Legendrian submanifolds from Bohr-Sommerfeld covers of monotone Lagrangian tori

TL;DR

This paper explores the construction of Legendrian submanifolds from Bohr-Sommerfeld covers of monotone Lagrangian tori, analyzing their Legendrian lifts, algebraic invariants, and cobordism properties within contact and symplectic geometry.

Contribution

It introduces a method to produce Legendrian submanifolds from monotone Lagrangian tori covers and computes their algebraic invariants, revealing new cobordism relations.

Findings

Legendrian lifts of monotone Lagrangian tori are not loose but admit exact Lagrangian cobordisms.

Computed Chekanov-Eliashberg algebras for these Legendrians.

Established existence of non-regular Lagrangian caps for the Legendrian submanifolds.

Abstract

By a result due to Ziltener, there exist no closed embedded Bohr-Sommerfeld Lagrangians inside $\mathbb CP^n$ for the prequantisation bundle whose total space is the standard contact sphere. On the other hand, any embedded monotone Lagrangian torus has a canonical nontrivial cover which is a Bohr-Sommerfeld immersion. We draw the front projections for the corresponding Legendrian lifts inside a contact Darboux ball of the threefold covers of both the two-dimensional Clifford and Chekanov tori (the former is the Legendrian link of the Harvey-Lawson special Lagrangian cone), and compute the associated Chekanov-Eliashberg algebras. Although these Legendrians are not loose, we show that they both admit exact Lagrangian cobordisms to the loose Legendrian sphere; they hence admit exact Lagrangian caps in the symplectisation, which are non-regular Lagrangian cobordisms. Along the way, we also compute bilinearised Legendrian contact homology of a general Legendrian surface in the standard contact vector space when all Reeb chords are of positive degree, as well as the augmentation variety in the case of tori.