A TQFT of Intersection Numbers on Moduli Spaces of Admissible Covers

We construct a two-level weighted TQFT whose structure coefficents are equivariant intersection numbers on moduli spaces of admissible covers. Such a structure is parallel (and strictly related) to the local Gromov-Witten theory of curves of Bryan-Pandharipande. We compute explicitly the theory using techniques of localization on moduli spaces of admissible covers of a parametrized projective line. The Frobenius Algebras we obtain are one parameter deformations of the class algebra of the symmetric group S_d. In certain special cases we are able to produce explicit closed formulas for such deformations in terms of the representation theory of S_d

Hodge-type integrals on moduli spaces of admissible covers

We study Hodge Integrals on Moduli Spaces of Admissible Covers. Motivation for this work comes from Bryan and Pandharipande's recent work on the local GW theory of curves, where analogouos intersection numbers, computed on Moduli Spaces of Relative Stable Maps, are the structure coefficients for a Topological Quantum Field Theory. Admissible Covers provide an alternative compactification of the Moduli Space of Maps, that is smooth and doesn't contain boundary components of excessive dimension. A parallel, yet different, TQFT, can then be constructed. In this paper we compute, using localization, the relevant Hodge integrals for admissible covers of a pointed sphere of degree 2 and 3, and formulate a conjecture for general degree. In genus 0, we recover the well-known Aspinwall Morrison formula in GW theory.Comment: This is the version published by Geometry & Topology Monographs on 21 September 200

A geometric perspective on the piecewise polynomiality of double Hurwitz numbers

We describe double Hurwitz numbers as intersection numbers on the moduli space of curves. Assuming polynomiality of the Double Ramification Cycle (which is known in genera 0 and 1), our formula explains the polynomiality in chambers of double Hurwitz numbers, and the wall crossing phenomenon in terms of a variation of correction terms to the {\psi} classes. We interpret this as suggestive evidence for polynomiality of the Double Ramification Cycle.Comment: 15 pages, 5 figure

Moduli spaces of rational weighted stable curves and tropical geometry

We study moduli spaces of rational weighted stable tropical curves, and their connections with the classical Hassett spaces. Given a vector w of weights, the moduli space of tropical w-stable curves can be given the structure of a balanced fan if and only if w has only heavy and light entries. In this case, we can express the moduli space as the Bergman fan of a graphic matroid. Furthermore, we realize the tropical moduli space as a geometric tropicalization, and as a Berkovich skeleton, of the classical moduli space. This builds on previous work of Tevelev, Gibney--Maclagan, and Abramovich--Caporaso--Payne. Finally, we construct the moduli spaces of heavy/light weighted tropical curves as fiber products of unweighted spaces, and explore parallels with the algebraic world.Comment: 26 pages, 8 TikZ figures. v3: Minor changes and corrections. Final version to appear in Forum of Mathematics, Sigm