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arXiv:hep-th/0009096

Logarithmic CFT on the Boundary and the World-Sheet

Alex Lewis

Published Sep 12, 2000• 7 citations
arXivHTMLPDF

Abstract

The correspondences between logarithmic operators in the CFTs on the boundary of AdS_3 and on the world-sheet and dipole fields in the bulk are studied using the free field formulation of the SL(2,C)/SU(2) WZNW model. We find that logarithmic operators on the boundary are related to operators on the world-sheet which are in indecomposable representations of SL(2). The Knizhnik-Zamolodchikov equation is used to determine the conditions for those representations to appear in the operator product expansions of the model.

DOI
10.48550/arxiv.hep-th/0009096

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The correspondences between logarithmic operators in the CFTs on the boundary of AdS_3 and on the world-sheet and dipole fields in the bulk are studied using the free field formulation of the SL(2,C)/SU(2) WZNW model. We find that logarithmic operators on the boundary are related to operators on the world-sheet which are in indecomposable representations of SL(2). The Knizhnik-Zamolodchikov equation is used to determine the conditions for those representations to appear in the operator product expansions of the model.
Work of the last few years has shown that the key algebraic features of Logarithmic Conformal Field Theories (LCFTs) are already present in some finite lattice systems (such as the XXZ spin-1/2 chain) before the continuum limit is taken. This has provided a very convenient way to analyze the structure of indecomposable Virasoro modules and to obtain fusion rules for a variety of models such as (boundary) percolation etc. LCFTs allow for additional quantum numbers describing the fine structure of the indecomposable modules, and generalizing the ‘b-number’ introduced initially by Gurarie for the c=0 case. The determination of these indecomposability parameters (or logarithmic couplings) has given rise to a lot of algebraic work, but their physical meaning has remained somewhat elusive. In a recent paper, a way to measure b for boundary percolation and polymers was proposed. We generalize this work here by devising a general strategy to compute matrix elements of Virasoro generators from the numerical analysis of lattice models and their continuum limit. The method is applied to XXZ spin-1/2 and spin-1 chains with open (free) boundary conditions. They are related to gl(n+m|m) and osp(n+2m|2m)-invariant superspin chains and to non-linear sigma models with supercoset target spaces. These models can also be formulated in terms of dense and dilute loop gas. We check the method in many cases where the results were already known analytically. Furthermore, we also confront our findings with a construction generalizing Gurarieʼs, where logarithms emerge naturally in operator product expansions to compensate for apparently divergent terms. This argument actually allows us to compute indecomposability parameters in any logarithmic theory. A central result of our study is the construction of a Kac table for the indecomposability parameters of the logarithmic minimal models LM(1,p) and LM(p,p+1).
Abstract The relationship between bulk and boundary properties is one of the founding features of (rational) conformal field theory (CFT). Our goal in this paper is to explore the possibility of having an equivalent relationship in the context of lattice models. We focus on models based on the Temperley–Lieb algebra, and use the concept of ‘braid translation’, which is a natural way, in physical terms, to ‘close’ an open spin chain by adding an interaction between the first and last spins using braiding to ‘bring’ them next to each other. The interaction thus obtained is in general non-local, but has the key feature that it is expressed solely in terms of the algebra for the open spin chain—the ‘ordinary’ Temperley–Lieb algebra and its blob algebra generalization. This is in contrast with the usual periodic spin chains which involve only local interactions, and are described by the periodic Temperley–Lieb algebra. We show that for the restricted solid-on-solid models, which are known to be described by minimal unitary CFTs (with central charge c < 1 ) in the continuum limit, the braid translation in fact does provide the ordinary periodic model starting from the open model with fixed (identical) boundary conditions on the two sides of the strip. This statement has a precise mathematical formulation, which is a pull-back map between irreducible modules of, respectively, the blob algebra and the affine Temperley–Lieb algebra. We then turn to the same kind of analysis for two models whose continuum limits are logarithmic CFTs (LCFTs)—the alternating g l ( 1 | 1 ) and s l ( 2 | 1 ) spin chains. We find that the result for minimal models does not hold any longer: braid translation of the relevant (in that case, indecomposable but not irreducible) modules of the Temperley–Lieb algebra does not give rise to the modules known to be present in the periodic chains. In the g l ( 1 | 1 ) case, the content in terms of the irreducibles is the same, as well as the spectrum, but the detailed structure (like logarithmic coupling) is profoundly different. This carries over to the continuum limit. The situation is similar for the s l ( 2 | 1 ) case. The problem of relating bulk and boundary lattice models for LCFTs thus remains open.
We construct boundary states in SU (2) k WZNW models using the bosonized Wakimoto free-field representation and study their properties. We introduce a Fock space representation of Ishibashi states which are coherent states of bosons with zero-mode momenta (boundary Coulomb-gas charges) summed over certain lattices according to Fock space resolution of SU (2) k . The Virasoro invariance of the coherent states leads to families of boundary states including the B-type D-branes found by Maldacena, Moore and Seiberg as well as the A-type corresponding to trivial current gluing conditions. We then use the Coulomb-gas technique to compute exact correlation functions of WZNW primary fields on the disk topology with A- and B-type Cardy states on the boundary. We check that the obtained chiral blocks for A-branes are solutions of the Knizhnik–Zamolodchikov equations.