# SUSY's Critical role in String Theory

In many popsci articles it is claimed that String Theory (ST) birthed SUSY. Yet ST was originally invented as a bosons-only theory, that later on brought fermions into the fold. This was only possible After SUSY was incorporated. Hence the old moniker, Superstrings. My understanding is that string theory (ST) is a house of cards' i.e., collapses & dies if SUSY is falsified. Is this true?

-
Essentially a duplicate of physics.stackexchange.com/q/6438 . Related: physics.stackexchange.com/q/9337 – Qmechanic Feb 14 '12 at 15:26
You could have a look at this summer student lecture: indico.cern.ch/getFile.py/… . I do not think that ST was a bosons only theory. – anna v Feb 14 '12 at 15:33
Anna thank U for that beautiful PDF ! In 1971, Ramond, Neveu, & Schwarz were the first to incorporate fermions into purely bosonic string theory via SUSY, rendering it Superstring theory'. SUSY had been discovered by Gelfand & Likhtman the same yr, but not applied to bosonic string theory. – Jimbo Feb 17 '12 at 18:50
Hi Jimbo, be careful not to confuse "low-scale" SUSY (superpartners at the TeV scale that we might see at particle colliders) with the role of SUSY in the different string theories. You cannot exclude the possibility of string theory at the Planck scale if we just have the Standard Model at the LHC - the latter is consistent with the former. – Vibert May 12 '13 at 14:48

It is crucially important to distinguish between local supersymmetry -- supergravity --and global supersymmetry -- a solution to supergravity which has at least one Killing spinor. What the LHC looked for is the latter. What string theory predicts is the former.

The remarkable story of the superstring goes like this:

First, one should observe that an ordinary spinning particle (e.g. an electron or a quark) in its worldline formulation already has local supersymmetry. Essentially, in 1+0 dimensions there is "not so much room" and putting both bosons and fermions on the worldline in the evident way just happens to enjoy (1+0)-dimensional local supersymmetry, whether one intends this or not, see the references here.

So local worldvolume supersymmetry has been experimentally observed since 1922, when Stern-Gerlach observed spin.

But the worldline supersymmetry of spinning particles turns out to have no implication on their effective second quantized background theories. This changes as we move up in dimension to the string.

Second, mimicking the worldline formulation of the spinning particle and increasing to worldsheet dimenion by one to 1+1, one obtains what was originally indeed called the spinning string, a worldsheet action with bosons and fermions. The latter are such that they induce fermions in the second quantized effective target space theory, which is what is necessary to match observation.

Now, as before for the particle, it turns out that the evident action involving bosons and fermions on the worldsheet already happens to be supersymmetric, whether one intends it or not, there is still not "too much room" in these low dimensions, in a sense. That is how supersymmetry was discovered in the West: people noticed that the "spinning string" worldsheet action just so happens to enjoy a super-extension of the local Poincare-symmetry. Ever since the spinning string is known as the superstring.

And now a miracle happens. As opposed to the spinning particle, the local worldsheet supersymmetry on the string, which is "automatic" in that it is hard to avoid, implies that also its second quantized effective target space theory is locally supersymmetric, that it is supergravity instead of plain gravity, as for the bosonic string.

And this is how string theory predicts local supersymmetry:

1. assume that there are fundamental strings;

2. they have to be taken to be spinning strings in order for their second quantized effective target space theory to contain the observed fermions;

3. then it follows that the second quantized effective target space theory is locally supersymmetric Einstein-Yang-Mills-Dirac theory with supergravity.

In short

• strings + fermions $\Rightarrow$ local supersymmetry .

So this is generic.

But now low energy global supersymmetry of the kind that the LHC is looking for is a rather different story. A global supersymmetry is a solution to local supersymmnetry, hence to supergravity, which has a global Killing spinor. As in the relation between global supersymmetry and Calabi-Yau manifolds.

The existence of a global Killing spinor in a solution to supergravity is something entirely non-generic. It is the direct super-analog of the existence of a Killing vector, of course. The existence of a Killining vector on spacetime means that spacetime looks the same no matter how far we translate in a certain direction. This is not what one expects to happen generically. The existence of a Killing spinor is like this, but even a bit stronger. There is no a priori reason for this to exist. It could, in some approximation maybe, but there is certainly no principle which would imply that it does.

So global supersymmetry as in the MSSM is a non-generic preservation of a global supersymmetry, which will not be enjoyed by the generic solution to a locally supersymmetric theory. So string theory, which predicts local supersymmetry, does not predict global supersymmetry. Some of its solutions do have global supersymmetry, but generically they do not.

Global supersymmetry instead was motivated phenomenologically. It is/was thought to be phenomenologically suggested by naturality and by gauge coupling unification. Since string theory predicts local supersymmetry, it is able to accomodate for such globally supersymmetric models. But it does not predict them.

For more on this see at String theory FAQ -- Does string theory predict supersymmetry?

-

"Falsifying supersymmetry" is a phrase that has to be properly qualified. Our ability to falsify with experiment is limited. We can rule out the existence of supersymmetry only at accessible energy/distance/density scales. LHC, for example, is not able to resolve physics at distance scales much smaller than $\frac{\hbar c}{7\mbox{ TeV}} \simeq 300000000000000000 \mbox{ Planck lengths}$. It isn't seeing any supersymmetry, but that doesn't prove that physics isn't supersymmetric at about $1$ Planck length. (It does prove that some theoretical physicists made wrong predictions. Too bad for them. But that's actually what theorists normally do. You're doing quite well if you're right once.)

The existing versions of string theory require supersymmetry at roughly 1 Planck Length. We are not able to do the kind of experiment that rules out supersymmetry at this scale. (I personally find somewhat unconvincing the arguments that the string theories we know of are the only possible kind.)

-

This isn't true. A counterexample can be constructed as a twist of the heterotic string, including discrete torsion. The resulting theory is a nonsupersymmetric theory, but has no tachyons and is therefore consistent. This is described in Chapter 11 of Polchinski.

-