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New boundary conformal field theories indexed by the simply laced Lie algebras

A.F. Yegulalp

Published Jan 1, 1995• 3 citations
OpenAlex

Abstract

We consider the field theory of N massless bosons which are free except for an interaction localized on the boundary of their 1+1 dimensional world. The boundary action is the sum of two pieces: a periodic potential and a coupling to a uniform abelian gauge field. Such models arise in open string theory and dissipative quantum mechanics, and possibly in edge state tunneling in the fractional quantized Hall effect. We explicitly show that conformal invariance is unbroken for certain special choices of the gauge field and the periodic potential. These special cases are naturally indexed by semi-simple, simply laced Lie algebras. For each such algebra, we have a discrete series of conformally invariant theories where the potential and gauge field are conveniently given in terms of the weight lattice of the algebra. We compute the exact boundary state for these theories, which explicitly shows the group structure. The partition function and correlation functions are easily computed using the boundary state result.

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We consider the field theory of N massless bosons which are free except for an interaction localized on the boundary of their 1+1 dimensional world. The boundary action is the sum of two pieces: a periodic potential and a coupling to a uniform abelian gauge field. Such models arise in open string theory and dissipative quantum mechanics, and possibly in edge state tunneling in the fractional quantized Hall effect. We explicitly show that conformal invariance is unbroken for certain special choices of the gauge field and the periodic potential. These special cases are naturally indexed by semi-simple, simply laced Lie algebras. For each such algebra, we have a discrete series of conformally invariant theories where the potential and gauge field are conveniently given in terms of the weight lattice of the algebra. We compute the exact boundary state for these theories, which explicitly shows the group structure. The partition function and correlation functions are easily computed using the boundary state result.
The theory of a massless two-dimensional scalar field with a periodic boundary interaction is considered. At a critical value of the period this system defines a conformal field theory and can be reexpressed in terms of free fermions, which provide a simple realization of a hidden SU(2) symmetry of the original theory. The partition function and the boundary S matrix can be computed exactly as a function of the strength of the boundary interaction. We first consider open strings with one interacting and one Dirichlet boundary, and then with two interacting boundaries. The latter corresponds to motion in a periodic tachyon background, and the spectrum exhibits an interesting band structure which interpolates between free propagation and tight binding as the interaction strength is varied.
We study the conformal field theory of a free massless scalar field living on the half line with interactions introduced via a periodic potential at the boundary. An SU(2)SU(2)SU(2) current algebra underlies this system and the interacting boundary state is given by a global SU(2)SU(2)SU(2) rotation of the left-moving fields in the zero-potential (Neumann) boundary state. As the potential strength varies from zero to infinity, the boundary state interpolates between the Neumann and the Dirichlet values. The full S-matrix for scattering from the boundary, with arbitrary particle production, is explicitly computed. To maintain unitarity, it is necessary to attribute a hidden discrete ``soliton'' degree of freedom to the boundary. The same unitarity puzzle occurs in the Kondo problem, and we anticipate a similar solution.
This dissertation discusses some applications of conformal field theories (CFTs) in topological phases of matter. The first part is devoted to a discussion of coupled wire constructions of some novel quantum Hall systems. Through a theoretical coupled wire model, we construct strongly correlated electronic integer quantum Hall states with filling factor 16. The edge state is a bosonic chiral (E8)1 CFT, which is closely related to topological paramagnets in (3+1)d. As a distinguishing feature, these states support electric and thermal Hall transport violating the Wiedemann-Franz law. We further construct two descendant non-Abelian quantum Hall states at filling 8, each carrying bosonic chiral (G2)1 or (F4)1 edge theories, and hosting Fibonacci anyonic excitations in the bulk. Finally, we discover a new notion of particle-hole conjugation based on the E8 state that relates the G2 and F4 Fibonacci states, which is reminiscent of similar physics in half-filled Landau level. The second part is focused on the surface