Classical solitons in anti-self-dual Yang-Mills theory

Tristan McLoughlin

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arXiv · 2610.27074

Classical solitons in anti-self-dual Yang-Mills theory

Tristan McLoughlin

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Abstract

We study classical soliton solutions in split-signature anti-self-dual Yang-Mills theory. Using Zakharov's dressing method, we construct rational lump solitons, which are localised near two-dimensional surfaces, and wall solitons, which have codimension one. We further describe multi-phase solutions which have multiple walls meeting at trivalent junctions and form four-dimensional foam-like structures. Two-pole solutions describe soliton interactions and we find non-trivial colour-dressed phase shifts for wall interactions. We consider the four-dimensional analogue of Ward's lump scattering for the restricted kinematics of coincident poles and show that there are non-trivial interactions in the collinear limit. Finally, we show that the determinant of the soliton matrix provides a $τ$-function which acts as a potential for gauge-invariant scalar quantities such as the Yang-Mills Lagrangian density.

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Tristan McLoughlin. 2026-10-19. Classical solitons in anti-self-dual Yang-Mills theory. https://arxiv.org/abs/2610.27074

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