Supergeometry of gauge PDE and AKSZ sigma models Maxim Grigoriev - - PowerPoint PPT Presentation

Supergeometry of gauge PDE and AKSZ sigma models

Maxim Grigoriev Based on:

M.G., 1606.07532 M.G., A. Verbovetsky, to appear

• K. Alkalaev, M.G. 2013

Glenn Barnich, M.G. 2010 M.G. 2010,2012

June 27, 2016, Bialowieza, Poland

Motivations

• Batalin-Vilkovisky (BV) approach to gauge systems

(or its generalizations) is probably the most powerful.

Batalin, (Fradkin), Vilkovisky, 1981 . . .

• For various topological models their BV formulation

can be cast into the form of AKSZ sigma model.

Alexandrov, Kontsevich, Schwartz, Zaboronsky, 1994.

• In so doing the equations of motion, gauge symmetries,
• etc. are encoded in a homological vector field Q on the

target space, which is a Q-manifold (or QP-manifold in Lagrangian case).

• A natural question is weather the same can be done for

non-topological systems? How the gauge symmetries, Lagrangians, etc. are encoded in the geometry of the target space?

• For a local gauge theory AKSZ-like formulation has

certain advantages over the usual jet-space version of the BV formalism

Henneaux; Barnich, Brandt, Henneaux

This has to do with the manifest background indepen- dence of AKSZ, which can be employed in studying boundary values, manifest realization symmetries etc.

Batalin-Vilkovisky formalism:

Given equations Ta, gauge symmetries Ri

α, reducibility re-

lations,.... 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 system. Equations of motion and gauge symmetries can be read

• ff 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]]

AKSZ sigma models

M - supermanifold (target space) with coordinates ΨA: Ghost degree – gh() (odd)symplectic structure σ, gh(σ) = n − 1 and hence (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

Φ : X → M. Fields ΨA(x, θ) ≡ Φ∗(ΨA). BV master action SBV =

(Φ∗(χ))(d) + Φ∗(SM) ,

gh(SBV ) = 0 χ is potential for σ = dχ. In components: SBV =

• dnxdnθ
• χA(Ψ(x, θ))dΨA(x, θ) + SM(Ψ(x, θ))
• BV antibracket
• F, G
• =
• dnxdnθ
• δRF

δΨA(x, θ)σAB δG δΨB(x, θ)

• ,

gh

• ,
• = 1

σAB(Ψ) – components of the Poisson bivector. Master equation:

• SBV , SBV
• = 0 ,

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

• ΨA, SM
• Natural lift of Q and d to the space of maps.

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) is dynamical.

AKSZ equations of motion σAB(dΨA + QA) = 0 , ⇒

dΨA(x, θ) + QA(Ψ(x, θ)) = 0

(recall: σAB is invertible) AKSZ at the level of equations of motion (nonlagrangian) {, } , SM ⇒ Q = QA

∂ ∂ΨA

Q2 = 0 . I.e. target is a generic Q manifold. target doesn’t know dim X! (Recall gh(SM) = n = dim X) If gh(ΨA) 0 ∀ ΨA then BV-BRST extended FDA. Otherwise BV-BRST extended FDA with constraints.

Examples:

Chern-Simons:

AKSZ, 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 = 1 6C, [C, C] ,

• Ci, Cj

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

µ(x) + θµθνA∗i µν + (θ)3c∗i

BV action SBV =

• (1

2C, dC+1

6C, [C, C]) =

1

2A, dA+1

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

1d AKSZ systems

Target space M – Extended phase space: {, } – Poisson bracket, SM = Ω − θH, Ω – BRST charge, H - BRST invariant Hamiltonian Source space X = T[1](R1), coordinates t, θ BV action

M.G., Damgaard, 1999

SBV =

• dtdθ(χAdψA + Ω − θH)

Integration over θ gives 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)) +

• xµ(t, θ) = xµ(t) + θpµ

∗(t) ,

• pµ(t, θ) = pµ(t) + θx∗

µ(t) ,

• c(t, θ) = c(t) + θλ(t) ,

. . .

