Bootstrapping N_f=4 conformal QED_3
Soner Albayrak
Department of Physics, Yale University, New Haven, CT 06511, USA
Institute of Physics, University of Amsterdam, Amsterdam, 1098 XH, The Netherlands
Rajeev S. Erramilli
Department of Physics, Yale University, New Haven, CT 06511, USA
Zhijin Li
Department of Physics, Yale University, New Haven, CT 06511, USA
David Poland
Department of Physics, Yale University, New Haven, CT 06511, USA
Yuan Xin
Department of Physics, Yale University, New Haven, CT 06511, USA

We present the results of a conformal bootstrap study of the presumed unitary IR fixed point of quantum electrodynamics in three dimensions (QED$_3$) coupled to $N_f=4$ two-component Dirac fermions. Specifically, we study the four-point correlators of the $SU(4)$ adjoint fermion bilinear $r$ and the monopole of lowest topological charge $\mathcal{M}_{1/2}$. Most notably, the scaling dimensions of the fermion bilinear $r$ and the monopole $\mathcal{M}_{1/2}$ are found to be constrained into a closed island with a combination of spectrum assumptions inspired by the $1/N_f$ perturbative results as well as a novel interval positivity constraint on the next-lowest-charge monopole $\mathcal{M}_1$. Bounds in this island on the $SU(4)$ and topological $U(1)_t$ conserved current central charges $c_J$, $c_J^t$, as well as on the stress tensor central charge $c_T$, are comfortably consistent with the perturbative results. Together with the scaling dimensions, this suggests that a part of estimates from the $1/N_f$ expansion — even at $N_f=4$ — provide a self-consistent solution to the bootstrap crossing relations, despite some of our assumptions not being strictly justified.

§ INTRODUCTION

Quantum electrodynamics in three dimensions (QED$_3$) has been extensively studied over the past decades, partially motivated by qualitative similarities with four dimensional quantum chromodynamics. The gauge coupling in QED$_3$ has positive mass dimension, and so the theory is asymptotically free and strongly coupled in the infrared (IR) limit. The IR phase of QED$_3$ depends on the number of electrons $N_f$.[In this work, the flavor number refers to $N_f$ two-component Dirac fermions, and we will assume $N_f$ is even to avoid the parity anomaly.]
In the large $N_f$ limit, the theory can be solved using a $1/N_f$ expansion, which suggests a renormalization group flow to an IR fixed point [1, 2]. The pure $U(1)$ gauge theory with $N_f=0$ is expected to confine in the IR due to a proliferation of monopoles [3, 4].  Schwinger-Dyson equation analysis [5, 6] and some lattice simulations [7] suggest that at small $N_f$, the IR phase of QED$_3$ has its chiral symmetry spontaneously broken ($\chi$SB),

\begin{equation*}
SU(N_f) \rightarrow SU(N_f/2)\times SU(N_f/2)\times U(1) \,,
\end{equation*}

due to the dynamical generation of a fermion mass. It is expected that there is a critical flavor number $N_f^*$ which separates the conformal phase from the $\chi$SB phase.

QED\(_3\) also has various fundamental applications in condensed matter physics. In particular, $N_f=4$ QED$_3$ has been utilized to describe high-temperature superconductors, or more generally Dirac spin liquids [8, 9, 10, 11, 12, 13]. $N_f=2$ QED$_3$ has been proposed to be part of the 3d fermion-boson duality web and is an effective theory for the deconfined quantum critical point, see [14] for a comprehensive review.

A crucial unanswered question in these studies is the value of the critical flavor number $N_f^*$ of QED$_3$.
Various approaches have been used to estimate $N_f^*$ [1, 2, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45];[Some of these studies focused on QED$_3$ with a noncompact gauge group $\R$, in which the monopole contributions have been suppressed. At small $N_f$ its low energy dynamics may be different from compact QED$_3$.] however, there is no general consensus to what the actual value should be. Estimates range from $0$ all the way up to $10$.[See [36] for more details on this discrepancy.]
The problem is made worse by the fact that the theory is actually strongly coupled near $N_f^*$, rendering the estimations of perturbative approaches unreliable.
Lattice simulations do offer a nonperturbative approach, but their results remain inconsistent between each other. In particular, some lattice simulations indicate that there is no $\chi$SB for any $N_f>0$ [41, 42, 43, 44] and that the low energy limit of QED$_3$ coupled with massless fermions is always conformal. This assertion stands in contrast to other lattice results which observed $\chi$SB at $N_f=2$ and a conformal phase at $N_f\geqslant4$ [37, 38, 39, 40]. A subtle issue in the study of  \(N_f=2\) QED$_3$ by lattice simulations is the violation of conformality by a small non-unitary factor, as this could not be distinguished from the standard conformal phase due to the finiteness of practical lattice simulations. For instance, [43] measures the scaling dimension of the monopole with lowest unit of topological charge in $N_f=2$ QED$_3$, and according to the bootstrap result [45], their data requires a weakly relevant singlet scalar, indicating that the theory is slightly below the conformal window in the so-called merger and annihilation scenario for the loss of conformality in QED$_3$ [18, 23, 46, 47, 24, 48].[
In the merger and annihilation scenario for the loss of conformality in QED$_3$, we expect the fixed point of QED$_3$ to merge with another fixed point as we continuously vary $N_f$ down to $N_f^*$ from above, and these points annihilate one another below $N_f^*$. A candidate theory for the other fixed point is the so-called QED Gross Neveu Yukawa (QED$_3$-GNY) fixed point.

The modern conformal bootstrap [49, 50] provides a powerful non-perturbative approach to study conformal QED$_3$, free of the subtleties of the perturbative and lattice computations, and poised to be able to answer puzzles such as the value of the critical \(N_f^*\).
Bootstrap studies of have been initiated in [51, 52]  by focusing on the monopole operators in QED$_3$. In 3d, $U(1)$ gauge theories have a unique property of admitting a topological symmetry $U(1)_t$, whose non-trivial representations are constructed by the monopole operators.  From bootstrap point of view, the power of the monopole operators is that they let us distinguish QED\(_3\) from e.g. the $SU(N_c)$ QCD\(_3\). Moreover,
monopole operators are known to play important roles in QED$_3$ with small $N_f$. For instance, in $N_f=2$ QED$_3$ which is a part of the 3d boson-fermion duality web  [14], the monopoles provide dual descriptions of the gauge invariant composite operators made from elementary fermions.
In [51], the authors obtained bootstrap bounds on the scaling dimensions of the leading charge \(q=1/2\) and \(q=1\) monopoles close to saturation, but these bounds were quite sensitive to the gap assumptions, especially to what the authors refer to \(\Delta_2\) (which we will refer to as \(\Delta_{S_{(220)}}\)), which will also play an important role in our study.

Other encouraging results towards bootstrapping conformal QED$_3$ have been obtained by bootstrapping $SU(N_f)$ adjoint fermion bilinear scalars [53]; these operators are the leading gauge-invariant operators with a nontrivial $SU(4)$ representation, and therefore can give us a view into the flavor symmetry of this theory. The study [53] found bootstrap bounds with sharp kinks for $N_f>2$: for large $N_f$, the location of the kink approaches free fermion theory; for large but finite $N_f$, the location is close to the perturbative predictions of conformal QED$_3$; and for sufficiently small \(N_f\) the kink disappears, implying some critical $N_f^*\in(2,3)$.
The lowest singlet operator approaches marginality condition near $N_f^*$, consistent with the merger and annihilation mechanism [18, 23, 46, 47, 24, 48] for the loss of conformality in QED$_3$.
However, it has been proved in [54] that the kinks in the singlet bounds are wholly $SO(N)$ symmetric and can not literally be identified with conformal QED$_3$, while they may correspond to the conformal QED$_3$ through $SO(N)$ symmetry enhancement in the bootstrap bounds [54, 55].
Another set of studies focused on the $SU(N)$ adjoint bilinears in scalar  [56, 57], with similarly promising results (including isolated regions at large $N_f$ or in $d = 2 + \epsilon$ which may contain the scalar solution [56]).

A natural next step would be to bootstrap crossing equations of both the monopoles and the $SU(4)$ adjoint fermion bilinears; this was recently pursued in [58]. The authors make assumptions inspired by the constraints of lattice implementations, based on which they obtain lower bounds on the dimension of the leading monopole \(\Delta_{\cM_{1/2}}\) in order to reach the IR fixed point of $N_f=4$ on a triangular lattice \(\Delta_{\cM_{1/2}}>1.046\) or kagome lattice \(\Delta_{\cM_{1/2}}>1.105\). The bounds are consistent with recent Monte Carlo estimates [41, 59, 43] but they exclude the large \(N_f\) expansion prediction \(\Delta_{\cM_{1/2}}\approx 1.022\).

In this work we will provide a more comprehensive bootstrap study for $N_f=4$ conformal QED$_3$. An important element of our analysis is that the crossing equations of single correlators with the $SU(4)$-adjoint fermion bilinear operator $r$ and the monopole operator with lowest unit of topological charge $\cM_{1/2}$ have enhanced $SO(15)$ and $SO(12)$ symmetry, respectively. A direct consequence of the $SO(N)$ symmetry enhancement of the crossing equations is that suitable gap assumptions are necessary to obtain bootstrap results for non-$SO(N)$ symmetric theories, e.g., conformal QED$_3$. We will use the fermion bilinear bootstrap to demonstrate the gap-dependence of the bootstrap bounds, and show that interesting results for $N_f=4$ conformal QED$_3$ can be obtained after introducing gap assumptions inspired by the perturbative results.
Our most interesting results are obtained from the monopole bootstrap, presented in <ref>, in which the scaling dimensions of operators $r$ and $\cM_{1/2}$ are restricted into a closed island after introducing an interval positivity assumption, along with some input about gaps in the monopole spectrum. Parity symmetry also plays a critical role in generating the monopole bootstrap results. Our bootstrap results suggest that part of the perturbative CFT data of $N_f=4$ conformal QED$_3$ provides a consistent solutions to the crossing equations.

The paper is organized as follows. In Section <ref> we briefly review the perturbative results on conformal QED$_3$, which provide useful guides for our bootstrap studies. In Section <ref> we explain the gap-dependence of the $SU(4)$ adjoint bootstrap bounds caused by the $SO(15)$ symmetry enhancement in the crossing equations and show that interesting results can be obtained after introducing gap assumptions inspired by the perturbative results. In Section <ref> we revisit the monopole bootstrap. We explain the $SO(12)$ symmetry enhancement in the crossing equations and show that parity symmetry can help to restrict the CFT data in a closed region consistent with the results from $1/N_f$ expansions.
In Section <ref> we study the mixed correlator bootstrap with fermion bilinear operator $r$ and the monopole $\cM_{1/2}$. Conclusions and discussions are given in Section <ref>. Technical details related to the bootstrap studies are provided in Appendices.

§ PERTURBATIVE RESULTS FOR CONFORMAL QED3

can be understood pertubatively in the large limit, where one can identify a conformal fixed point and solve conformal analytically in a 1/expansion. At small this expansion breaks down and the theory becomes strongly coupled, making perturbative estimates of the theory as well as the critical flavor number \(N_f^*\) harder to calculate. One of the main objectives of this work is to test whether the results from perturbative computations can be consistent with constraints from the conformal bootstrap.

In Euclidean signature, the QED$_3$ action, i.e. the action of a $U(1)$ gauge theory coupled to $N_f$ massless charged two-component Dirac fermions, is
=∫d^3x(1/4e^2F^μνF_μν - ∑_i=1^N_f   ψ_i σ^μ(∂_μ+i A_μ)ψ^i),
where $e$ is the $U(1)$ gauge coupling constant, $A_\mu$ is the gauge field with field strength $F_{\mu\nu}=\partial_\mu A_{\nu}-\partial_\nu A_{\mu}$, and $\psi^i$ are the $N_f$ fermions in the fundamental representation of the flavor symmetry $SU(N_f)$.
The gamma matrices associated with two-component Dirac fermions are given by the Pauli matrices $\sigma^\mu, ~\mu=1,2,3$. Besides the flavor symmetry $SU(N_f),$ the theory also has a $U(1)_t$ global symmetry associated with a conserved current
The current $J_\mu^t$ is conserved  due to the Bianchi identity of the $U(1)$ gauge field, i.e. $dF=0$. The local operators charged under $U(1)_t$ are the monopole operators corresponding to the non-trivial topology of the $U(1)$ gauge field. The $U(1)_t$ charges $q$ of the monopole operators are quantized according to the Dirac quantization condition. We will follow the normalization of monopole operators in [51, 52], in which $2q\in \Z$.

Due to the contributions from fermionic zero modes in the topological gauge field configurations, the monopole operators also construct nontrivial representations of the flavor symmetry $SU(N_f)$.
According to their charges under topological $U(1)_t$, the local gauge invariant operators in QED$_3$ can be separated into two parts: the $U(1)_t$ charged monopole operators, and the composite operators made from products of fundamental fields which are neutral under topological $U(1)_t$.

§.§ Scaling dimensions of low-lying gauge invariant operators with U(1)t charge q=0

In large $N_f$ QED$_3$, a set of  local gauge invariant operators can be constructed out of the fundamental fields $\psi_i$, $A_\mu$ and their derivatives. These operators do not correspond to any nontrivial topology of $U(1)$ gauge field and are neutral ($q=0$) under the topological $U(1)_t$; however, they form non-trivial representations of the flavor symmetry $SU(N_f)$. In this work, we will be interested in the fermion bilinear operator $r\equiv \psb_i\psi^j-\frac{1}{N_f} \delta_i^j\psb_k\psi^k$ with $N_f=4$, which forms an $SU(4)$ adjoint  representation. The OPE of $r\times r$ can be decomposed into $SU(N_f)$ irreducible representations (irreps):

\begin{multline}
(211) \bigotimes (211) =  (000)^+\bigoplus (211)^+\bigoplus (211)^- \bigoplus (220)^+\\\bigoplus(310)^-\bigoplus(332)^- \bigoplus(422)^+, \label{OPE}
\end{multline}

where the $i-$th number in the vector $(abc)$ denotes the number of boxes in the $i-$th line in the Young diagram of the representation, e.g. $(211)$ is the adjoint representation. The superscripts $+/-$ denote even/odd spin selection rules. Since $r$ forms a real representation of $SU(4)$, only real representations can appear in the right hand side (RHS) of above equation; for instance, only the real combination of $(310)$ and $(332)$ can appear in the $r\times r$ OPE, which will be denoted by $(310)_R$ throughout this paper.

Another important fact to take into account is the parity symmetry. The fermion bilinear scalar is parity odd, and so all the operators in the RHS of (<ref>) are parity even. The lowest parity-even operators in these sectors are constructed from fermion quadrilinear operators or their mixing with the gauge kinetic operator $F^2$. These four-fermion operators play important roles in solving the conformal QED$_3$ crossing equations. The scaling dimensions of these operators have been computed using $1/N_f$ expansion in previous studies [1, 2, 60, 61, 62, 63, 12, 64, 65], which we now summarize.

The scaling dimension of the parity odd $SU(N_f)$ adjoint fermion bilinear scalar has been computed to the order $1/N_f^2$ [61]:
Δ_(211)=2-64/3π^2N_f+256(28-3π^2)/9π^4N_f^2 .
The $SU(N_f)$ singlet four-fermion operator $(\psb_i\psi^i)^2$ has scaling dimension 4 at tree level, identical to the $U(1)$ gauge kinetic term $F_{\mu\nu}F^{\mu\nu}$. They can mix with each other through quantum loop corrections; the scaling dimensions of the resultant two operators at order $1/N_f$ are
Δ_(000)^±= 4+64(2±√(7))/3π^2N_f .
We expect that the singlet operator with negative anomalous dimension $\Delta_S<4$ plays an important role in the loss of conformality in QED$_3$. For sufficiently small $N_f$, $\Delta_S$ approaches the marginality condition $\Delta_S=3$ from above and eventually generates an RG flow, dissolving the IR fixed point of QED$_3$ below $N_f^*$. Above $N_f^*$, the singlet four-fermion coupling can also generate a UV fixed point, whose UV completion is given by the QED$_3$-Gross-Neveu-Yukawa model. In this work, we will only focus on the QED$_3$ IR fixed point, and assume that $N_f^*<4$, as indicated by previous bootstrap studies and some lattice simulations.

The scaling dimension of the lowest scalar in the parity even $(220)$ sector has been computed in [63, 65] at the order $1/N_f$ to be
Δ_(220)=4-64/π^2N_f ;
this operator will play an important role in bootstrap computations. Meanwhile, the scaling dimension of the parity even adjoint scalar[We remind the reader that adjeven is for the parity even scalar, whereas the result in fb is for the parity odd operator $r$, which is also in the $(211)$ sector.] is
Δ_(211)=4+8(25±√(2317))/3π^2N_f ,
where the two operators differ by the contraction of the flavor indices at tree level [65]. Note that with $N_f=4$ the two operators have scaling dimensions about 2.44 and 8.94, respectively! Clearly, these first order corrections to the scaling dimensions of these four-fermion operators are significant, and so these results should be taken cautiously: it would be interesting to know if the higher order corrections can improve the behavior of these perturbative expansions. In [63] the author also computed the scaling dimension of lowest parity even scalar in $(422)$ sector
Δ_(422)=4+64/3π^2N_f .

