Integration of Dg Symplectic Manifolds Via Mapping Spaces

INTEGRATION OF DG SYMPLECTIC MANIFOLDS VIA MAPPING SPACES

RAJAN MEHTA

Abstract. Severa and Roytenberg observed that Courant algebroids are in one-to-one correspondence with diﬀerential graded (DG) symplectic manifolds of degree 2. I will describe this correspondence, as well as an integration procedure (due to Severa, following Sullivan) involving mapping spaces. The result of the integration procedure is a symplectic 2-groupoid, but it is inﬁnite-dimensional. Nonetheless, the integration can be explicitly described in the case of exact Courant algebroids, and this construction helps to explain why (twisted) Dirac structures integrate to (twisted) presymplectic groupoids. This talk is based on joint work with Xiang Tang.

Outline: (1) Warmup: Integration of Poisson manifolds from the DG perspective (2) Correspondence between Courant algebroids and degree 2 symplectic NQ-manifolds (3) Integrating exact Courant algebroids (4) (twisted) Dirac structures

1. Warmup: Poisson manifolds

1.1. Poisson structures in the DG language. Let (M, π) be a Pois- son manifold. Because of the integrability condition [π, π] = 0, we have • a diﬀerential dπ := [π, ·] on the algebra X (M) = Γ(∧TM) of multivector ﬁelds:

dπ dπ C∞(M) / X(M) / X2(M) / ···

This differential, together with the Schouten bracket, gives X•(M) the structure of a differential graded Poisson algebra (i.e. a differential graded Lie algebra where the differential and Lie bracket are derivations). We can recover the Poisson bracket as a derived bracket:

{f, g} = [dπf, g]. 1 2 RAJAN MEHTA

It’s worth noting that all of the content of dπ is contained in the first map taking functions to their Hamiltonian vector fields. The action of dπ in higher degrees is completely determined by the compatibility conditions. From the perspective of graded geometry, we can view X•(M) as the “smooth functions” on the shifted cotangent bundle T ∗[1]M, the Schouten bracket as the Poisson bracket corresponding to the canonical symplectic structure, π as a degree 2 function, and the differential as its Hamiltonian vector field. Conversely, it can be shown that any symplectic graded manifold with co- ordinates in degrees 0 and 1 is canonically isomorphic to T ∗[1]M for some manifold M, giving a correspondence between Poisson manifolds and “degree 1 symplectic NQ-manifolds”.

1.2. Integration via mapping spaces. Given a Poisson manifold M, one can build a simplicial manifold in the following way: the k-simplices are Q- manifold maps T [1]∆k → T ∗[1]M, and the face and degeneracy maps come from the natural maps between simplices. We can translate this into non-super language, since all the Q-manifolds correspond to Lie algebroids: the k-simplices are Lie algebroid morphisms from T ∆k to T ∗M. Now we can give a more concrete description for low k ∗ degrees. We use the notation Xk := HomLA(T ∆ ,T M).

•X0 = M.

•X1 is the space of smooth cotangent paths.

•X2 consists of “triangular homotopies” between cotangent paths. To obtain a groupoid, we truncate this simplicial manifold at level 1, by identifying elements of X1 that are homotopic (with respect to X2). Specif- ically, given a pair η1, η2 ∈ X2 that agree on two of the three face maps, we identify the respective images of η1 and η2 under the third face map. The result is (up to integrability issues) a Lie groupoid G ⇒ M. It is essentially equivalent to the (CF)2 integration, and it works equally well with arbitrary Lie algebroids in the place of T ∗M.

1.3. Symplectic structure of the integration. Now we can try to see the symplectic structure on the integration of a Poisson manifold. Fix γ ∈ Xk, and let v1, v2 be tangent vectors in TγXk. The symplectic structure on ∗ T [1]M provides a way to pair v1 and v2, giving a degree 1 function on T [1]∆k, i.e. a 1-form on ∆k. When k = 1, the 1-form can be integrated.

Thus, X1 is equipped with a 2-form, and it is both closed and multiplicative. Furthermore, the kernel of the 2-form precisely coincides with the homotopies, so that it descends to multiplicative symplectic form on the quotient G ⇒ M. INTEGRATION OF DG SYMPLECTIC MANIFOLDS VIA MAPPING SPACES 3

2. Courant algebroids in the DG language

Let E → M be a Courant algebroid (we use the non-skew deﬁnition, where the bracket is a Leibniz bracket). There is an associated DG algebra in this setting as well:

D L C∞(M) / Γ(E) / o(E) / ··· Here, •D is given by hDf, ei = ρ(e)(f), • o(E) is the space of ﬁrst-order skew-symmetric operators on Γ(E), and

•L is given by Le1 e2 = e1, e2 . J K There is also a degree −2 Lie bracket {, }, where

•{e1, e2} coincides with the inner product for e1, e2 ∈ Γ(E), •{φ, e} = φ(e) and {φ, f} = σ(φ)(f) for φ ∈ o(E), e ∈ Γ(E), and f ∈ C∞(M) (here σ is the symbol map),

