Exact 3D Conformal Blocks from Fractional Calculus

Abstract

We uncover a striking connection between conformal blocks and fractional calculus. By employing a modified form of half-derivates, we derived explicitly the exact form of the three-dimensional conformal block, expressed as the product of two hypergeometric 4F3 functions. This result provides a rigorous proof of Hogervorst's formula, conjectured nearly a decade ago. Furthermore, we demonstrate its implications for the conformal bootstrap, potentially leading to new analytical techniques and numerical tools that deepen our understanding of conformal field theory.

Figures (3)

• Figure 1: Numerical test of the crossing symmetry relation Eq. \ref{['eq:crossing']}. The function $\tilde{C}'(\rho,\bar{\rho})$ is evaluated along the diagonal $\rho=\bar{\rho}$ with spin truncations $\ell_\mathrm{max}=0,2,\ldots,10$. Results are shown for the $s$-channel (left) and $t$-channel (right), using OPE data from Ref. simmons2017lightcone. Rapid convergence is observed as $\ell_\mathrm{max}$ increases. The thick dashed brown curve shows the isolated identity contribution $C_0'(\rho)C_0'(\bar{\rho})$, which dominates at large $\rho$, while the dashed gray curve reproduces the $s$-channel result with $\ell_\mathrm{max}=10$ for comparison.
• Figure 2: Direct numerical comparison of the scalar conformal block along the diagonal $z=\bar{z}$. Our ${}_4F_3$ formula (solid line) agrees with the standard numerical evaluation within machine precision.
• Figure 3: Numerical test of crossing symmetry for the four-point correlator, obtained using the inverse transform of our ${}_4F_3$ block representation. The sum is truncated at spins $\ell_{\max}=0,2,4,\ldots,10$. The OPE data are taken from Ref. simmons2017lightcone.