The frame bundle picture of Gaussian wave packet dynamics in semiclassical mechanics

Recently Ohsawa (Lett Math Phys 105:1301–1320, 2015) has studied the Marsden–Weinstein–Meyer quotient of the manifold T∗Rn×T∗R2n2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^*\mathbb {R}^n\times T^*\mathbb {R}^{2n^2}$$\end{document} under a O(2n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {O}(2n)$$\end{document}-symmetry and has used this quotient to describe the relationship between two different parametrisations of Gaussian wave packet dynamics commonly used in semiclassical mechanics. In this paper, we suggest a new interpretation of (a subset of) the unreduced space as being the frame bundle F(T∗Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {F}}(T^*\mathbb {R}^n)$$\end{document} of T∗Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^*\mathbb {R}^n$$\end{document}. We outline some advantages of this interpretation and explain how it can be extended to more general symplectic manifolds using the notion of the diagonal lift of a symplectic form due to Cordero and de León (Rend Circ Mat Palermo 32:236–271, 1983).


Motivation
The Gaussian wave packet ansatz is frequently used in the study of the time-dependent Schrödinger equation and its semiclassical limit. In particular, the wave function parametrised by (q, p) ∈ T * R n , φ, δ ∈ R and A + iB ∈ Σ n = {W ∈ M n (C) | W = W , Im W > 0}, is well known to be an exact solution of the Schrödinger equation for quadratic Hamiltonians, provided the parameters satisfy certain ODEs [6]. Faou and Lubich [4,9] have shown that these parameter ODEs constitute a Hamiltonian system, and recently, Ohsawa and Leok [13] have clarified the symplectic structure underlying these Hamiltonian dynamics. Their main observation is that Σ n is a symplectic manifold, the well-studied Siegel upper half plane [17].
In [5], Hagedorn introduced another parametrisation of Gaussian wave functions, by replacing A + iB ∈ Σ n by P Q −1 , where Q, P ∈ M n (C) satisfy certain algebraic relations. In this parametrisation, the ODEs governing the parameter evolution are somewhat simpler. In [12], Ohsawa explained how to interpret the relation between Q, P and A + iB as an instance of symplectic reduction on the symplectic manifold T * R n × T * R n 2 under a certain right O(2n)-action. The matrices Q, P are seen to be coordinates on a level set of the momentum map corresponding to the O(2n)-action, and symplectic reduction at this momentum level gives the Siegel upper half plane Σ n . The dynamics in the unreduced and reduced spaces are then related by the quotient map in the usual way, explaining how the Hagedorn ODEs can also be interpreted as a Hamiltonian system.

