An action principle for Vasiliev’s 4D equations Nicolas Boulanger Universit´ e de Mons, Belgium 11 April 2012, ESI Based on 1102.2219[hep-th] in collaboration with P. Sundell and a work to appear with P.S. and N. Colombo N. Boulanger (UMONS) An action principle for Vasiliev’s 4D equations ESI 1 / 33
Plan 1 Introduction 2 Classical off-shell unfolding 3 Brief review of Vasiliev’s 4D equations 4 A proposal for an action with QP structure 5 Conclusions and outlook N. Boulanger (UMONS) An action principle for Vasiliev’s 4D equations ESI 2 / 33
The gauge principle [H. Weyl, 1929] Classical Field Theory has witnessed a major achievement with Vasiliev’s formulation of fully nonlinear field equations four higher-spin gauge fields in four space-time dimensions [M. A. Vasiliev, 1990 – 1992] and in D space-time dimensions [hep-th/0304049] . Some salient features are Manifest diffeomorphism invariance, no explicit reference to a metric Manifest Cartan integrability ⇒ gauge invariance under infinite-dimensional HS algebra Formulation in terms of two infinite-dimensional unitarizable modules of so (2 , D − 1) : The adjoint and twisted-adjoint representations � master 1-form and master zero -form, resp. N. Boulanger (UMONS) An action principle for Vasiliev’s 4D equations ESI 3 / 33
Unfolded equations and FDA A free (graded commutative, associative) differential algebra R is sets { X α } of a priori independent variables that are locally-defined differential forms obeying first-order equations of motion whereby d X α are equated to algebraic functions of all the variables expressed entirely using the exterior algebra, viz. � R α = d X α + Q α ( X ) ≈ 0 , β 1 ...β n X β 1 · · · X β n . Q α ( X ) = f α n The nilpotency of d and the integrability condition d R α ≈ 0 require Q β ∂ L Q α ≡ 0 . ∂X β [ p α ] with p α > 0 , gauge transformation preserving R α ≈ 0 : For X α δ ǫ X α = d ǫ α − ǫ β ∂ L ∂X β Q α . N. Boulanger (UMONS) An action principle for Vasiliev’s 4D equations ESI 4 / 33
Why an action principle ? At least three reasons why to search for action principles : At the classical level → explore non-perturbative aspects, different phases of the theory ֒ At the quantum level ֒ → try and find a consistent and suitable quantization scheme To shed a different light on Vasiliev’s equations. N. Boulanger (UMONS) An action principle for Vasiliev’s 4D equations ESI 5 / 33
A prejudice : a QP -structure We address this issue by using the fully non-linear and background-independent Vasiliev equations in four spacetime dimensions. These possess an algebraic structure that enables one to construct a generalized Hamiltonian action with nontrivial QP -structures in a manifold with boundary ; a geometric structure which allows to construct additional boundary deformations [ − → Part II by Per Sundell]. N. Boulanger (UMONS) An action principle for Vasiliev’s 4D equations ESI 6 / 33
Manifold : bulk with non-empty boundary Like for the Cattaneo–Felder model (nonlinear Poisson sigma-model), we introduce a bulk with non-empty boundary, and add extra momentum-like variables. Impose boundary conditions compatible with a globally well-defined action principle [the action should be invariant, the Lagrangian picks up a total derivative under general variation] Here we focus on the bulk part of the Hamitonian action. Various classically marginal deformations on submanifolds will be presented by Per Sundell in Part II. N. Boulanger (UMONS) An action principle for Vasiliev’s 4D equations ESI 7 / 33
Classical action principle (1) Starting from { X α } defined locally on B ξ (where the base manifold B ˆ p = ∪ ξ B ξ ) satisfying some unfolded constraints with given Q -structure, → off-shell extensions based on sigma models with maps ֒ φ ξ : T [1] B ξ → M ˆ p , between two N -graded manifolds, from the parity-shifted tangent bundle T [1] B ˆ p to a target space M ˆ p that is a differential N -graded symplectic manifold with two-form O , Q -structure Q and Hamiltonian H with the following degrees : deg( O ) = ˆ p + 2 , deg( Q ) = 1 , deg( H ) = ˆ p + 1 . N. Boulanger (UMONS) An action principle for Vasiliev’s 4D equations ESI 8 / 33
