Deformations of Pre-Symplectic Structures: a Dirac Geometry Approach

We explain the geometric origin of the $L_{\infty}$-algebra controlling deformations of pre-symplectic structures.


Introduction
A pre-symplectic form is just a closed 2-form of constant rank. For instance, the restriction of a symplectic form to a coisotropic submanifold (such as the zero level set of a moment map) is pre-symplectic. Given a pre-symplectic from η of rank k, we constructed in [7] an algebraic structure that encodes the deformations of η, i.e. the 2-forms nearby η (in the C 0 -sense) which are both closed and of constant rank k. As in many deformation problems, this algebraic structure is an L ∞ -algebra, which we call Koszul L ∞ -algebra of η. Its construction -which is somewhat involved due to the simultaneous presence of the closedness and constant rank condition -relies on a certain BV ∞ -algebra structure on the differential forms and builds on the work of Fiorenza-Manetti [1]. The Koszul L ∞ -algebra has the property that its Maurer-Cartan elements are in bijection with the pre-symplectic deformations of η.
Given that pre-symplectic forms are geometric objects, it is natural to ask for a geometric derivation of the algebraic structure that governs their deformations (the Koszul L ∞ -algebra). The present note provides an answer to this question. The idea is the following: instead of restricting oneself to the realm of 2-forms, work in the larger class of almost Dirac structures, and consider deformations of graph(η) := {(v, η(v, ·)) | v ∈ T M } ⊂ T M ⊕ T * M within the Dirac structures satisfying a constant rank condition. This is explained in Subsection 2.2, which is the heart of this note.
The first step in [7] is to provide a parametrization of the constant rank forms nearby η in terms of (an open subset in) a vector space. This parametrization is obtained naturally by taking the point of view of Dirac linear algebra in Subsection 2.3.
The second step in [7] was to show that the closedness condition translates into a Maurer-Cartan equation for a suitable L ∞ -algebra. In Subsection 2.4 we re-obtain these results by showing that the L ∞ -algebra governing deformations of Dirac structures, in the case at hand and upon a suitable restriction, is the Koszul L ∞ -algebra. There we also improve slightly a result of [7], see our Corollary 1.9.
The Koszul L ∞ -algebra depends on an auxiliary choice of distribution transverse to ker(η). In the Dirac-geometric interpretation, this translates into a suitable choice of complement of graph(η) in T M ⊕ T * M . One of the achievements of [3] is to establish a general framework to control the effects of changing the complement, exhibiting explicit canonical L ∞ -isomorphisms between the corresponding L ∞ -algebras. A consequence of this note and of [3] is that the Koszul L ∞ -algebra of (M, η) is well-defined up to L ∞ -isomorphisms.

1.2.
A parametrization of constant rank 2-forms. Let V be a finite-dimensional, real vector space. Any bivector Z ∈ ∧ 2 V can be encoded by the linear map We denote by I Z the open neighborhood of 0 ⊂ ∧ 2 V * consisting of those elements β for which the map id + Z ♯ β ♯ : V → V is invertible. We consider the map F : This map is clearly non-linear, and it is smooth. The map F is a diffeomorphism from I Z to I −Z , which keeps the origin fixed. Fix η ∈ ∧ 2 V * of rank k. We now use F to construct submanifold charts for the space (∧ 2 V * ) k of skew-symmetric bilinear forms on V of rank k. We fix a subspace G ⊂ V , which is complementary to the kernel K = ker(η ♯ ). Since the restriction of η to G is non-degenerate, there is a unique element Z ∈ ∧ 2 G ⊂ ∧ 2 V determined by the requirement that The Dirac exponential map exp η of η (and for fixed G) is the mapping Let r : ∧ 2 V * → ∧ 2 K * be the restriction map; we have the natural identification ker(r) ∼ = ∧ 2 G * ⊕ (G * ⊗ K * ). The following theorem [7,Thm. 2.6] asserts that the restriction of exp η to ker(r) is a submanifold chart for ( (i) Let β ∈ I Z . Then exp η (β) lies in (∧ 2 V * ) k if, and only if, β lies in ker(r) = ( Then exp η (β) is the unique skew-symmetric bilinear form on V with the following properties: • its restriction to G equals (η + F (σ))| ∧ 2 G • its kernel is the graph of the map The Dirac exponential map exp η : I Z → ∧ 2 V * restricts to a diffeomorphism Remark 1.4. We notice that the construction of exp η can be readily extended to the case of vector bundles. In particular, given a pre-symplectic manifold (M, η), the choice of a complementary subbundle G to the kernel K of η yields a fibrewise map which maps the zero section to η, and an open neighborhood thereof into the space of 2-forms of rank equal to that of η. As a consequence, we can parametrize deformations of η inside Pre-Sym k (M ) by sections (µ, σ) ∈ Γ(K * ⊗ G * ) ⊕ Γ(∧ 2 G * ) ∼ = Ω 2 hor (M ) which are sufficiently close to the zero section, and which satisfy d((exp η )(µ, σ)) = 0, with d the de Rham differential.

