To systematize the geometric aspects of quantization in arbitrary symplectic manifolds, we now introduce the symplectic category S. As objects of S we take the class of all smooth, finite-dimensional symplectic manifolds. Given two objects (P, ω) (Q, ω0) of S, we define their product as the symplectic manifold (P × Q, π1∗ω +π2∗ω0), where π1, π2 denote the cartesian projections. The symplectic dual of an object (P, ω) is the object (P,−ω).
From Lemma 3.14, we recall that a smooth diffeomorphism from a symplectic manifold P to a symplectic manfoldQis a symplectomorphism if and only if its graph is a lagrangian submanifold of Q×P. More generally, an immersed lagrangian submanifold of Q×P is called a canonical relation from P to Q. The morphism set Hom(P, Q) is then defined to consist of all canonical relations in Q×P. Since immersed lagrangian submanifolds of the product Q×P coincide with those of its dual, we can therefore define theadjoint of a canonical relationL⊂Hom(P, Q) as the element L∗ ∈Hom(Q, P) represented by the same equivalence class L of immersions into the productP ×Q.
Composition of morphisms is unfortunately not defined for all L1 ∈ Hom(P, Q) and L2 ∈ Hom(Q, R), and so S is therefore not a true category. Nevertheless, we can describe
a sufficient condition for the composability of two canonical relations as follows. By Exam-ple 5.8, the product R×∆Q×P is a reducible coisotropic submanifold of R×Q×Q×P, where ∆Q denotes the diagonal inQ×Q. The product L2×L1 of the canonical relationsL1
and L2 is a well-defined immersed lagrangian submanifold of R×Q×Q×P, and we will call L2×L1 cleanif (R×∆Q×P , L2×L1) is a reducible pair. Applying Theorem 5.12, we then obtain
Proposition 5.28 If L2×L1 is clean, then L2◦L1 is an immersed lagrangian submanifold of R×P, i.e. L2◦L1 ∈Hom(P, R).
2 Associativity of compositions holds in the symplectic category in the sense that for canonical relations L1, L2, L3, the equation
L1 ◦(L2◦L3) = (L1◦L2)◦L3 is valid provided that both sides are defined.
Among the members of S, there is a minimal objectZ, the zero-dimensional symplectic manifold consisting of a single point ∗ equipped with the null symplectic structure. Mor-phisms from Z to any other object P ∈ S identify naturally with immersed lagrangian submanifolds of P. Thus, the “elements” of P are its immersed lagrangian submanifolds, and in particular, we may identify the set Hom(P, Q) of morphisms with the “elements” of Q×P for any P, Q∈S.8
A canonical relation L ∈ Hom(P, Q) is said to be a monomorphism if the projection of L onto P is surjective and the projection of L onto Q is injective, in the usual sense.
Equivalently, L is a monomorphism if L∗ ◦L = idP. Dually, one defines an epimorphism in the symplectic category as a canonical relation L for which L◦L∗ = idP. From these definitions we see that a canonical relation is an isomorphism if and only if it is the graph of a symplectomorphism.
Canonical lifts of relations
The full subcategory of Sconsisting of cotangent bundles possesses some special properties due to the Schwartz transform SM,N : T∗N ×T∗M → T∗(M ×N), which enables us to identify canonical relations in Hom(T∗M, T∗N) with immersed lagrangian submanifolds in T∗(M ×N).
Example 5.29 A smooth relation between two manifoldsM, N is a smooth submanifold S of the product M ×N. Under the Schwartz transform, the conormal bundle of S identifies with a canonical relation LS ∈ Hom(T∗N, T∗M) called thecotangent lift of S. In view of Example 3.27, this definition generalizes tangent lifts of diffeomorphisms. Note in particular
8Another point of view is to define Hom(P, Q), not as a set, but rather as the object Q×P in the symplectic category; then the composition Hom(P, Q)×Hom(Q, R)→Hom(P, R) is the canonical relation which is the product of three diagonals.
that although a smooth map f : M → N does not in general give rise to a well-defined transformationT∗N →T∗M, its graph in M×N nevertheless generates a canonical relation Lf ∈Hom(T∗N, T∗M).
