Exact Solutions for Loewner Evolutions

In this note, we solve the Loewner equation in the upper half-plane with forcing function ξ(t), for the cases in which ξ(t)has a power-law dependence on time with powers 0, 1/2, and 1. In the first case the trace of singularities is a line perpendicular to the real axis. In the second case the trace of singularities can do three things. If ξ(t)=2√kt, the trace is a straight line set at an angle to the real axis. If ξ(t)=2√k(1-t), as pointed out by Marshall and Rohde,⁽¹²⁾ the behavior of the trace as t approaches 1 depends on the coefficient κ. Our calculations give an explicit solution in which for κ<4 the trace spirals into a point in the upper half-plane, while for κ>4 it intersects the real axis. We also show that for κ=9/2 the trace becomes a half-circle. The third case with forcing ξ(t)=t gives a trace that moves outward to infinity, but stays within fixed distance from the real axis. We also solve explicitly a more general version of the evolution equation, in which ξ(t) is a superposition of the values ±1.     

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     

Stochastic Evolutions in Superspace and Superconformal Field Theory

Some stochastic evolutions of conformal maps can be described by SLE and may be linked to conformal field theory via stochastic differential equations and singular vectors in highest-weight modules of the Virasoro algebra. Here we discuss how this may be extended to superconformal maps of N=1 superspace with links to superconformal field theory and singular vectors of the N=1 superconformal algebra in the Neveu–Schwarz sector.     

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.     

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.     

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.     

Multiple Schramm–Loewner Evolutions and Statistical Mechanics Martingales

A statistical mechanics argument relating partition functions to martingales is used to get a condition under which random geometric processes can describe interfaces in 2d statistical mechanics at criticality. Requiring multiple SLEs to satisfy this condition leads to some natural processes, which we study in this note. We give examples of such multiple SLEs and discuss how a choice of conformal block is related to geometric configuration of the interfaces and what is the physical meaning of mixed conformal blocks. We illustrate the general ideas on concrete computations, with applications to percolation and the Ising model     

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.     

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.     

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.     

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.     

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.     

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     

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.     

Localization for Linear Stochastic Evolutions

We consider a discrete-time stochastic growth model on the d-dimensional lattice with non-negative real numbers as possible values per site. The growth model describes various interesting examples such as oriented site/bond percolation, directed polymers in random environment, time discretizations of the binary contact path process. We show the equivalence between the slow population growth and a localization property in terms of “replica overlap”. The main novelty of this paper is that we obtain this equivalence even for models with positive probability of extinction at finite time. In the course of the proof, we characterize, in a general setting, the event on which an exponential martingale vanishes in the limit.     

c = 2 Rational Toroidal Conformal Field Theories via the Gauss Product

We find a concise relation between the moduli , of a rational Narain lattice (,) and the corresponding momentum lattices of left and right chiral algebras via the Gauss product. As a byproduct, we find an identity which counts the cardinality of a certain double coset space defined for isometries between the discriminant forms of rank two lattices.     

Conformal invariance and stochastic Loewner evolution processes in two-dimensional Ising spin glasses

We present numerical evidence that the techniques of conformal field theory might be applicable to two-dimensional Ising spin glasses with Gaussian bond distributions. It is shown that certain domain wall distributions in one geometry can be related to that in a second geometry by a conformal transformation. We also present direct evidence that the domain walls are stochastic Loewner (SLE) processes with kappa approximately 2.1. An argument is given that their fractal dimension d(f) is related to their interface energy exponent theta by d(f) - 1 = 3/[4(3 + theta)], which is consistent with the commonly quoted values d(f) approximately 1.27 and theta approximately -0.28.     

Short Time Conservation of Gibbsianness Under Local Stochastic Evolutions

We prove that a Gibbs measure with a finite range interaction evolved under a general reversible local stochastic dynamics remains Gibbsian for a short interval of time. This generalizes previous results for Glauber dynamics.