THE GEOMETRY OF B-FIELDS. Nigel Hitchin (Oxford) Odense November 26th 2009

Transcription

1 THE GEOMETRY OF B-FIELDS Nigel Hitchin (Oxford) Odense November 26th 2009

2 THE B-FIELD IN PHYSICS B = i,j B ij dx i dx j flux: db = H a closed three-form Born-Infeld action: det(g ij + B ij ) complexified Kähler moduli : [B + iω] H 2 (M, C)

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5 A: VECTOR POTENTIAL Maxwell B = 0 B = A

6 TOPOLOGY

7 φ = m/r, 2 φ = 0 B = φ on R 3 \{0} what is A?

8 φ = m/r, 2 φ = 0 B = φ on R 3 \{0} } what is A? on a contractible open set U α, find A α with A α = B

9 φ = m/r, 2 φ = 0 B = φ on R 3 \{0} } what is A? on a contractible open set U α, find A α with A α = B... on U α U β, A β A α = f αβ

10 φ = m/r, 2 φ = 0 B = φ on R 3 \{0} } what is A? on a contractible open set U α, find A α with A α = B... on U α U β, A β A α = f αβ f αβ + f βγ + f γα = const. = c αβγ on U α U β U γ

11 Suppose φ = k/2r, k Z, (Dirac quantization condition)...then find f αβ such that c αβγ 2πZ Define g αβ = exp(if αβ ), then... g αβ g βγ g γα = 1

12 Stick together U α S 1 and U β S 1 using g αβ to get a principal S 1 -bundle P.

13 Stick together U α S 1 and U β S 1 using g αβ to get a principal S 1 -bundle P. A α is a connection on P : A = i A i dx i A β A α = g 1 αβ dg αβ/i da α = da β = curvature F

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16 GERBES g αβγ : U α U β U γ S 1 (g αβγ = g 1 βαγ =...) δg = g βγδ g 1 αγδ g αβδg 1 αβγ = 1 on U α U β U γ U δ

17 CONNECTIONS ON GERBES

18 CONNECTIONS ON GERBES Connective structure: A αβ + A βγ + A γα = g 1 αβγ dg αβγ/i

19 CONNECTIONS ON GERBES Connective structure: A αβ + A βγ + A γα = g 1 αβγ dg αβγ/i Curving: B β B α = da αβ db β = db α = H Uα global three-form H

20 principal circle bundle: connection A α A Ω 1, A α A + A α new connection, F F + da

21 principal circle bundle: connection A α A Ω 1, A α A + A α new connection, F F + da gerbe: connective structure + curving A αβ, B α B Ω 2, B α B + B α new curving, H H + db

22 ALGEBRA OF BRACKETS

23 BRACKETS Invariant vector field on P = M S 1 X + f section of T 1 [X + f, Y + g] = [X, Y ] + Xg Y f

24 BRACKETS Invariant vector field on P = M S 1 X + f section of T 1 [X + f, Y + g] = [X, Y ] + Xg Y f (X, f) (X, f + i X A) If da = 0 bracket unchanged

25 BRACKETS Invariant vector field on P = M S 1 X + f section of T 1 [X + f, Y + g] = [X, Y ] + Xg Y f (X, f) (X, f + i X A) If da = 0 bracket unchanged A = g 1 dg gauge transformation

26 X + ξ, Y + η sections of T T Courant bracket: [X + ξ, Y + η] = [X, Y ] + L X η L Y ξ 1 2 d(i Xη i Y ξ)

27 X + ξ, Y + η sections of T T Courant bracket: [X + ξ, Y + η] = [X, Y ] + L X η L Y ξ 1 2 d(i Xη i Y ξ)

28 X + ξ, Y + η sections of T T Courant bracket: [X + ξ, Y + η] = [X, Y ] + L X η L Y ξ 1 2 d(i Xη i Y ξ) B two-form (X, ξ) (X, ξ + i X B) If db = 0, the bracket is unchanged

