Nigel Hitchin (Oxford) Poisson 2012 Utrecht. August 2nd 2012

Transcription

1 GENERALIZED GEOMETRY OF TYPE B n Nigel Hitchin (Oxford) Poisson 2012 Utrecht August 2nd 2012

2 generalized geometry on M n T T inner product (X + ξ,x + ξ) =i X ξ SO(n, n)-structure

3 generalized geometry on M n T T inner product (X + ξ,x + ξ) =i X ξ SO(n, n)-structure type D n

4 extended generalized geometry on M n T 1 T inner product (X + λ + ξ,x + λ + ξ) =i X ξ + λ 2 SO(n +1,n)-structure type B n

5 C M Hull, Generalised geometry for M-theory, JHEP 07 (2007) D Baraglia, Leibniz algebroids, twistings and exceptional generalized geometry, arxiv: P Bouwknegt, Courant algebroids and generalizations of geometry, String-Math 2011 Roberto Rubio Mario Garcia Fernandez

6 so(n +1,n)=Λ 2 T + T + End T + T + Λ 2 T A T,B Λ 2 T

7 so(n +1,n)=Λ 2 T + T + End T + T + Λ 2 T A T,B Λ 2 T X + λ + ξ X +(λ + i X A)+(ξ 2λA i X AA) X + λ + ξ X + λ + ξ + i X B

8 ACTION ON VECTORS so(n +1,n)=Λ 2 T + T + End T + T + Λ 2 T A T,B Λ 2 T X + λ + ξ X +(λ + i X A)+(ξ 2λA i X AA) X + λ + ξ X + λ + ξ + i X B (A, B).(A,B )=(A + A,B+ B 2A A )

9 SPINORS exterior algebra Λ T τ(ϕ) =( 1) deg ϕ ϕ Clifford multiplication (X + λ + ξ) ϕ = i X ϕ + λτϕ + ξ ϕ exp B(ϕ) =e B ϕ exp A(ϕ) =e Aτ ϕ = ϕ A τϕ

10 SPINORS exterior algebra Λ T τ(ϕ) =( 1) deg ϕ ϕ Clifford multiplication (X + λ + ξ) ϕ = i X ϕ + λτϕ + ξ ϕ exp B(ϕ) =e B ϕ exp A(ϕ) =e Aτ ϕ = ϕ A τϕ

11 AUTOMORPHISMS generalized diffeomorphisms GDiff(M) =Diff(M) Ω 2+1 cl Ω 2+1 cl = {(A, B) Ω 1 Ω 2 : da = db =0} (A, B).(A,B )=(A + A,B+ B 2A A ) 0 Ω 2 cl Ω2+1 cl Ω 1 cl 0 central extension

12 generalized vector field u = X + λ + ξ u X dλ dξ gdiff(m) Lie derivative on T +1+T = Dorfman bracket Lie derivative on spinors d(u ϕ)+u dϕ

13 skew-symmetrize action on T +1+T, get a Courant bracket [u, fv] =f[u, v]+(xf)v (u, v)df [[u, v],w]+...= 1 d(([u, v],w)+...) 3 Z Chen, M Stienon and P Xu, On regular Courant algebroids, arxiv:

14 WHAT S NEW?

15 G 2 2 -STRUCTURES TWISTING AND COHOMOLOGY CONNECTIONS, TORSION AND CURVATURE

16 G2 2 MANIFOLDS 22

17 G 2 2 SO(4, 3) stabilizer of a non-null spinor ρ Λ T, ρ = ρ 0 + ρ 1 + ρ 2 + ρ 3 ρ, ρ = ρ 0 ρ 3 ρ 1 ρ 2 =0

18 G 2 2 SO(4, 3) stabilizer of a non-null spinor ρ Λ T, ρ = ρ 0 + ρ 1 + ρ 2 + ρ 3 ρ, ρ = ρ 0 ρ 3 ρ 1 ρ 2 =0 G 2 2 manifold: M 3 + closed form ρ with ρ, ρ =0

19 {f, g} = df dg ρ 1 ρ, ρ Poisson structure

20 EXAMPLES ρ =1+ρ 3, ρ 3 volume form. ρ 0 =0 ρ 1 ρ 2 =0 non-vanishing closed 1-form ρ 1 M 3 fibres over a circle. p : M S 1, ρ 1 λp dθ <, λ Q

21 ρ = ρ 1 + ρ 2 unique vector field X such that i X ρ 2 =0,i X ρ 1 =1

22 ρ = ρ 1 + ρ 2 unique vector field X such that i X ρ 2 =0,i X ρ 1 =1 p : M S 1, p dθ = ρ 1 ρ 2 = symplectic form on fibre, L X ρ 2 = d(i X ρ 2 )=0 mapping torus of a symplectic diffeomorphism of surface

23 QUESTIONS IF ρ, ρ ARE COHOMOLOGOUS AND SUFFICIENTLY CLOSE, ARE THEY EQUIVALENT UNDER A GENERALIZED DIF- FEOMORPHISM? WHICH COHOMOLOGY CLASSES CONTAIN G 2 2 STRUC- TURES?

