Charting the space of 3D CFTs with a continuous global symmetry

We study correlation functions of a conserved spin-1 current Jμ in three dimensional Conformal Field Theories (CFTs). We investigate the constraints imposed by permutation symmetry and current conservation on the form of three point functions 〈JμJνO∆,`〉 and the four point function 〈JμJνJρJσ〉 and identify the minimal set of independent crossing symmetry conditions. We obtain a recurrence relation for conformal blocks for generic spin-1 operators in three dimensions. In the process, we improve several technical points, facilitating the use of recurrence relations. By applying the machinery of the numerical conformal bootstrap we obtain universal bounds on the dimensions of certain light operators as well as the central charge. Highlights of our results include numerical evidence for the conformal collider bound and new constraints on the parameters of the critical O(2) model. The results obtained in this work apply to any unitary, three dimensional CFT.


Introduction
The classification of all Conformal Field Theories (CFTs) is the ambitious dream that drives the systematic development of the conformal bootstrap program [1,2]. Almost ten years ago it was observed that the constraint of crossing symmetry can be recast into an infinite set of linear and quadratic equations, whose feasibility can be studied numerically [3][4][5][6]. Since then, the numerical conformal bootstrap has been successfully applied to four-point functions of scalar operators in several spacetime dimensions [7][8][9][10][11][12][13][14], and to spin-1 2 operators in three dimensions [15,16]. This has led to spectacular results, such as the most precise determination of the three dimensional Ising model critical exponents and its spectrum [17][18][19], a partial classification of O(N )-models in three dimensions [20], and interesting insights on superconformal theories with and without Lagrangian formulation [21][22][23][24][25][26][27][28][29][30][31]; see [32] for a review and a comprehensive list of references. In this paper, we consider the next step in complexity: correlators of vector operators. In particular we study the four-point function of a conserved current.
Any local CFT with a continuous global symmetry contains a conserved current J µ , whose flux through the boundary of a region B measures the total charge inside this region. 1 This property is encoded in the Ward identity, where n µ is the unit normal to the boundary of the region B and q i are the charges of the local operators O i . We shall study the four-point function of J µ which is an observable that exists in any CFT with a continuous global symmetry. This will allow us to constrain the spectrum of operators that appear in the Operator Product Expansion (OPE) of two currents. In three spacetime dimensions, these neutral operators can be classified by their 1 The existence of a conserved current follows from the Noether theorem in any Lagrangian CFT. However, we do not know of a more general (bootstrap) proof of this statement. scaling dimension ∆, SO(3) spin and parity. 2 More precisely, we will study the conformal block decomposition where λ (p) JJO are the coefficients of the operator O in the OPE of two currents. The index p, q run over a finite range, which depends on the spin and parity of the operator O. The symbol stands for the conformal blocks that are labeled by p, q and the quantum numbers ∆, and parity of O. This is described in detail in section 2. Following the usual bootstrap strategy, we then impose crossing symmetry of this four-point function.
However, due to current conservation, not all crossing equations are linearly independent.
In section 2, we explain how to select a minimal set of independent crossing equations to be imposed numerically. With these ingredients and assuming unitarity, we applied the usual bootstrap semi-definite programming method (SDPB) to constrain the spectrum of neutral operators and some OPE coefficients λ (p) JJO . In figure 1, we show our result for the excluded region of the plane (∆ + 0 , ∆ − 0 ), where ∆ ± denotes the scaling dimension of the lightest parity even/odd neutral spin-operator. This curve was calculated using up to Λ = 23 derivatives of the crossing equations at the crossing symmetric point (451 components). The parameter Λ is defined in eq. (2.61). In this plot, we represented several known theories to verify that they all fall inside the allowed region. On one hand, the theories of a free Dirac fermion and of a free complex scalar field lie well within the allowed region. On the other hand, the critical O(2)-model JHEP05(2019)098 We impose that the first spin-2 operator after T µν has dimension larger than 3.5 (see section 3 for explanation). and the generalized free theory (GFVF) of a current seem to play an important role in determining the boundary of the allowed region. Our results suggest that these theories sit at kinks of the optimal boundary corresponding to Λ = ∞. The stress-energy tensor appears in the OPE of two currents, t αβ µν ( x) + 12γ t αβ µν ( x) T αβ (0) + . . . (1.3) where x µ = x µ |x| and the dots represent the contributions from all other operators besides the identity and the stress tensor T αβ . There are two independent tensor structures 3 compatible with conservation and permutation symmetry. The conformal Ward identities relate the overall coefficient to the OPE coefficient of the identity operator (C J ) but the relative coefficient γ is an independent parameter that characterises the CFT. In particular, it controls the high frequency/low temperature behaviour of the conductivity [33]. In the holographic context, γ corresponds to a higher derivative coupling between two photons and a graviton in the bulk. In particular, γ vanishes for Einstein-Maxwell theory. The conformal collider analysis of [34] gives rise to the bounds −1 ≤ 12γ ≤ 1 (see also [35][36][37]). This bound was recently proven using only unitarity and convergence of the OPE expansion [38] (also see [39] for an alternative approach). The bound is saturated by free complex bosons (12γ = −1) and free fermions (12γ = 1).

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lines we plot the conformal collider bounds and the value of the central charge C T of the minimal theories that saturate them: a free complex scalar and a free Dirac fermion. It is encouraging to notice that the lower bound on C T grows very rapidly outside the region −1 ≤ 12γ ≤ 1. We suspect it diverges when Λ → ∞. On the other hand, for −1 < 12γ < 1 the lower bound on C T seems to be converging to a finite curve as we increase Λ. Figures 1 and 2 are just an appetiser for the results presented in section 3. To facilitate the interpretation of our results we listed in appendix A some known 3D CFTs with a continuous global symmetry. In section 2, we summarize the steps involved in setting up the numerical conformal bootstrap approach to the four point function of a conserved current, leaving many details to appendices B, C, D, E and F. Finally, we conclude in section 4 with a discussion of future work.

Setup
In this section we define our notation for three and four point correlation functions of spin 1 currents. We will often work in general spacetime dimension d and specialize to d = 3 at the end. Through this section we will work in embedding space, see [40] for a detailed review.
In the embedding formalism each operator O ∆, is associated to a field Φ ∆, (P, Z), polynomial in the (d + 2)-dimensional polarization vector Z, such that We fix the normalization of the operators such that: where P ij ≡ −2P i · P j . The quantity H 12 entering the above equation, together with V i,jk are the building blocks needed to construct higher point correlation functions. They are defined as: In order to decompose the four-point function JJJJ in conformal blocks, we need to understand the structure of the OPE of two currents J ×J. This is equivalent to classifying all the conformal invariant three-point functions JJO ± ∆, . Since we are assuming the CFT is parity preserving, the three-point functions JJO + ∆, will not involve the -tensor while the three-point functions JJO − ∆, will do.

