Eingenvalues of the Dirac operator on compact Kähler manifolds

Kählerian twistor operators are introduced to get lower bounds for the eigenvalues of the Dirac operator on compact spin Kähler manifolds. In odd complex dimensions, manifolds with the smallest eigenvalues are characterized by an over determined system of differential equations similar to the Riemannian case. In these dimensions, we show the existence of a unique natural Kählerian twistor operator. It is also proved that, on a Kähler manifold with nonzero scalar curvature, the space of Riemannian twistor-spinors is trivial.This work has been partially supported by the EEC programme GADGET Contract Nr. SC1-0105     

Optimal Eigenvalue Estimate for the Dirac–Witten Operator on Bounded Domains with Smooth Boundary

Eigenvalue estimate for the Dirac–Witten operator is given on bounded domains (with smooth boundary) of spacelike spin hypersurfaces satisfying the dominant energy condition, under four natural boundary conditions (MIT, APS, modified APS and chiral conditions). Roughly speaking, any eigenvalue of the Dirac–Witten operator satisfies

Some remarks on discrete aperiodic Schrödinger operators

We consider Schrödinger operators onl ²( ) with deterministic aperiodic potential and Schrödinger operators on the l²-space of the set of vertices of Penrose tilings and other aperiodic self-similar tilings. The operators onl ²( ) fit into the formalism of ergodic random Schrödinger operators. Hence, their Lyapunov exponent, integrated density of states, and spectrum are almost-surely constant. We show that they are actually constant: the Lyapunov exponent for one-dimensional Schrödinger operators with potential defined by a primitive substitution, the integrated density of states, and the spectrum in arbitrary dimension if the system is strictly ergodic. We give examples of strictly ergodic Schrödinger operators that include several kinds of almost-periodic operators that have been studied in the literature. For Schrödinger operators on Penrose tilings we prove that the integrated density of states exists and is independent of boundary conditions and the particular Penrose tiling under consideration.     

Some Remarks on the Smoluchowski–Kramers Approximation

According to the Smoluchowski–Kramers approximation, solution q _t of the equation , where is the White noise, converges to the solution of equation as µ 0. Many asymptotic problems for the last equation were studied in recent years. We consider relations between asymptotics for the first order equation and the original second order equation. Homogenization, large deviations and stochastic resonance, approximation of Brownian motion W _t by a smooth stochastic process, stationary distributions are considered.     

Spectrum of the Dirac operator on complex projective space P 2q-1(ℂ)

We give explicit formulas for the spectrum of the Dirac operator on complex projective space P _2q-1().Aspirant au Fonds National de la Recherche Scientifique.     

Solution of the Dirac Equation on the Bertotti–Robinson Metric

It is shown that each component of the Dirac field satisfies a decoupled equation, which admits separable solutions, when the background spacetime is the Bertotti–Robinson metric, which is a solution of the Einstein vacuum field equations with a cosmological constant. Furthermore, the seperated functions appearing in the solutions are shown to obey identities of the Teukolsky–Starobinsky type and the separable solutions are shown to be eigenfunctions of a certain differential operator.     

Some remarks about the “efficiency” of polyautomata

A proposal is made to measure the efficiency of cellular algorithms, implemented in cellular automata, roughly speaking by the ratio of the number of proper state changes and the product of time, and number of single automata. Such a definition is discussed in some detail. For some cellular algorithms lower bounds for their efficiency are given.     

Some remarks on variable range hopping—in particular in the quantum Hall regime

An expression for the heat conductivity _xx is derived in the effective medium approximation. Mott type formulas are obtained for _xx and the Peltier coefficient _xx . Using percolation theory in a three-dimensional system the Wiedemann-Franz ratio was found to depend on the temperature like . The Mott type formulas were evaluated in a similar way for a two-dimensional system in the quantum Hall regime within the high-field percolation model. In contrast to previous calculations of the high field hopping conductivity _xx , the results are fully consistent with the experimental data on _xx and the density of states at the Fermi level. Finally, _xx is estimated which together with _xx and _xy =ie ²/h(i=0,1,2,...), determines both thermopower coefficients _xx and _xy .Dedicated to Professor W. Brenig on the occasion of his 60th birthday     

A golden age of general relativity? Some remarks on the history of general relativity

This article deals with the concepts “renaissance” and “low water mark between 1925 and 1955” of general relativity suggested in the literature. By empirical data, it is shown that no such period did exist. Research on general relativity continued continuously since the 1920s interrupted only by the second world war. On a broad scale, research on general relativity started only after 1945.     

