Pos(HRMS)028 Duality ) Symmetry

PoS(HRMS)028 duality ) symmetry. R , http://pos.sissa.it/ n 2 ( symmetry of the Sp off-shell off-shell off-shell is in that case an ) R , 2 ( SL electric-magnetic duality is an ) 2 ( SO symmetry. We finally indicate further possible extensions to twisted -forms, including Chern-Simons terms and Pauli couplings, as well as p 7 , 7 coset space, showing that E ) 2 ( SO / ∗ ) electric-magnetic duality as R , 2 ) ( SL R , n [email protected] [email protected] 2 We also show how the resultsymmetry can - be or generalized a tofour many subgroup dimensions Maxwell of with fields it, and recovering in particular the case of maximal supergravity in Speaker. linearized gravity, which will be treated in depth elsewhere. self-duality equations for free Maxwell theory, i.e., that itWe leaves review invariant here the that action analysis and and noton extend just the it the to equations the of motion. Maxwell field coupled to scalar fields defined It was established long ago that ( ∗ Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c

Quarks, Strings and the Cosmos -August Héctor 09-11, Rubinstein 2010 Memorial Symposium (AlbaNova, Stockholm) Sweden symmetry of interacting electromagnetic and scalar fields Marc Henneaux Université Libre de Bruxelles and InternationalB-1050 Solvay Brussels, Institutes, Belgium; ULB-Campus Plaine CP231, Centro de Estudios Científicos (CECS), CasillaE-mail: 1469, Valdivia, Chile Claudio Bunster Centro de Estudios Científicos (CECS), CasillaE-mail: 1469, Valdivia, Chile Sp PoS(HRMS)028 ] 1 (1.8) (1.5) (1.6) (1.7) (1.2) (1.3) (1.4) below is ] following k 2 B Marc Henneaux 1. is = 2 123 ε , 0123 ε − ) ) . q q = µ and do not asume that the electric field is E µν µν 1 . F p 0 µ F F = A mn ∗ ∗ ∂ 0 (1.1) ∂ is the electric field, while ( µν F p 1 α α k = F ∂ 0123 − ( E ε kmn ρσ 1 µν sin µν 4 ε F cos − 1 2 F xF − ∗ + 4 4 kpq µ = d µν µνρσ ∂ µν k βε kpq . Here, ε F 2 Z F B 1 2 β = α 1 4 ] in that we keep βε α 0; 2 k , = − k + A = 0 1 sin cos k δ F µν B µν ] = F − β µ ∗ , F → → 0 A − µ = [ ∂ S k = = E µν µν 0 k A F F E ∗ δ δ is defined through ) implies µν ’s signature for the Minkoswki metric and 1.6 F + ∗ -Duality ) 2 ( The explicit check of the invariance of the Maxwell action under duality transformations is done in [ In order to establish the duality invariance of the Maxwell action, We use the mostly The vacuum Maxwell equations Although it is often incorrectly stated that electric-magnetic duality is only an “on-shell" sym- SO 2 1 ]. Our formulas are slightly different from those of [ 1 the magnetic field, [ that this transformation is in fact aor, genuine as symmetry transformation one that says leaves the in actionis brief, invariant quite is important an since “off-shell it symmetry". permitsalso, The its it discussion fact enables at that one, the already duality quantum at is level the an through classical off-shell the level, symmetry path to integral use and the Noether method. one must first extend “off-shell" theical duality variables, transformations i.e., and the express components them of in the terms potential. the One dynam- appropriate extension for infinitesimal duality rotations of parameter The transformation ( transverse off-shell. metry, i.e., only a symmetry of the equations of motion and not of the action, it was shown in [ Electric-Magnetic Duality 1. Introduction 1.1 Here, the dual field 1.2 Invariance of the Maxwell Action are invariant under electric-magneticdimensional duality plane transformations, of i.e., the internalread electric rotations and in magnetic the fields. two- In covariant form, these transformations PoS(HRMS)028 , 1 ) mn − . Z F k , 4 (1.9) E A (1.10) ), one mn 0 F ∂ ) ) 1.6 = ( 4 n a / E 1 A m ∂ − Marc Henneaux ( ( 1 − . This step gets rid i 4 Z 2 (1.11) , . . 