Quantum deformation of planar amplitudes

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. (This was proven by N. Arkani-Hamed, J. Bourjaily, F. Cachazo, A. Goncharov, A. Postnikov, and J. Trnka.) 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 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.