Rational Conformal Field Theories and Complex Multiplication

We study the geometric interpretation of two dimensional rational conformal field theories, corresponding to sigma models on Calabi-Yau manifolds. We perform a detailed study of RCFTs corresponding to the T² target and identify the Cardy branes with geometric branes. The T²s leading to RCFTs admit complex multiplication which characterizes Cardy branes as specific D0-branes. We propose a condition for the conformal sigma model to be RCFT for arbitrary Calabi-Yau n-folds, which agrees with the known cases. Together with recent conjectures by mathematicians it appears that rational conformal theories are not dense in the space of all conformal theories, and sometimes appear to be finite in number for Calabi-Yau n-folds for n>2. RCFTs on K3 may be dense. We speculate about the meaning of these special points in the moduli spaces of Calabi-Yau n-folds in connection with freezing geometric moduli.     

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.     

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     

Z/NZ Conformal Field Theories

We compute the modular properties of the possible genus-one characters of some Rational Conformal Field Theories starting from their fusion rules. We show that the possible choices ofS matrices are indexed by some automorphisms of the fusion algebra. We also classify the modular invariant partition functions of these theories. This gives the complete list of modular invariant partition functions of Rational Conformal Field Theories with respect to theA _N ⁽¹⁾ level one algebra.Unité propre de Recherche du Centre National de la Recherche Scientifique, associée à l'Ècole Normale Supérieure et à l'Université de Paris-Sud     

Logarithmic Conformal Field Theories Near a Boundary

We consider logarithmic conformal field theories near a boundary and derive the general form of one and two point functions. We obtain results for arbitrary and two dimensions. Application to two-dimensional magnetohydrodynamics is discussed.     

Conformal Field Theories of Stochastic Loewner Evolutions

Stochastic Loewner evolutions (SLE) are random growth processes of sets, called hulls, embedded in the two dimensional upper half plane. We elaborate and develop a relation between SLE evolutions and conformal field theories (CFT) which is based on a group theoretical formulation of SLE processes and on the identification of the proper hull boundary states. This allows us to define an infinite set of SLE zero modes, or martingales, whose existence is a consequence of the existence of a null vector in the appropriate Virasoro modules. This identification leads, for instance, to linear systems for generalized crossing probabilities whose coefficients are multipoint CFT correlation functions. It provides a direct link between conformal correlation functions and probabilities of stopping time events in SLE evolutions. We point out a relation between SLE processes and two dimensional gravity and conjecture a reconstruction procedure of conformal field theories from SLE data. Member of the CNRS     

Topological Conformal Field Theories from the Noncompact Coset Models

It is shown in this letter that in the bosonic coset models G_kl×G_k2/G_kl+_k2 associated with noncompact Kac-Moody algebra there exist two kinds of topological points, _k2=0 and _kl+_k2-2g = 0. At these points, the coset models may be interpreted as twisted versions of noncompact counterparts of the Kazama-Suzuki models.     

Limits and Degenerations of Unitary Conformal Field Theories

In the present paper, degeneration phenomena in conformal field theories are studied. For this purpose, a notion of convergent sequences of CFTs is introduced. Properties of the resulting limit structure are used to associate geometric degenerations to degenerating sequences of CFTs, which, as familiar from large volume limits of non- linear sigma models, can be regarded as commutative degenerations of the corresponding quantum geometries. As an application, the large level limit of the A-series of unitary Virasoro minimal models is investigated in detail. In particular, its geometric interpretation is determined.Acknowledgements It is a pleasure to thank Gavin Brown, Jarah Evslin, José Figueroa-OFarrill, Matthias Gaberdiel, Maxim Kontsevich, Werner Nahm, Andreas Recknagel, Michael Rösgen, Volker Schomerus, Gérard Watts and the referee for helpful comments or discussions. We also wish to thank the Abdus Salam International Center for Theoretical Physics for hospitality, since part of this work was performed there.D. R. was supported by DFG Schwerpunktprogramm 1096 and by the Marie Curie Training Site Strings, Branes and Boundary Conformal Field Theory at Kings College London, under EU grant HPMT-CT-2001-00296. K. W. was partly supported under U.S. DOE grant DE-FG05-85ER40219, TASK A, at the University of North Carolina at Chapel Hill.     

Current Algebra Associated with Logarithmic Conformal Field Theories

We propose a general framework for deriving the OPEs within a logarithmic conformal field theory (LCFT). This naturally leads to the emergence of a logarithmetic partner of the energy momentum tensor within an LCFT and implies that the current algebra associated with an LCFT is expanded. We derive this algebra for a generic LCFT and discuss some of its implications. We observe that two constants arise in the OPE of the energy-momentum tensor with itself. One of these is the usual central charge.     

On the Equivalence¶of Certain Coset Conformal Field Theories

We demonstrate the equivalence of Kazama–Suzuki cosets G(m, n, k) and G(k, n, m) based on complex Grassmannians by proving that the corresponding conformal precosheaves are isomorphic. We also determine all the irreducible representations of the conformal precosheaves. Received: 24 August 2001 / Accepted: 5 December 2001     

c=3, N=2 超共形场论的Z2 Orbifold-Prime模型

讨论了二维环面上中心荷c=3, N=2 的超共形场论. 特别给出该理论的配分函数. 进一步,为了产生新的模型,回顾了一般的orbifold方法. 然后构造了模不变的Z2 Orbifold-Prime模型.     

