Asymptotic $\mathcal O(r)$ gauge symmetries and gauge-invariant Poincar\'e generators in higher spacetime dimensions

TL;DR

This paper extends the understanding of asymptotic symmetries in higher-dimensional electromagnetism, revealing a rich algebraic structure with central charges and a method to define gauge-invariant Poincaré generators across all dimensions $d \,\geq\, 4$.

Contribution

It introduces a consistent set of angle-dependent $u(1)$ gauge transformations in higher dimensions and constructs gauge-invariant Poincaré generators by decoupling $u(1)$ charges from the Poincaré algebra.

Findings

Asymptotic symmetry algebra includes a six-fold set of angle-dependent $u(1)$ transformations.

Presence of central charges in the algebra of asymptotic symmetries.

A nonlinear redefinition of Poincaré generators achieves gauge invariance in all $d\geq4$.

Abstract

The asymptotic symmetries of electromagnetism in all higher spacetime dimensions $d>4$ are extended, by incorporating consistently angle-dependent $u(1)$ gauge transformations with a linear growth in the radial coordinate at spatial infinity. Finiteness of the symplectic structure and preservation of the asymptotic conditions require to impose a set of strict parity conditions, under the antipodal map of the $(d-2)$-sphere, on the leading order fields at infinity. Canonical generators of the asymptotic symmetries are obtained through standard Hamiltonian methods. Remarkably, the theory endowed with this set of asymptotic conditions turns out to be invariant under a six-fold set of angle-dependent $u(1)$ transformations, whose generators form a centrally extended abelian algebra. The new charges generated by the $\mathcal O(r)$ gauge parameter are found to be conjugate to those associated to the now improper subleading $O(r^{-d+3})$ transformations, while the standard $\mathcal O(1)$ gauge transformations are canonically conjugate to the subleading $\mathcal{O}(r^{-d+4})$ transformations. This algebraic structure, characterized by the presence of central charges, allows us to perform a nonlinear redefinition of the Poincar\'e generators, that results in the decoupling of all of the $u(1)$ charges from the Poincar\'e algebra. Thus, the mechanism previously used in $d=4$ to find gauge-invariant Poincar\'e generators is shown to be a robust property of electromagnetism in all spacetime dimensions $d\geq 4$.

Equations

$$I_{H}[A_{i},\pi^{i},A_{0},\pi^{0};\psi,\lambda]=\int dt\int d^{d-1}x\Big{[}\pi^{i}\dot{A}_{i}+\pi^{0}\dot{A}_{0}-(\mathcal{H}+\tilde{\epsilon}\mathcal{G}+\tilde{\mu}\pi^{0})-\psi\mathcal{G}-\lambda\pi^{0}\Big{]}+B_{\infty}\,,\\$$

$$\mathcal{H}=\frac{1}{2\sqrt{g}}{\pi }^i{\pi }_i+\frac{\sqrt{g}}{4}{F}^{ij}{F}_{ij}+A_0\mathcal{G}-{\partial }^i{\pi }^0{A}_i.$$

$$\mathcal{G}=-{\partial }_i{\pi }^i=0,{\pi }^0=0,$$

$${\delta }_{ϵ}{A}_i={\partial }_iϵ,{\delta }_i{A}_0=\mu .$$

$$\xi$$

$${\xi }^i$$

$${\delta }_{\xi ,{\xi }^i}{A}_i$$

$${\delta }_{\xi ,{\xi }^i}{\pi }^i$$

$${\delta }_{\xi ,{\xi }^i}{A}_0$$

$${\delta }_{\xi ,{\xi }^i}{\pi }^0$$

$${\mathcal{L}}_{\xi }{A}_i$$

$${\mathcal{L}}_{\xi }{\pi }^i$$

$$\widetilde{ϵ}$$

$$\tilde{\mu }$$

$${\widetilde{ϵ}}^{(d-4)}$$

$${\tilde{\mu }}^{(d-4)}$$

$${\tilde{\mu }}^{(d-3)}$$

$$A_0$$

$$A_i$$

$$\Xi$$

$$\Phi$$

$$A_0$$

$$A_r$$

$$A_A$$

$${\pi }^0$$

$${\pi }^r$$

$${\pi }^A$$

$$\psi =\frac{{\psi }^{(1)}}{r}+\mathcal{O}\left(r^{-2}\right),\lambda =\frac{{\lambda }^{(1)}}{r^2}+\mathcal{O}\left(r^{-3}\right),$$

$$\text{vbox}{\text{hrule}height=0.5ptA}_r$$

$$\text{vbox}{\text{hrule}height=0.5ptA}_A$$

$${\Psi }_{\text{lin}}$$

$${\partial }_A\text{vbox}{\text{hrule}height=0.5pt\pi }^A=0,{\partial }_A{\pi }_{(2)}^A-{\pi }_{(2)}^r=0.$$

$$B_{\infty }=\int dt{\oint }_{S_{d-2}^{\infty }}{d}^{d-2}x\left(\mathcal{S}-\mathcal{B}\right),$$

$$\mathcal{S}$$

$$\mathcal{B}$$

$$\displaystyle\int dtd^{d-1}x\,\pi^{\mu}\dot{A}_{\mu}=\int dtdr\oint d^{d-2}x\Big{[}$$

$$\displaystyle+\frac{1}{r}\left(\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{A}\partial_{A}\dot{\Phi}+\pi_{(2)}^{r}\dot{\Phi}_{\text{lin}}+\pi_{(2)}^{A}\partial_{A}\dot{\Phi}_{\text{lin}}\right)+\mathcal{O}\left(r^{-2}\right)\Big{]}\,.$$

$${\dot{\pi }}^i=-\sqrt{g}{\nabla }_i{F}^{ij}+{\nabla }^i{\pi }^0.$$

$${\dot{\Psi }}_{\text{lin}}$$

Full text

1 Introduction

In light of the very recent results for gravity and electromagnetism Fuentealba:2022xsz ; Fuentealba:2023rvf , where it was possible to extend the asymptotic symmetries of the theories by consistently accommodating logarithmic branches and a linear $u(1)$ symmetry (namely, generated by an $\mathcal{O}(r)$ gauge parameter) at spatial infinity, we address the direct question about whether similar extensions can be found in the case of electromagnetism in higher spacetime dimensions $d>4$.

