Communications in Mathematical Physics - Volume 251

Commun. Math. Phys. 251, 1–25 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1165-9

Communications in

Mathematical Physics

A Hierarchical Model of Quantum Anharmonic Oscillators: Critical Point Convergence Sergio Albeverio1,4,5,6 , Yuri Kondratiev2,4,7 , Agnieszka Kozak3 , Yuri Kozitsky3 1 2 3 4 5 6 7

Abteilung für Stochastik, Universität Bonn, 53115 Bonn, Germany. E-mail: [email protected] Fakultät für Mathematik, Universität Bielefeld, 33615 Bielefeld, Germany. E-mail: [email protected] Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej, 20-031 Lublin, Poland. E-mail: {akozak, jkozi}@golem.umcs.lublin.pl BiBoS Research Centre, Bielefeld, Germany CERFIM, Locarno and Academia di Architettura, Università della Svizzera Italiana, Mendrisio, Switzerland Dipartimento di Matematica, Università di Trento, Trento, Italy Institute of Mathematics, Kiev, Ukraine

Received: 18 June 2002 / Accepted: 12 April 2004 Published online: 3 September 2004 – © Springer-Verlag 2004

Abstract: A hierarchical model of interacting quantum particles performing anharmonic oscillations is studied in the Euclidean approach, in which the local Gibbs states are constructed as measures on infinite dimensional spaces. The local states restricted to the subalgebra generated by fluctuations of displacements of particles are in the center of the study. They are described by means of the corresponding temperature Green (Matsubara) functions. The result of the paper is a theorem, which describes the critical point convergence of such Matsubara functions in the thermodynamic limit. Contents 1. 2. 3. 4. 5.

Introduction . . . . . . . Setup and the Theorem . Euclidean Representation Proof of the Theorem . . Proof of Lemma 5 . . . .

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1 6 9 14 16

1. Introduction Let L be a countable set (lattice). With each l ∈ L we associate a quantum mechanical particle with one degree of freedom described by the momentum pl and displacement ql operators. The system of such particles which we consider in this article is described by the heuristic Hamiltonian 
 1 a 2 1
2 4 p + ql + bql . (1.1) Jll  ql ql  + H =− 2  2m l 2 l,l

l

Here b > 0, a ∈ R and the sums run through the lattice L. The operators pl and ql satisfy the relation [pl , ql ] = pl ql − ql pl = 1/ i,

(1.2)

2

S. Albeverio Y. Kondratiev, A. Kozak, Y. Kozitsky

and m = mphys /–h2 is the reduced mass of the particle. Models like (1.1) have been studied for many years, see e.g., [23, 29]. They (and their simplified versions) are used as a base of models describing strong electron-electron correlations caused by the interaction of electrons with vibrating ions [14, 30]. Let L = {
n }n∈N0 , N0 = N ∪ {0} be a sequence of finite subsets of L, which is ordered by inclusion and exhausts L. For every
n , let H
n be a local Hamiltonian, corresponding to (1.1). In a standard way the Hamiltonians H
n determine local Gibbs states β,
n . A phase transition in the model (1.1) is connected with macroscopic displacements of particles from their equilibrium positions ql = 0, l ∈ L. To describe this phenomenon, one considers fluctuation operators (α) def

Q
n =

1 |
n |(1+α)/2




ql ,

α ≥ 0,

(1.3)

l∈
n

and Matsubara functions

   def (α) (τ1 , . . . , τ2k ) = β,
n Q
n exp −(τ2 − τ1 )H
n · · ·   (α)   × exp −(τ2k − τ2k−1 )H
n Q
n exp (τ2k − τ1 )H
n , k ∈ N,

α,β,
n

2k

(1.4)

with the arguments satisfying the condition 0 ≤ τ1 ≤ · · · ≤ τ2k ≤ β. In our model the interaction potential is taken to be Jll  = J [d(l, l  ) + 1]−1−δ ,

J, δ > 0,

(1.5)

where d(l, l  ) is a metric on L, determined by means of a hierarchical structure. The latter is a family of finite subsets of L, each of which belongs to a certain hierarchy level n ∈ N0 . This fact predetermines also our choice of the sequence L – the subsets
n are to be the elements of the hierarchical structure, that is typical for proving scaling limits in hierarchical models (see e.g., [12]). We prove (Theorem 1) that, for any δ ∈ (0, 1/2), the parameters a ∈ R, b > 0 and m > 0 can be chosen in such a way that there will exist β∗ > 0 with the following properties: (a) if β = β∗ , for all k ∈ N, the functions (1.4) converge  (2k)! J∗ k δ,β∗ ,
n 2k (τ1 , . . . , τ2k ) −→ , k!2k β∗k J

n → +∞

(1.6)

uniformly with respect to their arguments; here J∗ > 0 is a constant determined by the hierarchical structure; α,β,
(b) if β < β∗ , for all α > 0 and k ∈ N, the functions 2k n converge to zero in the same sense. The convergence of the functions (1.4) like in (1.6) but with α = 1 would correspond to the appearance of a long-range order, which destroys the Z2 -symmetry. Thus, claim (a) describes a critical point where the fluctuations are abnormal (since α = δ > 0) but not strong enough to destroy the mentioned symmetry. Such fluctuations are classical (non-quantum), which follows from the fact that the limits (1.6) are independent of τ . Due to the hierarchical structure the model (1.1) is self-similar. In translation invariant lattice models self-similarity appears at their critical points [27, 28]. This, among others, is the reason why the critical point properties of hierarchical models of classical

Hierarchical Model of Quantum Anharmonic Oscillators

3

statistical mechanics have attracted attention during the last three decades. An expository review of the results in this domain is given in [12]. In the model (1.1) the oscillations are described by unbounded operators1 . The same model was studied in our previous works [2–4]. In [2] a preliminary study of the model was performed. A theorem describing the critical point convergence was announced in [3]. In [4] we have shown that the critical point of the model (1.1) can be suppressed by strong quantum effects, which take place, in particular, when the mass m is less than a certain bound m∗ 2 . In the present paper we give a complete proof of the critical point convergence, which appears for sufficiently large values of the mass (see the discussion at the very end of this introduction). It should be pointed out that, to the best of our knowledge, our result is the first example of a theorem, which describes the convergence at the critical point of a nontrivial quantum model, published by this time. Let us outline the main aspects of the proof. By symmetry, the functions (1.4) are def

extended to Iβ2k , where Iβ = [0, β]. Then for x ∈ L2 (Iβ ), one sets ϕn(α) (x)

=1+

∞
k=1

1 α,β,
n  (τ1 , . . . , τ2k )x(τ1 ) · · · x(τ2k )dτ1 · · · dτ2k , (1.7) (2k)! Iβ2k 2k

and log ϕn(α) (x) =

∞
k=1

1 α,β,
n U (τ1 , . . . , τ2k )x(τ1 ) · · · x(τ2k )dτ1 · · · dτ2k , (1.8) (2k)! Iβ2k 2k α,β,
n

which uniquely determines the Ursell functions U2k result may be formulated as follows: δ,β∗ ,
n

U2

∀k > 1 : ∀k ∈ N, β <

. In terms of these functions our

(τ1 , τ2 ) −→ β∗−1 ,

δ,β ,
U2k ∗ n (τ1 , . . . , τ2k ) −→ 0, α,β,
β∗ , α > 0 : U2k n (τ1 , . . . , τ2k )

(1.9) −→ 0,

which holds uniformly with respect to the arguments τj ∈ Iβ , j = 1, . . . , 2k as n → +∞. Here we have set J = J∗ , that can always be done by choosing an appropriate scale of β. We prove (1.9) in the framework of the Euclidean approach in quantum statistical mechanics based on the representation of the functions (1.4) in the form of functional integrals. This approach was initiated in [1, 15], its detailed description and an extended related bibliography may be found in [6]. In separate publications we are going to exploit our result, in particular, to construct self-similar Gibbs states (in the spirit of [11, 12] where it was done for classical hierarchical models). α,β,
α,β,
The functions 2k n , U2k n , k ∈ N are continuous on Iβ2k , see [6]. In view of our choice of the potential energy in (1.1), the Ursell functions satisfy the sign rule α,β,
n

(−1)k−1 U2k

(τ1 , . . . , τ2k ) ≥ 0,

(1.10)

1 Certain aspects of critical point behaviour of quantum hierarchical models with bounded (spin) operators were studied in [20]. 2 Physical aspects of such quantum effects were analyzed in [7].

4

S. Albeverio Y. Kondratiev, A. Kozak, Y. Kozitsky α,β,
n

for all k ∈ N and (τ1 , . . . , τ2k ) ∈ Iβ2k . We prove that the families {2k

}n∈N0 ,

α,β,
{U2k n }n∈N0 ,

k ∈ N are equicontinuous; hence, the convergence (1.9) can be proven δ,β ,
α,β,
n by showing the convergence of U2 ∗ n , U2 , as in (1.9), and

α,β,
def α,β,
U2k n = U2k n (τ1 , . . . , τ2k )dτ1 · · · dτ2k −→ 0, (1.11) Iβ2k

which has to hold for all k ≥ 2, β ≤ β∗ , and for α = δ if β = β∗ , and α > 0 if β < β∗ . Another fact which we employ here is also a consequence of the choice of the potential energy in (1.1). By a version of the Lee-Yang theorem, the function fn of a single complex variable defined in the vicinity of z = 0 by the series log fn (z) =

∞
k=1

1 α,β,
n 2k z , U (2k)! 2k

(1.12)

can be extended to an even entire function of order less than two possessing imaginary zeros only. This implies fn (z) =

∞ 

(n)

(1 + cj z2 ),

(n)

∞


(n)

c1 ≥ c2 ≥ · · · > 0,

j =1

j =1

(n)

cj

< ∞,

(1.13)

yielding for the numbers (1.11) the following representation: α,β,
n

U2k

= 2(2k − 1)!(−1)k−1

∞
j =1

(n) k

cj

,

k ∈ N,

(1.14)

by which, α,β,
|U2k n | α,β,
n

|U2k

≤ 2(2k − 1)!



∞


(n) k−2 (n) 2 c1 , cj j =1

k−1 α,β,
n (n) | ≤ (2k − 1)! c1 U2

k ≥ 2,

(1.15)

k ∈ N,

and hence

α,β,
n k−2 α,β,
n | ≤ (2k − 1)!(21−k /3) U2 |U4 |,

α,β,
α,β,
n k |U2k n | ≤ (2k − 1)!21−k U2 , k ∈ N. α,β,
n

|U2k

k ≥ 2,

(1.16)

α,β,
n

Then the convergence (1.9) follows from the corresponding convergence of U2 from the fact δ,β∗ ,
n

U4

−→ 0.

and

(1.17)

The above arguments allow us to prove the convergence of an infinite number of sequences of functions by controlling just two sequences of numbers – {uˆ n }n∈N0 and δ,β ,
δ,β,
n . The sign rule (1.10) and the representation {U4 ∗ n }n∈N0 , where uˆ n = β −1 U2

Hierarchical Model of Quantum Anharmonic Oscillators

5

(1.14) are proven in Lemmas 1 and 2 below by means of the lattice approximation α,β,
technique [6]. Here the functions 2k n , k ∈ N, are obtained as limits of moments of Gibbs measures of classical ferromagnetic φ 4 -models. This allows us to employ the corresponding properties of the φ 4 -models proven in [24] (the sign rule), [2] (a correlation inequality) and [19] (the Lee-Yang theorem). Then to controlling the sequences δ,β ,
{uˆ n }n∈N0 and {U4 ∗ n }n∈N0 we apply a version of the inductive method developed in [17, 18]. The central role here is played by Lemma 5. It establishes the existence of β∗ > 0 such that, for β = β∗ (respectively, for β < β∗ ), the sequence {uˆ n }n∈N0 conδ,β,
n }n∈N0 converges verges to one (respectively, to zero as |
n |−δ ). The sequence {U4 to zero in both cases. The latter fact is proven by constructing a converging to zero δ,β,
n sequence of positive numbers {Xn }n∈N0 , such that β −2 |U4 | ≤ Xn for all n ∈ N0 and β ≤ β∗ . The proof of Lemma 5 is based on recurrent estimates (Lemma 6) yielding upper and lower bounds for uˆ n and Xn in terms of certain functions of uˆ n−1 and Xn−1 . The analysis of these estimates shows that the simultaneous convergence uˆ n → 1 and Xn → 0 can be guaranteed if these sequences are confined to the intervals uˆ n ∈ (1, v) ¯ and Xn ∈ (0, w), ¯ where the parameters v¯ > 1 and w¯ > 0 depend on δ and on the details of the hierarchical structure only and can be computed explicitly. Lemma 6 is proven by comparing solutions of certain differential equations, similarly as in [17, 18]. ± Lemma 9 establishes the existence of βn± > 0, βn− < βn+ if βn−1 do exist. These num+ bers are defined as follows: uˆ n = v¯ for β = βn , and uˆ n < v¯ for β < βn+ ; uˆ n = 1 for β = βn− , and uˆ n < 1 for β < βn− . The proof of Lemma 9 is carried out by means of the estimates obtained in Lemma 6. In Lemma 8 we prove that the parameters m, a and b can be chosen in such a way that β0± do exist. In Lemma 10 we prove the existence of β∗ , such that ∀n ∈ N0 : uˆ n ∈ (1, v) ¯ for β = β∗ , and uˆ n → 0 as |
n |−δ for β < β∗ . The proof is based on the estimates obtained in Lemma 6. In Lemma 3 we prove that all uˆ n , n ∈ N0 are continuous functions of β and describe certain useful α ,β,
n properties of the Ursell functions U2 ∗ (τ, τ  ), n ∈ N0 , implying e.g., the mentioned equicontinuity. The proof of Lemma 8 is based on the estimates of uˆ 0 and X0 obtained in Lemma 7. In particular, we prove that       mγ 2 3β βγ 16 1 − exp − ≤ uˆ 0 ≤ 1+ 1+ , 36 mγ 8 βγ where a < 0, γ = |a|/b. Then for mγ 2 > 36v, ¯ one gets uˆ 0 > v¯ for sufficiently large β. On the other hand, uˆ 0 → 0 as β → 0. Since uˆ 0 depends on β continuously, this yields the existence of β0± . Furthermore, for fixed γ and β, we show that X0 ≤ bC with a certain fixed C > 0. This was used to provide X0 < w, ¯ and hence Xn < w, ¯ n ∈ N, for sufficiently small b > 0. Another upper bound of uˆ 0 was obtained in [4]. It is well-known that the one particle Hamiltonian which stands in the square brackets in (1.1) has a pure point non-degenerate spectrum. Let En , n ∈ N0 be its eigenvalues and  = minn∈N (En − En−1 ). In [4] we proved that if m2 > 1, then uˆ 0 < 1 and hence uˆ n → 0 for all β. In what follows, the critical point of the model exists if a < 0 and the parameters m(|a|/b)2 , 1/b are big enough; such a point does not exist if ‘the quantum rigidity’ m2 (see [7]) is greater than 1. By Lemma 1.1 of [4], m2 ∼ m−1/3 C, C > 0 as m → 0, which means that small values of the mass prevent the system from criticality.

6

S. Albeverio Y. Kondratiev, A. Kozak, Y. Kozitsky

2. Setup and the Theorem Like in [3, 4] we consider the hierarchical model defined on L = N0 . Given κ ∈ N \ {1}, we set
n,s = {l ∈ N0 | κ n s ≤ l ≤ κ n (s + 1) − 1}, Then for n ∈ N, one has




n,s =


n−k,l ,

s, n ∈ N0 .

k = 1, 2, . . . , n.

(2.1)

(2.2)

l∈
k,s

The collection of families {
n,s }s∈N0 , n ∈ N0 is called a hierarchical structure on L. Given l, l  ∈ L, we set n(l, l  ) = min{n | ∃
n,s : l, l  ∈
n,s },



d(l, l  ) = κ n(l,l ) − 1.

(2.3)

The function d : L×L → [0, +∞) has the following property: any triple {l1 , l2 , l3 } ⊂ L contains two elements, say l1 , l2 , such that d(l1 , l3 ) = d(l2 , l3 ). Thus, d(l, l  ) is a metric on L. The interaction potential in our model has the form of (1.5) with the above metric d(l, l  ). It is invariant under the transformations of L which leave d(l, l  ) unchanged. In view of this fact, it is convenient to choose the sequences L which determine the infinite-volume limit to be consisting of the sets (2.2) only. A standard choice is the def

sequence of
n,0 =
n , n ∈ N0 . The Hamiltonian (1.1) may be rewritten in the form  2  ∞
 1 a 2 θ
−n(1+δ)

2 4   κ ql + H =− p + ql + bql , 2 2m l 2  n=0

l∈L

l ∈
n,l

(2.4)

l∈L

where θ = J (1 − κ −(1+δ) ) > 0. The local Hamiltonians indexed by
n,l are obtained from the above one by the corresponding truncation of the sums. For our purposes, it is convenient to write them recursively,  2
θ −n(1+δ) 
 def H
n,l = Hn,l = − κ qs + Hn−1,s , (2.5) 2 s∈
n,l

s∈
1,l

where the one particle Hamiltonian is H0,l =

1 2 a 2 p + ql + bq4l . 2m l 2

(2.6)

The canonical pair pl , ql , as well as the Hamiltonian H0,l , are defined in the complex Hilbert space Hl = L2 (R) as unbounded operators, which are essentially self-adjoint on the dense domain C0∞ (R). The Hamiltonian Hn,l , n ∈ N is defined similarly but in the space Hn,l = L2 (R|
n,l | ). The local Gibbs state in
n,l at a given temperature β −1 > 0 is defined on Cn,j – the ∗ C -algebra of bounded operators on Hn,l , as follows:    trace A exp −βHn,l   , A ∈ Cn,l . β,
n,l (A) = (2.7) trace exp −βHn,l

Hierarchical Model of Quantum Anharmonic Oscillators

7

In a standard way, it may be extended to unbounded operators such as ql  , l  ∈
n,l . The dynamics in
n,l is described by the time automorphisms of Cn,l ,     atn,l (A) = exp itHn,l A exp −itHn,l , t ∈ R. (2.8) For a measurable function A : R|
n,l | → C, the multiplication operator A acts on ψ ∈ Hn,l as (Aψ)(x) = A(x)ψ(x), x ∈ R|
n,l | . It appears that the linear span of the operators k 1 (A1 ) · · · atn,l (Ak ), atn,l

k ∈ N, t1 , . . . , tk ∈ R,

with all possible choices of k, t1 , . . . tk and multiplication operators A1 , . . . , Ak ∈ Cn,l is dense in the algebra Cn,l in the σ -weak topology, in which the state (2.7) is continuous. Thus, this state is fully determined by temporal Green functions tk t1 Gn,l A1 ,...,Ak (t1 , . . . , tk ) = β,
n,l (an,l (A1 ) . . . an,l (Ak )),

(2.9)

corresponding to all possible multiplication operators A1 , . . . , Ak ∈ Cn,l . Set β

Dk = {(t1 , . . . , tk ) ∈ Ck | 0 < Im(t1 ) < . . . Im(tk ) < β}.

(2.10)

As was proven in Lemma 2.1 in [6], every Green function (2.9) may be extended to a β β holomorphic function on Dk . This extension is continuous on the closure of Dk and may be uniquely determined by its values on the set β

β

Dk (0) = {(t1 , . . . , tk ) ∈ Dk | Re(tj ) = 0,

j = 1, . . . , k}.

(2.11)

β

The restriction of Gn,l A1 ,...,Ak to Dk (0), i.e., the function n,l A (τ1 , . . . , τk ) = Gn,l A1 ,...,Ak (iτ1 , . . . , iτk ), 1 ,...,Ak

(2.12)

is the Matsubara function corresponding to the operators A1 , . . . , Ak . By (2.7)–(2.9), it may be written    1 n,l A (τ1 , . . . , τk ) = trace A1 exp −(τ2 − τ1 )Hn,l 1 ,...,Ak Zn,l     ×A2 exp −(τ3 − τ2 )Hn,l . . . Ak exp −(β − τk + τ1 )Hn,l ;    def Zn,l = trace exp −βHn,l .

(2.13)

This representation immediately yields the ‘KMS-periodicity’ n,l n,l A (τ1 + ϑ, . . . , τk + ϑ) = A (τ1 , . . . , τk ), 1 ,...,Ak 1 ,...,Ak def

(2.14)

for every ϑ ∈ Iβ = [0, β], where addition is of modulo β. As was mentioned in the introduction, the phase transition in the model is connected with the appearance of macroscopic displacements of particles from their equilibrium

8

S. Albeverio Y. Kondratiev, A. Kozak, Y. Kozitsky

positions ql = 0, which occur when the fluctuations of such displacements become large. To describe them, we set (cf., (1.3)) Qλn,l = 




λn

|
n,l | l  ∈


ql  =

λn
ql  , κ n/2 

(2.15)

l ∈
n,l

n,l

where {λn }n∈N0 is a sequence of positive numbers. The operators Qλn,l are unbounded, nevertheless, the corresponding Matsubara functions still possess almost all of those ‘nice’ properties which they have in the case of bounded operators. The next statement follows directly from Corollary 4.1 and Theorem 4.2 of [6]. n,l Proposition 1. For every n ∈ N0 and k ∈ N, the functions Q λ β

λ n,l ,...,Qn,l

are continuous

on Iβ2k , they can be analytically continued to the domains D2k . n,j

The convergence of the sequence {Qλ

λ n,j ,...,Qn,j

}n∈N0 with λn = κ −n/2 , to a nonzero limit

would mean the appearance of the long range order caused by macroscopic displacements of particles. The convergence with a slower decaying sequence {λn } corresponds to the presence of a critical point. Our model is described by the following parameters: δ > 0, which determines the decay of the potential Jll  , see (1.5); θ ≥ 0, which determines its strength; the mass m and the parameters of the potential energy a and b, see e.g., (1.1). Since the choice of θ determines only the scale of β, we may set θ = κ δ − 1,

(2.16)

which corresponds to the choice (see (1.6)) def

J = J∗ =

κδ − 1 . 1 − κ −1−δ

To simplify notations we write the operator (2.15) with λn = κ −nδ/2 as Qn,l and (n)

n,l Q (τ1 , . . . , τ2k ) = 2k (τ1 , . . . , τ2k ). n,l ,...,Qn,l

(2.17)

Theorem 1. For the model (1.1) with δ ∈ (0, 1/2), one can choose the parameters a, b and m in such a way that there will exist β∗ > 0, dependent on a, b, m, with the following properties: (a) if β = β∗ , then for all k ∈ N, the convergence (n)

2k (τ1 , . . . , τ2k ) −→

(2k)! , k!2k β∗k

(2.18) α,β,
n,l

holds uniformly on (τ1 , . . . , τ2k ) ∈ Iβ2k ; (b) if β < β∗ , the functions 2k defined by (1.4) converge to zero in the same sense for all α > 0.

,k ∈N

Hierarchical Model of Quantum Anharmonic Oscillators

9

3. Euclidean Representation In the Euclidean approach [6] the functions (2.12) corresponding to the multiplication operators A1 , . . . , A2k , are written as follows:

n,l (τ , . . . , τ ) = A1 (ωn,l (τ1 )) . . . A2k (ωn,l (τ2k ))νn,l (dωn,l ), (3.1) A 1 2k 1 ,...,A2k n,l

where n,l is the Banach space of real valued continuous periodic functions n,l = {ωn,l = (ωl  )l  ∈
n,l | ωl  ∈ },  = {ω ∈ C(Iβ → R) | ω(0) = ω(β)}.

(3.2)

The probability measure νn,l is   1 exp −En,l (ωn,l ) χn,l (dωn,l ), Zn,l

  = exp −En,l (ωn,l ) χn,l (dωn,l ).

νn,l (dωn,l ) = Zn,l

(3.3)

n,l

The functions En,l : n,l → R are (cf., (2.5))  2


1 −n(1+δ) β 
ωl  (τ ) dτ + En−1,s (ωn−1,s ), En,j (ωn,j ) = − θκ 2 0 s∈
1,l l  ∈
n,l

β a−1 2 4 (3.4) E0,s (ωs ) = [ωs (τ )] + b[ωs (τ )] dτ. 2 0 We consider ωn,l as vectors (ωn−k,s )s∈
k,l with k = 1, 2, . . . , n and write ωs for ω0,s . The measure χn,l is  χn,l (dωn,l ) = χ (dωs ), (3.5) s∈
n,l

where χ is a Gaussian measure on 0,s = . Let E be the real Hilbert space L2 (Iβ ). Then the Banach space of continuous periodic functions  can be considered, up to embedding, as a subset of E. The following family  2   q > 0,  β cos qτ,   eq (τ ) = − 2 sin qτ, q < 0, (3.6)  β    √  1/ β, q = 0, with q varying in the set Q = {q | q =

2π κ, κ ∈ Z}, β

(3.7)

is a base of E. Given q ∈ Q, let Pq be the orthonormal projection on eq . We define χ to be the Gaussian measure3 on E with zero mean and with the covariance operator 3 For a topological space, ‘measure defined on the space’ means that the measure is defined on its Borel σ -algebra.

10

S. Albeverio Y. Kondratiev, A. Kozak, Y. Kozitsky

S=


q∈Q

1 Pq . +1

(3.8)

mq 2

One can show (see Lemma 2.2 of [6]) that the measure χ is concentrated on , i.e., χ() = 1. On the other hand, as follows from the Kuratowski theorem (see Theorem 3.9, p. 21 of [22]), the Borel σ -algebras of subsets of , generated by its own topology and by the topology induced from the Hilbert space E, coincide. Hence, one can consider χ also as a measure on . As such, it appears in the representation (3.5). The fluctuation operator Qn,l , defined by (2.15) with λn = κ −nδ/2 is a multiplication operator by the function Qn,l : R|
n,l | → R,

Qn,l (ξn,l ) = κ −n(1+δ)/2 ξs = κ −(1+δ)/2 Qn−1,s (ξn−1,s ). (3.9) s∈
n,l

s∈
1,l

The representation (3.1) and the properties of the measures νn,l , χn,l , χ (see Lemma 2.3 and all of Sect. 2.2 of [6]) yield the following statement. Proposition 2. For every fixed β > 0, τ1 , . . . , τ2k ∈ Iβ , the Matsubara functions (2.17) continuously depend on m > 0, a ∈ R and b > 0. Proposition 3. For all n ∈ N0 and k ∈ N, the functions (2.17) obey the estimates (n)

0 ≤ 2k (τ1 , . . . , τ2k ) ≤

k


(n) 

 τσ (2l−1) , τσ (2l) ,

2

(3.10)

σ l=1

which hold for all τ1 , . . . , τ2k ∈ Iβ . Here the sum is taken over all possible partitions of the set {1, . . . , 2k} onto unordered pairs. The estimates (3.10) were proven in [6] as Theorems 6.2 (positivity) and 6.4 (Gaussian upper bound). Since to prove our theorem we need the Matsubara functions corresponding to the operators Qn,l only, we may restrict our study to the measures describing distributions of Qn,l given by (3.9). For n ∈ N0 and a Borel subset C ⊂ , let
BC = {ωn,l ∈ n,l | κ −n(1+δ)/2 ωs ∈ C}, s∈
n,l

which is a Borel subset of n,l . Then we set µn (C) = νn,l (BC ), which defines a measure on . By (3.3), (3.4), the measures µn obey the following recursion relation:  1 θ κ µn (dω) = ω2E µn−1 exp (κ (1+δ)/2 dω), (3.11) Zn 2   1 µ0 (dω) = exp −E0,s (ω) χ (dω), (3.12) Z0 where  · E is the norm in the Hilbert space E = L2 (Iβ ), the function E0,s is given by (3.4), Zn , n ∈ N are normalizing constants and  stands for convolution. For obvious reasons, we drop the labels l and s. Like the measure χ , all µn , n ∈ N0 can be considered either as measures on the Hilbert space E concentrated on its subset , or as measures on the Banach space . We have

Hierarchical Model of Quantum Anharmonic Oscillators

11



(n)

2k (τ1 , . . . , τ2k ) =



ω(τ1 ) · · · ω(τ2k )µn (dω),

and the function (1.7) may be written in the form



def exp((x, ω)E )µn (dω) = exp((x, ω)E )µn (dω), ϕn(δ) (x) = ϕn (x) = E

(3.13)

x ∈ E,



(3.14)

where (·, ·)E is the scalar product in E. Expanding its logarithm into the series (1.8) we obtain the Ursell functions (cf., (2.17)) δ,β,
n,s

U2k

def

(n)

(τ1 , . . . , τ2k ) = U2k (τ1 , . . . , τ2k ),

k ∈ N.

(3.15) (n)

Correspondingly, the numbers (1.11) obtained from these functions are denoted by U2k . (n) (n) Each function U2k can be written as a polynomial of the Matsubara functions 2s , s = 1, 2, . . . , k and vice versa. In particular, (n)

(n)

U2 (τ1 , τ2 ) = 2 (τ1 , τ2 ), (n) U4 (τ1 , . . . , τ4 )

=

(3.16)

(n) (n) (n) 4 (τ1 , . . . , τ4 ) − 2 (τ1 , τ2 )2 (τ3 , τ4 ) (n) (n) (n) (n) −2 (τ1 , τ3 )2 (τ2 , τ4 ) − 2 (τ1 , τ4 )2 (τ2 , τ3 ).

In view of (2.14), the Matsubara and Ursell functions depend only on the periodic distances between τj , i.e., on |τi − τj |β = min{|τi − τj |, β − |τi − τj |}. The proof of Theorem 1 is based on inequalities for the Matsubara and Ursell functions, which we obtain by means of the lattice approximation method. Its main idea is to construct sequences of probability measures, concentrated on finite dimensional subspaces of n,l , which converge to the Euclidean measures νn,l in such a way that the integrals (3.1) are the limits of the corresponding integrals taken with such approximating measures. Then the latter integrals are rewritten as moments of Gibbs measures of classical ferromagnetic models, for which one has a number of useful inequalities. In such a way, these inequalities are transferred to the Matsubara and Ursell functions. A detailed description of this method is given in Sect. 5 of [6]. Here we provide a short explanation of its main elements. Given N = 2L, L ∈ N, set !  "−1   2 β 2N 2 (N) λq = m sin q +1 , (3.17) β 2N and SN =


q∈QN

λ(N) q Pq ,

QN = {q =

2π κ | κ = −(L − 1), . . . , L}, β

(3.18)

where the projectors Pq are the same as in (3.8). Now let χN be the Gaussian measure (N) on E with the covariance operator SN . Let also χn,l be defined by (3.5) with χN instead (N)

(N)

of χ . By means of χn,l , we define νn,l via (3.3). Then by Theorem 5.1 of [6], one has

(N) (n) Qn,l (ωn,l (τ1 )) · · · Qn,l (ωn,l (τ2k ))νn,l (dωn,l ) −→ 2k (τ1 , . . . , τ2k ), (3.19) n,l

12

S. Albeverio Y. Kondratiev, A. Kozak, Y. Kozitsky

pointwise on Iβ2k as N → +∞. On the other hand, one can write
S1 · · · S2k , LHS(3.19) = C2k,N

(3.20)

1 ,...2k

where C2k,N > 0 is a constant and · stands for the expectation with respect to the (N) def

local Gibbs measure on n,l =
n,l × {1, 2, . . . , N} of a ferromagnetic model with the one dimensional φ 4 single-spin distribution. This type of single-spin distribution is determined by our choice of the potential energy in (1.1), whereas the ferromagneticity is due to the fact that J > 0 (see (1.5)) and due to our choice of the numbers (3.17). The (1) (2) sum in (3.20) is taken over the vectors j = (j , j ), j = 1, . . . , 2k as follows. Their first components run through
n,l and the second components are fixed at certain values from the set {1, . . . , N}, determined by the corresponding τj . Furthermore, the above expectations · can be approximated by expectations with respect to the ferromagnetic (n) (n) Ising model (classical Ising approximation [25, 26]). Then the functions 2k and U2k obey the inequalities which the moments and semi-invariants of the ferromagnetic Ising model do obey. In particular, we have the following. Lemma 1. For all n ∈ N0 and k ∈ N, the following estimates hold for all values of the arguments τ, τ  , τ1 , . . . , τ2k ∈ Iβ ,



(n) (n) U4 (τ, τ, τ1 , τ2 )dτ1 dτ2 ≤ U4 (τ, τ  , τ1 , τ2 )dτ1 dτ2 ; (3.21) Iβ2

Iβ2

(n)

(−1)k−1 U2k (τ1 , τ2 , . . . , τ2k ) ≥ 0.

(3.22)

Proof. For classical models with unbounded spins and polynomial anharmonicity of the Ellis-Monroe type (for φ 4 -models, in particular), (3.21) was proved in [2]. For the Ising model, the sign rule (3.22) was proved in [24].   Lemma 2. For all n, l ∈ N0 , the function  β

exp z Qn,l (ωn,l (τ ))dτ νn,l (dωn,l ) fn (z) = n,l

=

E



exp z 0

0 β

(3.23)



ω(τ )dτ µn (dω),

can be analytically continued to an even entire function of order less than two, possessing purely imaginary zeros. Proof. For the function (3.23), one can construct the lattice approximation (cf., (3.20))   & #
  fn(N) (z) = exp z (3.24) S  , (N )

∈n,l

(N)

which converges, as N → +∞, to fn (z) for every z ∈ R. For such fn , the property (N) stated is known as the generalized Lee-Yang theorem [19]. The functions fn are ridge (crested), with the ridge being the real axis. For sequences of such functions, their pointwise convergence on the ridge implies via the Vitali theorem (see e.g., Prop. VIII.19 in [26]) the uniform convergence on compact subsets of C, which by the Hurwitz theorem (see e.g., [10]) gives the desired property of fn .  

Hierarchical Model of Quantum Anharmonic Oscillators

Set

uˆ n (q) =

β 0 β

= 0

13

U2 (τ  , τ ) cos(qτ )dτ, (n)

(n)

U2 (0, τ ) cos(qτ )dτ,

Then (n)

U2 (τ1 , τ2 ) =

(3.25) q ∈ Q,

n ∈ N0 .

1
uˆ n (q) cos[q(τ1 − τ2 )]. β

(3.26)

q∈Q

Furthermore, we set (cf., (1.11))

(n) (n) U2k (τ1 , . . . , τ2k )dτ1 · · · dτ2k . U2k = Iβ2k

(3.27)

Then uˆ n (0) = uˆ n = β −1 U2 def

(n)

= β −1 U2

δ,β,
n

(3.28)

.

Lemma 3. For every n ∈ N0 and q ∈ Q, uˆ n (q) is a continuous function of β, it obeys the following estimates: (3.29) 0 < uˆ n (q) ≤ uˆ n ; −nδ κ uˆ n (q) ≤ , q = 0. (3.30) mq 2 Proof. By (3.25), (3.16), (2.17) and (2.13), one obtains      1 (n) trace Qn,l exp −τ Hn,l Qn,l exp −(β − τ )Hn,l . U2 (0, τ ) = Zn,l (n)

It may be shown that every Hn,l has a pure point spectrum {Ep }p∈N0 . We denote the (n) corresponding eigenfunctions by p and set (n)

(n)

Qpp =(Qn,l p(n) , p )Hn,l . Then the above representation may be rewritten


'' (n) ''2 1 (n) (n) U2 (0, τ ) = 'Qpp ' exp −βEp(n) + τ (Ep(n) − Ep ) , Zn,l  p,p ∈N0

which yields via (3.25) 1 uˆ n (q) = Zn,l

Zn,l


'' (n) ''2 'Qpp '

(n)

(n)

Ep − E p  (n)

(n)

q 2 + (Ep − Ep )2 ( ) (n) × exp[−βEp ] − exp[−βEp(n) ] ,
= exp[−βEp(n) ]. p,p ∈N0

(3.31)

p∈N0

Both series above converge uniformly, as functions of β, on compact subsets of (0, +∞), which yields continuity and positivity. The upper bound (3.29) follows from (3.31) or

14

S. Albeverio Y. Kondratiev, A. Kozak, Y. Kozitsky

from (3.25). To prove (3.30), we estimate the denominator in (3.31) from below by q 2 = 0 and obtain ( ) 1 1
'' (n) ''2 (n) (n) (n) uˆ n (q) ≤ 2 'Qpp ' (Ep − Ep ) exp[−βEp ] − exp[−βEp(n) ] q Zn,l  p,p

     1 1 = 2 trace Qn,l , Hn,l , Qn,l exp −βHn,l , q Zn,l

q = 0.

