Representation Theory, Calabi-Yau Manifolds, and Mirror Symmetry

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This award will support the participation of researchers at the workshop Representation Theory, Calabi-Yau Manifolds, and Mirror Symmetry to he held November 28-December 1, 2022 at the Center of Mathematical Sciences and Applications (CMSA) at Harvard University. Quantum field theory has led to a unification of many seemingly disparate areas of mathematics during the past half century, including geometry, representation theory, algebra, combinatorics, and number theory. The purpose of this conference is to bring together active researchers in these areas with the goal of strengthening ties, fostering communication, and generating new ideas at the interface of physics and various areas of mathematics. This proposal is part of an ongoing effort directed at understanding all aspects of topological mirror symmetry and string theory at mathematical levels of rigor, transmitting this understanding to both mathematicians and physicists, and creating the infrastructure for intensive interactions between these two groups within a single, flourishing subject. Mirror symmetry, discovered by physicists in the 1990s, reveals a deep duality between symplectic and complex geometries. It transforms Gromov-Witten invariants, which are difficult to compute in general, to period integrals in complex analysis, which are computable and satisfy certain PDEs. For a given Calabi-Yau manifold, it is a longstanding problem to construct its mirror. The celebrated Strominger-Yau-Zaslow program proposed that a Calabi-Yau manifold has a special Lagrangian torus fibration, and its mirror can be constructed by taking the dual of the torus fibers. Vertex operator algebra (VOAs) are algebraic structures that arose from conformal field theory, and were axiomatized by Borcherds. Certain VOAs can be realized geometrically as algebras of operators on the cohomology rings of moduli spaces. A key example is the action of the Y-algebras of Gaiotto and Rapcak on the moduli space of spiked instantons of certain toric Calabi-Yau threefolds. A vast generalization of this picture has been proposed by Rapcak, Soibelman, Yang, and Zhao, and is based on the action of the Hall algebra on the cohomology of moduli spaces. On the other hand, the quiver formulation of some toric Calabi-Yau threefolds and their nc deformations can be constructed from the Fukaya category via mirror symmetry. Thus the Hall algebra action has an interesting meaning in symplectic geometry. This conference will provide a forum for researchers in these fields to communicate in the hope of resolving these conjectures and finding further connections between mirror symmetry and representation theory. The workshop web page is https://cmsa.fas.harvard.edu/homological-mirror-symmetry-workshop/.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.