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Solutions to the Witten-Dijkgraaf-Verlinde-Verlinde equations of associativity and their applications


The main subject of the proposed research is the theory of Frobenius manifolds, which unifies different areas of mathematics such as the theory of singularities, quantum cohomologies of algebraic varieties, and Hurwitz spaces. From the point of view of physics, some Frobenius structures describe the moduli space of topological conformal field theories. They also appear in other areas of active current research - from random matrices to interface dynamics.

The first goal of this project is to construct, starting with an arbitrary Frobenius manifold, an associated Frobenius manifold whose dimension is twice as large as the dimension of the given manifold. Such a construction was found in the previous work of the applicant for Frobenius structures on Hurwitz spaces (spaces of meromorphic functions on a Riemann surface). A generalization of the construction to an arbitrary Frobenius manifold would be of great interest due appearance of Frobenius structures in aforementioned topics. The second goal is to find new Frobenius structures on Hurwitz spaces which in the simplest case correspond to Hitchin's two-parametric solution to the Painlev\'e-6 equation.

This problem has already been partially addressed in the work of the applicant: new Frobenius structures related to a one-parameter subfamily of Hitchin's solutions were found. However, the problem of introducing two parameters into the simplest Hurwitz Frobenius structures remains an intriguing open question. Finally, we plan to investigate the meaning of new Frobenius structures from the point of view of applications. In particular, we shall explore a recently established relationship between Frobenius structures on Hurwitz spaces and random matrix theory. We plan to use the obtained Frobenius structures to find new random matrix models.

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