Minimal Gromov–Witten rings

We construct an abstract theory of Gromov–Witten invariants of genus 0 for quantum minimal Fano varieties (a minimal class of varieties which is natural from the quantum cohomological viewpoint). Namely, we consider the minimal Gromov–Witten ring: a commutative algebra whose generators and relations are of the form used in the Gromov–Witten theory of Fano varieties (of unspecified dimension). The Gromov–Witten theory of any quantum minimal variety is a homomorphism from this ring to $\mathbb C$. We prove an abstract reconstruction theorem which says that this ring is isomorphic to the free commutative ring generated by ‘prime two-pointed invariants’. We also find solutions of the differential equation of type $DN$ for a Fano variety of dimension $N$ in terms of the generating series of one-pointed Gromov–Witten invariants.