Moduli spaces of instantons on toric noncommutative manifolds

Abstract

We study analytic aspects of U(n) gauge theory over a toric noncommutative manifold M θ . We analyse moduli spaces of solutions to the selfdual Yang-Mills equations on U(2) vector bundles over four-manifolds M θ , showing that each such moduli space is either empty or a smooth Hausdorff manifold whose dimension we explicitly compute. In the special case of the four-sphere S 4 θ we find that the moduli space of U (2) instantons with fixed second Chern number k is a smooth manifold of dimension 8k − 3. e-print archive:

Scalar curvature in conformal geometry of Connes–Landi noncommutative manifolds

“…defines a deformed operator π θ (T ). The associativity of the × θ -multiplication turns out to be a functorial property for the deformation, whose categorical description can be found in [5]. As a special case, we see that the whole representation C ∞ (M ) ⊂ B(H) is deformed in the following sense.…”

“…The deformation extends to toric equivariant vector bundles (cf. [5]) over M . It simply means that we can deform the C ∞ (M )-module structure on the smooth sections to C ∞ (M θ )-bimodules using the same formula (2.4).…”

Modular curvature for toric noncommutative manifolds

“…An interesting class of examples comes from deformation of classical Riemannian manifolds, such is the Connes-Landi deformations (cf. [13,12]), also called toric noncommutative manifolds in [5]. The underlying deformation theory, called θ-deformation in the literature, was first developed in Rieffel's work [35].…”

“…In this section, we will provide the functional analytic framework which is necessary for our later discussion on toric noncommutative manifold. We refer to Rieffel's monograph [35] for further details, also [24], [5] and [43]. All the topological vectors spaces appeared in this paper are over the field of complex numbers.…”

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].…”