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I’ll be working a bit on supersymmetry.
Zoran, you had once left two query boxes there with complaints. The second one is after this bit of the original entry (this will change any minute now)
The theory of supergravity is, as a classical field theory, an action functional on functions on a supermanifold $X$ which is invariant under the super-diffeomorphism group of $X$.
where you say
Zoran: action functional is on paths, even paths in infinitedimensional space, but not on point-functions.
I think you got something mixed up here. If $X$ is spacetime, a field on $X$ is the “path” that you want to see. The statement as given is correct, but I’ll try to expand on it.
The second complaint is after where the original entry said
many models that suggest that the familiar symmetry of various action functionals should be enhanced to a supersymmetry in order to more properly describe fundamental physics.
You wrote:
This is doubtful and speculative. There are many models which have supersymmetry which is useful in their theoretical analysis, but the same models can be treated in formalisms not knowing about supersymmetry. Wheather the fundamental physics needs a model which has nontrivial supersymmetry is a speculative statement, and I disagree with equating theoretical physics with one direction in “fundamental physics”. I do not understand how can a model suggest supersymmetry; it is rather experimental evidence or problems with nonsupersymmetric models. Also one should distinguish the supersymmetry at the level of Lagrangean and the supersymmetry which holds only for each solution of the equation of motion.
I’ll rephrase the original statement to something less optimistic, but i do think that supersymmetry is suggsted more by looking at the formal nature of models than by lookin at the nature of nature. If you have a gauge theory for some Lie algebra (gravity, Poincaré Lie algebra) and the super extension of the Lie algebra has an interesting classification theory (the super Poincar´ algebra) then it is more th formalist in us who tends to feel compelled to investigate this than the phenomenologist. Supersymmetry is studied so much because it looks compelling on paper. Not because we have compelling phenomenological evidence. On the contrary.
So, if you don’t mind, I will remove both your query boxes and slightly polish the entry. Let’s have any further discussion here.
Okay, I have edited the entry.
I do not see any present mention of word “action functional” or of ’path” in supersymmetry. Are you referring to another entry ?
Peter Woit discusses the negative preliminary findings about supersymmetry at LHC
Yes, I have been following this.
I was already making plans for a blogpost trying to explain the distinction between worldvolume and target space supersymmetry.
Because, you know, the Lagrangian for the spinning particle is supersymmetric, one cannot help it. So on the worldline supersymmetry is experimentally verified long ago.
Similarly, the superstring was not originally designed to be supersymmetric. It just so happens that in dim 2 the evident fermion action makes the thing supersymmetric. That’s how supersymmetry was found in the west in the first place.
So it’s misleading to say that anyone “throws supersymmetry out of the window”. What is thrown out here is the idea of global supersymmetry in the effective target space theory. This is some very special model, and not “supersymmetry” as such.
Interesting to note in this context is the article by Dienes that somebody mentioned: that gives an argument (which is heuristically evident) that the moduli space of heterotic string backgrounds has almost not points corresponding to supersymmetric target space theory. But: it’s the heterotic string! So it’s worldsheet supersymmetric in any case.
We almost got into this discussion here on the $n$Café
Saying that the superstring implies global spacetime supersymmetric is mixing up two different things: Superstring theory (and super-particle theory for that matter) is perfectly consistent with there being no global susy. In fact, generically the target space theory of a susy worldvolume theory is not globally susy. This is precisely the same kind of statement as in the non.susy case: the target space theory of a locally Lorentz invariant worldvolume theory (say the relativistic particle) is of course in general not globally Lorentz invariant (it may be an arbitrary pseudo-Riemannian manifold). (This is so obvious that nobody ever talks about it, but it is precisely the same as for the super case.)
Wait a second, supersymmetry was needed in string theory for cancellation of anomalies. Now let us go to low energy regime. Are you saying that in “low” energy regime you may not have superpartners anyway ?
All the particle physics, experiment and media care if there are superparticles or not. Not the phrasing of abstract theory. Can you get cancellations of anomaly due supersymmetry without ever seeing superpartners or not ?
Or you are saying that superparticles of another kind which does not correspond to any of the models tested at LHC, as they ponder global supersymmetry ? I mean maybe not superpartners in the usual sense but some other effective quanta at low energy ? Those should still interact if they exist at some energy scale…Interaction responsible for cancellation should eventually be seen via particle processes. Is there any model which does that and could eventually be compared to pheno ?
