B-FIELDS, GERBES AND GENERALIZED GEOMETRY

B-FIELDS, GERBES AND GENERALIZED GEOMETRY Nigel Hitchin (Oxford) Durham Symposium July 29th 2005 1

PART 1 (i) BACKGROUND (ii) GERBES (iii) GENERALIZED COMPLEX STRUCTURES (iv) GENERALIZED KÄHLER STRUCTURES 2

BACKGROUND 3

BASIC SCENARIO 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 4

BASIC SCENARIO 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 5

BASIC SCENARIO 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 6

TRANSFORMATIONS exponentiate B: X + ξ X + ξ + i X B B Ω 2... the B-field natural group Diff(M) Ω 2 (M) 7

SPINORS Take S = Λ T S = S ev S od Define Clifford multiplication by (X + ξ) ϕ = i X ϕ + ξ ϕ (X + ξ) 2 ϕ = i X ξϕ = (X + ξ, X + ξ)ϕ exp B(ϕ) = (1 + B + 1 2 B B +...) ϕ 8

DERIVATIVES Lie bracket: 2i [X,Y ] α = d([i X, i Y ]α) + 2i X d(i Y α) 2i Y d(i X α) + [i X, i Y ]dα A = X + ξ, B = Y + η use Clifford multiplication A α to define a bracket [A, B]: 2[A, B] α = d((a B B A) α) + 2A d(b α) 2B d(a α) + (A B B A) dα COURANT bracket [X + ξ, Y + η] = [X, Y ] + L X η L Y ξ 1 2 d(i Xη i Y ξ) 9

DERIVATIVES Lie bracket: 2i [X,Y ] α = d([i X, i Y ]α) + 2i X d(i Y α) 2i Y d(i X α) + [i X, i Y ]dα A = X + ξ, B = Y + η use Clifford multiplication A α to define a bracket [A, B]: 2[A, B] α = d((a B B A) α) + 2A d(b α) 2B d(a α) + (A B B A) dα COURANT bracket [X + ξ, Y + η] = [X, Y ] + L X η L Y ξ 1 2 d(i Xη i Y ξ) 10

DERIVATIVES Lie bracket: 2i [X,Y ] α = d([i X, i Y ]α) + 2i X d(i Y α) 2i Y d(i X α) + [i X, i Y ]dα A = X + ξ, B = Y + η use Clifford multiplication A α to define a bracket [A, B]: 2[A, B] α = d((a B B A) α) + 2A d(b α) 2B d(a α) + (A B B A) dα COURANT bracket [X + ξ, Y + η] = [X, Y ] + L X η L Y ξ 1 2 d(i Xη i Y ξ) 11

Apply a 2-form B... d e B de B = d + db [X + ξ, Y + η] [X + ξ, Y + η] 2i X i Y db Diff(M) Ω 2 closed (M) preserves inner product, exterior derivative and Courant bracket. 12

GENERALIZED GEOMETRIC STRUCTURES SO(n, n) compatibility integrability d or Courant bracket transform by Diff(M) Ω 2 closed (M) 13

RIEMANNIAN METRIC Riemannian metric g ij X g(x, ) : g : T T graph of g = V T T T T = V V 14

T * graph of g T graph of g 15

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 16

GERBES 17

GERBES: ČECH 2-COCYCLES 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 This defines a gerbe. 18

CONNECTIONS ON GERBES Connective structure: A αβ + A βγ + A γα = g 1 αβγ dg αβγ Curving: B β B α = da αβ db β = db α = H Uα global three-form H J.-L. Brylinski, Characteristic classes and geometric quantization, Progr. in Mathematics 107, Birkhäuser, Boston (1993) 19

TWISTING T T da αβ + da βγ + da γα = d[g 1 αβγ dg αβγ] = 0 identify T T on U α with T T on U β by X + ξ X + ξ + i X da αβ defines a vector bundle E 0 T E T 0 with... an inner product and a Courant bracket. 20

