Yang-Mills Instanton Sheaves

TL;DR

This paper explores the construction of instanton sheaves on complex projective space CP^3 using SL(2,C) Yang-Mills instanton solutions, revealing new geometric structures linked to singularities absent in SU(2) cases.

Contribution

It demonstrates that SL(2,C) instantons can be used to explicitly construct instanton sheaves on CP^3, extending the geometric understanding of these solutions beyond traditional bundle frameworks.

Findings

SL(2,C) instantons satisfy complex ADHM equations.

Instanton sheaves on CP^3 can be explicitly constructed from these solutions.

Existence of sheaves is related to singularities of instantons on S^4.

Abstract

The SL(2,C) Yang-Mills instanton solutions constructed recently by the biquaternion method were shown to satisfy the complex version of the ADHM equations and the Monad construction. Moreover, we discover that, in addition to the holomorphic vector bundles on CP^3 similar to the case of SU(2) ADHM construction, the SL(2,C) instanton solutions can be used to explicitly construct instanton sheaves on CP^3. Presumably, the existence of these instanton sheaves is related to the singularities of the SL(2,C) instantons on S^4 which do not exist for SU(2) instantons.