Skip to main content
SpringerLink
Log in

Towards a bihamiltonian structure for the double ramification hierarchy

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

We propose a remarkably simple and explicit conjectural formula for a bihamiltonian structure of the double ramification hierarchy corresponding to an arbitrary homogeneous cohomological field theory. Various checks are presented to support the conjecture.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Buryak, A.: Double ramification cycles and integrable hierarchies. Commun. Math. Phys. 336(3), 1085–1107 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  2. Buryak, A., Dubrovin, B., Guéré, J., Rossi, P.: Tau-structure for the double ramification hierarchies. Commun. Math. Phys. 363(1), 191–260 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  3. Buryak, A., Dubrovin, B., Guéré, J., Rossi, P.: Integrable systems of double ramification type. Int. Math. Res. Not. (2016). https://doi.org/10.1093/imrn/rnz029

    Article  MATH  Google Scholar 

  4. Buryak, A., Guere, J.: Towards a description of the double ramification hierarchy for Witten’s \(r\)- spin class. J. Math. Pures Appl. 106(5), 837–865 (2016)

    Article  MathSciNet  Google Scholar 

  5. Buryak, A., Guéré, J., Rossi, P.: DR/DZ equivalence conjecture and tautological relations. Geom. Topol. 23(7), 3537–3600 (2019)

    Article  MathSciNet  Google Scholar 

  6. Buryak, A., Posthuma, H., Shadrin, S.: On deformations of quasi-Miura transformations and the Dubrovin–Zhang bracket. J. Geom. Phys. 62(7), 1639–1651 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  7. Buryak, A., Posthuma, H., Shadrin, S.: A polynomial bracket for the Dubrovin–Zhang hierarchies. J. Differ. Geom. 92(1), 153–185 (2012)

    Article  MathSciNet  Google Scholar 

  8. Buryak, A., Rossi, P.: Recursion relations for double ramification hierarchies. Commun. Math. Phys. 342(2), 533–568 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  9. Buryak, A., Rossi, P.: Double ramification cycles and quantum integrable systems. Lett. Math. Phys. 106(3), 289–317 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  10. Buryak, A., Shadrin, S., Spitz, L., Zvonkine, D.: Integrals of \(\psi \)- classes over double ramification cycles. Am. J. Math. 137(3), 699–737 (2015)

    Article  MathSciNet  Google Scholar 

  11. Carlet, G., Kramer, R., Shadrin, S.: Central invariants revisited. J. l’École Polytech. Math. 5, 149–175 (2018)

    Article  MathSciNet  Google Scholar 

  12. Carlet, G., Posthuma, H., Shadrin, S.: Deformations of semisimple Poisson pencils of hydrodynamic type are unobstructed. J. Differ. Geom. 108(1), 63–89 (2018)

    Article  MathSciNet  Google Scholar 

  13. Carlet, G., Dubrovin, B., Zhang, Y.: The extended Toda hierarchy. Mosc. Math. J. 4(2), 313–332 (2004)

    Article  MathSciNet  Google Scholar 

  14. Dickey, L.A.: Soliton equations and Hamiltonian systems, 2nd edn. World Scientific, Singapore (2003)

    Book  Google Scholar 

  15. du Crest, A., de Villeneuve, P., Rossi, P.: Quantum \(D_4\) Drinfeld–Sokolov hierarchy and quantum singularity theory. J. Geom. Phys. 141, 29–44 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  16. Dubrovin, B., Liu, S.-Q., Zhang, Y.: On Hamiltonian perturbations of hyperbolic systems of conservation laws I: Quasi-triviality of bihamiltonian perturbations. Commun. Pure Appl. Math. 59(4), 559–615 (2006)

    Article  Google Scholar 

  17. Dubrovin, B., Zhang, Y.: Bihamiltonian hierarchies in \(2\) D topological field theory at one-loop approximation. Commun. Math. Phys. 198(2), 311–361 (1998)

    Article  ADS  Google Scholar 

  18. Dubrovin, B., Zhang, Y.: Frobenius manifolds and Virasoro constraints. Sel. Math. New Ser. 5(4), 423–466 (1999)

    Article  MathSciNet  Google Scholar 

  19. Dubrovin, B., Zhang,Y.: Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov–Witten invariants. arXiv:math/0108160

  20. Faber, C., Pandharipande, R.: Logarithmic series and Hodge integrals in the tautological ring. With an appendix by Don Zagier. Dedicated to William Fulton on the occasion of his 60th birthday. Mich. Math. J. 48, 215–252 (2000)