TWISTED COHOMOLOGY identify Λ T on U α with Λ T on U β by ϕ e da αβϕ defines a vector bundle S = spinor bundle for E d : C (S) C (S) well-defined ker d/ im d = twisted cohomology. 21

... WITH A CURVING ϕ α = e da αβϕ β dϕ α = 0 Curving: B β B α = da αβ e B αϕ α = e B βϕ β = ψ dψ + H ψ = 0 22

... WITH A CURVING ϕ α = e da αβϕ β dϕ α = 0 Curving: B β B α = da αβ e B αϕ α = e B βϕ β = ψ dψ + H ψ = 0 23

... WITH A CURVING ϕ α = e da αβϕ β dϕ α = 0 Curving: B β B α = da αβ e B αϕ α = e B βϕ β = ψ dψ + H ψ = 0 24

... WITH A CURVING ϕ α = e da αβϕ β dϕ α = 0 Curving: B β B α = da αβ e B αϕ α = e B βϕ β = ψ dψ + H ψ = 0 25

Definition: A generalized metric is a subbundle V E such that rk V = dim M and the inner product is positive definite on V. V T = 0 splitting of the sequence 0 T E T 0 V E another splitting difference Hom(T, T ) = T T = Riemannian metric 26

SPLITTINGS IN LOCAL TERMS splitting: C α C (U α, T T ) : C β C α = da αβ Sym(C α ) = Sym(C β ) = metric Alt(C α ) = B α = curving of the gerbe H = db α closed 3-form 27

two splittings V and V of 0 T E T 0 X vector field, lift to X + C (V ) and X C (V ) in E Courant bracket [X +, Y ], Lie bracket [X, Y ] [X +, Y ] [X, Y ] is a one-form 28

two splittings V and V of 0 T E T 0 X vector field, lift to X + C (V ) and X C (V ) in E Courant bracket [X +, Y ], Lie bracket [X, Y ] [X +, Y ] [X, Y ] is a one-form 29

two splittings V and V of 0 T E T 0 X vector field, lift to X + C (V ) and X C (V ) in E Courant bracket [X +, Y ], Lie bracket [X, Y ] [X +, Y ] [X, Y ] is a one-form 30

two splittings V and V of 0 T E T 0 X vector field, lift to X + C (V ) and X C (V ) in E Courant bracket [X +, Y ], Lie bracket [X, Y ] [X +, Y ] [X, Y ] + is a one-form 31

[X +, Y ] [X, Y ] + = 2g( + X Y ) + Riemannian connection with skew torsion H [X, Y + ] [X, Y ] + = 2g( XY ) has skew torsion H 32

[X +, Y ] [X, Y ] + = 2g( + X Y ) + Riemannian connection with skew torsion H [X +, Y ] [X, Y ] + = 2g( XY ) has skew torsion H 33

[X +, Y ] [X, Y ] + = 2g( + X Y ) + Riemannian connection with skew torsion H [X, Y + ] [X, Y ] = 2g( XY ) has skew torsion H 34

T * graph of g T graph of g 35

EXAMPLE: the Levi-Civita connection [ x i g ik dx k, ] + g jk dx k x j [ x i, x j ] + = = ( gjk + g ik g ) ij dx k = 2g lk Γ l ij x i x j x k 36

GENERALIZED COMPLEX STRUCTURES 37

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 M. Gualtieri:Generalized complex geometry math.dg/0401221 38

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,...] 39

EXAMPLE: HOLOMORPHIC POISSON MANIFOLDS σ = σ ij z i z j [σ, σ] = 0 J = i :..., z j,..., d z k + l σ kl,... z l 40

GENERALIZED KÄHLER MANIFOLDS 41

Kähler complex structure + symplectic structure complex structure J 1 on T T symplectic structure J 2 on T T compatibility (ω Ω 1,1 ): J 1 J 2 = J 2 J 1 42

