3d Abelian Gauge Theories at the Boundary

A four-dimensional Abelian gauge field can be coupled to a 3d CFT with a $U(1)$ symmetry living on a boundary. This coupling gives rise to a continuous family of boundary conformal field theories (BCFT) parametrized by the gauge coupling $\tau$ in the upper-half plane and by the choice of the CFT in the decoupling limit $\tau \to \infty$. Upon performing an $SL(2,\mathbb{Z})$ transformation in the bulk and going to the decoupling limit in the new frame, one finds a different 3d CFT on the boundary, related to the original one by Witten's $SL(2, \mathbb{Z})$ action [1]. In particular the cusps on the real $\tau$ axis correspond to the 3d gauging of the original CFT. We study general properties of this BCFT. We show how to express bulk one and two-point functions, and the hemisphere free-energy, in terms of the two-point functions of the boundary electric and magnetic currents. We then consider the case in which the 3d CFT is one Dirac fermion. Thanks to 3d dualities this BCFT is mapped to itself by a bulk $S$ transformation, and it also admits a decoupling limit which gives the $O(2)$ model on the boundary. We compute scaling dimensions of boundary operators and the hemisphere free-energy up to two loops. Using an $S$-duality improved ansatz, we extrapolate the perturbative results and find good approximations to the observables of the $O(2)$ model. We also consider examples with other theories on the boundary, such as large-$N_f$ Dirac fermions --for which the extrapolation to strong coupling can be done exactly order-by-order in $1/N_f$-- and a free complex scalar.


Introduction
The objective of this paper is to study conformal invariant boundary conditions for free Abelian gauge theory in four-dimensions. A striking property of these BCFTs is that they are typically well-defined on some open patch in the space of the four-dimensional gauge coupling.
The simplest way to produce such boundary conditions is to couple the fourdimensional gauge fields to a three-dimensional CFT with a U(1) global symmetry. This is sometimes called a "modified Neumann" boundary condition [2]. Assuming that certain mild conditions are satisfied, one obtains a BCFT which is well-defined as long as the four-dimensional gauge coupling is sufficiently small [3][4][5][6][7][8][9]. The conformal data of the BCFT can be computed from the data of the original CFT by perturbation theory in the four-dimensional gauge coupling.
Conversely, there is a general expectation that any BCFT B defined at arbitrarily small 4d gauge coupling will be either a Dirichlet boundary condition or a modified Neumann boundary condition associated to some 3d CFT T ∞ [B] with a U(1) symmetry. Because of electric-magnetic duality, the same statement applies to any other "cusp" C in the space of the complexified gauge coupling, where some dual description of the four-dimensional gauge field becomes arbitrarily weakly coupled. If the BCFT B is defined around the cusp C, we can associate to it another 3d CFT T C [B], which is obtained from T ∞ [B] by applying the SL(2, Z) transformation [1] that maps the cusp at infinity to C. Therefore, the theories living at the other cusps can be thought of as 3d Abelian gauge theories obtained by gauging the U(1) global symmetry of T ∞ [B].
In the absence of phase transitions, a given BCFT B can be defined on the whole space of 4d gauge couplings and is thus associated to an infinite family T * [B] of 3d CFTs. The conformal data of the BCFT will admit a similar collection of perturbative expansions in the neighbourhood of each cusp.
In the first part of this paper we study general properties of this family of BCFT's. A universal feature is the presence in the spectrum of boundary operators of two conserved U(1) currents, the electric and the magnetic currents, that arise as a consequence of the electric and magnetic one-form symmetries in the bulk [10]. The endpoints of bulk line operators carry charge under this U(1) × U(1) symmetry, while all the local boundary operators are neutral. By matching the bulk and boundary OPE expansions of correlators of the bulk field strength, we show that several BCFT observables -including non-local ones such as the free-energy on a hemisphere background -can be obtained in terms of the coefficients c ij in the two-point correlators of these currents, and of the coefficient CD of the two-point function of the displacement operator. The latter relations hold for any -1 -JHEP05(2019)091 τ , provided B exists. We also show that the leading perturbative corrections to c ij and CD around a cusp are captured universally in terms of the two-point function of the U(1) current of the 3d CFT living at the cusp, in the decoupling limit.
In the second part of this paper we turn these abstract considerations into a very concrete computational strategy: if some T C is simple enough for perturbative computations to be feasible, we may study the properties of other T * theories by re-summing the perturbation theory. If we happen to know, or conjecture, that there are two cusps C and C such that T C and T C are both simple, we may be able to implement an enhanced re-summation which uses both piece of data to predict the properties of the other T * theories.
This approach gives a new approximation scheme, orthogonal to previously known perturbative approaches to 3d Abelian gauge theories such as the -expansion [11][12][13][14][15][16][17][18] or the large-N expansion (see e.g. [19][20][21][22][23][24][25] for recent results and the review [26]). We will apply this strategy to a very nice boundary condition for a U (1) gauge theory, which is conjecturally associated to a free Dirac fermion at two distinct cusps and to the O(2) model at two other cusps [27][28][29][30][31]. The fact that these theories appear at the cusps can be seen as a consequence of the recently discovered 3d dualities [27,32,33], and it entails the existence of a Z 2 action on τ that leaves B(τ,τ ) invariant. We will do a two-loop calculation at the free-fermion cusp and then extrapolate to the O(2) cusp, finding good agreement with the known data of the O(2) model.
We also consider other applications: taking the boundary degrees of freedom to be an even number 2N f of free Dirac fermions, setting the gauge coupling to g 2 = λ/N f and taking N f to infinity with λ fixed, we argue that the theory admits a 1/N f -expansion, which interpolates between the free theory at λ = 0 and large-N f QED 3 at λ = ∞. The exact λ dependence can be easily obtained order-by-order in the 1/N f expansion. Applying the general strategy to compute the hemisphere partition function to this case, and taking the limit λ → ∞, we obtain the 1/N f correction to the sphere partition function of large-N f QED 3 . Another example with a Z 2 duality acting on τ is conjecturally obtained in the case where the theory on the boundary is a free complex scalar, or equivalently the U(1) Gross-Neveu model [34,35]. We consider perturbation theory around the free-scalar cusp, and show the existence of a stable fixed point for the classically marginal sextic coupling on the boundary at large τ . We also discuss an example with two bulk gauge fields coupled to two distinct Dirac fermions on the boundary. We show how to obtain QED 3 with 2 fermionic flavors starting with this setup, using the extended electric-magnetic duality group Sp(4, Z) that acts on the two bulk gauge fields.

Structure of the paper
We start in section 2 by reviewing the non-interacting boundary conditions for a Maxwell field in four dimensions. We then define the family of interacting boundary conditions B(τ,τ ). We derive the general relations that we described above for the bulk two-and three-point functions of the field strength, and obtain the leading corrections in perturbation theory around the cusps in the τ plane. In section 3 we obtain similar results for a different observable, the hemisphere partition function of B(τ,τ ). In particular we show how to recover the S 3 partition function for the 3d CFTs in the decoupling limit. In sec--2 -JHEP05(2019)091 tion 4 we put this machinery at work in the example of the boundary condition defined by the O(2) model / a free Dirac fermion. Section 5 contains the other applications that we consider: large-N f fermions, a complex scalar, and two bulk gauge fields coupled to two Dirac fermions. We conclude in section 6 by discussing some future directions. Several appendices include the details of calculations, and some supplementary material, e.g. a calculation of the anomalous dimension of the boundary stress-tensor using multiplet recombination in appendix F, and an explanation of the technique that we used to evaluate the two-loop integrals in appendix G.
2 Boundary conditions for 4d Abelian gauge field

Generalities
Boundary Conformal Field Theories for a free d-dimensional bulk quantum field theory are interesting theoretical objects. On one hand, the correlation functions of bulk local operators are controlled by the free equations of motion. In particular, they are fully determined by their behaviour near the boundary, which is encoded in some very simple bulk-to-boundary OPE for the bulk free fields.
The free bulk-to-boundary OPE essentially identifies some special boundary local operators as the boundary values of the bulk free fields and their normal derivatives. The correlation functions of these boundary operators determine all correlation functions of bulk operators. These boundary correlation functions, though, can in principle be as complicated as those of any CFT in (d − 1) dimensions.
The case of four-dimensional free Abelian gauge theory (with compact gauge group) is particularly interesting because the bulk theory has an exactly marginal gauge coupling. 1 Furthermore, a BCFT defined for some value of the bulk gauge coupling can typically be deformed to a BCFT defined at a neighbouring value of the bulk gauge coupling by conformal perturbation theory in the gauge coupling. The leading order obstruction is the presence of marginal boundary operators in the bulk-to-boundary OPE of the bulk Lagrangian operators F 2 and F ∧ F , which can lead to a logarithmic divergence as the bulk perturbation approaches the boundary. Generically, no such operators will be present and the BCFT can be deformed.
In this section, we will discuss the properties of some standard BCFT's which can be defined in an arbitrarily weakly-coupled gauge theory, starting with free boundary conditions and then including interacting degrees of freedom at the boundary. On general grounds, we expect that any BCFT which can be defined at arbitrarily weak coupling will take this form. 1 If the gauge group is compact, say U(1), the gauge field has an intrinsic normalization and thus the coefficient in front of the bulk Lagrangian is canonically defined even if the bulk theory is free. Local interactions between the gauge fields and any other degrees of freedom localized in non-zero co-dimension obviously cannot renormalize the bulk gauge coupling. Furthermore, the strength of the interactions between the gauge fields and such other degrees of freedom is controlled by the bulk gauge coupling and by quantized gauge charges and thus cannot get renormalized. The only possible beta functions involve gauge-invariant boundary local operators. This fact is often obfuscated in perturbative treatments and then proven with the help of Ward identities, in a manner analogous to the non-renormalization of gauge charges in QED [3][4][5][6][7][8][9].

Free boundary conditions and SL(2, Z) action
Consider a U(1) gauge field A µ on R 3 × R + . We adopt Euclidean signature, and use coordinates x = ( x, y) where x 4 ≡ y ≥ 0 is the coordinate on R + , and x are the coordinates on R 3 . We denote the components of x as x µ , µ = 1, 2, 3, 4, and those of x as x a , a = 1, 2, 3. The field strength is F µν = ∂ µ A ν − ∂ ν A µ , its Hodge dual isF µν = 1 2 ρσ µν F ρσ and the selfdual/anti-self-dual components are F ± µν = 1 2 (F µν ±F µν ). They satisfy 1 2 ρσ µν F ± ρσ = ±F ± µν . In the absence of interactions with boundary modes, by varying the action we find the bulk equation of motion 1 g 2 ∂ µ F µν = 0 and the boundary term Our convention for the orientation is abcy = abc . In equations (2.2)-(2.4) we combined g and θ in the complex coupling τ = θ 2π + 2πi g 2 . From eq. (2.4) we see that the possible boundary conditions for the gauge field when no boundary modes are present are • Dirichlet: δA a | y=0 = 0, which is equivalent to (F − ya − F + ya )| y=0 = −F ya | y=0 = 0 ; (2.5) • Neumann: Equivalently, introducing we can write this condition as (F ya + iγF ya )| y=0 = 0, in particular for γ = 0 it simplifies to the standard Neumann condition F ya | y=0 = 0.
It is convenient to introduce the boundary currents 2πiĴ a = τ F − ya ( x, y = 0) −τ F + ya ( x, y = 0) , 2πiÎ a = F − ya ( x, y = 0) − F + ya ( x, y = 0) . (2.8) in terms of which the Dirichlet condition isÎ = 0, and the Neumann condition isĴ = 0. On R 4 this theory enjoys an SL(2, Z) duality group τ → τ = aτ + b cτ + d , a, b, c, d ∈ Z , ad − bc = 1 . The duality group acts on the fields as (2.10) When the boundary is introduced, the group SL(2, Z) also acts on the boundary conditions. From (2.10) we see that the action on the boundary currents iŝ J a → aĴ a + bÎ a , I a → cĴ a + dÎ a . (2.11) The Dirichlet and Neumann boundary conditions above are exchanged under the S transformation τ → − 1 τ , i.e. electric-magnetic duality. Indeed, the S transformation ex-changesĴ andÎ.
However, comparing eqs. (2.5)-(2.6) and eqs. (2.10)-(2.11) we see that the general SL(2, Z) transformation does not act within the set of boundary conditions that we described above. This is because we assumed that no degrees of freedom are present on the boundary, while the generic SL(2, Z) transformation requires the introduction of topological degrees of freedom on the boundary, namely 3d gauge-fields with Chern-Simons (CS) actions, coupled to the bulk gauge field through a topological U(1) current [1,36,37]. Note that even in the presence of these topological degrees of freedom the theory is still free, because the action is quadratic. Taking this into account, one finds that the most general free boundary condition for the U(1) gauge field is pĴ a + qÎ a = 0 , (2.12) where p, q ∈ Z. This set of boundary conditions is closed under the action (2.11) of SL(2, Z). We will refer to this more general free boundary condition as "(p, q) boundary condition". The (0, 1) and (1, 0) boundary conditions correspond to the Dirichlet and Neumann boundary conditions above, respectively. When we impose the (p, q) condition, the unconstrained components of the gauge fields give a current operator on the boundary p Ĵ a + q Î a (2.13) with pq − p q = 1, whose correlators are just computed by Wick contraction, i.e. the boundary theory is a mean-field theory for this current. We can always shift (p , q ) by a multiple of (p, q), and this gives rise to the same current thanks to the boundary condition. Since the above boundary conditions preserve conformal symmetry, we can regard this system as a free boundary conformal field theory, and rephrase the boundary conditions in terms of a certain bulk-to-boundary OPE of the field strength F µν . Using the equation of motion and the Bianchi identity one finds that the only primary boundary operators that can appear in the bulk-to-boundary OPE of F µν are conserved currents, see appendix B for a derivation. The free boundary conditions described above correspond to having only -5 -

JHEP05(2019)091
one conserved current in this OPE, that can be identified with p Ĵ a + q Î a . For instance, for the Dirichlet (0, 1) boundary condition (2.14) where the dots denote subleading descendant terms, and the square brackets denote antisymmetrization. The general (p, q) case can be obtained from the Dirichlet case by acting with an SL(2, Z) transformation (2.10)-(2.11).

