Rabinowitz Floer homology for Legendrian submanifolds in prequantization bundles

Abstract

Let $Y$ be a prequantization bundle over an integral symplectic manifold $(\Sigma ,\omega )$. Let $L$ be a closed monotone Lagrangian submanifold that admits a Legendrian lift $\mathcal{L}$ in $Y$. Under the assumption that the minimal Maslov number $N_L$ of $L$ is greater than 2, we define the Rabinowitz Floer homology of $\mathcal{L}$. We then establish an isomorphism between the ${\mathbb{Z}}_d$-equivariant Rabinowitz Floer homology of $\mathcal{L}$ and the quantum homology of $L$, where $d$ is the degree of the covering map $\mathcal{L}\to L$. Under a more restrictive condition on $N_L$, we show that this map is a ring isomorphism. Using this isomorphism, we compute the quantum homology ring of Lagrangian spheres in quadrics and two-step flag manifolds. Furthermore, we investigate the implications of the quantum invertibility of $\omega$ for the vanishing of the quantum homology of $L$ and the obstructions to topologically simple fillings of $\mathcal{L}$. We also show that if $(\Sigma ,\omega )$ admits a polarization and $L$ is disjoint from the Lagrangian trace, the quantum homology of $L$ vanishes.

74 pages, 5 figures