Abstract
We compute the partition function of 2D2D Jackiw-Teitelboim (JT) gravity at finite cutoff in two ways: (i) via an exact evaluation of the Wheeler-DeWitt wavefunctional in radial quantization and (ii) through a direct computation of the Euclidean path integral. Both methods deal with Dirichlet boundary conditions for the metric and the dilaton. In the first approach, the radial wavefunctionals are found by reducing the constraint equations to two first order functional derivative equations that can be solved exactly, including factor ordering. In the second approach we perform the path integral exactly when summing over surfaces with disk topology, to all orders in perturbation theory in the cutoff. Both results precisely match the recently derived partition function in the Schwarzian theory deformed by an operator analogous to the T\bar TTT‾ deformation in 2D2D CFTs. This equality can be seen as concrete evidence for the proposed holographic interpretation of the T\bar TTT‾ deformation as the movement of the AdS boundary to a finite radial distance in the bulk.
Highlights
We compute the partition function of 2D Jackiw-Teitelboim (JT) gravity at finite cutoff in two ways: (i) via an exact evaluation of the Wheeler-DeWitt wavefunctional in radial quantization and (ii) through a direct computation of the Euclidean path integral
In previous work [9], a particular deformation of the Schwarzian quantum mechanics was shown to be classically equivalent to JT gravity with Dirichlet boundary conditions for the metric and dilaton
The above results obtained through the Wheeler-de Witt (WdW) constraint are non-perturbative in both L and φb(u)
Summary
Can the holographic dictionary of AdS/CFT be generalized to gravitational theories defined on a finite patch of spacetime? This question has recently attracted renewed attention due to the discovery of a new class of solvable irrelevant deformations of two-dimensional conformal field theory, known as the T Tdeformation [1,2,3]. A key piece of evidence in support of this conjectured duality is that the conformal Ward identity of the CFT gets deformed into a second order functional differential equation that formally matches with the Wheeler-DeWitt equation of AdS3 gravity This relationship is akin to the familiar duality between Chern-Simons field theory and WessZumino-Witten conformal field theory [6], and points to the possible identification between the wave functionals of gravitational theories in D + 1-dimensions and partition functions of a special class of D-dimensional QFTs.. Using integrability properties of the Schwarzian theory, we manage to exactly compute the partition function to all orders in a perturbative expansion in the cutoff Both the canonical and path integral approach use widely different techniques to compute the finite cutoff partition function, yet, as expected from the equivalence between the two quantisation procedures, the results agree. Before diving in the computations, we first review (for completeness and later reference) this one-dimensional analog of T Tand present a more detailed summary of our results
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