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Quantum Riemann surfaces, 2D gravity and the geometrical origin of minimal models

M. Matone
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Abstract

Based on a recent paper by Takhtajan, we propose a formulation of 2D quantum gravity whose basic object is the Liouville action on the Riemann sphere $Σ_{0,m+n}$ with both parabolic and elliptic points. The identification of the classical limit of the conformal Ward identity with the Fuchsian projective connection on $Σ_{0,m+n}$ implies a relation between conformal weights and ramification indices. This formulation works for arbitrary $d$ and admits a standard representation only for $d\le 1$. Furthermore, it turns out that the integerness of the ramification number constrains $d=1-24/(n^2-1)$ that for $n=2m+1$ coincides with the unitary minimal series of CFT.

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M. Matone. 1994-02-15. Quantum Riemann surfaces, 2D gravity and the geometrical origin of minimal models. https://doi.org/10.1142/s0217732394002719

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