 ADHM construction

Take $V={\mathbb{C}}^k$, $W={\mathbb{C}}^N$, $B_1,B_2\in \mathrm{End}(V)$, $I:W\to V$, and $J:V\to W$, satisfyingRegular 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.

The ADHM construction, which stands for Atiyah-Drinfeld-Hitchin-Manin construction, is a mathematical framework used in theoretical physics and geometry, particularly in the study of instantons in gauge theory. It is a method for constructing solutions to the self-duality equations of gauge fields in four-dimensional Euclidean space, which are fundamental in the study of Yang-Mills theory.