In 'Non-abelian Bosonization in Two Dimensions', Witten writes down the commutation relations between currents and fields of the WZW model in equation (33):
\begin{equation} \bigg[\frac{1}{2\pi}\bigg(\frac{dg}{d\sigma}g^{-1}(x)\bigg)^{ij},g^k_l(y)\bigg]=-i\delta(x-y)(\delta^{jk}g^i_l(y)-\delta^{ik}g^j_l(y)), \end{equation}
\begin{equation} \bigg[\frac{1}{2\pi}\bigg(g^{-1}\frac{dg}{d\sigma}(x)\bigg)^{ij},g^k_l(y)\bigg]=-i\delta(x-y)(\delta^{jl}g^k_i(y)-\delta^{il}g^k_j(y)). \end{equation}
He explains that these follow since $\int dx \frac{dg}{d\sigma}g^{-1}$ and $\int dx g^{-1}\frac{dg}{d\tau}$ generate the chiral $O(N) \times O(N)$ symmetry, implying that these equations hold up to total derivatives that vanish after integrating over $x$.
He then says that there can be no such terms via locality and dimensional analysis, since $g$ is dimensionless. How does one show this?
I cannot seem to show that, e.g., a term of the form $$ \partial_x(g(x)^{ij}g(y)^{k}_l) $$ does not violate locality or dimensional analysis.
