Green-Schwarz superstring on the lattice

  • L. Bianchi
  • M.S. Bianchi
  • V. Forini
  • B. Leder
  • E. Vescovi
Open Access
Regular Article - Theoretical Physics


We consider possible discretizations for a gauge-fixed Green-Schwarz action of Type IIB superstring. We use them for measuring the action, from which we extract the cusp anomalous dimension of planar \( \mathcal{N}=4 \) SYM as derived from AdS/CFT, as well as the mass of the two AdS excitations transverse to the relevant null cusp classical string solution. We perform lattice simulations employing a Rational Hybrid Monte Carlo (RHMC) algorithm and two Wilson-like fermion discretizations, one of which preserves the global SO(6) symmetry the model. We compare our results with the expected behavior at various values of \( g=\frac{\sqrt{\lambda }}{4\pi } \). For both the observables, we find a good agreement for large g, which is the perturbative regime of the sigma-model. For smaller values of g, the expectation value of the action exhibits a deviation compatible with the presence of quadratic divergences. After their non-perturbative subtraction the continuum limit can be taken, and suggests a qualitative agreement with the non-perturbative expectation from AdS/CFT. Furthermore, we detect a phase in the fermion determinant, whose origin we explain, that for small g leads to a sign problem not treatable via standard reweigthing. The continuum extrapolations of the observables in the two different discretizations agree within errors, which is strongly suggesting that they lead to the same continuum limit. Part of the results discussed here were presented earlier in [1].


