Towards an orbifold generalization of Zvonkine’s 𝑟-ELSV formula

Kramer, R.; Lewanski, D.; Popolitov, A.; Shadrin, S.

doi:10.1090/tran/7793

Towards an orbifold generalization of Zvonkine’s $r$-ELSV formula
HTML articles powered by AMS MathViewer

by R. Kramer, D. Lewanski, A. Popolitov and S. Shadrin PDF

Trans. Amer. Math. Soc. 372 (2019), 4447-4469 Request permission

Abstract:

We perform a key step towards the proof of Zvonkine’s conjectural $r$-ELSV formula that relates Hurwitz numbers with completed $(r+1)$-cycles to the geometry of the moduli spaces of the $r$-spin structures on curves: we prove the quasi-polynomiality property prescribed by Zvonkine’s conjecture. Moreover, we propose an orbifold generalization of Zvonkine’s conjecture and prove the quasi-polynomiality property in this case as well. In addition to that, we study the $(0,1)$- and $(0,2)$-functions in this generalized case, and we show that these unstable cases are correctly reproduced by the spectral curve initial data.

References

A. Alexandrov, D. Lewanski, and S. Shadrin, Ramifications of Hurwitz theory, KP integrability and quantum curves, J. High Energy Phys. 5 (2016), 124, front matter+30. MR 3521843, DOI 10.1007/JHEP05(2016)124
Vincent Bouchard, Daniel Hernández Serrano, Xiaojun Liu, and Motohico Mulase, Mirror symmetry for orbifold Hurwitz numbers, J. Differential Geom. 98 (2014), no. 3, 375–423. MR 3263522
Vincent Bouchard and Marcos Mariño, Hurwitz numbers, matrix models and enumerative geometry, From Hodge theory to integrability and TQFT tt*-geometry, Proc. Sympos. Pure Math., vol. 78, Amer. Math. Soc., Providence, RI, 2008, pp. 263–283. MR 2483754, DOI 10.1090/pspum/078/2483754
Alessandro Chiodo, Towards an enumerative geometry of the moduli space of twisted curves and $r$th roots, Compos. Math. 144 (2008), no. 6, 1461–1496. MR 2474317, DOI 10.1112/S0010437X08003709
Norman Do, Oliver Leigh, and Paul Norbury, Orbifold Hurwitz numbers and Eynard-Orantin invariants, Math. Res. Lett. 23 (2016), no. 5, 1281–1327. MR 3601067, DOI 10.4310/MRL.2016.v23.n5.a3
P. Dunin-Barkowski, M. Kazarian, N. Orantin, S. Shadrin, and L. Spitz, Polynomiality of Hurwitz numbers, Bouchard-Mariño conjecture, and a new proof of the ELSV formula, Adv. Math. 279 (2015), 67–103. MR 3345179, DOI 10.1016/j.aim.2015.03.016
P. Dunin-Barkowski, D. Lewanski, A. Popolitov, and S. Shadrin, Polynomiality of orbifold Hurwitz numbers, spectral curve, and a new proof of the Johnson-Pandharipande-Tseng formula, J. Lond. Math. Soc. (2) 92 (2015), no. 3, 547–565. MR 3431649, DOI 10.1112/jlms/jdv047
P. Dunin-Barkowski, N. Orantin, S. Shadrin, and L. Spitz, Identification of the Givental formula with the spectral curve topological recursion procedure, Comm. Math. Phys. 328 (2014), no. 2, 669–700. MR 3199996, DOI 10.1007/s00220-014-1887-2
Torsten Ekedahl, Sergei Lando, Michael Shapiro, and Alek Vainshtein, Hurwitz numbers and intersections on moduli spaces of curves, Invent. Math. 146 (2001), no. 2, 297–327. MR 1864018, DOI 10.1007/s002220100164
Bertrand Eynard, Intersection numbers of spectral curves, arXiv:1104.0176 (2011).