We would like to briefly comment on the convergence of the perturbative results in eq: perturbative results.
For the parity odd fermion bilinear adjoint operator, the second order correction is quite small, being only $5.4\%$ of the first order correction.
Meanwhile, for the lowest scalars in the parity even $(220)$ and $(211)$ sectors, the first order corrections at $N_f=4$ are nearly $40\%$ of the tree level results. The $1/N_f$ perturbative results obtained in [65] suggest that the convergence becomes worse for composite operators with more fermions. It is currently unclear how well the leading order results can estimate scaling dimensions of four fermion operators in these sectors: as noted previously, it would be quite useful to compute higher order corrections to clarify this issue. For the $SU(N_f)$ adjoint fermion bilinear scalar, Monte Carlo simulations in [41, 59] have computed the scaling dimension of $r$, which is consistent with the leading order results even for $N_f=4$; however, there are significant error bars in the estimates, which cannot exclude a potentially notable correction to the current result.

§.§ Conserved charges in conformal QED3

Conserved currents play fundamental roles in the study of CFTs. In  conformal QED$_3$,  there are three such currents: the stress tensor $T_{\mu\nu}$ and two global symmetry currents, $J_{\mu,i}^j$ and $J^t_\mu$, the latter of which are associated with the $SU(N_f)$ flavor symmetry and the topological $U(1)_t$  symmetry, respectively. The two-point functions of these conserved currents (in the normalization of [65]) are
T_μν(x_1) T_λρ(x_2)    =c_T 3/16π^2(x_12^2)^3  I_μν,λρ(x_12) ,

J_μ,i^j(x_1) J_ν, k^l(x_2)    =c_J 1/8π^2(x_12^2)^2  I_μν(x_12)( δ_i^lδ_k^j-1/N_f δ_i^jδ_k^l),

J^t_μ(x_1) J^t_ν(x_2)    =c_J^t  1/8π^2(x_12^2)^2 I_μν(x_12) ,
where $c_x$ are the central charges and the tensor structures are defined through
I_μν(x)    ≡δ_μν-2x_μx_ν/x^2 ,

I_μν,λρ(x)    ≡1/2(   I_μλ(x) I_νρ(x)+I_μρ(x) I_νλ(x) )-1/3δ_μνδ_λρ .
for convenience. The above central charges have been computed to sub-leading order in the $1/N_f$ expansion in [26]:[These central charges  have also been studied in [66, 67].]
c_T    =c_T0 (1+0.7193/N_f+O(1/N_f^2) ),

c_J    =c_J0 (1+0.1429/N_f+O(1/N_f^2)),

c_J^t    = 6.4846/N_f (1-0.1429/N_f +O(1/N_f^2)).
Here $c_{T0}$ and $c_{J0}$ are the contributions from the free fermions to the central charges, which are equal to $1$ in our normalization.

It is worth mentioning one other result from [26], on $c_T$ and $c_J$ in QCD$_3$ with an $SU(N_c)$ Yang-Mills gauge field coupled with quarks in the fundamental representation of the color group:
c_T   =N_c c_T0(1+0.7193/N_fN_c^2-1/N_c +O(1/N_f^2)),

c_J   =N_c c_J0(1+0.1429/N_fN_c^2-1/N_c +O(1/N_f^2)).
Compared with QED$_3$, $c_T$ and $c_J$ in QCD$_3$ with gauge group $SU(N_c)$ have an additional factor of $N_c$, due to the color degrees of freedom carried by the fermions. The non-Abelian gauge fields also contain more degrees of freedom than the $U(1)$ gauge field, which increase the subleading order corrections in $c_T$ and $c_J$.
This provides a key differentiation between QED$_3$ and QCD$_3$, which otherwise might be hard to distinguish in bootstrap studies just by looking at their low-lying spectrum.

§.§ Large Nf expansion of the monopole spectrum in QED3

Monopole operators in  QED$_3$ have been studied in various works [68, 69, 70, 52, 71].
Their quantum numbers $(\Delta_i, q_i, R_i)$ consist of their scaling dimension $\Delta_i$, their topological charge $q_i$ under $U(1)_t$ symmetry, and their $SU(4)$ representation $R_i$.
We will be particularly interested in the monopoles $\cM_{1/2}$ and $\cM_1$ carrying the lowest topological charges $q=1/2$ and $q=1$,[In this work we will follow the conventions and the normalization used in [69, 70].]
which respectively sit in $(110)$ and $(220)$ representations of $SU(4)$. The scaling dimensions of these monopole operators were computed in [70, 71] to subleading order in the large $N_f$ expansion. The authors computed the free energy on $S^2\times \mathbb{R}$ in the presence of a monopole flux in the IR limit $e^2 N_f\rightarrow \infty$. The scaling dimensions of the monopole operators on $\mathbb{R}^3$ are then given by the energies of the monopole states on $S^2\times \mathbb{R}$ through the state-operator correspondence. For the monopoles $\cM_{1/2}$ and $\cM_1$, their scaling dimensions are [71]
Δ__1/2=   0.26510N_f - 0.038138(5)+O(1/N_f) ,

Δ__1=   0.67315N_f-0.19340(3)+O(1/N_f) .
At $N_f=4$, the above formulas give $\Delta_{\cM_{1/2}}\simeq1.022$, $\Delta_{\cM_1}\simeq2.499$. The subleading corrections are fairly small compared with the leading terms, even at small $N_f=4$.

The OPE of the monopole operators $\cM_{1/2}\times \cM_{1/2}$ plays a key role in our bootstrap study. There are an infinity family of operators  with topological charge $q=1$ appearing in this OPE. Like the monopoles $\cM_{1/2}$ and $\cM_1$, these operators can be constructed by applying fermionic creation operators on the monopole vacuum with $4\pi$ background magnetic flux. Our bootstrap study will make important use of the topological charge 1 spectrum appearing in the $\cM_{1/2}\times \cM_{1/2}$ OPE, which we discuss in more detail below.

States or operators with topological charge $q$ can be explicitly constructed in the free theory limit $e^2 N_f\rightarrow 0$ using a formalism developed in [52]. To construct these states, one first chooses a monopole vacuum with background magnetic flux $4\pi q$ uniformly distributed in the 2D sphere of the Lorentizan spacetime $S^2\times \mathbb{R}$. Then the spectrum with topological charge $q$ can be obtained by constructing the gauge invariant states of free massless fermions $\psi^i$ in this background. The building blocks of a generic state are the fermionic modes in the classical solutions of the fermion field, which can be obtained by solving the Dirac equation  $(i\slashed{\nabla}+\mathcal{A})\psi=0$ in the monopole vacuum, giving a mode expansion

ψ^i(t,x) =    ∑_m=1/2-q^q-1/2
c_q-1/2,m^i,† C_q,q-1/2,m(x)

a_jm^i,† A_qjm(x)e^iλ_j t
b_jm^i B_qjm(x)e^-iλ_j t
)   ,
where $q$ is an overall label of the family of states on the same monopole background, and each fermion mode is labeled by the flavor indices $i$ and total angular momentum quantum numbers $j$ and $m$.
The operators $a_{jm}^{i,\dagger}$, $b_{jm,i}^{\dagger}$, and $c_{q-1/2,m}^{i,\dagger}$ are fermion creation operators, and their corresponding coefficients $A_{qjm}$, $B_{qjm}$, and $C_{q,q-1/2,m}$, are spinor spherical harmonics. Specifically,  $a_{jm}^{i,\dagger}$ ( $b_{jm,i}^{\dagger}$) corresponds to (anti-)particles whereas $c_{q-1/2,m}^{i,\dagger}$ corresponds to fermion zero modes; furthermore, each
$a_{jm}^{i,\dagger}$ and $c_{q-1/2,m}^{i,\dagger}$ ($b_{jm,i}^{\dagger}$) mode transforms in the (anti-)fundamental representation of the $SU(N)$ group, and carry gauge $U(1)$ charge $+1$ ($-1$).
The quantum numbers of the creation operators are given in Table <ref>.[The energy of the bare monopole is the Casimir energy of the fermion fields

\Delta_{\rm bare} = -N \sum_{j=q-1/2}^{\infty} d_j \lambda_j\, ,

where $d_j = 2j+1$ is the degeneracy. The infinite sum is treated by $\zeta$-function regularization to give a finite answer.] See [52] for more details on the monopole vacuum and fermionic creation operators.

Quantum numbers of the bare monopole with topological charge $q$ and the fermionic creation operators, adapted from [52].

energy/scaling dim.         spin         $\begin{aligned}
\textbf{gauge}\\\textbf{charge}
\end{aligned}$        $\begin{aligned}
\end{aligned}$

$a^{i,\dagger}_{j m}$     $\sqrt{(j+1/2)^2-q^2}$     $j\;(\geqslant q+1/2)$     $+1$     $\mathbf{N}$

$b^{i,\dagger}_{j m,i}$     $\sqrt{(j+1/2)^2-q^2}$     $j\;(\geqslant q+1/2)$     $-1$    $\overline{\mathbf{N}}$

$c^{i,\dagger}_{q-1/2, m}$     $0$     $q-1/2$     $+1$     $\mathbf{N}$

$M_\text{bare}$     $\Delta_{\rm bare}$      $0$     $-qN$     $\mathbf{1}$

The UV properties of matter modes and of the bare monopole in QED$_3$.

energy     spin     gauge charge    $SU(N)$ irrep   degeneracy

$a^{i,\dagger}_{j m}$     $\lambda_{j}$     $j$     $+1$     $\mathbf{N}$   $Nd_j$

$b^{i,\dagger}_{j m,i}$     $\lambda_{j}$     $j$     $-1$    $\overline{\mathbf{N}}$     $Nd_j$

$c^{i,\dagger}_{q-1/2, m}$     $0$     $q-1/2$     $+1$     $\mathbf{N}$    $Nd_{q-1/2}$

$M_\text{bare}$     $\Delta_{\rm bare}$     $0$     $-qN$     $\mathbf{1}$    1

In principle the fermionic creation operators in Table <ref>  allow us to construct any states or operators in the topological charge $q$ sector.
There is a subtle issue that the above microstate construction is based on the free fermions in the UV limit $e^2N_f\rightarrow 0$ of QED$_3$, while the theory we are interested in corresponds to its IR fixed point, which relates to the $e^2N_f\rightarrow \infty$ limit. Nevertheless, there is evidence from the thermal computation which suggests that the states have significant overlaps between the two different limits [52].

We then set out to construct as completely as possible the low-lying states of $N_f=4$ QED$_3$. Our strategy is the following:

* We first set a maximum energy threshold $\Delta_{\rm max}$, and exhaust all possible combinations of creation operators $a_{jm}^{i,\dagger}$, $b_{jm,i}^{\dagger}$, and $c_{q-1/2,m}^{i,\dagger}$, with the constraint that the net gauge charge is zero.

* We decompose states created by each sequence of $a_{jm}^{i,\dagger}$, $b_{jm,i}^{\dagger}$, and $c_{q-1/2,m}^{i,\dagger}$ operators into irreps of the product group of spin and flavor symmetries $SU(2)\times SU(4)$.

* Within the sectors of the same $SU(2)\times SU(4)$ irreps, we anti-symmetrize the fermion creation operators, and collect the linearly independent states.

* After obtaining all possible states created by the fermion modes, it is straightforward to get the scaling dimension, spin, $SU(4)$ irreps and parity of the corresponding operators.

More details of our procedure can be found in Appendix <ref>, and we present the results in Table <ref>. Here, we would like to briefly comment on the data in Table <ref>: it describes the low-lying spectrum predicted by the large $N_f$ mode expansion, where some entries are improved wherever results about subleading corrections in $1/N_f$ are available. Additionally, there is a possible caveat of the above procedure that it does not include operators created by gauge fields. Therefore we need to add the operators constructed from gauge fields by hand. Pure gauge field operators include the topological current $J^{t\mu}$, $F^{\mu\nu} F_{\mu\nu}$, and their composite operators. $J^{t\mu}$ is already added to the table by hand, whereas $F^{\mu\nu} F_{\mu\nu}$ mixes with the $SU(4)$ singlet four-fermion operator. It is of course also possible to have composite operators between $J^{t\mu}$ and operators constructed from the fermion modes, which are annotated with a $^*$. We will frequently refer to this table when introducing assumptions on the spectrum in our bootstrap equations.

§ SU(4) ADJOINT FERMION BILINEAR BOOTSTRAP

The fermion bilinear scalar $r_i^j\equiv\psb_i\psi^j-\frac{1}{N_f}\delta_i^j\psb_k\psi^k$ is one of the lowest-dimension gauge-invariant operators in QED$_3$, making it a natural candidate for bootstrap studies of IR fixed points of gauge theories coupled with fermions; see e.g. [72, 73, 74, 53, 54].
A main challenge in the fermion bilinear bootstrap comes from the $SO(N_f^2-1)$ symmetry enhancement in the crossing equations  [54, 55]. To bootstrap conformal QED$_3$ with a proper $SU(N_f)$ symmetry, one has to resolve the $SO(N_f^2-1)$ symmetry enhancement in the crossing equations. In this section, we will describe how the $SO(N_f^2-1)$ symmetry enhancement can be slightly broken by introducing gap assumptions inspired by the perturbative $N_f=4$ conformal QED$_3$ spectrum, and the resulting bootstrap bounds have kinks which could conjecturally be connected to $N_f=4$ conformal QED$_3$. Nevertheless, the positions of the kinks are sensitive to the gap assumptions, so even under this conjecture more input needs to be given or more constraints need to be imposed in order to extract the physical solution of .

§.§ Crossing equations with different symmetries and gap-dependence

In certain theories, there exists an $SO(N)$ symmetry enhancement of the crossing equations which affect general single correlator bootstrap bounds [75, 54, 55]. In particular, there is a unique map up to normalization which transforms the $SU(N_f)$ adjoint crossing equation into the $SO(N_f^2-1)$ vector crossing equations; see [55] and [57]. Here we will follow [54] and provide a more detailed study of its effect on the bootstrap bounds.

The operators that can appear in the $r\times r$ OPE are provided in (<ref>). The crossing equations of the four-point correlator $\langle r(x_1)r(x_2)r(x_3)r(x_4) \rangle$ can be written in the vector form

∑_∈ℓ^+ λ_^2V^+_(000)+∑_∈ℓ^+ λ_^2V^+_(211)+∑_∈ℓ^- λ_^2V^-_(211)
+∑_∈ℓ^- λ_^2V^-_(310)_R+∑_∈ℓ^+ λ_^2V^+_(220)+∑_∈ℓ^+ λ_^2V^+_(422)=0 ,
where the vector $\Vec{V}^\pm_{R}$ is a $6$-component vector corresponding to the $SU(4)$ representation $R$ with even/odd spin.[The vector
$\Vec{V}_{(310)_R}$ corresponds to the real combination of $\Vec{V}_{(310)}$ and $\Vec{V}_{(332)}$.
] The crossing equations can be captured by a $6\times 6$ matrix:

0     0     0     -F     F     F

0     1/2 F    0     0     -1/2 F    1/6F

0     -F     -F     1/4 F    1/2F     1/6 F

F     -4 F     0     0     16/3F     16 /15F

H     -H     0     -H     -2/3  H     -14/15  H

0     H     -H     1/4 H    1/2H     -7/6  H

) ,
where the columns of the matrix correspond to the vectors $\Vec{V}^\pm_{R}$ in the order
_SU(4)-ad=( V^+_(000), V^+_(211), V^-_(211),
V^-_(310)_R,  V^+_(220), V^+_(422))_SU(4)-ad ,
and the variables $F,H$ denote the symmetrized/anti-symmetrized conformal blocks

A summary of estimates for the low-lying spectrum appearing in our bootstrap crossing equations obtained using the large $N_f$ expansion. The $SO(2)$ irrep, $SU(4)$ irrep, spin, the lowest 2 or 3 scaling dimensions, and the OPE channels that the operators contribute to are shown for each type of operators. The dimensions correspond to the scaling dimension of operators constructed using the fermion mode creation operators, $J^t$, and their composition. Whenever subleading order corrections are available in the literature, we added them as well. The dimension is annotated with $^*$ if the corresponding operator is a composite operator involving $J^t$. Note that the $SO(2)$ irrep encodes both the U$(1)$ charge and the parity information: the $SO(2)$ irreps $S$ and $A$ have U$(1)$ charge $q=0$ and are parity even and odd, respectively, whereas the $SO(2)$ irreps $V$ and $T$ have the respective U$(1)$ charges $q=1/2$ and $q=1$ while they can have either parity. Special operators are highlighted in the table using square brackets.

