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Extended manifolds and extended equivariant cohomology Shengda Hu a , Bernardo Uribe b,∗ a

Département de Mathématiques et de Statistique, Université de Montréal, CP 6128 succ Centre-Ville, Montréal, QC H3C 3J7, Canada

b

Department of Mathematics, University of Los Andes, Carrera 1 n. 18A -10, Bogotá, Colombia

article

a b s t r a c t

info

Article history: Received 20 June 2007 Received in revised form 25 September 2008 Accepted 13 October 2008 Available online 19 October 2008

We define the category of manifolds with extended tangent bundles, we study their symmetries and we consider the analogue of equivariant cohomology for actions of Lie groups in this category. We show that when the action preserves the splitting of the extended tangent bundle, our definition of extended equivariant cohomology agrees with the twisted equivariant de Rham model of Cartan, and for this case we show that there is localization at the fixed point set, à la Atiyah–Bott. © 2008 Elsevier B.V. All rights reserved.

MSC: 55N91 37K65 Keywords: Exact Courant algebroids Hamiltonian actions Twisted equivariant cohomology

1. Introduction The study of the geometry on the generalized tangent bundle TM = TM ⊕ T ∗ M starts with the paper [15], in which Hitchin introduced the notion of generalized complex structures. The framework of generalized complex geometry was first developed by Gualtieri in his thesis [13] and much more has been done since (cf. [5,7,8,10,16,17,19,21,27]). The algebraic structure underlying the considerations of generalized geometry is the structure of a Courant algebroid on TM (cf. Definition 3.1.1), a notion which was firstly introduced in [22]; this is the main object of study in this article. The natural group of symmetries for the tangent bundle TM is the diffeomorphism group Diff(M ). In contrast, the natural group of symmetries for TM is given by G = Diff(M ) n Ω 2 (M ). The action of B ∈ Ω 2 (M ), which is also called a Btransformation, is given by eB (X + ξ ) = X + ξ + ιX B,

for X ∈ Γ (TM ) and ξ ∈ Ω 1 (M ).

In this article, we take this approach one step further and take the following point of view. We see the B-transformations as part of an extended change of coordinates. As in classical differential geometry where intrinsic quantities do not depend on the choice of coordinates, here we ask for independence on the choice of extended coordinates. With this point of view, the splitting of TM into a direct sum is only a choice of extended coordinates, in which applying a B-transformation corresponds to a different choice of coordinates. Along the same line, the embedding of T ∗ M into TM is invariant with respect to the extended change of coordinates. Thus we are led to the consideration of the extension sequence: 0 → T ∗ M → T M → TM → 0,

∗

Corresponding author. Tel.: +57 1 33999x3563. E-mail addresses: [email protected] (S. Hu), [email protected] (B. Uribe).

0393-0440/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.geomphys.2008.10.004

S. Hu, B. Uribe / Journal of Geometry and Physics 59 (2009) 104–131

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where T M is isomorphic to TM but not canonically. The Courant algebroid T M of this type is called exact [25] and we will use the name extended tangent bundle to emphasize the relation with the geometry of the manifold, as well as the absence of a preferred splitting. In the following, we will use the notion extended manifold to denote a manifold M with an extended tangent bundle T M. This paper accomplishes the following. In the first part (Sections 2 and 3), starting with linear algebra (Section 2) we describe the category of extended manifolds, (Section 3). In the second part (Sections 4 and 5), we consider the symmetries of Courant algebroids and their induced action on the differential forms (viewed as spinors, see Section 4) and we describe a new equivariant cohomology (Section 5) which involves non-trivially the extended part of the symmetries, i.e. the Btransformations. In Section 5 we also consider some special cases and examples. In the last part (Sections 6 and 7), we study the properties of the equivariant cohomology we introduced (Section 6) and we show that, in certain cases, the new equivariant cohomology allows localization to fixed point sets (Section 7), in the sense of Atiyah and Bott. In the first part, in order to define the category, we are mainly concerned with the definition of morphisms (cf. Definition 3.5.1). There are two ingredients in the definition, reversed structure (cf. Definition 3.4.1) and isotropic extended submanifold (cf. Definition 3.2.2). Recall that the structure of a Courant algebroid on T M is given by the datum (∗, h, i, a) (cf. Definition 3.1.1), where h, i is a non-degenerate symmetric pairing. The reversed structure of Courant algebroid on T M is then given by the datum (∗, −h, i, a), which we denote by −T M. A standard reversion is by definition an isomorphism of Courant algebroids r : T M → −T M, covering the identity map on TM. Standard reversions can be seen as a more intrinsic way of representing isotropic splittings of T M, as they are in one-to-one correspondence to each other. Then a morphism between (M , T M ) and (N , T N ) is a map f : M → N whose graph is an isotropic extended submanifold of (M × N , −π1∗ T M ⊕π2∗ T N ). We show that this definition of morphism naturally encodes the action of Ω 2 (M ). Summarizing, we have: The category E Smth of extended manifolds is defined as follows. An object is a manifold M together with an extended tangent bundle T M. A morphism f˜ : (M , T M ) → (N , T N ) is given by a smooth map f : M → N and an isotropic extended structure E˜ on the graph of f . When we fix splittings of the extended tangent bundles, a morphism is given equivalently by a smooth map f : M → N and a 2-form bf˜ ∈ Ω 2 (M ) so that HM = f ∗ HN − dbf˜ , where HM (resp. HN ) is the twisting form defined by the chosen splitting and [HM ] (resp. [HN ]) is the Ševera class of the extended tangent bundles. In this way, the composition of two morphisms f˜ = (f , bf˜ ) and g˜ = (g , bg˜ ) is given by h˜ = (g ◦ f , bf˜ + f ∗ bg˜ ). Although the definition of composition as above uses the splittings of extended tangent bundles, we show that the morphism thus obtained does not depend on the choice of splitting. The second part discusses equivariant cohomology, in the extended setting. Recall that the space of spinors for T M with pairing h, i can be identified with Ω • (M ). From our point of view, the identification is not canonical, rather, it depends on the choice of extended coordinates. We thus arrive at the notion of abstract spinor space S • (M ) (cf. Definition 3.7.1). For g a Lie algebra, we introduce two notions of g-actions on T M. The first is the generalized action of g, given by a Lie algebra homomorphism g → XT , where XT is the Lie algebra of infinitesimal symmetries of T M. The second is when δ

κT

the generalized action factors through the composition g → Γ (T M ) → XT , where the map κT sends the bracket of the Courant algebroid T M to the Lie bracket on XT . We then say that this generalized action is an extended action and we obtain the following result. Fix a splitting s of T M. For X = s(X ) + ξ ∈ Γ (T M ), let LTX = LX + (dξ − ιX H )∧, ιX = ιX + ξ ∧ and dT = d − H ∧, where H is the twisting form defined by s. Then these operators are independent of the choice of splitting s. Let δ : g → Γ (T M ) be an extended g-action that is isotropic. Then the following is a chain complex

X Cg• (T M ) = ρ ∈ S • (M ) ⊗ b S (g∗ ) LTδ(τ ) ρ = 0 for all τ ∈ g with differential dT ,δ = dT − uj ιδ(τj ) , j

where S (M ) is the spinor space and b S (g ) is the a-adic completion of the polynomial algebra S (g ) with respect to the ideal a of polynomials with zero constant term (cf. Section 5). The cohomology HG• (T M ) of (Cg• (T M ), dT ,δ ) is called the extended g-equivariant cohomology. We note that even when the generalized action of g on T M is trivial, the map δ : g → Γ (T M ) may be non-trivial and the above cohomology is different from the tensor product of H ∗ (M ) and b S (g∗ ). In fact, as the examples in Section 5 show, the extended equivariant cohomology can be genuinely different from the ordinary equivariant cohomology. The last part concerns the case when the g-action is integrable to an action of a compact Lie group G. In this case, let F be the locus of fixed points. Then the induced extended tangent bundle T F naturally embeds into T M. The naturality of this embedding does not hold for general submanifolds. When the g-action is an extended action that preserves a splitting, we show that the extended g-equivariant cohomology localizes to the fixed point set in the sense of Atiyah–Bott. Along the way, we show that the axioms of cohomology theory are satisfied by the extended equivariant cohomology and that there is a Thom isomorphism between the cohomology of the base and the vertically compactly supported cohomology of a vector bundle. The latter is the cornerstone for the localization argument. We note that the terminology of extended action that has been introduced in [7] refers to extensions of the Lie algebra morphism g → Γ (TM ) to a Courant algebra morphism a → Γ (T M ) where a is a Courant algebra over g. The extended actions of this paper are what Bursztyn, Cavalcanti and Gualtieri called trivially extended actions (see [7, Def. 2.12]). •

∗

∗

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Various authors have described the twisted cohomology H • (M ; H ) in detail [23,1] and the idea of twisting an equivariant cohomology theory by a 3-class is also known [12]. As far as the authors know, the extended complex introduced in this article for the extended actions has not been discussed before in the literature. 2. Extended linear algebra 2.1. Extended linear spaces Let V be a (R-)linear space of dimension n. An extension of V is a triple (V, b V , h, i) where V is a (R-)linear space of dimension 2n, with a non-degenerate pairing h, i of signature (n, n), which fits into the exact sequence a∗

a

0 → V∗ − →V− → V → 0,

(2.1)

∗

∗

∗

where a is the dual of a via the pairing 2h, i. The subspace b V ⊂ V is the image of V under a . More precisely, the pairing provides the identification V∗ ' V and the following equation defines the map a∗ , where we use X to denote a(X):

ξ (X ) = 2ha∗ (ξ ), Xi for all ξ ∈ V ∗ and X ∈ V. '

We use b to denote the isomorphism V ∗ − →b V : ξ 7→ b ξ := a∗ (ξ ) and use ˇ to denote the inverse of b. A splitting s : V → V of the exact sequence (2.1) is called isotropic if the image is isotropic with respect to the pairing. The set IV of isotropic splittings s is a torsor over ∧2 V ∗ . In fact, for s0 ∈ IV , we have

h(s0 − s)(X ), Y i = −hX − s0 (X ), s0 (Y )i + hX − s(X ), s(Y )i = −hX , (s0 − s)(Y )i. 2 ∗ Let B(X , Y ) = h(s0 − s)(X ), Y i, then s0 (X ) = s(X ) + ιc X B. In particular, we may define the action of B ∈ ∧ V on IV by ∧2 V ∗ × IV → IV : (B, s) 7→ B ◦ s,

where (B ◦ s)(X ) = s(X ) + ιc X B.

(2.2)

2.2. Extended subspaces Let i : W ⊂ V be a linear subspace. An extended structure E on W is a subspace of V which fits into the following diagram: a|E

0 → C ,→ E − → W → 0,

for some C ⊂ b V.

(2.3)

The extended subspace W = (W , E ) is isotropic (respectively, non-degenerate) if the restriction of h, i to E vanishes (respectively, is non-degenerate). In the following, we will only consider maximal extended structures, i.e. those that are b b not contained in any other extended structures of the same type. Let K = Annb V W , W = V /K and W = Ann(K )/K , then we b , h, iW ). have an induced extension of W given by (W, W 2.2.1. Isotropic Let E be a (maximal — we will drop maximal in the following, as they are the only kind we consider here) isotropic extended structure on W , then C = K . The extension sequence (2.3) descends to W and we have 0 → E /K → W → 0 i.e. E /K is an isotropic splitting of W. On the other hand, for any splitting sW ∈ IW , the preimage of sW (W ) in Ann(K ) under the quotient map gives an isotropic extended structure on W . It follows that the space of isotropic extended structures on W is a torsor over ∧2 W ∗ . In fact, choose an isotropic splitting, we can explicitly write down the two form corresponding to E. Lemma 2.2.1. Let E be an isotropic extended structure on i : W ⊂ V and s ∈ IV a splitting of V. Define for X , Y ∈ W Ts (X , Y ) = 2hs(i(X )), i = η(i(X )), where X = s(i(X )) + b ξ and

= s(i(Y )) + b η belong to E. Then Ts ∈ ∧2 W ∗ and

E = {s(i(X )) + b ξ |X ∈ W and i∗ ξ = −ιX Ts }. 2

∗

0

(2.4) ∗

Let B ∈ ∧ V and consider s = B ◦ s, then Ts0 = Ts + i B. Proof. We note that hX, i = 0 because E is isotropic. Write it out and we have 1 hs(i(X )) + b ξ , s(i(Y )) + b ηi = (ξ (i(X )) + η(i(Y ))) = 0. 2

That Ts is well defined follows from the fact that different choices of b η differ by an element in K . Thus Ts ∈ ∧2 W ∗ .

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By definition, we have ιX Ts = −i∗ ξ where s(i(X )) + b ξ ∈ E. The ‘‘⊂’’ in (2.4) follows. On the other hand, E is maximal, and the right hand side of (2.4) is also an isotropic extended structure on W . The ‘‘=’’ then follows. Suppose that X = s(i(X )) + b ξ = s0 (i(X )) + b ξ 0 , then b ξ0 = b ξ − ι[ i(X ) B. Now

ιX Ts0 = −i∗ ξ 0 = −i∗ (ξ − ιi(X ) B) = ιX (Ts + i∗ B). It follows that Ts0 = Ts + i∗ B.