Two-point function in the free theory
In this section we compute the two-point function F µν (x 1 )F ρσ (x 2 ) on R 3 × R + in the free theory. We use that the two-point function is a Green function, i.e. it satisfies the equations of motion 15) and the Bianchi identity on y ≥ 0, and it also satisfies the boundary conditions at y = 0. We are denoting To start with, the Green function on R 4 (i.e. without a boundary) is Starting from (2.17) and using the method of images we can easily write down the two-point function in the presence of the boundary. The calculation is showed in the appendix A.
In the case γ = 0 we find for Dirichlet (s = 1) and Neumann (s = −1) conditions. Here X i µ are the conformally covariant vectors [38] and ξ is the conformally invariant cross-ratio

JHEP05(2019)091
For the more general Neumann boundary condition with γ = 0 we find Even though not manifest, it can be verified that Bose symmetry is satisfied in this expression. From now on we will drop the subscript R 3 × R + .
It is also useful to rewrite this two point function in terms of the selfdual/antiselfdual components. The selfdual/antiselfdual projectors are (2.24) We introduce the following notation 26) and similarly for the structure H. The following identities hold Recalling the definition (2.7) of γ, we obtain The result above is the field-strength two-point function in the free theory with Neumann boundary conditions. As we argued in section 2.2, the result for the (p, q) boundary conditions (2.12) simply follows from an SL(2, Z) transformation (2.10)-(2.11). As an example, for Dirichlet boundary conditions one finds

Coupling to a CFT on the boundary
Consider now a 3d CFT living on the boundary at y = 0. We assume that the CFT has a U(1) global symmetry, with associated currentĴ CFT a . We take the Neumann boundary condition for the gauge field, which corresponds to a mean-field current operatorÎ a on the boundary. The two sectors can be coupled in a natural way, simply by gauging the U(1) symmetry via the y → 0 limit of the bulk gauge field. This amounts to adding the boundary coupling y=0 d 3 xĴ a CFT A a + seagulls , (2.37) and restricting the spectrum of local boundary operators to the U(1) invariant ones. Charged boundary operators can be made gauge-invariant by attaching to them bulk Wilson lines. Therefore, it still makes sense to consider them after the gauging, but as endpoints of line operators rather than as local boundary operators. The boundary coupling modifies the boundary condition of the gauge field to the "modified Neumann" conditionĴ a ≡Ĵ CFT a . (2.38) Hence as a consequence of the interactions bothÎ a andĴ a are nontrivial operators. As we explained above τ is an exactly marginal coupling, but we should worry about quantum effects breaking the boundary conformal symmetry by generating beta functions for boundary interactions. If the original 3d CFT has no marginal operators, these boundary beta functions start at linear order in the coupling and can be cancelled order-by-order in perturbation theory by turning on extra boundary interactions of order τ −1 . 2 Barring other non-perturbative phenomena such as the emergence of a condensate, we expect a BCFT to exist for sufficiently large τ , with conformal data perturbatively close to that of the original CFT. We denote this BCFT with B(τ,τ ).
If the original 3d CFT has marginal operators the situation is more subtle: turning on boundary couplingsλ will produce a beta function of orderλ 2 for the marginal operators. This may or not have the correct sign to cancel the τ −1 contributions. If it does not, we do not expect any unitary BCFT to exist, though one may be able to produce some non-unitary "complex" BCFT with complex couplings.
Conversely, suppose that we are given a BCFT B(τ,τ ) defined continuously for arbitrarily weak gauge coupling. If B(τ,τ ) is an interacting boundary condition, we expect that if we take the gauge coupling to 0 the properties of B(τ,τ ) will approach those of a 3d CFT with a U(1) global symmetry.
As we will discuss later in this section, the bulk correlation functions are determined by the boundary correlation functions of the two conserved boundary currentÎ a andĴ a defined in eq. (2.8). Due to the boundary condition (2.38), at weak coupling,Ĵ a is inherited from the boundary degrees of freedom and the corresponding charge is carried by the endpoints of bulk Wilson lines ending at the boundary. On the other hand,Î a is analogue to the 2 E.g. if the theory on the boundary is a free scalar field, loop corrections can generate the operator φ 2 on the boundary with coefficient ∼ τ −1 Λ 2 U V , where ΛUV is the cutoff, but the only implication of this term is that the tuning of m 2 needs to be adjusted at order τ −1 .

JHEP05(2019)091
"topological" charge in three-dimensional U(1) gauge theories and the corresponding charge is carried by the endpoints of bulk 't Hooft lines ending at the boundary.
When the coupling is turned off, the conformal dimension of endpoints of 't Hooft lines blows up and the Î aÎa correlation functions go to zero. TheÎ a current decouples from the BCFT correlation functions as they collapse to the correlation functions of the underlying 3d CFT T 0,1 [B] (this is the CFT that we denoted with T ∞ [B] in the introduction).

Boundary propagator of the photon
In order to compute corrections to boundary correlators and beta functions of boundary couplings in perturbation theory at large τ , we need the propagator of the gauge field between two points on the boundary. Since we are perturbing around the decoupling limit, this can be readily obtained from the knowledge of the two-point function in the free theory (2.23). Recall from the discussion around eq. (2.14) that in the free theory F µν has a non-singular bulk-to-boundary OPE. So the boundary two-point function of the operator F ab is obtained by specifying the indices to be parallel in eq. (2.23), and then taking the limit in which both insertion points approach the boundary. When taking this limit, we need to pay attention to possible contact terms that can arise due the following nascent delta-functions and its derivatives. Even though usually we only compute correlators up to contact terms, these kind of contact terms in the two-point functions of 3d currents do actually contain physical information [39]. In this context, they encode the θ-dependence of the boundary two-point function of F ab . Relatedly, they are also needed to obtain the correct boundary propagator of the photon.
To obtain the (ab, cd) components of the two-point function (2.23) we need the components (ab, cd) and (ya , cd) of the structures G and H. The structure G gives Here G 3d ab,cd denotes the same structure as in eq. (2.17) with the replacement of I µν by the 3d analogue On the other hand the only non-zero component of the structure H in the limit y 1,2 → 0 is H ya,yb , hence the H structure completely drops in the calculation of the propagator. The result is

JHEP05(2019)091
It is convenient to go to momentum space, by applying a Fourier transform with respect to the boundary coordinates We obtain We can finally determine the propagator of the gauge field between two-points in the boundary by imposing that the exterior derivative reproduces the two-point function (2.45). The result is The parameter ξ is not fixed by requiring consistency with eq. (2.45), and parametrizes a choice of gauge. From the structure of the propagator we see that the natural perturbative limit is g 2 → 0 with γ fixed, which means τ → ∞ with a fixed ratio γ between the real and the imaginary part. Observables are expressed as a power series in g 2 1+γ 2 with coefficients that are themselves polynomials in γ, more precisely the coefficient of the order O g 2 1+γ 2 n is a polynomial in γ of degree n.

Relations to large-k and large-N f perturbation theories
Recall that a 3d Abelian gauge field a with CS action i k 4π a ∧ da has propagator (up to gauge redundancy) We see that the contact term in eq. (2.45) produced a term in the boundary propagator (2.46) that is identical to the CS one. In particular, from the perturbation theory that we will consider one can immediately recover results for large-k perturbation theory in Abelian 3d gauge theories, simply by setting (recall that γ = g 2 θ 4π 2 ) (2.49) Indeed, in the limit g 2 → ∞ only the θ-term is left in the bulk action, and the model that we are considering is equivalent to a CS theory on the boundary, with k = θ 2π . The only role played by the bulk in this case is to allow generic real values of the CS coupling.
We can also compare to the limit of large number of matter flavors N f , in which observables at the IR fixed point of 3d Abelian gauge theories can be computed perturbatively in -10 -

JHEP05(2019)091
1/N f . In this regime, after resumming bubble diagrams, one finds the following "effective" propagator (again, up to gauge redundancy) The proportionality constant depends on the details of the theory. The resulting "nonlocal" propagator has precisely the same form of the boundary propagator (2.46) in the case γ = 0. 3 Hence, once again, the two types of perturbation theories inform each other, and results for one case can be applied in the other case as well. Compared to the large-k perturbation theory, here additional care is needed, because the order at which we are computing a certain observable in the 1/N f -expansion does not coincide with the number of internal photon lines in the corresponding diagram, owing to the fact that diagrams with a larger number of internal photon lines can get an enhancement by a positive power of N f from loops of matter fields. Nevertheless, single diagrams computed in one context can be used in the other context, and we will see an application of this observation later. A generalization of the large-N f limit is obtained by taking both N f and k large, with a fixed ratio, and was studied recently in [20]. In this case one finds a propagator that contains both terms in eq. (2.46), and the same comments about the relation of the two types of perturbation theory apply.

Exploring strong coupling
As the coupling is increased, the two currentsÎ a andĴ a should be treated on an even footing. Indeed, they are rotated into each other by the SL(2, Z) group of electric-magnetic dualities of the bulk theory. Assuming no phase transitions, as we approach cusps τ → − q p where the dual gauge coupling becomes weak in some alternative duality frame, we expect dual statements to be true: the pĴ +qÎ current should decouple from the BCFT correlation functions as they collapse to the correlation functions of a new 3d CFT T p,q [B], which gives the dual weakly coupled description of the original BFCT. Using the notion of duality walls [36,37], one can argue that T p,q should be obtained from T 0,1 by Witten's SL(2, Z) action on 3d CFTs equipped with a U(1) global symmetry [1]. This involves coupling T 0,1 to a certain collection of 3d Abelian gauge fields with appropriate Chern-Simons couplings. This statement requires some care and several caveats about the absence of phase transitions as we vary τ .
In an optimal situation where these phase transitions are absent, this picture implies that the data of B(τ,τ ) will approach the data of an infinite collection of 3d CFTs T p,q as τ → − q p , sitting in the same universality classes as certain 3d Abelian gauge theories coupled to T 0,1 . This is depicted in figure 1. If we "integrate out" the bulk and restrict our attention to the 3d boundary, what we just described can be stated as the existence of a JHEP05(2019)091 At the cusp at infinity the currentÎ a decouples and we are left with the local 3d theory T 0,1 on the boundary, with U(1) current J a . Approaching this cusp from T -translations of the fundamental domain amounts to adding a CS contact term to the 3d theory, or equivalently to redefine the currentĴ a by multiples of the current I a that is decoupling. This is the T operation on T 0,1 in the sense of [1]. In the favorable situation in which no phase transitions occur, the BCFT continuously interpolate to the cusps at the rational points of the real axis τ = −q/p, where again the bulk and the boundary decouple and we find new 3d CFTs T p,q . These theories are obtained from T 0,1 with a more general SL(2, Z) transformation, that involves coupling the original U(1) global symmetry to a 3d dynamical gauge field.
family of non-local 3d conformal theories (i.e. with no stress-tensor in the spectrum) that continuously interpolate between different local 3d CFTs. More precisely, in the decoupling limit the 3d theory is a direct product of a 3d CFT and a non-local sector associated to the boundary condition of the free bulk field. This is reminiscent of the construction of [40][41][42] in the context of the long-range Ising model. Let us mention a possible mechanism for a phase transition. As we change continuously τ from the neighbourhood of the "ungauged cusp" T 0,1 towards the "gauged cusps" T p,q , the dimension of boundary operators are nontrivial functions of τ . A scalar boundary operatorÔ might become marginal at a certain codimension 1 wall in the τ -plane. This possibility is depicted in figure 2. In perturbation theory in the vicinity of the wall, we can repeat the logic that we used in the subsection 2.4 when discussing perturbation theory around T 0,1 in presence of boundary marginal operators. Namely, the boundary marginal couplingλ will generically have a non-trivial beta function, which depends both onλ and τ , and whose leading contributions are 4

E j a m i j J j E y r Z E L z V k 9 e h f V 3 1 L D / W K v W 7 P I 4 i n M E 5 X I I H N 1 C H B 2 h A C x h M 4 B l e 4 c 1 J n B f n 3 f l Y t h a c f O Y U / s j 5 / A F x x Y 7 4 < / l a t e x i t >
real B(τ,τ )

< l a t e x i t s h a 1 _ b a s e 6 4 = " a O f y c x 1 M 2 0 y 9 o m 1 5 6 U p p d T K N T Q Q = " > A A A C C H i c d Z D L S g M x F I Y z X m u 9 V V 2 6 M F i E C l I m R W y 7 K 3 X j s o K 9 Q K e U T J q 2 o Z k L y R m x D H X n x l d x 4 0 I R t z 6 C O 9 / G T F t B R Q 8 E P v 7 / H E 7 O 7 4 Z S a L D t D 2 t h c W l 5 Z T W 1 l l 7 f 2 N z a z u z s N n Q Q K c b r L J C B a r l U c y l 8 X g c B k r d C x a n n S t 5 0 R + e J 3 7 z m S o v A v 4 J x y D s e H f i i L x g F I 3 U z B w 7 w G 4 j N i J z c V n M O 0 O g E O y 5 V c Y K T 4 2 4 m a + d t 2 y a E 4 A R I 8 c w 2 U C 6 X C q S E S W K Z y q J 5 1 b q Z d 6 c X s M j j P j B J t W 4 T O 4 R O T B U I J v k k 7 U S a h 5 S N 6 I C 3 D f r U 4 7 o T T w + Z 4 C O j 9 H A / U O b 5 g K f q 9 4 m Y e l q P P d d 0 e h S G + r e X i H 9 5 7 Q j 6 p U 4 s / D A C 7 r P Z o n 4 k M Q Q 4 S Q X 3 h O I M 5 N g A Z U q Y v 2 I 2 p I o y M N m l T Q h f l + L / o V H I E 8 O X p 9 l K d R 5 H C u 2 j Q 5 R D B B V R B V 2 g G q o j h u 7 Q
A 3 p C z 9 a 9 9 W i 9 W K + z 1 g V r P r O H f p T 1 9 g k c 9 Z o G < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " a O f y c x 1 M 2 0 y 9 o m 1 5

