Grassmann Integral Topological Invariants

TL;DR

This paper demonstrates that certain Grassmann integrals representing 2D statistical models are topological invariants depending only on loop winding numbers, enabling the evaluation of partition functions and topological order parameters.

Contribution

It introduces a novel topological interpretation of Grassmann integrals in 2D statistical models, linking them to winding numbers and topological invariants.

Findings

Grassmann integrals depend only on loop winding numbers

Partition functions can be evaluated using topological invariants

Topological order parameters are derived from these invariants

Abstract

Partition functions of some two-dimensional statistical models can be represented by means of Grassmann integrals over loops living on two-dimensional torus. It is shown that those Grassmann integrals are topological invariants, which depend only on the winding numbers of the loops. The fact makes possible to evaluate the partition functions of the models and the statistical mean values of certain topological characteristics (indices) of the configurations, which behave as the (topological) order parameters.