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arxiv: 2604.19868 · v2 · pith:LM6BV2POnew · submitted 2026-04-21 · ✦ hep-th · cond-mat.stat-mech

Crosscap Defects

Nadav Drukker , Shota Komatsu , Anders Wallberg This is my paper

Pith reviewed 2026-05-19 17:33 UTC · model grok-4.3

classification ✦ hep-th cond-mat.stat-mech
keywords crosscap defectsconformal field theoryZ2 quotientdefect CFTO(N) modelcrossing equationsconformal blocksepsilon expansion
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The pith

Crosscap defects arise from Z2 quotients of spacetime and generalize CFT on real projective space to higher codimensions.

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

The paper establishes a new type of defect in conformal field theories called crosscap defects. These come from quotienting the spacetime manifold by a Z2 symmetry. They generalize the well-known setup of CFT on real projective space to defects of arbitrary codimension. This construction leads to specific symmetry preservation and multiple operator product channels that must satisfy crossing relations. Studying them in the O(N) model provides concrete examples where certain operators are absent.

Core claim

We introduce a novel class of defects, termed crosscap defects, in conformal field theory (CFT) in general dimensions. These arise from quotienting the spacetime by a Z_2 automorphism, and provide higher-codimension generalisations of CFT on real projective space (RP^d). Crosscap defects extend along a p-dimensional fixed locus of the Z_2 action and preserve an SO(p+1,1)×PO(d-p) subgroup of the conformal group. The two-point functions of operators in this setup exhibit three operator product expansion channels: bulk, image, and defect. These lead to several crosscap crossing equations, which we present.

What carries the argument

Crosscap defect from Z2 spacetime automorphism with p-dimensional fixed locus preserving SO(p+1,1) × PO(d-p) subgroup

If this is right

  • Two-point functions exhibit three OPE channels (bulk, image, defect) that yield crosscap crossing equations.
  • Conformal blocks match defect CFT blocks exactly after redefining cross ratios.
  • In the O(N) model, explicit CFT data at Gaussian and Wilson-Fisher points can be computed in the epsilon expansion as a function of p.
  • Displacement and tilt operators are absent for generic p, yielding defect conformal manifolds without exactly marginal operators.

Where Pith is reading between the lines

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

  • The construction may extend to other quotient symmetries or orbifold backgrounds in CFT.
  • Absence of displacement operators could simplify bootstrap analyses of such defects in additional models.
  • Numerical checks in low dimensions or lattice realizations could verify the predicted CFT data.
  • These defects might connect to existing work on defects in curved or non-orientable spacetimes.

Load-bearing premise

The Z2 automorphism admits a p-dimensional fixed locus that preserves an SO(p+1,1)×PO(d-p) subgroup of the conformal group.

What would settle it

Demonstrating that the conformal blocks fail to match defect CFT blocks after cross-ratio redefinition, or finding inconsistent crossing equations among the bulk, image, and defect channels, would invalidate the construction.

Figures

Figures reproduced from arXiv: 2604.19868 by Anders Wallberg, Nadav Drukker, Shota Komatsu.

