The Equivariant Cohomology Theory of Twisted Generalized Complex Manifolds

It has been shown recently by Kapustin and Tomasiello that the mathematical notion of Hamiltonian actions on twisted generalized Kähler manifolds is in perfect agreement with the physical notion of general (2, 2) gauged sigma models with three-form fluxes. In this article, we study the twisted equivariant cohomology theory of Hamiltonian actions on H-twisted generalized complex manifolds. If the manifold satisfies the ${\overline{\partial} \partial}It has been shown recently by Kapustin and Tomasiello that the mathematical notion of Hamiltonian actions on twisted generalized K?hler manifolds is in perfect agreement with the physical notion of general (2, 2) gauged sigma models with three-form fluxes. In this article, we study the twisted equivariant cohomology theory of Hamiltonian actions on H-twisted generalized complex manifolds. If the manifold satisfies the -lemma, we establish the equivariant formality theorem. If in addition, the manifold satisfies the generalized K?hler condition, we prove the Kirwan injectivity in this setting. We then consider the Hamiltonian action of a torus on an H-twisted generalized Calabi-Yau manifold and extend to this case the Duistermaat-Heckman theorem for the push-forward measure. As a side result, we show in this paper that the generalized K?hler quotient of a generalized K?hler vector space can never have a (cohomologically) non-trivial twisting. This gives a negative answer to a question asked by physicists whether one can construct (2, 2) gauged linear sigma models with non-trivial fluxes.     

Understanding anomalous delays in a model of intracellular calcium dynamics

In many cell types, oscillations in the concentration of free intracellular calcium ions are used to control a variety of cellular functions. It has been suggested [J. Sneyd et al., "A method for determining the dependence of calcium oscillations on inositol trisphosphate oscillations," Proc. Natl. Acad. Sci. U.S.A. 103, 1675-1680 (2006)] that the mechanisms underlying the generation and control of such oscillations can be determined by means of a simple experiment, whereby a single exogenous pulse of inositol trisphosphate (IP(3)) is applied to the cell. However, more detailed mathematical investigations [M. Domijan et al., "Dynamical probing of the mechanisms underlying calcium oscillations," J. Nonlinear Sci. 16, 483-506 (2006)] have shown that this is not necessarily always true, and that the experimental data are more difficult to interpret than first thought. Here, we use geometric singular perturbation techniques to study the dynamics of models that make different assumptions about the mechanisms underlying the calcium oscillations. In particular, we show how recently developed canard theory for singularly perturbed systems with three or more slow variables [M. Wechselberger, "A propos de canards (Apropos canards)," Preprint, 2010] applies to these calcium models and how the presence of a curve of folded singularities and corresponding canards can result in anomalous delays in the response of these models to a pulse of IP(3).     

The Optimal Sink and the Best Source in a Markov Chain

It is well known that the distributions of hitting times in Markov chains are quite irregular, unless the limit as time tends to infinity is considered. We show that nevertheless for a typical finite irreducible Markov chain and for nondegenerate initial distributions the tails of the distributions of the hitting times for the states of a Markov chain can be ordered, i.e., they do not overlap after a certain finite moment of time. If one considers instead each state of a Markov chain as a source rather than a sink then again the states can generically be ordered according to their efficiency. The mechanisms underlying these two orderings are essentially different though. Our results can be used, e.g., for a choice of the initial distribution in numerical experiments with the fastest convergence to equilibrium/stationary distribution, for characterization of the elements of a dynamical network according to their ability to absorb and transmit the substance (“information”) that is circulated over the network, for determining optimal stopping moments (stopping signals/words) when dealing with sequences of symbols, etc.     

Newtonian Cosmology in Lagrangian Formulation: Foundations and Perturbation Theory

The Newtonian theory of spatially unbounded, self-gravitating, pressureless continua in Lagrangian form is reconsidered. Following a review of the pertinent kinematics, we present alternative formulations of the Lagrangian evolution equations and establish conditions for the equivalence of the Lagrangian and Eulerian representations. We then distinguish open models based on Euclidean space R³ from closed models based (without loss of generality) on a flat torus T³. Using a simple averaging method we show that the spatially averaged variables of an inhomogeneous toroidal model form a spatially homogeneous background model and that the averages of open models, if they exist at all, in general do not obey the dynamical laws of homogeneous models. We then specialize to those inhomogeneous toroidal models whose (unique) backgrounds have a Hubble flow, and derive Lagrangian evolution equations which govern the (conformally rescaled) displacement of the inhomogeneous flow with respect to its homogeneous background. Finally, we set up an iteration scheme and prove that the resulting equations have unique solutions at any order for given initial data, while for open models there exist infinitely many different solutions for given data.     

