Bootstrapping Boundary QED Part I

We use the numerical conformal bootstrap to study boundary quantum electrodynamics, the theory of a four dimensional photon in a half space coupled to charged conformal matter on the boundary. This system is believed to be a boundary conformal field theory with an exactly marginal coupling corresponding to the strength of the interaction between the photon and the matter degrees of freedom. In part one of this project, we present three results. We show how the Maxwell equations put severe constraints on boundary three-point functions involving two currents and a symmetric traceless tensor. We use semi-definite programming to show that any three dimensional conformal field theory with a global U(1) symmetry must have a spin two gap less than about 1.05. Finally, combining a numerical bound on an OPE coefficient and some Ward identities involving the current and the displacement operator, we bound the displacement operator two-point function above. This upper bound also constrains a boundary contribution to the anomaly in the trace of the stress tensor for these types of theories.


Introduction
Four dimensional quantum electrodynamics (three spatial dimensions and one time) is an important example of quantum field theory, underpinning our understanding of the interaction between light and charged particles in the world around us.Three dimensional QED has had a more limited experimental impact but nonetheless has featured prominently in theoretical progress over the years.For example, by tuning the number of electron flavors N f , the low energy limit of this theory passes from a conformal fixed point at large N f to a theory with a mass gap at small N f , providing a model of confinement [1][2][3].This work is about a mixed version of these two theories, that we will call boundary QED, where the photon is four dimensional but the charged matter is confined to a 2+1 dimensional plane.In a sense that we will make more precise, the model is like graphene.Less well studied than its pure three dimensional and four dimensional cousins, it has inspired a small but fruitful line of inquiry over the years, with early references here [4][5][6].
As argued by refs.[7][8][9], boundary QED is a boundary conformal field theory (bCFT).The main purpose of this work is the numerical conformal bootstrap [10] of boundary QED.While numerical bootstrap of bulk correlation functions in bCFT face positivity issues [11], exceptionally there is a way around this problem for boundary QED.The strategy we employ here was inspired by previous work [12][13][14] where instead of a free photon, a free scalar is coupled to degrees of freedom on the boundary.The two ingredients are a restriction of the boundary spectrum by the bulk free equations of motions, and a locality constraint on bulk-reconstructed operators, which puts severe constraints on the boundary correlation functions.It is these constraints ultimately that we hope to impose in the numerical bootstrap.
In our case, we find that the Maxwell equations imply that there are exactly two currents J (1) and J (2) in the boundary operator expansion (BOE) of the bulk Maxwell field F µν .Of these currents, one has even and one has odd spatial parity, and as a result, we often refer to these as an electric current E and a magnetic current B. Ultimately, we will be using our constraints in the numerical conformal bootstrap of ⟨J (a 1 ) (x 1 )J (a 2 ) (x 2 )J (a 3 ) (x 3 )J (a 4 ) (x 4 )⟩.Before we get there, however, we first need to provide a more detailed description of boundary QED and also a discussion of its two and three-point correlation functions.In section 2, we start with a perturbative approach to the theory, to set up the problem and define the objects of interest.Of particular relevance to the numerical bootstrap, we review how the gauge field coupling gives rise to an anomalous dimension of the boundary stress tensor.This allows us to use the dimension of the leading boundary spin two operator as a substitute for the coupling in our numerical analysis.We also discuss the displacement operator, an operator with protected scaling dimension that every bCFT must contain in its spectrum.
In section 3, we move on to a discussion of two-point functions.We compute the BOE of a free Maxwell field F µν by using a Taylor series expansion into the bulk.The result can be expressed entirely in terms of the currents J (a) and their derivatives.Using the BOE, we compute all the two-point functions between F µν and the J (a) .These twopoint functions are informed by the symmetries of the theory and tell us about the moduli space of possible boundary conditions of the Maxwell field.By imposing unitarity, we find that this moduli space can be parametrized by a unit disk cross the half line, and discuss its physical significance.In particular, the ⟨J (a) (x)J (b) (y)⟩ two-point functions determine the conductive response of the system to an oscillating electric field.To the extent that boundary QED is like graphene, some constraints on these two-point functions may help resolve a longstanding puzzle about the optical conductivity of graphene, as we discuss in more detail in the Discussion in section 6.
After discussing the two-point functions, we move to three-point functions in section 4. For the numerical bootstrap of the correlation function of four currents ⟨J (a 1 ) (x 1 )J (a 2 ) (x 2 )J (a 3 ) (x 3 )J (a 4 ) (x 4 )⟩ , we need to compute three-point functions of the form ⟨J (a) (x 1 )J (b) (x 2 )T ℓ (x)⟩.Here, T ℓ is a symmetric traceless tensor of spin ℓ that can appear in the OPE of two of the currents J (a) .Similar to what happens for a free in the bulk scalar [12][13][14], 1 fixing T ℓ , we find that the data for ⟨J (a) (x 1 )J (b) (x 2 )T ℓ (x)⟩ is generically determined by the data for any single one of them, for example ⟨J (1) (x 1 )J (1) (x 2 )T ℓ (x)⟩.The relation comes from a regularity constraint on the bulk-boundary-boundary correlation function ⟨F µν (x 1 , x ⊥ )J (a) (x 2 )T ℓ (x)⟩.Consider the case T 0 .The three-point functions ⟨J (a) (x 1 )J (b) (x 2 )T 0 (x)⟩ are fixed by conformal symmetry up to constants which we call γ ab and γ ab , where the tilde denotes odd parity.The new result is then the relation between these constants: With a better grasp on the theory, we turn to the numerical conformal bootstrap in section 5.While we intend ultimately to enforce constraints such as (1.1) between the OPE coefficients, for this paper we restrict the bootstrap to four-point functions of a single current, which in this boundary QED context could be either ⟨E(x 1 )E(x 2 )E(x 3 )E(x 4 )⟩ or ⟨B(x 1 )B(x 2 )B(x 3 )B(x 4 )⟩.Of course, without constraints, we are looking at any 3d CFT with a global U(1) symmetry.In order to refine the search space, we focus on two special operators, an even parity scalar of dimension four and the lowest dimension spin two operator.In any bCFT, we are guaranteed the existence of a displacement operator D, which is the operator sourced by the location of the boundary.As the operator can be identified with the boundary limit of the normal-normal component of the bulk stress tensor, it must have protected scaling dimension equal to the bulk space-time dimension and even parity (in theories with parity symmetry).While before coupling the free fermions to the bulk, they have their own stress tensor, once we introduce g ̸ = 0, energy can leak off the boundary and the divergence of this boundary stress tensor no longer vanishes.In particular, its dimension must float up from the unitarity bound of three.These are exactly the two operators which we explore.
In our bootstrap analysis, we find two results associated with these operators.Let the spin ℓ gap be defined as the scaling dimension of the smallest spin ℓ operator minus the corresponding unitarity bound.The first result is that there appear to be no three dimensional CFTs with a spin 2 gap slightly larger than one, 1.05 to be precise.Thus having coupled any boundary CFT with a U(1) symmetry to the bulk Maxwell field, the operator that used to be the boundary stress tensor can acquire an anomalous dimension that is no larger than about one.Moreover, we have some indication that once the spin 2 gap is large enough, the only possible CFT is a generalized free vector field (GFVF), which has a lowest spin two operator of dimension 4. (A GFVF is defined to be a CFT constructed from a conserved vector V i assuming that all the correlation functions follow from Wick's Theorem.)We speculate that with a more sophisticated numerical bootstrap analysis, for example using a system of mixed correlators, the bound of 1.05 could be pushed down closer to one.Indeed, it would be interesting to see if the bound could be proven analytically using extremal functionals, along the lines of [17].
The second result is more specific to our boundary QED setup.There is a Ward identity [9] which relates ⟨J (a) J (b) D⟩ to ⟨J (a) J (b) ⟩ and ⟨DD⟩.Note by conformal invariance, all of these correlation functions are completely determined by a handful of constants.Using this Ward identity, we are able to bound the constant C D that determines ⟨DD⟩.We find (numerically) C D ≤ 1.728 C f ree D where C f ree D is the value for D = V µ V µ in the GFVF.This result has added significance as C D determines also certain boundary terms in the stress tensor trace anomaly for bCFTs [7,18].Thus by bounding the displacement two-point function, we are also bounding a boundary contribution to the trace anomaly.
Lastly, we include some additional information in our appendices.Appendix A proves that a certain class of boundary conditions for the field F µν must result in a bulk-boundary decoupling.In appendix B, we discuss the technical details of our numerics.In appendix C, we provide a brief overview of the GFVF, the free scalar, and the free fermion in 3d.Finally, in appendix D, we review a Ward identity argument constraining the three-point function of two currents with the displacement operator.

