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

In the paper [L. Fei et al., J. High Energy Phys. 09 (2015) 076JHEPFG1029-847910.1007/JHEP09(2015)076] a cubic field theory of a scalar field σ and two anticommuting scalar fields, θ and θ¯, was formulated. In 6-ϵ 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 nonlinear sigma model with hyperbolic target space H0|2 described by a pair of intrinsic anticommuting coordinates. It also describes the q→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 dc(M)=2[(2M+1)/(2M-1)]. They contain 2M anticommuting scalar fields, θi, θ¯i, and one commuting one, with interaction g(σ2+2θiθ¯i)(2M+1)/2. In dc(M)-ϵ 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 H0|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.