The generalized Moyal–Nahm and continuous Moyal–Toda equations

Abstract

We present in detail a class of solutions to the 4D SU(∞) Moyal–anti-self-dual Yang–Mills (ASDYM) equations (an effective 6D theory) that are related to reductions of the generalized Moyal–Nahm equations using the Ivanova–Popov ansatz. The former yields solutions to the ASDYM/SDYM equations for arbitrary gauge groups in four dimensions. A further dimensional reduction of the above effective 6D equations yields solutions to the Moyal–anti-self-dual gravitational equations in four dimensions. The self-dual Yang–Mills/self-dual gravity case requires a separate study. The SU(2) Toda lattice and SU(∞) (continuous) Moyal–Toda lattice equations are derived from the Moyal–Nahm equations. An explicit map taking the Moyal heavenly form (after a rotational Killing symmetry reduction of the Moyal heavenly equations) into the SU(2) Toda lattice field is found. Finally, the generalized Moyal–Nahm equations are conjectured that contain the (continuous) SU(∞) Moyal–Toda lattice equations, after a suitable reduction process. Embeddings of the different types of Moyal–Toda lattice equations into the Moyal–Nahm equations are described.