New Paths Towards Quantum Gravity (Lecture Notes in Physics, 807)

5 Lectures on Cohomology, T-Duality, and Generalized Geometry 303〈s 2 X ◦ ρ ∗ B(Y ), s 2 Z〉=X (B(Y, Z)) − 2〈ρ ∗ B(Y ), s 2 X ◦ s 2 Z〉= X (B(Y, Z)) − B(Y, [X, Z]),2〈ρ ∗ B(X) ◦ s 2 Y, s 2 Z〉=−2〈s 2 Y ◦ ρ ∗ B(X), s 2 Z〉+4〈D〈ρ ∗ B(X), s 2 Y 〉, s 2 Z〉=−Y (B(X, Z)) + B(X, [Y, Z]) + 2〈d(B(X, Y )), Z〉=−Y (B(X, Z)) + B(X, [Y, Z]) + Z(B(X, Y )) .Combining all terms proves the claim, i.e.H 1 (X, Y, Z) − H 2 (X, Y, Z) = X (B(Y, Z)) − Y (B(X, Z)) + Z(B(X, Y ))−B([X, Y ], Z) + B([X, Z], Y ) − B([Y, Z], X)= dB(X, Y, Z).Finally, let us now, given a representative of a class [H] ∈H 3 (M, R), explicitlyconstruct the Courant algebroid. We choose an open cover and a representativeof [H] in H 3 D (M, R) ∼ = H 3 dR (M, R), i.e. a four-tuple (Λ αβγ , A αβ , B α , H) in theČech – de Rham complex (over R). We then construct E by means of the clutchingconstructionE = ∐ α(TM⊕ T ∗ M) |Uα / ∼,where we identify X + ξ ∈ Ɣ(TM⊕ T ∗ M) |Uα with Y + η ∈ Ɣ(TM⊕ T ∗ M) |Uβ onoverlaps U αβ iff Y = X and η = ξ + ı X dA αβ . Consistency on triple overlaps U αβγfollows from dA βγ − dA αγ + dA αβ = (dδ A) αβγ = (d 2 Λ) αβγ = 0. On TM |Uα thesplitting is given bys |Uα : X ↦→ X + ı X B α (123)and consistency on overlaps follows from B β − B α = dA αβ . Moreover, since (123)is just a B-transform (cf. (106)) we see that the Courant bracket on E is simply theH-twisted Courant bracket (108).⊓⊔5.3.5 Generalized Complex GeometryWe briefly introduce some of the concepts of generalized complex geometry withthe aim of showing that T-duality acts naturally on all these structures.5.3.5.1 Generalized Complex StructuresWe recall that an almost complex structure on a manifold M is an endomorphismJ : TM → TM, such that J 2 =−1. Given an almost complex structure we candecompose the complexified manifold TM C = TM⊗ C as