Integration of Exact Courant Algebroids

In this paper, we describe an integration of exact Courant algebroids to symplectic 2-groupoids, and we show that the differentiation procedure from [26] inverts our integration.


Introduction
A Courant algebroid is a Lie 2-algebroid paired with a compatible symplectic structure [26][27][28][29][30][31]. Therefore, in the study of Courant algebroids two immediate questions are Question 1: What is the global object "integrating" a Courant algebroid? Question 2: How do you "differentiate" the global object to recover the Courant algebroid? Question 1 was first addressed in [31], where, given a Courant algebroid, a construction of a 2-groupoid carrying a symplectic form on the 2-simplices is sketched (see also [35]). Unfortunately, this construction was infinite dimensional. A procedure to differentiate Lie n-groupoids to Lie n-algebroids is described in [32], giving an answer to Question 2. However, this differentiation procedure does not invert the integration procedure described in [31] "on the nose". 1 Although the answer to the Question 1 might be somewhat complicated, it has a simple solution for some of the most popular Courant algebroids, namely exact Courant algebroids [33]. For instance, an elegant solution for exact Courant algebroids with trivial characteristic class is presented in [24] in terms of the "bar" construction. In this paper, we present a solution for arbitrary exact Courant algebroids. Our construction results in a (local) Lie 2-groupoid carrying a symplectic form on the space of 2-simplices. Furthermore, we show that if one differentiates our construction (as in [32]), one recovers the original Courant algebroid.
The idea of our construction is as follows. As a differential graded manifold, the standard Courant algebroid over a manifold M is T [1]T * [1]M . Since it is of the form T [1]X for some graded manifold X (X = T * [1]M ), the problem of its integration to a local 2-groupoid K • has a simple solution (see Remark 5). Next we construct a symplectic form on K 2 making K • into a (local) 2-symplectic 2-groupoid which differentiates to the standard Courant algebroid.
We then prove that an arbitrary exact Courant algebroid over M is isomorphic, as a differential graded manifold, to the standard one. We can therefore use the same 2-groupoid K • as its integration. The isomorphism is not, however, a symplectomorphism. We compute the modification to the symplectic form on T [1]T * [1]M and modify the symplectic form on K 2 correspondingly.
In [34], a more conceptual explanation for our construction is given, related to the work of Mehta, Gracia-Saz, Arias Abad, Crainic and Schaetz on actions up to homotopy [1,2,15,16] (see also [35]). We also thank the referee for their valuable recommendations which have made this paper considerably easier to read.

Background
Remark 1 (A note on Lie groupoids and Lie algebroids). We use the definitions of (local) Lie n-groupoids given by Andre Henriques and Chenchang Zhu [17,37], in terms of simplicial manifolds. We take the definition of Lie n-algebroids given in [32] in terms of N Q-manifolds.
In this section, we recall the differentiation procedure described in [32] which takes Lie n-groupoids to Lie n-algebroids.

Simplicial Manifolds.
For n ∈ N, let [n] be the category generated by the directed graph 0 → 1 → · · · → n, and ∆ the full subcategory of Cat (the category of small categories) generated by the objects [0], [1], [2], . . . . We have the distinguished functors [14] s j (0 → 1 → · · · → n + 1) (that is, we insert the identity in the j th place), and (that is, we compose i − 1 → i → i + 1). We denote the corresponding maps in ∆ op by s j and d i .
Let Man be the category of smooth manifolds. Then Man ∆ op is the category of simplicial manifolds. More generally, if C is any category, then C ∆ op is the category of simplicial objects (in C).
As a quick word on notation, if X ∈ C ∆ op , then it is conventional to write X n := X([n]).
Example 1. The standard n-simplex, ∆ n , is the contravariant functor (where sets are viewed as discrete manifolds).
We may think of Λ n k as the boundary of the standard n-simplex ∆ n with the k th face removed.
The following definition is due to Henriques [17].
In particular, a Lie 1-groupoid is the nerve of a Lie groupoid. For instance, the nerve of the pair groupoid (defined explicitly in the following example) is a Lie 1-groupoid. We will be interested in the local version of n-groupoids, as introduced by Zhu [37]. However, as suggested by a referee, we will reformulate it in terms of the microfolds introduced by Cattaneo, Dherin, and Weinstein [9] and Blohmann, Fernandes, and Weinstein [4] following Milnor [25].