Main results and outline
The purpose of the present paper is to reinterpret Ohsawa's results and to suggest some possible extensions. Specifically, we interpret an open subset of Ohsawa's unreduced space T * R n × T * R 2n 2 as the frame bundle F(T * R n ) T * R n × GL(2n, R) of T * R n and give an intrinsic description of the symplectic structure on F(T * R n ). There are several advantages to this picture: -there exists a dual pair structure on GL(2n, R), generated by left multiplication by Sp(2n, R) and right multiplication by O(2n) . The existence of this dual pair allows one to realise the symplectic reduced space of F(T * R n ) at all values of O(2n)-momentum in terms of adjoint orbits of sp(2n, R). An understanding of all reduced spaces will be important in any attempt to deduce the semiclassical theory described here using ideas from geometric quantisation. 1 -the physical picture of moving frames on T * R n is intuitive and gives insight into the definition of the Hamiltonian on the unreduced space. Through it, we see that the dynamics of the semiclassical wave function are determined by the underlying classical dynamics in two ways: The variables z := (q, p) ∈ T * R n in the semiclassical wave function evolve according to an underlying classical Hamiltonian H on T * R n , modified by an order correction for non-quadratic Hamiltonians.
Meanwhile, the variables E := Re Q Im Q Re P Im P ∈ GL(2n, R) evolve according to the linearisation of the Hamiltonian flow due to H about z. This view of the E-dynamics as linearised symplectic flow depends crucially on the interpretation of E as parametrising the possible frames at z. -the picture described in the previous point also has overlap with other work on semiclassical mechanics, in particular the nearby orbit approximation of Littlejohn [7]. -the frame bundle picture allows a natural interpretation of the conserved quantity associated with the O(2n)-invariance of the theory as simply the collection of symplectic products . . E 2n ∈ GL(2n, R). In particular, reduction at the particular value of momentum considered in this paper leads to conservation of symplectic frame. -the frame bundle picture can be extended to more general symplectic manifolds than T * R n , using the results of Cordero et al. [3] on the lifting of symplectic structures to frame bundles.
We now outline the structure of the paper. In Sect. 2, we describe a symplectic structure on the set M 2n (R) of 2n × 2n-dimensional real matrices, demonstrate that the obvious left Sp(2n, R)and right O(2n)-actions are Hamiltonian, and calculate the corresponding momentum maps and their Lie algebra-valued counterparts. In Sect. 3, we review some properties of dual pairs that will be needed in the sequel. In Sect. 4, we restrict our symplectic manifold to GL(2n, R) ⊂ M 2n (R) and demonstrate that the two group actions give a dual pair structure on this restricted space. In Sect. 5, we discuss reduction of GL(2n, R) under the O(2n)-action at a particular value of momentum, use the results of Sect. 3 to describe the reduced space as an adjoint orbit in sp(2n, R), and give a geometric description of this adjoint orbit as the set of ω R 2n -compatible complex structures, where ω R 2n is the canonical symplectic structure on R 2n . In Sect. 6, we use the standard isomorphism of ω R 2n -compatible complex structures with the positive Lagrangian Grassmannian to introduce coordinates on the adjoint orbit, and demonstrate that in these coordinates the symplectic projection and reduced symplectic form agree with those obtained by Ohsawa. In Sect. 7, starting with the cotangent symplectic structure ω = n α=1 dq α ∧ dp α and an arbitrary Hamiltonian H on T * R n , we describe the -diagonal symplectic structure Ω and -lifted Hamiltonian H on F(T * R n ), which agree with those defined by Ohsawa on their common domain. We give a geometric interpretation of H , demonstrate that for quadratic H one obtains the dynamics of Hagedorn, and emphasise that the O(2n)-invariance of H leads to conservation of symplectic frame. We also describe the dynamics on the reduced space and show it reproduces those of Heller. Finally in Sect. 8, we explain how the results of Cordero et al. [3] can be used to extend the construction of a symplectic structure on the frame bundle to arbitrary symplectic manifolds.

The symplectic geometry of M 2n (R)
In this section, we outline the symplectic structure on M 2n (R), describe two natural symplectic group actions on M 2n (R) and calculate the momentum maps associated with these actions.

The symplectic form on M 2n (R)
Take R 2n with its canonical symplectic structure where J := 0 n I n −I n 0 n . The 2n-fold direct sum of (R 2n , ω R 2n ) is naturally isomorphic to the space M 2n (R) of real 2n × 2n matrices with the symplectic form where E a denotes the ath column of E ∈ M 2n (R), considered as a vector in R 2n . We can consider (Ω, M 2n (R)) as a symplectic manifold, using the canonical isomorphism T E M 2n (R) M 2n (R), and we denote the induced symplectic form by Ω also.

Commuting symplectic actions
it is straightforward to check that L and R define left and right symplectic actions, respectively, i.e. for any E ∈ M 2n (R), The momentum maps corresponding to these two symplectic actions have a standard form.
Proof Both results follow from the general expression for the momentum map of a linear symplectic action on a symplectic vector space-see, for example, [11,Section 12.4, Example (a)].

Remark 2.2
Both momentum maps are easily seen to be equivariant, (again, a general result for linear symplectic actions).

Lie algebra-valued momentum maps
For any (real) Lie subalgebra g ⊂ gl(N , C), consider the trace form ·, · : g×g → C defined by If g is invariant under conjugate transpose, then ·, · is non-degenerate, as is its real part, since In particular, it is non-degenerate and real-valued on sp(2n, R) and o(2n). Using the trace form to identify Lie algebras with their duals, we can write down Lie algebravalued versions of the momentum maps discussed in the previous section.

Remark 2.4
Again, the Lie algebra-valued momentum maps are equivariant and

Mutually transitive actions and dual pairs
In Sect. 4, we will demonstrate that the left Sp(2n, R)-action and right O(2n)-action define a dual pair structure on a suitable subset of M 2n (R). In anticipation, we here introduce the notion of mutually transitive actions. We then explain how mutual transitivity allows us to view reduced spaces of one action as coadjoint orbits of the other. Finally, we indicate the relation of mutual transitivity to the more standard notion of a dual pair. A fuller treatment of dual pairs and related concepts can be found in [8,Section IV.7], [14,Chapter 11], and [2]. We also refer to [18] for a more thorough discussion of mutual transitivity.

