On Vanishing of Gromov--Witten Invariants

TL;DR

This paper investigates the computational complexity of determining whether certain Gromov--Witten invariants vanish on partial flag varieties, establishing their placement within the polynomial hierarchy under GRH.

Contribution

It proves that the zero-invariant decision problem for 3-pointed, genus zero cases is in the class AM assuming GRH, extending the Nullstellensatz for the reduction.

Findings

The problem is in the class AM under GRH.

Constructs explicit polynomial systems for the problem.

Extends Parametric Hilbert's Nullstellensatz for the reduction.

Abstract

We consider the decision problem of whether a particular Gromov--Witten invariant on a partial flag variety is zero. We prove that for the $3$-pointed, genus zero invariants, this problem is in the complexity class ${\sf AM}$ assuming the Generalized Riemann Hypothesis (GRH), and therefore lies in the second level of polynomial hierarchy ${\sf PH}$. For the proof, we construct an explicit system of polynomial equations through a translation of the defining equations. We also need to prove an extension of the Parametric Hilbert's Nullstellensatz to obtain our central reduction.