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1 ASIAN J. MATH. c 2016 International Press Vol. 20, No. 3, pp , July DEFORMATIONS OF COISOTROPIC SUBMANIFOLDS IN LOCALLY CONFORMAL SYMPLECTIC MANIFOLDS HÔNG VÂN LÊ AND YONG-GEUN OH Abstract. In this paper, we study deformations of coisotropic submanifolds in a locally conformal symplectic manifold. Firstly, we derive two equivalent equations that govern C deformations of coisotropic submanifolds. Using the first equation we define the corresponding C -moduli space of coisotropic submanifolds modulo the Hamiltonian isotopies. Secondly, we prove that the formal deformation problem is governed by an L -structure which is a b-deformation of strong homotopy Lie algebroids introduced in [OP] in the symplectic context. Then we study deformations of locally conformal symplectic structures and their moduli space. Using the second equation we study the corresponding bulk (extended deformations of coisotropic submanifolds. Finally we revisit Zambon s obstructed infinitesimal deformation [Za] in this enlarged context and prove that it is still obstructed. Key words. Locally conformal symplectic manifold, coisotropic submanifold, b-twisted differential, bulk deformation, Zambon s example. AMS subject classifications. Primary 53D Introduction. Symplectic manifolds (M,ω have been of much interest in global study of Hamiltonian dynamics, and symplectic topology via analysis of pseudoholomorphic curves. In this regard closedness of the two-form ω plays an important role in relation to the dynamics of Hamiltonian diffeomorphisms and the global analysis of pseudoholomorphic curves. On the other hand when one takes the coordinate chart definition of symplectic manifolds and implements the covariance property of Hamilton s equation, there is no compulsory reason why one should require the twoform to be closed. Indeed in the point of view of canonical formalism in Hamiltonian mechanics and construction of the corresponding bulk physical space, it is more natural to require the locally defined canonical symplectic forms n ω α = dqi α dp α i to satisfy the cocycle condition i=1 ω α = λ βα ω β, λ βα const. (1.1 with λ γβ λ βα = λ γα as the gluing condition. (See introduction [V2] for a nice explanation on this point of view The corresponding bulk constructed in this way naturally becomes locally conformal symplectic manifolds (abbreviated as l.c.s manifolds whose definition we first recall. For the consistency of the definition, we will mostly assume dim M>2inthispaper. Definition 1.1. An l.c.s. manifold is a triple (X, ω, b whereb is a closed one-form and ω is a nondegenerate 2-form satisfying the relation dω + b ω =0. (1.2 Received June 22, 2013; accepted for publication March 31, Institute of Mathematics of ASCR, Zitna 25, Praha 1, Czech Republic (hvle@math.cas.cz. The first named author is partially supported by RVO: Center for Geometry and Physics, Institute for Basic Sciences (IBS, 77 Cheongam-ro, Namgu, Pohang, Korea, 37673; and Department of Mathematics, POSTECH, Pohang, Korea, (yongoh1@postech.ac.kr. The second named author is supported by IBS # IBS-R003-D1. 553

2 554 H. V. LÊ AND Y.-G. OH We refer to [V2], [HR], [Ba1], [Ba2] for more detailed discussion of general properties of l.c.s. manifolds and non-trivial examples. Locally by choosing b = dg for a local function g : U R on an open neighborhood U, (1.2 is equivalent to d(e g ω = 0 (1.3 and so the local geometry of l.c.s manifold is exactly the same as that of symplectic manifolds. In particular one can define the notion of Lagrangian submanifolds, isotropic submanifolds, and coisotropic submanifolds in the same way as in the symplectic case since the definitions require only nondegeneracy of the two-form ω. The main questions of our interest in this paper are following. Firstly, we like to know whether the global geometry of coisotropic submanifolds is any different from that of symplectic case. Secondly, we like to treat the problem of extended deformation of coisotropic submanifolds posed by Oh-Park [OP, p.355], i.e. both the symplectic form ω and the coisotropic submanifold are allowed to vary, in even a larger setting of l.c.s. manifolds. Note that the problem of extended deformations of coisotropic submanifolds of certain Poisson manifolds have been recently considered by Schaetz and Zambon [SZ], see also Remark We recall that by the results from [Za], [OP], deformation theory of coisotropic submanifolds in symplectic manifolds is generally obstructed. In particular, the set of coisotropic submanifolds with a given rank does not form a smooth Frechet manifold [Za], and the relevant (formal deformation theory thereof is described by an L - structure called strong homotopy Lie algebroids [OP]. In the present paper, we show that Oh-Park s deformation theory naturally extends to that of l.c.s. manifolds, once appropriate normal form theorem of canonical neighborhoods of coisotropic submanifolds (Theorem 4.2 and the theory of bulk-deformed strong homotopy Lie algebroids (sections 9, 11 are developed. For this purpose, we need to prove the l.c.s analog of Darboux-type theorem [We] and develop the l.c.s. analog to Moser s trick, for which usage of Novikov-type cohomology instead of the ordinary de-rham cohomology is essential. (See [HR] for relevant exposition of this cohomology theory. We derive two equivalent equations that govern C -deformations of coisotropic submanifolds (Theorems 6.2, 8.1 and develop a theory of bulk deformations of l.c.s. forms and of coisotropic submanifolds in this larger context of l.c.s. manifolds (sections 11, 12. Some more motivations of the present study are in order. First of all, we would like to see if the obstructed example of Zambon [Za] in the symplectic context is still obstructed in this enlarged deformations of coisotropic submanifolds together with bulk deformations of l.c.s. structures with replacement of closedness of ω by the Novikov-closedness of b-twisted differential. We then prove that Zambon s example still remains obstructed even under this enlarged setting of bulk deformations (Theorem Another source of motivation comes from the study of J-holomorphic curves in this enlarged bulk of l.c.s. manifolds. Again all the local theory of J-holomorphic curves go through without change. The only difference lies in the global geometry of J-holomorphic curves and it is not completely clear at this moment whether Novikovclosedness of l.c.s. structure (X, ω, b would give reasonably meaningful implication to the study of moduli problem of J-holomorphic curves in the context of closed strings or open strings attached to suitably physical D-branes. We refer to [KaOr] for some physical motivation of coisotropic D-branes and to [CF] for a generalization of study of deformations of coisotropic submanifolds in the Poisson context.

