MORE EVIDENCE FOR THE WDVV EQUATIONS IN ${\mathcal N} = 2$ SUSY YANG–MILLS THEORIES

Abstract

We consider 4d and 5d N = 2 supersymmetric theories and demonstrate that in general their Seiberg-Witten prepotentials satisfy the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. General proof for the Yang-Mills models (with matter in the first fundamental representation) makes use of the hyperelliptic curves and underlying integrable systems. A wide class of examples is discussed, it contains few understandable exceptions. In particular, in perturbative regime of 5d theories in addition to naive field theory expectations some extra terms appear, like it happens in heterotic string models. We consider also the example of the Yang-Mills theory with matter hypermultiplet in the adjoint representation (related to the elliptic Calogero-Moser system) when the standard WDVV equations do not hold.

Whitham Deformations of Seiberg–witten Curves for Classical Gauge Groups

“…The relation between the Seiberg-Witten theory and d < 1 topological strings has been studied from several aspects, such as the WDVV equations [46,47], flat coordinates and Gauss-Manin systems [48,49], etc. Of course the very notion of prepotentials itself is a bridge connecting the two worlds.…”

Picard–Fuchs ordinary differential systems in N=2 supersymmetric Yang–Mills theories

“…Reduction of such massive differential was also recognized by Marshakov et al 28 in their construction of massive WDVV equations.…”

Second Order Reductions of the WDVV Equations Related to Classical Lie Algebras

“…The duality transformations comprise a group of contact symmetries acting as constant symplectic transformations on a vector containing the coordinates x i as one half of its indices and the first order derivatives F i of the prepotential as the other half. It is readily observed that the reductions (6), (7) and (14), (15) are not invariant under general linear coordinate changes nor general symplectic transformations. This implies among other things that the choice of basis in the root space is relevant for the reduction one obtains.…”

“…Generalizations, not requiring F 1 to be constant, have been introduced and studied in the context of four-and five-dimensional N = 2 supersymmetric gauge theory (see e.g. [6,7], [9].…”