Generalized Geometry and Double Field Theory: a toy Model Patrizia - PowerPoint PPT Presentation

Geometric formulation of the rigid rotator on configuration space SU (2) g − 1 ˙ g = i ( y 0 ˙ y i − y i ˙ y 0 + ǫ i jk y j ˙ y k ) σ i = i ˙ Q i σ i Since the Lagrangian reads 2 ( y 0 ˙ y j − y j ˙ y 0 + ǫ j kl y k ˙ y l )( y 0 ˙ y r − y r ˙ y 0 + ǫ r pq y p ˙ Q j ˙ 2 ˙ L 0 = 1 y q ) δ ir = 1 Q r δ jr Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

Geometric formulation of the rigid rotator on configuration space SU (2) g − 1 ˙ g = i ( y 0 ˙ y i − y i ˙ y 0 + ǫ i jk y j ˙ y k ) σ i = i ˙ Q i σ i Since the Lagrangian reads 2 ( y 0 ˙ y j − y j ˙ y 0 + ǫ j kl y k ˙ y l )( y 0 ˙ y r − y r ˙ y 0 + ǫ r pq y p ˙ Q j ˙ 2 ˙ L 0 = 1 y q ) δ ir = 1 Q r δ jr Tangent bundle coordinates: ( Q i , ˙ Q i ) Q i = 0 ¨ d g − 1 dg � � Equations of motion or , = 0 dt dt Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

Geometric formulation of the rigid rotator Cotangent bundle T ∗ SU (2) - Coordinates: ( Q i , I i ) Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

Geometric formulation of the rigid rotator Cotangent bundle T ∗ SU (2) - Coordinates: ( Q i , I i ) with I i the conjugate momenta I j = ∂ L 0 Q j = δ jr ˙ Q r ∂ ˙ Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

Geometric formulation of the rigid rotator Cotangent bundle T ∗ SU (2) - Coordinates: ( Q i , I i ) with I i the conjugate momenta I j = ∂ L 0 Q j = δ jr ˙ Q r ∂ ˙ Hamiltonian H 0 = 1 2 I i I j δ ij PB’s: { y i , y j } = 0 k I k { I i , I j } = ǫ ij j y 0 + ǫ i { y i , I j } − δ i jk y k = { g , I j } = − i σ j g or g − 1 ˙ ˙ EOM: I i = 0 , g = iI i σ i Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

Geometric formulation of the rigid rotator Cotangent bundle T ∗ SU (2) - Coordinates: ( Q i , I i ) with I i the conjugate momenta I j = ∂ L 0 Q j = δ jr ˙ Q r ∂ ˙ Hamiltonian H 0 = 1 2 I i I j δ ij PB’s: { y i , y j } = 0 k I k { I i , I j } = ǫ ij j y 0 + ǫ i { y i , I j } − δ i jk y k = { g , I j } = − i σ j g or g − 1 ˙ ˙ EOM: I i = 0 , g = iI i σ i Fiber coordinates I i are associated to the angular momentum components and the base space coordinates ( y 0 , y i ) to the orientation of the rotator. I i are constants of the motion, g undergoes a uniform precession. Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

The cotangent bundle T ∗ SU (2) Remarks: As a group T ∗ SU (2) ≃ SU (2) ⋉ R 3 with Lie algebra [ L i , L j ] = ǫ k [ L i , T j ] = ǫ k ij L k [ T i , T j ] = 0 ij T k Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

The cotangent bundle T ∗ SU (2) Remarks: As a group T ∗ SU (2) ≃ SU (2) ⋉ R 3 with Lie algebra [ L i , L j ] = ǫ k [ L i , T j ] = ǫ k ij L k [ T i , T j ] = 0 ij T k The non-trivial Poisson bracket is the Kirillov-Souriau Konstant bracket on g ∗ Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

The cotangent bundle T ∗ SU (2) Remarks: As a group T ∗ SU (2) ≃ SU (2) ⋉ R 3 with Lie algebra [ L i , L j ] = ǫ k [ L i , T j ] = ǫ k ij L k [ T i , T j ] = 0 ij T k The non-trivial Poisson bracket is the Kirillov-Souriau Konstant bracket on g ∗ In [Marmo Simoni Stern ’93] the carrier space of the dynamics has been generalized to SL (2 , C ), the Drinfeld double of SU (2). Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

The cotangent bundle T ∗ SU (2) Remarks: As a group T ∗ SU (2) ≃ SU (2) ⋉ R 3 with Lie algebra [ L i , L j ] = ǫ k [ L i , T j ] = ǫ k ij L k [ T i , T j ] = 0 ij T k The non-trivial Poisson bracket is the Kirillov-Souriau Konstant bracket on g ∗ In [Marmo Simoni Stern ’93] the carrier space of the dynamics has been generalized to SL (2 , C ), the Drinfeld double of SU (2). In [Rajeev ’89 , Rajeev, Sparano P.V. ’93] the same has been done for chiral & WZW model Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

The cotangent bundle T ∗ SU (2) Remarks: As a group T ∗ SU (2) ≃ SU (2) ⋉ R 3 with Lie algebra [ L i , L j ] = ǫ k [ L i , T j ] = ǫ k ij L k [ T i , T j ] = 0 ij T k The non-trivial Poisson bracket is the Kirillov-Souriau Konstant bracket on g ∗ In [Marmo Simoni Stern ’93] the carrier space of the dynamics has been generalized to SL (2 , C ), the Drinfeld double of SU (2). In [Rajeev ’89 , Rajeev, Sparano P.V. ’93] the same has been done for chiral & WZW model Here we introduce a dual dynamical model on the dual group of SU (2) and generalize to field theory. Only there, the duality transformation will be a symmetry. Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

