4,854 research outputs found
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
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
Isomonodromic tau-function of Hurwitz Frobenius manifolds and its applications
In this work we find the isomonodromic (Jimbo-Miwa) tau-function
corresponding to Frobenius manifold structures on Hurwitz spaces. We discuss
several applications of this result. First, we get an explicit expression for
the G-function (solution of Getzler's equation) of the Hurwitz Frobenius
manifolds. Second, in terms of this tau-function we compute the genus one
correction to the free energy of hermitian two-matrix model. Third, we find the
Jimbo-Miwa tau-function of an arbitrary Riemann-Hilbert problem with
quasi-permutation monodromy matrices. Finally, we get a new expression (analog
of genus one Ray-Singer formula) for the determinant of Laplace operator in the
Poincar\'e metric on Riemann surfaces of an arbitrary genus.Comment: The direct proof of variational formulas on branched coverings is
added. The title is modified due to observed coincidence of isomonodromic
tau-function of Hurwitz Frobenius manifolds with Bergman tau-function on
Hurwitz spaces introduced by the author
On computing Belyi maps
We survey methods to compute three-point branched covers of the projective
line, also known as Belyi maps. These methods include a direct approach,
involving the solution of a system of polynomial equations, as well as complex
analytic methods, modular forms methods, and p-adic methods. Along the way, we
pose several questions and provide numerous examples.Comment: 57 pages, 3 figures, extensive bibliography; English and French
abstract; revised according to referee's suggestion
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