Perturbative Perspective
While our approach is ultimately non-perturbative, from a pedagogical standpoint and to fix notation, it is worthwhile to start by discussing the decoupled case where the photon does not interact with anything on the boundary.Working in Euclidean space, truncated to x n > 0, consider a free 4d photon A µ , described by and we insist that this setup preserves the symmetry of a bCFT.In this notation Greek indices µ, ν are bulk while lower case Roman indices i, j are reserved for the boundary.The index n is the direction normal to the boundary.The Maxwell field strength is as usual We define the (Hodge) dual field strength F µν ≡ 1 2 ϵ µνλρ F λρ .The theory is free, the path integral is Gaussian, and the saddle point given by a solution to the equations of motion which are just Maxwell's equations.Because of the boundary, there is an extra boundary term in a variational analysis which leads to two boundary conditions, both consistent with conformal symmetry.From δA i = 0, we have the Dirichlet type condition F ij = 0 = F ni , often called absolute boundary conditions.This boundary condition does not allow us to couple the system to boundary degrees of freedom.Alternately, we have the more interesting Neumann type condition, often called relative boundary conditions, Defining the complexified gauge coupling the proportionality constant in the boundary condition depends only on the phase of τ .This relative boundary conditions suggest a rephrasing in terms of boundary currents.We define the "electric" and "magnetic" fields, In the absence of boundary interactions, the natural boundary conditions amount to forcing a linear relation between these two fields, halving the degrees of freedom.Field insertions of E or B can be rewritten in terms of some underlying unit-normalised current V , whose correlation functions are computed by Wick contractions.This theory is sometimes called a generalized free vector field (GFVF).Its field spectrum is made up of single-trace primaries created by normal ordering the product of fields decomposed over parity, spin ℓ and conformal dimension ∆2 where p and q are non-negative integers and z i is a polarization vector.
Returning to the free Maxwell field with boundary, this theory contains a bulk stresstensor.In the index free formalism, it is given by We are in particular interested in its boundary limit, from which we can identify two protected boundary primary operators, the displacement operator and the flux vector, (2.9) One recognizes the energy density and Poynting vector from electromagnetism.The operator P i is problematic and must be taken out of the spectrum to ensure the conformal invariance is preserved (see e.g.[7,13,[19][20][21]).In the free setting, vanishing is automatic because the electric and magnetic fields are parallel.Using Wick contractions, it is straightforward to evaluate the norm of the displacement operator (2.10) The constant C D has an interpretation in terms of a boundary conformal anomaly coefficient [7,18,22].It forms a crucial piece of CFT data, which we will use as a probe of the theory as we turn on interactions.
Having fixed the CFT data in the free case, one can introduce deformations.The idea is to pick any 3d CFT possessing a U(1) current J .Gauging this theory through coupling to the bulk F µν , one can compute correlation functions in the interacting theory through conformal perturbation theory, where the ellipsis are possible seagull terms.The non-trivial aspect of this system is that the boundary interaction does not (generally) induce a renormalisation of the bulk coupling τ .If we take the particular examples of massless fermions or complex scalars, gauging the U(1) vector symmetry by coupling to the bulk Maxwell field, the relative normalization of the boundary kinetic term and the vertex is fixed by gauge invariance.As in regular QED, the only way for the gauge coupling to develop a beta function is through wave function renormalization of the photon.However, the photon kinetic term is a bulk term in the Lagrangian, and there is no way for the "tail to wag the dog" -for the boundary to renormalize bulk quantities.Thus not only is the theory conformal, but it also has an exactly marginal coupling, the gauge coupling [7][8][9].Indeed, this argument generalizes to include coupling any free Maxwell field to a boundary CFT with a U(1) global symmetry [9], with important caveats.If the boundary theory already has a marginal coupling, then the addition of the gauge coupling g may lead to nonzero beta functions which cannot be tuned away.Further, the theory may become unstable if the coupling gets sufficiently large; for example, there could be spontaneous generation of a mass gap.Assuming the coupling is tuned to be in this conformal window, the ensuing interacting conformal manifold is our object of study.The existence of this space of boundary conditions is non-trivial.Although conformal manifolds are easy to construct for defect CFTs through symmetry breaking, see e.g.[23][24][25][26], our example is genuinely interacting.Moreover, QED with conformal matter on defects with codimension greater than 1 is likely trivial [27], making boundary QED special in this regard.
Turning on interactions means that the currents E and B no longer satisfy a linear relation.For simplicity, we will focus on the parity-preserving case θ = 0.The modified variational principle gives E = g 2 J instead of E = 0. 3  Current two-point functions are determined up to a constant by conformal symmetry.Let τ J J , τ EE , and τ BB be the constants associated with the two-point functions of J , E, and B respectively.The identification E = g 2 J along with a sum rule [9] we review in section 3 allows us to deduce the relations (2.12) While these relations are exact, τ JJ receives corrections from its decoupled-value, making it a function of the coupling τ .To visualize the moduli space of boundary conditions for F µν , we will find it convenient to introduce the reflection coefficient The current two-point functions are not the only interesting quantities.Some perturbative results are available for a couple of other quantities that will be important in this work [9]. 4  The norm of the displacement operator (and an associated contribution to the trace anomaly) changes as a function of coupling, making it an interesting observable of the interacting theory, which we will bound in this work.The displacement two-point function is corrected at leading order to A further observable is the leading primary T 2 with spin two and smallest conformal dimension.When the coupling is zero, this operator can be identified with the boundary stress tensor.However, as we turn on the interaction, this leading primary acquires an anomalous dimension, which can be expressed in terms of the central charge c of the 3d theory as [9] For reference c for the Ising model is 3/16π 2 .This fact motivates us to use the dimension of the leading boundary spin 2 operator as a substitute for the coupling in our numerical analysis.We will find evidence that the dimension of T 2 is bounded by that in the GFVF theory, i.e. four.Indeed, one may expect to find the GFVF in the infinite coupling limit.
In the limit g → ∞, the complexified coupling τ approaches the real axis, which can also be reached by an SL(2, Z) transformation of the free case.
3 More generally, 2πJ = 2π g 2 E + θ 2π B, and we can also define 2πI = B.The pair (J , I) transform under SL(2, Z) as a doublet and can be identified with the boundary values of where F ± µν ≡ 1 2 (Fµν ± Fµν ). 4 See also [28,29] for the particular case of coupling to free massless fermions on the boundary.
Another observable, which is accessible to our numerical setup, is the current-currentdisplacement three-point function coefficients.In a convention specified in eq.(D. 16), (2.16) One can also look at the normalised OPE coefficient, which shows up in the numerical bootstrap, (2.17) 3 The Two-Point Function of the Maxwell Field The goal of this section is to characterize and constrain the boundary conditions on F µν in the presence of charged degress of freedom on the boundary.We derive the bulk-to-boundary operator expansion (BOE) [20] of F µν , which we then use to compute ⟨F µν (x)F ρλ (y)⟩ in terms of a moduli space specified by a triplet of numbers.These numbers parametrize R + × D where D is a disk with unit radius.
The content can largely be found in [9] although our development and perspective is different.In particular, our discussion of the moduli space and its relation to symmetries is new.

Boundary Operator Expansion
To compute the BOE of a free Maxwell field, we begin with a Taylor series expansion of F µν .The resulting boundary objects are then decomposed in terms of boundary irreducible representations, using the building blocks that are the normal vector n µ to the boundary, and the boundary induced metric h µν = δ µν − n µ n ν , Note that x ⊥ ≡ n • x.The E i and B i currents of section 2 have reappeared as primary operators in this boundary decomposition.We work in Euclidean signature throughout, and the factor of i is inherited from the Wick rotation from Lorentzian to Euclidean.
The electromagnetic fields are defined on the boundary, and to rewrite (3.1) in terms of a BOE, we must trade the normal derivatives for tangential derivatives.To do so, we use the free Maxwell equations The divergence conditions on the electric and magnetic fields are precisely the conservation conditions, allowing us to reinterpret E and B as conserved currents.Using the differential relations (3.2), we gather Collecting factors and reshuffling slightly, we go from (3.1) to our BOE expressed purely in terms of the primary operators E and B, which are conserved currents, and their descendants.
The expression (3.5) can be massaged into a more palatable form.We define the following two differential operators Relating the 4d and 3d Levi-Civita tensors through the relation ϵ αβρλ n λ = ϵ αβρ , the final form of the boundary OPE becomes These structures are divided clearly between the normal and tangential directions.In the way we chose to write things here, it is clear that all parity-odd structures will come from resummations involving an odd number of magnetic fields, and that the different OPE channels are exchanged under Hodge-duality.Parity odd data will be pure imaginary, as it should be.

