Moduli Spaces of Non-Commutative Instantons: Gauging Away Non-Commutative Parameters

Moduli spaces of instantons on toric noncommutative manifolds

“…As in previous works [5,6], the crucial tool in the present paper will be a functorial deformation procedure to derive the noncommutative geometry of M θ from the classical geometry of M in a systematic way, by deforming along an action of the N -torus T N . This "quantisation functor" constitutes the foundation upon which our construction is built, in the sense that it explains very precisely which aspects of the classical geometry are preserved by the deformation.…”

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.…”

“…The precise construction of the projection P k; goes exactly as in [5], as does the proof of the fact that the Grassmann family of connections ∇ = P k; • (id ⊗ d) has anti-self-dual curvature and hence defines a family of instantons on R 4 . For each point x ∈ M(k; ) there is an evaluation map…”

“…On the other hand, the noncommutative geometry of instantons and the ADHM construction on the Connes-Landi plane R 4 θ was investigated in [15,16,4,5]. As already mentioned, however, this abstract geometric characterisation of the instanton construction is in need of a more concrete description.…”

Mapping spaces and automorphism groups of toric noncommutative spaces

“…Examples of noncommutative principal bundles (i.e., Hopf-Galois extensions [9,10]) in this framework were studied in [20], and these constructions were subsequently abstracted and generalized in [1]. In applications to noncommutative gauge theory, moduli spaces of instantons on toric noncommutative spaces were analyzed in [7,8,11,13], while analogous moduli spaces of self-dual strings in higher noncommutative gauge theory were considered by [25].…”