U p p d T K N T Q Q = " > A A A C C H i c d Z D L S g M x F I Y z X m u 9 V V 2 M F i E C l I m R W y 7 K 3 X j s o K 9 Q K e U T J q 2 o Z k L y R m x D H X n x l d x 4 0 I R t z C O 9 / G T F t B R Q 8 E P v 7 / H E 7 O 7 4 Z S a L D t D 2 t h c W l 5 Z T W 1 l l 7 f 2 N z a z u z s N n Q Q K c b r L J C B a r l U c y l 8 X g c B k r d C x a n n S t 5 0 R + e J 3 7 z m S o v A v 4 J x y D s e H f i i L x g F I 3 U z B w 7 w G 4 j N i J z c V n M O 0 O g E O y 5 V c Y K T 4 2 4 m a + d t 2 y a E 4 A R I 8 c w 2 U C X C q S E S W K Z y q J 5 1 b q Z d c X s M j j P j B J t W 4 T O 4 R O T B U I J v k k 7 U S a h 5 S N I C 3 D f r U 4 7 o T T w + Z 4 C O j 9 H A / U O b 5 g K f q 9 4 m Y e l q P P d d 0 e h S G + r e X i H 9 5 7 Q j p U 4 s / D A C 7 r P Z o n 4 k M Q Q 4 S Q X 3 h O I M 5 N g A Z U q Y v 2 I 2 p I o y M N m l T Q h f l + L / o V H I E 8 O X p 9 l K d R 5 H C u 2 j Q 5 R D B B V R B V 2 g G q o j h u 7 Q
A 3 p C z 9 a 9 9 W i 9 W K + z 1 g V r P r O H f p T 1 9 g k c 9 Z o G < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " a O f y c x 1 M 2 0 y 9 o m 1 5

U p p d T K N T Q Q = " > A A A C C H i c d Z D L S g M x F I Y z X m u 9 V V 2 M F i E C l I m R W y 7 K 3 X j s o K 9 Q K e U T J q 2 o Z k L y R m x D H X n x l d x 4 0 I R t z C O 9 / G T F t B R Q 8 E P v 7 / H E 7 O 7 4 Z S a L D t D 2 t h c W l 5 Z T W 1 l l 7 f 2 N z a z u z s N n Q Q K c b r L J C B a r l U c y l 8 X g c B k r d C x a n n S t 5 0 R + e J 3 7 z m S o v A v 4 J x y D s e H f i i L x g F I 3 U z B w 7 w G 4 j N i J z c V n M O 0 O g E O y 5 V c Y K T 4 2 4 m a + d t 2 y a E 4 A R I 8 c w 2 U C X C q S E S W K Z y q J 5 1 b q Z d c X s M j j P j B J t W 4 T O 4 R O T B U I J v k k 7 U S a h 5 S N I C 3 D f r U 4 7 o T T w + Z 4 C O j 9 H A / U O b 5 g K f q 9 4 m Y e l q P P d d 0 e h S G + r e X i H 9 5 7 Q j p U 4 s / D A C 7 r P Z o n 4 k M Q Q 4 S Q X 3 h O I M 5 N g A Z U q Y v 2 I 2 p I o y M N m l T Q h f l + L / o V H I E 8 O X p 9 l K d R 5 H C u 2 j Q 5 R D B B V R B V 2 g G q o j h u 7 Q
A 3 p C z 9 a 9 9 W i 9 W K + z 1 g V r P r O H f p T 1 9 g k c 9 Z o G < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " a O f y c x 1 M 2 0 y 9 o m 1 5

U p p d T K N T Q Q = " > A A A C C H i c d Z D L S g M x F I Y z X m u 9 V V 2 M F i E C l I m R W y 7 K 3 X j s o K 9 Q K e U T J q 2 o Z k L y R m x D H X n x l d x 4 0 I R t z C O 9 / G T F t B R Q 8 E P v 7 / H E 7 O 7 4 Z S a L D t D 2 t h c W l 5 Z T W 1 l l 7 f 2 N z a z u z s N n Q Q K c b r L J C B a r l U c y l 8 X g c B k r d C x a n n S t 5 0 R + e J 3 7 z m S o v A v 4 J x y D s e H f i i L x g F I 3 U z B w 7 w G 4 j N i J z c V n M O 0 O g E O y 5 V c Y K T 4 2 4 m a + d t 2 y a E 4 A R I 8 c w 2 U C X C q S E S W K Z y q J 5 1 b q Z d c X s M j j P j B J t W 4 T O 4 R O T B U I J v k k 7 U S a h 5 S N I C 3 D f r U 4 7 o T T w + Z 4 C O j 9 H A / U O b 5 g K f q 9 4 m Y e l q P P d d 0 e h S G + r e X i H 9 5 7 Q j p U 4 s / D A C 7 r P Z o n 4 k M Q Q 4 S Q X 3 h O I M 5 N g A Z U q Y v 2 I 2 p I o y M N m l T Q h f l + L / o V H I E 8 O X p 9 l K d R 5 H C u 2 j Q 5 R D B B V R B V 2 g G q o j h u 7 Q
A 3 p C z 9 a 9 9 W i 9 W K + z 1 g V r P r O H f p T 1 9 g k c 9 Z o G < / l a t e x i t > real or complex B(τ,τ ) < l a t e x i t s h a 1 _ b a s e 6 4 = " . We might be unable to find real fixed points for the marginal coupling. In such a situation, B(τ,τ ) can only be defined as a complex BCFT. Assuming that we were able to define B(τ,τ ) as a real BCFT in perturbation theory around τ → ∞ by continuity such a real BCFT is ensured to exist in the full region above the wall, but we might be unable to continue it beyond the wall without introducing complex couplings (or breaking conformality).
Here we are perturbing around a point τ 0 on the wall, the coefficient b's and C are (up to numerical factors) the bulk-to-boundary OPE coefficients [43], and the OPE coefficient of the boundary conformal theory, respectively. These OPE coefficients are functions of τ 0 . Depending on τ 0 and on the various OPE coefficients, setting βλ = 0 one might or might not be able to find a real solution forλ. If a real solution can be found perturbing away from the wall in a certain direction, by continuity B(τ,τ ) defines a real BCFT in a region of the τ plane on that side of the wall. Otherwise, on a side of the wall B(τ,τ ) exists only as a non-unitary "complex" BCFT.

Two-point function from the boundary OPE
In section 2.3 we computed the two-point function of the field strength in free theory using the method of images. We will now compute it in the more general case with interactions on the boundary. We will see that it can be fixed completely in terms of the coefficient of the two-point function of the boundary currents. The method that we will use is an explicit resummation of the bulk-to-boundary OPE. As a consequence of the interaction, the bulk-to-boundary OPE of the field strength contains two independent primary boundary operators, both of them conserved currents, rather than just one like in the free case. The leading terms in this OPE are The complete form of the above (including all descendants) can be found in (B.4). The boundary currentsV 1 andV 2 can be expressed in terms ofĴ a andÎ a as followŝ If the 3d CFT that the gauge field couples to has parity symmetry (i.e. symmetry under reflection of one of the coordinates) then the full boundary CFT B(τ,τ ) admits such a symmetry when restricted to θ = 0. Under this symmetry V 1 transforms like an ordinary vector, while V 2 transforms like an axial vector. We can extend this symmetry to the more general case θ = 0 by viewing it as a spurionic symmetry that flips the sign of θ.
Plugging the bulk-to-boundary OPE in the two-point function, one obtains the boundary channel decomposition. In this case, since only two boundary primaries appear in the OPE, we can explicitly resum the contributions from all the descendants. The result can be written in terms of the structures defined in (2.23) with coefficients We see that eq. (2.55) is written explicitly in terms of data of the boundary conformal theory. For the time being we can ignore the contact term in the current two-point function because it cannot contribute to the two-point function of F µν at separated points.
To make the action of SL(2, Z) more transparent we will also rewrite the above results in the selfdual/antiselfdual components. The bulk-to-boundary OPE takes the following form whereV An SL(2, Z) transformation acts onV ± in the same way as it acts on F ± . In particular under an S transformationV + →τV + andV − → τV − . Using the structures introduced in In this basis the SL(2, Z) action on the above two-point functions can be immediately read from (2.10). While in this subsection we discussed the two-point function of F µν , clearly a similar computational strategy could be used for an arbitrary n-point function, therefore reducing any such bulk correlation functions to correlators of the boundary currentsĴ,Î. Of course generically for n > 2 these correlation function are not just captured by the coefficients c ij , because they are sensitive to the full spectrum of boundary operators entering in the OPE of the currents.

One-point functions from the bulk OPE
When x 2 12 y 2 we can expand the two-point function (2.55) in the bulk OPE limit, which is controlled by the OPE of free Maxwell theory where we neglected spinning bulk primaries (since they do not acquire vev) and descendants, and we used the shorthand notation F 2 ≡ F µν F µν and FF ≡ F µνF µν .
Plugging the bulk OPE into the l.h.s. of (2.55) one obtains the following bulk channel decomposition of the two-point function where . . . denote subleading descendant terms, and we parametrized bulk one-point functions as Comparing (2.66) and (2.55) (see appendix C for details) we obtain a constraint from the contribution of the identity This shows that the bulk one-point functions of F 2 and FF are determined by the constants c ij . Note that these relations are compatible with the (spurionic) parity symmetry, because a FF and c 12 are odd, while all the other coefficients are even. What we discussed here is a very simple example of the use of the crossing symmetry constraint on bulk two-point functions to determine data of BCFTs [44]. The constraint can be solved exactly in this case because the bulk theory is gaussian. Equivalently, in selfdual/antiselfdual components Note that due to the constraint in eq. (2.68), the three entries of the matrix c ij actually only contain two independent functions of the coupling. In the appendix D we show how to express c ij (and also the possible contact terms in (2.57)) in terms of two real functions c J and κ J of (τ,τ ), which are the coefficients in the two-point function ofĴ.

c ij (τ,τ ) in perturbation theory
Having derived the bulk one-point and two-point functions in terms of the coefficients c ij (τ,τ ) in the two-point function of the boundary currents, we will now give the leading order results for these coefficients in perturbation theory in τ −1 .
Note that thanks to the modified Neumann condition, at leading orderĴ is identified with the U(1) currentĴ CFT , whose two-point function can be parametrized as Using the expression for c ij (τ,τ ) in appendix D, and plugging c J = c We observe that, to this order, An explanation of this relation, and also a reason why it must hold to all orders in perturbation theory, will be provided in section 3.
Going to higher orders in τ −1 , the correlators ofĴ, and in particular the coefficients c J and κ J , will start deviating from those of the CFT. When the CFT is free, these corrections can be computed by ordinary Feynman diagrams on the boundary. We will see examples of this in the following. In the more general case of an interacting CFT, these correction can be computed in conformal perturbation theory, by lowering an insertion of the bulk Lagrangian (2.2) integrated over the region y ≥ 0. It would be interesting to characterize the CFT observables that enter the subleading orders of this perturbation theory. We leave this problem for the future.

Displacement operator
In every BCFT with d-dimensional bulk there exists a boundary scalar operator of protected scaling dimension d, the so-called displacement operator. It can be defined as the only scalar primary boundary operator that appears in the bulk-to-boundary OPE of the bulk stress tensor There is a Ward Identity associated to this operator, namely that fixes the normalization of the operator. In this normalization its two-point function is 78) and the quantity CD is an observable of the BCFT. It follows from (2.76) that the displacement operator is the restriction of the component T yy of the stress-tensor to the boundary. In the theory that we are considering the bulk stress-tensor is the usual Maxwell stress-tensor Writing T yy (y = 0) in terms of the currentsÎ andĴ leads to the following expression for the displacement operator  The right-hand side of (2.80) contains products of two boundary operators at the same point, that are defined through a point-splitting procedure, similarly to the products on the right-hand side of (2.79). Such a point splitting makes sense for arbitrary τ even though generically the boundary currents are not generalized free fields. This is because their dimension and the dimension ofD are protected, and the contribution ofD in their OPE is non-singular, so after subtracting the contribution of the identity and possibly of additional operators of scaling dimension < 4 we can always take the coincident-point limit.
We can use the expression (2.80) to obtain the first two orders in the perturbative expansion of CD universally in terms of the two-point function of the CFT current (2.72). The leading order contribution to CD at large τ comes from the contraction of theÎ currents in theÎ 2 term, and is therefore proportional to the square of c 22 at leading order. At nextto-leading order there is a contribution from the correction to c 22 , and a contribution from theÎĴ term. See figure 3. The result is Even though the 3d CFT sector decouples from the bulk in the limit τ → ∞, and in particular it has a conserved 3d stress tensor, the displacement operator still exists within the sector of boundary operators coming from the free boundary condition of the bulk Maxwell field, and in particular C D is finite in this limit. Plugging Re τ = 0 and the value of c (0) J for a theory of two Dirac fermions, namely c (0) J = 1 4π 2 , we find perfect agreement with [8].

Three-point function V iVjD
Some of the distinctive features of the conformal theory living on the boundary of B(τ,τ ) are • the presence of a scalar operator of dimension 4, the displacement operatorD; this feature is common to all conformal boundary conditions; • the presence of the two U(1) currentsV 1 andV 2 .

