Connection between Space-Time Supersymmetry and Non-Commutative Geometry

The non-commutative geometric construction of A. Connes [1-5] has been successful in giving a geometrical interpretation of the standard model as well as some grand unification models. Our lack of ability to quantize the non-commutative actions forced on us to quantize the classical actions resulting from the non-commutative ones by adopting the usual rules. The symmetries that might be present in a noncommutative action are then lost since they are not respected by the quantization scheme. On the other hand, theories with space-time supersymmetry has many desirable properties which are well known [6]. It is then tempting to construct noncommutative actions whose classical part has space-time supersymmetry. Since all the particle physics models constructed using non-commutative geometric methods correspond to non-commutative spaces of a four-dimensional manifold times a discrete set of points, it is natural to think of extending this to a supermanifold times a discrete set of points. However, it proved that there are many mathematical issues that must be settled before this approach could become acceptable. An alternative way is to consider supersymmetric theories in their component form. As supersymmetry transformations relate the fermionic fields to the bosonic fields and vice versa, it is possible to start with a fermionic action to recover the bosonic one. The simplest example is provided by the N = 1 super Yang-Mills theory in four dimensions . The action is given by [7]: I = ∫ dx ( − 4 F a μνF μνa + 1 2 λaγDμλ a ) , (1) where λ is a Majorana spinor in the adjoint representation of a gauge group G, F a μν is the field strength of the gauge field Aμ and Dμ is a gauge covariant derivative. The action (1) is invariant under the supersymmetry transformations δλ = − 2 γF a μνǫ, (2) δAaμ = ǫγμλ , (3) It is possible to derive the supersymmetric action (1) using the Noether’s method by starting with the free fermionic part of (1) and the supersymmetry transformations (2) and (3). To reformulate the action (1) using the methods of non-commutative geometry [1], we first define the triple (A, h,D) where h is the Hilbert space L(M, τ,√gd4x)⊗ C of spinors on a four-dimensional spin manifold M , A is the involutive algebra A = C(M)⊗Mn(C) of n× n matrix valued functions, and D the Dirac operator D = / ∂ ⊗ 1n on h. The free part of the fermionic action is written as 1 2 (λ, [/ ∂, λ]),