it realizes an “intermediate” closed exchange algebra, contrary to the situation at c = −N , where t(z) immediately lies in the center. This stands in contrast with the construction in [21] where the quantized q-WN algebras must be reconstructed by an independent quantization of the deformed classical bosons in the Cartan algebra. We see here the key role of the initial 3-parameter structure b )c ) compared to the 2-parameter quantum algebra Uq (sl(N b )c ) used in [21]. It Aq,p (sl(N 1

allows for an intermediate 2-parameter step at (−p 2 )N M = q −c−N , where the generators t(z) themselves close an exchange algebra. Hence it provides at the same time the classical q-deformed WN algebra and its (q, p)-deformed quantization. Theorem 7. Setting q N h = p1−β for any integer h 6= 0, the h-labeled Poisson structure defined by: 1 t(z)t(w) − t(w)t(z) (5.1) {t(z), t(w)}(h) = lim β→0 β has the following expression: {t(z), t(w)}(h) = fh (w/z) t(z) t(w),

(5.2)

Deformed WN Algebras from Elliptic sl(N ) Algebras

713

where

 X E( N2M )(E( N2M ) + 1) fh (x) = 2N h ln q  `≥0

x2 q 2N `+2 x2 q 2N `−2 2x q − − × 1 − x2 q 2N ` 1 − x2 q 2N `+2 1 − x2 q 2N `−2 2 2N `+N 2x q x2 q 2N `+N +2 x2 q 2N `+N −2 N M +1 2 − − + E( 2 ) 1 − x2 q 2N `+N 1 − x2 q 2N `+N +2 1 − x2 q 2N `+N −2 2 2 2 2x x q x2 q −2 −1 1 NM NM − − − (x ↔ x ) − 2 E( 2 )(E( 2 ) + 1) 1 − x2 1 − x2 q 2 1 − x2 q −2 for h odd, (5.3a)  X 2x2 q 2N ` x2 q 2N `+2 x2 q 2N `−2 − − = N 2 M (N M + 1)h ln q  1 − x2 q 2N ` 1 − x2 q 2N `+2 1 − x2 q 2N `−2 `≥0 2x2 x2 q 2 x2 q −2 −1 1 − (x ↔ x ) − − for h even. (5.3b) −2 1 − x2 1 − x2 q 2 1 − x2 q −2

2 2N `

Here the notation E(n) means the integer part of the number n. Proof. Direct calculation.

The factors 2N h for h odd and N 2 M h(N M + 1) for h even are inessential and can be reabsorbed into the definition of the classical limit as β → −N hβ for h odd and β → − 21 N 2 M (N M + 1)hβ for h even. Provided this redefinition is done, the formula (5.3b) in the case h even coincides exactly with the Poisson structure (3.25) of the center b )−N ). The formula (5.3) to the contrary in the case h odd gives rise to a of Aq,p (sl(N new Poisson structure. As before, Eqs. (5.3) lead to infinite families of Poisson brackets for the modes of t(z) defined by (3.26). For h even, the labeling is identical to the c = −N case. For h 1

1

odd, the singularities lie at ±q ±N ` , ±q ±N `±1 and ±q ±N `± 2 N , ±q ±N `± 2 N ±1 , giving rise to extra triplets of poles lying halfway between the initial ones. Proposition 5. One gets for the Poisson bracket in the k = 0 sector: X [(N − 1)r]q [r]q (h) −1 E( N2M )(E( N2M ) + 1) {tn , tm } = −2 ln q(q − q ) [N r]q r∈Z ! [r]2q +1 2 −E( N M ) for h odd, (5.4a) tn−2r tm+2r 2 [N r]q X [(N − 1)r]q [r]q tn−2r tm+2r for h even. (5.3b) = −2 ln q(q − q −1 ) [N r]q r∈Z

A realization of higher spin generators is again achieved by the formula (3.31) with i = 1, . . . , N − 1. Its justification will be the same as in Sect. 3. The generators s1 (z), . . . , sN −1 (z) close a Poisson algebra with the following Poisson brackets (i, j = 1, . . . , N − 1):

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J. Avan, L. Frappat, M. Rossi, P. Sorba

X

(i−1)/2

{si (z), sj (w)}(h) =

X

(j−1)/2

u=−(i−1)/2 v=−(j−1)/2

w si (z) sj (w). fh q v−u z

(5.5)

The singularity structure here is as follows. Singularities of the function fh (x) occur at x = ±q ±N `+u , ±q ±N `±1+u , where ` ∈ N and u ∈ 1 − 21 (i + j), 21 (i + j) − 1 when h 1

is even. Additional singularities occur halfway between those ones, at ±q ±N `± 2 N +u , 1

±q ±N `± 2 N ±1+u when h is odd. The sector structure for the Poisson brackets of {si (z), sj (w)} is easily deduced from these results. Easiest to compute are the Poisson brackets in the sector k = 0 for all couples of indices (i, j). A simpler Poisson structure in this sector is defined by furthermore taking i h R1 ∈ 1, |q|−1/2 in all Poisson brackets a symmetrized double contour integral with R2 (or 1, |q|−1 depending on the parity of N and i + j). (Note that the symmetrized form is actually not required here either when i 6= j, but it leads to nicer formulae). One gets: {si (n), sj (m)}(h) = −2 ln q(q − q −1 )

X [(N − max(i, j))r]q [min(i, j)r]q si (n − 2r)sj (m + 2r) [N r]q r∈Z

for h even,

(5.6a)

{si (n), sj (m)}

(h)

= −2 ln q(q − q

−1

)

X

E( N2M )(E( N2M ) + 1)

r∈Z

[(N − max(i, j))r]q [min(i, j)r]q [N r]q

N M +1 2 [ir]q [jr]q −E( 2 ) si (n − 2r)sj (m + 2r) [N r]q for h odd and i + j ≤ N , {si (n), sj (m)}(h)

