Serberg Witten equation requires a given orthogonal frame bundle $P_{SO} \rightarrow X$, we give a $Spin^{c}$ lifting $P_{Spin^{c}}$ and the assoiated spinor bundle $S=P_{Spin^c}\times _{\rho}\Delta_{n}$ where ${\rho}$ is the representation of $Spin^c$.
The exsistence of $Spin^c$ will give rise to the the exsistence of determinant line bundle $L$ with mod 2 reduction. And if I undertsand correctly, the spinor bundle can be decomposed as $S^{+}=W^+ \otimes L^{1/2}$ (respectively for the minus sign.) although the latter bundle may not exsit(but their product exsits) unless $X$ admits a spin structure[$1$]. In many references I see, $L^{1/2}$ is called virtual bundle.[$2$]
So our Seiberg Witten equation is a pair $(A,\psi)$, where $A$ is a unitary connection on the virtual connection, although this may not exist but the connection $2A$ on det bundle $L$ exist. $\psi$ is a section of $S^{+}=W^+ \otimes L^{1/2}$.
Now given the Dirac operator $D_{A}^{+}$ from positve to negative spinor bundle, our Seiberg Witten equation becomes $$D_{A}^{+}=0,\qquad F_{A}^{+}=\sigma(\psi)=\psi \otimes \psi^* -1/2|\psi|^2id$$
But this confuses me, the virtual connection $A$ on $L^{1/2}$ may not exist, how can one ensure the corresponding Dirac operator and self dual 2-form form $A$ exist [2] ?
I think the connection should come from determinant bundle $L$ and this is the equation which appear on [$3$] and [$4$].
Reference:
[$1$] Lawson: Spin geometry
[$2$] Salamon: Spin geometry and Seiberg Witten invariants
[$3$] J.Morgan: The Seiberg Witten equations and the applications to the topology of smooth four-manifolds
[$4$] J.D Moore: Lectures on Seiberg Witten invariants
Second edition:
I think this is mainly depends on the notation of the reference, since we have the realtionship $2A$ and $2F_{A}^{+}$ on det bundle, as explained in [4]. A more suggested way is to choose the connection on det bundle directly, this will not cause confusion (I think).