# Supercharge transformation rules

Consider $${\cal N}=2$$ supersymmetry with $$SU(2)$$ global symmetry group. Then both supercharges $$Q_{ai},\bar{Q}_{\dot{a}\dot{j}}$$ transform by 2 dimensional representation of $$SU(2)$$. Denote $$SU(2)_I$$ as the aforementioned group.

Now consider flat $$R^4$$ with rotation $$Spin(4)\cong SU(2)_L\times SU(2)_R$$. Then supercharges transform like $$(2,1,2)\oplus (1,2,2)$$ respectively. I guess we mean in Euclidean field theory sense in this part.

Consider $$SU(2)_L\times SU(2)_D\cong Spin(4)\subset SU(2)_L\times SU(2)_R\times SU(2)_I$$ where $$SU(2)_D$$ denotes diagonal part of $$SU(2)_R\times SU(2)_I$$.

Now the book says supercharges transform as $$(2,2)\oplus (1,3)\oplus(1,1)$$ under $$SU(2)_L\times SU(2)_D$$.

I will discount the trivial representations. $$(2,2)$$ should correspond to $$Q_{ai}$$'s part. It seems that $$(1,3)\oplus (1,1)$$ is mixture of $$\bar{Q}_{\dot{a}\dot{j}}$$ as $$1/2\otimes 1/2=1\oplus 0$$ where $$1/2,1,0$$ denote total spin.

$$\textbf{Q:}$$ Is above correct? It seems that I do not have a systematic way to deal with this splitting. What would be a systematic way of dealing with this?

• @Qmechanic Sorry my bad. It should be arxiv.org/abs/hep-th/9408074. It is pg 8 top and bottom paragraphs. – user45765 Dec 17 '18 at 20:44
• pg 8 in PDF reader; pg 7 in text. – Qmechanic Dec 17 '18 at 21:04

## 1 Answer

Yes, OP is right. Ref. 1 is considering topological twisting. Let $$G:=SU(2)$$ to simplify notation. Let $$K:=G_L\times G_R$$ be the (double cover of the) group of Euclidean spacetime rotations. Let $$G_D\to H:=G_R\times G_I$$ be the diagonal embedding $$g\mapsto (g,g)$$. An irreducible representation of $$H$$ corresponds to a (possibly reducible) tensor product of representations of $$G_D$$. Then:

• The supercharge $$Q_{\alpha i}$$ transforms in the $${\bf 2}_L\otimes{\bf 1}_R\otimes{\bf 2}_I$$ of $$G_L\times G_R\times G_I$$, and hence in the $${\bf 1}_R\otimes{\bf 2}_I$$ of $$H$$, which corresponds to the $${\bf 1}_D\otimes{\bf 2}_D\cong {\bf 2}_D$$ of $$G_D$$.

• The supercharge $$\bar{Q}_{\dot{\alpha} i}$$ transforms in the $${\bf 1}_L\otimes{\bf 2}_R\otimes{\bf 2}_I$$ of $$G_L\times G_R\times G_I$$, and hence in the $${\bf 2}_R\otimes{\bf 2}_I$$ of $$H$$, which corresponds to the $${\bf 2}_D\otimes{\bf 2}_D\cong {\bf 3}_D\oplus {\bf 1}_D$$ of $$G_D$$.

References:

1. C. Vafa & E. Witten, arXiv:hep-th/9408074; p. 7.
• Was there a systematic treatment of the procedure(decomposition of representations) as described? Say a textbook or a reference book? Thanks. – user45765 Dec 17 '18 at 22:16
• I updated the answers with some links. – Qmechanic Dec 17 '18 at 22:29