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Parity and Spin CFT with boundaries and defects

by Ingo Runkel, Lóránt Szegedy, Gérard M. T. Watts

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Authors (as registered SciPost users): Lóránt Szegedy · Gerard Watts
Submission information
Preprint Link:  (pdf)
Date submitted: 2023-01-05 12:54
Submitted by: Szegedy, Lóránt
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
  • Mathematical Physics


This paper is a follow-up on [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.

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Reports on this Submission

Anonymous Report 2 on 2023-3-6 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2210.01057v1, delivered 2023-03-06, doi: 10.21468/SciPost.Report.6845


The authors discuss the structures of RCFTs graded by parity and in the presence of spin structures. The paper builds upon previous work by the authors and incorporates the most general RCFT observables that involve defects and boundaries. The exposition is comprehensive, clear and skillfully illustrated. It clarifies various subtleties in the literature on such RCFTs. This will be a very useful reference to learn about these RCFTs and certainly deserves to be published.

Some minor comments:

1. While the authors focus on RCFT in this paper, it maybe worthwhile to briefly comment on how some of the constructions can be generalized to general 2d CFT which may not be rational (e.g. gauging a special Frobenius algebra F which does not rely on starting with an underlying modular tensor category) .

2. The authors mention in the abstract that the construction (gauging F) is like "as gauging a possibly non-invertible symmetry". This was not mentioned in the main text. It would be good to mention and clarify this point (e.g. when and how it becomes non-invertible).

3. The main general construction of type 4 spin CFT comes from gauging F in a type 2 CFT as stated in Sec 1.5. This is related but different from the usual fermionization construction (which starts from type 1). Just for clarity, in discussing the Ising example (similarly for the other examples), it may be good to fill in the empty boxes in the table below (7.8).

4. Usual fermionization map from bosonic to fermionic (spin) CFT comes with a choice (e.g. by stacking with fermionic SPT). It would be good to mention where that choice is contained in the construction here (e.g. in the F algebra).

  • validity: high
  • significance: high
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: excellent

Author:  Lóránt Szegedy  on 2023-03-20  [id 3497]

(in reply to Report 2 on 2023-03-06)
answer to question

Thank you for your helpful comments. We will address your points in a revised version once invited to do so by the editor. One remark regarding the empty boxes in the table below (7.8) we can make already now: The table answers the question "Which types of CFT do we get starting from Ising, or from Ising with defects enhanced by parity, by taking the algebra 1 + eps, or 1 + Pi(eps)?" And the answer is the two filled places (type 1 and type 4), the empty spots cannot be reached in this way. We will try to make this more clear in a revised version.

Anonymous Report 1 on 2023-2-12 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2210.01057v1, delivered 2023-02-12, doi: 10.21468/SciPost.Report.6721


This work gives a detailed discussion of four types of 2d CFTs depending on whether they have spin structure and/or parity dependence on the local fields. Following the notations of eq.(1.2), the diagonal cases 1 and 4 are better understood, and the off-diagonal cases 2 and 3 are new to this work. The presentation are clear, rigorous, and contains enough details. This result is definitely worth to be published, and the referee recommends so.

It would be nice if the authors can clarify further on the following points:

In the case of Z2 symmetry, the authors clarified in Sec. 7.1 that the CFTs of off-diagonal types, i.e. 2 and 3, are non-unitary, while the diagonal types 1 and 4 are unitary. It is known, e.g. in [KTT] and [HNT] as cited in the current work, that starting with 1, gauging Z2 while introducing the spin structure dependence (via including an Arf term), leads to 4. This is known as the fermionization map and is indeed described in Sec.7.2. However, in the introduction Sec.1.5, the authors commented that starting with 2 and performing a (possibly different) gauging of topological symmetry, one leads to 4. It would be useful if the authors could clarify the difference between the normal fermionization map 1->4 and the “new gauging” 2->4. In particular, how to understand a non-unitary theory 2 is mapped to a unitary theory 4? This would help the readers to understand the relations between four types of CFTs better.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Author:  Lóránt Szegedy  on 2023-03-01  [id 3415]

(in reply to Report 1 on 2023-02-12)
answer to question

Thank you very much for your report and suggestions. We would prefer to delay implementing any changes or additions until invited to do so by the editor and will detail these in a separate reply. Here we just wanted to comment quickly about unitarity: Theories of type 2 and 3 cannot be unitary as they violate spin-statistics. Theories of type 1 and 4 can be unitary, but need not be unitary, so there is no inherent contradiction in going from type 2 to type 4.

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