In Chern Simons Gauge Theory as a String Theory, Witten derives the general coordinate variation of a Wilson loop, i.e., equation 3.11. My question is, how does one derive this? I only managed to obtain a similar result by doing a direct variation and integration by parts, the result I obtained had \begin{equation} F_{IJ}=\partial_I A_J-\partial_J A_I. \end{equation} This is clearly incorrect, since he specifically says that $F=dA+A\wedge A$. I believe I am supposed to vary the path somehow, or that the path ordering somehow gives rise to the term $A\wedge A$, which is just the commutator $[A_I,A_J]$. However, I cannot see how this happens.

  • $\begingroup$ This post (v2) is essentially a duplicate of this Phys.SE question, since Witten's eq. (3.11) is the same as Polyakov's eq. (7.25). $\endgroup$
    – Qmechanic
    Commented Mar 3, 2016 at 23:13