Lectures on String Theory

– 101 –
transformations as the spin-vector: 19
δ l χ a = −lɛ b
a χ b + 1 2 l¯ρχ a .
Condition ρ a χ a = 0 remains invariant under these transformations, i.e. ρ a δ l χ a = 0.
Decomposition of the gravitino into the ρ-trace and ρ-traceless part is decomposition
into two irreducible representations of the Lorentz group corresponding to helicities
±3/2 and ±1/2 respectively. This decomposition is orthogonal w.r.t. the scalar
product (φ|ψ) = ∫ d 2 σ ¯φ α ψ α .
The local supersymmetry transformation for the gravitino filed can be also decomposed
into the traceless- and the trace-parts:
Here we defined the operator
δ ɛ χ α = 2D α ɛ = 2(Πɛ) α + ρ α ρ β D β ɛ
} {{ }
trace part
(Πɛ) α = 1 2 ρβ ρ α D β ɛ , =⇒ ρ α (Πɛ) α = 0 .
Locally one can show that there always exists a spinor κ such that ˜χ α = ρ β ρ α D β κ.
Comparing this with the supersymmetry transformation for χ α we conclude that κ
can always be eliminated (locally!) by a supersymmetry variation. The possibility
to eliminate κ globally depends on the existence of a globally defined spinor ɛ which
solves the equation
(Πɛ) α = τ α
for arbitrary τ α satisfying the condition ρ α τ α =0. Global solvability of the last expression
relies on the absence of zero modes of the operator Π † : (Π † τ) = −2D α τ α .
This equation is the supercousin of the bosonic equation
(P ξ) αβ = t αβ
whose global solvability relies on the absence of zero modes of P † . According to our
discussion of the bosonic case it makes sense to call
and also
19 The element ɛ b
a
dim kerP † = moduli
dim kerΠ † = supermoduli
dim kerP = conformal Killing vectors
dim kerΠ = conformal Killing spinors .
( ) ( −1 0 0 1
= η ac ɛ cb is the following matrix ɛa b =
0 1 −1 0
)
=
( ) 0 − 1
.
−1 0