SYZ mirror symmetry for hypertoric varieties (Q2297310)

This paper concerns SYZ mirror symmetry from the Floer theoretic perspective, following the paradigm of \textit{D. Auroux} [J. Gökova Geom. Topol. GGT 1, 51--91 (2007; Zbl 1181.53076); Surv. Differ. Geom. 13, 1--47 (2009; Zbl 1184.53085)]. According to the näive SYZ conjecture [\textit{A. Strominger} et al., Nucl. Phys., B 479, No. 1--2, 243--259 (1996; Zbl 0896.14024)], given a Calabi-Yau manifold $X$ one attempts to find a (special) Lagrangian torus fibration $X\to B$, and construct the mirror $X^\vee$ by forming the moduli space of these Lagrangian tori equipped with rank 1 unitary local systems. This näive mirror has a natural complex structure away from singular fibres. However, to account for instanton effects produced by singular fibres and boundary divisors, one should correct this picture by analyzing holomorphic Maslov index 2 disks with boundary on these Lagrangian fibres. The enumerative invariants for these disks can jump when the Lagrangian fibres also bound a Maslov index 0 disk, leading to wall crossing phenomena. Typically this happens along real codimension 1 locus $W$ in $B$, which divides the complement $B\setminus W$ into chambers where the Lagrangian fibres are weakly unobstructed. The jumping of counts should be encoded by coordinate changes under wall crossing. This paper studies this procedure for hypertoric manifolds $X$, which are hyperkähler quotients of $T^*\mathbb{C}^n$ by the action of a subtorus $K\subset T^n$. Important examples include $T^*\mathbb{CP}^n$ and minimal resolutions of $A_n$ singularities. These hypertoric manifolds can be analyzed using the remaining torus action $T^n/K$, leading to certain combinatorial data called hyperplane arrangements. The authors construct a piecewise smooth Lagrangian torus fibration on $X$ using smoothing techniques and the logarithm map (which is not quite a special Lagrangian fibration), and give an explicit combinatorial analysis of the wall crossing phenomenon, by computing the Maslov index 0 and 2 disks with boundaries on the Lagrangian fibres. They then show the surjectivity of the linearization operators, leading to a computation of the generating function of the open Gromov-Witten invariants. The general theory predicts that such generating functions are invariant under wall crossing, so furnish global holomorphic functions on the mirror $X^\vee$. To obtain enough such functions, the authors use a partial compactification of $X$ to include more boundary divisors; the heuristic is to stretch the holomorphic disks to meet the divisors at infinity. This allows the identification of the global coordinate ring $R$ of the mirror $X^\vee$. The true mirror is a resolution of $\text{Spec}(R)$, as the wall crossing phenomenon prevents certain local coordinates on $X^\vee$ from a global extension; this mirror is identified in detail. The authors conclude by observing an alternative resolution of $\text{Spec}(R)$ provided by a multiplicative hypertoric variety, which is birational to $X^\vee$.