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Supersymmetry to the rescue?

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 infinite-dimensional Virasoro algebra

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 Fn (the worldsheet analog of the supercharge Q). Ramond discovered the super-Virasoro algebra

Super-Virasoro algebra, Ramond sector

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 super-Virasoro 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 two-dimensional 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 super-Virasoro 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 super-Poincaré 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 mid-seventies 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 mid-seventies, 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

Supercharge operator

   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

Supersymmetry variation

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 QI, 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

Gravitino variation

   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 GSO-projected, tachyon-free 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 talked-about 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 M-theory.
   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 non-supersymmetric 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 >>

String Theory Origins of Supersymmetry by John H. Schwarz
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A brief history // The Standard Model // Unification // Supersymmetry to the rescue? // Extra dimensions


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