Michael Müger

From subfactors to categories and topology I: Frobenius algebras in and Morita equivalence of tensor categories

We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground fields, and the proof of new results about topological invariants in three dimensions. The central notion is that of a Frobenius algebra in a tensor category A, which reduces to the classical notion if A = F-Vect, where F is a field. An object X ∈ A with two-sided dual X gives rise to a Frobenius algebra in A, and under weak additional conditions we prove a converse: There exists a bicategory E with ObjE = {A, B} such that EndE(A) ⊗ ≃ A and such that there are J, J : B ⇋ A producing the given Frobenius algebra. Many properties (additivity, sphericity, semisimplicity, . . . ) of A carry over to the bicategory E. We define weak monoidal Morita equivalence of tensor categories, denoted A ≈ B, and establish a correspondence between Frobenius algebras in A and tensor categories B ≈ A. While considerably weaker than equivalence of tensor categories, weak monoidal Morita equivalence A ≈ B has remarkable consequences: A and B have equivalent (as braided tensor categories) quantum doubles (‘centers’) and (if A, B are semisimple spherical or ∗-categories) have equal dimensions and give rise the same state sum invariant of closed oriented 3-manifolds as recently defined by Barrett and Westbury. An instructive example is provided by finite dimensional semisimple and cosemisimple Hopf algebras, for which we prove H − mod ≈ ˆ H − mod. The present formalism permits a fairly complete analysis of the center of a semisimple spherical category, which is the subject of the companion paper math.CT/0111205.

Multi-Interval Subfactors and Modularity of Representations in Conformal Field Theory

Abstract: We describe the structure of the inclusions of factors ?(E)⊂?(E′)′ associated with multi-intervals E⊂ℝ for a local irreducible net ? of von Neumann algebras on the real line satisfying the split property and Haag duality. In particular, if the net is conformal and the subfactor has finite index, the inclusion associated with two separated intervals is isomorphic to the Longo–Rehren inclusion, which provides a quantum double construction of the tensor category of superselection sectors of ?. As a consequence, the index of ?(E)⊂?(E′)′ coincides with the global index associated with all irreducible sectors, the braiding symmetry associated with all sectors is non-degenerate, namely the representations of ? form a modular tensor category, and every sector is a direct sum of sectors with finite dimension. The superselection structure is generated by local data. The same results hold true if conformal invariance is replaced by strong additivity and there exists a modular PCT symmetry.

On the Structure of Modular Categories

For a braided tensor category C and a subcategory K there is a notion of centralizer CC(K), which is a full tensor subcategory of C. A pre-modular tensor category [7] is known to be modular in the sense of Turaev iff the center Z2(C) ≡ CC(C) (not to be confused with the center Z1 of a tensor category, related to the quantum double) is trivial, i.e. consists only of multiples of the tensor unit, and dim C 6 0. Here dim C = P i d(Xi) 2 , the Xi being the simple objects. We prove several structural properties of modular categories. Our main technical tool is the following double centralizer theorem. Let C be a modular category and K a full tensor subcategory closed w.r.t. direct sums, subobjects and duals. Then CC(CC(K)) = K

The Witt group of non-degenerate braided fusion categories

Abstract 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.

Superselection Structure of Massive Quantum Field Theories in 1+1 Dimensions

We show that a large class of massive quantum field theories in 1+1 dimensions, characterized by Haag duality and the split property for wedges, does not admit locally generated superselection sectors in the sense of Doplicher, Haag and Roberts. Thereby the extension of DHR theory to 1+1 dimensions due to Fredenhagen, Rehren and Schroer is vacuous for such theories. Even charged representations which are localizable only in wedge regions are ruled out. Furthermore, Haag duality holds in all locally normal representations. These results are applied to the theory of soliton sectors. Furthermore, the extension of localized representations of a non-Haag dual net to the dual net is reconsidered. It must be emphasized that these statements do not apply to massless theories since they do not satisfy the above split property. In particular, it is known that positive energy representations of conformally invariant theories are DHR representations.

