Augmentations and immersed Lagrangian fillings

Abstract

For a Legendrian link Λ ⊂ 𝐽 1 𝑀 with 𝑀 = ℝ or 𝑆 1 , immersed exact Lagrangian fillings 𝐿 ⊂ Symp(𝐽 1 𝑀) ≅ 𝑇 * (ℝ >0 × 𝑀) of Λ can be lifted to conical Legendrian fillings Σ ⊂ 𝐽 1 (ℝ >0 × 𝑀) of Λ. When Σ is embedded, using the version of functoriality for Legendrian contact homology (LCH) from Pan and Rutherford [J. Symplectic Geom. 19 (2021), no. 3, 635-722], for each augmentation 𝛼 ∶ (Σ) → ℤ∕2 of the LCH algebra of Σ, there is an induced augmentation 𝜖 (Σ,𝛼) ∶ (Λ) → ℤ∕2. With Σ fixed, the set of homotopy classes of all such induced augmentations, 𝐼 Σ ⊂ 𝐴𝑢𝑔(Λ)∕∼, is a Legendrian isotopy invariant of Σ. We establish methods to compute 𝐼 Σ based on the correspondence between MCFs and augmentations. This includes developing a functoriality for the cellular differential graded algebra from Rutherford and Sullivan [Adv. Math. 374 (2020), 107348, 71 pp.] with respect to Legendrian cobordisms, and proving its equivalence to the functoriality for LCH. For arbitrary 𝑛 ⩾ 1, we give examples of Legendrian torus knots with 2𝑛 distinct conical Legendrian fillings distinguished by their induced augmentation sets. We prove that when 𝜌 ≠ 1 and Λ ⊂ 𝐽 1 ℝ, every 𝜌-graded augmentation of Λ can be induced in this manner by an immersed Lagrangian filling. Alternatively, this is viewed as a computation of cobordism classes for an appropriate notion of 𝜌-graded augmented Legendrian cobordism.M S C 2 0 2 0 53D42 (primary)

Lagrangian skeleta and plane curve singularities

“…An exact Lagrangian filling 16 defines an object in the category Aug + (Λ), and the morphisms between two such objects are given by (a linearized version of) Lagrangian Floer homology. In fact, there is a sense in which any object in Aug + (Λ) comes from a Lagrangian filling [88,89], possibly immersed, and thus ob(Aug + (Λ)) is a natural candidate for a moduli space of Lagrangian fillings. The algebra A(f ) is known to be a cluster algebra [51] in characteristic two.…”