The triviality of representations of the Virasoro algebra with vanishing central element and L0 positive

A simple and direct proof is presented that the vanishing of the central element in a representation of the Virasoro algebra with L₀ bounded below implies that all the Virasoro generators vanish. The case c ≠ 0, although inconclusive, is considered also.     

Toward classification of conformal theories

By studying the representations of the mapping class groups which arise in 2D conformal theories we derive some restrictions on the value of the conformal dimension h_i of operators and the central charge c of the Virasoro algebra. As a simple application we show that when there are a finite number of operators in the conformal algebra, the h_i and c are all rational.     

The algebra of the energy-momentum tensor and the Noether currents in classical non-linear sigma models

The recently derived current algebra of classical non-linear sigma models on arbitrary Riemannian manifolds is extended to include the energy-momentum tensor. It is found that in two dimensions the energy-momentum tensor , the Noether currentj associated with the global symmetry of the theory and the composite fieldj appearing as the coefficient of the Schwinger term in the current algebra, together with the derivatives ofj andj, generate a closed algebra. The subalgebra generated by the light-cone components of the energy-momentum tensor consists of two commuting copies of the Virasoro algebra, with central chargec=0, reflecting the classical conformal invariance of the theory, but the current algebra part and the semidirect product structure are quite different from the usual Kac-Moody/Sugawara type construction.Supported by the Deutsche Forschungsgemeinschaft, Contract No. Ro 864/2-1Supported by the Studienstiftung des Deutschen Volkes     

Indecomposable fusion products

We analyse the fusion products of certain representations of the Virasoro algebra for c = −2 and c = −7 which are not completely reducible. We introduce a new algorithm which allows us to study the fusion product level by level, and we use this algorithm to analyse the indecomposable components of these fusion products. They form novel representations of the Virasoro algebra which we describe in detail.We also show that a suitably extended set of representations closes under fusion, and indicate how our results generalise to all (1, q) models.     

Quantum Group as Semi-infinite Cohomology

We obtain the quantum group SL _q (2) as semi-infinite cohomology of the Virasoro algebra with values in a tensor product of two braided vertex operator algebras with complementary central charges c+[`(c)]=26{c+\bar{c}=26}. Each braided VOA is constructed from the free Fock space realization of the Virasoro algebra with an additional q-deformed harmonic oscillator degree of freedom. The braided VOA structure arises from the theory of local systems over configuration spaces and it yields an associative algebra structure on the cohomology. We explicitly provide the four cohomology classes that serve as the generators of SL _q (2) and verify their relations. We also discuss the possible extensions of our construction and its connection to the Liouville model and minimal string theory.     

<Emphasis Type="Italic">C</Emphasis><Subscript>2</Subscript>-Cofiniteness of the 2-Cycle Permutation Orbifold Models of Minimal Virasoro Vertex Operator Algebras

In this article, we give a sufficient and necessary condition for the C ₂-cofiniteness of \({\widetilde{V} = (V\otimes V)^\sigma}\) for a C ₂-cofinite vertex operator algebra V and the 2-cycle permutation σ of \({V\otimes V}\) . As an application, we show that the 2-cycle permutation orbifold model of the simple Virasoro vertex operator algebra L(c, 0) of minimal central charge c is C ₂-cofinite.     

Ultra-hard ceramic coatings fabricated through microarc oxidation on aluminium alloy

Ultra-hard ceramic coatings with microhardness of 2535 Hv have been synthesized on the Al alloy substrate by microarc oxidation (MAO) technique. The effects of anodic current density (j_a) and the ratio of cathodic to anodic current density (j_c/j_a) on the mechanical and corrosion resistance properties of MAO coatings have been studied by microhardness and pitting corrosion tests, respectively. In addition, the phase composition and microstructure of the coatings were analyzed by X-ray diffraction (XRD) and scanning electron microscopy (SEM), respectively. The results show that the coatings prepared at high anodic current density consist mainly of α-Al₂O₃, while those fabricated at low anodic current density are almost composed of γ-Al₂O₃. Microhardness test shows that the coatings have high microhardness, and the highest one is found in the coating formed at j_a = 15 A/dm² and j_c/j_a = 0.7. Pitting corrosion test shows that the structure of coatings is strongly influenced by the varying j_c/j_a.     

Conformal invariance,the KdV equation and coadjoint orbits of the Virasoro algebra

We use the transformation properties of the energy-momentum tensor of 2-dimensional conformal field theories to study the symplectic geometry of DiffS¹/S¹ together with the other coadjoint orbits of the Virasoro algebra in the space of quadratic differentials. Also, we examine orbits of the N = 1 and N = 2 super-Virasoro algebras and investigate the connections with nonlinear differential equations of KdV type. The importance that the dual of the Virasoro algebra may have in the geometric formulation of string theory is discussed as well.     

