Instanton moduli space (${\mathcal{M}}_k=\{A:\ast F_A=\pm F_A,c_2=k\}/\mathcal{G}$)

For a fixed principal bundle, orientation and instanton sign, quotient the Yang-Mills instanton solutions by gauge equivalence of principal connections. Boundary or framing conditions determine the gauge group $\mathcal{G}$ on a noncompact base. At an anti-self-dual solution, infinitesimal deformations obey $P_{+}{D}_{A}a=0$ modulo $a\mapsto a+D_A\epsilon$. Combining this with $D_A^{\ast }a=0$ gives an elliptic deformation operator. This description does not assert that every quotient is a smooth manifold: reducible connections can have nontrivial stabilizers.

The scale $\rho >0$ in a BPST instanton changes the width of its action density while leaving total action unchanged. This reflects conformal invariance of four-dimensional classical Yang-Mills action. As $\rho \to 0$, gauge curvature concentrates at the center and the smooth family approaches a bubbling boundary, not another smooth zero-size instanton.

Fix a trivialization at infinity and divide smooth finite-action instantons only by gauge transformations tending to the identity there. Global gauge orientations remain genuine parameters. For regular $SU(2)$ instantons of charge magnitude $k$, the real dimension is $8k$ by the ADHM construction. Forgetting the framing divides by the three-dimensional global group action of $SU(2)$, giving $8k-3$ on the irreducible locus. For one instanton the framed parameters are four center coordinates, one positive size, and three orientation coordinates.