Soliton Resonances in Four Dimensional Wess-Zumino-Witten Model

TL;DR

This work addresses soliton resonances in the four-dimensional ASDYM/WZW$_4$ setting on the ultrahyperbolic space $\mathbb{U}$ with unitary data by developing a unitary quasi-Grammian (Cauchy matrix) framework that links to the binary Darboux transformation via quasideterminants. It provides explicit one-, two-, and multi-soliton solutions and analyzes resonance limits, yielding double-pole and V-shaped solitons, while showing the NL$\sigma$M action densities are real for unitary solutions and exhibit KP-like asymptotics; the Wess-Zumino term vanishes in these asymptotics. The approach uses simpler input data than previous Darboux-based methods and demonstrates that quasi-Grammian solutions reproduce the same action densities as quasi-Wronskian constructions, suggesting a unifying classification path for ASDYM solitons and potential insights into open $N=2$ string theory. The results highlight resonance dynamics as building blocks for soliton interactions and hint at non-Grassmannian geometric structures underlying ASDYM solitons, with implications for non-perturbative aspects of related string theories.

Abstract

We present two kinds of resonance soliton solutions on the Ultrahyperbolic space $\mathbb{U}$ for the G=U(2) Yang equation, which is equivalent to the anti-self-dual Yang-Mills (ASDYM) equation. We reveal and illustrate the solitonic behaviors in the four-dimensional Wess-Zumino-Witten (WZW$_4$) model through the sigma model action densities. The Yang equation is the equation of motion of the WZW$_4$ model. In the case of $\mathbb{U}$, the WZW$_4$ model describes a string field theory action of open N=2 string theories. Hence, our solutions on $\mathbb{U}$ suggest the existence of the corresponding classical objects in the N=2 string theories. Our solutions include multiple-pole solutions and V-shape soliton solutions. The V-shape solitons suggest annihilation and creation processes of two solitons and would be building blocks to classify the ASDYM solitons, like the role of Y-shape solitons in classification of the KP (line) solitons. We also clarify the relationship between the Cauchy matrix approach and the binary Darboux transformation in terms of quasideterminants. Our formalism can start with a simpler input data for the soliton solutions and hence might give a suitable framework for the classification of the ASDYM solitons.

Figures (8)

• Figure 1: 2D slice of two-soliton NL$\sigma$M action density for $(w, \widetilde{w})=(0, 0)$. (a) The case of $\tilde{\delta}>0$. (b) The case of $\tilde{\delta}<0$.
• Figure 2: Plots of the 2D slice of the double-pole soliton NL$\sigma$M action density with $\lambda_1=-1+i,\alpha=0.5-i,\beta=-0.7-1.4i$, $(z,\tilde{z})\in[-20,20]\times[-10,10]$, and $(w,\tilde{w})=(0,0)$. (a) Shape of the 2D slice of NL$\sigma$M action density. (b) Density plot with four red curves of $Z_{\pm} \mp \delta=0.$
• Figure 3: Plots of the 2D slice of two-soliton NL$\sigma$M action density with $\lambda_1=0.5+0.5i,\alpha_1=0.5-0.5i,\beta_1=-0.7-1.4i$, $(z,\tilde{z})\in[-8,8]\times[-8,8]$, and $(w,\tilde{w})=(0,0)$. (a) Shape of the 2D slice of NL$\sigma$M action density. (b) Density plot of the 2D slice of NL$\sigma$M action density.
• Figure 4: Plots of the 2D slice of two-soliton NL$\sigma$M action density with $\lambda_1=-1+i,\lambda_2=0.5+0.5i,\alpha_1=\alpha_2=0.5-0.5i,\beta_1=\beta_2=-0.7-1.4i$, $(z,\tilde{z})\in[-8,8]\times[-8,8]$, and $(w,\tilde{w})=(0,0)$. (a) Shape of the 2D slice of NL$\sigma$M action density. (b) Density plot of the 2D slice of NL$\sigma$M action density.
• Figure 5: Plots of the 2D slice of double-pole soliton NL$\sigma$M action density with $\lambda_1=-1+i,\alpha=0.5-0.5i,\beta=-0.7-1.4i$, $(z,\tilde{z})\in[-8,8]\times[-8,8]$, and $(w,\tilde{w})=(0,0)$. (a) Shape of the 2D slice of NL$\sigma$M action density. (b) Density plot of the 2D slice of NL$\sigma$M action density.

Theorems & Definitions (5)

• Lemma 3.1
• Lemma 3.2
• Theorem 3.3
• Theorem 3.4
• Corollary 3.5