New aspects of duality in electromagnetism, linearized gravity and higher spin gauge fields

Transcription

1 4th Sakharov Conference Tamm Theory Department Lebedev Physics Institute May 20, 2009 New aspects of duality in electromagnetism, linearized gravity and higher spin gauge fields Glenn Barnich Physique théorique et mathématique Université Libre de Bruxelles & International Solvay Institutes

2 Electric-magnetic duality Motivation Special type of symmetries in modern theoretical physics Simplest context : electromagnetism with electric and magnetic sources Application to thermodynamics of black holes dyons New formalism making EM duality manifest Dualities are a special kind of symmetries that are very much studied in modern theoretical and mathematical physics. It is therefore interesting to really understand them deeply in the simplest possible context which is electromagnetism in the presence of both electric and magnetic charges. My personal interest in the subject originates from black holes which carry both electric and magnetic charges and the problem of how to discuss their thermodynamics. In order to do so, it turns out to be extremely convenient to construct a new formalism that makes duality manifestly a symmetry of the action. It is this formalism that I will review in this talk and show how it applies to the black hole problem.

3 Black hole dyons First law for Reissner-Nordstrøm BH RN dyon ds 2 = N 2 dt 2 + N 2 dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ), N = 1 2M + Q2 + P 2 r r 2, A = Q r dt + P (1 cos θ)dφ, infer thermodynamics for parameter variations from purely electric case using duality δm = κ 8π δa + φ HδQ + ψ H δp = M 2 (Q 2 + P 2 ), κ = r + r 2r 2 + r ± = M ±, A =8π [ M 2 Q2 + P M ], φ H = Q r +, ψ H = P r + Problem: excluded in action based derivations of 1st law and Euclidean approaches because of string singularity and absence of magnetic potential Consider for instance the Reissner-Nordstrøm dyon. It is a black hole solution to the Einstein-Maxwell equations with both electric and magnetic charge. The first law of black hole mechanics for variations of parameters can easily be inferred from the well-known purely electric case by using duality. It states that the variation of the parameter M, later to be identified with the mass of the black hole, is proportional to the variation of the area of the horizon, with a geometric factor called surface gravity and later to be identified with temperature, plus terms that are proportional to the variation of electric and magnetic charges with factors that depend on the electric or magnetic potentials on the horizon. In a geometric derivation of the first law, for discussing uniqueness theorems and in Euclidean approaches one needs derivations starting from an action principle. The case of dyons is usually excluded in these discussions because of the string singularity of the vector potential and the absence of the magnetic analog of the field A_0, whose value on the horizon could play the role of chemical potential for magnetic charge. The idea is then to use a formulation where duality is manifestly a symmetry of the action.

4 Of course, the standard action principle for electromagnetism is invariant under gauge transformations, transformations that depend on an arbitary function on spacetime. EM duality No sources equations of motion field strength { df =0 d F =0 F = 1 2 F µνdx µ dx ν F 0i = E i F ij = ɛ ijk B i invariance under ( F F ) M ( F F ) detm 0 action principle df =0= F = da S[A] = 1 F F 2 gauge invariance δ Λ A = dλ, δ Λ S =0 The equations of motion of electromagnetism in the absence of sources are conveniently expressed in terms of a 2-form F as the closure of this form and its dual. The components of the 2-form encode the electric and magnetic fields in the usual way. The equations of motions are invariant under general linear transformations that exchange electric and magnetic fields. In order to quantize the theory, one requires the equations to derive from an action principle. For this, one solves half of the equations in terms of potential 1-forms. The other half then follow from the requirement that the action be stationary under variations of the dynamical variables A_\mu. By construction F and {}^*F are treated asymmetrically and the previous invariance is no longer manifest at the level of the action. It is important to have symmetries manifest at the level of the action because one then knows how to implement them in the quantum theory, either in the Hilbert space or through identities among correlation functions.

