Algebraic Coset Conformal Field Theories

All unitary Rational Conformal Field Theories (RCFT) are conjectured to be related to unitary coset Conformal Field Theories, i.e., gauged Wess–Zumino–Witten (WZW) models with compact gauge groups. In this paper we use subfactor theory and ideas of algebraic quantum field theory to approach coset Conformal Field Theories. Two conjectures are formulated and their consequences are discussed. Some results are presented which prove the conjectures in special cases. In particular, one of the results states that a class of representations of coset W _N (N≥ 3) algebras with critical parameters are irreducible, and under the natural compositions (Connes' fusion), they generate a finite dimensional fusion ring whose structure constants are completely determined, thus proving a long-standing conjecture about the representations of these algebras. Received: 5 November 1998 / Accepted: 18 October 1999     

Markov traces and II1 factors in conformal field theory

Using the duality equations of Moore and Seiberg we define for every primary field in a Rational Conformal Field Theory a proper Markov trace and hence a knot invariant. Next we define two nested algebras and show, using results of Ocneanu, how the position of the smaller algebra in the larger one reproduces part of the duality data. A new method for constructing Rational Conformal Field Theories is proposed.     

Duality and quantum groups

We show that the duality properties of Rational Conformal Field Theories follow from the defining relations and the representation theory of quantum groups. The fusion and braiding matrices are q-analogues of the 6j-symbols and the modular transformation matrices are obtained from the properties of the co-multiplication. We study in detail the Wess-Zumino-Witten models and the rational gaussian models as examples, but carry out the arguments in general. We point out the connections with the Chern-Simons approach. We give general arguments of why the general solution to the polynomial equations of Moore and Seiberg describing the duality properties of Rational Conformal Field Theories defines a Quantum Group acting on the space of conformal blocks. A direct connection between Rational Theories and knot invariants is also presented along the lines of Jones' original work.     

The Nonchiral Fusion Rules in Rational Conformal Field Theories

We introduce a general method in order to construct the nonchiral fusion rules which determine the operator content of the operator product algebra for rational conformal field theories. We are particularly interested in the models of the complementary D-like solutions of the modular invariant partition functions with cyclic center Z _N . We find that the nonchiral fusion rules have a Z _N -grading structure.     

Spectra of Wess-Zumino-Witten models with arbitrary simple groups

We consider the Wess-Zumino-Witten two-dimensional sigma models with fields taking values in an arbitrary connected (but not necessarily simply connected) simple Lie groupG. The quantum states of the theory are realized geometrically as sections of a line bundle over the loop groupLG. The action of the current algebra of the theory is decomposed into highest weight representations by explicit construction of the highest weight states. This solves for the spectrum of the models. As a by-product, we obtain modular invariant partition functions of the theory on tori. The present paper extends the results of [7] where the casesG=SU(2) andSO(3) were treated.Ce rapport a été publié en partie grâce à une subvention du Fonds FCAR pour l'aide et le soutien à la rercherche     

Operator coproduct-realization of quantum group transformations in two-dimensional gravity I

A simple connection between the universalR matrix ofU _q(sl(2)) (for spins 1/2 andJ) and the required form of the coproduct action of the Hilbert space generators of the quantum group symmetry is put forward. This leads us to an explicit operator realization of the coproduct action on the covariant operators. It allows us to derive the expected quantum group covariance of the fusion and braiding matrices, although it is of a new type: the generators depend upon worldsheet variables, and obey a new central extension of theU _q(sl(2)) algebra realized by (what we call) fixed point commutation relations. This is explained by showing on a general ground that the link between the algebra of field transformations and that of the coproduct generators is much weaker than previously thought. The central charges of our extendedU _q(sl(2)) algebra, which includes the Liouville zero-mode momentum in a non-trivial way, are related to Virasoro-descendants of unity. We also show how our approach can be used to derive the Hopf algebra structure of the extended quantum-group symmetry related to the presence of both of the screening charges of 2D gravity.Partially supported by the EC contracts CHRXCT920069 and CHRXCT920035.Unité Propre du Centre National de la Recherche Scientifique, associée à l'École Normale Supérieure et à l'Université de Paris-Sud.     

Boundary conditions in rational conformal field theories

We develop further the theory of Rational Conformal Field Theories (RCFTs) on a cylinder with specified boundary conditions emphasizing the role of a triplet of algebras: the Verlinde, graph fusion and Pasquier algebras. We show that solving Cardy's equation, expressing consistency of a RCFT on a cylinder, is equivalent to finding integer valued matrix representations of the Verlinde algebra. These matrices allow us to naturally associate a graph G to each RCFT such that the conformal boundary conditions are labelled by the nodes of G. This approach is carried to completion for sl(2) theories leading to complete sets of conformal boundary conditions, their associated cylinder partition functions and the A-D-E classification. We also review the current status for WZW sl(3) theories. Finally, a systematic generalisation of the formalism of Cardy–Lewellen is developed to allow for multiplicities arising from more general representations of the Verlinde algebra. We obtain information on the bulk–boundary coefficients and reproduce the relevant algebraic structures from the sewing constraints.     