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Let us start by writing the most general form of the three-point function between two equal vector operators (of dimension d − 1) and a parity even operator, JJO + are undetermined constants and we used the notation In this expression, we only imposed conformal and permutation symmetry. To impose conservation of J(P 1 , Z 1 ), it is enough to demand that the embedding space differential operator ∂ ∂P 1 · ∂ ∂Z 1 annihilates the three-point function. This further reduces the number of independent constants. In the case of a scalar operator, conservation implies λ JJO + leaving only one free constant. In the case of odd spin, conservation implies λ JJO + = 0, which means that a parity even, spin odd operator cannot appear in the OPE J × J. Finally, in the case of even spin ≥ 2 we find λ (4) which reduces the number of independent structures down to 2.
Let us now turn our attention to the three point function of two conserved currents and one parity odd operator O − ∆, . In d ≤ 4 one can use the -tensor to make parity odd conformally invariant three point functions. Indeed, any parity odd structure can be obtained by multiplying parity even structures by -tensors. In d = 3, there are three parity odd building blocks: Conformal invariance and permutation symmetry restricts the tensor structures to be: 4 4 As explained in appendix B, the structure 12V l 3 is not independent of the ones we used for ≥ 2. Structures involving an tensor contracted with three polarization vectors can also be expressed in terms of ij .
for ≥ 2. In the case = 0, conservation is automatic and λ JJO − is a free parameter. In the case = 1, conservation implies that λ (2) JJO − = 0. In other words, no spin 1 operator that can appear in the OPE of two equal currents.
In summary, the number of independent constants in the three-point function JJO ± ∆, is given in the following table: Finally, let us comment on the special cases when O saturates the unitarity bound. For ≥ 1 this happens when O is a conserved current with ∆ = + d − 2. It is easy to check that the three-point function (2.4) of O + are automatically conserved at P 3 if we set ∆ = + d − 2. On the other hand, conservation of O − (P 3 ) implies that the three-point functions (2.7) vanish for > 2. For = 2 conservation follows from ∆ = 3. This is consistent with the fact that it is possible to couple the currents to the stress tensor with a parity odd three-point function in theories that violate parity. For = 0, one should impose ∂ 2 O = 0 when ∆ = 1 2 . This implies that the three-point function JJO must vanish for both ± parity.

Special case: J J T µν
Let us study in detail the three point function of two identical conserved currents and the energy momentum tensor. As discussed in the previous section there are only two independent structures. The three-point function is given by (2.4) with = 2, ∆ = d and is related via the conformal Ward identity to the current two-point function (2.12) The symbol S d = 2π is the volume of a (d − 1)-dimensional sphere and γ is an independent parameter that appears in the OPE (1.3). The parameter γ controls the -6 -JHEP05(2019)098 anisotropy of the energy correlator of a state created by the current [34,36,37]. Positivity of this energy correlator implies the bounds , (2.13) which are saturated by free scalars and free fermions, respectively. This bound was recently proven relying only on unitarity and OPE convergence [38]. The parameter γ also has a nice physical meaning from the perspective of the dual AdS description. The current three-point function can be computed from the bulk action where L is the AdS radius, W is the Weyl tensor and F is the field strength of the bulk gauge field dual to the current. In this form, it is clear that γ does not contribute to the two-point function of the current in the vacuum.
In the conformal bootstrap analysis of the four-point function JJJJ we normalize all operators to have unit two-point function. Recall that the stress tensor has a natural normalization due to the Ward identities, That means that we should multiply the OPE coefficients (2.10) by This shows that C J is not accessible in the bootstrap analysis of JJJJ . On the other hand, C T does affect the OPE coefficients of normalized operators. For comparison, we recall the values of C T for free theories [41]. Each real scalar field contributes Each Dirac field contributes is the dimension of the Dirac γ-matrices in d spacetime dimensions. Notice that in d = 3, a complex scalar contributes the same as a Dirac fermion This is the minimal matter content of free theories with a U(1) global symmetry. The general structure of the four point function is 5

Crossing symmetry
The crossing symmetry 1234 → 2134 sends the cross ratios (u, v) into u v , 1 v . As usual in the conformal bootstrap analysis, this crossing symmetry follows automatically from the conformal block expansion in the (12)(34) channel associated to the three-point functions studied in section 2.1.
On the other hand, the crossing symmetry 1234 → 3214 is not satisfied by the conformal blocks in the (12)(34) channel and gives rise to non-trivial constraints on the operator spectrum and OPE coefficients. The crossing symmetry 1234 → 3214 leads to 6  In other words, we have 8 odd and 11 even functions under the crossing symmetry u ↔ v.
We will see that the functions f 18 and f 19 will disappear in 3 dimensions, hence the choice to put them at the end of the list. 5 The factor of v 1−d is convenient to make the crossing equations simpler. 6 These equations are derived for √ u+ √ v = 1 where all the 43 tensor structures are linearly independent. By continuity, the equations also hold for any u and v. This is indeed the case for the free theory examples discussed in appendix A.

Conservation
In the numerical conformal bootstrap approach one writes the four point function as a sum of conformal blocks and imposes (a truncated version) of the 19 crossing equations (2.24). Fortunately, we can use conservation of the external currents to reduce this large number of crossing equations. Imposing conservation directly on the four point function produces a set of differential constraints that the functions f s (u, v) must satisfy. The four point function of three vectors and one scalar operator contains 14 independent tensor structures (in any dimension). As a consequence, each conservation condition will produce 14 first order differential equations of the form where i = 1, . . . , 14. The first important observation is that the conformal block decomposition 7 automatically satisfies these equations. 8 The second observation is that the equations (2.25) are crossing symmetric. In other words, applying the crossing symmetry u ↔ v to (2.25) and using (2.24) we obtain an equivalent set of differential equations. This means that if we use these differential equations to determine the functions f s evolving from a crossing symmetric "initial condition", then crossing symmetry is guaranteed everywhere. Therefore, if we start from a conformal block decomposition, it is sufficient to impose crossing symmetry on a minimal set of data about the functions f s that determines these functions everywhere via the differential equations (2.25).
To make this idea more precise it is convenient to introduce new coordinates [ One can check that the matrix [K t ] has rank 12. 9 That means that we can evolve 12 functions f s starting from an initial time slice, which we choose to be t = 0. Since the functions f s are either even or odd under t → −t, crossing symmetric boundary conditions are obtained by simply imposing the odd ones to vanish on the line t = 0, while the even ones are left unconstrained. One can explicitly check that the (7 dimensional) Kernel of 7 See for instance (2.54) in the next section. 8 In fact, we used this to cross check the computation of the conformal blocks. 9 In fact, this is true for a generic choice of time coordinate around the point u = v = 1/4. The exception being the coordinate y. In this special case, the rank of [K y ] is 10.  [K t ] decomposes in two orthogonal subspaces (of dimension 5 and 2) associated to the eigenvalues ±1 of the crossing symmetry matrix [ P ] defined in (2.23). This means that we can evolve 8 − 2 = 6 odd functions and 11 − 5 = 6 even functions. One possible choice is f 1 , . . . , f 6 and f 8 , f 9 , f 10 , f 11 , f 12 , f 14 . Hence, by using 12 out of the 14 conservation equations we reduced to the set of crossing symmetry conditions: Note that the boundary condition on the line doesn't constrain the even functions: any initial condition f s , s = 8, 9, 10, 11, 12, 14 will be automatically evolved into a crossing symmetric function. In fact, this is still not the minimal set of data where we can impose crossing symmetry. We will use the two remaining conservation equations to reduce further the set of crossing symmetry equations. The remaining conservation equations are not evolution equations. They are two constraint equations on the initial data at t = 0. One can check that at t = 0, the first constraint equation only involves odd functions and the second only involves even functions. More precisely, the first constraint equation can be written as where the sum runs over the odd functions (s = 1, 2, . . . , 7 and s = 19) and the coefficients A s (y) and B s (y) are regular at the crossing symmetric point y = 0. This means that -10 -

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it is sufficient to impose f 3 (t = 0, y = 0) = 0 because this equation will ensure that f 3 (t = 0, y) = 0 for any y. Since the second constraint equation only involves even functions, which are unconstrained at the initial surface t = 0 it is not useful to further reduce the crossing symmetry constraints.
In the end the minimal set of crossing symmetry conditions is: where we went back to the original coordinates u and v. In agreement with [42] in general d there are 7 equations in the "bulk"; additionally there are five constraints on the line and one at a crossing-symmetric point. We remark that our analysis of the conservation equations is valid only in a local neighbourhood of the crossing symmetric point u = v = 1/4. However, this is sufficient for the numerical bootstrap algorithm where we only consider a finite number of derivatives of the crossing equations at u = v = 1/4.