– Background-independent – AKSZ is both Lagrangian and Hamiltonian AKSZ model: (M, SM, {, }) and (X, d). Let X = XS × R1

Barnich, M.G, 2003

ΩBFV =

• XS

(Φ∗(χ))(d) + Φ∗(SM) ,

gh(ΩBFV ) = 1 { · , · }BFV =

• dn−1xdn−1θ { · , · }

{ΩBFV , ΩBFV }BFV = 0 . – Higher BRST charges Similarly: Xk ⊂ X – dimension-k submanifold ΩXk =

• Xk

(Φ∗(χ))(d) + Φ∗(SM)) In particular, ΩBFV = ΩXS , SBV = ΩX

– At the level of equations of motion AKSZ is a gen- eralization of so-called unfolded formalism independently developed in the context of HS theories

Vasiliev 1988,. . . .

– At the level of equations of motion the same target space gives an AKSZ model for any Xk ⊂ X or even different X. Useful for “replacing space-time”. E.g.

Vasiliev 2002

(asymptotic) boundary values, e.g. in the context of AdS/CFT For higher-spin fields

Vasiliev, 2012; Bekaert M.G. 2012

– Locally in X and M:

Barnich, M.G. 2009

Hg(sAKSZ, local functionals) ∼ = Hg+n(Q, C∞(M)) Isomorphism sends f ∈ C∞(M) to functional F =

f.

Compatible with the bracket. – If M finite dimensional and n > 1 – the model is topo-

• logical. What about non-topological?

AKSZ form of PDE

Jet-bundle: Fiber-bundle F → X (for simplicity: direct product of Rn ×RN): base space (independent variables or space-time coordinates): xa, a = 1, . . . , n. Fiber coordinates (dependent variables or fields) φi. Jet- bundle: J0(F) : xa, φi, J1(F) : xa, φi, φi

a ,

J2(F) xa, φi, φi

a, φi ab ,

. . . Projections: . . . → JN(F) → JN−1(F) → . . . → J1(F) → J0(F) = F Useful to work with J∞(F). A local diff. form on J∞(F) – a form on JN(F) pulled back to J∞(F).

J∞ is equipped with the total derivative ∂T

a = ∂ ∂xa + φi a ∂ ∂φi + φi ab ∂ ∂φi

b

+ . . . For a given section φi = si(x) and local function f[φ] (∂T

a f)

• φ=s,φa=∂as,... =

∂ ∂xa(f

• φ=s,φa=∂as,φab=∂a∂bs,...)

Space time differentials dxa. Horizontal differential dh ≡ dxa∂T

a ,

d2

h = 0 .

Differential forms: α = α(x, dx, φ, φa, . . .)I1...IkdvφI1 . . . dvφIk , φI = φi

a1...am

Vertical differential: dv ≡ d − dh = dvφI ∂

∂φI

Variational bicomplex (Vinogradov’s C-spectral sequence): d2

v = 0,

dvdh + dhdv = 0 , d2

h = 0

Bidegree (l, p). On the jet space H>0(dv) = 0 = H<n(dh) (unless global geometry!). Hn(dh) = local functionals

A system of partially differential equations (PDE) is a col- lection of local functions on J∞(F) Eα[φ, x] . The equation manifold (stationary surface): E ⊂ J∞(F) singled out by: ∂T

a1 . . . ∂T alEα = 0 ,

l = 0, 1, 2, . . . understood as the algebraic equations in J∞(F). It is usu- ally assumed that xa, φi are not constrained, e.g. E is a bundle over the space-time. ∂T

a are tangent to E and hence restricts to E. So do the

differentials dh and dv. ∂T

a |E determine a dim-n integrable

distribution (Cartan distribution).

Definition:

[Vinogradov] A PDE is a manifold E equipped

with an integrable distribution. In addition one typically assumes regularity, constant rank, and that E is a bundle over the spacetime. Use notation (E, dh). It is clear when PDEs are to be considered equivalent. Differential forms on E form the variational bicomplex of

• E. Note that in general Hk(dh) = 0 for k < n.

Scalar field Example: Start with: L = 1

2ηabφaφb − V (φ) ,

∂a∂aφ + ∂V

∂φ = 0 .

E is coordinatized by xa, φ, φa, φab, . . .. Already φab are not independent. One can e.g. take φabc... traceless. The dh-differential on E reads as dhxa = dxa , dhφ = dxaφa, , dhφa = dxb(φab − 1 nηab

∂V ∂φ ) ,

. . . So if the system is nonlinear, i.e.

