CoChat
Sign Up
Cited by 27

The DOI matches its Crossref record; 5 of the 5 references it cites were found in a registry or an index.

Multiboundary wormholes and OPE statistics

de Boer, Jan, Liška, Diego, Post, Boris·2024·Journal of High Energy Physics
DOI 10.1007/jhep10(2024)207
Open this paper in CoChat — free account

Abstract

A bstract: We derive higher moments in the statistical distribution of OPE coefficients in holographic 2D CFTs, and show that such moments correspond to multiboundary Euclidean wormholes in pure 3D gravity. The n th cyclic non-Gaussian contraction of heavy-heavy-light OPE coefficients follows from crossing symmetry of the thermal n -point function. We derive universal expressions for the cubic and quartic moments and demonstrate that their scaling with the microcanonical entropy agrees with a generalization of the Eigenstate Thermalization Hypothesis. Motivated by this result, we conjecture that the full statistical ensemble of OPE data is fixed by three premises: typicality, crossing symmetry and modular invariance. Together, these properties give predictions for non-factorizing observables, such as the generalized spectral form factor. Using the Virasoro TQFT, we match these connected averages to new on-shell wormhole topologies with multiple boundary components. Lastly, we study and clarify examples where the statistics of heavy operators are not universal and depend on the light operator spectrum. We give a gravitational interpretation to these corrections in terms of Wilson loops winding around non-trivial cycles in the bulk.

Overview

The claims and statistics of this paper haven’t been checked yet.

Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.

A few free checks a day · papers already verified open instantly · results are public

Integrity signals

Registry

Crossref · 10.1007/jhep10(2024)207

Cites

5 of 5 references verified0 citation errors · 0 not checkable

Statistics

Figures

Image 1
CC-BY · From the PDF
Image 2
CC-BY · From the PDF
Figure 1 The three basic crossing moves that relate conformal block decompositions in different channels. From left to right, the moves are called fusion, modular S-transform and braiding.
Figure 1 The three basic crossing moves that relate conformal block decompositions in different channels. From left to right, the moves are called fusion, modular S-transform and braiding.CC-BY · From the PDF
Image 4
CC-BY · From the PDF
Figure 2 The sequence of moves relating the necklace and OPE channel for the torus four-point function. Depicted below each surface is the corresponding trivalent diagram representing the pair-of-pants decomposition.
Figure 2 The sequence of moves relating the necklace and OPE channel for the torus four-point function. Depicted below each surface is the corresponding trivalent diagram representing the pair-of-pants decomposition.CC-BY · From the PDF
Image 6
CC-BY · From the PDF
Image 7
CC-BY · From the PDF
Figure 3 Glueing along the inner torus boundaries in blue produces the punctured-torus wormhole.
Figure 3 Glueing along the inner torus boundaries in blue produces the punctured-torus wormhole.CC-BY · From the PDF
Figure 4 Heegaard splitting of the three-boundary wormhole.
Figure 4 Heegaard splitting of the three-boundary wormhole.CC-BY · From the PDF
Figure 5 The Heegaard splitting described in this section leads to a three-boundary torus wormhole, whose conical defect lines meet in a bulk three-point vertex. The bulk has non-trivial topology (not shown in this figure), arising from the glueing of genus-two handlebodies.
Figure 5 The Heegaard splitting described in this section leads to a three-boundary torus wormhole, whose conical defect lines meet in a bulk three-point vertex. The bulk has non-trivial topology (not shown in this figure), arising from the glueing of genus-two handlebodies.CC-BY · From the PDF
Image 11
CC-BY · From the PDF
Image 12
CC-BY · From the PDF
Figure 6 Heegaard splitting of the four-boundary wormhole.
Figure 6 Heegaard splitting of the four-boundary wormhole.CC-BY · From the PDF
Image 14
CC-BY · From the PDF
Figure 7 The 'skyline' channel decomposition of the genus-three partition function (a) can be thought of, diagrammatically, as a three-dimensional tetrahedron (b) with edges labelled by Liouville momenta P i , i = 1 , . . . , 6. The smooth function appearing in the quartic non-Gaussianity (4.4) has the same symmetries as this tetrahedron. The diagram can equivalently be drawn as a crossing partition of an 'exchange' diagram (c) between four operators, modulo phases.
Figure 7 The 'skyline' channel decomposition of the genus-three partition function (a) can be thought of, diagrammatically, as a three-dimensional tetrahedron (b) with edges labelled by Liouville momenta P i , i = 1 , . . . , 6. The smooth function appearing in the quartic non-Gaussianity (4.4) has the same symmetries as this tetrahedron. The diagram can equivalently be drawn as a crossing partition of an 'exchange' diagram (c) between four operators, modulo phases.CC-BY · From the PDF
Figure 8 Heegaard splitting of the non-Gaussian genus-two wormhole.
Figure 8 Heegaard splitting of the non-Gaussian genus-two wormhole.CC-BY · From the PDF
Figure 9 Crossing moves relating the genus-two block in the dumbbell (left) and sunset (right) channels.
Figure 9 Crossing moves relating the genus-two block in the dumbbell (left) and sunset (right) channels.CC-BY · From the PDF
Image 18
CC-BY · From the PDF
Image 19
CC-BY · From the PDF
Image 20
CC-BY · From the PDF
Figure 10 A handlebody with a genus-two boundary whose filling is the outside region of the three-sphere. On the left, the cycle labeled by P 2 is contractible in the bulk. On the right, the same cycle is confined by the Wilson line P χ .
Figure 10 A handlebody with a genus-two boundary whose filling is the outside region of the three-sphere. On the left, the cycle labeled by P 2 is contractible in the bulk. On the right, the same cycle is confined by the Wilson line P χ .CC-BY · From the PDF
Image 22
CC-BY · From the PDF
Image 23
CC-BY · From the PDF
Image 24
CC-BY · From the PDF
Image 25
CC-BY · From the PDF
Image 26
CC-BY · From the PDF
Image 27
CC-BY · From the PDF

