Towards bootstrapping QED 3

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1 Introduction and summary Quantum electrodynamics in 2 + 1 spacetime dimensions (QED 3 ) could be regarded as a toy model for the real world quantum chromodynamics in 3 + 1 dimensions because it is an asymptotically free theory that may also exhibit analogs of chiral symmetry breaking [1] and confinement [2,3]. In a Lagrangian description, the field content of QED 3 consists of a U(1) gauge field possibly coupled to several flavors of charged fermions. When there is no charged matter, the theory confines [2,3]. When the number N of (two-component complex) fermion flavors is large, it can be argued using 1/N perturbation theory that -1 -JHEP08(2016)019 the infrared physics is described by an interacting conformal field theory [4,5]. When N is small but non-zero, the precise dynamics remains uncertain, however, because the theory is strongly coupled, and there are only very few non-perturbative tools available. 1 It is believed that in this regime the theory may exhibit analogs of both chiral symmetry breaking and confinement.
In this work, we aim to initiate a study of QED 3 at small N using the conformal bootstrap technique [11], with the goal of eventually shedding light on the behavior of the theory in this regime. The conformal bootstrap is a non-perturbative technique that has yielded quite impressive results in other non-supersymmetric examples, such as the 3d Ising model [12,13], the critical O(N ) vector model [13][14][15], or, more recently, the Gross-Neveu models [16], so it is natural to ask whether it can also be used to learn about 3d gauge theories as well. In its numerical implementation in terms of semi-definite programming, the conformal bootstrap makes use of unitarity and associativity of the operator algebra as applied to 4-point functions of certain operators in a conformal field theory.
In this paper, we assume that the conformal fixed point of QED 3 seen in 1/N perturbation theory extends to all values of N , and study this CFT using the conformal bootstrap. Explicitly, we derive and study numerically the crossing relations of four monopole operators (to be defined more precisely shortly) for N = 2, 4, and 6. What we find are rigorous bounds on the scaling dimensions of these monopole operators and of some of the operators appearing in their OPE. We find that these bounds come close to the large N results when extrapolated to small N . In addition, we find certain features in our bounds that are similar to those that appeared in the bounds of the lowest-dimension operators in 3d CFTs with global Z 2 symmetry when looking at the single 4-point function of Z 2 odd operators. In that case, examining the crossing equation of a system of mixed correlators yielded an allowed region in the form of an island centered around the 3d Ising CFT. It would be interesting to see if a study of mixed correlators of monopole operators also yields an island-shaped allowed region, though such an analysis is of a numerical complexity beyond what is currently feasible.
Before we delve into the details of our analysis, let us comment on our choice of studying the crossing equations of monopole operators as opposed to those of other operators in the theory. QED 3 with N unit charged fermions ψ i has SU(N ) × U(1) flavor symmetry. The fermions transform as a fundamental of SU(N ) and are uncharged under U(1). The monopole operators have non-zero U(1) charge and also transform in fairly complicated representations of SU(N ). In implementing the conformal bootstrap program, one option would have been to consider the 4-point function of the simplest non-monopole scalar operators, the bilinearsψ i ψ j transforming in the adjoint of SU(N ). The crossing equations for such a four-point function were worked out in [17], and it should be straightforward to study the constraints they imply numerically using computer programs such as SDPB [18]. The disadvantage of studying this four-point function by itself, however, is that besides QED 3 , there are other theories such as scalar QED, QCD 3 or supersymmetric analogs that 1 Recently, the -expansion was used to argue that spontaneous chiral symmetry breaking should occur when N ≤ 4 [6]. The F -theorem suggests that it occurs when N ≤ 8 [7]. Lattice studies suggest that it occurs when N = 2 [8,9] or N = 0 [10], but the situation at larger N is unclear.
all have SU(N ) flavor adjoint operators with similar properties, and thus from an abstract CFT point of view, it may be hard a priori to distinguish these theories from one another.
What is specific to QED 3 and is not shared by its QCD or supersymmetric analogs is indeed the spectrum of monopole operators, and this is why we focus on them. It can be shown [19,20] that the monopole operator M q that carries U(1) charge q ∈ Z/2 also transforms under SU(N ) as an irreducible representation given by the Young diagram (1.1) This feature makes QED 3 different from the other similar theories for which the lowestdimension non-monopole scalars are also SU(N ) adjoints. Note that without any Chern-Simons interactions, N is required to be even in order to avoid a parity anomaly [19], so the Young diagram (1.1) is indeed well-defined.
Monopole operators are interesting to study not just so that we can distinguish QED 3 from other theories. More generally, they are quite important for the dynamics of gauge theories in 2 + 1 dimensions. The simplest example is pure U(1) gauge theory, where it was shown by Polyakov that their proliferation provides a mechanism for confinement [2]. If one adds a sufficiently large number N of charged matter fields (bosons or fermions), the infrared physics is believed to be governed by an interacting conformal field theory (CFT), where, in certain condensed matter realizations, monopole operators can act as order parameters for quantum phase transitions that evade the Ginzburg-Landau paradigm [21][22][23][24][25][26][27][28][29][30][31][32][33][34]. In these interacting CFTs, the only available method 2 for studying the properties of the monopole operators is the 1/N expansion, which so far has been used to compute their scaling dimensions to next-to-leading order in 1/N [19,20,[36][37][38][39][40]. Going to higher orders in the 1/N expansion appears to be very challenging with current techniques. It is nevertheless desirable to learn about monopole operators away from the large N limit, which serves as further motivation for studying them using the conformal bootstrap.
The rest of this paper is organized as follows. In section 2, we review some known facts about 3d QED and monopole operators. Sections 3 and 4 represent the main part of this paper, in the former we compute the crossing equations for the monopole operators in 3d QED, including explicit crossing relations for the cases N = 2 , 4 , 6, and in the latter we present the results of our numerical bootstrap. In section 5 we conclude and discuss further directions. In the appendix we include the crossing relations for the cases N = 8 , 10 , 12 , 14.

