Lax-Pair for principal chiral model This question concerns Eq. (2.10) of the paper https://arxiv.org/abs/hep-th/0305116 by Bena, Polchinski and Roiban.
In section 2.1 they are showing that the infinite number of conserved quantities for the principal chiral model
\begin{equation}
 L = \frac{1}{2\alpha_0} \mathrm{Tr}(\partial_\mu g^{-1}\partial_\mu g)
\end{equation}
are given by the fixed-time Wilson lines $U(\infty,t;-\infty,t)$
where
\begin{equation}
 U(x;x_0) = \mathrm{P}\, e^{-\int_{\mathcal{C}}a}
\end{equation}
and $a$ is a 1-parameter family of flat connections given by Eq. (2.3).
My question is what becomes of the last two terms (i.e. $-a_0a_1 +a_1a_0$) in the second line of Eq. (2.10). Do they cancel? I don't see why the should because the $a$'s are non-commuting (Lie algebra-valued).
 A: To simplify, take the notation : $U_y(x)= U(y,t;x,t)$, $U^{-1}_z(x)=  U(x,t;z,t)$, $a_i(x) = a_i(x,t)$
Note that you have (on the spatial choosen path $ C = \int  dx^1 = \int  dx $) : 
$\partial_x U^{-1}_z(x)=-a_1(x) U^{-1}_z(x)$, and $\partial_x U_y(x)= U_y(x)a_1(x)$
The minus sign difference can be understood because $\partial_x (U_y U^{-1}_y)=0$
Now, the last line of $2.10$ is : 
$a_0(y,t)U(y,t;z,t) − U(y,t;z,t)a_0(z,t)$
With our notations, we have : 
$a_0(y) U^{-1}_z(y) - U_y(z)a_0(z) \\=
U_y(y)a_0(y) U^{-1}_z(y)- U_y(z)a_0(z)U^{-1}_z(z)
\\ =[U_y(x)a_0(x) U^{-1}_z(x)]_z^y
\\ = \int_z^y ~dx~\partial_x(U_y(x)a_0(x) U^{-1}_z(x))
\\ = \int_z^y ~dx~(\partial_x U_y(x))a_0(x) U^{-1}_z(x)   +
\int_z^y ~dx~U_y(x)(\partial_x a_0(x)) U^{-1}_z(x) + \int_z^y ~dx~ U_y(x)a_0(x) (\partial_x U^{-1}_z(x))
\\ = \int_z^y ~dx~U_y(x) (a_1(x) a_0(x) + a'_0(x) -a_0(x)a_1(x)) U^{-1}_z(x)$
So, this is the second line of $2.10$. There is a global minus sign difference, I think it is because here the integration is going from $z$ to $y$, while the first line of $2.10$ uses the integration from $y$ to $z$
A: Defining the Wilson loop without the minus sign in the exponent gives
\begin{align}
 \partial_t U(y,t;z,t)
  & = \partial_t \mathrm{P} \, e^{\int_{(z,t)}^{(y,z)} dx^\mu a_\mu} \\
  & = \partial_t \mathrm{P} \, e^{\int_z^y dx a_1} \\
  & = \int_z^y dx \, U(x,t;z,t)\dot{a}_1(x,t)U(y,t;x,t) \\
  & = \int_z^y dx \, U(x,t;z,t)[a_0' - a_0a_1 + a_1a_0]_{(x,t)}U(y,t;x,t) \\
  & = \int_z^y dx \partial_x \left[ U(x,t;z,t) a_0(x,t)U(y,t;x,t) \right] \\
\end{align}
