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 ${\varphi }^n$-Cubic universality classes.

Accepted for publication on JHEP

References in corpus (5)

• Six-loop $\epsilon$ expansion study of three-dimensional $n$-vector model with cubic anisotropy
• Scaling Solutions in Continuous Dimension
• Surprises in the $O(N)$ models: nonperturbative fixed points, large $N$ limit and multi-criticality
• New universality class in three dimensions: The critical Blume-Capel model
• Platonic Field Theories

Cited by in corpus (7)

• Monte Carlo study of a generalized icosahedral model on the simple cubic lattice
• Heavy Handed Quest for Fixed Points in Multiple Coupling Scalar Theories in the $\epsilon$ Expansion
• A multicritical Landau-Potts field theory
• Platonic Field Theories
• Multi-critical multi-field models: a CFT approach to the leading order
• The tricritical Ising CFT and conformal bootstrap
• Anomalous Dimensions in Hypercubic Theories