Collaborative Research: Factorization Homology, Deformation Theory, and Duality

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This project develops the geometric study of models of space-time, toward mathematical forms of fundamental concepts in particle physics. Two principal concepts from the theory of quantum fields are those of locality and duality. Locality generalizes the idea that there should not be so-called action at a distance in a complete quantum theory of nature. Duality occurs in two seemingly different quantum theories being fundamentally the same. The present project develops mathematical formulations of locality and duality in terms of a new theory of factorization homology. Factorization homology provides a mathematical means of articulating a physical theory on all of space-time in terms of physics in very small subregions, within which physics can be simplified in terms of combinatorics and higher-dimensional graph theory.This project develops the theory of factorization homology for variants of higher categories, and applications thereof to mathematical physics and differential topology. This theory can be thought of as the study of sheaves on moduli spaces of stratifications. One goal is to use factorization homology with adjoints to prove the cobordism hypothesis of Baez--Dolan and Lurie, which asserts that topological quantum field theories in the sense of Atiyah's axioms are uniquely determined by their value on a point. This would show that such a topological quantum field theory on space-time X comes from a sheaf on the moduli space of stratifications of X, and thus that the notion of locality can be understood in terms of the topology of such moduli spaces. A second goal is to show that this moduli space of stratifications of a manifold X satisfies an infinite-dimensional form of Poincar? duality. This would then give rise to a duality among topological physical theories on X, once expressed as sheaves on the aforementioned moduli space. This duality is a form of Koszul duality for topological physical theories.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.