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

CHAPTER 8. COMPUTATIONS 98Hence,Using Eq. 8.8, we are finally left withwhere we have defined C = λ2πA .(4π 2 α 2 ) 2 Fατ 2 = λ2 (1 + 4π 2A 2 α 2 Fατ2 )⇓]4π 2 α 2 Fατ[4π 2 2 α 2 − λ2A 2 = λ2A 2⇓4π 2 α 2 Fατ 2 λ 2=4π 2 α 2 A 2 − λ 2⇓λ2πF ατ =α √ (8.11)4πα 2 A 2 − λ2. 2πF ρτ = C√ α 2 − 1α √ α 2 − C 2 =8.3 Field strength variation√ C tanh ρ(8.12)chρ 2 − C2, Since our D2-brane is spread out over the entire cigar, we cannot perturb it in the ρand τ directions. We can however perturb the Maxwell field that lives on it. Therefore,considerA µ −→ A µ + δα µ (t, ρ, τ). (8.13)The variation <strong>of</strong> the field strength thus becomesDenoting ˜F ατ = F ατ + δF ατ , we find that⎡−1kg αβ + 2π ˜F αβ =⎢⎣− 2π∂ tα ρ√α 2 − 1Up to second order, we find that− det(g αβ + 2π ˜F)αβδF ρτ = ∂ ρ α τ − ∂ τ α ρ = f. (8.14)2π∂ t α ρ√α 2 − 11α 2 − 1−2π∂ t α τ − 2π √α 2 − 1 [F ρτ + f]= 1α 2 k − 4π2 (∂ t α τ ) 2α 2 − 14π 2k (α 2 − 1)2π∂ t α τ2π√α 2 − 1 [F ρτ + f]α 2 − 1α 2⎤. (8.15)⎥⎦− 4π2 (∂ t α ρ ) 2α 2 +[F2ρτ + 2F ρτ f + f 2] . (8.16)