$SO(2)$ rep    $SU(4)$ rep     spin-$j$     $\Delta_1$     $\Delta_2$     OPE

S     ($000$) ($\mathsf{Singlet}$)      0     $4 +\frac{64(2\pm\sqrt{7})}{3\pi^2N_f}= \substack{6.510\\ 3.651}$     $5.00^*$     $\lambda_{rrO}$, $\lambda_{MMO}$

S     ($211$) ($\mathsf{Adj}$)      0     $4 + \frac{8(25\pm\sqrt{2317})}{3\pi^2 N_f} = \substack{8.940\\ 2.437}$     $5.00^*$     $\lambda_{rrO}$

S     ($211$) ($\mathsf{Adj}$)      1     2.00 [$J_f$]     4.00     $\lambda_{rrO}$, $\lambda_{MMO}$

S     ($220$) ($\mathsf{A\bar A}$)      0     $4 -\frac{64}{\pi^2 N_f} = 2.379$     6.00     $\lambda_{rrO}$, $\lambda_{MMO}$

S     $(310)_R$ ($\mathsf{S\bar A}$)     1     5.00     6.00     $\lambda_{rrO}$

S     ($422$) ($\mathsf{S\bar S}$)     0     $4+\frac{64}{3\pi^2N_f} = 4.540$     6.00     $\lambda_{rrO}$

A     ($000$) ($\mathsf{Singlet}$)      1     2.00 [$J^t$]     3.00     $\lambda_{MMO}$

A     ($211$) ($\mathsf{Adj}$)      0     $2 -\frac{64}{3\pi^2 N_f} +\frac{256(28-3\pi^2)}{9\pi^4 N_f^2}= 1.43$ [$r$]     4.00     $\lambda_{MMO}$

A     ($220$) ($\mathsf{A\bar A}$)      1     4.00     6.00     $\lambda_{MMO}$

T     ($000$) ($\mathsf{Singlet}$)      0     4.424     6.156     $\lambda_{MMO}$

T     ($211$) ($\mathsf{Adj}$)      1     2.692     4.424     $\lambda_{MMO}$

T     ($220$) ($\mathsf{A\bar A}$)      0     $2.499$ [$M_1$]     6.156     $\lambda_{MMO}$

V     ($110$) ($\mathsf{Anti}$)      0     $1.022$ [$M_{1/2}$]     3.888     $\lambda_{rMO}$

V     ($110$) ($\mathsf{Anti}$)      1     2.474     $3.060^*$     $\lambda_{rMO}$

V     ($200$) ($\mathsf{Sym}$)      0     3.888     $4.474^*$     $\lambda_{rMO}$

V     ($200$) ($\mathsf{Sym}$)      1     2.474     3.888     $\lambda_{rMO}$

V     ($321$) ($\mathsf{AAdj}$)      0     3.888     5.303     $\lambda_{rMO}$

V     ($321$) ($\mathsf{AAdj}$)      1     3.888     4.924     $\lambda_{rMO}$

$SO(2)$     $SU(4)$     $j$     $\Delta_1$     $\Delta_2$     OPE

S     ($000$) ($\mathsf{singlet}$)      0     $4 +\frac{64(2\pm\sqrt{7})}{3\pi^2N}= \substack{6.510\\ 3.651}$     $5.00^*$     $\lambda_{rrO}$, $\lambda_{MMO}$

S     ($211$) ($\mathsf{Adj}$)      0     $4 + \frac{8(25\pm\sqrt{2317})}{3\pi^2 N} = \substack{8.940\\ 2.437}$     $5.00^*$     $\lambda_{rrO}$

S     ($211$) ($\mathsf{Adj}$)      1     2.00 [$J_f$]     4.00     $\lambda_{rrO}$, $\lambda_{MMO}$

S     ($220$) ($\mathsf{A\bar A}$)      0     $4 -\frac{64}{\pi^2 N} = 2.379$     6.00     $\lambda_{rrO}$, $\lambda_{MMO}$

S     ($310$) ($\mathsf{S\bar A}$)     1     5.00     6.00     $\lambda_{rrO}$

S     ($422$) ($\mathsf{S\bar S}$)     0     $4+\frac{64}{3\pi^2N_f} = 4.540$     6.00     $\lambda_{rrO}$

A     ($000$) ($\mathsf{singlet}$)      1     2.00 [$J^t$]     3.00     $\lambda_{MMO}$

A     ($211$) ($\mathsf{Adj}$)      0     $2 -\frac{64}{3\pi^2 N} = 1.460$ [$r$]     4.00     $\lambda_{MMO}$

A     ($220$) ($\mathsf{A\bar A}$)      1     4.00     6.00     $\lambda_{MMO}$

T     ($000$) ($\mathsf{singlet}$)      0     4.424     6.156     $\lambda_{MMO}$

T     ($211$) ($\mathsf{Adj}$)      1     2.692     4.424     $\lambda_{MMO}$

T     ($220$) ($\mathsf{A\bar A}$)      0     $0.673N - 0.194 = 2.499$ [$M_1$]     6.156     $\lambda_{MMO}$

V     ($110$) ($\mathsf{M}$)      0     $0.265N - 0.0383 = 1.022$ [$M_{1/2}$]     3.888     $\lambda_{rMO}$

V     ($110$) ($\mathsf{M}$)      1     2.474     $3.060^*$     $\lambda_{rMO}$

V     ($200$) ($\mathsf{S}$)      0     3.888     $4.474^*$     $\lambda_{rMO}$

V     ($200$) ($\mathsf{S}$)      1     2.474     3.888     $\lambda_{rMO}$

V     ($321$) ($\mathsf{AAdj}$)      0     3.888     5.303     $\lambda_{rMO}$

V     ($321$) ($\mathsf{AAdj}$)      1     3.888     4.924     $\lambda_{rMO}$

A summary of low-lying spectrum to the best of knowledge using large $N$ expansion. The $SO(2)$ irrep, $SU(4)$ irrep, spin, the lowest 2 or 3 scaling dimensions and the OPE channels that the operators contribute to are shown for each type of operators. The dimensions correspond to the scaling dimension of operators constructed from the fermion mode expansion, $J^t$, and their composition and some numbers are corrected where subleading order corrections are available in the literature. The dimension is annotated with $^*$ if the corresponding operator is a composite operator involving $J^t$.
The $SO(2)$ irrep encodes both the U$(1)$ charge and the parity information. The $SO(2)$ irreps $S$ and $A$ have U$(1)$ charge $q=0$ and parity even and odd, respectively. The $SO(2)$ irreps $V$ and $T$ have U$(1)$ charge $q=1/2$ and $q=1$, respectively, and they can have both parity. Special operators are highlighted in the table.

[eq:conformal block functions]
F    = v^Δ_rg_Δ,ℓ(u,v)-u^Δ_rg_Δ,ℓ(v,u) ,

H    = v^Δ_rg_Δ,ℓ(u,v)+u^Δ_rg_Δ,ℓ(v,u) .
A notable property of the above $SU(4)$ adjoint crossing equations is that they admit a unique (up to normalization) transformation $\cT_{SU(4)\textrm{-ad}}$

1     226/119     4/7     0     0     0

-1     894/119     -4/7     1     0     0

0     0     0     0     1     4/7

) ,
which maps the $SU(4)$ adjoint crossing equations to the $SO(15)$ vector crossing equations
_SO(15)= (V^+_S, V^+_T, V^-_A   )_SO(15)=

0     F     -F

F     13/15F     F

H     -17/15 H     -H

) ,
in which $S,T,A$ represent the singlet, traceless symmetric, and antisymmetric representations of $SO(15)$, respectively.

The action of $\cT_{SU(4)\textrm{-ad}}$ is

0     45 /119F     -4/7 F    -6/7 F     40 /119F     24 /17F

F     39 /119F     4 /7F     6 /7F     104 /357F     104 /85F

H     -3/7  H     -4/7  H     -6/7  H     -8/21  H     -8/5  H

) ,
which can be briefly expressed in a vector form
_SU(4)-ad·( V^+_(000),  V^+_(211),  V^-_(211),
V^-_(310)_R,   V^+_(220),  V^+_(422))_SU(4)-ad

=( V^+_S,  x_1V^+_T,  x_2V^-_A,
x_3V^-_A,  x_4V^+_T,  x_5V^+_T)_SO(15),
with positive coefficients $x_i$
x=(45/119,  4/7, 6/7,  40/119,  24/17) .
We will show that the positivity of these coefficients is critical in the bootstrap algorithm.

We can summarize the above by saying that the the transformation $\cT_{SU(4)\textrm{-ad}}$ maps the channels of the $SU(4)$ adjoint crossing equations $\cM_{SU(4)\textrm{-ad}}$ to the channels of the $SO(15)$ vector crossing equations $\cM_{SO(15)}$ through the branching rules
𝐒𝐎(15)        𝐒𝐔(4)

S   ⟷(000)^+ ,

T   ⟷(211)^+⊕(220)^+⊕(422)^+ ,

A   ⟷(211)^-⊕(310)_R^- .

The goal of the conformal bootstrap algorithm is to find linear functionals
$$\Vec{\beta}\equiv (\beta_1, \, \beta_2,\, \beta_3, \, \beta_4, \, \beta_5, \, \beta_6)$$
whose action on the crossing equations $\cM_{SU(4)\textrm{-ad}}$ satisfies
β·_SU(4)-ad= (β_(000)^+,   β_(211)^+,  β_(211)^-,   β_(310)_R^-,   β_(220)^+,   β_(422)^+)≽0_1×6 ,
∀Δ⩾Δ_R_i, ℓ^* ,
where $\Delta_{R_i, \ell}^*$ is the assumed minimum scaling dimension of any spin $\ell$ operator in the $R_i$ representation of $SU(4)$.[$\Delta_{R_i, \ell}^*$ is either the unitary bound or a specific value above the unitary bound.] Due to the algebraic relation (<ref>), any linear functional $\Vec{\alpha}\equiv (\alpha_1,\, \alpha_2, \, \alpha_3)$ satisfying the $SO(15)$ bootstrap equations
α·_SO(15)=α·(V^+_S, V^+_T, V^-_A   )_SO(15)= (α_S^+,   α_T^+,  α_A^-)≽0_1×3 ,
∀Δ⩾Δ_S/T/A, ℓ^* ,
can be used to construct linear functionals in the $SU(4)$ adjoint bootstrap
β_1×6=α_1×3·(_SU(4)-ad)_3×6 ,
which also satisfies the $SU(4)$ adjoint bootstrap equations

β·_SU(4)-ad   =(α·_SU(4)-ad)·_SU(4)-ad

=α·( V^+_S,   x_1V^+_T,  x_2V^-_A,
x_3V^-_A,   x_4V^+_T,  x_5V^+_T)_SO(15)

=( α_S^+,   x_1α_T^+,  x_2α_A^-, x_3α_A^-,   x_4α_T^+,  x_5α_T^+  )≽0_1×6 ,
∀Δ⩾Δ_R_i, ℓ^* ,
given that the gap assumptions $\Delta_{R_i, \ell}^*$ are consistent with those in the $SO(15)$ vector bootstrap $\Delta_{S/T/A, \ell}^*$ following the branching rules (<ref>). Note in the second line we have employed the identity (<ref>) and the positivity condition in the third line follows from the positivity of $\alpha_{S/T/A}^\pm$ owing to the positive coefficients $x_i$.

The relation (<ref>) suggests that
the bounds from $SU(4)$-adjoint bootstrap cannot be weaker than that from the $SO(15)$ vector bootstrap, i.e.
$\Delta^*_{SO(15)-v}\geq \Delta^*_{SU(4)-ad}$,
because any linear functional that excludes some CFT data in the $SO(15)$ vector bootstrap must exclude the same data in the $SU(4)$-adjoint bootstrap.

On the other hand, because any four-point correlator of the $SO(15)$ vectors can be decomposed into four-point correlators of the $SU(4)$ adjoint scalar, the inverse is true, i.e. $\Delta^*_{SO(15)-v}\leq \Delta^*_{SU(4)-ad}$. Therefore we have exactly the same bounds from $SO(15)$ vector bootstrap and $SU(4)$-adjoint bootstrap computations, $\Delta^*_{SO(15)-v}= \Delta^*_{SU(4)-ad}$, under the condition that sectors on both sides that are related by the branching rules (<ref>) have the same gap assumptions.

The above arguments show
that due to the algebraic relation (<ref>), the $SU(4)$ adjoint bootstrap problem with suitably related gap assumptions is equivalent to the $SO(15)$ vector bootstrap and admits the same solutions. The differences between the two bootstrap setups come from the gap assumptions $\Delta_{R_i, \ell}^*$.
To illustrate, let us consider the upper bounds on the scaling dimensions of the lowest non-identity singlet scalar $\Delta_0$, without imposing any gap assumptions besides the unitary bounds in other sectors; i.e., our assumptions are
Δ_(000),ℓ=0⩾Δ_0 ,  Δ_other sectors⩾unitary bounds
in the $SU(4)$ adjoint bootstrap and
Δ_S,ℓ=0⩾Δ_0 ,  Δ_other sectors⩾unitary bounds
in the $SO(15)$ vector bootstrap. The two sets of assumptions are consistent with the $SO(15)\rightarrow SU(4)$ branching rules (<ref>). In consequence the singlet bounds obtained from the $SU(4)$ adjoint bootstrap and $SO(15)$ vector bootstrap are exactly the same.

Another interesting example is given by the upper bound on the scaling dimension of the lowest $SO(15)$ traceless symmetric scalar $\Delta_{1}$ obtained from the $SO(15)$ vector bootstrap. Without imposing any extra gap assumptions, the assumptions are
Δ_T,ℓ=0⩾Δ_1 ,  Δ_other sectors⩾unitary bounds .
In the $SU(4)$ adjoint bootstrap, if we want to get the upper bound on the scaling dimension of the lowest scalar in a sector like the $(422)$ representation without imposing extra gap assumptions, the assumptions are
Δ_(422),ℓ=0⩾Δ_1 ,  Δ_other sectors⩾unitary bounds .
According to the branching rule (<ref>), the \(SO(15)\) assumptions in (<ref>) are actually equivalent to
Δ_(422),ℓ=0 ⩾Δ_1 ,

Δ_(220),ℓ=0 ⩾Δ_1 ,

Δ_(211),ℓ=0 ⩾Δ_1 ,

Δ_other sectors⩾unitary bounds ,
which is stronger than the assumptions (<ref>) in the $SU(4)$ adjoint bootstrap.  Consequently, the upper bound on the scaling dimensions of the lowest $(422)$ scalar in the $SU(4)$ adjoint bootstrap is weaker than the bound on the lowest traceless symmetric scalar in the $SO(15)$ vector bootstrap.[In principle, it is possible that there could be no solution to the crossing equations between the two gap sets (<ref>) and (<ref>). In this case, the two bootstrap boundary conditions (<ref>) and (<ref>) can actually generate the same bootstrap bound. In this work, we find the bootstrap bounds with such different boundary conditions are indeed different at finite derivative order $\Lambda$.] Nevertheless, the two bounds coincide with each other if we impose the assumption that the scalars in the three sectors $(422)$, $(220)$, and $(211)$ all have the same minimum scaling dimension $\Delta_1$.

The symmetry enhancement (<ref>) thus leads to a surprising fact, that in the single correlator bootstrap, although the crossing equations admit $SU(4)$ symmetry, it
cannot be distinguished from an $SO(15)$ symmetry at the crossing equation level. The constraints specific to $SU(4)$ symmetric theories can only be obtained from the gap assumptions that break the $SO(15)$ symmetry explicitly.
This suggests that the gap assumptions in the bootstrap conditions are the only ingredients that we may resort to to carve out the parameter space of non-$SO(N)$ symmetric CFTs, while the role of the non-$SO(N)$ symmetric crossing equations is to provide access to individual sectors branched from the $SO(N)$ representations. Our bootstrap bounds for non-$SO(N)$ symmetric theories are obtained based on non-$SO(N)$ symmetric gap assumptions, and the bounds directly rely on the magnitudes of gaps in certain sectors, i.e., they are gap-dependent.

For the ambitious bootstrap dream which aims to completely solve the IR fixed points of gauge theories, this gap-dependence could be a serious problem.
One hopes that the bootstrap bounds can provide numerical solutions of targeted theories with only few reliable and general assumptions.
On the other hand, the gap-dependence of the bootstrap bounds indicates that the physical solutions may not saturate the bootstrap bounds unless there are sufficiently precise gaps input to the bootstrap equations. Below we will show several examples of the the gap-dependence of the bootstrap bounds and study possible approaches to partially resolve this problem.

§.§ SU(4) adjoint bootstrap results

In this section we study the constraints on the CFT data of $N_f=4$ conformal QED$_3$ by bootstrapping the $SU(4)$ adjoint fermion bilinear scalars.
The main results are that the bootstrap approach indeed can provide nontrivial constraints on the putative CFT data of the theory, and after imposing certain gaps inspired by the QED$_3$ spectrum, there are prominent kinks in the bootstrap bounds on scaling dimensions of operators in different $SU(4)$ representations, indicating the existence of a special solution to the crossing equations containing an $SU(4)$ adjoint scalar. Notably, the dimension of this scalar is near the perturbative and lattice results of $N_f=4$ QED$_3$. However, as discussed above, the precise locations of these kinks are gap-dependent, and consequently we need more information or constraints to pin down the underlying theories of these kinks using the conformal bootstrap and to firmly establish their connection to .

The fermion bilinear scalar $r$ is parity odd in QED$_3$ and the operators appearing in the $r\times r$ OPE are parity even. The lowest scalars on the RHS of (<ref>) are parity even four-fermion operators, which have scaling dimensions $4\pm O(1/N_f)$ and break $SO(N_f^2-1)$ symmetry by their $1/N_f$ corrections, see Table <ref> for details on the subleading corrections of the scaling dimensions of these four-fermion operators. Another notable factor breaking the $SO(15)$ symmetry appears on the RHS of (<ref>): in the $(211)^-$ sector, the lowest operator is the spin-$1$ conserved current corresponding to the $SU(4)$ symmetry, while in the $(310)_R^-$ sector, the lowest spin-$1$ operator has scaling dimension $5\pm O(1/N_f)$. Its subleading correction is not known yet, while the scaling dimension of this operator is expected to be notably higher than the unitary bound.