3d QED and monopole operators
The Lagrangian for 3d QED with N complex two-component fermions is where ψ i are the fermion fields, A µ is a U(1) gauge field with field strength F µν , and e is the gauge coupling. In the following discussion we restrict to the case where N is even so that 2 For fermions, preliminary 4 − expansion results are discussed in [35].  we may preserve parity and time reversal symmetry [19]. At large N one can show that this theory flows to an interacting CFT in the infrared where the Maxwell term in (2.1) is irrelevant [41,42]. At small N the theory is strongly coupled and difficult to study, although lattice gauge theory studies [8,9,43] and other arguments [6,44] suggest that there is a critical value estimated around N crit = 2 below which the theory no longer flows to an interacting CFT. As mentioned in the introduction, in this paper we will work under the assumption that the IR dynamics is governed by a non-trivial interacting CFT whose properties are the same as those derived from the large N expansion extrapolated to finite N . At the CFT fixed point, one can define gauge-invariant order operators built from the fields in the Lagrangian, as well as disorder operators (monopole operators) defined through boundary conditions on these fields.

Lowest dimension monopole operators M q
A monopole operator M q with topological charge q at the conformal fixed point of 3d QED with N flavors must transform as a representation of the global symmetry group, which includes the conformal group SO (3,2), the flavor symmetry group SU(N ), and the U(1) "topological" symmetry generated by the topological current where (λ ν 1 1 , λ ν 2 2 , . . . ) denotes a Young tableau with ν i rows of length λ i . There are thus 1 + N/2 SU(N ) irreps in both the q = ±1 and q = 0 sectors. Because of Bose symmetry, only operators with certain spins can appear in each such irrep, as will be discussed in detail in section 3. In this bootstrap study, we will be interested primarily in bounding the scaling dimension of the lowest scalar q = 1 monopole operator M 1 , which according to (2.3) transforms under SU(N ) as 2 N/2 .