5 EM duality No sources Hamiltonian action principle S H [A µ,e i ]= d 4 x( Ėi A i H A 0 i E i ) canonical structure {A i (t, x), E j (t, y)} = δ j i δ3 (x y) Hamiltonian density H = 1 2 (Ei E i + B i B i ) B i = ɛ ijk j A k solve Gauss constraint i E i =0= E i = ɛ ijk j Z k reduced action principle S R [Z i T,A T i ]= d 4 x( ɛ ijk j Z k A i H) invariant under duality rotations { δd A i = Z i, δ D Z i = A i, δ D S R =0 Deser & Teitelboim Phys. Rev. D 13 (1976) 1592 It is in the Hamiltonian formalism that duality invariance becomes manifest at the level of the action. Due to gauge invariance, the Hamiltonian action principle involves a constraint, Gauss s law with A_0 the associated Lagrange multiplier. The electric field is minus the canonical momenta associated to the vector potential and the Hamiltonian density is half the sum of the electric field squared plus the magnetic field squared. Furthermore, the magnetic field is the curl of the vector potential. One can solve the Gauss constraint, which says that the divergence of the electric field vanishes. This implies that the electric field is also given by the curl of a potential, call it Z_i. The action principle for the reduced phase space, then effectively depends only on the physical variables given by the transverse part of the vector potentials. This action principle is easily seen to be invariant not under general linear transformations but under duality rotations among the fundmental variables of the variational principle. Indeed, the Hamiltonian is manifestly invariant, while invariance of the kinetic term follows by appropriate integrations by parts. This was pointed out in 1976 by Deser and Teitelboim.

6 EM duality No sources doublet notation A a i = ( Ai Z i ) B ai = ɛ ijk j A a k = ( B i E i ) δ D A ia = ɛ ab A b i manifestly invariant action S SS [A a µ]= 1 2 d 4 x [ ɛ ab B ai ( 0 A b i i A b 0) δ ab B ai Bi b ] A a 0 spurious Schwarz & Sen Nucl. Phys. B411 (1994) 35 In the mid-nineties, there had been a string-inspired surge of interest in the subject of EM duality. Schwarz and Sen gave an independent rederivation of this result. They collected the 2 vector potentials in an SO(2) doublet and used only the two SO(2) invariant tensor delta_ab and epsilon_ab to make duality invariance even more transparent. With respect to Deser & Teitelboim, they write the kinetic term in the form half (dot q p - dot p q) instead of dot q p, which amounts to modifying the Lagrangian density by a total time derivative. Then, because electric and magnetic fields are divergence free, the doublet of scalar potentials A_0^a that they introduce only occurs in a total derivative in their action. It is thus spurious and completely drops out of their formalism.

7 EM duality Point-particle sources duality requires both electric and magnetic sources magnetic pole qn a =(g n, 0) electron qn a = (0,e n ) dyon current j aµ (x) = qn a δ (4) (x z n )dz n µ n Γ n q a n =(g n,e n ) pole at origin singular potential regularize j µ M (x) =gδµ 0 δ(3) (x) A = g (cos θ + 1)dϕ 4π A i A ɛ i = g 4π y R(R z) x R(R z) 0 B i = g 4π x i r 3 A i = g 4π y r(r z) x r(r z) 0 R = r 2 + ɛ 2 magnetic field: semi-infinite solenoid The really interesting case is of course when there are some kind of interactions, for instance with external sources. Duality in the presence of sources requires both electric and magnetic sources. Let us concentrate on the case of point particles. They can carry carry either magnetic charge g and no electric charge and are called poles by Dirac, or just electric charge, electrons, or both types of charges, in which case they are called dyons. The magnetic and electric currents produced by such an object depends on the world-line Gamma the point-particle sweeps out in space-time. Consider the case of a single magnetic pole sitting at the origin. The monopole field it produces can be generated by a vector potential, albeit one that is singular along, say, the positive z axis. An action principle for the electrodynamics of point particles carrying electric or magnetic charges has been devised by Dirac. To generate the monopole field, he first replaces the singular potential by a regularized one describing a thin solenoid with no net magnetic charge.

