WDVV Equations in Seiberg-Witten theory and associative algebras1

1. What is WDVV. More than two years ago N.Seiberg and E.Witten [1] proposed a new way to deal with the low-energy effective actions of N = 2 four-dimensional supersymmetric gauge theories, both pure gauge theories (i.e. containing only vector supermultiplet) and those with matter hypermultiplets. Among other things, they have shown that the low-energy effective actions (the end-points of the renormalization group flows) fit into universality classes depending on the vacuum of the theory. If the moduli space of these vacua is a finite-dimensional variety, the effective actions can be essentially described in terms of system with finite-dimensional phase space (# of degrees of freedom is equal to the rank of the gauge group), although the original theory lives in a many-dimensional space-time. These effective theories turn out to be integrable. Integrable structure behind the Seiberg-Witten (SW) approach has been found in [2] and later examined in detail in [3].