Conformal Boundary Conditions and what they teach us

Abstract

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.

Expanded version of lectures given at the Summer School and Conference Nonperturbative Quantum Field Theoretic Methods and their Applications, August 2000, Budapest, Hungary. 35 pages

Cited by in corpus (22)

• Duality and defects in rational conformal field theory
• Permeable conformal walls and holography
• TFT construction of RCFT correlators III: Simple currents
• WZW branes and gerbes
• Reflection and Transmission for Conformal Defects
• TFT construction of RCFT correlators IV: Structure constants and correlation functions
• TFT construction of RCFT correlators II: Unoriented world sheets
• Permutation Branes
• Conformal boundary loop models
• Bosonization and Scale Invariance on Quantum Wires
• Non-compact String Backgrounds and Non-rational CFT
• Abelian and non-Abelian branes in WZW models and gerbes
• Lorentzian Dynamics and Factorization Beyond Rationality
• Crossing Kernels for Boundary and Crosscap CFTs
• Boundary States of c=1 and 3/2 Rational Conformal Field Theories
• Fermionization and boundary states in 1+1 dimensions
• Lattice regularisation of a non-compact boundary conformal field theory
• Ising model conformal boundary conditions from open string field theory
• Symmetry-breaking boundary states for WZW models
• Topological defects in periodic RSOS models and anyonic chains
• Disk one-point function for non-rational conformal theories
• The Boundary Conformal Field Theories of the 2D Ising critical points