Deformations of Pre-Symplectic Structures: a Dirac Geometry Approach
We explain the geometric origin of the L∞-algebra controlling deformations of pre-symplectic structures.
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| Цитувати: | Deformations of Pre-Symplectic Structures: a Dirac Geometry Approach / F. Schätz, M. Zambon // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 9 назв. — англ. |
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| author_facet | Schätz, F. Zambon, M. |
| citation_txt | Deformations of Pre-Symplectic Structures: a Dirac Geometry Approach / F. Schätz, M. Zambon // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 9 назв. — англ. |
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| description | We explain the geometric origin of the L∞-algebra controlling deformations of pre-symplectic structures.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 128, 12 pages
Deformations of Pre-Symplectic Structures:
a Dirac Geometry Approach
Florian SCHÄTZ † and Marco ZAMBON ‡
† University of Luxembourg, Mathematics Research Unit, Maison du Nombre 6,
avenue de la Fonte L-4364 Esch-sur-Alzette, Luxembourg
E-mail: florian.schaetz@gmail.com
‡ KU Leuven, Department of Mathematics, Celestijnenlaan 200B box 2400,
BE-3001 Leuven, Belgium
E-mail: marco.zambon@kuleuven.be
Received September 24, 2018, in final form November 27, 2018; Published online December 06, 2018
https://doi.org/10.3842/SIGMA.2018.128
Abstract. We explain the geometric origin of the L∞-algebra controlling deformations of
pre-symplectic structures.
Key words: pre-symplectic geometry; deformation theory; Dirac geometry
2010 Mathematics Subject Classification: 17B70; 53D17; 58H15
1 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 C0-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 the Koszul L∞-algebra of η. Its construction – which
is somewhat involved due to the simultaneous presence of the closedness and constant rank
conditions – 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 ∈ TM} ⊂ TM ⊕ T ∗M
within the Dirac structures satisfying a constant rank condition. This is explained in Section 3.2,
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 Section 3.3.
The second step in [7] was to show that the closedness condition translates into a Maurer–
Cartan equation for a suitable L∞-algebra. In Section 3.4 we re-obtain this result, and further
we improve slightly a result of [7], see our Corollary 2.9.
mailto:florian.schaetz@gmail.com
mailto:marco.zambon@kuleuven.be
https://doi.org/10.3842/SIGMA.2018.128
2 F. Schätz and M. Zambon
Combining these results, in Section 3.5 we recover the fact that the L∞-algebra governing
deformations of Dirac structures, in the case at hand and upon a suitable restriction, is the
Koszul L∞-algebra.
The Koszul L∞-algebra depends on an auxiliary choice of a distribution transverse to ker(η).
In the Dirac-geometric interpretation, this translates into a suitable choice of a complement of
graph(η) in TM ⊕ 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.
2 Review: deformations of pre-symplectic structures
We review the results on deformations of pre-symplectic structures obtained in the first three
sections of [7].
2.1 Pre-symplectic structures
Fix a smooth manifold M .
Definition 2.1. A 2-form η on M is called pre-symplectic if
1) η is closed,
2) the vector bundle map η] : TM → T ∗M, v 7→ ιvη = η(v, ·) has constant rank.
A pre-symplectic manifold is a pair (M,η) consisting of a manifold M and a pre-symplectic
structure η on M . We denote the space of all pre-symplectic structures of rank k on M
by Pre-Symk(M).
A pre-symplectic manifold (M,η) gives rise to a distribution
K := ker
(
η]
)
.
This distribution is involutive since η is closed, hence K is tangent to a foliation of M . Denote
by r : Ω(M) → Γ(∧K∗) the restriction map. We define the horizontal differential forms as the
elements of
Ωhor(M) := ker(r).
They form a subcomplex of the de Rham complex Ω(M), since the de Rham differential com-
mutes with the pullback of differential forms. The subcomplex Ωhor(M) is the multiplicative
ideal of Ω(M) generated by Γ(K◦), where K◦ ⊂ T ∗M denotes the annihilator of K.
2.2 A parametrization of constant rank 2-forms
In this subsection we fix a finite-dimensional, real vector space V . Recall that a bivector Z ∈ ∧2V
is encoded by the induced linear map
Z] : V ∗ → V, ξ 7→ ιξZ = Z(ξ, ·).
Define
IZ :=
{
β ∈ ∧2V ∗ : idV + Z]β] is invertible
}
,
Deformations of Pre-Symplectic Structures: a Dirac Geometry Approach 3
an open neighborhood of the origin in ∧2V ∗. Let F : IZ → ∧2V ∗ be the map determined by
(F (β))] = β]
(
id + Z]β]
)−1
. (2.1)
The map F is non-linear and smooth. It is a diffeomorphism from IZ to I−Z , which keeps the
origin fixed.
