ADHM construction

In mathematical physics, the ADHM construction or monad construction is the construction of all instantons using methods of linear algebra by Michael Atiyah, Vladimir Drinfeld, Nigel Hitchin, Yuri I. Manin in their paper "Construction of Instantons."

The ADHM construction uses the following data:

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

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 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.

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