Figure 1
Figure 1. Figure 1: The three different operator product expansion channels. We indicate [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Plot of the one-point function of SJ (4.15) in d = 4 as a function of J, in the limit N → ∞ with N− held fixed. The exact expression is pictured in purple, while the large J asymptotics (4.17) is given by the dashed grey line. This statement also holds for the interacting O(N) model at all orders in the ε expansion. We expect this to be true more generally, beyond the examples studied in this paper. Namely… view at source ↗
Figure 3
Figure 3. Figure 3: The interacting Feynman diagrams contributing to the one point func [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The interacting Feynman diagram can be represented (on the left) in [PITH_FULL_IMAGE:figures/full_fig_p028_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The counterterm (4.3) for p = 2 gives the additional Feynman diagrams represented on the left in the covering space and on the right in the quotient. in [PITH_FULL_IMAGE:figures/full_fig_p030_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The first order correction in ε to the conformal dimension (5.40) of the transverse spin-0 operator Oˆ + ı 0 , as defined in (5.36). This is a function of p and shown for N → ∞ with N− fixed. We clearly observe the pole in the anomalous dimension at p = 2. -1 1 2 3 -0.5 -0.4 -0.3 -0.2 -0.1 -1 1 2 3 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 [PITH_FULL_IMAGE:figures/full_fig_p033_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The first order correction in ε to the conformal dimensions ∆ (5.40) and bulk–defect couplings B2 (5.41), as defined in (5.36). Those are evaluated for N → ∞ with N− fixed. The left plot are made for sˆ = {1, . . . , 7} (the case of sˆ = 0 is in [PITH_FULL_IMAGE:figures/full_fig_p033_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The first order correction in ε to the one-point function coefficient of the scalar operator S (5.46) is plotted on the left, while the corresponding correction for the operator SJ (5.49) with J = {2, 4, . . . , 14} is on the right. both are plotted as a function of p in the limit N → ∞ with N− fixed. The more blue the colour, the larger the spin J. We again see a divergence at p = 2, in this case in the o… view at source ↗
Figure 9
Figure 9. Figure 9: The large J asymptotics (5.52) are pictured by the dashed grey line in the limit N → ∞ with N− fixed. The purple lines are the range of [PITH_FULL_IMAGE:figures/full_fig_p037_9.png] view at source ↗
read the original abstract

We introduce a novel class of defects, termed crosscap defects, in conformal field theory (CFT) in general dimensions. These arise from quotienting the spacetime by a $Z_2$ automorphism, and provide higher-codimension generalisations of CFT on real projective space ($RP^{d}$). Crosscap defects extend along a $p$-dimensional fixed locus of the $Z_2$ action and preserve an $SO(p+1,1)\times PO(d-p)$ subgroup of the conformal group. The two-point functions of operators in this setup exhibit three operator product expansion channels: bulk, image, and defect. These lead to several crosscap crossing equations, which we present. We analyse conformal block decompositions and show that the blocks are identical to defect CFT blocks up to a redefinition of cross ratios. As concrete examples, we study crosscap defects in the $O(N)$ model at the Gaussian and Wilson--Fisher fixed points in the $\varepsilon$-expansion. We compute explicitly the associated CFT data as a function of $p$ and find that, unlike standard defects, displacement and tilt operators are absent for generic $p$. They provide examples of defect conformal manifolds without exactly marginal operators.

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

3 major / 2 minor

Summary. The manuscript introduces a novel class of defects termed crosscap defects in CFTs in general dimensions. These are constructed by quotienting spacetime by a Z2 automorphism with a p-dimensional fixed locus, preserving an SO(p+1,1)×PO(d-p) subgroup of the conformal group. The setup yields three OPE channels (bulk, image, defect) and associated crossing equations. Conformal blocks are shown to match those of defect CFT up to cross-ratio redefinition. Explicit computations of CFT data as a function of p are performed in the O(N) model at the Gaussian and Wilson-Fisher fixed points in the ε-expansion, with the result that displacement and tilt operators are absent for generic p and that these defects realize conformal manifolds without exactly marginal operators.

Significance. If the symmetry preservation and consistency of the quotient construction can be established, the work offers a higher-codimension generalization of CFT on RP^d that may prove useful for classifying defects with reduced conformal symmetry. The block equivalence simplifies future calculations, and the O(N) results supply concrete, p-dependent data together with the distinctive feature that displacement and tilt operators are absent. The absence of exactly marginal operators on the defect conformal manifold is also noteworthy.

