Platonic Field Theories

Abstract: We study renormalization group (RG) fixed points of scalar field theories endowed with the discrete symmetry groups of regular polytopes. We employ the functional perturbative renormalization group (FPRG) approach and the $\epsilon$-expansion in $d=d_c-\epsilon$. The upper critical dimensions relevant to our analysis are $d_c = 6,4,\frac{10}{3},3,\frac{14}{5},\frac{8}{3},\frac{5}{2},\frac{12}{5}$; in order to get access to the corresponding RG beta functions, we derive general multicomponent beta functionals $\beta_V$ and $\beta_Z$ in the aforementioned upper critical dimensions, most of which are novel. The field theories we analyze have $N=2$ (polygons), $N=3$ (Platonic solids) and $N=4$ (hyper-Platonic solids) field components. The main results of this analysis include a new candidate universality class in three physical dimensions based on the symmetry group $\mathbb{D}_5$ of the Pentagon. Moreover we find new Icosahedron fixed points in $d<3$, the fixed points of the $24$-Cell, multi-critical $O(N)$ and $\phi^n$-Cubic universality classes.