Upper bounds ($\Lambda=31$) on the scaling dimension of the lowest scalar in the $(422)$ representation under various conditions: no gaps (lightest blue region), gaps $2.4$ in the $(211)^+$ and $(220)^+$ sectors and $3.0$ in the $(422)^+$ and  $(310)_R^-$ sectors (light blue),  gaps $2.8$ in the $(211)^+$ and $(220)^+$ sectors and $3.0$ in the $(422)^+$ and  $(310)_R^-$ sectors (dark blue). The green line denotes the upper bound on the scaling dimension of the lowest $SO(15)$ traceless symmetric scalar obtained from the $SO(15)$ vector bootstrap, which is identical to the $SU(4)$ adjoint bootstrap bound on the scaling dimensions of the four-fermion scalars on the RHS of <ref> with the assumption that these four-fermion scalars have the same scaling dimension. In the physical spectrum of $N_f=4$ QED$_3$, this assumption is violated by subleading $1/N_f$ corrections. The kink near $(\Delta_{\bar{\psi}\psi}\simeq 1.35, ~ \Delta_{S\bar{S}}\simeq 3.7)$ in the green dashed line  remains in the $SU(4)$ adjoint bootstrap bound after introducing different gaps inspired by the $1/N_f$ perturbative results. Nevertheless, the position of this kink depends on the gaps. The red dot denotes the $1/N_f$ perturbative results on the scaling dimensions of the $SU(4)$ adjoint fermion bilinear and the lowest scalar in the $(422)$ representation.

In Fig. <ref> we show the bootstrap bounds on the scaling dimension of the lowest scalar in the $S\bar{S}$  sector. The lightest blue shadowed region denotes the bootstrap bound obtained from the $SU(4)$ adjoint bootstrap without imposing any gap assumptions. The bootstrap bound is smooth without any significant structure, nevertheless, it is already quite interesting even without any extra input information specific to QED$_3$. The red dot represents the $1/N_f$ perturbative results for the scaling dimensions of the fermion bilinear $r$ (at order $1/N_f^2$) and the leading scalar in the $S\bar{S}$ sector (at order $1/N_f$). The perturbative data is notably above the bootstrap bounds and cannot belong to a unitary CFT, which suggests that at least one of the operators will receive significant higher order corrections.

The green dashed line gives the bootstrap bound on the lowest traceless symmetric scalar from the $SO(15)$ vector bootstrap.
The same bound appears in the $SU(4)$ adjoint bootstrap if the sectors on the RHS of (<ref>) have the same gap assumptions, due to the bound coincidence explained previously.
The bootstrap bound shows a sharp kink near $\Delta_{r}\sim 1.35$, close to the expected scaling dimension of the $SU(4)$ adjoint fermion bilinear scalar in $N_f=4$ QED$_3$. Comparing with the lightest blue shadowed region, the gap assumption helps to rule out the regions on the left of the kink, while the bootstrap bound to the right of the kink is only mildly modified.
This shows heuristically how the gap assumptions help in forming the kink structure in the $SU(4)$ adjoint bootstrap bound, and it indicates that a special solution stands out under the constraints posed by the gap assumptions.

The $SO(15)$ vector bootstrap bounds can be obtained in the $SU(4)$ adjoint bootstrap with the $SO(15)$ symmetric gap assumptions given in (<ref>). In $N_f=4$ QED$_3$, this is only satisfied  by the tree level scaling dimensions of four-fermion operators on the RHS of (<ref>).
In the physical spectrum of $N_f=4$ QED$_3$, these four-fermion scalars have different higher order corrections, which are summarized in Table <ref>. After taking this difference into account, the gap assumptions in (<ref>) need to be sightly modified and the bootstrap bound, especially the position of the kink will be shifted.

According to the $1/N_f$ perturbative results in Table <ref>, at order $O(1/N_f)$ the lowest scalars in the $(211)$ and $(220)$ representations have scaling dimensions $\Delta\sim2.4$, while the higher order corrections are expected to be significant, as shown in Fig. <ref> for the leading scalar in the $(422)$ representation. In Fig. <ref> we tested the gaps $\Delta>2.4$ (light blue region) and $\Delta>2.8$ (dark blue region) in both the $(211)$ and $(220)$ sectors.[A natural choice of the gaps in these sectors is the irrelevance condition $\Delta>3$, which can affect whether can be realized in lattice models [58]. However, for reasons that will be explained in our monopole bootstrap study, we chose to a make a slightly more conservative gap assumption $\Delta>2.8$ instead. The bounds with gaps $\Delta>3$ in the $(211)$ and $(220)$ sectors have slightly stronger but similar shapes as the bounds shown in this work.] In addition, we also imposed the gaps $\Delta>3$ for the lowest operators in the parity even singlet and $S\bar{A}$ sectors. In the new bootstrap bounds with these gaps there are vertical left cuts caused by the gaps $\Delta>2.4$ or $\Delta>2.8$ in the $(211)$ and $(220)$ sectors. The prominent kinks remain in the new bootstrap bounds, while their positions are slightly shifted in comparison with the kink in the $SO(15)$ vector bootstrap bound.

In Fig. <ref> we show more bootstrap bounds on the scaling dimensions of operators in different representations of $SU(4)$. Generally the bootstrap bounds of non-singlet operators show prominent kinks near the kink of the $SO(15)$ vector bootstrap bound, and the positions of the kinks depend on the gaps. Note the upper-left plot of Fig. <ref> gives an upper bound on the lowest spin 1 operator in the $S\bar{A}$ sector. Its branching rule is given in (<ref>), which is part of the spin 1 operator in the anti-symmetric representation of $SO(15)$ symmetry. So its bound is independent of the bound of the $SO(15)$ traceless-symmetric scalar given by the green line. Interestingly, it still shows a sharp kink with $\Delta_r$ close to the kink in the green line.

\begin{aligned}
\includegraphics[width=0.4\linewidth]{figs/ASb-bd}
\includegraphics[width=0.4\linewidth]{figs/Adj-bd}\\
\includegraphics[width=0.4\linewidth]{figs/AAb-bd}
\includegraphics[width=0.4\linewidth]{figs/Sig-bd1}
\end{aligned}

The (light) blue regions give bootstrap bounds ($\Lambda=31$) on the scaling dimensions of lowest operators in the $S\bar{A}$ (upper left), $Adj_{\ell=0}$ (upper right), $A\bar{A}$ (lower left), and singlet (lower right) representations of $SU(4)$ with certain gap assumptions.  The green lines denote the upper bound on the scaling dimension of the lowest $SO(15)$ traceless symmetric scalar obtained from the $SO(15)$ vector bootstrap. The red line in the lower right plot represents the $SU(4)$ singlet upper bound without any gap assumptions, which coincides with the singlet upper bound from the $SO(15)$ vector bootstrap. The red dots denote the large $N_f$ perturbative results. In the gap assumptions, we require the lowest operators in the $R = \{S\bar{A}$, $S\bar{S}, \textrm{singlet}\}$ sectors are all irrelevant $\Delta_R > 3$; while the lowest scalars in the $R = \{A\bar{A}, \textrm{Adj}_{\ell=0}\}$ sectors are above $\Delta_R>2.4$ (light blue) or $\Delta_R>2.8$ (blue).

The kink in the singlet sector (right bottom) is less significant in comparison with the non-singlet sectors. Nevertheless, a mild kink-like structure appears in the dark blue shadowed region, obtained after imposing gaps $\Delta>2.8$ in the $Adj$ and $A\bar{A}$ sectors. An interesting fact here is that after imposing gaps $\Delta>3$ in the $S\bar{A}$ and $S\bar{S}$ sectors, the singlet upper bound decreases significantly in comparison to the singlet upper bound without any gap assumptions (red line). The singlet upper bound has been observed to be significantly weaker than the expected value $\Delta_{\text{singlet}} \in (3, 4)$ in interesting physical theories. By introducing gaps inspired by the QED$_3$ spectrum which break the $SU(4)\rightarrow SO(15)$ symmetry enhancement (<ref>), the singlet bound can be notably improved. The gap in the $S\bar{A}$ sector is especially helpful to resolve the $SO(15)$ symmetry enhancement in that its dimension is much higher than the unitary bound of spin 1 currents which forbids a conserved current for $SO(15)$ symmetry. According to the large $N_f$ spectrum in Table <ref> and the bootstrap bounds in Fig. <ref>, we expect a stronger gap in this sector is also allowed and that the singlet upper bound can potentially be improved further.

We emphasize that gap assumptions, even those such in Fig. <ref> and <ref> which only slightly break the $SO(15)\rightarrow SU(4)$ relations (<ref>), play a critical role in bootstrapping a specific theory such as conformal $N_f=4$ QED$_3$. With insufficient gap assumptions, many undesired potential solutions to the $SU(4)$ or $SO(15)$ crossing equations may still be around, obscuring a physical solution (which may relate to a kink structure). Recently the authors of [58] observed that the kink in the $SU(4)$ adjoint scalar bootstrap singlet bound smooths out and perhaps disappears when one imposes a gap in only the spin-1 $S\bar{A}$ sector. We do not view this as a major surprise since it is not clear that a single $S\bar{A}$ gap is sufficient to pick out the conformal $N_f=4$ QED$_3$ solution. For several sectors shown
in Fig. <ref> and <ref>, when we use gaps inspired by the perturbative expectations for $N_f=4$ QED$_3$, the kinks remain and some become even sharper compared with those first found in [53].

\begin{aligned}
\Delta_{S_{(000)}}^{\ell=0}\geq3.0 ,~
\Delta_{S_{(310)}}^{\ell=0}\geq3.0 ,~
\Delta_{S_{(220)}}^{\ell=0} &\geq 2.8 ,~
\Delta_{S_{(211)}}^{\ell=0}\geq2.8 ,~
\Delta_{S\bar{S}'}\geq4.5
\\
\includegraphics[width=0.45\linewidth]{figs/SSb-T2OPE} &
\includegraphics[width=0.45\linewidth]{figs/SSb-J1OPE}
\end{aligned}
Contour plots of the stress tensor central charge $c_T$ (left panel) and the $SU(4)$ conserved current central charge $c_J$ (right panel). The bounds are obtained at $\Lambda=21$ with the gap assumptions: $\Delta_R>3$ for the lowest operators in the $SU(4)$ representations $R = \{\text{singlet},S\bar{A}\}$, $\Delta_R >2.8$ for the lowest scalars in the $R = \{A\bar{A}, \text{Adj}\}$ representations, and the second lowest scalar in the $S\bar{S}$ sector is above $4.5$. The green contours denote the values of $c_T$ and $c_J$ in $N_f=4$ conformal QED$_3$ obtained from the $1/N_f$ expansion.

Fig. <ref> shows the bootstrap bounds on another two important physical quantities in CFTs, the stress tensor central charge $c_T$ and the $SU(4)$ conserved current central charge $c_J$. In the plot we have imposed the gap assumptions $\Delta>2.8$ in the $Adj$ and $A\bar{A}$ sectors and $\Delta>3$ in the singlet and $S\bar{A}$ sectors. Besides, we also assume the second $S\bar{S}$ scalar satisfies $\Delta_{S\bar{S}'}>4.5$, which leads to a lower cut in the bootstrap bound. The second lowest $S\bar{S}$ scalar has scaling dimension 6 in the large $N_f$ limit, see Table <ref>. The gap $\Delta_{S\bar{S}}'>4.5$ for the second $S\bar{S}$ scalar is slightly above the scaling dimension of the lowest $S\bar{S}$ scalar near the kink at $\Delta_{\bar{\psi}\psi}\sim1.35$.  This gap introduced a lower cut in the bound on the scaling dimension of the lowest scalar in the $S\bar{S}$ sector.
Contours denoting the $1/N_f$ perturbative results on $c_T$ and $c_J$ given in Eqs. <ref> are highlighted in Fig. <ref>, which are remarkably close to the bootstrap lower bounds on $c_T$ and $c_J$ near the kink.

The bounds on $c_T$ and $c_J$ shown in Fig. <ref> are especially interesting for bootstrap studies of conformal QED$_3$. A widely recognized difficulty in bootstrapping conformal gauge theories is how to distinguish theories with different gauge groups and matter representations. The conformal bootstrap focuses on gauge invariant operators, in which information about the gauge group has been obscured and the low-lying operators can be similar in different gauge theories. For instance, the $SU(N_f)$ adjoint fermion bilinear operators and four-fermion operators also exist in $SU(N_c)$ gauge theories coupled to $N_f$ fundamental fermions. Without extra constraints on the gauge interactions, it is difficult for the bootstrap algorithm to distinguish the scaling dimensions of operators in conformal QED$_3$ from those in other gauge theories. In this sense, it is not surprising that by introducing a gap on the second $S\bar{S}$ scalar, the lower region is not carved out significantly. Information about the gauge group actually appears in the central charges $c_T$ and $c_J$. The central charges can be viewed as rough measures of the number of degrees of freedom, which are significantly affected by the rank of gauge groups and their representations.

In [26] the central charges $c_T$ and $c_J$ in $SU(N_c)$ gauge theories coupled to $N_f$ fundamental fermions were computed perturbatively, which gives at leading order $N_c$ times of the central charges of $N_f$ flavor QED$_3$.  Therefore the central charges $c_T$ and $c_J$  provide critical parameters to distinguish QED$_3$ from 3d Yang-Mills theories. Going back to the bootstrap bounds on $c_T$ and $c_J$ in Fig. <ref>, the lower bounds on $c_T$ and $c_J$ near the kink are close to the perturbative results of QED$_3$, while significantly lower than the central charges of QCD$_3$, giving evidence that the underlying theory of the kink could be QED$_3$ or a similar $U(1)$ gauge theory. Moreover, near the lower cut caused by the gap for the second $S\bar{S}$ scalar, $c_T$ and $c_J$ have much stronger lower bounds. This region may be excluded at higher $\Lambda$ and does not clearly appear to correspond to any physical theories. Alternatively this region may contain solutions of certain Yang-Mills gauge theories with scaling dimensions $(\Delta_r, \Delta_{S\bar{S}})$ in between the kink and lower cut, which cannot be excluded by the gap $\Delta_{S\bar{S}'}>4.5$ for the second $S\bar{S}$ scalar and have central charges $c_T$ and $c_J$ significantly larger than those of QED$_3$. By inputting upper bounds on the central charges, the bootstrap solutions near the lower cut of the $\Delta_{S\bar{S}}$ allowed region can be excluded. The central charges may thus play a more efficient role in excluding Yang-Mills gauge theory solutions compared with imposing gap assumptions in the spectrum.

In consideration of the special role that the central charges play in the bootstrap bounds, it would be very interesting to bootstrap mixed correlators between $SU(4)$ adjoint fermion bilinears and $SU(4)$ conserved currents. The roles of conserved currents in the 3d numerical bootstrap have been studied in [76, 77, 78]. Another motivation to study mixed correlators involving the $SU(4)$ conserved currents is that they may play an interesting role in resolving the bootstrap bound coincidence caused by the algebraic relation between the crossing equations of $SU(4)$ adjoint scalars and the $SO(15)$ vector scalars (see [54] for more discussions). We leave this direction for future study.

§ MONOPOLE SINGLE CORRELATOR BOOTSTRAP REVISITED

As noted in the introduction, monopole operators are particularly interesting in studying conformal , since the topological $U(1)_t$ symmetry provides an opportunity to distinguish conformal QED$_3$ from QCD$_3$ with $SU(N_c)$ gauge interactions, which are otherwise difficult for the bootstrap to distinguish just based on their flavor symmetries and the perturbative gauge invariant spectrum.[QCD$_3$ theories which contain $U(1)$ factors, e.g. $U(N_c)$ QCD$_3$, also contain monopole operators charged under the topological $U(1)^t$ generated by the current $J_\mu^t=\epsilon_{\mu\nu\rho}\mathrm{tr}(F^{\nu\rho})$. Both the monopole spectrum and central charges have a strong dependence on $N_c$ [70], which can be useful for distinguishing these theories.
Bootstrap studies of the monopole four-point correlator in this theory were performed previously in [51, 52]. The key results were that after imposing certain gaps, the bootstrap bounds show kink-like structures. Nevertheless, the kinks are gap dependent, meaning it may be hard to pin down the conformal QED$_3$ solution with the monopole bootstrap and just a few reliable and general assumptions. We will focus our attention on a less ambitious but still nontrivial task, which is to test the perturbative and lattice results of conformal QED$_3$ using the monopole bootstrap.

Along the way, we will show an algebraic relation between the crossing equations of the four-point functions of the monopole operator $\cM_{1/2}$ and the crossing equations of the $SO(12)$ vector scalar, which in turn leads to a coincidence of bootstrap bounds between the monopole bootstrap and the $SO(12)$ vector bootstrap. We find that gaps inspired by the perturbative spectrum which take advantage of parity symmetry can play an important role in resolving this $SO(12)$ symmetry enhancement in the bootstrap bounds and in carving out peninsula structures. Based on these, we will then introduce interval positivity constraints in the bootstrap setup, with which the allowed parameter space can be further isolated into a closed island.

