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Multipoint Conformal Blocks in the Comb Channel

Vladimir Rosenhaus

Abstract

Conformal blocks are the building blocks for correlation functions in conformal field theories. The four-point function is the most well-studied case. We consider conformal blocks for $n$-point correlation functions. For conformal field theories in dimensions $d=1$ and $d=2$, we use the shadow formalism to compute $n$-point conformal blocks, for arbitrary $n$, in a particular channel which we refer to as the comb channel. The result is expressed in terms of a multivariable hypergeometric function, for which we give series, differential, and integral representations. In general dimension $d$ we derive the $5$-point conformal block, for external and exchanged scalar operators.

Multipoint Conformal Blocks in the Comb Channel

Abstract

Conformal blocks are the building blocks for correlation functions in conformal field theories. The four-point function is the most well-studied case. We consider conformal blocks for -point correlation functions. For conformal field theories in dimensions and , we use the shadow formalism to compute -point conformal blocks, for arbitrary , in a particular channel which we refer to as the comb channel. The result is expressed in terms of a multivariable hypergeometric function, for which we give series, differential, and integral representations. In general dimension we derive the -point conformal block, for external and exchanged scalar operators.

Paper Structure

This paper contains 25 sections, 131 equations, 5 figures.

Table of Contents

  1. Introduction
  2. One and Two Dimensions
  3. Summary
  4. Definitions
  5. Conformal partial wave
  6. Result for $n$-point conformal blocks
  7. Conformal blocks for small $n$
  8. Four-point block
  9. Five-point block
  10. $n$-point block
  11. Analysis
  12. OPE limit
  13. Casimir equations
  14. $d$ Dimensions
  15. Four-point block, scalar exchange
  16. ...and 10 more sections

Figures (5)

  • Figure 1: (a) $4$-point conformal block. (b) $n$-point conformal block, in the comb channel.
  • Figure 2: The $n$-point conformal partial wave in the comb channel. Each vertex denotes a conformal three-point function. Each line denotes a position. Internal lines denote positions that are integrated over. The $h_i$ denote the dimensions of the external operators, and the $\mathbbm h_i$ are the dimensions of the exchanged operators.
  • Figure 3: The (a) $4$-point (b) $5$-point (c) $6$-point conformal partial waves. These are special cases of Fig. \ref{['nptF']} for $n=4,5,6$, respectively.
  • Figure 4: The $5$-point partial wave, defined in Eq. \ref{['5ptW']}.
  • Figure 5: The six-point conformal partial wave in the comb channel, represented as a Feynman diagram. Each line is a particular propagator, $1/x^2$ to some power involving the operator dimensions, as specified by the definition of the partial wave, and the three internal points are integrated over. There are six external points which are held fixed. The $n$-point partial wave will look like this figure, but with $n{-}2$ triangles.