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 T a , gauge symmetries R i α , reducibility re- lations,.... the BRST differential: s 2 = 0 , s = δ + γ + . . . , gh( s ) = 1 ∂ ∂ ∂ ∂ P a + Z a γ = c α R i δ = T a ∂φ i + . . . . A P a ∂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 system. 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 ,

T i = δS 0 If the theory is Lagrangian then: 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 +

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

Φ : X → M . Fields Ψ A ( x, θ ) ≡ Φ ∗ (Ψ A ). BV master action � � (Φ ∗ ( χ ))( d ) + Φ ∗ ( S M ) � , S BV = gh( S BV ) = 0 χ is potential for σ = dχ . In components: � � � d n xd n θ χ A (Ψ( x, θ )) d Ψ A ( x, θ ) + S M (Ψ( x, θ )) S BV = BV antibracket � � � � � � � δ R F δG d n xd n θ δ Ψ A ( x, θ ) σ AB = gh = 1 F, G , , δ Ψ B ( x, θ ) σ AB (Ψ) – components of the Poisson bivector. Master equation: � � = 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 Natural lift of Q and d to the space of maps. 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 ) is dynamical.

AKSZ equations of motion σ AB ( d Ψ A + Q A ) = 0 , d Ψ A ( x, θ ) + Q A (Ψ( x, θ )) = 0 ⇒ (recall: σ AB is invertible) AKSZ at the level of equations of motion (nonlagrangian) ∂ Q 2 = 0 . Q = Q A { , } , S M ⇒ ∂ Ψ A I.e. target is a generic Q manifold. target doesn’t know dim X ! (Recall gh( S M ) = n = dim X ) If gh(Ψ A ) � 0 ∀ Ψ A 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. e i –basis in g , C i – coordinates on g , gh( C i ) = 1, C = C i e i � C i , C j � S M = 1 = � e i , e j � − 1 6 � 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 ) 3 c ∗ i C i ( x, θ ) = c i ( x ) + θ µ A i BV action � � 1 2 � C, d C � +1 2 � A, d A � +1 ( 1 S BV = 6 � C, [ C, C ] � ) = 6 � A, [ A, A ] � )+ . . .

1d AKSZ systems Target space M – Extended phase space: { , } – Poisson bracket, S M = Ω − θH , Ω – BRST charge, H - 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 = Integration over θ gives BV for the Hamiltoninan action Fisch, Henneaux, 1989, Batalin, Fradkin 1988 . c, � x µ , � Example: coordinates on M : P , � p µ , BRST charge � p 2 − m 2 ), Ω = � c ( � � � x µ + � c ( p 2 − m 2 )) = x µ + λ ( p 2 + m 2 )) + S BV = dtdθ ( � c + � dt ( p µ ˙ p µ d � P d � x µ ( t, θ ) = x µ ( t ) + θp µ p µ ( t, θ ) = p µ ( t ) + θx ∗ ∗ ( t ) , µ ( t ) , � � c ( t, θ ) = c ( t ) + θλ ( t ) , � . . .

– Background-independent – AKSZ is both Lagrangian and Hamiltonian AKSZ model: ( M, S M , { , } ) and ( X , d ). Let X = X S × R 1 Barnich, M.G, 2003 � � , � (Φ ∗ ( χ ))( d ) + Φ ∗ ( S M ) Ω BFV = gh(Ω BFV ) = 1 X S � d n − 1 xd n − 1 θ { · , · } { · , · } BFV = { Ω BFV , Ω BFV } BFV = 0 . – Higher BRST charges Similarly: X k ⊂ X – dimension- k submanifold � (Φ ∗ ( χ ))( d ) + Φ ∗ ( S M )) Ω X k = X k In particular, Ω BFV = Ω X S , S BV = Ω X

– 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 X k ⊂ 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 H g ( s AKSZ , local functionals) ∼ = H g + n ( Q, C ∞ ( M )) � f . Isomorphism sends f ∈ C ∞ ( M ) to functional 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 R n × R N ): base space (independent variables or space-time coordinates): x a , a = 1 , . . . , n . Fiber coordinates (dependent variables or fields) φ i . Jet- bundle: J 0 ( F ) : J 1 ( F ) : J 2 ( F ) x a , φ i , x a , φ i , φ i x a , φ i , φ i a , φ i a , ab , . . . Projections: . . . → J N ( F ) → J N − 1 ( F ) → . . . → J 1 ( F ) → J 0 ( F ) = F Useful to work with J ∞ ( F ). A local diff. form on J ∞ ( F ) – a form on J N ( F ) pulled back to J ∞ ( F ).

J ∞ is equipped with the total derivative ∂ ∂ ∂ ∂ T ∂x a + φ i ∂φ i + φ i a = + . . . a ab ∂φ i b For a given section φ i = s i ( x ) and local function f [ φ ] � � ∂ � � ( ∂ T a f ) � φ = s,φ a = ∂ a s,... = ∂x a ( f � φ = s,φ a = ∂ a s,φ ab = ∂ a ∂ b s,... )

Space time differentials dx a . Horizontal differential d h ≡ dx a ∂ T d 2 h = 0 . a , Differential forms: φ I = φ i α = α ( x, dx, φ, φ a , . . . ) I 1 ...I k d v φ I 1 . . . d v φ I k , a 1 ...a m Vertical differential: d v ≡ d − d h = d v φ I ∂ ∂φ I Variational bicomplex ( Vinogradov’s C -spectral sequence): d 2 d 2 v = 0 , d v d h + d h d v = 0 , h = 0 Bidegree ( l, p ). On the jet space H > 0 ( d v ) = 0 = H <n ( d h ) (unless global geometry!). H n ( d h ) = local functionals

A system of partially differential equations (PDE) is a col- lection of local functions on J ∞ ( F ) E α [ φ, x ] . The equation manifold (stationary surface): E ⊂ J ∞ ( F ) singled out by: ∂ T a 1 . . . ∂ T a l E α = 0 , l = 0 , 1 , 2 , . . . understood as the algebraic equations in J ∞ ( F ). It is usu- ally assumed that x a , φ i 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 d h and d v . ∂ 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 , d h ). It is clear when PDEs are to be considered equivalent. Differential forms on E form the variational bicomplex of E . Note that in general H k ( d h ) � = 0 for k < n .