Quantum deformation of planar amplitudes

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. (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.

Total positivity is a quantum phenomenon: the grassmannian case

“…The results of this paper have potential applications in theoretical physics, via a recently established connection between scattering amplitudes and the totally nonnegative grassmannian, see, for example, [4], and the very recent paper by Movshev and Schwarz, [45], which takes this connection onwards to the quantum grassmannian as a result of our work.…”

“…Notice here that by [42,Lemma 3.1.4(v)], the automorphism σ multiplies each m i,j ((i, j) ∈ L γ ) by q. The action of H on (O q (G mn (F))/ Π γ )[γ −1 ] passes to S o (γ)[y ±1 ; σ] via the isomorphism (46) and this action of H on S o (γ)[y ±1 ; σ] restricts to the action of H on S o (γ) described in (45). In particular, the isomorphism ( 46) is H-equivariant where H acts on S o (γ) as in (45) and each (α 1 , .…”