Geodesic
WKB expansion for a Yang–Yang generating function and the Bergman tau function
Teoretičeskaâ i matematičeskaâ fizika, Tome 206 (2021) no. 3, pp. 295-338 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study symplectic properties of the monodromy map of second-order equations on a Riemann surface whose potential is meromorphic with double poles. We show that the Poisson bracket defined in terms of periods of the meromorphic quadratic differential implies the Goldman Poisson structure on the monodromy manifold. We apply these results to a WKB analysis of this equation and show that the leading term in the WKB expansion of the generating function of the monodromy symplectomorphism (the Yang–Yang function introduced by Nekrasov, Rosly, and Shatashvili) is determined by the Bergman tau function on the moduli space of meromorphic quadratic differentials.
Keywords: Riemann surface, symplectic map generating function, tau function, Goldman bracket.
Mots-clés : monodromy map 
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M. Bertola; D. A. Korotkin. WKB expansion for a Yang–Yang generating function and the Bergman tau function. Teoretičeskaâ i matematičeskaâ fizika, Tome 206 (2021) no. 3, pp. 295-338. http://geodesic.mathdoc.fr/item/TMF_2021_206_3_a1/

[1] N. S. Hawley, M. Schiffer, “Half-order differentials on Riemann surfaces”, Acta Math., 115 (1966), 199–236 | DOI | MR

[2] A. Beilinson, V. Drinfeld, Opers, arXiv: math/0501398

[3] J. D. Fay, Theta-Functions on Riemann Surfaces, Lecture Notes in Mathematics, 352, Springer, Berlin, New York, 1973 | DOI | MR

[4] M. Bertola, D. Korotkin, C. Norton, “Symplectic geometry of the moduli space of projective structures in homological coordinates”, Invent. Math., 210:3 (2017), 759–814 | DOI | MR

[5] D. Gaiotto, J. Teschner, “Irregular singularities in Liouville theory and Argyres–Douglas type gauge theories”, JHEP, 12 (2012), 50, 76 pp., arXiv: 1203.1052 | DOI

[6] D. Korotkin, “Periods of meromorphic quadratic differentials and Goldman bracket”, Topological Recursion and Its Influence in Analysis, Geometry, and Topology, Proceedings of Symposia in Pure Mathematics, 100, eds. M. L. Chiu–Chu, M. Mulase, AMS, Providence, RI, 2016, 491–516 | DOI | MR

[7] D. G. L. Allegretti, T. Bridgeland, “The monodromy of meromorphic projective structures”, Trans. Amer. Math. Soc., 373:9 (2020), 6321–6367 | DOI | MR

[8] D. G. L. Allegretti, “Voros symbols as cluster coordinates”, J. Topol., 12:4 (2019), 1031–1068 | DOI | MR

[9] D. Kvillen, “Determinanty operatorov Koshi–Rimana na rimanovykh poverkhnostyakh”, Funkts. analiz i ego pril., 19:1 (1985), 37–41 | DOI | MR | Zbl

[10] A. Kokotov, D. Korotkin, “Tau-functions on spaces of Abelian differentials and higher genus generalization of Ray–Singer formula”, J. Differ. Geom., 82:1 (2009), 35–100 | DOI | MR | Zbl

[11] V. G. Knizhnik, “Analytic fields on Riemann surfaces. II”, Commun. Math. Phys., 112:4 (1987), 567–590 | DOI | MR

[12] V. G. Knizhnik, “Mnogopetlevye amplitudy v teorii kvantovykh strun i kompleksnaya geometriya”, UFN, 159:11 (1989), 401–53 | DOI | MR

[13] B. Eynard, N. Orantin, “Invariants of algebraic curves and topological expansion”, Commun. Number Theory Phys., 1:2 (2007), 347–452 | DOI | MR

[14] D. Korotkin, P. Zograf, “Tau-function and Prym class”, Algebraic and Geometric Aspects of Integrable Systems and Random Matrices, Contemporary Mathematics, 593, eds. A. Dzhamay, K. Maruno, V. U. Pierce, AMS, Providence, RI, 2013, 241–261 | DOI | MR | Zbl

[15] M. Kontsevich, “Intersection theory on the moduli space of curves and the matrix Airy function”, Commun. Math. Phys., 147:1 (1992), 1–23 | DOI | MR