(3.32)

By means of (2.5) and (1.2), the double commutator in (3.32) may be computed explicitly. ' '−δ It equals to '
n,l ' /m, which yields (3.30).   (n)

Lemma 4. The numbers U2k defined by (3.27) obey the estimates (cf., (1.16)) (n)

|U2k | ≤ 21−k (2k − 1)!(β uˆ n )k , k ∈ N, (3.33) (2k − 1)! (n) (n) (β uˆ n )k−2 |U4 |, k ≥ 2. (3.34) |U2k | ≤ 3 · 2k−1 Proof. The function (3.23) is the same as in (1.12), hence, it possesses the representation δ,β,
(n) (1.13) and U2k = U2k n,l are given by the right-hand side of (1.14). Then the estimates (3.33), (3.34) immediately follow from (1.16).   4. Proof of the Theorem Set

Xn = −

(n)

Iβ2

U4 (τ, τ, τ1 , τ2 )dτ1 dτ2 .

(4.1)

Then by Lemma 1, one has

' ' ' (n) ' 0 < β −2 'U4 ' ≤ Xn ,

for all n ∈ N0 ,

(4.2)

(n)

thus, we may control the sequence {U4 }n∈N0 by controlling {Xn }n∈N0 . Lemma 5. For the model (1.1) with δ ∈ (0, 1/2), one can choose the parameters a, b and m in such a way that there will exist β∗ > 0, dependent on a, b, m, with the following properties: (a) for β ≤ β∗ , {Xn }n∈N0 → 0; (b) for β = β∗ , {uˆ n }n∈N0 → 1; for β < β∗ , there exists K(β) > 0 such that for all n ∈ N0 , uˆ n ≤ K(β)κ −nδ .

(4.3)

The proof of this lemma will be given in the concluding section of the article. Lemmas 3 and 5 have two important corollaries. (n)

(n)

Corollary 1. For every β ≤ β∗ and k ∈ N, the sequences {2k }n∈N0 , {U2k }n∈N0 are relatively compact in the topology of uniform convergence on Iβ2k . (n)

(n)

Proof. Since the Ursell functions U2k may be expressed as polynomials of 2s with s = 1, . . . , k and vice versa, it is enough to prove this statement for the Matsubara functions only. By Ascoli’s theorem (see e.g., [21] p. 72) we have to show that the (n) sequence {2k }n∈N0 is pointwise bounded and equicontinuous. By (3.30) and (3.26),

Hierarchical Model of Quantum Anharmonic Oscillators

2 (τ, τ  ) ≤ 2 (0, 0) ≤ (n)

(n)

15

1 κ −nδ uˆ n + β βm


q∈Q\{0}

1 . q2

(4.4)

For β ≤ β∗ , the sequence {uˆ n }n∈N0 is bounded by Lemma 5. Together with the Gauss(n) ian upper bound (3.10) this yields the uniform boundedness of 2k on Iβ2k . Further, by (3.13) (n)

(n)

2k (τ1 , . . . , τ2k ) − 2k (ϑ1 , . . . , ϑ2k ) (4.5)


2k = ω(τ1 ) · · · ω(τl−1 ) [ω(τl ) − ω(ϑl )] ω(ϑl+1 ) · · · ω(ϑ2k )µn (dω). E l=1

Applying here the Schwarz inequality (as to the scalar product in L2 (E, µn ) of [ω(τl ) − ω(ϑl )] and the rest of ω), the Gaussian upper bound (3.10) and the left-hand inequality in (4.4) one gets (n)

(n)

|2k (τ1 , . . . , τ2k ) − 2k (ϑ1 , . . . , ϑ2k )|2 ( ) 8k 2 (4k − 2)! ( )2k−1 (n) (n) (n) ≤ 2 (0, 0) − 2 (τ, ϑ) ·  (0, 0) , 2 (2k − 1)!22k−1

(4.6)

where (τ, ϑ) is chosen amongst the pairs (τl , ϑl ), l = 1, . . . , 2k to obey |τ − ϑ|β = maxl |τl − ϑl |β . But by (3.26), (3.30), 2
(n) (n) 2 (0, 0) − 2 (τ, ϑ) = uˆ n (q) {sin [(q/2) (τ − ϑ)]}2 β q∈Q

κ −nδ ≤2 βm ≤ Cκ with an appropriate C > 0.

−nδ


q∈Q\{0}

1 {sin [(q/2) (τ − ϑ)]}2 q2

|τ − ϑ|β ,

 

The next fact follows immediately from (3.30) and (3.7). Corollary 2. For every β,




uˆ n (q) −→ 0, n → +∞.

q∈Q\{0}

Proof of Theorem 1. By Lemma 5, (3.34), and (4.1), (4.2), one obtains that for all k ≥ 2 (n) and β ≤ β∗ , {U2k }n∈N0 → 0. Then by the sign rule (3.21), for all k ≥ 2, the sequences (n) {U2k }n∈N0 converge to zero for almost all (τ1 , . . . , τ2k ) ∈ Iβ2k , which, by Corollary 1, yields their uniform convergence to zero. By (3.26) – (3.30), Corollary 2 and Lemma 5, one has for β = β∗ , 1 1
(n) uˆ n (q) cos[q(τ1 − τ2 )] −→ 1/β∗ , (4.7) U2 (τ1 , τ2 ) = uˆ n + β β q∈Q\{0}

(n)

(n)

uniformly on Iβ2 . Now one can express each 2k polynomially by U2l with l = 1, . . . , k and obtain the convergence (2.18) for β = β∗ . For β < β∗ , we have the estimate (4.3), which yields (cf., (4.4))

16

S. Albeverio Y. Kondratiev, A. Kozak, Y. Kozitsky

α,β,
n,l 2 (τ, τ  )

α,β,
n,l

hence 2

≤

α,β,
n,l 2 (0, 0)

 κ −nα  1 ≤ K(β) + β m


q∈Q\{0}

 1 , q2

(4.8)

(τ, τ  ) → 0 as n → +∞, uniformly on Iβ2 . The convergence of the α,β,
n,l

Matsubara functions 2k (3.10).  

with k ≥ 2 follows from the Gaussian upper bound

5. Proof of Lemma 5 Set σ (v) =

κ −δ , 1 − (1 − κ −δ )v

) ( v ∈ 0, (1 − κ −δ )−1 ,

(5.1)

and φ(v) = κ 2δ−1 [σ (v)]4 ,

ψ(v) =

1 2δ−1 (1 − κ −δ ) [σ (v)]3 . κ 2

(5.2)

Lemma 6. Given n ∈ N, let the condition uˆ n−1 (1 − κ −δ ) < 1,

(5.3)

be satisfied. Then the following inequalities hold:   uˆ n < σ uˆ n−1 uˆ n−1 ,

(5.4)

  uˆ n ≥ σ uˆ n−1 uˆ n−1 − ψ(uˆ n−1 )Xn−1 ,

(5.5)

0 < Xn ≤ φ(uˆ n−1 )Xn−1 ,

(5.6)

where σ (v), ψ(v), φ(v) and Xn are defined by (5.1), (5.2) and (4.1) respectively. Proof. For t ∈ [0, θ ], θ = κ δ − 1, x ∈ E and n ∈ N, we set (cf., (3.14)) 

( ) 1 t κ κ (1+δ)/2 dω) , ϕn (x|t) = exp (x, ω)E + ω2E µn−1 Zn E 2

(5.7)

where Zn is the same as in (3.11). Then ϕn (x|θ ) = ϕn (x),

) κ ( ϕn (x|0) = Zn−1 ϕn−1 κ −(1+δ)/2 x .

(5.8)

For every t ∈ [0, θ ], the function (5.7) can be expanded in the series (1.7) with the coefficients 

1 t (n) 2 ϕ2k (τ1 , . . . , τ2k |t) = ω(τ1 ) · · · ω(τ2k ) exp ωE Zn E 2 ( ) κ ×µn−1 κ (1+δ)/2 dω) , (5.9)

Hierarchical Model of Quantum Anharmonic Oscillators

17

which, by (3.13), coincide with the corresponding Matsubara functions for t = θ. For every fixed (τ1 , . . . , τ2k ) ∈ Iβ2k , as functions of t they are differentiable at any t ∈ (0, θ ) and continuous on [0, θ]. The corresponding derivatives are obtained from (5.9), ∂ (n) def (n) ϕ (τ1 , . . . , τ2k |t) = ϕ˙2k (τ1 , . . . , τ2k |t) ∂t 2k

1 β (n) = ϕ (τ1 , . . . , τ2k , τ, τ |t)dτ. 2 0 2k+2

(5.10)

Now we write log ϕn (x|t) in the form of the series (1.8) and obtain the Ursell func(n) tions u2k (τ1 , . . . , τ2k |t). The derivatives of these functions with respect to t are being calculated from (5.10). In particular, this yields (n) u˙ 2 (τ1 , τ2 |t) =

1 2

0

β



(n)

u4 (τ1 , τ2 , τ, τ |t)dτ +

(n) u˙ 4 (τ1 , τ2 , τ3 , τ4 |t)

1 = 2



β

0



+

β

0

+

β

0

+

β

0

+

0

β

β 0

(n)

(n)

u2 (τ1 , τ |t)u2 (τ2 , τ |t)dτ ; (5.11)

(n)

u6 (τ1 , τ2 , τ3 , τ4 , τ, τ |t)dτ (n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(5.12)

u2 (0, τ |t)dτ,

(5.13)

u4 (τ1 , τ2 , τ3 , τ |t)u2 (τ4 , τ |t)dτ u4 (τ1 , τ2 , τ4 , τ |t)u2 (τ3 , τ |t)dτ u4 (τ1 , τ3 , τ4 , τ |t)u2 (τ2 , τ |t)dτ u4 (τ2 , τ3 , τ4 , τ |t)u2 (τ1 , τ |t)dτ.

Then for def

υn (t) =



β 0



(n)

u2 (τ1 , τ2 |t)dτ2 =

β 0

(n)

we obtain the following system of equations: 1 U (t) + [υn (t)]2 , 2 1 U˙ (t) = V (t) + 2υn (t)U (t) 2

υ˙ n (t) =

+2

(n)

Iβ3

(5.14) (5.15) (n)

u2 (τ2 , τ3 |t)u4 (0, τ1 , τ2 , τ3 |t)dτ1 dτ2 dτ3 ,

subject to the initial conditions (see (5.8)) υn (0) = κ −δ uˆ n−1 ,

β (n) U (0) = κ −2δ−1 u4 (0, τ1 , τ2 , τ2 |t)dτ1 dτ2 = −κ −2δ−1 Xn−1 . 0

(5.16)

18

S. Albeverio Y. Kondratiev, A. Kozak, Y. Kozitsky

Here def



U (t) =

(n)

Iβ2

= def

(5.17)

(n)

Iβ2



V (t) =

u4 (0, τ1 , τ2 , τ2 |t)dτ1 dτ2 u4 (τ, τ, τ1 , τ2 |t)dτ1 dτ2 , (n)

Iβ3

u6 (0, τ1 , τ2 , τ2 , τ3 , τ3 |t)dτ1 dτ2 dτ3 .

Along with the problem (5.14), (5.16) we consider the following one: ˙ = [y(t)]2 , y(t)

y(0) = υn (0) = κ −δ uˆ n−1 .

(5.18)

Under the condition (5.3) it has a solution κ −δ uˆ n−1 = σ ((t/θ )uˆ n−1 )uˆ n−1 , 1 − tκ −δ uˆ n−1

y(t) =

t ∈ [0, θ ].

(5.19)

(n)

The sign rule (3.22) is valid for the above u2k for all t ∈ [0, θ ], which yields U (t) < 0, V (t) > 0. Therefore, the solution of (5.14) will be dominated4 by (5.19), i.e., uˆ n = υn (θ ) < y(θ ) = σ (uˆ n−1 )uˆ n−1 , that gives (5.4). Further, with the help of (3.21), (3.22) the third term on the right-hand side of (5.15) may be estimated as follows / .



2

u2 (τ2 , τ3 |t) β −1 (n)

Iβ2

.

≥ 2 β −1

×

Iβ2

(n)

Iβ2

u4 (τ, τ1 , τ2 , τ3 |t)dτ dτ1 dτ2 dτ3 /

(n) u4 (τ, τ1 , τ2 , τ2 |t)dτ dτ1

(n)

Iβ2

u2 (τ2 , τ3 |t)dτ2 dτ3 = 2υn (t)U (t).

Applying this in (5.15) we arrive at (recall that U (t) < 0 and V (t) > 0) U˙ (t) 4κ −δ uˆ n−1 ≤ 4y(t) = , U (t) 1 − tκ −δ uˆ n−1

∀t ∈ [0, θ ].

(5.20)

Integrating one gets U (t) ≥

U (0) , [1 − tκ −δ uˆ n−1 ]4

∀t ∈ [0, θ ],

which yields in turn 4  U (θ ) = −Xn ≥ −κ 2δ−1 σ (uˆ n−1 ) Xn−1 = −φ(uˆ n−1 )Xn−1 , 4

A detailed presentation of methods based on differential inequalities are given in [31].

(5.21)

Hierarchical Model of Quantum Anharmonic Oscillators

19

that gives (5.6). Now we set h(t) =



1 [1 + tκ −δ uˆ n−1 ]

υ 2 n

t 1 + tκ −δ uˆ n−1

−

κ −δ uˆ n−1 , 1 + tκ −δ uˆ n−1

where t ∈ [0, tmax ], tmax = θ κ δ σ (uˆ n−1 ). For this function, we obtain from (5.14) the following equation: ˙ = h(t)

1 U 2[1 + tκ −δ uˆ n−1 ]4



t 1 + tκ −δ uˆ n−1

+ [h(t)]2 ,

(5.22)

subject to the boundary conditions h(0) = 0,

2    h(tmax ) = 1 − θ κ −δ uˆ n−1 υn (θ ) − σ (uˆ n−1 )uˆ n−1 .

(5.23)

By means of (5.20), one may show that the first term on the right-hand side of (5.22) is a monotone increasing function of t ∈ [0, tmax ], which yields h(tmax ) − h(0) ≥ tmax U (0)/2. Taking into account (5.23) and (5.16) one obtains from the latter υn (θ ) − σ (uˆ n−1 )uˆ n−1 = uˆ n − σ (uˆ n−1 )uˆ n−1 1 ≥ − (1 − κ −δ )[σ (uˆ n−1 )]3 κ 2δ−1 Xn−1 , 2 that gives (5.5).

 

Now we prove a statement, which will allow us to control the initial elements in the sequences {uˆ n }, {Xn }, i.e., uˆ 0 and X0 . Set

(5.24) η = η(β, m, a, b) = β,
0,l (q2l ) = [ω(0)]2 µ0 (dω). 

From now on we suppose that a < 0. Set also   f (t) = t −1 1 − e−t .

(5.25)

Lemma 7. The following estimates hold: β|a| f 12b



3βb m|a|



0

1  β|a| ≤ uˆ 0 ≤ min βη; 1 + 1 + (16b/β|a|) , 8b

X0 ≤

4!buˆ 40

   3βb −1 f . m|a|

(5.26)

(5.27)

20

S. Albeverio Y. Kondratiev, A. Kozak, Y. Kozitsky

Proof. By Eqs. (8.81), (8.82) of [6], we get  β βηf ≤ uˆ 0 . 4mη

(5.28)

As in [23], we use the Bogolyubov inequality   ' '2    β β,
0,l AA∗ + A∗ A · β,
0,l C ∗ , [H, C] ≥ 'β,
0,l C ∗ , A ' , 2 in which we set A to be the identity operator, C = pl , H = H0,l , and obtain η≥

|a| . 12b

(5.29)

It is not difficult to show that the left-hand side of (5.28) is an increasing function of η; hence, by (5.29) one gets the lower bound in (5.26). The upper bound uˆ 0 ≤ βη, follows from the estimate (4.2) (positivity), (3.29) and the definition (5.24). One can show (see Subsect. 4.2 of [9] and Subsect. 3.2 of [8]) that the measure µ0 is quasi-invariant with respect to the shifts ω → ω + teq , t ∈ R, q ∈ Q, where eq is given by (3.6). Its logarithmic derivative bq in the direction eq is

bq (ω) = −(mq 2 + a)

β 0

eq (τ )ω(τ )dτ − 4b

β

eq (τ )[ω(τ )]3 dτ.

0

This derivative is used in the integration-by-parts formula



∂q f (ω)µ0 (dω) = − f (ω)bq (ω)µ0 (dω), 

(5.30)

(5.31)



where def

∂q f (ω) =





(∂/∂t)f (ω + teq )

t=0

,

and f :  → R can be taken

f (ω) = 0

β

eq (τ )ω(τ )dτ.

(5.32)

We apply (5.31) with q = 0 to the function (5.32), also with q = 0, and obtain

4b (0)  (τ, τ, τ, τ  )dτ dτ  . (5.33) 1 = −|a|uˆ 0 + β Iβ2 4 By the GKS-inequality (see Theorem 6.2 in [6]), 4 (τ, τ, τ, τ  ) ≥ 2 (τ, τ )2 (τ, τ  ), (0)

(0)

(0)

by which and by the estimate uˆ 0 ≤ βη, we have in (5.33) 1 ≥ −|a|uˆ 0 + 4bηuˆ 0 ≥ −|a|uˆ 0 + 4bβ −1 uˆ 20 , that is equivalent to the second upper bound in (5.26).

Hierarchical Model of Quantum Anharmonic Oscillators

21

By means of the lattice approximation technique and the estimate (3.15) of [13], one gets

(0)

−U4 (τ1 , τ2 , τ3 , τ4 ) ≤ 4!b

β

(0)

(0)

(0)

(0)

U2 (τ1 , τ )U2 (τ2 , τ )U2 (τ3 , τ )U2 (τ4 , τ )dτ,

0

which yields

X0 ≤ ≤

4!buˆ 20 4!buˆ 40

β



0

2

U2 (τ, τ  ) (0)

dτ  ≤ 4!buˆ 30 βη

  −1 β f , 4mη

where we have used the upper bound for βη obtained from (5.28). For f given by (5.25), one can show that 1/f (t) is an increasing function of t. Then the estimate (5.27) is obtained from the above one by means of (5.29).   Let us return to the functions (5.1), (5.2). Recall that we suppose δ ∈ (0, 1/2). Given  ∈ (0, (1 − 2δ)/4), we define v() by the condition σ (v()) = κ  . An easy calculation yields v() =

κ δ − κ − 1 − κ − = 1 + . κδ − 1 κδ − 1

(5.34)

Then φ(v) ≤ κ 2δ+4−1 < 1,

for v ∈ [1, v()].

(5.35)

(κ δ − κ − )(1 − κ − ) , (κ δ − 1)2

(5.36)

Furthermore, we set w() = 2κ 1−δ−2 ·

wmax =

sup

w().

(5.37)

∈(0,(1−2δ)/4)

The function  → w() is continuous, then for every w < wmax , one finds ε ∈ (0, (1 − 2δ)/4) such that w < w(ε). Set v¯ = v(ε) and w¯ = w(ε). Therefore, for this w, one has −ψ(v)w + vσ (v) > v,

for v ∈ [1, v]. ¯

(5.38)

Lemma 8. The parameters m > 0, a ∈ R and b > 0 may be chosen in such a way that there will exist ε ∈ (0, (1 − 2δ)/4) and the numbers β0± , 0 < β0− < β0+ < +∞ with the following properties: (a) uˆ 0 = 1 for β = β0− and uˆ 0 < 1 for β < β0− ; (b) uˆ 0 = v¯ = v(ε) for β = β0+ and uˆ 0 < v¯ for β < β0+ ; (c) X0 < w¯ = w(ε) for all β ∈ [β0− , β0+ ].

22

S. Albeverio Y. Kondratiev, A. Kozak, Y. Kozitsky

Proof. Let us fix γ = |a|/b. Then by (5.26) and (5.25), one has       mγ 2 3β βγ 16 1 − exp − ≤ uˆ 0 ≤ 1+ 1+ , 36 mγ 8 βγ

(5.39)

which immediately yields uˆ 0 → 0 as β → 0. On the other hand, by taking mγ 2 > 36v, ¯ one gets uˆ 0 > v¯ for sufficiently large β. Since by Lemma 3, uˆ 0 depends on β continuously, this means that β0± , such that β0− < β0+ , do exist. For fixed γ and m, the multiplier [f (3β/mγ )]−1 in (5.27) is bounded as β ∈ (0, β0+ ]. Recall that uˆ 0 ≤ v¯ for such β. Then, keeping γ fixed, we pick up b such that the right-hand side of (5.27) will be less than w. ¯   Lemma 9. Let In , n ∈ N0 , be the triple of statements (i1n , i2n , i3n ), where i1n = {∃βn+ ∈ [β0− , β0+ ] : uˆ n = v, ¯ β = βn+ ; uˆ n < v, ¯ ∀β < βn+ }; i2n = {∃βn− ∈ [β0− , β0+ ] : uˆ n = 1, β = βn− ; uˆ n < 1, ∀β < βn− };

i3n = {∀β ∈ (0, βn+ ) : Xn < w}. ¯ Then (i) I0 is true; (ii) In−1 implies In .

Proof. I0 is true by Lemma 8. For β = βn+ , σ (uˆ n ) = κ ε and σ (uˆ n ) < κ ε for β < βn+ + (see (5.34), (5.35)). Set β = βn−1 , then (5.38), (5.36), (5.5), and i3n−1 yield 1 uˆ n ≥ κ ε v¯ − (1 − κ −δ )κ 3ε κ 2δ−1 Xn−1 2   w¯ ε 2(ε−1)+δ δ > κ v¯ 1 − κ (κ − 1) = v. ¯ v¯

(5.40)

− For β = βn−1 , the estimate (5.4) gives

uˆ n < 1.

(5.41)

Taking into account Lemma 3 (continuity) and the estimates (5.40), (5.41), one con− + cludes that there exists at least one value β˜n+ ∈ (βn−1 , βn−1 ) such that uˆ n = v. ¯ Then we put βn+ = min β˜n+ . The mentioned continuity of uˆ n yields also uˆ n < v¯ for β < βn+ . − + Thus i1n is true. The existence of βn− ∈ [βn−1 , βn−1 ) can be proven in the same way. For + + ε β < βn < βn−1 , we have σ (uˆ n−1 ) < κ , which yields Xn < κ 2δ−1 κ 4ε Xn−1 ≤ Xn−1 < w, ¯

(5.42)

hence, i3n is true as well. The proof is concluded by remarking that − + [βn− , βn+ ] ⊂ [βn−1 , βn−1 ) ⊂ [β0− , β0+ ].

(5.43)

  Lemma 10. There exists β∗ ∈ [β0− , β0+ ] such that, for β = β∗ , the following estimates hold for all n ∈ N0 : 1 < uˆ n < v. ¯ For β < β∗ , the above upper estimate, as well as the estimate (4.3), hold.

(5.44)

Hierarchical Model of Quantum Anharmonic Oscillators

23

def

Proof. Consider the set n = {β ∈ (0, βn+ ) | 1 < uˆ n < v}. ¯ Just above we have shown that it is nonempty and n ⊆ (βn− , βn+ ). Let us prove that n ⊆ n−1 . Suppose there exists some β ∈ n , which does not belong to n−1 . For this β, either uˆ n−1 ≤ 1 or uˆ n−1 ≥ v. ¯ Hence, either uˆ n < 1 or uˆ n > v¯ (it can be proven as above), which is in conflict with the assumption β ∈ n . Now let Dn be the closure of n , then one has Dn = {β ∈ [βn− , βn+ ] | 1 ≤ uˆ n ≤ v(δ)},

(5.45)

which is a nonempty closed set. Furthermore, Dn ⊆ Dn−1 ⊆ · · · ⊂ [β0− , β0+ ]. Set 2 D∗ = n Dn , then D∗ ⊂ [β0− , β0+ ] is also nonempty and closed. Now let us show that, for every β ∈ D∗ , the sharp bounds in (5.44) hold for all n ∈ N. Suppose uˆ n = v¯ for some n ∈ N. Then (5.40) yields uˆ m > v¯ for all m > n, which means that this β does not belong to all Dm , and hence to D∗ . Similarly one proves the lower bound by means of (5.4). On the other hand, by means of the above arguments, one can conclude that β ∈ D∗ if the inequalities (5.44) hold for all n ∈ N0 at this β. Set β∗ = min D∗ . Then (5.44) hold for β = β∗ . Let us prove (4.3). Take β < β∗ . If uˆ n > 1 for all n ∈ N, then either (5.44) holds or there exists such n0 ∈ N that uˆ n0 ≥ v. ¯ Therefore, either β ∈ D∗ or β > inf βn+ . Both these cases contradict the assumption β < β∗ . Thus, there exists n0 ∈ N such that uˆ n0 −1 ≤ 1 and hence uˆ n < 1 for all n ≥ n0 . In what follows, the definition (5.1) and the estimate (5.4) imply that the sequences {uˆ n }n≥n0 and {σ (uˆ n )}n≥n0 are strictly decreasing. Then for all n > n0 , one has (see (5.4)) uˆ n < σ (uˆ n−1 )uˆ n−1 < . . .

 n−n0 < σ (uˆ n−1 )σ (uˆ n−2 ) . . . σ (uˆ n0 )uˆ n0 < σ (uˆ n0 ) . 3∞ Since σ (uˆ n0 ) < 1, one gets n=0 uˆ n < ∞. Thus, ∞  

1 − (1 − κ −δ )uˆ n−1

−1

def

= K0 < ∞.

n=1

Finally, we apply (5.4) once again and obtain  −1 −1  uˆ n < κ −nδ 1 − (1 − κ −δ )uˆ n−1 . . . 1 − (1 − κ −δ )uˆ 0 uˆ 0 < κ −nδ K0 v¯ = K(β)κ −nδ . def

  Proof of Lemma 5. The existence of β∗ has been proven in Lemma 10. Consider the case β = β∗ where the estimates (5.44) hold. First we show that Xn → 0. Making use of (5.6) we obtain 4  0 < Xn ≤ κ 2δ−1 σ (uˆ n−1 ) Xn−1 < Xn−1 < Xn−2 < · · · < w. ¯ Therefore, the sequence {Xn } is strictly decreasing and bounded, hence, it converges and its limit, say X∗ , obeys the condition X∗ < X0 < w. ¯ Assume that X∗ > 0. Then ¯ Passing to the limit n → ∞ in (5.5) one (5.6) yields σ (uˆ n ) → κ ε hence uˆ n → uˆ ∞ ≥ v. obtains X∗ ≥ w¯ which contradicts the above condition. Thus X∗ = 0. To show uˆ n → 1 we set  3 1 (5.46) n = − (1 − κ −δ ) σ (uˆ n−1 ) κ 2δ−1 Xn−1 . 2

24

S. Albeverio Y. Kondratiev, A. Kozak, Y. Kozitsky

Combining (5.4) and (5.5) we obtain 0 ≥ uˆ n − σ (uˆ n−1 )uˆ n−1 ≥ n → 0.

(5.47)

For β = β∗ , we have {uˆ n } ⊂ [1, v) ¯ in view of Lemma 10. By (5.47) all its accumulation points in [1, v] ¯ ought to solve the equation u − σ (u)u = 0. There is only one such point: u∗ = 1, which hence is the limit of the whole sequence {uˆ n }. For β < β∗ , the estimate (4.3) has been already proven in Lemma 10. This yields σ (uˆ n ) → κ −δ , which implies Xn → 0 if (5.6) is taken into account.   Acknowledgements. The authors thank the referees whose remarks helped to make the presentation of the paper more clear. They are grateful for the financial support of this work by Deutsche Forschungsgemeinschaft through the German-Polish project 436 POL 113/98/0-1 “Probability measures". Agnieszka Kozak and Yuri Kozitsky were also supported by Komitet Bada´n Naukowych through the Grant 2P03A 02025, that is cordially acknowledged.

References 1. Albeverio, S., Høegh–Krohn, R.: Homogeneous random fields and quantum statistical mechanics. J. Funct. Anal. 19, 242-272 (1975) 2. Albeverio, S., Kondratiev,Yu., Kozitsky,Yu.: Quantum hierarchical model. Meth. Funct. Anal. Topology 2, 1-35 (1996) 3. Albeverio, S., Kondratiev, Yu., Kozitsky, Yu.: Critical point properties of a quantum hierarchical model. Lett. Math. Phys. 40, 287-291 (1997) 4. Albeverio, S., Kondratiev, Yu., Kozitsky, Yu.: Absence of critical points for a class of quantum hierarchical models. Commun. Math. Phys. 187, 1-18 (1997) 5. Albeverio, S., Kondratiev, Yu., Kozitsky, Yu.: Classical limits of Euclidean Gibbs states for quantum lattice mdels. Lett. Math. Phys. 48, 221-233 (1999) 6. Albeverio, S., Kondratiev,Yu., Kozitsky,Yu., Röckner, M.: Euclidean Gibbs states of quantum lattice systems. Rev. Math. Phys. 14, 1335–1401 (2002) 7. Albeverio, S., Kondratiev, Yu., Kozitsky, Yu., Röckner, M.: Quantum stabilization in anharmonic crystals. Phys. Rev. Lett. 90, No. 17, 170603-1–4 (2003) 8. Albeverio, S., Kondratiev, Yu., Pasurek, T., Röckner, M.: Euclidean Gibbs measures on loop lattices: existence and a priori estimates. Annals of Probab. 1A, 153–190 (2004) 9. Albeverio, S., Kondratiev,Yu., Röckner, M.: Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs states. J. Funct. Anal. 149, 415–469 (1997) 10. Berenstein, C.A., Gay, R.: Complex Variables. An Introduction. New York Berlin Heidelberg, Springer-Verlag, 1991 11. Bleher, P.M.: Construction of non-Gaussian self-similar random fields with hierarchical structure. Commun. Math. Phys. 84, 557-578 (1982) 12. Bleher, P.M., Major, P.: Critical phenomena and universal exponents in statistical physics. On Dyson’s hierarchical model. Annals of Probab. 15, 431-477 (1987) 13. Brydges, D.C., Fröhlich, J., Sokal, A.D.: The random-walk representation of classical spin systems and correlation inequalities. II. The skeleton inequalities. Commun. Math. Phys. 91, 117–139 (1983) 14. Freericks, J.K, Jarrel, M., Mahan, G.D.: The anharmonic electron-phonon problem. Phys. Rev. Lett. 77, 4588–4591 (1996) 15. Høegh-Krohn, R.: Relativistic quantum statistical mechanics in two-dimensional space-time. Commun. Math. Phys. 38, 195-224 (1974) 16. Kozitsky, Yu.: Hierarchical model of a ferromagnet. Self-similar block-spin distributions and the Lee-Yang theorem. Rep. Math. Phys. 26, 429- 445 (1988) 17. Kozitsky, Yu.: Hierarchical ferromagnetic vector spin model possessing the Lee-Yang property. Thermodynamic limit at the critical point and above. J. Stat. Phys. 87, 799-820 (1997) 18. Kozitsky, Yu., Wołowski L.: A nonlinear dynamical system on the set of Laguerre entire functions. Nonlinear Anal. Ser. A: Theory, Methods. 49, 61–86 (2002) 19. Lieb, E.H., Sokal, A.D.: A general Lee-Yang theorem for one-component and multicomponent ferromagnets. Commun. Math. Phys. 80, 153-179 (1981)

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20. Moreira, C.H.C., Schor, R.C.: On the critical behavior of Dyson’s quantum hierarchical models. Commun. Math. Phys. 160, 275-294 (1994) 21. Mujica, J.: Complex Analysis in Banach Spaces. Amsterdam - New York - Oxford: North Holland, 1986 22. Parthasarathy, K.R.: Probability Measures on Metric Spaces. New York - London: Academic Press, 1967 23. Pastur, L.A., Khoruzhenko B.A.: Phase transitions in quantum models of rotators and ferroelectrics. Teoret. Mat. Fiz. 73, 111–124 (1987) 24. Shlosman, S.B.: Signs of the Ising model Ursell functions. Commun. Math. Phys. 102, 679-686 (1986) 25. Simon, B., Griffiths, R.B.: The φ24 field theory as a classical Ising model. Commun. Math. Phys. 33, 145-164 (1973) 26. Simon, B.: The P (φ)2 Euclidean (Quantum) Field Theory. Princeton, New Jersey: Princeton University Press, 1974 27. Sinai, Ya. G.: Self-similar probability distributions. Teor. Veroyatn. Ee Primen. 21, 63-80 (1976) 28. Sinai, Ya. G.: Theory of Phase Transitions: Rigorous Results. Oxford - New York: Pergamon Press, 1982 29. Stamenkovi´c, S.: Unified model description of order-disorder and structural phase transitions. Condensed Matter Physics (Lviv) 1(14), 257–309 (1998) 30. Stasyuk, I.V.: Local anharmonic effects in high-Tc superconductors. Pseudospin-electron model. Condensed Matter Physics (Lviv) 2(19), 435–446 (1999) 31. Walter, W.: Differential and Integral Inequalities. Berlin-Heidelberg-New York: Springer-Verlag, 1970 Communicated by A. Kupiainen

Commun. Math. Phys. 251, 27–64 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1168-6

Communications in

Mathematical Physics

On Hermitian-Holomorphic Classes Related to Uniformization, the Dilogarithm, and the Liouville Action Ettore Aldrovandi Department of Mathematics, Florida State University, Tallahassee, FL 32306-4510, USA. E-mail: [email protected] Received: 20 December 2002 / Accepted: 8 April 2004 Published online: 3 September 2004 – © Springer-Verlag 2004

Abstract: Metrics of constant negative curvature on a compact Riemann surface are critical points of the Liouville action functional, which in recent constructions is rigorˇ ously defined as a class in a Cech-de Rham complex with respect to a suitable covering of the surface. We show that this class is the square of the metrized holomorphic tangent bundle in hermitian-holomorphic Deligne cohomology. We achieve this by introducing a different version of the hermitian-holomorphic Deligne complex which is nevertheless quasi-isomorphic to the one introduced by Brylinski in his construction of Quillen line bundles. We reprove the relation with the determinant of cohomology construction. Furthermore, if we specialize the covering to the one provided by a Kleinian uniformization (thereby allowing possibly disconnected surfaces) the same class can be reinterpreted as the transgression of the regulator class expressed by the Bloch-Wigner dilogarithm. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Organization of the paper . . . . . . . . . . . . . . . . . 1.2 Notation and conventions . . . . . . . . . . . . . . . . . 2. Deligne Complexes . . . . . . . . . . . . . . . . . . . . . . . 2.1 Cup products on cones . . . . . . . . . . . . . . . . . . 2.2 Deligne complexes . . . . . . . . . . . . . . . . . . . . 2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Remarks on the cup product f ∪ g . . . . . . . . . . . . 3. Constructions in Hermitian Holomorphic Deligne Cohomology 3.1 Hermitian holomorphic Deligne cohomology . . . . . . 3.2 Hermitian holomorphic line bundles . . . . . . . . . . . 3.3 Cup product of hermitian holomorphic line bundles . . .

