On the monodromy problem for the four-punctured sphere

Abstract

We consider the monodromy problem for the four-punctured sphere in which the character of one composite monodromy is fixed, by looking at the expansion of the accessory parameter in the modulus $x$ directly, without taking the limit of the quantum conformal blocks for infinite central charge. The integrals which appear in the expansion of the Volterra equation, involve products of two hypergeometric functions to first order and up to four hypergeometric functions to second order. It is shown that all such integrals can be computed analytically. We give the complete analytical evaluation of the accessory parameter to first and second order in the modulus. The results agree with the evaluation obtained by assuming the exponentiation hypothesis of the quantum conformal blocks in the limit of infinite central charge. Extension to higher orders is discussed.

14 pages LaTeX, 1 figure. Notation improved; Sec.4 extended to include the complete second order computation

References in corpus (4)

• Classical Conformal Blocks and Painleve VI
• Deforming SW curve
• Riemann-Hilbert treatment of Liouville theory on the torus
• Hyperbolic deformation of the strip-equation and the accessory parameters for the torus

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• Solving Heun's equation using conformal blocks
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