Showing papers on "Logarithmic conformal field theory published in 2013"

Logarithmic conformal field theory: beyond an introduction

Abstract: 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 Conformal Field Theory: a Lattice Approach

Abstract: Logarithmic Conformal Field Theories (LCFT) play a key role, for instance, in the description of critical geometrical problems (percolation, self avoiding walks, etc.), or of critical points in several classes of disordered systems (transition between plateaus in the integer and spin quantum Hall effects). Much progress in their understanding has been obtained by studying algebraic features of their lattice regularizations. For reasons which are not entirely understood, the non semi-simple associative algebras underlying these lattice models - such as the Temperley-Lieb algebra or the blob algebra - indeed exhibit, in finite size, properties that are in full correspondence with those of their continuum limits. This applies to the structure of indecomposable modules, but also to fusion rules, and provides an `experimental' way of measuring couplings, such as the `number b' quantifying the logarithmic coupling of the stress energy tensor with its partner. Most results obtained so far have concerned boundary LCFTs, and the associated indecomposability in the chiral sector. While the bulk case is considerably more involved (mixing in general left and right moving sectors), progress has also been made in this direction recently, uncovering fascinating structures. This article provides a short general review of our work in this area.

Logarithmic conformal field theory: a lattice approach

Abstract: Logarithmic conformal field theories (LCFT) play a key role, for instance, in the description of critical geometrical problems (percolation, self-avoiding walks, etc), or of critical points in several classes of disordered systems (transition between plateaux in the integer and spin quantum Hall effects). Much progress in their understanding has been obtained by studying algebraic features of their lattice regularizations. For reasons which are not entirely understood, the non-semi-simple associative algebras underlying these lattice models—such as the Temperley–Lieb algebra or the blob algebra—indeed exhibit, in finite size, properties that are in full correspondence with those of their continuum limits. This applies not only to the structure of indecomposable modules, but also to fusion rules, and provides an ‘experimental’ way of measuring couplings, such as the ‘number b’ quantifying the logarithmic coupling of the stress–energy tensor with its partner. Most results obtained so far have concerned boundary LCFTs and the associated indecomposability in the chiral sector. While the bulk case is considerably more involved (mixing in general left and right moving sectors), progress has also recently been made in this direction, uncovering fascinating structures. This study provides a short general review of our work in this area.

Logarithmic conformal field theory

Abstract: Conformal field theory (CFT) has proven to be one of the richest and deepest subjects of modern theoretical and mathematical physics research, especially as regards statistical mechanics and string theory. It has also stimulated an enormous amount of activity in mathematics, shaping and building bridges between seemingly disparate fields through the study of vertex operator algebras, a (partial) axiomatisation of a chiral CFT. One can add to this that the successes of CFT, particularly when applied to statistical lattice models, have also served as an inspiration for mathematicians to develop entirely new fields: Schramm-Loewner evolution and Smirnov’s discrete complex analysis being notable examples. When the energy operator fails to be diagonalisable on the quantum state space, the CFT is said to be logarithmic. Consequently, a logarithmic CFT is one whose quantum space of states is constructed from a collection of representations which includes reducible but indecomposable ones. This qualifier arises because of the consequence that certain correlation functions will possess logarithmic singularities, something that contrasts with the familiar case of power law singularities. While such logarithmic singularities and reducible representations were noted by Rozansky and Saleur in their study of the U(1|1) Wess-Zumino-Witten model in 1992, the link between the non-diagonalisability of the energy operator and logarithmic singularities in correlators is usually ascribed to Gurarie’s 1993 article (his paper also contains the first usage of the term “logarithmic conformal field theory”). The class of CFTs that were under control at this time was quite small. In particular, an enormous amount of work from the statistical mechanics and string theory communities had produced a fairly detailed understanding of the (so-called) rational CFTs. However, physicists from both camps were well aware that applications from many diverse fields required significantly more complicated non-rational theories. Examples include critical percolation, supersymmetric string backgrounds, disordered electronic systems, sandpile models describing avalanche processes, and so on. In each case, the non-rationality and non-unitarity of the CFT suggested that a more general theoretical framework was needed. Driven by the desire to better understand these applications, the mid-nineties saw significant theoretical advances aiming to generalise the constructs of rational CFT to a more general class. In 1994, Nahm introduced an algorithm for computing the fusion product of representations which was significantly generalised two years later by Gaberdiel and Kausch who applied it to explicitly construct (chiral) representations upon which the energy operator acts non-diagonalisably. Their work made it clear that underlying the physically relevant correlation functions are classes of reducible but indecomposable representations that can be investigated mathematically to the benefit of applications. In another direction, Flohr had meanwhile initiated the study of modular properties of the characters of logarithmic CFTs, a topic which had already evoked much mathematical interest in the rational case. Since these seminal theoretical papers appeared, the field has undergone rapid development, both theoretically and with regard to applications. Logarithmic CFTs are now known to describe non-local observables in the scaling limit of critical lattice models, for example percolation and polymers, and are an integral part of our understanding of quantum strings propagating on supermanifolds. They are also believed to arise as duals of three-dimensional chiral gravity models, fill out hidden sectors in non-rational theories with non-compact target spaces, and describe certain transitions in various incarnations of the quantum Hall effect. Other physical