∂V ∂φ nonlinear in φ, dh is

also nonlinear.

Intrinsic (unfolded) realization

Given PDE (E, dh) defined intrinsically one can always find a jet space J such that (E, Q) can be realized as a station- ary surface of some Eα[u, x]. There is an intrinsic way to realize (E, dh) explicitly. If xa, ψA coordinates on E (e.g. ψA = {φ, φa, φab, . . .}) pro- mote ψA to fields ψA(x) = of a new theory and subject them to EOM’s

d(ψA(x)) = (dhψA)(x)

components:

∂ ∂xaψA(x) = (∂T a ψA)(x)

Proposition: The original PDE (E, dh) is equivalent to

dψA = dhψA

Comments:

• Version of the unfolded formulation (though only zero

forms). Unfolded form of gauge systems involves gauge form-fields.

Vasiliev, 1987,. . .

• Generalized version of the Proposition involving gauge

forms and BRST extension was formulated and proved us- ing BRST technique and Koszule-Tate differential.

Bar- nich, M.G.,Semikhatov, Tipunin 2004, Barnich, M.G 2010

New jet-space

Because E is a bundle over spacetime, take J new ≡ J∞(E). More precisely, if xa, dxa, ψA are coordinates on E then xa, dxa, ψA, ψA

b ,

ψA

bc,

ψA

bcd,

. . . are coordinates on J new. New jet space is equipped with its own horizontal differen- tial: Dh = dxa( ∂

∂xa + ψA a ∂ ∂ψA + ψA ab ∂ ∂ψA

b

+ . . .) “Old” differential dh on E extends to J new by [DH, Q] = 0. In the new jet space J new consider the following PDE DhψA = dhψA In this form the new PDE is manifestly isomorphic to (E, Q) (because manifolds are isomorphic and horizontal differentials are equal by construction)

AKSZ form and reparametrization invariance

Consider dxa as ghosts ξa, change notation xa → za and extend E into a supermanifold with coordinates ΨA = {za, ξa, φi, φi

a, φi ab, . . .}. It is a Q-manifold:

Q = −dh = −ξA∂T

a

Take X = T[1]X with coordinates xµ, θµ and consider AKSZ model with source (X, d) and target (E, Q). Note that now za is promoted to a field za(x) and ξa to ea

µ(x)dxµ.

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

µ(x) – frame field components.

Gauge transf. for za: δza = ξa. Q is the BRST differential implementing repatametrization invariance. Gauge condition za = δa

µxµ give un-parametrized version:

dΨA + QA(Ψ) = 0

⇒

dΨA(x) − θa(∂T

a ΨA)(x, θ) = 0

Recall: ∂T

a – total derivative (vector field in the target).

Extension to gauge theories

If (E, dh) has gauge symmeries there are parameters ǫα which are arbitary space time functions. Promote them to ghost variables cα and consider the extension ¯ E of E by the jet-space for cα: CI = {cα, cα

a,

cα

ab,

. . .} The gauge symmetry is encoded in vector field γ satisfying [dh, γ] = 0 , γ2 = 0 , gh(γ) = 1 It can be written as γ = CIRA

I (ψ) ∂ ∂ψA − 1 2CICJUK IJ(ψ) ∂ ∂CK

Vector fields RI determine an integrable distribution on E (gauge-distribution), compatible with Cartan distribution.

AKSZ form

Consider AKSZ model with source (X, d) and the target (¯ E, Q), where Q = −dh + γ Total differential familiar in the local BRST cohomology

Stora, 1983, Batnich, Brandt, Henneaux 1993,. . . .

Equivalent to the parametrized version of gauge system. In addition to ea

µ(x)dxµ new 1-form fields AI µ(x)dxµ associ-

ated to CI. The equivalence was proved using

Barnich, M.G. 2010

• s = −dh + δ + γ + . . .

where δ is the Koszule–Tate differential of the stationary surface.