Classical action principle (2) ֒ → Classical action principle of Hamiltonian type : � � � � S cl L cl µ φ ∗ bulk [ φ ] = = ξ ( ϑ − H ) , ξ B ξ B ξ ξ ξ where ϑ is the pre-symplectic form, defined locally on M ˆ p . 2 d Z i d Z j � 2 d Z i O ij d Z j and defining → Writing ϑ = d Z i ϑ i , O = 1 O ij = 1 ֒ p ] = ( − 1) ˆ p + i +1) A ∂ i A P ik ∂ j B { A, B } [ − ˆ p +(ˆ where P ik O kj = ( − 1) ˆ p δ i j , then ... N. Boulanger (UMONS) An action principle for Vasiliev’s 4D equations ESI 9 / 33
Classical action principle (3) ... the variation of the Lagrangian : � � δZ i R j � δ L cl δZ i ϑ i = O ij + d , bulk where generalized curvatures and Hamiltonian vector field d Z i + Q i , Q i = ( − 1) ˆ R i p +1 P ij ∂ j H , = − → deg( − → Q i � Q = ∂ i , Q ) = 1 . R i ≈ 0 , whose Cartan integrability on Variational principle = ⇒ shell requires − → Q to be a Hamiltonian Q -structure − → Q j ∂ j Q i ≡ 0 p ] ≡ 0 . ∂ i { H , H } [ − ˆ L − Q ≡ 0 ⇔ ⇔ → Q N. Boulanger (UMONS) An action principle for Vasiliev’s 4D equations ESI 10 / 33
Classical action principle (4) Nilpotency of − → Q with suitable boundary conditions on the fields and gauge parameters ensure invariance of the action under d ǫ i − ǫ j ∂ j Q i + 1 2 ǫ k R l ∂ l � O kj P ji , δ ǫ Z i = K ǫ = ǫ i R j � δ ǫ L cl O ij + δ ǫ Z i ϑ i , = d K ǫ , bulk Closure of gauge transformations : δ ε 12 Z i − − → [ δ ε 1 , δ ε 2 ] Z i R ε i = 12 , where − → R = R i ∂ i and 2 [ − → ε 1 , − → ε 2 ] Q i . ε i 12 = − 1 N. Boulanger (UMONS) An action principle for Vasiliev’s 4D equations ESI 11 / 33
Classical action principle (5) Under certain extra assumptions on ϑ and H , the action can be defined globally by gluing together the locally defined fields and gauge parameters along chart boundaries using gauge transitions δ t Z i and δ t ǫ i with parameters { t i } = t ξ ξ ′ defined on overlaps. Assumptions : ∂ j ∂ k − → t Q i = 0 , ( i ) δ t K ǫ = 0 , ( ii ) ( iii ) K t ≡ 0 . ⇒ cancellation of contributions to δ ǫ S cl Assumption ( i ) = bulk from chart boundaries in the interior of B , s.t. the variational principle implies the BC ϑ i | ∂B = 0 . N. Boulanger (UMONS) An action principle for Vasiliev’s 4D equations ESI 12 / 33
Classical action principle (6) Assumption ( ii ) ensures compatibility between gauge transformations and gauge transitions in the sense that performing a transition transformation on fields and gauge parameters between two adjacent charts and moving along the gauge orbit are two operations that should commute. Assumption ( iii ) selects the subalgebra of Cartan transformations that preserve the Lagrangian density, i.e. selects the transitions. Assuming there are no constants of total degree ˆ p + 2 on M ˆ p , the p ] ≡ 0 is equivalent to the structure equation condition ∂ i { H , H } [ − ˆ p +1) ∂ i H P ij ∂ j H { H , H } [ − ˆ p ] ( − 1) i (ˆ ≡ 0 ⇔ ≡ 0 . N. Boulanger (UMONS) An action principle for Vasiliev’s 4D equations ESI 13 / 33
Brief review of Vasiliev’s 4D equations (1) The master fields are locally-defined (chart index ξ ) operators O ξ ( X M ξ , d X M ξ ; Z α , d Z α ; Y α ; K ) , where [ Y α , Y β ] = 2 iC αβ , [ Z α , Z β ] = − 2 iC αβ , with charge conjugation matrix C αβ = ǫ αβ , C ˙ α ˙ β = ǫ ˙ α ˙ β and where K = ( k, ¯ k ), are two outer Kleinian operators. The operators are represented by symbols f [ O ξ ] obtained by going to specific bases for the operator algebra � ordering prescriptions . N. Boulanger (UMONS) An action principle for Vasiliev’s 4D equations ESI 14 / 33
Brief review of Vasiliev’s 4D equations (2) One may think of the symbols as functions f ( X, Z ; d X, d Z ; Y ) on a correspondence space C � C = C ξ , C ξ = B ξ × Y , B ξ = M ξ × Z ξ equipped with a suitable associative star-product operation ⋆ which reproduces, in the space of symbols, the composition rule for operators. � The exterior derivative on B is given by q = d Z α ∂ α . d = d X M ∂ M + q , N. Boulanger (UMONS) An action principle for Vasiliev’s 4D equations ESI 15 / 33
Brief review of Vasiliev’s 4D equations (3) The master fields of the minimal bosonic model are an adjoint one-form A = W + V , W = d X M W M ( X, Z ; Y ) , = d Z α V α ( X, Z ; Y ) , V and a twisted-adjoint zero-form Φ = Φ( X, Z ; Y ) . Generically, start with locally-defined differential forms of total degree p � ∞ f [ p ] ( X M , d X M ; Z α , d Z α ; Y α ; k, ¯ f = k ) , p =0 f [ p ] ( λ d X M ; λ d Z α ) = λ p f [ p ] (d X M ; dZ α ) , λ ∈ C . N. Boulanger (UMONS) An action principle for Vasiliev’s 4D equations ESI 16 / 33
Recommend
More recommend
