There are two types of particles in nature - fermions
and bosons. A fundamental theory of nature
must contain both of these types. When we include fermions in the worldsheet
theory of the string, we automatically get a new type of symmetry called
supersymmetry which relates bosons
and fermions. Fermions and bosons are grouped together into supermultiplets
which are related under the symmetry. This is the reason for the "super"
A consistent quantum field theory of superstrings exists only in 10
spacetime dimensions! Otherwise there are quantum effects which
render the theory inconsistent or 'anomalous'. In 10 spacetime dimensions
the effects can precisely cancel leaving the theory anomaly free. It may
seem to be a problem to have 10 spacetime dimensions instead of the 4 spacetime
dimensions that we observe, but we will see that in getting from 10 to
4 we actually find some interesting physics.
In terms of weak coupling perturbation theory there appear to be only
five different consistent superstring theories known as Type
I SO(32), Type IIA, Type IIB, SO(32) Heterotic
and E8 x E8 Heterotic.
||E8 x E8 Heterotic
||Type I SO(32)
|10d Gauge groups
||E8 x E8
We see that the Heterotic theories don't contain D-branes. They do however
contain a fivebrane soliton which is not a D-brane. The IIA and IIB theories
also contain this fivebrane soliton in addition to the D-branes. This fivebrane
is usually called the "Neveu-Schwarz fivebrane" or "NS fivebrane".
Type I SO(32):
This is a theory which contains open superstrings. It has one (N=1)
supersymmetry in 10 dimensions. Open strings can carry gauge degrees of
freedom at their endpoints, and cancellation of anomalies uniquely constrains
the gauge group to be SO(32).
It contains D-branes with 1, 5, and 9 spatial dimensions.
This is a theory of closed superstrings which has two (N=2) supersymmetries
in ten dimensions. The two gravitini (superpartners to the graviton) move
in opposite directions on the closed string world sheet and have opposite
chiralities under the 10 dimensional Lorentz group, so this is a non-chiral
theory. There is no gauge group. It contains D-branes with 0, 2, 4, 6,
and 8 spatial dimensions.
This is also a closed superstring theory with N=2 supersymmetry. However
in this case the two gravitini have the same chiralities under the 10 dimensional
Lorentz group, so this is a chiral theory. Again there is no gauge group,
but it contains D-branes with -1, 1, 3, 5, and 7 spatial dimensions.
This is a closed string theory with worldsheet fields moving in one
direction on the world sheet which have a supersymmetry and fields moving
in the opposite direction which have no supersymmetry. The result is N=1
supersymmetry in 10 dimensions. The non-supersymmetric fields contribute
massless vector bosons to the spectrum which by anomaly cancellation are
required to have an SO(32) gauge symmetry.
E8 x E8 Heterotic:
This theory is identical to the SO(32) Heterotic string, except that
the gauge group is E8 X E8 which is the only other gauge group allowed
by anomaly cancellation.
It is worthwhile to note that the E8 x E8 Heterotic string has
historically been considered to be the most promising string theory for
describing the physics beyond the Standard Model. It was discovered
in 1987 by Gross, Harvey, Martinec, and Rohm and for a long time it was
thought to be the only string theory relevant for describing our universe.
This is because the SU(3) x SU(2) x U(1) gauge group of the standard model
can fit quite nicely within one of the E8 gauge groups. The matter
under the other E8 would not interact except through gravity, and might
provide a answer to the Dark Matter problem in astrophysics.
Due to our lack of a full understanding of string theory, answers to questions
such as how is supersymmetry broken and why are there only 3 generations
of particles in the Standard Model have remained unanswered. Most
of these questions are related to the issue of compactification (discussed
on the next page). What we have learned is that string theory contains
all the essential elements to be a successful unified theory of particle
interactions, and it is virtually the only candidate which does so.
However, we don't yet know how these elements specifically come together
to describe the physics that we currently observe.
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