Field representations of the conformal group with continuous mass spectrum

The discrete series of the conformal groupSU(2, 2) is realized on a Hilbert space of holomorphic functions over a bounded domain or the field theoretic tube domain. The boundary values of these functions form Hilbert spaces of distributions. For the realization over the tube domain the boundary distributions transform like classical spinorial fields with a continuous mass spectrum extending from zero to infinity. The reduction of these field realizations of the whole discrete series into unitary irreducible representations of the inhomogeneous Lorentz group is explicitly given.     

Leptoquarks in lepton-quark collisions

Low-energy experiments permit the existence of leptoquarks with masses of order 100 GeV and couplings to quark-lepton pairs as large as gauge couplings. We study systematically the signatures of all possible scalar and vector leptoquarks in electron (positron)-proton collisions. Clear evidence for leptoquarks would be narrow peaks in the x-distributions of inclusive neutral and charged current processes. At HERA one will be able to explore the mass range up to 300 GeV through direct production, and even somewhat beyond the CM energy of 314 GeV through virtual effects. Conversely, leptoquarks with masses of 200 GeV can be discovered for couplings as small as 10⁻³ α_em.     

Broken conformal symmetry and hadronic scaling

High energy, large momentum transfer hadronic reactions are studied in the framework of broken conformal symmetry. Hadrons lie on almost linear Regge trajectories. Inclusive cross section scale as $\text{～s}{}^{\mathrm{?2}}\text{(}\text{q}{}^2{}_{\mathrm{<mtext>T</mtext>}}\text{s)}{}^{\mathrm{?4}}$. Large rates of heavy particle production are predicted.     

Modular invariance and uniqueness of conformal characters

We show that the conformal characters of various rational models ofW-algebras can be already uniquely determined if one merely knows the central charge and the conformal dimensions. As a side result we develop several tools for studying representations of SL(2,) on spaces of modular functions. These methods, applied here only to certain rational conformal field theories, may be useful for the analysis of many others.     

The fifteen-parameter conformal group

The space of lines in a Hermitean quadric of signature (2, 2) in complex projective three-space is a quadric of signature (2, 4) in real projective five-space, the conformal compactification of Minkowski space. This geometric fact leads to the classical isomorphism ofPSU(2, 2) and the identity component ofPO(2, 4; ), the 15-parameter conformal group. In this paper it is shown how the geometry and the isomorphism, for all components ofPO(2, 4; ), arise naturally from a real form of the Clifford algebra, and its associated spin groups, of a certain complex vector space determined by skew-symmetric 4×4 matrices and having their Pfaffian as quadratic form.     

Gauging the conformal group

It is demonstrated that Kibble’s method of gauging the Poincaré group can be applied to the gauging of the conformal group. The action of the gauge transformations is the action of general spacetime diffeomorphisms (or coordinate transformations) combined with a local action of an 11-parameter subgroup of SO(4,2). Because the translational subgroup is not an invariant subgroup of the conformal group the appropriate generalisation of the derivative of a physical field is not a covariant derivative in the usual sense, but this does not lead to any inconsistencies.     

Anisotropic scaling and generalized conformal invariance at Lifshitz points

The behaviour of the 3D axial next-nearest-neighbor Ising model at the uniaxial Lifshitz point is studied using Monte Carlo techniques. A new variant of the Wolff cluster algorithm permits the analysis of systems far larger than in previous studies. The Lifshitz point critical exponents are alpha = 0.18(2), beta = 0.238(5), and gamma = 1.36(3). Data for the spin-spin correlation function are shown to be consistent with the explicit scaling function derived from the assumption of local scale invariance, which is a generalization of conformal invariance to the anisotropic scaling at the Lifshitz point.     

Asymptotic mass degeneracies in conformal field theories

By applying a method of Hardy and Ramanujan to characters of rational conformal field theories, we find an asymptotic expansion for degeneracy of states in the limit of large mass which isexact for strings propagating in more than two uncompactified space-time dimensions. Moreover we explore how the rationality of the conformal theory is reflected in the degeneracy of states. We also consider the one loop partition function for strings, restricted to physical states, for arbitrary (irrational) conformal theories, and obtain an asymptotic expansion for it in the limit that the torus degenerates. This expansion depends only on the spectrum of (physical and unphysical) relevant operators in the theory. We see how rationality is consistent with the smoothness of mass degeneracies as a function of moduli.     

Representing functionally the two-dimensional conformal group

Projective representations of the two-dimensional conformal group are explicitly constructed in terms of propagation kernels. The representation functionals are then used to study the effect of conformal transformations on the states of a free massless scalar field theory.     