24 QUESTIONS IF ρ, ρ ARE COHOMOLOGOUS AND SUFFICIENTLY CLOSE, ARE THEY EQUIVALENT UNDER A GENERALIZED DIF- FEOMORPHISM? WHICH COHOMOLOGY CLASSES CONTAIN G 2 2 STRUC- TURES?

25 ρ ρ = dϕ ρ t = ρ + tdϕ find generalized vector field u t such that u t ρ t = ϕ... integrate to a generalized diffeomorphism (Moser argument)

26 u T +1+T, ρ Λ T π : Λ T Λ T /Λ 3 T ρ 0 = 0 then u π(u ρ) is invertible

27 ρ 0 = 0? Assume ρ 1 = p dθ image u u ρ is {ψ : ρ, ψ =0} ϕ 2 ϕ 2 + λρ 2 ρ, ϕ = ρ 1 ϕ 2 + ρ 2 ϕ 1 ρ 3 ϕ 0 = dα

28 ρ 0 = 0? Assume ρ 1 = p dθ image u u ρ is {ψ : ρ, ψ =0} ϕ 2 ϕ 2 + λρ 2 ρ, ϕ = ρ 1 ϕ 2 + ρ 2 ϕ 1 ρ 3 ϕ 0 = dα integrate over fibres, α fρ 2 = dβ + ρ 1 γ dα = df ρ 2 ρ 1 dγ

29 QUESTIONS IF ρ, ρ ARE COHOMOLOGOUS AND SUFFICIENTLY CLOSE, ARE THEY EQUIVALENT UNDER A GENERALIZED DIF- FEOMORPHISM? WHICH COHOMOLOGY CLASSES CONTAIN G 2 2 STRUC- TURES?

30 ρ 0 = 0, use A = ρ 1 /ρ 0,B = ρ 2 /ρ 0 transform to ρ 0 +(ρ 3 ρ 1 ρ 2 /ρ 0 ) ρ equivalent to ρ 0 (1 + vol)

31 ρ 0 = 0, use A = ρ 1 /ρ 0,B = ρ 2 /ρ 0 transform to ρ 0 +(ρ 3 ρ 1 ρ 2 /ρ 0 ) ρ equivalent to ρ 0 (1 + vol) ρ 0 = 0: which classes in H 1 (M 3, R) are represented by nonvanishing 1-forms? Answer: cone on faces of the unit ball in the Thurston norm.

32 TWISTING

33 D n generalized geometry, T + T identify T + T U with T + T V over U V X + ξ X + ξ + i X B UV B UV 1-cocycle in Ω 2 cl

34 D n generalized geometry, T + T identify T + T U with T + T V over U V X + ξ X + ξ + i X B UV B UV 1-cocycle in Ω 2 cl Courant algebroid 0 T E T 0

35 B n generalized geometry, T +1+T identify with (A UV,B UV ) 1-cocycle in Ω 2+1 cl Courant algebroid 0 T E L L T 0

36 B n generalized geometry, T +1+T identify with (A UV,B UV ) 1-cocycle in Ω 2+1 cl Courant algebroid 0 T E L L T 0 L Lie algebroid (like TP/U(1) for P a U(1)-bundle over M)

37 [(A UV,B UV )] H 1 (M, Ω 2+1 cl ) 0 Ω 2 cl Ω2+1 cl Ω 1 cl 0 sheaves H 1 (M, Ω 2 cl ) H1 (M, Ω 2+1 cl ) H 1 (M, Ω 1 cl ) δ H 2 (M, Ω 2 cl )

38 [(A UV,B UV )] H 1 (M, Ω 2+1 cl ) 0 Ω 2 cl Ω2+1 cl Ω 1 cl 0 sheaves H 1 (M, Ω 2 cl ) H1 (M, Ω 2+1 cl ) H 1 (M, Ω 1 cl ) δ H 2 (M, Ω 2 cl ) H 3 (M, R) H 2 (M, R) H 4 (M, R) δ(x) =x 2

39 [(A UV,B UV )] H 1 (M, Ω 2+1 cl ) 0 Ω 2 cl Ω2+1 cl Ω 1 cl 0 sheaves H 1 (M, Ω 2 cl ) H1 (M, Ω 2+1 cl ) H 1 (M, Ω 1 cl ) δ H 2 (M, Ω 2 cl ) H 3 (M, R) H 2 (M, R) H 4 (M, R) δ(x) =x 2 F Ω 2 cl F 2 = dh

40 Λ T U, Λ T V, identify with exp( A UV τ + B UV ) spinor bundle S, differential d

