Search

We prove that the quantum moduli algebra associated to a possibly punctured compact oriented surface and a complex semisimple Lie algebra $\mathfrak{g}$ is a Noetherian and finitely generated ring; if the surface has punctures, we prove also that it has no non-trivial zero divisors. Moreover, we show that the quantum moduli algebra is isomorphic to the skein algebra of the surface, defined by means of the Reshetikhin-Turaev functor for the quantum group $U_q(\mathfrak{g})$, and which coincides with the Kauffman bracket skein algebra when $\mathfrak{g}=\mathfrak{sl}_2$., Comment: V1: 60 pages, 26 figures; V2: 75 pages, 37 figures, with typos corrected, section 6. 3 rewritten with simpler arguments and a new result (Corollary 6.12), and section 7 added, with results about the quantum reduction

An analytical model of open-cavity second sound resonators is presented and validated against simulations and experiments in superfluid helium using a new resonator design that achieves unprecedented resolution. The model incorporates diffraction, geometrical misalignments, and flow through the cavity, and is validated using cavities with aspect ratios close to unity, operated up to their 20th resonance in superfluid helium.An important finding of this study is that resonators can be optimized to selectively sense either the quantum vortex density carried by the throughflow -- as typically done in the literature -- or the mean velocity of the throughflow. We propose two velocity probing methods: one that takes advantage of geometrical misalignments between the tweezers plates, and another that drives the resonator non-linearly, beyond a threshold that results in the self-sustainment of a vortex tangle within the cavity.A new mathematical treatment of the resonant signal is proposed to adequately filter out parasitic signals, such as temperature and pressure drift, and accurately separate the quantum vorticity signal. This elliptic method consists in a geometrical projection of the resonance in the inverse complex plane. Its effectiveness is demonstrated over a wide range of operating conditions.The resonator model and elliptic method are being utilized to characterize a new design of second-sound resonator with high resolution thanks to miniaturization and design optimization. These second-sound tweezers are capable of providing time-space resolved information similar to classical local probes in turbulence, down to sub-millimeter and sub-millisecond scales. The principle, design, and micro-fabrication of second sound tweezers are being presented in detail, along with their potential for exploring quantum turbulence.

We prove that the quantum graph algebra and the quantum moduli algebra associated to a punctured sphere and complex semisimple Lie algebra $\mathfrak{g}$ are Noetherian rings and finitely generated rings over $\mathbb{C}(q)$. Moreover, we show that these two properties still hold on $\mathbb{C}\big[q,q^{-1}\big]$ for the integral version of the quantum graph algebra. We also study the specializations $\mathcal{L}_{0,n}^\epsilon$ of the quantum graph algebra at a root of unity $\epsilon$ of odd order, and show that $\mathcal{L}_{0,n}^\epsilon$ and its invariant algebra under the quantum group $U_\epsilon(\mathfrak{g})$ have classical fraction algebras which are central simple algebras of PI degrees that we compute.

The density fluctuations of quantum vortex lines are measured in a turbulent flow of superfluid He, at temperatures corresponding to superfluid fraction of 16%, 47% and 81%. The probe is a micro-fabricated second sound resonator that allows for local and small-scale measurements in the core of the flow, at a 10-mesh-size behind a grid. Remarkably, all the vortex power spectra collapse on a single master curve, independently from the superfluid fraction and the mean velocity. By contrast with previous measurements, we report an peculiar shape of the power spectra. The vortex density probability distributions are found to be strongly skewed, similarly to the vorticity distributions observed in classical turbulence. Implications of those results are discussed.

