Nonassociative differential geometry and gravity c Marija - PowerPoint PPT Presentation

10th MATHEMATICAL PHYSICS MEETING: School and Conference on Modern Mathematical Physics 9-14 September 2019, Belgrade, Serbia Nonassociative differential geometry and gravity c ´ Marija Dimitrijevi´ Ciri´ c University of Belgrade, Faculty of Physics, Belgrade, Serbia based on: Aschieri, Szabo arXiv: 1504.03915; Aschieri, MDC, Szabo arXiv: 1710.11467.

Noncommutativity & Nonassociativity NC historically: -Heisenberg, 1930: regularization of the divergent electron self-energy, coordinates are promoted to noncommuting operators x ν ≥ 1 x ν ] = i Θ µν ⇒ ∆ˆ x µ , ˆ x µ ∆ˆ 2Θ µν . [ˆ -First model of a NC space-time [ Snyder ’47 ]. More recently: -mathematics: Gelfand-Naimark theorems ( C ∗ -algebras of functions encodes information on topological Hausdorff spaces), -string theory (open string in a constant B -field), -new effects in QFT (UV/IR mixing), -quantum gravity (discretisation of space-time).

NA historically: -Jordan quantum mechanics [ Jordan ’32 ]: hermitean observables do not close an algebra (standard composition, commutator). New composition, A ◦ B is hermitean, commutative but nonassociative: A ◦ B = 1 2(( A + B ) 2 − A 2 − B 2 ) . -Nambu mechanics: Nambu-Poisson bracket (Poisson bracket) { f , g , h } and the fundamental identity (Jacobi identity) [ Nambu’73 ]; quantization still an open problem. f { g , h , k } + { f , h , k } g = { fg , h , k } { f , g , { h 1 , h 2 , h 3 }} + · · · = 0 . More recently: -mathematics: L ∞ algebras [ Stasheff ’94; Lada, Stasheff ’93 ]. -string field theory: symmetries of closed string field theory close a strong homotopy Lie-algebra, L ∞ algebra [ Zwiebach ’15 ], NA geometry of D-branes in curved backgrounds (NA ⋆ -products), closed strings in locally non-geometric backgrounds (low energy limit is a NA gravity). -magnetic monopoles, NA quantum mechanics.

NC/NA geometry and gravity Early Universe, singularities of BHs ⇒ QG ⇒ Quantum space-time NC/NA space-time = ⇒ Gravity on NC/NA spaces. General Relativity (GR) is based on the diffeomorphism symmetry. This concept (space-time symmetry) is difficult to generalize to NC/NA spaces. Different approaches: NC spectral geometry [ Chamseddine, Connes, Marcolli ’07; Chamseddine, Connes, Mukhanov ’14 ]. Emergent gravity [ Steinacker ’10, ’16 ]. Frame formalism, operator description [ Buri´ c, Madore ’14; Fritz, Majid ’16 ]. Twist approach [ Wess et al. ’05, ’06; Ohl, Schenckel ’09; Castellani, Aschieri ’09; Aschieri, Schenkel ’14; Blumenhagen, Fuchs ’16; Aschieri, MDC, Szabo, ’18 ]. NC gravity as a gauge theory of Lorentz/Poincar´ e group [ Chamseddine ’01,’04, Cardela, Zanon ’03, Aschieri, Castellani ’09,’12; Dobrski ’16 ].

Overview NA gravity General NA differntial geometry R -flux induced cochain twist NA tensor calculus NA differential geometry NA deformation of GR Levi-Civita connection NA vacuum Einstein equations NA gravity in space-time Discussion

NA gravity: General NA gravity is based on: -locally non-geometric constant R -flux. Cochain twist F and associator Φ with Φ ( F ⊗ 1) (∆ ⊗ id ) F = (1 ⊗ F ) ( id ⊗ ∆) F . -equivariance (covariance) under the twisted diffeomorphisms (quasi-Hopf algebra of twisted diffeomorphisms). -twisted differential geometry in phase space. In particular: connection, curvature, torsion. Projection of phase space (vacuum) Einstein equations to space-time. -more general, categorical approach in [ Barnes, Schenckel, Szabo ’14-’16 ]. Our goals: -construct NA differential geometry of phase space. -consistently construct NA deformation of GR in space-time: NA Einstein equations and action; investigate phenomenological consequences. -understand symmetries of the obtained NA gravity.