In the case of four-dimensional gravity in Fuentealba:2022xsz , by following the approach in Henneaux:2018cst ; Henneaux:2018hdj ; Henneaux:2019yax , a logarithmic behaviour in the fall-off of the metric was included. This enabled us to extend the renowned standard BMS algebra Bondi:1962px ; Sachs:1962wk ; Sachs:1962zza by a new family of angle-dependent logarithmic supertranslations. The latter together with standard supertranslations, were shown to satisfy an abelian algebra with non-trivial central charges. Now, it is precisely the presence of these central charges which allowed, through a suitable nonlinear redefinition of the Lorentz generators, to disentangle the whole set of pure supertranslations (standard and logarithmic ones) from the Lorentz algebra, finding in this manner a resolution to the angular momentum ambiguity under supertranslations at spatial infinity. Note that different mechanisms have been developed at null infinity Mirbabayi:2016axw ; Bousso:2017dny ; Javadinezhad:2018urv ; Javadinezhad:2022hhl (see also Chen:2021szm ; Chen:2021zmu ; Compere:2023qoa and references therein).

Analogue enhancements were explored in electromagnetism. Concretely, a relaxation of the asymptotic conditions of Henneaux:2018gfi was performed in Fuentealba:2023rvf . As a result of this analysis, it was consistently accommodated not only a logarithmic but a linear growth (associated to the subleading soft photon theorems Lysov:2014csa ; Campiglia:2016hvg ; Conde:2016csj ) for the gauge potential at infinity. As in the case of gravity, central charges emerged between different types of angle-depedent gauge transformations, to wit, linear, logarithmic, standard and subleading $u(1)$ transformations. It was then possible to redefine the Poincaré generators, in such a way that the asymptotic symmetry algebra becomes the direct sum of the Poincaré algebra and a centrally extended abelian algebra associated to the $u(1)$ symmetries.

Then, it is natural to wonder whether a similar extension of the asymptotic symmetries can be performed in higher dimensions. Spatial infinity yields the perfect arena to address this question, since one can avoid the complications associated to the fractional decay in the radial coordinate of the radiative branch of the fields (in odd dimensions), which could lead to problems with the conformal definition of null infinity Hollands:2003ie ; Hollands:2004ac ; Tanabe:2011es . Indeed, in the exhaustive study by Henneaux and Troessaert in Henneaux:2019yqq , it was shown how to extend the Hamiltonian formulation of electromagnetism in dimensions $d>4$, such that a two-fold family of angle-dependent $u(1)$ transformations emerges at spatial infinity (see also Esmaeili:2019hom , where stronger conditions are imposed, such that only one angle-dependent $u(1)$ symmetry is found). Unlike the $d=4$ case, parity conditions on the fields are not needed (being not inconsistent to impose them though, according to Henneaux:2019yqq ). Indeed, they explicitly integrated the equations of motion at spatial infinity and then performed the match with null infinity, which led to non divergent terms in the asymptotic behaviour of the fields without imposing any type of parity conditions. This latter fact enabled them to find generalizations of the matching conditions between the fields at the past of future null infinity $\mathscr{I}^{+}_{-}$ and the future of past null infinity $\mathscr{I}^{-}_{+}$.

Here, we report a set of boundary conditions for electromagnetism in higher dimensions that are invariant under an improper (or large) $u(1)$ gauge parameter growing linearly at spatial infinity (by “improper” we refer to an asymptotic gauge symmetry with a non vanishing charge, according to the terminology introduced in Benguria:1976in ). This is achieved by imposing parity conditions on the asymptotic behaviour of the fields under the antipodal map of the $(d-2)$-sphere. We have found that the asymptotic symmetries are given by a six-fold family of angle-dependent $u(1)$ transformations, whose canonical generators satisfy an abelian algebra endowed with non-zero central charges.

We recall that in the analysis of Henneaux:2019yqq , absence of magnetic sources was assumed in order to tame the Lorentz boosts. However, we shall show that this last assumption can be relaxed by taking a different boundary contribution to the symplectic form, which is necessary for the further extension that allows the linear growth in the gauge potential. In fact, in the absence of the linear behaviour, this reformulation of the theory leads to a four-fold family of angle-dependent $u(1)$ transformations, since subleading transformations generated by a parameter with decay $\sim r^{-d+4}$ become improper (in contrast with the two angle-dependent $u(1)$ symmetries found in Henneaux:2019yqq ). The corresponding canonical generators are shown to form a centrally extended abelian algebra. It must be mentioned that the role of the logarithmic gauge transformations in $d=4$ found in Fuentealba:2023rvf is now played by the improper subleading $\mathcal{O}(r^{-d+4})$ gauge transformations in all higher dimensions $d>4$ (note that these subleading $u(1)$ symmetries turn out to be analogue to the subleading supertranslations in higher dimensions found in Fuentealba:2021yvo ; Fuentealba:2022yqt ).

