On equivalence between solution structures of the anti-self-dual Yang--Mills equations
Abstract
We establish an exact correspondence between the quasi-Wronskian and quasi-Grammian constructions for the anti-self-dual Yang--Mills (ASDYM) equations with complex 2×2 matrix-valued fields. Starting from vacuum wave functions associated with 2M pairwise distinct nonzero constant spectral parameters, we restrict attention to a nonempty local domain on which the corresponding interpolation matrix is invertible. We show that both constructions are governed by the same uniquely solvable matrix-polynomial interpolation problem. In particular, the quasi-Wronskian Darboux polynomial T(λ) and the quasi-Grammian rational dressing matrix W(λ) satisfy T(λ)=qξ(λ)W(λ), where qξ(λ) is the monic scalar polynomial whose zeros are the prescribed poles of W(λ). This identity yields explicit relations between the corresponding Yang and potential matrices and shows that the two constructions generate identical ASDYM gauge potentials in the same gauge.
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BibTeX
@misc{li2026equivalence,
title = {{On equivalence between solution structures of the anti-self-dual Yang--Mills equations}},
author = {Shangshuai Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14469},
eprint = {2610.14469},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14469},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On equivalence between solution structures of the anti-self-dual Yang--Mills equations
AU - Shangshuai Li
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14469
UR - https://arxiv.org/abs/2610.14469
AB - We establish an exact correspondence between the quasi-Wronskian and quasi-Grammian constructions for the anti-self-dual Yang--Mills (ASDYM) equations with complex 2×2 matrix-valued fields. Starting from vacuum wave functions associated with 2M pairwise distinct nonzero constant spectral parameters, we restrict attention to a nonempty local domain on which the corresponding interpolation matrix is invertible. We show that both constructions are governed by the same uniquely solvable matrix-polynomial interpolation problem. In particular, the quasi-Wronskian Darboux polynomial T(λ) and the quasi-Grammian rational dressing matrix W(λ) satisfy T(λ)=q_ξ(λ)W(λ), where q_ξ(λ) is the monic scalar polynomial whose zeros are the prescribed poles of W(λ). This identity yields explicit relations between the corresponding Yang and potential matrices and shows that the two constructions generate identical ASDYM gauge potentials in the same gauge.
ER -