Fermionic CFTs and classifying algebras

Abstract

We study fermionic conformal field theories on surfaces with spin structure in the presence of boundaries, defects, and interfaces. We obtain the relevant crossing relations, taking particular care with parity signs and signs arising from the change of spin structure in different limits. We define fermionic classifying algebras for boundaries, defects, and interfaces, which allow one to read off the elementary boundary conditions, etc. As examples, we define fermionic extensions of Virasoro minimal models and give explicit solutions for the spectrum and bulk structure constants. We show how the $A$- and $D$-type fermionic Virasoro minimal models are related by a parity-shift operation which we define in general. We study the boundaries, defects, and interfaces in several examples, in particular in the fermionic Ising model, i.e. the free fermion, in the fermionic tri-critical Ising model, i.e. the first unitary $N=1$ superconformal minimal model, and in the supersymmetric Lee-Yang model, of which there are two distinct versions that are related by parity-shift.

arXiv comment40 pages, 3 figures

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• A classifying algebra for boundary conditions
• Boundary S-matrix for the Tricritical Ising Model
• Edge Critical Behaviour of the 2-Dimensional Tri-critical Ising Model
• Structure constants for the D-series Virasoro minimal models
• Local Magnetization in Critical Ising Model with Boundary Magnetic Field
• Defects in the Tri-critical Ising model
• Supersymmetry in the boundary tricritical Ising field theory
• The classifying algebra for defects
• Superconformal minimal models and admissible Jack polynomials
• The scaling supersymmetric Yang-Lee model with boundary
• The three-dimensional origin of the classifying algebra
• Boundary renormalisation group flows of the supersymmetric Lee-Yang model and its extensions
• Critical Ising Model with Boundary Magnetic Field: RG Interface and Effective Hamiltonians

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• Two More Fermionic Minimal Models
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• Fermionization of conformal boundary states
• Bootstrapping Fermionic Rational CFTs with Three Characters
• Open spin chain realization of topological defect on 1d Ising model and boundary and bulk symmetry
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• Parafermionization, bosonization, and critical parafermionic theories
• Fermionic Non-Invertible Symmetries in (1+1)d: Gapped and Gapless Phases, Transitions, and Symmetry TFTs
• Boundary RG Flows for Fermions and the Mod 2 Anomaly
• Is there supersymmetric Lee-Yang fixed point in three dimensions?
• The non-rational limit of D-series minimal models
• The fate of non-supersymmetric Gross-Neveu-Yukawa fixed point in two dimensions
• Gauging or extending bulk and boundary conformal field theories: Application to bulk and domain wall problem in topological matter and their descriptions by (mock) modular covariant
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• Operator-state correspondence in simple current extended conformal field theories: Toward a general understanding of chiral conformal field theories and topological orders
• Fusion rule in conformal field theories and topological orders: A unified view of correspondence and (fractional) supersymmetry and their relation to topological holography
• Information metric on the boundary
• Emergent Supersymmetry on the Edges
• Non-invertible defects from the Conway SCFT to K3 sigma models I: general results
• Ising model in a boundary magnetic field with random discontinuities
• Deconstruction hierarchies and locality diagrams of conformal models