The cotangent bundle T ∗ SU (2) Remarks: As a group T ∗ SU (2) ≃ SU (2) ⋉ R 3 with Lie algebra [ L i , L j ] = ǫ k [ L i , T j ] = ǫ k ij L k [ T i , T j ] = 0 ij T k The non-trivial Poisson bracket is the Kirillov-Souriau Konstant bracket on g ∗ In [Marmo Simoni Stern ’93] the carrier space of the dynamics has been generalized to SL (2 , C ), the Drinfeld double of SU (2). In [Rajeev ’89 , Rajeev, Sparano P.V. ’93] the same has been done for chiral & WZW model Here we introduce a dual dynamical model on the dual group of SU (2) and generalize to field theory. Only there, the duality transformation will be a symmetry. Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT The algebra is spanned by e i = σ i / 2 , b i = ie i [ e i , e j ] = i ǫ k [ e i , b j ] = i ǫ k [ b i , b j ] = − i ǫ k ij e k , ij b k , ij e k Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT The algebra is spanned by e i = σ i / 2 , b i = ie i [ e i , e j ] = i ǫ k [ e i , b j ] = i ǫ k [ b i , b j ] = − i ǫ k ij e k , ij b k , ij e k Non-degenerate invariant scalar products: < u , v > = 2 Im ( Tr ( uv )) , ∀ u , v ∈ sl (2 , C ) Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT The algebra is spanned by e i = σ i / 2 , b i = ie i [ e i , e j ] = i ǫ k [ e i , b j ] = i ǫ k [ b i , b j ] = − i ǫ k ij e k , ij b k , ij e k Non-degenerate invariant scalar products: < u , v > = 2 Im ( Tr ( uv )) , ∀ u , v ∈ sl (2 , C ) and ( u , v ) = 2 Re ( Tr ( uv )) , ∀ u , v ∈ sl (2 , C ) Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT The algebra is spanned by e i = σ i / 2 , b i = ie i [ e i , e j ] = i ǫ k [ e i , b j ] = i ǫ k [ b i , b j ] = − i ǫ k ij e k , ij b k , ij e k Non-degenerate invariant scalar products: < u , v > = 2 Im ( Tr ( uv )) , ∀ u , v ∈ sl (2 , C ) and ( u , v ) = 2 Re ( Tr ( uv )) , ∀ u , v ∈ sl (2 , C ) w.r.t. the first one (Cartan-Killing) we have two maximal isotropic subspaces e j > = 0 , e j > = δ j e i , ˜ < e i , e j > = < ˜ < e i , ˜ i e i = b i − ǫ ij 3 e j . with ˜ Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT The algebra is spanned by e i = σ i / 2 , b i = ie i [ e i , e j ] = i ǫ k [ e i , b j ] = i ǫ k [ b i , b j ] = − i ǫ k ij e k , ij b k , ij e k Non-degenerate invariant scalar products: < u , v > = 2 Im ( Tr ( uv )) , ∀ u , v ∈ sl (2 , C ) and ( u , v ) = 2 Re ( Tr ( uv )) , ∀ u , v ∈ sl (2 , C ) w.r.t. the first one (Cartan-Killing) we have two maximal isotropic subspaces e j > = 0 , e j > = δ j e i , ˜ < e i , e j > = < ˜ < e i , ˜ i e i = b i − ǫ ij 3 e j . { e i } , { ˜ e i } both subalgebras with with ˜ e k + ie k f ki [ e i , e j ] = i ǫ k e i , e j ] = i ǫ i e i , ˜ e j ] = if ij e k ij e k , [˜ jk ˜ j , [˜ k ˜ e i } span the Lie algebra of SB (2 , C ), the dual group of SU (2) with { ˜ f ij k = ǫ ijl ǫ l 3 k Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT The algebra is spanned by e i = σ i / 2 , b i = ie i [ e i , e j ] = i ǫ k [ e i , b j ] = i ǫ k [ b i , b j ] = − i ǫ k ij e k , ij b k , ij e k Non-degenerate invariant scalar products: < u , v > = 2 Im ( Tr ( uv )) , ∀ u , v ∈ sl (2 , C ) and ( u , v ) = 2 Re ( Tr ( uv )) , ∀ u , v ∈ sl (2 , C ) w.r.t. the first one (Cartan-Killing) we have two maximal isotropic subspaces e j > = 0 , e j > = δ j e i , ˜ < e i , e j > = < ˜ < e i , ˜ i e i = b i − ǫ ij 3 e j . { e i } , { ˜ e i } both subalgebras with with ˜ e k + ie k f ki [ e i , e j ] = i ǫ k e i , e j ] = i ǫ i e i , ˜ e j ] = if ij e k ij e k , [˜ jk ˜ j , [˜ k ˜ e i } span the Lie algebra of SB (2 , C ), the dual group of SU (2) with { ˜ f ij k = ǫ ijl ǫ l 3 k Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT ( su (2), sb (2 , C )) is a Lie bialgebra Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT ( su (2), sb (2 , C )) is a Lie bialgebra The role of su (2) and its dual algebra can be interchanged Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT ( su (2), sb (2 , C )) is a Lie bialgebra The role of su (2) and its dual algebra can be interchanged The triple ( sl (2 , C ) , su (2) , sb (2 , C )) is called a Manin triple Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT ( su (2), sb (2 , C )) is a Lie bialgebra The role of su (2) and its dual algebra can be interchanged The triple ( sl (2 , C ) , su (2) , sb (2 , C )) is called a Manin triple ⊳ g ∗ , D is the Drinfeld double, G , G ∗ are dual groups Given d = g ⊲ Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT ( su (2), sb (2 , C )) is a Lie bialgebra The role of su (2) and its dual