OSp[D+1,3 | 2] unifies the conformal symmetry and BRST

We show that the conformal transformations on the invariant metric of OSp[D,2 | 2], which is the group unifying Lorentz and BRST symmetries, are generated by OSp[D+1, 3 | 2], which thus unifies the conformal and BRST symmetries. We explicitly construct the generators for the scalar and for the spinning particle. We express the generators in terms of the energy-momentum tensor and study their action on the fields. We also study the linear realization of OSp[D+1, 3 | 2] on a space of D+4 bosonic and 2 fermionic variables. Field quantization is also examined.     

A new type of conformal dynamics

We consider the Lagrangian particle model introduced in [Ann. Phys. 260 (1997) 224] for zero mass but nonvanishing second central charge of the planar Galilei group. Extended by a magnetic vortex or a Coulomb potential the model exhibits conformal symmetry. In the former case we observe an additional SO(2,1) hidden symmetry. By either a canonical transformation with constraints or by freezing scale and special conformal transformations at t=0 we reduce the six-dimensional phase-space to the physically required four dimensions. Then we discuss bound states (bounded solutions) in quantum dynamics (classical mechanics). We show that the Schrödinger equation for the pure vortex case may be transformed into the Morse potential problem thus providing us with an explanation of the hidden SO(2,1) symmetry.     

Ramond-Neveu-Schwarz string with new boundary conditions

For the Ramond-Neveu-Schwarz string in D space-time dimensions we seek boundary conditions which preserve Poincaré invariance in d dimensions, d<D. We obtain twisted closed and twisted open strings preserving Gervais-Sakita supersymmetry. Covariant BRST quantization yields D=10. For some boundary conditions, partition functions exhibit space-time supersymmetry.     

Electric Chern–Simons term, enlarged exotic Galilei symmetry and noncommutative plane

The extended exotic planar model for a charged particle is constructed. It includes a Chern–Simons-like term for a dynamical electric field, but produces usual equations of motion for the particle in background constant uniform electric and magnetic fields. The electric Chern–Simons term is responsible for the noncommutativity of the boost generators in the 10-dimensional enlarged exotic Galilei symmetry algebra of the extended system. The model admits two reduction schemes by the integrals of motion, one of which reproduces the usual formulation for the charged particle in external constant electric and magnetic fields with associated field-deformed Galilei symmetry, whose commuting boost generators are identified with the nonlocal in time Noether charges reduced on-shell. Another reduction scheme, in which electric field transmutes into the commuting space translation generators, extracts from the model a free particle on the noncommutative plane described by the twofold centrally extended Galilei group of the nonrelativistic anyons.     

Non-relativistic conformal symmetries in fluid mechanics

The symmetries of a free incompressible fluid span the Galilei group, augmented with independent dilations of space and time. When the fluid is compressible, the symmetry is enlarged to the expanded Schrödinger group, which also involves, in addition, Schrödinger expansions. While incompressible fluid dynamics can be derived as an appropriate non-relativistic limit of a conformally invariant relativistic theory, the recently discussed conformal Galilei group, obtained by contraction from the relativistic conformal group, is not a symmetry. This is explained by the subtleties of the non-relativistic limit.     

Abelian 3-form gauge theory: Superfield approach

We discuss a D-dimensional Abelian 3-form gauge theory within the framework of Bonora-Tonin’s superfield formalism and derive the off-shell nilpotent and absolutely anticommuting Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry transformations for this theory. To pay our homage to Victor I. Ogievetsky (1928–1996), who was one of the inventors of Abelian 2-form (antisymmetric tensor) gauge field, we go a step further and discuss the above D-dimensional Abelian 3-form gauge theory within the framework of BRST formalism and establish that the existence of the (anti-)BRST invariant Curci-Ferrari (CF) type of restrictions is the hallmark of any arbitrary p-form gauge theory (discussed within the framework of BRST formalism).     

Galilean conformal and superconformal symmetries

Firstly we discuss briefly three different algebras named as nonrelativistic (NR) conformal: Schr?dinger, Galilean conformal, and infinite algebra of local NR conformal isometries. Further we shall consider in some detail Galilean conformal algebra (GCA) obtained in the limit c???? from relativistic conformal algebraO(d+1, 2) (d-number of space dimensions). Two different contraction limits providing GCA and some recently considered realizations will be briefly discussed. Finally by considering NR contraction of D = 4 superconformal algebra the Galilei conformal superalgebra (GCSA) is obtained, in the formulation using complexWeyl supercharges.     

