Prepotential and the Seiberg-Witten theory

Abstract

Construction of the prepotential in the Seiberg-Witten theory/the Whitham hierarchy is presented. Consideration begins from the notion of quasi-classical τ-functions, which uses as an input a family of complex spectral curves with a meromorphic differential dS subject to the constraint ∂dS/ ∂(moduli) = holomorphic, and which gives as an output a homogeneous prepotential on extended moduli space. The reversed construction is then discussed, which is straightforwardly generalizable from spectral curves to certain complex manifolds of dimension d > 1 (like K3 and CY families). Examples of particular N = 2 supersymmetric gauge models are considered from the point of view of this formalism. We discuss the similarity between the WP 1,1,2,2,6 12 Calabi-Yau model with h 21 = 2 and the 1d SL(2) Calogero/Ruijsenaars model by deriving the respective Picard-Fuchs equations. We, however, stop short of the claim that they belong to the same Whitham universality class beyond the conifold limit.