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Area of Science:

  • Theoretical Physics
  • Mathematical Physics

Background:

  • Conformal field theory (CFT) describes systems with scale and special conformal symmetries.
  • Conformal blocks are fundamental building blocks in CFT calculations.
  • Hogervorst's formula for the three-dimensional conformal block was previously a conjecture.

Purpose of the Study:

  • To establish a rigorous connection between conformal blocks and fractional calculus.
  • To derive the explicit form of the three-dimensional conformal block.
  • To provide a proof for Hogervorst's formula and explore its implications.

Main Methods:

  • Utilized a modified form of half derivatives from fractional calculus.
  • Derived the three-dimensional conformal block as a product of two hypergeometric _{4}F_{3} functions.

Main Results:

  • Established a novel link between conformal blocks and fractional calculus.
  • Provided an explicit, rigorously proven form for the three-dimensional conformal block.
  • Confirmed Hogervorst's formula, conjectured ten years prior.

Conclusions:

  • The derived formula offers a new perspective on conformal field theory.
  • The connection to fractional calculus may unlock advanced analytical and numerical techniques.
  • This work advances the understanding of the conformal bootstrap and CFT.