AdS 3 wormholes from a modular bootstrap
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Springer
Received: August 5, 2020 Accepted: October 10, 2020 Published: November 12, 2020
Jordan Cotlera and Kristan Jensenb a
Society of Fellows, Harvard University, Cambridge, MA 02138, U.S.A. b Department of Physics & Astronomy, San Francisco State University, San Francisco, CA 94132, U.S.A.
E-mail: [email protected], [email protected] Abstract: In recent work we computed the path integral of three-dimensional gravity with negative cosmological constant on spaces which are topologically a torus times an interval. Here we employ a modular bootstrap to show that the amplitude is completely fixed by consistency conditions and a few basic inputs from gravity. This bootstrap is notably for an ensemble of CFTs, rather than for a single instance. We also compare the 3d gravity result with the Narain ensemble. The former is well-approximated at low temperature by a random matrix theory ansatz, and we conjecture that this behavior is generic for an ensemble of CFTs at large central charge with a chaotic spectrum of heavy operators. Keywords: AdS-CFT Correspondence, Conformal Field Theory, Conformal and W Symmetry ArXiv ePrint: 2007.15653
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP11(2020)058
JHEP11(2020)058
AdS3 wormholes from a modular bootstrap
Contents 1 Introduction
1
2 Modular bootstrap
3 7 9 10 11 14
4 Towards bootstrapping CFT ensembles
15
1
Introduction
In [1] we obtained the path integral of three-dimensional gravity with negative cosmological constant on spaces that are topologically a torus times an interval. These spaces are Euclidean wormholes, which smoothly connect two asymptotic regions with torus conformal boundaries. Those tori have independent complex structures τ1 and τ2 . The result was ZT2 ×I (τ1 , τ¯1 , τ2 , τ¯2 ) =
X 1 Im(τ1 )Im(γτ2 ) Z (τ , τ ¯ )Z (τ , τ ¯ ) , 0 1 1 0 2 2 2π 2 |τ1 + γτ2 |2 γ∈PSL(2;Z)
1 Z0 (τ, τ¯) = p , Im(τ )|η(τ )|2
(1.1)
+b where γτ = aτ cτ +d , ad − bc = 1, and η(τ ) is the Dedekind eta function. It is difficult to interpret Euclidean wormholes within the standard AdS/CFT framework [2]. Inspired by the duality [3] between Jackiw-Teitelboim gravity and a random matrix ensemble, we [1] (see also [4–7]) have made the working hypothesis that pure 3d gravity is indeed a consistent theory of quantum gravity, dual to an ensemble of conformal field theories (CFTs). Under that hypothesis, the connected ensemble average of torus partition functions is equated with a sum over Euclidean geometries which connect two asymptotic regions with torus boundary,
(1.2) or, in an equation, hZ(τ1 , τ¯1 )Z(τ2 , τ¯2 )iensemble, conn. = ZT2 ×I (τ1 , τ¯1 , τ2 , τ¯2 ) + · · · ,
–1–
(1.3)
JHEP11(2020)058
3 Universality in random CFT? 3.1 The Narain ensemble 3.1.1 Primary counting partition functions 3.1.2 Low-temperature limit 3.2 Comparison and comments
–2–
JHEP11(2020)058
where the dots refer to other connected geometries with more complicated topology. Pure 3d gravity does not have a coupling constant which suppresses fluctuat
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