Closed and Open Conformal Field Theories and Their Anomalies

We describe a formalism allowing a completely mathematical rigorous approach to closed and open conformal field theories with general anomaly. We also propose a way of formalizing modular functors with positive and negative parts, and outline some connections with other topics, in particular elliptic cohomology.The authors were supported by the NSF.     

Factorization of Correlation Functions in Coset Conformal Field Theories

We use the conformal Ward identities to study the structure of correlation functions in coset conformal field theories. For a large class of primary fields of arbitrary g/h theory, a factorization anzatz is found. Corresponding correlation functions are explicitly expressed in terms of correlation functions of two independent WZNW theories for g and h.     

Extended U(1) Conformal Field Theories and Zk-Parafermions

A constructive approach is developed for studying local chiral algebras generated by a pair of oppositely charged fields Ψ (z, ±g) such that the operator product expansion (OPE) of Ψ(z₁, g) Ψ(z₂, −g) involves a U(1) current. The main tool in the study is the factorization property of the charged fields (exhibited in [PT 2, 3]) for Virasoro central charge c < 1 into U(1)-vertex operators tensored with ZAMOLODCHIKOV-FATEEV [ZF1] (generalized) Z_k-parafermions. The case Δ₂ = 4 (Δ₁ − 1), where Δ_v = Δ_k−v(Δ₀ = 0) are the conformaldimensions of the parafermionic currents, is studied in detail. For Δ_v = 2v(1 − v/k) the theory is related to GEPNER'S [Ge] Z₂ [so (k)] parafermions and the corresponding quantum field theoretic (QFT) representations of the chiral algebra are displayed. The Coulomb gas method of [CR] is further developed to include an explicit construction of the basic parafermionic current Ψ of weight Δ = Δ₁. The characters of the positive energy representations of the local chiral algebra are written as sums of products of Kac's string functions and classical θ-functions.     

Conjectured Z2-Orbifold Constructions of Self-Dual Conformal Field Theories at Central Charge 24 – the Neighborhood Graph

By considering constraints on the dimensions of the Lie algebra corresponding to the weight 1-states of Z₂ and Z₃ orbifold models arising from imposing the appropriate modular properties on the graded characters of the automorphisms on the underlying conformal field theory, we propose a set of constructions of all but one of the 71 self-dual meromorphic bosonic conformal field theories at central charge 24. In the Z₂ case, this leads to an extension of the neighborhood graph of the even self-dual lattices in 24 dimensions to conformal field theories, and we demonstrate that the graph becomes disconnected.     

Affine Orbifolds and Rational Conformal Field Theory Extensions of W 1+∞

Chiral orbifold models are defined as gauge field theories with a finite gauge group Γ. We start with a conformal current algebra associated with a connected compact Lie group G and a negative definite integral invariant bilinear form on its Lie algebra. Any finite group Γ of inner automorphisms or (in particular, any finite subgroup of G) gives rise to a gauge theory with a chiral subalgebra of local observables invariant under Γ. A set of positive energy modules is constructed whose characters span, under some assumptions on Γ, a finite dimensional unitary representation of . We compute their asymptotic dimensions (thus singling out the nontrivial orbifold modules) and find explicit formulae for the modular transformations and hence, for the fusion rules. As an application we construct a family of rational conformal field theory (RCFT) extensions of W _1+∞ that appear to provide a bridge between two approaches to the quantum Hall effect. Received: 5 December 1996 / Accepted: 1 April 1997     

Modular Group Representations and Fusion in Logarithmic Conformal Field Theories and in the Quantum Group Center

The SL(2, ℤ)-representation π on the center of the restricted quantum group at the primitive 2p^th root of unity is shown to be equivalent to the SL(2, ℤ)-representation on the extended characters of the logarithmic (1, p) conformal field theory model. The multiplicative Jordan decomposition of the ribbon element determines the decomposition of π into a ``pointwise' product of two commuting SL(2, ℤ)-representations, one of which restricts to the Grothendieck ring; this restriction is equivalent to the SL(2, ℤ)-representation on the (1, p)-characters, related to the fusion algebra via a nonsemisimple Verlinde formula. The Grothendieck ring of at the primitive 2p^th root of unity is shown to coincide with the fusion algebra of the (1, p) logarithmic conformal field theory model. As a by-product, we derive q-binomial identities implied by the fusion algebra realized in the center of .     

Renormalizable N = 2 Supersymmetric Gauge Theories

Renormalizable supersymmetric models of matter multiplets coupled to a Yang Mills multiplet are studied in the conventional N = 2 superspace. It is found that a consistent formulation of the Feynman rules in superspace requires the introduction of a compensator superfield in the matter sector. If the quantization is performed within the background superfield approach, the use of the compensator is confined to the quantum corrections. The ensuing ghost structure is analyzed and the non-renormalization beyond one-loop is demonstrated. We report an N = 2 superfield calculation of the one-loop β-function via the supercurrent anomaly and thereby derive as a byproduct the finiteness criterion for N = 2 non-Abelian gauge theories.     

BPS Quivers and Spectra of Complete {\mathcal{N} = 2} Quantum Field Theories

We study the BPS spectra of ${\mathcal{N}=2}$ N = 2 complete quantum field theories in four dimensions. For examples that can be described by a pair of M5 branes on a punctured Riemann surface we explain how triangulations of the surface fix a BPS quiver and superpotential for the theory. The BPS spectrum can then be determined by solving the quantum mechanics problem encoded by the quiver. By analyzing the structure of this quantum mechanics we show that all asymptotically free examples, Argyres-Douglas models, and theories defined by punctured spheres and tori have a chamber with finitely many BPS states. In all such cases we determine the spectrum.