2.3. Reversion Let (V, b V , h, i) be an extension of V , then the reversed extension is defined to be (V, b V , −h, i). When we abbreviate V as the extension, the reversed extension will be denoted −V. A standard reversion of V is an isomorphism τ : V → −V which covers the identity on V . Let s ∈ IV , then the map Rs : V → −V : X = s(X ) + b ξ 7→ s(X ) − b ξ defines a standard reversion: 1 hRs (X), Rs ( )i = hs(X ) − b ξ , s( Y ) − b ηi = − (ξ (Y ) + η(X )) = −hX, i. 2

On the other hand, for any standard reversion R, the sub-space VR fixed by R is non-empty and maximal isotropic, since hX, i = 0 for all X, ∈ VR . In fact, we see that isotropic splittings and standard reversions are in one-to-one correspondence. It follows that the set of standard reversions is a torsor over ∧2 V ∗ . 2.4. Extended complex structures A linear extended complex structure on V is defined as a linear map J : V → V so that J2 = −1 and hJ·, J•i = h·, •i. J

Let P : V ∗ → V − → V → V be the composition of the natural maps, then from the definition of J we have P ∗ = −P. It follows that P ∈ ∧2 V . An extended Lagrangian (resp. complex) subspace W ⊂ V is an isotropic (resp. non-degenerate) extended subspace (W , E ) so that E is preserved by J. When W is extended Lagrangian, K = Annb V W is isotropic with respect to P, since we have the restriction P |K : K → W = AnnV K . In this sense, we say that W is co-isotropic with respect to P. We may also see J as a linear extended complex structure on −V. Let R : V → −V be a standard reversion, then JR = R ◦ J ◦ R−1 : V → V is the R-reversed linear extended complex structure (which corresponds to the twisted structure in [5]). 2.5. Direct sums and morphisms

b , h, iW ) be two extended linear spaces. The direct sum extension of V ⊕ W is defined as Let (V, b V , h, iV ) and (W, W b , h, iV ⊕ h, iW ). (V ⊕ W, b V ⊕W Definition 2.5.1. A morphism between V and W is given by a pair (φ, E ), where φ : V → W is a linear map and E ⊂ −W ⊕ V is an isotropic extended structure over graph(φ) ⊂ W ⊕ V , i.e. there is an exact sequence 0 → Kφ ,→ E → graph(φ) → 0,

b ⊕b for some Kφ ⊂ W V.

We write φ˜ = (φ, E ) and say that φ˜ is an extension of φ . Let us choose and fix splittings sW ∈ IW and sV ∈ IV , then we have the direct sum splitting s := sW ⊕ sV ∈ I−W⊕V . According to Lemma 2.2.1 the morphism φ˜ can also be represented by a pair (φ, bφ˜ ) where bφ˜ ∈ ∧2 W ∗ . Proposition 2.5.2. Let φ : W → V be a linear map. Choose splittings sW ∈ IW and sV ∈ IV respectively, then any morphism φ˜ = (φ, E ) : W → V extending φ is determined by an element bφ˜ ∈ ∧2 W ∗ , where E = {sW (X ) + b φ(b η) + ι[ η|X ∈ W and b η∈b V }. X bφ˜ + sV (φ(X )) + b

(2.5)

0 2 ∗ 2 ∗ ∗ (η). Let B where φ ∗ is the dual of φ and b φ(b η) := φ\ W ∈ ∧ W and BV ∈ ∧ V and consider splittings sW := BW ◦ sW and

s0V := BV ◦ sV (Y ). Then the element b0˜ = bφ˜ − BW + φ ∗ BV ∈ ∧2 W ∗ determines the same morphism φ˜ . φ

Proof. Simply let bφ˜ = Ts as in Lemma 2.2.1, where s is the direct sum splitting. We only have to note that the reversing of the structure on W introduces the negative sign in the formula of b0˜ . φ

Proposition 2.5.2 implies that in particular that when choosing BW = bφ˜ and BV = 0, we have b0˜ = 0. In this case, with φ

the identifications W = W ⊕ W ∗ and V = V ⊕ V ∗ under the new splittings, the morphism is defined by the graphs of φ : W → V and φ ∗ : V ∗ → W ∗ .

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2.5.1. Automorpisms In the case that W = V and φ : V → V is an isomorphism, there is a useful alternative presentation of φ˜ as a pair (φ, Bφ˜ ), where Bφ˜ ∈ ∧2 V ∗ (cf. Section 4.1). Because φ is an isomorphism, we can choose to represent the morphism φ˜ using two forms on the target instead of the domain. More precisely, instead of (2.5), we write

b b E = {sV (X ) + b ξ + sV (φ(X )) + b φ −1 (b ξ + ι[ X Bφ˜ )|X ∈ V and ξ ∈ V }, where sV ∈ IV is a splitting of V. Direct computation shows that Bφ˜ = −Ts for the direct sum splitting s := sV ⊕ sV . The main advantage of this presentation is that the extended structure E is the graph of a genuine map

V → V : X = sV (X ) + b ξ 7→ sV (φ(X )) + b φ −1 (b ξ + ι[ X Bφ˜ ). 2.6. Composition

˜ = (ψ, F ) : V → U be two morphisms of extended linear spaces. We consider the Let φ˜ = (φ, E ) : W → V and ψ composition λ = ψ ◦φ : W → U and define the induced extended structure on graph(λ). Choose and fix isotropic splittings of W, V and U, then the morphisms can be represented by the pairs (φ, bφ˜ ) and (ψ, bψ˜ ), where bφ˜ ∈ ∧2 W ∗ and bψ˜ ∈ ∧2 V ∗ . The composition is defined as follows: ˜ = ψ˜ ◦ φ˜ is given by λ = ψ ◦ φ Definition 2.6.1. With the chosen splittings of the extended linear spaces, the composition λ and bλ˜ = bφ˜ + φ ∗ bψ˜ ∈ ∧2 W ∗ . We show that the composition is well-defined, i.e. it does not depend on the choice of splittings. Let BW ∈ ∧2 W ∗ , BV ∈ ∧2 V ∗ ˜ are given by and BU ∈ ∧2 U ∗ define different splittings of the extended linear spaces. Then the same morphisms φ˜ and ψ b0φ˜ = bφ˜ − BW + φ ∗ BV

and b0ψ˜ = bψ˜ − BV + ψ ∗ BU . Under the new splittings, the definition gives b0λ˜ = b0φ˜ + φ ∗ b0ψ˜ = bφ˜ + φ ∗ bψ˜ − BW + λ∗ BU = bλ˜ − BW + λ∗ BU ,

˜ . It is easy to check that the following equality of compositions of which by Proposition 2.5.2 defines the same morphism λ extended morphisms holds φ˜ 1 ◦ (φ˜ 2 ◦ φ˜ 3 ) = (φ˜ 1 ◦ φ˜ 2 ) ◦ φ˜ 3 . 2.7. The category Now the category E Vect of extended linear spaces is defined. An object in E Vect is given by the triple (V, b V , h, i) in the exact sequence 0→b V → V → V → 0, so that h, i is symmetric, non-degenerate of split signature and b V is maximal isotropic. We will usually denote such an object by V. A morphism φ˜ : W → V in E Vect is given by an isotropic extended structure E ⊂ −W ⊕ V on the graph of φ ∈ Hom(W , V ) (cf. Definition 2.5.1). The composition of morphisms is well defined (cf. Definition 2.6.1). In particular, we have the following fibration sequence:

∧2 W ∗ → HomE Vect (W, V) → Hom(W , V ). When W = V, we may consider the group of invertible morphisms AutE Vect (V), which fits into the extension sequence 0 → ∧2 V ∗ → AutE Vect (V) → GL(V ) → 1. The identity element in AutE Vect (V) is given by the diagonal 1 ⊂ −V ⊕ V, which is a morphism extending id ∈ GL(V ). 2.8. Spinors Let Cl(V) be the Clifford algebra of V, i.e. Cl(V) = ⊗∗ V/(X ⊗ X − hX, Xi). The contraction ιX by X ∈ V, which is defined below, generates an action of Cl(V) on IV × ∧∗ b V:

b ρ ), ιX (s,b ρ ) := (s, ιc Xρ + ξ ∧ b

where b ξ = X − s(X ).

Here and in the following, we use ρ to denote an element in ∧V ∗ and b ρ to denote the corresponding element in ∧∗ b V

∗b under the obvious identification. With this understood, ιc X ρ is then the element in ∧ V corresponding to ιX ρ . The action of B ∈ ∧2 V ∗ on ∧∗ b V defined by: b B ∧ ρ. B ◦b ρ := e−B ∧ b ρ = e−\

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Then on IV × ∧∗ b V we have the anti-diagonal action of B ∈ ∧2 V ∗ : B ◦ (s,b ρ ) := (B ◦ s, (−B) ◦ b ρ ). where the action B ◦ s is as defined in (2.2). Lemma 2.8.1. Let X ∈ V and B ∈ ∧2 V ∗ then ιX and B◦ commute on IV × ∧∗ b V. Proof. We only need to show it for the component ∧∗ b V and we abbreviate the notations accordingly. Let s0 = B ◦ s and b ρ 0 = (−B) ◦ b ρ where we use ·0 to denote elements in {s0 } × ∧∗ b V , e.g. 0 b0 ρ 0 , ιXb ρ 0 = ιd Xρ + ξ ∧ b

where b ξ 0 = X − s0 (X ).

Then we have 0 \ 0 B ιX ((−B) ◦ b ρ ) = ιd ρ 0 = ιX (\ eB ∧ ρ) + (X − s(X ) − ιc X ρ + (X − s (X )) ∧ b X B) ∧ e ∧ ρ b b b B B B c = ιc ρ + e ∧ ιc ρ XB ∧ e ∧ b X ρ + (X − s(X ) − ιX B) ∧ e ∧ b

= eB ∧ (ιc ρ) X ρ + (X − s(X )) ∧ b = (−B) ◦ ιXb ρ. b

Definition 2.8.2. The abstract spinor space SV for the triple (V, b V , h, i) is the quotient IV ×∧2 V ∗ ∧∗ b V := (IV ×∧∗ b V )/ ∧2 V ∗ . The bundle IV × ∧∗ b V over SV with fiber ∧2 V ∗ provides identification of SV with the section {s} × ∧∗ b V upon a choice of splitting s ∈ IV . The action of Cl(V) on IV × ∧∗ b V descends to an action on SV , which we will again denote it ιX for X ∈ V. Lemma 2.8.3. Let φ˜ : W → V be a morphism of extended linear spaces. Then it induces a natural pull-back map of spinor spaces: φ˜ • : SV → SW . Proof. Choose splittings sW ∈ IW and sV ∈ IV and represent φ˜ by the pair φ : W → V and bφ˜ ∈ ∧2 W ∗ . Let πV be the identification: =

πV : ∧∗ b V − → { sV } × ∧ ∗ b V → SV . b . The map φ˜ • : SV → SW is defined by πW is similarly defined for ∧∗ W φ˜ • (πV (b ρ )) = πW (bφ˜ ◦ b φ(b ρ )),

for b ρ ∈ ∧∗ b V,

(2.6)

∗ (ρ) does not depend on the splitting. where b φ(b ρ ) = φ\ Let BW ∈ ∧2 W ∗ and BV ∈ ∧2 V ∗ respectively and let

s0W = BW ◦ sW

s0V = BV ◦ sV .

and

=

V → SV , then we have Let πV0 : ∧∗ b V − → {s0V } × ∧∗ b

πV−1 πV0 (b ρ ) = e−BV ∧ b ρ. b

0 πW can be similarly defined and a similar relation holds. Recall that under the splittings s0V and s0W , the morphism φ˜ is represented by φ and b0˜ = bφ˜ − BW + φ ∗ BV . Now we compute φ b φ˜ • (πV0 (b ρ )) = φ˜ • (πV (πV−1 πV0 )(b ρ )) = φ˜ • (πV (e−BV ∧ b ρ )) ∗ \ b = πW (bφ˜ ◦ b φ(e−BV ∧ b ρ )) = πW ((b0φ˜ + BW − φ ∗ BV ) ◦ (e−φ BV ∧ b φ(b ρ )))

0 φ(b ρ ))) = πW (b0φ˜ ◦ b φ(b ρ )). = πW (e−BW ∧ (b0φ˜ ◦ b b

It follows that φ˜ • given by (2.6) is independent of the choice of splittings.

3. Category of extended manifolds Starting from this section, the notations in the rest of the article will follow the Convention 3.1.5.

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3.1. Courant algebroids Following [20], we have the following definition of a Courant algebroid: Definition 3.1.1. Let E → M be a vector bundle. A Loday bracket ∗ on Γ (E ) is an R-bilinear map satisfying the Jacobi identity, i.e. for all X, , Z ∈ Γ (E ),

X ∗ ( ∗ Z) = (X ∗ ) ∗ Z +

∗ (X ∗ Z).

(3.1)

E is a Courant algebroid if it has a Loday bracket ∗ and a non-degenerate symmetric pairing h, i on E, with an anchor map a : E → TM which is a vector bundle homomorphism so that a(X)h , Zi = hX,

∗Z+Z∗ i

(3.2)

a(X)h , Zi = hX ∗ , Zi + h , X ∗ Zi.

(3.3)

The bracket in the definition is not skew-symmetric in general and the skew-symmetrization [X, ] = X ∗ − ∗ X is usually called the Courant bracket of the Courant algebroid. The above definition is equivalent to the definition as given in, for example, [22] or [13]. We rephrase the definition of generalized complex structure [13,15]: Definition 3.1.2. An extended tangent bundle T M is a Courant algebroid (cf. Definition 3.1.1) which fits into the following extension: ∗

a a 0 → T ∗M − →TM − → TM → 0,

where a∗ is the dual map of a under the identification of T M with its dual given by 2h, i. The extended tangent bundle T M is split if the extension is split by some s : TM → T M, so that the image is isotropic. An extended almost complex structure J on T M is a bundle morphism J : T M → T M such that J 2 = −1 and that is also orthogonal with respect to the pairing h, i. Furthermore, J is integrable and is called an extended complex structure if the +i-eigensubbundle L of J is involutive with respect to the Courant bracket [, ]. From (3.2), it is easy to see that for X,

[X , ] = X ∗

∈ Γ (T M ) we have

− dhX, i.

In particular, the Courant and Loday brackets coincide on isotropic subspaces of Γ (T M ). An example of extended tangent bundle is TM ⊕ T ∗ M with the natural projection and inclusion, the natural pairing, the bracket that Courant discovered [9], namely 1

[X + ξ , Y + η] = [X , Y ] + LX η − LY ξ − d(ιX η − ιY ξ ), 2

and the Loday bracket

(X + ξ ) ∗ (Y + η) = [X , Y ] + LX η − ιY dξ . Remark 3.1.3. An extended tangent bundle T M is also known with the name of exact Courant algebroid [25,7]. The Loday bracket for an extended tangent bundle is also known as the Dorfman bracket [11]. Now, the sequence ∗

a a 0 → T ∗M − →TM − → TM → 0

is always split in the sense of the Definition 3.1.2. Such a map s is also called a connection in [25]. The choice of isotropic splitting determines a closed 3-form Hs ∈ Ω 3 (M ): Hs (X , Y , Z ) = 2hs(X ), [s(Y ), s(Z )]i and we have 1

[s(X ) + ξ , s(Y ) + η] = s([X , Y ]) + LX η − LY ξ − d(ιX η − ιY ξ ) + ιY ιX Hs . 2

(3.4)

The space of isotropic splittings, which will be denoted I(M ), is a torsor over Ω 2 (M ) and different choices give cohomologous 3-forms. The action of B ∈ Ω 2 (M ) on s ∈ I(M ) is defined by

(B ◦ s)(X ) = s(X ) + ιX B, and direct computation shows that HB◦s = Hs + dB. The class [Hs ] = [HB◦s ] ∈ H 3 (M , R) is the Ševera class of T M [25,26].

