Abstract
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.
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Bilal, A., de Lacroix, C. 2D gravitational Mabuchi action on Riemann surfaces with boundaries. J. High Energ. Phys. 2017, 154 (2017). https://doi.org/10.1007/JHEP11(2017)154
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DOI: https://doi.org/10.1007/JHEP11(2017)154