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Dimensional Reduction for Conformal Blocks

Matthijs Hogervorst

TL;DR

This work introduces a dimensional-reduction framework that decomposes a $d$-dimensional conformal block into an infinite sum of $(d-1)$-dimensional blocks with computable coefficients, enabling explicit representations in odd dimensions via ${}_2F_1$ functions. The key result is a closed-form expression for the reduction coefficients $\\mathcal{A}_{n,j}( abla,\ell)$, together with proven positivity and exponential decay leading to rapid convergence of the expansion. For $d=3$ and $d=5$, the blocks can be written entirely in terms of ${}_2F_1$ functions, providing practical tools for analytic and numerical bootstrap in these dimensions and offering a bridge to the 2d lightcone expansion. The method is extendable to nonzero external dimensions and to more general Lorentz-representation blocks, with potential applications to the analytic bootstrap and to superconformal blocks.

Abstract

We consider the dimensional reduction of a CFT, breaking multiplets of the d-dimensional conformal group SO(d+1,1) up into multiplets of SO(d,1). This leads to an expansion of d-dimensional conformal blocks in terms of blocks in d-1 dimensions. In particular, we obtain a formula for 3d conformal blocks as an infinite sum over 2F1 hypergeometric functions with closed-form coefficients.

Dimensional Reduction for Conformal Blocks

TL;DR

This work introduces a dimensional-reduction framework that decomposes a -dimensional conformal block into an infinite sum of -dimensional blocks with computable coefficients, enabling explicit representations in odd dimensions via functions. The key result is a closed-form expression for the reduction coefficients , together with proven positivity and exponential decay leading to rapid convergence of the expansion. For and , the blocks can be written entirely in terms of functions, providing practical tools for analytic and numerical bootstrap in these dimensions and offering a bridge to the 2d lightcone expansion. The method is extendable to nonzero external dimensions and to more general Lorentz-representation blocks, with potential applications to the analytic bootstrap and to superconformal blocks.

Abstract

We consider the dimensional reduction of a CFT, breaking multiplets of the d-dimensional conformal group SO(d+1,1) up into multiplets of SO(d,1). This leads to an expansion of d-dimensional conformal blocks in terms of blocks in d-1 dimensions. In particular, we obtain a formula for 3d conformal blocks as an infinite sum over 2F1 hypergeometric functions with closed-form coefficients.

Paper Structure

This paper contains 7 sections, 49 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Dimensional reduction
  3. Recursion relation for coefficients
  4. Convergence
  5. Comparison to 2$d$ expansion
  6. Discussion
  7. Free field theory

Figures (1)

  • Figure 1: Comparison of different conformal block expansions. Horizontal axis: the order of truncation $N$, vertical axis: relative error in the numerical value of the conformal block --- notice the logarithmic scale. Solid orange: dimensional reduction with $n \leq N$ terms; dashed green: $\rho$-series with $n \leq N$ terms; dotted blue: $z$-series with $n \leq 2N$ terms. The points are joined by lines to guide the eye. The left plot shows the 3$d$ scalar block at the point $u=v=1/4$ with $\Delta=1$, the right plot corresponds to $\Delta = 25$.