algebra can be interchanged The triple ( sl (2 , C ) , su (2) , sb (2 , C )) is called a Manin triple ⊳ g ∗ , D is the Drinfeld double, G , G ∗ are dual groups Given d = g ⊲ For f ij D → T ∗ G k = 0 For c k ij = 0 D → T ∗ G ∗ Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT ( su (2), sb (2 , C )) is a Lie bialgebra The role of su (2) and its dual algebra can be interchanged The triple ( sl (2 , C ) , su (2) , sb (2 , C )) is called a Manin triple ⊳ g ∗ , D is the Drinfeld double, G , G ∗ are dual groups Given d = g ⊲ For f ij D → T ∗ G k = 0 For c k ij = 0 D → T ∗ G ∗ Therefore D generalizes both the cotangent bundle of SU (2) and of SB (2 , C ); Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT ( su (2), sb (2 , C )) is a Lie bialgebra The role of su (2) and its dual algebra can be interchanged The triple ( sl (2 , C ) , su (2) , sb (2 , C )) is called a Manin triple ⊳ g ∗ , D is the Drinfeld double, G , G ∗ are dual groups Given d = g ⊲ For f ij D → T ∗ G k = 0 For c k ij = 0 D → T ∗ G ∗ Therefore D generalizes both the cotangent bundle of SU (2) and of SB (2 , C ); The bi-algebra structure induces Poisson structures on the double group manifold [ , ] sb (2 , C ) → ( F ( SU (2)) , ˜ [ , ] su (2) → ( F ( SB (2 , C )) , Λ); Λ) Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT ( su (2), sb (2 , C )) is a Lie bialgebra The role of su (2) and its dual algebra can be interchanged The triple ( sl (2 , C ) , su (2) , sb (2 , C )) is called a Manin triple ⊳ g ∗ , D is the Drinfeld double, G , G ∗ are dual groups Given d = g ⊲ For f ij D → T ∗ G k = 0 For c k ij = 0 D → T ∗ G ∗ Therefore D generalizes both the cotangent bundle of SU (2) and of SB (2 , C ); The bi-algebra structure induces Poisson structures on the double group manifold [ , ] sb (2 , C ) → ( F ( SU (2)) , ˜ [ , ] su (2) → ( F ( SB (2 , C )) , Λ); Λ) which reduce to KSK brackets on coadjoint orbits of G , G ∗ when f ij k = 0 , c k ij = 0 resp. Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT ( su (2), sb (2 , C )) is a Lie bialgebra The role of su (2) and its dual algebra can be interchanged The triple ( sl (2 , C ) , su (2) , sb (2 , C )) is called a Manin triple ⊳ g ∗ , D is the Drinfeld double, G , G ∗ are dual groups Given d = g ⊲ For f ij D → T ∗ G k = 0 For c k ij = 0 D → T ∗ G ∗ Therefore D generalizes both the cotangent bundle of SU (2) and of SB (2 , C ); The bi-algebra structure induces Poisson structures on the double group manifold [ , ] sb (2 , C ) → ( F ( SU (2)) , ˜ [ , ] su (2) → ( F ( SB (2 , C )) , Λ); Λ) which reduce to KSK brackets on coadjoint orbits of G , G ∗ when f ij k = 0 , c k ij = 0 resp. Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Poisson brackets What are these Poisson brackets? Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Poisson brackets What are these Poisson brackets? The double group SL (2 , C ) can be endowed with PB’s which generalize both those of T ∗ SU (2) and of T ∗ SB (2 C ) [[Semenov-Tyan-Shanskii ’91, Alekseev-Malkin ’94]] { γ 1 , γ 2 } = − γ 1 γ 2 r ∗ − r γ 1 γ 2 whith γ 1 = γ ⊗ 1 , γ 2 = 1 ⊗ γ 2 ; Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Poisson brackets What are these Poisson brackets? The double group SL (2 , C ) can be endowed with PB’s which generalize both those of T ∗ SU (2) and of T ∗ SB (2 C ) [[Semenov-Tyan-Shanskii ’91, Alekseev-Malkin ’94]] { γ 1 , γ 2 } = − γ 1 γ 2 r ∗ − r γ 1 γ 2 whith γ 1 = γ ⊗ 1 , γ 2 = 1 ⊗ γ 2 ; e i ⊗ e i , r ∗ = − e i ⊗ ˜ e i r = ˜ is the classical Yang Baxter matrix Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Poisson brackets What are these Poisson brackets? The double group SL (2 , C ) can be endowed with PB’s which generalize both those of T ∗ SU (2) and of T ∗ SB (2 C ) [[Semenov-Tyan-Shanskii ’91, Alekseev-Malkin ’94]] { γ 1 , γ 2 } = − γ 1 γ 2 r ∗ − r γ 1 γ 2 whith γ 1 = γ ⊗ 1 , γ 2 = 1 ⊗ γ 2 ; e i ⊗ e i , r ∗ = − e i ⊗ ˜ e i r = ˜ is the classical Yang Baxter matrix The group D equipped with the Poisson bracket is also called the Heisenberg double Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Poisson brackets What are these Poisson brackets? The double group SL (2 , C ) can be endowed with PB’s which generalize both those of T ∗ SU (2) and of T ∗ SB (2 C ) [[Semenov-Tyan-Shanskii ’91, Alekseev-Malkin ’94]] { γ 1 , γ 2 } = − γ 1 γ 2 r ∗ − r γ 1 γ 2 whith γ 1 = γ ⊗ 1 , γ 2 = 1 ⊗ γ 2 ; e i ⊗ e i , r ∗ = − e i ⊗ ˜ e i r = ˜ is the classical Yang Baxter matrix The group D equipped with the Poisson bracket