Geometric formulation of the rigid rotator on configuration space SU (2) g β 1 Λ g = i ( y 0 Λ y i β y i Λ y 0 + Η« i jk y j Λ y k ) Ο i = i Λ Q i Ο i Since the Lagrangian reads 2 ( y 0 Λ y j β y j Λ y 0 + Η« j kl y k Λ y l )( y 0 Λ y r β y r Λ y 0 + Η« r pq y p Λ Q j Λ 2 Λ L 0 = 1 y q ) Ξ΄ ir = 1 Q r Ξ΄ jr Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model
Geometric formulation of the rigid rotator on configuration space SU (2) g β 1 Λ g = i ( y 0 Λ y i β y i Λ y 0 + Η« i jk y j Λ y k ) Ο i = i Λ Q i Ο i Since the Lagrangian reads 2 ( y 0 Λ y j β y j Λ y 0 + Η« j kl y k Λ y l )( y 0 Λ y r β y r Λ y 0 + Η« r pq y p Λ Q j Λ 2 Λ L 0 = 1 y q ) Ξ΄ ir = 1 Q r Ξ΄ jr Tangent bundle coordinates: ( Q i , Λ Q i ) Q i = 0 Β¨ d g β 1 dg οΏ½ οΏ½ 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 β 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 β SU (2) - Coordinates: ( Q i , I i ) with I i the conjugate momenta I j = β L 0 Q j = Ξ΄ jr Λ Q r β Λ Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model
Geometric formulation of the rigid rotator Cotangent bundle T β SU (2) - Coordinates: ( Q i , I i ) with I i the conjugate momenta I j = β L 0 Q j = Ξ΄ jr Λ Q r β Λ Hamiltonian H 0 = 1 2 I i I j Ξ΄ ij PBβs: { y i , y j } = 0 k I k { I i , I j } = Η« ij j y 0 + Η« i { y i , I j } β Ξ΄ i jk y k = { g , I j } = β i Ο j g or g β 1 Λ Λ EOM: I i = 0 , g = iI i Ο i Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model
Geometric formulation of the rigid rotator Cotangent bundle T β SU (2) - Coordinates: ( Q i , I i ) with I i the conjugate momenta I j = β L 0 Q j = Ξ΄ jr Λ Q r β Λ Hamiltonian H 0 = 1 2 I i I j Ξ΄ ij PBβs: { y i , y j } = 0 k I k { I i , I j } = Η« ij j y 0 + Η« i { y i , I j } β Ξ΄ i jk y k = { g , I j } = β i Ο j g or g β 1 Λ Λ EOM: I i = 0 , g = iI i Ο 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 β SU (2) Remarks: As a group T β SU (2) β SU (2) β R 3 with Lie algebra [ L i , L j ] = Η« k [ L i , T j ] = Η« 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 β SU (2) Remarks: As a group T β SU (2) β SU (2) β R 3 with Lie algebra [ L i , L j ] = Η« k [ L i , T j ] = Η« 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 β Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model
The cotangent bundle T β SU (2) Remarks: As a group T β SU (2) β SU (2) β R 3 with Lie algebra [ L i , L j ] = Η« k [ L i , T j ] = Η« 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 β In [Marmo Simoni Stern β93] 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 β SU (2) Remarks: As a group T β SU (2) β SU (2) β R 3 with Lie algebra [ L i , L j ] = Η« k [ L i , T j ] = Η« 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 β In [Marmo Simoni Stern β93] the carrier space of the dynamics has been generalized to SL (2 , C ), the Drinfeld double of SU (2). In [Rajeev β89 , Rajeev, Sparano P.V. β93] the same has been done for chiral & WZW model Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model
