Liouville theory and matrix models: a Wheeler DeWitt perspective
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Springer
Received: April Revised: July Accepted: August Published: September
14, 24, 23, 18,
2020 2020 2020 2020
P. Betziosa and O. Papadoulakib a
Crete Center for Theoretical Physics, Institute for Theoretical and Computational Physics, Department of Physics, P.O. Box 2208, University of Crete, 70013, Heraklion, Greece b International Centre for Theoretical Physics, Strada Costiera 11, Trieste 34151 Italy
E-mail: [email protected], [email protected] Abstract: We analyse the connections between the Wheeler DeWitt approach for two dimensional quantum gravity and holography, focusing mainly in the case of Liouville theory coupled to c = 1 matter. Our motivation is to understand whether some form of averaging is essential for the boundary theory, if we wish to describe the bulk quantum gravity path integral of this two dimensional example. The analysis hence, is in a spirit similar to the recent studies of Jackiw-Teitelboim (JT)-gravity. Macroscopic loop operators define the asymptotic region on which the holographic boundary dual resides. Matrix quantum mechanics (MQM) and the associated double scaled fermionic field theory on the contrary, is providing an explicit “unitary in superspace” description of the complete dynamics of such two dimensional universes with matter, including the effects of topology change. If we try to associate a Hilbert space to a single boundary dual, it seems that it cannot contain all the information present in the non-perturbative bulk quantum gravity path integral and MQM. Keywords: 2D Gravity, Matrix Models, Models of Quantum Gravity, AdS-CFT Correspondence ArXiv ePrint: 2004.00002
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP09(2020)125
JHEP09(2020)125
Liouville theory and matrix models: a Wheeler DeWitt perspective
Contents 1 Introduction
2
2 Liouville theory
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4 One macroscopic loop 4.1 The WdW equation and partition function 4.2 The density of states of the holographic dual 4.2.1 Comparison with minimal models and JT gravity 4.2.2 The one sided Laplace transform
13 13 16 18 19
5 The 5.1 5.2 5.3 5.4
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case of two asymptotic regions Density two-point function Spectral form factor due to disconnected geometries Euclidean wormholes and the loop correlator Spectral form factor due to connected geometries
6 Comments on the cosmological wavefunctions
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7 Conclusions
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A Matrix models for minimal models A.1 Conformal maps and Integrable hierarchies A.2 The (p, q) minimal models A.3 Deformations
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B Properties of the WdW equation
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C Parabolic cylinder functions C.1 Even-odd basis C.2 Complex basis C.3 Resolvent and density of states C.4 Dual resolvent and density of states C.5 Density correlation functions
36 36 36 37 38 39
D Correlation functions from the fermionic field theory
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E Correlation functions for a compact boson
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–1–
JHEP09(2020)125
3 Matrix quantum mechanics and fermionic field theory
F Steepest descent F.1 Stationary phase approximation
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42 43
Introduction
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