Ali H. Chamseddine

The Spectral Action Principle

Abstract:We propose a new action principle to be associated with a noncommutative space . The universal formula for the spectral action is where is a spinor on the Hilbert space, is a scale and a positive function. When this principle is applied to the noncommutative space defined by the spectrum of the standard model one obtains the standard model action coupled to Einstein plus Weyl gravity. There are relations between the gauge coupling constants identical to those of SU(5) as well as the Higgs self-coupling, to be taken at a fixed high energy scale.

Gravity and the standard model with neutrino mixing

ARASON H, 1992, PHYS REV D, V46, P3945, DOI 10.1103-PhysRevD.46.3945; Atiyah M.F., 1967, K THEORY; Avramidi I. G., 1986, THESIS MOSCOW U; BARRETT JW, HEPTH0608221; Carminati L, 1999, EUR PHYS J C, V8, P697; Casas JA, 2000, NUCL PHYS B, V573, P652, DOI 10.1016-S0550-3213(99)00781-6; Chamseddine AH, 1996, PHYS REV LETT, V77, P4868, DOI 10.1103-PhysRevLett.77.4868; CHAMSEDDINE AH, 1992, PHYS LETT B, V296, P109, DOI 10.1016-0370-2693(92)90810-Q; Chamseddine AH, 2006, J MATH PHYS, V47, DOI 10.1063-1.2196748; Chamseddine AH, 1997, COMMUN MATH PHYS, V186, P731, DOI 10.1007-s002200050126; CHANG D, 1985, PHYS REV D, V31, P1718, DOI 10.1103-PhysRevD.31.1718; CODELLO A, HEPTH0607128; Coleman S., 1985, ASPECTS SYMMETRY; CONNES A, HEPTH0608226; Connes A, 1996, COMMUN MATH PHYS, V182, P155, DOI 10.1007-BF02506388; Connes A., 1994, NONCOMMUTATIVE GEOME; CONNES A, 1995, J MATH PHYS, V36, P6194, DOI 10.1063-1.531241; Dabrowski L., 2003, BANACH CTR PUBLICATI, V61, P49; DONOGHUE JF, 1994, PHYS REV D, V50, P3874, DOI 10.1103-PhysRevD.50.3874; EINHORN MB, 1992, PHYS REV D, V46, P5206, DOI 10.1103-PhysRevD.46.5206; Feynman R.P., 1995, FEYNMAN LECT GRAVITA; FIGUEROA H, 2000, ELEMENTS NONCOMMUTAT; Frohlich J., 1994, CRM P LECT NOTES, V7, P57; GILKEY P, 1984, INVARIANCE THEORY EQ; Gracia-Bondia JM, 1998, PHYS LETT B, V416, P123, DOI 10.1016-S0370-2693(97)01310-5; HOLMAN R, 1991, PHYS REV D, V43, P3833, DOI 10.1103-PhysRevD.43.3833; Inagaki T, 2004, J HIGH ENERGY PHYS; Knecht M, 2006, PHYS LETT B, V640, P272, DOI 10.1016-j.physletb.2006.06.052; Kolda C, 2000, J HIGH ENERGY PHYS, DOI 10.1088-1126-6708-2000-07-035; Lawson H.B., 1989, PRINCETON MATH SERIE, V38; Lazzarini S, 2001, PHYS LETT B, V510, P277, DOI 10.1016-S0370-2693(01)00595-0; Lizzi F, 1997, PHYS REV D, V55, P6357, DOI 10.1103-PhysRevD.55.6357; Mohapatra R. N., 2004, MASSIVE NEUTRINOS PH; vanNieuwenhuizen P, 1996, PHYS LETT B, V389, P29, DOI 10.1016-S0370-2693(96)01251-8; PARKER L, 1984, PHYS REV D, V29, P1584, DOI 10.1103-PhysRevD.29.1584; PERCACCI R, HEPTH0409199; PILAFTSIS A, 2002, PHYS REV D, V29; Ramond P., 1990, FIELD THEORY MODERN; REINA L, HEPTH0512377; ROSS G, 1985, FRONTIERS PHYS SERIE, V60; SHER M, 1989, PHYS REP, V179, P273, DOI 10.1016-0370-1573(89)90061-6; Veltman M., 1994, DIAGRAMMATICA PATH F; Weinberg S., 1972, GRAVITATION COSMOLOG

