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Journal of High Energy Physics

, 2018:22 | Cite as

The Schwarzian theory — a Wilson line perspective

  • Andreas Blommaert
  • Thomas G. MertensEmail author
  • Henri Verschelde
Open Access
Regular Article - Theoretical Physics

Abstract

We provide a holographic perspective on correlation functions in Schwarzian quantum mechanics, as boundary-anchored Wilson line correlators in Jackiw-Teitelboim gravity. We first study compact groups and identify the diagrammatic representation of bilocal correlators of the particle-on-a-group model as Wilson line correlators in its 2d holographic BF description. We generalize to the Hamiltonian reduction of SL+(2, ℝ) and derive the Schwarzian correlation functions. Out-of-time ordered correlators are determined by crossing Wilson lines, giving a 6j-symbol, in agreement with 2d CFT results.

Keywords

2D Gravity Chern-Simons Theories Field Theories in Lower Dimensions Topological Field Theories 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    A. Kitaev, Hidden correlations in the Hawking radiation and thermal noise, Talk given at the Fundamental Physics Prize Symposium, November 10, 2014, https://www.youtube.com/watch?v=OQ9qN8j7EZI.
  2. [2]
    A. Kitaev, Hidden correlations in the Hawking radiation and thermal noise, KITP seminar, February 12, 2015, http://online.kitp.ucsb.edu/online/joint98/kitaev/.
  3. [3]
    A. Kitaev, A simple model of quantum holography (part 1), talk at KITP, April 7, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev/.
  4. [4]
    A. Kitaev, A simple model of quantum holography (part 2), talk at KITP, May 27, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev2/.
  5. [5]
    S. Sachdev and J. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
  6. [6]
    J. Polchinski and V. Rosenhaus, The Spectrum in the Sachdev-Ye-Kitaev Model, JHEP 04 (2016) 001 [arXiv:1601.06768] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    A. Jevicki, K. Suzuki and J. Yoon, Bi-Local Holography in the SYK Model, JHEP 07 (2016) 007 [arXiv:1603.06246] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
  9. [9]
    A. Jevicki and K. Suzuki, Bi-Local Holography in the SYK Model: Perturbations, JHEP 11 (2016) 046 [arXiv:1608.07567] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    J.S. Cotler et al., Black Holes and Random Matrices, JHEP 05 (2017) 118 [Erratum ibid. 09 (2018) 002] [arXiv:1611.04650] [INSPIRE].
  11. [11]
    G. Turiaci and H. Verlinde, Towards a 2d QFT Analog of the SYK Model, JHEP 10 (2017) 167 [arXiv:1701.00528] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    D.J. Gross and V. Rosenhaus, The Bulk Dual of SYK: Cubic Couplings, JHEP 05 (2017) 092 [arXiv:1702.08016] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    D.J. Gross and V. Rosenhaus, All point correlation functions in SYK, JHEP 12 (2017) 148 [arXiv:1710.08113] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    S.R. Das, A. Jevicki and K. Suzuki, Three Dimensional View of the SYK/AdS Duality, JHEP 09 (2017) 017 [arXiv:1704.07208] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    S.R. Das, A. Ghosh, A. Jevicki and K. Suzuki, Space-Time in the SYK Model, JHEP 07 (2018) 184 [arXiv:1712.02725] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    A. Kitaev and S.J. Suh, The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual, JHEP 05 (2018) 183 [arXiv:1711.08467] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    M. Berkooz, P. Narayan and J. Simon, Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction, JHEP 08 (2018) 192 [arXiv:1806.04380] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    R. Jackiw, Lower Dimensional Gravity, Nucl. Phys. B 252 (1985) 343 [INSPIRE].
