Exact information in N=2 theories

TL;DR

This paper discusses how to compute an exact supersymmetric index in two-dimensional N=2 theories, revealing insights into soliton spectra and establishing connections with integrable models and classical equations.

Contribution

It introduces an exact method to compute the supersymmetric index in 2D N=2 theories, linking it to soliton spectra and integrable systems.

Findings

Exact supersymmetric index computed for N=2 theories

Index relates to soliton spectrum and polymer loop partition function

Equivalence between integral equations and classical sinh-Gordon and Toda equations

Abstract

This is intended to be a simple discussion of work done in collaboration with S. Cecotti, K. Intriligator and C. Vafa; and with H. Saleur. I discuss how $ Tr F (-1)^F e^{-\beta H}$ can be computed exactly in any N=2 supersymmetric theory in two dimensions. It gives exact information on the soliton spectrum of the theory, and corresponds to the partition function of a single self-avoiding polymer looped once around a cylinder of radius $\beta$. It is independent of almost all deformations of the theory, and satisfies an exact differential equation as a function of $\beta$. For integrable theories it can also be computed from the exact S-matrix. This implies a highly non-trivial equivalence of a set of coupled integral equations with the classical sinh-Gordon and the affine Toda equations.