Jackiw-Teitelboim Gravity in the Second Order Formalism

TL;DR

This work formulates JT gravity in the second order formalism, computing path integrals for AdS$_2$ and dS$_2$ on disk and double trumpet topologies while incorporating conformal matter. It demonstrates agreement with the first-order formalism in the asymptotic AdS/dS limit and clarifies the role of large diffeomorphisms, the Schwarzian boundary dynamics, and determinant factors arising from the dilaton and Liouville sectors. A key finding is a Casimir-type divergence on wormhole necks in the double trumpet when bosonic matter is present, which fermionic boundary conditions can sometimes avoid, with important implications for the interpretation of wormholes and the connected two-boundary correlator. In de Sitter space, the no-boundary wavefunction is analyzed via analytic continuation, yielding one-loop Schwarzian-dominated fluctuations and revealing subtle determinant-dependent corrections, all within a carefully defined contour framework. Overall, the paper exposes both technical and conceptual facets of JT gravity in the second order formalism, including how matter and topology shape quantum gravity in two dimensions.

Abstract

We formulate the path integral for Jackiw-Teitelboim gravity in the second order formalism working directly with the metric and the dilaton. We consider the theory both in Anti-de Sitter(AdS) and de Sitter space(dS) and analyze the path integral for the disk topology and the "double trumpet" topology with two boundaries. We also consider its behavior in the presence of conformal matter. In the dS case the path integral evaluates the wavefunction of the universe which arises in the no-boundary proposal. In the asymptotic AdS or dS limit without matter we get agreement with the first order formalism. More generally, away from this limit, the path integral is more complicated due to the presence of modes from the gravity-dilaton sector and also matter sector with short wavelengths along the boundary that are smaller than the AdS or dS scales. In the double trumpet case, for both AdS and dS, we find that bosonic matter gives rise to a diverging contribution in the moduli space integral rendering the path integral ill-defined. The divergence occurs when the size of the wormhole neck vanishes and is related to the Casimir effect. For fermions this divergence can be avoided by imposing suitable boundary conditions. In this case, in dS space the resulting path integral gives a finite contribution for two disconnected universes to be produced by quantum tunneling.

Figures (3)

• Figure 1: Analytic continuations in the complex r-plane for the no-boundary wavefunction
• Figure 2: dS double trumpet from AdS double trumpet
• Figure 3: dS double trumpet continuations from AdS double trumpet