Example 5.30 The diagonal embedding ∆ : M → M ×M induces a canonical relation L∆∈Hom(T∗(M ×M), T∗M) which in local coordinates assumes the form
L∆ ={((x, α),(x, x, β, γ)) :x∈M and α, β, γ ∈Tx∗M satisfyα=β+γ}.
In other words,L∆is the graph of addition in the cotangent bundle. IfL1, L2 ∈Hom(∗, T∗M) and L2×L1 is identified with an element of Hom(∗, T∗(M ×M)) by the usual symplecto-morphism T∗M ×T∗M → T∗(M ×M), then the sum L1 +L2 ∈ Hom(∗, T∗M) is defined as
L1+L2 =L∆◦(L2×L1).
Note that if L1, L2 coincide with the images of closed 1-forms ϕ1, ϕ2 on M, then L1 +L2 equals the image of ϕ1+ϕ2.
The dual of the isomorphism T∗M×T∗M →T∗(M×M) identifies L∆ with an element L0∆∈Hom(T∗M,Hom(T∗M, T∗M)). This canonical relation satisfiesL1+L2 =L0∆(L1)(L2), and if L is the image of a closed 1-form ϕ onM, then
L0∆(L) =fϕ,
where fϕ is the fiberwise translation mapping introduced in Section 3.2.
4 Example 5.31 (The Legendre transform) As a particular example of this situation, let V be a smooth manifold, and consider the fiber product TV ×V T∗V along with its natural inclusion ι:TV ×V T∗V ,→TV ×T∗V and “evaluation” function ev:TV ×V T∗V →R given by
ev((x, v),(x, p)) =hp, vi.
If L ⊂ T∗(TV ×V T∗V) is the lagrangian submanifold given by the image of the differential d(ev), then the push-forward Lleg =ι∗L defines an isomorphism in Hom(T∗(TM), T∗(T∗M)) given in local coordinates by
((x, v),(ξ, η))7→((x, η),(ξ,−v)).
As noted in [57], this canonical relation can be viewed as a geometric representative of the Legendre transform in the following sense: If L:TM →R is a hyperregular Lagrangian function, in the sense that its fiber-derivative defines a diffeomorphism TM → T∗M, then the composition of Lleg with the image of dL equals the image of dH, where H: T∗M →R is the classical Legendre transform of L.
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Morphisms associated with coisotropic submanifolds
If C ⊂ P is a reducible coisotropic submanifold, then the reduction relation RC ∈ Hom(P, C/C⊥) is defined as the composition of the quotient map C →C/C⊥ and the inclu-sion C/C⊥→P. Somewhat more concretely, the relation RC is the subset
{([x], x) :x∈C}
of C/C⊥ ×P, where [x] is the leaf of the characteristic foliation passing through x ∈ C.
Evidently RC is an epimorphism, and so RC ◦R∗C = idP. On the other hand, RC is not a monomorphism unless C =P, and we define the projection relation KC ∈Hom(P, P) as the compositionR∗C◦RC, i.e.
KC ={(x, y) :x, y ∈C,[x] = [y]}.
By associativity, we have KC = KC ◦KC, and KC∗ =KC. Thus, KC is like an orthogonal projection operator.
These relations give a simple interpretation of Theorem 5.12 in the symplectic category.
IfL is an immersed lagrangian submanifold of P, i.e. L∈Hom(Z, P), then Lis composable with bothRC and KC, provided that it intersects C cleanly. In this case, we have
LC =RC ◦L LC =KC ◦L.
In particular, KC fixes any lagrangian submanifold of C.
Example 5.32 Suppose thatF is a foliation on a manifoldB whose leaf spaceBF is smooth.
In the notation of Example 5.17, we then find that ifT∗(B/F) is canonically identified with the reduced space E/C⊥, then the reduction relation RE is the Schwartz transform of the conormal bundle to the graph of the projection B →B/F.
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