Riemann-Hilbert problem associated to Frobenius manifold structures on Hurwitz spaces: irregular singularity

Solutions to the Riemann-Hilbert problems with irregular singularities naturally associated to semisimple Frobenius manifold structures on Hurwitz spaces (moduli spaces of meromorphic functions on Riemann surfaces) are constructed. The solutions are given in terms of meromorphic bidifferentials defined on the underlying Riemann surface. The relationship between different classes of Frobenius manifold structures on Hurwitz spaces (real doubles, deformations) is described on the level of the corresponding Riemann-Hilbert problems.Comment: 41 page, 11 figure

Enumeration of N-rooted maps using quantum field theory

A one-to-one correspondence is proved between the N-rooted ribbon graphs, or maps, with e edges and the (e-N+1)-loop Feynman diagrams of a certain quantum field theory. This result is used to obtain explicit expressions and relations for the generating functions of N-rooted maps and for the numbers of N-rooted maps with a given number of edges using the path integral approach applied to the corresponding quantum field theory.Comment: 27 pages, 7 figure

Feynman diagrams, ribbon graphs, and topological recursion of Eynard-Orantin

We consider two seemingly unrelated problems, the calculation of the WKB expansion of the harmonic oscillator wave functions and the counting the number of Feynman diagrams in QED or in many-body physics and show that their solutions are both encoded in a single enumerative problem, the calculation of the number of certain types of ribbon graphs. In turn, the numbers of such ribbon graphs as a function of the number of their vertices and edges can be determined recursively through the application of the topological recursion of Eynard-Orantin to the algebraic curve encoded in the Schr\"odinger equation of the harmonic oscillator. We show how the numbers of these ribbon graphs can be written down in closed form for any given number of vertices and edges. We use these numbers to obtain a formula for the number of N-rooted ribbon graphs with e edges, which is the same as the number of Feynman diagrams for 2N-point function with e+1-N loops.Comment: 29 pages, 7 figure

Riemann-Hilbert problem for Hurwitz Frobenius manifolds: regular singularities

In this paper we study the Fuchsian Riemann-Hilbert (inverse monodromy) problem corresponding to Frobenius structures on Hurwitz spaces. We find a solution to this Riemann-Hilbert problem in terms of integrals of certain meromorphic differentials over a basis of an appropriate relative homology space, study the corresponding monodromy group and compute the monodromy matrices explicitly for various special cases.Comment: final versio

Frobenius structures, integrable systems and Hurwitz spaces

This thesis consists of two main parts. In the first part a new family of integrable systems related to Hurwitz spaces of elliptic coverings with simple branch points is constructed. The integrable systems are closely related to Takasaki's version of the Schlesinger system on an elliptic surface. A trigonometric degeneration of the integrable systems is presented. The trigonometric version of an auxiliary system of differential equations for the images of branch points of the covering under a uniformization map with respect to branch points is derived. This system is applied to solving the Boyer-Finley equation (self-dual Einstein equation with a rotating Killing vector). Thereby, a class of implicit solutions to the Boyer-Finley equation is found in terms of objects related to the Hurwitz spaces. The second part presents two classes of new semisimple Frobenius structures on Hurwitz spaces (spaces of ramified coverings of ([Special characters omitted.] P 1 ). The original construction of Hurwitz Frobenius manifolds by Dubrovin is described in terms of the normalized meromorphic bidifferential W of the second kind on a Riemann surface. In Dubrovin's construction, the branch points {{470} m } of the covering play the role of canonical coordinates on the Hurwitz Frobenius manifolds. We find new Frobenius structures on Hurwitz spaces with coordinates {{470} m ; n m } in terms of the Schiffer and Bergman kernels (bidifferentials) on a Riemann surface. We call these structures the "real doubles" of the Hurwitz Frobenius manifolds of Dubrovin. To construct another class of new Frobenius structures on Hurwitz spaces, we introduce a g ( g + 1)/2-parametric deformation of the bidifferential W , where g is the genus of the corresponding Riemann surface. Analogously to the bidifferential W its deformation defines Frobenius structures on Hurwitz spaces; these structures give a g ( g + 1)/2-parametric deformation of Dubrovin's Hurwitz Frobenius manifolds. Similarly, we introduce the deformations of the Schiffer and Bergman kernels which define Frobenius structures on the Hurwitz spaces with coordinates {{470} m ; n m }. Thereby we obtain deformations of the real doubles of the Hurwitz Frobenius manifolds of Dubrovin. Each new Frobenius structure gives a new solution to the WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) system. For the simplest Hurwitz space in genus one, the corresponding solutions are found explicitly, together with the corresponding G -function