Monotone Lagrangians in cotangent bundles of spheres

TL;DR

This paper investigates the structure of the monotone Fukaya category of cotangent bundles of spheres, showing it is generated by the zero-section and certain monotone Lagrangian tori, leading to non-displaceability results.

Contribution

It identifies the generators of the monotone Fukaya category of cotangent bundles of spheres and establishes non-displaceability of certain monotone Lagrangians.

Findings

The Fukaya category is split-generated by the zero-section and monotone Lagrangian tori.

Any monotone Lagrangian with non-trivial Floer cohomology is non-displaceable from these generators.

For $T^*S^3$, the tori can be replaced by a family of monotone $T^3$.

Abstract

We study the compact monotone Fukaya category of $T^*S^n$, for $n\geq 2$, and show that it is split-generated by two classes of objects: the zero-section $S^n$ (equipped with suitable bounding cochains) and a 1-parameter family of monotone Lagrangian tori $(S^1\times S^{n-1})_\tau$, with monotonicity constants $\tau>0$ (equipped with rank 1 unitary local systems). As a consequence, any closed orientable spin monotone Lagrangian (possibly equipped with auxiliary data) with non-trivial Floer cohomology is non-displaceable from either $S^n$ or one of the $(S^1\times S^{n-1})_\tau$. In the case of $T^*S^3$, the monotone Lagrangians $(S^1\times S^2)_\tau$ can be replaced by a family of monotone tori $T^3_\tau$.