aa r X i v : . [ m a t h . QA ] S e p ODD TRANSGRESSION FOR COURANT ALGEBROIDS
PAUL BRESSLER AND CAMILO RENGIFO
Abstract.
The “odd transgression” introduced in (Bressler and Rengifo, 2018) is appliedto construct and study the inverse image functor in the theory of Courant algebroids. Introduction
The goal of this note is to demonstrate applications of “odd transgression” introduced in(Bressler and Rengifo, 2018) to the theory of Courant algebroids.The “odd transgression” functor associates to a Courant algebroid Q a differential-graded(DG) Lie algebroid τ Q over the de Rham complex equipped with a central section of degree-2 which we refer to as a marking . Conversely, a marked DG Lie algebroid A over the deRham complex satisfying certain natural vanishing conditions gives rise, by way of taking itscomponent in degree -1, to a Courant algebroid Q ( A ). In particular, the Courant algebroid Q is recovered in this manner from its transgression τ Q . Using the functors τ and Q one canproject various standard constructions in Lie algebroids to the, perhaps less familiar, settingof Courant algebroids.In this note we apply the above idea to the functor of inverse image of a Lie (respectively,Courant) algebroid under a map manifolds. The construction of inverse image for Courant al-gebroids has appeared in the literature, at least in special cases; see (Li-Bland and Meinrenken,2009; ˇSevera and Valach, 2018; Vysoky, 2019). We study the naturality properties of theinverse image functor as well as its behavior with respect to some standard constructions.We formulate and prove descent for Lie (respectively, Courant) algebroids along a surjectivesubmersion.In addition, we take the opportunity to relate the inverse image for Courant algebroids tothe notion of Dirac structure with support and of Courant morphisms of (Alekseev and Xu,2002) (see also (Bursztyn et al., 2009)).The paper is organized as follows. In Section 2 we briefly review the requisite notionsfrom the theory of DG manifolds. In Section 3 we recall the “odd transgression” for Courantalgebroids of (Bressler and Rengifo, 2018) and further develop its properties. In Section 4we construct the inverse image functor for O -modules equipped with an “anchor” map tothe tangent bundle and study its behavior in compositions of maps. Section 5 is devotedto setting up the descent problem. In Section 6 we apply previously obtained results to thesetting of DG Lie algebroids. In Section 7 we construct the inverse image functor for Courantalgebroids and study its relationship to various notions of Courant algebroid theory. Notation.
In order to simplify notations in numerous signs we will write “ a ” insteadof “deg( a )” in expressions appearing in exponents of −
1. For example, ( − ab − stands for( − deg( a ) · deg( b ) − .Throughout the paper “manifold” means a C ∞ , real analytic or complex manifold. Fora manifold X we denote by O X the corresponding structure sheaf of real or complex valued C ∞ , respectively analytic or holomorphic functions. We denote by T X (respectively, by Ω kX )the sheaf of real or complex valued vector fields (respectively, differential forms of degree k )on X .For a sheaf of algebras R on X , let R − Mod denotes the (abelian) category of sheaves of R -modules on X .For a map of manifolds f : Y → X and E ∈ O X − Mod , we denote by f ∗ E := O Y ⊗ f − O X f − E the induced O Y -module, where f − denotes the sheaf-theoretic inverse image. The assign-ment E 7→ f ∗ E extends to a functor f ∗ : O X − Mod → O Y − Mod . For a sheaf F on X , by a ∈ F we mean that a is a local section of F , i.e. there is an openset U of X such that a ∈ Γ( U ; F ).For a Z -graded object A and i ∈ Z we denote by A i the graded component of A of degree i . 2. DG manifolds
In what follows we use notation introduced in (Bressler and Rengifo, 2018).2.1.
The category of DG manifolds.
For the purposes of the present note a differential-graded manifold (DG-manifold) is a pair X := ( X, O X ), where X is a manifold and O X is asheaf of commutative differential-graded algebras (CDGA) on X locally isomorphic to oneof the form O X ⊗ S ( E ), where S ( E ) is the symmetric algebra of a finite-dimensional gradedvector space E .Let X = ( X, O X ) and Y = ( Y, O Y ) be DG-manifolds. A morphism φ : X → Y is amorphism of ringed spaces, which is to say a map φ : X → Y of manifolds together with amorphism of differential-graded algebras φ ∗ : φ − O Y → O X compatible with the canoncicalmap φ − O Y → O X .We denote the category of DG-manifolds by dg Man . Let dg
Man + denote the full subcat-egory of DG-manifolds X = ( X, O X ) such that O i X = 0 if i < N for some N ∈ Z .For X ∈ dg Man we denote by O X − Mod the category of sheaves of differential-gradedmodules over the structure sheaf O X . Example . An ordinary manifold is an example of a DG-manifold with the structure sheafconcentrated in degree zero. Each ordinary manifold X determines a DG-manifold X ♯ ∈ dg Man + defined by X ♯ = ( X, Ω • X , d ) and frequently denoted by T [1] X in the literature. DD TRANSGRESSION FOR COURANT ALGEBROIDS 3
There is a canonical morphism X → X ♯ of DG-manifolds defined by the canonical mapΩ • X → O X . Example . Let ~ t denote the DG-manifold with the underlying space consisting of one pointand the DG-algebra of functions O ~ t = C [ ǫ ], the free graded commutative algebra with onegenerator ǫ of degree − ∂ ǫ : ǫ
1. Note that ~ t ∈ dg Man + .The category dg Man + has finite products ((Bressler and Rengifo, 2018), Lemma 2.1). For X = ( X, O X ) , Y = ( Y, O Y ) ∈ dg Man + the product is given by ( X × Y, O X × Y ) with O X × Y := O X × Y ⊗ pr − X O X ⊗ pr − Y O Y pr − X O X ⊗ pr − Y O Y . The odd path space.
For a manifold X the mapping space X ~ t is represented by theDG manifold X ♯ of Example 1 ((Bressler and Rengifo, 2018), Theorem 2.1). The evaluationmap ev : X ♯ × ~ t → X corresponds to the morphism of “pull-back of functions” ev ∗ : O X → O X ♯ [ ǫ ] := O X ♯ × ~ t given by f f + df · ǫ .There is a short exact sequence of graded O X ♯ -modules0 → O X ♯ ⊗ O X T X [1] → T X ♯ → O X ♯ ⊗ O X T X → , where O X ♯ ⊗ O X T X [1] ∼ = T X ♯ /X .2.3. Immersions and submersions.Proposition 3.
If a map of manifolds f : Y → X is an immersion (respectively, submer-sion), then the induced map f ♯ : Y ♯ → X ♯ is an immersion (respectively, submersion).Proof. The derivative df ♯ : T Y ♯ → f ♯ ∗ T X ♯ gives rise to the map of short exact sequences0 −−−→ O Y ♯ ⊗ O Y T Y [1] −−−→ T Y ♯ −−−→ O Y ♯ ⊗ O Y T Y −−−→ id ⊗ df y df ♯ y y id ⊗ df −−−→ O Y ♯ ⊗ f − O X T X [1] −−−→ T X ♯ −−−→ O Y ♯ ⊗ f − O X T X −−−→ (cid:3) Transgression for O -modules. We denote by pr : X ♯ × ~ t → X ♯ the canonical projec-tion. The diagram X ♯ × ~ t ev −−−→ X pr y X ♯ gives rise to the functor(2.4.1) pr ∗ ev ∗ : O X − Mod → O X ♯ − Mod . PAUL BRESSLER AND CAMILO RENGIFO
Since the underlying space of both X ♯ and X ♯ × ~ t is equal to X , the functor ev ∗ : O X − Mod →O X ♯ × ~ t − Mod is given by ev ∗ E = O X ♯ [ ǫ ] ⊗ O X E and the effect of the functor pr ∗ amounts torestriction of scalars along the unit map O X ♯ → O X ♯ [ ǫ ].2.5. Lie algebroids. An O X -Lie algebroid structure on an O X -module A consists of(1) a structure of a sheaf of C -Lie algebras [ , ] : A ⊗ C A → A ;(2) an O X -linear map σ : A → T X of Lie algebras called the anchor map .These data are required to satisfy the compatibility condition (Leibniz rule)[ a, f · b ] = σ ( a )( f ) · b + f · [ a, b ]for a, b ∈ A and f ∈ O X .A morphism of O X -Lie algebroids φ : A → A is an O X -linear map of Lie algebras whichcommutes with respective anchor maps.With the above definition of morphisms O X -Lie algebroids form a category denoted O X − LieAlgd .The notion of Lie algebroid generalizes readily to the DG context.2.6.
Transgression for Lie algebroids.
Suppose that A is an O X -Lie algebroid as in2.5. It is shown in (Bressler and Rengifo, 2018), 3.5, that the O X ♯ -module pr ∗ ev ∗ A admitsa canonical structure of a O X ♯ -Lie algebroid, denoted henceforth by A ♯ . Moreover, theassignment A 7→ A ♯ extends to a functor( ) ♯ : O X − LieAlgd → O X ♯ − LieAlgd which preserves terminal objects, i.e. the canonical map T ♯X → T X ♯ is an isomorphism, andproducts. 3. Transgression for Courant algebroids
Marked Lie algebroids.