Three dimensions
In three dimensions not all 43 tensor structures of the four point function are linearly independent. The easiest way to see this is to consider the embedding space tensor can be written as a linear combination of the 43 tensor structures Q s that form a basis for four-point functions of vector primary operators. Therefore, in d = 3 this gives rise to a linear relation between the 43 tensor structures Q s . Using the 3 invariants W (12) (34) , W (13) (24) and W (14)(23) we obtain 2 independent relations between the structures Q s in d = 3. 10 These constraints can be found in appendix F. We use these to express the structures Q 31 and Q 40 in terms of the other Q s . According to the definitions in appendix D, this corresponds to the functions f 18 and f 19 . The entire argument about the conservation equations proceeds in the same way just dropping these two functions.
In the end the minimal set of crossing symmetry conditions in d = 3 is as follows. It includes five equations in the "bulk" [42], five constraints on a line, and one at a point:

Conformal blocks
In this work we computed the conformal blocks (CB) for four external currents using the recurrence relation of [43][44][45]. The existence of such recurrence relation comes from the study of the analytic properties of the CBs as functions of the conformal dimension ∆ of the exchanged operator O. To see this, it is convenient to rewrite the CBs in radial quantization as follows have poles 11 at ∆ = ∆ A because of the contribution of all the null states in H O A . All these contributions together form a conformal block associated to the exchange of O A . Accordingly, the residue at the pole ∆ = ∆ A is proportional to the conformal block where the (R A ) pqp q are coefficients which depend on the representation of the operator O.
The previous discussion explains the pole structure of the conformal blocks. To complete the recurrence relation we also need to obtain the asymptotic of conformal blocks when ∆ → ∞. To this end it is convenient to write the conformal blocks in the basis of four-point function tensor structures as we did in (2.20), Here r and η ≡ cos θ are the radial coordinates of [46], defined by . The conformal blocks are not regular at ∆ → ∞ because of the essential singularity g O (r, η) ∝ (4r) ∆ , however we can factor it out and define a new function h O which is well behaved ∞O,s (r, η) . (2.43) So far the discussion was schematic and valid for any conformal block. We now want to give more details for the case of four external vectors in three dimensions. We shall construct the conformal blocks for generic external vector operators and only at the end we will specialize to the particular case of equal conserved currents. The goal is to find the conformal blocks where 12 s = 1, . . . 43. We obtain a set of recurrence relations for the conformal blocks 11 In [44] it was shown that there can exist only simple poles in odd dimensions. In even dimensions higher order poles can appear. However the CBs for even dimensions can be obtained by analytic continuation from the odd dimensional case. 12 The actual independent structures are 41, but we find it more convenient to work in the 43−dimensional space and project out the final result into the 41−dimensional space.

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which are diagonal in the label s but which couple the labels p, q, Here the label A stands for (T, n) where T is one of the four types I, II, III, IV and n is an integer belonging to the set S T which can be finite or infinite depending on T . In particular we have We present here a table which specifies the labels A, Further details about the table (2.49) can be found in the appendix E. The conformal blocks at large dimension h ∞ are computed exactly by solving the Casimir equation at leading order in the large ∆ expansion as explained in appendix E.4. The coefficients R can be conveniently written in terms of three contributions where the coefficient Q and the matrix M arise because of the different normalization of the two and three point functions involving the primary descendants O A . Schematically, In appendix E.3 we further detail how to obtain the coefficients R.
Notice that with formulas (2.46)-(2.47) one can obtain all the blocks correspondent to four generic external vector operators. In this work however we only need the blocks for -14 -

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conserved and equal currents. To obtain them we contract the labels p, q of the blocks with some matrices m ± which come from the conservation of the 3-point function of JJO ± explained in section 2.1. We further contract the index s with a matrix [ M ] which simplifies the crossing equation of 4 equal vector operators as explained in section 2.2, (2.53) Here the matrix m + is 2 × 5 while the matrix m − is 1 × 4, therefore p, q = 1, 2 for the parity even case and p, q = 1 for the parity odd case. In appendix E.5 we give the precise form of such matrices. The matrix [ M ] is 19 × 43 and it is defined in appendix D. It is worth to stress that since the equations (2.46)-(2.47) are diagonal in s, it is possible to compute only some structures, without having to compute the others. In the following sections we will drop the tilde symbol above the labels p, q, s. Using the OPE channel (12)(34), one obtains the following conformal block expansion 13 where the functions f were defined in section 2.2.2. For further details we refer the reader to appendix E.

Bootstrap equations
Plugging the conformal blocks decomposition (2.54) into the three dimensional crossing equations (2.35) we explicitly obtain 11 conditions which can be nicely written in vector notation as JJO − for higher . In particular, for the stress energy tensor we have: Finally, V ∆+ , V ∆ − are 11-dimensional vectors and V ∆ + is a 11-vector of 2 × 2 matrices. Introducing the (anti)symmetric combination of conformal blocks defined in (2.53), (2.57) 13 In appendix E, we compute the conformal blocks in a three-point function basis which is different from the one of section 2.1, therefore the coefficients λ -15 -

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we have For any 2 × 2 matrix M , s[M ] ≡ (M + M T )/2 selects its symmetric part. In the above expression we omitted the (p, q) upper indices when only one conserved conformal block is allowed, namely for parity even scalars and parity odd operators.

Setting up the semi-definite programming
The feasibility of the above set of equations can be constrained using semidefinite programming (SDP). We refer to [12] for details. To rule out a hypothetical CFT spectrum, we must find a linear functional α such that for the identity operator, for all parity even operators in the spectrum with even, for all parity odd operators in the spectrum with any = 1.
Here, the notation " 0" means "is positive semidefinite". Since the 11 crossing equations have a different dependence on the conformal invariants u, v, it's worth spelling out the explicit form of the linear functional α we consider in this work. Let us remind the reader the definition of the usual coordinates z, z: Then, we define the family of linear functionals α acting on an 11-dimensional vector, whose entries are functions of z, z Although we didn't write it explicitly, the linear functionals are parametrized by the integer Λ, which indicates the order of derivatives considered. Notice that the action of the functional on a vector of matrices results in a matrix, while its action on a vector of scalar functions produces a number. The existence of such a functional for a hypothetical CFT spectrum implies the inconsistency of this spectrum with crossing symmetry. In addition to any explicit assumptions placed on the allowed values of ∆, we impose that all operators must satisfy the unitarity bound The more information about the spectrum we use in (2.59), the easier it is to find a functional α that excludes the putative CFT. In this work we mainly focus on assumptions about the minimal values of operator dimensions in given sectors and the value of parameter γ defined in section 2.1.1.
We will review the exact SDP problem to solve case by case in the next section.

Results
In this section we present the results of our numerical investigations. In what follows ∆ ± will denote the dimension of the first parity even/odd neutral spin operator. We will also use (∆ ± ) to denote the second operator in the same sector.

