We present new evidence for the conjecture that BPS correlation functions in the N = 2 supersymmetric gauge theories are described by an auxiliary two dimensional conformal field theory. We study deformations of the N = 2 supersymmetric gauge theory by all gauge-invariant chiral operators. We calculate the partition function of the N = 2 theory on R 4 with appropriately twisted boundary conditions. For the U (1) theory with instantons (either noncommutative, or D-instantons, depending on the construction) the partition function has a representation in terms of the theory of free fermions on a sphere, and coincides with the tau-function of the Toda lattice hierarchy. Using this result we prove to all orders in string loop expansion that the effective prepotential (for U (1) with all chiral couplings included) is given by the free energy of the topological string on CP 1 .Gravitational descendants play an important rôle in the gauge fields/string correspondence. The dual string is identified with the little string bound to the fivebrane wrapped on the two-sphere. We also discuss the theory with fundamental matter hypermultiplets.February 2003 † On leave of absence from: ITEP, Moscow, 117259, Russia INTRODUCTIONThe Holy Grail of the theoretical physics is the nonperturbative theory which includes quantum gravity, sometimes called M-theory [1]. The current wisdom says there is no fundamental coupling constant. Whatever (string) perturbation theory is used depends on the particular solution one expands about. The expansion parameter is one of the geometric characteristics of the background. It is obviously interesting to look for simplified string and field theoretic models, which have string loop expansion, and where the string coupling constant has a geometric interpretation. String expansion in gauge theoryLarge N gauge theories are the most popular, and the most elusive models with string representation. In the gauge/string duality [2][3] one matches the connected correlation functions of the gauge theory observables with the partition function of the string theory in the bulk. The closed string dual has 1 N 2 as a string coupling constant. Advances in the studies of the type II string compactifications on Calabi-Yau manifolds led to another class of models, which in the low-energy limit reduce to N = 2 supersymmetric gauge theories, with a novel type of string loop expansion. Namely, certain couplings F g in the low-energy effective action are given by the genus g partition function of the topologically twisted string on Calabi-Yau. The gauge group of the N = 2 theory does not have to be U (N ) with large N . It is determined by the geometry of Calabi-Yau manifold [4][5] [6].The rôle of effective string coupling is played by the vev of the graviphoton field strength [7], which is usually assumed to be constant [8]. Generalized Scherk-Schwarz constructionIn this paper we shall explain that there exists another, natural from the gauge theory point of view, way to flesh out these couplings. The idea is to put the t...

We discuss topological theories, arising from the general N = 2 twisted gauge theories.We initiate a program of their study in the Gromov-Witten paradigm. We re-examine the low-energy effective abelian theory in the presence of sources and study the mixing between the various p-observables. We present the twisted superfield formalism which makes duality transformations transparent. We propose a scheme which uniquely fixes all the contact terms. We derive a formula for the correlation functions of p-observables on the manifolds of generalized simple type for 0 ≤ p ≤ 4 and on some manifolds with b + 2 = 1. We study the theories with matter and explore the properties of universal instanton. We also discuss the compactifications of higher dimensional theories. Some relations to sigma models of type A and B are pointed out and exploited.

We investigate a 4D analog of 2D WZW theory. The theory turns out to have surprising finiteness properties and an infinite-dimensional current algebra symmetry. Some correlation functions are determined by this symmetry. One way to define the theory systematically proceeds by the quantization of moduli spaces of holomorphic vector bundles over algebraic surfaces. We outline how one can define vertex operators in the theory. Finally, we define four-dimensional "conformal blocks" and present an analog of the Verlinde formula.

Abstract. We propose a few tests of Seiberg-Witten solutions of N = 2 supersymmetric gauge theories by the instanton calculus in twisted gauge theories. We re-examine the low-energy effective abelian theory in the presence of sources and present the formalism which makes duality transformations transparent and easily fixes all the contact terms in a broad class of theories. We also discuss ADHM integration and its relevance to the stated problems.