Some problems in algebraic geometry and string theory

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Problems to be investigated in the proposed research include the determination and properties of correlation functions and quantum sheaf cohomology arising from string theories and gauge theories associated to a smooth projective variety and a vector bundle satisfying the anomaly cancellation conditions. Particular attention will be paid to toric complete intersections and the mathematical formulation of the nonlinear sigma model version of quantum sheaf cohomology. A related problem is the determination of the quantum product in the ordinary quantum cohomology of a toric variety. Another theme is the study of algebro-geometric invariants related to topological string amplitudes: Gromov-Witten, Gopakumar-Vafa, and stable pair invariants, including motivic stable pair invariants. These techniques will be applied to put the definition of Gopakumar-Vafa invariants from string theory on a more firm mathematical foundation. Techniques will be developed for the computation of stable pair invariants in the non-toric setting, and will be applied to verify predictions of mirror symmetry. Problems in the area of F-theory and gauge/gravity duality in string theory will be investigated as well.The research funded by this grant has two modes: bringing current developments in algebraic geometry to bear on fundamental problems of particle physics as modeled by string theory, and bringing physical ideas and intuition to bear on current problems in geometry. The questions to be investigated are of pressing current interest in mathematics and physics. The mathematical techniques under development could make it possible to exactly compute the predicted masses of particles in increasingly realistic string models, and better understand how the observed forces and interactions in nature can arise from grand unified theories which include quantum gravity. Ideas of string theory provide key insight which are expected to directly lead to the solution of several unsolved problems in three-dimensional algebraic geometry. Graduate students will be trained in geometry and physics, and their interaction.This award is jointly funded by the Algebra and Number Theory and the Geometric Analysis programs.