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Geometric Properties of Semiclassically Quantized Strings

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Perturbative and Non-perturbative Approaches to String Sigma-Models in AdS/CFT

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Abstract

The geometric properties of string worldsheets embedded in a higher-dimensional space-time, and of linearized perturbations above them, have been object of various studies since the seminal observation on the relevance of quantizing string models [1]. In the framework of the AdS/CFT correspondence, a particularly important setting where these analyses have been performed is the non-linear sigma-model on the curved \(AdS_5\times S^5\).

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Notes

  1. 1.

    See also [4, 5] and references therein, where this analysis has been exploited for the description of QCD strings or stability effects for membrane solutions.

  2. 2.

    The AdS/CFT correspondence prescribes a conformal compactification of \(AdS_5\times S^5\) and the dual \(\mathbb {R}^{1,3}\) space, as explained in the review [8]. We thank Hagen Münkler for pointing out to us the reference [9] where such construction for \(\mathbb {R}^{p,q}\) is rigorously spelt out.

  3. 3.

    Note that the “shape” of a minimal surface is deeply influenced by the signature of the 10d \(G_{MN}\) and 2d metric \(h_{ij}\). This also implies that a rigorous mathematical treatment of the differential equations (3.10)—e.g. existence and uniqueness theorems of its solutions—is substantially different in a Riemannian or pseudo-Riemannian ambient space. We acknowledge an elucidating discussion with Thomas Klose about this point.

  4. 4.

    If we reintroduce the string tension T / 2 in the Polyakov action, then we need to rescale \(\delta X^M\rightarrow \sqrt{\frac{2}{T}}\delta X^M\) and the effective expansion parameter \(\sqrt{\frac{2}{T}}\zeta ^M\) is small in semiclassical approximation \(T\rightarrow \infty \).

  5. 5.

    This corresponds to the absence of non-trivial Beltrami differentials at genus 0.

  6. 6.

    This follows from the fact that we have just two independent components of the extrinsic curvature (e.g. \(K^M_{11}\) and \(K^M_{12}\)). Alternatively, we can argue this result, in a covariant way, from the following matrix relation \( \mathcal{F}^4=\frac{1}{2}\mathrm {tr}(\mathcal {F}^2) \mathcal{F}^2 \) satisfied by \(\mathcal F\).

  7. 7.

    The matrix elements are labelled by \(\underline{i},\underline{j}=2,\dots 9\) in our index conventions.

  8. 8.

    An alternative form for the flux [25] is

    figure a

    with which the corresponding part of the gauge-fixed Lagrangian reads

    figure b

    Its equivalence with (3.75) and (3.86) below is manifest in the \(5+5\) basis of [25], see also discussion in [3].

  9. 9.

    A widely used alternative to (3.78) is the light-cone gauge-fixing \(\Gamma ^+\Psi ^I=0\), where the light-cone might lie entirely in \(S^5\) [27, 28] or being shared between \(AdS_5\) and \(S^5\) [29] (and [30] for further references therein). One of the obvious advantages of the “covariant” gauge-fixing (3.78) is preservation of global bosonic symmetries of the action. A more general choice is \(\Psi _1=k\,\Psi _2\equiv k \, \Psi \) where k is a real parameter whose dependence is expected to cancel in the effective action, see discussion in [31]. Yet another \(\kappa \)-symmetry gauge-fixing, albeit equivalent to (3.78) [3], has been used for studying stringy fluctuations in \(AdS_5\times S^5\) in [32].

  10. 10.

    We will encounter an example of string background with non-flat connection in (5.29), see the induced fermionic Lagrangian in (5.53) and comments below. This was first noticed in (5.35) of [6].

  11. 11.

    This corresponds to the \(a_2\) coefficient in (3.102), compare in [41] the zeta-function regularization (5.51) with the proper-time cutoff scheme (5.74).

  12. 12.

    At variance with the notation in [6], from \(a^{(F)}_2\) we stripped off a minus sign and the factor of 1 / 2 of the Majorana condition. We also included the integration in the SDW coefficients as it is common in literature.

References

  1. A.M. Polyakov, Quantum geometry of bosonic strings. Phys. Lett. B 103, 207 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  2. C.G. Callan Jr., L. Thorlacius, Sigma models and string theory. Proceedings, Particles, strings and supernovae 2, 795–878 (1988). (in Providence)

    ADS  Google Scholar 

  3. 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 

  4. R. Capovilla, J. Guven, Geometry of deformations of relativistic membranes. Phys. Rev. D 51, 6736 (1995). arXiv:gr-qc/9411060

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. K. Viswanathan, R. Parthasarathy, String theory in curved space-time. Phys. Rev. D 55, 3800 (1997). arXiv:hep-th/9605007

    Article  ADS  MathSciNet  Google Scholar 

  6. V. Forini, V.G.M. Puletti, L. Griguolo, D. Seminara, E. Vescovi, Remarks on the geometrical properties of semiclassically quantized strings. J. Phys. A48, 475401 (2015). arXiv:1507.01883

