Structures in Non-linear Sigma Model I debated whether this belongs here, or on Physics SE, but all my questions correspond to the algebraic/complex geometry, not physics, so hopefully it's okay here.  
The non-linear sigma model considers maps $\phi : \Sigma \to X$, where $\Sigma$ is a compact, oriented Riemann surface, and $X$ is a Kahler manifold with Kahler form $g_{i, \bar{j}}$.  My confusion is about certain confusing structures in the "action" and exactly which of the two spaces they live on.  I'll write just a portion of the "action":
$$\int_{\Sigma} d^{2}z\bigg\{ g_{i \bar{j}} \big( \partial_{z}\phi^{i} \partial_{\bar{z}}\phi^{\bar{j}}+ \partial_{\bar{z}}\phi^{i} \partial_{z}\phi^{\bar{j}}\big) + \ldots + i g_{i \bar{j}} \psi^{\bar{j}} D \psi^{i}\bigg\},$$
where $D\psi^{i}= \partial\psi^{i} + \partial \phi^{j} \Gamma^{i}_{jk}\psi^{k}$ is the covariant derivative, and $\psi^{i}$ is a section of the bundle on $\Sigma$, $\overline{K}^{1/2} \otimes \phi^{*}T_{X}$, with $T_{X}$ the holomorphic tangent bundle, and $K$ the canonical line bundle on $\Sigma$.
My questions correspond to which spaces the above structures live on:
(I) Certainly $(z,\bar{z})$ are local complex coordinates on $\Sigma$, while $\phi^{i}$ correspond to the map $\phi$ represented in local coordinates on $X$, and presumably the Kahler metric is pulled back to $\Sigma$.  So how does it make sense to have structures on $\Sigma$, with indices from $X$, being differentiated with respect to local coordinates on $\Sigma$?  
(II) Same sort of confusion about the covariant derivative: $\Gamma$ is presumably the Levi-Civita connection on $X$, and yet it arises in differentiating section of bundles on $\Sigma$.  How does this make sense?
(III) Finally, when we consider holomorphic maps $\phi$, we see things like "$\bar{\partial}\phi=0$" or "$\bar{\partial}\phi^{i}=0$."  Are these derivatives with respect to coordinates on the worldsheet, or target spaces?
I think probably all these confusions arise from inferior understanding of holomorphic maps and pullbacks, but I'd really appreciate a few hints!
 A: $\newcommand{\Reals}{\mathbf{R}}\newcommand{\dd}{\partial}\newcommand{\Cpx}{\mathbf{C}}$Your three questions boil down to the nature of pullbacks under a mapping.
(I): A metric on $X$ determines a corresponding structure on $\Sigma$. (Away from critical points of $\phi$, the pullback $\phi^{*}g$ is a metric on $\Sigma$.) Particularly, the equation
$$
\phi^{*}g(\dd_{z}, \dd_{\bar{z}})
  = g(\phi_{*}\dd_{z}, \phi_{*}\dd_{\bar{z}})
  = g_{i\bar{\jmath}}\frac{\dd \phi^{i}}{\dd z}\, \frac{\dd \phi^{\bar{\jmath}}}{\dd \bar{z}}
$$
defines a metric $\phi^{*}g$ on $\Sigma$: To evaluate on a pair of tangent vectors to $\Sigma$, use $\phi_{*}$ to "push forward" the vectors to tangent vectors on $X$, then evaluate $g$. In local coordinates $w^{i} = \phi^{i}(z, \bar{z})$, the chain rule gives
\begin{align*}
\phi_{*}\frac{\dd}{\dd z}
  &= \frac{\dd \phi^{i}}{\dd z}\, \frac{\dd}{\dd w^{i}}
   + \frac{\dd \phi^{\bar{\jmath}}}{\dd z}\, \frac{\dd}{\dd w^{\bar{\jmath}}}, \\
\phi_{*}\frac{\dd}{\dd \bar{z}}
  &= \frac{\dd \phi^{i}}{\dd \bar{z}}\, \frac{\dd}{\dd w^{i}}
   + \frac{\dd \phi^{\bar{\jmath}}}{\dd \bar{z}}\, \frac{\dd}{\dd w^{\bar{\jmath}}}.
\end{align*}
(II): Similarly, a connection $D$ in $TX$ induces a connection $\phi^{*}D$ in $\phi^{*}TX$, via
$$
(\phi^{*}D)_{\dd_{z}}(\phi^{*}\psi) = D_{\phi_{*}(\dd_{z})} \psi.
$$
The local expression for this connection naturally involves the Christoffel symbols for connection on $X$ because the covariant derivative of a section $\phi^{*}\psi$ of $\phi^{*}TX$ in the direction of a tangent vector $\dd_{z}$ is defined by pushing $\dd_{z}$ to $TX$ by $\phi_{*}$ and using this vector to differentiate $\psi$.
(III): The derivatives are taken with respect to coordinates on $\Sigma$. An $X$-valued mapping $\phi$ is holomorphic if each component function (with respect to some holomorphic coordinate system on $X$) is an ordinary holomorphic function. In symbols, $\bar{\dd} \phi^{i} = 0$, or
$$
\frac{\dd \phi^{i}}{\dd \bar{z}} = 0.
$$

In case it helps, you've secretly been doing the same types of calculation ever since you learned vector calculus. Think of polar coordinates,
$$
(x, y) = \phi(r, \theta) = (r\cos\theta, r\sin\theta).
$$
By the chain rule, the mapping $\phi$ sends the coordinate vector fields on the $(r, \theta)$-plane $\Sigma$ to the fields
\begin{align*}
\phi_{*}\frac{\dd}{\dd r}
  &= \frac{\dd x}{\dd r}\, \frac{\dd}{\dd x}
   + \frac{\dd y}{\dd r}\, \frac{\dd}{\dd y}, \\
\phi_{*}\frac{\dd}{\dd \theta}
  &= \frac{\dd x}{\dd \theta}\, \frac{\dd}{\dd x}
   + \frac{\dd y}{\dd \theta}\, \frac{\dd}{\dd y}, \\
\end{align*}
on the $(x, y)$-plane $X$, and (in case this is useful to you elsewhere) pulls back the coordinate $1$-forms on $X$ to the $1$-forms
\begin{align*}
\phi^{*} dx &= \frac{\dd x}{\dd r}\, dr + \frac{\dd x}{\dd \theta}\, d\theta, \\
\phi^{*} dy &= \frac{\dd y}{\dd r}\, dr + \frac{\dd y}{\dd \theta}\, d\theta
\end{align*}
on $\Sigma$.
