Minimal Gromov--Witten ring

Abstract

We build the abstract theory of Gromov-Witten invariants of genus 0 for quantum minimal Fano varieties (a minimal natural (with respect to Gromov-Witten theory) class of varieties). In particular, we consider ``the minimal Gromov-Witten ring'', i. e. a commutative algebra with generators and relations of the form used in the Gromov-Witten theory of Fano variety (of unspecified dimension). Gromov-Witten theory of any quantum minimal variety is a homomorphism of this ring to $\mathbb C$. We prove the Abstract Reconstruction Theorem which states the particular isomorphism of this ring with a free commutative ring generated by ``prime two-pointed invariants''. We also find the solutions of the differential equations of type DN for a Fano variety of dimension N in terms of generating series of one-pointed Gromov-Witten invariants.