Dirac constraint analysis and symplectic structure of anti-self-dual Yanga€“Mills equations

摘要

We present the explicit form of the symplectic structure of anti-self-dual Yanga€“Mills (ASDYM) equations in Yang's e???- and e???-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 Yanga€“Mills theory. We construct the Dirac brackets and the symplectic 2-forms for both e???- and e???-gauges. The covariant symplectic structure of ASDYM equations is obtained using the Wittena€“Zuckerman formalism. We show that the appropriate component of the Wittena€“Zuckerman closed and conserved 2-form vector density reduces to the symplectic 2-form obtained from Dirac's theory. Finally, we present the B?¤cklund transformation between the e???- and e???-gauges in order to apply Magri's theorem to the respective two Hamiltonian structures.