Consistency does not require any parity condition so far. However, once the linear relaxation of the gauge potential is included, finiteness of the symplectic structure requires to impose parity conditions on the leading order fields. In this formulation (with the linear growth), the gauge parameter with subleading decay $\sim r^{-d+3}$ possesses a nonvanishing charge. Hence, the system is invariant under a six-fold family of angle-dependent $u(1)$ transformations. The algebra of the asymptotic symmetries acquires an additional nonvanishing central charge along the bracket of the new linear $u(1)$ charges and those generated by subleading $O(r^{-d+3})$ gauge transformations. Note that the required parity conditions do not discard any known solution of the theory.

Our generalized set of boundary conditions enables us to apply the mechanism that decouples all of the improper $u(1)$ gauge transformations (linear, standard and the subleading ones) from the Poincaré algebra (given the presence of invertible central charges among the $u(1)$ canonical generators). This explicitly shows that this decoupling, which transforms the asymptotic symmetry algebra in the direct sum form, is a general property of electromagnetism in all higher spacetime dimensions rather than an accident in $d=4$.

The plan of the paper goes as follows. In Section 2, the extended Hamiltonian formulation for electromagnetism in higher dimensions is presented. This comprises the asymptotic behaviour and parity conditions of the fields that ensure a well-defined action of the Poincaré symmetry. Finiteness of the symplectic structure and equations of motion are also discussed. We are then on solid grounds to apply standard Hamiltonian methods for the computation of the canonical generators of the gauge symmetries, which lead to finite and integrable charges, an analysis that is performed in Section 3. The canonical realization of the Poincaré symmetry is carried out in Section 4, where we obtain that the asymptotic symmetry algebra is given by the semi-direct sum of the Poincaré algebra and a centrally extended abelian algebra. In Section 5, we decouple all of the gauge transformations from the Poincaré algebra through an appropriate redefinition of the Poincaré generators. Section 6 is devoted to some concluding remarks. In Appendix A, we construct the symplectic form of the theory and prove Poincaré invariance. Finally, in Appendix B, we give the complete list of the transformation laws of the fields obtained from the preservation of the asymptotic conditions under the Poincaré symmetry.

2 Action principle and boundary conditions

In order to accomplish a consistent relaxation of the boundary conditions that are invariant under angle-dependent $u(1)$ transformations, we consider from the beginning the extended Hamiltonian formulation of the theory. Indeed, this is the action that one obtains after direct application of the Dirac formalism for systems with gauge symmetries Dirac1967 ; Henneaux:1992ig . This has been shown to be useful in previous treatments of the problem in the context of electromagnetism, see e.g. Henneaux:2018gfi ; Henneaux:2019yqq ; Fuentealba:2023rvf . The extended Hamiltonian action principle in $d$ spacetime dimensions on a Minkowski background reads

[TABLE]

where

[TABLE]

The phase space is described by the conjugate canonical pairs $(A_{i},\pi^{i})$ and $(A_{0},\pi^{0})$. The surface term at spatial infinity $B_{\infty}$ depends on the boundary conditions and its precise form will be shown below (once we provide our set of asymptotic conditions). Variation with respect to the Lagrange multipliers $\psi$ and $\lambda$ enforces the constraints

[TABLE]

respectively. The theory is thus invariant under the following gauge transformations

[TABLE]

The vector field that generates Poincaré transformations is

[TABLE]

The arbitrary constants $b_{i}$, $b_{ij}=-b_{ji}$, $a^{\perp}$ and $a^{i}$ parametrize Lorentz boosts, spatial rotations and standard translations, respectively. Note that the term $-b^{i}x^{0}$ that comes from $\xi^{i}$ can be eliminated by a spatial translation (at any given time).

The infinitesimal transformation laws of the fields under Poincaré deformations Henneaux:2019yqq (see also Henneaux:2018gfi ) are given by

[TABLE]

where the spatial Lie derivatives on the fields along $\xi^{i}$ read

[TABLE]

Note that the action of the Poincaré group is canonical only if we apply the additional corrective gauge transformations generated by $\epsilon_{(\xi,\xi^{i})}$ and $\mu_{(\xi,\xi^{i})}$. The precise form of these parameters will be shown when we discuss the canonical realization of the Poincaré symmetry in Section 4.

Consistency of the formulation for the boundary conditions imposed here, requires that Hamiltonian density in (1) is given by the sum of the usual term $\mathcal{H}$ in (2) and additional terms proportional to the constraints $\mathcal{G}$ and $\pi^{0}$, where

[TABLE]

with

[TABLE]

Concretely, these terms must be added due to the corrective gauge transformations necessary to have a canonical action of the Poincaré symmetry. The coefficients \hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0ptA\kern 0.0pt}}}_{A}, \hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0ptA\kern 0.0pt}}}_{r}, $\Phi^{(d-5)}$, $\Psi^{(d-5)}$ and $\Psi^{(d-5)}$ are arbitrary functions of the $(d-2)$-sphere at spatial infinity, which appear in the radial fall-off of the canonical fields as we shall show below.

