David Ridout

D-branes on group manifolds and fusion rings

In this paper we compute the charge group for symmetry preserving D-branes on group manifolds for all simple, simply-connected, connected compact Lie groups G.

From percolation to logarithmic conformal field theory

Abstract The smallest deformation of the minimal model M ( 2 , 3 ) that can accommodate Cardys derivation of the percolation crossing probability is presented. It is shown that this leads to a consistent logarithmic conformal field theory at c = 0 . A simple recipe for computing the associated fusion rules is given. The differences between this theory and the other recently proposed c = 0 logarithmic conformal field theories are underlined. The discussion also emphasises the existence of invariant logarithmic couplings that generalise Guraries anomaly number.

On staggered indecomposable Virasoro modules

In this article, certain indecomposable Virasoro modules are studied. Specifically, the Virasoro mode L0 is assumed to be nondiagonalizable, possessing Jordan blocks of rank 2. Moreover, the module is further assumed to have a highest weight submodule, the “left module,” and that the quotient by this submodule yields another highest weight module, the “right module.” Such modules, which have been called staggered, have appeared repeatedly in the logarithmic conformal field theory literature, but their theory has not been explored in full generality. Here, such a theory is developed for the Virasoro algebra using rather elementary techniques. The focus centers on two different but related questions typically encountered in practical studies: How can one identify a given staggered module, and how can one demonstrate the existence of a proposed staggered module. Given just the values of the highest weights of the left and right modules, themselves subject to simple necessary conditions, invariants are define...

MODULAR DATA AND VERLINDE FORMULAE FOR FRACTIONAL LEVEL WZW MODELS I

Abstract The modular properties of fractional level sl ˆ ( 2 ) -theories and, in particular, the application of the Verlinde formula, have a long and checkered history in conformal field theory. Recent advances in logarithmic conformal field theory have led to the realisation that problems with fractional level models stem from trying to build the theory with an insufficiently rich category of representations. In particular, the appearance of negative fusion coefficients for admissible highest weight representations is now completely understood. Here, the modular story for certain fractional level theories is completed. Modular transformations are derived for the complete set of admissible irreducible representations when the level is k = − 1 2 or k = − 4 3 . The S-matrix data and Verlinde formula are then checked against the known fusion rules with complete agreement. Finally, an infinite set of modular invariant partition functions is constructed in each case.

Logarithmic conformal field theory: beyond an introduction

This article aims to review a selection of central topics and examples in logarithmic conformal field theory. It begins with the remarkable observation of Cardy that the horizontal crossing probability of critical percolation may be computed analytically within the formalism of boundary conformal field theory. Cardy’s derivation relies on certain implicit assumptions which are shown to lead inexorably to indecomposable modules and logarithmic singularities in correlators. For this, a short introduction to the fusion algorithm of Nahm, Gaberdiel and Kausch is provided. While the percolation logarithmic conformal field theory is still not completely understood, there are several examples for which the formalism familiar from rational conformal field theory, including bulk partition functions, correlation functions, modular transformations, fusion rules and the Verlinde formula, has been successfully generalized. This is illustrated for three examples: the singlet model , related to the triplet model , symplectic fermions and the fermionic bc ghost system; the fractional level Wess–Zumino–Witten model based on at , related to the bosonic βγ ghost system; and the Wess–Zumino–Witten model for the Lie supergroup , related to at and 1, the Bershadsky–Polyakov algebra and the Feigin–Semikhatov algebras . These examples have been chosen because they represent the most accessible, and most useful, members of the three best-understood families of logarithmic conformal field theories. The logarithmic minimal models , the fractional level Wess–Zumino–Witten models, and the Wess–Zumino–Witten models on Lie supergroups (excluding ). In this review, the emphasis lies on the representation theory of the underlying chiral algebra and the modular data pertaining to the characters of the representations. Each of the archetypal logarithmic conformal field theories is studied here by first determining its irreducible spectrum, which turns out to be continuous, as well as a selection of natural reducible, but indecomposable, modules. This is followed by a detailed description of how to obtain character formulae for each irreducible, a derivation of the action of the modular group on the characters, and an application of the Verlinde formula to compute the Grothendieck fusion rules. In each case, the (genuine) fusion rules are known, so comparisons can be made and favourable conclusions drawn. In addition, each example admits an infinite set of simple currents, hence extended symmetry algebras may be constructed and a series of bulk modular invariants computed. The spectrum of such an extended theory is typically discrete and this is how the triplet model arises, for example. Moreover, simple current technology admits a derivation of the extended algebra fusion rules from those of its continuous parent theory. Finally, each example is concluded by a brief description of the computation of some bulk correlators, a discussion of the structure of the bulk state space, and remarks concerning more advanced developments and generalizations. The final part gives a very short account of the theory of staggered modules, the (simplest class of) representations that are responsible for the logarithmic singularities that distinguish logarithmic theories from their rational cousins. These modules are discussed in a generality suitable to encompass all the examples met in this review and some of the very basic structure theory is proven. Then, the important quantities known as logarithmic couplings are reviewed for Virasoro staggered modules and their role as fundamentally important parameters, akin to the three-point constants of rational conformal field theory, is discussed. An appendix is also provided in order to introduce some of the necessary, but perhaps unfamiliar, language of homological algebra.

