The beginning of the endpoint bootstrap for conformal line defects

This work uses the numerical conformal bootstrap and perturbation theory to place bounds on CFT data associated to line defects in general, and in particular to the defect obtained by coupling a magnetic field to a one dimensional submanifold in the Ising CFT.

The main novelty of the paper lies in the choice of observables to constrain via crossing symmetry and unitarity. The authors consider a set of correlators involving local bulk operators and defects with two endpoints. This allows to bootstrap defect data like the dimension of local operators on the defect, their OPE coefficients and the defect $g$ function, and at the same time input information about the bulk CFT.

The topic is timely and the specific results on the pinning field defect are remarkably strong in comparison to the typical results in the bootstrap literature on defect CFT. The authors complement the numerical results about the three dimensional theory with analytic computations in two dimensions and perturbative computations in $4-\epsilon$ dimensions. More importantly, the paper can be considered a breakthrough in the numerical bootstrap applied to defect CFT, because the new method has wide applicability and many generalizations are possible, as the authors note in the discussion session.

I am therefore happy to recommend this paper for publication. Before that, I would like that the authors consider the following minor observations:

• In figure 4, it would be useful to emphasize (in the caption or in the text) that the larger D allowed regions are strictly included in the lower D ones, and in particular that the upper bounds in the left figure coincide for all D (excluding the island of course). While the authors do point out that "they expect" this to happen, below 1.1, plots should be easy to parse when taken together with their captions.
• It might be worth pointing out that the content of footnote 4 dictates some qualitative features of the plot in figure 4, left. In particular, there cannot be an upper bound on $\Delta^{0+},$ which is reflected in the monotonically increasing (decreasing) nature of upper (lower) bounds.
• For the same reasons of clarity, the caption in figure 5 should mention that the corresponding bounds and fuzzy sphere results refer to the pinning field defect in the $3d$ Ising CFT.
• At the beginning of page 8, it might be worth adding a citation to reference 91 as well. It is also unclear why reference 54 is cited in a footnote rather than in the text.
• On page 14, the authors point out as "interesting" the observation that a defect $a$ which obeys $g\cdot a=a$ is acted upon trivially by the symmetry "as an object that can be inserted in a correlation function". It is unclear to me what the "object" is (the topological operator associated to the symmetry?), why this observation is interesting rather than tautological, and why the qualification $g_a>0$ is necessary. The authors might want to explain this point more extensively, if it is important, or remove the sentence altogether.
• Later on the same page, $R^\tau$ rather than $\hat{R}^\tau$ appears, possibly due to a typo.
• I find the presentation of the long-range order defect a bit obscure. It is introduced in a sentence on page 3 (but the name "long range ordered defect" only appears in footnote 2). On page 8, the acronym LRO is introduced, and for the first time the authors point out they will "explain in more detail later" the relation between long range order and explicit breaking of $\mathbb{Z}_2$. It would be better to refer the reader to section 2.2 already on page 3, and perhaps summarize the physics of this defect in a sentence, the first time it is mentioned. Then, in section 2.2, the definition of the LRO defect is crucially influenced by the requirement that it is conformal. This forces it to be non-simple, because of eq. 2.3 together with scale invariance. On the other hand, one often thinks of spontaneous symmetry breaking on a defect as a flow between a symmetry preserving defect at short distances and a conformal defect that breaks the symmetry explicitly at long distances (see e.g. [17] by one of the authors). In this case, the defect in the IR is simple, and local operators which transform under the symmetry do get a one-point function. It would be useful to clarify the relation between this flow (which seems physically more relevant at first sight) and the flow between the LRO defect and the pinning field one: why is the first flow fine-tuned if the LRO defect is unstable? Explicit comparison with the familiar difference between a "mixture" of ordered ground states and genuine spontaneous symmetry breaking might be useful as well.
• On page 20, the notation $\mathcal{B}^{q,i}$ is used, but the same notation is only explained on page 29.
• The tricks used in 2.6 to compute CFT data in 2d are analogous to the holomorphic maps used by Cardy-Calabrese for the computation of the data associated to twist operators. It might be worth adding a citation to the relevant paper (although I am not sure if this kind of tricks are older still).
• In 2.44-2.46 the upper labels are $0++0$: should they be $0+00$ instead?
• The trick used in eq. 3.29 to impose positivity on a compact interval was used before in 1702.00423, see also footnote 17 in 2312.11604. It would be worth citing some of these works.
• On pag 41, the authors refer to footnote 14 as a possible reason for discrepancies in the pinning field defect spectrum between the extremal functional and fuzzy sphere. However, on page 32, it is claimed that "the error introduced by fixing the values of these quantities" is "insignificant for the purpose of this work". The two considerations seem at odd with each other.
• In the discussion around eq. 4.29 and the role of $\phi_1^{++}$, it would be useful to make explicit the relation between these dimension 1 operators and the existence of a defect stress tensor, in non-generic cases.

