ADHM

ADHM refers to the Atiyah–Drinfeld–Hitchin–Manin construction, a foundational method in differential geometry and mathematical physics for describing instantons on four-dimensional Euclidean space. Developed in 1978, it provides a complete algebraic parametrization of anti-self-dual (ASD) Yang–Mills connections with gauge group U(N) and given instanton number k on R^4, translating a nonlinear partial differential equation problem into a finite-dimensional algebraic one.

The construction starts with linear data consisting of two complex vector spaces: V of dimension k and W of dimension N. The corresponding ADHM data are linear maps B1, B2 in End(V), I in Hom(W,V), and J in Hom(V,W). These must satisfy the ADHM equations: the complex equation [B1,B2] + IJ = 0, and, in the usual real presentation, the real moment-map equation [B1,B1†] + [B2,B2†] + II† − J†J = 0. The unitary group U(V) acts on this data, and the moduli space of framed instantons is the hyper-Kähler quotient μ−1(0)/U(V), i.e., the set of solutions to the equations modulo gauge transformations.

From the ADHM data one constructs a rank-N vector bundle with a connection on R^4 whose curvature is anti-self-dual. The instanton number equals k, the dimension of V. The ADHM construction thus reduces solving the ASD Yang–Mills equations to linear-algebraic data and stability conditions.

Significance and extensions: the framework underpins instanton moduli spaces, connects to hyper-Kähler geometry and quiver varieties (Nakajima), and has been generalized to noncommutative spaces and other gauge groups. It also informs string theory and brane constructions via gauge–/gravity dualities and dualities in supersymmetric theories.