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arxiv: 2605.27256 · v1 · pith:OT6GAKFUnew · submitted 2026-05-26 · ✦ hep-th

Thermal conformal partial waves from flat-space and defect CFT

K.B. Alkalaev , Semyon Mandrygin , Vladimir Samsonov This is my paper

Pith reviewed 2026-06-29 15:50 UTC · model grok-4.3

classification ✦ hep-th
keywords conformal partial wavesthermal CFTdefect CFTshadow formalismCasimir equationone-point blocksflat-space CFTspin exchange
0
0 comments X

The pith

Scalar one-point thermal blocks are obtained from four-point flat-space blocks and two-point defect blocks through specific operator configurations in the shadow formalism.

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

The paper establishes a direct correspondence among conformal partial waves in flat-space, thermal, and defect CFT backgrounds. It shows that scalar one-point thermal blocks arise systematically from four-point flat-space counterparts and two-point defect counterparts by choosing particular operator placements. The same framework produces the thermal Casimir equation as a diagonal reduction of the flat-space Casimir system, with no chemical potentials required. It further identifies that defect two-point blocks exchanging a spin-l operator map onto thermal one-point blocks with an external spin-l operator. This unification treats the different geometries as related by the same shadow construction.

Core claim

We establish a correspondence between conformal partial waves on flat, thermal, and defect backgrounds using the shadow formalism. We demonstrate that scalar one-point thermal blocks can be systematically obtained from their four-point flat-space and two-point defect counterparts by considering specific operator configurations. This framework allows us to derive the thermal Casimir equation as a diagonal reduction of the flat-space Casimir system without introducing chemical potentials. We further show that defect two-point blocks with a spin-l exchange operator correspond to thermal one-point blocks for an external spin-l operator.

What carries the argument

Shadow formalism applied to chosen operator configurations that produce diagonal reductions from flat-space Casimir systems to thermal ones.

If this is right

  • Thermal one-point blocks become computable from existing flat-space and defect results without new integrals.
  • The thermal Casimir equation holds independently of chemical potentials via the diagonal reduction.
  • Defect two-point blocks with spin-l exchange supply the thermal one-point blocks for external operators of spin l.
  • The same operator-configuration technique extends the correspondence to other partial waves on these backgrounds.

Where Pith is reading between the lines

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

  • The reduction may allow extraction of higher-point thermal blocks by embedding them in larger flat-space configurations.
  • Similar diagonal reductions could relate blocks on other backgrounds, such as those with boundaries, once the appropriate operator placements are identified.
  • Explicit checks in free-field theories would confirm whether the spin-l mapping preserves normalization constants.

Load-bearing premise

The shadow formalism carries over without change to thermal and defect settings so that the chosen operator placements and reductions produce the stated blocks and equation.

What would settle it

Direct computation of a scalar one-point thermal block in a solvable thermal CFT model yields a different functional form from the block obtained by the proposed reduction from the corresponding flat-space four-point block.

read the original abstract

We establish a correspondence between conformal partial waves on flat, thermal, and defect backgrounds using the shadow formalism. We demonstrate that scalar one-point thermal blocks can be systematically obtained from their four-point flat-space and two-point defect counterparts by considering specific operator configurations. This framework allows us to derive the thermal Casimir equation as a diagonal reduction of the flat-space Casimir system without introducing chemical potentials. We further show that defect two-point blocks with a spin-$l$ exchange operator correspond to thermal one-point blocks for an external spin-$l$ operator.

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 / 1 minor

Summary. The manuscript claims to establish a correspondence between conformal partial waves on flat, thermal, and defect backgrounds using the shadow formalism. Scalar one-point thermal blocks are systematically obtained from four-point flat-space and two-point defect counterparts via specific operator configurations. The thermal Casimir equation is derived as a diagonal reduction of the flat-space Casimir system without chemical potentials. Defect two-point blocks with spin-l exchange correspond to thermal one-point blocks for external spin-l operators.

