# Stacks in Einstein Gravity and a Stacky Equivalence of 3D Quantum Gravity with Gauge Theory

## TL;DR

This paper explores the stacky structures underlying Einstein gravity, constructs moduli stacks of solutions, and establishes an equivalence between 3D quantum gravity and gauge theory through isomorphic phase spaces and moduli stacks.

## Contribution

It introduces a stack-theoretic framework for Einstein's gravity and demonstrates a stacky equivalence between 3D quantum gravity and gauge theory.

## Key findings

- Constructed moduli stacks of Einstein solutions.
- Defined a stack encoding Einstein gravity on families of manifolds.
- Proved that the equivalence of phase spaces induces an isomorphism of moduli stacks.

## Abstract

In this paper, we examine stacky structures in Einstein's theory of gravity. In brief, we first give a construction of the moduli stack of solutions to (vacuum) Einstein field equations on $n$-dimensional spacetimes, with vanishing cosmological constant. Using a similar approach, we also study Einstein's gravity on families of manifolds and define another stack encoding this situation as well. Secondly, we focus on the gauge theoretical interpretation of 3D gravity and the concept of equivalence of 3D quantum gravity with gauge theory. By equivalence, we essentially mean the existence of an isomorphism between the phase spaces of 3D gravity and the associated gauge theory. In this regard, we show that once it exists, the equivalence induces an isomorphism between the corresponding moduli stacks.