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

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.