Deconstructing Monopoles and Instantons

We give a unifying description of the Dirac monopole on the 2-sphere S2, of a graded monopole on a (2, 2)-supersphere S2,2 and of the BPST instanton on the 4-sphere S4, by constructing a suitable global projector p via equivariant maps. This projector determines the projective module of finite type of sections of the corresponding vector bundle. The canonical connection ∇ = p◦d is used to compute the topological charge which is found to be equal to −1 for the three cases. The transposed projector q = p gives the value +1 for the charges; this showing that transposition of projectors, although an isomorphism in K-theory, is not the identity map. We also study the invariance under the action of suitable Lie groups. This work is dedicated to Matteo 1 Preliminaries and Introduction It is well known since the early sixties that vector bundles can be though of as projective modules of finite type (finite projective modules ‘for short’). The Serre-Swan’s theorem [18] states that there is 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 fact, in [18] the correspondence is stated in the continuous category, meaning for topological manifolds and vector bundles and for functions and sections which are continuous. However, the equivalence can be extended to the smooth case [8]. This correspondence was already used in [12] 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 [7] 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. Furthermore, since the creation of noncommutative geometry, finite projective modules are increasingly being used among (mathematical)-physicists. In this paper we present a finite-projective-module description of the basic topologically non trivial gauge configurations, namely monopoles and instantons. This will be done by constructing a suitable global projector p ∈ MN(C(M)), the latter being the algebra of N × N matrices whose entries are elements of the algebra C(M) of smooth functions defined over the base space. That p is a projector is expressed by the conditions p = p = p. The module of sections of the vector bundles on which monopoles or instantons live is identified with the image of p in the trivial module C(M) (corresponding to the trivial rank N -vector bundle over M), i.e. as the right module p(C(M)) . Now, not all the projectors that we construct are new. The Dirac monopole [9] projector is already present in [11] (in fact, the monopole projector is well know among physicists), while the BPST instanton [4] is present in the ADHM analysis [1], albeit in a local form. Our presentation is a global one which does not use any local chart or partition of unity and it 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. It is this reduction that has motivated the word ‘deconstructing’ in our title. For the time being, we present only the projectors carrying the lowest values of the charges, i.e. ±1. Now, when the sphere S is regarded as the complex projective space CP 1 or the compactified plane C∞ = C ∪ {∞} the monopole projector translates into the Bott projector (see for instance [20]). Thus, for the sphere S and the supersphere S the projectors we construct could be considered as analogues of the Bott projector for S. These three projectors will then give a generator of the reduced K-theory groups [11] K̃(S), K̃(S) and K̃(S) respectively. The construction of global (i.e. without partition of unity and local charts) projectors for all values of the charges as well as for projective spaces will be the content of a paper in preparation [14].