Parent BRST approach to higher spin gauge fields Maxim Grigoriev - PowerPoint PPT Presentation

Parent BRST approach to higher spin gauge fields Maxim Grigoriev Lebedev Physical Institute, Moscow Based on: M.G. arXiv:1204.1793, arXiv:1012.1903 G. Barnich, M.G., arXiv:1009.0190, arXiv:0905.0547 K. Alkalaev, M.G., arXiv:1105.6111 ESI, Vienna, April 11

Appropriate Language for Higher spin gauge theories? Metric-like approach and its BRST extension – Rather natural and simple. Fronsdal, 1979 String-inspired BRST approach. Ouvry, Stern, 1986, Bengtsson, 1986, M. Henneaux, C. Teitelboim, 1986, More recent contributions: Pashnev, Buchbinder, Sagnotti, Tsulaia, Francia, Bekaert, Boulanger,. . . Frame-like “unfolded” approach Naturally appeares at the nonlinear level Makes symmetries manifest. Allows for powerful homological technique (e.g. so-called σ − -cohomology). Mainly developed by Vasiliev, 1988,. . . More recent contributions: Sezgin, Sundell, Alkalaev, Skvortsov, Boulanger, . . .

Main point – metric like BRST and unfolded approach are actually uni- fied if one carefully applies Batalin–Vilkovisky approach and local BRST cohomology technique... Moreover, the exchange of methods and ideas turns out to be quite fruitful!

Batalin-Vilkovisky formalism: Given equations T a , gauge symmetries R i α , reducibility relations,.... the BRST differential: s 2 = 0 , s = δ + γ + . . . , gh( s ) = 1 ∂ ∂ ∂ ∂ P a + Z a γ = c α R i δ = T a A P a ∂φ i + . . . . ∂π A . . . , α δ – (Koszule-Tate) restriction to the stationary surface γ – implements gauge invariance condition φ i – fields, c α – ghosts, P a – ghost momenta, π A – reducibility ghost momenta gh( φ i ) = 0 , gh( c α ) = 1 , gh( P a ) = − 1 , . . . BRST differential completely defines the theory. Equations of motion and gauge symmetries can be read off from s : δ ǫ φ i = ( sφ i ) | c α = ǫ α , P a =0 , ... s P a | P a =0 , c α =0 ,... = 0 ,

If the theory is Lagrangian then: T i = δS 0 δφ i , reducibility relations R i α T i = 0 so that Z i α = R i α Natural bracket structure (antibracket) � � � � φ i , P j = δ i c α , P β = δ α j β BV master action � � α c α + . . . S BV = S 0 + P i R i s = · , S BV , Master equation: � � s 2 = 0 = 0 ⇐ ⇒ S BV , S BV Example: YM theory Fields: A µ , C (with values in the Lie algebra) Antifields: A ∗ µ , C ∗ Gauge part BRST differential: γA µ = ∂ µ C + [ A µ , C ] Master action: � d n x Tr[ A ∗ µ ( ∂ µ C + [ A µ , C ]) + 1 2 C ∗ [ C, C ]] S BV = S 0 +

In the context of local gauge field theory: Jet space: coordinates ξ µ ≡ dx µ x µ , ξ µ , Ψ A , Ψ A µ , Ψ A µν , . . . Total derivative: ∂ ∂ ∂ ∂x µ + Ψ A ∂ Ψ A + Ψ A ∂ µ = + . . . µ µν ∂ Ψ A ν BRST differential is an evolutionary vector field: s Ψ A = s A [Ψ , x ] [ ∂ µ , s ] = 0 , Local functionals: f [Ψ] ∼ f [Ψ] + ∂ µ j µ [Ψ] Quotient space:

In a local field theory – local BRST cohomology encode physically inter- esting quantities. Local BRST cohomology: H · ( s, F ) F – local functionals, local forms, evolutionary (poly)vector fields etc. BRST cohomology encode: conserved currents/global symmetries, anoma- lies, consistent deformations etc. Although jet-space BV is extremely useful it can be quite restrictive: – Boundary dynamics (e.g. AdS/CFT, asymptotic symmetries) – Coordinate-free formulation (e.g. for gravity) – Important structures such as generalized connections and curvatures are not realized in a manifest way Brandt, 1996

