Instanton calculus in R-R background and the topological string

TL;DR

This work constructs a string-theoretic realization of four-dimensional ${\mathcal{N}}=2$ gauge theory instantons using a D3/D(--1) system at a $\mathbb{Z}_2$ orbifold, connecting ADHM moduli to D-brane degrees of freedom. It shows that a constant self-dual graviphoton background induces the same localization as Nekrasov's Omega-background, enabling exact computation of multi-instanton partition functions and the non-perturbative prepotential, including gravitational F-terms. By promoting the graviphoton deformation to a Weyl superfield, the authors derive an expansion of the prepotential in powers of $W^+$ whose coefficients match higher-genus topological-string amplitudes, thus linking gauge theory non-perturbative effects to topological string theory. Overall, the paper provides a unified, string-based derivation of Nekrasov's results and their topological-string interpretation, with implications for geometric engineering and non-perturbative dynamics in ${\mathcal{N}}=2$ theories.

Abstract

We study a system of fractional D3 and D(-1) branes in a Ramond-Ramond closed string background and show that it describes the gauge instantons of N=2 super Yang-Mills theory and their interactions with the graviphoton of N=2 supergravity. In particular, we analyze the instanton moduli space using string theory methods and compute the prepotential of the effective gauge theory exploiting the localization methods of the instanton calculus showing that this leads to the same information given by the topological string. We also comment on the relation between our approach and the so-called Omega-background.