A continuous associahedron of type A

TL;DR

This work constructs a convex, continuous analogue of the type A associahedron from a representation-theoretic viewpoint, built on a Krull–Schmidt triangulated category and its continuous mesh relations. It develops quilting to define continuous deformed mesh relations, introduces zigzags and the heart $\mathcal D^{\heartsuit}$, and establishes a cluster-like framework via $\mathbf T$-clusters and mutations. The authors prove convexity of the continuous associahedron and show that cluster-like boundary points correspond to $\mathbf T$-clusters, with mutations realized as hyperplane intersections; they also embed all finite type $A_n$ generalized associahedra into the continuous object. The construction unifies representation-theoretic techniques with physics-inspired amplituhedron ideas, yielding a continuum version of cluster polytopes and a rich theory of mutations in a continuous setting.

Abstract

Taking a representation-theoretic viewpoint, we construct a continuous associahedron motivated by the realization of the generalized associahedron in the physical setting. We show that our associahedron shares important properties with the generalized associahedron of type A. Our continuous associahedron is convex and manifests a cluster theory: the points which correspond to the clusters are on its boundary, and the edges that correspond to mutations are given by intersections of hyperplanes. This requires development of several methods that are continuous analogues of discrete methods. We conclude the paper by showing that there is a sequence of embeddings of type A generalized associahedra into our continuous associahedron.

Figures (12)

• Figure 1: Augmented Auslander--Reiten quiver for $1\leftarrow 2\leftarrow 3\rightarrow 4\leftarrow 5$.
• Figure 2: For $n=2$, the kinematic space is $\mathbb{R}^5$. The associahedron is realized inside a $2$-dimensional affine plane determined by a system of inequalities in terms of the mesh relations. The facets correspond to diagonals of the pentagon, and vertices correspond to its triangulations.
• Figure 3: Morphisms between indecomposable objects in $\mathcal{D}$.
• Figure 4: An example of a tilting rectangle corresponding to the distinguished triangle $(x,y)\to T\oplus B\to (x',y')\to (x+\pi,-y)$.
• Figure 5: A schematic of distinguished triangles in $\mathcal{D}$ and the corresponding rectangles in $\mathbb R\times(-\frac{\pi}{2},\frac{\pi}{2})$.

Theorems & Definitions (96)

• Definition : Definition \ref{['def:continuous associahedron']}
• Theorem A: Theorem \ref{['thm:convex']}
• Remark
• Theorem B: Theorems \ref{['thm:clusters are extremal']} and \ref{['thm:mutation on associahedron']}
• Theorem C: Theorem \ref{['thm:big finite embedding']}
• Example 2.1
• Remark 2.2
• Theorem 2.3
• Remark 2.4
• Definition 2.5