The topological life of Dynkin indices: universal scaling and matter selection
Mboyo Esole, Monica Jinwoo Kang
TL;DR
This work identifies a universal topological mechanism behind Dynkin embedding indices: under an embedding $f:G\hookrightarrow H$ of simple, simply-connected Lie groups, a single integer $j_f$ rescales the fundamental topological data that classify instantons, Chern–Simons/WZW levels, and the suspended K-theory class. By integrating topology with K-theory via the $\beta$-construction and Harris’ degree-3 formula along with Naylor’s degree-4 refinement, the authors derive a universal scaling theorem and show $\pi_3(H/G)\cong \mathbb{Z}/j_f\mathbb{Z}$, linking embedding indices to coset instanton sectors. They connect these ideas to F-theory, arguing that index-one embeddings are generically favored in minimal Tate enhancements, providing a genericity heuristic for matter selection and a framework for charge matching across UV and IR theories. The paper thus unifies representation-theoretic indices, topological invariants, and string-theoretic engineering into a conservation-law perspective on topological charges and their dynamics under subgroup embeddings. This has broad implications for understanding when index-one matter appears naturally and how higher-index embeddings arise in non-generic settings with additional structure or constraints.
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 $π_3$ and on the canonical generators of $H^3$, $H^4(B{-})$, and $H^4(Σ{-})$. 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 $β$-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.