The cotangent bundle T β SU (2) Remarks: As a group T β SU (2) β SU (2) β R 3 with Lie algebra [ L i , L j ] = Η« k [ L i , T j ] = Η« 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 β In [Marmo Simoni Stern β93] the carrier space of the dynamics has been generalized to SL (2 , C ), the Drinfeld double of SU (2). In [Rajeev β89 , Rajeev, Sparano P.V. β93] 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 β SU (2) Remarks: As a group T β SU (2) β SU (2) β R 3 with Lie algebra [ L i , L j ] = Η« k [ L i , T j ] = Η« 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 β In [Marmo Simoni Stern β93] the carrier space of the dynamics has been generalized to SL (2 , C ), the Drinfeld double of SU (2). In [Rajeev β89 , Rajeev, Sparano P.V. β93] 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 = Ο i / 2 , b i = ie i [ e i , e j ] = i Η« k [ e i , b j ] = i Η« k [ b i , b j ] = β i Η« 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 = Ο i / 2 , b i = ie i [ e i , e j ] = i Η« k [ e i , b j ] = i Η« k [ b i , b j ] = β i Η« k ij e k , ij b k , ij e k Non-degenerate invariant scalar products: < u , v > = 2 Im ( Tr ( uv )) , β u , v β 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 = Ο i / 2 , b i = ie i [ e i , e j ] = i Η« k [ e i , b j ] = i Η« k [ b i , b j ] = β i Η« k ij e k , ij b k , ij e k Non-degenerate invariant scalar products: < u , v > = 2 Im ( Tr ( uv )) , β u , v β sl (2 , C ) and ( u , v ) = 2 Re ( Tr ( uv )) , β u , v β 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 = Ο i / 2 , b i = ie i [ e i , e j ] = i Η« k [ e i , b j ] = i Η« k [ b i , b j ] = β i Η« k ij e k , ij b k , ij e k Non-degenerate invariant scalar products: < u , v > = 2 Im ( Tr ( uv )) , β u , v β sl (2 , C ) and ( u , v ) = 2 Re ( Tr ( uv )) , β u , v β sl (2 , C ) w.r.t. the first one (Cartan-Killing) we have two maximal isotropic subspaces e j > = 0 , e j > = Ξ΄ j e i , Λ < e i , e j > = < Λ < e i , Λ i e i = b i β Η« ij 3 e j . with 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 The algebra is spanned by e i = Ο i / 2 , b i = ie i [ e i , e j ] = i Η« k [ e i , b j ] = i Η« k [ b i , b j ] = β i Η« k ij e k , ij b k , ij e k Non-degenerate invariant scalar products: < u , v > = 2 Im ( Tr ( uv )) , β u , v β sl (2 , C ) and ( u , v ) = 2 Re ( Tr ( uv )) , β u , v β sl (2 , C ) w.r.t. the first one (Cartan-Killing) we have two maximal isotropic subspaces e j > = 0 , e j > = Ξ΄ j e i , Λ < e i , e j > = < Λ < e i , Λ i e i = b i β Η« ij 3 e j . { e i } , { Λ e i } both subalgebras with with Λ e k + ie k f ki [ e i , e j ] = i Η« k e i , e j ] = i Η« i e i , Λ e j ] = if ij e k ij e k , [Λ jk Λ j , [Λ k Λ e i } span the Lie algebra of SB (2 , C ), the dual group of SU (2) with { Λ f ij k = Η« ijl Η« 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 = Ο i / 2 , b i = ie i [ e i , e j ] = i Η« k [ e i , b j ] = i Η« k [ b i , b j ] = β i Η« k ij e k , ij b k , ij e k Non-degenerate invariant scalar products: < u , v > = 2 Im ( Tr ( uv )) , β u , v β sl (2 , C ) and ( u , v ) = 2 Re ( Tr ( uv )) , β u , v β sl (2 , C ) w.r.t. the first one (Cartan-Killing) we have two maximal isotropic subspaces e j > = 0 , e j > = Ξ΄ j e i , Λ < e i , e j > = < Λ < e i , Λ i e i = b i β Η« ij 3 e j . { e i } , { Λ e i } both subalgebras with with Λ e k + ie k f ki [ e i , e j ] = i Η« k e i , e j ] = i Η« i e i , Λ e j ] = if ij e k ij e k , [Λ jk Λ j , [Λ k Λ e i } span the Lie algebra of SB (2 , C ), the dual group of SU (2) with { Λ f ij k = Η« ijl Η« 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 β³ g β , D is the Drinfeld double, G , G β are dual groups Given d = g β² 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 β³ g β , D is the Drinfeld double, G , G β are dual groups Given d = g β² For f ij D β T β G k = 0 For c k ij = 0 D β T β G β 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 β³ g β , D is the