1. Algebraic structures in physics

(A. Stolin, G. Ferretti, J. Fuchs, B. Nilsson, H. Rosengren, L. Turowska) 

Algebra and representation theory provide important tools in physics. Traditional applications involve finite groups or finite-dimensional Lie groups and algebras , but nowadays many more structures are known to be relevant, such as Kac-Moody algebras, quantum groups, Yangians, Lie bialgebras, or vertex operator algebras. They have been the basis of major progress in string theory, conformal and topological quantum field theory, and integrable systems. Conversely, ideas from these areas have led to new developments in mathematics; e.g. quantum groups originate from the study of integrable systems, and vertex algebras formalize chiral symmetries in conformal field theory. The project aims at identifying further algebraic structures playing a role in physical problems and, when necessary, develop their theory so as to suit the particular needs of the physical context. Some issues that we plan to study are:

- Non-rational conformal field theories, including e.g. models relevant to cosmological solutions in string theory, for which non-compact real Lie algebras and the modular representation theory of algebraic groups are relevant.

- Relations between string dualities, automorphic forms, and generalized Kac-Moody algebras.

- The possible relationship between N=4 super Yang-Mills theory and the geometric Langlands programme.
- Aspects of three-dimensional topological field theories which are expected to be relevant to topological quantum computing and string/M theory.”