Instanton Number Calculus on Noncommutative R^4

Details Instanton Number Calculus on Noncommutative R^4 Instanton Number Calculus on Noncommutative R^4

2002-01-24

arxiv.org/abs/hep-th/0201196v4

JHEP 0208 (2002) 028

Physics

High Energy Physics

High Energy Physics - Theory

29 pages, 7 figures, some statements in Sec.4.3 corrected

Scientific paper

10.1088/1126-6708/2002/08/028

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. Here we define ``Instanton number'' by the size of $B_{\alpha}$ in the ADHM construction. We show the analytical derivation of the noncommuatative U(1) instanton number as an integral of Pontrjagin class (instanton charge) with the Fock space representation. Our approach is for the arbitrary converge noncommutative U(1) instanton solution, and is based on the anti-self-dual (ASD) 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 ASD conditions, we conclude that the instanton charge is equivalent to the instanton number.

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