Linear Connections on Fuzzy Manifolds

Linear connections are introduced on a series of noncommutative geometries which have commutative limits. Quasicommutative corrections are calculated.Comment: 10 pages PlainTex; LPTHE Orsay 95/42; ESI Vienna 23

Fuzzy Surfaces of Genus Zero

A fuzzy version of the ordinary round 2-sphere has been constructed with an invariant curvature. We here consider linear connections on arbitrary fuzzy surfaces of genus zero. We shall find as before that they are more or less rigidly dependent on the differential calculus used but that a large number of the latter can be constructed which are not covariant under the action of the rotation group. For technical reasons we have been forced to limit our considerations to fuzzy surfaces which are small perturbations of the fuzzy sphere.Comment: 11 pages, Late

High Resolution Rotation Curves of Low Surface Brightness Galaxies

High resolution Halpha rotation curves are presented for five low surface brightness galaxies. These Halpha rotation curves have shapes different from those previously derived from HI observations, probably because of the higher spatial resolution of the Halpha observations. The Halpha rotation curves rise more steeply in the inner parts than the HI rotation curves and reach a flat part beyond about two disk scale lengths. With radii expressed in optical disk scale lengths, the rotation curves of the low surface brightness galaxies presented here and those of HSB galaxies have almost identical shapes. Mass modeling shows that the contribution of the stellar component to the rotation curves may be scaled to explain most of the inner parts of the rotation curves, albeit with high stellar mass-to-light ratios. On the other hand, well fitting mass models can also be obtained with lower contributions of the stellar disk. These observations suggest that the luminous mass density and the total mass density are coupled in the inner parts of these galaxies.Comment: Accepted for publication in ApJ Letter

Linear connections on matrix geometries

A general definition of a linear connection in noncommutative geometry has been recently proposed. Two examples are given of linear connections in noncommutative geometries which are based on matrix algebras. They both possess a unique metric connection.Comment: 14p, LPTHE-ORSAY 94/9

Topology at the Planck Length

A basic arbitrariness in the determination of the topology of a manifold at the Planck length is discussed. An explicit example is given of a `smooth' change in topology from the 2-sphere to the 2-torus through a sequence of noncommuting geometries. Applications are considered to the theory of D-branes within the context of the proposed $M$(atrix) theory.Comment: Orsay Preprint 97/34, 17 pages, Late

Linear Connections in Non-Commutative Geometry

A construction is proposed for linear connections on non-commutative algebras. The construction relies on a generalisation of the Leibnitz rules of commutative geometry and uses the bimodule structure of $\Omega^1$. A special role is played by the extension to the framework of non-commutative geometry of the permutation of two copies of $\Omega^1$. The construction of the linear connection as well as the definition of torsion and curvature is first proposed in the setting of the derivations based differential calculus of Dubois- Violette and then a generalisation to the framework proposed by Connes as well as other non-commutative differential calculi is suggested. The covariant derivative obtained admits an extension to the tensor product of several copies of $\Omega^1$. These constructions are illustrated with the example of the algebra of $n \times n$ matrices.Comment: 15 pages, LMPM ../94 (uses phyzzx

Detection of the Red Giant Branch Stars in M82 Using the Hubble Space Telescope

We present color-magnitude diagrams and luminosity functions of stars in two halo regions of the irregular galaxy in M82, based on F555W and F814W photometry taken with the Hubble Space Telescope and Wide Field Planetary Camera 2. The I-band luminosity function shows a sudden jump at I~23.95 mag, which is identified as the tip of the red giant branch (TRGB). Adopting the Lee et al. (1993) calibration of the TRGB based on the RR Lyrae distances to Galactic globular clusters, we obtain the distance modulus of (m-M)_0=27.95 +- 0.14 (random) +- 0.16 (systematic) mag. This corresponds to a linear distance of 3.9 +- 0.3 (random) +- 0.3 (systematicf) Mpc, which agrees well with the distance of M81 deteremined from the HST observations of the Cepheid variable stars. In addition, we observe a significant number of stars apparently brighter than the TRGB. However, with the current data, we cannot rule out whether these stars are blends of fainter stars, or are indeed intermediate-age asymptotic giant branch stars.Comment: 8 figure

On Curvature in Noncommutative Geometry

A general definition of a bimodule connection in noncommutative geometry has been recently proposed. For a given algebra this definition is compared with the ordinary definition of a connection on a left module over the associated enveloping algebra. The corresponding curvatures are also compared.Comment: 16 pages, PlainTe

Quantum Spacetimes in the Year 1

We review certain emergent notions on the nature of spacetime from noncommutative geometry and their radical implications. These ideas of spacetime are suggested from developments in fuzzy physics, string theory, and deformation quantisation. The review focuses on the ideas coming from fuzzy physics. We find models of quantum spacetime like fuzzy $S^4$ on which states cannot be localised, but which fluctuate into other manifolds like $CP^3$ . New uncertainty principles concerning such lack of localisability on quantum spacetimes are formulated.Such investigations show the possibility of formulating and answering questions like the probabilty of finding a point of a quantum manifold in a state localised on another one. Additional striking possibilities indicated by these developments is the (generic) failure of $CPT$ theorem and the conventional spin-statistics connection. They even suggest that Planck's `` constant '' may not be a constant, but an operator which does not commute with all observables. All these novel possibilities arise within the rules of conventional quantum physics,and with no serious input from gravity physics.Comment: 11 pages, LaTeX; talks given at Utica and Kolkata .Minor corrections made and references adde

Noncommutative Geometry as a Regulator

We give a perturbative quantization of space-time $R^4$ in the case where the commutators $C^{{\mu}{\nu}}=[X^{\mu},X^{\nu}]$ of the underlying algebra generators are not central . We argue that this kind of quantum space-times can be used as regulators for quantum field theories . In particular we show in the case of the ${\phi}^4$ theory that by choosing appropriately the commutators $C^{{\mu}{\nu}}$ we can remove all the infinities by reproducing all the counter terms . In other words the renormalized action on $R^4$ plus the counter terms can be rewritten as only a renormalized action on the quantum space-time $QR^4$ . We conjecture therefore that renormalization of quantum field theory is equivalent to the quantization of the underlying space-time $R^4$ .Comment: Latex, 30 pages, no figures,typos corrected,references added . Substantial amount of rewriting of the last section . Final interesting remarks added at the end of the pape