[16] M. Bertola, D. Korotkin, “Hodge and Prym tau functions, Strebel differentials and combinatorial model of ${\mathcal M}_{g,n}$”, Commun. Math. Phys., 378:2 (2020), 1279–1341 | DOI | MR

[17] D. Korotkin, “Bergman tau function: from Einstein equations and Dubrovin–Frobenius manifolds to geometry of moduli spaces”, Integrable Systems and Algebraic Geometry, v. 2, London Mathematical Society Lecture Note Series, 459, eds. R. Donagi, T. Shaska, Cambridge Univ. Press, Cambridge, 2020, 215–287, arXiv: 1812.03514 | DOI

[18] W. Goldman, “The symplectic nature of fundamental groups of surfaces”, Adv. Math., 54:2 (1984), 200–225 | DOI | MR

[19] N. Nekrasov, A. Rosly, S. Shatashvili, “Darboux coordinates, Yang–Yang functional, and gauge theory”, Nucl. Phys. B. Proc. Suppl., 216:1 (2011), 69–93, arXiv: 1103.3919 | DOI | MR

[20] M. Bertola, D. Korotkin, Tau-function ang monodromy symplectomorphisms, arXiv: 1910.03370

[21] D. Gaiotto, G. W. Moore, A. Neitzke, “Wall-crossing, Hitchin systems, and the WKB approximation”, Adv. Math., 234 (2013), 239–403, arXiv: 0807.4723 | DOI | MR

[22] M. Bertola, D. Korotkin, Extended Goldman symplectic structure in Fock–Goncharov coordinates, arXiv: arXiv/1910.06744

[23] L. O. Chekhov, “Simplekticheskie struktury na prostranstvakh Teikhmyullera $\mathfrak T_{g,s,n}$ i klasternye algebry”, Tr. MIAN, 309 (2020), 99–109 | DOI | DOI

[24] J. D. Fay, Kernel Functions, Analytic Torsion and Moduli Spaces, Memoirs of the American Mathematical Society, 96, no. 464, AMS, Providence, RI, 1992 | DOI | MR

[25] C. Kalla, D. Korotkin, “Baker–Akhiezer spinor kernel and tau-functions on moduli spaces of meromorphic differentials”, Commun. Math. Phys., 331:3 (2014), 1191–1235, arXiv: 1307.0481 | DOI | MR

[26] T. Kawai, Y. Takei, Algebraic Analysis of Singular Perturbation Theory, Translations of Mathematical Monographs, 227, AMS, Providence, RI, 2005 | DOI | MR

[27] V. Vazov, Asimptoticheskie razlozheniya reshenii obyknovennykh differentsialnykh uravnenii, Mir, M., 1968 | MR

[28] P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, X. Zhou, “Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory”, Commun. Pure Appl. Math., 52:11 (1999), 1335–1425 | 3.0.CO;2-1 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR

[29] M. Abramovits, I. Stigan (red.), Spravochnik po spetsialnym funktsiyam s formulami, grafikami i matematicheskimi tablitsami, Nauka, M., 1979 | MR | MR | Zbl

[30] L. Chekhov, B. Eynard, N. Orantin, “Free energy topological expansion for the 2-matrix model”, JHEP, 12 (2006), 053, 31 pp., arXiv: math-ph/0603003 | DOI | MR

[31] T. Bridgeland, “Riemann–Hilbert problems from Donaldson–Thomas theory”, Invent. Math., 216:1 (2019), 69–124 | DOI | MR

[32] W. P. Thurston, The Geometry and Topology of Three-Manifolds, Princeton Lecture Notes, 2002 http://library.msri.org/books/gt3m/

[33] V. V. Fock, Dual Teichmüller spaces, arXiv: dg-ga/9702018

[34] L. Chekhov, Lecture notes on quantum Teichmüller theory, arXiv: 0710.2051

[35] K. Iwaki, T. Nakanishi, “Exact WKB analysis and cluster algebras”, J. Phys. A: Math. Theor., 47:47 (2014), 474009, 98 pp. | DOI | MR

[36] K. Iwaki, T. Nakanishi, “Exact WKB analysis and cluster algebras II: Simple poles, orbifold points, and generalized cluster algebras”, Internat. Math. Res. Not., 2016:14 (2016), 4375–4417 | DOI | MR

[37] T. Bridgeland, I. Smith, “Quadratic differentials as stability conditions”, Publ. Math. Inst. Hautes Études Sci., 121:1 (2015), 155–278 | DOI | MR