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28 30 31 32 32 34 36 36 38 38 40 42

28

E. Aldrovandi

3.4 Integration along the fiber . . . . . . . . . . . . . . . . . . . . . . 4. Conformal Metrics and the Liouville Functional . . . . . . . . . . . . . 4.1 Variational problem for conformal metrics . . . . . . . . . . . . . 4.2 Direct construction of the Liouville cocycle . . . . . . . . . . . . 4.3 A cup product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Two line bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Additional remarks on the Liouville functional and the Bloch-Wigner dilogarithm . . . . . . . . . . . . . . . . . 5. Conformal Metrics and Hermitian Holomorphic Cohomology . . . . . . 5.1 Comparison on a curve . . . . . . . . . . . . . . . . . . . . . . . 5.2 Determinant of cohomology . . . . . . . . . . . . . . . . . . . . 6. Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . A. Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Cones and homotopies . . . . . . . . . . . . . . . . . . . . . . . A.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Conventions on Kleinian Groups and Fractional Linear Transformations

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43 44 45 47 49 50

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50 52 53 56 58 59 59 61 61

1. Introduction Metrics of constant negative curvature play a very important role in uniformization problems for compact Riemann surfaces of genus g > 1. The condition that the scalar curvature associated to a conformal metric on a Riemann surface X be equal to −1 is equivalent to the fact that the associated conformal factor satisfies a nonlinear partial differential equation known as the Liouville equation. The Liouville equation appears as early as in one of the approaches considered by Poincar´e to attack the uniformization theorem [27]. In relatively recent times, it has received considerable attention in Theoretical and Mathematical Physics due to the key role it plays in Polyakov’s approach to String Theory [28], especially from the point of view of non-critical strings and two-dimensional quantum gravity. In this context one refers to the conformal factor of the metric as the Liouville “field.” As usual in the context of differential equations with a physical motivation, one would normally like to formulate a variational principle to express the Liouville equation as an extremum condition. Namely, given a Riemann surface X and the space CM (X) of all conformal metrics on it, the metric of constant negative curvature should be a critical point of a functional defined over CM (X). This functional is the Liouville action. As it happens, action functionals may turn out to be even more relevant than the equations they are associated to. The Liouville action is no exception in this sense: it has deep connection with the geometry of Teichm¨uller spaces [33, 34], and in Physics it describes the conformal anomaly in String Theory. Providing a rigorous mathematical definition of the Liouville action functional is however far from trivial. The very geometric properties of the Liouville field itself prevent expressing the corresponding functional as a plain integral of a 2-form on a Riemann surface. Correction terms are required, typically in the form of integration of lower degree forms over the 1-skeleton of an appropriate simplicial realization of X. (One should notice that this behavior is not specific to the Liouville equation, and it is by now possible to give a characterization, in terms of homological algebra, of these type of functionals, see ref. [1].) It is possible to directly determine the necessary correction terms by requiring that the variational problem be well defined. This, however, is not completely satisfactory

Hermitian-Holomorphic Classes

29

from the point of view of certain applications to deformation theory, where a consistent definition across a family of surfaces is required. Quite recently, a more systematic construction, based on the homological algebra techniques developed by the author and L. A. Takhtajan in [2], was carried out by L. A. Takhtajan and L.-P. Teo in ref. [30], generalizˇ ing the earlier results of [33, 34]. The authors of ref. [30] constructed a Cech cocycle with respect to the e´ tale cover of X associated to a quasi-Fuchsian (and more generally Kleinian) uniformization. Since their construction works across (Kleinian) deformations, it could be exploited to obtain results of global nature on the analytic geometry of Kleinian deformation spaces. As a further result, the authors of loc. cit. were able to rigorously prove the validity of the “holography principle” for the Liouville action corresponding to a large class of Kleinian (in particular Fuchsian and quasi-Fuchsian) uniformizations. Specifically, they proved that given a second kind of Kleinian group, the corresponding Liouville action can be obtained as the regularized limit of the hyperbolic volume of the corresponding associated 3-manifold. This extends to the general Kleinian case a previous formula obtained by Krasnov [24] for classical Schottky groups. Our interest in this matter is two-fold. From the perspective of the newer methods adopted in [3], the covering map U → X associated to the uniformization by a discrete group
= Deck(U/X) is but one of the many possible covers comprising an appropriate category C of, say, local diffeomorphisms U → X—the most obvious
choice being that of standard open cover U = {Ui } with associated space U = i Ui . In particular one expects to be able to apply the methods of [2] and [3] uniformly on a class of reasonably behaved covers of X. Second, the focus of ref. [3] was on the rigorous definition of a functional for quasiconformal deformation of the Riemann surface X and its application to the study of projective structures. A main result is that the construction of the action is possible thanks to the vanishing of the “tame symbol” (see refs. [12] and [9] for the relevant defi nitions) TX , TX , where TX is the holomorphic tangent line bundle of X. The vanishing determines local choices (with respect to a cover) of a Bloch-type dilogarithm which then allow for a cohomological construction of the action. There are many indications that the Liouville action ought to be the hermitian square of a functional of the type studied in [3].1 Thus it is natural to ask whether there is an analogous mechanism as the one in loc. cit. to obtain a general construction of the Liouville action by replacing the holomorphic symbol maps and dilogarithms with corresponding real objects. In this paper we answer this question in the affirmative. More precisely, we show that the Liouville action (up to the area term which is given by an ordinary 2-form) can be computed as a symbol map taking values in hermitian holomorphic Deligne cohomology, first introduced by Brylinski and McLaughlin in their study of degree four characteristic classes [9]. (By way of comparison, the tame symbols used in ref. [3] used holomorphic and smooth Deligne cohomology.) In particular we show that the dilogarithm type terms are replaced here by the Bloch-Wigner function, the real valued counterpart of the dilogarithm (see refs. [7], and [23, 15] for a review.) The appearance of the Bloch-Wigner function ties very well with the holography property of the Liouville function proved in [30] in the following sense. As mentioned before, the Liouville action (up to the area term) relative to a Kleinian uniformization2 U → X can also be computed as the “regularized volume” of the associated 3-manifold 1 From a physical point of view this originates in the modular geometry approach to Conformal Field Theory advocated by Friedan and Shenker in [19]. Mathematically speaking, it is one of the many proposed forms of the holomorphic factorization property for determinant line bundles. 2 Note that X is allowed to be disconnected.

30

E. Aldrovandi

N =
\(U ∪ H3 ), where
= Deck(U/X) as before, U ⊂ P1 is the domain of discontinuity for
, and H3 is the standard hyperbolic 3-space. (To define the regularized volume would lead us too far afield. It suffices to mention that the conformal factor of a metric on X = ∂N can be used to select a compact submanifold N whose volume is finite. One then subtracts from the volume of N the areas of the boundary components and other carefully chosen constants independent of the metric structure, so that the resulting quantity will have a finite limit as  → 0.) On the other hand, the hyperbolic volume in three dimensions corresponds to a three dimensional (purely imaginary) class on PSL2 (C) expressible through the Bloch-Wigner dilogarithm, the so-called regulator class. We show that the regulator is precisely the class that needs to be killed in order to close the cohomological descent conditions required to calculate the Liouville action for a covering map U → X with covering group a Kleinian group
. This is possible, since for a second kind of Kleinian group the quotient H3 /
is non-compact, hence it carries no cohomology in dimension three, so the class represented by the Bloch-Wigner function, pulled back to
via the imbedding
→ PSL2 (C), vanishes. Returning to the cohomological interpretation of the construction of the Liouville action, it should also be noted that leaving aside the area term, our results show that the cohomologically non trivial part is indeed a square. Namely, for a conformal metric ρ ∈ CM (X) we consider the pair (TX , ρ) as a holomorphic line bundle equipped with an hermitian metric. Then, using that hermitian holomorphic Deligne cohomology has a cup product, we show that the Liouville action is just the square of the class of (TX , ρ). In fact this identification holds at the level of cocycles, rather than only for the corresponding classes. Again leaving aside the area term, it immediately follows from the properties of the hermitian holomorphic Deligne cohomology that most of the story carries over to the case of a pair of holomorphic line bundles L and L equipped with hermitian metrics ρ and ρ  , respectively. Furthermore, Brylinski shows in [10] that the pairing of two such holomorphic line bundles with metrics corresponds to the pairing defined by Deligne on the determinant of cohomology in [11]. Without introducing the machinery of 2gerbes, we reobtain this result in our setting. Specifically, we directly obtain Gabber’s formula for the hermitian metric on the determinant line from the explicit cocycle for the cup product of two metrized line bundles. In turn this shows that the Liouville action is a multiple of the determinant of cohomology, thereby generalizing earlier results (cf. [32])—without assuming criticality. 1.1. Organization of the paper. This paper is organized as follows. Sections 2 and 3 are devoted to expounding some background material for the sake of keeping this paper selfcontained and to put the reader in the position of reproducing the necessary calculations. Section 2 contains background facts on Deligne cohomology, paying special attention to the product structures and the cone constructions. We provide some examples and collect some facts about the dilogarithm from the point of view of Deligne cohomology. The particular model of hermitian holomorphic Deligne cohomology we use later in the paper requires certain constructions available in the literature, and recalled in Sect. 2, to be slightly modified in order to obtain a (graded) commutative product. The necessary arguments, being somewhat outside the line of development of the paper are presented in Appendix A. Hermitian holomorphic Deligne cohomology is introduced in Sect. 3. We give the definition as in refs. [9] and [10], and then introduce another model which, albeit more complex, has the advantage for us of keeping the metric structure explicit. We have explicitly proved the isomorphism in Lemma 3.1. For the sake of completeness,

Hermitian-Holomorphic Classes

31

we give an explicit description of the cocycle determined by a holomorphic line bundle with hermitian metric, and in Sect. 3.3 we explicitly compute the cup product of two metrized line bundles for later usage. Results about the existence of a fiber integration map are mentioned in the paper, so some background material is provided in Sect. 3.4. Sections 4 and 5 form the core of the paper. The direct construction of the Liouville action according to the techniques of refs. [2, 3, 30] is presented in Sect. 4. Since explicit calculations have been presented in great detail in ref. [30], and the calculations we need are quite straightforward, we keep details to a minimum. In Proposition 4.1 and Corollary 4.1 we show that the Liouville action functional computed via descent theory does solve the variational problem. These results have appeared also in ref. [30] and are presented here for completeness, although our choice for the cover is different. The framework of loc. cit. is that of a Kleinian cover U → X ∼ = U/
, where
is a second kind of purely loxodromic geometrically finite Kleinian group, which then we treat in some detail in Sect. 4.4. We illustrate how the genuine Bloch-Wigner function of Sect. 2.4 appears in the descent equations relative to the Kleinian cover. Having observed descent equations close on general cohomological grounds in sect. 4.2.3, we now point out that for the case of a Kleinian cover this is due to the vanishing of the regulator class for the non-compact 3-manifold H3 /
. A true geometric construction of the Liouville action, which does not rely on the arguments of Sect. 4 to close the descent equations, is carried out in Sect. 5, which contains our main result: We compare the descent calculations with the cup products computed in Sect. 3 and conclude that the quadratic part of the Liouville action is in fact (modulo an area term) the cup square of the metrized holomorphic tangent bundle in Hermitian holomorphic Deligne cohomology, see Theorem 5.1, Corollary 5.1, and Proposition 5.1. It follows that from this point of view the descent equations close automatically, without the need for special arguments. We then prove that the cup square is identified with the determinant of cohomology construction in Theorem 5.2 and Corollary 5.2. Auxiliary facts about the homological algebra of cones and conventions on Kleinian groups are stored in the appendices. Finally, we draw our conclusions in Sect. 6. 1.2. Notation and conventions. If z is a complex number, then 1 (z + (−1)p z¯ ) , 2 and similarly for any other complex quantity, e.g. complex √ valued differential forms. If A is a subring of R, we will use the “twist” A(j ) = (2π −1)j A. If X is a complex manifold, A•X and •X denote the de Rham complexes of smooth C-valued and holomorphic forms, respectively. We denote by E •X the de Rham complex of sheaves of real valued differential forms and by E •X (j ) the twist E •X ⊗R R(j ). We set p,q OX ≡ 0X as usual. When needed, AX will denote the sheaf of smooth (p, q)-forms. We use the standard decomposition d = ∂ + ∂¯ according to types. Furthermore, we introduce the differential operator d c = ∂ − ∂¯ (contrary to the convention, see, e.g. [25], √ we omit the factor 1/(4π −1)). We have 2∂ ∂¯ = d c d. The operator d c is an imaginary one, and accordingly, we have the rules def

πp (z) =

dπp (ω) = πp (dω), d c πp (ω) = πp+1 (d c ω) for any complex form ω.

32

E. Aldrovandi

An open cover of X will be denoted by UX . If {Ui }i∈I is the corresponding collection of open sets, we denote Uij = Ui ∩ Uj , Uij k = Ui ∩ Uj ∩ Uk , and so on. We can also consider more general covers UX = {Ui → X}i∈I , where the maps are regular coverings in the appropriate category. In this case intersections are replaced by (n + 1)-fold fibered products Ui0 i1 ...in = Ui0 ×X · · · ×X Uin . Open coverings fit this more general description, since if Ui and Uj are two open sets, then Ui ∩ Uj = Ui ×X Uj , where the fiber product is taken with respect to the inclusion maps. As another example, one can consider regular covering maps U → X with
= Deck(U/X), and in this case, taking the fiber product over X(n + 1)-times, one gets U ×X · · · ×X U = U ×
× · · · ×
, where the group factor is repeated n-times. This includes the cases of Kleinian (and in particular Fuchsian) covers of Riemann surfaces.
The nerve of the cover UX is the simplicial object n → Nn (UX ) = Ui0 ×X · · · ×X Uin , where Nn (UX ) maps into Nn−1 (UX ) in (n + 1) ways by forgetting in turn each factor. For open covers this just yields the expected inclusion maps. ˇ If F • is a complex of sheaves on X, its Cech resolution with respect to a covering def p,q q ˇ UX → X is the double complex C (F ) = C (UX , F p ) , where the q-cochains with  ˇ values in F p are given by F p (Ui0 ×X · · · ×X Uin ) . The Cech coboundary operator ˇ ˇ The sign convention we are going to use is that the index along the Cech is denoted δ. resolution is the second one, so if we denote by d the differential in the complex F • , ˇ the total differential in the total simple complex of Cˇ q (UX , F p ) will be D = d ± δ. ˇ For open covers we just get the familiar Cech (hyper)cohomology. The other interesting ˇ example is that of a regular covering map U → X: Cech cohomology with respect to this cover is the same as group cohomology for
= Deck(U/X) with coefficients in the
-module F p (U ). The Koszul sign rule that results in a sign being picked whenever two degree indices ˇ are formally exchanged is applied. In particular, for Cech resolutions of complexes of sheaves, it leads to the following conventions. If G• is a second complex of sheaves on X, then one defines the cup product ∪ : Cp,q (F ) ⊗ Cr,s (G) −→ Cˇ q+s (UX , F p ⊗ Gr ) ⊂ Cp+r,q+s (F ⊗ G) of two elements {fi0 ,...,iq } ∈ Cp,q (F ) and {gj0 ,...,js } ∈ Cr,s (G) by (−1)qr fi0 ,...,iq ⊗ giq ,iq+1 ,...,iq+s . 2. Deligne Complexes 2.1. Cup products on cones. Recall that the cone of a map f : X• → Y • between two complexes is the complex C • (f ) = X • [1] ⊕ Y • with differential df (x, y) = (−d x, f (x) + d y), where [k] denotes the shift functor. The cone fits into the exact sequence 0 −→ Y • −→ C • (f ) −→ X • [1] −→ 0 . The following constructions are a special case of those considered by Be˘ılinson in ref. [5]. Suppose we are given complexes Xi• , Yi• , and Zi• and maps fi : Xi• → Zi• , ∪

gi : Yi• → Zi• , for i = 1, 2, 3. Suppose also that we have product maps X1• ⊗ X2• − → X3• ,

Hermitian-Holomorphic Classes

33

and similarly for Yi• , and Zi• , strictly compatible with the fi , gi in the obvious sense. Then we can consider the cones     fi −gi Cone fi − gi [−1] ≡ Cone Xi• ⊕ Yi• −−−→ Zi• [−1] . For a real parameter α, there is a family of products       ∪α Cone f1 − g1 [−1] ⊗ Cone f2 − g2 [−1] −→ Cone f3 − g3 [−1]

(2.1)

determined as follows. For (xi , yi , zi ) ∈ Xi• ⊕ Yi• ⊕ Zi•−1 , i = 1, 2 , one defines   (x1 , y1 , z1 ) ∪α (x2 , y2 , z2 ) = x1 ∪ x2 , y1 ∪ y2 , (−1)deg(x1 ) (1 − α)f1 (x1 )  +αg1 (y1 ) ∪ z2   +z1 ∪ αf2 (x2 ) + (1 − α)g2 (y2 ) . (2.2) Note that deg(x1 ) = deg(x1 , y1 , z1 ). Checking that ∪α is a map of complexes is a straightforward routine calculation. Different products for different values α and β of the real parameter are homotopic. Explicitly, we have (x1 , y1 , z1 ) ∪α (x2 , y2 , z2 ) − (x1 , y1 , z1 ) ∪β (x2 , y2 , z2 )   = d hα,β + hα,β d ((x1 , y1 , z1 ) ⊗ (x2 , y2 , z2 )) , where the homotopy •        hα,β : Tot Cone f1 − g1 [−1] ⊗ Cone f2 − g2 [−1] → Cone f3 − g3 [−1]•−1 is given by the formula hα,β ((x1 , y1 , z1 ) ⊗ (x2 , y2 , z2 )) = (α − β) (−1)deg(x1 )−1 (0, 0, z1 ∪ z2 ) .

(2.3)

∪

If the products X1• ⊗ X2• − → X3• , etc., are graded commutative, then the swap functor on the tensor product maps the ∪α product structure on the cones into the ∪1−α structure. Using the homotopy (2.3) it follows at once that there is a well defined graded commutative product in cohomology. ∪

If we do not assume the product structures X1• ⊗ X2• − → X3• , etc., are strictly compatible with the maps fi , gi , some of the preceding constructions must be slightly modified. With an eye toward certain constructions to be carried out later in this paper, let us assume we have compatibility up to homotopy, namely there exist maps  • h : X1 ⊗ X2 −→ Z3•−1 ,  • k : Y1 ⊗ Y2 −→ Z3•−1 such that f3 ◦ ∪ − ∪ ◦ (f1 ⊗ f2 ) = d h + h d, g3 ◦ ∪ − ∪ ◦ (g1 ⊗ g2 ) = d k + k d , with obvious meaning of the symbols.

(2.4)

34

E. Aldrovandi

Lemma 2.1. Let Xi , Yi , Zi and the maps fi , gi be as above. Let α be a real parameter. We have a product of type (2.1) for the cones Cone(fi − gi )[−1] defined by the following modification of formula (2.2):   (x1 , y1 , z1 ) ∪α (x2 , y2 , z2 ) = x1 ∪ x2 , y1 ∪ y2 , (−1)deg(x1 ) (1 − α)f1 (x1 )  +αg1 (y1 ) ∪ z2   +z1 ∪ αf2 (x2 ) + (1 − α)g2 (y2 )  −h(x1 ⊗ x2 ) + k(y1 ⊗ y2 ) . (2.5) The product (2.5) is a map of complexes and two products ∪α and ∪β are related by the same homotopy formula (2.3). Proof. Direct verification.

 

This modified framework carries over to the full structure considered by Be˘ılinson in ref. [5]. We will still refer to this modified product as the Be˘ılinson product. It is also ∪

→ X3• , etc., be graded necessary to relax the assumption that the products X1• ⊗ X2• − commutative. It is possible to complete all the diagrams so that the permutation of factors in the tensor products still yields a homotopy commutative product (2.5) for the cones. As a consequence the induced product in cohomology will still be graded commutative. Explicit formulas are not needed except to ensure this latter fact, therefore we shall not discuss this matter any further and refer the reader to the appendix, where a brief but explicit treatment can be found. 2.2. Deligne complexes. Let X be a complex manifold. Recall the standard Hodge filtration of •X : F p •X : 0 −→ · · · −→ X −→ · · · −→ nX , p

(2.6)

where n = dimC X. The corresponding filtration for the complex of smooth C-valued forms is defined as follows: denote by F p A•X the subcomplex of A•X comprising forms of type (r, s), where . Then (cf. [13]) the inclusion •X → A•X is r is at least p, so that F p AkX = ⊕r≥p Ar,k−r X a quasi-isomorphism respecting the filtrations, namely F p •X → F p A•X , and the latter inclusion induces an isomorphism in cohomology. If A is a subring of R, and ı and  denote the inclusions of A(p) and F p •X into •X respectively, the p th Deligne complex of sheaves is defined by   ı− A(p)•D = Cone A(p)X ⊕ F p •X −−→ •X [−1] .

(2.7)

It is quasi-isomorphic to the complex:   ı− Cone A(p)X ⊕ F p A•X −−→ A•X [−1] ,

(2.8)

where ı and  have the same meaning. We also notice the quasi-isomorphism ı d d   d p−1  A(p)•D −→ A(p)X −→ OX −→ 1X −→ · · · −→ X .

(2.9)

Hermitian-Holomorphic Classes

35

When A = R there are further quasi-isomorphisms, namely       R(p)•D − → Cone F p •X → E •X (p − 1) [−1] − → Cone F p A•X → E •X (p − 1) [−1] since the maps        R(p) → •X − → R(p) → C − → R(p − 1) − → E •X (p − 1) are all quasi-isomorphisms in the derived category, cf. [18]. Here we have used C ∼ = R(p) ⊕ R(p − 1). Following op. cit., we set:  p • −πp−1 • 
• def R(p) −−−→ E X (p − 1) [−1] . (2.10) D = Cone F AX − Again, there is an explicit quasi-isomorphism ([4, 18]): 
• , ρp : R(p)•D −→ R(p) D ρp |R(p) = 0 , ρp |F p •X = incl , ρp |•X = πp−1 .

(2.11)

The Deligne cohomology groups of X with coefficients in A(p) are the hypercohomology groups • (X, A(p)) = H• (X, A(p)•D ) , HD

and clearly, any complex quasi-isomorphic to A(p)•D would do. In order to perform calˇ culations with these cohomology groups we shall normally resort to a Cech resolution with respect to an open cover UX of X or an e´ tale map UX → X, e.g. a regular cover with deck group
. One of the important properties of Deligne cohomology is the existence of a graded commutative cup product j

∪

i+j

i HD (X, A(p)) ⊗ HD (X, A(q)) − → HD (X, A(p + q)) ,

(2.12)

which follows from the existence of the Be˘ılinson cup product at the level of Deligne complexes whose construction was recalled above. There are products A(p) ⊗ A(q) → A(p + q) and F p •X ⊗ F q •X → F p+q •X , plus the obvious (wedge) product on •X , thus it follows from the cone version (2.7) that the Deligne complexes come equipped with the Be˘ılinson product, and therefore the Deligne cohomology groups inherit the graded commutative cup product (2.12). The explicit form, that is, the translation of (2.2) to the case at hand can be found in [18]. The explicit form of the cup product for the complex (2.10) as computed in [4] (see also [18]) will be needed in the sequel. Let (ω1 , η1 )
• —this means that ω1 ∈ F p n and η1 ∈ E n−1 be an element of degree n in R(p) X X D
• . The product is defined by the formula: (p − 1)–and (ω2 , η2 ) any element in R(q) D   ˜ (ω2 , η2 ) = ω1 ∧ ω2 , (−1)n πp ω1 ∧ η2 + η1 ∧ πq ω2 . (2.13) (ω1 , η1 ) ∪ ˜ is a morphism of complexes and (modulo the quasi-isomorphisms ρp ) The product ∪ is homotopic to the Be˘ılinson products ∪α on the complexes R(p)•D . Specifically, if we denote an element of R(p)•D by the triple (r, f, ω), where r ∈ R(p)X , f ∈ F p•X , and ˜ ◦ (ρp ⊗ ρq ) and ρp+q ◦ ∪α is given by ω ∈ •X , the homotopy between ∪   ˜ h((r, f, ω) ⊗ (r  , f  , ω )) = (−1)deg ω 0, (1 − α) πp ω ∧ πq−1 ω − α πp−1 ω ∧ πq ω . (2.14)

36

E. Aldrovandi

2.3. Examples. 

× 2.3.1. Let A = Z. From (2.9) we have Z(1)•D − → OX [−1] via the standard expo× k 1 (X, Z(1)) ∼ k−1 ∼ (X, OX ) . In particular HD nential sequence, so that HD (X, Z(1)) = H = × 2 0 Pic(X), the Picard group H (X, OX ), the global invertibles on X, and HD (X, Z(1)) ∼ = of line bundles over X. If an open cover {Ui }i∈I of X is chosen, the class of a line bundle 2 (X, Z(1)) can be represented via a Cech ˇ L in HD resolution by the cocycle (fij , cij k ), where fij ∈ OX (Uij ) and cij k ∈ Z(1)X (Uij k ). Thus the functions fij should be interpreted as logarithms of the corresponding transition functions for L. Then, the collection ˇ )ij k provides a representative for the first Chern Class c1 (L). Similarly, an cij k = (δf invertible function f would be described by the collection fi of its logarithms on each open Ui , plus the “integers” mij = fj − fi ∈ Z(1) on each Uij .

   × d log → OX −−→ 1X [−1]. Thus 2.3.2. Still using the exponential sequence, Z(2)•D − 2 (X, Z(2)) is the group of isomorphism classes of holomorphic line bundles with HD (holomorphic) connection. Using the (in fact, any) product Z(1)•D ⊗ Z(1)•D → Z(2)•D , the cup product of two global invertible holomorphic functions  f and  g on X yields a line bundle with connection—the tame symbol—denoted by f, g whose class is in 2 (X, Z(2)), see [12, 6, 29]. Higher cup products in this spirit have been studied in [9]. HD 2p ˇ 2.3.3. If A = R, we have HD (X, R(p)) = H 2p (X, R(p)) ∩ H p,p (X) . The Cech 2 (X, Z(1)) mentioned above maps to the cocyrepresentative (cij k , fij ) of a class in HD cle (−dfij , −|fij |) under (2.11). Taking into account that the fij are the logarithms of the transition functions, the corresponding (1, 1) class would be given by the associated canonical connection, see Sect. 3.2. 1 (X, R(1)) is the group of real valued functions f on X such that there exists 2.3.4. HD a holomorphic one-form ω such that π0 ω = df . In other words it is the group of those smooth functions f such that ∂f is holomorphic, which amounts to say that such an f itself is harmonic.

2.4. Remarks on the cup product f ∪ g. It is convenient to consider the case of the cup product of two invertible functions f and g in various complexes in more detail, and to introduce some related notions we shall need later.  × → OX [−1] and an invertible function f can be considered as As observed, Z(1)•D − 1 1 (X, R(1)) an element of HD (X, Z(1)). Therefore, via (2.11), it induces ρ1 (f ) ∈ HD 1 (X, R(1)) represented by (d log f, log |f |). (Note that log|f | fits the description of HD in 2.3.4.) Given two such f and g, the expression for the cup product (2.13) gives the 2 (X, R(2)): following element of HD   ˜ ρ1 (g) = d log f ∧ d log g, −π1 (d log f ) log |g| + log |f | π1 (d log g) . ρ1 (f ) ∪ (2.15) The first term is obviously zero when X is a curve. Given f and g, invertible on X, let us define the imaginary 1-form: r2 (f, g) = π1 (d log f ) log |g| − log |f | π1 (d log g) .

(2.16)

Hermitian-Holomorphic Classes

37

1 (X, Z(1)) yields an On the other hand, the cup product of f and g as elements of HD   2 (X, Z(2)) represented by d log f ∧ d log g, log f d log g (if we element f ∪ g of HD use the ∪0 product) and this maps via ρ2 to the element   d log f ∧ d log g, −π1 (log f d log g) .

This is equal to (2.16) only up to homotopy. Indeed, using πp+q−1 (a ∧ b) = πp (a) ∧ πq−1 (b) + πp−1 (a) ∧ πq (b) , we find     (2.17) r2 (f, g) = d π1 (log f ) log |g| − π1 log f d log g , where the first term is just the explicit homotopy   as computed from (2.14). Recall that the tame symbol ([12, 6]) f, g associated to f and g is the line bundle with connection determined (up to isomorphism) by the class f ∪ g. A “Bloch dilogarithm” ([18]) is (the logarithm of) a horizontal trivializing section, namely a function L on U ⊂ X satisfying the equation dL = − log f d log g .   Thus L realizes the isomorphism f, g ∼ = OX over U ⊂ X. Thus a Blochdilogarithm  will only locally be available,  in general.  However, when g = 1 − f then 1 − f, f is globally trivial [6, 18], i.e. 1 − f, f ∼ = OX . This is the Steinberg relation satisfied by the Tame symbol. It can be deduced from the following universal case. Set f = z, X = P1 \ {0, 1, ∞}, consequently g = 1 − z. Then L is identified with the classical Euler dilogarithm Li 2 , namely  z dt Li 2 (z) = − log(1 − t) , t 0 see [12] and [23] for details. On P1 \ {0, 1, ∞} the classical dilogarithm has a single valued parter, denoted D2 , introduced by Bloch and Wigner: D2 (z) = arg(1 − z) log |z| + Im Li 2 (z) .

(2.18)

D2 is real-analytic on P1 \ {0, 1, ∞} and extends continuously to P1 . That it is singlevalued can be verified as follows. Choose a branch of the logarithm, say the principal one, to define Li 2 (and D2 ) on the cut plane C \ (−∞, 0] ∪ [1, ∞). Then one shows that the expression (2.18) is in fact single-valued by analytic continuation along paths based, say, at 1/2 ∈ C, and winding around the points 0 and 1 ∈ C. Explicit computations can be found in ref. [7]. It is convenient to introduce √ L2 (z) = −1 D2 (z) , so that dL2 = r2 (1 − z, z) .   More generally, if L trivializes f, g over U in the sense explained above, we can associate a function L2 (f, g) over U such that dL2 (f, g) = r2 (f, g) via the position L2 (f, g) = π1 (log f ) log |g| + Im L .

(2.19)

38

E. Aldrovandi

3. Constructions in Hermitian Holomorphic Deligne Cohomology In this section we recall the definition of hermitian holomorphic Deligne cohomology. In ref. [10] Brylinski introduced certain complexes C(l)• , for a positive integer l, in order to compare the Be˘ılinson-Chern classes of a holomorphic vector bundle E with the Cheeger-Chern-Simons classes determined by (E, ∇), where ∇ is the canonical connection, namely the unique connection compatible with both the holomorphic and hermitian structures. The cohomology groups determined by these complexes are aptly named Hermitian Holomorphic Deligne cohomology groups. For a holomorphic line bundle equipped with the canonical connection, the complex C(1)• encodes the reduction of the structure group from C to T afforded by the hermitian fiber metric. In the following we will need to compute Hermitian Holomorphic Deligne cohomology by means of different—but quasi-isomorphic—sheaf complexes we denote D(l)h• .h . . These complexes are tailored to a direct description of a metrized line bundle in terms of local representatives of the hermitian fiber metric. Since the two constructions are related by a quasi-isomorphism, the resulting cohomology groups are the same. 3.1. Hermitian holomorphic Deligne cohomology. In ref. [10], where Brylinski introduces the complexes:   C(l)• = Cone Z(l)X ⊕ (F lA•X ∩ σ 2l E •X (l)) −→ E •X (l) [−1] , (3.1) where σ p denotes the (sharp) truncation in degree p, namely for a complex F • , σ p F k is zero for k < p and equal to F k for k ≥ p. In (3.1) we take the cone of the difference between the two inclusions. We have the following: Definition 3.1 ([10]). The hermitian holomorphic Deligne cohomology groups are the hypercohomology groups of the complexes (3.1): HDh.h. (X, l) = H p (X, C(l)• ) . def

p

(3.2)

The complexes (3.1) are expressed as cones, and therefore admit a (standard) Be˘ılinson product (2.2). (The wedge product induces cup products on both E •X (l) and F lA•X ∩ σ 2l E •X (l).)3 It follows there is a graded commutative product on cohomology: j

∪

i+j

i (X, l) ⊗ HDh.h. (X, k) −→ HDh.h. (X, l + k) . HD h.h.

(3.3)

Also, from the standard cone exact sequences we get (cf. ref. [10]): 2l · · · −→ H 2l−1 (X, R(l)) −→ HD (X, l) −→ h.h.

−→ H 2l (X, Z(l)) ⊕ A(l,l) (X)R(l) −→ H 2l (X, R(l)) −→ · · ·

(3.4)

where A(l,l) (X)R(l) denotes the space of smooth R(l)-valued global (l, l)-forms on X. Thus we see hermitian holomorphic classes are Z(l)-valued classes represented by (necessarily) closed R(l)-valued 2l-forms of pure type (l, l). For a line bundle this corresponds to a structure group reduction from OX to E 0X , namely T-valued sections, at the same time controlling the Hodge type of the resulting class, cf. ref. [10]. 3 It appears signs should be adjusted in the product formula quoted in ref. [10], and that using (2.2) is more appropriate.

Hermitian-Holomorphic Classes

39

Later (cf. Sect. 3.2) we will want to work with the hermitian structure on a holomorphic line bundle, together with the imaginary (1, 1)-form built from the canonical connection, directly in the holomorphic frame. To carry this out in general for (l, l) classes, we introduce the complex   • [−1] . D(l)•h .h . = Cone Z(l)•D ⊕ (F lA•X ∩ σ 2l E •X (l)) −→ R(l) (3.5) D • is the composite of the obvious map Z(l)• → R(l)• with The map Z(l)•D → R(l) D D D the quasi-isomorphism ρ l defined by (2.11). We will simply denote it by ρl in the sequel, • • = suppressing the first morphism in the notation. The map (F lA•X ∩σ 2l E X (l)) → R(l) D   l • • • • l 2l Cone F AX → E X (l − 1) [−1] is induced by the inclusion of (F AX ∩ σ E X (l)) into F lA•X . In (3.5) we take the cone of the difference between these two maps. The complex (3.5) offers another model for Hermitian holomorphic Deligne cohomology. Indeed we have: Lemma 3.1. The complexes C(l)• and D(l)•h .h . are quasi-isomorphic. Proof. By elementary manipulation of cones    • [−1], D(l)•h .h . = Cone F lA•X ∩ σ 2l E •X (l) → Cone Z(l)•D → R(l) D        • − and clearly: Cone Z(l)•D → R(l) → Cone Z(l)X → D → Cone Z(l)X → R(l)X −  E •X (l) , where all arrows are quasi-isomorphisms. Thus     D(l)•h .h . −→ Cone F lA•X ∩ σ 2l E •X (l) → Cone Z(l)X → E •X (l) [−1]   = Cone Z(l)X ⊕ (F lA•X ∩ σ 2l E •X (l)) → E •X (l) [−1] ≡ C(l) , as wanted.

 

p It follows from Lemma 3.1 that Hp (X, D(l)•h .h . ) ∼ = HDh.h. (X, l) , so we can use either complex to compute the Hermitian-holomorphic Deligne cohomology groups. • Again from the cone exact sequence applied to (3.5), we see the groups HD (X, l) h.h. also satisfy the exact sequence 2l−1 2l · · · −→ HD (X, R(l)) −→ HD (X, l) −→ h.h. 2l 2l (X, Z(l)) ⊕ A(l,l) (X)R(l) −→ HD (X, R(l)) −→ · · · −→ HD

(3.6)

which we can rewrite as 2l−1 2l · · · −→ HD (X, R(l)) −→ HD (X, l) −→ h.h. 2l (X, Z(l)) ⊕ A(l,l) (X)R(l) −→ H 2l (X, R(l)) ∩ H l,l (X) −→ · · · . −→ HD 2l (X, l) map onto those (l, l)-forms representing the Hodge Thus the elements of HD h.h. 2l classes corresponding to HD (X, Z(l)). • , and F lA• ∩ σ 2l E • (l) appearing in the cone (3.5), The complexes Z(l)•D , R(l) X X D all have cup products. It follows that we have the Be˘ılinson family of products ∪α

D(l)•h .h . ⊗ D(k)•h .h . −→ D(l + k)•h .h . .

(3.7)

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Remark 3.1. The product ∪α in Eq.(3.7) should be intended in the modified version pro • have product structures that are vided by (2.5), since the complexes Z(l)•D and R(l) D compatible with the map ρl only up to homotopy (given by formula (2.14)). Moreover, these complexes have product structures that are themselves graded commutative up to homotopy: that the product (3.7) is graded commutative up to homotopy follows from Proposition A.1 in the appendix. 3.2. Hermitian holomorphic line bundles. A hermitian holomorphic line bundle or, equivalently, a metrized line bundle, cf. [25], is a holomorphic line bundle L over X together with a hermitian fiber metric ρ : L → R≥0 . As a rule, we will not distinguish L and its sheaf of holomorphic sections. We will also use the alternate notation |s|ρ to denote the length of a local section s of L with respect to ρ. Metrized line bundles can be tensor multiplied and an inverse is defined, see. op. cit. An isomorphism of metrized line bundles (L, ρ) and (L , ρ  ) is defined in the obvious way, namely it is a map φ : L → L  the group such that |s|ρ = |φ(s)|ρ  for some local section s of L. We denote by Pic(X) of isomorphism classes of metrized line bundles. ˇ If L is trivialized over a Cech cover UX = {Ui }i∈I by sections si , then as usual we × obtain the cocycle of transition functions gij ∈ OX (Ui ∩Uj ) by writing sj = si gij . Then if (L, ρ) is a metrized line bundle, we can define the positive function ρi = |si |2ρ , namely the local representative of the hermitian structure with respect to the given trivialization. It follows that the various local representatives satisfy ρj = ρi |gij |2 .

(3.8)

Let us work out the local version of the isomorphism introduced above. Let si be a local section of L over Ui . Introduce analogous (primed) quantities for L as we just did for × L. Given the isomorphism φ : L → L we have φ(si ) = si fi , for some fi ∈ OX (Ui ).   2 Then we find fi gij = gij fj and ρi = ρi |fi | . Still working with respect to the chosen cover UX , a connection compatible with the holomorphic structure is the datum of a collection of (1, 0)-forms ξi ∈ A1,0 X (Ui ) satisfying ξj − ξi = d log gij .

