On Vanishing of Gromov--Witten Invariants

2025-08-21

AI Summary

This paper addresses the vanishing problem for three-point, genus-zero Gromov–Witten (GW) invariants on partial flag manifolds. Methodologically, it recasts the vanishing conditions of GW invariants as explicit polynomial systems and integrates the defining equations of flag manifolds with a generalized parametric Hilbert’s Nullstellensatz. This enables, for the first time, the geometric vanishing problem to be formulated within computational complexity theory. Under the Generalized Riemann Hypothesis (GRH), the problem is shown to lie in the interactive proof class AM—located at the second level of the polynomial hierarchy. The work establishes a deep connection between the vanishing behavior of GW invariants in quantum cohomology and computational complexity, providing the first complexity-theoretic foundation and algorithmic pathway for computing quantum cohomology rings of flag manifolds.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Quantum Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Sustainability of Web economics
• Security and Privacy: Applications of cryptography

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.

Problem

• Determining vanishing of Gromov-Witten invariants on flag varieties
• Solving decision problem for 3-pointed genus zero invariants
• Establishing complexity class AM under GRH assumption

Innovation

• Polynomial equations system construction
• Parametric Hilbert's Nullstellensatz extension
• Complexity class AM reduction under GRH