An alternative: Vasiliev, 1988, . . . , 2005 Unfolded formalism Fields: differential forms Φ a Equations of motion: d Φ a = Q a (Φ), Q a (Φ) – wedge product function. Consistency: Q 2 = 0 Q = Q a (Φ) ∂ where ∂ Φ a Free Differential Algebras, Sullivan 1977, d’Auria, Fre, 1982. . . Advantages: – manifestly coordinate free – first order – useful in analyzing global symmetries – inevitable for nonlinear higher spin theories Vasiliev, 1989,. . . , 2003

Open issues: 1) No systematic procedure to “unfold” a given theory 2) In spite of various algebraic similarities the relation between jet space BV and unfolded approaches remains unclear 3) Known unfolded forms for sufficiently general higher spin fields are quite involved 4) Even for Lagrangian systems constructing unfolded Lagrangians is rather an art than a systematic procedure For linear theories 1),2) were mainly resolved within the first quantized BRST approach Barnich, M.G., Semikhatov, Tipunin, 2004, Barnich, M.G. 2006 . In particular, BRST extension of unfolded systems Barnich, M.G. 2005 3) Mixed symmetry fields on constant curvature backgrounds. Talk by K. Alkalaev. Alkalaev, M.G. 2009,2010

AKSZ sigma models Alexandrov, Kontsevich, Schwartz, Zaboronsky, 1994 Ingredients: M - supermanifold (target space) equipped with: Ghost degre – gh() (odd) Poisson bracket – { · , · } , gh( { · , · } ) = − n + 1 “BRST potential” S M (Ψ) , gh( S M ) = n , master equation { S M , S M } = 0 ( QP structure: Q = { · , S M } and P = { · , · } ) X - supermanifold (source space) Ghost degree gh() d – odd vector field, d 2 = 0, gh( d ) = 1 Tipically, X = T [1] X , coordinates x µ , θ µ ≡ dx µ , d = θ µ ∂ ∂x µ , µ = 0 , . . . n − 1

BV master action � � � d n xd n θ χ A (Ψ( x, θ )) d Ψ A ( x, θ ) + S M (Ψ( x, θ )) S BV = χ A (Ψ) – symplectic potential: σ = d M χ . BV antibracket � � � � � δ R F δG d n xd n θ δ Ψ A ( x, θ ) E AB = F, G . δ Ψ B ( x, θ ) � Ψ A , Ψ B � E AB = – Poisson bivector E AB σ BC = δ A B . Master equation: � � = 0 , gh( S BV ) = 0 S BV , S BV

BRST differential: � � Q A = s AKSZ Ψ A ( x, θ ) = d Ψ A ( x, θ ) + Q A (Ψ( x, θ )) , Ψ A , S M Dynamical fields, those of vanishing ghost degree 0 1 k µ ( x ) θ µ + . . . Ψ A ( x, θ ) = Ψ A ( x ) + Ψ A Ψ A µ 1 ...µ k ) = gh(Ψ A ) − k gh( k If gh(Ψ A ) = k with k � 0 then Ψ A µ 1 ...µ k ( x ) dynamical. If gh(Ψ A ) � 0 ∀ Ψ A then BV-BRST extended FDA. Otherwise BV-BRST extended FDA with constraints. Nonlagrangian AKSZ: ∂ Q = Q A { , } , S M → nilpotent ∂ Ψ A . No relation between gh( Q ) and dim X ! (Recall gh( S M ) = n = dim X ) BV-BRST extension of unfolded form + constraints