is also called the Heisenberg double On writing γ as γ = ˜ gg it can be shown that these brackets are compatible with { ˜ g 1 , ˜ g 2 } = − [ r , ˜ g 1 ˜ g 2 ] , g 2 r ∗ g 1 { ˜ g 1 , g 2 } = − ˜ g 1 rg 2 , { g 1 , ˜ g 2 } = − ˜ { g 1 , g 2 } = [ r ∗ , g 1 g 2 ] , Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Poisson brackets What are these Poisson brackets? The double group SL (2 , C ) can be endowed with PB’s which generalize both those of T ∗ SU (2) and of T ∗ SB (2 C ) [[Semenov-Tyan-Shanskii ’91, Alekseev-Malkin ’94]] { γ 1 , γ 2 } = − γ 1 γ 2 r ∗ − r γ 1 γ 2 whith γ 1 = γ ⊗ 1 , γ 2 = 1 ⊗ γ 2 ; e i ⊗ e i , r ∗ = − e i ⊗ ˜ e i r = ˜ is the classical Yang Baxter matrix The group D equipped with the Poisson bracket is also called the Heisenberg double On writing γ as γ = ˜ gg it can be shown that these brackets are compatible with { ˜ g 1 , ˜ g 2 } = − [ r , ˜ g 1 ˜ g 2 ] , g 2 r ∗ g 1 { ˜ g 1 , g 2 } = − ˜ g 1 rg 2 , { g 1 , ˜ g 2 } = − ˜ { g 1 , g 2 } = [ r ∗ , g 1 g 2 ] , Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Poisson brackets e i ⊗ e i , In the limit λ → 0, with r = λ ˜ Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Poisson brackets e i ⊗ e i , ˜ g ( λ ) = 1 + i λ I i e i + O ( λ 2 ) In the limit λ → 0, with r = λ ˜ Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Poisson brackets e i ⊗ e i , ˜ g ( λ ) = 1 + i λ I i e i + O ( λ 2 ) In the limit λ → 0, with r = λ ˜ g = y 0 σ 0 + iy i σ i Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Poisson brackets e i ⊗ e i , ˜ g ( λ ) = 1 + i λ I i e i + O ( λ 2 ) In the limit λ → 0, with r = λ ˜ g = y 0 σ 0 + iy i σ i we obtain ǫ k { I i , I j } = ij I k Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Poisson brackets e i ⊗ e i , ˜ g ( λ ) = 1 + i λ I i e i + O ( λ 2 ) In the limit λ → 0, with r = λ ˜ g = y 0 σ 0 + iy i σ i we obtain ǫ k { I i , I j } = ij I k { I i , y 0 } iy j δ ij { I i , y j } = iy 0 δ j i − ǫ j ik y k = Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Poisson brackets e i ⊗ e i , ˜ g ( λ ) = 1 + i λ I i e i + O ( λ 2 ) In the limit λ → 0, with r = λ ˜ g = y 0 σ 0 + iy i σ i we obtain ǫ k { I i , I j } = ij I k { I i , y 0 } iy j δ ij { I i , y j } = iy 0 δ j i − ǫ j ik y k = { y 0 , y j } { y i , y j } = 0 + O ( λ ) = which reproduce correctly the canonical Poisson brackets on the cotangent bundle of SU (2). Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Poisson brackets e i ⊗ e i , ˜ g ( λ ) = 1 + i λ I i e i + O ( λ 2 ) In the limit λ → 0, with r = λ ˜ g = y 0 σ 0 + iy i σ i we obtain ǫ k { I i , I j } = ij I k { I i , y 0 } iy j δ ij { I i , y j } = iy 0 δ j i − ǫ j ik y k = { y 0 , y j } { y i , y j } = 0 + O ( λ ) = which reproduce correctly the canonical Poisson brackets on the cotangent bundle of SU (2). Consider now r ∗ as an independent solution of the Yang Baxter equation ρ = µ e k ⊗ e k and expand g ∈ SU (2) as a function of the parameter µ : g = 1 + i µ ˜ Ie i + O ( µ 2 ) By repeating the same analysis as above we get back the canonical Poisson structure on T ∗ SB (2 , C ), with position coordinates and momenta now interchanged. In particular we note { ˜ I i , ˜ I j } = f ij k ˜ I k Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Poisson brackets Last but not least, it is possible to consider a different Poisson structure on the double [Semenov] , given by 2 [ γ 1 ( r ∗ − r ) γ 2 − γ 2 ( r ∗ − r ) γ 1 ] ; { γ 1 , γ 2 } = λ Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Poisson brackets Last but not least, it is possible to consider a different Poisson structure on the double [Semenov] , given by 2 [ γ 1 ( r ∗ − r ) γ 2 − γ 2 ( r ∗ − r ) γ 1 ] ; { γ 1 , γ 2 } = λ This is the one that correctly dualizes the bialgebra structure on d when evaluated at the identity of the group D : e i + i λ ˜ I i e i and rescale r , r ∗ by the same Expand γ ∈ D as γ = 1 + i λ I i ˜ parameter λ = ⇒ k I k ; { I i , I j } = ǫ ij Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Poisson brackets Last but not least, it is possible to consider a different Poisson structure on the double [Semenov] , given by 2 [ γ 1 ( r ∗ − r ) γ 2 − γ 2 ( r ∗ − r ) γ 1 ] ; { γ 1 , γ 2 } = λ This is the one that correctly dualizes the bialgebra structure on d when evaluated at the identity of the group D : e i + i λ ˜ I i e i and rescale r , r ∗ by the same Expand γ ∈ D as γ = 1 + i λ I i ˜ parameter λ = ⇒ k I k ; { ˜ I i , ˜ I j } = f ij k ˜ I k { I i , I j } = ǫ ij Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Poisson