Quantum deformation of planar amplitudes

TL;DR

This paper introduces a quantum deformation framework for planar scattering amplitudes in supersymmetric gauge theories, linking positroid cell contributions to Hochschild homology classes of quantum algebras.

Contribution

It establishes a novel connection between volume forms on positroids and Hochschild homology, enabling quantization of amplitude contributions.

Findings

Quantum deformation relates amplitude volume forms to Hochschild homology.

Explicit formulas for positroid cell contributions in the classical limit.

Quantization of amplitude contributions using quantum algebra structures.

Abstract

In maximally supersymmetric four-dimensional gauge theories planar on-shell diagrams are closely related to the positive Grassmannian and the cell decomposition of it into the union of so called positroid cells \cite{A}. We establish that volume forms on positroids used to express scattering amplitudes can be $q$-deformed to Hochschild homology classes of corresponding quantum algebras. The planar amplitudes are represented in \cite {A} as sums of contributions of some set of positroid cells; we quantize these contributions. In classical limit our considerations allow us to obtain explicit formulas for contributions of positroid cells to scattering amplitudes.