Connection problem for the sine-Gordon/Painlevé III tau function and irregular conformal blocks

Abstract

The short-distance expansion of the tau function of the radial sine-Gordon/Painlevé III equation is given by a convergent series which involves irregular $c=1$ conformal blocks and possesses certain periodicity properties with respect to monodromy data. The long-distance irregular expansion exhibits a similar periodicity with respect to a different pair of coordinates on the monodromy manifold. This observation is used to conjecture an exact expression for the connection constant providing relative normalization of the two series. Up to an elementary prefactor, it is given by the generating function of the canonical transformation between the two sets of coordinates.

18 pages, 1 figure

References in corpus (5)

• Classical Conformal Blocks and Painleve VI
• Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function
• Isomonodromic tau-functions from Liouville conformal blocks
• Bilinear equations on Painleve tau functions from CFT
• Painlevé VI connection problem and monodromy of c=1 conformal blocks

Cited by in corpus (25)

• On Painlevé/gauge theory correspondence
• Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions
• Non-perturbative approaches to the quantum Seiberg-Witten curve
• Monodromy dependence and connection formulae for isomonodromic tau functions
• Irregular conformal blocks and connection formulae for Painlevé V functions
• Black hole perturbation theory and multiple polylogarithms
• Q-deformed Painleve tau function and q-deformed conformal blocks
• Irregular conformal blocks, with an application to the fifth and fourth Painlevé equations
• On the analysis of incomplete spectra in random matrix theory through an extension of the Jimbo-Miwa-Ueno differential
• Analyticity of Nekrasov Partition Functions
• On determinant representation and integrability of Nekrasov functions
• Exact relativistic Toda chain eigenfunctions from Separation of Variables and gauge theory
• On the connection problem for Painlevé I
• Tracy-Widom distributions in critical unitary random matrix ensembles and the coupled Painlevé II system
• Gap probability of the circular unitary ensemble with a Fisher-Hartwig singularity and the coupled Painlevé V system
• Quantum spectral problems and isomonodromic deformations
• Spectral determinants and quantum theta functions
• Instantons to the people: the power of one-form symmetries
• Connection problem for the tau-function of the Sine-Gordon reduction of Painlevé-III equation via the Riemann-Hilbert approach
• On self-similar solutions of the vortex filament equation
• Massive vector field perturbations in the Schwarzschild spacetime from supersymmetric gauge theory
• Short distance asymptotics for a generalized two-point scaling function in the two-dimensional Ising model
• Symplectic aspects of the tt*-Toda equations
• Painlevé Kernels and Surface Defects at Strong Coupling
• AGT correspondence, (q-)Painlevè equations and matrix models