Notes on noncommutative instantons Original Research Article

We study in detail the ADHM construction of U(N) instantons on noncommutative Euclidean space–time RNC4 and noncommutative space RNC2×R2. We point out that the completeness condition in the ADHM construction could be invalidated in certain circumstances. When this happens, regular instanton configuration may not exist even if the ADHM constraints are satisfied. Some of the existing solutions in the literature indeed violate the completeness condition and hence are not correct. We present alternative solutions for these cases. In particular, we show for the first time how to construct explicitly regular U(N) instanton solutions on RNC4 and on RNC2×R2. We also give a simple general argument based on the Corriganʹs identity that the topological charge of noncommutative regular instantons is always an integer.