×

Riemann-Hilbert problem associated to Frobenius manifold structures on Hurwitz spaces: Irregular singularity. (English) Zbl 1157.34065

The Hurwitz space is the moduli space of equivalence classes of pairs that consist of a genus \(g\) Riemann surface and a degree \(N\) meromorphic function on it. The latter function defines a realization of the Riemann surface as an \(N\)-sheeted branched covering of the Riemann sphere with \(M\) ramification points. On the Hurwitz spaces, there are three classes of Frobenius manifold structures (basically, the so-called rotational coefficients of the invariant metric), the first of which was found by B. Dubrovin, and two others were recently discovered by the author of the paper. All these Frobenius structures can be recovered from a fundamental solution of certain isomonodromic problem.
The author considers a linear non-Fuchsian \(n\times n\) matrix ODE in \(\mathbb{C}\mathbb{P}^1\), \(\Psi_z\Psi^{-1}=U+z^{-1}V\), where \(U=\text{diag}(\lambda_1,\dots,\lambda_n)\), and \(V=[\Gamma,U]\) is diagonalizable. The parameters \(\lambda_i\) serve as canonical coordinates on the Frobenius manifold, while the matrix \(\Gamma\) is constructed from the rotational coefficients for this manifold and appears in the asymptotic expansion of the function \(\Psi\) at infinity. The monodromy data of the above linear ODE are preserved as \(\lambda_i\), \(i=1,\dots,n\), vary. The author presents, firstly, a class of explicit exact solutions to this linear ODE expressible in terms of Laplace-type integrals of the canonical meromorphic bidifferentials on the above Riemann surface, and, secondly, Schlesinger transformations which relate the solutions \(\Psi\) corresponding to the three classes of Frobenius manifold structures on the Hurwitz spaces with each other. The latter transformations give rise to relationships between different Frobenius manifold structures on the Hurwitz spaces. The presentation is illustrated by computation of the monodromy data of the above solutions corresponding to particular examples of the Hurwitz spaces.

MSC:

34M40 Stokes phenomena and connection problems (linear and nonlinear) for ordinary differential equations in the complex domain
34M50 Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] W. Balser, W. B. Jurkat, and D. A. Lutz, Birkhoff invariants and Stokes’ multipliers for meromorphic linear differential equations , J. Math. Anal. Appl. 71 (1979), 48–94. · Zbl 0415.34008 · doi:10.1016/0022-247X(79)90217-8
[2] A. A. Bolibruch [Bolibrukh], On orders of movable poles of the Schlesinger equation , J. Dynam. Control Systems 6 (2000), 57–73. · Zbl 0961.34077 · doi:10.1023/A:1009569605675
[3] -, On the tau-function for the Schlesinger equation of isomonodromic deformations , Math. Notes 74 (2003), 177–184. · Zbl 1068.34082 · doi:10.1023/A:1025048023068
[4] P. A. Deift, A. R. Its, and X. Zhou, A Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics , Ann. of Math. (2) 146 (1997), 149–235. JSTOR: · Zbl 0936.47028 · doi:10.2307/2951834
[5] R. Dijkgraaf, H. Verlinde, and E. Verlinde, “Notes on topological string theory and \(2\)D quantum gravity” in String Theory and Quantum Gravity (Trieste, Italy, 1990) , World Sci., River Edge, N.J., 1991, 91–156. · Zbl 0985.81681
[6] B. Dubrovin, “Geometry of \(2D\) topological field theories” in Integrable Systems and Quantum Groups (Montecatini Terme, Italy, 1993) , Lecture Notes in Math. 1620 , Springer, Berlin, 1996, 120–348. · Zbl 0841.58065 · doi:10.1007/BFb0094793
[7] -, “Painlevé transcendents in two-dimensional topological field theory” in The Painlevé Property , CRM Ser. Math. Phys., Springer, New York, 1999, 287–412. · Zbl 1026.34095
[8] M. A. Evgrafov, Asymptotic Estimates and Entire Functions , Russian Tracts on Adv. Math. and Phys. 4 , Gordon and Breach, New York, 1961. · Zbl 0121.30202
[9] J. Fay, Kernel functions, analytic torsion, and moduli spaces , Mem. Amer. Math. Soc. 96 (1992), no. 464. · Zbl 0777.32011
[10] C. Hertling, personal communication, 2005.
[11] A. R. Its, The Riemann-Hilbert problem and integrable systems , Notices Amer. Math. Soc. 50 (2003), 1389–1400. · Zbl 1053.34081
[12] M. Jimbo, T. Miwa, and K. Ueno, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients, I: General theory and \(\tau\)-function , Phys. D 2 (1981), 306–352. · Zbl 1194.34167 · doi:10.1016/0167-2789(81)90013-0
[13] A. Kokotov and D. Korotkin, On \(G\)-function of Frobenius manifolds related to Hurwitz spaces , Int. Math. Res. Not. 2004 , no. 7, 343–360. · Zbl 1079.53134 · doi:10.1155/S1073792804131024
[14] -, Isomonodromic tau-function of Hurwitz Frobenius manifolds and its applications , Int. Math. Res. Not. 2006 , no. 18746. · Zbl 1098.53066 · doi:10.1155/IMRN/2006/18746
[15] -, A new hierarchy of integrable systems associated to Hurwitz spaces , Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 366 (2008), 1055–1088. · Zbl 1153.37414 · doi:10.1098/rsta.2007.2061
[16] D. Korotkin, Solution of matrix Riemann-Hilbert problems with quasi-permutation monodromy matrices , Math. Ann. 329 (2004), 335–364. · Zbl 1059.32002 · doi:10.1007/s00208-004-0528-z
[17] I. M. Krichever, Algebraic-geometrical methods in the theory of integrable equations and their perturbations , Acta Appl. Math. 39 (1995), 93–125. · Zbl 0840.35095 · doi:10.1007/BF00994629
[18] B. Malgrange, “Sur les déformations isomonodromiques, I: Singularités régulières” in Mathématique et Physique (Paris, 1979/1982) , Progr. Math. 37 , Birkhäuser, Boston, 1983, 401–426. · Zbl 0528.32017
[19] Y. I. Manin, Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces , Amer. Math. Soc. Colloq. Publ. 47 , Amer. Math. Soc., Providence, 1999. · Zbl 0952.14032
[20] H. E. Rauch, Weierstrass points, branch points, and moduli of Riemann surfaces , Comm. Pure Appl. Math. 12 (1959), 543–560. · Zbl 0091.07301 · doi:10.1002/cpa.3160120310
[21] V. Shramchenko, Deformations of Frobenius structures on Hurwitz spaces , Int. Math. Res. Not. 2005 , no. 6, 339–387. · Zbl 1069.53061 · doi:10.1155/IMRN.2005.339
[22] -, Real doubles of Hurwitz Frobenius manifolds , Comm. Math. Phys. 256 (2005), 635–680. · Zbl 1098.81084 · doi:10.1007/s00220-005-1321-x
[23] W. Wasow, Asymptotic Expansions for Ordinary Differential Equations , Pure Appl. Math. 14 , Interscience, Wiley, New York, 1965. · Zbl 0133.35301
[24] E. Witten, On the structure of the topological phase of two-dimensional gravity , Nuclear Phys. B 340 (1990), 281–332. · doi:10.1016/0550-3213(90)90449-N
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.