Multi-soliton dynamics of anti-self-dual gauge fields

• Published in: The journal of high energy physics Vol. 2022; no. 1; pp. 1 - 19
• Main Authors: Hamanaka, Masashi, Huang, Shan-Chi
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2022; Springer Nature B.V; SpringerOpen
• Subjects: Classical and Quantum Gravitation; Density; Duality in Gauge Field Theories; Elementary Particles; High energy physics; Hyperplanes; Integrable Field Theories; Integrable Hierarchies; Mathematical analysis; Phase shift; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; Solitary waves; Solitons Monopoles and Instantons; String Theory; Symmetry; Walls; Integrable Field Theories; Duality in Gauge Field Theories; Integrable Hierarchies; Solitons Monopoles and Instantons
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP01(2022)039

A bstract We study dynamics of multi-soliton solutions of anti-self-dual Yang-Mills equations for G = GL(2 , ℂ) in four-dimensional spaces. The one-soliton solution can be interpreted as a codimension-one soliton in four-dimensional spaces because the principal peak of action density localizes on a three-dimensional hyperplane. We call it the soliton wall. We prove that in the asymptotic region, the n -soliton solution possesses n isolated localized lumps of action density, and interpret it as n intersecting soliton walls. More precisely, each action density lump is essentially the same as a soliton wall because it preserves its shape and “velocity” except for a position shift of principal peak in the scattering process. The position shift results from the nonlinear interactions of the multi-solitons and is called the phase shift. We calculate the phase shift factors explicitly and find that the action densities can be real-valued in three kind of signatures. Finally, we show that the gauge group can be G = SU(2) in the Ultrahyperbolic space 𝕌 (the split signature (+ , + , −, − )). This implies that the intersecting soliton walls could be realized in all region in N=2 string theories. It is remarkable that quasideterminants dramatically simplify the calculations and proofs.