Dirac operator and Ising model on a compact 2D random lattice

TL;DR

This paper explores the formulation of fermionic fields and the Ising model on a randomly triangulated 2D manifold, addressing topological issues, duality, and the role of GSO projection in a lattice setting.

Contribution

It provides an explicit construction of fermionic fields on a random lattice and connects this to the Ising model, including an exact realization of Kramers-Wannier duality.

Findings

Explicit construction of fermionic fields on a random 2D lattice

Demonstration of Kramers-Wannier duality in this setting

Highlighting the necessity of GSO projection for duality

Abstract

Lattice formulation of a fermionic field theory defined on a randomly triangulated compact manifold is discussed, with emphasis on the topological problem of defining spin structures on the manifold. An explicit construction is presented for the two-dimensional case and its relation with the Ising model is discussed. Furthermore, an exact realization of the Kramers-Wannier duality for the two-dimensional Ising model on the manifold is considered. The global properties of the field are discussed. The importance of the GSO projection is stressed. This projection has to be performed for the duality to hold.