Schedule for: 16w5049 - Modular Categories--Their Representations, Classification, and Applications

A welcome drink will be served at the hotel.

In this talk I will review known results and open questions about modular tensor categories related with quantum groups at roots of unity. We will emphasize the structural results and module categories.

A physical system is said to be in topological phase if at low energies and long wavelengths the physical observables are invariant under smooth deformations. These physical systems have applications in a wide range of disciplines, especially in quantum information science. A quantum computer based on such systems are topologically protected from decoherence. This fault-tolerance removes the need for expensive error--correcting codes required by the qubit model. Topological phases of matter can be studied through their algebraic manifestations, modular categories. Thus, a complete classification of these categories would provide a taxonomy of admissible topological phases. In this talk we will discuss connections between modular categories and number theory. This connection allows one to show a finiteness result for modular categories that makes classification tractable, and provides new tools for classification. Time permitting we will cover a generalization of these finiteness results to premodular categories. Information Release: PNNL-SA-120321

Modular categories (or non-degenerate braided fusion categories) can be thought of as generalisations of quadratic spaces. All essential features of the theory of quadratic spaces have analogues for modular categories. In particular it is possible to define Witt equivalence for modular categories. Witt classes form a group. This group is a convenient tool for classifying modular categories. Higher category structure of defects in 3d TFTs and 2d RCFTs indicate to the existence of a 3-category with modular categories as objects. This 3-category provides a higher categorification of the Witt group of modular categories.

In this talk we will give a panorama about the state of the problem of classification of weakly integral modular categories at the moment. We will present some of the known results for low rank and for specific dimensions, like 4m and 8m, with m a square-free odd integer. We will also explain some of the techniques that we found useful to push further the classification.

A conformal field theory is a quantum field theory with extra symmetries (namely the conformal group). It can also be thought of as an invariant attached to a quantum field theory which captures its limiting behaviour at long distances or at short distances. Several quantum field theory can lead to the same conformal field theory at long (or short) distances, a phenomenon called "universality". Similarly, conformal field theories appear in the description of long range correlation functions of critical lattice models. In this overview talk I would like to illustrate the above ideas in the special case of two dimensions and explain the use of the extended symmetries in understanding field theories.

This talk will report our resent work on orbifod theory. The Schur-Weyl duality, generalized moonshine and classification of irreducible modules for the orbifold theory will be discussed.

We present an operator algebraic formulation of chiral conformal field theory and show how it is related to subfactor theory. Appearance of a modular tensor category through representation theory and the role of alpha-induction machinery are explained. We also exhibit the current status of the relations between our operator algebraic approach and the one based on vertex operator algebras.

Given a fusion category C, the Brauer-Picard group BrPic(C) is the group of equivalence classes of invertible C-bimodule categories. It is an important invariant of C and appears as a key ingredient in group extensions of fusion categories. From a physics point of view, this group is also interesting, since it is the symmetry group of certain 3-dimensional topological quantum field theories. In this talk, I would like to present an approach to calculate the Brauer-Picard group of the representation category of a finite group by providing a natural decomposition.

Given a factorizable finite ribbon category D, by work of Lyubashenko one can associate to any punctured surface M a functor Bl_M from a tensor power of D to the category of finite-dimensional vector spaces. The so obtained vector spaces Bl_M(-) carry representations of the mapping class groups Map(M) and are compatible with sewing, in much the same way as the spaces of conformal blocks of a (semisimple) rational conformal field theory. I will present a natural construction which, given any object F of D, selects vectors in all space Bl_M(F,...,F) (i.e. when all punctures on M are labeled by F). If and only if the object F carries a structure of a 'modular' commutative symmetric Frobenius algebra in D, the vectors obtained by this construction are invariant under the mapping class group actions and are mapped to each other upon sewing. Thereby they are natural candidates for the bulk correlators of a conformal field theory with bulk state space given by F. (Joint with C. Schweigert, arxiv 1604.01143)

I discuss a class of topological quantum field theories that I call "positive TQFTs". In these the objects associated to hypersurfaces are partially ordered vector spaces and the morphisms associated to cobordisms are completely positive linear maps. I describe a functorial construction that coverts "ordinary" TQFTs into positive TQFTs by "taking the modulus square". The mathematical exploration of positive TQFTs is wide open. Physically, positive TQFTs arise from an operational description of both classical statistical field theory and quantum statistical field theory. There, the "modulus square" construction converts a quantum field theory in the sense of Segal into a mixed state counterpart. The latter is of interest for the foundations of quantum theory and for quantum gravity.

Izumi classified the non-commutative near-group fusion categories, that is, fusion categories with one non-invertible simple object and a non-abelian group of invertible objects. He showed, in particular, that the groups of invertible simple objects must be extra-special two groups. I will discuss joint work with Izumi toward computing the Frobenius-Schur indicators for these categories. We also establish some realizations of these categories as representation categories for bi-crossed product quasi-Hopf algebras.

Quantum objects and their noncommutative algebras of functions have been ubitiquous in mathematics and physics– thus, an appropriate notion of symmetry is needed. As explained by Drinfel′d in his 1986 ICM address, replacing group actions with actions of Hopf algebras is an effective approach. For instance, one can deform group actions on classical objects (resp. commutative function algebras) to obtain (co)actions of Hopf algebras on quantum objects (resp. noncommutative function algebras). But this certainly does not account for all instances of Quantum Symmetry. In this talk, I will provide a survey of recent results on Hopf (co)actions on algebras. Classification results, including some from the analytic context, and many examples will be included.

We will discuss invariance of the order of the antipode, and traces of the powers of the antipode, under gauge equivalence. In particular, we will see that these values are in fact gauge invariants for Hopf algebras with the Chevalley property (e.g. Taft algebras and duals of pointed Hopf algebras). If time permits we will discuss how our study relates to recent efforts of Shimizu to produce a categorial approach to the indicators of a non-semisimple tensor category. This is joint work with Richard Ng.