§.§ Single correlator crossing equations of the monopole operator M1/2

The crossing equations for the monopole four-point correlator were computed in  [51].
The monopole $\cM_{1/2}$ with lowest $U(1)_t$ charge $q=\frac{1}{2}$ forms the $(110)$ representation of $SU(4)$. This monopole operator is not parity-definite: parity flips the sign of the $U(1)$ gauge flux and maps the monopole operator $\cM_{1/2}$ to the anti-monopole $\cM_{-1/2}$. It is convenient to rewrite the $U(1)_t$ charged monopole ($\cM_{1/2}$) and anti-monopole ($\cM_{-1/2}$) operators in an $SO(2)$ vector form $\cM_{1/2}^a$ with
_1/2^a=1=(_1/2+_-1/2)/2 ,  _1/2^a=2=-i(_1/2-_-1/2)/2 ,
where the $SU(4)$ indices have been assumed implicitly. Note that these are now parity-definite. Our crossing equations are of the monopole four-point correlator
⟨_1/2^a (x_1) _1/2^b (x_2)_1/2^c (x_3)_1/2^d (x_4)  ⟩ .
There are $9$ sectors with different $SU(4)\times SO(2)$ representations or parity charge which appear in the OPE of $\cM_{1/2}^a\times \cM_{1/2}^b$.
We can understand the algebraic structure of the crossing equations from (<ref>) with the tensor product of the monopole's $SU(4)$ and $SO(2)$ representations:

\begin{equation}\label{eq:repsMM}
\begin{aligned}
SU(4):\;& (110) \bigotimes (110)= (000) \bigoplus (211) \bigoplus (220)\;, \\
SO(2):\;& V \bigotimes V= S \bigoplus T \bigoplus A\;,
\end{aligned}
\end{equation}

where $V,S, T, A$ denote vector, singlet, traceless-symmetric tensor, and antisymmetric tensor representations of $SO(2)$. The $S$ and $A$ sectors are isomorphic for $SO(2)$, but they have different spin selection rules and parity charges; see Table <ref>.

Spin selection rules $(\ell^\pm)$ and parity charges $(P^\pm)$ for the monopole crossing equations. There are no definite parity charges in the $T$ sectors. Sectors of the same colors correspond to the same sectors in the $SO(12)$ vector crossing equations.

(000)         (211)         (220)

$S$     $l^+, P^+$                      $\ell^-,P^+ $             $\ell^+, P^+$

$A$     $\ell^-, P^-$       $\ell^+,P^-$              $\ell^-, P^-$

$T$      $\ell^+$                    $\ell^-$                      $\ell^+$

The crossing equations can be summarized by the vector equation

∑_,i λ_^2V_S_i^±+∑_,i λ_^2V_A_i^±+∑_,i λ_^2V_T_i^±=0 ,
in which the vector $\Vec{V}_{R_i}^\pm$ has an even/odd spin selection rule and its subscript $R_i$ denotes a sector with $SO(2)$ representation $R=S/A/T$ and $SU(4)$ representation $i=(000), (211)$, or $(220)$. The vectors $\Vec{V}$ have $9$ components and the crossing equations
_monopole≡( V^+_S_(000), V^-_S_(211), V^+_S_(220),V^-_A_(000), V^+_A_(211),
A^-_A_(220),V^+_T_(000), V^-_T_(211), V^+_T_(220))_monopole
can be written in a $9\times9$ square matrix form, as expected in the single correlator bootstrap with general global symmetries [79]:

0     0     0     0     F     -F     0     -F     F

0     0     0     -F     -F     -2/3F     F     F     2/3F

0     -F     F     0     -F     F     0     0     0

F     F     2 /3F     F     F     2/3 F     0     0     0

F     -F     -4/3F     -F     F     4 /3 F    -2 F     2 F     8 /3 F

0     0     0     -H     H     4 /3H     H     -H    - 4/3H

H     -H     -4/3 H     H     -H     -4/3H     0     0     0

0     -H     H     0     H     -H     0     2 H     -2 H

H     H     2 /3 H    -H     -H     -2/3 H     -2 H     -2 H     -4/3 H

) ,
where $F/H$ are the symmetrized/anti-symmetrized conformal block functions (<ref>).

It turns out that there is a relation which maps the above crossing equations (<ref> onto the much simpler \(SO(12)\) vector crossing equations which was not noted in previous monopole bootstrap works [51, 52].  Following the procedure discovered in [54], there is a $3\times9$ matrix

1     19/154     75/154     5/308     -5/308     0     0     0     0

0     40/77     12/77     62/77     15/77     0     0     0     0

0     0     0     0     0     5/11     15/22     1/11     7/22

) ,
which can transform the monopole crossing equations into the $SO(12)$ vector four-point crossing equations
_SO(12)= (V^+_S, V^+_T, V^-_A   )_SO(12)=

0     F     -F

F     5 /6F     F

H     -7/6 H     -H

) .
Its action on the monopole crossing equations gives

\begin{multline}
(\cT\cdot \cM)_{\text{monopole}}=
\\\left(
\begin{array}{ccccccccc}
0 & -\frac{5}{11} F& \frac{40 }{77} F & -\frac{1}{11}F & \frac{30  }{77}F & -\frac{20}{33}  F & \frac{12 }{77}F & -\frac{10}{11}  F & \frac{80 }{77}F \\
F & \frac{5 }{11}F & \frac{100 }{231}F & \frac{1}{11}F & \frac{25 }{77}F & \frac{20 }{33}F & \frac{10 }{77}F & \frac{10 }{11}F & \frac{200 }{231}F \\
H & -\frac{5}{11}  H & -\frac{20}{33}  H & -\frac{1}{11}H & -\frac{5}{11}  H & -\frac{20}{33}  H & -\frac{2}{11}  H & -\frac{10}{11}  H & -\frac{40}{33}  H \\
\end{array}
\right)\,,
\end{multline}

which can be briefly written in the vector form

\begin{multline}
\hspace*{-2em}\left[\cT\cdot\left( \Vec{V}^+_{S_{(000)}}, \Vec{V}^-_{S_{(211)}}, \Vec{V}^+_{S_{(220)}},\Vec{V}^-_{A_{(000)}}, \Vec{V}^+_{A_{(211)}}, \Vec{A}^-_{A_{(220)}},\Vec{V}^+_{T_{(000)}}, \Vec{V}^-_{T_{(211)}}, \Vec{V}^+_{T_{(220)}}\right)\right]_{\text{monopole}}\\
=\left( \Vec{V}^+_{S}, ~ x_1\Vec{V}^-_{A},~ x_2\Vec{V}^+_{T},~
x_3\Vec{V}^-_{A}, ~ x_4\Vec{V}^+_{T},~ x_5\Vec{V}^-_{A},~ x_6\Vec{V}^+_{T}, ~ x_7\Vec{V}^-_{A},~ x_8\Vec{V}^+_{T}\right)_{SO(12)}\,, \label{mp-SO}
\end{multline}

with positive coefficients $x_i$
x={5/11, 40/77, 1/11, 30/77, 20/33, 12/77, 10/11, 80/77} .
Therefore the transformation $\cT_{\text{monopole}}$ maps the monopole crossing equations into the $SO(12)$ vector crossing equations,
combined with the $SO(12)\rightarrow SU(4)\times SO(2)$ branching rules
[eq:s012 branching rules]
𝐒𝐎(12)        𝐒𝐔(4)×𝐒𝐎(2)

S   ⟷S_(000) ,

T   ⟷S_(220)⊕A_(211)⊕T_(000)⊕T_(220) ,

A   ⟷S_(211)⊕A_(000)⊕A_(220)⊕T_(211) .
Note that only even (odd) spins appear in the $S, T$ ($A$) sectors of $SO(12)$, consistent with the spin selection rules of the different $SU(4)\times SO(2)$ representations shown in Table <ref>.

Positivity of $x_i$ implies that the coefficients in the $N_f=4, q=1/2$ monopole crossing equations have the same positivity properties as in the $SO(12)$ vector crossing equations.
This agrees with the results in [54, 55], that in general for a scalar in a representation with $N^*$ degrees of freedom, its four-point crossing equations relate to the $SO(N^*)$ vector crossing equations through a unique linear transformation.
As proved in [54] and the section <ref> of this paper, this relation combined with suitable boundary conditions can lead to coincidences between the monopole and $SO(12)$ vector bootstrap bounds.
Indeed one can show that the bootstrap bound on the lowest non-unit scalar in the $S_{(000)}$ sector coincides with the singlet bound in $SO(12)$ vector bootstrap.
Such a bound coincidence can be broken using non-$SO(N^*)$ symmetric boundary conditions in the bootstrap implementation.

It is very interesting to compare the branching rules in the monopole crossing equations (<ref>) with those in the $SU(4)$ adjoint fermion bilinear crossing equations (<ref>). A major difference is that in (<ref>) all the operators on the RHS are parity even, while in (<ref>), the $SU(4)\times SO(2)$ representations branched from $SO(12)$ $A$ or $T$ sectors contain both parity even and parity odd representations, as well as $T_{\Vec{x}}$ monopole sectors without a definite parity charge. Specifically, the lowest scalar in the $S_{(220)}$ sector is a parity even four-fermion operator while the lowest scalar in the $A_{(211)}$ sector is just the parity odd fermion bilinear $r$, which have quite different scaling dimensions. The lowest scalars in the $T_{(000)}$ and $T_{(220)}$ sectors also have rather different scaling dimensions at leading order, see Table <ref>. Similar differences appear in the branching rule of the $SO(12)$ $A$ sector (<ref>). This is different from the fermion bilinear $r$ crossing equations (<ref>), in which the $SO(15)$ symmetry enhancement is broken only at the subleading order $O(1/N_f)$. Therefore, the monopole crossing equations perhaps provide the strongest way to break the $SO(N)$ symmetry enhancement appearing in bootstrap studies for gauge theories with smaller symmetry.

Based on the above facts, it is possible to introduce highly restrictive gap assumptions in the QED$_3$ monopole bootstrap. Perturbative calculations can provide valuable guidance on the possible gaps in different sectors. However, one needs to use this information carefully as the CFT data may receive notable higher order corrections. On the other hand, the monopole bootstrap can provide a nonperturbative check on whether the perturbative (or lattice) results can be consistent with conformality and unitarity.

§.§ Monopole bootstrap bounds with gaps inspired by QED3 spectrum

In this section we explore bootstrap constraints from the crossing equations of the four-point correlator  $\langle \cM_{1/2}\cM_{1/2}\cM_{1/2}\cM_{1/2}\rangle$. The symmetry enhancement of the crossing equations (<ref>) strongly affects the monopole bootstrap bounds. Both singlet and non-singlet bounds coincide with the $SO(12)$ vector bootstrap results unless the symmetry is strongly broken by gap assumptions. However, interesting bootstrap results can be obtained after introducing gap assumptions inspired by the perturbative spectrum of QED$_3$, shown in Table <ref>.

\includegraphics[width=0.53\linewidth]{figs/aab}
\includegraphics[width=0.53\linewidth]{figs/T2}
\end{aligned}$

Bounds on the scaling dimensions of the lowest scalar in the $S_{(220)}$ sector (left) and the charge 1 monopole in the $T_{(220)}$ sector (right) at $\Lambda=31$.
The green dots denote the $1/N_f$ perturbative predictions.
The axes highlighted with red color are positioned at the intersection of the bounds and $\Delta_{A\bar{A}},\Delta_{M_1} = 3$. Note that the perturbative results are ruled out if we assume $\Delta\geq 3$ in either sector.

In Fig. <ref>, we show the bootstrap bounds on the scaling dimensions of the lowest scalars in the $S_{(220)}$ (left panel) and $T_{(220)}$ (right panel) sectors  without imposing any gap assumptions.
The bootstrap bounds are close to straight lines in the regions away from the unitary bound $\Delta=1/2$.
A direct consequence is that by imposing a gap $\Delta^*$ for the lowest   scalar in $S_{(220)}$ sector: $\Delta_{S_{(220)}}\geqslant \Delta^*$, there will be a minimal $\Delta_{\cM_{1/2}}$  in  the bootstrap allowed region proportional to the gap $\Delta^*$.
This explains the $S_{(220)}$-gap dependent bootstrap bounds observed in [51].
The red dots in Fig. <ref> denote the $1/N_f$ perturbative results, which locate in the physically allowed regions and are well consistent with the bootstrap bounds without imposing any gap assumptions.

The lowest scalar in the $S_{(220)}$ sector is the four-fermion operator with scaling dimension $\Delta\simeq2.4$ at subleading order in the $1/N_f$ expansion, see Table <ref>. Its scaling dimension is expected to receive notable corrections from higher order terms.
An interesting question is whether this operator is relevant or not. Assuming the lowest scalar in the $S_{(220)}$ sector is irrelevant, the bootstrap bound in Fig. <ref> introduces a lower cut on the scaling dimension of the monopole $\cM_{1/2}$: $\Delta_{\cM_{1/2}}>1.05$ at $\Lambda=31$. This is consistent with the lattice result [43] but excludes the perturbative prediction at subleading order $\Delta_{\cM_{1/2}}\simeq 1.022$. We do not have solid evidence on the relevance of the lowest scalar in the $S_{(220)}$ sector and we will adopt a weaker gap assumption in the $S_{(220)}$ sector with which the perturbative result on $\Delta_{\cM_{1/2}}$ is still in the allowed region of the bootstrap bound.

Due to the gap dependence of the bootstrap bound, it is likely too much to hope that our current bootstrap setup can solve the $N_f=4$ conformal QED$_3$ as a special solution saturating the bootstrap bound. However, it is still interesting to know whether by imposing gaps inspired by the perturbative monopole spectrum, will the bootstrap bounds converge to the region near perturbative CFT data of $N_f=4$ conformal QED$_3$ or completely exclude it?
In the monopole spectrum,
the subleading order corrections on the scaling dimensions of the low-lying monopole operators have been shown to be small: only $3.6\%$  ($7.2\%$ ) of the leading term for $\cM_{1/2}$ ($\cM_{1}$). If this is also true for higher order corrections, i.e. the large $N_f$ expansion is still converging, then the current perturbative results should be close to the physical spectrum. In contrast, subleading order corrections of the four-fermion operators are more significant and the perturbative results have been shown in Fig. <ref> to be not reliable. The readers should be reminded that our assumptions on the gap $2.8$ in the $S_{(220)}$ sector and the monopole spectrum have not been strictly established yet and the bootstrap computations should be considered as numerical experiments before more solid evidence on these assumptions can be obtained.

In Fig. <ref> we show the bootstrap bound on the scaling dimension of the lowest parity odd $SU(4)$ adjoint scalar $r$ in the $A_{(211)}$ sector. To obtain the result, we have imposed gaps $\Delta_{S_{(000)}}\geq3.0$, $\Delta_{S_{(220)}}\geq2.8$, $\Delta_{T_{(000)}}\geq4.0$, and $\Delta_{A'_{(211)}}\geq3.0$ for the second lowest scalar in the $A_{(211)}$ sector. In the conformal phase of QED$_3$, the lowest parity even singlet scalar is expected to be irrelevant. The gap $\Delta_{S_{(220)}}\geq2.8$ is  weaker than the marginality condition and it can generate a lower cut on $\Delta_{\cM_{1/2}}$ below the perturbative result $1.022$. The gap in $\Delta_{T_{(000)}}$ can affect the upper bound in Fig. <ref>.  A weaker gap in this sector gives a higher upper bound on $\Delta_r$. According to the $1/N_f$ expansion results in Table <ref>, the leading order result gives $\Delta_{T_{(000)}}\simeq 4.42$, so the gap $\Delta_{T_{(000)}}\geqslant 4.0$ actually assumes the higher order corrections will not reduce the scaling dimension drastically. The next scalar in the parity odd $A_{(211)}$ sector can be constructed by contracting the spin indices of the $SU(4)$ conserved current and the topological $U(1)_t$ conserved current $J^{f}_{\mu}J^{t\mu}$, which has scaling dimension 4 in the large $N_f$ limit. We assume this operator remains irrelevant at $N_f=4$. The gap in this sector can affect the lower bound on $\Delta_r$.

\begin{aligned}
\Delta_{S_{(000)}}^{\ell=0} &\geq 3.0 \\
\Delta_{S_{(220)}}^{\ell=0} &\geq 2.8 \\
% \Delta_{A_{(211)},{\rm 2nd}}^{\ell=0} &\geq 3.0 \\
\Delta_{r'} &\geq 3.0 \\
\Delta_{T_{(000)}}^{\ell=0} &\geq 4.0
\end{aligned}

Bootstrap bound on the scaling dimensions of the monopole $\cM_{1/2}$ and adjoint fermion bilinear operator $r$ at $\Lambda=31$. To get this bound, we assumed the scaling dimensions of the lowest parity even singlet scalar and second parity odd $SU(4)$ adjoint scalar are irrelevant, the lowest scalar in the $A_{(220)}$ sector is above $\Delta>2.8$, and the lowest scalar in the $T_{(000)}$ sector has scaling dimension $\Delta>4.0$. The green dot denotes the large $N_f$ expansion estimate.

The three sectors with gaps, $S_{(220)}$, $A_{(211)}$, and $T_{(000)}$, together with the isolated operator $r$, appear in the $SO(12)\rightarrow SU(4)\times SO(2)$ branching rule (<ref>). In the physical spectrum of $N_f=4$ QED$_3$, the lowest scalars in these four sectors have rather different scaling dimensions, as they carry different charges under the parity symmetry. Therefore, the  spectrum in the monopole bootstrap strongly breaks the enhanced $SO(12)$ symmetry in the algebraic relation (<ref>)!
In contrast, in the $SU(4)$ adjoint crossing equations,  all the operators appearing in the $SO(15)\rightarrow SU(4)$ branching rule (<ref>) are parity even and the lowest operators in these sectors have the same scaling dimensions at the leading order. The $SO(15)$ symmetry is only broken mildly by the higher order $1/N_f$ corrections.

