added the following (here) to the History-section:

On the origin of the “super”-terminology, from Kane-Shifman 01, p. 4:

Often students ask where the name “super-symmetry” came from? It seems that it was coined in the paper by Salam and Strathdee where these authors constructed supersymmetric Yang-Mills theory. This paper was received by the editorial office on June 6, 1974, exactly eight months after that of Wess and Zumino. Super-symmetry (with a hyphen) is in the title, while in the body of the paper Salam and Strathdee use both, the old version of Wess and Zumino, “super-gauge symmetry”, and the new one. An earlier paper of Ferrara and Zumino (received by the editorial officec on May 27, 1974) where the same problem of super-Yang-Mills was addressed, mentions only supergauge invariance and supergauge transformations.

Accounts of the early history of the concept of supersymmetry:

{#KaneShifman00} Gordon Kane, Mikhail Shifman,

*The Supersymmetric World: The Beginnings of The Theory*, World Scientific 2000 (doi:10.1142/4611)- {#KaneShifman01} Gordon Kane, Mikhail Shifman,
*Introduction to “The Supersymmetric world”*(arXiv:hep-ph/0102298), chapter in: Kane-Shifman 00

- {#KaneShifman01} Gordon Kane, Mikhail Shifman,

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added DOI to

- Steven Shnider,
*The superconformal algebra in higher dimensions*, Letters in Mathematical Physics November 1988, Volume 16, Issue 4, pp 377-383 (doi:10.1007/BF00402046)

and to

- Shiraz Minwalla,
*Restrictions imposed by superconformal invariance on quantum field theories*, Adv. Theor. Math. Phys. 2, 781 (1998) (arXiv:hep-th/9712074, doi:10.4310/ATMP.1998.v2.n4.a4)

added DOI to

- {#Nahm78} Werner Nahm,
*Supersymmetries and their Representations*, Nucl.Phys. B135 (1978) 149 (spire:120988, doi:10.1016/0550-3213(78)90218-3)

added to the History-section these two references on hadron supersymmetry

Hironari Miyazawa,

*Baryon Number Changing Currents*, Prog. Theor. Phys. 36 (1966) 6, 1266-1276 (spire:1235194, doi:10.1143/PTP.36.1266)Hironari Miyazawa,

*Spinor Currents and Symmetries of Baryons and Mesons*, Phys. Rev. 170, 1586 (1968) (doi:10.1103/PhysRev.170.1586)

which pre-date the Neveu-Ramond-Schwarz and Golfsnd-Likhtman articles

]]>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)

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)

**Edit to**: supersymmetry by Urs Schreiber at 2018-04-01 01:18:05 UTC.

**Author comments**:

added textbook reference

]]>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*.)

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.

]]>Will do at some point, most likely when getting to a better connection to consult some additional sources.

]]>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.

]]>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.

]]>Oh no! Now the new edit-bug makes editing supersymmetry impossible.

Darn. But maybe I should take a break anyway.

]]>Here is a decent reference:

- Mariana Graña, Ruben Minasian, Michela Petrini, Alessandro Tomasiello,
*Generalized structures of $N=1$ vacua*(arXiv:hep-th/0505212)

I have added at supergravity a new section Solutions with global supersymmetry with some remarks along these lines.

]]>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.

]]>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.

]]>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?).

]]>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”.

]]>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! :-)

]]>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.

]]>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.

]]>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

]]>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

]]>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.

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