Advertisement

Exact Bremsstrahlung functions in ABJM theory

  • Lorenzo Bianchi
  • Michelangelo Preti
  • Edoardo Vescovi
Open Access
Regular Article - Theoretical Physics
  • 8 Downloads

Abstract

In this paper we study the Bremsstrahlung functions for the \( \frac{1}{6}\mathrm{B}\mathrm{P}\mathrm{S} \) and the \( \frac{1}{2}\mathrm{B}\mathrm{P}\mathrm{S} \) Wilson lines in ABJM theory. First we use a superconformal defect approach to prove a conjectured relation between the Bremsstrahlung functions associated to the geometric (B 1/6 φ ) and R-symmetry (B 1/6 θ ) deformations of the \( \frac{1}{6}\mathrm{B}\mathrm{P}\mathrm{S} \) Wilson line. This result, non-trivially following from a defect supersymmetric Ward identity, provides an exact expression for B 1/6 θ based on a known result for B 1/6 φ . Subsequently, we explore the consequences of this relation for the \( \frac{1}{2}\mathrm{B}\mathrm{P}\mathrm{S} \) Wilson line and, using the localization result for the multiply wound Wilson loop, we provide an exact closed form for the corresponding Bremsstrahlung function. Interestingly, for the comparison with integrability, this expression appears particularly natural in terms of the conjectured interpolating function h(λ). During the derivation of these results we analyze the protected defect supermultiplets associated to the broken symmetries, including their two- and three-point correlators.

Keywords

Chern-Simons Theories Supersymmetric Gauge Theory Wilson, ’t Hooft and Polyakov loops 

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]
    V. Pestun et al., Localization techniques in quantum field theories, J. Phys. A 50 (2017) 440301 [arXiv:1608.02952] [INSPIRE].MathSciNetMATHGoogle Scholar
  2. [2]
    J.A. Minahan and K. Zarembo, The Bethe ansatz for N = 4 super Yang-Mills, JHEP 03 (2003) 013 [hep-th/0212208] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    D. Correa, J. Henn, J. Maldacena and A. Sever, An exact formula for the radiation of a moving quark in N = 4 super Yang-Mills, JHEP 06 (2012) 048 [arXiv:1202.4455] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    D. Correa, J. Maldacena and A. Sever, The quark anti-quark potential and the cusp anomalous dimension from a TBA equation, JHEP 08 (2012) 134 [arXiv:1203.1913] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    N. Drukker, Integrable Wilson loops, JHEP 10 (2013) 135 [arXiv:1203.1617] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    M. Cooke, A. Dekel and N. Drukker, The Wilson loop CFT: Insertion dimensions and structure constants from wavy lines, J. Phys. A 50 (2017) 335401 [arXiv:1703.03812] [INSPIRE].MathSciNetMATHGoogle Scholar
  7. [7]
    S. Giombi, R. Roiban and A.A. Tseytlin, Half-BPS Wilson loop and AdS 2 /CFT 1, Nucl. Phys. B 922 (2017) 499 [arXiv:1706.00756] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  8. [8]
    B. Fiol, B. Garolera and A. Lewkowycz, Exact results for static and radiative fields of a quark in N = 4 super Yang-Mills, JHEP 05 (2012) 093 [arXiv:1202.5292] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    V. Pestun, Localization of the four-dimensional N = 4 SYM to a two-sphere and 1/8 BPS Wilson loops, JHEP 12 (2012) 067 [arXiv:0906.0638] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    J.K. Erickson, G.W. Semenoff and K. Zarembo, Wilson loops in N = 4 supersymmetric Yang-Mills theory, Nucl. Phys. B 582 (2000) 155 [hep-th/0003055] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    N. Drukker and D.J. Gross, An exact prediction of N = 4 SUSYM theory for string theory, J. Math. Phys. 42 (2001) 2896 [hep-th/0010274] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    N. Gromov, F. Levkovich-Maslyuk and G. Sizov, Analytic solution of Bremsstrahlung TBA II: turning on the sphere angle, JHEP 10 (2013) 036 [arXiv:1305.1944] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    N. Gromov and F. Levkovich-Maslyuk, Quantum spectral curve for a cusped Wilson line in \( \mathcal{N}=4 \) SYM, JHEP 04 (2016) 134 [arXiv:1510.02098] [INSPIRE].ADSMATHGoogle Scholar
