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

Abstract

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