I am reading Witten's paper on topological string, and I found some mathematical notation is hard to understand for me.
Consider the nonlinear sigma model in 2 dimensions governed by maps
$\Phi : \Sigma \rightarrow X$ with $\Sigma$ being a Riemann surface and $X$ a Riemann manifold of metric $g$. $z, \bar{z}$ are local coordinate on $\Sigma$ and $\phi^I$ is coordinate on $X$. $K$ and $\bar{K}$ are canonical and anticanonical line bundles of $\Sigma$ (the bundle of one forms of types (1,0) and (0,1) repectively), and let $K^{1/2}$ and $\bar{K}^{1/2}$.The fermi fields of the model are $\psi_{+}^I$, a section of $K^{1/2}\otimes\Phi^*(TX)$.

I can not understand the sections of $K^{1/2}$, $\Phi^*(TX)$ and $K^{1/2}\otimes\Phi^*(TX)$.

From my point of view, the element of $K$ should be of the following form $\alpha_z dz\in K$, and what is the element of $K^{1/2}$? the pull back of tangent space should be of form $\Phi^*(\beta^i \frac{\partial}{\partial \phi^i})=\beta^i \frac{1}{\frac{\partial \phi^i}{\partial z} }\frac{\partial}{\partial z}$. But in some notes the author seems give that the (0,1) form $\psi_-^i$ with values in $\Phi^*(T^{1,0} X)$ can be written as $\psi_{\bar{z}}^i$ satisfying $\psi \supset \psi_{\bar{z}}^id\bar{z}\otimes \frac{\partial}{\partial \phi^i}$. This contradicts with my naive point of view.
where did I make mistakes? How to understand the sections of $K^{1/2}$, $\Phi^*(TX)$ and $K^{1/2}\otimes\Phi^*(TX)$?

Thanks in advance.

This post imported from StackExchange Physics at 2014-04-18 16:22 (UCT), posted by SE-user Craig Thone