major comments (3)
  1. [Construction of crosscap defects] The explicit form of the Z2 involution, its action on coordinates, and the direct verification that the fixed locus is precisely p-dimensional while the preserved subalgebra is exactly SO(p+1,1)×PO(d-p) are not supplied. This verification is load-bearing for the definition of the defect and the claimed symmetry reduction.
  2. [Crossing equations] The derivation of the three OPE channels and the associated crossing equations from the quotient is stated but not derived in detail; in particular, consistency of the operator algebra along the fixed locus (absence of extra singularities or anomalies) is not checked explicitly.
  3. [O(N) model computations] In the O(N) model section, the ε-expansion results for the CFT data are given as functions of p, yet no error estimates, comparison with known limits (e.g., p=0 or p=d-1), or explicit checks confirming the absence of displacement and tilt operators for generic p are provided.
minor comments (2)
  1. [Symmetry discussion] Clarify the precise meaning of PO(d-p) in the preserved symmetry group, as the notation can be ambiguous between projective and parity-including orthogonal groups.
  2. [Introduction] Add a brief comparison table or limiting-case discussion showing how the crosscap defect reduces to the standard RP^d case when p=d-1.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments on our manuscript. We address each of the major comments in detail below and outline the revisions we intend to implement.

read point-by-point responses

  1. Referee: The explicit form of the Z2 involution, its action on coordinates, and the direct verification that the fixed locus is precisely p-dimensional while the preserved subalgebra is exactly SO(p+1,1)×PO(d-p) are not supplied. This verification is load-bearing for the definition of the defect and the claimed symmetry reduction.

    Authors: We agree with the referee that providing an explicit construction would clarify the setup. In the revised manuscript, we will include a new subsection that specifies the Z2 involution explicitly in coordinates, demonstrates that the fixed locus is p-dimensional, and verifies the preserved symmetry subgroup by computing the action on the conformal generators. revision: yes

  2. Referee: The derivation of the three OPE channels and the associated crossing equations from the quotient is stated but not derived in detail; in particular, consistency of the operator algebra along the fixed locus (absence of extra singularities or anomalies) is not checked explicitly.

    Authors: We acknowledge that the derivation could benefit from more detail. We will expand the relevant section to derive the three OPE channels (bulk, image, and defect) step by step from the quotient. For the consistency along the fixed locus, we will add a discussion explaining why no extra singularities or anomalies arise, based on the properties of the Z2 quotient and the conformal invariance. revision: yes

  3. Referee: In the O(N) model section, the ε-expansion results for the CFT data are given as functions of p, yet no error estimates, comparison with known limits (e.g., p=0 or p=d-1), or explicit checks confirming the absence of displacement and tilt operators for generic p are provided.

    Authors: We appreciate this suggestion for improving the robustness of our results. In the revised version, we will add error estimates to the ε-expansion computations, include comparisons with the special cases p=0 and p=d-1 where possible, and provide explicit checks or arguments (such as through Ward identities or coefficient vanishing) to confirm the absence of displacement and tilt operators for generic p. revision: yes

Circularity Check

0 steps flagged

No circularity: construction and block analysis are self-contained

full rationale

The paper defines crosscap defects by quotienting spacetime by a Z2 automorphism with a p-dimensional fixed locus, derives the preserved subgroup SO(p+1,1)×PO(d-p), presents three OPE channels and associated crossing equations, and shows that the conformal blocks match standard defect blocks after cross-ratio redefinition. These steps are direct mathematical consequences of the quotient construction and standard CFT kinematics; they do not reduce to fitted parameters or prior self-citations. Explicit computations in the O(N) model at Gaussian and Wilson-Fisher fixed points are perturbative expansions that produce new CFT data (including absence of displacement/tilt operators for generic p) without circular reuse of the target results. No load-bearing step relies on a self-citation chain or renames a known result as a derivation.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The paper rests on standard CFT axioms and introduces the crosscap defect as the primary new element; p is treated as a free parameter.

free parameters (1)
  • p
    Dimension of the fixed locus of the Z2 action, chosen freely in the construction and varied in the O(N) examples.
axioms (1)
  • standard math Standard axioms of conformal field theory including conformal invariance and the existence of operator product expansions.
    Invoked to define the bulk, image, and defect channels and to equate blocks to defect CFT blocks.
invented entities (1)
  • crosscap defect no independent evidence
    purpose: Higher-codimension generalization of RP^d obtained by Z2 quotient.
    Newly postulated in this work; no independent evidence outside the paper is provided.