  15. [15]
    M. Bonini, L. Griguolo, M. Preti and D. Seminara, Bremsstrahlung function, leading Lüscher correction at weak coupling and localization, JHEP 02 (2016) 172 [arXiv:1511.05016] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  16. [16]
    O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    N. Gromov and G. Sizov, Exact slope and interpolating functions in N = 6 supersymmetric Chern-Simons theory, Phys. Rev. Lett. 113 (2014) 121601 [arXiv:1403.1894] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    D. Gaiotto, S. Giombi and X. Yin, Spin chains in N = 6 superconformal Chern-Simons-Matter theory, JHEP 04 (2009) 066 [arXiv:0806.4589] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    G. Grignani, T. Harmark and M. Orselli, The SU(2) × SU(2) sector in the string dual of N =6 superconformal Chern-Simons theory, Nucl. Phys. B 810 (2009) 115 [arXiv:0806.4959] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    T. Nishioka and T. Takayanagi, On type IIA Penrose limit and N = 6 Chern-Simons theories, JHEP 08 (2008) 001 [arXiv:0806.3391] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    J.A. Minahan, O. Ohlsson Sax and C. Sieg, Magnon dispersion to four loops in the ABJM and ABJ models, J. Phys. A 43 (2010) 275402 [arXiv:0908.2463] [INSPIRE].MathSciNetMATHGoogle Scholar
  22. [22]
    J.A. Minahan, O. Ohlsson Sax and C. Sieg, Anomalous dimensions at four loops in N = 6 superconformal Chern-Simons theories, Nucl. Phys. B 846 (2011) 542 [arXiv:0912.3460] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    M. Leoni et al., Superspace calculation of the four-loop spectrum in N = 6 supersymmetric Chern-Simons theories, JHEP 12 (2010) 074 [arXiv:1010.1756] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    T. McLoughlin, R. Roiban and A.A. Tseytlin, Quantum spinning strings in AdS 4 × CP 3 : testing the Bethe ansatz proposal, JHEP 11 (2008) 069 [arXiv:0809.4038] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    M.C. Abbott, I. Aniceto and D. Bombardelli, Quantum strings and the AdS 4 /CFT 3 interpolating function, JHEP 12 (2010) 040 [arXiv:1006.2174] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  26. [26]
    C. Lopez-Arcos and H. Nastase, Eliminating ambiguities for quantum corrections to strings moving in AdS 4 × 3, Int. J. Mod. Phys. A 28 (2013) 1350058 [arXiv:1203.4777] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  27. [27]
    L. Bianchi et al., Two-loop cusp anomaly in ABJM at strong coupling, JHEP 10 (2014) 013 [arXiv:1407.4788] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    A. Cavaglià, N. Gromov and F. Levkovich-Maslyuk, On the exact interpolating function in ABJ theory, JHEP 12 (2016) 086 [arXiv:1605.04888] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    J.M. Maldacena, Wilson loops in large N field theories, Phys. Rev. Lett. 80 (1998) 4859 [hep-th/9803002] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    D. Berenstein and D. Trancanelli, Three-dimensional N = 6 SCFTs and their membrane dynamics, Phys. Rev. D 78 (2008) 106009 [arXiv:0808.2503] [INSPIRE].ADSGoogle Scholar
  31. [31]
    N. Drukker, J. Plefka and D. Young, Wilson loops in 3-dimensional N = 6 supersymmetric Chern-Simons Theory and their string theory duals, JHEP 11 (2008) 019 [arXiv:0809.2787] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    B. Chen and J.-B. Wu, Supersymmetric Wilson Loops in N = 6 super Chern-Simons-Matter theory, Nucl. Phys. B 825 (2010) 38 [arXiv:0809.2863] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    N. Drukker and D. Trancanelli, A supermatrix model for N = 6 super Chern-Simons-matter theory, JHEP 02 (2010) 058 [arXiv:0912.3006] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  34. [34]
