Quantum Black Holes, Elliptic Genera and Spectral Partition Functions

TL;DR

This paper explores the connections between quantum black hole partition functions, elliptic genera, and spectral functions within the framework of M-theory, D-branes, and the AdS/CFT correspondence, revealing modular properties and proposing a link to elliptic cohomology.

Contribution

It introduces new calculations of elliptic genera on Calabi-Yau threefolds and D-branes, and conjectures a novel connection between black hole partition functions and elliptic cohomology.

Findings

New supergravity elliptic genus calculations on Calabi-Yau threefolds.

Elliptic genera for equivariant D-branes on toric singularities.

Proposed link between black hole partition functions and elliptic cohomology.

Abstract

We study M-theory and D-brane quantum partition functions for microscopic black hole ensembles within the context of the AdS/CFT correspondence in terms of highest weight representations of infinite-dimensional Lie algebras, elliptic genera, and Hilbert schemes, and describe their relations to elliptic modular forms. The common feature in our examples lie in the modular properties of the characters of certain representations of the pertinent affine Lie algebras, and in the role of spectral functions of hyperbolic three-geometry associated with q-series in the calculation of elliptic genera. We present new calculations of supergravity elliptic genera on local Calabi-Yau threefolds in terms of BPS invariants and spectral functions, and also of equivariant D-brane elliptic genera on generic toric singularities. We use these examples to conjecture a link between the black hole partition functions and elliptic cohomology.