Family Floer SYZ mirror algorithm for the Grassmannian $Gr(2,4)$

Abstract: We give an explicit non-archimedean SYZ construction for the Landau-Ginzburg mirror of $Gr(2,4)$. This work is complementary to the approach of Hong-Kim-Lau \cite{hong2023immersed} to SYZ mirror symmetry for Grassmannians, while we focus on a more concrete fibration-level realization of the SYZ picture. Starting from a Lagrangian fibration on the A-side, we explicitly construct a non-archimedean analytic mirror fibration inside the Berkovich analytification of the Langlands dual Grassmannian on the B-side. We show that the two fibrations have identical smooth and singular loci and induce the same integral affine structure on the smooth locus. Moreover, the natural disk-counting Landau-Ginzburg superpotential agrees with the Marsh-Rietsch superpotential. While the construction is guided by the family Floer viewpoint, the proof proceeds mainly through explicit geometric constructions and does not rely on Floer-theoretic arguments. Thus, the Langlands-dual mirror and its superpotential are realized explicitly within a single framework, providing concrete geometric evidence for the SYZ principle.

• The paper presents an explicit SYZ mirror construction for Gr(2,4) by dualizing a Lagrangian torus fibration, recovering matching analytic and Lie-theoretic superpotentials.
• It employs explicit cluster coordinates and wall-crossing formulas to compute holomorphic disk counts and establish an integral affine structure on the base.
• The construction bridges Floer theory and Lie-theoretic approaches, providing a blueprint for extending mirror symmetry methods to higher Grassmannians and flag varieties.

Family Floer SYZ Mirror Algorithm for the Grassmannian $Gr(2,4)$

Introduction and Background

This work provides a concrete and explicit SYZ (Strominger-Yau-Zaslow) mirror construction for the Grassmannian $Gr(2,4)$, focusing on an explicit fibration-level realization of mirror symmetry within the non-archimedean analytic category. The approach bridges the gap between Lie-theoretic (Landau-Ginzburg) mirror models and the geometric SYZ philosophy by explicitly constructing both the mirror fibration and the superpotential, sidestepping abstract Floer-theoretic machinery in favor of constructive geometric arguments.

The construction centers on dualizing a Lagrangian torus fibration on the A-side (symplectic geometry of $Gr(2,4)\setminus D_{ac}$, $D_{ac}$ being the anti-canonical divisor) to produce a non-archimedean analytic mirror fibration on the B-side (analytified Langlands dual Grassmannian, realized as $Gr(2,4)_{\Lambda}^{an}\setminus D_{ac}$), and demonstrates that the corresponding bases, singular loci, and integral affine structures match exactly. Furthermore, the Landau-Ginzburg superpotential constructed via disk-counting agrees precisely with the Marsh-Rietsch superpotential previously obtained by Lie-theoretic approaches.

Lagrangian Fibrations on the Grassmannian

A Lagrangian torus fibration is constructed over the open part $X = Gr(2,4) \setminus D_{ac}$ via a Hamiltonian $T^3$-action, where the moment map image is the interior of the $(2,4)$-hypercube, $\Delta_{2,4}^\circ$—an octahedron in $\mathbb{R}^4$. The fibers are classified as smooth or singular depending on their intersections with specific divisors, and their explicit description leverages cluster chart coordinates and Plücker coordinates.

Figure 1: The octahedral hypersimplex $Gr(2,4)$0, which forms the moment map image for the Lagrangian fibration on $Gr(2,4)$1.

Singular loci are precisely described: the subset $Gr(2,4)$2 within the base $Gr(2,4)$3 is identified with points where $Gr(2,4)$4 and $Gr(2,4)$5. Wall loci $Gr(2,4)$6 and $Gr(2,4)$7 correspond to codimension-1 strata where Maslov index zero disks appear.

The construction employs two explicit cluster charts $Gr(2,4)$8, corresponding to $Gr(2,4)$9 and $Gr(2,4)\setminus D_{ac}$0 respectively. The overlap of these charts underpins the non-trivial wall-crossing structure and necessitates explicit gluing in the analytic mirror.

Topological and Holomorphic Disks: Monodromy and Affine Structures

All local systems $Gr(2,4)\setminus D_{ac}$1 and $Gr(2,4)\setminus D_{ac}$2 are calculated, their monodromies computed explicitly via intersection data with divisors, and an integral basis of disk classes $Gr(2,4)\setminus D_{ac}$3 (on $Gr(2,4)\setminus D_{ac}$4) and $Gr(2,4)\setminus D_{ac}$5 (on $Gr(2,4)\setminus D_{ac}$6) is identified.

