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

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.

40 pages

References in corpus (10)

• A Farey Tail for Attractor Black Holes
• Comments on AdS2 solutions of D=11 Supergravity
• AdS_3 Partition Functions Reconstructed
• Fluxes, Brane Charges and Chern Morphisms of Hyperbolic Geometry
• Truncated Heat Kernel and One-Loop Determinants for the BTZ Geometry
• Partition Functions of Three-Dimensional Quantum Gravity and the Black Hole Entropy
• Expository Remarks on Three-Dimensional Gravity and Hyperbolic Invariants
• Homology and K--Theory Methods for Classes of Branes Wrapping Nontrivial Cycles
• AdS Solutions in Gauge Supergravities and the Global Anomaly for the Product of Complex Two-Cycles
• Topological Quantum Field Theories, Strings, and Orbifolds