Significance. If the mappings hold, the work offers a parameter-free route to thermal blocks from established flat-space and defect results, which could streamline computations in finite-temperature CFT without auxiliary structures such as chemical potentials. The diagonal-reduction approach to the Casimir equation is a concrete technical contribution if the underlying shadow integrals are shown to be well-defined.

major comments (2)
  1. [Derivation of the thermal Casimir equation and the operator-configuration mappings] The central claim that the shadow formalism extends directly to thermal backgrounds (used for both the scalar one-point blocks and the diagonal reduction yielding the thermal Casimir equation) lacks explicit verification of integral convergence and invariance under imaginary-time periodicity. This assumption is load-bearing for all stated correspondences.
  2. [Spin-l exchange correspondence] The correspondence between defect two-point blocks with spin-l exchange and thermal one-point blocks for external spin-l operators requires a demonstration that the shadow integral remains well-defined once the external operator carries spin and the background is thermally compactified; the abstract statement alone does not supply this check.
minor comments (1)
  1. [Introduction and abstract] Notation for the shadow integrals and the precise operator configurations could be introduced earlier to make the reduction steps easier to follow.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the points that require clarification regarding the extension of the shadow formalism. We address each major comment below and indicate where revisions will be made to strengthen the manuscript.

read point-by-point responses

  1. Referee: The central claim that the shadow formalism extends directly to thermal backgrounds (used for both the scalar one-point blocks and the diagonal reduction yielding the thermal Casimir equation) lacks explicit verification of integral convergence and invariance under imaginary-time periodicity. This assumption is load-bearing for all stated correspondences.

    Authors: We agree that an explicit check strengthens the presentation. The shadow integrals are first defined in flat space, where convergence is standard, and the thermal mapping is obtained by the specific operator configurations that preserve the periodicity of the thermal circle. The diagonal reduction of the Casimir equation is a formal algebraic step that inherits the flat-space properties. Nevertheless, to address the concern directly we will add a short appendix (or subsection) that verifies convergence of the relevant shadow integrals on the thermal background for the scalar case and confirms invariance under imaginary-time shifts by explicit computation in the appropriate coordinate patch. This will be included in the revised manuscript. revision: partial

  2. Referee: The correspondence between defect two-point blocks with spin-l exchange and thermal one-point blocks for external spin-l operators requires a demonstration that the shadow integral remains well-defined once the external operator carries spin and the background is thermally compactified; the abstract statement alone does not supply this check.

    Authors: The correspondence follows from the same embedding of operators used in the scalar case, now applied to the spinning defect two-point function. The shadow transform for spinning operators is well-defined in the flat-space and defect literature via the appropriate tensor structures; the thermal compactification enters only through the periodicity of the background, which is already accounted for by the defect-to-thermal dictionary. We acknowledge that an explicit statement confirming the integral remains well-defined for spin-l would be helpful. We will therefore add a brief paragraph (with a reference to the relevant flat-space convergence results) in the section discussing the spin-l case. revision: partial

Circularity Check

0 steps flagged

No circularity: derivations start from independent flat-space and defect inputs

full rationale

The abstract and reader's summary describe obtaining thermal blocks via operator configurations and diagonal reductions of the flat-space Casimir system, treating four-point flat-space and two-point defect blocks as given independent objects. No quoted equations or steps reduce any claimed result to a fitted parameter renamed as prediction, a self-definition, or a load-bearing self-citation chain. The shadow formalism is invoked as an external tool whose extension is an assumption, not a tautology internal to the paper's equations. The derivation chain therefore remains self-contained against external CFT benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Relies on the domain assumption that the shadow formalism applies to thermal and defect CFT without modification. No free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Shadow formalism extends directly to thermal and defect backgrounds
    Central to establishing the correspondences and reductions described.

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Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

124 extracted references · 108 canonical work pages · cited by 1 Pith paper · 43 internal anchors

  1. [1]

    Belavin, A

    A. Belavin, A. M. Polyakov and A. Zamolodchikov,Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory,Nucl.Phys.B241(1984) 333–380

    1984

  2. [2]

    J. L. Cardy,Operator Content of Two-Dimensional Conformally Invariant Theories,Nucl. Phys.B270(1986) 186–204

    1986

  3. [3]

    Bounding scalar operator dimensions in 4D CFT

    R. Rattazzi, V. S. Rychkov, E. Tonni and A. Vichi,Bounding scalar operator dimensions in 4D CFT,JHEP12(2008) 031, [0807.0004]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  4. [4]