Fix η ∈ ∧2V ∗ of rank k. We now use F to construct submanifold charts for the space
(
∧2V ∗
)
k
of skew-symmetric bilinear forms on V of rank k. Fix a subspace G ⊂ V such that K ⊕G = V ,
where K = ker
(
η]
)
. The restriction of η to G is a non-degenerate skew bilinear form, therefore
there is a unique Z ∈ ∧2G ⊂ ∧2V such that
Z] : G∗ → G, ξ 7→ ιξZ = Z(ξ, ·)
equals −
(
η|]G
)−1
.
Definition 2.2. The Dirac exponential map expη of η (and for fixed G) is the mapping
expη : IZ → ∧2V ∗, β 7→ η + F (β).
Let r : ∧2V ∗ → ∧2K∗ be the restriction map; we have the natural identification ker(r) ∼=
∧2G∗ ⊕ (G∗ ⊗K∗). By the following theorem [7, Theorem 2.6], the restriction of expη to ker(r)
is a submanifold chart for
(
∧2V ∗
)
k
⊂ ∧2V ∗.
Theorem 2.3 (parametrizing constant rank forms).
(i) Let β ∈ IZ . Then expη(β) lies in
(
∧2V ∗
)
k
if, and only if, β lies in ker(r) = (K∗ ⊗G∗)⊕
∧2G∗.
(ii) Let β = (µ, σ) ∈ IZ ∩
(
(K∗ ⊗ G∗) ⊕ ∧2G∗
)
. Then expη(β) is the unique skew-symmetric
bilinear form on V with the following properties:
• its restriction to G equals (η + F (σ))|∧2G;
• its kernel is the graph of the map Z]µ] = −
(
η|]G
)−1
µ] : K → G.
(iii) The Dirac exponential map expη : IZ → ∧2V ∗ restricts to a diffeomorphism
IZ ∩
(
(K∗ ⊗G∗)⊕ ∧2G∗
) ∼=−→
{
η′ ∈
(
∧2V ∗
)
k
| ker
(
(η′)]
)
is transverse to G
}
onto an open neighborhood of η in
(
∧2V ∗
)
k
.
Remark 2.4. By the above linear algebra construction, given a pre-symplectic manifold (M,η),
choosing a subbundle G complementary to K = ker
(
η]
)
, one obtains a map
expη : IZ ∩
(
(K∗ ⊗G∗)⊕
(
∧2G∗
))
→ ∧2T ∗M. (2.2)
It is a not a vector bundle morphism but just a smooth fiberwise map. It maps the zero section
to η, and its image is an open neighborhood of η in the space of 2-forms having the same rank
as η. The map expη allows to parametrize deformations of η inside Pre-Symk(M) by means of
sections (µ, σ) ∈ Γ(K∗⊗G∗)⊕Γ
(
∧2G∗
) ∼= Ω2
hor(M) which are sufficiently small in the C0-sense
and for which the 2-form (expη)(µ, σ) is a closed.
4 F. Schätz and M. Zambon
2.3 An L∞-algebra associated to a bivector field
In this subsection we canonically associate an L∞-algebra to any bivector field Z on a mani-
fold M .
Definition 2.5. Let Z be a bivector field on M . The Koszul bracket associated to Z is the
operation
[·, ·]Z : Ωr(M)× Ωs(M)→ Ωr+s−1(M),
[α, β]Z := (−1)|α|+1
(
LZ(α ∧ β)− LZ(α) ∧ β − (−1)|α|α ∧ LZ(β)
)
.
Here LZ = ιZ◦d−d◦ιZ , where ιZ denotes contraction with Z and d is the de Rham differential.
On 1-forms α and β, the Koszul bracket reads [α, β]Z = LZ]αβ − LZ]βα− d〈Z,α ∧ β〉.
In general the Koszul bracket of Z does not satisfy the graded Jacobi identity (it does only
when Z is a Poisson bivector-field). We will see in Proposition 2.7 that nevertheless there is
a well-behaved algebraic structure associated to Z. To this aim, recall that a differential form
α ∈ Ωr(M) induces by contraction a linear map
α] : TM → ∧r−1T ∗M, v 7→ ιvα,
and, following [2, Section 2.3], we extend this definition to a collection of forms α1, . . . , αn by
setting
α]1 ∧ · · · ∧ α
]
n : ∧n TM → ∧|α1|+···+|αn|−nT ∗M,
v1 ∧ · · · ∧ vn 7→
∑
σ∈Sn
(−1)|σ|α]1(vσ(1)) ∧ · · · ∧ α
]
n(vσ(n)).
