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

CHAPTER 8. COMPUTATIONS 998.4 Regularized LagrangianBefore varying the action containing the perturbed field strength, and ultimatly computingthe equations <strong>of</strong> motion for the fluctuation, we note that we will again need toinvoke the Lagrange multiplier. As such, we first computeL <strong>DBI</strong> −λf√k (α 2 − 1) . (8.17)To this end, the first thing to do is to expand the Lagrangian density,√L <strong>DBI</strong> = A −α 2 det(g αβ + 2π ˜F)αβ{ 1= Ak − 4π2 α 2α 2 − 1 (∂ tα τ ) 2 − 4π 2 (∂ t α ρ ) 2 + 4π2 α 2k (α 2 − 1) F ρτ2}+ 8π2 α 2k (α 2 − 1) F ρτf + 4π2 α 2k (α 2 − 1) f2= A√(α 2 − 1) + 4π 2 α 2 F 2 ρτk (α 2 − 1){4π 2 α 2 k1 −(α 2 − 1) + 4π 2 α 2 Fρτ2 (∂ t α τ ) 24π 2 ( α 2 − 1 ) k−(α 2 − 1) + 4π 2 α 2 Fρτ2 (∂ t α ρ ) 2 8π 2 α 2+(α 2 − 1) + 4π 2 α 2 Fρτ2 F ρτ f4π 2 α 2+(α 2 − 1) + 4π 2 α 2 Fρτ2 f}. 2 (8.18)For the expansion, we find up to second order:√{(α 2 − 1) + 4π 2 α 2 Fρτ2 2π 2 α 2 kL <strong>DBI</strong> ≈ Ak (α 2 1 −− 1) (α 2 − 1) + 4π 2 α 2 Fρτ2 (∂ t α τ ) 22π 2 ( α 2 − 1 ) k−(α 2 − 1) + 4π 2 α 2 Fρτ2 (∂ t α ρ ) 2 4π 2 α 2+(α 2 − 1) + 4π 2 α 2 Fρτ2 F ρτ f[]2π 2 α 28π 4 α 4 Fρτ2 +(α 2 − 1) + 4π 2 α 2 Fρτ2 − ( )(α 2 − 1) + 4π 2 α 2 Fρτ2 2f}. 2 (8.19)Looking back at Eq. 8.17, we see we only need to consider the term ∼ F ρτ in thisexpansion when computing the effect <strong>of</strong> the Lagrange multiplier. We will want bothterms to annul each other, as otherwise the equations <strong>of</strong> motion will contain constantterms (i.e. without f) we do not want. Anticipating this result, in order not to clutterpages with unreadable derivations, we work out the ∼ F ρτ term when filling in the