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Tau-Structure for the Double Ramification Hierarchies

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In this paper we continue the study of the double ramification hierarchy of Buryak (Commun Math Phys 336(3):1085–1107, 2015). After showing that the DR hierarchy satisfies tau-symmetry we define its partition function as the (logarithm of the) tau-function of the string solution and show that it satisfies various properties (string, dilaton, and divisor equations plus some important degree constraints). We then formulate a stronger version of the conjecture from Buryak (2015): for any semisimple cohomological field theory, the Dubrovin–Zhang and double ramification hierarchies are related by a normal [i.e. preserving the tau-structure (Dubrovin et al. in Adv Math 293:382–435, 2016)] Miura transformation which we completely identify in terms of the partition function of the CohFT. In fact, using only the partition functions, the conjecture can be formulated even in the non-semisimple case (where the Dubrovin–Zhang hierarchy is not defined). We then prove this conjecture for various CohFTs (trivial CohFT, Hodge class, Gromov–Witten theory of \({\mathbb{CP}^1}\), 3-, 4- and 5-spin classes) and in genus 1 for any semisimple CohFT. Finally we prove that the higher genus part of the DR hierarchy is basically trivial for the Gromov–Witten theory of smooth varieties with non-positive first Chern class and their analogue in Fan–Jarvis–Ruan–Witten quantum singularity theory (Fan et al. in Ann Math 178(1):1–106, 2013).

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References

  1. Brini A., Carlet G., Romano S., Rossi P.: Rational reductions of the 2D-Toda hierarchy and mirror symmetry. J. Eur. Math. Soc. 19(3), 835–880 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Buryak A., Guéré 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  MATH  Google Scholar 

  4. 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  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. 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  MATH  Google Scholar 

  9. Candelas, P., Ossa, X.C.de la , Green, P.S., Parkes, L.: A pair of Calabi–Yau manifolds as an exactly soluble superconformal theory. Nucl. Phys. B 359(1): 21–74 (1991)

  10. Chiodo A., Iritani H., Ruan Y.: Landau–Ginzburg/Calabi–Yau correspondence, global mirror symmetry and Orlov equivalence. Publ. Math. Inst. Hautes Études Sci. 119, 127–216 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chiodo A., Ruan Y.: Landau–Ginzburg/Calabi–Yau correspondence for quintic three-folds via symplectic transformations. Invent. Math. 182(1), 117–165 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Chiodo A., Ruan Y.: LG/CY correspondence: the state space isomorphism. Adv. Math. 227(6), 2157–2188 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chiodo A., Ruan Y.: A global mirror symmetry framework for the Landau–Ginzburg/Calabi–Yau correspondence. Ann. Inst. Fourier (Grenoble) 61(7), 2803–2864 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dimca A.: Singularities and Topology of Hypersurfaces. Universitext. Springer, New York (1992)

    Book  MATH  Google Scholar 

  15. Dubrovin,B.:Differential geometry of the space of orbits of a Coxeter group. In: Surveys in Differential Geometry: Integral Systems (Integrable Systems), Surveys in Differential Geometry, vol. 4, pp. 181–211, International Press, Boston, MA (1998)

  16. Dubrovin B.A., Liu S.-Q., Yang D., Zhang Y.: Hodge integrals and tau-symmetric integrable hierarchies of Hamiltonian evolutionary PDEs. Adv. Math. 293, 382–435 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dubrovin B.A., Novikov S.P.: Hamiltonian formalism of one-dimensional systems of hydrodynamic type, and the Bogolyubov–Whitham averaging method. Sov. Math. Dokl. 27, 665–669 (1983)

    MATH  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  19. Dubrovin, B., Zhang, Y.: Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov–Witten invariants. A new 2005 version of arXiv:math/0108160

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Fan H., Francis A., Jarvis T., Merrell E., Ruan Y.: Witten’s D 4 Integrable Hierarchies Conjecture. Chin. Ann. Math. 37(2), 175–192 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Fan, H., Jarvis, T., Ruan, Y.: The Witten equation and its virtual fundamental cycle. arXiv:0712.4025