By the way it would be good that you write a physics discussion of what you wrote above into $n$Lab. I mean some toy model, toy equations illustrating your point. The way physicist understands. I understand the distinctions you are making, but do not see how the cancellation reflects at low energy (unless there is some huge breaking of symmetries and additional mechanism of balancing fine tuning the extraeffects of it) in your scenario of no target space supersymmetry.
Wait a second, supersymmetry was needed in string theory for cancellation of anomalies.
No! The step from the bosonic string to the superstring removes the tachyon excitation in the plain Minkowski background. Both are anomaly free in their critical dimension (26 and 10, respectively, for trivial dilaton backgrounds).
Are you saying that in “low” energy regime you may not have superpartners anyway ?
No, wait, let’s recall how it works:
the superstring has local supersymmetry on the worldsheet, which really just means that the worldsheet theory is worldsheet supergravity, hence a $\mathfrak{superPoincare}(1,1)$-gauge theory.
A background for this is a classical solution of supergravity (heterotic, type II, or type I, depending on which critical superstring we are looking ar) in 10d.
That solution may or may not have global supersymmetries. What does this mean? This is the analog of saying: a given solution to the bosonic Einstein equations has a Killing vector. A global supersymmetry is a “Killing spinor” called a “covariant constant spinor”.
It is an ansatz – motivated not from some deep theory but from phenomenological model building – that one demands this solutions to be of the form $M^4 \times Y_6$ and demands a single Killing spinor. Because this gives global $N = 1$ superymmetry for the effective target space theory. Solving this ansatz yields the condition that $Y_6$ is a Calabi-Yau and the rest is history. That’s where all these CYs etc. come from. This is not theory, this is model buidling.
In particular, at the level of rigour available it is clear that the generic solution will have no Killing spinor at all. Just as the generic solutions to the bosonic Einstein equations has no Lorentz symmetry left: no Killing vector.
Nevertheless, no matter what the background is, the worldsheet theory is always locally Poincaré-symmetric (in the bosonic case) or super-Poincaré-symmetric (in the super case).
And this has nothing particularly to do with the idea of strings: take the ordinary relativistic spinning particle – the stuff that you, me and everyhting around us is made of!! – write out its worldline Lagrangian. This is supersymmetric on the worldline. Period. Just like the Klein-Gordon particle is given by 1-dimensional gravity coupled to matter on the worldline, the spinning particle is given by 1-dimensional supergravity on the worldline. Still, the background that this propagates in may have but generically will not have any global supersymmetry.
I’ll write that out in an $n$Lab entry now.
Hi Zoran,
I have collected now a bunch of references that discuss how the action functional for ordinary Dirac spinors (electrons, quarks: experimentally well-verified stuff) is supersymmetric on the wordline: at spinning particle.
I have now added discussion along the line of the comments that I made above in the Idea-section at supersymmetry.
while probably not a wise idea to spend my night with this, I did now make a blog post about this here
Urs, I know that there is certain cancellation of anomalies for bosonic string as well, this is the first thing I learnd about it 20 years ago; I said it wrong way. Still once you are in supersymmetric theory the symmetry enables often the low energy effective theory to have various cancellations already at the level of perturbation theory (features like renormalizabiliy, especially question of the cosmological constant etc.).
Now you are telling me about hidden worldline symmetry of various models. I have no problem with this, and was for short doing research when graduate student related to some formal models like that. The Wess-Zumino model has been introduced as a non-supersymmetric model and only a laborious calculation by its authors revealed its hidden supersymmetry.
Now, you are selling us the story that nobody meant in supersymmetry theory superpartners, that it is just a wolrd-sheet supersymmetry and no space-time supersymmetry. This may be true, and it is a theoretical possibility. But this is NOT the conventional wisdom. The words-sheet is in Ramond-Schwarz-Neveu formalism. After GSO projection people got even there equal number of fermions and bosons at each mass level; this is not a proof but indication that there should be a space-time supersymmetry. Moreover, it is considered equivalent to Green-Schwarz approach which has manifest supersymmetry. I do not know about heterotic case from this point of view but all 5 string theories (and matrix theory) do give supergravity action in low energy limit. Supergravity is about space-time supersymmetry isn’t it ??
10 Urs, I was asking about a model related to string theory. I know the kind of examples you are presenting here. I said I know the distinctions you are making, I just say that appealing features of string theory require finiteness results which are not in other theories and which require cancellations due supersymmetry. Those arguments are often made at perturbative level and I do not see them without superpartners.
Urs, I know that there is certain cancellation of anomalies for bosonic string as well, this is the first thing I learnd about it 20 years ago;
Okay, sorry.
