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 Ta, gauge symmetries Ri

α, reducibility relations,....

the BRST differential: s = δ + γ + . . . , s2 = 0 , gh(s) = 1 δ = Ta

∂ ∂Pa + Za APa ∂ ∂πA . . . ,

γ = cαRi

α ∂ ∂φi + . . . .

δ – (Koszule-Tate) restriction to the stationary surface γ – implements gauge invariance condition φi – fields, cα – ghosts, Pa – ghost momenta, πA – reducibility ghost momenta gh(φi) = 0 , gh(cα) = 1 , gh(Pa) = −1 , . . . BRST differential completely defines the theory. Equations of motion and gauge symmetries can be read off from s: sPa|Pa=0, cα=0 ,... = 0 , δǫφi = (sφi)|cα=ǫα, Pa=0, ...

If the theory is Lagrangian then: Ti = δS0

δφi , reducibility relations Ri αTi = 0

so that Zi

α = Ri α

Natural bracket structure (antibracket)

• φi, Pj
• = δi

j

• cα, Pβ
• = δα

β

BV master action s =

• ·, SBV
• ,

SBV = S0 + PiRi

αcα + . . .

Master equation:

• SBV , SBV
• = 0

⇐ ⇒ s2 = 0 Example: YM theory Fields: Aµ, C (with values in the Lie algebra) Antifields: A∗µ, C∗ Gauge part BRST differential: γAµ = ∂µC + [Aµ, C] Master action: SBV = S0 +

• dnxTr[A∗µ(∂µC + [Aµ, C]) + 1

2C∗[C, C]]

In the context of local gauge field theory: Jet space: coordinates xµ, ξµ, ΨA, ΨA

µ , ΨA µν, . . .

ξµ ≡ dxµ Total derivative: ∂µ =

∂ ∂xµ + ΨA µ ∂ ∂ΨA + ΨA µν ∂ ∂ΨA

ν

+ . . . BRST differential is an evolutionary vector field: [∂µ, s] = 0 , sΨA = sA[Ψ, x] Local functionals: Quotient space: f[Ψ] ∼ f[Ψ] + ∂µjµ[Ψ]

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 Φa Equations of motion: dΦa = Qa(Φ), Qa(Φ) – wedge product function. Consistency: Q2 = 0 where Q = Qa(Φ) ∂

∂Φa

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”SM(Ψ) , gh(SM) = n, master equation {SM, SM} = 0 (QP structure: Q = { · , SM} and P = { · , · }) X - supermanifold (source space) Ghost degree gh()

d – odd vector field, d2 = 0, gh(d) = 1

Tipically, X = T[1]X, coordinates xµ, θµ ≡ dxµ, d = θµ ∂

∂xµ , µ = 0, . . . n − 1

BV master action SBV =

• dnxdnθ
• χA(Ψ(x, θ))dΨA(x, θ) + SM(Ψ(x, θ))
• χA(Ψ) – symplectic potential: σ = dMχ.

BV antibracket

• F, G
• =
• dnxdnθ
• δRF

δΨA(x, θ)EAB δG δΨB(x, θ)

• .

EAB =

• ΨA, ΨB

– Poisson bivector EABσBC = δA

B.

Master equation:

• SBV , SBV
• = 0 ,

gh(SBV ) = 0

BRST differential: sAKSZΨA(x, θ) = dΨA(x, θ) + QA(Ψ(x, θ)) , QA =

• ΨA, SM
• Dynamical fields, those of vanishing ghost degree

ΨA(x, θ) = ΨA(x) +

1

ΨA

µ (x)θµ + . . .

gh(

k

ΨA

µ1...µk) = gh(ΨA) − k

If gh(ΨA) = k with k 0 then

k

ΨA

µ1...µk(x) dynamical.

If gh(ΨA) 0 ∀ ΨA then BV-BRST extended FDA. Otherwise BV-BRST extended FDA with constraints. Nonlagrangian AKSZ: {, } , SM → nilpotent Q = QA

∂ ∂ΨA .

