PoS(Modave2019)003 . https://pos.sissa.it/ XV Modave Summer School in Mathematical Physics ∗ and asymptotically flat spacetimes are covered. This enables us to discuss the different 4 [email protected] Speaker. These notes are angeneral introduction relativity to in four asymptotic dimensions. symmetries We explainditions in how in to gauge the impose gauge consistent theories, fixing sets approach with of anddifferent boundary a how con- procedures to focus derive to the on obtain asymptotic symmetry thegeneral parameters. associated concepts, The charges the examples are of presented. four-dimensional(A)dS general As relativity an in asymptotically illustration (locally) ofextensions these of the Bondi-Metzner-Sachs-van der Burg (BMS)raphy, group soft and gravitons their theorems, relevance memory for effects, holog- andare black based hole on information paradox. lectures given These at notes the ∗ Copyright owned by the author(s) under the terms of the Creative Commons c Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). XV Modave Summer School in Mathematical8-14 Physics September - 2019 Modave2019 Modave, Belgium Romain Ruzziconi Université Libre de Bruxelles and InternationalCP Solvay 231, Institutes B-1050 Brussels, Belgium E-mail: Asymptotic Symmetries in the Gauge Fixing Approach and the BMS Group PoS(Modave2019)003 Romain Ruzziconi contains some important definitions B , after some digressions through the Noether 4 1 , we focus on the gauge fixing approach. We explain the 3 is a quick summary of the matching between Bondi and A and asymptotically flat spacetimes. These examples are interesting because , we review some recent applications of asymptotic symmetries in the con- 4 5 , we briefly mention the different main frameworks to study asymptotic symmetries 2 Many reviews related to asymptotic symmetries complementary to these notes exist in the Gauge theories are of major importance in physics since they are involved in the fundamental The study of asymptotic symmetries in gauge theories is an old subject that has recently known Several approaches exist regarding asymptotic symmetries in gauge theories and the construc- In section text of holography andnied the by infrared two sector appendices. of gauge Appendix theories. Finally, these notes are accompa- literature: see, for example, [19, 26, 63, 134, 137, 145, 153, 162]. and conventions about the jet bundles and homotopy operators widely used in the text. in gauge theories. Thereafter, in section conditions under which a given gaugediscuss fixing how is to suitable to impose study consistentderive asymptotic boundary the symmetries. conditions, asymptotic Then, the symmetry we associated algebra. solutionprocedure In space, to section and construct how charges to associatedof with this global construction symmetries, for we gaugecussed symmetries explain and is. what related the to In the analogue particular, covariant the phasetheories. Barnich-Brandt space methods prescription In in is the section dis- context of diffeomorphism-invariant Fefferman-Graham gauges in general relativity. Appendix description of nature. Theity Standard are Model two of examples particleuniverse. of physics Furthermore, gauge and gauge theories theories the provide offering theory astand never-equaled mathematical of the framework observational general deepest that predictions enables foundations relativ- us of for to our our under- physical theories. renewed interest. A first direction istotic motivated symmetries by of the the AdS/CFT gravity theory correspondence in wherethe the the dual bulk asymp- quantum spacetime correspond field to theory thecontrol through global on the symmetries asymptotic holographic of symmetries dictionary allows [3, us 126,second 165, to 167, direction investigate 177]. is new driven interesting A by holographic strong the dualities. recently-establishedsoft A connections theorems between and asymptotic symmetries, memory effects [162].infrared structure These of connections quantized furnish gauge crucialstanding theories. information problem about of In the black gravity, hole they information may paradox be [101, 102, relevant 105–107]. to solve the long- tion of associated charges.how The to aim of impose these boundary notesalgebra, conditions is and in to compute a provide the generic a associatedapproach. gauge self-consistent surface charges. introduction theory, Indeed, on derive We the discuss despite theseerences asymptotic this points discussing symmetry approach in the the being complete gauge widelydefinitions procedure fixing and used for relevant in results, a we the general discussically gauge literature, in (locally) detail theory. there (A)dS the examples are To of illustrate few general the relativity ref- in abstract asymptot- Asymptotic Symmetries in gauge theories 1. Introduction they involve all the subtleties oflectures the given procedure. at the The XV notes Modave aim Summer to School be in pedagogical Mathematical and Physics. are based on PoS(Modave2019)003 0 = (2.1) weak β µ ∂ µ ∂ transforms as µ Romain Ruzziconi A , that boundary conditions are , regardless of the exact form α definition of the asymptotic symmetry a posteriori (where appropriate will be defined below) stronger 2 0. This gauge can always be reached using the gauge = µ A always admits a solution for µ Trivial gauge transformations ), where the trivial gauge transformations are now the gauge ∂ ν is a function of the spacetime coordinates satisfying A β 2.1 ν ∂ appropriate gauge fixing − = , where α β µ Gauge transformation preserving the boundary conditions µ ∂ ∂ is a function of the spacetime coordinates) under a gauge transformation. The µ = ∂ + α G ( µ A α µ → ∂ µ + A µ A . However, residual gauge transformations remain that preserve the Lorenz gauge. These are A gauge theory has redundant degrees of freedom. The gauge fixing approach consists in using The geometric approach was essentially used in gravity theory and led to much progress in The advantage of this approach is that it is manifestly gauge invariant, since we do not refer to Several frameworks exist to impose boundary conditions in gauge theories. Some of them are The geometric approach of boundary conditions was initiated by Penrose, who introduced the µ → A is then defined as the quotient: µ still allows some redundancy. ForA example, in electrodynamics, the gauge field Lorenz gauge is defined by setting the gauge freedom of the theorytient to the impose field some constraints space on to the eliminateFor fields. some a This of given enables the gauge one unphysical theory, to or quo- an pure gauge redundancies in the theory. group is given by the quotient ( defined in this framework, after having obtained the results in2.2 coordinates. Gauge fixing approach the study of symmetries and symplectic structures[6, 7, for 83, asymptotically 103, 114] flat and spacetimes at spatial nullspacetimes infinity infinity [8–11]. [4, 5]. Moreover, It this was frameworkand also was associated considered recently phase to spaces applied study on to asymptotically null study (A)dS hypersurfaces boundary [56]. conditions any particular coordinate system tointerpretation impose of the the boundary symmetries conditions. isconditions Furthermore, transparent. is rigid. the The It geometric weak is a point non-trivialwork is task to to that highlight modify the a new given asymptotic definition set symmetries. of of boundary It boundary conditions is in often this frame- transformations that have associated vanishing charges. where the trivial gauge transformationson are the the boundary. gauge In transformationsgauge other that transformations words, reduce induced the to on asymptotic the thedefinition symmetry identity boundary of group which the is preserve asymptotic isomorphic the symmetry to given group. data. the group A This of is the redundancy, since mentioned next. 2.1 Geometric approach techniques of conformal compactification totimes study at null general infinity relativity [144,146]. in Accordingby asymptotically to requiring flat this that space- perspective, certain the boundary dataG conditions on are a defined fixed boundary be preserved. The asymptotic symmetry group Asymptotic Symmetries in gauge theories 2. Definitions of asymptotics of given by PoS(Modave2019)003 (2.3) (2.2) , this allowed # spacetimes. Indeed, 3.4 Romain Ruzziconi 1 + d # Virasoro superrotations [37,38]. × ). Therefore, fixing the gauge is similar 2.1 is trivial [69]. 3 strong G ), and the two definitions of asymptotic symmetry groups ), and 1 − 2.1 d S Residual gauge diffeomorphisms preserving the boundary conditions. × " R = Diff( . 1 weak = Residual gauge diffeomorphisms preserving the boundary conditions with associated non-vanishing charges. G " weak weak G G = ⊆ version of the definition of the asymptotic symmetry group is given by strong G strong weak , we study the gauge fixing approach and provide some examples related to gravity in G . Some explicit examples are discussed below. 3 µν g The advantage of the gauge fixing approach is that it is highly flexible to impose boundary One of the most striking examples of the difference between the weak and the strong definitions of the asymptotic Some alternative approaches exist that are also powerful in practice. For example, in the Then, the boundary conditions are imposed on the fields of the theory written in the chosen 1 conditions, since we are workinggroup with explicit in expressions four in dimensions coordinates.gauge was For fixing first example, is the discovered a BMS in localfore, this consideration framework (i.e. the [43, global it 154, 155]. holds considerationsthereby in related allowing a Furthermore, further to coordinate a flexibility. the patch For example, of topology as the are we spacetime). will not discuss There- directly in subsection relevant in this analysis, symmetry group is given by considering Neumann boundary conditions in asymptotically AdS Notice that asymptotically flat and asymptotically (A)dS spacetimes. 2.3 Hamiltonian approach Hamiltonian formalism, asymptotically flat [151] andat AdS spatial [47, 110] infinity. spacetimes have Furthermore, beenusing studied the twisted global parity conditions BMS [111–113]. groupordinate was In system this recently making framework, identified the the split at computations between spatial are space infinity done and in time a explicit, co- without performing any gauge coincide in most of the practicalasymptotic situations. symmetry As group in exists the and geometric is approach, given a by stronger version of the to consider singular extensions of theThese BMS new group: asymptotic the symmetries Virasoro are well-definedtial locally; sphere. however, they have In poles the onboundary the geometric to celes- approach, allow one these would superrotations haveThe by weakness to considering of modify this some the approach punctured is topology celestialfixing that of approach sphere it the is is [21, 164]. spacetime not often manifestly preferred gauge toric invariant. unveil approach Hence, new is even boundary complementary if conditions and the and necessary gauge symmetries,In to the section make geomet- the gauge invariance of the results manifest. in this situation, we have Asymptotic Symmetries in gauge theories (see, e.g., [119]). The samephisms phenomenon can occurs be in performed general to relativitymetric reach where a spacetime particular diffeomor- gauge defined by some conditions imposedgauge. on The the Intuitively, the gauge fixingnamely, the procedure trivial gauge eliminates transformations defined part underto ( of taking the the quotient as pure in gauge equation ( degrees of freedom, PoS(Modave2019)003 , (3.4) (3.2) (3.3) (3.5) (3.1) that are taken ) local functions α Romain Ruzziconi f are ) k = ( gauge transformation µ -dimensional spacetime ... F 1 n µ ( α R ... + , α , x ] f n ν d ∂ g ,... µ − Φ ∂ ) ν , √ α ∂ ) F f µν µ ( α Λ , ? k ∂ µ R 2 F , ∧ ∂ B + − Φ F d µ α R ... M 4 ∂ f ( 1 = are the fields of the theory. A , Z µ µ M ) ∂ Φ L i ∂ Z [ ) F k φ µ α ] = L µ δ G R A [ ... π M 1 is the cosmological constant. It can be checked that the 1 = ( S + Z µ α ( 16 Λ ), the Lagrangian transforms as α Φ ] R f ] = 0 F [ α 3.2 ] = ≥ Φ ∑ k R R [ g [ S S = = = Φ F δ . µ is a 1-form. It is straightforward to check that the gauge transformation ) x is an arbitrary function of the coordinates, is a symmetry of the theory. respectively, and A 1 are the scalar curvature and the square root of minus the determinant associated − α n µν g d is the Lagrangian and g and ( − x µ F n A √ We illustrate this definition by providing some examples. First, consider classical B d of the theory if, under ( d L [Gauge symmetry] Let us start with a Lagrangian theory in a , where = = and α = F d F B R L = Now, consider the general relativity theory We now focus on the aforementioned gauge fixing approach of asymptotic symmetries in A symmetry α δ where Examples vacuum electrodynamics the infinitesimal gauge transformation of thenamely fields. functions In this of expression, the coordinates,a the fields, and their derivatives. The gauge transformation is where where with the metric gauge theories. We illustratecific the focus on different asymptotically definitions flat and and asymptoticallyrelativity. concepts (A)dS spacetimes using in four-dimensional examples, general with a spe- 3.1 Gauge fixing procedure Definition Asymptotic Symmetries in gauge theories fixing. Then, the asymptotic symmetryformations preserving group the is boundary defined conditions as andmeans the the that quotient trivial the gauge between associated transformations, the charges where gaugethe are trivial trans- asymptotic identically symmetry vanishing group on corresponds the to phase the space. strong definition This in definition the of two3. first approaches. Asymptotic symmetries in the gauge fixing approach M is a transformation acting on theto fields, be and arbitrary which functions depends of the on spacetime parameters coordinates. We write where PoS(Modave2019)003 ) = µ x ν ) ∂ ), α g f − 3.2 is an (3.6) √ ρ = ( ( ν F ∂ generating is a vector g ) − 1 1 and µ α √ ξ f − 0, and the axial = n Romain Ruzziconi = µ = ( x De Donder (or har- 0 F A  ,..., 1 , where ρ = ξ Fefferman-Graham gauge a µ ) is of the form ∂ ) with gauge symmetry ( 3.2 ρν g 3.1 0 is the + , where . In this configuration, there are still some 6= ) ρ a , ξ Λ x α ν , f ∂ ρ µ µρ ∂ g µ 0, the temporal gauge α = ( R 0 (3.7) µ + = x i + µν ] = A α 5 i g f Φ partial gauge fixing ∂ ρ [ α ∂ be harmonic functions, namely, G R ρ µ ξ x , is equal to the number of independent gauge parameters = , which means that the number of independent conditions ). n 2 = Φ F 3.2 µν δ g ξ ) is inferior or equal to the number of independent parameters L 3.7 = µν g ξ 0, the Coulomb gauge δ ) has to be equal to the number of independent gauge parameters = µ 3.7 A generating the gauge transformation. procedure involves imposing some algebraic or differential constraints on the fields µ ) ∂ requires that the coordinates α in the gauge transformation. In electrodynamics, several gauge fixings are commonly used. Let us mention the ) is superior or equal to the number of independent parameters f 0. As previously discussed, the Lorenz gauge can always be reached by performing a [Gauge fixing] Starting from a Lagrangian theory ( α = 3.7 = ( 3 A the gauge transformations. in ( It has to use alltransformations of to the reach the available gauge freedom of the arbitrary functions parametrizing the gauge F It has to bependent reachable conditions by in a ( gauge transformation, which means that the number of inde- Another important gauge fixing in configurations where If the available freedom is not used, we talk about In gravity, many gauge fixings are also used in practice. For example, the Notice that in these examples, the transformation of the fields ( 2 • • gauge fixing . This gauge condition is suitable for studying gravitational waves in perturbation theory (see, µ [89,90,139,157,158]. We write the coordinates as 0. Notice that the number of constraints, arbitrary functions of the coordinates in the parameters of the residual gauge transformations. ξ e.g., [41]). Considering these two requirementsing together conditions tells in us ( thatinvolved the in number the fields of transformation independent ( gauge fix- gauge gauge transformation. We can checkNotice that the that same these statement holds gauge forparameter fixing all conditions the other involve gauge only fixings. one constraint, as there is only one free Examples Lorenz gauge the monic) gauge in order to eliminate (part of) the redundancy in the description of the theory. We write Asymptotic Symmetries in gauge theories gauge transformation namely they involve at most first order derivatives of the parameters. Definition a generic gauge fixing condition. This gauge has to satisfy two conditions: field generating a diffeomorphism, is a symmetry of the theory. PoS(Modave2019)003 , α µ (3.9) ∂ (3.13) (3.11) 0. The 0. The , where = ) = µ 6= A x A a , , where the Λ ρ r α ) , g δ a u ξ ( . ( ) F c L Romain Ruzziconi x , ) = 0 -dimensional metric 0. The most general u ) ρ F d ( 2 > B 0 and ab − U Λ g = . 0 n is the luminosity distance 0 -celestial sphere, the Bondi b ( − ρ ) r x ρ B ρρ d 2 d g 0 (3.10) a ρ dx ). Let us mention that the Bondi − ξ 0 is in the bulk). It is defined by x 0 , namely gauge transformations labels null hypersurfaces in the n d 0 (3.8) = Z )( L ( ) > u u c 1 (3.12) σ  = d 3.10 x ρ b ) − , A a ∂ 2 AB ρ ρ ) − U ) in general relativity with = g ( g n 2 ( − ab 2 ru − 3.8 r det g A g , 0 and timelike for n x )  + Λ d 2 )( r ( 2 < , [29, 135] involving only algebraic conditions: 2 1 ∂ 2 0 − ρ AB 2 Λ ρ have to satisfy − 0 in electrodynamics. As we discussed earlier, the n d g 6 = ρ , n ) µ )( coordinates. = ( Λ 0 + 2 1 ξ rA ) 2 r µ g 1 = − d − A ) + u n µ n − b rA d Λ , ( x ∂ )( g n β 2 0 ( ( 1 2 − a 0 configurations. Writing the coordinates as 0 e = ξ − , 2 = 6= 0 n rr = − residual gauge transformations ( is spacelike for g Λ ρρ 2 = a g − u ρ ξ will be relevant for us in the following [43, 154, 155]. This gauge rr d Newman-Unti gauge 0. They are local functions parametrized as g = r V 2 , 0 and s β 0 is at the spacetime boundary, and ρ 2 ] = d ) = e = a Φ [ x Λ = ρ ( G 2 are arbitrary functions of the coordinates, and the σ F s are the transverse angular coordinates on the δ r d V ) = 0. 2 Bondi gauge ρ − labels null geodesics inside a null hypersurface, and n = ξ and are arbitrary functions of θ Consider the Lorenz gauge A α A x a [Residual gauge transformation] After having imposed a gauge fixing as in equation µ U ∂ ,..., , µ 1 β ∂ θ Similarly, consider the Fefferman-Graham gauge ( Finally, the satisfies the determinant condition in the third equation of ( ), there usually remain some conditions). These conditions tell us that, geometrically, = ( conditions). The coordinate conditions). have to satisfy AB A 3.7 n n n solutions to these equations are given by g gauge is closely related to the residual gauge transformations generated by ( Definition ( preserving the gauge fixing condition. Formally, theF residual gauge transformations with generators parameters Examples residual gauge transformations for thewhere Lorenz gauge are the gauge transformations Asymptotic Symmetries in gauge theories expansion parameter ( the following conditions: ( metric takes the form where gauge is defined by the following conditions: ( spacetime, along the null geodesics. The most general metric takes the form fixing is valid for both x PoS(Modave2019)003 . n ) A ) in x , . (3.14) (3.15) . The u a ) ( 3.11 x A x , u ( , have to satisfy 3. There exist ) 1 µ ≥ Romain Ruzziconi − ξ r is associated with n ( coordinates A , O ) ) D g 1 + u ∂ − 1 AB n − ( D f g + of 2 1 a 0 ) has been fixed, we can impose AB ξ + ) in dimension , ) in the literature. For all of them, ) rC B 3.7 ∞ AB + and 3.10 fU g → β is an arbitrary function of B σ AB r 2 ∂ q ( e 2 ω ( r − 0 r A d = . I ) A ∞ A r AB D x 7 Z g , f , where u + ( B ω , ω ∂ [30]. The covariant derivative ) 4 2 1 − ) ( = − o AB = A arbitrary functions defined in equation ( g A AB = ( Y I n , or conditions on the leading functions in the expansion. This g A A 3 -dimensional symmetric tensors, which are functions of . The residual gauge transformations are parametrized by the ξ ) det D U , ( coordinates 2 L AB A 2 ) = g I − AB 1 , r g ), the residual gauge transformations generated by − n g ) + : is kept free at this stage. ( 2 − n A r , ∞ n ( f AB − ( Y 3.10 o Bondi gauge are q → , and = = = of 0 and = A r AB asymptotic flatness at null infinity r u A for the theory by requiering some constraints on the fields in a neighbourhood A Y r ξ D ξ = r V ξ Y ∂ rA , g = and ) and ξ 1 0 f ( AB Let us give some examples of boundary conditions in general relativity theory. Many L , o [Boundary conditions] Once a gauge condition ( C = -dimensional metric , ω ) 0, f = 2 r AB ∂ = β q − rr n Notice that the asymptotic region could be taken not only at (spacelike, null or timelike) infinity, but also in other Let us consider the In the Bondi gauge ( 3 ( g ξ spacetime regions, such as near a black hole horizon [77–79, 81, 99, 101, 102, 106, 107]. functions 3.2 Boundary conditions Definition L where the boundary conditions of a given spacetime region. Mostin of the those considered boundary asymptotic conditions are region fall-offchoice conditions on of the boundary fields conditions is motivatedof by boundary the conditions physical is model usually that consideredsymmetry we to group want be and to interesting consider. if solution A it(finite, space, generically set provides non-vanishing, exhibiting non-trivial integrable asymptotic and interesting conserved; properties seeare below). for If too the the boundary strong, associated conditions the charges Furthermore, asymptotic the symmetry solution group spacethe will will associated be not surface contain trivial, charges any with willshould solution be therefore vanishing be of divergent. surface located interest. between Consistent charges. these and two If interesting extreme they boundary situations. are conditions Examples too weak, examples of boundary conditions for[2, 30, other 55, 75, gauge 109, 124, 160]). theories can be found in the literatureseveral (see definitions e.g. of we require the following preliminary boundarythe asymptotic conditions region on the functions of the metric ( Notice in particular that Asymptotic Symmetries in gauge theories These solutions are parametrized by solutions to these equations are given by where PoS(Modave2019)003 ) - 0 ) ( ab 1 g Fef- − (3.21) (3.20) (3.17) (3.18) (3.19) (3.16) n ( asymptot- ). In addi- ) tend to the -dimensional ) 3.15 to have a fixed 2 3.11 in both the Romain Ruzziconi , − AB ¯ q n q ) ( 2 ) and we demand that − n 3.15 Λ )( 2 1 − 4, this condition can always be n ( = -dimensional boundary metric n ) 1 ) = . ) B − 0 x ( ab n ). We require ( d g ( ( A , ) x ) 2 -sphere metric, namely, . , d 2 AB 3.17 det ) − ¯ r ˚ ). A preliminary boundary condition, usually q -sphere metric, namely, q 2 AB ( ) ρ AB ˚ ϕ ˚ q √ ( 2 q 2 , − O 2 e 8 0 O 3.10 r asymptotically (A)dS spacetimes n − = = ( = ( n = q = = + ( AB ) r AB √ q 0 ab AB d ( tA g ): g q u g . Notice that the 3.9 2d ) 2 , , the leading orders of the spacetime metric ( − ) − . ∞ ) be the unit 2 2 ρ 2 u : ( r Bondi gauge − d o → ) ( AB 0 n o q r ab ( − + Λ g )( ) 2 + -sphere metric [36]. Note that for 0 = 1 ab ( ) ) and 2 g AB 2 − s 2 q n 3.8 − 2 − ( ( r ρ n ( = = = ) , that is, for 0 ( tt AB ab g g g ). All the conditions are the same, except that we require that the transverse boundary be conformally related to the unit is the unit is a fixed volume element (which may possibly depend on time) on the AB AB 3.15 q q q ). In addition to these fall-off conditions, we demand the following constraints on the A first definition of asymptotically (A)dS spacetime (AAdS1) is a sub-case of the definition A third definition of asymptotic flatness at null infinity (AF3), which is the historical one A first definition of asymptotic flatness at null infinity (AF1) is a sub-case of ( A second definition of asymptotic flatness at null infinity (AF2) is another sub-case of the def- Let us now present several definitions of 3.19 is kept free in this[130]. preliminary In set the Bondi of gauge, boundary asthat conditions, we will thus see justifying below, these the fall-off adjective conditions “locally" are equivalent to demand dimensional boundary metric ( or, equivalently, called the asymptotically locally (A)dStions condition, of the requires Fefferman-Graham the metric following ( conditions on the func- [43, 154, 155], is a sub-case of the second definition ( tion to all these fall-offdeterminant, conditions, namely, we require the transverse boundary metric where ˚ reached by a coordinate transformation, sincemally every metric flat on (but a even in two dimensional this surface case, is as confor- we will see below, this restricts the form of the symmetries). Note that this definition of asymptoticically flatness Minkowskian is the only one thatMinkowski has line the element property d to be or, equivalently, metric inition ( Asymptotic Symmetries in gauge theories where ¯ transverse space [51, 52, 65, 94]. the transverse boundary metric ferman Graham gauge PoS(Modave2019)003 )) of )) of South pole (3.24) (3.23) (3.22) ), while 1 ), except 3.18 3.16 .