Verification

Claims the paper makes

The claims and statistics of this paper haven’t been checked yet.

Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.

A few free checks a day · papers already verified open instantly · results are public

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

The self-check needs the paper’s full text.

Citations

Cites

Research works printed but not linked: 87

  1. P. Saad, S.H. Shenker and D. Stanford, JT gravity as a matrix integral, arXiv:1903.11115 [INSPIRE]. · arXiv
  2. J.M. Maldacena and L. Maoz, Wormholes in AdS, JHEP 02 (2004) 053 [hep-th/0401024] [INSPIRE]. · DOI · Paper page
  3. D. Stanford and E. Witten, JT gravity and the ensembles of random matrix theory, Adv. Theor. Math. Phys. 24 (2020) 1475 [arXiv:1907.03363] [INSPIRE]. · DOI · Paper page
  4. G.J. Turiaci and E. Witten, N = 2 JT supergravity and matrix models, JHEP 12 (2023) 003 [arXiv:2305.19438] [INSPIRE]. · DOI · Paper page
  5. E. Witten, Matrix models and deformations of JT gravity, Proc. Roy. Soc. Lond. A 476 (2020) 20200582 [arXiv:2006.13414] [INSPIRE]. · arXiv
  6. A. Altland and J. Sonner, Late time physics of holographic quantum chaos, SciPost Phys. 11 (2021) 034 [arXiv:2008.02271] [INSPIRE]. · DOI
  7. A. Altland et al., Quantum chaos in 2D gravity, SciPost Phys. 15 (2023) 064 [arXiv:2204.07583] [INSPIRE]. · DOI · Paper page
  8. S. Collier, L. Eberhardt, B. Muehlmann and V.A. Rodriguez, The Virasoro minimal string, SciPost Phys. 16 (2024) 057 [arXiv:2309.10846] [INSPIRE]. · DOI
  9. A. Altland et al., From operator statistics to wormholes, Phys. Rev. Res. 3 (2021) 033259 [arXiv:2105.12129] [INSPIRE]. · DOI
  10. A. Belin and J. de Boer, Random statistics of OPE coefficients and Euclidean wormholes, Class. Quant. Grav. 38 (2021) 164001 [arXiv:2006.05499] [INSPIRE]. · DOI · Paper page
  11. J. Chandra, S. Collier, T. Hartman and A. Maloney, Semiclassical 3D gravity as an average of large-c CFTs, JHEP 12 (2022) 069 [arXiv:2203.06511] [INSPIRE]. · DOI
  12. S. Collier, L. Eberhardt and M. Zhang, Solving 3d gravity with Virasoro TQFT, SciPost Phys. 15 (2023) 151 [arXiv:2304.13650] [INSPIRE]. · DOI
  13. S. Collier, L. Eberhardt and M. Zhang, 3d gravity from Virasoro TQFT: holography, wormholes and knots, arXiv:2401.13900 [INSPIRE]. · arXiv
  14. M. Sasieta, Wormholes from heavy operator statistics in AdS/CFT, JHEP 03 (2023) 158 [arXiv:2211.11794] [INSPIRE]. · DOI · Paper page
  15. V. Balasubramanian, A. Lawrence, J.M. Magan and M. Sasieta, Microscopic origin of the entropy of astrophysical black holes, Phys. Rev. Lett. 132 (2024) 141501 [arXiv:2212.08623] [INSPIRE]. · DOI
  16. B. Freivogel, D. Nikolakopoulou and A.F. Rotundo, Wormholes from averaging over states, SciPost Phys. 14 (2023) 026 [arXiv:2105.12771] [INSPIRE]. · DOI
