n -dimensional manifold M with T := TM

n -dimensional manifold M with T := TM T

n -dimensional manifold M with T := TM T frame bundle GL ( R n , T ) is a principal GL ( n )-bundle

n -dimensional manifold M with T := TM T + T ∗ frame bundle GL ( R n , T ) is a principal GL ( n )-bundle

n -dimensional manifold M with T := TM T + T ∗ pairing � X + ξ, X + ξ � = i X ξ frame bundle GL ( R n , T ) is a principal GL ( n )-bundle

n -dimensional manifold M with T := TM T + T ∗ pairing � X + ξ, X + ξ � = i X ξ frame bundle generalized frame bundle O ( R n + ( R n ) ∗ , T + T ∗ ) GL ( R n , T ) is a principal GL ( n )-bundle

n -dimensional manifold M with T := TM T + T ∗ pairing � X + ξ, X + ξ � = i X ξ frame bundle generalized frame bundle O ( R n + ( R n ) ∗ , T + T ∗ ) GL ( R n , T ) is a is a principal GL ( n )-bundle principal O ( n , n )-bundle

n -dimensional manifold M with T := TM T + T ∗ pairing � X + ξ, X + ξ � = i X ξ frame bundle generalized frame bundle O ( R n + ( R n ) ∗ , T + T ∗ ) GL ( R n , T ) is a is a principal GL ( n )-bundle principal O ( n , n )-bundle Lie bracket [ , ]

n -dimensional manifold M with T := TM T + T ∗ pairing � X + ξ, X + ξ � = i X ξ frame bundle generalized frame bundle O ( R n + ( R n ) ∗ , T + T ∗ ) GL ( R n , T ) is a is a principal GL ( n )-bundle principal O ( n , n )-bundle Lie bracket [ , ] Courant bracket [ , ]

Generalized almost complex structures (Hitchin,Gualtieri)

Generalized almost complex structures (Hitchin,Gualtieri) J : T + T ∗ → T + T ∗ such that J 2 = − Id

Generalized almost complex structures (Hitchin,Gualtieri) J : T + T ∗ → T + T ∗ such that J 2 = − Id orthogonal w.r.t. the pairing �J v , J w � = � v , w �

Generalized almost complex structures (Hitchin,Gualtieri) J : T + T ∗ → T + T ∗ such that J 2 = − Id orthogonal w.r.t. the pairing �J v , J w � = � v , w � Examples: � − J � 0 0 J ∗ for J almost cplx. str.

Generalized almost complex structures (Hitchin,Gualtieri) J : T + T ∗ → T + T ∗ such that J 2 = − Id orthogonal w.r.t. the pairing �J v , J w � = � v , w � Examples: � − J � 0 � − ω − 1 � 0 0 J ∗ ω 0 for J almost cplx. str. for ω presymplectic.

Generalized almost complex structures (Hitchin,Gualtieri) J : T + T ∗ → T + T ∗ such that J 2 = − Id orthogonal w.r.t. the pairing �J v , J w � = � v , w � Examples: � − J � 0 � − ω − 1 � 0 0 J ∗ ω 0 for J almost cplx. str. for ω presymplectic. Constraint: M must admit almost cplx. str. → n = 2 m even.

Generalized almost complex structures (Hitchin,Gualtieri) J : T + T ∗ → T + T ∗ such that J 2 = − Id orthogonal w.r.t. the pairing �J v , J w � = � v , w � Examples: � − J � 0 � − ω − 1 � 0 0 J ∗ ω 0 for J almost cplx. str. for ω presymplectic. Constraint: M must admit almost cplx. str. → n = 2 m even. Almost cplx. str.: reduction from GL (2 m , R ) to GL ( m , C ).

Generalized almost complex structures (Hitchin,Gualtieri) J : T + T ∗ → T + T ∗ such that J 2 = − Id orthogonal w.r.t. the pairing �J v , J w � = � v , w � Examples: � − J � 0 � − ω − 1 � 0 0 J ∗ ω 0 for J almost cplx. str. for ω presymplectic. Constraint: M must admit almost cplx. str. → n = 2 m even. Almost cplx. str.: reduction from GL (2 m , R ) to GL ( m , C ). Generalized ones: reduction from O (2 m , 2 m ) to U ( m , m ).

