Week 1: "Infinite-dimensional Algebras, Conformal Field Theory and Integrable Systems"

Some of the most spectacular applications of representation theory of infinite-dimensional algebras are related to recent developments in mathematical physics.

Conformal quantum field theory (CFT) in two dimensions and string theory have a multitude of deep connections to the representation theory of infinite algebras. In a sense the two subjects have been joined at the hip since their inception in the late 1960's. The early physics papers on string theory introduced representations of the Virasoro Lie algebra and quantum fields known as vertex operators. By the early 1980's it was realized that the Virasoro algebra is a universal symmetry algebra of a 2D CFT, and that affine Kac-Moody algebras (central extensions of loop algebras) are symmetry algebras of CFTs of a more special type (Wess-Zumino-Witten models). Eventually, Borcherds succeeded in formalizing the general notion of a symmetry algebra of a CFT in the concept of vertex algebra. More recently, Beilinson and Drinfeld have introduced a geometric version of vertex algebras which they call chiral algebras. Chiral algebras give rise to some novel concepts and techniques in algebraic geometry, in particular, in the geometric Langlands correspondence.

Physicists have recently begun to study perturbations of conformal field theories in the hopes of using the rich structure of CFT for understanding more general integrable quantum field theories. The study of conformal field theory in dimensions greater than two is also under way, and surprising connections have been found between these theories and string theory (AdS/CFT correspondence).

Infinite-dimensional algebras have also been used in the recent years in the theory of exactly solved models. It turned out that quantum deformations of the enveloping algebras of the affine Kac-Moody algebras, the Virasoro algebra, and the W-algebras may be realized as dynamical symmetry algebras of various lattice models, and hence their representation theory has important applications. Infinite-dimensional Lie algebras also play a central role in the study of integrable systems, such as those associated to the KdV equation and other soliton hierarchies. These integrable systems appear in the study of the Gromov-Witten invariants of algebraic manifolds. The Virasoro conjecture, in particular, indicates the relevance of infinite-dimensional algebras in this area.

In this workshop we will discuss the areas mentioned above and interactions between them.

Week 2: "Supersymmetry in Mathematics and Physics"

Supersymmetry is now considered as one of the most fundamental concepts in physics. Lie superalgebras in particular have increasingly been playing central role in recent studies of quantum field theory and string theory. There has also been an important development in mathematics, like structure and representation theory of Lie superalgebras, "supersymmetric" proofs of the index theorems, the theory of superconnections, etc. In this workshop, we hope to discuss the recent advances in "superphysics" and "supermathematics."
        

Group photo of participants

Funding

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