) Cst

0 = u 3.19 > . . ) and ( ) and ( ¯ Λ q Romain Ruzziconi + − dS dS ( AB I I ˚ q √ 3.15 3.15 Cosmological horizon = = q AB √ q , B , , ) translate into

x

) ) pole North d 1 1 A ( ( be fixed [110]. These conditions x o o 3.21 ) . d 0 ( ab = Λ = g 0 AB i A ˚ A q (AAdS2) is a sub-case of the definition U U + 4 -dimensional space (which may possibly . dS case 2 ) + − ) t I I ,

, 2

0

d Cst

)

) = u 2 ) 2 − = r 2 r ). In the Bondi gauge, the boundary conditions ( n ( ) ( Λ o o − A -sphere metric (as in the Bondi gauge, the upper ( + − x ) 9 i i n + , + 2 Λ t )( ) ) 2 1 − 2 2 = ( n Cst Bondi gauge for any − − − ( = a + u − n n I I Λ n x Λ . One usually chooses the cylinder metric as the boundary ( 2 2 )( )( r r 1 1 2 = 2 b 0 − i , and − x n n n Figure 1: d ( ( a x -dimensional boundary metric = ) d = ) 1 0 r r ( ab V 0 AdS V − I g

. Flat case n

Cst ) (

,

, = ) u 0 ) 1 1 ( run from 3 to ( < o o Λ ( = = ,... + AdS − AdS i i B β β , A ) [66]. In the Bondi gauge, the boundary conditions ( are the components of the unit t Dirichlet boundary conditions Cst is a fixed volume form for the transverse =

AB u AdS case q q 0 ) translate into ). We require the same conditions as in the preliminary boundary condition ( AdS This choice is less relevant for asymptotically dS spacetimes, since it strongly restricts the Cauchy problem and As we see it, the Bondi gauge is well-adapted for each type of asymptotics (see figure A second definition of asymptotically AdS spacetime I 4 3.23 3.19 the bulk spacetime dynamics [9, 10]. Notice the similarity of these conditions to the definition (AF3) (equations ( asymptotically flat spacetime. asymptotically flat spacetime. ( where ˚ case indices ( the Fefferman-Graham gauge is only defined in asymptotically (A)dS spacetimes. depend on Asymptotic Symmetries in gauge theories where ¯ Notice the similarity of these conditions to the definition (AF1) (equations ( that we demand that the are called metric, namely, PoS(Modave2019)003 ) of are b 3.19 Euler (3.27) (3.29) (3.25) (3.26) (3.28) (3.30) (3.31) (3.32) solution Romain Ruzziconi in asymptotically , ) 2 ρ , where the parameters ( ) b O ( ). The set of all solutions of the ˜ Φ + ). Solving the Einstein equations ) , B.5 3 = 0 ( ab ), that is, (AAdS1), the solution space . In summary, the solution space of g ) (7 functions). Finally, for Dirichlet ) ˜ = Φ 3.19 0 ρ ( ab , ) ) and boundary conditions, a g ab 3 0, the boundary conditions, and the 3.21 + 0 , , ( ab 3.29 0 0 ) T , Fefferman-Graham gauge g 2 3.7 ) , = 6= 6= | ( ab 0 0 0 ] = Λ Λ ( a g G Λ 6= } } ˜ | = µν D Φ π Λ + [ 3 g ab ab ˜ } Φ ) 10 G T T

