The Thirring-model current group and its representations

The Gel'fand-Araki method is used to construct universally covariant representations of the Thirring-model current group. In this representation the currents are recovered as infinitesimal generators of the corresponding one-parametric subgroups. The determination of the generators of the two-dimensional Poincaré group is discussed and the existence of a selfadjoint Hamiltonian is shown. The possibility of determining the charges and their connection with the quantities defining the representation of the Thirring-model current group (algebra) is considered.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 57–62, November, 1975.     

Second order Hamiltonian formalism and constraints algebra for non-polynomial supergravities

The second order Hamiltonian formalism for a non-polynomial N = 1D = 10 supergravity coupled to super Yang-Mills theory is developed. This is done by starting from the first order canoncial covariant formalism on group manifold. The Hamiltonian, generator of time evolution, is found as a functional of the first class constraints of this coupled system. These contraints close the constraint algebra and they are the generators of all the Hamiltonian gauge symmetries.     

Gauge invariant formulation of non-Abelian chiral gauge theories in two dimensions

We study the gauge invariant version of a chiral non-Abelian gauge theory. A local bosonic effective action is obtained and the covariant conservation of the gauge current is verified. A Hamiltonian analysis of the model and of its constraints is performed. We show that the constraints are first class and that no anomalous term appears in the commutators of the gauge group generators. The current algebra of the model is obtained and the gauge fixing is analyzed.     

Thermostatistics of the multi-dimensional q-deformed fermionic Newton oscillators

The algebraic and representative properties of the multi-dimensional q-deformed fermionic Newton oscillator algebra are discussed. This algebra is covariant under the undeformed group U(n). The high- and low-temperature thermostatistical properties of a gas of the multi-dimensional q-deformed fermionic Newton oscillators are obtained.     

Yang–Mills Algebra

Some unexpected properties of the cubic algebra generated by the covariant derivatives of a generic Yang–Mills connection over the (s+1)-dimensional pseudo Euclidean space are pointed out. This algebra is Koszul of global dimension 3 and Gorenstein but except for s=1 (i.e. in the two-dimensional case) where it is the universal enveloping algebra of the Heisenberg Lie algebra and is a cubic Artin–Schelter regular algebra, it fails to be regular in that it has exponential growth. We give an explicit formula for the Poincaré series of this algebra and for the dimension in degree n of the graded Lie algebra of which is the universal enveloping algebra. In the four-dimensional (i.e. s=3) Euclidean case, a quotient of this algebra is the quadratic algebra generated by the covariant derivatives of a generic (anti) self-dual connection. This latter algebra is Koszul of global dimension 2 but is not Gorenstein and has exponential growth. It is the universal enveloping algebra of the graded Lie algebra which is the semi-direct product of the free Lie algebra with three generators of degree one by a derivation of degree one.     

Properties of the covariant formulation of superstring theories

The covariant two-dimensional action principle that describes the dynamics of free superstrings in a Minkowski background is reviewed. Covariant gauge conditions are formulated, which simplify the equations of motion of the superspace coordinates to free equations. In this gauge there are bosonic and fermionic constraints whose generators give a supersymmetric generalization of the Virasoro algebra. As in certain supersymmetric field theories, closure of the algebra requires using the equations of motion. Covariant constrained bracket relations are obtained for the classical theory, but it is very difficult to extend them to quantum mechanical commutation relations. Interaction vertices satisfying supersymmetry and the necessary gauge conditions are constructed. They reduce in a special frame to ones found in earlier work in the light-cone gauge, and then can be interpreted quantum mechanically.     

Quantum Minkowski Space and Generators of Quantum Lorentz Group

From the basic 4 × 4 R matrix associated with the quantum Lorentz group SL_q(2, C) and its various fusion matrices, the covariant differential calculus on the quantum Minkowski space and the R commutation relation for the covariant generators of quantum Lorents group are presented.     

Fractional Super-Multi-Virasoro Algebra

An n-dimensional fractional supersymmetry theory is algebraically constructedon the n-dimensional multicomplex space M _n. By emphasizing its appearanceas a special case of generalized Clifford algebra (GCA), we formulate the fractional superspace FM _n through a generalized Grassmann algebra (GGA) and constructthe generators and the covariant derivative of FSUSY on FM _n . The generatorsof FSUSY are extended to get n copies of the fractional centerlesssuper-Virasoro algebra.     

Operator coproduct-realization of quantum group transformations in two-dimensional gravity I

A simple connection between the universalR matrix ofU _q(sl(2)) (for spins 1/2 andJ) and the required form of the coproduct action of the Hilbert space generators of the quantum group symmetry is put forward. This leads us to an explicit operator realization of the coproduct action on the covariant operators. It allows us to derive the expected quantum group covariance of the fusion and braiding matrices, although it is of a new type: the generators depend upon worldsheet variables, and obey a new central extension of theU _q(sl(2)) algebra realized by (what we call) fixed point commutation relations. This is explained by showing on a general ground that the link between the algebra of field transformations and that of the coproduct generators is much weaker than previously thought. The central charges of our extendedU _q(sl(2)) algebra, which includes the Liouville zero-mode momentum in a non-trivial way, are related to Virasoro-descendants of unity. We also show how our approach can be used to derive the Hopf algebra structure of the extended quantum-group symmetry related to the presence of both of the screening charges of 2D gravity.Partially supported by the EC contracts CHRXCT920069 and CHRXCT920035.Unité Propre du Centre National de la Recherche Scientifique, associée à l'École Normale Supérieure et à l'Université de Paris-Sud.     

