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arxiv: 2605.28933 · v1 · pith:KK4S2UXKnew · submitted 2026-05-27 · ✦ hep-th · cond-mat.stat-mech

Analytic Bootstrap for O(N) Boundary Conformal Field Theories with Interacting Boundaries

Xinyu Sun , Shao-Kai Jian , Hong Yao This is my paper

Pith reviewed 2026-06-29 10:45 UTC · model grok-4.3

classification ✦ hep-th cond-mat.stat-mech
keywords O(N) BCFTanalytic bootstrapboundary interactionsepsilon expansionfixed pointstricritical O(N) modelconformal dataoperator expansions
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The pith

Universal constraints from analytic bootstrap express infinitely many operator expansions in O(N) boundary CFTs using only a finite set of inputs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper applies the analytic bootstrap to O(N) boundary conformal field theories that have additional interactions on the boundary. It derives constraints that tie together many operator expansions, reducing them to dependence on a small number of parameters. Combined with perturbative renormalization group methods, this identifies previously unknown boundary fixed points in four minus epsilon dimensions, some of which are non-unitary and arise from a cubic boundary interaction. The same framework yields the first computation of boundary conformal data for the tricritical O(N) model in three minus epsilon dimensions.

Core claim

By deriving universal constraints on conformal data, the analytic bootstrap shows that infinitely many operator expansions can be expressed in terms of a finite set of inputs. Complementing this with perturbative renormalization-group analysis identifies totally new boundary fixed points in d=4-ε, including non-unitary ones generated by a boundary cubic coupling, with their conformal data computed to leading order. The solution in d=3-ε extracts the boundary conformal data for the tricritical O(N) model for the first time.

What carries the argument

Universal constraints on conformal data derived from the analytic bootstrap, which reduce infinitely many boundary operator expansions to a finite set of inputs.

If this is right

  • New boundary fixed points exist in d=4-ε generated by boundary cubic coupling, including non-unitary ones.
  • Conformal data for these fixed points is computed to leading order in epsilon.
  • Boundary conformal data for the tricritical O(N) model is obtained in d=3-ε.
  • Infinitely many operator expansions are determined by finite inputs via the constraints.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may extend to other interacting boundary theories beyond O(N) symmetry.
  • Non-unitary fixed points could connect to studies of non-unitary conformal field theories in lower dimensions.
  • Future work might test these predictions against lattice simulations of boundary models.

Load-bearing premise

The analytic bootstrap and epsilon-expansion remain valid when applied to O(N) theories with an added boundary cubic coupling without extra constraints or inconsistencies arising from the boundary interactions.

What would settle it

A lattice simulation or numerical study of the tricritical O(N) model in three dimensions that measures boundary scaling dimensions or OPE coefficients differing from the epsilon-expansion predictions would falsify the extracted conformal data.

Figures

Figures reproduced from arXiv: 2605.28933 by Hong Yao, Shao-Kai Jian, Xinyu Sun.

Figure 1
Figure 1. Figure 1: FIG. 1. The crossing symmetry equation in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The branch cut of (a) [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The procedure of solving bootstrap equations and OPE/BOE coefficients. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Feynman diagrams for (a) [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. RG flow phase diagrams for [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (a) Collision and subsequent separation of the spe [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
read the original abstract

We investigate $O(N)$ boundary conformal field theories (BCFTs) with boundary interactions in $d=4-\epsilon$ and $d=3-\epsilon$ employing the analytic bootstrap. By deriving universal constraints on conformal data, we show that infinitely many operator expansions can be expressed in terms of a finite set of inputs. Complementing the analytic bootstrap with a perturbative renormalization-group analysis, we identify totally new boundary fixed points in $d=4-\epsilon$, including non-unitary ones, generated by a boundary cubic coupling, and compute their conformal data to leading order. Moreover, we leverage our solution in $d=3-\epsilon$ to extract, for the first time, the boundary conformal data for the tricritical $O(N)$ model. Altogether, our approach provides a unified prescription for BCFTs with interacting boundaries and streamlines the determination of bulk and boundary operator expansions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates O(N) boundary conformal field theories with boundary interactions using the analytic bootstrap in d=4-ε and d=3-ε. It derives universal constraints on conformal data showing that infinitely many operator expansions reduce to a finite set of inputs, complements this with perturbative RG analysis to identify new boundary fixed points (including non-unitary ones) generated by a boundary cubic coupling and compute their data to leading order, and extracts boundary conformal data for the tricritical O(N) model in d=3-ε, providing a unified prescription for such BCFTs.

Significance. If the derivations hold, this supplies a streamlined method for determining bulk and boundary operator expansions in interacting BCFTs, with the first extraction of tricritical boundary data and identification of new fixed points representing concrete advances. The combination of analytic bootstrap constraints with RG matching is a methodological strength that could extend to other boundary deformations.

major comments (2)
  1. [Abstract and §3 (assumed bootstrap setup)] The central claim that the analytic bootstrap yields universal constraints reducing infinitely many OPEs to finite inputs (as stated in the abstract) requires explicit verification that the crossing equations close consistently under the added boundary cubic deformation; without a dedicated section deriving the modified crossing kernel or showing the reduction explicitly, it is unclear if this holds beyond the free-boundary case.
  2. [§4 (RG analysis and fixed-point identification)] For the new fixed points in d=4-ε, the leading-order conformal data computation via combined bootstrap and RG must specify how the boundary cubic coupling affects the beta functions and anomalous dimensions; if the non-unitary points rely on an unperturbed spectrum assumption, this needs explicit justification as it is load-bearing for the claim of 'totally new' fixed points.
minor comments (2)
  1. Notation for boundary operators and their dimensions should be standardized across sections to avoid ambiguity between bulk and boundary quantities.
  2. [Abstract] The abstract would benefit from a brief mention of the key technical inputs (e.g., the form of the crossing equations or the epsilon-expansion order) to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment of the work and for the detailed, constructive comments. We respond to each major comment below, indicating where revisions will be made for clarity.

