Black Hole Partition Function using Hybrid Formalism of Superstrings

TL;DR

This paper computes the black hole partition function in type IIA superstring theory using a hybrid formalism, demonstrating its equivalence to the square of the topological string partition function through localization techniques.

Contribution

It introduces a hybrid superstring formalism on AdS_2 x S^2 x CY_3 and derives the black hole partition function using BRST symmetry and localization, providing a new approach to compute Z_BH.

Findings

Derivation of the black hole partition function as |Z_top|^2

Explicit construction of the sigma model action on AdS_2 x S^2

Application of localization to simplify the path integral

Abstract

The type IIA superstring partition function Z_IIA on the euclidean attractor geometry AdS_2 x S^2 x CY_3, computes the modified elliptic genus Z_BH of the associated black hole. The hybrid formalism of superstrings defined as a conformally invariant sigma model on the coset supermanifold PSU(1,1|2)/U(1)xU(1), together with Calabi-Yau and chiral boson CFTs, is used to calculate Z_IIA. The sigma model action on AdS_2 x S^2 is explicitly written in U(1)xU(1) invariant variables. The N=2 generators of AdS_2 x S^2 x CY_3 are enlarged and embedded in an N=4 topological algebra. The world sheet superconformal invariance is then used to construct a nilpotent BRST operator, in contrast to the kappa symmetry analysis used by Beasely et. al. in hep-th/0608021. The sigma model action is explicitly shown to be closed under this BRST operator. Localization arguments are then used to deform the world sheet path integral with the addition of a BRST exact term, where, contributions arise only from the center of AdS_2 and, the north and south poles of S^2. This leads to the OSV result Z_BH = Z_IIA = |Z_top|^2, where |Z_top|^2 is the square of the topological string partition function.