Positivity and Cluster Structures in Landau Analysis

TL;DR

This paper explores the geometric and algebraic structures underlying Landau singularities in planar N=4 super Yang-Mills theory, revealing positivity and cluster algebra patterns through a novel recursive framework.

Contribution

It introduces a recursive mechanism for Landau singularities and proves positivity and cluster structure in LS discriminants at all loop orders.

Findings

Positivity and cluster structures are proven for LS discriminants.

A recursive framework governs Landau singularities.

The approach explains the emergence of algebraic structures in scattering amplitudes.

Abstract

Landau analysis in momentum twistor space can be formulated as the study of varieties of lines in three-dimensional projective space, together with their projections and discriminants. Within this framework, we define enumerative invariants (LS degrees) that count leading singularities. Leading Landau singularities (LS discriminants) arise as discriminants detecting the collision of leading singularities. We uncover a recursive mechanism underlying Landau singularities, governed by substitution maps between Grassmannians. Applying this framework, we prove positivity and factorization into cluster variables for the LS discriminant of a large class of Landau diagrams at arbitrary loop order. This provides a first-principles explanation for the emergence of positivity and cluster algebra structures in the singularities of planar N=4 super Yang-Mills theory.