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Global torus blocks in the necklace channel

Mikhail Pavlov

TL;DR

This work develops global torus conformal blocks in the necklace channel for multi-point correlators in a $2$-D CFT, expressing necklace blocks through $(N+2)$-point comb-channel blocks and enforcing Casimir equations. By imposing special conditions on the conformal dimensions, the authors obtain explicit polynomial block functions, starting from the $1$-pt case and building up to $N$-pt blocks. They show that the necklace blocks satisfy the Casimir equations and relate the necklace construction to OPE/comb-channel decompositions, highlighting structure such as factorization in simple cases and polynomial corrections controlled by a finite set of parameters. The results provide a concrete handle on torus necklace blocks and open avenues for AdS/CFT applications and generalizations to degenerate or supersymmetric blocks.

Abstract

We continue studying of global conformal blocks on the torus in a special (necklace) channel. Functions of such multi-point blocks are explicitly found under special conditions on the blocks' conformal dimensions. We have verified that these blocks satisfy the Casimir equations, which were derived in previous studies.

Global torus blocks in the necklace channel

TL;DR

This work develops global torus conformal blocks in the necklace channel for multi-point correlators in a -D CFT, expressing necklace blocks through -point comb-channel blocks and enforcing Casimir equations. By imposing special conditions on the conformal dimensions, the authors obtain explicit polynomial block functions, starting from the -pt case and building up to -pt blocks. They show that the necklace blocks satisfy the Casimir equations and relate the necklace construction to OPE/comb-channel decompositions, highlighting structure such as factorization in simple cases and polynomial corrections controlled by a finite set of parameters. The results provide a concrete handle on torus necklace blocks and open avenues for AdS/CFT applications and generalizations to degenerate or supersymmetric blocks.

Abstract

We continue studying of global conformal blocks on the torus in a special (necklace) channel. Functions of such multi-point blocks are explicitly found under special conditions on the blocks' conformal dimensions. We have verified that these blocks satisfy the Casimir equations, which were derived in previous studies.
Paper Structure (20 sections, 65 equations, 2 figures)

This paper contains 20 sections, 65 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Torus global conformal blocks
  3. $sl(2)$ representation theory and correlation functions on the torus.
  4. Torus matrix elements.
  5. The torus block in the necklace channel.
  6. Explicit block functions
  7. $1$-pt torus block
  8. $N$-pt necklace block - preliminaries and assumptions
  9. 2-pt and 3-pt necklace blocks
  10. 2-pt block.
  11. 3-pt block.
  12. $N$-pt necklace block
  13. Casimir equations
  14. Lower-point blocks.
  15. The $N$-pt necklace block.
  16. ...and 5 more sections

Figures (2)

  • Figure 1: Visualisation of the relation \ref{['maintool']}. The diagram on the left corresponds to the $N$-pt necklace block. One is represented as a sum of the $(N+2)$-pt comb channel blocks with two additional operators $\partial^m \mathcal{O}_{\alpha}$. Red dots are depicted derivatives with respect to $z_0$ and $z_{N+1}$.
  • Figure 2: The $N$-pt necklace channel block under conditions $h_{\alpha} + \tilde{h}_0 - h_1 = h_{\alpha} + \tilde{h}_{_{N-2}} - h_{_N} =0,$ which are shown by the green dots in the left picture. According to the \ref{['SNpt']}, this block factorizes into the product of the function \ref{['Pfunction']} (red) and the $N$-pt comb block with specific dimensions depicted by blue dots.