  19. [19]
    C. Teitelboim, Gravitation and Hamiltonian Structure in Two Space-Time Dimensions, Phys. Lett. B 126 (1983) 41 [INSPIRE].
  20. [20]
    R. Jackiw, Gauge theories for gravity on a line, Theor. Math. Phys. 92 (1992) 979 [Teor. Mat. Fiz. 92 (1992) 404] [hep-th/9206093] [INSPIRE].
  21. [21]
    A. Almheiri and J. Polchinski, Models of AdS 2 backreaction and holography, JHEP 11 (2015) 014 [arXiv:1402.6334] [INSPIRE].
  22. [22]
    K. Jensen, Chaos in AdS 2 Holography, Phys. Rev. Lett. 117 (2016) 111601 [arXiv:1605.06098] [INSPIRE].
  23. [23]
    J. Maldacena, D. Stanford and Z. Yang, Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space, PTEP 2016 (2016) 12C104 [arXiv:1606.01857] [INSPIRE].
  24. [24]
    J. Engelsöy, T.G. Mertens and H. Verlinde, An investigation of AdS 2 backreaction and holography, JHEP 07 (2016) 139 [arXiv:1606.03438] [INSPIRE].
  25. [25]
    M. Cvetič and I. Papadimitriou, AdS 2 holographic dictionary, JHEP 12 (2016) 008 [Erratum ibid. 01 (2017) 120] [arXiv:1608.07018] [INSPIRE].
  26. [26]
    G. Mandal, P. Nayak and S.R. Wadia, Coadjoint orbit action of Virasoro group and two-dimensional quantum gravity dual to SYK/tensor models, JHEP 11 (2017) 046 [arXiv:1702.04266] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    P. Nayak, A. Shukla, R.M. Soni, S.P. Trivedi and V. Vishal, On the Dynamics of Near-Extremal Black Holes, JHEP 09 (2018) 048 [arXiv:1802.09547] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  28. [28]
    T.G. Mertens, G.J. Turiaci and H.L. Verlinde, Solving the Schwarzian via the Conformal Bootstrap, JHEP 08 (2017) 136 [arXiv:1705.08408] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    T.G. Mertens, The Schwarzian theory — origins, JHEP 05 (2018) 036 [arXiv:1801.09605] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    H.T. Lam, T.G. Mertens, G.J. Turiaci and H. Verlinde, Shockwave S-matrix from Schwarzian Quantum Mechanics, arXiv:1804.09834 [INSPIRE].
  31. [31]
    D. Bagrets, A. Altland and A. Kamenev, Sachdev-Ye-Kitaev model as Liouville quantum mechanics, Nucl. Phys. B 911 (2016) 191 [arXiv:1607.00694] [INSPIRE].
  32. [32]
    D. Bagrets, A. Altland and A. Kamenev, Power-law out of time order correlation functions in the SYK model, Nucl. Phys. B 921 (2017) 727 [arXiv:1702.08902] [INSPIRE].
  33. [33]
    A. Kitaev and S.J. Suh, Statistical mechanics of a two-dimensional black hole, arXiv:1808.07032 [INSPIRE].
  34. [34]
    Z. Yang, The Quantum Gravity Dynamics of Near Extremal Black Holes, arXiv:1809.08647 [INSPIRE].
  35. [35]
    D. Stanford and E. Witten, Fermionic Localization of the Schwarzian Theory, JHEP 10 (2017) 008 [arXiv:1703.04612] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    R.A. Davison, W. Fu, A. Georges, Y. Gu, K. Jensen and S. Sachdev, Thermoelectric transport in disordered metals without quasiparticles: The Sachdev-Ye-Kitaev models and holography, Phys. Rev. B 95 (2017) 155131 [arXiv:1612.00849] [INSPIRE].
  37. [37]
    J. Yoon, SYK Models and SYK-like Tensor Models with Global Symmetry, JHEP 10 (2017) 183 [arXiv:1707.01740] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    S. Choudhury, A. Dey, I. Halder, L. Janagal, S. Minwalla and R. Poojary, Notes on melonic O(N)q−1 tensor models, JHEP 06 (2018) 094 [arXiv:1707.09352] [INSPIRE].
  39. [39]
