A Sheaf-Theoretic Approach to M5-Brane Geometry

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The project aims to establish and strengthen rich connections between mathematics and theoretical physics which will serve to advance both fields. In physics, M-theory is an approach to unifying the different string theories, which themselves provide a theoretical framework for unifying the forces of nature: gravity and particle theory. The PI's recent mathematical advances will provide rigorous calculations and predictions of quantities of physical interest. In addition, the PI will continue to supervise young mathematicians, run seminars and disseminate and speak widely about the research. Beyond this, the PI will serve the broader community by continuing to run a math circle and by continuing to play an central role in the summer Bridge program of Northwestern University.PI Zaslow, in a number of works, spearheaded the use of sheaf theory in symplectic geometry and mirror symmetry. His recent work on invariants of Legendrian knots and Lagrangian surfaces found new connections to cluster theory. In the present work, Zaslow will extend these linkages to the dimension of great interest to the physics of M5-branes in string theory: three-dimensional Lagrangians and two-dimensional Legendrian surfaces. New and interesting phenomena occur at this critical dimension. Zaslow will do the following: 1. find superpotentials encoding counts of holomorphic disks bounding Lagrangian fillings of Legendrian surfaces and establish Ooguri-Vafa integrality in all framings;2. explain how such counts are determined by (Seiberg-like) mutations from cluster theory (at each genus, Donaldson-Thomas transformations relate them to simple building blocks);3. establish "framing duality" (i.e., prove that the genus-g moduli space computes, via different framings, DT invariants of all symmetric quivers with g nodes);4. extend these results in complex three-space to the nonexact setting of the resolved conifold by incorporating Q-deformations into the sheaf theory via twisted sheaves;5. apply this machinery to knot conormals to explain large-N duality using sheaf theory;6. prove the Lagrangians found by Zaslow-Treumann have special-Lagrangian embeddings;7. explain the Goncharov-Kenyon-Beauville integrable system via mirror symmetry.