3 DEFORMATIONS OF COISOTROPIC SUBMANIFOLDS 555 We would like to thank Yoshihiro Ohnita for inviting us to the Pacific Rim Geometry Conference in 2011 where the first named author gave a talk on l.c.s. manifolds, which triggered our collaboration. We thank Marco Zambon for alerting their preprint [SZ] to us. We also thank Luca Vitagliano for a useful communication on the normalization theorem. 2. Locally conformal pre-symplectic manifolds. Suppose Y (X, ω X, b is a coisotropic submanifold. Then the restriction (Y,ω,b satisfies the same equation d b ω := dω + b ω = 0 (2.1 except that ω is no longer nondegenerate but has constant rank. This gives rise to the notion of locally conformal pre-symplectic manifolds, abbreviated as l.c.p-s. manifold. Definition 2.1. Atriple(Y,ω,b is called an l.c.p-s. manifold if b is a closed one-form and ω is a two-form with constant rank that satisfy d b ω := dω + b ω =0. (2.2 Remark 2.2. If the rank of ω is at least 4, then the wedge product with ω defines a linear injective map from Ω 1 (Y toω 3 (Y. Hence b is defined uniquely by the equation (2.2. If the rank of ω is 2, then the restriction of b to the null space TY ω of ω is defined by (2.2. The kernel of the wedge product Ω 1 (Y Ω 3 (Y, γ ω γ, is the two-dimensional annihilator T F of TY ω. In particular, if rank ω is 2 and (Y,ω,b is an l.c.p-s. manifold, then (Y,ω,b + b is also an l.c.p-s. manifold for any b T F such that db =0. From now on, we consider a general l.c.p-s. manifold (Y,ω,b. We next introduce morphisms between l.c.p-s. manifolds and automorphisms of (Y,ω,b generalizing those of l.c.s. manifolds (see [HR] for the corresponding definitions for the l.c.s. case. Definition 2.3. Let (Y,ω,b and(y,ω,b be two l.c.p-s. manifolds. A diffeomorphism φ : Y Y is called l.c.p-s. if there exists a C (Y,R \{0} such that φ ω =(1/aω, φ b = b + d(ln a. By setting a = e tu, it is easy to check that the following is the infinitesimal version of Definition 2.3. Definition 2.4. Let (Y,ω,b be a l.c.p-s. manifold. A vector field ξ on Y is called l.c.p-s. if there exists a function u C (Y such that L ξ ω = uω, L ξ b = du We denote by Diff(Y,ω,b the set of l.c.p-s. diffeomorphisms. Definition 2.5. We call any such function u C (Y that appears in Definition 2.4 is called an l.c.p-s. function. We denote by C (Y ; ω, b the set of l.c.p-s. functions. It is easy to see that C (Y ; ω, b is a vector subspace of C (X. We say an l.c.p-s. diffeomorphism (resp. vector field an l.c.s. diffeomorphism (resp. vector field, if (Y,ω,b is an l.c.s. manifold.

4 556 H. V. LÊ AND Y.-G. OH 3. Canonical neighborhoods of coisotropic submanifolds. In this section, we develop the l.c.s. analog to Gotay s coisotropic neighborhood theorem [Go] in the symplectic case. As in the symplectic case, we denote E =(TY ω the characteristic distribution on Y. The following lemma is one of the important ingredients that enables us to develop deformation theory of coisotropic submanifolds in l.c.s. manifolds in a way similar to the symplectic case as done in [OP]. Lemma 3.1. The distribution E on Y is integrable. Proof. This is an immediate consequence (2.2 which shows that the ideal generated by ω is a differential ideal. We call the corresponding foliation the null foliation on Y and denote it by F. We now consider the dual bundle π : E Y of E. The bundle TE Y where Y E is the zero section of E carries the canonical decomposition TE Y = TY E. In the standard notation in the foliation theory, E and E are denoted by T F and T F and called the tangent bundle (respectively cotangent bundle of the foliation F. Remark 3.2. When Y is a coisotropic submanifold of an l.c.s. manifold (X, ω X, b and(y,ω,b is the associated l.c.p-s. structure, then it is easy to check that the canonical isomorphism ω 0 : TX T X maps E = TY ω to the conormal N Y T X and so its adjoint ( ω 0 : TX T X induces an isomorphism between NY = TX/TY and E where E =(TY ω. We choose a splitting and denote by TY = G E, E =(TY ω, (3.1 p G : TY E the projection to E along G in the splitting (3.1. Using this splitting, we can write a conformal symplectic form on a neighborhood of the zero section Y E in the following way similarly as in the symplectic case. We have the bundle map TE Tπ TY pg E. (3.2 Let α E and ξ T α E. We define the one-form θ G on E by its value at each α E. Then the two form θ G, α (ξ := α(p G Tπ(ξ (3.3 ω G := π ω dθ G π b θ G (3.4

5 DEFORMATIONS OF COISOTROPIC SUBMANIFOLDS 557 is nondegenerate in a neighborhood U E of the zero section (See the coordinate expression (7.6 of dθ G and ω G. Later, we use ω G and ω U interchangeably depending on context. Then a straightforward computation proves Proposition 3.3. Then the pair (U, ω U, b U with b U := π b U defines an l.c.s. structure. Remark 3.4. If Y is Lagrangian, then E = TY, E Y = T Y, and hence p G = Id. For this special case, Proposition 3.3 is known as Example 3.1 in [HR], where θ G is the Liouville 1-form on T Y. In the next section, we will prove that this provides a general normal form of the l.c.s. neighborhood of the triple (Y,ω,b which depends only on (Y,ω,b andthe splitting (3.1, and that this normal form is unique up to diffeomorphism. We call the pair (U, ω U, b U a(canonical l.c.s. thickening of the l.c.p-s. manifold (Y,ω,b. 4. Normal form theorem of coisotropic submanifolds in l.c.s. manifold. Let Y be a compact coisotropic submanifold in a l.c.s. manifold (X, ω X, b. Denote by (Y,ω,b the induced l.c.p-s. structure given by ω = i ω X,b= i b, where i : Y X is the canonical embedding. We denote by (Ti the associated bundle map T X T Y. Consider the bundle π : E Y where E =(TY ω = T F as in the previous section. By Remark 3.2, the adjoint isomorphism induces an isomorphism ( ω 0 :TX T X ω X : NY = TX/TY E. (4.1 More precisely, we have the following lemma Lemma 4.1. The nondegenerate two-form ω X induces a canonical bundle isomorphism ω X : TX Y /T Y E given by (4.2. Proof. We define the bundle map ω XY : TX Y T Y, v (Ti (v ω X. Denote by j : T Y E the adjoint of j : E TY. Then E =kerω implies that TY ker(j ω XY. Hence j ω XY descends to a bundle map ω X : NY = TX Y /T Y E by setting [v] NY j ω XY (v. (4.2 The nondegeneracy of ω X implies that ω X induces a canonical bundle isomorphism. Using this isomorphism, we identify the pair (NY,Ywith(E,o E.

