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 $L_{\mathrm{\Delta }}\subseteq \mathbb{C}{\mathbb{P}}^3$. For example, we equip $L_{\mathrm{\Delta }}$ with a particular rank 2 local system $W$ over $\mathbb{Z}/2\mathbb{Z}$ for which 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 $L_{\mathrm{\Delta }}$ cannot be disjoined from $\mathbb{R}{\mathbb{P}}^3$ by a Hamiltonian isotopy.

Momchil Konstantinov

ML Science/Engineering, Maths PhD