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

CHAPTER 2. BOSONIC STRINGS 30World SheettheoryConformal GaugeQuantizeLigth-ConeGaugeVirasoroConstraintsQuantize<strong>String</strong>sFigure 2.1: Conformal vs. light-cone quantization.with ξ α an infinitesimal reparameterization parameter and Λ and infinitesimal Weylrescaling parameter, leave our physics, and in particular our previous gauge choice,unchanged. To see this, first note that we could rewrite the local world sheet symmetries(Eqs. 2.7 and 2.8) in infinitesimal form asδh αβ = ξ γ ∂ γ h αβ − ∂ γ ξ α h γβ − ∂ γ ξ β h αγ , (2.108)δh αβ = Λh αβ , (2.109)where the first line represents an infinitesimal reparameterization, and the second linean infinitesimal Weyl rescaling. Replacing h αβ with η αβ , we see that we could chooseparameters such that Eq. 2.107 is indeed satisfied, i.e. we could perform a reparameterizationthat in fact has the same effect as a Weyl rescaling, and then cancel thisrescaling by applying the “inverse” Weyl rescaling.Defining ξ ± = ξ 0 ± ξ 1 , and using our old friends σ ± , Eq. 2.107 tells us thatwhich can be solved by stating that∂ + ξ + = ∂ − ξ + = 0,∂ − ξ − = ∂ + ξ − = 0,ξ + = ξ + ( σ +) ; ξ − = ξ − ( σ −) . (2.110)This implies that this residual gauge freedom allows us to apply reparameterizations <strong>of</strong>the formσ ± −→ ˜σ ± ( σ ±) , (2.111)and in particular τ = 1 2 (σ+ + σ − ) and σ = 1 2 (σ+ − σ − ) get transformed according to