Miniature heaters are immersed in flows of quantum fluid and the efficiency of heat transfer is monitored versus velocity, superfluid fraction and time. The fluid is $^4$He helium with a superfluid fraction varied from 71% down to 0% and an imposed velocity up to 3 m/s, while the characteristic sizes of heaters range from 1.3 $\mu$m up to few hundreds of microns. At low heat fluxes, no velocity dependence is observed. In contrast, some velocity dependence emerges at larger heat flux, as reported previously, and three non-trivial properties of heat transfer are identified. First, at the largest superfluid fraction (71%), a new heat transfer regime appears at non-null velocities and it is typically 10% less conductive than at zero velocity. Second, the velocity dependence of the mean heat transfer is compatible with the square-root dependence observed in classical fluids. Surprisingly, the prefactor to this dependence is maximum for an intermediate superfluid fraction or temperature (around 2 K). Third, the heat transfer time series exhibit highly conductive short-lived events. These \textit{cooling glitches} have a velocity-dependent characteristic time, which manifest itself as a broad and energetic peak in the spectrum of heat transfer time series, in the kHz range. After showing that the velocity dependence can be attributed to the breaking of superfluidity within a thin shell surrounding heaters, an analytical model of forced heat transfer in a quantum flow is developed to account for the properties reported above. We argue that large scale flow patterns must form around the heater, having a size proportional to the heat flux (here two decades larger than the heater diameter) and resulting in a turbulent wake. The observed spectral peaking of heat transfer is quantitatively consistent with the formation of a Von K\'arm\'an vortex street in the wake of a bluff body., Comment: 23 pages, 16 figures

Let $\Sigma$ be a finite type surface, and $G$ a complex algebraic simple Lie group with Lie algebra $\mathfrak{g}$. The quantum moduli algebra of $(\Sigma,G)$ is a quantization of the ring of functions of $X_G(\Sigma)$, the variety of $G$-characters of $\pi_1(\Sigma)$, introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche in the mid '90s. It can be realized as the invariant subalgebra of so-called graph algebras, which are $U_q(\mathfrak{g})$-module-algebras associated to graphs on $\Sigma$, where $U_q(\mathfrak{g})$ is the quantum group corresponding to $G$. We study the structure of the quantum moduli algebra in the case where $\Sigma$ is a sphere with $n+1$ open disks removed, $n\geq 1$, using the graph algebra of the "daisy" graph on $\Sigma$ to make computations easier. We provide new results that hold for arbitrary $G$ and generic $q$, and develop the theory in the case where $q=\epsilon$, a primitive root of unity of odd order, and $G={\rm SL}(2,{\mathbb C})$. In such a situation we introduce a Frobenius morphism that provides a natural identification of the center of the daisy graph algebra with a finite extension of the coordinate ring $\mathcal{O}(G^n)$. We extend the quantum coadjoint action of De-Concini-Kac-Procesi to the daisy graph algebra, and show that the associated Poisson structure on the center corresponds by the Frobenius morphism to the Fock-Rosly Poisson structure on $\mathcal{O}(G^n)$. We show that the set of fixed elements of the center under the quantum coadjoint action is a finite extension of ${\mathbb C}[X_G(\Sigma)]$ endowed with the Atiyah-Bott-Goldman Poisson structure. Finally, by using Wilson loop operators we identify the Kauffman bracket skein algebra $K_{\zeta}(\Sigma)$ at $\zeta:={\rm i}\epsilon^{1/2}$ with this quantum moduli algebra specialized at $q=\epsilon$.

Let $F$ be a local field, let ${\mathcal O}$ be its integer ring and $\varpi$ a uniformizer of its maximal ideal. To an irreducible complex finite dimensional smooth representation $\pi$ of $GL(2,{\mathcal O})$ is associated a pair of positive integers $k, k'$ called the level and the sublevel of $\pi.$ The level is the smallest integer $k$ such that $\pi$ factorizes through the finite group $GL(2,{\mathcal O}/\varpi^k {\mathcal O})$, whereas the sublevel is the smallest integer $k'\leq k$ such that there exists $\chi,$ one dimensional representation of $GL(2,{\mathcal O}),$ such that $\pi\otimes \chi$ factorizes through the finite group $GL(2,{\mathcal O}/\varpi^{k'} {\mathcal O}).$ A representation of $GL(2,{\mathcal O})$ is said strongly primitive if the level and sublevel are equal. The classification of smooth finite dimensional representations of $GL(2,{\mathcal O})$ is equivalent to the classification of strongly primitive irreducible representations of $GL(2,{\mathcal O}).$ In this first article we describe explicitely the even level strongly primitive irreducible finite dimensional complex representations of $GL(2,{\mathcal O})$ along the lines of A.Stasinski and R.Barrington-Leigh, G.Cliff and Q.Wen using Clifford theory. In the case where the characteristic $p$ of the residue field is not equal to $2,$ we give exact formulas for the characters of these representations in most cases by reducing them to the evaluation of Gauss sums, Kloosterman sums and Sali\'e sums for the finite ring ${\mathcal O}/\varpi^k{\mathcal O}.$ It generalizes the work of R.Barrington-Leigh, G.Cliff and Q.Wen which was devoted to $F={\mathbb Q}_p.$ A second article will give the evaluation of characters in the odd level case and the exact expressions for certain generalized Zeta function representations of $PGL(2,{\mathcal O})$., Comment: 38 pages

