Symmetry group at future null infinity II: Vector theory

In this paper, we reduce the electromagnetic theory to future null infinity and obtain a vector theory at the boundary. We compute the Poincar\'e flux operators which could be generalized. We quantize the vector theory, and impose normal order on the extended flux operators. It is shown that these flux operators generate the supertranslation and superrotation. When working out the commutators of these operators, we find that a generalized electromagnetic duality operator should be included as the generators to form a closed symmetry algebra.


Introduction
The study of the gravitational waves led to the famous Bondi-Metzner-Sachs (BMS) group [1][2][3] at future null infinity (I + ) in asymptotically flat spacetime.Classically, the BMS group is the semi-direct product of the Lorentz group and supertranslations where C ∞ (S 2 ) denotes the smooth functions on the unit sphere S 2 .Supertranslation is an extension of the global spacetime translation in Minkowski spacetime.It generates an angledependent transformation of the retarded time at I + .Over the past decade, the BMS group has been extended to include the so-called superrotation transformations [4][5][6][7].Just like supertranslation, superrotation is a direct extension of the Lorentz rotation in Minkowski spacetime.
The Barnich-Troessaert (BT) superrotations are generated by local conformal Killing vectors on the celestial sphere while the Campiglia-Laddha (CL) superrotations [8,9] are generated by smooth vectors on the celestial sphere.
In [46], we present a systematic method to overcome these obstacles.This is achieved by projecting the massless quantum field theory of flat spacetime to I + and constructing the phase space of radiation solution.The outgoing Poincaré fluxes are completely determined by the radiation degrees of freedom.We can generalize the flux operators to define the corresponding supertranslation and superrotation generators.The supertranslation and superrotation form an infinite-dimensional group which could be identified as Carrollian diffeomorphism defined in [46][47][48] classically.The BMS algebra is recovered in the soft limit.Moreover, there is a higher dimensional Virasoro algebra with non-trivial central charge which follows from the time-dependent supertranslations.
In this work, we obtain a vector field theory by projecting the electromagnetic theory to I + .The vector field has only two independent propagating degrees of freedom A θ , A ϕ , where (θ, ϕ) are the spherical coordinates.We find the corresponding Poincaré flux operators and define the supertranslation and superrotation generators.Interestingly, to make the definition of the superrotation generators reasonable, one should generalize the Lie derivative variation to a covariant variation which is compatible with the metric at I + .We find the symmetry algebra at I + using the generators.In contrast to the real scalar field theory, one should include a generalized electromagnetic duality (EM duality) operator O g to form a closed symmetry algebra.
This paper is organized as follows.In section 2 we review the BMS group and explain the terminology used in this paper.In section 3, we introduce the ten Poincaré fluxes radiated to I + for electromagnetic theory.We quantize the theory at I + in the following section.In section 5, we introduce the concept of covariant variation and find a closed Lie algebra by including the new operator O g .In section 6, we interpret the new operator as a generalized EM duality operator.We discuss the antipodal matching condition in section 7 and conclude in section 8. Properties for the tensors on the sphere, canonical quantization of the vector field at I + , details about the calculation of commutators and Green's functions in electromagnetic theory are relegated to several appendices.

Review of the formalism
In this work, the Minkowski spacetime can be described in Cartesian coordinates x µ = (t, x i ), where µ = 0, 1, 2, 3 are spacetime coordinates and i = 1, 2, 3 are space coordinates.We also use the retarded coordinate system (u, r, θ, ϕ) and write the Minkowski spacetime as ds 2 = −du 2 − 2dudr + r 2 γ AB dθ A dθ B , A, B = 1, 2. (2.1) The future null infinity I + is a three-dimensional Carrollian manifold with a degenerate metric though the manifold I + should be described by three coordinates (u, θ A ).The spherical coordinates θ A = (θ, ϕ) are used to describe the unit sphere S 2 whose metric is 3) The covariant derivative ∇ A is adapted to the metric γ AB , while the covariant derivative ∇ µ is adapted to the Minkowski metric.Besides the metric (2.2), there is also a null vector which is to generate the retarded time direction.We will use the notation Ḟ ≡ ∂ u F for any function F .
In an asymptotically flat spacetime, transformations generated by the vector are called supertranslations.Similarly, transformations generated by the vector are called superrotations.Usually, the function f and (2.7) The metric of I + is preserved by supertranslations while is deformed by superrotations where There are six independent global solutions for the conformal Killing equation and they correspond to the Lorentz transformations in Minkowski spacetime.
The BMS transformation including (2.5) and (2.6) at the Carrollian manifold (2.12) Interestingly, the null structure of I + is preserved by a more general vector field [46] where f (u, Ω) could depend on the retarded time.The finite transformation corresponding to the vector field (2.13) is called Carrollian diffeomorphism.

