Graded Geometry and Gravity Interaction via deformation Peter - PowerPoint PPT Presentation

Graded Geometry and Gravity Interaction via deformation Peter Schupp Jacobs University Bremen based on: Eugenia Boffo, PS: Deformed graded Poisson structures, Generalized Geometry and Supergravity arXiv:1903.09112 (JHEP), Boffo, Pinkwart-Walker, PS: in preparation , and earlier work with Brano Jurco, Fech-Scen Khoo, Jan Vysoky Solvay Workshop on Higher Spin Gauge theories, topological field theory and deformation quantization Brussels, February 2020

Outline ◮ Interaction via deformation, monopoles, aspects of quantization ◮ Graded/generalized geometry and gravity ◮ Deformation, gauge theory and Moser’s lemma, non-associativity, non-metricity, gravitipols ◮ Remarks on higher spin in a graded setting

Interaction via deformation “Beyond gauge theory” ◮ gravity = free fall in curved spacetime → extend this idea to all forces! ◮ free Hamiltonian, interaction via deformation: deformed symplectic structure (or operator algebra) ◮ gauge theory recovered via Moser’s lemma: deformation maps are not unique ⇒ gauge symmetry ◮ somewhat more general than gauge theory and just as powerful as the good old gauge principle

Interaction via deformation � � pdq − H ( p , q ) dt ” : Hamiltonian (first order) action “ S H = � S H = α − H ( X ) d τ + d λ vary with δ X = 0 at boundary L δ X ( α − Hd τ ) = i δ X d α + d ( i δ X α ) − ( i δ X dH ) d τ ω ( − , ˙ → X ) = dH where ω = d α X = θ ( − , dH ) → ˙ ˙ θ = ω − 1 ↔ f = { f , H } where interaction, coupling to gauge field: ◮ either deform H (“minimal substitution”): H ′ = H ( p − A , q ) ◮ or deform ω and hence { , } : α ′ = � pdq + A → ω ′ = ω + dA

Interaction via deformation example: relativistic particle in einbein formalism � 1 � � x ν − 1 � p µ = 1 x µ ˙ 2 em 2 + A µ ( x )˙ x µ x ν + A µ S = d τ 2 e g µν ( x )˙ e g µν ˙ � � ( p µ − A µ ) 2 + m 2 � p µ dx µ − 1 S H = 2 e d τ ← p µ : canonical momentum � � µ + m 2 � ( p µ + A µ ) dx µ − 1 p 2 S H = 2 e d τ ← p µ : physical momentum � � ω ′ = d ∧ dx µ � p µ + A µ { p µ , p ν } ′ = F µν , { x µ , p ν } ′ = δ µ ν , { x µ , x ν } ′ = 0 { p λ , { p µ , p ν } ′ } ′ + cycl. = ( dF ) λµν = ( ∗ j m ) λµν ← magnetic 4-current magnetic sources ⇔ non-associativity

Interaction via deformation Quantization ◮ path integral � ◮ deformation quantization � ( → details later) ◮ canonical? depends. . . ( � ) : Deformed CCR: [ x µ , p ν ] = i � δ µ [ x µ , x ν ] = 0 , [ γ µ , γ ν ] + = 2 g µν [ p µ , p ν ] = i � F µν , ν , 2 p 2 � correct coupling of fields to spin Let p = γ µ p µ and H = 1 � � 2 p 2 − i � H = 1 [ γ µ , γ ν ] + [ p µ , p ν ] + + [ γ µ , γ ν ][ p µ , p ν ] = 1 2 S µν F µν 8 Lorentz-Heisenberg equations of motion (ignoring spin) x ν + ˙ x ν = i p µ = i � [ H , p µ ] = 1 x ν F µν ) � [ H , x ν ] = p ν ˙ 2 ( F µν ˙ with ˙ this formalism allows dF � = 0: magnetic sources, non-associativity

