Picard–Fuchs equations and Whitham hierarchy in $N=2$ supersymmetric $SU(r+1)$ Yang–Mills theory

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 one-form of the Whitham hierarchy on the Seiberg–Witten curve is represented by a finite linear combination of meromorphic one-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 one-form, their consistency conditions are found to give the Picard–Fuchs equations for the Seiberg–Witten periods.