Geometry & Topology

DR/DZ equivalence conjecture and tautological relations

Alexandr Buryak, Jérémy Guéré, and Paolo Rossi

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/gt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We present a family of conjectural relations in the tautological ring of the moduli spaces of stable curves which implies the strong double ramification/Dubrovin–Zhang equivalence conjecture introduced by the authors with Dubrovin (Comm. Math. Phys. 363 (2018) 191–260). Our tautological relations have the form of an equality between two different families of tautological classes, only one of which involves the double ramification cycle. We prove that both families behave the same way upon pullback and pushforward with respect to forgetting a marked point. We also prove that our conjectural relations are true in genus 0 and 1 and also when first pushed forward from ̄g,n+m to ̄g,n and then restricted to g,n for any g,n,m0. Finally we show that, for semisimple CohFTs, the DR/DZ equivalence only depends on a subset of our relations, finite in each genus, which we prove for g2. As an application we find a new formula for the class λg as a linear combination of dual trees intersected with kappa- and psi-classes, and we check it for g3.

Article information

Source
Geom. Topol., Volume 23, Number 7 (2019), 3537-3600.

Dates
Received: 5 May 2018
Accepted: 4 November 2018
First available in Project Euclid: 7 January 2020

Permanent link to this document
https://projecteuclid.org/euclid.gt/1578366034

Digital Object Identifier
doi:10.2140/gt.2019.23.3537

Mathematical Reviews number (MathSciNet)
MR4059088

Zentralblatt MATH identifier
07152164

Subjects
Primary: 14H10: Families, moduli (algebraic)
Secondary: 37K10: Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.)

Keywords
moduli space of curves cohomology double ramification cycle partial differential equations

Citation

Buryak, Alexandr; Guéré, Jérémy; Rossi, Paolo. DR/DZ equivalence conjecture and tautological relations. Geom. Topol. 23 (2019), no. 7, 3537--3600. doi:10.2140/gt.2019.23.3537. https://projecteuclid.org/euclid.gt/1578366034


Export citation

References

  • A Buryak, Double ramification cycles and integrable hierarchies, Comm. Math. Phys. 336 (2015) 1085–1107
  • A Y Buryak, New approaches to hierarchies of topological type, Uspekhi Mat. Nauk 72 (2017) 63–112 In Russian; translated in Russian Math. Surveys 72 (2017) 841–887
  • A Buryak, B Dubrovin, J Guéré, P Rossi, Tau-structure for the double ramification hierarchies, Comm. Math. Phys. 363 (2018) 191–260
  • A Buryak, B Dubrovin, J Guéré, P Rossi, Integrable systems of double ramification type, Int. Math. Res. Notices (online publication February 2019) art. id. rnz029
  • A Buryak, J Guéré, Towards a description of the double ramification hierarchy for Witten's $r$–spin class, J. Math. Pures Appl. 106 (2016) 837–865
  • A Buryak, H Posthuma, S Shadrin, A polynomial bracket for the Dubrovin–Zhang hierarchies, J. Differential Geom. 92 (2012) 153–185
  • A Buryak, P Rossi, Double ramification cycles and quantum integrable systems, Lett. Math. Phys. 106 (2016) 289–317
  • A Buryak, P Rossi, Recursion relations for double ramification hierarchies, Comm. Math. Phys. 342 (2016) 533–568
  • A Buryak, S Shadrin, L Spitz, D Zvonkine, Integrals of $\psi$–classes over double ramification cycles, Amer. J. Math. 137 (2015) 699–737
  • B Dubrovin, Y Zhang, Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov–Witten invariants, preprint (2001)
  • C Faber, Chow rings of moduli spaces of curves, I: The Chow ring of $\bar{\mathscr M}_3$, Ann. of Math. 132 (1990) 331–419
  • C Faber, R Pandharipande, Hodge integrals and Gromov–Witten theory, Invent. Math. 139 (2000) 173–199
  • C Faber, R Pandharipande, Relative maps and tautological classes, J. Eur. Math. Soc. 7 (2005) 13–49
  • E Getzler, Intersection theory on $\bar{\mathscr M}_{1,4}$ and elliptic Gromov–Witten invariants, J. Amer. Math. Soc. 10 (1997) 973–998
  • E Getzler, Topological recursion relations in genus $2$, from “Integrable systems and algebraic geometry” (M-H Saito, Y Shimizu, K Ueno, editors), World Sci., River Edge, NJ (1998) 73–106
  • R Hain, Normal functions and the geometry of moduli spaces of curves, from “Handbook of moduli, I” (G Farkas, I Morrison, editors), Adv. Lect. Math. 24, International, Somerville, MA (2013) 527–578
  • E-N Ionel, Topological recursive relations in $H^{2g}(\mathcal M_{g,n})$, Invent. Math. 148 (2002) 627–658
  • F Janda, Relations on $\bar M_{g,n}$ via equivariant Gromov–Witten theory of $\mathbb P^1$, preprint (2015)
  • F Janda, R Pandharipande, A Pixton, D Zvonkine, Double ramification cycles on the moduli spaces of curves, Publ. Math. Inst. Hautes Études Sci. 125 (2017) 221–266
  • M Kontsevich, Y Manin, Gromov–Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994) 525–562
  • S Marcus, J Wise, Stable maps to rational curves and the relative Jacobian, preprint (2013)
  • R Pandharipande, A Pixton, D Zvonkine, Relations on $\bar{\mathscr M}_{g,n}$ via $3$–spin structures, J. Amer. Math. Soc. 28 (2015) 279–309
  • P Rossi, Integrability, quantization and moduli spaces of curves, Symmetry Integrability Geom. Methods Appl. 13 (2017) art. id. 60