X

(5.6b)

[(N − max(i, j))r]q [min(i, j)r]q [N r]q r∈Z +1 2 [ir]q [jr]q +1 2 −E( N M ) − E( N M ) [(N − i − j)r]q si (n − 2r)sj (m + 2r) 2 2 [N r]q for h odd and i + j > N . (5.6c)

= −2 ln q(q − q −1 )

E( N2M )(E( N2M ) + 1)

The Poisson bracket (5.6a) is again identical to the “core” contribution in [21] (i.e. without lower-spin extensions). Equation (5.6a–c) however is a completely new type of quadratic q-deformed classical WN -algebra with the two extra terms. It would be interesting to know about its possible explicit constructions. 6. Quantum Exchange Algebra The exchange algebras (4.11) are now understood as natural quantizations of the classical q-deformed WN algebras, including the initial algebra [21]. Moreover it follows that

Deformed WN Algebras from Elliptic sl(N ) Algebras

715

the exchange algebra (4.11) now provides us with building blocks for new deformed quantum Wq,p (sl(N )) algebras with an extra integer parameter M . We wish to describe here an explicit formulation in terms of modes of quantum generating operators si (z) defined as in (3.31), although with a required notion of ordering between individual t(z)-generators: x Y

si (z) =

t(q u z).

(6.1)

i−1 i−1 2 ≥u≥− 2

Again justification of this definition as giving genuine q-WN algebras will come from arising sectors in the individual t(z)-t(w) exchange algebra, which eventually combine in a non-trivial way in the product formula to give rise to a new algebraic structure. The exchange algebra from (6.1) and (4.11) takes the form: i−1

si (z)sj (w) =

2 Y

u=− i−1 2

j−1

w sj (w)si (z) , YN,p,q,M q v−u z j−1

2 Y

v=−

(6.2)

2

for any choice of ordering in (6.1). Furthermore it follows from (6.1) and (4.12) that sN (z) commutes with all other generators; hence we are justified in restricting i to 1, . . . , N − 1. Once we choose an exchange function in (4.12) by choosing the integer M , the next step in the procedure consists in factorizing the exchange functions in (6.2) into a function analytic around w/z = 0 and a function analytic around z/w = 0. More precisely, and following [31], one defines a Riemann problem: −1 −1 (x ) YN,p,q,M ;(i,j) (x) = Y+ (x)Y−

(6.3)

in the neighborhood of a circle C of radius R. Y+ and Y− are respectively analytic for |x| < R and |x| > R. Varying the values of R with respect to the position of zeroes and poles of Y leads to different factorizations; moreover even with fixed R the solution of (6.3) is not unique. This leads to a large choice of acceptable factorizations. It may be possible to choose Y+ = Y− in the sense of analytic continuation, for instance for i = j = 1, when R takes 1 1 any value between |p| 2 and |p|− 2 ; this case is treated in detail below. The third step is specific to our approach. It consists in promoting the exchange relation deduced from (6.2) and (6.3) to the level of an analytic extension to the full complex plane of z/w: z w t(z)t(w) = Y+ t(w)t(z) . (6.4) Y− w z Singularities of Y± now play a crucial role as in the classical case, when we define mode expansions of (6.4). The fields si (z) are considered as abstract generating operatorvalued functionals with modes defined by contour integrals: I dz −n (6.5) z si (z) . si (n) = C 2πiz They obey no particular supplementary relations, which may eventually arise from explicit realizations of t(z) as in [27, 31].

716

J. Avan, L. Frappat, M. Rossi, P. Sorba

As a consequence, we need to introduce a similar notion of “sectors”, determined by the singularities of Y± , and defined as the regions in the complex plane for z/w, where Y+ and Y− are given by a particular convergent series expansion (necessarily unique in each domain since Y± are meromorphic). A choice of relative positions for the contours C1 for z and C2 for w thus gives the unique formal series expansions for Y+ and Y− to be inserted in the double contour integral, which eventually gives an exchange relation for the modes. We first illustrate this on the simplest example i = j = 1 (i.e. si (z) = sj (z) ≡ t(z)). To extract from (4.11) an exchange relation between the modes tn (3.26) of t(z), we rewrite (4.11) in the form: z w t(z) t(w) = t(w) t(z) fN,p,q,M . (6.6) fN,p,q,M w z A consistent choice for fN,p,q,M (x) is then: fN,p,q,M (x) =

S(M Y)

(1 − x2 )2

k=1 ∞ Y

×

n=0

(6.7)

(1 − x2 p−k q 2N (1+n) )2 (1 − x2 pk q 2+2N n )(1 − x2 pk q −2+2N (1+n) ) , (1 − x2 pk q 2N n )2 (1 − x2 p−k q −2+2N (1+n) )(1 − x2 p−k q 2+2N n )

where S(M ) = N M for M > 0 and S(M ) = N |M | − 1 for M < 0. In the hypothesis: |p| < 1,

|p|S(M )+1 > |q|2 ,

|q| < 1,

(6.8)

− 21

fN,p,q,M (x) is analytic for |x| < |p| . Then the choice Y+ = Y− = fN,p,q,M solves 1 1 the Riemann problem (6.3) with |p| 2 < R < |p|− 2 . Exchange relations for the modes of t(z) are then obtained by double contour integrals. We shall not give here explicit expressions of the expansion coefficients of fN,p,q,M (x) but we shall describe the connections between the different sector expansions. Theorem 8. Let f (x) be a meromorphic function of x2 having only simple and double −1

poles, denoted by ±αj 2 , j ∈ N. Suppose that |α1 |− 2 > 1 and |αj |− 2 < |αj 0 |− 2 if 1 j < j 0 and denote fl the coefficients of the Taylor expansion of f (x) for |x| < |α1 |− 2 : 1

f (x) =

∞ X

1

fl x2l .