Flux operators
In this section, we will study the radiation fluxes of electromagnetic theory.The electromagnetic vector potential is denoted as a U (1) gauge field a µ .The electric and magnetic fields are combined into an antisymmetric tensor More explicitly, the electric field e i and magnetic field b i are where the symbol ϵ ijk denotes the Levi-Civita tensor in three dimensions.We use the convention ϵ 123 = 1 in Cartesian coordinates.The action is where the last term involves a source j µ coupled to the field a µ and the source causes the electromagnetic radiation.The Maxwell equations are Usually, the source is located at a finite region of space.Therefore, we may set it to zero near I + .

Equations of motion
To solve the equations (3.4), we may impose the fall-off conditions for the vector potential near In terms of the retarded coordinates, We have used the following abbreviations A . (3.9) Switching to the retarded coordinates, we find The vector n i is the unit normal vector of S 2 by embedding it into the Euclidean space R 3 , The vectors Y A i are the three strictly CKVs of S 2 whose explicit form can be found in Appendix A.1.They are orthogonal to the normal vector and project the vector to the transverse direction.The equations (3.10) can be solved reversely This is equivalent to In this work, we only need the leading terms and the subleading terms A i (u, Ω). (3.20)In reverse, they are equivalent to and A (u, Ω). (3.24) The electric and magnetic fields are expanded asymptotically where The vector Y A i is related to the Killing vector Y A jk defined in Appendix A.1 by In the following, we will also call E i , E i , • • • the electric fields and B i , B i , • • • the magnetic fields at I + .Since the electric field e i and magnetic field b i are gauge invariant under gauge transformations, the electric fields E i , E i , • • • and magnetic fields B i , B i , • • • are also gauge invariant quantities at I + .Using the following properties we find that the leading electric field is orthogonal on shell to the leading magnetic field The same is true for the subleading order fields E i (u, Ω) and B i (u, Ω).
We can also write the electromagnetic fields in retarded coordinates From the equation of motion, we find The first equation (3.42) fixes the radial component of the vector potential up to a timeindependent function For the second equation (3.43), we may integrate it where φ(Ω) is also time-independent.Therefore, the on-shell electric and magnetic fields are Using the orthogonality relation in Appendix A.1, the electric field E i and magnetic field B i are orthogonal to the normal vector of S 2 at I + On the other hand, at the subleading order, the electric field E (2) i and magnetic field B (2) i are not orthogonal to the normal vector n i .We may define their radial components for later convenience.

Poincaré fluxes
The electromagnetic theory is invariant under Poincaré transformations, so there are ten corresponding Poincaré fluxes which are related to the conservation laws where The stress-energy tensor is It is easy to derive the following ten Poincaré fluxes.
• Energy flux • Momentum fluxes • Angular momentum fluxes At the last step, we have used the definition of E (2) and discarded the term φ(Ω) since it does not affect the total angular momentum radiated to null infinity.
• Center-of-mass fluxes At the last step, we have discarded the term involving φ(Ω) with the same reason as the case of the angular momentum fluxes.
From the Poincaré fluxes, we read out the energy flux density operator and the angular momentum flux density operator where we have defined a rank 4 tensor Useful properties of the tensor P ABCD are collected in Appendix B.
The data of the fluxes depend on angular directions as well as retarded time.Therefore, we may construct two smeared operators on without losing any information.Classically, the two operators are related to the fluxes radiated to I + .More explicitly, Obviously, T f should be regarded as a generalized Fourier transformation of the energy flux density T (u, Ω) on I + .It encodes the same information as the energy flux density operator when f can be any smooth function on I + .The test vector function Y A in M Y can also depend on the retarded time.Note that the test functions f and Y A here are not related to supertranslations and superrotations defined in the context of asymptotic symmetry analysis so far.However, we may use the terminology in [46] and distinguish the following four cases Special supertranslation (SST) Note that there is an ambiguity in the definition of the angular momentum flux density operator (3.61).To illustrate this problem, we may define a family of angular momentum flux density operators and the corresponding smeared operator To reproduce the angular momentum fluxes, we set Y A to be a Killing vector of S 2 which is independent of u.Using integration by parts the smeared operator M Y (λ) is independent of λ.The one-parameter family of the operators (3.70) shares the same classical meaning.We will fix the choice of λ in the next section.