Following the approach introduced in Henneaux:2018hdj ; Henneaux:2019yax (see also Henneaux:2018gfi ; Fuentealba:2023rvf for the Maxwell theory in $d=4)$, we will deform the usual decay of the gauge potential $(\tilde{A}_{0},\tilde{A}^{i})\sim(r^{-1},r^{-d+3})$ (obtained from the fall-off of the electric and magnetic fields at spatial infinity $\sim r^{-d+2}$), by improper gauge transformations, namely,

[TABLE]

where we will impose that

[TABLE]

Thus, in polar coordinates this fall-off reads

[TABLE]

while the one for the conjugate momenta (of weight density one) goes as follows

[TABLE]

The coefficients in the radial expansions of the fields are functions of the angles \hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0ptx\kern 0.0pt}}}^{A} of the round unit $(d-2)$-sphere at spatial infinity. The indices $A,B,C,\dots$ are raised and lowered by the metric \hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0ptg\kern 0.0pt}}}_{AB} on the $(d-2)$-sphere, where \hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0ptD\kern 0.0pt}}}_{A} stands for its associated covariant derivative.

The fall-off of the Lagrange multipliers is chosen to be the same as the proper gauge transformations (which are established in Section 3), namely,

[TABLE]

with $\psi^{(d-4)}=\psi^{(d-3)}=0$ and $\lambda^{(d-4)}=\lambda^{(d-3)}=0$.

Finiteness of the symplectic structure and preservation under Poincaré symmetry require that the coefficients in the asymptotic behaviour of the fields must satisfy the following parity conditions under the antipodal map of the $(d-2)$-sphere (\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0ptx\kern 0.0pt}}}^{A}\rightarrow-\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0ptx\kern 0.0pt}}}^{A}),

[TABLE]

Note that in addition to the strict parity conditions obeyed by the leading order fields, the “core” fields, i.e., leading order fields for the usual decay (already denoted by $(\tilde{A}_{0},\tilde{A}^{i})$ in (18) and (19), respectively), are subject to a class of twisted parity conditions as \hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0ptA\kern 0.0pt}}}_{r} and \hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0ptA\kern 0.0pt}}}_{A} do not have a definite parity. If we set $d=4$, \hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0ptA\kern 0.0pt}}}_{r} would be purely odd, while the function \hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0ptA\kern 0.0pt}}}_{A} remains as a strict parity (even) function deformed by a gauge transformation that generates a term of the opposite parity, recovering thus the parity properties of the fields in Henneaux:2018gfi .

Our set of boundary conditions is completed by imposing the faster fall-off for the Gauss constraint $\partial_{i}\pi^{i}\sim r^{-3}$ (in polar coordinates), which implies the following asymptotic relations

[TABLE]

We are now in position to bring the specific form of the surface term at infinity $B_{\infty}$ of the action principle in (1). This is given by two different terms $\mathcal{S}$ and $\mathcal{B}$. The former is linear in time derivatives on the canonical variables, contributing then to the symplectic two-form $\Omega$ of the theory. The second one, $\mathcal{B}$, is quadratic in the canonical variables, without time derivatives on the fields. Thus, we have that

[TABLE]

where

[TABLE]

Here, we must mention the difference with the boundary conditions adopted in Henneaux:2019yqq . In absence of linear fields $(\Phi_{\text{lin}}=\Psi_{\text{lin}}=0)$, instead of asking for a faster fall-off of the magnetic field through the condition \hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0ptA\kern 0.0pt}}}_{A}=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0ptD\kern 0.0pt}}}_{A}\Theta in order to tame the Lorentz boosts, we managed to solve it by introducing the field $\Psi^{(d-4)}$ (associated to a subleading coefficient of $A_{0}$ in (122)) in the boundary contribution to the kinetic term. This field is conjugate to the leading order field

$\Phi$

. The introduction of $\Psi^{(d-4)}$ will have a direct implication in the subleading gauge symmetry associated to the shifts in \hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0ptA\kern 0.0pt}}}_{r}, which becomes improper (as shown in Section 3). Note that the boundary term $\mathcal{B}$ vanishes in this case. Then, if we switch on the linear growth, similar terms appear in $\mathcal{S}$, where the linear fields are conjugate to sub-subleading coefficients in the radial expansion of the gauge field $(A_{r}^{(2)},\Psi^{(d-3)})$. In this case, the boundary term $\mathcal{B}$ is necessary in order to have a well-defined action principle. The explicit construction of the boundary contribution $\mathcal{S}$ to the symplectic form is performed in Appendix A.

Given the full set of boundary conditions, we proceed to show the finiteness of the symplectic structure. Using the fall-off of the canonical fields, a rather direct calculation implies that

[TABLE]

It is clear that the linear divergence vanishes by virtue of the parity conditions as the term \hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt\pi\kern 0.0pt}}}^{r}\dot{\Phi}_{\text{lin}}+\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt\pi\kern 0.0pt}}}^{A}\partial_{A}\dot{\Phi}_{\text{lin}} is odd, and it is then zero under the integral on the $(d-2)$-sphere. The remaining terms, along the logarithmic divergence, vanish (after integration by parts) by virtue of the faster fall-off of the Gauss constraint (after use of relations (32)).

Note that in absence of the linear growth, parity conditions are not needed. This is in agreement with Henneaux:2019yqq .

We have now all the elements to analyse the equations of motion for the extended Hamiltonian formulation of the theory with this enlarged set of boundary conditions.

• •

As aforementioned, variation of the Hamiltonian action (1) with respect to the Lagrange multipliers $\psi$ and $\lambda$ enforce the constraints $\mathcal{G}=-\partial_{i}\pi^{i}$ and $\pi^{0}=0$.

