July 27, 2011

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

spinning particle

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:

1. 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;

2. 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.

Posted at July 27, 2011 2:10 AM UTC

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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.

Posted by: Alejandro Rivero on July 27, 2011 10:53 AM | Permalink | Reply to this

Re: Local and Global Supersymmetry

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.

Posted by: Urs Schreiber on July 27, 2011 12:43 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

(ashamed) OK, sorry, yes, it is not SUGRA but MSSM what is being killed. I think my slip (note that in the other post I used MSSM) was because I seriously wonder if it is the same thing, or at least the same underlying structure. Lets reformulate “It is a pity that MSSM is being killed now that it happens to have the same number of helicities than D=11 SUGRA”.

By the way, probably one source of the misunderstanding is that in the original models to get the gauge bosons out of D=11 via Kaluza Klein you need to involve gravity -or at least diffeomorfisms and isometries-, independently of the kind of susy you have, or of having susy at all.

Posted by: Alejandro Rivero on July 27, 2011 2:28 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

Hmm so the fundamental concept here is the Killing spinor. Just I do not remember… how are the global generators out of the Killing spinors related to the local ones in the uncompactified thing?

Posted by: Alejandro Rivero on July 27, 2011 2:33 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

Hmm OK I see, I had really forgot a lot of this things. Thanks for you post, Urs, and for your effort in the wiki entry too.

Posted by: Alejandro Rivero on July 27, 2011 5:55 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

Just I do not remember… how […]

The canonical textbook to have within reach for these things is

Deligne et al. Quantum Fields and Strings

The material you are after here is in the chapter that starts in vol II on page 1091: supersymmetry and Calabi-Yau manifolds .

(This is becoming anachronistic as we speak as far as phenomenology goes. But it is still important to understand the logic here.)

Killing spinors and their relation to global supersymmetry are first explained on p. 907.

If you have problems getting ahold of a copy, drop me an email.

Posted by: Urs Schreiber on July 27, 2011 6:07 PM | Permalink | Reply to this

SSM under the MSSM

(To keep order, I am doing separate comments for separate topics)

It could be useful when considering higgsless, or higgs beyond MSSM, models, to keep in mind what I call the SSM plainly: the minimal set you need to build supermultiplets where the W and Z particles are massive. I am not sure if it is possible to move the mass continously toward zero (I believe to remember that there was some cases in QFT where it was conceptually different) but if you can, then you see that each massive gauge supermultiplet is the joining of a massless gauge plus a chiral supermultiplet giving you two scalars and an extra spin 1/2 “ino”. For SU(2)xU(1) breaking it meanse six extra scalars, which usually appear as a piece of the higgs(es) in the MSSM, but they are really previous, or independent, of the Higgs mechanism.

To fix ideas: the MSSM has 128+128, the SSM has 126+126.

Posted by: Alejandro Rivero on July 27, 2011 11:05 AM | Permalink | Reply to this

Re: Local and Global Supersymmetry

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.

Re: Local and Global Supersymmetry

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.

Posted by: Urs Schreiber on July 27, 2011 6:13 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

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.

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?

Posted by: mozibur ullah on July 27, 2011 10:52 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

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.

Posted by: Urs Schreiber on July 28, 2011 10:32 AM | Permalink | Reply to this

Re: Local and Global Supersymmetry

Urs Schreiber wrote:

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.

Which is kind of confusing, to me at least: when I hear “spinning string”, my first thought is of a string rotating around its midpoint, which a bosonic string can do just fine. (The rigidly rotating bosonic string is also a historically significant picture, since the angular momenta of its excitations scale with the square of their energy, i.e., a Regge trajectory.)

Posted by: Blake Stacey on July 28, 2011 1:05 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

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.

Which is kind of confusing

Maybe spinorial string would have been a better choice of terminology. But the problem here is really rooted in the term “spin” itself: I guess for a completely uninitiated novice it is also confusing to learn that an electron can “spin” but not “rotate”, right? I guess that’s one of these points about which they say that if a student isn’t confused about it at least once, he or she hasn’t understood it ;-)

In any case, it does not matter so much, as the term “spinning string” is essentially out of use anyway. Everybody says “superstring”. I just like to revive the old terminology as it is makes a point that is sometimes forgotten: the supersymmetry on the string worldsheet follows already from making it spin like an electron spins, just as the supersymmetry on the electron’s worldline follows this way.

Posted by: Urs Schreiber on July 28, 2011 3:01 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

But the problem here is really rooted in the term “spin” itself: I guess for a completely uninitiated novice it is also confusing to learn that an electron can “spin” but not “rotate”, right?

Right.

I see the nLab entry on the spinning string has already been modified to clarify this — great!

