ADHM construction

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The ADHM construction or monad construction is the construction of all instantons using method of linear algebra by Michael Atiyah, Vladimir G. Drinfel'd, Nigel. J. Hitchin, Yuri I. Manin in their paper Construction of Instantons.

[edit] ADHM data

An ADHM data consists of the following objects:

complex vector spaces V and W of dimension k and N,
k × k matrix B₁, B₂, k × N matrix I and N × k matrix J,
$\mu_r = [B_1,B_1^\dagger]+[B_2,B_2^\dagger]+II^\dagger-J^\dagger J$ ,
$\displaystyle\mu_c = [B_1,B_2]+IJ$ .

Then ADHM construction claims that, given certain regularity conditions,

Given B₁, B₂, I, J such that $μ r = μ c = 0$ , an Anti-Self-Dual instanton of charge N and instanton number k can be constructed,
All Anti-Self-Dual instantons can be obtained in this way.

[edit] The construction formula

Let x be the 4-dimensional Euclidean spacetime coordinates written in quaternionic notation $x_{ij}=\begin{pmatrix}z_2&z_1\\-\bar{z_1}&\bar{z_2}\end{pmatrix}$ .

Consider the 2k × (N+2k) matrix

$\Delta= \begin{pmatrix}I&B_2+z_2&B_1+z_1\\J^\dagger&-B_1^\dagger-\bar{z_1}&B_2^\dagger+\bar{z_2}\end{pmatrix}$ .

Then the conditions $\displaystyle\mu_r=\mu_c=0$ are equivalent to the factorization condition

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

Then a hermition projection operator P can be constructed as

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

The nullspace of Δ(x) is of N dimension for generic x. The basis vector 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 guaranteed the completeness condition

$P+UU^\dagger=1$

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

$A_m=U^\dagger \partial_m U$ .

[edit] References

Construction of Instantons, Michael Atiyah, Vladimir G. Drinfel'd, Nigel. J. Hitchin, Yuri I. Manin, Phys. Lett. A65 (1978) 185-187
Instantons in Gauge Theory by M. Shifman.
On the Construction of Monopoles