29 GENERALIZED GEOMETRY

30 53C99 None of the above, but in this section 53Dxx Symplectic geometry, contact geometry [See also 37Jxx, 70Gxx, 70Hxx] 53D05 Symplectic manifolds, general 53D10 Contact manifolds, general 53D12 Lagrangian submanifolds; Maslov index 53D15 Almost contact and almost symplectic manifolds 53D17 Poisson manifolds; Poisson groupoids and algebroids 53D18 Generalized geometries (à la Hitchin) 53D20 Momentum maps; symplectic reduction 53D22 Canonical transformations 53D25 Geodesic flows Pro etc. 54Axx Gen 54A05 Top (clo 54A10 Sev top tice 54A15 Syn 54A20 Con (seq spa 54A25 Car tion

31 manifold M n replace T by T T inner product of signature (n, n) (X + ξ, X + ξ) = i X ξ

32 manifold M n replace T by T T inner product of signature (n, n) (X + ξ, X + ξ) = i X ξ skew adjoint transformations: End T Λ 2 T Λ 2 T... in particular B Λ 2 T

33 B-FIELDS two-form B X + ξ X + ξ + i X B... is an orthogonal transformation of T T... which preserves the Courant bracket if B is closed

34 Generalized symmetry group G = Diff(M) Ω 2 closed (M) X + ξ X + dξ g [X + ξ, Y + η] [X + dξ, Y + dη]

35 Riemannian metric g ij X g(x, ) : g : T T graph of g = V T T X + gx V, inner product (X + gx, X + gx) = g(x, X)

36 T * graph of g T graph of!g

37 GENERALIZED RIEMANNIAN METRIC V T T positive definite rank n subbundle = graph of g + B : T T g + B T T : g symmetric, B skew

38 GENERALIZED RIEMANNIAN METRIC V T T positive definite rank n subbundle = graph of g + B : T T g + B T T : g symmetric, B skew T T natural structure group SO(n, n) generalized metric: reduction to maximal compact SO(n) SO(n)

39 T * graph of g T graph of!g

40 T * graph of g X T graph of!g

41 T * graph of g X + X T X graph of!g

42 [X, Y + ] defines the Levi-Civita connection X Y 2g lk Γ l ij dx k = = [ x i g ik dx k, ( gjk + g ik g ) ij dx k x i x j x k ] + g jk dx k x j

43 B-FIELD TRANSFORM 2g lk Γ l ij dx k = = [ x i g ik dx k + B ik dx k, ( gjk + g ik g ) ij dx k + x i x j x k ] + g jk dx k + B jk dx k x j ( Bjk x i B ) ik dx k. x j H ijk = B jk x i B ik x j Riemannian connection with skew torsion H = db

44 B-FIELD TRANSFORM 2g lk Γ l ij dx k = = [ x i g ik dx k + B ik dx k, ( gjk + g ik g ) ij dx k + x i x j x k ] + g jk dx k + B jk dx k x j ( Bjk x i B ) ik dx k. x j H ijk = B jk x i B ik x j Riemannian connection with skew torsion H = db flux

45 THE B-FIELD IN PHYSICS B = i,j B ij dx i dx j flux: db = H a closed three-form Born-Infeld action: det(g ij + B ij ) complexified Kähler moduli : [B + iω] H 2 (M, C)

46 THE B-FIELD IN PHYSICS B = i,j B ij dx i dx j flux: db = H a closed three-form Born-Infeld action: det(g ij + B ij ) complexified Kähler moduli : [B + iω] H 2 (M, C)

47 GENERALIZED COMPLEX STRUCTURES

48 A generalized complex structure is: J : T T T T, J 2 = 1 (JA, B) + (A, JB) = 0 if JA = ia, JB = ib then J[A, B] = i[a, B] (Courant bracket)

49 A generalized complex structure is: J : T T T T, J 2 = 1 (JA, B) + (A, JB) = 0 if JA = ia, JB = ib then J[A, B] = i[a, B] (Courant bracket) U(m, m) SO(2m, 2m) structure on T T

50 EXAMPLES complex manifold J = ( I 0 0 I ) J = i : [... / z i...,... d z i...] symplectic manifold J = ( 0 ω 1 ω 0 ) J = i : [..., / x j + i ω jk dx k,...]

