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

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.