Two-Parameter Deformation of the Oscillator Algebra and (p,q)–Analog of Two-Dimensional Conformal Field Theory

Abstract

The two-parameter deformation of canonical commutation relations is discussed. The self-adjointness property of the (p, q)-deformed position Q and momentum P operators is investigated. The (p, q)-analog of two-dimensional conformal field theory based on the (p, q)-deformation of the su(1, 1) subalgebra of the Virasoro algebra is presented.     

Multi-Interval Subfactors and Modularity¶of Representations in Conformal Field Theory

We describe the structure of the inclusions of factors ?(E)⊂?(E′)′ associated with multi-intervals E⊂ℝ for a local irreducible net ? of von Neumann algebras on the real line satisfying the split property and Haag duality. In particular, if the net is conformal and the subfactor has finite index, the inclusion associated with two separated intervals is isomorphic to the Longo–Rehren inclusion, which provides a quantum double construction of the tensor category of superselection sectors of ?. As a consequence, the index of ?(E)⊂?(E′)′ coincides with the global index associated with all irreducible sectors, the braiding symmetry associated with all sectors is non-degenerate, namely the representations of ? form a modular tensor category, and every sector is a direct sum of sectors with finite dimension. The superselection structure is generated by local data. The same results hold true if conformal invariance is replaced by strong additivity and there exists a modular PCT symmetry. Received: 7 July 1999 / Accepted: 13 January 2001     

Teichmüller Groupoids,and Monodromy in Conformal Field Theory

We study monodromy representations of the Teichmüller groupoid for the moduli space of pointed compact Riemann surfaces of any genus with first-order infinitesimal structure. To calculate these representations, using arithmetic Schottky-Mumford uniformization theory we construct a real orbifold in the moduli space consisting of fusing and simple moves which gives tangential base points. For a certain vector bundle on the moduli space with projectively flat connection, we show that the monodromy of each fusing move can be expressed as a connection matrix, and give the relations to the monodromy of simple moves. Furthermore, we describe the monodromy representation associated with Tsuchiya-Ueno-Yamadas conformal field theory, and show that this representation can be expressed as the monodromy of the Wess-Zumino-Witten model.     

Spectral Determinants for Schrödinger Equation and Q-Operators of Conformal Field Theory

Relation between the vacuum eigenvalues of CFT Q-operators and spectral determinants of one-dimensional Schrödinger operator with homogeneous potential, recently conjectured by Dorey and Tateo for special value of Virasoro vacuum parameter p, is proven to hold, with suitable modification of the Schrödinger operator, for all values of p.     

On gl（2｜2）（2）k Current Superalgebra and Twisted Conformal Field Theory

Motivated by the recently discovered hidden symmetry of the type IIB Green-Schwarz superstring on certain background, the non-semisimple Kac-Moody twisted superalgebra gl（2｜2）（2）k is investigated by means of the vector coherent state method and boson-fermion realization. The free field realization of the twisted current superalgebra at general level k is constructed. The corresponding Conformai Field Theory （CFT） has zero central charge. According to the classification theory, this CFT is a nonunitary field theory. After projecting out a U（1） factor and an outer automorphism operator, we get the free field representation of psl（2｜2）（2）k, which is the a/gebra of gl（2｜2）（2）k modulo the Z4-outer automorphism, the CFT has central charge -2.     

A New Holographic Entropy Bound from Conformal Field Theory at the Killing Horizon

A new holographic entropy bound is obtained by using conformal field theory at the Killing horizon.The entropy bound is tighter than the well-known bounds,such as the Bekenstein,Bekenstein-Mayo and‘t Hooft bounds.The result shows that the entropy of a system decreases when quantum effects are included.Therefore,the quantum effect will increase the degree of order of the system.     

Absence of Subsystems for the Haag–Kastler Net Generated by the Energy-Momentum Tensor in Two-Dimensional Conformal Field Theory

We show that if A is the Haag–Kastler net generated by the energy-momentum tensor in a chiral quantum field theory, then every subsystem B A which is covariant under the action of SL(2,R given on A must coincide with A. The result is valid for all the allowed values of the central charge and is obtained using scaling limit techniques.     

Vacuum Stability of the Wrong Sign (−ϕ 6) Scalar Field Theory

We apply the effective potential method to study the vacuum stability of the bounded from above (?? ⁶) (unstable) quantum field potential. The stability (?E/?b = 0) and the mass renormalization (? ² E/?b ² = M ²) conditions force the effective potential of this theory to be bounded from below (stable). Since bounded from below potentials are always associated with localized wave functions, the algorithm we use replaces the boundary condition applied to the wave functions in the complex contour method by two stability conditions on the effective potential obtained. To test the validity of our calculations, we show that our variational predictions can reproduce exactly the results in the literature for the \(\mathcal {PT}\) -symmetric ? ⁴ theory. We then extend the applications of the algorithm to the unstudied stability problem of the bounded from above (?? ⁶) scalar field theory where classical analysis prohibits the existence of a stable spectrum. Concerning this, we calculated the effective potential up to first order in the couplings in d space-time dimensions. We find that a Hermitian effective theory is instable while a non-Hermitian but \(\mathcal {PT}\) -symmetric effective theory characterized by a pure imaginary vacuum condensate is stable (bounded from below) which is against the classical predictions of the instability of the theory. We assert that the work presented here represents the first calculations that advocates the stability of the (?? ⁶) scalar potential.     

