From strings to superstrings
In the West, physicists working on dual resonance
models were beginning to understand them in terms of vibrating strings
whose modes of vibration were solutions to the wave equation on the
worldsheet swept out by the string as it propagated in spacetime. The
modes of oscillation all have integer spin, and so these dual resonance
models described bosonic string theory.
In bosonic string theory, negative and zero norm states are eliminated
from the spectrum by fixing the spacetime dimension D=26, which gives
a conformally invariant worldsheet theory
and a Poincaré invariant theory in spacetime.
The infinitedimensional Virasoro algebra
was discovered to be the string worldsheet analog of the Poincaré
algebra in spacetime, except for the last term, called the "central
extension", which arises from quantum effects and is canceled when
the dimension of spacetime is 26.
But in order to describe Nature, a
theory must contain fermions. Physicist Pierre Ramond began to
investigate solutions to the Dirac equation on the string worldsheet,
and found that this led to a much larger symmetry algebra than the Virasoro
algebra, one that included anticommuting operators F_{n} (the
worldsheet analog of the supercharge Q). Ramond discovered the superVirasoro
algebra
or the algebra of a supersymmetric version of conformal invariance,
which is called superconformal invariance.
At
the same time, John Schwarz and André Neveu were working on a
new bosonic string theory that had an anticommuting field with half
integral boundary conditions on the world sheet. They also found a superVirasoro
algebra, but one that looked slightly different from what Ramond had
found. It was soon realized that the theories developed by Ramond and
by Neveu and Schwarz fit together into two sectors of the same theory,
called the RNS model after the initials of the founders. In this
case, the central extension cancels for d=10.
Physicists Gervais and Sakita put the two pictures
together into a theory described by a twodimensional worldsheet action
and noted that this action was invariant under a global (that is, independent
of position) symmetry that transformed bosons into fermions and vice
versa. In other words, string theories with fermions were supersymmetric
theories.
But the supersymmetry they uncovered was confined
to the two dimensional surface swept out by the string as it propagated
through spacetime. The superVirasoro algebra
represents an extension of the worldsheet symmetry
of the theory from conformal invariance
to superconformal invariance. What wasn't
understood yet was whether this worldsheet supersymmetry led to supersymmetry
in the spacetime in which the string propagates. Or in other words,
whether there was an analogous extension of the spacetime
symmetry of the theory from Poincaré
invariance to superPoincaré
invariance.
The biggest problem with bosonic string theory (aside
from the lack of fermions) is that the lowest energy state was a tachyon,
or a particle mode with negative mass squared. This means the vacuum
state of the theory is unstable.
In the midseventies Gliozzi, Scherk and Olive realized
that they could implement a rule to consistently discard certain states
from the RNS model, and after this truncation, known as the GSO projection,
was made on the string spectrum in ten spacetime dimensions, the ground
state was massless, and the theory was tachyon free.
But string theory was out of favor by the midseventies,
and as the number of physicists working in the field dropped, the pace
of work on the theory slowed. It took another five years for John Schwarz
and Mike Green to get together to reformulate the RNS description in
a way such that the spacetime supersymmetry of the theory is visible
and obvious. So in 1981 superstring theory was born.
From supersymmetry to supergravity
One
of the complicating factors in string theory is that one cannot avoid
gravity. And gravity complicates supersymmetry. It changes the supersymmetry
from a global to a local
symmetry.
First
let's discuss ordinary spacetime supersymmetry in a bit more detail.
Remember that the supercharge Q acts on bosonic and fermionic states
as
The
operator Q is a spinor with spin 1/2. A supersymmetric field theory
can be constructed by studying the variation of some field f
by an infinitesimal spinor x
in the Q direction such that
Then the appropriate terms in an action for the field can be constructed
by demanding that the action be invariant under a variation by x.
If
x is a constant spinor, i.e.
not a function of spacetime position x(x),
then the supersymmetry is a global symmetry. One can take the usual
scalar, spinor and gauge fields, such as those present in the Standard
Model, add some number of supercharges Q^{I}, figure out how
each field in the action transforms under a variation by x,
and then figure out what terms to add to the action to cancel the overall
variation variation by x and
make the theory globally supersymmetric. For one supercharge, the theory
is called N=1 supersymmetry. If there are two supercharges, it is N=2
supersymmetry, etc.
The
result of this exercise for a single supercharge is called the Minimal
Supersymmetric Standard Model, or MSSM,
and this will be discussed in the next
section. The new fields in the MSSM have funny names. Higgsinos
and gauginos are the names of the fermionic superpartners of the Higgs
scalars and gauge bosons respectively. The scalar superpartners of quarks
and electrons are called squarks and selectrons. Grand Unified Theories
can also be turned into supersymmetric theories, and this will also
be discussed in the next section.
If
x is not a constant spinor, in
order words x = x(x),
then the picture changes. The loss of global Poincaré invariance
means there is a dynamic spacetime geometry,
i.e. gravity, rather than the rigid
flat spacetime upon which the Standard Model is based. In this case,
instead of mere supersymmetry, we have supergravity.
There is a new gauge field for this new local symmetry, although since
x(x) is a spinor, the new gauge
field has spin 3/2. It's called the gravitino because it is the superpartner
of the graviton. The infinitesimal variation of the gravitino under
the spinor x(x) can be written
Superstring
theories invariably contain gravity. Therefore the low energy effective
field theory that one gets when looking at a string theory at an energy
scale so low that the strings look just like their massless particle
modes is generally a supergravity theory. However, the topic of supergravity
was developed independently from string theory, because eventually particle
theorists began to look for quantum field theories that had larger symmetry
groups than the Standard Model or Grand Unified Theories.
By
the time Green and Schwarz realized that their GSOprojected, tachyonfree
fermionic string theories had spacetime supersymmetry as well as the
worldsheet variety, there was already a community at work understanding
the implications of supersymmetry for particle physics. In 1984, when
Green and Schwarz discovered the anomaly cancellation for Type I superstrings
based on the gauge group SO(32), the most talkedabout candidate for
a unified field theory was a quantum field theory based on N=1 supergravity
in eleven spacetime dimensions. Now both theories are a part of a larger
framework that some people call Mtheory.
If
supersymmetry is a prediction of superstring theory, and whatever larger
theory that may encompass it, then it is important to know:
a.
How is supersymmetry broken to give the nonsupersymmetric world we
see so far?
b.
What are the signs of supersymmetry that might show up in particle physics
experiments?
Next:
Supersymmetry and particle physics >>
