Non-perturbative geometries for planar $\mathcal{N}=4$ SYM amplitudes

TL;DR

This paper constructs polytopal realizations of configuration space compactifications that connect cluster variables and algebraic functions, providing a geometric framework for understanding amplitude symbol letters and algebraic functions in planar T SYM theory.

Contribution

It introduces a geometric approach using stringy canonical forms to realize finite polytopes associated with configuration spaces, linking cluster variables and algebraic functions in amplitude analysis.

Findings

Polytopal realizations are finite for all k and n.

Certain facets correspond to cluster variables, others to algebraic functions.

Square roots are related to overpositive functions of kinematical invariants.

Abstract

There is a remarkable well-known connection between the G$(4,n)$ cluster algebra and $n$-particle amplitudes in $\mathcal{N}=4$ SYM theory. For $n \ge 8$ two long-standing open questions have been to find a mathematically natural way to identify a finite list of amplitude symbol letters from among the infinitely many cluster variables, and to find an explanation for certain algebraic functions, such as the square roots of four-mass-box type, that are expected to appear in symbols but are not cluster variables. In this letter we use the notion of "stringy canonical forms" to construct polytopal realizations of certain compactifications of (the positive part of) the configuration space Conf${}_n(\mathbb{P}^{k-1}) \cong {\rm G}(k,n)/T$ that are manifestly finite for all $k$ and $n$. Some facets of these polytopes are naturally associated to cluster variables, while others are naturally associated to algebraic functions constructed from Lusztig's canonical basis. For $(k,n) = (4,8)$ the latter include precisely the expected square roots, revealing them to be related to certain "overpositive" functions of the kinematical invariants.