Posted by: Blake Stacey on July 28, 2011 6:23 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

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.)

Posted by: Tim Silverman on July 28, 2011 10:33 AM | Permalink | Reply to this

Re: Local and Global Supersymmetry

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.

Posted by: Urs Schreiber on July 28, 2011 11:10 AM | Permalink | Reply to this

Re: Local and Global Supersymmetry

Don’t forget the Killing-Yano tensor.

Posted by: Mitchell Porter on July 28, 2011 11:35 AM | Permalink | Reply to this

Re: Local and Global Supersymmetry

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.)

Posted by: Urs Schreiber on July 28, 2011 12:20 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

Quick stupid-sounding question: How can an experiment done inside a relatively small chamber at CERN make a statement about global supersymmetry? If I understand your relativity analogy correctly, I would have the same issue there: it seems unlikely that an experiment done at small length scales (“locally”) could rule out Minkowski space (“globally Poincare-invariant theory”). On the other hand, I guess that’s exactly what the Cavendish experiment does?

Posted by: Bruce Bartlett on July 28, 2011 5:00 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

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.

Posted by: Urs Schreiber on July 28, 2011 5:12 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

Do I understand it correctly then: the culprit who isn’t admitting supersymmetry, the one who’s causing problems, is not 4-dimensional spacetime $X$ on universal scales (which we can approximate for CERN experiments by $M^4$), but the tiny little Riemannian manifold, the “$Y^6$” piece?

And presumably, that $Y^6$ piece is “so small” that one can never claim (like we did for $X$) that the LHC chamber is only observing a “local fraction” of it, which would look like $R^6$?

In other words, as I understand it, the fact that these extra space dimensions $Y^6$ are so tiny (one of the requirements as I have it, else we would have seen them!) means that a global issue on that $Y^6$ (like not being able to admit a globally defined covariantly constant spinor), the kind of issue we normally ignore for small-scale experiments, is forced to become a local issue in the LHC chamber?

Posted by: Bruce Bartlett on July 31, 2011 1:05 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

That’s right, yes.

If we are talking about a phenomenology where the 4-dimensional physics comes via such a “Kaluza-Klein mechanism” from a higher dimensional supergravity this way, then the particle spectrum seen in $M^4$ is that of 0-modes on $Y$ and the failure of the configuration on $M^4 \times Y$ to admit a global supersymmetry transformation (the super-analog of a global translation/rotation symmetry) is what makes the effective theory seen at CERN look non-supersymmetric.

Posted by: Urs Schreiber on July 31, 2011 1:35 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

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

Posted by: Arun on July 29, 2011 9:25 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

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.

Posted by: Urs Schreiber on July 30, 2011 1:32 AM | Permalink | Reply to this

Re: Local and Global Supersymmetry

Dirac particles are fundamental. That they have a supersymmetric world line description might be just a neat trick like the quaternion representation of Maxwell’s equations.

Posted by: Arun on July 30, 2011 3:31 AM | Permalink | Reply to this

Re: Local and Global Supersymmetry

Dirac particles are fundamental. That they have a supersymmetric world line description might be just a neat trick like the quaternion representation of Maxwell’s equations.

Well, as I said, the worldline supersymmetry comes down to the defining equation for the Dirac operator. That defining equation is the characterization of the $d= 1$ translation super Lie algebra. So it seems hard for me to imagine what it might mean that worline supersymmetry is a “trick” while $D^2 = H$ is not. Do you see what I mean?

Posted by: Urs Schreiber on July 30, 2011 2:59 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

Does world line SUSY survive beyond the free particle?

Posted by: Arun on July 30, 2011 11:00 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

Does world line SUSY survive beyond the free particle?

Yes, it’s robust. At spinning particle are explicitly pointed out references that demonstrate worldline supersymmetry for all kinds of background interactions.

Posted by: Urs Schreiber on July 30, 2011 11:47 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

Still not clear to me that worldline supersymmetry corresponds to our world. e.g., is the usual electrodynamics obtained? (or do we need a new “generalized Lorentz force”?)

Posted by: Arun on August 6, 2011 3:00 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

Still not clear to me that worldline supersymmetry corresponds to our world.

I am not quite sure what you mean by saying this, sorry. I am getting the impression that there is a general misunderstanding here about the statement “The worldline theory of a fermion is supersymmetric”.

My point would be: This is just a fact. Since there are fermions in our world, there is worldline supersymmetry in our world.

But maybe you are looking for something else? Let’s see. You write:

e.g., is the usual electrodynamics obtained?

Maybe you need to tell me what you are looking for when you say “is XYZ obtained?” Obtained by doing what?