1.
3. An L ∞ -algebra associated to a bivector field. In this subsection, we introduce an L ∞ -algebra, which is naturally attached to a bivector field Z on a manifold M . Definition 1.5. Let Z be a bivector field on M . The Koszul bracket associated to Z is the operation Here the Lie derivative is defined as the (graded) commutator of the contraction by Z with the de Rham differential, i.e. L Z = ι Z • d − d • ι Z . When applied to 1-forms α, β, the Koszul bracket can be written as Unless we assume that Z is Poisson, i.e. that it commutes with itself under the Schouten-Nijenhuis bracket, the Koszul bracket will fail to satisfy the graded version of the Jacobi identity, however the failure can be controlled. As a preparation, we introduce some notation: for a differential form α ∈ Ω r (M ), we have and, following [2, §2.3], we extend this definition to a collection of forms α 1 , . . . , α n by setting Definition 1.6. We define the trinary bracket [·, ·, ·] Z : These brackets endow Ω(M ) [2] with an L ∞ [1]-algebra structure, extending results of Fiorenza and Manetti [5]. The following is [7, Prop. 3.5]: [2]). Let Z be a bivector field on M . The multilinear maps λ 1 , λ 2 , λ 3 on the graded vector space Ω(M ) [2] given by (1) λ 1 the de Rham differential d, and define the structure of an L ∞ [1]-algebra on Ω(M ) [2].
We now turn to the geometry encoded by the L ∞ [1]-algebra (Ω(M ) [2], λ 1 , λ 2 , λ 3 ). To this end, recall that we can naturally associate the following equation to such a structure: Recall that in Equation (1) we introduced a map F :
In Section 2.4 we will show that as open subset U one can choose the whole of I Z .
1.4. The Koszul L ∞ -algebra of a pre-symplectic manifold. We return to the pre-symplectic setting, i.e. suppose η is a pre-symplectic structure on M . Let us fix a complementary subbundle G to the kernel K ⊂ T M of η and let Z be the bivector field on M determined by [2] associated to the bivector field Z, see Proposition 1.7, maps Ω hor (M ) [2] to itself. The subcomplex Ω hor (M ) [2] ⊂ Ω(M ) [2] therefore inherits the structure of an L ∞ [1]-algebra, which we call the Koszul L ∞ [1]-algebra of (M, η).
We denote by MC(η) the set of Maurer-Cartan elements of the Koszul L ∞ [1]-algebra of (M, η). In view of the above theorem, the following result [7,Thm. 3.19] is an immediate consequence of Thm. 1.3 and Cor. 1.9.
The following statements are equivalent: (1) β is a Maurer-Cartan element of the Koszul L ∞ [1]-algebra Ω hor (M ) [2] of (M, η), which was introduced in Theorem 1.10. (2) The image of β under the map exp η , which is introduced in Def. 1.2, is a pre-symplectic structure of the same rank as η.
The above Thm. 1.11 is the main result of [7], as it states that the Koszul L ∞ [1]-algebra governs the deformations of the pre-symplectic structure η. More precisely, rephrasing the above result, the fibrewise map restricts, on the level of sections, to an injective map with image the pre-symplectic structures of rank equal to the rank of η and with kernel transverse to G.

Dirac geometric interpretation
In the remainder of this note we explain the geometric framework that underlies the results of Section 1 recalled from [7]. We recover naturally the statements made there and provide some alternative and more geometric proofs.
2.1. Background on Dirac geometry. We first review some notions from Dirac linear algebra. Let V be a finite-dimensional, real vector space. We denote by V the direct sum V ⊕ V * and by ·, · the following non-degenerate pairing on V: (v, ξ), (w, χ) := ξ(w) + χ(v).
Given an element Z ∈ ∧ 2 V , we defined the linear map Z ♯ : V * → V in Subsection 1.2, and we can consider the Lagrangian subspace graph(Z) : Every β ∈ ∧ 2 V * defines an orthogonal transformation t β of (V, ·, · ), by Similarly, every Z ∈ ∧ 2 V gives rise to an orthogonal transformation t Z , which takes (v, ξ) to (v + Z ♯ (ξ), ξ). In particular, elements of ∧ 2 V * and ∧ 2 V act on the set of Lagrangian subspaces of V.
Remark 2.2. Suppose L, R are transverse Lagrangian subspaces of V. There is a canonical isomorphism R ∼ = L * , r → r, · | L . Since R is transverse to L, any subspace of V transverse to R is the graph of a linear map L → R. Any Lagrangian subspace transverse to R is the graph of a linear map L → R such that, composing with the canonical isomorphism above, we obtain a skew-symmetric linear map L → L * (i.e. the sharp map associated to an element of ∧ 2 L * ).
Together with the projection to T M , this makes TM into an example of Courant algebroid. . We now recall a result of Liu-Weinstein-Xu [4] establishing when such an almost Dirac structure is Dirac. Recall that every Dirac structure, with the restricted Dorfman bracket and anchor, is a Lie algebroid. Since L is a Lie algebroid, it induces a differential d L on Γ(∧L * ). Further 1 , since L * ∼ = R is a Lie algebroid, it induces a graded Lie bracket [·, ·] L * on Γ(∧L * ) [1]. Together with d L and [·, ·] L * , the graded vector space Γ(∧L * )[1] becomes a differential graded Lie algebra. The main result of [4] is: for all ε ∈ Γ(∧ 2 L * ), the graph L ε = {v + ι v ε : v ∈ L} is a Dirac structure iff ε satisfies the Maurer-Cartan equation, that is

