Prepotentials of $N=2$ Supersymmetric Gauge Theories and Soliton Equations

TL;DR

This paper derives a fundamental identity relating the prepotential's scaling behavior to soliton equations in N=2 supersymmetric gauge theories, applicable across various models with massless matter.

Contribution

It introduces a new identity connecting the prepotential's scaling violation to soliton equations in N=2 theories, expanding understanding of their Coulomb branch dynamics.

Findings

Derived a basic identity for the scaling violation of the prepotential.

Established the applicability of this identity in all N=2 theories with massless matter.

Connected soliton equations to the structure of supersymmetric gauge theories.

Abstract

Using recently proposed soliton equations we derive a basic identity for the scaling violation of $N=2$ supersymmetric gauge theories $\sum_i a_i\partial F/\partial a_i-2F=8 \pi i b_1 u$. Here $F$ is the prepotential, $a_i$'s are the expectation values of the scalar fields in the vector multiplet, $u=1/2\, {\rm Tr}\, \langle\phi^2\rangle$ and $b_1$ is the coefficient of the one-loop $\beta$-function. This equation holds in the Coulomb branch of all $N=2$ supersymmetric gauge theories coupled with massless matter.