Skip to main content
SpringerLink
Log in

Polyakov’s formulation of \(2d\) bosonic string theory

  • Published:
Publications mathématiques de l'IHÉS Aims and scope Submit manuscript

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. P. Albin, F. Rochon and D. Sher, Analytic torsion and R-torsion of Witt representations on manifolds with cusps, Duke Math. J., 167 (2018), 1883–1950.

    MathSciNet  MATH  Google Scholar 

  2. P. Albin, F. Rochon and D. Sher, Resolvent, heat kernel and torsion under degeneration to fibered cusps, Mem. Am. Math. Soc., in press.

  3. O. Alvarez, Theory of strings with boundaries: Fluctuations, topology and quantum geometry, Nucl. Phys. B, 216 (1983), 125–184.

    MathSciNet  Google Scholar 

  4. J. Ambjorn, B. Durhuus and T. Jonsson, Quantum Geometry: a statistical field theory approach, Cambridge Monographs on Mathematical Physics, 2005.

    MATH  Google Scholar 

  5. J. Ambjorn, J. Barkley and T. Budd, Roaming moduli space using dynamical triangulations, Nucl. Phys. B, 858 (2012), 267–292.

    MathSciNet  MATH  Google Scholar 

  6. A. A. Belavin and V. G. Knizhnik, Algebraic geometry and the geometry of quantum strings, Phys. Lett. B, 168 (1986).

    MathSciNet  MATH  Google Scholar 

  7. E. A. Bender and E. R. Canfield, The asymptotic number of rooted maps on a surface, J. Comb. Theory, Ser. A, 43 (1986), 244–257.

    MathSciNet  MATH  Google Scholar 

  8. N. Berestycki, An elementary approach of Gaussian multiplicative chaos, Electron. Commun. Probab., 22 (2017)

  9. A. Bilal and F. Ferrari, Multi-Loop Zeta function regularization and spectral cutoff in curved spacetime, arXiv:1307.1689.

  10. D. Borthwick, Spectral theory of infinite-area hyperbolic surfaces, Progress in Mathematics, vol. 256, Birkhäuser Boston, Inc., Boston, 2007, xii+355 pp.

    MATH  Google Scholar 

  11. M. Burger, Asymptotics of small eigenvalues of Riemann surfaces, Bull. Am. Meteorol. Soc., 18 (1988), 39–40.

    MathSciNet  MATH  Google Scholar 

  12. M. Burger, Small eigenvalues of Riemann surfaces and graphs, Math. Z., 205 (1990), 395–420.

    MathSciNet  MATH  Google Scholar 

  13. N. Curien, A glimpse of the conformal structure of random planar maps, Commun. Math. Phys., 333 (2015), 1417–1463.

    MathSciNet  MATH  Google Scholar 

  14. F. David, Conformal Field Theories Coupled to 2-D Gravity in the Conformal Gauge, Mod. Phys. Lett. A, 3 (1988), 1651–1656.

    MathSciNet  Google Scholar 

  15. F. David, A. Kupiainen, R. Rhodes and V. Vargas, Liouville Quantum Gravity on the Riemann sphere, Commun. Math. Phys., 342 (2016), 869–907.

    MathSciNet  MATH  Google Scholar 

  16. F. David, R. Rhodes and V. Vargas, Liouville Quantum Gravity on complex tori, J. Math. Phys., 57, 022302 (2016).

    MathSciNet  MATH  Google Scholar 

  17. E. D’Hoker and P. S. Kurzepa, 2-D quantum gravity and Liouville Theory, Mod. Phys. Lett. A, 5 (1990), 1411–1421.

    MathSciNet  MATH  Google Scholar 

  18. E. D’Hoker and D. H. Phong, Multiloop amplitudes for the bosonic Polyakov string, Nucl. Phys. B, 269 (1986), 205–234.

    MathSciNet  Google Scholar 

  19. E. D’Hoker and D. H. Phong, On determinants of Laplacians on Riemann surfaces, Commun. Math. Phys., 104 (1986), 537–545.