1

(6.9)

l=0

Then the relation: I C1

dz 2πiz

I C2

dw + 2πiw

I C2

dz 2πiz

I C1

dw 2πiw

z −n w−m

(6.10)

× [ f (z/w) t(z) t(w) − f (w/z) t(w) t(z) ] = 0, where C1 and C2 are circles of radii respectively R1 and R2 defines a family of exchange 1 relations for the modes tn (3.26) of t(z), depending on R1 /R2 . If R1 /R2 ∈ [1, |α1 |± 2 [ one has:

Deformed WN Algebras from Elliptic sl(N ) Algebras ∞ X

717

fl (tn−2l tm+2l − tm−2l tn+2l ) = 0,

(6.11)

l=0

where fl is given by (6.9), while in the regions R1 /R2 ∈]|αj0 |± 2 , |αj0 +1 |± 2 [ , the exchange relation becomes: 1

X l∈Z

1

fl(j0 ) (tn−2l tm+2l − tm−2l tn+2l ) = 0,

(6.12)

where fl(j0 ) (l ∈ Z and j0 here is an index, not a power coefficient) is obtained by adding to fl the contributions: 2 1 l α [(1 − α x )f (x)] , (6.13) j 2 j −1/2

x=αj

−1/2

for every simple pole αj

such that 1 ≤ j ≤ j0 , and the contributions:

− 21 (l + 1)αjl [(1 − αj x2 )2 f (x)]

l− 1 d [(1 − αj x2 )2 f (x)] + 41 αj 2 dx −1/2

for every double pole αj

−1/2

x=αj

(6.14)

−1/2

x=αj

such that 1 ≤ j ≤ j0 .

Proof. If R1 /R2 ∈ [1, |α1 |± 2 [ , in the integrand of (6.10) both |w/z| and |z/w| are 1 smaller than |α1 |− 2 : therefore one can develop both f (z/w) and f (w/z) in Taylor series according to Eq. (6.9). Equation (6.11) then follows immediately. 1 1 −1/2 • Suppose now R1 /R2 ∈ ]|αj0 |± 2 , |αj0 +1 |± 2 [ , where αj0 is a simple pole. Without 1 1 losing generality we can suppose R1 /R2 ∈]|αj0 |− 2 , |αj0 +1 |− 2 [ and rewrite Eq. (6.10) in the following way: 1

I C1

dw 2πiw

I C2

dz −n −m z w [f (z/w)t(z)t(w) − f (w/z)t(w)t(z)] − (n ↔ m) = 0. 2πiz (6.15)

In the domain of integration, f (z/w) can be expanded in a convergent power series of z/w according to formula (6.9), while f (w/z) admits the factorization: f

w z

=

jY 0 −1 j=1

1

2 m(j)

w 1 − αj 2 z

1 w 1 − αj0 2 z

2

f (j0 )

w z

,

(6.16)

where m(j) = 1, 2 is the order of the j th pole and f (j0 ) (w/z) is an analytic function. The first two factors in the r.h.s. of (6.16) can be expanded in a convergent series of z/w:

718

J. Avan, L. Frappat, M. Rossi, P. Sorba jY 0 −1 j=1

w2 1 − αj 2 z 1 − αj0

−m(j) =

jY 0 −1

∞ X

j=1

s=1

gs(j) αj−s

z 2s w

, (6.17)

2 −1

w z2

=−

∞ X

αj−k 0

k=1

z 2k w

(gs(j) = −1 if m(j) = 1, gs(j) = s − 1 if m(j) = 2), while f (j0 ) (w/z) has a convergent series expansion in w/z: f (j0 )

w z

=

∞ X l=0

fl(j0 )

w 2l z

.

(6.18)

Therefore if R1 /R2 ∈]|αj0 |± 2 , |αj0 +1 |± 2 [ the exchange relation (6.10) can be written, in the sense of a formal power series: 1

∞ X

I fl tn−2l tm+2l −

l=0

×

∞ X

−αj−k 0

1

dw 2πiw

C1

∞ z 2k X

w

k=1

l=0

fl(j0 )

I C2

j0 −1 ∞ dz −n −m Y X (j) −s z 2s gs αj z w 2πiz w j=1

w 2l

s=1

t(w) t(z) − (n ↔ m) = 0.

z

(6.19)

Let us concentrate on the terms: −

∞ X

αj−k 0

∞ z 2k X

w

k=1

l=0

fl(j0 )

w 2l z

.

(6.20)

This double sum can be rewritten as a sum of two contributions: −

∞ XX h∈Z l=0

fl(j0 ) αjh−l 0

w 2h z

+

∞ X h X h=0 l=0

fl(j0 ) αjh−l 0

w 2h z

.

(6.21)

In the second sum of (6.21) the term: h X l=0

fl(j0 ) αjh−l 0

(6.22)

is just the coefficient fh(j0 −1) of the Taylor expansion of the function f (j0 −1) (w/z) (ana 1 w lytic for < |αj0 |− 2 ) defined by: z f

w z

=

jY 0 −1 j=1

1 w2 1 − αj 2 z

m(j)

f (j0 −1)

w z

Using this result, we can reinsert Eq. (6.21) in (6.19). We obtain:

.

