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Towards $W_3$ classical blocks with semi-degenerate operators

V. Belavin, Mikhail Pavlov

TL;DR

This work addresses the computation of 4-point classical blocks in the $\mathcal{W}_3$ CFT when external operators are semi-degenerate at level 1 or 2. It develops BPZ-type equations for auxiliary 5-point blocks with a fully degenerate operator and solves them using heavy-light perturbation theory, thereby determining the accessory parameters that fix the blocks. The authors obtain explicit expressions for non-identity classical blocks in two configurations: (i) one non-degenerate and three level-1 semi-degenerate operators, and (ii) one non-degenerate, one level-2 and two level-1 semi-degenerate operators. This work extends the monodromy method beyond the identity intermediate case in $\mathcal{W}_3$ and suggests potential AdS/CFT applications and extensions to broader $\mathcal{W}_N$ settings.

Abstract

We consider 4-point $W_3$ classical blocks focusing on the blocks level-1 and level-2 semi-degenerate operators. We derive BPZ-type equations for the auxiliary 5-point blocks with one additional fully degenerate operator. The monodromy properties of these equations are encoded by the accessory parameters, related to the 4-point $W_3$ classical blocks. We solve the BPZ-type equations via heavy-light perturbation theory and find the accessory parameters, which allows us to obtain the explicit expressions for the considered class of classical blocks.

Towards $W_3$ classical blocks with semi-degenerate operators

TL;DR

This work addresses the computation of 4-point classical blocks in the CFT when external operators are semi-degenerate at level 1 or 2. It develops BPZ-type equations for auxiliary 5-point blocks with a fully degenerate operator and solves them using heavy-light perturbation theory, thereby determining the accessory parameters that fix the blocks. The authors obtain explicit expressions for non-identity classical blocks in two configurations: (i) one non-degenerate and three level-1 semi-degenerate operators, and (ii) one non-degenerate, one level-2 and two level-1 semi-degenerate operators. This work extends the monodromy method beyond the identity intermediate case in and suggests potential AdS/CFT applications and extensions to broader settings.

Abstract

We consider 4-point classical blocks focusing on the blocks level-1 and level-2 semi-degenerate operators. We derive BPZ-type equations for the auxiliary 5-point blocks with one additional fully degenerate operator. The monodromy properties of these equations are encoded by the accessory parameters, related to the 4-point classical blocks. We solve the BPZ-type equations via heavy-light perturbation theory and find the accessory parameters, which allows us to obtain the explicit expressions for the considered class of classical blocks.
Paper Structure (19 sections, 81 equations, 2 figures)

This paper contains 19 sections, 81 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The $\mathcal{W}_3$ algebra and its representations
  4. Algebra.
  5. HW representations.
  6. Semi- and fully degenerate representations.
  7. Classical limit.
  8. Conformal blocks and the Ward identities in $\mathcal{W}_3$ CFT
  9. BPZ-type equations
  10. Blocks with one non-degenerate and 3 level-1 semi-degenerate operators
  11. Blocks with one non-degenerate, 2 level-1 and one level-2 operators
  12. The monodromy method and the HL perturbation theory
  13. The monodromy equation for the 4-pt block with one non-degenerate and 3 level-1 operators
  14. The monodromy equation for the 4-pt block with one non-degenerate, two level-1 and one level-2 operators
  15. The classical $4$-pt blocks
  16. ...and 4 more sections

Figures (2)

  • Figure 1: The auxiliary $5$-pt block $\Psi (y, z_i)$ for the $4$-pt block with three level-1 degenerate operators and one non-degenerate operator. The fully degenerate operator is depicted in red.
  • Figure 2: The auxiliary $5$-pt block $\tilde{\Psi} (y, z_i)$ for the $4$-pt block with two level-1 semi-degenerate operators, one level-2 semi-degenerate operator and one non-degenerate operator. The fully degenerate operator is in red.