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arXiv · 1509.04266

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

Xiao Chen·Apoorv Tiwari·Shinsei Ryu
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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.

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Xiao Chen, Apoorv Tiwari, Shinsei Ryu. 2016-07-20. Bulk-boundary correspondence in (3+1)-dimensional topological phases. https://doi.org/10.1103/physrevb.94.045113

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