Factorization homology and applications (introductory lecture)

Factorization algebras, and factorization homology, began in the work of Beilinson-Drinfeld, as an algebro-geometric/coordinate-free approach to vertex algebras and conformal blocks, respectively. They were re-interpreted by Costello-Gwilliam as a framework for algebras of observables in quantum field theory. A special class, the so-called "locally constant" factorization algebras received special attention from Lurie, Ayala-Francis, and Scheimbauer in the context of fully extended topological field theories. In the first lecture I shall recall this history, define factorization homology in the mold of Ayala-Francis, and recall the key property of excision, which both uniquely determines factorization homology as a functor, and gives an effective mechanism for its computation. In the second lecture, I will turn to examples in geometry and representation theory, following Ben-Zvi-Francis-Nadler, and our works with Ben-Zvi-Brochier and Brochier-Snyder. Specializing the "coefficients" to lie in presentable k-linear categories (the natural home of algebraic geometry and representation theory), one recovers character varieties, and their canonical quantizations, as a computation in factorization homology.Non UBCUnreviewedAuthor affiliation: University of EdinburghResearche