Prepotentials of $N=2$ Supersymmetric Gauge Theories and Soliton Equations

Abstract

Using recently proposed soliton equations we derive a basic identity for the scaling violation of $N=2$ supersymmetric gauge theories ${\sum }_i{a}_i\partial F/\partial a_i-2F=8\pi i{b}_{1}u$. Here $F$ is the prepotential, $a_i$'s are the expectation values of the scalar fields in the vector multiplet, $u=1/2\mathrm{Tr}⟨{\varphi }^2⟩$ and $b_1$ is the coefficient of the one-loop $\beta$-function. This equation holds in the Coulomb branch of all $N=2$ supersymmetric gauge theories coupled with massless matter.

10 pages, latex, no figures, a reference added

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