Prepotential and Instanton Corrections in N=2 Supersymmetric SU(N_1)xSU(N_2) Yang Mills Theories

TL;DR

This paper analyzes Seiberg-Witten curves for N=2 supersymmetric SU(N_1) x SU(N_2) gauge theories, developing recursive methods to compute instanton corrections to the prepotential and extending results to other gauge groups and matter representations.

Contribution

It introduces a recursive approach to calculate instanton corrections in non-hyperelliptic Seiberg-Witten theories with product gauge groups, including explicit formulas up to third order.

Findings

Derived explicit formulas for instanton corrections up to third order.

Extended methods to SU(N) theories with symmetric and antisymmetric matter.

Validated results through non-trivial consistency checks.

Abstract

In this paper we analyse the non-hyperelliptic Seiberg-Witten curves derived from M-theory that encode the low energy solution of N=2 supersymmetric theories with product gauge groups. We consider the case of a SU(N_1)xSU(N_2) gauge theory with a hypermultiplet in the bifundamental representation together with matter in the fundamental representations of SU(N_1) and SU(N_2). By means of the Riemann bilinear relations that hold on the Riemann surface defined by the Seiberg--Witten curve, we compute the logarithmic derivative of the prepotential with respect to the quantum scales of both gauge groups. As an application we develop a method to compute recursively the instanton corrections to the prepotential in a straightforward way. We present explicit formulas for up to third order on both quantum scales. Furthermore, we extend those results to SU(N) gauge theories with a matter hypermultiplet in the symmetric and antisymmetric representation. We also present some non-trivial checks of our results.