Definition 2.
A microfold is an equivalence class of pairs (M, S) of manifolds such that S ⊆ M is a closed submanifold. Two such pairs (M 1 , S 1 ) and (M 2 , S 2 ) are said to be equivalent if S 1 = S 2 = S and there exists a third pair (U, S) such that U is simultaneously an open subset of both M 1 and M 2 . We denote the equivalence class by [M, S], and refer to S as the microfold core of [M, S].
A morphism between microfolds is a germ of maps between representatives which takes the source microfold core to the target microfold core. Such a morphism is said to be a surjective submersion (resp. a diffeomorphism) if it is a surjective submersion (resp. a diffeomorphism) for a suitable choice of representatives.
We denote the category of microfolds by Mfold. There is a forgetful functor F core : Mfold → Man which takes a microfold [M, S] to its microfold core S. Working with microfolds in place of manifolds, one obtains direct analogues of Henriques' Kan conditions [17] for local simplicial manifolds. Thus one obtains the following definition of a local Lie n-groupoid, essentially a reformulation of the one by Zhu [37]. . It restricts to a functor from Lie n-groupoids to local Lie n-groupoids.

N Q-manifolds.
An N Q-manifold is a differential non-negatively graded manifold. We recall a reformulation of this definition from [32].
We let SMan N Q and SMan N denote the categories of N Q-manifolds and N -manifolds, respectively. Definitions 2 and 3 also extend in the obvious way to define the categories SMfold N Q , SMfold N , SMan N Q   Suppose that [M • ] ∈ Man ∆ op loc , then (following [32]) we define Remark 2. Note that 1-Jet is not a faithful functor. In general, for n > 1, there are more morphisms between local Lie n-groupoids than between their corresponding N Q-manifolds (see Remark 4 for a relevant example). Furthermore, the composition from Lie n-groupoids to N Q-manifolds already fails to be full for n = 1 (see the work Crainic and Fernandes [12]).
Definition 6. We say that a local Lie n-groupoid Example 5. Since the functor E : Man → Man ∆ op from Example 3 is full and faithful, Example 6. If G is a Lie group and g is its Lie algebra, then the nerve of G integrates the N Q-manifold g [1]. More generally, if Γ is a Lie groupoid and A the corresponding Lie algebroid, then (the nerve of) Γ integrates A[1].
Example 7. Let Vect f (R) denote the category of finite dimensional vector spaces over R, Ch + (Vect f (R)) the positively graded chain complexes and V ∈ Vect f (R). The Dold-Kan correspondence , which we view as a simplicial manifold. Concretely, and for any monotone map h : The following picture can be useful. An element of K(V, n) k is a labelling of the n dimensional faces of the standard k simplex by elements of V , so that the alternating sum around any n + 1 dimensional face is zero. Note that we have a diffeomorphism V ∼ = K(V, n) n given by 2.3.1. Multiplicative Forms. Following [5-8, 11, 13, 18] we make the following definition of multiplicative forms on a simplicial manifold.
Definition 7. Let M • be a (local) simplicial manifold. We say that a k-form α ∈ Ω k (M n ) is multiplicative if for any 0 ≤ i < n, s * i α = 0, and Dα = 0, where is the simplicial differential.
In the spirit of [5,6,22,23,36] we would like to interpret Definition 7 in terms of morphisms of simplicial manifolds. But first we need to point out that one can extend the functor 1-Jet to (local) simplicial N Q-manifolds, Specifically, in the formula End(R 0|1 ) × (1, 0) acts directly on the factor X • and (1, 0) × End(R 0|1 ) acts directly on the factor R 0|1 . Just as certain (local) simplicial manifolds integrate N Q-manifolds, certain (local) simplicial N Q-manifolds integrate bi-N Q-manifolds.
We will be interested in the following example. Recall that a k-form α ∈ Ω k (M ) can be thought of as a (grading-preserving) function α : Lemma 1. Let M • be a (local) simplicial manifold, and α a k-form α ∈ Ω k (M n ). The corresponding map extends to a simplicial map to the Eilenberg-Mac Lane object (2.3), if and only if α is multiplicative. In this case, the extension (2.7) is unique.
Proof. First, we will compute the unique extensionα (assuming it exists). Using (2.4), we see that for x ∈ T [1]M n , (2.7) is given bŷ Let y ∈ . It follows thatα must be defined by Using (2.10), we see that for any g : Comparison with Example 7 shows that (2.10) defines a simplicial map if and only if s * i α = 0 and Dα = 0, that is α is multiplicative. As a consequence, if M • integrates an N Q-manifold X, and is a multiplicative k-form, then is a morphism of bi-graded Q-manifolds. That is, 1-Jet(α) defines a Q-invariant k-form of degree n on X. Moreover, 1-Jet(α) is closed whenever α is. Remark 3 (Some Shorthand Notation). As a shorthand, we will denote a point with coordinates as another shorthand, where Σ k denotes the permutation group of {1, . . . , k} and sign : Σ k → Z 2 is the unique non-trivial group morphism.
