I would like to know if the Hirzebruch-Riemann-Roch theorem exists for bundles over Riemann surfaces with a boundary. I am asking this because the Hirzebruch-Riemann-Roch theorem is used in the following paper (https://arxiv.org/pdf/0707.2786v2.pdf) on page 10 to compute the index of the following differential operators defined over fiber bundles on a Riemann surface $\Sigma$, \begin{align} (\nabla^A)^{0,1} &: \Omega^0 (\Sigma ; \mathfrak{g}_P) \longrightarrow \Omega^{0,1} (\Sigma ; \mathfrak{g}_P) \label{2.7} \\ (\phi^{\ast}\nabla^A)^{0,1} &: \Omega^0 (\Sigma ; \phi^\ast \ker \textrm{d} \pi_E) \longrightarrow \Omega^{0,1} (\Sigma ; \phi^\ast \ker \textrm{d} \pi_E) \ , \end{align} where the definitions of the fiber bundles $\mathfrak{g}_P$ and $\phi^\ast \ker \textrm{d} \pi_E$ are given on page 6 of the aforementioned paper.

The indices for these operators are said to be easily obtained from the Hirzebruch-Riemann-Roch theorem, and the result for a general compact $\Sigma$ with no boundary is \begin{align} \rm{index} (\nabla^A)^{0,1} &= c_1 (\mathfrak{g}_P \rightarrow \Sigma) + ({\rm dim}G)(1-g) \label{2.8} \\ \rm{index} (\phi^{\ast}\nabla^A)^{0,1} &= c_1 (\phi^\ast \ker \textrm{d} \pi_E \rightarrow \Sigma) + ({\rm dim}_\mathbb{C} X)(1-g) \ . \end{align}

I would like to know the generalizations of these formulae for general $\Sigma$ *with* boundary.

I suspect, based on the ordinary Riemann-Roch theorem, that the answer is replacing $(1-g)$ in the expressions above to $(1-g-\frac{b}{2})$, where $b$ is the number of boundaries. This is because the Euler characteristic for the Riemann surface is $\chi =2-2g-b$. Is this correct, and why? References would be highly appreciated.