Definition 2.6. We define the trinary bracket [·, ·, ·]Z : Ωr(M)×Ωs(M)×Ωk(M)→Ωr+s+k−3(M)
associated to the bivector field Z to be
[α, β, γ]Z :=
(
α] ∧ β] ∧ γ]
)(
1
2 [Z,Z]
)
.
These brackets endow Ω(M)[2] with an L∞[1]-algebra structure, extending the results of
Fiorenza and Manetti [5]. The following is [7, Proposition 3.5]:
Proposition 2.7 (the L∞[1]-algebra Ω(M)[2]). Let Z be a bivector field on M . The multilinear
maps λ1, λ2, λ3 on the graded vector space Ω(M)[2] given by
1) λ1 the de Rham differential d,
2) λ2(α[2] � β[2]) = −
(
LZ(α ∧ β) − LZ(α) ∧ β − (−1)|α|α ∧ LZ(β)
)
[2] = (−1)|α|([α, β]Z)[2],
and
3) λ3(α[2]� β[2]� γ[2]) = (−1)|β|+1
(
α] ∧ β] ∧ γ]
(
1
2 [Z,Z]
))
[2],
define the structure of an L∞[1]-algebra on Ω(M)[2].
We now explain the geometric relevance of the L∞[1]-algebra (Ω(M)[2], λ1, λ2, λ3). As for
any L∞[1]-algebra, it comes with distinguished elements:
Definition 2.8. An element β ∈ Ω2(M) is a Maurer–Cartan element of (Ω(M)[2], λ1, λ2, λ3) if
it satisfies the Maurer–Cartan equation
d(β[2]) + 1
2λ2(β[2]� β[2]) + 1
6λ3(β[2]� β[2]� β[2]) = 0.
Recall that at the beginning of Section 2.2 we defined an open subset IZ ⊂ ∧2T ∗M and
a map F : IZ → ∧2T ∗M . The following is [7, Corollary 3.9].
Corollary 2.9 (Maurer–Cartan elements of Ω(M)[2]). There is an open subset U ⊂ IZ , which
contains the zero section of ∧2T ∗M , such that a 2-form β ∈ Γ(U) is a Maurer–Cartan element
of (Ω(M)[2], λ1, λ2, λ3) if, and only if, the 2-form F (β) is closed.
In Section 3.4 we will show that for the open subset U one can choose the whole of IZ .
Deformations of Pre-Symplectic Structures: a Dirac Geometry Approach 5
2.4 The Koszul L∞-algebra of a pre-symplectic manifold
Let again η be a pre-symplectic structure on a manifold M . Fix a subbundle G ⊂ TM which is
complementary to the kernel K of η. Consider the bivector field Z satisfying Z] = −
(
η|]G
)−1
.
The following is [7, Theorem 3.17].
Theorem 2.10 (the Koszul L∞[1]-algebra). The L∞[1]-algebra structure on Ω(M)[2] associ-
ated to the bivector field Z, see Proposition 2.7, maps Ωhor(M)[2] to itself. The subcomplex
Ωhor(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, Theorem 3.19] is an immediate conse-
quence of Theorem 2.3 and Corollary 2.9.
Theorem 2.11 (Maurer–Cartan elements of the Koszul L∞[1]-algebra). Let (M,η) be a pre-
symplectic manifold. The choice of a complement G to the kernel of η determines a bivector
field Z by requiring Z] = −
(
η|]G
)−1
. Suppose β is a 2-form on M , which lies in IZ . The
following statements are equivalent:
1. β is a Maurer–Cartan element of the Koszul L∞[1]-algebra Ωhor(M)[2] of (M,η), which
was introduced in Theorem 2.10.
2. The image of β under the map expη, which is introduced in Definition 2.2, is a pre-
symplectic structure of the same rank as η.
The above Theorem 2.11 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: the fibrewise
map expη as in (2.2), on the level of sections, restricts to a map
expη : Γ(IZ) ∩MC(η)→ Pre-Symk(M)
which is injective and whose image consists of the pre-symplectic structures of rank equal to the
rank of η and with kernel transverse to G.
3 Dirac geometric interpretation
In the remainder of this note we explain the geometric framework that underlies the results of
Section 2 recalled from [7]. We recover naturally the statements made there and provide some
alternative and more geometric proofs.
3.1 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).
Definition 3.1. A subspace W ⊂ V is called Lagrangian if for all w,w′ ∈W we have 〈w,w′〉 = 0
and dim(W ) = dim(V ). Two subspaces W and W̃ ⊂ V are transverse, if W ⊕ W̃ = V.