No relation between gh(Q) and dim X ! (Recall gh(SM) = 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. ei –basis in g, Ci – coordinates on g, gh(Ci) = 1, C = Ciei SM = C, [C, C] ,

• Ci, Cj

= ei, ej−1 Source space: X = T[1]X, X – 3-dim manifold. Fied content Ci(x, θ) = λi(x) + θµAi

µ(x) + θµθνA∗i µν + θµθνθρλ∗i µνρ

BV action SBV =

• d3xd3θ(1

2C, dC + 1

6C, [C, C]) =

1

2A, dA + 1

6A, [A, A]) + . . .

Hamiltonian BFV-BV

Target space M: BFV extended phase space, {, } –Poisson bracket, SM = Ω – BRST charge, {Ω, Ω} = 0 – BFV master equation, in addition: function H, {H, Ω} = 0 – BRST invariant Hamiltonian Source space X = T[1](R1), coordinates t, θ BV action

M.G., Damgaard, 1999

SBV =

• dtdθ(χAdψA + Ω − θH)

BV for the Hamiltoninan action

Fisch, Henneaux, 1989, Batalin, Fradkin 1988.

Example: coordinates on M: c, P, xµ, pµ, BRST charge Ω = c( p2 − m2), SBV =

• dtdθ(

pµd xµ + Pd c + c(p2 − m2)) =

• dt(pµ ˙

xµ + λ(p2 + m2)) + . . .

• c(t, θ) = c(t) + θλ(t),

xµ(t, θ) = xµ(t) + θpµ

∗(t), . . .

– If M, SM, {, } and T[1]X, d define AKSZ sigma model and X = Xspace×R1 ΩBFV =

• dn−1xdn−1θ
• χA(Ψ(x, θ))dΨA(x, θ) + SM(Ψ(x, θ))
• { · , · }BFV =
• dn−1xdn−1θ { · , · }

{ΩBFV , ΩBFV }BFV = 0 . AKSZ is neither Lagrangian nor Hamiltonian

Barnich, M.G, 2003

• Moreover. Higher BRST charges. χdΨ + SM – integrand of SBV con-

sidered as inhomogeneous form on X, Xk ⊂ X – dimension-k submanifold ΩXk =

• Xk

LAKSZ =

• Xk

(χdΨ + SM) In particular, ΩBFV = ΩXspace , SBV = ΩX

– At the level of equations of motion one induces AKSZ sigma model on any X0 ⊂ 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(sAKSZ, local functionals) ∼ = H(Q, C∞(M)) Function F on M, QF = 0 gives a conserved charge

• Xk F

Vasiliev 2005.

Map I : C∞(M) → local functionals: IF =

dnxdnθF(Ψ(x, θ)) is quasi-

isomorphism and

Barnich, M.G., 2009

• IF, IG
• = I {F, F}

– 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.: ψI(x) Jet space M for BV formulation: coordinates ΨA = {za, ξa ≡ dza, ψI

(a)}

(short-hand ψI

(a) = {ψI, ψI a, ψI a1a2, . . .})

Horizontal differential: dH = ξa∂a BRST differential: s – vector field on M, [dH, s] = 0 Basic object s = −dH + s

Brandt, 1997

Parent formulation AKSZ sigma model: – target space M equipped with s = −dH + s – source space xµ, θµ. Fields: ΨA(x, θ) = {ψI

(a)(x, θ),

za(x, θ), ξa(x, θ)} Dynamical fields (gh() = 0):

k

ψI

(a)µ1...µk(x)

gh(ψI) = k 0, za(x) = 0 za(x) , ea

µ(x) = 1

ξa

µ(x)

BRST differential sPΨA(x, θ) = (d + s)ΨA(x, θ)

In fact: we are dealing with parametrized version. za(x) – space-time coordinates understood as fields ea

µ – frame field components.

Gauge transf. for za: δza = ξa. Fixing gauge symmetry za = δa

µxµ equations of motion imply ea µ(x) = −δa µ.

Unarametrized version: sPΨA(x, θ) = (d − θa∂a + s)ΨA(x, θ) Recall: ∂a – target space total derivative.

