The other day, Florian mentioned to me that a heuristic he frequently employed to check whether some definition or construction was the right way forward was to see if it could be easily reconstructed from memory. This is an incredibly powerful general-purpose heuristic, one which I myself have extensively used without explicitly realising what I am doing (as have many mathematicians, I’m certain). Of course, heuristics are not iron-clad propositions, and sometimes in order to get to our destination, we do have to get our hands dirty and make nasty complicated arguments. But the wonderful thing about the memorability heuristic is that even when you do have to make such arguments, it is often possible to find a formulation which is far easier to contain in your mind. As such, the memorability heuristic is not only useful to see if a concept you are introducing is “natural” but also to test whether you have really understood an existing concept, which is to say, whether you are able to see its “naturality.”
For instance, the implicit function theorem in multivariable calculus is notorious enough for being hard to get right that it is the go-to bow that applicants to faculty positions have to string in order to demonstrate their pedagogical chops (or so my co-advisor Bernd Siebert tells me). But one of the early rewards of introducing the idea of a manifold is that it allows you to repackage the theorem into the statement that if you are given a smooth map between two manifolds then the preimage of any regular value in the target manifold is a submanifold of the domain manifold. In fact, it would be rather fair to say that the reason we care about the low-brow formulation of the implicit function theorem is so that we can have nice clean result like the one above. And it is only when one has seen this version of the theorem that one can claim to have truly understood it.
Florian’s remark was prompted by my annoyance at having to repeatedly look up the definition of a symplectic reduction, which suggested two things at once: (a) that I had to look it up repeatedly suggested a certain lack of clarity as to why it is defined the way it is—more precisely, it wasn’t clear why moment maps had to enter the picture at all—and (b) that I had to look it up repeatedly suggested that this lack of clarity wasn’t due to a corresponding lack of naturality but due to a fault in my own understanding. Indeed, this has been the case and now I believe that I have figured it out to my satisfaction, so you all can now scarper off, good day to you and goodbye!
Just kidding, just kidding, don’t go away, here’s the reason why a symplectic reduction has to involve moment maps. In §1, I recall the basic notions I am going to be talking about, i.e. group actions of symplectic manifolds, moment maps, and the definition of a symplectic reduction; in §2, I illustrate these with examples; in §3, I revisit the notion of symplectic reductions and note that the reason they appear opaque is that our choice of category for symplectic manifolds is bad; and finally, in §4, I fix this by constructing the right category for symplectic manifolds. It goes without saying, none of this is original to me.
1. Symplectic manifolds, group actions, and moment maps
Symplectic manifolds are (smooth) manifolds equipped with a closed nondegenerate -form called the symplectic form. Note that when has finite dimension, this forces it to be even. A symplectic map between two symplectic manifolds and is a smooth map such that . When the underlying smooth map is a diffeomorphism, the symplectic map is said to be a symplectomorphism. Symplectomorphisms are isomorphisms in the naive category of symplectic manifolds (whose objects are symplectic manifolds and whose morphisms are symplectic maps).
The set of symplectomophisms from a symplectic manifold to itself forms a Lie group under composition, and its associated Lie algebra is the Lie algebra of symplectic vector fields, defined to be vector fields along which the Lie derivative of vanishes, with the usual vector field commutator being the Lie bracket. Given a Lie group homomorphism , the Lie group is said to act on via the action . This induces a Lie algebra homomorphism between the associated Lie algebras.
Let us, for sake of notational convenience, denote as , where . Now, for any vector field , Cartan’s magic formula tells us:
So for symplectic vector fields in particular, we have that is closed. If it is furthermore exact, the vector field is said to be Hamiltonian; if for all , the associated vector fields are Hamiltonian, the action is said to be Hamiltonian. The reason for this terminology is that in physics symplectic manifolds represent the phase space of a dynamical system and so admit a natural action of the Lie group via time evolution. There is only one generator in this case ( being time) and turns out to be where is the Hamiltonian.
The Hamiltonian function in the case of time evolution may be generalised to any Hamiltonian group action . In the general case we have a smooth map called a moment map, which linearly assigns to every , a smooth function satisfying . We shall see in the next section that things like linear and angular momentum can be regarded as special instances of the moment map construction.
For a sufficiently “nice” value (such as a regular value), the preimage is a submanifold of . Assuming that the group is connected, the preimage is in fact fixed by the action of . (The more general case where has multiple connected components may be handled by replacing by its identity component in the subsequent discussion.) In particular, this means that it makes sense to take the (topological) quotient . Now may not be a manifold and if you hope to do string theory, this is something you just have to live with. But in order to avoid a long digression into developing the theory of orbifolds, I am instead going to assume that the action of restricted to has only one orbit type and so is an honest-to-goodness manifold.
