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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?

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  • $\begingroup$ @Qmechanic Sorry my bad. It should be arxiv.org/abs/hep-th/9408074. It is pg 8 top and bottom paragraphs. $\endgroup$ – user45765 Dec 17 '18 at 20:44
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    $\begingroup$ pg 8 in PDF reader; pg 7 in text. $\endgroup$ – Qmechanic Dec 17 '18 at 21:04
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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.
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  • $\begingroup$ Was there a systematic treatment of the procedure(decomposition of representations) as described? Say a textbook or a reference book? Thanks. $\endgroup$ – user45765 Dec 17 '18 at 22:16
  • $\begingroup$ I updated the answers with some links. $\endgroup$ – Qmechanic Dec 17 '18 at 22:29

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