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We describe Lagrangian algebras in twisted Drinfeld centres for finite groups. Using the full centre construction, we establish a 1-1 correspondence between Lagrangian algebras and module categories over pointed fusion categories. [ABSTRACT FROM AUTHOR]

A modular invariant for a chiral conformal field theory is physical if there is a full conformal field theory with the given chiral halves realising the modular invariant. The easiest modular invariants are the charge conjugation and the diagonal modular invariants. While the charge conjugation modular invariant is always physical there are examples of chiral CFTs for which the diagonal modular invariant is not physical. Here we give (in group theoretical terms) a necessary and sufficient condition for diagonal modular invariants of G -orbifolds of holomorphic conformal field theories to be physical. Mathematically a physical modular invariant is an invariant of a Lagrangian algebra in the product of (chiral) modular categories. The chiral modular category of a G -orbifold of a holomorphic conformal field theory is the so-called (twisted) Drinfeld centre Z ( G , α ) of the finite group G . We show that the diagonal modular invariant for Z ( G ) is physical if and only if the group G has a double class inverting automorphism, that is an automorphism ϕ : G → G with the property that for any commuting x , y ∈ G there is g ∈ G such that ϕ ( x ) = g x − 1 g − 1 , ϕ ( y ) = g y − 1 g − 1 . Groups without double class inverting automorphisms are abundant and provide examples of chiral conformal field theories for which the diagonal modular invariant is unphysical. [ABSTRACT FROM AUTHOR]

The notion of the center of an algebra over a field k has a far reaching generalization to algebras in monoidal categories. The center then lives in the monoidal center of the original category. In this paper, we study functorial properties of the center. We show that it gives rise to a 2-functor from the bicategory of semisimple indecomposable module categories over a fusion category to the bicategory of commutative algebras in the monoidal center of this fusion category. Morphism spaces of the latter bicategory are extended from algebra homomorphisms to certain categories of cospans. We conjecture that the above 2-functor arises from a lax 3-functor between tricategories, and that in this setting one can relax the conditions from fusion categories to finite tensor categories. Our construction is motivated by two-dimensional conformal field theory enriched by defects of all codimensions. [ABSTRACT FROM AUTHOR]

Abstract: Starting with a self-dual Hopf algebra in a braided monoidal category we construct a -graded monoidal category . The degree zero component is the category of representations of and the degree one component is the category . The extra structure on needed to define the associativity isomorphisms is a choice of self-duality map and cointegral, subject to certain conditions. We also describe rigid, braided and ribbon structures on in Hopf algebraic terms. Our construction permits a uniform treatment of Tambara–Yamagami categories and categories related to symplectic fermions in conformal field theory. [Copyright &y& Elsevier]

Abstract: By studying NIM-representations we show that the Fibonacci category and its tensor powers are completely anisotropic; that is, they do not have any non-trivial separable commutative ribbon algebras. As an application we deduce that a chiral algebra with the representation category equivalent to a product of Fibonacci categories is maximal; that is, it is not a proper subalgebra of another chiral algebra. In particular the chiral algebras of the Yang–Lee model, the WZW models of and at level 1, as well as their tensor powers, are maximal. [Copyright &y& Elsevier]

Abstract: Motivated by algebraic structures appearing in Rational Conformal Field Theory we study a construction associating to an algebra in a monoidal category a commutative algebra (full centre) in the monoidal centre of the monoidal category. We establish Morita invariance of this construction by extending it to module categories. As an example we treat the case of group-theoretical categories. [Copyright &y& Elsevier]

Abstract: We classify indecomposable commutative separable (special Frobenius) algebras and their local modules in (untwisted) group-theoretical modular categories. This gives a description of modular invariants for group-theoretical modular data. As a bi-product we provide an answer to the question when (and in how many ways) two group-theoretical modular categories are equivalent as ribbon categories. [Copyright &y& Elsevier]