Finally, we will also invoke Einstein's summation convention, summing over repeated upper and lower indices.
Using Einstein's summation convention, the symplectic form can be written (2.11) ω := dp a dx a + dη a dξ a , and the homological vector field is The corresponding Poisson bracket is defined on coordinates by (and the Poisson bracket of any other pair of coordinates is equal to zero).
(with both terms vanishing identically). Therefore, with X κ := {q * κ, ·}, Consequently, the time-1 Hamiltonian flow e X β defines an isomorphism of Courant algebroids In particular, if β is closed, e X β defines an automorphism of Courant algebroids. Furthermore, (2.13) shows that the isomorphism class of (T *   This suggests that a more flexible definition of (local) symplectic 2-groupoids is needed, which interprets diagrams such as (2.15) in terms of appropriate 2-morphisms. We plan to address this issue in a future paper.
The remainder of this paper will be devoted to constructing strictly-2-symplectic (local) 2-groupoids integrating standard and exact Courant algebroids.
Remark 5. If we ignore the symplectic structure, there is a very simple construction of a simplicial manifold integrating the standard and exact Courant algebroids. However, integrating the symplectic form is slightly more challenging, and the bulk of the work in this note will be spent doing this.
There is a natural isomorphism of N Q-manifolds T * As explained by Mehta and Tang [24], this construction has the following more conceptual explanation: V can be viewed as a bundle of non-negatively graded chain complexes (with trivial differential). Using the Dold-Kan correspondence pointwise (cf. Example 7), we get a bundle Γ(V ) • of simplicial vector spaces over M . Next we note that the bisimplicial manifold E For t ∈ [0, 1] let q t : U → M be given by q t (x, y) = γ x,y (t). We define W := q * 1/2 T * M . Notice that, using the parallel transport, we can identify W with W t := q * t T * M for any t ∈ [0, 1].
We shall suppose that U is such that m is an injective local diffeomorphism. Similarly, we let U n ⊂ M n+1 = E n M be given by For any monotone map f : In the sequel, when describing an element of T * n M , we will often omit the terms w ii ≡ 0, since they are always 0. We can visualize T * • M graphically as in Figure 1. It is clear that [T * • M ] is a local 2-groupoid since simplices are fully determined by their 0 and 1 dimensional faces.
Remark 6. If one replaces the vector bundle W := W 1 2 with T * M × M ⊇ W 0 , the construction of T * • M described above is formally identical to the construction of the simplicial manifold I • described in Remark 5. Since parallel transport defines an isomorphism W ∼ = W 0 , we get a canonical embedding of simplicial manifolds T * • M ⊆ I • . Note that I • is precisely the Lie 2-groupoid described by Mehta and Tang [24].
where VBund is the category of vector bundles.
We may view our simplicial manifold T * • M as a simplicial submanifold of VBund(T |∆ • |, T * M ). Indeed, Meanwhile, we can realize elements of T * n M (n > 0) inductively as harmonic maps 1 in VBund(T |∆ n |, T * M ) which satisfy the following boundary conditions • they restrict to elements of T * n−1 M ⊂ VBund(T |∆ n−1 |, T * M ) along the boundary T ∂|∆ n | ⊂ T |∆ n |, and • they map horizontal vectors at points in T ∂|∆ n | ⊥ ⊂ T |∆ n | to horizontal vectors (horizontal with respect to the Levi-Cevita connections). In the construction of T * • M , we only considered 'harmonic' simplicies in X • which are suitably close to being degenerate. However, given any element of VBund T |∆ n |, T * M , one could first deform it to a homotopic 'energy minimizing' map, and then look at nearby 'harmonic' simplices. Patching these neighbourhoods together, one might hope to construct a global integration of T * The symplectic form ω T on T * 2 M is defined to be the pullback of the symplectic form on T * M 3 .
To check that ω T is multiplicative, i.e. that Dω = 0, it suffices to show that ω T = Dα for some α ∈ Ω 2 (T * 1 M ). Letq 1/2 : T * 1 M → T * M denote the map defined by (x, y; w) → (q 1/2 (x, y), w). We can use α =q  For v ∈ T [1] x M , the corresponding map ev v : E 1 R 0|1 → E 1 M is given by We would like to look at all possible maps E • R 0|1 → T * • M lying over the map ev v . Now a map F v : If we would like to extend F v : E 1 R 0|1 → T * 1 M to a map of simplicial (super)-manifolds, the extension must be equivariant with respect to the action of the category ∆ op . In particular, we must have However (3.3) is the only obstruction to extending F v . A routine check shows that is the unique extension of (3.2) to a map  • M ]) is given by (2.12). The natural action of R 0|1 on E • R 0|1 is θ · (θ 0 , . . . , θ n ) = (θ 0 + θ, . . . , θ n + θ).
Notice that, unlike the construction in [24], T * • M : ∆ op → Man extends naturally to a functor FSets op → Man (where FSets is the category of finite sets).

