Abstract

Anti-self-dual Yang-Mills (ASDYM) equations have played important roles in guantum field theory(QFT), geometry and integrable systems for more than 50 years. In particular, instantons, globalsolutions of them, have revealed nonperturbative aspects of QFT ['t Hoof....] and have given a newinsight into the study of the four-dimensional geometry iDonaldsonl. Furthermore, it is well knownas the Ward conjecture that the ASDYM equations can be reduced to many integrable systemssuch as the KdV eg. and Toda eq. Integrability aspects of them can be understood from theviewpoint of the twistor theory [Mason-Woodhouse.,... The ASDYM eguation is realized as theeguation of motion of the four-dimensional Wess-Zumino-Witten (4dWZW) model in Yang's formThe 4dWZW model is analogous to the two dimensional WZW model and possesses aspects ofconformal field theory and twistor theory [Losev-Moore-Nekrasov-Shatashvili,...].On the other hand, 4d Chern-Simons (CS) theory has connections to many solvable models suchas 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" [CostelloBittleston-Skinnerl.

This suggests a nontrivial duality correspondence between the 4dWZW model and the 4dCstheory. We note that the Ward conjecture holds mostly in the split signature (+,+.) and then the4dWZW model describes the open N=2 string theory in the four-dimensional space-time. Hence aunified theory of integrable systems (6dCS-->4dCS/4dWZW) can be proposed in this context withthe split signature.

In this talk, l would like to discuss integrability aspects of the ASDYM equation and constructsoliton/instanton solutions of it by the Darboux/ADHM procedures, respectively. We calculate the4dWZW action density of the solutions and found that the soliton solutions behaves as the KP-typesolitons, that is, the one-soliton solution has localized action (energy) density on a 3d hyperplane in4-dimensions (soliton wall) and the N-soliton solution describes N intersecting soliton walls withphase shifts. Our soliton solutions would describe a new-type of physical objects (3-brane) in theN=2 string theory. lf time permits, l would mention reduction to lower-dimensions and extension tononcommutative spaces

This talk is based on our works: [arXiv:2212.11800,2106.01353, 2004.09248, 2004.01718] andforthcoming papers.