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

TL;DR

The paper proves that the Chekanov-Eliashberg DGA of a horizontally displaceable Legendrian sphere is a $(n+1)$-Calabi-Yau DGA by constructing a Calabi–Yau quasi-isomorphism to the inverse dualizing bimodule, using a Rabinowitz DG-bimodule realized as a cone between 2-copy bimodules. It develops a detailed cone description with maps $\mathrm{CY}$ and $\mathrm{CY}[-n-1]$ and shows acyclicity of the Rabinowitz complex, enabling the CY quasi-isomorphism, which extends to higher $A_\infty$-structures via CY$_d$ maps. The work also builds Hochschild (co)homology via cyclic bimodules, defines compatible product structures, and demonstrates functorial CY behavior, including explicit computations in the unknot case. These results connect to Ganatra’s Calabi–Yau framework and provide a computable approach through gradient-flow-tree methods, enriching the interaction between Legendrian contact geometry and Calabi–Yau duality.

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.

Figures (16)

• Figure 1: Pseudo-holomorphic discs contributing to the map $\mathop{\mathrm{CY}}\nolimits$.
• Figure 2: Types of pseudo-holomorphic buildings in the boundary of moduli spaces of bananas with a positive asymptotic at $\gamma_{10}$. When the connecting Reeb chords between the two components of the leftmost building is a long chord or $y_{01}$, then it contributes algebraically to \ref{['ba4']}; and if it is the chord $x_{01}$, then it contributes algebraically to \ref{['ba2']}. The middle building schematizes the contributions of type \ref{['ba3']} and the rightmost building contributes to \ref{['ba1']}.
• Figure 3: Pseudo-holomorphic disc contributing to $\widehat{\mathop{\mathrm{\mathfrak{m}}}\nolimits}(\gamma_{10}a_1\dots a_k)$. Observe that the "bubble" at the top is not a pseudo-holomorphic disc but a way to write the cyclic word $\gamma_{10}a_1\dots a_k$. The contribution $\beta_{10}\boldsymbol{\delta}_0a_1\dots a_k\boldsymbol{\delta}_1$ of the disc is given by the output mixed chord $\beta_{10}$ followed by a word of Reeb chords as they appear along the boundary of the disc when following it counterclockwise from $\beta_{10}$.
• Figure 4: Pseudo-holomorphic building with boundary on $\mathbb R\times(\Lambda_0\cup\Lambda_1\cup\Lambda_2)$ contributing to the product on $\widecheck{C}_-^{cyc}(\Lambda_0,\Lambda_1)$.
• Figure 5: Pseudo-holomorphic buildings contributing to the product $\widehat{\mathop{\mathrm{\mathfrak{m}}}\nolimits}_2$ on $\widehat{C}^{cyc}_+(\Lambda_0,\Lambda_1)$.

Theorems & Definitions (43)

• Theorem 1.1
• Example 2.1
• Remark 2.1
• Definition 2.1
• Definition 2.2
• Remark 3.1
• Remark 3.2
• Remark 4.1
• Remark 4.2
• Theorem 4.1