  17. J. Chandra and T. Hartman, Coarse graining pure states in AdS/CFT, JHEP 10 (2023) 030 [arXiv:2206.03414] [INSPIRE]. · DOI
  18. J. Chandra, Euclidean wormholes for individual 2d CFTs, JHEP 04 (2024) 051 [arXiv:2305.07183] [INSPIRE]. · DOI
  19. J. Chandra and T. Hartman, Toward random tensor networks and holographic codes in CFT, JHEP 05 (2023) 109 [arXiv:2302.02446] [INSPIRE]. · DOI
  20. A. Belin et al., Approximate CFTs and random tensor models, JHEP 09 (2024) 163 [arXiv:2308.03829] [INSPIRE]. · DOI
  21. D.L. Jafferis, D.K. Kolchmeyer, B. Mukhametzhanov and J. Sonner, Jackiw-Teitelboim gravity with matter, generalized eigenstate thermalization hypothesis, and random matrices, Phys. Rev. D 108 (2023) 066015 [arXiv:2209.02131] [INSPIRE]. · DOI
  22. D.L. Jafferis, D.K. Kolchmeyer, B. Mukhametzhanov and J. Sonner, Matrix models for eigenstate thermalization, Phys. Rev. X 13 (2023) 031033 [arXiv:2209.02130] [INSPIRE]. · arXiv
  23. J. de Boer, D. Liska, B. Post and M. Sasieta, A principle of maximum ignorance for semiclassical gravity, JHEP 02 (2024) 003 [arXiv:2311.08132] [INSPIRE]. · DOI
  24. J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE]. · DOI
  25. M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50 (1994) 888 [cond-mat/9403051] [INSPIRE]. · DOI · Paper page
  26. M. Rigol, V. Dunjko and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature 452 (2008) 854 [arXiv:0708.1324] [INSPIRE]. · DOI
  27. L. Foini and J. Kurchan, Eigenstate thermalization hypothesis and out of time order correlators, Phys. Rev. E 99 (2019) 042139 [arXiv:1803.10658] [INSPIRE]. · DOI
  28. C. Murthy and M. Srednicki, Bounds on chaos from the eigenstate thermalization hypothesis, Phys. Rev. Lett. 123 (2019) 230606 [arXiv:1906.10808] [INSPIRE]. · DOI
  29. S. Jackson, L. McGough and H. Verlinde, Conformal bootstrap, universality and gravitational scattering, Nucl. Phys. B 901 (2015) 382 [arXiv:1412.5205] [INSPIRE]. · DOI
  30. J. Cardy, A. Maloney and H. Maxfield, A new handle on three-point coefficients: OPE asymptotics from genus two modular invariance, JHEP 10 (2017) 136 [arXiv:1705.05855] [INSPIRE]. · DOI
  31. S. Collier, A. Maloney, H. Maxfield and I. Tsiares, Universal dynamics of heavy operators in CFT2, JHEP 07 (2020) 074 [arXiv:1912.00222] [INSPIRE]. · DOI · Paper page
  32. A. Belin, J. de Boer and D. Liska, Non-Gaussianities in the statistical distribution of heavy OPE coefficients and wormholes, JHEP 06 (2022) 116 [arXiv:2110.14649] [INSPIRE]. · DOI · Paper page
  33. T. Anous, A. Belin, J. de Boer and D. Liska, OPE statistics from higher-point crossing, JHEP 06 (2022) 102 [arXiv:2112.09143] [INSPIRE]. · DOI
  34. N. Lashkari, A. Dymarsky and H. Liu, Eigenstate thermalization hypothesis in conformal field theory, J. Stat. Mech. 1803 (2018) 033101 [arXiv:1610.00302] [INSPIRE]. · DOI