(6.23)

Deformed WN Algebras from Elliptic sl(N ) Algebras ∞ X

I fl tn−2l tm+2l −

C1

l=0

×

jY 0 −1

∞ X

j=1

s=1

C1

dw 2πiw

I + ×

jY 0 −1

∞ X

j=1

s=1

C2

I

∞ z 2s X

gs(j) αj−s I

dw 2πiw

w

l=0

C2

dz −n −m z w 2πiz

fl(j0 −1)

dz −n −m z w 2πiz ∞ z 2s X X

gs(j) αj−s

w

719

h∈Z l=0

w 2l z

fl(j0 ) αjh−l 0

t(w) t(z)

w 2h z

t(w) t(z) − (n ↔ m) = 0. (6.24)

The first line gives the exchange relation for R1 /R2 ∈]|αj0 −1 |± 2 , |αj0 |± 2 [ . The second line is the extra contribution (11j0 )n,m coming from the crossing of the simple singularity 1

−1/2

at w/z = αj0

. The summation over h and l gives:

dz −n −m w2 z w = δ αj0 2 (6.25) z C1 C2 2πiz jY ∞ 0 −1 X z 2s −1/2 t(w) t(z) − (n ↔ m). gs(j) αj−s f (j0 ) αj0 × w I

(11j0 )n,m

1

dw 2πiw

j=1

I

s=1

Owing to the properties of the δ-distribution, the series expansions in (6.26) are all convergent, since |αj0 /αj | < 1 for j < j0 . Reinserting their sums in (6.26), one recognizes the expression: jY 0 −1 j=1

1 αj 1− αj0

−1/2 (j ) , m(j) f 0 αj0

(6.26)

−1/2

which, by comparison with (6.16), is just the residue of f (x) at x = αj0 extra contribution is: I (11j0 )n,m

= C1

dw 2πiw

I C2

. Then the

dz −n −m w2 z w δ αj0 2 [(1 − αj0 x2 )f (x)] −1/2 2πiz z x=αj 0

× t(w) t(z) − (n ↔ m),

(6.27)

or after integration, using the formal power series expansion of δ(x): (11j0 )n,m =

X l∈Z

αjl 0 [(1 − αj0 x2 )f (x)]

−1/2

x=αj

0

(tm−2l tn+2l − tn−2l tm+2l ). (6.28)

720

J. Avan, L. Frappat, M. Rossi, P. Sorba

• Similar but rather longer calculations can be performed in the case of a double pole. The extra contribution (12j0 )n,m is now: I I dw dz −n −m 0 δ z w (12j0 )n,m = C1 2πiw C2 2πiz w2 × αj0 2 [(1 − αj0 x2 )2 f (x)] t(w) t(z) −1/2 z x=αj 0 I I dw dz −n −m 1 δ z w −2 C1 2πiw C2 2πiz w2 −1/2 d 2 2 [(1 − αj0 x ) f (x)] × αj0 2 αj0 −1/2 z dx x=α j0

× t(w) t(z) − (n ↔ m), where: δ 0 (x) =

X

(6.29)

h xh−1 .

(6.30)

h∈Z

Integrating (6.29) one has: (12j0 )n,m

=

X l∈Z



(l + 1)αjl [(1 − αj0 x2 )2 f (x)] 0

l− 1 − 21 αj0 2

d 2 2 [(1 − αj0 x ) f (x)] dx

−1/2

x=αj



0

 −1/2

x=αj

0

(tm−2l tn+2l − tn−2l tm+2l ).

(6.31)

Comparing (6.31,6.28) to the first line of (6.24), one obtains that (6.13,6.14) are precisely the quantities one has to add to fl when crossing simple and double singularities. Remark 1. Theorem 8 gives the mode exchange algebra between two identical fields t(z) and t(w) using symmetrized integration contours, see formula (6.10). A similar result can be obtained for the mode exchange algebra between two different fields si (z) and sj (w) but with more complicated formulae. The Riemann problem (6.3) is then generically solved by two functions Y+ 6= Y− ; the sector structure involves poles of both functions; the separate contributions of the poles of these two functions to the difference between adjacent sectors are given by (6.13) and (6.14) applied to the corresponding functions. The mode expansion cannot of course be factored out as in (6.12) but takes the generic form ∞ X

Y− (l) si (n − 2l) sj (m + 2l) − Y+ (l) sj (m − 2l) si (n + 2l) = 0.

(6.32)

l=0

Finally let us emphasize that the symmetrization of the integration contours is actually not required in the quantum case, even in the case of identical fields, but it leads to nicer formulae for the coefficients Y± (l).

Deformed WN Algebras from Elliptic sl(N ) Algebras

721

Remark 2. It may happen that either function Y± , solution of (6.3), have a multiple pole α of order κ > 2. In this case, formulae analogous to (6.13) and (6.14) hold: the supplementary contribution to the coefficient fl is then given by a sum of κ terms, each term being proportional to the nth -derivative of (1 − αx2 )κ f (x) taken at x = α−1/2 , where n = 0, . . . , κ − 1. Remark 3. The condition (6.8) is incompatible with the classical limit (where q N h = p1−β with β → 0). In order to get a classical limit in the case N h > 2, it is necessary to change (6.8) in such a way that the Riemann problem (6.3) is solved by two functions Y+ 6= Y− (in the sense of analytic continuations). Remark 4. Finally, let us comment on the classical limit when using non-symmetrized integration contours for the quantum case. One starts with the exchange formula between two different fields si (z) and sj (w): Y− (z/w) si (z) sj (w) = sj (w) si (z) Y+ (w/z).

(6.33)

When using non-symmetrized integration contours C1 and C2 , one obtains for the modes ∞ X

Y− (l) si (n − 2l) sj (m + 2l) − Y+ (l) sj (m − 2l) si (n + 2l) = 0.

