Modular Conjugation and the Implementation of Supersymmetry

Any -graded C ^*-dynamical system with a self-adjoint graded-Kubo-Martin-Schwinger (KMS) functional on it can be represented (canonically) as a -graded algebra of bounded operators on a -graded Hilbert space, so that the grading of the latter is compatible with the functional. The modular conjugation operator plays a crucial role in this reconstruction. The results are generalized to the case of an unbounded graded-KMS functional having as dense domain the union of a net of C ^*-subalgebras. It is shown that the modulus of such an unbounded graded-KMS functional is KMS.      

Classification of three-dimensional covariant differential calculi on Podles' quantum spheres and on related spaces

Covariant differential calculi on the quantum space for the quantum group SL _q (2) are classified. Our main assumptions are thatq is not a root of unity and that the differentials de _j of the generators of form a free right module basis for the first-order forms. Our result says, in particular, that apart from the two casesc =c(3), there exists a unique differential calculus with the above properties on the space which corresponds to Podles' quantum sphereS _qc ^/2 .     

Moment Bounds for the Smoluchowski Equation and their Consequences

We prove bounds on moments of the Smoluchowski coagulation equations with diffusion, in any dimension d ≥ 1. If the collision propensities α(n, m) of mass n and mass m particles grow more slowly than , and the diffusion rate is non-increasing and satisfies for some b ₁ and b ₂ satisfying 0 ≤ b ₂ < b ₁ < ∞, then any weak solution satisfies for every and T ∈(0, ∞), (provided that certain moments of the initial data are finite). As a consequence, we infer that these conditions are sufficient to ensure uniqueness of a weak solution and its conservation of mass. This work was performed while A.H. held a postdoctoral fellowship in the Department of Mathematics at U.B.C. This work is supported in part by NSF grant DMS0307021.     

Symmetries and Modular Intersections of Von Neumann Algebras

Let be von Neumann algebras acting on a Hilbert space and let be a common cyclic and separating vector. We say that have the modular intersection property with respect to if(1) -half-sided modular inclusions,(2) (If (1) holds the strong limit exists.) We show that under these conditions the modular groups of and generate a 2-dim. Lie group.This observation is the basis for obtaining group representations of Sl(2, )/Z ₂ generated by modular groups.     

Chevalley Cohomology for linear graphs

The space of linear polyvector fields on is a Lie subalgebra of the (graded) Lie algebra , equipped with the Schouten bracket. In this paper, we compute the cohomology of this subalgebra for the adjoint representation in , restricting ourselves to the case of cochains defined with purely aerial Kontsevich’s graphs, as in Pac. J. Math. 218(2):201–239, 2005. We find a result which is very similar to the cohomology for the vector case Pac. J. Math. 229(2):257–292, 2007. This work was supported by the CMCU contract 06 S 1502. W. Aloulou and R. Chatbouri thank the Université de Bourgogne and D. Arnal the Faculté des Sciences de Monastir for their kind hospitalities during their stay.     

Cardy Condition for Open-Closed Field Algebras

Let V be a vertex operator algebra satisfying certain reductivity and finiteness conditions such that , the category of V-modules, is a modular tensor category. We study open-closed field algebras over V equipped with nondegenerate invariant bilinear forms for both open and closed sectors. We show that they give algebras over a certain -extension of the so-called Swiss-cheese partial dioperad, and we can obtain Ishibashi states easily in such algebras. The Cardy condition can be formulated as an additional condition on such open-closed field algebras in terms of the action of the modular transformation on the space of intertwining operators of V. We then derive a graphical representation of S in the modular tensor category . This result enables us to give a categorical formulation of the Cardy condition and the modular invariance condition for 1-point correlation functions on the torus. Then we incorporate these two conditions and the axioms of the open-closed field algebra over V equipped with nondegenerate invariant bilinear forms into a tensor-categorical notion called the Cardy -algebra. In the end, we give a categorical construction of the Cardy -algebra in the Cardy case.     

Exact Results for Topological Strings on Resolved <Emphasis Type="Italic">Y</Emphasis><Superscript><Emphasis Type="Italic"> p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Superscript> Singularities

We obtain exact results in α′ for open and closed A-model topological string amplitudes on a large class of toric Calabi-Yau threefolds by using their correspondence with five dimensional gauge theories. The toric Calabi-Yaus that we analyze are obtained as minimal resolution of cones over Y ^p,q manifolds and give rise via M-theory compactification to SU(p) gauge theories on . As an application we present a detailed study of the local case and compute open and closed genus zero Gromov-Witten invariants of the orbifold. We also display the modular structure of the topological wave function and give predictions for higher genus amplitudes. The mirror curve in this case is the spectral curve of the relativistic A ₁ Toda chain. Our results also indicate the existence of a wider class of relativistic integrable systems associated to generic Y ^p,q geometries.     

