Open orbifold Gromov-Witten invariants of \({[\mathbb{C}^3/\mathbb{Z}_n]}\): Localization and mirror symmetry. (English) Zbl 1236.14046
Compared to the Gromov-Witten invariants of closed holomorphic curves invariants of their open counterparts are poorly understood and a satisfactory general definition of them is still lacking. These invariants are supposed to count holomorphic curves in a variety \(X\) with the boundaries on a Lagrangian submanifold \(L\subset X\). In addition to geometry their values depend on “framing” of \(L\), which remains mysterious in general. Most explicit computations rely on physical heuristics and involve mirror symmetry and/or large \(N\) duality. However, in some special cases open invariants can be defined ad hoc and computed mathematically. One of the computable cases, considered by S. Katz and C.-C. M. Liu [Adv. Theor. Math. Phys. 5, No. 1, 1–49 (2001; Zbl 1026.32028)], is when \(X\) is a Calabi-Yau threefold and \(L\) is a fixed locus of an antiholomorphic involution with a suitable torus action. Framing can then be identified with a choice of the action. Katz and Liu postulated a perfect obstruction theory for this case, analogous to the closed case, and computed the open invariants by virtual localization for the resolved conifold \(X=\mathcal{O}(-1)\oplus\mathcal{O}(-1)\) with a natural Lagrangian in it.
The paper under review presents a similar computation for the Calabi-Yau orbifolds \(\mathbb{C}^3/\mathbb{Z}_n\) with suitable Lagrangians. The fixed loci of the torus action comprise a compact curve with twisted marked points and attached orbi-discs. It is the disc contributions, reflecting the \(\mathbb{Z}_n\) action on \(\mathbb{C}^3\), that are essentially new compared to the Katz-Liu case. All open invariants are reduced recursively to the degree \(0\) closed Gromov-Witten invariants with descendants, and a special function that the authors name the disc function. Although the approach works for any \(n\), explicit computations are performed for \(n=3,4\) with two different choices for the Lagrangian when \(n=4\), symmetric and asymmetric.
Large part of the paper compares the results to mirror symmetry predictions based on \(B\)-model computations. The authors perform them within the paper, and one of the sections is an exposition of the \(B\)-model aimed at mathematicians. In particular, the mirror pair \((\widehat{X},\widehat{L})\) is constructed explicitly following Aganagic-Vafa, and Eynard-Orantin recursions for higher genus \(B\)-model invariants are reviewed in detail. For \(\mathbb{C}^3/\mathbb{Z}_3\) and the asymmetric Lagrangian in \(\mathbb{C}^3/\mathbb{Z}_4\) both the disc and the annulus potentials agree with the mirror computations. In the latter case this confirms that the annulus potential is a quasi-modular form of \(\Gamma(2)\subset \mathrm{SL}(2,\mathbb{Z})\). However, for the symmetric Lagrangian in \(\mathbb{C}^3/\mathbb{Z}_4\) there is a phase discrepancy. The authors speculate that \(\mathbb{Z}_2\)-isotropy on the fixed circle of the Lagrangian produces a non-trivial normalizing factor.
The paper under review presents a similar computation for the Calabi-Yau orbifolds \(\mathbb{C}^3/\mathbb{Z}_n\) with suitable Lagrangians. The fixed loci of the torus action comprise a compact curve with twisted marked points and attached orbi-discs. It is the disc contributions, reflecting the \(\mathbb{Z}_n\) action on \(\mathbb{C}^3\), that are essentially new compared to the Katz-Liu case. All open invariants are reduced recursively to the degree \(0\) closed Gromov-Witten invariants with descendants, and a special function that the authors name the disc function. Although the approach works for any \(n\), explicit computations are performed for \(n=3,4\) with two different choices for the Lagrangian when \(n=4\), symmetric and asymmetric.
Large part of the paper compares the results to mirror symmetry predictions based on \(B\)-model computations. The authors perform them within the paper, and one of the sections is an exposition of the \(B\)-model aimed at mathematicians. In particular, the mirror pair \((\widehat{X},\widehat{L})\) is constructed explicitly following Aganagic-Vafa, and Eynard-Orantin recursions for higher genus \(B\)-model invariants are reviewed in detail. For \(\mathbb{C}^3/\mathbb{Z}_3\) and the asymmetric Lagrangian in \(\mathbb{C}^3/\mathbb{Z}_4\) both the disc and the annulus potentials agree with the mirror computations. In the latter case this confirms that the annulus potential is a quasi-modular form of \(\Gamma(2)\subset \mathrm{SL}(2,\mathbb{Z})\). However, for the symmetric Lagrangian in \(\mathbb{C}^3/\mathbb{Z}_4\) there is a phase discrepancy. The authors speculate that \(\mathbb{Z}_2\)-isotropy on the fixed circle of the Lagrangian produces a non-trivial normalizing factor.
Reviewer: Sergiy Koshkin (Houston)
MSC:
| 14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |
| 14J33 | Mirror symmetry (algebro-geometric aspects) |
| 81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
| 53D12 | Lagrangian submanifolds; Maslov index |
Keywords:
open Gromov-Witten invariants; Calabi-Yau orbifold; Lagrangian submanifold; mirror symmetry; B-model; Givental’s J-function; Eynard-Orantin recursionCitations:
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