(3.5)

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With this point of view, we see that the twisting form is defined only when a splitting is chosen. On the other hand, the relative twisting form between two extended tangent bundles can be defined, and is not dependent on the specific choice of splitting. Let Tj M = (T M , T ∗ M , h, i, [, ]j ) for j = 1, 2 be two extended tangent bundles over M, where [, ]1 and [, ]2 denote the two Courant brackets respectively. Definition 3.1.4. The relative twisting form δ H (T2 M , T1 M ) of T2 M with respect to T1 M is the difference Hs,2 − Hs,1 , where s ∈ I(M ) is a splitting and for j = 1, 2, Hs,j is the corresponding twisting form of Tj M. Convention 3.1.5 (Notational). In the following, unless otherwise specified, we will always use s together with various subscripts to denote isotropic splittings of extended tangent bundles. The twisting form defined by a splitting s will be denoted Hs . For example, sM denotes a splitting in I(M ) and the twisting form defined by sM is denoted HsM . The notations X, and Z always denote sections of extended tangent bundles, and X , Y and Z denotes the image of the respective sections under the anchor map. When a splitting s is chosen, then we always write ξ for X − s(X ), η for − s(Y ) and ζ for Z − s(Z ). We use Ω0k (M ) to denote the space of closed differential k-forms. There will be many actions of Ω 2 (M ) on various spaces, we denote them all by B◦ for B ∈ Ω 2 (M ). The spaces that are acted upon are often clear from the context. Similarly, we use [, ] for all the skew-symmetric brackets involved, and when confusion might arise, such as two brackets on the same space, we use subscripts to distinguish them. When a Lie algebra g acts on M (or T M), the infinitesimal action of τ ∈ g is denoted by Xτ , Yτ and etc. (or Xτ , τ and etc. respectively). In the case that we choose base {τj } for g and dual base {uj } for g∗ , the infinitesimal action of the base element τj are denoted Xj (on M) or Xj (on T M). 3.2. Extended submanifolds Let i : F ⊂ M be a submanifold. Then the embedding defines an extended tangent bundle T F by T F = Ann(K )/K where K = AnnT ∗ M TF ⊂ i∗ T ∗ M. The fact that the structure of an extended tangent bundle descends is a special case of more general reduction constructions, e.g. in [18] (Appendix A) and references therein. We obtain the roof diagram Ann(K )

TF

nn π nnn n n nnn v nn n

Qu QQ QQQ QQQ QQQ Q(

(3.6)

i∗ T M

where π is the projection. We note that a priori the submanifold F is not associated with an embedding T F → i∗ T M. Lemma 3.2.1. Let [HM ] ∈ H 3 (M ) be the Ševera class of T M, then i∗ [HM ] ∈ H 3 (F ) is the Ševera class of T F . Proof. Choose a splitting sM ∈ I(M ) which defines the twisting form HsM . The image of TF under sM must lie in Ann(K ). Because sM (TF ) ∩ K = {0}, we see that sM descends to sF : TF → T F , which must also be an isotropic splitting. The twisting form HsF defined by sF is simply i∗ HsM . Definition 3.2.2. Let i : F → M be a submanifold of M and K = AnnT ∗ M TF . An extended structure E on F is an involutive subbundle E ⊂ Ann(K ), with respect to the bracket [, ], which also fits into the following exact sequence a |E

0 → C ,→ E −→ TF → 0,

for some subbundle C ⊂ i∗ T ∗ M .

(3.7)

An extended submanifold F is the pair (F , E ) where E is an extended structure over F . We say that F is an isotropic (respectively, non-degenerate) extended submanifold if the restriction of h, i to E vanishes (respectively, is non-degenerate). By definition, E descends to an involutive subbundle EF = E /(K ∩ E ) ⊂ T F , which will be called the reduced structure. The set of extended structures on F form a partially ordered set ordered by the inclusion of E . We will consider in the following only maximal extended structures in the partial order, and will often drop the adjective maximal. We show in the following that maximal isotropic and non-degenerate structures correspond respectively to the notion of generalized tangent bundle in [13] and of split submanifold in [5]. Lemma 3.2.3. F admits a (maximal) isotropic extended structure if and only if the Ševera class of T F is 0. For such F , the space IF of (maximal) isotropic extended structures is a torsor over Ω02 (F ). Proof. Let E be an isotropic extended structure, then C = K in (3.7), and (3.7) descends to T F : 0 → EF := E /K → TF → 0. Since E is isotropic and involutive, we see that EF is too. It follows that the splitting sF : TF → EF ⊂ T F , defined as the inverse of the non-trivial map in the above sequence, gives a twisting form HsF (X , Y , Z ) = 2hsF (X ), [sF (Y ), sF (Z )]F iF = 0, In particular, the Ševera class of T F is 0.

where X , Y , Z ∈ TF .

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Conversely, suppose that T F has Ševera class 0 and let sF : TF → T F be an isotropic splitting, which defines the twisting form HF = 0. In particular, sF (TF ) is involutive. Consider the preimage E = π −1 (sF (TF )), where π is the projection in (3.6), then the corresponding C in (3.7) is K . For any X, ∈ E , we have

π ([X, ]) = [π (X), π ( )]F and hX, i = hπ (X), π ( )iF . Since EF is isotropic and involutive, we see that E is isotropic and involutive as well. We note that E as constructed is maximal among the isotropic extended structures. Let E 0 be another maximal isotropic extended structure on F , which induces the same splitting EF of T F . Let X ∈ E and 0 X ∈ E 0 so that a(X) = a(X0 ) = X ∈ TF , then η = X0 − X ∈ T ∗ M and hη, TF i = 0. It follows that η ∈ K and X0 ∈ E . Thus E is uniquely determined by the reduced structure EF . Since the subspace of I(F ) with vanishing twisting forms is a torsor over Ω02 (F ), the same is true for IF . The above lemma shows that the class i∗ [HM ] vanishes for isotropic extended submanifold F . The next lemma gives an explicit construction at the form level: Lemma 3.2.4. Let (F , E ) be an isotropic extended submanifold of M and s ∈ I(M ) a splitting of T M. Let Hs ∈ Ω 3 (M ) be the twisting form defined by s. Define

βs (X , Y ) = 2hs(i∗ (X )), i = ιX η where X = s(i∗ (X )) + ξ and

= s(i∗ (Y )) + η are elements in E . Then βs ∈ Ω 2 (F ),

E = {s(i∗ (X )) + ξ |X ∈ TM and −i∗ ξ = ιX βs } and i∗ Hs − dβs = 0. Proof. We only have to show the last equality. The rest follows from the linear case in Lemma 2.2.1. Suppose that s0 ∈ I(M ) be another splitting of T M so that s0 = B ◦ s for some B ∈ Ω 2 (M ), then the twisting form defined by s0 is Hs0 = Hs + dB and we have τs0 = βs + i∗ B. It follows that i∗ Hs − dβs is independent of the choice of splitting s. Let sF : TF → T F be the splitting induced from E , then it is involutive, i.e. sF ([X , Y ]) = [sF (X ), sF (Y )]F . Let π be the projection in (3.6) and choose s ∈ I(M ) so that s(TF ) ⊂ E , then sF (X ) = π (X) = π (s(i∗ (X )) + ξ ) = π (s(i∗ (X ))) − ιX βs . Note also that Li∗ (X ) η − ιi∗ (Y ) dξ ∈ K for X , Y ∈ TF and η, ξ ∈ K . We compute

[sF (X ), sF (Y )]F = [π (s(i∗ (X ))) − ιX βs , π (s(i∗ (Y ))) − ιY βs ]F = π ([s(i∗ (X )), s(i∗ (Y ))]) − LX ιY βs + ιY dιX βs + ιY ιX i∗ Hs = sF ([X , Y ]) + ιY ιX (i∗ Hs − dβs ). It then follows that i∗ Hs − dβs = 0.

Lemma 3.2.5. A (maximal) non-degenerate extended structure on the submanifold F is equivalent to an embedding of the induced extended tangent bundles (cf. (3.6)) i∗ : T F → T M covering the embedding i∗ : TF → TM. Proof. Suppose that E is a maximal non-degenerate extended structure on F , then C ' T ∗ F via the restricted pairing 2h, i. Since E is involutive, it is an extended tangent bundle over F with the induced Courant algebroid structure. The projection π|E : E → T F is an isomorphism of extended tangent bundles and we obtain an embedding of extended tangent bundles 1 i∗ = π |− E : T F → T M. The other direction is obvious. 3.3. Product Let (M , T M ) and (N , T N ) be two smooth manifolds with extended tangent bundles. Then the product M × N admits a natural extended tangent bundle T (M × N ) defined as follows. Let πi be the projection of M × N onto the i-th factor for i = 1, 2. Then as bundles, we have natural identifications

T (M × N ) = π1∗ T M ⊕ π2∗ T N

and

T (M × N ) = π1∗ TM ⊕ π2∗ TN .

The structure of Courant algebroid on T (M × N ) is then given by declaring that the bracket and pairing all vanish between π1∗ T M and π2∗ T N. The axioms are easy to check and the extended tangent bundle T (M × N ) is defined.

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3.4. Reversion Similar to the linear case in Section 2.3, we have Definition 3.4.1. Let M be a smooth manifold with an extended tangent bundle T M, on which the structure of a Courant algebroid is given by (∗, h, i, a), then the reversed extended tangent bundle −T M is the same bundle with the structure of Courant algebroid given by (∗, −h, i, a). A bundle isomorphism r : T M → T M is called a standard reversion if it gives an isomorphism of Courant algebroids r : T M → −T M and covers the identity map on TM. We recall that the inclusion a∗ : T ∗ M → T M is the dual of the projection a : T M → TM when T M is identified with its dual under the pairing 2h, i. Thus, for −T M, the inclusion T ∗ M → −T M is given by −a∗ . Recall that s ∈ I(M ) defines the twisting form by Hs (X , Y , Z ) = 2hs(X ), [s(Y ), s(Z )]i and we have

1 [X, ] = s([X , Y ]) + LX η − LY ξ − d(ιX η − ιY ξ ) + ιY ιX Hs . 2

Since a reversion of T M only changes the sign of the pairing, with the same splitting s, the twisting form of −T M is −Hs . Let rs : T M → T M : s(X ) + ξ 7→ s(X ) − ξ , then it is a standard reversion: 1

[rs (X), rs ( )]−T = s([X , Y ]) − (LX η − LY ξ − d(ιX η − ιY ξ ) + ιY ιX (−Hs )) = rs ([X, ]T ).

2

As in Section 2.3, we see that there is one-to-one correspondence between I(M ) and the set of standard reversions. It follows that the set of standard reversions is a torsor over Ω 2 (M ). Suppose that J is an extended complex structure on T M, then it is also an extended complex structure on −T M. When we want to be clear as to which extended tangent bundle we are looking at, we use −J to denote the one on −T M. 3.5. Morphisms Let (M , T M ) and (N , T N ) be two smooth manifolds with extended tangent bundles. Let f : M → N be a smooth map ˜ N ), and id × f : M → M × N its graph. On the product M × N, we may define another natural extended tangent bundle T (M × which is given by the product of (M , −T M ) and (N , T N ). An extended structure on the map f is defined to be an extended structure E˜ on its graph (id × f )(M ). More specifically, we have 0 → C˜ → E˜ → TM → 0,

˜ N )) = −T M ⊕ f ∗ T N is an involutive subbundle. where E˜ ⊂ (id × f )∗ (T (M × Definition 3.5.1. A morphism between (M , T M ) and (N , T N ) is a pair f˜ = (f , E˜ ) where f : M → N is a smooth map and E˜ is a maximal isotropic extended structure on the graph of f . We also call E˜ an extended structure on f and f˜ an extension of f .

˜ N ) is given by π2∗ [HN ] − π1∗ [HM ], then by Lemma 3.2.3, we see that f admits an We note that the Ševera class of T (M × extension into a morphism f˜ if and only if f ∗ [HN ] = [HM ]. Similar to Proposition 2.5.2, we have the following Proposition 3.5.2. Let f : M → N be a smooth map between extended manifolds, so that f ∗ [HN ] = [HM ]. Choose splittings sM and sN of T M and T N respectively, then any morphism f˜ : (M , T M ) → (N , T N ) extending f is determined by an element bf˜ ∈ Ω 2 (M ) so that HM = f ∗ HN − dbf˜ . Let BM ∈ Ω 2 (M ) and BN ∈ Ω 2 (N ) and consider splittings s0M = BM ◦ sM and s0N = BN ◦ sN . Then the morphism f˜ is determined in the new splittings by (f , b0˜ ), where b0˜ = bf˜ − BM + f ∗ BN ∈ Ω 2 (M ). In particular, choose f

f

BM = bf˜ and BN = 0, we see that b0˜ = 0. f Proof. The direct sum splitting s := sM ⊕ sN gives rise to the twisting form π1∗ HM + π2∗ HN . Under the splitting s, the isotropic extended structure E˜ corresponds to the two form βs defined in Lemma 3.2.4. Take bf˜ := βs then the rest of statement follows from Lemma 3.2.4. As an example, let F ⊂ M and let T F be defined by the diagram (3.6), then the inclusion i : F → M can be extended to a morphism ˜i. In particular, choosing the splittings as given in the Lemma 3.2.1, the two form part b˜i vanishes. Definition 3.5.3. Let f˜ : (M , T M ) → (N , T N ) and g˜ : (N , T N ) → (P , T P ) be two morphisms. Fix splittings of the extended tangent bundles, then the composition h˜ = g˜ ◦ f˜ is defined by h = g ◦ f and bh˜ = bf˜ + f ∗ (bg˜ ) ∈ Ω 2 (M ).

