Bulk-boundary correspondence in (3+1)-dimensional topological phases

Xiao Chen, Apoorv Tiwari, Shinsei Ryu

Abhijeet Alase

Daniel Bulmash, Maissam Barkeshli

Kyle Kawagoe, Michael Levin

Kyle Kawagoe, Michael Levin

Xiao Chen, Apoorv Tiwari, Shinsei Ryu

We discuss (2+1)-dimensional gapless surface theories of bulk (3+1)-dimensional topological phases, such as the BF theory at level $\mathrm{K}$, and its generalization. In particular, we put these theories on a flat (2+1)-dimensional torus ${T}^{3}$ parameterized by its modular parameters, and compute the partition functions obeying various twisted boundary conditions. We show the partition functions are transformed into each other under $SL(3,\mathbb{Z})$ modular transformations, and furthermore establish the bulk-boundary correspondence in (3+1) dimensions by matching the modular $\mathcal{S}$ and $\mathcal{T}$ matrices computed from the boundary field theories with those computed in the bulk. We also propose the three-loop braiding statistics can be studied by constructing the modular $\mathcal{S}$ and $\mathcal{T}$ matrices from an appropriate boundary field theory.