In my lecture I consider integrals over moduli spaces of supersymmetric gauge field configurations (instantons, Higgs bundles, torsion free sheaves ).The applications are twofold: physical and mathematical; they involve supersymmetric quantum mechanics of D-particles in various dimensions, direct computation of the celebrated Seiberg-Witten prepotential, sum rules for the solutions of the Bethe ansatz equations and their relation to the Laumon's nilpotent cone. As a by-product we derive some combinatoric identities involving the sums over Young tableaux.

We review the generalization of field theory to space-time with noncommuting coordinates, starting with the basics and covering most of the active directions of research. Such theories are now known to emerge from limits of M theory and string theory, and to describe quantum Hall states. In the last few years they have been studied intensively, and many qualitatively new phenomena have been discovered, both on the classical and quantum level. Submitted to Reviews of Modern Physics.

We study N = 2 supersymmetric four-dimensional gauge theories, in a certain N = 2 supergravity background, called the -background. The partition function of the theory in the -background can be calculated explicitly. We investigate various representations for this partition function: a statistical sum over random partitions, a partition function of the ensemble of random curves, and a free fermion correlator.These representations allow us to derive rigorously the Seiberg-Witten geometry, the curves, the differentials, and the prepotential.We study pure N = 2 theory, as well as the theory with matter hypermultiplets in the fundamental or adjoint representations, and the five-dimensional theory compactified on a circle.

Abstract. We study four dimensional N = 2 supersymmetric gauge theory in the Ω-background with the two dimensional N = 2 super-Poincare invariance. We explain how this gauge theory provides the quantization of the classical integrable system underlying the moduli space of vacua of the ordinary four dimensional N = 2 theory. The ε-parameter of the Ω-background is identified with the Planck constant, the twisted chiral ring maps to quantum Hamiltonians, the supersymmetric vacua are identified with Bethe states of quantum integrable systems. This four dimensional gauge theory in its low energy description has two dimensional twisted superpotential which becomes the Yang-Yang function of the integrable system. We present the thermodynamic-Bethe-ansatz like formulae for these functions and for the spectra of commuting Hamiltonians following the direct computation in gauge theory. The general construction is illustrated at the examples of the many-body systems, such as the periodic Toda chain, the elliptic Calogero-Moser system, and their relativistic versions, for which we present a complete characterization of the L 2 -spectrum. We very briefly discuss the quantization of Hitchin system.

We conjecture an equivalence between the Gromov-Witten theory of 3-folds and the holomorphic Chern-Simons theory of Donaldson and Thomas. For Calabi-Yau 3-folds, the equivalence is defined by the change of variables e iu = −q, where u is the genus parameter of Gromov-Witten theory and q is the Euler characteristic parameter of Donaldson-Thomas theory. The conjecture is proven for local Calabi-Yau toric surfaces.

Extending recent work of Kachru and Silverstein, we consider "orbifolds" of 4-dimensional N = 4 SU (n) super-Yang-Mills theories with respect to discrete subgroups of the SU (4) R-symmetry which act nontrivially on the gauge group. We show that for every discrete subgroup of SU (4) there is a canonical choice of imbedding of the discrete group in the gauge group which leads to theories with a vanishing one-loop beta-function. We conjecture that these give rise to (generically non-supersymmetric) conformal theories. The gauge group is ⊗ i SU (N n i ) where n i denote the dimension of the irreducible representations of the corresponding discrete group; there is also bifundamental matter, dictated by associated quiver diagrams. The interactions can also be read off from the quiver diagram. For SU (3) and SU (2) subgroups this leads to superconformal theories with N = 1 and N = 2 respectively. In the N = 1 case we prove the vanishing of the beta functions to two loops.

We show that the resolution of moduli space of ideal instantons parameterizes the instantons on non-commutative IR 4 . This moduli space appears as a Higgs branch of the theory of k D0-branes bound to N D4-branes by the expectation value of the B field. It also appears as a regularized version of the target space of supersymmetric quantum mechanics arising in the light cone description of (2, 0) superconformal theories in six dimensions. 02/98It seems natural to study all possible solutions U i to the consistency equations for the compactification of the matrix fields 1 [8].Recently, the non-commutative torus emerged as one of the solutions to (1.1) [9]. It has been argued that the parameter of non-commutativity is related to the flux of the B-field through the torus. It has been further shown in [10] that the compactification on a non-commutative torus can be thought of as a T -dual to a limit of the conventional compactification on a commutative torus. See [11] for further developments in the studies of compactifications on low-dimensional tori.On the other hand, the modified self-duality equations on the matrices in the Matrix description of fivebrane theory has been used in [12] in the study of quantum mechanics on the instanton moduli space. The modification is most easily described in the framework of ADHM equations. It makes the moduli space smooth and allows to define a six dimensional theory decoupled from the eleven-dimensional supergravity and all others M -theoretic degrees of freedom. The heuristric reason for the possibility of such decoupling is the fact that the Higgs branch of the theory is smooth and there is no place for the Coulomb branch to touch it.In this paper we propose an explanation of the latter construction in terms of noncommutative geometry. We show, that the solutions to modified ADHM equations parameterize (anti-)self-dual gauge fields on non-commutative IR 4 .1 The conjecture of [8] is that the non-abelian tensor fields in six dimensions would also appear as such solutions

This note is a short announcement of some results of a longer paper where the supersymmetric vacua of two dimensional N = 4 gauge theories with matter, softly broken by the twisted masses down to N = 2, are shown to be in one-to-one correspondence with the eigenstates of integrable spin chain Hamiltonians. The Heisenberg SU (2) XXX spin chain is mapped to the two dimensional U (N ) theory with fundamental hypermultiplets, the XXZ spin chain is mapped to the analogous three dimensional super-Yang-Mills theory compactified on a circle, the XY Z spin chain and eight-vertex model are related to the four dimensional theory compactified on T 2 .The correspondence extends to any spin group, representations, boundary conditions, and inhomogeneity, it includes Sinh-Gordon and non-linear Schrödinger models as well as the dynamical spin chains such as the Hubbard model. Compactifications of four dimensional N = 2 theories on a two-sphere lead to the instanton-corrected Bethe equations. We propose a completely novel way for the Yangian, quantum affine, and elliptic algebras to act as a symmetry of a union of quantum field theories.a On leave of absence from ITEP, Moscow, Russia for discussions. The results of this note, as well as those in [14], were presented at various conferences and workshops 3 and we thank the organizers for the opportunity to present our results. We thank various agencies and institutions 4 for supporting this research. The gauge theoryHere we give a brief review of the relevant gauge theories. 3 The IHES seminars and the theoretical physics conference dedicated to the 50th anniversary of IHES (Bures-sur-Yvette,