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2

General comment to the question (v3): Non-abelian YM [such as, e.g., YM with gauge group $SU(2)$ or $SU(3)$] has besides quartic gauge boson interactions also cubic gauge boson interactions, while abelian YM (aka. QED) has neither. This is because the Feynman-rules for the cubic (quartic) gauge boson vertices are linear (quadratic) in the Lie algebra ...

5

Let us start with $U(1)$ electromagnetism and see why it does not have such interactions. The field strength tensor is given by $F_{\mu\nu}=\partial_\mu A_\nu - \partial_\nu A_\mu$, and the relevant part of the QED Lagrangian is proportional to $F_{\mu\nu}F^{\mu\nu}$. This means that the Lagrangian has only terms that are at most quadratic in the gauge field ...

0

The ten-dimensional case is explained in some detail in appendix 4.A of volume one of Green-Schwarz-Witten. Let me therefore consider the 4d case here. The calculation works essentially the same in 3d, 6d and 10d, though you can sometimes make use of Majorana and/or Weyl conditions of the spinors to simplify things. We want to check that the expression ...

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