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


SIGMA 11 (2015), 036, 13 pages      arXiv:1501.05205      https://doi.org/10.3842/SIGMA.2015.036
Contribution to the Special Issue on Algebraic Methods in Dynamical Systems

The Stokes Phenomenon and Some Applications

Marius van der Put
University of Groningen, Department of Mathematics, P.O. Box 407, 9700 AK Groningen, The Netherlands

Received January 21, 2015, in final form April 21, 2015; Published online May 01, 2015

Abstract
Multisummation provides a transparent description of Stokes matrices which is reviewed here together with some applications. Examples of moduli spaces for Stokes matrices are computed and discussed. A moduli space for a third Painlevé equation is made explicit. It is shown that the monodromy identity, relating the topological monodromy and Stokes matrices, is useful for some quantum differential equations and for confluent generalized hypergeometric equations.

Key words: Stokes matrices; moduli space for linear connections; quantum differential equations; Painlevé equations.

pdf (411 kb)   tex (25 kb)

References

  1. Babbitt D.G., Varadarajan V.S., Local moduli for meromorphic differential equations, Astérisque (1989), 1-217.
  2. Balser W., From divergent power series to analytic functions. Theory and application of multisummable power series, Lecture Notes in Math., Vol. 1582, Springer-Verlag, Berlin, 1994.
  3. Balser W., Formal power series and linear systems of meromorphic ordinary differential equations, Universitext, Springer-Verlag, New York, 2000.
  4. Balser W., Jurkat W.B., Lutz D.A., On the reduction of connection problems for differential equations with an irregular singular point to ones with only regular singularities. I, SIAM J. Math. Anal. 12 (1981), 691-721.
  5. Balser W., Jurkat W.B., Lutz D.A., On the reduction of connection problems for differential equations with an irregular singular point to ones with only regular singularities. II, SIAM J. Math. Anal. 19 (1988), 398-443.
  6. Cano J., Ramis J.-P., Théorie de Galois différentielle, multisommabilité et phénomenes de Stokes, in preparation.
  7. Cruz Morales J.A., van der Put M., Stokes matrices for the quantum differential equations of some Fano varieties, Eur. J. Math. 1 (2015), 138-153, arXiv:1211.5266.
  8. Dubrovin B., Geometry of $2$D topological field theories, in Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120-348, hep-th/9407018.
  9. Dubrovin B., Geometry and analytic theory of Frobenius manifolds, in Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math. (1998), Extra Vol. II, 315-326, math.AG/9807034.
  10. Dubrovin B., Painlevé transcendents in two-dimensional topological field theory, in The Painlevé Property, CRM Ser. Math. Phys., Springer, New York, 1999, 287-412, math.AG/9803107.
  11. Duval A., Mitschi C., Matrices de Stokes et groupe de Galois des équations hypergéométriques confluentes généralisées, Pacific J. Math. 138 (1989), 25-56.
  12. Guest M.A., From quantum cohomology to integrable systems, Oxford Graduate Texts in Mathematics, Vol. 15, Oxford University Press, Oxford, 2008.
  13. Guest M.A., Its A., Lin C.-S., Isomonodromy aspects of the $tt^{\ast}$ equations of Cecotti and Vafa I. Stokes data, arXiv:1209.2045.
  14. Guest M.A., Sakai H., Orbifold quantum D-modules associated to weighted projective spaces, Comment. Math. Helv. 89 (2014), 273-297, arXiv:0810.4236.
  15. Guzzetti D., Stokes matrices and monodromy of the quantum cohomology of projective spaces, Comm. Math. Phys. 207 (1999), 341-383, math.AG/9904099.
  16. Iritani H., An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math. 222 (2009), 1016-1079, arXiv:0903.1463.
  17. Jimbo M., Miwa T., Ueno K., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and $\tau $-function, Phys. D 2 (1981), 306-352.
  18. Jimbo M., Miwa T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, Phys. D 2 (1981), 407-448.
  19. Loday-Richaud M., Stokes phenomenon, multisummability and differential Galois groups, Ann. Inst. Fourier (Grenoble) 44 (1994), 849-906.
  20. Loday-Richaud M., Pourcin G., On index theorems for linear ordinary differential operators, Ann. Inst. Fourier (Grenoble) 47 (1997), 1379-1424.
  21. Marcolli M., Tabuada G., From exceptional collections to motivic decompositions via noncommutative motives, J. Reine Angew. Math. 701 (2015), 153-167, arXiv:1202.6297.
  22. Mitschi C., Differential Galois groups of confluent generalized hypergeometric equations: an approach using Stokes multipliers, Pacific J. Math. 176 (1996), 365-405.
  23. Ramis J.-P., Stokes phenomenon: historical background, in The Stokes Phenomenon and Hilbert's 16th Problem (Groningen, 1995), World Sci. Publ., River Edge, NJ, 1996, 1-5.
  24. Stokes G.G., Early letters to lady Stokes, March 17, 1857, in Memoir and Scientific Correspondence of the Late Sir George Gabriel Stokes, Bart, Cambridge University Press, Cambridge, 2010, 50-76.
  25. Tanabé S., Invariant of the hypergeometric group associated to the quantum cohomology of the projective space, Bull. Sci. Math. 128 (2004), 811-827, math.AG/0201090.
  26. Tanabé S., Ueda K., Invariants of hypergeometric groups for Calabi-Yau complete intersections in weighted projective spaces, Commun. Number Theory Phys. 7 (2013), 327-359, arXiv:1305.1659.
  27. Ueda K., Stokes matrices for the quantum cohomologies of Grassmannians, Int. Math. Res. Not. 2005 (2005), 2075-2086, math.AG/0503355.
  28. Ueda K., Stokes matrix for the quantum cohomology of cubic surfaces, math.AG/0505350.
  29. van der Put M., Saito M.-H., Moduli spaces for linear differential equations and the Painlevé equations, Ann. Inst. Fourier (Grenoble) 59 (2009), 2611-2667, arXiv:0902.1702.
  30. van der Put M., Singer M.F., Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften, Vol. 328, Springer-Verlag, Berlin, 2003.
  31. van der Put M., Top J., Geometric aspects of the Painlevé equations ${\rm PIII}({\rm D}_6)$ and ${\rm PIII}({\rm D}_7)$, SIGMA 10 (2014), 050, 24 pages, arXiv:1207.4023.

Previous article  Next article   Contents of Volume 11 (2015)