The n-Category Café

It is a pity to try to kill SUGRA just now that

It is also a bit of a pity that you say this in reply to an entry that was pointing out the difference between local supersymmetry = supergravity and global supersymmetry = QFT with global supersymmetries. Because what is currently being killed is not the former, but the latter.

The world can still very well be described by, say, a KK-compactification of heterotic supergravity, and we’d still see no global susy in our accelerator experiments if the compactified geometry that gives the particle spectrum does not admit a Killing spinor, hence a global supersymmetry. And as I tried to point out, this is in fact the generic situation to expect.

And even if our spacetime theory is not sugra, we know for sure already that our worldline theory is. For whatever that’s worth.

Following discussion with Zoran Škoda here over at the $n$Forum, I have by now expanded the entry supersymmetry a bit further in an attempt to bring this out more clearly.

When the result of the Michelson-Morley experiment has just arrived, it is of course important to stress that ether theory is the correct theory of sound.

Indeed, I think it is: when people get mixed up and discard the theory of sound wave propagation in media because they found that light propagates without a medium, it is important to stress what’s really going on.

Generally, I think it is important to keep the concepts straight. Discussion elsewhere gives justification for a little post like this one here, it seems to me.

Thanks Urs for pointing out that little known fact of local 1d worldline SUSY. I had thought there was no hard evidence of any form of SUSY.

Thanks for saying this. Yes, I think it is something worth thinking about.

I believe one can usefully further expand on this point. For instance one could also say the following:

One distinction that is important but often not made is that between supergeometry itself and supersymmetry . Namely one can argue that any QFT with fermions naturally lives in the context of supergeometry: where the boson fields are maps between ordinary manifolds (sections of ordinary bundles, say) fermion fields are odd components of maps between supermanifolds. This is what is behind the skew-commutativity of fermion fields. So it is fully justified to say: we have perfect experimental evidence that the world is described by supergeometry instead of just plain differential geometry.

From this perspective then the worldline supersymmetry of spinning particles looks as follows: we see that for a QFT with supergeometric fields in small dimension, it is hard to avoid it exhibiting super-Poincaré-symmetry. The obvious action with fermions just happens to have this.

It may be useful to recall that the same kind of “inevitable worldvolume supersymmetry” was the reason that people eventually considered the superstring. Because initially what Neveu-Schwarz and Ramond wrote down was simply a model for the spinning string, the stringy analog of a fermion. And in the original articles from the 1970s it was called this way: the spinning string. But, much as for the spinning particle, it then turns out that the evident action functional for the spinning string just so happens to exhibit supersymmetry. The historical review by John Schwarz which I have linked to at spinning string is worth reading.

So just as for the point particle, the supersymmetry appears here “by itself” in a way. Of course for the spinning string (as opposed to the spinning particle) something remarkable then happens: the inevitable supersymmetry that appears in low dimension on the worldsheet then induces local supersymmetry on the effective target space theory in high dimension (once one applies the GSO projection, which however is essentially forced by consistency.)

So this may be summarized as follows: while spinning particles, despite their worldline supersymmetry, may be quanta in a background field theory which is not supersymmetric, spinning strings have to be quanta of a target space theory that exhibits local supersymmetry. This is the way in which the assumption of both strings and fermions indeed “predicts” supersymmetry. But local supersymmetry. Unfortunately at this point the standard textbooks often drop the local and just say “predicts supersymmetry”. Which eventually leads to some mess in the public perception of what string theory does and does not.

is this medium to be taken as one that is mechanically conceived

Yes, the aether theory of the second half of the nineteenth century conceived of everything (matter, electromagnetism) as, ultimately, activities of a kind of mechanical fluid, viz. the aether—albeit one which was required to have increasingly bizarre properties as knowledge of physics improved. Towards the end of the nineteenth century, this theory started to collapse, and the revolutions in physics at the beginning of the twentieth century overthrew it completely.

Anyway, the ‘medium’ of electromagnetic propagation that people talk about means the aether of the aether theory. The idea that everything was ultimately activities of fields, rather than a mechanical fluid, replaced the aether theory and involved a very different mindset.

(In fact, for a period, some people thought everything, including matter, could be considered as activities of the just the electromagnetic field, but that idea also failed as knowledge about atoms and electrons improved.)

Also, surely the ‘medium’ in which light propagates is the electromagnetic field? When its said that light does not propagate in a medium, is this medium to be taken as one that is mechanically conceived, like say sound through the medium of air?

Yes, the hypothetical “aether” was meant to be a medium in which light waves are mechanical waves much like sound waves (apparently already Newton pointed out that it could not be just like sound waves). Have a look at the Wikipedia entry luminiferous aether. That claims for instance that

Maxwell himself proposed several mechanical models of aether based on wheels and gears,

But, maybe let’s leave the discussion of aether at that. I don’t think Thomas’ attempt at an analogy to the Michelson-Morely experiment really worked, and this is getting quite off-topic.

