A procedure, allowing to calculate the coefficients of the SW prepotential in the framework of the instanton calculus is presented. As a demonstration explicit calculations for 2, 3 and 4− instanton contributions are carried out. poghos@moon.yerphi.am 1 R.F. thanks Francesco Fucito for a discussion of this point. 2 Localization without regularization renders a vanishing residuum at the corresponding critical surface.

We generalize the previously established connection between the off-shell Bethe ansatz equation for inhomogeneous SU(2) lattice vertex models in the quasiclassical limit and the solutions of the SU(2) Knizhnik-Zamolodchikov equations to the case of simple Lie algebras of higher rank.

The prepotential in N = 2 SUSY Yang-Mills theories enjoys remarkable properties. One of the most interesting is its relation to the coordinate on the quantum moduli space u = Tr ϕ 2 that results into recursion equations for the coefficients of the prepotential due to instantons. In this work we show, with an explicit multi-instanton computation, that this relation holds true at arbitrary winding numbers. Even more interestingly we show that its validity extends to the case in which gravitational corrections are taken into account if the correlators are suitably modified. These results apply also to the cases in which matter in the fundamental and in the adjoint is included. We also check that the expressions we find satisfy the chiral ring relations for the gauge case and compute the first gravitational correction.

We construct Drinfel'd twists for the rational sl(n) XXX-model giving rise to a completely symmetric representation of the monodromy matrix. We obtain a polarization free representation of the pseudoparticle creation operators figuring in the construction of the Bethe vectors within the framework of the quantum inverse scattering method. This representation enables us to resolve the hierarchy of the nested Bethe ansatz for the sl(n) invariant rational Heisenberg model. Our results generalize the findings of Maillet and Sanchez de Santos for sl(2) models.♮ on leave of absence from the Institute

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