Conformal Blocks in the Large D Limit

TL;DR

The paper develops conformal blocks in the large-D limit using a 1/D expansion, introducing new variables y_+ and y_- that render the conformal Casimir separable. It provides closed-form expressions for scalar and higher-spin blocks in terms of hypergeometric functions, and demonstrates that these large-D blocks approximate exact finite-D blocks surprisingly well, even at D=4. The authors argue that in the strict D→∞ limit, large-spin anomalous dimensions vanish when the limit is taken after other limits, though the limits do not commute, hinting at subtle high-dimensional CFT behavior. This work offers analytic tools for the conformal bootstrap at large D and aligns with EFT expectations from AdS/CFT about suppressed interactions in high dimensions.

Abstract

We derive conformal blocks in an inverse spacetime dimension expansion. In this large D limit, the blocks are naturally written in terms of a new combination of conformal cross-ratios. We comment on the implications for the conformal bootstrap at large D.

Figures (2)

• Figure 1: Comparison of the absolute value of the exact conformal blocks $G^{(D)}_{\Delta,\ell}$ in $D=4$ (black, dashed line) with the large $D$ approximation formula ${\cal G}^{(D)}_{\Delta,\ell}$ evaluated at $D=4$ (red, solid line). Plots are shown as a function of the cross-ratio $u$ at fixed values of $v$. One can generate similar plots and find what appears to be excellent agreement even when $D=2$.
• Figure 2: These plots show partial sums of the absolute value of the bootstrap crossing function $F(z, \bar{z})$ with the mean field theory conformal block coefficients from equation (\ref{['eq:MFTCoeffs']}), comparing the exact 4D conformal blocks (black, dashed line) with the large $D$ conformal blocks evaluated at $D=4$ (red, solid line), as a function of $z$ with $\bar{z} = 0.3$. In the plots on the left (right) we have $\Delta_e = 6.2 (2.2)$. The plots on the top have partials sums up to a maximum $n_{\rm max} = 10$ and $\ell_{\rm max} = 14$. The plots on the bottom have $n_{\rm max}$ and $\ell_{\rm max}$ large enough that the sums have numerically converged over the values plotted, so $F(z,\bar{z})=1$ everywhere for the exact 4D conformal blocks. In all cases we see extremely good agreement considering that here the large $D$ blocks neglect terms formally of $O(1/4)$.