Suppose that X = ( X, O X ) is a DG-manifold.A marked O X -Lie algebroid is a pair ( A , c ), where A is a O X -Lie algebroid and c ∈ Γ( X ; A )is a homogeneous central section (i.e. [ c , A ] = 0). The section c is called a marking.It is easy to see ((Bressler and Rengifo, 2018), Lemma 3.6) that any (homogeneous) centralsection belongs to the kernel of the anchor map.A morphism φ : ( A , c ) → ( A , c ) is a morphism of Lie algebroids φ : A → A such that φ ( c ) = c . In particular, c and c have the same degree.With the above definitions marked O X -Lie algebroids and morphisms thereof form a cate-gory denoted O X − LieAlgd ⋆ . The full subcategory of marked O X -Lie algebroids ( A , c ) withdeg c = n is denoted O X − LieAlgd ⋆n .For a marked Lie algebroid ( A , c ) with deg c = n the Lie algebroid structure on A descendsto A := coker( O X [ n ] · c −→ A )((Bressler and Rengifo, 2018), Lemma 3.7). The assignment ( A , c )
7→ A extends to a functor( ) : O X − LieAlgd ⋆ → O X − LieAlgd . DD TRANSGRESSION FOR COURANT ALGEBROIDS 5
Example . The structure sheaf O X [ n ] has a canonical structure of a marked O X -Lie alge-broid, whose non trivial part is the marking id : O X [ n ] → O X [ n ]. In particular, the markingis a morphism of marked Lie algebroids and the O X -Lie algebroid O X [ n ] is zero.3.2. O X [ n ] -extensions. Suppose that B is a O X -Lie algebroid.An O X [ n ]-extension of B is a marked O X -Lie algebroid ( A , c ) with deg c = n together with a morphism A → B such that the sequence0 → O X [ n ] · c −→ A → B → O X [ n ]-extensions of B is a morphism of marked Lie algebroids whichinduces the identity map on B . Such a map is necessarily an isomorphism. We denote thecategory (groupoid) of O X [ n ]-extensions of B by O [ n ]Ext( B ).The category O [ n ]Ext( B ) has a canonical structure of a ‘ C -vector space in categories’(hence, in particular, that of a Picard groupoid). Namely, given extensions A , . . . , A m andcomplex numbers λ , . . . , λ m the ‘linear combination’ λ A ∔ · · · ∔ λ m A m is defined by thepush-out diagram O X [ n ] × · · · × O X [ n ] −−−→ A × B · · · × B A m y y O X [ n ] −−−→ λ A ∔ · · · ∔ λ m A m where the left vertical arrow is given by ( α , . . . , α m ) P i λ i α i . The bracket on λ A ∔ · · · ∔ λ m A m is characterized by the fact that the right vertical map is a morphism of Lie algebras.3.3. Courant algebroids.
Courant algebroids were introduced in (Liu et al., 1997), (Roytenberg,2002) and (Bressler and Chervov, 2005). For comparison of the following definition with theone encountered in the literature see Remark 5.A Courant algebroid is an O X -module Q equipped with(1) a structure of a Leibniz C -algebra { , } : Q ⊗ C Q → Q ;(2) an O X -linear map of Leibniz algebras (the anchor map) π : Q → T X ;(3) a symmetric O X -bilinear pairing h , i : Q ⊗ O X Q → O X ;(4) an O X -linear map (the co-anchor map) π † : Ω X → Q . not assumed to be non-degenerate PAUL BRESSLER AND CAMILO RENGIFO
These data are required to satisfy π ◦ π † = 0(3.3.1) { q , f q } = f { q , q } + π ( q )( f ) q (3.3.2) h{ q, q } , q i + h q , { q, q }i = L π ( q ) h q , q i (3.3.3) { q, π † ( α ) } = π † ( L π ( q ) ( α ))(3.3.4) h q, π † ( α ) i = ι π ( q ) α (3.3.5) { q , q } + { q , q } = π † ( d h q , q i )(3.3.6)for f ∈ O X and q, q , q ∈ Q .A morphism φ : Q → Q of Courant algebroids on X is an O X -linear map of Leibniz C -algebras such that the diagramΩ X π † −−−→ Q π −−−→ T X (cid:13)(cid:13)(cid:13) φ y (cid:13)(cid:13)(cid:13) Ω X π † −−−→ Q π −−−→ T X is commutative.With the above definitions Courant algebroids on X and morphisms thereof form a cate-gory henceforth denoted CA ( X ). Remark . The orginal definition of Courant algebroid is due to (Liu et al., 1997), based on(Courant, 1990) and (Dorfman, 1993), where the additional assumption of non-degeneracy ofthe pairing is made. Under the latter assumption the co-anchor map is uniquely determinedby the anchor map and, hence, does not appear as a separate item. The broader conceptof Courant algebroid as described by the definition was advanced in (Bressler, 2007) andpermits, for example, regarding the kernel of the anchor map as a Courant algebroid withtrivial anchor.Suppose that Q is a Courant algebroid on X . Let Q = coker( π † ) . The Courant algebroid structure on Q descends to a structure on a Lie algebroid on Q . Inwhat follows we refer to the Lie algebroid Q as the Lie algebroid associated to the Courantalgebroid Q .The assignment Q 7→ Q extends to a functor( ) : CA ( X ) −→ O X − LieAlgd . For a Courant algebroid Q the opposite Courant algebroid, denoted Q op , has the sameunderlying O X -module as Q , the same Leibniz bracket (i.e. { , } op = { , } ), same anchormap π op = π , the symmetric pairing h , i op = −h , i and the co-anchor map π op † = − π † . DD TRANSGRESSION FOR COURANT ALGEBROIDS 7
Suppose that Q and Q are Courant algebroids. The Courant algebroid Q ∔ Q is definedby the push-out square Ω X × Ω X −−−→ Q × T X Q y y Ω X −−−→ Q ∔ Q and equipped with the component-wise operations.3.4. Courant extensions.
Suppose that A is a O X -Lie algebroid. A Courant extension of A is a Courant algebroid Q together with the identification Q ∼ = A such that the sequence(3.4.1) 0 → Ω X π † −→ Q → A → φ : Q → Q of Courant extensions of A is a morphism of Courant algebroidswhich is compatible with the identifications Q i ∼ = A . We denote the category of Courantextension of A by C Ext( A ). A morphism in C Ext( A ) induces a morphism of associated shortexact sequences (3.4.1), hence is an isomorphism of underlying O X -modules. It is easy to seethat the inverse map is, in fact, a morphism of Courant algebroids. Consequently, C Ext( A )is a groupoid.3.5. Linear algebra.
The category C Ext( A ) has a canonical structure of a ‘ C -vector spacein categories’ (hence, in particular, that of a Picard groupoid). Namely, given extensions Q , . . . , Q n and complex numbers λ , . . . , λ n the ‘linear combination’ λ Q ∔ · · · ∔ λ n Q n isdefined by the push-out diagramΩ X × · · · × Ω X −−−→ Q × A · · · × A Q n y y Ω X −−−→ λ Q ∔ · · · ∔ λ n Q n where the left vertical arrow is given by ( α , . . . , α n ) P i λ i α i . The Leibniz bracket on λ Q ∔ · · · ∔ λ n Q n is characterized by the fact that the right vertical map is a morphism ofLeibniz algebras.3.6. Exact Courant algebroids.
A Courant algebroid Q is called exact if the map Q → T X is an isomorphism. Thus, an exact Courant algebroid is a Courant extension of T X . Wedenote the category of exact Courant algebroids by E CA ( X ) := C Ext( T X ).3.7. Transitive Courant algebroids.
A Courant algebroid is called transitive if the as-sociated Lie algebroid is, which is to say, the anchor map is an epimorphism. If Q is atransitive Courant algebroid, the sequence0 → Ω X → Q → Q → PAUL BRESSLER AND CAMILO RENGIFO
Transgression for Courant algebroids.
We denote by(3.8.1) R : pr ∗ ev ∗ Ω X = O X ♯ [ ǫ ] ⊗ O X Ω X → O X ♯ [2] . the map of O X ♯ -modules whose component of degree − Q the marked Lie algebroid ( τ Q , c ) ∈ O X − LieAlgd ⋆ is given bythe O X ♯ -module τ Q := coker( pr ∗ ev ∗ Ω X ( R , − pr ∗ ev ∗ ( π † )) −−−−−−−−−→ O X ♯ [2] ⊕ pr ∗ ev ∗ Q ) , where R is the map (3.8.1). In other words, the square(3.8.2) pr ∗ ev ∗ Ω X pr ∗ ev ∗ ( π † ) −−−−−−→ pr ∗ ev ∗ Q R y y O X ♯ [2] −−−→ τ Q is cocartesian.The anchor map is induced by pr ∗ ev ∗ ( π ) and the marking c ∈ Γ( X ; τ Q − ) is the image of1 ∈ Γ( X ; ( O X ♯ [2]) − ) under the bottom horizontal map in (3.8.2).The bracket is the extension by Leibniz rule of(1) [ c , τ Q ] = 0(2) [ q · ǫ, β ] = − ( − | β | [ β, q · ǫ ] = ι π ( q ) β (3) [ q, β ] = − [ β, q ] = L π ( q ) β (4) [ q · ǫ, q · ǫ ] − , − = h q , q i ∈ ( O X ♯ [2]) − = O X (5) [ q , q · ǫ ] , − = { q , q } · ǫ (6) [ q · ǫ, q ] − , = − d h q , q i + { q , q } · ǫ The marked O X ♯ -Lie algebroid τ Q enjoys the following properties:(1) The natural map Q ♯ → τ Q is an isomorphism.(2) The canonical map Q → τ Q − is an isomorphism.(3) Q is a Courant extension of A if and only if τ Q is a O X ♯ [2]-extension of A ♯ .The assignment Q 7→ τ Q extends to a functor τ : CA ( X ) −→ O X ♯ − LieAlgd ⋆ . Suppose that ( B , c ) ∈ O X ♯ − LieAlgd ⋆ satisfies B i = 0 for i −
2. Then, the derivedbracket and the ( − , − O X ♯ [2] · c −→ B endow B − with a structure of a Courantalgebroid (see (Bressler and Rengifo, 2018), Lemma 6.1). We denote this Courant algebroidstructure on B − by Q ( B , c ).For any Courant algebroid Q the marked Lie algebroid τ Q satisfies the above requirements.Moreover, Q ( τ Q ) = Q . DD TRANSGRESSION FOR COURANT ALGEBROIDS 9
Proposition 6.