Potentials with Convergent Schwinger–DeWitt Expansion

Convergence of the Schwinger–DeWittexpansion for the evolution operator kernel for specialclass of potentials is studied. It is shown that thisexpansion, which is in the general case asymptotic,converges for the potentials considered (widely used, inparticular, in one-dimensional many-body problems), andthat convergence takes place only for definite discretevalues of the coupling constant. For other values of the charge, a divergent expansiondetermines the kernels having essential singularity atthe origin (beyond the usual -function). If oneconsiders only this class of potentials, then one can avoid many problems connected withasymptotic expansions, and one gets a theory withdiscrete values of the coupling constant that is incorrespondence with the discreteness of charge innature. This approach can be applied to quantum fieldtheory.     

Strongly correlated non-equilibrium steady states with currents – quantum and classical picture

In this minireview we will discuss recent progress in the analytical study of current-carrying non-equilibrium steady states (NESS) that can be constructed in terms of a matrix product ansatz. We will focus on one-dimensional exactly solvable strongly correlated cases, and will study both quantum models, and classical models which are deterministic in the bulk. The only source of classical stochasticity in the time-evolution will come from the boundaries of the system. Physically, these boundaries may be understood as Markovian baths, which drive the current through the system. The examples studied include the open XXZ Heisenberg spin chain, the open Hubbard model, and a classical integrable reversible cellular automaton, namely the Rule 54 of A. Bobenko et al. [A. Bobenko et al., Commun. Math. Phys. 158, 127 (1993)] with stochastic boundaries. The quantum NESS can be at least partially understood through the Yang–Baxter integrability structure of the underlying integrable bulk Hamiltonian, whereas for the Rule 54 model NESS seems to come from a seemingly unrelated integrability theory. In both the quantum and the classical case, the underlying matrix product ansatz defining the NESS also allows for construction of novel conservation laws of the bulk models themselves. In the classical case, a modification of the matrix product ansatz also allows for construction of states beyond the steady state (i.e., some of the decay modes – Liouvillian eigenvectors of the model). We hope that this article will help further the quest to unite different perspectives of integrability of NESS (of both quantum and classical models) into a single unified framework.     

A cellular-automaton fluid model with simple rules in arbitrarily many dimensions

A new cellular-automaton model for fluid dynamics is introduced. Unlike the conventional FHP-type models, the model uses easily implementable, deterministic pair interaction rules which work on arbitrary-dimensional orthogonal lattices. The statistical and hydrodynamic theory of the model is developed, and the Navier-Stokes-like hydrodynamic equations that describe the macroscopic behavior of the model are derived. It turns out that the unwanted anisotropic convection behavior can be eliminated in the incompressible limit by suitable choice of the mass density. An explicit expression for the viscosity tensor is calculated from a Boltzmann-type approximation. Unfortunately, the viscosity turns out to be anisotropic, which is a drawback as against the conventional FHP and FCHC models. Nevertheless, the new model could become interesting for fluid dynamic problems with additional variables (e.g., free surfaces), especially in two dimensions, since its simple rules could relatively easily be extended for such cases.     