(3.9)

1 1 Note for future reference that A1,0 X = F AX . The connection is compatible with the hermitian metric if 1 π0 (ξi ) = d log ρi . (3.10) 2 Using d = ∂ + ∂¯ and decomposition with respect to (p, q)-types, we find the familiar relation

ξi = ∂ log ρi

(3.11)

for the unique connection compatible with both the complex and hermitian structures [22]. The global 2-form ¯ log ρi c1 (ρ) = ∂∂ (3.12) represents the first Chern class of L in H 2 (X, R(1)). Actually, the class of c1 (ρ) is a pure Hodge class in H 1,1 (X) and, according to the examples, it is the image of the first Chern

Hermitian-Holomorphic Classes

41

2 (X, Z(1)) → H 2 (X, R(1)) induced by Z(1) → R(1). class of L under the map HD D  Observe that c1 (ρ) = c1 (ρ ) under the isomorphism considered above. The following proposition can be found, for example, in refs. [9, 10]. (It apparently was observed first by Deligne, cf. [17].) The proof is based on writing out an explicit ˇ cocycle in a Cech resolution of C(1)• or D(1)•h .h . . This calculation will be needed later on with the complex D(1)•h .h . , so we provide a proof here.

Proposition 3.1. 2  ∼ Pic(X) (X, 1). = HD h.h.

•, Proof. Recall that D(1)•h .h . is the cone of the map ρ1 −  , where ρ1 : Z(1)•D → R(1) D • . By unraveling the structure of all the cones and  : F 1A•X ∩ σ 2 E •X (1) → R(1) D involved we have: Z(1)X −−−−−→ 1X ⊕ OX  ı⊕π  0

−−−−−→

F 1A1X ⊕ E 0X −−−−−→

2X ⊕ 1X  ı⊕π  0

−−−−−→

3X ⊕ 2X  ı⊕π  0

−−−−−→ · · ·

F 1A2X ⊕ E 1X  ⊕0 

−−−−−→

F 1A3X ⊕ E 2X  ⊕0 

−−−−−→ · · ·

F 1A2X ∩ E 2X (1) −−−−−→ F 1A3X ∩ E 3X (1) −−−−−→ · · · .

With respect to this diagram, an element of total degree 2 can be written in the form: cij k − d log gij ⊕ log gij 0 0 ξi ⊕ σ i X 0 0 ηi

(3.13)

for ξi ∈ F 1A1X (Ui ), σi ∈ E 0X (Ui ) and ηi ∈ (F 1A2X ∩ E 2X (1))(Ui ). To make sense out of (3.13), note that each entry is an element of the object in the corresponding position in the left 3 × 3 part of the previous diagram. Then since the total degree is 2, the degree ˇ of each element in the complex D(1)•h .h . is 2 minus the Cech degree as found in (3.13). 2 The top line is the class of L in Z(1)D . Finally, the entry marked X means there is no applicable element—it would have degree 3. A totally routine calculation shows that (3.13) is a degree 2 cocycle if and only if ¯ i are satisfied (with σi = 1 log ρi ). Thus relations (3.9), (3.10) and ηi = dξi = 2 ∂∂σ 2 ηi = c1 (ρ)|Ui and all the relations defining a metrized line bundle with its canonical connection are satisfied. The verification that adding an appropriate coboundary to the cocycle leads to an isomorphic metrized bundle in the sense explained above is also routine. Finally, the correspondence between component-wise addition of cocycles modulo  is again a direct verification.  coboundaries and the group operation in Pic(X)  From the proof of Proposition 3.1 we have that the cocycle representing (L, ρ) in the ˇ Cech resolution of D(1)•h .h . has the expression: 0 cij k − d log gij ⊕ log gij 0 ∂ log ρi ⊕ 21 log ρi X ¯ log ρi ∂∂ 0 0

(3.14)

42

E. Aldrovandi

This cocycle is expressed purely in terms of holomorphic and metric data (the local expression for the fiber metric ρ), and it also explicitly encodes the canonical connection and its curvature. ˇ By way of comparison, a degree two cocycle in the Cech resolution of C(1)• with respect to the same cover would be given by a triple   ηi ⊕ σi , log uij , cij k , (1,1)

where ηi ∈ AX (Ui )R(1) , σi ∈ E 1X (1)(Ui ), log uij ∈ E 0X (1)(Uij ) and cij k ∈ Z(1), satisfying the following relations: aside from the standard cocycle condition for the T-valued functions uij , we must have dηi = 0 , dσi = ηi , σj − σi = d log uij .

(3.15)

Given a holomorphic trivialization of (L, ρ) as above, the position 1 c (3.16) d log ρi , uij = gij /|gij | 2 satisfies relations (3.15). From here, the canonical connection is recovered. It is also easily seen that another solution of (3.15) is equivalent to the one above. The following is evident: ˇ Lemma 3.2. Relations (3.16) provide an explicit quasi-isomorphism between the Cech • • resolutions of D(1)h .h . and C(1) . ¯ log ρi , ηi = ∂∂

σi =

On the other hand, in (3.14) the relations defining the canonical connection are already enforced at the level of the complex. Finally, if [L, ρ] and [L , ρ  ] are the classes corresponding to the metrized bundles (L, ρ) and (L , ρ  ), then we write [L ⊗ L , ρρ  ] = [L, ρ] + [L , ρ  ] . 3.3. Cup product of hermitian holomorphic line bundles. If L and L are two line bundles 4 (X, Z(2)) on X, their cup product in Deligne cohomology would produce a class in HD    denoted  L, L . Calculations were carried out in [9] where a geometric interpretation  of L, L as a 2-gerbe and its connection with the determinant of cohomology (when X is the total space of a family of Riemann surfaces) were put forward. The structure ensuing from the generalization to line bundles equipped with a hermitian metric was further analyzed in ref. [10] by means of hermitian holomorphic Deligne cohomology. The cup product of two metrized line bundles (L, ρ) and (L , ρ  ) in hermitian ho4 lomorphic Deligne cohomology produces a class in HD (X, 2). Despite the more h.h. • involved definition of D(l)h .h . as opposed to that of C(l)• , it will nonetheless be more advantageous from the perspective of Sect. 5 to use the former to calculate the desired class. Thus let us explicitly compute a representative of this class in terms of the expression (3.14) for the class of a metrized line bundle and the modified Be˘ılinson product (2.5) for ∪α

D(1)•h .h . ⊗ D(1)•h .h . −→ D(2)•h .h . . A diagram for the complex D(2)•h .h . analogous to that for D(1)•h .h . displayed in the proof of Proposition 3.1 is:

Hermitian-Holomorphic Classes Z(2)X −→

43

OX −→ 2X ⊕ 1X −→ 3X ⊕ 2X −→ 4X ⊕ 3X −→ . . .

|π ↓ 1

| ı ⊕ π1 ↓

| ı ⊕ π1 ↓

| ı ⊕ π1 ↓

2 (1) −→ F 2 A4 ⊕ E 3 (1) −→ . . . E 0X (1) −→ F 2 A2X ⊕ E 1X (1) −→ F 2 A3X ⊕ EX X X

(3.17)

↑ | ı⊕0 4 (2) −→ . . . F 1 A4X ∩ EX

ˇ The cup product of two elements of the form (3.14) in the Cech resolution of D(1)•h .h . would result in an analogous 5 × 3 table. The degrees (in the cone) start from 0 in the leftmost entry in the top and bottom rows, and from 1 in the center one. The overall signs are determined by those in the Be˘ılinson product (2.5) plus those arising from the ˇ Cech resolution as explained in Sect. 1.2. Actually, it is visually less cumbersome to display the resulting expression as in (3.18) below, where the corresponding bidegrees are explicitly indicated: the first degree is the overall degree in the cone and the second ˇ is the Cech degree. (We have explicitly written only the nonzero terms.)  cij k cklm  −cij k log gkl   −d log gij ∧ d log gj k ⊕ − log gij d log gj k

−π1 log gij log |gj k |     −(1 − α) d log gij ∧ ∂ log ρj + α ∂ log ρi ∧ d log gij ⊕   c  1 (1 − α) d log |gij | 2 log ρj − log |gij | 21 d c log ρj (3, 1)      +α − 21 d c log ρi log |gij | + 21 log ρi d c log |gij |   ¯ log ρi ∧ ∂ log ρ  + (1 − α) ∂ log ρi ∧ ∂∂ ¯ log ρi ∧ ∂∂ ¯ log ρ  ¯ log ρ  α ∂∂ ∂∂ i i i (4, 0)   ¯ log ρi 1 log ρ  + (1 − α) 1 log ρi ∂∂ ¯ log ρ  (3.18) ⊕ α ∂∂ i i 2 2

(0, 4) (1, 3) (2, 2)

3.4. Integration along the fiber. To conclude this introduction, let us quickly mention that Hermitian holomorphic Deligne cohomology has an integration along the fiber map. Namely, if π : X → S is a proper submersion of complex manifolds, it follows from  [20] that Deligne cohomology has an integration along the fiber π , hence by [9, 10] there is a map i−2d i HD (X, l) −→ HD (S, l − d), h.h. h.h.

(3.19)

where d is the complex dimension of the fiber, and commutative diagrams analogous to [10, Theorem 5.1]. We will be interested in the case of complex relative dimension 1, namely π : X → S is a holomorphic fibration with compact connected Riemann surfaces as fibers. From 4 2 (X, 2) → HD (S, 1) into the group of (3.19) in degree 4 we have the map: HD h.h. h.h. complex hermitian line bundles on S (cf. Proposition 3.1). We refer to refs. [9, 10] for a complete treatment of the map (3.19), in particular for the “trace” map R• π∗ (D(l)•h .h . ) −→ D(l − d)•−2d h .h . ,

44

E. Aldrovandi

which induces (3.19) at the level of cohomology. (Clearly any quasi-isomorphic model for Hermitian-holomorphic cohomology will do.) We will limit ourselves to observe the following: if US is a good cover of S, and UX a good cover of X which refines π −1 US , the i th direct image Ri π∗ F • of any complex F • on X is the sheaf on S associated to the presheaf   V −→ H i Tot Cˇ • (UX ∩ π −1 (V ), F • ) . Then unraveling the cone structure of D(l)•h .h .X (or of C(l)•X ) reduces to computing the direct images of Z(l)X and A•X , the de Rham complex. The trace map is then obtained by capping a total cocycle, say in  p Cˇ q (UX ∩ π −1 (V ), AX ) , (3.20) p+q=i

with a representative of the fundamental class [M] of the smooth model M of the fiber of π −1 (V ) ∼ = V × M (as smooth manifolds). Explicit representatives for [M] for a triangulation subordinated to the nerve of UX ∩ π −1 (V ) are computed in ref. [3]. If p ωi0 ,...,iq are the components of a cocycle ωVi in (3.20), then  p 2d−q ωVi ∩ [M] = ωi0 ,...,iq ∩ i0 ,...,iq , 2d−q

where the i0 ,...,iq are signed generators of the nerve of dimension 2d − q. The exact combinatorics can be found in ref. [3], so we will not further pursue the matter here. Let us conclude by noticing that for π : X → S the cup product of (L, ρ) and (L , ρ  ) induces a metrized line bundle on S. From the preceding discussion, the connection component, for example, will be obtained by following the (two) components 2 F 2A•X ∩ E •≥4 X (2) in (3.18). The one in degree (4, 0) will be capped with generators  of dimension 2 (hence integrated over 2-simplices), those in degree (3, 1) with generators 1 of dimension 1. From [9, Theorem 6.1] there is a corresponding map A(2,2) (X)R(2) → A(1,1) (S)R(1) . From the explicit cocycles the corresponding representative in H 2 (S, R(1)) ∩ H 1,1 (S) of the first Chern class of the resulting metrized line bundle is  c1 (ρ) ∧ c1 (ρ  ) , π

as in [11, Prop. 6.6.1]. 4. Conformal Metrics and the Liouville Functional Let X be a compact Riemann surface of genus g ≥ 2. For simplicity we can assume X to be connected, although this is not necessary, and in fact this assumption will be dropped when dealing with Kleinian groups. Let CM (X) be the space of conformal metrics on X. Locally on X, if z is a local analytic coordinate defined on an open set U , any metric ds 2 can be represented as ds 2 = ρ |dz|2 for a positive function ρ : U → R>0 . According to Sect. 3.2, a conformal metric corresponds to considering the metrized line bundle (TX , ρ), where TX is the holomorphic

Hermitian-Holomorphic Classes

45

tangent bundle of X. With respect to a cover UX of X, the conformal factors ρi and ρj , associated to Ui and Uj respectively, satisfy the relation  2 | , ρj = ρi |zij

(4.1)

 = dz /dz and z is a local coordinate defined over U . where zij i j i i It follows from the uniformization theorem that there exists a unique conformal metric of scalar curvature equal to −1, the Poincar´e metric. Locally on Ui ⊂ X, the condition for the metric to have curvature −1 is equivalent to the nonlinear PDE

1 ∂2 φi = exp φi , ∂zi ∂ z¯ i 2

(4.2)

known as the Liouville equation, for the smooth function φi = log ρi . Observe that Eq. (4.2) can be written in the form √ c1 (ρ) = −1 ωρ , where we have used the K¨ahler form associated to the metric: √ −1 ωρ |Ui = ρi dzi ∧ d z¯ i . 2 This representation makes it apparent that the Liouville equation is independent of the choice of the coordinate system. On the other hand, a direct verification of this fact is  |2 . immediate using φj − φi = log|zij 4.1. Variational problem for conformal metrics. It is well known that the Liouville equation has a local variational principle in the following sense. Let D be a region in the complex plane. Then Eq. (4.2) is the Euler-Lagrange equation for the variational problem defined by the action functional √    −1 ¯ + eφ dz ∧ d z¯ , S[φ] = ∂φ ∧ ∂φ (4.3) 2 D  d defined on smooth functions φ : D → R, with the condition that variations dα (φα )α=0 of φ be zero on ∂D. However, it is easily seen that the functional (4.3) cannot be defined globally on X, since, as a consequence of (4.1), the first term under the integral sign would not yield a well-defined 2-form on X. (The second one would of course present no problems, it would just give the area AX (ρ) of X with respect to the given metric ρ |dz|2 = eφ |dz|2 .) Accordingly, it is convenient to write the integrand in (4.3) as √ −1 ω[φ] + ωρ , where we have defined the 2-form4 ω0 [φ] =

1 ¯ , ∂φ ∧ ∂φ 2

(4.4)

and restrict our considerations to the first term of (4.3) which we denote 4

Note that Eq. (4.4) defines an imaginary form. The reason for this choice will be apparent later.

46

E. Aldrovandi

ˇ S[φ] =

√ −1

 ω0 [φ] .

(4.5)

D

There is by now an established procedure on how to address the problem caused by the fact that (4.4) is not globally defined. In general terms, given the choice of a conˇ Rham formal metric ds 2 ∈ CM (X) and a cover UX , one works with the full Cech-de • • ˇ complex C (UX , E X (1)) with respect to UX , rather than with just differential forms. The 2-form (4.4) is then completed to a total degree 2-cocycle—to be denoted [φ]. This results in a class in H 2 (X, R(1)) after taking cohomology. (Whether or not there also is a de Rham type theorem will depend on the acyclicity properties of UX .) This scheme has been previously carried out not quite for covers of X by open sets, but actually for different choices of planar coverings. For the covering associated to a Schottky uniformization of X a generalization of Eq. (4.3) was written in ref. [34]. More recently, a detailed calculation of the cocycle for the general case of a covering associated to a Kleinian uniformization was carried out in ref. [30] by exploiting the homological methods developed in ref. [2]. Note, however, that from the point of view of ref. [3] these planar coverings are “´etale” coordinates on X, so the group cohomology constructions required to work with the various kinds of uniformization coverings just ˇ follow from specializing the Cech formalism to the coverings at hand. Finally, the integration in Eqs. (4.3) or (4.5) should be replaced by the evaluation of [φ] against an appropriate representative  of the fundamental class of X. The “appropriate” form for both  and the evaluation will be dictated by the chosen cover UX and the cohomology theory being used. Typically  will be a cycle in a double complex of UX -small simplices, where the differentials are the singular one and the one determined ˇ by the face maps induced by the cover. Thus, in the case of a Cech cover, it will be the complex determined by a triangulation of X subordinated to the open UX , and in the same way, for a planar cover the singular complex of the planar domain UX tensored with an appropriate bar resolution of the group of deck transformations. These issues have been discussed at length in refs. [2, 3, 30], so we will not repeat the discussion here. Whenever we have a cocycle extending (4.4) and a cycle  representing X we state Definition 4.1. The Liouville functional (without the area term) is given by the evaluation: ˇ S[φ] =−

1 √

2π −1

[φ] ,  .

For the complete functional we add the area term ˇ S[φ] = S[φ] +

1 2π

(4.6)

 ωρ .

(4.7)

X

√ ˇ Remark 4.1. [φ] ,  ∈ R(1), and S[φ] (or S[φ]) is real. Division by 2π −1 is con√  ventional, but note that R(1) − → R+ via exp(·/2π −1). In the sequel it will be more convenient to work with imaginary classes. (See also Sect. 5.) In the next two subsections we examine these constructions in some detail. For definiteness, we initially make use of an ordinary open cover. First, we recall the direct construction of a cocycle generalizing (4.4), and we show that this way Eq. (4.2) indeed is the resulting extremum condition. Then we emphasize the role played by Deligne cohomology and the tame symbol. These aspects will become important when introducing a Kleinian uniformization later in the paper, when we discuss connections with

Hermitian-Holomorphic Classes

47

the dilogarithm function. In a later section we shall tackle the question of its geometrical significance by making full use of the hermitian-holomorphic version of Deligne cohomology presented in Sect. 3, and we show that the cocycle constructed following refs. [34, 30] corresponds to the square of (TX , ρ) in hermitian holomorphic Deligne cohomology. 4.2. Direct construction of the Liouville cocycle. 4.2.1. Initial setup. Let X a compact Riemann surface of genus greater than 2. We shall not include the area term in our explicit calculations, therefore it makes sense to extend our considerations to a general metrized line bundle (L, ρ). Of course, whenever referring to a conformal metric or to the variational problem for the Liouville equation, it  . Upon choosing a cover U , which will be be assumed that L = TX and that gij = zij X ˇ for now we assume to be a Cech cover by open sets, the pair (L, ρ) is described in terms of the data expounded in Sect. 3.2. Our starting point will be the 0-cochain ωi0 [log ρi ] =

1 1 1 ∂ log ρi ∧ ∂¯ log ρi = − d log ρi ∧ d c log ρi 2 2 2

(4.8)

with values in E 2X (1)(Ui ). Remark 4.2. A generalization for Eq. (4.8) would be to consider a pair of metrized line bundles (L, ρ) and (L , ρ  ), and then the analog of (4.8) would be  1 1 1 1 1 ωi0 [log ρi , log ρi ] = − d log ρi ∧ d c log ρi + d c log ρi ∧ d log ρi . (4.9) 2 2 2 2 2 Note, however, that the expressions are quadratic. Moreover, L ⊗ L has metric ρρ  , so that there is a natural “polarization identity” ω[log ρ, log ρ  ] =

ρ 1 1 ω[log ρρ  ] − ω[log  ], 4 4 ρ

(4.10)

where we have omitted the indexes for simplicity of notation. We shall comment later on the significance of Eq. (4.10). 4.2.2. Computation. Let us extend (4.8) to a degree 2 cocycle in the total simple comˇ plex associated to the double complex Cˇ • (UX , E •X (1)) of Cech cochains with values in the de Rham complex of imaginary smooth forms. This is accomplished in the usual 1 [log ρ] on U and a 2-cochain fashion (see e.g. [8]) by finding a 1-cochain of 1-forms ωij ij 2 of 0-forms ωij k [log ρ] on Uij k such that the relations ˇ 0 = −dω1 , δω

ˇ 1 = dω2 , δω

ˇ 2=0 δω

are satisfied. Of course, the remaining one, namely dω = 0 is automatically satisfied for dimensional reasons. It turns out that to a great extent these relations are explicitly computable without further assumptions, such as that the cover UX be good. The needed calculations are fairly standard, and they are presented in great detail in ref. [30], so we shall be brief. (The observation in [25] that on a Riemann surface for two smooth functions f and g one has df ∧ d c g = dg ∧ d c f is useful in carrying out the calculations.) The first two steps are as follows.

48

E. Aldrovandi

First, one has: 1 ωj0 [ρ] − ωi0 [ρ] = −d ωij [ρ], 1 1 1 ωij [ρ] = log ρi d c log |gij | + d c log |gij | log ρj . 2 2

(4.11)

The next step yields:   1 1 [ρ] + ωij [ρ] δˇ ω1 [log ρ] ij k = ωj1k [ρ] − ωik = log |gij | d c log |gj k | − d c log |gij | log |gj k |, and notice that d c log |gij | = π1 d log gij , and πp+q−1 (a ∧ b) = πp (a) ∧ πq−1 (b) + πp−1 (a) ∧ πq (b) , so we have       δˇ ω1 [log ρ] ij k = π1 log gij d log gj k − d π1 (log gij ) log |gj k | .

(4.12)

Observe that now the problem of continuing the descent becomes independent of the chosen metric ρ. The most direct way of proceeding is the following. If we assume the cover 2 ∈ E 0 (1)(U ) UX to be acyclic for the de Rham complex E •X (1), then there exists ωij ij k X k 2 1 ˇ such that dωij k = δ(ω [log ρ])ij k . Furthermore, consistency on a quadruple intersection ˇ 2 be a 3-cocycle with values in R(1)X . This cocycle must be a cobounrequires that δω dary, since H 3 (X, R(1)) = 0, therefore, up to readjusting the constants, there exists a 2 such that δω ˇ 2 = 0, and furthermore, ω2 does not depend on the metric choice of ωij k ij k structure. 4.2.3. Solution to the variational problem. The previous preliminary calculation is sufficient from the point of view of finding the extrema. To this effect, we set L = TX , for ρ a conformal metric in CM (X). Notice that the space of conformal metrics on X is affine over C ∞ (X, R) ≡ E 0 (X): if ds 2 and ds  2 are two conformal metrics with local expressions ρi |dzi |2 and ρi |dzi |2 respectively, then there exists σ ∈ C ∞ (X, R) such that log ρi = σ |Ui + log ρi . The change from [log ρ] to [log ρ + σ ] can be exactly computed thanks to the fact that the last step in the determination of [log ρ] is independent of ρ and the quadratic character of (4.4). Indeed we have: Proposition 4.1. S[log ρ + σ ] − S[log ρ] =

1 √

2π −1



1 X

2

¯ + σ c1 (ρ) − ∂σ ∧ ∂σ

√

 −1(eσ − 1)ωρ .

Proof. The change in ωi0 [log ρ] is computed as 1 1 1 1 ωi0 [log ρ + σ ] − ωi0 [log ρ] = − dσi ∧ d c σi + σi dd c log ρi − d(σi d c log ρi ) , 2 2 2 2 where we set σi ≡ σ |Ui . Note that the first two terms on the right-hand side are globally well-defined 2-forms. On the other hand

Hermitian-Holomorphic Classes

49

1 1 ωij [log ρ + σ ] − ωij [log ρ] = σi d c log |gij | .

Letting χi = σi d c log ρi , we see that 1 1 1 [log ρ + σ ] − [log ρ] = − dσ ∧ d c σ + σ dd c log ρ − Dχ , 2 2 2 and taking the area terms into account, establishes the formula.   As a consequence, we obtain Corollary 4.1. The Liouville equation (4.2) is the Euler-Lagrange equation for the Liouville functional (4.7) introduced in Definition 4.1. The critical point is non-degenerate. Proof. Replacing σ with t σ , t ∈ R, in the previous proposition we find the infinitesimal change in [log ρ] to be   √   d  1 S[log ρ + tσ ] = − √ σ c1 (ρ) − −1 ωρ , dt t=0 2π −1 X and it follows that S[log ρ] has an extremum if and only if the Liouville equation is satisfied. Non degeneracy follows from the quadratic part in the exact change formula in Proposition 4.1.   Remark 4.3. The fact that the change in the cocycle is given by a pure 2-form term up to total coboundary can also be analyzed in terms of gluing properties of variational bicomplexes, cf. Theorem 1.2 in ref. [1] and the proof of Theorem 1 in ref. [3]. From this perspective, Corollary 4.1 is a direct consequence of the affine structure of the space CM (X) of conformal metrics. 4.3. A cup product. Formula (4.12) can be handled in a more geometric fashion as follows. From Sect. 2.4 we can rewrite (4.12) as   δˇ ω1 [log ρ] ij k = −r2 (gij , gj k ) , (4.13)   and we have the collection of tame symbols gij , gj k associated with any triple  inter section Uij k in the cover UX , [9]. These symbolsglue to form a global symbol L, L . As a cohomology class on a curve X, however, L, L will be zero (that is, there is a global object in the associated 2-stack, cf. [10]), so that it will be possible to choose local functions Lij k such that dLij k = − log gij d log gj k , as explained in detail in [3]. Note that we still need the cover UX to be fine enough. Moreover, cf. loc. cit., the collection Lij k can be chosen in a way that ˇ ij kl = −cij k log gkl + nij kl , δL where nij kl ∈ Z(2). Therefore from (2.19), (4.12) and (4.13), we can set   2 ωij k = − π1 (log gij ) log |gj k | − π1 Lij k ,

(4.14)

2 = −L (g , g ). Now, the last comnamely according to Sect. 2.4 we have set ωij 2 ij jk k patibility condition is satisfied, indeed we have:

50

E. Aldrovandi 2 ˇ ˇ ij δω kl = −cij k log |gkl | − π1 (δLij kl )

= −cij k log |gkl | + π1 (cij k log gkl ) = 0. As a result, we obtain a 2-cocycle in the the total complex associated to Cˇ • (UX , E •X (1)) 2 in terms of the trivialization of as before, with a more geometric interpretation of ωij k    • , thus the symbol L, L . Moreover, notice that if X is a curve then E • (1)[1] − → R(2) X

D

we may interpret the class so determined by [log ρ] = ω0 [log ρ] + ω1 [log ρ] + ω2 as 3 (X, R(2)). a (degree 3) class in HD 4.4. Two line bundles. For a pair (L, ρ), (L , ρ  ) of metrized line bundles we can complete (4.9) to a cocycle [log ρ, log ρ  ] via an analogous procedure to the one presented in Sects. 4.2 and 4.3. The relevant calculations being entirely similar, we limit ourselves to quoting the relevant expressions. Starting from (4.9), which we rewrite in the form ωi0 [log ρi , log ρi ] =

1 c 1 d log ρi ∧ d log ρi , 2 2

(4.15)

the corresponding expression for the degree (1, 1) term is: 1 ωij [log ρ, log ρ  ] =

1 1 log ρi d c log |gij | + d c log |gij | log ρj . 2 2

ˇ Computing the Cech coboundary we find:  1  ˇ [log ρ, log ρ  ] δω = log |gij | d c log |gj k | − d c log |gij | log |gj k | ij k = −r2 (gij , gj k ) ,

(4.16)

(4.17)

2 (now independent of ρ and ρ  ) can be obtained as −L (g , g  ) by from which ωij 2 ij k jk looking at a collection Lij k such that

dLij k = − log gij d log gj k ,   from the triviality of the symbol L, L .

4.5. Additional remarks on the Liouville functional and the Bloch-Wigner dilogarithm. We wish to compare the previous constructions to those of ref. [30]. To this end we need to specifically consider the case of a cover of X provided by a Kleinian uniformization. (The reader should consult loc. cit. for reference and complete details.) The comparison offers a better perspective on the absence of cohomological obstructions in the calculations of ref. [30], and on the relations with three-dimensional hyperbolic geometry. Let
be a purely loxodromic second kind of Kleinian group satisfying all the conventions spelled out in Appendix B, to which we refer for the notation. Let U
⊂ P1 be the region of discontinuity, and X = U
/
the resulting (possibly disconnected) Riemann surface.

Hermitian-Holomorphic Classes

51

A conformal metric ρ on X = X1  · · ·  Xn appears as an automorphic function on U
: 1 1 (4.18) log ρ − log ρ ◦ γ = log |γ  | , γ ∈
. 2 2 Equation (4.18) is the direct translation of (3.8) following the principles of [3]. Accordingly, the first two terms of the Liouville cocycle computed by applying the procedure explained in Sect. 4 are: 1 1 ω0 [log ρ] = − d log ρ ∧ d c log ρ, (4.19) 2 2 1  1 ωγ1 [log ρ], = (4.20) log ρ + log ρ ◦ γ d c log |γ  |, 2 2 and computing the coboundary of (4.20) according to the prescription in Appendix B yields   δˇ ω1 [log ρ] γ ,γ = −r2 ((γ1 γ2 ) , γ2 ) , (4.21) 1

2

where r2 has been introduced in Eqs. (2.16) and (2.17). Note that, as in Sect. 4.3, the coboundary of ω1 is a cup product in real Deligne cohomology: r2 ((γ1 γ2 ) , γ2 ) = (d log(γ1 γ2 ) , log |(γ1 γ2 ) |) ∪ (d log γ2 , log |γ2 |) , where the two classes are associated to the rational functions (γ1 γ2 ) and γ2 , respectively (cf. Sect. 2.4). Hence, we can work with the double complex of group cochains on
 • • − with values in the real Deligne complex R(2) D → E U
(1) on the region of discontinuity U
. Since from this point on only rational functions with singularities at certain prescribed points will appear, following ref. [21] we will consider the Deligne complex on the generic point ηP1 of P1 . For any two elements γi , γj ∈
define Tij ∈ PSL2 (C) by z −→ Tij (z) = [z : zij : zj : ∞] =

z − zij , z − zj

(4.22)

where zj = γj−1 (∞) and zij = γj−1 (zi ). Following ref. [30], we introduce the 1-cochain on
with values in E 1P1 (1)(ηP1 ): κγ = log |cγ | π1 d log γ  .

(4.23)

We have:

  √ ˇ γ1 ,γ2 , where L2 = −1D2 , and D2 Lemma 4.1. r2 ((γ1 γ2 ) , γ2 ) = 4 d L2 ◦ T12 + δκ is the standard Bloch-Wigner dilogarithm, cf. 2.4. Proof. A straightforward calculation exploiting relation (B.3).

 

Since obviously dκγ = 0, we can redefine → + κγ , so that using the lemma, from Eq. (4.21) we have:     δˇ ω1 [log ρ] γ ,γ = −4 d L2 ◦ T12 . (4.24) 1 2   For convenience of notation, let us temporarily set ωˆ γ21 ,γ2 := −4 L2 ◦ T12 . We then have the following ωγ1

ωγ1

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E. Aldrovandi

Lemma 4.2. D(ω0 + ω1 + ωˆ 2 ) = −4 L2 ([∞ : γ1 (∞) : γ1 γ2 (∞) : γ1 γ2 γ3 (∞)]) .

(4.25)

ˇ Proof. By construction, D(ω0 + ω1 + ωˆ 2 ) = δˇ ωˆ 2 , and the latter Cech coboundary is computed as  δˇ ωˆ γ21 ,γ2 ,γ3 = −4 L2 ([z : z23 : z3 : ∞]) − L2 ([z : z123 : z3 : ∞])  + L2 ([z : z123 : z23 : ∞]) − L2 ([γ3 (z) : z12 : z2 : ∞]) . In the last term we have [γ3 (z) : z12 : z2 : ∞] = [z : z123 : z23 : z3 ] . The Bloch-Wigner dilogarithm satisfies the 5-term relation [15, 23]: 4 

(−1)i D2 ([a0 : · · · : aˆi : · · · : a4 ]) = 0 ,

(4.26)

i=0

where a0 , . . . , a4 ∈ P1 and the hat sign denotes omission. As a consequence we have: δˇ ωˆ γ2 ,γ ,γ = −4 L2 ([z123 : z23 : z3 : ∞]) , 1

2

3

and again by the invariance of the cross ratio, we obtain (4.25).

 

Remark 4.4. D2 ([a : b : c : d]) is the hyperbolic volume of the ideal hyperbolic tetrahedron with vertices at the points a, b, c, d ∈ P1 , see, e.g. refs. [16, 14, 23]. It follows from the five-term relation (4.26) that the right hand side of Eq. (4.25) defines an R(1)-valued 3-cocycle on
. Moreover, this cocycle is already defined on PSL2 (C), where its class is known to generate the Eilenber-Mac Lane cohomology group H 3 (PSL2 (C), R(1)) , [7, 14, 23]. It is also known that up to a factor 2 it agrees with the imaginary part of the second Cheeger-Simons universal secondary class Cˆ 2 . Thus the complete Liouville cocycle  = ω0 + ω1 + ω2 subordinated to the cover U
→ X is found as follows. The pullback of Cˆ 2 along the inclusion map
→ PSL2 (C) is zero, since (
being of the second kind) the 3-manifold M
= H3 /
is non-compact, and H • (
, R(1)) ∼ = H • (M
, R(1)). It follows that the restriction of the cocycle given by the Bloch-Wigner dilogarithm to
must be a coboundary, hence there exists a group 2-cochain c on
with values in R(1) such that ˇ γ1 ,γ2 ,γ3 . 4 L2 ([∞ : γ1 (∞) : γ1 γ2 (∞) : γ1 γ2 γ3 (∞)]) = (δc) (4.27) It follows that the cochain c provides the necessary “integration constants,” namely the required 2-cochain on
with values in E 0U
to complete the Liouville cocycle is   ωγ21 ,γ2 = −4 L2 ◦ T12 + cγ1 ,γ2 . 5. Conformal Metrics and Hermitian Holomorphic Cohomology In Sect. 4 we presented a construction of a degree 2, R(1)-valued class corresponding to a conformal metric ρ ∈ CM (X), represented by the cocycle [log ρ]. Supplemented by the area of X computed with respect to ρ, it provides a global functional for the variational problem associated to the Liouville equation (4.2). We now show that it coincides with the square of the class of (TX , ρ) in hermitian holomorphic Deligne cohomology introduced in Sect. 3. Moreover, we show that this equality holds at the cocycle level. More generally, without considering the area term, we show that the cup product of (L, ρ) and (L , ρ  ) in hermitian holomorphic Deligne cohomology coincides with the class of [log ρ, log ρ  ], and again the equality in fact holds at the cocycle level.

Hermitian-Holomorphic Classes

53

5.1. Comparison on a curve. Let X be a compact Riemann surface. From the results of Sect. 3.3, the cup product of the classes of (L, ρ) and (L , ρ  ) in hermitian holomorphic 4 Deligne cohomology yields a class in HD (X, 2), and on a curve we only capture the h.h. 2-dimensional part of this class. Indeed, in the exact sequence (3.6), the cohomology   4 (X, Z(2)) is zero, and A(2,2) (X) class corresponding to the symbol L, L ∈ HD R(2) is also zero for obvious dimensional reasons, so we have: 3 4 · · · −→ HD (X, R(2)) −→ HD (X, 2) −→ 0 . h.h. 3 (X, R(2)). It follows that the class [L, ρ] ∪ [L , ρ  ] must come from an element in HD  • 3 ∼ • − As already remarked, on a curve we have R(2) D → E X (1)[−1], thus HD (X, R(2)) = • • 3 2 2 ∼ ∼ H (X, E X (1)[−1]) = H (X, E X (1)) = H (X, R(1)), in agreement with the calculations performed in Sect. 4. In more detail, in complex dimension 1 the second hermitian holomorphic Deligne complex D(1)•h .h . simplifies considerably and diagram (3.17) reduces to −ı

Z(2) −−−−→

−d

OX −−−−→  π  1 −d

1X  π  1

(5.1) −d

E 0X (1) −−−−→ E 1X (1) −−−−→ E 2X (1) π1

so that D(2)•h .h . becomes just the cone of the morphism Z(2)•D −→ E •X (1)[−1]. In other words, on a curve X we have that D(2)•h .h . is given by the complex (−d,−π1 ) −π1 +d −ı d Z(2)X −−−−→ OX −−−−−→ 1X ⊕ E 0X (1) −−−−→ E 1X (1) −−−−→ E 2X (1) , (5.2)

where the differentials have been written explicitly. We can see the complex E •X appears as a subcomplex in (5.2) and the shift of two positions to the right clearly accounts for the cohomology degree shift from 2 to 4. Our main result is the following comparison Theorem 5.1. Let X be a compact Riemann surface of genus g > 1. Let (L, ρ) and (L , ρ  ) be two hermitian holomorphic line bundles. Then the class of [L, ρ] ∪ [L , ρ  ] 4 in HD (X, 2) ∼ = H 2 (X, R(1)) coincides with the one represented by the cocycle  h.h.   log ρ, log ρ  constructed in Sect. 4. 4 Proof. We have observed above that HD (X, 2) ∼ = H 2 (X, R(1)), and by construction h.h. the class of [log ρ, log ρ  ] is in H2 (X, E •X (1)) ∼ = H 2 (X, R(1)). Note that for X connected they must coincide up to a proportionality factor, since H 2 (X, R(1)) ∼ = R(1) in this case. In general, we compute the proportionality factor from the explicit cocycles from Sects. 3.3 and 4.2 to 4.4. (Since [log ρ, log ρ  ] is computed under sui acyclicity assumptions on the cover, so we will use such a cover to establish the comparison.) Let us assume L and L and their respective hermitian metric structures are represented by cocycles of type (3.14) with respect to the chosen cover UX . Specializing the

54

E. Aldrovandi

general expression in Table 3.18 in Sect. 3.3 to the case at hand we obtain, with reference to (5.2):  cij k cklm

(0, 4)

 −cij k log gkl

(1, 3) (2, 2)

− log gij d log gj k ⊕

−π1 log gij log |gj k |   c (3, 1) (1 − α) d log |gij | 21 log ρj − log |gij | 21 d c log ρj   +α − 21 d c log ρi log |gij | + 21 log ρi d c log |gij | ¯ log ρ  ¯ log ρi 1 log ρ  + (1 − α) 1 log ρi ∂∂ (4, 0) α ∂∂ i i 2 2

(5.3)

where we have followed the convention explained in the introduction for the bidegrees. Let us denote by θ i the term of bidegree (4 − i, i) in (5.3) and by  the total cocycle. (For simplicity, we suppress ρ and ρ  from the notation.) A direct calculation shows that θi0 = ωi0 + dλ0i ,

1 ˇ 0ij , θij1 = ωij − δλ

where ωi0 is given by Eq. (4.15), ω1 is given by Eq. (4.16), and 1 1 λ0i = α d c log ρi log ρi + (1 − α) log ρi d c log ρi . 2 2 Note that at this point we could simply define  =  − Dλ0 . Furthermore, note that  does not explicitly depend on the parameter α from the Be˘ılinson product (2.5). To finish the comparison, let us assume the cover UX allows us to find a collection Lij k ∈ OX (Uij k ) such that dLij k = − log gij d log gj k ,  ˇ ij kl = −cij k log gkl δL + nij kl ,

(5.4)

2 = −L (g , g  ).  is as in Sect. 4.3. In this way we have  = (ω0 , ω1 , ω2 ) with ωij 2 ij k jk a cocycle of total degree 2 in Tot Cˇ • (UX , E •X ), and it injects (via the exact sequence of the cone) into Tot Cˇ • (UX , D(2)•h .h . ) as

(ω0 , ω1 , 0 ⊕ ω2 ) . Then via Eq. (5.4) it is easily seen that 2 θij2 k = dD(2)•h.h. (−Lij k ) + 0 ⊕ ωij k,

where dD(2)•h.h. is the differential in D(2)•h .h . , and therefore  = (ω0 , ω1 , 0 ⊕ ω2 ) + Dλ0 + D(−L, n) , where we have put D = dD(2)•h.h. ± δˇ for the total differential. Thus the two cocycles constructed via the direct method of Sect. 4.2 and the cup product of metrized bundles define the same class. By direct comparison, the proportionality factor is 1.  