Mutually transitive actions
Let (N , Ω) be a symplectic manifold, and let Φ 1 : G 1 × N → N and Φ 2 : G 2 × N → N be symplectic actions. We assume N , G 1 and G 2 are all finite-dimensional. Definition 3. 1 We say the actions Φ 1 , Φ 2 are mutually transitive if the following three properties hold: There exists a one-to-one correspondence between coadjoint orbits in J 1 (N ) and J 2 (N ) given by

The relation between coadjoint orbits and reduced spaces
From now on, we assume that all group actions Φ i are both free and proper (see [18] for a discussion when these conditions do not hold). For μ 1 ∈ g * 1 , let (G 1 ) μ 1 denote the coadjoint stabiliser of μ 1 and N μ 1 the quotient space J −1 1 (μ 1 )/(G 1 ) μ 1 at μ 1 . It is well known that N μ 1 may be given a smooth structure making σ 1 : J −1 1 (μ 1 ) → N μ 1 a submersion and a symplectic structure Ω N μ 1 satisfying For mutually transitive actions, it turns out that the reduced space under one action is symplectomorphic to a coadjoint orbit of the other action: Then any reduced space under the G 1 -action is symplectomorphic to a coadjoint orbit in J 2 (N ) ⊂ g * 2 and similarly with 1 and 2 interchanged. Explicitly, for x ∈ N , via a resp. G 2 -and G 1 -equivariant symplectomorphism.
In fact, the symplectomorphism from N J 1 (x) to O J 2 (x) has a simple interpretation: Since the G 1 -and G 2 -actions on N commute, the G 2 -action drops to the reduced space N J 1 (x) . The resulting momentum mapJ 2 : and can be shown to be a symplectomorphism (in particular,J 2 is G 2 -equivariant).

The relation to dual pairs
In this subsection, we make contact with the notion of dual pair (in the sense of Weinstein [19]).

Definition 3.5 ([1, Appendix E])
Let N be a symplectic manifold and P 1 , P 2 the Poisson manifolds. A pair of Poisson maps is called a full dual pair if J 1 , J 2 are submersions, and Proposition 3.6 If Φ 1 , Φ 2 are mutually transitive and free, then their momentum maps J 1 , J 2 form a full dual pair.
A standard result says that the former equals (ker T x J 1 ) Ω . Also freeness of Φ 1 , Φ 2 implies that J 1 , J 2 are submersions onto their images [14, Corollary 4.5.13]. Then is an embedded submanifold [14, Section 1.1.13], and so, The result follows.

The dual pair structure on GL(2n, R) ⊂ M 2n (R)
The subset GL(2n, R) of M 2n (R) is open, and hence, Ω restricts to an non-degenerate form on GL(2n, R). Denote the restriction of Ω and the momentum maps to GL(2n, R) by the same symbols for simplicity.
With this restriction, we can demonstrate that the left Sp(2n, R)and right O(2n)actions are mutually transitive (Definition 3.1). For this, it is more convenient to work with the equivalent double fibration Proof (i) Using Proposition 2.3, we have that for any E, E ∈ GL(2n, R), (ii) Similar. Proof We prove the result for the Sp(2n, R)-action only; the O(2n) case is similar. Freeness is trivial. To prove properness, suppose we have two convergent

Remark 4.3 For any
). However, restriction to GL(2n, R) ⊂ M 2n (R) is necessary to get equality in the second inclusion. For example, and this is not simply an orbit of Sp(2n, R) (since, for example, the Sp(2n, R)-action preserves the dimension of span E). It is, however, possible to prove that With the restriction from M 2n (R) to GL(2n, R), the momentum map j Sp acquires a nice interpretation. As before, let E a denote the ath column of E, so {E 1 , . . . E 2n } defines a basis of R 2n . Let g(E) denote a metric on R 2n with respect to which the basis {E a } is orthonormal. Then any basis related to E by the right O(2n)-action defines the same metric, i.e.
for all O ∈ O(2n), and in fact, the set of metrics on R 2n is in one-to-one correspondence with the O(2n)orbits of GL(2n, R). We then have the following result: The metric g(E) and canonical symplectic form ω R 2n are related by From this, it follows that Inverting this identity to give we see that j Sp (E) defines the metric g(E), and so essentially corresponds to it. In the next section, we will consider this correspondence in the particular case that the ordered basis (E a ) forms a symplectic frame of (R 2n , ω R 2n ).