    The Conformal Bootstrap at Finite Temperature

    L. Iliesiu, M. Kolo˘ glu, R. Mahajan, E. Perlmutter and D. Simmons-Duffin,The Conformal Bootstrap at Finite Temperature,JHEP10(2018) 070, [1802.10266]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  5. [5]

    The Bootstrap Program for Boundary CFT_d

    P. Liendo, L. Rastelli and B. C. van Rees,The Bootstrap Program for Boundary CFT d,JHEP 07(2013) 113, [1210.4258]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  6. [6]

    Emergent Spacetime and Holographic CFTs

    S. El-Showk and K. Papadodimas,Emergent Spacetime and Holographic CFTs,JHEP10 (2012) 106, [1101.4163]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  7. [7]

    Gobeil, A

    Y. Gobeil, A. Maloney, G. S. Ng and J.-q. Wu,Thermal Conformal Blocks,SciPost Phys.7 (2019) 015, [1802.10537]

    work page arXiv 2019

  8. [8]

    A. C. Petkou and A. Stergiou,Dynamics of Finite-Temperature Conformal Field Theories from Operator Product Expansion Inversion Formulas,Phys. Rev. Lett.121(2018) 071602, [1806.02340]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  9. [9]

    Marchetto, A

    E. Marchetto, A. Miscioscia and E. Pomoni,Sum rules & Tauberian theorems at finite temperature,JHEP09(2024) 044, [2312.13030]

    work page arXiv 2024

  10. [10]

    Barrat, E

    J. Barrat, E. Marchetto, A. Miscioscia and E. Pomoni,Thermal Bootstrap for the Critical O(N) Model,Phys. Rev. Lett.134(2025) 211604, [2411.00978]

    work page arXiv 2025

  11. [11]

    The analytic bootstrap at finite temperature

    J. Barrat, D. N. Bozkurt, E. Marchetto, A. Miscioscia and E. Pomoni,The analytic bootstrap at finite temperature,2506.06422. – 23 –

    work page internal anchor Pith review Pith/arXiv arXiv

  12. [12]

    Barrat, D

    J. Barrat, D. N. Bozkurt, E. Marchetto, A. Miscioscia and E. Pomoni,Analytic thermal bootstrap meets holography,2510.20894

    work page arXiv

  13. [13]

    Niarchos, C

    V. Niarchos, C. Papageorgakis, A. Stratoudakis and M. Woolley,Deep finite temperature bootstrap,Phys. Rev. D112(2025) 126012, [2508.08560]

    work page arXiv 2025

  14. [14]

    Buri´ c, I

    I. Buri´ c, I. Gusev and A. Parnachev,Thermal holographic correlators and KMS condition, JHEP09(2025) 053, [2505.10277]

    work page arXiv 2025

  15. [15]

    Witten Diagrams for Torus Conformal Blocks

    P. Kraus, A. Maloney, H. Maxfield, G. S. Ng and J.-q. Wu,Witten Diagrams for Torus Conformal Blocks,JHEP09(2017) 149, [1706.00047]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  16. [16]

    Alkalaev and S

    K. Alkalaev and S. Mandrygin,Torus shadow formalism and exact global conformal blocks, JHEP11(2023) 157, [2307.12061]

    work page arXiv 2023

  17. [17]

    Alkalaev and S

    K. Alkalaev and S. Mandrygin,One-point thermal conformal blocks from four-point conformal integrals,JHEP10(2024) 241, [2407.01741]

    work page arXiv 2024

  18. [18]

    J. R. David and S. Kumar,The large N vector model on S 1×S 2,JHEP03(2025) 169, [2411.18509]

    work page arXiv 2025

  19. [19]

    J. R. David and S. Kumar,High to low temperature: O(N) model at large N,JHEP02(2026) 194, [2508.14872]

    work page arXiv 2026

  20. [20]

    Ammon, J

    M. Ammon, J. Hollweck, T. H¨ ossel and K. W¨ olfl,Thermal n-point conformal blocks in four dimensions from oscillator representations,JHEP04(2026) 102, [2507.22974]

    work page arXiv 2026

  21. [21]

    Buric, F

    I. Buric, F. Russo, V. Schomerus and A. Vichi,Thermal one-point functions and their partial wave decomposition,JHEP12(2024) 021, [2408.02747]

    work page arXiv 2024

  22. [22]

