FRG: Collaborative Research: Complex Lagrangians, Integrable Systems, and Quantization

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Mirror symmetry was discovered over three decades ago in theoretical physics. Since then, it has been a deep mystery appearing in many frontiers of mathematics. In general, it appears to be a hidden relation between symplectic geometry, which is the natural framework for describing classical mechanics, and complex algebraic geometry. More recent discoveries link it with several other areas of mathematics and physics. Still, as of today, there is no systematic understanding of mirror symmetry. The main goal of this focused research group is to establish a universal tool to study mirror symmetry and related geometric questions arising in modern frontiers of geometry and topology. In particular, the project aims at constructing a large class of concrete models that would demonstrate all aspects of mirror symmetry. The investigators bring expertise from different areas of mathematics and will use a variety of techniques. They will also organize workshops, summer schools, and conferences, aimed at training early career researchers in this area, disseminating recent results and facilitating further advances.In the original context, mirror symmetry relates counting problems in one space with complex differential equations on its mirror symmetric space. As understanding has evolved, mirror symmetry has been identified as the Laplace transform in certain cases, which provides an effective mechanism of computing quantum invariants of a space from the complex geometry of its mirror. In general, it is a higher categorical relation between symplectic geometry and complex algebraic geometry. In a gauge theoretic context, mirror symmetry appears in the form of the Langlands duality among algebraic groups. More recent discoveries link it with quantum knot invariants. The main goal of the project is to establish complex Lagrangian geometry as a universal tool to study geometric questions and quantization arising in modern frontiers of geometry and topology and to construct models in which we can see all aspects of mirror symmetry, including the real symplectic-complex algebraic duality, the Langlands duality, the Laplace transform, and a relation to three-manifold invariants. In Lagrange's formulation many non-linear problems can be solved via integrable systems, an approach which is widely applicable, extending far beyond its classical origins to high-energy particle physics and string theory. This project aims to advance understanding of the complex integrable systems that arise in such situations, together with their quantum mechanical counterparts, by considering concrete models constructed on Hitchin integrable systems. The work will employ a generalization of the Laplace transform known as topological recursion that is expected to provide a conjectural new mechanism for holomorphic quantization of these spaces and shed new light on mirror symmetry.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.