Still once you are in supersymmetric theory the symmetry enables often the low energy effective theory to have various cancellations already at the level of perturbation theory (features like renormalizabiliy, especially question of the gravitational constant etc.).
Sure, right.
The Wess-Zumino model has been introduced as a non-supersymmetric model and only a laborious calculation by its authors revealed its hidden supersymmetry.
Yes, exactly. I have just created an entry spinning string to record precisely this fact.
Do you know which article exactly in the 1970s first pointed out that the NSR model is supersymmetric? I mean the very first one? I would like to cite that, but am not sure which one it is right now.
Now, you are selling us the story that nobody meant in supersymmetry theory superpartners,
No, why do you say this? Also local supersymmetric theory (sugra) has superpartners, of course.
that it is just a wolrd-sheet supersymmetry and no space-time supersymmetry. This may be true, and it is a theoretical possibility. But this is NOT the conventional wisdom.
I think this is exactly the conventional wisdom. It’s only that part of this story tend to be forgotten as people talk and talk. There is just as much literature on local supersymmetry as there is on global one.
After GSO projection people got even there equal number of fermions and bosons at each mass level; this is not a proof but indication that there should be a space-time supersymmetry
Yes, of course. This does prove that the space-time theory is also supersymmetric. But locally supersymmetric not necessarily globally: it is known that the effective target space theory of the superstring is supergravity+extra fields. Of course. But the assumption that the standard model sits in a theory of supergravity does not imply that we see superpartners in the accelerators. Because in these models the particles that we do see that KK-modes (or maybe open string modes) of the truly fundamental excitations of the sugra theory, and these have superpartners only if the 10-dimensional spacetime geometry has a global Killing spinor. But in general it will not.
I do not know about heterotic case from this point
That, too, has a supergravity theory as its target space theory. Called “heterotic supergravity”.
Urs said at cafe:
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.
I agree absolutely with that for a general supersymmetric sigma model. But a general sigma model does not have good properties of string theory. There are formulations of string theory which are manifestly space-time invariant. I am not saying it is proved exactly that they are the same as Ramond-Neveu-Schwarz but it is enough circumstancial evidence that they give the same, and especially in low limit.
Now you say some guy had in 2008 paper in which he says something different about the space of vacua of heterotic string. I do understand that this is possible, but this is going against the mainstream in string theory.
John Schwarz, Becker and Becker say in their textbook:
Space-time supersymmetry is one of the major predictions of superstring theory.
In the same paragraph earlier they talk about local supersymmetry. They know all you say of course and still say that the theory predicts space-time supersymmetry.
Oh, now I learned something substantial. You are saying if I understood that (10d)supergravity is not giving the effective global supersymmetry in low energy limit (and in 4d)either. This was my misconception.
Do you know which article exactly in the 1970s first pointed out that the NSR model is supersymmetric?
I don’t. By storytelling I know that the first evidence was circumstancial counting fermions and bosons after using GSO projection, but I am not sure and never looked into original GSO articles if already they discuss this. If you can look, these are the candidates:
F. Gliozzi, J. Scherk, D. I. Olive, Supergravity and its spinor dual model, Phys. Letters B65, 282, 1976
F. Gliozzi, J. Scherk, D. I. Olive, Supersymmetry, supergravity theories and the dual spinor model, Nucl. Phys, B122, 253, 1977
This does prove that the space-time theory is also supersymmetric. But locally supersymmetric not necessarily globally: it is known that the effective target space theory of the superstring is supergravity+extra fields. Of course. But the assumption that the standard model sits in a theory of supergravity does not imply that we see superpartners in the accelerators. Because in these models the particles that we do see that KK-modes (or maybe open string modes) of the truly fundamental excitations of the sugra theory, and these have superpartners only if the 10-dimensional spacetime geometry has a global Killing spinor. But in general it will not.
Urs, this is very useful. This is kind of arguments I would like to learn in more detail (maybe it is good to eventually organize the discussion and entries to have first zeroth level about the concepts, like the examples of local vs global supersymmetry and then at next level things which are specific to superstrings in 10 dim, and then separately to heterotic string; my weak point of understanding is 10d sugra). The requirement about global Killing spinor, can be probably at least weakened, regarding that we have compactification to 4 dimensions ?
