Polyakov’s formulation of $2d$ bosonic string theory

Colin Guillarmou; Rémi Rhodes; Vincent Vargas

doi:10.1007/s10240-019-00109-6

Article

Polyakov’s formulation of

2 d

bosonic string theory

Colin Guillarmou ¹; Rémi Rhodes ¹; Vincent Vargas ¹

¹

Publications Mathématiques de l'IHÉS, Volume 130 (2019), pp. 111-185

Received: 2017-03-29
Accepted: 2019-05-27
Online First: 2019-06-26
Published online: 2019-12-07

MR

DOI: 10.1007/s10240-019-00109-6

Author's affiliations:

Colin Guillarmou ¹; Rémi Rhodes ¹; Vincent Vargas ¹

¹

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     author = {Colin Guillarmou and R\'emi Rhodes and Vincent Vargas},
     title = {Polyakov{\textquoteright}s formulation of $2d$ bosonic string theory},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {111--185},
     year = {2019},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {130},
     doi = {10.1007/s10240-019-00109-6},
     mrnumber = {4028515},
     language = {en},
     url = {https://pmihes.centre-mersenne.org/articles/10.1007/s10240-019-00109-6/}
}

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Colin Guillarmou; Rémi Rhodes; Vincent Vargas. Polyakov’s formulation of $2d$ bosonic string theory. Publications Mathématiques de l'IHÉS, Volume 130 (2019), pp. 111-185. doi: 10.1007/s10240-019-00109-6

References
Cited by

[ARS1] P. Albin; F. Rochon; D. Sher Analytic torsion and $R$ -torsion of Witt representations on manifolds with cusps, Duke Math. J., Volume 167 (2018), pp. 1883-1950 | MR | Zbl | DOI

[ARS2] P. Albin, F. Rochon and D. Sher, Resolvent, heat kernel and torsion under degeneration to fibered cusps, Mem. Am. Math. Soc., in press. | MR

[Al] O. Alvarez Theory of strings with boundaries: Fluctuations, topology and quantum geometry, Nucl. Phys. B, Volume 216 (1983), pp. 125-184 | MR | DOI

[ADJ] J. Ambjørn; B. Durhuus; T. Jonsson Quantum Geometry: a statistical field theory approach, 2005 | Zbl | MR

[ABB] J. Ambjørn; J. Barkley; T. Budd Roaming moduli space using dynamical triangulations, Nucl. Phys. B, Volume 858 (2012), pp. 267-292 | MR | Zbl | DOI

[BeKn] A.A. Belavin; V.G. Knizhnik Algebraic geometry and the geometry of quantum strings, Physics Letters B, Volume 168 (1986), pp. 201-206 | MR | Zbl | DOI

[BeCa] E. A. Bender; E. R. Canfield The asymptotic number of rooted maps on a surface, J. Comb. Theory, Ser. A, Volume 43 (1986), pp. 244-257 | MR | Zbl | DOI

[Be] N. Berestycki, An elementary approach of Gaussian multiplicative chaos, Electron. Commun. Probab., 22 (2017) | MR

[BiFe] A. Bilal and F. Ferrari, Multi-Loop Zeta function regularization and spectral cutoff in curved spacetime, | arXiv | Zbl

[Bo] D. Borthwick Spectral theory of infinite-area hyperbolic surfaces, 256, Birkhäuser Boston, Inc., Boston, 2007 (xii+355 pp.) | Zbl | MR

[Bu1] M. Burger Asymptotics of small eigenvalues of Riemann surfaces, Bull. Am. Meteorol. Soc., Volume 18 (1988), pp. 39-40 | MR | Zbl | DOI

[Bu2] M. Burger Small eigenvalues of Riemann surfaces and graphs, Math. Z., Volume 205 (1990), pp. 395-420 | MR | Zbl | DOI

[Cu] N. Curien A glimpse of the conformal structure of random planar maps, Commun. Math. Phys., Volume 333 (2015), pp. 1417-1463 | MR | Zbl | DOI

[Da] F. David Conformal Field Theories Coupled to $2 - D$ Gravity in the Conformal Gauge, Mod. Phys. Lett. A, Volume 3 (1988), pp. 1651-1656 | MR | DOI

[DKRV] F. David; A. Kupiainen; R. Rhodes; V. Vargas Liouville Quantum Gravity on the Riemann sphere, Commun. Math. Phys., Volume 342 (2016), pp. 869-907 | MR | Zbl | DOI