Interaction via deformation: monopoles 3 [ p λ , [ p µ , p ν ]] dx λ dx µ dx ν = � 2 dF = � 2 ∗ j m 1 local non-associativity: j m � = 0 ⇔ no operator representation of the p µ ! spacetime translations are still generated by p µ , but magnetic flux Φ m leads to path-dependence with phase e i φ ; where φ = iq e Φ m / � globally: � � Φ m = F = ↔ non-commutativity A S ∂ S � � � Φ m = F = dF = ∗ j m = q m ↔ non-associativity ∂ V V V global associativity requires φ ∈ 2 π Z ⇒ q e q m 2 π � ∈ Z Dirac quantization non-relativistic version of this: Jackiw 1985, 2002

Aspects of quantization � ⋆ The operator-state formulation of QM cannot handle non-associative structures. . . Phase-space formulation of QM ◮ Observables and states are (real) functions on phase space. ◮ Algebraic structure introduced by a star product, traces by integration. � ◮ State function (think: “density matrix”): S ρ ≥ 0, S ρ = 1. 1 � ◮ Expectation values �O� = O ⋆ S ρ . odinger equation H ⋆ S ρ − S ρ ⋆ H = i � ∂ S ρ ◮ Schr¨ ∂ t ◮ “Stargenvalue” equation: H ⋆ S ρ = S ρ ⋆ H = E S ρ . 1 Wick-Voros formulation yields non-negative state function; Moyal-Weyl leads instead to Wigner quasi-probability function that can be negative in small regions.

Aspects of quantization � ⋆ Popular choices of star products ◮ Moyal-Weyl (symmetric ordering, Wigner quasi-probability function) Weyl quantization associates operators to polynomial functions via symmetric ordering: x µ � ˆ x µ , x µ x ν � 1 x ν + x ν ˆ x µ ˆ x µ ), etc. 2 (ˆ extend to functions, define star product � f 1 ⋆ f 2 := � f 1 � f 2 . ◮ Wick-Voros (normal ordering, coherent state quantization) QHO states in Wick-Voros formulation: ◮ xp-ordered star product: ⋆ -exponential ≡ ordinary path integral

Aspects of quantization � ⋆ Deformation quantization of the point-wise product in the direction of a Poisson bracket { f , g } = θ ij ∂ i f · ∂ j g : f ⋆ g = fg + i � 2 { f , g } + � 2 B 2 ( f , g ) + � 3 B 3 ( f , g ) + . . . , with suitable bi-differential operators B n . There is a natural (local) gauge symmetry: “equivalent star products” ⋆ �→ ⋆ ′ , Df ⋆ Dg = D ( f ⋆ ′ g ) , with Df = f + � D 1 f + � 2 D 2 f + . . . Dynamical non-associative star product: � � � � i � i � 2 R ijk p k ∂ i ⊗ ∂ j e ∂ i ⊗ ˜ ∂ i − ˜ ∂ i ⊗ ∂ i f ⋆ p g = · ( f ⊗ g ) e 2

Aspects of quantization θ ( x ) � ⋆ Kontsevich formality and star product U n maps n k i -multivector fields to a (2 − 2 n + � k i ) -differential operator � U n ( X 1 , . . . , X n ) = w Γ D Γ ( X 1 , . . . , X n ) . The graphs and hence the integrals factorize. The basic graph Γ ∈ G n θ 1 p 1 ψ 1 The star product for a given bivector θ is: ∞ � ( i � ) n f ⋆ g = U n (Θ , . . . , Θ)( f , g ) n ! n =0 � � θ ij ∂ i f · ∂ j g − � 2 = f · g + i θ ij θ kl ∂ i ∂ k f · ∂ j ∂ l g 2 4 �� � − � 2 θ ij ∂ j θ kl ( ∂ i ∂ k f · ∂ l g − ∂ k f · ∂ i ∂ l g ) + . . . 6 Kontsevich (1997)