Drinfeld double, G , G β are dual groups Given d = g β² For f ij D β T β G k = 0 For c k ij = 0 D β T β G β 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 β³ g β , D is the Drinfeld double, G , G β are dual groups Given d = g β² For f ij D β T β G k = 0 For c k ij = 0 D β T β G β 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 ) β ( F ( SU (2)) , Λ [ , ] su (2) β ( F ( SB (2 , C )) , Ξ); Ξ) 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 β³ g β , D is the Drinfeld double, G , G β are dual groups Given d = g β² For f ij D β T β G k = 0 For c k ij = 0 D β T β G β 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 ) β ( F ( SU (2)) , Λ [ , ] su (2) β ( F ( SB (2 , C )) , Ξ); Ξ) which reduce to KSK brackets on coadjoint orbits of G , G β 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 β³ g β , D is the Drinfeld double, G , G β are dual groups Given d = g β² For f ij D β T β G k = 0 For c k ij = 0 D β T β G β 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 ) β ( F ( SU (2)) , Λ [ , ] su (2) β ( F ( SB (2 , C )) , Ξ); Ξ) which reduce to KSK brackets on coadjoint orbits of G , G β 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 PBβs which generalize both those of T β SU (2) and of T β SB (2 C ) [[Semenov-Tyan-Shanskii β91, Alekseev-Malkin β94]] { Ξ³ 1 , Ξ³ 2 } = β Ξ³ 1 Ξ³ 2 r β β r Ξ³ 1 Ξ³ 2 whith Ξ³ 1 = Ξ³ β 1 , Ξ³ 2 = 1 β Ξ³ 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 PBβs which generalize both those of T β SU (2) and of T β SB (2 C ) [[Semenov-Tyan-Shanskii β91, Alekseev-Malkin β94]] { Ξ³ 1 , Ξ³ 2 } = β Ξ³ 1 Ξ³ 2 r β β r Ξ³ 1 Ξ³ 2 whith Ξ³ 1 = Ξ³ β 1 , Ξ³ 2 = 1 β Ξ³ 2 ; e i β e i , r β = β e i β Λ e i r = Λ 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 PBβs which generalize both those of T β SU (2) and of T β SB (2 C ) [[Semenov-Tyan-Shanskii β91, Alekseev-Malkin β94]] { Ξ³ 1 , Ξ³ 2 } = β Ξ³ 1 Ξ³ 2 r β β r Ξ³ 1 Ξ³ 2 whith Ξ³ 1 = Ξ³ β 1 , Ξ³ 2 = 1 β Ξ³ 2 ; e i β e i , r β = β e i β Λ e i r = Λ 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 PBβs which generalize both those of T β SU (2) and of T β SB (2 C ) [[Semenov-Tyan-Shanskii β91, Alekseev-Malkin β94]] { Ξ³ 1 , Ξ³ 2 } = β Ξ³ 1 Ξ³ 2 r β β r Ξ³ 1 Ξ³ 2 whith Ξ³ 1 = Ξ³ β 1 , Ξ³ 2 = 1 β Ξ³ 2 ; e i β e i , r β = β e i β Λ e i r = Λ is the classical Yang Baxter matrix The group D equipped with the Poisson bracket is also called the Heisenberg double On writing Ξ³ as Ξ³ = Λ gg it can be shown that these brackets are compatible with { Λ g 1 , Λ g 2 } = β [ r , Λ g 1 Λ g 2 ] , g 2 r β g 1 { Λ g 1 , g 2 } = β Λ g 1 rg 2 , { g 1 , Λ g 2 } = β Λ { g 1 , g 2 } = [ r β , 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 PBβs which generalize both those of T β SU (2) and of T β SB (2 C ) [[Semenov-Tyan-Shanskii β91, Alekseev-Malkin β94]] { Ξ³ 1 , Ξ³ 2 } = β Ξ³ 1 Ξ³ 2 r β β r Ξ³ 1 Ξ³ 2 whith Ξ³ 1 = Ξ³ β 1 , Ξ³ 2 = 1 β Ξ³ 2 ; e i β e i , r β = β e i β Λ e i r = Λ is the classical Yang Baxter matrix The group D equipped with the Poisson bracket is also called the Heisenberg double On writing Ξ³ as Ξ³ = Λ gg it can be shown that these brackets are compatible with { Λ g 1 , Λ g 2 } = β [ r , Λ g 1 Λ g 2 ] , g 2 r β g 1 { Λ g 1 , g 2 } = β Λ g 1 rg 2 , { g 1 , Λ g 2 } = β Λ { g 1 , g 2 } = [ r β , 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 β e i , In the limit Ξ» β 0, with r = Ξ» Λ Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model