JHEP05(2019)091
We will now show that the displacement operatorD appears in the OPE of the currents, with a matrix of OPE coefficients that can be fixed in terms of the coefficients of the bulk one-point functions a F 2 and a FF , and the coefficient CD.
To show this, we consider the three-point correlator between the field strength and the displacement operator (2.82) We compute this three-point function in two OPE channels for F µν (x 1 )F ρσ (x 2 ). In the boundary channel y 1,2 → 0, using the OPE (2.52) this three-point function can be fixed in terms of the OPE coefficients V iVjD that we want to determine. On the other hand, in the bulk OPE channel For simplicity we placed the displacement at infinity. λ are respectively the parity-even/odd OPE coefficients in the conventions of [46], andx a = x a /| x|. Recall that under parityV 1 is a vector whileV 2 is an axial vector, hence the coefficients λ are odd under a spurionic parity transformation, while the others are even.
The details of the calculation are showed in the appendix E, and here we will just give the final result The parity-odd three-point structures are all set to zero. The spurionic parity symmetry is again satisfied because λ are even. Going to the basis in which the matrix of current-current 2-pt functions is the identity Recall that a F 2 ∈ R and a FF ∈ iR, so A is real and ≥ 0. Seemingly the upper entry has a pole at A = 3 4π 2 , which corresponds to the value at the decoupling limit. However recall from (2.81) that CD → 6 π 4 in the decoupling limit, so that actually the entry is finite in the limit.
The upshot of this analysis is that the OPE coefficients between two currents and the displacement can be completely characterized in terms of the two positive quantities A and CD, that can be taken to effectively parametrize the position on the conformal manifold. It would be interesting to derive these relations from more standard analytic bootstrap techniques, along the lines of [47][48][49].

Free energy on a hemisphere
In this section we study the hemisphere free energy for the conformal boundary conditions of the U(1) gauge field.
Following [50], to any given conformal boundary condition for a CFT 4 we can assign a boundary free energy F ∂ , defined as Z S 4 denotes the sphere partition function of the CFT, while Z HS 4 denotes the partition function of the theory placed on an hemisphere, with the chosen boundary condition on the boundary S 3 . In writing (3.1) we discarded power-law UV divergences, and focused on the universal finite term. Conformal symmetry ensures that the coupling to the curved background can be defined via Weyl rescaling. In our setup the bulk theory is a U(1) gauge-field with action (2.2), so we have Using a Weyl transformation the one-point functions can be obtained from those on R 3 × R + as Here x is a point on the hemisphere, Ω(x) is the Weyl factor induced by the stereographic projection, and u(x) denotes the chordal distance between the point x and the boundary S 3 . The shifts A andÃ stand for a scheme-dependent contribution to the one-point function, due to the ambiguity in the definition of the theory on the curved background: we can always add local counterterms given by a scalar density of dimension four built -20 -

JHEP05(2019)091
out of the background curvature, multiplied by the real or imaginary part of the marginal coupling τ , and integrated in the interior of the hemisphere. On the other hand, if we compute the partition function on S 4 in the same scheme, the one-point functions on S 4 receive contribution only from those counterterms, because on R 4 one-point functions must vanish, and there is a relative factor of two because in this case the counterterm is integrated over the whole sphere. Hence such that the ambiguity precisely cancels in (3.2). Here we see the virtue of the choice of normalization in (3.1). The remaining integral on HS 4 has a UV divergence when the point x approaches the boundary S 3 . We introduce a UV regulator 1 and restrict the integral to the region u(x) > . The result is As implicit in the definition of F ∂ , we will neglect the power-law UV divergent term and focus on the universal finite piece. Hence we finally obtain We used the relations (2.68) to rewrite the result in terms of the two-point functions of the conserved currents. A consequence of this equation is that the relation (2.75) must be valid to all orders in perturbation theory, or more generally whenever F ∂ is well-defined. Plugging (2.73)-(2.74) in (3.6)-(3.7) and solving the equations we find the following leading behavior of F ∂ at large τ The first term, which diverges for τ → ∞, is the value of F ∂ for a free Maxwell field with Neumann boundary conditions [50]. Matching eq. (3.8) with the value of F ∂ for a decoupled system of a Maxwell field with Neumann conditions and a 3d CFT on the boundary, we find that the constant C, that remained undetermined by the differential constraint, is in fact the S 3 free energy F 0,1 of the theory T 0,1 .
Using an SL(2, Z) transformation, the same asymptotic behavior holds in the vicinity of any cusp point, upon replacing τ with the transformed variable τ that goes to ∞ at the selected cusp, and identifying C with the S 3 free energy of the decoupled 3d CFT living at the cusp. Near the cusp where the current pĴ + qÎ decouples from the 3d theory T p,q , we have where τ = aτ +b pτ +q , with aq − bp = 1, and F p,q is the S 3 free energy of T p,q . Hence the function F ∂ (τ,τ ) contains information about the S 3 free energies of an infinite family of 3d Abelian gauge theories, namely all the theories obtained by applying SL(2, Z) transformations to T 0,1 . We note in passing that the shift by 1 2 log |q| in eq. (3.10) has a nice interpretation in terms of the S 3 free energy for a pure CS theory. Indeed, starting with a 4d gauge field with Neumann condition, applying the transformation ST k , i.e. τ = − 1 τ +k , and taking the decoupling limit τ → ∞, we are left with a pure CS theory at level k on the boundary. Hence, the free energy F ∂ in this limit should be the sum of the contribution of the decoupled 4d gauge field with Neumann boundary condition, and the contribution from the CS theory at level k, which is 1 2 log |k|. This is precisely what eq. (3.10) gives. Similarly eq. (3.11) can be interpreted by starting with a 4d gauge field with Dirichlet boundary condition, whose partition function is the left-hand side of (3.11), applying ST k S, i.e. τ = τ −kτ +1 , and going to the decoupling limit. Again, we find a decoupled 4d gauge field with Neumann boundary condition, and a CS theory at level k on the boundary. The shift by 1 2 log |p| in eq. (3.11) precisely reproduces the 1 2 log |k| contribution of the CS theory.

A minimal phase transition
In this section we will study a non-trivial BCFT which conjecturally describes a second order (boundary) phase transition between two free boundary conditions (p, q) and (p , q ) of the 4d gauge field, with pq − p q = 1. In particular, the conjectural BCFT should have a single relevant boundary operator, which can be turned on to flow to either of these free boundary conditions in the IR, depending on the sign of the coupling. We will assume that -22 -

JHEP05(2019)091
this BCFT exists for all values of the gauge coupling τ , with no further phase transitions as a function of τ . Without loss of generality, we can pick two canonical duality frames where the phase transition interpolates between Dirichlet and Neumann boundary conditions or viceversa. We can also pick two duality frames where the phase transition interpolates between Neumann and (1, 1) boundary conditions or viceversa.
• If we go to weak coupling in the former duality frames, the boundary degrees of freedom should describe a phase transition between phases with spontaneously broken or unbroken U(1) global symmetry. We expect that to be described by a critical O(2) model.
• If we go to weak coupling in the latter duality frames, the boundary degrees of freedom should describe a phase transition between two gapped phases with unbroken U(1) global symmetry, but background Chern-Simons coupling which differs by one unit.
We expect that to be described by a massless Dirac fermion.
Keeping track of the duality transformations between the different frames, we can assemble an overall picture.
• Denote as τ DN the gauge coupling associated to the description as a phase transition between Dirichlet and Neumann boundary conditions, so that one "O(2) cusp" is at τ DN → ∞.
In the following we will do most of our calculations in a perturbative expansion around a "Dirac fermion" cusp. The correct boundary theory is actually a Dirac fermion dressed by half a unit of background Chern-Simons coupling [51,52]. It is convenient to absorb that background Chern-Simons coupling into an improperly-quantized shift of the θ angle, so that the gauge coupling is denoted as τ = τ N N − 1 2 = 1 2 1+τ DN 1−τ DN . Therefore, denoting with ψ the Dirac fermion, the action that we consider is

O(2) model < l a t e x i t s h a 1 _ b a s e 6 4 = " Z S r n b o 8 t V t Q 7 W F a L n 5 i g I n E y y V g = " > A A A B + n i c d V D L S g N B E J y N r x h f i R 6 9 D A Y h X s L u G p J 4 C 3 j x Z g S j g S S E 2 U k n G T L 7 Y K Z X D W s + x Y s H R b z 6 J d 7 8 G 2 c 1 g o o W N B R V 3 X R 3 e Z E U G m 3 7 z c o s L C 4 t r 2 R X c 2 v r G 5 t b + c L 2 h Q 5 j x a H F Q x m q t s c 0 S B F A C w V K a E c K m O 9 J u P Q m x 6 l / e Q V K i z A 4 x 2 k E P Z + N A j E U n K G R + v l C F + E G k 9 O S e 0 D 9 c A B y 1 s 8 X 7 f J R v e p W q t Q u 2 3 b N c Z 2 U u L X K Y Y U 6 R k l R J H M 0 + / n X 7 i D k s Q 8 B c s m 0 7 j h 2 h L 2 E K R R c w i z X j T V E j E / Y C D q G B s w H 3 U s + T p / R f a M M 6 D B U p g K k H + r 3 i Y T 5 W k 9 9 z 3 T 6 D M f 6 t 5 e K f 3 m d G I f 1 X i K C K E Y I + O e i Y S w p h j T N g Q 6 E A o 5 y a g j j S p h b K R 8 z x T i a t H I m h K 9 P 6 f / k w i 0 7 h p 9 V i o 3 6 P I 4 s 2 S V 7 p E Q c U i M N c k K a p E U 4 u S Z 3 5 I E 8 W r f W v f V k P X + 2 Z q z 5 z A 7 5 A e v l H c i + k 6 k = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Z S r n b o 8 t V t Q 7 W F a L n 5 i g I n E y y V g = " > A A A B + n i c d V D L S g N B E J y N r x h f i R 6 9 D A Y h X s L u G p J 4 C 3 j x Z g S j g S S E 2 U k n G T L 7 Y K Z X D W s + x Y s H R b z 6 J d 7 8 G 2 c 1 g o o W N B R V 3 X R 3 e Z E U G m 3 7 z c o s L C 4 t r 2 R X c 2 v r G 5 t b + c L 2 h Q 5 j x a H F Q x m q t s c 0 S B F A C w V K a E c K m O 9 J u P Q m x 6 l / e Q V K i z A 4 x 2 k E P Z + N A j E U n K G R + v l C F + E G k 9 O S e 0 D 9 c A B y 1 s 8 X 7 f J R v e p W q t Q u 2 3 b N c Z 2 U u L X K Y Y U 6 R k l R J H M 0 + / n X 7 i D k s Q 8 B c s m 0 7 j h 2 h L 2 E K R R c w i z X j T V E j E / Y C D q G B s w H 3 U s + T p / R f a M M 6 D B U p g K k H + r 3 i Y T 5 W k 9 9 z 3 T 6 D M f 6 t 5 e K f 3 m d G I f 1 X i K C K E Y I + O e i Y S w p h j T N g Q 6 E A o 5 y a g j j S p h b K R 8 z x T i a t H I m h K 9 P 6 f / k w i 0 7 h p 9 V i o 3 6 P I 4 s 2 S V 7 p E Q c U i M N c k K a p E U 4 u S Z 3 5 I E 8 W r f W v f V k P X + 2 Z q z 5 z A 7 5 A e v l H c i + k 6 k = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Z S r n b o 8 t V t Q 7 W F a L n 5 i g I n E y y V g = " > A A A B + n i c d V D L S g N B E J y N r x h f i R 6 9 D A Y h X s L u G p J 4 C 3 j x Z g S j g S S E 2 U k n G T L 7 Y K Z X D W s + x Y s H R b z 6 J d 7 8 G 2 c 1 g o o W N B R V 3 X R 3 e Z E U G m 3 7 z c o s L C 4 t r 2 R X c 2 v r G 5 t b + c L 2 h Q 5 j x a H F Q x m q t s c 0 S B F A C w V K a E c K m O 9 J u P Q m x 6 l / e Q V K i z A 4 x 2 k E P Z + N A j E U n K G R + v l C F + E G k 9 O S e 0 D 9 c A B y 1 s 8 X 7 f J R v e p W q t Q u 2 3 b N c Z 2 U u L X K Y Y U 6 R k l R J H M 0 + / n X 7 i D k s Q 8 B c s m 0 7 j h 2 h L 2 E K R R c w i z X j T V E j E / Y C D q G B s w H 3 U s + T p / R f a M M 6 D B U p g K k H + r 3 i Y T 5 W k 9 9 z 3 T 6 D M f 6 t 5 e K f 3 m d G I f 1 X i K C K E Y I + O e i Y S w p h j T N g Q 6 E A o 5 y a g j j S p h b K R 8 z x T i a t H I m h K 9 P 6 f / k w i 0 7 h p 9 V i o 3 6 P I 4 s 2 S V 7 p E Q c U i M N c k K a p E U 4 u S Z 3 5 I E 8 W r f W v f V k P X + 2 Z q z 5 z A 7 5 A e v l H c i + k 6 k = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Z S r n b o 8 t V t Q 7 W F a L n 5 i g I n E y y V g = " > A A A B + n i c d V D L S g N B E J y N r x h f i R 6 9 D A Y h X s L u G p J 4 C 3 j x Z g S j g S S E 2 U k n G T L 7 Y K Z X D W s + x Y s H R b z 6 J d 7 8 G 2 c 1 g o o W N B R V 3 X R 3 e Z E U G m 3 7 z c o s L C 4 t r 2 R X c 2 v r G 5 t b + c L 2 h Q 5 j x a H F Q x m q t s c 0 S B F A C w V K a E c K m O 9 J u P Q m x 6 l / e Q V K i z A 4 x 2 k E P Z + N A j E U n K G R + v l C F + E G k 9 O S e 0 D 9 c A B y 1 s 8 X 7 f J R v e p W q t Q u 2 3 b N c Z 2 U u L X K Y Y U 6 R k l R J H M 0 + / n X 7 i D k s Q 8 B c s m 0 7 j h 2 h L 2 E K R R c w i z X j T V E j E / Y C D q G B s w H 3 U s + T p / R f a M M 6 D B U p g K k H + r 3 i Y T 5 W k 9 9 z 3 T 6 D M f 6 t 5 e K f 3 m d G I f 1 X i K C K E Y I + O e i Y S w p h j T N g Q 6 E A o 5 y a g j j S p h b K R 8 z x T i a t H I m h K 9 P 6 f / k w i 0 7 h p 9 V i o 3 6 P I 4 s 2 S V 7 p E Q c U i M N c k K a p E U 4 u S Z 3 5 I E 8 W r f W v f V k P X + 2 Z q z 5 z A 7 5 A e v l H c i + k 6 k = < / l a t e x i t >
O(2) model