6 F. Schätz and M. Zambon
Given an element Z ∈ ∧2V , we defined the linear map Z] : V ∗ → V in Section 2.2, and we can
consider the Lagrangian subspace graph(Z) :=
{(
Z]ξ, ξ
)
| ξ ∈ V ∗
}
⊂ V. Similarly, for β ∈ ∧2V ∗
we define β] : V → V ∗ and consider graph(β).
Every β ∈ ∧2V ∗ defines an orthogonal transformation tβ of (V, 〈·, ·〉), by
(v, ξ) 7→
(
v, ξ + β](v)
)
.
Similarly, every Z ∈ ∧2V gives rise to an orthogonal transformation tZ , which takes (v, ξ) to(
v+Z](ξ), ξ
)
. In particular, elements of ∧2V ∗ and ∧2V act on the set of Lagrangian subspaces
of V.
Remark 3.2. Suppose L, R are transverse Lagrangian subspaces of V. There is a canonical
isomorphism
R ∼= L∗, r 7→ 〈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 ∧2L∗).
Let us now briefly recall the basic constituencies of Dirac geometry. Consider the generalized
tangent bundle TM = TM ⊕ T ∗M . It comes equipped with a non-degenerate pairing
〈(X,α), (Y, β)〉 := α(Y ) + β(X)
and the Dorfman bracket
[[(X,α), (Y, β)]] = ([X,Y ],LXβ − ιY dα).
Together with the projection to TM , this makes TM into an example of Courant algebroid.
Definition 3.3. An almost Dirac structure on M is a Lagrangian subbundle L ⊂ (TM, 〈·, ·〉).
A Dirac structure is an almost Dirac structure whose space of sections is closed with respect to
the Dorfman bracket [[·, ·]].
Remark 3.4. Let L, R be transverse Dirac structures on M . As seen in Remark 3.2, almost
Dirac structures transverse to R are in bijection with elements of Γ
(
∧2L∗
)
. 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 dL on Γ(∧L∗). Further1, since
L∗ ∼= R is a Lie algebroid, it induces a graded Lie bracket [·, ·]L∗ on Γ(∧L∗)[1]. Together with dL
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
(
∧2L∗
)
, the graph Lε = {v + ιvε : v ∈ L} is a Dirac structure
iff ε satisfies the Maurer–Cartan equation, that is
dLε+ 1
2 [ε, ε]L∗ = 0.
1The Lie algebroid structures on L and L∗ are compatible in the sense that the pair (L,L∗) forms a Lie
bialgebroid.
Deformations of Pre-Symplectic Structures: a Dirac Geometry Approach 7
3.2 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
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 complement2 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 3.7 below. (Notice that
when η is symplectic, the new complement is just TM , hence we are deforming η by viewing it
as a Poisson structure, just as in [7, Section 1.3].)
We first state two lemmas about the effect of applying the orthogonal transformation t−η of
TM ⊕ T ∗M , given by (v, ξ) 7→
(
v, ξ − η](v)
)
.
Lemma 3.5. Denote by Z ∈ Γ
(
∧2G
)
the bivector field such that Z] is the inverse of −(η|G)].
Then t−η maps G⊕K∗ to graph(Z).
Proof. tη(graph(Z)) =
{(
Z]ξ, ξ|K
)
: ξ ∈ T ∗M
}
= G⊕K∗. �
Lagrangian subbundles nearby graph(η) can be written, for some β̄ ∈ Γ(∧2(graph(η))∗), as
the graph of the map
β̄] : graph(η)→ (graph(η))∗ ∼= G⊕K∗,
by Remark 3.2. We denote this graph as ΦG⊕K∗
(
β̄
)
. Moreover, let β ∈ Ω2(M) be the 2-form
corresponding to β̄ under the isomorphism graph(η) ∼= TM, v + ιvη 7→ v and denote by ΦZ(β)
the graph of the map β] : TM → T ∗M ∼= graph(Z).
Lemma 3.6. t−η maps ΦG⊕K∗
(
β̄
)
to ΦZ(β).
Proof. t−η preserves the pairing on TM ⊕ T ∗M , clearly maps graph(η) to TM , and maps
G⊕K∗ to graph(Z) by Lemma 3.5. 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.
Proposition 3.7. Let β̄ ∈ Γ
(
∧2(graph(η))∗
)
.