Two-Point Correlators From the BOE
The point of this section is to discuss and compute two-point functions, ⟨F µν (x)E i (y)⟩ , ⟨F µν (x)B i (y)⟩ , ⟨F µν (x)F λρ (y)⟩ . (3.9) We will exploit the BOE developed in the previous section along with the constraints of conformal symmetry.
Although they are primary fields, E and B need not be orthonormal fields.In general, conformal invariance constrains the two-point function of two conserved currents to take the form where I ij (x) is the inversion tensor δ ij − 2x i x j /x 2 .The two-point functions of our currents E and B are thus completely characterized by the three numbers τ EE , τ BB and by Bose symmetry τ EB = τ BE .One consequence of parity is that if we require the theory be parity symmetric, then ⟨E i (x)B j (y)⟩ and its corresponding coefficient τ EB must vanish, i.e. the currents are orthogonal.
To write down the bulk-boundary two-point functions ⟨F µν (x)E i (y)⟩ and ⟨F µν (x)B j (y)⟩, we need some additional tensorial building blocks compatible with conformal symmetry [20].We will use the inversion tensor I µν defined below (3.10) along with its index contracted form I µ ≡ I µν z ν where z µ is a boundary polarization vector satisfying z • n = 0. We need two additional structures as well5 The most general bulk-boundary correlation function between F µν and a boundary vector J (a) (y, z) = J i (a) (y)z i takes the form The Maxwell equations for F µν impose that ∆ J = 2 (as does the condition that J µ is a conserved current).The Maxwell field is special in this respect; curiously, bulk-symmetrictensor operators have a zero two-point function with boundary conserved currents (see e.g.[21]).Now comparing the BOE of F µν (3.7), the two-point function of a conserved current (3.10), and the above expression, we see that Next, we consider ⟨F µν (x)F λρ (y)⟩.Bulk two-point functions can depend on an invariant cross ratio The additional indices require more complicated tensor structures, still constructed from the previously defined building-blocks: 6 αβ,ρσ ϵ αβ µν . (3.16) It is now straightforward to write the general form of the bulk-to-bulk correlator, µν,ρσ + f 2 (ξ)F (2) µν,ρσ + f 2 (ξ)L (2) µν,ρσ (3.17) The functions f i and f i are constrained by the conservation and Bianchi identities, with the generic solutions ) That the solutions for f i and f i are so similar follows from the fact that F µν and F µν obey the same equations of motion.
We can fix the constants c i and c i by matching to the leading terms in the BOE of the Maxwell field.The result is (3.20) In fact we have checked that the full expression for ⟨F µν (x)F µν (y)⟩ can be recovered by using the BOE of F µν twice on the current-current correlation functions of E and B and resumming, but we omit the details.The equations of motion allow for a constant term c 1 in f 1 , which is never realised.The vanishing of c 1 can be explained as follows.Far away from the boundary, this correlator must reduce to pure free Maxwell.In this limit, only the structures F (1) and L (1) persist, and one must take the cross-ratio ξ to zero.As the free two-point function without a boundary is parity-invariant (the θ term is a total derivative), c 1 vanishes and only c 1 contributes.
which fixes c 1 in terms of the gauge coupling.We find Via (3.20), we find in turn a constraint on τ EE +τ BB which we will discuss briefly in section 3.3 and at length in section 6. 7  In what follows, we will swap the c i , c i , and τ ij for the new variables κ, χ e , and χ o where With these new variables, we can write the F µν two-point function in the form µν,ρσ − F (2) µν,ρσ with xµ ≡ x µ − 2x ⊥ n µ .The tensorial structures have an interpretation as an (even or odd) reflection of the initial bulk structure.The first term corresponds to the free two-point function, and so to the exchange of the identity in the bulk channel.The two remaining terms are due to the exchange of the bulk scalar and pseudo-scalar in the bulk OPE of which expectation values define a moduli space.We refer to χ e and χ o as reflection coefficients.
The system of two-point functions of E and B must be reflection positive for a unitary CFT, which imposes the following two constraints on the τ ij : The constraint τ EE +τ BB ≥ 0 we now see is equivalent to the positivity of g 2 via (3.20) and (3.21).The second constraint, however, is less trivial and restricts the conformal manifold: In other words, the moduli space of possible Maxwell field two-point functions can be parametrized by a disk of unit radius (corresponding to the possible values of χ e and χ o ) cross the half line (corresponding to the possible values of g 2 or equivalently κ).
In the limiting case χ 2 e + χ 2 o = 1, the matrix of current two-point functions becomes degenerate and E and B are the same up to rescaling, i.e. they are parallel.We discussed 7 The Fourier transformed form of the current correlation function is Defining σ ab ≡ τ ab π 2 2 , the sum rule is slightly nicer in Fourier transformed language, σEE + σBB = g 2 .
in section 2 that in free theories, some linear combination of the currents E and B must vanish.Thus we conclude that all free theories must lie on the circle χ 2 e + χ 2 o = 1.The converse is in fact also true, that if a theory lies on χ 2 e + χ 2 o = 1, then the correlation functions of the remaining vector field must all follow from Wick's Theorem, as we will demonstrate later in appendix A.

Moduli Space & Symmetries
The moduli space coordinatized by (χ e , χ o , κ) describes a conformal manifold of theories.We now discuss how symmetries and dualities relate to features of the moduli space.
Recall that under parity, E is even, while B is a pseudo-vector, hence (χ e , χ o ) P → (χ e , −χ o ).In parity preserving theories, to which we will specialise later on, τ EB and correspondingly χ o must vanish, i.e.E and B are perpendicular.Theories with parity are restricted to the meridian line −1 ≤ χ e ≤ 1 and χ o = 0.At the point χ e = 1, τ EE and the electric current must vanish, while at the point χ e = −1, τ BB and the magnetic current must vanish.Restoring χ o , we see that it encodes the strength of the parity breaking.In the maximally parity breaking cases (χ o = ±1 and χ e = 0), the currents E and B become the same up to a choice of sign.Along the perimeter χ 2 o + χ 2 e = 1, the E and B fields are parallel, which is realised clearly through the θ angle appearing in the proportionality constant between the currents in the case where the bulk decouples from the boundary degrees of freedom.The story is summarized in fig. 1.It is interesting to consider the action of SL(2, Z) on this conformal manifold.We can deduce how SL(2, Z) acts on (κ, χ e , χ o ) by looking at how it acts on the field strength ⟨:F 2 :⟩ and ⟨: F 2 :⟩ one-point functions, which we now review.Let Introduce the self dual and anti-self dual combinations of the field strength: 24), the one-point functions take the form defining a complexified reflection coefficient χ = χ e + iχ o .The field-strengths F ± have a simple transformation law, which translates into an induced action on the moduli space: In other words, SL(2, Z) acts as a phase rotation on the disk parametrized by χ and a rescaling of κ.Under the action of S-duality, τ → −1/τ , the origin of the disk χ = 0 is a fixed point while κ gets rescaled by the coupling κ → |τ | 2 κ.On the other hand, under T : τ → τ + b, both κ and χ are fixed.In the limit where we stay on the perimeter of the disk, this transformation is related to the SL(2, Z) action on the 3d CFT of [33], see [9].Before closing, it is important to emphasize the physical significance of these constraints EB .The quantities τ EE and τ BB determine the charge conductivities of the matter degrees of freedom with respect to the two currents E and B. From these constraints, we deduce that in any theory of this type where the interactions are mediated by the bulk photon, both conductivities are positive numbers and that their magnitudes are bounded above by the coupling constant 2g 2 /π 2 .This sum rule was already noted in [9] and a complexified version of it was rediscovered from localization in [34].
We can push the reasoning a bit further and investigate what can happen for self-dual theories that map back to themselves under an element of SL(2, R).Suppose we are in a situation where the theory maps back to itself under a T ST transformation, where T acts by a translation τ → τ + t.Two such theories were considered in [34], one with t = 0 and the other with t = 1/2 (see also [35]).Under such a transformation, τ = i √ 1 − t 2 is a fixed point for 0 ≤ t < 1, κ is left unchanged, and χ rotates by a nontrivial phase given by cos −1 (1 − 2t 2 ).Thus to be self-dual, we must fix χ = 0.At the center of the disk, the E and B fields are on equal footing and have the same two-point functions, τ EE = τ BB .Thus we conclude that at this proposed self-dual point, it must be that In other words, the constraint of self-duality is enough to completely fix the conductivities of these two currents.A version of this result, formulated for scalar and fermionic matter degrees of freedom, can be found in [36,37].

A Bulk Constraint on Boundary Data
As our ultimate goal is to bootstrap the correlation function of four currents, as a preliminary step we need to describe the three-point functions of two currents with the operators that can be exchanged in the four-point correlator.The exchanged operators are symmetric traceless tensors (STTs) T ℓ where ℓ is the spin.Thus we begin this section by writing down the ⟨JJT ℓ ⟩ three-point functions compatible with conformal symmetry.The currents in our problem are the boundary values of the same bulk field F µν , and it turns out that regularity of the parent ⟨F µν JT ℓ ⟩ correlation function puts nontrivial constraints on the boundary three-point functions.In the second part of this section, we explain these constraints.

The ⟨JJT ℓ ⟩ Three-Point Function
We first specify our basis choice for the different ⟨JJT ℓ ⟩ correlation functions.We follow conventions close to [38].

Boundary Structures in ⟨JJT ℓ ⟩
We are interested in the configuration with two, potentially different, currents, and one STT operator of generic spin ℓ.We will comment later on the specific cases ℓ = 0, 1.We write the correlator in the index-free formalism [31].We define where we use m and n to index the insertion point.The even parity building blocks, in the embedding and real-space picture, take the form and all the structures in both frames indeed correspond exactly to one another under projection.We pick the shorthand convention The parity-odd structures are more involved.In the embedding space they are It turns out these three building blocks are enough to generate all the parity-odd structures in the three-point functions.In fact for ℓ ≥ 2, we can trade ϵ 12 for the other two building blocks.

