Liouville theory, N=2 gauge theories and accessory parameters

Abstract

The correspondence between the semiclassical limit of the DOZZ quantum Liouville theory and the Nekrasov-Shatashvili limit of the N = 2 (Omega-deformed) U(2) super-Yang-Mills theories is used to calculate the unknown accessory parameter of the Fuchsian uniformization of the 4-punctured sphere. The computation is based on the saddle point method. This allows to find an analytic expression for the N_f = 4, U(2) instanton twisted superpotential and, in turn, to sum up the 4-point classical block. It is well known that the critical value of the Liouville action functional is the generating function of the accessory parameters. This statement and the factorization property of the 4-point action allow to express the unknown accessory parameter as the derivative of the 4-point classical block with respect to the modular parameter of the 4-punctured sphere. It has been found that this accessory parameter is related to the sum of all rescaled column lengths of the so-called 'critical' Young diagram extremizing the instanton 'free energy'. It is shown that the sum over the 'critical' column lengths can be rewritten in terms of a contour integral in which the integrand is built out of certain special functions closely related to the ordinary Gamma function.

26 pages, published version

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• Classical conformal blocks
• Classical limit of irregular blocks and Mathieu functions
• On a singular Fredholm-type integral equation arising in N=2 super Yang-Mills theories
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• Classical Liouville Three-point Functions from Riemann-Hilbert Analysis
• Solving Heun's equation using conformal blocks
• On a path integral representation of the Nekrasov instanton partition function and its Nekrasov--Shatashvili limit
• The continuation method and the real analyticity of the accessory parameters: the general elliptic case
• The continuation method and the real analyticity of the accessory parameters: the parabolic case
• Solvable spectral problems from 2d CFT and N=2 gauge theories