Equivalently, one could define J by giving:

Equivalently, one could define J by giving: � � ∂ ∂ • the analogue to (1 , 0)-vectors, span ∂ z 1 , . . . , , i.e., ∂ z m

Equivalently, one could define J by giving: � � ∂ ∂ • the analogue to (1 , 0)-vectors, span ∂ z 1 , . . . , , i.e., ∂ z m the + i -eigenspace of J in ( T + T ∗ ) C , i.e.,

Equivalently, one could define J by giving: � � ∂ ∂ • the analogue to (1 , 0)-vectors, span ∂ z 1 , . . . , , i.e., ∂ z m the + i -eigenspace of J in ( T + T ∗ ) C , i.e., a maximal isotropic subbundle L ⊂ ( T + T ∗ ) C such that L ∩ L = 0.

Equivalently, one could define J by giving: � � ∂ ∂ • the analogue to (1 , 0)-vectors, span ∂ z 1 , . . . , , i.e., ∂ z m the + i -eigenspace of J in ( T + T ∗ ) C , i.e., a maximal isotropic subbundle L ⊂ ( T + T ∗ ) C such that L ∩ L = 0. • the analogue to the (local) form d ¯ z 1 ∧ . . . ∧ d ¯ z m ,

Equivalently, one could define J by giving: � � ∂ ∂ • the analogue to (1 , 0)-vectors, span ∂ z 1 , . . . , , i.e., ∂ z m the + i -eigenspace of J in ( T + T ∗ ) C , i.e., a maximal isotropic subbundle L ⊂ ( T + T ∗ ) C such that L ∩ L = 0. • the analogue to the (local) form d ¯ z 1 ∧ . . . ∧ d ¯ z m , which is a form ϕ ∈ Ω • ( M ) C such that ( ϕ, ϕ ) � = 0 for ( α, β ) = [ α T ∧ β ] top .

Equivalently, one could define J by giving: � � ∂ ∂ • the analogue to (1 , 0)-vectors, span ∂ z 1 , . . . , , i.e., ∂ z m the + i -eigenspace of J in ( T + T ∗ ) C , i.e., a maximal isotropic subbundle L ⊂ ( T + T ∗ ) C such that L ∩ L = 0. • the analogue to the (local) form d ¯ z 1 ∧ . . . ∧ d ¯ z m , which is a form ϕ ∈ Ω • ( M ) C such that ( ϕ, ϕ ) � = 0 for ( α, β ) = [ α T ∧ β ] top . For the action ( X + ξ ) · ϕ = i X ϕ + ξ ∧ ϕ , we want Ann( ϕ ) = L , the maximal isotropic subbundle.

Equivalently, one could define J by giving: � � ∂ ∂ • the analogue to (1 , 0)-vectors, span ∂ z 1 , . . . , , i.e., ∂ z m the + i -eigenspace of J in ( T + T ∗ ) C , i.e., a maximal isotropic subbundle L ⊂ ( T + T ∗ ) C such that L ∩ L = 0. • the analogue to the (local) form d ¯ z 1 ∧ . . . ∧ d ¯ z m , which is a form ϕ ∈ Ω • ( M ) C such that ( ϕ, ϕ ) � = 0 for ( α, β ) = [ α T ∧ β ] top . For the action ( X + ξ ) · ϕ = i X ϕ + ξ ∧ ϕ , we want Ann( ϕ ) = L , the maximal isotropic subbundle. As ( X + ξ ) 2 · ϕ = i X ξϕ = � X + ξ, X + ξ � ϕ ,