[DRV] F. David; R. Rhodes; V. Vargas Liouville Quantum Gravity on complex tori, J. Math. Phys., Volume 57 (2016) | MR | Zbl | DOI

[DhKu] E. D’Hoker; P. S. Kurzepa $2 - D$ quantum gravity and Liouville Theory, Mod. Phys. Lett. A, Volume 5 (1990), pp. 1411-1421 | MR | Zbl | DOI

[DhPh] E. D’Hoker; D. H. Phong Multiloop amplitudes for the bosonic Polyakov string, Nucl. Phys. B, Volume 269 (1986), pp. 205-234 | MR | DOI

[DhPh2] E. D’Hoker; D. H. Phong On determinants of Laplacians on Riemann surfaces, Commun. Math. Phys., Volume 104 (1986), pp. 537-545 | MR | Zbl | DOI

[DhPh3] E. D’Hoker; D. H. Phong The geometry of string perturbation theory, Rev. Mod. Phys., Volume 60 (1988), pp. 917-1065 | MR | DOI

[dFMS] P. Di Francesco; P. Mathieu; D. Senechal Conformal Field Theory, Springer, Berlin, 1997 | Zbl | MR | DOI

[DiKa] J. Distler; H. Kawai Conformal Field Theory and $2 - D$ Quantum Gravity or Who’s Afraid of Joseph Liouville?, Nucl. Phys. B, Volume 321 (1989), pp. 509-517 | MR | DOI

[Du1] J. Dubédat SLE and the Free Field: partition functions and couplings, J. Am. Math. Soc., Volume 22 (2009), pp. 995-1054 | MR | Zbl | DOI

[DuSh] B. Duplantier; S. Sheffield Liouville Quantum Gravity and KPZ, Invent. Math., Volume 185 (2011), pp. 333-393 | MR | Zbl | DOI

[DRSV1] B. Duplantier; R. Rhodes; S. Sheffield; V. Vargas Critical Gaussian multiplicative chaos: convergence of the derivative martingale, Ann. Probab., Volume 42 (2014), pp. 1769-1808 | MR | Zbl | DOI

[DRSV2] B. Duplantier; R. Rhodes; S. Sheffield; V. Vargas Renormalization of Critical Gaussian Multiplicative Chaos and KPZ formula, Commun. Math. Phys., Volume 330 (2014), pp. 283-330 | Zbl | MR | DOI

[DMS] B. Duplantier, J. Miller and S. Sheffield, Liouville quantum gravity as a mating of trees, | arXiv

[FKZ] F. Ferrari; S. Klevtsov; S. Zelditch Random Kähler metrics, Nucl. Phys. B, Volume 869 (2012), pp. 89-110 | Zbl | MR | DOI

[Ga] K. Gawȩdzki Lectures on conformal field theory, Quantum fields and strings: A course for mathematicians, Vols. 1, 2 (1999), pp. 727-805 | MR | Zbl

[GuZw] L. Guillopé; M. Zworski Upper bounds on the number of resonances for non-compact Riemann surfaces, J. Funct. Anal., Volume 129 (1995), pp. 364-389 | MR | Zbl | DOI

[HRV] Y. Huang; R. Rhodes; V. Vargas Liouville quantum field theory in the unit disk, Ann. Inst. Henri Poincaré Probab. Stat., Volume 54 (2018), pp. 1694-1730 | MR | Zbl | DOI

[Ji] L. Ji The asymptotic behavior of Green’s functions for degenerating hyperbolic surfaces, Math. Z., Volume 212 (1993), pp. 375-394 | MR | Zbl | DOI

[Ka] J-P. Kahane Sur le chaos multiplicatif, Ann. Sci. Math. Qué., Volume 9 (1985), pp. 105-150 | MR | Zbl

[Kleb] I. Klebanov, String theory in two dimensions, | arXiv

[KlZe] S. Klevtsov; S. Zelditch Heat kernel measures on random surfaces, Adv. Theor. Math. Phys., Volume 20 (2016), pp. 135-164 | MR | Zbl | DOI

[KPZ] V. G. Knizhnik; A. M. Polyakov; A. B. Zamolodchikov Fractal structure of $2 D$ -quantum gravity, Mod. Phys. Lett. A, Volume 3 (1988), pp. 819-826 | MR | DOI

