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Conformal blocks in conformal field theory are linked to Calogero-Sutherland models. This connection allows new derivations of conformal field theory results using superintegrable systems and hypergeometric functions.

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Area of Science:

  • Theoretical Physics
  • Quantum Field Theory
  • Mathematical Physics

Background:

  • Conformal field theory (CFT) describes systems with scale and special conformal symmetries.
  • Conformal blocks are fundamental building blocks of correlation functions in CFT.
  • Calogero-Sutherland models are exactly solvable quantum many-body systems.

Purpose of the Study:

  • To establish a novel connection between conformal blocks and Calogero-Sutherland Hamiltonians.
  • To leverage the mathematical framework of Calogero-Sutherland models for CFT.
  • To derive new results for conformal blocks.

Main Methods:

  • Mapping conformal blocks of scalar four-point functions to eigenfunctions of a hyperbolic Calogero-Sutherland Hamiltonian.
  • Utilizing the superintegrable properties of the two-particle Pöschl-Teller system.
  • Employing Heckman-Opdam hypergeometric functions for explicit constructions.

Main Results:

  • A direct correspondence between conformal blocks and eigenfunctions of a specific Calogero-Sutherland model was identified.
  • The interaction strength's smooth dependence on dimension 'd' was highlighted.
  • An explicit construction of conformal blocks was achieved using Heckman-Opdam hypergeometric functions.

Conclusions:

  • The established link provides a powerful new tool for analyzing conformal field theories.
  • The superintegrability of the Calogero-Sutherland model has significant implications for conformal block theory.
  • Future research can explore further consequences of integrability in CFT.