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

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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

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

4 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

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

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

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

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

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

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

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)

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

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

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

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 )

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

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

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 )

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

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

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.