8 EM duality Dirac strings add dynamical string G µν Σ (x) =g Σ =Γ M Σ δ (4) (x y)dy µ dy ν y µ (τ, σ), y µ (τ, 0) = z µ M (τ) property d G Σ = G Σ = j M modified field strength F D = da + G Σ df D = j M action principle S D [A µ (x),y µ (τ, σ),z e (τ)] = 1 2 Dirac veto: electron cannot touch string F D F D + e m e Γ e A dz µ e dz µe m M Γ e Γ M dz µ M dz µm quantization condition eg =2πN Dirac Phys. Rev. 74 (1948) 817 He then adds a dynamical string into the theory which brings the magnetic charge. This string comes from infinity and ends on the position of magnetic pole. In space-time this gives a 2 dimensional sheet extending to infinity and ending on the world-line of the pole. The defining property of the form G associated with this sheet is that the exterior derivative of its dual just gives the dual of magnetic current. So Dirac modifies the usual field strength so as to include the dual of the form defined by the string. The appropriate modification of Maxwell s equations in the presence of magnetic sources then follows kinematically. Dirac then goes on to show that if one uses this modified field strength in an action principle that only involves minimal coupling of A_\mu to the electric particle and appropriate kinetic terms for the pole and the electron, one gets all field equations of the system correctly. This means both Maxwell s equations and the equations of motion for the particles under the influence of the Lorentz force. There is one important proviso which stipulates that an electron can never touch a magnetic string. This is the Dirac veto and leads to the famous quantization condition between electric and magnetic charge. It shows another aspect of electric magnetic duality which is very much at the heart of the matter today: when the electric charge is weak, the magnetic one is strong and vice-versa.

9 EM duality Remarks on Dirac strings (i) asymmetric treatment of both types of sources, manifest Poincaré invariance but no duality (ii) non trivial U(1) fiber bundles, no strings Wu & Yang Phys. Rev. D12 (1975) 3845, D14 (1976)437 (iii) Dirac strings related to de Rham currents and Poincaré duality De Rham, Variétés différentiables, Hermann, 1960 (i) The Dirac treatment of both types of sources is asymmetric, Poincaré invariance is manifest, while duality invariance at the level of the action is not. (ii) The formulation of Dirac theory without strings but in terms of non trivial U(1) fiber bundles has been discussed in detail in the mid-seventies by Wu and Yang. (iii) There is of course a relation of Dirac strings to de Rham currents, differential forms with coefficients that are distributions and Poincaré duality as discussed for instance in the book by de Rham or the recent paper by Lechner and Marchetti.

10 BH dyons Duality complicated proof of duality between electric and magnetic BH partition function through semi-classical evaluation of Euclidean path integral Z(β, P ) vs Z(β, φ H ) Hawking & Ross Phys. Rev. D52 (1995) 5685 additional Legendre transformation needed to compare The asymmetric treatment causes a complication in the BH context, where Hawking and Ross have shown that there is a duality between an electrically and magnetically charged black hole. They computed the partition function through a semi-classical approximation of the Euclidean path integral. In the magnetic case, this is done at fixed charge, but in the electric case at fixed potential, so they need an additional Legendre transformation in order to be able to compare both results and show that they are duality symmetric. Because the formalism does not easily allow for it, Hawking and Ross are not considering dyons.

11 Dyons and duality Extended formulation in flat space New construction dynamical longitudinal fields and non spurious scalar potentials A a µ (A µ, Z µ ) C a (C, Y ) B a ( B, E) B a = A a + C a external sources action principle µ j aµ = 0 I[A a µ,c a ]=I M [A a µ,c a ]+I I [A a µ; j aµ ], I I [A a µ; j aµ ]= I M [A a µ,c a ]= 1 2 d 4 x [ɛ ab ( B a + C a ) ( 0 A b A b 0) B a B a ], d 4 x ɛ ab A a µj bµ. Maxwell s equations { A a 0 : B a 2 C a = j 0a C a : 2 C a = ɛ ab ( 0 A b 2 A b 0) A a : ɛ ab 0 B b + B a = ɛ ab j b. no strings, duality manifest G.B. & Gomberoff Phys. Rev. D78 (2008) In order to tackle these problems, we have extended the action principle of Schwarz and Sen to the case of conserved external electric and magnetic sources. As we have seen, the reason why the scalar potentials were spurious in the source free case was that the electric and magnetic fields were transversal. We thus introduce potentials for the longitudinal parts of these fields and generalise the action principle in such a way as to give the correct Maxwell equations in presence of both type of sources. Variation with respect to the scalar potential gives the electric and magnetic Gauss law. Variation with respect to the potentials for longitudinal fields allows to solve the scalar potentials A_0^a in terms of the other fields. Hence all four scalar potentials are auxiliary. Finally, variation with respect to the vector potentials gives Maxwell s equations for electric and magnetic fields. In this formulation, there are thus no strings to produce the monopole fields but longitudinal potentials, duality invariance is manifest while Lorentz invariance is not.