Tensor categories and the mathematics of rational and logarithmic conformal field theory

Abstract: We review the construction of braided tensor categories and modular tensor categories from representations of vertex operator algebras, which correspond to chiral algebras in physics. The extensive and general theory underlying this construction also establishes the operator product expansion for intertwining operators, which correspond to chiral vertex operators, and more generally, it establishes the logarithmic operator product expansion for logarithmic intertwining operators. We review the main ideas in the construction of the tensor product bifunctors and the associativity isomorphisms. For rational and logarithmic conformal field theories, we review the precise results that yield braided tensor categories, and in the rational case, modular tensor categories as well. In the case of rational conformal field theory, we also briefly discuss the construction of the modular tensor categories for the Wess–Zumino–Novikov–Witten models and, especially, a recent discovery concerning the proof of the fundamental rigidity property of the modular tensor categories for this important special case. In the case of logarithmic conformal field theory, we mention suitable categories of modules for the triplet -algebras as an example of the applications of our general construction of the braided tensor category structure.

Vertex operator (super)algebras and LCFT

Abstract: We review some of the developments in logarithmic conformal field theory from the vertex algebra point of view. Several important examples of vertex operator (super)algebras of the triplet type are discussed, including their representation theory. Particular emphasis is put on C2-cofiniteness of these vertex algebras, a description of Zhu’s algebras and the construction of logarithmic modules.

Infinitely extended Kac table of solvable critical dense polymers

Abstract: Solvable critical dense polymers is a Yang?Baxter integrable model of polymers on the square lattice. It is the first member of the family of logarithmic minimal models . The associated logarithmic conformal field theory admits an infinite family of Kac representations labelled by the Kac labels r, s = 1, 2, ?. In this paper, we explicitly construct the conjugate boundary conditions on the strip. The boundary operators are labelled by the Kac fusion labels (r, s) = (r, 1)?(1, s) and involve a boundary field ?. Tuning the field ? appropriately, we solve exactly for the transfer matrix eigenvalues on arbitrary finite-width strips and obtain the conformal spectra using the Euler?Maclaurin formula. The key to the solution is an inversion identity satisfied by the commuting double-row transfer matrices. The transfer matrix eigenvalues are classified by the physical combinatorics of the patterns of zeros in the complex spectral-parameter plane. This yields selection rules for the physically relevant solutions to the inversion identity which takes the form of a decomposition into irreducible blocks corresponding combinatorially to finitized characters given by generalized q-Catalan polynomials. This decomposition is in accord with the decomposition of the Kac characters into irreducible characters. In the scaling limit, we confirm the central charge c = ?2 and the Kac formula for the conformal weights for r, s = 1, 2, 3, ? in the infinitely extended Kac table.

Logarithmic Conformal Field Theory: Beyond an Introduction

Abstract: This article aims to review a selection of central topics and examples in logarithmic conformal field theory. It begins with a pure Virasoro example, critical percolation, then continues with a detailed exposition of symplectic fermions, the fractional level WZW model on SL(2;R) at level -1/2 and the WZW model on the Lie supergroup GL(1|1). It concludes with a general discussion of the so-called staggered modules that give these theories their logarithmic structure, before outlining a proposed strategy to understand more general logarithmic conformal field theories. Throughout, the emphasis is on continuum methods and their generalisation from the familiar rational case. In particular, the modular properties of the characters of the spectrum play a central role and Verlinde formulae are evaluated with the results compared to the known fusion rules. Moreover, bulk modular invariants are constructed, the structures of the corresponding bulk state spaces are elucidated, and a formalism for computing correlation functions is discussed.