Equivalently, one could define J by giving: � � ∂ ∂ • the analogue to (1 , 0)-vectors, span ∂ z 1 , . . . , , i.e., ∂ z m the + i -eigenspace of J in ( T + T ∗ ) C , i.e., a maximal isotropic subbundle L ⊂ ( T + T ∗ ) C such that L ∩ L = 0. • the analogue to the (local) form d ¯ z 1 ∧ . . . ∧ d ¯ z m , which is a form ϕ ∈ Ω • ( M ) C such that ( ϕ, ϕ ) � = 0 for ( α, β ) = [ α T ∧ β ] top . For the action ( X + ξ ) · ϕ = i X ϕ + ξ ∧ ϕ , we want Ann( ϕ ) = L , the maximal isotropic subbundle. As ( X + ξ ) 2 · ϕ = i X ξϕ = � X + ξ, X + ξ � ϕ , Ω • ( M ) C are (up to scaling) spinors , and ϕ must be pure .

Pure spinors are either even or odd (the spin representation splits).

Pure spinors are either even or odd (the spin representation splits). E.g., ϕ = ϕ 0 + ϕ 2 + ϕ 4 + . . .

Pure spinors are either even or odd (the spin representation splits). E.g., ϕ = ϕ 0 + ϕ 2 + ϕ 4 + . . . The type at a point p is the least index j for which ϕ j ( p ) � = 0.

Pure spinors are either even or odd (the spin representation splits). E.g., ϕ = ϕ 0 + ϕ 2 + ϕ 4 + . . . The type at a point p is the least index j for which ϕ j ( p ) � = 0. Cplx. str. are of type m ; symplectic ones are of type 0 ( ϕ = e i ω ).

Pure spinors are either even or odd (the spin representation splits). E.g., ϕ = ϕ 0 + ϕ 2 + ϕ 4 + . . . The type at a point p is the least index j for which ϕ j ( p ) � = 0. Cplx. str. are of type m ; symplectic ones are of type 0 ( ϕ = e i ω ). In a 4-manifold we can have ϕ 0 + ϕ 2 + ϕ 4 , with ϕ 0 vanishing at a codimension 2 submanifold.

Pure spinors are either even or odd (the spin representation splits). E.g., ϕ = ϕ 0 + ϕ 2 + ϕ 4 + . . . The type at a point p is the least index j for which ϕ j ( p ) � = 0. Cplx. str. are of type m ; symplectic ones are of type 0 ( ϕ = e i ω ). In a 4-manifold we can have ϕ 0 + ϕ 2 + ϕ 4 , with ϕ 0 vanishing at a codimension 2 submanifold. For example, in R 4 ∼ = C 2 , ϕ = z 1 + dz 1 ∧ dz 2

Pure spinors are either even or odd (the spin representation splits). E.g., ϕ = ϕ 0 + ϕ 2 + ϕ 4 + . . . The type at a point p is the least index j for which ϕ j ( p ) � = 0. Cplx. str. are of type m ; symplectic ones are of type 0 ( ϕ = e i ω ). In a 4-manifold we can have ϕ 0 + ϕ 2 + ϕ 4 , with ϕ 0 vanishing at a codimension 2 submanifold. For example, in R 4 ∼ = C 2 , ϕ = z 1 + dz 1 ∧ dz 2 When looking at integrable structures ( L involutive w.r.t. [ · , · ]):

Pure spinors are either even or odd (the spin representation splits). E.g., ϕ = ϕ 0 + ϕ 2 + ϕ 4 + . . . The type at a point p is the least index j for which ϕ j ( p ) � = 0. Cplx. str. are of type m ; symplectic ones are of type 0 ( ϕ = e i ω ). In a 4-manifold we can have ϕ 0 + ϕ 2 + ϕ 4 , with ϕ 0 vanishing at a codimension 2 submanifold. For example, in R 4 ∼ = C 2 , ϕ = z 1 + dz 1 ∧ dz 2 When looking at integrable structures ( L involutive w.r.t. [ · , · ]): there are compact generalized complex manifolds that do not admit neither complex nor symplectic structures (3 C P 2 #19 C P 2 by Gualtieri/Cavalcanti, and more by Rafael Torres, Wed 18:40).

B n -generalized geometry

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