Lowest dimension scalar
In our bootstrap study, it would be useful to make use of more information on the operators in the M ±1/2 × M ±1/2 and M ±1/2 × M ∓1/2 OPEs, such as their scaling dimensions.
For simplicity, let us focus on the Lorentz scalars with q = 0 appearing in the M ±1/2 × M ∓1/2 OPE. For a given index n > 0, for which the SU(N ) irrep is 1 N −2n , 2 n , let us denote the lowest dimension primary by O n , the next lowest by O n , and so on. As mentioned above, all these operators can be built from gauge invariant combinations of ψ i and A µ because they have zero topological charge.
As will be explained in more detail in [45], the operator O n has the form where α m = 1, 2 are Lorentz spinor indices. This operator is parity even (odd) depending on whether n is even (odd). Its scaling dimension is [45] etc. (2.5) Note that in this expansion N is taken to infinity before all other quantities. In particular, the results corresponding to the n channel may break down when N is comparable to n. The next two operators O n and O n have opposite parity from O n and can be constructed from n + 1 ψ's and n + 1ψ's. Their scaling dimensions can also be calculated in the 1/N expansion and take the form ∆ n = 2(n + 1) + O(1/N ) and ∆ n = 2(n + 1) + O(1/N ). The previous results are only for n > 0. For n = 0, i.e. the SU(N ) singlet case, the lowest dimension parity odd operator is O 0 ∝ψ i ψ i , whose scaling dimension is given by [25] For the lowest dimension parity even SU(N ) singlet, we must consider the mixing between (ψ i ψ i )(ψ j ψ j ) and F 2 µν , which gives [45] (See also [46].) These central charges have been computed to next to leading order in [47] as well as [48,49]. In our normalization (2.8) we have (2.9)

Crossing equations
We now show how to set up the conformal bootstrap for the four point function of monopole operators in 3d QED. We will focus on the four-point function of two q = 1/2 monopole operators and two q = −1/2 antimonopole operators, which as mentioned previously transform in the (1 N/2 ) representation of SU(N ), i.e. they are completely antisymmetric tensors of SU(N ) with N/2 indices. Let M I 1/2 denote the monopole operator, where I = {i 1 , . . . , i N/2 } and i = 1, . . . , N are SU(N ) fundamental indices. It is convenient to recast U(1) as SO(2) by writing M I 1/2 = M 1I 1/2 +iM 2I 1/2 and M I −1/2 = M 1I 1/2 −iM 2I 1/2 and working with M aI 1/2 , where a = 1, 2 is a fundamental SO(2) index. We consider the four-point function: the index (R, n). Here, R labels the SO(2) representation, and it can take the values: R = S for SO(2) singlets; R = A for rank-two anti-symmetric tensors 5 of SO(2); and R = T for rank-two traceless symmetric tensors. (In terms of the topological charge q, we have that R = S, A correspond to q = 0 and R = T corresponds to q = ±1.) For SU(N ), we see from (2.3) that we have representations 1 N −2n , 2 n where n = 0, . . . , N/2. We will show shortly that for each (R, n) only operators with either even or odd can appear in the M aI where we combined the contribution from each conformal multiplet into a conformal block, and where f abcd , and g ∆, (u, v) are defined as follows. The f abcd R are SO(2) 4-point tensor structures corresponding to exchanging operators in representation R of SO (2). They are given by [50] f abcd The t IJKL n are 4-point tensor structures corresponding to exchanging operators in 1 N −2n , 2 n of SU(N ). The s R,n are very important signs (s R,n = +1 or −1) that are determined by unitarity, as we will discuss in section 3.3. The λ 2 are the squares of the OPE coefficients that must be positive by unitarity. (We can normalize the OPE coefficient of the identity operator λ Id =1.) Lastly, g ∆, (u, v) are conformal blocks corresponding to the exchange of the operator O ∆, (R,n) , normalized, for concreteness, as in [13]. Swapping (1, I, a) ↔ (3, K, c) in the four point function (3.1) yields crossing equations of the form where O runs over all conformal primaries in the M aI The number of components is determined according to [50] by the number of SO(2) × SU(N ) representations R, n that occur in the M aI 1/2 × M bJ 1/2 OPE, where representations with both odd and even spins contribute twice.) The components of the crossing function are explicit functions of the conformally-invariant cross-ratios u = and v = . The form of d R,n ∆, depends only on the dimension of both the external monopole operator ∆ M 1/2 and on the dimension ∆ O , Lorentz spin , 5 The singlet (S) and rank-two antisymmetric tensor (A) representations of SO(2) are of course isomorphic, but it is convenient to keep track of whether O ∆, (R,n) appears in the symmetric (S) or anti-symmetric (A) product of two SO(2) fundamentals. As will be explained, the operators in S and A have spins of opposite parity, with those in S having spins of the same parity as that of the operators in T .