Maybe you mean: when writing the worldline action functional for a charged fermion that propagates in an electromagnetic background field subject to the usual Lorentz force, is the action functional still supersymmetric?

The answer is: yes. I gave the references that discuss this in detail above.

But if you mean something else, maybe you have to help me. Sorry. Can you say in detail what it is that you would like to see?

Posted by: Urs Schreiber on August 6, 2011 3:15 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

The references seem to give very generalized supersymmetric worldline actions that
a. seem to include terms not observed in practice.
b. are not clear that they can be reduced in particular cases to only terms observed in practice.

Neilsen even predicts a new effect that might be measurable in a muon atom.

Posted by: Arun on August 12, 2011 5:44 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

The references seem to give very generalized supersymmetric worldline actions

Are we talking for instance about van Holten’s D=1 supergravity and spinning particles ?

The “generalized Lorentz force” mentioned there after the equations of motion (37) is the correct Lorentz force for spinors. It is called “generalized” as compared to that for the bosonic particle, because as opposed to that case it contains the spin-coupling to the electromagnetic field. For instance reviewed here.

Posted by: Urs Schreiber on August 12, 2011 10:10 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

Maybe this here is a better reference for discussion of the spin-coupling contribution to the Lorentz force law

J.W. van Holten, Relativistic Dynamics of Spin in Strong External Fields (arXiv:hep-th/9303124)

(see around equation (14)).

Posted by: Urs Schreiber on August 13, 2011 12:35 AM | Permalink | Reply to this

Re: Local and Global Supersymmetry

Thanks for this post! I wasn’t aware of this stuff, but it’s exciting to hear about! I have some very naive questions, based on knowing some of the mathematics of gauge field theory but not a lot about super-stuff.

At supersymmetry it says first that in “global” supersymmetry, the “particles” are representations of the super Poincaré group (rather than the ordinary Poincaré group), with the new odd-graded representations being the superpartners, while in “local” supersymmetry, the theory of supergravity is a gauge theory of connections in the super Poincaré Lie algebra.

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 (which comes with the adjoint representation of the group, making gauge bosons into particles). So electrons are a representation of U(1) and the electromagnetic field (or, I suppose, the electromagnetic potential) is a u(1)-connection, and so on. And the relationship is that the particle fields are sections of a bundle whose structure group is the gauge group, so that in order to differentiate them and express their field equations, we have to incorporate a connection on that bundle, which is the gauge field. 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.

I guess that even if the gravity gauge field lives in the super Poincaré Lie algebra, nothing requires that the (other) particles be “nontrivially super” representations of the super Poincaré group? Does that mean that even if current experiments rule out some super-partners coming from the odd-graded representations of the super Poincaré group, there could be some other super-partners arising from its adjoint representation on the super Lie algebra?

Regardless of the answer to that, what is the relationship of either of these to global symmetry of spacetime? I thought that even the fact of particles being representations of some gauge group was an expression of local symmetry; isn’t “promoting a global symmetry to a local symmetry” the whole point of gauge theory?

Posted by: Mike Shulman on July 30, 2011 7:49 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

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

More replies to come…

Posted by: Urs Schreiber on July 30, 2011 11:56 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

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.”

Posted by: Urs Schreiber on July 31, 2011 11:26 AM | Permalink | Reply to this

Re: Local and Global Supersymmetry

Thanks, this helps. I’m making progress, but I’m not quite there yet.

The first thing I gather from your answer is that what’s being described is a purely quantum-field-theoretic phenomenon—it has no analogue for classical gauge field theory. Is that correct?

The second thing I gather is that it sounds like the answer to my last question, about why what you called “global” is related to the global symmetry of spacetime, is that in QFT, rather than just a field that is a section of some G-vector bundle over spacetime, for which we could talk about the gauge group acting locally, we have a global “state” which lives in some Hilbert space and is like a “wavefunction over the space of all possible fields on all of spacetime”. And somehow the action of the Poincaré group on that Hilbert space is what enables us to interpret that state as consisting of “particles”.

When I first learned about spontaneous symmetry breaking, I was told that it meant that the theory as a whole has a symmetry, but a particular state may lack that symmetry—the classic example being a ball rolling down a circularly symmetric hill into a moat and coming to rest somewhere in the moat, breaking the global circular symmetry by having chosen a particular position along that circle. Is this an instance of that? There’s a theory that has a global supersymmetry, but the universe might be currently in a state which doesn’t exhibit that symmetry, hence no superpartners are visible? That’s what you seem to be saying at the end; but how is that related to the idea of “the only way to make sense of this… is to interpret it as an effective QFT that approximates some deeper, unknown theory”?