In fact, more happens to be true: the manifold inherits a symplectic structure from . More precisely, if is the inclusion and the canonical quotient, then it turns out that there is a unique -form such that . This -form is in fact a symplectic form, and so is a symplectic manifold. This is said to be a symplectic reduction. I say a symplectic reduction, because it’s actually sensitive to the choice of the regular value (or equivalently, the choice of the moment map , which can always be shifted by a constant). In fact, the (smooth) topology of the submanifold (and consequently, the quotient ) can jump as we vary through a critical value, since the gradient flow induced by ceases to be a diffeomorphism in that case.
Now let’s get our hands on some examples.
2. Cotangent bundles
Cotangent bundles are an important class of examples of manifolds with a natural symplectic structure associated with them. The story outlined in the previous section happens to play out particularly nicely for cotangent bundles, so they’re an obvious choice of starting point for gaining an understanding into what is actually going on in the above construction.
Given the cotangent bundle over a manifold , we can define the Liouville -form on as follows. For a point and a cotangent vector , we define to be the pullback . Then, we have the following claim.
Claim. The -form is a symplectic form on . □
Proof. That it is closed is obvious. To show that it is also nondegenerate takes a little more work. Observe that for any vector bundle we have the following short exact sequence in the category of vector bundles over :
Now, the fibres of the vertical bundle consist of tangent vectors to the fibres of , which themselves are vector spaces. Since the tangent spaces of a vector space may be canonically identified with the vector space itself, there is a canonical identification . Thus, we may write:
It actually turns out that this exact sequence splits, which is to say there exists an identification:
One way to see this is to first establish the splitting on the restriction of the exact sequence above to the zero section in —the zero section map induces a map which is right-inverse to —and then use the fact that the restriction to the zero section is a deformation retract of the bundles involved in the exact sequence to extend this splitting to all of .
This extension and is in fact equivalent to choice of a connection on . Such a connection induces a pullback connection on . Then if we let be the tautological section of , given by , we get as a left-inverse to the injective map . This gives us the required splitting.
Now we specialise to the case of the cotangent bundle . We choose a torsion-free connection on , which induces a connection on , also denoted , and therefore an identification . It is then just a matter of following definitions to see that for any and , we have via the above (-dependent) identification:
As we can locally guarantee that for any nonzero vector there is a covector whose evaluation on the vector is nonzero (and vice versa), it follows that is indeed nondegenerate and that is a symplectic manifold. ■
Since the symplectic structure on was induced by nothing more than the smooth structure on , it is reasonable to guess that diffeomorphisms of induce symplectomorphisms of . Actually, we can wring out a stronger result from this.
Claim. Let be a diffeomorphism and let be the diffeomorphism it induces on the cotangent bundles. Then, , where (respectively ) is the Liouville form on (respectively ). □
Proof. Concretely, is given by the following map:
The induced tangent action is therefore given by:
A straighforward substitution therefore gives us:
In particular, is a symplectomorphism. This may be shown either via the naturality of the exterior derivative or by direct substitution into the expression for the symplectic form on a cotangent bundle that we derived above. ■
The above turns out to have a neat consequence.
Claim. Let be a group action on . Then the induced group action is Hamiltonian. □
Proof. Our previous result implies that the image of consists of symplectomorphisms that preserve not only the symplectic form, but the Liouville form as well. So, the image of the differential consists of symplectic vector fields such that . Cartan’s formula then tells us that:
So, is exact and the action is Hamiltonian. ■
The above proof suggests a natural choice for the moment map , namely:
Now, we want to allow for nontrivial stabilisers at every point (but require that they vary smoothly and isomorphically). Unforunately, this means that there are no regular values in the image of since the image of a fibre of the cotangent bundle would be i.e. the space of elements of which vanish on the Lie subalgebra corresponding to the stabiliser of under the action. But the situation can be salvaged. The assignment of the fibres to gives us a (not necessarily trivial) vector bundle over . In some local patch of we may then choose some local trivialisation and use that trivialisation to think of local sections as maps from to the typical fibre , where is (noncanonically) isomorphic to the stabilisers which are all assumed to be isomorphic. These local maps are submersions (check this!) and so the preimage of any point in is a submanifold of . In order to be able to patch all these together into a global manifold, we look for a global section that corresponds to an image point in independently of the choice of local trivialisations. The canonical choice is the zero section. And so it follows that even though may not be regular, is nevertheless a manifold.