0 1 kmn A ) coincide on-shell with k ε ) = A mn m + n b 1.6 F . E , A a m ) and thus it is a total derivative, m ) ∂ kmn n ∂ , k m ε E and the symmetric operator ∂ ρσ E m ( b F kmn k k m ∂ ) 1 ε ∂ B ( ∂ E − m · 1 λ µ ( 1 2 1 E F and the potential term a − 0 4 − m ∂ k + B β 4 0 ∂ k k 4 k ab + F k − A E δ λ µρσ ∂ k ) derived from ( as independent variables in the variational 0 k B ( 0 ε k ∂ i 1 − E F n kmn A k 1.9 ) − b A β 0 ε A 2 ∂ ˙ is divergence-free, replaced by A ∂ 4 m / · n k k 3 k ∂ 0 1 = β a A B A ∂ B − B − ( ) and ( kmn − ) = k kmn ab ε m k ε ε E 1.8 . In terms of the two spatial vector potential E B = ) = 0 β ) with m ). n x δ k , introducing a second vector potential m A ∂ i 3 k 0 E = ∂ k conjugate to F π d B 1.3 m k ∂ i 1.10 0 = ∂ ( ], the invariance of the action is easily verified to remain true B mn π 0 1 k 1 F δ dx ∂ ) = − A ( 0 1 Z 4 mn kmn ) and ( ∂ − ] n ε 1 2 F 3 k A 2 1 4 1.2 , k mn B m 1 F E ] = ∂ = ( a i k δ A kmn kmn [ E 1 4 k ε ε inv B S − = k ]. First, one goes to the Hamiltonian formalism (which was actually the starting point . These somewhat awkward features can be remedied by following the procedure 0 1 µ F k A 0 F 2, the action reads [ , ]), introducing the momenta As it was already shown in [ It would seem fair to state that, in the discussion given above, the invariance of the Maxwell To verify that the Maxwell action is invariant under the duality transformations ( δ 1 1 2 1 . Second, one solves Gauss’law for = i The last term has the same form as ( of [ Hence, the infinitesimal transformations ( are separately equal to total derivatives. We start with the potential term. One has inserted and is thus also a total derivative. devised in [ principle. These momenta coincide inE the case of theat pure the Maxwell same time theory of with the time thea component electric field action cannot be said to be manifest.transformations This are is non in part local due in topotential the space fact (but that the local electric-magnetic in duality time) when expressed in terms of the vector when the coupling to gravity is switched on. 1.3 Manifestly duality-invariant action (the infinitesimal form of) ( checks that the variations of the kinetic term Electric-Magnetic Duality and The first term is proportional to the characteristic class whereas The second term is also a total derivative since The proof of the invariance of the kinetic term proceeds similarly. One finds explicitly PoS(HRMS)028 is E (1.14) (1.12) (1.13) ) but the are invariant 1.11 Marc Henneaux ab δ and ab . ε N B 1). The duality rotations in · = M B 12 , ε . MN             2 2 δ while the standard electric field . . . A A 1 − 1 0 α α B . . . -numbers. N 0 0 0 1 0 0 0 0 0 0 c ˙ -symbol is replaced by the antisymmetric A ≡ sin · ε a cos . . . B ··· ··· − ··· ··· ··· ··· A − M + 1 . . . B 1 is × ), A A 1 0 n 4 B ∇ MN . . . α 2 α − σ , = . . . a x cos sin ··· 3 B , 1 0 0 0 . . . d 0 00 0 0 0 0 0 0 1 0 0 0 00 0 0 1 1 − 0 ) is a mere rewriting of the standard Maxwell action. The → → values and the = dx             n N 1 2 , 1.11 = Z A A 1 2 M σ theories, which gives additional weight to its adoption as a sound ( ] = MN M i σ A [ inv S interacting . The standard magnetic field ) is the Levi-Civita tensor in 2 dimensions (with ) is clearly invariant under these transformations since both 2 ( ba ε 1.11 SO Maxwell fields, the manifestly duality invariant action takes the form ( − ]. There, had it not been for the insistence in raising electric-magnetic duality invariance . Although manifestly duality invariant, this reformulation of the Maxwell action is not -Duality n 4 2 ) = n B ( ab For It is important to realize that ( The possibility of introducing a second potential in order to make the electric-magnetic duality ε U ≡ − which gives internal indices run now fromcanonical 1 symplectic form to 2 physical principle. 