6 558 H. V. LÊ AND Y.-G. OH Now let g be a Riemannian metric on X. Theng gives rise to a splitting We also have a canonical isomorphism Combining (4.3 with Lemma 4.1 we get TX Y = TY NY. TE Y = TY E. (4.3 TE oe = To E E = TY NY = TX Y. Through this identification, we regard a neighborhood U 1 X of Y as a neighborhood of the zero section o E = Y E. For any open set U X we denote the restriction of ω X (resp. b tou also by ω X (resp. by b. In this section, we prove the following normal form theorem. Theorem 4.2 (Normal form. Assume Y is compact. There exist an open neighborhood U X of Y, a neighborhood V E of the zero section Y, and a l.c.s. diffeomorphism φ :(U, ω X, b (V,(ω G V,π b such that φ Y = Id and dφ NY = ω X under the identification (4.3. More specifically, φ satisfies φ (π ω dθ G π b θ G =e f ω X for some f C (U. Proof. Assume that U 1 be a neighborhood of Y in X which can be identified with a neighborhood W 1 of Y in NY via the exponential map Exp Y : NY U 1.Set ω 1 := Exp Y (ω X, b 1 := Exp Y (b. (4.4 Then (W 1,ω 1, b 1 is a l.c.s. manifold. Since the restriction of dexp Y identity, Y is also a coisotropic submanifold in (W 1,ω 1, b 1. Let i X : Y X be the inclusion. Set to Y is equal to V := ω X (W 1, b := i X b Ω1 (Y, ω 1 := ( ω 1 X (ω 1 Ω 2 (V, b1 := ( ω 1 X (b 1 Ω 1 (V. (4.5 Denote by i E : Y E the inclusion as the zero section and by Hb (Y the cohomology group ker d b /im d b. Lemma 4.3. The embedding i E : Y E induces an isomorphism between Hb (Y and H π b (E. In particular, there exists a one-form η Ω 1 (E such that ω 1 π (ω 1 Y =d π b (η. Proof. DenotebyS the following locally constant sheaf on Y U S(U :={f C (U, R d b U f =0}. It is known that H b (Y =H(Y,S, see e.g. [HR, Remark 1.10]. The first assertion of Lemma 4.3 follows from the homotopy invariant property of cohomology with values

7 DEFORMATIONS OF COISOTROPIC SUBMANIFOLDS 559 in locally constant sheaf. The second assertion of Lemma 4.3 is a consequence of the first assertion. Since H 1 (E, R =H 1 (Y,R there exists a function f C (E such that η = π (i η+df.thene f ω is an l.c.s. form on E with the Lee form b 1 df = π (b Y [L]. Now we apply Moser s deformation to the normal form. Set By (3.4 and (4.4 we have ω 0 := ω G V. ω 0 (y = ω 1 (y for all y Y. (4.6 Since H π b (V = H b1 (V, there exists a one form σ on V such that Set ω 1 ω 0 = d π b σ. ω t := ω 0 + td π b σ = π (ω 1 Y d π b (θ G tσ. By making V smaller if necessary, taking into account (4.6 and the compactness of Y, we assume that ω t are nondegenerate for all t [0, 1]. To prove Theorem 4.2, it suffices to solve the equation ψ t ( ω t=e ft(x ω 0 (4.7 for a family of diffeomorphism ψ t of V and a function f t with f 1 = f. Letξ t be the generating vector field of ψ t i.e. d dt ψ t = ξ t (ψ t,ψ 0 = Id. Differentiating (4.7, we obtain ( d ψt dt ω t + L ξt ω t = f t ω 0 which is equivalent to d dt ω t + L ξt ω t = f t ψ 1 t. But by definition of ω t and Cartan s formula, this becomes whichinturnbecomes In other words, we obtain d π b σ + ξ t d ω t + d(ξ t ω t =g t, g t (x = f t (ψ 1 t (x, g t = d π b σ ξ t (π b ω t +d π b (ξ t ω t π b (ξ t ω t. (g t π b(ξ t ω t = d π b (σ + ξ t ω t.

8 560 H. V. LÊ AND Y.-G. OH Using non-degeneracy of ω t,wefirstsolve for ξ t on V and then define g t by σ + ξ t ω t =0 g t = π b(ξ t for all (t, x [0, 1] V again shrinking V, if necessary. We denote by ψ t the flow of ξ t which then determines f t by f t = g t ψ t. This proves Theorem 4.2. Remark 4.4. A careful examination of the above proof in fact shows that the compactness hypothesis of Y is not necessary as in the case of Weinstein s normal form theorem in [We]. We thank L. Vitagliano for pointing this out. 5. Geometry of the null foliation of l.c.p-s. manifold. Let (Y,ω,b bean l.c.p-s. manifold of dimension n+k and denote by F the associated null foliation. Set 2n := dim X, n k := dim F, l := 2k. We now formulate a uniqueness statement in the symplectic thickening of (Y,ω (Proposition 5.1, extending an analogous result in [OP]. We also prove the existence of a transverse l.c.s. form (Proposition 5.2, which is important for later sections. Recalling that the l.c.s. form ω G of (3.4 depends on the choice of the splitting Π, in this section we re-denote by ω Π the l.c.s. form ω G associated to the splitting Π. Proposition 5.1. (cf. [OP, Proposition 5.1] For given two splittings Π, Π, there exist neighborhoods U, U of the zero section Y E and a diffeomorphism φ : U U andafunctionf : U R such that 1. φ ω Π = e f ω Π, 2. φ Y id, andtφ TY E id where T Y E is the restriction of TE to Y. Proof. SinceA E (TY is contractible, we can choose a smooth family {Π t } 0 t 1, Π 0 =Π, Π 1 =Π. Denoting ω t := ω Πt, applying the isomorphism H 1 b (Y = H 1 π b (E, we have From the definition, we have ω t ω 0 = d π b σ t. σ t TY E 0. for all 0 t 1. With these, we imitate the proof of Theorem 4.2 to finish off the proof. For the study of the deformation problem of l.c.p-s. structures it is crucial to understand the transverse geometry of the null foliation. First we note that the l.c.ps. form ω carries a natural transverse l.c.p-s. form. This defines the l.c.s. analog to the transverse symplectic form to the null foliation of pre-symplectic manifold. (See [Go], [OP], for example. Proposition 5.2. (cf. [OP, Proposition 5.2] Let F be the null foliation of the l.c.p-s. manifold (Y,ω,b. Then it defines a transverse l.c.s. form on F in the following sense:

