Kac-Ward operators, Kasteleyn operators, and s-holomorphicity on arbitrary surface graphs

TL;DR

This paper extends the concept of s-holomorphicity to arbitrary weighted surface graphs, connecting it with Kac-Ward, Kasteleyn, and Dirac operators, and generalizes key Ising model results beyond planar isoradial graphs.

Contribution

It introduces a new generalized s-holomorphicity framework for surface graphs and links it to fundamental operators, broadening the scope of Ising model analysis.

Findings

s-holomorphicity criteria involving Kac-Ward, Kasteleyn, and Dirac operators

Extension of fermionic observables' properties to general surface graphs

Duality result for Kac-Ward determinants on arbitrary graphs

Abstract

The conformal invariance and universality results of Chelkak-Smirnov on the two-dimensional Ising model hold for isoradial planar graphs with critical weights. Motivated by the problem of extending these results to a wider class of graphs, we define a generalized notion of s-holomorphicity for functions on arbitrary weighted surface graphs. We then give three criteria for s-holomorphicity involving the Kac-Ward, Kasteleyn, and discrete Dirac operators, respectively. Also, we show that some crucial results known to hold in the planar isoradial case extend to this general setting: in particular, spin-Ising fermionic observables are s-holomorphic, and it is possible to define a discrete version of the integral of the square of an s-holomorphic function. Along the way, we obtain a duality result for Kac-Ward determinants on arbitrary weighted surface graphs.