Skip to main content
SpringerLink
Log in

Introduction

  • Chapter
  • First Online:
Book cover Perturbative and Non-perturbative Approaches to String Sigma-Models in AdS/CFT

Part of the book series: Springer Theses ((Springer Theses))

  • 390 Accesses

Abstract

Modern theoretical physics has been written in the language of two major scientific paradigms.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    In principle, one can also consider the instanton angle \(\theta \) which combines with the YM coupling constant into a complex coupling \(\tau =\frac{\theta }{2\pi }+\frac{4\pi i}{g_\mathrm{YM}^2}\). Through the AdS/CFT correspondence (Sect. 1.1), the angle \(\theta \) equals the expectation value of the axion field in the spectrum of the dual Type IIB superstring, e.g. [19].

  2. 2.

    Twist-two (Wilson) operators play an important role in deep inelastic scattering in QCD as much as in \(\mathcal {N}=4\). Their anomalous dimension for large spin is governed by the so-called scaling function of the theory in question, see Sect. 7.1. The maximal transcendentality principle conjectured in [27] states that the \(\mathcal {N}=4\) SYM scaling function has uniform degree of transcendentality \(2l-2\) at loop order l and can be extracted from the QCD expression by removing the terms that are not of maximal transcendentality. A brief account of the subject and references are in [28, 29].

  3. 3.

    Among the many reviews on the topic, we suggest [19, 34,35,36,37,38] and the excellent textbook [39].

  4. 4.

    We will make clear the distinction between the ’t Hooft parameter \(\lambda _\mathrm{YM}\) of \(\mathcal {N}=4\) SYM and the one \(\lambda _\mathrm{ABJM}\) of ABJM when necessary, namely in Chap. 6.

  5. 5.

    Supersymmetry is enhanced to \(\mathcal {N}=8\) at Chern–Simons level is \(k=1,2\) [43, 47]. We can disregard this exception since we will be working in planar limit.

  6. 6.

    We suggest [53, 54] for an extensive discussion of integrable systems and also [55, 56] for a focus on AdS/CFT.

  7. 7.

    A transparent and concise introduction to the subject is in [60,61,62].

  8. 8.

    Higher point functions decompose into these elementary constituents [9].

  9. 9.

    The integrability of the string in the \(AdS_5\times S^5\) background has been mostly studied in the supercoset description [70] (e.g. in [77]) than in the pure spinor version. Some integrable properties in the former formalism will be discussed to some extent in Sects. 2.1.1, 2.1.2 and 2.1.3, while we refer the reader to a non-exhaustive selection of relevant references in the latter formalism in [78, 79] and in the reviews [80, 81]. Arguments that support the quantum integrability of the pure spinor action were given in [82].

  10. 10.

    This is a function undetermined by the symmetries of the theory, but constrained by physical requirements such as crossing symmetry and unitarity, see [92] for a short review.

  11. 11.

    More references on the subject are below (2.35) in Sect. 2.2.

  12. 12.

    We will test the strong-coupling expansion of the interpolating function in Chap. 6.

  13. 13.

    For a general curved target space, the string equations of motion are nonlinear and the right and left oscillator modes of the string interact with themselves and with each other [155], see also [156] for further issues. As for the quantization of a generic field theory in curved spacetime, a good initial reference is the textbook [157].

  14. 14.

    The string sigma-model of [70] is nonlinear because the curvature of the target space brings field-dependent coefficients of the kinetic terms. Expanding the path-integral around a classical solution generates standard quadratic kinetic terms (and interaction terms) for the fluctuation fields that make the sigma-model tractable.

  15. 15.

    One should also remember that a rigorous definition takes into account some factors associated with conformal Killing vectors and/or Teichmüller moduli. We defer the discussion to [158, 159] and the textbooks [156, 160].

  16. 16.

    See Sect. 5.1. Subsequent steps were made to construct further generalization called super Maldacena-Wilson loops [180]. The field theoretical description is explored in [142, 181] while the complementary view at strong coupling in [182].

  17. 17.

    The Green–Schwarz action is known to quadratic order in fermions for any general type II supergravity background [201] and recently up to fourth order [202].

  18. 18.

    The net contribution comes only from the kinetic operators \(\mathcal {O}_B\) and \(\mathcal {O}_F\) because we suppose that the determinant of the diffeomorphism ghosts cancels the one of the “unphysical” (longitudinal) bosons, see comments below (3.46) and (3.98), and we also ignore the caveats in footnote 15.

  19. 19.

    The three spaces listed here are maximally symmetric because their metrics possess the maximal number \(d(d+1)/2\) of Killing vectors in d dimensions.

  20. 20.

    We are referring to (4.11) and (4.27) for the two examples of spinning strings studied in this thesis, (5.38) and (5.53) for the worldsheet dual to a 1/4-BPS latitude Wilson loop.

  21. 21.

    This geometrically manifests as an axial symmetry of the minimal surface (5.25) in Fig. 5.3.

  22. 22.

    For the spinning strings of Chap. 4, this is the invariance of the string surface under shifts of the AdS global time t, see text above (4.1) and formula (4.22).

  23. 23.

    This is of course just one possible route. It is a well-known result [221] that the logarithm of the determinant is equivalently computed by the on-shell vacuum energy, as obtained by summing over the frequency spectrum of the operator, e.g. [198], under certain assumptions. Alternatively, one can also employ the phaseshift method [222,223,224], see [225] for a recent application. We are grateful to Xinyi Chen-Lin and Daniel Medina Rincón for long discussions about this point.

  24. 24.

    We thank Amit Dekel for informing us about this reference.

  25. 25.

    In the first two-loop calculation of [238] the conformal gauge was used, in which propagators are non-diagonal, implying the evaluation of a larger number of two-loop diagrams.

  26. 26.

    In regularizing our integrals all manipulations of tensor structures in the Feynman integrands are however carried out strictly in two dimensions.

  27. 27.

    In literature scaling function and cusp anomalous dimension are essentially synonyms, as we will explain in Sect. 7.1.

References

  1. C.M. Will, The confrontation between general relativity and experiment. Living Rev. Rel. 17, 4 (2014). arXiv:1403.7377

    Article  MATH  Google Scholar 

  2. Virgo, LIGO Scientific Collaboration, B.P. Abbott et al., Observation of gravitational waves from a binary black hole merger. Phys. Rev. Lett. 116, 061102 (2016). arXiv:1602.03837

  3. Virgo, LIGO Scientific Collaboration, B.P. Abbott et al., GW151226: observation of gravitational waves from a 22-solar-mass binary black hole coalescence. Phys. Rev. Lett. 116, 241103 (2016). arXiv:1606.04855

  4. K.G. Wilson, J.B. Kogut, The Renormalization group, the epsilon expansion. Phys. Rept. 12, 75 (1974)

    Article  ADS  Google Scholar 

  5. K. Symanzik, Small distance behaviour in field theory and power counting. Commun. Math. Phys. 18, 227 (1970)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. C.G. Callan Jr., Broken scale invariance in scalar field theory. Phys. Rev. D 2, 1541 (1970)

    Article  ADS  Google Scholar 

  7. K. Symanzik, Small-distance-behaviour analysis and Wilson expansions. Commun. Math. Phys. 23, 49 (1971)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. M.E. Peskin, D.V. Schroeder, An Introduction to Quantum Field Theory, (Addison-Wesley Publishing Company, 1995)

    Google Scholar 

  9. A.A. Belavin, A.M. Polyakov, A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B 241, 333 (1984)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. F. Englert, R. Brout, Broken symmetry and the mass of gauge vector mesons. Phys. Rev. Lett. 13, 321 (1964)

    Article  ADS  MathSciNet  Google Scholar 

  11. P.W. Higgs, Broken symmetries and the masses of gauge bosons. Phys. Rev. Lett. 13, 508 (1964)

    Article  ADS  MathSciNet  Google Scholar 

  12. G.S. Guralnik, C.R. Hagen, T.W.B. Kibble, Global conservation laws and massless particles, 5. Phys. Rev. Lett. 13, 585 (1964)

    Article  ADS  Google Scholar 

  13. D.J. Gross, F. Wilczek, Ultraviolet behavior of nonabelian gauge theories. Phys. Rev. Lett. 30, 1343 (1973)

    Article  ADS  Google Scholar 

  14. K.A. Olive et al., Particle Data Group Collaboration. Review of Particle Physics. Chin. Phys. C38, 090001 (2014)

    Google Scholar 

  15. C.M.S. Collaboration, S. Chatrchyan et al., Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC. Phys. Lett. B 716, 30 (2012). arXiv:1207.7235

