Hyperbolic deformation of the strip-equation and the accessory parameters for the torus

Abstract

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.

20 pages LaTeX, corrected typos

References in corpus (2)

• Differential equation for four-point correlation function in Liouville field theory and elliptic four-point conformal blocks
• Riemann-Hilbert treatment of Liouville theory on the torus

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