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3.6. The category We are now able to describe the category E Smth of extended manifolds. An object in E Smth is a pair (M , T M ), a smooth manifold with an extended tangent bundle. The Ševera class of T M will be denoted [HM ]. A morphism f˜ : (M , T M ) → (N , T N ) is given by an isotropic extended structure on the graph of a smooth map f : M → N (cf. Definition 3.5.1). The composition of morphisms is well defined (cf. Definition 3.5.3). We have the following fibration sequence:

Ω02 (M ) → HomE Smth (M , N ) → CT∞ (M , N ) where CT∞ (M , N ) denotes the space of smooth maps f : M → N such that f ∗ [HN ] = [HM ]. In particular, when M = N as extended manifolds, we may consider the group GT of symmetries of T M, consisting of invertible morphisms, which fits in the following sequence (see Section 4.1): 0 → Ω02 (M ) → GT → Diff[HM ] (M ), where Diff[HM ] (M ) denotes the subspace of Diff(M ) preserving [HM ]. When the Ševera class vanishes, we use the subscript 0 in the various notations. It is shown in [13] that G0 = Diff(M ) n Ω02 (M ). In Section 4, we will have a more detailed discussion about the group GT and its Lie algebra. Note also that E Smth is a category with an involution, where the involution is given by the reversion T M 7→ −T M. 3.7. Spinors and cohomology Let Cl(T M ) be the bundle of Clifford algebras associated to T M, defined from the pairing h, i. The spinorial action of the algebra of sections of Cl(T M ) on I(M ) × Ω • (M ) is defined from the following contraction ιX by sections X of T M:

ιX (s, ρ) := (s, ιX ρ + ξ ∧ ρ) where ξ = X − s(X ).

(3.8)

Recall that the action of Ω (M ) on I(M ) is given by (3.5). We define the action of B ∈ Ω (M ) on the differential form ρ ∈ Ω • (M ) by 2

2

B ◦ ρ = e−B ∧ ρ. As in Lemma 2.8.1, the contraction commutes with the anti-diagonal action of Ω 2 (M ) on I(M ) × Ω • (M ). Analogous to the linear case in Section 2.8, we have Definition 3.7.1. The abstract spinor space for T M is

S • (M ) = I(M ) ×Ω 2 (M ) Ω • (M ), where Ω 2 (M ) acts in the anti-diagonal fashion. Again, a choice of s ∈ I(M ) gives an identification of S • (M ) to Ω • (M ). Note that the space S • (M ) is no longer graded by Z, instead, it only retains the induced Z2 -grading. Lemma 3.7.2. The abstract spinor space S • (M ) is naturally a differential graded module over Ω • (M ). Proof. We define the operations on I(M ) × Ω • (M ) and show that they commute with the action of Ω 2 (M ). Let (s, ρ) ∈ I(M ) × Ω • (M ), then the action of B ∈ Ω 2 (M ) is given by B ◦ (s, ρ) = (B ◦ s, (−B) ◦ ρ) = (B ◦ s, eB ∧ ρ). The wedge product ∧α for α ∈ Ω • (M ) is given by

(s, ρ) ∧ α := (s, ρ ∧ α), which obviously commute with the Ω 2 (M ) action. The differential dT is defined as dT (s, ρ) := (s, dHs ρ) where dHs ρ = dρ − Hs ∧ ρ. It is easy to see that d2T (s, ρ) = (s, 0). We only have to show that dT commutes with the action of Ω 2 (M ) on the Ω • (M )component: d(eB ∧ ρ) − HB◦s ∧ (eB ∧ ρ) = eB (dB ∧ ρ + dρ) − eB ∧ (Hs + dB) ∧ ρ

= eB ∧ (dρ − Hs ∧ ρ). Thus the wedge product and differential descend to S • (M ) and we use the same notations for the corresponding operations on S • (M ). It is obvious that the wedge product makes S • (M ) a graded Ω • (M )-module. For the differential, we note that d2T = 0 and compute dHs (ρ ∧ α) = d(ρ ∧ α) − Hs ∧ ρ ∧ α = dHs ρ ∧ α + (−1)|ρ| ρ ∧ dα.

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Definition 3.7.3. The de Rham cohomology of T M is the cohomology of the complex (S • (M ), dT ), i.e. H • (T M ) := H • (S • (M ), dT ). From the above lemma, we see that H • (T M ) is a graded module over H • (M ). When we use s ∈ I(M ) to identify S • (M ) with Ω • (M ), we see that the de Rham cohomology of T M is simply the Hs -twisted cohomology H • (M ; Hs ) of M. In particular, we see that H • (M ; H ) depends only on the cohomology class [H ] ∈ H 3 (M ). Definition 3.7.4. Let f˜ : (M , T M ) → (N , T N ) be a morphism. Choose splittings of T M and T N, so that f˜ is represented by f : M → N and bf˜ ∈ Ω 2 (M ). The pull-back f˜ • : S • (N ) → S • (M ) is given by f˜ • (ρ) = bf˜ ◦ f ∗ (ρ) = e

−bf˜

∧ f ∗ (ρ).

The pull-back is independent of the choice of splitting as in the linear case (Lemma 2.8.3). Proposition 3.7.5. f˜ • is a chain map, and therefore it induces a homomorphism f˜ • : H • (T N ) → H • (T M ). Proof. Choose splittings sM ∈ I(M ) and sN ∈ I(N ) and let the twisting forms be HM and HN respectively. Then HM = f ∗ HN − dbf˜ for some bf˜ ∈ Ω 2 (M ) and −bf˜

dHM (e

−bf˜

∧ f ∗ (ρ)) = e

−b ∧ −dbf˜ ∧ f ∗ (ρ) + df ∗ HN f ∗ (ρ) + dbf˜ ∧ f ∗ (ρ) = e f˜ ∧ f ∗ (dHN ρ).

3.8. Thom isomorphism Let π : V → M be an oriented real vector bundle of rank k. In the classical situation, the Thom isomorphism is as follows (cf. [6]). Let Hc∗v (V ) denote the de Rham cohomology of forms with compact support along the fibers, then in Hckv (V ), there is a Thom class [Θ ], which is the unique class that restricts to the orientation class [Θx ] ∈ Hck (Vx ). The Thom isomorphism is therefore defined by wedge product with [Θ ]: ∧[Θ ]

Th : H ∗ (M ) −−→ Hc∗v (V ). In the category E Smth, a vector bundle π˜ : (V , T V ) → (M , T M ) is a morphism extending a classical vector bundle π . It follows that the Ševera class of T V is given by [HV ] = π ∗ [HM ]. Then splittings may be chosen so that the morphism π˜ is given by π together with bπ˜ = 0. We fix such splittings and drop them from the notations. It follows that HV = π ∗ HM . By the local to global principle we can show that the extended de Rham cohomology also has a Thom isomorphism, via wedge product with the Thom class. The ingredients in the argument are discussed in Section 6 for the equivariant case. Let U ⊂ M be a contractible open subset and VU → U be the restriction of V on U. The Poincaré lemma then states that ∧[Θ |U ]

HU = HM |U = dBU for some BU ∈ Ω 2 (U ). Then it follows that T hU : H ∗ (U ; HU ) −−−→ Hc∗v (VU ; π ∗ HU ) is an isomorphism because we have

T hU = e−π

∗B

U

◦ Th|U ◦ eBU .

Applying the Mayer–Vietoris sequence to open sets U , W : H ∗ (U ∩ W ; HU ∩W ) ∼ = ∧[θU ∩V ]

Hc∗v (VU ∩W ; π ∗ HU ∩W )

/ H ∗ (U ∪ W ; HU ∪W ) ∧[θU ∪V ]

/ Hc∗v (VU ∪W ; π ∗ HU ∪W )

/ H ∗ (U ; HU ) ⊕ H ∗ (W ; HW ) ∼ = ∧[θU ]⊕∧[θV ]

/ Hc∗v (VU ; π ∗ HU ) ⊕ Hc∗v (VW ; π ∗ HW )

and with the use of the five-lemma, we have that the Thom isomorphism on U and W implies the Thom isomorphism on U ∪ W . Therefore by induction on the open contractible sets that cover M, we have the Thom isomorphism in twisted cohomology, namely ∧[Θ ]

H • (M ; H ) −−→ Hc•v (V ; π ∗ H ). Then we can conclude: Proposition 3.8.1. Let π˜ be a vector bundle in the category E Smth. Then wedge product by the Thom class induces an isomorphism ∧[Θ ]

T h : H • (T M ) −−→ Hc•v (T V ).

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4. Symmetries and actions 4.1. Generalized symmetries We first consider the symmetries of the triple (T M , T ∗ M , h, i). With a choice of splitting s ∈ I(M ), T M is identified with TM ⊕ T ∗ M and the group of symmetries is given by G = Diff(M ) n Ω 2 (M ), where the composition is given by

(µ, β) ◦ (λ, α) = (µ ◦ λ, λ∗ β + α) for µ, λ ∈ Diff(M ) and α, β ∈ Ω 2 (M ). More precisely, we have the following (compare with the alternative description of automorphisms in Section 2.5.1):

I(M ) × G × T M → T M : (s, (λ, α), X) 7→ (λ, α) ◦s X := s(λ∗ (X )) + λ∗ (ξ + ιX α), where λ ∈ Diff(M ), α ∈ Ω 2 (M ), X = a(X), X = s(X ) + ξ and λ∗ on the forms is defined to be (λ−1 )∗ = (λ∗ )−1 . It is easy to check that the following defines an action of Ω 2 (M ) on G (cf. Proposition 3.5.2) B ◦ (λ, α) = (λ, α − B + λ∗ B) for B ∈ Ω 2 (M ) and (λ, α) ∈ G.

(4.1)

Direct computation gives Lemma 4.1.1. Ω 2 (M ) acts on G by group automorphisms and we have

((−B) ◦ (λ, α)) ◦B◦s X = (λ, α) ◦s X. We then have the following definition Definition 4.1.2. The abstract group of symmetries G of (T M , T ∗ M , h, i) is G := I(M ) ×Ω 2 (M ) G = I(M ) × G/Ω 2 (M ),

where the action of B ∈ Ω 2 (M ) on (s, (λ, α)) ∈ I(M ) × G is: B ◦ (s, (λ, α)) := (B ◦ s, (−B) ◦ (λ, α)) = (B ◦ s, (λ, α + B − λ∗ B)). A choice of splitting s ∈ I(M ) gives the identification of G with the section {s} × G of the bundle I(M ) × G → G with fiber Ω 2 (M ). The action of an element λ˜ ∈ G on T M is defined by such an identification. In particular, we see that the group G fits into the exact sequence: p

0 → Ω 2 (M ) → G − → Diff(M ) → 1, where the first map is simply α 7→ (1, α) and the map p is induced by the projection

I(M ) × G → Diff(M ) : (s, (λ, α)) 7→ λ. Let T M be an extended tangent bundle and GT the group of symmetries of T M, as described in Section 3.6. Choose a splitting s ∈ I(M ) and let Hs ∈ Ω 3 (M ) be the corresponding twisting form. We describe a presentation of GT as a subgroup of {s} × G. According to (3.4), the action of (λ, α) ∈ G under the splitting s preserves the Courant bracket is equivalent to

[s(λ∗ (X )) + λ∗ (ξ + ιX α), s(λ∗ (Y )) + λ∗ (η + ιY α)] 1 = s(λ∗ ([X , Y ])) + λ∗ LX η − LY ξ − d(ιX η − ιY ξ ) + ιY ιX Hs + ι[X ,Y ] α . 2

Straightforward computation using (3.4) shows that the above holds iff

λ∗ Hs + dα = Hs . Definition 4.1.3. The abstract group of symmetries GT of the extended tangent bundle T M is the image of {s} × GHs in G under the quotient I(M ) × G → G where

GHs := {(λ, α)|λ∗ Hs + dα = Hs } ⊂ G. The action of a Lie group G on T M is given by a Lie group homomorphism σ˜ : G → GT . The action σ˜ is said to preserve the splitting s ∈ I(M ) if the image of σ˜ lies in the image of {s} × (GHs ∩ Diff(M )) under the quotient. Let Diff[HM ] (M ) be the subgroup of Diff(M ) preserving the Ševera class [HM ], then we note img(p|GT : GT → Diff(M )) = Diff[HM ] (M ) and

ker(p|GT ) = Ω02 (M ).

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Proposition 4.1.4. Let G be a compact Lie group and σ˜ an action of G on T M, then there exists s ∈ I(M ) which is preserved by σ˜ . In particular, the corresponding twisting form Hs is G-invariant. Proof. Let us start with a random splitting s0 ∈ I(M ), from which we obtain σ˜

G− → G → {s0 } × G : g 7→ (s0 , (λg , αg )). We are looking for a 2-form B ∈ Ω 2 (M ) such that

αg + B − λ∗g B = 0 for all g ∈ G. Consider the action of G on Ω 2 (M ) induced by σ˜ : g (B) := λg ∗ (αg + B). We check that

(gh)(B) = λgh∗ (αgh + B) = λg ∗ λh∗ (λ∗h αg + αh + B) = g (h(B)), i.e. it is indeed a well-defined action. Because G is compact, consider

Z

h(0)dµ(h) =

B :=

Z

G

αh dµ(h), G

where µ is the Haar measure normalized so that the volume of G is 1. Obviously we have for any g ∈ G g (B) = λg ∗ (αg + B) =

Z

gh(0)dµ(h) = B. G

It follows that the splitting s = B ◦ s0 is preserved by σ˜ . The invariance of Hs is obvious from the definition.

Now we go through the same process for the Lie algebras. The Lie algebra of G is X = Γ (TM ) ⊕ Ω (M ) with the Lie bracket 2

[(X , A), (Y , B)] = ([X , Y ], LX B − LY A). The infinitesimal action of (X , A) ∈ X on

(X , A) ◦s

(4.2)

= s(Y ) + η ∈ Γ (T M ) under the splitting s ∈ I(M ) is given by

= −s([X , Y ]) − LX η + ιY A.