    Article  ADS  Google Scholar 

  16. ATLAS Collaboration, G. Aad et al., Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC. Phys. Lett. B 716, 1 (2012). arXiv:1207.7214

  17. K. Ramachandran, G. Deepa, K. Namboori, Computational Chemistry and Molecular Modeling: Principles And Applications (Springer Science & Business Media, 2008)

    Google Scholar 

  18. L. Brink, J.H. Schwarz, J. Scherk, Supersymmetric yang-mills theories. Nucl. Phys. B 121, 77 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  19. E. D’Hoker, D.Z. Freedman, Supersymmetric gauge theories and the AdS/CFT correspondence. arXiv:hep-th/0201253

  20. C. Montonen, D. Olive, Magnetic monopoles as gauge particles? Phys. Lett. B 72, 117 (1977)

    Article  ADS  Google Scholar 

  21. P. Goddard, J. Nuyts, D.I. Olive, Gauge theories and magnetic charge. Nucl. Phys. B 125, 1 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  22. E. Witten, D.I. Olive, Supersymmetry algebras that include topological charges. Phys. Lett. B 78, 97 (1978)

    Article  ADS  Google Scholar 

  23. M.B. Green, J.H. Schwarz, L. Brink, \(\cal{N}=4\) Yang-Mills and \(\cal{N}=8\) supergravity as limits of string theories. Nucl. Phys. B 198, 474 (1982)

    Article  ADS  Google Scholar 

  24. S. Mandelstam, Light cone superspace and the ultraviolet finiteness of the \(\cal{N}=4\) model. Nucl. Phys. B 213, 149 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  25. L. Brink, O. Lindgren, B.E.W. Nilsson, The ultraviolet finiteness of the \(\cal{N}=4\) yang-mills theory. Phys. Lett. B 123, 323 (1983)

    Article  ADS  Google Scholar 

  26. N. Seiberg, Supersymmetry and nonperturbative beta functions. Phys. Lett. B 206, 75 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  27. A.V. Kotikov, L.N. Lipatov, DGLAP and BFKL equations in the \({\cal N\it } =4\) supersymmetric gauge theory, Nucl. Phys. B 661, 19 (2003). arXiv:hep-ph/0208220, [Erratum: Nucl. Phys. B685,405(2004)]

  28. L. Freyhult, Review of AdS/CFT Integrability, Chapter III.4: twist states and the cusp anomalous dimension. Lett. Math. Phys. 99, 255 (2012). arXiv:1012.3993

  29. G.P. Korchemsky, Review of AdS/CFT Integrability, Chapter IV.4: Integrability in QCD and \({\cal{N}}<4\) SYM. Lett. Math. Phys. 99, 425 (2012). arXiv:1012.4000

  30. L.J. Dixon, J.M. Henn, J. Plefka, T. Schuster, All tree-level amplitudes in massless QCD. JHEP 1101, 035 (2011). arXiv:1010.3991

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. J.M. Maldacena, The Large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231 (1998). arXiv:hep-th/9711200

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. S.S. Gubser, I.R. Klebanov, A.M. Polyakov, Gauge theory correlators from non-critical string theory. Phys. Lett. B 428, 105 (1998). arXiv:hep-th/9802109

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. E. Witten, Anti-de Sitter space and holography. Adv. Theor. Math. Phys. 2, 253 (1998). arXiv:hep-th/9802150

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri, Y. Oz, Large N field theories, string theory and gravity. Phys. Rept. 323, 183 (2000). arXiv:hep-th/9905111

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. A. Zaffaroni, Introduction to the \(AdS/CFT\) correspondence. Class. Quant. Grav. 17, 3571 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. J. Plefka, Spinning strings and integrable spin chains in the AdS/CFT correspondence. Living Rev. Rel. 8, 9 (2005). arXiv:hep-th/0507136

    Article  MATH  Google Scholar 

  37. H. Nastase, Introduction to \(AdS/CFT\). arXiv:0712.0689

  38. N. Beisert et al., Review of AdS/CFT integrability: an overview. Lett. Math. Phys. 99, 3 (2012). arXiv:1012.3982

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. M. Ammon, J. Erdmenger, Gauge/Gravity Duality (Cambridge University Press, Cambridge, UK, 2015)

    Book  MATH  Google Scholar 

  40. G. ’t Hooft, A planar diagram theory for strong interactions. Nucl. Phys. B 72, 461 (1974)

    Google Scholar 

  41. A. Gustavsson, Algebraic structures on parallel M2-branes. Nucl. Phys. B 811, 66 (2009). arXiv:0709.1260

    Article  ADS  MATH  Google Scholar 

  42. J. Bagger, N. Lambert, Gauge symmetry and supersymmetry of multiple M2-branes. Phys. Rev. D 77, 065008 (2008). arXiv:0711.0955

    Article  ADS  MathSciNet  Google Scholar 

  43. O. Aharony, O. Bergman, D.L. Jafferis, J. Maldacena, \(\cal{N} = 6\) superconformal Chern-Simons-matter theories, M2-branes and their gravity duals. JHEP 0810, 091 (2008). arXiv:0806.1218

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. I.R. Klebanov, G. Torri, M2-branes and AdS/CFT. Int. J. Mod. Phys. A 25, 332 (2010). arXiv:0909.1580 (In: Crossing the boundaries: Gauge dynamics at strong coupling. Proceedings, Workshop in Honor of the 60th Birthday of Misha Shifman, (Minneapolis, USA, May 14–17, 2009), p. 332–350)

  45. S. Deser, R. Jackiw, S. Templeton, Three-dimensional massive gauge theories. Phys. Rev. Lett. 48, 975 (1982)

    Article  ADS  Google Scholar 

  46. E. Witten, Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351 (1989)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. A. Gustavsson, S.-J. Rey, Enhanced \({\cal{N}} =8\) supersymmetry of ABJM theory on \({\mathbb{R}}^8\) and \({\mathbb{R}}^8{/}{\mathbb{Z}}_2\). arXiv:0906.3568

  48. T. McLoughlin, R. Roiban, A.A. Tseytlin, Quantum spinning strings in \(AdS_4\times \mathbb{CP}^3\): testing the Bethe Ansatz proposal. JHEP 0811, 069 (2008). arXiv:0809.4038

    Article  ADS  Google Scholar 

  49. O. Bergman, S. Hirano, Anomalous radius shift in \(AdS_4/CFT_3\). JHEP 0907, 016 (2009). arXiv:0902.1743

    Article  ADS  Google Scholar 

  50. O. Aharony, O. Bergman, D.L. Jafferis, Fractional M2-branes. JHEP 0811, 043 (2008). arXiv:0807.4924

    Article  ADS  MathSciNet  Google Scholar 

  51. A. Sfondrini, Towards integrability for \(AdS_2/CFT_3\). J. Phys. A 48, 023001 (2015). arXiv:1406.2971

    Article  ADS  MathSciNet  MATH  Google Scholar 

  52. A. Pittelli, Dualities and Integrability in Low Dimensional AdS/CFT, Ph.D Thesis. http://epubs.surrey.ac.uk/812577/

  53. O. Babelon, D. Bernard, M. Talon, Introduction to Classical Integrable Systems (Cambridge University Press, 2003)

    Google Scholar 

  54. G. Mussardo, Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics (Oxford University Press, 2010)

    Google Scholar 

  55. N. Beisert, Integrability in QFT and AdS/CFT, Lecture notes. http://edu.itp.phys.ethz.ch/hs14/14HSInt/IntAdSCFT14Notes.pdf

  56. A. Torrielli, Lectures on Classical Integrability. arXiv:1606.02946

  57. L.N. Lipatov, High energy asymptotics of multi-colour QCD and exactly solvable lattice models. JETP Lett. 59, 596 (1994). arXiv:hep-th/9311037, [Pisma Zh. Eksp. Teor. Fiz. 59, 571(1994)]

  58. L.D. Faddeev, G.P. Korchemsky, High-energy QCD as a completely integrable model. Phys. Lett. B 342, 311 (1995). arXiv:hep-th/9404173

    Article  ADS  Google Scholar 

  59. V.M. Braun, G.P. Korchemsky, D. Mueller, The Uses of conformal symmetry in QCD. Prog. Part. Nucl. Phys. 51, 311 (2003). arXiv:hep-ph/0306057

    Article  ADS  Google Scholar 

  60. M. Staudacher, Review of AdS/CFT Integrability, Chapter III.1: Bethe Ansátze and the R-matrix formalism. Lett. Math. Phys. 99, 191 (2012). arXiv:1012.3990

