Multi-Soliton Dynamics of Anti-Self-Dual Gauge Fields

TL;DR

This paper investigates multi-soliton solutions of anti-self-dual Yang-Mills equations, revealing their interpretation as intersecting soliton walls, calculating phase shifts, and demonstrating their realization in various signatures and string theories.

Contribution

It introduces a detailed analysis of multi-soliton dynamics, including explicit phase shift calculations and the realization of soliton walls in different signatures, using quasideterminants for simplification.

Findings

Multi-soliton solutions form intersecting soliton walls with preserved shape and velocity.

Explicit phase shift factors are calculated for soliton interactions.

Soliton walls can be realized in all regions of N=2 string theories with different signatures.

Abstract

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=U(2) in the Ultrahyperbolic space (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.