Tau-functions for Quiver Gauge Theories

TL;DR

This paper explores the structure of prepotentials in quiver supersymmetric gauge theories, showing their relation to tau-functions and isomonodromic deformations, with explicit computations for SU(2) cases.

Contribution

It establishes the connection between quiver gauge theory prepotentials and tau-functions, proving gradient formulas via Riemann bilinear relations and analyzing their dependence on bare couplings.

Findings

Prepotentials are quasiclassical tau-functions depending on UV parameters and IR condensates.

Dependence on bare couplings matches tau-function derivatives for isomonodromic deformations.

Explicit computations for SU(2) theories with matter are provided and analyzed.

Abstract

The prepotentials for the quiver supersymmetric gauge theories are defined as quasiclassical tau-functions, depending on two different sets of variables: the parameters of the UV gauge theory or the bare compexified couplings, and the vacuum condensates of the theory in IR. The bare couplings are introduced as periods on the UV base curve, and the consistency of corresponding gradient formulas for the tau-functions is proven using the Riemann bilinear relations. It is shown, that dependence of generalised prepotentials for the quiver gauge theories upon the bare couplings turns to coincide with the corresponding formulas for the derivatives of tau-functions for the isomonodromic deformations. Computations for the SU(2) quiver gauge theories with bi- and tri-fundamental matter are performed explicitly and analysed in the context of 4d/2d correspondence.