Partition functions for quantum gravity, black holes, elliptic genera and Lie algebra homologies

There is a remarkable connection between quantum generating functions of field theory and formal power series associated with dimensions of chains and homologies of suitable Lie algebras. We discuss the homological aspects of this connection with its applications to partition functions of the minimal three-dimensional gravities in the space-time asymptotic to AdS₃, which also describe the three-dimensional Euclidean black holes, the pure N=1 supergravity, and a sigma model on N-fold generalized symmetric products. We also consider in the same context elliptic genera of some supersymmetric sigma models. These examples can be considered as a straightforward application of the machinery of modular forms and spectral functions (with values in the congruence subgroup of SL(2,Z)) to partition functions represented by means of formal power series that encode Lie algebra properties.     

Strings fromN=2 gauged Wess-Zumino-Witten models

We present an algebraic approach to string theory. An embedding ofsl(2|1) in a super Lie algebra together with a grading on the Lie algebra determines a nilpotent subalgebra of the super Lie algebra. Chirally gauging this subalgebra in the corresponding Wess-Zumino-Witten model, breaks the affine symmetry of the Wess-Zumino-Witten model to some extension of theN=2 superconformal algebra. The extension is completely determined by thesl(2|1) embedding. The realization of the superconformal algebra is determined by the grading. For a particular choice of grading, one obtains in this way, after twisting, the BRST structure of a string theory. We classify all embeddings ofsl(2|1) into Lie super algebras and give a detailed account of the branching of the adjoint representation. This provides an exhaustive classification and characterization of both all extendedN=2 superconformal algebras and all string theories which can be obtained in this way.     

Affine Lie algebras, string junctions and 7-branes

We consider the realization of affine ADE Lie algebras as string junctions on mutually non-local 7-branes in Type 1113 string theory. The existence of the affine algebra is signaled by the presence of the imaginary root junction δ, which is realized as a string encircling the 7-brane configuration. The level k of an affine representation partially constrains the asymptotic (p, q) charges of string junctions departing the configuration. The junction intersection form reproduces the full affine inner product, plus terms in the asymptotic charges.     

Integrable Highest Weight Modules over Affine Superalgebras and Appell's Function

We classify integrable irreducible highest weight representations of non-twisted affine Lie superalgebras. We give a free field construction in the level 1 case. The analysis of this construction shows, in particular, that in the simplest case of the sℓ (2|1) level 1 affine superalgebra the characters are expressed in terms of the Appell elliptic function. Our results demonstrate that the representation theory of affine Lie superalgebras is quite different from that of affine Lie algebras. Received: 17 April 2000 / Accepted: 7 July 2000     

String junctions for arbitrary Lie-algebra representations

We consider string junctions with endpoints on a set of branes of IIB string theory defining an ADE-type gauge Lie algebra. We show how to characterize uniquely equivalence classes of junctions related by string/brane crossing through invariant charges that count the effective number of prongs ending on each brane. Each equivalence class defines a point on a lattice of junctions. We define a metric on this lattice arising from the intersection pairing of junctions, and use self-intersection to identify junctions in the adjoint and fundamental representations of all ADE algebras. This information suffices to determine the relation between junction lattices and the Lie-algebra weight lattices. Arbitrary representations are built by allowing junctions with asymptotic (p,q) charges, on which the group of conjugacy classes of representations is represented additively. One can view the (p, q) asymptotic charges as Dynkin labels associated to two new fundamental weight vectors.     

Hadron potentials within the gauge/string correspondence

It is known, since the 70s, that the large N 't Hooft limit of gauge theories is related to string theories. In 1998, J. M. Maldacena identified precisely such a relation: the so-called AdS/CFT correspondence which speculates a duality between a large N strongly-coupled supersymmetric and conformal Yang-Mills theory in four dimensions and a weakly-coupled string theory defined in a five-dimensional anti-de Sitter AdS₅ space-time. This review aims at introducing concepts and methods used to derive, in the framework of the gauge/string correspondence, the interaction potentials of mesons and baryons at zero and finite temperature. The dual string configurations associated with the different kinds of hadrons are described and their behaviours at short and large distances are understood. Although the application of Maldacena's AdS/CFT conjecture to QCD is not straightforward, QCD being neither supersymmetric nor conformal, the AdS/QCD correspondence approach attempts to identify the dual theory of QCD. Especially, the study of heavy quark-antiquark bound-states leads to establish general dual criteria for the confinement.     

A Weyl group for the Virasoro andN=1 super-Virasoro algebras

We introduce a Weyl group for the highest weight modules over the Virasoro algebra and the Neveu-Schwarz and Ramond superalgebras. Using this group we rewrite the character formulae for the irreducible highest weight modules over these algebras in the form of the classical Weyl character formula for the finite-dimensional irreducible representations of semi-simple Lie algebras (and also of the Weyl-Kac character formula for the integrable highest weight modules over affine Kac-Moody algebras). This is the same group we introduced recently in order to rewrite in a similar manner the characters of the singular highest weight modules over the affine Kac-Moody algebraA ₁ ⁽¹⁾ .     

