Multi-Component Open/Relative/Local Correspondence

For a toric Calabi-Yau 3-orbifold relative to s Aganagic-Vafa outer branes, we prove a correspondence among the genus-zero open Gromov-Witten invariants with maximal winding at each brane and: (i) closed invariants of a toric Calabi-Yau (3+s)-orbifold; (ii) formal relative invariants of a formal toric Calabi-Yau (FTCY) 3-orbifold with maximal tangency to s divisors; (iii) formal relative invariants of a sequence of FTCY intermediate geometries interpolating dimensions 3 and 3+s. The correspondence provides examples of the log/local principle of van Garrel-Graber-Ruddat in the multi-component setting and the refined conjecture of Brini-Bousseau-van Garrel via intermediate geometries. It also establishes the multi-component case of the open/closed correspondence proposed by Lerche-Mayr and studied by Liu-Yu. As an application, we obtain examples of the conjecture of Klemm-Pandharipande on the integrality of BPS invariants of higher-dimensional toric Calabi-Yau manifolds. Along the way, we set the basic stages of the relative Gromov-Witten theory of higher-dimensional FTCY orbifolds, generalizing the case of smooth 3-folds by Li-Liu-Liu-Zhou.

💡 Research Summary

The paper establishes a comprehensive correspondence between genus‑zero open Gromov‑Witten (GW) invariants of a toric Calabi‑Yau (CY) 3‑orbifold X equipped with s Aganagic‑Vafa outer branes and three other types of invariants: (i) closed GW invariants of a toric CY (3 + s)‑orbifold 𝔛̂, (ii) formal relative GW invariants of a formal toric CY (FTCY) 3‑orbifold (Ŷ,Ď) with maximal tangency to s formal divisors, and (iii) formal relative invariants of a chain of intermediate FTCY geometries (Ŷ(ℓ),Ď(ℓ)) interpolating dimensions from 3 to 3 + s. The authors construct these objects explicitly: starting from (X,L) with a chosen framing f∈ℚ, they build the relative FTCY 3‑orbifold (Ŷ,Ď) whose divisor components correspond to the branes, and a toric CY (3 + s)‑orbifold 𝔛̂. Then, for each ℓ=0,…,s they define a higher‑dimensional FTCY (3 + ℓ)‑orbifold Ŷ(ℓ) by formally adding line bundles Ô(−Ď_i) and removing one relative condition at each step, thereby creating a sequence of “intermediate” geometries.

The main results are two theorems. Theorem 1.1 (Theorem 6.1) asserts an exact equality (up to a sign determined by the framing and winding numbers) between the open GW invariants ⟨γ₁,…,γ_n⟩{X,L,β′,d}, the formal relative invariants ⟨γ^{(0)}₁,…,γ^{(0)}{n+s}⟩{Ŷ/Ď,β̂,k}, and the closed local invariants ⟨\tildeγ₁,…,\tildeγ{n+s}⟩{𝔛̂,β̃}. The sign (−1)^{∑{i=1}^s(⌈d_i a_i⌉−1)} matches the prediction of the log/local correspondence generalized to the orbifold setting; the open/closed correspondence itself carries no extra sign. Theorem 1.2 (Theorem 6.2) shows that the formal relative invariants of successive intermediate geometries are related by the same sign factor, i.e. each step of replacing a relative condition by a local condition (the log/local move) introduces precisely the factor (−1)^{⌈d_{ℓ+1} a_{ℓ+1}⌉−1}. Consequently, the whole chain of correspondences factorizes the multi‑component open/closed correspondence into a sequence of log/local correspondences, confirming the refined conjecture of Brini‑Bousseau‑van Garrel in this multi‑brane context.

Technically, the authors extend the theory of relative FTCY orbifolds from dimension three to arbitrary r≥3 and incorporate orbifold structures. They define FTCY graphs with rational edge weights and local embeddings into ℝ^{r−1}, and they restrict relative conditions to univalent vertices, which correspond to the divisor components. The virtual localization formula of Graber‑Pandharipande is adapted to the orbifold setting, with careful treatment of Chen‑Ruan cohomology insertions, maximal tangency conditions, and cyclic stabilizer groups on each divisor component. The proof involves comparing the contributions of fixed loci across the different geometries, tracking framing‑dependent weights, and verifying that the combinatorial factors match the predicted signs.

As an application, the paper uses the established open/closed correspondence to study BPS integrality for higher‑dimensional toric CY orbifolds. By applying the Gopakumar‑Vafa resummation in genus zero and its generalization by Klemm‑Pandharipande to dimensions four and above, the authors obtain explicit BPS invariants for several examples (including the non‑compact case C³ and its orbifold quotients). They verify that these invariants are indeed integers, providing new evidence for the Klemm‑Pandharipande integrality conjecture in the multi‑brane, higher‑dimensional setting.

The final section presents a concrete example with X=C³, where all constructions can be made non‑formal. The authors compute open, relative, and local invariants directly, illustrating the entire correspondence and confirming the sign factors. They also discuss future directions, such as extending the FTCY framework to non‑regular graphs with bivalent vertices, handling higher genus, and developing a full relative GW theory for arbitrary dimensions.

In summary, this work unifies several strands of modern enumerative geometry—open/closed GW correspondence, log/local principle, and BPS integrality—within a single, multi‑component framework, extending known results from single‑brane cases to arbitrary numbers of branes and from three‑folds to higher‑dimensional toric CY orbifolds. It provides rigorous proofs of conjectures that previously existed only at the heuristic or physical level, and it opens new avenues for exploring the interplay between GW theory, mirror symmetry, and string‑theoretic BPS counting.