A Note on Colored HOMFLY Polynomials for Hyperbolic Knots from WZW Models

Using the correspondence between Chern-Simons theories and Wess-Zumino-Witten models, we present the necessary tools to calculate colored HOMFLY polynomials for hyperbolic knots. For two-bridge hyperbolic knots we derive the colored HOMFLY invariants in terms of crossing matrices of the underlying Wess-Zumino-Witten model. Our analysis extends previous works by incorporating non-trivial multiplicities for the primaries appearing in the crossing matrices, so as to describe colorings of HOMFLY invariants beyond the totally symmetric or anti-symmetric representations of SU(N). The crossing matrices directly relate to 6j-symbols of the quantum group \({\mathcal{U}_{q}su(N)}\). We present powerful methods to calculate such quantum 6j-symbols for general N. This allows us to determine previously unknown colored HOMFLY polynomials for two-bridge hyperbolic knots. We give explicitly the HOMFLY polynomials colored by the representation {2, 1} for two-bridge hyperbolic knots with up to eight crossings. Yet, the scope of application of our techniques goes beyond knot theory; e.g., our findings can be used to study correlators in Wess-Zumino-Witten conformal field theories or—in the limit to classical groups—to determine color factors for Yang Mills amplitudes.     

Scalar modifications to gravity from unparticle effects may be testable

Interest has focused recently on low energy implications of a nontrivial scale invariant sector of an effective field theory with an IR fixed point, manifest in terms of "unparticles" with peculiar properties. If unparticle stuff exists it could couple to the stress tensor and mediate a new "fifth" force ("ungravity"). Under the assumption of strict conformal invariance in the hidden sector down to low energies, we compute the lowest order ungravity correction to the Newtonian gravitational potential and find scale invariant power law corrections of type (R_(G)/r)(2d)_(U)(-1), where d_(U) is an anomalous unparticle dimension and R_(G) is a characteristic length scale where the ungravity interactions become significant. It is shown that a discrimination between extra dimension models and ungravity is possible in future improved submillimeter tests of gravity.     

Covariant Thermodynamics of Quantum Systems: Passivity, Semipassivity, and the Unruh Effect

According to the Second Law of Thermodynamics, cycles applied to thermodynamic equilibrium states cannot perform any work (passivity property of thermodynamic equilibrium states). In the presence of matter this can hold only in the rest frame of the matter, as moving matter drives, e.g., windmills and turbines. If, however, a homogeneous and stationary state has the property that no cycle can perform more work than an ideal windmill, then it can be shown that there is some inertial frame where the state is a thermodynamic equilibrium state. This provides a covariant characterization of thermodynamic equilibrium states. In the absence of matter, cycles should perform work only when driven by nonstationary inertial forces caused by the observer's motion. If a (pure) state of a relativistic quantum field theory behaves this way, it satisfies the spectrum condition and exhibits the Unruh effect.     

The time evolution of fluctuation algebra in mean field theories

Fluctuation algebra is defined for equilibrium states of mean field theories. The time evolution is calculated; and, in contrast to interactions with finite range, the fluctuation algebra is not in a KMS state with respect to this time evolution though the underlying quasilocal state. If the system is coupled to an other system mimicking a laser then the evolution depends on the underlying mean field theory and shows varying large time behaviour, so that the fluctuations either rotate or increase linearly or exponentially in time, in correspondence to the stability of the underlying quasilocal state.     

Landau-Ginzburg orbifolds,mirror symmetry and the elliptic genus

We compute the elliptic genus for arbitrary two-dimensional N = 2 Landau-Ginzburg orbifolds. This is used to search for possible mirror pairs of such models. We show that if two Landau-Ginzburg models are conjugate to each other in a certain sense, then to every orbifold of the first theory corresponds an orbifold of the second theory with the same elliptic genus (up to a sign) and with the roles of the chiral and anti-chiral rings interchanged. These orbifolds thus constitute a possible mirror pair. Furthermore, new pairs of conjugate models may be obtained by taking the product of old ones. We also give a sufficient (and possibly necessary) condition for two models to be conjugate, and show that it is satisfied by the mirror pairs proposed by one of the authors and Hübsch.     