Suppose that A is a O X -Lie algebroid A . The functor τ restricts to anequivalence of categories τ : C Ext( A ) −→ O [2]Ext( A ♯ ) whose quasi-inverse is given by the functor Q : O [2]Ext( A ♯ ) → C Ext( A ) . Transgression and linear algebra.
Suppose that A is a O X -Lie algebroid locally freeof finite rank over O X . Then, so is any Courant extension of A . Suppose that Q , . . . , Q n ∈C Ext( A ), λ , . . . , λ n ∈ C . For α , . . . α n ∈ Ω X let ℓ ( α , . . . α n ) = P i λ i α i .The diagram pr ∗ ev ∗ (Ω X × · · · × Ω X ) −−−→ pr ∗ ev ∗ Ω X × · · · × pr ∗ ev ∗ Ω X y y pr ∗ ev ∗ ( Q × A · · · × A Q n ) −−−→ pr ∗ ev ∗ Q × A ♯ · · · × A ♯ pr ∗ ev ∗ Q n is commutative with horizontal maps isomorphisms. Both squares in the diagram pr ∗ ev ∗ (Ω X × · · · × Ω X ) −−−→ pr ∗ ev ∗ ( Q × A · · · × A Q n ) pr ∗ ev ∗ ( ℓ ) y y pr ∗ ev ∗ Ω X −−−→ pr ∗ ev ∗ ( ∔ ni =1 λ i Q i ) R y y O X ♯ [2] −−−→ τ ( ∔ ni =1 λ i Q i )are co-Cartesian, hence so is the outer one and the same holds for the diagram pr ∗ ev ∗ Ω X × · · · × pr ∗ ev ∗ Ω X −−−→ pr ∗ ev ∗ Q × A ♯ · · · × A ♯ pr ∗ ev ∗ Q n R ×···× R y y O X ♯ [2] × · · · × O X ♯ [2] −−−→ τ Q × A ♯ · · · × A ♯ τ Q nℓ y y O X ♯ [2] −−−→ ∔ ni =1 λ i τ Q i Since the diagram pr ∗ ev ∗ (Ω X × · · · × Ω X ) −−−→ pr ∗ ev ∗ Ω X −−−→ O X ♯ [2] y y y pr ∗ ev ∗ Ω X × · · · × pr ∗ ev ∗ Ω X −−−→ O X ♯ [2] × · · · × O X ♯ [2] −−−→ O X ♯ [2]is commutative with vertical maps isomorphisms it follows that there is a canonical isomor-phism τ ( ∔ ni =1 λ i Q i ) ∼ = ∔ ni =1 λ i τ Q i . We leave the details of the proof of the following lemma to the reader.
Proposition 7.
The functors τ : C Ext( A ) −→ O [2]Ext( A ♯ ) : Q are morphisms of C -vector spaces in categories (and, in particular, of Picard groupoids). The inverse image functor
Inverse image and fiber product for O -modules. Suppose that φ : Y → X is amorphism of DG manifolds and A s −→ C t ←− B are morphisms in O X − Mod . The sequence0 → A × C B → A × B s − t −−→ C is exact. Assume furthermore that A , B and C are locally free of finite rank. In what followswe will consider the following condition on morphisms A s −→ C t ←− B :(C) A × C B = ker( s − t ) and coker( s − t ) are locally freeCondition (C) is trivially fulfilled if at least one of the two maps s and t is an epimorphism. Lemma 8.
Suppose that the morphisms A s −→ C t ←− B satisfy the condition (C) . Then,the canonical morphism φ ∗ ( A × C B ) → φ ∗ A × φ ∗ C φ ∗ B is an isomorphism and the maps φ ∗ A φ ∗ ( s ) −−−→ φ ∗ C φ ∗ ( t ) ←−−− φ ∗ B satisfy the condition (C) .Proof. Applying the functor φ ∗ to the exact sequence0 → A × C B → A × B s − t −−→ C → coker( s − t ) → O X -modules we obtain the commutative diagram with exact rows0 −−−→ φ ∗ ( A × C B ) −−−→ φ ∗ ( A × B ) −−−−→ φ ∗ ( s − t ) φ ∗ C −−−→ φ ∗ coker( s − t ) −−−→ y y y id y −−−→ φ ∗ A × φ ∗ C φ ∗ B −−−→ φ ∗ A × φ ∗ B −−−−−→ φ ∗ s − φ ∗ t φ ∗ C −−−→ coker( φ ∗ s − φ ∗ t ) −−−→ φ ∗ ( A × B ) → φ ∗ A × φ ∗ B is an isomorphism it follows that so is the map φ ∗ ( A × C B ) → φ ∗ A × φ ∗ C φ ∗ B . (cid:3) O -modules over T . Suppose that X = ( X, O X ) is a DG manifold.We denote by O X − Mod / T X the category of O X -modules over T X , i.e. the category of pairs( E , π ), where E ∈ O X − Mod and π : E → T X is a morphism O X − Mod in referred to as theanchor .A morphism t : ( E , π ) → ( E , π ) in O X − Mod / T X is a morphism of O X -modules t : E → E such that π ◦ t = π .The category O X − Mod / T X has a terminal object, namely T X := ( T X , id). The product( E , π ) × ( E , π ) is represented by E × T X E → T X .We shall call ( E , π ) transitive if the anchor map is surjective. DD TRANSGRESSION FOR COURANT ALGEBROIDS 11
Inverse image.
Suppose that φ : Y → X is a morphism of DG manifolds.For E := ( E , π ) ∈ O X − Mod / T X the object φ + E ∈ O Y − Mod / T Y is defined as the left verticalarrow in the pull-back square(4.3.1) φ + E f dφ −−−→ φ ∗ E φ + ( π ) y y φ ∗ ( π ) T Y dφ −−−→ φ ∗ T X which is to say φ + E = T Y × φ ∗ T X φ ∗ E .A morphism t : ( E , π ) → ( E , π ) in O X − Mod / T X induces the morphism φ + ( t ) : φ + E → φ + E in O Y − Mod / T Y and the above construction extends to a functor, called pull-back or inverse image under φ φ + : O X − Mod / T X → O Y − Mod / T Y . The functor of inverse image preserves terminal objects, that is to say φ + T X = T Y .The functor of inverse image preserves transitivity: if E := ( E , π ) is transitive then so is φ + E .By the universal property of φ ∗ there is a unique map φ ∗ ( E × φ ∗ T X E ) → φ ∗ E × T Y φ ∗ E .Since fiber products commute with products there is a unique isomorphism T Y × φ ∗ T X ( φ ∗ E × φ ∗ E ) → ( T Y × φ ∗ T X φ ∗ E ) × T Y ( T Y × φ ∗ T X φ ∗ E ).The composition T Y × φ ∗ T X φ ∗ ( E × T X E ) → T Y × φ ∗ T X ( φ ∗ E × φ ∗ T X φ ∗ E ) → ( T Y × φ ∗ T X φ ∗ E ) × T Y ( T Y × φ ∗ T X φ ∗ E )gives rise to the map(4.3.2) φ + ( E × T X E ) → φ + E × T Y φ + E Lemma 9.
Suppose that ( E i , π i ) ∈ O X − Mod / T X , i = 1 , , are locally free. Assume thatthe maps E π −→ T X π ←− E satisfy condition (C) . Then, the canonical map (4.3.2) is anisomorphism.Proof. Lemma 8 implies that the canonical morphism φ ∗ ( E × T X E ) → φ ∗ E × T Y φ ∗ E is anisomorphism. Then the map (4.3.2) is an isomorphism. Moreover, the diagrams φ + ( E × T X E ) / / (cid:15) (cid:15) φ ∗ ( E × T X E ) (cid:15) (cid:15) φ + E × T Y φ + E / / φ ∗ E × φ ∗ T X φ ∗ E φ + ( E × T X E ) / / & & ▲▲▲▲▲▲▲▲▲▲▲ φ ∗ E × φ ∗ T X φ ∗ E x x qqqqqqqqqqqq T Y φ ∗ ( E × T X E ) / / & & ▼▼▼▼▼▼▼▼▼▼▼ φ ∗ E × T Y φ ∗ E x x ♣♣♣♣♣♣♣♣♣♣ φ ∗ T X are commutative and the result follows. (cid:3) Inverse image and composition.
