Generating Function of Monodromy Symplectomorphism for 2 × 2 Fuchsian Systems and Its WKB Expansion


  • Marco Bertola Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve W., Montréal, Québec, Canada H3G 1M8
  • Dmitry Korotkin Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve W., Montréal, Québec, Canada H3G 1M8
  • Fabrizio del Monte Centre de recherches mathématiques, Université de Montréal, C. P. 6128, succ. centreville, Montréal, Québec, Canada H3C 3J7


Ключові слова:

системи Фукса, вiдображення монодромiї, генерувальна функцiя, тау-функцiя, ВКБ розвинення


Ми вивчаємо ВКБ розвинення 2 × 2 системи лiнiйних диференцiальних рiвнянь з фуксовими сингулярностями. Основна увага сфокусована на генерувальнiй функцiї монодромного симплектоморфiзму, яка, вiдповiдно до недавньої роботи [10], є тiсно пов’язаною з тау-функцiєю Джимбо–Мiви. Ми обчислюємо першi три члени ВКБ розвинення генерувальної функцiї та встановлюємо її зв’язок з тау-функцiєю Бергмана.

Mathematical Subject Classification 2020: 34M60, 53D22, 34M56


A.Y. Alekseev and A.Z. Malkin, The hyperbolic moduli space of flat connections and the isomorphism of symplectic multiplicity spaces, Duke Math. J. 93 (1998), 575--595.

D.Allegretti and T.Bridgeland, The monodromy of meromorphic projective structures, Trans. Amer. Math. Soc. 373 (2020), 6321--6367.

D.Allegretti, Voros symbols as cluster coordinates, J. Topol. 12 (2019), 1031--1068.

O.Babelon, D.Bernard, and M.Talon, Introduction to Classical Integrable Systems, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2003.

M.Bershtein, P.Gavrylenko, and A.Grassi, Quantum spectral problems and isomonodromic deformations, Comm. Math. Phys. 393 (2022), 347--418.

M.Bertola and D.Korotkin, Spaces of Abelian differentials and Hitchin's spectral covers, Int. Math. Res. Not. IMRN 2021 (2021), 11246--11269.

M.Bertola and D.Korotkin, Hodge and Prym tau functions, Strebel differentials and combinatorial model of $M_{g,n}$, Commun. Math. Phys. 378 (2020), 1279--1341.

M.Bertola and D.A. Korotkin, WKB expansion for a Yang-Yang generating function and the Bergman tau function, Theor. Math. Phys.206 (2021), 258--295.

M.Bertola and D.Korotkin, Extended Goldman symplectic structure in fock-goncharov coordinates, J. Diff. Geom., to appear (2021),

M.Bertola and D.Korotkin, Tau-functions and monodromy symplectomorphisms, Commun. Math. Phys. 388 (2021), 245--290.

G.Bonelli, O.Lisovyy, K.Maruyoshi, A.Sciarappa, and A.Tanzini, On Painlevé/gauge theory correspondence, Lett. Math. Phys. 107 (2017), 2359--2417.

T.Bridgeland, Riemann-Hilbert problems from Donaldson-Thomas theory, Invent. Math. 216 (2019), 69--124.

T.Bridgeland and D.Masoero, On the monodromy of the deformed cubic oscillator, Math. Annalen, 385 (2022), 193--258.

T.Bridgeland and I.Smith, Quadratic differentials as stability conditions, Publ. Math. Inst. Hautes Études Sci. 121 (2015), 155--278.

I.Coman, P.Longhi, and J.Teschner, From quantum curves to topological string partition functions II, Comm. Math. Phys. 399 (2023), 1501--1548.

F.Del Monte, H.Desiraju, and P.Gavrylenko, Isomonodromic tau functions on a torus as Fredholm determinants, and charged partitions, Comm. Math. Phys. 398 (2023), 1--56.

F.Del Monte, H.Desiraju and P.Gavrylenko, Monodromy dependence and symplectic geometry of isomonodromic tau functions on the torus, preprint,

H.Dillinger, E.Delabaere, and F.Pham, Résurgence de Voros et périodes des courbes hyperelliptiques, Ann. Inst. Fourier (Grenoble) 43 (1993), 163--199.

B.Dubrovin, Integrable systems and Riemann surfaces, Lecture Notes, 2009. Available from:

J.D. Fay, Theta functions on Riemann surfaces, 352, Springer, 2006.

V.Fock and A.Goncharov, Moduli spaces of local systems and higher teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006), 1--211.