  61. J.A. Minahan, K. Zarembo, The Bethe ansatz for \({\cal{N}} =4\) superYang-Mills. JHEP 0303, 013 (2003). arXiv:hep-th/0212208

    Article  ADS  Google Scholar 

  62. F. Levkovich-Maslyuk, Lectures on the Bethe Ansatz. arXiv:1606.02950

  63. H. Bethe, Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette. Z. Phys. 71, 205 (1931). (in German)

    Article  ADS  MATH  Google Scholar 

  64. L.D. Faddeev, L.A. Takhtajan, Spectrum and scattering of excitations in the one-dimensional isotropic Heisenberg model. J. Sov. Math. 24, 241 (1984). [Zap. Nauchn. Semin. 109, 134(1981)]

    Google Scholar 

  65. N.Y. Reshetikhin, A method of functional equations in the theory of exactly solvable quantum systems. Lett. Math. Phys. 7, 205 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  66. N.Y. Reshetikhin, Integrable models of quantum one-dimensional magnets with \(O(N)\) and \(Sp(2k)\) symmetry. Theor. Math. Phys. 63, 555 (1985)

    Article  Google Scholar 

  67. N. Beisert, M. Staudacher, The \({\cal{N}} =4\) SYM integrable super spin chain. Nucl. Phys. B 670, 439 (2003). arXiv:hep-th/0307042

    Article  ADS  MathSciNet  MATH  Google Scholar 

  68. N. Beisert, The complete one loop dilatation operator of \({\cal{N}}=4\) superYang-Mills theory. Nucl. Phys. B 676, 3 (2004). arXiv:hep-th/0307015

    Article  ADS  MathSciNet  MATH  Google Scholar 

  69. N. Beisert, C. Kristjansen, M. Staudacher, The Dilatation operator of conformal \({\cal{N}} =4\) superYang-Mills theory. Nucl. Phys. B 664, 131 (2003). arXiv:hep-th/0303060

    Article  ADS  MATH  Google Scholar 

  70. R. Metsaev, A.A. Tseytlin, Type IIB superstring action in \(AdS_5\times S^5\) background. Nucl. Phys. B 533, 109 (1998). arXiv:hep-th/9805028

    Article  ADS  MATH  Google Scholar 

  71. I. Bena, J. Polchinski, R. Roiban, Hidden symmetries of the \(AdS_5 \times S^5\) superstring. Phys. Rev. D 69, 046002 (2004). arXiv:hep-th/0305116

    Article  ADS  MathSciNet  Google Scholar 

  72. B.C. Vallilo, Flat currents in the classical \(AdS_5\times S^5\) pure spinor superstring. JHEP 0403, 037 (2004). arXiv:hep-th/0307018

    Article  MathSciNet  Google Scholar 

  73. N. Berkovits, Super Poincare covariant quantization of the superstring. JHEP 0004, 018 (2000). arXiv:hep-th/0001035

    Article  ADS  MathSciNet  MATH  Google Scholar 

  74. N. Berkovits, O. Chandia, Superstring vertex operators in an \(AdS_5\times S^5\) background. Nucl. Phys. B 596, 185 (2001). arXiv:hep-th/0009168

    Article  ADS  MATH  Google Scholar 

  75. B.C. Vallilo, One loop conformal invariance of the superstring in an \(AdS_5 \times S^5\) background. JHEP 0212, 042 (2002). arXiv:hep-th/0210064

    Article  MathSciNet  Google Scholar 

  76. N. Berkovits, BRST cohomology and nonlocal conserved charges. JHEP 0502, 060 (2005). arXiv:hep-th/0409159

    Article  ADS  MathSciNet  Google Scholar 

  77. G. Arutyunov, S. Frolov, Foundations of the \(AdS_5 \times S^5\) Superstring. Part I. J. Phys. A 42, 254003 (2009). arXiv:0901.4937

    ADS  MATH  Google Scholar 

  78. A. Mikhailov, S. Schafer-Nameki, Algebra of transfer-matrices and Yang-Baxter equations on the string worldsheet in \(AdS_5 \times S^5\). Nucl. Phys. B 802, 1 (2008). arXiv:0712.4278

    Article  ADS  MATH  Google Scholar 

  79. R. Benichou, First-principles derivation of the AdS/CFT Y-systems. JHEP 1110, 112 (2011). arXiv:1108.4927

    Article  ADS  MathSciNet  MATH  Google Scholar 

  80. N. Berkovits, ICTP lectures on covariant quantization of the superstring. arXiv:hep-th/0209059 (In: Superstrings and related matters. Proceedings, Spring School (Trieste, Italy, March 18–26, 2002), p. 57–107. http://www.ictp.trieste.it/~pub_off/lectures/lns013/Berkovits/Berkovits.pdf)

  81. Y. Oz, The pure spinor formulation of superstrings. Class. Quant. Grav. 25, 214001 (2008). arXiv:0910.1195 (In: Strings, supergravity and gauge theories. Proceedings, European RTN Winter School (CERN, Geneva, Switzerland, January 21–25, 2008), p. 214001)

  82. N. Berkovits, Quantum consistency of the superstring in \(AdS_5\times S^5\) background. JHEP 0503, 041 (2005). arXiv:hep-th/0411170

    Article  ADS  Google Scholar 

  83. V. Kazakov, A. Marshakov, J. Minahan, K. Zarembo, Classical/quantum integrability in AdS/CFT. JHEP 0405, 024 (2004). arXiv:hep-th/0402207

    Google Scholar 

  84. N. Beisert, V. Dippel, M. Staudacher, A Novel long range spin chain and planar \({\cal{N}}=4\) super Yang-Mills. JHEP 0407, 075 (2004). arXiv:hep-th/0405001

    Article  ADS  MathSciNet  Google Scholar 

  85. N. Beisert, M. Staudacher, Long-range \({\mathfrak{psu}} (2,2|4)\) Bethe Ansatze for gauge theory and strings. Nucl. Phys. B 727, 1 (2005). arXiv:hep-th/0504190

    Article  ADS  MathSciNet  MATH  Google Scholar 

  86. G. Arutyunov, S. Frolov, M. Staudacher, Bethe ansatz for quantum strings. JHEP 0410, 016 (2004). arXiv:hep-th/0406256

    Google Scholar 

  87. R. Hernandez, E. Lopez, Quantum corrections to the string Bethe ansatz. JHEP 0607, 004 (2006). arXiv:hep-th/0603204

    Article  ADS  MathSciNet  Google Scholar 

  88. R.A. Janik, The \(AdS_5\times S^5\) superstring worldsheet S-matrix and crossing symmetry. Phys. Rev. D 73, 086006 (2006). arXiv:hep-th/0603038

    Article  ADS  MathSciNet  Google Scholar 

  89. N. Beisert, R. Hernandez, E. Lopez, A Crossing-symmetric phase for \(AdS_5\times S^5\) strings. JHEP 0611, 070 (2006). arXiv:hep-th/0609044

    Article  ADS  Google Scholar 

  90. Z. Bern, M. Czakon, L.J. Dixon, D.A. Kosower, V.A. Smirnov, The four-loop planar amplitude and cusp anomalous dimension in maximally supersymmetric Yang-Mills theory. Phys. Rev. D 75, 085010 (2007). arXiv:hep-th/0610248

    Article  ADS  MathSciNet  Google Scholar 

  91. N. Beisert, T. McLoughlin, R. Roiban, The four-loop dressing phase of \(\cal{N}=4\) SYM. Phys. Rev. D 76, 046002 (2007). arXiv:0705.0321

    Article  ADS  MathSciNet  Google Scholar 

  92. P. Vieira, D. Volin, Review of AdS/CFT Integrability, Chapter III.3: the dressing factor, Lett. Math. Phys. 99, 231 (2012). arXiv:1012.3992

  93. C. Sieg, A. Torrielli, Wrapping interactions and the genus expansion of the 2-point function of composite operators. Nucl. Phys. B 723, 3 (2005). arXiv:hep-th/0505071

    Article  ADS  MathSciNet  MATH  Google Scholar 

  94. J. Ambjorn, R.A. Janik, C. Kristjansen, Wrapping interactions and a new source of corrections to the spin-chain/string duality. Nucl. Phys. B 736, 288 (2006). arXiv:hep-th/0510171

    Article  ADS  MathSciNet  MATH  Google Scholar 

  95. R.A. Janik, T. Lukowski, Wrapping interactions at strong coupling: the Giant magnon. Phys. Rev. D 76, 126008 (2007). arXiv:0708.2208

    Article  ADS  Google Scholar 

  96. R.A. Janik, Review of AdS/CFT Integrability, Chapter III.5: luscher corrections. Lett. Math. Phys. 99, 277 (2012). arXiv:1012.3994

