Będlewo. October 19, 2007 Glenn Barnich Physique théorique et mathématique Université Libre de Bruxelles & International Solvay Institutes
Algebraic structure of gauge systems: Theory and Applications Why gauge theories? BRST differential for finite-dimensional toy model Field theory: locality and examples Anomalies, divergences, consistent deformations Characteristic cohomology, central extensions in gravitational theories
Theory 1: Finite-dimensional toy model Symetries & Stationary surface function S 0 [φ i ] on manifold F action vector fields S v, vs 0 = 0 symmetries stationary surface Σ F : S 0 φ i = 0 shell symmetries induce well-defined vector fields S Σ on-shell symmetries regularity conditions imply that S Σ Γ(T Σ) de Rham differential on Σ : γ longitudinal differential NB: dim F= N rank 2 S 0 φ i φ j = N M dim(σ) = M aim of BRST construction: off-shell description of H(γ).
Theory 1: Finite-dimensional toy model Koszul-Tate resolution symmetries e α = Rα i on F generating Γ(T Σ) φ i generating set all symmetries contain trivial ones S v = v = f α e α + µ [ij] S 0 φ j φ i on-shell closure of generating set [ e α, e β ] f γ αβ e γ dual one-forms C α ghosts longitudinal differential γ = C α e α 1 2 Cα C β f γ αβ C γ, γ2 0 additional generators φ i, C α φ i, C α antifields δ = S 0 φ i φ i + φ i R i α C α H ( δ, C (F ) (φ i, C α) ) = C (Σ) Koszul-Tate resolution
Theory 1: Finite-dimensional toy model BV complex homological perturbation theory s = δ + γ +..., s 2 = 0 H ( s, C (F ) (C a, φ i, C α) ) = H(γ, C (Σ) (C a )) BV complex Gerstenhaber algebra (A, B) = R A φ A L B φ A (φ φ ) φ A (φ i, C α ) antibracket (degree 1) canonical generator (degree 0) s = (S, ), 1 2 (S, S) = 0 S = S 0 + φ i R i αc α + 1 2 C γf γ αβ Cα C β +... solution of classical master equation Henneaux & Teitelboim, Quantization of gauge systems
Theory 1: Finite-dimensional toy model Deformation theory antibracket in cohomology (, ) M : H g 1 H g 2 H g 1+g 2 +1 ([A], [B]) M = [(A, B)] deformation theory S = (0) S + (1) S + (2) S +... non trivial infinitesimal deformations [ (1) S ] H 0 (s) no obstruction if 1 2 ([(1) S ], [ (1) S ]) M = [0] H 1 (s) 1 ((0) S, (0) S ) = 0 2
Theory 2: Field theory Jet-bundles classical mechanics: action functional S 0 [q] = dynamics determined by Euler-Lagrange derivatives Jet-bundle of order 1: local coordinates t1 t 0 δl δq t, q, q dt L(q, q) L q d dt L q = 0 total derivative d dt = t + q q + q q Local functions : finite order in derivatives Field theory E M φ i, x µ J (E) M φ i (µ) φi µ 1...µ k, x µ dim(m) = n total derivative ν = x ν + φi (µ)ν φ i (µ) Euler-Lagrange derivative δ = ( ) µ δφ i (µ) φ i (µ) µ1...µ k = µ1... µk
Theory 2: Field theory Variational bicomplex de Rham differential on jet-bundle d = dx µ x µ + dφi (µ) φ i (µ) = d H + d V horizontal or total differential d H = dx µ µ vertical differential d V = (dφ i (µ) dxν φ i (µ)ν ) φ i (µ) infinitesimal field variation bicomplex (Ω r,s, d H, d V ) ω r,s = 1 r!s! ω µ 1...µ k i 1 (ν 1 )...i s (ν s )dx µ 1... dx µ k d V φ i 1 (ν1 )... d V φ i s (νs ) local function
variational bicomplex local functional forms F s = Ω n,s /d H Ω n 1,s Euler-Lagrange complex locally exact E = d V φ i δ δφ i ω n = d H η n 1 δωn δφ i = 0
Globally Anderson, The variational bicomplex horizontal complex I F 0 0
Theory 2: Field theory Symmetries & Stationary surface action functional F 0 S 0 = [Ld n x] = U (Ld n x) φ(x) Lie algebra of symmetries S δ Q = (µ) Q i φ i (µ), δ Q [Ld n x] = [0] δ Q L = µ k µ [Q 1, Q 2 ] i = δ Q1 Q i 2 (1 2) stationary surface Σ : (µ) δl δφ i 0 NB: det 2 L q i q j 0 Σ : t, q i, q i Noether operator N +i = N +i(µ) (µ), N +i ( δl δφ i ) = 0 initial conditions associated symmetry N i = ( ) (µ) N +i(µ) δ N [Ld n x] = [0] gauge symmetries global symmetries G S S/G Lie ideal