    P. Narayan and J. Yoon, Supersymmetric SYK Model with Global Symmetry, JHEP 08 (2018) 159 [arXiv:1712.02647] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    A. Gaikwad, L.K. Joshi, G. Mandal and S.R. Wadia, Holographic dual to charged SYK from 3D Gravity and Chern-Simons, arXiv:1802.07746 [INSPIRE].
  41. [41]
    W. Fu, D. Gaiotto, J. Maldacena and S. Sachdev, Supersymmetric Sachdev-Ye-Kitaev models, Phys. Rev. D 95 (2017) 026009 [Addendum ibid. D 95 (2017) 069904] [arXiv:1610.08917] [INSPIRE].
  42. [42]
    H.A. González, D. Grumiller and J. Salzer, Towards a bulk description of higher spin SYK, JHEP 05 (2018) 083 [arXiv:1802.01562] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    A. Blommaert, T.G. Mertens and H. Verschelde, Edge dynamics from the path integral — Maxwell and Yang-Mills, JHEP 11 (2018) 080 [arXiv:1804.07585] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    E. Witten, (2+1)-Dimensional Gravity as an Exactly Soluble System, Nucl. Phys. B 311 (1988) 46 [INSPIRE].
  45. [45]
    M. Bershadsky and H. Ooguri, Hidden SL(n) Symmetry in Conformal Field Theories, Commun. Math. Phys. 126 (1989) 49 [INSPIRE].
  46. [46]
    H.L. Verlinde, Conformal Field Theory, 2-D Quantum Gravity and Quantization of Teichmüller Space, Nucl. Phys. B 337 (1990) 652 [INSPIRE].
  47. [47]
    P. Forgacs, A. Wipf, J. Balog, L. Feher and L. O’Raifeartaigh, Liouville and Toda Theories as Conformally Reduced WZNW Theories, Phys. Lett. B 227 (1989) 214 [INSPIRE].
  48. [48]
    J. Balog, L. Feher, L. O’Raifeartaigh, P. Forgacs and A. Wipf, Toda Theory and W Algebra From a Gauged WZNW Point of View, Annals Phys. 203 (1990) 76 [INSPIRE].
  49. [49]
    E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    S. Elitzur, G.W. Moore, A. Schwimmer and N. Seiberg, Remarks on the Canonical Quantization of the Chern-Simons-Witten Theory, Nucl. Phys. B 326 (1989) 108 [INSPIRE].
  51. [51]
    G.W. Moore and N. Seiberg, Taming the Conformal Zoo, Phys. Lett. B 220 (1989) 422 [INSPIRE].
  52. [52]
    L. Iliesiu, S. Pufu, Y. Wang, to appear.Google Scholar
  53. [53]
    S. Cordes, G.W. Moore and S. Ramgoolam, Lectures on 2-D Yang-Mills theory, equivariant cohomology and topological field theories, Nucl. Phys. Proc. Suppl. 41 (1995) 184 [hep-th/9411210] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    E. Witten, On quantum gauge theories in two-dimensions, Commun. Math. Phys. 141 (1991) 153 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    A.A. Migdal, Recursion Equations in Gauge Theories, Sov. Phys. JETP 42 (1975) 413 [Zh. Eksp. Teor. Fiz. 69 (1975) 810] [INSPIRE].
  56. [56]
    E. Witten, Two-dimensional gauge theories revisited, J. Geom. Phys. 9 (1992) 303 [hep-th/9204083] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. [57]
    W. Donnelly and G. Wong, Entanglement branes in a two-dimensional string theory, JHEP 09 (2017) 097 [arXiv:1610.01719] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    M.S. Marinov and M.V. Terentev, Dynamics on the group manifolds and path integral, Fortsch. Phys. 27 (1979) 511 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  59. [59]
    M.-f. Chu and P. Goddard, Quantization of a particle moving on a group manifold, Phys. Lett. B 337 (1994) 285 [hep-th/9407116] [INSPIRE].
  60. [60]
    D.J. Gross and W. Taylor, Twists and Wilson loops in the string theory of two-dimensional QCD, Nucl. Phys. B 403 (1993) 395 [hep-th/9303046] [INSPIRE].
  61. [61]
    A. Blommaert, T.G. Mertens and H. Verschelde, Fine Structure of Jackiw-Teitelboim Quantum Gravity, arXiv:1812.00918 [INSPIRE].
  62. [62]
    A. Giveon, D. Kutasov and N. Seiberg, Comments on string theory on AdS3, Adv. Theor. Math. Phys. 2 (1998) 733 [hep-th/9806194] [INSPIRE].
