Higher rank local systems in Lagrangian Floer theory

Abstract

We extend Floer theory for monotone Lagrangians to allow coefficients in local systems of arbitrary rank. Unlike the rank 1 case, this is often obstructed by Maslov 2 discs. We study exactly what the obstruction is and define some natural unobstructed subcomplexes. To illustrate these constructions we do some explicit calculations for the Chiang Lagrangian $L_{\Delta }\subseteq \mathbb{C}{P}^3$. For example, we equip $L_{\Delta }$ with a particular rank 2 local system $W$ over the field with 2 elements such that the resulting Floer complex $C{F}^{\ast }(W,W)$ is unobstructed despite the presence of Maslov 2 discs. We compute that the cohomology $H{F}^{\ast }(W,W)$ is non-zero and deduce that the Chiang Lagrangian cannot be disjoined from $\mathbb{R}{P}^3$ by a Hamiltonian isotopy.

70 pages, 13 figures; Comments welcome; v2 typos fixed, references added

References in corpus (3)

• Quantum Structures for Lagrangian Submanifolds
• The Lagrangian Floer-quantum-PSS package and canonical orientations in Floer theory
• Constraints on exact Lagrangians in cotangent bundles of manifolds fibred over the circle

Cited by in corpus (1)

• Floer homology of fibrations I: Representing flow lines in Moore path spaces