Let Hs be the twisting form defined by s, then the Lie algebra of GHs is the following sub-algebra of X:

XHs := {(X , A)|LX Hs + dA = 0} ⊂ X. We write down the 1-parameter subgroup of G generated by (X , A): e

t (X ,A)

:= (λt , αt ) where λt = e and αt = tX

Z

t

λ∗u Adu.

0

From (4.1), we obtain the induced action of Ω 2 (M ) on X: B ◦ (X , A) = (X , A + LX B) for B ∈ Ω 2 (M ) and (X , A) ∈ X. As in Lemma 4.1.1, Ω 2 (M ) acts on X by Lie algebra automorphisms. The following is analogous to Definitions 4.1.2 and 4.1.3: Definition 4.1.5. The Lie algebra of abstract infinitesimal symmetries of the triple (T M , T ∗ M , h, i) is the quotient X := I(M ) ×Ω 2 (M ) X = I(M ) × X/Ω 2 (M ),

where the action of Ω 2 (M ) is the anti-diagonal action on the factors. The Lie algebra XT of abstract infinitesimal symmetries of the extended tangent bundle T M is the image of {s} × XHs in X under the quotient map. Let X˜ ∈ X which is identified to (s, (X , A)) ∈ {s} × X, then the 1-parameter subgroup generated by X˜ , denoted et X is the image of (s, et (X ,A) ) ∈ {s} × G in G. ˜

Note that X (respectively XT ) is the Lie algebra of G (respectively GT ). Correspondingly, we have the exact sequence exhibiting X as a Lie algebra extension: p∗

0 → Ω 2 (M ) → X − → Γ (TM ) → 0. The map p in the above sequence is induced by the natural projection {s} × X → Γ (TM ). We note that ker(p∗ |XT : XT → Γ (TM )) = Ω02 (M ).

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We describe the relation among the Lie sub-algebras XT ⊂ X when the Courant bracket on T M varies. Let T1 M and T2 M be two extended tangent bundles on M and H = δ H (T2 M , T1 M ) the relative twisting form of T2 M with respect to T1 M. Let XTj denote the respective abstract Lie algebra of infinitesimal symmetries of Tj M for j = 1, 2. On X , we may define the H-twisted Lie bracket as following:

[X˜ , Y˜ ]H := [X˜ , Y˜ ] + dιY ιX H , where X˜ , Y˜ ∈ X , p(X˜ ) = X and p(Y˜ ) = Y . We have Lemma 4.1.6. The map ψH : (X , [, ]) → (X , [, ]H ) defined by

ψH (X˜ ) = X˜ + ιX H ,

where X˜ ∈ X and X = p(X˜ ),

is an isomorphism of Lie algebras. Here ιX H ∈ Ω 2 (M ) is an element of X by the natural inclusion. Furthermore, the image of XT2 under ψH is XT1 . Proof. ψH is obviously an isomorphism of linear spaces. Choose and fix a splitting s ∈ I(M ) and identify X with {s} × X. We will drop s from the notations. In this identification, we have ΨH (X , A) = (X , A + ιX H ) and

[ψH (X , A), ψH (Y , B)]H = = = =

[(X , A + ιX H ), (Y , B + ιY H )] + (0, dιY ιX H ) ([X , Y ], LX (B + ιY H ) − LY (A + ιX H ) + dιY ιX H ) ([X , Y ], LX B − LY A + ι[X ,Y ] H ) ψH ([(X , A), (Y , B)])

Let Hs,1 and Hs,2 denote the respective twisting forms defined by s for T1 M and T2 M, then H = Hs,2 − Hs,1 and XTj is identified with {s} × XHs,j . Let (X , A) ∈ XHs,2 , then

LX Hs,1 + d(A + ιX H ) = LX Hs,2 + dA = 0. It follows that ψH (XT2 ) = XT1 .

Definition 4.1.7. Let H ∈ Ω03 (M ) be a closed 3-form. Let X˜ ∈ X and X˜ ◦

denote the action of X˜ on

∈ Γ (T M ). Then the

H-twisted infinitesimal action of X˜ on T M is given by X˜ ◦H

:= ψH−1 (X˜ ) ◦

= X˜ ◦

− ιY ιX H where Y = a( ).

4.2. Extended symmetries Choose a splitting s ∈ I(M ), which defines the twisting form Hs , then we define:

κs : Γ (T M ) → XHs : X = s(X ) + ξ 7→ κs (X) = (X , dξ − ιX Hs ). The action of X on

X ◦s where X ∗

= s(Y ) + η ∈ Γ (T M ) via κs is then

:= −s([X , Y ]) − LX η + ιY (dξ − ιX Hs ) = −X ∗ , is the Loday bracket and is independent of the splitting. It is easy to check the following holds

κB◦s (X) = (−B) ◦ κs (X), and κs induces a map

κT : Γ (T M ) → XT . Direct computation shows Proposition 4.2.1. ker(κT ) = Ω01 (M ) and κT ([X, ]) = κT (X ∗ ) = [κT (X), κT ( )].

Definition 4.2.2. The image XE T := img(κT ) is the Lie algebra of infinitesimal extended symmetries of T M. The 1-parameter subgroup generated by X ∈ Γ (T M ), denoted expT (t X), is the 1-parameter subgroup of GT generated by κT (X). Definition 4.2.3. The action of a Lie algebra g on T M is given by a Lie algebra homomorphism σ˜ : g → XT . We say that g acts by extended symmetries if σ˜ factors through κT , i.e. there is a linear map δ : g → Γ (T M ) so that σ˜ = κT ◦ δ . Such an action is called isotropic if the image of δ is isotropic with respect to h, i and δ is a Lie algebra homomorphism onto the image. The action σ˜ is integrable to a G-action, where G is a connected Lie group, if it is induced by a Lie group homomorphism G → GT and g is the Lie algebra of G. Then G acts by extended symmetries, or the G-action is isotropic, if the corresponding g-action is so.

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4.3. Hamiltonian action Let (M , J ) be an extended complex manifold, which is necessarily of even dimension 2n. It is well known that the extended complex structure J induces a natural Poisson structure Π on M (cf. [9,13]). Let G be a connected Lie group acting on M via a homomorphism σ : G → GT , which is induced by a Lie algebra homomorphism σ˜ : g → XE T . The action σ is Hamiltonian with moment map µ : M → g∗ if the geometrical action of G is Hamiltonian with respect to the Poisson structure Π , with equivariant moment map µ, and the extended action is generated by J (dµ), i.e. σ˜ = κT ◦ δ where δ(τ ) := J (dhµ, τ i) for τ ∈ g (cf. [17]). 4.4. Extended action on spinors Let X ∈ T M. Recall that the contraction ιX is defined in (3.8). By definition, ιX : S • (M ) → S • (M ) does not depend on the Courant (or Loday) bracket on T M, in particular, it does not depend on [HM ]. ˜ ∈ G acts on the abstract spinor space S • (M ) via pull back, as in Definition 3.7.4. More explicitly, the action The element λ of G on S • (M ) is induced from

I(M ) × G × Ω • (M ) → Ω • (M ) : (s, (λ, α), ρ) 7→ eα ∧ λ∗ ρ. Note the sign difference between the above and Definition 3.7.4, because here we use the alternative description of automorphisms as in Section 2.5.1. The infinitesimal action of X on S • (M ) is then induced from

I(M ) × X × Ω • (M ) → Ω • (M ) : (s, (X , A), α) 7→ LX ρ + A ∧ ρ. The special case of action by extended symmetries is of importance: Definition 4.4.1. Let X ∈ Γ (T M ) and ρ˜ ∈ S • (M ). The following defines the extended Lie derivative of ρ˜ in the direction of X

LTX ρ˜ :=

dt d

(expT (t X))• ρ. ˜ t =0

Choose s ∈ I(M ) which determines the twisting form Hs . Let ρ˜ ∈ S • (M ), which is represented by ρ ∈ {s} × Ω • (M ). Then the extended Lie derivative LTX ρ˜ is given explicitly by

LX ρ + (dξ − ιX Hs ) ∧ ρ ∈ {s} × Ω • (M ). Definition 4.4.2. The system of Cartan operators associated with T M on the space of spinors S • (M ) is the collection LTX , ιX and dT defined as above and Lemma 3.7.2. Theorem 4.4.3. Let X ∗

be the Loday bracket for the extended tangent bundle T M. Then we have that as graded operators

[dT , ιX ] = LTX

(4.3)

[LTX , LT

(4.4)

]=

LTX∗

[ιX , ι ] = 2hX, i

(4.5)

[LTX , ι

] = ιX∗

(4.6)

[dT , LTX ] = 0,

(4.7)

[dT , dT ] = 0.

(4.8)

where [, ] denotes the graded commutator of operators (see [14]).

˜ ∈ G and ρ˜ ∈ S • (M ). Proof. Eqs. (4.3) and (4.5) are straightforward calculations, while (4.8) is shown in Lemma 3.7.2. Let λ Then direct computation (by choosing a splitting s ∈ I(M )) shows dT (µ ˜ • ρ) ˜ =µ ˜ • (dT ρ) ˜

ιµ◦ ˜ • ρ) ˜ =µ ˜ • (ιX ρ) ˜ ˜ X (µ Lµ◦ ˜ • ρ) = ˜ X (µ T

(4.9)

µ ˜ • (LTX ρ) ˜

where µ ˜ • denotes the push-forward action on S • (M ), i.e. µ ˜ • = (µ ˜ −1 )• . Now let µ ˜ t = expT (t ) be the 1-parameter subgroup of extended symmetries generated by ∈ Γ (T M ). Then (4.7), (4.6) and (4.4) follows from differentiate the equations in (4.9) (in that order) with respect to t.

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5. Extended equivariant cohomology: Definition and properties In Section 5.1, we define the extended equivariant cohomology through the analogue of the Cartan complex. In Sections 5.2–5.5 we will show that the extended equivariant cohomology satisfies all the expected properties of a cohomology theory, and we will prove in Section 5.6 the existence of a Thom isomorphism when the action preserves a splitting, in particular, when the group is compact. We will not assume G to be compact unless explicitly stated. In the next section, Section 6, we consider some interesting special cases and examples. 5.1. Cartan complex Consider an extended action σ˜ of a Lie algebra g on T M as in Definition 4.2.3. This means that there is a map δ : g → Γ (T M ) so that σ˜ = κT ◦ δ . Then we can generalize the usual definition of the Cartan complex to the extended case by considering the algebra S • (M ) ⊗ b S (g∗ ) of formal series on g with values in S • (M ). The algebra b S (g∗ ) is the a-adic completion of the polynomial algebra S (g∗ ) where a is the ideal generated by all polynomials with zero constant term (we refer to chapter 10 of [3] for the definition of the a-adic completion). In practice what it means is that if {uj } is a base for g∗ and S (g∗ ) = R[u1 , u2 , . . .], then the a-adic completion is the algebra of formal series in the uj ’s:

b S (g∗ ) = R[[u1 , u2 , . . .]]. The parity of the elements is assigned according to the usual rule, i.e. the forms are even or odd according to their degree, while the formal series part b S (g∗ ) is always even. There are various reasons why one needs to consider the algebra of formal series and not the polynomial algebra. One of them being the fact that the B-transformations send forms to forms via multiplications with exponentials (e.g. Proposition 5.2.7). These exponentials are in general not polynomials and therefore by considering only polynomials we would be reducing the change of coordinates transformations to a small group which is not the one we are interested in. Most of the algebraic properties of the Cartan model hold also for the completed model. For ρ ∈ S • (M ) ⊗ b S (g∗ ) we define dT ,δ ρ by its value at τ ∈ g as follows: dT ,δ : S • (M ) ⊗ b S (g∗ ) → S • (M ) ⊗ b S (g∗ )

(dT ,δ ρ)(τ ) := dT ρ(τ ) − ιδ(τ ) ρ(τ ). One may check that dT ,δ is an odd operator and that

[d2T ,δ (ρ)](τ ) = −LTδ(τ ) ρ(τ ) + hδ(τ ), δ(τ )iρ(τ ). Choosing dual bases {τj } and {uj } of the Lie algebra g and its dual g∗ , we may rewrite the above equation in coordinates: dT ,δ ρ = dT ρ −

X

uj ιXj ρ,

and

j

d2T ,δ ρ = −

X

uj LTXj ρ +

X

j

uj uk hXj , Xk iρ,

j ,k

where Xj := δ(τj ). The operators dT and ιX only act on the factor S • (M ) while LTX acts on both factors of S • (M ) ⊗ b S (g∗ ). T 2 Notice that if the extended action σ˜ is isotropic and if Lδ(τ ) ρ = 0, then dT ,δ ρ = 0. Thus, following the definition of the Cartan complex for equivariant cohomology, we propose Definition 5.1.1. Let σ˜ be an isotropic extended action of a Lie algebra g on T M that factors through δ : g → Γ (T M ). The extended g-equivariant Cartan complex is Cg• (T M ; δ) := {ρ ∈ S • (M ) ⊗ b S (g∗ )|LTδ(τ ) ρ = 0 for all τ ∈ g},

(5.1)

with the odd differential dT ,δ . The cohomology Hg (T M ; δ) of the complex Cg (T M ; δ) is the extended g-equivariant de Rham cohomology of T M under the extended action σ˜ defined by δ . •

•

Recall that the (completed) ordinary equivariant Cartan complex is Cg• (M ) = {ρ ∈ Ω • (M ) ⊗ b S (g∗ )|LXτ ρ = 0 for all τ ∈ g}, where Xτ is the infinitesimal action of τ , and the differential dg is given by

(dg ρ)(τ ) = dρ(τ ) − ιXτ ρ(τ ). Similar to Lemma 3.7.2 we have Lemma 5.1.2. The complex Cg• (T M ) is a differential Z2 -graded module over the ordinary Cartan complex Cg• (M ). It follows that the cohomology HG• (T M ) is a Z2 -graded module over the ordinary equivariant cohomology HG• (M ).

