The present proposal deals with several algebraic structures motivated by two-dimensional conformal field theory, as well as with higher-dimensional analogs related to quantum field theory in space-time dimension higher than two. The structures investigated are: vertex algebras, (non-linear) Lie conformal algebras, Lie pseudoalgebras, and vertex (Lie) algebras in higher dimensions. Three main projects are proposed. The first is to classify certain non-linear Lie conformal algebras with three generators, which can be viewed as non-linear deformations of affine sl(2). This will lead to the discovery of new interesting vertex algebras. The second project is to classify the irreducible representations of simple Lie pseudoalgebras of type H and K, extending previous work of the PI and collaborators that settled types W and S. This project is closely related to the representation theory of Lie-Cartan algebras of vector fields, and it involves pseudoalgebra versions of the hamiltonian and contact de Rham complexes. The third project develops further the theory of vertex algebras in higher dimensions introduced previously by N.M. Nikolov, with the long-term goal of constructing examples that lead to nontrivial models of axiomatic quantum field theory. The PI will continue his investigation of the vertex Lie algebras in higher dimensions introduced by him and Nikolov. Another approach proposed here is to reconstruct a conformal vertex algebra in higher dimensions from its one-dimensional restriction and the action of the Lie algebra of infinitesimal conformal transformations.
Lie groups, named after the 19th century Norwegian mathematician Sophus Lie, provide a mathematical description of the notion of continuous symmetry and play a prominent role in mathematics and physics. Related objects, Lie algebras, describe infinitesimal transformations. The PI investigates infinite-dimensional Lie algebras and related algebraic structures, which first appeared in an algebraic approach to quantum field theory. The goal of this research is to develop further the mathematical theory of such algebras and at the same time discover new concrete examples, which will lead to the construction of new models in quantum field theory and thus will deepen our understanding of the nature.
