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Given some spacetime dimensions $D$, are there only certain allowed supersymmetry charge nunbers $\mathcal{N}$?

What are the relations of $\mathcal{N}$ and $D$ for the following cases:

  1. When the theory is conformal.

  2. When the theory does not have to be conformal.

  3. When the theory is Lorentz invariant.

  4. When the theory does not have to be Lorentz invariant.

Possible other situations are worthwhile to comment on relating $\mathcal{N}$ and $D$?

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All the questions are basically answered in the classic paper "Supersymmetries and their representations". See also the wonderful talk: What's new with Q?.

1.- When the theory is conformal:

In $D=2$ $N=(1,0)$ (heterotic and type I strings), $N=(1,1)$ (type $IIB$ string), $N=(2,0)$ (type $IIA$ string), $N=(2,2)$ ( N=2 strings), $N=(2,1)$ ($N=2$ Heterotic strings) and $N=4$ strings are alloweed.

For the remaining I change the notation to enumerate the number of possible supercharges. In $D=3$ $N=2,4,6,8,10,12,16$ are alloweed. $D=4$ has $N=4,8,12,16$. $D=5$ $N=8$ is the only option and for $D=6$ the options are $N=8$ and 16 supercharges.

2.- No satisfactory answer can exist (to my poor knowledge). See https://arxiv.org/abs/hep-th/9409111 and https://arxiv.org/abs/hep-th/9506101 for interesting subtleties in $D=3$.

To answer 3) and 4): Supersymmetry is the "square root of the Poincaré group". Supersymmetry enforces Poincaré invariance. And basically all the possibilities are the number of supercharges of all string theories and the eleven dimensional supergravity. You can check the precise answers in The String Landscape, the Swampland, and the Missing Corner (page 5).

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    $\begingroup$ See physics.stackexchange.com/q/587895/42982 . is this consistent and familiar then? $\endgroup$
    – ann marie cœur
    Oct 18, 2020 at 3:15
  • $\begingroup$ Yes, perfectly consistent. The notation is confusing, but the results agree. The notation $\mathcal{N}=(p,q)$ means that there are p left-handed Majorana spinors and q right-handed ones in a given dimension; on the other hand, the notation $N=c$ means that the total number of spinor components of both chiralities is c; see page 5 in arxiv.org/pdf/1711.00864.pdf for a dictionary in some cases. Let's check some: The worldvolume theory of the $D5$-brane has $\mathcal{N}=(1,1)$ in $D=6$ in your notation; the number of supercharges is $N$=8(1+1)=16, a possibility I had enumerated. $\endgroup$
    – Ramiro Hum-Sah
    Oct 18, 2020 at 4:10
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    $\begingroup$ then can you answer my other question then? $\endgroup$
    – ann marie cœur
    Oct 18, 2020 at 4:15
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    $\begingroup$ thanks I voted up. You may want to look at the Table in Sec 3.8 p.15 of paper I found scipost.org/10.21468/SciPostPhys.7.5.058 $\endgroup$
    – ann marie cœur
    Oct 18, 2020 at 4:17
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    $\begingroup$ Your definition "Supersymmetry is the "square root of the Poincaré group"" is unnecessarily restrictive. It is perfectly fine to have non-relativistic theories with odd symmetries, in which case it is better to say that SUSY is the square root of Bargmann, for example. Or even in situations where some isometries are broken, so you have the square root of some subgroup of Poincare/Bargmann. This is precisely the situation OP has in mind in 3-4. $\endgroup$
    – AccidentalFourierTransform
    Oct 18, 2020 at 22:36

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