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Reference graph

Works this paper leans on

75 extracted references · 75 canonical work pages · 36 internal anchors

  1. [1]

    The Conformal Bootstrap: Theory, Numerical Techniques, and Applications

    D. Poland, S. Rychkov, and A. Vichi, “The conformal bootstrap: Theory, numerical techniques, and applications,” Rev. Mod. Phys. 91 (2019) 015002 , arXiv:1805.04405

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  2. [2]

    The Bootstrap Program for Boundary CFT_d

    P. Liendo, L. Rastelli, and B. C. van Rees, “The bootstrap program for boundary CFTd,” JHEP 07 (2013) 113 , arXiv:1210.4258

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  3. [3]

    Boundary and Interface CFTs from the Conformal Bootstrap,

    F. Gliozzi, P. Liendo, M. Meineri, and A. Rago, “Boundary and interface CFTs from the conformal bootstrap,” JHEP 05 (2015) 036 , arXiv:1502.07217. [Erratum: JHEP 12, 093 (2021)]

    work page arXiv 2015

  4. [4]

    Defects in conformal field theory

    M. Billò, V. Gonçalves, E. Lauria, and M. Meineri, “Defects in conformal field theory,” JHEP 04 (2016) 091 , arXiv:1601.02883

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  5. [5]

    The Conformal Bootstrap at Finite Temperature

    L. Iliesiu, M. Koloğlu, R. Mahajan, E. Perlmutter, and D. Simmons-Duffin, “The conformal bootstrap at finite temperature,” JHEP 10 (2018) 070 , arXiv:1802.10266

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  6. [6]

    Bootstrapping critical Ising model on three-dimensional real projective space

    Y. Nakayama, “Bootstrapping critical Ising model on three-dimensional real projective space,” Phys. Rev. Lett. 116 no. 14, (2016) 141602 , arXiv:1601.06851

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  7. [7]

    $\epsilon$-Expansion in Critical $\phi^3$-Theory on Real Projective Space from Conformal Field Theory

    C. Hasegawa and Y. Nakayama, “ ϵ-expansion in critical ϕ3-theory on real projective space from conformal field theory,” Mod. Phys. Lett. A 32 no. 07, (2017) 1750045 , arXiv:1611.06373. 46

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  8. [8]

    Crossing Kernels for Boundary and Crosscap CFTs

    M. Hogervorst, “Crossing kernels for boundary and crosscap CFTs,” arXiv:1703.08159

    work page internal anchor Pith review Pith/arXiv arXiv

  9. [9]

    Three ways to solve critical $\phi^4$ theory on $4-\epsilon$ dimensional real projective space: perturbation, bootstrap, and Schwinger-Dyson equation

    C. Hasegawa and Y. Nakayama, “Three ways to solve critical ϕ4 theory on 4 −ϵ dimensional real projective space: perturbation, bootstrap, and Schwinger-Dyson equation,” Int. J. Mod. Phys. A 33 no. 08, (2018) 1850049 , arXiv:1801.09107

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  10. [10]

    Giombi, H

    S. Giombi, H. Khanchandani, and X. Zhou, “Aspects of CFTs on real projective space,” J. Phys. A 54 no. 2, (2021) 024003 , arXiv:2009.03290

    work page arXiv 2021

  11. [11]

    Klein Bottles and Simple Currents

    L. R. Huiszoon, A. N. Schellekens, and N. Sousa, “Klein bottles and simple currents,” Phys. Lett. B 470 (1999) 95–102 , hep-th/9909114

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  12. [12]

    Boundaries, crosscaps and simple currents

    J. Fuchs, L. R. Huiszoon, A. N. Schellekens, C. Schweigert, and J. Walcher, “Boundaries, crosscaps and simple currents,” Phys. Lett. B 495 (2000) 427–434 , hep-th/0007174

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  13. [13]

    Notes on Orientifolds of Rational Conformal Field Theories

    I. Brunner and K. Hori, “Notes on orientifolds of rational conformal field theories,” JHEP 07 (2004) 023 , hep-th/0208141

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  14. [14]

    New crosscap states,

    W. Harada, J. Kaidi, Y. Kusuki, and Y. Liu, “New crosscap states,” arXiv:2508.18357

    work page arXiv

  15. [15]