OPE Coefficients and Conventions for ⟨JJT ℓ ⟩
In this subsection we specify our basis choice for the enumeration of the different structures.
Our setup is that of two conserved currents E and B which we collectively refer to as J (a) and a STT T ℓ : We now define the different structures E (I) and O (I) .We also specify the consequences of conservation and Bose symmetry on the OPE coefficients γ I ab and γ I ab .
Primary with ℓ ≥ 2 : Given the V i , H ij and ϵ ij building blocks above, we can parametrise the three-point function as and Note that we got rid of the ϵ 12 structure, as it is redundant for ℓ ≥ 2, as follows from the identity Conservation imposes the constraints (4.10) and Using Bose symmetry, one can swap the two currents to relate the matrix of coefficients: This crossing equation reduces the amount of data we have to consider.To wit, if we consider ⟨EET ℓ ⟩, we find that ℓ = 2n implies The constraints for ⟨BBT ℓ ⟩ are the same.These results will be important for the bootstrap application later because they mean that for each even spin ℓ ≥ 2, there are three independent OPE coefficients, which we can take to be γ 1 EE , γ 2 EE and γ 1 EE .However for odd spin ℓ ≥ 3, there is just γ 1 EE .In view of the bootstrap application to come, we introduce γ + EE and γ − EE where (4.13)Note one needs to be a bit careful with these relations.While the map from γ ± to γ I is generically full rank, it can be that for certain specific values of ∆, the rank decreases from two to one, in which case SDPB does not function well. 8We have designed the transformations such that the rank is always two for ∆ in the unitary region.
For a mixed current bootstrap, we will ultimately need the constraints γ I EB = (−1) ℓ γ I BE , and ( Primary with ℓ = 1 : We now note the modification when ℓ = 1.We have fewer structures in both sectors, but lose the linear relation that got rid of ϵ 12 .In this setting we use the basis and which defines what we mean by γ I and γ I when ℓ = 1.Imposing conservation imposes the relations Bose symmetry in turn implies the properties Now, the resulting simplification is even more straightforward, as γ I EE = 0 = γ I EE , and γ I EB = −γ I BE , γ I EB = − γ I BE , a degenerate case of the ℓ ≥ 2 conditions.In other words, in the bootstrap application to come of four identical currents, we will not need to consider spin one exchange.
Primary with ℓ = 0 : In this simplest of the cases, we have the independent structures Conservation again brings restriction on the data, Bose symmetry implies which goes to zero on the boundary.There are six structures that we use to represent the six tensorial components of F µν in this three-point function.To build the tensor structures, we first consider the bulk-to-boundary building blocks as well as the analogue of H ij with one index freed, From these building blocks are constructed the six allowed even parity objects which we antisymmetrise with 2X One has their odd counterpart as well, which we define by acting with i 2 ϵ µναβ .We can now define the correlator of interest where the functions f b,I and g b,I are called form factors and will be our main interest in what follows.For simplicity, we focus on the parity-even sector f b,I .The constraints on the parity even and parity odd sector are essentially the same because the Bianchi identity and equation of motion for F µν are exchanged under S-duality.In total, there are nine different parity even objects that can enter in this correlator.They are given by The nine parity odd objects K (I) µν = i 2 ϵ µναβ S (I)αβ follow from contraction with the epsilontensor.Note that if ℓ = 1, one must set f b,6 (v) = 0 and f b,8 (v) = 0, and likewise for the odd components.If ℓ = 0, we set f b,2 , f b,5 , f b,7 , and f b,9 and their odd counterparts to zero as well.
The form factors f a,I and g b,J can be decomposed into a sum of boundary conformal blocks which are functions of the cross ratio v.If the field theory is moved to hyperbolic space through a Weyl transformation then when v = 1, the bulk insertion lies on a geodesic joining the two boundary insertions.In general, this point v = 1 marks the end of the region of convergence of the sum of conformal blocks.However, in our case, the sum over conformal blocks is finite.Only the two currents E and B are in the BOE of the Maxwell field.There is no possibility to eliminate any naive divergences at v = 1 through an infinite sum.Instead, divergences at v = 1 have to be cured by imposing additional constraints on the boundary data of the CFT.It is these constraints we now aim to extract.Similar behavior was exploited for a free in the bulk scalar in the series of publications [12][13][14].This feature is generic, and gives rises to sum rules which have recently been studied in AdS for scalars and stress-tensor insertions [15,16].

Resummation and OPE Constraint
We followed two separate pathways to obtain the bulk regularity constraints on the boundary OPE coefficients.Both led to the same results.The first pathway was to apply Maxwell's equations and conservation to the ⟨F µν J (a) T ℓ ⟩ three-point function and solve the resulting coupled system of differential equations for the form factors f a,I and g b,J .The second was to apply the BOE (3.7) to the boundary three-point function ⟨J (a) J (b) T ℓ ⟩ to reconstitute ⟨F µν J (a) T ℓ ⟩.The first approach requires some cleverness to reduce the system of differential equations to a smaller manageable subset, and still leaves a number of integration constants that can only be fixed by referring to the BOE.However, looking at just the first few terms of the BOE is sufficient to fix all the undetermined quantitites.The second approach leads to the answer directly but requires some cleverness to handle the infinite sums.We will describe only the second approach below.

General Approach for Resummation
To express the form factors using the boundary data, we need to act with the previously defined BOE differential operators (3.6) on the boundary three-point function.This highly technical endeavour we automated with Mathematica.Intermediate stages of the computation involve sums of thousands of hypergeometric functions along with tensorial structures.In this subsection, we sketch the procedure and give the resulting constraint obtained.
Our starting point is the following kinematical configuration lim leaving the first point unfixed so that it can be acted upon by the differential operators in the BOE.In this setting, the different building blocks previously defined take the simplified form with ẑ3 = z 3 − 2(θ • z 3 )θ entering in all these correlators, θ encodes the direction at infinity along which we send the third point.Since this vector satisfies ẑ2 3 = 0 (because θ 2 = 1), it is a bona-fide polarisation vector for the third point, and we will drop the hat on it from now on.We also defined xi = x i √ x•x .One can play the same game with the bulk structures to match the final results to the form factors.
This configuration makes it straightforward to reproduce most of the OPE constraints we already stated.The swapping property is implemented by sending (x, z 1 , z 2 ) → (−x, z 2 , z 1 ) .
Imposing conservation on the first insertion can be done by taking the divergence, and on the second one by performing the swap and then repeating the same operation.The new OPE constraints we want to derive however, require more work.
Our goal is now to act on this correlator (4.28) with the boundary OPE of (3.7).We break the problem apart into smaller pieces and initially determine how the k th power of the Laplacian acts on these building blocks.The numerators in our tensor structures take the form for some number m and constant vectors p and q.In these expressions and the following ones, x stands for the tangential components of the position, and so does x • x and x 2 , except when appearing as x ⊥ .Consider for example which can be computed using the identity The □ k operator has returned an expression of the same general form but with shifted exponent in the denominator.The remaining resummation is of hypergeometric type, and similar to other conformal block computations, see e.g.[13,14,20,30,[39][40][41][42].This gives There is a similar result from D i acting on (x • z 3 ) m /|x| 2α : Indeed, the result of acting on these rational expressions with D and D i will be linear combinations of hypergeometric function multiplied by power laws.Looking already at (4.32), there is clearly a potential divergence at v = 1, in this case both from the power law prefactor and the 2 F 1 itself.The extra constraints on the OPE coefficients γ I and γ I are needed to cure these potential divergences.
With the pieces (4.32), (4.33), (4.34), and (4.35) in hand, it becomes an algorithmic although tedious task to bring together these different elements, perform the resummation, and identify the different form factors.As they are sums of multiple hypergeometric functions with shifted arguments, their explicit expressions are not particularly enlightening.We performed this summation using Mathematica, and will now only quote the result for the constraint coming from bulk regularity.Note that in all of these different cases, there are multiple non-trivial checks, as there are more functions than variables to be constrained, and their Taylor series around the problematic v = 1 points admit multiple poles, which all must cancel after fixing 1 or 2 linear relations.Moreover, there are no approximations made at any steps of our procedure; this is a purely analytical computation.

Results for ℓ = 0
The resummation yields three independent form factors that depend on two parameters, γ 1 ab and γ 1 ab .A single linear relation is enough to make all even functions regular, while the constraint coming from the odd form factors can be obtained by using an Hodgetransformation to swap E → B, B → −E, giving the dual relation We stress that we do not ask Hodge-duality to be a symmetry of our theory; we merely use it as a convenient tool to extract the consequences of the odd sector from the ones of the even sector.This relation coupled with Bose symmetry allows us to express all the OPE coefficients involving a B field in terms of the OPE coefficients involving only the E field: The net effect is that the bootstrap of ⟨EEEE⟩ is left unchanged by the existence of the bulk.However the data of ⟨EEEB⟩, ⟨EEBB⟩, ⟨EBBB⟩, and ⟨BBBB⟩ are non-trivially related to ⟨EEEE⟩, which may generate further bounds.It is non-trivial that the data of ⟨EEEE⟩ can, provided these identifications, also satisfy crossing symmetry for these other correlators.As a quick example, we see that the bootstrap of ⟨BBBB⟩ is equivalent to the one of ⟨EEEE⟩, since the OPE coefficients appear squared.
For specific values of ∆, f (∆) may vanish or diverge, in which case one of the OPE coefficients in the relation must vanish.These special cases occur for k a non-negative integer and In the cases ∆ is even, the odd parity γ coefficients vanish.In the cases ∆ is odd, the even parity γ coefficients vanish.In either of these cases, we lose the glue that relates γ EE to γ BB (correspondingly γ EE to γ BB ); we cannot necessarily conclude that for the remaining nonzero OPE coefficients γ EE = −γ BB or γ EE = − γ BB .

