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

CHAPTER 5. BRANES 71The embedding <strong>of</strong> the Dp-brane in space-time can be achieved, just as for the string,by first defining a set <strong>of</strong> coordinates σ n with n ∈ {0, . . .,p} on the brane. This allowsus to write the pullback <strong>of</strong> the space-time metric g µν to the brane, and write down thefirst part <strong>of</strong> our action asS p = −T p∫d p+1 σe −Φ √− det g µν (X)∂ α X µ ∂ β X ν , (5.37)where as usual ∂ α,β denotes derivation with respect to σ α,β . We need to include a factore −Φ ∼ gs−1 in this action because this is in fact an open string tree level action.Moving on to the B µν an A µ fields, recall that the Kalb-Ramond coupling <strong>of</strong> a stringwas given by Eq. 4.16 which we repeat for convenience,S B = − 1 ∫2πls2 d 2 σǫ αβ B µν (X)∂ α X µ ∂ β X ν , (5.38)and use that the coupling <strong>of</strong> A µ to the world sheet boundary ∂M is given by∫S A = dτA µ ∂ t X µ . (5.39)∂MIf we want to mimick this behaviour for the D-brane, we are not free to take anycombination as we please. Space-time gauge invariance imposes a choice on us. Thepoint is that S A is invariant under a space-time gauge transformation A µ −→ A µ +∂ µ f,but when applying a space-time gauge transformation B µν −→ B µν + ∂ µ Λ ν − ∂ ν Λ µ wepick up a boundary termδS B = − 1 ∫2πls2 dτΛ µ ∂ τ x µ . (5.40)∂MTo cancel this boundary term and restore gauge invariane, we see that A µ should transformsimultaneously according toA µ −→ A µ − 12πls2 Λ µ . (5.41)From this we can conclude that the only way to inlucde both fields in our action is byusing the combinationB µν + 2πα ′ F µν (5.42)which remains invariant under both symmetries.By considering the case <strong>of</strong> a D2-brane in Minkowski space-time, extended in directionsX 1 and X 2 (i.e. ( σ 0 , σ 1 , σ 2) = ( t, X 1 , X 2) ) <strong>of</strong> which one dimension is compactified,say X 2 , one can show that upon T-dualizing the compactified dimension, the remainingD1-brane is tilted with respect to the X 1 direction, namely the dualized direction X ′2satisfiesX ′2 = 2πα ′ X 1 F 12 , (5.43)