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Monodromic vs geodesic computation of Virasoro classical conformal blocks

K. B. Alkalaev, V. A. Belavin

TL;DR

This work computes the 5-point Virasoro classical conformal block in a heavy–light–superlight setup using the monodromy method and demonstrates its exact match with a bulk geodesic computation in AdS$_3$/CFT$_2$ up to third order in the superlight expansion. The authors develop a perturbative scheme around a known 4-point seed to solve the monodromic equations and compare the resulting accessory parameters with bulk angular momenta and geodesic actions, establishing a common root between boundary and bulk descriptions. They provide explicit formulas for the block and the bulk action up to third order, show how the two approaches are related by a precise mapping between boundary punctures and bulk attachment points, and discuss the implications for the AdS/CFT correspondence and Liouville theory’s role in encoding the semiclassical dynamics. The results reinforce a strong, dual description of Liouville-type dynamics in the semiclassical regime and clarify how multi-particle geodesic configurations encode boundary conformal data in the heavy-light limit.

Abstract

We compute 5-point classical conformal blocks with two heavy, two light, and one superlight operator using the monodromy approach up to third order in the superlight expansion. By virtue of the AdS/CFT correspondence we show the equivalence of the resulting expressions to those obtained in the bulk computation for the corresponding geodesic configuration.

Monodromic vs geodesic computation of Virasoro classical conformal blocks

TL;DR

This work computes the 5-point Virasoro classical conformal block in a heavy–light–superlight setup using the monodromy method and demonstrates its exact match with a bulk geodesic computation in AdS/CFT up to third order in the superlight expansion. The authors develop a perturbative scheme around a known 4-point seed to solve the monodromic equations and compare the resulting accessory parameters with bulk angular momenta and geodesic actions, establishing a common root between boundary and bulk descriptions. They provide explicit formulas for the block and the bulk action up to third order, show how the two approaches are related by a precise mapping between boundary punctures and bulk attachment points, and discuss the implications for the AdS/CFT correspondence and Liouville theory’s role in encoding the semiclassical dynamics. The results reinforce a strong, dual description of Liouville-type dynamics in the semiclassical regime and clarify how multi-particle geodesic configurations encode boundary conformal data in the heavy-light limit.

Abstract

We compute 5-point classical conformal blocks with two heavy, two light, and one superlight operator using the monodromy approach up to third order in the superlight expansion. By virtue of the AdS/CFT correspondence we show the equivalence of the resulting expressions to those obtained in the bulk computation for the corresponding geodesic configuration.

Paper Structure

This paper contains 16 sections, 70 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Classical conformal block and monodromy problem
  3. Heavy-light approximation
  4. Contour integration and monodromy
  5. Solving the monodromic equations
  6. Bulk geodesic approach
  7. Monodromy vs geodesic approach
  8. 4-point case
  9. 5-point case
  10. Exact relations.
  11. Expanding over the root.
  12. Classical conformal block vs multi-particle action
  13. Classical conformal block
  14. Multi-particle action
  15. Conclusion
  16. ...and 1 more sections

Figures (1)

  • Figure 1: The $5$-point classical heavy-light conformal block. Two bold lines on the right represent heavy operators. As usual the projective invariance is used to fix three insertion positions as $z_1 =0$, $z_4= 1$, $z_5 = \infty$.