New feature: contractibe pairs for Q: if by local invertible change of coordinates: Qwa = va, Qψα = Qα(ψ) then wa, va are contractibe pairs. Their elimination results in the reduced Q-manifold (Q, E). Eliminating all such triv- ial pairs one arrives at “minimal” Q-manifold associated the gauge system

Brandt, 1996

The manifold of generalized connections and tensor fields. For the AKSZ model trivial pairs give rise to generalized auxiliary fields. Lagrangian:

Dresse, Gr´ egoire, Henneaux, 1990

EOM:

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

Their elimination is an equivalence of the respective AKSZ models.

Example of Einstein gravity

For diffeomorphism-invariant theory parameterization brings

• nothing. It follows xa, ξa can be eliminated together with

dh, giving Q = γ. After elimination the contractible pairs of Q manifold E: ea, ωab, W cd

ab ,

W cd

ab|e,

W cd

ab|e...

– ghosts associated to frame field and spin connection and Weyl tensor and its independent covariant derivatives. Qea = ωac ec , Qωab = ωac ωcb + 1

2ecedW ab cd ,

QW = eW + ωW + . . . Minimal BRST complex (Q-manifold) for gravity. Gives minimal AKSZ formulation (unfolded formulation).

Variational (Lagrangian) equations

Let us get back to equations Ei[φ, x] = 0 on the jet space J∞(F). These are said variational (Lagrangian) if Ei = δELL

δφi , δELF[u, x] δφi

≡ ∂F

∂φi − ∂T a ∂F ∂φi

a

+ ∂T

a ∂T b ∂F ∂φi

ab

− . . . for some local function L = L[φ, x]. It is convenient to work in terms of Lagrangian density L = (dx)n L. Here and below (dx)n = dx1 . . . dxn , (dx)n−1

a

= 1 (n − 1)!ǫab2...bndxb1 . . . dxbn The notion of Lagrangian is explicitly based on the realiza- tion of the equation (E, dh) in terms of a jet space J . For instance it’s possible that E ⊂ J is variational while E ⊂ J ′ is not. Naive invariant object – the restriction of L to E, does not make much sense.

Presymplectic structure

It is well-known that L = (dx)nL[x, φ] induces an invariant

• bject on

Crnkovic, Witten, 1987, Hydon 2005,. . .

dvL = dφi Ei (dx)n − dh χ , components: δELL δφi = ∂L

∂φi+∂T a (

χa

i )

for some 1 form χ = χidvφi + χiadvφi

a + . . . of degree n − 1,

called presymplectic potential. For χ = χ|E we have dhσ = 0 , σ = dχ So we have conserved closed 2-form on E. It’s called canonical presymplectic structure.

As an example consider L(φ, φa, φab). One finds: χ =

• ( ∂L

∂φa − ∂T b ∂L ∂φab)dvφ + ∂L ∂φabdvφb

• E

(dx)n−1

a

In particular, for a scalar field with L = 1

2ηabφaφb − V (φ)

χ = φadvφ (dx)n−1

a

, σ = dvφa dvφ (dx)n−1

a

More generally: Definition: A 2-form σ of degree n − 1 on (E, dh) is called compatible presymplectic structure if dhσ = 0 , dσ = 0.

. . . ,Khavkin 2012, Alkalaev, M.G. 2013

Such form in general can be considered irrespective of any realization in terms of jet-space and/or Lagrangian.

Symmetries and conservation laws

A well-known fact: both symmetries and conservation laws can be defined in terms of the equation manifold (E, dh). Recall: a vector field V on J is a symmetry if it is evolu- tionary i.e. [dh, V ] = 0 and tangent to E ⊂ J . Intrinsic terms: a vector field V

• n (E, dh) satisfying is

called symmetry if [dh, V ] = 0 (typically one also requires V xa = 0). If E ⊂ J is variational then variational symmetries restricted to E satisfy in addition LV σ = dhdvα for some (n − 2, 1)-form α.

Conservation law (conserved curent) is a degree n − 1 0- form K on E such that dhK = 0. K of the form K = dhM is trivial. Any compatible presymplectic structure determines a map from symmetries to conserved currents according to dK = iV σ − dhα , Note: diV σ = LV σ = 0. Trivial symmetries are mapped to trivial conserved currents. In the Lagrangian case this is usual Noether theorem. General case was also discussed recently

Sharapov 2016.

