The proposal is dedicated to the development of three interrelated areas in the representation theory of infinite dimensional Lie algebras and their applications to quantum field theory (QFT). The first part of the proposal is focused on the vertex operator algebra structure of a modification of regular representations of affine Lie algebras. The author plans to prove, in particular, his conjecture that the zeroth semi-infinite cohomology of the reduced vertex operator algebra yields the famous Verlinde algebra of the Wess-Zumino-Novikov-Witten model of 2 dimensional QFT. The second part of the proposal consists of a cohomological and K-theoretic realization of various algebraic constructions of the first part by means of 4 dimensional classical Yang-Mills theory. The author proposes to use the geometric construction due to Nakajima of representations of affine Lie algebras based on Yang-Mills instantons on asymptotically locally Euclidean spaces, and also a new geometric construction related to the first one by the level-rank duality discovered by the author in the algebraic context in his earlier work. Finally, the third part is dedicated to a new direction in the representation theory of 3 dimensional convolution and current algebras presently developed by the author. It is based on a new noncommutative geometry of quantum Minkowski space-time recently introduced by Jardim and the author in their study of a noncommutative version of the Yang- Mills theory. The three parts of the proposal are related to each other in that they constitute a new program: a representation theoretic approach to 4 dimensional QFT. This is a far reaching extension of the representation theoretic approach to 2 dimensional QFT previously developed by the author and other mathematicians and mathematical physicists. The proposal is devoted to problems at the frontiers of modern representation theory: the field of mathematics that studies symmetries and their realizations. By its nature representation theory underlies and unifies the structures of diverse areas in mathematics and physics, and in the past 25 years the author has made a number of important contributions to this field. The present proposal contains a new program of development for the representation theory of infinite dimensional symmetries which is especially relevant to the most advanced part of the existing theoretical physics that is supported by experimental physics; namely, 4 dimensional quantum field theory. Progress in this direction will enrich our understanding of the fundamental mathematical structures that lie at the foundation of the physical universe.
