Instanton on toric singularities and black hole countings

TL;DR

This paper calculates the instanton partition functions for ${ m ext{N}=4}$ U(N) gauge theories on toric varieties, providing microscopic insights into black hole entropy from D-brane bound states, including regular and fractional instanton contributions.

Contribution

It introduces explicit formulas for instanton partition functions on toric singularities, connecting microscopic gauge theory calculations with black hole entropy in string theory.

Findings

Partition functions include regular and fractional instantons.

Fractional instantons match recent 2d SYM results.

Large charge limit reproduces supergravity entropy formulas.

Abstract

We compute the instanton partition function for ${\cal N}=4$ U(N) gauge theories living on toric varieties, mainly of type $\R^4/\Gamma_{p,q}$ including $A_{p-1}$ or $O_{\PP_1}(-p)$ surfaces. The results provide microscopic formulas for the partition functions of black holes made out of D4-D2-D0 bound states wrapping four-dimensional toric varieties inside a Calabi-Yau. The partition function gets contributions from regular and fractional instantons. Regular instantons are described in terms of symmetric products of the four-dimensional variety. Fractional instantons are built out of elementary self-dual connections with no moduli carrying non-trivial fluxes along the exceptional cycles of the variety. The fractional instanton contribution agrees with recent results based on 2d SYM analysis. The partition function, in the large charge limit, reproduces the supergravity macroscopic formulae for the D4-D2-D0 black hole entropy.