  97. G. Arutyunov, S. Frolov, On string S-matrix. Bound states and TBA, JHEP 0712, 024 (2007). arXiv:0710.1568

    MATH  Google Scholar 

  98. G. Arutyunov, S. Frolov, String hypothesis for the \(AdS_5\times S^5\) mirror. JHEP 0903, 152 (2009). arXiv:0901.1417

    Article  ADS  Google Scholar 

  99. N. Gromov, V. Kazakov, P. Vieira, Exact spectrum of anomalous dimensions of planar \(\cal{N}=4\) supersymmetric Yang-Mills theory. Phys. Rev. Lett. 103, 131601 (2009). arXiv:0901.3753

    Article  ADS  MathSciNet  Google Scholar 

  100. D. Bombardelli, D. Fioravanti, R. Tateo, Thermodynamic Bethe Ansatz for planar AdS/CFT: a proposal. J. Phys. A 42, 375401 (2009). arXiv:0902.3930

    Article  ADS  MathSciNet  MATH  Google Scholar 

  101. N. Gromov, V. Kazakov, A. Kozak, P. Vieira, Exact spectrum of anomalous dimensions of planar \(\cal{N}=4\) supersymmetric Yang-Mills theory: TBA and excited states. Lett. Math. Phys. 91, 265 (2010). arXiv:0902.4458

    Article  ADS  MathSciNet  MATH  Google Scholar 

  102. G. Arutyunov, S. Frolov, Thermodynamic Bethe Ansatz for the \(AdS_5 \times S^5\) mirror model. JHEP 0905, 068 (2009). arXiv:0903.0141

    Article  ADS  Google Scholar 

  103. Z. Bajnok, Review of AdS/CFT Integrability, Chapter III.6: Thermodynamic Bethe Ansatz. Lett. Math. Phys. 99, 299 (2012). arXiv:1012.3995

  104. S.J. van Tongeren, Introduction to the thermodynamic Bethe ansatz. arXiv:1606.02951

  105. L.F. Alday, J. Maldacena, A. Sever, P. Vieira, Y-system for scattering amplitudes. J. Phys. A 43, 485401 (2010). arXiv:1002.2459

    Article  MathSciNet  MATH  Google Scholar 

  106. D. Correa, J. Maldacena, A. Sever, The quark anti-quark potential and the cusp anomalous dimension from a TBA equation. JHEP 1208, 134 (2012). arXiv:1203.1913

    Article  ADS  Google Scholar 

  107. N. Drukker, Integrable Wilson loops. JHEP 1310, 135 (2013). arXiv:1203.1617

    Google Scholar 

  108. N. Gromov, A. Sever, Analytic solution of Bremsstrahlung TBA. JHEP 1211, 075 (2012). arXiv:1207.5489

    Google Scholar 

  109. N. Gromov, V. Kazakov, S. Leurent, D. Volin, Quantum spectral curve for planar \(\cal{N} =4\) super-Yang-Mills theory. Phys. Rev. Lett. 112, 011602 (2014). arXiv:1305.1939

    Article  ADS  Google Scholar 

  110. N. Gromov, V. Kazakov, S. Leurent, D. Volin, Quantum spectral curve for arbitrary state/operator in \(AdS_{5}/CFT_{4}\). JHEP 1509, 187 (2015). arXiv:1405.4857

    Article  ADS  MATH  Google Scholar 

  111. N. Gromov, F. Levkovich-Maslyuk, Quantum spectral curve for a cusped Wilson line in \({\cal N\it } =4 \) SYM. JHEP 1604, 134 (2016). arXiv:1510.02098

  112. N. Gromov, F. Levkovich-Maslyuk, Quark-anti-quark potential in \({\cal{N}} =4\) SYM. arXiv:1601.05679

  113. T. Klose, Review of AdS/CFT Integrability, Chapter IV.3: \({\cal{N}} =6\) Chern-Simons and strings on \(AdS_4\times {\mathbb{C}P}^3\). Lett. Math. Phys. 99, 401 (2012). arXiv:1012.3999

  114. J. Minahan, K. Zarembo, The Bethe ansatz for superconformal Chern-Simons. JHEP 0809, 040 (2008). arXiv:0806.3951

    Article  ADS  MathSciNet  MATH  Google Scholar 

  115. D. Bak, S.-J. Rey, Integrable spin chain in superconformal Chern-Simons theory. JHEP 0810, 053 (2008). arXiv:0807.2063

    Article  ADS  MathSciNet  MATH  Google Scholar 

  116. J.A. Minahan, W. Schulgin, K. Zarembo, Two loop integrability for Chern-Simons theories with \(\cal{N}=6\) supersymmetry. JHEP 0903, 057 (2009). arXiv:0901.1142

    Article  ADS  MathSciNet  Google Scholar 

  117. B.I. Zwiebel, Two-loop integrability of planar \(\cal{N}=6\) superconformal Chern-Simons theory. J. Phys. A 42, 495402 (2009). arXiv:0901.0411

    Article  MathSciNet  MATH  Google Scholar 

  118. N. Gromov, P. Vieira, The all loop \(AdS_4/CFT_3\) Bethe ansatz. JHEP 0901, 016 (2009). arXiv:0807.0777

    Article  ADS  MATH  Google Scholar 

  119. N. Gromov, P. Vieira, The \(AdS_4\)/\(CFT_3\) algebraic curve. JHEP 0902, 040 (2009). arXiv:0807.0437

    Article  ADS  MATH  Google Scholar 

  120. G. Arutyunov, S. Frolov, Superstrings on \(AdS_4\times \mathbb{CP}^3\) as a Coset Sigma-model. JHEP 0809, 129 (2008). arXiv:0806.4940

    Article  ADS  MATH  Google Scholar 

  121. D. Bombardelli, D. Fioravanti, R. Tateo, TBA and Y-system for planar \({AdS_4/CFT_3}\). Nucl. Phys. B 834, 543 (2010). arXiv:0912.4715

    Article  ADS  MathSciNet  MATH  Google Scholar 

  122. N. Gromov, F. Levkovich-Maslyuk, Y-system, TBA and Quasi-Classical strings in \(AdS_4\times \mathbb{CP}^3\). JHEP 1006, 088 (2010). arXiv:0912.4911

    Article  ADS  MATH  Google Scholar 

  123. A. Cavaglia, D. Fioravanti, N. Gromov, R. Tateo, The quantum spectral curve of the ABJM theory. Phys. Rev. Lett. 113, 021601 (2014). arXiv:1403.1859

    Article  ADS  Google Scholar 

  124. N. Gromov, G. Sizov, Exact slope and interpolating functions in \(\cal{N}=6\) supersymmetric Chern-Simons theory. Phys. Rev. Lett. 113, 121601 (2014). arXiv:1403.1894

    Article  ADS  Google Scholar 

  125. A. Cavaglia, N. Gromov, F. Levkovich-Maslyuk, On the exact interpolating function in ABJ theory. JHEP 1612, 086 (2016). arXiv:1605.04888

  126. V.G. Drinfeld, Hopf algebras and the quantum Yang-Baxter equation. Sov. Math. Dokl. 32, 254 (1985). [Dokl. Akad. Nauk Ser. Fiz. 283 1060(1985)]

    Google Scholar 

  127. L.F. Alday, J.M. Maldacena, Gluon scattering amplitudes at strong coupling. JHEP 0706, 064 (2007). arXiv:0705.0303

    Article  ADS  MathSciNet  Google Scholar 

  128. N. Beisert, R. Ricci, A.A. Tseytlin, M. Wolf, Dual superconformal symmetry from \(AdS_5 \times S^5\) superstring integrability. Phys. Rev. D 78, 126004 (2008). arXiv:0807.3228

    Article  ADS  MathSciNet  Google Scholar 

  129. I. Adam, A. Dekel, Y. Oz, On integrable backgrounds self-dual under fermionic T-duality. JHEP 0904, 120 (2009). arXiv:0902.3805

    Article  ADS  MathSciNet  Google Scholar 

  130. L.F. Alday, Review of AdS/CFT Integrability, Chapter V. 3: scattering amplitudes at strong coupling. Lett. Math. Phys. 99, 507 (2012). arXiv:1012.4003

  131. M.C. Abbott, J. Murugan, S. Penati, A. Pittelli, D. Sorokin, P. Sundin, J. Tarrant, M. Wolf, L. Wulff, T-duality of Green-Schwarz superstrings on \(AdS_d \times S^d \times M^{10-2d}\). JHEP 1512, 104 (2015). arXiv:1509.07678

