A Factor-Graph Approach to Algebraic Topology, With Applications to Kramers--Wannier Duality

TL;DR

This paper introduces a factor-graph framework for algebraic topology and applies it to provide an alternative proof of Kramers--Wannier duality in the Ising model, extending to 3D and Potts models.

Contribution

It presents a novel factor-graph approach to algebraic topology and offers an alternative proof of Kramers--Wannier duality, connecting topological concepts with statistical physics models.

Findings

Factor-graph methods effectively represent algebraic topology concepts.

An alternative proof of Kramers--Wannier duality is provided.

Extensions to 3D Ising and Potts models are discussed.

Abstract

Algebraic topology studies topological spaces with the help of tools from abstract algebra. The main focus of this paper is to show that many concepts from algebraic topology can be conveniently expressed in terms of (normal) factor graphs. As an application, we give an alternative proof of a classical duality result of Kramers and Wannier, which expresses the partition function of the two-dimensional Ising model at a low temperature in terms of the partition function of the two-dimensional Ising model at a high temperature. Moreover, we discuss analogous results for the three-dimensional Ising model and the Potts model.