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.