Feynman Path Integral Formulation

1.11 Supersymmetric Strings 49One way of introducing fermionic degrees of freedom is to look for a supersymmetricextension of the bosonic string action of Eq. (1.195) (Brink and Schwarz,1977),∫I [g, χ,X,ψ] = 1 2d 2 σ e[ g ab ∂ a X μ ∂ b X μ + ¯ψ μ iγ a ∂ a ψ μ+ ¯χ a γ b γ a (∂ b X μ + 1 2 χ b ψ μ )ψ μ ] . (1.229)Here again X μ (σ,τ) (μ = 1...d) parametrizes the surface, ψ μ (σ,τ) is a twocomponentMajorana spinor, χ a (σ,τ) a spin- 3 2gravitino field (a two-componentMajorana spinor and a world-sheet vector), and ea α (σ,τ) a zweibein for the metricg ab , such that √ g = e. In order to ensure local supersymmetry, g ab and χ a have to betreated as independent variables. The action of Eq. (1.229) now has a much largerinvariance, which consists of the local supersymmetry transformationsδX μ = ¯ε ψ μ δψ μ = −iγ a ε (∂ a X μ − ¯ψ μ χ a )δea α = −2i ¯ε γ α χ a δχ a = ∇ a ε , (1.230)with ε(x) an arbitrary fermionic function. In addition there is the local Weyl (orconformal) symmetry, already present in the bosonic string,δX μ = 0 δψ μ = − 1 2 Λψμδea α = Λ ea α δχ a = 1 2 Λχ a , (1.231)with Λ(x) a real function, as well as the purely fermionic local symmetryδX μ = 0 δψ μ = 0δea α = 0 δχ a = iγ a η , (1.232)with η(x) an arbitrary Majorana spinor. The resulting invariance under ε, Λ and ηtransformations is denoted as superconformal.Just as conformal invariance of the bosonic string restricted the Virasoro algebrafor the quantities L m , here the corresponding superconformal symmetry will restrictthe structure of the commutation relations for the quantities L m , F m and G r .Onenow finds that the theory is ghost-free provides d = 10 and a = 1/2 in the bosonicsector, and a = 0 in the fermionic one.Alternatively one can treat the theory using covariant functional integral methods.First one needs to fix the superconformal gauge by the choiceg ab (σ,τ) =e 2 (σ,τ) δ ab χ a (σ,τ) =γ a χ(σ,τ) , (1.233)after which one can integrate out the ψ and X fields (Polyakov, 1981a,b) as wasdone in the bosonic case. This gives an effective action for the e and χ fields justdefined,∫e −S(e,χ) = [dψ][dX]e −I [g,χ,X,ψ] . (1.234)