  35. G. Turiaci and H. Verlinde, On CFT and quantum chaos, JHEP 12 (2016) 110 [arXiv:1603.03020] [INSPIRE]. · DOI
  36. Y. Hikida, Y. Kusuki and T. Takayanagi, Eigenstate thermalization hypothesis and modular invariance of two-dimensional conformal field theories, Phys. Rev. D 98 (2018) 026003 [arXiv:1804.09658] [INSPIRE]. · DOI
  37. P. Basu, D. Das, S. Datta and S. Pal, Thermality of eigenstates in conformal field theories, Phys. Rev. E 96 (2017) 022149 [arXiv:1705.03001] [INSPIRE]. · DOI
  38. P. Kraus and A. Maloney, A Cardy formula for three-point coefficients or how the black hole got its spots, JHEP 05 (2017) 160 [arXiv:1608.03284] [INSPIRE]. · DOI
  39. B. Ponsot and J. Teschner, Liouville bootstrap via harmonic analysis on a noncompact quantum group, hep-th/9911110 [INSPIRE]. · arXiv
  40. B. Ponsot and J. Teschner, Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of Uq(sl(2, R)), Commun. Math. Phys. 224 (2001) 613 [math/0007097] [INSPIRE]. · DOI
  41. L. Eberhardt, Notes on crossing transformations of Virasoro conformal blocks, arXiv:2309.11540 [INSPIRE]. · arXiv
  42. J. Wang et al., Emergence of unitary symmetry of microcanonically truncated operators in chaotic quantum systems, Phys. Rev. E 110 (2024) L032203 [arXiv:2310.20264] [INSPIRE]. · DOI
  43. S. Pappalardi, L. Foini and J. Kurchan, Eigenstate thermalization hypothesis and free probability, Phys. Rev. Lett. 129 (2022) 170603 [arXiv:2204.11679] [INSPIRE]. · DOI
  44. A. Maloney and E. Witten, Quantum gravity partition functions in three dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [INSPIRE]. · DOI
  45. C.A. Keller and A. Maloney, Poincaré series, 3D gravity and CFT spectroscopy, JHEP 02 (2015) 080 [arXiv:1407.6008] [INSPIRE]. · DOI
  46. N. Benjamin, S. Collier and A. Maloney, Pure gravity and conical defects, JHEP 09 (2020) 034 [arXiv:2004.14428] [INSPIRE]. · DOI
  47. H. Maxfield and G.J. Turiaci, The path integral of 3D gravity near extremality; or, JT gravity with defects as a matrix integral, JHEP 01 (2021) 118 [arXiv:2006.11317] [INSPIRE]. · DOI · Paper page
  48. G. Di Ubaldo and E. Perlmutter, AdS3 pure gravity and stringy unitarity, Phys. Rev. Lett. 132 (2024) 041602 [arXiv:2308.01787] [INSPIRE]. · DOI
  49. T. Hartman, C.A. Keller and B. Stoica, Universal spectrum of 2d conformal field theory in the large c limit, JHEP 09 (2014) 118 [arXiv:1405.5137] [INSPIRE]. · DOI
  50. F.M. Haehl, C. Marteau, W. Reeves and M. Rozali, Symmetries and spectral statistics in chaotic conformal field theories, JHEP 07 (2023) 196 [arXiv:2302.14482] [INSPIRE]. · DOI
  51. C. Yan, More on torus wormholes in 3d gravity, JHEP 11 (2023) 039 [arXiv:2305.10494] [INSPIRE]. · DOI