  7. A. Faraggi, W. Mueck, L.A. Pando, Zayas, one-loop effective action of the holographic antisymmetric wilson loop. Phys. Rev. D 85, 106015 (2012). arXiv:1112.5028

    Article  ADS  Google Scholar 

  8. 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 

  9. M. Schottenloher, A Mathematical Introduction to Conformal Field Theory (Springer, Berlin Heidelberg, 2008)

    MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. J.L. Lagrange, Essai d’une nouvelle methode pour determiner les maxima: et les minima des formules integrales indefinies. Miscellanea Taurinensia II 173 (1760–1761)

    Google Scholar 

  12. N. Drukker, S. Giombi, R. Ricci, D. Trancanelli, Supersymmetric Wilson loops on \(S^3\). JHEP 0805, 017 (2008). arXiv:0711.3226

    Article  ADS  MathSciNet  Google Scholar 

  13. H. Dorn, Wilson loops at strong coupling for curved contours with cusps. J. Phys. A49, 145402 (2016). arXiv:1509.00222

  14. L. Alvarez-Gaume, D.Z. Freedman, S. Mukhi, The background field method and the ultraviolet structure of the supersymmetric nonlinear sigma model. Ann. Phys. 134, 85 (1981)

    Article  ADS  Google Scholar 

  15. S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, vol. II (Interscience, New York, 1963)

    MATH  Google Scholar 

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

    Google Scholar 

  17. J. Polchinski, String Theory. Vol. 1: An Introduction to the Bosonic String (Cambridge University Press, Cambridge, 2007)

    Google Scholar 

  18. A.R. Kavalov, I.K. Kostov, A.G. Sedrakian, Dirac and weyl fermion dynamics on two-dimensional surface. Phys. Lett. B 175, 331 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  19. A. Sedrakian, R. Stora, Dirac and weyl fermions coupled to two-dimensional surfaces: determinants. Phys. Lett. B 188, 442 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  20. F. Langouche, H. Leutwyler, Two-dimensional fermion determinants as Wess-Zumino actions. Phys. Lett. B 195, 56 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  21. F. Langouche, H. Leutwyler, Weyl fermions on strings embedded in three-dimensions. Phys. C 36, 473 (1987)

    Article  MathSciNet  Google Scholar 

  22. F. Langouche, H. Leutwyler, Anomalies generated by extrinsic curvature. Phys. C 36, 479 (1987)

    Article  MathSciNet  Google Scholar 

  23. D.R. Karakhanian, Induced Dirac operator and smooth manifold geometry

    Google Scholar 

  24. 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 

  25. 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 

  26. R. Kallosh, J. Rahmfeld, A. Rajaraman, Near horizon superspace. JHEP 9809, 002 (1998). arXiv:hep-th/9805217

    Google Scholar 

  27. 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 

  28. 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 

  29. J. Callan, G. Curtis, H.K. Lee, T. McLoughlin, J.H. Schwarz, I. Swanson et al., Quantizing string theory in \(AdS_5 \times S^5\): beyond the pp wave. Nucl. Phys. B 673, 3 (2003). arXiv:hep-th/0307032

    Article  ADS  MATH  Google Scholar 

  30. 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 

  31. 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 

  32. 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 

  33. P.B. Wiegmann, Extrinsic geometry of superstrings. Nucl. Phys. B 323, 330 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  34. K. Lechner, M. Tonin, The cancellation of world sheet anomalies in the \(D = 10\) Green-Schwarz heterotic string sigma model. Nucl. Phys. B 475, 535 (1996). arXiv:hep-th/9603093

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. 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 

  36. N. Drukker, B. Fiol, All-genus calculation of Wilson loops using D-branes. JHEP 0502, 010 (2005). arXiv:hep-th/0501109

    Article  ADS  MathSciNet  Google Scholar 

  37. 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 

  38. J.S. Dowker, R. Critchley, Effective lagrangian and energy momentum tensor in de sitter space. Phys. Rev. D 13, 3224 (1976)

    Article  ADS  Google Scholar 

  39. S.W. Hawking, Zeta function regularization of path integrals in curved space-time. Commun. Math. Phys. 55, 133 (1977)

    Article  ADS  MATH  Google Scholar 

  40. E. Elizalde, Ten Physical Applications of Spectral Zeta Functions (Springer, New York, 2012)

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  42. J. Schwinger, Casimir effect in source theory II. Lett. Math. Phys. 24, 59 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  44. P.B. Gilkey, The spectral geometry of a riemannian manifold. J. Differ. Geom. 10, 601 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  45. P.B. Gilkey, Invariance Theory, the Heat Equation and the Atiyah-Singer Index Theorem (CRC Press, Boca Raton, 1995)

    MATH  Google Scholar 

  46. D. Friedan, Introduction to Polyakov’s string theory, in Les Houches Summer School in Theoretical Physics: Recent Advances in Field Theory and Statistical Mechanics Les Houches, August 2-September 10 (France, 1982)

    Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Edoardo Vescovi

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Vescovi, E. (2017). Geometric Properties of Semiclassically Quantized Strings. 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_3

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