    MathSciNet  MATH  Google Scholar 

  20. E. D’Hoker and D. H. Phong, The geometry of string perturbation theory, Rev. Mod. Phys., 60 (1988), 917–1065.

    MathSciNet  Google Scholar 

  21. P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Graduate Texts in Contemporary Physics, Springer, Berlin, 1997.

    MATH  Google Scholar 

  22. J. Distler and H. Kawai, Conformal Field Theory and 2-D Quantum Gravity or Who’s Afraid of Joseph Liouville? Nucl. Phys. B, 321 (1989), 509–517.

    Google Scholar 

  23. J. Dubédat, SLE and the Free Field: partition functions and couplings, J. Am. Math. Soc., 22 (2009), 995–1054.

    MathSciNet  MATH  Google Scholar 

  24. B. Duplantier and S. Sheffield, Liouville Quantum Gravity and KPZ, Invent. Math., 185 (2011), 333–393.

    MathSciNet  MATH  Google Scholar 

  25. B. Duplantier, R. Rhodes, S. Sheffield and V. Vargas, Critical Gaussian multiplicative chaos: convergence of the derivative martingale, Ann. Probab., 42 (2014), 1769–1808.

    MathSciNet  MATH  Google Scholar 

  26. B. Duplantier, R. Rhodes, S. Sheffield and V. Vargas, Renormalization of Critical Gaussian Multiplicative Chaos and KPZ formula, Commun. Math. Phys., 330 (2014), 283–330.

    MATH  Google Scholar 

  27. B. Duplantier, J. Miller and S. Sheffield, Liouville quantum gravity as a mating of trees, arXiv:1409.7055.

  28. F. Ferrari, S. Klevtsov and S. Zelditch, Random Kähler metrics, Nucl. Phys. B, 869 (2012), 89–110.

    MATH  Google Scholar 

  29. K. Gawedzki, Lectures on conformal field theory, in Quantum fields and strings: A course for mathematicians, Vols. 1, 2, Princeton, NJ, 1996/1997, pp. 727–805, Amer. Math. Soc., Providence, 1999.

    Google Scholar 

  30. L. Guillopé and M. Zworski, Upper bounds on the number of resonances for non-compact Riemann surfaces, J. Funct. Anal., 129 (1995), 364–389.

    MathSciNet  MATH  Google Scholar 

  31. Y. Huang, R. Rhodes and V. Vargas, Liouville quantum field theory in the unit disk, Ann. Inst. Henri Poincaré Probab. Stat., 54 (2018), 1694–1730.

    MathSciNet  MATH  Google Scholar 

  32. L. Ji, The asymptotic behavior of Green’s functions for degenerating hyperbolic surfaces, Math. Z., 212 (1993), 375–394.

    MathSciNet  MATH  Google Scholar 

  33. J-P. Kahane, Sur le chaos multiplicatif, Ann. Sci. Math. Qué., 9 (1985), 105–150.

    MathSciNet  MATH  Google Scholar 

  34. I. Klebanov, String theory in two dimensions, arXiv:hep-th/9108019.

  35. S. Klevtsov and S. Zelditch, Heat kernel measures on random surfaces, Adv. Theor. Math. Phys., 20 (2016), 135–164.

    MathSciNet  MATH  Google Scholar 

  36. V. G. Knizhnik, A. M. Polyakov and A. B. Zamolodchikov, Fractal structure of 2D-quantum gravity, Mod. Phys. Lett. A, 3 (1988), 819–826.

    Google Scholar 

  37. N. E. Mavromatos and J. L. Miramontes, Regularizing the functional integral in 2D-quantum gravity, Mod. Phys. Lett. A, 04 (1989), 1847–1853.

    Google Scholar 

  38. R. B. Melrose and X. Zhu, Boundary behaviour of Weil–Petersson and fiber metrics for Riemann moduli spaces, Int. Math. Res. Not., rnx264 (2017). https://doi.org/10.1093/imrn/rnx264.

