The minimal superconformal (p,0) supercurrents in two dimensions

Superconformal transformations in rigid (p, q) superspace are defined. For the (p, 0) case it is shown how the general superspace Noethercurrent for any given model can be reduced to a minimal supercurrent. This generalization of the energy-momentum tensor consists of two (p, 0) superfields, and the conservation equations show that no auxiliary currents are present in their components.     

The superconformal algebra in higher dimensions

The superconformal algebra for 4/4N-dimensional super-Minkowski space (d=4) can be identified with the simple superalgebra su (2,2/N). For even-dimension d=5,6 the superconformal algebra can be identified with a real form of the simple superalgebras F(4), D(4,1) respectively in Kac's classification. For even-dimension d>-7 it is impossible to define a superconformal algebra satisfying three natural conditions: (1) it acts as infinitesimal automorphisms on super-Minkowski space; (2) this action extends the natural action of the super-Poincaré algebra; (3) when the action of the even part of the superconformal algebra is reduced to an infinitesimal action on ordinary Minkowski space, it extends the natural action of the conformal algebra so (2, d).     

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.     

Correlation functions of conserved currents in four dimensional conformal field theory

We derive a generating function for all the 3-point functions of higher spin conserved currents in four dimensional conformal field theory. The resulting expressions have a rather surprising factorized form which suggests that they can all be realized by currents built from free massless fields of arbitrary (half-)integer spin s  . This property is however not necessarily true also for the higher-point functions. As an illustration we analyze the general 4-point function of conserved abelian U(1)$U(1)$ currents of scale dimension equal to three and find that apart from the two free field realizations there is a unique possible function which may correspond to an interacting theory. Although this function passes several non-trivial consistency tests, it remains an open challenging problem whether it can be actually realized in an interacting CFT.     

Gauge theory of the conformal and superconformal group

We show that the locally scale invariant Weyl theory of gravity is the gauge theory of the conformal group. Proper conformal transformations are gauged by a non-propagating gauge field.A gauge theory for the superconformal group is obtained which is locally scale, Lorentz, and chiral invariant but not locally supersymmetric despite remarkable cancellations.     

The conserved currents for the Maxwellian field

We classify the conserved currents for the Maxwellian field. There are four families: (1) the classical currents derived using Noether's theorem from conformal invariance (2) certain Noetherian currents based on translations in field space, (3,4) two more kinds not equivalent to any Noetherian form.     

Persistent currents in mesoscopic rings and conformal invariance

The effect of point defects on persistent currents in mesoscopic rings is studied in a simple tight-binding model. Using an analogy with the treatment of the critical quantum Ising chain with defects, conformal invariance techniques are employed to relate the persistent current amplitude to the Hamiltonian spectrum just above the Fermi energy. From this, the dependence of the current on the magnetic flux is found exactly for a ring with one or two point defects. The effect of an aperiodic modulation of the ring, generated through a binary substitution sequence, on the persistent current is also studied. The flux-dependence of the current is found to vary remarkably between the Fibonacci and the Thue-Morse sequences. Received: 4 March 1998 / Revised: 20 April 1998 / Accepted: 30 April 1998     

Structure of conformal supergravities in two dimensions

We examine constraints on curvatures inN=1 andN=2 conformal supergravities in two dimensions. We show that all curvatures should vanish in order that, the whole conformal supergravity algebra closes on all gauge fields. On the other hand we find some closed sub-algebra of the conformal supergravity one.     

Canonical dimensions in asymptotically conformal invariant theories

It is shown that canonical dimensions in the product of currents near the light-cone imply a free field structure for the Wightman functions of currents in the underlying skeleton theory.     

Identically conserved currents and their charges

We examine, at a classical level, the features that distinguish dynamically conserved currents from identically conserved ones. Then we prove that, in four spacetime dimensions, the quantum charges corresponding to identically conserved currents always annihilate the vacuum state. We illustrate our discussion with several examples.     

The conformal anomaly and the neutral currents sector of the Standard Model

We elaborate on the structure of the graviton–gauge–gauge vertex in the electroweak theory, obtained by the insertion of the complete energy–momentum tensor (T  ) on 2-point functions of neutral gauge currents (VV^′$V{V}^{\prime }$). The vertex defines the leading contribution to the effective action which accounts for the conformal anomaly and related interaction between the Standard Model and gravity. The energy–momentum tensor is derived from the curved spacetime Lagrangian in the linearized gravitational limit, and with the inclusion of the term of improvement of a conformally coupled Higgs sector. As in the previous cases of QED and QCD, we find that the conformal anomaly induces an effective massless scalar interaction between gravity and the neutral currents in each gauge invariant component of the vertex. This is described by the exchange of an anomaly pole. We show that for a spontaneously broken theory the anomaly can be entirely attributed to the poles only for a conformally coupled Higgs scalar. In the exchange of a graviton, the trace part of the corresponding interaction can be interpreted as due to an effective dilaton, using a local version of the effective action. We discuss the implications of the anomalous Ward identity for the TVV^′$TV{V}^{\prime }$ correlator for the structure of the gauge/gauge/effective dilaton vertex in the effective action. The analogy between these effective interactions and those related to the radion in theories with large extra dimensions is pointed out.     

The conformal bootstrap equations for the four fermi interaction in arbitrary dimensions

We derive the conformal bootstrap equations for theO(N) Gross Neveu model in arbitrary dimensions and solve them at \(\mathcal{O}(1/N^2 )\) to obtain the fermion anomalous dimension.     

Model-independent improvement of conformal and Poincaré Noether currents

It is shown, in a model independent way, how the most general conformal or Poincaré Noether current can be improved applying the shift technique.     

Infinite conformal symmetry of critical fluctuations in two dimensions

We study the massless quantum field theories describing the critical points in two dimensional statistical systems. These theories are invariant with respect to the infinite dimensional group of conformal (analytic) transformations. It is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of the Virasoro algebra. Exactly solvable theories associated with degenerate representations are analized. In these theories the anomalous dimensions are known exactly and the correlation functions satisfy the system of linear differential equations.Professor A. B. Zamolodchikov was unable to attend the conference to present this invited paper personally.     

Models of G time variations in diverse dimensions

A review of different cosmological models in diverse dimensions leading to a relatively small time variation in the effective gravitational constant G is presented. Among them: the 4-dimensional (4-D) general scalar-tensor model, the multidimensional vacuum model with two curved Einstein spaces, the multidimensional model with the multicomponent anisotropic “perfect fluid”, the S-brane model with scalar fields and two form fields, etc. It is shown that there exist different possible ways of explaining relatively small time variations of the effective gravitational constant G compatible with present cosmological data (e.g. acceleration): 4-dimensional scalar-tensor theories or multidimensional cosmological models with different matter sources. The experimental bounds on ? may be satisfied either in some restricted interval or for all allowed values of the synchronous time variable.     

On local one-particle approximations and locally conserved currents

Local one-particle approximations are constructed for matrix elements of two local field operators. If one of the fields is a locally conserved current the approximation is extended in such a way that both locality and current conservation are valid in the approximation.