[Paper Review] Amplituhedron cells and Stanley symmetric functions

Thomas Lam|arXiv (Cornell University)|Aug 23, 2014

Black Holes and Theoretical Physics7 references4 citations

TL;DR

This paper establishes a connection between the cohomology of amplituhedron cells in Grassmannians and truncated affine Stanley symmetric functions. It shows that the cohomology class of a tree amplituhedron subvariety under a generic linear map is given by the truncation of an affine Stanley symmetric function, providing a Schubert calculus framework for scattering amplitudes in N=4 super Yang-Mills theory.

ABSTRACT

The amplituhedron was recently introduced in the study of scattering amplitudes in $N=4$ super Yang-Mills. We compute the cohomology class of a tree amplituhedron subvariety of the Grassmannian to be the truncation of an affine Stanley symmetric function.

Motivation & Objective

• To understand the cohomology class of amplituhedron subvarieties arising from the image of positroid cells under a linear map Z.
• To establish a connection between the geometry of the amplituhedron and symmetric function theory, particularly affine Stanley symmetric functions.
• To analyze the behavior of the map Z on positroid cells, especially the dimension and degree of the induced map on real nonnegative parts.
• To provide a Schubert calculus interpretation of scattering amplitudes via the cohomology of amplituhedron varieties.

Proposed method

• Uses the stratification of the Grassmannian by positroid varieties Π_f indexed by bounded affine permutations f.
• Applies the known result that the cohomology class of Π_f is given by the affine Stanley symmetric function F̃_f.
• Considers the rational map Z_Gr: Gr(k,n) → Gr(k,k+m) induced by a generic linear map Z: C^n → C^{k+m}.
• Defines Y_f as the closure of the image of Π_f under Z_Gr, calling it an amplituhedron variety.
• Computes the cohomology class [Y_f] in H^*(Gr(k,k+m), Z) as the truncation τ_{k+m}(F̃_f) of the affine Stanley symmetric function.
• Employs algebraic geometry tools including fiber dimension, Zariski density, and quasi-finite morphisms to analyze the degree and dimension of the map Z_f.

Experimental results

Research questions

• RQ1 What is the cohomology class of the amplituhedron variety Y_f = Z_Gr(Π_f) in the Grassmannian Gr(k,k+m)?
• RQ2 How does the degree of the map Z_f: Π_f → Y_f behave, and what determines its dimension drop or degree greater than one?
• RQ3 Under what conditions does the real nonnegative part (Π_f)_{≥0} map dominantly to (Y_f)_{≥0}, and how does this relate to kinematical support?
• RQ4 Can the truncation τ_{k+m}(F̃_f) be given a direct combinatorial interpretation in terms of cyclically decreasing factorizations of f?

Key findings

• The cohomology class of the amplituhedron variety Y_f is equal to the truncation τ_{k+m}(F̃_f) of the affine Stanley symmetric function associated to f.
• When f has kinematical support, the real dimension of the totally nonnegative part (Y_f)_{≥0} equals the complex dimension of Y_f, implying Zariski density.
• The degree of the map Z_f: Π_f → Y_f is bounded by the degree of Z_f' for any Π_f' ⊂ ∂Π_f with kinematical support, with d_f' ≤ d_f.
• The map Z_f restricted to (Π_f)_{≥0} is not necessarily generically d_f-to-1, indicating nontrivial real geometry in the nonnegative part.
• For generic Z, the image Z_Gr((Π_f)_{≥0}) may have dimension less than expected, even when dim(Π_f) = dim(Gr(k,k+m)).
• The cohomology class [Y_f] is Schur-positive, as τ_{k+m}(F̃_f) is a monomial-positive truncation of an affine Stanley symmetric function.