Partition Functions for Quantum Gravity, Black Holes, Elliptic Genera and Lie Algebra Homologies

TL;DR

This paper explores the deep connections between quantum field theory generating functions, Lie algebra homologies, and modular forms, with applications to three-dimensional quantum gravity, black holes, supergravity, and sigma models.

Contribution

It introduces a homological framework linking partition functions in quantum gravity and black hole physics to Lie algebra structures and modular forms, expanding the mathematical understanding of these theories.

Findings

Homological interpretation of partition functions in 3D gravity and black holes.

Application of modular forms and spectral functions to quantum field theory.

Connections between elliptic genera, Lie algebra homologies, and supersymmetric sigma models.

Abstract

There is a remarkable connection between quantum generating functions of field theory and formal power series associated with dimensions of chains and homologies of suitable Lie algebras. We discuss the homological aspects of this connection with its applications to partition functions of the minimal three-dimensional gravities in the space-time asymptotic to AdS3, which also describe the three-dimensional Euclidean black holes, the pure N = 1 supergravity, and a sigma model on N-fold generalized symmetric products. We also consider in the same context elliptic genera of some supersymmetric sigma models. These examples can be considered as a straightforward application of the machinery of modular forms and spectral functions (with values in the congruence subgroup of SL(2, Z)) to partition functions represented by means of formal power series that encode Lie algebra properties.