By using a simple BS.cklund-like transformation which linearizes the GL(N, C) self-dual Yang-Mills equation, an infinite number of local conservation laws for this equation are constructed. In the SL(N, C) case, the currents become trivial, which explains why these currents are not found in SU(N) gauge theory.
AMS subject classifications (1980). 81E13; 53B50.
As is well known, the self-dual Yang-Mills (SDYM) equation, when properly formulated, displays many of the typical characteristics of an 'integrable' system, such as parametric B/icklund transformations [1-4], infinite number of nonlocal conservation laws [5,6], linear system [7,8,6], Painlev6 property [9, 10], etc. An infinite number of local conservation laws were recently constructed by these authors [11] by applying infinitesimal B~cklund transformations [4] on SDYM (which is itself in the form of a local conservation law). However, the problem of finding all conservation laws for SDYM is far from being solved.
Indeed, the search for local currents for SDYM has been long and frustrating. It seems that the best we can do is write densities which are local in the Yang-Brihaye-Pohlmeyer function J [ 12,13,6]. But the latter is nonlocal in the potentials A,, and it is these potentials that are regarded as fundamental in the theory. Even in the J formulation there has been little progress in constructing local conservation laws.
Perhaps one of the reasons is that the search has been mostly confined to SU(N) [or, more generally, SL(N, C)] gauge theory, which requires unit determinant. This restriction ceases to exist in GL(N, C) theory. There the determinant itself becomes a field and, as we show in this Letter, can be used to produce new local conservation laws for the SDYM equation. In the limiting case of SL(N, C), the densities become zero and the conservation laws trivial, which explains why these objects are not found in SU(N) gauge theory.
The construction of the GL(N, C) conservation laws is based on a simple, known B/icklund-like transformation that 'linearizes' the SDYM equation to the Laplace