Logarithmic angle-dependent gauge transformations are symmetries of electromagnetism that are canonically conjugate to the standard $$ \mathcal{O}(1) $$ angle-dependent $\textit{u}$(1) transformations. They were exhibited a few years ago at spatial infinity. In this paper, we derive their explicit form at null infinity. We also derive the expression there of the associated “conserved” surface integrals. To that end, we provide a comprehensive analysis of the behaviour of the electromagnetic vector potential $\textit{A}$ in the vicinity of null infinity for generic initial conditions given on a Cauchy hypersurface. This behaviour is given by a polylogarithmic expansion involving both gauge-invariant logarithmic terms also present in the field strengths and gauge-variant logarithmic terms with physical content, which we identify. We show on which explicit terms, and how, do the logarithmic angle-dependent gauge transformations act. Other results of this paper are a derivation of the matching conditions for the Goldstone boson and for the conserved charges of the angle-dependent $\textit{u}$(1) asymptotic symmetries, as well as a clarification of a misconception concerning the non-existence of these angle-dependent $\textit{u}$(1) charges in the presence of logarithms at null infinity. We also briefly comment on higher spacetime dimensions.

Belgium

Chile

Oscar Fuentealba

France

Marc Henneaux

Logarithmic angle-dependent gauge transformations at null infinity

The asymptotic structure of three-dimensional Carroll gravity with negative cosmological constant is studied. We formulate a consistent set of boundary conditions preserved by an infinite-dimensional extension of the AdS Carroll algebra, which turns out to be isomorphic to a precise generalized BMS algebra. This is described by four independent functions of the circle at infinity, generating spatial superrotations, Carroll superboosts, spatial supertranslations and time supertranslations. Remarkably, this asymptotic symmetry algebra contains as subalgebras to BMS (generated by spatial superrotations and time supertranslations) and the two-dimensional conformal algebra (spanned by spatial superrotations and spatial supertranslations). We also introduce a new solution — endowed with a Carroll extremal surface — that fulfills this set of asymptotic conditions. By taking advantage of the Chern-Simons formulation of the theory, Carroll thermal properties, obtained from regularity conditions, and entropy of the configuration are also addressed.

Luis Avilés

Iceland

Diego Hidalgo

Pablo Rodríguez

An enhanced version of the conformal BMS algebra is presented. It is shown to emerge from the asymptotic structure of an extension of conformal gravity in 3D by Pope and Townsend that consistently accommodates an additional spin-2 field, once it is endowed with a suitable set of boundary conditions. The canonical generators of the asymptotic symmetries then span a precise nonlinear W algebra, whose central extensions and coefficients of the nonlinear terms are completely determined by the central charge of the Virasoro subalgebra. The wedge algebra corresponds to the conformal group in four dimensions SO(4, 2) and therefore, enhanced conformal BMS can also be regarded as an infinite-dimensional nonlinear extension of the AdS algebra with nontrivial central extensions. It is worth mentioning that our boundary conditions might be considered as a starting point in order to consistently incorporate either a finite or an infinite number of conformal higher spin fields.

Austria

Iva Lovrekovic

David Tempo

Ricardo Troncoso

Enhanced conformal BMS

Matching conditions relating the fields at the future of past null infinity with the fields at the past of future null infinity play a central role in the analysis of asymptotic symmetries and conservation laws in asymptotically flat spacetimes. These matching conditions can be derived from initial data given on a Cauchy hypersurface by integrating forward and backward in time the field equations to leading order in an asymptotic expansion, all the way to future and past null infinities. The standard matching conditions considered in the literature are valid only in the case when the expansion near null infinity (which is generically polylogarithmic) has no dominant logarithmic term. The absence of dominant logarithmic term, in turn, holds only when the leading order of the initial conditions on a Cauchy hypersurface (which contains no logarithm) fulfills definite parity conditions under the antipodal map of the sphere at infinity. One can consistently consider opposite parity conditions. While these do not conflict with the asymptotic symmetry group, they lead to a very different asymptotic behaviour near null infinity, where the expansion starts now with logarithmic terms that are no longer subdominant (even though such logarithmic terms are absent in the initial data), which implies different matching conditions.

