Symplectic Manifolds

The canonical example of the n-symplectic manifold is that of the frame bundle, so the question is whether this formalism can be generalized to other principal bundles, and distinguished from the quantization arising from symplectic geometry on the prototype manifold, the bundle of linear frames, a good place to motivate the formalism.

Let us start with an n-dimensional manifold M, and let π : LM → M be the space of linear frames over a base manifold M, the set of pairs (m,e_k), where m ∈ M and {e_k},k = 1,···,n is a linear frame at m. This gives LM dimension n(n + 1), with GL(n,R) as the structure group acting freely on the right. We define local coordinates on LM in terms of those on the manifold M – for a chart on M with coordinates {xⁱ}, let

qⁱ(m,e_k) = xⁱ ◦ π(m,e_k) = xⁱ(m)

π_jⁱ(m,e_k) = e^j ∂/∂xj

where {e^j} denotes the coframe dual to {e_j}. These coordinates are analogous to those on the cotangent bundle, except, instead of a single momentum coordinate, we now have a momentum frame. We want to place some kind of structure on LM, which is the prototype of n-symplectic geometry that is similar to symplectic geometry of the cotangent bundle T∗M. The structure equation for symplectic geometry

df= _| X dθ

gives Hamilton’s equations for the phase space of a particle, where θ is the canonical symplectic 2-form. There is a naturally defined Rⁿ-valued 1-form on LM, the soldering form, given by

θ(X) ≡ u⁻¹[π∗(X)] ∀X ∈ T_uLM

where the point u = (m,e_k) ∈ LM gives the isomorphism u : Rⁿ → T_π(u)M by ξⁱr_i → ξⁱe_i, where {r_i} is the standard basis of Rⁿ. The Rⁿ-valued 2-form dθ can be shown to be non-degenerate, that is,

X _| dθ = 0 ⇔ X = 0

where we mean that each component of X dθ is identically zero. Finally, since there is also a structure group on LM, there are also group transformation properties. Let ρ be the standard representation of GL(n, R) on Rn. Then it can be shown that the pullback of dθ under right translation by g ∈ GL (n,R) is R_g^∗ dθ = ρ(g−1) · dθ.

Thus, we have an Rⁿ-valued generalization of symplectic geometry, which motivates the following definition.

Let P be a principal fiber bundle with structure group G over an m-dimensional manifold M . Let ρ : G → GL(n, R) be a linear representation of G. An n-symplectic structure on P is a Rⁿ-valued 2-form ω on P that is (i) closed and non-degenerate, in the sense that

X _| ω = 0 ⇔ X = 0

for a vector field X on P, and (ii) ω is equivariant, such that under the right action of G, R_g^∗ ω = ρ(g−1) · ω. The pair (P, ω) is called an n-symplectic manifold.

Here, we have modeled n-symplectic geometry after the frame bundle by defining the general n-symplectic manifold as a principal bundle. There is no reason, however, to limit ourselves to this, since we can let P be any manifold with a group action defined on it. One example of this would be to look at the action of the conformal group on R⁴. Since this group is locally isomorphic to O(2, 4), which is not a subgroup of GL(4, R), then forming a O(2,4) bundle over R⁴ cannot be thought of as simply a reduction of the frame bundle.

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