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

CHAPTER 8. COMPUTATIONS 97and the induced metric g αβ equalsg αβ =⎡⎢⎣⎤−k −1 0 00 (∂ α ρ) 2 + th 2 ρ (∂ α τ) 2 ∂ α ρ∂ β ρ + th 2 ρ∂ α τ∂ β τ ⎥⎦ . (8.4)0 ∂ α ρ∂ β ρ + th 2 ρ∂ α τ∂ β τ (∂ β ρ) 2 + th 2 ρ (∂ β τ) 2Using parameterization invariance to set{α= chρ,β = τ,(8.5)we find the induced metric to be⎡−1⎤0 0kg αβ =1⎢0⎣ α 2 0− 1 ⎥α 2 ⎦ , (8.6)− 10 0α 2and the induced field strengthF αβ =⎡⎢⎣0 0 00 0 F ατ0 −F ατ 0⎤⎥⎦ , (8.7)withF ατ =1√α 2 − 1 F ρτ. (8.8)8.2 Classical solutionTo find the embedding <strong>of</strong> the D2-brane, we use a Lagrange multiplier, as shown in [15],§4.2. We need to minimizeS D2 − √ λ (∫ ) ∫ √ ( ) 1 1F − N = A dαdτ αk k√2 α 2 + 4π2 Fατ2 − √ λ ∫Fdαdτ √ ατk α 2 − 1= √ A ∫ [ √1dαdτ + α 2 π 2 Fατ 2 − λ ]F√ ατ, (8.9)k A α 2 − 1with A = T 2 e −Φ 0. We find thatδ (S − λ . . .)δF ατ= A √k∫[4π 2 α 2 F ατdαdτ √ − λ ]δF. (8.10)1 + 4π 2 α 2 F ατ A