Collaborative Research: Factorization Homology and the Cobordism Hypothesis

PROJECT SUMMARY DAVID AYALA AND JOHN FRANCIS Proposed Activities This project involves the development of the theory of factorization homology with coe�cients in (1; n)-categories. The initial proposed use of this theory is to prove the cobordism hypothesis of Baez-Dolan and Lurie, which asserts that topological quantum �eld theories valued in a symmetric monoidal (1; n)-category are in bijection with fully dualizable object of that category { in particular, that a �eld theory is determined by its value on a point. A second application is an unexpected re�nement of the cobordism hypothesis using a formulation of the state-operator correspondence in terms of factorization homology. This re�ned cobordism hypothesis produces numerical invariants of manifolds under a strictly weaker condition than the full dualizability proposed by Baez-Dolan and Lurie. These invariants satisfy only a weaker version of locality, which does not allow them to be computed from handlebody presentations. Ayala and Francis further propose to relate their construction to sigma models by way of a conjectural analogue of Tannakian duality for factorization homology, and to employ these techniques to study what physical duality results from a form of Poincar�e duality for factorization homology. Intellectual Merit The cobordism hypothesis of Baez-Dolan and Lurie is the structural crux in the current math- ematical view of topological quantum �eld theory. It informs the work of every mathematician in this �eld, but a complete proof has not, to this point, been written. The present proposal o�ers both a complete proof of the cobordism hypothesis as well as a conceptually natural construction of the correspondence therein. It also o�ers a new basis for locality in topological quantum �eld theory centered around the observables of a physical theory rather than the state spaces. This is based on the theory of strati�cations rather than on Morse theory and handlebody presentations, which have formed the basis for locality since Atiyah's axioms. This modi�cation potentially allows for new manifold invariants and new examples from physics which did not conform to the original cobordism hypothesis. The manifold topology required for the foundations of their theory is signi�cant: the existence of factorization homology necessitates that the collection of disk-strati�cations of manifold can be assembled into an 1-category { the requisite lifting condition uses a coherently functorial con- struction of resolutions of singularities. Likewise, to establish even the most basic property of factorization homology involves showing that the classifying space of disk-strati�cations of mani- fold is contractible { this is an analogue of Whitehead's theorem, that smooth manifolds have an essentially unique piecewise linear structure. Similarly, the essential local-to-global principle for factorization homology requires an excision result for di�eomorphism groups which is new and of independent interest. Broader Impact Ayala and Francis have a number of graduate students, and the scope and design of their agenda is well-suited toward incorporating young researchers, certainly including graduate students, into the work. They have previously collaborated with and mentored graduate students. 1