Anti-Self-Dual Yang-Mills Equations and a Unification of Integrable Systems

Abstract: Anti-self-dual Yang-Mills (ASDYM) equations have played important roles in quantum field theory (QFT), geometry and integrable systems for more than 50 years. In particular, instantons, global solutions of them, have revealed nonperturbative aspects of QFT ['t Hooft,...] and have given a new insight into the study of the four-dimensional geometry [Donaldson]. Furthermore, it is well known as the Ward conjecture that the ASDYM equations can be reduced to many integrable systems, such as the KdV eq. and Toda eq. Integrability aspects of them can be understood from the viewpoint of the twistor theory [Mason-Woodhouse,...]. The ASDYM equation is realized as the equation of motion of the four-dimensional Wess-Zumino-Witten (4dWZW) model in Yang's form. The 4dWZW model is analogous to the two dimensional WZW model and possesses aspects of conformal field theory and twistor theory [Losev-Moore-Nekrasov-Shatashvili,...].

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 this context with the split signature.

In this talk, I would like to discuss integrability aspects of the ASDYM equation and construct soliton/instanton solutions of it by the Darboux/ADHM procedures, respectively. We calculate the 4dWZW action density of the solutions and found that the soliton 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 soliton solutions would describe a new-type of physical objects (3-brane) 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.

mathematical physicsdynamical systemsquantum algebrarepresentation theorysymplectic geometry

Audience: general audience

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BIMSA Integrable Systems Seminar

Series comments: The aim is to bring together experts in integrable systems and related areas of theoretical and mathematical physics and mathematics. There will be research presentations and overview talks.

Audience: Graduate students and researchers interested in integrable systems and related mathematical structures, such as symplectic and Poisson geometry and representation theory.

The zoom link will be distributed by mail, so please join the mailing list if you are interested in attending the seminar.

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