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Proof. Choose a splitting s ∈ I(M ) and let Hs be the twisting form. We only need to compute the operators LTδ(τ ) and dT ,δ for ρ ∧ α where ρ ∈ Cg• (T M ), α ∈ Cg• (M ) and τ ∈ g. We write Xτ := δ(τ ) then

LTXτ (ρ ∧ α) = LXτ (ρ ∧ α) + (dξτ − ιXτ Hs ) ∧ (ρ ∧ α) = LTδ(τ ) ρ ∧ α + ρ ∧ LXτ α = 0, dT ,δ (ρ ∧ α) = d(ρ ∧ α) − Hs ∧ (ρ ∧ α) −

X

uj (ιXj + ξj ∧)(ρ ∧ α)

j

= dT ,δ ρ ∧ α + (−1)|ρ| ρ ∧ dg α. where Xτ = s(Xτ ) + ξτ . The lemma follows.

Notation: To use the standard notation of equivariant cohomology we will assume that the action σ˜ is integrable to an action by a connected Lie group G, and denote the extended g-equivariant de Rham cohomology as HG• (T M ; δ). Furthermore, we will often drop the δ from the notations when no confusion should arise. Thus the extended g-equivariant de Rham cohomology will be denoted as HG• (T M ). Assumption 5.1.3. All the extended actions in the following are isotropic. 5.2. Functoriality Let (M , T M ) be an extended G-manifold, i.e. a manifold M with extended tangent bundle T M and an extended G-action defined by δM : g → Γ (T M ). Let (N , T N ) be another such manifold and consider the morphism f˜ = (f , E˜ ) : (M , T M ) → (N , T N ), where E˜ ⊂ −T M ⊕ f ∗ T N is the extended structure on f , then Definition 5.2.1. f˜ is an extended G-equivariant morphism if f is G-equivariant in the usual sense and E˜ is preserved by the diagonal G-action on −T M ⊕ f ∗ T N. Proposition 5.2.2. Let sM ∈ I(M ) and sN ∈ I(N ) be splittings of T M and T N, so that the morphism f˜ is given by f and bf˜ ∈ Ω 2 (M ). Then f˜ is equivariant if and only if dξτ = d(f ∗ ητ + ιXτ bf˜ ) for all τ ∈ g,

(5.2)

where Xτ = δM (τ ) = sM (Xτ ) + ξτ ∈ Γ (T M ) and τ = δN (τ ) = sN (Yτ ) + ητ ∈ Γ (T N ). The following is equivalent to (5.2)

LTXτ (f˜ • (ρ)) = f˜ • (LT ρ), τ

for all ρ ∈ S • (N ) ⊗ b S (g∗ ) and τ ∈ g.

Proof. To prove that the action of G preserves E˜ , we only need to prove that the infinitesimal action of g preserves E˜ . Then (5.2) follows by directly computing the elements of the form

Xτ ∗M (X + f ∗ η + ιX bf˜ ) +

τ

∗N (f∗ (X ) + η),

for X ∈ TM and η ∈ T ∗ N ,

where ∗M and ∗N denote the respective Loday brackets on T M and T N. The map of linear spaces f˜ • : S • (N ) ⊗ b S (g∗ ) → • ∗ b S (M ) ⊗ S (g ) is induced by the pull-back map of Definition 3.7.4. The last statement follows from direct computation and we leave it to the reader. Lemma 5.2.3. Let f˜ be an equivariant morphism. Then dT ,δM (f˜ • (ρ)) = f˜ • (dT ,δN ρ),

for ρ ∈ S • (N ) ⊗ b S (g∗ )

if and only if the following stronger version of (5.2) holds

ξτ = f ∗ ητ + ιXτ bf˜ for all τ ∈ g. Proof. Direct computation and we leave it to the reader.

(5.3)

We remark that one checks easily that the Eq. (5.3) (as well as (5.2)) does not depend on the choice of splittings sM and sN . Definition 5.2.4. An extended G-equivariant morphism f˜ is an extended strictly G-equivariant morphism if the Eq. (5.3) holds. Lemma 5.2.3 implies the functoriality of the extended equivariant de Rham cohomology with respect to extended strictly equivariant morphisms f˜ , and we have

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Corollary 5.2.5. Let f˜ be an extended strictly G-equivariant morphism f˜ : (M , T M ) → (N , T N ), then it induces a morphism f˜ • : HG• (T N , δN ) → HG• (T M , δM ).

We discuss some more implications of a morphism f˜ being equivariant. Lemma 5.2.6. Let f˜ be an extended equivariant morphism, then there exists sM ∈ I(M ) and sN ∈ I(N ) so that the 2-form bf˜ in the presentation of f˜ = (f , bf˜ ) is an invariant form. Proof. Let s0M ∈ I(M ) and sN ∈ I(N ) be splittings and HM and HN are the respective twisting forms. Let (f , b0˜ ) be the f

presentation of f˜ then db0f˜ = f ∗ HN − HM

and dιXτ b0f˜ = −f ∗ dητ − dξτ ,

where Xτ = s0M (Xτ ) + ξτ and τ = sN (Yτ ) + ητ are the infinitesimal actions defined by τ ∈ g. It follows that

LXτ b0f˜ = dιXτ b0f˜ + ιXτ db0f˜ = (dξτ − ιXτ HM ) − f ∗ (dητ − ιYτ HN ).

(5.4)

Let sM = b0˜ ◦ s0M , then under the splittings sM and sN , the morphism is given by (f , bf˜ ) and we have LXτ bf˜ = 0 for all τ ∈ g. f We have a somewhat more general form of functoriality, where we consider the (ordinary) equivariant extension of bf˜ above: Proposition 5.2.7. Let f˜ : (M , T M ) → (N , T N ) be an extended strictly G-equivariant morphism. Let (f , bf˜ ) be the presentation of f˜ under the splittings sM ∈ I(M ) and sN ∈ I(N ), so that bf˜ is invariant. Let bf˜ ,g be an (ordinary) equivariant extension of bf˜ , then f˜g := (f , bf˜ ,g ) induces a natural morphism 0 f˜g• : HG• (T N , δN ) → HG• (T M , δM ), 0 where δM is determined by bf˜ ,g . We have a natural isomorphism 0 HG• (T M , δM ) ' HG• (T M , δM ).

Proof. An equivariant extension of bf˜ is of the form bf˜ ,g = bf˜ − h, where h ∈ (C ∞ (M ) ⊗ g∗ )g is invariant. For τ ∈ g, we write hτ for hh, τ i. Then

δM0 (τ ) := δM (τ ) − dhτ . 0 We note that σ˜ M = κ ◦ δM = κ ◦ δM . From

(dT ,δM0 ρ)(τ ) = (dT ,δM ρ)(τ ) + dhτ ∧ ρ(τ ) we have dT ,δM (eh ρ) = eh dT ,δ 0 ρ M

for all ρ ∈ S • (M ) ⊗ b S (g∗ ).

The last isomorphism is simply give by eh . At the chain level, we define the equivariant pull-back by (cf. Definition 3.7.4): −b 0 f˜g• : Cg• (T N , δN ) → Cg• (T M , δM ) : ρ 7→ e f˜ ,g f ∗ ρ = eh f˜ • (ρ).

It is a chain map and induces the stated morphism on cohomology.

We note that h ∈ (C ∞ (M ) ⊗ g∗ )g is independent of all the other choices made. The above lemma can also be thought of as a statement on g-invariant functions. 0 0 Definition 5.2.8. Two extended actions δM and δM are equivalent if there exists h ∈ (C ∞ (M ) ⊗ g∗ )g so that δM = δM − dh.

Proposition 5.2.7 then states in particular that the equivariant cohomology of equivalent extended actions are isomorphic. An example of non-equivalent exended actions, which generate the same generalized action σ˜ , having different equivariant cohomology is given in Example 6.3.1.

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5.3. Mayer–Vietoris Let U1 and U2 be open subsets of M with U1 ∪ U2 = M and let us denote by U12 the intersection U1 ∩ U2 . Then, it is clear that the sequence j

k

0 → (Cg• (T M ), dT ,δM ) → (Cg• (T U1 ), dT ,δU ) ⊕ (Cg• (T U2 ), dT ,δU ) → (Cg• (T U12 ), dT ,δU ) → 0 1

2

12

is exact, where j(ω) = (ω|U1 , ω|U2 ) and k(α, β) = α|U12 − β|U12 . Then it induces a long exact sequence in cohomology j

HG• (T M )

O

o

HG•+1 (T U12 )

k

/ HG• (T U1 ) ⊕ HG• (T U2 )

HG•+1 (T U1 ) ⊕ HG•+1 (T U2 )

/ HG• (T U12 )

k

o

HG•+1 (T M )

j

5.4. Exact sequence for a pair Let F ⊂ M be a submanifold and suppose that there are extended G-actions σ˜ F and σ˜ M on both F and M. Let ˜i : (F , T F ) → (M , T M ) be a morphism extending the embedding i : F ⊂ M. Then ˜i is an equivariant embedding if the extended action σ˜ F coincides with the action induced from the roof (3.6), which is equivalent to the following: δM |F

π

δM |F : g → Γ (Ann(K )) ⊂ Γ (i∗ T M ) and δF : g −−→ Γ (Ann(K )) − → Γ (T F ), where K = AnnT ∗ M TF . In particular, when we only consider the geometrical action, F is a G-submanifold of M. By Lemma 2.8.3, we have the induced map of equivariant cohomology ˜i• : HG• (T M ) → HG• (T F ), which exists at the chain level. Now, performing the cone construction of ˜i• , one can define the relative complex as Cg• (T M , T F ) := Cg• (T M ) ⊕ Cg•−1 (T F ),

dT ,δ (ω, θ ) = (dT M ,δM ω, ˜i• ω + dT F ,δF θ )

that induces a short exact sequence of complexes 0 → Cg•−1 (T F ) → Cg• (T M , T F ) → Cg• (T M ) → 0. Then we get the long exact sequence in cohomology

/ HG• (T M )

HG• (T M , T F )

O

HG•+1 (T N )

o

˜i•

HG•+1 (T M )

˜i•

o

/ HG• (T N )

HG•+1 (T M , T F )

which is known as the exact sequence for a pair. 5.5. Excision Consider the triple A ⊂ Y ⊂ X . We obtain the isomorphism HG• (T (X − A), T (Y − A)) ∼ = HG• (T X , T Y ) using the Mayer–Vietoris sequence for the sets X − A and Y , and the long exact sequences for the pairs (X , Y ) and (X − A, Y − A). This is an exercise in algebraic topology. 5.6. Thom isomorphism This section is not completely satisfactory because we were not able to prove the Thom isomorphism in the generality of the extended equivariant cohomology. Nevertheless we will show in what follows the Thom isomorphisms for extended actions that preserve a splitting (see Proposition 6.1.2), for which the extended equivariant cohomology is the twisted equivariant cohomology.

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5.6.1. Thom isomorphism for twisted equivariant cohomology Let π : Z → M be a G equivariant real vector bundle of rank k. Then integration along the fibers of the map π on the Cartan complexes

g g π∗ : Ωc•v (Z ) ⊗ S (g∗ ) −→ Ω • (M ) ⊗ S (g∗ ) , where Ωc•v (Z ) denotes differential forms with vertical compact support, is a chain map and induces an isomorphism in equivariant cohomology (see [14, Thm 10.6.1]) ∼ =

π∗ : HGl ,c v (Z ) −→ HGl−k (M ). This isomorphism is what is known as the Thom isomorphism. Its inverse is obtained by wedge product with the Thom class [Θ ] ∈ HGk,c v (Z ) ∼ =

HGl−k (M ) → HGl ,c v (Z )

α 7→ [Θ ] ∧ π ∗ α. Let HG be a closed equivariant three form on M and Let us consider the HG -twisted equivariant Cartan complex of M and the π ∗ HG -twisted equivariant Cartan complex of Z with vertical compact support. Proposition 5.6.1. The map π∗ is a chain map of complexes

π∗ :

g Ωc•v (Z ) ⊗ b S (g∗ ) , dG − π ∗ HG ∧ −→

g Ω • (M ) ⊗ b S (g∗ ) , dG − HG ∧

and it induces an isomorphism of twisted equivariant cohomologies ∼ =

π∗ : HG∗,c v (Z ; π ∗ HG ) −→ HG∗ (M ; HG ). Proof. Let us filter by degree the complexes

g Ωc•v (Z ) ⊗ b S (g∗ ) , dG − π ∗ HG ∧ g Ω • (M ) ⊗ b S (g∗ ) , dG − HG ∧

Cg,c v (Z ; π ∗ HG ) := Cg (M ; HG ) := p

with F Cg (M , HG ) the equivariant forms of degree ≥ p and F p Cg,c v (Z ; π ∗ HG ) the equivariant forms of degree ≥ p + k. The homomorphism π∗ is a chain map of twisted complexes because

π∗ (dG ρ − π ∗ HG ∧ ρ) = dG π∗ ρ − π∗ (π ∗ HG ∧ ρ) = dG π∗ ρ − HG ∧ π∗ ρ, and moreover, it induces a morphism of filtered differential graded modules

π∗ : (Cg,c v (Z ; π ∗ HG ), dG − π ∗ HG ∧; F ) → (Cg (M ; HG ), dG − HG ∧; F ). Thus π∗ induces a homomorphism on the spectral sequences associated with the filtrations ∗,∗

π∗ : Ek∗,∗ → E k

whose first level are the equivariant differential forms

π∗ : E1∗,∗ = Ωc•v (Z ) ⊗ S (g∗ )

g

∗,∗

→ E 1 = Ω • (M ) ⊗ S (g∗ )

g

and and whose differential is the equivariant derivative δ1 = dG . Therefore the second level is the equivariant cohomology and π∗ induces an isomorphism ∼ =

∗,∗

π∗ : E2∗,∗ = HG•,c v (Z ) → E 2 = HG• (M ). Now, we also have that the twisted cohomology is complete with respect to the filtration, i.e. HG•,c v (Z ; π ∗ HG ) = lim HG•,c v (Z ; π ∗ HG )/F p HG•,c v (Z ; π ∗ HG ). ←p

This last statement holds because of two facts: first because the filtration by degree F p HG•,c v (Z ; π ∗ HG ) is equivalent to the filtration ap HG•,c v (Z ; π ∗ HG ) given by the a-adic topology, where a is the ideal of S (g∗ ) generated by polynomials with zero constant term; and second because the twisted cohomology HG•,c v (Z , π ∗ HG ) is complete with respect to the a-adic completion, as it is a finitely generated b S (g∗ )-module. The facts that the twisted cohomologies are complete, that the filtrations are exhaustive and weakly convergent (because the filtrations are by degree), and that at the second level we have an isomorphism, imply by Theorem 3.9 of [24] that π∗ induces an isomorphism of twisted equivariant cohomologies ∼ =

π∗ : HG•,c v (Z ; π ∗ HG ) → HG• (M ; HG ).

