  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.