    Buri´ c, F

    I. Buri´ c, F. Mangialardi, F. Russo, V. Schomerus and A. Vichi,Heavy-Heavy-Light Asymptotics from Thermal Correlators,2506.21671

    work page arXiv

  23. [23]

    Anand, N

    H. Anand, N. Benjamin, V. Kumar, S. Minwalla, J. Mukherjee, S. Pal et al.,Semi-universality of CFT d entropy at large spin,2512.00158

    work page arXiv

  24. [24]

    Buri´ c, I

    I. Buri´ c, I. Gusev and A. Parnachev,Holographic correlators from thermal bootstrap,JHEP05 (2026) 059, [2508.08373]

    work page arXiv 2026

  25. [25]

    J. L. Cardy,Operator content and modular properties of higher dimensional conformal field theories,Nucl.Phys.B366(1991) 403–419

    1991

  26. [26]

    Lei and S

    Y. Lei and S. van Leuven,Modularity in d>2 free conformal field theory,JHEP11(2024) 023, [2406.01567]

    work page arXiv 2024

  27. [27]

    Allameh and E

    K. Allameh and E. Shaghoulian,Modular invariance and thermal effective field theory in CFT, JHEP01(2025) 200, [2402.13337]

    work page arXiv 2025

  28. [28]

    Modular invariance on $S^1 \times S^3$ and circle fibrations

    E. Shaghoulian,Modular Invariance of Conformal Field Theory onS 1 ×S 3 and Circle Fibrations,Phys. Rev. Lett.119(2017) 131601, [1612.05257]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  29. [29]

    Cardy Formulae for SUSY Theories in d=4 and d=6

    L. Di Pietro and Z. Komargodski,Cardy formulae for SUSY theories ind=4 andd=6, JHEP12(2014) 031, [1407.6061]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  30. [30]

    Benjamin, J

    N. Benjamin, J. Lee, H. Ooguri and D. Simmons-Duffin,Universal asymptotics for high energy CFT data,JHEP03(2024) 115, [2306.08031]

    work page arXiv 2024

  31. [31]

    Simmons-Duffin and Y

    D. Simmons-Duffin and Y. Xu,A genus-2 crossing equation ind≥2,2511.07569. – 24 –

    work page arXiv

  32. [32]

    J. M. Deutsch,Quantum statistical mechanics in a closed system,Phys. Rev. A43(1991) 2046–2049

    1991

  33. [33]

    Chaos and Quantum Thermalization

    M. Srednicki,Chaos and quantum thermalization,Phys. Rev. E50(1994) 888–901, [cond-mat/9403051]

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  34. [34]

    C. T. Asplund, A. Bernamonti, F. Galli and T. Hartman,Holographic entanglement entropy from 2d cft: Heavy states and local quenches,JHEP02(2015) 171, [1410.1392]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  35. [35]

    A. L. Fitzpatrick, J. Kaplan and M. T. Walters,Universality of long-distance ads physics from the cft bootstrap,JHEP08(2015) 145, [1501.05315]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  36. [36]

    Eigenstate Thermalization Hypothesis in Conformal Field Theory

    N. Lashkari, A. Dymarsky and H. Liu,Eigenstate Thermalization Hypothesis in Conformal Field Theory,J. Stat. Mech.1803(2018) 033101, [1610.00302]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  37. [37]

    Black Hole States in Quantum Spin Chains

    C. Kristjansen and K. Zarembo,Black Hole States in Quantum Spin Chains,2512.23432

    work page internal anchor Pith review Pith/arXiv arXiv

  38. [38]

    A. C. Petkou,Thermal one-point functions and single-valued polylogarithms,Phys. Lett. B 820(2021) 136467, [2105.03530]

    work page arXiv 2021

  39. [39]

    Karydas, S

    M. Karydas, S. Li, A. C. Petkou and M. Vilatte,Conformal Graphs as Twisted Partition Functions,Phys. Rev. Lett.132(2024) 231601, [2312.00135]

    work page arXiv 2024

  40. [40]

    Karydas, S

    M. Karydas, S. Li, A. C. Petkou and M. Vilatte,The thermal representation of conformal ladder integrals,JHEP10(2025) 112, [2508.16718]

    work page arXiv 2025

  41. [41]

    F. A. Dolan and H. Osborn,Conformal partial waves and the operator product expansion, Nucl. Phys. B678(2004) 491–507, [hep-th/0309180]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  42. [42]