Covariant (hh′)-Deformed Bosonic and Fermionic Algebras as Contraction Limits of q-Deformed Ones

GL_h(n) ×GL_h(m)-covariant (hh)-bosonic[or (hh)-fermionic] algebras are built in terms of thecorresponding R_h and -matrices by contracting theGL_q(n) × -covariant q-bosonic (or q-fermionic) algebras , = 1, 2.When using a basis of wherein theannihilation operators are contragredient to thecreation ones, this contraction procedure can be carried out for any n, m values. Whenemploying instead a basis wherein the annihilationoperators, like the creation ones, are irreducibletensor operators with respect to the dual quantumalgebra U_q(gl(n)) , a contraction limit only exists forn, m {1, 2, 4, 6, . . .}. For n = 2, m = 1, andn = m = 2, the resulting relations can be expressed interms of coupled (anti)commutators (as in the classical case), by usingU_h(sl(2)) [instead of s1(2)] Clebsch-Gordancoefficients. Some U_h(sl(2)) rank-1/2irreducible tensor operators recently constructed byAizawa are shown to provide a realization of (2, 1).     

(Co)cyclic (Co)homology of Bialgebroids: An Approach via (Co)monads

For a (co)monad T _l on a category , an object X in , and a functor , there is a (co)simplex in . The aim of this paper is to find criteria for para-(co)cyclicity of Z ^*. Our construction is built on a distributive law of T _l with a second (co)monad T _r on , a natural transformation , and a morphism in . The (symmetrical) relations i and w need to satisfy are categorical versions of Kaygun’s axioms of a transposition map. Motivation comes from the observation that a (co)ring T over an algebra R determines a distributive law of two (co)monads and on the category of R-bimodules. The functor Π can be chosen such that is the cyclic R-module tensor product. A natural transformation is given by the flip map and a morphism is constructed whenever T is a (co)module algebra or coring of an R-bialgebroid. The notion of a stable anti-Yetter-Drinfel’d module over certain bialgebroids, the so-called  ×  _R -Hopf algebras, is introduced. In the particular example when T is a module coring of a  ×  _R -Hopf algebra and X is a stable anti-Yetter-Drinfel’d -module, the para-cyclic object Z _* is shown to project to a cyclic structure on . For a -Galois extension , a stable anti-Yetter-Drinfel’d -module T _S is constructed, such that the cyclic objects and are isomorphic. This extends a theorem by Jara and Ştefan for Hopf Galois extensions. As an application, we compute Hochschild and cyclic homologies of a groupoid with coefficients in a stable anti-Yetter-Drinfel’d module, by tracing it back to the group case. In particular, we obtain explicit expressions for (coinciding relative and ordinary) Hochschild and cyclic homologies of a groupoid. The latter extends results of Burghelea on cyclic homology of groups.     

A Space–Time Integral Estimate For A Large Data Semi-linear Wave Equation on the Schwarzschild Manifold

We consider the wave equation with in . The wave equation on a spherically symmetric manifold with a single closed geodesic surface or on the exterior of the Schwarzschild manifold can be reduced to this form. Using a smoothed Morawetz estimate which does not require a spherical harmonic decomposition, we show that there is decay in for initial data in the energy class, even if the initial data is large. This requires certain conditions on the potentials V, V _L and f. We show that a key condition on the weight in the smoothed Morawetz estimate can be reduced to an ODE condition, which is verified numerically.      

On the Global Wellposedness to the 3-D Incompressible Anisotropic Navier-Stokes Equations

Corresponding to the wellposedness result [2] for the classical 3-D Navier-Stokes equations (NS _ν) with initial data in the scaling invariant Besov space, here we consider a similar problem for the 3-D anisotropic Navier-Stokes equations (ANS _ν), where the vertical viscosity is zero. In order to do so, we first introduce the Besov-Sobolev type spaces, and Then with initial data in the scaling invariant space we prove the global wellposedness for (ANS _ν) provided the norm of initial data is small enough compared to the horizontal viscosity. In particular, this result implies the global wellposedness of (ANS _ν) with high oscillatory initial data (1.2).     

Influence of a strong longitudinal static electric field on free carrier faraday effect in an n-type InSb at room temperature at submillimeter wave frequencies

The expression for free carrier Faraday rotation and for ellipticity , as the function of the applied parallel static electric field and static magnetic field for a given value of wave angular frequency and electron concentration N₀, are obtained and theoretically analyzed with the aid of one-dimensional linearized wave theory and Kane's non-parabolic isotropic dispersion law. It is shown that the maximum Faraday rotation occurs near the cyclotron resonance condition, which can be expressed as , where , , and . Here m^* and e denote the effective mass and charge of electron, respectively. _g is the forbidden bandgap of semiconductor. v₀ is the carrier drift velocity, which is a non-linear function of E₀ in high field condition. A possibility of a simple way of determining the non-linear v₀ vs E₀ characteristics of semiconductors by the measurement of Faraday rotation is also discussed.     

