Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 026, 36 pages      arXiv:2206.10578      https://doi.org/10.3842/SIGMA.2023.026

On Generalized WKB Expansion of Monodromy Generating Function

Roman Klimov
Department of Mathematics and Statistics, Concordia University,1455 de Maisonneuve W., Montreal, QC H3G 1M8, Canada

Received June 22, 2022, in final form April 11, 2023; Published online April 28, 2023

Abstract
We study symplectic properties of the monodromy map of the Schrödinger equation on a Riemann surface with a meromorphic potential having second-order poles. At first, we discuss the conditions for the base projective connection, which induces its own set of Darboux homological coordinates, to imply the Goldman Poisson structure on the character variety. Using this result, we extend the paper [Theoret. and Math. Phys. 206 (2021), 258-295, arXiv:1910.07140], by performing generalized WKB expansion of the generating function of monodromy symplectomorphism (the Yang-Yang function) and computing its first three terms.

Key words: WKB expansion; moduli spaces; tau-functions.

pdf (614 kb)   tex (39 kb)  

References

  1. Allegretti D.G.L., Voros symbols as cluster coordinates, J. Topol. 12 (2019), 1031-1068, arXiv:1802.05479.
  2. Allegretti D.G.L., Bridgeland T., The monodromy of meromorphic projective structures, Trans. Amer. Math. Soc. 373 (2020), 6321-6367, arXiv:1802.02505.
  3. Baraglia D., Huang Z., Special Kähler geometry of the Hitchin system and topological recursion, Adv. Theor. Math. Phys. 23 (2019), 1981-2024, arXiv:1707.04975.
  4. Bertola M., Harnad J., Hurtubise J., Hamiltonian structure of rational isomonodromic deformation systems, arXiv:2212.06880.
  5. Bertola M., Korotkin D., Hodge and Prym tau functions, Strebel differentials and combinatorial model of $\mathcal{M}_{g,n}$, Comm. Math. Phys. 378 (2020), 1279-1341, arXiv:1804.02495.
  6. Bertola M., Korotkin D., Spaces of Abelian differentials and Hitchin's spectral covers, Int. Math. Res. Not. 2021 (2021), 11246-11269, arXiv:1812.05789.
  7. Bertola M., Korotkin D., Tau-functions and monodromy symplectomorphisms, Comm. Math. Phys. 388 (2021), 245-290, arXiv:1910.03370.
  8. Bertola M., Korotkin D., WKB expansion for a Yang-Yang generating function and the Bergman tau function, Theoret. and Math. Phys. 206 (2021), 258-295, arXiv:1910.07140.
  9. Bertola M., Korotkin D., Norton C., Symplectic geometry of the moduli space of projective structures in homological coordinates, Invent. Math. 210 (2017), 759-814, arXiv:1506.07918.
  10. Birkhoff G.D., The generalized Riemann problem for linear differential equations and the allied problems for linear difference and $q$-difference equations, Proc. Amer. Acad. Arts Sci. 49 (1913), 521-568.
  11. Bridgeland T., Riemann-Hilbert problems from Donaldson-Thomas theory, Invent. Math. 216 (2019), 69-124, arXiv:1611.03697.
  12. Bridgeland T., Masoero D., On the monodromy of the deformed cubic oscillator, Math. Ann. 385 (2023), 193-258, arXiv:2006.10648.
  13. Chekhov L.O., Symplectic structures on Teichmüller spaces $ \mathfrak{T}_{g,s,n}$ and cluster algebras, Proc. Steklov Inst. Math. 309 (2020), 87-96, arXiv:1912.11862.
  14. Chen D., Gendron Q., Towards a classification of connected components of the strata of $k$-differentials, Doc. Math. 27 (2022), 1031-1100, arXiv:2101.01650.
  15. Douady A., Hubbard J., On the density of Strebel differentials, Invent. Math. 30 (1975), 175-179.
  16. Eriksson D., Montplet G., Wentworth R., Complex Chern-Simons bundles in the relative setting, arXiv:2109.02033.
  17. Eynard B., Orantin N., Invariants of algebraic curves and topological expansion, Commun. Number Theory Phys. 1 (2007), 347-452, arXiv:math-ph/0702045.
  18. Fock V., Goncharov A., Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. (2006), 1-211, arXiv:math.AG/0311149.
  19. Gaiotto D., Moore G.W., Neitzke A., Wall-crossing, Hitchin systems, and the WKB approximation, Adv. Math. 234 (2013), 239-403, arXiv:0907.3987.
  20. Goldman W.M., The symplectic nature of fundamental groups of surfaces, Adv. Math. 54 (1984), 200-225.
  21. Hawley N.S., Schiffer M., Half-order differentials on Riemann surfaces, Acta Math. 115 (1966), 199-236.
  22. Iwaki K., Koike T., Takei Y., Voros coefficients for the hypergeometric differential equations and Eynard-Orantin's topological recursion: Part II: For confluent family of hypergeometric equations, J. Integrable Syst. 4 (2019), xyz004, 46 pages, arXiv:1810.02946.
  23. Kawai S., The symplectic nature of the space of projective connections on Riemann surfaces, Math. Ann. 305 (1996), 161-182.
  24. Kawai T., Takei Y., Algebraic analysis of singular perturbation theory, Transl. Math. Monogr., Vol. 227, Amer. Math. Soc., Providence, RI, 2005.
  25. Klimov R., Variational formulas on spaces of ${\rm SL}(2)$ Hitchin's spectral covers, arXiv:2108.12911.
  26. Kokotov A., Korotkin D., Tau-functions on spaces of abelian differentials and higher genus generalizations of Ray-Singer formula, J. Differential Geom. 82 (2009), 35-100, arXiv:math.SP/0405042.
  27. Kontsevich M., Zorich A., Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math. 153 (2003), 631-678, arXiv:math.GT/0201292.
  28. Korotkin D., Periods of meromorphic quadratic differentials and Goldman bracket, in Topological Recursion and its Influence in Analysis, Geometry, and Topology, Proc. Sympos. Pure Math., Vol. 100, Amer. Math. Soc., Providence, RI, 2018, 491-515, arXiv:1702.04705.
  29. Korotkin D., Bergman tau-function: from Einstein equations and Dubrovin-Frobenius manifolds to geometry of moduli spaces, in Integrable Systems and Algebraic Geometry, Vol. 2, London Math. Soc. Lecture Note Ser., Vol. 459, Cambridge University Press, Cambridge, 2020, 215-287, arXiv:1812.03514.
  30. Korotkin D., Zograf P., Tau function and the Prym class, in Algebraic and Geometric Aspects of Integrable Systems and Random Matrices, Contemp. Math., Vol. 593, Amer. Math. Soc., Providence, RI, 2013, 241-261, arXiv:1302.0577.
  31. Nekrasov N., Rosly A., Shatashvili S., Darboux coordinates, Yang-Yang functional, and gauge theory, Nuclear Phys. B Proc. Suppl. 216 (2011), 69-93, arXiv:1103.3919.
  32. Teschner J., Quantization of the Hitchin moduli spaces, Liouville theory and the geometric Langlands correspondence I, Adv. Theor. Math. Phys. 15 (2011), 471-564, arXiv:1005.2846.
  33. Thurston W., The geometry and topology of 3-manifolds, Princeton University Notes, 1980.
  34. Voros A., The return of the quartic oscillator: the complex WKB method, Ann. Inst. H. Poincaré Sect. A (N.S.) 39 (1983), 211-338.

Previous article  Next article  Contents of Volume 19 (2023)