Classical conformal blocks and isomonodromic deformations

TL;DR

This paper links the classical asymptotics of Virasoro conformal blocks on punctured spheres to Garnier systems, revealing how conformal field theory quantizes isomonodromic deformation problems.

Contribution

It provides a complete characterization of the leading classical asymptotics of Virasoro conformal blocks via Garnier systems and clarifies their relation to quantization of isomonodromic deformations.

Findings

Classical asymptotics described by Garnier systems

Conformal blocks relate to monodromy preserving deformations

Quantization of isomonodromic problems explained

Abstract

The leading classical asymptotics of Virasoro conformal blocks on the Riemann sphere with n generic and n-3 "heavy" degenerate field insertions can be described in terms of the geometry of Garnier system describing the monodromy preserving deformations of second order Fuchsian differential equations on an n-punctured sphere. This allows us to characterise the leading classical asymptotics of Virasoro conformal blocks completely, and to clarify in which sense conformal field theory represents a quantisation of the isomonodromic deformation problem.