Spin Foams and Noncommutative Geometry

We extend the formalism of embedded spin networks and spin foams to include topological data that encode the underlying three-manifold or four-manifold as a branched cover. These data are expressed as monodromies, in a way similar to the encoding of the gravitational field via holonomies. We then describe convolution algebras of spin networks and spin foams, based on the different ways in which the same topology can be realized as a branched covering via covering moves, and on possible composition operations on spin foams. We illustrate the case of the groupoid algebra of the equivalence relation determined by covering moves and a 2-semigroupoid algebra arising from a 2-category of spin foams with composition operations corresponding to a fibered product of the branched coverings and the gluing of cobordisms. The spin foam amplitudes then give rise to dynamical flows on these algebras, and the existence of low temperature equilibrium states of Gibbs form is related to questions on the existence of topological invariants of embedded graphs and embedded two-complexes with given properties. We end by sketching a possible approach to combining the spin network and spin foam formalism with matter within the framework of spectral triples in noncommutative geometry.Comment: 48 pages LaTeX, 30 PDF figure

Lignin biomarkers as tracers of mercury sources in lakes water column

This study presents the role of specific terrigenous organic compounds as important vectors of mercury (Hg) transported from watersheds to lakes of the Canadian boreal forest. In order to differentiate the autochthonous from the allochthonous organic matter (OM), lignin derived biomarker signatures [Lambda, S/V, C/V, P/(V ? S), 3,5-Bd/V and (Ad/Al)v] were used. Since lignin is exclusively produced by terrigenous plants, this approach can give a non equivocal picture of the watershed inputs to the lakes. Moreover, it allows a characterization of the source of OM and its state of degradation. The water column of six lakes from the Canadian Shield was sampled monthly between June and September 2005. Lake total dissolved Hg concentrations and Lambda were positively correlated, meaning that Hg and ligneous inputs are linked (dissolved OM r2 = 0.62, p\0.0001; particulate OM r2 = 0.76, p\0.0001). Ratios of P/(V ? S) and 3,5-Bd/V from both dissolved OM and particulate OM of the water column suggest an inverse relationship between the progressive state of pedogenesis and maturation of the OM in soil before entering the lake, and the Hg concentrations in the water column. No relation was found between Hg levels in the lakes and the watershed flora composition—angiosperm versus gymnosperm or woody versus non-woody compounds. This study has significant implications for watershed management of ecosystems since limiting fresh terrestrial OM inputs should reduce Hg inputs to the aquatic systems. This is particularly the case for largescale land-use impacts, such as deforestation, agriculture and urbanization, associated to large quantities of soil OM being transferred to aquatic systems

Trace metal budgets for forested catchments in Europe – Pb, Cd, Hg, Cu and Zn

Input/output budgets for cadmium (Cd), lead (Pb) and mercury (Hg) in the years 1997 monitored and determined for 14 small forest-covered catchments across Europe as part of the Integrated Monitoring program on the effects of long-range pollutants on ecosystems. Metal inputs were considered to derive from bulk deposition, throughfall and litterfall. Outputs were estimated from run-off values. Litterfall plus throughfall was taken as a measure of the total deposition of Pb and Hg (wet+dry) on the basis of evidence suggesting that, for these metals, internal circulation is negligible. The same is not true for Cd. Excluding a few sites with high discharge, between 74 and 94 % of the input Pb was retained within the catchments; significant Cd retention was also observed. High losses of Pb (>1.4 mgm (>0.15 mgm Central European sites with high water discharge. All other sites had outputs below or equal to 0.36 and 0.06 mgm Almost complete retention of Hg, 86 was reported in the Swedish sites. These high levels of metal retention were maintained even in the face of recent dramatic reductions in pollutant loads.–2011 were−2year−1) and Cd−2year−1) were observed in two mountainous−2year−1, respectively, for the two metals.–99 % of input