On the tau function of the hypergeometric equation

Abstract

The monodromy map for a rank-two system of differential equations with three Fuchsian singularities is classically solved by the Kummer formulæ for Gauss' hypergeometric functions. We define the tau-function of such a system as the generating function of the extended monodromy symplectomorphism, using an idea recently developed. This formulation allows us to determine the dependence of the tau-function on the monodromy data. Using the explicit solution of the monodromy problem, the tau-function is then explicitly written in terms of Barnes $G$-function. In particular, if the Fuchsian singularities are placed to $0$, $1$ and $\infty$, this gives the structure constants of the asymptotical formula of Iorgov-Gamayun-Lisovyy for solutions of Painlevé VI equation.

arXiv comment15 pages, 1 figure

References in corpus (13)

• Structure Constants and Conformal Bootstrap in Liouville Field Theory
• Two and three-point functions in Liouville theory
• On the Liouville three-point function
• Symplectic structure of the moduli space of flat connections on a Riemann surface
• Symplectic Manifolds and Isomonodromic Deformations
• On Correlation Functions for Non-critical Strings with c<1 but d>1
• Quasi-Hamiltonian Geometry of Meromorphic Connections
• Isomonodromic tau-functions from Liouville conformal blocks
• Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions
• Monodromy dependence and connection formulae for isomonodromic tau functions
• The dependence on the monodromy data of the isomonodromic tau function
• Zeros of the Jimbo, Miwa, Ueno tau function
• Trinion Conformal Blocks from Topological strings

Cited by in corpus (1)

• Monodromy dependence and symplectic geometry of isomonodromic tau functions on the torus