[Paper Review] Classical conformal blocks and isomonodromic deformations

J. Teschner|arXiv (Cornell University)|Jan 1, 2017

Black Holes and Theoretical Physics9 references12 citations

TL;DR

This paper establishes that the classical limit of Virasoro conformal blocks in 2D CFT corresponds to the generating function of a canonical coordinate transformation between two Darboux coordinate systems—Hamiltonian and complex Fenchel-Nielsen—on the moduli space of flat SL(2,C)-connections. It demonstrates that conformal field theory quantizes the isomonodromic deformation problem underlying the Garnier system, with classical conformal blocks arising as generating functions for monodromy data in the c→∞ limit.

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.

Motivation & Objective

• To characterize the classical limit (c→∞) of Virasoro conformal blocks in terms of isomonodromic deformations.
• To identify the classical conformal blocks as generating functions for canonical coordinate transformations between Hamiltonian and complex Fenchel-Nielsen coordinates on the moduli space of flat connections.
• To clarify the role of conformal field theory as a quantization of the isomonodromic deformation problem.
• To establish a precise correspondence between the semiclassical limit of CFT and the geometry of the Garnier system via monodromy data.

Proposed method

• Derives the classical limit of Virasoro conformal blocks using the isomonodromic deformation problem for second-order ODEs with regular singularities.
• Uses the Garnier system as the Hamiltonian formulation of isomonodromic deformations, with coordinates (u,v) as Darboux coordinates.
• Introduces complex Fenchel-Nielsen coordinates (l,k) parameterizing monodromy data, derived from pants decompositions of Riemann surfaces.
• Shows that the classical conformal block generating function W(l,z) equals the real part of a holomorphic function Wu(z,¯z), which satisfies ∂zr Re(Wu) = Hr.
• Establishes that the Liouville action functional SL(z,¯z) coincides with Re(Wu) up to a constant, linking CFT to Teichmüller theory.
• Constructs a quantum deformation of the Garnier system via b-deformation of the Hamiltonians, showing that the resulting equations reproduce the null vector decoupling equations of Virasoro conformal blocks.

Experimental results

Research questions

• RQ1 How does the classical limit of Virasoro conformal blocks relate to the isomonodromic deformation problem?
• RQ2 What is the geometric meaning of the classical conformal block as a generating function in the context of moduli spaces of flat connections?
• RQ3 How are the Hamiltonian and complex Fenchel-Nielsen coordinates related on the moduli space of monodromy data?
• RQ4 What is the role of the Liouville action in the semiclassical limit of 2D CFT?
• RQ5 How does conformal field theory emerge as a quantization of the classical Garnier system?

Key findings

• The classical limit of Virasoro conformal blocks is identified as the generating function for a canonical transformation between Hamiltonian coordinates (u,v) and complex Fenchel-Nielsen coordinates (l,k) on the moduli space of flat SL(2,C)-connections.
• The function W(l,z) generating the coordinate change satisfies ∂zr Re(Wu(z,¯z)) = Hr, where Hr are the Hamiltonians of the Garnier system.
• The Liouville action functional SL(z,¯z) coincides with Re(Wu(z,¯z)) up to a constant, establishing a direct link between CFT and Teichmüller theory.
• The classical conformal blocks are shown to be the generating function for the wave function in the Fenchel-Nielsen representation, with the wave function Φ(l) related to the conformal block via a Fourier-type transform.
• The quantum deformation of the Garnier system, with b-deformed Hamiltonians, reproduces the null vector decoupling equations of Virasoro conformal blocks, confirming that CFT quantizes the classical isomonodromic problem.
• The classical limit of Verlinde loop operators is shown to complete the duality between the Hamiltonian and Fenchel-Nielsen representations, providing a full characterization of classical conformal blocks as generating functions.