Theory 2: Field theory Longitudinal differential (irreducible) generating set of Noether opertor trivial operators irreducibility commutator is a gauge smmetry Q i αβ = ( ) (µ) Q +i(µ) αβ R +i α = R +i(µ) α (µ), R +i α ( δl δφ i ) = 0, M +i = M +[j(ν)i(λ)] (ν) δl δφ j (λ) Z +α R +i α 0 Z +α 0 δ Rα R i β (α β) = Q i αβ Q +i αβ f +γ αβ R+i γ R+i(µ) α / 0 N +i ( δl δφ i ) = 0 N +i = Z +α R +i α + M +i additional generators (µ) C α longitudinal differential γ = δ γ 2 Rα (C α ) 1 0 2 (µ)f γ αβ (Cα C β ), C γ (µ)
Theory 2: Field theory Koszul-Tate & BV antifields (µ) φ i, (µ) C α δ = (µ) δl δφ i φ i(µ) + (ν) R +iα (φ i ) C α(ν) resolution (Ω r,s (Σ), d H, d V ) = H(δ, (Ω r,s, (E A ), d H, d V )) HPT s = δ + γ +..., s 2 = 0 H(s, Ω(E AC )) = H(γ, Ω(Σ C )) antibracket (, ) : F g 1 F g 2 F g 1+g 2 +1 (A, B) = [d n x( δr a δφ A δ L b δφ A (φ φ ))] A = [d n x a], B = [d n x b] master equation 1 2 (S, S) = 0 S = [d n x (L + φ i R i α(c α ) + 1 2 C γf γ αβ (Cα C β ) +... )]
Theory 2: Field theory Local BRST cohomology generator in modified bracket s = (S, ) alt (, ) alt : F Ω Ω (A, ) alt = (µ) δ R a δφ i L φ (µ) (φ φ ) local BRST cohomology {s, d H } = 0 H(s, F) applications! generated in standard antibracket sa = (S, A) deformation theory in the space of local functionals
Theory 2: Field theory Examples scalar field theory S = [d n x ( 1 2 µφ µ φ + 1 2 m2 φ 2 + 1 4! φ4 )] Yang-Mills theory S = [d n x ( 1 4 F a µνf µν a + A µ a D µ C a + 1 2 C c f c abc a C b )] general relativity S = [d n x ( g R + g µν L ξ g µν ξ µ ν ξ µ )] Poisson Sigma model computation of H(s, F)!
Applications 1: Quantum field theory Perturbation theory perturbative expansion of Green s functions free quadratic action non trivial gauge invariance: not invertible because of zero eigenvalues aim: make quadratic part invertible while still retaining consequences of gauge invariance gauge fixation generator of canonical transformation Ψ[φ] φ A = φ A + δr Ψ δφ A φ A = φ A gauge fixing fermion gauge fixed action S gf [ φ, φ ] = S[ φ, φ + δψ δφ ] 1 2 (S gf, S gf ) eφ, e φ = 0
Applications 1: Quantum field theory Anomalies connected Green s functions W [J, φ ] = ln Z[J, φ ] Z[0, φ ] Legendre transform φj, e φ = δw δj J = J e φ, e φ effective action Γ[ φ, φ ] = (W Jφ) J=J e φ, e φ = S gf + Γ (1) +... Zinn-Justin equation 1 not a local functional consistency condition 2 (Γ, Γ) = A Γ, A Γ = A + O( ) local functional (Γ, (Γ, Γ)) = 0 (Γ, A Γ) = 0 (S, A) = 0 trivial anomalies absorbed through counterterm A = (S, B) S S B nontrivial anomalies [A] H 1 (s, F) SU(3) YM theory T rc[d(ada + 1 2 A3 )] Adler-Bardeen anomaly
Applications 1: Quantum field theory Counterterms divergences in effective action consistency condition Γ (1) = 1 ɛ Γ(1) 1 + finite 1 2 (Γ, Γ) = A Γ (S, Γ(1) 1 ) = 0 local functional counterterm S (1) = S ɛ Γ(1) 1, (S (1), S (1) ) = O( 2 ) trivial divergence Γ (1) 1 = (S, Ξ) can be absorbed by canonical field antifield redefinition renormalizability if [Γ (1) 1 ] H 0 (s, F) can be absorbed by modifying coupling constants 4d semi-simple YM H 0 (s, F) = [d 4 x P ] P : consequence: group invariant polynomial in S 0 = [d 4 x 1 4g F a µνf µν a ] S 0 = [d 4 xp ] Y a A = D µ1... D µk F a νρ is renormalizable (powercounting, Lorentz invariance) renormalizable in the modern sense
Applications 2: NC field theory Seiberg-Witten map non-commutative U(N) YM theory Weyl-Moyal star product deformation of solution of master equation for standard Yang-Mills, controlled by H 0 (s, F) = [d 4 x P ] no antifield dependence consequence: Seiberg-Witten map
Applications 3: Classical field theory Consistent deformations Start form free quadratic gauge theories Construction of interactions preserving gauge invariance? computation of H 0 (s, F) obstructions? uniqueness results on YM construction or general relativity massless spin 2 fields gauge transformations only possible deformation
