On Liouville's Equation, Accessory Parameters, and the Geometry of Teichmüller Space for Riemann Surfaces of Genus 0

Zograf, Peter; Takhtadzhyan, L. A.

doi:10.1070/sm1988v060n01abeh003160

Cited by 108 publications

(193 citation statements)

References 11 publications

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“…In the case of the parabolic singularities on n-punctured Riemann sphere a rigorous proof based on the theory of quasiconformal mappings was given by Zograf and Takhtajan [11]. Other proofs, valid both in the case of parabolic and general elliptic singularities, were proposed in [12] and [4].…”

Section: Dρdρmentioning

confidence: 99%

Liouville theory and uniformization of four-punctured sphere

Hadasz

Jaskólski

2006

Journal of Mathematical Physics

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Few years ago Zamolodchikov and Zamolodchikov proposed an expression for the 4-point classical Liouville action in terms of the 3-point actions and the classical conformal block [1]. In this paper we develop a method of calculating the uniformizing map and the uniformizing group from the classical Liouville action on n-punctured sphere and discuss the consequences of Zamolodchikovs conjecture for an explicit construction of the uniformizing map and the uniformizing group for the sphere with four punctures.

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Section: Dρdρmentioning

confidence: 99%

Liouville theory and uniformization of four-punctured sphere

Hadasz

Jaskólski

2006

Journal of Mathematical Physics

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“…(26) is more complicated than eq. (27), is more apt to give a larger lower bound on the convergence radius.…”

Section: The Harmonic Case: the Squarementioning

confidence: 99%

“…Using such a technique Keen, Rauch and Vaquez [19] found that the accessory parameter for the torus with one parabolic singularity (puncture) is a real-analytic functions of the modulus; in addition in [19] some numerical investigation of the accessory parameter was performed. Zograf and Takhtajan [27] treated the case of parabolic singularities on a Riemann surface of genus 0. The result of [19,27] is that the accessory parameters are real-analytic functions of the moduli.…”

Section: Introductionmentioning

confidence: 99%

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Accessory parameters for Liouville theory on the torus

Menotti

2012

J. High Energ. Phys.

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We give an implicit equation for the accessory parameter on the torus which is the necessary and sufficient condition to obtain the monodromy of the conformal factor. It is shown that the perturbative series for the accessory parameter in the coupling constant converges in a finite disk and give a rigorous lower bound for the radius of convergence. We work out explicitly the perturbative result to second order in the coupling for the accessory parameter and to third order for the one-point function. Modular invariance is discussed and exploited. At the non perturbative level it is shown that the accessory parameter is a continuous function of the coupling in the whole physical region and that it is analytic except at most a finite number of points. We also prove that the accessory parameter as a function of the modulus of the torus is continuous and real-analytic except at most for a zero measure set. Three soluble cases in which the solution can be expressed in terms of hypergeometric functions are explicitly treated.

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“…Remarkably, in considering the tree level of (4) one uses [1][2][3] the well-known relation between the accessory parameters and the classical Liouville action [4] …”

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confidence: 99%

Quantum Riemann Surfaces, 2-D Gravity and the Geometrical Origin of Minimal Models

Matone

1994

Mod. Phys. Lett. A

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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 ≤ 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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On Liouville's Equation, Accessory Parameters, and the Geometry of Teichmüller Space for Riemann Surfaces of Genus 0

Cited by 108 publications

References 11 publications

Liouville theory and uniformization of four-punctured sphere

Liouville theory and uniformization of four-punctured sphere

Accessory parameters for Liouville theory on the torus

Quantum Riemann Surfaces, 2-D Gravity and the Geometrical Origin of Minimal Models

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