    Half-BPS crosscap defects in N = 4 super Yang–Mills

    N. Drukker, S. Komatsu, and A. Wallberg, “Half-BPS crosscap defects in N = 4 super Yang–Mills. ” In progress

    work page

  16. [16]

    Spinning Conformal Correlators

    M. S. Costa, J. Penedones, D. Poland, and S. Rychkov, “Spinning conformal correlators,” JHEP 11 (2011) 071 , arXiv:1107.3554

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  17. [17]

    Dynamical derivation of vacuum operator-product expansion in Euclidean conformal quantum field theory,

    V. K. Dobrev, V. B. Petkova, S. G. Petrova, and I. T. Todorov, “Dynamical derivation of vacuum operator-product expansion in Euclidean conformal quantum field theory,” Physical Review D 13 no. 4, (Feb., 1976) 887–912

    work page 1976

  18. [18]

    Line defects in the 3d Ising model

    M. Billó, M. Caselle, D. Gaiotto, F. Gliozzi, M. Meineri, and R. Pellegrini, “Line defects in the 3d Ising model,” JHEP 07 (2013) 055 , arXiv:1304.4110

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  19. [19]

    Bootstrapping the 3d Ising twist defect

    D. Gaiotto, D. Mazac, and M. F. Paulos, “Bootstrapping the 3d Ising twist defect,” JHEP 03 (2014) 100 , arXiv:1310.5078

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  20. [20]

    The $\epsilon$-expansion of the codimension two twist defect from conformal field theory

    S. Yamaguchi, “The ϵ-expansion of the codimension two twist defect from conformal field theory,” PTEP 2016 no. 9, (2016) 091B01 , arXiv:1607.05551

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  21. [21]

    Giombi, E

    S. Giombi, E. Helfenberger, Z. Ji, and H. Khanchandani, “Monodromy defects from hyperbolic space,” JHEP 02 (2022) 041 , arXiv:2102.11815

    work page arXiv 2022

  22. [22]

    Entanglement entropy and conformal field theory

    P. Calabrese and J. Cardy, “Entanglement entropy and conformal field theory,” J. Phys. A 42 (2009) 504005 , arXiv:0905.4013

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  23. [23]

    Continuous family of conformal field theories and exactly marginal operators,

    S. Komatsu, Y. Kusuki, M. Meineri, and H. Ooguri, “Continuous family of conformal field theories and exactly marginal operators,” arXiv:2512.11045

    work page arXiv

  24. [24]

    Gimenez-Grau, Probing magnetic line defects with two-point functions , 2212.02520

    A. Gimenez-Grau, “Probing magnetic line defects with two-point functions,” arXiv:2212.02520

    work page arXiv

  25. [25]

    Local bulk operators in AdS/CFT: A holographic description of the black hole interior

    A. Hamilton, D. N. Kabat, G. Lifschytz, and D. A. Lowe, “Local bulk operators in AdS/CFT: A Holographic description of the black hole interior,” Phys. Rev. D 75 (2007) 106001 , hep-th/0612053. [Erratum: Phys.Rev.D 75, 129902 (2007)]. 47

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  26. [26]

    A Stereoscopic Look into the Bulk

    B. Czech, L. Lamprou, S. McCandlish, B. Mosk, and J. Sully, “A Stereoscopic Look into the Bulk,” JHEP 07 (2016) 129 , arXiv:1604.03110

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  27. [27]

    Operator product expansion for conformal defects

    M. Fukuda, N. Kobayashi, and T. Nishioka, “Operator product expansion for conformal defects,” JHEP 01 (2018) 013 , arXiv:1710.11165

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  28. [28]

    Ghosh and E

    K. Ghosh and E. Trevisani, “A tale of two uplifts: Parisi-Sourlas with defects,” JHEP 03 (2026) 067 , arXiv:2508.18356

    work page arXiv 2026

  29. [29]

    Conformal Field Theories Near a Boundary in General Dimensions

    D. M. McA vity and H. Osborn, “Conformal field theories near a boundary in general dimensions,” Nucl. Phys. B 455 (1995) 522–576 , cond-mat/9505127

