Condensation inversion and Witt equivalence via generalised orbifolds

Abstract

In Mulevi\v{c}ius-Runkel, arXiv:2002.00663, it was shown how a so-called orbifold datum $\mathbb{A}$ in a given modular fusion category (MFC) $\mathcal{C}$ produces a new MFC $\mathcal{C}_{\mathbb{A}}$. Examples of these associated MFCs include condensations, i.e. the categories $\mathcal{C}_B^\circ$ of local modules of a separable commutative algebra $B\in\mathcal{C}$. In this paper we prove that the relation $\mathcal{C} \sim \mathcal{C}_{\mathbb{A}}$ on MFCs is the same as Witt equivalence. This is achieved in part by providing one with an explicit construction for inverting condensations, i.e. finding an orbifold datum $\mathbb{A}$ in $\mathcal{C}_B^\circ$ whose associated MFC is equivalent to $\mathcal{C}$. As a tool used in this construction we also explore what kinds of functors $F\colon\mathcal{C}\rightarrow\mathcal{D}$ between MFCs preserve orbifold data. It turns out that $F$ need not necessarily be strong monoidal, but rather a `ribbon Frobenius' functor, which has weak monoidal and weak comonoidal structures, related by a Frobenius-like property.