Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Non-perturbative Quantum Effects

Boundary conditions in conformal and integrable theories

Jean-Bernard Zuber, V. B. Petkova

Published Sep 25, 2000• 1 citation
Publisher

Abstract

The study of boundary conditions in rational conformal field theories is not only physically important.It also reveals a lot on the structure of the theory "in the bulk".The same graphs classify both the torus and the cylinder partition functions and provide data on their hidden "quantum symmetry".The Ocneanu triangular cells -the 3j-symbols of these symmetries, admit various interpretations and make a link between different problems.

DOI
10.22323/1.006.0038

Related papers

Perturbations of boundary conformal field theories

A. Recknagel, Volker Schomerus

We have gathered quite a large amount of technology to construct conformal boundary conditions and study their properties. So far, most of the tools relied on detailed knowledge of the underlying algebraic structure of conformal field theories (CFTs), such as representations of symmetry algebras and modular transformations. In almost all cases, this route will lead to a restricted class of boundary conditions only, favouring those that preserve a lot of symmetry. In “real life applications” of boundary CFT, arising in string theory, in condensed matter physics, or in the study of integrable models, we are often faced with situations where we start from a simple given boundary condition and then “deform” by “turning on perturbations”. In this chapter, we will discuss methods aimed at making this procedure well defined and at determining the perturbed boundary theory. Most of the time, we will restrict ourselves to pure boundary perturbations, which leave the bulk CFT untouched. We will sometimes be led to new boundary conditions that would have been difficult to find directly. The total space of boundary CFTs takes the form of a fibration over the moduli space of bulk theories, with the fibre over each point made up from all boundary conditions (branes) that exist for that bulk theory. Marginal boundary deformations, if they exist, will generate, starting from a single boundary condition, whole continuous families within this boundary moduli space.

Cambridge University Press eBooks•Nov 5, 2013•0 citations

CONFORMAL BOUNDARY CONDITIONS: and what they teach us

V. B. Petkova, Jean-Bernard Zuber

The question of boundary conditions in conformal field theories is discussed, in the light of recent progress. Two kinds of boundary conditions are examined, along open boundaries of the system, or along closed curves or ``seams''. Solving consistency conditions known as Cardy equation is shown to amount to the algebraic problem of finding integer valued representations of (one or two copies of) the fusion algebra. Graphs encode these boundary conditions in a natural way, but are also relevant in several aspects of physics ``in the bulk''. Quantum algebras attached to these graphs contain information on structure constants of the operator algebra, on the Boltzmann weights of the corresponding integrable lattice models etc. Thus the study of boundary conditions in Conformal Field Theory offers a new perspective on several old physical problems and offers an explicit realisation of recent mathematical concepts.

arXiv:hep-th/0103007•Non-Perturbative QFT Methods and Their Applications•May 1, 2001•45 citations

Finite size effects in perturbed boundary conformal field theories

Patrick Dorey, Mathias Pillin, Pocklington, A. +3 more

We discuss the finite-size properties of a simple integrable quantum field theory in 1+1 dimensions with non-trivial boundary conditions. Novel off-critical identities between cylinder partition functions of models with differing boundary conditions are derived.

arXiv:hep-th/0010278•Oct 30, 2000•4 citations

Finite size effects in perturbed boundary conformal field theories

Roberto Tateo, Patrick Dorey, Mathias Pillin +3 more

We discuss the finite-size properties of a simple integrable quantum field theory in 1+1 dimensions with non-trivial boundary conditions.Novel off-critical identities between cylinder partition functions of models with differing boundary conditions are derived.

Non-perturbative Quantum Effects•Sep 25, 2000•6 citations

Boundary Conditions in Conformal Field Theory

John Cardy

<!-- *** Custom HTML *** --> We consider conformal field theories on manifolds with a boundary, and the constraints placed by modular invariance on their partition functions. In particular, the partition functions on an annulus with particular boundary conditions are given by the fusion rules. This leads to a simple derivation of the Verlinde formula. We note the remarkable fact that, for some integrable models, these partition functions have the same form away from criticality, with the modular parameter $q$ of the annulus replaced by a temperature-like variable, and give a partial explanation of this in the case of the Ising model.

Advanced studies in pure mathematics•Jun 17, 2018•17 citations

Fusion Algebras and Differential Equation Approach to Rational Conformal Field Theories

F. Ravanini

Constraints on conformal dimensions and central charge of Rational Conformal Field Theories (RCFTs) coming from analytic closure of 4-point functions on the sphere and modular invariance of partition function on the torus are reviewed. These constraints, together with modular invariance ones and a deeper understanding of the structure of Fusion Algebras (FAs) can help a lot in the classification of RCFTs. In particular we show how FAs with 2 and 3 primary operators can be completely classified.

•Jan 1, 1990•0 citations
The study of boundary conditions in rational conformal field theories is not only physically important.It also reveals a lot on the structure of the theory "in the bulk".The same graphs classify both the torus and the cylinder partition functions and provide data on their hidden "quantum symmetry".The Ocneanu triangular cells -the 3j-symbols of these symmetries, admit various interpretations and make a link between different problems.
We have gathered quite a large amount of technology to construct conformal boundary conditions and study their properties. So far, most of the tools relied on detailed knowledge of the underlying algebraic structure of conformal field theories (CFTs), such as representations of symmetry algebras and modular transformations. In almost all cases, this route will lead to a restricted class of boundary conditions only, favouring those that preserve a lot of symmetry. In “real life applications” of boundary CFT, arising in string theory, in condensed matter physics, or in the study of integrable models, we are often faced with situations where we start from a simple given boundary condition and then “deform” by “turning on perturbations”. In this chapter, we will discuss methods aimed at making this procedure well defined and at determining the perturbed boundary theory. Most of the time, we will restrict ourselves to pure boundary perturbations, which leave the bulk CFT untouched. We will sometimes be led to new boundary conditions that would have been difficult to find directly. The total space of boundary CFTs takes the form of a fibration over the moduli space of bulk theories, with the fibre over each point made up from all boundary conditions (branes) that exist for that bulk theory. Marginal boundary deformations, if they exist, will generate, starting from a single boundary condition, whole continuous families within this boundary moduli space.
The question of boundary conditions in conformal field theories is discussed, in the light of recent progress. Two kinds of boundary conditions are examined, along open boundaries of the system, or along closed curves or ``seams''. Solving consistency conditions known as Cardy equation is shown to amount to the algebraic problem of finding integer valued representations of (one or two copies of) the fusion algebra. Graphs encode these boundary conditions in a natural way, but are also relevant in several aspects of physics ``in the bulk''. Quantum algebras attached to these graphs contain information on structure constants of the operator algebra, on the Boltzmann weights of the corresponding integrable lattice models etc. Thus the study of boundary conditions in Conformal Field Theory offers a new perspective on several old physical problems and offers an explicit realisation of recent mathematical concepts.
<!-- *** Custom HTML *** --> We consider conformal field theories on manifolds with a boundary, and the constraints placed by modular invariance on their partition functions. In particular, the partition functions on an annulus with particular boundary conditions are given by the fusion rules. This leads to a simple derivation of the Verlinde formula. We note the remarkable fact that, for some integrable models, these partition functions have the same form away from criticality, with the modular parameter qqq of the annulus replaced by a temperature-like variable, and give a partial explanation of this in the case of the Ising model.