brackets Last but not least, it is possible to consider a different Poisson structure on the double [Semenov] , given by 2 [ γ 1 ( r ∗ − r ) γ 2 − γ 2 ( r ∗ − r ) γ 1 ] ; { γ 1 , γ 2 } = λ This is the one that correctly dualizes the bialgebra structure on d when evaluated at the identity of the group D : e i + i λ ˜ I i e i and rescale r , r ∗ by the same Expand γ ∈ D as γ = 1 + i λ I i ˜ parameter λ = ⇒ k I k ; { ˜ I i , ˜ I j } = f ij k ˜ I k { I i , I j } = ǫ ij { I i , ˜ jk I k − ˜ I j } I k ǫ ki j = − f i which is the Poisson bracket induced by the Lie bi-algebra structure of the double; Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Poisson brackets Last but not least, it is possible to consider a different Poisson structure on the double [Semenov] , given by 2 [ γ 1 ( r ∗ − r ) γ 2 − γ 2 ( r ∗ − r ) γ 1 ] ; { γ 1 , γ 2 } = λ This is the one that correctly dualizes the bialgebra structure on d when evaluated at the identity of the group D : e i + i λ ˜ I i e i and rescale r , r ∗ by the same Expand γ ∈ D as γ = 1 + i λ I i ˜ parameter λ = ⇒ k I k ; { ˜ I i , ˜ I j } = f ij k ˜ I k { I i , I j } = ǫ ij { I i , ˜ jk I k − ˜ I j } I k ǫ ki j = − f i which is the Poisson bracket induced by the Lie bi-algebra structure of the double; I j play a symmetric role; We see that the fiber coordinates I i and ˜ Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Poisson brackets Last but not least, it is possible to consider a different Poisson structure on the double [Semenov] , given by 2 [ γ 1 ( r ∗ − r ) γ 2 − γ 2 ( r ∗ − r ) γ 1 ] ; { γ 1 , γ 2 } = λ This is the one that correctly dualizes the bialgebra structure on d when evaluated at the identity of the group D : e i + i λ ˜ I i e i and rescale r , r ∗ by the same Expand γ ∈ D as γ = 1 + i λ I i ˜ parameter λ = ⇒ k I k ; { ˜ I i , ˜ I j } = f ij k ˜ I k { I i , I j } = ǫ ij { I i , ˜ jk I k − ˜ I j } I k ǫ ki j = − f i which is the Poisson bracket induced by the Lie bi-algebra structure of the double; I j play a symmetric role; We see that the fiber coordinates I i and ˜ I i appears in the expansion of g , it Moreover, since the fiber coordinate ˜ can also be thought of as the fiber coordinate of the tangent bundle TSU (2), so that the couple ( I i , ˜ I i ) identifies the fiber coordinate of the generalized bundle T ⊕ T ∗ over SU (2). Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Poisson brackets Last but not least, it is possible to consider a different Poisson structure on the double [Semenov] , given by 2 [ γ 1 ( r ∗ − r ) γ 2 − γ 2 ( r ∗ − r ) γ 1 ] ; { γ 1 , γ 2 } = λ This is the one that correctly dualizes the bialgebra structure on d when evaluated at the identity of the group D : e i + i λ ˜ I i e i and rescale r , r ∗ by the same Expand γ ∈ D as γ = 1 + i λ I i ˜ parameter λ = ⇒ k I k ; { ˜ I i , ˜ I j } = f ij k ˜ I k { I i , I j } = ǫ ij { I i , ˜ jk I k − ˜ I j } I k ǫ ki j = − f i which is the Poisson bracket induced by the Lie bi-algebra structure of the double; I j play a symmetric role; We see that the fiber coordinates I i and ˜ I i appears in the expansion of g , it Moreover, since the fiber coordinate ˜ can also be thought of as the fiber coordinate of the tangent bundle TSU (2), so that the couple ( I i , ˜ I i ) identifies the fiber coordinate of the generalized bundle T ⊕ T ∗ over SU (2). Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT Relation to Generalized Geometry: We can consider TSU (2) ⊕ T ∗ SU (2) ≃ T ∗ SB (2 , C ) ⊕ T ∗ SU (2): Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT Relation to Generalized Geometry: We can consider TSU (2) ⊕ T ∗ SU (2) ≃ T ∗ SB (2 , C ) ⊕ T ∗ SU (2): Fiber coordinates are of the form P I = (˜ I i , I i ) with Poisson brackets given by the KSK brackets on the coadjoint orbits of SL (2 , C ); Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT Relation to Generalized Geometry: We can consider TSU (2) ⊕ T ∗ SU (2) ≃ T ∗ SB (2 , C ) ⊕ T ∗ SU (2): Fiber coordinates are of the form P I = (˜ I i , I i ) with Poisson brackets given by the KSK brackets on the coadjoint orbits of SL (2 , C ); They are induced by the bialgebra structure of sl (2 , C ) Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT Relation to Generalized Geometry: We can consider TSU (2) ⊕ T ∗ SU (2) ≃ T ∗ SB (2 , C ) ⊕ T ∗ SU (2): Fiber coordinates are of the form P I = (˜ I i , I i ) with Poisson brackets given by the KSK brackets on the coadjoint orbits of SL (2 , C ); They are induced