Noncommutative Families of Instantons

The Adhm Construction of Instantons on Noncommutative Spaces

“…In this section we recall the basic theory of anti-self-dual connections on Euclidean space R 4 from the point of view of noncommutative geometry. Following [18,5], we then generalise this by recalling what it means to have a family of anti-self-dual connections on R 4 and when such families are gauge equivalent. These notions will pave the way for the algebraic formulation of the ADHM construction to follow.…”

Noncommutative instantons come in families

“…A noncommutative principal fibration A(S 4 θ ) → A(S 7 θ ) was introduced in [11] and infinitesimal instantons on it were constructed in [12] using infinitesimal conformal transformations; a global version is in [10] with the construction of a noncommutative family of instantons. We refer to these papers for a detailed description of the inclusion A(S 4 θ ) → A(S 7 θ ) as a noncommutative principal fibration (with classical SU(2) as structure group) and for its use for noncomutative instantons.…”

“…A parallel construction of instantons on A(S 4 θ ) is in [10], using a quantum group SL θ (2, H) and its quantum subgroup SO θ (2). An infinitesimal construction was proposed in [12] where a deformed dual enveloping algebra U θ (so(5, 1)) was used to generate infinitesimal instantons ('the tangent space to the moduli space') by acting on the basic instanton described above.…”

“…The deformation from U(g) to U θ (g) is implemented with a twist of Drinfel'd type [9] -in fact constructed in [14] for the cases at hand. The enveloping algebra U θ (so(5, 1)) was explicitly constructed in [12] while the dual Hopf algebra A (SL θ (2, H)) and its quotient A(SO θ (2)) are given in [10]. We briefly review them here.…”

“…In [12] infinitesimal instantons -'the tangent space to the moduli space' -were constructed using infinitesimal conformal transformations, that is elements in a quantized enveloping algebra U θ (so(5, 1)). In [10] we looked at a global construction and obtain generic charge 1 instantons by 'quantizing' the action of the Lie groups SL(2, H) and SO(2) on the basic instanton which enter the classical construction [1]. We review all this here.…”