\includegraphics[width=0.4\linewidth]{figs/pltExtrapolationMLeft} \quad
\includegraphics[width=0.4\linewidth]{figs/pltExtrapolationRUpDown} \\
\includegraphics[width=0.4\linewidth]{figs/pltExtrapolationCJ} \quad
\includegraphics[width=0.4\linewidth]{figs/pltExtrapolationCJT}
\end{aligned}$

Upper left: Extrapolations of the bootstrap bounds on scaling dimensions of $\Delta_{M_{1/2}}$ with fixed $\Delta_r=1.43$.
Upper right: Extrapolations of the bootstrap bounds on scaling dimensions of $\Delta_r$ with fixed $\Delta_{\cM_{1/2}}=1.022$.
Lower left: Extrapolations of the central charge of the $SU(4)$  conserved current at fixed $\Delta_{\cM_{1/2}}=1.022$ and $\Delta_r=1.43$.
Lower right: Extrapolations of the central charge of topological $U(1)_t$  conserved current at fixed $\Delta_{\cM_{1/2}}=1.022$ and $\Delta_r=1.43$.
The red dots denote the extrapolation's prediction at $\Lambda=\infty$, and black dashed lines denote the large-$N_f$ perturbation theory prediction.

With the above gap assumptions, the bootstrap bound on the scaling dimension of the lowest parity odd $SU(4)$ adjoint scalar forms an interesting peninsula structure and the $1/N_f$ expansion results locate near the tip of the peninsula. Due to the special role of parity symmetry, the monopole bootstrap combined with gap assumptions inspired by the large $N_f$ QED$_3$ spectrum is more effective at carving out the CFT parameter space as compared with the $SU(4)$ fermion bilinear bootstrap shown in Fig. <ref>.

We would like to make two remarks about the results in Fig. <ref>. Firstly the bound has a clear gap-dependence. The boundary in different directions is determined by the gaps in certain sectors. Due to this fact, our current bootstrap setup cannot be used to solve the target theory without extra specific input. We think this is a general problem for the bootstrap studies of non-$SO(N)$ vector scalars with scaling dimensions notably above the unitary bound. Another fact that the readers should keep in mind is that though we have pushed the bootstrap numerical precision to $\Lambda=31$, the bound is far from converged. This fact can be seen in Fig. <ref>. In the top two panels of Fig. <ref>, we show the extrapolations of the bootstrap bounds at different $\Lambda$ with fixed $\Delta_r=1.43$ or $\Delta_{\cM_{1/2}} = 1.022$.
It requires much higher $\Lambda$ to have the lower or upper bounds close to the optimal bounds in the linear extrapolations.  In the lower two panels of Fig. <ref>, we show the extrapolations of the lower bounds on the central charges of the $SU(4)$ conserved current and the topological $U(1)_t$ conserved current with fixed $\{\Delta_{\cM_{1/2}},\Delta_r\}=\{1.022, 1.43\}$.
Interestingly, the $SU(4)$ conserved current central charge bound has a large $\Lambda$ extrapolation at $c_J\simeq 0.95$, not far from the $1/N_f$ perturbative prediction $c_J\simeq 1.04$. Similarly, the $U(1)_t$ conserved current central charge has a large $\Lambda$ extrapolation at $c_J^t\simeq 1.50$, and  the $1/N_f$ expansion at subleading order predicts $c_J\simeq 1.56$. Extrapolation of the stress tensor central charge goes to $c_T\simeq 0.89$ at large $\Lambda$, which is somewhat lower than the $1/N_f$ expansion result $c_T\simeq 1.18$. This is consistent with the observation that the bootstrap bounds in the singlet sectors are relatively weaker than those in the non-singlet sectors.

§.§ Closed islands from monopole single correlator bootstrap with interval positivity assumptions

In the last section we have shown that non-trivial peninsula structures show up if we break the $SO(12)\rightarrow SU(4)\times U(1)_t$ symmetry enhancement by physically inspired gap assumptions. There we did not impose any assumptions on the spectrum in the $T_{(220)}$ sector.
As a part of the branching rule (<ref>), the spectrum in this sector also plays an important role in the monopole bootstrap.

The $N_f=4$ QED$_3$ spectrum in the $T_{(220)}$  sector is shown in Table <ref>. According to the $1/N_f$ expansion at sub-leading order, the lowest charge 1 monopole operator in the $T_{(220)}$
sector has scaling dimension $\Delta_{\cM_1}\simeq 2.5$. An interesting fact is
that the second scalar in this sector has a significantly higher scaling dimension at leading order $\Delta_{\cM'_1}^0\simeq 6.16$. To take advantage of this big gap while still allowing uncertainty about the precise value of $\Delta_{\cM_1}$, we employ an interval positivity assumption, namely, we assume an upper bound on the dimension of the lowest charge 1 monopole operator, $\Delta_{\cM_1}\leqslant\Delta_{\cM_1}^{\rm max}$, together with a lower bound on the dimension of the next operator in the same channel $\Delta_{\cM'_1}\geqslant\Delta_{\cM_1'}^{\rm min}>\Delta_{\cM_1}^{\rm max}$. Assumptions of this type can be efficiently studied with a modification to the bootstrap algorithm, see Appendix <ref> for more details.
A quick-and-dirty setup for the interval positivity constraints is to sample the interval $\Delta_{\cM_1}\in [\Delta_{\cM_1}^{\rm min},\Delta_{\cM_1}^{\rm max}]$ with many isolated points, and refine the sampling until the bounds stop changing. A more rigorous setup is to redefine

$$\Delta_{\cM_1} = \Delta_{\cM_1}^{\rm min} + \frac{x}{1+x} (\Delta_{\cM_1}^{\rm max}-\Delta_{\cM_1}^{\rm min})$$

such that the interval is mapped to $x>0$. In practice both methods give reliable results. The interval positivity constraints is much more efficient than isolating $\Delta_{\cM_1}$ and scanning its value.].

We refer to the perturbative results given in Table <ref> when making assumptions on $\Delta_{\cM_1}^{\rm max}$ and $\Delta_{\cM_1'}^{\rm min}$. Specifically, we take $\Delta_{\cM_1'}^{\rm min}=5.0<\Delta_{\cM'_1}^0\simeq 6.16$, and will test gaps $\Delta_{\cM_1}^{\rm max}=2.5, 2.6$, which are inspired by the $1/N_f$ expansion result $\Delta_{\cM_1}\simeq 2.5$.
The  interval positivity assumptions can provide surprisingly strong constraints on the CFT data. We will then compare the bootstrap results with the perturbative and lattice CFT data of $N_f=4$ QED$_3$.

1.0in $
\begin{aligned}
\Delta_{S_{(000)}}^{\ell=0} &\geq 3.0 \\
\Delta_{S_{(220)}}^{\ell=0} &\geq 2.8 \\
% \Delta_{A_{(211)},{\rm 2nd}}^{\ell=0} &\geq 3.0 \\
\Delta_{r'} &\geq 3.0 \\
\Delta_{T_{(000)}}^{\ell=0} &\geq 4.0 \\
\Delta_{\cM_1'} &\geq 5.0
\end{aligned}$

Bounds on the scaling dimensions of $(\Delta_{\cM_{1/2}}, \Delta_r)$ with the same gaps as in Fig. <ref> along with the interval positivity assumptions: $\Delta_{\cM'_1}\geqslant 5.0$ and $\Delta_{\cM_1}\leqslant 2.5, 2.6$. We used $\Lambda=39$ in the bootstrap computations.

Bootstrap results with these different interval positivity assumptions are shown in Fig. <ref>. Remarkably, with these gap assumptions inspired by the perturbative $N_f=4$ QED$_3$ spectrum, the CFT data $(\Delta_{\cM_{1/2}}, \Delta_r)$ can be restricted into closed islands!
The shapes of the islands are gap-dependent, and become very small if we take $\Delta_{\cM_1}^{\rm max}=2.4$ and disappear with smaller $\Delta_{\cM_1}^{\rm max}$. The island is still closed at $\Delta_{\cM_1}^{\rm max}=2.65 \,(\Lambda=31)$, extending to a maximum $\Delta_{\cM_{1/2}}\simeq 1.4$.
Note that the bounds shown in the plot are computed with relatively high numerical precision ($\Lambda=31$), however, they are not well converged yet and are actually affected by the issue of slow convergence. This can be qualitatively seen through the linear extrapolation of the bound to the large $\Lambda$ limit. In Fig. <ref>, we show the maximum values of $\Delta_{\cM_{1/2}}$ at fixed $\Delta_r=1.43$ in the islands computed at different values of $\Lambda$, and their linear extrapolation to $\Lambda=\infty$. Surprisingly, if we set the gap $\Delta_{\cM_1}^{\rm max}$ at the perturbative estimate $\Delta_{\cM_1}^{\rm max}=2.5~(\simeq \Delta_{\cM_1})$, the upper bound on $\Delta_{\cM_{1/2}}$ extrapolates to $\Delta_{\cM_{1/2}}\simeq1.04$, close to the perturbative result $\Delta_{\cM_{1/2}}\simeq1.02$. The left part of the island coincides with the tip of the peninsula structure in Fig. <ref>, in which the minimum $\Delta_{\cM_{1/2}}$ with fixed $\Delta_r=1.43$ extrapolates to $\Delta_{\cM_{1/2}}\simeq1.02$, as shown in Fig. <ref>.
Therefore with the interval positivity assumptions $\Delta_{\cM_{1}}\leqslant 2.5$, $\Delta_{\cM'_{1}}\geqslant 5.0$, our bootstrap implementation gives a closed island in $(\Delta_{\cM_{1/2}}, \Delta_r)$, which shrinks to a rather small region consistent with the perturbative predictions.

Here we would like to remind the readers that the gap $\Delta_{A_{(220)}} \geqslant 2.8$, which effectively determines the minimum $\Delta_{\cM_{1/2}}$, is chosen by hand (but without tuning), and the agreement between the linear extrapolation of the left edge of the bootstrap result and the perturbative result could be considered accidental. On the other hand, the assumed maximum value $2.5$ for $\Delta_{\cM_1}$, which affects the maximum $\Delta_{\cM_{1/2}}$ in the island, is coming from the perturbative result at subleading order. These gaps together conspiratorially restrict the CFT data close to the perturbative QED$_3$ spectrum. If we relax the maximum of $\Delta_{\cM_1}$ to $2.6$, the left part of the closed island remains the same, while its right side increases to $\Delta_{\cM_{1/2}}\simeq 1.25$ ($\Lambda=31$), which overlaps with the lattice results enclosed by the red dashed rectangle in Fig. <ref>. However, the right part of the island shrinks a lot at higher $\Lambda$. The linear extrapolation of the maximum $\Delta_{\cM_{1/2}}$ at fixed $\Delta_r=1.43$ in the islands gives the estimate $\Delta_{\cM_{1/2}}\simeq 1.14$ at $\Lambda=\infty$, which marginally excludes the lattice results.

\includegraphics[width=0.75\linewidth]{figs/pltExtrapolationInterval}\\
\includegraphics[width=0.75\linewidth]{figs/pltExtrapolationIntervalRUpDown}
\end{aligned}$
Top panel: extrapolations of the maximum $\Delta_{\cM_{1/2}}$ at fixed $\Delta_r=1.43$ in the islands with gaps $\Delta_{\cM_1}\leqslant 2.5, 2.6$. Bottom panel: extrapolations of the upper and lower bounds on the scaling dimension $\Delta_r$ in the island with $\Delta_{\cM_1}^{\rm max}=2.6$, at fixed $\Delta_{\cM_{1/2}}=1.08$, which is the center of the range $\Delta_{\cM_{1/2}}\in (1.02, 1.14)$ obtained from the large $\Lambda$ extrapolation. The red dots denote the $1/N_f$ perturbative results for $\Delta_{\cM_{1/2}}$ and $\Delta_r$.

\begin{aligned}
\Delta_{S_{(000)}}^{\ell=0} \geq 3.0, ~
\Delta_{S_{(220)}}^{\ell=0} \geq 2.8, ~
\Delta_{r'}\geq 3.0,
\Delta_{T_{(000)}}^{\ell=0} \geq 4.0, ~
\Delta_{\cM_1} \leq 2.6, ~
\Delta_{\cM_1'} \geq 5.0 \hspace{4cm}
\\
\includegraphics[width=0.33\linewidth]{figs/cJ_Island_2pt60}
\includegraphics[width=0.33\linewidth]{figs/ctJ_Island_2pt60}
\includegraphics[width=0.33\linewidth]{figs/cT_Island_2pt60}
\end{aligned}
Lower bounds ($\Lambda=31$) on the $SU(4)$ conserved current central charge $c_J$ (left), topological $U(1)_t$ conserved current central charge $c_J^t$ (middle), and the stress tensor central charge $c_T$ (right), inside the island of Fig. <ref> with the interval positivity assumption $\Delta_{\cM_1}^{\rm max}=2.60$ (other gap assumptions are the same as in Fig. <ref>). The green contours denote the $1/N_f$ perturbative results at subleading order: $c_J=1.04$, $c_J^t=1.56$, and $c_T=1.18$. For the $(\Delta_{\cM_{1/2}},\Delta_{r})$ inside the green contours, the bounds on the central charges are consistent with the perturbative results, while in the right part of the island, they are significantly higher than the perturbative results.

More restrictive constraints come from the lower bounds on the central charges $c_J$, $c_J^t$, and $c_T$, which are shown in Fig. <ref>.
The large $N_f$ perturbative results on the central charges are given by the green contours. Inside the contours the central charges have lower bounds below the perturbative results.
In the right part of the island with $\Delta_{\cM_{1/2}}>1.15$, the lower bounds on conserved current central charges quickly increase to the range $c_J>1.5$ and $c_J^t>2.5$, significantly above the $1/N_f$ perturbative results at subleading order $c_J\simeq 1.04$ and $c_J^t\simeq 1.56$. Such big discrepancies are unlikely to be explained by the higher order corrections, which indicate the bootstrap bounds in Fig. <ref> are inconsistent with the lattice results on $N_f=4$ QED$_3$.
Nevertheless, this contradiction  should not be simply interpreted to exclude the lattice results, as our bootstrap bounds are gap-dependent.
By relaxing the gap assumptions, e.g., using an interval positivity assumption with $\Delta_{\cM_1}^{\rm max} > 2.6$ in the bootstrap implementation, one can obtain weaker bootstrap bounds in which the lattice results locate in the allowed region. In the next subsection we will study additional bootstrap bounds with different gap assumptions which provide some necessary conditions for the lattice results to be physical. Here the roles of central charges are quite reminiscent of their roles in Fig. <ref>, where in comparison with the allowed parameter space of the operator scaling dimensions, bounds on the central charges provide more restrictive constraints for conformal QED$_3$.

The above numerical experiment is surprising to us in two aspects. From the bootstrap point of view, it is a welcome surprise that the bootstrap algorithm, though affected by the gap-dependence problem, can effectively capture a special solution which is rather close to the perturbative estimates of $N_f=4$ QED$_3$. Note that due to the parity symmetry, operators in different sectors have diversified scaling dimensions; the conserved current central charges also have notable differences both in their physical meanings and  magnitudes.  Therefore it is highly nontrivial that several of these properties can be simultaneously satisfied by the bootstrap constraints. From the QED$_3$ side, we do not have solid evidence on the gap $2.8$ in the $A_{(220)}$ sector, and the current perturbative results on the monopole spectrum and central charges may still receive notable higher order corrections. In this sense, it is surprising that the perturbative CFT data taken at face value can seemingly provide a consistent solution to the bootstrap equations.

Since our bootstrap results are gap-dependent, their physical relevance relies on the validity of the gap assumptions in our bootstrap implementation.
Given our gap assumptions are consistent or close to the physical spectrum, then our bootstrap results are closely relevant to the physical solution of $N_f=4$ conformal QED$_3$, which have significant meanings both for understanding the IR phases of QED$_3$ and its applications in condensed matter systems.
On the other hand, we cannot exclude the possibility, although less likely, that few of our gap assumptions  strongly violate the physical spectrum, and the coincidences between our bootstrap results and perturbative CFT data are purely accidental.
To verify the two possibilities, we suggest to compute the CFT data using other nonperturbative approaches, e.g., the lattice simulations. The scaling dimensions of the lowest scalar in $A_{(220)}$ and the charge 1 monopole operators are especially important in our bootstrap setup. Reliable estimations of these operators can verify whether our assumptions are consistent with the physical spectrum.

§.§ Bound on the charge 1 monopole operator M1 and the lattice results

In this subsection we study the bootstrap bounds on the scaling dimension of charge 1 monopole operator $\cM_1$ and the bounds on the central charges in the resulting allowed region. The results will explain why the interval positivity assumptions can generate closed islands. We will additionally provide more comparisons between the bootstrap bounds and the lattice results. Since our bootstrap results are gap-dependent and the gap assumptions are not strictly established yet, our results cannot verify or exclude the lattice results by themselves. Nevertheless, they can provide strong necessary conditions for the lattice results to be physical.

Bootstrap bounds on the scaling dimension of the lowest charge 1 monopole $\cM_{1}$ at $\Lambda=31$.
To get the bounds, we have employed gap assumptions on the lowest parity even singlet scalar $\Delta_{S_{(000)}}\geqslant 3.0$, the second lowest charge 1 monopole operator $\cM_{1}'$ in the $T_{(220)}$ sector $\Delta_{\cM_1'}\geqslant 5.0$, and the lowest scalar in the $S_{(220)}$ sector $\Delta_{A\bar{A}}\geqslant 2.4, 2.8, 3.0,3.2$.