Quantization
In the previous section, we found the Poincaré fluxes at I + .The densities T (u, Ω), M A (u, Ω) are classical objects so far.In this section, we will use the covariant phase space method [49,50] to quantize the densities T (u, Ω) and M A (u, Ω).For simplicity, we will use the radial gauge a r = 0.
The variation of the action (3.3) is given by a bulk term which is proportional to the equation of motion, and a boundary term where the volume form The presymplectic potential form is We can obtain the presymplectic form Using the fall off condition (3.6)-(3.8), the presymplectic form becomes Now it is straightforward to work out the fundamental commutators at where the Dirac function on the sphere reads out explicitly as and the function α(u − u ′ ) is defined as The commutators (4.6)-(4.8)have already been found in the literature [11,[51][52][53].In Appendix C, we use the standard canonical quantization method [54] to obtain the same answer.Similar to the scalar case, we find the following correlators ) We have defined a divergent function From this divergent function, we could find a finite result by considering the following difference For more details of the function β(u − u ′ ), we refer readers to [46].
After quantization, the densities T (u, Ω), M A (u, Ω) are quantum operators.We may refine their definition by using normal ordering Now the vacuum expectation values of these flux operators become zero Using the normal ordering, we find the following two-point functions The two-point functions have similar structure as those in the scalar theory.The divergent constant δ (2) (0) is the Dirac function (4.9) on the sphere with the argument equalling to 0. The tensor Λ (1) We use the subscript (1) to distinguish from the tensor Λ which has been defined in the scalar theory.The vanishing of the two-point function (4.21) indicates that M A is orthogonal to T .This corresponds to the operator (3.69) with For any other value of λ, the energy flux density operator is not orthogonal to the angular momentum flux density operator.The orthogonality condition fixes the value of λ uniquely.
5 Symmetry algebra at I + In this section, we first obtain the variation of the field A A generated by supertranslation and superrotation.The variation from superrotation is not compatible with the metric of I + .This motivates us to define a covariant variation of A A under superrotation so that we can identify the operators T f and M Y as supertranslation and superrotation generators, respectively.The symmetry algebra can be found in the last part of this section.

Covariant variation
The Lie derivative of the (co-)vector field a µ is From the fall-off conditions we can find the variation of the radiation degrees of freedom A A on I + .The result is collected as follows.
• When ξ = ξ f , the supertranslation variation of the vector field A A is (5.4) • When ξ = ξ Y , the superrotation variation of the vector field A A is Now the Lie derivative of the vector field a µ is From the fall-off conditions where we can also find the variation of (5.12) The supertranslation is compatible with the metric (5.13)However, the superrotation variation of the field A A is not covariant since its indices cannot be raised or lowered by the metric of S 2 This is expectable since the metric of S 2 is not invariant even for SSRs, which has been shown in equation (2.9).Now we try to define a covariant variation δ / for the field on I + .This is denoted by for supertranslations and for superrotations.The • • • in parenthesis is any well-defined field on I + .For example, the supertranslation and superrotation for the scalar field Σ in [46] are respectively.For the vector field, these are denoted as We use a slash to distinguish it from the original variation induced by Lie derivative.From (5.13), there is no need to modify the variation of the vector field under supertranslation For superrotations, the covariant variation should satisfy the following conditions • Linearity.For any vector fields Y A and Z A and any constants c 1 , c 2 , Also, for any two fields F 1 and F 2 of the same type, we require (5.21) • Leibniz rule.For any two fields F 1 and F 2 on I + , their tensor product should obey the Leibniz rule (5.22) • Metric compatibility.The covariant variation of the metric should be zero (5.23) • For the scalar field Σ, the variation is the variation induced by Lie derivative From the linearity condition, a possible definition of δ / Y A A may be The rank 2 tensor Γ C A (Y ) should be linear in Y and independent of A A .Now using the Leibniz rule and the condition (5.24), we should find (5.27) We have defined Γ AB (Y ) with lower indices as (5.28) The metric compatibility condition (5.23) implies (5.29) We decompose the connection into symmetric and antisymmetric part (5.30) The metric compatibility condition fixes the symmetric part (5.31) The antisymmetric part should be proportional to the Levi-Civita tensor ϵ AB where Υ(Y ) is an arbitrary linear function of Y .To remove this ambiguity, we may require the connection Γ AB (Y ) to be symmetric Then the connection is uniquely fixed to (5.34) The symmetric connection is a rank 2 traceless tensor.Therefore, one can use the metric γ AB to raise and lower its indices.For example, In the following, we will use the covariant variation whose connection is symmetric.As a consequence of the definition, we find δ / Y γ AB = 0, δ / Y ϵ AB = 0. (5.36) Due to the nice property of the covariant variation, we may regard the transformation δ / Y A A as the "real" superrotation of the vector field.The variation δ Y A A induced by diffeomorphisms is only partial variation of the superrotations.