Universal Formula for Noncommutative Geometry Actions: Unification of Gravity and the Standard Model

A universal formula for an action associated with a noncommutative geometry, defined by a spectal triple (A,H, D), is proposed. It is based on the spectrum of the Dirac operator and is a geometric invariant. The new symmetry principle is the automorphism of the algebra A which combines both diffeomorphisms and internal symmetries. Applying this to the geometry defined by the spectrum of the standard model gives an action that unifies gravity with the standard model at a very high energy scale. PACS numbers: 02.40, 04.62, 12.10.

Gravity in non-commutative geometry

We study general relativity in the framework of non-commutative differential geometry. As a prerequisite we develop the basic notions of non-commutative Riemannian geometry, including analogues of Riemannian metric, curvature and scalar curvature. This enables us to introduce a generalized Einstein-Hilbert action for non-commutative Riemannian spaces. As an example we study a space-time which is the product of a four dimensional manifold by a two-point space, using the tools of non-commutative Riemannian geometry, and derive its generalized Einstein-Hilbert action. In the simplest situation, where the Riemannian metric is taken to be the same on the two copies of the manifold, one obtains a model of a scalar field coupled to Einstein gravity. This field is geometrically interpreted as describing the distance between the two points in the internal space.

Topological gauge theory of gravity in five and all odd dimensions

Abstract A topological gauge theory of gravity in five dimensions is presented. This is based on the Chern-Simons five-form and the SO(1, 5) gauge group. The action contains a Gauss-Bonnet term, an Einstein term and a cosmological constant. Quantization and renormalizability of the theory are discussed. Indications of how to generalize to arbitrary odd dimensions are given.

Topological gravity and supergravity in various dimensions

Topological theories of gravity are constructed in odd-dimensional space-times of dimensions 2n + 1, using the Chern-Simons (2n + 1)-forms and with the gauge groups ISO(1, 2n) or SO(1, 2n + 1) or SO(2, 2n). In even dimensions the presence of a scalar field in the fundamental representation of the gauge group is needed, besides the gauge field. Supersymmetrization of the de Sitter groups can be performed up to a maximal dimension of seven, but there is no limit on the super-Poincare groups. The different phases of the topological theory are investigated. It is argued that these theories are finite. It is shown that the graviton propagates in a perturbative sense around a non-trivial background.

NonAbelian solitons in N=4 gauged supergravity and leading order string theory

We study static, spherically symmetric, and purely magnetic solutions of the N=4 gauged supergravity in four dimensions. A systematic analysis of the supersymmetry conditions reveals solutions which preserve 1/4 of the supersymmetries and are characterized by a BPS-monopole-type gauge field and a globally hyperbolic, everywhere regular geometry. We show that the theory in which these solutions arise can be obtained via compactification of ten-dimensional supergravity on the group manifold. This result is then used to lift the solutions to ten dimensions.

Mimetic Dark Matter

A bstractWe modify Einstein’s theory of gravity, isolating the conformal degree of freedom in a covariant way. This is done by introducing a physical metric defined in terms of an auxiliary metric and a scalar field appearing through its first derivatives. The resulting equations of motion split into a traceless equation obtained through variation with respect to the auxiliary metric and an additional differential equation for the trace part. As a result the conformal degree of freedom becomes dynamical even in the absence of matter. We show that this extra degree of freedom can mimic cold dark matter.

Deforming Einstein's gravity

Abstract A deformation of Einstein gravity is constructed based on gauging the noncommutative ISO (3,1) group using the Seiberg–Witten map. The transformation of the star product under diffeomorphism is given, and the action is determined to second order in the deformation parameter.

Cosmology with Mimetic Matter

We consider minimal extensions of the recently proposed Mimetic Dark Matter and show that by introducing a potential for the mimetic non-dynamical scalar field we can mimic nearly any gravitational properties of the normal matter. In particular, the mimetic matter can provide us with inflaton, quintessence and even can lead to a bouncing nonsingular universe. We also investigate the behaviour of cosmological perturbations due to a mimetic matter. We demonstrate that simple mimetic inflation can produce red-tilted scalar perturbations which are largely enhanced over gravity waves.