First let us consider the bootstrap bounds on the scaling dimension of the lowest charge 1 monopole operator $\cM_{1}$. The results are shown in Fig. <ref>.
To get the bounds we used the gap assumptions $\Delta_{S_{(000)}}\geqslant3.0$ and $\Delta_{\cM_1'}\geqslant 5.0$ for the second lowest charge 1 monopole $\cM_1'$. The bootstrap bounds change notably with different gaps $\Delta_{A\bar{A}}\geqslant \Delta^*$ for the lowest scalar in the $S_{(220)}$ sector.
The most interesting point in  Fig. <ref> is that the bootstrap allowed region forms a wave structure when the $S_{(220)}$ gap is in the range $\Delta^*\leqslant 3$.
The $1/N_f$ perturbative results locate near the tip of the bootstrap bound associated with the gap $\Delta^*=2.8$. For larger gaps $\Delta^*\geqslant 3.2$ the wave structure disappears.
The wave structures in the $\cM_1$ bootstrap bounds are reminiscent of the bootstrap bound on the 3d critical Ising model with a gap on the second $\mathbb{Z}_2$ even scalar [80],
while the gaps imposed here are not fully justified as our knowledge of $N_f=4$ QED$_3$ is limited. Due to the wave structure in Fig. <ref>, the interval positivity condition $\Delta_{\cM_1}\leqslant \Delta_{\cM_1}^{\rm max}$ truncates the tip of the wave structure below $\Delta_{\cM_1}^{\rm max}$, which disconnects from the right part of the bulk region and forms a closed island. Bootstrap bounds on the scaling dimension of the monopole $\cM_1$ were first presented in [51], in which the authors introduced a weaker gap assumption on the scaling dimension of the second lowest monopole $\cM'_1$ and the bootstrap bound shows a weaker peninsula structure. The sharp wave structure we see here can not appear unless the stronger gap assumption on  $\Delta_{\cM'_{1}}$ is imposed.

In Fig. <ref>, we also present the lower bound on the topological conserved current central charge $c_J^t$ inside the wave structure, where its $1/N_f$ perturbative prediction is given by the green contour. Similar to the results in Fig. <ref>, the central charge $c_J^t$ has a much higher lower bound in the right part of the allowed region. According to the $c_J^t$ lower bound, it requires the scaling dimension of the monopole $\cM_1$ to be above $\Delta_{\cM_1}\geqslant 2.67$, or even higher values for the $\Delta_{\cM_{1/2}}$ in the range predicted by the lattice results [43].

\begin{aligned}
\Delta_{S_{(000)}}^{\ell=0}\geq3.0 ,
% \Delta_{S_{(310)}}^{\ell=0}\geq3.0 ,~
\Delta_{S_{(220)}}^{\ell=0}\geq \mathbf{2.8} ,
\Delta_{\cM_1'}\geq5.0  \hspace{1.2cm}
\\
\includegraphics[width=0.45\linewidth]{figs/M1boundgap5-cJt}
\end{aligned}
\;\;
\begin{aligned}
\Delta_{S_{(000)}}^{\ell=0}\geq3.0 ,
% \Delta_{S_{(310)}}^{\ell=0}\geq3.0 ,~
\Delta_{S_{(220)}}^{\ell=0}\geq\mathbf{3.0} ,
\Delta_{\cM_1'}\geq5.0  \hspace{1.2cm}
\\
\includegraphics[width=0.45\linewidth]{figs/M1boundgap5-cJt-aab3.0}
\end{aligned}

Lower bounds on the  topological $U(1)_t$ central charge $c_J^t$ with different gap assumptions at $\Lambda=31$.
The green dot denotes the perturbative results of the monopole scaling dimensions $(\Delta_{\cM_{1/2}},\Delta_{\cM_1})\simeq(1.022, 2.499)$, and the green line shows the contour $c_J^t=1.56$ predicted by the $1/N_f$ expansion. The lattice value $\Delta_{\cM_{1/2}}=1.252(84)$ [43] is given by the pink shaded region.

§ BOOTSTRAPPING MIXED CORRELATORS WITH M1/2 AND R

We have shown that the single correlator bootstrap results can provide strong constraints on the conformal $N_f=4$ QED$_3$.
To improve the the bootstrap results, the key is to find a more restrictive bootstrap implementation. A straightforward generalization of our work is to bootstrap mixed correlators with multiple operators in $N_f=4$ QED$_3$. In this section, we perform a mixed correlator bootstrap study of conformal $N_f=4$ QED$_3$ with emphasis on the two low-lying scalars $r$ and $\cM_{1/2}$. We will show that this bootstrap setup indeed can significantly improve the lower cuts of the closed islands obtained in <ref>.[
A similar study of the same mixed correlators was also performed in [58], which obtained general constraints on the possible stable critical phases of Dirac spin liquids on triangular and kagome lattices.]

Bounds on the scaling dimensions of the operators $\cM_{1/2}$ and $r$ obtained from the $r-\cM_{1/2}$ mixed correlator bootstrap are shown in Fig. <ref>. Details on the mixed correlator bootstrap implementation are presented in Appendix <ref>. In the mixed correlator bootstrap, we used the same gap assumptions as in the monopole single correlator bootstrap, including the interval positivity assumption. In addition, we also required that the lowest scalar in the $S\bar{S}$, i.e. $(422)$, representation of the $SU(4)$ flavor symmetry is irrelevant. Compared with the single correlator bootstrap bound, the mixed correlator bootstrap significantly improves the lower bound on the scaling dimension of $\Delta_r$ in the closed island: $\Delta_r\geqslant 1.12$ at $\Lambda=27$.

\begin{aligned}
\includegraphics[width=0.8\linewidth]{figs/island-single-vs-mixed-narrow}
\end{aligned}
\;\;
\begin{aligned}
\Delta_{S_{(000)}}^{\ell=0} &\geq 3.0 \\
\Delta_{S_{(220)}}^{\ell=0} &\geq 2.8 \\
\Delta_{r'} &\geq 3.0 \\
\Delta_{T_{(000)}}^{\ell=0} &\geq 4.0 \\
\Delta_{\cM_1} &\leqslant 2.6 \\
\Delta_{\cM_1'} &\geq 5.0  \\ \\
\textrm{mixed:} \\
\Delta_{S\bar{S}}^{\ell=0} &\geq 3.0
\end{aligned}
Bounds on the scaling dimensions of $(\Delta_{\cM_{1/2}}, \Delta_r)$ with the interval positivity assumption $\Delta_{\cM_1}\leqslant 2.60$,
comparing monopole single correlator bootstrap results with the monopole-adjoint mixed correlator bootstrap results at $\Lambda=27$.

The large $N_f$ prediction and fermion bilinear bootstrap bounds on the scaling dimension of the lowest scalar in the $S\bar{S}$ sector were shown in Fig. <ref>, from which we expect the gap $3.0$ is a reliable assumption. Moreover, the fermion bilinear bootstrap bound on $\Delta_{S\bar{S}}$ in Fig. <ref> explains why the we can obtain a stronger minimum on $\Delta_{r}$ after introducing the gap $\Delta_{S\bar{S}}\geqslant3.0$: the upper bound on $\Delta_{S\bar{S}}$ cannot be higher than 3 for $\Delta_r<1.12$. This provides a nice example which illustrates how the mixed correlator bootstrap can help to get stronger bounds with reliable assumptions. The bootstrap bounds in certain sectors are more restrictive and the mixed correlator bootstrap implementation can help to exploit the constraints in these sectors. We expect there are extra sectors, especially in the $\cM_{1/2}$ and $\cM_1$ mixed correlator setup, which can provide strong constraints on the CFT data with reliable gap assumptions. We hope to give a more systematic study of these constraints in our next work.

In the bootstrap studies of $N_f=4$ conformal QED$_3$, several operators in different sectors play important roles. Their scaling dimensions relate to higher dimensional structures in the parameter space of CFT data. In Fig. <ref> we make a first attempt to map out such a higher dimensional structure at \(\Lambda=19\). Specifically we show the closed 3d allowed region in the space $(\Delta_r,\Delta_{\cM_1},\Delta_{A\bar{A}})$ with different fixed scaling dimensions of $\cM_{1/2}$, $\Delta_{\cM_{1/2}}=0.98, 1.02, 1.06$, making a set of plausible gap assumptions. The bootstrap allowed regions are 3d slices of a more complicated higher dimensional geometric structure and so have interesting shapes. The ranges of the islands with different fixed $\Delta_{\cM_1/2}$ have been summarized in Table <ref>. In particular, by taking the large $N_f$ result $\Delta_{\cM_1/2}\simeq 1.02$, the perturbative predictions of $\Delta_r\simeq1.43$ and $\Delta_{\cM_1}\simeq2.50$ are located inside the 3d island. The large $N_f$ prediction of $\Delta_{A\bar{A}}\simeq2.38$ is slightly outside of the island, while the gap $\Delta_{A\bar{A}}\geqslant 2.8$ used in other sections is consistent with the range of $\Delta_{A\bar{A}}$ in the 3d island.

\begin{aligned}
\includegraphics[width=0.8\linewidth]{figs/islands-3d}
\end{aligned}
\;\;
\begin{aligned}
\Delta_{\cM_{1/2}'} &\geqslant 3.0 \\
\Delta_{r'} &\geqslant 3.0 \\
\Delta_{\cM_1'} &\geqslant 4.5 \\
\Delta_{A\bar{A}'}^{\ell=0} &\geqslant 4.5 \\
\Delta_{S_{(000)}}^{\ell=0} &\geqslant 3.0 \\
\Delta_{S\bar{S}}^{\ell=0} &\geqslant 3.5 \\
\Delta_{V_{(200)}}^{\ell=0} &\geqslant 3.0 \\
\Delta_{V_{(321)}}^{\ell=0} &\geqslant 3.0 \\
\Delta_{V_{(321)}}^{\ell=1} &\geqslant 3.0 \\
\Delta_{T_{(000)}}^{\ell=0} &\geqslant 4.0 \\
\end{aligned}
Bounds on the scaling dimension of $(\Delta_r, \Delta_{A\bar{A}}, \Delta_{\cM_{1}})$ from the monopole-adjoint mixed correlator bootstrap ($\Lambda=19$) at fixed $\Delta_{\cM_{1/2}}$. The islands in the plot, from the largest to the smallest, correspond to $\Delta_{\cM_{1/2}} = 1.06, 1.02,$ and $0.98$, respectively. The full dynamical version of this 3-dimension plot is included in the attached athematica notebook.

§ CONCLUSIONS AND DISCUSSIONS

The broad goal of the conformal bootstrap project is to find and classify CFTs.
On the other hand, non-supersymmetric gauge theories have so far shown resistance to being solved numerically using bootstrap methods.[However, remarkable progress towards numerically solving conformal gauge theories with extended supersymmetry has been made in the recent work [81, 82].]
In this work we have attempted to study the presumed IR fixed point of $N_f=4$ QED$_3$ using the conformal bootstrap. Most notably, we found that after imposing some assumptions inspired by perturbative computations for \(N_f=4\) we can obtain a closed island in parameter space. The ranges of the islands under different bootstrap setups are summarized in Table <ref>.
Promisingly, bounds in this island on the scaling dimensions $\Delta_{\cM_{1/2}}$, $\Delta_{\cM_1}$, $\Delta_r$ as well as on the central charges $c_J$, $c_J^t$ and $c_T$ are consistent with their $1/N_f$ perturbative results, which in turn are close to saturating the bootstrap bounds. It is important to be clear that the physical relevance of these results relies on the validity of the gap assumptions used in our bootstrap computations, but nevertheless we believe our work has progressed our understanding of $N_f=4$ conformal QED$_3$.

A major challenge in getting precise results from the bootstrap is the notable sensitivity of the bounds to assumed gaps in the spectrum, closely connected to the symmetry-enhancement phenomena discussed in [54, 55]. The crossing equations of the single four-point correlators $\langle rrrr\rangle$ and $\langle  \cM_{1/2}\cM_{1/2}\cM_{1/2}\cM_{1/2}\rangle$ have positivity properties that can be mapped to the crossing equations of $SO(15)$ and $SO(12)$ vectors, respectively. To bootstrap non-$SO(N)$ symmetric theories, one has to impose gap assumptions which explicitly break the $SO(N)$ symmetries, and intriguing kinks and peninsulas which appear in the bootstrap bounds show a clear dependence on these gap assumptions. Despite this gap sensitivity, we believe that these discontinuities could still be of physical relevance to our understanding of , in the sense that they could be directly connected to the physical solution through larger geometrical structures in scaling-dimension space. We have gained some confidence in this interpretation by inputting a set of gap assumptions inspired by the perturbative spectrum, and seeing that lower bounds on the stress tensor and current central charges near these kinks are nicely compatible with their estimated values from $1/N_f$ perturbation theory. In particular, this makes it seem unlikely that the kinks are related to non-Abelian gauge theories, which have significantly larger values of the central charges.

The ranges of scaling dimensions of the adjoint fermion bilinear $r$, lowest charge 1 monopole $\CM_1$, and the lowest $A\bar{A}$-rep scalar, in the 3-dimensional islands of fixed $\Delta_{\CM_{1/2}}$ values. The islands and the gap assumptions are shown in Figure <ref>. The island corresponding to the large-$N_f$ value of $\Delta_{\CM_{1/2}}$ is highlighted in bold font. See the body of this paper for further discussion of these values.
\hline\hline\\[-1em]
\Delta_{\CM_{1/2}}  \hspace*{2em}& \Delta_{r} \hspace*{5em}& \Delta_{\CM_1} \hspace*{4.5em}& \Delta_{A\bar{A}} \\
0.98 & (1.36, 1.52)~~ & (2.27, 2.46)~~ & (2.43, 2.76)~~ \\
\bf 1.02 & \bf(1.30, 1.66)~~ & \bf(2.28, 2.60)~~ & \bf(2.39, 2.91)~~\\
1.06 & (1.26, 1.79)~~ & (2.29, 2.75)~~ & (2.33, 3.06)~~\\
\end{array}$

The parity symmetry of $N_f=4$ QED$_3$ makes the monopole bootstrap particularly effective in separating from other solutions to crossing equations. Operators appearing in the $\cM_{\pm1/2}\times\cM_{\pm1/2}$  OPE carry different parity charges depending on their representations of $SU(4)\times U(1)_t$, and their scaling dimensions strongly break the $SU(4)\times U(1)_t\rightarrow SO(12)$ relation between the crossing equations. In contrast, in the fermion bilinear bootstrap, operators in different sectors branched from $SO(N)$ representations have the same parity charges and their scaling dimensions only differ by loop corrections in the $1/N_f$ expansion. By inputting gap assumptions inspired by the $N_f=4$ QED$_3$ perturbative spectrum, in particular an expected large gap until the second charge $1$ monopole, we are able to find a sharp peninsula structure in \((\Delta_{\cM_{1/2}},\Delta_{\cM_1})\) whose narrow tip coincides neatly with the perturbative estimates of the theory. The peninsula structure remains gap dependent, and the gap assumption in the $S_{(2,2,0)}$ sector is particularly important as it determines the minimum value of $\Delta_{\cM_{1/2}}$. As emphasized recently in [58], the leading operator in this sector is also physically important because its relevance or irrelevance determines whether can be reached in lattice systems. We found that a gap $\Delta_{S_{(2,2,0)}}\geqslant 2.8$ allows for a nice consistency with $1/N_f$ perturbation theory, while irrelevance of this operator implies that uncomputed $1/N_f$ corrections to \((\Delta_{\cM_{1/2}},\Delta_{\cM_1})\) should be of the same order as computed ones. It will be important in future work to determine which of these scenarios is correct.

A summary of the ranges of scaling dimensions of the lowest charge-1/2 monopole $\cM_{1/2}$ and the $SU(4)$ adjoint fermion bilinear $r$ in different setups. The shared gap assumptions are shown in Figure <ref>, and the assumptions specific to the different setups are presented in the first row.
\hline\hline\\[-1em]
\textbf{setup} \hspace*{7em}&\Delta_{\cM_{1/2}}\textbf{ range} \hspace*{3em}& \Delta_{r}\textbf{ range}
\\\hline\\[-1em]
\begin{aligned}
\Lambda=39~\text{single} \\ \Delta_{\cM_1} \leq 2.50
\end{aligned}
&(1.00, 1.10)&(1.03, 1.61)
\\\hline\\[-1em]
\begin{aligned}
\Lambda=39~\text{single} \\ \Delta_{\cM_1} \leq 2.60
\end{aligned}
&(1.00, 1.22)&(0.73, 1.81)
\\\hline\\[-1em]
\begin{aligned}
\bigcap \makecell{\Lambda=39~\text{single}\\\Lambda=27~\text{mixed}} \\
\left\{\makecell{\Delta_{\cM_1} \leq 2.60\\\Delta_{S\bar{S}} \geq 3.50 } \right.
\end{aligned}
&(1.00, 1.22)&(1.12, 1.81)
\\\hline\\[-1em]
\begin{aligned}
\Lambda=31~\text{single} \\
\left\{\makecell{\Delta_{\cM_1} \leq 2.60\\c_J^t\leq 2 } \right.
\end{aligned}
&(1.00, 1.14)&(0.98, 1.77)
\\\hline\hline
\end{array}$

\hline\hline\\[-1em]
~~\text{\bf setup} ~ & ~
\makecell{\Lambda=39~\text{single} \\ \Delta_{\cM_1} \leq 2.50} ~&~
\makecell{\Lambda=39~\text{single} \\ \Delta_{\cM_1} \leq 2.60} ~ &~
\makecell{
\bigcap \makecell{\Lambda=39~\text{single}\\\Lambda=27~\text{mixed}} \\
\left\{\makecell{\Delta_{\cM_1} \leq 2.60\\\Delta_{S\bar{S}} \geq 3.50 } \right.
} ~&~
\makecell{\Lambda=31~\text{single} \\
\left\{\makecell{\Delta_{\cM_1} \leq 2.60\\c_J^t\leq 2 } \right.
}~ \\
\bf \Delta_{\cM_{1/2}}~\text{range} & ~~(1.00, 1.10)~~ & ~~(1.00, 1.22)~~ & ~~(1.00, 1.22)~~ & ~~(1.00, 1.14)~~ \\
\bf \Delta_{r}~\text{range} & ~~(1.03, 1.61)~~ & ~~(0.73, 1.81)~~ & ~~(1.12, 1.81)~~ & ~~(0.98, 1.77)~~ \\
\end{array}$

A summary of the ranges of scaling dimensions of the lowest charge-1/2 monopole $\cM_{1/2}$ and the $SU(4)$ adjoint fermion bilinear $r$ in different setups. The shared gap assumptions are shown in Figure <ref>, and the assumptions specific to the different setups are presented in the first row.

