The 16-fold way conjecture for super-modular categories

Equivalent formulations 2

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. The 16-fold way conjecture for super-modular categories

   Let $\mathcal{B}$ be a super-modular category, meaning a braided fusion category whose Müger center is equivalent to the category of super vector spaces. A minimal unitary modular extension is a unitary modular category extending $\mathcal{B}$ with minimal global dimension.

   16-fold way conjecture. Let $\mathcal{B}$ be super-modular. Then $\mathcal{B}$ has precisely $16$ minimal unitary modular extensions up to ribbon equivalence.

   The conjecture expresses the expected sixteen possible gaugings of fermion parity for a fermionic topological phase. The supplied text presents it as a conjecture and gives no resolution evidence.

   source: Paul Bruillard, Cesar Galindo, Tobias Hagge, Siu-Hung Ng, Julia Yael Plavnik, Eric C. Rowell and Zhenghan Wang, “Fermionic Modular Categories and the 16-fold Way”, arXiv:1603.09294 (2017).

2. The 16-fold way conjecture for super-modular categories

   Let $\mathcal{B}$ be a super-modular category, meaning a unitary pre-modular category whose Müger center is equivalent to $\operatorname{sVec}$. 16-fold way conjecture. $\mathcal{B}$ has precisely $16$ minimal unitary modular extensions. The source explains this as the expected unobstructed gauging of fermion parity and does not give a proof.

   source: Eric C. Rowell and Zhenghan Wang, “Mathematics of Topological Quantum Computing”, arXiv:1705.06206 (2017).