Platonic Field Theories

Understand

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.