Moduli Space Tilings and Lie-Theoretic Color Factors

TL;DR

This paper introduces a Lie-theoretic approach to tilings of moduli spaces that simplifies the understanding of color factors in biadjoint scalar amplitudes, addressing a complex problem in mathematical physics.

Contribution

It presents a novel Lie-theoretic realization of color factors using moduli space tilings, resolving an open question and simplifying the realization space complexity.

Findings

Constructed a collection of tilings with simple topologies

Each amplitude in the collection satisfies $U(1)$ decoupling

Superposing all amplitudes reveals the essential complexity

Abstract

A detailed understanding of the moduli spaces $X(k,n)$ of $n$ points in projective $k-1$ space is essential to the investigation of generalized biadjoint scalar amplitudes, as discovered by Cachazo, Early, Guevara and Mizera (CEGM) in 2019. But in math, conventional wisdom says that it is completely hopeless due to the arbitrarily high complexity of realization spaces of oriented matroids. In this paper, we nonetheless find a path forward. We present a Lie-theoretic realization of color factors for color-dressed generalized biadjoint scalar amplitudes, formulated in terms of certain tilings of the real moduli space $X(k,n)$ and collections of logarithmic differential forms, resolving an important open question from recent work by Cachazo, Early and Zhang. The main idea is to replace the realization space decomposition of $X(k,n)$ with a large class of overlapping tilings whose topologies are individually relatively simple. So we obtain a collection of color-dressed amplitudes, each of which satisfies $U(1)$ decoupling separately. The essential complexity appears when they are all superposed.