  63. [63]
    J. Teschner, The minisuperspace limit of the SL(2, ℂ)/SU(2) WZNW model, Nucl. Phys. B 546 (1999) 369 [hep-th/9712258] [INSPIRE].
  64. [64]
    J. Teschner, Operator product expansion and factorization in the H+(3) WZNW model, Nucl. Phys. B 571 (2000) 555 [hep-th/9906215] [INSPIRE].
  65. [65]
    J.M. Maldacena and H. Ooguri, Strings in AdS 3 and SL(2, ℝ) WZW model 1.: The spectrum, J. Math. Phys. 42 (2001) 2929 [hep-th/0001053] [INSPIRE].
  66. [66]
    R. Dijkgraaf, H.L. Verlinde and E.P. Verlinde, String propagation in a black hole geometry, Nucl. Phys. B 371 (1992) 269 [INSPIRE].
  67. [67]
    S. Carlip, Dynamics of asymptotic diffeomorphisms in (2+1)-dimensional gravity, Class. Quant. Grav. 22 (2005) 3055 [gr-qc/0501033] [INSPIRE].
  68. [68]
    G. Compère, L. Donnay, P.-H. Lambert and W. Schulgin, Liouville theory beyond the cosmological horizon, JHEP 03 (2015) 158 [arXiv:1411.7873] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  69. [69]
    O. Coussaert, M. Henneaux and P. van Driel, The asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant, Class. Quant. Grav. 12 (1995) 2961 [gr-qc/9506019] [INSPIRE].
  70. [70]
    A. Alekseev and S.L. Shatashvili, Path Integral Quantization of the Coadjoint Orbits of the Virasoro Group and 2D Gravity, Nucl. Phys. B 323 (1989) 719 [INSPIRE].
  71. [71]
    A. Alekseev and S.L. Shatashvili, From geometric quantization to conformal field theory, Commun. Math. Phys. 128 (1990) 197 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  72. [72]
    Y. Hikida and V. Schomerus, H+(3) WZNW model from Liouville field theory, JHEP 10 (2007) 064 [arXiv:0706.1030] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  73. [73]
    B. Ponsot and J. Teschner, Liouville bootstrap via harmonic analysis on a noncompact quantum group, hep-th/9911110 [INSPIRE].
  74. [74]
    B. Ponsot and J. Teschner, Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of U(q)(SL(2, ℝ)), Commun. Math. Phys. 224 (2001) 613 [math/0007097] [INSPIRE].
  75. [75]
    N.Y. Vilenkin, Special Functions and the Theory of Group Representations, American Mathematical Society, (1968).Google Scholar
  76. [76]
    N.Y. Vilenkin and A.U. Klimyk, Representation of Lie Groups and Special Functions: Volume 1, Kluwer Academic Publishers, (1991).Google Scholar
  77. [77]
    I. Tsutsui and L. Feher, Global aspects of the WZNW reduction to Toda theories, Prog. Theor. Phys. Suppl. 118 (1995) 173 [hep-th/9408065] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  78. [78]
    C.B. Thorn, Liouville perturbation theory, Phys. Rev. D 66 (2002) 027702 [hep-th/0204142] [INSPIRE].
  79. [79]
    E. Braaten, T. Curtright, G. Ghandour and C.B. Thorn, A Class of Conformally Invariant Quantum Field Theories, Phys. Lett. B 125 (1983) 301 [INSPIRE].
  80. [80]
    E. Braaten, T. Curtright, G. Ghandour and C.B. Thorn, Nonperturbative Weak Coupling Analysis of the Quantum Liouville Field Theory, Annals Phys. 153 (1984) 147 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  81. [81]
    N. Seiberg, Notes on quantum Liouville theory and quantum gravity, Prog. Theor. Phys. Suppl. 102 (1990) 319 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  82. [82]
    A. Gervois and H. Navelet, Some Integrals Involving Three Modified Bessel Functions. 2., J. Math. Phys. 27 (1986) 688 [INSPIRE].
  83. [83]
    A. Gervois and H. Navelet, Some Integrals Involving Three Modified Bessel Functions. 1., J. Math. Phys. 27 (1986) 682 [INSPIRE].
  84. [84]