Results for ℓ = 1
There are seven form factors with potential divergences v = 1.A single linear relation is enough to get rid of the poles: Bose symmetry puts tight constraints on the OPE coefficients in the ℓ = 1 case, which in combination with the regularity constraint now impose As a result, the mixed coefficients γ EB and γ EB can be nonzero only in the special cases where f 1 (∆) vanishes or diverges, with k a non-negative integer.

Results for ℓ ≥ 2
This is the most general situation where all nine form factors enter the game.The constraint now takes the form of a linear relation between vectors, i.e. a matrix relation, and we will now use matrix notation.The tangent arises through a special ratio of Γfunctions.This matrix has no null vector, but it may wholly be zero or infinite provided In these cases, one of the vectors of OPE coefficients must vanish.
To investigate the relation between the different OPE coefficients of E and B, we can use a chain of relations previously highlighted in the scalar case, noting that the S-duality argument still holds.There are in principle four different vectors when a B field is present, γ BB , γ BB , γ BE , γ BE .They are related using (in matrix notation) It is convenient to differentiate even and odd spin.For odd spin, γ EE = 0, and the discussion is similar to what happened for ℓ = 1.The γ EB and γ BB coefficients are then also sent to zero as well, provided 0 < f ℓ (∆) < ∞.For even spin, provided A ℓ is invertible, these relations hold, reducing the system to the data of the electric channel.We can note that the matrix M satisfies M 2 = 1, and one can pick a basis of OPE coefficients which diagonalise it.
In conclusion, we have derived a set of relations between the OPE coefficients of the electric and magnetic fields.We should keep in mind that the E-B basis of currents is not orthonormal, giving rise to a non-trivial mixing problem which inputs the values of (χ e , χ o ) into these relations.For each point on the conformal manifold of our theory, these constraints form a non-trivial set of equations relating the OPE coefficients of one current to the other.

The Bootstrap Problem
While in future work we intend to impose directly the regularity constraints on the threepoint functions and bootstrap the full (E, B) system, in this work we will content ourselves by looking at a bootstrap of a single current, which could be E or B, and the much weaker constraints that come from assuming the presence of a displacement operator and a gap in the spectrum of spin two operators.We furthermore restrict to theories with a parity symmetry to further constrain the theory.
In general, in the OPE of two currents, we expect there to be a tower of symmetric traceless tensor operators T ℓ .The four-point function can be decomposed into a sum over the conformal blocks associated with these operators and their OPE coefficients.Schematically, we find crossing equations of the form .
In this notation for the OPE coefficients γ ℓ,∆ and conformal blocks V ℓ,∆ , the presence of a tilde indicates odd parity and its absence even parity.For identical currents, no operator of spin one is exchanged, but all other non-negative spins are allowed.Of course these are tensorial objects, and in fact in appendix B we detail how this single schematic crossing equation is actually 5 distinct crossing equations for the different tensor components.

Finding Gaps
As a sanity check, we first reproduce an exclusion plot from the original four current bootstrap paper [43].The setup of ref. [43] has a couple of major technical differences.The first is that the conformal blocks of ref. [43] were generated expressly for that work while ours were generated using Blocks 3d [44].A second is that the authors expressed the crossing equations in terms of the (u, v) cross ratios while our system works with z and z where z z = u and ( In this exclusion plot, the authors [43] ask SDPB the question, does a CFT exist with no even parity scalar below ∆ + and no odd parity scalar below ∆ − .Our (very similar) version of their plot is figure 2. The different curves correspond to increasing the number of linear functionals that SDPB uses in examining the crossing constraints.The number of linear functionals scales roughly as the square of the parameter Λ, which in turn is the maximal derivative order used in constructing these functionals.As the number of functionals increase, the bounds become tighter, and the allowed CFTs are below the curves.The exclusion plot for the four current bootstrap with gaps for the even and odd parity scalars on the axes.The allowed region lies below the curves.Our plot is very similar to fig. 5 of [43] and gives us confidence that the numerics is working.The curves correspond to exclusion runs with Λ = 11, 15, 17, and 25.The points correspond to known theories: the free scalar, the generalized free vector field, and the free fermion.The vertical line at ∆ + = 1.5117 corresponds to the lightest even parity operator in the O(2) model.The blue wedge at the bottom is scraped from the standard O(2) CFT bootstrap of four scalars.
As expected, the free scalar, free fermion, and GFVF are all allowed.(Some details about these trivial theories are reviewed in appendix C.) The one obvious corner or cusp in the plot seems to be associated with the GFVF.It is possible that if Λ were increased further, there may also be corners associated with the free scalar and the O(2) model.For the O(2) model, the smallest odd parity scalar has not yet been determined; the vertical line corresponds to the placement of the lowest even parity scalar.
In the context of boundary CFT, the vertical section of the curves near ∆ + = 4 is noteable.All boundary theories with a four dimensional bulk should have a displacement operator of scaling dimension 4. It seems plausible from this plot that indeed quite generally all CFTs with a global U(1) symmetry must have an even parity scalar gap of at most four.(The foot extending to larger values of ∆ + can be ruled out by bootstrap of four scalars in the O(2) model [45].) As the boundary theory should not have its own stress tensor, which is a spin two operator saturating the unitarity bound ∆ 2 = 3, we next ask what happens to this exclusion plot fig. 2 if we introduce a spin 2 gap.The results, which are new, are shown in fig. 3.As expected, the free scalar, which has a stress tensor, is excluded once the gap becomes large enough.(Computing the exclusion plot with a smaller value of Λ, the free scalar is excluded at a larger value of the gap.)The GFVF on the other hand does not have a stress tensor but does have a spin two operator of dimension 4, namely V µ V ν − 1 3 δ µν V 2 .The GFVF survives even when the spin 2 gap is increased to one.The free fermion appears to be allowed according to this plot but presumably only because it is protected inside the convex hull of the region fixed by the presence of the GFVF.From top to bottom, the values are 0, 1/2, and 1.The plot is consistent with the hypothesis that at large enough spin gap, the only theory left is the GFVF and theories trivially related.
If we continue to increase the spin 2 gap, eventually all CFTs are ruled out, even if no gap assumptions are made for the scalar sector, by which we mean the scalar dimensions can saturate the unitarity bound of one half.What this critical value of the spin 2 gap is, depends on how we set the number of linear functionals or equivalently Λ.In fig.4, we show the maximal allowed value for the spin 2 gap as a function of the inverse number of linear functionals.In this way, we can try to fit the points to a curve and extrapolate the result to the infinite Λ limit.The result is that there are no CFTs once the spin 2 gap exceeds 1.05, which is suspiciously close to one.We are tempted to speculate that in fact the true upper bound is one, that the unique CFT with a spin 2 gap of one is the GFVF or something constructed from it in a trivial way, for example by taking products of the GFVF and orbifolding, and that there is no CFT with a larger spin 2 gap.The maximal allowed spin 2 gap vs. the inverse number of linear functionals used in SDPB.Note that the number of linear functionals scales roughly as the derivative order squared, Λ 2 .We also have provided a fit to the data of the form ∆ 2 − 3 = an α + b where the best fit parameters for these seven data points were a = 7530, b = 1.05 and α = −2.16,suggesting the convergence is quadratic in the number of functionals and that there are no CFTs for a spin 2 gap a little bit bigger than one.