  52. G.W. Moore and N. Seiberg, Classical and quantum conformal field theory, Commun. Math. Phys. 123 (1989) 177 [INSPIRE]. · DOI · Paper page
  53. J. Sonner and M. Vielma, Eigenstate thermalization in the Sachdev-Ye-Kitaev model, JHEP 11 (2017) 149 [arXiv:1707.08013] [INSPIRE]. · DOI
  54. S. Collier, Y. Gobeil, H. Maxfield and E. Perlmutter, Quantum Regge trajectories and the Virasoro analytic bootstrap, JHEP 05 (2019) 212 [arXiv:1811.05710] [INSPIRE]. · DOI
  55. B. Mukhametzhanov and A. Zhiboedov, Modular invariance, tauberian theorems and microcanonical entropy, JHEP 10 (2019) 261 [arXiv:1904.06359] [INSPIRE]. · DOI
  56. D. Das, S. Datta and S. Pal, Charged structure constants from modularity, JHEP 11 (2017) 183 [arXiv:1706.04612] [INSPIRE]. · DOI
  57. S. Pal and Z. Sun, Tauberian-Cardy formula with spin, JHEP 01 (2020) 135 [arXiv:1910.07727] [INSPIRE]. · DOI
  58. D. Das, Y. Kusuki and S. Pal, Universality in asymptotic bounds and its saturation in 2D CFT, JHEP 04 (2021) 288 [arXiv:2011.02482] [INSPIRE]. · DOI
  59. S. Pal, Bound on asymptotics of magnitude of three point coefficients in 2D CFT, JHEP 01 (2020) 023 [arXiv:1906.11223] [INSPIRE]. · DOI
  60. J. Qiao and S. Rychkov, A tauberian theorem for the conformal bootstrap, JHEP 12 (2017) 119 [arXiv:1709.00008] [INSPIRE]. · DOI
  61. C. Scarinci and K. Krasnov, The universal phase space of AdS3 gravity, Commun. Math. Phys. 322 (2013) 167 [arXiv:1111.6507] [INSPIRE]. · DOI
  62. K. Krasnov and J.-M. Schlenker, Minimal surfaces and particles in 3-manifolds, Geom. Dedicata 126 (2007) 187 [math/0511441] [INSPIRE]. · DOI
  63. E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE]. · DOI · Paper page
  64. H.L. Verlinde, Conformal field theory, 2D quantum gravity and quantization of Teichmüller space, Nucl. Phys. B 337 (1990) 652 [INSPIRE]. · DOI
  65. J. Ellegaard Andersen and R. Kashaev, A TQFT from quantum Teichmüller theory, Commun. Math. Phys. 330 (2014) 887 [arXiv:1109.6295] [INSPIRE]. · DOI
  66. J. Ellegaard Andersen and R. Kashaev, A new formulation of the Teichmüller TQFT, arXiv:1305.4291 [INSPIRE]. · arXiv
  67. J.E. Andersen and R. Kashaev, The Teichmüller TQFT, in the proceedings of the International congress of mathematicians, World Scientific, Singapore (2018), p. 2541 [https://doi.org/10.1142/9789813272880_0149] [arXiv:1811.06853] [INSPIRE]. · DOI
  68. K. Hikami, Hyperbolic structure arising from a knot invariant, Int. J. Mod. Phys. A 16 (2001) 3309. · DOI
  69. K. Hikami, Generalized volume conjecture and the A-polynomials: the Neumann-Zagier potential function as a classical limit of quantum invariant, J. Geom. Phys. 57 (2007) 1895 [math/0604094] [INSPIRE]. · DOI
  70. J. Cotler and K. Jensen, AdS3 gravity and random CFT, JHEP 04 (2021) 033 [arXiv:2006.08648] [INSPIRE]. · DOI
