&  D_1\psi_1\pm D_1 \psi_1\\ 0 & \vert\psi\vert^2-1 & -(D_2\psi_2\pm D_2 \psi_2) & D_2\psi_1\pm D_2 \psi_1\\ -(D_1\psi_2\pm D_1 \psi_2) & -(D_2\psi_2\pm D_2\psi_2) & (1\pm\frac{1}{2})(\vert\psi\vert^2-1)+V_kV_k & 0 \\ D_1\psi_1\pm D_1 \psi_1 & D_2\psi_1\pm D_2 \psi_1 & 0 & (1\pm\frac{1}{2})(\vert\psi\vert^2-1)+V_kV_k
\end{array}\right). \label{vspot}$$]{} Use of the high-temperature heat trace asymptotic expansions, $${\rm Tr}_{L^2}{\rm exp}(-\beta{\cal H}^\pm)\simeq \frac{e^{-\beta}}{4\pi} \sum_{n=1}^\infty \, {\rm tr} [ c_n({\cal H}^\pm)]\beta^{n-1}$$ leads to estimate the index in the form: $${\rm ind}\,{\cal D}=\lim_{\beta\to 0} \left({\rm Tr}_{L^2}\, e^{-\beta {\cal H}^+} - {\rm Tr}_{L^2}\, e^{-\beta {\cal H}^-}  \right)=\frac{1}{4\pi}\left({\rm tr} [ c_1({\cal H}^+)]-{\rm tr}[ c_1({\cal H}^-)] \right).$$ Here ${\rm tr}$ refers to the conventional $(4\times 4)$-matrix trace and the divergent ${\rm tr} [c_0({\cal H}^\pm)]$ terms have been discarded because they cancel each other in the index formula. We shall see that $${\rm tr}[ c_1({\cal H}^\pm)]=-{\rm tr} \int_{\mathbb{R}^2} \, d^2x \, U^\pm(\vec{x}) \, \, , \quad  \, \, \mbox{henceforth}, \, \,  \, \, \quad
{\rm ind}\,{\cal D}=\frac{1}{2 \pi} \int_{\mathbb{R}^2} \, d^2x \, \left(1-\vert \psi\vert^2\right)= 2 N \label{vind} \, \, .$$ But ${\cal H}^-$ is a positive definite operator such that ${\rm dim}\,{\rm Ker}\,{\cal D}^\dagger=0$, which means that ${\cal H}^+$ has $2N$ zero modes.

BPS vortex heat kernel asymptotic expansion: impact of zero modes
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One of the 