From symplectic cohomology to Lagrangian enumerative geometry

Abstract

We build a bridge between Floer theory on open symplectic manifolds and the enumerative geometry of holomorphic disks inside their Fano compactifications, by detecting elements in symplectic cohomology which are mirror to Landau-Ginzburg potentials. We also treat the higher Maslov index versions of the potentials. We discover a relation between higher disk potentials and symplectic cohomology rings of smooth anticanonical divisor complements (themselves conjecturally related to closed-string Gromov-Witten invariants), and explore several other applications to the geometry of Liouville domains.

arXiv comment48 pages, 13 figures; v2: reference fixes, minor corrections; v3: minor changes, accepted version

References in corpus (24)

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• Rigid subsets of symplectic manifolds
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• String topology with gravitational descendants, and periods of Landau-Ginzburg potentials
• Disjoinable Lagrangian spheres and dilations
• Lagrangian spheres in Del Pezzo surfaces
• Floer cohomology in the mirror of the projective plane and a binodal cubic curve
• Exact Lagrangians in plumbings
• A spectral sequence for symplectic homology
• Intrinsic mirror symmetry and punctured Gromov-Witten invariants

Cited by in corpus (7)

• String topology with gravitational descendants, and periods of Landau-Ginzburg potentials
• Intrinsic mirror symmetry and categorical crepant resolutions
• Disk counting and wall-crossing phenomenon via family Floer theory
• Wall-crossing from Lagrangian Cobordisms
• Nonexistence of exact Lagrangian tori in affine conic bundles over ${\mathbb{C}}^n$
• Orbifold Kodaira-Spencer maps and closed-string mirror symmetry for punctured Riemann surfaces
• Asymptotic behavior of Vianna's exotic Lagrangian tori $T_{a,b,c}$ in ${\mathbb{CP}}^2$ as $a+b+c\to \infty$