Notes on Noncommutative Instantons

We study in detail the ADHM construction of U(N) instantons on noncommutative Euclidean space-time NCR^4 and noncommutative space NCR^2*R^2. 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 noncommutative space-time. However, it turns out that in the case of NCR^2*R^2 there is a conceptual obstacle in finding non-singular instanton configurations. We also give a simple general argument based on the Corrigan's identity that the topological charge of noncommutative regular instantons on NCR^4 is always an integer. Our results are different from previous ones which reported non-integral topological charge.