Jean-Loup Gervais

Novel triangle relation and absence of tachyons in Liouville string field theory

Abstract We construct the quantum mechanical field operator of the two-dimensional Liouville theory in a finite box. This leads us to the discovery of a new type of triangle relation which does not reduce to the already known ones. We apply our result to the construction of the string model in an arbitrary number of space-time dimensions D . We show that there are no tachyons in −∞ D ⩽1 , and we indicate how to continue to D >1, which is a strong-coupling region for the Liouville field theory.

FIELD THEORY INTERPRETATION OF SUPERGAUGES IN DUAL MODELS.

Abstract Possible new invariances of generalized dual models are discussed in the context of the functional integral formulation. The operators relevant to new gauges of those models, such as those obtained by Neveu and Schwarz, are derived as infinitesimal generators of new field transformations which leave the action integral invariant.

Systematic Approach to Conformal Systems with Extended Virasoro Symmetries

Abstract The Toda field theories, which exist for every simple Lie group, are shown to give realizations of extended Virasoro algebras that involve generators of spins higher than or equal to two. They are uniquely determined from the canonical lagrangian formalism. The quantization of the Toda field theories gives a systematic treatment of generalized conformal bosonic models. The well-known pattern of conformal field theories with non-extended Virasoro algebra, appears to be repeated for any simple group, leading to a “periodic table”, parallel to the mathematical classification of simple Lie groups.

Dual string spectrum in Polyakov's quantization (II). Mode separation

The analysis of our first paper is completed by (a) explicitly working out the set of ground state classical motions, (b) separating the modes into equally spaced harmonic oscillators. The Virasoro generators are obtained as quadratic expressions of these modes providing a spectrum generating algebra. The quantum spectrum is shown to be similar to a standard string spectrum with a new kind of zero mode associated with a representation of a dynamical SU(2) group.

The dual string spectrum in Polyakov's quantization (I)

Abstract We consider the quantization of Liouvilles equation in a box, with appropriate boundary conditions, as comes out of Polyakovs analysis of the Nambu action. The answer to this problem gives the spectrum of the dual string in this new approach. In this first paper, we concentrate mostly on the classical theory, including the inverse scattering method, which are needed to separate the modes, a prerequisite to an exact canonical quantization. In particular, we point out the important role of a parameter appearing in the boundary conditions, which depends on the space-time dimension, and we show that the system reduces to an involved set of harmonic oscillators.

Infinite component field theory of interacting relativistic strings and dual theory

Using light-cone quantization, the multistring formalism is unambiguously established from operator formalism, by introducing an infinite component field Φ which is a ket vector in the Fock space of transverse oscillators and the Φ3 interaction corresponding to the dual vertex of ADDF. A functional multistring field is naturally obtained as the component of Φ in the basis {x} where the string operator X is diagonal. It differs from Kaku and Kikkawas one by an infinite phase factor which does not allow them to take the value of the intercept into account. In this way, we recover the functional approach to the dual amplitude with a well-defined functional measure which could not have been guessed a priori, and which is determined from our previously established expression of the vertex in the {x} basis. The scattering amplitudes are deduced from field-theory asymptotic conditions and perturbation theory. On-shell amplitudes are shown to agree with the form proposed by Mandelstam integrated over time differences instead of Koba-Nielsen variables. Two prescriptions for off-shell amplitudes are discussed. One of them agress with Mandelstams while the other one is deduced from reduction formulae. The latter prescription is time-translation invariant whereas the former is not. For the four-string amplitude, it is proved that the part of the integrand which does not depend on the external states is at 26 space-time dimensions, precisely the one required to change variables from time differences to Koba-Nielsen variables. This allows us to determine unambiguously the Φ4 term which is needed to reproduce the complete tree dual amplitude.

Point canonical transformations in the path integral

We discuss point canonical transformations using the path integral quantization procedure. First, employing the Feynman diagram technique we demonstrate how a formal treatment of point transformations leads to erroneous results. Then we present the correct treatment of this problem using a more precise definition of path integrals. We show how this more careful treatment leads to additional potential terms in the action as compared to the formal treatment. We also demonstrate the equivalence of our results with the corresponding operator method discussion. Furthermore, we investigate the consequences of these results on the path integral collective coordinate method. Giving the improved treatment on the one-soliton sector example we establish the path integral collective coordinate method of the same level of rigor as the parallel operator approaches.

Systematic Construction of Conformal Theories with Higher Spin Virasoro Symmetries

Abstract The conformally invariant Toda theories are quantized following our earlier proposal [1] and starting from our classical mode separation [2]. As part of our general derivation of conformal theories associated with simple Lie algebras, the following new results are presented: 1. (a) The operator realizations of the extended (higher-spin) Virasoro algebras for arbitrary values of the Virasoro central charge C, and their highest-weight representations. 2. (b) The basic family of intertwining operators and the associated generalized Kac formulae, both for simply-laced and non-simply-laced Lie algebras. 3. (c) The novel features of the rational theories associated with non-simply-laced Lie algebras. 4. (d) The family of equivalent free fields and the associated Galois canonical transformations. 5. (e) The covariance properties of the intertwining operators under the extended Virasoro transformations. 6. (f) The operator differential equations satisfied by these primary operators, and the corresponding c-number differential equations for their Green functions. 7. (g) The Coulomb gas picture and the reduction to the hypergeometric type of the two-point function. 8. (h) The solution of the Yang-Baxter equations describing the braid properties of the conformal field theories so constructed.

Extended C = ∞ conformal systems from classical toda field theories

Abstract In a recent article [1] we showed that the bosonic Toda field theories obey extended Virasoro symmetries which involve generators of spins higher than two; and that their quantization gives a systematic treatment of generalized conformal bosonic models. Their Virasoro central charges are such that they become infinite in the classical limit. This latter situation is studied in detail in the present paper, where a simple form of the general solution of the Toda field equations is given, that allows one to separate the modes and to study the Poisson bracket structure of the generators of the extended symmetry in a systematic way. Besides its relevance to the study of integrable classical systems this paves the way to the quantum case, already discussed by the authors and to be worked out in full detail in a separate publication.

New quantum treatment of Liouville field theory

Abstract We exhibit a new treatment of the quantum Liouville theory in a box. In this treatment, the central charge of the Virasoro algebra is finite at the critical dimension of the associated string model, and we show how to reconstruct conformally covariant quantum field operators, in terms of a set of equally spaced harmonic oscillators.