  132. J.M. Drummond, G.P. Korchemsky, E. Sokatchev, Conformal properties of four-gluon planar amplitudes and Wilson loops. Nucl. Phys. B 795, 385 (2008). arXiv:0707.0243

    Article  ADS  MathSciNet  MATH  Google Scholar 

  133. J.M. Drummond, Review of AdS/CFT Integrability, Chapter V. 2: dual superconformal symmetry. Lett. Math. Phys. 99, 481 (2012). arXiv:1012.4002

  134. J.M. Drummond, J. Henn, G.P. Korchemsky, E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in \(\cal{N}=4\) super-Yang-Mills theory. Nucl. Phys. B 828, 317 (2010). arXiv:0807.1095

    Article  ADS  MathSciNet  MATH  Google Scholar 

  135. A. Brandhuber, P. Heslop, G. Travaglini, A Note on dual superconformal symmetry of the \(\cal{N}=4\) super Yang-Mills S-matrix. Phys. Rev. D 78, 125005 (2008). arXiv:0807.4097

    Article  ADS  MathSciNet  Google Scholar 

  136. J.M. Drummond, J.M. Henn, J. Plefka, Yangian symmetry of scattering amplitudes in \({\cal N\it } =4\) super Yang-Mills theory. JHEP 0905, 046 (2009). arXiv:0902.2987 (In: Strangeness in quark matter. Proceedings, International Conference, SQM 2008 (Beijing, P.R. China, October 5–10, 2008), p. 046)

  137. T. Bargheer, N. Beisert, W. Galleas, F. Loebbert, T. McLoughlin, Exacting \(\cal{N}=4\) superconformal symmetry. JHEP 0911, 056 (2009). arXiv:0905.3738

    Article  ADS  MathSciNet  Google Scholar 

  138. N. Beisert, J. Henn, T. McLoughlin, J. Plefka, One-loop superconformal and Yangian symmetries of scattering amplitudes in \(\cal{N}=4\) super Yang-Mills. JHEP 1004, 085 (2010). arXiv:1002.1733

    Article  ADS  MathSciNet  MATH  Google Scholar 

  139. S. Caron-Huot, S. He, Jumpstarting the all-loop S-matrix of planar \(\cal{N}=4\) super Yang-Mills. JHEP 1207, 174 (2012). arXiv:1112.1060

    Article  ADS  MathSciNet  Google Scholar 

  140. L. Dolan, C.R. Nappi, E. Witten, A relation between approaches to integrability in superconformal Yang-Mills theory. JHEP 0310, 017 (2003). arXiv:hep-th/0308089

    Article  ADS  MathSciNet  Google Scholar 

  141. D. Mueller, H. Muenkler, J. Plefka, J. Pollok, K. Zarembo, Yangian symmetry of smooth Wilson loops in \(\cal{N}= \) 4 super Yang-Mills theory. JHEP 1311, 081 (2013). arXiv:1309.1676

    Article  ADS  MathSciNet  MATH  Google Scholar 

  142. N. Beisert, D. Mueller, J. Plefka, C. Vergu, Integrability of smooth Wilson loops in \({\cal N\it } =4\) superspace. JHEP 1512, 141 (2015). arXiv:1509.05403

  143. B. Basso, A. Sever, P. Vieira, Spacetime and Flux tube S-matrices at finite coupling for \(\cal{N}=4\) supersymmetric Yang-Mills theory. Phys. Rev. Lett. 111, 091602 (2013). arXiv:1303.1396

    Article  ADS  Google Scholar 

  144. B. Basso, A. Sever, P. Vieira, Space-time S-matrix and Flux tube S-matrix II. Extracting and matching data. JHEP 1401, 008 (2014). arXiv:1306.2058

    Google Scholar 

  145. B. Basso, A. Sever, P. Vieira, Space-time S-matrix and Flux-tube S-matrix III. The two-particle contributions. JHEP 1408, 085 (2014). arXiv:1402.3307

    Google Scholar 

  146. B. Basso, A. Sever, P. Vieira, Space-time S-matrix and Flux-tube S-matrix IV. Gluons and fusion. JHEP 1409, 149 (2014). arXiv:1407.1736

    Google Scholar 

  147. B. Basso, J. Caetano, L. Cordova, A. Sever, P. Vieira, OPE for all helicity amplitudes. JHEP 1508, 018 (2015). arXiv:1412.1132

    Google Scholar 

  148. B. Basso, S. Komatsu, P. Vieira, Structure constants and integrable bootstrap in planar \({\cal N\it } =4\) SYM theory. arXiv:1505.06745

  149. V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops. Commun. Math. Phys. 313, 71 (2012). arXiv:0712.2824

    Article  ADS  MathSciNet  MATH  Google Scholar 

  150. A. Neveu, J.H. Schwarz, Factorizable dual model of pions. Nucl. Phys. B 31, 86 (1971)

    Article  ADS  Google Scholar 

  151. P. Ramond, Dual theory for free fermions. Phys. Rev. D 3, 2415 (1971)

    Article  ADS  MathSciNet  Google Scholar 

  152. J.H. Schwarz, Covariant field equations of chiral \(\cal{N}=2\) \(D=10\) supergravity. Nucl. Phys. B 226, 269 (1983)

    Article  ADS  Google Scholar 

  153. M.B. Green, J.H. Schwarz, Covariant description of superstrings. Phys. Lett. B 136, 367 (1984)

    Article  ADS  Google Scholar 

  154. M.B. Green, J.H. Schwarz, Properties of the covariant formulation of superstring theories. Nucl. Phys. B 243, 285 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  155. H.J. de Vega, N. G. Sanchez, Lectures on string theory in curved space-times. arXiv:hep-th/9512074 (In: String gravity and physics at the Planck energy scale. Proceedings, NATO Advanced Study Institute, Erice, Italy, September 8–19, 1995)

  156. J. Polchinski, String theory, in An Introduction to the Bosonic String, vol. 1 (Cambridge University Press, 2007)

    Google Scholar 

  157. S.A. Fulling, Aspects of quantum field theory in curved space-time. Lond. Math. Soc. Stud. Texts 17, 1 (1989)

    MathSciNet  MATH  Google Scholar 

  158. O. Alvarez, Theory of strings with boundaries: fluctuations, topology, and quantum geometry. Nucl. Phys. B 216, 125 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  159. H. Luckock, Quantum geometry of strings with boundaries. Ann. Phys. 194, 113 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  160. M. Nakahara, Geometry, Topology and Physics (CRC Press, 2003)

    Google Scholar 

  161. L. Bianchi, V. Forini, B. Hoare, Two-dimensional S-matrices from unitarity cuts. JHEP 1307, 088 (2013). arXiv:1304.1798

    Google Scholar 

  162. O.T. Engelund, R.W. McKeown, R. Roiban, Generalized unitarity and the worldsheet \(S\) matrix in \(AdS_n \times S^n \times M^{10-2n}\). JHEP 1308, 023 (2013). arXiv:1304.4281

    Article  ADS  MATH  Google Scholar 

  163. L. Bianchi, B. Hoare, \(AdS_3 \times S^3 \times M^4\) string S-matrices from unitarity cuts. JHEP 1408, 097 (2014). arXiv:1405.7947

    Article  ADS  Google Scholar 

  164. S. Frolov, A.A. Tseytlin, Multispin string solutions in \(AdS_5 \times S^5\). Nucl. Phys. B 668, 77 (2003). arXiv:hep-th/0304255

    Article  ADS  MATH  Google Scholar 

  165. S. Frolov, A.A. Tseytlin, Quantizing three spin string solution in \(AdS_5\times S^5\). JHEP 0307, 016 (2003). arXiv:hep-th/0306130

    Article  ADS  Google Scholar 

  166. S. Frolov, I. Park, A.A. Tseytlin, On one-loop correction to energy of spinning strings in \(S^5\). Phys. Rev. D 71, 026006 (2005). arXiv:hep-th/0408187

    Article  ADS  Google Scholar 

  167. I. Park, A. Tirziu, A.A. Tseytlin, Spinning strings in \(AdS_5\times S^5\): One-loop correction to energy in SL(2) sector. JHEP 0503, 013 (2005). arXiv:hep-th/0501203

    Article  ADS  Google Scholar 

  168. N. Beisert, A.A. Tseytlin, K. Zarembo, Matching quantum strings to quantum spins: one-loop versus finite-size corrections. Nucl. Phys. B 715, 190 (2005). arXiv:hep-th/0502173

    Article  ADS  MATH  Google Scholar 

  169. B. Hoare, Y. Iwashita, A.A. Tseytlin, Pohlmeyer-reduced form of string theory in \(AdS_5 \times S^5\): semiclassical expansion. J. Phys. A 42, 375204 (2009). arXiv:0906.3800

