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”