In reference [1], the index of the linearized operator for the symplectic vortex equations is computed on page 27-28.

The first step of the proof says that the operator \begin{equation}\tag{1} \Omega^1(\Sigma,\mathfrak{g}_P)\rightarrow\Omega^0(\Sigma,\mathfrak{g}_P)\oplus \Omega^0(\Sigma,\mathfrak{g}_P):\alpha\mapsto(-d^*_A\alpha,*d_A\alpha) \end{equation} has index $-\chi\textrm{ dim }G$. How does one show that this is true?

My attempt to understand this is as follows. In reference [2], the index of the Cauchy-Riemann operator \begin{equation} (\nabla^A)^{0,1}:\Omega^0(\Sigma,\mathfrak{g}_P)\rightarrow\Omega^{0,1}(\Sigma,\mathfrak{g}_P) \end{equation} is shown to be equal to $c_1(\mathfrak{g}_P\rightarrow\Sigma)+(\textrm{dim }G)(1-g)$ via Hirzebruch-Riemann-Roch theorem, and is just \begin{equation} \textrm{dim }G(1-g)=\frac{\chi}{2}\textrm{dim }G \end{equation} for compact $G$. Hence, it seems to me that the operator in equation $(1)$ is equal to the adjoint of $2(\nabla^A)^{0,1}$. This seems to be supported by the statement below equation (24) of reference [2]. Is my understanding correct?

**References**

[1] "The symplectic vortex equations and invariants of Hamiltonian group actions" (https://arxiv.org/abs/math/0111176)

[2] "Twisting gauged nonlinear sigma models" (https://arxiv.org/abs/0707.2786)