Exact 3D Conformal Blocks from Fractional Calculus

We uncover a striking connection between conformal blocks and fractional calculus. By employing a modified form of half derivatives, we derive explicitly the exact form of the three-dimensional conformal block, expressed as the product of two hypergeometric <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mmultiscripts> <a:mrow> <a:msub> <a:mrow> <a:mi>F</a:mi> </a:mrow> <a:mrow> <a:mn>3</a:mn> </a:mrow> </a:msub> </a:mrow> <a:mprescripts/> <a:mrow> <a:mn>4</a:mn> </a:mrow> <a:none/> </a:mmultiscripts> </a:math> 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.