Projective Modules of Finite Type over the Supersphere S

In the spirit of noncommutative geometry we construct all inequivalent vector bundles over the (2, 2)-dimensional supersphere S2,2 by means of global projectors p via equivariant maps. Each projector determines the projective module of finite type of sections of the corresponding ‘rank 1’ supervector bundle over S2,2. The canonical connection ∇ = p ◦ d is used to compute the Chern numbers by means of the Berezin integral on S2,2. The associated connection 1-forms are graded extensions of monopoles with not trivial topological charge. Supertransposed projectors gives opposite values for the charges. We also comment on the K-theory of S2,2. This work is dedicated to Anna 1 Preliminaries and Introduction The Serre-Swan’s theorem [23, 8] constructs 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 . In the context of noncommutative geometry [7], where a noncommutative algebra A is the analogue of the algebra of smooth functions on some ‘virtual noncommutative space’, finite projective (left/right) modules over A have been used as algebraic substitutes for vector bundles, notably in order to construct noncommutative gauge and gravity theories (see for instance, [6, 7, 9, 13, 19]). In fact, in noncommutative geometry there seems to be more that one possibility for the analogue of the category of vector bundles [14]. On the other hand, there is a generalization of ordinary geometry which loosely speaking goes under the name of supergeometry. Supergeometry can hardly be considered noncommutative geometry and, indeed, one usually labels it graded commutative geometry. In this paper we present a finite-projective-module description of all not trivial monopoles configurations on the (2, 2)-dimensional supersphere S. This will be done by constructing a suitable global projector p in the graded matrix algebra M|n|,|n|+1(G(S )), n being the value of the topological charge, while G(S) denotes the graded algebra of superfunctions on S. In the spirit of Serre-Swan theorem, the projector p determines the G(S)-module E of sections of the supervector bundles on which monopoles live, as its image in the trivial module G(S) (corresponding to the trivial rank (2|n|+1) supervector bundle over S), i.e. E = p(G(S)). The value of the topological charge is computed by taking the Berezin integral on S of a suitable form. These monopoles will be also called Grassmann (or graded) monopoles. A description of a Grassmann monopole on a supersphere, as a strong connection in the framework of the theory of Hopf-Galois extensions, is in [11]. We refer to [15] for a friendly approach to modules of several kind (including finite projective). Throughout the paper we shall avoid writing explicitly the exterior product symbol for forms. 2 A Few Elements of Graded Algebra and Geometry For us, graded will be synonymous of Z2-graded with the grading denoted as follows. If M =M0⊕M1, then |m| = j means m ∈Mj . The element m is said to be homogeneous if either m ∈ M0 or m ∈ M1. Elements of M0 (resp. of M1) are called even (resp. odd). A morphism φ :M → N of graded structures is said to be even [ resp. odd ] if φ(Mj) ⊆ Nj [ resp. φ(Mj) ⊆ Nj+1 , mod. 2 ]. With BL = (BL)0 + (BL)1 we shall indicate a real Grassmann algebra with L generators. For simplicity we shall assume that L < ∞; mild assumptions (on the linear span of the products of odd elements) would allow to treat the case L = ∞ as well. Here BL