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Isomonodromic tau-functions from Liouville conformal blocks

N. Iorgov, O. Lisovyy, J. Teschner

TL;DR

This work connects isomonodromic tau-functions to Liouville conformal blocks at $c=1$ by solving the Riemann-Hilbert problem via infinite linear combinations of Virasoro blocks. It develops an explicit construction, shows the emergence of commutative monodromy data in the $c\to1$ limit, and derives tau-functions as correlators in the Liouville/free-fermion framework. The approach reproduces Jimbo’s asymptotics for Painlevé VI and yields algebro-geometric Schlesinger solutions from Ashkin–Teller blocks, while outlining implications for $\mathcal{N}=2$ gauge theories and Verlinde operator quantization. Overall, it provides a bosonized, $c=1$ perspective on isomonodromic deformations, offering new tools to study tau-functions through Liouville conformal blocks.

Abstract

The goal of this note is to show that the Riemann-Hilbert problem to find multivalued analytic functions with $SL(2,\mathbb{C})$-valued monodromy on Riemann surfaces of genus zero with $n$ punctures can be solved by taking suitable linear combinations of the conformal blocks of Liouville theory at $c=1$. This implies a similar representation for the isomonodromic tau-function. In the case $n=4$ we thereby get a proof of the relation between tau-functions and conformal blocks discovered in \cite{GIL}. We briefly discuss a possible application of our results to the study of relations between certain $\mathcal{N}=2$ supersymmetric gauge theories and conformal field theory.

Isomonodromic tau-functions from Liouville conformal blocks

TL;DR

This work connects isomonodromic tau-functions to Liouville conformal blocks at by solving the Riemann-Hilbert problem via infinite linear combinations of Virasoro blocks. It develops an explicit construction, shows the emergence of commutative monodromy data in the limit, and derives tau-functions as correlators in the Liouville/free-fermion framework. The approach reproduces Jimbo’s asymptotics for Painlevé VI and yields algebro-geometric Schlesinger solutions from Ashkin–Teller blocks, while outlining implications for gauge theories and Verlinde operator quantization. Overall, it provides a bosonized, perspective on isomonodromic deformations, offering new tools to study tau-functions through Liouville conformal blocks.

Abstract

The goal of this note is to show that the Riemann-Hilbert problem to find multivalued analytic functions with -valued monodromy on Riemann surfaces of genus zero with punctures can be solved by taking suitable linear combinations of the conformal blocks of Liouville theory at . This implies a similar representation for the isomonodromic tau-function. In the case we thereby get a proof of the relation between tau-functions and conformal blocks discovered in \cite{GIL}. We briefly discuss a possible application of our results to the study of relations between certain supersymmetric gauge theories and conformal field theory.

Paper Structure

This paper contains 27 sections, 98 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Riemann-Hilbert problem
  3. Formulation of the Riemann Hilbert problem
  4. Trace coordinates
  5. Isomonodromic deformations and tau-function
  6. Chiral vertex operators and conformal blocks
  7. Chiral vertex operators
  8. Degenerate fields
  9. Braiding and fusion of degenerate fields
  10. Monodromy action on spaces of conformal blocks
  11. Geometrical set-up
  12. The algorithm
  13. Solving the Riemann-Hilbert problem
  14. The construction
  15. Existence of classical monodromies
  16. ...and 12 more sections

Figures (4)

  • Figure 1: A sphere with three holes. The arrows indicate our orientation conventions.
  • Figure 2: Basis of loops of $\pi_1(C_{0,4})$ and the decomposition $C_{0,4}=C_{0,3}^L\cup C_{0,3}^R$.
  • Figure 3: Labeling of pairs of pants for the conformal block $\mathcal{B}\left({m}\,|\,{p},{p}'\,|\,{z}\right)$.
  • Figure 4: Canonical homology basis on $\Sigma$.

Theorems & Definitions (1)

  • Remark 1