# Tensor network formulation of two dimensional gravity

## TL;DR

This paper formulates a lattice gauge theory for 2D quantum gravity using tensor networks, enabling analysis at strong coupling and revealing critical points related to phase transitions, with potential extensions to Polyakov or Liouville actions.

## Contribution

It introduces a tensor network formulation of 2D quantum gravity on a lattice, allowing strong coupling analysis and exact solutions, and explores critical behavior and topological dependence.

## Key findings

- Exact solution of the tensor network in 2D
- Identification of critical points at imaginary coupling
- Extension to Polyakov or Liouville actions

## Abstract

We show how to formulate a lattice gauge theory whose naive continuum limit corresponds to two-dimensional (Euclidean) quantum gravity including a positive cosmological constant. More precisely the resultant continuum theory corresponds to gravity in a first-order formalism in which the local frame and spin connection are treated as independent fields. Recasting this lattice theory as a tensor network allows us to study the theory at strong coupling without encountering a sign problem. In two dimensions this tensor network is exactly soluble and we show that the system has a series of critical points that occur for pure imaginary coupling and are associated with first order phase transitions. We then augment the action with a Yang-Mills term which allows us to control the lattice spacing and show how to apply the TRG to compute the free energy and look for critical behavior. Finally we perform an analytic continuation in the gravity coupling in this extended model and show that its critical behavior in a certain scaling limit depends only on the topology of the underlying lattice. We also show how the lattice gauge theory can be naturally generalized to generate the Polyakov or Liouville action for two dimensional quantum gravity.