Gromov-Witten theory, Hurwitz theory, and completed cycles

Abstract

We establish an explicit equivalence between the stationary sector of the Gromov-Witten theory of a target curve X and the enumeration of Hurwitz coverings of X in the basis of completed cycles. The stationary sector is formed, by definition, by the descendents of the point class. Completed cycles arise naturally in the theory of shifted symmetric functions. Using this equivalence, we give a complete description of the stationary Gromov-Witten theory of the projective line and elliptic curve. Toda equations for the relative stationary theory of the projective line are derived.

55 pages, no figures

Cited by in corpus (15)

• Three questions in Gromov-Witten theory
• Matrix Models for Random Partitions
• The equivariant Gromov-Witten theory of P^1
• On Microscopic Origin of Integrability in Seiberg-Witten Theory
• The uses of random partitions
• Random surfaces enumerating algebraic curves
• Cohomology of moduli spaces
• Gromov--Witten Theory of CP^1 and Integrable Hierarchies
• Virasoro constraints for target curves
• Multipoint series of Gromov-Witten invariants of CP^1
• Higher-genus Gromov-Witten invariants as genus 0 invariants of symmetric products
• Les Houches lecture notes on moduli spaces of Riemann surfaces
• Twisted vertex operators and Bernoulli polynomials
• The Gromov-Witten and Donaldson-Thomas correspondence for trivial elliptic fibrations
• Periodic Schur process and cylindric partitions