Microlocal Sheaves in Geometric Representation Theory

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

Differential equations provide basic tools for studying quantities that vary in space and time. A system of linear differential equations can be encoded in an algebraic structure known as a D-module; this encoding allows us to isolate algebraic properties of the system of equations from questions about features of its solutions. In the presence of spatial symmetry, those D-modules preserved by the symmetry possess a special structure reflecting a corresponding decomposition of space itself into simpler pieces. The focus of this research is to refine these structural features and apply them in mathematics and physics. D-modules provide fundamental objects in geometric representation theory. They and their more flexible symplectic counterparts, which we call "microlocal sheaves," have been systematically mined in the subject for several decades. In recent work, the PI (jointly with McGerty) has established the existence of a new structural feature, categorical Morse decomposition, for G-equivariant D-modules on a variety X, or, generalizing the quotient of X by G, on a more general class of algebraic stacks. This categorical decomposition mirrors, one categorical level higher, long-anticipated Morse-theoretic assertions for the geometry of the equivariant cotangent bundle of X (i.e., the cotangent bundle of the quotient stack). This research places the structure of categorical Morse decomposition at the center of a web of interconnected directions and problems informed by, and with applications to, representation theory, topology, geometry, and the mathematics of supersymmetric field theories. "Peeling off the top layer" of the categorical Morse decomposition leads to applications to the representation theory of algebras that are realized by quantum Hamiltonian reduction (examples include many symplectic reflection algebras). Decategorifying leads to applications to classical topological invariants of hyperkahler and algebraic symplectic quotients. Applying the framework to algebraic varieties that arise as moduli of vacua in supersymmetric gauge theories will yield insight into mathematical structures of these physically-defined spaces.