ADHM construction


In mathematical physics and gauge theory, 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."

ADHM data

The ADHM construction uses the following data:
Then the ADHM construction claims that, given certain regularity conditions,

Noncommutative instantons

In a noncommutative 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. In this case instantons exist even when the gauge group is U. 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 . The generalization to greater numbers of flavors appeared in . 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
Consider the 2k × matrix
Then the conditions are equivalent to the factorization condition
Then a hermitian projection operator P can be constructed as
The nullspace of Δ is of dimension N for generic x. The basis vectors for this null-space can be assembled into an × N matrix U with orthonormalization condition U†U = 1.
A regularity condition on the rank of Δ guarantees the completeness condition
The anti-selfdual connection is then constructed from U by the formula