9 DEFORMATIONS OF COISOTROPIC SUBMANIFOLDS ker(ω x =T x F for any x Y,and 2. L ξ ω = b(ξω for any vector field ξ on Y tangent to F. Proof. The first statement is trivial by definition of the null foliation and the second is an immediate consequence of the Cartan identity L ξ ω = d(ξ ω+ξ dω. The first term vanishes since X is tangent to the null foliation F. On the other hand, the second term becomes ξ dω = ξ (b ω = b(ξω + b (ξ ω = b(ξω which finishes the proof. One immediate consequence of the presence of the transverse l.c.p-s. form above is that any transverse section T of the foliation F carries a natural l.c.s. form: in any foliation coordinates, it follows from E =kerω =span{ q } α 1 α n k that we have π ω = 1 2 2k i>j 1 where ω ij = ω( y i, y j is skew-symmetric and invertible. The condition (2 above implies ω ij dy i dy j, (5.1 ω ij;β = b β ω ij (5.2 where b = j b jdy j + β b βdq β in the foliation coordinates (y 1,,y 2k,q 1, q n k. Note that this expression is independent of the choice of splitting as long as y 1,,y 2k are those coordinates that characterize the leaves of E by y 1 = c 1,,y l = c l, c i s constant. (5.3 By the closedness of the one-form b, (5.2 gives rise to the following proposition. Proposition 5.3. Let L be a leaf of the null foliation F on (Y,ω, λ apathin L, andlett and S be transverse sections of F with λ(0 T and λ(1 S. Then the holonomy map hol S,T (λ :(T,λ(0 (S, λ(1 defines the germ of a l.c.s. diffeomorphism. In particular, each transversal T to the null foliation carries a natural holonomy-invariant l.c.s. structure. 6. Master equation and equivalence relations; classical part. Let us recall the proof of the fact that a graph of a 1-form s Ω 1 (L is Lagrangian with respect to the canonical symplectic form on TL if and only if ds =0. Thisfactisadirect consequence of the following formula which is obtained by s (θ =s, s (θ,δq = θ, s (δq = s(π s δq =s(δq

10 562 H. V. LÊ AND Y.-G. OH where δq stands for the infinitesimal variation of q. Similarly we will derive the second equation for the graph Γ s of a section s : Y E = NY to be coisotropic with respect to ω G (Theorem 6.2. We also call the corresponding equation the classical part of the master equation (cf. Theorem 8.1. We will study the full (local moduli (with respect to different equivalence relations problem of coisotropic submanifolds by analyzing the condition that the graph of a section s : Y U in the symplectic thickening U is to be coisotropic with respect to ω G (Lemmas 6.9, Description of coisotropic Granssmannian. In this subsection, we recall some basic algebraic facts on the coisotropic subspace C (with real dimensions n + k where 0 k n inc n.wedenotebyc ω the ω-orthogonal complement of C in R 2n and by Γ k the set of coisotropic subspaces of (R 2n,ω. In other words, Γ k =Γ k (R 2n,ω=:{C Gr n+k (R 2n C ω C}. (6.1 From the definition, we have the canonical flag, 0 C ω C R 2n for any coisotropic subspace. We call (C, C ω acoisotropic pair. Combining this with the standard complex structure on R 2n = C n,wehavethesplitting C = H C C ω (6.2 where H C is the complex subspace of C. Next we give a parametrization of all the coisotropic subspaces near given C Γ k. Up to the unitary change of coordinates we may assume that C is the canonical model C = C k R n k. We denote the (Euclidean orthogonal complement of C by C = ir n k which is canonically isomorphic to (C ω via the isomorphism ω : C n (C n. Then any nearby subspace of dimension dim C that is transverse to C can be written as the graph of the linear map i.e., has the form Denote A = A H A I where A : C C = (C ω C A := {(x, Ax C C = R 2n x C}. (6.3 A H : H = C k C = (C ω, A I : C ω = R n k C = (C ω. Note that the symplectic form ω induce the canonical isomorphism ω H : C k (C k, ω I : R n k = C ω (C ω = C = ir n k. With this identification, the symplectic form ω has the form n k ω = π ω 0,k + dx i dy i, (6.4 i=1

11 DEFORMATIONS OF COISOTROPIC SUBMANIFOLDS 563 where ω 0,k is the standard symplectic form in C k, π : C n C k the projection, and (x 1,,x n k the standard coordinates of R n k and (y 1,,y n k its dual coordinates of (R n k. We also denote by π H :(C k C k the inverse of the above mentioned canonical isomorphism ω H. The following statements are fundamental in our work. Proposition 6.1. Let C A be as in ( The subspace C A is coisotropic if and only if A H and A I satisfies A I (A I + A H π H (A H =0. ( The subspace C A is coisotropic, if and only if ω k+1 CA =0. Proof. The first assertion of Proposition 6.1 is Proposition 2.2 in [OP]. The second assertion of Proposition 6.1 follows from the observation that C A is coisotropic, if and only the restriction ω CA is maximally degenerate, i.e. rank(ω CA =rankπ ω 0,k = k The equation for coisotropic sections. Note that the projection p G : TY E induces a bundle map p G : E T Y by p G (s,δq := s, p G(δq for any s E and δq T π(s Y. As before, assume that m =dimy =dime (n k =n + k. Theorem 6.2. The graph Γ s of a section s : Y (U E,ω G,π b is coisotropic if and only if the 2-form ω G (s :=ω Y d b (p G (s Ω2 (Y is maximally degenerate, i.e. (ω G (s k+1 =0. Proof. By Proposition 6.1, Γ s is coisotropic if and only if the restriction of ω G to Γ s is maximally degenerate, or equivalently (ω G k+1 Γ s =0. Sinceω G s = s (ω G we get By (3.4 we have (ω G k+1 Γs =0 (s (ω G Y k+1 =0. (6.6 s (ω G Y = s (π (ω Y d π b θ G =ω Y s (d π b θ G =ω Y d b (s θ G. (6.7 Using (3.3 we obtain for any y Y and any q T y Y s (θ G,δq y = θ G,s (q = s(p G π s δq =s(p G (δq. (6.8 It follows from (6.7 and (6.8 Theorem 6.2 follows immediately from (6.9. s (ω G Y =(ω Y d b (p G (s. ( (Pre-Hamiltonian equivalence and infinitesimal equivalence. In this section, we clarify the relation between the intrinsic pre-hamiltonian equivalence (resp. intrinsic l.c.p-s. equivalence between the l.c.p-s. structures (ω, b and the extrinsic Hamiltonian equivalence (resp. extrinsic l.c.s. equivalence between coisotropic embeddings in (U, ω G,π b. The intrinsic pre-hamiltonian equivalence is

12 564 H. V. LÊ AND Y.-G. OH provided by the pre-hamiltonian diffeomorphisms (Definition 6.5 on the l.c.p-s. manifold (Y,ω,b and the extrinsic ones by Hamiltonian deformations of its coisotropic embedding into (U, ω U,π b (Definition 6.8. The intrinsic l.c.p-s. equivalence is provided by l.c.p-s. diffeomorphisms and the extrinsic ones by l.c.s. deformations of its coisotropic embedding into (U, ω U,π b. The infinitesimal ((pre-hamiltonian equivalence is a natural infinitesimal version of the intrinsic/extrinsic ((pre-hamiltonian equivalence. First we shall prove Lemma 6.3. Avectorfieldξ on an l.c.p-s. manifold (Y,ω,b is l.c.p-s. if and only if on each connected component of Y d b (ξ ω =cω for some c R. (6.10 Proof. Assume that ξ is l.c.p-s. vector field on Y. To prove (6.10 we note that a l.c.p-s. vector field ξ on a l.c.p-s. manifold (Y,ω,b satisfies the following equation for some smooth function u C (Y (see Definition 2.4 By Definition (2.4 L ξ b = du, or equivalently L ξ ω = uω ξ dω + d(ξ ω = uω b(ξ ω + d b (ξ ω = uω d b (ξ ω =( u + b(ξ ω. (6.11 d(b(ξ u =0. Comparing this with (6.11 we obtain (6.10 immediately. Now assume that a vector field ξ on Y satisfies (6.10. Set u := b(ξ c. Then (6.11 holds. The above computations yield L ξ ω = uω. Since L ξ b = d(b(ξ = du, we conclude that ξ is l.c.p-s. This completes the proof of Lemma 6.3. We define a b-deformed Lie derivative as follows L b ξ (φ :=db i ξ + i ξ d b. (6.12 The following statements are direct consequences of Lemma 6.3, hence we omit their proof. Corollary 6.4. Let (Y,ω,b be an l.c.p-s. manifold. 1. Assume that [ω] 0 Hb 2 (Y,R. Then any l.c.p-s. vector field ξ on (Y,ω,b satisfies d b (ξ ω =0(the constant c in (6.10 is zero, equivalently, L b ξ (ω = Assume that ω = d b θ for some θ Ω 1 (Y. Then ξ ω = cθ + α for some α ker d b Ω 1 (Y. In this case L b ξ ω = cω.

