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

Xiao Chen, Apoorv Tiwari, Shinsei Ryu

TL;DR

This work extends the bulk-boundary correspondence from $(2+1)$d to $(3+1)$d topological phases by analyzing boundary theories on the spatial torus $T^3$ under $SL(3,\mathbb{Z})$ modular transformations. It demonstrates that modular $\mathcal{S}$ and $\mathcal{T}$ matrices computed from gapless boundary theories of $(3+1)$d BF theories precisely match those from bulk considerations, thereby linking boundary data to bulk topological invariants. The paper further explores a BF theory with a spatially varying $\Theta$ term, showing the boundary theory acquires twisted quantization and modular non-closure, and introduces a coupled two-copy BF framework to realize three-loop braiding statistics, with dimensional reduction illuminating connections to $(2+1)$d topological orders. Together, these results establish a boundary-based route to bulk topological data in $(3+1)$d and propose concrete continuum models for novel three-loop braiding phenomena.

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.

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

TL;DR

This work extends the bulk-boundary correspondence from d to d topological phases by analyzing boundary theories on the spatial torus under modular transformations. It demonstrates that modular and matrices computed from gapless boundary theories of d BF theories precisely match those from bulk considerations, thereby linking boundary data to bulk topological invariants. The paper further explores a BF theory with a spatially varying term, showing the boundary theory acquires twisted quantization and modular non-closure, and introduces a coupled two-copy BF framework to realize three-loop braiding statistics, with dimensional reduction illuminating connections to d topological orders. Together, these results establish a boundary-based route to bulk topological data in d and propose concrete continuum models for novel three-loop braiding phenomena.

Abstract

We discuss (2+1)-dimensional gapless surface theories of bulk (3+1)-dimensional topological phases, such as the BF theory at level , and its generalization. In particular, we put these theories on a flat (2+1) dimensional torus 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 modular transformations, and furthermore establish the bulk-boundary correspondence in (3+1) dimensions by matching the modular and 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 and matrices from an appropriate boundary field theory.

Paper Structure

This paper contains 28 sections, 221 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. The compactified free boson in $(2+1)$d
  4. Modular transformations on $T^3$
  5. The partition function on $T^3$
  6. Modular invariance
  7. The surface theory of the (3+1)d BF theory
  8. Bulk and surface theories
  9. The bulk field theory
  10. The boundary theory
  11. The surface theory and quantization
  12. The partition functions and modular transformations
  13. Entropy of the boundary theory
  14. The surface theory of the $(3+1)$d BF theory with the $\Theta$ term
  15. The BF theory with the $\Theta$-term in (3+1)d
  16. ...and 13 more sections

Figures (1)

  • Figure 1: (a) The presence of a point-like quasiparticle in the bulk (solid torus) induces a fractional flux on the spatial boundary $\Sigma$ (torus). (b) The presence of a quasivortex line in the bulk twists the boundary conditions of the surface theory. Here and in (c), the bulk is presented as a filled cylinder where the top and the bottom of the cylinder are identified. The shaded surface is a sheet of the branch cut which emanates from the quasivortex, and intersects with the spatial boundary (depicted by a wavy line). The surface excitations experience a twisted boundary condition as they go through the branch cut. (c) Similar to (b), a bulk quasivortex, which creates a branch cut on the surface which now goes along a different cycle of the surface, is depicted.