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.