The Calabi–Yau criterion for the Maurer–Cartan algebra of an immersed Lagrangian

Let $\mathbb{L}$ be a compact, relatively spin, graded, oriented immersed Lagrangian of dimension $n$ in a symplectic manifold $M$, and let $\mathcal{A}_{\mathbb{L}}$ be its Maurer–Cartan algebra formed by degree-one elements. Let $\mathscr{F}^{(\mathbb{L},\boldsymbol{b})}(\mathbb{L},\boldsymbol{b})$ denote the associated Lagrangian Floer complex. Calabi–Yau criterion. It is expected that the following are equivalent: (1) $\mathscr{F}^{(\mathbb{L},\boldsymbol{b})}(\mathbb{L},\boldsymbol{b})$ provides a projective resolution of $\mathcal{A}_{\mathbb{L}}$ as an $\mathcal{A}_{\mathbb{L}}$-bimodule; (2) $\mathcal{A}_{\mathbb{L}}$ is an $n$-Calabi–Yau algebra. This expectation is motivated by cyclic $A_\infty$ structures and existing results on Calabi–Yau algebras and categories. The source provides no resolution, so the assertion remains open.