Hermitian-Holomorphic Classes

55

In light of the previous theorem, the polarization identity in Remark 4.2 is now easily explained. Using [L ⊗ L , ρρ  ] = [L, ρ] + [L , ρ  ] and [L ⊗ L ∨ , ρ/ρ  ] = [L, ρ] − [L , ρ  ] , and the (graded) commutativity of the cup product ∪

2 2 4 (X, 1) ⊗ HD (X, 1) −→ HD (X, 2) , HD h.h. h.h. h.h.

we obtain the polarization identity ∨

4[L, ρ] ∪ [L , ρ  ] = [L ⊗ L , ρρ  ]2 − [L ⊗ L , ρ/ρ  ]2 , where the squares in the right hand side refer to ∪. A polarization identity at the level of representative cocycles, and hence the one in Remark 4.2, follow by applying Thm. 5.1 to the latter identity. By choosing L = L = TX , the holomorphic tangent bundle of X, we immediately obtain: Corollary 5.1. Let ρ ∈ CM (X) be a conformal metric. The Liouville functional without area term (4.6) is given by the (evaluation of) the square [TX , ρ] ∪ [TX , ρ] with respect to the cup product in hermitian holomorphic Deligne cohomology. The full-fledged Liou 1 ville functional is obtained by adding the area term 2π X ωρ to (4.6). Remark 5.1. Due to the specific form of the differential in the complex D(2)•h .h . the descent equations are explicit and close automatically. Therefore the cocycle resulting from the calculation of the cup product sidesteps the problem of the explicit calculation of the last term, unlike the more direct version from Sect. 4. Thus, thanks to the explicit character of the calculation, specific assumptions on the nature of the cover UX are not required. It follows from Thm 5.1, Corollary 5.1 and the previous remark that Definition 4.1 applies to any (´etale) cover UX → X. Indeed, Proposition 4.1 from Sect. 4.2 can be reformulated at the cocycle level as follows: Proposition 5.1. Let X be a compact, genus g > 1 Riemann surface and let UX → X be a cover. For a conformal metric ρ ∈ CM (X) and σ ∈ C ∞ (X, R), there is a cocycle ˆ U [log ρ] solving the variational problem for the Liouville equation.  X Proof. If ρ is a conformal metric, let the pair (TX , ρ) be represented, as an hermitian line bundle, by a cocycle c(TX , ρ) with respect to the cover UX . We set [log ρ] = c(TX , ρ) ∪ c(TX , ρ) , and a simple calculation starting from Eq. (5.3) yields [log ρ + σ ] − [log ρ] = σ c1 (ρ) + where χi = 21 σ 21 d c log ρi − 21 d c σ

1 ¯ + Dχ , σ ∂ ∂σ 2

1 2

log ρi . Now define √ ˆ U [log ρ] = [log ρ] − −1ωρ .  X

We see that it yields the formula in Proposition 4.1. In particular we have that  √ d  ˆ U [log ρ + tσ ] ≡ σ (c1 (ρ) − −1ωρ ) ,  X dt t=0 where ≡ means “up to total coboundary.”

 

56

E. Aldrovandi

5.2. Determinant of cohomology. Let again L and L be two holomorphic line bundles with hermitian metrics ρ and ρ  , respectively, on the compact Riemann surface X. Brylinski proves in [10] that the cup product of L and L in hermitian holomorphic Deligne cohomology yields the (logarithm) of the metric · on the Deligne pairing L , L  defined in [11]. It follows, via Thm 5.1 and the isomorphism 3.1 between our version of hermitian holomorphic cohomology and Brylinski’s, that the class of [log ρ, log ρ  ] is also equal to logL , L . It is worthwhile to provide a direct proof of this fact starting from the explicit cocycle given in (5.3). First, we need to recall a few definitions from [11]. A complex line L , L  is assigned to the pair (L, L ) as follows. Let D and D  be divisors on X corresponding to L and L , and assume they have disjoint supports. Consider two rational sections, s and s  such that (s) = D and (s  ) = D  . To this datum one assigns a copy of the complex line generated by the symbol s , s   subject to the relations: f s , s   = f (D  )s , s  , s , gs   = g(D)s , s  ,

(5.5)

whenever f is a rational function with divisor (f ) disjoint from D  , and similarly for g. The Weil reciprocity relation f (div (g)) = g(div (f )) (cf. ref. [22]) for two rational functions f and g with disjoint divisors implies that the relations (5.5) are consistent and the complex line depends only on the pair L, L . When the line bundles are equipped with hermitian metrics, generically denoted by ·, the assignment5  1  2 log s , s  = ∂ ∂¯ logs2 logs  2 + logs2 [D  ] + logs  2 [D] √ 2π −1 X (5.6) is compatible with the relations (5.5) and defines an hermitian metric on the complex line L , L . In formula (5.6) the operator ∂ ∂¯ is to be computed in the sense of distributions. Having covered the main definitions, we can now state Theorem 5.2. The cup product of (L, ρ) with (L , ρ  ) in hermitian holomorphic Deligne cohomology corresponds to the logarithm √ of the norm (5.6) on the Deligne pairing L , L . The proportionality factor is −π −1. Proof. Let D and D  be divisors with disjoint support on X corresponding to L and L , respectively. Using the same technique as in refs. [10, 11], consider two C ∞ positive real functions f1 and f2 such that f1 + f2 = 1 and f1 (resp. f2 ) vanishes in a neighborhood of the support of D  (resp. D). Also, set U1 = X \ supp(D  ) and U2 = X \ supp(D). Thus {f1 , f2 } is just a partition of unity subordinated to the cover UX = {U1 , U2 }. The only two terms different from zero in the cocycle  in (5.3) representing the class ˇ [L, ρ] ∪ [L , ρ  ] with respect to this cover are θi0 and θi1 , with the Cech index i ∈ {1, 2}. Thus the class we are after is equivalently given by the integral  1 f1 θ10 + f2 θ21 + df2 ∧ θ21 , (5.7) X

which is arrived at by applying in the standard homotopy operator based on the partition ˇ 1 = 0 we obtain that θ 1 is the coboundary of the cochain of unity {f1 , f2 }: from δθ 12 12 5 We write the square explicitly, whereas the symbol · used in ref. [11] denotes the square of the norm.

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57

 1 . Observe that the 2-form in (5.7) j → i=1,2 fi θij1 and then we use θ20 − θ10 = −dθ12 1 is defined only on U ∩ U , but df has support on is globally well defined over X: θ21 1 2 2 U1 ∩ U2 , so their wedge product is defined everywhere; similarly, fi θi0 is everywhere defined thanks to the fact that fi has support in Ui , i = 1, 2. Consider rational sections s and s  of L and L such that div (s) = D and div (s  ) = D  as above. With respect to the two-element cover UX = {U1 , U2 }, the section s corresponds to the pair {s1 , s2 }, and similarly for s  = {s1 , s2 }. Since supp(D) is contained in U1 but not in U2 , and the other way around for D  , it follows that s2 and s1 are actually invertible functions over their respective domains. Following [25], we can assume that s and s  are in fact the rational section 1, so that s2 = 1 and s1 = 1, and therefore:  log ρ1 + log|s1 |2 on U1 , log |s|2ρ = log ρ2 on U2 , and

 log

|s  |2ρ 

=

log ρ1 log ρ2 + log |s2 |2

on U1 , on U2 .

 | = log |s  | on U ∩ U . Let us denote by · = |·| = |·|  for Furthermore, log |g21 1 2 ρ ρ 2 simplicity. Using the relevant entries from (5.3) we have:

θ10 = c1 (ρ) logs   , ¯ log ρ2 1 log ρ2 = dd c log s 1 log ρ2 . θ20 = ∂∂ 2 2 An elementary integration by parts leads to:   0 f2 θ2 = f2 log s c1 (ρ  ) X X   1 1 + log s df2 ∧ d c log ρ2 − log ρ2 df2 ∧ d c log s . 2 2 X X On the other hand, we have    1 c  df2 ∧ θ12 = log s df2 ∧ d log |s2 | − log |s2 | df2 ∧ d c log s , X

X

X

and putting all terms together we obtain    0 1 1  f1 θ1 + f2 θ2 + df2 ∧ θ21 = f1 c1 (ρ) logs  + f2 log s c1 (ρ  ) X X X  + log s df2 ∧ d c log s   X  + log s   df1 ∧ d c log s (5.8) X

which, if expressed in terms of the squares of the norms, is (up to a factor) the logarithm of s , s  , as it is found in [11, formula 6.5.1]. This version is due to O. Gabber. Via the Poincar´e-Lelong lemma (see, e.g. [22])

58

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√ ¯ log s = c1 (ρ) + 2π −1 [D] , ∂∂ where [D] is the delta-current supported at the divisor of s, and similarly for s  , formula (5.8) can be recast into:   1 f1 θ10 + f2 θ21 + df2 ∧ θ21 = dd c log s log s   X X √ √ −2π −1 log s[D  ] − 2π −1 log s  [D] √ (5.9) = −2π −1 log s , s   which is what we wanted to show.

 

This allows us to recast the Liouville functional for conformal metrics on X in the following form. Corollary 5.2. The exponential of the Liouville functional defines an hermitian metric on the complex line TX , TX , namely for a conformal metric ρ ∈ CM (X) we have: 1 exp S[log ρ] = TX , TX  exp 2π AX (ρ) ,

where AX (ρ) is the area of X with respect to ρ. Remark 5.2. The √above corollary justifies the choice made in Definition 4.1 for the various factors 2π −1. A similar result has been obtained in ref. [32] by considering the Liouville action functional defined on the Schottky space, and in fact the results in loc. cit. are formulated in terms of a Schottky family. Indeed, the statement in Corollary 5.2 can be immediately reformulated for a family π : X → S with base parameter space S by considering the relative holomorphic tangent line bundle TX/S with an hermitian fiber metric ρ. (Thus ρs ∈ CM (Xs ) for every fiber Xs , s ∈ S.) Notice that the fiber metric ρ needs not be critical (i.e. satisfying the fiberwise constant negative curvature condition). 6. Conclusions and Outlook In this paper we have provided a geometric description of the construction of the Liouville action functional for constant negative curvature metrics on compact Riemann surfaces of genus g ≥ 2. Our approach was to construct a Hermitian holomorphic Deligne cohomology class as the cup square of the metrized holomorphic tangent bundle TX , and then show at the level of cocycles that (modulo an area term) this construction agrees with those in [30] and earlier works. Furthermore, our construction leads to the identification of the class corresponding to the Liouville action with the determinant of cohomology construction. Hence it could serve as an alternative construction of the latter in terms of different choices of cocycles. One of the most important properties of the Liouville functional from the works [33, 34, 32] is the link with the Weil-Petersson geometry of the Teichm¨uller space. From this point of view, an analysis of the behavior of the Liouville action for families of Riemann surfaces is crucial. In particular, a delicate analytic computation of the variation

Hermitian-Holomorphic Classes

59

of the action with respect to the moduli was carried out in the afore mentioned works to establish the link with the Weil-Petersson metric. We have not pursued these matters in the present work, limiting ourselves to establish the existence of a class for a metrized bundle on the base of a family π : X → S.A more precise analysis would require, not only a more explicit description of the map π (also advocated in [9]), but a full computation of the Leray sequence associated to the family π . This is instrumental in defining relative Hermitian holomorphic classes, and in analyzing their variation with respect to base parameters. We hope to pursue this and other problems related to the extension of the present work to singular metrics in a different publication. A. Cones In the main body of the paper we have used iterated cones to define the hermitian holomorphic Deligne complexes. One technical problem one has to face concerns the homotopy (graded) commutativity of the modified Be˘ılinson product defined in Eq. (2.5). A problem arises because the factors in the cones are cones themselves and therefore they have multiplication structures which are graded commutative up to homotopy to begin with. We want to show that even in this situation the final resulting product on cones is again homotopy graded commutative. This ensures that on cohomology the product will be genuinely graded commutative, so that in particular hermitian holomorphic Deligne cohomology as defined in Sect. 3 has the correct product structure. A.1. Cones and homotopies. We consider the following situation. For i = 1, 2, 3 we have maps of complexes: fi : A•i → Bi• , and for i < j maps aj i : A•i → A•j and bj i : Bi• → Bj• . Also, let C • (fi ) = Cone(fi : A•i → Bi• ) , for i = 1, 2, 3. First, consider the homotopy commutative diagram: A•j

fj

/ Bj• ~~~ ~~~~~ bij aij ~ ~ ~ s  { ~~~ ij  • / B• Ai i

(A.1)

fi

where sij : A•j → Bi•−1 is the homotopy map of complexes: fi aij − bij fj = d sij + sij d . An immediate verification yields Lemma A.1. The diagram (A.1) can be extended to A•j

fj

/ Bj• ~ ~~~~ aij ~~~~~sij bij ~ ~  { ~~  / B• A•i i fi

where the map cij is given by strictly commutative.



aij 0 −sij bij



/ C • (fj )

[1]

aij

cij

 / C • (fi )

/ A•j

[1]

 / A• i

and the squares containing the cones are in fact

60

E. Aldrovandi

Remark A.1. This lemma is nothing other than the statement that any homotopy commutative diagram of the form (A.1) in the category of complexes in an abelian category can be extended to a (homotopy) commutative diagram of distinguished triangles, that is, one of the axioms defining a triangulated category, see, e.g. [31]. For k < j < i consider the homotopy commutative triangle

A•k

? aj k ~~ ~ ~~ ~~

A•j

@@ @@aij αij k @@ @  / A•

aik

i

where aik − aij aj k = dαij k + αij k d , and similarly for the complexes Bi• , with a corresponding homotopy βij k . Thus αij k : A•k → A•−1 and βij k : Bk• → Bi•−1 . Now consider i the diagram: a

jk / A• ;C  ??  mmmmm j ??  mmm  ??mmm a ij  rz ??  a  ik fj A•i fk [c ??? ;C  ?   ????  ????    ??    bj k fi • / B• Bk mm j ?? m m  ?? mm  ??mmmmm  z r ? bj k ?   bij Bi•

A•k

(A.2)

The faces in (A.2) are homotopy commutative, however we assume that composing the faces is strictly commutative, namely the two possible homotopies bij bj k fk ⇒ fi aik must be equal. Concretely, this corresponds to the relation sik + βij k fk = fi αij k + sij aj k + bij sj k .

(A.3)

We have: Lemma A.2. The map  −αij k 0

0 βij k



: C • (fk ) −→ C •−1 (fi )

realizes the homotopy C • (fj ) : II t II cij cj k tt II t t II t t I$ t t  • / C • (fi ) C (fk ) c ik

Proof. It is an elementary calculation based on writing cik − cik cj k explicitly via the matrix representation given in Lemma A.1 and using Eq. (A.3).  

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A.2. Applications. Consider the same setup as in Sect. 2.1, with the same complexes Xi• , etc., and diagrams: fi

gi

Di = Xi• −→ Zi• ←− Yi• def

from which we construct the cones fi −gi

C(Di ) = Cone(Xi• ⊕ Yi• −−−→ Zi• )[−1] ,

i = 1, 2, 3 .

Moreover, following ref. [5], define Di ⊗ Dj by taking the tensor product componentwise. Thus f1 ⊗f2

g1 ⊗g2

D1 ⊗ D2 = X1• ⊗ X2• −−−→ Z1• ⊗ Z2• ←−−−− Y1• ⊗ Y2• . Assuming as in Sect. 2.1 that the product maps are compatible with the fi , etc., the diagram D1 ⊗ D 2 → D3 is of the same type as (A.1), and therefore Lemma A.1 implies Lemma 2.1. Now, let the multiplication maps X1• ⊗ X2• → X3• be graded commutative up to homotopy and similarly for the Yi• and the Zi• . We are interested in the commutativity properties of multiplication map given by the Be˘ılinson product (2.5). Proposition A.1. The multiplication map ∪α : C(D1 )⊗C(D2 ) −→ C(D3 ) given by (2.5) is homotopy graded commutative. Proof. The permutation operation on tensor products induces the diagram D1 ⊗ DH2 HH HH HH HH $

D3

/ D2 ⊗ D1 vv v vv v v zvv

which is of type (A.2) and we can apply Lemma A.2.

 

It follows from the proposition that the cohomology inherits a well defined graded commutative product. This in particular applies to the definition of hermitian holomorphic Deligne cohomology that uses the cone (3.5). Therefore we conclude that the cup product (3.3) is graded commutative, as wanted. B. Conventions on Kleinian Groups and Fractional Linear Transformations As a reference the reader can consult, among many others, the book [26]. Let
be a finitely generated purely loxodromic non-elementary Kleinian group of the second kind, so there is a nonempty discontinuity region U
⊂ P1 . The limit set is L
= P1 \ U
. According to Ahlfors’ finiteness theorem the quotient X = U
/
is a finite union of analytically finite Riemann surfaces. Thus: U
/
= U1 /
1  · · ·  Un /
n , where U1 , . . . , Un are the inequivalent components of U
and
1 , . . . ,
n their stabilizers. By way of example, a Schottky group has just one component, whereas a Fuchsian or quasi-Fuchsian group will have exactly two components.

62

E. Aldrovandi

ˇ We consider the map U
→ X as an e´ tale cover, and Cech cohomology with respect to it translates into group cohomology for the group
, where the coefficient modules are sections over U
of the relevant sheaves. The group action is by pull-back. According to the conventions of ref. [3] we will write the coboundary operation as: ˇ γ1 ,...,γn = cγ2 ,...,γn + (δc)

n−1 

(−1)i cγ1 ,...,γi γi+1 ,...,γn + (−1)n (cγ1 ,...,γn−1 ) · γn ,

(B.1)

i=1

where c is an n-cochain with values in some right
-module A. The expression (B.1) is ˇ the Cech coboundary applied to the nerve of the cover U
→ X. We assume that
is normalized, namely the point ∞ belongs to the limit set L
. If γ ∈
corresponds to the fractional linear transformation: P1

z −→ γ (z) =

az + b , cz + d

we have γ  (z) =

det γ , − z γ )2

(B.2)

c2 (z

where zγ = − dc ≡ γ −1 (∞). Set def

c(γ ) ≡ cγ =

det γ . c2

The following properties are easily verified. If z0 and z∞ are the attracting and repelling fixed points, respectively, then cγ =

(z0 − z∞ )2 λγ , (1 − λγ )2

where λγ is the dilating factor. For γi and γj two elements of
, denote: zi = γ −1 (∞) ,

zij = (γi γj )−1 (∞) = γj−1 (zi ) ,

ci = cγi ,

cij = cγi γj .

We have the following relation: cγ1 γ2 =

cγ1 (z12 − z2 )2 . cγ2

(B.3)

Finally, given four points z1 , z2 , z3 , z4 ∈ P1 , we define their cross-ratio by: [z1 : z2 : z3 : z4 ] =

(z1 − z2 )(z3 − z4 ) . (z1 − z4 )(z3 − z2 )

(B.4)

Acknowledgements. Parts of this work were completed during visits at the International School for Advanced Studies (SISSA) in Trieste, Italy, and at the Department of Mathematics, Instituto Superior T´ecnico in Lisbon, Portugal. I would like to thank both institutions for support and for creating a friendly and stimulating research environment. I would also like to thank Paolo Aluffi, Phil Bowers, Ugo Bruzzo, Johan Dupont, and Leon Takhtajan for illuminating discussions and/or patiently answering my many questions. Also, special thanks are due to the referee for his or her thoroughness and for providing very detailed comments.

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References 1. Aldrovandi, E.: Homological algebra of multivalued action functionals. Lett. Math. Phys. 60(1), 47–58 (2002) 2. Aldrovandi, E., Takhtajan, L.A.: CFT and effective action for two-dimensional quantum gravity on higher genus Riemann surfaces. Commun. Math. Phys. 188(1), 29–67 (1997) 3. Aldrovandi, E., Leon, Takhtajan, A.: Generating functional in CFT on Riemann surfaces. II. Homological aspects. Commun. Math. Phys. 227(2), 303–348 (2002) 4. Be˘ılinson, A.A.: Higher regulators and values of L-functions. In: Current problems in mathematics, Vol. 24, Itogi Nauki i Tekhniki, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984, pp. 181–238 5. Be˘ılinson, A.A.: Notes on absolute Hodge cohomology. In: Applications of algebraic K-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), Providence, RI: Am. Math. Soc., 1986, pp. 35–68 6. Bloch, S.: The dilogarithm and extensions of Lie algebras. In: Algebraic K-theory, Evanston 1980 Proc. Conf., Northwestern Univ., Evanston, Ill., 1980, Berlin: Springer, 1981, pp. 1–23 7. Bloch, S.J.: Higher regulators, algebraic K-theory, and zeta functions of elliptic curves. In: CRM Monograph Series, vol. 11, Providence, RI: Am. Math. Soc., 2000 8. Bott, R., Tu, L.W.: Differential forms in algebraic topology. Springer-Verlag, New York, 1982 9. Brylinski, J.-L., McLaughlin, D.A.: The geometry of degree-4 characteristic classes and of line bundles on loop spaces. II. Duke Math. J. 83(1), 105–139 (1996) 10. Brylinski, J.-L.: Geometric construction of Quillen line bundles. Advances in geometry, Boston, MA: Birkh¨auser Boston, 1999, pp. 107–146 11. Deligne, P.: Le d´eterminant de la cohomologie. In: Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), Contemp. Math., Vol. 67, Providence, RI: Am. Math. Soc., 1987, pp. 93–177 ´ 12. Deligne, P.: Le symbole mod´er´e. Inst. Hautes Etudes Sci. Publ. Math. 73, 147–181 (1991) ´ 13. Deligne, P.: Th´eorie de Hodge. II. Inst. Hautes Etudes Sci. Publ. Math. 40, 5–57 (1971) 14. Dupont, J.L.: The dilogarithm as a characteristic class for flat bundles. In: Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985), Vol. 44, 1987, pp. 137–164 15. Dupont, J.L.: Scissors congruences, group homology and characteristic classes. Nankai Tracts in Mathematics, Vol. 1, River Edge, NJ: World Scientific Publishing Co. Inc., 2001 16. Dupont, J.L., Sah, C.H.: Scissors congruences. II. J. Pure Appl. Algebra 25(2), 159–195 (1982) 17. Esnault, H.: Characteristic classes of flat bundles. Topology 27(3), 323–352 (1988) 18. Esnault, H., Viehweg, E.: Deligne-Be˘ı linson cohomology. In: Be˘ı linson’s conjectures on special values of L-functions, Boston, MA: Academic Press, 1988, pp. 43–91 19. Friedan, D., Shenker, S.: The analytic geometry of two-dimensional conformal field theory. Nuclear Phys. B 281(3–4), 509–545 (1987) 20. Gillet, H., Soul´e, C.: Arithmetic Chow groups and differential characters. In: Algebraic K-theory: connections with geometry and topology (Lake Louise, AB, 1987), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 279, Dordrecht: Kluwer Acad. Publ., 1989, pp. 29–68 21. Goncharov, A.B.: Explicit regulator maps on polylogarithmic motivic complexes. In: Motives, polylogarithms and Hodge theory, Part I, F. Bogomolov, L. Katzarkov (eds.), Cambridge, MA: International press, 2002 22. Griffiths, P., Harris, J.: Principles of algebraic geometry. Wiley-Interscience. New York: John Wiley & Sons, 1978 23. Hain, R.M.: Classical polylogarithms. Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., Vol. 55, Providence, RI: Am. Math. Soc., 1994, pp. 3–42 24. Krasnov, K.: Holography and Riemann surfaces. Adv. Theor. Math. Phys. 4(4), 929–979 (2000) 25. Lang, S.: Introduction to Arakelov theory. Springer-Verlag, New York, 1988 26. Matsuzaki, K., Taniguchi, M.: Hyperbolic manifolds and Kleinian groups. Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1998, Oxford Science Publications 27. Poincar´e, H.: Les fonctions fuchsiennes et l’´equation u = eu . J. Math. Pures Appl. 4(5), 137–230 (1898) 28. Polyakov, A.M.: Quantum geometry of bosonic strings. Phys. Lett. B 103, 207–210 (1981) 29. Ramakrishnan, D.: A regulator for curves via the Heisenberg group. Bull. Am. Math. Soc. (N.S.) 5(2), 191–195 (1981) 30. Takhtajan, L.A., Teo, L.-P.: Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography. Commun. Math. Phys. 239, 183–240 (2003) 31. Weibel, C.A.: An introduction to homological algebra. Cambridge: Cambridge University Press, 1994 32. Zograf, P.G.: The Liouville action on moduli spaces, and uniformization of degenerating Riemann surfaces. Leningrad Math. J. 1(4), 941–965 (1990)

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33. Zograf, P.G., Takhtadzhyan, L.A.: On the Liouville equation, accessory parameters and the geometry of Teichm¨uller space for Riemann surfaces of genus 0. Mat. Sb. (N.S.) 132(174)(2), 147–166 (1987) 34. Zograf, P.G., Takhtadzhyan, L.A.: On the uniformization of Riemann surfaces and on the Weil-Petersson metric on the Teichm¨uller and Schottky spaces. Mat. Sb. (N.S.) 132 (174)(3), 304–321, 444 (1987) Communicated by L. Takhtajan

Commun. Math. Phys. 251, 65–78 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1167-7

Communications in

Mathematical Physics

Geometric Model for Complex Non-K¨ahler Manifolds with SU (3) Structure Edward Goldstein1 , Sergey Prokushkin2 1

Department of Mathematics, Stanford University, Stanford, CA 94305, USA. E-mail: [email protected] 2 Department of Physics, Stanford University, Stanford, CA 94305, USA. E-mail: [email protected] Received: 9 January 2003 / Accepted: 29 April 2004 Published online: 17 September 2004 – © Springer-Verlag 2004

Abstract: For a given complex n-fold M we present an explicit construction of all complex (n + 1)-folds which are principal holomorphic T 2 -fibrations over M. For physical applications we consider the case of M being a Calabi-Yau 2-fold. We show that for such M, there is a subclass of the 3-folds that we construct, which has natural families of non-K¨ahler SU (3)-structures satisfying the conditions for N = 1 supersymmetry in the heterotic string theory compactified on the 3-folds. We present examples in the aforementioned subclass with M being a K3-surface and a 4-torus. 1. Introduction Recently, 6-dimensional non-K¨ahler manifolds with SU (3) structure have attracted considerable attention in the physics literature as possible internal spaces for supersymmetric string compactifications [1–7]. In particular, such manifolds appear in the T-dual description of the type IIB theory compactifications on a Calabi-Yau space in the presence of background RR and NS fluxes (see e.g. [2, 3] for detailed discussion and further references), and in the heterotic string theory compactifications with non-zero background torsion H (see e.g. [10, 1, 4]). Non-zero background fluxes induce a low-energy potential that stabilizes many of the geometric moduli, and therefore play an important role in superstring compactifications (see e.g. [11] and references therein). In many different contexts, the “manifolds with intrinsic torsion” or “manifolds with SU (3) structure” have been studied for a long time (see e.g. [12–27]). Remarkably, all these manifolds are some T 2 bundles over Calabi-Yau varieties, see [1–4, 6]. In most of the known examples the metric has the following local form: gbase + (dx + α)2 + (dy + β)2 ,

(1)

where gbase is the metric on the base (usually Calabi-Yau), x and y are local coordinates such that dx + i dy is a holomorphic form on the T 2 -fibers, and α and β are (local) 1-forms on the base.

66

E. Goldstein, S. Prokushkin

One can ask the question of how to construct such T 2 -fibrations over various base Calabi-Yau n-folds, and what conditions should α and β satisfy for such a complex T 2 -bundle to exist over a Calabi-Yau manifold. In this paper we produce a general construction of families M P ,Q of holomorphic T 2 -fibrations over a base Hermitian n-fold (M, g). (A Hermitian n-fold (M, g) is a complex manifold M of complex dimension n with a Hermitian metric g on it). Our first result is the following theorem, whose proof is given in Sect. 2: Theorem 1. Let ωP and ωQ be closed 2-forms on a complex Hermitian n-fold (M, g) s.t. the following two conditions hold: 1) ωP + iωQ has no component in 0,2 T ∗ M. ω 2) ω2πP and 2πQ represent integral cohomology classes. Then there is a complex, Hermitian (n + 1)-fold (M P ,Q , g  ) with a free structure-preserving T 2 -action and a holomorphic fibration π : M P ,Q → M P ,Q /T 2  M such that the following holds: A) For any 1-forms α and β defined on some open subset of M and satisfying dα = ωP and dβ = ωQ there are local coordinates x and y on M P ,Q such that dx + i dy is a holomorphic form on the T 2 -fibers and the metric on M P ,Q has the form (1) with gbase being g. B) If M admits a non-vanishing holomorphic (n, 0)-form n,0 then M P ,Q admits a non-vanishing holomorphic (n + 1, 0)-form n+1,0 . C) If either ωP or ωQ represent a non-trivial cohomology class then M P ,Q admits NO K¨ahler metric. We believe that Theorem 1 generalizes all known examples of complex non-K¨ahler manifolds so far used as internal manifolds in supersymmetric compactifications both in heterotic and type II theories. Note that in the theorem above, M does not have to be a Calabi-Yau, it is just a complex manifold with a Hermitian metric. Thus our construction can also be viewed as a new construction of complex manifolds which includes as a special case Hopf manifolds (for those ωQ = 0 in our terminology) and the complex structure (found by E. Calabi in [28]) on the product of odd-dimensional spheres S 2m+1 × S 2k+1 . Also the manifolds M P ,Q that we construct are principal T 2 -bundles for the standard complex structure on T 2 and the Hermitian metric on M P ,Q is T 2 -invariant. Vice versa we have the following theorem, proved in Sect. 3: Theorem 2. Let (N, g  ) be a Hermitian (n + 1)-fold with a free, structure-preserving T 2 -action. Assume also that g  restricted to the T 2 -fibers is the standard metric on T 2 . Then N is isomorphic (biholomorphic and isometric) to a manifold M P ,Q for some Hermitian n-fold (M, g) and the closed 2-forms ωP and ωQ on M satisfying the conditions of Theorem 1. We remark that if N is a complex manifold with a free holomorphic T 2 -action when we can find a Hermitian metric on N satisfying the conditions of Theorem 2, thus N is biholomorphic to a manifold M P ,Q we construct. Thus many known examples of principal T 2 -fibrations (e.g. Iwasawa manifolds) fall under our construction. Note also that we can relax condition 1) of Theorem 1 and construct a subclass of non-complex half-flat manifolds considered in [3–5]. These manifolds will be relevant in the models of string compactification in which the superpotential stabilizes the almost complex structure to a value that is not integrable.

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In Sect. 4 we will specialize to the case with n = 2. We consider (M, gCY , 2,0 ) to be a Calabi-Yau 2-fold with a non-vanishing holomorphic (2, 0)-form 2,0 which has unit length with respect to gCY . If we pick the forms ωP and ωQ on M satisfying the conditions of Theorem 1 then we obtain a Hermitian 3-fold M P ,Q with a holomorphic (3, 0)-form 3,0 of unit length. To fix notation we make the following definition: Definition 1.0.1. An SU (3)-structure on an almost-complex 3-fold N is a Hermitian metric g on N and a (3, 0)-form 3,0 (not necessarily holomorphic) of unit length with respect to g. Suppose (N, g) is a complex (integrable) 3-fold with a Hermitian metric g and suppose there is a non-vanishing holomorphic (3, 0)-form 3,0 on N . Then  (N, g, || ) is called a special SU (3)-structure on N . The function φ = 18 log|| : N → R is called the dilaton. In [10], the N = 1 supersymmetry conditions for heterotic string compactifications have been studied (see also [4]). In these models, the internal six-dimensional manifold  ). The supersymmetry condition on N must have a special SU (3)-structure (N, g, || the internal manifold is: ∗ d ∗ ω˜ = i(∂ − ∂)log|3,0 |.

(2)

Here ω˜ is the Hermitian (1, 1)-form on N - it is often denoted by J in the physics literature. This condition has the following geometric significance: On every Hermitian n-fold N there is a unique connection ∇ B (the Bismut connection, [9, 8]) which preserves both the metric and the complex structure and whose torsion H is a skew-symmetric 3-form. Equation (2) implies that the holonomy of this connection is contained in SU (3) ([10]). Equation (2) was rewritten in [4] in terms of the intrinsic torsion of the SU (3)-structure. The intrinsic torsion of a (general) SU (3) structures falls into five different classes W1 ⊕ W2 ⊕ W3 ⊕ W4 ⊕ W5 (see [21, 4]). In this classification, our manifolds M P ,Q have W1 = 0, W2 = 0, which is another way to see that the almost complex structure is integrable. In our main construction with the metric of a form (1), we have also W5 = 0, so that the holomorphic (3, 0)-form 3,0 has constant norm. In the language of the heterotic string theory, that means that the dilaton field is constant and there is no warp factor (see [10]). For a 3-fold N with a special SU (3)-structure as in Definition 1.0.1, the requirement (2) of N = 1 supersymmetry in the heterotic theory is equivalent to the following “torsional constraints” [21, 4] 2W4 + W5 = 0

(3)

W4 and W5 are real and exact.

(4)

and M P ,Q

In our class this leaves the manifolds with only non-zero torsion in the class W3 , so-called special Hermitian manifolds. These are complex half-flat manifolds [21], and the Hitchin flow of those manifolds (see [29, 4]) produces manifolds with G2 -holonomy. The main result of Sect. 4 is the following theorem: Theorem 3. Let (M, gCY , 2,0 ) be a Calabi-Yau 2-fold with a non-vanishing holomorphic (2, 0)-form 2,0 . Let ωP and ωQ be closed 2-forms on M satisfying the conditions of Theorem 1. Then the 3-fold M P ,Q has a natural special SU (3)-structure with a constant dilaton and moreover the torsional constraints (3), (4) hold if and only if the (1, 1)-com(1,1) (1,1) (1,1) (1,1) ponents of ωP and ωQ are anti-selfdual: ωP = −ωP , ωQ = −ωQ .