Posted by: Mike Shulman on July 31, 2011 6:50 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

it sounds like the answer to my last question, about why what you called “global” is related to the global symmetry of spacetime, is that in QFT, rather than just a field that is a section of some G-vector bundle over spacetime, for which we could talk about the gauge group acting locally, we have a global “state” which lives in some Hilbert space and is like a “wavefunction over the space of all possible fields on all of spacetime”. And somehow the action of the Poincaré group on that Hilbert space is what enables us to interpret that state as consisting of “particles”.

Yes!

The “global” refers to the fact that the theory of perturbations about a classical solution $(X,g)$ of (super)gravity has translation symmetries given by the the (super)isometry group $ISOM(X,g)$: its local net of observables is $ISOM(X,g)$-covariant and hence the space of states of this theory – which you could call “global states”, yes – decomposes into irreducible $ISOM(X,g)$-(super-)representations.

There’s a theory that has a global supersymmetry, but the universe might be currently in a state which doesn’t exhibit that symmetry, hence no superpartners are visible?

Yes!

But beware that this kind of statement applies at two different stages of the story. It’s a somewhat convoluted story, but that’s typical of all “model building” in physics:

first we have a theory of supergravity where each field has its superpartner. Then we either consider that around some classical solution which has no “super-isometry” left and so the resulting theory has no superpartner states.

Or – what used to be done since the 90s and maybe ending now – you assume that it has one “super-isometry” left, which means that there are superpartner states. If in addition there is ordinary Poincaré-symmetry left it means that all particles now come in supermultiplets.

BUT being supermultiplets, the superpartners at this point will have the same mass as their bosonic partners. So in common phenomenological models one repeats the process, assumes now a classical solution to the new theory (that is itself an approximation to the classical solution of the original supergravity theor) exists which in turn spontaneously breaks that remaining supersymmetry.

All that is done in order to end up with a model where the superpartner particles are visible, but not of equal mass as their bosonic partners, anymore. This second step is what in 99% of all literature is called “the breaking of supersymmetry”. But if we assume that fundamentally there is supergravity at work, then strictly speaking this breaking is only the second of two steps of breaking superymmetry in stages.

how is that related to the idea of “the only way to make sense of this… is to interpret it as an effective QFT that approximates some deeper, unknown theory”?

That was a parenthetical remark just about quantization of these theories that involve gravity. Most of what we are discussing here has its analogs in classical field theory (even fermion fields, which are sometimes said to only exist in the quantum theory), so maybe for the point of this discussion here this is not so important. I just thought that in case some bystander is wondering how I am talking about quantization here in the context of supergravity, I point out that it is to be understood in the sense of effective quantum gravity (at the sufficiently low energies that need to concern us here). Which is well understood.

Posted by: Urs Schreiber on August 1, 2011 1:35 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

Thanks! That helps.

Posted by: Mike Shulman on August 3, 2011 4:59 AM | Permalink | Reply to this

Re: Local and Global Supersymmetry

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.

Posted by: Urs Schreiber on July 31, 2011 11:38 AM | Permalink | Reply to this

Re: Local and Global Supersymmetry

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

Yes, I think that’s what I meant by the parenthetical:

(which comes with the adjoint representation of the group, making gauge bosons into particles)

Posted by: Mike Shulman on July 31, 2011 6:22 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

Bruce, Mike,

in an attempt to provide further details in reply to your questions, I have started an entry

spontaneous symmetry breaking

on the $n$Lab. Still very sketchy, though, but less sketchy than the discussion here.

The lecture by Strominger pointed to there might be useful to look at (even though it is not, despite its advertizement, “a course for mathematicians” ;-).

Posted by: Urs Schreiber on July 31, 2011 4:17 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

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

Posted by: Jeffery Winkler on August 5, 2011 1:39 AM | Permalink | Reply to this

Re: Local and Global Supersymmetry

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.

Posted by: Urs Schreiber on August 5, 2011 9:12 AM | Permalink | Reply to this

Re: Local and Global Supersymmetry

Urs, you comment that a generic solution to supergravity will not be globally supersymmetric. By analogy to ordinary sigma models with a global symmetry, such as an O(N) sigma model, in that case many solutions spontaneously break the symmetry but this goes along with having massless Goldstone modes. So, if you have a solution to supergravity which is not globally supersymmetric, does this mean that there will also be massless modes in analogy to the Goldstone modes, or do they get eaten by a Higgs-like mechanism and if so is there necessarily any other observable consequence of a non-supersymmetric solution?

Posted by: matt on August 26, 2011 9:37 PM | Permalink | Reply to this

Re: Local and Global Supersymmetry

Thanks for sharing.

Posted by: Kavi raj on September 18, 2018 12:30 PM | Permalink | Reply to this

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