String Theory and M-Theory

5
Strings with space-time supersymmetry
After the GSO projection the spectrum of the ten-dimensional RNS superstring
has an equal number of bosons and fermions at each mass level. This is
strong circumstantial evidence that the theory has space-time supersymmetry,
even though this symmetry is extremely obscure in the RNS formalism.
This suggests that there should exist a different formulation of the theory
in which space-time supersymmetry becomes manifest. This chapter begins
by describing the Green–Schwarz (GS) formulation of superstring theory,
which achieves this.
Since the bosonic string theory is defined in terms of maps of the string
world sheet into space-time, a natural supersymmetric generalization to consider
is based on maps of the string world sheet into superspace, so that the
basic world-sheet fields are
X µ (σ, τ) and Θ a (σ, τ). (5.1)
This is the approach implemented in the GS formalism.
The GS formalism has advantages and disadvantages compared to the
RNS formalism. The basic disadvantage of the GS formalism stems from the
fact that it is very difficult to quantize the world-sheet action in a way that
maintains space-time Lorentz invariance as a manifest symmetry. However,
it can be quantized in the light-cone gauge. This is sufficient for analyzing
the physical spectrum. It is also sufficient for studying tree and one-loop
amplitudes. An advantage of the GS formalism is that the GSO projection
is automatically built in without having to make any truncations, and
space-time supersymmetry is manifest. Moreover, in contrast to the RNS
formalism, the bosonic and fermionic strings are unified in a single Fock
space.
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