Aspects of quantization θ ( x ) � ⋆ Formality condition The U n define a quasi-isomorphisms of L ∞ -DGL algebras and satisfy � � � d · U n ( X 1 , . . . , X n )+ 1 ε X ( I , J ) U |I| ( X I ) , U |J | ( X J ) 2 G I⊔J =(1 ,..., n ) I , J � = ∅ � � � ( − 1) α ij U n − 1 [ X i , X j ] S , X 1 , . . . , � X i , . . . , � = X j , . . . , X n , i < j relating Schouten brackets to Gerstenhaber brackets. 1 This implies in particular Φ( d Θ Θ) = i � d ⋆ Φ(Θ), i.e. θ (non-)Poisson ⇔ ⋆ (non-)associative

Aspects of quantization θ ( x ) � ⋆ Poisson sigma model 2-dimensional topological field theory, E = T ∗ M � � � ξ i ∧ d X i + 1 S (1) 2 Θ ij ( X ) ξ i ∧ ξ j AKSZ = , Σ 2 with Θ = 1 2 Θ ij ( x ) ∂ i ∧ ∂ j , ξ = ( ξ i ) ∈ Ω 1 (Σ 2 , X ∗ T ∗ M ) perturbative expansion ⇒ Kontsevich formality maps valid on-shell ([Θ , Θ] S = 0) as well as off-shell, e.g. twisted Poisson Kontsevich (1997) Cattaneo, Felder (2000)

Graded spacetime mechanics Now try to do the same for gravity! Deformation maybe fine for curvature R µν , however, the metric g µν is symmetric but { , } is not. ◮ use graded geometry, i.e. odd variables and/or odd brackets ◮ or consider derived brackets g µν ∼ {{ x µ , H } , x ν } , { H , H } = 0 ◮ � algebraic approach to the geodesic equation, connections, curvature, etc. Properties like metricity follow from associativity. Local inertial coordinates are reinterpreted as Darboux charts ◮ the classical formulation requires graded variables ( ∼ differentials), quantization leads to γ -matrices and Clifford algebras classical ↔ quantum θ µ γ µ ↔ θ µ θ ν = − θ ν θ µ 1 2 [ γ µ , γ ν ] − ↔ 1 2 { θ µ , θ ν } = g µν 1 2 [ γ µ , γ ν ] + = g µν ↔

Graded spacetime mechanics Graded Poisson algebra µ , θ ν } = 2 g µν ν } = δ ν { θ ( x ) { p µ , x { p µ , f ( x ) } = ∂ µ f ( x ) µ a a 0 0 0 c Since g µν ( x ) has degree 0, the Poisson bracket must have degree b = − 2 a for θ µ of degree a , i.e. it is an even bracket. ! Since g µν ( x ) is symmetric, we must have − ( − 1) b + a 2 = +1, i.e. a is odd. θ µ are Grassmann variables of degree 1, wlog: { , } is of degree b = − 2, θ µ θ ν = − θ ν θ µ , and the momenta p µ have degree c = − b = 2 ⇔ a metric structur on TM and natural symplectic structure on T ∗ M , shifted in degree and combined into a graded Poisson structure on T ∗ [2] ⊕ T [1] θ µ M p µ x µ

Graded spacetime mechanics Graded Poisson algebra µ , θ ν } = 2 g µν ν } = δ ν { θ ( x ) { p µ , x { p µ , f ( x ) } = ∂ µ f ( x ) µ 1 1 0 0 0 2 Jacobi identity (i.e. associativity) ⇔ metric connection β =: ∇ µ θ α α } = Γ α { p µ , θ µβ θ 1 1 2 { p µ , { θ α , θ β }} = 2 ∂ µ g αβ = {{ p µ , θ α } , θ β } + { θ α , { p µ , θ β }} and curvature {{ p µ , p ν } , θ α } = [ ∇ µ , ∇ ν ] θ α = θ β R β α µν ν } = 1 β θ α R βαµν ⇒ { p µ , p 4 θ 1 1 2 2