Logarithmic operator intervals in the boundary theory of critical percolation

Abstract: We consider the sub-sector of the c = 0 logarithmic conformal field theory (LCFT) generated by the boundary condition changing (bcc) operator in two dimensional critical percolation. This operator is the zero weight Kac operator ?1, 2 ? ?, identified with the growing hull of the SLE6 process. We identify percolation configurations with the significant operators in the theory. We consider operators from the first four bcc operator fusions: the identity and ?; the stress-tensor and its logarithmic partner; ?? and its logarithmic partner; and the pre-logarithmic operator ?1, 3. We construct several intervals in the percolation model, each associated to one of the LCFT operators we consider, allowing us to calculate crossing probabilities and expectation values of crossing cluster numbers. We review the CG, which we use as a method of calculating these quantities when the number of bcc operator makes a direct solution to the system of differential equations intractable. Finally we discuss the case of the six-point correlation function, which applies to crossing probabilities between the sides of a conformal hexagon. Specifically we introduce an integral result that allows one to identify the probability that a single percolation cluster touches three alternating sides a hexagon with free boundaries. We give results of the numerical integration for the case of a regular hexagon.

Logarithmic operator intervals in the boundary theory of critical percolation

Abstract: We consider the sub-sector of the $c=0$ logarithmic conformal field theory (LCFT) generated by the boundary condition changing (bcc) operator in two dimensional critical percolation. This operator is the zero weight Kac operator $\phi_{1,2}$, identified with the growing hull of the SLE$_6$ process. We identify percolation configurations with the significant operators in the theory. We consider operators from the first four bcc operator fusions: the identity and bcc operator; the stress tensor and its logarithmic partner; the derivative of the bcc operator and its logarithmic partner; and the pre-logarithmic operator $\phi_{1,3}$. We construct several intervals in the percolation model, each associated to one of the LCFT operators we consider, allowing us to calculate crossing probabilities and expectation values of crossing cluster numbers. We review the Coulomb gas, which we use as a method of calculating these quantities when the number of bcc operator makes a direct solution to the system of differential equations intractable. Finally we discuss the case of the six-point correlation function, which applies to crossing probabilities between the sides of a conformal hexagon. Specifically we introduce an integral result that allows one to identify the probability that a single percolation cluster touches three alternating sides a hexagon with free boundaries. We give results of the numerical integration for the case of a regular hexagon.

On one-loop partition functions of three-dimensional critical gravities

Abstract: We calculate the one-loop partition function of three-dimensional parity even tricritical gravity. Agreement with logarithmic conformal field theory single-particle partition functions on the field theory side is found and we furthermore discover a partially massless limit of linearized six-derivative parity even gravity. Then we define a 'truncation' of the critical theory, at the level of the partition function, by calculating black hole determinants via summation over quasi-normal mode spectra and discriminating against those modes which are not present in the physical spectrum. This 'truncation' is applied to critical new massive gravity and three-dimensional parity even tricritical gravity.

Brst quantization of a sixth-order derivative scalar field theory

Abstract: We study a sixth-order derivative scalar field model in Minkowski spacetime as a toy model of higher-derivative critical gravity theories. This model is consistently quantized when using the Becchi–Rouet–Stora–Tyutin (BRST) quantization scheme even though it does not show gauge symmetry manifestly. Imposing a BRST quartet generated by two scalars and ghosts, there remains a nontrivial subspace with positive norm. This might be interpreted as a Minkowskian dual version of the unitary truncation in the logarithmic conformal field theory.

Indecomposability in field theory and applications to disordered systems and geometrical problems

Abstract: Logarithmic Conformal Field Theories (LCFTs) are crucial for describing the critical behavior of a variety of physical systems. These include phase transitions in disordered non-interacting electronic systems (such as the transition between plateaus in the integer quantum Hall effect), disordered critical points in classical statistical models (such as the random bond Ising model), or critical geometrical models (such as polymers and percolation). LCFTs appear when one has to give up the unitarity condition, which is natural in particle physics applications, but not in statistical mechanics and condensed matter physics. Without unitarity, the powerful algebraic approach of conformal invariance encounters formidable technical difficulties due to 'indecomposability'. This in turn yields logarithmic corrections to the power-law correlations at the critical point, and prevents the use of general classification techniques that have proven so powerful in the unitary case. The goal of this thesis is to understand LCFTs by studying their lattice regularizations, the crucial point being that most algebraic complications due to indecomposability occur in finite size systems as well. Our approach is thus to consider critical statistical models with non-diagonalizable transfer matrices, or gapless quantum spin chains with non-diagonalizable hamiltonians, and to study their scaling limit by utilizing a variety of algebraic, numerical and integrable techniques. We show how to measure numerically universal parameters that characterize the indecomposable representations of the Virasoro algebra which emerge in the thermodynamic limit. An extensive understanding of a wide class of lattice models allows us to conjecture a tentative classification of all possible (chiral) LCFTs with Virasoro symmetry only. This approach is partially extended to the bulk case, for which we discuss how the long-standing bulk CFT formulation of percolation can be tackled along these lines. We also argue that boundary and periodic lattice models can be related algebraically only in the case of minimal models, and we work out the consequences for the underlying boundary and bulk field theories. Several concrete applications to disordered systems and geometrical problems are discussed, and we uncover a large class of geometrical observables in the Potts model that behave logarithmically at the critical point.

