CAREER: Algebra and Representation Theory for Non-Semisimple Topological Field Theory

This award will support research in the field of algebra and representation theory with connections to theoretical physics. Representation theory is the branch of mathematics concerned with the study of symmetry using techniques from linear algebra, such as vectors and matrices. Classical examples of symmetry include rotations of a circle and reflections of a square. Modern research in mathematics naturally encounters more abstract notions of symmetry and requires the development of more advanced techniques for their study. The PI will identify and study concrete instances of these abstract symmetries and apply the results to advance the mathematical understanding of quantum field theory in dimensions two and three. The research will also develop new connections between algebra, representation theory, and low dimensional topology. This project integrates education, research, and career training opportunities for high school, undergraduate, and graduate students. Examples include creating a series of videos which will improve the algebra education of Utah's high school mathematics teachers-in-training, and organizing the graduate student focused Moab Topology Conference. In more detail, the PI will engage in three related research projects. Each project is motivated by and has direct implications for the long-standing conjecture that Rozansky-Witten theory, a three dimensional supersymmetric quantum field theory, defines a non-semisimple topological field theory. In the first project the PI will further develop and extend the theory of relative modular categories and their associated non-semisimple topological field theories to find new connections with homological blocks of 3-manifolds. In the second project the PI will develop the representation theory of affine L-infinity algebras and unrolled quantum groups and use the results to construct abelian approximations of Rozansky-Witten theory. In the third project the PI will extend his work on twisted equivariant matrix factorizations to construct non-semisimple, possibly unoriented, topological field theories for Landau-Ginzburg theory and explore their interpretation as Rozansky-Witten defects. The physical origins of the representation theories considered suggest many non-trivial interrelations. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.