Instead, if we want to consider theory/experiment relations that have good analogy to the supersymmetry theory/LHC experiment situation today, then I suggest considering situtations as the following:

After Einstein had the theory of gravity, he proposed a phenomenological model of the cosmos based on this theory: called the Einstein steady state model or the like. This model was motivated from a phenomenological prejudice he had: that the cosmos is not shrinking or expanding.

Then Hubble’s experiment discovered that this model was wrong: the cosmos is expanding. So Einstein’s phenomenological model was ruled out. But his theory of gravity was not ruled out: it could easily accomodate models that fitted the observation well, now called the FRW models and their variants.

I think we see something very analogous happening here: based on the theory of higher supergravity people suggested a phenomenological model in the theory: compactifications to four dimensions that repect precisely one global supersymmetry. But there is a strong prejudice motivating these models: nothing at all in the theory demands solutions with one or any global supersymmetry. On the contrary, in the “model space” of the theory these are pretty exceptional points. (As was, for that matter, Einstein’s steady-state model.)

Now with recent experimental insights, this class of models with one global supersymmetry is looking increasingly unlikely as realistic models. So they will probably be discarded. But that does not mean that the theory of higher supergravity is wrong. (Of course it can well be wrong! But this here is not the proof or even a hint that it is.) Instead, just as after Einstein’s steady state model was ruled out, people will swiftly come up with other phenomenological models within the theory of higher supergravity.

Don’t forget the Killing-Yano tensor.

Just to amplify: this is about enhanced worldline supersymmetry for special backgrounds:

the spinning particle $\sigma$-model for generic target spacetime geometry $X$ has a single supersymmetry ($N = 1$). If however $X$ exhibits special structure, there may appear more supersymmetry on the wordline. In analogy to the more familiar statement that every Killing tensor of $X$ gives rise to one constant of motion of the bosonic particle on $X$ – one operator commuting with the Laplace operator – every Killing-Yano tensor on $X$ gives rise to an extra operator commuting also with the Dirac operator.

So one finds for instance that the ordinary spinning article propagating in a spacetime of any charged and rotating black hole has not just one worldline supersymmetry, but an enhancement thereof.

So one can strengthen one of the punchlines of the above entry from: “Worldline supersymmetry is an experimentally verified fact since 1922.” to: “Also enhanced worldline supersymmetry is experimentally verified.” Just for the sake of the argument.

One can also characterize Kähler and hyper-Kähler geometry structure on $X$ in terms of the supersymmetry of the spinning particle on $X$ this way. This is also the way spectral non-commutative Kähler and hyper-Kähler geometry is formulated in Connes’ spectral geometry.

And, last not least, all this is of course a shadow of the corresponding situation in higher dimensions: it is the low dimensional analog of how the worldsheet theory of the spinning string has enhanced supersymmetry when the target is Calabi-Yau.

Concerning forgetting the Killing-Yano tensor: incidentally, I had used this a lot in my master thesis on supersymmetric quantum cosmology, back then. There I considered supersymmetric quantum cosmological models as worldline $N=1$-susy dynamics on Wheeler-deWitt configuration space and looked for Killing-Yano tensors on Wheeler-deWitt space in order to find the super-constants of motion of the cosmological model. (This setup is in principle just as in Hermann Nicolai’s supergravity quantum billiards.)

How can an experiment done inside a relatively small chamber at CERN make a statement about global supersymmetry?

The (hyper)cube of space(time) that CERN sits in is well-approximated by a Minkowski space $M^4$. The quantum field theory needed is that on this $M^4$. The supersymmetry that is being tested (and not found so far) is that which applies globally over this $M^4$.

The issue here of “local globality” is not special to supersymmetry. It applies already to the fact that they speak about particles in the first place.

For quantum field theory on cosmological scales, on a curved spacetime $X$ there is in general no concept of “particle excitation” at all. This concept exists only on Minkowski space. Any statement CERN makes about QFT is always about the QFT over some $M^4$ inside which CERN sits. It is always the “globally Minkowskian” QFT over this “local piece” of spacetime.

Or, if you want to extend the discussion to the models of higher compactified supergravity: we can also imagine CERN sits inside an $M^4 \times Y^6$ where $Y^6$ is a tiny compact Riemannian manifold. Then a globally defined covariant constant spinor on this $M^4 \times Y^6$ gives a global supersymmetry of the 10-dimensional supergravity theory compactified on this $Y^6$. This says nothing about “cosmologically globally” defined constant spinors. Even if there were one on CERN-scales, there’d hardly be one on cosmological scales. But also, that does not matter.

Finally notice that the “local” in the “local Poincaré-supersymmetry” of supergravity is a bit of physics jargon: it just refers the fact that the configurations of the theory are connections on a super-Poincaré-principal bundle.