Logarithmic matching between past infinity and future infinity: The massless scalar field in Minkowski space

We analyse the asymptotic symmetries of electromagnetism non-minimally coupled to scalar fields, with non-minimal couplings of the Fermi type that occur in extended supergravity models. Our study is carried out at spatial infinity where minimal and non-minimal couplings exhibit very different asymptotic properties: while the former generically cannot be neglected at infinity, the latter can. Electromagnetic non-minimal couplings are in that respect similar to gravitational minimal couplings, which are also asymptotically subdominant. Because the non-minimally interacting model is asymptotic to the free one, its asymptotic symmetries are the same as the ones of the free theory, i.e., described by angle-dependent u(1) gauge transformations. We also analyse the duality symmetry and show that it is broken to its compact subgroup by the asymptotic conditions. Finally, we consider logarithmic gauge transformations and use them to simplify the symmetry algebra.

Jules Mas

Non-minimal couplings to U(1)-gauge fields and asymptotic symmetries

p-form electrodynamics in d ≥ 2 dimensions is shown to emerge as the edge modes of a topological field theory with a precise set of boundary conditions, through the Hamiltonian reduction of its action. Electric and magnetic charges correspond to Noether ones in the topological field theory. For chiral p-forms, the topological action can be consistently truncated, so that the Henneaux-Teitelboim action is recovered from a pure Chern-Simons theory, with a manifestly covariant stress-energy tensor at the boundary. Topologically massive p-form electrodynamics as well as axion couplings are also shown to be described through this mechanism by considering suitable (self-)interaction terms in the topological theory.

p-form electrodynamics as edge modes of a topological field theory

We show that the non-linear BMS5 symmetry algebra of asymptotically flat Einstein gravity in five dimensions, as well as the super-BMS4 superalgebra of asymptotically flat supergravity, can be redefined so as to take a direct sum structure. In the new presentation of the (super-)algebra, angle-dependent translations and angle-dependent supersymmetry transformations commute with the (super-)Poincaré generators. We also explain in detail the structure and charge-integrability of asymptotic symmetries with symmetry parameters depending on the fields (through the charges themselves), a topic relevant for nonlinear asymptotic symmetry algebras.

Simplifying (super-)BMS algebras

The asymptotic symmetry algebra of four-dimensional Einstein gravity in the asymptotically flat context has been shown recently to be the direct sum of the Poincaré algebra and of an infinite-dimensional Abelian algebra (with central charge) that includes the Bondi-Metzner-Sachs supertranslations. This result, obtained within the Hamiltonian formalism, yields a supertranslation-invariant definition of the Lorentz generators (angular momentum and boosts). Definitions of Lorentz generators free from the “supertranslation ambiguities” have also been proposed recently at null infinity. We prove the equivalence of the two approaches for redefining the charges.

Cédric Troessaert

Asymptotic Symmetry Algebra of Einstein Gravity and Lorentz Generators

The asymptotic symmetries of electromagnetism in all higher spacetime dimensions d &gt; 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   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (r) gauge parameter are found to be conjugate to those associated to the now improper subleading O(r  −d+3) transformations, while the standard   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (1) gauge transformations are canonically conjugate to the subleading   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (r  −d+4) transformations. This algebraic structure, characterized by the presence of central charges, allows us to perform a nonlinear redefinition of the Poincaré generators, that results in the decoupling of all of the u(1) charges from the Poincaré algebra. Thus, the mechanism previously used in d = 4 to find gauge-invariant Poincaré generators is shown to be a robust property of electromagnetism in all spacetime dimensions d ≥ 4.

Asymptotic   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (r) gauge symmetries and gauge-invariant Poincaré generators in higher spacetime dimensions

We extend the asymptotic symmetries of electromagnetism in order to consistently include angle-dependent u(1) gauge transformations ϵ that involve terms growing at spatial infinity linearly and logarithmically in r, ϵ ~ a(θ, φ)r + b(θ, φ) ln r + c(θ, φ). The charges of the logarithmic u(1) transformations are found to be conjugate to those of the   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (1) transformations (abelian algebra with invertible central term) while those of the   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (r) transformations are conjugate to those of the subleading   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (r  −1) transformations. Because of this structure, one can decouple the angle-dependent u(1) asymptotic symmetry from the Poincaré algebra, just as in the case of gravity: the generators of these internal transformations are Lorentz scalars in the redefined algebra. This implies in particular that one can give a definition of the angular momentum which is free from u(1) gauge ambiguities. The change of generators that brings the asymptotic symmetry algebra to a direct sum form involves non linear redefinitions of the charges. Our analysis is Hamiltonian throughout and carried at spatial infinity.