Representation theory in unoriented and non-semisimple physics

This project is in the field of 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. Classical examples of symmetry in geometry include reflections of a square and rotations of a sphere. Modern research in mathematics naturally encounters more abstract notions of symmetry and so requires the development of more advanced techniques for their study. This project 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 (via Landau-Ginzburg models) and three (via Rozansky-Witten models). Two novel features of these quantum field theories are that they are defined on non-orientable geometries or involve non-semisimple categories of line operators; both features make them particularly challenging to study. This project provides research and career training opportunities for high school, undergraduate and graduate students. More specifically, the PI will engage in three related research projects, all of which involve students. Through the first project the PI will develop a theory of real equivariant matrix factorizations, using techniques from categorical representation theory, and apply the results to formulate a mathematical theory of Landau-Ginzburg orientifolds. In the second project the PI will extend prior work on the representation theory of cohomological Hall algebras and its applications to orientifold Donaldson-Thomas theory. In the third project the PI will use the representation theory of quantum supergroups to construct the equivariant Rozansky-Witten theory of a holomorphic symplectic manifold. The first two projects involve physical theories which are unoriented; all three are non-semisimple. 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.