Minimal nondegenerate extensions

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 $\mathcal {Z}({_{}\mathrm {Mod}\text {-}\mathcal {B}})$ of the fusion $2$-category ${_{}\mathrm {Mod}\text {-}\mathcal {B}}$ of module categories of a braided fusion $1$-category $\mathcal {B}$. We show that minimal nondegenerate extensions of $\mathcal {B}$ correspond to certain trivializations of $\mathcal {Z}({_{}\mathrm {Mod}\text {-}\mathcal {B}})$. In the slightly degenerate case, such trivializations are obstructed by a class in $H^5(K(\mathbb {Z}_2, 2); \mathbb {k}^\times )$ 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 $\mathcal {Z}({_{}\mathrm {Mod}\text {-}\mathcal {B}})$ in terms of braided $\mathcal {B}$-module categories, we develop a computationally useful model in terms of certain algebra objects in $\mathcal {B}$. We construct an $S$-matrix pairing for any braided fusion $2$-category, and show that it is nondegenerate for $\mathcal {Z}({_{}\mathrm {Mod}\text {-}\mathcal {B}})$. As a corollary, we identify components of $\mathcal {Z}({_{}\mathrm {Mod}\text {-}\mathcal {B}})$ with blocks in the annular category of $\mathcal {B}$ and with the homomorphisms from the Grothendieck ring of the Müger centre of $\mathcal {B}$ to the ground field.