B. Eynard, Invariants of spectral curves and intersection theory of moduli spaces of complex curves, Commun. Number Theory Phys. 8 (2014), no. 3, 541–588. MR 3282995, DOI 10.4310/CNTP.2014.v8.n3.a4
Bertrand Eynard, Motohico Mulase, and Bradley Safnuk, The Laplace transform of the cut-and-join equation and the Bouchard-Mariño conjecture on Hurwitz numbers, Publ. Res. Inst. Math. Sci. 47 (2011), no. 2, 629–670. MR 2849645, DOI 10.2977/PRIMS/47
B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion, Commun. Number Theory Phys. 1 (2007), no. 2, 347–452. MR 2346575, DOI 10.4310/CNTP.2007.v1.n2.a4
F. Janda, R. Pandharipande, A. Pixton, and D. Zvonkine, Double ramification cycles on the moduli spaces of curves, Publ. Math. Inst. Hautes Études Sci. 125 (2017), 221–266. MR 3668650, DOI 10.1007/s10240-017-0088-x
P. Johnson, R. Pandharipande, and H.-H. Tseng, Abelian Hurwitz-Hodge integrals, Michigan Math. J. 60 (2011), no. 1, 171–198. MR 2785870, DOI 10.1307/mmj/1301586310
Paul D. Johnson, Equivariant Gromov-Witten theory of one dimensional stacks, arXiv:0903.1068v1 [math.AG] (2009).
Paul Johnson, Double Hurwitz numbers via the infinite wedge, Trans. Amer. Math. Soc. 367 (2015), no. 9, 6415–6440. MR 3356942, DOI 10.1090/S0002-9947-2015-06238-2
Serguei Kerov and Grigori Olshanski, Polynomial functions on the set of Young diagrams, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 2, 121–126 (English, with English and French summaries). MR 1288389
Reinier Kramer, Danilo Lewanski, and Sergey Shadrin, Quasi-polynomiality of monotone orbifold Hurwitz numbers and Grothendieck’s dessins d’enfants, arXiv:1610.08376 (2016).
Danilo Lewanski, On ELSV-type formulae, Hurwitz numbers and topological recursion, arXiv:1703.06435 (2017).
Danilo Lewanski, Alexandr Popolitov, Sergey Shadrin, and Dimitri Zvonkine, Chiodo formulas for the $r$-th roots and topological recursion, Lett. Math. Phys. 107 (2017), no. 5, 901–919. MR 3633029, DOI 10.1007/s11005-016-0928-5
M. Mulase, S. Shadrin, and L. Spitz, The spectral curve and the Schrödinger equation of double Hurwitz numbers and higher spin structures, Commun. Number Theory Phys. 7 (2013), no. 1, 125–143. MR 3108774, DOI 10.4310/CNTP.2013.v7.n1.a4
Motohico Mulase and Naizhen Zhang, Polynomial recursion formula for linear Hodge integrals, Commun. Number Theory Phys. 4 (2010), no. 2, 267–293. MR 2725053, DOI 10.4310/CNTP.2010.v4.n2.a1
A. Okounkov and R. Pandharipande, Gromov-Witten theory, Hurwitz theory, and completed cycles, Ann. of Math. (2) 163 (2006), no. 2, 517–560. MR 2199225, DOI 10.4007/annals.2006.163.517
A. Okounkov and R. Pandharipande, The equivariant Gromov-Witten theory of $\textbf {P}^1$, Ann. of Math. (2) 163 (2006), no. 2, 561–605. MR 2199226, DOI 10.4007/annals.2006.163.561
S. Shadrin, L. Spitz, and D. Zvonkine, On double Hurwitz numbers with completed cycles, J. Lond. Math. Soc. (2) 86 (2012), no. 2, 407–432. MR 2980918, DOI 10.1112/jlms/jds010
S. Shadrin, L. Spitz, and D. Zvonkine, Equivalence of ELSV and Bouchard-Mariño conjectures for $r$-spin Hurwitz numbers, Math. Ann. 361 (2015), no. 3-4, 611–645. MR 3319543, DOI 10.1007/s00208-014-1082-y
Dimitri Zvonkine, A preliminary text on the \texorpdfstring{$r$}r-ELSV formula, preprint (2006).