A theory of reparameterizations for AdS3 gravity

Abstract

We rewrite the Chern-Simons description of pure gravity on global AdS 3 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 AdS 2 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. arXiv:1808.03263v3 [hep-th] 20 Sep 2018 7 Supersymmetry 58 8 Discussion 65 1. Eq. (1.4) is an ultraviolet-complete two-dimensional theory in its own right. It is a perfectly sensible CFT in all respects except one: it is not modular invariant. In fact, the Hilbert space of the model is just the vacuum module, which can be established by computing the torus partition function. The latter is one-loop exact on account of the localization formula, and is simply the vacuum character. (Maloney and Witten obtained the same Hilbert space from their geometric quantization.)2. The action (1.4) may be understood as a Wess-Zumino term for the Weyl and gravitational anomalies of a chiral theory with chiral central charge C. The field φ also contributes 13 to the anomaly (rather than 1, due to the P SL(2; R) quotient, which effectively removes the first oscillator), and we find a total chiral central charge c = C +13 as we explain in Section 5. Moreover, 1/C plays the role of the coupling constant of the model, so that it is weakly coupled at large central charge. Excitations of the φ field correspond to boundary gravitons, and so the model computes loop-level Witten diagrams with exchanges of the gravitational field. The shift c = C + 13 may then be understood as a one-loop renormalization of the central charge, which in fact is already visible in the direct gravitational computation of [2]. (There is no contradiction with the statement that the cosmological constant of AdS 3 gravity is unnrenormalized in the Chern-Simons formulation [6]. The shift comes from the boundary excitations rather than the renormalization of a bulk divergence.)3. The only P SL(2; R)-invariant local operators of the model are ...

Geometric actions and flat space holography

“…For the global Minkowski spacetime our results match with the partition function computed in [25] by heat kernel methods only if no regularization is performed. Using the zeta function regularization applied for the AdS 3 case in [63], we find agreement with the highest-weight characters of [81] and a quantum shift of c 1 , while c 2 ∼ 1/G N does not receive any corrections.…”

“…This is in essence the uniformizing transformation that proved useful in computing Virasoro blocks in the heavy-light limit from AdS 3 geometry [75]. It is also possible to compute the identity Virasoro blocks directly using the AS action, as was demonstrated in [63]. Hence this is a useful framework for AdS 3 holography, which is in our opinion still underexplored.…”

“…In this section we compute the flat space torus partition function from the geometric action on the BMS 3 coadjoint orbit. The approach taken here follows the one taken in [63] for AdS 3 gravity and extends their analysis to flat space. We will see that the result matches BMS 3 characters obtained in [79] (see also [80,81]) both for the one-loop contribution around the Minkoswki vacuum and for the flat space cosmologies.…”

“…There is another way of looking at possible quantum shifts of c 1 and c 2 . The authors in [63] used arguments from heat kernel computations in AdS 3 performed in [6] in order to argue that the zeta function regularization they used is consistent with the one-loop result of [6]. This basically boils down to keeping track of the divergences caused by the infinite volume near the conformal boundary.…”

Asymptotic dynamics of AdS3 gravity with two asymptotic regions

“…We have seen that the general solution of the constraint is given by (2.12), i.e., A r = ∂ r µ and A ϕ = ∂ ϕ µ for some function µ(r, ϕ). If A ϕ (r 1 , ϕ) (without zero mode) is given, the only allowed gauge transformations are µ → µ + Λ(r, ϕ) , (2.20) where the gauge parameter is constrained to reduce to a constant at the boundary, 21) in order to match the given A ϕ (r = r 1 , ϕ). But this is a proper gauge transformation since the corresponding charge vanishes,…”