Reduction
Following Ohsawa [12], we consider in this section the Marsden-Weinstein quotient for the right O(2n)-action at a particular value of momentum. Using the correspondence between reduced spaces and coadjoint orbits provided by the dual pair structure, we will give a geometric characterisation of this space. Using this characterisation, we will explain how to naturally introduce a global coordinate chart on the reduced space, reproducing Ohsawa's description of the space as the Siegel upper half plane.

Reduction through a general point
It will be more convenient to work with the Lie algebra-valued momentum maps j O and j Sp . For ξ ∈ g, now let G ξ denote the adjoint stabiliser of ξ and O ξ ⊂ g the adjoint orbit through ξ . Writing N = GL(2n, R), the reduced space through E ∈ GL(2n, R) is Using the trace form to transfer the Poisson structure from sp(2n, R) * to sp(2n, R), Proposition 3.4 tells use there exists a Sp(2n, R)-equivariant symplectomorphism

Reduction through the identity
We consider the reduced space for the left O(2n)-action through I = I 2n ∈ GL(2n, R). From Proposition 2.3(ii), we see that the level set of j O through I is the set which corresponds to the set of symplectic frames of (R 2n , ω R 2n ). Meanwhile, and we see that the reduced space N j O (I ) is isomorphic to Sp(2n, R)/U(n), as originally obtained by Ohsawa [12].

Geometric interpretation of the adjoint orbit O j Sp (I)
As explained above, there is a Sp(2n, R)-equivariant map We now give an alternative geometric characterisation of O j Sp (I ) in terms of complex structures on R 2n .

Definition 5.1
Let j : R 2n → R 2n be a complex structure on R 2n , i.e. an R-linear map satisfying j 2 = −I . The map j is called ω R 2n -compatible if it satisfies the two properties (i) j is symplectic (i.e. j ∈ Sp(2n, R)), and (ii) the bilinear form g j (u, v) := ω R 2n (u, jv) is positive definite.
Denote the set of ω R 2n -compatible complex structures on R 2n by J (R 2n , ω R 2n ).

The upper half plane coordinates
As demonstrated in the previous section, the adjoint orbit O j Sp (I ) can be naturally viewed as the space of ω R 2n -compatible complex structures J (R 2n , ω R 2n ). It is a standard result that J (R 2n , ω R 2n ) has another natural interpretation, that of the negative Lagrangian Grassmannian G − (C 2n , ω R 2n ). In this section, we outline this correspondence and explain how it can be used to introduce a coordinate system on O j Sp (I ) .

The negative Lagrangian Grassmannian
Consider the complexification (R 2n )⊗ R C C 2n of R 2n . Real linear functions on R 2n extend to complex linear functions on C 2n in the obvious way, and we will generally use the same symbol to denote both a function and its complex extension. Define a sesquilinear form s : C 2n × C 2n → C by s(w, z) := −iω R 2n (w, z).

Definition 6.1
The negative Lagrangian Grassmannian G − (C 2n , ω R 2n ) is the set of (complex) Lagrangian subspaces of C 2n on which s restricts to a negative definite form,

Coordinates on G − (C 2n , ! R 2n )
In order to introduce coordinates on G − (C 2n , ω R 2n ), we first introduce the two complex Lagrangian subspaces of C 2n where {e 1 , . . . , e 2n } is the canonical basis of R 2n . Note that C 2n = Γ 1 ⊕ Γ 2 , and denote the direct sum projections by p i : C 2n → Γ i . We use the following lemma.
In light of Lemma 6.3, for any Γ ∈ G − (C 2n , ω R 2n ) we may define a map W Γ : Then for any w ∈ Γ , It follows from identity (6.2) that the map Γ → W Γ is injective. We now use this representation of G − (C 2n , ω R 2n ) as linear maps to introduce coordinates on G − (C 2n , ω R 2n ). The following lemma is straightforward to verify. Then The set arising in Lemma 6.4 is traditionally referred to as the Siegel upper half plane. It has a well-known symplectic structure. In Sect. 6.5, we show this structure is the pushforward of the usual Kostant-Kirillov-Souriau symplectic structure on O j Sp (I ) under the bijection Γ · of diagram (6.1).