D.Gaiotto, G.W. Moore, and A.Neitzke, Wall-crossing, Hitchin systems, and the WKB approximation, Adv. Math. 234 (2013), 239--403.

O.Gamayun, N.Iorgov, and O.Lisovyy, Conformal field theory of Painlevé VI, J. High Energ. Phys. 2012 (2012), 38.

P.Gavrylenko and O.Lisovyy, Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions, Commun. Math. Phys. 363 (2018), 1-58.

P.Gavrylenko, A.Marshakov, and A.Stoyan, Irregular conformal blocks, Painlevé III and the blow-up equations, J. High Energ. Phys. 2020 (2020), 125.

W.M. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. Math. 54 (1984), 200--225.

N.Hitchin, Frobenius manifolds, Gauge theory and symplectic geometry, (Eds. J.Hurtubise, F.Lalonde, and G.Sabidussi), Nato Science Series C, 488, Springer, 1997, 69--112.

N.Iorgov, O.Lisovyy, and J.Teschner, Isomonodromic tau-functions from Liouville conformal blocks, Commun. Math. Phys. 336 (2015), 671--694.

N.Iorgov, O.Lisovyy, and Y.Tykhyy, Painlevé VI connection problem and monodromy of c=1 conformal blocks, J. High Energ. Phys. 2013 (2013), 029.

A.R. Its, O.Lisovyy, and A.Prokhorov, Monodromy dependence and connection formulae for isomonodromic tau functions, Duke Math. J. 167 (2018), 1347--1432.

A.Its, O.Lisovyy, and Y.Tykhyy, Connection problem for the sine-Gordon/Painlevé III tau function and irregular conformal blocks, Int. Math. Res. Not. IMRN 2015 (2015), 8903--8924.

A.Its and A.Prokhorov, Connection problem for the tau-function of the sine-Gordon reduction of Painlevé-iii equation via the Riemann-Hilbert approach, Int. Math. Res. Not. IMRN 2016 (2016), 6856--6883.

K.Iwaki, 2-Parameter τ-function for the first Painlevé equation: Topological recursion and direct monodromy problem via exact WKB analysis, Commun. Math. Phys. 377 (2020), 1047--1098.

S.Jeong and N.Nekrasov, Riemann-Hilbert correspondence and blown up surface defects, J. High Energ. Phys. 2020 (2020), 6.

M.Jimbo, T.Miwa, and K.Ueno, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients: I. General theory and τ-function, Physica D: Nonlinear Phenomena 2 (1981), 306--352.

C.Kalla and D.Korotkin, Baker-Akhiezer spinor kernel and tau-functions on moduli spaces of meromorphic differentials, Comm. Math. Physics 331 (2014), 1191--1235.

T.Kawai and Y.Takei, Algebraic Analysis of Singular Perturbation Theory, 227, Amer. Math. Soc., Providence, RI, 2005.

A.Kokotov and D.Korotkin, Tau-functions on spaces of Abelian differentials and higher genus generalizations of Ray-Singer formula, J. Differential Geom. 82 (2009), 35--100.

D.Korotkin, Solution of matrix Riemann-Hilbert problems with quasi-permutation monodromy matrices, Math. Ann. 329 (2004), 335--364.

D.Korotkin and H.Samtleben, Quantization of coset space sigma models coupled to two-dimensional gravity, Commun. Math. Phys. 190 (1997), 411--457.

D.Korotkin, Bergman tau-function: From Einstein equations and Dubrovin-Frobenius manifolds to geometry of moduli spaces, Integrable Systems and Algebraic Geometry, 2, (Eds. R.Donagi and T.Shaska), LMS Lecture Notes, 459, 2020, 215--287.

D.Korotkin and P.Zograf, Tau function and the Prym class, Algebraic and Geometric Aspects of Integrable Systems and Random Matrices (Eds. A.Dzhamay, K.Maruno, and V.Pierce), Contemp. Math., 593, 2013, 241--262.

N.Nekrasov, Blowups in BPS/CFT correspondence, and Painlevé VI, preprint,

A.Voros, The return of the quadratic oscillator. The complex WKB method, Ann. IHP Phys. Théor. 39 (1983), 211--338.


Як цитувати

Bertola, M.; Korotkin, D.; del Monte, F. Generating Function of Monodromy Symplectomorphism for 2 × 2 Fuchsian Systems and Its WKB Expansion. Журн. мат. фіз. анал. геом. 2023, 19, 301-338.





Дані завантаження ще не доступні.