Open Collections

Reconstruction

Banff International Research Station for Mathematical Innovation and Discovery

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.

Mathematics; Number theory; Algebraic geometry

video/mp4

Author affiliation: University of Alberta

2017-04-01

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61084

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/