In Strathdeee's "Extended Poincare Supersymmetry", the first entry on page 16 lists the massless multiplets of 6d $\mathcal{N} = (1,0)$ supersymmetry as

  • $2^2 = (2,1; 1) \oplus (1,1; 2)$. This is the half-hyper (matter) multiplet.
  • $(2,1;1) \otimes 2^2 = (3, 1;1) \oplus (1,1; 1) \oplus (2,1; 2)$. This is the tensor multiplet.
  • $(1,2;2) \otimes 2^2 = (2,2;1) \oplus (1,2;2)$. This is the vector (Yang-Mills) multiplet.
  • $(2,3;1) \otimes 2^2 = (3,3;1) \oplus (1,3;1) \oplus (2,3;2)$. This is the gravity multiplet.

where the entries specify representations of the little group $SO(4) \simeq SU(2) \times SU(2)$ and the R-symmetry group $USp(2) \simeq SU(2)$.

But there's also another entry:

  • $(1,2;3) \otimes 2^2 = (2,2; 3) \oplus (1,2;2) \oplus (1,2;4)$

which consists of (1) a vector transforming in the adjoint of the R-symmetry, (2) a Weyl spinor transforming in the doublet of R-symmetry, and (3) another Weyl soinor transforming in the 4-dimensional representation of the R-symmetry group.

What is this fifth multiplet? Is there some reason why it doesn't feature in discussions about 6d $\mathcal{N} = (1,0)$ theories, even in papers from the 90s by Seiberg and others?

  • $\begingroup$ Is it not just the vector multiplet? $\endgroup$ – Elliot Schneider Mar 23 '17 at 13:32
  • $\begingroup$ Thanks @user81003 for your comment. I just realized that I had forgotten to write the Yang-Mills (vector) multiplet. In response to your question, no, the mysterious fifth multiplet isn't just the vector multiplet. Hence my question. $\endgroup$ – leastaction Mar 23 '17 at 13:36
  • $\begingroup$ Maybe it is a multiplet that is specific to theories with gravity, and therefore it is not present in the (non-gravitational) analysis of Seiberg and others ? $\endgroup$ – Antoine Mar 23 '17 at 19:13
  • $\begingroup$ What kind of multiplet would that be? The gravity multiplet includes a gravitino. $\endgroup$ – leastaction Mar 23 '17 at 19:18
  • $\begingroup$ It would be a 3-form multiplet. However I can't find any precise reference about it. $\endgroup$ – Antoine Mar 23 '17 at 19:50

In Free equations of motion for all D = 6 supermultiplets, page 224:

For example $$(1,2;3) \otimes 2^2 = (2,2; 3) \oplus (1,2; 2) \oplus (1,2; 4)$$ can be interpreted as an $USp(2)$ $(2,0)$-Yang-Mills multiplet, its component field strengths being $(\lambda_a^{A\alpha}, F^{A\beta}_{\alpha})$ where A is an $USp(2)$ adjoint index. Notice also that any multiplet with extended supersymmetry can be thought of as being composed of simple (i.e. $(2,0)$ and $(0,2)$) multiplets, with appropriate assignments to extended $USp(N)$ irreps.


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