Some augmentations of the Legendrian DGA come from smooth fillings; in “augmentations are sheaves” we showed that, at least for Legendrian knots in the standard contact 3-space, all augmentations are geometric in the sense that the category of augmentations is equivalent to a geometrically defined category of sheaves.

I will outline a strategy for making this into the perhaps more viscerally satisfying result that all augmentations are geometric in the sense of coming from the Floer theory of an immersed Lagrangian.

 

1. By the augmentations are sheaves result + usual facts about the Kashiwara-Schapira sheaf, the category of augmentations is the global sections of a sheaf of categories on the union of a cylinder on the Legendrian knot with the front plane.
2. By Nadler’s arborealization, this singular Lagrangian can be deformed to one with arboreal singularities, in such a way that this sheaf of categories comes along for the ride.
3. Each arboreal singularity can be approximated by an immersed Lagrangian.  To construct the immersed Lagrangian, recall that the arboreal singularity is glued together from a certain number of things which are homeomorphic to $\mathbb{R}^n$; deform each to a smooth one and perturb.  Paradigmatic examples of the sort of immersed object which appears mean appear in Seidel’s treatment of the genus 2 curve and Sheridan’s treatment of pairs of pants.  Also see this paper of Cho, Hong, and Lau.
4. There should be a local argument showing that the local contribution to the Fukaya category of this Lagrangian is enough to capture the local contributions of the arboreal singularity.
5. This construction should glue.

The above should both produce both the desired exact Lagrangian, and a way of deducing from an “augmentations are sheaves” result that its Floer theory exhausts the augmentations of its boundary.

Similarly, one could go from a result localizing the Fukaya category to sheaves on an arboreal skeleton to a result which recovers the Fukaya category from the Floer theory of a single immersed Lagrangian.

 

Remark.  For Legendrian knots, the only relevant singularities are the A1 and A2 arboreal singularities.  These both appear already in the skeleton for the 4d symplectic pair of pants, so the desired local model of immersed Lagrangian most likely already appears in Sheridan’s work.