On Soliton Automorphisms in Massive and Conformal Theories

For massive and conformal quantum field theories in 1+1 dimensions with a global gauge group we consider soliton automorphisms, viz. automorphisms of the quasilocal algebra which act like two different global symmetry transformations on the left and right spacelike complements of a bounded region. We give a unified treatment by providing a necessary and sufficient condition for the existence and Poincare' covariance of soliton automorphisms which is applicable to a large class of theories. In particular, our construction applies to the QFT models with the local Fock property -- in which case the latter property is the only input from constructive QFT we need -- and to holomorphic conformal field theories. In conformal QFT soliton representations appear as twisted sectors, and in a subsequent paper our results will be used to give a rigorous analysis of the superselection structure of orbifolds of holomorphic theories.Comment: latex2e, 20 pages. Proof of Thm. 3.14 corrected, 2 references added. Final version as to appear in Rev. Math. Phy

Monoids, Embedding Functors and Quantum Groups

We show that the left regular representation \pi_l of a discrete quantum group (A,\Delta) has the absorbing property and forms a monoid (\pi_l,\tilde{m},\tilde{\eta}) in the representation category Rep(A,\Delta). Next we show that an absorbing monoid in an abstract tensor *-category C gives rise to an embedding functor E:C->Vect_C, and we identify conditions on the monoid, satisfied by (\pi_l,\tilde{m},\tilde{\eta}), implying that E is *-preserving. As is well-known, from an embedding functor E: C->\mathrm{Hilb} the generalized Tannaka theorem produces a discrete quantum group (A,\Delta) such that C is equivalent to Rep_f(A,\Delta). Thus, for a C^*-tensor category C with conjugates and irreducible unit the following are equivalent: (1) C is equivalent to the representation category of a discrete quantum group (A,\Delta), (2) C admits an absorbing monoid, (3) there exists a *-preserving embedding functor E: C->\mathrm{Hilb}.Comment: Final version, to appear in Int. Journ. Math. (Added some references and Subsection 1.2.) Latex2e, 21 page

Cleft Extensions and Quotients of Twisted Quantum Doubles

Given a pair of finite groups $F, G$ and a normalized 3-cocycle $\omega$ of $G$, where $F$ acts on $G$ as automorphisms, we consider quasi-Hopf algebras defined as a cleft extension $\Bbbk^G_\omega\#_c\,\Bbbk F$ where $c$ denotes some suitable cohomological data. When $F\rightarrow \overline{F}:=F/A$ is a quotient of $F$ by a central subgroup $A$ acting trivially on $G$, we give necessary and sufficient conditions for the existence of a surjection of quasi-Hopf algebras and cleft extensions of the type $\Bbbk^G_\omega\#_c\, \Bbbk F\rightarrow \Bbbk^G_\omega\#_{\overline{c}} \, \Bbbk \overline{F}$. Our construction is particularly natural when $F=G$ acts on $G$ by conjugation, and $\Bbbk^G_\omega\#_c \Bbbk G$ is a twisted quantum double $D^{\omega}(G)$. In this case, we give necessary and sufficient conditions that Rep($\Bbbk^G_\omega\#_{\overline{c}} \, \Bbbk \overline{G}$) is a modular tensor category.Comment: LaTex; 14 page

Lagrangian subcategories and braided tensor equivalences of twisted quantum doubles of finite groups

We classify Lagrangian subcategories of the representation category of a twisted quantum double of a finite group. In view of results of 0704.0195v2 this gives a complete description of all braided tensor equivalent pairs of twisted quantum doubles of finite groups. We also establish a canonical bijection between Lagrangian subcategories of the representation category of a twisted quantum double of a finite group G and module categories over the category of twisted G-graded vector spaces such that the dual tensor category is pointed. This can be viewed as a quantum version of V. Drinfeld's characterization of homogeneous spaces of a Poisson-Lie group in terms of Lagrangian subalgebras of the double of its Lie bialgebra. As a consequence, we obtain that two group-theoretical fusion categories are weakly Morita equivalent if and only if their centers are equivalent as braided tensor categories.Comment: 26 pages; several comments and references adde

