Ising Model and $N=2$ Supersymmetric Theories

TL;DR

This paper reveals a deep connection between the massive Ising model and N=2 supersymmetric quantum field theories in two dimensions, linking spin correlations to geometric indices and extending known mathematical invariants.

Contribution

It establishes a direct correspondence between Ising model equations and N=2 theories, introducing a new index related to Ray-Singer torsion and geometric structures.

Findings

Identifies the same equations govern spin correlations and vacuum geometry.

Reinterprets the Ising tau-function as a new supersymmetric index.

Connects the index to Ray-Singer torsion and loop space geometry.

Abstract

We establish a direct link between massive Ising model and arbitrary massive $N=2$ supersymmetric QFT's in two dimensions. This explains why the equations which appear in the computation of spin-correlations in the non-critical Ising model are the same as those describing the geometry of vacua in $N=2$ theories. The tau-function appearing in the Ising model (i.e., the spin correlation function) is reinterpreted in the $N=2$ context as a new `index'. In special cases this new index is related to Ray-Singer analytic torsion, and can be viewed as a generalization of that to loop space of K\"ahler manifolds.