(i) The rank of
ΦG⊕K∗
(
β̄
)
∩ TM (3.1)
equals the rank of
{v ∈ K : ιvβ ∈ G∗}. (3.2)
2Indeed, for every v ∈ TM we have ιvη ∈ K◦ = G∗, so requiring that ιvη lies in K∗ implies ιvη = 0. This
means that v ∈ K, so requiring that v lies in G implies v = 0.
8 F. Schätz and M. Zambon
(ii) Assume that ΦG⊕K∗
(
β̄
)
is the graph of a 2-form. Then the rank of this 2-form equals
rank(η) iff β lies in the vector space Ω2
hor(M) of horizontal 2-forms.
Proof. (i) Applying the transformation t−Z ◦ t−η to ΦG⊕K∗
(
β̄
)
, by Lemma 3.6 we obtain
t−Z(ΦZ(β)) = graph(β). Applying it to TM we obtain
{(
v+Z]ιvη,−ιvη
)
| v ∈ TM
}
= K⊕G∗.
Hence applying the transformation to the intersection (3.1) we obtain
graph(β) ∩ (K ⊕G∗),
which is isomorphic to (3.2).
(ii) Denote by η′ the 2-form whose graph is ΦG⊕K∗
(
β̄
)
. The kernel of η′ is given by (3.1), and
the assertion follows immediately from (i). Recall that the vector space Ω2
hor(M) of horizontal
2-forms was defined in Section 2.1, as the space of 2-forms that vanish on ∧2K. �
Remark 3.8. Since t−η is actually an automorphism of the standard Courant algebroid TM ⊕
T ∗M , the following two deformation problems of Dirac structures are equivalent:
• deformations of graph(η), using the complement G⊕K∗,
• deformations of TM , 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 Section 3.4.
TM
graph(Z)
ΦZ(β)
graph(η)
G⊕K∗ ΦG⊕K∗
(
β̄
)
Figure 1. The Dirac structures graph(η) and TM , together with the complementary Lagrangian sub-
bundles we use to deform them.
3.3 Dirac-geometric interpretation of Section 2.2
Using Dirac linear algebra, we explain and re-prove the results recalled in Section 2.2.
3.3.1 Revisiting the map F from formula (2.1)
Let V be a finite-dimensional, real vector space. We fix a bivector Z ∈ ∧2V . Recall that IZ
consists of elements β ∈ ∧2V ∗ such that id +Z]β] is invertible. In formula (2.1), we defined the
map F : IZ → ∧2V ∗ given by
(F (β))] = β]
(
id + Z]β]
)−1
.
The following lemma provides a geometric explanation of the map F .
Deformations of Pre-Symplectic Structures: a Dirac Geometry Approach 9
Lemma 3.9. Fix Z ∈ ∧2V .
(i) Taking graphs with respect to the decompositions V = V ⊕ V ∗ resp. V = V ⊕ graph(Z),
yields bijections
Φ0 : ∧2 V ∗
∼=−→ {Lagrangian subspace of V transverse to V ∗},
α 7→ {(v, ιvα) | v ∈ V },
ΦZ : ∧2 V ∗
∼=−→ {Lagrangian subspace of V transverse to graph(Z)},
β 7→
{(
v + Z](ιvβ), ιvβ
)
| v ∈ V
}
.
(ii) Given β ∈ ∧2V ∗, the Lagrangian subspace ΦZ(β) is transverse to V ∗ ⊂ V if, and only if
β ∈ IZ .
(iii) The map
Φ−10 ◦ ΦZ : IZ → ∧2V ∗
is well-defined and coincides with F .
In particular, the map F is characterized by the property that
graph(F (β)) = ΦZ(β) (3.3)
for all β ∈ IZ . 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 3.2, any Lagrangian subspace transverse to V ∗ is the graph of
a skew-symmetric linear map V → V ∗, and therefore can be written as {(v, ιvα) | v ∈ V } for some
α ∈ ∧2V ∗. Similarly, graph(Z) is transverse to V and the induced isomorphism graph(Z) ∼= V ∗
is just (Z](ξ), ξ) 7→ ξ. Hence any Lagrangian subspace transverse to graph(Z) can be written
as {(v, 0) + (Z](ιvβ), ιvβ) | v ∈ V } for some β ∈ ∧2V ∗.
