Ja n 20 03 N = 1 SUSY inspired Whitham prepotentials and WDVV

This brief review deals with prepotentials inspired by N = 1 SUSY considerations due to Cachazo, Intrilligator, Vafa. These prepotentials associated with matrix models following Dijkgraaf and Vafa should be considered as given on an enlarged moduli space that includes Whitham times (couplings of the superpotential). This moduli space is nothing but the whole moduli space of (decorated) hyperelliptic curves. Corresponding prepotentials are logarithms of (quasiclassical) τ-functions and satisfy the WDVV equations. 1. Introductory remarks. It was realized during last years that supersymmetric gauge theories in the low-energy limit [1, 2] are described by integrable structures and, besides, can be also associated with Whitham structures [3, 4, 5, 6]. In particular, the low-energy effective action of N = 2 SUSY theory is described by a single function called prepotential [1], while the superpotential of N = 1 SUSY theory is also described by a single function [2] associated [6] with the partition function of Hermitian one-matrix model in a planar limit [7]. Both these functions are associated with the τ-functions of Whitham hierarchy and play a key role in all related integrable/Whitham etc. structures which we briefly describe below. For the sake of brevity, we always call these τ-functions prepotentials. 2. Algebraic-geometrical setup of SUSY theories. With SUSY theories in the low-energy limit, one can associate a family of auxiliary Riemann surfaces so that its moduli characterize physical moduli/parameters of SUSY theories including v.e.v.'s (vacua) and coupling constants (correlation function). Both N = 2 and N = 1 SUSY theories are characterized by two types of variables. One set of variables, {ξ i } is associated with v.e.v's of various (generally composite) fields, the dependence on this set being usually mostly concentrated on. These variables correspond to A-periods of a meromorphic 1-form dS given on the auxiliary Riemann surface. On the integrable side, these variables are nothing but the action variables of a proper integrable system so that the Jacobian of the auxiliary Riemann surface is the Liouville torus of the integrable system. For the genus g Riemann surface there are totally g such variables, therefore, one deals with an integrable system with g degrees of freedom. The defining property of the differential dS is that its variations w.r.t. moduli are holomorphic, δdS δmoduli = holomorphic (1)