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In bosonic string theory there are two bold problems. The first one is the presence
of tachyons in the spectrum. The second one is that there are no spacetime fermions.
Here is where superstrings come to the rescue. A superstring is described, despite of
the usual bosonic fields , by
fermionic fields
on the worldsheet . Which satisfy anticommutation rules and
where
the and denote the left and right worldsheet chirality respectively. The action for the
superstring is given by

(25) 
where and are the superpartners of and the tetrad field
respectively.
In the superconformal gauge (
and ) and in lightcone
coordinates it can be reduced to

(26) 
In analogy to the bosonic case, the local dynamics
of the worldsheet metric is manifestly conformal anomaly free at the quantum level if
the critical
spacetime dimension is 10. Thus the string
oscillates in the 8 transverse dimensions. The action (25) is
invariant under: worldsheet supersymmetry, Weyl transformations,
superWeyl transformations, Poincaré transformations and
Worldsheet reparametrizations. The equation of motion for the fields is
the same that in the bosonic case (Laplace equation) and
whose general solution is given by Eqs. (5) or (6). Equation of
motion for the fermionic field is the Dirac equation in two dimensions.
Constraints here are more involved and they are called the superVirasoro
constraints. However in the lightcone gauge, everything simplifies and
the transverse coordinates (eight coordinates) become the bosonic physical degrees
of freedom together with their corresponding supersymmetric partners. Analogously
to the bosonic case,
massless states of the spectrum come into representations of the little group SO(8)
which is a subgroup of SO, while that the massive states lie into representations of the little
group SO.
For the closed string there are two possibilities for the boundary conditions of fermions:
periodic
boundary conditions (Ramond (R) sector)
and
antiperiodic boundary conditions
(NeveuSchwarz (NS) sector)
.
Solutions of Dirac equation satisfying these boundary conditions are

(27) 
where
and are fermionic modes of left and right movers
respectively.
In the case
of the fermions in the R sector
is integer and it is semiinteger in the NS sector.
The quantization of the
superstring come from the promotion of the fields and to operators
whose oscillator variables are operators satisfying the relations
and
where and
stand for commutator and anticommutator respectively.
The zero modes of are diagonal in the Fock space and its
eigenvalue can be identified with its momentum. For the NS sector there is no
fermionic zero modes but they can exist for the R sector and they satisfy a Clifford
algebra
. The Hamiltonian for the
closed superstring is given by
. For the
NS sector , while for the R sector . The mass is given by
with
.
There are five consistent superstring theories: Type IIA, IIB, Type I, SO(32) and
heterotic strings, represented by HO and HE respectively. In what follows of this
section we briefly describe the spectrum
in
each one of them.
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Type II Superstring Theories
In this case the theory consist of closed strings only. They are theories with
spacetime supersymmetry. There are 8 scalar fields (representing
the 8 transverse coordinates to the string) and one WeylMajorana spinor. There are
8 leftmoving and 8 rightmoving fermions.
In the NS sector there is still a tachyon in the ground state. But in the
supersymmetric case this problem can be solved through the introduction of
the called GSO projection. This projection eliminates the tachyon in the NS sector
and it acts in the R sector as a tendimensional spacetime chirality operator. That means
that the
application of the GSO projection operator defines the chirality of a
massless spinor in the R sector. Thus from the left and right
moving sectors, one can construct states in
four different sectors: NSNS,
NSR, RNS and RR. Taking into account the two types of
chirality and one has two possibilities:
The GSO projections on the left and right fermions produce different
chirality in the ground state of the R sector (Type IIA).
GSO projection are equal in left and right sectors and the ground states
in the R sector, have the same chirality (Type IIB). Thus the spectrum for the Type IIA
and
IIB superstring theories is:
 Type IIA
The NSNS sector has a symmetric tensor field (spacetime metric), an
antisymmetric
tensor field
and a scalar field (dilaton). In the RR sector there is a vector field
associated with a 1form (
) and a rank 3
totally
antisymmetric tensor
and by Hodge duality in ten
dimensions also we have , and . In general the RR sector
consist of forms
(where are called RR fields) on the
tendimensional spacetime with even i.e.
. In the NSR and RNS sectors we have
two gravitinos with opposite
chirality and the supersymmetric partners of the mentioned bosonic fields.
 Type IIB
In the NSNS sector Type IIB theory has exactly the same spectrum that of Type IIA theory.
On the RR sector it has a scalar field
(the axion field), an
antisymmetric tensor
field
and a rank 4 totally antisymmetric tensor
whose field strength is selfdual i.e.,
with
. Similar than for the case of Type IIA
theory one has also the Hodge dual fields .
In general,
RR fields in Type IIB theory are given by forms
on the spacetime
with odd
i.e.
. The NSR and RNS sectors do contain two
gravitinos
with the
same chirality and the corresponding fermionic matter.
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20010115