(ii) The expression for ΦZ(β) in item (i) shows that ΦZ(β) ∩ V ∗ = {(0, ιvβ) | v ∈ V, v +
Z](ιvβ) = 0}. 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, ΦZ(β) is transverse to V ∗ by item (ii). By item (i) the
element Φ−10 (ΦZ(β)) is well-defined. In concrete terms, it is given by α ∈ ∧2V ∗ such that for all
v ∈ V , there is w ∈ V for which(
v + Z]β](v), β](v)
)
=
(
w,α](w)
)
holds. Equivalently, this means that α]
(
id + Z]β]
)
(v) = β](v) for all v ∈ V . This shows that
Φ−10 ◦ ΦZ agrees with F . �
3.3.2 Revisiting Theorem 2.3 (parametrizing constant rank forms)
Now let η ∈ ∧2V ∗ be of rank k, fix a complement G to K := ker(η), and denote by Z ∈ ∧2G
the bivector determined by Z] = −
(
η|]G
)−1
. In Section 3.2 we considered deformations of the
Dirac structure graph(η) using G⊕K∗ as a complement. They are graphs of 2-forms given by
the Dirac exponential map expη (see Definition 2.2). More precisely:
Lemma 3.10. For all β ∈ IZ we have
graph(expη(β)) = ΦG⊕K∗
(
β̄
)
. (3.4)
10 F. Schätz and M. Zambon
Proof. We have graph(expη(β)) = tη(ΦZ(β)) = ΦG⊕K∗
(
β̄
)
, where the first equality holds by
equation (3.3) and the second by Lemma 3.6. �
Using this we recover Theorem 2.3, in particular item (i) stating that expη(β) has rank equal
to k = dim(K) iff β is horizontal.
Alternative proof of Theorem 2.3. (i) Apply Proposition 3.7(ii) together with equa-
tion (3.4).
(ii) We only prove the statement about the kernel of expη(β). Write β = (µ, σ). By the proof
of Proposition 3.7(i), the intersection of the subspace (3.4) with V is (tη◦tZ)(graph(β)∩(K⊕G∗)),
which is precisely the image of K under id + Z]µ].
(iii) By Lemma 3.9(ii), the map ΦZ provides a bijection between IZ and Lagrangian subspaces
transverse to graph(Z) and to V ∗. Hence tη◦ΦZ provides a bijection between IZ and Lagrangian
subspaces transverse to tη(graph(Z)) = G ⊕ K∗ (see Lemma 3.5) and to V ∗. The latter are
exactly the graphs of elements η′ ∈ ∧2V ∗ so that the η′|∧2G is non-degenerate. Hence, by
the proof of Lemma 3.10, expη provides a bijection between IZ and such η′. We conclude
using (i). �
3.4 Dirac-geometric interpretation of Section 2.3
Using Dirac geometry and adapting results from [2], we explain and re-prove the results recalled
in Section 2.3. Fix a bivector field Z on M .
3.4.1 Revisiting Proposition 2.7 (the L∞[1]-algebra Ω(M)[2])
In Proposition 2.7, the L∞[1]-algebra (Ω(M)[2], λ1, λ2, λ3) was constructed out of a bivector
field Z. It can be recovered using Dirac geometry – or more precisely, the deformation theory
of Dirac structures – as a special case of the construction from [2, Section 2.2].
Proposition 3.11. Let L be a Dirac structure and R a complementary almost Dirac structure,
i.e., we have a vector bundle decomposition L ⊕ R = TM . Then Γ(∧L∗)[2] has an induced
L∞[1]-algebra structure, whose only non-trivial multibrackets are µ1, µ2, µ3 given as follows:
1) µ1 is the differential dL associated to the Lie algebroid L,
2) µ2(α[2]� β[2]) = −(−1)|α|[α, β]L∗ [2], where [·, ·]L∗ := prR([[·, ·]]) denotes the (extension of)
the bracket of the almost Lie algebroid R ∼= L∗,
3) µ3(α[2]�β[2]�γ[2]) = (−1)|β|
(
α]∧β]∧γ]
)
ψ[2], where ψ ∈ Γ
(
∧3L
)
is given by Γ
(
∧3L∗
)
→
C∞(M), ξ1 ∧ ξ2 ∧ ξ3 7→ 〈prL([[ξ1, ξ2]]), ξ3〉, where we made use of the identification R ∼= L∗.
More generally, Proposition 3.11 holds if 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 2.7. Let Z be a bivector field on M . We apply Proposi-
tion 3.11 for the case L = TM and R = graph(Z). In this case dL 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 3.11 are µ1 = λ1, µ2 = −λ2 and µ3 = λ3. Applying the automorphism −id to
Ω(M)[2] yields Proposition 2.7. �
Deformations of Pre-Symplectic Structures: a Dirac Geometry Approach 11
3.4.2 Revisiting Corollary 2.9 (Maurer–Cartan elements of Ω(M)[2])
We now turn to Maurer–Cartan elements. In Lemma 3.9(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
β 7→ ΦZ(β) =
{(
v + Z](ιvβ), ιvβ
)
| v ∈ TM
}
.
