Renzo Cavalieri

Crepant resolutions and open strings

Abstract In the present paper, we formulate a Crepant Resolution Correspondence for open Gromov–Witten invariants (OCRC) of toric Lagrangian branes inside Calabi–Yau 3-orbifolds by encoding the open theories into sections of Givental’s symplectic vector space. The correspondence can be phrased as the identification of these sections via a linear morphism of Givental spaces. We deduce from this a Bryan–Graber-type statement for disk invariants, which we extend to arbitrary topologies in the Hard Lefschetz case. Motivated by ideas of Iritani, Coates–Corti–Iritani–Tseng and Ruan, we furthermore propose (1) a general form of the morphism entering the OCRC, which arises from a geometric correspondence between equivariant K-groups, and (2) an all-genus version of the OCRC for Hard Lefschetz targets. We provide a complete proof of both statements in the case of minimal resolutions of threefold A n {A_{n}} -singularities; as a necessary step of the proof we establish the all-genus closed Crepant Resolution Conjecture with descendents in its strongest form for this class of examples. Our methods rely on a new description of the quantum D-modules underlying the equivariant Gromov–Witten theory of this family of targets.

Gerby localization, Z3-Hodge integrals and the GW theory of [C³/Z3]

We exhibit a set of recursive relations that completely determine all equivariant Gromov-Witten invariants of

Tropical compactification and the Gromov–Witten theory of \mathbb {P}^1

[{\Bbb C}^3/{\Bbb Z}_3]

Tautological pairings on moduli spaces of curves

. We interpret such invariants as

Hurwitz theory and the double ramification cycle

{\Bbb Z}_3

Geometric Perspective on Piecewise Polynomiality of Double Hurwitz Numbers

-Hodge integrals, and produce relations among them via Atiyah-Bott localization on moduli spaces of twisted stable maps to gerbes over~

Tropical Hurwitz numbers

{\Bbb P}^1

TROPICALIZING THE SPACE OF ADMISSIBLE COVERS

.

Polynomial families of tautological classes on Mg,nrt

We use tropical and non-archimedean geometry to study the moduli space of genus 0, stable maps to