Supertranslation and superrotation generators
There is another variation defined by the commutators where (5.39) The rank 4 tensor ρ BCDA is defined as (5.40) Interestingly, (5.37) is exactly the variation of the vector field A A under the supertranslation (5.4) up to a constant factor.Moreover, after some algebra, we could find When the vector Y A is time-independent, the non-local part vanishes.In this case, we find the confusing inequality This is contradictory to the scalar case where Fortunately, the problem is cured by the covariant derivative Therefore, we find the supertranslation and superrotation generators.
• Supertranslation generators T f .It is the smeared operator of the energy flux density operator T (u, Ω).
• Superrotation generators T 1 2 u∇ A Y A + M Y .Using the same convention as scalar theory, we will call M Y as the superrotation generator.It is a smeared operator of the angular momentum flux density operator M A (u, Ω).
We should emphasize that the covariant variation δ / is necessary for the identification.

Symmetry algebra of flux operators
It is straightforward to find the following commutators This is not a standard Lie algebra.There are two new operators and on the right-hand side of the commutators, where ϵ BC is the anti-symmetric tensor on the sphere The first operator Q g could compare to the one in the scalar theory.The second operator O g is new.Note that the operators Q and O disappear in (5.46) when Ẏ = 0.However, the operator O still appears on the right hand of (5.47) even for Ẏ = 0. To be more precise, the function is zero only when Y or Z is a CKV.We leave the discussion on this new operator in the next section.In this section, we just calculate its commutator with and the following two-point correlators ) ) where the operator O(u, Ω) = ϵ AB : ȦB A A :. Now it is straightforward to find3 We will discuss the commutators in the following.
• Central charges.The central extension terms can be derived from the two-point functions (4.20)-(4.22)and (5.53)-(5.55).There are four non-vanishing central extension terms where the function and the identity operator have already been defined in the scalar theory.We use a constant c to denote the divergent part c = δ (0) (0). (5.68) • Virasoro algebra.By transforming to the Fourier space, the equation (5.56) implies a higher dimensional Virasoro algebra (5.69) The constants c ℓ,m;ℓ ′ ,m ′ ;L,M are Clebsch-Gordan coefficients.There are two propagating degrees of freedom in the vector theory, hence the central term is two times compared to the real scalar theory.
• The reader can find more details in [46].

Closed algebra
Due to the existence of non-local terms and the physically meaningless operator Q h , the aforementioned algebra (5.56)-(5.61) is not closed.As we have shown, requiring (5.70) will make these annoying terms vanish, and hence lead to a closed algebra.
Truncation I.By imposing the condition (5.70), we find the following truncated algebra (5.76) This is an enlarged algebra compared to the one found in the scalar theory.The Jacobi identities are checked in Appendix A.2.In the scalar theory, the operator T f generates GSTs and M Y generates SSRs.The corresponding group is where the notation Diff(S 2 ) means that the vectors Y A (Ω) generate diffeomorphisms of S 2 and C ∞ (I + ) means that f is any smooth function on I + .In the vector theory, although the term i 2 O Ẏ A ∇ C f ϵ CA in (5.57) now vanishes, the appearance of operator O on the right-hand side of (5.73) indicates that the enhancement of the group (5.77) is unavoidable.
One may be interested in this new operator.As we will show in the next section, this operator can be derived from the electromagnetic duality transformations.It is amazing that the commutator of superrotations will produce a term reflecting internal symmetry.However, we must point out that the superrotation flux operators do not agree with ordinary variation when acting on vector fields.As is shown in (5.2), for Ẏ = 0, we have where ∆ A ′ (Y ; A; u ′ , Ω ′ ) is given by (5.41).Note that it is not the one induced by Lie derivative.Namely, the superrotation variation needs to be corrected according to the principle of covariant variation.Besides, O g generates original EM duality transformations when g is a constant.In this case, the right-hand side of (5.75) vanishes.It makes sense since original EM duality is expected to have nothing to do with geometric transformations.
Truncation II.To eliminate the operator O, we should require This implies that Y, Z are CKVs.In this case, the truncated algebra becomes (5.82)This is the Newmann-Unti group with a central charge (5.83) As one expects, we obtain a geometric algebra.It is consistent with the fact that the connection (5.34) vanishes for Y being a CKV.In other words, for Lorentz transformations, the covariant variation agrees with the one induced by Lie derivative, and there are no additional terms in the algebra.
Other truncations.One may further truncate the aforementioned Newmann-Unti group to the one of level k as has been done in [46].Besides, one can also demand ḟ = 0 and Y, Z to be CKVs, which leads to the BMS algebra.