1 16 Λ p , ( ab , , ab ) ) L 0 g satisfies ( Φ T + 0 0 1 ( ab ( AB = { δ δ − g g = ab µν { { ab ρ T ab T G + T is defined in equation ( ) ) satisfying 0 0 ab ( ab ( the boundary metric and we define g g ˜ . It is parametrized as Φ ) 2 0 ( ab − [89, 90, 139, 157, 158]. The only free data in this expansion are g ρ a coordinates. = x ) 1 ) (AAdS2), the solution space reduces to ab g − ) (11 functions). n 3.23 ( solution space 3.29 We now provide some examples of solution spaces of four-dimensional general rela- . All the other coefficients are determined in terms of these free data. Following the is the covariant derivative with respect to has a fixed determinant and ) [Solution space] Given a gauge fixing ( are functions of 3 Euler-Lagrange derivative ) satisfies ( ab ( ) 0 ) 0 i a ( g ( AB ( ab ab spacetimes with preliminary boundary conditions ( D g g T 4 Now, for the restricted set of boundary conditions ( and ) 0 ( ab (A)dS boundary conditions ( holographic dictionary, we call Examples tivity in different gauge fixings. We first consider the g arbitrary functions of where where general relativity in the Fefferman-Graham gaugeis with parametrized by the the preliminary set boundary of condition functions ( reduces to the theory is a field configuration where the theory is called the we obtain the following analytic fall-offs: as the stress energy tensor. From the Einstein equations, we have where where Asymptotic Symmetries in gauge theories 3.3 Solution space Definition Lagrange-equations PoS(Modave2019)003 (3.36) (3.38) (3.33) (3.34) (3.37) and its , i AB . 0 q ) β 5 A − ∂ r ( , , which imposes ) Romain Ruzziconi r CD O . In particular, we C / r C x 4 + , CD u i = ( , 2 C ) ) 3 AB 3 AB i 16 − AB g ) r F r AB C 3 ( − ∂ spacetimes with preliminary 2 − + D 1 AB O r r 4 AB i AB ( ) C AB 2 g + o ( B ) D x C ) (see also [66, 149]), we know that , + AB 1 B CD 1 u F 12 r yields 128 ( 3 A D C 2 ) ) r A 1 ln r 1 3 2 − + ( CD ) U . The determinant condition defining the h rA B + − C ) ( ) implies 3 AB 0 (3.35) ( x + 1 A 0 r , D x AB β r 1 1 u = , 32 B + ( ) E 3.10 u AB ) ∂ 0 and B A i r ( 1 − µν x L3 D , , U , ( g , = AB + ) + ( u i ) 1 11 CD Λ C ( 16 C + C AB ) x A AB AB D x BC 3 + 1 AB , , 1 − C ( r D C C U u u C in asymptotically (A)dS ) CD B ( C ( µν AB AB B + 1 D D q D 32 x G AB E ) + AB , , 1 2 1 2 B AB ) AB E D u − AB x 0 (indices are lowered and raised with the metric ( , + C + h rC AB + C ) A + . u 0 1 2 3 2 q = are functions of ( ( 1 3 + r β ( U CD 32 AB A CD 0 CD − B E AB + ∂ C ... C AB U D A det − , , ) + q B 0 F 4 CD Bondi gauge N h AB 2 = CD A CD r β D h 4 r C AB x C 1 0 0 AB A C A D r h , F h β β q ∂ u = 0 , 2 2 U AB 0 ). From the Fefferman-Graham theorem and the gauge matching between AB ( ) = β + AB e e β ) (5 functions). q 2 0 = q q AB 2 2 3 3 2 AB e 4 1 AB β 1 2 e 1 2 E g g AB − − 2 − 3.20 , ( = 3.29 E = = ) = AB A AB AB AB D AB x is an arbitrary function. The component ) = ) = ) = ) = q , , F E D B B B B ) r x x x x , = A AB , , , , u x u u u u C ( , ( ( ( ( AB satisfies ( , u ) β ) ) ) A A A A ( D 2 1 3 0 (we follow [66, 149]; see also [127] for the Newman-Penrose version). The component 0 ( ( ( L3 AB ab U U U U ( β T q AB 6= q Let us now consider the gives the following radial constraints on the Bondi functions: Λ ) rr where the functions appearing in the metriccan admit write an analytic expansion in powers of where with with Asymptotic Symmetries in gauge theories where boundary condition ( Bondi and Fefferman-Graham that is described in appendix Bondi gauge and appearing in thesuccessively third that equation det of ( inverse). We now sketch the results obtained by solving the Einstein equations for ( PoS(Modave2019)003 . 0 N is an leads ∈ of the (3.40) (3.41) (3.43) (3.39) ) ) , deter- ) n A angular , x ru is a fixed AB ) , ( AB , ; however, n of the Ein- u 0 coordinate. ( ) ) F the ( − u 1 3 ) r u − − is a set of two A ( M ∼ r r uA ) . Moreover, the ( ( N O ( 4 Romain Ruzziconi AB o AB ( O AB E g and g + , where and Λ ) AB M r ) q C 2 . x 0). The condition at the at order uu i , and ( 0 − ) . Morover, it indicates that = ) AB u i ( ... AB A q 0 Λ 0. The component A 0 x C β coordinate for the Bondi mass G 0 (3.42) , A N + ( U u u 6= ) ( ∂ C = . It shows that 1 0 Λ D C − 0 β 0 and n AB A ( AB − U 6= E ) ) ∂ g ) Λ u B C 0 4 u B } ∂ x ∂ D A , U + A 0 A ∼ ( N 0 ( = u 0 (but not for , ) C β ( , ) D 3 n A ( AB 0 D 6= ( AB 2 M M cancel between g , ∂ 2 g u , = A + Λ ) Λ ∂ + AB AB . We will not describe these equations explicitly D C are arbitrary functions. We call 12 E AB A AB E x 2 . Afterwards, we solve the components , or AB q , ) N D ) ) for , we see that the iterative solution of the components u B + 0. Finally, the components D l AB ( n x 3 AB Λ q C  , − , − C 6= AB u r D A u AB 0 ) 3.37 ( q C 0 n Λ ∂ A ( AB C , U ( U , ) g ) N h 0 AB C 0 0 A 0 + 0 is parametrized by the set of functions β x β C ≥ have constrained evolutions with respect to the , 2 { U n of that equation yields to 8 u A Λ AB 6= − and A ) ( ∑ e D 1 D at the next order turns out to be a constraint for 2 ) Λ N should have been fixed by the equation found at u − AB r B ] + = ) + ∂ r x E q Bondi mass aspect ( l [ , = ... AB , ( AB u and ) R r O E ( ) and is the scalar curvature associated with the metric C C  AB A + ( 0 x − ] 3 g 2 1 Λ M ) , q 2 U 3 [ h u r AB ( 3.33 0 ( 0 R g β β A 0 u 2 , 2 . Notice that, at this stage, logarithmic terms are appearing in the expansion e ∂ q e U 0. Using an iterative argument as in [149], we now make the following observa- ) , we get 3 , ) 3 √ − Λ = ) 2 6= − ) C r − ln 4 ( x = Λ r ( AB u , ( g ∂ O u r and the angular momentum aspect V . ( O Λ u 0 = β M l . The leading order of the Einstein equations organizes itself as ). However, we will see below that these terms vanish for In summary, the solution space for general relativity in the Bondi gauge with the preliminary ) AB g AB 3.37 value of where arbitrary function called the Einstein equations order by order, therebyof providing us with the constraints imposed on each order Going to which removes the logarithmic term in ( (11 functions), where Therefore, the characteristic initialgiven: value problem is well-defined when the following data are Asymptotic Symmetries in gauge theories In these expressions, momentum aspect to next order is trivial for tion. If we decompose ( ( Accordingly, the form of this is not the case, since both contributions of mined with other subleading data suchfree data as on the boundary, built upno from two more arbitrary data functions exist of tostein be equations uncovered give for some evolution constraints withaspect respect to the boundary condition ( equation here (see [66, 149]). PoS(Modave2019)003 , ) ) C x AB , ( 0 (3.49) (3.44) (3.45) (3.46) (3.48) (3.47) u ( . AB has a fixed A E N , AB ) q ) (AAdS2), the is a function of C and can be ex- gets a time evo- x Romain Ruzziconi , r . Furthermore, by ) are still valid by AB 0 AB 3.24 ˜ u Λ E q ( 0, that is, the solution . Therefore, by this , but this infinite tower 1 3.39 AB = − is not constrained at this r D ... Λ , , ) 0 to make these logarithmic AB , where C , AB ) D ) and ( x = 0 E C , coordinate, and = coordinate (5 functions). , x 0 ). From the previous analysis of is analytic in Λ AB , u u u r , } AB ( 3.38 , 0 D A u of the Einstein equations yield time A D AB = 0). This condition is slightly more 3.15 ( N g Λ ), ( ) ,... D , } , 6= AB ) 0 AB ˜ uA E C AB , 6= Λ ( ˜ . 3.36 x 0 F E Λ , 0 , , 6= } 0 ) + Λ A u , = AB AB and C } ( 0 N x E A ) D and the angular momentum aspect , , , M AB ) (AAdS1), the solution space reduces to 0, the subleading orders of the components , N u = ) becomes . Notice a subtle point here: by taking the flat , q A M uu , ( AB M 13 ) ) u , ( = l ) becomes trivial and N C µν M D AB , 3.22 x , AB , Λ 3.40 − G , D A E M 0 u in asymptotically flat spacetimes [36,43,65,66,93,154, AB , , 3.41 )) and all the terms proportional to u N ∂ E ( , ( have constrained time evolutions (infinite tower of inde- ) + AB { AB 0, we assumed that , the most general solution space can be written as q C M AB C r x , { 3.15 , q becomes unconstrained, and the metric ... , 6= , u , in AB AB ) ( AB Λ q ) C C AB , 1 c { C AB , − F AB u r rC Bondi gauge , ( ( q 0. The radial constraints ( { 0, the components is a fixed value of O AB AB ) + → E C 6= 0 ) is parametrized by the set of functions C , u 0, we can readily obtain the solution space with x Λ Λ , (see equation ( AB 6= u ) where analyticity is assumed only up to order 3.15 A ( 0 D Λ U , AB , have constrained evolutions with respect to the A have constrained evolutions with respect to the where 3.15 q 0 ) (this condition was not restrictive for 2 A N A β r )). One has to impose the additional condition , N ... N , M 3.33 ) ) = , 3.37 C and and C x x AB , , q 0 r M M , u In summary, the solution space for general relativity in the Bondi gauge with the preliminary Let us finally discuss the Notice that for the boundary conditions ( ( u ( AB AB limit of the solution space with F panded as ( boundary condition ( evolution constraints for the Bondi mass aspect of functions is otherwiseFinally, unconstrained as and for they the case become free parameters of the solution space. restrictive than ( flat limit procedure, weg only obtain aall the subsector coordinates of of order the most general solution space. Writing 155]. We first consider the preliminary boundary conditions ( pendent functions). Therefore, thefollowing data characteristic are initial given: value problem is well-defined when the where determinant [66] (7 functions). Finally, forsolution the space Dirichlet finally reduces boundary to conditions ( where of by taking the flat limit setting to zero order. In particular, this allows forequation ( the existence of logarithmic termsterms disappear. in Finally, the one Bondi can expansion see (see of that the for Einstein equations impose time evolution constraints on where Asymptotic Symmetries in gauge theories solution space with the same procedure, the constraint equation ( Therefore, the asymptotic shear lution constraint. Similarly, the equation ( PoS(Modave2019)003 = AB (3.51) (3.50) (3.52) (3.53) (3.54) q u ∂ of the form AB q Romain Ruzziconi . ) asymptotic Killing vec- C is fixed. In particular, if x , q r , √ 0 u ( , obey time evolution constraints. ] 1 AB 1 F , ˜ 2 E , F 0 )), the solution space reduces to ). AB [ ), we immediately see that F ˜ 0 , = E δ R 0 = Λ 2.2 , 2 Λ = } F and + 3.17 } Λ , 3.47 δ AB 2 ) ] } AB ˜ F A C D AB − E ] ˜ 1 x , AB ] E , 2 F ˜ , , 2 A E F δ 0 F , , AB u N [ ) with ( AB 1 − ( , D R AB F ) D , 1 [[ 2 AB , M F A D 14 F A 3.15 , , δ R D , N A )) yields the solution space N , , 1 AB ≈ , ) N F q M , ( C ]] = M , x 2 C , 3.18 , M F 0 , AB [ = AB u C R , and ( AB , C A , A ] , ] C AB 2 1 AB { ˜ ϕ N ). This agrees with results of [36]. Finally, taking the boundary F F E q ) with ( { , [ , obey time evolution constraints, and { ) 1 R [ C F is time-independent, from ( [ x 3.47 AB ˜ 3.15 , q E obey time evolution equations. This agrees with the historical results 0 obey time evolution equations. Notice that the metric u √ ( AB AB ˜ ˜ and 0 was assumed for technical reasons in [51, 52, 65], but this was actually not restrictive. E M E are defined as the residual gauge transformations preserving the boundary , = ) AB C and AB and D x q , , u 0 A )), we obtain of gauge symmetry generators is given by AB ∂ AB u ( N A D D ] tangent to the solution space. In practice, the requirement to preserve the boundary , , , 2 AB ] 3.16 A A F M q . In other words, the asymptotic symmetries considered on-shell are the gauge transfor- [Asymptotic symmetry] Given boundary conditions imposed in a chosen gauge, the [Asymptotic symmetry algebra] Once the asymptotic symmetries are known, we have is the trace-free part of , F 6 , , N N [ 1 ) means that this equality holds on-shell, i.e. on the solution space. In this expression, the , , F R AB C AB [ ˜ c M ≈ q M E , ) automatically satisfies ( ) with ( u We complete this set of examples by mentioning the restricted solution spaces in the different This is the weak definition of asymptotic symmetry, in the sense of ( The condition . ( 6 5 . For boundary conditions (AF2) (equations ( AB 5 3.17 3.15 definitions of asymptotic flatness( introduced above. For boundary conditions (AF1) (equations Now, the characteristic initial value problem is well-defined when the following data are given: ( we choose a branch where 0 where conditions (AF3) (equations ( Asymptotic Symmetries in gauge theories where where where of [43, 154, 155]. 