• •

The variation with respect to the spatial component of the gauge potential $A_{i}$ gives a bulk term and a boundary term, which should vanish independently. The bulk term imposes the following equation of motion for the spatial conjugate momentum

[TABLE]

Vanishing of the boundary term in turn implies the following dynamical equations

[TABLE]

If we now expand asymptotically the equation of motion for the spatial component of the conjugate momentum in (37), we obtain that

[TABLE]

From here we can see that

[TABLE]

• •

Variation of the action with respect to $\pi^{i}$ enforces the following equation of motion of the gauge potential

[TABLE]

where the asymptotic expansion of its radial component implies that

[TABLE]

The asymptotic expansion of the angular component of (44) gives the conditions $\partial_{A}\dot{\Phi}_{\text{lin}}=0$ (which is immediately satisfied) and

[TABLE]

• •

The variation with respect to the time component of the gauge potential $A_{0}$, gives also a bulk a term and a boundary term. However, the bulk term vanishes by virtue of the constraints $\mathcal{G}=-\partial_{i}\pi^{i}$ and $\pi^{0}=0$, while the boundary term is zero provided the following equations are satisfied

[TABLE]

These are in agreement with the asymptotic expansion of the equations of motion obtained from the variation with respect to the spatial conjugate momentum in (45), (46) and (47).

• •

Finally, we take the variation of the action with respect to $\pi^{0}$. This yields the following equation of motion for $A_{0}$

[TABLE]

Note that the radial expansion of the above equations leads to the equations of motion for the boundary fields obtained in (38), (39) and (40).

Thus, we have explicitly shown that our asymptotic conditions provide a true extremum of the extended Hamiltonian action, where the dynamical equations obtained from the boundary terms are fully compatible with the equations of the motion for the canonical variables.

3 A six-fold set of angle-depedent $u(1)$ gauge transformations

This new set of asymptotic conditions turns out to be preserved by gauge parameters with a relaxed fall-off, compared with Henneaux:2019yqq , that is to say

[TABLE]

where the parity conditions of the leading order fields are given by

[TABLE]

The remaining functions in the radial expansions of the parameters are no subject to parity conditions in principle. However, as we will see only the even component of $\epsilon^{(d-3)}$ and the odd component of $\mu^{(d-3)}$ generate improper gauge transformations. The subleading functions $\epsilon^{(d-4)}$ and $\mu^{(d-4)}$ (being completely arbitrary) also generate improper gauge transformations (see below).

The boundary conditions are preserved provided the following transformation laws are satisfied

[TABLE]

[TABLE]

[TABLE]

In order to compute the canonical generators $G_{\epsilon,\mu}$ of the gauge symmetries, we will make use of the (non-degenerate) symplectic form of the theory

[TABLE]

through the formula $\iota_{X}\Omega=-d_{V}G_{X}$ (where $X$ is a Hamiltonian vector field generating the phase space transformations). Thus, we obtain that

[TABLE]

where the surface term is given by

[TABLE]

with

[TABLE]

and

[TABLE]

Notably, as in the seminal work by Brown and Henneaux in the case of AdS3 gravity Brown:1986nw , the Poisson brackets of the canonical generators acquire non trivial central charges

[TABLE]

Similar central extensions can also be found in the context of supergravity Fuentealba:2021xhn and gravity in five dimensions Fuentealba:2022yqt (in addition to the aforementioned four-dimensional cases of gravity Fuentealba:2022xsz and electromagnetism Fuentealba:2023rvf ).

Some comments are in order:

• •

Preservation of the parity conditions on the fields implies that $\epsilon_{\text{lin}}$ is odd and $\mu_{\text{lin}}$ is even. Thus, one expects that the charge $Q_{\epsilon_{\text{lin}}}$, related to the electric dipole moment (for the $\ell=1$ component in the spherical harmonic expansion), together with its pair $Q_{\mu_{\text{lin}}}$ should combine to form an arbitrary angle-depedent $u(1)$ symmetry at null infinity. The latter combination of the charges should be connected to the Ward identities associated to the higher-dimensional subleading soft photon theorems found in He:2019pll . As in the $d=4$ case (for the standard $u(1)$ symmetry), matching conditions at null infinity for the $\mathcal{O}(r)$ gauge symmetries are expected to be a consequence of the asymptotic behaviour at spatial infinity.

• •

From the expressions for the charges, one can also see that only the even part of $\epsilon^{(d-3)}$ and the odd part of $\mu^{(d-3)}$ generate improper gauge transformations. Thus, this pair should also combine in an arbitrary subleading $u(1)$ symmetry at null infinity.

• •

In spite of \hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt\pi\kern 0.0pt}}}^{r} is a parity even function (as requested by parity conditions), both parities associated to the gauge parameter

$\epsilon$

generate improper gauge transformations (unlike the four-dimensional case where only the even part of

$\epsilon$

generates a non zero charge), and this is because of the parity odd field $\Psi^{(d-4)}$, which appears in the expression for the charge Q_{\hbox{\vbox{\hrule height=0.5pt\kern 0.90417pt\hbox{\kern 0.0pt\epsilon\kern 0.0pt}}}} in (62). A similar observation can be made for the parameter

$\mu$

, where both parities (of

$\mu$

) generate improper gauge transformations in all $d>4$, and in the $d=4$ case, it is only the parity odd part which span improper gauge symmetries (since \hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0ptA\kern 0.0pt}}}_{r} becomes purely odd in $d=4$, as can be seen in (29)). Indeed, if we put $d=4$, the set of charges ($G_{\epsilon^{(d-4)}}$, $G_{\mu^{(d-4)}}$) vanishes, recovering the results found in Fuentealba:2023rvf , but in absence of the logarithmic growth.