Dynamical Differential Equations Compatible with Rational <Emphasis Type="Italic">qKZ</Emphasis> Equations

For the Lie algebra _N we introduce a system of differential operators called the dynamical operators. We prove that the dynamical differential operators commute with the _N rational quantized Knizhnik–Zamolodchikov difference operators. We describe the transformations of the dynamical operators under the natural action of the _N Weyl group.Mathematics Subject Classifications (2000). 17B37, 17B80, 81R10.     

Local Asymptotic Normality in Quantum Statistics

The theory of local asymptotic normality for quantum statistical experiments is developed in the spirit of the classical result from mathematical statistics due to Le Cam. Roughly speaking, local asymptotic normality means that the family consisting of joint states of n identically prepared quantum systems approaches in a statistical sense a family of Gaussian state ϕ _u of an algebra of canonical commutation relations. The convergence holds for all “local parameters” such that parametrizes a neighborhood of a fixed point . In order to prove the result we define weak and strong convergence of quantum statistical experiments which extend to the asymptotic framework the notion of quantum sufficiency introduces by Petz. Along the way we introduce the concept of canonical state of a statistical experiment, and investigate the relation between the two notions of convergence. For the reader’s convenience and completeness we review the relevant results of the classical as well as the quantum theory. Dedicated to Slava Belavkin on the occasion of his 60th anniversary     

The Green-Kubo Formula for the Spin-Fermion System

The spin-fermion model describes a two level quantum system (spin 1/2) coupled to finitely many free Fermi gas reservoirs which are in thermal equilibrium at inverse temperatures β _j . We consider non-equilibrium initial conditions where not all β _j are the same. It is known that, at small coupling, the combined system has a unique non-equilibrium steady state (NESS) characterized by strictly sitive entropy production. In this paper we study linear response in this NESS and prove the Green-Kubo formula and the Onsager reciprocity relations for heat fluxes generated by temperature differentials. Dedicated to Jean Michel Combes on the occasion of his sixtyfifth birthday     

On the local implementations of gauge symmetries in local quantum theory

Under the general assumptions of quantum field theory in terms of local algebras of field operators fulfilling the split property, we prove that any two local coveriant implementations of the gauge group (or, in the case of a connected and simply connected Lie gauge group, any two choices of local current algebras) relative to a pair of double cones ₁, ₂, are related by a unitary equivalence induced by a unitary in the algebra of observables localized in ₂ which commutes with all fields localized in ₁, where ₁ is any double cone contained in the interior of ₁, and ₂ any double cone containing ₂ in its interior.     

Singular Spectrum Near a Singular Point of Friedrichs Model Operators of Absolute Type

In we consider a family of selfadjoint operators of the Friedrichs model: . Here is the operator of multiplication by the corresponding function of the independent variable , and (perturbation) is a trace-class integral operator with a continuous Hermitian kernel satisfying some smoothness condition. These absolute type operators have one singular point of order . Conditions on the kernel are found guaranteeing the absence of the point spectrum and the singular continuous one of such operators near the origin. These conditions are actually necessary and sufficient. They depend on the finiteness of the rank of a perturbation operator and on the order of singularity . The sharpness of these conditions is confirmed by counterexamples.     

Conformal properties of nonpeeling vacuum space-times

In a previous paper we investigated a class ofnonpeeling asymptotic vacuum solutions which were shown to admit finite expressions for the Winicour-Tamburino energy-momentum and angular momentum integrals. These solutions have the property that $$\psi _0 = O(r^{ - 3 - \in _0 } ), \in _0 \leqslant 2$$ and $$\psi _1 = O(r^{ - 3 - \in _1 } ), \in _1< \in _0 and \in _1< 1$$ withψ ₂,ψ ₃, andψ ₄ having the same asymptotic behavior as they do for peeling solutions. The above investigation was carried out in the physical space-time. In this paper we examine the conformal properties of these solutions, as well as the more general Couch-Torrence solutions, which include them as a subclass. For the Couch-Torrence solutions $$\begin{gathered} \psi _0 = O(r^{ - 2 - \in _0 } ) \hfill \\ \psi _1 = O(r^{ - 2 - \in _1 } ), \in _1< \in _0 {\text{ }}and \in _1 \leqslant 2 \hfill \\ \end{gathered} $$ and , $$\psi _2 = O(r^{ - 2 - \in _2 } ),{\text{ }} \in _2< \in _1 {\text{ }}and \in _2 \leqslant 1$$ withψ ₃ andψ ₄ behaving as they do for peeling solutions. It is our purpose to determine how much of the structure generally associated with peeling space-times is preserved by the nonpeeling solutions. We find that, in general, a three-dimensional null boundary (?⁺) can be defined and that the BMS group remains the asymptotic symmetry group. For the general Couch-Torrence solutions several physically and/or geometrically interesting quantities