Reconstruction of Rigidly Generated Symmetric W*-Multitensor Categories

Prove that every rigidly generated symmetric $W^*$-multitensor category is equivalent to the representation category of a symmetry algebra, thereby providing the reconstruction needed to construct a single reducible unitary quantum field theory extending a non-factorizing gravitational path integral.

Background

The paper seeks to extend the reconstruction known for multifusion categories to the more general setting of rigidly generated symmetric W∗W^*-multitensor categories. Such a result would produce a symmetry algebra whose representation category realizes the baby-universe category associated with a gravitational path integral.

This reconstruction is needed to obtain a single quantum field theory valued in a generalized representation category, with the nontrivial unit object encoding baby-universe sectors. The paper states that the conjecture is not established in general, although the multifusion case is known.

References

Conjecture \ref{Lemma} therefore collapses to the following Conjecture: Given a rigidly generated symmetric $W*$-multitensor category $\mathcal{C}$, there exists a symmetry algebra $H$ such that $\mathcal{C} = \text{Rep}(H)$. In general, the validity of this conjecture is the subject of ongoing analysis for rather technical reasons as anticipated in Section 10.3.

— Toward a Unique Filter for the Gravitational Path Integral  (2609.20697 - Klinger, 17 Sep 2026) in Conjecture 1, “Reconstruction for Multitensor Categories,” Section 3.1