1.4 to the majesty of aled principle to that should discover have those aimplemented potentials. manifest also expression, in As one would we hardly shall have see been below, this point of view can be successfully this formulation are E manifestly Lorentz invariant. and classical steps that go from onemechanically form in of the the Maxwell path actionstraints) to integral, or the Gaussian as other integration can either over beform the repeated insertion of quantum- momenta of the (to delta action). go from functions Inintegral the our measure (for first-order in case, including to the the second-order the the accompanying formalism, second-order determinant con- are factors, just which appear in the path manifest stems technically from the fact thatis the quite gauge remarkable, constraints and can by be no written means asgravity obvious, a that [ divergence. this can It also be achieved in the case of linearized Electric-Magnetic Duality where tensors for The action ( PoS(HRMS)028 ) ]. n 8 ( , 7 U (1.18) (1.17) (1.15) (1.16) , 6 , , namely, ) 5 [ 2 ( ) , which is the n always means duality group, ) Marc Henneaux ( SO ) n ) n U 2 spacetime vector n ( ( duality invariance ( n to O O ) U ( n n ∩ Maxwell actions. The ) ( )] ×···× ) n 2 n ) U ( ( R 2 , . , O ( , 0 n λ invariance, the scalar fields SO 2 ∈ [ M ) ( SO = + A Λ R I N , × Sp P n ) = λ 2 2 , known to be isomorphic to , ( ( Λ ) 0 MP R δ . Sp SO , = I n + N λ 2 = A ( M P + N Λ ]. For M λ I Sp T factors T Λ λ 14 PN must belong to n Λ , rotations in the space of the δ = 6 , ) Λ 5 , , 0 , M n 0 invariance (or a subgroup of it) if couplings to ap- ( σ 0 A 13 = . Invariance was established only at the level of the ) ) invariant if and only if they preserve the symplectic , ) = = SO R n → , N ( 12 Λ P σλ , while the second implies 1. We then consider the general case. n , ) 1.14 M σ U λ 2 + A / T R ( describing a single standard Maxwell field. But duality is = 11 ) , σ MP Λ n ) n R Sp k T σ 2 , 2 Maxwell actions is invariant under the full ( λ n A + n 2 , ( Sp invariance of the action, contrary to somewhat fatalistic widespread ) in second order form. It is just the sum of 1 M P − ∈ k λ Sp 2 1.14 Λ A PN ( σ off-shell , leave the action ( ) ]). 9 R , n 2 ( -duality ). Accordingly, the transformation ) ) for each pair R GL R , , ) n ∈ n 2 ] for general information on symmetries in the first and second order formulations, which 2 2 ( ( Λ ( 10 O By reversing the steps that lead from the second order formalism to the first order formalism, In infinitesimal form, the invariance condition reads, with It has been realized in the context of supergravity that the compact The duality group clearly contains in this case SO Sp one can rewrite the action ( equations of motion. The purposeagain, of in fact, this a paper proper is tofears. show We start that with this the extended simple duality case, invariance is here maximal compact subgroup of the(see, symplectic for group example, [ or in component form, first order and second order formsduality of symmetries, the although action some share the of(see same these [ symmetries, take in a particular, the non-local same form in the second order formalism parametrize the coset space in fact bigger, because onebelonging can to also different pairs. perform Hence linear the transformations duality that group mix is enlarged the from vector potentials are related by changes oftrary variables to and the introduction folklore, (or elimination) the of sum auxiliary of fields). So, con- one This can be seen as follows: linear transformations of the potentials where The first condition implies even though it is onlypotentials. manifestly so under 1.5 product and the scalar producttively), (in order to preserve the kinetic term and the Hamiltonian, respec- Electric-Magnetic Duality can be extended to a non-compact propriate scalar fields are introduced [ PoS(HRMS)028 . ) 0 is R ξ , χ (2.2) (2.1) (2.3) (2.4) (2.5) (2.6) (2.7) 2 ( ] = + SL ξ , − ξ [ Marc Henneaux ] that , − 14 ξ , 2 6 − . − -duality transformations, ] = , is