(6.34)

l=0

Let us set Y− (x) = 1 + βf (x) and Y+ (x) = 1 + βg(x). This definition is consistent since the Riemann-Hilbert problem (6.3) reads Y+ /Y− = 1 when β = 0. (6.34) becomes: X 1 [si (n), sj (m)] + fl si (n − 2l) sj (m + 2l) − gl sj (m − 2l) si (n + 2l) = 0. β (6.35) l∈Z In the classical limit β → 0, one gets therefore, when i 6= j: X (fl − g−l ) si (n − 2l) sj (m + 2l). {si (n), sj (m)} = −

(6.36)

l∈Z

This gives precisely the Poisson structures computed with non-symmetrized integration contours with the structure function f (z/w) − g(w/z). When i = j, (6.34) can be further decoupled into two exchange relations: ∞ X

(Y− (l) + Y+ (l)) (si (n − 2l) si (m + 2l) − si (m − 2l) si (n + 2l)) = 0, (6.37a)

l=0 ∞ X

(Y− (l) − Y+ (l)) (si (n − 2l) si (m + 2l) + si (m − 2l) si (n + 2l)) = 0. (6.37b)

l=0

The first equation leads in the classical case to the Poisson structure X (fl − g−l − f−l + gl ) si (n − 2l) si (m + 2l), {si (n), si (m)} = −

(6.38)

l∈Z

while the second relation has to be interpreted as supplementary constraint equations. Equation (6.38) gives precisely the antisymmetric Poisson structure obtained in (5.4) and (5.6) by using symmetrized contour integration. Of course, had we started with symmetrized contours in the quantum case, we would easily get the corresponding Poisson structure in the classical limit where the mode Poisson bracket is computed with symmetrized integration contours.

722

J. Avan, L. Frappat, M. Rossi, P. Sorba

7. Conclusions We study here exchange relations of the form f (z/w) t(z) h(w) = h(w) t(z) f (w/z), and their classical Poisson bracket limits. These relations were interpreted in terms of analytic continuations and lead us to sets of formal series expansions associated to each particular convergent expansion of f (x). What we have established here is a set of universal structures for q-deformed WN classical algebras and Wq,p (sl(N )) algebras.These structures still allow for supplementary, representation-dependent extensions of the type founded in [27] or more general! Nothing was assumed here concerning the fields t(z), h(w), treated as abstract objects with a priori no singular behaviour. Of course, had we constructed explicit realizations of these fields, we would at one strike in the mode expansion fix a particular sector (k) and add possible central, linear or generically lower-spin extensions as were constructed in [27, 31]. A closer look at these particular constructions indeed shows that they all follow from bosonic realizations of the exchange algebra [21, 27] and the extensions are generated either due to cancellations between otherwise a priori independent fields in the general quadratic expression [27], or (in a different framework, that of deformed chiral algebras) due to singularities in the operator products t(z) h(w) at the poles of f (z/w), resolved by using explicit bosonization formulae [31]. It must therefore be expected that extended structures, with lower-spin δ-type extensions, play the most important role in practical applications of these Wq,p (sl(N )) algebras. These are also crucial to define a reasonable representation theory for these algebras. Hence the developments from the general frame which we have established are twofold. One should look for explicit realizations (bosonizations ?) of the classical/quantum algebras, thereby also getting admissible (Jacobi or cocycle solutions) extensions. Alternatively one should also look systematically for allowed extensions of our structures by defining and solving the corresponding “cocycle” equations. We hope to report on this question in a forthcoming paper [28]. Acknowledgement. This work was supported in part by the CNRS, Foundation Angelo della Riccia and EC network n. FMRX-CT96-0012. L.F. is indebted to Centre de Recherches Math´ematiques of Universit´e de Montr´eal, where an early stage of this work was done, for its kind invitation and support. M.R. and J.A. wish to thank LAPTH for its kind hospitality. We also wish to thank M. Jimbo and T. Miwa for a number of precious comments and indications, and drawing our attention to Ref. [14].

Appendix A: Jacobi Theta Functions Let H = {z ∈ C | Imz > 0} be the upper half-plane and 3τ = {λ1 τ + λ2 | λ1 , λ2 ∈ Z, τ ∈ H} the lattice with basis (1, τ ) in the complex plane. One denotes the congruence ring modulo N by ZN ≡ Z/N Z with basis {0, 1, . . . , N − 1}. One sets ω = e2iπ/N . Finally, for any pairs γ = (γ1 , γ2 ) and λ = (λ1 , λ2 ) of numbers, we define the (skewsymmetric) pairing hγ, λi ≡ γ1 λ2 − γ2 λ1 . One defines the Jacobi theta functions with rational characteristics γ = (γ1 , γ2 ) ∈ 1 Z × N1 Z by: N X γ exp iπ(m + γ1 )2 τ + 2iπ(m + γ1 )(ξ + γ2 ) . (A.1) ϑ 1 (ξ, τ ) = γ2 m∈Z

Deformed WN Algebras from Elliptic sl(N ) Algebras

723

γ1 The functions ϑ (ξ, τ ) satisfy the following shift properties: γ2 γ γ + λ1 (ξ, τ ) = exp(2iπγ1 λ2 ) ϑ 1 (ξ, τ ), (A.2a) ϑ 1 γ2 + λ2 γ2 γ γ ϑ 1 (ξ + λ1 τ + λ2 , τ ) = exp(−iπλ21 τ − 2iπλ1 ξ) exp(2iπhγ, λi) ϑ 1 (ξ, τ ), γ2 γ2 (A.2b) where γ = (γ1 , γ2 ) ∈ N1 Z × N1 Z and λ = (λ1 , λ2 ) ∈ Z × Z. Moreover, for arbitrary λ = (λ1 , λ2 ) (not necessarily integers), one has the following shift exchange: γ1 + λ1 γ1 2 (ξ + λ1 τ + λ2 , τ ) = exp(−iπλ1 τ − 2iπλ1 (ξ + γ2 + λ2 )) ϑ (ξ, τ ). ϑ γ2 γ2 + λ2 (A.3) Considering the usual Jacobi theta function: 2p (z) = (z; p)∞ (pz −1 ; p)∞ (p; p)∞ ,

