Computations of instanton invariants

TL;DR

This paper introduces numerical algorithms implemented in Macaulay 2 to compute analytic invariants of vector bundles on non-compact spaces, revealing fine geometric properties and extending understanding of instanton invariants.

Contribution

It develops new computational methods for instanton invariants of vector bundles on non-compact spaces, specifically calculating the width and height invariants.

Findings

Algorithms successfully compute the invariants for given bundles.

The invariants relate to the local holomorphic Euler characteristic.

Provides tools for analyzing geometric properties of instantons.

Abstract

Motivated by newly discovered properties of instantons on non-compact spaces, we realised that certain analytic invariants of vector bundles detect fine geometric properties. We present numerical algorithms, implemented in Macaulay 2, to compute these invariants. Precisely, we obtain the direct image and first derived functor of the contraction map $\pi \colon Z \to X$, where $Z$ is the total space of a negative bundle over $\mathbb{P}^1$ and $\pi$ contracts the zero section. We obtain two numerical invariants of a rank-2 vector bundle $E$ on $Z$, the width $h^0\bigl(X; (\pi_*E)^{\vee \vee} \bigl/ \pi_*E\bigr)$ and the height $h^0\bigl(X; R^1 \pi_*E \bigr)$, whose sum is the local holomorphic Euler characteristic $\chi^\text{loc}(E)$.