Towards Unification of Integrable Systems -- from ASD Yang-Mills viewpoints

Abstract: Anti-Self-Dual (ASD) Yang-Mills equations have played important roles in quantum field theory, four-dimensional geometry and integrable systems. The ASD Yang-Mills equations have two potential formalisms described by the J-matrix and the K-matrix. These equations by J and K are equations of motion of the 4-dimensional Wess-Zumino-Witten (4dWZW) model and the Leznov-Mukhtarov-Parkes (4dLMP) model, respectively. Both models can be space-time actions of N=2 open string theories in (2+2) dimensions and hence solutions of the ASD Yang-Mills equations descrive classical physical objects in the N=2 open string theory. Furthermore, both 4dWZW and 4dLMP models can be obtained from six-dimensional Chern-Simons theory and hence can be one wing of the unification senario of integrable systems (6dCS-->4dCS/ASDYM) described in scope of this workshop.

In this talk, I review basic of ASD Yang-Mills equations and reduced equations in the framework of the Yang-Mills, 4dWZW and 4dLMP models and give soliton solutions of them with resonance processes clarifying difference with the classification theory of KP solitons by Yuji Kodama et al. Finally we discuss perspectives of the unification of integrable systems in the split signature, in noncommutative settings, and in homotopy algebra formulations. (I note that Xianghang Zhang is developing a homotopy algebra formutation of string field theory action of the N=2 open string theory [arXiv:2506.21247].)

This talk is partly based on collabolation with Shan-chi Huang, Hiroaki Kanno (Nagoya) and Shangshuai li (Ningbo) and Da-Jun Zhang (Shanghai): arXiv:2408.16554, arXiv:2501.08250 arXiv:2212.11800 and forthcoming papers.

high energy physicsmathematical physicsnonlinear sciences

Audience: researchers in the topic

Nagoya IAR workshop on Unification of Integrable Systems