### Local and Global Supersymmetry

#### Posted by Urs Schreiber

The field of *fundamental high energy physics* – that part of physics that deals with fundamental particles probed in particle accelerators – is witnessing interesting developments these days: after decades of only a minimum of new experimental observations of interest, finally plenty of data has been collected and now analyzed. And finally a multitude of theoretical models that have been developed over the years can be tested against experiment.

Apart from lots of new information about which mass the hypothetical *Higgs particle* – if it indeed exists – does *not* have, one of the striking experimental results is that they increasingly – and by now strongly – disfavour what are called *supersymmetric* extensions of the *standard model of particle physics* .
Well-informed discussion of these developments can for instance be found on this blog.

In the course of these developments, I see and hear a lot of discussion around me of whether the concept of “supersymmetry” as such is thus experimentally ruled out. There is an enormous amount of literature revolving around the concept of supersymmetry quite independently of the “supersymmetric standard model of particle physics”. Is all that now proven to be ill-conceived? Is “supersymmetry” being shown to play no role in nature?

We *almost* had a discussion of this kind also here on the blog recently. Since this is a widespread misunderstanding, I thought I’d try to say something about it here.

A little appreciated but important fact is this: there is a crucial distinction between what is called *local* supersymmetry and what is called *global supersymmetry* and between *target space* supersymmetry and *worldvolume* supersymmetry. I have tried to say a bit about this in the $n$Lab entry

Even less widely appreciated seems to be the following noteworthy fact: local worldline supersymmetry is experimentally verified since 1922 – when the *Stern-Gerlach experiment* showed that there are fundamental particles with a property called *spin* : these spinning fermion particles – the electrons and quarks that you, me, and everything around us is made of – happen to have *worldline supersymmetry* .

I have tried to give an indication of this in the entry

which also collects a bunch of original references and textbook chapters where this fact is discussed in detail.

So the assumption that there is local worldvolume supersymmetry in nature is not speculation, but experimental fact as soon as there is any spinor in the world. Of course this is not the global target space supersymmetry that is currently being experimentally ruled out at the LHC. So it is good to distinguish these concepts. And indeed, despite of what many people are on record as having said: nothing at all in sigma-model theory implies that a supersymmetric sigma-model (such as the spinning particle, or the spinning string, for that matter) has target space backgrounds that *generically* are globally supersymmetric. On the contrary: the generic background will not be!

This simple fact seems not to be widely appreciated, either. It is the direct analog of the following self-evident bosonic statement: while ordinary gravity is a *locally* Poincaré-invariant theory (a Poincaré-gauge theory) its generic solution – a given pseudo-Riemannian manifold – does *not* have a nontrivial action of the Poincaré group or of any of its nontrivial subgroups. It will only have such actions if it has flows of isometries given by Killing vectors. Analogously, the generic solution to a theory of supergravity – which is a locally super-Poincaré-invariant theory– does *not* have any covariantly constant spinor, hence the perturbative quantum field theory on this background does *not* have a global supersymmetry.

This has always been clear. Some more sophisticated discussion of this point is for instance in

Dienes, Lennek, Sénéchal, Wasnik, *Is SUSY natural?* (arXiv:0804.4718)

which is effectively a detailed expansion of the statement about generic absence of global symmetries in backgrounds.

There’d be much more to say (and there’d be need to expand the above $n$Lab entries much more), but I must stop here and take care of other tasks. The upshot is:

there is still all the reason in the world to believe that the concept of local supersymmetry (aka: supergravity) is fundamental for our world – not the least because 1-dimensional worldline supergravity is an experimentally observed fact;

the models of global supersymmetry that are currently being ruled out by experiment are not rooted in theory, but in phenomenological model building. The general

*theory*of supersymmetry is as unaffected by these models being ruled out as the theory of gravity is unaffected by a given cosmological model being ruled out.

## Re: Local and Global Supersymmetry

It is a pity to try to kill SUGRA just now that, with massive neutrinos, the MSSM happens to have 128 bosonic and 128 fermionic helicities. But, on the same token, it implies that any phenomenological effort should avoid new particles, or to have some mechanism to put them in different footing that the basic D=11 SUGRA supermultiplet. At most, it could be interesting to remove two helicities from the MSSM Higgs to make place for the graviton.