The quantum group structure associated with non-linearly extended Virasoro algebras

Recently, an infinite family of chiral Virasoro vertex operators, whose exchange algebra is given by the universalR-matrix ofSL(2) _q , has been constructed. In the present paper, the case of non-linearly (W-) extended Virasoro symmetries, related to the algebrasA _N,N>1, is considered along the same line. Contrary to the previous case (A ₁) the standardR-matrix ofSL(N+1)q does not come out, and a different solution of Yang and Baxter's equations is derived. The associated quantum group structure is displayed.Unité Propre du Centre National de la Recherche Scientifique, associée à l'École Normale Supérieure et à l'Université de Paris-Sud     

Complete classification of simple current modular invariants for RCFT's with a center (Z p ) k

Simple currents have been used previously to construct various examples of modular invariant partition functions for given rational conformal field theories. In this paper we present for a large class of such theories (namely those with a center that decomposes into factors Z _p ,p prime) thecomplete set of modular invariants that can be obtained with simple currents. In addition to the fusion rule automorphisms classified previously forany center, this includes all possible left-right combinations of all possible extensions of the chiral algebra that can be obtained with simple currents, for all possible current-current monodromies. Formulas for the number of invariants of each kind are derived. Although the number of invariants in each of these subsets depends on the current-current monodromies, the total number of invariants depends rather surprisingly only onp and the number ofZ _p factors.     

Modular transformations ofSU(N) affine characters and their commutant

We describe the algebra of matrices commuting with the action of the modular group on characters ofSU(N) _k integrable representations. Using methods of finite quantum mechanics we find a canonical basis for this commutant over and prove the existence of an equivalent basis over with integral matrix elements. A final section is devoted to the case ofSU(3).Laboratoire de l'Institut de Recherche Fondamentale du Commissariat à l'Energie Atomique     

Modular invariant partition functions in the quantum Hall effect

We study the partition function for the low-energy edge excitations of the incompressible electron fluid. On an annular geometry, these excitations have opposite chiralities on the two edges; thus, the partition function takes the standard form of rational conformal field theories. In particular, it is invariant under modular transformations of the toroidal geometry made by the angular variable and the compact Euclidean time. The Jain series of plateaus have been described by two types of edge theories: the minimal models of the W_1+∞ algebra of quantum area-preserving diffeomorphisms, and their non-minimal version, the theories with affine algebra. We find modular invariant partition functions for the latter models. Moreover, we relate the Wen topological order to the modular transformations and the Verlinde fusion algebra. We find new, non-diagonal modular invariants which describe edge theories with extended symmetry algebra; their Hall conductivities match the experimental values beyond the Jain series.     

The A-D-E classification of minimal andA 1 (1) conformal invariant theories

We present a detailed and complete proof of our earlier conjecture on the classification of minimal conformal invariant theories. This is based on an exhaustive construction of all modular invariant sesquilinear forms, with positive integral coefficients, in the characters of the Virasoro or of theA ₁ ⁽¹⁾ Kac-Moody algebras, which describe the corresponding partition functions on a torus. A remarkable correspondence emerges with simply laced Lie algebras.     

Conformal correlation functions,Frobenius algebras and triangulations

We formulate two-dimensional rational conformal field theory as a natural generalization of two-dimensional lattice topological field theory. To this end we lift various structures from complex vector spaces to modular tensor categories. The central ingredient is a special Frobenius algebra object A in the modular category that encodes the Moore–Seiberg data of the underlying chiral CFT. Just like for lattice TFTs, this algebra is itself not an observable quantity. Rather, Morita equivalent algebras give rise to equivalent theories. Morita equivalence also allows for a simple understanding of T-duality.We present a construction of correlators, based on a triangulation of the world sheet, that generalizes the one in lattice TFTs. These correlators are modular invariant and satisfy factorization rules. The construction works for arbitrary orientable world sheets, in particular, for surfaces with boundary. Boundary conditions correspond to representations of the algebra A. The partition functions on the torus and on the annulus provide modular invariants and NIM-reps of the fusion rules, respectively.     

W-geometry of the Toda systems associated with non-exceptional simple Lie algebras

The present paper describes theW-geometry of the Abelian finite non-periodic (conformal) Toda systems associated with theB, C andD series of the simple Lie algebras endowed with the canonical gradation. The principal tool here is a generalization of the classical Plücker embedding of theA-case to the flag manifolds associated with the fundamental representations ofB _n ,C _n andD _n , and a direct proof that the corresponding Kähler potentials satisfy the system of two-dimensional finite non-periodic (conformal) Toda equations. It is shown that theW-geometry of the type mentioned above coincide with the differential geometry of special holomorphic (W) surfaces in target spaces which are submanifolds (quadrics) ofCP ^N with appropriate choices ofN. In addition, theseW-surfaces are defined to satisfy quadratic holomorphic differential conditions that ensure consistency of the generalized Plücker embedding. These conditions are automatically fulfilled when Toda equations hold.Unité Propre du Centre National de la Recherche Scientifique, associée à l'École Normale Supérieure et à l'Université de Paris-Sud.     