Generalized auxiliary fields and equivalent reductions

At the lagrangian level: χi, χ∗

i are generalized auxiliary fields for SBV if they are conjugate in the

antibracket and equations δSBV

δχi

• χ∗

i =0 = 0 can be algebraically solved for

χi.

Dresse, Gr´ egoire, Henneaux, 1990

At the level of equations of motion: ϕα, va, wa (swa)|wa=0 = 0 ⇔ va = V a[ϕ] va, wa – generalized auxiliary fields. Barnich, M.G., Semikhatov, Tipunin, 2004 Reduced system: sRφα = sφα|w=0,v=V [φ] , (sR)2 = 0 Can be seen as reduction to the surface: wa = 0, va − V a[ϕ] = 0

Equivalence = Elimination of generalized auxiliary fields (Local) BRST cohomology is invariant. E.g. observables, global symmetries, consistent interactions, anomalies, possible Lagrangians, are isomorphic. Parent formulation is equivalent to the starting point one. All the fields ψI

(a)µ1..µk(x) save for

ψI(x) are generalized auxiliary. Simple algebraic reason: In terms of extra variables ya all the fields can be packed into generating function

• ψI(y, θ) =
• m,k

k

ψI

b1...bm|a1...akyb1 . . . ybmθa1 . . . θak

For polynomials in ya, θa there is a basis 1, fi, gi such that θa ∂

∂yafi = gi.

In the representation

• ψI =

ψI + F ifi + Gigi fields F i, Gi are generalized auxiliary.

Reduction to unfolded formulation

BRST differential decomposition: s = δ + γ + . . ., where δ implements equations of motion. For a regular theory new coordinates on M φλ, T i, Pi such that φλ are coordinates on the stationary surface and δPi = T i. Fields T i, Pi are generalized auxiliary fields for the parent formulation. sP

• n−shellφλ(x, θ) = (d +

γ)φλ(x, θ) ,

• γ = γon−shell − dH

As gh(φλ) 0 the equations of motion and gauge symmetries are that of some FDA. General prescription to unfold a given gauge theory. However: 1) Not a standard Vasiliev unfolded form but usually a nonminimal one. 2) Parametrized version

Parent Lagrangians

Starting point theory: Fields, ghosts, ghosts for ghosts (but no antifields!): ψI(y) Gauge part of BRST differential: γ (for simplicity γ2 = 0) Lagrangian: L[ψ, y], γL = ∂µjµ[ψ, y]. Parent Lagrangian Jet space N with coordinates Ψα = {ψI

(a), ya, ξa}.

Equipped with: ghost degree, dH = ξa∂a, γ = −dH + γ Lagrangian potential L(ψ, y, ξ):

• L = Ln + Ln−1 + . . . + L0 ,

where Ln = ξn−1 . . . ξ0L[ψ, y]

Ln−1, . . . , L0 through “Descent equation” (−dH + ¯ γ) L = 0: γLn = dHLn−1 γLn−1 = dHLn−2 . . . = . . . γL0 = 0

• L represents Lagrangian as a

γ = −dH + γ cohomology class. Introduce antifields Λα = {Λ(a)

I

, πa, ρa} and the canonical (anti)bracket: gh(Λ(a)

I

) = n − 1 − gh(ψI

(a)) ,

gh(πa) = n − 1 , gh(ρa) = n − 2

• ψI

(a), Λ(b) J

• = δI

Jδb a ,

{ya, πb} = δa

b ,

{ξa, ρb} = δa

b

Supermanifold M = T ∗[n − 1]N.

Lagrangian parent formulation Target space: M = T ∗[n − 1]N, canonical degree 1 − n bracket, BRST potential SM = Λα γΨα + L(Ψ) BV master action: SBV =

• dnxdnθ[Λα(d +

γ)Ψα + L(φ)] SBV satisfies master equation

• SBV , SBV
• = 0 .

Λα(x, θ) – sources for parent BRST transformation. Unify momenta, Lagrange multipliers, BV antifields.

