In the Mirror Symmetry monograph (http://www.claymath.org/library/monographs/cmim01c.pdf), on page 297, the index theorem is used for a two-dimensional twisted Dirac operator. Below equation 13.37, it is claimed that the number of $\psi_-$ zero modes is equal to the number of $\overline{\psi}_+$ zero modes, and the number of $\overline{\psi}_-$ zero modes is equal to the number of ${\psi}_+$ zero modes. How does one show that this is true?

In addition, on page 811, the same problem is re-considered for a two-dimensional surface with boundary. Here it is claimed in equation 39.213 that the index of the twisted Dirac operator is \begin{equation} \textrm{Index }\mathcal{D}=\#[(\psi_-,\overline{\psi}_+)\textrm{ zero modes}]-\#[(\overline{\psi}_-,{\psi}_+)\textrm{ zero modes}]. \end{equation} However, as far as I understand there is no well-defined index for $\mathcal{D}$, but only well-defined indices for its chiral or antichiral parts $D$ and $\overline{D}$, where \begin{equation} \mathcal{D}=\bigg(\begin{array}{cc} 0 & D \\ \overline{D} & 0 \\ \end{array} \bigg). \end{equation} What exactly does equation 39.213 mean?