One-loop partition functions of 3D gravity

Abstract

We consider the one-loop partition function of free quantum field theory in locally Anti-de Sitter space-times. In three dimensions, the one loop determinants for scalar, gauge and graviton excitations are computed explicitly using heat kernel techniques. We obtain precisely the result anticipated by Brown and Henneaux: the partition function includes a sum over "boundary excitations" of AdS 3 , which are the Virasoro descendants of empty Anti-de Sitter space. This result also allows us to compute the one-loop corrections to the Euclidean action of the BTZ black hole as well its higher genus generalizations.

Comments on Rényi entropy in AdS3/CFT2

“…Because the higher spin currents do not contribute at O(c) as described earlier, this is in fact their leading contribution. Following [18], we compute these 1-loop determinants in the small interval expansion using known formulas for handlebodies [21] and spin-s gauge fields [22], and find complete agreement with CFT. That is, if S (s) n is the contribution to the CFT Rényi entropy from the pair of spin-s currents, and S (s) n (M) is the holographic contribution to the Rényi entropy obtained from linearized spin-s gauge 2 The analog of (1.2) was only proven in [2,3] to hold for a noncompact CFT in its ground state and, implicitly, for all states related by conformal transformations.…”

“…In the bulk, the idea is straightforward: AdS/CFT instructs us to compute 1-loop determinants on the handlebody geometries, for whatever bulk fields are present. The formula for such determinants is known for a generic handlebody solution [21]:…”

“…There are already two proposals [12,13] for computing single interval entanglement entropy in generic backgrounds of sl(N,R) × sl(N,R) Chern-Simons theories. 21 These prescriptions are quite natural from a Chern-Simons perspective: they are given in terms of Wilson lines for the bulk connection. An extension to Rényi entropies and to multiple intervals begs to be discovered.…”

“…In some regions of the moduli space, the conclusions herein may hold, but not in other regions where the CFT fails to obey our assumptions. It would be instructive to try 21 They appear to be distinct, but give the same answers for the handful of backgrounds in which they have been evaluated explicitly. This coincidence has yet to be understood.…”

The heat kernel on AdS 3 and its applications

“…As an immediate application of these results we are able to evaluate the one loop contribution from the physical spin 3 2 gravitino in, for example, N = 1 supergravity on thermal AdS 3 . This one loop result together with the answer for the spin two graviton combines into left-and right-moving super-Virasoro characters for the identity representation…”