Diffeomorphism invariance Non-parametrized version (gauge za = xa).

• ψI

(a)(x, θ) , Λ(b) J (x′, θ′)

• = δI

Jδ(b) (a)δ(n)(x − x′)δ(n)(θ − θ′)

The BV master action SBV =

• dnxdnθ
• Λ(a)

I

(d − θa∂T

a + γ)ψI (a) +

L(ψ(x, θ), x, θ)

• Genuine diff. invariance – redefinition of ghosts makes za(x, θ), ξa(x, θ))-

generalized auxiliary. This amounts to γ → γ so that SBV =

• dnxdnθ[Λ(a)

I

(d + γ)ψI

(a) +

L(ψ(x, θ))]

First quantized BRST picture

Linear gauge theories ∼ = BRST first quantized systems Pack all fields, ghosts, antifields into “string field” Ψ(y) = ΨA(y)eA , gh(eA) = −gh(ΨA) , yµ – space-time H - graded vector space with basis eA Ω = ΩA

B(y, ∂ ∂y) – BRST operator: gh(Ω) = 1 and ΩΩ = 0 defined through

ΩΨ = sΨ Ψ(y) = ΨA(y)eA = . . . + Ψ−1 + Ψ0 + Ψ1 + . . . gh(Ψi) = −i . Ψ0 – physical fields, Ψ1 – gauge parameters (ghosts), . . . Equations of motion, gauge symmetries, . . . : ΩΨ(0) = 0 , Ψ(0) ∼ Ψ(0) + Ωχ(1) , . . .

Extension analogous to that used in Fedosov quantization:

Fedosov (1994)

new variables: xµ new constraints:

∂ ∂xµ − ∂ ∂yµ = 0

new ghosts: θµ ≡ dxµ

Barnich, M.G., Semikhatov, Tipunin (2004)

Φ(y) → Φ(x, y, θ) , Ω → Ωparent Ωparent = d − σ + ¯ Ω , ¯ Ω = Ω(x + y, ∂

∂y)

d = θµ ∂

∂xµ,

σ = θµ ∂

∂yµ

Fields: ΨA(x) − → ΨA

(µ1...)[ν1...](x)

Fronsdal fields

Fields and Ghosts (gauge parameters): φa1...as , Ca1...as−1 φ = 1 s!pa1 . . . pasφa1...as , C = 1 (s − 1)!pa1 . . . pas−1Ca1...as−1 . TTφ = 0 , TC = 0 , T ≡

∂ ∂pa ∂ ∂pa

Gauge part of the BRST differential γφ = pa∂aC

Target space coordinates za, ξa ≡ dza, φ(p, y), C(p, y) φ(p, y) = φ(p) + φa(p)ya + 1

2φab(p)yayb + . . . ,

C(p, y) = C(p) + Ca(p)ya + 1

2Cab(p)yayb + . . . .

On-shell version φ, C-totally traceless: SC = C = Sφ = φ = 0 , S =

∂ ∂ya ∂ ∂pa ,

=

∂ ∂ya ∂ ∂ya .

Dynamical fields and ghosts C(x, θ|y, p) = λ(x|y, p) + θaAa(x|y, p) + . . . , φ(x, θ|y, p) = F(x|y, p) + θa1 φ(x|y, p) + . . . , Equations and gauge symmetries:

Barnich, M.G., Semikhatov, Tipunin, 2004

(d − σ)A = 0 (d − σ)F + S†A = 0 , δA = (d − σ)λ, δF = S†λ , σ = ξa ∂

∂ya S† = pa ∂ ∂ya

Cohomological results

Barnich, M.G., Semikhatov, Tipunin, 2004

All variables are contractible pairs for γ save for za, ξa and Generalized connections C(y, P) ⊂ C(y, p) S†C = 0 ,

p y

· · · · · · · · · Generalzed curvatures (de Witt - Freedman) R(y, p) ⊂ φ(y, p) ya ∂

∂paR = 0 ,

y p

· · · · · · · · ·

• γ = dH + γ reduces to
• γredC = σC + Πσ¯

σR ,

• γredR = ΠσR ,

HS Russian formula HS version of the familiar YM

Stora, 1983,.....