    MathSciNet  Google Scholar 

  39. G. Miermont, Tessellations of random maps of arbitrary genus, Ann. Sci. Ec. Norm. Super., 42 (2009), 725–781.

    MathSciNet  MATH  Google Scholar 

  40. B. Osgood, R. Phillips and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal., 80 (1988), 148–211.

    MathSciNet  MATH  Google Scholar 

  41. S. J. Patterson, The Selberg zeta function of a Kleinian group, in Number Theory, Trace Formulas and Discrete Groups, pp. 409–441, Academic Press, Boston, 1989.

    Google Scholar 

  42. J. Polchinski, What is string theory? Lectures presented at the 1994 Les Houches Summer School “Fluctuating Geometries in Statistical Mechanics and Field Theory.” arXiv:hep-th/9411028.

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

    MathSciNet  Google Scholar 

  44. D. B. Ray and I. M. Singer, R-torsion and the Laplacian on Riemannian manifolds, Adv. Math., 7 (1971), 145–210.

    MathSciNet  MATH  Google Scholar 

  45. R. Rhodes and V. Vargas, Gaussian multiplicative chaos and applications: a review, Probab. Surv., 11 (2014), 315–392.

    MathSciNet  MATH  Google Scholar 

  46. R. Rhodes and V. Vargas, Lecture notes on Gaussian multiplicative chaos and Liouville Quantum Gravity, in Les Houches summer school proceedings, arXiv:1602.07323.

  47. R. Robert and V. Vargas, Gaussian multiplicative chaos revisited, Ann. Probab., 38 (2010), 605–631.

    MathSciNet  MATH  Google Scholar 

  48. P. Sarnak, Determinants of Laplacians, Commun. Math. Phys., 110 (1987), 113–120.

    MathSciNet  MATH  Google Scholar 

  49. R. Schoen, S. Wolpert and S. T. Yau, Geometric bounds on low eigenvalues of a compact Riemann surface, Geometry of the Laplace operator, in Proc. Sympos. Pure Math, vol. 36, pp. 279–285, AMS, Providence, 1980.

    Google Scholar 

  50. M. Schulze, On the resolvent of the Laplacian on functions for degenerating surfaces of finite geometry, J. Funct. Anal., 236 (2006), 120–160.

    MathSciNet  MATH  Google Scholar 

  51. A. Shamov, On Gaussian multiplicative chaos, J. Funct. Anal., 270, 3224–3261.

    MathSciNet  MATH  Google Scholar 

  52. S. Sheffield, Gaussian free fields for mathematicians, Probab. Theory Relat. Fields, 139 (2007), 521–541.

    MathSciNet  MATH  Google Scholar 

  53. L. Takhtajan and L-P. Teo, Quantum Liouville Theory in the Background Field Formalism I. Compact Riemann Surfaces, Commun. Math. Phys., 268 (2006), 135–197.

    MathSciNet  MATH  Google Scholar 

  54. A. J. Tromba, Teichmüller theory in Riemannian Geometry, Lectures in Math., ETH Zürich, Birkhäuser Verlag, Basel, 1992.

    MATH  Google Scholar 

  55. S. Wolpert, On the Weil–Petersson geometry of the moduli space of curves, Am. J. Math., 107 (1985), 969–997.

    MathSciNet  MATH  Google Scholar 

  56. S. Wolpert, Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces, Commun. Math. Phys., 112 (1987), 283–315.

    MathSciNet  MATH  Google Scholar 

  57. S. Wolpert, The hyperbolic metric and the geometry of the universal curve, J. Differ. Geom., 31 (1990), 417–472.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques, Université Paris-Sud, CNRS, 91400, Orsay, France

    Colin Guillarmou

  2. Aix Marseille Université, CNRS, Centrale Marseille, I2M, Marseille, France

    Rémi Rhodes

  3. DMA, U.M.R. 8553 CNRS, École Normale Superieure, 45 rue d’Ulm, 75230, Paris cedex 05, France

    Vincent Vargas

Authors
  1. Colin Guillarmou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rémi Rhodes
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vincent Vargas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rémi Rhodes.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guillarmou, C., Rhodes, R. & Vargas, V. Polyakov’s formulation of \(2d\) bosonic string theory. Publ.math.IHES 130, 111–185 (2019). https://doi.org/10.1007/s10240-019-00109-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10240-019-00109-6

Search

Navigation