Quantization of non-geometric flux backgrounds Dionysios Mylonas - PowerPoint PPT Presentation

Quantization of non-geometric flux backgrounds Dionysios Mylonas School of Mathematical and Computer Science Heriot-Watt university, Edinburgh EMPG November 14, 2012 Based on: D.M, P. Schupp and R. Szabo, JHEP 1209 (2012) 012, [arXiv:1207.0926]

Background and motivation σ -models Deformation quantization Convolution product quantization Outline The past: Background and motivation. From a membrane σ -model to string theory on the boundary. Deformation quantization a l´ a Kontsevich. Convolution product quantization. Recap. The future: ? Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ -models Deformation quantization Convolution product quantization This is the end... You have... 36d 9hr and 25min left to the END OF THE WORLD! Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ -models Deformation quantization Convolution product quantization This is the end... You have... 36d 9hr and 25min left to the END OF THE WORLD! “Two things are infinite: the universe and human stupidity; and I’m not sure about the the universe.” Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation Open and closed string NCity σ -models Toroidal backgrounds Deformation quantization CFT analysis Convolution product quantization Open and closed strings Open string with constant B-field � � X i ( τ, σ ) , X j ( τ, σ ′ ) σ = σ ′ =0 , 2 π = i θ ij where θ = − 2 π α ′ (1 + F 2 ) − 1 F and F = B − F . (Seiberg & Witten, Chu & Ho) Commutator not well defined for closed strings. Jacobiator 3-bracket: �� � � [ X i , X j , X k ] := lim X i ( τ, σ 1 ) , X j ( τ, σ 2 ) , X k ( τ, σ 3 ) +cyclic σ i → σ For the linearized SU (2) WZW model with H = dB � = 0 [ X i , X j , X k ] ∼ H ijk , i.e. the target space is non-associative . (Blumenhagen & Plauschinn, 1010.1263) Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation Open and closed string NCity σ -models Toroidal backgrounds Deformation quantization CFT analysis Convolution product quantization T-duality frames Consider T 3 with non-vanishing H -flux. T x 3 T x 2 T x 1 f -flux Q -flux H -flux R -flux (non-geometric) (nilmanifold) (T-fold) � � � � X i ( τ, σ ) , X j ( τ, σ ′ ) X i ( τ, σ ) , X j ( τ, σ ′ ) = 0 � = 0 T x 2 � � � � − → X i ( τ, σ ) , X ∗ X i ( τ, σ ) , X ∗ j ( τ, σ ′ ) � = 0 j ( τ, σ ′ ) = 0 where X ∗ i = X i,L − X i,R ∈ M ∗ . Should use doubled geometry . In this context the same type of nonassociativity was found and a NA target space algebra on R -space was proposed. (L¨ ust, 1010.1361) Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation Open and closed string NCity σ -models Toroidal backgrounds Deformation quantization CFT analysis Convolution product quantization Boundary conditions Closed str on M × M ∗ e 2 π i θ X ( τ, σ ) X ( τ, σ + 2 π ) = e 2 π i θ X ∗ ( τ, σ ) X ∗ ( τ + 2 π, σ ) = where θ = − n H and n ∈ Z the dual momentum along the T-dualised direction. The situation resembles the open string case: NCFT on D-branes T → FT on intersecting D-branes Are there some kind of closed string “D-branes”? Are they dynamical solutions in doubled gravity? NA SW map? (L¨ ust, 1010.1361) Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation Open and closed string NCity σ -models Toroidal backgrounds Deformation quantization CFT analysis Convolution product quantization Linearized CFT Flat space with constantl H -flux [ X i , X j , X k ] = i α θ ijk where θ ijk ∼ H . α = 0 for the H -flux background α = 1 after an odd number of T-duality transformations. 