# n -dimensional manifold M with T := TM n -dimensional manifold M with - PowerPoint PPT Presentation

## 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

1. n -dimensional manifold M with T := TM

2. n -dimensional manifold M with T := TM T

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

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

5. 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

6. 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

7. 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

8. 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 [ , ]

9. 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 [ , ]

10. Generalized almost complex structures (Hitchin,Gualtieri)

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

12. 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 �

13. 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.

14. 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.

15. 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.

16. 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 ).

17. 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 ).

18. Equivalently, one could define J by giving:

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

20. 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.,

21. 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.

22. 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 ,

23. 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 .

24. 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.

25. 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 ,

26. 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 .

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

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

29. 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.

30. 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 ).

31. 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.

32. 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

33. 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. [ · , · ]):

34. 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).

35. B n -generalized geometry

Recommend

More recommend