• •

As aforementioned, for $d>4$ the parameters (

$\epsilon$

,

$\mu$

, $\epsilon^{(d-4)}$ and $\mu^{(d-4)}$) are not subject to any additional condition, generating all improper gauge symmetries. This explains why instead of having eight towers of infinite-dimensional abelian symmetries, the system is actually invariant under a six-fold set of angle-dependent $u(1)$ transformations.

• •

We stress again that in the case without the linear growth in the gauge potential, the presence of the subleading field $\Psi^{(d-4)}$ of the radial expansion of $A_{0}$ in the symplectic (58) form renders improper the subleading gauge transformations generated by $\epsilon^{(d-4)}$ and $\mu^{(d-4)}$. These new improper symmetries enable us to obtain central charges between the standard $u(1)$ canonical generators (G_{\hbox{\vbox{\hrule height=0.5pt\kern 0.90417pt\hbox{\kern 0.0pt\epsilon\kern 0.0pt}}}}, G_{\hbox{\vbox{\hrule height=0.5pt\kern 0.90417pt\hbox{\kern 0.0pt\mu\kern 0.0pt}}}}) and the subleading ones ($G_{\epsilon^{(d-4)}}$, $G_{\mu^{(d-4)}}$). A similar effect occurs when one relaxes the asymptotic conditions with the linear growth. The term along the subleading field $\Psi^{(d-3)}$ (of the radial expansion of $A_{0}$) in the boundary integral of the symplectic form (58) makes improper the subleading gauge transformations generated by $\epsilon^{(d-3)}$ and $\mu^{(d-3)}$, finding hence central charges betwee the canonical generators ($G_{\epsilon_{\text{lin}}}$, $G_{\mu_{\text{lin}}}$) and the subleading generators ($G_{\epsilon^{(d-3)}}$, $G_{\mu^{(d-3)}}$).

4 Canonical realization of the Poincaré symmetry

In order to obtain a canonical action of the Poincaré symmetry, the symplectic form must be invariant, i.e., $\mathcal{L}_{X_{\xi,\xi^{i}}}\Omega=0$. As noted in the case of the $u(1)$ gauge transformations treated in the latter section, this means that $X_{\xi,\xi^{i}}$ is a Hamiltonian vector field generating canonical phase space transformations, which implies

[TABLE]

where $P_{\xi,\xi^{i}}$ is the canonical generator. The Poincaré Killing vector in polar coordinates reads

[TABLE]

where the boost parameter $b$ and the spatial translations $W$ are both in the kernel of the differential operator (\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0ptD\kern 0.0pt}}}_{A}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0ptD\kern 0.0pt}}}_{B}+\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0ptg\kern 0.0pt}}}_{AB}), while time translations satisfy that $\partial_{A}T=0$.

As we explicitly show in Appendix A, the purely bulk symplectic form fails to be Poincaré invariant. This can be cured by adding a very precise boundary term to the symplectic form (which explains the boundary term $\mathcal{S}$ in (34)) and applying additional corrective gauge transformations (by finding $\epsilon_{(\xi,\xi^{i})}$ and $\mu_{(\xi,\xi^{i})}$ in (7) and (9), respectively). Parity conditions play a key role by taking care of divergences in $\mathcal{L}_{X_{\xi,\xi^{i}}}\Omega$ (see Appendix A for details).

We find that the corrective parameters are asymptotically given by

[TABLE]

where

[TABLE]

Note that these field-dependent corrective gauge transformations (required by integrability of the canonical generators) could lead to modifications of the transformation laws of the fields in equations (7), (8), (9) and (10), by constraint terms. However, due to their fast decay, these terms do not change our results.

Using the transformation laws of the fields listed in appendix B , we obtain that the canonical generator of Poincaré symmetry is given by

[TABLE]

where $\mathcal{H}$ is given in (2) and

[TABLE]

with

[TABLE]

By making use of the transformation laws of the fields listed in Appendix B, we find that the Poisson brackets of the canonical generators are given by

[TABLE]

where the Poincaré Killing vectors transform as

[TABLE]

and the transformation laws of the improper gauge parameters read

[TABLE]

The central charges $C_{\{\mu_{1},\epsilon_{1};\mu_{2},\epsilon_{2}\}}$, that appear in the bracket between the gauge canonical generators in (88), can be found in Section 3.

One can check that all the Jacobi identities are identically satisfied.

The charges associated to $(Q_{\epsilon_{\text{lin}}},Q_{\mu_{\text{lin}}})$ and $(Q_{\epsilon^{(d-4)}},Q_{\mu^{(d-4)}})$ (which play the role of the logarithmic $u(1)$ symmetries in $d=4$) are not conserved in time (under the assumption of non explicit time dependence). This can be readily checked from the integrand of these charges (given in Section 3) and using the equations of motion found in Section 2, specifically in (43), (46), (39) and (49), respectively. This is indeed consistent with the brackets of these canonical generators with the Hamiltonian $H$ (as can be directly read from the transformation law of the parameters). Similar phenomenon occurs in the case of $d=4$, where linear and logarithmic $u(1)$ charges are not conserved in time (with no explicit time dependence on the parameters) Fuentealba:2023rvf .

As we shall see in the next section, this specific algebraic structure allows to redefine the Poincaré canonical generators, so that all of the $u(1)$ charges transform in the trivial representation of the Lorentz group. The direct sum form of the algebra has as direct consequence the conservation in time of all $u(1)$ charges.

