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Scattering Amplitudes from Intersection Theory.

Sebastian Mizera1

  • 1Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada.

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This study introduces a novel formula for intersection numbers using Picard-Lefschetz theory. These numbers correspond to scattering amplitudes in quantum field theory when applied to moduli spaces of Riemann spheres.

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Area of Science:

  • Algebraic Geometry
  • Mathematical Physics
  • Topology

Background:

  • Picard-Lefschetz theory provides a powerful framework for studying topological and geometric structures.
  • Intersection numbers are fundamental quantities in algebraic geometry and have connections to various fields.
  • Moduli spaces of Riemann spheres are central objects in complex analysis and string theory.

Purpose of the Study:

  • To derive a new formula for calculating intersection numbers of twisted cocycles.
  • To explore the connection between these intersection numbers and quantum field theory amplitudes.
  • To investigate the specific case of hyperplane arrangements yielding moduli spaces of punctured Riemann spheres.

Main Methods:

  • Application of Picard-Lefschetz theory.
  • Calculation of intersection numbers for twisted cocycles.
  • Analysis of hyperplane arrangements and their associated moduli spaces.

Main Results:

  • A new formula for intersection numbers of twisted cocycles is established.
  • In a specific geometric context, these intersection numbers are shown to be equivalent to tree-level scattering amplitudes.
  • The Cachazo-He-Yuan formulation of quantum field theory is utilized.

Conclusions:

  • The study reveals a deep connection between algebraic geometry and quantum field theory.
  • The new formula offers a novel computational tool for intersection numbers.
  • The findings provide insights into the structure of scattering amplitudes through geometric methods.