Journal of High Energy Physics

, 2017:154 | Cite as

2D gravitational Mabuchi action on Riemann surfaces with boundaries

Open Access
Regular Article - Theoretical Physics


We study the gravitational action induced by coupling two-dimensional non-conformal, massive matter to gravity on a Riemann surface with boundaries. A small-mass expansion gives back the Liouville action in the massless limit, while the first-order mass correction allows us to identify what should be the appropriate generalization of the Mabuchi action on a Riemann surface with boundaries. We provide a detailed study for the example of the cylinder. Contrary to the case of manifolds without boundary, we find that the gravitational Lagrangian explicitly depends on the space-point, via the geodesic distances to the boundaries, as well as on the modular parameter of the cylinder, through an elliptic θ-function.


2D Gravity Models of Quantum Gravity 


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  1. [1]
    A.M. Polyakov, Quantum geometry of bosonic strings, Phys. Lett. B 103 (1981) 207 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    T. Mabuchi, K-energy maps integrating Futaki invariants, Tôhuku Math. J. 38 (1986) 575.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    T. Mabuchi, Some symplectic geometry on compact Kähler manifolds, Osaka J. Math. 24 (1987) 227.MathSciNetMATHGoogle Scholar
  4. [4]
    S. Semmes, Complex Monge-Ampère and symplectic manifolds, Amer. J. Math. 114 (1992) 495.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    D.H. Phong and J. Sturm, Lectures on stability and constant scalar curvature, arXiv:0801.4179.
  6. [6]
    F. Ferrari, S. Klevtsov and S. Zelditch, Gravitational actions in two dimensions and the Mabuchi functional, Nucl. Phys. B 859 (2012) 341 [arXiv:1112.1352] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    A. Bilal and L. Leduc, 2D quantum gravity on compact Riemann surfaces with non-conformal matter, JHEP 01 (2017) 089 [arXiv:1606.01901] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    F. Ferrari, S. Klevtsov and S. Zelditch, Random Kähler metrics, Nucl. Phys. B 869 (2013) 89 [arXiv:1107.4575] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  9. [9]
    S. Klevtsov and S. Zelditch, Stability and integration over Bergman metrics, JHEP 07 (2014) 100 [arXiv:1404.0659] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    A. Bilal, F. Ferrari and S. Klevtsov, 2D quantum gravity at one loop with Liouville and Mabuchi actions, Nucl. Phys. B 880 (2014) 203 [arXiv:1310.1951] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    A. Bilal and L. Leduc, Liouville and Mabuchi quantum gravity at two and three loops, to appear.Google Scholar
  12. [12]
    C. de Lacroix, H. Erbin and E.E. Svanes, Mabuchi spectrum from the minisuperspace, Phys. Lett. B 758 (2016) 186 [arXiv:1511.06150] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  13. [13]
    C. de Lacroix, H. Erbin and E. Svanes, Minisuperspace computation of the Mabuchi spectrum, to appear.Google Scholar
  14. [14]
    S. Klevtsov, Random normal matrices, Bergman kernel and projective embeddings, JHEP 01 (2014) 133 [arXiv:1309.7333] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    T. Can, M. Laskin and P. Wiegmann, Fractional quantum Hall effect in a curved space: gravitational anomaly and electromagnetic response, Phys. Rev. Lett. 113 (2014) 046803 [arXiv:1402.1531] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    F. Ferrari and S. Klevtsov, FQHE on curved backgrounds, free fields and large-N, JHEP 12 (2014) 086 [arXiv:1410.6802] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    S. Klevtsov and P. Wiegmann, Geometric adiabatic transport in quantum Hall states, Phys. Rev. Lett. 115 (2015) 086801 [arXiv:1504.07198] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    S. Klevtsov, X. Ma, G. Marinescu and P. Wiegmann, Quantum Hall effect and Quillen metric, Commun. Math. Phys. 349 (2017) 819 [arXiv:1510.06720] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    S. Klevtsov, Geometry and large-N limits in Laughlin states, Trav. Math. 24 (2016) 63 [arXiv:1608.02928] [INSPIRE].MathSciNetMATHGoogle Scholar
  20. [20]
    X.G. Wen and A. Zee, Shift and spin vector: new topological quantum numbers for the Hall fluids, Phys. Rev. Lett. 69 (1992) 953 [Erratum ibid. 69 (1992) 3000] [INSPIRE].
  21. [21]
    A. Bilal and F. Ferrari, Multi-loop zeta function regularization and spectral cutoff in curved spacetime, Nucl. Phys. B 877 (2013) 956 [arXiv:1307.1689] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    D.M. McAvity and H. Osborn, A DeWitt expansion of the heat kernel for manifolds with a boundary, Class. Quant. Grav. 8 (1991) 603 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    A. Erdelyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Bateman manuscript project, higher transcendental functions, volume 2, McGraw-Hill U.S.A., (1953).Google Scholar

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© The Author(s) 2017

Authors and Affiliations

  1. 1.Laboratoire de Physique Théorique de l’ École Normale SupérieurePSL Research University, CNRS, Sorbonne Universités, UPMC Paris 6Paris Cedex 05France

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