Andreas Wipf

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.

Zeta functions and the Casimir energy

Abstract We use zeta function techniques to give a finite definition for the Casimir energy of an arbitrary ultrastatic spacetime with or without boundaries. We find that the Casimir energy is intimately related to, but not identical to, the one-loop effective energy. We show that in general the Casimir energy depends on a normalization scale. This phenomenon has relevance to applications of the Casimir energy in bag models of QCD. Within the framework of Kaluza-Klein theories we discuss the one-loop corrections to the induced cosmological and Newton constants in terms of a Casimir like effect. We can calculate the dependence of these constants on the radius of the compact dimensions, without having to resort to detailed calculations.

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.

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.

Supersymmetry and the Dirac Equation

Abstract We discuss in detail two supersymmetries of the 4-dimensional Dirac operator / kD 2 where / kD = ∂ − ieA , namely the usual chiral supersymmetry and a separate complex supersymmetry. Using SUSY methods developed to categorize solvable potentials in 1-dimensional quantum mechanics we systematically study the cases where the spectrum, eigenfunctions, and S -matrix of / kD 2 can be obtained analytically. We relate these solutions to the solutions of the ordinary massive Dirac equation in external fields. We show that whenever a Schrodinger equation for a potential V ( x ) is exactly solvable, then there always exists a corresponding static scalar field ϕ ( x ) for which the Jackiw-Rebbi type (1 + 1)-dimensional Dirac equation is exactly solvable with V ( x ) and ϕ ( x ) being related by V ( x ) = ϕ 2 ( x ) + ϕ ′( x ). We also discuss and exploit the supersymmetry of the path integral representation for the fermion propagator in an external field.

Low-dimensional supersymmetric lattice models☆

We study and simulate N=2 supersymmetric Wess-Zumino models in one and two dimensions. For any choice of the lattice derivative, the theories can be made manifestly supersymmetric by adding appropriate improvement terms corresponding to discretizations of surface integrals. In one dimension, our simulations show that a model with the Wilson derivative and the Stratonovitch prescription for this discretization leads to far better results at finite lattice spacing than other models with Wilson fermions considered in the literature. In particular, we check that fermionic and bosonic masses coincide and the unbroken Ward identities are fulfilled to high accuracy. Equally good results for the effective masses can be obtained in a model with the SLAC derivative (even without improvement terms). In two dimensions we introduce a non-standard Wilson term in such a way that the discretization errors of the kinetic terms are only of order O(a^2). Masses extracted from the corresponding manifestly supersymmetric model prove to approach their continuum values much quicker than those from a model containing the standard Wilson term. Again, a comparable enhancement can be achieved in a theory using the SLAC derivative.

Generalized Toda theories and W-algebras associated with integral gradings

A general class of conformal Toda theories associated with integral gradings of Lie algebras is investigated. These generalized Toda theories are obtained by reducing the Wess--Zumino--Novikov--Witten (WZNW) theory by first--class constraints, and thus they inherite extended conformal symmetry algebras, generalized W--algebras, and current dependent Kac--Moody (KM) symmetries from the WZNW theory, which are analysed in detail in a non--degenerate case. We recover an

Classical gauge vacua as knots

sl(2)