Norton's Trace Formulae for the Griess Algebra¶of a Vertex Operator Algebra with Larger Symmetry

Formulae expressing the trace of the composition of several (up to five) adjoint actions of elements of the Griess algebra of a vertex operator algebra are derived under certain assumptions on the action of the automorphism group. They coincide, when applied to the moonshine module V , with the trace formulae obtained in a different way by S. Norton, and the spectrum of some idempotents related to 2A, 2B, 3A and 4A elements of the Monster is determined by the representation theory of the Virasoro algebra at c= 1/2, the W ₃ algebra at c= 4/5 or the W ₄ algebra at c= 1. The generalization to the trace function on the whole space is also given for the composition of two adjoint actions, which can be used to compute the McKay-Thompson series for a 2A involution of the Monster. Received: 24 July 2000 / Accepted: 15 June 2001     

Effect of structural water on the critical characteristics of highly textured YBa2Cu3O6.9

The effect of low-temperature treatment (200°C) in a humid argon atmosphere and subsequent annealing (930°C) on the critical parameters of a highly textured YBa₂Cu₃O_6.9 has been studied. During annealing at T = 200°C, the absorbed water is incorporated into the structure of the compound, which is accompanied by the deterioration of its superconducting properties. However, after the recovery annealing at T = 930°C and subsequent oxidation, the superconducting characteristics (j _c , B _1c , and F _p ) are improved. This is explained by the formation of 124-type planar defects, which are effective pinning centers, especially in high fields applied perpendicular to the c axis (⊥ c). The optimum conditions of double annealing substantially increasing the critical current density (j _c ≥ 10⁴ A/cm²) in an external magnetic field up to 10 T and also the first critical fields have been found. In fields up to ～3 T, the critical current density j _c is isotropic despite the conservation of high texture in the samples.     

Dynamical properties of superconducting nanowires

The dynamical properties of thin superconducting wires (nanowires) are studied using numerical simulations based on a one-dimensional time-dependent Ginzburg-Landau equation, which is modified by introducing an order parameter u characterizing the “purity” of the superconductor material. It is established that relatively long nanowires (with lengths much greater than the coherence length) made of a “pure” superconductor (u > 1) are characterized by two critical current density values: j _c1 and j _c2. For j < j _c1, the total current is entirely superconducting, whereas for j > j _c2, the current is purely normal. In the intermediate region of current densities, j _c1 < j < j _c2, the total current contains both superconducting and normal components (mixed state) and the nanowire exhibits the generation of high-frequency electromagnetic waves. The current-voltage characteristics are constructed and the radiation spectrum is obtained. The properties of short superconducting nanowires (with lengths on the order of the coherence length) coincide with those of the Josephson junction. In the case of an “impure” superconductor (u < 1), the nanowire is characterized by a single critical current density.     

Einfluß der Bestrahlung mit hochenergetischen Sauerstoffionen auf die kritische Stromdichte supraleitender Nb3Sn-Schichten

Nb₃Sn diffusion layers were irradiated with 24 MeV oxygen ions at fluences from 3.2×10¹³ up to 1.6×10¹⁵ cm^?2. The enhancement of the superconducting critical current density Δj _c has been measured as a function of fluence and of the external magnetic fieldH _a(j _c⊥H _a). The thermal annealing treatment of the defects concerning thej _c and induced by irradiation, has been investigated in the temperature region from 200 to 800 °C. The results are compared with the measurements of irradiation of Nb₃Sn with protons and deuterons. The measured data are discussed in connection with size of defects, cluster distance, fluxline distance and pinning-force.     

Boundary Structure and Module Decomposition of the Bosonic Z2 Orbifold Models with R=1/2k

The Z₂ bosonic orbifold models with compactification radius R²=1/2k are examined in the presence of boundaries. Demanding the extended algebra characters to have definite conformal dimension and to consist of an integer sum of Virasoro characters, one arrives at the right splitting of the partition function. This is used to derive a free field representation of a complete, consistent set of boundary states, compatible with the modular transformations of the characters. Finally the modules of the extended symmetry algebra that correspond to the finitely many characters are identified inside the direct sum of Fock modules that constitute the space of states of the theory.     

New algebraic structures in the Cλ-extended Hamiltonian system

A realization of various algebraic structures in terms of the C_λ-extended oscillator algebras is introduced. In particular, the C_λ-extended oscillator algebras realization of the Fairlie–Fletcher–Zachos (FFZ) algebra is given. This latter lead easily to the realization of the quantum U_t(sl(2)) algebra. The new deformed Virasoro algebra is also presented.     

