Quantum Geometry of Moduli Spaces and Motives

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This project concerns research at the crossroads of representation theory, algebraic geometry and mathematical physics. The principle of symmetry has played a crucial role in mathematics since Galois's study of roots of polynomials, and in physics since Einstein’s development of special relativity. Over the past 50 years, new instances of the very idea of symmetry have been discovered: supersymmetry, and later quantum symmetry, expressed in the form of the quantum groups. Representation theory comes into the picture as the way to understand the notion of quantum symmetry, but the very idea of quantum symmetry and its geometric origins remain elusive. ALong these lines, this project concerns the study on systems of linear differential equations of one complex variable, with algebraic coefficients and arbitrary singularities. One of the main objects of study is the space of solutions of such differential equations (and their generalizations), Stokes data, reflecting asymptotic properties of solutions. The PI will investigate quantization of these moduli spaces and its connections with representation theory and theoretical physics. Quantum geometry of moduli spaces of Stokes data provides a new geometric approach to quantum symmetries. It embeds quantum groups and their key properties into a much more general and geometric framework of the quantized moduli spaces of Stokes data. This project will provide training and research opportunities for graduate students in this area of research.In more detail, the PI will study the quantum geometry of various moduli spaces, using the theory cluster Poisson varieties as the main tool, and applications to algebraic geometry, representation Theory, and mathematical physics. In addition, the PI will study classical and quantum polylogarithms and quantum Hodge field theory. The main priorities of the project are the following: a) To give a comprehensive treatment of the cluster structure of moduli spaces of meromorphic connections with possibly irregular singularities on Riemann surfaces and apply the results to quantization of these moduli spaces. b) To develop cluster quantization at roots of unity with applications to representations of DeConcini-Kac quantum groups and invariants of threefolds. c) To develop the theory of quantum multiple polylogarithms, providing a quantum deformation of the periods of the pro-unipotent completion of the motivic fundamental group of the punctured projective line. d) To further develop quantum Hodge 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.