Bounds on operator dimensions
We begin our journey in the space of CFTs with global symmetries by inspecting the constraints imposed by crossing symmetry on the spectrum of scalar operators. As reviewed in section 2.1, the OPE J × J contains both parity even and parity odd scalars. The first issue we want to address is how large can the dimensions of these operators be.
To answer this question we solved the semi-definite problem (2.59) with the assumption that all scalar parity-even/odd operators have dimension larger than ∆ ± 0 correspondingly. The allowed region is shown in figure 5. The very first surprising result is that crossing symmetry is able to constrain the plane ∆ + 0 , ∆ − 0 into a closed region, meaning that all CFTs with global symmetry must have parity even and parity odd scalar operators. This is completely universal: this result is only based on unitarity and associativity of the OPE. To our knowledge this is the first completely general result for 3D unitary CFT with global symmetry. 14 Let us now describe the shape of figure 5. If we regard the boundary of the allowed region as a function (∆ − 0 ) max of ∆ + 0 , then it can only be a monotonic non-increasing function. 15 Hence we expect the allowed region to be shaped by existing CFTs with the 14 All previous results in the bootstrap literature assumed at least the presence of a scalar or fermion operator with a given fixed dimension; theories with extended supersymmetry represent an exception: scalars are contained in certain protected super-multiplets. 15 If we can not exclude a theory with ∆ + 0 = a and ∆ − 0 = b then we cannot exclude theories with ∆ + 0 ≤ a and ∆ − 0 ≤ b. largest gap in the scalar sector. There are three solvable models that we can place in the ∆ + 0 , ∆ − 0 plane: a free massless complex scalar field φ, a free massless 3d Dirac fermion ψ and a Generalized Free Vector Field (GFVF). In the free scalar field case, the U(1) current OPE schematically reads: In the free fermion case, we find Finally, the GFVF is equivalent to a free photon in AdS 4 . From the three dimensional point of view it corresponds to a conserved current with a standard 2 point function, and all higher point correlators satisfying Wick theorem. In this case the lightest scalar operators are given by

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Notice that the GFVF is technically a so called dead-end CFT, since it doesn't contain relevant scalar operators. On the other hand it doesn't contain a local energy momentum tensor either, since it corresponds to a U(1) gauge theory on a fixed AdS background (infinite central charge C T and no dynamical gravity). These solvable CFTs are marked in figure 5 as described in the caption. While the boundary of the allowed region is close to the GFVF point, it is quite far from the point of free boson theory. Instead it starts at higher values ∆ − 0 and after a small plateau it displays a kink for values of ∆ + 0 seemingly in correspondence to the interacting O(2) model. To our knowledge the dimension of the leading parity odd scalar in this model is not known, neither in the ε-expansion nor in the 1/N expansion. Accordingly, we conjecture that the lightest parity odd operator in (3.1) in critical O(2) theory acquires a positive anomalous dimension.
One additional feature of figure 5 is the region extending to values ∆ + 0 larger than 4 but requiring at the same time parity odd scalars with small dimension. Let us call φ − the parity odd scalar operator with smallest dimension. The OPE of φ − with itself would contain a parity even scalar operator 16 , for some function f . This bounds has been already obtained in past works focused on the three dimensional Ising model [11,12,17]. For this work we extended these results to larger values of ∆ − 0 . The blue shading in figure 5 represents the disallowed region. We expect that the use of mixed correlators of scalars and conserved currents will shed light on the fate of this region.
The existence of a CFTs with large gaps in the scalar sector, namely the GFVF, shapes the bound shown in figure 5 for 1.6 ∆ + 0 4 and could potentially hide other theories in the bulk of the allowed region. In order to better probe this region we explored the constraints on theories with a finite value of the central charge. To do that, we modified the conditions (2.59) and looked for a linear functional that satisfies the following requirements: Compared to (2.59) we have modified the normalization condition in order to input a specific value of γ and we have used (2.56). It is straightforward to show that the bound obtained with a functional satisfying (3.4) only applies to CFTs with C T ≤ C max T . 16 Unless there is symmetry argument preventing this from happening, this operator must coincide with the smallest dimension parity even scalar operator entering the J × J OPE. Figure 6. Allowed region assuming that all parity-even scalars appearing in the OPE J × J have dimension larger than ∆ + 0 and all parity-odd scalars have dimensions larger than ∆ − 0 . We also impose small central charge C T ≤ C free  More interestingly, one can observe that varying the parameter γ the bounds smoothly interpolates two very different regimes. For γ 1/12, the free fermion theory drives the shape of the bound, while as we decrease γ, the allowed region is entirely concentrated at smaller ∆ + 0 but large ∆ − 0 . Notice also that the maximum of ∆ − 0 is not reached at the free boson theory but at slightly larger values of γ and ∆ + 0 . These results are also shown as a 3D plot in figure 7.
In figure 8 we show the upper bound on the dimension of the second lightest parity even scalar operator (∆ + 0 ) as a function of ∆ + 0 . We performed the analysis with and without forbidding relevant scalar parity odd operators. Next, in figure 9 we show the bound on the dimension of the first non conserved spin-2 parity even operator (∆ + 2 ) . Notice that ∆ + 2 = 3 because the dimension of the stress tensor is fixed. 17 Interestingly, both bounds display a kink structure in the proximity of the location of the O(2) model. On the other hand both the maximal allowed values of (∆ + 0 ) and (∆ + 2 ) at the kink are much larger than the ones of the free O(2) model. It would be surprising if the interacting critical O(2) model displayed such large anomalous dimensions. At this stage, it is unclear if the kink feature is related to the O(2) model. It would be interesting to include correlations functions of charged operators in our bootstrap study to further explore this region. We leave this mixed correlator analysis for the future. Finally, notice that in figure 9 the region ∆ + 0 4.52 is excluded if we also take into account the constraints coming from the four point function of the lightest parity-odd scalar appearing in J × J (see figure 5). 17 However, we do not exclude solutions where the OPE coefficient of the stress tensor vanishes (CT = ∞).

