arXiv:0912.3200v1 [math.CO] 16 Dec 2009 GRAPHS CRITICALLY EMBEDDED ON RIEMANN SURFACES AND IHARA-SELBERG FUNCTIONS: GENUS ONE CASE MARTIN LOEBL AND PETR SOMBERG Abstract. Let G =(V,E) be a bipartite graph embedded in the torus, each degree of G being two or four, and let the embedding be critical. We define, using the geometric information on this embedding, (|V |×|V |) matrix Δ. We show that the generating function of the even sets of edges of G (i.e., the Ising partition function) is a linear combination of the square roots of 4 signed Ihara-Selberg functions determined by Δ and the spin structures of the torus. 1. Introduction It is well understood (see e.g. [m], [k], and below) how to define the notion of critical embedding of a finite graph on an orientable surface, in a way so that one can derive, from the collection of the angles of the embedding called discrete conformal structure and graphs G principally studied in statistical physics (e.g. square grid), the critical values of coupling constants (edge-weights) of the dimer problem and Ising problems on G. It is an attractive task to see whether some properties associated with the notion of criticality may also be derived from the collection of data attached to critical embedding. The aim of this paper is to propose a discrete analog of the claim made by Alvarez-Gaume et all (see [a]) that the partition function of free fermion on a closed Riemann surface of genus g is a linear combination of 2 2g Pfaffians of Dirac operators. 1.1. Critical embedding. We follow [m] in this subsection. Let us consider a triple (X, H, ϕ), where X is a smooth closed Riemann surface, H a graph and ϕ : H → X an embeding. It defines a CW decomposition of X and X \ ϕ(H ) is a disjoint union of (open) faces. We fix an atlas {U i ,ϕ i : U i → C|  i U i = X } on X . We choose the realization of X as follows. For a Riemann surface X of genus g and a finite set of points {P j } j on X there exists a flat metric with conical singularities at {P j } j such that the cone angles θ Pj fulfill the Gauss-Bonet formula χ(X )= ∑ j (1 − θP j 2π ). Notice that, due to the eveness of Euler characteristic χ(X ), we can choose θ Pj to be an odd multiple of 2π for each j . Recall that the local model for a neighborhood of a conical singularity in P j is the standard cone C(θ) := {(r, β)|r ∈ R + ,β ∈ R/θR}/{(0,β) ∼ (0,β ′ )} with the metric g C(θ) := (dr) 2 + r 2 (dβ) 2 . An embedding of a graph H on a surface X defines the dual graph H ∗ (of the embedding). Note that H ∗ is an abstract graph and it has a natural embedding on X , so that each vertex of H ∗ lies on the face of the embedding of H it represents. A basic notion is that of the diamond graph: Given simultaneous embeddings of H and H ∗ , the diamond graph H ′ has the vertex-set equal to V (H ) ∪ V (H ∗ ) and the edges connecting the end-vertices of each dual pair of edges e, e ∗ into a (facial) cycle F (e) which is a 4−gon (see Figure 1). The diagonals of such a 4−gon are formed by the corresponding dual pair of edges. We are ready to define a critical embedding. Definition 1.1. The embedding ϕ of H ′ is called critical embeding if each of its faces F (e),e ∈ E(H ) is rhombus, i.e., the following conditions hold true (for all i) with respect to the induced conformal class of metrics on X : (1) The diagonals of each rhombus F (e),e ∈ E(H ) (in Im(ϕ i ◦ ϕ)) are perpendicular, (2) The lengths of sides of all rhombi F (e),e ∈ E(H ) (in Im(ϕ i ◦ ϕ)) are the same. Date : This edition: December 16, 2009 First edition: September, 2009. M.L. gratefully acknowledges partial support by Basal project Ctr. Modelamiento Matem´ atico, U. Chile. Key words and phrases: discrete conformal structure, critical embedding, Ihara-Selberg function . 1