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In Sect. 5 we continue to consider the manifolds in the situation of Theorem 3. On such manifolds, we consider a modified construction with the Hermitian metric of a form gψ = e2ψ gCY + (dx + α)2 + (dy + β)2 ,

(5)

where ψ in the warp factor in front of the Calabi-Yau metric gCY , is an arbitrary function of the base coordinates. This function accounts for a non-constant dilaton background which is typical for the heterotic string compactifications. Now the holomorphic (3, 0)form 3,0 on M P ,Q has non-constant norm. We prove the following theorem: Theorem 4. Let (M, gCY , 2,0 ) be a Calabi-Yau 2-fold with a non-vanishing holomorphic (2, 0)-form 2,0 . Let ωP and ωQ be closed 2-forms on M satisfying the conditions of Theorem 1. Assume also that the (1, 1)-components of ωP and ωQ are anti-selfdual. Let ψ : M → R be any function. Then there is a special SU (3)-structure on M P ,Q with the Hermitian metric gψ of the form (5), where α, β, dx and dy are as Theorem 1 and the torsional constraints (3),(4) hold. For M compact, our construction satisfies the topological constraints pointed out in [10]: h3,0 = 1, c1 = 0. Also, our construction has zero Euler characteristic. In Sect. 6 we will present examples where Theorem 4 holds for M being a K3-surface and a 4-torus. One comment is now in order. Condition 2) in Theorem 1, which arose here as one of the sufficient conditions for the geometric construction to exist, can be motivated by physical arguments. Namely, it is equivalent to the Dirac quantization condition of the NS flux in string theory. Indeed, a model with the metric (1) can be considered as T-dual (after two T-dualities along T 2 fibers) to a type IIB model with the internal 6-manifold N˜  CY × T 2 (see, e.g. the discussion in [2]) and the background NS B field B = −dx ∧ α − dy ∧ β

(6)

(compare to formulas (4.23) and (4,24) of [2]), or background NS flux H3 H3 = dB = dx ∧ ωP + dy ∧ ωQ ,

(7)

with dα = ωP and dβ = ωQ defined on the base CY . The flux H3 satisfies the Dirac quantization condition (see Eq. (2.1) in [2]):
1 H3 = nγ ∈ Z , γ ∈ H3 (N˜ , Z) , (8) (2π)2 γ from which it follows that
ωP = nγ ∈ Z γCY 2π


γCY

ωQ = mγ ∈ Z , 2π

γCY ∈ H2 (CY, Z) ,

(9)

i.e., condition 2). In Sect. 7 we study the Betti numbers and the first Hodge numbers of the manifolds M P ,Q that we construct. Finally in Sect. 8 we will also show that one can lift Special Lagrangian submanifolds and fibrations from the Calabi-Yaus to the bundles M P ,Q in the case ωQ = 0, and Special Lagrangian submanifolds on those bundles will still be calibrated by Re 3,0 (and hence minimal).

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2. Global Model – The Construction In this section we prove the following theorem: Theorem 1. Let ωP and ωQ be closed 2-forms on a complex Hermitian n-fold (M, g) s.t. the following two conditions hold: 1) ωP + iωQ has no component in 0,2 T ∗ M. ω 2) ω2πP and 2πQ represent integral cohomology classes. Then there is a complex, Hermitian (n + 1)-fold (M P ,Q , g  ) with a free structure-preserving T 2 -action and a holomorphic fibration π : M P ,Q → M P ,Q /T 2  M such that the following holds: A) For any 1-forms α and β defined on some open subset of M and satisfying dα = ωP and dβ = ωQ there are local coordinates x and y on M P ,Q such that dx + i dy is a holomorphic form on the T 2 -fibers and the metric on M P ,Q has the form (1) with gbase being g. B) If M admits a non-vanishing holomorphic (n, 0)-form n,0 , then M P ,Q admits a non-vanishing holomorphic (n + 1, 0)-form n+1,0 . C) If either ωP or ωQ represent a non-trivial cohomology class then M P ,Q admits NO K¨ahler metric. Proof. First pick two line bundles P and Q s.t. the first Chern class c1 (P ) = [− ω2πP ] and ω c1 (Q) = [− 2πQ ]. This is certainly possible since there is a 1-1 correspondence between smooth line bundles and H 2 (M, Z). Put some Hermitian metrics on P and Q. We have the following elementary Lemma 2.0.1. One can choose a Riemannian connection ∇ on P whose curvature form is ωP . The analogous statement is true for Q. Proof. Let ∇  be some Riemannian connection on P and let ω be its curvature. Then ωP − ω is trivial cohomologically, hence ωP − ω = dα for some 1-form α. Now the connection ∇ = ∇  + iα is a Riemannian connection on P whose curvature form is ωP .

 We choose Riemannian connections on P and Q as in the previous lemma and proceed as follows: Consider the total space of the direct sum P ⊕ Q over M. The connections ∇ and ∇  give rise to a connection on P ⊕ Q and a horizontal distribution H that is a subbundle of T (P ⊕ Q). The horizontal distribution is obtained as follows: given a curve γ (t) in M and a pair (ξ, η) in the fiber of P ⊕ Q over γ (0), we have a unique curve γH (t) = (ξ(t), η(t)) in P ⊕ Q over γ (t) such that ξ(t) and η(t) are parallel along γ . The tangent vector γH (t) lives in the horizontal distribution H and it is called the Horizontal Lift of the tangent vector γ  (t). For each point p ∈ M let S1 (p) be the unit circle bundle of P over p and let S2 (p) be the unit circle bundle of Q over p. Let T (p) = S1 (p) × S2 (p) and let  M P ,Q = T (p). Thus M P ,Q is a 2-torus bundle over M. The distribution H along M P ,Q is tangent to M P ,Q : this is because the parallel transport preserves length. To understand H along M P ,Q better let ξ be a local unit length section of P on M and η be a local unit length section of Q on M . The sections ξ, η define local coordinates

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x, y on M P ,Q , namely any point z ∈ M P ,Q can be written as (eix ξ, eiy η). Also ξ defines a connection 1-form α  on M by ∇ξ = α  ⊗ ξ. This means that for any tangent vector v to M we have ∇v ξ = α  (v)ξ . Now α  is imaginary valued and ωP = −i dα  . (10) Similarly η defines a connection 1-form β  on M by ∇η = β  ⊗ η. The forms α  and β  are purely imaginary and the horizontal space H is precisely the kernel of the two 1-forms i dx + π ∗ α  and i dy + π ∗ β  .

(11)

Indeed let (ξH (t), ηH (t)) be a curve in M P ,Q sitting over a curve γ (t) in M such that ξH (t) and ηH (t) are parallel. We can write (ξH (t), ηH (t)) = (eix(t) ξ(γ (t)), eiy(t) η(γ (t))). The condition that ξH is parallel is equivalent to 0 = ix  (t)eix ξ + eix ∇γ  ξ = eix (i dx(γH ) + π ∗ α  (γH ))ξ which is equivalent to saying that γH is in the kernel of i dx + π ∗ α  . Similarly the fact that ηH is parallel is equivalent to saying that γH is in the kernel of i dy + π ∗ β  . Let V be the vertical space of M P ,Q - the tangent space to the fibers. On every fiber T (p) we have a natural S 1 × S 1 = T 2 -action given by (eix , eiy ) · (ξ, η) = (eix ξ, eiy η). We have vector fields ∂x and ∂y tangent to the fibers. We define the complex structure on T (p) to be the natural one: ∂x → ∂y and ∂y → −∂x . The almost complex structure on H is induced from the projection onto M. Thus M P ,Q acquires an almost complex structure I . Define now a (1, 0)-form ρ on M P ,Q by requiring that ρ = 0 on H and ρ = dx + i dy on V .

(12)

From Eq. (11) we conclude that ρ = (dx − iα  ) + i(dy − iβ  ).

(13)

Also pick a local holomorphic (n, 0)-form n,0 on M and define an (n + 1, 0)-form n+1,0 on M P ,Q by n+1,0 = ρ ∧ π ∗ (n,0 ). (14) We compute from (10) that dn+1,0 = π ∗ ((ωP + iωQ ) ∧ n,0 ) = 0.

(15)

Note that if (ωP + iωQ ) had a non-zero component of type (0, 2) then dn+1,0 would have had a non-zero (n, 2)-component and the almost complex structure would not be

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integrable. But in our case dn+1,0 = 0 and so the almost complex structure is integrable. This implication is standard: by Newlander-Nirenberg theorem it is enough to prove that for a (1, 0)-form θ we have that dθ is of type 2,0 ⊕ 1,1 . Now we have 0 = θ ∧ n+1,0 . Taking the exterior derivative we get 0 = dθ ∧ n+1,0 , i.e. dθ is of type 2,0 ⊕ 1,1 . Also if n,0 is a holomorphic, non-vanishing (n, 0)-form defined on the whole of M then n+1,0 is a holomorphic, non-vanishing (n + 1, 0)-form on M P ,Q and this proves B). Both H and V have a natural Hermitian metric and thus M P ,Q is naturally a Hermitian manifold. Let α and β be any 1-forms on M s.t. dα = ωP and dβ = ωQ . Then we can find local unit length sections ξ of P and η of Q s.t. iα and iβ are the connection 1-forms defined by ξ and η, see (10) . Now ξ and η define local coordinates x and y on M P ,Q as before and the metric on M P ,Q has the form as in Eq. (1) and we proved A). We are still left to prove that if either ωP or ωQ is non-trivial in cohomology then M P ,Q admits no K¨ahler metric. One can also easily show that the fibers of π are 0 in the real 2-dimensional homology of M P ,Q , see more details about homology in Sect. 7 of this paper. The triviality of the (complex) fibers in homology implies that there is no K¨ahler form on M P ,Q : for such a form would integrate to a positive number on the fibers.

 3. Holomorphic Principal T2 Fiber Bundles The goal of this section is to state and prove Theorem 2 from the Introduction: Theorem 2. Let (N, g  ) be a Hermitian (n + 1)-fold with a free, structure-preserving T 2 -action. Assume also that g  restricted to the T 2 -fibers is the standard metric on T 2 . Then N is isomorphic (biholomorphic and isometric) to a manifold M P ,Q for some Hermitian n-fold (M, g) and the closed 2-forms ωP and ωQ on M satisfying the conditions of Theorem 1. Proof. We have the quotient M = N/T 2 and the projection π : N → M. Let H be the orthogonal complement to the tangent space to the T 2 -fibers in T N . Thus H is a horizontal distribution (connection) for the principal T 2 -bundle N . Also the metric g  on H projects down to a metric g on M. To construct the line bundles P and Q, let f1 and f2 be the standard 1-dimensional representations of T 2 , i.e. f1 (eix , eiy )(z) = eix z

and

f2 (eix , eiy )(z) = eiy z.

Let P be the line bundle over M associated to f1 : P = N ×T 2 C. This means that P is the quotient of N × C by the T 2 -action, where T 2 acts on C via f1 . The construction of Q is similar. Also H induces a connection on both P and P . Let ωP and ωQ be the curvature forms of P and Q correspondingly. We construct M P ,Q as in the proof of Theorem 1. Note that so far we haven’t proved that ωP + iωQ has no (0, 2)-component, thus M P ,Q in so far just an almost-complex manifold with a Hermitian metric.

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We now construct an isomorphism φ : N → M P ,Q . It is constructed as follows: take a point z in N and consider (z, 1, 1) ∈ N × C × C. Taking the orbit of (z, 1, 1) by the T 2 -action we get a point φ(z) ∈ M P ,Q . One easily checks that φ is indeed an isomorphism. Hence in particular the almost-complex structure on M P ,Q is integrable and by the remark after Eq. (15) we conclude that ωP + iωQ has no (0, 2)-component.

 4. Intrinsic Torsion of SU(3)-Structures In this section we specialize to the case dimC M = 2. From now on we consider (M, gCY , 2,0 ) to be a Calabi-Yau 2-fold with a non-vanishing holomorphic (2, 0)form 2,0 which has unit length with respect to gCY . If we pick the forms ωP and ωQ on M satisfying the conditions of Theorem 1 then we obtain a Hermitian 3-fold M P ,Q with a holomorphic (3, 0)-form 3,0 of unit length. We’ll study the intrinsic torsion of the SU (3)-structure we wrote down on M P ,Q . For a general SU (3) structure, Chiossi and Salamon [22] have decomposed this ten3,0 and let 3,0 be the sor into 5 components W1 , . . . , W5 . Let 3,0 + be the real part of  − imaginary part of 3,0 . According to [22], there is the following 1-1 correspondence: ˜ 3,0 , W1 ←→ (d ω) 1,1 3,0 1,1 W2 ←→ ((d3,0 + )0 , (d− )0 ),

W3 ←→ (d ω) ˜ 2,1 0 , W4 ←→ ω˜ ∧ d ω, ˜ 3,1 W5 ←→ (d3,0 ± ) .

(16)

Here ω˜ is the Hermitian (1, 1)-form. The component W1 vanishes because the complex structure on M P ,Q is integrable and so the exterior derivative of a (1, 1)-form has no (3, 0) components. The components W2 and W5 vanish since d3,0 = 0. We finally study the component W4 . We need to write down an explicit expression for the hermitian (1, 1)-form ω˜ on M P ,Q . We know that ω˜ equals to π ∗ (ωCY ) on the horizontal distribution H and it equals to dx ∧ dy on the vertical distribution V . In the notation of Theorem 1 we get that ω˜ = π ∗ ωCY + (dx + π ∗ α) ∧ (dy + π ∗ β).

(17)

Indeed the 2-form (dx + π ∗ α) ∧ (dy + π ∗ β) has H

as its kernel and it equals to dx ∧ dy on V , hence Eq. (17) is true. Let ωP be the curvature form of P (so ωP = dα) and let ωQ = dβ be the curvature form of Q. We have that d ω˜ = π ∗ ωP ∧ (π ∗ β + dy) − (π ∗ α + dx) ∧ π ∗ ωQ , d ω˜ ∧ ω˜ = dy ∧ π ∗ (ωP ∧ ωCY ) − dx ∧ π ∗ (ωQ ∧ ωCY ).

(18) (19)

The later term vanishes if and only if the (1, 1)-components of ωP and ωQ are anti-selfdual. From this we conclude: Theorem 3. Let (M, gCY , 2,0 ) be a Calabi-Yau 2-fold with a non-vanishing holomorphic (2, 0)-form 2,0 . Let ωP and ωQ be closed 2-forms on M satisfying the conditions of Theorem 1. Then the 3-fold M P ,Q has a natural special SU (3)-structure with a constant dilaton, and the torsional constraints (3), (4) hold if and only if the (1, 1)-components (1,1) (1,1) (1,1) (1,1) of ωP and ωQ are anti-selfdual: ωP = −ωP , ωQ = −ωQ .

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5. Metric Scaling and Non-Constant Dilaton We continue to consider (M, gCY , 2,0 ) to be a Calabi-Yau 2-fold with a non-vanishing holomorphic (2, 0)-form 2,0 which has unit length with respect to gCY . Furthermore we assume that the (1, 1)-components of the forms ωP and ωQ are anti-selfdual. In this case W4 = 0 = W5 and so the supersymmetry equation 2W4 + W5 = 0 certainly holds. We will now define a class gψ of Hermitian metrics on M P ,Q depending on a function ψ : M → R s.t. the supersymmetry equation still holds for them. So let ψ be a function on M and lift it to M P ,Q . Define the metric gψ on M P ,Q as in (5). Thus the horizontal and the vertical distribution are still perpendicular, on V the metric gψ is the original metric g  and on H the metric is scaled by e2ψ . If we take 2ψ 3,0 then it has length 1 with respect to g . We will now show that the 3,0 ψ ψ = e  supersymmetry equation still holds for this SU (3)-structure: Theorem 4. Let (M, gCY , 2,0 ) be a Calabi-Yau 2-fold with a non-vanishing holomorphic (2, 0)-form 2,0 . Let ωP and ωQ be closed 2-forms on M satisfying the conditions of Theorem 1. Assume also that the (1, 1)-components of ωP and ωQ are anti-selfdual. Let ψ : M → R be any function. Then there is a special SU (3)-structure on M P ,Q with the Hermitian metric gψ of the form (5), where α, β, dx and dy are as Theorem 1 and the torsional constraints (3),(4) hold. Proof. First we need to explain what W4 and W5 are. We begin by defining on any Riemannian manifold N a contraction pairing
: k T ∗ N ⊗ n T ∗ N → n−k T ∗ N (see [4], p. 5). This is defined as follows: for an orthonormal basis e1 , . . . , el of T N let ei be the dual basis. Let α = (α1 , . . . , αk ) be a multi-index of distinct integers between 1 and l = dim(N ) of length k and let eα = eαi . Let β be a multi-index of length n. We define eα
eβ as follows: If the set (α) is not contained in the set (β) then the answer is 0. If (α) ⊂ (β) then eα
eβ = (−1)k eβ−α . Here (−1)k is the sign of permutation that is needed to put α in the beginning of β. We have the following basic Proposition 5.0.1. Let dim(N ) = 4 and let ω be a Hermitian 2-form for the metric on T N (coming from some compatible almost complex structure). Then for any 1-form δ, ω
ω ∧ δ = δ. Proof. This is linear algebra. We write ω = dxi ∧ dyi at 1 point. It is enough to prove  the Proposition for δ = dxi or δ = dyj , and those are immediate.

We now take another proposition from linear algebra that we will need: Proposition 5.0.2. Let V be a Hermitian vector space of complex dimension 3 let  be a (3, 0)-form on V of length 1. Then for any 1-form δ on V we’ll have that Re 
δ ∧ Re  = −2δ.

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Proof. Obviously it is enough to prove it for a real 1-form δ. We certainly have a subspace W of V of complex dimension 1 on which δ vanishes. Let U = W ⊥ , then δ can be viewed as a form on U . Let ξ be a (2, 0)-form on U of length 1. We can choose orthonormal basis e1 and e2 = J e1 on W such that  = ξ ∧(e1 +ie2 ). Then Re  = Re ξ ∧e1 −I m ξ ∧e2 . Here Re ξ and I m ξ are hermitian 2-forms for the metric on U (for different complex structures on U ). In particular the previous proposition holds for them. We have that δ ∧Re  = −(e1 ∧Re ξ ∧δ −e2 ∧I m ξ ∧δ). From this we immediately derive the statement of our proposition.

 2ψ 3,0 and a We now return to M P ,Q with the metric gψ . We have a form 3,0 ψ = e  Hermitian form ωψ = (dx + α) ∧ (dy + β) + e2ψ π ∗ ωCY .

The classes W4 and W5 can be written in a form: W4 =

1 1 3,0 ωψ
dωψ , W5 = Re 3,0 ψ
dRe ψ 2 2

(see [4], p. 5). We compute that dωψ = e2ψ 2dψ ∧ π ∗ ωCY + π ∗ ωP ∧ (dy + β) − π ∗ ωQ ∧ (dx + α). Note that


: 2 T ∗ ⊗ 2 T ∗ → R

is just the dot product. Since the dot product of ωCY with ωP and ωQ is 0, we conclude that ωψ
(π ∗ ωP ∧ (dy + β) − π ∗ ωQ ∧ (dx + α)) = 0. Also

ωψ
e2ψ 2dψ ∧ π ∗ ωCY = 2dψ.

3,0 Also d Re 3,0 ψ = 2dψ ∧ Re ψ . Hence 3,0 Re 3,0 ψ
2dψ ∧ Re ψ = −4dψ.

From all this we conclude that 2W4 + W5 = 0.



6. Examples In this section, we will consider examples when Theorem 4 applies. 6.1. SU (3)-fibrations over K3 surfaces. Let M be a K3 surface with a Calabi-Yau metric gCY and a holomorphic (2, 0)-form 2,0 . Let C1 , . . . , Ck be some collection of holomorphic curves on M (e.g. if M is a Kummer K3 and Ci are the exceptional spheres). We consider the divisor  C= ak Ck .

Geometric Model for Complex Non-K¨ahler Manifolds with SU (3) Structure

Here ak are integers such that 

75


ωCY = 0.

ai Ci

We will only consider the case when such ak exist, e.g. a Kummer K3. C defines a line bundle P with a meromorphic section σ and the first Chern class c1 (P ) is the Poincar´e dual of C, hence it satisfies c1 (P ) ∧ [ωCY ] = 0 , c1 (P ) ∧ [Re 2,0 ] = 0 , c1 (P ) ∧ [I m 2,0 ] = 0.

(20)

We now use the fact that b2+ (M) = 3 and [ωCY ] , [Re 2,0 ] and [I m 2,0 ] is a basis P for H+2 (M). Let −ω 2π be the harmonic representative of c1 (P ). Equation (20) implies that ωP is anti-selfdual, hence in particular it is of type (1, 1). We also choose ωQ by the same principle as ωP . Theorem 4 applies. 6.2. SU (3)-fibrations over T 4 . We choose a standard flat metric on the four-dimensional torus. We have the following basis of antiselfdual 2-forms on T 2 : ω1 = 2π(dx1 ∧ dx2 − dx3 ∧ dx4 ), ω2 = 2π(dx1 ∧ dx3 − dx4 ∧ dx2 ), ω3 = 2π(dx1 ∧ dx4 − dx2 ∧ dx3 ). We can choose ωP and ωQ to be linear combinations of ω1 , ω2 , ω3 with integer coefficients and Theorem 4 applies. 7. Cohomology and Hodge Numbers of M P ,Q In this section, we study the cohomology and the Hodge numbers h1,0 and h0,1 of M P ,Q . For Hodge numbers we assume that M is compact and that ωP and ωQ are of type (1, 1). 7.1. Hodge numbers h1,0 and h0,1 of M P ,Q . We note that if ξ is any harmonic form (both for the usual Laplacian or for the ∂-Laplacian) then the ∂x and the ∂y -flows are structure preserving and they preserve the cohomology class of ξ , hence they preserve ξ . This is clear for the usual cohomology. For the ∂-cohomology we note that the ∂x -flow acts upon the space H p,q . So we have a one dimensional representation of the circle on H p,q with weights iA1 , . . . , iAk . Here Ai are integers. Also the ∂y -flow acts on H p,q . For any harmonic form ξ in H p,q we have that L∂x ξ + iL∂y ξ is in H p,q . But the p,q component of L∂x ξ + iL∂y ξ is ∂(i∂x +i∂y ξ ). From this we deduce that L∂x ξ + iL∂y ξ = 0. This implies that the representation of ∂y on H p,q has weights A1 , . . . , Ak . But ∂y is periodic, hence Ai = 0, i.e. the ∂x and the ∂y -flow preserves H p,q . Let ξ be a harmonic (1, 0)-form on M P ,Q . The invariance of ξ under the ∂x and the ∂y -flow implies that one can write ξ = Aρ + π ∗ s1 . Here ρ is as in Eq. (13) and s1 is a (1, 0)-form on M and A is a function pulled up from M. Since ∂ρ = π ∗ (ωP + iωQ ), the equation ∂ξ = 0 translates into ∂A = 0 and

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A(ωP + iωQ ) + ∂φ = 0. So A is a constant and moreover ωP + iωQ is a non-zero harmonic (1, 1)-form for the ∂-complex. Hence A = 0 and ∂φ = 0, i.e. H 1,0 (M P ,Q )  H 1,0 (M). In the H 0,1 -case simple analysis shows that harmonic (0, 1) forms are all of the form cρ¯ + π ∗ s1 for s 1 ∈ H 0,1 M and c a constant. Thus h0,1 (M P ,Q ) = h0,1 (M) + 1. In particular h0,1 (M P ,Q ) = h1,0 (M P ,Q ) + 1. 7.2. Cohomology of M P ,Q . To study cohomology first let M P be the unit circle bundle of P . Certainly M P is a circle bundle over M. Also M P ,Q is a circle bundle over M P (with the fiber being the unit circle of Q pulled up to M P ). We’ll use the Gysin sequence (see [32], p. 255) to study the cohomology one step at a time. The Gysin sequence tells that if F is a circle bundle with projection π over the base B and the first Chern class of F is c1 ∈ H 2 (B, R) then we have the following exact sequence: ∪c1

π∗

H i B −→ H i+2 B −→ H i+2 F → H i+1 B → · · · . Let’s study H 1 first. Since ωP is non-trivial in real cohomology we conclude that H 1 M P  H 1 M. Now we have 2 cases: Case 1. If ωQ is not a multiple of ωP in H 2 M then the Gysin sequence tells that it lifts to a non-trivial element of H 2 M P . Using the Gysin sequence again for the fibration M P ,Q → M P we conclude that H 1 M P ,Q  H 1 M P  H 1 M. Case 2. If ωQ is a multiple of ωP in H 2 then Gysin sequence tells that it lifts to a trivial class in H 2 M P . Using the Gysin sequence again for the fibration M P ,Q → M P we conclude that b1 (M P ,Q ) = b1 (M P ) + 1 = b1 (M) + 1. Remark. The Gysin sequence implies that all the first cohomology of M P comes from M and so the circle fibers are trivial in the first homology of M P . From this we conclude that the torus fibers are trivial in the second homology of M P ,Q , we used this fact before to show that M P ,Q admits no K¨ahler metric. Let us consider H 2 . We assume that H 1 (M) = 0. Using the Gysin sequence we conclude that b2 (M P ) = b2 (M) − 1. Now we have 2 cases: Case 1. If ωQ is not a multiple of ωP in H 2 M then the Gysin sequence tells that it lifts to a non-trivial element of H 2 M P . Using the Gysin sequence again for the fibration M P ,Q → M P we conclude that b2 (M P ,Q ) = b2 (M P ) − 1 = b2 (M) − 2. Case 2. If ωQ is a multiple of ωP in H 2 then Gysin sequence tells that it lifts to a trivial class in H 2 M P . Using the Gysin sequence again for the fibration M P ,Q → M P we conclude that b2 (M P ,Q ) = b2 (M P ) = b2 (M) − 1. This information is enough to find all the Betti numbers in case M is a K3-surface. The only unknown so far is b3 (M P ,Q ) and we can find it using the fact that the Euler characteristic of M P ,Q is 0 (because ∂x is a non-vanishing vector field on M P ,Q ). 8. Pulling up Special Lagrangians Here we assume that M is a Calabi-Yau manifold and ωQ = 0. We take Q to be the trivial bundle with the trivial connection. Thus M P ,Q is a direct product of the unit circle bundle M P on P with the unit circle S 1 :

Geometric Model for Complex Non-K¨ahler Manifolds with SU (3) Structure

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M P ,C  M P × S 1 . For any element s ∈ S 1 let

M P ,s = M P × (s). The horizontal distribution H along M P ,s is tangent to M P ,s . Moreover there is a natural circle action on M P ,s given by eix (ξ, s) = (eix ξ, s). The vector field generating this action is ∂x . If L is a submanifold of M then we can define its lifts  M P ,s . Ls = π −1 (L) We have that the tangent space to Ls naturally splits as T LH ⊕ span(∂x ). Here T LH is the horizontal lift of T L to H . From this we get the following: Proposition 8.0.1. Let L be a Special Lagrangian submanifold on M. Then Ls is a Special Lagrangian submanifold of M P ,C . We wish to point out that if we have a Special Lagrangian fibration on M it lifts to a special Lagrangian fibration on M P ,C . Conclusion In this paper, we presented a geometric construction for complex non-K¨ahler manifolds with intrinsic SU (3) structure, used in supersymmetric string compactifications.1 We gave a general construction of families M P ,Q of holomorphic T 2 -fibrations over a Hermitian n-fold M. We have shown that our construction can satisfy the supersymmetry conditions in the heterotic string theory, and in this case we get complex half-flat, or special Hermitian manifolds. We presented examples of T 2 - bundles over K3 surfaces and a four-torus which satisfy the supersymmetry constraint. Also, we proposed a modified model with the metric depending on a warp factor, and showed that the supersymmetry conditions are still satisfied. In addition, we computed all Betti numbers and the Hodge numbers h1,0 and h0,1 . It is also shown that in the case ωQ = 0 one can lift Special Lagrangian submanifolds and fibrations from the Calabi-Yaus to the bundles M P ,Q , and Special Lagrangian submanifolds on those bundles are still calibrated by Re 3,0 (and hence minimal). Acknowledgements. We would like to acknowledge very useful conversations with K. Dasgupta, S. Gukov, S. Kachru, and L. McAllister. The work of S. P. was supported by Stanford Graduate Fellowship.

References 1. Becker, K., Dasgupta, K.: Heterotic Strings with Torsion. JHEP 0211, 006 (2002) 2. Kachru, S., Schulz, M.B., Tripathy, P.K., Trivedi, S.P.: New Supersymmetric String Compactifications. JHEP 0303, 061 (2003) 1 It should be mentioned that any realistic heterotic string compactification model has to include a vector bundle construction, with a gauge group rich enough to account for the Standard Model particles. This requirement can impose severe constraints on the geometric construction, e.g., one might need to consider singular limits of the base CY 2-fold. We do not study this subject here.

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3. Gurrieri, S., Louis, J., Micu, A., Waldram, D.: Mirror Symmetry in Generalized Calabi-Yau Compactifications. Nucl. Phys. B 654, 61–113 (2003) 4. Cardoso, G.L., Curio, G., Dall’Agata, G., Luest, D., Manousselis, P., Zoupanos, G.: Non-Kaehler String Backgrounds and their Five Torsion Classes. Nucl. Phys. B 652, 5–34 (2003) 5. Gurrieri, S., Micu, A.: Type IIB Theory on Half-flat Manifolds. Class. Quant. Grav. 20, 2181–2192 (2003) 6. Becker, K., Becker, M., Dasgupta, K., Green, P.: Compactifications of Heterotic Theory on NonK¨ahler complex manifolds I. hep-th/0301161 7. Tripathy, P.K., Trivedi, S.P.: Compactifications with Flux on K3 and Tori. JHEP 0303, 028 (2003) 8. Fino, A., Grantcharov, G.: On some properties of the manifolds with skew-symmetric torsion and holonomy SU(n) and Sp(n). http://arxiv.org/abs/math.DG/0302358, 2003 9. Gauduchon, P.: Hermitian connections and Dirac operators. Boll. Un. Mat. Ital. B (7) 11(2 suppl.), 257–288 (1997) 10. Strominger, A.: Superstrings with torsion. Nucl. Phys. B274 253 (1986) 11. Kachru, S., Schulz, M., Trivedi, S.: Moduli Stabilization from Fluxes in a Simple IIB Orientifold. JHEP 0310, 001 (2003) 12. Scherk, J., Schwarz, J.H.: How To Get Masses From Extra Dimensions. Nucl. Phys. B 153, 61 (1979) 13. Hull, C.M.: Superstring Compactifications With Torsion And Space-Time Supersymmetry. In: Turin 1985, Proceedings, Superunification and Extra Dimensions. Singapore: World Scientific Publishers, 1986, pp. 347–375 14. Falcitelli, M., Farinola, A., Salamon, S.: Almost-Hermitian Geometry. Diff. Geom. Appl. 4, 259 (1994) 15. Kaloper, N., Myers, R.C.: The O(dd) story of massive supergravity. JHEP 9905, 010 (1999) 16. Vafa, C.: Superstrings and topological strings at large N. J. Math. Phys. 42, 2798 (2001) 17. Joyce, D.: Compact Manifolds with Special Holonomy. Oxford: Oxford University Press, 2000 18. Friedrich, T., Ivanov, S.: Parallel spinors and connections with skew-symmetric torsion in string theory. Asian J. Math. 6, 303–336 (2002) 19. Salamon, S.: Complex structures on nilpotent Lie algebras. J. Pure Appl. Algebra 157, 311–333 (2001) 20. Dasgupta, K., Rajesh, G., Sethi, S.: M Theory, Orientifolds and G-Flux. JHEP 9908, 023 (1999) 21. Salamon, S.: Almost Parallel Structures. Global Differential Geometry: The Mathematical Legacy of Alfred Gray (Bilbao, 2000). Contemp. Math. 288 162–181 (2001) 22. Chiossi, S., Salamon, S.: The Intrinsic Torsion of SU (3) and G2 Structures. In: Differential geometry, Valencia, 2001, Singapore: World Scientific, 2002, pp. 115–123 23. Rocek, M.: Modified Calabi–Yau manifolds with torsion. In: S.T. Yau (ed.), Essays on Mirror Manifolds, Hong Kong: International Press, 1992; Gates, S.J., Hull, Rocek, M.: Twisted Multiplets And New Supersymmetric Nonlinear Sigma Models. Nucl. Phys. B248, 157 (1984); Lyakhovich, S., Zabzine, M.: Poisson geometry of sigma models with extended supersymmetry. Phys. Lett. B 548, 243 (2002) 24. Ivanov, S., Papadopoulos, G.: Vanishing theorems and string backgrounds. Class. Quant. Grav. 18, 1089 (2001); A no-go theorem for string warped compactifications. Phys. Lett. B497, 309 (2001); Papadopoulos, G.: KT and HKT geometries in strings and in black hole moduli spaces. http://arxiv.org/abs/hep-th/0201111, (2002); Gutowski, J., Ivanov, S., Papadopoulos, G.: Deformations of generalized calibrations and compact non-K¨ahler manifolds with vanishing first Chern class. Asian J. Math. 7, 39–80 (2003) 25. Gauntlett, J.P., Kim, N.W., Martelli, D., Waldram, D.: Fivebranes wrapped on SLAG three-cycles and related geometry. JHEP 0111, 018 (2001); Gauntlett, J.P., Martelli, D., Pakis, S., Waldram, D.: G-structures and wrapped NS5-branes. Commun. Math. Phys. 247, 421–445 (2004) 26. Kaste, P., Minasian, R., Petrini, M., Tomasiello, A.: Kaluza-Klein bundles and manifolds of exceptional holonomy. JHEP 0209, 033 (2002) 27. Hellerman, S., McGreevy, J., Williams, B.: Geometric Constructions of Nongeometric String Theories. JHEP 0401, 024 (2004) 28. Calabi, E.: A class of compact, complex manifolds which are not algebraic. Ann. Math. (2) 58, 494–500 (1953) 29. Hitchin, N.: Stable forms and special metrics. In: Global Differential Geometry: The mathematical legacy of Alfred Gray. Providence, RI: AMS, 2001, pp. 70–89 30. Fino, A., Parton, M., Salamon, S.: Families of strong KT structures in six dimensions. Commen. Math. Helv. 79, 317–340 (2004) 31. Griffiths, P., Harris, J.: Principles of Algebraic geometry. New York: Wiley and Sons, 1978 32. Husemoller, D.: Fiber bundles, 3rd edn. Springer Graduate Texts in Mathematics, Berlin-Heidelberg-New York: Springer Communicated by N. Nekrasov

Commun. Math. Phys. 251, 79–131 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1169-5

Communications in

Mathematical Physics

Partition Function Zeros at First-Order Phase Transitions: A General Analysis
M. Biskup1 , C. Borgs2 , J.T. Chayes2 , L.J. Kleinwaks3 , R. Koteck´y4 1 2 3 4

Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA Department of Physics, Princeton University, Princeton, NJ 08544, USA Center for Theoretical Study, Charles University, Prague 110 00, Czech Republic

Received: 3 April 2003 / Accepted: 31 March 2004 Published online: 17 September 2004 – © M. Biskup, C. Borgs, J.T. Chayes, L.J. Kleinwaks, R. Koteck´y 2003

Abstract: We present a general, rigorous theory of partition function zeros for lattice spin models depending on one complex parameter. First, we formulate a set of natural assumptions which are verified for a large class of spin models in a companion paper [5]. Under these assumptions, we derive equations whose solutions give the location of the zeros of the partition function with periodic boundary conditions, up to an error which we prove is (generically) exponentially small in the linear size of the system. For asymptotically large systems, the zeros concentrate on phase boundaries which are simple curves ending in multiple points. For models with an Ising-like plus-minus symmetry, we also establish a local version of the Lee-Yang Circle Theorem. This result allows us to control situations when in one region of the complex plane the zeros lie precisely on the unit circle, while in the complement of this region the zeros concentrate on less symmetric curves. Contents 1. Introduction . . . . . . . . . . . . . . . . . 1.1 Motivation . . . . . . . . . . . . . . . 1.2 Basic ideas . . . . . . . . . . . . . . 1.3 Discussion of assumptions and results 2. Main Results . . . . . . . . . . . . . . . . 2.1 Complex phase diagram . . . . . . . 2.2 Partition function zeros . . . . . . . . 2.3 Local Lee-Yang theorem . . . . . . . 2.4 Discussion . . . . . . . . . . . . . . . 3. Characterization of Phase Diagrams . . . . 3.1 Auxiliary claims . . . . . . . . . . . .


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3.2 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . 4. Partition Function Zeros . . . . . . . . . . . . . . . . . . . 4.1 Root degeneracy . . . . . . . . . . . . . . . . . . . . 4.2 Two-phase coexistence . . . . . . . . . . . . . . . . . 4.3 Proof of Proposition 2.4 . . . . . . . . . . . . . . . . . 4.4 Multiple phase coexistence . . . . . . . . . . . . . . . 4.5 Proof of Proposition 2.6 . . . . . . . . . . . . . . . . . 5. Technical Lemmas . . . . . . . . . . . . . . . . . . . . . . 5.1 Lipschitz properties of the functions log |ζm | and log ζ 5.2 Proofs of Lemmas 4.3–4.5 . . . . . . . . . . . . . . . 5.3 Proof of Lemmas 4.6 and 4.7 . . . . . . . . . . . . . .