5 Gauge-invariant Poincaré generators

The general argument in Fuentealba:2022xsz gives a mechanism to disentangle all pure supertranslations from the Poincaré algebra through a very precise nonlinear automorphism111By nonlinear automorphism we refer to a nonlinear map of the generators that preserves a given Lie algebra. In the case of infinite-dimensional Lie algebras this sort of redefinitions has shown to play a central role in the mapping between different two-dimensional models Rodriguez:2021tcz ; Tempo:2022ndz ; Bagchi:2022nvj . See also Section 5.1 in Fuentealba:2022yqt for a recent discussion on nonlinear redefinitions (of Lie algebra generators) and Poisson manifolds. of the Poincaré canonical generators. This procedure works due to the existence of invertible central charges in the abelian algebra of all supertranslations, for which logarithmic supertranslations were crucial (central charges appear in the bracket between standard and logarithmic supertranslations).

The mechanism was then applied to the case of electromagnetism in Fuentealba:2023rvf , which it was possible to carry out due to the relaxation of the asymptotic conditions by logarithmic terms in the gauge potential. Again, the presence of invertible central charges, allowed us to perform an appropriate nonlinear Poincaré automorphism that decouple all the $u(1)$ symmetries from the Poincaré algebra. Finding in this way a definition for the angular momentum free of $u(1)$ ambiguities (in line with the Coleman-Mandula theorem Coleman:1967ad ).

Given the form of the asymptotic symmetry algebra found in this paper (with invertible central charges), we can find the precise nonlinear automorphism of the Poincaré algebra. This is implemented through a suitable improper gauge transformation with the parameters $\epsilon^{\text{extra}}_{(\xi,\xi^{i})}$ and $\mu^{\text{extra}}_{(\xi,\xi^{i})}$, such that the new Poincaré generator

[TABLE]

satisfies the following Poisson brackets

[TABLE]

We find that the extra surface integral in $G_{\epsilon^{\text{extra}}_{(\xi,\xi^{i})},\mu^{\text{extra}}_{(\xi,\xi^{i})}}$ reads

[TABLE]

The asymptotic behaviour of the field-dependent parameters $\epsilon^{\text{extra}}_{(\xi,\xi^{i})}$ and $\mu^{\text{extra}}_{(\xi,\xi^{i})}$ generating the above charge, is determined by the following parameters

[TABLE]

Thus, the gauge-invariant Poincaré generator reads

[TABLE]

We have thus shown that the decoupling of the $u(1)$ charges from the Poincaré algebra is a general property of electromagnetism in all spacetime dimensions $d\geq 4$.

6 Concluding remarks

In this paper, we have extended the results recently obtained in Fuentealba:2023rvf to all higher spacetime dimensions. In particular, we have shown how to consistently relax the asymptotic conditions in electromagnetism, by accommodating a linear growth of the gauge potential at spatial infinity. The linear relaxation was done by adopting a different set of boundary conditions with respect to the ones in Henneaux:2019yqq , which leads to a slower decay of the magnetic field. Thus, solutions that possess magnetic sources with monopole decay at infinity are included. Consistency of Poincaré invariance with finiteness of the symplectic structure imposes a set of strict parity conditions on the leading order fields. This is in contrast with the analysis performed in Henneaux:2019yqq where no parity conditions were needed. Notably, the boundary conditions introduced in this work render improper a set of subleading gauge symmetries, which mimic the subleading supertranslations found in the context of gravity in higher dimensions Fuentealba:2021yvo ; Fuentealba:2022yqt . The asymptotic symmetries are given by a six-fold family of angle-dependent $u(1)$ transformations. We have also found that the canonical generators associated to these gauge symmetries obey an abelian algebra with non zero central charges. Finally, we have made use of the mechanism proposed in Fuentealba:2022xsz to decouple all the $u(1)$ gauge transformations from the Poincaré algebra. This is realized through a nonlinear automorphism of the Poincaré algebra, which makes the canonical generators of the improper gauge symmetries Poincaré scalars.

One could wonder if extensions with higher powers of $r$ or even containing a logarithmic branch can be accommodated in the asymptotic condition of the gauge field. Nonetheless, preliminary results suggest that the former growth would lead to divergences in the symplectic structure which are difficult to cure by using parity conditions. The logarithmic growth on the other hand does not seem to contribute to the surface integral of the canonical generator, being associated then to a proper gauge symmetry (in $d>4$). Thus, a connection to a would-be higher-dimensional generalization of the results obtained in Peraza:2023ivy seems to be non trivial. Indeed, it would be of interest to analyse to what extent this sort of symmetries emerges either in the light-cone formulation for electromagnetism in higher dimensions, along the lines of Majumdar:2022fut , or in the light-front (where the flat space is foliated by retarded/advanced time) Gonzalez:2023yrz . This could shed some light on how to understand (or even circumvent) the obstructions, that we have faced so far, in order to find further consistent relaxations (with higher powers of $r$) respecting both finiteness and exact invariance of the symplectic form. These requirements are essential to guarantee the direct application of standard Hamiltonian methods to obtain well-defined canonical generators, whose algebra does satisfy all the necessary Jacobi identities by construction.

It must be emphasized that the linear behaviour of the gauge potential requires, in addition to the parity conditions on the leading order fields, to impose twisted parity conditions on the “core” fields (which are associated to the Weinberg’s leading soft theorems). Thus, it would be also interesting to connect these results to null infinity, where the necessary matching conditions in He:2019pll ; He:2019jjk are expected to be consequence of the asymptotic behaviour at spatial infinity. This particular problem for all dimensions is harder (due to the more intricate definition of null infinity in odd dimensions). Work along these lines is currently in progress Fuentealba:matchHD .

