ADHM construction

ADHM Construction

ADHM construction

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Description:
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.

ADHM data

The ADHM construction uses the following data:

  • complex vector spaces V and W of dimension k and N,
  • k &times; k complex matrices B<sub>1</sub>, B<sub>2</sub>, a k &times; N complex matrix I and a N &times; k complex matrix J,
  • a real moment map <math>mu_r = ++II^dagger-J^dagger J</math>,
  • a complex moment map <math>displaystylemu_c = +IJ</math>.


Then ADHM construction claims that, given certain regularity conditions,
  • Given B<sub>1</sub>, B<sub>2</sub>, I, J such that <math>mu_r=mu_c=0</math>, an Anti-Self-Dual instanton in a SU 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 <math>vecmu </math> is set equal to the...
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