13 DEFORMATIONS OF COISOTROPIC SUBMANIFOLDS 565 Lemma 6.3 motivates the following definition Definition 6.5. A vector field ξ on an l.c.p-s. manifold (Y,ω,b is called pre- Hamiltonian, if ξ ω = d b f for some f C (Y. A diffeomorphism φ is called pre-hamiltonian, if it is generated by a time-dependent pre-hamiltonian vector field. Remark If Y is an l.c.s. manifold, our definition of a pre-hamiltonian vector field coincides with the definition of a Hamiltonian vector field given by Vaisman [V2, (2.3]. For b = 0, our definition of a pre-hamiltonian vector field agrees with the definition in [OP, Definition 3.3]. 2. Clearly, any vector field ξ on Y tangent to F is pre-hamiltonian with the Hamiltonian f = 0. Using Lemma 6.3 we obtain again the second assertion of Proposition 5.2, noting that the corresponding constant c is zero. The following Theorem is an extension of Theorem 8.1 in [OP]. Theorem 6.7. Any l.c.p-s. (resp. pre-hamiltonian vector field ξ on an l.c.p-s. manifold (Y,ω,b can be extended to an l.c.s. (resp. Hamiltonian vector field on (U, ω G,π b. Proof. Let ξ be a l.c.p-s. (resp. pre-hamiltonian vector field on (Y,ω,b. We decompose ξ = ξ G + ξ E where ξ G G and ξ E is tangent to F. By Remark 6.6 (2, ξ E is pre-hamiltonian, hence it suffices to show that 1. ξ E extends to a Hamiltonian vector field on (U, ω G,π b; 2. ξ G extends to a l.c.s. (resp. Hamiltonian vector field on (U, ω G,π b. To prove (1, we define a Hamiltonian function f on (U E,ω G,π b as follows It is straightforward to check that f( α := α, ξ E. f Y =0=f, and (d b f Y =(d f Y. (6.13 Denote by (d π b f#ωg the associated Hamiltonian vector field on U. Using (6.13, (7.5 and the coordinate expression of ω G in (7.8, we obtain easily that ξ E (y =(d π b f#ωg (y for all y Y.Thisproves(1. Now we shall show (2. Since ω G F =0,wehave (ξ G ω G (y =(ξ G ω(y (6.14 for all y Y. Suppose [ω] =0 H 2 b (Y =H2 π b (U. Then ω G = d π b θ U for some 1-form θ U on Y.Since (d π b θ U Y = d b (θ U Y,

14 566 H. V. LÊ AND Y.-G. OH the one-form θ := (θ U Y (6.15 satisfies the condition in Corollary 6.4 (2. Using Corollary 6.4 (2, formulas (6.14, (6.15 and the non-degeneracy of ω U, we observe that the extendability of ξ G is equivalent to the extendability of the one-form α associated to ξ G as in Corollary 6.4 (2 to a one-form α U on U satisfying the following condition: d π b α U =0and(α U Y = α. (6.16 Set α U := π (α. Then α U satisfies (6.16. This proves Theorem 6.7 for the case [ω] =0 Hb 2 (Y. Now assume that [ω] 0 Hb 2(Y =H2 π b (U. By Corollary 6.4(1 db (ξ ω =0, or equivalently, ξ G ω = γ, whered b γ =0. Sinceω U is nondegenerate, using (6.14, we note that the required extendability of ξ G is equivalent to the extendability γ to aone-formγ U on U such that d π b γ U = 0. Clearly γ U := π (γ satisfies the required condition. This proves (2 and completes the proof of Theorem 6.7. Now we study the geometry of the master equation for coisotropic sections s E. By Proposition 6.1, the coisotropic condition for s is given by (ω d b (p G sk+1 =0 Ω 2k+2 (Y. (6.17 Abbreviate p G s as s G and p G (E asg.notethatg T Y is the annihilator of G. We rewrite the master equation (6.17 in the following form s G G, (ω d b s G k+1 =0. (6.18 Since p G E : E G is a bundle isomorphism, Y is also a coisotropic submanifold in (p G (U G, (p G 1 E (ω G. Abusing the notation, we abbreviate (p G 1 E (ω G asω G and p G (U asu. Clearly, the linearized equation of (6.18 at the zero section is ω k d b α =0. (6.19 Since ω k G 0andω k+1 = 0, the linearized equation of (6.17 is equivalent to the following equation d b F α = 0 (6.20 for a section α E.Here b denote the restriction of b to F. Definition 6.8. Two sections s 0,s 1 : Y U G are called Hamiltonian equivalent (resp. l.c.s. equivalent, if there exists a family of Hamiltonian diffeomorphisms (resp. l.c.s. diffeomorphisms ψ t of (U, ω G and a family of diffeomorphisms g t Diff(Y, t [0, 1], such that g 0 = Id Y, ψ 0 = Id U and s 1 = ψ 1 s 0 g 1. Two sections ξ 0,ξ 1 : Y U are called infinitesimally Hamiltonian equivalent (resp. infinitesimally l.c.s. equivalent, if ξ 0 ξ 1 is the vertical (fiber component of a Hamiltonian (resp. l.c.s. vector field on (U, ω G. Clearly, if s 0 and s 1 are (Hamiltonian equivalent sections, and s 0 is a coisotropic section, then s 1 is also a coisotropic section.

