Kinematic Varieties for Massless Particles

TL;DR

The paper develops an algebraic-geometry framework for the kinematics of $n$ massless particles in $d$ dimensions by encoding momenta with spinor brackets derived from ${\rm Cl}(1,d-1)$. It constructs and analyzes the massless-momentum ideal $I_{d,n}$, proves it is a prime complete intersection for ${\max(n,d)\ge 4}$, and introduces Mandelstam invariants $s_{ij}$ and the Mandelstam variety $V(M_{d,n})$, relating their dimensions to the ambient action of ${\rm O}(1,d-1)$. It then builds spinor-bracket structures, presenting two families of kinematic varieties: a matrix-space family $\mathcal{K}^{(2)}_{d,n}$ (rank-based) and a tensor-space family $\mathcal{K}^{(3)}_{d,n}$ (through the tensor $ST$), with explicit descriptions in low dimensions (e.g., $d=3$ yields ${\rm Gr}(2,n)$ and $d=4,5$ give the first secant variety of ${\rm Gr}(2,n)$). The paper provides concrete results and conjectures for higher dimensions, including a detailed prime-ideal description for the $d=5,n=5$ case, illustrating the rich algebraic structure underlying massless kinematics and its potential connections to spinor-helicity formalisms in higher dimensions.

Abstract

We study algebraic varieties that encode the kinematic data for $n$ massless particles in $d$-dimensional spacetime subject to momentum conservation. Their coordinates are spinor brackets, which we derive from the Clifford algebra associated to the Lorentz group. This was proposed for $d=5$ in the recent physics literature. Our kinematic varieties are given by polynomial constraints on tensors with both symmetric and skew symmetric slices.

Theorems & Definitions (33)

• Theorem 2.1
• proof
• Example 2.2
• Remark 2.3
• Remark 2.4
• Theorem 2.5
• proof
• Remark 2.6
• Proposition 3.1
• Example 3.2: $k = 1,2,3$