Partition functions for quantum gravity, black holes, elliptic genera and Lie algebra homologies

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 AdS3AdS3, which also describe the three-dimensional Euclidean black holes, the pure N=1N=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)SL(2,Z)) to partition functions represented by means of formal power series that encode Lie algebra properties.