OPE Maximisation and the Displacement
In addition to checking whether CFTs with certain gaps in the operator spectrum are allowed, SDPB is also useful for computing bounds on OPE coefficients.In this next plot, we examine the OPE coefficient of two currents with an even parity, dimension four operator, which could be the displacement operator in our boundary CFT.An upper bound for this coefficient is plotted in fig. 5. We normalised with respect to the result obtained when the spin two gap becomes equal to one, which should correspond to the value in the GFVF.In this plot, we have assumed that the scalar operators satisfy the unitarity bound ∆ ± ≥ 1 2 .Once the spin 2 gap reaches one, our hypothesis is that the only theory left is the GFVF.Note that fig. 5 was computed by a linear extrapolation of the results for runs at three lower values of Λ.The extrapolation is pictured in fig.6.
To provide some evidence that the theory with ∆ 2 − 3 ≥ 1 that is saturating the OPE bounds is the GFVF, we take a another section through the OPE coefficient space.This time, we gradually increase the spin 2 gap from zero but assuming that there are no even parity scalars with ∆ + < 4 and no odd parity scalars with dimension ∆ − < 2. These gap assumptions are designed to allow the free fermion theory when the spin 2 gap is vanishing.The corresponding plot is shown as fig.7. The plot is again normalized such that the OPE coefficient bound at ∆ 2 = 4 is equal to one.The ratio between the value at ∆ 2 = 3 and the value at ∆ 2 = 4 is very close to 3  2 , which is the ratio between the free fermion and the GFVF calculated in appendix C. (For an SDPB run with Λ = 25, the precise value obtained was 1.50067.)It should also be emphasized that although the bound computed JJD when the gap is zero is approximately 1.728.The slope of the line at that point is −1.873.The slope of the line when the gap is one is −0.139.Dashed lines with these slopes have been plotted as a guide to the eye.The curve was computed by using linear extrapolation for runs with Λ = 11, 17 and 25.The bounds from these three runs sit very nearly on a line when plotted against the inverse number of linear functionals input to SDPB (117, 243, and 473 respectively).Our last plot is the one most constrained by the notion that there is a bulk Maxwell field coupled to a CFT with a global U(1) symmetry.Using the Ward identity of [9], we relate the OPE coefficient γ JJD to the two-point functions of the currents and the displacement operator: where J can be either E or B. This result follows directly from (D.16), and the derivation is reviewed in appendix D.
The upper bound on γ JJD then translates into bounds on τ EE , τ BB , and C D , pictured in fig.8.As the spin 2 gap is increased, the blue region shrinks down to the red region.(The results of perturbation theory reviewed in section 2 are shown as the dashed lines, which lie comfortably inside the allowed region.)The upper bound of 1.728 in fig. 5 translates into an upper bound on the displacement two-point function, namely that C D ≤ 1.728 C f ree D .This result is doubly interesting as C D is well known to control boundary contributions to the anomalous trace of the stress tensor [7,18], as we review in the discussion.

Discussion
To conclude, we discuss three facets of this work.First, we discuss the two-point function sum rule and a possible interpretation that may shed some light on the puzzle surrounding the optical conductivity of graphene.Second, we further develop the bound on the displacement two-point function and its relation to conformal anomalies.Finally, we describe the next steps.
There is a curious relation between boundary QED and graphene.Effective field theory models of graphene typically involve massless Dirac fermions with a linear dispersion relation E = vp, where the velocity v ≈ c/300 is small compared to that of light, small enough that magnetic interactions can be effectively ignored.The speed v however develops a negative beta function, and thus becomes larger at low energies.Researchers have argued [5] that boundary QED can be thought of as the ultimate IR fixed point of graphene, where v → c and magnetic interactions are restored.
The optical conductivity puzzle starts with a calculation in the free field limit of graphene.Using the current J µ = e ψγ µ ψ with two-component fermions, the conductivity can be calculated from Wick's Theorem to be For graphene, the expectation is that N f = 4, and remarkably this result matches experiment at the percent level for electric field frequencies in the optical range [46,47] .The result follows from a Kubo formula where k = (ω, k) and G µν (k) is the Fourier transform of the current two-point function This conductivity is that of the zero temperature limit, which in practice means that the ratio ω/T should be taken large.In other words, one is measuring the current response of graphene to an oscillating electric field, for example that in a beam of light.
The puzzle is that this free field calculation is suspect; the effective fine structure constant in graphene, because of the relatively low value of v, is order one, suggesting Coulomb interactions should be important.Researchers have tried to address these concerns perturbatively by calculating the first order correction to the conductivity.In an expansion in α g , the first order correction turns out to be nummerically small and positive [48], A similar calculation has also been performed in boundary QED, where it is also small and positive [28,29] Of course these are just perturbative results, and one would like some additional nonperturbative input.A result of [9] that we rederived in section 3 can be rephrased as a sum rule for the conductivities of the two currents, one of which is the charge current just discussed, 9  σ As both conductivities need to be positive by reflection positivity, we learn that σ ≤ 1.A further consequence is that some of the higher order perturbative corrections to σ must be 9 Earlier, we used a different normalization for the current two-point functions.To make the translation, there is a factor of π 2 /2 that comes from the Fourier transform (see footnote 7) and an additional factor of g 2 that is removed in translating the electric current E to a charge current J which, unlike J used previously, includes a factor of the gauge coupling in its definition.Note this sum rule was also found in a supersymmetric context from localization [34].
negative.If the theory is self-dual under swapping the two currents, it follows also that σ = σ ′ = 1 2 at the self-dual point, generalizing a result of [36,37].Regarding the trace anomaly, famously four dimensional CFTs have two scheme independent contributions to the trace of the stress tensor proportional to the Euler density and the Weyl curvature squared, with coefficients called respectively a and c.Less well known is that in the presence of a boundary, there are two additional terms.(That these are the only Wess-Zumino consistent terms was shown in [49].)Schematically, the full trace anomaly takes the form where E is the Euler density and E bry its boundary contribution, W µνλρ the Weyl curvature, K µν the extrinsic curvature of the boundary and Kµν its traceless part.It is known from [7,18] that b 2 is proportional to the displacement operator two-point function coefficient c D and b 1 to the displacement operator three-point function coefficient, essentially because for a slightly deformed planar boundary, the extrinsic curvature is proportional to the Hessian of the boundary's position.For many years, people have speculated on possible constraints on these coefficients.It is immediately clear from reflection positivity that b 2 ≥ 0 [18].Fursaev [50] noted a relationship between b 2 and the bulk coefficient c for free theories, b 2 = 8c, while one of us [18] demonstrated that b 2 changes as a function of the marginal coupling g in boundary QED while c (which is determined by its value for free Maxwell theory) does not.Here at last we have a strict upper bound on b 2 through our result that C D ≤ 1.728 C f ree D , at least for boundary CFTs with a global U(1) that are coupled to free Maxwell theory.We find that b 2 ≤ 13.8 c .
It will be interesting to see if this bound applies more generally and also to see if it can be tightened further.
The coefficient b 1 depends on a displacement three-point function and is not direclty accessible through this four current bootstrap.One analytic result that is known is a sum rule that relates b 1 to b 2 and also all the spin two operators that appear in the OPE of two displacement operators [21].It would be interesting to try to constrain ⟨DDD⟩ by bootstrapping a system of four scalar operators of dimension four, assuming a spin 2 gap above the unitarity bound in order to restrict to theories without a boundary stress tensor.Such a project should be interesting and probably much simpler than the higher spin bootstrap performed here and the more complicated bootstrap project we have in mind next.
As part two of this project, we intend to look at the full system of four current correlation functions, involving both the electric E and magnetic B currents, imposing the constraints on the OPE coefficients we derived in section 4. We hope to be able to follow the changes to the free fermion and free scalar once they are coupled to the bulk Maxwell field, using the spin 2 gap as a proxy for the coupling strength.For small values of the coupling, the results are known in perturbation theory, which we can then hopefully match to our bootstrap results.With confidence in the bootstrap established, we can then use the bootstrap in a nonperturbative regime to compute anomalous dimensions and OPE coefficients.That, anyway, is the plan.and so they are genuine free fields.The proof follows by noting that all the discontinuities of the correlation functions are encoded in the two-point function.Inductively, a generic (n + 2)-point function can be reduced to its discontinuity on branch-cuts, reducing it to a product of a two-point function and an n-point correlator, thus ultimately reducing it to a sum over products of two-point functions.
We have established that the ring is solely made up of free theories, which are precisely those we found from the Lagrangian picture.

