Liouville theory and uniformization of four-punctured sphere

Hadasz, Leszek; Jaskólski, Zbigniew

doi:10.1063/1.2234272

Cited by 29 publications

(39 citation statements)

References 15 publications

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“…Motivations to study classical blocks were, for a long time, mainly confined to applications in pure mathematics, in particular, to the celebrated uniformization problem of Riemann surfaces [28,29] which is closely related to the monodromy problem for certain ordinary differential equations [30,31,32,33,34,35,36,37,38]. 10 The importance of the classical blocks is not only limited to the uniformization theorem, but gives also information about the solution of the Liouville equation on surfaces with punctures.…”

Section: Introductionmentioning

confidence: 99%

Classical irregular block, N $$ \mathcal{N} $$ = 2 pure gauge theory and Mathieu equation

Piątek

Pietrykowski

2014

J. High Energ. Phys.

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Combining the semiclassical/Nekrasov-Shatashvili limit of the AGT conjecture and the Bethe/gauge correspondence results in a triple correspondence which identifies classical conformal blocks with twisted superpotentials and then with Yang-Yang functions. In this paper the triple correspondence is studied in the simplest, yet not completely understood case of pure SU p2q super-Yang-Mills gauge theory. A missing element of that correspondence is identified with the classical irregular block. Explicit tests provide a convincing evidence that such a function exists. In particular, it has been shown that the classical irregular block can be recovered from classical blocks on the torus and sphere in suitably defined decoupling limits of classical external conformal weights. These limits are "classical analogues" of known decoupling limits for corresponding quantum blocks. An exact correspondence between the classical irregular block and the SU p2q gauge theory twisted superpotential has been obtained as a result of another consistency check. The latter determines the spectrum of the 2-particle periodic Toda (sin-Gordon) Hamiltonian in accord with the Bethe/gauge correspondence. An analogue of this statement is found entirely within 2d CFT. Namely, considering the classical limit of the null vector decoupling equation for the degenerate irregular block a celebrated Mathieu's equation is obtained with an eigenvalue determined by the classical irregular block. As it has been checked this result reproduces a well known weak coupling expansion of Mathieu's eigenvalue. Finally, yet another new formulae for Mathieu's eigenvalue relating the latter to a solution of certain Bethe-like equation are found. 1

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Section: Introductionmentioning

confidence: 99%

Classical irregular block, N $$ \mathcal{N} $$ = 2 pure gauge theory and Mathieu equation

Piątek

Pietrykowski

2014

J. High Energ. Phys.

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“…The real-valued accessory parameter for C 0,4 can be found by taking the classical limit of the NVD equations obeyed by the physical Liouville 5-point function on the sphere with V α=− b 2 , cf. [7,8,52,53]. The latter is nothing but the projected correlation function discussed above integrated over the Liouville field theory spectrum ∆ = ∆ α , α = 1 2 Q + iR + with the structure constant C(∆ 3 , ∆ 2 , ∆ 1 ) identified as the Liouville 3-point function [7,54].…”

Section: Classical Limit Of Bpz Equation For the Degenerate Five-poinmentioning

confidence: 97%

Solving Heun's equation using conformal blocks

Piątek

Pietrykowski

2019

Nuclear Physics B

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It is known that the classical limit of the second order BPZ null vector decoupling equation for the simplest two 5-point degenerate spherical conformal blocks yields: (i) the normal form of the Heun equation with the complex accessory parameter determined by the 4-point classical block on the sphere, and (ii) a pair of the Floquet type linearly independent solutions. A key point in a derivation of the above result is the classical asymptotic of the 5-point degenerate blocks in which the so-called heavy and light contributions decouple. In the present work the semi-classical heavylight factorization of the 5-point degenerate conformal blocks is studied. In particular, a mechanism responsible for the decoupling of the heavy and light contributions is identified. Moreover, it is shown that the factorization property yields a practical method of computation of the Floquet type Heun's solutions. Finally, it should be stressed that tools analyzed in this work have a broad spectrum of applications, in particular, in the studies of spectral problems with the Heun class of potentials, sphere-torus correspondence in 2d CFT, the KdV theory, the connection problem for the Heun equation and black hole physics. These applications are main motivations for the present work. *

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“…Several conjectures and proposals [1,2,3,4,5], [6], [7,8,9], [10], [11] for the computation of such accessory parameters have been put forward e.g. taking the semiclassical limit of the quantum correlation functions in conformal theories, the 4-point function on the sphere and the 1-point function on the torus .…”

Section: Introductionmentioning

confidence: 99%

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

Menotti

2013

J. High Energ. Phys.

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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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Liouville theory and uniformization of four-punctured sphere

Cited by 29 publications

References 15 publications

Classical irregular block, N $$ \mathcal{N} $$ = 2 pure gauge theory and Mathieu equation

Classical irregular block, N $$ \mathcal{N} $$ = 2 pure gauge theory and Mathieu equation

Solving Heun's equation using conformal blocks

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

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