arXiv:1011.2706v1 [hep-th] 11 Nov 2010 Noncommutative Geometry and Supergravity. Jos´ e Luis L´ opez, ∗ O.Obreg´on, † and M. Sabido ‡ Departamento de F´ ısica de la Universidad de Guanajuato, A.P. E-143, C.P. 37150, Le´ on, Guanajuato, M´ exico M. P. Ryan § Instituto de Ciencias Nucleares Universidad Nacional Aut´ onoma de M´ exico, A.P. 70-543, M´ exico D.F. 04510,M´ exico. (Dated: November 12, 2010) A spectral action associated with an Einstein-Cartan formulation of supergravity is proposed. To construct this action we make use of the Seeley-DeWitt coefficients in a Riemann-Cartan space. For consistency in its construction the Rarita-Schwinger action is added to the resulting spectral action. PACS numbers: 04.65.+e,02.40.Gh,11.10.Nx I. INTRODUCTION The equivalence principle and gauge invariance are fun- damental pillars of the two most successful theories in physics, general relativity and Yang-Mills theory. By means of them the basic interactions in our Universe can be understood, however these theories seem to be incom- patible at the quantum level. This incompatibility might suggest that they are theories arising from some other more fundamental formulation. One of the most inter- esting proposals in the literature is the spectral action of noncommutative geometry. It involves new spectral geometry consistent with the physical measurements of distances. The usual emphasis on the points x ∈M on a geometric space is replaced by the spectrum Σ of the Dirac operator D. It is assumed that the spectral action depends only on Σ. This is the spectral action principle. The spectrum is a geometric invariant that replaces dif- feomorphism invariance. By applying this basic principle to the noncommutative geometry defined by the standard model it has been shown [1] that the dynamics of all in- teractions, including gravity are given by the spectral action. Its heat kernel expansion in terms of the Seeley- De Witt coefficients a n results in an effective action up to the coefficient considered. For the gravitational sector of the spectral action, the first three terms on the expan- sion correspond to a constant, the usual Einstein-Hilbert action plus Weyl gravity and a Gauss-Bonnet topological invariant. The Dirac operator considered in the spectral action principle proposal [1–4] is constructed with the Ricci ro- tation coefficients, which are assumed to be an explicit * jl lopez@fisica.ugto.mx † octavio@fisica.ugto.mx ‡ msabido@fisica.ugto.mx § ryanmex2002@yahoo.com; Present address: 30101 Clipper Lane, Millington MD 21651 USA. function of the tetrads and their derivatives, namely the ones resulting from solving the Riemannian torsion-free condition in standard general relativity. However, if the Ricci rotation coefficients were considered as independent variables, one would need to write the action up to the desired order of approximation in the Seeley-De Witt co- efficients and then vary the action with respect to both the connection and the metric (or tetrads) independently, as in the Palatini first order formulation of general rela- tivity. The Ricci connections would then be complicated functions of the tetrads and would depend on the param- eters that appear at each order in the action. The Rie- mann tensor and tensors derived from it will depend on these generalized connections and not on the usual Rie- mannian connection of standard general relativity. It is also well known that theories with higher order terms in the Riemann, Ricci tensors or Ricci scalar do not provide the same equations of motion obtained from the second as opposed to those from the first order formalism [5]. The gravity action arising from the spectral action is usually given in terms of functions of the metric (or tetrads) and its derivatives obtained from using the Ricci rotation co- efficients as explicit functions of the metric or tetrads and their derivatives as in the standard second order formu- lation of general relativity [5]. In [1], the authors derive this gravity spectral action. It is constructed as a func- tion of the metric and its derivatives, by means of the Ricci rotation coefficients corresponding to the standard Riemannian connection. Gravity theories with torsion have been of interest for many years [6]. Torsion is usually associated with some matter fields in the action. The Ricci rotation coefficients in the theory are modified by adding the contorsion to the standard connection. One associates this new connec- tion with a Dirac operator. The presence of matter in the action yields, after solving the torsion constraint, a par- ticular dependence of the torsion on the matter content of the theory. On the other hand, Seeley-De Witt coeffi- cients have been calculated for connections in a Riemann- Cartan manifold [7], by means of them the expansion of