B Details About the Numerics
Our numerical bootstrap pipeline has three major pieces.The first and last piece consist of pre-existing software: Blocks 3d [44] and SDPB [51,52], while the third was purpose built for this project, adapting the output of Blocks 3d for input into SDPB.
Blocks 3d [44] approximates conformal blocks for three dimensional conformal field theories to arbitrary precision.The external and exchanged operators can be in arbitrary representations of SO(3) and carry arbitrary conformal dimension.Often in numerical bootstrap applications, the conformal dimension of the external operators needs to be adjusted for each run of SDPB.Our case was simpler in this respect as the external currents are protected operators with conformal dimension equal to two.More than making up for this simplification however were two additional considerations: the tensor structures that accompany spinning operators and the conservation condition that the currents enjoy.
A limitation of Blocks 3d is that the conformal blocks do not satisfy conservation conditions for the external operators.A major part of the second piece of pipeline that we built was the construction of the conserved blocks from linear combinations of the output of Blocks 3d.As the algebra required was nontrivial, we implemented this part of the pipeline in Mathematica, following procedures detailed in [38].
The other parts of the second piece of pipeline that we built were a collection of C programs and bash scripts to feed the conserved conformal blocks into SDPB.While we did not need to adjust the dimension of the external operators for each run of SDPB, it was useful to be able adjust the gap assumptions quickly and frequently and also to alter the norm and objective.Thus we have some C routines that introduce these gaps into the conserved conformal blocks and compute the norm and objective in specially tailored ways.
The blocks have to be assembled into crossing equations that are the input for SDPB.We have C routines that pack different parts of the SDPB input file: some that are run once for input that does not change from run to run, others that pack the scalar or spin two sector whose gap assumptions change frequently, that need to be run prior to each instance of SDPB.
We have a C routine that removes x/x pieces from the conserved conformal blocks.It turns out that our conserved blocks, which are approximated as ratios of polynomials in x = ∆ − ∆ 0 , where ∆ 0 is the unitarity bound for the exchanged operator, often have factors of the form x/x. In particular our even parity, even spin exchanged blocks, J ≥ 2 all have this x/x factor, as does the odd parity J = 2 block.It was suggested that for improved behavior of SDPB, these factors should be removed. 10inally, we have many bash scripts, some for setting up the static portion of the input file for SDPB, others for running a single instance of SDPB, others to implement binary search routines to search for where a particular behavior of SDPB changes.
In what follows we sketch the technical details involved in constructing the conserved conformal blocks and constructing the crossing equations.Our procedure for constructing the conserved blocks follows closely ref. [38] as does our construction of the crossing equations.(For a more detailed account of how to set up the crossing equations in a closely related case, the stress tensor bootstrap, the reader may wish to consult [53] from which both ref. [38] and we draw heavily.) B.1 Three-Point Functions: Vector to q-Basis To construct the conserved blocks, we begin with the three-point functions.Our initial "vector" basis of three-point function tensor structures was described already in section 4.1.However, to interface with Blocks 3d, it is necessary to change to an SO(3) basis.
Consider a conformal block where external operators with SO(3) spins j 1 and j 2 merge to form an exchanged operator with spin j 3 .Blocks 3d uses an SO(3) basis that involves states with quantum numbers |j 1 , j 2 , j 3 , j 12 , j 123 , m⟩.The numbers j 12 and j 123 are the principle quantum numbers corresponding to the angular momenta J 1 +J 2 and J 1 +J 2 +J 3 .Constructing the conserved conformal blocks from the output of Blocks 3d involves several steps.The first we have already accomplished in section 4.1, figuring out the restrictions on the allowed tensor structures from conservation.The next step is to convert this "vector" basis to the SO(3) basis |j 1 , m 1 ; j 2 , m 2 ; j 3 , m 3 ⟩.Then we use Clebsch-Gordan coefficients twice to go to the final |j 1 , j 2 , j 3 , j 12 , j 123 , m⟩ basis.This procedure tells us how to change the basis of the three-point functions.Of course, we really want to know how to reassemble the conformal blocks, but knowing the rules for the three-point function is enough to deduce what happens to the full conformal block; we perform the change of basis twice, one for each pair of external operators in the conformal block.
In more detail, the first step is to relate our "vector" basis of embedding space structures to the |j 1 , m 1 ; j 2 , m 2 ; j 3 , m 3 ⟩ states.Note that by angular momentum conservation, we assume i m i = 0. We fix the insertions in the three-point function to x 1 = (0, 0, 0), x 2 = (0, 0, 1), and x 3 = (0, 0, ∞).A prescription was outlined in appendix F of [44] (available in the published version).One makes the identifications , P 2 = (0, 0, 1, 0, 1) , P 3 = 0, 0, 0, − and then reads off the SO(3) states by expanding the embedding space structure in terms of ξ a and η a and using the definition (2.28) of [44] |j 1 , m 1 ; j 2 , m 2 ; j 3 , m 3 ⟩ = (−1) j 1 −j 3 +m 2 Parity in this basis corresponds to the action of the σ 1 Pauli matrix which swaps ξ a and η a , and hence sends m a → −m a .Thus the even and odd parity combinations are symmetric and antisymmetric combinations of pairs of states related by this swap.Note [000] is its own conjugate, and so we get 5 parity even states and 4 parity odd states in general.For l = 1, we keep only the first 7 of these states, yielding 4 parity even and 3 parity odd.For l = 0, we keep only the first 3, giving 2 parity even and 1 parity odd state.These results match the counting from our "vector" basis construction.

B.2 From the Conformal Blocks to the Crossing Equations
In addition to the (j 12 , j 120 , j 34 , j 340 ) quantum numbers, the conformal blocks are labeled by four azimuthal angular momenta [q 1 q 2 q 3 q 4 ], with no constraint on the sum and q a ∈ {−j a , −j a + 1, . . ., j a }.In what follows, we will focus primarily on these four numbers [q 1 q 2 q 3 q 4 ].The map from embedding and/or real space to these q-states is similar to what we saw above for the three-point function.We will fix x 1 = (0, 0, 0), x 2 = (z, z, 0), x 3 = (0, 1, 0) and x 4 at infinity, where we are using complex coordinates to parametrize the first two dimensions.More details can be found in Appendix F of [44].The basic idea is to decompose the four-point function into a sum over conformal blocks along with their tensor structures.Our four-point function of four currents can be decomposed into conformal blocks ⟨JJJJ⟩ ∼ [q 1 q 2 q 3 q 4 ]g [q 1 q 2 q 3 q 4 ] (z, z) . (B.9) Given that for the current four-point function j a = 1, there are then 81 states.We take parity to be given by the action of the σ 3 Pauli matrix (instead of σ 1 as we did in the three-point function case) which sends ξ a → ξ a and η a → −η a .The parity is thus the parity of a (j a − q a ).There are 41 parity even and 40 parity odd states.
For all identical currents, we can group the q-states together under orbits of the Z 2 ×Z 2 permutation group that leaves the cross ratios z and z invariant.There are 17 parity even orbits and 10 parity odd orbits To be consistent with Bose symmetry, the conformal blocks assemble into permutation invariant combinations ⟨q 1 q 2 q 3 q 4 ⟩ = [q 1 q 2 q 3 q 4 ] + n(1 − z) −q 1 +q 2 +q 3 −q 4 [q 2 q 1 q 4 q 3 ] + n(z) q 1 +q 2 −q 3 −q 4 [q 4 q 3 q 2 q 1 ] +n(z) q 1 +q 2 −q 3 −q 4 n(1 − z) −q 1 +q 2 +q 3 −q 4 [q 3 q 4 q 1 q 2 ] (B.10) where n(z) = z/z.The four-point function can be expressed then as with the same conformal block coefficients as before.This discussion can be repeated for ⟨JJJ ′ J ′ ⟩ for which there is only a Z 2 permutation symmetry, 25 even parity orbits and 20 odd parity orbits.It remains to enforce current conservation.Current conservation can be expressed in terms of the Todorov differential operator D w and the generator of rotations L 23 in the 23-plane: These expressions make use of the polarization notation ω = w z = w 1 + iw 2 , ω = w z = w 1 − iw 2 , ω 0 = w 3 .The q-variables are the exponents of the complex variable ω, where [1] = ω, [0] = ω 0 , and [−1] = ω.
In the case of four identical currents and a parity preserving theory, the current conservation operator can be thought of as a map from a 17 dimensional subset of the full 81 dimensional q-state space to a smaller q-state space involving a scalar and three identical currents.This 27 dimensional image space has 14 even parity states and 13 odd parity ones.
We followed a brute force approach and solved the conservation equation perturbatively in a series expansion near z = z = 1 2 .The results depend sensitively on whether the currents are identical or not and whether or not the theory preserves parity.
In the simplest case of four identical currents and where parity is preserved, we can find a solution if we provide initial data for five conformal blocks everywhere and two more conformal blocks "along a line".More specifically, we must specify the value of five conformal blocks and all their derivatives in ∂ z and ∂ z at z = z = 1 2 .We must further specify for a sixth and seventh conformal block their values at z = z = 1  2 along with a subset of their derivatives, which we may take to be ∂ n z ∂ n−1 z and ∂ n z ∂ n z for all positive n.The point of this exercise is that it shows most of the 17 conformal blocks in this case are redundant and can be deduced from current conservation.Thus we are well advised to construct crossing equations only from the remaining set.
This conservation constraint is sufficient when all the operators are identical.However, to apply conservation to the ⟨JJJ ′ J ′ ⟩ four-point function, we can use the above expression on the second operator, but to ensure J ′ is conserved as well, the strategy is first to perform the swap 1 ↔ 4 and 2 ↔ 3 and then act with the same conservation equation, which gives us a second set of constraints.
One last wrinkle is that the output of Blocks 3d is actually linear combinations of the conformal blocks so far discussed: The basic crossing symmetry constraint is that g [q 1 q 2 q 3 q 4 ] (z, z) = g [q 3 q 2 q 1 q 4 ] (1 − z, 1 − z) .
Given this relation, we find the following cases when all the currents are identical.We will leave an exposition of the more general case of two currents for part two of this project.

⟨JJJJ⟩ Even Sector
Conservation leaves 5 conformal blocks which are unconstrained, and two conformal blocks for which data needs to be provided just along a line.One can consistently take ⟨1111⟩, ⟨0110⟩, ⟨0101⟩, ⟨0011⟩ and ⟨0000⟩ to be the five bulk blocks.⟨−1−111⟩ and ⟨−111−1⟩ are a convenient choice for the line blocks, which furthermore exchange under 1 ↔ 3. The crossing equations are Note that keeping purely holomorphic or antiholomorphic derivatives in the line constraint will fail to provide enough data to satisfy the conservation condition.Curiously, while we find a line constraint, we were not able to implement successfully a line constraint in SDPB in all cases.We found that for Λ > 18, we were unable to increase the maximal n in the line constraint beyond 18.There could be a reason we have not understood that makes the higher order line equations degenerate.Indeed, our crossing equations are different from [38], where they find a point constraint instead of a line constraint.In the numerics, we found we were able to get tighter bounds by working with a line constraint rather than a point constraint although the difference between the two decreases with increasing Λ, presumably because the number of linear functionals from the five bulk constraints grows quadratically while the number of line constraint functionals grows only linearly.

