CAREER: Factorization Algebras in Quantum Field Theory

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Mathematics and physics have sustained an ongoing, ever-expanding dialogue for centuries, enriching both subjects by the swapping of problems and insights. In the last few decades, this conversation has led to a fruitful exchange centered on the role of higher algebra (such as category theory and homotopical algebra) in quantum field theory (QFT). It has transformed and enlarged our view of QFT, and thus it plays an active role not only in the farther reaches of particle theory, but also in the concrete discoveries of condensed matter physics, notably in topological states and phases of matter. A recent innovation from this exchange is factorization algebras, which appear naturally in physics as the observables of field theories (both classical and quantum) but originally appeared in mathematics. This project explores the power of this tool in the setting of 4-dimensional gauge theories. To pursue goals of this research, the Principal Investigator will collaborate with both mathematicians and physicists. This kind of interdisciplinary effort is an inspiring and motivating aspect of this area of research, but effective communication is often difficult because, despite the long-running relationship of mathematics and physics, each community has its own prerogatives and modes of discourse. A key component of this project is thus to offer chances for researchers at all levels to become fluent in speaking to both disciplines and, moreover, to build direct personal bridges. At the graduate and postdoctoral level, the project will run annual summer schools for both mathematicians and theoretical physicists, focused on topics of mutual interest. In addition, each academic year, it will produce high-quality, online masterclasses by experts about such topics, with lecture notes and exercises. Finally, the project will support summer research for undergraduates, tackling problems between mathematics and physics, from the University of Massachusetts and nearby Five Colleges.In more detail, this project orbits around two foci. The first is the study of holomorphic theories on complex surfaces, particularly those related to moduli of holomorphic G-bundles. The PI will pursue analogs of relationships between affine Lie algebras, moduli of bundles, and chiral CFT, which involve holomorphic theories on Riemann surfaces. The long-term target is a holomorphic version of Seiberg duality (arising from N = 1 supersymmetric gauge theories), which bears natural analogies to mirror symmetry. This first project uses a toolkit, developed by the PI with collaborators, for constructing BV quantization of holomorphic field theories and then analyzing their factorization algebras of observables. The second focus involves topological theories on manifolds with boundaries and corners, where the long-term target is a rigorous demonstration of the Frenkel-Gaiotto conjecture about how vertex algebras and ribbon categories arise from the Kapustin-Witten gauge theories, which play a key role in the physical approach to the geometric Langlands correspondence. This second project will approach the conjecture using both global methods (by developing, via derived algebraic geometry, the AKSZ procedure with manifolds with boundaries and corners, and thence a higher categorical deformation quantization) and perturbative methods (building upon an extension of the BV/factorization package for field theories on manifolds with boundary, which yields stratified factorization algebras of observables).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.