15 DEFORMATIONS OF COISOTROPIC SUBMANIFOLDS 567 Lemma 6.9. Two solutions of the linearized equation (6.20 are infinitesimally Hamiltonian equivalent if and only if they are cohomologous as elements in Ω 1 b (Y,ω. Consequently, the set of equivalence classes of the infinitesimally Hamiltonian equivalent solutions of the linearized equations is Hb 1(Y,ω. Proof. It suffices to prove Lemma 6.9 for the case where one of the two solutions is the zero section. For f C (U denoteby(d π b f #ωg the associated Hamiltonian vector field on U. We identify Y with the zero section of G U and for y Y we denote by Ty ver G the vertical (fiber component of T y G = T y Y G y. For any V T y G denote by V ver the vertical component of V. Now assume that a section ξ : Y U G is infinitesimally Hamiltonian equivalent to the zero section, i.e. there exists f C (U such that ξ =(d π b f ver #ω G. Abusing notation we denote by π the projection G Y. Using (7.5 and (7.8, we obtain for all y Y (d π b f ver #ω G (y =(d π b f π 1 (F ver f β dp β (y =d b F f(y, (6.21 where (d π b f π 1 (F ver fβ dp (y denotes the vertical (fiber component of the vector β in T y (π 1 (F = E(y E (y that is dual to the one-form d π b f π 1 (F(y with respect to the nondegenerate two-form β f β dp β(y Λ 2 Ty (π 1 (F. Hence [ξ] =0 Hb 1(Y,ω. Now assume that ξ = d b F f where f Ω 0 (U. By (6.21 ξ is infinitesimally Hamiltonian equivalent to the zero section. This proves the first assertion of Lemma 6.9. The second assertion is an immediate consequence of the first one and (6.20. This completes the proof of Lemma 6.9. Next, let us consider the case where ξ is infinitesimally l.c.s. equivalent to the zero section, i.e. there is a l.c.s. vector field ξ on U such that ξ(y isthevertical (fiber component of ξ(y for all y Y. 1. The case with [ω G ] 0 Hπ 2 b (U, R =H2 b (Y,R: Corollary 6.4 (1 implies that d b( ξ ω G = 0. The same argument as in the proof of Lemma 6.9 yields that for all y Y ξ(y =( ξ ω G F (y E (y. This leads to specify a subgroup H 1 b,ext (F of the group H 1 b (F whose elements are the restriction of d π b -closed one-forms on U. It is easy to see that H 1 b,ext (F =i (Hb 1 (Y, where i : F Y is the natural inclusion. 2. The case with ω G = d b θ U for some θ U Ω 1 (U: Clearly d b F (θ U F =0. Corollary 6.4 (2 implies that ( ξ ω G F = cθ U F + α F,whereα Ω 1 (U and d π b α = 0. Using the argument in the proof of Lemma 6.9 we get ξ(y =cθ U F (y+α F (y E (y. The discussion above immediately yields Lemma Denote Hb 1(Y,ω = H1 b (F. The set of the infinitesimal l.c.s. equivalence classes of the solutions ξ of the linearized equation (6.20 has one-one correspondence with 1. Hb 1(Y,ω/i (Hb 1(Y if [ω] 0in H2 b (Y and 2. Hb 1(Y,ω/(i (Hb 1(Y + θ F R if ω = d b θ.

16 568 H. V. LÊ AND Y.-G. OH 7. Geometry of the l.c.s. thickening of a l.c.p-s. manifold. In this section, imitating the scheme performed for the pre-symplectic manifolds in [OP], we introduce special coordinates in the l.c.s. thickening (U, ω U,π b of a l.c.p-s. manifold and we compute important geometric structures in these coordinates ((7.4, (7.8, (7.11, preparing for our study of deformations of compact coisotropic submanifolds in l.c.s. manifolds in the next two sections. Again we start with a splitting TY = G E, the associated bundle projection Π : TY TY, the associated canonical one form θ G, and the l.c.s. form on U E.Let ω U = π ω d π b θ G (7.1 (y 1,,y 2k,q 1,,q n k be coordinates on Y adapted to the null foliation on an open subset V Y. choosing the frame By {f 1,,f n k} of E that is dual to the frame { q,, } of E, we introduce the canonical coordinates on E by writing an element α E as a linear combination of {f1,,fn k 1 q n k } and taking α = p β f β, (y 1,,y 2k,q 1,,q n k,p 1,,p n k as the associated coordinates. For a given splitting Π : TY = G T F, there exists the unique splitting of TU that satisfies TU = G Tπ 1 (F (7.2 G =(T α π 1 (F ωu (7.3 for any α U, which is invariant under the action of l.c.s. diffeomorphisms on (U, ω U,π b that preserve the leaves of π 1 (F. Definition 7.1. We call the above unique splitting the leafwise l.c.s. connection of U Y compatible to the splitting Π:TY = G T F or simply a leafwise l.c.s. Π-connection of U Y. We would like to emphasize that this connection is not a vector bundle connection of E although U is a subset of E, which reflects nonlinearity of this connection. We refer the reader to subsection 13.1 for more detailed explanation. Note that the splitting Π naturally induces the splitting Π : T Y =(T F G.

17 DEFORMATIONS OF COISOTROPIC SUBMANIFOLDS 569 For the given splitting TY = G E we can write, as in [OP, (4.5], { } m l G x =span y i + Ri α q α α=1 1 i l for some Ri α s, which are uniquely determined by the splitting and the given coordinates. To derive the coordinate expression of θ G, we compute ( ( θ G y i = α p G Tπ( ( y i = α p G ( y i ( = p β fβ Ri α q α = p α Ri α, ( ( θ G q β = p β, θ G =0. p β Hence we derive Here we note that θ G = p β (dq β R β i dyi. (7.4 (dq β R β i dyi Gx 0. This shows that if we identify E = T F with G, then we may write the dual frame on T F as Motivated by this, we write and f β = dq β R β i dyi. (7.5 dθ G = dp β (dq β R β i dyi p β dr β i dyi = dp β (dq β R β i dyi (dq γ R γ R β j dyj i p β q γ dyi (R β i p β y j R β Rγ i j q γ dy j dy i (7.6 π b θ G =(b γ dq γ + b i dy i p δ (dq δ R δ jdy j. (7.7 Combining (7.6, (7.7 and (5.1, we derive ω U = 1 ( ω ij p β F β ij dy i dy j 2 ( (dp ν + p ν (b γ dq γ + b i dy i R β i +p β q ν dy i (dq ν Rj ν dyj = 1 ( ω ij p β F β ij dy i dy j 2 ( ( ( dp ν + p ν b γ dq γ R γ i dyi + p ν (b i + b γ R γ i +p R β i β q ν dy i (dq ν R ν j dyj (7.8

18 570 H. V. LÊ AND Y.-G. OH similarly as in the derivation of [OP, (6.8], where F β ij are the components of the transverse Π-curvature of the null-foliation given by (13.5 in the Appendix. Note that we have { Tπ 1 (F =span q 1,, q n k, p 1,, } p n k which is independent of the choice of the above induced foliation coordinates of TU. Now we compute G =(Tπ 1 (F ωu in TU in terms of these induced foliation coordinates. We will determine when the expression a j ( y j + Rα j q α +dβ q β + c γ p γ satisfies ω U ( a j ( y j + Rα j q α +dβ q β + c γ,tπ 1 (F =0. p γ It is immediate to see by pairing with Next we study the equation p μ d β =0, β =1,,n k. (7.9 0=ω U ( a j ( y j + Rα j q α +c γ, p γ q ν for all ν =1,,n k. A straightforward check provides a (p j ν (b i + b γ R γ i +p R β i β q ν + c ν = 0 (7.10 for all ν and j. Combining (7.9 and (7.10, we have obtained (Tπ 1 (F ωu { =span y j + Rα j ( q α p ν (b i + b γ R γ i +p R β i β q ν } p ν 1 j 2k. (7.11 Remark 7.2. Just as we have been considering Π : TY = G T F as a connection over the leaf space, we may consider the splitting Π : TU = G T (π 1 Fas the leaf space connection canonically induced from Π under the fiber-preserving map π : U Y over the same leaf space Y/ : Note that the space of leaves of F and π 1 F are canonically homeomorphic.

