Bootstrapping the mathcal{N} = 1 Wess-Zumino models in three dimensions

Junchen Rong,Ning Su

doi:10.1007/jhep06(2021)153

Junchen Rong, Ning Su

Open Access

https://doi.org/10.1007/jhep06(2021)153

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Abstract

Using numerical bootstrap method, we determine the critical exponents of the minimal three-dimensional mathcal{N} = 1 Wess-Zumino models with cubic superpotetential mathcal{W}sim {d}_{ijk}{Phi}^i{Phi}^j{Phi}^k . The tensor dijk is taken to be the invariant tensor of either permutation group SN, special unitary group SU(N), or a series of groups called F4 family of Lie groups. Due to the equation of motion, at the Wess-Zumino fixed point, the operator dijkΦjΦk is a (super)descendant of Φi. We observe such super-multiplet recombination in numerical bootstrap, which allows us to determine the scaling dimension of the super-field ∆Φ.

Highlights

In generalized free theories, dijkΦjΦk is a superconformal primary. Such a phenomenon should be viewed as a supersymmetric version of the multiplet recombination that appeared in 3d Ising CFT, where σ3 recombines with σ at the fixed point [9]
We have shown that by studying the supermultiplet recombination phenomena associated N =1 Wess-Zumino model with a superpotential W ∼ dijkΦiΦjΦk, we were able to determine the scaling dimension of the superfield Φ numerically
One important motivation to bootstrap N =1 superconformal field theories is to study the various N =1 dualities proposed in [8, 25, 26]. Such a supermultiplet recombination phenomenon may help us identify some of the Wess-Zumino models

Summary

General discussion

Conformal multiplets in N =1 superconformal field theories group themselves into supermultiplets. A generic super-multiplet contains four conformal multiplets, suppose the superconformal primary has spin l and scaling dimension ∆0, there are two level-1 (super)descendant with. For fermionic super-fields F±j , remember the multiplet contains two level-1 superdescendant (but conformal primary) fields, with spin j ± 1. Since the bootstrap bound only constrains the scaling dimension of superconformal conformal primaries, we get a much higher bound, which corresponding to the dimension of the B+l=,n0 operator next to dijkΦjΦk. Such jumps are observed in all the flavor symmetry groups that we have considered. Since our SCFT is an infra-red fixed point of an N =1 renormalization group flow, this is a natural condition

SN bootstrap

N2 5N 2

F 4 family of Lie groups’ bootstrap

Discussion

A Fake primal effect

C SUSY crossing equation for ΦiΦjΦkΦl