(A.4)

where the infinite multiple products are defined by: Y (1 − zpn1 1 . . . pnmm ), (z; p1 , . . . , pm )∞ =

(A.5)

ni ≥0

the Jacobi theta functions with rational characteristics (γ1 , γ2 ) ∈ N1 Z × expressed in terms of the 2p function as: 1 2 1 γ ϑ 1 (ξ, τ ) = (−1)2γ1 γ2 p 2 γ1 z 2γ1 2p (−e2iπγ2 pγ1 + 2 z 2 ), γ2

1 NZ

can be

(A.6)

where p = e2iπτ and z = eiπξ . Appendix B: Proof of Formula (3.34) Let us start with Eq. (3.32): X

(i−1)/2

{si (z), sj (w)} =

X

(j−1)/2

u=−(i−1)/2 v=−(j−1)/2

w si (z) sj (w), f q v−u z

(B.1)

where the function f (x) is given by (3.25). • Let us first suppose that i < j. Then, (3.32) can be rewritten as: X

(i+j)/2−1

{si (z), sj (w)} =

u=1−(i+j)/2

w w si (z) sj (w) ≡ fij si (z) sj (w), η(u)f q u z z (B.2)

where the function η(u) is given by the following graph:

724

J. Avan, L. Frappat, M. Rossi, P. Sorba

η(u) 6 i @

@

@ @ -

0 − i+j 2

i−j 2

j−i 2

u.

i+j 2

Therefore, one has for the mode Poisson bracket, using symmetrized integration contours (with m, n ∈ Z): I I I I dz dw dz dw 1 + {si (n), sj (m)} = 2 C1 2πiz C2 2πiw C2 2πiz C1 2πiw w si (z) sj (w). (B.3) × z −n w−m fij z One has X

(i+j)/2−1

fij (x) = −2 ln q

η(u)

u=1−(i+j)/2

X

x2 q 2N `+2u+2 x2 q 2N `+2u−2 2x2 q 2N `+2u − − 1 − x2 q 2N `+2u 1 − x2 q 2N `+2u+2 1 − x2 q 2N `+2u−2 `≥0 X 2x−2 q 2N `−2u x−2 q 2N `−2u+2 x−2 q 2N `−2u−2 − − − 1 − x−2 q 2N `−2u 1 − x−2 q 2N `−2u+2 1 − x−2 q 2N `−2u−2 `≥0

x2 q 2u+2 x2 q 2u−2 x−2 q −2u x2 q 2u 1 1 + + + 1 − x2 q 2u 2 1 − x2 q 2u+2 2 1 − x2 q 2u−2 1 − x−2 q −2u x−2 q −2u+2 x−2 q −2u−2 1 1 . −2 −2 1 − x−2 q −2u+2 1 − x−2 q −2u−2

−

(B.4)

In the following, one has to distinguish the cases i + j odd and i + j even. In the sector k = 0, the integration contours C1 and C2 of radii R1 and R2 correspond to the choice h R1 i 1/2 R1 ∈ |q| , |q|−1/2 when i + j is odd and ∈ |q|, |q|−1 when i + j is even (recall R2 R2 that |q| < 1). Hence, in the sector k = 0, the singularities of the structure function lie at x = q ±u , q ±u±1 . a) case i + j odd. In that case, u takes only half-integer values. One has therefore   (i+j)/2−1 XX X η(u) (q r − q −r )2 (x2r q 2N `r+2ur fij (x) = −2 ln q −  `≥1 r≥1

u=1−(i+j)/2

−2r 2N `r−2ur

−x

q

)+

X r≥1

−

X r≥1

x−2r

X u<0

x

2r

X

2η(u)q

u>0

2η(u)q −2ur −

X u<1

2ur

−

X

η(u)q

2ur−2r

u>1

η(u)q −2ur+2r −

−

X

! η(u)q

u>−1

X u<−1

η(u)q −2ur−2r

2ur+2r

!  

. (B.5)

Deformed WN Algebras from Elliptic sl(N ) Algebras

725

Then resumming the series over `, (B.5) can be rewritten as: fij (x) = −2 ln q

X [r]q (q r − q −r ) [N r]q r≥1

(

q

× x

2r

Nr

X

η(u)q

+q

2ur

−N r

u<0

−x

−2r

q

−N r

X

η(u)q

2ur

−

u>0

X

η(u)q

−2ur

+q

Nr

u<0

X

η(u)q

[N r]q η( 21 ) [r]q

−2ur

u>0

−

! (B.6)

[N r]q η( 21 ) [r]q

!) .

From the shape of the function η(u) (see the figure above) it is easy to calculate the sums occurring in Eq. (B.7). One gets X

η(u)q 2ur =

u>0

X

X

η(u)q −2ur =

u<0

η(u)q −2ur =

u>0

X

q rj (q ri − q −ri ) − i(q r − q −r ) , (q r − q −r )2

η(u)q 2ur = −

u<0

q −rj (q ri − q −ri ) − i(q r − q −r ) . (q r − q −r )2

(B.7)

Inserting then Eq. (B.7) into Eq. (B.7), one obtains fij (x) = −2 ln q(q − q −1 )

X

−x2r

r∈Z

[(N − j)r]q [ir]q . [N r]q

(B.8)

Hence, from Eqs. (B.3) and (B.8), formula (3.34) immediately follows. b) case i + j even. In that case, u takes only integer values; in particular it can take the value u = 0. One has fij (x) =   −2 ln q −  +

X

x

2r

−

r≥1

x−2r

η(u)

X

2η(u)q

X u<0

XX

(q r − q −r )2 (x2r q 2N `r+2ur − x−2r q 2N `r−2ur )

`≥1 r≥1

u=1−(i+j)/2

u>0

r≥1

X

X

(i+j)/2−1

2ur

−

X

η(u)q

u>1

2η(u)q −2ur −

2ur−2r

−

!