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By the same argument as in the untwisted case, the inverse map of π∗ is given by wedge product with the equivariant Thom form Θ . We can conclude that the following is an isomorphism ∧[Θ ]

Th : HG• (M ; HG ) −−→ HG•,c v (Z ; π ∗ HG ). 5.6.2. Thom isomorphism when the action preserves a splitting Let π˜ : (Z , T Z ) → (M , T M ) be a vector bundle in the category E Smth, where a Lie group G acts on (M , T M ) by generalized (resp. extended) symmetries. The bundle π˜ is an equivariant bundle if there is a generalized (resp. extended) G-action on (Z , T Z ) lifting the one on M. It means that the action on Z is fiberwise linear and the action on M is induced from that on Z by restricting to the 0-section. When the G action on T M preserves a splitting, we see that HG• (T M ) ∼ = HG∗ (M ; HG ) and HG•,c v (T Z ) ∼ = HG∗,c v (Z ; π ∗ HG ). As ∼ =

Th : HG• (M ; HG ) → HG•,c v (Z ; π ∗ HG ) we have that it induces an isomorphism of extended equivariant cohomologies ∼ =

Th : HG• (T M ) → HG•,c v (T Z ). 6. Extended equivariant cohomology: Special cases and examples 6.1. Preserving a splitting Let G act by extended symmetries on T M and suppose that the action preserves a splitting s ∈ I(M ), i.e. dξτ − ιXτ Hs = 0

where δ(τ ) = s(Xτ ) + ξτ for τ ∈ g,

and Hs is the twisting form determined by s. Write Xj = s(Xj )+ξj for the P infinitesimal action of the basis element τj ∈ g. Then we have that LTδ(τ ) = LXτ and dT ,G = dG − Hs,G ∧ where dG = d − uj ιXj is the equivariant differential of the (ordinary) P Cartan model, and Hs,G := Hs + j uj ξj is a closed equivariant 3-form also in the (ordinary) Cartan model. We recall Definition 6.1.1. The cohomology of the equivariant Cartan complex

(Ω • (M ) ⊗ b S (g∗ ))g = {ρ ∈ Ω • (M ) ⊗ b S (g∗ )|LXτ ρ = 0 for all τ ∈ g} with differential dG − HG ∧ will be called the HG -twisted equivariant cohomology of M, and will be denoted by HG• (M ; HG ) (cf. [12]). Thus we obtain Proposition 6.1.2. If the extended action σ˜ is isotropic and preserves the splitting s, then HG• (T M ) = HG• (M ; Hs,G ). Proof. We only need to check that the following vanishes: dG Hs,G = dHs +

X

uj (dξj − ιXτj Hs ) −

j

X hXτj + ξj , Xτl + ξl iuj ul . j ,l

The first term vanishes because Hs is closed, the second because the action preserves splitting and the third because the action is isotropic. Corollary 6.1.3. Let G be a compact Lie group, which acts on T M by extended symmetries, then HG• (T M ) is isomorphic to HG• (M ; HG ) for some closed equivariant 3-form HG . Proof. This corollary follows from Proposition 4.1.4.

6.2. Factor through a splitting In general, an ordinary closed G-invariant form H (with respect to the geometrical action) does not necessarily lift to an equivariantly closed form HG with respect to dG . From our construction, when the action is, in a certain sense, compatible with the twisting, we can still obtain a cohomology encoding the action that will be twisted by the (not necessarily invariant) ordinary closed 3-form H. We note that when the cohomology class [HM ] fails to lift to an equivariant class, according to Corollary 6.1.3, the g-action does not integrate to an action of a compact Lie group. An extended g-action factors through a splitting s ∈ I(M ) if the map δ : g → Γ (T M ) factors through s. For such an action, the condition for δ being Lie algebra homomorphism is equivalent to ιXτ ιXω Hs = 0 for all τ , ω ∈ g, where Hs is the twisting form defined by s. Then the extended g-equivariant Cartan complex becomes Cg• (M ; Hs ) = ρ ∈ Ω • (M ) ⊗ b S (g∗ )|LXτ ρ − ιXτ Hs ∧ ρ = 0 for all τ ∈ g ,

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with differential dG,Hs = dHs − following definition:

P

j

uj ιX τj = dG − Hs ∧. The corresponding cohomology is denoted by HH•s ,G (M ). We make the

Definition 6.2.1. Let M be an ordinary G-manifold and H ∈ Ω03 (M ) so that ιXτ ιXω H = 0 for all τ , ω ∈ g, the H-twisted equivariant cohomology of M is defined to be HH• ,G (M ) as above. 6.3. Trivial action We consider the trivial action of G on T M, where G can be taken as a compact Lie group. This is an example of action preserving a splitting described in Section 6.1, while it is of independent interest in the discussion of localization. In the ordinary case, the trivial action of G on M gives the equivariant cohomology HG∗ (M ; 0) = H ∗ (M ) ⊗ b H ∗ (BG), where • ∗ ∗ g b b H (BG) = S (g ) . Here, although the action of G on T M is trivial, the cohomology HG (T M ) may be different from HG∗ (M ; 0). Consider a linear map δ : g → Ω01 (M ) = ker(κT ) (see Section 4.2). The induced action of G is then trivial on T M and the extended g-equivariant Cartan complex becomes: Cg• (T M ) = Ω • (M ) ⊗ b S (g∗ ) and

dT ,δ ρ = dT ρ −

X

uj ξj ∧ ρ,

j

where ξj := δ(τj ). We choose a splitting s ∈ I(M ), which defines a twisting form Hs , and we rewrite the differential as:

! dT ,δ ρ = dρ −

Hs +

X

uj ξj

∧ ρ = dρ − Hs,G ∧ ρ,

j 3 where Hs,G := Hs + j uj ξj is seen as representing a cohomology class in HG (M ) because dξj = 0 and Xτj = 0. Thus, the • extended equivariant cohomology HG (T M ) is the Hs,G -twisted equivariant cohomology HG• (M ; Hs,G ).

P

Example 6.3.1. Take M = S 1 and G = S 1 with the trivial G action in M, and consider the extended action δ : R → Ω01 (M ), 1 7→ dθ with T M = TM ⊕ T ∗ M. The extended equivariant Cartan complex becomes Cg• (T M ) = Ω • (M ) ⊗ R[[u]] with differential dT ,G = d − udθ ∧. An element α ∈ Cg• (T M ) is of the following form, where fi and gj ’s are functions over S 1 :

α=

X

f i ui + d θ

X

gj uj

and dT ,G α = df0 +

i >0

j

i

X (dfi − fi−1 dθ )ui .

Thus α is closed if and only if df0 = 0 and dfi = fi−1 dθ for all i > 0, which implies that fi = 0 for all i, i.e. the closed forms are

α = dθ

X

gj uj ∈ Cgod (T M ).

j

If we consider the form β = R i hi ui then α = dβ is equivalent to the set of equations dh0 = g0 dθ , dhi = (gi + hi−1 )dθ , i > 0 have a solution whenever S 1 g0 dθ = 0. So we can conclude that

P

HGev (T S 1 ) = 0

and

HGod (T S 1 ) = R.

6.4. Circle bundle over surfaces Let π : M → Σ be an S 1 -principle bundle over a closed surface Σ of genus g and choose H ∈ Ω 3 (M ) to be an invariant volume form. We compute the cohomology group HG• (M ; kH ), for k 6= 0, as defined in Definition 6.2.1, which is a special case of the equivariant cohomology HG• (T M ) for T M with Ševera class [kH ]. We note that H cannot be lifted to an equivariantly closed form in the usual Cartan model. Suppose that HG = H + uξ were such a lifting, then we see that the equation dG HG = 0 1 ∗ 1 implies dRξ = ιX H and R ιX ξ = 0, where X is the infinitesimal action of S . It follows that ξ = π λ for some λ ∈ Ω (Σ ). Then we have M H = a Σ dλ = 0 for certain a 6= 0, which is a contradiction as H is a volume form. The fact that H fails to lift to an equivariantly closed form implies that the geometrical action of S 1 cannot be lifted to an isotropic extended action of S 1 on TM, i.e. the action under consideration is in fact an extended action of R1 . We consider the complex: Cg• (M ; kH ) = {ρ ∈ Ω • (M ) ⊗ b S (g∗ )|LX ρ − kιX H ∧ ρ = 0} and 1

Because G = S is abelian, we have Cg• (M ; kH ) = {ρ ∈ Ω • (M )|LX ρ − kιX H ∧ ρ = 0} ⊗ b S (g∗ ).

dG,H = d − kH ∧ −uιX .

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Let θ be a connection form on M and write ρ ∈ Ω • (M ) as

ρ = ρ0 + θ ∧ ρ1 ,

where ρi = ai + ρi1 + ρi2 and ιX ρi = 0,

j

with ai ∈ Ω 0 (M ), ρi ∈ Ω j (M ). Then ρ ∈ Cg• (M ; kH ) if and only if

LX ai = 0, LX ρi1 = 0 and LX ρi2 = kai ιX H . By ιX ρi = 0, we see that ιX (dρi2 − ai kH ) = 0, from which it follows that dρi2 − ai kH descends to Σ , i.e. dρi2 = ai kH. Again, a volume form argument implies that ai = 0. In all, we have

ρ ∈ Cg• (M ; kH ) ∩ Ω • (M ) if and only if ρ = π ∗ (α1 + α2 ) + θ ∧ π ∗ (β1 + β2 ). where α1 , β1 ∈ Ω 1 (Σ ) and α2 , β2 ∈ Ω 2 (Σ ). Applying dG,kH we get dG,kH ρ = π ∗ (dα1 ) − uπ ∗ (β1 + β2 ) − dθ1 ∧ π ∗ (dβ1 ). For a general form ρ =

P

j

dG,kH ρ = −dθ1 ∧ π ∗

uj ρj in the tensor product we compute

X

uj dβ1,j − π ∗

X

j =0

uj (β1,j−1 + (β2,j−1 − dα1,j )) + π ∗ dα1,0 .

j =1

It follows that dG,kH ρ = 0 is equivalent to

β1,j = 0,

β2,j = dα1,j+1 for all j > 0 and dα1,0 = 0

and therefore ker dG,kH is equal to the set

( ρ = dθ1 ∧ π

) ∗

X j =0

u dα1,j+1 + π j

∗

X

u (α1,j + α2,j ), with dα1,0 = 0 . j

j=0

To conclude we find that the equivariant cohomology HG• (M ; kH ) in the case of k 6= 0 is always the truncated de Rham cohomology of Σ = M /G:

n

d

o

HG• (M ; kH ) ' H • 0 → Ω 1 (M /G) − → Ω 2 (M /G) → 0 , which, of course, maps to the usual de Rham cohomology of Σ = M /G. For k = 0, the cohomology HG• (M ; 0) is simply the usual equivariant cohomology, which is isomorphic to the de Rham cohomology of Σ = M /G. 6.5. Non-free action on S 3 Let S 3 ⊂ C2 be the unit sphere. We consider the standard coordinates z = (z1 , z2 ) = (x1 + iy1 , x2 + iy2 ) = (x1 , y1 , x2 , y2 ) as well as the polar coordinates (z1 , z2 ) = r (eiφ1 sin λ, eiφ2 cos λ) on C2 , where r 2 = |z1 |2 +|z2 |2 , λ ∈ [0, π2 ) and φj ∈ [0, 2π ) for j = 1, 2. Let H = − sin(2λ)dλ ∧ dφ1 ∧ dφ2 and consider the extended tangent bundle T M of M = C2 \ {(0, 0)} with Ševera class [H ] (note that [H ] 6= 0) with its corresponding splitting. Now, the embedding i : S 3 → M induces the extended structure T S 3 with nontrivial Ševera class and with the chosen splitting, the twisting form is i∗ H. We consider the action of G = S 1 on S 3 induced by rotating the first coordinate z1 :

σ˜ : R1 → Γ (TS 3 ⊕ T ∗ S 3 ) : 1 7→ X =

∂ − cos2 λdφ2 , ∂φ1

which preserves the splitting. Thus by Proposition 6.1.2, the extended S 1 -equivariant cohomology HS•1 (T S 3 ) is given by the twisted equivariant cohomology HS•1 (S 3 ; HS 1 ) with HS 1 = i∗ H − u cos2 λdφ2 . 6.6. Calabi–Yau manifold We first rephrase the notion of a generalized Calabi–Yau manifold, in the sense of [15]. For each maximal isotropic subbundle L of TC M := T M ⊗ C, there is an associated spinor line bundle UL ⊂ ∧• T ∗ M, so that L = AnnUL under Clifford multiplication. An extended complex manifold (M , J ) is called extended Calabi–Yau in the sense of [15], if there is a dT -closed non-vanishing section of ULJ , where LJ is the +i-eigensubbundle of J . We show the following lemma, which is analogous to and a generalization of the corresponding one in symplectic geometry: Lemma 6.6.1. Let (M , J ; ρ) be an extended Calabi–Yau manifold, i.e. ρ ∈ Ω • (M ) is non-vanishing section of ULJ with dT ρ = 0, Suppose that there is a Hamiltonian G-action on M with moment map µ, then ρ admits an equivariant extension which is closed under dT ,δ where δ(τ ) := J (dhµ, τ i) for τ ∈ g (see Section 4.3).