The action of Sp(2n, R) in upper half plane coordinates
There is a natural action of Sp(2n, R) on G − (C 2n , ω R 2n ), given by (S, Γ ) → SΓ . We wish to express this action in terms of upper half plane coordinates, i.e. to express W SΓ in terms of S and W Γ . Since Γ 1 = span C {e 1 , . . . , e n }, we see from Eq. (6.2) that a basis for Γ ∈ G − (C 2n , ω R 2n ) is given by Now recall that the coordinates W SΓ of SΓ are simply the matrix elements of It follows that for S = A B C D ,

The projection in upper half plane coordinates
Referring again to diagram (6.1), we have demonstrated the existence of an Sp(2n, R)equivariant map is simply Sp(2n, R), and we have shown that the Siegel upper half plane provides a global coordinate chart on the manifold G − (C 2n , ω R 2n ). Using the results of the previous section, we can derive an expression for the projection Γ j Sp (·) in terms of the coordinates on its domain and range manifolds.
First note that Γ j Sp (I ) = 1 2 (I + i j Sp (I ))R 2n = 1 2 (I − iJ)R 2n has a basis It follows easily from Eq. (6.2) that By equivariance of Γ j Sp (·) , we have that Γ j Sp (S) = SΓ j Sp (I ) for any S ∈ j −1 Sp (j Sp (I )), and so, W Γ j Sp (S) = W SΓ j Sp (I ) . Writing S = Q 1 Q 2 P 1 P 2 for the coordinates on j −1 Sp (j Sp (I )), and using Eq. (6.3), this becomes In summary, the projection written in (global) coordinates is

Upper half plane coordinates on O j Sp (I)
we may instead interpret (6.4) as the coordinate expression for the momentum map j Sp : This is in agreement with [12,Equation (21)], which was obtained by a method involving the Iwasawa decomposition of Sp(2n, R).

Dynamics
In this section, we finally come to the main point of the previous constructions, namely to describe the dynamics of semiclassical Gaussian wave packets in terms of Hamiltonian dynamics on the frame bundle of T * R n , and its symplectic reduction.

The Gaussian wave packet ansatz
Consider the time-dependent Schrödinger equation on R n , where the potential V (x) is at most quadratic in x. Following [9], we recall here the Gaussian wave packet ansatz, in both the Heller [6] and Hagedorn [5] parametrisations. In Heller's parametrisation, the wave function is written with (q, p) ∈ T * R n R 2n , φ a real phase, and A + iB ∈ Σ n with A, B real. Substitution of the ansatz into Eq. (7.1) shows that it produces a solution provided the following equations are satisfieḋ Here, D 2 V denotes the Hessian of V . In Hagedorn's parametrisation, the wave function is for some appropriate choice of the square root in the denominator, with (q, p) ∈ R 2n , S a real phase, and Q, P ∈ M n (C) satisfying Substitution into Eq. (7.1) yieldṡ The two parametrisations (7.2) and (7.4) are related by

Description as Hamiltonian dynamics on the frame bundle
Given a symplectic manifold (M, ω), recall that the a frame at z ∈ M is an ordered basis of T z M, and the frame bundle F(M) consists of all such ordered bases as z ranges over M. F(M) is naturally a right principal GL(2n, R)-bundle, with group action We say that a frame (v a ) at z ∈ M is symplectic if and a local section s : U ⊂ M → F(M) of the frame bundle is symplectic if s(z) is a symplectic frame for each z ∈ U . Consider now the case when M = T * R n and ω = n α=1 dq α ∧ dp α is the usual cotangent bundle symplectic form. We will write the natural coordinates on T * R n as either (z 1 , . . . , z 2n ) or (q 1 , . . . , q n , p 1 , . . . , p n ), depending on situation. The bundle F(T * R n ) has a global symplectic frame s : T * R n → F(T * R n ), given by which induces a global trivialisation Λ : T * R n × GL(2n, R) → F(T * R n ) of the frame bundle, given by We define the -diagonal lifted symplectic form Ω on T * R n × GL(2n, R) by By analogy with [12], for any Hamiltonian H : T * R n → R we define the lifted Hamiltonian on the frame bundle H : where D 2 z H denotes the Hessian of H evaluated at z. We can give a more geometric interpretation to (7.9) as follows: For any E ∈ GL(2n, R), let g(E) denote the metric on T * R n with respect to which the global frame Λ(z, E) = s(z) · E is orthonormal, i.e. the vector fields form an orthonormal basis at each point. Explicitly and so, (7.9) can be alternatively written as To find the Hamiltonian vector field corresponding to H , note that where d 1 , d 2 denote the exterior derivatives in T * R n and GL(2n, R), respectively. The first term can be written The second term satisfies Here we have used the identity ζ · E = (ζ E) E . Overall i.e.
In the situation where H : T * R n → R is at most quadratic in the coordinates  (q 1 , . . . , q n , p 1 , . . . , p n ), the Hessian D 2 z H , and hence Δ g(E) H , will be constant, and the Hamiltonian vector field X H simplifies to In this case, motion on the frame bundle consists of classical motion on the base space T * R n and the corresponding induced linearised motion on the frame. This picture of semiclassical wave mechanics has much in common with the nearby orbit approximation discussed by Littlejohn [7, Section 7]-semiclassical motion consists of the classical motion, plus motion of a frame moving along the classical trajectory, and describing to first order the classical flow relative to it. In particular, taking the standard Hamiltonian and writing E = Q 1 Q 2 P 1 P 2 yields the first four equations of the Hagedorn equations (7.5).
In the general case (7.12) where H is not quadratic, there is an additional correction term in the motion on the base space, leading to a deviation from strictly classical motion. This deviation has recently been proposed as a means of introducing quantum tunnelling into semiclassical quantum mechanics [13].

