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From global to heavy-light: 5-point conformal blocks

K. B. Alkalaev, V. A. Belavin

TL;DR

This work advances the understanding of conformal blocks at large central charge by explicitly constructing the 5-point global block and validating its connection to heavy-light and linearized classical blocks via the Fitzpatrick–Kaplan–Walters framework. Using projection techniques and Casimir equations, it derives a two-variable Horn-hypergeometric representation and verifies two Casimir PDEs in the 5-point case, including two vacuum configurations. The heavy-light analysis maps the problem to a transformed global block in the $w$-plane and shows that the linearized classical block coefficients are precisely the homogeneous degree-1 terms of the heavy-light block, reproducing prior monodromy results. These results generalize the FKW method to higher-point blocks and lay groundwork for future closed-form constructions and holographic interpretations in AdS$_3$/CFT$_2$.

Abstract

We consider Virasoro conformal blocks in the large central charge limit. There are different regimes depending on the behavior of the conformal dimensions. The most simple regime is reduced to the global sl(2, C) conformal blocks while the most complicated one is known as the classical conformal blocks. Recently, Fitzpatrick, Kaplan, and Walters showed that the two regimes are related through the intermediate stage of the so-called heavy-light semiclassical limit. We study this idea in the particular case of the 5-point conformal block. To find the 5-point global block we use the projector technique and the Casimir operator approach. Furthermore, we discuss the relation between the global and the heavy-light limits and construct the heavy-light block from the global block. In this way we reproduce our previous results for the 5-point perturbative classical block obtained by means of the monodromy method.

From global to heavy-light: 5-point conformal blocks

TL;DR

This work advances the understanding of conformal blocks at large central charge by explicitly constructing the 5-point global block and validating its connection to heavy-light and linearized classical blocks via the Fitzpatrick–Kaplan–Walters framework. Using projection techniques and Casimir equations, it derives a two-variable Horn-hypergeometric representation and verifies two Casimir PDEs in the 5-point case, including two vacuum configurations. The heavy-light analysis maps the problem to a transformed global block in the -plane and shows that the linearized classical block coefficients are precisely the homogeneous degree-1 terms of the heavy-light block, reproducing prior monodromy results. These results generalize the FKW method to higher-point blocks and lay groundwork for future closed-form constructions and holographic interpretations in AdS/CFT.

Abstract

We consider Virasoro conformal blocks in the large central charge limit. There are different regimes depending on the behavior of the conformal dimensions. The most simple regime is reduced to the global sl(2, C) conformal blocks while the most complicated one is known as the classical conformal blocks. Recently, Fitzpatrick, Kaplan, and Walters showed that the two regimes are related through the intermediate stage of the so-called heavy-light semiclassical limit. We study this idea in the particular case of the 5-point conformal block. To find the 5-point global block we use the projector technique and the Casimir operator approach. Furthermore, we discuss the relation between the global and the heavy-light limits and construct the heavy-light block from the global block. In this way we reproduce our previous results for the 5-point perturbative classical block obtained by means of the monodromy method.

Paper Structure

This paper contains 18 sections, 77 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Global conformal block
  3. Projection technique
  4. Horn's classification
  5. 4-point global block
  6. Vacuum global blocks
  7. Vacuum block I.
  8. Vacuum block II.
  9. Small $\Delta_3$ expansion
  10. Global conformal block: the Casimir equations
  11. 4-point global block
  12. 5-point global conformal block
  13. Heavy-light and linearized classical blocks
  14. 5-point heavy-light block
  15. 4-point case.
  16. ...and 3 more sections

Figures (1)

  • Figure 1: Comb diagram related to the $5$-point conformal block. The wavy lines correspond to the contributions of the two highest weight representations, $\widetilde{\Delta}_{1}$ and $\widetilde{\Delta}_{2}$, in the intermediate channels.