Suppose given a map ψ : Z → Y . Applying ψ ∗ to(4.3.1) we obtain the commutative diagram(4.4.1) ψ ∗ φ + E ψ ∗ ( f dφ ) −−−−→ ( φ ◦ ψ ) ∗ E ψ ∗ ψ + ( π ) y y ( φ ◦ ψ ) ∗ ( π ) ψ ∗ T Y ψ ∗ ( dφ ) −−−−→ ( φ ◦ ψ ) ∗ T X which is not Cartesian in general. Combining the definition of ψ + with (4.4.1) we obtain thecommutative diagram(4.4.2) ψ + φ + E f dψ −−−→ ψ ∗ φ + E ψ ∗ ( f dφ ) −−−−→ ( φ ◦ ψ ) ∗ E ψ + φ + ( π ) y ψ ∗ φ + ( π ) y y ψ ∗ φ ∗ ( π ) T Z dψ −−−→ ψ ∗ T Y ψ ∗ ( dφ ) −−−−→ ( φ ◦ ψ ) ∗ T X On the other hand, the inverse image functor under the map φ ◦ ψ is defined by thepullback diagram ( φ ◦ ψ ) + E ^ d ( φ ◦ ψ ) −−−−→ ( φ ◦ ψ ) ∗ E ( φ ◦ ψ ) + ( π ) y y ( φ ◦ ψ ) ∗ ( π ) T Z d ( φ ◦ ψ ) −−−−→ ( φ ◦ ψ ) ∗ T X Since ψ ∗ ( dφ ) ◦ dψ = d ( φ ◦ ψ ) the universal property of pull-back provides the canonical map(4.4.3) c + φ,ψ : ψ + φ + E → ( φ ◦ ψ ) + E which satisfies ^ d ( φ ◦ ψ ) ◦ c + φ,ψ = ψ ∗ ( f dφ ) ◦ f dψ ( φ ◦ ψ ) ∗ ( π ) ◦ c + φ,ψ = ψ + φ + ( π ) . In general the canonical map c + φ,ψ is not an isomorphism. Lemma 10.
Suppose that ( E , π ) ∈ O X − Mod / T X is locally free and the maps T Y dφ −→ φ ∗ T X φ ∗ ( π ) ←−−− φ ∗ E satisfy condition (C) . Then, the map (4.4.3) is an isomorphism.Proof. The assumptions and Lemma 8 imply that the diagram (4.4.1) is Cartesian. Thus,both small squares in (4.4.2) are Cartesian hence so is their composition. (cid:3)
DD TRANSGRESSION FOR COURANT ALGEBROIDS 13
Suppose given yet another map ξ : W → Z . In this case both compositions W ψ ◦ ξ −−→ Y φ −→ X , W ξ −→ Z φ ◦ ψ −−→ X provide respectively the canonical maps(4.4.4) c + φ,ψ ◦ ξ : ( ψ ◦ φ ) + φ + E → ( φ ◦ ψ ◦ ξ ) + E (4.4.5) c + φ ◦ ψ,ξ : ξ + ( φ ◦ ψ ) + E → ( φ ◦ ψ ◦ ξ ) + E These two maps satisfy similar compatibility conditions as the map (4.4.3), namely ^ d ( φ ◦ ψ ◦ ξ ) ◦ c + φ,ψ ◦ ξ = ( ψ ◦ ξ ) ∗ ( f dφ ) ◦ ^ d ( ψ ◦ ξ )( φ ◦ ψ ◦ ξ ) + ( π ) ◦ c + φ,ψ ◦ ξ = ( ψ ◦ ξ ) + φ + ( π )and ^ d ( φ ◦ ψ ◦ ξ ) ◦ c + φ ◦ ψ,ξ = ξ ∗ ( ^ d ( φ ◦ ψ )) ◦ g d ( ξ )( φ ◦ ψ ◦ ξ ) + ( π ) ◦ c + φ ◦ ψ,ξ = ξ + ( φ ◦ ψ ) + ( π )Applying the functor ξ + to the map (4.4.3) we obtain the map(4.4.6) ξ + ( c + φ,ψ ) : ξ + ψ + φ + E → ξ + ( φ ◦ ψ ) + E which satisfies ξ + ψ + φ + ( π ) = ξ + ( φ ◦ ψ ) + ( π ) ◦ ξ + ( c + φ,ψ ) . The inverse image functor ξ + on the object ( ψ + φ + E , ψ + φ + ( π )) ∈ O Z − Mod / T Z is defined bythe pullback square ξ + ψ + φ + E e dξ −−−→ ξ ∗ ψ + φ + E ξ + ψ + φ + ( π ) y y ξ ∗ ψ + φ + ( π ) T W dξ −−−→ ξ ∗ T Z The commutative diagram, ξ + ψ + φ + E ξ ∗ ψ + φ + E ξ ∗ ψ ∗ φ + E ξ ∗ ψ ∗ φ ∗ E ∼ = ( φ ◦ ψ ◦ ξ ) ∗ ET W ξ ∗ T Z ξ ∗ ψ ∗ T Y ξ ∗ ψ ∗ φ ∗ T X ∼ = ( φ ◦ ψ ◦ ξ ) ∗ T X e dξξ + ψ + φ + ( π ) ξ ∗ ψ + φ + ( π ) ξ ∗ ( f dψ ) ξ ∗ ψ ∗ g ( dφ ) ξ ∗ ψ ∗ φ + ( π ) ( φ ◦ ψ ◦ ξ ) ∗ ( π ) dξ ξ ∗ ( dψ ) ξ ∗ ψ ∗ ( dφ ) and the universal property of pullback provide the canonical map ξ + ψ + φ + E → ( φ ◦ ψ ◦ ξ ) + E . Lemma 11.
The identity c + φ ◦ ψ,ξ ◦ ξ + ( c + φ,ψ ) = c + φ,ψ ◦ ξ ◦ c + ψ,ξ : ξ + ψ + φ + E → ( φ ◦ ψ ◦ ξ ) + E holds.Proof. It is enough to check that both compositions make the following diagram ξ + ψ + φ + E ( φ ◦ ψ ◦ ξ ) + E ( φ ◦ ψ ◦ ξ ) ∗ ET W ( φ ◦ ψ ◦ ξ ) ∗ T X ξ ∗ ψ ∗ ( f dφ ) ◦ ξ ∗ ( f dψ ) ◦ e dξξ + ψ + φ + ( π ) ^ d ( φ ◦ ψ ◦ ξ )( φ ◦ ψ ◦ ξ ) + ( π ) ( φ ◦ ψ ◦ ξ ) ∗ ( π ) d ( φ ◦ ψ ◦ ξ ) commutative. The computations ^ d ( φ ◦ ψ ◦ ξ ) ◦ c + φ,ψ ◦ ξ ◦ c + ψ,ξ = ξ ∗ ψ ∗ ( f dφ ) ◦ ^ d ( ψ ◦ ξ ) ◦ c + ψ,ξ = ( ψ ◦ ξ ) ∗ ( f dφ ) ◦ ξ ∗ e dξ and ^ d ( φ ◦ ψ ◦ ξ ) ◦ c + φ ◦ ψ,ξ ◦ ξ + ( c + φ,ψ ) = ξ ∗ ( ^ d ( φ ◦ ψ )) ◦ e dξ ◦ ξ + ( c + φ,ψ )= ξ ∗ ψ ∗ ( f dφ ) ◦ ξ ∗ ( f dψ ) ◦ e dξ show that ^ d ( φ ◦ ψ ◦ ξ ) ◦ c + φ,ψ ◦ ξ ◦ c + ψ,ξ = ^ d ( φ ◦ ψ ◦ ξ ) ◦ c + φ ◦ ψ,ξ ◦ ξ + ( c + φ,ψ ). Similarly, the calculations( φ ◦ ψ ◦ ξ ) + ( π ) ◦ c + φ,ψ ◦ ξ ◦ c + ψ,ξ = ( ψ ◦ ξ ) + ◦ φ + ( π ) ◦ c + ψ,ξ = ξ + ψ + φ + ( π )and ( φ ◦ ψ ◦ ξ ) + ( π ) ◦ c + φ ◦ ψ,ξ ◦ ξ + ( c + φ,ψ = ξ + ( φ ◦ ψ ) + ( π ) ◦ c + ψ,ξ = ξ + ψ + φ + ( π )show that ( φ ◦ ψ ◦ ξ ) + ( π ) ◦ c + φ,ψ ◦ ξ ◦ c + ψ,ξ = ( φ ◦ ψ ◦ ξ ) + ( π ) ◦ c + φ ◦ ψ,ξ ◦ ξ + ( c + φ,ψ ). Therefore c + φ ◦ ψ,ξ ◦ ξ + ( c + φ,ψ ) = c + φ,ψ ◦ ξ ◦ c + ψ,ξ . (cid:3) Localization and descent
Our exposition is inspired by (Beilinson and Bernstein, 1993).
DD TRANSGRESSION FOR COURANT ALGEBROIDS 15
General framework.