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• A new idea how to incorporate bulk information into bootstrap analysis of endable conformal defects.
• Demonstration of this idea for the case of 3D Ising pinning defect, showing as an application that the LRO defect is RG unstable. -Potential future applications include Wilson lines in conformal gauge theories.

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Requested changes

1) On page 4, “and others that have studied the properties of the pinning field defect more broadly [23, 29-33]”. In particular [Gimenez-Grau 2022], [Gimenez-Grau, Lauria, Liendo, van Vliet 2022], [Nishioka, Okuyama, Shimamori 2022] also study this defect in $d=4-\varepsilon$. Also, later on, “ … using so-called bulk-to-defect/boundary crossing symmetry…”, one should include [Meineri, Radhakrishnan 2025]. Lastly, right after “... while others have studied the consequences of the usual crossing symmetry adapted to defect/boundary operators”, some contributions that are missing are e.g. [Cavaglia, Gromov, Julius, Preti 2021] and [Bartlett-Tisdale, Herzog, Schaub 2023.]

2) In section 2.4, right above section 2.5: the g-function has also been bounded in the setup considered in [Meineri, Radhakrishnan 2025]. A comment would be in place.

3) In section 2.5 the analysis of bulk operators as SL(2,R) primaries is done for D = 3, specifying representations of $SO(3)$ and $SO(2)_T$. Can this discussion not be generalized to arbitrary D? Even though D = 3 is the case of interest for the numerics, the following paragraph discusses D = 2, and the perturbative section focuses on computations in $D = 4-\varepsilon$.

4) In section 3.1 on page 30: “ Next, we have assumed that the ratio of OPE coefficients $r_{\sigma \varepsilon}$ … is known exactly, …” in combination with eq. (3.5). It looks like that in the end, you input both the numerical values for $\lambda_{\varepsilon \varepsilon \varepsilon}$ and $\lambda_{\varepsilon \sigma \sigma}$ given in Table 4. However, the wording in this section 3.1 (“...remove the dependence on $\lambda_{\varepsilon \varepsilon \varepsilon}$... “) makes it look like you are treating these two OPE coefficients differently. Also, in the last paragraph of section 3.1: “... it is not possible to completely eliminate it …”. Why would you expect you could eliminate this OPE coefficient if it is a piece of CFT data present in the crossing equation?

5) In figure 13: you are only scanning over the values for $\Delta_{0}^{0+}$ which produce the island in figure 12. The caption of this figure could contain a comment that there is also a continent that is not shown here. Did you look at this continent and does it contain any robust features under increasing $\Lambda$, in contrast to figures appearing earlier in the paper? In addition, the comment about the robust kink in the zoomed-in plot in figure 13 that is now mentioned in the text could also be repeated in the caption so that it becomes immediately obvious to the reader while studying the figure why this region is highlighted.

