Effective string theory and nonlinear Lorentz invariance

  • Ofer Aharony
  • Matthew Dodelson


We study the low-energy effective action governing the transverse fluctuations of a long string, such as a confining flux tube in QCD. We work in the static gauge where this action contains only the transverse excitations of the string. The static gauge action is strongly constrained by the requirement that the Lorentz symmetry, that is spontaneously broken by the long string vacuum, is nonlinearly realized on the Nambu-Goldstone bosons. One solution to the constraints (at the classical level) is the Nambu-Goto action, and the general solution contains higher derivative corrections to this. We show that in 2 + 1 dimensions, the first allowed correction to the Nambu-Goto action is proportional to the squared curvature of the induced metric on the worldsheet. In higher dimensions, there is a more complicated allowed correction that appears at lower order than the curvature squared. We argue that this leading correction is similar to, but not identical to, the one-loop determinant \( \sqrt {{ - h}} R{\square^{ - 1}}R \) computed by Polyakov for the bosonic fundamental string.


Long strings Confinement Space-Time Symmetries Spontaneous Symmetry Breaking 


  1. [1]
    A.A. Abrikosov, On the magnetic properties of superconductors of the second group, Sov. Phys. JETP 5 (1957) 1174 Google Scholar
  2. [2]
    H.B. Nielsen and P. Olesen, Vortex line models for dual strings, Nucl. Phys. B 61 (1973) 45 [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    M. Lüscher and P. Weisz, String excitation energies in SU(N) gauge theories beyond the free-string approximation, JHEP 07 (2004) 014 [hep-th/0406205] [INSPIRE].CrossRefGoogle Scholar
  4. [4]
    H.B. Meyer, Poincaré invariance in effective string theories, JHEP 05 (2006) 066 [hep-th/0602281] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    O. Aharony and E. Karzbrun, On the effective action of confining strings, JHEP 06 (2009) 012 [arXiv:0903.1927] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    O. Aharony, Z. Komargodski and A. Schwimmer, The effective action on long strings, talk given at Strings 2009, Rome Italy, June 22-26, 2009, Strings09.ppt, and at the ECT* Workshop on Confining Flux Tubes and Strings, Trento Italy, July 5-9, 2010, 05 07 2010/Aharony.ppt, work in progress.
  7. [7]
    J.D. Cohn and V. Periwal, Lorentz invariance of effective strings, Nucl. Phys. B 395 (1993) 119 [hep-th/9205026] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    O. Aharony and M. Field, On the effective theory of long open strings, JHEP 01 (2011) 065 [arXiv:1008.2636] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    O. Aharony and N. Klinghoffer, Corrections to Nambu-Goto energy levels from the effective string action, JHEP 12 (2010) 058 [arXiv:1008.2648] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    A. Athenodorou, B. Bringoltz and M. Teper, Closed flux tubes and their string description in D = 2 + 1 SU(N) gauge theories, JHEP 05 (2011) 042 [arXiv:1103.5854] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    M. Teper, Large-N and confining flux tubes as stringsa view from the lattice, Acta Phys. Polon. B 40 (2009) 3249 [arXiv:0912.3339] [INSPIRE].Google Scholar
  12. [12]
    P. Giudice, F. Gliozzi and S. Lottini, The confining string beyond the free-string approximation in the gauge dual of percolation, JHEP 03 (2009) 104 [arXiv:0901.0748] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  13. [13]
    M. Caselle and M. Zago, A new approach to the study of effective string corrections in LGTs, Eur. Phys. J. C 71 (2011) 1658 [arXiv:1012.1254] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    M. Billó, M. Caselle, V. Verduci and M. Zago, New results on the effective string corrections to the inter-quark potential, PoS(Lattice 2010)273 [arXiv:1012.3935] [INSPIRE].
  15. [15]
    S. Jaimungal, G.W. Semenoff and K. Zarembo, Universality in effective strings, JETP Lett. 69 (1999) 509 [hep-ph/9811238] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  16. [16]
    F. Gliozzi, Dirac-Born-Infeld action from spontaneous breakdown of Lorentz symmetry in brane-world scenarios, Phys. Rev. D 84 (2011) 027702 [arXiv:1103.5377] [INSPIRE].ADSGoogle Scholar
  17. [17]
    R. Casalbuoni, J. Gomis and K. Kamimura, Space-time transformations of the Born-Infeld gauge field of a D-brane, Phys. Rev. D 84 (2011) 027901 [arXiv:1104.4916] [INSPIRE].ADSGoogle Scholar
  18. [18]
    J. Polchinski and A. Strominger, Effective string theory, Phys. Rev. Lett. 67 (1991) 1681 [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  19. [19]
    J. Polchinski, Strings and QCD?, hep-th/9210045 [INSPIRE].
  20. [20]
    J.M. Drummond, Universal subleading spectrum of effective string theory, hep-th/0411017 [INSPIRE].
  21. [21]
    O. Aharony, M. Field, N. Klinghoffer, The effective string spectrum in the orthogonal gauge, arXiv:1111.5757 [INSPIRE].
  22. [22]
    A.M. Polyakov, Quantum geometry of bosonic strings, Phys. Lett. B 103 (1981) 207 [INSPIRE].MathSciNetADSGoogle Scholar
  23. [23]
    I. Low and A.V. Manohar, Spontaneously broken space-time symmetries and Goldstone’s theorem, Phys. Rev. Lett. 88 (2002) 101602 [hep-th/0110285] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    J. Polchinski, String theory. Volume 1: an introduction to the bosonic string, Cambridge University Press, Cambridge U.K. (1998) [INSPIRE].Google Scholar
  25. [25]
    P.O. Mazur and V.P. Nair, Strings in QCD and θ vacua, Nucl. Phys. B 284 (1987) 146 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  26. [26]
    A.M. Polyakov, Quantum geometry of fermionic strings, Phys. Lett. B 103 (1981) 211 [INSPIRE].MathSciNetADSGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  1. 1.Department of Particle Physics and AstrophysicsWeizmann Institute of ScienceRehovotIsrael
  2. 2.Department of PhysicsBrown UniversityProvidenceUSA

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