Critical Field Theories with OSp$(1|2M)$ Symmetry

Abstract

In the paper [L. Fei et al., JHEP 09 (2015) 076] a cubic field theory of a scalar field $\sigma$ and two anticommuting scalar fields, $\theta$ and $\bar{\theta }$, was formulated. In $6-\epsilon$ dimensions it has a weakly coupled fixed point with imaginary cubic couplings where the symmetry is enhanced to the supergroup OSp$(1|2)$. This theory may be viewed as a "UV completion" in $2<d<6$ of the non-linear sigma model with hyperbolic target space H${}^{0|2}$ described by a pair of intrinsic anticommuting coordinates. It also describes the $q\to 0$ limit of the critical $q$-state Potts model, which is equivalent to the statistical mechanics of spanning forests on a graph. In this letter we generalize these results to a class of OSp$(1|2M)$ symmetric field theories whose upper critical dimensions are $d_c(M)=2\frac{2M+1}{2M-1}$. They contain $2M$ anticommuting scalar fields, ${\theta }^i,{\bar{\theta }}^i$, and one commuting one, with interaction $g{\left({\sigma }^2+2{\theta }^i{\bar{\theta }}^i\right)}^{(2M+1)/2}$. In $d_c(M)-\epsilon$ dimensions, we find a weakly coupled IR fixed point at an imaginary value of $g$. We propose that these critical theories are the UV completions of the sigma models with fermionic hyperbolic target spaces H${}^{0|2M}$. Of particular interest is the quintic field theory with OSp$(1|4)$ symmetry, whose upper critical dimension is $10/3$. Using this theory, we make a prediction for the critical behavior of the OSp$(1|4)$ lattice system in three dimensions.

12 pages; v2: minor improvements, references added; v3: corrections to some beta functions; main conclusions not affected; v4: some anomalous dimension formulae corrected

References in corpus (7)

• Critical $O(N)$ Models in $6-\epsilon$ Dimensions
• Three Loop Analysis of the Critical $O(N)$ Models in $6-\epsilon$ Dimensions
• Five loop renormalization of ${\varphi }^3$ theory with applications to the Lee-Yang edge singularity and percolation theory
• Ferromagnetic phase transition for the spanning-forest model (q \to 0 limit of the Potts model) in three or more dimensions
• Grassmann Integral Representation for Spanning Hyperforests
• Critical exponents from five-loop scalar theory renormalization near six-dimensions
• New universality class in three dimensions: The critical Blume-Capel model

Cited by in corpus (5)

• Ginzburg-Landau Description and Emergent Supersymmetry of the $(3,8)$ Minimal Model
• Sphere free energy of scalar field theories with cubic interactions
• Percolation transition for random forests in $d\ge 3$
• Dualities between fermionic theories and the Potts model
• Renormalization of supersymmetric chiral theories in rational spacetime dimensions