Lectures on String Theory

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to pick up the signs for ¯Λ (αβ) such that relation (6.2) is satisfied. When it is possible,
the corresponding manifold M is said to admit the spin structure and it is called the
spin manifold. Note that M may admit several inequivalent spin structures.
String theory contains world-sheet fermions and therefore it can be defined on
spin-manifolds. It turns out that in 2 and 3dim any orientable manifold is the spin
manifold. It is not so in 4dim and higher. Finally, we state without proof that on a
Riemann surface of genus g there are 2 2g inequivalent spin structures.
6.2 Superstring action and its symmetrices
The superstring action is based on two multiplest of 2dim supersymmetry. The first
one is the matter multiplet (X µ , ψ µ , F µ )
.
Here X µ is a bosonic field of the Polyakov string, ψ µ is the Majorana spinor in 2dim
and F µ is the real scalar, which is an auxiliary field to guarantee the equality of
the bososnic and fermionic degrees of freedom off-shell (i.e. without usage of the
equations of motion). Also the target space-time index µ is just a label so that for
µ = 0, . . . , d−1 we have d matter multiplets. the second multiplet is the supergravity
multiplet ( )
e a α, χ α , A .
Here χ α is the gravitino, i.e. the Majorana spinor which is also a vector of the 2dim
world-sheet. The field A is an auxiliary scalar field which is needed to guarantee the
equality of the bososnic and fermionic degrees of freedom off-shell. We note that the
kinetic term for the gravitino in any dimension is ¯χ α γ αβγ D β χ γ and it is absent in
two dimensions (because there are only two ρ -matrices while the anti-symmetrized
product γ αβγ requires at least three to exist). Finally we introduce e ≡ |dete a α| = √ h.
The superstring action is a generalization of the bosonic Polyakov action to include
the include the world-sheet fermionic degrees of freedom in the supersymmetric
way. It has the following structure
S = − 1
8π
∫
d 2 σe
(h αβ ∂ α X µ ∂ β X µ + 2i ¯ψ µ ρ α ∂ α ψ µ − i¯χ α ρ β ρ α ψ µ (
∂ β X µ − i 4 ¯χ βψ µ
) ) .
This action has five local symmetries
1. Local supersymmetry. Let ɛ is the Majorana spinor. We consider it as the
infinitezimal parameter of local supersymmetry transformations
δ ɛ X µ = i¯ɛψ µ , δ ɛ e a α = i 2¯ɛρa χ α ,
δ ɛ ψ µ = 1 (
2 ρα ∂ α X µ − i 2 ¯χ αψ
)ɛ µ , δ ɛ χ α = 2D α ɛ .