Anti-self-dual Research Articles

NONCOMMUTATIVE INSTANTONS AND THE INFORMATION METRIC

By using the so-called Information Metric on the moduli space of an anti-self-dual (ASD) Instanton in a self-dual (SD) noncommutative background, we investigate the geometry of moduli space. The metric is evaluated perturbatively in noncommutativity parameter and we show that by putting a cutoff in the location of instanton in the definition of Information Metric we can recover the five-dimensional space–time in the presence of a B-field. This result shows that the noncommutative YM-Instanton Moduli corresponds to D-Instanton Moduli in the gravity side where the radial and transverse location of D-Instanton correspond to YM-Instanton size and location, respectively. The match is shown in the first order of noncommutativity parameter.

Gluing theorems for complete anti-self-dual spaces

1.1 Summary. One of the special features of 4-dimensional differential geometry is the existence of objects with self-dual (SD) or anti-self-dual (ASD) curvature. The objects in question can be connections in an auxiliary bundle over a 4-manifold, leading to the study of instantons in Yang– Mills theory [DK], or as in this paper, Riemannian metrics or conformal structures. Although such ASD conformal structures give absolute minima of the functional c �→ � W (c)� 2 ,w hereW (c) denotes the Weyl tensor of the conformal structure c, variational methods are not well suited to the study of this problem, essentially because of its conformal invariance. For this reason, gluing theorems provide a very important source of information about ASD conformal structures. Our purpose in this paper is to give some new and rather general gluing theorems for ASD and HermitianASD conformal structures, following the method suggested by Floer in [F]. The prototypical gluing theorem takes a pair (Xj ,c j )( j =1 , 2) of compact conformally ASD 4-manifolds and analyzes the problem of finding an ASD conformal structure c on X = X1�X 2 that is ‘close to’ cj in suitable subsets Xj\Bj ⊂ X1�X 2. In this situation there exist finite-dimensional vector spaces (the obstruction spaces) H 2 cj (Xj) whose vanishing is sufficient to guarantee the existence of c with the desired properties. (If H 2 cj (Xj) � , then the gluing theorem yields a map from another finite-dimensional vector space into H 2 c1 (X1) ⊕ H 2 c2 (X2), the zeroes of which yield ASD conformal structures on X1�X 2.) The result just stated (gluing for compact conformally ASD spaces) was proved by Donaldson and Friedman [DF] and in a very special case by Floer [F]. The approach of [DF] was to exploit the twistor description [P], [AtHS] of conformally ASD spaces, to translate the gluing problem into one of deformation theory of complex singular spaces. Floer, on the other hand, worked directly with the 4-manifolds and used some tools from the theory of elliptic operators on non-compact manifolds with cylindrical ends.

Anti-self-dual Maxwell solutions on hyper-Kähler manifold and N = 2 supersymmetric Ashtekar gravity

Anti-self-dual (ASD) Maxwell solutions on four-dimensional hyper-Kähler manifolds are constructed. The N = 2 supersymmetric half-flat equations are derived in the context of the Ashtekar formulation of N = 2 supergravity. These equations show that the ASD Maxwell solutions have a direct connection with the solutions of the reduced N = 2 supersymmetric ASD Yang-Mills equations with a special choice of gauge group. Two examples of the Maxwell solutions are presented.

Canonical structures on anti-self-dual four-manifolds and the diffeomorphism group

An unusual and attractive system is studied that arises from the anti-self-dual (ASD) Yang–Mills equations with maximal translational symmetry and with gauge group the volume preserving diffeomorphisms of an auxiliary four-manifold ℳ. The resulting equations lead to a system consisting of a volume form together with four independent vector fields on ℳ satisfying three simple Lie bracket relations. This structure is shown to give rise to a two-sphere’s worth of closed simple two-forms which in turn lead to the standard hyper-Kahler structure of an anti-self-dual metric on ℳ. The system determines not only the ASD metric, but also a frame that is proportional to an orthonormal frame. It is shown that the freedom in the choice of frame is related to a pair of solutions of a modified Laplacian and can always be chosen so that the proportionality factor is unity. The Plebanski first and second heavenly forms for general ASD metrics are written out in terms of these structures (the first being the standard description in terms of a complex structure and Kahler scalar). One of the scalars is interpreted as the generating function for the diffeomorphisms (symplectomorphism) in line with the origin of the system as the ASD Yang–Mills equations with the volume preserving diffeomorphisms as gauge group.