AdS-CFT Correspondence Lattice Quantum Field Theory Sigma Models 


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]
    V. Forini, L. Bianchi, M.S. Bianchi, B. Leder and E. Vescovi, Lattice and string worldsheet in AdS/CFT: a numerical study, PoS(LATTICE 2015) 244 [arXiv:1601.04670] [INSPIRE].
  2. [2]
    N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    A.A. Tseytlin, Review of AdS/CFT integrability, chapter II.1: classical AdS 5 × S 5 string solutions, Lett. Math. Phys. 99 (2012) 103 [arXiv:1012.3986] [INSPIRE].
  5. [5]
    T. McLoughlin, Review of AdS/CFT integrability, chapter II.2: quantum strings in AdS 5 × S 5, Lett. Math. Phys. 99 (2012) 127 [arXiv:1012.3987] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    R. Roiban, A. Tirziu and A.A. Tseytlin, Two-loop world-sheet corrections in AdS 5 × S 5 superstring, JHEP 07 (2007) 056 [arXiv:0704.3638] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    R. Roiban and A.A. Tseytlin, Spinning superstrings at two loops: strong-coupling corrections to dimensions of large-twist SYM operators, Phys. Rev. D 77 (2008) 066006 [arXiv:0712.2479] [INSPIRE].ADSMathSciNetGoogle Scholar
  8. [8]
    S. Giombi, R. Ricci, R. Roiban, A.A. Tseytlin and C. Vergu, Quantum AdS 5 × S 5 superstring in the AdS light-cone gauge, JHEP 03 (2010) 003 [arXiv:0912.5105] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    S. Giombi, R. Ricci, R. Roiban and A.A. Tseytlin, Quantum dispersion relations for excitations of long folded spinning superstring in AdS 5 × S 5, JHEP 01 (2011) 128 [arXiv:1011.2755] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    S. Giombi, R. Ricci, R. Roiban and A.A. Tseytlin, Two-loop AdS 5 × S 5 superstring: testing asymptotic Bethe ansatz and finite size corrections, J. Phys. A 44 (2011) 045402 [arXiv:1010.4594] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  11. [11]
    L. Bianchi, M.S. Bianchi, A. Bres, V. Forini and E. Vescovi, Two-loop cusp anomaly in ABJM at strong coupling, JHEP 10 (2014) 013 [arXiv:1407.4788] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    L. Bianchi and M.S. Bianchi, Quantum dispersion relations for the AdS 4 × CP 3 GKP string, JHEP 11 (2015) 031 [arXiv:1505.00783] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    L. Bianchi and M.S. Bianchi, Worldsheet scattering for the GKP string, JHEP 11 (2015) 178 [arXiv:1508.07331] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    L. Bianchi and M.S. Bianchi, On the scattering of gluons in the GKP string, JHEP 02 (2016) 146 [arXiv:1511.01091] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    S.S. Gubser, I.R. Klebanov and A.M. Polyakov, A semiclassical limit of the gauge/string correspondence, Nucl. Phys. B 636 (2002) 99 [hep-th/0204051] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    S. Frolov and A.A. Tseytlin, Semiclassical quantization of rotating superstring in AdS 5 × S 5, JHEP 06 (2002) 007 [hep-th/0204226] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    S. Frolov and A.A. Tseytlin, Multispin string solutions in AdS 5 × S 5, Nucl. Phys. B 668 (2003) 77 [hep-th/0304255] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    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].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    M. Beccaria, V. Forini, A. Tirziu and A.A. Tseytlin, Structure of large spin expansion of anomalous dimensions at strong coupling, Nucl. Phys. B 812 (2009) 144 [arXiv:0809.5234] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    M. Beccaria, G.V. Dunne, V. Forini, M. Pawellek and A.A. Tseytlin, Exact computation of one-loop correction to energy of spinning folded string in AdS 5 × S 5, J. Phys. A 43 (2010) 165402 [arXiv:1001.4018] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  21. [21]
    N. Drukker and V. Forini, Generalized quark-antiquark potential at weak and strong coupling, JHEP 06 (2011) 131 [arXiv:1105.5144] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  22. [22]
    V. Forini, Quark-antiquark potential in AdS at one loop, JHEP 11 (2010) 079 [arXiv:1009.3939] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  23. [23]
    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].ADSMathSciNetzbMATHGoogle Scholar
