DBI Analysis of Open String Bound States on Non-compact D-branes

CHAPTER 5. BRANES 72where F 12 is the (12) component <strong>of</strong> the Maxwell field strength living on the D2-branewhich is taken to be constant. The tilting angle is given by θ = tan −1 (2πα ′ F 12 ). Because<strong>of</strong> this, the new pullback from the metric to the brane becomes⎡⎤−1 ∂ 1 X ′2 0 . . . 0∂ 1 X ′2 1 0 . . . 0g αβ =0 0 0 . . . 0(5.44).⎢ . . . .. . ⎥⎣⎦0 0 0 . . . 0resulting in the action for the D1-brane (we ignore the e −Φ factor)∫ √∫ √S D1 ∼ d 2 σ 1 + (∂ 1 X ′2 ) 2 = d 2 σ 1 + (2πα ′ F 12 ) 2 . (5.45)By boosting and rotating the D-brane, one can bring F µν in block-diagonal form, whichallows one to write the Dp-brane generalization <strong>of</strong> the above action as∫ √S Dp ∼ d p+1 σ − det(η µν + 2πα ′ F µν ). (5.46)This action is called the Born-Infeld action. By further applying T-dualizing arguments,one can obtain the full Dirac-Born-Infeld action which also incorporates non-trivialbackgrounds and the Kalb-Ramond field,∫S Dp = −T p d p+1 σe√− −Φ det(g αβ + B αβ + 2πα ′ F αβ ), (5.47)where indices αβ denote the pullbacks <strong>of</strong> the fields to the brane.The explicit form <strong>of</strong> T p can be computed by analysing the exchange <strong>of</strong> a closed loopbetween two parallel Dp-branes, and results inT p =Applying T-duality further results in a recursion relation,1(2π) p l p+1s g s. (5.48)T p−1 = (2πl s )T p . (5.49)On a final note, we point the reader to an alternative derivation <strong>of</strong> the <strong>DBI</strong> action,which consists <strong>of</strong> computing the β-function for the A µ gauge field, and solving itsequation <strong>of</strong> motion. Details can be found in [5], §2.3.3.1.