Fermionic CFTs and classifying algebras

TL;DR

This work develops a comprehensive framework for fermionic conformal field theories on spin surfaces with boundaries, defects and interfaces, introducing crossing relations that account for parity and spin-structure signs and defining fermionic classifying algebras that encode elementary boundary, defect, and interface data. It builds explicit fermionic Virasoro minimal models (A- and D-type) and implements a parity-shift operation in the Ramond sector, relating various theories and exposing when shifts yield equivalent vs distinct CFTs. The paper applies the framework to concrete models—the fermionic Ising model, fermionic TCIM, and the supersymmetric Lee-Yang model—computing bulk structure constants, boundary/defect/classifying algebras, and boundary states, and detailing how parity-shift affects these structures. The results provide a robust algebraic toolkit for organizing and classifying fermionic CFT data on spin surfaces, with implications for boundary flows, defects, and interfaces across supersymmetric and non-supersymmetric fermionic theories.

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.

Figures (3)

• Figure 1: Properties of the topological line defect $F$. a) A field $\phi \in {\cal H}_F$ sits at the start of $F$. b) The OPE of the weight zero defect field $\pi$ and a field $\phi$ multiplies $\phi$ by $\pm 1$, depending on its parity. c) The weight zero junction joining two $T$ defects into one. d) Pushing the defect field $\pi$ through the defect junction. e) Associativity relation for the junction field. f) Dragging the $F$ defect through a field $\phi$ of spin grade $\nu_\phi$ inserts $\pi^{\nu_\phi}$. g) Rotating $\phi \in {\cal H}_F^{\nu_\phi}$ by $2\pi$ can be traded for an insertion of $\pi^{\nu_\phi+1}$. In particular, one cannot just unwind an $F$-defect around $\phi$, instead the tangent at the insertion point of $\phi$ has to remain fixed.
• Figure 2: a) The arrangement of $F$-defects needed to substitute the OPE in the $a\to 0$ limit. b) The corresponding configuration required in the $b \to 0$ limit.
• Figure 3: Two bulk fields and one boundary field together with their defect lines as used in computing the crossing constraint linking bulk-boundary couplings and bulk structure constants. In the correlator $g(a,b)$, the boundary field $\psi_z$ inserted at $L$ is moved off to $\infty$.