12 Dyons and duality Extended formulation in flat space fixed point particle dyon at origin j aµ (x) = 4πQ a δ µ 0 δ3 (x) Coulomb-type solution A a = ɛab Q b r dt, C a = Qa r resolution of string singularity gauge invariance δ ɛ A a µ = µ ɛ a, δ ɛ C a = 0 magnetic charge is surface integral, no longer a topologically charge canonical pairs ( A T, Z T ), ( A L, Y )( Z L, C) gauge fixing the magnetic Gauss constraint gives back standard EM For a point particle dyon at the origin, the solution of the EOM is now described by two spherically symmetric Coulomb-type fields. The string singularity of the previous formulation has thus been resolved. This action has twice the gauge invariance of the usual Maxwell action. Since there are electric and magnetic Gauss constraints, magnetic charge now appears not as a topological charge but as a surface charge on a par with electric charge. Without magnetic sources, the associated action is of course equivalent to the standard action. In fact, by analysing the canonical pairs, one finds the 2 physical ones, in terms of the transverse part of the vector potential and the transverse part of the electric field. Besides the usual non physical pair given by the longitudinal part of the vector potential and the longitudinal part of the electric field, there is a second non physical pair in this formulation given by the longitudinal part of the second vector potential and the longitudinal part of the magnetic field. Of course, gauge fixing the magnetic Gauss constraint then gives back the standard formulation of electromagnetism.

13 Dyons and duality Extended formulation with dynamical dyons dynamical point particle dyons need strings for Lorentz force law G aµν = n total action q a n Σ n δ (4) (x y n )dy µ n dy ν n I M + I I + I k y µ n(σ n, τ n ), y µ n(0, τ n ) = z µ n(τ n ) d 4 x ɛ ab [ 2 i C a α b i β ai α b i + β ai T 0 γ b i ) ]. I k [z µ n]= n m n Γ n dz µ ndz nµ β ai = G a 0i α a i = 1 2 ɛ ijk G a jk β ai T = ɛijk j γ a k variation with respect to z µ n gives Lorentz force law veto: string attached to dyon n cannot cross any other dyon leads to standard quantization condition for dyons ɛ ab q a nq b m =2πN In the case of dynamical dyons, appropriate string terms for all dyons are needed in order to get the Lorentz force law correctly from a variation of the position of dyons and the strings. Besides the kinetic term for the dyons, these string terms only couple to the longitudinal fields and among themselves. The derivation of the Lorentz force law again needs a veto, that no dyon can touch the string of another dyon, which leads to the generalization of the quantization condition for dyons. The string terms are also needed to control Lorentz invariance, but they can be dropped in the case of a single dyon.