We present the second part of [2, Lemma 2.6], which is an extension of the work by Liu–
Weinstein–Xu recalled in Remark 3.4.
Proposition 3.12. Let L be given a Dirac structure and R a complementary almost Dirac struc-
ture. An element σ ∈ Γ
(
∧2L∗
)
[2] is a Maurer–Cartan element of the L∞[1]-algebra structure
given in Proposition 3.11 iff the graph
Γσ := {(X − ιXσ) : X ∈ L} ⊂ L⊕R
is a Dirac structure. (The above inclusion makes use of the identification R ∼= L∗.)
Corollary 2.9 states that for β ∈ Ω2(M) taking values in some sufficiently small neighbor-
hood U of the zero section in ∧2T ∗M – in particular taking values in IZ , i.e., id + Z]β] is
invertible, – β is a Maurer–Cartan element of (Ω(M)[2], λ1, λ2, λ3) iff F (β) is closed. We now
provide an alternative proof of this result, which also shows that one can choose U to equal IZ .
Alternative proof of Corollary 2.9. For any β ∈ Ω2(M), being a Maurer–Cartan element
of the L∞[1]-algebra (Ω(M)[2], λ1, λ2, λ3) is equivalent to ΦZ(β) being a Dirac structure. This
follows from applying Proposition 3.12 to the Dirac structure L = TM and to the almost
Dirac structure R = graph(Z), noticing that Γ−β = {(v + Z](ιvβ), ιvβ) | v ∈ TM} = ΦZ(β).
When β ∈ Γ(IZ), we know that ΦZ(β) can be written as the graph of the 2-form F (β), by
equation (3.3). Now use the fact that the graph of a 2-form is a Dirac structure if, and only if,
the 2-form is closed. �
Remark 3.13. In this subsection we recovered the L∞[1]-algebra Ω(M)[2] of Proposition 2.7
as the L∞[1]-algebra governing deformations of the Dirac structure TM taking graph(Z) as
a complement. By Remark 3.8, 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.
3.5 Dirac-geometric interpretation of Section 2.4
Theorem 2.10 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.
Proposition 3.14. In the setting of Proposition 3.11, let K be a subbundle of L and define
Γhor(∧L∗) as the kernel of the restriction map Γ(∧L∗) → Γ(∧K∗). Then the multibrackets µ1,
µ2, µ3 preserve Γhor(∧L∗)[2] iff K satisfies the following:
• K is a Lie subalgebroid of L,
• 〈[[ξ1, ξ2]], K+K◦〉 = 0 for all ξ1, ξ2 ∈ Γ(K◦), where we use the identification K◦ ⊂ L∗ ∼= R
and [[·, ·]] denotes the Dorfman bracket.
12 F. Schätz and M. Zambon
Proof. We will use the fact that µ1, µ2, µ3 are derivations w.r.t. the wedge product in each
entry. The Lie algebroid differential dL preserves Γhor(∧L∗) iff the subbundle K is involutive.
The bracket [·, ·]L∗ preserves Γhor(∧L∗) iff 〈[[ξ1, ξ2]],K〉 = 0 for all ξ1, ξ2 ∈ Γ(K◦). The trinary
bracket µ3 preserves Γhor(∧L∗) iff µ3(ξ1, ξ2, ξ3) = 0 for all ξi ∈ Γ(K◦), which in turn is equivalent
to 〈[[ξ1, ξ2]], ξ3〉 = 0. �
Finally, as mentioned earlier, Theorem 2.11 follows immediately from the other results pre-
sented.
Acknowledgements
We thank Stephane Geudens for comments on a draft of this note. M.Z. acknowledges partial
support by IAP Dygest, the long term structural funding – Methusalem grant of the Flemish
Government, the FWO under EOS project G0H4518N, the FWO research project G083118N
(Belgium).
References
[1] Fiorenza D., Manetti M., Formality of Koszul brackets and deformations of holomorphic Poisson manifolds,
Homology Homotopy Appl. 14 (2012), 63–75, arXiv:1109.4309.
[2] Frégier Y., Zambon M., Simultaneous deformations and Poisson geometry, Compos. Math. 151 (2015),
1763–1790, arXiv:1202.2896.
[3] Gualtieri M., Matviichuk M., Scott G., Deformation of Dirac structures via L∞ algebras, Int. Math. Res.
Not., to appear, arXiv:1702.08837.
[4] Liu Z.-J., Weinstein A., Xu P., Manin triples for Lie bialgebroids, J. Differential Geom. 45 (1997), 547–574,
dg-ga/9508013.