It is different from the Poisson (BV antibracket) bracket map from conservation laws to symmetries. The degener- ate version of the bracket is known as Hamilton/Lagrange structure

Kersten, Krasilshchik, Verbovetsky Kaparulin, Lyakhovich, Sharapov

Suppose that (E, dh, σ) is realized as E ⊂ J∞(F). Then σ determines a Lagrangian form L on J∞(F) such that the EL equations derived from L are in general consequences

• f those defining E.

Khavkine 2012, based on earlier: Bridges, Hydon, Lawson 2009, Hydon 2005

More precisely, if E′ is an equation manifold defined by L then E ⊂ E′. Even if σ is canonical (derived from a Lagrangian) there is no guarantee that constructed L is equivalent to the starting point Lagrangian.

Intrinsic Lagrangian

Given an equation manifold (E, dh, σ) equipped with the compatible presymplectic structure one can construct a natural Lagrangian in terms of the E-valued fields. First: define covariant Hamiltonian (better BRST charge) which is a conserved current associated to dh seen as a symmetry of E. Degree n function H on E defined by dhH = idhσ , components:

∂ ∂ψAH = σABdhψB

If σ originates from L: H = χAdhψA−L|E E.g. in the simple case where L = (dx)nL(φ, φa) χ = ( ∂L

∂φadvφ)

• E(dx)n−1

a

, H = ( ∂L

∂φaφa − L)

• E(dx)n
• cf. de Donder-Weyl covariant Hamiltonian. The intrinsic

Lagrangian

More invariant language: It follows from dσ = 0 that σ = d(χ + l) l is an (n, 0)-form (1) For canonical σ l = L|E. New jet-space J∞(E). Intrinsic Lagrangian LC = H(π∗(χ + l)) , (2) where π : J∞(M) → M. H denotes horizontalization map:

H(dxa) = dxa, H(dψA

ab...) = DhψA ab... .

(3)

Switching to the language of Q-manifolds: dH = iQσ The respective action can be seen as presymplectic gener- alization

Alkalaev, M.G. 2013

SC = χAdψA(x) − H(ψ(x))

• f AKSZ action. Its equations of motion read as

σAB(dψB(x) − QB(x)) = 0 , and hence are consequences of the original dψB − QB = 0. For a local theory LC does not depend on most of the fields ψA. These can be treated as pure-gauge variables with algebraic (shift) gauge transformations. With this interpretation and under certain assumptions we can prove that starting point L and LC are equivalent for a wide class

• f systems (but not all).

Examples

Scalar field: Start with: L = 1

2ηabφaφb − V (φ)

E is coordinatized by xa, φ, φa, φab, . . .. take φabc... traceless. The dh differential dhxa = dxa , dhφ = dxaφa, , dhφa = dxb(φab − 1 nηab

∂V ∂φ )

The presymplectic potential and 2-form: χ =

• ( ∂L

∂φa − ∂T c ∂L ∂φca)dvφ)

• E

(dx)n−1

a

= (dx)n−1

a

φadvφ , The Hamiltonian obtained from dH − iQσ = 0: H = (dx)n(φaφa − L|E) = 1

2φaφa + V (φ)

The intrinsic Larangian: Schwinger Lc = (dx)n

• φa(∂aφ − 1

2φa) − V (φ)

Polywave equation

The simplest genuine higher derivative example is L =

1 2φφ = 1 2φaaφbb

(here and below φaa = ηabφab). Presymplectic potential: χ = (−φaccdvφ + φccdvφa)(dx)n−1

a

Hamiltonian H = (dx)n(−φaccφa + 1

2φccφaa) .

The intrinsic action takes the form SC =

• dnx(−φacc(∂aφ − φa) + φcc∂aφa − 1

2φaaφcc) .

Note that the action depends on only the following vari- ables φ, φa, φaa, φacc but NOT on the traceless component

• f φab and φabc.

It is equivalent to

φaaφcc.

Indeed, varying φa and φacc gives φa = ∂aφ and φacc = ∂aφcc resulting in

• dnx(φcc∂a∂aφ − 1

2φaaφcc)

YM theory

The YM field is Aa taking values in a Lie algebra g equipped with an invariant inner product , . We will use notation Aa

b1...bl for ∂T b1 . . . ∂T

• blAa. The Lagrangian:

L = 1 4Fab, Fab , Fab := Ab

a − Aa b + [Aa, Ab] .