Tensor categories and the mathematics of rational and logarithmic conformal field theory

Abstract: We review the construction of braided tensor categories and modular tensor categories from representations of vertex operator algebras, which correspond to chiral algebras in physics. The extensive and general theory underlying this construction also establishes the operator product expansion for intertwining operators, which correspond to chiral vertex operators, and more generally, it establishes the logarithmic operator product expansion for logarithmic intertwining operators. We review the main ideas in the construction of the tensor product bifunctors and the associativity isomorphisms. For rational and logarithmic conformal field theories, we review the precise results that yield braided tensor categories, and in the rational case, modular tensor categories as well. In the case of rational conformal field theory, we also briefly discuss the construction of the modular tensor categories for the Wess-Zumino-Novikov-Witten models and, especially, a recent discovery concerning the proof of the fundamental rigidity property of the modular tensor categories for this important special case. In the case of logarithmic conformal field theory, we mention suitable categories of modules for the triplet \mathcal{W}-algebras as an example of the applications of our general construction of the braided tensor category structure.

Aging logarithmic conformal field theory: a holographic view

Abstract: We consider logarithmic extensions of the correlation and response functions of scalar operators for the systems with aging as well as Schrodinger symmetry. Aging is known to be the simplest nonequilibrium phenomena, and its physical significances can be understood by the two-time correlation and response functions. Their logarithmic part is completely fixed by the bulk geometry in terms of the conformal weight of the dual operator and the dual particle number. Motivated by recent experimental realizations of Kardar-Parisi-Zhang universality class in growth phenomena and its subsequent theoretical extension to aging, we investigate our two-time correlation functions out of equilibrium, which show several qualitatively different behaviors depending on the parameters in our theory. They exhibit either growing or aging, i.e. power-law decaying, behaviors for the entire range of our scaling time. Surprisingly, for some parameter ranges, they exhibit growing at early times as well as aging at later times.

BRST quantization of a sixth-order derivative scalar field theory

Abstract: We study a sixth order derivative scalar field model in Minkowski spacetime as a toy model of higher-derivative critical gravity theories. This model is consistently quantized when using the Becchi-Rouet-Stora-Tyutin (BRST) quantization scheme even though it does not show gauge symmetry manifestly. Imposing a BRST quartet generated by two scalars and ghosts, there remains a non-trivial subspace with positive norm. This might be interpreted as a Minkowskian dual version of the unitary truncation in the logarithmic conformal field theory.

Born-Infeld Gravity Revisited

Abstract: In this paper, we investigate the behavior of linearized gravitational excitation in the Born–Infeld gravity in AdS3 space. We obtain the linearized equation of motion and show that this higher-order gravity propagate two gravitons, massless and massive, on the AdS3 background. In contrast to the R2 models, such as TMG or NMG, Born–Infeld gravity does not have a critical point for any regular choice of parameters. So the logarithmic solution is not a solution of this model, due to this one cannot find a logarithmic conformal field theory as a dual model for Born–Infeld gravity.

Massive logarithmic graviton in the critical generalized massive gravity

Abstract: We study the generalized massive gravity in three dimensional flat spacetime. A massive logarithmic mode is propagating in the flat spacetime at the critical point where two masses degenerate. Furthermore, we discuss the logarithmic extension of the Galilean conformal algebra (GCA) which may arise from the exotic and standard rank-2 logarithmic conformal field theory (LCFT) on the boundary of AdS3 spacetime.

Tricritical gravity waves in the four-dimensional generalized massive gravity

Abstract: We construct a generalized massive gravity by combining quadratic curvature gravity with the Chern–Simons term in four dimensions. This may be a candidate for the parity-odd tricritical gravity theory. Considering the AdS4 vacuum solution, we derive the linearized Einstein equation, which is not similar to that of the three dimensional (3D) generalized massive gravity. When a perturbed metric tensor is chosen to be the Kerr–Schild form, the linearized equation reduces to a single massive scalar equation. At the tricritical points where two masses are equal to −1 and 2, we obtain a log-square wave solution to the massive scalar equation. This is compared to 3D tricritical generalized massive gravity, whose dual is a rank-3 logarithmic conformal field theory.