Space-time CFTs from the Riemann sphere

  • Tim Adamo
  • Ricardo Monteiro
  • Miguel F. Paulos
Open Access
Regular Article - Theoretical Physics


We consider two-dimensional chiral, first-order conformal field theories governing maps from the Riemann sphere to the projective light cone inside Minkowski space — the natural setting for describing conformal field theories in two fewer dimensions. These theories have a SL(2) algebra of local bosonic constraints which can be supplemented by additional fermionic constraints depending on the matter content of the theory. By computing the BRST charge associated with gauge fixing these constraints, we find anomalies which vanish for specific target space dimensions. These critical dimensions coincide precisely with those for which (biadjoint) cubic scalar theory, gauge theory and gravity are classically conformally invariant. Furthermore, the BRST cohomology of each theory contains vertex operators for the full conformal multiplets of single field insertions in each of these space-time CFTs. We give a prescription for the computation of three-point functions, and compare our formalism with the scattering equations approach to on-shell amplitudes.


Conformal Field Models in String Theory Conformal Field Theory 


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.


  1. [1]
    H. Elvang and Y.-t. Huang, Scattering Amplitudes in Gauge Theory and Gravity, Cambridge University Press, (2015).Google Scholar
  2. [2]
    F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles in Arbitrary Dimensions, Phys. Rev. Lett. 113 (2014) 171601 [arXiv:1307.2199] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    F. Cachazo, S. He and E.Y. Yuan, Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM, JHEP 07 (2015) 149 [arXiv:1412.3479] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    D.B. Fairlie and D.E. Roberts, Dual Models without Tachyons - a New Approach, Durham Preprint (1972) PRINT-72-2440.Google Scholar
  5. [5]
    D.J. Gross and P.F. Mende, String Theory Beyond the Planck Scale, Nucl. Phys. B 303 (1988) 407 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    E. Witten, Parity invariance for strings in twistor space, Adv. Theor. Math. Phys. 8 (2004) 779 [hep-th/0403199] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    F. Cachazo, S. He and E.Y. Yuan, Scattering equations and Kawai-Lewellen-Tye orthogonality, Phys. Rev. D 90 (2014) 065001 [arXiv:1306.6575] [INSPIRE].ADSGoogle Scholar
  8. [8]
    L. Dolan and P. Goddard, Proof of the Formula of Cachazo, He and Yuan for Yang-Mills Tree Amplitudes in Arbitrary Dimension, JHEP 05 (2014) 010 [arXiv:1311.5200] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    L. Mason and D. Skinner, Ambitwistor strings and the scattering equations, JHEP 07 (2014) 048 [arXiv:1311.2564] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    R. Roiban, M. Spradlin and A. Volovich, On the tree level S matrix of Yang-Mills theory, Phys. Rev. D 70 (2004) 026009 [hep-th/0403190] [INSPIRE].ADSMathSciNetGoogle Scholar
  12. [12]
    K. Ohmori, Worldsheet Geometries of Ambitwistor String, JHEP 06 (2015) 075 [arXiv:1504.02675] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    E. Casali, Y. Geyer, L. Mason, R. Monteiro and K.A. Roehrig, New Ambitwistor String Theories, JHEP 11 (2015) 038 [arXiv:1506.08771] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    T. Adamo, E. Casali and D. Skinner, Ambitwistor strings and the scattering equations at one loop, JHEP 04 (2014) 104 [arXiv:1312.3828] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    T. Adamo and E. Casali, Scattering equations, supergravity integrands and pure spinors, JHEP 05 (2015) 120 [arXiv:1502.06826] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, Loop Integrands for Scattering Amplitudes from the Riemann Sphere, Phys. Rev. Lett. 115 (2015) 121603 [arXiv:1507.00321] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, One-loop amplitudes on the Riemann sphere, JHEP 03 (2016) 114 [arXiv:1511.06315] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, Two-Loop Scattering Amplitudes from the Riemann Sphere, Phys. Rev. D 94 (2016) 125029 [arXiv:1607.08887] [INSPIRE].ADSGoogle Scholar
  19. [19]
    J.J. Fourier, Theorie analytique de la chaleur, Firmin Didot, pere et fils, (1822).Google Scholar
  20. [20]
    T. Adamo, M. Bullimore, L. Mason and D. Skinner, A Proof of the Supersymmetric Correlation Function/Wilson Loop Correspondence, JHEP 08 (2011) 076 [arXiv:1103.4119] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  21. [21]
    T. Adamo, Correlation functions, null polygonal Wilson loops and local operators, JHEP 12 (2011) 006 [arXiv:1110.3925] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    D. Chicherin et al., Correlation functions of the chiral stress-tensor multiplet in \( \mathcal{N} \) = 4 SYM, JHEP 06 (2015) 198 [arXiv:1412.8718] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    D. Chicherin and E. Sokatchev, \( \mathcal{N} \) = 4 super-Yang-Mills in LHC superspace part II: non-chiral correlation functions of the stress-tensor multiplet, JHEP 03 (2017) 048 [arXiv:1601.06804] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    L. Koster, V. Mitev, M. Staudacher and M. Wilhelm, Composite Operators in the Twistor Formulation of N = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 117 (2016) 011601 [arXiv:1603.04471] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    L. Koster, V. Mitev, M. Staudacher and M. Wilhelm, All tree-level MHV form factors in \( \mathcal{N} \) = 4 SYM from twistor space, JHEP 06 (2016) 162 [arXiv:1604.00012] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    D. Chicherin and E. Sokatchev, Composite operators and form factors in \( \mathcal{N} \) = 4 SYM, J. Phys. A 50 (2017) 275402 [arXiv:1605.01386] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  27. [27]
