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.
Highlights
The studies of SYK and its low energy limit described by the one dimensional Schwarzian theory [1,2,3,4,5] revealed a holographic connection with a bulk two dimensional Jackiw-Teitelboim (JT) gravity theory
The argument supporting this connection, is that the usual double scaling limit of a single Hermitean matrix model can describe the (2, p) minimal models coupled to gravity, and the physics of JT gravity can be reached as a p → ∞ limit of these models
This is in accord with the duality between JT gravity and the Hermitean one matrix model of (1.1). From this point of view the D0 branes live in “superspace” and the second quantised fermionic field theory is a “third quantised” description of the dynamics of bulk universes. This means that Matrix quantum mechanics (MQM) and the associated fermionic field theory provide us with a specific non-perturbative completion of the c = 1 bulk quantum gravity path integral,5 as the one and two matrix models do in the simpler cases of JT-gravity and (p, q) minimal models
Summary
The studies of SYK and its low energy (hydrodynamic) limit described by the one dimensional Schwarzian theory [1,2,3,4,5] revealed a holographic connection with a bulk two dimensional Jackiw-Teitelboim (JT) gravity theory. From this point of view the D0 branes live in “superspace” and the second quantised fermionic field theory is a “third quantised” description of the dynamics of bulk universes This means that MQM and the associated fermionic field theory provide us with a specific non-perturbative completion of the c = 1 bulk quantum gravity path integral, as the one and two matrix models do in the simpler cases of JT-gravity and (p, q) minimal models. This discussion raises new interesting possibilities as well as questions. The acronyms used are DOS : for the density of states and SF F : for the spectral form factor
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