Integrable systems and supersymmetric gauge theory

Abstract

After the work of Seiberg and Witten, it has been seen that the dynamics of N=2 Yang-Mills theory is governed by a Riemann surface $\Sigma$. In particular, the integral of a special differential $\lambda_{SW}$ over (a subset of) the periods of $\Sigma$ gives the mass formula for BPS-saturated states. We show that, for each simple group $G$, the Riemann surface is a spectral curve of the periodic Toda lattice for the dual group, $G^\vee$, whose affine Dynkin diagram is the dual of that of $G$. This curve is not unique, rather it depends on the choice of a representation $\rho$ of $G^\vee$; however, different choices of $\rho$ lead to equivalent constructions. The Seiberg-Witten differential $\lambda_{SW}$ is naturally expressed in Toda variables, and the N=2 Yang-Mills pre-potential is the free energy of a topological field theory defined by the data $\Sigma_{\gg,\rho}$ and $\lambda_{SW}$.Comment: 20 pages, latex, 3 uuencoded figures (needs epsf.tex); minor errors corrected, references adde

Nonperturbative Superpotentials and Compactification to Three Dimensions

“…Therefore, it is in many ways natural to identify the gauge invariant quantities 1 k+1 Tr(Φ k+1 ) with the conserved quantities u k = 1 k+1 Tr(M k ) of the dynamical system. This is also what the original discussion of the role of the integrable system in N = 2 theories in four dimensions implies [7,8]. Therefore, we conjecture, following [20,21,22] that the quantum superpotential can be obtained from the classical one by the rule…”

“…In four dimensions the gauge symmetry was broken according to (8), and since the complex structure of the moduli space did not depend on R, we expect that for every finite value of R this remains true. Therefore, we expect that the moduli space collapses to a finite collection of tori of dimension 2k.…”

“…This integrable system is the periodic Toda chain, and its relation to the Seiberg-Witten curve and N = 2 theories in four dimensions was found in [7,8,9]. The periodic Toda chain is described by the Hamiltonian…”

“…It accordingly never appears in the superpotential. The remaining U(1) k−1 gauge symmetry that is left unbroken (see (8)) will emerge as k − 1 free parameters that are left after extremizing the superpotential. In other words, the superpotential will not have isolated vacua in general, but will have moduli spaces of vacua of real dimension 2(k − 1).…”

“…Another question is whether the matrix model approach, or a suitable modification thereof, is applicable to all possible gauge theories with all possible matter content. One may also wonder whether the integrable system that underlies the matrix model has any relation to the integrable system that underlies N = 2 gauge theories [7,8,9].…”