Central charge bounds
A well established feature of the conformal bootstrap is the possibility to place upper bounds on OPE coefficients, or equivalently a lower bound on C T [5,7,9]. In this section we investigate the minimal value of the central charge that a CFT with a continuous local global symmetry is allowed to have, as a function of the parameter γ. To find such a bound, we search for a functional α satisfying the properties: Notice that compared to (2.59) we have eliminated the assumption of the functional α being positive on the identity operator contribution. As shown later, we will instead minimize −α[ V 0 + ]. Also, by fixing the normalization we input a specific value of γ. Here JJT , λ each component being a linear function of γ, and we have used (2.56). Finally, we introduced a gap in the spin 2 even sector, and assume that, besides the energy momentum tensor, whose dimension saturates the unitarity bound, all the parity-even spin-2 operators satisfy [O =2 ] ≥ (∆ + 2 ) . We will come back to this assumption later. Applying the functional to the crossing equations (2.55) and using the results of section 2.1.1 one obtains Therefore, the optimal bounds on C T will be set by the functional minimizing −α[ V 0 + ], subject to the constraints (3.5).
In figure 2, presented in the introduction, we show our best bound on the central charge as a function of γ and how the bound improves when increasing the numerical power Λ. As expected, inside the interval |12γ| ≤ 1, the bound seems to converge to a finite value, while outside it improves by a order one factor at each step.
In figure 10 we display the zoomed version of the same plot. As discussed in section 2.1.1, the two extremes of the interval 12|γ| ≤ 1 are saturated by the free complex boson and the free fermion theory. In [47], it was shown that when γ assumes the extremal values, the CFT must necessarily be free, i.e. all the correlators of the CFT must be equal to those of a corresponding (bosonic or fermionic) free theory. One would therefore expect the bound to approach the value of the central charge of a free complex boson or free fermion given in eq. (2.19) at the extremes of the allowed interval. This doesn't appear to be the case with the current numerical power. Nevertheless we might hope to approach the optimal bound in the limit Λ → ∞. In figure 11 we show a linear extrapolation of the bounds computed at γ = ±1/12 for Λ = 11, . . . , 25. For γ = 1/12 a linear extrapolation (upper blue line in figure 11) is consistent with an asymptotic bound C T ≥ C free T . Extrapolating the bound for γ = −1/12 is trickier. Although we expect the bound to be C T ≥ C free T , the linear fit (bottom red line in figure 11) clearly gives an asymptotic value smaller than C free T . Most likely, the linear extrapolation in Λ simply does not capture the infinite number of derivatives limit. It is plausible that the apparent convergence of the bound to a value smaller that C free T is due to some hypothetical CFT with C T < C free T and γ close to −1/12. With the current numerical power we cannot make a conclusive statement confirming or ruling out such a theory.
An interesting feature of figure 10 is that the central charge bound is well below C free T not only near 12|γ| = 1 but in the whole region 12|γ| ≤ 1. Based on previous works on conformal bootstrap [11,17] we are keen to consider this as an indication that there might exist a number of CFTs whose central charge is smaller than the free theory one. A largely accepted lore suggests that the central charge measures the number of degree of freedom in the theory. 18 Accordingly we expect a CFT with the central charge smaller than C free T to have minimal possible gloabal symmetry, i.e. only a global U(1). The critical O(2)-model is the only known example of such a theory with C T ≈ 0.944. The other possible candidate, the N = 2 Gross-Neveu model is in fact expected to have a central charge larger than C free T (see appendix A for a review). The critical O(2)-model clearly can not explain the current shape of the bound. As the numerics improves, Λ → ∞, we expect the optimal bound to become significantly stronger and be saturated by the hypothetical new theories with C T ≤ C free T .
18 This is clearly the case for free theories and CFTs that are perturbatively away from a free theory. Extrapolation: γ=-1/12 (red),γ=1/12 (blue) Figure 11. Extrapolation of the bound on central charge normalized to C free T as a function of the number of derivatives included in the semidefinite-programming Λ. The upper (blue) lines corresponds to a linear fit of the bounds at γ = 1/12. In the limit Λ → ∞ the extrapolation approaches the value 1. The lower curves correspond to a linear (red continuous) and quadratic (red dashed) fit of the bounds at γ = −1/12. The linear fit predicts an asymptotic bound much smaller than the free theory one. Other fits may predict a value of C T closer to C free T . This is exemplified by the quadratic fit.
Let us now discuss the role of the gap in the spin-2 parity even sector. The key observation is that the proof of the conformal collider bound (2.13) elegantly obtained in [38] relies on the assumption of the existence of a single energy momentum tensor. If instead a CFT possesses several conserved spin-2 operators, the bound (2.13) must be replaced by a bound on a weighted sum over the corresponding γ's: Unfortunately, in our bootstrap analysis with a finite truncation parameter Λ, any parity even spin-2 operator of dimension close to 3 is almost indistinguishable from another stresstensor. This is precisely the role played by the gap (∆ + 2 ) in (3.5): imposing a single energy momentum tensor corresponds to input a gap strictly larger than 3. In figure 12 we show the impact of this gap on the lower bound on the central charge of the theory. As expected, the effect is stronger in the region disallowed by the bound (2.13) because the imposed gap on the spin-2 sector implies uniqueness of the stress tensor. On the other hand, imposing a gap like (∆ + 2 ) = 3.5 probably excludes most CFT's with global symmetry bigger than U(1). For example, consider the OPE of two conserved currents in the O(3)-model: The grey line corresponds to not imposing any gap between the energy momentum tensor and the next spin-2 parity even operator. The blue line shows how a gap (∆ + 2 ) = 3.5 impacts the strength of the bound. While inside the interval |12γ| ≤ 1 the bound is marginally affected, the effect outside the interval is dramatic. singlet of the U(1) generated by J 3 µ and we expect its dimension to be perturbatively close to the unitarity bound. A similar argument holds for all O(N > 2) models: generically there can be more than one spin-2 operator entering the J × J OPE, whose anomalous dimension is 1/N suppressed. We expect that to properly constraint these theories one has to bootstrap the four-point functions of full set of conserved currents.
A final comment regarding the comparison between our analysis and the case of bootstrapping the stress-tensor four-point function is in order. Since the 3 point function of three stress tensors is structurally different from the one of two stress tensor and a non conserved spin-2 operator, there is no contribution in the 4 point function that could fake a second energy momentum tensor. As a consequence, the uniqueness of T µν is automatic and in principle there is no need to impose a gap in the spin-2 even sector.

Central charge bounds with spectrum assumptions
In this section we investigate how the bounds on the central charge change when we introduce additional assumptions on the spectrum of scalar operators or in the spin-4 parity-even sector. We therefore replace the conditions (3.5) with the following conditions In figure 13a we show the impact of imposing the absence of relevant odd scalar operators in the J × J OPE. This amounts to set ∆ − 0 = 3 while keeping all the other gaps to their minimal value consistent with unitarity. As expected, the bound on the central charge increases for positive values of γ, excluding the free fermion theory, which is indeed ruled out by this assumption. Close to γ = −1/12 the bound is almost unaffected, consistent with the conjecture that the left part of the plot is driven by the free boson theory and possibly by the critical O(2)-model. Notice that in this analysis we haven't made any assumption about the parity-even spectrum, and in particular no assumption about the number of relevant parity-even scalars. A second investigation, shown in figure 13b, solely assumes that no relevant parity-even scalar operators are present. The impact of this assumption is more dramatic: very small room is left for theories with C T < C free T . Although we haven't performed a careful extrapolation we believe this window will close in the limit of infinite number of derivatives Λ → ∞.
Finally, in figure 14 we combine both assumptions to study the central charge limits for the case of dead-end CFTs, namely those CFTs without any relevant scalar operator. As the name suggests, these CFTs would be stable under any scalar deformation and therefore would represent an attractive point for all the renormalization group flows driven by rotation-preserving deformations. While we expect such CFTs with a large central charge (from weakly coupled abelian gauge theory in AdS 4 ), there are no known examples with small values of C T . Interestingly, at present, our limits do not preclude the existence of dead-end CFTs with C T /C free gap ∆ + 4 can be considered as a knob to interpolate from free theories to holographic CFTs. Indeed, the J ×J OPE in free CFTs contains a conserved spin-4 parity even operator. When going to interacting CFTs, its dimension must be lifted [48] and the operator acquires a positive anomalous dimension. On the other hand, in holographic CFTs the lightest spin 4 operator is the "double-trace" operator ∼ J (µ 1 ∂ µ 2 ∂ µ 3 J µ 4 ) of dimension 6, with corrections suppressed as 1/N . As we increase the value of the gap, we exclude more and more theories, and it is natural to expect that the only solution still consistent with crossing symmetry are those which have a large central charge. This behavior is indeed realized in figure 15a, where we show the lower bound on the central charge as a function of γ for several values of ∆ + 4 . As anticipated, the bound grows with the gap. By increasing the numerical power one can presumably make the bound much stronger. In figure 15b we performed an extrapolation in the number of derivatives of the central charge limit when ∆ + 4 = 6 for the central value γ = 0. The extrapolation suggests that ∆ + 4 = 6 implies C T = ∞, in agreement with the holographic interpretation. 19

