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Logarithmic operators in $c=0$ bulk CFTs
by Yifei He
Submission summary
| Authors (as registered SciPost users): | Yifei He |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2411.18696v2 (pdf) |
| Date accepted: | June 24, 2025 |
| Date submitted: | June 16, 2025, 10:36 a.m. |
| Submitted by: | Yifei He |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We study Kac operators (e.g. energy operator) in percolation and self-avoiding walk bulk CFTs with central charge $c=0$. The proper normalizations of these operators can be deduced at generic $c$ by requiring the finiteness and reality of the three-point constants in cluster and loop model CFTs. At $c=0$, Kac operators become zero-norm states and the bottom fields of logarithmic multiplets, and comparison with $c<1$ Liouville CFT suggests the potential existence of arbitrarily high rank Jordan blocks. We give a generic construction of logarithmic operators based on Kac operators and focus on the rank-2 pair of the energy operator mixing with the hull operator. By taking the $c\to 0$ limit, we compute some of their conformal data and use this to investigate the operator algebra at $c=0$. Based on cluster decomposition, we find that, contrary to previous belief, the four-point correlation function of the bulk energy operator does not vanish at $c=0$, and a crucial role is played by its coupling to the rank-3 Jordan block associated with the second energy operator. This reveals the intriguing way zero-norm operators build long-range higher-point correlations through the intricate logarithmic structures in $c=0$ bulk CFTs.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
We would like to thank you for considering the publication of the paper, and thank all the referees for careful reading of the manuscript, for writing the reports and for the comments that help improving the paper. Below we list the changes made as suggested by the referees and answer some of the questions.
List of changes
Referee 1: 1, We reworded the sentence below eq.(2.2). 2, From <TT>=0, by state-operator correspondence, the stress-energy tensor corresponds to a zero-norm state, and if it is orthogonal to all other states in the CFT, it will decouple from the CFT state space. In this case, it also does not have non-trivial three point functions with other operators, for example <O_1|O_2|T> would vanish by the OPE of O_1 and O_2. We added a footnote 4 to clarify this argument. 3, There is no typo in eq.(2.39). The b here is given in eq.(2.21) and the b_{1,2} here is given in eq.(2.34). To avoid confusion, we reordered the letters in the expression. 4, We have rewritten the sentence below eq.(5.5) which hopefully clarifies things. 5, The $\hat{\Phi}_{2,1}$ in eq. (5.48) is the Kac operator $\Phi_{2,1}$ (which satisfies the BPZ equation at generic c) but normalized as in eq.(4.25). The s-channel expansion in eq.(5.48) is the usual BPZ solution but taking into account this operator normalization. We included a reference to eq.(4.25) above eq.(5.48). 6, We have modified the expression following the referee's suggestion.
Referee 2: 1, We included the reference arXiv:1311.2055 in the second paragraph of the introduction.
Referee 3: 1, We specified the meaning of 'Kac operators' in the first sentence of the third paragraph in the introduction. 2, We made some clarifications and explanations of the cluster, dense and dilute loop models in the second paragraph of the introduction, we also included additional references on these models. 3, We fixed the citation error of [22] (previously [21]) to include the arXiv number.
Published as SciPost Phys. 19, 008 (2025)