Ramification Profiles Research Articles

Orbifold GW theory as the Hurwitz–Frobenius submanifold

We study the relation between the Frobenius manifolds of GW theory and the Hurwitz–Frobenius manifold. We prove that the Frobenius manifold given by the orbifold GW theory of P1(2,2,2,2) is isomorphic to the submanifold in the Hurwitz–Frobenius manifold of ramified coverings of the sphere by the genus 1 curve with the ramification profile (2,2,2,2) over ∞.

Polynomiality, wall crossings and tropical geometry of rational double Hurwitz cycles

We study rational double Hurwitz cycles, i.e. loci of marked rational stable curves admitting a map to the projective line with assigned ramification profiles over two fixed branch points. Generalizing the phenomenon observed for double Hurwitz numbers, such cycles are piecewise polynomial in the entries of the special ramification; the chambers of polynomiality and wall crossings have an explicit and “modular” description. A main goal of this paper is to simultaneously carry out this investigation for the corresponding objects in tropical geometry, underlining a precise combinatorial duality between classical and tropical Hurwitz theory.

Wall crossings for double Hurwitz numbers

Double Hurwitz numbers count covers of P 1 by genus g curves with assigned ramification profiles over 0 and ∞, and simple ramification over a fixed branch divisor. Goulden, Jackson and Vakil have shown double Hurwitz numbers are piecewise polynomial in the orders of ramification (Goulden et al., 2005) [10], and Shadrin, Shapiro and Vainshtein have determined the chamber structure and wall crossing formulas for g = 0 (Shadrin et al., 2008) [15]. This paper gives a unified approach to these results and strengthens them in several ways — the most important being the extension of the results of Shadrin et al. (2008) [15] to arbitrary genus. The main tool is the authorsʼ previous work (Cavalieri et al., 2010) [6] expressing double Hurwitz number as a sum over certain labeled graphs. We identify the labels of the graphs with lattice points in the chambers of certain hyperplane arrangements, which give rise to piecewise polynomial functions. Our understanding of the wall crossing for these functions builds on the work of Varchenko (1987) [17], and could have broader applications.

Chamber Structure For Double Hurwitz Numbers

Double Hurwitz numbers count covers of the sphere by genus $g$ curves with assigned ramification profiles over $0$ and $\infty$, and simple ramification over a fixed branch divisor. Goulden, Jackson and Vakil (2005) have shown double Hurwitz numbers are piecewise polynomial in the orders of ramification, and Shadrin, Shapiro and Vainshtein (2008) have determined the chamber structure and wall crossing formulas for $g=0$. We provide new proofs of these results, and extend them in several directions. Most importantly we prove wall crossing formulas for all genera. The main tool is the authors' previous work expressing double Hurwitz number as a sum over labelled graphs. We identify the labels of the graphs with lattice points in the chambers of certain hyperplane arrangements, which give rise to piecewise polynomial functions. Our understanding of the wall crossing for these functions builds on the work of Varchenko (1987). This approach to wall crossing appears novel, and may be of broader interest. This extended abstract is based on a new preprint by the authors. Les nombres de Hurwitz doubles dénombrent les revêtements de la sphère par une surface de genre $g$ avec ramifications prescrites en $0$ et $\infty$, et dont les autres valeurs critiques sont non dégénérées et fixées. Goulden, Jackson et Vakil (2005) ont prouvé que les nombres de Hurwitz doubles sont polynomiaux par morceaux en l'ordre des ramifications prescrites, et Shadrin, Shapiro et Vainshtein (2008) ont déterminé la structure des chambres et ont établis des formules pour traverser les murs en genre $0$. Nous proposons des nouvelles preuves de ces résultats, et les généralisons dans plusieurs directions. En particulier, nous prouvons des formules pour traverser les murs en tout genre. L'outil principal est le précédent travail des auteurs exprimant les nombres de Hurwitz doubles comme somme de graphes étiquetés. Nous identifions les étiquetages avec les points entiers à l'intérieur d'une chambre d'un arrangement d'hyperplans, qui sont connu pour donner une fonction polynomiale par morceaux. Notre étude des formules pour traverser les murs de ces fonctions se base sur un travail antérieur de Varchenko (1987). Cette approche paraît nouvelle, et peut être d'un large intérêt. Ce résumé élargi se base sur un papier nouveau des auteurs.