by the bialgebra structure of sl (2 , C ) They can be identified with the C -brackets of Generalized Geometry Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT Relation to Generalized Geometry: We can consider TSU (2) ⊕ T ∗ SU (2) ≃ T ∗ SB (2 , C ) ⊕ T ∗ SU (2): Fiber coordinates are of the form P I = (˜ I i , I i ) with Poisson brackets given by the KSK brackets on the coadjoint orbits of SL (2 , C ); They are induced by the bialgebra structure of sl (2 , C ) They can be identified with the C -brackets of Generalized Geometry [ C brackets are mixed brackets between vector fields and forms. They generalize Courant and Dorfmann brackets] Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT Relation to Generalized Geometry: We can consider TSU (2) ⊕ T ∗ SU (2) ≃ T ∗ SB (2 , C ) ⊕ T ∗ SU (2): Fiber coordinates are of the form P I = (˜ I i , I i ) with Poisson brackets given by the KSK brackets on the coadjoint orbits of SL (2 , C ); They are induced by the bialgebra structure of sl (2 , C ) They can be identified with the C -brackets of Generalized Geometry [ C brackets are mixed brackets between vector fields and forms. They generalize Courant and Dorfmann brackets] We can also consider SL (2 , C ) as configuration space for the dynamics and TSL (2 , C ) ≃ SL (2 , C ) × SL (2 , C ) as its tangent space; Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT Relation to Generalized Geometry: We can consider TSU (2) ⊕ T ∗ SU (2) ≃ T ∗ SB (2 , C ) ⊕ T ∗ SU (2): Fiber coordinates are of the form P I = (˜ I i , I i ) with Poisson brackets given by the KSK brackets on the coadjoint orbits of SL (2 , C ); They are induced by the bialgebra structure of sl (2 , C ) They can be identified with the C -brackets of Generalized Geometry [ C brackets are mixed brackets between vector fields and forms. They generalize Courant and Dorfmann brackets] We can also consider SL (2 , C ) as configuration space for the dynamics and TSL (2 , C ) ≃ SL (2 , C ) × SL (2 , C ) as its tangent space; In this case we have doubled configuration space coordinates = ⇒ DFT Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT Relation to Generalized Geometry: We can consider TSU (2) ⊕ T ∗ SU (2) ≃ T ∗ SB (2 , C ) ⊕ T ∗ SU (2): Fiber coordinates are of the form P I = (˜ I i , I i ) with Poisson brackets given by the KSK brackets on the coadjoint orbits of SL (2 , C ); They are induced by the bialgebra structure of sl (2 , C ) They can be identified with the C -brackets of Generalized Geometry [ C brackets are mixed brackets between vector fields and forms. They generalize Courant and Dorfmann brackets] We can also consider SL (2 , C ) as configuration space for the dynamics and TSL (2 , C ) ≃ SL (2 , C ) × SL (2 , C ) as its tangent space; In this case we have doubled configuration space coordinates = ⇒ DFT PB for the generalized momenta are again C -brackets Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT Relation to Generalized Geometry: We can consider TSU (2) ⊕ T ∗ SU (2) ≃ T ∗ SB (2 , C ) ⊕ T ∗ SU (2): Fiber coordinates are of the form P I = (˜ I i , I i ) with Poisson brackets given by the KSK brackets on the coadjoint orbits of SL (2 , C ); They are induced by the bialgebra structure of sl (2 , C ) They can be identified with the C -brackets of Generalized Geometry [ C brackets are mixed brackets between vector fields and forms. They generalize Courant and Dorfmann brackets] We can also consider SL (2 , C ) as configuration space for the dynamics and TSL (2 , C ) ≃ SL (2 , C ) × SL (2 , C ) as its tangent space; In this case we have doubled configuration space coordinates = ⇒ DFT PB for the generalized momenta are again C -brackets Notice that here C -brackets satisfy Jacobi identity because they stem from a Lie bi-algebra (the generalized tangent bundle is a Lie bi-algebroid) Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT We go back to scalar products on the Lie bi-algebra Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT We go back to scalar products on the Lie bi-algebra w.r.to the second scalar product we have another splitting ( e i , e j ) = − ( b i , b j ) = δ ij , ( e i , b j ) = 0 f ± 1 with maximal isotropic subspaces: = 2 ( e i ± b i ) √ i Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT We go back to scalar products on the Lie bi-algebra w.r.to the second scalar product we have another splitting ( e i , e j ) = − ( b i , b j ) = δ ij , ( e i , b j ) = 0 f ± 1 with maximal isotropic subspaces: = 2 ( e i ± b i ) √ i Remark : Both splittings can be related to two different complex structures on SL (2 , C ). Some connection with Gualtieri ’04 Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT We go back to scalar products on the Lie bi-algebra w.r.to the second scalar product we have another splitting ( e i , e j ) = − ( b i , b j ) = δ ij , ( e i , b j ) = 0 f ± 1 with maximal isotropic subspaces: = 2 ( e i ± b i ) √ i Remark : Both splittings can be related to two different complex structures on SL (2 , C ). Some connection with Gualtieri ’04 Introduce the doubled notation � e i � e i ∈ sb (2 , C ) , e I = , e i ∈ su (2) , ˜ e i ˜ Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT We go back to scalar products on the Lie bi-algebra w.r.to the second scalar product we have another splitting ( e i , e j ) = − ( b i , b j ) = δ ij , ( e i , b j ) = 0 f ± 1 with maximal isotropic subspaces: = 2 ( e i ± b i ) √ i Remark : Both splittings can be related to two different complex structures on SL (2 , C ). Some connection with Gualtieri ’04 Introduce the doubled notation � e i � e i ∈ sb (2 , C ) , e I = , e i ∈ su (2) , ˜ e i ˜ The first scalar product becomes δ j � � 0 i < e I , e J > = L IJ = δ i 0 j This is a O (3 , 3) invariant metric; Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT We go back to scalar products on the Lie bi-algebra w.r.to the second scalar product we have another splitting ( e i , e j ) = − ( b i , b j ) = δ ij , ( e i , b j ) = 0 f ± 1 with maximal isotropic subspaces: = 2 ( e i ± b i ) √ i Remark : Both splittings can be related to two different complex structures on SL (2 , C ). Some connection with Gualtieri ’04 Introduce the doubled notation � e i � e i ∈ sb (2 , C ) , e I = , e i ∈ su (2) , ˜ e i ˜ The first scalar product becomes δ j � � 0 i < e I , e J > = L IJ = δ i 0 j This is a O (3 , 3) invariant metric; ( O ( d , d ) metric is a fundamental structure in DFT) Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT The second scalar product yields � δ ij ǫ j 3 � i ( e I , e J ) = R IJ = δ ij − ǫ i k 3 ǫ j − ǫ i l 3 δ kl j 3 Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT The second scalar product yields � δ ij ǫ j 3 � i ( e I , e J ) = R IJ = δ ij − ǫ i k 3 ǫ j − ǫ i l 3 δ kl j 3 On denoting by C + , C − the two subspaces spanned by { e i } , { b i } respectively, we notice that the splitting d = C + ⊕ C − defines a positive definite metric on d via G = ( , ) C + − ( , ) C − Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT The second scalar product yields � δ ij ǫ j 3 � i ( e I , e J ) = R IJ = δ ij − ǫ i k 3 ǫ j − ǫ i l 3 δ kl j 3 On denoting by C + , C − the two subspaces spanned by { e i } , { b i } respectively, we notice that the splitting d = C + ⊕ C − defines a positive definite metric on d via G = ( , ) C + − ( , ) C − Indicate the Riemannian metric with double round brackets: (( e i , e j )) := ( e i , e j ); (( b i , b j )) := − ( b i , b j ); (( e i , b j )) := ( e i , b j ) = 0 which satisfies G T LG = L Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT The second scalar product yields � δ ij ǫ j 3 � i ( e I , e J ) = R IJ = δ ij − ǫ i k 3 ǫ j − ǫ i l 3 δ kl j 3 On denoting by C + , C − the two subspaces spanned by { e i } , { b i } respectively, we notice that the splitting d = C + ⊕ C − defines a positive definite metric on d via G = ( , ) C + − ( , ) C − Indicate the Riemannian metric with double round brackets: (( e i , e j )) := ( e i , e j ); (( b i , b j )) := − ( b i , b j ); (( e i , b j )) := ( e i , b j ) = 0 which satisfies G T LG = L G is a pseudo-orthogonal metric - the sum α L + β G is the generalized metric of DFT Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT Remark 1: Both scalar products have applications in theoretical physics to build invariant action functionals; two relevant examples 2+1 gravity with cosmological term as a CS theory of SL (2 , C ) [Witten ’88] Palatini action with Holst term [Holst, Barbero, Immirzi..] Remark 2: While the first product is nothing but the Cartan-Killing metric of the Lie algebra sl (2 , C ), the Riemannian structure G can be mathematically formalized in a way which clarifies its role in the context of generalized complex geometry [freidel ’17] : it can be related to the structure of para-Hermitian manifold of SL (2 , C ) and therefore generalized to even-dimensional real manifolds which are not Lie groups. Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

