I'm currently reading a paper written by John Baez [1]. At a certain point he argues, that the holonomy of a flat connection around a contractible loop is the identity, so its trace in the representation $ρ$ is dim($ρ$).
More explicit, $$ \operatorname{tr}\left(\rho\left(T e^{\oint_{e} A}\right)\right)\stackrel{A\, \text{is flat}}{=}\text{dim}{(\rho)}\,. $$
What I don't see is the exact proof for $$ e^{\oint_{e} A} =1 \;\;,\text { by flatness } $$
I imagine that it could be done with Stokes theorem, but I'm not sure...
[1] An Introduction to Spin Foam Models of BF Theory and Quantum Gravity. [https://arxiv.org/pdf/gr-qc/9905087v1.pdf]