
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 . 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.
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