Averaging over Narain moduli space
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Springer
Received: July 17, 2020 Accepted: September 27, 2020 Published: October 28, 2020
Alexander Maloneya and Edward Wittenb a
Department of Physics, McGill University, Montreal, QC H3A 2T8, Canada b Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, U.S.A.
E-mail: [email protected], [email protected] Abstract: Recent developments involving JT gravity in two dimensions indicate that under some conditions, a gravitational path integral is dual to an average over an ensemble of boundary theories, rather than to a specific boundary theory. For an example in one dimension more, one would like to compare a random ensemble of two-dimensional CFT’s to Einstein gravity in three dimensions. But this is difficult. For a simpler problem, here we average over Narain’s family of two-dimensional CFT’s obtained by toroidal compactification. These theories are believed to be the most general ones with their central charges and abelian current algebra symmetries, so averaging over them means picking a random CFT with those properties. The average can be computed using the Siegel-Weil formula of number theory and has some properties suggestive of a bulk dual theory that would be an exotic theory of gravity in three dimensions. The bulk dual theory would be more like U(1)2D Chern-Simons theory than like Einstein gravity. Keywords: AdS-CFT Correspondence, Conformal Field Theory, Gauge-gravity correspondence ArXiv ePrint: 2006.04855
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP10(2020)187
JHEP10(2020)187
Averaging over Narain moduli space
Contents 1 Introduction Siegel-Weil formula in genus one A practice case: D = 1 The Siegel-Weil formula for higher D Gravitational interpretation of the formula Adding more information about the CFT
7 7 9 11 13
3 Higher genus and disconnected boundaries 3.1 Higher genus 3.2 Disconnected boundaries 3.3 Genus zero
18 18 21 26
4 Path integrals in gauge theory 4.1 Preliminaries 4.2 Path integral on a handlebody 4.3 Disconnected boundaries 4.4 U(1)2D and R2D Chern-Simons theories
27 27 29 33 35
A Derivation of the Siegel-Weil formula at D > 1 and g > 1 A.1 Narain moduli space A.2 Geometry of Siegel upper half space A.3 The average CFT partition function at genus g
37 37 38 40
1
Introduction
A simple model of gravity in two dimensions — JT gravity — is dual to a random ensemble of quantum mechanical systems, rather than a specific quantum mechanical system [1]. It is natural to wonder if something similar happens in higher dimensions. For example, gravity is still relatively simple in three spacetime dimensions, at least from some points of view. Are there simple theories of gravity in three dimensions — maybe even pure Einstein gravity — that are dual in some sense to a random two-dimensional conformal field theory (CFT)? The difficulty here is that while a quantum mechanical system can be defined by specifying a Hamiltonian, the data required to specify a 2d CFT are far more complicated. Accordingly, it is far from clear what s
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