Collaborative Research: Derived Skein Modules

Electron fields confined to 2-dimensional surfaces accommodate an abundance of “spin” types, not merely “up/down” as in 3-dimensions. This feature accommodates ground-state ensembles of mutually entangled electrons; the irreducible such ensembles are “anyons”. Anyon trajectories exist in 2+1=3 dimensions. This project develops an object called "factorization homology", which is a sensitive mathematical construction that integrates a complicated object over 3-dimensional space. Factorization homology abstracts the global observables of an anyon system. Applications are intended for the study of knots in 3-dimensional spaces, as well as symmetries from quantum physics, the study of which has proved instrumental for classifying fundamental particle types and their interactions. Applications are further intended to important outstanding problems in the area. The project includes the organization of workshops directed to the community of early career researchers, mentoring post-docs and graduate students. This project develops the theory of factorization homology for 3-manifolds with coefficients in rigid braided infinity-categories, together with applications to low-dimensional topology, topological quantum field theory, and representation theory. The objectives include the analysis of a moduli space of graphs Graph(M) in a 3-manifold; the construction of a cosheaf on this moduli space given a rigid braided infinity-category; the proof of the Tangle Hypothesis for singular 1-dimensional tangles; the relation of this theory to that of skein modules; a corresponding E3-quantization of the stack of local systems on a 3-manifold; and a derived enhancement of a conjectured Langlands duality for skein modules. This opens the gates of derived category theory (i.e., infinity-category theory and moduli spaces of embedded manifolds) into quantum topology. For one, it establishes the moduli space Graph(M) of graphs in a 3-manifold M as the fundamental topological object over which quantum invariants of M can be analyzed by local-to-global techniques. Secondly, the tools of homological algebra, such as free resolutions and derived dualities (e.g., Koszul/Verdier duality on Graph(M)), are imported to braided category theory, allowing for new homotopical-algebraic maneuvers in the quantum-algebraic inputs for low-dimensional topology invariants. Specifically, successful completion of the project would result in applications to several conjectures in 3-dimensional topological quantum field theory, concerning Chern–Simons theory and the geometric Langlands program. 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.