    V. Cardinali, L. Griguolo, G. Martelloni and D. Seminara, New supersymmetric Wilson loops in ABJ(M) theories, Phys. Lett. B 718 (2012) 615 [arXiv:1209.4032] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    A.M. Polyakov, Gauge fields as rings of glue, Nucl. Phys. B 164 (1980) 171 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    G.P. Korchemsky and A.V. Radyushkin, Renormalization of the Wilson loops beyond the leading order, Nucl. Phys. B 283 (1987) 342 [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    N. Drukker, D.J. Gross and H. Ooguri, Wilson loops and minimal surfaces, Phys. Rev. D 60 (1999) 125006 [hep-th/9904191] [INSPIRE].ADSMathSciNetGoogle Scholar
  38. [38]
    N. Drukker and V. Forini, Generalized quark-antiquark potential at weak and strong coupling, JHEP 06 (2011) 131 [arXiv:1105.5144] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  39. [39]
    L. Griguolo et al., The generalized cusp in ABJ(M) N = 6 Super Chern-Simons theories, JHEP 05 (2013) 113 [arXiv:1208.5766] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  40. [40]
    D.H. Correa, J. Aguilera-Damia and G.A. Silva, Strings in AdS 4 × 3 Wilson loops in \( \mathcal{N}=6 \) super Chern-Simons-matter and bremsstrahlung functions,JHEP 06(2014) 139 [arXiv:1405.1396] [INSPIRE].ADSMATHGoogle Scholar
  41. [41]
    M.S. Bianchi et al., BPS Wilson loops and Bremsstrahlung function in ABJ(M): a two loop analysis, JHEP 06 (2014) 123 [arXiv:1402.4128] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  42. [42]
    M.S. Bianchi et al., Towards the exact Bremsstrahlung function of ABJM theory, JHEP 08 (2017) 022 [arXiv:1705.10780] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  43. [43]
    L. Bianchi, L. Griguolo, M. Preti and D. Seminara, Wilson lines as superconformal defects in ABJM theory: a formula for the emitted radiation, JHEP 10 (2017) 050 [arXiv:1706.06590] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  44. [44]
    M.S. Bianchi et al., Framing and localization in Chern-Simons theories with matter, JHEP 06 (2016) 133 [arXiv:1604.00383] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  45. [45]
    A. Lewkowycz and J. Maldacena, Exact results for the entanglement entropy and the energy radiated by a quark, JHEP 05 (2014) 025 [arXiv:1312.5682] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  46. [46]
    A. Kapustin, B. Willett and I. Yaakov, Exact results for Wilson loops in superconformal Chern-Simons theories with matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  47. [47]
    M. Mariño and P. Putrov, Exact results in ABJM theory from topological strings, JHEP 06 (2010) 011 [arXiv:0912.3074] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  48. [48]
    N. Drukker, M. Mariño and P. Putrov, From weak to strong coupling in ABJM theory, Commun. Math. Phys. 306 (2011) 511 [arXiv:1007.3837] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  49. [49]
    A. Klemm, M. Mariño, M. Schiereck and M. Soroush, Aharony-Bergman-Jafferis-Maldacena Wilson loops in the Fermi gas approach, Z. Naturforsch. A 68 (2013) 178 [arXiv:1207.0611] [INSPIRE].ADSGoogle Scholar
  50. [50]
    M.S. Bianchi and A. Mauri, ABJM θ-Bremsstrahlung at four loops and beyond, JHEP 11 (2017) 173 [arXiv:1709.01089] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  51. [51]
    M.S. Bianchi and A. Mauri, ABJM θ-Bremsstrahlung at four loops and beyond: non-planar corrections, JHEP 11 (2017) 166 [arXiv:1709.10092] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  52. [52]
    M. Billò, V. Gonçalves, E. Lauria and M. Meineri, Defects in conformal field theory, JHEP 04 (2016) 091 [arXiv:1601.02883] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  53. [53]
    C. Cordova, T.T. Dumitrescu and K. Intriligator, Multiplets of superconformal symmetry in diverse dimensions, arXiv:1612.00809 [INSPIRE].
