Rationality and categorical properties of the moduli of instanton bundles on the projective 3-space

TL;DR

This work proves that the moduli spaces of mathematical instanton bundles on ${\mathbb P^3}$ are irreducible and rational for arbitrary rank $r$ and charge $n$, including the rank-2 case. The authors develop a unified framework based on the Barth–Hulek monad, Koszul and Beilinson resolutions, and restriction-to-hyperplane techniques, reducing the global moduli problem to stable bundles on Hirzebruch surfaces and on ${\mathbb P^2}$ or quadrics. They establish that the moduli spaces form a self-dual monoidal category under tensor product, and they show the restriction maps to wedges and quadrics are birational, linking the ${\mathbb P^3}$-instantons to Yang–Mills instantons on ${\mathbb P^2}$. These results yield birational and sometimes open-immersion relations between various moduli, provide explicit rational parametrizations via extension data on Hirzebruch surfaces, and illuminate deep connections to physics via the ADHM/Penrose correspondences and the monoidal structure. The paper thus achieves a broad, constructive understanding of rationality/irreducibility and of the categorical structure governing instanton moduli, with concrete consequences for YM-instanton spaces and for potential multi-instanton physics.

Abstract

We prove the rationality and irreducibility of the moduli space of mathematical instanton vector bundles of arbitrary rank and charge on $\mathbb P^3$. In particular, the result applies to the rank-2 case. This problem was first studied by Barth, Ellingsrud-Stromme, Hartshorne, Hirschowitz-Narasimhan in the late 1970s. We also show that the mathematical instantons of variable rank and charge form a monoidal category. The proof is based on an in-depth analysis of the Barth-Hulek monad-construction and on a detailed description of the moduli space of (framed and unframed) stable bundles on Hirzebruch surfaces.

Theorems & Definitions (1)

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