Supersymmetry $\mathcal{N}$ constrained by spacetime dimensions $D$

Question

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 $D$?

This post imported from StackExchange Physics at 2020-12-07 19:33 (UTC), posted by SE-user annie marie heart

Answer 1

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).

This post imported from StackExchange Physics at 2020-12-07 19:33 (UTC), posted by SE-user Ramiro Hum-Sah

Comments

I recommend the appendix on spinors in the Polchinski's string theory textbook for conventions and facts on spinors in diverse dimensions, to workout all the remaining examples; except one, the $D=7$ case. The wolrdvolume theory of the $D6$ brane is a topological theory. Standard references that discuss the theory (and other $D>6$ cases) are arxiv.org/abs/hep-th/9705138,https://arxiv.org/abs/hep-th/…, arxiv.org/abs/hep-th/0404041 arxiv.org/abs/hep-th/0602087. First, please don't get confused. Those theories are topological.

This post imported from StackExchange Physics at 2020-12-07 19:34 (UTC), posted by SE-user Ramiro Hum-Sah

---

The classification of the theories I've offered is for conformal and interacting quantum field theories in diverse dimensions. Topological field theories does not have bulk interactions and does not obey Lorentz symmetry.

This post imported from StackExchange Physics at 2020-12-07 19:34 (UTC), posted by SE-user Ramiro Hum-Sah

---

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.

This post imported from StackExchange Physics at 2020-12-07 19:34 (UTC), posted by SE-user AccidentalFourierTransform

---

Thanks for your valuable comment @AccidentalFourierTransform. My knowledge of of the situations you mention is poor.

This post imported from StackExchange Physics at 2020-12-07 19:34 (UTC), posted by SE-user Ramiro Hum-Sah

---

What make differences for Topological field theories? I think there are SUSY also mentioned in the paper?

This post imported from StackExchange Physics at 2020-12-07 19:34 (UTC), posted by SE-user annie marie heart

---

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.

This post imported from StackExchange Physics at 2020-12-07 19:34 (UTC), posted by SE-user Ramiro Hum-Sah

---

then can you answer my other question then?

This post imported from StackExchange Physics at 2020-12-07 19:34 (UTC), posted by SE-user annie marie heart