Electromagnetic duality operator
The electromagnetic duality (EM duality) is a symmetry transformation for sourceless Maxwell equation [55][56][57][58][59]. Duality invariance of Maxwell equation leads to the introduction of magnetic monopole and the quantization of electric charge [60].It has been elaborated in non-Abelian gauge theories by [59].One can find more details in [61].
EM duality transformation exchanges the role of the electric and magnetic field where φ is a constant.The SO( 2) rotation (6.1) is equivalent to the following phase transformation where At I + , this reduces to with Since the duality transformation (6.1) is a continuous global symmetry, there may be a corresponding conserved current by Noether's theorem.However, it is easy to check that the original action is not invariant under the EM duality transformation.To find the corresponding current, we may introduce a dual EM vector field a µ and its corresponding EM field f µν The dual field f µν is invariant under the dual gauge transformation Following [62], we write a symmetric action for (sourceless) electromagnetic theory Treating the vector fields a µ , a µ as independent quantities, the equation of motions from the symmetric action (6.10) become .11)They are equivalent to the sourceless Maxwell equations with an additional constraint 12) The constraint relates the field f µν to the Hodge dual of the field f µν .Note that the symmetric action is not equal to the original action (6.7) when the constraint is imposed.Nevertheless, it is invariant under EM duality transformation and turns out to be useful to derive the corresponding conserved currents.The EM duality transformation may be expressed as The conserved current for the EM duality could be found in [62,63] by using Noether's theorem The conserved charge corresponding to the current is called optical helicity.At the microscopic level, this is the difference between the number of the photons with left helicity and right helicity.To obtain the relation between vector field and its dual, we may first use the dual gauge transformation (6.9) to fix a r = 0. (6.16) In retarded coordinates, we impose the fall-off condition for the dual field .17) To satisfy the constraint condition (6.12), we should identify This is a Hodge dual on the unit sphere.Now we may use the conservation of the EM duality current ∂ µ j µ em = 0 (6.20) to find the EM duality fluxes which is radiated to Obviously, this looks like the second new operator (5.49) and that is why we discuss the EM duality transformation.Actually, the EM duality transformation and its generator have been studied from other starting points, and we refer readers to [64][65][66][67].
Now we define the EM duality flux density operator and use it to construct the smeared operator .23)The commutators between O g and F i , Fi are [O g , Fi ] = g Fi + 1 2 ġ Ḡi .(6.25) where Ġi = F i , Ġi = Fi .(6.26) • When g is a constant, the transformation (6.24)-(6.25) is exactly the infinitesimal EM duality transformation (6.4).
• When g is a time-independent function the transformation (6.24)-(6.25)would be angle-dependent.This is a generalized EM duality transformation at I + .
• When g is time-dependent, the additional terms in (6.24)-(6.25)obscure the interpretation of the operator.

Antipodal matching condition
The symmetry group can also be discussed at past null infinity (I − ).The fall-off conditions (3.5) can also be expressed near where v = t+r is the advanced time.The supertranslation and superrotation generators depend on the first two leading orders of the vector potential.We may use the large-r expansion of the spherical Bessel function of the first kind and the mode expansion (C.2) to find with The creation and annihilation operators are related by In frequency space, we have where Ω P is the antipodal point of Ω The electric and magnetic fields near I − are expanded as where In Fourier space, the electric and magnetic fields ( i (ω, Ω), k (ω, Ω), Using the relations and the matching condition (7.12), the antipodal matching condition for the electric and magnetic fields is The antipodal matching condition can also be checked using Green's functions.One can find the details in Appendix E.

