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>)



Index of linearized operator for symplectic vortex equations
The operator $D$ in (1) is the related to (odd part of) the de Rham operator $d+d^\*$. If you omit the Hodge star in the last term, you end up in $\Omega^0\oplus\Omega^2$. The Atiyah-Singer index (or if you like, the twisted Gauß-Bonnet theorem) theorem gives
\begin{align\*}
\operatorname{ind}(D)&=-\operatorname{ind}(d+d^\*\colon\Omega^{\mathrm{even}}(\Sigma;\mathfrak g)\to\Omega^{\mathrm{odd}}(\Sigma;\mathfrak g))\\
&=-\int\_\Sigma e(T\Sigma)\,\operatorname{ch}(\mathfrak g)\\
&=-\int\_\Sigma e(T\Sigma)\,\dim\mathfrak g=-\chi(\Sigma)\,\dim\mathfrak g\;.
\end{align\*}
The Euler class is homogeneous of degree $\dim\Sigma=2$, hence only the zero degree term $\dim\mathfrak g$ of the Chern character survives. In particular, this index does not see if the principal bundle $P$ is nontrivial.