Two remarks on the painlevé test

Using the example of two previously discussed autonomous differential equations it is pointed out that the Painlevé test with an ansatz w = Σ_n=p^∞w_n(z − z₀)ⁿ, p < 0, integer, will not reveal movable branch points if all leading order contribution arise from terms with equal degree of nonlinearity. Just contrary to this case, for the reduced damped and driven φ⁴ equation with p = −1 two resonance conditions arise at order n = 3 which cannot be satisfied simultaneously.     

Fedosov observables on constant curvature manifolds and the Klein–Gordon equation

In this paper we construct the Fedosov star-algebra of observables on the phase–space of a single particle in the case of all (finite-dimensional) constant curvature manifolds imbeddable in a flat space with codimension one. This set of spaces includes the two-sphere and de Sitter (dS)/Anti-de Sitter (AdS) space–times. The algebra of observables was constructed by DQ techniques using, in particular, the algorithm provided by Fedosov.     

A Dirac–Dunkl Equation on S2 and the Bannai–Ito Algebra

The Dirac–Dunkl operator on the two-sphere associated to the \({{\mathbb{Z}_{2}^{3}}}\) reflection group is considered. Its symmetries are found and are shown to generate the Bannai–Ito algebra. Representations of the Bannai–Ito algebra are constructed using ladder operators. Eigenfunctions of the spherical Dirac–Dunkl operator are obtained using a Cauchy–Kovalevskaia extension theorem. These eigenfunctions, which correspond to Dunkl monogenics, are seen to support finite-dimensional irreducible representations of the Bannai–Ito algebra.     

Some Remarks on the Cohomology of Krichever–Novikov Algebras

We calculate the continuous cohomology of the Lie algebra of meromorphic vector fields on a compact Riemann surface from the cohomology of the holomorphic vector fields on the open Riemann surface pointed in the poles. This cohomology has been given by Kawazumi. Our result shows the Feigin–Novikov conjecture.     

Gromov–Witten invariants and rational curves on Grassmannians

We study the enumerative significance of the s$s$-pointed genus zero Gromov–Witten invariant on a homogeneous space X$X$. For that, we give an interpretation in terms of rational curves on X$X$.     

Asymptotic properties of the Hitchin–Witten connection

Letters in Mathematical Physics - We explore extensions to $${{\,\mathrm{SL}\,}}(n,{\mathbb {C}})$$ -Chern–Simons theory of some results obtained for $${{\,\mathrm{SU}\,}}(n)$$...     

Dirac Operator on the Podleś Sphere

The spectral problem for the Dirac operator on the Podle sphere is discussed and solved in one of its possible formulations. The standard constructions for the Dirac operator on classical symmetric spaces and the spectrum of the Dirac operator on the classical two-dimensional sphere are recalled. The problem of defining a spinor structure on a quantum space is discussed and the definitions of a classical spinor structure and Dirac operator according to Ðurdevich are sketched. The Dirac operator for the Podles´ quantum sphere treated as a quotient space of SU(2) is constructed using the Woronowicz left-covariant calculus over this quantum group. The spectrum of the operator is obtained. Disagreement of its asymptotic behavior with Connes' axiom of noncommutative spectral geometry is stressed.     

On the level-dependence of Wess–Zumino–Witten three-point functions

Three-point functions of Wess–Zumino–Witten models are investigated. In particular, we study the level-dependence of three-point functions in the models based on algebras su(3) and su(4). We find a correspondence with Berenstein–Zelevinsky triangles. Using previous work connecting those triangles to the fusion multiplicities, and the Gepner–Witten depth rule, we explain how to construct the full three-point functions. We show how their level-dependence is similar to that of the related fusion multiplicity. For example, the concept of threshold level plays a prominent role, as it does for fusion.     

Gauge Theoretical Equivariant Gromov–Witten Invariants and the Full Seiberg–Witten Invariants¶of Ruled Surfaces

Let F be a differentiable manifold endowed with an almost K?hler structure (J,ω), α a J-holomorphic action of a compact Lie group on F, and K a closed normal subgroup of which leaves ω invariant. The purpose of this article is to introduce gauge theoretical invariants for such triples (F,α,K). The invariants are associated with moduli spaces of solutions of a certain vortex type equation on a Riemann surface Σ. Our main results concern the special case of the triple