    D. Basu and K.B. Wolf, The Clebsch-Gordan Coefficients of the Three-dimensional Lorentz Algebra in the Parabolic Basis, J. Math. Phys. 24 (1983) 478 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  85. [85]
    I.S. Gradstein and I.M. Ryzhik, Tables of integrals, sums, series and derivatives, Izd. Fizmatgiz, (1963).Google Scholar
  86. [86]
    W. Groenevelt, The Wilson function transform, math/0306424.
  87. [87]
    W. Groenevelt, Wilson function transforms related to Racah coefficients, math/0501511.
  88. [88]
    A.L. Fitzpatrick, J. Kaplan, D. Li and J. Wang, Exact Virasoro Blocks from Wilson Lines and Background-Independent Operators, JHEP 07 (2017) 092 [arXiv:1612.06385] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  89. [89]
    Y. Hikida and V. Schomerus, Structure constants of the OSP(1—2) WZNW model, JHEP 12 (2007) 100 [arXiv:0711.0338] [INSPIRE].
  90. [90]
    M.R. Douglas, I.R. Klebanov, D. Kutasov, J.M. Maldacena, E.J. Martinec and N. Seiberg, A new hat for the c=1 matrix model, hep-th/0307195 [INSPIRE].
  91. [91]
    A. Gerasimov, S. Kharchev, A. Marshakov, A. Mironov, A. Morozov and M. Olshanetsky, Liouville type models in group theory framework. 1. Finite dimensional algebras, Int. J. Mod. Phys. A 12 (1997) 2523 [hep-th/9601161] [INSPIRE].
  92. [92]
    A. Bhatta, P. Raman and N.V. Suryanarayana, Holographic Conformal Partial Waves as Gravitational Open Wilson Networks, JHEP 06 (2016) 119 [arXiv:1602.02962] [INSPIRE].
  93. [93]
    M. Besken, A. Hegde, E. Hijano and P. Kraus, Holographic conformal blocks from interacting Wilson lines, JHEP 08 (2016) 099 [arXiv:1603.07317] [INSPIRE].
  94. [94]
    M. Guica, Bulk fields from the boundary OPE, arXiv:1610.08952 [INSPIRE].
  95. [95]
    A. Bhatta, P. Raman and N.V. Suryanarayana, Scalar Blocks as Gravitational Wilson Networks, arXiv:1806.05475 [INSPIRE].
  96. [96]
    G. Barnich, H.A. González and P. Salgado-Rebolledo, Geometric actions for three-dimensional gravity, Class. Quant. Grav. 35 (2018) 014003 [arXiv:1707.08887] [INSPIRE].
  97. [97]
    F. Falceto and K. Gawedzki, Lattice Wess-Zumino-Witten model and quantum groups, J. Geom. Phys. 11 (1993) 251 [hep-th/9209076] [INSPIRE].
  98. [98]
    I. C-H. Ip, Representation of the quantum plane, its quantum double and harmonic analysis on GL q+(2, R), Sel. Math. New Ser. 19 (2013) 987.Google Scholar
  99. [99]
    H. Jacquet, Fonctions de Whittaker associees aux groupes de Chevalley, Bull. Soc. Math. Fr. 95 (1967) 243.Google Scholar
  100. [100]
    G. Schiffmann, Integrales d’entrelacement et fonctions de Whittaker, Bull. Soc. Math. Fr. 99 (1971) 3.Google Scholar
  101. [101]
    M. Hashizume, Whittaker models for real reductive groups, J. Math. Soc. Jap. 5 (1979) 394.Google Scholar
  102. [102]
    M.Hashizume, Whittaker functions on semisimple Lie groups, Hiroshima Math. J. 12 (198 ) 259.Google Scholar
  103. [103]
    S. Kharchev, D. Lebedev and M. Semenov-Tian-Shansky, Unitary representations of U(q) (sl(2, R)), the modular double and the multiparticle q deformed Toda chains, Commun. Math. Phys. 225 (2002) 573 [hep-th/0102180] [INSPIRE].
  104. [104]
    A. Kitaev, Notes on \( \overset{\sim }{\mathrm{SL}}\left(2,\mathbb{R}\right) \) representations, arXiv:1711.08169 [INSPIRE].
  105. [105]
    A. Chervov, Raising operators for the Whittaker wave functions of the Toda chain and intertwining operators, math/9905005.

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Andreas Blommaert
    • 1
  • Thomas G. Mertens
    • 1
    Email author
  • Henri Verschelde
    • 1
  1. 1.Department of Physics and AstronomyGhent UniversityGentBelgium

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