3.4 Asymptotic symmetry algebra Definition asymptotic symmetries conditions mations conditions gives some constraints ontions. the In functions gravity, the parametrizing generators the of residual asymptotic symmetries gauge are transforma- often called tors Definition where bracket C PoS(Modave2019)003 0 ). )) = ) do ) 3.13 0 (3.58) (3.60) (3.55) (3.56) 3.21 (3.57) ( tA g 3.14 ξ L where 0, ) , A ) = ˆ l Romain Ruzziconi Y t ) , where , 0 0 ˆ ) . f ξ Fefferman-Graham ( tt ) A 0 ( g ˆ 2 , + ξ . ξ ξ 0 (3.59) ) ) together with ( , . β A 2 t 0 2 0 ) f L ↔ ˆ = = ξ ξ 2 l ) in the ( ν 1 A 0 ↔ 3.19 , ( ω ( A ∂ ξ ) )] ↔ 1 D 2 A − ( 2 1 = ( ... , 2 ( Y , 1 1 2 f A , − A 0 ↔ ), the constraints on the parame- ν 2 − 2 B )] ξ f 1 ∂ , 1 ) = t 2 ∂ 0 A ξ ( , ( 1 α 2 1 , A ) means that the infinitesimal action 2 ξ 0 A f 0 ξ Y σ ) ξ 3.22 AB ξ ξ ) − spacetimes k , 1 , , δ , A q ) satisfies the Jacobi identity, i.e. the A 1 ) µ 1 0 ) does not impose any constraint on the 2 , 4 2 t Y 0 ξ f , A ∂ , 3 , 3.53 1 , Λ + t ξ 0 1 ) 1 t ( , f 0 Y 2 t A ξ 0 1 ( ξ , − ξ 3.54 ... ( ξ ξ Y 3.19 1 ξ , δ ( B 1 1 f 1 ξ δ ξ ξ f = ∂ µ ) is given by , ( ( ) − δ δ ∂ ) 0 A ξ − ξ ) 2 1 AB ( ) is due to the possible field-dependence of the l [ , , δ − − 2 t A Y 0 ν g 0 f u 3.54 2 2 ξ ξ , ··· − 3 ∂ 15 B Λ A ξ 1 , 0 B 1 ∂ ν 3.53 A 1 ξ ∂ 2 , ξ − ) ) t )( 0 Y 0 0 k , AB ξ ( A AB ( A L µ [36]. Therefore, the asymptotic symmetry algebra can A = in ( q ( g ] D 1 D ··· . The statement ( Y 1 A ξ 1 f = 1 0 1 , 1 [ , A t . Let us mention that a Lie algebroid structure is showing asymptotically AdS f 0 F µ t ξ αβ 0 A ( 2 [ 3 Λ ξ u ] D Φ ξ 2 1 F ] 2 ∂ C 1 3 2 2 δ Λ 1 2 ). 0 ξ − F + , , ≥ A 1 + l 1 2 = 2 − + , ∑ , F f [ 2 ξ k Y 2 2 A f 0 [ , , 3.20 δ F B A A t 0 0 ξ 1 ∂ ∂ ) ξ ξ  F 0 = B A l ) = 1 1 A B δ ( A 2 1 2 ∂ ∂ Y Y D Φ − 1 1 F ] , , , A B 1 2 2 = = 0 0 − 1 F ξ ξ ˆ u f A F δ = ∂ ( ˆ , modified Lie bracket Y = = t 1 0 .  C F t ] 0 ξ A 0 ˆ δ ξ ˆ . In this case, the Lie bracket ( [ F ξ 0. This leads to the following constraints on the parameters: ¯  [ q ) preserving the boundary conditions, namely, satisfying l . The preliminary boundary condition ( is a skew-symmetric bi-differential operator [21, 26] R = 1 2 √ ) ) 2 0 → ln 3.13 Let us start by considering − ab ( F asymptotic symmetry algebra u , u g F ∂ 1 ∂ ξ F  ( = L ) l C 0 ab ( Now, let us consider the boundary conditions (AAdS1) (equation ( g Bondi gauge given in ( µ In the Bondi gauge with corresponding boundary conditions ( The presence of the terms and where ters are given by and the asymptotic symmetry algebra is written as asymptotic symmetry generators. We can verify that ( generators of the residual gauge diffeomorphismsSimilarly, of the the generators Fefferman-Graham of gauge the given residual in gauge ( diffeomorphisms in Bondi gauge given in ( in the Fefferman-Graham gauge.ξ The asymptotic symmetries are generated by the vectors fields Asymptotic Symmetries in gauge theories where asymptotic symmetry generators form a (solutionis space-dependent) called Lie the algebra for thisof bracket. the It gauge symmetries on themetry fields generators: forms a representation of the Lie algebra of asymptotic sym- not get further constraints with ( and is referred as the be worked out and is given explicitly by up at this stage [21, 21,Lie 72]. algebra The is base the manifold Lie is algebrais of given the asymptotic by map symmetry the generators solution introduced space, above the and field-dependent the anchor and Examples PoS(Modave2019)003 , ) ) 4 = AB 3.60 3.56 (3.63) (3.64) (3.66) (3.65) (3.61) (3.62) g ) in the ξ L reduces to 3.57 4 . Furthermore, and global dS ) become AB 4 Romain Ruzziconi are called the su- algebra [66]. The , -BMS q 4 ) A Λ B , 3.59 Y x A 0 is a conformal Killing ( are the residual gauge ξ A A ) . The asymptotic Killing 0 µ -BMS 0 Y , ξ ( A ξ 0 Λ = D , = 1 2 A 0 ). In the Bondi gauge, Dirichlet Y ω = = σ , . A , 3.58 t  0 Y . AB Bondi gauge u ξ )] ˚ , q ∂ t ). The equations ( B 0 AB 0 [30] (see also Appendix A of [66]). C ω ˚ 0 ∂ ξ q 2 ξ B C > ) , , while the additional constraint AB . The asymptotic symmetry algebra ( ∂ 0 3.24 in the ˚ Y − ) ( q C ω Λ 0 C AB A AB AB ( 3 D Λ q ˚ q D − q Y A − A = , which indicates that 3 Λ = ) 0 D Y A 2 = ( A − 16 A ξ − 4 1 Y ) A D 0 B ρ = − ( B follows closely [36]. Y ( 1 2 asymptotic Killing vectors q D u A o [ D 7 ). Compared to the above situation, the equations ( 0 ∂ algebra for Y = + u u + = ) d f B ∂ 0 ) B 4 ) together with ( . The computation of the modified Lie bracket ( , u Y 3.23 0 , 4 ξ 0 A AB (  A ) 1 l Z g , 0 ). The Y ( D A ( ξ 2 1 A A 3.20 1 2 0 D ξ D L SO − ) -BMS 3.15 0 2 1 u A ( ) + Λ ∂ A D = x  1 2 ( f 0 and T u asymptotically flat spacetimes = ∂  < t 0 1 4 ) with the following constraints on the parameters: ) together with ( ξ q Λ u is a conformal Killing vector of ˚ ∂ = . These equations can be readily solved and the solutions are given by 3.14 3.19 A f q Y √ are called the supertranslation generators, while the parameters ln , namely, f u is the covariant derivative associated with the fixed unit sphere metric ˚ AB ∂ is the covariant derivative with respect to ˚ ) q algebra for 0 A ( A = ) D l D 2 gives , This completes the results obtained in [66] where the asymptotic symmetry algebra was obtained by pullback Let us now consider Let us consider the Fefferman-Graham gauge with Dirichlet boundary conditions ) 3 7 2 ( r ( o where This means that methods. remains of the same form.SO It can be shownTherefore, that we the see asymptotic how symmetry the algebra infinite-dimensionalthese corresponds finite-dimensional asymptotic algebras, to symmetry which algebra are the symmetry algebras of global AdS vectors can be derived into a proceed similar is way to to takethe that the expressions flat in limit obtained the of previous by the examples.liminary following previous these boundary results Another two conditions obtained way in equivalent in ( the procedures. which Bondi gauge. First, We consider sketch the pre- perrotation generators. These namesasymptotic symmetry will algebra be justified below when studying the flat limit of this reduce to there is an additional constraint: vector of ˚ boundary conditions are given by ( respectively. diffeomorphisms ( where (AAdS2), that is, ( Asymptotic Symmetries in gauge theories This asymptotic symmetry algebra is infinite-dimensional (in particular,diffeomorphisms it as contains a the subgroup) area-preserving and field-dependent, andparameters it is called the Bondi gauge for these boundary conditions where The asymptotic symmetry algebra remains of the same form as ( PoS(Modave2019)003 ) and (3.67) (3.69) (3.72) (3.70) (3.71) . Fur- A to not be Y 3.19 A ), we obtain Y 3.57 Romain Ruzziconi ), i.e. considering (AF1), . If we allow ). The additional constraint AB are the smooth supertransla- 3.15 q ). This extension of the global algebra reduces in the flat limit are the abelian Weyl rescalings , S 3.15 algebra (see below) is called the ) 4 2 R 4 3.67 and , ↔ A 1 . -BMS AB , and ( Y ˚ q R Λ T C − , × , Y , A 2 ) ) superrotation generators. Notice that there is C Y ∗ S [36]. Finally, as a sub-case of this one, con- 2 S A D A 0 (3.68) n S Y n D ↔ = 1 ) ) f = is the algebra of the globally well-defined confor- 2 n 1 1 A 17 , ( S ) 1 2 Y ω 3 B 1 ), i.e. (AF3), and allowing only globally well-defined where ( − Vir , algebra algebra. + D ) 3 A 2 2 4 4 × ( ˆ Diff( f SO ω Y + [43, 154, 155], which is given by 3.18 , A B B A SO ∂ ∂ ) to the preliminary condition ( Vir Y ˆ A B ( Y . 1 1 A , 0 Y Y ˆ f D 3.16 ( = = = ξ algebra ˆ f A ˆ ω 4 ˆ at this stage. Computing the modified Lie bracket ( = Y ) of the BMS A A extended BMS Y [51,52,65,94]. Therefore, the )] . Note that the supetranslations also contain singular elements since they 2 ) to the preliminary boundary condition ( 3.69 ω ω , A 2 3.17 . This extension of the original global BMS Y , global BMS T algebra 2 f 4 ( ξ are the supertranslations, parametrized by ) are the smooth superrotations generated by , ∗ 2 ) Vir is the direct product of two copies of the Witt algebra, parametrized by 1 S S ω are the supertranslations and × , are called supertranslation generators and A 1 is a conformal Killing vector of the unit round sphere metric ˚ algebra is called the S T , parametrized by Y A , 4 )). The asymptotic symmetry algebra reduces to the semi-direct product Now, we discuss the two relevant sub-cases of boundary conditions in asymptotically flat The other sub-case of boundary conditions for asymptotically flat spacetimes (AF2) is given 1 AB Y f , we recover the q ( A 3.21 ξ to the smooth extension ( spacetimes. Adding the condition ( tions generated by generalized BMS BMS sidering the more restrictive constraints ( mal Killing vectors of theLorentz group unit in 2-sphere four metric, dimensions. which is isomorphic to the proper orthocronous by adding condition ( gives the additional constraint ( where Diff( on the parameters is now given by where Note that this case corresponds exactly to the flat limit of the (AAdS1) case (equations ( are related to the singular superrotations through the algebra ( Asymptotic Symmetries in gauge theories where [ of ˚ no additional constraint on i.e. globally well-defined on the 2-sphere, then the asymptotic symmetry algebra has the structure Here, Vir thermore, Y PoS(Modave2019)003 ) 2 , , the 3 (4.1) , ( (3.73) on the BC , SO ) C a A asymptoti- ( Y F A of the theory D = AC B Romain Ruzziconi C ) by inserting the D F C BC ) as 3.2 D . f N 1 8 8 symmetry B 3.52 + D 1 2 AB ) is a local function. In agree- N + , BC B Q f AB BC asymptotically flat spacetime fC . C f D C C ) B D A ab f , with generators BC T C D D C C D )) of 1 2  f N D c D 0 C AB A ( + ξ ˚ q ) A D D f 3.18 0 , B c ( D B + 3 Q D , 16 D 8 3 f D )), the asymptotic symmetry algebra B 1 3 A B + BC d − f D D f ) + C + ) with ( 18 A = B C c 3.23 A 0 with Dirichlet boundary conditions for AB of the fields, where BC ξ D D D L N 2 ) C 2 Q Q L 3.15 ( C 1 4 MD δ − AB  3 1 4 D ˚ = q + A f AB = + ) with ( + Φ D C M C A ] ] Q ab D N ), this transformation is said to be a − δ CD C C T ] C c 3.19 0 C Y Y C ξ AC D C C 3.3 δ Y CD C 1 2 D D C ) as C C 3 2 2 1 D B − D f B Y + − + 3.32 B B D . The form of this action can be deduced from ( Y Y Y ( ) D D f b L L L A (AAdS2) (( A ( B D D + + + ˜ [36]. For the action of the associated asymptotic symmetry group on these Φ D ( u u u Fefferman-Graham gauge with definition (AF3) (( 1 1 32 4 2 3 ∂ ∂ ∂ = action of the asymptotic symmetry algebra AB f f f + − − C ˜ Φ u and a transformation = [ = [ = [ ∂ algebra acts on the leading functions of the solution space ( ] In the A 4 [Global symmetry] Let us consider a Lagrangian theory with Lagrangian density spacetimes = M AB [Action on the solution space] Given boundary conditions imposed in a chosen gauge, N ) ) C 4 ,... Y ) , Y AB , f Φ Y f , ( N ( f µ Bondi gauge δ ( δ ∂ This action is usually not linear. However, in three-dimensional general relativity, this action is precisely the In this section, we review how to construct the surface charges associated with gauge sym- δ , 8 Φ [ if coadjoint representation of the asymptotic symmetry algebra [27, 28, 33, 34, 138]. ment with the above definition ( metries. After recalling some resultsBrandt about prescription global to symmetries obtain and the Noethercussed. surface currents, We charges the illustrate in Barnich- this the construction(A)dS context with and of asymptotically the asymptotic flat example spacetimes. symmetries of The isphase general relation space dis- relativity between methods with this is asymptotically prescription established. and the covariant 4.1 Global symmetries and Noether’s first theorem Definition L solution spaces, see [35]. 4. Surface charges global BMS where Asymptotic Symmetries in gauge theories Definition there is a natural solution space solution space and the explicit form of the asymptotic symmetry generators In the Examples cally AdS acts on the solution space ( PoS(Modave2019)003 0. ≈ (4.8) (4.2) (4.5) (4.6) (4.3) (4.4) j . The ] F [ R . In particular, global symme- Romain Ruzziconi -form, i.e. d ) 1 for this equivalence  ] − j ... [ n ( + is just a symmetry that de- ) µ Q j Φ L µ µ ∂ ∂ ∂ global symmetries ( ∂ ≡ . We define an on-shell equivalence ] Q .  ) together, we obtain F x  [ n ... µ R d 4.6 ∂ ) , + = µ Q , ... ] + )  gauge symmetry . B , 0 (4.7) ] Q F is an on-shell closed + Φ µ  j [ [ K L ... j µ ∂ ) ) and ( R d ∂ ∂ ), a is an equivalence class ... ] = ... + Φ ( + µ L + , i.e. 4.5 µ = ( ) 1-1 ∂ + 19 + ) ∂ j 3.2 ∂ ∂ ←→ Q , ) Q Φ α ( ) Q ] between the currents as L f Q µ ∼ B ∂ Φ ∼ Φ Q δ ∂ ∂ − j L d [ [ µ L ∼ µ ( Φ µ Q ∂ ∂ Q ∂ ∂ = ( ∂ µ ( = ( B ∂ ∂ F Q ∂ L  Q µ ). Putting (  Q δ Q µ ∂ δ µ µ ∂ + ∂ ∂ − B.5 conserved current = Noether current L Φ + + L Φ , which can be written as , we obtain ∂ ] ∂ L Φ ∂ ∂ x j L L Φ Φ [ δ n δ Φ ∂ δ  δ ∂ d Q Q L δ Q Q Q . Then, as defined in ( = for this equivalence relation are called the µ = = = = ) ] -form. A L x L ) Q 1 Q [ 2 − δ n − d ( n ( µ Q B Noether currents This theorem also enables us to construct explicit representatives of the Noether current [Noether current] A between the symmetries of the theory as = [Noether’s first theorem] A one-to-one correspondence exists between is a Q ∼ and K B and, in the last equality, we used ( i.e. two symmetries areequivalence equivalent classes if they differ, on-shell, by a gauge transformation Furthermore, writing where, in the second line, we used pends on arbitrary spacetime functions relation a gauge symmetry is a trivial global symmetry. Definition We define an on-shell equivalence relation where In particular, Noether currents associated with gaugeof symmetries are this trivial. theorem Recent can demonstrations for example be found inRemark [22, 26]. for a given global symmetry. We have Asymptotic Symmetries in gauge theories where relation. Theorem tries Q PoS(Modave2019)003 ), 4.9 (4.9) ), and (4.11) (4.14) (4.15) (4.10) (4.12) (4.13) B.7 x ) together n  d 4.1 L Φ  δ δ L Φ k ), we obtain Romain Ruzziconi µ δ δ 1 ... , 1 B.8 2 Q µ Q ∂ Φ  ∂ 1 , µ Q  T k  through the correspondence . µ µ 0 L ∂ Φ ∂ Q δ δ = + ... k µ 1  x µ 1 ... n  1 ∂ L Φ d Q k µ L Φ δ ) δ ∂  Φ δ 1 δ k ) together results in , ] . µ ∂ , 2 L 2 Φ 2 − 2 1 ... Q ( Q δ Q δ  , 1 j 0 j , Q 1 x Q k , 1 µ 1 4.14 ≥ n Q Q 2 µ ∑ ∂ Q  k [ Q j Φ d j ∂ ). In the last equality, we used ( δ 1 2 Q δ d ∂ Q d  Q  = ... = T  1 B.9 L Φ 1 20 ≈ } ≈ µ − } ) and ( 2 , Q µ 2 δ 2 − δ 2 L ) Φ T ∂ Q Q ] Q  , Q δ k 2 j µ k δ 0 holds on-shell. Hence, we have obtained a represen- L j 2 j , ) µ 1 , Q ∂ L on the left-hand side and the right-hand side of ( 4.12 Φ d , 1 Q 1 1 ∂ Q Q ≈ 1 1 δ . In other words, the Noether currents form a representa- δ Q 1 L Φ Q + δ Q Q j  − 1 j [ , Q Q ( δ δ ( ... j δ { 1 j 2 { δ Q L 1 0 µ Φ 1 Q Φ 2 Q Q δ µ ≥ δ − δ T ∑ Q k δ δ ∂ 2 )  µ δ 2 2 2 0 1 Q 1 ∂ j ≥ Q Q Q Q Q − 1 ∑ k ). Putting ( 2 + 2 Q T − + + Q δ − d Q L δ Φ 4.9  1 L L L Φ Φ Φ δ + δ L Φ Q d − δ δ δ ] ] δ δ δ . On the right-hand side, using the first equation of ( δ δ δ 2 2 2 2 2 2 2 2 Q , Q Q Q Q Q Q 1 . In particular, d , 1 Q 1 1 1 ] = Q 1 1 µ Q [ Q Q Q 2 ) j Q is an expression vanishing on-shell. In the second equality, we used ( δ δ δ δ Q x Q δ d 1 ,  = = = 1 − L = [ = ( = = Φ n δ δ Q  d [  , ( 2 L Φ µ Q L Φ j Q δ δ δ δ 2  [Noether representation theorem] Defining the bracket as 2 ). Now, using Leibniz rules in the second term of the right-hand side, we get = 1 is replaced by Q µ Q Q Q  Q j T  B.6 1 1), where 1 To prove this theorem, we apply Q ). Q δ > δ n 4.4 tion of the symmetries. ( where with ( in the last equality, we used ( we have where On the left-hand side, we have where, to obtain the second equality, we used ( tative of the Noether current associated with the global symmetry Theorem Asymptotic Symmetries in gauge theories or, equivalently where ( PoS(Modave2019)003 if by n 0 α (4.19) (4.16) (4.17) (4.18) (4.20) ∼ ) yields n α 4.15 is given by ) is conserved -form is exact, p dR n Romain Ruzziconi H 4.18 Noether charge . , as ] Σ Φ [ ∂ 0 and because Noether . However, this cannot 9 Q ). Notice that in classical ≈ H for the operator d defined in  , 4.11 ) L , Φ ] = d η δ δ ( K Φ d , [ p Σ 2 0 Q + ∂ H Q Z n H   1 < L Φ . ] + Q , we can construct a , with boundary δ n p 0 δ n ] T Φ j Σ , [ [ ≤ . 2 = = . < Q j 0 p 0 Q p p H Σ = <  ≈ Z if if -forms for the equivalence relation 0, we have p 1 ] n ] Q ) = if Φ n = ] = [ 21 T K α Q 0 if 0 R [ 0R if 0 Φ d K [ + H Σ ] ( Q + 2      ∂ d j dt H Q R = . , ( 1 Σ p dR Q ). Applying the algebraic Poincaré lemma to ( ) = -forms, we obtain the result ( [ Z ) j H d ( 1 p = ] = 4.11 − 2 ), constant central extensions may appear in the current algebra. H Φ n Q [ ( j 0 Q 1 )) that takes into account the field-dependence. In this context, we Q 4.16 H δ B.3 -dimensional spacelike surface ) 1 -form. Therefore, on-shell, since − ) 1), from ( 2 n [26]. ( = 1 − n − algebraic Poincaré lemma n n ( β d designates the equivalence classes of [Noether charge] Given a Noether current ] is a + [Conservation and algebra of Noether charges] The Noether charge ( n [Algebraic Poincaré lemma] The cohomology class n 0 α η [ α The Poincaré lemma states that in a star-shaped open subset, the de Rham cohomology class Le us go back to the proof of ( ) is given by 9 = n B.3 mechanics (i.e. where we used the Stokes theorem. Since Remark ( where where currents are defined up to exact Definition integrating it on a If we assume thatdepend the on the currents representative of and the their Noether current. ambiguities Indeed, vanish at infinity, this definition does not in time, that is, have to use the Lemma Asymptotic Symmetries in gauge theories We know from Poincaré lemmabe that the locally, case every in closedand Lagrangian form therefore, is field there exact would theories. notoperator be d In that any we fact, possibility are of this usingjet non-trivial is would not dynamics. bundle the imply (see standard Let that definition exterior us derivative, ( every but remark a that horizontal derivative the in the α PoS(Modave2019)003 . α f (4.23) (4.27) (4.22) (4.26) (4.21) (4.24) (4.25) Romain Ruzziconi  ... + on this equation. Since ,  . In the second equality, is a set of arbitrary func- α 0 2 0. We have f L . Φ Σ F L Φ } ≈ δ  δ = δ ) δ Q 2 ... j t  and µν d ( α . + 1 1 Since 2 ≡ R ... Σ Q Σ 2 ,  j −  ) + 2 1 Σ ... L ν µ Φ F Σ Q α ∂ , δ S Z δ + δ ] f ) for a gauge symmetry: µ 2 ) Σ ν  − ∂ . = Q Z , . ∂ L µν Φ µ 0 and 1 µ F F 4.9 ( α 0 Q j µ + j δ ) with respect to δ Q j = [ R ∂ ) ( µ = 1 ) µ  F 2 = µ H ∂ Σ  S 1 µν Q B.5 ∂ t {z L Z µν ( α Φ ν ≡  ( α H = δ ∂ R δ 1 ≡ = − R L Φ } ≈ 22 α  µ α Q L 2 1 Φ f Q δ δ δ  ν + R j Q δ Σ δ ∂ 2  ] α L −  Φ H = Σ µ f , † α δ F µ δ Z ∂ 1 [ } L Φ µ R ∂ 2 Q R  δ ∂ . There is one identity for each independent generator δ + Q H µ − α † α ] = α  H { R f R , ). Φ L L Φ Φ 1 [ α δ µ α δ 1 δ + δ Q f t Q R µ α α α H |  H 4.11 f R { R ) form a representation of the algebra of global symmetries, i.e. µ  α −  ∂ ] ) can be rewritten as µ R α ∂ f +  Φ 4.18 [ 2 − t = = Q 4.24 Noether identity L Φ H L δ Φ δ δ δ α ] R F [ = R is the spacetime volume encompassed between  1 L Φ Σ δ δ [Noether identities] Consider the relation (  − )) and we obtain 2 † α Σ R B.6 The Noether charges ( This is a direct consequence of ( we used the hypothesis that currents vanish at infinity and the Stokes theorem. This identity is called a Notice that these identities are satisfied off-shell. where the bracket of Noether charges is defined as where the right-hand side is a total derivative, it(see vanishes ( under the action of the Euler-Lagrange derivative where 4.2 Gauge symmetries and lower degree conservation law Definition The left-hand side can be worked out as Therefore, the equation ( tions, we can apply the Euler-Lagrange derivative ( Asymptotic Symmetries in gauge theories In fact, consider two spacelike hypersurfaces PoS(Modave2019)003 )). ) into 4.3 ), the (4.34) (4.28) (4.29) (4.32) (4.33) (4.30) 4.4 4.27 . The Euler- x . n ) is completely x d n g Romain Ruzziconi d ] − 4.34 g . √ ) is given by 0) that was defined in − F ) Λ √ K is obtained by following ), one may propose the ≈ ) 2 4.28 µ F Λ − . ξ S x 4.18 µν R n . ( g d µ 1 g ) + − appearing in ( x ) − 1 F µν G − √ n x π ) K G d n , ( Λ d ν 16  ), taking the Noether identity ( F )( g ξ 0, and hence, from the algebraic Poincaré µν L Λ K Φ , 2 − g [ = ( δ F Σ δ -form µ µν 4.25 ∂ ) √ 0 (4.31) + ) =  ∂ K L g ) Z 2 F d F = Λ − + µν S S − ≈ + x d G µν n − n µν F µν 23 ( F ( j g d 1 F = S G G j Σ g − µ ( + ( Z ) and Stokes’ theorem. This charge will be conserved ) = L − ν Φ ∇ ξ G δ µν δ = F √ ] j g π G G F 4.33 F ( µν − [ π 16 H ν ( G 8 R √ ξ µ µ − − ∇ ∇ ν = ), we have d 2 = ξ − 2 ξ µν L S 4.28 g = = δ δ is the weakly vanishing Noether current (i.e. µν L g -form. Therefore, as already stated in Noether’s first theorem ( δ µ ) and ( ) δ ) 2 0. The problem is that the x 1 ). This is a direct consequence of ( µν − ≈ − g 4.24 n n ): ξ F d ( j δ 4.25 ( ) µ F G S ), 4.25 Consider the theory of general relativity is a From ( π [Noether’s second theorem] We have = F 16 4.16 F ( -form. A natural question arises at this stage: is it possible to define a notion of conserved K S ) 1 − n where, in the second equality, we used ( the lines of ( Noether current associated with a gauge( symmetry is trivial, i.e. vanishingquantity on-shell, up for to gauge an exact symmetries?following definition Naively, for conserved following charge: the definition ( on-shell since d arbitrary. Indeed, the NoetherTherefore, we currents have are to find equivalence an classes appropriate procedure of to currents isolate a (see particular equation ( where the last line of ( account. Example The Noether identity associated with the diffeomorphism generated by Remark lemma ( where Asymptotic Symmetries in gauge theories Theorem Lagrange derivative of the Lagrangian is given by Therefore, the Noether identity is the Bianchi identity for the Einstein tensor and the weakly vanishing Noether current of Noether’s second theorem ( PoS(Modave2019)003 . - ) γ , if 2 0 − K (4.37) (4.36) (4.40) (4.41) (4.35) (4.38) (4.39) equiv- -form. n ) ∼ ( 2 K of interest. − . Note that 0 n ¯ Φ ( F as ] ≈ Romain Ruzziconi Φ ¯ [ F ) of the homotopy ¯ F ) as the fundamental K , if 0 satisfying the off-shell B.18 ¯ F 4.40 ) [18, 22]. From Noether’s x ∼ ( ¯ -form ¯ F ¯ ) F F 0 to the solution 2 − are parameters of gauge transfor- ] = n ¯ ( ¯ F Φ defined in ( [ ] ¯ F , . Φ ] S ¯ F δ are said to be equivalent, i.e. Φ S . ; 0 1 δ 0 ) and is defined up to an exact generically depends on the chosen path Φ ; Φ . − K [ ] n δ B ] ¯ ≈ . . Φ F I . [ ), using the property ( K 0 0 ¯ 0 Φ k [ ¯ F 10 F − [ and ¯ S such that k F ≈ = γ ] = ] 4.39 24 ¯ ¯ K K = . 1-1 Φ ] = F ¯ ¯ Z equivalence classes of on-shell conserved F F ←→ ¯ S [ [ F -form Φ ] d ) R R ¯ δ K F ] = 2 ) of ; [ and d ), we get ) is a matter of convention. Φ − Φ [ − -forms [ ¯ n ) F (see appendix ¯ are said to be equivalent, i.e. F 4.41 ( Reducibility parameters 2 4.39 4.16 1 0 k K , ¯ 2 F − ), we define − n n ) gives back ( ( ), we have Ω -form [23, 24]. and ) B.16 3 ¯ 4.41 F 4.36 − n . ( ] Φ [ -forms for given exact reducibility parameters ) and ( ¯ ) is a F 2 l K exact reducibility parameters − 4.28 is an element of n ( ] Φ where l δ The Barnich-Brandt procedure allows for the construction of explicit representatives of [Reducibility parameter] [Generalized Noether’s theorem] A one-to-one correspondence exists between ; d , which can be written as ] is a path on the solution space relating Φ + [ K 0 ¯ γ F [ K k The minus sign on the left-hand side of ( 10 ≈ K second theorem ( the conserved Remark In this statement, two conserved Theorem We will call them alence classes of reducibility parameters forms This Therefore, in practice, we consider the condition This enables us to find an explicit expression for the conserved where Applying the operator d onoperator. ( Notice that the expression ( object, rather than Two reducibility parameters for a large class of gaugedimensions theories superior (including or electrodynamics, equal Yang-Mills to and three general [22,26]),parameters relativity these in are equivalence classes determined of by asymptotic reducibility field-independent ordinary functions From the algebraic Poincaré Lemma ( Asymptotic Symmetries in gauge theories Definition mations satisfying Using the homotopy operator ( PoS(Modave2019)003 asso- ) and -form (4.44) (4.46) (4.45) (4.43) (4.42) ) F 2 3.6 k − n ( ) are the gen- -form ) Romain Ruzziconi 2 -form will not be 4.43 ) − 2 n ) becomes ( − n ( 4.44 , ) does not hold; therefore, the is essentially the same as the  . We can show that F F -form ( S 4.38 ) µν ) can be constructed explicitly and µ 2 h x ∂ . d − + , µ ∂ 4.41 n , , ) F ( 0 0 µν x S  ¯ 1 g 1 ) = = − Φ − , namely a gauge theory involving only first n = n Φ δ in the gauge transformations as in ( , the d I µ ) µν µν -form ( ( ∂ i ) ¯ is defined as ∂ ) g g µν is the weakly vanishing Noether current defined − α ( φ ¯ ¯ L 2 Φ ξ ξ g f F F ∂ δ δ . The prescription to construct the 25 S L L − , which satisfy ] = = ( α Φ ¯ n ξ f Φ = ( = = 3.4 ( δ Φ µ δ α 1 2 F ; ) and R µν µν − Φ g h [ ¯ ¯ = ξ ξ F δ δ . Note that for a generic metric, this equation does not admit F ] = B.16 k S Φ µν -form for asymptotic symmetries] The ) g δ 2 ; Φ first order gauge theory − [ n F ( k in the conservation law. ) is not valid anymore. Nonetheless, as we will see below, we still have a ). For a -sphere at infinity. This gives the ADM charges of linearized gravity [22]. ) 2 is taken to be the Minkowski metric, then the solutions of ( 4.39 4.25 − µν n g ( breaking Let us return to our example of general relativity. The exact reducibility parameters of . If ¯ is the homotopy operator ( associated with generators of asymptotic symmetries [Barnich-Brandt 1 ] µν Φ − Φ g n δ I δ , We now come to the case of main interest, where we are dealing with asymptotic symmetries Φ [ F where The simplicity of these expressions motivates thein study this of first context order [31, 32, formulations 95]. of gauge theories k one introduced above forconserved exact on-shell. reducibility Indeed, parameters. for a generic However, asymptotic this symmetry, ( ciated with asymptotic symmetries generated by in the last line of ( 4.3 Asymptotic symmetries and surface charges in the sense of the definition in subsection control on the erators of the Poincaré transformations.integrated The on a weak equality in ( order derivatives of the gaugefirst parameters order equations of motion for the fields where Definition Asymptotic Symmetries in gauge theories Example the theory are the diffeomorphism generators i.e. the exact reducibilityground parameters of ¯ the linearized theory are the Killing vectors of the back- i.e. they are the Killing vectorsany of solution. Hence, thelinearized previous general construction relativity is around a irrelevant background for this general case. Now, consider PoS(Modave2019)003 Φ δ (4.51) (4.48) (4.49) (4.50) (4.47) 0, since [22]. Notice ≈ µ µ F , h S Romain Ruzziconi )] δ = σ . ξ h µ  ∇ L Φ δ δ σ µ νσ ] h h F ν [ + ∇ R ν σ .  