Conclusion and discussion
In this paper, we reduce the electromagnetic field theory in Minkowski spacetime to future null infinity I + .The boundary vector theory is characterized by a single vector field A A with a non-trivial symplectic form.The ten Poincaré fluxes are totally determined by the field A A .We obtain the flux operators and interpret them as supertranslation and superrotation generators.Interestingly, one should define a covariant variation to identify the superrotation generators.
The supertranslation and superrotation flux operators do not form a closed algebra for GSTs and GSRs.In contrast to the scalar field theory, the GSTs and SSRs cannot form a closed group.One should introduce a new operator which corresponds to a generalized EM duality transformation at I + .By combining the GSTs, SSRs as well as the generalized EM duality transformations, we could find a closed group whose Lie algebra has been given in (5.71)-(5.76).
There is a no-go theorem which is presented in [38] recently.It states that the conformal symmetry of the holographic theory on the celestial sphere must not be extended to diffeomorphism symmetry since the Diff(S 2 ) implies the conservation of the conformal spin.Our work bypasses the no-go theorem in two ways.At first, the vector theory that we find is free at I + .Secondly, the diffeomorphism is intertwined with the EM duality transformation for the boundary vector theory.The EM duality symmetry is broken in the interacting theory.Therefore, the symmetry group we find in this work may break when there is interaction.It would be interesting to explore the interacting vector theory in the future.
There are various open questions in this direction.
• Covariant variation.The introduction of the covariant variation δ / is rather interesting.There is a natural variation δ at I + which is induced by the diffeomorphism from the bulk.The variation δ has a direct geometric meaning.However, it is not always metric compatible.To cure this problem, we define a connection Γ AB such that the covariant variation is δ / = δ + connections (8.1) schematically.This is quite similar to the definition of the covariant derivative ∇ in general relativity We note the connection term Γ AB is proportional to the variation of the metric γ and it is non-zero when the superrotation vector is not a CKV.The consequence of the covariant variation is that the commutator between two superrotation generators is not a superrotation generator.Equation (5.73) may be read schematically as [superrotation, superrotation] = superrotation + generalized EM duality.(8.3)Using the notation We have introduced a formal curvature tensor R(Y, Z) similar to the case of covariant derivative.It is rather interesting to understand why the generalized EM duality operator is related to the curvature tensor.
• Field theories on the Carroll manifold I + .The field theory on I + may provide an explicit realization of flat holography.Carrollian diffeomorphism has a direct geometric meaning which is enough for constructing scalar theory.Our result implies that Carrollian diffeomorphism Diff(S 2 ) ⋉ C ∞ (I + ) is not the end of the story for theories with non-zero spin.The most intriguing case would be to project the gravitational theory to its boundary.We will present the result in the near future.
• Large gauge transformation.Besides the diffeomorphism, the electromagnetic theory is also invariant under U (1) gauge transformation.The gauge invariance is broken at I + and part of the gauge transformations become large gauge transformations.As a consequence, there is an infinite-dimensional algebra [68][69][70][71][72][73][74][75][76] at the boundary.Our work shows that there is also an extended algebra coming from diffeomorphism.Therefore, it may be natural to combine the two results [77] in the future.
• Divergences.There are two kinds of divergences appearing in the context.The first one is about the correlation function of two fields, i.e. ⟨0|A A A B |0⟩.It is divergent due to the appearance of β(u − u ′ ).Taking time derivative will eliminate this divergence, and the same is true for taking the difference, as we have shown in (4.16).Actually, we could deal with the divergence of β(u − u ′ ) in two different ways.The first one is similar to dimensional regularization, i.e., adding an infinitesimal parameter κ such that to order O(κ 0 ), we have with γ E denoting Euler constant.The divergent term 1 4πκ and constant terms may be absorbed into a constant − 1 4π log ω 0 , and we find a finite result The second way is to introduce an infrared cutoff ω ′ 0 → 0 for the integral which is similar to the Pauli-Villars regularization.This makes sense since the divergence comes from integration in the region of little ω.It follows that to order O(1), we have We have absorbed the first term and constant terms to a constant − 1 4π log ω 0 again.These two ways lead to the same result, and they both agree with the fact that the time derivatives or the difference of β(u − u ′ ) are finite.
Secondly, there is a divergent factor c = δ (2) (0) in some central charges, which comes from two Dirac delta functions in the angular direction, appearing in the four-point correlators.As we have analyzed in the scalar theory [46], this function can be obtained from the summation of spherical functions with the same arguments, i.e., δ (2) Ω). Making use of the addition theorem, we find The denominator 4π equals the area of a unit sphere, and thus δ (2) (0) can be interpreted as the state density on unit sphere.Moreover, as a naive method, one may use Riemann zeta function to regularize δ (2) (0).From the classic evaluations ζ(−1) = −1/12, ζ(0) = −1/2, one gets a finite value δ (2) (0) = 1 12π .
We list the related properties in the following.
1. Commutators.The six CKVs form a closed lie algebra which is isomorphic to so(1, 3) 2. Relations.The three Killing vectors and the three strictly CKVs are related to each other by the identities The vector Y A i , Y A i are related by the Levi-Civita tensor From these, it is easy to find 3. The six CKVs are related to the metric γ AB and δ ij by 4. The vector Y A i is orthogonal to the normal vector n i For the vector Y A i , we also find 6.There are also some useful identities involving the Levi-Civita tensor • For any three smooth vectors X A , Y A , Z A on S 2 , we can find the following identity The function o(Y, Z) is defined in (5.51).This identity is useful to check the Jacobi identity

