[Paper Review] Parity and Spin CFT with boundaries and defects

Ingo Runkel, Lóránt Szegedy+1 more|arXiv (Cornell University)|Oct 3, 2022

Black Holes and Theoretical Physics4 citations

TL;DR

This paper introduces and classifies four types of two-dimensional conformal field theories (CFTs) based on their dependence on spin structures and parity symmetry, using a 3D topological field theory (TFT) framework with Frobenius algebras in ribbon fusion categories. It extends the TFT description to include boundaries and topological line defects, and provides a systematic construction for CFTs with spin without parity and parity without spin structures—two previously underexplored types—using Bershadsky-Polyakov models as key examples.

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.

Motivation & Objective

• To systematically classify two-dimensional conformal field theories (CFTs) based on their dependence on spin structures and parity symmetry.
• To develop a 3D topological field theory (TFT) framework that unifies the description of CFT correlators across all four types of CFTs.
• To extend the TFT description to include world sheets with boundaries and topological line defects, specifying the corresponding categories of boundary and defect labels.
• To provide a construction of CFTs that are either spin-dependent without parity or parity-symmetric without requiring spin structures—two novel classes not previously studied in depth.
• To demonstrate the construction using Bershadsky-Polyakov models at k = −1/2, which realize the two new types of CFTs.

Proposed method

• The paper uses a 3D TFT construction based on Reshetikhin-Turaev theory, with values in super-vector spaces (SVect), to define CFT correlators.
• It introduces four types of CFTs distinguished by whether they require a spin structure and/or possess a parity symmetry.
• For each type, the correlators are defined via a special Frobenius algebra F in a ribbon fusion category, where the Nakayama automorphism is the identity (oriented case) or squares to the identity (spin case).
• The theory incorporates boundaries and line defects by specifying appropriate categories of boundary labels and fusion categories of defect labels, using the TFT to ensure consistency with factorization and single-valuedness.
• The construction is interpreted as gauging a possibly non-invertible topological symmetry, with a detailed analysis of Z2-symmetries.
• Examples are constructed using the Bershadsky-Polyakov algebra at k = −1/2, where the even-graded subalgebra corresponds to a CFT with parity but no spin structure, and the half-integer-graded sector realizes a CFT with spin but no parity symmetry.

Experimental results

Research questions

• RQ1 What are the four distinct classes of 2D CFTs based on spin structure and parity symmetry, and how do they differ in their defining data?
• RQ2 How can a 3D TFT framework be extended to consistently describe CFT correlators with boundaries and topological line defects in the presence of spin and parity structures?
• RQ3 Can CFTs that are spin-dependent but lack parity symmetry, or parity-symmetric but not spin-dependent, be systematically constructed and classified?
• RQ4 How does the TFT construction via Frobenius algebras in ribbon fusion categories generalize to include spin and parity structures?
• RQ5 What is the role of topological line defects in realizing the gauging of non-invertible symmetries in such CFTs, and how is this reflected in the algebraic data?

Key findings

• The paper identifies and constructs two new classes of CFTs: those with spin structure but no parity symmetry, and those with parity symmetry but no spin structure—previously unexplored in detail.
• The correlators for all four types of CFTs are consistently defined via a 3D TFT using Frobenius algebras in ribbon fusion categories, with the Nakayama automorphism condition distinguishing the oriented (identity) and spin (square to identity) cases.
• For the Bershadsky-Polyakov model at k = −1/2, the even-graded subalgebra yields a CFT with parity but no spin structure, while the half-integer-graded sector yields a CFT with spin but no parity symmetry.
• The torus partition functions for all four spin structures in the spin CFT are computed and shown to be consistent with modular invariance and spectral flow symmetry.
• The characters of the NS and R sectors of the Bershadsky-Polyakov model at k = −1/2 are computed explicitly up to level 3, with leading terms matching known results and the S-matrix confirmed numerically and via comparison to su(3)2.
• The construction via TFT ensures factorization and single-valuedness of correlators, and the invariance under local moves (e.g., Dehn twists) is proven in the appendix, validating the consistency of the framework.