The quantitative nature of reduced Floer theory

We study the reduced symplectic cohomology of disk subbundles in negative symplectic line bundles. We show that this cohomology theory sees the spectrum of a quantum action on quantum cohomology. Precisely, quantum cohomology decomposes into generalized eigenspaces of the action of the first Chern class by quantum cup product. The reduced symplectic cohomology of a disk bundle of radius R sees all eigenspaces whose eigenvalues have size less than R, up to rescaling by a fixed constant. Similarly, we show that the reduced symplectic cohomology of an annulus subbundle between radii R-1 and R-2 captures all eigenspaces whose eigenvalues have size between R-1 and R-2, up to a rescaling. We show how local closed-string mirror symmetry statements follow from these computations (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

We investigate the reduced symplectic cohomology of disk subbundles in negative symplectic line bundles and its relation to the spectrum of a quantum action on quantum cohomology. By analyzing the quantum cohomology decomposed into generalized eigenspaces, we demonstrate the properties of eigenspaces in the reduced symplectic cohomology of disk and annulus subbundles. These computations lead to statements about local closed-string mirror symmetry.