The intermittency of turbulent superfluid helium is explored systematically in a steady wake flow from 1.28 K up to T>2.18K using a local anemometer. This temperature range spans relative densities of superfluid from 96% down to 0%, allowing to test numerical predictions of enhancement or depletion of intermittency at intermediate superfluid fractions. Using the so-called extended self-similarity method, scaling exponents of structure functions have been calculated. No evidence of temperature dependence is found on these scaling exponents in the upper part of the inertial cascade, where turbulence is well developed and fully resolved by the probe. This result supports the picture of a profound analogy between classical and quantum turbulence in their inertial range, including the violation of self-similarities associated with inertial-range intermittency.

We recall the relation between Zeta function representation of groups and two-dimensional topological Yang-Mills theory through Mednikh formula. We prove various generalisations of Mednikh formulas and define generalization of Zeta functions representations of groups. We compute some of these functions in the case of the finite group $GL(2, {\mathbb F}_q)$ and $PGL(2,{\mathbb F}_q).$ We recall the table characters of these groups for any $q$, compute the Frobenius-Schur indicator of their irreducible representations and give the explicit structure of their fusion rings, Comment: 27 pages

In this paper, we report the first evidence of an enhancement of the heat transfer from a heated wire by an external turbulent flow of superfluid helium. We used a standard Pt-Rh hot-wire anemometer and overheat it up to 21 K in a pressurized liquid helium turbulent round jet at temperatures between 1.9 K and 2.12 K. The null-velocity response of the sensor can be satisfactorily modeled by the counter flow mechanism while the extra cooling produced by the forced convection is found to scale similarly as the corresponding extra cooling in classical fluids. We propose a preliminary analysis of the response of the sensor and show that -contrary to a common assumption- such sensor can be used to probe local velocity in turbulent superfluid helium., Comment: 9 pages, 4 figures

Superfluid Turbulence is unusual and presents a challenge to fluid dynamicists because it consists of two coupled, inter penetrating turbulent fluids: the first is inviscid with quantised vorticity, the second is viscous with continuous vorticity. Despite this double nature, the observed spectra of the superfluid turbulent velocity at sufficiently large length scales are similar to those o ordinary turbulence. We present experimental, numerical and theoretical results which explain these similarities, and illustrate the limits of our present understanding of superfluid turbulence at smaller scales., Comment: 15 pages. 4 figs

The 4/5-law of turbulence, which characterizes the energy cascade from large to small-sized eddies at high Reynolds numbers in classical fluids, is verified experimentally in a superfluid 4He wind tunnel, operated down to 1.56 K and up to R_lambda ~ 1640. The result is corroborated by high-resolution simulations of Landau-Tisza's two-fluid model down to 1.15 K, corresponding to a residual normal fluid concentration below 3 % but with a lower Reynolds number of order R_lambda ~ 100. Although the K\'arm\'an-Howarth equation (including a viscous term) is not valid \emph{a priori} in a superfluid, it is found that it provides an empirical description of the deviation from the ideal 4/5-law at small scales and allows us to identify an effective viscosity for the superfluid, whose value matches the kinematic viscosity of the normal fluid regardless of its concentration., Comment: 6 pages, 7 figures

Rayleigh-B\'enard cells are one of the simplest systems to explore the laws of natural convection in the highly turbulent limit. However, at very high Rayleigh numbers (Ra > 1E12) and for Prandtl numbers of order one, experiments fall into two categories: some evidence a steep enhancement of the heat transfer while others do not. The origin of this apparent disagreement is presently unexplained. This puzzling situation motivated a systematic study of the triggering of the regime with an enhanced heat transfer, originally named the "Ultimate Regime" of convection. High accuracy heat transfer measurements have been conducted in convection cells with various aspect ratios and different specificities, such as altered boundary conditions or obstacles inserted in the flow. The two control parameters, the Rayleigh and Prandtl numbers have been varied independently to disentangle their relative influence. Among other results, it is found that i) most experiments reaching very high $Ra$ are not in disagreement if small differences in Prandtl numbers are taken into account, ii) the transition is not directly triggered by the large scale circulation present in the cell, iii) the sidewall of the cell have a significant influence on the transition. The characteristics of this Ultimate regime are summarized and compared with R. Kraichnan prediction for the asymptotic regime of convection., Comment: 25 pages, 14 figures

