Sign-fixed self-dual moduli and deformation setup

Let $X$ be a closed oriented Riemannian four-manifold and let $P\to X$ be a smooth principal $SU(2)$-bundle with $c_2(P)[X]=-1$. The Hodge star on two-forms is $\ast$, and $F_A^{-}=(F_A-\ast F_A)\mathrm{/}2$ denotes the anti-self-dual projection of the curvature of a connection $A$. The equation used in this item is $F_A^{-}=0$, so the curvature is self-dual; this convention and the negative second Chern number are fixed together for the positive-definite orientation. The self-dual moduli set is the set of smooth connections satisfying $F_A^{-}=0$, modulo the action of smooth gauge transformations of $P$. At a solution, the deformation complex is $0\to {\mathrm{\Omega }}^0(X;\mathrm{ad}P)\stackrel{d_A}{\to }{\mathrm{\Omega }}^1(X;\mathrm{ad}P)\stackrel{d_A^{-}}{\to }{\mathrm{\Omega }}^{-}(X;\mathrm{ad}P)\to 0$, and its gauge-fixed operator is $D_A=d_A^{\ast }\oplus d_A^{-}$. For index calculations, $D_A$ is completed from the first Sobolev space of one-forms to the direct sum of the zeroth Sobolev spaces of zero-forms and anti-self-dual two-forms.

Smooth principal bundle and its adjoint bundle

Connection and covariant derivative

Curvature of a connection

Hodge star and self-dual projections

Gauge transformations acting on connections