  24. [24]
    L. Bianchi, V. Forini and B. Hoare, Two-dimensional S-matrices from unitarity cuts, JHEP 07 (2013) 088 [arXiv:1304.1798] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  25. [25]
    V. Forini, V.G.M. Puletti, M. Pawellek and E. Vescovi, One-loop spectroscopy of semiclassically quantized strings: bosonic sector, J. Phys. A 48 (2015) 085401 [arXiv:1409.8674] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  26. [26]
    V. Forini, V.G.M. Puletti, L. Griguolo, D. Seminara and E. Vescovi, Remarks on the geometrical properties of semiclassically quantized strings, J. Phys. A 48 (2015) 475401 [arXiv:1507.01883] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  27. [27]
    V. Forini, V. Giangreco M. Puletti, L. Griguolo, D. Seminara and E. Vescovi, Precision calculation of 1/4-BPS Wilson loops in AdS 5 × S 5, JHEP 02 (2016) 105 [arXiv:1512.00841] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    D.B. Kaplan and M. Ünsal, A Euclidean lattice construction of supersymmetric Yang-Mills theories with sixteen supercharges, JHEP 09 (2005) 042 [hep-lat/0503039] [INSPIRE].
  29. [29]
    S. Catterall, D.B. Kaplan and M. Ünsal, Exact lattice supersymmetry, Phys. Rept. 484 (2009) 71 [arXiv:0903.4881] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    S. Catterall, D. Schaich, P.H. Damgaard, T. DeGrand and J. Giedt, N = 4 supersymmetry on a space-time lattice, Phys. Rev. D 90 (2014) 065013 [arXiv:1405.0644] [INSPIRE].ADSGoogle Scholar
  31. [31]
    A. Joseph, Review of lattice supersymmetry and gauge-gravity duality, Int. J. Mod. Phys. A 30 (2015) 1530054 [arXiv:1509.01440] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    D. Schaich, Aspects of lattice N = 4 supersymmetric Yang-Mills, PoS(LATTICE 2015) 242 [arXiv:1512.01137] [INSPIRE].
  33. [33]
    G. Bergner and S. Catterall, Supersymmetry on the lattice, arXiv:1603.04478 [INSPIRE].
  34. [34]
    T. Ishii, G. Ishiki, S. Shimasaki and A. Tsuchiya, N = 4 super Yang-Mills from the plane wave matrix model, Phys. Rev. D 78 (2008) 106001 [arXiv:0807.2352] [INSPIRE].ADSMathSciNetGoogle Scholar
  35. [35]
    G. Ishiki, S.-W. Kim, J. Nishimura and A. Tsuchiya, Deconfinement phase transition in N = 4 super Yang-Mills theory on R×S 3 from supersymmetric matrix quantum mechanics, Phys. Rev. Lett. 102 (2009) 111601 [arXiv:0810.2884] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    G. Ishiki, S.-W. Kim, J. Nishimura and A. Tsuchiya, Testing a novel large-N reduction for N = 4 super Yang-Mills theory on R×S 3, JHEP 09 (2009) 029 [arXiv:0907.1488] [INSPIRE].
  37. [37]
    M. Hanada, S. Matsuura and F. Sugino, Two-dimensional lattice for four-dimensional N = supersymmetric Yang-Mills, Prog. Theor. Phys. 126 (2011) 597 [arXiv:1004.5513] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  38. [38]
    M. Honda, G. Ishiki, J. Nishimura and A. Tsuchiya, Testing the AdS/CFT correspondence by Monte Carlo calculation of BPS and non-BPS Wilson loops in 4d N = 4 super-Yang-Mills theory, PoS(LATTICE 2011) 244 [arXiv:1112.4274] [INSPIRE].
  39. [39]
    M. Honda, G. Ishiki, S.-W. Kim, J. Nishimura and A. Tsuchiya, Direct test of the AdS/CFT correspondence by Monte Carlo studies of N = 4 super Yang-Mills theory, JHEP 11 (2013) 200 [arXiv:1308.3525] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    M. Hanada, Y. Hyakutake, G. Ishiki and J. Nishimura, Holographic description of quantum black hole on a computer, Science 344 (2014) 882 [arXiv:1311.5607] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    R.W. McKeown and R. Roiban, The quantum AdS 5 × S 5 superstring at finite coupling, arXiv:1308.4875 [INSPIRE].
  42. [42]
    G.P. Korchemsky and G. Marchesini, Structure function for large x and renormalization of Wilson loop, Nucl. Phys. B 406 (1993) 225 [hep-ph/9210281] [INSPIRE].
  43. [43]
