A proposal for 3d quantum gravity and its bulk factorization

TL;DR

The paper proposes a universal 3d pure gravity framework describing high-temperature holographic CFTs, in which bulk entropy is interpreted as entanglement entropy of gravitational edge modes. It develops a gauge-theory factorization program and extends it to gravity, arguing that the bulk edge sector is governed by the quantum semi-group SL^+_q(2,R)×SL^+_q(2,R) and, in the JT limit, by SL^+(2,R) edge modes. A doubled JT regime emerges at low temperatures, while a fully 3d gravity description requires an extended TQFT perspective, with factorization implemented via a q-deformed edge-category and a corresponding boundary category of Rep(SL^+_q(2,R)×SL^+_q(2,R)). The bulk entropy matches the Bekenstein-Hawking result and arises from the logarithm of the edge-state quantum dimensions, providing a microscopic-entanglement-based account of black hole microstates. The work situates gravity and gauge theory within an extended TQFT framework, linking edge-state factorization, the ER=EPR paradigm, and holographic entropy through a categorical, topological approach with potential connections to Teichmüller TQFT.

Abstract

Recent progress in AdS/CFT has provided a good understanding of how the bulk spacetime is encoded in the entanglement structure of the boundary CFT. However, little is known about how spacetime emerges directly from the bulk quantum theory. We address this question in an effective 3d quantum theory of pure gravity, which describes the high temperature regime of a holographic CFT. This theory can be viewed as a $q$-deformation and dimensional uplift of JT gravity. Using this model, we show that the Bekenstein-Hawking entropy of a two-sided black hole equals the bulk entanglement entropy of gravitational edge modes. In the conventional Chern-Simons description, these black holes correspond to Wilson lines in representations of $\PSL(2,\mathbb{R})\otimes \PSL(2,\mathbb{R}) $. We show that the correct calculation of gravitational entropy suggests we should interpret the bulk theory as an extended topological quantum field theory associated to the quantum semi-group $\SL^+_{q}(2,\mathbb{R})\otimes \SL^+_{q}(2,\mathbb{R})$. Our calculation suggests an effective description of bulk microstates in terms of collective, anyonic degrees of freedom whose entanglement leads to the emergence of the bulk spacetime.

Figures (17)

• Figure 1: Left: splitting a gauge theory Wilson line $W_{\text{bulk}}$ in left- and right Wilson lines $W_L$ and $W_R$. Right: splitting a gravitational Wilson line disconnects spacetime.
• Figure 2: Modular $S$-transform of the vacuum character leads to a sum (integral) of black hole states. The blue lines are Wilson loops in the interior of the torus. The labels $(p_+,p_-)$ correspond to defects that have a hyperbolic holonomy around them (as measured on a spatial $\varphi$-slice).
• Figure 3: Left: Black hole state defined by specifying its hyperbolic monodromy around a spatial slice, as measured by the red Wilson loop. Right: Spatial slice of the classical geometry containing a spatial wormhole.
• Figure 4: Modular $S$-transform of the non-vacuum character.
• Figure 5: The left figure shows the Penrose diagram for the two-sided AdS$_2$ geometry, with the red time slice denoting the wormhole. On the right, we show the Lorentzian evolution of the resulting Hartle-Hawking state.