14 Dyons and duality Extended formulation in curved space need to generalize first order ADM formulation of Einstein-Maxwell no non singular longitudinal magnetic field B i ADM = ɛ ijk j A k curved space matter action total action I[z, u] = B ai = ɛ ijk j A a k + g i C a I M [A a µ,c a,g ij, N, N i ]= 1 [ d 4 x (B ai + g i C a )ɛ ab ( 0 A b i i A b 8π 0) N B i g abi a ɛ ab ɛ ijk N i B aj B bk] d 4 x [a A (z) 0 z A u α γ α ] kinetic term a A (z) 0 z A = πij 16π 0g ij E i 4π 0A i + g i C 4π 0Z i Lagrange multipliers and constraints u α (N, N i,a a 0) γ α (H, H i, G a ) H = 1 16π (HADM + H mat ), H i = 1 16π (HADM i + H mat i ), G a = 1 4π ɛ ab i B bi gauge parameters smeared constraints ɛ α (ξ, ξ i, λ a ) Γ[ɛ] = d 3 x γ α ɛ α In flat space, our action is first order in time derivatives. In order to generalise it to curved spacetime, it is thus natural to generalize the first order ADM formulation of Einstein-Maxwell theory. In the latter formulation, the magnetic field is by construction purely transversal and does thus not support a nonsingular longitudinal magnetic field. In curved space, we now define electric and magnetic fields with suitable transverse and longitudinal potentials and suitably generalize our flat space action to curved space. As usual for generally covariant theories, the total action is then only a kinetic term, which determines the Poisson brackets and constraints, which now involve also electric and magnetic Gauss constraints.

15 Dyons and duality Extended formulation in curved space first class gauge algebra {Γ[ɛ 1 ], Γ[ɛ 2 ]} = Γ[[ɛ 1, ɛ 2 ]] {H[ξ],H[η]} = H[[ξ, η] SD ]+G[[ξ, η] B ] surface deformation algebra [ξ, η] SD = ξ i i η η i i ξ, [ξ, η] i SD = g ij (ξ j η η j ξ )+ξ j j η i η j j ξ i [ξ, η] a B = B ai ɛ ijk ξ j η k ɛac B ci g (ξ η i η ξ i ) diffeomorphisms ξ = Nη 0 ξ i = g iµ η µ L η g µν δ ξ g µν sources I J [A a µ,c a,y µ ]= 1 4π d 4 [ x ɛ ab A a µ j bµ + g i C a αi b 1 2 βai αi b 1 2 βai T 0 γi b ] β at i = ɛ ijk j γ a k 1-1 correspondence of solutions to those of covariant equations resolved RN dyon ds 2 = N 2 dt 2 + N 2 dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ), N = 1 2M + Q2 + P 2 r r 2, A a = ɛ ab Q b ( 1 r 1 r + )dt C a = Q a r dr r 2 N(r ) The first question is then about the gauge structure of this action. It turns out that all constraints are first class and that the usual algebra of the gravitational constraints only closes on the 2 Gauss laws with structure functions that involve both electric and magnetic fields. This constraint algebra is what guarantees general covariance in the Hamiltonian framework. Diffeomorphisms are obtained as usual by the gauge transformations generated by the geometric constraints with a suitable choice of gauge parameters. In the presence of sources suitable string terms are needed in order to show equivalence with the covariant approach, but can again be dropped in the case of a single dyon. Like in flat space, the new formulation allows to resolve the string singularity of the RN dyon: the singularity of all dynamical variables is at worst Coulomb-like. For later use, I have used a gauge where the potentials vanish on the horizon and thus have a value at infinity, which is their value at the horizon in the gauge where the potentials vanish at infinity.