Chiral partition functions characterizing rational conformal models

Rational conformal theories are characterized by a chiral (analytic in the modular parameter) “partition function” with values in the fusion rules algebra covariant with respect to a subgroup of the modular group.     

Asymptotically anti-de Sitter spaces

Asymptotically anti-de Sitter spaces are defined by boundary conditions on the gravitational field which obey the following criteria: (i) they are O(3, 2) invariant; (ii) they make the O(3, 2) surface integral charges finite; (iii) they include the Kerr-anti-de Sitter metric. An explicit expression of the O(3, 2) charges in terms of the canonical variables is given. These charges are shown to close in the Dirac brackets according to the anti-de Sitter algebra. The results are extended to the case ofN=1 supergravity. The coupling to gravity of a third-rank, completely antisymmetric, abelian gauge field is also considered. That coupling makes it possible to vary the cosmological constant and to compare the various anti-de Sitter spaces which are shown to have the same energy.On leave from Département de Physique, Université Libre de Bruxelles, BelgiumChercheur qualifié du Fonds National Belge de la Recherche Scientifique     

High nuclear polarization in3He and3He-4He gas mixtures by optical pumping with a laser diode

New diode lasers delivering 50 mW output power at 1083 nm are shown to be efficient sources for optical pumping of helium. They can polarize nuclei in a³He gas up toM = 50% over the pressure range 0.4–1.6 torr. Larger nuclear polarizationsM of³He nuclei, of order 80%, can be obtained in³He-⁴He mixtures when the laser frequency is tuned to a⁴He line. A standard optical measurement of nuclear polarizationM has been extended to the case of³He-⁴He mixtures. The effect of various parameters on the steady-state polarizationM and on the pumping timeT _p is discussed.Laboratoire associé au Centre National de la Recherche Scientifique et à l'Université P. M. Curie     

Construction of modular branching functions from Bethe's equations in the 3-state Potts chain

We use the single-particle excitation energies and the completeness rules of the 3-state antiferromagnetic Potts chain, which have been obtained from Bethe's equation, to compute the modular invariant partition function. This provides a fermionic construction for the branching functions of theD ₄ representation ofZ ₄ parafermions which complements the bosonic constructions. It is found that there are oscillations in some of the correlations and a new connection with the field theory of the Lee-Yang edge is presented.     

On α-Induction, Chiral Generators and¶Modular Invariants for Subfactors

We consider a type III subfactor N⊂N of finite index with a finite system of braided N-N morphisms which includes the irreducible constituents of the dual canonical endomorphism. We apply α-induction and, developing further some ideas of Ocneanu, we define chiral generators for the double triangle algebra. Using a new concept of intertwining braiding fusion relations, we show that the chiral generators can be naturally identified with the α-induced sectors. A matrix Z is defined and shown to commute with the S- and T-matrices arising from the braiding. If the braiding is non-degenerate, then Z is a “modular invariant mass matrix” in the usual sense of conformal field theory. We show that in that case the fusion rule algebra of the dual system of M-M morphisms is generated by the images of both kinds of α-induction, and that the structural information about its irreducible representations is encoded in the mass matrix Z. Our analysis sheds further light on the connection between (the classifications of) modular invariants and subfactors, and we will construct and analyze modular invariants from SU(n) _k loop group subfactors in a forthcoming publication, including the treatment of all SU(2) _k modular invariants. Received: 13 April 1999 / Accepted: 13 July 1999     

Spin effects in the electron structure functions

The electron structure functions are studied in polarized e ⁺ e ^- scattering. The formulae for longitudinally and transversely polarized electrons are presented. The smallness of the electron mass leads to negligible cross-sections and asymmetries in some cases. Positivity constraints on the structure functions and parton densities are constructed and discussed. The cross-section asymmetries at very high energies, where the inclusion of all elecroweak bosons is necessary, are calculated. Numerical examples, using the asymptotic solutions for the parton densities inside the electron, are presented.Received: 23 June 2004, Revised: 26 October 2004, Published online: 17 December 2004PACS: 12.15.Ji, 12.38.QkJ. Szwed: Work supported by the European Commission contract ICA1-CT-2002-70013/INCO Strategic action on training and excellence, 5FP.Unité Mixte de Recherche du CNRS et des Universités de Provence, de la Méditerranée et du Sud Toulon-Var - Laboratoire affilié a la FRUNAM - FR 2291, CNRS-Luminy, Case 907, 13288 Marseille Cedex 9, France.