Spectral equivalence of nearby Lagrangians

TL;DR

This paper develops a framework for the wrapped Donaldson--Fukaya category with coefficients in a commutative ring spectrum, proving a spectral equivalence for Lagrangian branes in cotangent bundles.

Contribution

It constructs the wrapped Donaldson--Fukaya category over a ring spectrum and proves spectral equivalence of Lagrangian branes with the zero section, extending classical results.

Findings

Wrapped Donaldson--Fukaya category constructed with ring spectrum coefficients

Any closed exact Lagrangian R-brane is isomorphic to a zero section brane

Floer homotopy type matches the stable homotopy type of the based loop space

Abstract

Let $R$ be a commutative ring spectrum. We construct the wrapped Donaldson--Fukaya category with coefficients in $R$ of any stably polarized Liouville sector. We show that any two $R$-orientable and isomorphic objects admit $R$-orientations so that their $R$-fundamental classes coincide. Our main result is that any closed exact Lagrangian $R$-brane in the cotangent bundle of a closed manifold is isomorphic to an $R$-brane structure on the zero section in the wrapped Donaldson--Fukaya category, generalizing a well-known result over the integers. To achieve this, we prove that the Floer homotopy type of the cotangent fiber is given by the stable homotopy type of the based loop space of the zero section.