{"ID":22964815,"CreatedAt":"2026-09-17T02:29:35.102872001Z","UpdatedAt":"2026-09-17T02:29:35.102872001Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.18884","arxiv_id":"2609.18884","title":"The Geometry of the CKM matrix, the Standard Model and RG fixed points","abstract":"We investigate the geometry of the Fermion mixing matrix in the Standard Model. The principal structure studied is the CKM matrix, $V$, viewed as an element of the double coset space $M$, where $M=T\\backslash SU(3)/T$, and $T=U(1)\\times U(1)$. The space $M$ is given a Riemannian metric, and has both point singularities, which it is shown correspond to the known renormalization group fixed points, and line singularities connecting the fixed points. A key discrete object inducing many of these properties is the Weyl group, $W$, of $SU(3)$, $W$ being the symmetric group $S_3$, which appears in the guise of the vertices of a hexagon in $V$.","short_abstract":"We investigate the geometry of the Fermion mixing matrix in the Standard Model. The principal structure studied is the CKM matrix, $V$, viewed as an element of the double coset space $M$, where $M=T\\backslash SU(3)/T$, and $T=U(1)\\times U(1)$. The space $M$ is given a Riemannian metric, and has both point singularities...","url_abs":"https://arxiv.org/abs/2609.18884","url_pdf":"https://arxiv.org/pdf/2609.18884v1","authors":"[\"Brian P. Dolan\",\"Charles Nash\"]","published":"2026-09-16T16:18:21Z","proceeding":"hep-th","tasks":"[\"hep-th\"]","methods":"[]","has_code":false}