Claim. The reduction is canonically symplectomorphic to . □
Proof. We shall first show that they are indeed diffeomorphic. Let be the canonical quotient map thought of as a fibre bundle. Since the fibres of are the orbits of the -action , the vertical subbundle of , which is to say the subbundle of vectors tangent to the fibres, is precisely the subbundle spanned by vector fields generated by the -action. As a result, we see that is precisely the horizontal subbundle of , which is to say the subbundle of covectors which vanish on the verticle subbundle. It is general fact about any fibre bundle that the horizontal covectors are precisely the covectors that are obtained by pulling back covectors on the base. So, we may write as follows:
Now let be a diffeomorphism in the image of . Then, since it sends a fibre to itself, we have . In particular, the pushforwards and satisfy . Therefore the symplectomorphism induced on by the diffeomorphism is given by:
Thus, the quotient may be (canonically) identified with at least at the level of manifolds via the map which is given by . All that remains to be shown is that this diffeomorphism is actually a symplectomorphism with respect to the symplectic structures we have defined on them. To this end, it will suffice to show that , where is the inclusion, and and are the Liouville forms of and respectively.
We follow the definitions. The Lioville forms associated with the cotangent bundles and are given by and respectively. Then and are given by:
But note that by the definition of , we have . So, indeed. ■
So we see that, symplectic reductions are indeed something natural! A few concrete illustrations of the above generalities before we go on to try and make sense of this:
Example. Let be a vector space and be a subspace that acts on via the action where and . The cotangent bundle is the product and the symplectomorphism induced on it is . The Lie algebra associated to is itself and the tangent bundles of and may be identified with and respectively. So the associated vector fields and are given by and respectively. The moment map is meanwile given by . In physical terms, this is the linear momentum in the direction , just as we expect. The preimage is a subbundle of given by , where is the subspace of consisting of covectors which vanish on . Note that this may be canonically identified with , so quotienting out the (symplectic) action (which, as noted above, doesn’t do anything within the fibres) gives us . This is the cotangent bundle . □
Example. Now equip the vector space with an inner product and let be and be with the standard action on . The cotangent bundle is the product and the symplectomorphism induced on it by is as follows:
where and denote the musical isomorphisms with respect to . The Lie algebra associated to is the Lie algebra of skew adjoint endomorphisms . This is generated by elements of the form , where , . The tangent bundles of and may be identified with and respectively. The vector fields and are given by:
The moment map is meanwhile given by:
In physical terms, this is the angular momentum in the plane spanned by the vectors and , again just as we expect. The preimage is a subbundle of whose points satisfy the equation . But this means that is parallel to , which is to say there exists a real number such that . In other words, . Quotienting out the action of gives us via the map . But this may be identified with the cotangent bundle . □
Example. We let again but this time without the inner product. Let be with the action given by where and . The induced action on is therefore . The Lie algebra associated to is and for , the associated vector fields and are given by and . The moment map is hence given by and the preimage is the subbundle whose quotient by the the action may be identified as the cotangent bundle . □
3. Friendship ended with symplectic maps…
Now that we have seen that symplectic reductions are actually doing something nice, let’s get back to the question we started off with. Why are symplectic reductions defined the way they are? It’s clear that they are performing some kind of quotient on a symplectic manifold, but why do we have to first define moment maps and take the quotient of a level set of this map instead of just going ahead and taking the quotient directly?
The problem is that symplectic maps, despite seeming very natural, are a very bad choice of morphisms for symplectic manifolds. Let’s unpack this assertion.
Say we try to take the quotient of a symplectic manifold directly with the canonical quotient map being a symplectic map with respect to some symplectic form on . Let be such that , the symplectic vector field on that it generates, does not identically vanish. Since is tangent to the orbits of the -action on , it must be in the kernel of . As a consequence, we have the following:
So, identically vanishes despite not identically vanishing. This is not supposed to happen since symplectic forms are nondegenerate by definition. In fact, we see from this argument that a proper submersion can never be a symplectic map and that the only way the quotient under a Lie group action be symplectic is if the action is locally trivial i.e. discrete.
How do we fix this? The first thing that comes to mind when one is confronted with an equation that cannot identically hold is to look for solutions to it. That is to say, we try to solve for vector fields . We have already seen in §1 that is closed. But this is the Frobenius integrability condition for the subbundle of vector fields satisfying . Locally, is the exterior derivative of some function (defined up to a constant of integration) and the vector fields must be tangent to the level sets of .