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and SO(2) × SU(N ) representation (R, n) of the operator O. In the rest of this section we provide an efficient algorithm to compute d R,n ∆, for any N , which we demonstrate explicitly for the cases N = 2, 4, 6. The cases N = 8, 10, 12, 14 are given in appendix A.
The Lorentz scalars M aI 1/2 transforms in the fundamental of SO(2) and in the representation 1 N/2 of SU(N ). The crossing equations of an operator such as M aI 1/2 that transforms under a product group can be expressed, roughly, as a tensor product of the crossing equations under each group factor. In this case, we rewrite (3.4) more explicitly as where d R,n ∆, are given by the O(2) fundamental crossing functions [50] with d ±,n ∆, being the crossing functions under SU(N ) that we will describe next. In (3.5), the notation + ( − ) means that we sum over the same (opposite) set of spins as the component SU(N ) crossing functions.

Known results for N = 2, 4
In the cases N = 2, 4, the crossing functions d ±,n ∆, appearing in (3.6) are already known. When N = 2, the representation (1 N/2 ) = (1) of the external operator is the fundamental representation of SU (2). The corresponding crossing functions are a reduced version of the general fundamental SU(N ) crossing functions written in [50], and they are given by 6 Here, the operators in the n = 0 singlet (n = 1 adjoint) representations can have odd (even) spins, and the functions F ± ∆, are defined in terms of the conformal blocks g ∆, (u, v), the conformal cross ratios u = , and the scaling dimension ∆ ext of the external operator: Recall that the external operator dimension in our case is ∆ ext = ∆ M 1/2 . For N = 4, the six dimensional (1 2 ) representation of SU(4) is isomorphic to the six dimensional fundamental representation of SO(6), so the crossing functions are given by 6 We multiplied d ∓,1 ∆, by an overall minus sign in order to agree with the conventions we use in section 3.2.1.
For now, we can think of this minus sign as a redefinition of the sR,1 coefficients in (3.6). These coefficients will be determined in section 3.3.

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the O(6) fundamental crossing functions [50]: Here, the operators in the singlet n = 0, antisymmetric n = 1, and traceless symmetric n = 2 representations of O(6) can have even, odd, and even spins, respectively. For N ≥ 6 there are no results in the literature for the crossing equations, but they can be efficiently derived using the algorithm described below. As a check on our algorithm, we recover the known results given above for N = 2, 4.

General algorithm
We begin by considering the four point function of operators O I where I = {i 1 , . . . , i N/2 } and i = 1, . . . , N are SU(N ) fundamental indices: where t IJKL n is the four-point tensor structure that corresponds to the exchange of a conformal multiplet whose primary transforms as 1 N −2n , 2 n for n = 0, . . . , N/2, and we will suppress the sets of SU(N ) indices IJKL for now on. Using explicit expressions for t n , it will be straightforward to implement the crossings (1, I) ↔ (3, K) and (1, I) ↔ (2, J). The former crossing will give us the crossing functions, while the latter will give us the allowed spins in each representation.
All the indices on the l.h.s. of (3.10) are fundamentals of SU(N ), which implies that t n can be written as where p ∈ {i 1 , . . . , i N/2 , j 1 , . . . , j N/2 , k 1 , . . . , k N/2 , l 1 , . . . , l N/2 } and b m form a basis for all tensor structures of this form. Our first step is to exchange (I) ↔ (K) or (I) ↔ (J) for each b m and express the result as a linear combination of b m 's: Our second step is to compute the matrix U m n that transforms between the bases t n and b m . For this purpose we will use the SU(N ) rank-2 Casimir, which we define in our case as where T (q) jq iq are fundamental SU(N ) generators for each index i q , so that C 2 acts on SU(N ) tensors with N/2 fundamental indices i q . C 2 acts on the (suppressed) first N fundamental SU(N ) indices of b m as (3.14) The eigenvectors (t n ) m of D n m are eigenvectors of C 2 The eigenvalues (c 2 ) n of an SU(N ) tensor in representation 1 N −2n , 2 n for n = 0, . . . , N/2 can be calculated by standard group theory formulae and are given by so that indexing t n by order of increasing (c 2 ) n is consistent with the original definition of t n in (3.10). Note that each t n as defined above can be multiplied by any real constant and still obeys (3.15). Here, we just make a choice of some t n that obey (3.15).
The transformation matrix U m n in (3.11) between the bases t n and b m is then given by where we compute ((t n ) m ) in (3.15).
Putting everything together, the crossing function d −,n ∆, for the exchange (1, I) ↔ (3, K) acting on the four point function (3.10) is an (N/2 + 1) × (N/2 + 1) matrix given by which we can rewrite in terms of F ± ∆, (u, v) using the definition (3.8). When expressing d −,n ∆, as a column vector, it is convenient to do so in a basis different from b m that is chosen such that some components involve only F + ∆, (u, v) and some only F − ∆, (u, v). The analogous equation for the exchange (1, I) ↔ (2, J), with X ↔ Y, will yield equations of form F ± ∆, (u, v)λ 2 On = 0 for each representation n, which for F − , F + imposes even, odd spins for that representation.
To demonstrate this algorithm, we will now perform it explicitly for the cases N = 2, 4, 6. The crossing functions for N = 8, 10, 12, 14 are given in appendix A.

