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Register a new userHyperbolic deformation of the strip-equation and the accessory parameters for the torus
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Authors
Pietro Menotti
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By applying an hyperbolic deformation to the uniformization problem for the infinite strip, we give a method for computing the accessory parameter for the torus with one source as an expansion in the modular parameter q. At O(q^0) we obtain the same equation for the accessory parameter and the same value of the semiclassical action as the one obtained from the b -> 0 limit of the quantum one point function. The procedure can be carried over to the full O(q^2) or even higher order corrections although the procedure becomes somewhat complicated. Here we compute to order q^2 the correction to the weight parameter intervening in the conformal factor and it is shown that the unwanted contribution O(q) to the accessory parameter equation cancel exactly.
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We consider the problem of the real analytic dependence of the accessory parameters of Liouville theory on the moduli of the problem, for general elliptic singularities. We give a simplified proof of the almost everywhere real analyticity in the case of a single accessory parameter as it occurs e.g. in the sphere topology with four sources or for the torus topology with a single source by using only the general analyticity properties of the solution of the auxiliary equation. We deal then the case of two accessory parameters. We use the obtained result for a single accessory parameter to derive rigorous properties of the projection of the problem on lower dimensional planes. We derive the real analyticity result for two accessory parameters under an assumption of irreducibility.
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