The topological life of Dynkin indices: universal scaling and matter selection

Abstract

For simple, simply-connected compact Lie groups, Dynkin embedding indices obey a universal scaling law with a direct topological meaning. Given an inclusion $f:G\hookrightarrow H$, the Dynkin embedding index $j_f$ is characterized equivalently by the induced maps on $\pi_3$ and on the canonical generators of $H^3$, $H^4(B{-})$, and $H^4(\Sigma{-})$. Consequently, $j_f$ controls instanton-number scaling, the quantization levels of Chern--Simons and Wess--Zumino--Witten terms, and the matching of gauge couplings and one-loop RG scales. We connect this picture to representation theory via the $\beta$-construction in topological $K$-theory, relating Dynkin indices to Chern characters through Harris' degree--$3$ formula and Naylor's suspended degree--$4$ refinement. Finally, we apply these results to F-theory to explain the prevalence of index-one matter: we propose a ``genericity heuristic'' where geometry favors regular embeddings (typically $j_f=1$) associated with minimal singularity enhancements, while higher-index embeddings require non-generic tuning.