The Factorization Problem in Jackiw-Teitelboim Gravity

TL;DR

This paper analyzes 1+1 dimensional Jackiw-Teitelboim gravity, constructing its phase space and Hilbert space, and discusses the factorization problem in AdS/CFT, showing that JT gravity lacks a CFT dual and exploring implications for pure Einstein gravity.

Contribution

It explicitly constructs the gauge-invariant phase space and Hilbert space of JT gravity, and discusses the implications for the factorization problem and CFT duals in low-dimensional gravity.

Findings

JT gravity has no CFT dual despite wormholes.

The Hilbert space factorizes on the CFT side, but this conflicts with bulk gauge constraints.

JT gravity is a consistent quantum theory without black hole microstates.

Abstract

In this note we study the $1+1$ dimensional Jackiw-Teitelboim gravity in Lorentzian signature, explicitly constructing the gauge-invariant classical phase space and the quantum Hilbert space and Hamiltonian. We also semiclassically compute the Hartle-Hawking wave function in two different bases of this Hilbert space. We then use these results to illustrate the gravitational version of the factorization problem of AdS/CFT: the Hilbert space of the two-boundary system tensor-factorizes on the CFT side, which appears to be in tension with the existence of gauge constraints in the bulk. In this model the tension is acute: we argue that JT gravity is a sensible quantum theory, based on a well-defined Lorentzian bulk path integral, which has no CFT dual. In bulk language, it has wormholes but it does not have black hole microstates. It does however give some hint as to what could be added to to rectify these issues, and we give an example of how this works using the SYK model. Finally we suggest that similar comments should apply to pure Einstein gravity in $2+1$ dimensions, which we'd then conclude also cannot have a CFT dual, consistent with the results of Maloney and Witten.