The Open/Closed Gromov-Witten/Hurwitz Correspondence and Localized World Sheets for Completed Cycles

TL;DR

This paper explores the open/closed Gromov-Witten/Hurwitz correspondence, clarifying the role of completed cycles, and introduces localized world sheet contributions in the context of algebraic geometry and string theory.

Contribution

It provides new insights into the duality between Gromov-Witten invariants and Hurwitz counts, emphasizing the importance of completed cycles and localized world sheet contributions.

Findings

Confirmed the necessity of completed cycles for the correspondence

Identified contributions of localized world sheets in equivariant amplitudes

Proposed a combinatorial picture involving localization diagrams and string interactions

Abstract

We discuss the open/closed version of the Gromov-Witten/Hurwitz correspondence. The duality equates the relative Gromov-Witten invariants and the count of covers of the target space with prescribed holonomies at boundaries. We clarify the projective large N limit as well as the role of the completed versus the ordinary cycles associated to the bulk and the boundary vertex operators respectively. We provide an example check of both the correspondence and the fact that cycles dual to closed strings need to be completed. Moreover, we identify the connected world sheets that contribute to an equivariantly localized amplitude in the bulk that is solely due to a completion term. We also propose a picture for the completed cycle combinatorics that involves a localization diagram glued to a cut-and-join string interaction.