O(2n)-invariance and conservation of symplectic frame
The right GL(2n, R)-action (7.7) on the bundle F(T * R n ) T * R n × GL(2n, R) restricts to a right O(2n)-action, and the lifted Hamiltonian H is easily seen to be invariant under this restricted action. By Noether's theorem, the corresponding momentum map j O •π 2 : T * R n ×GL(2n, R) → o(2n) is conserved under the dynamical evolution corresponding to H , i.e. Since H =Ȟ • (id × j Sp ), and id × j Sp is a Poisson map, the Hamiltonian vector fields Again specialising to quadratic Hamiltonians, these (id × j Sp )-related vector fields are and The unreduced dynamics, generated by X H and reproducing the Hagedorn equations (7.5), will drop to the reduced dynamics generated by XȞ , and these will reproduce the Heller equations

Generalisation to other symplectic manifolds
Now that we have outlined the construction of a Hamiltonian system on the frame bundle of T * R n , an obvious question is whether this construction can be extended to more general symplectic manifolds. In this section, we suggest a generalisation by employing the results of Cordero and de León [3]. Essentially they show that the frame bundle can be provided with a symplectic structure if it has a trivialisation compatible with the symplectic structure, encoded in the form of a symplectic connection. Their construction may be seen as the symplectic analogue of the Sasaki-Mok metric on the frame bundle of a Riemannian manifold [10].
In the definition of the horizontal lift, we have used the canonical isomorphism ⊕ 2n a=1 T z M V e F(M) (for z = π(e)) given by This is essentially the diagonal lift defined in [3], up to a factor of 2 . Denote the horizontal lift of a vector field X ∈ X(M) to F(M) by X hor and the infinitesimal generator on F(M) corresponding to ξ ∈ gl(2n, R) by ξ F (M) . Then it is straightforward to verify Using the non-degeneracy of ω to infer that ω z (e b , e c ) is an invertible matrix in the last identity, it follows that the 2-form ω is non-degenerate, and so defines an almostsymplectic form on F(M).

Conditions for the lifted form to be closed
In order for the almost-symplectic form ω to be a symplectic form, it must satisfy the additional condition dω = 0. The conditions for this to be the case are computed in [3]. Here, F A ∈ Ω 2 (F(M), gl(2n, R)) denotes the curvature 2-form of A, and ∇ A denotes the covariant derivative associated with A.
i.e. ω is closed if and only if ω is closed and A is a flat, symplectic (i.e. ω-preserving) connection.

Consequences of the condition
Assuming the conditions in Corollary 8.2 hold, the frame bundle F(M) has a particularly simple description. Firstly, F A = 0 implies that the horizontal distribution on F(M) is involutive and hence integrable. If we further assume the absence of monodromy, 2 this implies the existence of a global horizontal section s : M → F(M) through any point of F(M). Since the connection A is symplectic, the frames s(z) = (s 1 (z), . . . , s 2n (z)) are such that ω z (s a (z), s b (z)) is independent of z. Let us arrange for these frames to be symplectic, i.e. We use Λ to pull back the symplectic form ω . Firstly, Ω ver := Λ * ω ver = (π • Λ) * ω = π * 1 ω,