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

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 $q\rightarrow 0$ 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.