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.

## Contents

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 real moment map $\mu_r = [B_1,B_1^\dagger]+[B_2,B_2^\dagger]+II^\dagger-J^\dagger J$,
• a complex moment map $\displaystyle\mu_c = [B_1,B_2]+IJ$.

Then ADHM construction claims that, given certain regularity conditions,

• Given B1, B2, I, J such that μr = μc = 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 $\vec\mu$ 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.

### 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 $x_{ij}=\begin{pmatrix}z_2&z_1\\-\bar{z_1}&\bar{z_2}\end{pmatrix}$.

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 hermitian 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 UU=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

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

## 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

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• List of Russian people — The Millennium of Russia monument in Veliky Novgorod, featuring the statues and reliefs of the most celebrated people in the first 1000 years of Russian history …   Wikipedia

• Michael Atiyah — Sir Michael Atiyah Born 22 April 1929 (1929 04 22) (age 82) …   Wikipedia

• Nahm equations — The Nahm equations are a system of ordinary differential equations introduced by Werner Nahm in the context of the Nahm transform – an alternative to Ward s twistor construction of monopoles. The Nahm equations are formally analogous to the… …   Wikipedia

• Nigel Hitchin — Nigel Hitchin, 2004 Born 2 August 1946 ( …   Wikipedia

• List of mathematics articles (A) — NOTOC A A Beautiful Mind A Beautiful Mind (book) A Beautiful Mind (film) A Brief History of Time (film) A Course of Pure Mathematics A curious identity involving binomial coefficients A derivation of the discrete Fourier transform A equivalence A …   Wikipedia

• Wolf Barth — is a mathematician whose work on vector bundles has been important for the ADHM construction.External links* [http://www.mi.uni erlangen.de/ barth/Eindex.html homepage] References Construction of Instantons , Michael Atiyah, Vladimir G. Drinfel d …   Wikipedia

• Geoffrey Horrocks — is a British mathematician whose work on vector bundles has been important for the ADHM construction.He was a professor at Newcastle University until his retirement in 1998. ee also*Horrocks Mumford bundleReferences* Construction of Instantons ,… …   Wikipedia

• Instanton — An instanton or pseudoparticle is a notion appearing in theoretical and mathematical physics. Mathematically, a Yang Mills instanton is a self dual or anti self dual connection in a principal bundle over a four dimensional Riemannian manifold… …   Wikipedia

• Vladimir Drinfel'd — Born February 4, 1954 (1954 02 04) (age 57) Kharkiv, Ukrainian SSR, Soviet Union (currently in Ukraine) Nationality …   Wikipedia

• List of Russian mathematicians — Andrey Kolmogorov, a preeminent 20th century mathematician. This list of Russian mathematicians includes the famous mathematicians from the Russian Empire, the Soviet Union and the Russian Federation. This list is incomplete; you can help by …   Wikipedia