Conformal Orbifold Theories and Braided Crossed G-Categories

The aim of the paper is twofold. First, we show that a quantum field theory A living on the line and having a group G of inner symmetries gives rise to a category G–Loc A of twisted representations. This category is a braided crossed G-category in the sense of Turaev [60]. Its degree zero subcategory is braided and equivalent to the usual representation category Rep A. Combining this with [29], where Rep A was proven to be modular for a nice class of rational conformal models, and with the construction of invariants of G-manifolds in [60], we obtain an equivariant version of the following chain of constructions: Rational CFT modular category 3-manifold invariant.Secondly, we study the relation between G–Loc A and the braided (in the usual sense) representation category Rep AG of the orbifold theory AG. We prove the equivalence RepAG≃(G–Loc A)G, which is a rigorous implementation of the insight that one needs to take the twisted representations of A into account in order to determine Rep AG. In the opposite direction we have is the full subcategory of representations of AG contained in the vacuum representation of A, and ⋊ refers to the Galois extensions of braided tensor categories of [44, 48].Under the assumptions that A is completely rational and G is finite we prove that A has g-twisted representations for every g∈ G and that the sum over the squared dimensions of the simple g-twisted representations for fixed g equals dim Rep A. In the holomorphic case this allows to classify the possible categories G− Loc A and to clarify the rôle of the twisted quantum doubles Dω(G) in this context, as will be done in a sequel. We conclude with some remarks on non-holomorphic orbifolds and surprising counterexamples concerning permutation orbifolds.

Quantum Double Actions on Operator Algebras and Orbifold Quantum Field Theories

Abstract:Starting from a local quantum field theory with an unbroken compact symmetry group G in 1+1-dimensional spacetime we construct disorder fields implementing gauge transformations on the fields (order variables) localized in a wedge region. Enlarging the local algebras by these disorder fields we obtain a nonlocal field theory, the fixpoint algebras of which under the appropriately extended action of the group G are shown to satisfy Haag duality in every simple sector. The specifically 1+1 dimensional phenomenon of violation of Haag duality of fixpoint nets is thereby clarified. In the case of a finite group G the extended theory is acted upon in a completely canonical way by the quantum double D(G) and satisfies R-matrix commutation relations as well as a Verlinde algebra. Furthermore, our methods are suitable for a concise and transparent approach to bosonization. The main technical ingredient is a strengthened version of the split property which is expected to hold in all reasonable massive theories. In the appendices (part of) the results are extended to arbitrary locally compact groups and our methods are adapted to chiral theories on the circle.

Monoids, Embedding Functors and Quantum Groups*

We show th a t th e left regular representation n l of a discrete q uantum group (A , A) has the absorbing property and forms a monoid (ni,m ,fj) in th e representation category Rep(A , A). N ext we show th a t an absorbing monoid in an ab stra ct tensor ^-category C gives rise to an em bedding functor (or fiber functor) E : C ^ Vectc, and we identify conditions on th e monoid, satisfied by (nl,r h,fj), im plying th a t E is ^-preserving. As is well-known, from an em bedding functor E : C ^ Hilb the generalized T annaka theorem produces a discrete quantum group (A, A) such th a t C ~ R epƒ (A, A ). Thus, for a C*-tensor category C w ith conjugates and irreducible u n it th e following are equivalent: (1) C is equivalent to the representation category of a discrete quantum group (A, A ), (2) C adm its an absorbing monoid, (3) there exists a ^-preserving em bedding functor E : C ^ Hilb.

DISORDER OPERATORS, QUANTUM DOUBLES, AND HAAG DUALITY IN 1 + 1 DIMENSIONS ∗

Since the notion of the ‘quantum double’ was coined by Drinfel’d in his famous ICM lecture [8] there have been several attempts aimed at a clarification of its relevance to two dimensional quantum field theory. The quantum double appears implicitly in the work [3] on orbifold constructions in conformai field theory, where conformal quantum field theories (CQFTs) are considered whose operators are fixpoints under the action of a symmetry group on another CQFT. Whereas the authors emphasize that ‘the fusion algebra of the holomorphic G-orbifold theory naturally combines both the representation and class algebra of the group G’ the relevance of the double is fully recognized only in [4]. The quantum double also appears in the context of integrable quantum field theories, e.g. [1], as well as in certain lattice models (e.g. [18]). Common to these works is the role of disorder operators or ‘twist fields’ which are ‘local with respect to A up to the action of an element g ∈ G’ [3].