We denote by
Man the category of manifolds. For X ∈ Man we denote by X sm the full subcategory of Man /X with objects P π P −→ X , where π P is asubmersion. The category X sm is equipped with the Grothendieck topology (the “smooth”topology) with covers given by surjective submersions.We assume given a category P equipped with a functor p : P →
Man ; for a manifold X we denote by P X the corresponding fiber, i.e. the full subcategory with objects F ∈ P suchthat p ( F ) = X . We assume that the functor p makes P a category prefibered over Man .This means, by definition, that • For a morphism Y f −→ X in Man there is a functor f H : P X → P Y of inverse imagealong f • For a pair of composable morphisms Z g −→ Y f −→ X there is a morphism of functors c H f,g : g H f H → ( f ◦ g ) H which satisfy the associativity constraint c H f ◦ g,h ◦ h H ( c H f,g ) = c H f,g ◦ h ◦ c H g,h : h H g H f H F → ( f ◦ g ◦ h ) H F , for any triple of composable morphisms W h −→ Z g −→ Y f −→ X and any F ∈ P X .In addition, for each manifold X we assume given a full subcategory P ♭X of P X . Weassume that, if Y f −→ X is a submersion and F ∈ P ♭X then f H F ∈ P ♭Y , and for g arbitrarythe morphism c H f,g : g H f H F → ( f ◦ g ) H F is an isomorphism. Example . (1) Let ( O− Mod / T ) ♯ denote the category with objects pairs ( X, E ), where X is a manifoldand E ∈ O X ♯ − Mod / T X ♯ . A morphism u : ( Y, F ) → ( X, E ) is a pair u = ( f, t ), where f : Y → X is a map of manifolds and t : F → f ♯ + E is a morphism in O X ♯ − Mod / T X ♯ .The functor(5.1.1) ( O− Mod / T ) ♯ → Man : ( X, E ) X, ( f, t ) f makes ( O− Mod / T ) ♯ a prefibered category over the category Man of manifolds.Let ( O− Mod lf / T ) ♯X denote the full subcategory of ( O− Mod / T ) ♯X with objects lo-cally free of finite rank over O X ♯ .This is an example of the framework of 5.1 with P = ( O− Mod / T ) ♯ , P ♭X =( O− Mod lf / T ) ♯X and the functor of inverse image defined in 4.3.(2) Lie algebroids, see 6.4(3) Marked Lie algebroids, see 6.5(4) Courant algebroids, see 7.75.2. The category of descent data.
For P π P −→ X in X sm consider the diagram P × X P × X P π ij −−→−−→−−→ P × X P π i −−→−−→ P π P −→ X where the maps π i , i = 1 , π ij , 1 i < j
3) denote projections onto the i th (respectively, i th and j th ) factor.We denote by Desc( P ; P ♭ ) = Desc( P π P −→ X ; P ♭ ) the category with objects pairs ( F , g F ),where • F ∈ P ♭P ; • g F : π H F → π H F is an isomorphism which satisfies the cocycle condition π H ( g F ) ◦ π H ( g F ) = π H ( g F ).A morphism t : ( F , g F ) → ( F ′ , g F ′ ) in Desc( P ; P ♭ ) is a morphism t : F → F ′ in P ♭P whichsatisfies π ′ H ( t ) ◦ g F ′ = g F ◦ π H ( t ).The assignment ( F , g F )
7→ F defines a functor Desc( P ; P ♭ ) → P ♭P .5.3. Localization.
For
E ∈ P ♭X and P π P −→ X in X sm let E P := π H P E ∈ P ♭P . If Q f −→ P is amorphism in X sm , the map c π P ,f : E Q → E P is an isomorphism. Therefore, the assignment X sm ∋ P
7→ E P defines a Cartesian section of P /X sm .The functor π H P : P ♭X → P ♭P lifts to the functor(5.3.1) f π H P : P X → Desc( P ; P ♭ )defined by E 7→ ( E P , g E P ), where g E P is the composition of the canonical isomorphisms π H E P ∼ = E P × X P ∼ = π H E P .A morphism f : Q → P in X sm induces the functor f f H : Desc( P ; P ♭ ) → Desc( Q ; P ♭ )defined by ( F , g F ) ( f H F , ( f × X f ) H g F ).5.4. Descent.
We shall say that P π P −→ X ∈ X sm is classical if π P is a disjoint union of openembeddings.For any smooth cover P π P −→ X there exists a classical cover T π T −→ X such that Hom X sm ( T, P ) = ∅ , i.e. there exists a morphism f : T → P such that π T = f ◦ π P . In other words, everysmooth cover admits a classical refinement.We shall say that P ♭ has the smooth (respectively, classical) descent property if for anysmooth (respectively, classical) cover P π P −→ X the functor (5.3.1) is an equivalence. Theorem 13. If P ♭ has the classical descent property then it has the smooth descent property.Proof. Suppose that f : T → P is a morphism in X sm with T π T −→ X a classical cover. For( F , g F ) ∈ Desc( P ; P ♭ ) there exist E ∈ P ♭X and an isomorphism f π H T E ∼ = f f H ( F , g F ).Consider the diagram(5.4.1) P × X T pr T −−−→ T pr P y y π T P π P −−−→ X DD TRANSGRESSION FOR COURANT ALGEBROIDS 17
Note that pr P : P × X T → P a classical cover of P . Since π P ◦ pr P = π T ◦ pr T , pr ◦ (id × f ) = f ◦ pr T and pr ◦ (id × f ) = pr P there are isomorphisms(5.4.2) g pr H P f π H P E ∼ = g pr H T f π H T E ∼ = g pr H T f f H ( F , g F ) ∼ = ∼ = ^ (id × f ) H g pr H ( F , g F ) ^ (id × f ) H ( g F ) −−−−−−−→ ^ (id × f ) H g pr H ( F , g F ) ∼ = g pr H P ( F , g F )Since pr P : P × X T → P is a classical cover of P the functor g pr H P is an equivalence by as-sumption. Therefore, an isomorphism g pr H P ( F , g F ) ∼ = g pr H P f π H P E is equivalent to an isomorphism( F , g F ) ∼ = f π H P E . Thus, the functor f π H P is essentially surjective.Since f f H ◦ f π H P ∼ = f π H T is an equivalence, it follows that f π H P is faithful.Suppose that ( F , g F ) φ −→ ( F , g F ) is a morphism in Desc( P ; P ♭ ). Then, there exits amorphism E ψ −→ E in P ♭X and isomorphisms f π H T E i ∼ = f f H ( F i , g F i ) which intertwine φ and ψ .Proceeding as in the proof of essential subjectivity one obtains the commutative diagram g pr H P f π H P E ∼ = −−−→ g pr H P ( F , g F ) g pr H P f π H P ( ψ ) y y g pr H P ( φ ) g pr H P f π H P E ∼ = −−−→ g pr H P ( F , g F )Since the functor g pr H P is an equivalence by assumption it follows that φ = f π H P ( ψ ). Thereforethe functor f π H P is full. (cid:3) Corollary 14 (of Theorem 13) . ( O− Mod lf / T ) ♯ has the smooth descent property. Inverse image for Lie algebroids
The anchor map of an O X -Lie algebroid A renders the latter as an object of O X − Mod / T X .This correspondence extends to a functor O X − LieAlgd → O X − Mod / T X .6.1. Inverse image for Lie algebroids.
For an O X -Lie algebroid A and a map of DG-manifolds φ : Y → X the inverse image φ + A has a canonical structure of a Lie algebroid on Y . Namely, the bracket on φ + A is given by[( f ⊗ a, ξ ) , ( g ⊗ b, η )] = (( − ag f g ⊗ [ a, b ] + ξ ( g ) ⊗ b − ( − ηξ η ( f ) ⊗ a, [ ξ, η ]) , where f, g ∈ O Y , a, b ∈ A , ξ, η ∈ T Y are homogeneous elements respectively. Since ( f ⊗ a, ξ ) , ( g ⊗ b, η ) are homogeneous elements, a + f = ξ and g + b = η .For any composition of maps of DG-manifolds Z ψ −→ Y φ −→ X , a general element in ψ + φ + A is a finite sum of homogeneous elements of the form ( h ⊗ ( f ⊗ a, ξ ) , ρ ), where h ∈ O Z , f ∈ O Y , a ∈ A , ξ ∈ T Y and ρ ∈ T Z . In these terms, the canonical map c + φ,ψ : ψ + φ + A → ( φ ◦ ψ ) + A in the category O Z − Mod / T Z is equal to c + φ,ψ ( h ⊗ ( f ⊗ a, ξ ) , ρ ) = ( hψ ∗ ( f ) ⊗ a, ρ ) , where,(6.1.1) ( dψ )( ρ ) = h ⊗ ξ, see Subsection 4.4. Lemma 15.
