Determine alternative area spectra for other topological boundary conditions

Determine the spectra of the area operators associated with topological boundary conditions other than the rough boundary condition, including smooth and mixed charge-flux boundary conditions, for the factorization cut of topological tensor networks.

Background

The paper derives the area spectrum for a rough boundary condition, where pure charges condense at the cut. Other topological boundary conditions would change the superselection labels to the anyons that can terminate on the chosen boundary. The authors state that the entropy derivation should remain structurally valid, but the resulting spectra have not been explicitly determined.

References

We leave the explicit determination of these alternative spectra for future work.

— Entanglement entropy in topological tensor networks  (2609.10667 - Balasubramanian et al., 9 Sep 2026) in Section 5.2, “The area contribution”

Constructing that map is the natural next step. To construct this map, one would need the analogs of the nested algebras $\mathcal{B}R \supset \mathcal{A}_R \supset \mathcal{Z}(\mathcal{A}_R)$ for a closed cut, a measure on the label space eq:diagonalquotient defining a trace on each of them, and the operator valued weights relating those traces, following Sec.~\ref{sec:algebras}. As in Sec.~\ref{sec:algebras}, the mismatch in the measures for the analogs of $\mathcal{A}_R$ and $\mathcal{Z}(\mathcal{A}_R)$, generalized to include the full label set eq:diagonalquotient, is what would then fix the eigenvalues of the area operator along a closed curve. When $G$ is a finite group, these eigenvalues are known to be the logarithm of the ``quantum dimension'' of the anyons represented by the ribbon operators $F\gamma([t],\lambda)$ , so in principle there is no obstruction to determining this measure for the topological model of this paper. We leave this explicit determination for future work.

— Topological entanglement entropy in 3D gravity  (2609.19247 - Balasubramanian et al., 16 Sep 2026) in Appendix A, final subsection, “What labels the ribbons?”