Hunting the O(2)-model
So far we have investigated bounds on the central charge under very general assumptions on the spectrum of CFTs. However, they do not appear to be saturated by any known CFT. The extrapolation in the number of derivatives shown in figure 11 suggests that in this limit 19 Recall that the anomalous dimension ∆ + 4 − 6 ∼ 1/CT must be negative due to Nachtman's theorem [49,50]. we can make contact with a known result, namely the free fermion theory. On the other hand, theories such as the O(2) model seem to remain in the bulk of the regions allowed by crossing symmetry. In oder to understand the reason for this it is useful to inspect the solution of crossing along the boundary extracted with the extremal functional 20 method introduced in [51] and successfully used in [17,19] to extract the spectrum of the three dimensional Ising model. We observe that all the extremal solutions contain odd operators with ≥ 2 and dimension saturating the unitarity bound ∆ − = + 1 (or very close to it). On the contrary, all known theories display a larger gap. For instance, free theories and GFVF satisfy ∆ − = + 3 (see appendix A). Basically, the extra gap comes from the need to contract -tensor indices with derivatives.
It is natural to expect that the O(2) model also displays an extra gap for all parity odd operators with spin ≥ 2. Hence, in order to make contact with the O(2) model, we replace the conditions (3.5) with the following requirements: (3.10) 20 We remind the reader that on the boundary of the allowed region the solution of the truncated crossing equation is unique and it is given by the zeros of the linear functional α. The novelty in the above conditions consists in raising the twist of all parity odd operators to τ − all ≥ 1, and imposing that relevant parity even scalar must be confined in a narrow interval ∆ ∈ [∆ min , ∆ max ] = [1.5092, 1.5142], for which we take the rigorous bound from previous bootstrap studies [18]. In figure 16a we show the impact of varying τ − all from 1 to 3. Interestingly the bounds start developing more and more pronounced minima as we increase the value of τ − all . In addition, the left part of the bound is insensitive to this parameter, while the right part heavily depends on it. Although from figure 16a it would be tempting to set τ − all = 3, the large spin analysis discussed below suggests that this is not possible. Nevertheless we expect that τ − all = 2 is a safe assumption. With this choice in (3.10), we can obtain a rigorous bound on the parameter γ for theories with the central charge smaller than the free theory one: Let us comment on the consistency of our assumption that the dimension of the leading twist parity odd operators of spin ≥ 2 in the O(2) model is not too far from 3, which is the free theory value. 21 The leading correction to the dimension of these operators in the large spin expansion has been computed in [50] using analytic bootstrap techniques. It JHEP05(2019)098 was found that: Notice that the leading correction in the above formula is negative whenever γ satisfies the conformal collider bound. Moreover, our estimate γ O(2) ∼ −1/12 is compatible with the assumption of small anomalous dimension δ O − .

Comments on parity non preserving theories
While the main focus of this work was to obtain bounds on parity preserving theories, in this section we shall argue that many of the constraints that we found should also apply to theories which do not preserve parity. In a parity non preserving theory, it is useless to classify local operators according to their parity. Moreover, correlation functions do not transform in a definite way under parity. In order to understand some important feature of the bootstrap equations for parity non preserving theories we now explain how to generalize the discussion on three and four point functions of sections 2.1 and 2.2. To extend the discussion on three point function it is sufficient to say that both the parity even t + (2.4) and parity odd t − (2.7) structures can appear for the exchange of a given operator O, schematically (3.14) In order to characterize the four point function one needs to add to the set of tensor structures Q s of (2.20), new structures Q − s which are parity odd (and therefore proportional to the epsilon tensor), schematically Since the structures Q s and Q − s differ by an epsilon tensor, the crossing equations for f s and for f − s do not mix, namely 2. Finally, in order to conclude that the constraints obtained in this work also apply to parity breaking theories one needs to check that the functions f s (u, v) admit the same conformal block decomposition assumed here. This is indeed the case, since the crossed terms arising from the mixing of structures t + and t − in (3.14) only contributes to f − s , schematically: We conclude that the bounds that we obtained by studying crossing symmetry of f s are also valid for parity non preserving theories. To be precise the bounds only apply once we give up the notion of parity of operators, therefore all the gaps ∆ ± on the spectrum of parity ± operators of spin should be replaced by gaps ∆ on the spectrum of operators of spin ∆ ± → ∆ . The bounds on central charge of subsection 3.2 and 3.3 are left unchanged. Notice that every time we consider a different gap in the sector ∆ + and ∆ − (for a given spin ), we should consider that the plot is valid only for ∆ = max[∆ + , ∆ − ]. For instance the bound in figure 12 is understood with assumptions on the gap ∆ 2 on the first spin two operator after the stress tensor. Similarly figure 14 is the bound for parity non preserving theories with no relevant singlets (the plots 13a and 13b in this context descend trivially from 14).

Conclusions
In this work we have used the numerical conformal bootstrap to study the space of three dimensional conformal field theories with (at least) a global U(1) symmetry. We did this by analyzing the four point function of identical conserved currents. We have shown that, analogously to the case of the correlation function of 4 scalars or 4 fermions, unitarity and OPE associativity alone let us carve out the parameter space of CFTs. Inspecting the allowed values of scalar operator dimensions we found that any CFT with a conserved spin-1 current must contain both parity even and parity odd scalars. The boundary of the allowed region displays a non trivial structure with multiple features. In particular a kink appears close to the location of the O(2) model, providing an upper bound on the dimension -32 -

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of the first parity odd scalar ∆ − 0 ≤ 7.65 (1). A similar kink is present in the bound on the second spin-2 parity even operator. Also, we excluded the existence of dead-end CFTs with central charge smaller than twice the central charge of a free 3d Dirac fermion. We also explored bounds on the central charge with several assumptions on the CFT spectrum. In this case we observed a slower numerical converge. Nevertheless we found clear evidence of the conformal collider bounds for spin-1 currents [34][35][36][37].
The present work paves the way to many generalizations and extensions. Given the special role that the O(2) model seems to play in our exclusion bounds, it is natural to expect that a mixed scalar-current bootstrap analysis will allow to precisely determine the spectrum of the theory [52]. Similarly, one could consider multiple correlators including external fermionic charged operators in order to narrow down the location of the N = 2 Gross-Neveu model.
As mentioned several times, the results of this work are very general and apply to CFTs with a continuous global symmetry that admits a local conserved current. 22 On the other hand, by studying a single current inside a larger symmetry, we loose the ability to distinguish operators that are singlets under the entire global symmetry group from those that instead are only invariant under the specific U(1) considered. As an example, spin-2 operators with dimension close to the unitarity bound but not singlet under the full global symmetry are difficult to distinguish from the energy momentum tensor in the numerical analysis. As we have seen in section 3.2 this dramatically affects the bound on the central charge. Hence, in order to obtain numerical evidences of the conformal collider bounds we restricted to theories with a finite gap between the energy momentum tensor and the dimension of the next spin-2 operator. While we expect this merely represents a technical assumption for theories with global U(1) symmetry, it might not apply to CFTs with larger symmetry group. In this case, it will be important to bootstrap the full set of correlation functions J a 1 µ 1 J a 2 µ 2 J a 3 µ 3 J a 4 µ 4 , with a i spanning all the generators of the global symmetry. This set up would also allow to specify the global symmetry by inputting the group structure constants f abc and to put a bound on the current central charge C J . The analysis will require a minor modification of the present framework. All necessary conformal blocks required for this analysis have been already computed in the current work. The main difference will be represented by the higher number of crossing conditions. Finally, the same investigation presented in this work can be extended to higher dimensions with minor modifications. The recurrence relation presented in appendix E could be generalized in order to build conformal blocks in any dimension. Alternatively, the fundamental results obtained in [53][54][55] allows us to compute the conformal blocks in four dimensions in closed form. Moreover, the analysis of the crossing equations in section 2.2 is valid in any spacetime dimension. This direction would be of particular interest in presence of N = 1 4D supersymmetry. Indeed, the U(1) R-symmetry current J µ is embedded in the Ferrara-Zumino supermultiplet, which also contains the energy momentum tensor as a super-descendant. The study of J µ correlation functions will provide a universal handle 22 A trivial example of a theory with a global symmetry but no conserved current is a free complex field in AdS d+1 with mass strictly larger than −(d 2 − 4)/4, which is the dual of a Generalized Free Field in d spacetime-dimensions with scaling dimension ∆ > (d − 2)/2. on all local SCFTs, allowing in principle to discover theories we currently know nothing about [56].
This work represents a first exploration of an uncharted territory. Very much like 15th century navigators, we landed and explored the border of a whole new world. We created a first map of the landscape of CFTs with global symmetries which will serve as a roadmap for further investigations. We are confident that future expeditions will lead to finer understanding of this space.

