Calabi-Yau structure on the Chekanov-Eliashberg algebra of a Legendrian sphere

TL;DR

This paper proves that the Chekanov-Eliashberg algebra of a displaceable Legendrian sphere has a (n+1)-Calabi-Yau structure, revealing deep symmetries and dualities in Legendrian contact homology.

Contribution

It establishes a Calabi-Yau property for the Chekanov-Eliashberg algebra of Legendrian spheres, including the construction of $A_$ operations extending the Calabi-Yau isomorphism.

Findings

Chekanov-Eliashberg algebra is (n+1)-Calabi-Yau

Construction of $A_$ operations on cyclic bimodules

Extension of Calabi-Yau isomorphism to an $A_$-functor

Abstract

In this paper, we prove that the Chekanov-Eliashberg algebra of an horizontally displaceable n-dimensional Legendrian sphere in the contactisation of a Liouville manifold is a (n+1)-Calabi-Yau differential graded algebra. In particular it means that there is a quasi-isomorphism of DG-bimodules between the diagonal bimodule and the inverse dualizing bimodule associated to the Chekanov-Eliashberg algebra. On some cyclic version of these bimodules, which are chain complexes computing the Hochschild homology and cohomology of the Chekanov-Eliashberg algebra, we construct $A_\infty$ operations and show that the Calabi-Yau isomorphism extends to a family of maps satisfying the $A_\infty$-functor equations.