    Article  Google Scholar 

  21. Faber, C., Shadrin, S., Zvonkine, D.: Tautological relations and the \(r\)-spin Witten conjecture. Ann. Sci. l’École Norm. Supér. (4) 43(4), 621–658 (2010)

    Article  MathSciNet  Google Scholar 

  22. Hain, R.: Normal functions and the geometry of moduli spaces of curves. In: Handbook of Moduli, vol. I, pp. 527–578. International Press (2013)

  23. Janda, F., Pandharipande, R., Pixton, A., Zvonkine, D.: Double ramification cycles on the moduli spaces of curves. Publ. Math. 125, 221–266 (2017)

    Article  MathSciNet  Google Scholar 

  24. Kontsevich, M.: Intersection theory on the moduli space of curves and the matrix Airy function. Commun. Math. Phys. 147(1), 1–23 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  25. Kontsevich, M., Manin, Yu.: Gromov–Witten classes, quantum cohomology, and enumerative geometry. Commun. Math. Phys. 164(3), 525–562 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  26. Kruska, M., Miura, R., Gardner, C., Zabusky, N.: Korteweg–de Vries equation and generalizations. V. Uniqueness and nonexistence of polynomial conservation laws. J. Math. Phys. 11(3), 952–960 (1970)

    Article  ADS  MathSciNet  Google Scholar 

  27. Liu, S.-Q.: Lecture notes on Bihamiltonian Structures and their Central Invariants. In: B-Model Gromov–Witten theory, pp. 573–625. Birkhäuser, Cham (2018)

  28. Liu, S.-Q., Zhang, Y.: Deformations of semisimple bihamiltonian structures of hydrodynamic type. J. Geom. Phys. 54(4), 427–453 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  29. Liu, S.-Q., Zhang, Y.: Jacobi structures of evolutionary partial differential equations. Adv. Math. 227(1), 73–130 (2011)

    Article  MathSciNet  Google Scholar 

  30. Liu, S.-Q., Zhang, Y.: Bihamiltonian cohomologies and integrable hierarchies I: a special case. Commun. Math. Phys. 324(3), 897–935 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  31. Manin YI: Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces. American Mathematical Society Colloquium Publications, vol. 47. American Mathematical Society, Providence (1999)

  32. Marcus, S., Wise, J.: Stable maps to rational curves and the relative Jacobian. arXiv:1310.5981

  33. Pandharipande, R., Pixton, A., Zvonkine, D.: Relations on \({\overline{{\cal{M}}}}_{g, n}\) via \(3\)- spin structures. J. Am. Math. Soc. 28(1), 279–309 (2015)

    Article  MathSciNet  Google Scholar 

  34. Rossi, P.: Integrability, quantization and moduli spaces of curves. SIGMA. Symmetry, Integrability and Geometry: Methods and Applications, vol. 13 , Paper No. 060 (2017)

  35. Witten, E.: Two-dimensional gravity and intersection theory on moduli space. Surv. Differ. Geom. 1, 243–310 (1991)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We are grateful to A. Arsie and P. Lorenzoni for valuable remarks about the preliminary version of the paper. We would like to thank G. Carlet and F. Hernández Iglesias for useful discussions on closely related topics. We thank anonymous referees of our paper for valuable comments that allowed to improve the exposition of the paper. The work of A. B. (Sections 1 and 4) was supported by the Grant No. 20-11-20214 of the Russian Science Foundation. S. S. was supported by the Netherlands Organization for Scientific Research.

Author information

Authors and Affiliations

  1. Faculty of Mathematics, National Research University Higher School of Economics, 6 Usacheva str, Moscow, Russian Federation, 119048

    Alexandr Buryak

  2. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, GSP-1, Moscow, Russian Federation, 119991

    Alexandr Buryak

  3. Dipartimento di Matematica “Tullio Levi-Civita”, Università degli Studi di Padova, Via Trieste 63, 35121, Padua, Italy

    Paolo Rossi

  4. Korteweg–de Vries Instituut voor Wiskunde, Universiteit van Amsterdam, Postbus 94248, 1090GE, Amsterdam, The Netherlands

    Sergey Shadrin

Authors
  1. Alexandr Buryak
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paolo Rossi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sergey Shadrin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexandr Buryak.

Additional information

To the memory of Boris Dubrovin, our teacher and friend.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Buryak, A., Rossi, P. & Shadrin, S. Towards a bihamiltonian structure for the double ramification hierarchy. Lett Math Phys 111, 13 (2021). https://doi.org/10.1007/s11005-020-01341-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11005-020-01341-6

Keywords

Mathematics Subject Classification

Search

Navigation