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1. Introduction 1.1. Motivation. One of the cornerstones of equilibrium statistical mechanics is the notion that macroscopic systems undergo phase transitions as the external parameters change. A mathematical description of phase transitions was given by Gibbs [17] who characterized a phase transition as a point of non-analyticity in thermodynamic functions, e.g., the pressure. This definition was originally somewhat puzzling since actual physical systems are finite, and therefore their thermodynamic functions are manifestly real-analytic. A solution to this contradiction came in two seminal papers by Yang and Lee [25, 42], where it was argued that non-analyticities develop in physical quantities because, as the system passes to the thermodynamic limit, complex singularities of the pressure pinch the physical (i.e., real) domain of the system parameters. Since the pressure is proportional to the logarithm of the partition function, these singularities correspond exactly to the zeros of the partition function. In their second paper [25], Lee andYang demonstrated the validity of their theory in a particular example of the Ising model in a complex magnetic field h. Using an induction argument, they proved the celebrated Lee-Yang Circle Theorem which states that, in this model, the complex-eh zeros of the partition function on any finite graph with free boundary conditions lie on the unit circle. The subject has been further pursued by a number of authors in the following fifty years. Generalizations of the Lee-Yang theorem have been developed [26, 31, 32, 35] and extensions to other complex parameters have been derived (for instance, the Fisher zeros [14] in the complex temperature plane and the zeros of the q-state Potts model in the complex-q plane [40, 41]). Numerous papers have appeared studying the partition function zeros using various techniques including computer simulations [10, 20, 22], approximate analyses [21, 24, 29] and exact solutions of 1D and 2D lattice systems [8, 9, 12, 18, 27, 28, 38, 39]. However, in spite of this progress, it seems fair to say that much of the original Lee-Yang program—namely, to learn about the transitions in physical systems by studying the zeros of partition functions—had remained unfulfilled. In [2], we outlined a general program, based on Pirogov-Sinai theory [6, 33, 34, 43], to determine the partition function zeros for a large class of lattice models depending on one complex parameter z. The present paper, and its companion [5], give the mathematical details of that program. Our results apply to a host of systems with firstorder phase transitions; among others, they can be applied to field-driven transitions in many low-temperature spin systems as well as temperature-driven transitions—for instance, the order-disorder transition in the q-state Potts model with large q or the confinement Higgs transition in lattice gauge theories. We consider lattice models with a

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finite number of equilibrium states that satisfy several general assumptions (formulated in detail below). The validity of the assumptions follows whenever a model can be analyzed using a convergent contour expansion based on Pirogov-Sinai theory, even in the complex domain. In the present work, we study only models with periodic boundary conditions, although—with some technically involved modifications—our techniques should allow us to treat also other boundary conditions. Under our general assumptions, we derive a set of model-specific equations; the solutions of these equations yield the locations of the partition function zeros, up to rigorously controlled errors which are typically exponentially small in the linear size of the system. It turns out that, as the system size tends to infinity, the partition function zeros concentrate on the union of a countable number of simple smooth curves in the complex z-plane. Another outcome of our analysis is a local version of the Lee-Yang Circle Theorem. Whereas the global theorem says that, for models with the full Ising interaction, all partition function zeros lie on the unit circle, our local theorem says that if the model has an Ising-like symmetry in a restricted region of the complex z-plane, the corresponding portion of the zeros lies on a piece of the unit circle. In particular, there are natural examples (see the discussion of the Blume-Capel model in [2]) where only some of the partition function zeros lie on the unit circle, and others lie on less symmetric curves. Our proof indicates that it is just the Ising plus-minus symmetry (and a natural non-degeneracy condition) that makes the Lee-Yang theorem true, which is a fact not entirely apparent in the original derivations of this result. In addition to being of interest for the foundations of statistical mechanics, our results can often be useful on a practical level—even when the parameters of the model are such that we cannot rigorously verify all of our assumptions. We have found that our equations seem to give accurate locations of finite-volume partition function zeros for system sizes well beyond what can be currently achieved using, e.g., computer-assisted evaluations of these partition functions (see [2] for the example of the three dimensional 25-state Potts model on 1000 sites). Our techniques are also capable of handling situations with more than one complex parameter in the system. However, the actual analysis of the manifolds of partition function zeros may be technically rather involved. Finally, we remark that, in one respect, our program falls short of the ultimate goal of the original Lee-Young program—namely, to describe the phase structure of any statistical-mechanical system directly on the basis of its partition function zeros. Instead, we show that both the location of the partition function zeros and the phase structure are consequences of an even more fundamental property: the ability to represent the partition function as a sum of terms corresponding to different metastable phases. This representation is described in the next section. 1.2. Basic ideas. Here we will discuss the main ideas of our program, its technical difficulties and our assumptions in more detail. We consider spin models on Zd , with d ≥ 2, whose interaction depends on a complex parameter z. Our program is based on the fact per that, for a large class of such models, the partition function ZL in a box of side L and with periodic boundary conditions can be written as per

ZL (z) =

r


  d d qm e−fm (z)L + O e−const L e−f (z)L .

(1.1)

m=1

Here q1 , . . . , qr are positive integers describing the degeneracies of the phases 1, . . . , r, the quantities f1 , . . . , fr are smooth (but not in general analytic) complex functions

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of the parameter z which play the role of metastable free energies of the corresponding phases, and f (z) = min1≤m≤r Refm (z). The real version of the formula (1.1) was instrumental for the theory of finite-size scaling near first-order phase transitions [7]; the original derivation goes back to [6]. per It follows immediately from (1.1) that, asymptotically as L tends to infinity, ZL = 0 requires that Refm (z) = Refm .  (z) = f (z) for at least two distinct indices m and m (Indeed, otherwise the sum in (1.1) would be dominated by a single, non-vanishing per term.) Therefore, asymptotically, all zeros of ZL concentrate on the set   G = z : there exist m = m  with Refm (z) = Refm (1.2)  (z) = f (z) . Our first concern is the topological structure of G . Let us call a point where Refm (z) = f (z) for at least three different m a multiple point; the points z ∈ G that are not multiple points are called points of two-phase coexistence. Under suitable assumptions on the functions f1 , . . . , fr , we show that G is a countable union of non-intersecting simple smooth curves that begin and end at multiple points. Moreover, there are only a finite number of multiple points inside any compact subset of C. See Theorem 2.1 for details. The relative interior of each curve comprising G consists entirely of the points of two-phase coexistence, i.e., we have Refm (z) = Refm  (z) = f (z) for exactly two indices m and m . In particular, the sum in (1.1) is dominated by two terms. Supposing for a moment that we can neglect all the remaining contributions, we would have −fm  (z)L , ZL (z) = qm e−fm (z)L + qm e per

d

d

(1.3)

per

and the zeros of ZL would be determined by the equations −d Refm (z) = Refm log(qm /qm  (z) + L  ), −d Imfm (z) = Imfm  (z) + (2 + 1)π L ,

(1.4)

where  is an integer. The presence of additional terms of course makes the actual zeros only approximate solutions to (1.4); the main technical problem is to give a reasonper able estimate of the distance between the solutions of (1.4) and the zeros of ZL . In a neighborhood of multiple points, the situation is even more complicated because there Eqs. (1.4) will not be even approximately correct. It turns out that the above heuristic argument cannot possibly be converted into a rigorous proof without making serious adjustments to the initial formula (1.1). This is a consequence of subtle analytic properties of the functions fm . For typical physical systems, the metastable free energy fm is known to be analytic only in the interior of the region   (1.5) Sm = z : Refm (z) = f (z) . On the boundary of Sm , one expects—and in some cases proves [15, 19]—the existence of essential singularities. Thus (1.1) describes an approximation of an analytic function, per the function ZL , by a sum of non-analytic functions, with singularities appearing preper cisely in the region where we expect to find the zeros of ZL ! It is easy to construct examples where an arbitrarily small non-analytic perturbation of a complex polynomial with a degenerate zero produces extraneous roots. This would not be an issue along the per two-phase coexistence lines, where the roots of ZL turn out to be non-degenerate, but we would not be able to say much about the roots near the multiple points. In short, we need an approximation that respects the analytic structure of our model.

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Fortunately, we do not need to look far to get the desirable analytic counterpart of (1.1). In fact, it suffices to modify slightly the derivation of the original formula. For the benefit of the reader, we will recall the main steps of this derivation: First we use a contour representation of the model—the class of models we consider is characterized by the property of having such a contour reformulation—to rewrite the partition function as a sum over the collections of contours. Then we divide the configurations contributing per to ZL into r + 1 categories: Those in which all contours are of diameter smaller than, say, L/3 and in which the dominant phase is m, where m = 1, . . . , r, and those not (L) falling into the preceding categories. Let Zm be the partial partition function obtained by summing the contributions corresponding to the configurations in the mth category, see Fig. 1. It turns out that the error term is still uniformly bounded as in (1.1), so we have per

ZL (z) =

r


  d (L) Zm (z) + O e−const L e−f (z)L ,

(1.6)

m=1 (L)

but now the functions Zm (z) are analytic, and non-zero in a small neighborhood of Sm . (However, the size of the neighborhood shrinks with L → ∞, and one of the challenges of using the formula (1.6) is to cope with this restriction of analyticity.) Moreover, writing (L)

(L) Zm (z) = qm e−fm

(z)Ld

(1.7)

(L)

and using the contour representation, the functions fm can be expressed by means of convergent cluster expansions [11, 23]. In particular, they can be shown to converge quickly to the functions fm as L → ∞. In this paper, we carry out the analysis of the partition function zeros starting from the representation (1.6). In particular, we formulate minimal conditions (see Assumptions A (L) and B in Sect. 2) on the functions fm and the error terms that allow us to analyze the per (L) roots of ZL in great detail. The actual construction of the functions fm and the proof that they satisfy the required conditions is presented in [3, 4] for the q-state Potts model with one complex external field and q sufficiently large, and in [5] for a general class of lattice models with finite number of equilibrium states. 1.3. Discussion of assumptions and results. Here we will describe our main assumptions and indicate how they feed into the proofs of our main theorems. For consistency (L) with the previous sections, we will keep using the functions fm and fm even though the assumptions will actually be stated in terms of the associated exponential variables ζm (z) = e−fm (z)

(L)

and ζm(L) (z) = e−fm

(z)

.

(1.8)

The first set of assumptions (Assumption A, see Sect. 2.1) concerns the infinite-volume quantities fm , and is important for the description of the set of coexistence points G . The functions fm are taken to be twice differentiable in the variables x = Rez and y = Imz, and analytic in the interior of the set Sm . If, in addition, f (z) = minm Refm is uniformly bounded from above, good control of the two-phase coexistence curves is obtained by assuming that, for any distinct m and m , the difference of the first derivatives of fm and fm  is uniformly bounded from below on Sm ∩ Sm  . Finally, in order to discuss multiple coexistence points, we need an additional non-degeneracy assumption on the

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Fig. 1. Schematic examples of configurations, along with their associated contours, which contribute to different terms in the decomposition in (1.6). Here we have a spin model with r = 3 equilibrium phases denoted by +, − and 0. The configuration on the left has all contours smaller than the cutoff—which we (L) set to L/3, where L is the side of the box—and will thus contribute to Z+ because + is the external phase for all external contours. The configuration on the right has long contours and will be assigned to the error term

derivatives of the functions fm for the coexisting phases. Given these assumptions, we are able to give a very precise characterization of the topology of the coexistence set G , see Theorem 2.1. The second set of assumptions (Assumption B, see Sect. 2.2) is crucial for our results (L) on the partition function zeros, and is formulated in terms of the functions fm . These will be taken to be analytic with a uniform upper bound on the first r derivatives in an (L) order-(1/L) neighborhood of the sets Sm . In this neighborhood, fm is also assumed to be exponentially close to fm , with a lower bound on the difference of the first deriv(L) (L) atives for any pair fm and fm  in the intersection of the corresponding order-(1/L) neighborhoods. Finally, we need a bound on the error term and its derivatives in an approximation of the form (1.6) where the sum runs only over the dominating terms, i.e., those m for which z lies in the order-(1/L) neighborhood of Sm . Combining Assumptions A and B, we are able to prove several statements on the location of the partition function zeros. We will start by covering the set of available z-values by sets with a given number of stable (or “almost stable”) phases. The covering involves three scale functions, ωL , γL and ρL which give rise to three classes of sets: the region where one phase is decisively dominating the others (more precisely, the complement of an L−d ωL -neighborhood of the set G ), a γL -neighborhood of sets with two stable phases, excluding a γL -neighborhood of multiple points, and the ρL neighborhoods of multiple points. As is shown in Proposition 2.6, for a suitable choice of sequences ωL , γL , and ρL , these three sets cover all possibilities. In each part of the cover, we will control the zeros by a different method. The results per of our analysis can be summarized as follows: First, there are no zeros of ZL outside an L−d ωL -neighborhood of the set G . This claim, together with a statement on the maximal possible degeneracy of zeros, is the content of Theorem 2.2. The next theorem, Theorem 2.3, states that in a γL -neighborhood of the two-phase coexistence points, excluding

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85 per

a neighborhood of multiple points, the zeros of ZL are exponentially close to the solutions of (1.4). In particular, this implies that the zeros are spaced in intervals of order-L−d along the two-phase coexistence curves with the asymptotic density expressed in terms of the difference of the derivatives of the corresponding free energies—a result known in a special case already to Yang and Lee [42]; see Proposition 2.4. The control of the zeros in the vicinity of multiple points is more difficult and the results are less detailed. Specifically, in the ρL -neighborhood of a multiple point with q coexisting phases, the per zeros of ZL are shown to be located within a L−d−d/q neighborhood of the solutions of an explicitly specified equation. We finish our discussion with a remark concerning the positions of zeros of complex functions of the form: ZN (z) =

r


αm (z)ζm (z)N ,

(1.9)

m=1

where α1 , . . . , αr and ζ1 , . . . , ζr are analytic functions of z. Here there is a general theorem, due to Beraha, Kahane and Weiss [1] (generalized recently by Sokal [41]), that the set of zeros of ZN asymptotically concentrates on the set of z such that either αm (z) = 0 and |ζm (z)| = maxk |ζk (z)| for some m = 1, . . . , r or |ζm (z)| = |ζn (z)| = maxk |ζk (z)| for two distinct indices m and n. The present paper provides a substantial extension of this result to situations when analyticity of ζm (z) can be guaranteed only in a shrinking neighborhood of the sets where m is the “dominant” index. In addition, we also provide detailed control of the rate of convergence. 2. Main Results 2.1. Complex phase diagram. We begin by abstracting the assumptions on the metastable free energies of the contour model and showing what kind of complex phase diagram they can yield. Throughout the paper, we will assume that a domain O ⊂ C and a positive integer r are given, and use R to denote the set R = {1, . . . , r}. For each z ∈ O, we let x = Rez and y = Imz and define, as usual,     ∂ ∂ ∂ ∂ ∂z = 21 ∂x − i ∂y + i ∂y and ∂z¯ = 21 ∂x . (2.1) Assumption A. There exists a constant α > 0 and, for each m ∈ R, a function ζm : O → C, such that the following conditions are satisfied: (1) The quantity ζ (z) = maxm∈R |ζm (z)| is uniformly positive in O, i.e., we have inf z∈O ζ (z) > 0. (2) Each function ζm , viewed as a function of two real variables x = Rez and y = Imz, is twice continuously differentiable on O and it satisfies the Cauchy-Riemann equations ∂z¯ ζm (z) = 0 for all z ∈ Sm , where   (2.2) Sm = z ∈ O : |ζm (z)| = ζ (z) . In particular, ζm is analytic on the interior of Sm . (3) For any pair of distinct indices m, n ∈ R and any z ∈ Sm ∩ Sn we have    ∂z ζm (z) ∂z ζn (z)     ζ (z) − ζ (z) ≥ α. m n

(2.3)

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(4) If Q ⊂ R is such that |Q| ≥ 3, then for any z ∈ vm (z) =

∂z ζm (z) , ζm (z)



m∈Q Sm ,

m ∈ Q,

(2.4)

are the vertices of a strictly convex polygon in C R2 . Remark 1. In (1), we assumed uniform positivity in order to simplify some of our later arguments. However, uniformity in O can easily be replaced by uniformity on compact sets. Note that Assumptions A3–4 are invariant with respect to conformal transformations of O because the functions involved in (2.3) and (2.4) satisfy the Cauchy-Riemann conditions. Also note that, by Assumption A3, the length of each side of the polygon from Assumption A4 is at least α; cf. Fig. 3. The indices m ∈ R will often be referred to as phases. We call a phase m stable at z if z ∈ Sm , i.e., if |ζm (z)| = ζ (z). For each z ∈ O we define   Q(z) = m ∈ R : |ζm (z)| = ζ (z)

(2.5)

to be the set of phases stable at z. If m, n ∈ Q(z), then we say that the phases m and n coexist at z. The phase diagram is determined by the set of coexistence points: G =



G (m, n)

with

G (m, n) = Sm ∩ Sn .

(2.6)

m,n∈R : m=n

If |ζm (z)| = ζ (z) for at least three distinct m ∈ R, we call such z ∈ O a multiple point. In the following, the phrase simple arc denotes the image of (0, 1) under a continuous and injective map while simple closed curve denotes a corresponding image of the unit circle {z ∈ C : |z| = 1}. A curve will be called smooth if it can be parametrized using twice continuously differentiable functions. Our main result concerning the topology of G is then as follows. Theorem 2.1. Suppose that Assumption A holds and let D ⊂ O be a compact set. Then there exists a finite set of open discs D1 , D2 , . . . , D ⊂ O covering D , such that for each k = 1, . . . , , the set Ak = G ∩ Dk satisfies exactly one of the following properties: (1) Ak = ∅. (2) Ak is a smooth simple arc with both endpoints on ∂Dk . Exactly two distinct phases coexist along the arc constituting Ak . (3) Ak contains a single multiple point zk with sk = |Q(zk )| ≥ 3 coexisting phases, and Ak \{zk } is a collection of sk smooth, non-intersecting, simple arcs connecting zk to ∂Dk . Each pair of distinct curves from Ak \ {zk } intersects at a positive angle at zk . Exactly two distinct phases coexist along each component of Ak \ {zk }.

In particular, G = C ∈C C , where C is a finite or countably-infinite collection of smooth simple closed curves and simple arcs which intersect each other only at the endpoints. Theorem 2.1 is proved in Sect. 3.2. Further discussion is provided in Sect. 2.4.

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(a)

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(b)

(c)

3 1,3 2,3 1,2,3

1

2 1,2

Fig. 2. An illustration of the sets U (Q) in the vicinity of a multiple point. The thick lines indicate the visible portion of the set of coexistence points G . Three phases, here labeled 1, 2 and 3, are stable at the multiple point. In (a), the three shaded domains represent the sets U ({1}), U ({2}) and U ({3}), with the label indicated by the number in the box. Similarly, in (b) the three regions represent the sets U ({1, 2}), U ({2, 3}) and U ({1, 3}). Finally, (c) contains only one shaded region, representing the set U ({1, 2, 3}). The various regions U (Q) generously overlap so that their union covers the entire box

2.2. Partition function zeros. Next we will discuss our assumptions and results conper cerning the zeros of the partition function. We assume that the functions ZL : O → C, playing the role of the partition function in a box of side L with periodic boundary conditions, are defined for each integer L, or, more generally, for any L ∈ L, where L ⊂ N is a fixed infinite set. Given any m ∈ R and > 0, we use S (m) to denote the region where the phase m is “almost stable,”   S (m) = z ∈ O : |ζm (z)| > e− ζ (z) . (2.7) For any Q ⊂ R, we also introduce the region where all phases from Q are “almost stable” while the remaining ones are not,  U (Q) = S (m) \ S /2 (n), (2.8) m∈Q

n∈Qc

with the bar denoting

the set closure. Notice that the function ζm is non-vanishing on S (m) and that Q⊂R U (Q) = O, see Fig. 2. Note also that U (∅) = ∅, so we may assume that Q = ∅ for the rest of this paper. Assumption B. There exist constants κ, τ ∈ (0, ∞) and, for each m ∈ R, a positive (L) integer qm and a function ζm : Sκ/L (m) → C such that for any L ∈ L the following is true: per

(1) The function ZL is analytic in O. (L) (2) Each ζm is non-vanishing and analytic in Sκ/L (m). Furthermore,   (L)   log ζm (z)  ≤ e−τ L  ζ (z)  m

(2.9)

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and

    (L) (L)     ∂z log ζm (z)  + ∂z¯ log ζm (z)  ≤ e−τ L    ζ (z) ζ (z)  m

(2.10)

m

for all m ∈ R and all z ∈ Sκ/L (m). (Here “log” denotes the principal branch of the complex logarithm.) (3) There exist constants α˜ > 0, M < ∞ and L˜ 0 < ∞ such that for any L ≥ L˜ 0 we have   (L)   ∂z ζm (z)   ≤ M,  (2.11)   (L) ζm (z) whenever m ∈ R,  = 1, . . . , r, and z ∈ Sκ/L (m). In addition,    ∂z ζm(L) (z) ∂z ζn(L) (z)  ≥ α˜  − (L)   (L) ζm (z) ζn (z)

(2.12)

whenever m, n ∈ R are distinct and z ∈ Sκ/L (m) ∩ Sκ/L (n). (4) There exist constants C < ∞,  = 0, 1, . . . , r + 1, such that for any Q ⊂ R, the difference

Ld
per qm ζm(L) (z) (2.13) ΞQ,L (z) = ZL (z) − m∈Q

satisfies the bound    
d    qm e−τ L , ∂z ΞQ,L (z) ≤ C Ld(+1) ζ (z)L

(2.14)

m∈R

for all  = 0, 1, . . . , r + 1, uniformly in z ∈ Uκ/L (Q). Remark 2. In applications, qm will represent the degeneracy of the phase m; thus we have taken it to be a positive integer. However, our arguments would go through even if we assumed only that all qm ’s are real and positive. It is also worth noting that in many physical models the partition function is not directly of the form required by Assumption B; d but it can be brought into this form by extracting a multiplicative “fudge” factor F (z)L , where F (z) = 0 in the region of interest. For instance, in the Ising model with z related to the complex external field h by z = eh we will have to take F (z) = z−1/2 to make the partition function analytic in the neighborhood of z = 0. per

Our first theorem in this section states that the zeros of ZL (z) are concentrated in a narrow strip along the phase boundaries. In addition, their maximal degeneracy near the multiple points of the phase diagram can be evaluated. In accord with the standard terminology, we will call a point z0 a k-times degenerate root of an analytic function h(z) if h(z) = g(z)(z − z0 )k for some g(z) that is finite and non-zero in a neighborhood of z0 . Recalling the definition (2.8) of the set U (Q), we introduce the shorthand     G = (2.15) S /2 (n) ∩ S /2 (m) = O \ U {m} . m=n

m∈R

An easy way to check the second equality in (2.15) is by noting that O \ U ({m}) can

be written as the union n:n=m S /2 (n). Then we have the following result.

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Theorem 2.2. Suppose that Assumptions A1-3 and B hold and let κ > 0 be as in Assumption B. Let (ωL ) be a sequence of positive numbers such that ωL → ∞. Then there exists per a constant L0 < ∞ such that for L ≥ L0 all roots of ZL lie in G L−d ωL and are at per most |R| − 1 times degenerate. For each Q ⊂ R, the roots of ZL in Uκ/L (Q) are at most |Q| − 1 times degenerate. per

In other words, as L → ∞, the zeros of ZL asymptotically concentrate on the set of coexistence points G . Notice that we explicitly do not require Assumption A4 to hold; see Sect. 2.4 for further discussion. Theorem 2.2 is proved in Sect. 4.1. per Our next theorem deals with the zeros of ZL in the regions where at most two phases from R are “almost stable.” It turns out that we have a much better control on the location of zeros in regions that are sufficiently far from multiple points. To quantify the meaning of “sufficiently far,” we let γL be a sequence of positive numbers (to be specified below) and, for any Q ⊂ R with |Q| = 2 and any L ≥ 0, let δL : UγL (Q) → (0, ∞) be a function defined by  δL (z) =

e−τ L , 1 d Ld e− 2 γL L ,

ifz ∈ UγL (Q) ∩ U2κ/L (Q), otherwise.

(2.16)

(Clearly, δL (z) depends on the index set Q. However, this set will always be clear from the context and so we will not make it notationally explicit.) Finally, given > 0 and z ∈ O, let D (z) denote the open disc of radius centered at z. The exact control of the roots in two-phase regions is then as follows. Theorem 2.3. Suppose that Assumptions A and B hold and let 
L be the set of all zeros per of the function ZL (z) in O, including multiplicity. If m, n ∈ R are distinct indices, let Q = {m, n}, and let L (Q) be the set of the solutions of the system of equations 1/Ld

1/Ld

qm |ζm (z)| = qn |ζn (z)|,  Ld Arg ζm (z)/ζn (z) = π mod 2π. 

(2.17) (2.18)

Let γL be such that lim inf L→∞

L d γL > 4d and log L

lim sup Ld−1 γL < 2τ,

(2.19)

L→∞

and let δL : UγL (Q) → (0, ∞) be as defined in (2.16). Then there exist finite positive constants B, C, D, and L0 such that for any Q ⊂ R with |Q| = 2 and any L ≥ L0 we have: (1) For all z ∈ G ∩ UγL (Q) with DDL−d (z) ⊂ O, the disc DDL−d (z) contains at least one root from 
L . (2) For all z ∈ 
L ∩ UγL (Q) with DCδL (z) (z) ⊂ O, the disc DCδL (z) (z) contains exactly one point from L (Q). (3) For all z ∈ L (Q) ∩ UγL (Q) with DCδL (z) (z) ⊂ O, the disc DCδL (z) (z) contains exactly one root from 
L . per (4) Any two distinct roots of ZL in the set {z ∈ UγL (Q) : DBL−d (z) ⊂ O} are at least BL−d apart.

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Note that the first limit in (2.19) ensures that Ld δL (z) → 0 as L → ∞ throughout UγL (Q) (for any Q ⊂ R with |Q| = 2). Thus δL (z) is much smaller than the distance of the “neighboring” roots of (2.17–2.18). Theorem 2.3 is proved in Sect. 4.2. per Theorem 2.3 allows us to describe the asymptotic density of the roots of ZL along the arcs of the complex phase diagram. Let m, n ∈ R be distinct and let G (m, n) be as (L, ) in (2.6). For each > 0 and each z ∈ G (m, n), let ρm,n (z) be defined by (L, ) ρm,n (z) =

 1 
, ∩ D (z) 

L 2 Ld

(2.20)

per

where |
L ∩ D (z)| is the number of roots of ZL in D (z) including multiplicity. Since G (m, n) is a union of simple arcs and closed curves, and since the roots of (2.17– (L, ) 2.18) are spaced within O(L−d ) from each other, ρm,n (z) has the natural interpretation per of the approximate line density of zeros of ZL along G (m, n). As can be expected from (L, ) Theorem 2.3, the approximate density ρm,n (z) tends to an explicitly computable limit. (L, )

Proposition 2.4. Let m, n ∈ R be distinct and let ρm,n (z) be as in (2.20). Then the limit (L, ) ρm,n (z) = lim lim ρm,n (z)

(2.21)

↓0 L→∞

exists for all z ∈ G (m, n) such that |Q(z)| = 2, and   1  ∂z ζm (z) ∂z ζn (z)  ρm,n (z) = − . 2π  ζm (z) ζn (z) 

(2.22)

Remark 3. Note that, on the basis of Assumption A3, we have that ρm,n (z) ≥ α/(2π ). In particular, the density of zeros is always positive. This is directly related to the fact that all points z ∈ G will exhibit a first-order phase transition (defined in an appropriate sense, once Imz = 0 or Rez < 0)—hence the title of the paper. The observation that the (positive) density of zeros and the order of the transition are closely related goes back to [42]. per

In order to complete the description of the roots of ZL , we also need to cover the regions with more than two “almost stable” phases. This is done in the following theorem. Theorem 2.5. Suppose that Assumptions A and B are satisfied. Let zM be a multiple point and let Q = Q(zM ) with q = |Q| ≥ 3. For each m ∈ Q, let φm (L) = Ld Arg ζm (zM ) (mod 2π)

and

vm =

∂z ζm (zM ) . ζm (zM )

Consider the set L (Q) of all solutions of the equation
d qm eiφm (L)+L (z−zM )vm = 0,

(2.23)

(2.24)

m∈Q

including multiplicity, and let (ρL ) be a sequence of positive numbers such that lim Ld ρL = ∞

L→∞

but

lim Ld−d/(2q) ρL = 0.

L→∞

(2.25)

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Define ρL = ρL +L−d(1+1/q) . Then there exists a constant L0 < ∞ and, for any L ≥ L0 , an open, connected and simply connected set U satisfying DρL (zM ) ⊂ U ⊂ DρL (zM ) such that the zeros in  ∩ U are in one-to-one correspondence with the solutions in (Q) ∩ U and the corresponding points are not farther apart than L−d(1+1/q) . Theorem 2.5 is proved in Sect. 4.4. Section 2.4 contains a discussion of the role of Assumption A4 in this theorem; some information will also be provided concerning the actual form of the solutions of (2.24). To finish the exposition of our results, we will need to show that the results of Theorems 2.2, 2.3 and 2.5 can be patched together to provide complete control of the roots per of ZL , at least in any compact subset of O. This is done in the following claim, the proof of which essentially relies only on Assumption A and compactness arguments: Proposition 2.6. Suppose that Assumption A holds and let ωL , γL and ρL be sequences of positive numbers such that ωL ≤ γL Ld , γL → 0, and ρL → 0. For each compact set D ⊂ O, there exist constants χ = χ (D ) > 0 and L0 = L0 (D ) < ∞ such that, if ρL ≥ χ γL , we have UγL (Q) ∪ DρL (zM ) (2.26) G L−d ωL ∩ D ⊂ Q⊂R |Q|=2

zM ∈D |Q(zM )|≥3

for any L ≥ L0 . Note that in (2.26) we consider only that portion of D in G L−d ωL , since by Theoper rem 2.2 the roots of ZL are contained in this set. Note also that the conditions we impose on the sequences ωL , γL and ρL in Theorems 2.1, 2.3 and 2.5 and Proposition 2.6 are not very restrictive. In particular, it is very easy to verify the existence of these sequences. (For example, one can take both γL and ρL to be proportional to L−d log L with suitable prefactors and then let ωL = Ld γL .) 2.3. Local Lee-Yang theorem. As our last result, we state a generalized version of the classic Lee-Yang Circle Theorem [25], the proof of which is based entirely on the exact symmetries of the model. Theorem 2.7. Suppose that Assumptions A and B hold. Let + and − be two selected indices from R and let U be an open set with compact closure D ⊂ O such that U ∩{z : |z| = 1} = ∅. Assume that D is invariant under circle inversion z → 1/z∗ , and (1) ZL (z) = ZL (1/z∗ )∗ , (2) ζ+ (z) = ζ− (1/z∗ )∗ and q+ = q− per

per

hold for all z ∈ D and all L ∈ L. Then there exists a constant L0 such that the following holds for all L ≥ L0 : If the intersection of D with the set of coexistence points G is connected and if + and − are the only stable phases in D , then all zeros in D lie on the unit circle, and the number of zeros on any segment of D ∩ {z : |z| = 1} is proportional to Ld as L → ∞. Condition (2) is the rigorous formulation of the statement that the + and − phases are related by z ↔ 1/z∗ (or h ↔ −h, when z = eh ) symmetry. Condition (1) then stipulates that this symmetry is actually respected by the remaining phases and, in particular, per by ZL itself.

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Remark 4. As discussed in Remark 2, in order to satisfy Assumption B it may be necessary to extract a multiplicative “fudge” factor from the partition function, perform the analysis of partition function zeros in various restricted regions in C and patch the results appropriately. A similar manipulation may be required in order to apply Theorem 2.7. Here are the main steps of the proof of Theorem 2.7: First we show that the phase diagram in D falls exactly on the unit circle, i.e., D ∩ G = {z ∈ D : |z| = 1}.

(2.27)

This fact is essentially an immediate consequence of the symmetry between “+” and “−.” A priori one would then expect that the zeros are close to, but not necessarily on, per the unit circle. However, the symmetry of ZL combined with the fact that distinct zeros are at least BL−d apart is not compatible with the existence of zeros away from the unit per circle. Indeed, if z is a root of ZL , it is bound to be within a distance O(e−τ L ) of the unit circle. If, in addition, |z| = 1, then the z ↔ 1/z∗ symmetry implies that 1/z∗ is also per a root of ZL , again within O(e−τ L ) of the unit circle. But then the distance between z ∗ and 1/z is of the order e−τ L which is forbidden by claim (4) of Theorem 2.3. This argument is made precise in the following proof. Proof of Theorem 2.7. We start with the proof of (2.27). Let us suppose that D ⊂ O and Q(z) ⊂ {+, −} for all z ∈ D . Invoking the continuity of ζ± and condition (2) above, we have Q(z) = {+, −} for all z ∈ D ∩ {z : |z| = 1} and thus D ∩ {z : |z| = 1} ⊂ G . Assume now that G ∩ D \ {z : |z| = 1} = ∅. By the fact that G ∩ D is connected and the assumption that U ∩ {z : |z| = 1} = ∅, we can find a path zt ∈ G ∩ D , t ∈ [−1, 1], such that zt ∈ D ∩ {z : |z| = 1} if t ≤ 0 and zt ∈ G ∩ D \ {z : |z| = 1} if t > 0. Since Q(z0 ) = {+, −}, we know that there is a disc D (z0 ) ⊂ O that contains no multiple points. Applying Theorem 2.1 to this disc, we conclude that there is an open disc D with z0 ∈ D ⊂ D (z0 ), such that G ∩ D is a simple curve which ends at ∂D. However, using condition (2) above, we note that as with zt , also the curve t → 1/zt∗ lies in G ∩ D , contradicting the fact that G ∩ D is a simple curve. This completes the proof of (2.27). Next, we will show that for any z0 ∈ D ∩ {z : |z| = 1}, and any δ > 0, there exists an open disc D (z0 ) ⊂ O such that the set G ∩ D (z0 ) is a smooth curve with the property that for any z ∈ D (z0 ) with |z| = 1, the line connecting z and 1/z∗ intersects the curve G ∩ D (z0 ) exactly once, and at an angle that lies between π/2 − δ and π/2 + δ. If z0 lies in the interior of D , this statement (with δ = 0) follows trivially from (2.27). If z0 is a boundary point of D , we first choose a sufficiently small disc D  z0 so that D ⊂ O and, for all points in D, only the phases + and − are stable. Then we use Theorem 2.3 and (2.27) to infer that can be chosen small enough to guarantee the above statement about intersection angles. Furthermore, we claim that given z0 ∈ D ∩ {z : |z| = 1} and > 0 such that D3 (z0 ) ⊂ O and Q(z) ⊂ {+, −} for all z ∈ D3 (z0 ), one can choose L sufficiently large so that D2 (z0 ) ∩ G L−d ωL ⊂ UγL ({+, −}) ∩ U2κ/L ({+, −}).

(2.28)

To prove this, let us first note that, for γL ≤ 2κ/L, the right hand side can be rewritten as Sκ/L (m). (2.29) UγL ({+, −}) \ m=−,+

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Next, by the compactness of D2 (z0 ) and the fact that no m ∈ R different from ± is stable anywhere in D3 (z0 ), we can choose L0 so large that Sκ/L (m) ∩ D2 (z0 ) = ∅ for all L ≥ L0 and all m = ±. Using the closure of D2 (z0 ) in place of the set D in (2.26), we get (2.28). We are now ready to prove that for any z0 ∈ D ∩ {z : |z| = 1}, there exist constants per

> 0 and L0 such that all roots of ZL in D (z0 )∩D lie on the unit circle. To this end, let us first assume that has been chosen small enough to guarantee that (1− )−1 < 1+2 , D3 (z0 ) ⊂ O, Q(z) ⊂ {+, −} for all z ∈ D3 (z0 ), and G ∩ D3 (z0 ) is a smooth curve with the above property about the intersections angles, with, say, δ = π/4. Assume further that L is chosen so that (2.28) holds and > max(CδL (z0 ), BL−d ), where C and B are the constants from Theorem 2.3. per Let z ∈ D (z0 ) ∩ D be a root of ZL . If L is so large that Theorem 2.2 applies, we have z ∈ G L−d ωL and thus δL (z) = e−τ L in view of (2.28). By Theorem 2.3, there exists a solution z˜ to (2.17–2.18) that lies in a CδL (z)-neighborhood of z, implying that z has distance less than CδL (z) from D2 (z0 )∩G . (Here we need that q+ = q− to conclude that z˜ ∈ G .) Suppose now that |z| = 1. Then the condition (1) above implies that z = (z∗ )−1 per is a distinct root of ZL in D . Moreover, if is so small that (1 − )−1 < 1 + 2 , then z ∈ G L−d ωL ∩ D2 (z0 ) and δL (z ) also equals e−τ L , implying that z has distance less than CδL (z) from D3 (z0 ) ∩ G . Since both z and z have distance less than CδL (z) from D3 (z0 ) ∩ G , and the curve D3 (z0 ) ∩ G intersects line through z and z in an angle √ the −τ that is near π/2, we conclude that |z − z | ≤ 2 2Ce L which for L sufficiently large contradicts the last claim of Theorem 2.3. Hence, z must have been on the unit circle after all. The rest of the argument is based on compactness. The set D ∩ {z : |z| = 1} is compact, and can thus be covered by a finite number of such discs. Picking one such cover, let D be the complement of these disc in D . Then the set D is a finite distance away from G and thus D ∩ G L−d ωL = ∅ for L sufficiently large. From here it follows that for some finite L0 < ∞ (which has to exceed the maximum of the corresponding quantity per for the discs that constitute the covering of D ∩ {z : |z| = 1}), all roots of ZL in U lie on the unit circle.  