ξ ξ  ] ,  Φ σ ] ), we deduce the explicit expression L + Φ n δ Φ  ∇ . I L Φ Φ δ δ ] δ δ δ L σν Φ − δ , µσ Φ Φ F , h 4.32 δ δ ] h Φ Φ and its inverse, and δ S δ ] = σ , F δ d δ ] [ +  F ∇ ; [  F Φ µν  F R µ µ F n Φ δ [ S R g Φ ; n I L Φ δ Φ [ S ξ 1 ξ R I n δ  ] δ d Φ δ ν ; Φ − I − [ + W n F δ Φ Φ [ ] ∇ 2 1 I Φ Φ F n n δ δ h [ R F h W d I I [ i 26 S δ µ is a solution of the Euler-Lagrange equations and ( R 1 2 − δ i W 2 1  ≈ ∇ ] = ] Φ ] ν F = ≈ ≈ ≈ − ≈ − ≈ Φ + [ invariant presymplectic current as ξ Φ ] = [ n R δ I δ , Φ ; µν Φ δ ) Φ ; . x δ ] = [ obtained in equation ( ] ). In the third equality, we used the fact that 2 ; F Φ Φ ξ [ − Φ k Φ n δ S [ F δ d , d , k ( B.18 W d Φ Φ g G δ δ ; − ; π 8 Φ √ Φ [ [ , it is implied that ). W ≈ conservation law W ] ] = ] F ). In the fifth equality, we used h F [ [ ; R 4.48 R i g i . Indices are lowered and raised by [ 4.28 ξ µν k ] = g δ Φ Let us consider the theory of general relativity. Applying the homotopy operator on the δ = , [Conservation law] Define the ] F µν [ h R ; The proof of this proposition involves the properties of the operators introduced in appendix is a solution of the linearized Euler-Lagrange equations. Furthermore, we use the notation Φ [ . We have Φ The proof of this statement canused be the found definition in ( appendix A.5 of [25]. Finally, in the sixth equality, we is a solution of the linearizedsecond Euler-Lagrange theorem equations. ( In the fourth equality, we used Noether’s B We have the following where, in the equality δ W In the second equality, we used ( Asymptotic Symmetries in gauge theories Example weakly vanishing Noether current where that this expression has also beenin derived [31] in (see the also first [32]). order Cartan formulation of generalTheorem relativity PoS(Modave2019)003 , ] ¯ ¯ Φ Φ [ N (4.56) (4.52) (4.55) (4.53) (4.54) . Φ )). Furthermore, Romain Ruzziconi 4.54 , ] is independent from the -dimensional boundary. to the solution ¯ , and divergences when Φ ) ] . [ r integrated surface charge ¯ 2 -exact, i.e. if . 0 Φ Φ ] N . It satisfies the 2-cocycle δ [ ] − Φ ≈ F ¯ n Φ . ] δ ] + [ ] ( H , 1 ¯ Φ ] Φ F Φ [ ; . 2 its 2 2 F ] δ if it is F [ F F ; ; ¯ δ Σ Φ A R K [ ; ] Φ ; ∂ 1 2 [ F F Φ , ; F Φ [ 3 1 [ . 1 ] = k F F F [ Σ ¯ Σ W K Φ k K ∂ [ ∂ Σ 2 Σ Z + F integrable Z ∂ ; γ A ] + 1 . Two types of divergences may occur: di- Z ] Z F 2 as ¯ ≈ Φ F K [ , ] ] = 1 1 -surface and 1 F ] = F Φ Φ 27 [ ; ) finite F [ -surface ¯ A δ 1 Φ ] ) H F H , which depends only on the reference solution [ ; 3 2 ] 2 F H − , F , i.e. N Φ ¯ 2 / Φ [ 2 ) emphasizes that the infinitesimal surface charge is not δ δ − n [ F [ } ≈ ( F F 2 is said to be ). Integrability implies that the charges form a representa- n 2 F ] + = k K ( ; ] . As explained above, this criterion enables us to define in- F 1 Σ Φ 4.52 } F H Φ [ ∂ 2 [ , and be a ] + 4.53 K F F Z 1 F ¯ F 1 in ( Σ Φ H H H [ F , H 3 / δ / δ 1 δ ] = F { F γ ; A Φ Z ] H [ integrable 2 { F F , 1 ] = H F [ / δ Φ K [ F H . The symbol ] Φ [ [63]. infinitesimal surface charge F γ H In the literature, there are several criteria based on properties of the surface charges, that [Surface charges] Let δ [Charge representation theorem] Assuming integrability, the integrated surface charges is a path in the solution space, going from a reference solution ] = as γ tion of the asymptotic symmetry algebra, upthe to integrated a charges central generate extension the (see symmetries ( on the solution space. tegrated surface charges as in ( The charges have to be The charges are usually required to be vergences in the expansionperforming parameter the defining integration on asymptotics, the say ] Φ For the proof of this theorem, see e.g. section 1.4 of [63]. [ Φ • • F [ F H / We define the Remark make a choice of boundary conditions interesting. The main properties are the following: necessarily integrable. If it is actuallyH integrable, then we can define the Therefore, the integrated charges form aa representation central of extension the [22, 25]. asymptotic symmetry algebra, up to Asymptotic Symmetries in gauge theories Definition The infinitesimal surface charge δ is antisymmetric with respect to Furthermore, the central extension condition where is a chosen value ofNotice that the for charge integrable infinitesimal for charge, this the integrated reference charge solution, which is not fixed by the formalism. chosen path In this expression, the integrated charges bracket is defined as Theorem satisfy the algebra PoS(Modave2019)003 . can be as a cut- r r , encircling a r Romain Ruzziconi Bondi mass loss contain some interesting physical ] Φ δ ; . Indeed, since the integrated surface Φ [ at a constant finite value of may also contain relevant information (see F k Σ r ∂ in terms of the expansion parameter ). leads to an infinite volume; therefore, it does not 28 ∞ -form 4.49 ) 2 → r − in time when the integration is performed on a spacelike n ( at infinity. This statement is not guaranteed a priori because not be finite of the charges contains some important information on the Σ ∂ in the generically non-vanishing r conserved of the charges may be circumvented by different procedures to isolate ). 2.3 non-conservation -dimensional surface ) 2 non-integrability − n The fact that the charges may charges generate the symmetry, identicallythe vanishing solution charges space. would imply In trivial particular,charges action the on asymptotic identically symmetries vanish for are whichsymmetry associated considered group integrated ( as trivial in the strongThe charges definition have of to be asymptotic ( The charges have to be of the breaking in the conservation law ( expected when the asymptotic regionoff. is taken It to makes be sense at to infinity. integrate on Indeed, a consider surface finite volume. Then, taking the limit come as a surprise thatthat quantities subleading diverge. orders Furthermore, in it has recently been shown in [97] This tells us that theation mass through decreases the in boundary. time at Hence,information future the on null non-conservation the infinity of dynamics because the of of charges the contains system. a important flux of radi- information, such as the 10reasonable conserved Newman-Penrose to charges think [133]. that Therefore, overleading it orders seems in physics. For example, in asymptotically flatin spacetimes time at null of infinity, the the non-conservation charges associated with time translations is known as the an integrable part in thefinal expression integrated of surface the charges obtained chargesthat by (see integrable these e.g. charges procedures would [172] have. do and In notically [65]). particular, have hold. the all Another However, representation philosophy the the theorem is does properties to notmaking keep gener- working any with specific non-integrable choice expressions, without forstill the possible to integrable define part a ofasymptotic modified the symmetry bracket algebra, charges. for up the to In charges, aever, some leading no 2-cocycle situations, general which to representation may a it theorem depend representation is exists on of inmade fields this the [170]. [39, context, 65]. even How- if some progress has been Finally, the e.g. [64, 70]). The However, even if these requirements seem reasonable, in practice, some of them may not • • • • • be satisfied. Indeed, asfinite, we neither integrable, will nor see conserved below,requirements: [39]. the BMS We now charges discuss in the four violation dimensions of are some of not the always above Asymptotic Symmetries in gauge theories Even if the charges have thesecould pathologies, be they seen still as offer important interestingproperties insights in combinations on their of the transformation the system. (see e.g. elements They [34, of 35]). the solution space that enjoy some PoS(Modave2019)003 . In- (4.62) (4.61) (4.63) (4.58) (4.59) (4.60) (4.57) ) (see, 3.4 ) results 4.53 4.47 Romain Ruzziconi . )] -form ( A  1 ) . We have the explicit 2 Y )) ∞ A S − D CB n C C ( D . The result is given by CB B ∞ C ) does not hold. Nevertheless, D S ( . 2 ] A f , , ] g ∂ 11 ] 1 g 4.54 g [ − ξ 1 2 16 δ δ A ξ , 2 ; ; ; ) 1 Y  g + t g ξ [ A [ a 2 A K ξ AB T ξ D N a 0 Vir superrotations. C Θ C spacetimes with Dirichlet boundary condi- Θ . 2 ξ δ ] + ( 0 D ( 4 × g A B ] + [ ] + AB ) of asymptotically flat spacetimes in four di- )), the associated solution derived in subsection ≈ A Ω Y g D g ] N [ ] 2 [ 2 1 1 ξ f f + ξ g d 2 , [ ξ J 29 ( 1 ∞ 3.18 J 3.23  ξ ξ )) and the asymptotic Killing vectors discussed in S δ [ 2 f M ξ BC H Z J Ω δ ≈ 4 C d 2 [ dt ] ≈  3.52 d = ] = g ∗ group with Vir Ω ∞ g ∗ δ } ), and then integrating over Ω ) with ( 4 [ S 2 . Therefore, we can construct an integrated surface charge ; 2 } 2 ξ Z d 0 ξ 2 g d [ J ξ ∞ H ρ , J G ∞ 2-cocycle given explicitly by ξ S 4.47 3.15 1 , S Z π ξ 1 1 H Z J ξ / δ G J { 16 G ) together with ( { π 1 is taken to be the 2-sphere at infinity, written π 1 Σ 32 ] = 16 (see equation ( 3.19 ∂ g [39]. As mentioned above, the infinitesimal surface charges are not inte- δ ] = ; ] = g 3.3 g [ [ AB g [ 2 ξ C ξ ξ ; field-dependent u )), and the associated asymptotic Killing vectors derived in subsection Θ J 1 ∂ ξ K = is a into the expression ( 3.32 ] g We now provide explicit examples of surface charge constructions in four-dimensional AB [ is the integration measure on the 2-sphere (see e.g. [69]). These charges are finite and 2 3.4 N ξ ; 1 Ω ξ 2 K ) where the 2-surface Notice that this 2-cocycle is zero for globally well-defined conformal transformations on the 2-sphere. It becomes Now, we consider definition ( (equation ( 11 4.53 where We can show that non-trivial when considering the extended BMS we can define the following modified bracket [39]: grable. Therefore, we cannot unambiguously define an integrated surface charge as in ( serting this solution space and thesein asymptotic an Killing vectors integrable into expression the at( order where and where however, [65, 172]). In particular, the representation theorem ( mensions (AF3). The surface chargesderived are in obtained subsection by insertingsubsection the corresponding solution space 3.3 Asymptotic Symmetries in gauge theories Examples general relativity. First, consider asymptotically AdS expression where d generically non-vanishing. Furthermore, we can easily show that they are conserved in time, i.e. tions (AAdS2) (condition ( PoS(Modave2019)003 , 4 ] g [ ξ (4.66) (4.65) (4.67) (4.68) (4.64) N = . These are N ) not being AB C 4.60 u ∂ Romain Ruzziconi = for some AB ) and ( N N . δ ] Bondi mass loss formula + AB 4.59 . C introduced below. However, as we Θ . A ] ] δ 0 2 ) is that they are not conserved. g = ξ , ∧ δ 0 , ≈ 1 , ] ξ g [ Θ , g AB ] 4.60 δ [ N ; 2 g N g ). We can consider many other exam- ξ [ ; δ δ + and 1 [ , 0 ω ξ 2 g ξ N K ξ Ω 4.68 ) and ( N 2 δ 1 − ; d ξ ] + J g δ ∞ [ g 4.59 and the news function S [ cyclic (1,2,3) = + Z A ] W 0 1 2 AB + ξ ∞ J ξ G , 2 C S 30 1 ξ N π 