How many String theories are there?

How many String theories are there?

The bosonic string world sheet action

Bosonic string world sheet action

for a string propagating in flat 26-dimensional spacetime with coordinates Xm(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 M2=-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.

Bosonic strings, d=26
Open (plus closed)YesScalar tachyon, massless antisymmetric tensor, graviton and dilaton
Open (plus closed)NoScalar tachyon, massless graviton and dilaton
ClosedYesScalar tachyon, massless vector boson, antisymmetric tensor, graviton and dilaton
ClosedNoScalar 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 Xm, a set of anticommuting coordinates qA 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

Anticommutation relations

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 q1 and q2 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 q1 and q2 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
TypeOpen or closed?Oriented?NDetails
IOpen (plus closed)No1Graviton, no tachyon, SO(32) gauge symmetry, charges are attached to the ends of the strings
IIAClosedNo2Graviton, no tachyon, only a U(1) gauge symmetry
IIBClosedYes2Graviton, no tachyon, no gauge symmetry
Heterotic E8XE8ClosedYes1Graviton, no tachyon, E8XE8 gauge symmetry
Heterotic SO(32)ClosedYes1Graviton, 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 q1 and q2 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 q1 and q2 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 q1 and q2 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 q1 and q2 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 q1 and q2 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 q1 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 E8XE8. The E8XE8 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.

Recommended Books:

1. String Theory and M-Theory (A Modern Introduction)

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2. The Little Book of String Theory

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