Thomas L. Curtright

TORSION AND GEOMETROSTASIS IN NONLINEAR SIGMA MODELS

Abstract We discuss some general effects produced by adding Wess-Zumino terms to the actions of nonlinear sigma models, an addition which may be made if the underlying field manifold has appropriate homological properties. We emphasize the geometrical aspects of such models, especially the role played by torsion on the field manifold. For general chiral models, we show explicitly that the torsion is simply the structure constant of the underlying Lie group, converted by vielbeine into an antisymmetric rank-three tensor acting on the field manifold. We also investigate in two dimensions the supersymmetric extensions on nonlinear sigma models with torsion, showing how the purely results carry over completely. We consider in some detail the renormalization effects produced by the Wess-Zumino terms using the background field method. In particular, we demonstrate to two-loop order the existence of geometrostasis, i.e. fixed points in the renormalized geometry of the field manifold due to parallelism.

An exact operator solution of the quantum Liouville field theory

Exact, closed form results are given expressing the quantum Liouville field theory in terms of a canonical free pseudoscalar field. The classical conformal transformation properties and Baechlund transformation of the Liouville model are briefly reviewed and then developed into explicit operator statements for the quantum theory. This development leads to exact expressions for the basic operator functions of the Liouville field: partial/sub ..mu../Phi, and e/sup n/phi. An operator product analysis is then used to construct the Liouville energy-momentum tensor operator, which is shown to be equal to that of a free pseudoscalar field. Dynamical consequences of this equilvalence are discussed, including the relation between the Liouville and free field energy eigenstates. Liouville correlation functions are partially analyzed, and remaining open questions are discussed.

Quantum Bäcklund transformation for the Liouville theory

Abstract The classical Backlund transformation, which maps the Liouville model onto a free field theory, is developed into an explicit operator relation for quantum fields. Conformal transformation properties and canonical commutation relations are preserved by this operator mapping.

Nonperturbative Weak Coupling Analysis of the Quantum Liouville Field Theory

Abstract Two independent weak-coupling expansions are developed for the Liouville quantum field theory on a circle. In the first, the coupling of the nonzero modes is treated as a perturbation on the exact solution to the zero-mode problem (quantum mechanics with an exponential potential). The second approach is a weak-coupling approximation to an explicit operator solution which expresses various Liouville operators as functions of a free massless field using a Backlund transformation. It is shown that the free state space associated with the latter solution must be restricted to the sector which is odd with respect to a type of “parity.” Various matrix elements are computed to order g10 using both approaches, yielding identical results.

Quantum algebra deforming maps, Clebsch–Gordan coefficients, coproducts, R and U matrices

Quantum algebra deforming maps explicitly define comultiplications that differ from the usual noncocommutative coproducts. Map‐induced coproducts are connected to the usual ones by similarity transformations U that may be expressed either in terms of Clebsch–Gordan coefficients, or in a universal operator form. The product of two such U matrices yields the R matrix for a fixed value of the spectral parameter, which bears on the Yang–Baxterization of U as well as R. All this is explicitly illustrated for the tensor product 1/2⊗j of SU(2)q using several deforming maps whose coproducts are continuously connected by similarity transformations to form a two‐parameter manifold. Some observations are made on the general structure of U and R matrices, and of coproduct manifolds, based on the solutions of hierarchies of partial difference equations. Applications of deforming maps and U matrices to the physics of spin‐chains are outlined.

Charge renormalization and high spin fields

Abstract A simple physical interpretation of charge renormalization in non-abelian theories permits the rapid determination of s (one-loop) in models containing high spins ( S > 1). For any possible choice of spins, all N > 4 extended supersymmetry multiplets are “doubly-finite” and give s (one-loop) = 0 for either their SO( N ) or SU( N ) internal symmetry groups.

Symmetry patterns in the mass spectra of dual string models

Abstract We develop techniques for analyzing the rotation and gauge group representation content of string models. We explicitly construct all rotation group representations at low mass levels. We then obtained simple approximate formulas for the degeneracy of any given rotation or gauge group representation in the limit of very high mass. We also derive generating functions that give exactly the degeneracy of a representation at any mass level. Finally, we numerically study these results in several cases, and then abstract from the numerical data some simple empirical rules relating different representations.

Geometrostasis and torsion in covariant superstrings

Abstract We give a geometric argument to understand the relative strength of the metric and torsion terms that constitute the covariant actions for freely propagating superstrings. We show the relative strength is precisely that for which the torsion flattens the underlying superspace manifold, i.e. for which geometrostasis occurs, thereby yielding trivially integrable systems on the world-sheet, in complete analogy with conventional two-dimensional σ-models. We fully discuss free heterotic superstrings, and give partial results for N = 2 superstrings.

The effective potential in quantum mechanics

General properties of the effective potential are discussed for quantum mechanical systems with a single degree of freedom. These properties are illustrated using specific one‐dimensional potential models. In particular, it is stressed that the ground state for a system can exist even when the effective potential decreases monotonically towards a unique finite minimum at infinite 〈x〉.

Spin content of the bosonic string

Abstract A new form is derived for the generating function, χ [ λ ] ( x ) = Σ n =0 ∞ x n χ n [ λ ] , which counts the number of times, χ n [ λ ] , that the irreducible O( D − 1) rotation group representation [λ] appears at the n th mass level for the bosonic string. The derivation goes through for arbitrary old spatial dimensions, D − 1 = 2 v + 1. Simplifications of the results are noted for large v and the relation to a previously obtained formula for χ [ λ ] ( x ) is explained.