  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.