l a t e x i t s h a 1 _ b a s e 6 4 = " Q U a a G j f e x M l A s g f Z 6 D 9 o B S W F w 6 U = " > A A A B 8 H i c d Z D L S g M x F I Y z 9 V b r r e r S T b A I r k p S x E 5 3 B V 2 4 k g r 2 I u 1 Q M m n a h i a Z I c k I Z e h T u H G h i F s f x 5 1 v Y 6 a t o K I / B D 7 + / x x y z g l j w Y 1 F 6 M P L r a y u r W / k N w t b 2 z u 7 e 8 X 9 g 5 a J E k 1 Z k 0 Y i 0 p 2 Q G C a 4 Y k 3 L r W C d W D M i Q 8 H a 4 e Q i y 9 v 3 T B s e q V s 7 j V k g y U j x I a f E O u u u Z 0 n S T y + v Z / 1 i C Z U R Q h h j m A G u n i M H t Z p f w T 7 E W e R U A k s 1 + s X 3 3 i C i i W T K U k G M 6 W I U 2 y A l 2 n I q 2 K z Q S w y L C Z 2 Q E e s 6 V E Q y E 6 T z g W f w x D k D O I y 0 e 8 r C u f u 9 I y X S m K k M X a U k d m x + Z 5 n 5 V 9 Z N 7 N A P U q 7 i x D J F F x 8 N E w F t B L P t 4 Y B r R q 2 Y O i B U c z c r p G O i C b X u R g V 3 h K 9 N 4 f / Q q p S x 4 5 u z U t 1 f n i M P j s A x O A U Y V E E d X I E G a A I K J H g A T + D Z 0 9 6 j 9 + K 9 L k p z 3 r L n E P y Q 9 / Y J F S q Q j w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Q U a a G j f e x M l A s g f Z 6 D 9 o B S W F w 6 U = " > A A A B 8 H i c d Z D L S g M x F I Y z 9 V b r r e r S T b A I r k p S x E 5 3 B V 2 4 k g r 2 I u 1 Q M m n a h i a Z I c k I Z e h T u H G h i F s f x 5 1 v Y 6 a t o K I / B D 7 + / x x y z g l j w Y 1 F 6 M P L r a y u r W / k N w t b 2 z u 7 e 8 X 9 g 5 a J E k 1 Z k 0 Y i 0 p 2 Q G C a 4 Y k 3 L r W C d W D M i Q 8 H a 4 e Q i y 9 v 3 T B s e q V s 7 j V k g y U j x I a f E O u u u Z 0 n S T y + v Z / 1 i C Z U R Q h h j m A G u n i M H t Z p f w T 7 E W e R U A k s 1 + s X 3 3 i C i i W T K U k G M 6 W I U 2 y A l 2 n I q 2 K z Q S w y L C Z 2 Q E e s 6 V E Q y E 6 T z g W f w x D k D O I y 0 e 8 r C u f u 9 I y X S m K k M X a U k d m x + Z 5 n 5 V 9 Z N 7 N A P U q 7 i x D J F F x 8 N E w F t B L P t 4 Y B r R q 2 Y O i B U c z c r p G O i C b X u R g V 3 h K 9 N 4 f / Q q p S x 4 5 u z U t 1 f n i M P j s A x O A U Y V E E d X I E G a A I K J H g A T + D Z 0 9 6 j 9 + K 9 L k p z 3 r L n E P y Q 9 / Y J F S q Q j w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Q U a a G j f e x M l A s g f Z 6 D 9 o B S W F w 6 U = " > A A A B 8 H i c d Z D L S g M x F I Y z 9 V b r r e r S T b A I r k p S x E 5 3 B V 2 4 k g r 2 I u 1 Q M m n a h i a Z I c k I Z e h T u H G h i F s f x 5 1 v Y 6 a t o K I / B D 7 + / x x y z g l j w Y 1 F 6 M P L r a y u r W / k N w t b 2 z u 7 e 8 X 9 g 5 a J E k 1 Z k 0 Y i 0 p 2 Q G C a 4 Y k 3 L r W C d W D M i Q 8 H a 4 e Q i y 9 v 3 T B s e q V s 7 j V k g y U j x I a f E O u u u Z 0 n S T y + v Z / 1 i C Z U R Q h h j m A G u n i M H t Z p f w T 7 E W e R U A k s 1 + s X 3 3 i C i i W T K U k G M 6 W I U 2 y A l 2 n I q 2 K z Q S w y L C Z 2 Q E e s 6 V E Q y E 6 T z g W f w x D k D O I y 0 e 8 r C u f u 9 I y X S m K k M X a U k d m x + Z 5 n 5 V 9 Z N 7 N A P U q 7 i x D J F F x 8 N E w F t B L P t 4 Y B r R q 2 Y O i B U c z c r p G O i C b X u R g V 3 h K 9 N 4 f / Q q p S x 4 5 u z U t 1 f n i M P j s A x O A U Y V E E d X I E G a A I K J H g A T + D Z 0 9 6 j 9 + K 9 L k p z 3 r L n E P y Q 9 / Y J F S q Q j w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Q U a a G j f e x M l A s g f Z 6 D 9 o B S W F w 6 U = " > A A A B 8 H i c d Z D L S g M x F I Y z 9 V b r r e r S T b A I r k p S x E 5 3 B V 2 4 k g r 2 I u 1 Q M m n a h i a Z I c k I Z e h T u H G h i F s f x 5 1 v Y 6 a t o K I / B D 7 + / x x y z g l j w Y 1 F 6 M P L r a y u r W / k N w t b 2 z u 7 e 8 X 9 g 5 a J E k 1 Z k 0 Y i 0 p 2 Q G C a 4 Y k 3 L r W C d W D M i Q 8 H a 4 e Q i y 9 v 3 T B s e q V s 7 j V k g y U j x I a f E O u u u Z 0 n S T y + v Z / 1 i C Z U R Q h h j m A G u n i M H t Z p f w T 7 E W e R U A k s 1 + s X 3 3 i C i i W T K U k G M 6 W I U 2 y A l 2 n I q 2 K z Q S w y L C Z 2 Q E e s 6 V E Q y E 6 T z g W f w x D k D O I y 0 e 8 r C u f u 9 I y X S m K k M X a U k d m x + Z 5 n 5 V 9 Z N 7 N A P U q 7 i x D J F F x 8 N E w F t B L P t 4 Y B r R q 2 Y O i B U c z c r p G O i C b X u R g V 3 h K 9 N 4 f / Q q p S x 4 5 u z U t 1 f n i M P j s A x O A U Y V E E d X I E G a A I K J H g A
T + D Z 0 9 6 j 9 + K 9 L k p z 3 r L n E P y Q 9 / Y J F S q Q j w = = < / l a t e x i t >   l / e Q V K i z A 4 x 2 k E P Z + N A j E U n K G R + v l C F + E G k 9 O S e 0 D 9 c A B y 1 s 8 X 7 f J R v e p W q t Q u 2 3 b N c Z 2 U u L X K Y Y U 6 R k l R J H M 0 + / n X 7 i D k s Q 8 B c s m 0 7 j h 2 h L 2 E K R R c w i z X j T V E j E / Y C D q G B s w H 3 U s + T p / R f a M M 6 D B U p g K k H + r 3 i Y T 5 W k 9 9 z 3 T 6 D M f 6 t 5 e K f 3 m d G I f 1 X i K C K E Y I + O e i Y S w p h j T N g Q 6 E A o 5 y a g j j S p h b K R 8 z x T i a t H I m h K 9 P 6 f / k w i 0 7 h p 9 V i o 3 6 P I 4 s 2 S V 7 p E Q c U i M N c k K a p E U 4 u S Z 3 5 I E 8 W r f W v f V k P X + 2 Z q z 5 z A 7 5 A e v l H c i + k 6 k = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Z S r n b o 8 t V t Q 7 W F a L n 5 i g I n E y y V g = " > A A A B + n i c d V D L S g N B E J y N r x h f i R 6 9 D A Y h X s L u G p J 4 C 3 j x Z g S j g S S E 2 U k n G T L 7 Y K Z X D W s + x Y s H R b z 6 J d 7 8 G 2 c 1 g o o W N B R V 3 X R 3 e Z E U G m 3 7 z c o s L C 4 t r 2 R X c 2 v r G 5 t b + c L 2 h Q 5 j x a H F Q x m q t s c 0 S B F A C w V K a E c K m O 9 J u P Q m x 6 l / e Q V K i z A 4 x 2 k E P Z + N A j E U n K G R + v l C F + E G k 9 O S e 0 D 9 c A B y 1 s 8 X 7 f J R v e p W q t Q u 2 3 b N c Z 2 U u L X K Y Y U 6 R k l R J H M 0 + / n X 7 i D k s Q 8 B c s m 0 7 j h 2 h L 2 E K R R c w i z X j T V E j E / Y C D q G B s w H 3 U s + T p / R f a M M 6 D B U p g K k H + r 3 i Y T 5 W k 9 9 z 3 T 6 D M f 6 t 5 e K f 3 m d G I f 1 X i K C K E Y I + O e i Y S w p h j T N g Q 6 E A o 5 y a g j j S p h b K R 8 z x T i a t H I m h K 9 P 6 f / k w i 0 7 h p 9 V i o 3 6 P I 4 s 2 S V 7 p E Q c U i M N c k K a p E U 4 u S Z 3 5 I E 8 W r f W v f V k P X + 2 Z q z 5 z A 7 5 A e v l H c i + k 6 k = < / l a t e x i t > O(2) model < l a t e x i t s h a 1 _ b a s e 6 4 = " Z S r n b o 8 t V t Q 7 W F a L n 5 i g I n E y y V g = " > A A A B + n i c d V D L S g N B E J y N r x h f i R 6 9 D A Y h X s L u G p J 4 C 3 j x Z g S j g S S E 2 U k n G T L 7 Y K Z X D W s + x Y s H R b z 6 J d 7 8 G 2 c 1 g o o W N B R V 3 X R 3 e Z E U G m 3 7 z c o s L C 4 t r 2 R X c 2 v r G 5 t b + c L 2 h Q 5 j x a H F Q x m q t s c 0 S B F A C w V K a E c K m O 9 J u P Q m x 6 l / e Q V K i z A 4 x 2 k E P Z + N A j E U n K G R + v l C F + E G k 9 O S e 0 D 9 c A B y 1 s 8 X 7 f J R v e p W q t Q u 2 3 b N c Z 2 U u L X K Y Y U 6 R k l R J H M 0 + / n X 7 i D k s Q 8 B c s m 0 7 j h 2 h L 2 E K R R c w i z X j T V E j E / Y C D q G B s w H 3 U s + T p / R f a M M 6 D B U p g K k H + r 3 i Y T 5 W k 9 9 z 3 T 6 D M f 6 t 5 e K f 3 m d G I f 1 X i K C K E Y I + O e i Y S w p h j T N g Q 6 E A o 5 y a g j j S p h b K R 8 z x T i a t H I m h K 9 P 6 f / k w i 0 7 h p 9 V i o 3 6 P I 4 s 2 S V 7 p E Q c U i M N c k K a p E U 4 u S Z 3 5 I E 8 W r f W v f V k P X + 2 Z q z 5 z A 7 5 A e v l H c i + k 6 k = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Z S r n b o 8 t V t Q 7 W F a L n 5 i g I n E y y V g = " > A A A B + n i c d V D L S g N B E J y N r x h f i R 6 9 D A Y h X s L u G p J 4 C 3 j x Z g S j g S S E 2 U k n G T L 7 Y K Z X D W s + x Y s H R b z 6 J d 7 8 G 2 c 1 g o o W N B R V 3 X R 3 e Z E U G m 3 7 z c o s L C 4 t r 2 R X c 2 v r G 5 t b + c L 2 h Q 5 j x a H F Q x m q t s c 0 S B F A C w V K a E c K m O 9 J u P Q m x 6 l / e Q V K i z A 4 x 2 k E P Z + N A j E U n K G R + v l C F + E G k 9 O S e 0 D 9 c A B y 1 s 8 X 7 f J R v e p W q t Q u 2 3 b N c Z 2 U u L X K Y Y U 6 R k l R J H M 0 + / n X 7 i D k s Q 8 B c s m 0 7 j h 2 h L 2 E K R R c w i z X j T V E j E / Y C D q G B s w H 3 U s + T p / R f a M M 6 D B U p g K k H + r 3 i Y T 5 W k 9 9 z 3 T 6 D M f 6 t 5 e K f 3 m d G I f 1 X i K C K E Y I + O e i Y S w p h j T N g Q 6 E A o 5 y a g j j S p h b K R 8 z x T i a t H I m h K 9 P 6 f / k w i 0 7 h p 9 V i o 3 6 P I 4 s 2 S V 7 p E Q c U i M N c k K a p E U 4 u S Z 3 5 I E 8 W r f W v f V k P X + 2 Z q z 5 z A 7 5 A e v l H c i + k 6 k = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Z S r n b o 8 t V t Q 7 W F a L n 5 i g I n E y y V g = " > A A A B + n i c d V D L S g N B E J y N r x h f i R 6 9 D A Y h X s L u G p J 4 C 3 j x Z g S j g S S E 2 U k n G T L 7 Y K Z X D W s + x Y s H R b z 6 J d 7 8 G 2 c 1 g o o W N B R V 3 X R 3 e Z E U G m 3 7 z c o s L C 4 t r 2 R X c 2 v r G 5 t b + c L 2 h Q 5 j x a H F Q x m q t s c 0 S B F A C w V K a E c K m O 9 J u P Q m x 6 l / e Q V K i z A 4 x 2 k E P Z + N A j E U n K G R + v l C F + E G k 9 O S e 0 D 9 c A B y 1 s 8 X 7 f J R v e p W q t Q u 2 3 b N c Z 2 U u L X K Y Y U 6 R k l R J H M 0 + / n X 7 i D k s Q 8 B c s m 0 7 j h 2 h L 2 E K R R c w i z X j T V E j E / Y C D q G B s w H 3 U s + T p / R f a M M 6 D B U p g K k H + r 3 i Y T 5 W k 9 9 z 3 T 6 D M f 6 t 5 e K f 3 m d G I f 1 X i K C K E Y I + O e i Y S w p h j T N g Q 6 E A o 5 y a g j j S p h b K R 8 z x T i a t H I m h K 9 P 6 f / k w i 0 7 h p 9 V i o 3 6 P I 4 s 2 S V 7 p E Q c U i M N c k K a p E U 4 u S Z 3 5 I E 8 W r f W v f V k P X + 2 Z q z 5 z A 7 5 A e v l H c i + k 6 k = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Z S r n b o 8 t V t Q 7 W F a L n 5 i g I n E y y V g = " > A A A B + n i c d V D L S g N B E J y N r x h f i R 6 9 D A Y h X s L u G p J 4 C 3 j x Z g S j g S S E 2 U k n G T L 7 Y K Z X D W s + x Y s H R b z 6 J d 7 8 G 2 c 1 g o o W N B R V 3 X R 3 e Z E U G m 3 7 z c o s L C 4 t r 2 R X c 2 v r G 5 t b + c L 2 h Q 5 j x a H F Q x m q t s c 0 S B F A C w V K a E c K m O 9 J u P Q m x 6 l / e Q V K i z A 4 x 2 k E P Z + N A j E U n K G R + v l C F + E G k 9 O S e 0 D 9 c A B y 1 s 8 X 7 f J R v e p W q t Q u 2 3 b N c Z 2 U u L X K Y Y U 6 R k l R J H M 0 + / n X 7 i D k s Q 8 B c s m 0 7 j h 2 h L 2 E K R R c w i z X j T V E j E / Y C D q G B s w H 3 U s + T p / R f a M M 6 D B U p g K k H + r 3 i Y T 5 W k 9 9 z 3 T 6 D M f 6 t 5 e K f 3 m d G I f 1 X i K C K E Y I + O e i Y S w p h j T N g Q 6 E A o 5 y a g j j S p h b K R 8 z x T i a t H I m h K 9 P 6 f / k w i 0 7 h p 9 V i o 3 6 P I 4 s 2 S V 7 p E Q c U i M N c k K a p E U 4 u S Z 3 5 I E 8 W r f W v f V k P X + 2 Z q z 5 z A 7 5 A e v l H c i + k 6 k = < / l a t e x i t > τ = τ NN 0 −      Figure 5. The upper-half plane of the gauge coupling τ = τ NN − 1 2 , i.e. in the duality frame in which at τ → ∞ we find a free Dirac fermion on the boundary. Thanks to fermionic particle-vortex duality, the cusp in the origin τ = 0 also gives a free Dirac fermion on the boundary. Thanks to the boson-fermion duality between U(1) 1 2 coupled to a Dirac fermion and the O(2) model, the cusps at τ = ± 1 2 give the O(2) model.