    Article  MathSciNet  MATH  Google Scholar 

  170. S.S. Gubser, I.R. Klebanov, A.M. Polyakov, A semi-classical limit of the gauge/string correspondence. Nucl. Phys. B 636, 99 (2002). arXiv:hep-th/0204051

    Article  ADS  MathSciNet  MATH  Google Scholar 

  171. S. Frolov, A.A. Tseytlin, Semiclassical quantization of rotating superstring in \(AdS_5 \times S^5\). JHEP 0206, 007 (2002). arXiv:hep-th/0204226

    Article  ADS  Google Scholar 

  172. M. Beccaria, G.V. Dunne, V. Forini, M. Pawellek, A.A. Tseytlin, Exact computation of one-loop correction to energy of spinning folded string in \(AdS_5 \times S^5\). J. Phys. A 43, 165402 (2010). arXiv:1001.4018

    Article  ADS  MathSciNet  MATH  Google Scholar 

  173. V. Forini, Quark-antiquark potential in AdS at one loop. JHEP 1011, 079 (2010). arXiv:1009.3939

    Article  ADS  MATH  Google Scholar 

  174. N. Drukker, V. Forini, Generalized quark-antiquark potential at weak and strong coupling. JHEP 1106, 131 (2011). arXiv:1105.5144

    Article  ADS  MATH  Google Scholar 

  175. V. Forini, V.G.M. Puletti, O. Ohlsson, Sax, The generalized cusp in \(AdS_4 \times \mathbb{CP}^3\) and more one-loop results from semiclassical strings. J. Phys. A 46, 115402 (2013). arXiv:1204.3302

    Article  ADS  MathSciNet  MATH  Google Scholar 

  176. S. Frolov, A. Tirziu, A.A. Tseytlin, Logarithmic corrections to higher twist scaling at strong coupling from AdS/CFT. Nucl. Phys. B 766, 232 (2007). arXiv:hep-th/0611269

    Article  ADS  MathSciNet  MATH  Google Scholar 

  177. R. Roiban, A. Tirziu, A.A. Tseytlin, Two-loop world-sheet corrections in \(AdS_5 \times S^5\) superstring. JHEP 0707, 056 (2007). arXiv:0704.3638

    Article  ADS  Google Scholar 

  178. N. Drukker, D.J. Gross, H. Ooguri, Wilson loops and minimal surfaces. Phys. Rev. D 60, 125006 (1999). arXiv:hep-th/9904191

    Article  ADS  MathSciNet  Google Scholar 

  179. K.G. Wilson, Confinement of quarks. Phys. Rev. D 10, 2445 (1974) [,45(1974)]

    Google Scholar 

  180. H. Ooguri, J. Rahmfeld, H. Robins, J. Tannenhauser, Holography in superspace. JHEP 0007, 045 (2000). arXiv:hep-th/0007104

    Google Scholar 

  181. N. Beisert, D. Mueller, J. Plefka, C. Vergu, Smooth Wilson loops in \({\cal N\it } =4 \) non-chiral superspace. JHEP 1512, 140 (2015). arXiv:1506.07047

  182. H. Muenkler, J. Pollok, Minimal surfaces of the \({AdS}_{5}\times {S}^{5}\) superstring and the symmetries of super Wilson loops at strong coupling. J. Phys. A 48, 365402 (2015). arXiv:1503.07553

  183. J.M. Maldacena, Wilson loops in large N field theories. Phys. Rev. Lett. 80, 4859 (1998). arXiv:hep-th/9803002

    Article  ADS  MathSciNet  MATH  Google Scholar 

  184. S.-J. Rey, J.-T. Yee, Macroscopic strings as heavy quarks in large N gauge theory and anti-de Sitter supergravity. Eur. Phys. J. C 22, 379 (2001). arXiv:hep-th/9803001

    Article  ADS  MathSciNet  MATH  Google Scholar 

  185. G.W. Semenoff, K. Zarembo, Wilson loops in SYM theory: from weak to strong coupling. Nucl. Phys. Proc. Suppl. 108, 106 (2002). arXiv:hep-th/0202156 (In: Light cone physics: Particles and strings. Proceedings, International Workshop (Trento, Italy, September 3–11, 2001), p. 106–112, [,106(2002)])

  186. Y. Makeenko, A brief introduction to Wilson loops, large N, Phys. Atom. Nucl. 73, 878 (2010). arXiv:0906.4487 (In: 12th International Moscow School of Physics and 37th ITEP Winter School of Physics Moscow (Russia, February 9–16, 2009), p. 878–894)

  187. R. Roiban, A.A. Tseytlin, Strong-coupling expansion of cusp anomaly from quantum superstring. JHEP 0711, 016 (2007). arXiv:0709.0681

    Article  ADS  MathSciNet  MATH  Google Scholar 

  188. S. Giombi, R. Ricci, R. Roiban, A.A. Tseytlin, C. Vergu, Generalized scaling function from light-cone gauge \(AdS_5 \times S^5\) superstring. JHEP 1006, 060 (2010). arXiv:1002.0018

    Article  ADS  MATH  Google Scholar 

  189. T. McLoughlin, Review of AdS/CFT Integrability, Chapter II.2: quantum strings in \(AdS_5 \times S^5\). Lett. Math. Phys. 99, 127 (2012). arXiv:1012.3987

  190. A.A. Tseytlin, Review of AdS/CFT Integrability, Chapter II.1: classical \(AdS_5 \times S^5\) string solutions. Lett. Math. Phys. 99, 103 (2012). arXiv:1012.3986

  191. S. Forste, D. Ghoshal, S. Theisen, Stringy corrections to the Wilson loop in \({\cal{N}} =4\) superYang-Mills theory. JHEP 9908, 013 (1999). arXiv:hep-th/9903042

    Article  ADS  MATH  Google Scholar 

  192. N. Drukker, D.J. Gross, A.A. Tseytlin, Green-Schwarz string in \(AdS_5 \times S^5\): semiclassical partition function. JHEP 0004, 021 (2000). arXiv:hep-th/0001204

    Article  ADS  MATH  Google Scholar 

  193. M. Beccaria, V. Forini, A. Tirziu, A.A. Tseytlin, Structure of large spin expansion of anomalous dimensions at strong coupling. Nucl. Phys. B 812, 144 (2009). arXiv:0809.5234

    Article  ADS  MathSciNet  MATH  Google Scholar 

  194. T. Klose, T. McLoughlin, R. Roiban, K. Zarembo, Worldsheet scattering in AdS\(_5 \times S^5\). JHEP 0703, 094 (2007). arXiv:hep-th/0611169

    Article  ADS  MathSciNet  Google Scholar 

  195. R. Roiban, P. Sundin, A. Tseytlin, L. Wulff, The one-loop worldsheet S-matrix for the \(AdS_{n} \times S^{n} \times T^{10-2n}\) superstring. JHEP 1408, 160 (2014). arXiv:1407.7883

    Article  ADS  Google Scholar 

  196. M. Kruczenski, A. Tirziu, Matching the circular Wilson loop with dual open string solution at 1-loop in strong coupling. JHEP 0805, 064 (2008). arXiv:0803.0315

    Article  ADS  MathSciNet  Google Scholar 

  197. E.I. Buchbinder, A.A. Tseytlin, One-loop correction to the energy of a wavy line string in \(AdS_{5}\). J. Phys. A 46, 505401 (2013). arXiv:1309.1581

    Article  MathSciNet  MATH  Google Scholar 

  198. J. Aguilera-Damia, D.H. Correa, G.A. Silva, Semiclassical partition function for strings dual to Wilson loops with small cusps in ABJM. JHEP 1503, 002 (2015). arXiv:1412.4084

    Article  ADS  Google Scholar 

  199. T. Kameyama, K. Yoshida, Generalized quark-antiquark potentials from a \(q\)-deformed \(AdS_5 \times S^5\) background. PTEP 2016, 063B01 (2016). arXiv:1602.06786

  200. R. Kallosh, J. Kumar, Supersymmetry enhancement of D-p-branes and M-branes. Phys. Rev. D 56, 4934 (1997). arXiv:hep-th/9704189

    Article  ADS  MathSciNet  Google Scholar 

  201. M. Cvetic, H. Lu, C.N. Pope, K.S. Stelle, T duality in the Green-Schwarz formalism, and the massless/massive IIA duality map. Nucl. Phys. B 573, 149 (2000). arXiv:hep-th/9907202