1 , Z ξ ), we recover the famous 2 0 [ 1 ξ ξ J 32 δ K and ] = 3 = − 4.67 ] − ξ g [ ∗ g 2 δ automatically satisfies the generalized 2-cocycle condition 1 ξ ξ } , ξ 2 ] = 1 + A 0 H ξ δ ] ξ g 3 / 2 J in [65]. We have ; δ 0 in ( ξ K ξ , δ , , g 1 , [ 1 A d 12 0 ] ξ 2 = ξ du g = 2 [ J 0 ξ ξ δ 2 A , { N ; ξ Θ 1 ; Y ξ g 1 [ + [ 0 ξ 2 K ] + K ξ W g [ N ∞ 1 1 ). In particular, defining S 0 ξ ξ 1 and Z J δ . This was a striking argument for the existence of gravitational waves at 2 = ξ ) to make sense, its form should not depend on the particular choice of inte- + + 4.60 δ was computed f 1 ] I ξ = g 4.62 in ( N ∗ δ 2 ] , ξ } g 2 g [ δ , we can show that some of the supertranslation charges diverge for the Kerr solution 0 ξ r ξ δ J − J A non-trivial relation seems to exist between conservation and integrability of the sur- ; , 1 g [ 0 ξ J { W ) [39]. Another property of the surface charges ( More precisely, in [65], we computed the presymplectic potential 12 4.64 Notice that we obtain where ( precisely the functions involved in the right-hand side of ( [43, 154, 155]. This formularadiation indicates through that the mass is decreasing in time because of the leak of the non-linear level of the theory. Finally, despite the BMS charges ( Indeed, where Notice that taking divergent in [39]. Remark spacetimes (AAdS2) considered above,conserved. we Reciprocally, see there that is a thesurface relation surface charges. between For charges non-conservation example, and are in non-integrability both the ofintegrability asymptotically is the integrable contained flat in and case the (AF3), asymptotic shear we see that the source of non- ples where this phenomenon appears.of the Therefore, charges. non-integrability We is will see relatednon-conservation below and to that integrability non-conservation for is diffeomorphism-invariant transparent theories, in the the relation covariant between phase space formalism. face charges. For example, in the case of Dirichlet boundary conditions in asymptotically AdS grable part Asymptotic Symmetries in gauge theories It satisfies the generalized 2-cocycle condition will see, this is equal to the invariant presymplectic current in the Bondi gauge. For the algebra ( PoS(Modave2019)003 ) IW ξ ] = k Φ 4.69 δ (4.74) (4.69) (4.70) (4.71) (4.72) (4.73) ; Φ [ Noether- -form IW ξ ) k 2 L − Φ n n δ Romain Ruzziconi ( I = is called the µ ] ) x ) is given by Φ 1 ξ − n Iyer-Wald L d 4.70 ; , ( , µ ] Φ  ) [  x Φ θ 1 ... 1 ... δ − ; . − ... + n . ] n + ξ ] Φ d I ) [ + Φ ) Φ is Grassmann odd, the equation ( )( in front of the Noether-Wald charge does not see − Φ θ δ ) , Φ δ h ] L , ξ µ δ L ; δ µ ι Φ µ ∂ ∂ Φ Φ L ∂ ∂ . Strictly speaking, one should write µ is defined as ( Φ ] = [ ∇ ( [ δ µ ∂ ∂ ∂ ; ] + Φ ξ ( ∂ ξ θ [ − δθ d ∂ Φ Φ Φ ξ Φ [ [ δ µν ξ µ Φ − δ Q h ] =  µ 31 θ Q  ν L Φ µ µ Φ δ δ ∇ δ δ∂ ∂ ∂ = δ ( − , Φ µ + + + g G ) Φ δ x π L L L − Φ Φ Φ ] = δ 1 . Finally, we relate this prescription to the Barnich-Brandt ; δ ∂ δ = δ δ ∂ − √ Φ 16 Φ n Φ δ L ξ [ Φ Φ Φ d ; δ δ δ δ ω ]( L ] = Φ [ h Φ = = = ≡ ; is defined as δ IW ξ g L ; [ -form for asymptotic symmetries] The . Taking into account that Φ k ω δ . In this expression, ) θ . Φ F ] [ 2 13 δ µ Φ − δ θ ; n and ( Φ ), we assumed that the variational operator [ -form ] = θ ) ξ ξ 2 Φ ι 4.73 ≡ δ − − ; ] F n Φ Φ ( [ [ θ δξ For general relativity theory, the presymplectic potential ( [Presymplectic form] Consider a diffeomorphism-invariant theory with Lagrangian [Iyer-Wald Q presymplectic form . Let us perform an arbitrary variation of the Lagrangian. Using a similar procedure as in x ] + n presymplectic potential Φ d surface charge. [ L In the definition ( In this subsection, we briefly discuss the covariant phase space formalism leading to the Iyer- ξ ), we obtain 13 Q = δ 4.6 can be rewritten as the possible field-dependence of the asymptotic Killing vectors prescription presented in detail in the previous section. Definition L ( where is the Now, the − Wald prescription for surface chargesfor [104, 117, diffeomorphism-invariant 171, theories 172]. (including Notice generalries. that relativity), In this practice, and this method not means is for thatgenerators, valid the any i.e. parameters only gauge of the theo- asymptotic symmetries are diffeomorphisms Asymptotic Symmetries in gauge theories 4.4 Relation between Barnich-Brandt and Iyer-Wald procedures associated with asymptotic symmetries generated by Wald Example Definition up to an exact PoS(Modave2019)003 ] = Φ (4.82) (4.77) (4.76) (4.80) (4.75) (4.81) (4.78) δ , . Φ ξ µν . From this by an exact ) µ L x θ µ ; 2 ) is defined up h − Φ ) yields n ) is modified as [ Romain Ruzziconi = d ω ( 4.73 h 4.73  4.72 µ ξ , ] ρ : , µν -form ( ] ∇ Φ ) ) δ x Φ 2 ρν , 2 δ h − ; − Φ , n ] ξ − n Φ d , , [ ( ] ] Φ ( µ L g ν Y δ ; ) (see e.g. [63] for a detailed proof). ξ Φ ] ; δ ] ξ δ ν δ Φ , g µ [ d ] , Φ Φ g ∇ 0 (4.79) and its inverse, and [ δ g ∇ h ω 4.28 , δ Φ δ Y δ , ; g g ] + Σ ξ 1 2 [ , µν d G g ∂ ] = conservation law Φ [ g − Φ g L [ Z π g [ + − ; δ δθ ω 8 √ ] δ θ , IW ξ ξ ξ ≈ , Φ ξ µσ ι ι [ Φ and is an obstruction for the integrability. k ] g Φ h ] g δ δ δι − − ω δ δ ] = ν ; 0, if and only if g 32 ; ; δ is a solution of the Euler-Lagrange equations and Σ Σ Σ Σ g ∇ , g δ ≈ ∂ ∂ ∂ ∂ Φ ξ [ ). In fact, we have the freedom to shift Φ [ , g σ [ ] Z Z Z Z Φ [ ] = g ξ ω θ L ω Φ ; ξ δ = = = Φ IW ξ ι + , [ g δ 4.70 ] = [ → k ; g ξ Σ h → . [ ] d Φ θ ∂ ] ν ] Φ H [ 1 Σ Z [ ω Φ / ξ Φ ∇ ∂ Φ − δ . Finally, inserting these expression into ( n ξ µ ξ Z δδ H IW ξ δ ; I / , ξ k L Φ − d , δδ [ Φ ) in the second and the fourth equality, respectively. The surface ] = − θ δ Φ Φ ; [ δ ] = νσ ; ξ Φ g -form. This implies that the presymplectic form ( 4.72 h [ [ ) Φ H σ ξ 2 [ ω , it is implied that / δ ∇ Q ω d − ≈ µ n − Komar charge ξ ) and ( (  -form. However, there is another source of ambiguity here coming from the . Indices are lowered and raised by ) ] = g 2 G is a 4.73 µν Φ − is integrable only if ] π g − δ 8 ] √ , Φ δ n Φ ( δ Φ [ As in the Barnich-Brandt procedure, the Iyer-Wald In the covariant phase space formalism, the relation between non-integrability and non- ; = ξ [Conservation law] We have the following δ ] = ; Φ h H [ µν ; / Φ δ -form as h Y [ g ) [ 1 ω This can be proved using Noether’s second theorem ( is a solution of the linearized Euler-Lagrange equations. Furthermore, IW ξ Φ − ξ k Φ n L where, in the equality δ i definition of the presymplectic potential ( Theorem the non-conservation is controlled by Remark to an exact ( charge and we recognize the Remark Therefore, from conservation mentioned in the previous subsection is clear. Indeed, Asymptotic Symmetries in gauge theories where where expression, the Noether-Wald charge can be computed; we obtain where we used ( PoS(Modave2019)003 − + n dis- L ( Φ (4.83) (4.90) (4.86) (4.87) (4.85) (4.88) (4.89) (4.84) n δ IW ξ I k δ differs from ] Φ δ Romain Ruzziconi ; Φ as . [ ] ξ . ] Φ k W Φ δ . ), we obtain the relation ] ; ξ Φ Φ , L . [ ] δ , . . -form ambiguity for 4.85 -form L , ) by Y Φ ) . ) Φ θ θ Φ ] d ξ 2 Φ 2 1 1 n δ δ ξ I Φ δ ; ι Φ Φ − − L − , 1 − ; 4.73 n n ξ δ δ , ] Φ I I Φ − − n n [ Φ n d d L ( Φ ] δ Φ ) and ( [ ( ), we obtain I , E θ ξ δ E Φ d − − d 1 2 ; Φ d ξ L Φ δ  Φ ; 4.48 − n 4.77 δ [ ; L ] + I δθ ; Φ ] + L Φ E = [ given in ( Φ Φ [ − − δ δ Φ Φ Y [ δ θ E δ , ] +   IW F 1 Φ Y , -form which is involved in the relation between ) and ( k Φ − Φ δ ) Φ ] + L L δ Φ Φ n δ Φ ξ 2 I δ  33 δ δ δ δ ) of the variation of the Lagrangian. We apply the Φ δ ; − [ 2 1 L ; Φ − ] 4.49 ξ Φ Φ ; n Φ δ n [ Φ − I Φ δ δ [ Q ( Φ 4.71 ξ [ δ   k = ; are Grassmann odd. The Noether-Wald charge becomes W ] = → Φ Φ ). Now, using ( W by the term Φ n n ] ≈ δ δ δ [ Φ ] I I ] δθ ] = Φ δ Φ [ ] = IW ξ , Φ = = 4.70 + Φ ξ results in and the invariant presymplectic current δ k Φ δ Φ ; δ L L ; Q Φ δ , ω δ ξ Φ δ δ , → 0, the left-hand side of the last equality can be rewritten as ; Φ [ Φ [ ] L Φ Φ Φ i Φ n n δ δ δ = IW ξ [ ξ Φ I I ; IW ξ 2 k E δ k L Φ δ ; [ ; Φ ω Φ [ [ -form which can be reabsorbed in the -form. Therefore, the Barnich-Brandt ) -form ) IW ξ ω ) 2 2 k 2 where we used ( − − − 0 because n n ω n ( ( 2 ( Let us introduce an important ] = = We now relate the Barnich-Brandt and the Iyer-Wald prescriptions to construct the Φ n δ I , δθ δ 2 [ = -form. Let us start from the expression ( ) cussed above. Therefore, this ambiguity modifies up to an exact up to an exact 2 homotopy operator on each side of the equality. We have Therefore, Since between the presymplectic form Asymptotic Symmetries in gauge theories where we used the fact that both d and Remark δθ Contracting this relation with Finally, using the on-shell conservation laws ( the Iyer-Wald the Barnich-Brandt and Iyer-Wald prescriptions discussed in the remark below. We define Definition PoS(Modave2019)003 - d , the (5.1) 5 -form (4.91) (4.92) ) 2 is dual to − 5 n S ( × ) by comparing 5 Romain Ruzziconi , 4.90 µν ) x . . Furthermore, a simple 2 µ −    µν n . ξ ) d x ( 2 − σν n of these theories are exactly the ) d g ( δ σν )( ) µ g ξ δ ( σ ∇ ∧ σ + µ σ ) ξ ) vanishes in both the Fefferman-Graham gauge g 34 µ ⇐⇒ δ ( ∇ global symmetries ( 4.91 g G ). We obtain g G π − which tells us that the gravitational theory living in the π ). Notice that the difference between the Barnich-Brandt − √ 32 4.85 √ 16 -form ( ) vanishes for a Killing vectors 4.76 − ) ] = 2 g δ being compactified. A first extension of this original holographic states that quantum gravity can be described in terms of lower- ). Therefore, the Barnich-Brandt and the Iyer-Wald prescriptions ] = − 4.92 , g 5 Gauge symmetries in the bulk theory g n ξ ) and ( S ( δ ; 3.10 Global symmetries in the boundary theory L , this leads to g , [ g    g 4.47 ξ E δ L -form. This expression can also be obtained from ( ; i ) g 2 [ E − n ( can be computed using ( AdS/CFT correspondence ] We illustrate these concepts with the case of general relativity. The Φ 4 supersymmetric Yang-Mills theory living on the four-dimensional spacetime boundary δ holographic principle , -dimensional asymptotically AdS spacetime (AAdS2) is dual to a CFT living on the = Φ ) δ 1 Asymptotic symmetries have a wide range of applications in theoretical physics. We briefly The ; ) and the Bondi gauge ( N + Φ [ d 3.8 When contracted with up to an exact the explicit expressions ( and the Iyer-Wald definitionscomputation ( shows that the More specifically for us, considercorrespondence a tells us given that bulk a solutionsolutions, set space of such with quantum asymptotic that field symmetries. in theories exist the The that UV are regime, associated with the the bulk mention two of them and explain why the formalism presented above5.1 is relevant Holography in these contexts. ( lead to the same surfacenot charges coincide, in see these for gauges. instance, [12]. For an example where the two prescriptions5. do Applications dimensional dual quantum fieldprinciple theories asserts that [165, 167]. the type IIB A string concrete theory realization living in of the the bulk spacetime holographic AdS Asymptotic Symmetries in gauge theories Examples E the [126]. The gravitational theoryfive is dimensions effectively of living the in factor duality the five-dimensional is spacetime the AdS dimensional boundary. Other holographic dualitiesbeen with studied. different A types of holographicterms asymptotics dictionary of have enables the also one dual tobetween boundary interpret the theory. symmetries properties of of the For the two example, theories: bulk the theory dictionary in imposes the following relation ( PoS(Modave2019)003 a (5.3) (5.4) (5.2) ). Brown -particles n 5.1 (see [162] for spacetime with 3 Romain Ruzziconi Brown-Henneaux , ) q ( , ) O 0 q [48, 98, 118, 123, 175]. They ( memory effects ) + q n O p and ) + n ,... p 1 ), the dual theory would have BMS as p that has been partially studied in these CFT [159], this reproduces exactly the ( 5.1 n ,... d 1 p M ( ) [96,122,173,174,178]. These theorems state ) n , 1 ( ` G M S soft theorems 3 2 ) , 0 35 + ( = ) S particles involved in the process. Furthermore, some 0 c ( n S ) = n p soft theorems ) = ( ). n 2 p have been established for the different types of soft particles ,... asymptotic symmetries 1 p , ,... 