The turbulence of superfluid helium is investigated numerically at finite temperature. Direct numerical simulations are performed with a "truncated HVBK" model, which combines the continuous description of the Hall-Vinen-Bekeravich-Khalatnikov equations with the additional constraint that this continuous description cannot extend beyond a quantum length scale associated with the mean spacing between individual superfluid vortices. A good agreement is found with experimental measurements of the vortex density. Besides, by varying the turbulence intensity only, it is observed that the inter-vortex spacing varies with the Reynolds number as $Re^{-3/4}$, like the viscous length scale in classical turbulence. In the high temperature limit, Kolmogorov's inertial cascade is recovered, as expected from previous numerical and experimental studies. As the temperature decreases, the inertial cascade remains present at large scales while, at small scales, the system evolves towards a statistical equipartition of kinetic energy among spectral modes, with a characteristic $k^2$ velocity spectrum. The accumulation of superfluid excitations on a range of mesoscales enables the superfluid to keep dissipating kinetic energy through mutual friction with the residual normal fluid, although the later becomes rare at low temperature. It is found that most of the superfluid vorticity can concentrate on these mesoscales at low temperature, while it is concentrated in the inertial range at higher temperature. This observation should have consequences on the interpretation of decaying turbulence experiments, which are often based on vortex line density measurements., Comment: 6 pages, 5 figures

This is a review of the present status of loop and spin foam approaches to quantization of four-dimensional general relativity. It aims at raising various issues which seem to challenge some of the methods and the results often taken as granted in these domains. A particular emphasis is given to the issue of diffeomorphism and local Lorentz symmetries at the quantum level and to the discussion of new spin foam models. We also describe modifications of these two approaches which may overcome their problems and speculate on other promising research directions., Comment: 75 pages

Recently, an exact description of instanton corrections to the moduli spaces of 4d N=2 supersymmetric gauge theories compactified on a circle and Calabi-Yau compactifications of Type II superstring theories was found. The equations determining the instanton contributions turn out to have the form of Thermodynamic Bethe Ansatz. We explore further this relation and, in particular, we identify the contact potential of quaternionic string moduli space with the free energy of the integrable system and the Kahler potential of the gauge theory moduli space with the Yang-Yang functional. We also show that the corresponding S-matrix satisfies all usual constraints of 2d integrable models, including crossing and bootstrap, and derive the associated Y-system. Surprisingly, in the simplest case the Y-system is described by the MacMahon function relevant for crystal melting and topological strings., Comment: 25 pages, 1 figure

A new regime of turbulent convection has been reported nearly one decade ago, based on global heat transfer measurements at very high Rayleigh numbers. We examine the signature of this "Ultimate Regime" from within the flow itself. A systematic study of probe-size corrections shows that the earlier temperature measurements within the flow were altered by an excessive size of thermometer, but not according to a theoretical model proposed in the literature. Using a probe one order of magnitude smaller than the one used previously, we find evidence that the transition to the Ultimate Regime is indeed accompanied with a clear change in the statistics of temperature fluctuations in the flow., Comment: Submitted for publication

To model isotropic homogeneous quantum turbulence in superfluid helium, we have performed Direct Numerical Simulations (DNS) of two fluids (the normal fluid and the superfluid) coupled by mutual friction. We have found evidence of strong locking of superfluid and normal fluid along the turbulent cascade, from the large scale structures where only one fluid is forced down to the vorticity structures at small scales. We have determined the residual slip velocity between the two fluids, and, for each fluid, the relative balance of inertial, viscous and friction forces along the scales. Our calculations show that the classical relation between energy injection and dissipation scale is not valid in quantum turbulence, but we have been able to derive a temperature--dependent superfluid analogous relation. Finally, we discuss our DNS results in terms of the current understanding of quantum turbulence, including the value of the effective kinematic viscosity.