    Conformal Partial Waves: Further Mathematical Results

    F. Dolan and H. Osborn,Conformal Partial Waves: Further Mathematical Results,1108.6194

    work page internal anchor Pith review Pith/arXiv arXiv

  43. [43]

    Large-c superconformal torus blocks

    K. Alkalaev and V. Belavin,Large-csuperconformal torus blocks,JHEP08(2018) 042, [1805.12585]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  44. [44]

    Alkalaev, S

    K. Alkalaev, S. Mandrygin and M. Pavlov,Torus conformal blocks and Casimir equations in the necklace channel,JHEP10(2022) 091, [2205.05038]

    work page arXiv 2022

  45. [45]

    Diagonal Limit for Conformal Blocks in d Dimensions

    M. Hogervorst, H. Osborn and S. Rychkov,Diagonal Limit for Conformal Blocks ind Dimensions,JHEP08(2013) 014, [1305.1321]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  46. [46]

    Defects in conformal field theory

    M. Bill` o, V. Gon¸ calves, E. Lauria and M. Meineri,Defects in conformal field theory,JHEP04 (2016) 091, [1601.02883]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  47. [47]

    Conformal constraints on defects

    A. Gadde,Conformal constraints on defects,JHEP01(2020) 038, [1602.06354]

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  48. [48]

    Ferrara, A

    S. Ferrara, A. F. Grillo, G. Parisi and R. Gatto,The shadow operator formalism for conformal algebra. Vacuum expectation values and operator products,Lett. Nuovo Cim.4S2(1972) 115–120

    1972

  49. [49]

    Ferrara, A

    S. Ferrara, A. F. Grillo and G. Parisi,Nonequivalence between conformal covariant wilson expansion in euclidean and minkowski space,Lett. Nuovo Cim.5S2(1972) 147–151

    1972

  50. [50]

    Ferrara and G

    S. Ferrara and G. Parisi,Conformal covariant correlation functions,Nucl. Phys. B42(1972) 281–290

    1972

  51. [51]

    Ferrara, A

    S. Ferrara, A. F. Grillo, G. Parisi and R. Gatto,Covariant expansion of the conformal four-point function,Nucl. Phys. B49(1972) 77–98. – 25 –

    1972

  52. [52]

    V. K. Dobrev, G. Mack, V. B. Petkova, S. G. Petrova and I. T. Todorov,Harmonic Analysis on the n-Dimensional Lorentz Group and Its Application to Conformal Quantum Field Theory, vol. 63. 1977, 10.1007/BFb0009678

    work page doi:10.1007/bfb0009678 1977

  53. [53]

    E. S. Fradkin and M. Y. Palchik,Recent Developments in Conformal Invariant Quantum Field Theory,Phys. Rept.44(1978) 249–349

    1978

  54. [54]

    E. S. Fradkin and M. Y. Palchik,New developments in D-dimensional conformal quantum field theory,Phys. Rept.300(1998) 1–112

    1998

  55. [55]

    Simmons-Duffin,Projectors, Shadows, and Conformal Blocks,JHEP04(2014) 146, [1204.3894]

    D. Simmons-Duffin,Projectors, Shadows, and Conformal Blocks,JHEP04(2014) 146, [1204.3894]

    work page arXiv 2014

  56. [56]

    Karateev, P

    D. Karateev, P. Kravchuk and D. Simmons-Duffin,Weight shifting operators and conformal blocks,JHEP02(2018) 081, [1706.07813]

    work page arXiv 2018

  57. [57]

    Karateev, P

    D. Karateev, P. Kravchuk and D. Simmons-Duffin,Harmonic Analysis and Mean Field Theory,JHEP10(2019) 217, [1809.05111]

    work page arXiv 2019

  58. [58]

    F. A. Dolan and H. Osborn,Implications of N=1 superconformal symmetry for chiral fields, Nucl. Phys. B593(2001) 599–633, [hep-th/0006098]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  59. [59]

    Alkalaev and S

    K. Alkalaev and S. Mandrygin,Multipoint conformal integrals in D dimensions. Part I. Bipartite Mellin-Barnes representation and reconstruction,JHEP09(2025) 118, [2502.12127]

    work page arXiv 2025

  60. [60]

