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(Sd) which is fixed to be Z(Sd)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(S4)χ(M)/2 can be extracted if and only if σ(M)=0. This is due to the fact that the first Pontryagin class p1 has a local expression and thus eiθσ(M)=ei/3θ∫M p1 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)∫ w2w3 we compute the multi-invariants exactly, and two graphs with identical local data, encoding the Dold manifold and ℂℙ2× S1, give opposite signs.
Related
- Multipartite entanglement spreads
- Asymptotics for partition functions of random normal matrix models with singularities
- The Calkin--Wilf problems on ℓ-ary partition functions
- Efficient Computation and Congruences for Colored Partition Functions
- The β=2 partition function of Dirichlet L-functions and typical maxima
BibTeX
@misc{zotto2026multipartite,
title = {{Multipartite Entanglement, TQFT Partition Functions and PL Bordism}},
author = {Michele Del Zotto and Abhijit Gadde and Pavel Putrov},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07000},
eprint = {2611.07000},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07000},
}
RIS
TY - UNPB
TI - Multipartite Entanglement, TQFT Partition Functions and PL Bordism
AU - Michele Del Zotto
AU - Abhijit Gadde
AU - Pavel Putrov
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.07000
UR - https://arxiv.org/abs/2611.07000
AB - 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^i/3θ∫_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)^∫ w_2w_3 we compute the multi-invariants exactly, and two graphs with identical local data, encoding the Dold manifold and ℂℙ^2× S^1, give opposite signs.
ER -