Platonic Field Theories

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.