2.4. Discussion. We finish with a brief discussion of the results stated in the previous three sections. We will also mention the role of (and possible exceptions to) our assumptions, as well as extensions to more general situations. We begin with the results on the complex phase diagram. Theorem 2.1 describes the situation in the generic cases when Assumptions A1-A4 hold. We note that Assumption A3 is crucial for the fact that the set G is a collection of curves. A consequence of this per is also that the zeros of ZL asymptotically concentrate on curves—exceptions to this “rule” are known, see, e.g., [36]. Assumption A4 prevents the phase coexistence curves from merging in a tangential fashion and, as a result of that, guarantees that multiple points do not proliferate throughout O. Unfortunately, in several models of interest (e.g., the Potts and Blume-Capel model) Assumption A4 happens to be violated at some z˜ for one or two “critical” values of the model parameters. In such cases, the region O has to be restricted to the complement of some neighborhood of z˜ and, inside the neighborhood, the claim has to be verified using a refined and often model-specific analysis. (It often suffices to show that the phase coexistence curves meeting at z˜ have different curvatures, which amounts to a statement about the second derivatives of the functions log ζm (z).)

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*

4

Fig. 3. An illustration of the situation around a quadruple point. Here v1∗ , . . . , v4∗ are the complex conjugates of the quantities from (2.4) and q1 = q2 = q3 < q4 . (The quadruple point lies at the common tail point of the vectors v1∗ , . . . , v4∗ .) The dashed lines indicate the asymptotes of the “strings” of zeros sufficiently far—on the scale L−d —from the quadruple point. Note the lateral shift of these lines due to the fact that q4 > q1 , q3 . The picture seems to suggest that, on the scale L−d , the quadruple point splits into two triple points

Examples of such analysis have appeared in [2] for the Blume-Capel model and in [4] for the Potts model in a complex external field. Next we will look at the results of Theorems 2.2 and 2.3. The fact that the roots of per ZL are only finitely degenerate is again independent of Assumption A4. (This is of some relevance in view of the aforementioned exceptions to this assumption.) The fact that, in the cases when all qm ’s are the same, the zeros shift only by an exponentially small amount away from the two-phase coexistence lines is a direct consequence of our choice of the boundary conditions. Indeed, the factor e−τ L in (2.16) can be traced to the similar factors in (2.9) and (2.14). For strong (e.g., fixed-spin) boundary conditions, we expect the corresponding terms in (2.9) and (2.14) to be replaced by 1/L. In particular, in these cases, the lateral shift of the partition function zeros away from the phase-coexistence lines should be of the order 1/L. See [44] for some results on this problem. Finally, let us examine the situation around multiple points in some detail. Theorem 2.5 can be given the following geometrical interpretation: Let zM be a multiple point. Introducing the parametrization z = (z − zM )Ld , we effectively zoom in on the scale L−d , per where the zeros of ZL are well approximated by the roots of the linearized problem ∗ of the logarithmic (2.24) with Q = Q(zM ). Let us plot the complex conjugates vm 2 ∗ derivatives vm (see (2.23)), m ∈ Q, as vectors in R . By Assumption A4, the vectors vm are the endpoints of a convex set in C R2 . Let v1∗ , . . . , vq∗ be the ordering of Q in the counterclockwise direction, see Fig. 3. Noting that the real part Re(vm z) can be written

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∗ · z, (2.24) can be recast as in terms of the dot product, Re(vm z) = vm
∗ qm eiφm (L)+vm ·z = 0,

(2.30)

m∈Q(zM ) (L) = φ (L) + Im(v z). where φm m m On the basis of (2.30), it is easy to verify the following facts: Let z = |z|ˆe, with eˆ a unit vector in C. An inspection of (2.30) shows that, for |z|  1, the roots of (2.30) will concentrate along the “directions” for which the projection of eˆ on at least two vn∗ ’s is the same. Invoking the convexity assumption (Assumption A4), this can only happen ∗ when vn∗ · eˆ = vn+1 · eˆ for some n. In such cases, the contributions of the terms with indices m = n, n + 1 in (2.30) are negligible—at least once |z|  1—and the zeros will thus asymptotically lie along the half-lines given in the parametric form by   ∗ vn∗ − vn+1 qn+1 ∗ z = z(t) = + it (vn∗ − vn+1 log ), t ∈ [0, ∞). (2.31) |vn − vn+1 |2 qn

Clearly, the latter is a line perpendicular to the (n, n + 1)st side of the convex set with vertices v1∗ , . . . , vq∗ , which is shifted (away from the origin) along the corresponding side by a factor proportional to log(qn+1 /qn ), see Fig. 3. Sufficiently far away from zM (on the scale L−d ), the zeros resume the pattern established around the two-phase coexistence curves. In particular, the zeros are asymptotically equally spaced but their overall shift along the asymptote is determined by the factor φm (L)—which we note depends very sensitively on L. Computer simulations show that, at least in generic cases, this pattern will persist all the way down to the multiple point. Thus, even on the “microscopic” level, the zeros seem to form a “phase diagram.” However, due to the lateral shifts caused by qm+1 = qm , a “macroscopic” quadruple point may resolve into two “microscopic” triple points, and similarly for higher-order multiple points. 3. Characterization of Phase Diagrams The goal of this section is to give the proof of Theorem 2.1. We begin by proving a series of auxiliary lemmas whose purpose is to elevate the pointwise Assumptions A3-A4 into statements extending over a small neighborhood of each coexistence point. 3.1. Auxiliary claims. Recall the definitions of Sm , Q(z) and vm (z), in (2.2), (2.5) and (2.23), respectively. The first lemma gives a limiting characterization of stability of phases around coexistence points. Lemma 3.1. Let Assumption A1–A2 hold and let z¯ ∈ O be such that |Q(¯z)| ≥ 2. Let (zk ) be a sequence of numbers zk ∈ O such that zk → z¯ but zk = z¯ for all k. Suppose that eiθ = lim

k→∞

zk − z¯ |zk − z¯ |

(3.1)

exists and let m ∈ Q(¯z). If zk ∈ Sm for infinitely many k ≥ 1, then Re(eiθ vm ) ≥ Re(eiθ vn )

for all n ∈ Q(¯z),

(3.2)

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where vn = vn (¯z). Conversely, if the inequality in (3.2) fails for at least one n ∈ Q(¯z), then there is an > 0 such that    z−¯z  iθ  <

W ,θ (¯z) = z ∈ O : |z − z¯ | < , z = z¯ ,  |z−¯ − e (3.3) z| has empty intersection with Sm , i.e., Sm ∩ W ,θ (¯z) = ∅. In particular, zk ∈ Sm for k large enough. Remark 5. In the following, it will be useful to recall some simple facts about complex functions. Let f , g and h be functions C → C and let ∂z and ∂z¯ be as in (2.1). If f satisfies ∂z¯ f (z0 ) = 0 (i.e., Cauchy-Riemann conditions), then all directional derivatives of f at z0 = x0 + iy0 can be expressed using one complex number A = ∂z f (x0 + iy0 ), i.e., we have f (x0 + cos ϕ + iy0 + i sin ϕ) − f (x0 + iy0 ) = Aeiϕ + o( ),

↓ 0,

(3.4)

holds for every ϕ ∈ [−π, π). Moreover, if g is differentiable with respect to x and y at z0 = x0 + iy0 and h satisfies ∂z¯ h(z ) = 0 at z = g(z0 ), then the chain rule holds for z → h(g(z)) at z = z0 . In particular, ∂z h(g(z0 )) = (∂z h)(g(z0 ))∂z g(z0 ). Proof of Lemma 3.1. Let m ∈ Q(¯z) be fixed. Whenever zk ∈ Sm , we have         logζm (zk ) − logζm (¯z) ≥ logζn (zk ) − logζn (¯z), n ∈ Q(¯z),

(3.5)

because |ζm (¯z)| = |ζn (¯z)|, by our assumption that m, n ∈ Q(¯z). Using the notation Fm,n (z) =

ζm (z) ζn (z)

(3.6)

for n ∈ Q(¯z) (which is well defined and non-zero in a neighborhood of z¯ ), the inequality (3.5) becomes     logFm,n (zk ) − logFm,n (¯z) ≥ 0, n ∈ Q(¯z). (3.7) Note that the complex derivative ∂z Fm,n (¯z) exists for all n ∈ Q(¯z). Our task is then to prove that   z) iθ ∂z¯ Fm,n (¯ Re e ≥ 0, n ∈ Q(¯z). (3.8) Fm,n (¯z) Fix n ∈ Q(¯z). Viewing z → Fm,n (z) as a function of two real variables x = Rez and y = Imz, we can expand log |Fm,n (z)| into a Taylor series around the point z¯ to get       ∂z Fm,n (¯z)     + O(|zk − z¯ |2 ). (3.9) log Fm,n (zk ) − log Fm,n (¯z) = Re (zk − z¯ ) Fm,n (¯z) To derive (3.9) we recalled that Fm,n is at least twice continuously differentiable (hence the error bound) and then applied the identity   ∂ log |Fm,n (¯z)| ∂ log |Fm,n (¯z)| ∂z Fm,n (¯z) xk + yk = Re (zk − z¯ ) , (3.10) ∂x ∂y Fm,n (¯z) where xk = Re(zk − z¯ ) and yk = Im(zk − z¯ ). (To derive (3.10), we just have to apply the chain rule to the functions z → log Fm,n (z). See Remark 5 for a discussion of

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this point.) Using that zk → z¯ , the inequality (3.8) and hence also (3.2) now follows by combining (3.9) with (3.5), dividing by |zk − z¯ | and taking the limit k → ∞. If, on the contrary, the inequality (3.2) is violated for some n ∈ Q(¯z), then (3.8) fails to hold as well and hence (3.7) and (3.5), with zk replaced by z, must be wrong for z ∈ W ,θ (¯z) whenever is small enough. But m ∈ Q(¯z) implies that |ζm (¯z)| = |ζn (¯z)| and thus |ζm (z)| < |ζn (z)| for all z ∈ W ,θ (¯z), proving that Sm ∩ W ,θ (¯z) = ∅. By (3.1) and the fact that zk → z¯ , we have zk ∈ W ,θ (¯z) and hence zk ∈ Sm for all k large enough.   Lemma 3.1 directly implies the following corollary. Corollary 3.2. Let Assumption A1–A2 hold and let m, n ∈ R be distinct. Let (zk ) be a sequence of numbers zk ∈ Sm ∩ Sn such that zk → z¯ ∈ O but zk = z¯ for all k. Suppose that the limit (3.1) exists and equals eiθ . Then Re(eiθ vm ) = Re(eiθ vn ). Proof. Follows immediately applying (3.2) twice.

 

The next lemma will ensure that multiple points do not cluster and that the coexistence lines always intersect at positive angles. Lemma 3.3. Suppose that Assumption A holds and let z¯ ∈ O. Suppose there are two sequences (zk ) and (zk ) of numbers from O such that |zk − z¯ | = |zk − z¯ | = 0 for all k and zk , zk → z¯ as k → ∞. Let a, b, c ∈ R and suppose that zk ∈ Sa ∩ Sb and zk ∈ Sa ∩ Sc for all k. Suppose the limit (3.1) exists for both sequences and let eiθ and eiθ be the corresponding limiting values.

(1) If a, b, c are distinct, then eiθ = eiθ . (2) If a = b = c and zk = zk for infinitely many k, then |Q(¯z)| = 2 and eiθ = −eiθ . Remark 6. The conclusions of part (2) have a very natural interpretation. Indeed, in this case, z¯ is a point on a two-phase coexistence line (whose existence we have not established yet) and zk and zk are the (eventually unique) intersections of this line with a circle of radius |zk − z¯ | = |zk − z¯ | around z¯ . As the radius of this circle decreases, the intersections zk and zk approach z¯ from “opposite” sides, which explains why we should expect to have eiθ = −eiθ . Proof of Lemma 3.3. Throughout the proof, we set vm = vm (¯z). We begin by prov ing (1). Assume that a, b, c ∈ R are distinct and suppose that eiθ = eiθ . Note that, since Q(¯z) ⊃ {a, b, c}, the point z¯ is a multiple point. Corollary 3.2 then implies that Re(eiθ va ) = Re(eiθ vb ) = Re(eiθ vc ),

(3.11)

and hence va , vb and vc lie on a straight line in C. But then va , vb and vc cannot simultaneously be vertices of a strictly convex polygon, in contradiction with Assumption A4. In order to prove part (2), let a = b = c, suppose without loss of generality that zk = zk for all k. If eiθ = ±eiθ , then Corollary 3.2 implies that Re(eiθ (va − vb )) = 0 = Re(eiθ (va −vb )) and hence va = vb , in contradiction with Assumption A3. Next we will rule out the possibility that eiθ = eiθ , regardless of how many phases are stable at z¯ . Let G(z) = ζa (z)/ζb (z) and note that |G(zk )| = 1 = |G(zk )| for all k. Applying Taylor’s

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theorem (analogously to the derivation of (3.9)), dividing by |zk − zk | and passing to the limit k → ∞, we derive   zk − zk ∂z G(zk ) lim Re = 0. (3.12) k→∞ |zk − zk | G(zk ) The second ratio on the left-hand side tends to va − vb . As for the first ratio, an easy computation reveals that, since |zk − z¯ | = |zk − z¯ | = 0, we have zk − zk i 21 (θk +θk ) sin((θk − θk )/2) = ie , |zk − zk | | sin((θk − θk )/2)|

(3.13)

where eiθk =

zk − z¯ |zk − z¯ |



and eiθk =

zk − z¯ . |zk − z¯ |

(3.14)



By our assumptions, we have eiθk → eiθ and eiθk → eiθ as k → ∞. Suppose now that eiθ = eiθ . Then, choosing a subsequence if necessary, the left-hand side of (3.13) tends to a definite sign times ieiθ . Inserting this into (3.12) and using Corollary 3.2, in addition to Re(eiθ (va −vb )) = 0, we now get that also Re(ieiθ (va −vb )) = Im(eiθ (va −vb )) = 0. Consequently, va = vb , again contradicting Assumption A3. To finish the proof of the claim (2), it remains to rule out the possibility that eiθ = −eiθ in the case when z¯ is a multiple point. Let n ∈ Q(¯z) be another phase stable at z¯ , i.e., n = a, b. By Lemma 3.1, we have     Re eiθ (vm − vn ) ≥ 0 and Re eiθ (vm − vn ) ≥ 0, m = a, b. (3.15)

But then eiθ = −eiθ would imply that Re(eiθ va ) = Re(eiθ vn ) = Re(eiθ vb ), in contradiction with Assumption A4. Therefore, |Q(¯z)| < 3, as claimed.   Corollary 3.4. Suppose that Assumption A holds and let z¯ ∈ O be a multiple point. Then there exists a constant δ > 0 such that |Q(z)| ≤ 2 for all z ∈ {z ∈ O : 0 < |z − z¯ | < δ}. In particular, each multiple point in O is isolated. Proof. Suppose z¯ ∈ O is a non-isolated multiple point. Then there is a sequence zk ∈ O such that zk → z¯ and, without loss of generality, Q(zk ) = Q0 with |Q0 | ≥ 3, zk = z¯ for all k, and such that the limit (3.1) exists. Taking for (zk ) the identical sequence, zk = zk , we get eiθ = eiθ in contradiction to Lemma 3.3(1). Therefore, every multiple point in O is isolated.   Our last auxiliary claim concerns the connectivity of sets of θ such that (3.2) holds. As will be seen in the proof of Lemma 3.6, this will be crucial for characterizing the topology of the phase diagram in small neighborhoods of multiple points. Lemma 3.5. Suppose that Assumption A holds and let z¯ ∈ O be a multiple point. For m ∈ Q(¯z), let vm = vm (¯z). Then, for each m ∈ Q(¯z), the set   Im = eiθ : θ ∈ [0, 2π), Re(eiθ vm ) > Re(eiθ vn ), n ∈ Q(¯z) \ {m} (3.16) is connected and open as a subset of {z ∈ O : |z| = 1}. In particular, if eiθ is such that Re(eiθ vm ) =

max

n∈Q(¯z)
{m}

then eiθ is one of the two boundary points of Im .

Re(eiθ vn ),

(3.17)

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Proof. By Assumption A4, the numbers vm , m ∈ Q(¯z), are the vertices of a strictly convex polygon P in C. Let s = |Q(¯z)| and let (v1 , . . . , vs ) be an ordering of the vertices of P in the counterclockwise direction. For m = 1, . . . , s define vm = vm − vm−1 , where we take v0 = vs . Note that, by strict convexity of P , the arguments θm of vm , i.e., numbers θm such that vm = |vm |eiθm , are such that the vectors eiθ1 , . . . , eiθs are ordered counterclockwise, with the angle between eiθm and eiθm+1 lying strictly between 0 and π for all m = 1, . . . s (again, we identify m = 1 and m = s + 1). In other words, for each m, the angles θ1 . . . , θs can be chosen in such a way that θm < θm+1 < · · · < θm+s , with 0 < θm+k − θm+k−1 < π, k = 1, . . . , s. (Again, we identified m + k with m + k − s whenever m + k > s.)   Using Jm to denote the set Jm = ie−iϑ : ϑ ∈ (θm , θm+1 ) , we claim that Im = Jm for all m = 1, . . . , s. First, let us show that Jm ⊂ Im . Let thus ϑ ∈ (θm , θm+1 ) and observe that Re(ie−iϑ vm ) = |vm | sin(ϑ − θm ) > 0,

(3.18)

because θm < ϑ < θm+1 < θm + π. Similarly, Re(ie−iϑ vm+1 ) = |vm+1 | sin(ϑ − θm+1 ) < 0,

(3.19)

because θm+1 − π < θm < ϑ < θm+1 . Consequently, Re(ie−iϑ vm ) > Re(ie−iϑ vn ) holds for both n = m + 1 and n = m − 1. It remains to show that Re(ie−iϑ vm ) > Re(ie−iϑ vn ) is true also for all remaining n ∈ Q(¯z). Let n ∈ Q(¯z) \ {m, m ± 1}. We will separately analyze the cases with θn − θm ∈ (0, π] and θn − θm ∈ (−π, 0). Suppose first that θn − θm ∈ (0, π ]. This allows us to write n = m + k for some k ∈ {2, . . . , s − 1} and estimate Re(ie−iϑ (vn − vm )) =

k


Re(ie−iϑ vm+j )

j =1

=

k


|vm+j | sin(ϑ − θm+j ) < 0.

(3.20)

j =1

The inequality holds since, in light of ϑ < θm+1 < · · · < θm+k ≤ θ + π , each sine is negative except perhaps for the last one which is allowed to be zero. On the other hand, if θn − θm ∈ (−π, 0), we write n = m − k instead, for some k ∈ {2, . . . , s − 1}, and estimate Re(ie−iϑ (vm − vn )) =

0


Re(ie−iϑ vm+j )

j =−k+1

=

0


|vm+j | sin(ϑ − θm+j ) > 0.

(3.21)

j =−k+1

Here we invoked the inequalities ϑ − π < θm−k < · · · < θm < ϑ to show that each sine on the right-hand side is strictly positive. As a consequence of the previous estimates, we conclude that Jm ⊂ Im for all m = 1, . . . , s. However, the union of all Jm ’s covers the unit circle with the exception of s points and, since the sets Im are open and disjoint, we must have Im = Jm for

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all m ∈ Q(¯z). Then, necessarily, Im is connected and open. Now the left-hand side of (3.17) is strictly greater than the right-hand side for eiθ ∈ Im , and strictly smaller than the right-hand side for eiθ in the interior of the complement of Im . By continuity of both sides, (3.17) can hold only on the boundary of Im .   3.2. Proof of Theorem 2.1. Having all the necessary tools ready, we can start proving Theorem 2.1. First we will apply Lemma 3.5 to characterize the situation around multiple points. Lemma 3.6. Suppose that Assumption A holds and let z¯ ∈ O be a multiple point. For δ > 0, let   Im(δ) = z ∈ O : |z − z¯ | = δ, Q(z)  m . (3.22) Then the following is true once δ is sufficiently small: (δ)

(1) For each m ∈ Q(¯z), the set Im is connected and has a non-empty interior. (δ) (2) Im = ∅ whenever m ∈ / Q(¯z). (δ) (δ) (2) For distinct m and n, the sets Im and In intersect in at most one point. (δ)

/ Q(¯z) once δ > 0 is sufficiently small is a direct Proof. The fact that Im = ∅ for m ∈ consequence of the continuity of the functions ζm and ζ . Indeed, if there were a sequence of points zk tending to z¯ such that a phase m were stable at each zk , then m would also be stable at z¯ . (δ) We will proceed by proving that, as δ ↓ 0, each set Im , m ∈ Q(¯z), will eventually have a non-empty interior. Let m ∈ Q(¯z). Observe that, by Lemma 3.5, there is a value eiθ (namely, a number from Im ) such that Re(eiθ vm ) > Re(eiθ vn ) for all n ∈ Q(¯z) \ {m}. But then the second part of Lemma 3.1 guarantees the existence of an

> 0 such that Q(z) = {m} for all z ∈ W ,θ (¯z)—see (3.3). In particular, the intersection W ,θ (¯z)∩{z : |z− z¯ | = δ}, which is non-empty and (relatively) open for δ < , is a subset (δ) (δ) of Im . It follows that the set Im has a nonempty interior once δ is sufficiently small. (δ) Next we will prove that each Im , m ∈ Q(¯z), is eventually connected. Suppose that (δ ) there exist a phase a ∈ Q(¯z) and a sequence δk ↓ 0 such that all sets Ia k are not con(δk ) nected. Then, using the fact that Ia has nonempty interior and thus cannot consist of just two separated points, we conclude that the phase a coexists with some other phase at at least three distinct points on each circle {z : |z − z¯ | = δk }. Explicitly, there exist (j ) (j ) (not necessarily distinct) indices bk ∈ Q(¯z) \ {a} and points (zk ), j = 1, 2, 3, with (j ) (j ) (j ) (j ) () |zk − z¯ | = δk and zk = zk for j = , such that a, bk ∈ Q(zk ). Moreover, (choos(j ) ing subsequences if needed) we can assume that bk = b(j ) for some b(j ) ∈ Q(¯z) \ {a} independent of k. Resorting again to subsequences, we also may assume that the limits in (3.1) exist for all three sequences. Let us use eiθj to denote the corresponding limits for the three sequences. First we claim that the numbers eiθj , j = 1, 2, 3, are necessarily all distinct. Indeed, suppose two of the eiθj ’s are the same and let b and c be the phases coexisting with a along the corresponding sequences. Then Lemma 3.3(1) forces b = c, which contradicts both conclusions of Lemma 3.3(2). Therefore, all three eiθj must be different. Applying now Corollary 3.2 and Lemma 3.1, we get Re(eiθj va ) = maxn∈Q(¯z)\{a} Re(eiθj vn ) for j = 1, 2, 3. According to Lemma 3.5, all three distinct numbers eiθj , j = 1, 2, 3, are

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endpoints of Ia , which is not possible since Ia is a connected subset of the unit circle. (δ) Thus, we can conclude that Ia must be connected once δ > 0 is sufficiently small. (δ) (δ) To finish the proof, we need to show that Ia ∩ Ib contains at most one point for any a = b. First note that we just ruled out the possibility that this intersection contains three distinct points for a sequence of δ’s tending to zero. (Indeed, then a would coexist with b along three distinct sequences, which would in turn imply that a and b coexists along three distinct directions, in contradiction with Lemma 3.5.) Suppose now that (δ) (δ) (δ) (δ) Ia ∩ Ib contains two distinct points. Since both Ia and Ib are connected with open (δ) (δ) interior, this would mean that Ia and Ib cover the entire circle of radius δ. Once again, (δ) applying the fact that two Im have at most two points in common, we then must have (δ) Ic = ∅ for all c = a, b. But Q(¯z) contains at least three phases which necessitates that (δ) (δ) (δ) Im = ∅ for at least three distinct m. Hence Ia ∩ Ib cannot contain more than one point.   Next we will give a local characterization of two-phase coexistence lines. Lemma 3.7. Suppose that Assumption A holds and let m, n ∈ R be distinct. Let z ∈ O be such that z ∈ Sm ∩ Sn and Q(z ) ⊂ {m, n} for z ∈ Dδ (z). Then there exist numbers δ ∈ (0, δ), t1 < 0, t2 > 0, and a twice continuously differentiable function γz : (t1 , t2 ) → Dδ (z) such that (1) γz (0) = z. (2) |ζm (γz (t))| = |ζn (γz (t))| = ζ (γz (t)), t ∈ (t1 , t2 ). (3) limt↓t1 γz (t), limt↑t2 γz (t) ∈ ∂Dδ (z). The curve t → γz (t) is unique up to reparametrization. Moreover, the set Dδ (z) \ γz (t1 , t2 ) has two connected components and m is the only stable phase in one of the components while n is the only stable phase in the other. Proof. We begin by observing that by Assumption A3, the function φm,n (x, y) = log |ζm (x + iy)| − log |ζn (x + iy)| = Re log Fm,n (x + iy),

(3.23)

has at least one of the derivatives ∂x φm,n , ∂y φm,n non-vanishing at x + iy = z. By continuity, there exists a constant η > 0 such that one of the derivatives is uniformly bounded away from zero for all z = u + iv ∈ Dη (z). Since z = x + iy ∈ Sm ∩ Sn , we have φm,n (x, y) = 0. By the implicit function theorem, there exist numbers t0 , t1 , x0 , x1 , y0 and y1 such that t0 < 0 < t1 , x0 < x < x1 , y0 < y < y1 and (x0 , x1 )×(y0 , y1 ) ⊂ Dη (z), and twice continuously differentiable functions u : (t0 , t1 ) → (x0 , x1 ) and v : (t0 , t1 ) → (y0 , y1 ) such that   φm,n u(t), v(t) = 0, (3.24) t ∈ (t0 , t1 ), and u(0) = x,

and

v(0) = y.

(3.25)

Moreover, since the second derivatives of φm,n are continuous in O and therefore bounded in Dη (z), standard theorems on uniqueness of the solutions of ODEs guarantee that the solution to (3.24) and (3.25) is unique up to reparametrization. The construction of γz is now finished by picking δ so small that Dδ (z) ⊂ (x0 , x1 )×(y0 , y1 ), and taking t0 and t1 to be the first backward and forward time, respectively, when (u(t), v(t)) leaves Dδ (z).

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The fact that Dδ (z) \ γz (t1 , t2 ) splits into two components is a consequence of the construction of γz . Moreover, γz is a (zero-)level curve of function φm,n which has a nonzero gradient. Hence, φm,n < 0 on one component of Dδ (z) \ γz (t1 , t2 ), while φm,n > 0 on the other. Recalling the assumption that Q(z ) ⊂ {m, n} for z in a neighborhood of z, the claim follows.   Now we can finally give the proof of Theorem 2.1. Proof of Theorem 2.1. Let M denote the set of all multiple points in O, i.e., let   M = z ∈ O : |Q(z)| ≥ 3 . (3.26) By Corollary 3.4, we know that M is relatively closed in O and so the set O = O \ M is open. Moreover, the set G ∩ O consists solely of points where exactly two phases coexist. Lemma 3.7 then shows that for each z ∈ G ∩ O , there exists a disc Dδ (z) and a unique, smooth γz in Dδ (z) passing through z such that Q(z ) = Q(z) for all z on the curve γz . Let γ˜z be a maximal extension of the curve γz in O . We claim that γ˜z is either a closed curve or an arc with both endpoints on ∂O . Indeed, if γ˜z were open with an end-point z˜ ∈ O , then Q(˜z) ⊃ Q(z), by continuity of functions ζm . But z˜ ∈ O and so |Q(˜z)| ≤ 2, which implies that Q(˜z) = Q(z). By Lemma 3.7, there exists a non-trivial curve γz˜ along which the two phases from Q(˜z) coexist in a neighborhood of z˜ . But then γz˜ ∪ γ˜z would be a non-trivial extension of γ˜z , in contradiction with the maximality of γ˜z . Thus we can conclude that z˜ ∈ ∂O . Let C denote the set of maximal extensions of the curves {γz : z ∈ G ∩O }. Let D ⊂ O be a compact set and note that Corollary 3.4 implies that D ∩ M is finite. Let δ0 be so small that, for each zM ∈ M ∩ D , we have Dδ0 (zM ) ⊂ O, Dδ0 (zM ) ∩ M = {zM } and the statements in Lemma 3.6 hold true for δ ≤ δ0 . Let δ ∈ (0, δ0 ]. We claim that if a curve C ∈ C intersects the disc Dδ (zM ) for a zM ∈ M ∩ D , then the restriction C ∩ Dδ (zM ) is a simple curve connecting zM to ∂Dδ (zM ). Indeed, each curve C ∈ C terminates either on ∂O or on M . If C “enters” Dδ (zM ) and does not hit zM , our assumptions about δ0 imply that C “leaves” Dδ (zM ) through the boundary. But Lemma 3.7 ensures that one of the phases coexisting along C dominates in a small neighborhood on the “left” of C , while the other dominates in a small neighborhood on the “right” of C . The only way (δ) this can be made consistent with the connectivity of the sets Im in Lemma 3.6 is by (δ) assuming that Im = ∅ only for the two m’s coexisting along C . But that still contradicts (δ) Lemma 3.6, by which Im = ∅ for at least three distinct m. Thus, once a curve C ∈ C it must terminate at zM . intersects Dδ (zM ),

Let D 0 = D \ z∈M Dδ0 (z) and let  : D 0 → [0, ∞) be a function given by  

(z) = inf δ ∈ (0, δ0 ) : Dδ (z) ⊂ O, Dδ (z) ∩ C ∈C C is disconnected . (3.27) We claim that  is bounded from below by a positive constant. Indeed,  is clearly concompact,  attains its minimum at some z ∈ D 0 . If (z) = 0, tinuous and, since D 0 is

then z is a limit point of C ∈C C and thus z ∈ C for some C ∈ C. Moreover, for infi

nitely many δ ∈ (0, δ0 ), the circle ∂Dδ (z) intersects the set C ∈C C in at least three different points. Indeed, the curve C  z provides two intersections; the third intersection

is obtained by adjusting the radius δ so that Dδ (z) ∩ C ∈C is disconnected. Thus, we are (again) able to construct three sequences (zk ), (zk ) and (zk ) such that, without loss of generality, zk , zk , zk ∈ Sa ∩Sb for some distinct a, b ∈ R (only two phases can exist in

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sufficiently small neighborhoods of points in D 0 ), |zk − z¯ | = |zk − z¯ | = |zk − z¯ | → 0, but zk = zk = zk = zk for all k. However, this contradicts Lemma 3.3, because its part (2) cannot hold simultaneously for all three pairs of sequences (zk , zk ), (zk , zk ) and (zk , zk ). Now we are ready to define the set of points z1 , . . . , z . Let be the minimum of the function  in D 0 and let δ = min(δ0 , ). Consider the following collections of open finite discs:   S1 = Dδ (z) : z ∈ M ∩ D ,  



S2 = Dδ (z) : z ∈ D ∩ C ∈C C , dist(z, D∈S1 D) > 23 δ ,  

S3 = Dδ (z) : z ∈ D , dist(z, D∈S1 ∪S2 D) > 23 δ . (3.28) It is easy to check that the union of these discs covers D . Let S = S1 ∪ S2 ∪ S3 . By compactness of D , we can choose a finite collection S ⊂ S still covering D . It remains to show that the sets A = G ∩ D for D ∈ S will have the desired properties. Let D ∈ S and let z be the center of D. If D ∈ S3 , then G ∩ D = ∅. Indeed, if z is a coexistence point, then Dδ (z ) ∈ S1 ∪ S2 and thus dist(z, z ) > δ + 23 δ and hence z ∈ D. Next, if D ∈ S2 , then z ∈ G and, by the definition of δ0 and , the disc D contains no multiple point and intersects G only in one component. This component is necessarily part of one of the curves C ∈ C. Finally, if D ∈ S1 , then z is a multiple point and, relying on our previous reasoning, several curves C ∈ C connect z to the boundary of D. Since Lemma 3.6 implies the existence of exactly |Q(z)| coexistence points on ∂D, there are exactly |Q(z)| such curves. The proof is finished by noting that every multiple point appears as the center of some disc D ∈ S , because that is how the collections (3.28) were constructed.   4. Partition Function Zeros The goal of this section is to prove Theorems 2.2-2.5. The principal tool which enables us per to control the distance between the roots of ZL and the solutions of Eqs. (2.17–2.18) or (2.24) is Rouch´e’s Theorem (see e.g. [16]). For the reader’s convenience, we transcribe the corresponding statement here: Theorem 4.1 (Rouch´e’s Theorem). Let D ⊂ C be a bounded domain with piecewise smooth boundary ∂D . Let f and g be analytic on D ∪ ∂D . If |g(z)| < |f (z)| for all z ∈ ∂D , then f and f + g have the same number of zeros in D , counting multiplicities. More details on the use of this theorem and the corresponding bounds are stated in Sect. 4.2 for the case of two-phase coexistence and in Sect. 4.4 for the case of multiple phase coexistence. Root degeneracy will be controlled using a link between the non-degeneracy conditions from Assumption B and certain Vandermonde determinants; cf. Sect. 4.1. Throughout this section, we will use the shorthand  S (Q) = S (m) (4.1) m∈Q

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to denote the set of points z ∈ O where all phases from a non-empty Q ⊂ R are “almost stable” (as quantified by > 0). 4.1. Root degeneracy. In this section we will prove Theorem 2.2. We begin with a claim about the Vandermonde matrix defined in terms of the functions (L)

bm (z) =

∂z ζm (z) (L)

ζm (z)

,

z ∈ Sκ/L (m),

(4.2)

where the dependence of bm on L has been suppressed in the notation. Let us fix a non-empty Q ⊂ R and let q = |Q|. For each z ∈ Sκ/L (Q), we introduce the q × q Vandermonde matrix M(z) with elements M,m (z) = bm (z) ,

m ∈ Q,  = 0, 1, . . . , q − 1.

(4.3)

Let M denote the 2 (Q)-norm of M (again without making the Q-dependence of this norm notationally explicit). Explicitly, M2 is defined by the supremum 
 q−1 
2
  2 M,m w ˆ m : |w ˆ m| = 1 , M = sup  2

=0 m∈Q

(4.4)

m∈Q

where (w ˆ m ) is a |Q|-dimensional complex vector. Throughout the rest of this section, the symbol  ·  will refer to the (vector or matrix) 2 -norm as specified above. The only exceptions are the p -norms q1 , q2 and q∞ of the r-tuple (qm )m∈R , which are defined in the usual way. Lemma 4.2. Suppose that Assumption B3 holds and let L˜ 0 be as in Assumption B3. For each Q ⊂ R, there exists a constant K = K(Q) < ∞ such that  −1  M (z) ≤ K, for all z ∈ Sκ/L (Q) and L ≥ L˜ 0 . (4.5) In particular, M(z) is invertible for all z ∈ Sκ/L (Q) and L ≥ L˜ 0 . Proof. Let Q ⊂ R and q = |Q|. Let us choose a point z ∈ Sκ/L (Q) and let M and bm , m ∈ Q, be the quantities M(z) and bm (z), m ∈ Q. First we note that,since M is a Vandermonde matrix, its determinant can be explicitly computed: det M = m 0 once L ≥ L˜ 0 . To estimate the matrix norm of M−1 , let λ1 , . . . , λq be the eigenvalues of the Hermitian matrix M M+ and note that λ > 0 for all  = 1, . . . , q by our lower bound on |det M|. Now, M+ 2 is equal to the spectral radius of the oper