NA differntial geometry: Review of twist deformation Symmetry algebra g and the universal covering algebra Ug . A well defined way of deforming symmetries: the twist formalism. Twist F (introduced by Drinfel’d in 1983-1985) is: -an invertible element of Ug ⊗ Ug -fulfills the 2-cocycle condition (ensures the associativity of the ⋆ -product). F ⊗ 1(∆ ⊗ id ) F = 1 ⊗ F ( id ⊗ ∆) F . (2.1) -additionaly: F = 1 ⊗ 1 + O ( h ); h -deformation parameter.

NA differntial geometry: R -flux induced cochain twist Phase space M : x A = ( x µ , ˜ ∂ µ = ∂ µ , ˜ ∂ � � x µ = p µ ), ∂ A = . ∂ p µ 2 d dimensional, A = 1 , . . . 2 d . The twist F : 2 R µνρ ( p ν ∂ ρ ⊗ ∂ µ − ∂ µ ⊗ p ν ∂ ρ ) � � 2 ( ∂ µ ⊗ ˜ ∂ µ − ˜ ∂ µ ⊗ ∂ µ ) − i κ − i � F = exp , (2.2) with R µνρ totally antisymmetric and constant, κ := ℓ 3 6 � . s Does not fulfill the 2-cocycle condition Φ ( F ⊗ 1) (∆ ⊗ id ) F = (1 ⊗ F ) ( id ⊗ ∆) F . (2.3) The associator Φ: � κ R µνρ ∂ µ ⊗ ∂ ν ⊗ ∂ ρ � � =: φ 1 ⊗ φ 2 ⊗ φ 3 = 1 ⊗ 1 ⊗ 1 + O ( � κ ) . (2.4) Φ = exp Notation: F = f α ⊗ f α , F − 1 = ¯ f α ⊗ ¯ f α , Φ − 1 =: ¯ φ 1 ⊗ ¯ φ 2 ⊗ ¯ φ 3 , Braiding: R = F − 2 =: R α ⊗ R α , R − 1 = F 2 =: R α ⊗ R α .

Hopf aglebra of infinitesimal diffeomorphisms U Vec( M ): [ u , v ] = ( u B ∂ B v A − v B ∂ B u A ) ∂ A , ∆( u ) = 1 ⊗ u + u ⊗ 1 , ǫ ( u ) = 0 , S ( u ) = − u . Quasi-Hopf algebra of infinitesimal diffeomorphisms U Vec F ( M ): -algebra structure does not change -coproduct is deformed: ∆ F ξ = F ∆ F − 1 -counit and antipod do not change: ǫ F = ǫ, S F = S . On basis vectors: ∆ F ( ∂ µ ) = 1 ⊗ ∂ µ + ∂ µ ⊗ 1 , ∂ µ + ˜ ∂ µ ⊗ 1 + i κ R µνρ ∂ ν ⊗ ∂ ρ . ∆ F (˜ 1 ⊗ ˜ ∂ µ ) =

NA differential geometry: NA tensor calculus Guiding principle: Differential geometry on M is covatiant under U Vec( M ). NA differential geometry on M should be covariant under U Vec F ( M ). In pactice: U Vec( M )-module algebra A (functions, forms, tensors) and a , b ∈ A , u ∈ Vec( M ) u ( ab ) = u ( a ) b + au ( b ) , Lie derivative, Leibinz rule (coproduct) . The twist: U Vec( M ) → U Vec F ( M ) and A → A ⋆ with ab → a ⋆ b = f α ( a ) · f α ( b ) . Then A ⋆ is a U Vec F ( M )-module algebra: ξ ( a ⋆ b ) = ξ (1) ( a ) ⋆ ξ (2) ( b ) , for ξ ∈ U Vec F ( M ) and using the twisted coproduct ∆ F ξ = ξ (1) ⊗ ξ (2) .

Commutativity: a ⋆ b = f α ( a ) · f α ( b ) = R α ( b ) ⋆ R α ( a ) =: α b ⋆ α a Associativity: ( a ⋆ b ) ⋆ c = φ 1 a ⋆ ( φ 2 b ⋆ φ 3 c ). Functions: C ∞ ( M ) → C ∞ ( M ) ⋆ f α ( f ) · f α ( g ) f ⋆ g = (2.5) + i κ R µνρ p ν ∂ ρ f · ∂ µ g + · · · , ∂ µ f · ˜ ∂ µ g − ˜ ∂ µ f · ∂ µ g f · g + i � � � = 2 , x ν ] = 2 i κ R µνρ p ρ , [ x µ ⋆ [ x µ ⋆ , p ν ] = i � δ µν , [ p µ ⋆ , p ν ] = 0, , x ρ ] = ℓ 3 [ x µ ⋆ , x ν ⋆ s R µνρ .