16 Dyons and duality 1st law of black hole mechanics surface integrals solution of constraints δ z (γ α ɛ α )=δz A δ(γ α ɛ α ) δz A z A s i k i ɛ solution of linearized constraints Regge & Teitelboim Ann. Phys. 88 (1974) 286 δz A s time independent solution z a s,u α s 0 z A s =0 δa B δz 0z A A 0 a A = δ(γ αu α ) δz A implies i ku i s [zs A, δzs A ] = 0 d 3 x i ku i s [zs A, δzs A ]= S r1 d 3 x i ku i s [zs A, δzs A ]. S r2 Stokes theorem standard gravitational part kɛ[z i A ; δz A ]=kɛ grav,i [g ij, π ij ; δg ij, δπ ij ]+kɛ mat,i [z A ; δz A ] kɛ grav,i = 1 ] [G ljki (ξ k δg lj k ξ δg lj )+2ξ k δπ ki + (2ξ k π ji ξ i π jk )δg jk, 16π G ljki = 1 2 g(g lk g ji + g il g jk 2g lj g ki ) new matter part includes electric and magnetic contributions kɛ mat,i = 1 ( ξ ɛ ijk B a 4π g j δa ak ξ B ai δc a + ɛ ab (ξ k B ai ξ i B ak )δa b k ɛ ab gg il ɛ ljk ξ j B ak δc b + ɛ ab ( ) g i λ a δc b λ a δb bli ) for RN dyon S r d n 1 x i k mat,i u = 1 4π π 0 dθ 2π 0 dφ ɛ ab A a 0δB bli We finally have all the ingredients to discuss the thermodynamics of BH dyons in the Hamiltonian approach with fixed potentials of both kinds instead of fixed charges. It is of course clear that every derivation people have done before in the purely electric case can now be done in the dyonic case in a manifestly duality invariant way. In the Hamiltonian approach the all important surface integrals related to the conserved charges are computed from the variation of the constraints through integrations by parts. If the fields satisfy the constraints and the variations the linearized constraints the lhs vanishes. If the complete solution with Lagrange multipliers gives time independent canonical variables, so does the first term on the RHS. It follows that the divergence of k vanishes and Stokes theorem applies: the integral of k over a sphere does not depend on the radius. The total k splits into the standard gravitational part and a new matter part that now contains both electric and magnetic contributions. In particular, for variations around the RN dyon, the matter part reduces to the scalar potentials multiplying the variations of longitudinal electric and magnetic fields.

17 Dyons and duality 1st law of black hole mechanics first law S d 3 x i k i u[z,δz] = S r+ d 3 x i k i u[z,δz] at infinity S d 3 x i k i u[z,δz] =δ z M φ H δ z Q ψ H δ z P at horizon d n 1 x i ku grav,i S r+ = κ 8π δ za geometric derivation of first law for RN for arbitrary perturbations δ z M = κ 8π δ za + φ H δ z Q + ψ H δ z P The first law is now a consequence of applying Stokes theorem between infinity and the horizon. At infinity one finds from the gravitational part the variation of the ADM mass by definition of the ADM mass, and the electric potential on the horizon times the variation of electric charge. The new feature is now that there is also the magnetic potential on the horizon times the variation of magnetic charge. On the horizon, the matter part vanishes because we have chosen a gauge where the scalar potentials vanish on the horizon. Finally, one can argue that the gravitational part gives the required kappa over 8 pi times the variation of the area. This is the difficult part of the computation but it is unchanged so previous computations apply. This concludes the geometric derivation of the first law for arbitrary perturbations around the RN dyon black hole that are still solutions.

18 Dyons and duality Black hole thermodynamics Euclidean approach in the grand canonical ensemble improved constraints for suitable fall-off conditions contain surface terms H = d 3 x (H N + H i N i )+M Q = 1 φ c d 3 x (G 1 A 0 )+Q, P = 1 ψ c d 3 x (G 2 Z 0 )+P three commuting observables {H, Q} =0={H, P } = {Q, P } partition function Z[β, φ c, ψ c ] = Tr e β(c H φ c b Q ψ c b P ) = e Ψ G Massieu function Ψ G (β, βφ c, βψ c ) Let us now finish by sketching the Euclidean approach in the grand canonical ensemble. First of all, under the fall-off conditions under considerations, where the lapse goes to 1 at infinity and the electric and magnetic potentials go to constant values, the constraints need to be supplemented by suitable surface terms at infinity in order for Poisson brackets to be well defined. They are precisely the ADM mass, and the electric and magnetic surface integrals. We then have three commuting classical observables and would like to compute, in the quantum theory, the partition function, or its logarithm, which is the Massieu function for the appropriate variables.

