Towards exact holography in AdS3

Abstract

In this thesis, we develop the program of supersymmetric localization for the compu- tation of the functional integral of string theory on AdS3×S2. We are placed in the framework of off-shell 5d N = 2 supergravity coupled to vector multiplets. We first present how to set up a consistent Euclidean version of this theory. We then show how the condition of supersymmetry in the Euclidean H3/Z×S2 geometry naturally leads to a twist of the S2 around the time direction of AdS3. The twist gives us a five-dimensional Euclidean supergravity background which is dual to the elliptic genus of (0, 4) SCFT2 at the semiclassical level. On this background we set up the off-shell BPS equations for one of the Killing spinors, such that the functional inte- gral of five-dimensional Euclidean supergravity on H3/Z×S2 localizes to its space of solutions. We obtain a class of solutions to these equations by lifting known off-shell BPS solutions of four-dimensional Euclidean supergravity on AdS2×S2. In order to do this consistently, we construct and use a Euclidean version of the off-shell 4d/5d lift of arxiv:1112.5371, which could be of independent interest. We then assess the consistency of these localization solutions with the standard AdS3×S2 bound- ary conditions on which the functional integral is defined. We find that the off-shell gauge fields respect their usual conditions, but that the off-shell metric in the AdS3 directions is not compatible with the Brown-Henneaux conditions. We show instead that the metric fluctuations are consistent with a set of chiral boundary conditions recently constructed by Compere, Strominger and Song (CSS) in arxiv:1303.2662.
We subsequently use this observation to propose a partial set of boundary terms for the 5d supergravity derived from these boundary conditions. We evaluate the bulk ac- tion and these boundary terms on the localization solutions, which yields a finite and tractable expression. Lastly, we perform a numerical search for additional localization solutions in the space of asymptotic metrics obeying CSS or Brown-Henneaux bound- ary conditions, using recursive methods analogous to those employed in holographic renormalization.

Date of Award	1 Jul 2023
Original language	English
Awarding Institution	King's College London
Supervisor	Dionysios Anninos (Supervisor) & Sameer Murthy (Supervisor)