The quantitative nature of reduced Floer theory

Abstract

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.