We need to find a subbundle of vector fields such that the above holds for all . To ensure this, we choose a basis of generators and note that in every local patch if we make choices , then for , we have . Thus we can always choose the such that the assignment is linear. This gives us a map from an open set of to . When the group action is Hamiltonian, the open set may be taken to be the entire manifold itself. Thus, we see that the definition of a moment map automatically falls out like pulp in an overly ripe tomato, and that the only sensible quotient we can take is that of the level sets of the moment map since these are precisely the “submanifolds” (in quotes because they can have singularities) the tangent vector fields to which lie in the the kernel of .
In fact, we get something more. We find that the above argument works just as fine for group actions which are not Hamiltonian. We just have to stitch the level sets in different patches together to get a “global level set.” Here’s an illustration.
Example. Consider again the first example in §2, namely the action of on but we make a small modification. Let be such that there is at least one satisfying . Then we take our symplectic manifold to be the discrete quotient with the symplectic structure as inherited from that on . If we let be the vector field , then the flow line associated to is a closed cycle and the integral of along this cycle is . So, the action is not Hamiltonian, yet the submanifold that we had considered earlier descends to this discrete quotient, thus allowing us to take its quotient under the action of and get a symplectic reduction. □
4. …now Lagrangian submanifolds are my best friend
Symplectic maps may have been a bad choice of morphism but the hope still is that if we could consruct an appropriate category of symplectic manifolds, then sympletic reductions might be described by some universal property. In particular, we would like to describe symplectic reductions as categorical quotients.
To recapitulate, given a group action on an object in some category and another object , then there is an induced group action on given by , where and . The fixed points of the induced action are said to be -invariant. The categorical quotient of under a given -action is then defined to be a -invariant morphism such that any -invariant morphism factors uniquely through .
Let’s make note of some intermediate desiderata. Firstly, whatever is our new notion of morphism between symplectic manifolds, symplectomophisms have to be special cases of that. Secondly, given that the the assignment of cotangent bundles to manifolds is functorial (with respect to the morphisms inherited from the category of star bundles, where maps on the bases go forward but maps on the fibres go backward) and that the categorical quotient of a manifold, when defined, is the usual quotient, we see that at least for cotangent bundles, the symplectic reduction is the categorical quotient. So, we expect morphisms between cotangent bundles to be realised as special cases of morphisms between symplectic manifolds.
Both these desiderata are satisfied by the following tentative notion of (pre-)morphisms introduced by Weinstein.
A Weinstein premorphism from a symplectic manifold to another symplectic manifold is a Langrangian submanifold of the symplectic manifold . Recall that a Lagrangian submanifold is a submanifold whose tangent spaces are all Lagrangian and a subspace is said to be Lagrangian if its symplectic perpendicular is exactly itself. Note that when a symplectic manifold has finite dimension, Lagrangian submanifolds have half the dimension of the full manifold. In particular, the restriction of the symplectic form to a Lagrangian submanifold vanishes.
(I will get to the reason for the prefix “pre-” in a moment.)
Claim. Let be a symplectomorphism. Then the graph submanifold is a Lagrangian submanifold of . □
Proof. The graph is indeed a submanifold by virtue of the implicit function theorem and its tangent space at the point can be canonically identified with the subspace of consisting of vectors of the form where . The goal is to show that is a Lagrangian subspace of .
To see that , note that for all , we have:
To see that , first note that because is a diffeomorphism, and in particular, . Then if are such that is symplectically perpendicular to all elements in , which is to say is symplectically perpendicular to for all , then we have:
Since this is true for all and is nondegenerate, it follows that which implies that . ■
Claim. Let and be the two cotangent bundles with a smooth map between the bases. Then is a Lagrangian submanifold of . □
Proof. As in the above case, the implicit function theorem tells us that is indeed a submanifold. Furthermore, its tangent space at the point may be canonically identified with the subspace of consisting of vectors of the form where and . We need to show is a Lagrangian subspace of .
To see that , note that for all and , we have:
To see that , let be symplectically perpendicular to all elements in , which is to say is symplectically perpendicular to for all and . Then, we have:
Since this is true for all and , it follows that and , which imply that . ■
So Weinstein premorphisms do fit the bill, except that they are not morphisms, at least not yet. In order to be morphisms, they need to be composable, and it’s not even clear whether the composition of two Lagrangian submanifolds even makes sense. But wait, the submanifolds are subsets of a Cartesian product and so they constitute a relation in set-theoretic terms. We know how to compose relations! If and are relations, then the composition is the set of pairs such that there exists a satisfying the conditions and . The question then is if are symplectic manifolds and and are Lagrangian submanifolds of and respectively, then whether is a Lagrangian submanifold of .
