L. O'Raifeartaigh

Spontaneous Symmetry Breaking for Chiral Scalar Superfields

Abstract The spontaneous breakdown of supersymmetry and internal symmetry is studied for the interaction of N chiral scalar superfields. A lemma is established which shows that the spontaneous breakdown of supersymmetry is a very rare occurrence, but that it can happen (in at least the tree approximation) is shown by the construction of a model. The model contains three superfields ( N = 3) and preserves parity. In contrast to supersymmetry itself, a spontaneous breakdown of internal symmetry triggered by the supersymmetry is found to be a frequent occurrence, and a general formalism for it is set up. A mass-formula is derived and examples given.

Toda Theory and W-Algebra from a Gauged WZNW Point of View

A new formulation of Toda theories is proposed by showing that they can be regarded as certain gauged Wess-Zumino-Novikov-Witten (WZNW) models. It is argued that the WZNW variables are the proper ones for Toda theory, since all the physically permitted Toda solutions are regular when expressed in these variables. A detailed study of classical Toda theories and their W-algebras is carried out from this unified WZNW point of view. We construct a primary field basis for the w-algebra for any group, we obtain a new method for calculating the W-algebra and its action on the Toda fields by constructing its Kac-Moody implementation, and we analyse the relationship between w-algebras and Casimir algebras. The w-algebra of G2 and the Casimir algebras for the classical groups are exhibited explicitly.

On Hamiltonian reductions of the Wess-Zumino-Novikov-Witten theories

The structure of Hamiltonian symmetry reductions of the Wess-Zumino-Novikov-Witten (WZNW) theories by first class Kac-Moody (KM) constraints is analysed in detail. Lie algebraic conditions are given for ensuring the presence of exact integrability, conformal invariance and W-symmetry in the reduced theories. A Lagrangean, gauged WZNW implementation of the reduction is established in the general case and thereby the path integral as well as the BRST formalism are set up for studying the quantum version of the reduction. The general results are applied to a number of examples. In particular, a W-algebra is associated to each embedding of sl(2) into the simple Lie algebras by using purely first class constraints. The primary fields of these W-algebras are manifestly given by the sl(2) embeddings, but it is also shown that there is an sl(2) embedding present in every polynomial and primary KM reduction and that the Wn l-algebras have a hidden sl(2) structure too. New generalized Toda theories are found whose chiral algebras are the W-algebras based on the half-integral sl(2) embeddings, and the W-symmetry of the effective action of those generalized Toda theories associated with the integral gradings is exhibited explicitly.

Liouville and Toda theories as conformally reduced WZNW theories

It is shown that Liouville theory can be regarded as an SL(2, o) Wess-Zumino-Novikov-Witten theory with conformal invariant constraints and that Polyakovs SL(2, o) Kac-Moody symmetry of induced two-dimensional gravity is just one side of the WZNW current algebra. Analogously, Toda field theories can be regarded as conformal-invariantly constrained WZNW theories for appropriate (maximally non-compact) groups.

Change of variables and equivalence theorems in quantum field theories

Abstract In a recent paper, Chisholm has proved the important theorem that the S-matrix in quantum field theories remains unchanged under any point transformation of field operators. In view of some unsatisfactory features in his argument, another proof of this theorem, within the framework of the conventional canonical formalism of field theories, is given. Further, it is pointed out that on the basis of Chisholms theorem, most of the ordinary equivalence theorems in field theories can be obtained by rather trivial changes of variables in the Lagrangians. The equivalence theorem for the Yang-Mills field is discussed in detail.

Consistency of string propagation on curved spacetimes. An SU(1, 1) based counterexample

String propagation on non-compact group manifolds is studied as an exactly solvable example of propagation on more general curved spacetimes. It is shown that for the only viable group SU(1, 1) × Gc string propagation is consistent classically but not quantum mechanically (unitarity is violated). This shows that conformal invariance of the corresponding σ-model (vanishing of the β-functions) is not sufficient to guarantee unitarity.

The Constraint Effective Potential

Because of the non-perturbative nature of the conventional effective potential Γ(Ω, ϕ) (for classical Higgs potentials and volume Ω) and because of the inconvenience of a Legendre transform for numerical computations, it is proposed to replace Γ(Ω, ϕ) by a “constraint” effective potential U(Ω, ϕ) , which has a direct intuitive meaning, which is very convenient for lattice computations, and from which Γ(Ω, ϕ) can immediately be recovered (as the convex hull). In particular, Γ(∞, ϕ) = U(∞, ϕ). Various properties of U(Ω, ϕ) , such as convexity properties, upper and lower bounds and volume dependence are established. It is computed directly for zero dimensions and by Monte Carlo simulations in one and four dimensions, with up to 160 and 8 4 lattice sites, respectively.

WEYL GAUGING AND CONFORMAL INVARIANCE

Abstract Scale-invariant actions in arbitrary dimensions are investigated in curved space to clarify the relation between scale, Weyl and conformal invariance on the classical level. The global Weyl group is gauged. Then the class of actions is determined for which Weyl gauging may be replaced by a suitable coupling to the curvature (Ricci gauging). It is shown that this class is exactly the class of actions which are conformally invariant in flat space. The procedure yields a simple algebraic criterion for conformal invariance and produces the improved energy-momentum tensor in conformally invariant theories in a systematic way. It also provides a simple and fundamental connection between Weyl anomalies and central extensions in two dimensions. In particular, the subset of scale-invariant Lagrangians for fields of arbitrary spin, in any dimension, which are conformally invariant is given. An example of a quadratic action for which scale invariance does not imply conformal invariance is constructed.

Effective potential for non-convex potentials

Abstract It is shown that the well-known relationship between the effective potential Γ and the vacuum graphs Σ of scalar QFT follows directly from the translational invariance of the measure, and that it holds for all values of the fields ϕ if, and only if, the classical potential is convex. In the non-convex case Σ appears to become complex for some values of ϕ, but it is shown that the complexity is only apparent and is due to the failure of the loop expansion. The effective potential actually remains real and well-defined for all ϕ, and reduces to Σ in the neighbourhood of the classical minima. A number of examples are considered, notably potentials which are spontaneously broken. In particular the mechanism by which a spontaneous breakdown may be generated by radiative corrections is re-investigated and some new insights obtained. Finally, it is shown that the renormalization group equations for the parameters may be obtained by inspection from the effective potential, and among the examples considered are SU( n ) fields and supermultiplets. In particular, it is shown that for supermultiplets the effective potential is not only real but positive.

Kac-Moody realization of W-algebras

Abstract By realizing the W -algebras of Toda field-theories as the algebras of gauge-invariant polynomials of constrained Kac-Moody systems we obtain a simple algorithm for constructing W -algebras without computing the W -generators themselves. In particular this realization yields an identification of a primary field basis for all the W -algebras, quadratic bases for the A, B, C-algebras, and the relation of W -algebras to Casimir algebras. At the quantum level it yields the general formula for the Virasoro centre in terms of the KM level.