N = 2
We choose the b m basis: The exchanges (I) ↔ (K) or (I) ↔ (J) yield the transformation matrices:  (2) crossing equations are consistent with the expected spin parities required by Bose symmetry, namely odd and even for t 0 (singlet) and t 1 (adjoint), respectively. 7

Reflection positivity
Reflection positivity is the Euclidean version of the unitarity constraints on a Lorentzian CFT. These constraints fix the sign of λ 2 O , by demanding that when we consider the fourpoint function of scalar operators , the coefficients multiplying the conformal blocks in the s-channel OPE should be positive [11]. SU(N ) has complex generators, so to enforce this condition in our case, we must define what we mean by the complex conjugate of an operator O aI transforming under SO(2) × SU(N ). In fact, we will consider O aI to be real under this notion of complex conjugation.
The subtlety in defining the reality properties of our operators comes from the fact that the SU(N ) irrep (1 N/2 ) under which these operators transform is real when N/2 is even and pseudo-real when N/2 is odd. We thus have two different reality conditions depending on whether N/2 is even or odd: (3.33) There are several ways of determining the signs s R,n appearing in (3.6). We choose to do so by looking at an example, namely the one where O aI represent free fields obeying (3.33) with ∆ O = 1/2. In this free theory, the four-point function can be obtained from Wick contractions using (3.33): (3.36) (3.38) where r, η are functions of u, v defined in [51]. We can now read off the signs multiplying the conformal blocks of each tensor structure from this example. These signs must be the same in all theories where the reality conditions (3.32) are satisfied. We now carry out this program explicitly for the cases N = 2 , 4 , 6.

SU(2)
Computing the inverse of U for SU (2) where the third equation follows as an identity. So  Using the relations (3.38), we express the four point function (3.37) for the N/2 odd case in terms of conformal blocks: (3.41) Table 2 follows. As a consistency check, the spin parities in this table match our computation in section 3.2.1.

SU(4)
Computing the inverse of U for SU(4) (3.26) we get Using the relations (3.38), we express the four point function (3.37) for the N/2 odd case in terms of conformal blocks: (3.44)

SU(6)
Computing the inverse of U for SU (6) Using the relations (3.38), we express the four point function (3.37) for the N/2 odd case in terms of conformal blocks: (3.47) Table 4 follows. As a consistency check, the spin parities in this table match our computation in section 3.2.3.

Constraints from space-time parity
As described in [19], space-time parity maps a monopole operator M q to an anti-monopole operator with opposite charge M −q . In terms of SO(2) indices, parity acts by sending 1 → 1 and 2 → −2, thus the S sector is parity even, the A sector is parity odd, and the T sector can transform as both even or odd for different operators.
To find the parity of the uncharged spin 0 operators in each SU(N ) sector, we must determine whether they are in the A or S sector. Operators appearing in the M aI 1/2 ×M bJ 1/2 OPE -16 -JHEP08(2016)019  Table 5. Composite fermion operator in representation 1 N −2n , 2 n with required parity for N/2 even or odd.
As described in section 2, the two lowest dimension spin 0 operators O n and O n in SU(N ) representations 1 N −2n , 2 n are composed of 2n and 2n + 2 fermions, respectively. The parity of a 2n fermion operator is even/odd for n even/odd, so the lowest dimension spin 0 operator O (n) in SU(N ) representations 1 N −2n , 2 n with the required parity depends on whether N/2 is even or odd. In table 5 we show which operator O n or O n is the lowest dimension operator with the required parity for each SU(N ) sectors for N/2 even or odd. The scaling dimensions of these operators presented in section 2 will be used to motivate the gaps we impose in the subsequent section 4.2.

