Chiral observables and S-duality in N = 2* U(N) gauge theories

TL;DR

This paper computes quantum expectation values of chiral ring elements in N=2* U(N) gauge theories using localization, revealing their modular properties under S-duality and providing explicit instanton contributions.

Contribution

It introduces a method to express chiral ring expectations as quasi-modular forms and confirms these results via spectral curve comparisons, offering new exact instanton formulas.

Findings

Chiral ring elements form quasi-modular expansions under S-duality.

Constructed combinations transform as modular forms of definite weight.

Provided an exact expression for the 1-instanton contribution.

Abstract

We study N = 2* theories with gauge group U(N) and use equivariant localization to calculate the quantum expectation values of the simplest chiral ring elements. These are expressed as an expansion in the mass of the adjoint hypermultiplet, with coefficients given by quasi-modular forms of the S-duality group. Under the action of this group, we construct combinations of chiral ring elements that transform as modular forms of definite weight. As an independent check, we confirm these results by comparing the spectral curves of the associated Hitchin system and the elliptic Calogero-Moser system. We also propose an exact and compact expression for the 1-instanton contribution to the expectation value of the chiral ring elements.