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  30. [30]

    Conformal Four Point Functions and the Operator Product Expansion

    F. A. Dolan and H. Osborn, “Conformal four point functions and the operator product expansion,” Nucl. Phys. B 599 (2001) 459–496 , hep-th/0011040

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  31. [31]

    Tr´ epanier,Surface defects in the O(N) model,JHEP09(2023) 074, [arXiv:2305.10486]

    M. Trépanier, “Surface defects in the O(N ) model,” JHEP 09 (2023) 074 , arXiv:2305.10486

    work page arXiv 2023

  32. [32]

    Giombi and B

    S. Giombi and B. Liu, “Notes on a surface defect in the O(N ) model,” JHEP 12 (2023) 004 , arXiv:2305.11402

    work page arXiv 2023

  33. [33]

    Phases of surface defects in S calar Field Theories,

    A. Raviv-Moshe and S. Zhong, “Phases of surface defects in scalar field theories,” JHEP 08 (2023) 143 , arXiv:2305.11370

    work page arXiv 2023

  34. [34]

    Transdimensional defects,

    E. de Sabbata, N. Drukker, and A. Stergiou, “Transdimensional defects,” arXiv:2411.17809

    work page arXiv

  35. [35]

    Lauria, P

    E. Lauria, P. Liendo, B. C. Van Rees, and X. Zhao, “Line and surface defects for the free scalar field,” JHEP 01 (2021) 060 , arXiv:2005.02413

    work page arXiv 2021

  36. [36]

    Bootstrapping boundary-localized interactions,

    C. Behan, L. Di Pietro, E. Lauria, and B. C. Van Rees, “Bootstrapping boundary-localized interactions,” JHEP 12 (2020) 182 , arXiv:2009.03336

    work page arXiv 2020

  37. [37]

    Cuomo, Z

    G. Cuomo, Z. Komargodski, and M. Mezei, “Localized magnetic field in the O(N ) model,” JHEP 02 (2022) 134 , arXiv:2112.10634

    work page arXiv 2022

  38. [38]

    Nishioka, Y

    T. Nishioka, Y. Okuyama, and S. Shimamori, “The epsilon expansion of the O(N ) model with line defect from conformal field theory,” JHEP 03 (2023) 203 , arXiv:2212.04076

    work page arXiv 2023

  39. [39]

    Analytic bootstrap for magnetic impurities,

    L. Bianchi, D. Bonomi, E. de Sabbata, and A. Gimenez-Grau, “Analytic bootstrap for magnetic impurities,” JHEP 05 (2024) 080 , arXiv:2312.05221

    work page arXiv 2024

  40. [40]

    Henriksson,The critical O(N) CFT: Methods and conformal data,Phys

    J. Henriksson, “The critical O(N ) CFT: Methods and conformal data,” Phys. Rept. 1002 (2023) 1–72 , arXiv:2201.09520

    work page arXiv 2023

  41. [41]

    A Lorentzian inversion formula for defect CFT

    P. Liendo, Y. Linke, and V. Schomerus, “A Lorentzian inversion formula for defect CFT,” JHEP 08 (2020) 163 , arXiv:1903.05222

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  42. [42]

    Bianchi, D

    L. Bianchi, D. Bonomi, and E. de Sabbata, “Analytic bootstrap for the localized magnetic field,” JHEP 04 (2023) 069 , arXiv:2212.02524

    work page arXiv 2023

  43. [43]

    Critical exponents in 3.99 dimensions,

    K. G. Wilson and M. E. Fisher, “Critical exponents in 3.99 dimensions,” Phys. Rev. Lett. 28 (1972) 240–243

    work page 1972

  44. [44]

    An Approach to the evaluation of three and four point ladder diagrams,

    N. I. Usyukina and A. I. Davydychev, “An Approach to the evaluation of three and four point ladder diagrams,” Phys. Lett. B 298 (1993) 363–370

    work page 1993

  45. [45]