Y M

• γred

C = 1

2[

C, C] + F ,

• γredF = . . .

GR

• γredξa = ξa

c ξc ,

γredξa

b = ξa c ξc b − 1 2ξcξdRa b cd ,

• γredR = . . .

Reduced system determied by sred = d + γred

(Lopatin, Vasiliev 1988):

(unfolded form) (d − Πσ) F = 0, (d − σ) A = −σ¯ σ Π F where C(x, θ|y, p) = λ(x|y, p) + θa Aa(x|y, p) + . . . , R(x, θ|y, p) = F(x|y, p) + θa1 r(x|y, p) + . . . ,

Frame-like Lagrangian

Fields and ghosts (gauge parameters): φa1...as , Ca1...as−1 φ = 1 s!pa1 . . . pasφa1...as , C = 1 (s − 1)!pa1 . . . pas−1Ca1...as−1 . TTφ = 0 , TC = 0 , T ≡

∂ ∂pa ∂ ∂pa

Gauge part of the BRST differential γφ = pa∂aC Fronsdal Larangian:

Fronsdal, 1979

L = 1

2φa, φa − 1 2¯

paφa, ¯ pbφb + paDa, ¯ pbφb − Da, Da − 1

2¯

paDa, ¯ pbDb, ¯ pa =

∂ ∂pa and D = Tφ.

If D is independent – “triplet” formulation

Ouvry, Stern, (1986), Bengtsson, (1986),Henneaux, Teitelboim (1986))

Also:

Pashnev, Buchbinder, Sagnotti, Tsulaia,. . .

Lagrangian Parent formulation:

Target space supermanifold: φ(y, p), C(y, p), za, ξa Simplification: eliminate contractible pairs for γ such that Tφ(y, p) = 0 , SC(y, p) = 0 , S =

∂ ∂pa ∂ ∂ya

Target space version of the “traceless gauge”

Alvarez, Blas, Garriga, Verdaguer (2006), Skvortsov, Vasiliev (2007)

Lagrangian:

Skvortsov, Vasiliev (2007)

L = 1

2φa, φa − 1 2Sφ, Sφ|y=0

The Lagrangian potential (−dH + γ) L = 0

• L = VL + VaJa + 1

2VabJab

Va1...ak = 1 (n − k)!ǫaa1...akb1...bn−kξb1 . . . ξbn−k Possible solution Ja = φ, paC|y=0 − φ, ∂aS†C|y=0 Jba = 1

2

• pbC, paC − ∂aS†C|y=0 − S†C, pb∂aC|y=0 − (a ↔ b)
• ,

All ingredients for the parent Lagrangian: Supermanifold φ, C, za, ξa, γ = −dH + γ, Lagrange potential L Equivalence:

• L → L +

γK.

Cohomological results

Bekaert, Boulanger, 2005:

All variables are γ-contractible pairs save for za, ξa HS connections C(y, p) ⊂ C(y, p), Off-shell HS curvatures R ⊂ φ(y, p), Fronsdal tensors F ⊂ φ(y, p) . F = independent components of (φ − S†S + S†S†T)φ(y, p) Few relations for Q = γred QC = ξbCb , QCa = ξbCba + ξaξc

∂ ∂pcF′ ,

. . . . F′ is linearly related to the F. Extra term ξaξc ∂

∂pcF′ related to “Einstein σ−-cohomology” Vasiliev, 2001

Better choice for L (s 2):

• L = 1

2Vab

• Ca, Cb − paCd, pbCd
• + M

where M vanishes when trivial pairs for γ are eliminated.