  71. J.-M. Schlenker and E. Witten, No ensemble averaging below the black hole threshold, JHEP 07 (2022) 143 [arXiv:2202.01372] [INSPIRE]. · DOI
  72. C.-M. Chang and Y.-H. Lin, Bootstrap, universality and horizons, JHEP 10 (2016) 068 [arXiv:1604.01774] [INSPIRE]. · DOI
  73. D. Berenstein, D. Grabovsky and Z. Li, Aspects of holography in conical AdS3, JHEP 04 (2023) 029 [arXiv:2205.02256] [INSPIRE]. · DOI
  74. J.-M. Schlenker and E. Witten, No ensemble averaging below the black hole threshold, JHEP 07 (2022) 143 [arXiv:2202.01372] [INSPIRE]. · DOI
  75. A. Belin, J. de Boer, P. Nayak and J. Sonner, Generalized spectral form factors and the statistics of heavy operators, JHEP 11 (2022) 145 [arXiv:2111.06373] [INSPIRE]. · DOI
  76. J.E. Andersen et al., On the Kontsevich geometry of the combinatorial Teichmüller space, arXiv:2010.11806 [INSPIRE]. · arXiv
  77. H. Maxfield, S. Ross and B. Way, Holographic partition functions and phases for higher genus Riemann surfaces, Class. Quant. Grav. 33 (2016) 125018 [arXiv:1601.00980] [INSPIRE]. · DOI
  78. D. Rolfsen, Knots and links, AMS Chelsea Publishing Series (2003). · DOI
  79. H. Murakami, An introduction to the volume conjecture, in Interactions between hyperbolic geometry, quantum topology and number theory 541, (2011), p. 1. · DOI
  80. E. Witten, Analytic continuation of Chern-Simons theory, AMS/IP Stud. Adv. Math. 50 (2011) 347 [arXiv:1001.2933] [INSPIRE]. · DOI
  81. D. Gabai, G.R. Meyerhoff and N. Thurston, Homotopy hyperbolic 3-manifolds are hyperbolic, Ann. Math. 157 (2003) 335 [math/9609207]. · DOI
  82. A. Castro, I. Coman, J.R. Fliss and C. Zukowski, Coupling fields to 3D quantum gravity via Chern-Simons theory, Phys. Rev. Lett. 131 (2023) 171602 [arXiv:2304.02668] [INSPIRE]. · DOI
  83. D.L. Jafferis, L. Rozenberg and G. Wong, 3d gravity as a random ensemble, arXiv:2407.02649 [INSPIRE]. · arXiv
  84. J. Cotler and K. Jensen, Wormholes and black hole microstates in AdS/CFT, JHEP 09 (2021) 001 [arXiv:2104.00601] [INSPIRE]. · DOI
  85. J. Cotler and K. Jensen, A precision test of averaging in AdS/CFT, JHEP 11 (2022) 070 [arXiv:2205.12968] [INSPIRE]. · DOI · Paper page
  86. G. Di Ubaldo and E. Perlmutter, AdS3/RMT2 duality, JHEP 12 (2023) 179 [arXiv:2307.03707] [INSPIRE]. · DOI
  87. N. Nemkov, On modular transformations of toric conformal blocks, JHEP 10 (2015) 039 [arXiv:1504.04360] [INSPIRE]. · DOI

Cited by

Sorted by citations

See all 27

Metadata

Authors
de Boer, J., Liška, D., & Post, B.
Venue
Journal of High Energy Physics
Publisher
Springer Science and Business Media LLC
Published
2024-10-29
Volume
2024
Issue
10
ISSN
1029-8479
DOI
10.1007/jhep10(2024)207
Cited by
27
References
5
Language
en
Licence

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

Are you an author, or do you see a problem with this page? Request a review