19 Dyons and duality Black hole thermodynamics path integral Z[β, φ, ψ] = DΦe IT e Euclidean action I T e = β 0 ( dτ i ) d 3 xa A (z) 0 z A (H φ c Q ψ c P ) + ghost terms leading contribution: action evaluated at Euclidean Reissner-Nordstrom dyon Ie mat (RND) =βφ H Q + βψ H P Ie grav (RND) = βm A total result Ψ G = βm A + βφ HQ + βψ H P The path integral representation of the partition function involves an action, that, after integration over the momenta corresponds to Euclidean geometries. The leading contribution to the path integral is given by the action evaluated at the classical solution satisfying the boundary conditions. In our case, this is the Euclidean version of the Reissner-Nordstrom dyon, where the chemical potentials coincide with the scalar potentials at the horizon. Because the solution is static and satisfies the constraints, the action reduces to boundary terms. The matter part directly gives what we want with both electric and magnetic contributions. The gravitational part is more difficult. It has been worked out before and gives minus beta times the mass plus the famous 1/4 of the area, with the sum the total result, as expected from duality arguments.

20 Conclusion construction of an explicitly duality invariant version of electromagnetism enhanced gauge invariance static dyon described by Coulomb fields without string singularities electric and magnetic charges are surface integrals applications in the context of thermodynamics of BH dyons To conclude, we have constructed a local, explicitly duality invariant formulation of electromagnetism for which the field of a static dyon is described by Coulomb type fields without string singularities. Furthermore, both electric and magnetic charges appear as surface integrals. Finally I have discussed applications of this formulation in the context of the thermodynamics of black hole dyons.

21 Spin 2 Reduced phase space duality first order formulation S PF [h mn, π mn,n m,n]= [ dt d 3 x ( π mn ḣ mn n m H m nh ) ] H PF, Hamiltonian constraints H PF [h mn, π mn ]= d 3 x ( π mn π mn 1 2 π r h mn r h mn 1 2 mh mn r h rn m h n h mn 1 4 m h m h ), H m = 2 n π mn, H = h m n h mn. decomposition φ mn = φ TT mn + φ T mn + φ L mn, φ L mn = m ψ n + n ψ m, φ T mn = 1 2 (δ mn m n ) ψ T. ψ m = 1 ( n φ mn m k l φ kl ), ψ T = 1 ( φ 1 m n φ mn ), conjugate pairs ( h TT mn(x), π kl TT( y) ), ( h L mn ( x), π kl L (y) ), ( h T mn (x), π kl T (y) ). solving constraints + gauge fixing H m =0=H h L mn = 0 = πt kl d 3 x reduced phase space H R = π kl L =0=h T mn ( πtt mn πmn TT + 1 ) 4 rh TT mn r h mn TT. introduce potentials to make reduced phase space manifestly duality invariant Henneaux & Teitelboim Phys. Rev. D71 (2005)

22 Spin 2 Extended gauge invariance extended gauge invariance to couple to both electric and magnetic sources degree of freedom count phase space variables # dof = 2 # fcc = 8 # cp = 10 2 (# cp) = 2 (# dof) + 2 (# fcc) z A =(H a mn,a a m,c a ). change of variables brackets h a mn =(h mn,h D mn) π mn a =(π mn D, π mn ) h a mn = ɛ mpq p H aq n + ɛ npq p H aq m + m A a n + n A a m (δ mn m n )C a π a mn = H a mn. ( Hmn 1TT (x), 2 ( OHTT 2 ) kl ) (y), ( A 2 m(x), 2 r H 1rn L ( C 1 (x), 1 ) 2 ( H2 T p q H 2pq T )(y), ( ) A 1 m(x), 2 r HL 2rn (y), ) ( (y), C 2 (x), 1 2 ( H1 T p q H 1pq T ). )(y) {h a mn(x), π bkl (y)} = ɛ ab 1 2 (δk mδ l n + δ k mδ l n)δ (3) (x, y). abelian gauge structure H am = 2ɛ ab n π b mn =2ɛ ab n H b mn, H a = h a m n h mn a = 2 C a.