A reason for fusion rules to be even

We show that certain tensor product multiplicities in semisimple braided sovereign tensor categories must be even. The quantity governing this behavior is the Frobenius-Schur indicator. The result applies in particular to the representation categories of large classes of groups, Lie algebras, Hopf algebras and vertex algebras.Comment: 6 pages, LaTe

On the extension of stringlike localised sectors in 2+1 dimensions

In the framework of algebraic quantum field theory, we study the category \Delta_BF^A of stringlike localised representations of a net of observables O \mapsto A(O) in three dimensions. It is shown that compactly localised (DHR) representations give rise to a non-trivial centre of \Delta_BF^A with respect to the braiding. This implies that \Delta_BF^A cannot be modular when non-trival DHR sectors exist. Modular tensor categories, however, are important for topological quantum computing. For this reason, we discuss a method to remove this obstruction to modularity. Indeed, the obstruction can be removed by passing from the observable net A(O) to the Doplicher-Roberts field net F(O). It is then shown that sectors of A can be extended to sectors of the field net that commute with the action of the corresponding symmetry group. Moreover, all such sectors are extensions of sectors of A. Finally, the category \Delta_BF^F of sectors of F is studied by investigating the relation with the categorical crossed product of \Delta_BF^A by the subcategory of DHR representations. Under appropriate conditions, this completely determines the category \Delta_BF^F.Comment: 36 pages, 1 eps figure; v2: appendix added, minor corrections and clarification

Rigid C^*-tensor categories of bimodules over interpolated free group factors

Given a countably generated rigid C^*-tensor category C, we construct a planar algebra P whose category of projections Pro is equivalent to C. From P, we use methods of Guionnet-Jones-Shlyakhtenko-Walker to construct a rigid C^*-tensor category Bim whose objects are bifinite bimodules over an interpolated free group factor, and we show Bim is equivalent to Pro. We use these constructions to show C is equivalent to a category of bifinite bimodules over L(F_infty).Comment: 50 pages, many figure

The fusion algebra of bimodule categories

We establish an algebra-isomorphism between the complexified Grothendieck ring F of certain bimodule categories over a modular tensor category and the endomorphism algebra of appropriate morphism spaces of those bimodule categories. This provides a purely categorical proof of a conjecture by Ostrik concerning the structure of F. As a by-product we obtain a concrete expression for the structure constants of the Grothendieck ring of the bimodule category in terms of endomorphisms of the tensor unit of the underlying modular tensor category.Comment: 16 page

Level-rank duality via tensor categories

We give a new way to derive branching rules for the conformal embedding (\asl_n)_m\oplus(\asl_m)_n\subset(\asl_{nm})_1. In addition, we show that the category \Cc(\asl_n)_m^0 of degree zero integrable highest weight (\asl_n)_m-representations is braided equivalent to \Cc(\asl_m)_n^0 with the reversed braiding.Comment: 16 pages, to appear in Communications in Mathematical Physics. Version 2 changes: Proof of main theorem made explicit, example 4.11 removed, references update

Bicategories for boundary conditions and for surface defects in 3-d TFT

We analyze topological boundary conditions and topological surface defects in three-dimensional topological field theories of Reshetikhin-Turaev type based on arbitrary modular tensor categories. Boundary conditions are described by central functors that lift to trivializations in the Witt group of modular tensor categories. The bicategory of boundary conditions can be described through the bicategory of module categories over any such trivialization. A similar description is obtained for topological surface defects. Using string diagrams for bicategories we also establish a precise relation between special symmetric Frobenius algebras and Wilson lines involving special defects. We compare our results with previous work of Kapustin-Saulina and of Kitaev-Kong on boundary conditions and surface defects in abelian Chern-Simons theories and in Turaev-Viro type TFTs, respectively.Comment: 34 pages, some figures. v2: references added. v3: typos corrected and biliography update