Integration of the symplectic form over these disk classes yields explicit action coordinates, globally defined on $Gr(2,4)\setminus D_{ac}$7. These coordinates are instrumental in constructing a concrete integral affine atlas, whose transition function is piecewise linear—incorporating quantum corrections from wall-crossing phenomena.

Figure 2: The six holomorphic disks bounded by a regular Lagrangian fiber in the chart $Gr(2,4)\setminus D_{ac}$8 over $Gr(2,4)\setminus D_{ac}$9.

Figure 3: The geometric interpretation of the action coordinates using cylinder areas between adjacent Lagrangian torus fibers.

Analytic Mirror Construction via Family Floer Theory

Working over the Novikov field $D_{ac}$0, affinoid charts are constructed via the Berkovich analytification. The mirror space $D_{ac}$1 is explicitly realized as the gluing of two charts $D_{ac}$2, with the transition map precisely determined by wall-crossing formulae reflecting quantum corrections from Maslov index zero disks.

The data on each chart is presented as follows:

• Coordinates $D_{ac}$3 (resp.\ $D_{ac}$4) parametrize the dual torus fibration.
• The Landau-Ginzburg superpotential is explicitly constructed using holomorphic disk enumeration:

$D_{ac}$5

with a corresponding expression on $D_{ac}$6.

Crucially, the analytic gluing map $D_{ac}$7 is determined by equating the superpotential across the wall to enforce the corrected mirror geometry, ensuring that the global analytic function $D_{ac}$8 is well-defined.

Embedding into the Langlands Dual and Superpotential Identification

A global analytic embedding $D_{ac}$9 is constructed, with explicit formulas matching cluster coordinates and Plücker coordinates. The fiberwise torus action on the mirror aligns the bases of the A-side and B-side fibrations, and the gluing is shown to be injective and compatible with the analytic structures.

The global superpotential $Gr(2,4)_{\Lambda}^{an}\setminus D_{ac}$0 on the analytic mirror is shown to coincide explicitly with the Marsh-Rietsch superpotential:

$Gr(2,4)_{\Lambda}^{an}\setminus D_{ac}$1

demonstrating that holomorphic disk enumeration via the SYZ construction recovers precisely the Lie-theoretic LG model.

Theoretical and Practical Implications

This explicit construction substantiates the conjectural SYZ framework for Grassmannians at the level of explicit topological and analytic data, showing that the dual of the Lagrangian fibration recovers the Langlands dual Grassmannian (with anti-canonical divisor removed) not just as an abstract variety, but as an analytic space with matching fibration, affine structure, and superpotential.

Practically, this provides:

• An effective algorithm for constructing mirrors to other similarly structured varieties (e.g., higher Grassmannians, flag varieties) using explicit affine charts and wall-crossing data.
• A bridge between enumerative techniques (disk-counting) and cluster/Lie-theoretic mirrors, with every phase of the construction laid bare for computation.
• An analytic apparatus for understanding mirrors over Novikov fields, which could facilitate the study of non-archimedean, tropical, and Berkovich geometry in the context of mirror symmetry.

On the theoretical side, it demonstrates that the wall-crossing corrections (quantum corrections in gluing) are sufficient to resolve singularities and precisely recover the expected mirror geometry and superpotential even in the presence of walls generated by Maslov-zero disks. The method would likely generalize, perhaps modulo additional complexity, to higher $Gr(2,4)_{\Lambda}^{an}\setminus D_{ac}$2 and to more complicated flag varieties.

Further, the approach makes the analytic topology—so often mysterious in mirror symmetry—entirely explicit in this case, giving a concrete foothold for future non-Archimedean studies, and suggests new avenues for explicit computations of Fukaya categories, disk potentials, and wall-crossing automorphisms in a rigorously controlled non-Archimedean analytic setting.

Conclusion

This paper demonstrates a fully constructive, explicit, non-archimedean analytic realization of SYZ mirror symmetry for $Gr(2,4)_{\Lambda}^{an}\setminus D_{ac}$3 by dualizing a Lagrangian fibration, carefully encoding holomorphic disk enumeration, wall-crossing, and integral affine base structures, and identifying the analytic mirror with the Landau-Ginzburg model of Marsh and Rietsch. The method establishes concrete evidence for the geometric SYZ principle in the context of Grassmannians and provides a clear blueprint for future generalizations and computational analyses in both symplectic and algebraic mirror symmetry frameworks.