19 DEFORMATIONS OF COISOTROPIC SUBMANIFOLDS Master equation in coordinates. We will derive the first equation for the graph of a section s : Y E = NY to be coisotropic with respect to ω U (Theorem 8.1, which is a natural extension of a similar equation in the symplectic setting obtained in [OP]. We call the corresponding equation the classical part of the master equation for the deformation theory of coisotropic submanifolds. Recall that an Ehresmann connection of U Y with a structure group H is a splitting of the exact sequence 0 VTU TU Tπ TY 0 that is invariant under the action of the group H. HereH is not necessarily a finite dimensional Lie group. In other words, an Ehresmann connection is a choice of decomposition TU = HTU VTU that is invariant under the fiberwise action of H. Recalling that there is a canonical identification V α TU = V α TE = Eπ( α, a connection can be described as a horizontal lifting HT α U of TY to TU at each point y Y and α U E with π( α =y. We denote by F # HTU the horizontal lifting of a subbundle F TY in general. Let (y 1,,y 2k,q 1,,q n k be foliation coordinates of F on Y and (y 1,,y 2k,q 1,,q n k,p 1,,p n k be the induced foliation coordinates of π 1 (F onu. ThenG # =(Tπ 1 (F ωu has the natural basis given by e j = y j + Rα j ( q α p ν (b i + b γ R γ i +p R β i β q ν which are basic vector fields of T (π 1 F. We also denote We define a local lifting of E f α = q α. The lifting (8.2 of E provides a local splitting p ν (8.1 } E =span {f 1,,f n k. (8.2 TU =(G E VTU TY and defines a locally defined Ehresmann connection on the bundle U Y.Fromthe expression (7.8 of ω U, it follows that G E is a coisotropic lifting of TY to TU. We denote by Π v : TU VTU the vertical projection with respect to this splitting. With this preparation, we are finally ready to derive the master equation. Let s : Y U E be a section and denote s := Π v ds (8.3

20 572 H. V. LÊ AND Y.-G. OH its locally defined covariant derivative. In coordinates (y 1,,y 2k,q 1,,q n k, we have ( ds y j = y j + s α y j p α ( = e j Rj α q α + s ν y j + s ν(b i + b γ R γ i +s R β i β q ν. p ν Therefore we have derived ( s y j = ( s ν y j + s ν(b i + b γ R γ i +s R β i β q ν p ν. (8.4 Similarly we compute and so ( ds q ν = q ν + s α q ν p α = q ν + s α q ν, p α ( s q ν = s α q ν. (8.5 p α Recalling that T α U =(E # α VT αu ωu E # α VT αu, we conclude that the graph of ds with respect to the frame { e 1,,e 2k,f 1,,f n k, canbeexpressedbythelinearmap where ( s y i ( s q β Finally we note that p 1,, } p n k A H :(E # VTU ωu VTU = E ; (A H i α = is α, A I : E # VTU = E ; (A I β α = β s α, =( i s α q α, =( β s α q α, is α := s α y j + s α(b i + b γ R γ i +s R β i β q α, βs α := s α q β. (8.6 ω U (s(e i,e j =w ij s β F β ij := ω ij and denote its inverse by ( ω ij. Note that ( ω ij isinvertibleifs β is sufficiently small, i.e., if the section s is C 0 -close to the zero section, or its image stays inside of U. Now Proposition 2.2 immediately implies Theorem 8.1. Let s be the vertical projection of ds as in (8.4. Then the graph of the section s : Y U is coisotropic with respect to ω U if and only if s satisfies i s α ω ij j s β = β s α α s β (8.7

21 DEFORMATIONS OF COISOTROPIC SUBMANIFOLDS 573 for all α>βor 1 2 ( is α ω ij j s β fα fβ =( β s α fα fβ (8.8 where fα is the dual frame of { q,, 1 q } defined by (7.5. n k Note that (8.8 involves terms of all order of s β because the matrix ( ω ij isthe inverse of the matrix ω ij = ω ij s β F β ij. There is a special case where the curvature vanishes i.e., satisfies F G = F β ij q β dyi dy j = 0 (8.9 in addition to (5.3. In this case, ω ij = ω ij which depends only on y i s and so does ω ij. Therefore (8.8 is reduced to the quadratic equation 1 2 ( is α ω ij j s β fα fβ =( β s α fα fβ. ( Deformation of strong homotopy Lie algebroids. In this section, we provide an invariant description of the master equation we have derived in the previous section. This can be regarded as the equation for the coisotropic submanifolds in the formal power series version of the equation or in the formal manifold in the sense of Kontsevich [K], [AKSZ] b-deformed Oh-Park s strong homotopy Lie algebroids [OP]. We start with the normal form (7.1 of the symplectic thickening. We also note that the discussion of leaf space connection and the curvature, in particular the one-form θ G does not depend on the closed one-form b but only depends on the conformal presymplectic form ω and the splitting TY = G E only. In this regard, we can view the normal form in (7.1 as a deformation of the nondegenerate two form π ω dθ G to a conformal symplectic form relative to π b. So from now on, we denote ω U = π ω dθ G and ω b U = ω U d π b θ G. (9.1 This deformation is responsible for the appearance of b-terms in (7.8 and then (7.11, (8.1 and eventually for the covariant derivative (8.4. Again we regard (8.6 as the deformation of the old covariant derivative formula appearing in [OP, (7.3] and denote the full covariant derivative b s = s + b F s + b R s (9.2 where b F is the restriction to the null-foliation of the one-form b and b R is the pairing of b and R which produces a one-form on G with values in T F. Now we give a deformed version of the notion of strong homotopy Lie algebroid introduced in [OP].