51 T T T is isotropic and closed under Courant bracket

52 T T T is isotropic and closed under Courant bracket iω = complex B-field (symplectic structure) X + i X (iω) spans +i-eigenspace of J

53 T T T is isotropic and closed under Courant bracket iω = complex B-field (symplectic structure) X + i X (iω) spans +i-eigenspace of J transform by any closed B: X + i X (B + iω) defines a generalized complex structure

54 T T T is isotropic and closed under Courant bracket iω = complex B-field (symplectic structure) X + i X (iω) spans +i-eigenspace of J transform by any closed B: X + i X (B + iω) defines a generalized complex structure

55 WHAT IS THE EFFECT OF A B-FIELD?

56 generalized geometric structure closed 2-form B transform to a new structure

57 GENERALIZED METRIC 2g lk Γ l ij dx k = = [ x i g ik dx k + B ik dx k, ( gjk + g ik g ) ij dx k + x i x j x k ] + g jk dx k + B jk dx k x j ( Bjk x i B ) ik dx k. x j H ijk = B jk x i B ik x j Riemannian connection with skew torsion H = db B closed no change

58 GENERALIZED COMPLEX SUBMANIFOLDS Submanifold Y M, 0 T Y T M Y NY 0 T Y (T Y ) o = T Y N Y (T M T M) Y J preserves T Y N Y generalized complex submanifold

59 COMPLEX THE COMPLEX CASE Generalized complex submanifold = complex submanifold

60 COMPLEX THE COMPLEX CASE Generalized complex submanifold = complex submanifold SYMPLECTIC ) ) ( 0 ω 1 ω 0 ) ( X ξ = ( ω 1 ξ ωx ω 1 ξ T Y Y coisotropic ωx N Y isotropic Y is Lagrangian submanifold

61 SYMPLECTIC + B-FIELD J ( X ξ ) = ( ω 1 ) (ξ + BX) (ω + Bω 1 B)X + Bω 1 ξ

62 SYMPLECTIC + B-FIELD J ( ) ( ( X ) ( ξ = ω 1 (ξ + BX) (ω + Bω 1 B)X + Bω 1 ξ ) X = 0 ω 1 ξ T Y Y coisotropic

63 SYMPLECTIC + B-FIELD J ( ) ( X ω 1 ) ( ) ( (ξ + BX) ) ξ = (ω + Bω 1 B)X + Bω 1 ξ X = 0 ω 1 ξ T Y Y coisotropic Im ω 1 ξ = foliation, transversally symplectic B + iω transverse holomorphic symplectic form

64 e.g. HOLOMORPHIC SYMPLECTIC MANIFOLD complex manifold M holomorphic symplectic form ω c = ω 1 + iω 2 = B + iω

65 e.g. HOLOMORPHIC SYMPLECTIC MANIFOLD complex manifold M holomorphic symplectic form ω c = ω 1 + iω 2 = B + iω a generalized complex submanifold is a coisotropic complex submanifold e.g. any complex hypersurface, or M itself.

66 GENERALIZED COMPLEX STRUCTURES complex manifold (T T ) 0,1 = [... / z i...,... d z i...] B type (1, 1), then z i + i / zi B = z i + B i j d z j (T T ) 1,0 B closed B-field which preserves the generalized complex structure

67 GENERALIZED COMPLEX STRUCTURES complex manifold (T T ) 0,1 = [... / z i...,... d z i...] B type (1, 1), then z i + i / zi B = z i + B i j d z j (T T ) 1,0 B closed B-field which preserves the generalized complex structure Hamiltonian f symmetry f

68 complex structure on T T df T T T f = (T T ) 0,1 component

69 complex structure on T T df T T T f = (T T ) 0,1 component complex (T T ) 0,1 = [... z i...,... d z i...], f = f z i d z i symplectic f = 1 2 x i ( dx i + iω ij x j )

