Non-associative Deformations of Geometry in Double Field Theory - PowerPoint PPT Presentation

Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT Non-associative Deformations of Geometry in Double Field Theory Michael Fuchs Max Planck Institut f¨ ur Physik M¨ unchen based on JHEP 04(2014)141 by R. Blumenhagen, MF, F. Haßler, D. L¨ ust, R. Sun 29th IMPRS Workshop July 7th, 2014

Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT Outline • Deformation Quantization • T-Duality and non-associativity in String Theory • Non-associative Deformations of Geometry in DFT

Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT Canonical Quantization Replace Poisson-bracket by commutator: { x , p } PB = 1 → [ x , p ] = i � p = − i � ∂ Fulfilled for instance by operators: ˆ ∂ x

Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT Deformation Quantization: No operators, instead change multiplication law: Replace f · g by � T � 0 � ∂ x f � � ∂ x g � f ⋆ g := f · g + i � 1 − 1 2 ∂ p f 0 ∂ p g Insert coordinate and momentum:  x ⋆ p = x · p + i � 2   [ x , p ] = x ⋆ p − p ⋆ x = i � p ⋆ x = p · x − i � 2

Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT Take home message: Commutation relations realized by deformed product 9709040 f ⋆ g := f · g + i � 2 ω ij ∂ i f ∂ j g + O ( � 2 )

Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT String Theory Fundamental objects not points, but strings Strings must live in 10D → compactify!

Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT T-Duality Closed strings wind around compactified dimensions: T-Duality p i ← → momenta p i winding momenta ˜ � � T-Duality coordinate x i ← → winding coordinate ˜ x i

Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT Non-geometric Fluxes T-Duality mixes G and B ⇒ change of geometry T k T j T i F ij k Q i jk R ijk ← → ← → ← → H ijk

Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT Non-associative Geometry Blumenhagen, L¨ ust, Plauschinn et alii: [ x a , x b ] ∼ = R abc p c “fuzzy” geometry due to Heisenberg uncertainty: ∆ x a ∆ x b ∼ = � [ x a , x b ] � � = 0

Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT non-vanishing Jacobi identity! ˆ = non-associative operators! Not possible in ordinary quantum mechanics! Deformed product vanishes for observables by momentum conservation! 1106.0316 by Blumenhagen, Deser, L¨ ust, Plauschinn, Rennecke Our work: Investigate in double field theory how non-associativity vanishes!

Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT Double Field Theory � normal � Combine in 2D vector winding � p i � � ∂ i � � x i � P M = ∂ M = X M = ˜ ˜ p i ∂ i x i ˜ ⇒ Coordinates and winding on equal footage! BUT: Constraints needed for consistency!

Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT Non-associative Deformations of Geometry in DFT Translate deformed product into DFT: ∂ A f ∂ B g ∂ C h f △ g △ h = f · g · h + F ABC � �� � contains H,f,Q,R We found: Deformation vanishes by consistency constraints! Deformation in physical situations (action) ˆ = integration: � � F ABC D A f D B g D C h PI Z AB = − f D A g D B h ���� DFT DFT Bianchi Z AB =0!

Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT Another deformation: DFT allows for another deformation: f △ g △ h = f · g · h + ˘ F ABC ∂ A f ∂ B g ∂ C h ⇒ Generalization of open strings in B-field background 9812219 No reason to vanish! Integral: � � F ABC D A f D B g D C h PI ˘ G AB = − f D A g D B h ���� DFT DFT eom : G AB =0!

Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT Conclusion associativity of observables preserved by crucial ingredients of double field theory ˘ F ABC ∂ A f ∂ B g ∂ C h F ABC ∂ A f ∂ B g ∂ C h equation of motion Bianchi identity continuity equation closure of algebra

Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT Outlook Future research directions: • Derive higher orders of the product (ongoing) • Non-associativity in Hamiltionian formalism