Abstract
We rewrite the Chern-Simons description of pure gravity on global AdS3 and on Euclidean BTZ black holes as a quantum field theory on the AdS boundary. The resulting theory is (two copies of) the path integral quantization of a certain coadjoint orbit of the Virasoro group, and it should be regarded as the quantum field theory of the boundary gravitons. This theory respects all of the conformal field theory axioms except one: it is not modular invariant. The coupling constant is 1/c with c the central charge, and perturbation theory in 1/c encodes loop contributions in the gravity dual. The QFT is a theory of reparametrizations analogous to the Schwarzian description of nearly AdS2 gravity, and has several features including: (i) it is ultraviolet-complete; (ii) the torus partition function is the vacuum Virasoro character, which is one-loop exact by a localization argument; (iii) it reduces to the Schwarzian theory upon compactification; (iv) it provides a powerful new tool for computing Virasoro blocks at large c via a diagrammatic expansion. We use the theory to compute several observables to one-loop order in the bulk, including the “heavy-light” limit of the identity block. We also work out some generalizations of this theory, including the boundary theory which describes fluctuations around two-sided eternal black holes.
Highlights
Power series expansion at small q with positive integer coefficients which count the number of states with a given energy and momentum
The resulting theory is the path integral quantization of a certain coadjoint orbit of the Virasoro group, and it should be regarded as the quantum field theory of the boundary gravitons
This prescription includes a sum over “topologies”: in order to specify a Euclidean AdS3 geometry with torus boundary one must choose a cycle of the boundary torus which is contractible in the bulk, and one sums over this choice
Summary
Consider a Lie group G with algebra g, and let b be a coadjoint vector, b ∈ g∗. Coadjoint vectors are linear maps b : g → R which take adjoint vectors v ∈ g to numbers. Recalling that the coadjoint orbit Mb is swept out under the action of the group G, consider any adjoint vector v. Denoting coordinates on the coadjoint orbit Mb as xi, one promotes the coordinates to functions of time and defines an action functional. Because this action is first-order in time derivatives, upon Legendre transformation we recover the Hamiltonian system we started with, with phase space Mb, symplectic form ω, and Hamiltonian HX. (These “gauge” transformations are “symplectomorphisms” — coordinate transformations on phase space which leave the symplectic form invariant.) the phase factor exp(iSX ) is well-defined only if the symplectic form ω has quantized periods, ω = 2πZ ,. In the remainder of this section we review the classification of coadjoint orbits of the Virasoro group and their path integral quantization. There are several recent papers related to coadjoint orbits, the Schwarzian quantum mechanics, and nearly AdS2 gravity [15, 44,45,46]
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