Coordinates on E: xa, Aa, Fab, Sab := Ab

a + Aa b, Aa bc, . . .

The one form χ is given by χ = ∂L

∂Ab

a

dAb(dx)n−1

a

= Fab, dAb(dx)n−1

a

The Hamiltonian H = ( ∂L

∂Ab

a

Ab

a − 1

4Fab, Fab)(dx) = 1

2Fab, 1 2Fab − [Aa, Ab]

The intrinsic action

1

2Fab, ∂aAb − ∂bAa − 1 2Fab, 1 2Fab − [Aa, Ab] =

1

2Fab, ∂aAb − ∂bAa + [Aa, Ab] − 1 2Fab

equivalent to the starting point action through the elimi- nation of Fab by its own equations of motion. Well-known first-order action for YM.

Example of gravity: frame like Lagrangian

Recall: reducwd Q-manifold E ea, ωab, W cd

ab ,

W cd

ab|e,

W cd

ab|e...

– ghosts to which frame field and spin connection are as- sociated and Weyl tensor and its covariant derivatives. Qea = ωac ec , Qωab = ωac ωcb + ecedW ab

cd ,

. . . , Presymplectic potential χ and form

Alkalaev, M.G. 2013

χ = 1 2ǫabcddωabeced , σ = dωabdecǫabcded Hamiltonian (term with Weyl tensor vanishes) H = QAχA = 1

2ωa cωcbǫabcdeced

Intrinsic action (frame-like GR action): SC =

• χA(dψA + QA) =
• (dωab + ωacωcb)ǫabcdeced

Conclusions

• A Lagrangian system can be defined in terms of its equa-

tion manifold E without refereeing to any particular real- ization of E in one or another set of fields and choice of the Lagrangian. While the structure of the equation is en- coded in the differential Q the Lagrangian is encoded in the compatible presymplectic structure σ.

• In particular, when looking for a Lagrangian for an equa-

tion E it is enough to study compatible presymplectic struc- tures on E. No need to study possible explicit realizations

• f E.
• Easy to see whether Lagrangian systems are equivalent
• r not.
• BRST extension to manifestly gauge systems. Intrinsic

Lagrangian = Frame-like Lagrangian.

• The presymplectic form can be seen to originate from the
• dd symplectic form of the Batalin-Vilkovisky formalism.

Parent Lagrangian

One way to understand where do the structure of the in- trinsic Lagrangian comes from is to consider “parent” ac- tion for L = L(φ, φa, φab): SP =

• (L(φ, φa, φab) + πa(∂aφ − φa) + πac(∂aφc − φac) + . . .) .

Its equations of motion read as

∂L ∂φ − ∂aπa = 0 ,

πa − ∂L

∂φa + ∂cπca = 0 ,

πab − ∂L

∂φab = 0 ,

πab... = 0 φa = ∂aφ , φab = ∂(aφb) , . . . Using the last line the derivatives in the first two lines can be replaced with the total derivatives. Using the second line the first equation becomes EL

∂L ∂φ − ∂T a ∂L ∂φa + ∂T c ∂T a ∂L ∂φca = 0 .

Introduce 1-form of degree n − 1: ¯ χ = (dx)n−1

a

(πadφ + πabdφb + . . .) ”parent” Hamiltonian ¯ H = (πaφa + πabφab + . . . − L(φ, φa, φab))(dx)n . The parent action can be written as SP =

• (¯

χAdΨA − ¯ H) , where ΨA stand for all the coordinates φ, φ..., π....

Consider the following submanifold of the space of xa, dxa, φ, π..., φ... πa − ∂L

∂φa + ∂T c ∂L ∂φca = 0 ,

πab − ∂L

∂φab = 0 ,

πab... = 0 , ∂T

a1 . . . ∂T ak(EL) = 0 ,

These are consequences of the parent action equations of motion. The submanifold they single out is E (equation manifold

• f L).

χ = ¯ χ|E Presymplectic potential for L One can show iQdσ = dH , H = ¯ H|E , σ = dχ

Furthermore, χ and H determine the intrinsic action SC[ψ] = χA(x, dxa, ψ)dψA − H(x, dxa, ψ)

• ,

where xa, ψA are coordinates on E. This can be indepen- dently arrived at by eliminating auxiliary fields starting from the parent action.