Quantum Invariants, Enhanced Moduli, and Integrable Systems

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This award supports research in the field of algebraic geometry. The research sets the stage for understanding the basic structure of fundamental problems in a way suitable for pragmatic use in a broad spectrum of applications. The results of the work are intended to be immediately relevant to questions in the theory of integrable systems, string theory, gauge theory, and the study of topological insulators. In addition, the research project aims to have concrete applications to field theory and quantization. The project also aims to build a new arsenal of geometric techniques applicable to the theory of algebraic cycles, symplectic topology, and high energy physics. This will be achieved by providing research opportunities on the interface of geometry and string theory for graduate students and postdoctoral associates in mathematics and physics, and by development of courses on formal localization and shifted quantization and on Calabi-Yau integrable systems and higher Donaldson-Thomas theory. The project integrates ideas from derived geometry and quantum field theory to unravel the hidden complexity of moduli problems, to extract new enumerative invariants of varieties, and to study completely integrable systems. The resolution of these questions will consolidate and demystify several existing quantization schemes in geometry, symplectic topology, and field theory. Three directions will be studied. The first aims to characterize those moduli spaces that admit a realization as the critical locus of a potential. The characterization requires the development of the formalism of shifted symplectic and Poisson structures in derived geometry and the theory of isotropic foliations. In the second project a new method will be investigated for constructing motivic orientation data on non-abelian cohomology by building explicit Lagrangian foliations and computing the associated potential functions. The project aims to construct higher Chern-Simons functionals, and sets the stage for extracting enumerative invariants from higher dimensional quantum field theory. The final project searches for Calabi-Yau integrable systems that realize the tame or wild meromorphic Hitchin fibrations for ADE structure groups.