23 Spin 2 Duality + sources first order action S G [z A,u α ]= d 4 x (a A [z]ż A u α γ α [z]) dt H[z], duality invariant Hamiltonian H = d 3 x ( H a mn H mn a 1 2 Ha H a Ca 2 C a ). interacting theory S J = d 4 x 1 2 ha µνt µν a, µ T µν a =0 h a 0m = n a m = h a m0, h a 00 = 2n a, S T [z A,u α ; T aµν ]= 1 16πG S G + S J linearized Taub-NUT T µν a (x) =δ µ 0 δν 0 M a δ (3) (x i ), M a =(M, N). H a = 16πGM a δ (3) (x). C a = GM a (2r), n a = GM a ( 1 r ),Aa m = GM a ( 1 2 x m r ),nam = H a mn =0, singularity resolved by additional pure gauge degrees of freedom h a mn = GM a ( 2x mx n r 3 ), π mn a =0.

24 Spin 2 Surface charges + global symmetries Regge-Teitelboim γ α ε α =( m ξ an + n ξ am )ɛ ab π b mn +(δ mn m n )ξ a h amn i ki ε [z] ki ε [z] = 2ξ a mɛ ab π bmi ξ a (δ mn i δ mi n )h amn + h amn (δ mn i δ ni m )ξ a { m ξ an s + n ξ am s (δ mn m n )ξ a s =0= 0 ξ am s m ξ a = 0 = 0 ξ a s, s, ξ a µs = ω a [µν] xν + a a µ, electric and magnetic ADM type conserved surface charges Q εs [z s ]= 1 16πG S d 2 x i ki εs [z s ]= 1 2 ωa µνj µν a a a µp µ a, dyon P a = M a 1 copy of global Poincaré transformations acting T µν a (x )=Λ µ αλ ν βta αβ (x) ξ ν as(x )=Λ µ αξ α as(x) = (Λω a Λ 1 x ) ν +(Λω a Λ 1 b + Λa a ) ν, Q ε s [z s]=q ε [z s ]. GB & Troessaert JHEP 01 (2009) 30

25 Spin 1 Duality generator as Chern-Simons term spin1reduced phase space description what about global symmetries? [ S R [A Ta i ]= dt d 3 x 1 2 ɛ ab(oa Ta ) i 0 A Tb i H 1 ], H 1 = 1 d 3 x (OA Ta ) i (OA T a ) i = 1 d 3 xa T ai A T 2 2 ai, curl (OA) i = ɛ ijk j A k d 3 xg i (Of) i = d 3 x (Og) i f i, (O 2 f T ) i = f Ti reduced phase space Poisson brackets {A Ta i (x),a T bj (y)} 1 = ɛ ab 1 ɛ jkl y (3) kδt il (x y) =ɛ ab 1( O y δ T (3) i (x y) ) j, vacuum Maxwell s equations 0 A T ai (x) ={A T ai (x),h 1 } 1 = ɛ ab (OA T b ) i (x), duality generator H 0 = 1 2 d 3 xa ai T (OA T a ) i, {H 0,H 1 } 1 =0. SO(2) Chern-Simons term Deser Henneaux Teitelboim Phys Rev D55 (1997) 826

26 Higher Spins Bi-Hamiltonian systems new simplified Poisson bracket {A Ta i (x),a Tb j (y)} 0 = ɛ ab δ T (3) (x y), ij duality generator as 2nd Hamiltonian {A T ai (x),h 1 } 1 = {A T ai (x),h 0 } 0 recursion operator R ai bj = δ a b (O) i j hierarchy K T ai p =( ) p ɛ ab (O p A T b ) i 0 A T ai (x) =Kp T ai (x) {A T ai (x),h p } 1 = {A T ai (x),h p 1 } 0 H p 1 = ( )p 2 d 3 xa Ta i (O p A T ) i infinity of Hamiltonians in involution {H n,h m } 1 =0={H n,h m } 0, n, m 0. generalizes directly to reduced phase space description of massless HS fields GB & Troessaert Phys Rev D55 (1997) 826

27 Talk based on G. Barnich and A. Gomberoff. Dyons with potentials: Duality and black hole thermodynamics. Phys. Rev. D, 78:025025, G. Barnich and C. Troessaert. Manifest spin 2 duality with electric and magnetic sources. JHEP, 01:030, G. Barnich and C. Troessaert. Duality and integrability: Electromagnetism, linearized gravity, and massless higher spin gauge fields as bi-hamiltonian systems. J. Math. Phys., 50:042301, 2009.