ADHM construction

Generalizations

Noncommutative instantons

In a noncommutative gauge theory, the ADHM construction is identical but the moment map ${\displaystyle \overrightarrow{\mu }}$ is set equal to the self-dual projection of the noncommutativity matrix of the spacetime times the identity matrix. In this case instantons exist even when the gauge group is U(1). The noncommutative instantons were discovered by Nikita Nekrasov and Albert Schwarz in 1998.

Vortices

Setting B₂ and J to zero, one obtains the classical moduli space of nonabelian vortices in a supersymmetric gauge theory with an equal number of colors and flavors, as was demonstrated in Vortices, instantons and branes. The generalization to greater numbers of flavors appeared in Solitons in the Higgs phase: The Moduli matrix approach. In both cases the Fayet–Iliopoulos term, which determines a squark condensate, plays the role of the noncommutativity parameter in the real moment map.

The construction formula

Let x be the 4-dimensional Euclidean spacetime coordinates written in quaternionic notation ${\displaystyle x_{ij}=\left(\begin{array}{cc}{z}_2& z_1\\ -\overline{z_1}& \overline{z_2}\end{array}\right).}$

Consider the 2k × (N + 2k) matrix

${\displaystyle \mathrm{\Delta }=\left(\begin{array}{ccc}I& B_2+z_2& B_1+z_1\\ J^{†}& -B_1^{†}-\overline{z_1}& B_2^{†}+\overline{z_2}\end{array}\right).}$

Then the conditions ${\displaystyle {\displaystyle {\mu }_r={\mu }_c=0}}$ are equivalent to the factorization condition

${\displaystyle \mathrm{\Delta }{\mathrm{\Delta }}^{†}=\left(\begin{array}{cc}{f}^{-1}& 0\\ 0& f^{-1}\end{array}\right)}$ where f(x) is a k × k Hermitian matrix.

Then a hermitian projection operator P can be constructed as

${\displaystyle P={\mathrm{\Delta }}^{†}\left(\begin{array}{cc}f& 0\\ 0& f\end{array}\right)\mathrm{\Delta }.}$

The nullspace of Δ(x) is of dimension N for generic x. The basis vectors for this null-space can be assembled into an (N + 2k) × N matrix U(x) with orthonormalization condition U^†U = 1.

A regularity condition on the rank of Δ guarantees the completeness condition

${\displaystyle P+U{U}^{†}=1.}$

The anti-selfdual connection is then constructed from U by the formula

${\displaystyle A_m=U^{†}{\mathrm{\partial }}_{m}U.}$