In Carqueville et al., arXiv:1809.01483, the notion of an orbifold datum A in a modular fusion category C was introduced as part of a generalised orbifold construction for Reshetikhin-Turaev TQFTs. In this paper, given a simple orbifold datum A in C, we introduce a ribbon category C A and show that it is again a modular fusion category. The definition of C A is motivated by properties of Wilson lines in the generalised orbifold. We analyse two examples in detail: (i) when A is given by a simple commutative ∆-separable Frobenius algebra A in C; (ii) when A is an orbifold datum in C = Vect, built from a spherical fusion category S. We show that in case (i), C A is ribbon-equivalent to the category of local modules of A, and in case (ii), to the Drinfeld centre of S. The category C A thus unifies these two constructions into a single algebraic setting.

An orbifold datum is a collection A of algebraic data in a modular fusion category C. It allows one to define a new modular fusion category C A in a construction that is a generalisation of taking the Drinfeld centre of a fusion category. Under certain simplifying assumptions we characterise orbifold data A in terms of scalars satisfying polynomial equations and give an explicit expression which computes the number of isomorphism classes of simple objects in C A .In Ising-type modular categories we find new examples of orbifold data which -in an appropriate sense -exhibit Fibonacci fusion rules. The corresponding orbifold modular categories have 11 simple objects, and for a certain choice of parameters one obtains the modular category for sl(2) at level 10. This construction inverts the extension of the latter category by the E 6 commutative algebra.

In [MR1] it was shown how a so-called orbifold datum A in a given modular fusion category (MFC) C produces a new MFC C A . Examples of these associated MFCs include condensations, i.e. the categories C • B of local modules of a separable commutative algebra B ∈ C. In this paper we prove that the relation C ∼ C 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 A in C • B whose associated MFC is equivalent to C. As a tool used in this construction we also explore what kinds of functors F : C → 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.

We study surface defects in three-dimensional topological quantum field theories which separate different theories of Reshetikhin–Turaev type. Based on the new notion of a Frobenius algebra over two commutative Frobenius algebras, we present an explicit and computable construction of such defects. It specialises to the construction in Carqueville et al. (Geom Topol 23:781–864, 2019. https://doi.org/10.2140/gt.2019.23.781. arXiv:1705.06085) if all 3-strata are labelled by the same topological field theory. We compare the results to the model-independent analysis in Fuchs et al. (Commun Math Phys 321:543–575, 2013. https://doi.org/10.1007/s00220-013-1723-0. arXiv:1203.4568) and find agreement.

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