for a string propagating in flat 26-dimensional spacetime with coordinates
X^{m}(s,t)
can give rise to four different quantum mechanically consistent string
theories, depending on the choice of boundary conditions used to solve
the equations of motion. The choices are divided into two categories:
A. Are the strings open (with free ends)
or closed (with ends joined together
in a loop)?
B. Are the strings orientable (you can
tell which direction you're traveling along the string) or unorientable
(you can't tell which direction you're
traveling along the string)?
There are four different combinations of options,
giving rise to the four bosonic string theories shown in the table below.
Notice in the table that open string theories also contain closed strings.
Why is this? Because an open string can sometimes join its two free
ends and become a closed string and then break apart again into an open
string. In pure closed string theory, the analog of that process does
not occur.
The bosonic string theories are all unstable because
the lowest excitation mode, or the ground state, is a tachyon
with M^{2}=-1/a'. The
massless particle spectrum always includes the graviton,
so gravity is always a part of any bosonic string theory. The vector
boson is similar to the photon of electromagnetism or the gauge fields
of any Yang-Mills theory. The antisymmetric tensor field carries a force
that is difficult to describe in this short space. The strings act as
a source of this field.

Scalar tachyon, massless antisymmetric
tensor, graviton and dilaton

Open (plus closed)

No

Scalar tachyon, massless graviton and
dilaton

Closed

Yes

Scalar tachyon, massless vector boson,
antisymmetric tensor, graviton and dilaton

Closed

No

Scalar tachyon, massless graviton and
dilaton

It's just as well that bosonic string theory is unstable,
because it's not a realistic theory to begin with. The real world has
stable matter made from fermions that satisfy the Pauli Exclusion Principle
where two identical particles cannot be in the same quantum state at
the same time.
Adding fermions to string theory introduces a new
set of negative norm states or ghosts,
to add to the ghost states that come from the bosonic sector described
on the previous page. String theorists learned that all of these bad
ghost states decouple from the spectrum when two conditions are satisfied:
the number of spacetime dimensions is 10, and theory is supersymmetric,
so that there are equal numbers of bosons and fermions in the spectrum.
Fermions have more complicated boundary conditions
than bosons, so unraveling the different possible consistent superstring
theories took researchers quite a bit of doing. The simplest way to
examine a superstring theory is to go to what is called superspace.
In superspace, in addition to the normal commuting coordinates X^{m},
a set of anticommuting coordinates q^{A}
are added. In superstring theories index A runs from 1 to 2 (an additional
spinor index is not shown). The anticommutation relations of the coordinates
are

The options of open vs closed, and oriented Vs unoriented
boundary conditions are still present, but there are also choices involving
fermions that distinguish one superstring theory from another. The superspace
coordinates q^{1} and
q^{2} behave like particles with spin 1/2 and zero mass,
which can only spin two ways -- with the spin axis in the same or opposite
direction as the momentum. This property is called handedness. So q^{1}
and q^{2} can have either
the same or the opposite handedness.
The resulting consistent string theories can be described
in terms of the massless particle spectrum and the resulting number
of spacetime supersymmetry charges, denoted by the letter N
in the table below. None of the theories below suffer from the tachyon
problem that plagues bosonic string theories. All of the theories below
contain gravity.

Superstrings, d=10

Type

Open or closed?

Oriented?

N

Details

I

Open (plus closed)

No

1

Graviton, no tachyon, SO(32)
gauge symmetry, charges are attached to the ends of the strings

IIA

Closed

No

2

Graviton, no tachyon, only
a U(1) gauge symmetry

IIB

Closed

Yes

2

Graviton, no tachyon, no
gauge symmetry

Heterotic E_{8}XE_{8}

Closed

Yes

1

Graviton, no tachyon, E_{8}XE_{8}
gauge symmetry

Heterotic SO(32)

Closed

Yes

1

Graviton, no tachyon, SO(32)
gauge symmetry

A supersymmetric theory has a fermionic partner for
every bosonic particle. The superpartner of a graviton is called a gravitino
and has spin 3/2. All of the theories above contain gravitons and gravitinos.
For open superstrings,
the choices turn out to be restricted by conditions too complicated
to explain here. It turns out that the only consistent theory has unoriented
strings, with q^{1} and
q^{2} having the same
handedness, with an SO(32) gauge symmetry included by attaching little
charges to the ends of the open string. These charges are called Chan
Paton factors. The resulting theory is called Type
I. Closed string oscillations
can be separated into modes that propagate around the string in different
directions, sometimes called left movers and right movers. If q^{1}
and q^{2} have opposite
handedness, then they also have opposite momentum, and hence travel
in opposite directions. Therefore they provide a way to tell which direction
one is traveling around the string. Therefore these strings are oriented.
This is called Type IIA superstring
theory.
Because q^{1}
and q^{2} have opposite
handedness, this theory winds up being too symmetric for real life.
Every fermion has a partner of the opposite handedness, which is not
what is observed in our world, where the neutrino comes in a left-handed
version but not a right-handed version. The real world seems to be chiral,
which means having a preferred handedness for massless fermions. But
Type IIA superstring theory is a nonchiral
theory. There is also no way to add a gauge symmetry to Type IIA superstrings,
so here also the theory fails as a model of the real world.
If q^{1}
and q^{2} have the same
handedness, and the string is oriented, then we get Type
IIB superstring theory. This theory is chiral,
and so there will be massless fermions that don't have partners of the
opposite handedness, as is observed in our world today. However, there
is no way to add a gauge symmetry to the Type IIB theory. So there isn't
a way to include any of the known forces other than gravity.
If q^{1}
and q^{2} have the same
handedness, but the string is unoriented, that turns out to just give
the closed string part of the Type I theory.
This seems to have exhausted all of the obvious options.
But there's actually something crazy that can be done with a closed
string that yields two more important superstring theories.
The left-moving and right-moving modes of a string
can be separated and treated as different theories. In 1984 it was realized
that consistent string theories could be built by combining a bosonic
string theory moving in one direction along the string, with a supersymmetric
string theory with a single q^{1}
moving in the opposite direction. These theories are called heterotic
superstring theories.
That sounds crazy -- because bosonic strings live
in 26 dimensions but supersymmetric string theories live in 10 dimensions.
But the extra 16 dimensions of the bosonic side of the theory aren't
really spacetime dimensions. Heterotic string theories are supersymmetric
string theories living in ten spacetime dimensions.
The two types of heterotic
theories that are possible come from the two types of gauge symmetry
that give rise to quantum mechanically consistent theories. The first
is SO(32) and the second is the more
exotic combination called E_{8}XE_{8}_{.}
The E_{8}XE_{8}_{}
heterotic theory was previously regarded as the only string theory that
could give realistic physics, until the mid-1990s, when additional possibilities
based on the other theories were identified.

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