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.

💡 Research Summary

The paper investigates an open/closed version of the celebrated Gromov‑Witten/Hurwitz correspondence, focusing on the role of “completed cycles” that appear on the closed‑string side. The authors begin by recalling that the grand‑canonical formulation of the symmetric‑orbifold gauge theory can be encoded in the semisimple algebra of partial permutations (P_n). This algebra, denoted (B_n=\mathbb{C}