The path integral of 3D gravity near extremality; or, JT gravity with defects as a matrix integral

TL;DR

The paper addresses spectral pathologies in pure 3D gravity near extremality by proposing nonperturbative topologies (Seifert manifolds) that contribute to the path integral. It shows that the near-extremal physics can be captured by a dimensionally reduced 2D theory—JT gravity with a gas of KK instantons—which is dual to a double-scaled matrix integral. This JT+defects framework yields a positive density of states and a nonperturbative shift of the BTZ extremality bound, resolving negativity without introducing light matter. The authors further develop a matrix-integral description at all genus for JT gravity with defects and discuss implications for a potential ensemble dual for 3D gravity, along with open questions about modular invariance, matter, and the full 3D path integral.

Abstract

We propose that a class of new topologies, for which there is no classical solution, should be included in the path integral of three-dimensional pure gravity, and that their inclusion solves pathological negativities in the spectrum, replacing them with a nonperturbative shift of the BTZ extremality bound. We argue that a two-dimensional calculation using a dimensionally reduced theory captures the leading effects in the near extremal limit. To make this argument, we study a closely related two-dimensional theory of Jackiw-Teitelboim gravity with dynamical defects. We show that this theory is equivalent to a matrix integral.

Figures (3)

• Figure 1: The first few topologies contributing to the expansion of $\langle Z(\beta)\rangle$, as in \ref{['eq:Zexpansion']} with $n=1$. The top row shows the topologies for the disc $n=1,g=0$, with some number of defects $k=0,1,2,\ldots$, of order $e^{S_0}\lambda^k$. The second row shows the $n=1,g=1$ contributions for $k=0,1,\ldots$, of order $e^{-S_0}\lambda^k$.
• Figure 2: Density of states of JT gravity with (black) and without (dashed blue) defects, for $2\gamma=1$. (a) For $\alpha=1/2<\alpha_c$ and $\lambda=-0.1<0$ we see $E_0>0$ as expected and the theory is fine (b) For $\lambda = 0.01$ smaller than $\lambda_c(\alpha=1/2)\approx 0.06$ we get $E_0<0$ and the theory is fine (c) For $\lambda=0.08>\lambda_c(1/2)$ the density of states becomes negative in a finite range of energies.
• Figure 3: Shape of the dilaton potential for (a) $\lambda<0$ with only a single zero at a positive value of $\phi$ (b) $0<\lambda<\lambda_c$ with two zeros at negative values and $\phi_0$ the largest and (c) $\lambda_c < \lambda$ with no zeros.