ar X iv : h ep - t h / 02 01 19 6 v 1 2 4 Ja n 20 02 Instanton Number Calculus on Noncommutative R 4

In noncommutative spaces it is unknown whether the Pontrjagin class gives integer, as well as, the relation between the instanton number and Pontrjagin class is not clear. Where we define “Instanton number” by the size of Bα in the ADHM construction. We show the analytical derivation of the noncommuatative U(1) instanton number as an integral of Pontrjagin class (instanton charge) in the Fock space representation. Our approach is for the arbitrary converge noncommutative U(1) instanton solution, and is based on the converge condition and the anti-self dual equation itself. We give the Stokes’ theorem for the number operator representation. The Stokes’ theorem on the noncommutative space shows that instanton charge is given by some boundary sum. Using the anti-self-dual (ASD) condition, we conclude that the instanton charge is equivalent to the instanton number.