The Algebra of Grand Unified Theories

TL;DR

The paper presents a purely algebraic, finite-dimensional representation-theoretic account of the Standard Model and three grand unified theories: $SU(5)$, Spin(10), and the Pati–Salam model. It builds explicit embeddings and maps between SM representations and GUT representations using the exterior algebra $\Lambda\mathbb{C}^5$ and spinor constructions, and proves commuting-square and commuting-cube diagrams that relate the different unification routes. A central result is that Spin(10) sits as a unifying node, with $G_{SM}/\mathbb{Z}_6$ identified as the intersection of $SU(5)$ and $Spin(4)\times Spin(6)$ inside $Spin(10)$, yielding a coherent algebraic reconciliation of the two historical GUT visions. The work emphasizes the beauty and structural insights of grand unification, while discarding dynamics, symmetry breaking, and gravity to highlight representation-theoretic structure and interrelationships among GUTs.

Abstract

The Standard Model of particle physics may seem complicated and arbitrary, but it has hidden patterns that are revealed by the relationship between three "grand unified theories": theories that unify forces and particles by extending the Standard Model symmetry group U(1) x SU(2) x SU(3) to a larger group. These three theories are Georgi and Glashow's SU(5) theory, Georgi's theory based on the group Spin(10), and the Pati-Salam model based on the group SU(2) x SU(2) x SU(4). In this expository account for mathematicians, we explain only the portion of these theories that involves finite-dimensional group representations. This allows us to reduce the prerequisites to a bare minimum while still giving a taste of the profound puzzles that physicists are struggling to solve.

Figures (2)

• Figure 1: The nucleons absorbing pions.
• Figure 2: A nucleon absorbs a pion.

Theorems & Definitions (9)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9