Small Instantons, Little Strings and Free Fermions

TL;DR

This paper provides evidence that BPS correlators in N=2 supersymmetric gauge theories can be described by a 2D conformal field theory, linking gauge theory, string theory, and integrable systems through explicit calculations and dualities.

Contribution

It demonstrates that the partition function of U(1) N=2 gauge theory with instantons corresponds to free fermions and the Toda hierarchy, establishing a connection to topological string theory and little strings.

Findings

Partition function matches free fermion theory on a sphere.

Effective prepotential equals topological string free energy on CP^1.

Dual string is identified with little string on fivebrane.

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 role 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.