Parity and Spin CFT with boundaries and defects

TL;DR

The paper extends 2d CFT/classification by introducing four CFT types distinguished by spin-structure dependence and parity of fields, and provides a unified framework to construct their correlators via a parity-enriched 3d TFT. Central to the construction are symmetric special Frobenius algebras in a ribbon fusion category extended by SVect, with Nakayama automorphisms constraining the oriented vs spin cases. Boundaries and defects are incorporated through module and bimodule data, and the whole framework is interpreted as gauging a possibly non-invertible topological symmetry, including a detailed Z2 example. The authors supply explicit realizations via Bershadsky–Polyakov models and related Ising/Potts theories, deriving torus partition functions for all spin structures and demonstrating consistency with gluing and monodromy. Collectively, the work generalizes fermionic CFTs beyond previous spin-structure approaches, linking topological defects, gauging, and modular data into a coherent, parity-aware CFT construction with broad applicability to rational and beyond-rational theories.

Abstract

This paper is a follow-up to [arXiv:2001.05055] in which two-dimensional conformal field theories in the presence of spin structures are studied. In the present paper we define four types of CFTs, distinguished by whether they need a spin structure or not in order to be well-defined, and whether their fields have parity or not. The cases of spin dependence without parity, and of parity without the need of a spin structure, have not, to our knowledge, been investigated in detail so far. We analyse these theories by extending the description of CFT correlators via three-dimensional topological field theory developed in [arXiv:hep-th/0204148] to include parity and spin. In each of the four cases, the defining data are a special Frobenius algebra $F$ in a suitable ribbon fusion category, such that the Nakayama automorphism of $F$ is the identity (oriented case) or squares to the identity (spin case). We use the TFT to define correlators in terms of $F$ and we show that these satisfy the relevant factorisation and single-valuedness conditions. We allow for world sheets with boundaries and topological line defects, and we specify the categories of boundary labels and the fusion categories of line defect labels for each of the four types. The construction can be understood in terms of topological line defects as gauging a possibly non-invertible symmetry. We analyse the case of a $\mathbb{Z}_2$-symmetry in some detail and provide examples of all four types of CFT, with Bershadsky-Polyakov models illustrating the two new types.

Figures (46)

• Figure 1: The choice of path to interchange the positions of two identical fields
• Figure 2: a) An open-closed bordism $\Sigma$. The boundary $\partial\Sigma$ decomposes as follows: $\partial^\mathrm{f} \Sigma$ consists of the components labelled $2,5,7$, $\partial^\mathrm{c}_\mathrm{in} \Sigma$ consists of $3,8$, $\partial^\mathrm{c}_\mathrm{out} \Sigma$ of $1$, $\partial^\mathrm{o}_\mathrm{in} \Sigma$ of $6$, $\partial^\mathrm{o}_\mathrm{out} \Sigma$ of $4$. b) The bordism obtained by gluing components 1 and 8, as well as 4 and 6. The location of the now erased gluing boundary is shown as dashed lines.
• Figure 3: Standard position of $n$ vertices (resp. $n+1$ vertices) and $n$ edges on the unit circle (resp. unit half-circle).
• Figure 4: Example of a marked polygonal decomposition. The red half-dots indicate chosen edges for the polygons, the red dotted arrows indicate the anticlockwise orientation around the vertices. Vertex $v$ is used in Figure \ref{['fig:node-counting']}.
• Figure 5: Illustration of the half-edges around the vertex $v$ showing which contribute to $D_v$. Here one has $H_v=\{h_1, h_2, h_3\}$, $|H_v|=3$, $D_v=\{h_1\}$$|D_v|=1$, $\widehat{s}_{h_1}=s_{e_1}$, $\widehat{s}_{h_2}=s_{e_2}$, $\widehat{s}_{h_3}=-1-s_{e_3}$. The consistency condition is $s_{h_1}+s_{h_2}-1-s_{h_3}\equiv 1-3+1{~~(\mathrm{mod}\ r )}$.

Theorems & Definitions (82)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Theorem 2.2
• Proposition 2.3
• Remark 4.1
• Lemma 5.1
• proof
• Remark 5.2
• Proposition 5.3