The Factorization Problem in Jackiw-Teitelboim Gravity

TL;DR

This work analyzes Jackiw-Teitelboim gravity in Lorentzian signature, constructing the gauge-invariant phase space and quantum Hilbert space, and computing the Hartle-Hawking state in dual bases. It demonstrates that the two-boundary JT Hilbert space does not factorize into left/right boundary theories, signaling the absence of a CFT dual despite a well-defined bulk path integral and wormhole solutions. The paper connects the bulk theory to SYK via a Lorentzian embedding, clarifying how factorization constraints arise and how matter can potentially rectify them. It also argues that similar non-dual behavior likely extends to pure Einstein gravity in 2+1 dimensions, echoing Maloney and Witten’s results.

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.

Figures (8)

• Figure 1: Different kinds of time evolution in the bulk and boundary: the endpoints of a bulk timeslice are moved up and down using the ADM Hamiltonian, while the interior of the slice is evolved using the Hamiltonian constraint. Since the Hamiltonian constraint is zero on physical states, the second two slices describe the same CFT state.
• Figure 2: The dilaton profile in the wormhole solution of Jackiw-Teitelboim gravity. Between the two vertical black lines the geometry is global $AdS_2$, and we indicate the value of the dilaton on various surfaces. In Reissner-Nordstrom language, the dashed black lines where $\Phi=\Phi_h$ are the "outer horizon", while the dashed red lines where $\Phi=-\Phi_h$ are the "inner horizon". The dynamical problem with boundary conditions \ref{['bc']} is well-defined only in the shaded green region. If we assume that the inner horizon is singular, this solution describes a wormhole connecting two asymptotically-$AdS$ boundaries.
• Figure 3: Different time slices of the wormhole solution can correspond to different initial data for the JT gravity. Here the first and second slices are different points in phase space, while the first and third are the same since they differ by evolution by $H_R-H_L$.
• Figure 4: Using a geodesic, shown in red, to measure $\delta$.
• Figure 5: The Hartle-Hawking state: we sum over geometries and dilaton configurations with an AdS boundary of length $r_c\beta/2$ and a "bulk" boundary $\Sigma$, which we interpret as a time-slice of the two-boundary system. The boundary conditions on $\Sigma$ depend on which basis we wish to compute the wave function in.