topological orders of 3D bulk topological systems. Symmetry protected and symmetry-enriched topological (SPT/SET) phases in three dimensions are quantum systems that support non-trivial two-dimensional surface states. These surface states develop finite excitation energy gaps when the relevant symmetries are broken. On the other hand, one-dimensional gapless modes can populate along interfaces that separate adjacent gapped surface domains with distinct symmetry-breaking orders. A surface strip pattern in general reduces the low-energy SPT/SET surface degrees of freedom onto a 2D array of gapless 1D channels. These channels can be coupled to one another by quasiparticle tunneling, and these inter-wire interactions collectively provide an effective description of the surface state. In this part, we study a general class of symmetry-preserving or breaking SPT/SET surface states that admit finite excitation energy gaps and Abelian topological orders via the coupled wire construction. In particular, we focus on the prototype Abelian surface topological orders that fall under the ADE classification of simply-laced Lie algebras. We also elaborate on the emergent symmetry and duality properties of the coupled wire models. The third part is to discuss the relation between the conformal boundary state and (2+1)d SPT phases. We propose a diagnostic tool for detecting nontrivial symmetry-protected topological (SPT) phases protected by a symmetry group G in 2 + 1 dimensions. Our method is based on directly studying the 1 + 1-dimensional anomalous edge conformal field theory (CFT) of SPT phases. We claim that if the CFT is the edge theory of an SPT phase, then there must be an obstruction to cutting it open. This obstruction manifests as the non-existence of boundary states in the CFT that preserves both the conformal symmetry and the global symmetry G. We discuss the relation between edgeability and gappability in the presence of G. We study several examples including time-reversal symmetric topological insulators, ZN symmetric bosonic SPT phases, and Z2 × Z2 symmetric topological superconductors.
Some results for commuting local integrals of motion for free bosonic theories are presented, which can be applied in construction of integrable boundary states. We consider a particular model of two-dimensional quantum field theory on a disk, whose bulk dynamics is that of a two-component free massless Bose field X = (X, Y), and interaction occurs at the boundary, where the boundary values (XB, YB) are constrained to special curve—the “paperclip brane”. The interaction breaks conformal invariance, but we argue that it preserves integrability. We propose exact expression for the disk partition function (and more general overlap amplitudes ⟨P| B⟩ of the boundary state with all primary states) in terms of solutions of certain ordinary linear differential equations. Usable program code for calculation of commuting integrals of motion is also introduced.
We formulate the ZN{Z}_{N}ZN​ group gauging operations (or group extensions) in (smeared) boundary conformal field theories (BCFTs) and bulk conformal field theories, and study their applications to various phenomena in topologically ordered systems. We apply the resultant theories to analyze the correspondence between the renormalization group (RG) flow of CFTs and the classification of topological quantum field theories using the testable information of general classes of partition functions. One can obtain the bulk topological properties of the 2+12+12+1 dimensional topological ordered phase corresponding to the massive RG flow of the 1+11+11+1 dimensional systems equivalent to the smeared BCFT. We present an obstruction to mass condensation for the smeared BCFT analogous to the Lieb-Shultz-Mattis theorem for noninvertible symmetry. Corresponding to the bulk topological degeneracies in 2+12+12+1 dimensions and gapped quantum phases in 1+11+11+1 dimensions, we construct a new series of BCFT. We also study the implications of the massless RG flow of the 1+11+11+1 dimensional CFTs corresponding to the domain wall or coupling problem in 2+12+12+1 dimensional topological orders, which corresponds to the earlier proposal by L. Kong and H. Zheng [Nucl. Phys. B 966, 115384 (2021)] closely related to the integer-spin simple current by Schellekens and Gato-Rivera. We reveal that two distinct kinds of massless RG flow exist, coming from anomaly cancellation and anomaly matching related to Witt equivalence of bulk CFTs. The anomaly cancellation flows produce a new series of modular invariants as coupled models, and the anomaly matching flows produce nontrivial domain walls changing the chirality of anyons. Our paper gives a unified direction for future theoretical and numerical studies of the topological phase based on the established data of classifications of conformal field theories or modular invariants (and the corresponding category theories).