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Algebraic aspects of the computation of partition functions for quantumgravity and black holes in $AdS_3$ are discussed. We compute the sub-leadingquantum corrections to the Bekenstein-Hawking entropy. It is shown that thequantum corrections to the classical result can be included systematically bymaking use of the comparison with conformal field theory partition functions,via the $AdS_3/CFT_2$ correspondence. This leads to a better understanding ofthe role of modular and spectral functions, from the point of view of therepresentation theory of infinite-dimensional Lie algebras. Besides, the sum ofknown quantum contributions to the partition function can be presented in aclosed form, involving the Patterson-Selberg spectral function. Thesecontributions can be reproduced in a holomorphically factorized theory whosepartition functions are associated with the formal characters of the Virasoromodules. We propose a spectral function formulation for quantum corrections tothe elliptic genus from supergravity states.

Quantum Corrections to Bekenstein-Hawking Black Hole Entropy and Gravity Partition Functions