22 574 H. V. LÊ AND Y.-G. OH Definition 9.1. Let E Y be a Lie algebroid. A b-deformed L -structure over the Lie algebroid is a structure of strong homotopy Lie algebra (l[1], m onthe associated b-deformed E-de Rham complex l =Ω (E =Γ(Λ (E such that m 1 is the E-differential E d b induced by the (deformed Lie algebroid structure on E as described in subsection We call the pair (E Y,m ab-deformed strong homotopy Lie algebroid. Here we refer to [NT] or Appendix 13.2 for the definition of E-differential used in this definition. With this definition of a b-deformed strong homotopy Lie algebroid, we will show that for given l.c.p-s. manifold (Y,ω,b each splitting Π : TY = G T F induces a canonical L -structure over the Lie algebroid T F Y. The following linear map and quadratic map are introduced in [OP] which play crucial roles in the construction of L -structure on the foliation de Rham complex: a linear map a quadratic map ω :Ω 1 (Y ;Λ E Γ(Λ +1 E =Ω +1 (F, (9.3, ω :Ω 1 (Y ;Λ l1 E Ω 1 (Y ;Λ l2 E Ω l1+l2 (F, (9.4 and the third map that is induced by the transverse Π-curvature, whose definition is now in order. We recall the definitions of those maps. The linear map ω is defined by ω(a :=(A E skew. (9.5 Here note that an element A Ω 1 (Y ;Λ k E is a section of T Y Λ k E. Restricting A to E for the first factor we get A E E Λ k E. Then (A E skew is the skewsymmetrization of A E. The quadratic map is defined by A, B ω := A π B B π A where π is the transverse Poisson bi-vector on N F associated to the transverse symplectic form ω on NF. We will denote F # := Fω 1 = F αj i dy i ( y j + Rβ j q β q α Γ(G G E, (9.6 where F αj i = Fik αωkj. NotethatwecanidentifyΓ(G G E with Γ(N F NF E via the isomorphism π G : G NF. For given ξ Ω l (F, we define and deformed bracket d b F (ξ :=( b ξ E skew, (9.7 {ξ 1,ξ 2 } b Π := b ξ 1, b ξ 2 ω = i<j ω ij ( b i ξ 1 ( b j ξ 2. (9.8 Here the map in (9.7 is nothing but the b-deformed leafwise differential of the null foliation which is indeed independent of the choice of splitting Π : TY = G T F

23 DEFORMATIONS OF COISOTROPIC SUBMANIFOLDS 575 but depends only on the foliation and the projection of the one-form b to F, seealso subsection By Remark 2.2 the obtained leafwise differential depends only on ω. We use d b F and d b F interchangeably. The second is a bracket in the transverse direction which is a b-deformation of the one given in [OP, (9.13]. Now we promote the maps d b F and {, }b to an infinite family of graded multilinear maps m b l =(Ω[1] (F l Ω[1] (F (9.9 so that the structure n k l[1] j ; {m b l} 1 l< j=0 defines a strong homotopy Lie algebroid on E = T F Y in the above sense. Here Ω[1] (F is the shifted complex of Ω (F, i.e., Ω[1] k (F =Ω k+1 (F andm 1 is defined by and m 2 is given by m b 1(ξ =( 1 ξ d b F (ξ m b 2 (ξ 1,ξ 2 =( 1 ξ1 ( ξ2 +1 {ξ 1,ξ 2 } b Π. On the un-shifted group l, d b F defines a differential of degree 1 and {, } ω is a graded bracket of degree 0 and m b l is a map of degree 2 l. We now define m b l for l 3. Here enters the transverse Π-curvature F = F Π of the splitting Π of the null foliation F. We define m b l(ξ 1,,ξ l := σ S l ( 1 σ b ξ σ(1, (F # ξ σ(2 (F # ξ σ(l 1 b ξ σ(l ω (9.10 where σ is the standard Koszul sign in the suspended complex. We have now arrived at our definition of strong homotopy Lie algebroid associated to the coisotropic submanifolds, which is a b-deformation of the one introduced in [OP, section 9], but which is applied after enlarging our category to that of locally conformal pre-symplectic two forms instead of pre-symplectic two forms. Theorem 9.2. Let (Y,ω,b be a l.c.p.s. manifold and Π:TY = G T F be a splitting. Then Π canonically induces a structure of strong homotopy Lie algebroid on T F in that the graded complex ( Ω[1] (F, {m b l } 1 l< defines the structure of strong homotopy Lie algebra. We denote by l (Y,ω,b;Π the corresponding strong homotopy Lie algebra. Proof. The proof of this theorem follows the strategy used in the proof of Theorem 9.4 [OP], which uses the formalism of super-manifolds and odd symplectic structure on the super tangent bundle T [1]U [AKSZ] of the l.c.p.s. thickening U of (Y,ω,b.

24 576 H. V. LÊ AND Y.-G. OH We changethe parity oft U alongthe fiber anddenote by T [1]U the corresponding super tangent bundle of U. One considers a multi-vector field on U as a (fiberwise polynomial function on T [1]U. For example, the bi-vector field P, inverse to the nondegenerate form ω U (cf. (9.1, defines a quadratic function, which we denote by H. This also coincides with the push-forward of the even function H : T [1]U R induced by ω U. Wedenoteby{, } Ω the (super-poisson bracket associated to the odd symplectic form Ω on T [1]U. Then the bracket operation Q := {H, } Ω defines a derivation on the set O T [1]U of functions on T [1]U: Here O T [1]U is the set of differential forms on U considered as fiberwise polynomial functions on T [1]U. We refer to [Gz] or [OP, Appendix] for the precise mathematical meaning for this correspondence. Therefore it defines an odd vector field. Restricting ourselves to a Darboux neighborhood of L = T F[1] T [1]U, we identify the neighborhood with a neighborhood of the zero section T [1]L. Usingthe fact that (9.14 depends only on ξ, not on the extension, we will make a convenient choice of coordinates to write H in the Darboux neighborhood and describe how the derivation Q = {H, } Ω acts on Ω (F in the canonical coordinates of T [1]L. In this way, we can apply the canonical quantization which provides a canonical correspondence between functions on the phase space T [1]L and the corresponding operators acting on the functions on the configuration space L, later when we find out how the deformed differential δ b (9.15 acts on Ω (F. We denote by (y i,q α,p α,yi,q α,p α the canonical coordinates T L associated with the coordinates (y i,q α,p α ofn F. Note that these coordinates are nothing but the canonical coordinates of N Y T U pulled-back to T F TU and its Darboux neighborhood, with the corresponding parity change: We denote the (super canonical coordinates of T [1]L associated with (y i,q α p α by ( y i, q α p α yi,. q α p α Here we note that the degree of y i,q α and p α are 0 while their anti-fields, i.e., those with in them have degree 1. And we want to emphasize that L is given by the equation y i = p α = p α = 0 (9.11 and (y i,y i, (p α,q α and(pα,qα are conjugate variables. In terms of these coordinates, [OP, (9.23] provides the formula Here, we define y # i y # i H = 1 2 ωij y # i y# j + pδ q δ. (9.12 to be ( := yi + Ri δ p δ p ν (b i + b γ R γ i +p R β i β q δ q δ arising from (8.1 similarly as in [OP, (9.23]. When ω is a closed symplectic form as in [OP], we have {H,H } = 0. However in the current l.c.s. case, this is no longer the case.