[5] Manetti M., On some formality criteria for DG-Lie algebras, J. Algebra 438 (2015), 90–118, arXiv:1310.3048.
[6] Roytenberg D., Courant algebroids, derived brackets and even symplectic supermanifolds, Ph.D. Thesis,
University of California at Berkeley, math.DG/9910078.
[7] Schätz F., Zambon M., Deformations of pre-symplectic structures and the Koszul L∞-algebra, Int. Math.
Res. Not., to appear, arXiv:1703.00290.
[8] Voronov T., Higher derived brackets and homotopy algebras, J. Pure Appl. Algebra 202 (2005), 133–153,
math.QA/0304038.
[9] Voronov T., Higher derived brackets for arbitrary derivations, in Travaux mathématiques. Fasc. XVI, Trav.
Math., Vol. 16, University of Luxembourg, Luxembourg, 2005, 163–186, math.QA/0412202.
https://doi.org/10.4310/HHA.2012.v14.n2.a4
https://arxiv.org/abs/1109.4309
https://doi.org/10.1112/S0010437X15007277
https://arxiv.org/abs/1202.2896
https://doi.org/10.1093/imrn/rny134
https://doi.org/10.1093/imrn/rny134
https://arxiv.org/abs/1702.08837
https://doi.org/10.4310/jdg/1214459842
https://arxiv.org/abs/dg-ga/9508013
https://doi.org/10.1016/j.jalgebra.2015.04.029
https://arxiv.org/abs/1310.3048
https://arxiv.org/abs/math.DG/9910078
https://doi.org/10.1093/imrn/rny123
https://doi.org/10.1093/imrn/rny123
https://arxiv.org/abs/1703.00290
https://doi.org/10.1016/j.jpaa.2005.01.010
https://arxiv.org/abs/math.QA/0304038
https://arxiv.org/abs/math.QA/0412202
1 Introduction
2 Review: deformations of pre-symplectic structures
2.1 Pre-symplectic structures
2.2 A parametrization of constant rank 2-forms
2.3 An L-algebra associated to a bivector field
2.4 The Koszul L-algebra of a pre-symplectic manifold
3 Dirac geometric interpretation
3.1 Background on Dirac geometry
3.2 Deformations of pre-symplectic structures: the point of view of Dirac geometry
3.3 Dirac-geometric interpretation of Section 2.2
3.3.1 Revisiting the map F from formula (2.1)
3.3.2 Revisiting Theorem 2.3 (parametrizing constant rank forms)
3.4 Dirac-geometric interpretation of Section 2.3
3.4.1 Revisiting Proposition 2.7 (the L[1]-algebra (M)[2])
3.4.2 Revisiting Corollary 2.9 (Maurer–Cartan elements of (M)[2])
3.5 Dirac-geometric interpretation of Section 2.4
References
|
| id | nasplib_isofts_kiev_ua-123456789-209876 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T15:43:30Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Schätz, F. Zambon, M. 2025-11-28T09:38:48Z 2018 Deformations of Pre-Symplectic Structures: a Dirac Geometry Approach / F. Schätz, M. Zambon // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 9 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B70; 53D17; 58H15 arXiv: 1807.10148 https://nasplib.isofts.kiev.ua/handle/123456789/209876 https://doi.org/10.3842/SIGMA.2018.128 We explain the geometric origin of the L∞-algebra controlling deformations of pre-symplectic structures. We thank Stephane Geudens for comments on a draft of this note. M.Z. acknowledges partial support by IAP Dygest, the long-term structural funding - Methusalem grant of the Flemish Government, the FWO under EOS project G0H4518N, and the FWO research project G083118N (Belgium). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Deformations of Pre-Symplectic Structures: a Dirac Geometry Approach Article published earlier |
| spellingShingle | Deformations of Pre-Symplectic Structures: a Dirac Geometry Approach Schätz, F. Zambon, M. |
| title | Deformations of Pre-Symplectic Structures: a Dirac Geometry Approach |
| title_full | Deformations of Pre-Symplectic Structures: a Dirac Geometry Approach |
| title_fullStr | Deformations of Pre-Symplectic Structures: a Dirac Geometry Approach |
| title_full_unstemmed | Deformations of Pre-Symplectic Structures: a Dirac Geometry Approach |
| title_short | Deformations of Pre-Symplectic Structures: a Dirac Geometry Approach |
| title_sort | deformations of pre-symplectic structures: a dirac geometry approach |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209876 |
| work_keys_str_mv | AT schatzf deformationsofpresymplecticstructuresadiracgeometryapproach AT zambonm deformationsofpresymplecticstructuresadiracgeometryapproach |