Abstract
We use Picard-Lefschetz theory to prove a new formula for intersection numbers of twisted cocycles associated with a given arrangement of hyperplanes. In a special case when this arrangement produces the moduli space of punctured Riemann spheres, intersection numbers become tree-level scattering amplitudes of quantum field theories in the Cachazo-He-Yuan formulation.
Highlights
Introduction.—Over the past years, study of scattering amplitudes revealed many unexpected connections to geometric structures [1,2,3,4], allowing us to understand physical properties of quantum field theories—such as locality or unitarity—from a different perspective
It has recently transpired that intersection theory plays an important role in string theory amplitudes, where, in particular, it provides a geometric interpretation of the Kawai-Lewellen-Tye (KLT) relations between open and closed string amplitudes, or—in the field-theory limit— Yang-Mills and Einstein gravity amplitudes [9,10]
We show that analogous structures appear directly in scattering amplitudes of ordinary quantum field theories. We find that they can be understood as intersection numbers of the so-called twisted cocycles [6,7,8], which are certain families of differential forms
Summary
Introduction.—Over the past years, study of scattering amplitudes revealed many unexpected connections to geometric structures [1,2,3,4], allowing us to understand physical properties of quantum field theories—such as locality or unitarity—from a different perspective. We find that they can be understood as intersection numbers of the so-called twisted cocycles [6,7,8], which are certain families of differential forms. These objects are examples of twisted cocycles, which, roughly speaking, are differential forms on X defined up to equivalence classes φ ∼ φ þ ω ∧ ξ for any d log form ξ.
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