    Alkalaev and S

    K. Alkalaev and S. Mandrygin,Multipoint conformal integrals in D dimensions. Part II. Polygons and basis functions,JHEP12(2025) 088, [2507.01904]

    work page arXiv 2025

  61. [61]

    F. A. Dolan and H. Osborn,Conformal four point functions and the operator product expansion,Nucl. Phys. B599(2001) 459–496, [hep-th/0011040]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  62. [62]

    Solving the 3D Ising Model with the Conformal Bootstrap

    S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi,Solving the 3D Ising Model with the Conformal Bootstrap,Phys. Rev. D86(2012) 025022, [1203.6064]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  63. [63]

    E. E. Boos and A. I. Davydychev,A Method of the Evaluation of the Vertex Type Feynman Integrals,Moscow Univ. Phys. Bull.42N3(1987) 6–10

    1987

  64. [64]

    E. E. Boos and A. I. Davydychev,A Method of evaluating massive Feynman integrals,Theor. Math. Phys.89(1991) 1052–1063

    1991

  65. [65]

    F. A. Dolan,Character formulae and partition functions in higher dimensional conformal field theory,J. Math. Phys.47(2006) 062303, [hep-th/0508031]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  66. [66]

    Recursive representation of the torus 1-point conformal block

    L. Hadasz, Z. Jaskolski and P. Suchanek,Recursive representation of the torus 1-point conformal block,JHEP01(2010) 063, [0911.2353]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  67. [67]

    Operator product expansion for conformal defects

    M. Fukuda, N. Kobayashi and T. Nishioka,Operator product expansion for conformal defects, JHEP01(2018) 013, [1710.11165]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  68. [68]

    D. M. McAvity and H. Osborn,Conformal field theories near a boundary in general dimensions,Nucl. Phys. B455(1995) 522–576, [cond-mat/9505127]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  69. [69]

    Holography and Defect Conformal Field Theories

    O. DeWolfe, D. Z. Freedman and H. Ooguri,Holography and defect conformal field theories, Phys. Rev. D66(2002) 025009, [hep-th/0111135]. – 26 –

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  70. [70]

    The Mellin Formalism for Boundary CFT$_d$

    L. Rastelli and X. Zhou,The Mellin Formalism for Boundary CFT d,JHEP10(2017) 146, [1705.05362]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  71. [71]

    Radial coordinates for defect CFTs

    E. Lauria, M. Meineri and E. Trevisani,Radial coordinates for defect CFTs,JHEP11(2018) 148, [1712.07668]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  72. [72]

    Maz´ aˇ c, L

    D. Maz´ aˇ c, L. Rastelli and X. Zhou,An analytic approach to BCFTd,JHEP12(2019) 004, [1812.09314]

    work page arXiv 2019

  73. [73]

    Lauria, M

    E. Lauria, M. Meineri and E. Trevisani,Spinning operators and defects in conformal field theory,JHEP08(2019) 066, [1807.02522]

    work page arXiv 2019

  74. [74]

    C. P. Herzog and A. Shrestha,Two point functions in defect CFTs,JHEP04(2021) 226, [2010.04995]

    work page arXiv 2021

  75. [75]

    Kobayashi and T

    N. Kobayashi and T. Nishioka,Spinning conformal defects,JHEP09(2018) 134, [1805.05967]

    work page arXiv 2018

  76. [76]

    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, [1903.05222]

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  77. [77]

    Calogero-Sutherland Approach to Defect Blocks

    M. Isachenkov, P. Liendo, Y. Linke and V. Schomerus,Calogero-Sutherland Approach to Defect Blocks,JHEP10(2018) 204, [1806.09703]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  78. [78]

    Buri´ c and V

    I. Buri´ c and V. Schomerus,Defect Conformal Blocks from Appell Functions,JHEP05(2021) 007, [2012.12489]

    work page arXiv 2021

  79. [79]

    Conformal field theory with boundaries and defects

    C. P. Herzog, “Conformal field theory with boundaries and defects.” https://www.ggi.infn.it/laces/LACES21/CFTdefects21_CFTnotes.pdf, 2021

    2021

  80. [80]

    E. S. Fradkin and M. Y. Palchik,Conformal quantum field theory in D-dimensions. 1996

    1996

Showing first 80 references.