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.     

Simple superconformal field theories in two dimensions

Superconformal field theories with first-order lagrangians, the super BC-systems, are considered on an arbitrary Riemann surface with an arbitrary gravitino field. The corresponding functional integrals are carefully defined, and the Weyl, supersymmetry, and holomorphic anomalies are computed. All the anomalies have a common coefficient. The holomorphic anomaly involves terms quadratic in the gravitino field.     

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.     

Study of N = 2 superconformal field theories in 4 dimensions

Making use of the exact solutions of the N = 2 supersymmetric gauge theories, we construct new classes of superconformal field theories (SCFTs) by fine-tuning the moduli parameters and bringing the theories to critical points. SCFTs we have constructed represent universality classes of the 4-dimensional N = 2 SCFTs.     

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     

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.     

Nonlocal current algebra and N = 2 superconformal field theory in two dimensions

We present a new construction of the unitary highest weight representations of the N = 2 superconformal algebras in two dimensions. This construction is based on the non-local current in the Z_N conformal theory and a free scalar field. It provides a physical realization of all the unitary N = 2 superconformal field theories by critical systems. The correlation functions of the theories can be calculated through this construction.     

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.     

Renormalisation group flows and conserved vector currents

Irreversibility of RG flows in two dimensions is shown using conserved vector currents. Out of a conserved vector current, a quantity decreasing along the RG flow is built up such that it is stationary at fixed points where it coincides with the constant coefficient of the two-current correlation function. For Wess-Zumino-Novikov-Witten models this constant coefficient is the level of their associated affine Lie algebra. Extensions to higher dimensions using the spectral decomposition of the two-current correlation function are studied.     

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.     

Broken conformal invariance and spectrum of anomalous dimensions in QCD

Employing the operator algebra of the conformal group and the conformal Ward identities, we derive the constraints for the anomalies of dilatation and special conformal transformations of the local twist-2 operators in Quantum Chromodynamics. We calculate these anomalies in the leading order of perturbation theory in the minimal subtraction scheme. From the conformal consistency relation we derive then the off-diagonal part of the anomalous dimension matrix of the conformally covariant operators in the two-loop approximation of the coupling constant in terms of these quantities. We deduce corresponding off-diagonal parts of the Efremov-Radyushkin-Brodsky-Lepage kernels responsible for the evolution of the exclusive distribution amplitudes and non-forward parton distributions in the next-to-leading order in the flavour singlet channel for the chiral-even parity-odd and -even sectors as well as for the chiral-odd one. We also give the analytical solution of the corresponding evolution equations exploiting the conformal partial wave expansion.     

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.     

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.