Acknowledgments

I would like to express my gratitude to Marc Henneaux and Ricardo Troncoso for their careful reading of this manuscript and valuable comments. Special thanks to Marc Henneaux for his encouragement to write this paper. I would also like to thank José Figueroa, Hernán González, Jakob Salzer and Cédric Troessaert for discussions. This work was partially supported by a Marina Solvay Fellowship, FNRS-Belgium (conventions FRFC PDRT.1025.14 and IISN 4.4503.15), as well as by funds from the Solvay Family.

Appendix A Solving the non-integrability of the Poincaré generators

In this appendix, we shall derive, in a constructive way, the boundary deformation of the symplectic form together with the necessary corrective gauge transformations (along the Poincaré parameters) in order to obtain a canonical action of the Poincaré symmetry ($\mathcal{L}_{\xi,\xi^{i}}\Omega=0$) for the relaxed set of asymptotic conditions presented in this work.

Let us first consider the pure bulk symplectic form for the usual Hamiltonian theory,

[TABLE]

The need of considering the extended Hamiltonian formulation will be explicit in the course of the calculation. This is due to the fact that $A_{0}$ carries boundary degrees of freedom, which are the key ingredients for having boundary conditions that are invariant under both Poincaré and angle-dependent $u(1)$ transformations.

In what follows, it is computed the change in the symplectic form by acting with a normal Poincaré diffeormorphism $\xi=br+T$. This suggests the boundary term that must be added to the bulk symplectic form (associated to $\mathcal{S}$) and the additional boundary degrees of freedom (ultimately related to $A_{0}$). As anticipated, additional corrective gauge transformations must be performed in order to obtain that $\mathcal{L}_{X_{(\xi,\xi^{i})}}\Omega=0$, where $\xi^{i}$ is the spatial Poincaré deformation \xi^{i}=(\xi^{r},\xi^{A})=(W,Y^{A}+\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0ptD\kern 0.0pt}}}^{A}W/r).

Change of $\Omega^{\text{bulk}}$ along normal Poincaré transformations

By using the transformation laws

[TABLE]

we obtain that the variation of the bulk symplectic form under $\xi$ becomes

[TABLE]

The asymptotic conditions for the gauge potential imply that

[TABLE]

The variation of $\Omega^{\text{bulk}}$ then becomes

[TABLE]

By making use of the parity conditions on the fields declared in section 2, we can eliminate the linear divergence that appears in (121) (the integrand becomes an odd function under the antipodal map).

The remaining finite terms in (121) are not zero and they cannot be eliminated by corrective gauge transformations. We shall solve this problem by following the approach developed in Henneaux:2018gfi ; Henneaux:2019yqq , namely, by introducing new boundary degrees of freedom and deforming the symplectic form by boundary terms.

We adopt in this point a different set of boundary conditions as the ones considered in Henneaux:2019yqq . To show how this works, let us switch off the linear fields for the moment. In Henneaux:2019yqq the problem was solved by introducing a new boundary field

$\Psi$

(related to $A_{0}$) and imposing that \hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0ptA\kern 0.0pt}}}_{A}=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0ptD\kern 0.0pt}}}_{A}\Theta (equivalent to a faster fall-off of the magnetic field), which makes the formalism to work perfectly, making the Lorentz boosts canonical transformations (see Henneaux:2019yqq ). However, we propose here another solution, by leaving \hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0ptA\kern 0.0pt}}}_{A} arbitrary, but at the price to introduce a new boundary field $\Psi^{(d-4)}$. The latter is nothing but a subleading term in the radial expansion of $A_{0}$ (see below). Now, let us switch on the linear behaviour again. To solve integrability problems, we introduce the new boundary fields $\Psi_{\text{lin}}$ and $\Psi^{(d-3)}$, such that the asymptotic behaviour of $A_{0}$ is given by

[TABLE]

Symplectic form for the extended phase space

We shall show now the invariance of the system under the Poincaré symmetry. For this we extend the phase space by the conjugate pair $(A_{0},\pi^{0})$, and consider the following symplectic form

[TABLE]

The form of the additional boundary term is suggested by the equation (121). In order to show the invariance of the theory, we apply the transformation rules already proposed in Henneaux:2018gfi ; Henneaux:2019yqq , to wit,

[TABLE]

which imply the following transformation laws for the boundary fields

[TABLE]

[TABLE]

A direct calculation shows that

[TABLE]

These terms can now be cancelled out by additional corrective gauge transformation. In this case we must perform subleading gauge transformations whose parameters read

[TABLE]

Thus, the change of the new symplectic form under $\xi$ is zero, i.e., $\mathcal{L}_{\xi}\Omega=0$.

On the other hand, the change of the new symplectic form $\Omega$ under spatial diffeomorphisms $\xi^{i}$ reveals that additional corrective subleading gauge transformations along spatial translations $W$ must be performed. Thus, we obtain that

[TABLE]

which is identically cancelled out by gauge transformations with parameters

[TABLE]

Therefore, due to $\mathcal{L}_{\xi,\xi^{i}}\Omega=0$, the theory is invariant under Poincaré transformations. This fact enables us to compute directly the Poincaré canonical generators, whose expressions are shown in Section 4.

Appendix B Transformation laws of the first terms in the asymptotic expansion of the fields

In this appendix we list the transformation laws of the first terms in the asymptotic expansion of the fields under Poincaré symmetry. These are obtained by asking for preservation of the asymptotic conditions.

• •

Leading orders:

[TABLE]

• •

Core fields:

[TABLE]

• •

Subleading orders:

[TABLE]

[TABLE]

The corrective gauge parameters are given by

[TABLE]