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