I guess the question is whether worldline supersymmetry is an accidental feature of our world or an essential ingredient.

The worldline supersymmetry of fermions comes down to the fact that their Hamiltonian(-constraint) operator $H$ has a square root: the Dirac operator $D$. Its defining equations

$$[D,D] = 2 D^2 = H\,,\,\,\,\, [H,H] = 0 \,,\,\,\,\, [D,H] = 0$$

characterize the $d = 1$ translation super-Lie algebra.

The Dirac equation in single particle dynamics and its analogs in QFT, in turn, appears to be a most essential ingredient of our world!: the fine structure of the atomic spectrum, the existence of solids, of antimatter,… In a world without Dirac particles, there’d be no matter, just a sea of gauge fields.

So I would say, to the extent that this kind of question makes sense: worldline supersymmetry is a fundamental feature of our world.

Thanks for these questions, Mike. I was wondering when somebody would ask these.

I will reply item-by-item. And since it’s already very late here, I may have to continue tomorrow. But here is the first reply:

nothing requires that the (other) particles be “nontrivially super” representations of the super Poincaré group?

I read that as asking if there are representations of the super-Poincaré group which contain only bosons without superpartners. (Otherwise maybe you need to say this again.)

The answer to that is: no. The irreducible representations of the super-Poincaré group are what is called supermultiplets : as representations of the underlying ordinary Poincaré group they are in general decomposable with the different irreducible pieces being the bosonic particle and its superpartners.

I have collected some useful references on this point now in the new $n$Lab entry

unitary representation of the super Poincaré group.

More replies to come…

So I don’t quite understand how the group that particles are a representation of, and the group whose Lie algebra the gauge field is a connection in, could be different.

Right, this is the crucial phenomenon here. I try to say again in words what is happening. With a little luck I may find the time to give a more technical description later.

There are two points to notice here:

1. backgrounds and their perturbations;

2. spontaneous symmetry breaking

The first point refers to considering configurations of a quantum theory (possibly with local gauge invariance) that are small quantum perturbations about a solution to its classical field equations. The second point refers to the fact that after this splitting, the QFT of the remaining small perturbations on the given fixed background in general does not exhibit the same symmetry as the former theory exhibited.

Specifically, for our purposes, consider the theory of ordinary gravity, with its local Poincaré-symmetry (meaning: its field configurations are Poincaré-connections). The only way to make sense of this as a quantum theory at “low” energies (short of embedding it into a UV completed finite theory such as string theory) is to regard it as an effective quantum field theory that approximates some deeper, unknown theory. A useful review is Donoghue 95. As discussed for instance there, this involves splitting the gravitational field as a classical part –some pseudo-Riemannian manifold – equipped with a bunch of quantum fields propagating on it.

This means that after the splitting, the remaining theory is a quantum field theory on a fixed curved background spacetime. Unless this background spacetime happens to be flat Minkowski space (or, actually, asymptotically such), this theory will not exhibit global Poincaré invariance: its Hilbert space of states will not decompose into irreducible representations of the Poincaré group. One says that the Poincaré symmetry of the theory is broken by one of its solutions.

For ordinary gravity this is a well-known phenomenon in the study of cosmology, where exactly this setup plays a role: in cosmology – as opposed to in collider particle physics – one cannot get away with pretending that the average field configuration of gravity is well-approximated by flat Minkowski space. Hence the ordinary particle picture breaks down on cosmological scales and one has to do something else.

For supergravity it is technically all the same kind of mechanism, only that here now it does also matter for the purposes of collider physics. Namely it can well happen that a classical solution to a locally super-Poincaré-symmetric theory (supergravity) displays global ordinary Poincaré-symmetry, but is not globally symmetric under extensions to the super-Poincaré-group. Hence while the particle picture exists, the super-multiplet picture breaks down and we do not see superpartners in the theory over this background.

This phenomenon is often discussed in the literature in a more complicated situation, where one considers the breaking of the symmetry here in two steps: first one considers a supergravity background which preserves some of the supersymmetry. And then one repeats the process, splits the remaining theory again into a background which in turn also breaks that remaining supersymmetry. This is what happens when people say things like: “Supersymmetry is broken at ordinary low energy scales, but at a high energy scale one supersymmetry may be unbroken.” And others may then add “And at the string scale we even see full supergravity.”

My general understanding of (non-super) gauge field theories is that the particles are representations of the symmetry group, while the gauge field is a connection valued in the Lie algebra of that group

Yes, only that, strictly speaking, the terminology is slightly different.

Both the connection and the sections of bundles on which it acts are fields and/or particles. The components of the connection are particles, too: those that transmit gauge forces: photons, gluons, $W$s, gravitons.

How about Split Supersymmetry? Even if we don’t detect supersymmetric particles at the LHC, you could still have split supersymmetry.

Many a thing is possible. I am glad that I am not a model builder.