Comments on Lorentzian topology change in JT gravity

TL;DR

This work defines a Lorentzian JT gravity path integral that incorporates topology-changing processes by analytic continuation from the Euclidean theory, using lightcone diagrams to organize the moduli space and degenerate metrics at interaction points. The Lorentzian amplitudes are formulated as $Z_{\mathrm{L}}=\int_{\rm moduli}{\rm d}\mu\ \int \mathcal{D}\omega\ \mathcal{D}\Phi\ e^{i I_{\mathrm{JT}}[e^{2\omega}\hat g,\Phi]}$, with degenerate points encoding topology changes and a Weyl-factor freedom distributing curvature while preserving a Gauss-Bonnet constraint; the approach closely follows the interacting-string picture for genus expansions and analyzes how Lorentzian amplitudes relate to Euclidean JT amplitudes, including the role of determinants and potential ambiguities from singular moduli. The paper discusses how Euclidean Weil-Petersson structures may translate to the Lorentzian setting, the subtleties of determinants on singular surfaces, and possible universality of wormhole-related features, while outlining avenues for extending the framework to asymptotically AdS boundaries and non-perturbative Lorentzian physics.

Abstract

We propose a definition for the Lorentzian Jackiw-Teitelboim (JT) gravity path integral that includes Lorentzian topology changing configurations. The construction is inspired by the bosonic string genus expansion on singular Lorentzian worldsheets, with geometries known as lightcone diagrams playing a prominent role. The Lorentzian path integral is defined through a suitable analytic continuation of the Euclidean path integral, and includes metrics that become degenerate at isolated points allowing for Lorentzian topology changing transitions. We discuss the relation between Euclidean JT amplitudes and the proposed Lorentzian amplitudes.

Figures (6)

• Figure 1: Lorentzian topology changing transition where two spatial circles evolves into three circles. The metric is Lorentzian everywhere except the splitting points (blue circles) where it is degenerate. At the splitting points the topology of spatial slices changes. The spacetime is an analytic continuation of a genus two Euclidean geometry with five circular boundaries.
• Figure 2: Euclidean/Lorentzian lightcone diagram of genus two with five boundaries. The geometry is flat except at interactions points $\tau_i$ with delta functions of curvature. A lightcone diagram is built by gluing pairs of pants together. The interaction times are labelled by $\tau$, the twist angles by $\theta$, and the free internal radius by $\rho$.
• Figure 3: Euclidean Pair of pants constructed by identification in the complex $w=\tau + i \sigma$ plane. The middle line (dashed) is infinitesimally split up to the interaction time $\tau$ and identified with the top and bottom segments as indicated. The singular point is indicated by the circle (blue). A closed loop around the singular point goes through angle $4\pi$.
• Figure 4: A genus three surface $\Sigma$ with two punctures. We can put a metric on $\Sigma$ given by \ref{['eqn:Wolpert_Metric']} to turn it into a Euclidean lightcone diagram with two asymptotic cylinders.
• Figure 5: A torus with two punctures mapped to a lightcone diagram with two interaction points and two asymptotic cylinders. The circles (blue) on the torus denote where the one-form $\omega$ has zeros. The two cycles on the torus are illustrated with one at constant time $\tau$ (orange). The loops $\gamma_i$ enclose the punctures which run off to infinity in the lightcone metric.