•{φ1, φ2} coincides with the commutator bracket, •{f, g} = {f, e} = 0. As in the Poisson case, all of the content is in this lower-degree part of the complex. The algebra is generated in degrees 0, 1, and 2, and the Lie bracket and diﬀerential can be uniquely extended to give a DG Poisson algebra. Again like the Poisson case, the anchor and Courant bracket are derived brackets:

ρ(e)(f) = {Le, f} = {Df, e}

e1, e2 = {Le1 , e2}. J K A slightly less trivial observation is that the Courant algebroid axioms are direct consequences of the DG Poisson algebra axioms. From the perspective of graded geometry, we view this DG Poisson algebra as the “smooth functions” on a degree 2 symplectic NQ-manifold E. Con- versely, any degree 2 symplectic NQ-manifold gives a Courant algebroid via the derived bracket construction, so there is a correspondence.

3. Integrating exact Courant algebroids

As in the Poisson case, one can build a simplicial manifold by taking k Xk := HomQ(T [1]∆ , E). 4 RAJAN MEHTA

The 2-form now appears on X2, because the symplectic structure on E is degree 2. But again, the general theory says that the 2-form is closed and multiplicative, and that it descends to a symplectic form on the 2-groupoid truncation G2 V G1 ⇒ M. But, even in simple cases (such as a Lie algebra with a bilinear form), this 2-groupoid is inﬁnite-dimensional. In the case of the standard Courant algebroid E = TM ⊕ T ∗M, we have E = T [1](T ∗[1]M), and we can make the identiﬁcation k ∗ k ∗ Xk = HomQ(T [1]∆ ,T [1](T [1]M)) = Hom(T [1]∆ ,T [1]M), where the last Hom is just in the category of graded manifolds. In other k ∗ words, Xk consists of bundle maps from T ∆ to T M. In low degrees, we have:

•X0 = M, ∗ •X1 can be identiﬁed with the space of paths in T M. It will not be aﬀected by the truncation, so G1 = X1.

• After truncation, the points of the quotient G2 := X2/ ∼ can be described by a homotopy class of maps α : ∆2 → M and maps ∗ ξ0, ξ1, ξ2 lifting each edge of α to T M. (The ξi are not required to agree at the vertices.)

Note that, in this case, there is a natural symplectic structure ω1 on G1 which is not part of the general theory. However, the simplicial coboundary (i.e. alternating sum of pullbacks by the face maps) of ω1 is equal to the symplectic form ω2 on G2 (which is part of the general theory). If the standard Courant bracket is twisted by a closed 3-form H, then the corresponding NQ-manifold E0 is isomorphic (but not symplectomorphic!) to the standard one. So, the integrating 2-groupoid is the same, but the sym- 0 plectic form ω2 is diﬀerent. Nonetheless, it is still the simplicial coboundary 0 0 ˆ ˆ of a 2-form ω1 on G1. Speciﬁcally, ω1 = ω1 + H, where H is the “transgres- 0 sion” of H to the path space. But now, ω1 isn’t closed. Instead, letting δ be the simplicial coboundary, we have 0 0 ω2 = δω1, 0 dω1 = δH, dH = 0.

4. integrating Dirac structures

In the DG picture, Dirac structures correspond to wide Lagrangian sub- NQ-manifolds of E. By “wide” I mean that the degree 0 part is all of M. The general theory then says that Dirac structures will integrate to wide Lagrangian sub-2-groupoids L2 V L1 ⇒ M. Let i be the inclusion into the symplectic 2-groupoid G2 V G1 ⇒ M. INTEGRATION OF DG SYMPLECTIC MANIFOLDS VIA MAPPING SPACES 5

∗ In the case of the standard Courant algebroid, L1 inherits the 2-form i ω1. ∗ ∗ It is obviously closed, and it is also multiplicative, since δi ω1 = i δω1 = ∗ i ω2 = 0, because L2 is Lagrangian. Therefore, it survives the 1-groupoid truncation and induces a closed, multiplicative (but possibly degenerate) 2-form on the Lie groupoid integrating the Dirac structure.

If the standard Courant bracket is twisted by a closed 3-form H, then L1 ∗ 0 inherits the 2-form i ω1. It is still multiplicative, but instead of being closed, it satisﬁes the equation ∗ 0 d(i ω1) = δH. In other words, it is “H-closed”. Remarks: This helps to explain why (twisted) Dirac structures integrate to (twisted) presymplectic groupoids. However, in the BCWZ paper there is an extra nondegeneracy condition that we should be able to see. Presumably this condition distinguishes sub-NQ-manifolds that are Lagrangian from those that are non-maximally isotropic. There is also the question of what happens in the case of general Courant algebroids. In general there doesn’t seem to be a 2-form on G1. Is there any geometric structure that the groupoids integrating Dirac structures can inherit?