Modular Classes of Lie Groupoid Representations up to Homotopy

We describe a construction of the modular class associated to a representation up to homotopy of a Lie groupoid. In the case of the adjoint representation up to homotopy, this class is the obstruction to the existence of a volume form, in the sense of Weinstein's ''The volume of a differentiable stack''

Differential graded contact geometry and Jacobi structures

We study contact structures on nonnegatively-graded manifolds equipped with homological contact vector fields. In the degree 1 case, we show that there is a one-to-one correspondence between such structures (with fixed contact form) and Jacobi manifolds. This correspondence allows us to reinterpret the Poissonization procedure, taking Jacobi manifolds to Poisson manifolds, as a supergeometric version of symplectization.Comment: 9 pages. v2: Added references, improved proof of Proposition 3.3. v3: Expanded introduction, clarifying remarks, some changes of sign conventions. Main results are unchanged. v4: Final version, implementing changes suggested by referee

Eigenvalue density of Wilson loops in 2D SU(N) YM

In 1981 Durhuus and Olesen (DO) showed that at infinite N the eigenvalue density of a Wilson loop matrix W associated with a simple loop in two-dimensional Euclidean SU(N) Yang-Mills theory undergoes a phase transition at a critical size. The averages of det(z-W), 1/det(z-W), and det(1+uW)/(1-vW) at finite N lead to three different smoothed out expressions, all tending to the DO singular result at infinite N. These smooth extensions are obtained and compared to each other.Comment: 35 pages, 8 figure

The embedding method beyond the single-channel case: Two-mode and Hubbard chains

We investigate the relationship between persistent currents in multi-channel rings containing an embedded scatterer and the conductance through the same scatterer attached to leads. The case of two uncoupled channels corresponds to a Hubbard chain, for which the one-dimensional embedding method is readily generalized. Various tests are carried out to validate this new procedure, and the conductance of short one-dimensional Hubbard chains attached to perfect leads is computed for different system sizes and interaction strengths. In the case of two coupled channels the conductance can be obtained from a statistical analysis of the persistent current or by reducing the multi-channel scattering problem to several single-channel setups.Comment: 14 pages, 13 figures, submitted for publicatio

Lectures on Chiral Disorder in QCD

I explain the concept that light quarks diffuse in the QCD vacuum following the spontaneous breakdown of chiral symmetry. I exploit the striking analogy to disordered electrons in metals, identifying, among others, the universal regime described by random matrix theory, diffusive regime described by chiral perturbation theory and the crossover between these two domains.Comment: Lectures given at the Cargese Summer School, August 6-18, 200

The interaction of a gap with a free boundary in a two dimensional dimer system

Let $\ell$ be a fixed vertical lattice line of the unit triangular lattice in the plane, and let \Cal H be the half plane to the left of $\ell$. We consider lozenge tilings of \Cal H that have a triangular gap of side-length two and in which $\ell$ is a free boundary - i.e., tiles are allowed to protrude out half-way across $\ell$. We prove that the correlation function of this gap near the free boundary has asymptotics $\frac{1}{4\pi r}$, $r\to\infty$, where $r$ is the distance from the gap to the free boundary. This parallels the electrostatic phenomenon by which the field of an electric charge near a conductor can be obtained by the method of images.Comment: 34 pages, AmS-Te

Statistics of Coulomb blockade peak spacings for a partially open dot

We show that randomness of the electron wave functions in a quantum dot contributes to the fluctuations of the positions of the conductance peaks. This contribution grows with the conductance of the junctions connecting the dot to the leads. It becomes comparable with the fluctuations coming from the randomness of the single particle spectrum in the dot while the Coulomb blockade peaks are still well-defined. In addition, the fluctuations of the peak spacings are correlated with the fluctuations of the conductance peak heights.Comment: 13 pages, 1 figur

The random phase property and the Lyapunov Spectrum for disordered multi-channel systems

A random phase property establishing in the weak coupling limit a link between quasi-one-dimensional random Schrödinger operators and full random matrix theory is advocated. Briefly summarized it states that the random transfer matrices placed into a normal system of coordinates act on the isotropic frames and lead to a Markov process with a unique invariant measure which is of geometric nature. On the elliptic part of the transfer matrices, this measure is invariant under the unitaries in the hermitian symplectic group of the universality class under study. While the random phase property can up to now only be proved in special models or in a restricted sense, we provide strong numerical evidence that it holds in the Anderson model of localization. A main outcome of the random phase property is a perturbative calculation of the Lyapunov exponents which shows that the Lyapunov spectrum is equidistant and that the localization lengths for large systems in the unitary, orthogonal and symplectic ensemble differ by a factor 2 each. In an Anderson-Ando model on a tubular geometry with magnetic field and spin-orbit coupling, the normal system of coordinates is calculated and this is used to derive explicit energy dependent formulas for the Lyapunov spectrum

Spectral Correlations from the Metal to the Mobility Edge

We have studied numerically the spectral correlations in a metallic phase and at the metal-insulator transition. We have calculated directly the two-point correlation function of the density of states $R(s,s')$. In the metallic phase, it is well described by the Random Matrix Theory (RMT). For the first time, we also find numerically the diffusive corrections for the number variance $$ predicted by Al'tshuler and Shklovski\u{\i}. At the transition, at small energy scales, $R(s-s')$ starts linearly, with a slope larger than in a metal. At large separations $|s - s'| \gg 1$, it is found to decrease as a power law $R(s,s') \sim - c / |s -s'|^{2-\gamma}$ with $c \sim 0.041$ and $\gamma \sim 0.83$, in good agreement with recent microscopic predictions. At the transition, we have also calculated the form factor $\tilde K(t)$, Fourier transform of $R(s-s')$. At large $s$, the number variance contains two terms $= B ^\gamma + 2 \pi \tilde K(0) where$\tilde{K}(0)$is the limit of the form factor for$t \to 0$.Comment: 7 RevTex-pages, 10 figures. Submitted to PR