    J.M. Maldacena, Wilson loops in large-N field theories, Phys. Rev. Lett. 80 (1998) 4859 [hep-th/9803002] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    S.-J. Rey and J.-T. Yee, Macroscopic strings as heavy quarks in large-N gauge theory and anti-de Sitter supergravity, Eur. Phys. J. C 22 (2001) 379 [hep-th/9803001] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    L.F. Alday and J.M. Maldacena, Comments on operators with large spin, JHEP 11 (2007) 019 [arXiv:0708.0672] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    L.F. Alday, D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, An operator product expansion for polygonal null Wilson loops, JHEP 04 (2011) 088 [arXiv:1006.2788] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    B. Basso, A. Sever and P. Vieira, Spacetime and flux tube S-matrices at finite coupling for N = 4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 111 (2013) 091602 [arXiv:1303.1396] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    B. Basso, A. Sever and P. Vieira, Space-time S-matrix and flux tube S-matrix II. Extracting and matching data, JHEP 01 (2014) 008 [arXiv:1306.2058] [INSPIRE].
  49. [49]
    D. Fioravanti, S. Piscaglia and M. Rossi, Asymptotic Bethe ansatz on the GKP vacuum as a defect spin chain: scattering, particles and minimal area Wilson loops, Nucl. Phys. B 898 (2015) 301 [arXiv:1503.08795] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    A. Bonini, D. Fioravanti, S. Piscaglia and M. Rossi, Strong Wilson polygons from the lodge of free and bound mesons, JHEP 04 (2016) 029 [arXiv:1511.05851] [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    Z. Bern, M. Czakon, L.J. Dixon, D.A. Kosower and V.A. Smirnov, The four-loop planar amplitude and cusp anomalous dimension in maximally supersymmetric Yang-Mills theory, Phys. Rev. D 75 (2007) 085010 [hep-th/0610248] [INSPIRE].ADSMathSciNetGoogle Scholar
  52. [52]
    N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. 0701 (2007) P01021 [hep-th/0610251] [INSPIRE].Google Scholar
  53. [53]
    M. Hanada, What lattice theorists can do for quantum gravity, arXiv:1604.05421 [INSPIRE].
  54. [54]
    R.R. Metsaev and A.A. Tseytlin, Superstring action in AdS 5 × S 5 . Kappa symmetry light cone gauge, Phys. Rev. D 63 (2001) 046002 [hep-th/0007036] [INSPIRE].
  55. [55]
    R.R. Metsaev, C.B. Thorn and A.A. Tseytlin, Light cone superstring in AdS space-time, Nucl. Phys. B 596 (2001) 151 [hep-th/0009171] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    R.R. Metsaev and A.A. Tseytlin, Type IIB superstring action in AdS 5 × S 5 background, Nucl. Phys. B 533 (1998) 109 [hep-th/9805028] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. [57]
    I. Montvay and G. Muenster, Quantum fields on a lattice, Cambridge University Press, Cambridge U.K. (1994).CrossRefGoogle Scholar
  58. [58]
    A.D. Kennedy, I. Horvath and S. Sint, A new exact method for dynamical fermion computations with nonlocal actions, Nucl. Phys. Proc. Suppl. 73 (1999) 834 [hep-lat/9809092] [INSPIRE].
  59. [59]
    M.A. Clark and A.D. Kennedy, The RHMC algorithm for two flavors of dynamical staggered fermions, Nucl. Phys. Proc. Suppl. 129 (2004) 850 [hep-lat/0309084] [INSPIRE].
  60. [60]
    ALPHA collaboration, U. Wolff, Monte Carlo errors with less errors, Comput. Phys. Commun. 156 (2004) 143 [Erratum ibid. 176 (2007) 383] [hep-lat/0306017] [INSPIRE].
  61. [61]
    B. Basso, Exciting the GKP string at any coupling, Nucl. Phys. B 857 (2012) 254 [arXiv:1010.5237] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  62. [62]
    M. Kruczenski, R. Roiban, A. Tirziu and A.A. Tseytlin, Strong-coupling expansion of cusp anomaly and gluon amplitudes from quantum open strings in AdS 5 × S 5, Nucl. Phys. B 791 (2008) 93 [arXiv:0707.4254] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  63. [63]
    S. Catterall, First results from simulations of supersymmetric lattices, JHEP 01 (2009) 040 [arXiv:0811.1203] [INSPIRE].ADSCrossRefGoogle Scholar
  64. [64]
    M. Lüscher, R. Narayanan, P. Weisz and U. Wolff, The Schrödinger functional: a renormalizable probe for non-Abelian gauge theories, Nucl. Phys. B 384 (1992) 168 [hep-lat/9207009] [INSPIRE].

Copyright information

© The Author(s) 2016

Authors and Affiliations

  • L. Bianchi
    • 1
    • 2
  • M.S. Bianchi
    • 3
  • V. Forini
    • 1
  • B. Leder
    • 1
  • E. Vescovi
    • 1
  1. 1.Institut für PhysikHumboldt-Universität zu Berlin, IRIS AdlershofBerlinGermany
  2. 2.II. Institut für Theoretische PhysikUniversität HamburgHamburgGermany
  3. 3.Queen Mary University of LondonLondonU.K.

Personalised recommendations