Gross fibrations, SYZ mirror symmetry, and open Gromov-Witten invariants for toric Calabi-Yau orbifolds (Q726784)

The SYZ (Storminger-Yau-Zaslow) mirror \(\hat{\mathcal{X}}\) [\textit{A. Strominger} et al., Nucl. Phys., B 479, No. 1--2, 243--259 (1996; Zbl 0896.14024)] of a toric CY (Calabi-Yau) orbifold \(\mathcal{X}\) (with a hypersurface removed) whose coarse moduli space is a semi-projective toric variety (alternatively \S4, Setting 4.2), equipped with the Gross fibration [\textit{M. Gross}, in: Symplectic geometry and mirror symmetry. Proceedings of the 4th KIAS annual international conference, Seoul, South Korea, August 14--18, 2000. Singapore: World Scientific. 81--109 (2001; Zbl 1034.53054)] is shown to be the family of non-compact CY manifolds \[ \{(u,v,z_1,\ldots,z_{n-1})\in\mathbb{C}^2\times(\mathbb{C}^\times)^{n-1}| uv=g(z_1,\ldots,z_{n-1})\}, \] with an explicit description of \(g\) (Theorem 1.1, \S5.2). The authors remark that the SYZ mirror of \(\mathcal{X}\) without removing a hypersurface is given by the Landau-Ginzburg model as in the manifold case [\textit{K. Chan} et al., J. Differ. Geom. 90, No. 2, 177--250 (2012; Zbl 1297.53061); erratum ibid. 99, No. 1, 165--167 (2015)]. As applications, denoting the maps defined in terms of genus-0 open orbifold invariants from the SYZ construction and the combinatorially defined mirror map [\textit{T. Coates} et al., Compos. Math. 151, No. 10, 1878--1912 (2015; Zbl 1330.14093)], hereafter referred to as in [\textit{M. Abouzaid} et al., Publ. Math., Inst. Hautes Étud. Sci. 123, 199--282 (2016; Zbl 1368.14056)] by \(\mathcal{F}^{\mathrm{SYZ}}\) and \(\mathcal{F}^{\mathrm{mirror}}\), it is shown that \[ \mathcal{F}^{\mathrm{SYZ}}=(\mathcal{F}^{\mathrm{mirror}})^{-1}, \] near the large volume limit \((q,\tau)=0\) of \(\mathcal{X}\). In particular, this holds for a semi-projective CY manifolds (Open mirror theorem for toric CY orbifolds: Theorem 1.5. \S7.2). Here, \(q\) and \(\tau\) are the Kähler and orbifold parameters in the complexified Kähler moduli space of \(\mathcal{X}\) (cf. \S7.1). Combining Theorem 1.5 and the analysis of relations between period integrals and the GKZ hypergeometric system associated to \(\mathcal{X}\), another version of the open mirror theorems are also obtained (Theorem 1.6, Corollary 1.7: cf. [\textit{K. Chan} et al., Adv. Math. 244, 605--625 (2013; Zbl 1286.14056)]). It is also shown that if \(\mathcal{X}'\) is a toric crepant partial resolution of \(\mathcal{X}\), then \[ \mathcal{F}^{\mathrm{SYZ}}_\mathcal{X}=\mathcal{F}^{\mathrm{SYZ}}_{\mathcal{X}'}, \] after analytic continuation and a change of variables (Open crepant resolution conjecture: Theorems 1.8. and 8.1). A review of the construction and of basic properties of toric orbifolds including the equivariant mirror theorem and closed mirror theorem for toric orbifold (Theorem 2.7 and 2.8. Proved in [K. Chan et al., 2013, loc. cit.]) are given in \S2. The coefficients of \(g(z_1,\ldots,z_{n-1})\) are generating functions of orbi-disk invariants of \((\mathcal{X},F_r)\), \(F_r\) is a Lagrangian torus fiber of the Gross fibration. \S3 reviews the construction of genus-0 open orbifold GW (Gromov-Witten) invariants of toric orbifolds (Orbi-disk invariants; cf. [\textit{C.-H. Cho} and \textit{M. Poddar}, J. Differ. Geom. 98, No. 1, 21--116 (2014; Zbl 1300.53077)]). The Gross fibration, a special Lagrangian torus fibration \(\mu:\mathcal{X}\to B\) of \(\mathcal{X}\), is defined and constructed with an explicit description of the discriminant locus \(\Gamma\subset B\) (Proposition 4.8). On \(B_0=B\setminus\Gamma\), \(\mu\) is a torus bundle and its dual bundle admits a complex structure producing the so-called semi-flat mirror of \(\mathcal{X}\). The Gross fibrations of \([\mathbb{C}^2/\mathbb{Z}_m]\) and \([\mathbb{C}^3/\mathbb{Z}_{2g+1}]\) are described as examples. The semi-flat mirror is not the genuine mirror of \(\mathcal{X}\) because the semi-flat complex structure can not be extended further due to nontrivial monodromy of the affine structure around the discriminant locus \(\Gamma\). According to the SYZ proposal, the semi-flat complex structure is deformed by instanton correction so that it becomes extendable. This is described in \S5, and Theorem 1.1 (as well as Proposition 5.2) is proved with exhibiting examples. In \S6, the genus-0 open orbifold GW invariants, which are relevant to the SYZ mirror construction, are computed via an open/closed equality (Theorem 1.3, Theorem 6.12). Here, an open/closed equality asserts equality between genus-0 open manifold GW invariants of \((\mathcal{X}, L)\), \(L\subset \mathcal{X}\) is a Lagrangian torus fiber of the moment map of \(\mathcal{X}\), and closed orbifold GW invariants of \(\bar{\mathcal{X}}\), where \(\bar{\mathcal{X}}\) is the toric partial compactification of \(\mathcal{X}\) depending on \(\beta\in\pi_2(\mathcal{X}, L)\), a holomorphic disk class of Chern-Weil-Maslov index 2 (cf. Construction 6.1). Sections 5 and 6 are the main parts of the paper. Applying the results of these sections, the open mirror theorems for toric CY orbifolds and the open crepant resolution conjecture are proved in Sections 7 and 8. In the appendix, the analytic continuation of mirror maps, which is used in the proof of Theorem 8.1, is explained [\textit{T. Coates} et al., Geom. Topol. 13, No. 5, 2675--2744 (2009; Zbl 1184.53086)].