Integrable Hierarchies and Contact Terms in u-plane Integrals of Topologically Twisted Supersymmetric Gauge Theories

TL;DR

This paper links contact terms in u-plane integrals of topologically twisted N=2 supersymmetric gauge theories to integrable hierarchies and their Whitham deformations, revealing a deep mathematical structure underlying physical observables.

Contribution

It introduces a novel interpretation of contact terms as multi-time tau functions of integrable hierarchies, extending previous single-time descriptions while preserving modular invariance.

Findings

Contact terms correspond to coefficients in Gaussian factors of tau functions.

Multi-time tau functions incorporate physical coupling constants as time variables.

The framework connects contact terms with derivatives of the prepotential in the theory.

Abstract

The $u$-plane integrals of topologically twisted $N = 2$ supersymmetric gauge theories generally contain contact terms of nonlocal topological observables. This paper proposes an interpretation of these contact terms from the point of view of integrable hierarchies and their Whitham deformations. This is inspired by Mari\~no and Moore's remark that the blowup formula of the $u$-plane integral contains a piece that can be interpreted as a single-time tau function of an integrable hierarchy. This single-time tau function can be extended to a multi-time version without spoiling the modular invariance of the blowup formula. The multi-time tau function is comprised of a Gaussian factor $e^{Q(t_1,t_2,...)}$ and a theta function. The time variables $t_n$ play the role of physical coupling constants of 2-observables $I_n(B)$ carried by the exceptional divisor $B$. The coefficients $q_{mn}$ of the Gaussian part are identified to be the contact terms of these 2-observables. This identification is further examined in the language of Whitham equations. All relevant quantities are written in the form of derivatives of the prepotential.