Atiyah covering index theorem for riemannian foliations

We use the symbol calculus for foliations developed in our previous paper to derive a cohomological formula for the Connes-Chern character of the semi-finite spectral triple. The same proof works for the Type I spectral triple of Connes-Moscovici. The cohomology classes of the two Connes-Chern characters induce the same map on the image of the maximal Baum-Connes map in K-theory, thereby proving an Atiyah $L^2$ covering index theorem

Enlargeability, foliations, and positive scalar curvature

We extend the deep and important results of Lichnerowicz, Connes, and Gromov-Lawson which relate geometry and characteristic numbers to the existence and non-existence of metrics of positive scalar curvature (PSC). In particular, we show: that a spin foliation with Hausdorff homotopy groupoid of an enlargeable manifold admits no PSC metric; that any metric of PSC on such a foliation is bounded by a multiple of the reciprocal of the foliation K-area of the ambient manifold; and that Connes' vanishing theorem for characteristic numbers of PSC foliations extends to a vanishing theorem for Haefliger cohomology classes.Comment: To appear in Inventiones Mathematicae. We have made a minor editing chang

Gerbes, simplicial forms and invariants for families of foliated bundles

The notion of a gerbe with connection is conveniently reformulated in terms of the simplicial deRham complex. In particular the usual Chern-Weil and Chern-Simons theory is well adapted to this framework and rather easily gives rise to `characteristic gerbes' associated to families of bundles and connections. In turn this gives invariants for families of foliated bundles. A special case is the Quillen line bundle associated to families of flat SU(2)-bundlesComment: 28 page

An interesting example for spectral invariants

In "Illinois J. of Math. {\bf 38} (1994) 653--678", the heat operator of a Bismut superconnection for a family of generalized Dirac operators is defined along the leaves of a foliation with Hausdorff groupoid. The Novikov-Shubin invariants of the Dirac operators were assumed greater than three times the codimension of the foliation. It was then showed that the associated heat operator converges to the Chern character of the index bundle of the operator. In "J. K-Theory {\bf 1} (2008) 305--356", we improved this result by reducing the requirement on the Novikov-Shubin invariants to one half of the codimension. In this paper, we construct examples which show that this is the best possible result.Comment: Third author added. Some typos corrected and some material added. Appeared in Journal of K Theory, Volume 13, in 2014, pages 305 to 31

2MASS Studies of Differential Reddening Across Three Massive Globular Clusters

J, H, and K_S band data from the Two Micron All-Sky Survey (2MASS) are used to study the effects of differential reddening across the three massive Galactic globular clusters Omega Centauri, NGC 6388, and NGC 6441. Evidence is found that variable extinction may produce false detections of tidal tails around Omega Centauri. We also investigate what appears to be relatively strong differential reddening towards NGC 6388 and NGC 6441, and find that differential extinction may be exaggerating the need for a metallicity spread to explain the width of the red giant branches for these two clusters. Finally, we consider the implications of these results for the connection between unusual, multipopulation globular clusters and the cores of dwarf spheroidal galaxies (dSph).Comment: 40 pages, 14 figures. Accepted for publication in Oct. 2003 A

Giant Molecular clouds: what are they made from, and how do they get there?

We analyse the results of four simulations of isolated galaxies: two with a rigid spiral potential of fixed pattern speed, but with different degrees of star-formation induced feedback, one with an axisymmetric galactic potential and one with a `live' self-gravitating stellar component. Since we use a Lagrangian method we are able to select gas that lies within giant molecular clouds (GMCs) at a particular timeframe, and to then study the properties of this gas at earlier and later times. We find that gas which forms GMCs is not typical of the interstellar medium at least 50 Myr before the clouds form and reaches mean densities within an order of magnitude of mean cloud densities by around 10 Myr before. The gas in GMCs takes at least 50 Myr to return to typical ISM gas after dispersal by stellar feedback, and in some cases the gas is never fully recycled. We also present a study of the two-dimensional, vertically-averaged velocity fields within the ISM. We show that the velocity fields corresponding to the shortest timescales (that is, those timescales closest to the immediate formation and dissipation of the clouds) can be readily understood in terms of the various cloud formation and dissipation mechanisms. Properties of the flow patterns can be used to distinguish the processes which drive converging flows (e.g.\ spiral shocks, supernovae) and thus molecular cloud formation, and we note that such properties may be detectable with future observations of nearby galaxies.Comment: 13 pages, 8 figures, accepted for publication in MNRA