A.1 Free scalar theories
The simplest example of CFT in 3 dimensions with U(1) global symmetry is the theory of free complex scalar field ϕ. The central charge of this theory C T = C free T was given in (2.19). The global U(1) current is given by the conventional expression J µ = iϕ∂ µ ϕ − iϕ∂ µ ϕ. The lightest parity even neutral scalar ϕϕ has dimension ∆ + 0 = 1. The lightest parity-odd scalar is more complicated. Normally, one can build a parity-odd scalar out of two vectors but in case of one complex field this combination vanishes. Hence the lightest parity-odd scalar has more derivatives and is of dimension ∆ − 0 = 7. A complex field ϕ can be decomposed into two real fields. One can consider a more general case of N free real fields ϕ i . This theory has O(N ) global symmetry, N (N − 1)/2 The lightest parity-even scalar ϕ i ϕ i still has the same dimension ∆ + 0 = 1. When N ≥ 4 one can combine two mutually commuting currents J [ij] µ and J [kl] ν with four distinct i, j, k, l into a dimension ∆ − 0 = 5 parity-odd scalar This operator is charged under full O(N ) but is neutral under some generators, including J [ij] µ and J [kl] µ . Depending on the choice of the generator J µ = ω ij J ij µ the OPE two identical J µ will or will not include (A.4). For example the OPE of two J µ = J [12] µ will remain the same as in the theory of one complex boson, with ∆ − 0 = 7, while the OPE of two J µ = (J  u). Finally, the 11 independent functions appearing in the above equation are:

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First few terms in the conformal block decomposition of (A.6) are summarized in the table below. In particular it shows that the second lightest parity-odd scalar appearing in the OPE of two currents has dimension (∆ + 0 ) = 4. This is because dimension 3 operator ∂ µ ϕ∂ µ ϕ is not a primary. Similarly, second lightest spin-2 operator ϕϕT µν also has dimension (∆ + 2 ) = 4. The OPE coefficients λ JJO ± in the above tables are defined in appendix E.5.

A.2 Critical O(N ) models
The spectrum of critical O(N ) models at large N is in many ways similar to that one of free theories. Including leading 1/N corrections the central charge is given by [57,58] (also see [43] for further references)

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The main difference is the dimension of the lightest parity even scalar. At large N its dimension approaches 2, The dimensions of the parity-odd scalars are less studied. For N = 2, 3, the lightest parity odd scalar appearing in the OPE of two currents has the same quantum numbers as (A.2) and is expected to have dimension ∆ − 0 ≈ 7. For N ≥ 4 there is also a parity-odd operator in the representation of (A.4). Thus for some generators J µ = ω ij J ij µ we still expect ∆ − 0 ≈ 7 while for others ∆ − 0 ≈ 5. For small N = 2, 3, . . . certain dimensions and central charge are known with a good precision from the conformal bootstrap and Monte-Carlo simulations [18,20,43,59]. We report some of them in the table below: Here , are the first and second singlet scalar operators appearing in the OPE φ a × φ b , while T ab is the leading scalar operator transforming in the tensor traceless representation of O(N ). Under a given U(1) ⊂ O(N ), T ab decomposes into neutral and charged components. The neutral ones are allowed to enter the OPE of the conserved current associated with the U(1). This means that

A.3 Free fermion theories
In three dimensions, a free Dirac fermion ψ is invariant under a global U(1) symmetry. This theory has the same central charge as a free complex scalar, C T = C free T . The lightest parity-odd scalar ψψ has dimension ∆ − 0 = 2, while lightest parity-even scalar (ψψ) 2 has dimension ∆ + 0 = 4. The four point function of the conserved current J µ = ψσ µ ψ can be easily calculated explicitly. The four point function contains two distinct contributions f s = f disc s − f con s /Υ where the disconnected piece f disc s is given by (A.23), while the connected one is given below. Also, Υ = Tr 1 denotes the trace of the identity in γ-matrix algebra in d-dimensions, 1 = (γ 1 ) 2 (Υ = 2 in 3 dimensions). Following the same conventions as -37 -JHEP05(2019)098 in (A.5), we have: A first few terms in the conformal block decomposition of (A.12) are summarized in the table below. The OPE coefficients λ JJO ± in the above tables are defined in appendix E.5.

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A.4 QED 3 A theory of N f Dirac fermions ψ i in 3d coupled to a U(1) gauge field A µ flows to a nontrivial IR fixed point if N f is sufficiently large. This theory has global SU(N f ) flavor symmetry, with the currents J a , with a = 1, 2, . . . , N 2 F − 1. Flavor symmetry might be spontaneously broken for small N f by chiral condensate. Besides this there is a topological U(1), with the topological current J top ∝ F . The operators charged under this U(1) are monopole operators. When N f is odd the theory is not parity-invariant [60]. Accordingly we consider only even N f such that the effective number of Majorana fermions N = 2N f is a multiple of four. For large N , the central charge is given by [61] For minimal possible value N = 4 this gives C T ≈ 2.72C free T . This result is valid only if there is no spontaneous symmetry breaking.
Identifying the lightest parity even and odd scalars appearing in the OPE of two currents requires consideration. Since monopole operators are charged under topological U(1) they are excluded from the OPE of both J top × J top and J a × J a . First, we consider OPE of two J top which contains only SU(N f ) singlets. For large N , the lightest parity-odd singlet scalar ψ i ψ i has dimension [62,63] while lightest parity-even scalar is a combination of (ψ i ψ i ) 2 and F 2 µν of dimension The OPE of two flavor currents include all fields charged in representations appearing in the product of two adjoints. In this case the lightest parity-odd scalar is in adjoint representation of SU(N f ), (O n=1 ) i j = ψ i ψ j . At leading order it has dimension [62,63] which is smaller than ∆ 0 . Similarly, the lightest parity-even operator is (O n=2 ) [ij] [kl] = ψ i ψ j ψ k ψ l of dimension which is smaller than ∆ 0,− = 4 + 128(2−

Gross-Neveu models
One Dirac spinor can be decomposed into two Majorana spinors. A theory of N ≥ 2 free Majorana fermions has O(N ) symmetry, while the dimension of lightest parity even and odd scalars remain the same for all N . Upon adding a parity-odd scalar field φ with quartic interaction which couples to Majorana fermions via Yukava coupling ψ i φψ i , the theory flows into an interacting fixed point characterized by O(N ) symmetry. The lowest parity-odd scalar φ has dimension [64][65][66] (also see [15] for further references) The lightest parity-even scalar φ 2 has dimension .20) while the central charge is given by [67] Below we compare C T and ∆ + 0 , ∆ − 0 for small N found using leading 1/N expansion and Pade extrapolation of -expansion in d = 2, 4 [67,68] 23 and bootstrap techniques [16]. It is worth noting that, similar to critical bosonic O(N ) theories, central charge of Gross-Neveu models even for small N is substantially close to the free theory counterpart.

A.6 Generalized free vector field
The generalized free vector field (GFVF) is a theory of a conserved current J µ (of dimension d − 1) with the standard two-point function and all higher-point correlation functions

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satisfying Wick theorem. In particular the four-point function of currents JJJJ includes only the disconnected piece (all other components are zero), This theory contains no stress-energy tensor, i.e. C −1 T = 0. The only operators present in the spectrum are those build of J µ . In particular the lightest parity even scalar J µ J µ has dimension ∆ + 0 = 4 and parity-odd scalar given by (A.1) has dimension ∆ − 0 = 5. GFVF is dual to U(1) gauge theory in AdS 4 in the limit of zero Newton constant, when only disconnected Witten diagrams contribute. In the table below, we list some OPE coefficients that we obtained from the conformal block expansion of JJJJ in d = 3 dimensions. The OPE coefficients λ JJO ± in the above tables are defined in appendix E.5.

