The ADHM construction or monad construction is the construction of all instanton

Instanton

An instanton 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 that plays the role of physical space-time in non-abelian gauge theory...

s using method of linear algebra by Michael Atiyah

Michael Atiyah

Sir Michael Francis Atiyah, OM, FRS, FRSE is a British mathematician working in geometry.Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at Oxford and Cambridge, and in the United States at the Institute for Advanced Study...

, Vladimir G. Drinfel'd, Nigel. J. Hitchin, Yuri I. Manin

Yuri I. Manin

Yuri Ivanovitch Manin is a Soviet/Russian/German mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics.-Biography:...

 in their paper Construction of Instantons.

ADHM data

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
  Real number
  In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
  
   moment map
  Moment map
  In mathematics, specifically in symplectic geometry, the momentum map is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The moment map generalizes the classical notions of linear and angular momentum...
  
   ,
• a complex
  Complex number
  A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
  
   moment map .

Then ADHM construction claims that, given certain regularity conditions,

• Given B₁, B₂, I, J such that , an Anti-Self-Dual instanton
  Instanton
  An instanton 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 that plays the role of physical space-time in non-abelian gauge theory...
  
   in a SU(N)
  Special unitary group
  The special unitary group of degree n, denoted SU, is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication...
  
   gauge theory
  Gauge theory
  In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
  
   with instanton
  Instanton
  An instanton 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 that plays the role of physical space-time in non-abelian gauge theory...
  
   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
  Adjoint representation
  In mathematics, the adjoint representation of a Lie group G is the natural representation of G on its own Lie algebra...
  
   and on I and J via the fundamental
  Fundamental representation
  In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group...
  
   and antifundamental representations
• The metric
  Metric (mathematics)
  In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
  
   on the moduli space
  Moduli space
  In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
  
   of instantons is that inherited from the flat metric on B, I and J.

Noncommutative instantons

In a noncommutative

Noncommutative geometry

Noncommutative geometry is a branch of mathematics concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative algebras of functions...

 gauge theory, the ADHM construction is identical but the moment map is set equal to the self-dual projection of the noncommutativity matrix of the spacetime times the identity matrix

Identity matrix

In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...

. In this case instantons exist even when the gauge group is U(1). The noncommutative instantons were discovered by Nikita Nekrasov

Nikita Nekrasov

-Background:Nerkrasov was born in Moscow, Russia and attended the Moscow State 57th School, graduating in 1995. He went on to graduate with honors from Moscow Physical Technical Institute in 1995. Nekrasov did his graduate work at Princeton University, receiving his PhD in 1996...

 and Albert Schwarz

Albert Schwarz

Albert S. Schwarz is a mathematician and a theoretical physicist educated in Soviet Union and now a Professor at the University of California-Davis. He is one of the pioneers of Morse theory and brought up the first example of a topological quantum field theory. Schwarz worked on some examples in...

 in 1998.

Vortices

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

Abrikosov vortex

In superconductivity, an Abrikosov vortex is a vortex of supercurrent in a type-II superconductor. The supercurrent circulates around the normal core of the vortex. The core has a size \sim\xi — the superconducting coherence length...

 in a supersymmetric

Supersymmetry

In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...

 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

Condensate

Condensate may refer to:* The liquid phase produced by the condensation of steam or any other gas* The product of a chemical condensation reaction, other than water* Natural gas condensate, in the natural gas industry- Quantum physics :...

, plays the role of the noncommutativity parameter in the real moment map.

The construction formula

Let x be the 4-dimensional Euclidean

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

 spacetime

Spacetime

In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

 coordinates written in quaternion

Quaternion

In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...

ic notation
.

Consider the 2k × (N+2k) matrix.

Then the conditions are equivalent to the factorization condition where f(x) is a k × k hermitian

Hermitian

A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite:*Hermitian adjoint*Hermitian connection, the unique connection on a Hermitian manifold that satisfies specific conditions...

 matrix.

Then a hermitian projection

Projection (linear algebra)

In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. It leaves its image unchanged....

 operator P can be constructed as.

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

The anti-selfdual connection

Connection (mathematics)

In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport...

is then constructed from U by the formula.