Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1183Here the embedding of the p−brane into dD space–time is given by functionsx µ (σ α ). The index α = 0, . . . , p labels the p + 1 coordinates σ α ofthe p−brane world–volume and the index µ = 0, . . . , d − 1 labels the dcoordinates x µ of the dD space–time. We have defined ∂ α x µ = ∂xµ∂σ.Theαdeterminant operation acts on the (p + 1) × (p + 1) matrix whose rows andcolumns are labelled by α and β. The tension T p is interpreted as the massper unit volume of the p−brane. For a 0−brane, it is just the mass.Let us now specialize to the string, p = 1. Evaluating the determinantgives the Nambu–Goto action (see subsection 6.5.4 above)∫S[x] = −T dσdτ √ ẋ 2 x ′2 − (ẋ · x ′ ) 2 ,where we have defined σ 0 = τ, σ 1 = σ , and ẋ µ = ∂xµ∂τ , x′µ = ∂xµ∂σ . Theabove action is equivalent to the actionS[x, h] = − T ∫d 2 σ √ −hh αβ η2µν ∂ α x µ ∂ β x ν , (6.299)where h αβ (σ, τ) is the world–sheet metric, h = det h αβ , and h αβ = (h αβ ) −1is the inverse of h αβ . The Euler–Lagrangian equations obtained by varyingh αβ areT αβ = ∂ α x · ∂ β x − 1 2 h αβh γδ ∂ γ x · ∂ δ x = 0.In addition to reparametrization invariance, the action S[x, h] has anotherlocal symmetry, called conformal invariance, or, Weyl invariance.Specifically, it is invariant under the replacementh αβ → Λ(σ, τ)h αβ , x µ → x µ .This local symmetry is special to the p = 1 case (strings).The two reparametrization invariance symmetries of S[x, h] allow us tochoose a gauge in which the three functions h αβ (this is a symmetric 2 × 2matrix) are expressed in terms of just one function. A convenient choice isthe conformally flat gaugeh αβ = η αβ e φ(σ,τ) .Here, η αβ denoted the 2D Minkowski metric of a flat world–sheet. However,h αβ is only ‘conformally flat’, because of the factor e φ . Classically,