ADHM construction

In

in their paper "Construction of Instantons."

The ADHM construction uses the following data:

• complex vector spaces V and W of dimension k and N,
• k × k complex matrices B₁, B₂, a k × N complex matrix I and a N × k complex matrix J,
• a
  moment map
  ${\displaystyle \mu _{r}=[B_{1},B_{1}^{\dagger }]+[B_{2},B_{2}^{\dagger }]+II^{\dagger }-J^{\dagger }J,}$
• a complex moment map ${\displaystyle \displaystyle \mu _{c}=[B_{1},B_{2}]+IJ.}$

Then the ADHM construction claims that, given certain regularity conditions,

• Given B₁, B₂, I, J such that ${\displaystyle \mu _{r}=\mu _{c}=0}$, an anti-self-dual instanton in a SU(N) gauge theory with instanton number k can be constructed,
• All anti-self-dual
  instantons can be obtained in this way and are in one-to-one correspondence with solutions up to a U(k) rotation which acts on each B in the adjoint representation and on I and J via the fundamental
  and antifundamental representations,
• The
  metric on the moduli space
  of instantons is that inherited from the flat metric on B, I and J.

Generalizations

Noncommutative instantons

In a noncommutative gauge theory, the ADHM construction is identical but the moment map ${\displaystyle {\vec {\mu }}}$ 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.

Setting B₂ and J to zero, one obtains the classical moduli space of nonabelian vortices in a

, 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 ${\displaystyle x_{ij}={\begin{pmatrix}z_{2}&z_{1}\\-{\bar {z_{1}}}&{\bar {z_{2}}}\end{pmatrix}}.}$

Consider the 2k × (N + 2k) matrix

${\displaystyle \Delta ={\begin{pmatrix}I&B_{2}+z_{2}&B_{1}+z_{1}\\J^{\dagger }&-B_{1}^{\dagger }-{\bar {z_{1}}}&B_{2}^{\dagger }+{\bar {z_{2}}}\end{pmatrix}}.}$

Then the conditions ${\displaystyle \displaystyle \mu _{r}=\mu _{c}=0}$ are equivalent to the factorization condition

${\displaystyle \Delta \Delta ^{\dagger }={\begin{pmatrix}f^{-1}&0\\0&f^{-1}\end{pmatrix}}}$ where f(x) is a k × k Hermitian matrix.

Then a hermitian projection operator P can be constructed as

${\displaystyle P=\Delta ^{\dagger }{\begin{pmatrix}f&0\\0&f\end{pmatrix}}\Delta .}$

The

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

${\displaystyle P+UU^{\dagger }=1.\,}$

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

${\displaystyle A_{m}=U^{\dagger }\partial _{m}U.}$

• Monad (homological algebra)
• Twistor theory

• MR 0554924
• MR 0598562
• Hitchin, N. (1983), "On the Construction of Monopoles", Commun. Math. Phys. 89, 145–190.