SL (2 , C ) as a Drinfeld double Poisson brackets e i β e i , Λ g ( Ξ» ) = 1 + i Ξ» I i e i + O ( Ξ» 2 ) In the limit Ξ» β 0, with r = Ξ» Λ Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model
SL (2 , C ) as a Drinfeld double Poisson brackets e i β e i , Λ g ( Ξ» ) = 1 + i Ξ» I i e i + O ( Ξ» 2 ) In the limit Ξ» β 0, with r = Ξ» Λ g = y 0 Ο 0 + iy i Ο i Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model
SL (2 , C ) as a Drinfeld double Poisson brackets e i β e i , Λ g ( Ξ» ) = 1 + i Ξ» I i e i + O ( Ξ» 2 ) In the limit Ξ» β 0, with r = Ξ» Λ g = y 0 Ο 0 + iy i Ο i we obtain Η« 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 β e i , Λ g ( Ξ» ) = 1 + i Ξ» I i e i + O ( Ξ» 2 ) In the limit Ξ» β 0, with r = Ξ» Λ g = y 0 Ο 0 + iy i Ο i we obtain Η« k { I i , I j } = ij I k { I i , y 0 } iy j Ξ΄ ij { I i , y j } = iy 0 Ξ΄ j i β Η« 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 β e i , Λ g ( Ξ» ) = 1 + i Ξ» I i e i + O ( Ξ» 2 ) In the limit Ξ» β 0, with r = Ξ» Λ g = y 0 Ο 0 + iy i Ο i we obtain Η« k { I i , I j } = ij I k { I i , y 0 } iy j Ξ΄ ij { I i , y j } = iy 0 Ξ΄ j i β Η« j ik y k = { y 0 , y j } { y i , y j } = 0 + O ( Ξ» ) = 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 β e i , Λ g ( Ξ» ) = 1 + i Ξ» I i e i + O ( Ξ» 2 ) In the limit Ξ» β 0, with r = Ξ» Λ g = y 0 Ο 0 + iy i Ο i we obtain Η« k { I i , I j } = ij I k { I i , y 0 } iy j Ξ΄ ij { I i , y j } = iy 0 Ξ΄ j i β Η« j ik y k = { y 0 , y j } { y i , y j } = 0 + O ( Ξ» ) = which reproduce correctly the canonical Poisson brackets on the cotangent bundle of SU (2). Consider now r β as an independent solution of the Yang Baxter equation Ο = Β΅ e k β e k and expand g β SU (2) as a function of the parameter Β΅ : g = 1 + i Β΅ Λ Ie i + O ( Β΅ 2 ) By repeating the same analysis as above we get back the canonical Poisson structure on T β SB (2 , C ), with position coordinates and momenta now interchanged. In particular we note { Λ I i , Λ I j } = f ij k Λ 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 [ Ξ³ 1 ( r β β r ) Ξ³ 2 β Ξ³ 2 ( r β β r ) Ξ³ 1 ] ; { Ξ³ 1 , Ξ³ 2 } = Ξ» 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 [ Ξ³ 1 ( r β β r ) Ξ³ 2 β Ξ³ 2 ( r β β r ) Ξ³ 1 ] ; { Ξ³ 1 , Ξ³ 2 } = Ξ» This is the one that correctly dualizes the bialgebra structure on d when evaluated at the identity of the group D : e i + i Ξ» Λ I i e i and rescale r , r β by the same Expand Ξ³ β D as Ξ³ = 1 + i Ξ» I i Λ parameter Ξ» = β k I k ; { I i , I j } = Η« 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 [ Ξ³ 1 ( r β β r ) Ξ³ 2 β Ξ³ 2 ( r β β r ) Ξ³ 1 ] ; { Ξ³ 1 , Ξ³ 2 } = Ξ» This is the one that correctly dualizes the bialgebra structure on d when evaluated at the identity of the group D : e i + i Ξ» Λ I i e i and rescale r , r β by the same Expand Ξ³ β D as Ξ³ = 1 + i Ξ» I i Λ parameter Ξ» = β k I k ; { Λ I i , Λ I j } = f ij k Λ I k { I i , I j } = Η« 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 [ Ξ³ 1 ( r β β r ) Ξ³ 2 β Ξ³ 2 ( r β β r ) Ξ³ 1 ] ; { Ξ³ 1 , Ξ³ 2 } = Ξ» This is the one that correctly dualizes the bialgebra structure on d when evaluated at the identity of the group D : e i + i Ξ» Λ I i e i and rescale r , r β by the same Expand Ξ³ β D as Ξ³ = 1 + i Ξ» I i Λ parameter Ξ» = β k I k ; { Λ I i , Λ I j } = f ij k Λ I k { I i , I j } = Η« ij { I i , Λ jk I k β Λ I j } I k Η« ki j = β 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 [ Ξ³ 1 ( r β β r ) Ξ³ 2 β Ξ³ 2 ( r β β r ) Ξ³ 1 ] ; { Ξ³ 1 , Ξ³ 2 } = Ξ» This is the one that correctly dualizes the bialgebra structure on d when evaluated at the identity of the group D : e i + i Ξ» Λ I i e i and rescale r , r β by the same Expand Ξ³ β D as Ξ³ = 1 + i Ξ» I i Λ parameter Ξ» = β k I k ; { Λ I i , Λ I j } = f ij k Λ I k { I i , I j } = Η« ij { I i , Λ jk I k β Λ I j } I k Η« ki j = β 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 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 [ Ξ³ 1 ( r β β r ) Ξ³ 2 β Ξ³ 2 ( r β β r ) Ξ³ 1 ] ; { Ξ³ 1 , Ξ³ 2 } = Ξ» This is the one that correctly dualizes the bialgebra structure on d when evaluated at the identity of the group D : e i + i Ξ» Λ I i e i and rescale r , r β by the same Expand Ξ³ β D as Ξ³ = 1 + i Ξ» I i Λ parameter Ξ» = β k I k ; { Λ I i , Λ I j } = f ij k Λ I k { I i , I j } = Η« ij { I i , Λ jk I k β Λ I j } I k Η« ki j = β 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 Λ I i appears in the expansion of g , it Moreover, since the fiber coordinate Λ can also be thought of as the fiber coordinate of the tangent bundle TSU (2), so that the couple ( I i , Λ I i ) identifies the fiber coordinate of the generalized bundle T β T β 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 [ Ξ³ 1 ( r β β r ) Ξ³ 2 β Ξ³ 2 ( r β β r ) Ξ³ 1 ] ; { Ξ³ 1 , Ξ³ 2 } = Ξ» This is the one that correctly dualizes the bialgebra structure on d when evaluated at the identity of the group D : e i + i Ξ» Λ I i e i and rescale r , r β by the same Expand Ξ³ β D as Ξ³ = 1 + i Ξ» I i Λ parameter Ξ» = β k I k ; { Λ I i , Λ I j } = f ij k Λ I k { I i , I j } = Η« ij { I i , Λ jk I k β Λ I j } I k Η« ki j = β 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 Λ I i appears in the expansion of g , it Moreover, since the fiber coordinate Λ can also be thought of as the fiber coordinate of the tangent bundle TSU (2), so that the couple ( I i , Λ I i ) identifies the fiber coordinate of the generalized bundle T β T β 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) β T β SU (2) β T β SB (2 , C ) β T β 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) β T β SU (2) β T β SB (2 , C ) β T β SU (2): Fiber coordinates are of the form P I = (Λ 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) β T β SU (2) β T β SB (2 , C ) β T β SU (2): Fiber coordinates are of the form P I = (Λ 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) β T β SU (2) β T β SB (2 , C ) β T β SU (2): Fiber coordinates are of the form P I = (Λ 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) β T β SU (2) β T β SB (2 , C ) β T β SU (2): Fiber coordinates are of the form P I = (Λ 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) β T β SU (2) β T β SB (2 , C ) β T β SU (2): Fiber coordinates are of the form P I = (Λ 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 ) β SL (2 , C ) Γ 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) β T β SU (2) β T β SB (2 , C ) β T β SU (2): Fiber coordinates are of the form P I = (Λ 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 ) β SL (2 , C ) Γ SL (2 , C ) as its tangent space; In this case we have doubled configuration space coordinates = β 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) β T β SU (2) β T β SB (2 , C ) β T β SU (2): Fiber coordinates are of the form P I = (Λ 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 ) β SL (2 , C ) Γ SL (2 , C ) as its tangent space; In this case we have doubled configuration space coordinates = β 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) β T β SU (2) β T β SB (2 , C ) β T β SU (2): Fiber coordinates are of the form P I = (Λ 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 ) β SL (2 , C ) Γ SL (2 , C ) as its tangent space; In this case we have doubled configuration space coordinates = β 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 ) = β ( b i , b j ) = Ξ΄ ij , ( e i , b j ) = 0 f Β± 1 with maximal isotropic subspaces: = 2 ( e i Β± b i ) β 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 ) = β ( b i , b j ) = Ξ΄ ij , ( e i , b j ) = 0 f Β± 1 with maximal isotropic subspaces: = 2 ( e i Β± b i ) β i Remark : Both splittings can be related to two different complex structures on SL (2 , C ). Some connection with Gualtieri β04 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 ) = β ( b i , b j ) = Ξ΄ ij , ( e i , b j ) = 0 f Β± 1 with maximal isotropic subspaces: = 2 ( e i Β± b i ) β i Remark : Both splittings can be related to two different complex structures on SL (2 , C ). Some connection with Gualtieri β04 Introduce the doubled notation οΏ½ e i οΏ½ e i β sb (2 , C ) , e I = , e i β su (2) , Λ e i 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 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 ) = β ( b i , b j ) = Ξ΄ ij , ( e i , b j ) = 0 f Β± 1 with maximal isotropic subspaces: = 2 ( e i Β± b i ) β i Remark : Both splittings can be related to two different complex structures on SL (2 , C ). Some connection with Gualtieri β04 Introduce the doubled notation οΏ½ e i οΏ½ e i β sb (2 , C ) , e I = , e i β su (2) , Λ e i Λ The first scalar product becomes Ξ΄ j οΏ½ οΏ½ 0 i < e I , e J > = L IJ = Ξ΄ 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 ) = β ( b i , b j ) = Ξ΄ ij , ( e i , b j ) = 0 f Β± 1 with maximal isotropic subspaces: = 2 ( e i Β± b i ) β i Remark : Both splittings can be related to two different complex structures on SL (2 , C ). Some connection with Gualtieri β04 Introduce the doubled notation οΏ½ e i οΏ½ e i β sb (2 , C ) , e I = , e i β su (2) , Λ e i Λ The first scalar product becomes Ξ΄ j οΏ½ οΏ½ 0 i < e I , e J > = L IJ = Ξ΄ 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 οΏ½ Ξ΄ ij Η« j 3 οΏ½ i ( e I , e J ) = R IJ = Ξ΄ ij β Η« i k 3 Η« j β Η« i l 3 Ξ΄ 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 οΏ½ Ξ΄ ij Η« j 3 οΏ½ i ( e I , e J ) = R IJ = Ξ΄ ij β Η« i k 3 Η« j β Η« i l 3 Ξ΄ kl j 3 On denoting by C + , C β the two subspaces spanned by { e i } , { b i } respectively, we notice that the splitting d = C + β C β defines a positive definite metric on d via G = ( , ) C + β ( , ) C β 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 οΏ½ Ξ΄ ij Η« j 3 οΏ½ i ( e I , e J ) = R IJ = Ξ΄ ij β Η« i k 3 Η« j β Η« i l 3 Ξ΄ kl j 3 On denoting by C + , C β the two subspaces spanned by { e i } , { b i } respectively, we notice that the splitting d = C + β C β defines a positive definite metric on d via G = ( , ) C + β ( , ) C β Indicate the Riemannian metric with double round brackets: (( e i , e j )) := ( e i , e j ); (( b i , b j )) := β ( 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 οΏ½ Ξ΄ ij Η« j 3 οΏ½ i ( e I , e J ) = R IJ = Ξ΄ ij β Η« i k 3 Η« j β Η« i l 3 Ξ΄ kl j 3 On denoting by C + , C β the two subspaces spanned by { e i } , { b i } respectively, we notice that the splitting d = C + β C β defines a positive definite metric on d via G = ( , ) C + β ( , ) C β Indicate the