Abstract
We use the 4 − ϵ expansion to search for fixed points corresponding to 2 + 1 dimensional mathcal{N} =1 Wess-Zumino models of NΦ scalar superfields interacting through a cubic superpotential. In the NΦ = 3 case we classify all SUSY fixed points that are perturbatively unitary. In the NΦ = 4 and NΦ = 5 cases, we focus on fixed points where the scalar superfields form a single irreducible representation of the symmetry group (irreducible fixed points). For NΦ = 4 we show that the S5 invariant super Potts model is the only irreducible fixed point where the four scalar superfields are fully interacting. For NΦ = 5, we go through all Lie subgroups of O(5) and use the GAP system for computational discrete algebra to study finite subgroups of O(5) up to order 800. This analysis gives us three fully interacting irreducible fixed points. Of particular interest is a subgroup of O(5) that exhibits O(3)/Z2 symmetry. It turns out this fixed point can be generalized to a new family of models, with NΦ = frac{mathrm{N}left(mathrm{N}-1right)}{2} − 1 and O(N)/Z2 symmetry, that exists for arbitrary integer N≥3.
Highlights
We use the 4 − expansion to search for fixed points corresponding to 2 + 1 dimensional N =1 Wess-Zumino models of NΦ scalar superfields interacting through a cubic superpotential
In the NΦ = 4 and NΦ = 5 cases, we focus on fixed points where the scalar superfields form a single irreducible representation of the symmetry group
We focus on the SUSY fixed points where the scalar superfields form a single irreducible representation of the symmetry group
Summary
We can use the explicit form of the polynomial I3 to study the S5 invariant fixed point. We use the GAP system [23] for computational discrete algebra and the Small Groups library [24] to search for finite groups with 5 dimensional faithful irreducible representations, similar to the study done in [25]. Theorem 2 Suppose a finite group G has a p dimensional faithful irreducible representation, this irreducible representation has a single character p in the character table. We found only 109 groups that have at least one 5 dimensional faithful irrep Among the 13 finite subgroups of O(5), only four of them have irreducible representations that preserve at least one degree three invariant polynomial. For a fixed NΦ, bigger groups tend to preserve less invariant polynomials, which leads us to conjecture that this list is complete.
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