1 q ( p 1 , + q n ( 1 asymptotic symmetries M + of the bulk solution space. Even if the AdS/CFT correspondence has not n 0 (that may be a photon, a gluon or a graviton), is equal to the radius. The AdS/CFT correspondence tells us that there is a set of two- M 3 → is the soft factor whose precise form depends on the nature of the soft particle q 1 -particles scattering amplitude involving a massless soft particle, namely a particle is given by − ) q 1 ) into the Cardy entropy formula valid for 2 ∼ + subleading soft theorems ) n is the AdS 0 ( 5.2 ( ` S unrelated, namely The holographic principle is believed to hold in all types of asymptotics. In particular, in The first corner is the area of A connection has recently been established between various areas of gauge theories that are We now mention a famous hint in favor of this correspondence using the relation ( . We have 0 so-called and they provide some informationtake about the the form subleading terms in involved. Taking as softphoton theorem, particle soft a gluon photon, theoremsoft and gluon factor soft is or graviton independent theorem. graviton of the will A spin remarkable respectively of property lead the is to that the the soft scattering amplitude without the soft particle, multipliedq by the soft factor, plus corrections of order where a review). These three fieldstriangle of of gauge research theories are (see often figure referred to as the three corners of the infrared with momentum that any priori notes. The second corner is the topic of asymptotically flat spacetimes, from the correspondence ( entropy of three-dimensional BTZ black hole solutions [16, 17]. and Henneaux showed that the asymptotic symmetry groupDirichlet for boundary asymptotically conditions AdS is given bytions the in infinite-dimensional two group dimensions. of conformal Furthermore, transforma- are they integrable, revealed and that the exhibit associated acentral surface non-trivial charge charges central are extension finite, in their algebra. This where dimensional dual conformal field theories.charge The ( remarkable fact is that, when inserting the central the global symmetry. Important steps have been(see taken e.g. in this [13, 14, direction 20, in 40, 73, three 86,asymptotically and 87, four 152] flat dimensions and spacetimes, references traces therein). ofof Furthermore, well-known two-dimensional in techniques CFT four-dimensional of seem the toNotice AdS/CFT that appear, correspondence global enabling [15, BMS 36, symmetry the 57, 74, can 82, use beis 121, seen 132, especially 141, as relevant 150]. a in conformal the Carroll context symmetry of [84,85,91], the fluid/gravity which correspondence5.2 [50, 54, 59, Infrared 60, 100, physics 116]. Asymptotic Symmetries in gauge theories asymptotic symmetries been proven yet, it has been verified in a number of situations and extended in various directions. PoS(Modave2019)003 ) (5.5) 3.15 refraction Romain Ruzziconi and the has the same effect as a burst effects AB Memory C occurs, for example, in the passage of spin memory effect and trigger an observable displacement (ordinary memory effect), a burst of grav- AB , C M AB C 36 ∝ ∆ A (Christodoulou effect), or a burst of null matter (null s symmetries Asymptotic ∆ AB N Infrared sector of gauge theories. of two inertial observers in the asymptotic region is dictated by A s displacement memory effect ∆ Figure 2: Soft theorems -matrix is invariant under the BMS symmetry [161], then the Ward identity S is the subleading soft factor. Proposals for sub-subleading soft theorems can also 0 . q . Three processes can turn on the field ∼ AB )), it can be shown that ) C 1 ( S 3.18 The third corner of the triangle is the topic of memory effects [42, 45, 46, 58, 88, 143, 168, We now briefly discuss the relation between these different topics. It has been shown that if with ( where gravitational waves. It can beof shown a that this pair produces of awhen inertial permanent the detectors. shift gravitational in wave the This is relativeBondi shift passing positions gauge through is in the controlled spacetime asymptotically by region flat a of spacetime field interest. with in Considering boundary the the conditions metric (AF3) that (equations is ( turned on be found [49, 76, 180]. 169, 176, 179]. In gravity, the Asymptotic Symmetries in gauge theories i.e. the angular displacement the field itational waves controlled by the news memory effect: a variation of the Bondi mass aspect memory effect) [63]. The(electromagnetic analogous memory memory effect) [140, effects 166] can andwhere in also Yang-Mills a be theory field (color established memory is in effect) turnedleading electrodynamics [143] on to as an a observable result phenomenon.gravity of [65, Notice a 80, 92, that burst 129, other 136, of 142, memory energy 147, effects 148], passing have including through been the the identified region in of interest, memory effect the quantum gravity associated with the supertranslations is equivalent tothe the displacement soft memory graviton theorem effect [108]. iscisely, Furthermore, equivalent the to action performing of a theof supertranslation supertranslation gravitational [163]. on waves the More passing memory pre- through field the region of interest. This can be understood as a vac- PoS(Modave2019)003 4 (A.2) (A.1) 0 trans- = ab T . ) 0 A a ( Romain Ruzziconi N D ) in the Fefferman- . , that have to be taken 3.19 AB q = ) are related to degrees of free- . 0 ( AB soft hairs g ab AB T E ∼ , 0 AB A T U [105]. Indeed, an infinite number of soft − ) associated with the preliminary boundary = , ) A 0 3.43 ( tA N g 37 ∼ and the angular momentum aspect tA M T , 0 C U C 0 , U can be expressed in terms of the functions of the Bondi gauge M + ) 0 ∼ 0 β ab ( 4 tt g e T 3 Λ = ) 0 black hole information paradox ( tt g ) associated with the preliminary boundary condition ( 3.30 ) in the Bondi gauge through this diffeomorphism. The components of the three- 3.20 coordinate for the Bondi mass aspect In this appendix, we establish some important frameworks and conventions. The aim of this Finally, let us mention that this understanding of the infrared structure of quantum gravity is I would like to thank Glenn Barnich, Luca Ciambelli, Geoffrey Compère, Adrien Fiorucci, The diffeomorphism between Bondi and Fefferman-Graham gauges in asymptotically (A)dS u as Graham gauge matches with thecondition solution ( space ( The precise relations can belated found in in the terms references. of the Noticethe functions that in the constraint the Bondi gauge gives the evolution constraints with respect to dimensional boundary metric formalism is to manipulate some localfollow expressions, [61, as 63]. this is convenient in field theory. We closely Furthermore, the degrees of freedom ofdom of the the stress-energy Bondi tensor gauge as B. Useful results and conventions relevant to tackle the Yannick Herfray, Kévin Nguyendiscussions and on Céline related Zwikel subjects. forSummer their School I comments 2019 would on for also the thisFNRS, like Belgium). stimulating manuscript to edition. and thank This the work participants is of supported the byA. XV the Diffeomorphism FRIA Modave between (F.R.S.- Bondi and Fefferman-Graham gauges Asymptotic Symmetries in gauge theories uum transition process [1, 62, 67, 68, 156].theorem Finally, with a the Fourier memory transform enables effect,structure us which of to closes the relate the theory the has triangle. soft alsomore, This been subleading triangle constructed infrared for controlling triangles other the have gauge beenIn infrared theories particular, uncovered the [125, and 128, Ward 143]. identities discussed of [53, Further- 65, superrotations 71,ing have 115, been 120, soft 125]. shown graviton to theorem. be equivalent Furthermore, tohave the the been sublead- spin related to memory the effect superrotations. and the refraction memory effect gravitons are produced in thedence, these process soft of gravitons black are related holeinto with account evaporation. surface in charges, the Through called information the storage [44, above 101, 102, correspon- 106, 107, 131]. Acknowledgements spacetime has been worked outsolution explicitly space in ( [66, 149]. In this appendix, we briefly recall how the PoS(Modave2019)003 . . δ ) δ + ,... (B.1) (B.5) (B.6) (B.2) (B.3) (B.4) . The d Φ µν = k µ , i.e. the ∂ Φ ∂ , Tot 0. Hence, µ ... Φ 1). The fields, = , 1 µ ∂ 2 − ∂ Φ is defined as Romain Ruzziconi δ , n µ = x consists in the fields k ( . µ k ,..., ν ... 0 1 ... µ 1 = ∂ ν Φ vertical derivative µ Φ ( . ∂ µ k  ν x jet space J k ... µ 1 , where ... f . , -forms with respect to the spacetime . 1 ) µν ˜ µ k p µ Φ ∂ Φ variational operator j µ Φ δ Φ 0 is defined as µ ,... ) ... . A section of this fiber bundle is a map k k 1 ∂ , i.e. a function on the total space of the ≥ ∂ J ν ∑ ∂ µ µ k µν  δ = × Φ . The k . + Φ ) µ , d f ∂ ... µ M ∂ k µ δ 1 x µ ∂ 1 µ ,... Φ ν − ∂ ( ...... , 1 δ 38 1 µν µ ⇔ = Φ µ = ( = ∂ Φ Φ δ µ k k δ δ k d ) ∂ , µ ν 0 0 . 1 µ local function f ...... ≥ structure ∑ 14 1 1 k − = Φ ν µ ( to be Grassmann odd, which implies that 0 ˜ f δ = Φ Φ Φ , ≥ horizontal derivative ∑ k δ ∂ δ ∂ δ Φ , is defined as ,... where as the fiber bundle with local trivialization ] = δ the set of functions that are ( µν f , q . The Φ ,... , Φ ) µ p δ δ δ ∂ µν , Ω µ µ ,... Φ x derivative of a ) , are Grassmann odd and they anti-commute, namely d Φ x jet bundle µ is defined as ( δ δ = Φ variational bicomplex , , µν d , are supposed to be Grassmann even. The Φ ) Φ -dimensional spacetime with local coordinates Φ i δ , , n x φ ) [ x f ( µ = ( = Φ be the f Φ , ) Euler-Lagrange x M ( Φ -forms with respect to the jet space ( Now, we define the One often refers to a In this subsection, we introduce some additional operators usedThe in the text and discuss their Let q 14 → Locally, the total space of the jet bundle looks like x In the jet space, the cotangentderivatives at space that at point, a namely point is generated by the variations of the fields and their is seen as an exterior derivative on the jet space. This operator satisfies We choose all the In this perspective, thederivative variational along the operator fibers. can The exterior also derivative on be the total seen space as can be the defined as d Notice that both d and jet bundle properties. B.2 Some operators Asymptotic Symmetries in gauge theories B.1 Jet bundles symmetrized derivative On the jet bundle, we write and written as and the symmetrized derivatives of the fields PoS(Modave2019)003 ] = 2 (B.8) (B.7) (B.9) Q (B.15) (B.18) (B.10) (B.13) (B.14) (B.16) (B.17) (B.11) (B.12) δ , 1 Q δ [ Romain Ruzziconi . . ) k . n µ k  ) is given by  ... − < n ν 1 Q f ν µ Φ x and satisfies p , ∂ x is defined as d δ δ 1 = ∂α Φ d ∂ k . ξ Q ] µ ∧ Φ 2 2 ν ∂δ k ... with respect to a characteristic Q Q Q ) , 1 µ Q . -forms δ δ 1 ... µ 1 ... Q n ∂ Q -forms ( − 1 . [ δ ∧ − k Φ i δ p µ k µ 1 k 2 ∂ µ ν µ (i.e. 1. We can check that Φ ... = . Q x ... , 1 ∂  1 δ 1 k 1 µ δ d k = µ µ Q k ) Q ∂ Q µ Φ i − ... 1 δ x ... Φ n δ 2 ∂ 1 1 1 δ − ν Q µ µ n . + ...  = ) δ k ∂ ... ∂δ ] = 1 ... µ k x 2 − 1 ν ) Q x µ k − k n 01 µ is defined as ∂ d Q ∂ µ + ( − Q 2 ε d ∂ , k 1 n ( δδ ∂ k Q ... k ] 1 . We use the notation µ d 1 + ) µ δ ... µ σ ( ∂ q µ 39 1 1 Q , 1 1 k ... M [ ξ ε 1 , Q 1 − µ µ − ! i k ∂ µ − Q ... ∂ ( ) ... µ = p 1 δ 1 0 k 1 Φ ... µ µ d ξ ≥ 1 Ω , ∑ ∂ ι ∂ k with respect to a vector field − µ + 1 α Q ( [ ) = 0 n k δ 1 − 7→ when acting on spacetime α Q ( = d ≥ ) k ! ∑ q k σ + when acting on spacetime + , = Q f µ k ∂ p α p δ k ∂ Q Φ Q = i n Ω = δ δ = Φ − interior product of a jet space form I Q ( : δ ... k p i δ d n µ 1 I α Φ Φ + µ 0 d δ d p ... δ − δ ∂ 1 ≥ δ ∑ ( µ k − 0 Q Φ ) δ i = d x ≥ δ = ∑ 1 k k f + Φ Φ − Φ α p n δ -form on the space-time = δ δ δ I Φ ) d f p δ k ( Q I = = δ Q − is completely antisymmetric and δ δ δ n n ( µ ... 1 . This operator satisfies the following relations µ q , be a ε p is a local function (for a proof, see e.g. section 1.2 of [26]). homotopy operator I variation under a transformation of characteristic Q α Ω µ j . A contracted variation of this type is Grassmann even and we have ] ∈ 2 The Let The interior product of a spacetime form Q , α 1 and the Euler-Lagrange derivative: as Q [ We also have the following relationQ between the variation under a transformation of characteristic δ where Notice that we can also define the Q It satisfies for The Lie bracket of characteristics is defined by Asymptotic Symmetries in gauge theories where and where The PoS(Modave2019)003 7→ q , p (B.20) (B.19) (B.21) Ω : J. Math. p F I ). -Form p 4.16 Class. Quant. Romain Ruzziconi .  is defined as ν x d ∂α ∂ ν k µ ... 1 δ µ , 114(11):111602, 2015. , D92(4):044011, 2015. F δ F . 1 .  k δ , 323:183–386, 2000. = µ Φ , 32(2):025004, 2015. ∂ , D92(10):104032, 2015. p p Phys. Rev. F δ I I d ... 40 = Phys. Rev. Lett. 1 + , D100(2):024042, 2019. µ Φ , 105:171601, 2010. ∂ d p δ 1 I 1 Phys. Rept. + δ Phys. Rev. p F + , A376:585–607, 1981. I k 1 + + Phys. Rev. p k Class. Quant. Grav. − n Phys. Rev. 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