    Boundaries and interfaces with localized cubic interactions in the O(N ) model,

    S. Harribey, I. R. Klebanov, and Z. Sun, “Boundaries and interfaces with localized cubic interactions in the O(N ) model,” JHEP 10 (2023) 017 , arXiv:2307.00072. 48

    work page arXiv 2023

  46. [46]

    Multiscalar critical models with localised cubic interactions,

    S. Harribey, W. H. Pannell, and A. Stergiou, “Multiscalar critical models with localised cubic interactions,” arXiv:2407.20326

    work page arXiv

  47. [47]

    Mellin space bootstrap for global symmetry

    P. Dey, A. Kaviraj, and A. Sinha, “Mellin space bootstrap for global symmetry,” JHEP 07 (2017) 019 , arXiv:1612.05032

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  48. [48]

    The Pomeranchuk singularity in nonabelian gauge theories,

    E. A. Kuraev, L. N. Lipatov, and V. S. Fadin, “The Pomeranchuk singularity in nonabelian gauge theories,” Sov. Phys. JETP 45 (1977) 199–204

    work page 1977

  49. [49]

    The Pomeranchuk singularity in quantum chromodynamics,

    I. I. Balitsky and L. N. Lipatov, “The Pomeranchuk singularity in quantum chromodynamics,” Sov. J. Nucl. Phys. 28 (1978) 822–829

    work page 1978

  50. [50]

    Line defect RG flows in the εexpansion,

    W. H. Pannell and A. Stergiou, “Line defect RG flows in the εexpansion,” JHEP 06 (2023) 186 , arXiv:2302.14069

    work page arXiv 2023

  51. [51]

    Comments on gluon scattering amplitudes via AdS/CFT

    L. F. Alday and J. Maldacena, “Comments on gluon scattering amplitudes via AdS/CFT,” JHEP 11 (2007) 068 , arXiv:0710.1060

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  52. [52]

    Wilson Loop Renormalization Group Flows

    J. Polchinski and J. Sully, “Wilson loop renormalization group flows,” JHEP 10 (2011) 059, arXiv:1104.5077

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  53. [53]

    Non-supersymmetric Wilson loop in N = 4 SYM and defect 1d CFT,

    M. Beccaria, S. Giombi, and A. Tseytlin, “Non-supersymmetric Wilson loop in N = 4 SYM and defect 1d CFT,” JHEP 03 (2018) 131 , arXiv:1712.06874

    work page arXiv 2018

  54. [54]

    A Study of Quantum Field Theories in AdS at Finite Coupling

    D. Carmi, L. Di Pietro, and S. Komatsu, “A study of quantum field theories in AdS at finite coupling,” JHEP 01 (2019) 200 , arXiv:1810.04185

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  55. [55]

    Giombi and H

    S. Giombi and H. Khanchandani, “CFT in AdS and boundary RG flows,” JHEP 11 (2020) 118 , arXiv:2007.04955

    work page arXiv 2020

  56. [56]

    Finite-coupling spectrum of O(N ) model in AdS,

    J. Dujava and P. Vaško, “Finite-coupling spectrum of O(N ) model in AdS,” JHEP 12 (2025) 036 , arXiv:2503.16345

    work page arXiv 2025

  57. [57]

    Taming Mass Ga p with Anti-de-Sitter Space,

    C. Copetti, L. Di Pietro, Z. Ji, and S. Komatsu, “Taming mass gaps with anti-de Sitter space,” Phys. Rev. Lett. 133 no. 8, (2024) 081601 , arXiv:2312.09277

    work page arXiv 2024

  58. [58]

    The Analytic Bootstrap and AdS Superhorizon Locality

    A. L. Fitzpatrick, J. Kaplan, D. Poland, and D. Simmons-Duffin, “The analytic bootstrap and AdS superhorizon locality,” JHEP 12 (2013) 004 , arXiv:1212.3616

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  59. [59]

    Convexity and Liberation at Large Spin

    Z. Komargodski and A. Zhiboedov, “Convexity and Liberation at Large Spin,” JHEP 11 (2013) 140 , arXiv:1212.4103

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  60. [60]

    Universality at large transverse spin in defect CFT

    M. Lemos, P. Liendo, M. Meineri, and S. Sarkar, “Universality at large transverse spin in defect CFT,” JHEP 09 (2018) 091 , arXiv:1712.08185

