“By simplest”, we mean the theory with the least degrees of freedom.

In particular, if the theory were conformal, then it would be expected to have an enhanced $O(4)$ symmetry [8, 9, 10, 11], but the bootstrap puts bounds on the scaling dimension of the order parameter that exclude the lattice estimate [12] by a huge margin.

In particular, the theory is believed to have an enhanced $SO(5)$ symmetry [17], but the lattice estimate for the order parameter is ruled out by the bootstrap if one assumes there is just one relevant $SU(2)\times U(1)$ singlet, i.e. no relevant $SO(5)$ singlets.

For odd $N$, one must have non-zero Chern-Simons coupling.

This theory is also referred to as non-compact QED3 or $NCC{P}^{N-1}$ model in condensed matter literature, which refers to the fact that the theory has an explicit $U(1)$ symmetry that forbids monopoles from being added to the action.

The theory is believe to be conformal for all $N\ge N_{\mathrm{crit}}$, so if $N_{\mathrm{crit}}=4$, then $N=3$ would not be conformal.

Another simulation with the JQ model claimed it was only conformal for $N=4$ and higher [27].

This evidence is summarized in Table II of [18], except that the $N=3$ value in Table II should be compared to [25] as shown in I, since [27] did not find conformality for $N=3$.

Note that for $N=2$, we argued in [18] that the theory is a CFT with two relevant singlets, with bootstrap results that compare well to the large $N$ expansion, as shown in Table I of that paper. Both the large $N$ and bootstrap results were later confirmed by a lattice study [19].

This evidence is summarized in Table III of [18].

We emphasize that physical theories do not exactly appear on the boundary of an allowed region formed by bootstrapping a finite amount of correlators, because this would then imply that adding further correlators to the bootstrap could not change the bound even slightly. Nonetheless, physical theories are very near the boundary, as the CFT data read off from the approximate solution to crossing at the boundary matches independent estimates to good accuracy in all the examples listed in the main text.

For instance, the critical $O(N)$ models [41, 42, 43], $O(N)\times O(2)$ models [44], $N=4$ fermionic QED3 [45, 46], $N=2$ bosonic QED3 [18], bosonic QED3 for very large $N$ [47], the 3-state Potts model [48, 49], the Gross-Neveu-Yukawa model [50, 51], and the $\mathcal{N}=1$ Ising model [52, 53, 54].

The large charge expansion for the critical $O(2)$ model was previously compared against lattice predictions for $\ell =0$ in [61].

One can also tune both $m^2$ and $u$ to zero to get a tricritical theory.

The scaling dimensions were also computed using the $4-\epsilon$ expansion [3, 71, 72], which is also not accurate for $\epsilon =1$.

We normalize $q$ as twice the value given in [18, 29].

These values can be most conveniently extracted from [57], in particular equations 24 and Appendix A of that paper.

This can be seen from the fact that $\sqrt{\ell (\ell +1)/2}=1$ for $\ell =1$.

One must also truncate the set of spins to some maximal value, but in practice this does not effect the numerical results as long as the value is large enough. For more details, see e.g. [78].

In [41], ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_0$ are denoted as ${\mathrm{\Delta }}_{\varphi }$ and ${\mathrm{\Delta }}_s$, respectively.

Recall that the $C{P}^1$ theory is believed to either have two $q=0$ relevant operators or to have a weakly first order transition.

Note that the large charge expansion for larger values of $\ell$ was worked out in [88], but these expansions do not match bootstrap data either for the critical $O(2)$ model or for our theory, which might be because these large $q$ expansions are known to fewer orders than the small $\ell$ regime expansion considered here.

Since the scaling dimensions of monopole operators grows linearly with $N$, it will require stronger constraints for CFTs with larger $N$ to saturate the bound.