Dirac constraint analysis and symplectic structure of anti-self-dual Yang–Mills equations

Pramana

Y Sucu³  Y Nutku⁴  Z Can¹  U Camci1²  D Yazici¹ 

[1] Department of Physics, Yildiz Technical University, 80270 Istanbul, Turkey$$;Department of Physics, Art and Science Faculty, Canakkale Onsekiz Mart University, 17100 Canakkale, Turkey$$;Department of Physics, Akdeniz University, 07058 Antalya, Turkey$$;Feza Gursey Institute, P.O. Box 6, Çengelkoy 81200 Istanbul, Turkey$$

关键词: Integrable equations in physics;    integrable field theories;    Dirac constraint analysis;    symplectic structure;    anti-self-dual Yang–Mills equations.;   

DOI  :  

学科分类：物理（综合）

来源: Indian Academy of Sciences

【 摘 要 】

We present the explicit form of the symplectic structure of anti-self-dual Yang–Mills (ASDYM) equations in Yang's 𝐽- and 𝐾-gauges in order to establish the bi-Hamiltonian structure of this completely integrable system. Dirac's theory of constraints is applied to the degenerate Lagrangians that yield the ASDYM equations. The constraints are second class as in the case of all completely integrable systems which stands in sharp contrast to the situation in full Yang–Mills theory. We construct the Dirac brackets and the symplectic 2-forms for both 𝐽- and 𝐾-gauges. The covariant symplectic structure of ASDYM equations is obtained using the Witten–Zuckerman formalism. We show that the appropriate component of the Witten–Zuckerman closed and conserved 2-form vector density reduces to the symplectic 2-form obtained from Dirac's theory. Finally, we present the Bäcklund transformation between the 𝐽- and 𝐾-gauges in order to apply Magri's theorem to the respective two Hamiltonian structures.

【 授权许可】

Unknown