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Recursion Relations for Double Ramification Hierarchies

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Abstract

In this paper we study various properties of the double ramification hierarchy, an integrable hierarchy of hamiltonian PDEs introduced in Buryak (CommunMath Phys 336(3):1085–1107, 2015) using intersection theory of the double ramification cycle in the moduli space of stable curves. In particular, we prove a recursion formula that recovers the full hierarchy starting from just one of the Hamiltonians, the one associated to the first descendant of the unit of a cohomological field theory. Moreover, we introduce analogues of the topological recursion relations and the divisor equation both for the Hamiltonian densities and for the string solution of the double ramification hierarchy. This machinery is very efficient and we apply it to various computations for the trivial and Hodge cohomological field theories, and for the r -spin Witten’s classes. Moreover, we prove the Miura equivalence between the double ramification hierarchy and the Dubrovin-Zhang hierarchy for the Gromov-Witten theory of the complex projective line (extended Toda hierarchy).

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References

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Buryak A.: Dubrovin-Zhang hierarchy for the Hodge integrals. Commun. Number Theory Phys. 9(2), 239–271 (2015)

    Article  MathSciNet  Google Scholar 

  3. Buryak A., Posthuma H., Shadrin S.: On deformations of quasi-Miura transformations and the Dubrovin-Zhang bracket. J. Geomet. Phys. 2012(2), 1639–1651 (2012)

    Article  ADS  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  5. 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 

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

    MathSciNet  MATH  Google Scholar 

  7. Cavalieri R., Marcus S., Wise J.: Polynomial families of tautological classes on \({\mathcal{M}^{rt}_ {g,n}}\). J. Pure Appl. Algebra 216(4), 950–981 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chiodo A.: The Witten top Chern class via K-theory. J. Algebraic Geom. 15(4), 681–707 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dubrovin, B.: Geometry of 2D topological field theory. In: Integrable Systems and Quantum Groups, Lecture Notes in Mathematics, vol. 1620, pp. 120–348 (1996)

  10. Dubrovin B., Zhang Y.: Virasoro symmetries of the extended Toda hierarchy. Commun. Math. Phys. 250(1), 161–193 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Dubrovin, B.A., Zhang, Y.: Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants, p. 295, a new 2005 version of arXiv:math/0108160v1

  12. Eliashberg, Y., Givental, A., Hofer, H.: Introduction to symplectic field theory. GAFA 2000 Visions in Mathematics special volume, part II, pp. 560–673 (2000)

  13. Faber C., Pandharipande R.: Hodge integrals, partition matrices, and the λ g -conjecture. Ann. Math. 157(1), 97–124 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Faber C., Shadrin S., Zvonkine D.: Tautological relations and the r -spin Witten conjecture. Annales Scientifiques de l’Ecole Normale Superieure 43(4), 621–658 (2010)

    MathSciNet  MATH  Google Scholar 

  15. Fabert O., Rossi P.: String, dilaton and divisor equation in Symplectic Field Theory. Int.Math. Res. Not. IMRN 19, 4384–4404 (2011)

    MathSciNet  Google Scholar 

  16. Gelfand I.M., Dikii L.A.: Fractional powers of operators and Hamiltonian systems. Funct. Anal. Appl. 10, 259–273 (1976)

    Article  Google Scholar 

  17. Getzler, E.: Topological recursion relations in genus 2. In: Integrable Systems and Algebraic Geometry (Kobe/Kyoto 1997), pp. 73–106. World Scientific Publishing, River Edge (1998)

  18. Getzler, E.: TheVirasoro conjecture for Gromov-Witten invariants, Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), 147–176, Contemp. Math., 241, Am. Math. Soc., Providence, RI (1999)

  19. Getzler E., Pandharipande R.: Virasoro constraints and the Chern classes of the Hodge bundle. Nucl. Phys. B 530(3), 701–714 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Givental A.: Gromov-Witten invariants and quantization of quadratic Hamiltonians. Moscow Math. J. 1(4), 551–568 (2001)

    MathSciNet  MATH  Google Scholar 

  21. Hain, R.: Normal functions and the geometry of moduli spaces of curves. In: Handbook of Moduli, vol. I, pp. 527–578, Adv. Lect. Math. (ALM), 24. Int. Press, Somerville (2013)

  22. Hori K.: Constraints for topological strings in D ≥  1. Nucl. Phys. B 439(1–2), 395–420 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

  25. Pandharipande R.: The Toda equations and the Gromov-Witten theory of the Riemann sphere. Lett. Math. Phys. 53(1), 59–74 (2000)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Polishchuk, A., Vaintrob, A.: Algebraic construction of Witten’s top Chern class. Advances in algebraic geometry motivated by physics (Lowell, MA, 2000), pp. 229–249, Contemp. Math., 276, Am. Math. Soc., Providence, RI (2001)

  28. Rossi, P.: Integrable systems and holomorphic curves. In: Proceedings of the Gökova Geometry-Topology Conference 2009, pp. 34–57. Int. Press, Somerville (2010)

  29. Rossi, P.: Nijenhuis operator in contact homology and descendant recursion in symplectic field theory. In: Proceedings of the Gökova Geometry-Topology Conference 2014. International Press (2015). arXiv:1201.1127

  30. Satsuma J., Ablowitz M.J., Kodama Y.: On an internal wave equation describing a stratified fluid with finite depth. Phys. Lett. A 73(4), 283–286 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  31. Witten, E.: Algebraic geometry associated with matrix models of two-dimensional gravity. In: Topological Methods in Modern Mathematics (Stony Brook, NY, 1991), pp. 235–269. Publish or Perish, Houston (1993)

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Authors and Affiliations

  1. Department of Mathematics, ETH Zurich, Ramistrasse 101, HG G 27.1, 8092, Zurich, Switzerland

    Alexandr Buryak

  2. IMB, UMR 5584 CNRS, Université de Bourgogne, 9, Avenue Alain Savary, 21078, Dijon Cedex, France

    Paolo Rossi

Authors
  1. Alexandr Buryak
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  2. Paolo Rossi
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Correspondence to Alexandr Buryak.

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Communicated by N. Reshetikhin

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Buryak, A., Rossi, P. Recursion Relations for Double Ramification Hierarchies. Commun. Math. Phys. 342, 533–568 (2016). https://doi.org/10.1007/s00220-015-2535-1

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