Strategy
After deriving the precise form of the crossing equations (3.4), in order to find bounds on the scaling dimensions of operators appearing in the M aI 1/2 × M bJ 1/2 OPE, one can consider linear functionals α satisfying the following conditions: where ∆ * R,n, are the assumed lower bounds for spin-conformal primaries (other than the identity) that appear in the M aI 1/2 × M bJ 1/2 OPE and transform in the SO(2) × SU(N ) representation (R, n). The existence of any such α would contradict (3.4), and thereby would allow us to find an upper bound on the lowest-dimension ∆ * R,n, of the spin-conformal primary in representation R, n. In particular, if we set ∆ * T,N/2,0 = ∆ M 1 and all other ∆ * R,n, equal to either their unitarity value or some gap value, then we can then find a disallowed region in the (∆ M 1/2 , ∆ M 1 ) plane for our chosen gap assumptions. The above procedure allows us to put gaps for operators that do not have both the same representation and spin as the operator we are bounding. If we would like to put a gap above the operator O (R ,n ), that we are bounding, then we must add the following condition: 3.
as well as make sure in condition (4.2) that ∆ * R ,n , > ∆ R ,n , . To find lower bounds on the central charges of conserved currents, we relate these charges to OPE coefficients of conformal primaries appearing in the M aI 1/2 × M bJ 1/2 OPE, for which we can find upper bounds using the bootstrap. On general grounds, the relation must take the form where the OPE coefficient λ R,n,∆, has R either S or A depending on which SO(2) representation gives the prescribed spin for the given SU(N ) representation, and n = 0 , 1 are the singlet, adjoint representations of SU(N ). The OPE coefficient λ 2 R,0,0,0 of the identity operator can be chosen to be equal to 1 as a normalization condition for the external operator. The coefficients of proportionality in (4.4) can be found from the free theory presented in section 3.3. A theory of free scalars transforming in representation R of SU(N ) and fundamental representation of SO(2), with the reality condition (3.32) has 8 where dim R is the dimension of R and C 2 (R) is the value of the quadratic Casimir of the representation. For us, R = (1 N/2 ), which has C 2 (R) = N (N + 1)/8 and with A 2 = 4, A 4 = 8, and A 6 = 2. Using (4.6), the lower bounds on the central charges can be recast as upper bounds on certain OPE coefficients. Upper bounds on the OPE coefficient of an operator O * can be determined by considering linear functionals α satisfying the following conditions: where ∆ * R,n, are the assumed lower bounds for spin-conformal primaries (other than the identity) that appear in the M aI 1/2 × M bJ 1/2 OPE and transform in the SO(2) × SU(N ) representation R. If such a functional α exists, then this α applied to (3.4) along with the positivity of all λ 2 O except, possibly, for that of λ 2 O * implies that provided that the scaling dimensions of each O = O satisfies ∆ ≥ ∆ * R,n, . We can choose the spectrum to only satisfy unitarity bounds, or impose gaps on various sectors. To obtain the most stringent upper bound on λ 2 O * , and therefore lower bound on its associated central charges, one should then minimize the r.h.s. of (4.9) under the constraints (4.8).
The numerical implementation of the above problems requires two truncations: one in the number of derivatives used to construct α and one in the range of spins that we consider, whose contributions to the conformal blocks are exponentially suppressed for large . We denote the maximum derivative order by Λ (as in [52]) and the maximum spin by max . The truncated constraint problem can be rephrased as a semidefinite programing problem using the method developed in [11]. This problem can than be solved efficiently using sdpb [18]. In this study, we set Λ = 19 and max = 25. We checked that increasing Λ and max did not change the values of ∆ M 1/2 or ∆ M 1 by more than .01 for N = 2, 4, and .02 for N = 6. In terms of computing time, sdpb took approximately 4 cpu hour for N = 2, 12 cpu hours for N = 3, and 18 cpu hours for N = 6.