70 A generalized holomorphic bundle is: a vector bundle V a differential operator D : C (V ) C (V (T T ) 0,1 ) such that D(fs) = f Ds + fs and D 2 = 0 C (V Λ 2 (T T ) 0,1 )

71 EXAMPLE complex structure: co-higgs bundle: (i) holomorphic structure on V (ii) holomorphic section φ H 0 (M, End V T ) (iii)... such that φ 2 = 0 H 0 (M, End V Λ 2 T ) Ds = ( s z µ + A µ s ) d z µ + φ ν s z ν

72 B-FIELD TRANSFORM Ds = ( ) s + A µ s d z µ + φ ν s z µ z ν Transform: Ds = ( ) s + A µ + φ ν B ν µ s d z µ + φ ν s z µ z ν New holomorphic structure B = + i φ B B φ = 0

73 EXAMPLE: DIMENSION ONE M = CP 1 tangent bundle T = O(2) generically V is trivial, φ H 0 (CP 1, O(2)) gl(m) φ = (φ 0 + φ 1 z + φ 2 z 2 ) d dz φ 2 = 0 trivially

74 spectral curve det(η φ) = η m + a 1 η m a m = 0 S T CP 1 p CP 1 a i H 0 (CP 1, O(2i)), η H 0 (T CP 1, p O(2)) tautological L line bundle on S V = p L, φ = p η : V V (2) direct image

75 DIRECT IMAGE SHEAF p : X Y U Y open set H 0 (U, p L) = H 0 (p 1 (U), L) p : S CP 1 m-fold covering V = p L rank m vector bundle

76 B Ω 1,1 (CP 1 ) change holomorphic structure on V by V V + i φ B

77 B Ω 1,1 (CP 1 ) change holomorphic structure on V by V V + i φ B change holomorphic structure on L by L L + i η B [i η B] H 1 (T CP 1, O) L L + ti η B linear flow on Jacobian of S

78 NAHM FLOW φ H 0 (CP 1, O(2)) gl(m) V trivial φ = (T 1 + it 2 ) + 2T 3 z (T 1 it 2 )z 2 Nahm equations: dt 1 dt = [T 2, T 3 ] dt 2 dt = [T 3, T 1 ] dt 3 dt = [T 1, T 2 ] NJH, On the construction of monopoles, Comm. Math. Phys. 89 (1983)

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81 line bundles gerbes 2-gerbes...

82 line bundles gerbes 2-gerbes p 1 (T ), Chern-Simons,...

83 T 1 T T T Λ 2 T

84 T 1 T T T Λ 2 T [X + ξ, Y + η] = [X, Y ] + L X η L Y ξ 1 2 d(i Xη i Y ξ)

85 C.M.Hull, Generalised geometry for M-theory, J.High Energy Phys. 07, (2007) P.P.Pacheco & D.Waldram, Exceptional generalised geometry and superpotentials, arxiv/hep-th:

86 n E n R SL(n, ) reps E 2 SL(2, ) E T Λ 2 T 3 SL(3, ) SL(2, ) (3, 2) E T Λ 2 T 4 SL(5, ) E T Λ 2 T 5 Spin(5, 5) E T Λ 2 T Λ 5 T 6 E 6(6) 27(+1) (+1) E T Λ 2 T Λ 5 T ( Λ 6 T ) 7 E 7(7) E T Λ 2 T Λ 5 T Λ 6 T

87 n E n R SL(n, ) reps E 2 SL(2, ) E T Λ 2 T 3 SL(3, ) SL(2, ) (3, 2) E T Λ 2 T 4 SL(5, ) E T Λ 2 T 5 Spin(5, 5) E T Λ 2 T Λ 5 T 6 E 6(6) 27(+1) (+1) E T Λ 2 T Λ 5 T ( Λ 6 T ) 7 E 7(7) E T Λ 2 T Λ 5 T Λ 6 T