I was reading John Morgan's book "The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds" and was a little confused by his statement of the linearization of the Seiberg-Witten functional.
As formulated, given a four-manifold $X$ we fix a $\text{Spin}^c$-structure with complex determinant line bundle $\mathcal{L}$ and spinor bundles $S^{\pm}$.
The Seiberg-Witten functional for a $U(1)$ connection $A$ on $\mathcal{L}$ and a section $\psi$ of the positive spinor bundle $S^+$ is then given by $$F(A, \psi) = (F_A^+ - q(\psi), D_A(\psi)).$$
Here $F_A^+$ denotes the self-dual part of the curvature of $A$, while $q(\psi)$ is an endomorphism of $S^+$ given by $\psi \otimes \psi^* - \frac{1}{2}|\psi|^2\text{Id}$ which with some work can be identified with an imaginary self-dual two-form.
The operator $D_A$ is the Dirac operator of $A$. Then, the claim is that the differential at $(A, \psi)$ is given by $$DF_{(A, \psi)} = \begin{pmatrix} P_+d & -Dq_\psi \\ \cdot \frac{1}{2}\psi & D_A \end{pmatrix}.$$
The entries in the right-hand column make sense to me since the top right is tautological and the Dirac operator is linear. Also, the bottom left entry should follow from the fact that $D_{A + \alpha} = D_A + \frac{1}{2}\alpha \cdot \psi$, where $\alpha$ is some imaginary one-form.
However, I'm not sure what "$P_+d$" is, and why it is the linearization of $F_A^+$ with respect to $A$. Any hints or explanations would be appreciated!
Update: P_+ denotes the projection of a 2-form into its self-dual component.