41 Λ T U, Λ T V, identify with exp( A UV τ + B UV ) spinor bundle S, differential d or... split extension, and get d + F τ + H : Ω Ω (d + F τ + H) 2 =0

42 WHAT IS THIS COHOMOLOGY?

43 Suppose F = curvature of a U(1)-connection connection form θ on principal bundle P d(h + θf )=0 P is T -dual to itself

44 T (α + θβ) =β θα T 1 (d + H + θf )T = (d + H + θf )

45 T (α + θβ) =β θα T 1 (d + H + θf )T = (d + H + θf ) (d + H + θf )(α + θτα) =(1+θτ)(dα + F τα+ Hα) τt (α + θτα) =τ(τα θα) =α + θτα

46 T τ = τt (τt ) 2 =1 H = H + θf closed 3-form (H, F )-twisted cohomology of M = τt -invariant part of d + H-cohomology of P.

47 T τ = τt (τt ) 2 =1 H = H + θf closed 3-form (H, F )-twisted cohomology of M = τt -invariant part of d + H-cohomology of P. Generalization of twisted K-theory?

48 CONNECTIONS AND TORSION

49 CONNECTIONS generalized connection on W = D : W E W D(fs)=fDs + df s (df T E) = ordinary connection + section of E End W u E,D u s =(Ds, u) W frame bundle, principal bundle...

50 TORSION generalized connection D : E E E torsion for affine connection : X Y Y X [X, Y ] Courant bracket [u, fv] =f[u, v]+(xf)v (u, v)df T (u, v, w) =(D uv D v u [u, v],w)+ 1 2 (D wu, v) 1 2 (D wv, u)

51 TORSION generalized connection D : E E E torsion for affine connection : X Y Y X [X, Y ] Courant bracket [u, fv] =f[u, v]+(xf)v (u, v)df T (u, v, w) =(D uv D v u [u, v],w)+ 1 2 (D wu, v) 1 2 (D wv, u) D preserves inner product torsion in Λ 3 E

52 T T T π Λ 2 T torsion for affine connection = p d

53 T T T π Λ 2 T torsion for affine connection = p d spinors S, D : S E S Dirac operator: S D E S Cliff S torsion = Cliff D d

54 METRICS generalized metric: reduction of SO(n +1,n) structure of T +1+T (or E) tos(o(n + 1) O(n)) rank n subbundle V E on which induced metric is negative definite. V positive definite e.g. V = {X gx T +1+T },V = {X + λ + gx}

55 PROBLEM: FIND A TORSION-FREE GENERALIZED CONNECTION WHICH PRESERVES THE GENERALIZED METRIC

56 PROBLEM: FIND A TORSION-FREE GENERALIZED CONNECTION WHICH PRESERVES THE GENERALIZED METRIC E = V V V = T,V = 1 T take Levi-Civita and trivial connection on 1 and add on terms so that Cliff D = d A Coimbra, C Strickland-Constable and D Waldram, Supergravity as generalized geometry I: Type II theories, arxiv:

57 V = {X = X gx},v + = {X + = X + gx},v 0 =1 add terms in Λ 2 V (V + V 0 + V + ) and Λ 2 (V 0 + V + ) (V + V 0 + V + ) so that Cliff D = d + F τ + H

58 dϕ = gx i i ϕ = 1 2 (X+ i X i ) iϕ H ϕ = 1 48 H ijk(x + i X i ) (X+ j X j ) (X+ k X k ) ϕ F τϕ = 1 8 F ij(x i + Xi ) (X+ j Xj ) X 0 ϕ

59 Λ 2 V (V + V 0 + V + ) (H ϕ) = 1 48 H ijkxi Xj X k ϕ (H ϕ) + = 1 16 H ijkx i X j X+ k ϕ (F τϕ) 0 = 1 8 F ijxi Xj X 0 ϕ

60 CURVATURE D u D v D v D u D [u,v]? Courant bracket [u, fv] =f[u, v]+(xf)v (u, v)df OK if (u, v) =0 Curvature End E V V

61 RICCI CONTRACTIONS (End V End V ) V V End V V V V V End V V V V V

62 RICCI CONTRACTIONS (End V End V ) V V End V V V V V End V V V V V generalized metric V E tangent vector Hom(V,V ) define generalized Ricci flow

63 RICCI CURVATURES End V V V V V V V = T (1 + T ) = Sym 2 T + Λ 2 T + T Sym 2 T : R i 1 4 H ijkh jk Λ 2 T : i H ijk

64 RICCI CURVATURES End V V V V V Sym 2 T : R i 1 4 H ijkh jk F ij F j Λ 2 T : i H ijk T : i F ij F ki H jki

65 RICCI CURVATURES End V V V V V Sym 2 T : R i 1 4 H ijkh jk F ij F j Λ 2 T : i H ijk T : i F ij F ki H jki 2 heterotic supergravity + U(1) gauge field