Integration of Exact Courant Algebroids
In this section, we show that as N Q-manifolds, any exact Courant algebroid over M is isomorphic to the standard Courant algebroid over M (see also [34]). Therefore once we modify the symplectic structure, the construction in § 3 also integrates exact Courant algebroids.
Lemma 2. Let F : SMan N Q → SMan N be the forgetful functor. Then T [1] : SMan N → SMan N Q is a right adjoint for F . In particular, for X ∈ SMan N Q and Z ∈ SMan N , Proof. Since T [1]Z represents the pre-sheaf SMan(· × R 0|1 , Z), we have It is clear that this bijection SMan N Q (X, T [1]Z) ∼ = SMan N (X, Z) is natural in both X and Z, which concludes the proof.
Let us describe the unit and co-unit of the adjunction explicitly. Note that the homological vector field Q X on X defines a section Q X : X → T [1]X. Furthermore, Q X : X → T [1]X is a N Q-morphism with respect to the de Rham vector field on T [1]X. The unit of the adjunction is the natural transformation Meanwhile, if q Z : T [1]Z → Z is the canonical projection, then the co-unit is the natural transformation In particular, if f : X → Z is any N -morphism, is the corresponding N Q-morphism. In the other direction, iff : X → T [1]Z is an N Qmorphism, the corresponding N -morphism is just q Z •f . Example 9 (The Anchor Map). Suppose X is any N Q-manifold over the base B(X) := (0, 0) · X (here (0, 0) ∈ End(R 0|1 ), see Definition 5). Multiplication by (0, 0) defines the canonical projection q X : X → B(X). It follows from Lemma 2 that there is a unique N Q-morphism a X : X → T [1]B(X) over q X : X → B(X), called the anchor map for X.
Furthermore, Lemma 2 implies that a X is a natural transformation from the identity functor on SMan N Q to T [1] • B : SMan N Q → SMan N Q . In fact T [1] : Man → SMan N Q is a right adjoint for B : SMan N Q → Man, with unit a X and co-unit the identity.

Let
where q M : T [1]M → M is the projection, and Q dR is the de Rham vector field on T [1]M . In particular, ψ κ is a diffeomorphism.