X

η(u)q

2ur+2r

u>−1

X u<1

η(u)q −2ur+2r −

X u<−1

η(u)q −2ur−2r

!  

. (B.9)

Although (B.9) has the same form as (B.5), one has to be careful in the derivation of this formula due to the peculiar role of the value u = 0 (in particular note that i < j and i + j even implies η(0) = η(±1)). When resumming the series over `, one obtains after some algebra:

726

J. Avan, L. Frappat, M. Rossi, P. Sorba

fij (x) = −2 ln q

X [r]q (q r − q −r ) [N r]q

(B.10)

r≥1

(

q

x

2r

Nr

X

η(u)q

+q

2ur

X

−N r

u<0

−x

−2r

q

−N r

X

η(u)q

[(N − 1)r]q − η(0) [r]q

2ur

u>0

η(u)q

−2ur

+q

X

Nr

u<0

η(u)q

−2ur

u>0

!

[(N − 1)r]q − η(0) [r]q

!) .

The sums over u are now given by: X

η(u)q 2ur =

u>0

X

X

q rj (q ri − q −ri ) − iq r (q r − q −r ) , (q r − q −r )2

η(u)q −2ur =

u<0

η(u)q 2ur =

u<0

X

η(u)q −2ur = −

u>0

q −rj (q ri − q −ri ) − iq −r (q r − q −r ) . (q r − q −r )2

(B.11)

Inserting (B.11) into (B.11), one gets the same formula (B.8) as in the case i + j odd. • We consider now the case i = j. Then the trapezoidal function η(u) degenerates into a triangle: η(u) 6 i @

@

0

@

−i

@ -

u .

i

The formulae (B.2)–(B.4) remain valid with the condition i = j. Performing the series expansions in the sector k = 0, one obtains  i−1  X XX η(u) (q r − q −r )2 (x2r q 2N `r+2ur − x−2r q 2N `r−2ur ) fii (x) = −2 ln q −  u=1−i `≥1 r≥1 ! X X X X 2r 2ur 2ur−2r 2ur+2r x 2η(u)q − η(u)q − η(u)q +1 (B.12) + u>0

r≥1

−

X r≥1

x−2r

u>1

X u<0

2η(u)q −2ur −

u>−1

X

η(u)q −2ur+2r −

u<1

X

η(u)q −2ur−2r + 1

u<−1

!  

.

Since now η(0) − η(±1) = 1, Eq. (B.13) leads again to formula (B.11). Using (B.11), which still holds with i = j, one gets finally fii (x) = −2 ln q(q − q −1 )

X r∈Z

which achieves the proof of (3.34).

−x2r

[(N − i)r]q [ir]q [N r]q

(B.13)