Finally

• Lred = 1

2Vab

• Ca, Cb − paCd, pbCd
• All fields but C(x, θ|p), Ca(x, θ|p) and their antifileds are generalized auxiliary

as they do not enter

• L. Elimination results in

SR[e, ω, Λ] =

• Λ, de − σω +

Lred(ω) , C(x, θ|p) = C(x|p) + θbeb(x, a) + θbθd . . . + . . . Ca(x, θ|p) = Ca(x|p) + θbωa|p(x, a) + θbθd . . . + . . . In fact ω is auxiliary as well. Paramerizing n−2 form Λ in terms of 1 form

• ω one gets:

Vasiliev, 1980

Sframe[e, ω] =

• dnx

ω, ya ∂

∂xae − 1 2

ω′ =

• dnθdnx Vcab ∂

∂pc

ωa, ∂

∂pb(de − 1 2σ

ω)

Off-shell nonlinear system

Recall (parent EOM’s):

Barnich, M.G., Semikhatov, Tipunin, 2004

(d − σ)A = 0 (d − σ)F + S†A = 0 , , A, F – totally traceless σ = ξa ∂

∂ya, S† = pa ∂ ∂ya

Off-shell version (A, F – unconstrained) is a linearization of: Vasiliev, 2005 dA + 1

2[A, A]∗ = 0 ,

dF + [A, F]∗ = 0 , around a particular solution A0 = θbpb, F0 = 1

2ηabpapb .

Indeed: [A0, ·]∗ = −σ, [·, F0]∗ = S†. [G, H]∗ = G ∗ H − (−1)|G||H|H ∗ G – Weyl ∗-commutator determined by [ya, pb]∗ = δa

b .

Can be seen as a BFV master equation Ω ∗ Ω = 0 for Ω = d + θµA(x|y, p) + cF(x|y, p) Parent form of the quantized scalar particle propagating in HS back- ground. In this way one also implements double-tracelessness condition M.G., 2006

Off-shell constraints and gauge symmetries for AdS HS fields

AdS geometry through embedding: X = {X ⊂ Rn+1 : ηABXAXB + 1 = 0} On TRn+1 flat o(n − 1, 2) metric η, metric connection ∇0, “tautological” vector field V0 = XA

∂ ∂XA.

By pulling back TRn+1 to X one gets: fiberwise metric, flat o(n − 1, 2) connection ∇, fixed section such that V dω + ωω = 0 , ηABV AV B + 1 = 0 . Frame eA

µ = ∂µV A + ωA µBV B.

Consider “formal” version of T ∗Rn+1. Formal power series in Y A and polynomials in PA. Weyl star product Y A ∗ PB − PB ∗ Y A = δA

B

In addition g[1] – ghosts νi for g = sp(2) with gh(νi) = 1 [e2, e1] = 2e1 , [e2, e3] = −2e3 , [e1, e3] = e2 sp(2) BRST operator q = −1

2νiνjUk ij ∂ ∂νk ,

[ei, ej] = Uk

ijek

Target space M: generating function Ψ, gh(Ψ) = |Ψ| = 1 for coordinates Ψ(c, Y, P) = C(Y, P) + νiFi(Y, P) + νiνjGij(Y, P) + νiνjνkGijk(Y, P) . gh(C) = 1, gh(Fi) = 0, gh(Gij) = −1, gh(Gijk) = −2 Odd vector field Q QΨ = qΨ + 1

2[Ψ, Ψ]∗

In components QFi = [Fi, C]∗ , QC = 1

2[C, C]∗ ,

QGij = 1

2[Fi, Fj]∗ − 1 2Uk ijFk + [Gij, C]∗ ,

. . . . AKSZ sigma model: M, Q and T[1]X, d Equations of motion

dA + 1

2[A, A]∗ = 0 ,

dFi + [A, Fi]∗ = 0 ,

[Fi, Fj]∗ − Uk

ijFk = 0

Gauge symmetries δλFi = [Fi, λ]∗ , δλA = dλ + [A, λ]∗ Component fields C(x, θ|Y, P) = λ(x|Y, P) + θµAµ(x|Y, P) + . . . Fi(x, θ|Y, P) = Fi(x|Y, P) + θµ . . .