⟨JJJJ⟩ Odd Sector
If we break parity, then we need to supply only two more conformal blocks.⟨0111⟩ and ⟨0001⟩ are a convenient choice.Despite some effort, we were never able to see dual jumps in the progress of SDPB, only primal jumps.As a result, for the exclusion plots, an allowed point is one for which we observe a primal jump but a disallowed point corresponds to creeping behavior where both the dual and primal errors gradually creep down.To distinguish these behaviors, we set the dualErrorThreshold higher than the primalErrorThreshold but still well below the values of primal error where the jumps were observed.

C Low-Lying Operators in 3d Free Theories
We describe the first few scalar primary operators and their correlation functions for the free three dimensional conformal field theories that make an appearance in this work: the generalized free vector field, the massless free scalar, and the free fermion.

C.1 Generalized Free Vector Field
The generalized free vector field V µ (x) obeys the conservation condition ∂ µ V µ = 0 and as a result has the two-point function ⟨V µ (x)V ν (0)⟩ = I µν (x)/|x| 4 .Correlation functions all follow from Wick's Theorem.Writing a scalar operator as O ∆,p where ∆ is its conformal dimension and p the parity, we are interested in the following three low-lying primary operators: The operators O 4,+ and O 5,− help us to identify the location of the GFVF near a cusp in the exclusion plot (fig.2).The dimension seven O 7,− is the lowest odd parity scalar for the free scalar theory but is also present for the GFVF.We find the following table of two and three-point function coefficients.The λ V V O have been defined using the three-point structures described in the text.
O 5,− 12 2 The normalized OPE coefficient is defined such that The results for O 4,+ are consistent with the results discussed for the free Maxwell field, where O 4,+ was identified with the displacement operator D, and V µ with the boundary current B µ .With the normalizations used there, we found that C D = 6 π 4 , the current twopoint function was normalized as τ BB = 2g 2 π 2 , and the three-point function is γ BBD = 2g 2 π 4 .The result λ 2 V V O = 1 6 for O 4,+ was crucial for ensuring the proper normalization of the OPE coefficient bounds in the text.The values for the odd parity operators we intend to make use of future work.

C.2 Free Scalar
Using the two-point function ⟨ϕ(x)ϕ(0)⟩ = 1 |x| , we can compute the OPE coefficients of some low lying primary operators in the spectrum.
The lightest odd parity scalar can be deduced from the result (C.1) for O 7,− above along with the equation of motion □ϕ = 0: The dimension five operator ϵJ∂J that we found for a GFVF vanishes for a single complex scalar field.Writing such an operator for free scalar fields requires at least two commuting currents, J = 1 √ 2 (J 1 + J 2 ).We see that the exclusion plot (fig.2) bends upward at small ∆ + to allow for an O 7,− operator.
We find the following table of two and three-point function coefficients: For the current operator J µ = ψγ µ ψ, the two-point function is the same as for the free scalar, ⟨J µ (x)J ν (0)⟩ = 2I µν (x)/|x| 4 .The mass breaks parity in three dimensions, and correspondingly the operator O 2,− = ψψ is parity odd.There is an even parity dimension four operator O 4,+ = ( ψψ) 2 .By a Fierz identity, ( ψψ) 2 ∼ J µ J µ .The two operators O 4,+ and O 2,− lie safely inside the exclusion plot (fig.2).The value of λ 2 JJO for the O 4,+ operator was recovered as an upper bound on an OPE coefficient in fig. 7.
D Current-Current-Displacement Three-Point Function While three-point functions in conformal field theory are normally fixed up to constants, the current-current-displacement three-point function in our set-up is special.It can be expressed in terms of the two-point functions of the currents and the displacement operator.As the relation is important for us, we reproduce here the derivation of ref. [9].
Expanding ⟨F µν F λρ D⟩ using the BOE (3.7), we can express this bulk-bulk-boundary three-point function in terms of the following purely boundary correlation functions: We compare this boundary limit with the coincident limit x → y.The bulk OPE of F µν with itself is fixed by the free theory, and there are only three operators in this OPE that will contribute to the ⟨F F D⟩ three-point function.They are F 2 , F F , and the stress tensor T µν = 1 g 2 (F µρ F ν ρ − 1 4 δ ab F 2 ) which come with coefficients Note other higher spin operators are present in the OPE of F µν with itself but they have zero overlap with the displacement operator.It is precisely this point which makes the calculation useful; in the more general case, there would be an infinite sum over bulk operators over which we would have less fine control.By the constraints of conformal invariance, the following bulk-boundary two-point functions are all fixed up to constants:

Figure 1 :
Figure 1: Space of boundary conditions for the two-point function ⟨F µν (x)F ρλ (y)⟩.The outer region is called the perimeter, and has E ∥ B. The center-point s is mapped to itself under S-duality.The poles m and e correspond to purely magnetic and purely electric boundary conditions.The parity invariant meridian satisfies E ⊥ B.
a, b, c, d ∈ Z and ab − cd = 1 be an element of SL(2, Z) acting on the gauge coupling τ in the usual way τ → aτ + b cτ + d .

Figure 2 :
Figure 2:The exclusion plot for the four current bootstrap with gaps for the even and odd parity scalars on the axes.The allowed region lies below the curves.Our plot is very similar to fig.5of[43] and gives us confidence that the numerics is working.The curves correspond to exclusion runs with Λ = 11, 15, 17, and 25.The points correspond to known theories: the free scalar, the generalized free vector field, and the free fermion.The vertical line at ∆ + = 1.5117 corresponds to the lightest even parity operator in the O(2) model.The blue wedge at the bottom is scraped from the standard O(2) CFT bootstrap of four scalars.

Figure 3 :
Figure3: Three different runs with Λ = 17 and increasing values of the spin 2 gap.From top to bottom, the values are 0, 1/2, and 1.The plot is consistent with the hypothesis that at large enough spin gap, the only theory left is the GFVF and theories trivially related.

20 Δ 2 - 3 Figure 4 :
Figure4: The maximal allowed spin 2 gap vs. the inverse number of linear functionals used in SDPB.Note that the number of linear functionals scales roughly as the derivative order squared, Λ 2 .We also have provided a fit to the data of the form ∆ 2 − 3 = an α + b where the best fit parameters for these seven data points were a = 7530, b = 1.05 and α = −2.16,suggesting the convergence is quadratic in the number of functionals and that there are no CFTs for a spin 2 gap a little bit bigger than one.

2 Figure 5 :
Figure 5: A bound on γ 2JJD as a function of the spin 2 gap.Allowed theories must lie below the curve.The OPE coefficient has been normalized such that for a gap of one,γ 2 JJD = 1.The value of γ 2JJD when the gap is zero is approximately 1.728.The slope of the line at that point is −1.873.The slope of the line when the gap is one is −0.139.Dashed lines with these slopes have been plotted as a guide to the eye.The curve was computed by using linear extrapolation for runs with Λ = 11, 17 and 25.The bounds from these three runs sit very nearly on a line when plotted against the inverse number of linear functionals input to SDPB (117, 243, and 473 respectively).

2 Figure 6 : 2 Figure 7 :
Figure 6: The linear extrapolation used to compute the OPE bound in fig. 5.For each final point in fig.5, SDPB was run with Λ = 11, 17 and 25, corresponding to 117, 243, and 473 independent linear functionals.The x-axis of the plot is the inverse number 1/n of linear functionals.From top to bottom, the lines correspond to spin 2 gaps from zero to one.

Figure 8 :
Figure 8: The allowed region in the r = C D /C f ree D vs. χ e plane.The larger region assumes γ 2 BB /τ 2 BB C D and γ 2 EE /τ 2 EE C D are bounded above by 1.728 × 1 6 .For the red region, the upper bound is instead the GFVF result 1 6 .The black lines in the corners correspond to perturbation theory.Note the plot gives a rigorous upper bound on the b 2 boundary anomaly coefficient.

⟨:F 2 2 .
(x):D(z)⟩ = b F 2 ,D |x − z| 8 , (D.9) ⟨:F F (x):D(z)⟩ = b F F ,D |x − z| 8 , (D.10) ⟨T µν (x)D(z)⟩ = b T,D X µ X ν − 1 4 δ µν |x − z| 8 .(D.11) That the boundary limit of T nn = D tells us that b T,D = 4 3 C D (D.12) where C D is the coefficient of the displacement two-point function.We can then use the Ward identity that for a general bulk operator O d d−1 y⟨O(x)D(y)⟩ = ∂ n ⟨O(x)⟩ , (D.13) which yields from (3.24) the following relations b F 2 ,D = − 12κχ e a F 2 π 2 , b F F ,D = − 12κχ o a F F π τ EE + τ BB = κ, the definitions (3.20) and (3.22), and the result (2.10) for C f ree D .It would be interesting to look at a more general version of this calculation where D is replaced in the three-point function with the boundary value of a higher spin primary in the bulk OPE of two Maxwell fields.