Superstrings from Hamiltonian reduction

In any string theory there is a hidden, twisted superconformal symmetry algebra, part of which is made up by the BRST current and the anti-ghost. We investigate how this algebra can be systematically constructed for strings with N − 2 supersymmetries, via quantum Hamiltonian reduction of the Lie superalgebras osp(N|2). The motivation is to understand how one could systematically construct generalized string theories from superalgebras. We also briefly discuss the BRST algebra of the topological string, which is a doubly twisted N = 4 superconformal algebra.     

Elementary Systems of (1 + 1) Kinematical Groups: Contraction and Quantization

We present the (algebra) group contraction chain SU(1, 1) → P(1, 1) → G(1, 1), where P(1, 1) and G(1, 1) are the Poincaré and the Galilei groups, respectively, in (1 + 1) dimensions. We have paid attention to the contraction of the pseudo-extended Poincaré group to the central extended Galilei group. Objects like group laws, coadjoint orbits and representations of the contracted groups have been obtained in terms of their noncontracted counterparts. As an application we study the Moyal quantization of classical systems, having those groups as symmetry groups, by means of the contraction of the so called Stratonovich-Weyl kernels which provide such quantization.     

Structure constants for the osp(1|2) current algebra

We study the free field realization of the two-dimensional osp(1|2) current algebra. We consider the case in which the level of the affine osp(1|2) symmetry is a positive integer. Using the Coulomb gas technique we obtain integral representations for the conformal blocks of the model. In particular, from the behaviour of the four-point function, we extract the structure constants for the product of two arbitrary primary operators of the theory. From this result we derive the fusion rules of the osp(1|2) conformal field theory and we explore the connections between the osp(1|2) affine symmetry and the N = 1 superconformal field theories.     

Unified field theories with U(N) internal symmetries: G

Gauge theories for extended SU(N) conformal supergravity are constructed which are invariant under local scale, chiral, proper conformal, supersymmetry and internal SU(N) transformations. The relation between intrinsic parity and symmetry properties of their generators of the internal vector mesons is established. These theories contain no cosmological constants, but technical problems inherent to higher derivative actions are pointed out.     

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.     

Group manifold analysis of the structure of relativistic quantum dynamics

We present a group law, derived as a contraction of the conformal group, from which we obtain by using a canonical procedure a relativistic quantum system with an invariant evolution parameter (the proper time) and where the position operator belongs to the Lie algebra of the group. The restriction of the theory to the mass shell breaks part of the symmetry; of the previous 15 generators, only 10 remain which generate an action of the Poincaré group defining an orbit in the former group manifold. Some comments on the relativistic position operator are also made.     

String center of mass operator and its effect on BRST cohomology

We consider the theory of bosonic closed strings on the flat background ℝ^25,1. We show how the BRST complex can be extended to a complex where the string center of mass operator,x ₀ ^μ is well defined. We investigate the cohomology of the extended complex. We demonstrate that this cohomology has a number of interesting features. Unlike in the standard BRST cohomology, there is no doubling of physical states in the extended complex. The cohomology of the extended complex is more physical in a number of aspects related to the zero-momentum states. In particular, we show that the ghost number one zero-momentum cohomology states are in one to one correspondence with the generators of the global symmetries of the backgroundi.e., the Poincaré algebra. Supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under cooperative agreement #DF-FC02-94ER40818     

Spinor algebras

We consider supersymmetry algebras in space–times with arbitrary signature and minimal number of spinor generators. The interrelation between super Poincaré and super conformal algebras is elucidated. Minimal super conformal algebras are seen to have as bosonic part a classical semisimple algebra naturally associated to the spin group. This algebra, the Spin(s,t)-algebra, depends both on the dimension and on the signature of space–time. We also consider maximal super conformal algebras, which are classified by the orthosymplectic algebras.     

Strings in the harmonic gauge

The bosonic string is quantized in a path-integral formulation in harmonic gauge. The BRST transformations are identified and the stress tensor is constructed. The conformal anomaly is computed, and has the standard (D−26) form. The ghost number current is shown to have no anomalies, even on a curved world sheet.     

Canonical BRST quantisation of worldsheet gravities

We reformulate the BRST quantisation of chiral Virasoro and W₃ worldsheet gravities. Our approach follows directly the classic BRST formulation of Yang-Mills theory in employing a derivative gauge condition instead of the conventional conformal gauge condition, supplemented by an introduction of momenta in order to put the ghost action back into first-order form. The consequence of these simple changes is a considerable simplification of the BRST formulation, the evaluation of anomalies and the expression of Wess-Zumino consistency conditions. In particular, the transformation rules of all fields now constitute a canonical transformation generated by the BRST operator Q, and we obtain in this reformulation a new result that the anomaly in the BRST Ward identity is obtained by application of the anomalous operator Q², calculated using operator products, to the gauge fermion.