the “dilaton" while ) 0 0 ξ , R φ , , − 2 ( ρσ ξ [ = + F , χ SL α + λ µ µ ∂ transformations that preserve these ξ . ∂ χ F 2 , − ) χ∂ ) . This is the stability subgroup of the 2 ε R µ χ under − ) χ , . It was shown in [ ∂ , ] = S + λ µρσ ξ 2 0 φ µ φ L ( − 2 + ξ ( A − e χε , φ ε χ + α 2 SL . The scalar field 2 1 + + 1 8 ξ ) and the appropriate scalar fields come in a pair V − ξ = ξ ∂ 2 [ e ∂ χ L α ) − + -invariant. ( ( 2 ε ) transformations reduce to χ 6 x φ ( δ χ ) µν 4 2 R + µ SO = , d R F / is achieved provided the couplings of the vector field , SO − 2 , ) ∂ 0 ( φ∂ Z ∂φ 2 µ µν δ χ ε R ' µ ( ∂ F , A χ 2 ∂ SL ∂φ ∂ is also invariant ) = φ ∂ χ 2 2 1 2 SL 1 2 , ( − S = ( ) e = = − = χ SL U 1 4 , , − 0 + δφ = ) φ ξ ξ ξ − ( S tangent to the coset space are explicitly given by R φ , L α = transformations, which are non linear and read in infinitesimal form α 2 ξ ξ ( V ) α L R ε SL , 2 = ' ( since there, the algebra in the Lie bracket, ) ) ) SL δφ R 0 subgroup. , R , ) , 2 0 1 2 ( 2 transformations are generated by ( ( = ) Sp . As we have already indicated, the SL 2 SO n ) ) = ( ( 1, 0 χ , , = 0 SO φ ( n parametrizing the coset space ) = ( ) The goal is to prove that the action For Turn now to the Lagrangian for the vector field χ χ , , φ φ Here the Killing vector fields values form a the “axion". The Lagrangian for the scalars is given by and is invariant under and fulfill the The action for the coupled system is and leads to equations of motion that are which are therefore off-shell symmetries.above for This the pure extends Maxwell to case,( the recovered coupled by choosing system a the particular analysis value made of the scalar fields, say invariance of the equations of motionto for the scalar fields are chosen as ( Electric-Magnetic Duality 2. The case 2.1 Second order Lagrangian The rigid “origin" PoS(HRMS)028 (2.9) (2.8) (2.11) (2.10) (2.14) (2.12) Marc Henneaux replaced by the 2 (2.13) , 1 ab δ = b mn , , i F a Z mn , F ) . b φ m B − Z · . e n a ∂ ! + B G φ ) 0 − φ e 1. i jk χ A e φ n , χ 2 ε e + Z − φ jk is invariant since the scalar term is mani- χ m − ( ( F ∂ V 2 1 ab χ H . i = φ xL G 1 2 + π − 4 − i φ i e d − mn + ∂ e ˙ mn A reduces at the origin of scalar field space. Note i b i R H χ 7 + F 0 − ˙ i π A ab φ F − π · = e , φ ), but with the Euclidean metric G a x 2 − 4 imn G B mn χ e in a form where this invariance is manifest, by going to d ε H ab φ − 1.11

V ε Z e imn = χ xL are = x ε i 4 ) = i 3 + 2 1 d π H i d A , ab , to which R V 0 π − i G S ab ( π dx = G φ e π Z -invariant. This invariance lifts to the original second order form of the 2 1 ) 1 2 R , ] = = 2 conjugate to a ( drops out and the vector action takes the form i A [ 0 π H SL A inv , V S has Euclidean signature and determinant equal to ab Because Gauss’ law takes exactly the same form as in the absence of scalar fields, it can be The momenta To prove invariance, it suffices to prove that G solved in exactly the same way, by introducing a second vector potential So we find again the fundamental feature that the gauge constraint is2.2.2 a divergence. Solving Gauss’ law where and with This is exactly the samescalar expression field as dependent ( metric that As stressed above, this isgrange the multiplier key to exhibit manifest duality invariance. When doing so, the La- action, just as in the pure Maxwell case, and for the same2.2.1 reasons indicated Hamiltonian in form of that the case. vector action festly so. To that end, wethe first rewrite order formalism and solvingindeed Gauss’ manifestly law. As we shall see, the resulting first-order action is Electric-Magnetic Duality 2.2 First order form of the vector Lagrangian from which one derives the Hamiltonian form of the vector action PoS(HRMS)028 . ) α in R X MN , λ (3.1) (3.2) (3.3) G 2 ( (2.17) (2.15) δ replaced SL . to n MN ) 2 δ 2 , Marc Henneaux ( ··· SO , 0 (3.4) 1 = = λ N . electric-magnetic duality , G ) ! M while the three matrices + R b , 0 (2.16) B G 0 1 0 0 2 , ( b T δ χ = a

