Let $(M^4,g)$ be a closed four-dimensional Riemannian manifold and $J$ be an almost complex structure on $M$. Then for normal coordinate $e_1,\dots e_4$ at a point $m,$ and for a section $\alpha$ of a $\mathrm{Spin}^\mathrm{c}$ bundle $V$ on $M$, one can define the operator: \begin{equation*} D_J(\alpha)=\sum_i J(e_i)\cdot \nabla_i\alpha \end{equation*}

This can be seen as: \begin{equation*} \Gamma(V)\xrightarrow{\nabla}\Omega^1(M)\otimes\Gamma(V)\xrightarrow{J}\Omega^1(M)\otimes\Gamma(V)\xrightarrow{\text{Cliff. mult.}}\Gamma(V) \end{equation*}

Does this operator appear in the literature? If $D$ denotes the usual Dirac operator and $D\alpha$ vanishes, can we say something about $D_J\alpha?$

Comment: At a first glance this reminded me about the twisted differential, one defines in Kähler geometry: \begin{equation*} d^c:=\sum_i Je_i\wedge \nabla_i \end{equation*} I am not sure if this helps in any way though.


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