Feynman Path Integral Formulation

1.11 Supersymmetric Strings 49
One way of introducing fermionic degrees of freedom is to look for a supersymmetric
extension of the bosonic string action of Eq. (1.195) (Brink and Schwarz,
1977),
∫
I [g, χ,X,ψ] = 1 2
d 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 twocomponent
Majorana spinor, χ a (σ,τ) a spin- 3 2
gravitino field (a two-component
Majorana spinor and a world-sheet vector), and ea α (σ,τ) a zweibein for the metric
g ab , such that √ g = e. In order to ensure local supersymmetry, g ab and χ a have to be
treated as independent variables. The action of Eq. (1.229) now has a much larger
invariance, which consists of the local supersymmetry transformations
δX 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 (or
conformal) symmetry, already present in the bosonic string,
δX μ = 0 δψ μ = − 1 2 Λψμ
δea 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 symmetry
δX μ = 0 δψ μ = 0
δea α = 0 δχ a = iγ a η , (1.232)
with η(x) an arbitrary Majorana spinor. The resulting invariance under ε, Λ and η
transformations is denoted as superconformal.
Just as conformal invariance of the bosonic string restricted the Virasoro algebra
for the quantities L m , here the corresponding superconformal symmetry will restrict
the structure of the commutation relations for the quantities L m , F m and G r .One
now finds that the theory is ghost-free provides d = 10 and a = 1/2 in the bosonic
sector, 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 choice
g 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 was
done in the bosonic case. This gives an effective action for the e and χ fields just
defined,
∫
e −S(e,χ) = [dψ][dX]e −I [g,χ,X,ψ] . (1.234)