Riemannian metric with double round brackets: (( e i , e j )) := ( e i , e j ); (( b i , b j )) := β ( 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 Ξ± L + Ξ² 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 β88] 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 β17] : 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 β88] 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 β17] : 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 β SB (2 , C ) we may define action functional S 0 = β 1 οΏ½ Λ g β 1 d Λ g β 1 d Λ T r (Λ g β§ β Λ g ) 4 R with Λ g : t β R β 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 β SB (2 , C ) we may define action functional S 0 = β 1 οΏ½ Λ g β 1 d Λ g β 1 d Λ T r (Λ g β§ β Λ g ) 4 R with Λ g : t β R β 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 β SB (2 , C ) we may define action functional S 0 = β 1 οΏ½ Λ g β 1 d Λ g β 1 d Λ T r (Λ g β§ β Λ g ) 4 R with Λ g : t β R β 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 β SB (2 , C ) we may define action functional S 0 = β 1 οΏ½ Λ g β 1 d Λ g β 1 d Λ T r (Λ g β§ β Λ g ) 4 R with Λ g : t β R β 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 := (( , )) = β the Lagrangian Q i ( Ξ΄ ij + Η« i Λ l 3 ) Ξ΄ kl Λ L 0 = 1 Λ Λ k 3 Η« j Λ Q j 2 Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model
The dual model On T β SB (2 , C ) we may define action functional S 0 = β 1 οΏ½ Λ g β 1 d Λ g β 1 d Λ T r (Λ g β§ β Λ g ) 4 R with Λ g : t β R β 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 := (( , )) = β the Lagrangian Q i ( Ξ΄ ij + Η« i Λ l 3 ) Ξ΄ kl Λ L 0 = 1 Λ Λ k 3 Η« j Λ Q j 2 Q i , Λ g = Λ g β 1 Λ Tangent bundle coordinates: ( Λ Λ Λ e i Q i ), with Λ Λ Q i Λ ( Ξ΄ ij + Η« i l 3 Ξ΄ kl ) Β¨ k 3 Η« j Λ Equations of motion Q j = 0 Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model
The dual model On T β SB (2 , C ) we may define action functional S 0 = β 1 οΏ½ Λ g β 1 d Λ g β 1 d Λ T r (Λ g β§ β Λ g ) 4 R with Λ g : t β R β 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 := (( , )) = β the Lagrangian Q i ( Ξ΄ ij + Η« i Λ l 3 ) Ξ΄ kl Λ L 0 = 1 Λ Λ k 3 Η« j Λ Q j 2 Q i , Λ g = Λ g β 1 Λ Tangent bundle coordinates: ( Λ Λ Λ e i Q i ), with Λ Λ Q i Λ ( Ξ΄ ij + Η« i l 3 Ξ΄ kl ) Β¨ k 3 Η« j Λ 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 β SB (2 , C ) - Coordinates: ( Λ Q j , Λ I j ) Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model
The dual model I j the Cotangent bundle T β SB (2 , C ) - Coordinates: ( Λ Q j , Λ I j ) with Λ conjugate momenta I j = β Λ L 0 Q r = β i = ( Ξ΄ jr + Η« jr 3 ) Λ g β 1 Λ Λ Λ e j )) 2((Λ Λ g , Λ β Λ Λ Q j Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model
The dual model I j the Cotangent bundle T β SB (2 , C ) - Coordinates: ( Λ Q j , Λ I j ) with Λ conjugate momenta I j = β Λ L 0 Q r = β i = ( Ξ΄ jr + Η« jr 3 ) Λ g β 1 Λ Λ Λ e j )) 2((Λ Λ g , Λ β Λ Λ Q j with Λ Λ 2 Η« jr 3 )Λ Q j = ( Ξ΄ jr β 1 I r Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model
The dual model I j the Cotangent bundle T β SB (2 , C ) - Coordinates: ( Λ Q j , Λ I j ) with Λ conjugate momenta I j = β Λ L 0 Q r = β i = ( Ξ΄ jr + Η« jr 3 ) Λ g β 1 Λ Λ Λ e j )) 2((Λ Λ g , Λ β Λ Λ Q j with Λ Λ 2 Η« jr 3 )Λ Q j = ( Ξ΄ jr β 1 I r Hamiltonian Λ 2 Λ q Ξ΄ kl )Λ H 0 = 1 I p ( Ξ΄ pq β 1 2 Η« k 3 p Η« l 3 I q Patrizia Vitale Generalized Geometry and Double Field Theory: a toy Model
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