Four-dimensional Wess-Zumino-Witten (4dWZW) models are analogous to the two dimensional WZW models and possesses aspects of conformal field theory and twistor theory [Losev-Moore-Nekrasov-Shatashvili, Inami-Kanno-Ueno-Xiong,...]. Equation of motion of the 4dWZW model is the Yang equation which is equivalent to the anti-self-dual Yang-Mills (ASDYM) equation. It is well known as the Ward conjecture that the ASDYM equations can be reduced to many classical integrable systems, such as the KdV eq. and Toda eq. [Ward, Mason-Woodhouse,...]. On the other hand, 4d Chern-Simons (CS) theory has connections to many solvable models such as spin chains and principal chiral models [Costello-Witten-Yamazaki, ...]. These two theories (4dCS and 4dWZW) have been derived from a 6dCS theory like a ``double fibration'' [Costello, Bittleston-Skinner]. This suggests a nontrivial duality correspondence between the 4dWZW model and the 4dCS theory. We note that the Ward conjecture holds mostly in the split signature (+,+,−,−) and then the 4dWZW model describes the open N=2 string theory in the four-dimensional space-time. Hence a unified theory of integrable systems (6dCS-->4dCS/4dWZW) can be proposed in the split signature.
In this talk, I would like to discuss the soliton/instanton solutions of the 4dWZW model. We calculate the 4dWZW action density of the soliton solutions and found that the solutions behaves as the KP-type solitons, that is, the one-soliton solution has localized action (energy) density on a 3d hyperplane in 4-dimensions (soliton wall) and the N-soliton solution describes N intersecting soliton walls with phase shifts. Our solutions would describe a new-type of physical objects in the N=2 string theory. If time permits, I would mention reduction to lower-dimensions and extension to noncommutative spaces.
This talk is based on our works: [arXiv:2212.11800, 2106.01353, 2004.09248, 2004.01718] and forthcoming papers.
