Picard-Fuchs Equations and Whitham Hierarchy in N=2 Supersymmetric SU(r+1) Yang-Mills Theory

TL;DR

This paper explores the connection between Picard-Fuchs equations and the Whitham hierarchy in N=2 supersymmetric SU(r+1) Yang-Mills theory, revealing how differential relations among Seiberg-Witten differentials lead to key equations governing the theory.

Contribution

It demonstrates how the Picard-Fuchs equations naturally arise from the differential relations among Whitham differentials in the finite Whitham hierarchy of N=2 SU(r+1) Yang-Mills theory.

Findings

Derivation of Picard-Fuchs equations from Whitham hierarchy relations

Finite system description of Whitham dynamics in N=2 theories

Identification of differential relations among Seiberg-Witten differentials

Abstract

In general, Whitham dynamics involves infinitely many parameters called Whitham times, but in the context of N=2 supersymmetric Yang-Mills theory it can be regarded as a finite system by restricting the number of Whitham times appropriately. For example, in the case of SU(r+1) gauge theory without hypermultiplets, there are r Whitham times and they play an essential role in the theory. In this situation, the generating meromorphic 1-form of the Whitham hierarchy on Seiberg-Witten curve is represented by a finite linear combination of meromorphic 1-forms associated with these Whitham times, but it turns out that there are various differential relations among these differentials. Since these relations can be written only in terms of the Seiberg-Witten 1-form, their consistency conditions are found to give the Picard-Fuchs equations for the Seiberg-Witten periods.