Multipartite Entanglement, TQFT Partition Functions and PL Bordism

Abstract

In a previous work we conjectured that the multi-invariants of the ground state of a gapped $(d-1)+1$-dimensional system, labeled by bipartite $(d+1)$-edge-colored graphs encoding closed oriented $d$-manifolds $M$, compute the partition functions $Z(M)$ of the low energy topological quantum field theory (TQFT), up to non-universal factors determined by the local structure of the triangulation. In this paper we determine which combinations of partition functions can be isolated from the non-universal factors. The answer is governed by oriented bordism. If $M$ is null-bordant, $Z(M)$ is obtained from an explicit, state-independent finite family of multi-invariants, up to a power of $Z(S^d)$ which is fixed to be $Z(S^d)^{χ(M)/2}$ for even $d$ and can be removed for odd $d$. Conversely, the obstruction is measured by a quotient of the group of "residues" of colored graphs which receives a surjective map from the oriented bordism group, with kernel precisely the torsion subgroup as far as smooth manifolds are concerned. In $d=4$ the obstruction is the signature: $Z(M)/Z(S^4)^{χ(M)/2}$ can be extracted if and only if $σ(M)=0$. This is due to the fact that the first Pontryagin class $p_1$ has a local expression and thus $e^{iθσ(M)}=e^{\frac{i}{3}θ\int_M p_1}$ cannot be separated from the non-universal factors. Torsion invariants, in contrast, are not obstructed: for the ground state of the $4+1$-dimensional invertible phase with partition function $(-1)^{\int w_2w_3}$ we compute the multi-invariants exactly, and two graphs with identical local data, encoding the Dold manifold and $\mathbb{CP}^2\times S^1$, give opposite signs.