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Exact Path Integral for 3D Quantum Gravity II

Continuing the work arXiv:1504.05991, we discuss various aspects of three dimensional quantum gravity partition function in AdS in the semi-classical limit. The partition function is holomorphic and is the one which we obtained by using the localization technique of Chern-Simons theory in arXiv:1504.05991. We obtain a good expression for it in the summation form over Virasoro characters for the vacuum and primaries. A key ingredient for that is an interpretation of boundary localized fermion. We also check that the coefficients in the summation form over Virasoro characters of the partition function are positive integers and satisfy the Cardy formula. These give physical interpretation that these coefficients represent the number of primary fields in the dual CFT in the large k limit.