Cluster varieties from Legendrian knots

Abstract

Many interesting spaces --- including all positroid strata and wild character varieties --- are moduli of constructible sheaves on a surface with microsupport in a Legendrian link. We show that the existence of cluster structures on these spaces may be deduced in a uniform, systematic fashion by constructing and taking the sheaf quantizations of a set of exact Lagrangian fillings in correspondence with isotopy representatives whose front projections have crossings with alternating orientations. It follows in turn that results in cluster algebra may be used to construct and distinguish exact Lagrangian fillings of Legendrian links in the standard contact three space.

47 pages

References in corpus (11)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Total positivity, Grassmannians, and networks
• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Symplectic Manifolds and Isomonodromic Deformations
• The twist for positroid varieties
• Microlocal condition for non-displaceablility
• On the combinatorics of exact Lagrangian surfaces
• Cluster X-varieties, amalgamation and Poisson-Lie groups
• Calabi-Yau structures on topological Fukaya categories
• Brane structures in microlocal sheaf theory
• The cardinality of the augmentation category of a Legendrian link

Cited by in corpus (38)

• Microlocal Morse theory of wrapped Fukaya categories
• On the combinatorics of exact Lagrangian surfaces
• Legendrian Weaves: N-graph Calculus, Flag Moduli and Applications
• The conormal torus is a complete knot invariant
• Lagrangian pairs of pants
• Morsifications and mutations
• Schubert puzzles and integrability I: invariant trilinear forms
• Calabi-Yau structures on topological Fukaya categories
• Augmentations, Fillings, and Clusters
• Brane structures in microlocal sheaf theory
• Sheaf quantization in Weinstein symplectic manifolds
• Domain wall formation and magnon localization in twisted chromium trihalides
• Infinitely many Lagrangian fillings
• Lagrangian submanifolds from tropical hypersurfaces
• Algebras of quantum monodromy data and decorated character varieties
• Weave Realizability for D-type
• Critical varieties in the Grassmannian
• Tropical Lagrangians in toric del-Pezzo surfaces
• Augmentations and immersed Lagrangian fillings
• Lagrangian fillings for Legendrian links of affine type
• Positive Braid Links with Infinitely Many Fillings
• Exotic Lagrangian tori in Grassmannians
• Fukaya category of Grassmannians: rectangles
• Wrapped sheaves
• Lagrangian Fillings in A-type and their Kalman Loop Orbits
• Augmentations and ruling polynomials for Legendrian graphs
• Integrability As Duality: The Gauge/YBE Correspondence
• A Lagrangian filling for every cluster seed
• On the equatorial Dehn twist of a Lagrangian nodal sphere
• Twisted spectral correspondence and torus knots
• A Legendrian Turaev torsion via generating families
• Categorification of Legendrian knots
• Smoothly non-isotopic Lagrangian disk fillings of Legendrian knots
• Variation of GIT and Variation of Lagrangian Skeletons I: Flip and Flop
• Dual boundary complexes of Betti moduli spaces over the two-sphere with one irregular singularity
• Symplectomorphisms of some Weinstein 4-manifolds
• Lagrangian cobordism of positroid links
• Non-fillable augmentations of twist knots