Whitham Deformations of Seiberg-Witten Curves for Classical Gauge Groups

TL;DR

This paper extends the construction of Whitham deformations of Seiberg-Witten curves from the $SU(N+1)$ case to all classical gauge groups and related systems, revealing connections to integrable hierarchies in topologically twisted theories.

Contribution

It generalizes the explicit construction of Whitham deformations to all classical gauge groups and links these deformations to integrable hierarchies in topological quantum field theories.

Findings

Constructed Whitham deformations for all classical gauge groups.

Linked the deformations to KdV-type integrable hierarchies.

Analyzed the $u$-plane integral in the context of these deformations.

Abstract

Gorsky et al. presented an explicit construction of Whitham deformations of the Seiberg-Witten curve for the $SU(N+1)$ $\calN = 2$ SUSY Yang-Mills theory. We extend their result to all classical gauge groups and some other cases such as the spectral curve of the $A^{(2)}_{2N}$ affine Toda Toda system. Our construction, too, uses fractional powers of the superpotential $W(x)$ that characterizes the curve. We also consider the $u$-plane integral of topologically twisted theories on four-dimensional manifolds $X$ with $b_2^{+}(X) = 1$ in the language of these explicitly constructed Whitham deformations and an integrable hierarchy of the KdV type hidden behind.