Logarithmic M(2,p) minimal models, their logarithmic couplings, and duality

Abstract A natural construction of the logarithmic extension of the M ( 2 , p ) (chiral) minimal models is presented, which generalises our previous model of percolation ( p = 3 ). Its key aspect is the replacement of the minimal model irreducible modules by reducible ones obtained by requiring that only one of the two principal singular vectors of each module vanish. The resulting theory is then constructed systematically by repeatedly fusing these building block representations. This generates indecomposable representations of the type which signify the presence of logarithmic partner fields in the theory. The basic data characterising these indecomposable modules, the logarithmic couplings, are computed for many special cases and given a new structural interpretation. Quite remarkably, a number of them are presented in closed analytic form (for general p). These are the prime examples of “gauge-invariant” data—quantities independent of the ambiguities present in defining the logarithmic partner fields. Finally, mere global conformal invariance is shown to enforce strong constraints on the allowed spectrum: It is not possible to include modules other than those generated by the fusion of the models building blocks. This generalises the statement that there cannot exist two effective central charges in a c = 0 model. It also suggests the existence of a second “dual” logarithmic theory for each p. Such dual models are briefly discussed.

sl(2)_{-1/2}: A case study

The construction of the non-logarithmic conformal field theory based on sl^(2)_{-1/2} is revisited. Without resorting to free-field methods, the determination of the spectrum and fusion rules is streamlined and the beta gamma ghost system is carefully derived as the extended algebra generated by the unique finite-order simple current. A brief discussion of modular invariance is given and the Verlinde formula is explicitly verified.

Convergence properties of gradient descent noise reduction

Gradient descent noise reduction is a technique that attempts to recover the true signal, or trajectory, from noisy observations of a non-linear dynamical system for which the dynamics are known. This paper provides the first rigorous proof that the algorithm will recover the original trajectory for a broad class of dynamical systems under certain conditions. The proof is obtained using ideas from linearisation theory. Since the first introduction of the algorithm it has been recognised that the algorithm can fail to recover the true trajectory, and it has been suggested that this is a practical or numerical limitation that is a consequence of near tangencies between stable and unstable manifolds. This paper demonstrates through numerical experiments and details of the proof that the situation is worse than expected in that near tangencies impose essential limitations on noise reduction, not just practical or numerical limitations. That is, gradient descent noise reduction will sometimes fail to recover the true trajectory, even with unlimited, perfect computation. On the other hand, the numerical experiments suggest that the gradient descent noise-reduction algorithm will always recover a trajectory that is entirely consistent with the evidence provided by the observations, that is, it attains the best that can be achieved given the observations. It is argued that near tangencies will therefore impose the same limitations on any noise-reduction algorithm.

The extended algebra of the minimal models

Abstract The minimal models M ( p ′ , p ) with p ′ > 2 have a unique (non-trivial) simple current of conformal dimension h = 1 4 ( p ′ − 2 ) ( p − 2 ) . The representation theory of the extended algebra defined by this simple current is investigated in detail. All highest weight representations are proved to be irreducible: There are thus no singular vectors in the extended theory. This has interesting structural consequences. In particular, it leads to a recursive method for computing the various terms appearing in the operator product expansion of the simple current with itself. The simplest extended models are analysed in detail and the question of equivalence of conformal field theories is carefully examined.

The extended algebra of the SU(2) Wess–Zumino–Witten models

Abstract The Wess–Zumino–Witten model defined on the group SU ( 2 ) has a unique (non-trivial) simple current of conformal dimension k / 4 for each level k . The extended algebra defined by this simple current is carefully constructed in terms of generalised commutation relations, and the corresponding representation theory is investigated. This extended algebra approach is proven to realise a faithful (“free-field-type”) representation of the SU ( 2 ) model. Subtleties in the formulation of the extended theory are illustrated throughout by the k = 1 , 2 and 4 models. For the first two cases, bases for the modules of the extended theory are given and rigorously justified.