Diffeomorphism invariant tensor networks for 3d gravity

TL;DR

The paper develops a diffeomorphism-invariant tensor-network framework for 3D gravity by extending topological tensor networks to transformable gauge groups, applied to $\text{SL}(2,\mathbb{R})\times\text{SL}(2,\mathbb{R})$ Chern-Simons theory. It constructs a pre-Hilbert space on a bulk graph with $L^2(G)$ legs, imposes electric and magnetic constraints to realize gauge invariance, and defines a physical Hilbert space $\mathcal{H}_{phys}(\Sigma)$ that encodes diffeomorphism-invariant states in metric variables via Wheeler-DeWitt and momentum constraints. The framework yields a bulk-to-boundary map with a multipartite entanglement structure controlled by intertwiners and a nontrivial area operator arising from ribbon operators, demonstrating noncommuting area operators for overlapping subregions. A coset extension toward Virasoro TQFT is proposed to address gravity measure and Liouville/CFT dual descriptions, outlining a path to incorporate loop corrections in the semiclassical limit and connect to the Virasoro TQFT/Chern-Simons formulation. The work sets the stage for constructing semi-classical gravitational states in 3D, exploring BTZ-like geometries, and extending to non-perturbative topology changes within a controlled tensor-network framework.

Abstract

Tensor networks prepare states that share many features of states in quantum gravity. However, standard constructions are not diffeomorphism invariant and do not support an algebra of non-commuting area operators. Recently, analogues of both problems were addressed in a tensor network discretization of topological field theories (TFT) with finite or compact gauge groups. Here, we extend this work towards gravity by generalizing to gauge groups that are discrete or continuous, compact or non-compact. Applied to $\text{SL}(2,\mathbb{R}) \times \text{SL}(2,\mathbb{R})$ Chern-Simons theory, our construction can be interpreted as building states of three dimensional gravity with a negative cosmological constant. Our tensor networks prepare states that satisfy the constraints of Chern-Simons theory. In metric variables, this implies that the states we construct satisfy the Wheeler-DeWitt equation and momentum constraints, and so are diffeomorphism invariant.

Figures (12)

• Figure 1: A lattice $\Lambda$ tessellating the disk $\Sigma$. The bulk vertices are in black, and the boundary vertices are in white.
• Figure 2: An example of a plaquette that $B_{(v,p)}(h)$ acts on in \ref{['eqn:Bdef']}.
• Figure 3: The most basic topological tensor network $\Lambda_1$. There is a single vertex, plaquette $p$ (shaded in grey), and leg $\ell = \partial p$.
• Figure 4: The action of $A_v(h)$ on an example vertex
• Figure 5: An example of move 1, which can be performed in either direction to add or remove a bulk vertex/leg from the graph $\Lambda$.