2.2.
Deformations of pre-symplectic structures: the point of view of Dirac geometry.
In this subsection we cast the deformations of pre-symplectic forms in the framework of Dirac geometry. Let η be a pre-symplectic form on M , with kernel K. The natural way to parametrize deformations of η is by 2-forms α such that η+α is again pre-symplectic, but this parametrization has a serious flaw: the space of such α's does not have a natural vector space structure, due to the constant rank condition. Taking the point of view of Dirac geometry, the above approach 1 The Lie algebroid structures on L and L * are compatible in the sense that the pair (L, L * ) forms a Lie bialgebroid.
to parametrize the deformations of η amounts to deforming the Dirac structure graph(η) using {0} ⊕ T * M as a complement.
A better way to parametrize the deformations of η in terms of Dirac geometry works as follows: Let us first choose a complement G to K. Then G ⊕ K * is a complement 2 of graph(η). We can now use G ⊕ K * -instead of {0} ⊕ T * M -to parametrize deformations of the Dirac structure graph(η). This choice of complement has the advantage of linearizing the constant rank condition, as we show in Proposition 2.7 below. (Notice that when η is symplectic, the new complement is just T M , hence we are deforming η by viewing it as a Poisson structure, just as in [7, Section 1.3].) To do so, we first state two lemmas about the effect of applying the orthogonal transformation Lemma 2.5. Denote by Z ∈ Γ(∧ 2 G) the bivector field such that Z ♯ is the inverse of −(η| G ) ♯ . Then t −η maps G ⊕ K * to graph(Z).
Proof. t −η preserves the pairing on T M ⊕T * M , clearly maps graph(η) to T M , and maps G⊕K * to graph(Z) by Lemma 2.5. Therefore the statement follows by functoriality.
Now we can explain why the choice of G ⊕ K * as a complement is a good one to describe pre-symplectic deformations.
(i) The rank of Hence applying the transformation to the intersection (3) we obtain which is isomorphic to (4). (ii) Denote by η ′ the 2-form whose graph is Φ G⊕K * (β). The kernel of η ′ is given by (3), and the assertion follows immediately from (i). Recall that the vector space Ω 2 hor (M ) of horizontal 2-forms was defined in Subsection 1.1, as the space of 2-forms that vanish on ∧ 2 K. Remark 2.8. Since t −η is actually an automorphism of the standard Courant algebroid T M ⊕ T * M , the following two deformation problems of Dirac structures are equivalent: • deformations of graph(η), using the complement G ⊕ K * , • deformations of T M , using the complement graph(Z).
The latter deformation problem is easier to handle, and the L ∞ [1]-algebra structure governing it will be recovered in Subsection 2.4.
Dirac-geometric interpretation of Subsection 1.2. Using Dirac linear algebra, we explain and re-prove the results recalled in Subsection 1.2, "A parametrization of constant rank 2-forms".

