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We consider a D3 brane within a D9 brane in the presence of a B-field whose polarization is transverse to the D3 brane. Using the Green-Schwarz formulation we study the supersymmetry of the configuration by examining the surface terms. It is found that there is residual supersymmetry only when the B-field satisfies a certain condition.
Then we turn to another description of the same system via an N=1 D=10 gauge theory. This is in the spirit of the works of Witten and Douglas who observed that the instanton solutions of a Dp brane world volume theory can be identified as D(p-4) branes within the Dp branes. In the present case the gauge theory becomes non-commutative (NC) in its six dimensions due to the presence of the B-field. Using the NC techniques one can write down, in abstract notations, the soliton solutions of the six dimensinal NC gauge theory, which can be uplifted to ten dimensions. We show that the supersymmetry preserved by the solutions is the same as before.
Therefore the result can be viewed as a generalization of the work of Witten and Douglas.
M. Mihailescu, I. Y. Park and T. A. Tran, “D-branes as solitons of an N=1, D=10 non-commutative gauge theory,” Phys. Rev. D 64, 046006 (2001) [hep-th/0011079]
E. Witten, “BPS bound states of D0-D6 and D0-D8 systems in a B-field,” hep-th/0012054.
