# Question about multiplets of 6d $\mathcal{N}=(1,0)$ SUSY

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?

• Is it not just the vector multiplet? – Elliot Schneider Mar 23 '17 at 13:32
• 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. – leastaction Mar 23 '17 at 13:36
• 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 ? – Antoine Mar 23 '17 at 19:13
• What kind of multiplet would that be? The gravity multiplet includes a gravitino. – leastaction Mar 23 '17 at 19:18
• It would be a 3-form multiplet. However I can't find any precise reference about it. – Antoine Mar 23 '17 at 19:50

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.