Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

BCFW tilings and cluster adjacency for the amplituhedron

Chaim Even-Zohar, Tsviqa Lakrec, Matteo Parisi, Melissa Sherman-Bennett, Ran Tessler, Lauren Williams

Abstract

In 2005, Britto, Cachazo, Feng and Witten gave a recurrence (now known as the BCFW recurrence) for computing scattering amplitudes in N=4 super Yang Mills theory. Arkani-Hamed and Trnka subsequently introduced the amplituhedron to give a geometric interpretation of the BCFW recurrence. Arkani-Hamed and Trnka conjectured that each way of iterating the BCFW recurrence gives a "triangulation" or "tiling" of the m=4 amplituhedron. In this article we prove the BCFW tiling conjecture of Arkani-Hamed and Trnka. We also prove the cluster adjacency conjecture for BCFW tiles of the amplituhedron, which says that facets of tiles are cut out by collections of compatible cluster variables for the Grassmannian Gr(4,n). Moreover we show that each BCFW tile is the subset of the Grassmannian where certain cluster variables have particular signs.

BCFW tilings and cluster adjacency for the amplituhedron

Abstract

In 2005, Britto, Cachazo, Feng and Witten gave a recurrence (now known as the BCFW recurrence) for computing scattering amplitudes in N=4 super Yang Mills theory. Arkani-Hamed and Trnka subsequently introduced the amplituhedron to give a geometric interpretation of the BCFW recurrence. Arkani-Hamed and Trnka conjectured that each way of iterating the BCFW recurrence gives a "triangulation" or "tiling" of the m=4 amplituhedron. In this article we prove the BCFW tiling conjecture of Arkani-Hamed and Trnka. We also prove the cluster adjacency conjecture for BCFW tiles of the amplituhedron, which says that facets of tiles are cut out by collections of compatible cluster variables for the Grassmannian Gr(4,n). Moreover we show that each BCFW tile is the subset of the Grassmannian where certain cluster variables have particular signs.

Paper Structure

This paper contains 8 sections, 9 theorems, 8 equations, 7 figures.

Table of Contents

  1. Main results.
  2. Further motivation.
  3. Background
  4. Results

Key Result

Theorem 14

Let ${\mathop{\mathrm{Gr}}\nolimits}_{k,n}^\circ$ be the open subset of the Grassmannian where the frozen variables don't vanish. Then the coordinate ring $\mathbb{C}[\widehat{\mathop{\mathrm{Gr}}\nolimits}_{k,n}^\circ]$ of the affine cone over ${\mathop{\mathrm{Gr}}\nolimits}_{k,n}^\circ$ is the cl

Figures (7)

  • Figure 1: Artistic illustration by Annabel Ma
  • Figure 2: The rectangle seed $\Sigma_{4,7}$. Mutable variables are in the colored box.
  • Figure 3: The BCFW product $S_L \bowtie S_R$ of $S_L$ and $S_R$ in terms of their plabic graphs.
  • Figure 4: Plabic graphs of a BCFW cell in $\mathop{\mathrm{Gr}}\nolimits^{\geq 0}_{1,n}$ (left) and in $\mathop{\mathrm{Gr}}\nolimits^{\geq 0}_{2,7}$ (right).
  • Figure 5: BCFW tiling for $\mathcal{A}_{n,k,4}$. On the right: the first term is obtained by tiling $\mathcal{A}_{[n] \setminus \{d\},k,4}$ (from $\mathcal{T}_{pre}$); the second term is the union over $b,k_L,k_R$ as in \ref{['def:BCFWoutput']} of the collections of tiles obtained by tiling $\mathcal{A}_{N_L,k_L,4}$ and $\mathcal{A}_{N_R,k_R,4}$ (from $\mathcal{T}_{k_L,k_R,b}$).
  • ...and 2 more figures

Theorems & Definitions (33)

  • Definition 1: Positive Grassmannian lusztigpostnikov
  • Definition 2
  • Definition 3: Operations on the Grassmannian
  • Definition 4: Amplituhedron
  • Definition 5: Tiles
  • Definition 6: Tilings
  • Definition 7: Facet of a cell and a tile
  • Definition 8: Twistor coordinates
  • Definition 9
  • Definition 10
  • ...and 23 more