Deformation quantization of instantons on R4

We study noncommutative instanton solutions which are given as deformation of instanton solutions on commutative Euclidean 4-space. The ingredients include observations of the instanton numbers for noncommutative instanton solutions and the spinor zero modes of the Dirac operator in this noncommutative instanton background. Here, we modify the index of the Dirac operator on the noncommutative space slightly. This derives noncommutative ADHM equations which coincide with the ones introduced by Nekrasov and Schwarz. A one-to-one correspondence between the instantons on noncommutative R-4 and ADHM data up to noncommutative gauge equivalence is shown. Deformation quantization of instantons in U(N) gauge theories is also discussed.