The proposal is dedicated to the development of new directions of representation theory of infinite-dimensional Lie algebras and related structures, and applications to models in theoretical physics. The proposal consists of three interrelated parts. The first part of the proposal outlines a new construction of the Monster vertex operator algebra via chiral three-dimensional quantum gravity, which is strongly suggested by the recent works of the author on Rademacher sums. The second part is dedicated to the continuation of the author?s work on the development of quaternionic analysis with expected applications to representations of three dimensional current algebras and relations to four-dimensional quantum field theory. Finally, the third part contains further steps towards realization of various structures in quaternionic analysis using the semi-infinite cohomology of the Virasoro algebra that recently was set forth by the author. It also links and reinforces the first and second part of the proposal.
The relation between representation theory and physical models was very fruitful in the twentieth century with the famous examples of representations of Lorentz and Poincare groups and special relativity, representations of the Heisenberg group and quantum mechanics, representations of the group SU(3) and quark composition of elementary particles. The author actively participated in the development of a recent relation between representation theory of infinite-dimensional Lie algebras and two-dimensional quantum field theory. His present proposal outlines a new representation theoretic approach to important areas of modern theoretical physics, namely, three dimensional quantum gravity and four dimensional quantum field theory, and has a profound relation to different areas of mathematics such as Monstrous Moonshine, absolute Galois group, quaternionic analysis, theory of instantons and non-commutative geometry.
