Lattice construction of exotic invertible topological phases

TL;DR

This paper introduces a lattice path integral framework for exotic invertible topological phases with time reversal symmetry, revealing a $bZ_8$ classification in 3+1 dimensions and connecting to known topological quantum field theories.

Contribution

It provides a novel state sum construction for exotic invertible phases that depend on Wu structures, extending the classification and linking to existing TQFTs and SPT phases.

Findings

Proposes a lattice construction for exotic phases with $bZ_8$ classification.

Connects the phase to bosonic Crane-Yetter TQFT on oriented spacetime.

Generalizes the Gu-Wen fermionic SPT classification.

Abstract

In this paper, we provide state sum path integral definitions of exotic invertible topological phases proposed in the recent paper by Hsin, Ji, and Jian. The exotic phase has time reversal ($T$) symmetry, and depends on a choice of the spacetime structure called the Wu structure. The exotic phase cannot be captured by the classification of any bosonic or fermionic topological phases, and thus gives a novel class of invertible topological phases. When the $T$ symmetry defect admits a spin structure, our construction reduces to a sort of the decorated domain wall construction, in terms of a bosonic theory with $T$ symmetry defects decorated with a fermionic phase that depends on a spin structure of the $T$ symmetry defect. By utilizing our path integral, we propose a lattice construction for the exotic phase that generates the $\mathbb{Z}_8$ classification of the (3+1)d invertible phase based on the Wu structure. This generalizes the $\mathbb{Z}_8$ classification of the $T$-symmetric (1+1)d topological superconductor proposed by Fidkowski and Kitaev. On oriented spacetime, this (3+1)d invertible phase with a specific choice of Wu structure reduces to a bosonic Crane-Yetter TQFT which has a topological ordered state with a semion on its boundary. Moreover, we propose a subclass of $G$-SPT phases based on the Wu structure labeled by a pair of cohomological data in generic spacetime dimensions. This generalizes the Gu-Wen subclass of fermionic SPT phases.