3-product conjecure for constant R -flux � π 2 � � � 2 θ ijk ∂ x 1 i ∂ x 2 j ∂ x 3 � ( f 1 ⋄ f 2 ⋄ f 3 )( x ) := exp f 1 ( x 1 ) f 2 ( x 2 ) f 3 ( x 3 ) � k x i = x (Blumenhagen et al, 1106.0316) Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation More background σ -models Model building Deformation quantization Reduction Convolution product quantization Boundary conditions Definitions Poisson manifold Given a bivector field Θ = 1 2 Θ ij ( x ) ∂ i ∧ ∂ j on a smooth manifold M a skew-symmetric bracket {− , −} Θ can be defined. This is a Poisson structure if the Schouten-Nijehuis bracket [Θ , Θ] S is zero. A quasi-Poisson structure doesn’t satisfy the Jacobi identity. Lie algebroid A Lie algebroid is a vector bundle E → M endowed with a Lie bracket [ − , − ] E on smooth sections of E and an anchor map ρ : E → T M . The tangent map to ρ is a Lie algebra homomorphism. Courant algebroid : E is further equipped with a metric �− , −� and a Jacobiator. Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation More background σ -models Model building Deformation quantization Reduction Convolution product quantization Boundary conditions Poisson σ -model A Poisson manifold M (= symplectic Lie 1 -algebroid with the canonical symplectic structure on T ∗ M ). A 2d string worldsheet Σ 2 . A differential form on Σ 2 is given by the embedding X = ( X i ) : Σ 2 → M and an auxilliary 1-form field on Σ 2 ξ = ( ξ i ) ∈ Ω 1 (Σ 2 , X ∗ T ∗ M ) , i ∈ { 1 , . . . , d } . Action � � � ξ i ∧ d X i + 1 S (1) = 2 Θ ij ( X ) ξ i ∧ ξ j Σ 2 This is a topological field theory on C ∞ ( T Σ 2 , T ∗ M ) . (Cattaneo & Felder, math.QA/0102108) Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation More background σ -models Model building Deformation quantization Reduction Convolution product quantization Boundary conditions Courant σ -model Courant algebroid (= symplectic Lie 2 -algebroid). A 3d membrane worldvolume Σ 3 . α = ( α I ) ∈ Ω 1 (Σ 3 , X ∗ E ) and φ = ( φ i ) ∈ Ω 2 (Σ 3 , X ∗ T ∗ M ) . Choose a local basis of sections { ψ I } of E → M s..t. the fibre metric h IJ := � ψ I , ψ J � is constant, I ∈ { 1 , . . . , 2 d } . Def. the anchor matrix ρ ( ψ I ) = P Ii ( x ) ∂ i , and the 3-form T IJK ( x ) := [ ψ I , ψ J , ψ K ] E . Action � � 1 S (2) = 2 h IJ α I ∧ d α J − P Ii ( X ) φ i ∧ α I + φ i ∧ d X i + Σ 3 6 T IJK ( X ) α I ∧ α J ∧ α K � 1 + Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation More background σ -models Model building Deformation quantization Reduction Convolution product quantization Boundary conditions H-space Standard Courant algebroid C = TM ⊕ T ∗ M twisted by 6 H ijk ( x ) d x i ∧ d x j ∧ d x k . H = 1 Structure maps: - Antisymmetrized H -twisted Courant-Dorfman bracket. - The usual pairing between TM and T ∗ M and ρ =trivial. = M × R d to keep only H -flux. Assume TM ∼ Write ( α I ) := ( α 1 , . . . , α d , ξ 1 , . . . , ξ d ) and integrate out φ i . H -twisted Poisson σ -model � � � ξ i ∧ d X i + 1 S (1) � 2 Θ ij ( X ) ξ i ∧ ξ j = Σ 2 � 1 6 H ijk ( X ) d X i ∧ d X j ∧ d X k + Σ 3 Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation More background σ -models Model building Deformation quantization Reduction Convolution product quantization Boundary conditions H-space Standard Courant algebroid C = TM ⊕ T ∗ M twisted by 6 H ijk ( x ) d x i ∧ d x j ∧ d x k . H = 1 Structure maps: - Antisymmetrized H -twisted Courant-Dorfman bracket. - The usual pairing between TM and T ∗ M and ρ =trivial. = M × R d to keep only H -flux. Assume TM ∼ Write ( α I ) := ( α 1 , . . . , α d , ξ 1 , . . . , ξ d ) and integrate out φ i . H -twisted Poisson σ -model � � � ξ i ∧ d X i + 1 S (1) � 2 Θ ij ( X ) ξ i ∧ ξ j = Σ 2 � 1 6 H ijk ( X ) d X i ∧ d X j ∧ d X k + Σ 3 Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation More background σ -models Model building Deformation quantization Reduction Convolution product quantization Boundary conditions H-space Standard Courant algebroid C = TM ⊕ T ∗ M twisted by 6 H ijk ( x ) d x i ∧ d x j ∧ d x k . H = 1 Structure maps: - Antisymmetrized H -twisted Courant-Dorfman bracket. - The usual pairing between TM and T ∗ M and ρ =trivial. = M × R d to keep only H -flux. Assume TM ∼ Write ( α I ) := ( α 1 , . . . , α d , ξ 1 , . . . , ξ d ) and integrate out φ i . H -twisted Poisson σ -model � � � ξ i ∧ d X i + 1 S (1) � 2 Θ ij ( X ) ξ i ∧ ξ j = Σ 2 � 1 6 H ijk ( X ) d X i ∧ d X j ∧ d X k + Σ 3 Dionysios Mylonas Quantization of non-geometric flux backgrounds