Bosonic Short Range Entangled states Beyond Group Cohomology classification
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.