On the Representation Theory of Virasoro Nets

We discuss various aspects of the representation theory of the local nets of von Neumann algebras on the circle associated with positive energy representations of the Virasoro algebra (Virasoro nets). In particular we classify the local extensions of the c=1 Virasoro net for which the restriction of the vacuum representation to the Virasoro subnet is a direct sum of irreducible subrepresentations with finite statistical dimension (local extensions of compact type). Moreover we prove that if the central charge c is in a certain subset of (1, ), including [2, ), and h(c–1)/24, the irreducible representation with lowest weight h of the corresponding Virasoro net has infinite statistical dimension. As a consequence we show that if the central charge c is in the above set and satisfies c25 then the corresponding Virasoro net has no proper local extensions of compact type.Supported in part by the Italian MIUR and GNAMPA-INDAM.     

The effects of anodic and cathodic processes on the characteristics of ceramic coatings formed on titanium alloy through the MAO coating technology

This paper reports on the investigation into the effects of the different anodic j_a and cathodic j_c current densities on the variations of the anodic and cathodic processes and the associated changes in the characteristics of the coatings synthesized on Ti-6Al-4V alloy substrate by microarc oxidation technique. The coated samples were subjected to coating thickness and cross-section fracture observation. Phase and elemental composition at different depth of the coatings were evaluated through X-ray diffraction and energy dispersive spectrum analyses. The experimental results indicate that the increase of j_a leads to the increasing slope of anodic voltage U₊ versus oxidation time plots, the larger coating thickness and the more coarse surface morphology of MAO coatings; while the aggrandizement of the cathodic process results in the lower growth rate and more uniform structure of coatings. The changes of the elements distribution from the interface towards the coating surface with variation of j_a and j_c are affected by the Ti anodic dissolution and the electromigration of electriferous particles, such as Al(OH)₄⁻, in electrolytes.     

Magnetoresistance hysteresis of granular YBa2Cu3O7−δ high-temperature superconductor in weak magnetic fields

The variation of the transverse magnetoresistance of YBa₂Cu₃O_～6.95 high-temperature superconducting ceramic with external magnetic field intensity H _ext first increasing from zero to ～500 Oe (Δρ⁺ / ρ_{273 K}) and then decreasing from about 500 Oe to zero (Δρ^? / ρ_{273 K}) is studied for transport current densities varying from j/j _c ≈ 0.01 to ≈0.99 (where j _c is the critical current density in the absence of the magnetic field) at 77.3 K. It is found that the field dependence of the magnetoresistance of YBa₂Cu₃O_～6.95 high-temperature superconductor is characterized by pronounced hysteresis, the difference Δρ⁺ / ρ_{273 K} ? Δρ^? / ρ_{273 K} increasing with j/j _c. As j/j _c grows, the effective critical fields of Josephson weak links, H _c2J , and the lower critical fields of superconducting grains, H _c1A , decline. When field H _ext rises, the critical fields are lower than when H _ext diminishes: H _c2J ⁺ < H _c2J ^? and H _c1A ⁺ < H _c1A ^? .     

Multiphoton interband transitions assisted by photoexcited free carriers

The rate of generation of electron-hole pairs by intense light is calculated within a three-band model of an insulator including a valence band v and conduction bands c and c ₁. It is assumed that the photon energy ?ω is small in comparison with the band gap E _g and generation occurs due to the transitions involving free electrons in the upper conduction band c ₁. It is shown that, at a light intensity j ～ 10¹¹ W/cm² and an electron concentration in the c ₁ band exceeding 10¹⁷ cm^?3, the rates of the transitions c ₁ +l ?ω → ccv (l = 2 or 3) exceed the rates of direct five-photon transitions between the v and the c bands. In this case, a multiphoton avalanche can be initiated.     

Flux creep in the Tl-1223 high temperature superconductor

The relaxation of the irreversible magnetization has been measured by a Faraday magnetometer for the weakly anisotropic polycrystalline Tl_0.5Pb_0.5Sr₂Ca₂Cu₃O₉ (Tl-1223) and a highly anisotropic Tl₂Ba₂Ca₂ Cu₃O₁₀ (Tl-2223) single crystal. From these data, the effective activation energy, U _eff(j)=U _l(j _c /j)^μ, is determined by the method of Maley. A comparison of the irreversibility data at 77 K for different high-temperature superconductors (HTSC) indicates that the irreversibility field is quickly decreasing with increasing distance between the superconducting CuO₂-layers. In agreement with this tendency, U _eff is higher for the one-intermediatelayer compound Tl-1223 than for the highly anisotropic two-intermediate-layer Tl-2223 compound. The exponent μ≡d log(U _eff)/d log(j) saturates for Tl-1223 at a value of μ≈1.4 near j _c and decreases with decreasing current down to 0.3. On the other hand μ increases for Tl-2223 with increasing current from 0.1 up to 2.5 and shows no tendency to saturate. A comparison with data for Y-123 and Bi-2212 taken from literature, indicates that the flux creep in Tl-1223 is 3-dimensional like in Y-123. Therefore it can be explained by the 3D collective creep model for not too small currents. For Tl-2223, 2-dimensional behaviour is observed.