Oh right, my guess in 18, about the reference you asked for, was correct, according to Schwarz:
The string theory introduced in early 1971 by Ramond, Neveu, and myself has two-dimensional world-sheet supersymmetry. This theory, developed at about the same time that Golfand and Likhtman constructed the four-dimensional super-Poincar'e algebra, motivated Wess and Zumino to construct supersymmetric field theories in four dimensions. Gliozzi, Scherk, and Olive conjectured the spacetime supersymmetry of the string theory in 1976, a fact that was proved five years later by Green and myself.
The second reference mentioned in the abstract is
There are formulations of string theory which are manifestly space-time invariant.
Yes, the Green-Schwarz string! But still, the spacetime theory is locally supersymmetric (being supergravity).
So what would you expect phenomenologically as a consequence of 10d sugra in 4d at low energies accessible to nearby range of pheno ?
I have split off superstring in order to collect the links to the various formulations.
Now you say some guy had in 2008 paper in which he says something different about the space of vacua of heterotic string. I do understand that this is possible, but this is going against the mainstream in string theory.
First, Dienes is not quite “some guy” but a well-known hep and susy researcher. He was organizer of annual susy conferences.
Second: the “main stream string theory” statements that you are probably thinking of here are all the usual non-technical statements in introductions and outlooks. There are very few texts who consider soberly the question as to how likely a vacuum with global supersymmetry is. And it is heuristically clearl that this is very unlikely. That article by Dienes only adds some quantitative sugar to this.
John Schwarz, Becker and Becker say in their textbook:
Space-time supersymmetry is one of the major predictions of superstring theory.
In the same paragraph earlier they talk about local supersymmetry. They know all you say of course and still say that the theory predicts space-time supersymmetry.
Yes, they all know it and still say it in a misleading fashion. The more careful statement is that string theory predicts local spacetime supersymmetry while everybody who talks particle phenomenology implicitly means global supersymmetry.
I know that the first evidence was circumstancial counting fermions and bosons after using GSO projection,
Right, thanks for listing those GSO references. I have included them now into spinning string, with a brief comment in the Idea-section
Oh right, my guess in 18, about the reference you asked for,
Thanks for these references! Very good. I have included them into a new History-section at supersymmetry and into the references at spinning string
Oh, now I learned something substantial. You are saying if I understood that (10d)supergravity is not giving the effective global supersymmetry in low energy limit (and in 4d)either.
Yes! Not generically, that it. Only if it is compactified on a certain geometry, such that there is a Killing spinor.
Urs, this is very useful. This is kind of arguments I would like to learn in more detail (maybe it is good to eventually organize the discussion and entries to have first zeroth level about the concepts, like the examples of local vs global supersymmetry and then at next level things which are specific to superstrings in 10 dim, and then separately to heterotic string; my weak point of understanding is 10d sugra).
Right, we should eventually expand on all that. I am just a bit worried that I won’t have much time. I am already spending time on this right now that I don’t actually have ;-)
The requirement about global Killing spinor, can be probably at least weakened, regarding that we have compactification to 4 dimensions ?
On a product spacetime of the form $M^4 \times Y^6$ the condition of a covariant spinor is non-trivial only on $Y^6$, and there it is solved by the CY conditon. So this is for supersymmetry in 4-dimensions: the compactified dimensions need to admit a global supersymmetry.
This argument is generalized when further background gauge fields are taken into account. For instance if the Kalb-Ramond gerbe has nontrivial background curvature then the spinor needs to be covariantly constant with respect to the metric with torsion whose torsion is that curvature 3-form. And so on. Some of these “generalized Calabi-Yau spaces” have therefore more recently been usefully formulated in terms of generalized complex geometry.
So what would you expect phenomenologically as a consequence of 10d sugra in 4d at low energies accessible to nearby range of pheno ?
I would not really expect anything at all! :-)
Are you saying in 28 that without a gerbe we need a CY and we do get the global supersymmetry in 4d, but if we are having a gerbe (with nontrivial curvature or whatever) this relaxes the Calabi-Yau condition to allow more general compactification and that only in that case the 10 d sugra gives 4d effective theory possibly without global supersymmetry ? If so, why would then the moduli space be dominated (for heterotic string vacua I guess) with curved gerby cases ?
I did not know who was Dienes in the subject, sorry for calling him “some guy”.
Oh, you also possibly meant in 28 that the first factor is not Minkowski $M^4$ and that this is generic (but should such a large scale feature affect crucially the phenomenology?).
Yes, if the KR field has nontrivial field strength, the compact space must be some “generalized Calabi-Yau” for there to be a covariantly constant spinor.
More generally, if there are also RR-fields, the dilaton, etc. then all these change the CY condition to something related but different.