Three-Dimensional Face Recognition Using Solid Harmonic Wavelet Scattering and Homotopy Dictionary Learning

Data representation has been one of the core topics in 3D graphics and pattern recognition in high-dimensional data. Although the high-resolution geometrical information of a physical object can be well preserved in the form of metrical data, e.g., point clouds/triangular meshes, from a regular data (e.g., image/audio) processing perspective, they also bring excessive noise in the course of feature abstraction and regression. For 3D face recognition, preceding attempts focus on treating the scan samples as signals laying on an underlying discrete surface (mesh) or morphable (statistic) models and by embedding auxiliary information, e.g., texture onto the regularized local planar structure to obtain a superior expressive performance to registration-based methods, but environmental variations such as posture/illumination will dissatisfy the integrity or uniform sampling condition, which holistic models generally rely on. In this paper, a geometric deep learning framework for face recognition is proposed, which merely requires the consumption of raw spatial coordinates. The non-uniformity and non-grid geometric transformations in the course of point cloud face scanning are mitigated by modeling each identity as a stochastic process. Individual face scans are considered realizations, yielding underlying inherent distributions under the appropriate assumption of ergodicity. To accomplish 3D facial recognition, we propose a windowed solid harmonic scattering transform on point cloud face scans to extract the invariant coefficients so that unrelated variations can be encoded into certain components of the scattering domain. With these constructions, a sparse learning network as the semi-supervised classification backbone network can work on reducing intraclass variability. Our framework obtained superior performance to current competing methods; without excluding any fragmentary or severely deformed samples, the rank-1 recognition rate (RR1) achieved was 99.84% on the Face Recognition Grand Challenge (FRGC) v2.0 dataset and 99.90% on the Bosphorus dataset.     

Quantization of closed mini-superspace models as bound states

The Wheeler-DeWitt equation is applied to closedk>0 Friedmann-Robertson-Walker metric with various combination of cosmological constant and matter (e.g., radiation or pressureless gas). It is shown that if the universe ends in the matter dominated era (e.g., radiation or pressureless gas) with zero cosmological constant, then the resulting Wheeler-DeWitt equation describes a bound state problem. As solutions of a nondegenerate bound state system, the eigen-wave functions are real (Hartle-Hawking). Furthermore, as a bound state problem, there exists a quantization condition that relates the curvature of the three space with the various energy densities of the universe. If we assume that our universe is closed, then the quantum number of our universe isN(Gk)^–110¹²². The largeness of this quantum number is naturally explained by an early inflationary phase which resulted in a flat universe we observe today. It is also shown that if there is a cosmological constant >0 in our universe that persists for all time, then the resulting Wheeler-DeWitt equation describes a non-bound state system, regardless of the magnitude of the cosmological constant. As a consequence, the wave functions are in general complex (Vilenkin).     

The CPT-theorem in two-dimensional theories of local observables

Let be a von Neumann algebra with cyclic and separating vector , and letU(a) be a continuous unitary representation ofR with positive generator and as fixed point. If these unitaries induce for positive arguments endomorphisms of then the modular group act as dilatations on the group of unitaries. Using this it will be shown that every theory of local observables in two dimensions, which is covariant under translation only, can be imbedded into a theory of local observables covariant under the whole Poincaré group. This theory is also covariant under the CPT-transformation.     

Mean-field theory of critical phenomenon for mutually repelling particles in complex plasmas

A mean-field theory of criticality for charged particles in complex plasmas is proposed. It is shown that the existence of the critical point and the liquid-vapor coexistence is fully consistent with a purely repulsive potential between particles; the cohesive field due to the plasma background drives these. The critical exponents, calculated by expanding the free energy near the critical point, are found to be classical. The phase coexistence curve, obtained by minimizing Gibbs potential, is similar to that of other mean-field models, e.g., van der Waals fluids, ionic fluids, etc. These results lend support to the concept of "universality" in widely different systems.     

On a conjecture of Alley and Alder for fluids and Lorentz models

We discuss a conjecture of Alley and Alder predicting a relation between the four-point and the two-point velocity autocorrelation functions for fluids and Lorentz models at sufficiently long times. If the conjecture is correct a modified Burnett coefficient can be defined, which has a finite value, contrary to the ordinary Burnett coefficient, which is divergent. The conjecture is tested for four classes of models with different methods: for three-dimensional fluids mode-coupling theory yields a negative result. The conjecture is confirmed for thed-dimensional deterministic Lorentz gas (d 2) and for a class ofd-dimensional stochastic Lorentz models (d 1) by low-density kinetic theory, as well as by rigorous results, available for one dimension. For yet another class of one-dimensional stochastic Lorentz models, which are exactly solvable in one dimension, the result is negative again. All four classes of models show long-time tails in the velocity autocorrelation function and have a finite diffusion coefficient.