2.3.1.
Revisiting the map F from formula (1). Let V be a finite-dimensional, real vector space. We fix a bivector Z ∈ ∧ 2 V . Recall that I Z consists of elements β ∈ ∧ 2 V * such that id + Z ♯ β ♯ is invertible. In formula (1), we defined the map F : I Z → ∧ 2 V * given by The following lemma provides a geometric explanation of the map F .
In particular, the map F is characterized by the property that graph(F (β)) = Φ Z (β) (5) for all β ∈ I Z . In other words, F (β) is obtained taking the graph of β w.r.t. the splitting V = V ⊕ graph(Z).
Proof. (i) According to Remark 2.2, any Lagrangian subspace transverse to V * is the graph of a skew-symmetric linear map V → V * , and therefore can be written as Similarly, graph(Z) is transverse to V and the induced isomorphism graph(Z) ∼ = V * is just (Z ♯ (ξ), ξ) → ξ. Hence any Lagrangian subspace transverse to graph(Z) can be written as . This intersection is trivial iff ker (id + Z ♯ β ♯ ) ⊂ ker(β ♯ ). In turn, this condition is equivalent to (id + Z ♯ β ♯ ) being injective, and thus invertible.
(iii) Finally, if id + Z ♯ β ♯ is invertible, Φ Z (β) is transverse to V * by item (ii). By item (i) the element Φ −1 0 (Φ Z (β)) is well-defined. In concrete terms, it is given by α ∈ ∧ 2 V * such that for all v ∈ V , there is w ∈ V for which holds. Equivalently, this means that
Using this we recover Thm. 1.3, in particular item (i) stating that exp η (β) has rank equal to k = dim(K) iff β is horizontal.
(iii) By Lemma 2.9 (ii), the map Φ Z provides a bijection between I Z and Lagrangian subspaces transverse to graph(Z) and to V * . Hence t η •Φ Z provides a bijection between I Z and Lagrangian subspaces transverse to t η (graph(Z)) = G ⊕ K * (see Lemma 2.5) and to V * . The latter are exactly the graphs of elements η ′ ∈ ∧ 2 V * so that the η ′ | ∧ 2 G is non-degenerate. Hence, by the proof of Lemma 2.10, exp η provides a bijection between I Z and such η ′ . We conclude using (i).

2.4.
Dirac-geometric interpretation of Subsection 1.3. Using Dirac geometry and adapting results from [2], we explain and re-prove the results recalled in Subsection 1.3, " An L ∞algebra associated to a bivector field". Fix a bivector field Z on M .
More generally, Proposition 2.11 holds replacing TM by any Courant algebroid.
Proof. The proof is a minor adaptation of the first part of the proof of [2, Lemma 2.6], setting ϕ = 0 there. We recall briefly the idea of the latter. By [6] there is a natural description of the Courant algebroid structure on TM in terms of graded geometry. One can use it to apply Voronov's Higher Derived Brackets construction (see [8,9]) and obtain an L ∞ [1]-algebra structure on Γ(∧L * ) [2]. The multibrackets obtained are the ones in the statement of the lemma, as one checks using [6] and via computations in local coordinates.
Alternative proof of Proposition 1.7. Let Z be a bivector field on M . We apply Proposition 2.11 choosing L = T M and R = graph(Z). In that case d L is the de Rham differential, and the bracket on R is given by the formula for the Koszul bracket. One checks that ψ is the trivector field − 1 2 [Z, Z], using [7, Lemma 1.6]. Hence the L ∞ [1]-brackets on Ω(M ) [2] given by Proposition 2.11 are µ 1 = λ 1 , µ 2 = −λ 2 and µ 3 = λ 3 . Applying the automorphism −id to Ω(M ) [2] yields Proposition 1.7.

2.4.2.
Revisiting Corollary 1.9 (Maurer-Cartan elements of Ω(M ) [2]). We now turn to Maurer-Cartan elements. In Lemma 2.9 (i), we gave a parametrization of all almost Dirac structures that are transverse to graph(Z) in terms of 2-forms β on M . This parametrization is given by We present the second part of [2, Lemma 2.6], which is an extension of the work by Liu-Weinstein-Xu recalled in Remark 2.4. Proposition 2.12. Let be given a Dirac structure L and a complementary almost Dirac structure R. An element σ ∈ Γ(∧ 2 L * ) [2] is a Maurer-Cartan element of the L ∞ [1]-algebra structure given in Proposition 2.11 iff the graph is a Dirac structure. (The above inclusion makes use of the identification R ∼ = L * .) Corollary 1.9 states that for β ∈ Ω 2 (M ) taking values in some sufficiently small neighborhood U of the zero section in ∧ 2 T * M -in particular taking values in I Z , i.e. id + Z ♯ β ♯ is invertible -, β is a Maurer-Cartan element of (Ω(M ) [2], λ 1 , λ 2 , λ 3 ) iff F (β) is closed. We now provide an alternative proof of this result, which also shows that one can choose U to equal I Z .
Remark 2.13. In this subsection we recovered the L ∞ [1]-algebra Ω(M ) [2] of Prop. 1.7 as the L ∞ [1]-algebra governing deformations of the Dirac structure T M taking graph(Z) as a complement. By Remark 2.8, this deformation problem is equivalent to the deformations of the Dirac structure graph(η) taking G⊕K * as the complement. This explains why the L ∞ [1]-algebra Ω(M ) [2] governs the latter deformation problem, and therefore is relevant for the deformations of pre-symplectic structures.
2.5. Dirac-geometric interpretation of Subsection 1.4. Thm. 1.10 can be deduced from a general statement about (almost) Dirac structures, however doing so amounts essentially to the same computations that were needed for the proof given in [7]. We include this general statement for the sake of completeness.
Finally, as mentioned earlier, Thm. 1.11 follows immediately from the other results presented.