Minimal Gromov--Witten ring

TL;DR

This paper develops an abstract algebraic framework for genus 0 Gromov-Witten invariants of quantum minimal Fano varieties, establishing a universal ring structure and solving related differential equations.

Contribution

It introduces the minimal Gromov-Witten ring and proves an isomorphism with a free ring generated by prime invariants, advancing the algebraic understanding of Gromov-Witten theory.

Findings

Constructed the minimal Gromov-Witten ring for quantum minimal Fano varieties.

Proved the Abstract Reconstruction Theorem establishing an isomorphism with a free ring.

Derived solutions to DN-type differential equations using generating series of invariants.

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.