Deformed WN Algebras from Elliptic sl(N ) Algebras

727

References 1. Foda, O., Iohara, K., Jimbo, M., Kedem, R. Miwa, T. Yan, H.: An elliptic quantum algebra for sbl2 . Lett. Math. Phys. 32, 259 (1994) 2. Jimbo, M., Konno, H., Miwa, T.: Massless XXZ model and degeneration of the elliptic algebra Aq,p (sbl(2)c ). hep-th/9610079, Proceedings of the conference “Deformation Theory, Symplectic Geometry and Applications”, Ascona, Switzerland, 1996 3. Khoroshkin, S., Lebedev, D., Pakuliak, S.: Elliptic algebra Aq,p (slˆ2 ) in the scaling limit. Commun. Math. Phys. 190, 597–627 (1998) 4. Konno, H.: Degeneration of the elliptic algebra Aq,p (slˆ2 ) and form factors in the sine-Gordon theory. hep-th/9701034, to appear in the CRM series in Mathematical Physics, Springer Verlag 5. Jimbo, M., Miwa, T., Nakayashiki, A.: Difference equations for correlation functions of the eightvertexmodel. J. Phys. A 26, 2199 (1993) 6. Jimbo, M., Kedem, R., Konno, H., Miwa, T., Weston, R.: Difference equations in spin chains with a boundary. Nucl. Phys. B 448, 429 (1995) 7. Konno, H.: An elliptic algebra Uq,p (sbl(2)c ) and the fusion RSOS model. Commun. Math. Phys. 195, 373–403 (1998) 8. Andrews, G., Baxter, R., Forrester, P.J.: Eight-vertex SOS model and generalized Rogers-Ramanujan type identities. J. Stat. Phys. 35, 193 (1984) 9. Date, E., Jimbo, M., Miwa, T., Okado, M.: Fusion of the eight-vertex SOS model. Lett. Math. Phys. 12, 209 (1986) 10. Avan, J., Babelon, O., Billey, E.: The Gervais–Neveu–Felder equation and quantum Calogero–Moser systems. Commun. Math. Phys. 178, 281 (1996) 11. Felder, G.: Proceedings of the ICM 1994. Basel–Boston: Birkh¨auser Eds., 1994, pp. 1247 12. Hou, B.Y., Yang, W.L. Dynamically twisted algebra Aq,p;πˆ (sbl(2)) as current algebra generalizing screening currents of q-deformed Virasoro algebra. q-alg/9709024 13. Fronsdal, C.: Publication RIMS Kyoto University 33, 91 (1997) and Lett. Math. Phys. 40, 117 (1997) 14. Jimbo, M., Konno, H., Odake, S., Shiraishi, J.: Quasi-Hopf twistors for elliptic quantum groups. qalg/9712029 15. Gervais, J.L., Neveu, A.: Novel triangle relation and absence of tachyons in Liouville string field theory. Nucl. Phys. B 238, 125 (1984) 16. Arnaudon, D., Buffenoir, E., Ragoucy, E., Roche, Ph.: Universal solutions of quantum dynamics Yang– Baxter equations. q-alg/9712037 17. Shiraishi, J., Kubo, H., Awata, H., Odake, S.: A Quantum Deformation of the Virasoro algebra and the Macdonald symmetric functions. Lett. Math. Phys. 38, 33 (1996) 18. Andric, I., Jevicki, A., Levine, H.: On the large N limit in symplectic matrix models. Nucl. Phys. B 215, 307 (1983) 19. Minahan, J.A., Polychronakos, A.P.: Density correlation functions in Calogero–Sutherland models. Phys. Rev. B 50, 4236 (1994) 20. Avan, J., Jevicki, A.: Classical integrability and higher symmetries of collective string field theory. Phys. Lett. B 266, 35 (1991) 21. Frenkel, E., Reshetikhin, N.: Quantum affine algebras and deformations of the Virasoro and W -algebras. Commun. Math. Phys. 178, 237 (1996) 22. Lukyanov, S.: A note on the deformed Virasoro algebra. Phys. Lett. B 367, 121 (1996) 23. Lukyanov, S., Pugai, Y.: Multipoint local height probabilities in the integrable RSOS model. Nucl. Phys. B 473, 631 (1996) 24. Jimbo, M., Shiraishi, J.: A coset-type construction for the deformed Virasoro algebra. q-alg/9709037 25. Avan, J., Frappat, L., Rossi, M., Sorba, P.: Poisson structures on the center of the elliptic algebra Aq,p (sbl(2)c ). Phys. Lett. A 235, 323 (1997) 26. Avan, J., Frappat, L., Rossi, M., Sorba, P.: New Wq,p (sl(2)) algebras from the elliptic algebra Aq,p (sbl(2)c ). Phys. Lett. A 239, 27 (1998) 27. Feigin, B., Frenkel, E.: Quantum W algebras and elliptic algebras. Commun. Math. Phys. 178, 653 (1996) 28. Avan, J., Frappat, L., Rossi, M., Sorba, P.: Central Extensions of Classical and Quantum q-Virasoro Algebras. Preprint LAPTH-680/98 and PAR-LPTHE 98-28, math. QA/9806065 29. Awata, H., Kubo, H., Odake, S., Shiraishi, J.: Quantum WN algebras and Macdonald polynomials. Commun. Math. Phys. 179, 401 (1996)

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30. Feigin, B., Jimbo, M., Miwa, T., Odesskii, A., Pugai, Ya.: Algebra of screening operators for the deformed WN Commun. Math. Phys. 191, 501–544 (1998) 31. Frenkel, E., Reshetikhin, N.Yu.: Towards deformed chiral algebras. Proceedings of the Quantum Group Symposium at the XXIth ICGTMP, H.D. Doebner and V.K. Dobrev eds., Sofia: Heron Press, 1997, p. 27 32. Asai, Y., Jimbo. M., Miwa, T., Pugai, Ya.: Bosonization of vertex operators for the A(1) N −1 face model. J. Phys. A 29, 6595 (1996) 33. Belavin, A.A.: Dynamical symmetry of integrable quantum systems. Nucl. Phys. B 180, 189 (1981) 34. Chudnovsky, D.V., Chudnovsky, G.V.: Completely X-symmetric S-matrices corresponding to theta functions. Phys. Lett. A 81, 105 (1981) 35. Tracy, C.A.: Embedded elliptic curves and the Yang–Baxter equation. Physica D 16, 203 (1985) 36. Richey, M.P., Tracy, C.A.: ZN Baxter model: Symmetries and the Belavin parametrization. J. Stat. Phys. 42, 311 (1986) 37. Reshetikhin, N.Yu., Semenov-Tian-Shansky, M.A.: Central extensions of quantum current groups. Lett. Math. Phys. 19, 133 (1990) 38. Mikhalkov, A.V.: JETP Lett. 30, 443 (1979) 39. Fadeev, L.D., Reshetikhin, N.Yu.: The Hamiltonian structures for integrable field models. Theor. Math. Phys. 56, 323 (1983) Communicated by T. Miwa

Commun. Math. Phys. 199, 729 (1999)

Communications in

Mathematical Physics © Springer-Verlag 1999

Erratum Universality of Correlation Functions of Hermitian Random Matrices in an External Field P. Zinn-Justin Department of Physics, Rutgers University, 126 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA Received: 30 September 1998 / Accepted: 23 October 1998 Commun. Math. Phys. 194, 631–650 (1998)

In the published version of the paper, Fig. 3 has been incorrectly reproduced, which made it look identical to Fig. 2. The correct figures appear below: poles

µ

λ

α

poles

β

λ

µ

λ

α

G(z)

β

λ a(z)

Fig. 2. Analytic structure of Gλ,µ and aλ,µ when λ and µ are on the same side of the cut [α, β]. When λ ∼ µ and the λ on the physical sheet (indicated by a small arrow) approaches the cut, it does not reach the pole at z = µ on the other sheet poles

poles

µ

µ λ

λ

α

G(z)

λ

β

α

λ

β

a(z)

Fig. 3. Analytic structure of Gλ,µ and aλ,µ when λ and µ are on opposite sides of the cut. This time the λ on the physical sheet, as it approaches the real axis, “sees” the pole at z = µ on the other sheet (through the cut) Communicated by A. Jaffe

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