  54. [54]
    P. Liendo and C. Meneghelli, Bootstrap equations for \( \mathcal{N}=4 \) SYM with defects, JHEP 01 (2017) 122 [arXiv:1608.05126] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  55. [55]
    R. Blumenhagen, N = 2 supersymmetric W algebras, Nucl. Phys. B 405 (1993) 744 [hep-th/9208069] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  56. [56]
    P. Di Vecchia, J.L. Petersen and H.B. Zheng, N = 2 extended superconformal theories in two-dimensions, Phys. Lett. B 162 (1985) 327.ADSMathSciNetCrossRefGoogle Scholar
  57. [57]
    G. Mussardo, G. Sotkov and M. Stanishkov, N = 2 superconformal minimal models, Int. J. Mod. Phys. A 4 (1989) 1135 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  58. [58]
    M. Cornagliotto, M. Lemos and V. Schomerus, Long multiplet bootstrap, JHEP 10 (2017) 119 [arXiv:1702.05101] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  59. [59]
    A.L. Fitzpatrick et al., Covariant approaches to superconformal blocks, JHEP 08 (2014) 129 [arXiv:1402.1167] [INSPIRE].ADSCrossRefGoogle Scholar
  60. [60]
    M.A. Bandres, A.E. Lipstein and J.H. Schwarz, Studies of the ABJM theory in a formulation with manifest SU(4) R-symmetry, JHEP 09 (2008) 027 [arXiv:0807.0880] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  61. [61]
    S.-J. Rey, T. Suyama and S. Yamaguchi, Wilson loops in superconformal Chern-Simons theory and fundamental strings in Anti-de Sitter supergravity dual, JHEP 03 (2009) 127 [arXiv:0809.3786] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  62. [62]
    K.-M. Lee and S. Lee, 1/2-BPS Wilson loops and vortices in ABJM model, JHEP 09 (2010) 004 [arXiv:1006.5589] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  63. [63]
    N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. 0701 (2007) P01021 [hep-th/0610251] [INSPIRE].Google Scholar
  64. [64]
    N. Gromov and P. Vieira, The all loop AdS 4 /CFT 3 Bethe ansatz, JHEP 01 (2009) 016 [arXiv:0807.0777] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  65. [65]
    M. Bonini, L. Griguolo, M. Preti and D. Seminara, Surprises from the resummation of ladders in the ABJ(M) cusp anomalous dimension, JHEP 05 (2016) 180 [arXiv:1603.00541] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  66. [66]
    M. Preti, WiLE: a Mathematica package for weak coupling expansion of Wilson loops in ABJ(M) theory, Comput. Phys. Commun. 227 (2018) 126 [arXiv:1707.08108] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  67. [67]
    V. Forini, V.G.M. Puletti and O. Ohlsson Sax, The generalized cusp in AdS 4 × CP 3 and more one-loop results from semiclassical strings, J. Phys. A 46 (2013) 115402 [arXiv:1204.3302] [INSPIRE].ADSMATHGoogle Scholar
  68. [68]
    B. Fiol, E. Gerchkovitz and Z. Komargodski, Exact Bremsstrahlung function in N = 2 superconformal field theories, Phys. Rev. Lett. 116 (2016) 081601 [arXiv:1510.01332] [INSPIRE].ADSCrossRefGoogle Scholar
  69. [69]
    L. Bianchi, M. Meineri, R.C. Myers and M. Smolkin, Rényi entropy and conformal defects, JHEP 07 (2016) 076 [arXiv:1511.06713] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  70. [70]
    L. Bianchi et al., Shape dependence of holographic Rényi entropy in general dimensions, JHEP 11 (2016) 180 [arXiv:1607.07418] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  71. [71]
    M. Lemos, P. Liendo, M. Meineri and S. Sarkar, Universality at large transverse spin in defect CFT, arXiv:1712.08185 [INSPIRE].