To analyse this, we follow Weinstein and reformulate the composition operation as the sequence of the following three operations:
- Take the Cartesian product .
- Intersect the submanifold with the submanifold where is the diagonal of .
- Project the intersection onto the component .
Things may go wrong at the second step since the intersection of two submanifolds is not necessarily a submanifold. The sufficient condition for this is transversality and given that intersects transversally, we may deduce that is indeed a submanifold.
In fact, this is the only obstruction there is to the composability of Lagrangian submanifolds in the above sense. In order to prove this, we require the following two lemmas:
Lemma. If and are two symplectic vector spaces with and Lagrangian subspaces, then is a Lagrangian subspace of . □
Proof. Let . Then we have:
This shows that .
Let be such that for all . Setting tells us that for all , which implies that . A similar argument tells us that . So, . This shows that . ■
Lemma. Let be a symplectic vector space with a Lagrangian subspace and a subspace of . Then, is a Lagrangian subspace of the symplectic vector space where is given by for any . □
Proof. First, we check that everything at least makes sense. Since identically, the symplectic perpendicular of must at least contain , so makes sense and is indeed a subspace of . Furthermore, is indeed well-defined (i.e. independent of the choice of the lifts and ) and nondegenerate since we quotient out by precisely the kernel of .
In order to prove that is a Lagrangian in , we need to show that for any , we have , and that if is such that for all , then . By definition of , this is the same as showing that that for any , we have , and that if is such that for all , then . But this just follows from the hypothesis that is Lagrangian. ■
Claim. If and are Lagrangian submanifolds of and respectively and intersects transversally, then the submanifold is Lagrangian. □
Proof. By applying the first lemma above to the tangent spaces of , we see that it must be a Lagrangian submanifold of .
Now let and let . Then note that is contained in . Moreover, if then it follows from the definitions that for all , we have . Since is nondegenerate, we gather that and consequently that .
Note that and that is given by:
Thus, by virtue of the second lemma we proved above, if we can show is Lagrangian in , then we are done. To show this, we make use of the fact that if are three subspaces of some vector space such that then:
Keeping the above in mind along with the observation that is an involutive lattice (exercise), we have the following chain of equalities:
So, is indeed Lagrangian in . ■
Transversality is a property that generically holds so Weinstein premorphisms are generically composable but not always. Weinstein discusses a few ways in which one might go about fixing this issue; the most natural one for our purposes is the prescription of Wehrheim and Woodward which basically amounts to enlarging the set of Weinstein premorphisms by adding in the kinky cases involving nontransversal intersections by hand. The way we fomally do this is akin to how we fill in the holes in the rational number line by thinking of numbers as sequences of Cauchy convergent sequences of rational numbers subject subject to an equivalence relation.
We define WWW morphisms from a symplectic manifold to a symplectic manifold to be, up to certain equivalence relations, finite tuples where is a Lagrangian submanifold of where runs from to and are some symplectic manifolds such that and . The equivalence relation in question is given by whenever the intersection is a submanifold. Note that this is more general than requiring transversality since tranversality is after all only a sufficient condition. In fact, if you notice, in the proof of the last claim, we never used the fact that the intersection is transversal, but only that it results in a submanifold.
So, to summarise, WWW morphisms are basically the morphisms generated by Weinstein premorphisms.
We thus have a bona fide category of sympelctic manifolds on our hands and we might naively expect symplectic reductions to be categorical quotients in this category but unfortunately, this is not the case. The problem is that there are too many morphisms between two given objects. In particular, the singleton object is not a terminal object; the hom-set is basically the set of (generalised, in the sense of Wehrheim and Woodward) Lagrangian submanifolds of .
Why is this a problem? Suppose we have a categorical quotient . Then this means that for every -invariant Lagrangian submanifold , there exists a unique Lagrangian submanifold such that . This is not obviously untrue, so here’s a concrete counterexample.
Example. Take to be the manifold with coordinates on and the group to be the group of translations along the direction. Then, we would expect to be with for some real constant . We see that for any Lagrangian submanifold , we have . So, for a Lagrangian submanifold , where , there is no such that . □
The problem is easily patched. We consider the slice category whose objects are WWW morphisms and whose morphisms are commuting triangles. In other words, our objects are symplectic manifolds with distinguished generalised Lagrangian submanifolds and our morphisms are WWW morphisms compatible with the distinguished generalised Lagrangian submanifolds. Note that in this category, the singleton manifold with itself as the distinguished Lagrangian submanifold is the terminal object.