) λ N transformations. Accordingly, so b ab 0 which must be imposed on SL N c α G B ) ∂ χ ) = + B ∂ · X = α R ( N + M G , X M λ + transformations. α X ( 2 δ λ transformations, B ( ε ) + ) ) i , ac . = δφ R T = SL R , 0 ϕ G ! ab transformation, , ( λ a M 2 ⇔ α G n ( ∂φ 1 ) = B B ) but with the Euclidean metric enlarges the symmetry from ε ), already without the scalar ﬁelds, since the ∂ 2 MN R − δ δ ( Sp + , ab 0 G = ), one ﬁnds 1 0 0 2 σλ a 8 G Sp 1.14 c ' ( 2.15 ; ; = − ) ab b +

) N 2.2 will extend to the Hamiltonian if the variation α N N SL G A Q R A σ ) X ˙ = b , A δ λ N ( T electromagnetic ﬁelds is straihtforward. The manifestly a M ) · 0 2 R i ) ( , λ n λ X cb M α ϕ n MQ ( X G 2 B SL = G , ( ( α α MN ε M ! + MN Sp ε G A σ M + Q δ = 0 0 1 0 λ x ab a 3 , G

d A QN ) δ 0 δ = G R , dx − + 2 ( X -invariance Z ) MN SL 2 1 is invariant under R G , ab δ 2 ( ] = ε M i SL A [ inv S The replacement of the Euclidean metric by It follows that the vector action is invariant under The extension to the general case of This is because under a general infinitesimal Similarly, the kinetic term is invariant under ( are the generators of by a scalar field dependent metric symplectic form Electric-Magnetic Duality 2.3 Manifest under which the scalar fields transform as in ( so that the Hamiltonian is invariant. Here, The kinetic term in the action is invariant under is also the total action, inis either first an order off-shell or symmetry. second order form. 3. The general case invariant form of the vector action reads as in ( with The coupling to the scalar fields doeskinetic not term. affect The this invariance property, under since the scalar fields do not enter the of the metric duefields, to i.e., the variation of the scalar fields compensates the variation of the magnetic This condition replaces the more restrictivethe condition absence of scalar fields. PoS(HRMS)028 , ) a n and Z (3.5) n ) ] per- rather i and to ∂ 1 ϕ ) . − ( n 3 MN conjugate a ( n ab G Z ai µ U m / π ], which also ∂ ) ]. Marc Henneaux ( R 17 , 14 imn n ε 2 ( 1 2 . Here, b ρσ ) F − n Sp ( a -forms [ µν = p U F / ai ) through the process of going π ]. The first-order form of the R ) , which are found to be subject i , µνρσ a i 13 n ϕ ε A , ( 2 ) duality of interacting electromag- i ( ) 12 ϕ MN [ ( SL R G ) , ab ∈ n R ) of the action, one can derive the first- ν i 2 , ; and (ii) integrating over the conjugate is given for instance in [ ( a 0 ϕ + ]. The group of dualities cannot be bigger 3.5 56 A ( SL MN 16 0 through the Lagrange multipliers method, µν electric-magnetic duality invariance is in fact conjugate to b G Sp i ) a ≈ F π R ⊂ 9 , a ai µν 2 7 π , F ( i 7 ) i ∂ E SL ϕ determined from − and repeats the above construction, one gets invariance ( i ) in second order form, ab ϕ ] R ] is, contrary to the existing folklore, an off-shell symmetry µ µ a , of dualities. This is the situation for maximal supergravity in n A 14 ) [ x 2 , . If one takes a smaller set of scalar fields, belonging to the coset 4 ( V R ) 6 ) that are kept, through the relation . The solution for d S , n , n ) n Sp ( , Z 2 R 13 U ( , , 1 4 ) is to take the scalar fields in the coset space / ··· n ) , 2 Sp − ] and extensively used in [ 12 1 . ( R 3.4 is a maximal symmetry of the theory, implemented when the scalar fields a , i ), which is then viewed as a set of equations that determine , ⊂ ) 15 = Z n Sp ] = 11 R 2 G a 3.4 µ a , ( ( ∈ n A a i [ 2 λ Sp A ( V (on-shell symmetry). But S from ( because the kinetic term does not (and cannot) depend on the scalar fields Sp ) ) by first introducing the momenta is a subgroup of ) R G , R 3.1 MN n , ) is given in [ 2 G n of the potentials (conventionally the even-numbered ones) for the momenta ( 2 are functions of the scalar fields potentials 3.1 n ( ) SL n i ], where they are obtained through the requirement that