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

Jan Troost

TL;DR

The paper establishes a rigorous open/closed Gromov-Witten/Hurwitz duality by formulating a grand canonical open/closed orbifold theory based on the inverse monoid of partial permutations and its holonomy interpretation. It demonstrates that bulk operators in GW correspond to completed cycles in Hurwitz theory and shows that the large $n$ limit yields an isomorphism with shifted symmetric functions, thereby unifying open/closed sectors via relative invariants and boundary holonomies. Through explicit equivariant localization calculations on ${f P}^1$, the authors identify localized world sheets that contribute to completed-cycle terms and develop a diagrammatic framework (left-right factors, symmetry factors, cut-and-join interactions) to compute two-point and one-point functions, including relative correlators. The results include concrete checks of the degree/dimension constraints, oscillator representations, and linear Hodge integral manipulations (ELSV-type formulas), culminating in a coherent picture of completed cycles as localized world-sheet contributions. The work provides a tractable combinatorial and operator-theoretic approach to the open/closed GW/Hurwitz correspondence, with implications for higher-genus amplitudes, relative theories, and potential links to integrable hierarchies and holographic interpretations.

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.

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

TL;DR

The paper establishes a rigorous open/closed Gromov-Witten/Hurwitz duality by formulating a grand canonical open/closed orbifold theory based on the inverse monoid of partial permutations and its holonomy interpretation. It demonstrates that bulk operators in GW correspond to completed cycles in Hurwitz theory and shows that the large limit yields an isomorphism with shifted symmetric functions, thereby unifying open/closed sectors via relative invariants and boundary holonomies. Through explicit equivariant localization calculations on , the authors identify localized world sheets that contribute to completed-cycle terms and develop a diagrammatic framework (left-right factors, symmetry factors, cut-and-join interactions) to compute two-point and one-point functions, including relative correlators. The results include concrete checks of the degree/dimension constraints, oscillator representations, and linear Hodge integral manipulations (ELSV-type formulas), culminating in a coherent picture of completed cycles as localized world-sheet contributions. The work provides a tractable combinatorial and operator-theoretic approach to the open/closed GW/Hurwitz correspondence, with implications for higher-genus amplitudes, relative theories, and potential links to integrable hierarchies and holographic interpretations.

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.
Paper Structure (65 sections, 161 equations, 7 figures)

This paper contains 65 sections, 161 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The Open/Closed Grand Canonical Orbifold
  3. The Open/Closed Inverse Monoid Theory
  4. The Invariant Orbifold Theory
  5. The Holonomy Perspective
  6. The Group Theory
  7. The Holonomy Perspective in Inverse Monoid Theories
  8. The Holonomy Perspective in the Grand Canonical Hurwitz Theory
  9. Extended Partition Functions
  10. The Relation to Hurwitz Numbers and Extended Characters
  11. The Large N Limit and the Correspondence
  12. The Projective Limit of Partial Permutation Algebras
  13. The Isomorphism with Shifted Symmetric Functions
  14. The Open/Closed Correspondence
  15. The Open/Closed Gromov-Witten Theory
  16. ...and 50 more sections

Figures (7)

  • Figure 1: The two diagrams contributing to the $\tau_2(\omega)$ one-point function. The edge is of degree $1$ and the vertices are labelled by genera. A right hand vertex is marked by the $\tau_2$ insertion.
  • Figure 2: The first set of 15 diagrams contributing to the two-point function.
  • Figure 3: The remaining set of 12 diagrams contributing to the two-point function.
  • Figure 4: The left hand diagrams for degree two contributions up to genus one
  • Figure 5: The left hand diagrams for degree $(1,1)$ contributions up to genus one.
  • ...and 2 more figures