From bordisms of three-manifolds to domain walls between topological orders

TL;DR

This paper establishes a mathematical correspondence between spin three-manifolds and bosonic abelian topological orders, extending it to domain walls via spin bordisms, linking topology with quantum order transitions.

Contribution

It introduces a novel correspondence between spin three-manifolds and topological orders, and constructs domain walls from spin bordisms, connecting topology with quantum phase boundaries.

Findings

Defines topological order $ ext{TO}_N$ from spin three-manifold $N$

Relates surgery presentations to Chern--Simons descriptions

Constructs domain walls from spin bordisms, combining gapped and gapless boundaries

Abstract

We study a correspondence between spin three-manifolds and bosonic abelian topological orders. Let $N$ be a spin three-manifold. We can define a $(2+1)$-dimensional topological order $\mathrm{TO}_N$ as follows: its anyons are the torsion elements in $H_1(N)$, the braiding of anyons is given by the linking form, and their topological spins are given by the quadratic refinement of the linking form obtained from the spin structure. Under this correspondence, a surgery presentation of $N$ gives rise to a classical Chern--Simons description of the associated topological order $\mathrm{TO}_N$. We then extend the correspondence to spin bordisms between three-manifolds, and domain walls between topological orders. In particular, we construct a domain wall $\mathcal{D}_M$ between $\mathrm{TO}_N$ and $\mathrm{TO}_{N'}$, where $M$ is a spin bordism from $N$ to $N'$. This domain wall unfolds to a composition of a gapped boundary, obtained from anyon condensation, and a gapless Narain boundary CFT.