A New Supersymmetric Index

TL;DR

This paper introduces a new supersymmetric index for N=2 theories in two dimensions, linking it to vacuum geometry and exact differential equations, with computational methods for integrable cases.

Contribution

It defines a novel supersymmetric index that remains invariant under deformations and connects it to geometric and integrable structures in the theory.

Findings

The index is independent of most deformations.

The index relates to Berry's curvature of vacua.

Exact differential equations govern the index as a function of eta.

Abstract

We show that ${\rm Tr}(-1)^F F e^{-\beta H}$ is an index for $N$=2 supersymmetric theories in two dimensions, in the sense that it is independent of almost all deformations of the theory. This index is related to the geometry of the vacua (Berry's curvature) and satisfies an exact differential equation as a function of $\beta$. For integrable theories we can also compute the index thermodynamically, using the exact $S$-matrix. The equivalence of these two results implies a highly non-trivial equivalence of a set of coupled integral equations with these differential equations, among them Painleve III and the affine Toda equations.