Enumerative Geometry of Hitchin Systems and Topological Quantum Field Theory

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This award supports research in mathematical physics. The topological and geometric structures of various kinds of spaces have captivated mathematicians and physicists for many years. The current proposal lies at the interface between algebra, geometry, combinatorics and analysis with applications to 2D-topological quantum field theory. In recent years the PI has organized several workshops that have specifically facilitated interactions between different research communities. The PI will continue to promote interactions between enumerative geometry, algebra and the theory of quantization in the direction emphasized in this project.The proposed project is aimed at understanding a new point of view utilizing edge contraction operations of ribbon graphs. These operations were originally used to give a recursion relation of the generalized Catalan numbers of arbitrary genus. This recursion implies the DVV formula for the intersection numbers of tautological cotangent classes on the moduli space of stable pointed curves. The PI and a collaborator have discovered, an alternative axiomatic formulation for 2D-topological quantum field theory by edge contraction operations of ribbon graphs. The set of rules based on edge contractions also represent the key structure of topological recursion. The edge contraction axioms reflect both the structure of a Frobenius algebra and the pair of pants decomposition of a topological surface. The pair of pants decomposition of a punctured Riemann surface was first used by Mirzakhani to give a recursion of Weil-Petersson volumes of the moduli space of hyperbolic surfaces with geodesic boundary components of fixed lengths. Using the multiplication and comultiplication of the Frobenius algebra we aim at giving alternative axiomatic definition of cohomological field theories in the same way. The goal of this project is to study the interplay between topological recursion, Mirzakhani recursion of Weil-Petersson volumes, the classification theorem of CohFT for semi-simple Frobenius algebra and character varieties.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.