We study spin Hurwitz numbers, which count ramified covers of the Riemann sphere with a sign coming from a theta characteristic. These numbers are known to be related to Gromov-Witten theory of Kähler surfaces and to representation theory of the Sergeev group, and are generated by BKP tau-functions. We use the latter interpretation to give polynomiality properties of these numbers and we derive a spectral curve which we conjecture computes spin Hurwitz numbers via a new type of topological recursion. We prove that this conjectural topological recursion is equivalent to an ELSV-type formula, expressing spin Hurwitz numbers in terms of the Chiodo class twisted by the 2-spin Witten class.
A. We derive explicit formulae for the generating series of mixed Grothendieck dessins d'enfant/monotone/simple Hurwitz numbers, via the semi-in nite wedge formalism. This reveals the strong piecewise polynomiality in the sense of Goulden-Jackson-Vakil, generalising a result of Johnson, and provides a new explicit proof of the piecewise polynomiality of the mixed case. Moreover, we derive wall-crossing formulae for the mixed case. These statements specialise to any of the three types of Hurwitz numbers, and to the mixed case of any pair. double case is still an active topic of research. A tropical approach for the monotone case is developed in [DK ] and in [Hah ].Grothendieck dessins d'enfant or strictly monotone Hurwitz numbers. Dessins d'enfant have been introduced by Grothendieck in [Gro ]. Their enumeration counts Hurwitz coverings of genus and degree d over the Riemann sphere, with two rami cations µ and ν over 0 and ∞, and a single further rami cation over 1, whose length is determined by the Riemann-Hurwitz formula.The CEO recursion for the r = 2 orbifold case (i.e. ν = (2) d /2 ) was proved in [Nor ; DMSS ] and is known as enumeration of ribbon graphs. In [DM ] the CEO recursion was conjectured for the general r -orbifold case (i.e. ν = (r ) d /r ), which was then proved in [DOPS ] by combinatorial methods. Moreover, CEO recursion was proved in [KZ ] for the case of two consecutive intermediate rami cations of xed lengths instead of one, together with a proof of the KP integrability and the Virasoro constraints for the same case. ELSV formulae for simple, orbifold or double cases are still not known. The connection between strictly monotone numbers and dessins d'enfant counting is explained in the following.Mixed cases. It is natural to interpolate several Hurwitz enumerative problems, by allowing di erent conditions on di erent blocks of intermediate rami cations. In fact hypergeometric tau functions for the D Toda integrable hierarchy have been proved to have several explicit combinatorial interpretations [HO ]-one of them is in terms of mixed double strictly monotone/weakly monotone Hurwitz numbers, another one involves a mixed case of combinatorial problems, in which the part relative to the strictly monotone rami cations can be interpreted in terms of Grothendieck dessins d'enfant. This implies indirectly that the enumeration of Grothendieck dessins and strictly monotone numbers coincide. A direct proof of this fact through the Jucys correspondence [Juc ] is derived in [ALS ].A combinatorial study of the mixed double monotone-simple case can be found in [GGN ], in which piecewise polynomiality is proved. A tropical interpretation providing an algorithm to compute the chamber polynomials and wall-crossing formulae via Erhart theory is developed in [Hah ]. Further developments on CEO topological recursion for general Hurwitz enumerative geometric problems appear in [ACEH ]. This study con rms the existence of an ELSV-type formula for mixed Hurwitz enumerative problems.
We perform a key step towards the proof of Zvonkine’s conjectural r r -ELSV formula that relates Hurwitz numbers with completed ( r + 1 ) (r+1) -cycles to the geometry of the moduli spaces of the r 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 ) (0,1) - and ( 0 , 2 ) (0,2) -functions in this generalized case, and we show that these unstable cases are correctly reproduced by the spectral curve initial data.
A. We prove the Zvonkine conjecture [Zvo ] that expresses Hurwitz numbers with completed cycles in terms of intersection numbers with the Chiodo classes [Chi a] via the so-called r -ELSV formula, as well as its orbifold generalization, the qr -ELSV formula, proposed recently in [KLPS ].
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