WDVV-like equations in N=2 SUSY Yang-Mills Theory

Abstract

The prepotential $F(a_i)$, defining the low-energy effective action of the $SU(N)$ $\mathcal{N}=2$ SUSY Yang-Mills theories, satisfies an enlarged set of the WDVV-like equations $F_i{F}_k^{-1}{F}_j=F_j{F}_k^{-1}{F}_i$ for any triple $i,j,k=1,\dots ,N-1$, where matrix $F_i$ is equal to $(F_i{)}_{mn}={\partial }^{3}F/\partial a_i\partial a_m\partial a_n$. The same equations are actually true for generic topological theories. In contrast to the conventional formulation, when $k$ is restricted to $k=0$, in the proposed system there is no distinguished ``first'' time-variable, and indices can be raised with the help of any ``metric'' ${\eta }_{mn}^{(k)}=(F_k{)}_{mn}$, not obligatory flat. All the equations (for all $i,j,k$) are true simultaneously. This result provides a new parallel between the Seiberg-Witten theory of low-energy gauge models in $4d$ and topological theories.

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