= ADHM construction
{c}
{title2=$\mu_{\mathbb C}=\mu_{\mathbb R}=0$}
{wiki}
Take $V=\mathbb C^k$, $W=\mathbb C^N$, $B_1,B_2\in\operatorname{End}(V)$, $I:W\to V$, and $J:V\to W$, satisfying
$$
[B_1,B_2]+IJ=0,\qquad [B_1,B_1^\dagger]+[B_2,B_2^\dagger]+II^\dagger-J^\dagger J=0.
$$
Regular data modulo $U(k)$ construct framed <Anti-self-dual Yang-Mills equations> solutions of charge magnitude $k$. The <ADHM factorization identity> gives an explicit <gauge curvature> proof. The two complex <matrices> and the two rectangular maps have $4k^2+4kN$ real parameters; three real <matrix> constraints remove $3k^2$, and quotient by $U(k)$ removes another $k^2$, giving dimension $4kN$ at regular points. Degenerate data can describe zero-size singular limits rather than smooth <instantons>.
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