[MaMi] N. E. Mavromatos; J. L. Miramontes Regularizing the functional integral in $2 D$ -quantum gravity, Mod. Phys. Lett. A, Volume 04 (1989), pp. 1847-1853 | MR | DOI

[MeZh] Richard Melrose; Xuwen Zhu Boundary Behaviour of Weil–Petersson and Fibre Metrics for Riemann Moduli Spaces, International Mathematics Research Notices, Volume 2019 (2017), pp. 5012-5065 | MR | DOI

[Mi] G. Miermont Tessellations of random maps of arbitrary genus, Ann. Sci. Ec. Norm. Super., Volume 42 (2009), pp. 725-781 | MR | Zbl | Numdam | DOI

[OPS] B. Osgood; R. Phillips; P. Sarnak Extremals of determinants of Laplacians, J. Funct. Anal., Volume 80 (1988), pp. 148-211 | MR | Zbl | DOI

[Pa] S. J. Patterson The Selberg zeta function of a Kleinian group, Number Theory, Trace Formulas and Discrete Groups, Academic Press, Boston, 1989, pp. 409-441 | MR | Zbl

[Pol] J. Polchinski, What is string theory? Lectures presented at the 1994 Les Houches Summer School “Fluctuating Geometries in Statistical Mechanics and Field Theory.” | arXiv | MR

[Po] A. M. Polyakov Quantum geometry of bosonic strings, Phys. Lett. B, Volume 103 (1981), p. 207 | MR | DOI

[RaSi] D. B. Ray; I. M. Singer $R$ -torsion and the Laplacian on Riemannian manifolds, Adv. Math., Volume 7 (1971), pp. 145-210 | MR | Zbl | DOI

[RhVa1] R. Rhodes; V. Vargas Gaussian multiplicative chaos and applications: a review, Probab. Surv., Volume 11 (2014), pp. 315-392 | MR | Zbl | DOI

[RhVa2] R. Rhodes and V. Vargas, Lecture notes on Gaussian multiplicative chaos and Liouville Quantum Gravity, in Les Houches summer school proceedings, | arXiv

[RoVa] R. Robert; V. Vargas Gaussian multiplicative chaos revisited, Ann. Probab., Volume 38 (2010), pp. 605-631 | MR | Zbl | DOI

[Sa] P. Sarnak Determinants of Laplacians, Commun. Math. Phys., Volume 110 (1987), pp. 113-120 | MR | Zbl | DOI

[SWY] R. Schoen; S. Wolpert; S. T. Yau Geometric bounds on low eigenvalues of a compact Riemann surface, Geometry of the Laplace operator, Proc. Sympos. Pure Math, 36, AMS, Providence, 1980, pp. 279-285 | MR | Zbl | DOI

[Sc] M. Schulze On the resolvent of the Laplacian on functions for degenerating surfaces of finite geometry, J. Funct. Anal., Volume 236 (2006), pp. 120-160 | MR | Zbl | DOI

[Sha] Alexander Shamov On Gaussian multiplicative chaos, Journal of Functional Analysis, Volume 270 (2016), pp. 3224-3261 | MR | Zbl | DOI

[She] S. Sheffield Gaussian free fields for mathematicians, Probab. Theory Relat. Fields, Volume 139 (2007), pp. 521-541 | MR | Zbl | DOI

[TaTe] L. Takhtajan; L-P. Teo Quantum Liouville Theory in the Background Field Formalism I. Compact Riemann Surfaces, Commun. Math. Phys., Volume 268 (2006), pp. 135-197 | MR | Zbl | DOI

[Tr] A. J. Tromba Teichmüller theory in Riemannian Geometry, ETH Zürich, Birkhäuser Verlag, Basel, 1992 | Zbl | MR | DOI

[Wo1] S. Wolpert On the Weil–Petersson geometry of the moduli space of curves, Am. J. Math., Volume 107 (1985), pp. 969-997 | MR | Zbl | DOI

[Wo2] S. Wolpert Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces, Commun. Math. Phys., Volume 112 (1987), pp. 283-315 | MR | Zbl | DOI

[Wo3] S. Wolpert The hyperbolic metric and the geometry of the universal curve, J. Differ. Geom., Volume 31 (1990), pp. 417-472 | MR | Zbl | DOI

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