Discrete Dirac Operators, Critical Embeddings and Ihara-Selberg Functions

作者：Loebl Martin^*; Somberg Petr

来源：Electronic Journal of Combinatorics, 2015, 22(1).

DOI：10.37236/3741

• discrete conformal structure
• critical embedding
• Ihara-Selberg function
• discrete Dirac operator
• Ising partition function

Tine aim the paper is to formulate a discrete analogue of the claim made by Alvarez-Gaume et al., ([1]), realizing the partition function of the free fermion on a closed Riemann surface of genus g as a linear combination of 2(2g) Pfaffians of Dirac operators. Let G = (V, E) be a finite graph embedded in a closed Riemann surface X of genus g, x(e) the collection of independent variables associated with each edge e of G (collected in one vector variable x) and Sigma the set of all 2(2g) spin-structures on X. We introduce 2(2g) rotations rot(s) and (2|E| x 2|E|) matrices Delta(s) (x), s is an element of Sigma, of the transitions between the oriented edges of G determined by rotations rot(s). We show that the generating function for the oven subsets of edges of G, i.e., the Ising partition function, is a linear combination of the square roots of 2(2g) Ihara-Selberg functions I(Delta(s)(x)) also called Feynman functions. By a result of Foata-Zeilberger holds I(Delta(s)(x)) = det (I - Delta'(s)(x)), where Delta'(s)(x) is obtained from Delta(s)(x) by replacing some entries by 0. Thus each Feynman function is computable in polynomial time. We suggest that in the case of critical embedding of a bipartite graph G, the Feynman functions provide suitable discrete analogues for the Pfaffians of discrete Dirac operators.

• 出版日期2015-1-9