The Maurer–Cartan algebra of a Lagrangian $L$ is the algebra that encodes the deformation of the Floer complex $CF(L,L;\Lambda)$ as an $A_\infty$-algebra. We identify the Maurer–Cartan algebra with the 0‑th cohomology of the Koszul dual dga of $CF(L,L;\Lambda)$. Making use of the identification, we prove that there exists a natural isomorphism between the Maurer–Cartan algebra of $L$ and a suitable subspace of the completion of the wrapped Floer cohomology of another Lagrangian $G$ when $G$ is dual to $L$ in the sense to be defined. In view of mirror symmetry, this can be understood as specifying a local chart associated with $L$ in the mirror rigid analytic space. We examine the idea by explicit calculation of the isomorphism for several interesting examples.
$\def\partialol{\bar{\partial}}$Let $(X, J)$ be a complex manifold with a non-degenerated smooth $d$-closed $(1,1)$-form $\omega$. Then we have a natural double complex $\partialol+ \partialol^\Lambda$, where $\partialol^\Lambda$ denotes the symplectic adjoint of the $\partialol$-operator. We study the Hard Lefschetz Condition on the Dolbeault cohomology groups of $X$ with respect to the symplectic form $\omega$. In [$\href{https://www.worldscientific.com/doi/abs/10.1142/S0129167X18500957}{29}$], we proved that such a condition is equivalent to a certain symplectic analogue of the $\partialol\partialol$-Lemma, namely the $\partialol\partialol^\Lambda$-Lemma, which can be characterized in terms of Bott–Chern and Aeppli cohomologies associated to the above double complex. We obtain Nomizu type theorems for the Bott–Chern and Aeppli cohomologies and we show that the $\partialol\partialol^\Lambda$-Lemma is stable under small deformations of $\omega$, but not stable under small deformations of the complex structure. However, if we further assume that $X$ satisfies the $\partialol\partialol$-Lemma then the $\partialol\partialol^\Lambda$-Lemma is stable.
Rabinowitz–Floer homology is the Morse–Bott homology in the sense of Floer associated with the Rabinowitz action functional introduced by Kai Cieliebak and Urs Frauenfelder in 2009. In our work, we consider a generalisation of this theory to a Rabinowitz–Floer homology of a Liouville automorphism. As an application, we show the existence of noncontractible periodic Reeb orbits on quotients of symmetric star-shaped hypersurfaces. In particular, our theory applies to lens spaces.
In this paper, we present new obstructions to the existence of Lagrangian cobordisms in $\mathbb{R}^4$ that depend only on the enriched knot diagrams of the boundary knots or links, using holomorphic curve techniques. We define enriched knot diagrams for generic smooth links. The existence of Lagrangian cobordisms gives a well-defined transitive relation on equivalence classes of enriched knot diagrams that is a strict partial order when restricted to exact enriched knot diagrams To establish obstructions we study 1-dimensional moduli spaces of holomorphic disks with corners that have boundary on Lagrangian tangles—an appropriate immersed Lagrangian closely related to embedded Lagrangian cobordisms. We adapt existing techniques to prove compactness and transversality, and compute dimensions of these moduli spaces. We produce obstructions as a consequence of characterizing all boundary points of such moduli spaces. We use these obstructions to recover and extend results about “growing” and “shrinking” Lagrangian slices. We hope that this investigation will open up new directions in studying Lagrangian surfaces in $\mathbb{R}^4$.