Instantons on noncommutative R⁴ and projection operators

The noncommutative version of ADHM construction of instantons, which was proposed by Nekrasov and Schwarz, is carefully studied. Noncommutative R⁴ is described by an algebra of operators acting in a Fock space. In the ADHM construction of instantons, one looks for zero-modes of the Dirac-like operator. The feature peculiar to the noncommutative case is that these zero-modes project out some states in the Fock space. The mechanism of these projections is clarified in the case that the gauge group is U(1). In U(N) cases, it is shown in some explicit examples that projections similar to those in the U(1) cases also appear. A physical interpretation of these projections in the IIB matrix model is also discussed.