NA tensor calculus Forms: Ω ♯ ( M ) → Ω ♯ ( M ) ⋆ ω ∧ ⋆ η = f α ( ω ) ∧ f α ( η ) , (2.6) f ⋆ d x A = d x C ⋆ δ A C f − i κ R AB � � C ∂ B f , with non-vanishing components R x µ , x ν x ρ = R µνρ . Basis 1-forms ˜ ( d x A ∧ ⋆ d x B ) ∧ ⋆ d x C = φ 1 ( d x A ) ∧ ⋆ � φ 2 ( d x B ) ∧ ⋆ φ 3 ( d x C ) � = d x A ∧ ⋆ ( d x B ∧ ⋆ d x C ) = d x A ∧ d x B ∧ d x C . Exterior derivative d : d 2 = 0 and the undeformed Leibniz rule d ( ω ∧ ⋆ η ) = d ω ∧ ⋆ η + ( − 1) | ω | ω ∧ ⋆ d η. (2.7) Duality, ⋆ -pairing: � f α ( ω ) , f α ( u ) � � ω , u � ⋆ = (2.8) .

NA tensor calculus: Lie derivative ⋆ -Lie drivative: L ⋆ u ( T ) = L f α ( u ) ( f α ( T )) , (2.9) ¯ ¯ ¯ ¯ L ⋆ u ( ω ∧ ⋆ η ) = L ⋆ φ 2 ω ) ∧ ⋆ φ 3 η + α ( φ 1 ¯ ϕ 1 ω ) ∧ ⋆ L ⋆ φ 3 ¯ ϕ 3 η ) , φ 1 u ( ϕ 2 u ) ( ¯ α ( ¯ φ 2 ¯ [ L ⋆ u , L ⋆ v ] • = [ f α L ⋆ u , f α L ⋆ v ] = L ⋆ [ u , v ] ⋆ , � f α ( u ) , f α ( v ) � with [ u , v ] ⋆ = and ¯ ¯ ¯ ¯ ¯ ¯ φ 1 u , φ 2 v ] ⋆ , φ 3 z � α ( φ 1 ¯ ϕ 1 v ) , [ α ( φ 2 ¯ ϕ 2 u ) , φ 3 ¯ ϕ 3 z ] ⋆ � � � � � u , [ v , z ] ⋆ ⋆ = [ ⋆ + ⋆ . Relation of L ⋆ u with diffeomorphism symmetry in space-time needs to be understood. ⋆ -Lie derivative generates ”twisted, braided” diffeomorphism symmetry. This symmetry has the L ∞ structure. Work in progress with G. Giotopoulos, V. Radovanovi´ c and R. Szabo.

NA differential geometry: connection, torsion, curvature ⋆ -connection: ∇ ⋆ : Vec ⋆ Vec ⋆ ⊗ ⋆ Ω 1 − → ⋆ ∇ ⋆ u , u �− → (2.10) � ¯ ¯ ¯ ∇ ⋆ ( u ⋆ f ) φ 1 ∇ ⋆ ( φ 2 u ) � φ 3 f + u ⊗ ⋆ d f , = (2.11) ⋆ the right Leibniz rule, for u ∈ Vec ⋆ and f ∈ A ⋆ . In particular: ∇ ⋆ ∂ A =: ∂ B ⊗ ⋆ Γ B A =: ∂ B ⊗ ⋆ (Γ B AC ⋆ d x C ) . (2.12) d ∇ ⋆ ( ∂ A ⊗ ⋆ ω A ) = ∂ A ⊗ ⋆ ( d ω A + Γ A B ∧ ⋆ ω B ) , for ω A ∈ Ω ♯ ⋆ . Torsion: T ⋆ := d ∇ ⋆ � ∂ A ⊗ ⋆ d x A � : Vec ⋆ ⊗ ⋆ Vec ⋆ → Vec ⋆ , T ⋆ ( ∂ A , ∂ B ) = ∂ C ⋆ (Γ C AB − Γ C BA ) =: ∂ C ⋆ T C AB . Torsion-free condition: Γ C AB = Γ C BA . Curvature: R ⋆ := d ∇ ⋆ • d ∇ ⋆ : Vec ⋆ − → Vec ⋆ ⊗ ⋆ Ω 2 ⋆ , R ⋆ ( ∂ A ) = ∂ C ⊗ ⋆ ( d Γ C A + Γ C B ∧ ⋆ Γ B A ) = ∂ C ⊗ ⋆ R C A ,