Numerical bounds for N = 2, 4, 6
We now present bounds on scaling dimensions and central charges using the numerical conformal bootstrap. The number of crossing equations, and therefore the numerical complexity, increases as 3(N/2 + 1), so we will only focus on the cases N = 2, 4, 6. We use the crossing functions and spin parities computed in the previous section. We will also impose gaps on operators in the uncharged U(1) sector, motivated by the operator scaling dimensions in section 2. The parity constraints discussed in section 3.4 require that for N = 2, 6 the lowest dimension operators in SU(N ) representation 1 N −n , 2 2n are the   (2n + 2)-fermion operators of dimension ∆ n , while for N = 4 they are the 2n-fermion operators of dimension ∆ n . In the singlet n = 0 sector, N = 2, 6 has the 2-fermion operator of dimension ∆ 0 , while N = 4 has the 4-fermion operator of dimension ∆ 0 . As the 1/N expansion for these values still seems rather large for N = 2, 3, 4, the precise numerical values obtained from the large N expansion will serve more as rough guides than exact inputs. with an associated ∆ M 1 value appears. This feature (kink) seems to depend linearly on this gap -see the dotted lines in figures 1 and 2. Moreover, the slope of this line of kinks has the same value, ≈ 3, for all of the values of N that we considered. It is a reassuring check on our crossing equations, which differ drastically in form, that all these plots show the same qualitative features. For the cases N = 4, 6 we mark the large N prediction listed in table 1 for (∆ M 1/2 , ∆ M 1 ) with a cross in the corresponding plots. 9 For N = 4, the large N extrapolation seems to lie almost exactly on the dotted line connecting the kinks, which implies that a certain value of the gap ∆ 2 will give a feature at exactly the predicted value in the (∆ M 1/2 , ∆ M 1 ) plane. We note that imposing reasonable gaps 10 in the other uncharged sectors for N = 4 does not noticeably change the plots. For N = 6, the large N value lies somewhat below the dotted line connecting the kinks. We found that for N = 6, unlike N = 4, imposing gaps in the other uncharged sectors does change the location of the kinks and brings the line joining the kinks down closer to the large N extrapolation value.
In figure 2, the righthand plot focuses on the gap ∆ 2 = 3 case, which from the lefthand plot seems to match the large N values of (∆ M 1/2 , ∆ M 1 ) best. The righthand plot puts an additional gap ∆ M 1 above ∆ M 1 . We find that any value of ∆ M 1 > ∆ M 1 creates a peninsular allowed region around the kink seen in the lefthand plot. In previous bootstrap studies [12,14], it was found that such a peninsula leads to islands once mixed correctors are used -see, for instance, figure 3 in [12]. It would be interesting to see whether a similar phenomenon occurs here.

Discussion
In this work, we studied constraints coming from crossing symmetry and unitarity in 3d above. Interpreted in the context of 3d QED, we obtained bounds on the scaling dimension of the doubly-charged monopole operators in terms of the scaling dimension of the singlycharged one (figures 1 and 2), and also on the coefficients c T , c t J , and c f J appearing in the two-point function of the canonically normalized stress tensor, U(1) flavor current, and SU(N ) flavor current (figures 3, 4, and 5).
We hope that our work represents the first steps toward a more systematic study of QED 3 using the conformal bootstrap. We observed that when we impose certain gaps in the operator spectrum, we obtain a kink in our scaling dimension bounds (figure 2) that is at the edge of an allowed region whose shape is similar to that seen in the study of theories with Z 2 global symmetry. In a further mixed correlator study, such a region turned into an island centered around the 3d Ising CFT, so it would be interesting to see if a mixed correlator study in the present setup would also lead to an island-shaped allowed region. In this study we also assumed that a CFT exists for all N , which is still an unsettled question. Perhaps by looking at mixed correlators one could exclude the existence of such a CFT for low N . We hope to report on such a mixed correlator study in an upcoming work. and n odd.