read point-by-point responses

  1. Referee: [Abstract and §3 (assumed bootstrap setup)] The central claim that the analytic bootstrap yields universal constraints reducing infinitely many OPEs to finite inputs (as stated in the abstract) requires explicit verification that the crossing equations close consistently under the added boundary cubic deformation; without a dedicated section deriving the modified crossing kernel or showing the reduction explicitly, it is unclear if this holds beyond the free-boundary case.

    Authors: In §3 we derive the universal constraints for the interacting-boundary case by solving the crossing equations after including the boundary operators generated by the cubic deformation. The reduction to a finite set of inputs follows from the structure of the modified crossing kernel, whose contributions from higher operators are fixed by the RG matching conditions at the fixed point. To make this explicit, we will add a dedicated subsection that derives the modified crossing kernel step by step and verifies the consistent closure of the equations. revision: yes

  2. Referee: [§4 (RG analysis and fixed-point identification)] For the new fixed points in d=4-ε, the leading-order conformal data computation via combined bootstrap and RG must specify how the boundary cubic coupling affects the beta functions and anomalous dimensions; if the non-unitary points rely on an unperturbed spectrum assumption, this needs explicit justification as it is load-bearing for the claim of 'totally new' fixed points.

    Authors: Section 4 computes the beta functions with the boundary cubic coupling included at leading order in ε and matches the resulting anomalous dimensions to the bootstrap data. The non-unitary fixed points are located by solving the full set of beta functions; their spectra receive perturbative corrections from the cubic term rather than assuming an unperturbed spectrum. We will expand the text to show the explicit dependence of the beta functions on the cubic coupling and to state the justification for the non-unitary cases. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central results follow from deriving crossing equations for O(N) BCFTs with boundary cubic interactions via the analytic bootstrap, then matching to perturbative RG flows in d=4-ε and d=3-ε. These steps rely on standard conformal block expansions and epsilon-expansion techniques applied to the deformed boundary action, without any load-bearing reduction to self-citations, fitted parameters renamed as predictions, or ansatze imported from prior author work. The new fixed points and conformal data are computed directly from the resulting equations, rendering the derivation self-contained against external CFT consistency conditions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no specific free parameters, axioms, or invented entities are identifiable. Arrays left empty.

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Works this paper leans on

117 extracted references · 61 canonical work pages · 26 internal anchors

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    We have used the discontinuity of the hypergeometric function, Eq

    P ( d 2 −1,0) n−1 − ξ+ 2 ξ , whereP (α,β) n denotes the Jacobi polynomial. We have used the discontinuity of the hypergeometric function, Eq. (A7), given in Appendix A 2. Therefore, using the orthogonality relation of Jacobi polynomials [67], Z 1 −1 dy(1−y) α(1 +y) βP (α,β) m (y)P (α,β) n (y) =δm,n 2α+β+1Γ(n+α+ 1)Γ(n+β+ 1) (2n+α+β+ 1)Γ(n+α+β+ 1)Γ(n+ 1) , ...

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    (52) For the first term in Eq. (49), we can first consider the expansion for ξ 1+ξ a , which serves as the building block. On the one hand, ξ 1+ξ a =ξ aP∞ l=0 (a)l l! (−1)lξl. On the other hand, the expansion in terms of the bulk conformal block reads ξ 1+ξ a =P∞ n=0 c′ a,nfb(2a+ 2n, ξ). We can further expand the bulk conformal blocks as power series ofξ,...

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    (53) Now, with these two results, Eqs. (52) and (53), it is straightforward to obtain the OPE coefficients for ⟨O(k)O(k)⟩ ˜λ2l∆ϕ+2n = Γ(k+ 1) Γ(l+ 1)Γ(k−l+ 1) slc′ l∆ϕ,n +δl,kαk,N ck∆ϕ,n,(54) withl= 0,1,· · ·, kandn= 0,1,· · ·. The expansion re- vealskdistinct families of conformal blocks correspond- ing to the scaling dimension ∆ = 2l∆ ϕ + 2n, where l= 1...

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    To avoid any ambiguity, we resum the conformal blocks inH i(ξ) using the BOE coefficients in Eq

    , (68) where B′ 3 =− ˆ∆(1) 0 + 2∆(1) ϕ − 1 2∆(1) 0 , B ′ 4 = ∆(1) ϕ .(69) 12 In general, we expect the operations of taking discon- tinuities, performing series expansions, and resumming to commute. To avoid any ambiguity, we resum the conformal blocks inH i(ξ) using the BOE coefficients in Eq. (68) and check whether the result reproducesH i(ξ) in Eq. (65...

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    RG calculation for theO(N)theory with cubic boundary interactions We consider anO(N) theory in ad-dimensional semi- infinite spaceR d + with a boundary atz= 0. In the bulk, theN-component field, denoted asϕ i,i= 1, ..., N, trans- forms in the usual way under theO(N) group. On the boundary, we consider anS N+1 invariant cubic interac- tion, whereϕ i fulfil...

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