Small Instantons, Little Strings and Free Fermions

Abstract

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...

The Topological Vertex

“…In the next section we explain how to interpret the free chiral boson as the relevant field for the Kodaira-Spencer theory of gravity [8] in this local context (related ideas have appeared in [23,24]). …”

BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters

“…It is based on the observation that the supersymmetric partition functions [94] are the remarkable special functions, which generalize all the known special functions given by the periods, matrix integrals, matrix elements of group, Kac-Moody, and quantum group representations etc. [67,84,94]. The particular implementations of this correspondence are well-known under the names of the AGT conjecture [2,120], and the Bethe/gauge correspondence [89,96] (see [43,44,74] for the prior work).…”

Gauge theories as matrix models

“…This form of the curve is especially natural from the point of view of comparison with the conformal representation [5] in terms of a theory of two-dimensional free scalar field…”

“…Simultaneously, they are beautiful objects of pure interest for the mathematical physics, and have been recently re-derived as quasiclassical limit of the Nekrasov instanton partition functions [4,5,6,7,8,11,12]. The latter ones have the form of statistical models, where summing is taken over the sets of random partitions or Young diagrams [4,13].…”