The three-dimensional origin of the classifying algebra
arXiv:0907.0685 · doi:10.1016/j.nuclphysb.2009.07.017
Abstract
It is known that reflection coefficients for bulk fields of a rational conformal field theory in the presence of an elementary boundary condition can be obtained as representation matrices of irreducible representations of the classifying algebra, a semisimple commutative associative complex algebra. We show how this algebra arises naturally from the three-dimensional geometry of factorization of correlators of bulk fields on the disk. This allows us to derive explicit expressions for the structure constants of the classifying algebra as invariants of ribbon graphs in the three-manifold S^2 x S^1. Our result unravels a precise relation between intertwiners of the action of the mapping class group on spaces of conformal blocks and boundary conditions in rational conformal field theories.
References in corpus (24)
- Vertex Algebras and Algebraic Curves
- TFT construction of RCFT correlators I: Partition functions
- Supersymmetric Standard Model Spectra from RCFT orientifolds
- Completeness Conditions for Boundary Operators in 2D Conformal Field Theory
- D-branes on the Quintic
- Instanton Induced Neutrino Majorana Masses in CFT Orientifolds with MSSM-like spectra
- TFT construction of RCFT correlators IV: Structure constants and correlation functions
- Symmetry breaking boundaries I. General theory
- The geometry of WZW branes
- Boundaries, crosscaps and simple currents
- The conformal boundary states for SU(2) at level 1
- Branes: from free fields to general backgrounds
- Conformal Correlation Functions, Frobenius Algebras and Triangulations
- A classifying algebra for boundary conditions
- Morita classes of algebras in modular tensor categories
- The action of outer automorphisms on bundles of chiral blocks
- Orbifold analysis of broken bulk symmetries
- Cardy condition for open-closed field algebras
- On Frobenius algebras in rigid monoidal categories
- Bundle Gerbes and Surface Holonomy
- Discriminating MSSM families in (free-field) Gepner Orientifolds
- Classifying algebras for boundary conditions and traces on spaces of conformal blocks
- Permutation Orientifolds of Gepner Models
- On Reducibility of Mapping Class Group Representations: The SU(N) Case