Ramanujan's 1/π Series and Conformal Field Theories.

Logarithmic operators and logarithmic conformal field theories are reviewed. Prominent examples considered here include c = −2 and c = 0 logarithmic conformal field theories. c = 0 logarithmic conformal field theories are especially interesting since they describe some of the critical points of a variety of longstanding problems involving a two dimensional quantum particle moving in a spatially random potential, as well as critical two dimensional self-avoiding random walks and percolation. Lack of classification of logarithmic conformal field theories remains a major impediment to progress towards finding complete solutions to these problems.

This thesis is devoted to the study of a new construction of two-dimensional topological conformal field theories by gauge fixing two-dimensional topological gauge theories. We study in detail the Lorenz-gauged abelian and non-abelian BF theory, which are topological conformal classically and on the quantum level. We find that the abelian model corresponds to Witten’s B-model with a parity shifted flat target space. It is therefore obtained by twisting a N = (2,2) supersymmetric model, while no such twist exists in the non-abelian case. Furthermore, we study an analogue of Gromov-Witten periods in the abelian model. Finally, we show that the non-abelian model allows non-trivial Jordan blocks of the Hamiltonian and thus defines a logarithmic conformal field theory. The existence of infinite-dimensional Jordan blocks allows an explicit construction of primary fields, whose conformal weights are subject to quantum corrections.

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.

Boundary states in c=−2 logarithmic conformal field theory

We describe the structure of the moduli space of $\ensuremath{\sigma}$-model couplings for the worldsheet description of a system of N D-particles, in the case that the couplings are represented by a pair of logarithmic recoil operators. We derive expressions for the canonical momenta conjugate to the D-particle couplings and the Zamolodchikov metric to the first few orders in $\ensuremath{\sigma}$-model perturbation theory. We show, using only very general properties of the operator product expansion in logarithmic conformal field theories, that the canonical dynamics on moduli space agree with the predictions of the non-Abelian generalization of the Born-Infeld effective action for D-particles with a symmetrized trace structure. We demonstrate that the Zamolodchikov metric naturally encodes the short-distance structure of spacetime, and from this we derive uncertainty relations for the D-particle coordinates directly from the quantum string theory. We show that the moduli space geometry naturally leads to new forms of spacetime indeterminancies involving only spatial coordinates of target space and illustrate the manner in which the open string interactions between D-particles lead to a spacetime quantization. We also derive appropriate non-Abelian generalizations of the string-modified Heisenberg uncertainty relations and the space-time uncertainty principle. The non-Abelian uncertainties exhibit decoherence effects suggesting the interplay of quantum gravity in multiple D-particle dynamics.

A comprehensive introduction to two-dimensional conformal field theory is given. The structure of the meromorphic subtheory is described in detail, and a number of examples are presented explicitly. Standard constructions such as the coset and the orbifold construction are explained. The concept of a representation of the meromorphic theory is introduced, and the role of Zhu's algebra in classifying highest weight representations is elucidated. The fusion product of two representations and the corresponding fusion rules are defined, and Verlinde's formula is explained. Finally, higher correlation functions are considered, and the polynomial relations of Moore and Seiberg and the quantum group structure of chiral conformal field theory are discussed. The treatment is relatively general and also allows for a description of less well known classes of theories such as logarithmic conformal field theories.

It is discussed how a limiting procedure of conformal field theories may result in logarithmic conformal field theories with Jordan cells of arbitrary rank. This extends our work on rank-two Jordan cells. We also consider the limits of certain three-point functions and find that they are compatible with known results. The general construction is illustrated by logarithmic limits of minimal models in conformal field theory. Characters of quasirational representations are found to emerge as the limits of the associated irreducible Virasoro characters.

We construct chiral algebras that centralize rank-2 Nichols algebras with at least one fermionic generator. This gives ‘logarithmic’ W-algebra extensions of a fractional-level algebra. We discuss crucial aspects of the emerging general relation between Nichols algebras and logarithmic conformal field theory (CFT) models: (i) the extra input, beyond the Nichols algebra proper, needed to uniquely specify a conformal model; (ii) a relation between the CFT counterparts of Nichols algebras connected by Weyl groupoid maps; and (iii) the common double bosonization U(X) of such Nichols algebras. For an extended chiral algebra, candidates for its simple modules that are counterparts of the U(X) simple modules are proposed, as a first step toward a functorial relation between U(X) and W-algebra representation categories.

We describe an approach to logarithmic conformal field theories as limits of sequences of ordinary conformal field theories with varying central charge c. Logarithmic behaviour arises from degeneracies in the spectrum of scaling dimensions at certain values of c. The theories we consider are all invariant under some internal symmetry group, and logarithmic behaviour occurs when the decomposition of the physical observables into irreducible operators becomes singular. Examples considered are quenched random magnets using the replica formalism, self-avoiding walks as the n → 0 limit of the O(n) model, and percolation as the limit Q → 1 of the Potts model. In these cases we identify logarithmic operators and pay particular attention to how the c → 0 paradox is resolved and how the b-parameter is evaluated. We also show how this approach gives information on logarithmic behaviour in the extended Ising model, uniform spanning trees and the O( − 2) model. Most of our results apply to general dimensionality. We also consider massive logarithmic theories and, in two dimensions, derive sum rules for the effective central charge and the b-parameter.

Logarithmic conformal field theory is investigated using the AdS/CFT correspondence and a novel method based on nilpotent weights. Using this device we add ghost fermions and point to a BRST invariance of the theory.

We examine the properties of two-dimensional conformal field theories(CFTs) with vanishing central charge based on the extended Kac table forc(9, 6) = 0 using a general ansatz for the stress energy tensor residing in a Jordan cell of rank two.Within this set-up we will derive the operator product expansions(OPEs) and two-point functions of the stress energy tensorT(z) and its logarithmicpartner field t(z) and illustrate this by a bosonic field realization.We will show why our approach may be more promising than those chosen in the literatureso far, including a discussion on properties of the augmented minimal model with vanishingcentral charge such as full conformal invariance of the vacuum as a state in an irreduciblerepresentation.Furthermore we will present a more general solution of another solution to the c→0 catastrophe based on a logarithmic CFT tensor model. Asan example, we consider a tensor product of the well knownc = −2 logarithmic CFT with a fourfold Ising model.We give an overview of the possible configurations and various consequencesfor the two-point functions and the OPEs of the stress energy tensorT(z) and its logarithmicpartner field t(z). We will motivate why, due to the full conformal invariance of the vacuum atc = 0, we should assume a Jordan cell for the identity, sincet(z) is presumably adescendant of a new h = 0 field.

We discuss several closely related concepts in the NSR formulation of superstring theory. We demonstrated that recently proposed NSR model for superstrings on AdS 5× S5 is described by the world sheet logarithmic conformal field theory (LCFT). The origin of LCFT on a world sheet is closely connected to the matter-ghost mixing in the structure of a brane-like vortex operators. We suggest a dynamical origin of M theory as a string theory with an extra dimension given by bosonized superconformal ghosts.

Logarithmic conformal field theories have a vast range of applications, from critical percolation to systems with quenched disorder. In this paper we thoroughly examine the structure of these theories based on their symmetry properties. Our analysis is model-independent and holds for any spacetime dimension. Our results include a determination of the general form of correlation functions and conformal block decompositions, clearing the path for future bootstrap applications. Several examples are discussed in detail, including logarithmic generalized free fields, holographic models, self-avoiding random walks and critical percolation.

Nonabelions in the fractional quantum hall effect

Logarithmic conformal field theory and Seiberg-Witten models