This brings us to the main result of today: symplectic reductions are indeed categorical quotients in the category we have defined above!
Proposition. Let be a symplectic manifold with a connected distinguished submanifold and let be a connected Lie goup acting on via a Hamiltonian -action that sends the distinguished Lagrangian submanifold to itself. Furthermore, let and be a choice of moment map and value such that is submanifold of , the intersection is nonempty, and is a (symplectic) manifold. Then, with distinguished generalised Lagrangian submanifold and , where is the canonical quotient map. □
Proof. In order to prove the above proposition, we need to show the following things:
- The Lagrangian submanifold is contained inside a level set of the moment map . In other words, we can choose can such that . We need this firstly to ensure that the choice of the level set is fixed once we are given and secondly for to even make sense.
- The submanifold is indeed Lagrangian in . Note that, since and could just as well be viewed as elements and respectively, composability of (generalised) Lagrangian submanifolds would then imply that , which at the level of manifolds is the same as , is automatically a generalised Lagrangian submanifold in .
- For any symplectic manifold with distinguished generalised Lagrangian submanifold and a -invariant generalised Lagrangian submanifold satisfying , we can construct a generalised Lagrangian submanifold such that and . Note that since Lagrangian submanifolds generate generalised Lagrangian submanifolds under (formal) composition, it is enough to consider the case where is a bona fide Lagrangian submanifold.
- Such a is unique, i.e. if was an element of satisfying , then .
The Lagrangian submanifold is contained inside a level set of the moment map :
Let us denote the bundle spanned by the vector fields generated by action of the Lie algebra as . Then since by hypothesis, is preserved under this action, must be a subbundle of . Furthermore, since is Lagrangian is a subbundle of . We saw in §3 that the level sets of the moment map may be characterised as . This implies that the tangent space of at a point is contained in the tangent space of the level set of containing the point (provided the tangent space is defined). Since is connected, it follows that it is therefore contained within a level set of .
The submanifold is Lagrangian in :
Given an , the tangent space of consists of vectors of the form where is a vector in . For any two vectors , we have:
Therefore, . Now, note that is a submersion, so every vector in is of the form where and . So assume that . This means that for all , we have:
Since this is true for all , it means that . But we have already mentioned that this is which is in fact the kernel of acting on . But this means, firstly that and secondly that which is in after all. So, and hence it follows that is Lagrangian.
For every -invariant Lagrangian submanifold satisfying , we can construct a such that and :
Let and be such that . Since is -invariant, is contained in . And since is Lagrangian, is contained in which, as we have argued above, is the same as for . So, every connected component of is contained in a level set of . Distinct connected components may be contained in distinct level sets, but we choose the one containing . Note that there will always be an such that since by hypothesis, so this is always possible. Let be the quotient map. Then we may set . Firstly note that if this was indeed a generalised Lagrangian submanifold, it would satisfy and . Secondly, to see that it is indeed a generalised Lagrangian submanifold, consider , which is a Lagrangian submanifold since is. Then composability implies that is a generalised Lagrangian submanifold.
If satisfies , then :
Note that is the right inverse of , meaning that . That has a right inverse implies that is right-cancellable in . ■
What we have now achieved is a much cleaner formulation of the idea of symplectic reductions which is much easier to memorise. In fact, we have in the process, ended up generalising things a little. We have already seen one way in which the categorical quotient version is a generalisation of symplectic reductions, as originally defined, at the end of §3. Since the Lagrangianness of submanifolds is a local property, we can drop the requirement that the group action be Hamiltonian. The other generalisation has to do with the fact we are taking into account not only Lagrangian submanifolds but generalised Lagrangian submanifolds as well. This allows us to drop the requirement that is a submanifold of . Thus, the following concluding exercise:
Exercise. Construct a symplectic manifold with distinguished submanifold and submanifolds and , which are Lagrangian in and respectively and are compatible with all the distinguished generalised Lagrangian submanifolds, such that holds at the level of set-theoretic relations whether or not is a submanifold. (Hint: Regularity of values in the image can be reformulated as a transversality condition.) □
Thanks to Murad Alim, Florian Beck, and Martin Vogrin for discussion and to Áron Szabo for help with the examples and inducting me into the upper echelons of the Church of Basis-Free Computations.