Geodesic
The Bi-Hamiltonian Structures of the DR and DZ Hierarchies in the Approximation up to Genus One
Funkcionalʹnyj analiz i ego priloženiâ, Tome 55 (2021) no. 4, pp. 22-39.

Voir la notice de l'article provenant de la source Math-Net.Ru

In a recent paper, given an arbitrary homogeneous cohomological field theory (CohFT), Rossi, Shadrin, and the first author proposed a simple formula for a bracket on the space of local functionals, which conjecturally gives a second Hamiltonian structure for the double ramification hierarchy associated to the CohFT. In this paper we prove this conjecture in the approximation up to genus $1$ for any semisimple CohFT and relate this bracket to the second Poisson bracket of the Dubrovin–Zhang hierarchy by an explicit Miura transformation.
Keywords: moduli space of curves, cohomology ring, partial differential equation. 
@article{FAA_2021_55_4_a1,
     author = {O. Brauer and A. Yu. Buryak},
     title = {The {Bi-Hamiltonian} {Structures} of the {DR} and {DZ} {Hierarchies} in the {Approximation} up to {Genus} {One}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {22--39},
     publisher = {mathdoc},
     volume = {55},
     number = {4},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2021_55_4_a1/}
}
TY  - JOUR
AU  - O. Brauer
AU  - A. Yu. Buryak
TI  - The Bi-Hamiltonian Structures of the DR and DZ Hierarchies in the Approximation up to Genus One
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2021
SP  - 22
EP  - 39
VL  - 55
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2021_55_4_a1/
LA  - ru
ID  - FAA_2021_55_4_a1
ER  - 
%0 Journal Article
%A O. Brauer
%A A. Yu. Buryak
%T The Bi-Hamiltonian Structures of the DR and DZ Hierarchies in the Approximation up to Genus One
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2021
%P 22-39
%V 55
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2021_55_4_a1/
%G ru
%F FAA_2021_55_4_a1
O. Brauer; A. Yu. Buryak. The Bi-Hamiltonian Structures of the DR and DZ Hierarchies in the Approximation up to Genus One. Funkcionalʹnyj analiz i ego priloženiâ, Tome 55 (2021) no. 4, pp. 22-39. http://geodesic.mathdoc.fr/item/FAA_2021_55_4_a1/

[1] A. Buryak, “Double ramification cycles and integrable hierarchies”, Comm. Math. Phys., 336:3 (2015), 1085–1107 | DOI | MR | Zbl

[2] A. Buryak, B. Dubrovin, J. Guéré, P. Rossi, “Tau-structure for the double ramification hierarchies”, Comm. Math. Phys., 363:1 (2018), 191–260 | DOI | MR | Zbl

[3] A. Buryak, B. Dubrovin, J. Guéré, P. Rossi, “Integrable systems of double ramification type”, Intern. Math. Res. Notices, 2020:24 (2020), 10381–10446 | DOI | MR | Zbl

[4] A. Buryak, J. Guéré, “Towards a description of the double ramification hierarchy for Witten's $r$-spin class”, J. Math. Pures Appl., 106:5 (2016), 837–865 | DOI | MR | Zbl

[5] A. Buryak, J. Guéré, P. Rossi, “DR/DZ equivalence conjecture and tautological relations”, Geom. Topol., 23:7 (2019), 3537–3600 | DOI | MR | Zbl

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

[7] A. Buryak, H. Posthuma, S. Shadrin, “A polynomial bracket for the Dubrovin–Zhang hierarchies”, J. Differential Geom., 92:1 (2012), 153–185 | DOI | MR | Zbl

[8] A. Buryak, P. Rossi, “Recursion relations for double ramification hierarchies”, Comm. Math. Phys., 342:2 (2016), 533–568 | DOI | MR | Zbl

[9] A. Buryak, P. Rossi, S. Shadrin, “Towards a bihamiltonian structure for the double ramification hierarchy”, Lett. Math. Physics, 111:1 (2021), 13 | DOI | MR | Zbl

[10] A. Buryak, S. Shadrin, L. Spitz, D. Zvonkine, “Integrals of $\psi$-classes over double ramification cycles”, Amer. J. Math., 137:3 (2015), 699–737 | DOI | MR | Zbl

[11] A. du Crest de Villeneuve, P. Rossi, “Quantum $D_4$ Drinfeld–Sokolov hierarchy and quantum singularity theory”, J. Geom. Phys., 141 (2019), 29–44 | DOI | MR | Zbl

[12] B. Dubrovin, “Geometry of 2D topological field theories”, Integrable systems and quantum groups (Lectures on the 1st session of CIME, Montecatini Terme, 1993), Lecture Notes in Math., no. 1620, Springer-Verlag, Berlin, 1996, 120–348 | DOI | MR | Zbl

[13] B. Dubrovin, Y. Zhang, “Bihamiltonian hierarchies in 2D topological field theory at one-loop approximation”, Comm. Math. Phys., 198:2 (1998), 311–361 | DOI | MR | Zbl

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

[15] B. Dubrovin, Y. Zhang, “Virasoro symmetries of the extended Toda hierarchy”, Comm. Math. Phys., 250:1 (2004), 161–193 | DOI | MR | Zbl

[16] C. Faber, S. Shadrin, D. Zvonkine, “Tautological relations and the $r$-spin Witten conjecture”, Ann. Sci. École Norm. Sup. (4), 43:4 (2010), 621–658 | DOI | MR | Zbl

[17] F. Hernández Iglesias, S. Shadrin, Bi-Hamiltonian recursion, Liu–Pandharipande relations, and vanishing terms of the second Dubrovin–Zhang bracket, arXiv: 2105.15138

[18] M. Kontsevich, “Intersection theory on the moduli space of curves and the matrix Airy function”, Comm. Math. Phys., 147:1 (1992), 1–23 | DOI | MR | Zbl

[19] M. Kontsevich, Yu. Manin, “Gromov–Witten classes, quantum cohomology, and enumerative geometry”, Comm. Math. Phys., 164:3 (1994), 525–562 | DOI | MR | Zbl

[20] S.-Q. Liu, Z. Wang, Y. Zhang, Linearization of Virasoro symmetries associated with semisimple Frobenius manifolds, arXiv: 2109.01846

[21] A. Okounkov, R. Pandharipande, “The equivariant Gromov–Witten theory of ${\mathbb P}^1$”, Ann. of Math., 163:2 (2006), 561–605 | DOI | MR | Zbl

[22] E. Witten, “Two dimensional gravity and intersection theory on moduli space”, Surv. Differential Geom., 1 (1991), 243–310 | DOI | MR | Zbl

[23] E. Witten, “Algebraic geometry associated with matrix models of two-dimensional gravity”, Topological methods in modern mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, 1993, 235–269 | MR | Zbl