Abstract
Abstract The correspondence between the semiclassical limit of the DOZZ quantum Liouville theory and the Nekrasov-Shatashvili limit of the $ \mathcal{N} = 2 $ (Ω-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.
Highlights
Instanton sector of the effective twisted superpotential [12].1 The latter quantity determines the low energy effective dynamics of the two-dimensional gauge theories restricted to the Ωbackground
A result of that duality is that the twisted superpotential is identified with the Yang’s functional [21] which describes the spectrum of the corresponding quantum integrable system
Afterwards, we briefly review the so-called geometric approach to quantum Liouville theory originally proposed by Polyakov and further developed by Takhtajan [29, 31, 33, 34]
Summary
Of the fundamental solutions (ψ1, ψ2) of the eq (2.1) with Wronskian ψ1ψ2′ − ψ1′ ψ2 = 1 and SL(2, R) monodromy with respect to all punctures It is a well known fact [28, 36] that ρ : C0,n ∋ z −→ τ (z) ∈ H is a multi-valued map from the n-punctured Riemann sphere to the upper half plane H = {τ ∈ C : Im τ > 0} with branch points z1, . The monodromy problem for the Fuchs equation (2.1) formulated above has been proposed by Poincare in order to construct the so-called uniformization map in the case of the n-punctured sphere with parabolic singularities. The inverse map ρ can be computed if the appropriate solution of the Liouville equation is available or, equivalently, if the accessory parameters in the Fuchs equation (2.1). The result (2.11) holds when the singularities are elliptic and, as before, the problem of calculating the map ρ is equivalent to that of finding the solution of the Liouville equation or solving the monodromy problem for the Fuchs equation (2.1)
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