323,269 research outputs found
Tautological relations and integrable systems
Full text linkWe present a family of conjectural relations in the tautological cohomologyof the moduli spaces of stable algebraic curves of genus with markedpoints. A large part of these relations has a surprisingly simple form: thetautological classes involved in the relations are given by stable graphs thatare trees and that are decorated only by powers of the psi-classes athalf-edges. We show that the proposed conjectural relations imply certainfundamental properties of the Dubrovin-Zhang (DZ) and the double ramification(DR) hierarchies associated to F-cohomological field theories. Our relationsnaturally extend a similar system of conjectural relations, which were proposedin an earlier work of the first author together with Gu\'er\'e and Rossi andwhich are responsible for the normal Miura equivalence of the DZ and the DRhierarchy associated to an arbitrary cohomological field theory. Finally, weprove all the above mentioned relations in the case and arbitrary using a variation of the method from a paper by Liu and Pandharipande, this canbe of independent interest. In particular, this proves the main conjecture fromour previous joined work together with Hern\'andez Iglesias. We also prove allthe above mentioned relations in the case and arbitrary .Comment: v3: final journal version, 44 page
Quasi-polynomiality of monotone orbifold Hurwitz numbers and Grothendieck's dessins d'enfants
Full text linkWe prove quasi-polynomiality for monotone and strictly monotone orbifold Hurwitz numbers. The second enumerative problem is also known as enumeration of a special kind of Grothendieck's dessins d'enfants or -hypermaps. These statements answer positively two conjectures proposed by Do-Karev and Do-Manescu. We also apply the same method to the usual orbifold Hurwitz numbers and obtain a new proof of the quasi-polynomiality in this case. In the second part of the paper we show that the property of quasi-polynomiality is equivalent in all these three cases to the property that the -point generating function has a natural representation on the -th cartesian powers of a certain algebraic curve. These representations are the necessary conditions for the Chekhov-Eynard-Orantin topological recursion. <br
Towards an orbifold generalization of Zvonkine's -ELSV formula
Full text linkWe perform a key step towards the proof of Zvonkine's conjectural -ELSV formula that relates Hurwitz numbers with completed -cycles to the geometry of the moduli spaces of the -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 - and -functions in this generalized case and we show that these unstable cases are correctly reproduced by the spectral curve initial data
Miura-reciprocal transformations and localizable Poisson pencils
Full text linkWe show that the equivalence classes of deformations of localizable semisimple Poisson pencils of hydrodynamic type with respect to the action of the Miura-reciprocal group contain a local representative and are in one-to-one correspondence with the equivalence classes of deformations of local semisimple Poisson pencils of hydrodynamic type with respect to the action of the Miura grou
Ramifications of Hurwitz theory, KP integrability and quantum curves
Full text linkIn this paper we revisit several recent results on monotone and strictly monotone Hurwitz numbers, providing new proofs. In particular, we use various versions of these numbers to discuss methods of derivation of quantum spectral curves from the point of view of KP integrability and derive new examples of quantum curves for the families of double Hurwitz numbers
Half-Spin Tautological Relations and Faber's Proportionalities of Kappa Classes
Full text linkWe employ the 1/2-spin tautological relations to provide a particular combinatorial identity. We show that this identity is a statement equivalent to Faber's formula for proportionalities of kappa-classes on M-g, g >= 2. We then prove several cases of the combinatorial identity, providing a new proof of Faber's formula for those cases
Special cases of the orbifold version of Zvonkine's r-ELSV formula
No full textWe prove the orbifold version of Zvonkine's r-ELSV formula in two special cases: the case of r=2 (complete 3-cycles) for any genus g≥0 and the case of any r≥1 for genus g=
Stable tree expressions with Omega-classes and double ramification cycles
Full text linkWe propose a new system of conjectural relations in the tautological ring of the moduli space of curves involving stable rooted trees with level structure decorated by Hodge and Ω-classes and prove these conjectures in different cases
Chiodo formulas for the r-th roots and topological recursion
No full textWe analyze Chiodo's formulas for the Chern classes related to the r-th roots of the suitably twisted integer powers of the canonical class on the moduli space of curves. The intersection numbers of these classes with -classes are reproduced via the Chekhov-Eynard-Orantin topological recursion. As an application, we prove that the Johnson-Pandharipande-Tseng formula for the orbifold Hurwitz numbers is equivalent to the topological recursion for the orbifold Hurwitz numbers. In particular, this gives a new proof of the topological recursion for the orbifold Hurwitz numbers
Polynomiality of orbifold Hurwitz numbers, spectral curve, and a new proof of the Johnson-Pandharipande-Tseng formula
Full text linkIn this paper, we present an example of a derivation of an ELSV-type formula using the methods of topological recursion. Namely, for orbifold Hurwitz numbers we give a new proof of the spectral curve topological recursion, in the sense of Chekhov, Eynard and Orantin, where the main new step compared to the existing proofs is a direct combinatorial proof of their quasi-polynomiality. Spectral curve topological recursion leads to a formula for the orbifold Hurwitz numbers in terms of the intersection theory of the moduli space of curves, which, in this case, appears to coincide with a special case of the Johnson-Pandharipande-Tseng formul
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