    Article  ADS  MathSciNet  MATH  Google Scholar 

  202. L. Wulff, The type II superstring to order \(\theta ^4\). JHEP 1307, 123 (2013). arXiv:1304.6422

    Article  ADS  MathSciNet  MATH  Google Scholar 

  203. V. Fock, The proper time in classical and quantum mechanics, Izv. Akad. Nauk USSR (Phys.) 4–5, 551 (1937)

    Google Scholar 

  204. J.S. Schwinger, On gauge invariance and vacuum polarization. Phys. Rev. 82, 664 (1951)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  205. B. DeWitt, Dynamical Theory of Groups and Fields (Gordon and Breach, 1965)

    Google Scholar 

  206. B.S. DeWitt, Quantum theory of gravity. 1. The canonical theory. Phys. Rev. 160, 1113 (1967)

    Article  ADS  MATH  Google Scholar 

  207. B.S. DeWitt, Quantum theory of gravity. 2. The manifestly covariant theory. Phys. Rev. 162, 1195 (1967)

    Article  ADS  MATH  Google Scholar 

  208. B.S. DeWitt, Quantum theory of gravity. 3. Applications of the covariant theory. Phys. Rev. 162, 1239 (1967)

    Article  ADS  MATH  Google Scholar 

  209. D.V. Vassilevich, Heat kernel expansion: User’s manual. Phys. Rept. 388, 279 (2003). arXiv:hep-th/0306138

    Article  ADS  MathSciNet  MATH  Google Scholar 

  210. V. Mukhanov, S. Winitzki, Introduction to Quantum Effects in Gravity (Cambridge University Press, 2007)

    Google Scholar 

  211. D. Fursaev, D. Vassilevich, Operators, Geometry and Quanta: Methods of Spectral Geometry in Quantum Field Theory (Springer, Verlag, 2011)

    Book  MATH  Google Scholar 

  212. R. Camporesi, Harmonic analysis and propagators on homogeneous spaces. Phys. Rept. 196, 1 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  213. R. Camporesi, A. Higuchi, Spectral functions and zeta functions in hyperbolic spaces. J. Math. Phys. 35, 4217 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  214. R. Camporesi, The Spinor heat kernel in maximally symmetric spaces. Commun. Math. Phys. 148, 283 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  215. R. Camporesi, A. Higuchi, On the eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces. J. Geom. Phys. 20, 1 (1996). arXiv:gr-qc/9505009

    Article  ADS  MathSciNet  MATH  Google Scholar 

  216. O. Calin, D.-C. Chang, K. Furutani, C. Iwasaki, Heat Kernels for Elliptic and Sub-Elliptic Operators: Methods and Techniques (Springer, 2011)

    Google Scholar 

  217. E. Elizalde, Zeta Regularization Techniques with Applications (World Scientific, 1994)

    Google Scholar 

  218. E.I. Buchbinder, A.A. Tseytlin, \(1/N\) correction in the D3-brane description of a circular Wilson loop at strong coupling. Phys. Rev. D 89, 126008 (2014). arXiv:1404.4952

    Article  ADS  Google Scholar 

  219. R. Bergamin, A.A. Tseytlin, Heat kernels on cone of \(AdS_2\) and \(k\)-wound circular Wilson loop in \(AdS_5 \times S^5\) superstring. J. Phys. A 49, 14LT01 (2016). arXiv:1510.06894

  220. I.M. Gelfand, A.M. Yaglom, Integration in functional spaces and it applications in quantum physics. J. Math. Phys. 1, 48 (1960)

    Article  ADS  MathSciNet  Google Scholar 

  221. R. Camporesi, A. Higuchi, Stress-energy tensors in anti-de Sitter spacetime. Phys. Rev. D 45, 3591 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  222. R.F. Dashen, B. Hasslacher, A. Neveu, Nonperturbative methods and extended-hadron models in field theory. I. Semiclassical functional methods. Phys. Rev. D 10, 4114 (1974)

    Article  ADS  Google Scholar 

  223. R.F. Dashen, B. Hasslacher, A. Neveu, Nonperturbative methods and extended-hadron models in field theory. II. Two-dimensional models and extended hadrons. Phys. Rev. D 10, 4130 (1974)

    Article  ADS  Google Scholar 

  224. R.F. Dashen, B. Hasslacher, A. Neveu, Nonperturbative methods and extended-hadron models in field theory. III. Four-dimensional non-Abelian models. Phys. Rev. D 10, 4138 (1974)

    Article  ADS  Google Scholar 

  225. X. Chen-Lin, D. Medina-Rincon, K. Zarembo, Quantum String test of nonconformal holography. arXiv:1702.07954

  226. R. Forman, Functional determinants and geometry. Invent. math. 88, 447 (1987)

    Article  Google Scholar 

  227. R. Forman, Functional determinants and geometry (Erratum). Invent. math. 108, 453 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  228. G.V. Dunne, K. Kirsten, Functional determinants for radial operators. J. Phys. A 39, 11915 (2006). arXiv:hep-th/0607066

    Article  ADS  MathSciNet  MATH  Google Scholar 

  229. V. Forini, A.A. Tseytlin, E. Vescovi, Perturbative computation of string one-loop corrections to Wilson loop minimal surfaces in \(AdS_5 \times S^5\). JHEP 1703, 003 (2017). arXiv:1702.02164

  230. A.M. Polyakov, Conformal fixed points of unidentified gauge theories, Mod. Phys. Lett. A 19, 1649 (2004). arXiv:hep-th/0405106, [1159(2004)]

  231. S. Giombi, R. Ricci, R. Roiban, A. Tseytlin, C. Vergu, Quantum \(AdS_5 \times S^5\) superstring in the AdS light-cone gauge. JHEP 1003, 003 (2010). arXiv:0912.5105

    Article  ADS  MATH  Google Scholar 

  232. S. Giombi, R. Ricci, R. Roiban, A. Tseytlin, Two-loop \(AdS_5 \times S^5\) superstring: testing asymptotic Bethe ansatz and finite size corrections. J. Phys. A 44, 045402 (2011). arXiv:1010.4594

    Article  ADS  MathSciNet  MATH  Google Scholar 

  233. R. Metsaev, A.A. Tseytlin, Superstring action in \(AdS_5\times S^5\). Kappa symmetry light cone gauge. Phys. Rev. D 63, 046002 (2001). arXiv:hep-th/0007036

    Article  ADS  MathSciNet  Google Scholar 

  234. R. Metsaev, C.B. Thorn, A.A. Tseytlin, Light cone superstring in AdS space-time. Nucl. Phys. B 596, 151 (2001). arXiv:hep-th/0009171

    Article  ADS  MathSciNet  MATH  Google Scholar 

  235. M. Kruczenski, A Note on twist two operators in \({\cal{N}} =4\) SYM and Wilson loops in Minkowski signature. JHEP 0212, 024 (2002). arXiv:hep-th/0210115

    Article  ADS  MathSciNet  Google Scholar 

  236. D. Uvarov, \(AdS_4\times \mathbb{CP}^3\) superstring in the light-cone gauge. Nucl. Phys. B 826, 294 (2010). arXiv:0906.4699

    Article  ADS  MATH  Google Scholar 

  237. D. Uvarov, Light-cone gauge Hamiltonian for \(AdS_4\times \mathbb{CP}^3\) superstring. Mod. Phys. Lett. A 25, 1251 (2010). arXiv:0912.1044

    Article  ADS  MATH  Google Scholar 

  238. R. Roiban, A.A. Tseytlin, Spinning superstrings at two loops: Strong-coupling corrections to dimensions of large-twist SYM operators. Phys. Rev. D 77, 066006 (2008). arXiv:0712.2479

    Article  ADS  MathSciNet  Google Scholar 

  239. L. Bianchi, M.S. Bianchi, Quantum dispersion relations for the \(Ad{S}_4\times {\mathbb{C}P}^3\) GKP string. JHEP 1511, 031 (2015). arXiv:1505.00783

  240. B. Basso, A. Rej, Bethe ansaetze for GKP strings. Nucl. Phys. B 879, 162 (2014). arXiv:1306.1741

    Article  ADS  MATH  Google Scholar 

  241. I. Montvay, G. Muenster, Quantum Fields on a Lattice (Cambridge University Press, 1994)

    Google Scholar 

  242. T. DeGrand, C.E. Detar, Lattice Methods for Quantum Chromodynamics (World Scientific, 2006)

    Google Scholar 

  243. C. Gattringer, C. Lang, Quantum Chromodynamics on the Lattice: An Introductory Presentation (Springer, Berlin Heidelberg, 2009)