6) Footnote 19 (page 41): what do you mean with “...not expect it to necessarily appear as a zero…”? If you include it in the discrete constraint, you explicitly exclude it from $V_{\Delta}^{0+}$. So you do not expect it to appear as a zero of $\alpha[V_{\Delta}^{0+}]$ at all, except if there is a degenerate operator. Did you end up seeing it appear as a zero or not? Furthermore, I would recommend not including it in the list of extracted low-lying spectra compared to fuzzy sphere predictions, because as you say, it is an external value which you choose and fix before extracting the spectrum from the extremal functional.

7) In section 4 you make several predictions for observables using a Padé approximation. While the paper is very detailed, here it seems there is minimal information about the calculation. At the top of page 50, two Padé approximations ([1,1] and [2,0]) are mentioned, but only one ([1,1]) result is actually given in the text. For other observables appearing earlier in this section, again only the [1,1] result is quoted. Why is this value chosen? Purely based on the agreement with fuzzy sphere results, or additionally because other Padés contain e.g. poles? I assume the Padé values quoted are for D=3, but this is not explicitly mentioned in the text. There is also inconsistent use of Padé and Pade (without the accent).

8) In section 4.3, the OPE coefficients $\lambda^{0+0}_{\sigma 00}$ and $\lambda^{0+0}_{\varepsilon 00}$ for $L \to \infty$ are the one-point functions of $\sigma, \varepsilon$ in the presence of a half-infinite line. Such one-point functions were computed perturbatively for the infinite pinning line defect in [Gimenez-Grau 2022]. Is there a relation between these one-point functions one can extract? Some integrals might also have been computed there already.

Minor points:

9) In section 2.2, on page 15: has such an example of an LRO defect been studied before? If so, where?

10) In section 2.3, eq. (2.5): define $Z^0$ as well.

11) In section 2.6, page 24, bottom line: the order $\tau_1 < \tau_2$ is flipped with respect to earlier conventions $\tau_i \geq \tau_{i+1}$.

12) On page 30, first equation, $\langle \phi^{+0} \phi^{0+} \phi^{+0} \phi^{0+}\rangle$ should this be $\pm$? If not, why are you not including $\langle \phi^{-0} \phi^{0-} \phi^{-0} \phi^{0-}\rangle$ but are including the mixed correlator $\langle \phi^{+0} \phi^{0-}\phi^{-0} \phi^{0+} \rangle$?

13) On page 38, on using the ratio of OPE coefficients as a continuous parameter. Has this been done before and could you provide a reference?

14) Figure 14: do the different colors have a certain meaning? Or are they just meant to better distinguish different lines?

Curiosities:

15) What about the pinning defect for O(N)? Generalizing the perturbative results should be straightforward, can you estimate how severely the reduction in precision of bulk CFT data is expected to affect the bounds? A small comment could be nice.

16) In section 2.2, on page 14: ".. a notion of SSB for line defects. We will first discuss the properties that we expect for such a line defect, and then show how we may construct ...". The properties are now discussed only for this specific construction. Are some of them valid for general SSB for line defects? Or is it hard to make general statements?

17) With regards to the continent in figure 12: you say there are no robust features with increasing $\Lambda$, but what about the kink around $\Delta_{0}^{0+} = 0.33?$ It seems to stay sharp for increasing $\Lambda$. Also, do you know of other theories that lie within the continent, not necessarily at the edge? Perhaps the pinning line defect in the Gross-Neveu-Yukawa CFT for example?

18) On page 33, middle: “ .. bulk $Z_2$ even tensor primaries with odd $\ell$ are kinematically excluded from these OPEs, and in principle our setup would be sensitive to such operators.” This could also be a feature. Did you check what happened if you did not a priori exclude them? Do the bounds just become weak or can you somehow rule them out or put gaps?

19) Did you do a spectrum extraction for the OPE maximization in section 3.4? If so, how did it compare to the spectrum extraction from g-minimization?

20) On page 38, at the bottom: could you think of a way to fix the uniqueness problem of $\sigma, \varepsilon$ for all components of the OPE vector given in eq. (3.21).

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