  23. Fan H., Jarvis T., Ruan Y.: The Witten equation, mirror symmetry, and quantum singularity theory. Ann. Math. (2) 178(1), 1–106 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Getzler E.: Intersection theory on \({\overline{\mathcal{M}}_{1,4}}\) and elliptic Gromov–Witten invariants. J. Am. Math. Soc. 10(4), 973–998 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  25. Getzler, E.: The Virasoro conjecture for Gromov–Witten invariants. In: Algebraic Geometry: Hirzebruch 70 (Warsaw, 1998). Contemporary Mathematics, vol. 241, pp. 147–176. American Mathematical Society, Providence, RI (1999)

  26. Givental A.: Equivariant Gromov–Witten invariants. Int. Math. Res. Not. 1996(13), 613–663 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  27. Givental, A.: Symplectic geometry of Frobenius structures. In: Frobenius Manifolds, Aspects Mathematics, vol. E36, 91–112. Friedr. Vieweg, Wiesbaden (2004)

  28. Guéré J.: theoremwithout concavity. Duke Math. J. 165(13), 2461–2527 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  29. Guéré J.: Hodge integrals in FJRW theory. Michigan Math. J. 66(4), 831–854 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  30. Guéré, J.: Maple computer program. on Jérémy Guéré’s website

  31. Hain, R.: Normal functions and the geometry of moduli spaces of curves. In: Handbook of Moduli, vol. I, Advanced Lectures in Mathematics (ALM), vol. 24, pp. 527–578, International Press, Somerville, MA (2013)

  32. 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  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Liu S.-Q., Ruan Y., Zhang Y.: BCFG Drinfeld–Sokolov hierarchies and FJRW-theory. Invent. Math. 201(2), 711–772 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

  36. Milanov T.: Analyticity of the total ancestor potential in singularity theory. Adv. Math. 255, 217–241 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. 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 

  38. Polishchuk A., Vaintrob A.: Matrix factorizations and cohomological field theories. J. Reine Angew. Math. 714, 1–122 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  39. Rossi, P.: Integrable systems and holomorphic curves. In: Proceedings of the Göa Geometry–Topology Conference 2009, pp. 34–57, International Press, Somerville, MA (2010)

  40. Rossi, P.: Nijenhuis operator in contact homology and descendant recursion in symplectic field theory. In: Proceedings of the Göa Geometry–Topology Conference 2014, pp. 156–191, Gökova Geometry/Topology Conference (GGT), Gökova (2015)

  41. Teleman C.: The structure of 2D semi-simple field theories. Invent. Math. 188(3), 525–588 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. Witten, E.: Two-dimensional gravity and intersection theory on moduli space. In: Surveys in Differential Geometry (Cambridge, MA, 1990), pp. 243–310, Lehigh University, Bethlehem, PA (1991)

  43. Witten E.: Phases of N =  2 theories in two dimensions. Nucl. Phys. B 403(1–2), 159–222 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to thank Andrea Brini, Guido Carlet, Rahul Pandharipande, Sergey Shadrin, and Dimitri Zvonkine for useful discussions. A. B. was supported by Grant ERC-2012-AdG-320368-MCSK in the group of R. Pandharipande at ETH Zurich, Grant RFFI-16-01-00409 and the Marie Curie Fellowship (Project ID 797635). B. D. was partially supported by PRIN 2010-11 Grant “Geometric and analytic theory of Hamiltonian systems in finite and infinite dimensions” of Italian Ministry of Universities and Researches. J. G. was supported by the Einstein foundation. P. R. was partially supported by a Chaire CNRS/Enseignement superieur 2012-2017 Grant. Part of the work was completed during the visits of B. D. and P. R to the Forschungsinstitut für Mathematik at ETH Zürich in 2014 and 2015.

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Authors and Affiliations

  1. School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK

    Alexandr Buryak

  2. SISSA, Via Bonomea 265, Trieste, 34136, Italy

    Boris Dubrovin

  3. Univ. Grenoble Alpes, CNRS, Institut Fourier, 38000, Grenoble, France

    Jérémy Guéré

  4. IMB, UMR5584 CNRS, Université de Bourgogne Franche-Comté, 21000, Dijon, France

    Paolo Rossi

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  1. Alexandr Buryak
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  2. Boris Dubrovin
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  3. Jérémy Guéré
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  4. Paolo Rossi
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Correspondence to Paolo Rossi.

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Communicated by H.-T. Yau

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Buryak, A., Dubrovin, B., Guéré, J. et al. Tau-Structure for the Double Ramification Hierarchies. Commun. Math. Phys. 363, 191–260 (2018). https://doi.org/10.1007/s00220-018-3235-4

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