Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 304 / 4 / a / Solution

For a short Wilson line, a gauge transformation givesUse $V(x+an)=1+i\alpha (x)+ia{n}^{\mu }{\partial }_{\mu }\alpha (x)+O(a^2,{\alpha }^2)$ and $U=1+iga{n}^{\mu }{A}_{\mu }+O(a^2)$. Keeping terms of order $a$, including $a\alpha$, givesSince the Lie bracket is encoded by the Lie algebra structure constants, $i[\alpha ,A_{\mu }{]}^a=f^{abc}{A}_{\mu }^b{\alpha }^c$, and thereforeThis is the infinitesimal form of the Wilson-line gauge transformation.