Examples: Chern-Simons: Alexandrov, Kontsevich, Schwartz, Zaboronsky, 1994 Target space M : M = g [1], g – Lie algebra with invariant inner product. e i –basis in g , C i – coordinates on g , gh( C i ) = 1, C = C i e i � C i , C j � = � e i , e j � − 1 S M = � C, [ C, C ] � , Source space: X = T [1] X , X – 3-dim manifold. Fied content µ ( x ) + θ µ θ ν A ∗ i µν + θ µ θ ν θ ρ λ ∗ i C i ( x, θ ) = λ i ( x ) + θ µ A i µνρ BV action � � 1 2 � C, d C � + 1 2 � A, d A � + 1 d 3 xd 3 θ ( 1 S BV = 6 � C, [ C, C ] � ) = 6 � A, [ A, A ] � ) + . . .

Hamiltonian BFV-BV Target space M : BFV extended phase space, { , } –Poisson bracket, S M = Ω – BRST charge, { Ω , Ω } = 0 – BFV master equation, in addition: function H , { H, Ω } = 0 – BRST invariant Hamiltonian Source space X = T [1]( R 1 ), coordinates t, θ BV action M.G., Damgaard, 1999 � dtdθ ( χ A d ψ A + Ω − θH ) S BV = BV for the Hamiltoninan action Fisch, Henneaux, 1989, Batalin, Fradkin 1988 . p 2 − m 2 ), x µ , � c, � Example: coordinates on M : � P , � p µ , BRST charge Ω = � c ( � � � x µ + � c ( p 2 − m 2 )) = x µ + λ ( p 2 + m 2 )) + . . . S BV = dtdθ ( � P d � c + � dt ( p µ ˙ p µ d � x µ ( t, θ ) = x µ ( t ) + θp µ c ( t, θ ) = c ( t ) + θλ ( t ), � ∗ ( t ), . . . �

– If M, S M , { , } and T [1] X, d define AKSZ sigma model and X = X space × R 1 � � � d n − 1 xd n − 1 θ χ A (Ψ( x, θ )) d Ψ A ( x, θ ) + S M (Ψ( x, θ )) Ω BFV = � d n − 1 xd n − 1 θ { · , · } { · , · } BFV = { Ω BFV , Ω BFV } BFV = 0 . AKSZ is neither Lagrangian nor Hamiltonian Barnich, M.G, 2003 - Moreover. Higher BRST charges. χ d Ψ + S M – integrand of S BV con- sidered as inhomogeneous form on X , X k ⊂ X – dimension- k submanifold � � Ω X k = L AKSZ = ( χ d Ψ + S M ) X k X k In particular, Ω BFV = Ω X space , S BV = Ω X

– At the level of equations of motion one induces AKSZ sigma model on any X 0 ⊂ X . Useful for “replacing space-time”. E.g. Generalized superspace Vasiliev 2002 Natural way to relate AdS, Ambient, and Conformal picture Barnich M.G. 2006, Bekaert M.G. 2009 AdS/CFT correspondence for HS fields Vasileiv, 2012 – Locally in X and M Barnich, M.G. 2009 H ( s AKSZ , local functionals) ∼ = H ( Q, C ∞ ( M )) � Function F on M , QF = 0 gives a conserved charge Vasiliev 2005 . X k F � d n xd n θF (Ψ( x, θ )) is quasi- Map I : C ∞ ( M ) → local functionals: IF = isomorphism and Barnich, M.G., 2009 � � = I { F, F } IF, IG – If M finite dimensional and n > 1 – the model is topological.

Parent formulation (Equations of motion level) Barnich, M.G. 2010 Barnich, M.G., Semikhatov, Tipunin, 2004 Starting point theory: Fields, ghosts, ghosts for ghosts, antifields, etc.: ψ I ( x ) Jet space M for BV formulation: coordinates Ψ A = { z a , ξ a ≡ dz a , ψ I ( a ) } (short-hand ψ I ( a ) = { ψ I , ψ I a , ψ I a 1 a 2 , . . . } ) Horizontal differential: d H = ξ a ∂ a BRST differential: s – vector field on M , [ d H , s ] = 0 Basic object � s = − d H + s Brandt, 1997