# (3+1)-dimensional topological phases and self-dual quantum geometries encoded on Heegard surfaces

## TL;DR

This paper extends (2+1)-dimensional topological quantum field theories to (3+1) dimensions, constructing state spaces and operators that facilitate understanding of quantum geometries and topological phases relevant to quantum gravity.

## Contribution

It introduces a method to lift (2+1)D TQFT state spaces to (3+1)D, creating a self-dual quantum geometry framework with applications to quantum gravity and topological phases.

## Key findings

- Constructed (3+1)D state spaces from (2+1)D theories.
- Developed a quantum deformed spin network basis.
- Provided a dual curvature basis for the Walker-Wang model.

## Abstract

We apply the recently suggested strategy to lift state spaces and operators for (2+1)-dimensional topological quantum field theories to state spaces and operators for a (3+1)-dimensional TQFT with defects. We start from the (2+1)-dimensional Turaev-Viro theory and obtain a state space, consistent with the state space expected from the Crane-Yetter model with line defects. This work has important applications for quantum gravity as well as the theory of topological phases in (3+1) dimensions. It provides a self-dual quantum geometry realization based on a vacuum state peaked on a homogeneously curved geometry. The state spaces and operators we construct here provide also an improved version of the Walker-Wang model, and simplify its analysis considerably. We in particular show that the fusion bases of the (2+1)-dimensional theory lead to a rich set of bases for the (3+1)-dimensional theory. This includes a quantum deformed spin network basis, which in a loop quantum gravity context diagonalizes spatial geometry operators. We also obtain a dual curvature basis, that diagonalizes the Walker-Wang Hamiltonian. Furthermore, the construction presented here can be generalized to provide state spaces for the recently introduced dichromatic four-dimensional manifold invariants.