Higher-dimensional TFT-type invariants from higher Segal objects

Construct topological field theory–type manifold invariants in dimensions greater than two from higher Segal objects equipped with additional symmetries, extending the two-dimensional correspondence-valued constructions based on Segal–Pachner moves to higher dimensions.

Background

The paper explains how lower and upper d-Segal maps yield correspondences that implement Segal–Pachner moves, providing invariance for certain triangulated manifolds. In two dimensions, this leads to a workable framework for constructing correspondence-valued topological field theories and related invariants.

The authors note that while a satisfactory resolution exists in dimension two, extending these constructions to produce useful higher-dimensional manifold invariants from higher Segal objects requires additional symmetries and remains unresolved.

References

In dimensions bigger than 2, the construction of useful analogous higher-dimensional TFT-type manifold invariants from higher Segal objects with additional symmetries is a very interesting open problem.

— Cyclic polytopes, orientals, and correspondences: some aspects of higher Segal spaces  (2505.02051 - Dyckerhoff, 4 May 2025) in Section 8, Further perspectives, paragraph “Coherent Pachner moves”

The resulting tensors would also provide a setting for testing the conjectured relation between genuine multipartite entanglement and TQFT partition functions proposed in. In the present (3+1)d setting, a concrete goal is to construct five-partite replica invariants associated with triangulated four-manifolds and determine whether their genuine entanglement signals reproduce |Z_{\mathcal C}(M4)|2 with the prescribed sphere normalization.

— Categorical spin-networks: state-sum invariants of 4-manifolds from higher-gauge theory  (2609.31365 - Chen et al., 25 Sep 2026) in Section 8, “Outlook,” final paragraph