2025/07/16 by Ben Gripaios, Gripaios, Ben, Khoi Le Nguyen Nguyen +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Physical sciences #Gas Dynamics and Kinetic Theory #High Energy Physics - Phenomenology (hep-ph) #High Energy Physics - Theory (hep-th)
paper · pdf · doi:10.48550/arxiv.2507.12348
openalex publication_date 2025/07/16 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28
We show how to classify the asymptotically-free gauge theories in four spacetime dimensions, focussing here on the case of purely fermionic matter. The classification depends on the fact (which we prove) that both the dimension and Dynkin index of irreducible representations of a simple Lie algebra are strictly increasing functions of each Dynkin label. This implies not only that the number of asymptotically-free representations of any one semisimple Lie algebra is finite, but also that they can be written down in a systematic fashion using tables for the asymptotically-free irreducible representations of simple Lie algebras, which we supply. These tables show that at most two out of a possible ten Dynkin labels can be non-zero and that no Dynkin label can exceed four. The extension to bosonic matter or supersymmetric theories is straightforward.