Applications 3: Classical field theory Characteristic cohomology standard techniques H g (s, F) = H n g (d H, Ω,0 (Σ)) descent equations consequence: characteristic cohomology for variational surface forms a graded Lie algebra characteristic cohomology g = 1 H n 1 (d H, Ω,0 (Σ)) [j] H 1 (s, F) = S/G { Lie algebra of global symmetries d H j 0, j j + d H k + t, t 0 conserved currents canonical form for symmetry X i δl δφ i dn x = d H j X complete version of Noether s theorem that deals with ambiguities Charges Q X [φ s ] = S j X [φ s ]
Applications 3: Classical field theory irreducible gauge theories (no 2,3-forms): Characteristic cohomology H g (s, F) =0 = H n g (dh, Ω,0 (Σ)) g!3 vanishing theorems for characteristic cohomology in low form degree g=2 H 2 (s, F)! [f ] α! i Rα (f α ) 0 f α f α + tα, tα 0 associated conserved n-2 forms [kfn 2 ] H n 2 (dh, Ω,0 (Σ)) surface charges: Qf [φs ] =! S n 2 kfn 2 [φs ] reducibility parameters
Applications 3: Classical field theory Characteristic cohomology derived bracket: K H 0 : 1 2 (K, K) = 0 H 3 = 0 F H 2, G F = (F, K) H 1 = H 2 is a Lie algebra with bracket [F 1, F 2 ] = (G F1, F 2 )
Applications 3: Classical field theory Surface charges Examples semi-simple YM theory: δ ɛ A a µ = D µ ɛ a = 0 = ɛ a = 0 δ ɛ A µ = µ ɛ = 0 = ɛ = cte EM: k n 2 = F electric charge Q = GR: δ ξ g µν = L ξ g µν = 0 = ξ µ = 0 S n 2 F linearized gravity: δ ξ h µν = L ξ ḡ µν = 0 = ξ µ Killing vector of ḡ µν Q ξ = S n 2 k ξ [h, ḡ] application: first law of BH mechanics S k t = H k t δm = κ 8π δa
Applications 3: Classical field theory Surface charges expand GR S GR = S 2 + S 3 +... global symmetry L ξ ḡ µν = 0 = δ 1 ξh µν = L ξ h µν ḡ µν = η µν Poincaré invariance of Pauli-Fierz theory derived bracket: Lie bracket of Killing vector fields Surface charges form a representation of the algebra of Killing vectors {Q ξ1, Q ξ2 } := δ 1 ξ 1 Q ξ2 = Q [ξ1,ξ 2 ] full GR, asymptotics 1 g µν = ḡ µν + O( r χ ) µν replace h µν = g µν ḡ µν at boundary charges r Q ξ = S k ξ [g ḡ, ḡ]
Applications 3: Classical field theory Algebra & asymptotics new feature: asymptotic Killing vectors L ξ ḡ µν 0 to leading order that preserve the fall-off conditions 1 L ξ g µν = O( r χ ) µν suitable tuning of fall-off conditions on metrics and asymptotic Killing vectors: centrally extended charge representation of algebra of asymptotic Killing vectors {Q ξ1, Q ξ2 } := δ ξ1 Q ξ2 = Q [ξ1,ξ 2 ] + K ξ1,ξ 2 K ξ1,ξ 2 = k ξ2 [L ξ1 ḡ, ḡ] S NB: central extension vanishes for exact symmetries of the background
Applications 3: Classical field theory Asymptotically AdS non trivial asymptotic Kvf= conformal Kvf of flat boundary metric n>3: so(n 1, 2) only exact Killing vectors of AdS, no central extension n=3: pseudo-conformal algebra in 2 dimensions, 2 copies of Wit algebra charge algebra: 2 copies of Virasoro cornerstone of AdS3/CFT2 correspondence similar results in de Sitter spacetimes at timelike infinity
Applications 3: Classical field theory Asymptotically flat conformal boundary in asymptotically flat spacetimes: null infinity bms n Y A (θ A ) T (θ A ) conformal Kvf of n-2 sphere supertranslations, arbitrary function on n-2 sphere
Applications 3: Classical field theory Asymptotically flat ξ = [ξ, ξ ] algebra: semi-direct product with abelian ideal i n 2 n>4: so(n 1, 1) i n 2 n=4: conformal algebra in 2d so(3, 1) i 2 Bondi-Metzner-Sachs (1962)
Central extensions: Asymptotically flat spacetimes n=3: no restriction on Y (θ) 1 copy of Wit algebra acting on i 1 iso(2, 1) Ashtekar et al. (1997) charge algebra: relation to AdS 3 similar to contraction between so(2, 2) iso(2, 1) L ± m = 1 2 ( lp ±m ± J ±m ) l
Selected references Reviews on BV
Locality, jet-bundles Deformation theory & BV
Asymptotic symmetries in gravity
Origin, QFT