  72. [72]
    S. Balakrishnan, T. Faulkner, Z.U. Khandker and H. Wang, A general proof of the quantum null energy condition, arXiv:1706.09432 [INSPIRE].
  73. [73]
    X. Dong, Shape dependence of holographic Rényi entropy in conformal field theories, Phys. Rev. Lett. 116 (2016) 251602 [arXiv:1602.08493] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  74. [74]
    S. Balakrishnan, S. Dutta and T. Faulkner, Gravitational dual of the Rényi twist displacement operator, Phys. Rev. D 96 (2017) 046019 [arXiv:1607.06155] [INSPIRE].ADSGoogle Scholar
  75. [75]
    C.P. Herzog and K.-W. Huang, Boundary conformal field theory and a boundary central charge, JHEP 10 (2017) 189 [arXiv:1707.06224] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  76. [76]
    J. Aguilera-Damia, D.H. Correa and G.A. Silva, Semiclassical partition function for strings dual to Wilson loops with small cusps in ABJM, JHEP 03 (2015) 002 [arXiv:1412.4084] [INSPIRE].ADSCrossRefGoogle Scholar
  77. [77]
    M.S. Bianchi, A note on multiply wound BPS Wilson loops in ABJM, JHEP 09 (2016) 047 [arXiv:1605.01025] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  78. [78]
    M.S. Bianchi et al., A matrix model for the latitude Wilson loop in ABJM theory, arXiv:1802.07742 [INSPIRE].
  79. [79]
    M. Mariño, Chern-Simons theory, matrix integrals and perturbative three manifold invariants, Commun. Math. Phys. 253 (2004) 25 [hep-th/0207096] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  80. [80]
    I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series, and products, Academic Press, U.S.A. (2007).MATHGoogle Scholar
  81. [81]
    E. Vescovi, Perturbative and non-perturbative approaches to string sigma-models in AdS/CFT, Springer, Germany (2017).CrossRefMATHGoogle Scholar
  82. [82]
    B. Basso, An exact slope for AdS/CFT, arXiv:1109.3154 [INSPIRE].
  83. [83]
    A. Cavaglià, D. Fioravanti, N. Gromov and R. Tateo, Quantum Spectral Curve of the \( \mathcal{N}=6 \) Supersymmetric Chern-Simons Theory, Phys. Rev. Lett. 113 (2014) 021601 [arXiv:1403.1859] [INSPIRE].ADSCrossRefGoogle Scholar
  84. [84]
    D. Bombardelli, D. Fioravanti and R. Tateo, TBA and Y-system for planar AdS 4 /CFT 3, Nucl. Phys. B 834 (2010) 543 [arXiv:0912.4715] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  85. [85]
    N. Gromov and F. Levkovich-Maslyuk, Y-system, TBA and quasi-classical strings in AdS 4 × CP 3, JHEP 06 (2010) 088 [arXiv:0912.4911] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  86. [86]
    D. Bombardelli et al., The full Quantum Spectral Curve for AdS 4 /CFT 3, JHEP 09 (2017) 140 [arXiv:1701.00473] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Lorenzo Bianchi
    • 1
    • 2
  • Michelangelo Preti
    • 3
    • 4
  • Edoardo Vescovi
    • 5
  1. 1.Institut für Theoretische PhysikUniversität HamburgHamburgGermany
  2. 2.Center for Research in String Theory — School of Physics and AstronomyQueen Mary University of LondonLondonU.K.
  3. 3.Laboratoire de Physique Théorique, Département de Physique de l’ENS, École Normale SupérieureParisFrance
  4. 4.PSL Universités, Sorbonne Universités, CNRSParisFrance
  5. 5.Institute of PhysicsUniversity of São PauloSão PauloBrazil

Personalised recommendations