Riemann–Hilbert problem for Hurwitz Frobenius manifolds: Regular singularities

Korotkin, Dmitry; Shramchenko, Vasilisa

doi:10.1515/crelle.2011.084

Cited by 4 publications

(5 citation statements)

References 35 publications

(79 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…So far the notion of Bergman tau-function was introduced for two kinds of spaces, namely, for Hurwitz spaces, and the spaces of holomorphic differentials over Riemann surfaces. In the context of Hurwitz spaces the Bergman tau-function coincides with the isomonodromic tau-function of matrix Riemann-Hilbert problem associated to Hurwitz Frobenius manifolds [7,26]. The Bergman tau-function is also a non-trivial ingredient of the Jimbo-Miwa tau-function of Riemann-Hilbert problems with quasi-permutation monodromy groups [25]; it appears also in large N expansion of partition function of hermitian matrix models [12].…”

Section: Introductionmentioning

confidence: 99%

Baker–Akhiezer Spinor Kernel and Tau-functions on Moduli Spaces of Meromorphic Differentials

Kalla

Korotkin²

2014

Commun. Math. Phys.

Self Cite

View full text Add to dashboard Cite

In this paper we study Baker-Akhiezer spinor kernel on moduli spaces of meromorphic differentials on Riemann surfaces. We introduce the Baker-Akhiezer tau-function which is related to both Bergman tau-function (which was studied before in the context of Hurwitz spaces and spaces of holomorphic and quadratic differentials) and KP tau-function on such spaces.In particular, we derive variational formulas of Rauch-Ahlfors type on moduli spaces of meromorphic differentials with prescribed singularities: we use the system of homological coordinates, consisting of absolute and relative periods of the meromorphic differential, and show how to vary the fundamental objects associated to a Riemann surface (the matrix of b-periods, normalized Abelian differentials, the Bergman bidifferential, the Szegö kernel and the Baker-Akhiezer spinor kernel) with respect to these coordinates. The variational formulas encode dependence both on the moduli of the Riemann surface and on the choice of meromorphic differential (variation of the meromorphic differential while keeping the Riemann surface fixed corresponds to flows of KP type).Analyzing the global properties of the Bergman and Baker-Akhiezer tau-functions we establish relationships between various divisor classes on the moduli spaces.DK was partially supported by NSERC, FQRNT and CURC.

show abstract

Section: Introductionmentioning

confidence: 99%

Baker–Akhiezer Spinor Kernel and Tau-functions on Moduli Spaces of Meromorphic Differentials

Kalla

Korotkin²

2014

Commun. Math. Phys.

Self Cite

View full text Add to dashboard Cite

show abstract

“…One of them if non-Fuchsian; it's solution was implicitly given in [16] later represented in a complete form in [69]. Another Riemann-Hilbert problem is Fuchsian; its monodromy group and solution were described in [56]. Monodromy matrices of such monodromy group always have integer values; therefore, they do not contain any parameters and represent isolated points in the spaces of all monodromy groups.…”

Section: Andmentioning

confidence: 99%

Bergman tau function: from Einstein equations and Dubrovin-Frobenius manifolds to geometry of moduli spaces

Korotkin¹

2018

Preprint

View full text Add to dashboard Cite

We review the role played by tau functions of special type -called Bergman tau functions in various areas: theory of isomonodromic deformations, solutions of Einstein's equations, theory of Dubrovin-Frobenius manifolds, geometry of moduli spaces and spectral theory of Riemann surfaces. These tau functions are natural generalizations of Dedekind's eta-function to higher genus. Study of their properties allows to get an explicit form of Einstein's metrics, obtain new relations in Picard groups of various moduli spaces and derive holomorphic factorization formulas of determinants of Laplacians in flat singular metrics on Riemann surfaces, among other things.

show abstract

“…Moreover, in our argument there is still some freedom for the choice of the matrix D. For example, one can look for the deformations corresponding to a matrix D represented as D := U D 0 , where U is an arbitrary g × g matrix and D 0 is some fixed g × n matrix that satisfies D 0 D t 0 = 0. It would give us a family of special deformations parametrized by two matrices, and the deformations of Hurwitz Frobenius manifolds considered in [6,Remark 4] fit into this scheme after a change of parametrization.…”

Section: Shramchenko's Formulasmentioning

confidence: 99%

A Remark on Deformations of Hurwitz Frobenius Manifolds

Buryak

Shadrin

2010

Lett Math Phys

View full text Add to dashboard Cite

Abstract. In this note, we use the formalism of multi-KP hierarchies in order to give some general formulas for infinitesimal deformations of solutions of the Darboux-Egoroff system. As an application, we explain how Shramchenko's deformations of Frobenius manifold structures on Hurwitz spaces fit into the general formalism of Givental-van de Leur twisted loop group action on the space of semi-simple Frobenius manifolds. Mathematics Subject Classification (2000). 37K10 (53D45, 37K30).

show abstract

Riemann–Hilbert problem for Hurwitz Frobenius manifolds: Regular singularities

Cited by 4 publications

References 35 publications

Baker–Akhiezer Spinor Kernel and Tau-functions on Moduli Spaces of Meromorphic Differentials

Baker–Akhiezer Spinor Kernel and Tau-functions on Moduli Spaces of Meromorphic Differentials

Bergman tau function: from Einstein equations and Dubrovin-Frobenius manifolds to geometry of moduli spaces

A Remark on Deformations of Hurwitz Frobenius Manifolds

Contact Info

Product

Resources

About