B Relations between parity odd structures
Parity odd conformally invariant three point functions can be construct using the -tensor. In d = 3, there are six parity odd building blocks: However, not all of them are independent. To see this we use the following identity

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where Ξ is an arbitrary 5 dimensional vector. The determinant vanishes automatically because the first row of the matrix is a linear combination of the other 5 rows. By choosing for instance Ξ = P 1 one gets Similarly one can get two more equations by choosing Ξ = P 2 , P 3 . All together these relations allow to express ij in terms of linear combination of ij only. In addition, one can find linear relations involving only the three ij . This follows immediately if we choose Ξ orthogonal to the three P 's. This is achieved with One can easily check that and conclude that (B.2) reduces to  All others can be expressed in terms of a linear combination of the above. For instance the following identity holds:

C Basis for four point function
Out of the above list one can construct 43 tensor structure. These are listed in table 1.

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As a first step we explain our convention for the labels p, q of the OPE. We define the leading OPE in terms of a linear combination of tensor structures where z µ are null polarization vectors. Here ∼ means that we are considering only the channel of the OPE in which O ± × J 1 exchanges the operator J 2 (therefore omitting all the other possible exchanged primaries), and taking into account only the leading term of the OPE for x µ → 0 (therefore omitting all the contribution of the descendant operators). We also define The various OPE coefficients c (q) 12O ± multiply the tensor structures t (q) ± (x, z, z 2 , z 3 ), which are Lorentz invariant and satisfy t (q) The sum over q in (E.2) runs from one to five for parity even operators, since we can build the following five structures t (1)

(E.6)
Notice that for = 0 only t (1) and t (2) survive and for = 1 all are allowed except t (5) . These structures are related by a simple linear transformation to the structures of the main text. To be more precise, the basis T (q) can be related to t (q)

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Similarly for parity odd operators with generic spin there are four allowed tensor structures which can be build by using the three dimensional epsilon tensor (see appendix E.5.2) Again it is clear that for = 0 there is only t (1) and for = 1 only t (1) , t (2) , t (3) . The basis in embedding space can be related to t for ≥ 2 by means of the following matrix

E.2 Null states
In this section we write all the possible primary descendant states that can be exchanged when the external operators are all vectors. In d = 3 the only irreducible representations of the rotation group are traceless and symmetric tensors of spin . We consider such a primary state of spin and we contract it with null polarization vectors z µ as follows It is possible to recover the expressions with the indices by acting with the differential operator D z of [40,69], (E.14) The primary descendant states are of four possible types which are additionally labeled by an integer n (n runs over all positive integers for type I and type II, and over a finite set for type III and IV). We define each descendant state |O A ; z by the action of an operator D A (built as a linear combination of many P µ , the generators of the translations) on a primary state |O; z

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The operators D A can be fixed by asking that K µ |O A ; z = 0 when ∆ = ∆ A . This is simply the requirement that when ∆ = ∆ A , the descendants |O A ; z become primaries. The operators D A can be written in the following compact form [44] D I,n ≡ c I,n (z · P ) n , (E.16) where V j and E are defined by The coefficients c are unimportant normalization constants that we are free to set to any value. For convenience, we choose c I,n = 1 = c IV,n , c II,n = 1 . (E.22) We want to stress that the operators of type I, II and III, do not change the parity of the state on which they are applied, while the one of type IV does, namely The residue R A is obtained using formula (2.50), where Q A are defined in (E.25).
The matrices M A can be defined by the action of differential operator D A on the tensor structures appearing in the leading OPE (E.2). For the first three types we have where the matrices M +A are 5 × 5 while the M −A are 4 × 4. The exponent α ±A is equal to α ± where we replace → A and ∆ → ∆ + n A . Moreover, we set ∆ = ∆ A . The type IV is slightly different since it changes the parity of the primary state, therefore In this case M +IV is a rectangular matrix 5 × 4 while M −IV is 4 × 5.
The definitions given here can be directly used to compute M . However there can be better strategies to implement this computation. One possible strategy is to act with each building block differential operator contained in D A (namely P · z, P · D z , V j and E) on the set of tensor structures, in order to obtain a correspondent building block matrix that rotates the tensor structures. The full result can be computed as products of these building block matrices, as detailed in [44]. A new strategy is explained in appendix E.6.1, where we show how to obtain M A for the types (I, n), (II, n), (III, n) in a closed form by doing a trivial computation.
It is worth commenting that by direct computation one can check that M IV,n vanishes for n > 2 which means that only two poles of type (IV, n) contribute, namely ∆ = 0, 1 as mentioned in appendix E.6.4.

E.4 Conformal block at large ∆
In this section we explain how to compute h ∞ , the large ∆ limit of the conformal blocks. To do so, we are going to solve the Casimir differential equation at the leading order for large ∆ with the appropriate initial condition for G (p,q) ∆ ± when x 12 , x 34 → 0. The Casimir equation can be schematically expressed as . We consider the leading order in ∆ of (E.28) and we substitute the definitions (2.41) and g where M is a 43 × 43 matrix of explicitly known rational functions of r and η and where we dropped all the labels of h s (which will be reintroduced when we will fix the initial condition of the Casimir equation to obtain a set of differential equations for the 43 functions P (s ) s . This ansatz, inspired by the solution of the scalar Casimir at large ∆, has the property of eliminating completely the dependence on ∆ 12 = ∆ 1 − ∆ 2 and ∆ 34 = ∆ 3 − ∆ 4 from the differential equation. Moreover it turns out that we can easily fix all the functions P (s ) s (r, η) since they are simply polynomials in r (of maximal degree 12) and η. Notice also that we are leaving d unfixed: in fact the solution that we find works in any dimension. We can further choose a basis such that 2. Next, we pass to the basis that diagonalizes crossing symmetry. This is done by defining the 17 functions f s shown in (D.2).
3. As described in [12], the problem of finding α satisfying (2.59) can be transformed into a semidefinite program. The form of the functional α is given in (2.61). The first step is to compute the derivatives of the vectors V + ,V + and V − defined in (2.58).
To do this, we started directly from the explicit form of the conformal blocks as a power series in the variable r defined in (2.42).
4. Once we take the derivatives and set r = 3 − 2 √ 2 and η = 1, these expressions reduce to rational approximations for conformal blocks in the variable ∆. Keeping only the polynomial numerator in these rational approximations, (2.59) becomes a "polynomial matrix program" (PMP), which can be solved with SDPB [71]. We use Mathematica to compute and store tables of derivatives of conformal blocks. Another Mathematica program reads these tables, computes the polynomial matrices corresponding to the V 's, and uses the package SDPB.m to write the associated PMP to an xml file. This xml file is then used as input to SDPB. Our settings for SDPB are given in table 2.
As discussed in section 2.2.2, the minimal crossing constraints consist in 5 bulk equations, 5 boundary equations and one constraint at a point. Once one considers derivatives of the crossing equations at a given point, the conservation equations (2.27) (and their derivatives at the crossing symmetric point u = v = 1/4) simply become a set of linear relations between various derivatives of the functions f s . We explicitly checked that the set of derivatives included in the numerical bootstrap has maximal rank, i.e. there is no linear dependence induced by the conservation equations. Also, we explicitly checked that the system made by the conservation equations and their derivatives at the crossing symmetric point can be used to determine neglected components.
Because the functions involved have definite symmetric properties under z → z, the number of non-vanishing derivatives included for a given Λ is: dim(α) = 5   Table 2. SDPB parameters for the computations of scaling dimension bounds in this work. For C T bounds we need to set all of the Boolean parameters in the table to False. In addition to that, we used dualityGapThreshold = 10 −6 , while all the rest of the parameters were kept at the same values as for the dimension bounds.
The degree of the numerator and denominator is controlled by the order of the conformal blocks expansion in r, or equivalently by the number of poles kept in the recursion relations (2.46)-(2.47). Contrary to previous conformal bootstrap works [12], we used the obtained expressions as they are, without employing any further approximation. Approximations might be useful to push to higher number of derivatives.
Finally, we must choose which spins to include in the PMP. We have chosen the number of spins to depend on Λ as follows