The map c + φ,ψ is a morphism of O Z -Lie algebroids.Proof. [ c + φ,ψ ( h ⊗ ( f ⊗ a, ξ ) , ρ ) , c + φ,ψ ( l ⊗ ( g ⊗ b, η ) , ν )]= [( hψ ∗ ( f ) ⊗ a, ρ ) , ( lψ ∗ ( g ) ⊗ b, ν )]= ( ρ ( l ) ψ ∗ ( g ) ⊗ b + ( − ρl lρ ( ψ ∗ ( g )) ⊗ b − ( − νρ ν ( h ) ψ ∗ ( f ) ⊗ a − ( − νρ + νh hν ( ψ ∗ ( f )) ⊗ a + ( − lf + a ( f + g ) hlψ ∗ ( f g ) ⊗ [ a, b ] , [ ρ, ν ]) . On the other hand, c + φ,ψ [( h ⊗ ( f ⊗ a, ξ ) , ρ ) , ( l ⊗ ( g ⊗ b, η ) , ν )]= c + φ,ψ ( ρ ( l ) ⊗ ( g ⊗ b, η ) − ( − νρ ν ( h ) ⊗ ( f ⊗ a, ξ ) + ( − lξ hl ⊗ [( f ⊗ a, ξ ) , ( g ⊗ b, η )] , [ ρ, ν ])= c + φ,ψ ( ρ ( l ) ⊗ ( g ⊗ b, η ) − ( − νρ ν ( h ) ⊗ ( f ⊗ a, ξ )+ ( − lξ hl ⊗ (( − ag f g ⊗ [ a, b ] + ξ ( g ) ⊗ b − ( − ηξ η ( f ) ⊗ a, [ ξ, η ]) , [ ρ, ν ])= ( ρ ( l ) ψ ∗ ( g ) ⊗ b − ( − νρ ν ( h ) ψ ∗ ( f ) ⊗ a + ( − lξ hl ( ψ ∗ ( ξ ( g )) ⊗ b − ( − lξ + ηξ ψ ∗ ( ν ( f )) ⊗ a + ( − lξ + ag ψ ∗ ( f g ) ⊗ [ a, b ]) , [ ρ, ν ]) . Comparing term by term the calculations of [ c + φ,ψ ( h ⊗ ( f ⊗ a, ξ ) , ρ ) , c + φ,ψ ( l ⊗ ( g ⊗ b, η ) , ν )] and c + φ,ψ [( h ⊗ ( f ⊗ a, ξ ) , ρ ) , ( l ⊗ ( g ⊗ b, η ) , ν )], it remains to show that hl ( ψ ∗ ( ξ ( g ))) ⊗ b = lρ ( ψ ∗ ( g )) ⊗ b,hl ( ψ ∗ ( η ( f ))) ⊗ a = hν ( ψ ∗ ( f )) ⊗ a. Applying the formula (6.1.1) in the first case one gets,( − lξ hl ( ψ ∗ ( ξ ( g ))) ⊗ b = ( − lh + lξ l ( hψ ∗ ( ξ ( g ))) ⊗ b = ( − lρ l ( h ⊗ ξ ( g ) ⊗ b ) = lρ ( ψ ∗ ( g )) ⊗ b. The second case is analogous. (cid:3)
DD TRANSGRESSION FOR COURANT ALGEBROIDS 19
Inverse image for marked Lie algebroids.
Suppose that Y φ −→ X is a map of DG-manifolds. Let ( A , c ) ∈ O X − LieAlgd ⋆n be a marked Lie algebroid on X , i.e. there is a mapof marked Lie algebroids O X [ n ] · c −→ A . Since φ + O X [ n ] = ker( dφ ) ⊕ O Y [ n ], there is a canonical map(6.2.1) O Y [ n ] → ker( dφ ) ⊕ O Y [ n ] = φ + O X [ n ]Let φ + ( c ) ∈ Γ( Y ; φ + A ) − n denote the image of 1 ∈ Γ( Y ; O Y ) under the composition O Y [ n ] (6.2.1) −−−→ φ + O X [ n ] φ + ( · c ) −−−→ φ + A . The pair ( φ + A , φ + ( c )) is a marked Lie algebroid denoted φ + ( A , c ). The map O Y [ n ] · φ + ( c ) −−−→ φ + A . is given by g ( g ⊗ c ,
0) for g ∈ O Y [ n ].The assignment ( A , c ) φ + ( A , c ) extends to a functor φ + : O X − LieAlgd ⋆n → O Y − LieAlgd ⋆n . For any composition of maps of DG-manifolds, Z ψ −→ Y φ −→ X , the morphism c + φ,ψ makesthe diagram O Z [ n ] · ψ + φ + ( c ) ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘❘ · ( φ ◦ ψ ) + ( c ) / / ( φ ◦ ψ ) + A ψ + φ + A c + φ,ψ O O commutative. Lemma 16.
The map c + φ,ψ is a morphism of marked Lie algebroids.Proof. Lemma 15 implies that c + φ,ψ is a morphism of O Y -Lie algebroids. For g ∈ O Z g · ( φ ◦ ψ ) + ( c ) = ( g ⊗ c , , by construction. On the other hand, the calculation c + φ,ψ ( g · ψ + φ + ( c )) = c + φ,ψ ( g ⊗ (1 ⊗ c , ,
0) = ( g ⊗ c , g · ( φ ◦ ψ ) + ( c ) = c + φ,ψ ( g · ψ + ψ + ( c )), i.e. c + φ,ψ preserves the marking. (cid:3) Inverse image for O [ n ] -extensions. Suppose that Y φ −→ X is a map of DG-manifoldsand B is a O X -Lie algebroid. Lemma 17.
Suppose that ( A , c ) is a O [ n ] -extension of B . Then, φ + ( A , c ) is a O [ n ] -extensionof φ + B .Proof. Since both small squares in the commutative diagram φ + B × φ ∗ B φ ∗ A −−−→ φ + B −−−→ T Y y y y φ ∗ A −−−→ φ ∗ B −−−→ φ ∗ T X are Cartesian, so is the large one. Therefore there is a canonical isomorphism φ + A ∼ = φ + B × φ ∗ B φ ∗ A . Hence, φ + ( A , c ) is a O [ n ]-extension of φ + B . (cid:3) Thus, the inverse image functor for marked Lie algebroids induces a functor(6.3.1) φ + : O [ n ]Ext( B ) → O [ n ]Ext( φ + B ) Proposition 18.
The functor (6.3.1) is a morphism of C -vector spaces in categories (and,in particular, of Picard groupoids).Proof. We prove the statement in the case λ A ∔ λ A . The general case is analogous andis left to the readerRecall that for any object A ∈ O [ n ]Ext( B ) there is the canonical isomorphism φ + A = φ + B × φ ∗ B φ ∗ A . Thus, for a pair of objects A , A ∈ O [ n ]Ext( B ) there is a sequence ofisomorphisms φ + A × φ + B φ + A ∼ = ( φ + B × φ ∗ B φ ∗ A ) × φ + B ( φ + B × φ ∗ B φ ∗ A ) ∼ = T Y × φ ∗ T X ( φ ∗ A × φ ∗ B φ ∗ A ) ∼ = T Y × φ ∗ T X φ ∗ ( A × B A ) ∼ = φ + ( A × B A ) . The inverse image functor φ + applied to the commutative diagram O X [ n ] × O X [ n ] −−−→ A × B A α ,α ) λ α + λ α y y O X [ n ] −−−→ λ A ∔ λ A and canonical isomorphisms φ + ( O X [ n ] ×O X [ n ]) ∼ = φ + O X [ n ] × φ + O X [ n ] and φ + A × φ + B φ + A ∼ = φ + ( A × B A ) give rise to the commutative diagram O Y [ n ] × O Y [ n ] ((6.2.1) , (6.2.1)) −−−−−−−−→ φ + O X [ n ] × φ + O X [ n ] −−−→ φ + A × φ + B φ + A α ,α ) λ α + λ α y y φ + (( α ,α ) λ α + λ α ) y O Y [ n ] (6.2.1) −−−→ φ + O X [ n ] −−−→ φ + ( λ A ∔ λ A ) . DD TRANSGRESSION FOR COURANT ALGEBROIDS 21
Therefore, there is a unique morphism in O [ n ]Ext( φ + B )(6.3.2) λ φ + A ∔ λ φ + A → φ + ( λ A ∔ λ A ) . (cid:3) Smooth localization for Lie algebroids.
Let ( O− LieAlgd ) ♯ denote the categorywith objects pairs ( X, A ), where X is a manifold and A ∈ O X ♯ − LieAlgd . A morphism u : ( Y, B ) → ( X, A ) is a pair u = ( f, t ), where f : Y → X is a map of manifolds and t : B → f ♯ + ( A is a morphism in O Y ♯ − LieAlgd . It follows from Lemma 15 that the forgetfulfunctor ( X, A ) X makes ( O− LieAlgd ) ♯ a category prefibered over Man .Let ( O− LieAlgd lf ) ♯X denote the full subcategory of ( O− LieAlgd ) ♯X with objects locallyfree of finite rank over O X ♯ .This is an example of the framework of 5.1 with P = ( O− LieAlgd ) ♯ , P ♭X = ( O− LieAlgd lf ) ♯X and the functor of inverse image defined in 6.1. Corollary 19 (of Theorem 13) . ( O− LieAlgd lf ) ♯ has the smooth descent property. Smooth localization for marked Lie algebroids.