Dirac fermion < l a t e x i t s h a 1 _ b a s e 6 4 = " G Y B a / r I i P p S l c z L t 9 K 6 1 y c Q t k a w = " > A A A B / X i c b Z D J S g N B E I Z 7 4 h b j N i 4 3 L 4 1 B 8 B R m R D D H g B 4 8 R j A L J E P o 6 d Q k T X o W u m v E O A R f x Y s H R b z 6 H t 5 8 G z v J H D T x h 4 a P v 6 q o 6 t 9 P p N D o O N 9 W Y W V 1 b X 2 j u F n a 2 t 7 Z 3 b P 3 D 5 o 6 T h W H B o 9 l r N o + 0 y B F B A 0 U K K G d K G C h L 6 H l j 6 6 m 9 d Y 9 K C 3 i 6 A 7 H C X g h G 0 Q i E J y h s X r 2 U R f h A b N r o R i n A a j Q u J O e X X Y q z k x 0 G d w c y i R X v W d / d f s x T 0 O I k E u m d c d 1 E v Q y p l B w C Z N S N 9 W Q M D 5 i A + g Y j F g I 2 s t m 1 0 / o q X H 6 N I i V e R H S m f t 7 I m O h 1 u P Q N 5 0 h w 6 F e r E 3 N / 2 q d F I O q l 4 k o S R E i P l 8 U p J J i T K d R 0 L 5 Q w F G O D T C u h L m V 8 i E z Q a A J r G R C c B e / v A z N 8 4 p r + P a i X K v m c R T J M T k h Z 8 Q l l 6 R G b k i d N A g n j + S Z v J I 3 6 8 l 6 s d 6 t j 3 l r w c p n D s k f W Z 8 / 4 t u V d Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " G Y B a / r I i P p S l c z L t 9 K 6 1 y c Q t k a w = " > A A A B / X i c b Z D J S g N B E I Z 7 4 h b j N i 4 3 L 4 1 B 8 B R m R D D H g B 4 8 R j A L J E P o 6 d Q k T X o W u m v E O A R f x Y s H R b z 6 H t 5 8 G z v J H D T x h 4 a P v 6 q o 6 t 9 P p N D o O N 9 W Y W V 1 b X 2 j u F n a 2 t 7 Z 3 b P 3 D 5 o 6 T h W H B o 9 l r N o + 0 y B F B A 0 U K K G d K G C h L 6 H l j 6 6 m 9 d Y 9 K C 3 i 6 A 7 H C X g h G 0 Q i E J y h s X r 2 U R f h A b N r o R i n A a j Q u J O e X X Y q z k x 0 G d w c y i R X v W d / d f s x T 0 O I k E u m d c d 1 E v Q y p l B w C Z N S N 9 W Q M D 5 i A + g Y j F g I 2 s t m 1 0 / o q X H 6 N I i V e R H S m f t 7 I m O h 1 u P Q N 5 0 h w 6 F e r E 3 N / 2 q d F I O q l 4 k o S R E i P l 8 U p J J i T K d R 0 L 5 Q w F G O D T C u h L m V 8 i E z Q a A J r G R C c B e / v A z N 8 4 p r + P a i X K v m c R T J M T k h Z 8 Q l l 6 R G b k i d N A g n j + S Z v J I 3 6 8 l 6 s d 6 t j 3 l r w c p n D s k f W Z 8 / 4 t u V d Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " G Y B a / r I i P p S l c z L t 9 K 6 1 y c Q t k a w = " > A A A B / X i c b Z D J S g N B E I Z 7 4 h b j N i 4 3 L 4 1 B 8 B R m R D D H g B 4 8 R j A L J E P o 6 d Q k T X o W u m v E O A R f x Y s H R b z 6 H t 5 8 G z v J H D T x h 4 a P v 6 q o 6 t 9 P p N D o O N 9 W Y W V 1 b X 2 j u F n a 2 t 7 Z 3 b P 3 D 5 o 6 T h W H B o 9 l r N o + 0 y B F B A 0 U K K G d K G C h L 6 H l j 6 6 m 9 d Y 9 K C 3 i 6 A 7 H C X g h G 0 Q i E J y h s X r 2 U R f h A b N r o R i n A a j Q u J O e X X Y q z k x 0 G d w c y i R X v W d / d f s x T 0 O I k E u m d c d 1 E v Q y p l B w C Z N S N 9 W Q M D 5 i A + g Y j F g I 2 s t m 1 0 / o q X H 6 N I i V e R H S m f t 7 I m O h 1 u P Q N 5 0 h w 6 F e r E 3 N / 2 q d F I O q l 4 k o S R E i P l 8 U p J J i T K d R 0 L 5 Q w F G O D T C u h L m V 8 i E z Q a A J r G R C c B e / v A z N 8 4 p r + P a i X K v m c R T J M T k h Z 8 Q l l 6 R G b k i d N A g n j + S Z v J I 3 6 8 l 6 s d 6 t j 3 l r w c p n D s k f W Z 8 / 4 t u V d Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " G Y B a / r I i P p S l c z L t 9 K 6 1 y c Q t k a w = " > A A A B / X i c b Z D J S g N B E I Z 7 4 h b j N i 4 3 L 4 1 B 8 B R m R D D H g B 4 8 R j A L J E P o 6 d Q k T X o W u m v E O A R f x Y s H R b z 6 H t 5 8 G z v J H D T x h 4 a P v 6 q o 6 t 9 P p N D o O N 9 W Y W V 1 b X 2 j u F n a 2 t 7 Z 3 b P 3 D 5 o 6 T h W H B o 9 l r N o + 0 y B F B A 0 U K K G d K G C h L 6 H l j 6 6 m 9 d Y 9 K C 3 i 6 A 7 H C X g h G 0 Q i E J y h s X r 2 U R f h A b N r o R i n A a j Q u J O e X X Y q z k x 0 G d w c y i R X v W d / d f s x T 0 O I k E u m d c d 1 E v Q y p l B w C Z N S N 9 W Q M D 5 i A + g Y j F g I 2 s t m 1 0 / o q X H 6 N I i V e R H S m f t 7 I m O h 1 u P Q N 5 0 h w 6 F e r E 3 N / 2 q d F I O q l 4 k o S R E i P l 8 U p J J i T K d R 0 L 5 Q w F G O D T C u h L m V 8 i E z Q a A J r G R C c B e / v A z N 8 4 p r + P a i X K v m c R T J M T k h Z 8 Q l l 6 R G b k i d N
Essentially by construction, the picture is compatible with a well-known duality web of particle-vortex, fermion-boson and fermion-fermion dualities [27], which inspired this investigation. In particular, thanks to the particle-vortex duality between the O(2) model and the gauged O(2) model [53,54], or equivalently thanks to its fermionic version [55], in this case we have a Z 2 subgroup of SL(2, Z) that is a duality of B(τ,τ ), i.e. it leaves -24 -

JHEP05(2019)091
invariant both the bulk and the boundary condition. This subgroup acts on τ = τ NN − 1 2 as It is interesting to note that the self-dual point τ = i 2 , i.e. τ DN = i, is an extreme of F ∂ . In our formalism, this is a straightforward consequence of the differential equations (3.6)-(3.7), once we set c 11 = c 22 = 2 π Imτ and c 12 = 0 -as dictated by self-duality and equation (2.68). 6 Before proceeding, let us mention some of the previous literature on this theory, and comment on the relation to the results that we will present in the rest of this section. The interplay between the 3d dualities and the 4d electric-magnetic duality in the setup with a 3d Dirac fermion coupled to a bulk gauge field was investigated in [27][28][29][30][31]. In particular [30,31] studied the transport properties of the boundary theory at the self-dual point. For the theory with an even number of Dirac fermions on the boundary, the twoloop two-point function of the boundary currentĴ was obtained in [3] (see also [4][5][6][7]) while the Weyl anomalies (or equivalently the two-and three-point functions of the displacement operator) were computed to next-to-leading order in [8,57] (for the supersymmetric version of the theory see [58]). The point of view of boundary conformal field theory was first adopted in this theory in [8,57], but these papers do not consider the action of electricmagnetic duality and the existence of multiple decoupling limits. Besides the transport coefficients and the Weyl anomalies, other boundary observables such as scaling dimensions of operators, or the hemisphere free-energy, were not studied before. Since the duality explained above only exists for the theory with one Dirac fermion, we will first concentrate on this case. Later we will also consider the theory with an even number 2N f of fermions, both at large N f and in the special case 2N f = 2.

