Deligne-Beilinson cohomology and abelian link invariants: torsion case

For the abelian Chern-Simons field theory, we consider the quantum functional integration over the Deligne-Beilinson cohomology classes and present an explicit path-integral non-perturbative computation of the Chern-Simons link invariants in $SO(3)\simeq\mathbb{R}P^3$, a toy example of 3-manifold with torsion

Influence of pumping operational schedule on solute concentrations at a well in randomly heterogeneous aquifers

We investigate the way diverse groundwater extraction strategies affect the history of solute concentration recovered at a pumping well while taking into account random spatial variability of the system hydraulic conductivity. Considering the joint effects of spatially heterogeneous hydraulic conductivity and temporally varying well pumping rates leads to a realistic evaluation of groundwater contamination risk at the pumping well location. We juxtapose the results obtained when the pumping well extracts a given amount of water operating (a) at a uniform pumping rate and (b) under a transient regime. The analysis is performed within a numerical Monte Carlo framework. Our results show that contaminant concentration breakthrough curves (BTCs) at the well are markedly affected by the transient pumping strategy according to which the well is operated. Our results document the occurrence in time of multiple peaks in the mean and variance of flux-averaged concentrations at the extraction well operating at a transient rate. Our findings suggest that lowest and largest values of mean and variance of flux-averaged concentration at the well tend to occur at the same time. We show that uncertainty associated with detected BTCs at the well increases for pumping regimes displaying a high degree of temporal variability. As such, the choice of the type of engineering control to the temporal sequence of pumping rates could represent a key factor to drive quantification of uncertainty of the contaminant concentration detected at the well. It is documented that pumping rate fluctuations induce a temporally oscillating risk pattern at the well, thus suggesting that the selection of a dynamic pumping regime has a clear influence on the temporal evolution of risk at the well

Three-manifold invariant from functional integration

We give a precise definition and produce a path-integral computation of the normalized partition function of the abelian U(1) Chern-Simons field theory defined in a general closed oriented 3-manifold. We use the Deligne-Beilinson formalism, we sum over the inequivalent U(1) principal bundles over the manifold and, for each bundle, we integrate over the gauge orbits of the associated connection 1- forms. The result of the functional integration is compared with the abelian U(1) Reshetikhin-Turaev surgery invariant

Solute concentration at a well in non-Gaussian aquifers under constant and time-varying pumping schedule

Our study is keyed to the analysis of the interplay between engineering factors (i.e., transient pumping rates versus less realistic but commonly analyzed uniform extraction rates) and the heterogeneous structure of the aquifer (as expressed by the probability distribution characterizing transmissivity) on contaminant transport. We explore the joint influence of diverse (a) groundwater pumping schedules (constant and variable in time) and (b) representations of the stochastic heterogeneous transmissivity (T) field on temporal histories of solute concentrations observed at an extraction well. The stochastic nature of T is rendered by modeling its natural logarithm, Y = ln T, through a typical Gaussian representation and the recently introduced Generalized sub-Gaussian (GSG) model. The latter has the unique property to embed scale-dependent non-Gaussian features of the main statistics of Y and its (spatial) increments, which have been documented in a variety of studies. We rely on numerical Monte Carlo simulations and compute the temporal evolution at the well of low order moments of the solute concentration (C), as well as statistics of the peak concentration (Cp), identified as the environmental performance metric of interest in this study. We show that the pumping schedule strongly affects the pattern of the temporal evolution of the first two statistical moments of C, regardless the nature (Gaussian or non-Gaussian) of the underlying Y field, whereas the latter quantitatively influences their magnitude. Our results show that uncertainty associated with C and Cpestimates is larger when operating under a transient extraction scheme than under the action of a uniform withdrawal schedule. The probability density function (PDF) of Cpdisplays a long positive tail in the presence of time-varying pumping schedule. All these aspects are magnified in the presence of non-Gaussian Y fields. Additionally, the PDF of Cpdisplays a bimodal shape for all types of pumping schemes analyzed, independent of the type of heterogeneity considered

Continuum-scale characterization of solute transport based on pore-scale velocity distributions

We present a methodology to characterize a continuum-scale model of transport in porous media on the basis of pore-scale distributions of velocities computed in three-dimensional pore-space images. The methodology is tested against pore-scale simulations of flow and transport for a bead pack and a sandstone sample. We employ a double continuum approach to describe transport in mobile and immobile regions. Model parameters are characterized through inputs resulting from the micron-scale reconstruction of the pore space geometry and the related velocity field. We employ the outputs of pore-scale analysis to (i) quantify the proportion of mobile and immobile fluid regions, and (ii) assign the velocity distribution in an effective representation of the medium internal structure. Our results (1) show that this simple conceptual model reproduces the spatial profiles of solute concentration rendered by pore-scale simulation without resorting to model calibration, and (2) highlight the critical role of pore-scale velocities in the characterization of the model parameters

Knots in SU(M∣N)SU\left(M|N\right) Chern-Simons Field Theory

Knots in the Chern-Simons field theory with Lie super gauge group $SU\left(M|N\right)$ are studied, and the $$ polynomial invariant with skein relations are obtained under the fundamental representation of $\mathfrak{su}\left(M|N\right)$.Comment: 15 pages, 5 figure

Higher dimensional abelian Chern-Simons theories and their link invariants

The role played by Deligne-Beilinson cohomology in establishing the relation between Chern-Simons theory and link invariants in dimensions higher than three is investigated. Deligne-Beilinson cohomology classes provide a natural abelian Chern-Simons action, non trivial only in dimensions $4l+3$, whose parameter $k$ is quantized. The generalized Wilson $(2l+1)$-loops are observables of the theory and their charges are quantized. The Chern-Simons action is then used to compute invariants for links of $(2l+1)$-loops, first on closed $(4l+3)$-manifolds through a novel geometric computation, then on $\mathbb{R}^{4l+3}$ through an unconventional field theoretic computation.Comment: 40 page

The Nucleon ``Tensor Charges'' and the Skyrme Model

The lowest moment of the twist-two, chiral-odd parton distribution $h_1(x)$ of the nucleon can be related to the so-called ``tensor charges'' of the nucleon. We consider the tensor charges in the Skyrme model, and find that in the large-$N_c$, SU(3)-symmetric limit, the model predicts that the octet isosinglet tensor charge, $g^8_T$, is of order $1/N_c$ with respect to the octet isovector tensor charge, $g^3_T$. The predicted $F/D$ ratio is then 1/3, in the large-$N_c$ limit. These predictions coincide with the Skyrme model predictions for the octet ${\it axial}$ charges, $g^8_A$ and $g^3_A$. (The prediction $F/D=1/3$ for the axial charges differs from the commonly quoted prediction of 5/9, which is based on an inconsistent treatment of the large-$N_c$ limit.) The model also predicts that the singlet tensor charge, $g^0_T$, is of order $1/N_c$ with respect to $g^3_T$.Comment: 9 single-spaced pages, no figures, MIT-CTP-212

On the Quantization of the Chern-Simons Fields Theory on Curved Space-Times: the Coulomb Gauge Approach

We consider here the Chern-Simons field theory with gauge group SU(N) in the presence of a gravitational background that describes a two-dimensional expanding ``universe". Two special cases are treated here in detail: the spatially flat {\it Robertson-Walker} space-time and the conformally static space-times having a general closed and orientable Riemann surface as spatial section. The propagator and the vertices are explicitely computed at the lowest order in perturbation theory imposing the Coulomb gauge fixing.Comment: 15 pp., Preprint LMU-TPW 93-5, (Plain TeX + Harvmac