Linearize around background solution: Ψ0 = θµA0

µ + νiF 0 i ,

where A0

µ = θµωB µA(x)(Y A + V A)PB ,

F 0

1 = 1 2P · P ,

F 0

2 = Y ′ · P ,

F 0

3 = −1 2Y ′ · Y ′ .

Y ′

A = YA+VA, ωB µA(x) – flat AdS connection, V A – compensator. “Twisted”

version of the familiar sp(2)- representation Represent linearized system as ΩΨ = 0 and δΨ = Ωλ where Ω = d + [A0, ·]∗ + νi[F 0

i , ·]∗ + q , Barnich, M.G., 2006

“states” – Ψ(x, θ|Y, P, c)

d + [A0, ·]∗ = d + θµωB

µA

• PB

∂ ∂PA − Y ′ ∂ ∂Y B

• ,

– covariant derivative νi[F 0

i , ·]∗ + q =

= −ν1P ·

∂ ∂Y A + ν2(P · ∂ ∂P − (Y + V ) · ∂ ∂Y ) − ν3(Y + V ) · ∂ ∂P + q (1)

– fiber part (BRST operator of sp(2) represented on Y, P variables). If A, Fi totally traceless. ΩΨ = 0 and δΨ = Ωλ is equivalent to Fronsdal

Ambient picture

Parent form of the ambient space equation (Y A + V A → XA) [Fi, Fj] = Uk

ijFk ,

δFi = [Fi, λ] Fi = Fi(X, P), λ = λ(X, P) around a particular solution F 0

1 = 1 2P · P, F 0 2 = X · P, F 0 3 = −1 2X · X

BFV master equation for a quntized particle on the ambient space. This constraint system is familiar in many contexts – Singleton on conformal boundary (quotient of the hyper-cone X2 = 0) – 2-time physics

Bars,. . .

– Observables – HS algebra (symmetries of singleton)

Vasiliev, Eastwood

– HS singletons and their symmetries

Bekaert, M.G., 2009

– Talk by

Waldron

It can be given either AdS or conformal interpretation.

– The off-shell nonlinear system is naturally defined as the AKSZ sigma- model whose target space is the direct sum of the Weyl algebra and sp(2) with shifted parity. – The formulation has well known sp(2) structure realized in a manifest way. – Can be seen as a version of the off-shell system proposed in M.G., 2006 – HS geometry? Connection A and multiplet of curvatures Fi?

Untouched topics

• Parent formalism can be used as a starting point to construct the

theory. Used for massive and (partially) massless mixed symmetry fields on constant curvature backgrounds. Talk by

Alkalaev

For symmetric fields

M.G., Waldoron, 2011

• Analogous constructions can be done for conformal fields. For bosonic

singletons – explicitly done symmetries classified

Bekaert, M.G. 2009

• Being of AKSZ form the parent Lagrangian formulation automatically

gives the BFV-BRST Hamiltonian description.

• Can be naturally interpreted in terms of polymomentum DeDonder-

Weyl covariant Hamiltonian formalism. Moreover, allows to system- atically derive such fomulations for general gauge systems.

Conclusions

• Fruitful exchange of ideas and methods between the local BV-BRST

cohomology methods, unfolded formalism, and various approaches to covariant Hamiltonian formalism.

• Provides set-up for the quantization problem along the BV quantiza-

tion method. However, gauge-fixing fermion, integration measure is still to be studied, celebrated ∆-operator etc.

• Systematic way to construct unfolded formulation starting from the

usual form. In particular, to generate frame-type action. Still to be done for McDowell-Mansouri-Stelle-West type HS Lagrangians.

• Generating procedure for new formulations. In particular, those that

manifest one or another structure. In some sense parent formula- tion and its reductions make the gauge and the BRST cohomology structure manifest. For instance, gravity as a gauge theory of dif- feomorphism algebra or bosonic string as a gauge theory for (regular part of) Virasoro algebra.

• As a tool to find a relevant geometry.

For instance starting from metric gravity one ends up with the Cartan formulation and finds relevant curvatures just by trying to compute BRST cohomology.

• Naturally incorporates nonlinear structure, at least at the level of

gauge-symmetries and off-shell constraints.