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Calabi-Yau manifolds and higher potential functions


Calabi-Yau manifolds arise naturally in algebraic geometry as objects with certain special structures. In theoretical physics these geometric objects define super symmetric models of the universe via string theory. String theory in turn constructs a hierarchy of functions on the parameter space of Calabi-Yau manifolds, the so-called higher potential functions. In the context of algebraic geometry, the mathematical understanding of potential functions is far from complete. The aim of this project is to develop the understanding of higher potential functions by analogy with the well-understood case of tree-level (level 0) functions. In more detail, the objectives of the project are the following:-Interpret mathematically the computations of closed string higher-level potential functions on Calabi-Yaumoduli spaces found in the physics literature; establish the connection of potential functions and categories defined by the geometric objects; apply the obtained mathematical constructions to the study of parameter spaces of geometric objects, and-analyse the correspondence between spaces of matrices and open strings on families of Calabi-Yaumanifolds, recently discovered by physicists; in particular, compare symmetries of the two sides. The project involves an interdisciplinary component, aiming to bridge a gap between mathematics and theoretical physics. It also involves a training component, developing expertise of the Researcher in areas where the host institution has particular strengths: matrix models, Hodge theory and representation theory.

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