SL (2 , C ) as a Drinfeld double Relation to generalized geometry and DFT Remark 1: Both scalar products have applications in theoretical physics to build invariant action functionals; two relevant examples 2+1 gravity with cosmological term as a CS theory of SL (2 , C ) [Witten ’88] Palatini action with Holst term [Holst, Barbero, Immirzi..] Remark 2: While the first product is nothing but the Cartan-Killing metric of the Lie algebra sl (2 , C ), the Riemannian structure G can be mathematically formalized in a way which clarifies its role in the context of generalized complex geometry [freidel ’17] : it can be related to the structure of para-Hermitian manifold of SL (2 , C ) and therefore generalized to even-dimensional real manifolds which are not Lie groups. Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

The dual model Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

The dual model On T ∗ SB (2 , C ) we may define action functional S 0 = − 1 � ˜ g − 1 d ˜ g − 1 d ˜ T r (˜ g ∧ ∗ ˜ g ) 4 R with ˜ g : t ∈ R → SB (2 , C ), T r a suitable trace over the Lie algebra Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

The dual model On T ∗ SB (2 , C ) we may define action functional S 0 = − 1 � ˜ g − 1 d ˜ g − 1 d ˜ T r (˜ g ∧ ∗ ˜ g ) 4 R with ˜ g : t ∈ R → SB (2 , C ), T r a suitable trace over the Lie algebra No non-degenerate invariant products on sb (2 , C ); Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

The dual model On T ∗ SB (2 , C ) we may define action functional S 0 = − 1 � ˜ g − 1 d ˜ g − 1 d ˜ T r (˜ g ∧ ∗ ˜ g ) 4 R with ˜ g : t ∈ R → SB (2 , C ), T r a suitable trace over the Lie algebra No non-degenerate invariant products on sb (2 , C ); We choose the non-degenerate one T r := (( , )) Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

The dual model On T ∗ SB (2 , C ) we may define action functional S 0 = − 1 � ˜ g − 1 d ˜ g − 1 d ˜ T r (˜ g ∧ ∗ ˜ g ) 4 R with ˜ g : t ∈ R → SB (2 , C ), T r a suitable trace over the Lie algebra No non-degenerate invariant products on sb (2 , C ); We choose the non-degenerate one T r := (( , )) = ⇒ the Lagrangian Q i ( δ ij + ǫ i ˙ l 3 ) δ kl ˙ L 0 = 1 ˜ ˜ k 3 ǫ j ˜ Q j 2 Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

The dual model On T ∗ SB (2 , C ) we may define action functional S 0 = − 1 � ˜ g − 1 d ˜ g − 1 d ˜ T r (˜ g ∧ ∗ ˜ g ) 4 R with ˜ g : t ∈ R → SB (2 , C ), T r a suitable trace over the Lie algebra No non-degenerate invariant products on sb (2 , C ); We choose the non-degenerate one T r := (( , )) = ⇒ the Lagrangian Q i ( δ ij + ǫ i ˙ l 3 ) δ kl ˙ L 0 = 1 ˜ ˜ k 3 ǫ j ˜ Q j 2 Q i , ˙ g = ˙ g − 1 ˙ Tangent bundle coordinates: ( ˜ ˜ ˜ e i Q i ), with ˜ ˜ Q i ˜ ( δ ij + ǫ i l 3 δ kl ) ¨ k 3 ǫ j ˜ Equations of motion Q j = 0 Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

The dual model On T ∗ SB (2 , C ) we may define action functional S 0 = − 1 � ˜ g − 1 d ˜ g − 1 d ˜ T r (˜ g ∧ ∗ ˜ g ) 4 R with ˜ g : t ∈ R → SB (2 , C ), T r a suitable trace over the Lie algebra No non-degenerate invariant products on sb (2 , C ); We choose the non-degenerate one T r := (( , )) = ⇒ the Lagrangian Q i ( δ ij + ǫ i ˙ l 3 ) δ kl ˙ L 0 = 1 ˜ ˜ k 3 ǫ j ˜ Q j 2 Q i , ˙ g = ˙ g − 1 ˙ Tangent bundle coordinates: ( ˜ ˜ ˜ e i Q i ), with ˜ ˜ Q i ˜ ( δ ij + ǫ i l 3 δ kl ) ¨ k 3 ǫ j ˜ Equations of motion Q j = 0 Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

The dual model Cotangent bundle Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

The dual model Cotangent bundle T ∗ SB (2 , C ) - Coordinates: ( ˜ Q j , ˜ I j ) Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

The dual model I j the Cotangent bundle T ∗ SB (2 , C ) - Coordinates: ( ˜ Q j , ˜ I j ) with ˜ conjugate momenta I j = ∂ ˜ L 0 Q r = − i = ( δ jr + ǫ jr 3 ) ˙ g − 1 ˙ ˜ ˜ e j )) 2((˜ ˜ g , ˜ ∂ ˙ ˜ Q j Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

The dual model I j the Cotangent bundle T ∗ SB (2 , C ) - Coordinates: ( ˜ Q j , ˜ I j ) with ˜ conjugate momenta I j = ∂ ˜ L 0 Q r = − i = ( δ jr + ǫ jr 3 ) ˙ g − 1 ˙ ˜ ˜ e j )) 2((˜ ˜ g , ˜ ∂ ˙ ˜ Q j with ˙ ˜ 2 ǫ jr 3 )˜ Q j = ( δ jr − 1 I r Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model

The dual model I j the Cotangent bundle T ∗ SB (2 , C ) - Coordinates: ( ˜ Q j , ˜ I j ) with ˜ conjugate momenta I j = ∂ ˜ L 0 Q r = − i = ( δ jr + ǫ jr 3 ) ˙ g − 1 ˙ ˜ ˜ e j )) 2((˜ ˜ g , ˜ ∂ ˙ ˜ Q j with ˙ ˜ 2 ǫ jr 3 )˜ Q j = ( δ jr − 1 I r Hamiltonian ˜ 2 ˜ q δ kl )˜ H 0 = 1 I p ( δ pq − 1 2 ǫ k 3 p ǫ l 3 I q Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model