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  61. [61]

    Analyticity in Spin in Conformal Theories

    S. Caron-Huot, “Analyticity in spin in conformal theories,” JHEP 09 (2017) 078 , arXiv:1703.00278

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  62. [62]

    Carmi and S

    D. Carmi and S. Caron-Huot, “A Conformal dispersion relation: correlations from absorption,” JHEP 09 (2020) 009 , arXiv:1910.12123

    work page arXiv 2020

  63. [63]

    Bianchi and D

    L. Bianchi and D. Bonomi, “Conformal dispersion relations for defects and boundaries,” SciPost Phys. 15 no. 2, (2023) 055 , arXiv:2205.09775

    work page arXiv 2023

  64. [64]

    Barrat, A

    J. Barrat, A. Gimenez-Grau, and P. Liendo, “A dispersion relation for defect CFT,” JHEP 02 (2023) 255 , arXiv:2205.09765

    work page arXiv 2023

  65. [65]

    Superintegrability of $d$-dimensional Conformal Blocks

    M. Isachenkov and V. Schomerus, “Superintegrability of d-dimensional Conformal Blocks,” Phys. Rev. Lett. 117 no. 7, (2016) 071602 , arXiv:1602.01858. 49

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  66. [66]

    Harmony of Spinning Conformal Blocks

    V. Schomerus, E. Sobko, and M. Isachenkov, “Harmony of spinning conformal blocks,” JHEP 03 (2017) 085 , arXiv:1612.02479

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  67. [67]

    Harmonic Analysis and Mean Field Theory,

    D. Karateev, P. Kravchuk, and D. Simmons-Duffin, “Harmonic analysis and mean field theory,” JHEP 10 (2019) 217 , arXiv:1809.05111

    work page arXiv 2019

  68. [68]

    Gaudin models and multipoint conformal blocks: general theory,

    I. Buric, S. Lacroix, J. A. Mann, L. Quintavalle, and V. Schomerus, “Gaudin models and multipoint conformal blocks: general theory,” JHEP 10 (2021) 139 , arXiv:2105.00021

    work page arXiv 2021

  69. [69]

    Classification of homogeneous conformally flat riemannian manifolds,

    D. V. Alekseevskii and B. N. Kimel’fel’d, “Classification of homogeneous conformally flat riemannian manifolds,” Mathematical notes of the Academy of Sciences of the USSR 24 (1978) 559–562

    work page 1978

  70. [70]

    Holographic Dual of BCFT

    T. Takayanagi, “Holographic dual of BCFT,” Phys. Rev. Lett. 107 (2011) 101602 , arXiv:1105.5165

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  71. [71]

    Aspects of AdS/BCFT

    M. Fujita, T. Takayanagi, and E. Tonni, “Aspects of AdS/BCFT,” JHEP 11 (2011) 043, arXiv:1108.5152

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  72. [72]

    Holography, Entanglement Entropy, and Conformal Field Theories with Boundaries or Defects

    K. Jensen and A. O’Bannon, “Holography, entanglement entropy, and conformal field theories with boundaries or defects,” Phys. Rev. D 88 no. 10, (2013) 106006 , arXiv:1309.4523

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  73. [73]

    Towards a $C$-theorem in defect CFT

    N. Kobayashi, T. Nishioka, Y. Sato, and K. Watanabe, “Towards a C-theorem in defect CFT,” JHEP 01 (2019) 039 , arXiv:1810.06995

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  74. [74]

    Holographic defect CFTs with Dirichlet end-of-the-world branes,

    H. Nakayama and T. Nishioka, “Holographic defect CFTs with Dirichlet end-of-the-world branes,” arXiv:2509.22270

    work page arXiv

  75. [75]

    Wei,Holographic dual of crosscap conformal field theory,JHEP03(2025) 086 [2405.03755]

    Z. Wei, “Holographic dual of crosscap conformal field theory,” JHEP 03 (2025) 086 , arXiv:2405.03755. 50

    work page arXiv 2025