Let ( O− LieAlgd ⋆n ) ♯ denote thecategory with objects pairs ( X, ( A , c )), where X is a manifold and ( A , c ) ∈ O X ♯ − LieAlgd ⋆n .A morphism u : ( Y, ( B , b )) → ( X, ( A , c )) is a pair u = ( f, t ), where f : Y → X is a mapof manifolds and t : ( B , b ) → f ♯ + A , c ) is a morphism in O Y ♯ − LieAlgd ⋆n . It follows fromLemma 15 and Lemma 16 that the forgetful functor ( X, ( A , c )) X makes ( O− LieAlgd ⋆n ) ♯ a category prefibered over Man .Let ( O− LieAlgd ⋆ lf n ) ♯X denote the full subcategory of ( O− LieAlgd ⋆n ) ♯X with objects locallyfree of finite rank over O X ♯ .This is an example of the framework of 5.1 with P = ( O− LieAlgd ⋆n ) ♯ , P ♭X = ( O− LieAlgd ⋆ lf n ) ♯X and the functor of inverse image defined in, 6.2. Corollary 20 (of Theorem 13) . ( O− LieAlgd ⋆n ) ♯ has the smooth descent property. Inverse image for Courant algebroids
The inverse image functor.
Suppose that f : Y → X is a map of manifolds. For Q ∈ CA ( X ) the functor of inverse image under f for Courant algebroids denoted f ++ : CA ( X ) → CA ( Y )is defined by f ++ Q := Q ( f + τ Q ) . in the notation of 3.8.7.2. Explicit description.
Suppose that Q is a Courant algebroid on X and f : Y → X is a map of manifolds. It transpires from the definition, that f ++ Q is a sub-quotient ofΩ Y ⊕ f ∗ Q ⊕ T Y : Ω Y ⊕ f ∗ Q ⊕ T Y ← ֓ Ω Y ⊕ f ∗ Q × f ∗ T X T Y → f ++ Q . The symmetric pairing.
Let h , i ′ denote the symmetric pairing on f ∗ Q ⊕ T Y suchthat(1) it restricts to the pairing induced by h , i on f ∗ Q ,(2) hT Y , T Y i = 0,(3) for α ∈ f ∗ Ω X , ξ ∈ T Y , h i ( α ) , ξ i ′ = ι ξ df ∨ ( α ).Let h , i ′′ denote the unique symmetric pairing on Ω Y ⊕ f ∗ Q × f ∗ T X T Y such that(1) it restricts to the pairing h , i ′ on f ∗ Q × f ∗ T X T Y ,(2) h Ω Y , Ω Y i = 0,(3) for α ∈ Ω Y , q ∈ f ∗ Q × f ∗ T X T Y , h α, q i ′′ = ι σ ( q ) α .Since h ( df ∨ ( α ) , − i ( α ) , , ( β, q, ξ ) i ′′ = 0for all α ∈ f ∗ Ω X , β ∈ Ω Y , ( q, ξ ) ∈ f ∗ Q × f ∗ T X T Y , it follows that the pairing h , i ′′ descendsto a symmetric pairing on f ++ Q which we will denote by h , i .7.2.2. The bracket.
Let [ , ] ′ denote the unique operation on Ω Y ⊕ f ∗ Q ⊕ T Y defined by theformula[( α, h ⊗ q, ξ ) , ( β, j ⊗ p, η )] ′ = ( − ι η dα + L ξ β + jdh h q, p i , hj ⊗ [ q, p ] − ι η dh ⊗ q + L ξ j ⊗ p, [ ξ, η ]) , where α, β ∈ Ω Y , h, j ∈ O Y , q, p ∈ Q and ξ, η ∈ T Y .For any γ ∈ f ∗ Ω X , the calculation[( df ∨ ( γ ) , − i ( γ ) , , ( β, j ⊗ p, η )] ′ = ( − ι η d ( df ∨ ( γ )) , − j ⊗ [ γ, p ] , − ι η d ( df ∨ ( γ )) , j ⊗ L π ( p ) γ − j ⊗ π † ( d h p, γ i ) , − ι η d ( df ∨ ( γ )) , j ⊗ L π ( p ) γ − j ⊗ dι π ( p ) ( γ ) , − ι η ( df ∨ ( dγ )) , j ⊗ ι π ( p ) ( dγ ) , , shows that the bracket [ , ] ′ descends to an operation on f ++ ( Q ) which we will denoted by[ , ]. Remark . The construction of inverse image for Courant algebroids specializes to that(Li-Bland and Meinrenken, 2009) under suitable transversality assumptions.7.3.
Twisting by a 3-form.
Suppose that Q is a Courant algebroid on X and H is a closed3-form on X . Recall that the H -twist of Q , denoted Q H , is the Courant algebroid whosesymmetric pairing and the anchor map coincide with the ones given on Q and the bracketis given by the [ q, p ] H = [ q, p ] + π † ( ι π ( p ) ι π ( q ) H ) , for any q, p ∈ Q .Suppose that f : Y → X is a map of manifolds. Lemma 22. ( f ++ Q ) f ∗ H = f ++ Q H . DD TRANSGRESSION FOR COURANT ALGEBROIDS 23
Proof.
The underlying O Y -modules of both objects coincide. The bracket on f ++ Q H is givenby the formula,[( α, h ⊗ q, ξ ) , ( β, j ⊗ p, η )]= ( − ι η dα + L ξ β + jdh h q, p i , hj ⊗ [ q, p ] H − ι η dh ⊗ q + L ξ j ⊗ p, [ ξ, η ])= ( − ι η dα + L ξ β + jdh h q, p i , hj ⊗ [ q, p ] − ι η dh ⊗ q + L ξ j ⊗ p, [ ξ, η ]) + (0 , hj ⊗ π † ( ι π ( p ) ι π ( q ) H ) , α, h ⊗ q, ξ ) , ( β, j ⊗ p, η )] + (0 , hj ⊗ π † ( ι π ( p ) ι π ( q ) H ) , . On the other hand, the bracket on ( f ++ Q ) f ∗ H is given by the formula[( α, h ⊗ q, ξ ) , ( β, j ⊗ p, η )] f ∗ H = [( α, h ⊗ q, ξ ) , ( β, j ⊗ p, η )] + ( ι ξ ι η f ∗ H, , α, h ⊗ q, ξ ) , ( β, j ⊗ p, η )] + (0 , π † ( H ( df ( ξ ) , df ( η ) , )) , α, h ⊗ q, ξ ) , ( β, j ⊗ p, η )] + (0 , hj ⊗ π † ( ι π ( p ) ι π ( q ) H ) , . (cid:3) Connections.
Recall that a connection ∇ on a Courant algebroid Q on X is splitting ∇ : T X → Q of the anchor map π : Q → T X isotropic with respect to the symmetric pairingon Q . We denote by C ( Q ) the sheaf of locally defined connections on Q .Suppose that Q is a Courant algebroid on X , and f : Y → X is a map of manifolds. Aconnection ∇ on Q induces a splitting f ∗ ( ∇ ) : f ∗ T X → f ∗ Q of f ∗ ( π ) and the splitting(7.4.1) T Y × f ∗ T X f ∗ ( ∇ ) : T Y → T Y × f ∗ T X f ∗ Q of T Y × f ∗ T X f ∗ ( π ).Let f ++ ( ∇ ) : T Y → f ++ Q denote the composition T Y (7.4.1) −−−→ T Y × f ∗ T X f ∗ Q → f ++ Q . Lemma 23.
The map f ++ ( ∇ ) is a connection on f ++ Q .Proof. For ξ ∈ T Y , f ++ ( ∇ )( ξ ) = (0 , f ∗ ( ∇ )( df ( ξ )) , ξ ). Then h f ++ ( ∇ )( ξ ) , f ++ ( ∇ )( η ) i = h (0 , f ∗ ( ∇ )( df ( ξ )) , ξ ) , (0 , f ∗ ( ∇ )( df ( η )) , η ) i = 0 . (cid:3) Therefore, the map f induces the morphism of sheaves(7.4.2) f ++ : f − C ( Q ) → C ( f ++ Q ) . Inverse image and linear algebra.
Suppose that f : Y → X is a map of manifoldsand A ∈ O X − LieAlgd . Lemma 24.
Suppose that Q is a Courant extension of A . Then, f ++ Q is a Courantextension of f + A .Proof. Follows from Proposition 6 and Lemma 17. (cid:3)
Thus, the inverse image functor for Courant algebroids induces a functor(7.5.1) f ++ : C Ext( A ) → C Ext( f + A ) Proposition 25.
The functor (7.5.1) is a morphism of C -vector spaces in categories (and,in particular, of Picard groupoids).Proof. Follows from Proposition 7 and Proposition 18. (cid:3)
Exact Courant algebroids.
Suppose that Q is an exact Courant algebroid on X .Recall ((Bressler, 2007), 3.7) that the sheaf C ( Q ) of locally defined connections on Q togetherwith the curvature map c : C ( Q ) → Ω ,clX is a (Ω X → Ω ,clX )-torsor and the assignment Q 7→ ( C ( Q ) , c ) defines an equivalence E CA ( X ) → (Ω X → Ω ,clX ) − torsors of C -vector spaces incategories.Suppose that f : Y → X is a map of manifolds. The map (7.4.2) is a morphism of torsorsrelative to the map f ∗ : f − Ω X → Ω Y , hence induces the morphism of Ω Y -torsors(7.6.1) f ++ : f ∗ C ( Q ) := Ω Y × f − Ω X f − C ( Q ) → C ( f ++ Q ) . Lemma 26.