A generalized Kähler structure: two commuting generalized complex structures J 1, J 2 such that (J 1 J 2 (X + ξ), X + ξ) is definite. GUALTIERI S THEOREM A generalized Kähler manifold is: two integrable complex structures I +, I on M a metric g, hermitian with respect to I +, I a 2-form B U(m) connections +, with skew torsion ±H = ±db 43

(J 1 J 2 ) 2 = 1 eigenspaces V, V, metric on V positive definite connections +, torsion ±db... also define on 0 T E T 0 44

J 1 = J 2 on V ±I + J 1 = J 2 on V ±I {J 1 = i} {J 2 = i} = V 1,0 is Courant integrable [X 1,0 + ξ, Y 1,0 + η] = Z 1,0 + ζ I + integrable 45

V Apostolov, P Gauduchon, G Grantcharov, Bihermitian structures on complex surfaces, Proc. London Math. Soc. 79 (1999), 414 428 S. J. Gates, C. M. Hull and M. Roček, Twisted multiplets and new supersymmetric nonlinear σ-models. Nuclear Phys. B 248 (1984), 157 186. 46

[J 1, J 2 ] = 0, [I +, I ] 0 in general g([i +, I ]X, Y ) = Φ(X, Y ) 2-form Φ Λ 2,0 + Λ 0,2 for both complex structures σ Λ 0,2 = Λ 2 T σ is holomorphic σ is a holomorphic Poisson structure 47

[J 1, J 2 ] = 0, [I +, I ] 0 in general g([i +, I ]X, Y ) = Φ(X, Y ) 2-form Φ Λ 2,0 + Λ 0,2 for both complex structures σ Λ 0,2 = Λ 2 T σ is holomorphic σ is a holomorphic Poisson structure 48

[J 1, J 2 ] = 0, [I +, I ] 0 in general g([i +, I ]X, Y ) = Φ(X, Y ) 2-form Φ Λ 2,0 + Λ 0,2 for both complex structures σ Λ 0,2 = Λ 2 T σ is holomorphic σ is a holomorphic Poisson structure 49

[J 1, J 2 ] = 0, [I +, I ] 0 in general g([i +, I ]X, Y ) = Φ(X, Y ) 2-form Φ Λ 2,0 + Λ 0,2 for both complex structures σ Λ 0,2 = Λ 2 T σ is holomorphic σ is a holomorphic Poisson structure 50

[J 1, J 2 ] = 0, [I +, I ] 0 in general g([i +, I ]X, Y ) = Φ(X, Y ) 2-form Φ Λ 2,0 + Λ 0,2 for both complex structures σ Λ 0,2 = Λ 2 T σ is holomorphic σ is a holomorphic Poisson structure 51

[J 1, J 2 ] = 0, [I +, I ] 0 in general g([i +, I ]X, Y ) = Φ(X, Y ) 2-form Φ Λ 2,0 + Λ 0,2 for both complex structures σ Λ 0,2 = Λ 2 T σ is holomorphic σ is a holomorphic Poisson structure 52

EXAMPLES T 4 and K3, σ = holomorphic 2-form (D. Joyce) CP 2, σ = / z 1 / z 2 CP 1 CP 1 = projective quadric. σ = Q/ z 3 [ / z 1 / z 2 ] moduli space of ASD connections on above. 53

J. Bogaerts, A. Sevrin, S. van der Loo, S. Van Gils, Properties of Semi-Chiral Superfields, Nucl.Phys. B562 (1999) 277-290 V Apostolov, P Gauduchon, G Grantcharov, Bihermitian structures on complex surfaces, Proc. London Math. Soc. 79 (1999), 414 428 NJH, Instantons, Poisson structures and generalized Kaehler geometry, math.dg/0503432 C. Bartocci and E. Macrì, Classification of Poisson surfaces, Communications in Contemporary Mathematics, 7 (2005), 1-7 54