The Whitham Deformation of the Dijkgraaf-Vafa Theory

TL;DR

This paper explores the Whitham deformation of the effective superpotential in Dijkgraaf-Vafa theory, linking it to hyperelliptic curves and analyzing how different meromorphic differentials induce various deformations, with applications to N=1* theory.

Contribution

It introduces the Whitham deformation framework for the DV superpotential and analyzes its effects through the elliptic case and hodograph solutions, providing new insights into the deformation parameters.

Findings

The superpotential deformation depends on chosen meromorphic differentials.

Different differentials lead to distinct superpotential deformations.

Physical interpretation of deformation parameters is provided.

Abstract

We discuss the Whitham deformation of the effective superpotential in the Dijkgraaf-Vafa (DV) theory. It amounts to discussing the Whitham deformation of an underlying (hyper)elliptic curve. Taking the elliptic case for simplicity we derive the Whitham equation for the period, which governs flowings of branch points on the Riemann surface. By studying the hodograph solution to the Whitham equation it is shown that the effective superpotential in the DV theory is realized by many different meromorphic differentials. Depending on which meromorphic differential to take, the effective superpotential undergoes different deformations. This aspect of the DV theory is discussed in detail by taking the N=1^* theory. We give a physical interpretation of the deformation parameters.