The diagram f − C ( Q ) f ++ −−−→ C ( f ++ Q ) f ∗ ( c ) y y c f − Ω ,clX f ∗ −−−→ Ω ,clY is commutative.Proof. By definition, for ∇ ∈ C ( Q ), the curvature c ( ∇ ) ∈ Ω ,clX is defined by c ( ∇ )( ξ, η ) = [ ∇ ( ξ ) , ∇ ( η )] − ∇ ([ ξ, η ]) , where ξ, η ∈ T X .For ξ, η ∈ T Y , the computation c ( f ++ ( ∇ ))( ξ, η ) = [ f ++ ( ∇ ( ξ )) , f ++ ( ∇ ( η ))] − f ++ ( ∇ ([ ξ, η ]))= (0 , [ f ∗ ( ∇ )( df ( ξ )) , f ∗ ( ∇ )( df ( η ))] , [ ξ, η ]) − (0 , f ∗ ( ∇ )( df ([ ξ, η ])) , [ ξ, η ])= (0 , [ f ∗ ( ∇ )( df ( ξ )) , f ∗ ( ∇ )( df ( η ))] − f ∗ ( ∇ )( df ([ ξ, η ])) , df ) ∨ ([ f ∗ ( ∇ )( df ( ξ )) , f ∗ ( ∇ )( df ( η ))] − f ∗ ( ∇ )( df ([ ξ, η ]))) , , f ∗ ( c ( ∇ ))( ξ, η )implies the claim. (cid:3) Proposition 27.
The diagram
E CA ( X ) ( C ,c ) −−−→ (Ω X → Ω ,clX ) − torsors f ++ y y f ∗ E CA ( Y ) ( C ,c ) −−−→ (Ω Y → Ω ,clY ) − torsors DD TRANSGRESSION FOR COURANT ALGEBROIDS 25 is commutative.Proof.
The map (7.6.1) provides the isomorphism f ∗ ◦ C ∼ = C ◦ f ++ . Lemma 26 says that itis a morphism of (Ω Y → Ω ,clY )-torsors. (cid:3) Smooth descent for Courant algebroids.
Let CA denote the category with objectspairs ( X, Q ), where X is a manifold and Q ∈ CA ( X ). A morphism u : ( Y, Q ′ ) → ( X, Q ) is apair u = ( f, t ), where f : Y → X is a map of manifolds and t : Q ′ → f ++ Q is a morphism in CA ( Y ). Lemma 15 and the definition of the inverse image functor imply that the forgetfulfunctor ( X, Q ) X makes CA a prefibered category over Man .Let CA lf ( X ) denote the full subcategory of CA ( X ) with objects locally free of finite rankover O X . This is an example of the framework of 5.1 with P = CA , P ♭X = CA lf ( X ) and thefunctor of inverse image defined in 7.1. Corollary 28 (of Theorem 13) . CA lf has the smooth descent property. Dirac structures with support.
Suppose that Z is a submanifold of X and let i : Z → X denote the embedding. Let Q be a Courant algebroid on X locally free offinite rank over O X . For an O X -module E and a submodule F ⊂ i ∗ E we denote by e F thesub-module of E defined by the Cartesian square e F −−−→ E y y i ∗ F −−−→ i ∗ i ∗ E Definition 29 ((Alekseev and Xu, 2002; Bursztyn et al., 2009)) . A Dirac structure in Q supported on Z is a sub-bundle K ⊂ i ∗ Q which satisfies(1) K is maximal isotropic with respect to the restriction of the symmetric pairing;(2) K is mapped to T Z under (the restriction of) the anchor map;(3) the sheaf e K is closed under the bracket on Q .We denote the collection of Dirac structures in Q supported on Z by Dir Z ( Q ) and setDir( Q ) := Dir X ( Q ). Remark . The second condition in Definition 29 is equivalent to
K ⊂ T Z × i ∗ T X i ∗ Q ⊂ i ∗ Q .Let N ∨ Z | X denote the conormal bundle defined by the exact sequence0 → N ∨ Z | X → i ∗ Ω X di ∨ −−→ Ω Z → , i.e. N ∨ Z | X = ann( T Z ).Suppose that K is a Dirac structure supported on Z . In view of Remark 30 K + π † ( N ∨ Z | X )is isotropic. Therefore, by maximality of K , K = K + π † ( N ∨ Z | X ), and π † ( N ∨ Z | X ) ⊂ K .Recall that, by definition, i ++ Q = T Z × i ∗ T X i ∗ Q /π † ( N ∨ Z | X ). Let i ++ K := K /π † ( N ∨ Z | X ) . Proposition 31. (1) i ++ K is a Dirac structure in i ++ Q . (2) The assignment
K 7→ i ++ K defines a bijection between the set of almost Dirac struc-tures in Q supported on Z and the set of almost Dirac structures in i ++ Q .Proof. Given such a D / ⊂ i ++ Q , its pre-image in i + Q , i.e. D / + π † ( N ∨ Z | X ), is a Dirac structuresupported on Z . (cid:3) For φ ∈ Hom CA ( X ) ( Q , Q ) the graph Γ φ is a subsheaf of Q × Q . Since, by definition, φ induces the identity map on T X , it follows that Γ φ ⊂ Q × T X Q . Since, by definition, φ restricts to the identity map on Ω X , it follows that Γ φ ∩ (Ω X × Ω X ) is the diagonal. Therefore,the restriction of the map Q × T X Q → Q ∔ Q to Γ φ is a monomorphism and (the image of) Γ φ is a Dirac structure in Q ∔ Q . Theassignment φ Γ φ defines a canonical mapHom CA ( X ) ( Q , Q ) → Dir( Q ∔ Q op ) . Courant algebroid morphisms ((Alekseev and Xu, 2002; Bursztyn et al., 2009)).
Suppose that f : Y → X is a map of manifolds, Q X ∈ CA ( X ), Q Y ∈ CA ( Y ). Let pr X : Y × X → X and pr Y : Y × X → Y denote the projections; let γ f : Y → Y × X denote the graph embedding y ( y, f ( y )).The sheaf pr ∗ Y Q Y ⊕ pr ∗ X Q opX is endowed with the canonical structure of a Courant algebroidon Y × X canonically isomorphic to pr ++ Y Q Y ∔ pr ++ X Q opX .According to Proposition 31 a Courant algebroid morphism ((Alekseev and Xu, 2002;Bursztyn et al., 2009))
K ∈
Dir γ f ( Y ) ( pr ++ Y Q Y ∔ pr ++ X Q opX ) corresponds to the Dirac structure γ f ( Y ) ++ K ∈
Dir( γ ++ f ( pr ++ Y Q Y ∔ pr ++ X Q opX )) ∼ = Dir( Q Y ∔ f ++ Q opX ) and there is a canonicalmap Hom CA ( Y ) ( Q Y , f ++ Q X ) → Dir γ f ( Y ) ( pr ++ Y Q Y ∔ pr ++ X Q opX ) . References
Alekseev A., and Xu P. “Derived brackets and Courant algebroids.”
Unpublished (2002).Bressler, P., and Chervov A. “Courant algebroids”
Journal of Mathematical Science , Vol128, no. 4, (2005): 3030 – 3053. https://arxiv.org/abs/hep-th/0212195.Bressler P. “The first Pontryagin class.”
Compositio Math.
Letters in Math.Phys.
Advances in Soviet mathe-matics
16, no. 1 (1993): 1–50.Bursztyn H., Iglesias Ponte D., and ˇSevera P. “Courant morphisms and moment maps.”
Math. Research Letters
16, no. 2 (2009): 215–232.Courant T., “Dirac manifolds”. Trans. Amer. Math. Soc., (2), 631–661 (1990).
DD TRANSGRESSION FOR COURANT ALGEBROIDS 27
Dorfman I., “Dirac structures and integrability of nonlinear evolution equations”. NonlinearScience: Theory and Applications, John Wiley and Sons Ltd., Chichester (1993).Liu Z.-J., Weinstein A., and Xu P., “Manin triples for Lie bialgebroids”. Journal of Differ-ential Geometry, , 547–574 (1997).Li-Bland D., and Meinrenken E. “Courant algebroids and Poisson geometry.” Int. Math.Research Notices
11, (2009): 2106–2145.Roytenberg D., “On the structure of graded symplectic supermanifolds and Courant alge-broids”. Quantization Poisson bracket and Beyond, Manchester (2001), Th Voronov (ed.),Contemp. Math., Vol. , Amer. Math. Soc., Providence, RI (2002).ˇSevera P., “Some title containing the words ”homotopy” and ”symplectic”, e.g. this one”.Travaux mathematiques. Fasc.
XVI , Univ. Luxemb. Luxembourg (2005).ˇSevera P., and Valach F. “Courant algebroids, Poisson-Lie T-duality, and type II supergrav-ities.” arXiv preprint arXiv:1810.07763, (2018).Vysoky J. “Hitchhiker’s Guide to Courant Algebroid Relations.” arXiv preprint arXiv:1910.05347, (2019).
Universidad de los Andes, Bogot´a, Colombia
E-mail address : [email protected] Universidad de La Sabana, Ch´ıa, Colombia
E-mail address ::