the equations of motion are invariant where Sp A scalar field dependence of the kinetic term would clash with the gauge invariances of the vector fields. Once the first-order action is completely known, one can go to its second order form by (i) In this article, we have explicitly shown that the A similar analysis applies to (twisted) self-duality equations for Conversely, if one knows the second order form ( One way to achieve ( The group ϕ 3 14 ( H , / ab 6 from the first order form to the[ second order form of the vector action. These functions are given in ν to which one must add the action for the scalar fields action ( derive from a (non manifestly spacetime covariant) variational principle contrary to widespread order form ( four dimensions where the duality group is than and not just a symmetry of the equations of motion. This property extends the analysis of [ trading construct to the and implementing the Gauss’the constraints Lagrange multipliers beingmomenta the to time get the components vector action under under a smaller group 4. Conclusions netic and scalar fields [ formed for one free Maxwell field.Gauss’ Crucial law takes in the the form existence of of asidered, total the divergence. duality but This symmetry is property is is preserved the lost by when factcouplings minimal Pauli that or coupling couplings, is Chern-Simons present con- terms, in all of supergravity, which as therefore well preserve as the Chapline-Manton symmetry. a bona fide off-shell invariance astransformations our of analysis the shows, vector although potentials in are the non-local second in order space formalism, and the somewhat awkward. Electric-Magnetic Duality than conditions on to Gauss’ constraints, andsecond then set solving of explicitly potentials the constraints through the introduction ofbelong to the the coset G PoS(HRMS)028 ], 8 [ ) R , , 221 n ] and con- 2 ( 193 8 Sp Marc Henneaux à la Yang-Mills , 154 (2001) , 024018 (2005) 101 71 , 1053 (1982). 15 ]. Since the obstruction to gaug- 21 , 80 (1997) [arXiv:hep-th/9702184]. -symmetry. It was shown in [ 400 ) R , ]. We shall return to this question elsewhere n 2 18 ( 10 Sp ]. 4 , 393 (1977). , 1592 (1976). 121 13 ] that one cannot gauge the electric-magnetic duality symmetry in the case of free 20 York: 1978) [hep-th]. [arXiv:hep-th/0103086]. [arXiv:gr-qc/0408101]. Nucl. Phys. B (1981). quantization in abelian N-form theories,” Phys. Lett. B Phys. Rev. D M. H. gratefully acknowledges support from the Alexander von Humboldt Foundation through Finally, a word about the gauging of the ]. Gravitational equations in diverse dimensions viewed as twisted self-duality equations will [9] S. Helgason, “Differential Geometry, Lie groups and Symmetric Spaces", Academic Press (New [8] C. Bunster and M. Henneaux, “Can (Electric-Magnetic) Duality Be Gauged?,” arXiv:1011.5889 [4] M. Henneaux and C. Teitelboim, “Duality in linearized gravity,” Phys. Rev. D [5] S. Ferrara, J. Scherk and B. Zumino, “Algebraic Properties Of Extended Supergravity[6] Theories,” M. K. Gaillard and B. Zumino, “Duality Rotations For Interacting Fields,” Nucl.[7] Phys. B B. de Wit, “Electric-magnetic duality in supergravity,” Nucl. Phys. Proc. Suppl. [3] S. Deser, A. Gomberoff, M. Henneaux and C. Teitelboim, “Duality, self-duality, sources and charge [1] S. Deser and C. Teitelboim, “Duality Transformations Of Abelian And Nonabelian Gauge[2] Fields,” S. Deser, “Off -shell electromagnetic duality invariance,” J. Phys. A 19 a Humboldt Research Award and support from theThe ERC Centro through de the Estudios “SyDuGraM" Advanced Científicos Grant. (CECS) isters funded of by Excellence the Chilean Base Government Financing throughported the Program by Cen- of IISN - Conicyt. Belgium The (conventionsPolicy 4.4511.06 Office work through and of the 4.4514.08), M. Interuniversity by Attraction H. 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