    Google Scholar 

  244. R.W. McKeown, R. Roiban, The quantum \(AdS_5 \times S^5\) superstring at finite coupling. arXiv:1308.4875

  245. N. Beisert, B. Eden, M. Staudacher, Transcendentality and crossing. J. Stat. Mech. 0701, P01021 (2007). arXiv:hep-th/0610251

    Google Scholar 

  246. M. Hanada, What lattice theorists can do for quantum gravity. arXiv:1604.05421

  247. A.A. Tseytlin, On semiclassical approximation and spinning string vertex operators in \(AdS_5\times S^5\). Nucl. Phys. B 664, 247 (2003). arXiv:hep-th/0304139

    Article  ADS  MATH  Google Scholar 

  248. R. Roiban, A.A. Tseytlin, On semiclassical computation of 3-point functions of closed string vertex operators in \(AdS_5 \times S^5\). Phys. Rev. D 82, 106011 (2010). arXiv:1008.4921

    Article  ADS  Google Scholar 

  249. A. Joseph, Review of lattice supersymmetry and gauge-gravity duality. Int. J. Mod. Phys. A 30, 1530054 (2015). arXiv:1509.01440

  250. G. Bergner, S. Catterall, Supersymmetry on the lattice. arXiv:1603.04478

  251. D.B. Kaplan, M. Unsal, A Euclidean lattice construction of supersymmetric Yang-Mills theories with sixteen supercharges. JHEP 0509, 042 (2005). arXiv:hep-lat/0503039

    Article  ADS  MathSciNet  Google Scholar 

  252. M. Unsal, Twisted supersymmetric gauge theories and orbifold lattices. JHEP 0610, 089 (2006). arXiv:hep-th/0603046

    Article  ADS  MathSciNet  Google Scholar 

  253. S. Catterall, From twisted supersymmetry to orbifold lattices. JHEP 0801, 048 (2008). arXiv:0712.2532

    Article  ADS  MathSciNet  Google Scholar 

  254. P.H. Damgaard, S. Matsuura, Geometry of orbifolded supersymmetric lattice gauge theories. Phys. Lett. B 661, 52 (2008). arXiv:0801.2936

    Article  ADS  MathSciNet  MATH  Google Scholar 

  255. S. Catterall, D.B. Kaplan, M. Unsal, Exact lattice supersymmetry. Phys. Rept. 484, 71 (2009). arXiv:0903.4881

    Article  Google Scholar 

  256. D. Schaich, T. DeGrand, Parallel software for lattice \(\cal{N}=4\) supersymmetric Yang-Mills theory. Comput. Phys. Commun. 190, 200 (2015). arXiv:1410.6971

    Article  ADS  MATH  Google Scholar 

  257. S. Catterall, D. Schaich, P.H. Damgaard, T. DeGrand, J. Giedt, \(\cal{N}=4\) Supersymmetry on a space-time lattice. Phys. Rev. D 90, 065013 (2014). arXiv:1405.0644

    Article  ADS  Google Scholar 

  258. S. Catterall, J. Giedt, D. Schaich, P.H. Damgaard, T. DeGrand, Results from lattice simulations of \({\cal N\it } =4\) supersymmetric Yang-Mills. PoS LATTICE2014, 267 (2014). arXiv:1411.0166 (In: Proceedings, 32nd International Symposium on Lattice Field Theory (Lattice 2014), p. 267)

  259. D. Schaich, Aspects of lattice \({\cal {N}} =4\) supersymmetric Yang-Mills, PoS LATTICE2015, 242 (2015). arXiv:1512.01137 (In: Proceedings, 33rd International Symposium on Lattice Field Theory (Lattice 2015), p. 242)

  260. T. Ishii, G. Ishiki, S. Shimasaki, A. Tsuchiya, \(\cal{N}=4\) super Yang-Mills from the plane wave matrix model. Phys. Rev. D 78, 106001 (2008). arXiv:0807.2352

    Article  ADS  MathSciNet  Google Scholar 

  261. G. Ishiki, S.-W. Kim, J. Nishimura, A. Tsuchiya, Deconfinement phase transition in \(\cal{N}=4\) super Yang-Mills theory on \(\mathbb{R} \times S^3\) from supersymmetric matrix quantum mechanics. Phys. Rev. Lett. 102, 111601 (2009). arXiv:0810.2884

    Article  ADS  Google Scholar 

  262. G. Ishiki, S.-W. Kim, J. Nishimura, A. Tsuchiya, Testing a novel large-N reduction for \(\cal{N}=4\) super Yang-Mills theory on \(\mathbb{R} \times S^3\). JHEP 0909, 029 (2009). arXiv:0907.1488

    Article  ADS  MathSciNet  Google Scholar 

  263. M. Hanada, S. Matsuura, F. Sugino, Two-dimensional lattice for four-dimensional \(\cal{N}=4\) supersymmetric Yang-Mills. Prog. Theor. Phys. 126, 597 (2011). arXiv:1004.5513

    Article  ADS  MATH  Google Scholar 

  264. M. Honda, G. Ishiki, J. Nishimura, A. Tsuchiya, Testing the AdS/CFT correspondence by Monte Carlo calculation of BPS and non-BPS Wilson loops in 4d \({\cal N\it } =4\) super-Yang-Mills theory. PoS LATTICE2011, 244 (2011). arXiv:1112.4274 (In: Proceedings, 29th International Symposium on Lattice field theory (Lattice 2011), p. 244)

  265. M. Honda, G. Ishiki, S.-W. Kim, J. Nishimura, A. Tsuchiya, Direct test of the AdS/CFT correspondence by Monte Carlo studies of \(\cal{N}=4\) super Yang-Mills theory. JHEP 1311, 200 (2013). arXiv:1308.3525

    Article  ADS  Google Scholar 

  266. M. Hanada, Y. Hyakutake, G. Ishiki, J. Nishimura, Holographic description of quantum black hole on a computer. Science 344, 882 (2014). arXiv:1311.5607

    Article  ADS  Google Scholar 

  267. V. Forini, V.G.M. Puletti, L. Griguolo, D. Seminara, E. Vescovi, Remarks on the geometrical properties of semiclassically quantized strings. J. Phys. A 48, 475401 (2015). arXiv:1507.01883

  268. V. Forini, V.G.M. Puletti, M. Pawellek, E. Vescovi, One-loop spectroscopy of semiclassically quantized strings: bosonic sector. J. Phys. A 48, 085401 (2015). arXiv:1409.8674

    Article  ADS  MathSciNet  MATH  Google Scholar 

  269. V. Forini, V. Giangreco M. Puletti, L. Griguolo, D. Seminara, E. Vescovi, Precision calculation of 1/4-BPS Wilson loops in \(AdS_5\times S^5\). JHEP 1602, 105 (2016). arXiv:1512.00841

  270. L. Bianchi, M.S. Bianchi, A. Bres, V. Forini, E. Vescovi, Two-loop cusp anomaly in ABJM at strong coupling. JHEP 1410, 13 (2014). arXiv:1407.4788

    Article  ADS  Google Scholar 

  271. V. Forini, L. Bianchi, M.S. Bianchi, B. Leder, E. Vescovi, Lattice and string worldsheet in AdS/CFT: a numerical study. PoS LATTICE2015, 244 (2015). arXiv:1601.04670 (In: Proceedings, 33rd International Symposium on Lattice Field Theory (Lattice 2015), p. 244)

  272. L. Bianchi, M.S. Bianchi, V. Forini, B. Leder, E. Vescovi, Green-Schwarz superstring on the lattice. JHEP 1607, 014 (2016). arXiv:1605.01726

    Google Scholar 

  273. J. Aguilera-Damia, A. Faraggi, L.A. Pando Zayas, V. Rathee, G.A. Silva, D. Trancanelli, E. Vescovi, in preparation

    Google Scholar 

  274. V. Forini, L. Bianchi, B. Leder, P. Toepfer, E. Vescovi, Strings on the lattice and AdS/CFT. PoS LATTICE2016, 206 (2016). arXiv:1702.02005 (In: Proceedings, 34th International Symposium on Lattice Field Theory (Lattice 2016, Southampton, UK, July 24–30, 2016), p. 206)

  275. V. Forini, L. Bianchi, B. Leder, P. Toepfer, E. Vescovi, in preparation

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Physics, University of São Paulo, São Paulo, São Paulo, Brazil

    Edoardo Vescovi

Authors
  1. Edoardo Vescovi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Edoardo Vescovi .

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Vescovi, E. (2017). Introduction. In: Perturbative and Non-perturbative Approaches to String Sigma-Models in AdS/CFT. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-63420-3_1

Download citation

Publish with us

Policies and ethics

Search

Navigation