Adopting the assumption that we should take perturbation theory at least somewhat seriously, our most notable results are obtained by imposing an interval positivity assumption $\Delta_{\cM_1} \leqslant 2.6$, that $\Delta_{\cM_1}$ required to be near or below its subleading perturbative estimate $\Delta_{\cM_1} \approx 2.5$, which in turn restricts the peninsula structure to a closed island. Notably, this gives a closed region for the fermion bilinear dimension $\Delta_{r}$ as well as for $\Delta_{\cM_{1/2}}$. Our bootstrap island at $\Lambda=31$ overlaps with previous lattice estimates for $\Delta_r$ and $\Delta_{\cM_{1/2}}$ [41, 59, 43]. However, by computing lower bounds on the central charges $c_J$, $c_J^t$ and $c_T$ inside this island, we see that the lattice estimates of $(\Delta_{\cM_{1/2}},\Delta_r)$ require $c_J$ and $c_J^t$ to be significantly higher than their $1/N_f$ perturbative estimates, suggesting that this region is likely unphysical. In contrast, the lower bounds on the central charges agree with their $1/N_f$ perturbative results in the region with $\Delta_{\cM_{1/2}}\in (1.0,1.1)$, compatible with the $1/N_f$ estimate $\Delta_{\cM_{1/2}}\simeq 1.022$. In fact, if we adopt the more restrictive assumption $\Delta_{\cM_1}\leqslant 2.5$, then a linear extrapolation of the bootstrap island suggests that it shrinks to a small range with $\Delta_{\cM_{1/2}}\in (1.02, 1.04)$, beautifully compatible with the perturbative results for both the scaling dimension and central charge data.

The results we have laid out so far give a potentially bright outlook for the future of bootstrapping , and we see several concrete directions for future work. The islands we obtained in this work rely on inputting an assumption which places either $\Delta_{\cM_1}$ or $\Delta_{\cM_{1/2}}$ near its perturbative value. It is important to find ways to get rid of this condition. Moreover, in this work some of the gap assumptions we made are not fully justified, and we hope that bootstrap results for can ultimately be established using a set of sufficiently general assumptions that are more firmly established. A key point to improving this situation is to find an even more restrictive bootstrap setup, and there are a number of concrete mixed-correlator setups that could be pursued.

The bounds on the scaling dimension of the lowest charge 1 monopole operator have an interesting wave structure, which explains why islands can be formed with interval positivity assumptions and generally provides strong constraints if one assumes that the lattice results [43] are reliable. This wave structure is reminiscent of a similar structure appearing in the bootstrap of the 3d Ising CFT, leading to the conjecture that the solution at the tip of the wave might be further isolated by bootstrapping mixed correlators of the monopoles $\cM_{1/2}$ and $\cM_1$. In this mixed correlator setup, we can get access to more representations of $SU(4)\times U(1)_t$ and further exploit the constraints from parity symmetry and gaps in the monopole spectrum (which reflect the underlying equations of motion), which have played crucial roles in generating the current bootstrap results. Besides the gaps explored in this work, there are likely to be other sectors which can also introduce strong constraints on the CFT data, especially the spin $1$ sectors appearing in the $SO(12)\rightarrow SU(4)\times U(1)_t$ branching rules. We hope to provide a more systematic exploration of these directions in future work.

§ ACKNOWLEDGEMENTS

We thank Thomas Appelquist, Meng Cheng, Shai Chester, George Fleming, Liam Fitzpatrick, Luca Iliesiu, Yin-Chen He, Ami Katz, Petr Kravchuk, Walter Landry, Silviu Pufu, Slava Rychkov, William Witczak-Krempa, David Simmons-Duffin, and Ning Su for discussions. The work of SA, RE, ZL, and DP is supported by Simons Foundation grant 488651 (Simons Collaboration on the Nonperturbative Bootstrap) and DOE grant no. DE-SC0020318. SA is also supported by a VIDI grant of the Netherlands Organisation for Scientific Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW). YX is supported by a Yale Mossman Prize Fellowship in Physics. Our bootstrap computations were carried out on the Yale Grace computing cluster, supported by the facilities and staff of the Yale University Faculty of Sciences High Performance Computing Center.

§ FURTHER DETAILS ON THE FERMION BILINEAR BOOTSTRAP

§.§ Conventions

In this appendix, we set the conventions to describe various operators in QED$_3$. We use the Minkowski metric in mostly plus signature $\eta_{\mu\nu}=\textrm{diag}(-1,1,1)$. We will denote a fermion in the fundamental representation and its dual with uppercase and lowercase indices respectively, i.e. $\psi^\a$ and $\psi_\a$ — and similarly for the complex conjugate representation $\psi_{\da}$ and its dual $\psi^{\da}$. Between these representations, we have the relation $(\psi^\a)^\dagger=(\psi^{\dagger})^{\da}$ and we have the intertwining operator $\gamma^0_{\da\a}$, so that $(\psi^{\a})^\dagger\gamma^0_{\da\b}$ transforms as the dual fermion $\psi_\b$. We will then define $ψ_(ψ^)̱^†γ^0_$, with which we can construct the invariant scalar $ψ_ψ^$.\footnote{
For Majorana fermions, we can convert all dotted indices to undotted ones, with which $\gamma^0_{\a\b}$ and $\psi^a$ can be interpreted as the symplectic tensor and fundamental representation of $\Sp(2,\R)$ group, as was done in [83, 84, 85, 86].}
Whenever there is no room for confusion, we will suppress spinor indices.\footnote{In our conventions, (un)dotted indices are contracted from north(south)-west to south(north)-east.}

We use an explicit real representation of the $γ^μ$ matrices, i.e.
\be
\gamma^0=\begin{pmatrix}
\end{pmatrix}\;,\quad
\gamma^1=\begin{pmatrix}
\end{pmatrix}\;,\quad
\gamma^2=\begin{pmatrix}
\end{pmatrix}\,,
\ee
with which the Lorentz generators acting on the Dirac spinors can be written as $γ_μν=i/4(γ_μγ_ν-γ_νγ_μ)$. We take the space parity transformation as the reflection $x^2→-x^2$, and we choose its action on the fermions as $ψ→γ^2ψ$.\footnote{As the double cover of the rotation group ($\mathrm{Pin}$ group) acts on $\psi$, both $\pm\gamma^2\psi$ are valid choices. Since we will always consider operators containing an even number of fermions, this does not pose any ambiguity.} This means $ψ→-ψγ^2$, indicating that $ψψ$ transforms as a parity-odd scalar.

We can also work out how space parity transformations act on the standard $|l,m$ basis from the theory of angular momentum. Its coordinate representation, the spherical harmonics $Y_lm(θ,ϕ)$, pick up a sign under our parity transformation: $Y_lm(θ,ϕ)→(-1)^l+mY_lm(θ,ϕ)$.\footnote{We note that this is different from the standard formula $\cP: Y_{lm}(\theta,\phi)\rightarrow(-1)^{l}Y_{lm}(\theta,\phi)$ because we took our parity transformation as the reflection $x^2\rightarrow -x^2$ instead of the inversion $x^i\rightarrow-x^i$.} In the presence of a magnetic flux $q$ with $l-q≥0$, we will instead resort to the \emph{scalar monopole spherical harmonics} introduced in [87, 88]:
\small
\be
\label{eq: scalar monopole spherical harmonics}
Y_{q,lm}(\theta,\phi) &\equiv \Theta_{q,lm} (\cos\theta) e^{i(m+\ka q)\phi}\,, \\
\Theta_{q,lm}(x) &\equiv 2^{m-1}
\sqrt{\frac{(2\ell+1)(\ell-m)!(\ell+m)!}{\pi(\ell-q)!(\ell+q)!}}
\sqrt{\frac{(1+x)^{q-m}}{(1-x)^{q+m}}} P_{\ell+m}^{-q-m,q-m}(x)\,,
\ee
\normalsize
where $P_n^α,β(x)$ is the Jacobi polynomial and the parameter $=±1$.\footnote{The value of $\ka$ depends on which coordinate chart we are using to describe $Y_{q,lm}$: if we choose the chart that includes the whole sphere minus the south (north) pole, then $\ka$ is 1 (-1). In the rest of the paper, we stick to $\ka=1$.} In our conventions,
\small
\be
\label{eq: transformation of scalar monopole spherical harmonics}
\text{Space Parity: }\quad Y_{q,lm}(\theta,\phi)\rightarrow(-1)^{l+m}e^{2iq\phi}Y_{-q,lm}(\theta,\phi)\;.
\ee
\normalsize

Our conventions for the global $SU(N)$ symmetry is analogous: we write indices in the (anti)fundamental representations (downstairs)upstairs, i.e. $^i$ vs $_i$. However, unlike the spacetime representations, these ones are actually conjugate, hence we have $(^i)^†=_i$. Similar to $γ^0$ for the Clifford algebra, we have the Levi-Civita tensor $ϵ$ which acts an intertwining operator between these conjugate representations,\footnote{
Let an operator $\cO^{i_1\dots i_n}$ transform under the $SU(N)$ action as \mbox{$\cO^{i_1\dots i_n}\rightarrow U^{i_1}_{\;\;j_1}\dots U^{i_n}_{\;\;j_n}\cO^{j_1\dots j_n}$}, or \mbox{$\cO^I\rightarrow U^I_{\;\;J}\cO^J$} as a shorthand notation. We similarly have $\cO_I\rightarrow\cO_J(U^\dagger)^J_{\;\;I}$ for the conjugate operator. Due to the identity \mbox{$\e_{IK}U^K_{\;\;J}=(U^\dagger)^L_{\;\;I}\e_{LJ}$}, $(\e_{IJ}\cO^J)$ transforms as an operator in the conjugate representation, i.e. $(\e_{KJ}\cO^J)(U^\dagger)^K_{\;\;I}$ --- similarly, the relation \mbox{$\e^{IK}U^J_{\;\;K}=(U^\dagger)^I_{\;\;L}\e^{LJ}$} implies the inverse, i.e. \mbox{$\left(\cO_K\e^{KI}\right)\rightarrow U^I_{\;\;J}\left(\cO_K\e^{KJ}\right)$}. Note that the identity \mbox{$\e_{i_1\dots i_n k_1\dots k_{\bar{n}}}U^{k_1}_{j_1}\cdots U^{k_{\bar{n}}}_{j_{\bar{n}}}=(U^\dagger)^{l_1}_{i_1}\cdots (U^\dagger)^{l_n}_{i_n}\e_{l_1\dots l_n j_1\dots j_{\bar{n}}}$} follows from $\det U =1$ condition and hence is valid only for the special unitary group.
} hence we choose\footnote{Our choice of normalization follows from the useful identity $\epsilon_{i_1\dots i_{n}j_1\dots j_{m}}\epsilon^{k_1\dots k_{n}j_1\dots j_{m}}=n! m! \delta^{k_1}_{[i_1}\cdots \delta^{k_n}_{i_n]}$.}
\be
\label{intertwining conjugate representations}
A_{i_1\dots i_{n}}=&\frac{1}{\sqrt{n!\bar{n}!}}\epsilon_{i_1\dots i_{n}j_1\dots j_{\bar{n}}}B^{j_1\dots j_{\bar{n}}}\;,\\
C^{i_1\dots i_m}=&\frac{1}{\sqrt{m!\widebar{m}!}}D_{j_1\dots j_{\widebar{m}}}\epsilon^{j_1\dots j_{\widebar{m}}i_1\dots i_{m}}\,,
\ee
for operators $A,B,C,$ and $D$ in representations $𝔪_1,𝔪_1^*,𝔪_2,$ and $𝔪_2^*$ respectively. Here, we defined the shorthand notation
\be
\bar{n}\equiv N-n
\ee
and similarly for $m$. If $m=m=n=n̅=N/2$, we can choose $B=C=$ and $A=D=^†$, which gives the reality conditions
\be
\label{eq: transformation of real operators}
\left(\cO^{i_1\dots i_n}\right)^\dagger=&\frac{1}{n!}\epsilon_{i_1\dots i_{n}j_1\dots j_{n}}\cO^{j_1\dots j_{n}}\;,\\
\cO^{i_1\dots i_n}=&\frac{1}{n!}\left(\cO^{j_1\dots j_n}\right)^\dagger\epsilon^{j_1\dots j_{n}i_1\dots i_{n}}\,.
\ee

The generalization of our notation to mixed tensors of $SU(N)$ is straightforward. For notational brevity, we'll take consecutive indices antisymmetrized, whereas groups of indices separated by lines are symmetrized; for instance, the Young diagram for the representation of $^il|jm|k$ reads as {\scriptsize \begin{ytableau}i& j&k\cr l& m\cr \end{ytableau}}.\footnote{We note that for the representation to correspond to a valid Young diagram, we have the constraint \mbox{$l_1\ge l_2\ge \dots \ge l_n$} for $\cO^{k_{11}\dots k_{1l_1}|k_{21}\dots k_{2l_2}|\cdots|k_{n1}\dots k_{nl_n}}$.}  The \equref{intertwining conjugate representations} then generalizes as
\small
\be
A_{k_{11}\dots k_{1n_1}|
k_{21}\dots k_{2n_2}|
\cdots |
k_{c1}\dots k_{cn_c}
\left(
\prod\limits_{a=1}^c
\frac{\epsilon_{k_{a1}\dots k_{an_a}l_{a1}\dots l_{a\bar{n}_a}}}{\sqrt{n_a!\bar{n}_a!}}
\right)
\\\x
B^{l_{c1}\dots l_{c\bar{n}_c}|
\cdots|
l_{21}\dots l_{2\bar{n}_2}|
l_{11}\dots l_{1\bar{n}_1}
\ee
\normalsize
where $A$ transforms in the Young diagram of $c$ columns, each column having $n_c$ boxes (and $B$ transforms as its dual). For instance, two operators $A$ and $B$ in the conjugate representations  $(𝖠𝖲̅)$ and $(𝖲𝖠̅)$ of $SU(4)$ would be related as
\small
\be
\left(A^{k_{11}k_{12}k_{13}|k_{21}k_{22}k_{23}|k_{31}k_{32}}\right)^\dagger=\frac{\e_{k_{11}k_{12}k_{13}l_{11}}\e_{k_{21}k_{22}k_{23}l_{21}}\e_{k_{31}k_{32}l_{31}l_{32}}}{\sqrt{3!3!2!2!}}B^{l_{31}l_{32}|l_{21}|l_{11}}\;.\label{key}
\ee
\normalsize

The final group that we should set our conventions for is the $SO(2)$ group under which the monopole operators transform in the fundamental representation, i.e. $M^I;a$ for $a=1,2$ ($I$ denotes the collective indices for $SU(N)$). We are interested in cases where the monopole operators are real, hence \equref{eq: transformation of real operators} generalizes as\footnote{As monopole operators have the Dynkin labels $[\bar{0},2\abs{q},\bar{0}]$ ($\bar{0}$ denoting the sequence of $\frac{N-2}{2}$ many $0$'s), they are pseudoreal if $2\abs{q}+1\in2\N^+$ and $N=4n-2\text{ for }n\in\N^+$ [89]. For such cases, one uses $\e_{ab}$ instead of $\de_{ab}$.}
\be
\left(\cO^{i_1\dots i_n;b}\right)^\dagger=&\frac{1}{n!}\delta_{ab}\epsilon_{i_1\dots i_{n}j_1\dots j_{n}}\cO^{j_1\dots j_{n};a}\;,\\
\cO^{i_1\dots i_n;b}=&\frac{1}{n!}\delta_{ab}\left(\cO^{j_1\dots j_n;a}\right)^\dagger\epsilon^{j_1\dots j_{n}i_1\dots i_{n}}\;.
\ee

\subsection{Index free notation for \texorpdfstring{$SU(N_f)$}{SU(N)} tensors}
\label{sec: index-free notation}
One can represent arbitrary mixed representations of the $SU(N)$ group as polynomials of a set of commuting and anticommuting variables by constructing explicit projector operators in the basis of fundamental indices.\footnote{Interested readers can consult [90, 91] for examples of such projectors and the related illustrative birdtrack notation.} Instead of doing this, we will follow a similar approach to [92] and use a less systematic yet more practical approach by mixing fundamental and antifundamental indices.