Wk structure of A SU(2)-level k from covariant construction of Kac-Moody algebras

The W_k structure underlying the transverse realization of affine SU(2) at level k is analyzed. The extension of the equivalence existing between the covariant and light-cone gauge realization of an affine Kac-Moody algebra to W_k algebras is given. Higher spin generators are extracted by the less singular terms in the operator product expansion of the parafermions constructed by means of the projection of the covariant on the light-cone gauge. These fields can be written in terms of only one free boson compactified on a circle.     

Quantum deformation of space-time symmetries and interactions

We discuss quantum deformations of Lie algebra as described by the noncoassociative modification of its coalgebra structure. We consider for simplicity the quantum D = 1 Galilei algebra with four generators: energy H, boost B, momentum P and central generator M (mass generator). We describe the nonprimitive coproducts for H and B and show that their noncocommutative and noncoassociative structure is determined by the two-body interaction terms. Further we consider the case of physical Galilei symmetry in three dimensions. Finally we discuss the noninteraction theorem for manifestly covariant two-body systems in the framework of quantum deformations of D = 4 Poincaré algebra and a possible way out.     

Center for Elliptic Quantum Group Eτ,η(sln)

We give the center of the elliptic quantum group in general cases. Based on the dynamical Yang-Baxter relation and the fusion method, we prove that the center commutes with all generators of the elliptic quantum group. Then for a kind of assumed form of these generators, we find that the coefficients of these generators form a new type of closed algebra. We also give the center for the algebra.     

Combinatorial Space from Loop Quantum Gravity

The canonical quantization of diffeomorphism invariant theories of connections in terms of loop variables is revisited. Such theories include general relativity described in terms of Ashtekar-Barbero variables and extension to Yang-Mills fields (with or without fermions) coupled to gravity. It is argued that the operators induced by classical diffeomorphism invariant or covariant functions are respectively invariant or covariant under a suitable completion of the diffeomorphism group. The canonical quantization in terms of loop variables described here, yields a representation of the algebra of observables in a separable Hilbert space. Furthermore, the resulting quantum theory is equivalent to a model for diffeomorphism invariant gauge theories which replaces space with a manifestly combinatorial object.     

Hidden quantum groups inside Kac-Moody algebra

A lattice analogue of the Kac-Moody algebra is constructed. It is shown that the generators of the quantum algebra with the deformation parameterq=exp(iπ/k+h) can be constructed in terms of generators of the lattice Kac-Moody algebra (LKM) with the central chargek. It appears that there exists a natural correspondence between representations of the LKM algebra and the finite dimensional quantum group. The tensor product for representations of the LKM algebra and the finite dimensional quantum algebra is suggested.     

Position operators as integrals with respect to Euclidean systems of covariance

Position operators (p.o.) for relativistic elementary quantum systems are constructed as operator-valued integrals with respect to Euclidean systems of covariance (ESC), i.e., positive operator-valued (POV) measures being covariant under the Euclidean group, and are expressed in terms of the generators of the Poincaré transformations. These p.o. are partly well-known in the literature where they are found by other methods.     

The algebraic structure of the fermionic string

The local structure of the product expansion algebra of the covariant NSR string is analyzed. An “on-shell” Kac-Moody like algebra is found to generate the BRST invariant part of the covariant lattice Γ_5,1. This algebra is a local version of ten-dimensional SUSY.     

Constrained systems,OSP(1,1|2) and string theory

The underlying OSP invariance of the Fradkin-Vilkovisky formalism is discussed. Ghost degrees of freedom are interpreted as negative dimensional phase space variables that eliminate unphysical degrees of freedom by the Parisi-Sourlas mechanism, ensuring manifest covariance. The formalism makes use of subsidiary constraints, extending the usual algebra of constraints. A relations between abelian and nonabelian constraint algebras is established, and exploited to construct a nonabelian representation of the OSP generators. For theories based entirely on constraints such as string theories, the natural Fradkin-Vilkovisky hamiltonian is a manifestly OSP invariant squared length of a graded phase space vector. As an application, the OSP covariant formulation of bosonic strings is discussed.     

Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics

Given the local observables in the vacuum sector fulfilling a few basic principles of local quantum theory, we show that the superselection structure, intrinsically determined a priori, can always be described by a unique compact global gauge group acting on a field algebra generated by field operators which commute or anticommute at spacelike separations. The field algebra and the gauge group are constructed simultaneously from the local observables. There will be sectors obeying parastatistics, an intrinsic notion derived from the observables, if and only if the gauge group is non-Abelian. Topological charges would manifest themselves in field operators associated with spacelike cones but not localizable in bounded regions of Minkowski space. No assumption on the particle spectrum or even on the covariance of the theory is made. However the notion of superselection sector is tailored to theories without massless particles. When translation or Poincaré covariance of the vacuum sector is assumed, our construction leads to a covariant field algebra describing all covariant sectors.Research supported by Ministero della Pubblica Istruzione and CNR-GNAFA     

Trefoil Symmetry IV: Basic Enhanced Superspace for the Minimal Vector Clover Extension

We construct a differential representation and covariant derivatives of the minimal vector clover extension of the Poincaré algebra. In analogous way as in the supersymmetric case, there arises an enhanced superspace which allows to define superfields. The action of group transformations on such superfields determines a representation out of which the covariant derivatives are obtained.