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

TL;DR

This paper explores the relationship between bulk (3+1)-dimensional topological phases and their (2+1)-dimensional boundary theories by analyzing partition functions on a torus and establishing a bulk-boundary correspondence through modular transformations.

Contribution

It demonstrates how to compute and match modular $ ext{S}$ and $ ext{T}$ matrices between boundary theories and bulk phases, establishing a bulk-boundary correspondence in higher dimensions.

Findings

Partition functions transform under $SL(3, ext{Z})$ modular transformations.

Bulk-boundary correspondence is established via matching modular matrices.

Proposes studying three-loop braiding statistics through boundary modular matrices.

Abstract

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.