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Global symmetry and conformal bootstrap in the two-dimensional $O(n)$ model
by Linnea Grans-Samuelsson, Rongvoram Nivesvivat, Jesper Lykke Jacobsen, Sylvain Ribault, Hubert Saleur
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Linnea Grans-Samuelsson · Jesper Lykke Jacobsen · Rongvoram Nivesvivat · Sylvain Ribault |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2111.01106v3 (pdf) |
| Code repository: | https://gitlab.com/s.g.ribault/Bootstrap_Virasoro/-/tree/precision |
| Date accepted: | 2022-04-06 |
| Date submitted: | 2022-03-23 11:01 |
| Submitted by: | Ribault, Sylvain |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We define the two-dimensional $O(n)$ conformal field theory as a theory that includes the critical dilute and dense $O(n)$ models as special cases, and depends analytically on the central charge. For generic values of $n\in\mathbb{C}$, we write a conjecture for the decomposition of the spectrum into irreducible representations of $O(n)$. We then explain how to numerically bootstrap arbitrary four-point functions of primary fields in the presence of the global $O(n)$ symmetry. We determine the needed conformal blocks, including logarithmic blocks, including in singular cases. We argue that $O(n)$ representation theory provides upper bounds on the number of solutions of crossing symmetry for any given four-point function. We study some of the simplest correlation functions in detail, and determine a few fusion rules. We count the solutions of crossing symmetry for the $30$ simplest four-point functions. The number of solutions varies from $2$ to $6$, and saturates the bound from $O(n)$ representation theory in $21$ out of $30$ cases.
Author comments upon resubmission
We are grateful to the Editor and reviewers for their work, and in particular to the author of Report 1 for the helpful suggestions. Following the four numbered suggestions, we have made the following changes:
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We have stated explicitly that the coefficients are $n$-independent.
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We have rewritten that discussion in order to clarify it and make the matrices more explicit.
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After Table (3.43), we now state that the excluded field may be chosen arbitrarily, and that we chose them by increasing values of the conformal dimension.
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The conjecture is now displayed as Eq. (4.5), and the argument for the conjecture is more detailed, including the new Table (4.6).
Moreover, Report 1 suggests that the paragraph on "Four-point $O(n)$ invariants" could be simplified. We do not see how to simplify it while keeping the needed results like Eq. (3.26). What we have done is to add a review of the notation Hom and its properties, see Eq. (3.20) and the preceding text. We hope that this clarifies the paragraph.
Published as SciPost Phys. 12, 147 (2022)