B Properties of the tensor P ABCD
The properties of the rank 4 tensor P ABCD are collected in the following.
• Symmetries P ABCD = P BADC = P BDAC = P DBCA = P CDAB = P DCBA = P ACBD = P CADB .(B.1) • Traces This identity follows from the Fierz identity • The tensor P ABCD can also be written as As a consequence, one can find

C Canonical quantization
In perturbative quantum field theory, by imposing the Lorenz gauge the electromagnetic field a µ may be quantized using annihilation and creation operators b where the vector k is the momentum and ω is the energy of the corresponding mode.For a massless particle, we have The indices µ denote the spacetime coordinates and α is the polarization index.The polarization vector ϵ α µ (k) has two physical degrees of freedom They also satisfy the completeness relation where with the normal vectors in momentum space The annihilation and creation operators satisfy the standard commutation relations By expanding the plane wave into spherical waves, the propagating modes A A are Then we find the commutators at Notice that the second term on the right-hand side of (C.15) will not contribute to the commutators due to n i Y A i = 0.In fact, the first term is just the usual δ(ω − ω ′ )δ ℓ,ℓ ′ δ m,m ′ .We therefore get the desired commutators.

D Commutators
We will show the derivation of the commutators (5.56)- (5.61).Take the commutator [M Y , M Z ] as an example.We rewrite the superrotation generator as where

E Green's functions
The antipodal matching condition can also be checked using Green's functions of Maxwell equation ∂ µ f µν = −j ν .(E.1) The vector potential can be solved in Lorenz gauge using retarded Green's function a µ (t, x) = a in µ (t, x) + a ret µ (t, x), (E.2) where the retarded solution is The ingoing wave a in (t, x) is determined by imposing the initial conditions at I − .The vector potential can also be represented in terms of advanced Green's function a µ (t, x) = a out µ (t, x) + a adv µ (t, x). (E.4) The advanced solution is and the outgoing wave is denoted as a out µ (t, x).The radiation field is the difference between the outgoing wave and the ingoing wave [78] a rad µ (t, x) = a out µ (t, x) − a in µ (t, x) = a ret µ (t, x) − a adv µ (t, x).where Ω P is the antipodal point of Ω Ω P = (π − θ, π + ϕ).

(E.15)
There is an antipodal matching condition in frequency space A µ (ω, Ω) = −A − µ (ω, Ω P ).(E.16) The radiation electric and magnetic fields are (ω, Ω P ), B i (ω, Ω) = B −(2) i (ω, Ω P ).(E.27) Non-local terms.There are three non-local terms in (5.59)-(5.61).The non-local terms introduce new operators in the commutators.It is understood that the new operators are also normal ordered.Interestingly, the non-local term in (5.59) has the same structure as scalar theory.It would be interesting to explore the physical origin of this fact.There is also an interesting truncation by setting Ẏ = Ż = ġ1 = ġ2 = 0. (5.70)In this case, all the non-local terms and the central terms C M , C O , C M O are vanishing.