Why Z_{BH} = |Z_{top}|^2

TL;DR

Using an M-theory lift and localization techniques, the paper demonstrates that the black hole partition function equals the squared magnitude of the topological string partition function, providing a direct derivation of this relation.

Contribution

The paper provides a new derivation of the relation Z_BH = |Z_top|^2 using M-theory and localization, connecting black hole entropy to topological string theory.

Findings

Derivation of Z_BH = |Z_top|^2 from M-theory perspective

Localization reduces the path integral to contributions from specific points

Explicit computation confirms the perturbative relation

Abstract

It is argued, using an M-theory lift, that the IIA partition function on a euclidean AdS_2 x S^2 x CY_3 attractor geometry computes the modified elliptic genus Z_BH of the associated black hole in a large charge expansion. The partition function is then evaluated using the Green-Schwarz formalism. After localizing the worldsheet path integral with the addition of an exact term, contributions arise only from the center of AdS_2 and the north and south poles of S^2. These are the toplogical and anti-topological string partition functions Z_top and {\bar Z_top} respectively. We thereby directly reproduce the perturbative relation Z_BH = |Z_top|^2.