Adel Bilal

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.

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.

The strong coupling spectrum of the Seiberg-Witten theory

Abstract We carefully study the global structure of the solution of the N = 2 supersymmetric pure Yang-Mills theory with gauge group SU(2) obtained by Seiberg and Witten. We exploit itsZsymmetry and describe the curve in moduli space where BPS states can become unstable, separating the strong-coupling from the weak-coupling region. This allows us to obtain the spectrum of stable BPS states in the strong-coupling region: we prove that only the two particles responsible for the singularities of the solution (the magnetic monopole and the dyon of unit electric charge) are present in this region. Our method also permits us to very easily obtain the weak-coupling spectrum, without using semiclassical methods. We discuss how the BPS states disintegrate when crossing the border from the weak- to the strong-coupling region.

(Weak) G2 holonomy from self-duality, flux and supersymmetry

The aim of this paper is two-fold. First, we provide a simple and pedagogical discussion of how compactifications of M-theory or supergravity preserving some four-dimensional supersymmetry naturally lead to reduced holonomy or its generalization, reduced weak holonomy. We relate the existence of a (conformal) Killing spinor to the existence of certain closed and co-closed p-forms, and to the metric being Ricci flat or Einstein. Then, for seven-dimensional manifolds, we show that octonionic self-duality conditions on the spin connection are equivalent to G_2 holonomy and certain generalized self-duality conditions to weak G_2 holonomy. The latter lift to self-duality conditions for cohomogeneity-one spin(7) metrics. To illustrate the power of this approach, we present several examples where the self-duality condition largely simplifies the derivation of a G_2 or weak G_2 metric.

Curves of marginal stability, and weak- and strong-coupling BPS spectra in N = 2 supersymmetric QCD

We explicitly determine the global structure of the SL(2, Z) bundle over the Coulomb branch of the moduli space of asymptotically free N = 2 supersymmetric Yang-Mills theories with gauge group SU(2) when massless hypermultiplets are present. For each relevant number of flavours, we show that there is a curve of marginal stability of the Coulomb branch, diffeomorphic to a circle, across which the BPS spectrum is discontinuous. We determine rigorously and completely the BPS spectra inside and outside the curve. In all cases, the spectrum inside the curve consists of only those BPS states that are responsible for the singularities of the low energy effective action (in addition to the massless abelian gauge multiplet which is always present). The predicted decay patterns across the curve of marginal stability are perfectly consistent with all quantum numbers carried by the BPS states. As a byproduct, we also show that the electric and magnetic quantum numbers of the massless states at the singularities proposed by Seiberg and Witten are the only possible ones.

Effective actions and N = 1 vacuum conditions from SU(3) ? SU(3) compactifications

We consider compactifications of type II string theory on general SU(3) × SU(3) structure backgrounds allowing for a very large set of fluxes, possibly nongeometric ones. We study the effective 4d low energy theory which is a gauged N = 2 supergravity, and discuss how its data are obtained from the formalism of the generalized geometry on T ⊕ T*. In particular we relate Hitchins special Kahler metrics on the spaces of even and odd pure spinors to the metric on the supergravity moduli space of internal metric and B-field fluctuations. We derive the N = 1 vacuum conditions from this N = 2 effective action, as well as from its N = 1 truncation. We prove a direct correspondence between these conditions and an integrated version of the pure spinor equations characterizing the N = 1 backgrounds at the ten dimensional level.

The BPS spectra and superconformal points in massive N = 2 supersymmetric QCD

Abstract We present a detailed study of the analytic structure, BPS spectra and superconformal points of the N = 2 susy SU(2) gauge theories with Nf = 1, 2, 3 massive quark hypermultiplets. We compute the curves of marginal stability with the help of the explicit solutions for the low-energy effective actions in terms of standard elliptic functions. We show that only a few of these curves are relevant. As a generic example, the case of Nf = 2 with two equal bare masses is studied in depth. We determine the precise existence domains for each BPS state, and show how they are compatible with the RG flows. At the superconformal point, where two singularities coincide, we prove that (for Nf = 2) the massless spectrum consists of four distinct BPS states and is S-invariant. This is due to the monodromy around the superconformal point being S, providing strong evidence for exact S-duality of the SCFT For all Nf, we compute the slopes ω of the β-functions at the fixed point couplings and show that they are related to the anomalous dimensions α of u = 〈trƒ 2 〉 by ω = 2(α − 1).

M(atrix) theory : a pedagogical introduction

I attempt to give a pedagogical introduction to the matrix model of M-theory as developed by Banks, Fischler, Shenker and Susskind (BFSS). In the first lecture, I introduce and review the relevant aspects of D-branes with the emergence of the matrix model action. The second lecture deals with the appearance of eleven-dimensional supergravity and M-theory in strongly coupled type IIA superstring theory. The third lecture combines the material of the two previous ones to arrive at the BFSS conjecture and explains the evidence presented by these authors. The emphasis is not on most recent developments but on a hopefully pedagogical presentation.

New critical dimensions for string theories

Abstract The open string dynamics which include the Liouville field theory following Polyakov and Gervais-Neveu are shown to enjoy the same remarkable properties as the celebrated string theories in D = 26 or D = 10 dimensions, provided D is changed to D = 7 or 13 for the purely bosonic string and to D = 3 or 5 for superstrings. For these new critical dimensions all four models have a free open string partition function which is such that the closed string singularities in the one-loop open string amplitude are poles. For superstrings, the partition functions in the Neveu-Schwarz and Ramond sectors are shown to coincide after the appropriate GOS projection has been carried out, and the new models are presumably space-time supersymmetric. These new critical dimensions are, in each model, the lowest two of the three special values already derived by Gervais and Neveu from their study of the Liouville dynamics per se.