ADHM construction
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In
mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developme ...
and gauge theory, the ADHM construction or monad construction is the construction of all
instanton An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. Mo ...
s using methods of linear algebra by
Michael Atiyah Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded th ...
,
Vladimir Drinfeld Vladimir Gershonovich Drinfeld ( uk, Володи́мир Ге́ршонович Дрінфельд; russian: Влади́мир Ге́ршонович Дри́нфельд; born February 14, 1954), surname also romanized as Drinfel'd, is a renowne ...
,
Nigel Hitchin Nigel James Hitchin FRS (born 2 August 1946) is a British mathematician working in the fields of differential geometry, gauge theory, algebraic geometry, and mathematical physics. He is a Professor Emeritus of Mathematics at the University o ...
, Yuri I. Manin in their paper "Construction of Instantons."


ADHM data

The ADHM construction uses the following data: * complex
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s ''V'' and ''W'' of dimension ''k'' and ''N'', * ''k'' × ''k'' complex matrices ''B''1, ''B''2, a ''k'' × ''N'' complex matrix ''I'' and a ''N'' × ''k'' complex matrix ''J'', * a
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
moment map In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the ac ...
\mu_r = _1,B_1^\dagger _2,B_2^\daggerII^\dagger-J^\dagger J, * a
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
moment map \displaystyle\mu_c = _1,B_2IJ. Then the ADHM construction claims that, given certain regularity conditions, * Given ''B''1, ''B''2, ''I'', ''J'' such that \mu_r=\mu_c=0, an anti-self-dual
instanton An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. Mo ...
in a SU(''N'') gauge theory with instanton number ''k'' can be constructed, * All anti-self-dual
instantons An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. Mo ...
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 In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is G ...
and on ''I'' and ''J'' via the
fundamental Fundamental may refer to: * Foundation of reality * Fundamental frequency, as in music or phonetics, often referred to as simply a "fundamental" * Fundamentalism, the belief in, and usually the strict adherence to, the simple or "fundamental" idea ...
and antifundamental representations * The
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...
on the moduli space of instantons is that inherited from the flat metric on ''B'', ''I'' and ''J''.


Generalizations


Noncommutative instantons

In a
noncommutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
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 Nikita Alexandrovich Nekrasov (russian: Ники́та Алекса́ндрович Некра́сов; born 10 April 1973) is a mathematical and theoretical physicist at the Simons Center for Geometry and Physics and C.N.Yang Institute for Th ...
and
Albert Schwarz Albert Solomonovich Schwarz (; russian: А. С. Шварц; born June 24, 1934) is a Soviet and American mathematician and a theoretical physicist educated in the Soviet Union and now a professor at the University of California, Davis. Early lif ...
in 1998.


Vortices

Setting ''B''2 and ''J'' to zero, one obtains the classical moduli space of nonabelian vortices in a
supersymmetric In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories ...
gauge theory with an equal number of colors and flavors, as was demonstrated i
Vortices, instantons and branes
The generalization to greater numbers of flavors appeared i
Solitons in the Higgs phase: The Moduli matrix approach
In both cases the Fayet–Iliopoulos term, which determines a squark
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 * ''Condensate'' (album) ...
, plays the role of the noncommutativity parameter in the real moment map.


The construction formula

Let ''x'' be the 4-dimensional Euclidean
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
coordinates written in quaternionic notation x_=\beginz_2&z_1\\-\bar&\bar\end. Consider the 2''k'' × (''N'' + 2''k'') matrix :\Delta= \beginI&B_2+z_2&B_1+z_1\\J^\dagger&-B_1^\dagger-\bar&B_2^\dagger+\bar\end. Then the conditions \displaystyle\mu_r=\mu_c=0 are equivalent to the factorization condition :\Delta\Delta^\dagger=\beginf^&0\\0&f^\end where ''f''(''x'') is a ''k'' × ''k''
Hermitian matrix In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -t ...
. Then a hermitian projection operator ''P'' can be constructed as :P=\Delta^\dagger\beginf&0\\0&f\end\Delta. The
nullspace In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kernel of ...
of Δ(''x'') is of dimension ''N'' for generic ''x''. The basis vectors for this null-space can be assembled into an (''N'' + 2''k'') × ''N'' matrix ''U''(''x'') with orthonormalization condition ''U''''U'' = 1. A regularity condition on the rank of Δ guarantees 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.


See also

* Monad (linear algebra) *
Twistor theory In theoretical physics, twistor theory was proposed by Roger Penrose in 1967 as a possible path to quantum gravity and has evolved into a branch of theoretical and mathematical physics. Penrose proposed that twistor space should be the basic are ...


References

* * * Hitchin, N. (1983)
"On the Construction of Monopoles"
''Commun. Math. Phys.'' 89, 145–190. Gauge theories Differential geometry Quantum chromodynamics {{Quantum-stub