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Proof. The main point is that hJ (dµ), J (dµ)i = hdµ, dµi = 0 which makes Theorem 4.4.3 suitable Pas an infinitesimal definition of a group action (in the sense of Cartan, see §2 of [14]). Let Xj = J (dµj ) where µ = j µj uj , then by the Hamiltonian condition, we have (Xj + idµj ) · ρ = 0, i.e. ιXj ρ = −idµj ∧ ρ . Let ρG = e−iµ ρ , then we compute that dT ,δ ρG = dT (e−iµ ρ) −

X

uj ιXj (e−iµ ρ) = 0.

j

7. Localization 7.1. Fixed point set of a generalized action Let us fix the splitting s ∈ I(M ) that is preserved by the action of the compact Lie group G (cf. Proposition 4.1.4). Then by definition, the image of the homomorphism

σ˜ : G → GT ' {s} × GH lies in GH ∩ Diff(M ). Let x ∈ F ⊂ M be a fixed point where F is a connected component of the fixed point set. Consider the induced representation of G on Tx M, which splits into sub-representations (via s) Tx M = Tx M ⊕ Tx∗ M. Furthermore, the representations Tx M and Tx∗ M split as follows: Tx M = Tx F ⊕ Nx

and Tx∗ M = Tx∗ F ⊕ Nx∗ ,

where we have Tx∗ F = Nx⊥ and Nx∗ = Tx F ⊥ with respect to the natural pairing between TM and T ∗ M. This can be seen explicitly as following. It is obvious that Tx M splits as such, where the Tx F component is simply the trivial subrepresentation, while Nx is the non-trivial part. For Tx∗ M, we choose dual bases {vi } and {ui } of Tx M and Tx∗ M respectively, so that Tx F = Span{vi | i = 1, . . . , k} and Nx = Span{vi | i = k + 1, . . . , n}. Let g ∈ G and we compute gij = hg ◦ ui , vj i = hui , g ◦ vj i =

Ik×k 0

0

∗

.

In particular, it follows that g ◦ ui = ui for i = 1, . . . , k. Let Tx∗ F = Span{ui | i = 1, . . . , k} and Nx∗ = Span{ui | i = k + 1, . . . , n}, then they are sub-representations of Tx∗ M and Nx∗ is the non-trivial part. Let Tx F = Tx F ⊕ Tx∗ F and Nx = Nx ⊕ Nx∗ , then the representation Tx M = Tx F ⊕ Nx naturally splits into trivial and non-trivial components. Let T F = ∪x∈F Tx F , then we show that Lemma 7.1.1. Γ (T F ) is closed under the Courant bracket. Proof. Because σ˜ fixes the splitting s, we have the induced homomorphism of Lie groups:

σ˜ : G → GT ' {s} × GH : g 7→ (λg , 0). and λg H = H for all g ∈ G. Then by definition, ∗

= s(Y ) + η ∈ Γ (T F ) ⇐⇒ λg ∗ Y = Y and λg ∗ η = η for all g ∈ G. Let Z ∈ Γ (T F ), straightforward computation gives λg ∗ [ , Z]H = [ , Z]H .

By Lemma 3.2.5, we see that T F is isomorphic to the induced extended tangent bundle in (3.6) as a Courant algebroid. Thus the Ševera class of T F is given by [HF ] = [i∗ HM ] ∈ H 3 (F ). Corollary 7.1.2. Let σ˜ : G → G be a proper extended action and F a fixed point set component of the geometrical action σ . Then the induced action on T F is trivial, as described in Section 6.3. 7.2. Localization in twisted equivariant cohomology In this section we prove the localization theorem in twisted equivariant cohomology following [2]. We choose and fix a splitting s ∈ I(M ) which is preserved by the G-action, and drop it from the notations. As we have seen in Proposition 6.1.2 the cohomology HG• (T M ) can be calculated using the twisted equivariant de Rham complex

(Ω • (M ) ⊗ b S (g∗ ))G = {ρ ∈ Ω • (M ) ⊗ b S (g∗ )|LXτ ρ = 0 for all τ ∈ g} 3 with differential dG − HG ∧. The equivariantly closed form α := HG = H + j uj ξτj defines a class [α] ∈ HG (M ) and the • twisted equivariant de Rham cohomology is denoted by HG (M , α). As HG• (M ; α) is a module over HG• (M ) (see Lemma 5.1.2) then HG• (M ; α) becomes a module over H • (BG) = S (g∗ )g . If i : F → M is the inclusion of the fixed point set of the geometrical action of G, we will show that the pullback i∗ and the pushout i∗ in extended equivariant cohomology are inverses of each other after inverting the equivariant Euler class

P

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129

of the normal bundle of F . For this we will mimic the proof of Atiyah and Bott of the localization theorem in equivariant cohomology [2,4]. For the sake of simplicity we will focus on the case that G is a torus and we will make use of complex coefficients. Having set up the hypothesis we can start. • ∗ Recall that for G an n-dimensional torus we have T that H (BG) = S (g ) = C[u1 , . . . un ] a polynomial ring in n variables. The support of a H • (BG)-module A is, supp(A) = {f |f ·A=0} Vf where Vf = {X ∈ g|f (X ) = 0}, then Lemma 7.2.1. supp(HG• (M ; α)) ⊂ supp(HG• (M )) Proof. If f · HG• (M ) = 0, by the definition of the H • (BG)-module structure in HG• (M ; α), one has that f · HG• (M ; α) = 0. The inclusion follows. Let F = M G be the fixed point set of the G-action, then from [4, Prop 5.2.5] we know that supp(HG• (M − F )) ⊂ H h where H describes the finite set of proper stabilizers of points in M − F , and h is the Lie algebra of H. Then we have that S supp(HG• (M − F ; α)) ⊂ H h, and therefore HG• (M − F ; α) is a torsion H • (BG)-module.

S

Lemma 7.2.2. The kernel and the cokernel of the map i∗ : HG• (M ; α) → HG• (F ; i∗ α) have support contained in H ( G runs over the stabilizers of points in M that are not fixed by G.

S

H

h where the

• Proof. Let U be an equivariant H h and also S tubular neighborhood of F . We know that supp(HG (M − U ; α)) ⊂ supp(HG• (∂(M − U ); α)) ⊂ H h. Using the long exact sequence for the pair (M − U , ∂(M − U )) we conclude that

S

supp(HG• (M − U , ∂(M − U ); α)) ⊂

[

h.

H

Let V be another equivariant tubular neighborhood containing U, such that V − U ∼ ∂(M − US ) = ∂ U. Then by excision HG• (M , F ; α) ∼ = HG• (M , V ; α) ∼ = HG• (M − U , ∂(M − U ); α), so in particular HG• (M , F ; α, ) ⊂ H h. From the long exact sequence for the pair (M , F ), i∗

HG• (M , F ; α, ) → HG• (M ; α) → HG• (F ; i∗ α, ) → HG• (M , F ; α) the lemma follows.

The Thom isomorphism in twisted cohomology is obtained by wedge product with the Thom class (see Section 5.6). So for π : ν → F the equivariant normal bundle of F in M of rank d = dim(M ) − dim(F ) (seen as a tubular neighborhood q : ν → M), and [Θ ] ∈ Hcdv,G (ν) the Thom class, the Thom isomorphism is Th : HG• (F ; i∗ α) → Hc•v,G (ν; π ∗ i∗ α) a 7→ π ∗ (a) ∧ [Θ ]. But we need to land in Hc•v,G (ν; q∗ α). Then we use the fact that π ∗ is in isomorphism in equivariant cohomology and therefore e−σ

there is an equivariant form σ on ν such that q∗ α = π ∗ i∗ α − dG σ . This gives us the isomorphism Hc•v,G (ν; π ∗ i∗ α) → Hc•v,G (ν; q∗ α), that precomposed with the Thom map is what we are going to call (by abuse of notation) the Thom isomorphism Th := e−σ ◦ th. The pushforward map i∗ : HG• (F ; i∗ α) → HG• (M ; α) is defined as the composition of the maps Th HG• (F ; i∗ α) → Hc•v,G (ν; q∗ α) ∼ = HG• (M , M − F ; α) → HG• (M ; α),

and recall that the equivariant Euler class eG (ν) is defined as the element in HGd (F ) such that π ∗ eG (ν) = j∗ [Θ ], where j∗ : Hc•v,G (ν) → HG• (ν) is the natural homomorphism Lemma 7.2.3. The composition i∗ i∗ is simply the multiplication by eG (ν) (using the HG• (F )-module structure), i.e. for a ∈ HG• (F ; i∗ α), we have i∗ i∗ (a) = a ∧ eG (ν). Proof. Consider the commutative diagram i∗

HG• (F ; i∗ α)

Th

/

+ / HG• (M ; α)

Hc•v,G (ν; q∗ α) j∗

HG• (ν; q∗ α) o

i∗

π∗

HG• (F ; i∗ α).

Then from the left-hand side j∗ Th(a) = π ∗ a ∧ j∗ [Θ ] = π ∗ (a ∧ eG (ν)), and from the right-hand side j∗ Th(a) = π ∗ (i∗ i∗ (a)), and as π ∗ is an isomorphism, one obtains that i∗ i∗ (a) = a ∧ eG (ν).

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Lemma 7.2.4. The kernel and the cokernel of the map i∗ : HG• (F ; i∗ α) → HG• (M ; α) have support contained in H ( G runs over the stabilizers of points in M that are not fixed by G.

S

H

h where the

Proof. Consider the cohomology exact sequence of the pair (M , M − F ) HG• (M − F ; α)

/ HG• (M , M − F ; α)

and as supp(HG• (M − F ; α)) ⊂

/ HG• (M ; α) nn7 nnn ∼ n = n n nnn i∗ • ∗ HG (F ; i α) S H

h the lemma follows.

/ HG• (M − F ; α)

For Z ⊂ F a connected component of the fixed point set there exists a polynomial fZ ∈ H • (BG) such that the Euler class • eG (νZ ) ofQν|Z is invertible in (HG• (F ))(fZ ) , the localization of HG• (F ) in the ideal generated by fZ as a H S (BG)-module. Therefore • for f = Z ⊂F fZ then eG (ν) is invertible in (HG (F ))(f ) and moreover the kernel of f is contained in H 6=G h. ∗ −1 Now, define the homomorphism Q : (HG• (M ; α))(f ) → (HG• (F ; i∗ α))(f ) by Q (a) := where Z ⊂F iZ (a) ∧ eG (νZ ) • −1 eG (νZ ) ∈ HG (F )(f ) . It turns out that Q is the inverse of i∗ after localizing at (f ): Q ◦ i∗ = 1 because of Lemma 7.2.3, and i∗ ◦ Q = 1 because the module structure is compatible with projections; namely for π : ν → F and a ∈ HG• (M ; α), one has that π ∗ (i∗ a) ∧ π ∗ (eG (ν)−1 ) = π ∗ (i∗ a ∧ eG (ν)−1 ). So, we can conclude:

P

Theorem 7.2.5 (Localization at Fixed Points). For all x ∈ HG• (M ; α) in a suitable localization, one has x=

X

iZ∗ (i∗Z (x)) ∧ eG (νZ )−1 .

Z ⊂F

From the localization theorem we get that HG• (T M )(f ) ∼ Section 6.3 we get = HG• (T F )(f ) , and from the results in sectionP • that HG (T F ) is isomorphic to the twisted equivariant cohomology of the fixed point set. So, for α = H + j uj ξj the twisting form in M, we have Corollary 7.2.6. If the G action is pure, then HG• (T M )(f ) ∼ = HG• (F ; i∗ α)(f ) and if the 1-forms i∗ ξj are all exact (say when H 1 (F ) = 0) then HG• (T M )(f ) ∼ H • (BG)(f ) = H • (F , i∗ H ) ⊗ C[[u1 , . . . , un ]](f ) . = H • (F , i∗ H ) ⊗ b Example 7.2.7. Let us consider the S 1 extended action on S 3 from section Section 6.5. A point in S 3 is a pair of complex numbers z = (z1 , z2 ) with |z1 |2 + |z2 |2 = 1 that could also be written in polar coordinates as z1 = eiφ1 sin λ and z2 = eiφ2 cos λ. The 3-form H is − sin(2λ)dλ ∧ dφ1 ∧ dφ2 , the S 1 action is defined by rotating the first coordinate z1 and the extended action is R1 → Γ (TS 3 ) : 1 7→ ∂φ∂ − cos2 λdφ2 . As the action preserves the splitting, the form α := H − u cos2 λdφ2 1

is equivariantly closed, and therefore defines a cohomology class [α] ∈ HS31 (S 3 ). Recall that H • (BS 1 ) = C[u]. The fixed point set of the circle action is the set F = {(z1 , z2 )|z1 = 0} ∩ S 3 which is also a circle. If i : F → S 3 is the inclusion, then i∗ α = −udφ2 ∈ HS31 (F ) = H 1 (F ) ⊗ H 2 (BS 1 ) and therefore we can apply the results of the Example 6.3.1. So

we have that HSod1 (F ; i∗ α) = R, while HSe1v (F ; i∗ α) = 0. As the normal bundle ν of F in S 3 is trivial, and the action of the circle in the fibers is by rotation, then the equivariant Euler class of the normal bundle is eS 1 (ν) = u. By the localization theorem, if we invert u we get the isomorphisms HSod1 (S 3 ; α)(u) ∼ = HSod1 (F ; i∗ α)(u) ∼ = R(u) = 0,

and HSe1v (S 3 ; α)(u) ∼ = HSe1v (F ; i∗ α)(u) = 0.

Hence, applying the localization theorem we can deduce that the equivariant twisted cohomology HS•1 (S 3 ; α) is a torsion C[u]-module. Acknowledgements The authors would like to thank H. Bursztyn, A. Cardona, G. Cavalcanti, R. Cohen, M. Crainic, N. Hitchin, K. Hori, E. Lupercio, V. Mathai, C. Teleman, for many fruitful conversations. Also the authors would like to thank the support of the Mathematical Sciences Research Institute, the Max Planck Institut and the Erwin Schrödinger Institut. Finally we would like to thank the referee for making a detailed list of corrections and queries that improved the paper dramatically. The first author was supported by funding from DMS at Université de Montréal. Part of the research was carried out at the Yantze Center of Mathematics at Sichuan University. The second author was partially supported by the ‘‘Fondo de apoyo a investigadores jovenes’’ from Universidad de los Andes and from COLCIENCIAS grant # PRE00405000138. Part of the research was carried out at the Max Planck Institute in Bonn and the Erwin Schrödinger Institute in Vienna.

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