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

Beginning on Sunday, August 14 and ending Friday August 19, 2016

All times in Oaxaca, Mexico time, CDT (UTC-5).

Sunday, August 14 | |
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14:00 - 23:59 | Check-in begins (Front desk at your assigned hotel) |

19:30 - 22:00 | Dinner (Restaurant Hotel Hacienda Los Laureles) |

20:30 - 21:30 |
Informal gathering ↓ A welcome drink will be served at the hotel. (Hotel Hacienda Los Laureles) |

Monday, August 15 | |
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07:30 - 09:00 | Breakfast (Restaurant at your assigned hotel) |

09:30 - 09:45 | Introduction and Welcome (Conference Room San Felipe) |

09:45 - 11:00 |
Victor Ostrik: Introduction to modular tensor categories and those arising from quantum groups ↓ 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. (Conference Room San Felipe) |

11:00 - 11:30 | Coffee Break (Conference Room San Felipe) |

11:30 - 12:30 | Xiao-Gang Wen: Applications of braided fusion category to classify topological orders in 2-dimensional matter (Conference Room San Felipe) |

12:30 - 13:00 |
Paul Bruillard: Rank Finiteness for Premodular 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 (Conference Room San Felipe) |

13:20 - 13:30 | Group Photo (Hotel Hacienda Los Laureles) |

13:30 - 15:00 | Lunch (Restaurant Hotel Hacienda Los Laureles) |

15:00 - 16:00 |
Alexei Davydov: Witt group of modular categories ↓ 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. (Conference Room San Felipe) |

16:00 - 16:30 | Coffee Break (Conference Room San Felipe) |

16:30 - 17:00 |
Julia Plavnik: On the classification of weakly integral 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. (Conference Room San Felipe) |

18:00 - 20:00 | Dinner (Restaurant Hotel Hacienda Los Laureles) |

20:00 - 21:00 | problem Session I (Conference Room San Felipe) |

Tuesday, August 16 | |
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07:30 - 08:30 | Breakfast (Restaurant at your assigned hotel) |

09:00 - 10:00 |
Ingo Runkel: Conformal field theory and universality ↓ 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. (Conference Room San Felipe) |

10:00 - 10:30 | Coffee Break (Conference Room San Felipe) |

10:30 - 11:30 |
Chongying Dong: On orbifold theory ↓ 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. (Conference Room San Felipe) |

11:30 - 12:30 |
Yasuyuki Kawahigashi: Subfactors, conformal field theory and modular tensor categories ↓ 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. (Conference Room San Felipe) |

12:30 - 13:00 |
Jan Priel: Decomposition of the Brauer-Picard group ↓ 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. (Conference Room San Felipe) |

13:00 - 14:30 | Lunch (Restaurant Hotel Hacienda Los Laureles) |

14:30 - 15:30 |
Jurgen Fuchs: Correlators for non-semisimple conformal field theories ↓ 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) (Conference Room San Felipe) |

15:30 - 16:00 |
Robert Oeckl: Positive TQFT ↓ 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. (Conference Room San Felipe) |

16:00 - 16:30 | Coffee Break (Conference Room San Felipe) |

16:30 - 17:00 |
Henry Tucker: Frobenius-Schur indicators and modular data ↓ 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. (Conference Room San Felipe) |

18:00 - 20:00 | Dinner (Restaurant Hotel Hacienda Los Laureles) |

Wednesday, August 17 | |
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07:30 - 08:30 | Breakfast (Restaurant at your assigned hotel) |

09:00 - 10:00 |
Chelsea Walton: Survey of Quantum Symmetry in the context of Hopf (co)actions ↓ 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. (Conference Room San Felipe) |

10:00 - 10:30 |
Cris Negron: Gauge invariants from the antipode for Hopf algebras with the Chevalley property ↓ 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. (Conference Room San Felipe) |

10:30 - 11:00 | Coffee Break (Conference Room San Felipe) |

11:00 - 12:00 |
Noah Snyder: Topological Field Theory and Modular Tensor Categories ↓ 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. (Conference Room San Felipe) |

12:00 - 12:30 |
Xingshan Cui: Higher Categories and Topological Quantum Field Theories ↓ 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. (Conference Room San Felipe) |

12:30 - 13:00 |
Patrick Gilmer: An application of TQFT to modular representation theory ↓ 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. (Conference Room San Felipe) |

13:00 - 14:30 | Lunch (Restaurant Hotel Hacienda Los Laureles) |

14:30 - 18:00 | Free Afternoon (Oaxaca) |

18:00 - 20:00 | Dinner (Restaurant Hotel Hacienda Los Laureles) |

20:00 - 21:00 | problem Session II (Conference Room San Felipe) |

Thursday, August 18 | |
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07:30 - 08:30 | Breakfast (Restaurant at your assigned hotel) |

09:00 - 10:00 |
Hans Wenzl: Centralizer Algebras for Quantum Groups ↓ 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. (Conference Room San Felipe) |

10:00 - 10:30 | Coffee Break (Conference Room San Felipe) |

10:30 - 11:30 |
Scott Morrison: Modular data for centres ↓ 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. (Conference Room San Felipe) |

11:30 - 12:00 |
Costel Bontea: Classification of non-semisimple pointed braided tensor categories and their Brauer-Picard groups ↓ 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 $\Gamma$. (Conference Room San Felipe) |

12:00 - 13:00 | Emily Peters (Conference Room San Felipe) |

13:00 - 14:30 | Lunch (Restaurant Hotel Hacienda Los Laureles) |

14:30 - 15:30 |
Terry Gannon: Vector-valued modular forms and modular tensor categories ↓ 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. (Conference Room San Felipe) |

15:30 - 16:00 |
James Tener: On classification of vertex operator algebras by their representation categories ↓ 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. (Conference Room San Felipe) |

16:00 - 16:30 | Coffee Break (Conference Room San Felipe) |

16:30 - 17:00 |
Juan Cuadra: On Frobenius 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. (Conference Room San Felipe) |

18:00 - 20:00 | Dinner (Restaurant Hotel Hacienda Los Laureles) |

Friday, August 19 | |
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07:30 - 08:30 | Breakfast (Restaurant at your assigned hotel) |

09:00 - 10:00 | Alain Bruguieres (Conference Room San Felipe) |

10:00 - 10:30 | Coffee Break (Conference Room San Felipe) |

10:30 - 11:30 | Vladislav Khartchenko: Multilinear quantum Lie operations (Conference Room San Felipe) |

11:30 - 13:00 | Discussion (Conference Room San Felipe) |

13:00 - 14:30 | Lunch (Restaurant Hotel Hacienda Los Laureles) |