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

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

1993-09-17

arxiv.org/abs/hep-th/9309096v3

Mod.Phys.Lett. A9 (1994) 2871-2878

Physics

High Energy Physics

High Energy Physics - Theory

DFPD/93/TH/62. Remarks on the d=1 barrier and references added

Scientific paper

10.1142/S0217732394002719

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 $\Sigma_{0,m+n}$ with both parabolic and elliptic points. The identification of the classical limit of the conformal Ward identity with the Fuchsian projective connection on $\Sigma_{0,m+n}$ 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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