Abstract: Topological quantum field theories are invariants of manifolds which can be computed via cutting and gluing. This can be rephrased as a symmetric monoidal functor from a bordism category to the category . This connection can be exploited in both directions: using algebra to construct topological invariants, or using topology to prove algebraic theorems. An important generalization of the notion of an extended topological field theory, where one now allows further cutting along lower dimensional submanifolds. This can be again rephrased in terms of a functor from a symmetric monoidal n-category to a more algebraic n-category. Modular tensor categories appear very naturally when studying 321 extended field theories. The connection between MTC and TQFTS is originally due to Reshetikhin--Turaev, and I will take an approach inspired by more recent work of Bartlett--Douglas--Schommer-Pries--Vicary. I will begin with some simpler examples in lower dimensions.

We give a construction of Turaev-Viro type (3+1)-dimensional Topological Quantum Field Theory out of a G-crossed braided spherical fusion category for G a finite group. The resulting invariant of 4-manifolds generalizes several known invariants such as the Crane-Yetter invariant and Yetter's invariant from homotopy 2-types. Some concrete examples will be provided to show the calculations. If the category is concentrated only at the sector indexed by the trivial group element, a co-cycle in H^4(G,U(1)) can be introduced to produce another invariant, which reduces to the twisted Dijkgraaf-Witten theory in a special case. It can be shown that with a G-crossed braided spherical fusion category, one can construct a monoidal 2-category with certain extra structure, but these structures do not satisfy all the axioms of a spherical 2-category given by M. Mackaay. Although not proven, it is believed that our invariant is strictly different from other known invariants. It remains to see if the invariant has the power to detect any smooth structures.

Abstract: For p>3 a prime, and g >2 an integer, we use Topological Quantum Field Theory (TQFT) to study a family of p-1 highest weight modules L_p(lambda) for the symplectic group Sp(2g,K) where K is an algebraically closed field of characteristic p. This permits explicit formulae for the dimension and the formal character of L_p(lambda) for these highest weights. This is joint work with Gregor Masbaum.

It has been well-known for some time that there are generalizations of Schur-Weyl duality between vector representations of quantum groups of classical Lie types and the braid groups. More recently, this has been extended to tensor powers of spinor representations, where it is more convenient to replace the braid groups by certain non-standard deformations of orthogonal groups. We review these results and discuss applications towards identifying and classifying categories for certain fusion rings.

The classification of small index subfactors has resulted in the discovery of some rather unusual fusion categories. Those coming from the extended Haagerup subfactor seem particularly interesting --- at this point we know of no relationship to any family or standard construction. As part of the effort to understand these unusual objects, we have computed the modular data for the centres of these fusion categories. As it turns out, we need to know remarkably little about the fusion categories; the conditions on modular data are so restrictive that we can leverage information about the Galois action and the representation theory of SL(2,Z) to completely determine the modular data. Time permitting, I'll indicate the range of examples we've since tested these techniques against.

A well-known result in the theory of tensor categories states that the category of braided equivalence classes of pointed braided fusion categories is equivalent to the category of pre-metric groups. Objects in this latter category are pairs $(\Gamma, q)$, where $\Gamma$ is a finite abelian group and $q$ is a quadratic form on $\Gamma$.

In my talk i'll explain how to find vector-valued modular forms whose multiplier is the modular data of a modular tensor category, and how that can help us reconstruct a rational vertex operator algebra from that category.

In this talk, I will present work in progress with Zhenghan Wang on the subject of classification of vertex operator algebras whose representation theory is given by a fixed modular tensor category. Perhaps the best studied problem in this area is the classification of "holomorphic" vertex operator algebras, i.e. those VOAs whose representation theory is simply Vec, with small central charge. However in this talk we will focus on a complementary problem, that of classifying, modulo holomorphic VOAs, those VOAs whose representation theory is given by a fixed non-trivial modular tensor category. Using techniques of Terry Gannon, we will explore the landscape of this classification problem for some small modular tensor categories.

A Hopf algebra H over a field k is called co-Frobenius if it possesses a nonzero right (or left) integral R : H → k. The existence of a nonzero integral amounts to each one of the following conditions on the category C of finite-dimensional right (or left) H-comodules: • C has a nonzero injective object. • C has injective hulls. • C has a nonzero projective object. • C has projective covers. A tensor category satisfying one of these (equivalent) conditions is called Frobenius. In this case, every injective object is projective and vice versa. Radford showed in [3] that a co-Frobenius Hopf algebra whose coradical is a subalgebra has finite coradical filtration. Andruskiewitsch and D˘asc˘alescu proved later in [2] that a Hopf algebra with finite coradical filtration is necessarily co-Frobenius and they conjectured that any co-Frobenius Hopf algebra has finite coradical filtration. In this talk we will show that this conjecture admits a categorical formulation and we will answer it in the affirmative. The idea of the proof is to provide a uniform bound on the composition length of the indecomposable injective objects in terms of the composition series of the injective hull of the unit object. This result is joint with Nicol´as Andruskiewitsch and Pavel Etingof and appears in [1]. References [1] N. Andruskiewitsch, J. Cuadra, and P. Etingof, On two finiteness conditions for Hopf algebras with nonzero integral. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) Vol. XIV, 401-440. [2] N. Andruskiewitsch and S. D˘asc˘alescu, Co-Frobenius Hopf algebras and the coradical filtration. Math. Z. 243 (2003), 145-154. [3] D. E. Radford, Finiteness conditions for a Hopf algebra with a nonzero integral. J. Algebra 46 (1977), 189-195.