Perturbative calculation of scaling dimensions
We will compute the anomalous dimensions of the first two fermion bilinear operators O s of spin s, namely up to two-loop level. Note that in the limit τ → ∞ of decoupling between bulk and boundary (O 2 ) ab becomes a conserved current, namely the stress-tensor of the 3d freefermion CFT. The anomalous dimension can be obtained from the renormalization of the 1PI correlator of the composite operator with two elementary fields O s (q = 0)ψ(−p)ψ(p) 1PI .
(4.5) 6 Alternatively, we can implement the reasoning of [56] to show that this property follows from the emergent Z2 symmetry of the system at the self-dual point. We employ dimensional regularization and minimal subtraction, i.e. we set d = 3 − 2 and keep the codimension fixed = 1, expand the dimensionally-continued loop integrals around = 0, and reabsorb the poles in the renormalization constants where the subscript B denotes the bare operators. Even though the correlator in (4.5) involves the operator ψ that is not gauge-invariant, it is still sensible to renormalize it. The resulting renormalized correlator, as well as the renormalization constant Z ψ , both depend on the choice of gauge-fixing, but the renormalization constant Z O of the gauge-invariant operator does not, hence we can extract physical information from it.
The renormalization constants admit the loopwise expansion (at small g 2 with fixed γ) where δZ (n) is a polynomial in γ of degree ≤ n, and furthermore by invariance under space reflections only even powers of γ are present. The n-loop term δZ (n) contains divergences up to −n , but the familiar RG argument constrains all the coefficients in terms of the ones at lower loop order, except that of the −1 divergence. The anomalous dimension is then given by The dependence on the renormalization scale µ is through the d-dimensional coupling In the latter equation we do not need to include a renormalization of the coupling because, as we explained in section 2.4, g does not run. Therefore we can rewrite To compute (4.5) we use the Feynman diagrams in figure 8, computed in momentum space, and for simplicity we take the composite operator to carry zero momentum. The Feynman rules given in figure 6.   We performed the calculation up to two loops. See appendix G for more details about the computation of the two-loop Feynman diagrams. The resulting renormalization constants are (4.13) where we denoted δZ s ≡ δZ Os . Note that indeed δZ 0 and δZ 2 do not depend on the gauge-fixing parameter. As a check, we also verified that the operator O 1 =ψγ a ψ does not get renormalized, i.e. we explicitly computed the renormalization up to two-loop order and found δZ 1 = 0, as expected for a conserved current. On the other hand, note that δZ 2 = 0. This is a manifestation of the fact that the boundary degrees of freedom do not define a local 3d theory once we couple them to the bulk: the conservation of the boundary wouldbe stress-tensor is violated at g = 0, and the system only admits a stress-tensor in the bulk. This means that the short operator of spin 2 must recombine into a long conformal multiplet. In the appendix F, we show that this mechanism can be used to compute the one-loop anomalous dimension, and we use this to check the Feynman diagram calculation. The resulting anomalous dimensions, expressed as a function of τ are  From these result we can immediately recover the anomalous dimensions for the 3d gauge theory U(1) k coupled to a Dirac fermion at large k as explained in section 2.5.1. Since this is a local 3d theory, we expect γ 2 = 0 and indeed this is what we obtain from (4.16). For the anomalous dimension of the scalar bilinear, that starts at two-loop order in this theory, we find 17) in agreement with [59]. 7

Perturbative F ∂
Thanks to the differential equation (3.6)-(3.7), and to the relations derived in appendix D, the computation of the hemisphere free energy is reduced to the computation of the twopoint functions of the boundary currentĴ. Up to next-to-leading order, we already wrote the universal formula (3.8) for the hemisphere free energy in terms of the two-point function of the currentĴ CFT of the unperturbed CFT. In this particular setup where the boundary theory at τ → ∞ is a free Dirac fermion we can do better without much effort, because the correction to the current two-point function, given by the two diagrams in figure 9, already exists in the literature. For the parity even part of the two-point function, we can either extract the value of these diagrams from the large-N f calculation of [19], using the similarities between the two perturbative expansions that we explained in 2.5.1, or alternatively use the computation performed directly in the mixed-dimensional setup in [3,6]. 8 The parity-odd part can be obtained from the large-k calculation in [62]. The sum of the diagrams in figure 9 is the next-to-leading order correction for the one-photon irreducible two-point function, which we denote by Σ, see appendix D for more details. Due to the shift in the real part of τ , i.e. τ = τ NN − 1 2 , we have that κ Σ vanishes at leading order in perturbation theory, or equivalently κ (0) J = 0. The results mentioned above give In [60] there appears to be a sign mistake in the two-loop diagram that we denoted with (b.2) in figure 8.
This mistake leads to the different result for this anomalous dimension given in [61]. Upon correcting that sign, we find perfect agreement with our result. We thank E. Stamou for helping us with this check. 8 In comparing with [3,6] one needs to take into account that they consider a 3d interface with the gauge field propagating on both sides, rather than a boundary. The propagator of the photon restricted to an interface has a factor of 1 2 compared to the case of the boundary.

Extrapolations to the O(2) model
We can now attempt to extrapolate the perturbative results obtained above around the Dirac fermion cusp to the O(2) cusp (see figure 5), to obtain the data of the O(2) model. The O(2) model, while being strongly coupled, is a well-studied theory via a variety of techniques, so that we can compare our extrapolations to the known data. Even though so far we only obtained the first two orders in perturbation theory, and one might be wary to already attempt an extrapolation, we will see that the results we obtain are compatible with the known data. We view this as an encouraging indication that the perturbative technique that we are presenting here can indeed be a useful tool to obtain data of 3d Abelian gauge theories, and as a motivation to try to obtain more precise predictions by going to higher orders.
In order to extrapolate, we need to apply a resummation technique. The nice property of our setup is the duality τ ↔ τ = − 1 4τ , which means that the perturbative expansions obtained above also tell us about the behavior of the observables around τ → ∞, i.e. the second Dirac fermion cusp. To leverage on this, the idea is to use a "duality-improved" Padé approximant, i.e. a function with a number of free parameters that we can fix by matching to the perturbative result at τ → ∞, and that is invariant under a duality transformation.
Similar resummations were studied in the context of perturbative string theory [64] and N = 4 super Yang-Mills (SYM) in [65]. In particular [65] introduced Padé-like approximants with the property of being invariant under a subgroup of SL(2, Z), and we will borrow their method. Note that the perturbative results of the previous subsections, -29 -JHEP05(2019)091 expressed in terms of g s = g 2 and θ, and expanded for small g s with θ fixed take the form 26) and f ∂ is the boundary free energy where the contributions from free gauge field as well as the constant term have been subtracted. The expressions above all have the pattern 27) which can be approximated by the manifestly duality-invariant interpolation functions written in [65]. At this order, there are two of their functions that we can use, the integralpower Padé F 1 (g s , θ) and half-integral-power Padé F 2 (g s , θ), defined by where S · g s is the new gauge coupling under the transformation τ → − 1 4τ , which reads explicitly S · g s = g 2 s θ 2 + 16π 4 π 2 g s . (4.30) The unconventional negative power in the above two Padé approximant was devised in [65] to remove the θ dependence in the Taylor expansion. This is appropriate to match our perturbative expansion up to the order we are considering, because the θ-dependence starts at the subleading order g 3 s . On the other hand, while the perturbative expansion of N = 4 SYM is independent of θ to all orders in perturbation theory, and therefore in that context it is desirable to have an ansatz whose Taylor expansion does not contain θ, in our setup observables do depend on θ even in perturbation theory. Indeed, by taking a different scaling such as g s small with γ = θgs 4π 2 fixed, rather than θ fixed, we would have a non-trivial dependence on γ already at the order we are considering, and with this scaling we could not match the observables with the Taylor expansion of the approximants (4.28)-(4.29). The upshot is that in order to use the duality-improved approximants from [65] we are forced to consider the expansion at small g s with θ fixed, and doing so we throw away some of the information contained in the perturbative calculation, namely the g 2 terms. It would be desirable to find an ansatz that is: (i) duality invariant; (ii) has a final limit to the real τ axis (or at least to the O(2) cusp); and (iii) can be matched with the perturbative expansion at small g s and γ fixed (at least up to the order g 2 s at which the observables are currently known).
1.406 3.635 1.039 F 2 (g s = ∞, θ = π) 1.560 3.391 0.166 Table 1. Comparison of the extrapolations with the known data: for the energy operator we are quoting the value from the conformal bootstrap [66], and from the -expansion [67]. For the sphere free energy we are comparing to the value from the -expansion in [68].  Figure 10. Extrapolations of the scaling dimensions from the Dirac fermion point (tan −1 (g s ) = 0) to the O(2) point (tan −1 (g s ) = π/2).
By matching the coefficients in the expansion up to the order g 2 s , we find the unknown coefficients h i to be In the table 1 we show the resulting values of the approximant extrapolated at the O(2) point. The fermion-mass operator is mapped to the energy operator of the O(2) model, whose dimension can be obtained for instance from the conformal bootstrap [66], or from the -expansion [67]. The spin 2 operator is expected to approach the conserved stressenergy tensor on the boundary in the decoupling limit, hence the dimension should approach the protected value ∆ 2 | cusp = 3. As for the hemisphere free energy, one needs to subtract a finite contribution coming from the decoupled gauge field at the O(2) cusp, and the remaining constant gives the sphere free energy of the O(2) model. To our knowledge this has only been computed using -expansion [68].
We see that both ansatzes give good approximations for the dimension of the energy operator, and in particular F 2 is quite close to the values obtained with the other methods. For the other two observables, we see that the ansatz F 2 also gives compatible results, while F 1 is not as good. In figure 10 we show the plots of the approximants at θ = π, i.e. the value of the O(2) cusp, as a function of g s from 0 to ∞.  In this section we consider the coupling of 2N f Dirac fermions to the bulk gauge fields, all with the same charge q = 1, and we take the limit of large N f with λ = g 2 N f fixed. For simplicity we take θ = 0. We will see that computing observables in 1/N f expansion, and later taking the limit λ → ∞, one can recover the 1/N f expansion in QED 3 . This would be the expected result if we would take g 2 → ∞ first, obtaining the decoupling limit in which on the boundary we have QED 3 with 2N f flavors, and later take N f large. Hence, the observation here is that these two limits commute. This is interesting because order by order in 1/N f we can explicitly follow observables as exact functions of λ, and see how they interpolate from the "ungauged cusp" at λ = 0 to the "gauged cusp" at λ = ∞.
To derive that the limits commute, it is sufficient to observe that in the limit of large N f with λ = g 2 N f fixed we can obtain an effective propagator for the photon by resumming the fermionic bubbles, see figure 12, obtaining (up to gauge redundancy) in a level-rank duality fashion, giving the so-called regular boson/critical fermion duality.
There is a large amount of evidence for this duality at large N , and its extension to finite N was recently studied in [78,79]. It is not clear whether the duality still holds when N = 1. Assuming it does, it would have a nice manifestation in our setup: by starting with a free complex scalar on the boundary, one would find that the cusp at τ = 1 corresponds to the Gross-Neveu CFT with 1 Dirac fermion. 9 One crucial new ingredient of the regular boson/critical fermion dualities is the existence of a additional sextic couplings that are classically marginal and potentially lead to multiple fixed points that can be mapped across the duality. With this motivation in mind, we will now consider the setup with a free complex scalar on the boundary, coupled to the bulk gauge field. The action is The couplings |φ| 2 and |φ| 4 are fine-tuned to zero. This fine-tuning might need to be adjusted as a function of the bulk gauge coupling. At least for τ large enough, these operators are relevant and correspondingly the beta function is linear in the couplings, so this adjustment is possible. On the other hand, the beta function for the classically marginal operator |φ| 6 will start quadratically in ρ and we need to check the existence of (real) fixed points. We list the Feynman rules in figure 13.
To compute the β function of ρ we need to renormalize the six-point vertex. We use the same approach as in the fermion case, i.e. we dimensionally regularize by continuing the dimension of the boundary to d = 3 − 2 , keeping the codimension fixed = 1. The boundary action in renormalized variables is where the subscript B denotes the bare variables. Figure 14 shows the diagrams that contribute to the wavefunction renormalization of the field φ, from which we obtain (5.11) 9 The Gross-Neveu CFT is expected to exist also for a small number N of Dirac fermion, the UV completion being provided by a Yukawa theory. See [80] for a recent study in -expansion.   There are three types of diagram contributing to the six-point vertex counterterm, showed in figure 15 and 16, from which we can compute The β function is Up to this order we find: a zero at ρ = ρ + * > 0 from the first two terms, and since ρ + * = O(g 2 ) the third term is negligible; and a zero at ρ = ρ − * from the second and the third therm, and since ρ The derivative of β ρ is positive at ρ + * and negative at ρ − * . Hence we have found that perturbatively around large τ there exists a fixed point ρ = ρ + * which is IR stable, and gives a scalar potential bounded from below. The fixed point ρ − * on the other hand is only physical for 1 − 3γ 2 < 0, because otherwise it gives the wrong sign of the scalar potential, and it is unstable in the RG sense.
Having checked the existence of the fixed point in perturbation theory, we proceed to consider the anomalous dimension of boundary operators in this theory, similarly to what we did in section (4.1) for the fermion case. We consider the mass-squared operator O = |φ| 2 . Its anomalous dimension can be obtained from the renormalization of the 1PI correlator of the composite operator with two elementary fields The one-loop (two-loop) diagrams contributing to the three-point function (5.16) are showed in figure 14 (figure 17, respectively).
At one loop, using (5.11), the renormalization constant of the operator is found to be and correspondingly the anomalous dimension is (5.18) Differently from the fermion case, we were not able to evaluate all of the dimensionallyregularized integrals coming from the two-loop diagrams of figure 17. See the appendix G for the details. Knowing the two-loop anomalous dimension would enable an extrapolation to τ = 1 that could be compared with the known estimates of the mass operator in the Gross-Neveu CFT. This is therefore an interesting direction left for the future.

