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?

• Minor comment to the post (v3): In the future please link to abstract pages rather than pdf files, e.g., arxiv.org/abs/0804.2902 – Qmechanic Mar 30 '18 at 19:35
• x-posted on math.OF: mathoverflow.net/q/296588/106114 – AccidentalFourierTransform Mar 30 '18 at 19:52
• @AccidentalFourierTransform My apologies, I was not aware that cross-posting between Physics.SE and MathOverflow is not encouraged. I have deleted the MathOverflow post. I would really appreciate it if the previous answer to my question is reinstated, as I was not able to read the answer before it was deleted, and I need this information urgently. – Mtheorist Mar 31 '18 at 6:14
• @Mtheorist cross-posting is ok, but it is preferred to say so explicitly (either in the post itself or in the comment section). – AccidentalFourierTransform Mar 31 '18 at 17:01
• @AccidentalFourierTransform Thank you for the clarification, and I appreciate you providing a screenshot of the deleted answer. Will keep in mind that it could be wrong. – Mtheorist Apr 1 '18 at 16:34

1 Answer

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.