# Dimension reduction of supersymmetric Yang-Mills from 10D to 4D

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I was recomputing the dimension reduction of the 10-dimensional SYM theory to 4-dimension in an old paper "Supersymmetric Yang-Mills theories" https://lib-extopc.kek.jp/preprints/PDF/1977/7703/7703036.pdf.

Explicit 10D gamma matrices $\Gamma^\mu,\mu=0,1,...,9$ were not given in that paper, instead, they were given in the form $\Gamma^{ij},i,j=1,2,3$(see equation (5.3) in the paper). So my question is what is the form of $\Gamma^\mu,\mu=1,2,...9$?

I have tried the so-called "friendly representation" of gamma matrices to produce 10D gamma matrices. In this representation one can obtian gamma matrices in $(2m)$D from the gamma matrices in $(2m-2)$D.

The method works as follows: Denote the gamma matrices in $(2m)$D dimensional spacetime as $\Gamma$, and $\gamma$ in $(2m-2)$D.
$$\Gamma^\mu=\gamma^\mu\otimes I_2\\ \Gamma^{2D-2}=\gamma^{2D-1}\otimes i\sigma_1\\ \Gamma^{2D-2}=\gamma^{2D-1}\otimes i\sigma_2\\ \Gamma^{2D+1}=\gamma^{2D-1}\otimes \sigma_3$$

So started from 2D gamma matrices in that paper which are $$\gamma^0=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) ,~\gamma^1=\left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right),~\gamma_3=\gamma^0\gamma^1=\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)$$ one can obtain the 4D,6D,8D,10D gamma matrices, and the explicit form of 10D matrices in terms of 4D gamma matrices are $$\Gamma_{(10)}^\mu=\Gamma_{(4)}^\mu\otimes I_2\otimes I_2\otimes I_2,~~\mu=0,1,2,3\\ \Gamma_{(0)}^0=\Gamma_{(4)}^0\otimes I_2\otimes I_2\otimes I_2=\sigma_3\otimes I_2\otimes I_2\otimes I_2\otimes I_2\\ \Gamma_{(1)}^1=\Gamma_{(4)}^1\otimes I_2\otimes I_2\otimes I_2=i\sigma_2\otimes I_2\otimes I_2\otimes I_2\otimes I_2\\ \Gamma_{(4)}^2=\Gamma_{(4)}^2\otimes I_2\otimes I_2\otimes I_2=\sigma_1\otimes i\sigma_1\otimes I_2\otimes I_2\otimes I_2\\ \Gamma_{(4)}^3=\Gamma_{(4)}^3\otimes I_2\otimes I_2\otimes I_2=\sigma_1\otimes i\sigma_2\otimes I_2\otimes I_2\otimes I_2\\ \Gamma_{(10)}^4=\Gamma_{(4)}^5\otimes i\sigma_1\otimes I_2\otimes I_2=\sigma_1\otimes\sigma_3\otimes i\sigma_1\otimes I_2\otimes I_2 \\ \Gamma_{(10)}^5 =\Gamma_{(4)}^5\otimes i\sigma_2\otimes I_2\otimes I_2 =\sigma_1\otimes\sigma_3\otimes i\sigma_2\otimes I_2\otimes I_2 \\ \Gamma_{(10)}^6 =\Gamma_{(4)}^5\otimes \sigma_3\otimes i\sigma_1\otimes I_2= \sigma_1\otimes\sigma_3\otimes \sigma_3\otimes i\sigma_1\otimes I_2 \\ \Gamma_{(10)}^7 =\Gamma_{(4)}^5\otimes \sigma_3\otimes i\sigma_2\otimes I_2 =\sigma_1\otimes\sigma_3\otimes \sigma_3\otimes i\sigma_2\otimes I_2\\ \Gamma_{(10)}^8 =\Gamma_{(4)}^5\otimes \sigma_3\otimes \sigma_3\otimes i\sigma_1 = \sigma_1\otimes\sigma_3\otimes \sigma_3\otimes \sigma_3\otimes i\sigma_1\\ \Gamma_{(10)}^9= \Gamma_{(4)}^5\otimes\sigma_3\otimes \sigma_3\otimes i\sigma_2= \sigma_1\otimes\sigma_3\otimes\sigma_3\otimes \sigma_3\otimes i\sigma_2 \\ \Gamma_{(10)}^{11} =\Gamma_{(4)}^5\otimes\sigma_3\otimes \sigma_3 \otimes \sigma_3= \sigma_1\otimes\sigma_3\otimes\sigma_3\otimes \sigma_3 \otimes \sigma_3\\$$ where the number in brakert denote the dimension of spactime.

However, this represetation is not consistent with the charge conjuation matrix (5.6) in that paper in the sense that not all $C\Gamma^\mu_{(10)}$ are symmetric.

Above is my computation. Does anyone know the explicit form for gamma matrices in 10D, or other representation different from this paper when one consider the dimensional reduction from 10D to 4D?

This post imported from StackExchange Physics at 2017-01-21 16:17 (UTC), posted by SE-user phys
When you say "the explicit form", are you asking what specific choice of $\Gamma$-matrices that paper made? Because there is no "the explicit form", since you can choose the basis of your vector space arbitrarily.
Yes, "the explicit form" means a specific choice of gamma matrices. I think there are two rensons that I want to know the explicit form(which may be wrong, so please correct me if so). First, in (5.3) of the paper a particular choice of gamma matrices labeled $ij$ were given, but what the relationship between $\Gamma_{ij}$ and $\Gamma^\mu$. Second, It seem from second line in (5.10), one can abstract $\Gamma^4,\Gamma^5,...,\Gamma^9$ if $\phi_{ij}$ is reexpressed as $A_4,A_5,...,A_9.$
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