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

CHAPTER 2. BOSONIC STRINGS 162.3 Equations <strong>of</strong> motionFrom Eq. 2.11, we obtain the equations <strong>of</strong> motion for X µ .δS = T ∫ {}d 2 σ∂Ẋ22 ∂Ẋ (∂ τδX) − ∂X′2∂X ′ (∂ σ δX)= T ∫) ( )d 2 (σ{2∂ τ(ẊδX − 2 ∂ τ Ẋ δX − 2∂ σ X ′ δX ) + 2 ( ∂ σ X ′) }δX2∫= T d 2 σ ( ∂σX 2 − ∂τX 2 ) δX, (2.12)from which we conclude that∂ 2 X µ∂σ 2 − ∂2 X µ∂τ 2 = ∂ α ∂ α X µ = 0. (2.13)We thus obtain a two-dimensional wave equation for every map X µ .2.4 <strong>Bound</strong>ary conditionsLooking back at Eq. 2.12, we see that we have assumed, as is usual when varying anaction, that the total derivatives vanish. The total time (actually, τ) derivative is, asis customary, eliminated by choosing initial and final states for which the variations arezero, i.e.δX µ (σ, τ i ) = δX µ (σ, τ f ) = 0, (2.14)with τ i and τ f representing the initial and final time respectively. This leaves only theσ derivative, ∫− T dτ [ X µδX ′ µ | σ=π − X µδX ′ µ ]| σ=0 = 0, (2.15)in which we further assume that σ ∈ [0, π], for open and closed strings alike (a differencewe will quantify in a second). Now, there are several ways to make this condition fit,and these different possibilities mathematically define different types <strong>of</strong> strings.2.4.1 Closed stringsA first possibility is to assume thatX µ (σ, τ) = X µ (σ + π, τ) . (2.16)This would <strong>of</strong> course imply that the string is periodic, or closed, and as a consequence,since we have just identified points whose σ values differ by an integer multiple <strong>of</strong> π,that alsoδX µ (σ, τ) = δX µ (σ + π, τ). (2.17)Hence, this condition defines a closed string.