Unitary positive energy representations of the gauge group

We investigate the positive energy representations (also called highest weight representations) of the gauge groupC (T ^v,G ₀),G ₀ being a compact simple Lie group, and discuss their unitarity, using the technique of Verma modules constructed from generalized loop algebras (a simple generalization of Kac-Moody affine Lie algebras). We show that the unitarity of the representation imposes severa restrictions in it. In particular, we show, as a part of a more general result, that the gauge group does not admit faithful unitary positive energy representations.Allocataire du MRT.     

Review of AdS/CFT Integrability,Chapter II.2: Quantum Strings in AdS<Subscript>5</Subscript> × S<Superscript>5</Superscript>

We review the semiclassical analysis of strings in AdS₅ × S⁵ with a focus on the relationship to the underlying integrable structures. We discuss the perturbative calculation of energies for strings with large charges, using the folded string spinning in AdS₃ ? AdS₅ as our main example. Furthermore, we review the perturbative light-cone quantisation of the string theory and the calculation of the worldsheet S-matrix.     

Fock Representations of Lie Algebras,Loop Algebras and Affine Lie Algebras

Explicit Fock representations of the classical Lie algebras in terms of boson creation and annihilation operators with an arbitrary highest weight are derived, and a general rule to construct Fock represen tations of a loop algebra from a boson realization ofits corresponding Lie algebra is establislted. A new kind of lowest weight represen tations of the affine Lie algebras attached to the classical Lie algebras, which require a zero center, is also presented. Based on these, a simple affinization procedure is given to obtain the Fock representations of level 1 of these affine Lie algebras.     

Quantum many-body problems and perturbation theory

We show that the existence of algebraic forms of exactly solvable A-B-C-D, G ₂, and F ₄ Olshanetsky-Perelomov Hamiltonians allows one to develop algebraic perturbation theory, where corrections are computed by purely algebraic means. A classification of perturbations leading to such a perturbation theory based on the theory of representations of Lie algebras is given. In particular, this scheme admits an explicit study of anharmonic many-body problems. Some examples are presented.     

Lie 2-Bialgebras

In this paper, we study Lie 2-bialgebras, paying special attention to coboundary ones, with the help of the cohomology theory of L _∞-algebras with coefficients in L _∞-modules. We construct examples of strict Lie 2-bialgebras from left-symmetric algebras (also known as pre-Lie algebras) and symplectic Lie algebras (also called quasi-Frobenius Lie algebras).     

Integrability of a family of quantum field theories related to sigma models

A method is introduced for constructing lattice discretizations of large classes of integrable quantum field theories. The method proceeds in two steps: The quantum algebraic structure underlying the integrability of the model is determined from the algebra of the interaction terms in the light-cone representation. The representation theory of the relevant quantum algebra is then used to construct the basic ingredients of the quantum inverse scattering method, the lattice Lax matrices and R-matrices. This method is illustrated with four examples: The sinh-Gordon model, the affine sl(3) Toda model, a model called the fermionic sl(2|1) Toda theory, and the N=2 supersymmetric sine-Gordon model. These models are all related to sigma models in various ways. The N=2 supersymmetric sine-Gordon model, in particular, describes the Pohlmeyer reduction of string theory on AdS₂×S², and is dual to a supersymmetric non-linear sigma model with a sausage-shaped target space.     

Fock space representations of affine lie algebras and integral representations in the Wess-Zumino-Witten models

Fock space representations of affine Lie algebras are studied. Explicit forms of correction terms adding to the currentsF _i (z) are determined. It is proved that the Sugawara energy-momentum tensor on the Fock spaces is quadratic in free bosons. Furthermore, screening operators are constructed. This implies the existence of generalized hypergeometric integrals satisfying the Knizhnik-Zamolodchikov equation.     

Vertex Operators Arising from the Homogeneous Realization for

We use the underlying Fock space for the homogeneous vertex operator representation of the affine Lie algebra to construct a family of vertex operators. As an application, an irreducible module for an extended affine Lie algebra of type A _{N −1} coordinatized by a quantum torus ℂ _q of 2 variables (or 3 variables) is obtained. Moreover, this module turns out to be a highest weight module which is an analog of the basic module for affine Lie algebras. Received: 16 August 1999 / Accepted: 18 January 2000     

Supersymmetric Wilson loops in super-Chern–Simons-matter theory

We study supersymmetric Wilson loop operators in ABJM theory from both sides of the AdS₄/CFT₃ correspondence. We first construct some supersymmetric Wilson loops. The perturbative computations are performed in the field theory side at the first two orders. A fundamental string solution ending on a circular loop is also studied.     

Boson expansion of lie algebras: The fermion pair algebra

A unitary boson expansion theory for Lie algebras with ladder representations is discussed and applied to the fermion pair algebra SO(2n).     

Solvable RSOS models based on the dilute BWM algebra

In this paper we present representations of the recently introduced dilute Birman-Wenzl-Murakami algebra. These representations, labelled by the level-l B_n⁽¹⁾, C_n⁽¹⁾ and D_n⁽¹⁾ affine Lie algebras, are baxterized to yield solutions to the Yang-Baxter equation. The thus obtained critical solvable models are RSOS counterparts of the, respectively, D_n+1⁽²⁾, A_2n⁽²⁾ and B_n⁽¹⁾R-matrices of Bazhanov and Jimbo. For the D_n+1⁽²⁾ and B_n⁽¹⁾ algebras the RSOS models are new. An elliptic extension which solves the Yang-Baxter equation is given for all three series of dilute RSOS models.