Equivariant Lagrangian displacements

TL;DR

This work develops a $\mathbb{Z}/2\mathbb{Z}$-equivariant version of Lagrangian quantum cohomology for monotone Lagrangians preserved by a symplectic involution and proves it is sensitive to equivariant displacements. By constructing a Borel-type pearl complex with an equivariant differential $d_{eq}$ and proving $d_{eq}^{2}=0$, the authors show that an equivariant displacement forces $\mathrm{QH}^{*}_{eq}(L)=0$. Central to the analysis is the BK-type neck-stretching framework, which yields an explicit computation of $\mathrm{QH}^{*}_{eq}(L)$ in terms of the Floer–Euler class $F$ and the reduced quantum cohomology $\mathrm{QH}^{*}(\bar L)$: $\mathrm{QH}^{*}_{eq}(L) \cong (\mathrm{QH}^{*}(\bar L) \otimes \mathbf{k}[e])/(e^{2}+F)$. In the case of free $\mathbb{Z}/2\mathbb{Z}$-actions, the theory simplifies to a direct relation between the equivariant and quotient quantum cohomologies, with the Floer–Euler class vanishing and a split $\mathrm{QH}^{*}_{eq}(L) \cong \mathrm{QH}^{*}(\bar L) \oplus \mathrm{QH}^{*-1}(\bar L)$. These results apply, in particular, to the Hopf-fibration pullbacks $L=\pi^{-1}(\bar L)$ in $\mathbb{C}^{n}$, yielding non-displaceability results when $\mathrm{QH}^{*}(\bar L)\neq 0$, and provide a quantum analogue of Gysin-type phenomena for equivariant cohomology. The framework opens several avenues for generalizing to other cyclic groups, relaxing monotonicity assumptions, or extending to generating-function techniques in the Euclidean setting.

Abstract

This paper proves that certain monotone Lagrangians in the standard symplectic vector space cannot be displaced by a Hamiltonian isotopy which commutes with the antipodal map. The method of proof is to develop a Borel equivariant version of the quantum cohomology of Biran and Cornea, and prove it is sensitive to equivariant displacements. The Floer--Euler class of Biran and Khanevsky appears as a term in the equivariant differential in certain cases.

Figures (5)

• Figure 1: $\mathscr{G}_k$ is the parameter space of strictly ordered $k$-tuples (shown with $k=3$).
• Figure 2: Illustration of a pearl trajectory $w$ with $k=3$ and asymptotics $x_{-},x_{+}$.
• Figure 3: The 1-manifold $\Pi^{x}_{2}$ and example boundary configurations. Here $y$ is in the unstable manifold of $x$ and $y'$ is in the stable manifold of $x$.
• Figure 4: Nodal curve lying at the interface of two moduli spaces of non-nodal curves.
• Figure 5: A non-simple pearl "shatters" into open holomorphic pieces separated by a certain graph $G$ by the results of lazzarini-1lazzarini-2; moreover, the interior of each holomorphic piece is a multiple cover of the interior of an underlying simple holomorphic disk. Therefore one can extract an underlying chain of simple disks as explained in biran-cornea-arXiv-2007.

Theorems & Definitions (59)

• Theorem 1
• Remark 2
• Remark 3
• Proposition 4
• Remark 5
• Remark 6: Invariance statement
• Theorem 7
• Theorem 8
• Remark 9
• Remark 10