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Towards quantum chiral algebras

Banff International Research Station for Mathematical Innovation and Discovery

Chiral algebras are extensively studied in many areas of both mathematics and physics, due to the wealth of examples from various classes of algebras generated by chiral fields (although precise axiomatic/mathematical definition of the concept is lacking in it full generality). Super vertex algebras constitute a class of chiral algebras, corresponding to the chiral part of a conformal quantum field theory on the complex plane, and their theory is well established. Nevertheless there is a variety of important examples of algebras generated by chiral fields that cannot be described by the concept of super vertex algebra. The most challenging case concerns the quantum vertex operators and the quantum chiral algebras they generate. This area of research, which by now is quite large and growing, was started with the fundamental problem posed by Igor Frenkel: to formulate and develop a theory of quantum vertex algebras incorporating as examples the Frenkel-Jing quantum vertex operators realizing the quantum affine algebras. This problem is still ultimately unsolved despite the comparative progress lately. In this talk we will discuss some of the issues that are encountered on the way to defining a suitable theory of quantum chiral algebras. As a guiding principle for the the mathematical description of any class of chiral algebras we will discuss the notable instances of certain special isomorphisms between chiral algebras of that class (such as the boson-fermion correspondences).

Mathematics; Nonassociative rings and algebras; Associative rings and algebras; Representation theory

video/mp4

Author affiliation: College of Charleston

2016-08-12

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/58769

Unreviewed

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