In 1997, a Rayleigh-B\'enard experiment evidenced a significant increase of the heat transport efficiency for Rayleigh numbers larger than $Ra \sim 10^{12}$ and interpreted this observation as the signature of the Kraichnan's ``Ultime Regime'' of convection. According to Kraichnan's 1962 prediction, the flow boundary layers above the cold and hot plates -in which most of the fluid temperature drop is localized- become unstable for large enough $Ra$ and this instability boosts the heat transport compared to the other turbulent regimes. Using the same convection cell as in the 1997 experiment, we show that the reported heat transport increase is accompanied with enhanced temperature fluctuations of the bottom plate, which was heated at constant power levels. Indeed, for $Ra < 10^{12}$, the bottom plate fluctuations can simply be accounted from those in the bulk of the flow. In particular, they share the same spectral density at low frequencies, as if the bottom plate was following the slow temperature fluctuations of the bulk, modulo a constant temperature drop across the bottom boundary layer. Conversely, to account for the plate's temperature fluctuations at higher $Ra$, we no-longuer can ignore the fluctuations of the temperature drop across the boundary layer. The negative skewness of fluctuations at high $Ra$ supports the picture of a boundary layer instability. These observations provide new evidence that the transition reported in 1997 corresponds to the triggering of the Ultimate Regime of convection., Comment: Submitted for publication

The equations of motion describing buoyant fluids are often simplified using a set of approximations proposed by J. Boussinesq one century ago. To resume, they consist in assuming constant fluid properties, incompressibility and conservation of calories during heat transport. Assuming fulfilment of the first requirement (constant fluid properties), we derive a set of 4 criteria for assessing the validity of the two other requirements in turbulent Rayleigh-B\'enard convection. The first criterion $\alpha \Delta \ll 1 $ simply results from the incompressibility condition in the thermal boundary layer ($\alpha$ and $\Delta$ are the thermal expansion coefficient and the temperature difference driving the flow). The 3 other criteria are proportional or quadratic with the density stratification or, equivalently with the temperature difference resulting from the adiabatic gradient across the cell $\Delta_{h}$. Numerical evaluations with air, water and cryogenic helium show that most laboratory experiments are free from such Boussinesq violation as long as the first criterion is fulfilled. In ultra high Rayleigh numbers ($Ra>10^{16}$) experiments in He, one of the stratification criteria, scaling with $\alpha \Delta_{h}$, could be violated. This criterion garanties that pressure fluctuations have a negligible influence both on the density variation and on the heat transfer equation through compression/expansion cycles. Extrapolation to higher $Ra$ suggests that strong violation of Boussinesq approximation could occur in atmospheric convection., Comment: Submitted to Phys.Fluids (oct 2007)

We discuss a recent experiment in which the spectrum of the vortex line density fluctuations has been measured in superfluid turbulence. The observed frequency dependence of the spectrum, $f^{-5/3}$, disagrees with classical vorticity spectra if, following the literature, the vortex line density is interpreted as a measure of the vorticity or enstrophy. We argue that the disagrement is solved if the vortex line density field is decomposed into a polarised field (which carries most of the energy) and an isotropic field (which is responsible for the spectrum)., Comment: Submitted for publication http://crtbt.grenoble.cnrs.fr/helio/GROUP/infa.html http://www.mas.ncl.ac.uk/~ncfb/

We construct a universal Vertex-IRF transformation between Vertex type universal solution and Face type universal solution of the quantum dynamical Yang-Baxter equation. This universal Vertex-IRF transformation satisfies the generalized coBoundary equation and is an extension of our previous work to the quantum affine $U_q(A^{(1)}_r)$ case. This solution has a simple Gauss decomposition which is constructed using Sevostyanov's characters of twisted quantum Borel algebras. We show that the evaluation of this universal solution in the evaluation representation of $U_q(A_1^{(1)})$ gives the standard Baxter's transformation between the 8-Vertex model and the IRF height model., Comment: 58 pages

The fluctuations of the vortex density in a turbulent quantum fluid are deduced from local second-sound attenuation measurements. These measurements are performed with a micromachined open-cavity resonator inserted across a flow of turbulent He-II near 1.6 K. The power spectrum of the measured vortex line density is compatible with a (-5/3) power law. The physical interpretation, still open, is discussed., Comment: Submitted to Europhys. Lett

