Projective Modules of Finite Type and Monopoles over S

We give a unifying description of all inequivalent vector bundles over the 2dimensional sphere S2 by constructing suitable global projectors p via equivariant maps. Each projector determines the projective module of finite type of sections of the corresponding complex rank 1 vector bundle over S2. The canonical connection ∇ = p ◦ d is used to compute the topological charges. Transposed projectors gives opposite values for the charges, thus showing that transposition of projectors, although an isomorphism in K-theory, is not the identity map. Also, we construct the partial isometry yielding the equivalence between the tangent projector (which is trivial in K-theory) and the real form of the charge 2 projector. This work is dedicated to Jacopo 1 Preliminaries and Introduction Since the creation of noncommutative geometry [1, 3] finite (i.e. of finite type) projective modules as substitutes for vector bundles are increasingly being used among (mathematical)-physicists. This substitution is based on the Serre-Swan’s theorem [12, 2] which construct a complete equivalence between the category of (smooth) vector bundles over a (smooth) compact manifold M and bundle maps, and the category of finite projective modules over the commutative algebra C(M) of (smooth) functions over M and module morphisms. The space Γ(M,E) of smooth sections of a vector bundle E → M over a compact manifold M is a finite projective module over the commutative algebra C(M) and every finite projective C(M)-module can be realized as the module of sections of some vector bundle over M . The correspondence was already used in [6] to give an algebraic version of classical geometry, notably of the notions of connection and covariant derivative. But it has been with the advent of noncommutative geometry that the equivalence has received a new emphasis and has been used, among several other things, to generalize the concept of vector bundles to noncommutative geometry and to construct noncommutative gauge and gravity theories. In this paper we present a finite-projective-module description of all monopoles configurations on the 2-dimensional sphere S. This will be done by constructing a suitable global projector p ∈ M|n|+1(C(S)), n ∈ Z being the value of the topological charge, which determines the module of sections of the vector bundles on which monopoles live, as the image of p in the trivial module C(S) (corresponding to the trivial rank (|n|+ 1)-vector bundle over S). Now, a local expression for projectors corresponding to monopoles was given in [11]. Our presentation is a global one which does not use any local chart or partition of unity. The price we pay for this is that the projector carrying charge n is a matrix of dimension (|n|+1)× (|n|+1) while in [11] the projectors were always 2× 2 matrices. Furthermore, our construction is based on a unifying description in terms of global equivariant maps. We express the projectors in terms of a more fundamental object, a vector-valued function of basic equivariant maps. In a sense, we may say that we ‘deconstruct’ the projectors [8]. The present construction will be generalized to supergeometry in [9] where we will report on a construction of ‘graded monopoles’ on the supersphere S. A friendly approach to modules of several kind (including finite projective) is in [7]. In the following we shall avoid writing explicitly the exterior product symbol for forms. 2 The General Construction Let us start by briefly describing the general scheme that will be given in details in the next Sections. Let π : S → S be the Hopf principal fibration over the sphere S with U(1) as structure group. We shall denote with BC =: C(S,C) the algebra of C-valued smooth functions on the total space S while AC =: C(S,C) will be the algebra of Cvalued smooth functions on the base space S. The algebra AC will not be distinguished from its image in the algebra BC via pullback. On C there are left actions of the group U(1) and they are labeled by an integer n ∈ Z,