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Notes on crossing transformations of Virasoro conformal blocks

Lorenz Eberhardt

Abstract

We carefully bootstrap the crossing kernels of Virasoro conformal blocks from first principles. Our approach emphasizes the Hilbert space structure of the space of Virasoro conformal blocks which makes the consistency of crossing transformations obvious. We give a pedagogical explanation of the necessary background about Virasoro conformal blocks and special functions. We also explain several applications and prove new results about the crossing kernels.

Notes on crossing transformations of Virasoro conformal blocks

Abstract

We carefully bootstrap the crossing kernels of Virasoro conformal blocks from first principles. Our approach emphasizes the Hilbert space structure of the space of Virasoro conformal blocks which makes the consistency of crossing transformations obvious. We give a pedagogical explanation of the necessary background about Virasoro conformal blocks and special functions. We also explain several applications and prove new results about the crossing kernels.
Paper Structure (119 sections, 250 equations, 10 figures)

This paper contains 119 sections, 250 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Omissions.
  3. Virasoro conformal blocks and the Moore-Seiberg construction
  4. Definition of conformal blocks
  5. Abstract characterization.
  6. Behaviour under degeneration.
  7. Explicit characterization for the four-punctured sphere.
  8. Explicit characterization for the once-punctured torus.
  9. Normalizability.
  10. Some Virasoro representation theory and degenerate blocks
  11. Degenerate fusion rules.
  12. Implications for conformal blocks.
  13. The explicit form of the inner product.
  14. Crossing transformations
  15. Definition of $\mathbb{F}$.
  16. ...and 104 more sections

Figures (10)

  • Figure 1: Left: A pair of pants decomposition of the four-punctured sphere. Right: The same pair of pants decomposition with additional decoration keeping track of the Dehn twists.
  • Figure 2: The two types of degenerations of surfaces. Pinching the blue curve on the left results in a once-punctured torus and a twice-punctured torus which are connected at a single point (node). Pinching the blue curve on the right results in a three-punctured torus.
  • Figure 3: Idempotency of $\mathbb{F}$.
  • Figure 4: The hexagon equation.
  • Figure 5: The pentagon equation.
  • ...and 5 more figures

Theorems & Definitions (3)

  • Conjecture 1
  • Conjecture 2
  • proof