QED 3 with two flavors
In this section we will discuss a realization in our setup of QED 3 coupled to two Dirac (complex two-component) fermions of charge 1. There are several reasons why this is an interesting theory: it is conjectured to describe the easy-plane version of the "deconfined" Néel-VBS quantum phase transition in antiferromagnets [81], and enjoy an emergent O(4) symmetry [82,83]; while initially believed to be a second-order transition, recent evidences from simulations of the spin system on the lattice [84] and from the conformal bootstrap [85] suggest that this is actually a weakly first-order transition, which can still be compatible with the QED description if the latter has a complex fixed point with O(4) symmetry (see section 5 of [73] and [22]); it is conjectured to enjoy a self-duality [33,82,83,86] and a fermion-boson duality [87].
A simple way to realize QED with two flavors in our setup would be to put the CFT of two Dirac fermions on the boundary, and couple a bulk gauge field to the U(1) symmetry that gives charge 1 to both of them. However in this case we only expect a weakly coupled cusp at τ → ∞. For the purpose of attempting an extrapolation from weak coupling, it would be desirable to have additional weakly coupled cusps, as in the example of section 4. With this idea in mind, a more promising approach is to consider a generalization of the former set-up in which we have two Maxwell gauge fields in the bulk and two Dirac fermions on the boundary, namely two decoupled copies of the theory of section 4. By performing an S-duality for either of the two gauge fields separately we find again two free Dirac fermions on the boundary. On the other hand using the larger electric-magnetic duality group that exists for a theory of two gauge fields, we can also go to a duality frame where in the decoupling limit we have precisely QED with two flavors on the boundary.
In the rest of this section we will first review electric-magnetic duality for multiple Maxwell fields, and then show how to get QED with two flavors starting with two copies of a bulk gauge field coupled to a boundary Dirac fermion. The task of performing perturbative calculations of observables in this theory is left for the future.

Multiple Maxwell fields
The action of free bulk U(1) n gauge theory is determined in terms of n Abelian gauge fields A I , such that F I = dA I and an n×n symmetric matrix of complexified gauge couplings τ IJ where τ IJ = θIJ 2π + 2πi and we introduced F ±,I µν = 1 2 (F I µν ± 1 2 µνρσ F I,ρσ ). This theory enjoys an Sp(2n, Z) duality group

JHEP05(2019)091
This duality group is generated by the three types of transformations obtained in [88,89], which we reproduce here 10 T-type: where I the n × n identity and B is a symmetric matrix that generates τ = τ + B, (5.23) S-type: where J = diag(j 1 , j 2 , . . . , j n ) and j i ∈ {0, 1}. This gauges those A i 's that have j i = 1.
where U ∈ SL(n, Z) generate the rotations A = U −1T A.
In the rest of this section we will be focusing on the case of n = 2. Following [88] we define the generators of Sp(4, Z) as Furthermore we use the succinct notation S[1, 0], S[0, 1] to denote the gauging of A 1 , A 2 (respectively) and T [m, n] for the introduction of the Chern-Simons terms mA 1 dA 1 + nA 2 dA 2 .

Targeting two-flavor QED
We now have all the tools to obtain two-flavour QED 3 via an Sp(4, Z) action from a theory of two free fermions. The action of two-flavour QED 3 is [83] 11 10 More precisely, these elements generate Sp(2n, Z)/ ∼, where we identify S ∼ −S. 11 Here we are using a different charge normalization compared to [83]. For example, the lowest charged gauge invariant operator is the mesonψiψj, which has charge 1 under gauge field A 1 in our case but charge 2 under the gauge field X in [83]. Our choice is necessary if we want to start from (5.30), because Sp(4, Z) respects the charge normalization. The difference between the charge-two theory and charge-one theory is that the former has fewer monopole operators. Starting with the charge-one theory, we can gauge Z2 ⊂ U(1)J , where U(1)J is the magnetic U(1) global symmetry. This has the effect of changing the gauge group G = U(1) toG such thatG/ Z2 = G. For example, in this case G = U(1), and we gauge Z2 ⊂ U(1)J , then the new gauge group isG = U(1) but with the replacement of the gauge field Aµ → 2Aµ, namely all the particle charges are multiplied by 2 [10,90]. In this way we obtain the charge-two theory.

JHEP05(2019)091
where A I=1,2 are bulk U(1) gauge fields while a is a 3d spin c connection. The gravitational term CS g is needed because which is well-defined for a spin c connection a. 12 We want to target this action via an Sp(4, Z) transformation from where A I=1,2 are spin c connections. To this end, we can start from a rotation of the gauge fields by performing a GL-type transformation with . 13 The resulting relation between τ and τ is The decoupling limit of (5.28) is  12 In the sense that this combination of boundary CS term is independent of the choices of different extensions of the boundary into bulk mod 2π Z 13 We follow the notation in [27] that the minus sign in S 2 = −1 denotes charge conjugation. 14 Note that here we are not shifting the definition of the bulk coupling τ by 1/2 as we did in (4.1). So the transformation is the same as the one presented in [27] instead of the transformation τ = −1/4τ that we had in the previous section. . Applying this to either A 1 or A 2 in (5.30), we obtain that the decoupling limits in the two following duality frames also correspond to two free Dirac fermions Hence, in the variable τ IJ the theory (5.30) has weakly coupled cusps at To summarize, we showed that the theory (5.30) of two bulk gauge fields coupled to two Dirac fermions has two additional duality frames (5.39) in which the boundary theory is still the free theory of two Dirac fermions, and a duality frame (5.34) in which the boundary theory is QED 3 with two flavors. Clearly, additional duality frames corresponding to QED 3 with two flavors can be obtained by applying the transformation (5.32) to either of the additional free-fermions points. This is a promising setup to study QED 3 with two flavors via an extrapolation from the weakly-coupled points.

Future directions
We conclude by discussing some directions for future investigation.
• A universal feature of the setup considered in this paper is the existence of bulk line operators, whose endpoints may be attached to boundary charged operators. It is possible to assign conformal dimensions to the local operators at the location where the line defect ends on the boundary, and these dimensions can be computed perturbatively. Similarly to cusp anomalous dimensions, they are functions of the angle between the defect and the boundary. Starting with the dimensions of the endpoints of 't Hooft lines (and 't Hooft-Wilson lines) around τ → ∞ with a certain CFT on the boundary, it would be interesting to attempt an extrapolation to the cusps on the real axis, where they approach the dimensions of local monopole operators in the gauged version of the initial CFT. Concretely, in the example of section 4, from the dimension of the endpoint of a 't Hooft line around the Dirac fermion point one can attempt to recover the scaling dimension of the spin operator of the O(2) model.
• It would be interesting to perform perturbative calculations of anomalous dimensions and of the free energy in the theory with two bulk gauge fields presented in section 5.3, and attempt an extrapolation to QED 3 with two flavors. In particular, it is possible to use our setup to test whether this theory exists as a real CFT, by studying the dimension of four-fermion operators and checking whether they cross marginality before we reach the QED cusp, leading to the "phase-transition" described in section 2.6.

JHEP05(2019)091
• In the model considered in section 4 we have only used the two-sided extrapolations to give estimates for the O(2) model. However there are infinitely many other cusps on the real axis where strongly-coupled CFTs live, and they are of course amenable to the same extrapolation technique. These theories typically take the form of QED-CS theories, and they also describe interesting phase transitions [91]. A direction for the future would be to use our method to give estimates for the observables of these theories.
• Finally, dualities analogous to the one considered in this paper exist for N = 2 gauge theory. One of the simplest examples is the so-called triality [92][93][94][95] generated by ST transformation [89,96], with (ST ) 3 = 1. It would be interesting to see how the triality can improve the extrapolation. Thanks to supersymmetric localization the boundary free energy and dimensions of chiral endpoints of line operators are exactly computable [50]. For many other interesting observables, such as the conformal dimensions of operators analogous to O 0 , which are non-protected, one has to resort to Feynman diagrams.

A Method of images
In this appendix we show how to compute the two-point function of F µν in the free theory using the method of images.
Reflections about the boundary are implemented by the matrix where n µ is the inward pointing vector normal to the boundary. Note that the reflection of the field strength has components (F R ya (x), F R ya (x)) = (−F ya (R x),F ya (R x)). Hence, the combination satisfies the equation of motion and Bianchi identity for y ≥ 0, and also satisfies the Dirichlet (Neumann with γ = 0) boundary condition upon choosing the sign s = 1 (s = −1, respectively). Even though Bose symmetry is not manifest in (A.3), it is satisfied because F µν (x 1 )F R ρσ (x 2 ) R 4 = F R µν (x 1 )F ρσ (x 2 ) R 4 . We can then rewrite the image term using the cross-ratio ξ and the vectors X i µ by means of the following identity In this way we find (2.19).
In the more general case of Neumann boundary condition with γ = 0, consider the combination (A.6) For F µν the problem is reduced to the Neumann boundary condition with γ = 0, so we have Note that

B Defect OPE of F µν
Let us consider what can appear as a primary inside the bulk-to-boundary OPE of the field strength F µν . By spin selection rules only vectors are admitted, with two possible structures, namely must be boundary descendants. This requires ∆ 1 = ∆ 2 = 2. We conclude that the only allowed boundary primaries are conserved currents.
To obtain the complete form of the bulk-to-boundary OPE of F (including all the descendants) we first need the exact FV correlator. This can be easily computed using the techniques of [97] to find where c ij (τ ) are defined in eq. (2.57). The bulk-to-boundary OPE of F can now be obtained by expanding both sides of (B.3) to find With the bulk-to-boundary OPE above, it is straightforward to obtain the F F 2-point function in terms of the defect CFT data as in (2.55).

C Bulk OPE limit of F µν F ρσ
Here we present some details of the bootstrap analysis presented in section 2.8. To simplify computations, it is convenient to start from a configuration where the two bulk operators lie at the same parallel distance from the defect, i.e. x 12 = 0. In this case some expressions in (2.55) simplify considerably, e.g.

JHEP05(2019)091
It is now a simple exercise to derive the bulk OPE limit of (2.55) F ab ( x, y 1 )F cy ( x, y 2 ) ∼ (D. 15) We see that all the coefficients c ij can be expressed in terms of the functions of the coupling c J and κ J (or equivalently c Σ and κ Σ ). As a check, note that the first identity in (2.69), that was derived from the contribution of the identity in the bulk OPE and relates c 11 and c 22 , is identically satisfied.

E Calculation of V iVjD
We start by computing the three-point function using the boundary channel. At leading order in the boundary OPE limit the three-point function becomes which upon placing the displacement operator at infinity simplifies to [45,46] V a i ( abc .

(E.3)
From the boundary OPE-channel we find where the ellipses denote the descendant contributions from the second term of (E.3), which are proportional to λ (1) ijD− and will not play any role in the following. Next, we compute the three-point function using the bulk OPE channel. The Lorentz spin and scaling dimensions of the full set of operators appearing in the OPE of two F 's can be found in [98] -see eq. (2.12) therein -where they are discussed in the context of the so-called minimal type-C higher spin theory on AdS 5 , the bulk dual to the free Maxwell CFT 4 . All the operators with scaling dimension ∆ > 4 in this OPE are higher-spin conserved currents (there is both a family of symmetric traceless tensors and a family of mixed-symmetry ones), and in addition there is the identity operator and a few operators of scaling dimension ∆ = 4: the scalar operators F 2 and FF , the stress tensor T µν = ( 1 g 2 F µρ F ρ ν − trace), and a non-conserved operator in the representation (2, 0) ⊕ (0, 2) of rotations, i.e. a tensor with four indices and the same symmetry and trace properties of a Weyl tensor, for this reason we will denote it as W µνρσ . The three-point function in the bulk OPE channel is written as a sum of the bulk-boundary two-point functions between these operators and the displacement operator. Let us analyze which of these two-point functions can contribute. First of all, it is easy to see that two-point function between the conserved higher-spin currents and the displacement operator must vanish. This is an instance of the more general statement that in boundary CFTs bulk conserved currents J can only have non-zero two-point functions with a scalar boundary operatorÔ that has the same scaling dimension. The latter statement can be easily proved by placing the boundary operator at infinity, because in this case invariance under scaling and parallel translations force the two-point function to take the schematic form J(y, x)Ô(∞) = b JÔ 1 y ∆ J −∆Ô (structure) , (E.7) where "structure" denotes an appropriate tensor built out of the δ µν , the unit normal vector n µ and possibly epsilon tensors. Clearly when ∆ J = ∆Ô this two-point function cannot be compatible with current conservation unless the coefficient b JÔ vanishes. Moreover, rotational invariance (2.77) implies that also the operator W µνρσ has vanishing two-point function with the displacement. 15 Therefore, the only bulk operators that can contribute to the three-point function are the scalar operators and the stress-tensor. When the displacement is placed at infinity, the corresponding two-point functions are Since the two-point function between Wµνρσ(x) andD(∞) is a constant, the allowed structures are obtained by acting with this projector on constant four-tensors built out of δ and , such as: δ µ ρ δ ν σ , δ µ ρ δ ν y δ σ y , µ ν ρ σ , µ ν ρ y δ σ y . Applying the projector to any of these structures we find 0.

JHEP05(2019)091
Using the OPE (2.76) and the Ward identity (2.77) we can express the above two-point function coefficients in terms of the one-point function of the scalar operators, and of the coefficient CD in the two-point function of the displacement, namely [38,99,100] b F 2 ,D = − 32a F 2 π 2 , (E.12) (E.14) Since the two-point functions are constant, we can simply plug in the three-point function the leading bulk OPE, ignoring the descendants (and also ignoring the singular contribution from the identity that drops from the three-point function)

F Dimension of the boundary pseudo stress tensor
In section 4 we mentioned that the conservation of the stress tensor of the 3d CFT is violated at g = 0 due to multiplet recombination. At g = 0 we will call this operator boundary pseudo stress tensor. This is expected from the Ward identities derived in [100]. In this appendix we exploit this idea, to reproduce the one loop result of (4.16). We start from the boundary Lagrangian of a 3d Dirac fermion ψ where D a ψ = (∂ a − iA a )ψ and D aψ = (∂ a + iA a )ψ. The algebra of gamma matrices is {γ a , γ b } = 2δ ab . The pseudo boundary stress tensor is where the symmetrization includes a factor of 1/2. Note that the above operator is traceless as a consequence of the equations of motion: Requiring the divergence to cancel with −δ((Z ψ ) 2 )/ k, we obtain eq. (4.12).
• Anomalous dimension of O 0 : summing over all possible insertions in the given topology, the diagrams give Requiring the divergence to cancel with δ((Z ψ ) 2 Z 0 ), we obtain eq. (4.13).
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.