We establish an equivalence between the Hamiltonian formulation of the Plebanski action for general relativity and the covariant canonical formulation of the Hilbert-Palatini action. This is done by comparing the symplectic structures of the two theories through the computation of Dirac brackets. We also construct a shifted connection with simplified Dirac brackets, playing an important role in the covariant loop quantization program, in the Plebanski framework. Implications for spin foam models are also discussed., Comment: 18 pages

In a recent preprint (cond-mat/0601398), D. Funfschilling and G. Ahlers describe a new effect, that they interpret as non-Boussinesq, in a convection cell working with ethane, near its critical point. They argue that such an effect could have spoiled the Chavanne {\it et al.} (Phys. Rev. Lett. {\bf 79} 3648, 1997) results, and not the Niemela {\it et al.} (Nature, {\bf 404}, 837, 2000) ones, which would explain the differences between these two experiments. We show that:-i)Restricting the Chavanne's data to situations as far from the critical point than the Niemela's one, the same discrepancy remains.-ii)The helium data of Chavanne show no indication of the effect observed by D. Funfschilling and G. Ahlers., Comment: comment on cond-mat/0601398

A transition to Kraichnan ultimate regime of convection has been reported in very high Rayleigh numbers experiments, but not in all of them. These apparently contradictory results can be explained by a recent phenomenological model which accounts for the non-ideality of the plate thermal properties [Chill\`{a} et al, Physics of Fluids 16:2452 (2004)]. In this paper, we present a direct test of this model, using a low conductivity plate. We found an unaltered transition, not compatible with the model's predictions., Comment: submitted to Phys.Fluids

Finding signatures of quantum gravity in cosmological observations is now actively pursued both from the theoretical and the experimental side. Recent work has concentrated on finding signatures of light-cone fluctuations in the CMB. Because in inflationary scenarios a Gravitational Wave Background (GWB) is always emitted much before the CMB, we can ask, in the hypothesis where this GWB could be observed, what is the imprint of light cone fluctuations on this GWB. We show that due to the flat nature of the GWB spectrum, the effect of lightcone fluctuations are negligible., Comment: 10 pages, references added

We calculate the distribution of current fluctuations in two simple exclusion models. Although these models are classical, we recover even for small systems such as a simple or a double barrier, the same distibution of current as given by traditionnal formalisms for quantum mesoscopic conductors. Due to their simplicity, the full counting statistics in exclusion models can be reduced to the calculation of the largest eigenvalue of a matrix, the size of which is the number of internal configurations of the system. As examples, we derive the shot noise power and higher order statistics of current fluctuations (skewness, full counting statistics, ....) of various conductors, including multiple barriers, diffusive islands between tunnel barriers and diffusive media. A special attention is dedicated to the third cumulant, which experimental measurability has been demonstrated lately., Comment: Submitted to Eur. Phys. J. B

We study the quantum deformation of the Barrett-Crane Lorentzian spin foam model which is conjectured to be the discretization of Lorentzian Plebanski model with positive cosmological constant and includes therefore as a particular sector quantum gravity in de-Sitter space. This spin foam model is constructed using harmonic analysis on the quantum Lorentz group. The evaluation of simple spin networks are shown to be non commutative integrals over the quantum hyperboloid defined as a pile of fuzzy spheres. We show that the introduction of the cosmological constant removes all the infrared divergences: for any fixed triangulation, the integration over the area variables is finite for a large class of normalization of the amplitude of the edges and of the faces., Comment: 37 pages, 7 figures included

We describe an efficient construction of a canonical non-commutative deformation of the algebraic functions on the moduli spaces of flat connections on a Riemann surface. We show that this algebra, which is a variant of the quantum moduli algebra introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche, has a trace functional which is related to the canonical trace in the formal index theory of Fedosov and Nest-Tsygan via the Verlinde formula.

We give an integrable extension of the lattice models recently considered by I.Kostov in his study of strings in discrete space. These models are IRF models with spin variables living in any connected graph, the vertex model underlying these models is the Izergin-Korepin model. When the graph is taken to be a simply laced Dynkin diagram, it is conjectured that these models possess critical regimes which are the dilute phase of SOS models of ADE type., Comment: #8