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UBC cIRcle BIRS Workshop Lecture Videos Logo
Moonshine â old and new
Banff International Research Station for Mathematical Innovation and Discovery
Gannon, Terry — 2019-03-06 Non UBC Unreviewed Author affiliation: University of Alberta Faculty http://creativecommons.org/licenses/by-nc-nd/4.0/
UBC cIRcle BIRS Workshop Lecture Videos Logo
Fantastic beasts and where to find them
Banff International Research Station for Mathematical Innovation and Discovery
Gannon, Terry — 2019-03-27 In recent years, subfactor methods have constructed hundreds of new fusion categories, with strong evidence these live in infinite families. Taking their doubles, we obtain hundreds (and probably infinite families) of new modular tensor categories. These modular tensor categories have a distinctive appearance. Abstracting this appearance, it is natural to guess that there is a new construction, which we call the smashed-sum, combining old modular tensor categories into new ones. It is tempting to guess the smashed-sum construction also lives in the VOA world, in fact this may be its natural home. An example of such a VOA would be the mythical Haagerup VOA. Non UBC Unreviewed Author affiliation: University of Alberta Faculty http://creativecommons.org/licenses/by-nc-nd/4.0/
UBC cIRcle BIRS Workshop Lecture Videos Logo
Vector-valued modular forms and modular tensor categories
Banff International Research Station for Mathematical Innovation and Discovery
Gannon, Terry — 2017-02-17 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. Non UBC Unreviewed Author affiliation: University of Alberta Faculty http://creativecommons.org/licenses/by-nc-nd/4.0/
UBC cIRcle BIRS Workshop Lecture Videos Logo
Reconstruction
Banff International Research Station for Mathematical Innovation and Discovery
Gannon, Terry — 2017-04-01 The most standard mathematical reformulations of conformal field theory are vertex operator algebras (Wightman axioms) and conformal nets (Haag-Kastler axioms). A conformal field theory lives in the representations of such structures. Representation theory is most elegantly formulated in the language of categories, and the relevant categories for both vertex operator algebras and conformal nets (in the best-understood=semisimple case) are called modular tensor categories. These are the categories which give rise to knot and link invariants for 3-manifolds. One of the deepest open questions in the theory, with the most ramifications, is Reconstruction: which modular tensor categories arise from vertex operator algebras or conformal nets? This is the analogue here of Tannaka-Krein duality: if you look like the category of representations of a compact group, then you ARE the category of representations of a compact group. In my talk I'll explain why this is such an important question, and review the little that is currently understood about it. Then I'll explain the fundamental role which modular forms play in attacking this problem. Non UBC Unreviewed Author affiliation: University of Alberta Faculty http://creativecommons.org/licenses/by-nc-nd/4.0/
UBC cIRcle BIRS Workshop Lecture Videos Logo
Quantum Doubles of Quadratic Systems II
Banff International Research Station for Mathematical Innovation and Discovery
Gannon, Terry — 2014-10-14 Non UBC Unreviewed Author affiliation: University of Alberta Faculty http://creativecommons.org/licenses/by-nc-nd/2.5/ca/
UBC cIRcle BIRS Workshop Lecture Videos Logo
The theory of C2-cofinite VOAs
Banff International Research Station for Mathematical Innovation and Discovery
Gannon, Terry — 2016-08-10 Rational VOAs are by now quite well understood: their representation theory is captured by a modular tensor category; their characters define a vector-valued modular form for $SL(2,Z)$; etc. The class of C2-cofinite (logarithmic) VOAs is the natural generalisation of rationality, but their theory is still much less clear. This talk reviews and contributes to this theory. It is joint work with Thomas Creutzig. Non UBC Unreviewed Author affiliation: University of Alberta Faculty http://creativecommons.org/licenses/by-nc-nd/4.0/

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