Classify topological boundary conditions for continuous and non-compact groups

Classify the topological boundary conditions of the TQFTs associated with topological tensor networks whose gauge groups are continuous and possibly non-compact, including the boundary conditions relevant to factorization maps.

Background

The entropy construction depends on imposing a topological boundary condition at the factorization cut. For finite groups, such boundary conditions are classified by subgroup and cohomological data, but the paper extends the rough and smooth boundary conditions to continuous and non-compact groups only by analogy. A complete classification would determine the allowed factorization maps and the corresponding superselection sectors and area spectra.

References

The situation is much less understood for the TQFTs of Sec.~\ref{sec:themodel}, whose gauge groups are continuous and possibly non-compact. There, the topological boundary conditions have not been classified, and we will not attempt a complete classification here.

— Entanglement entropy in topological tensor networks  (2609.10667 - Balasubramanian et al., 9 Sep 2026) in Appendix A, Section A.1, “Topological boundary conditions”

We add, however, that these results are established rigorously for categories with finitely many simple objects, whereas for a general transformable group $G$, the fusion of the legs is a direct integral over $\widehat{G}$ rather than a finite sum. When $G$ is finite, the identification of the two presentations carries a normalization by the total quantum dimension, which diverges for continuous and/or non-compact groups. However, this is the same obstruction which required us to use the co-invariant inner product to define the physical Hilbert space, rather than the divergent invariant inner product, so we expect that these results can be adapted to our setting in some form. We leave a more careful treatment of this to future work.

— Topological entanglement entropy in 3D gravity  (2609.19247 - Balasubramanian et al., 16 Sep 2026) in Section 2.2.4, immediately after the discussion comparing the two factorization maps