You see, this comes from the following kind of argument:
Supersymmetry transformations change fields by the covariant derivative $\nabla \xi$ of the “susy parameter” $\xi$, which is some section of the spinor bundle (compare with the analogous formula for infinitesimal diffeomorphisms). So to have a global symmetry we want this to vanish.
But $\nabla$ here in general contains not just the metric, but contributions from all the background fields. It is a “generalized” covariant derivative. If it only depends on the metric, then the condition that $ker \nabla$ is 1-dimensional on a 6d space is exactly the CY condition. Otherwise it is some deformation theoreof.
I have included a pointer to a lecture note pdf that discusses this a bit in the references at supergravity. I should eventially add more canonical references.
I have included a pointer to a lecture note pdf that discusses this a bit in the references at supergravity.
Sorry, no, I haven’t! I thought I did. Will do so now.
Here is a decent reference:
I have added at supergravity a new section Solutions with global supersymmetry with some remarks along these lines.
Oh no! Now the new edit-bug makes editing supersymmetry impossible.
Darn. But maybe I should take a break anyway.
Interesting historical remarks about the precursors of supersymmetry considerations in physics and mathematics in late 1960s, before Volkov and Akulov.
Rosenberg was telling me that he was in a conference where Kac and some physicist had presentations on the subject somewhere in central asia (or Caucasus) in 1969, when he was student, and this fits with this account on Georgii Stavraki. The book can be downloaded at https://theor.jinr.ru/~belyov/books/Bagger_SUSYEnciclopedia.pdf.
That’s a great passage to read! We should add it to the history-section at supersymmetry. Can you edit the entry? If so – or when so – please do. I may have to go offline now.
Will do at some point, most likely when getting to a better connection to consult some additional sources.
started a section Supersymmetry – Classification.
Worked a bit on putting content into the further subsection Classification – Superconformal symmetry.
I have there now spelled out the statement and proof of the classification theorem which states that the only superconformal super Lie algebras above dim 2 are
$d$ | $N$ | superconformal super Lie algebra | R-symmetry | brane worldvolume theory |
---|---|---|---|---|
3 | $2k+1$ | $B(k,2) \simeq$ osp$(2k+1,4)$ | $SO(2k+1)$ | |
3 | $2k$ | $D(k,2)\simeq$ osp$(2k,4)$ | $SO(2k)$ | M2-brane |
4 | $k+1$ | $A(3,k)$ | $U(k+1)$ | D3-brane |
5 | 1 | $F(4)$ | $SO(3)$ | |
6 | $k$ | $D(4,k) \simeq$ osp$(8,2k)$ | $Sp(k)$ | M5-brane |
Despite what I thought before actually looking at it, this is not exactly the statement in (Nahm 78). (I suppose this doesn’t restrict attention to genuine spinor reps of the even subalgebra on the odd one.) The statement is actually due to (Shnider 88) which emphasizes the non-existence of superconformal algebras in dimension $d \gt 6$ as its main theorem and has discussion of the above table somewhat tersely and more as a side-remark on its last two pages. Useful review of this that also adds the discussion of unitary supermultiplets (which finally gives the actual classification of relevance in physics) in Minwalla 98, section 4.
If anyone has further pertinent references, let me know.
I have added an actual Definition section to the entry supersymmetry.
(This is taken now from the more comprehensive lecture notes at geometry of physics – supersymmetry.)
Edit to: supersymmetry by Urs Schreiber at 2018-04-01 01:18:05 UTC.
Author comments:
added textbook reference
added (re-)publication data to
{#GolfandLikhtman72} Yuri Golfand, Evgeny Likhtman,On the Extensions of the Algebra of the Generators of the Poincaré Group by the Bispinor Generators, in: Victor Ginzburg et al. (eds.) I. E. Tamm Memorial Volume Problems of Theoretical Physics, (Nauka, Moscow 1972), page 37,
translated and reprinted in: Mikhail Shifman (ed.) The Many Faces of the Superworld pp. 44-53, World Scientific (2000) (doi:10.1142/9789812793850_0006)
added these pointers:
{#Likhtman01} Evgeny Likhtman, Around SuSy 1970, talk at “30 years of supersymmetry”, Nucl.Phys.Proc.Suppl. 101 (2001) 5-14 (arxiv:hep-ph/0101209)
Pierre Ramond, SuSy: The Early Years (1966-1976), Eur. Phys. J. C (2014) 74: 2698 (arxiv:1401.5977)
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