Biquaternions and ADHM Construction of Non-Compact SL(2,C) Yang-Mills Instantons

TL;DR

This paper extends quaternion methods to biquaternions in the ADHM construction, producing new non-compact SL(2,C) instanton solutions with detailed analysis of their singularities and moduli spaces.

Contribution

It introduces biquaternion-based ADHM construction for SL(2,C) instantons, revealing their moduli count and singularity structures, expanding the understanding of complex Yang-Mills instantons.

Findings

Constructed SL(2,C) instantons using biquaternions.

Found the moduli space size to be 16k-6 for k-instantons.

Analyzed singularity structures and their mathematical consistency.

Abstract

We extend quaternion calculation in the ADHM construction of Sp(1) (=SU(2)) self-dual Yang-Mills (SDYM) instantons to the case of biquaternion. We use the biconjugate operation of biquaternion first introduced by Hamilton to construct the non-compact SL(2,C) k-instantons. The number of moduli for SL(2,C) k-instantons is found to be twice of that of Sp(1), 16k-6. These new SL(2,C) instanton solutions contain the SL(2,C) (M,N) instanton solutions constructed previously as a subset. The structures of singularities or jumping lines of the complete SL(2,C) k=1,2,3 instantons with 10,26,42 moduli parameters are particularly investigated. The existence of singular structures of the SL(2,C) k-instantons is mathematically consistent with recent results of solutions of complex ADHM equations. It may also help to clearify the long standing global singularity problems associated with Backlund transformations of SU(2) instantons.