Coupling of Two Conformal Field Theories and Nakajima–Yoshioka Blow-Up Equations

We study the conformal vertex algebras which naturally arise in relation to the Nakajima–Yoshioka blow-up equations.     

On Dynamics of Brans–Dicke Theory of Gravitation

We study longstanding problem of cosmological clock in the context of Brans–Dicke theory of gravitation. We present the Hamiltonian formulation of the theory for a class of spatially homogeneous cosmological models. Then, we show that formulation of the Brans–Dicke theory in the Einstein frame allows how an identification of an appropriate cosmological time variable, as a function of the scalar field in the theory, can be emerged in quantum cosmology. The classical and quantum results are applied to the Friedmann–Robertson–Walker cosmological models.     

An Assessment of Evans’ Unified Field Theory I

Evans developed a classical unified field theory of gravitation and electromagnetism on the background of a spacetime obeying a Riemann-Cartan geometry. This geometry can be characterized by an orthonormal coframe ϑ ^α and a (metric compatible) Lorentz connection Γ ^{α β} . These two potentials yield the field strengths torsion T ^α and curvature R ^{α β} . Evans tried to infuse electromagnetic properties into this geometrical framework by putting the coframe ϑ ^α to be proportional to four extended electromagnetic potentials ; these are assumed to encompass the conventional Maxwellian potential A in a suitable limit. The viable Einstein-Cartan (-Sciama-Kibble) theory of gravity was adopted by Evans to describe the gravitational sector of his theory. Including also the results of an accompanying paper by Obukhov and the author, we show that Evans’ ansatz for electromagnetism is untenable beyond repair both from a geometrical as well as from a physical point of view. As a consequence, his unified theory is obsolete.     

The Complete Solution of the Classical SL(2,ℝ/U(1) Gauged WZNW Field Theory

We prove that any gauged WZNW model has a Lax pair representation, and give explicitly the general solution of the classical equations of motion of the SL(2,ℝ/U(1) theory. We calculate the symplectic structure of this solution by solving a differential equation of the Gelfand–Dikii type with initial state conditions at infinity, and transform the canonical physical fields non-locally onto canonical free fields. The results will, finally, be collected in a local B?cklund transformation. These calculations prepare the theory for an exact canonical quantization. Received: 9 June 1998 / Accepted: 7 March 1999     

On the Poincaré Gauge Theory of Gravitation

We present a compact, self-contained review of the conventional gauge theoretical approach to gravitation based on the local Poincaré group of symmetry transformations. The covariant field equations, Bianchi identities and conservation laws for angular momentum and energy-momentum are obtained.     

An Assessment of Evans’ Unified Field Theory II

Evans attempted to develop a classical unified field theory of gravitation and electromagnetism on the background of a spacetime obeying a Riemann-Cartan geometry. In an accompanying paper I, we analyzed this theory and summarized it in nine equations. We now propose a variational principle for a theory that implements some of the ideas that have been (imprecisely) indicated by Evans and show that it yields two field equations. The second field equation is algebraic in the torsion and we can resolve it with respect to the torsion. It turns out that for all physical cases the torsion vanishes and the first field equation, together with Evans’ unified field theory, collapses to an ordinary Einstein equation.     

Classical Electromagnetic Field Theory in the Presence of Magnetic Sources

Using two new well-defined four-dimensional potential vectors,we formulate the classical Maxwell field theory in a form which has manifest Lorentz covariance and SO(2) duality symmetry in the presence of magnetic sources.We set up a consistent Lagrangian for the theory.Then from the action principle we obtain both Maxwell‘s equation and the equation of motion of a dyon moving in the electromagnetic field.     

Collective-Coordinate Analysis of Inhomogeneous Nonlinear Klein–Gordon Field Theory

Two different sets of collective-coordinate equations for solitary solutions of nonlinear Klein–Gordon (NKG) model are introduced. The collective-coordinate equations are derived using different approaches for adding the inhomogeneities as external potentials to the soliton equation of motion. The interaction of the NKG field with a local inhomogeneity like a delta function potential wall or a delta-function potential well is investigated using the presented collective-coordinate equations, and the results of the two different models are compared. Most of the characters of the interaction are derived analytically. The analytical results are also compared to the results of numerical simulations.