Domain Wall in MQCD and Supersymmetric Cycles in Exceptional Holonomy Manifolds

It was conjectured by Witten that a BPS-saturated domain wall exists in the M-theory fivebrane version of QCD (MQCD) and can be represented as a supersymmetric three-cycle in the sense of Becker et al with an appropriate asymptotic behavior. We derive the differential equation which defines an associative cycle in $G_2$ holonomy seven-manifold corresponding to the supersymmetric three-cycle and show that it contains a sum of the Poisson brackets. We study solutions of the differential equation with prescribed asymptotic behavior.