    S. He and Y. Zhang, Connected formulas for amplitudes in standard model, JHEP 03 (2017) 093 [arXiv:1607.02843] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    A. Brandhuber, E. Hughes, R. Panerai, B. Spence and G. Travaglini, The connected prescription for form factors in twistor space, JHEP 11 (2016) 143 [arXiv:1608.03277] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    S. He and Z. Liu, A note on connected formula for form factors, JHEP 12 (2016) 006 [arXiv:1608.04306] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    L. Koster, V. Mitev, M. Staudacher and M. Wilhelm, On Form Factors and Correlation Functions in Twistor Space, JHEP 03 (2017) 131 [arXiv:1611.08599] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  31. [31]
    B. Eden, P. Heslop and L. Mason, The Correlahedron, arXiv:1701.00453 [INSPIRE].
  32. [32]
    R. Marnelius, Manifestly Conformal Covariant Description of Spinning and Charged Particles, Phys. Rev. D 20 (1979) 2091 [INSPIRE].ADSGoogle Scholar
  33. [33]
    I. Bars and C. Kounnas, Theories with two times, Phys. Lett. B 402 (1997) 25 [hep-th/9703060] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    I. Bars and C. Kounnas, String and particle with two times, Phys. Rev. D 56 (1997) 3664 [hep-th/9705205] [INSPIRE].ADSMathSciNetGoogle Scholar
  35. [35]
    I. Bars, C. Deliduman and O. Andreev, Gauged duality, conformal symmetry and space-time with two times, Phys. Rev. D 58 (1998) 066004 [hep-th/9803188] [INSPIRE].ADSGoogle Scholar
  36. [36]
    I. Bars, Conformal symmetry and duality between free particle, H-atom and harmonic oscillator, Phys. Rev. D 58 (1998) 066006 [hep-th/9804028] [INSPIRE].ADSMathSciNetGoogle Scholar
  37. [37]
    I. Bars, Survey of two time physics, Class. Quant. Grav. 18 (2001) 3113 [hep-th/0008164] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    M.B. Green, World Sheets for World Sheets, Nucl. Phys. B 293 (1987) 593 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    P.A.M. Dirac, Wave equations in conformal space, Annals Math. 37 (1936) 429 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    G. Mack and A. Salam, Finite component field representations of the conformal group, Annals Phys. 53 (1969) 174 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    S. Ferrara, R. Gatto and A.F. Grillo, Conformal algebra in space-time and operator product expansion, Springer Tracts Mod. Phys. 67 (1973) 1 [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    S. Weinberg, Six-dimensional Methods for Four-dimensional Conformal Field Theories, Phys. Rev. D 82 (2010) 045031 [arXiv:1006.3480] [INSPIRE].ADSGoogle Scholar
  44. [44]
    S. Weinberg, Six-dimensional Methods for Four-dimensional Conformal Field Theories II: Irreducible Fields, Phys. Rev. D 86 (2012) 085013 [arXiv:1209.4659] [INSPIRE].ADSGoogle Scholar
  45. [45]
    W. Siegel, Embedding versus 6D twistors, arXiv:1204.5679 [INSPIRE].
  46. [46]
    M.F. Paulos, Towards Feynman rules for Mellin amplitudes, JHEP 10 (2011) 074 [arXiv:1107.1504] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    W.D. Goldberger, W. Skiba and M. Son, Superembedding Methods for 4d N = 1 SCFTs, Phys. Rev. D 86 (2012) 025019 [arXiv:1112.0325] [INSPIRE].ADSGoogle Scholar
  48. [48]
    D. Simmons-Duffin, Projectors, Shadows and Conformal Blocks, JHEP 04 (2014) 146 [arXiv:1204.3894] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    M.F. Paulos, Loops, Polytopes and Splines, JHEP 06 (2013) 007 [arXiv:1210.0578] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    B. DeWitt, Supermanifolds, second edition, Cambridge University Press, (1992).Google Scholar
  51. [51]
    J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP 03 (2011) 025 [arXiv:1011.1485] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    T. Adamo, E. Casali and D. Skinner, A Worldsheet Theory for Supergravity, JHEP 02 (2015) 116 [arXiv:1409.5656] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  53. [53]
    N.A. Nekrasov, Lectures on curved beta-gamma systems, pure spinors and anomalies, hep-th/0511008 [INSPIRE].
  54. [54]
    E. Witten, On quantum gauge theories in two-dimensions, Commun. Math. Phys. 141 (1991) 153 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    M.F. Paulos, to appear.Google Scholar
  56. [56]
    Y. Geyer, A.E. Lipstein and L.J. Mason, Ambitwistor Strings in Four Dimensions, Phys. Rev. Lett. 113 (2014) 081602 [arXiv:1404.6219] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Tim Adamo
    • 1
  • Ricardo Monteiro
    • 2
    • 3
  • Miguel F. Paulos
    • 3
  1. 1.Theoretical Physics GroupBlackett Laboratory, Imperial College LondonLondonUnited Kingdom
  2. 2.Centre for Research in String TheoryQueen Mary University of LondonLondonUnited Kingdom
  3. 3.Theoretical Physics DepartmentCERNGeneva 23Switzerland

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