Bosonic Short Range Entangled states Beyond Group Cohomology classification

TL;DR

This paper constructs and analyzes a new class of bosonic short-range entangled states in higher dimensions, generalizing the 2D Kitaev $E_8$ state, revealing their boundary properties, gravitational anomalies, and relations to other topological states.

Contribution

It introduces a systematic construction of higher-dimensional BSRE states beyond group cohomology, extending the $E_8$ state framework and exploring their boundary and bulk properties.

Findings

Bulk is fully gapped and nondegenerate

Boundaries are described by self-dual tensor gauge fields and are gapless with gravitational anomalies

These states relate to bosonic SPT states and can be derived from fermionic states via confinement

Abstract

We explore and construct a class of bosonic short range entangled (BSRE) states in all $4k+2$ spatial dimensions, which are higher dimensional generalizations of the well-known Kitaev's $E_8$ state in 2d. These BSRE states share the following properties: (1) their bulk is fully gapped and nondegenerate; (2) their $(4k+1)d$ boundary is described by a "self-dual" rank-$2k$ antisymmetric tensor gauge field, and it is guaranteed to be gapless without assuming any symmetry; (3) their $(4k+1)d$ boundary has intrinsic gravitational anomaly once coupled to the gravitational field; (4) their bulk is described by an effective Chern-Simons field theory with rank-$(2k+1)$ antisymmetric tensor fields, whose $K^{IJ}$ matrix is identical to that of the $E_8$ state in $2d$; (5) The existence of these BSRE states lead to various bosonic symmetry protected topological (BSPT) states as their descendants in other dimensions; (6) These BSRE states can be constructed by confining fermionic degrees of freedom from 8 copies of $(4k+2)d$ SRE states with fermionic $2k-$branes; (7) After compactifying the $(4k+2)d$ BSRE state on a closed $4k$ dimensional manifold, depending on the topology of the compact $4k$ manifold, the system could reduce to nontrivial $2d$ BSRE states.