Filtrations on equivariant quantum cohomology and Hilbert-Poincaré series

We prove that Floer theory induces a filtration by ideals on equivariant quantum cohomology of symplectic manifolds equipped with a ${\mathbb{C}}^{\ast }$-action. In particular, this gives rise to Hilbert-Poincaré polynomials on ordinary cohomology that depend on Floer theory. En route, the paper develops structural properties of filtrations on three versions of equivariant Floer cohomology. We obtain an explicit presentation for equivariant symplectic cohomology in the Calabi-Yau and Fano settings.