Space-times admitting a three-dimensional conformal group

Perfect fluid space-times admitting a three-dimensional Lie group of conformal motions containing a two-dimensional Abelian Lie subgroup of isometries are studied. Demanding that the conformal Killing vector be proper (i.e., not homothetic nor Killing), all such space-times are classified according to the structure of their corresponding three-dimensional conformal Lie group and the nature of their corresponding orbits (that are assumed to be non-null). Each metric is then explicitly displayed in coordinates adapted to the symmetry vectors. Attention is then restricted to the diagonal case, and exact perfect fluid solutions are obtained in both the cases in which the fluid four-velocity is tangential or orthogonal to the conformal orbits, as well as in the more general tilting case.     

Finite-size scaling and the renormalization group

Renormalization group calculations ind = 4 andd = 4 – are performed for a system of finite size. A form of mean-field theory is used which yields a rounded transition for a finite system, and this allows a sensible expansion in fluctuations. A combination of Ewald and Poisson sum techniques is used to produce explicit numerical results for the specific heat ind = 4 which, with the setting of two nonuniversal metrical factors and the fourth-order coupling constant may be compared with simulations. The numerical visibility of logarithmic corrections is investigated. The universal scaling function for the specific heat to relativeO() is also evaluated numerically.     

Functional representation for the two-dimensional conformal group

A functional projective representation for the two-dimensional conformal group is explicitly constructed. The representation functionals are propagation kernels for self-dual fields.     

Modular invariance and the spectrum of two dimensional conformal field theories

Modular invariance has recently emerged as a powerful tool in conformal field theory. In conjunction with the representation theory of infinite dimensional Lie algebras, the study of modular invariance gave the spectrum of several families of theories. These include the minimal conformal models (Cardy and others), WZW theories which describe string propagation on group manifolds (Gepner and Witten) and parafermionic field theories (Gepner and Qiu). The minimal conformal models models were shown to be a product of two SU(2) WZW theories (Gepner). These results represent a step towards a complete classification of conformal field theories, an important goal both for the study of critical phenomena and string theory.     

Dilogarithm identities in conformal field theory and group homology

Recently, Rogers' dilogarithm identities have attracted much attention in the setting of conformal field theory as well as lattice model calculations. One of the connecting threads is an identity of Richmond-Szekeres that appeared in the computation of central charges in conformal field theory. We show that the Richmond-Szekeres identity and its extension by Kirillov-Reshetikhin (equivalent to an identity found earlier by Lewin) can be interpreted as a lift of a generator of the third integral homology of a finite cyclic subgroup sitting inside the projective special linear group of all 2×2 real matrices viewed as adiscrete group. This connection allows us to clarify a few of the assertions and conjectures stated in the work of Nahm-Recknagel-Terhoven concerning the role of algebraic K-theory and Thurston's program on hyperbolic 3-manifolds. Specifically, it is not related to hyperbolic 3-manifolds as suggested but is more appropriately related to the group manifold of the universal covering group of the projective special linear group of all 2×2 real matrices viewed as a topological group. This also resolves the weaker version of the conjecture as formulated by Kirillov. We end with a summary of a number of open conjectures on the mathematical side.To Professor C. N. Yang for his 70^th birthdayThis work was partially supported by grants from the Statens Naturvidenskabelige Forskningsraad and the Paul and Gabriella Rosenbaum Foundation     

Renormalization group operators for maps and universal scaling of universal scaling exponents

Renormalization group (RG) methods provide a unifying framework for understanding critical behaviour, such as transition to chaos in area-preserving maps and other dynamical systems, which have associated with them universal scaling exponents. Recently, de la Llave et al. (2007) [10] have formulated the Principle of Approximate Combination of Scaling Exponents (PACSE for short), which relates exponents for different criticalities via their combinatorial properties. The main objective of this paper is to show that certain integrable fixed points of RG operators for area-preserving maps do indeed follow the PACSE.     

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.     

Phenomenology of local scale invariance: from conformal invariance to dynamical scaling

Statistical systems displaying a strongly anisotropic or dynamical scaling behaviour are characterized by an anisotropy exponent θ or a dynamical exponent z. For a given value of θ (or z), we construct local scale transformations, which can be viewed as scale transformations with a space–time-dependent dilatation factor. Two distinct types of local scale transformations are found. The first type may describe strongly anisotropic scaling of static systems with a given value of θ, whereas the second type may describe dynamical scaling with a dynamical exponent z. Local scale transformations act as a dynamical symmetry group of certain non-local free-field theories. Known special cases of local scale invariance are conformal invariance for θ=1 and Schrödinger invariance for θ=2.The hypothesis of local scale invariance implies that two-point functions of quasiprimary operators satisfy certain linear fractional differential equations, which are constructed from commuting fractional derivatives. The explicit solution of these yields exact expressions for two-point correlators at equilibrium and for two-point response functions out of equilibrium. A particularly simple and general form is found for the two-time autoresponse function. These predictions are explicitly confirmed at the uniaxial Lifshitz points in the ANNNI and ANNNS models and in the aging behaviour of simple ferromagnets such as the kinetic Glauber–Ising model and the kinetic spherical model with a non-conserved order parameter undergoing either phase-ordering kinetics or non-equilibrium critical dynamics.