# Gamma matrices in Gaiotto-Witten analysis of N=4 Super Yang-Mills boundary conditions

In the paper Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory by Gaiotto and Witten, an in-depth analysis of supersymmetric boundary conditions in N=4 Super Yang-Mills in four dimensions in performed. One of the main points in this analysis is the breaking of the R-symmetry at the boundary from $SO(6)$ to $SO(3)\times SO(3)$, which is explained on page 6.

My main interest is the set of operators given in equation (2.7). It is claimed that the action of $W=SO(1,2)\times SO(3)\times SO(3)$ commutes with the operators $$B_0=\Gamma_{456789}\\B_1=\Gamma_{3456}\\B_2=\Gamma_{3789},$$ (where the subscripts on right-hand-sides indicate completely antisymmetrized products of 10D gamma matrices. Note that $SO(1,2)$ acts on the indices $012$, and the two $SO(3)$ groups act on $456$ and $789$ respectively.)

How does one prove that this is true?

The $$B_0$$ is the chirality matrix of $$SO(6)\rightarrow SO(3)_X\times SO(3)_Y$$, namely $$\Gamma_{456789}$$.
The $$B_1$$ is the product between the gamma matrices of $$SO(3)_{X}$$, namely $$\Gamma_{456}$$, with $$\Gamma_3$$.
The $$B_{2}$$ is the product between the gamma matrices of $$SO(3)_{Y}$$, namely $$\Gamma_{789}$$, with $$\Gamma_3$$
You need to prove that $$\Gamma_{456789}$$, $$\Gamma_{3456}$$ and $$\Gamma_{3789}$$ commutes with $$\Gamma_{\mu\nu}$$ for $$\mu$$ and $$\nu$$ equal $$0,1,2$$ xor $$4,5,6$$ xor $$7,8,9$$. This is the same thing as showing that the number of indices that are equal between $$\Gamma_{\mu\nu}$$ and $$\Gamma_{456789}$$ is always even. That the number of indices that are equal between $$\Gamma_{\mu\nu}$$ and $$\Gamma_{3456}$$ are always even. That the number of indices that are equal between $$\Gamma_{\mu\nu}$$ and $$\Gamma_{3789}$$ are always even.