Fields in the action of the Non-linear Sigma Model (WZW)

Question

I am trying to understand the action of the nonlinear sigma model in the context of understanding WZW-models. On Wikipedia, its action is given as

$S_k\left(\gamma\right)=-\frac{k}{8\pi}\int_{S^2}\textrm{d}^2x\,\mathcal{K}\left(\gamma^{-1}\partial^\mu\gamma,\gamma^{-1}\partial_\mu\gamma\right)+2\pi k S^{\text{WZ}}\left(\gamma\right)$.

Here, $G$ is a compact simply-connected Lie group with $g$ its Lie algebra, $\gamma$ is a $G$-valued field and $\mathcal{K}$ is the Killing form on $g$. My difficulty is in understand how the products of the $\gamma$'s in the Killing form are supposed to work.

To my understanding, for the arguments in the Killing form to make sense, we require $\gamma^{-1}\partial^\mu\gamma$ and $\gamma^{-1}\partial_\mu\gamma$ to be elements of the Lie algebra $g$. I do not see, however, how this is supposed to be true.

To make sense of $\gamma^{-1}$, I would expect it to be the inverse of the element $\gamma$ maps to, i.e. for $z\in S^2$, we would have $\gamma\left(z\right)^{-1}\in G$. Furthermore, the derivative of $\gamma$ as seen from a manifold point of view would be a map from $TS^2$ to $TG$, if I am not mistaken. How then, would this be 'multiplied' by $\gamma^{-1}$? And how is it the resulting product to take arguments from $S^2$ into an element of $g$?

I have been following the book on Conformal Field Theory by Philippe Di Francesco et al. (http://www.springer.com/jp/book/9780387947853) and have been trying to find answers in other sources, but to no avail as of yet.

Answer 1

That $g^{-1}\mathrm{d} g$ is Liealgebra-valued for a Lie group-valued function $g$ has nothing to do with the particular model or with physics, it is true for all matrix groups. Write $g(x) = \exp(k(x))$, where $k(x)$ is now Lie algebra-valued and $\exp$ is the usual power series in the case of a matrix group. Then $\partial_\mu g = \partial_\mu k\exp(k)$ by the chain rule, and multiplying this with $g^{-1} = \exp(-k)$ gives $g^{-1}\partial_\mu g = \partial_\mu k$, which is Lie algebra-valued.

For Lie groups which are not matrix groups, writing $g^{-1}\mathrm{d}g$ is non-sensical, and you actually need to use the pullback of the Cartan-Maurer form by the function into the group.