The Stringy Instanton Partition Function

TL;DR

This paper computes the exact partition function of a D1-D5 brane system on a resolved A_1 singularity, revealing connections to Nekrasov functions, Gromov-Witten invariants, and quantum cohomology.

Contribution

It provides a novel exact localization computation linking string theory, gauge theory, and algebraic geometry, including world-sheet instanton corrections and their mathematical interpretations.

Findings

Partition function reduces to Nekrasov's in the blow-down limit.

Finite radius introduces instanton corrections related to Gromov-Witten invariants.

Connections established between D-brane systems and advanced enumerative invariants.

Abstract

We perform an exact computation of the gauged linear sigma model associated to a D1-D5 brane system on a resolved A_1 singularity. This is accomplished via supersymmetric localization on the blown-up two-sphere. We show that in the blow-down limit C^2/Z_2 the partition function reduces to the Nekrasov partition function evaluating the equivariant volume of the instanton moduli space. For finite radius we obtain a tower of world-sheet instanton corrections, that we identify with the equivariant Gromov-Witten invariants of the ADHM moduli space. We show that these corrections can be encoded in a deformation of the Seiberg-Witten prepotential. From the mathematical viewpoint, the D1-D5 system under study displays a twofold nature: the D1-branes viewpoint captures the equivariant quantum cohomology of the ADHM instanton moduli space in the Givental formalism, and the D5-branes viewpoint is related to higher rank equivariant Donaldson-Thomas invariants of P^1 x C^2.