Minimal nondegenerate extensions
Theo Johnson-Freyd, David Reutter
TL;DR
The paper resolves a long-standing problem in braided fusion categories by proving that every slightly degenerate braided fusion category admits a minimal nondegenerate extension, using the theory of fusion 2-categories and the Drinfel'd center of their suspensions. It develops explicit models for the Drinfel'd center via half-braided algebras and braided module categories, and leverages higher Morita theory to relate centers to nondegenerate extensions through Lagrangian objects. A key technical achievement is showing the obstruction in $H^5(K(\mathbb Z_2,2);k^\times)$ vanishes in the slightly degenerate case, leading to a constructive procedure for obtaining the extension; the work also extends to pseudo-unitary settings, yielding pseudo-unitary minimal modular extensions for pseudo-unitary super-modular categories. The methods provide concrete computational tools, including an $S$-matrix-like pairing and eta-traces, and connect to physical interpretations in 3+1D topological phases through obstruction theory and Klein bottle invariants. Overall, the results complete the characterization of minimal nondegenerate extensions, offer explicit construction techniques, and establish new higher-categorical machinery for analyzing modular extensions in braided fusion categories.
Abstract
We prove that every slightly degenerate braided fusion category admits a minimal nondegenerate extension, and hence that every pseudo-unitary super modular tensor category admits a minimal modular extension. This completes the program of characterizing minimal nondegenerate extensions of braided fusion categories. Our proof relies on the new subject of fusion 2-categories. We study in detail the Drinfel'd centre Z(Mod-B) of the fusion 2-category Mod-B of module categories of a braided fusion 1-category B. We show that minimal nondegenerate extensions of B correspond to certain trivializations of Z(Mod-B). In the slightly degenerate case, such trivializations are obstructed by a class in $\mathrm{H}^5(K(\mathbb{Z}_2, 2); k^\times)$ and we use a numerical invariant -- defined by evaluating a certain two-dimensional topological field theory on a Klein bottle -- to prove that this obstruction always vanishes. Along the way, we develop techniques to explicitly compute in braided fusion 2-categories which we expect will be of independent interest. In addition to the model of Z(Mod-B) in terms of braided B-module categories, we develop a computationally useful model in terms of certain algebra objects in B. We construct an S-matrix pairing for any braided fusion 2-category, and show that it is nondegenerate for Z(Mod-B). As a corollary, we identify components of Z(Mod-B) with blocks in the annular category of B and with the homomorphisms from the Grothendieck ring of the Müger centre of B to the ground field.