This project concerns hyperbolic Kac-Moody groups, which are infinite dimensional Lie groups that have yet been extensively studied. We are interested in the Lie groups of infinite dimensional hyperbolic Kac-Moody algebras which contain affine Kac-Moody subalgebras. Some of the proposed problems are physically motivated. Our study of physical theories such as supergravity,a theory that incorporates both supersymmetry and general relativity, has revealed a number of compelling and intriguing mathematical problems, consistent with open problems in Kac-Moody group theory. The proposed work is the first mathematical initiative that aims to apply the symmetry properties of hyperbolic Kac-Moody groups to the study of supergravity models in theoretical physics. While some of these correspondences of hyperbolic Kac-Moody symmetry with supergravity theories are conjectural, developing a full mathematical theory of hyperbolic Kac-Moody groups and their symmetric spaces amenable to computation will have a significant impact on the understanding of open problems concerning the symmetry groups of supergravity.
The objective of this project is to advance understanding in the study of algebraic symmetries underlying high energy theoretic physics. Almost all finite dimensional semisimple Lie groups and Lie algebras occur in space-time symmetries and the development of the Standard Model of particle physics, which could not have progressed without an understanding of symmetries and group transformations. Infinite dimensional generalizations, known as Kac-Moody algebras and their associated groups, naturally form two distinct classes, namely affine and hyperbolic. By the 1980's the class of affine Kac-Moody algebras was shown to have wide applications in physical theories such as elementary particle theory, quantum field theory, gauge theory, conformal field theory, gravity and string theory. This project concerns Lie groups of infinite dimensional hyperbolic Kac-Moody algebras which contain affine Kac-Moody subalgebras.
The objective of this program is to advance understanding in the study of algebraic symmetries underlying high energy theoretical physics. Our study of physical theories such as supergravity, a theory that incorporates both supersymmetry and general relativity, has revealed a number of compelling and intriguing mathematical problems. While some of the mathematical correspondences with supergravity theories are conjectural, developing a full mathematical theory amenable to computation will have a significant impact on the understanding of open problems concerning the symmetry groups of supergravity. Due to the nature and status of the subject, we approached the open mathematical questions on all levels; abstract, concrete and computational. Our outcomes have contributed in part to the understanding of open problems concerning the symmetry groups of supergravity theories in theoretical physics. In particular, we made a number of breakthroughs in our long term goal to understand hyperbolic mathematical symmetries, and to develop algebraic tools needed to study applications of these symmetries in theories of supergravity, black hole spacetimes and cosmology in theoretical physics. We trained 6 graduate students who participated in the projects in the proposal, and we mentored 6 postdoctoral researchers with whom we collaborated. We gave seminar and conference talks about the research findings around the US and in Australia, France and Japan.