Spitzer and HHT observations of starless cores: masses and environments

We present Spitzer observations of a sample of 12 starless cores selected to have prominent 24 micron shadows. The Spitzer images show 8 and 24 micron shadows and in some cases 70 micron shadows; these spatially resolved absorption features trace the densest regions of the cores. We have carried out a 12CO (2-1) and 13CO (2-1) mapping survey of these cores with the Heinrich Hertz Telescope (HHT). We use the shadow features to derive optical depth maps. We derive molecular masses for the cores and the surrounding environment; we find that the 24 micron shadow masses are always greater than or equal to the molecular masses derived in the same region, a discrepancy likely caused by CO freeze--out onto dust grains. We combine this sample with two additional cores that we studied previously to bring the total sample to 14 cores. Using a simple Jeans mass criterion we find that ~ 2/3 of the cores selected to have prominent 24 micron shadows are collapsing or near collapse, a result that is supported by millimeter line observations. Of this subset at least half have indications of 70 micron shadows. All cores observed to produce absorption features at 70 micron are close to collapse. We conclude that 24 micron shadows, and even more so the 70 micron ones, are useful markers of cloud cores that are approaching collapse.Comment: 41 pages, 28 figures, 5 tables; accepted by Ap

The ISM in spiral galaxies: can cooling in spiral shocks produce molecular clouds?

We investigate the thermodynamics of the ISM and the formation of molecular hydrogen through numerical simulations of spiral galaxies. The model follows the chemical, thermal and dynamical response of the disc to an external spiral potential. Self-gravity and magnetic fields are not included. The calculations demonstrate that gas can cool rapidly when subject to a spiral shock. Molecular clouds in the spiral arms arise through a combination of compression of the ISM by the spiral shock and orbit crowding. These results highlight that local self-gravity is not required to form molecular clouds. Self-shielding provides a sharp transition density, below which gas is essentially atomic, and above which the molecular gas fraction is >0.001. The timescale for gas to move between these regimes is very rapid (<~1 Myr). From this stage, the majority of gas generally takes between 10 to 20 Myr to obtain high H$_{2}$ fractions (>50 %). Although our calculations are unable to resolve turbulent motions on scales smaller than the spiral arm and do not include self-gravity. True cloud formation timescales are therefore expected to be even shorter. The mass budget of the disc is dominated by cold gas residing in the spiral arms. Between 50 and 75 % of this gas is in the atomic phase. When this gas leaves the spiral arm and drops below the self-shielding limit it is heated by the galactic radiation field. Consequently, most of the volume in the interarm regions is filled with warm atomic gas. However, some cold spurs and clumps can survive in interarm regions for periods comparable to the interarm passage timescale. Altogether between 7 and 40% of the gas in our disc is molecular, depending on the surface density of the calculation, with approximately 20% molecular for a surface density comparable to the solar neighbourhood.Comment: 16 pages, 19 figures, accepted for publication in MNRA

Magnetic Fields in Large Diameter HII Regions Revealed by the Faraday Rotation of Compact Extragalactic Radio Sources

We present a study of the line-of-sight magnetic fields in five large-diameter Galactic HII regions. Using the Faraday rotation of background polarized radio sources, as well as dust-corrected H-alpha surface brightness as a probe of electron density, we estimated the strength and orientation of the magnetic field along 93 individual sight-lines through the HII regions. Each of the HII regions displayed a coherent magnetic field. The magnetic field strength (line-of-sight component) in the regions ranges from 2 to 6 microgauss, which is similar to the typical magnetic field strength in the diffuse interstellar medium. We investigated the relationship between magnetic field strength and electron density in the 5 HII regions. The slope of magnetic field vs. density in the low-density regime (0.8 < n_e < 30 per cubic cm) is very slightly above zero. We also calculated the ratio of thermal to magnetic pressure, beta_th, for each data point, which fell in the range 1.01 < beta_th < 25. Finally, we studied the orientation of the magnetic field in the solar neighborhood (d < 1.1 kpc) using our data from 5 HII regions along with existing measurements of the line-of-sight magnetic field strength from polarized pulsars whose distances have been determined from their annual parallax. We identify a net direction for the magnetic field in the solar neighborhood, but find no evidence for a preferred vertical direction of the magnetic field above or below the Galactic plane.Comment: Accepted to the Astrophysical Journal, June 4th 201