Search

We introduce the notion of n-commutativity (0 ≤ n ≤ ∞) for cosimplicial monoids in a symmetric monoidal category V, where n = 0 corresponds to just cosimplicial monoids in V, while n = ∞ corresponds to commutative cosimplicial monoids. When V has a monoidal model structure, we endow (under some mild technical conditions) the total object of an n-cosimplicial monoid with a natural and very explicit En+1-algebra structure. Our main applications are to the deformation theory of tensor categories and tensor functors. We show that the deformation complex of a tensor functor is a total complex of a 1 -commutative cosimplicial monoid and, hence, has an E2-algebra structure similar to the E2-structure on Hochschild complex of an associative algebra provided by Deligne's conjecture. We further demonstrate that the deformation complex of a tensor category is the total complex of a 2-commutative cosimplicial monoid and, therefore, is naturally an E3-algebra. We make these structures very explicit through a language of Delannoy paths and their noncommutative liftings. We investigate how these structures manifest themselves in concrete examples. [ABSTRACT FROM AUTHOR]

We establish an action of the representations of N = 2-superconformal symmetry on the category of matrix factorisations of the potentials xd and xd - yd, for d odd. More precisely we prove a tensor equivalence between (a) the category of Neveu-Schwarz-type representations of the N = 2 minimal super vertex operator algebra at central charge 3-6/d, and (b) a full subcategory of graded matrix factorisations of the potential xd - yd. The subcategory in (b) is given by permutation-type matrix factorisations with consecutive index sets. The physical motivation for this result is the Landau-Ginzburg/conformal field theory correspondence, where it amounts to the equivalence of a subset of defects on both sides of the correspondence. Our work builds on results by Brunner and Roggenkamp [BR], where an isomorphism of fusion rules was established. [ABSTRACT FROM AUTHOR]

It was observed recently that for a fixed finite group G, the set of all Drinfeld centres of G twisted by 3-cocycles form a group, the so-called group of modular extensions (of the representation category of G), which is isomorphic to the third cohomology group of G. We show that for an abelian G, pointed twisted Drinfeld centres of G form a subgroup of the group of modular extensions. We identify this subgroup with a group of quadratic extensions containing G as a Lagrangian subgroup, the so-called group of Lagrangian extensions of G. We compute the group of Lagrangian extensions, thereby providing an interpretation of the internal structure of the third cohomology group of an abelian G in terms of fusion categories. Our computations also allow us to describe associators of Lagrangian algebra in pointed braided fusion categories. [ABSTRACT FROM AUTHOR]

We describe the group Aut¹br(Ƶ(G)) of braided tensor autoequivalences of the Drinfeld centre of a finite group G isomorphic to the identity functor (just as a functor). We prove that the semi-direct product Out2-cl(G) ⋉ B(G) of the group of double class preserving automorphisms and the Bogomolov multiplier of G is a subgroup of Aut¹br(Ƶ(G)) . An automorphism of G is double class preserving if it preserves conjugacy classes of pairs of commuting elements in G. The Bogomolov multiplier B(G) is the subgroup of its Schur multiplier H²(G, k*) of classes vanishing on abelian subgroups of G. We show that elements of Aut¹br(Ƶ(G)) give rise to different realisations of the charge conjugation modular invariant for G-orbifolds of holomorphic conformal field theories. [ABSTRACT FROM AUTHOR]

We give a characterization of Drinfeld centers of fusion categories as non-degenerate braided fusion categories containing a Lagrangian algebra. Further we study the quotient of the monoid of non-degenerate braided fusion categories modulo the submonoid of the Drinfeld centers and show that its formal properties are similar to those of the classical Witt group. [ABSTRACT FROM AUTHOR]

We describe (braided-) commutative algebras with non-degenerate multiplicative form in certain braided monoidal categories, corresponding to abelian metric Lie algebras (so-called Drinfeld categories). We also describe local modules over these algebras and classify commutative algebras with a finite number of simple local modules. [ABSTRACT FROM PUBLISHER]

We apply the full centre construction, defined in [4], to algebras in and module categories over categories of representations of Hopf algebras. We obtain a compact formula for the full centre of a module algebra over a Hopf algebra. [ABSTRACT FROM PUBLISHER]