NSF-BSF: Derived and quantum corrected structures on arithmetic and geometric moduli

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This is a research in algebraic geometry. The field of algebraic geometry studies geometric models by distilling and encoding their essential complexity in polynomial equations. The project integrates ideas from quantum physics and the physical study of symmetries of the fundamental laws of nature to extract new and unexpected information about the geometry of spaces and the deep properties of number systems. The project will focus on unraveling the structure of hidden and broken symmetries of quantum fields and to capture this structure in computable invariants. These invariants will give a new mathematical tool for understanding and proving various empirically observed physics dualities, and also mysterious arithmetic dualities, which are expected to identify a priori unrelated quantum theories. The project will unify the analytic and geometric properties of parameter spaces of representations in arbitrary dimension and sets the stage for understanding the basic local symmetries of moduli problems in a way suitable for pragmatic use in a broad spectrum of applications. The work will be immediately relevant to deep questions in symplectic geometry, number theory, string theory and quantum field theory. This project provides research training opportunities for graduate students.Three directions will be studied. The first is to construct geometric realizations of arithmetic duality maps and will use geometry to produce new insights into the complexity of Galois representations. In the second project a new method is proposed for understanding maps between coisotropic branes, parametrizing the deformations of the composition laws for such maps, and extracting new symplectic information from such deformations. The final project uses the geometry of higher stacks to develop a duality and decomposition formalism for canceling anomalies and understanding secondary quantum symmetries in quantum field theory.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.