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

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