Geometric quantization with metaplectic-c structures
Karshon, Yael
In the classical geometric quantization procedure with the "half-form correction", one cannot quantize a complex projective space of even complex dimension (there is no "half form bundle"), and one cannot equivariantly quantize any symplectic toric manifold (there is no "equivariant half form bundle"). I will describe a geometric quantization procedure that uses metaplectic-c structures to incorporate the "half form correction" into the prequantization stage and that does apply to these examples. This follows work of Harald Hess from the late 1970s, with recent contributions of Jennifer Vaughan.
 Thank you, organizer, Eckhardt, Xenon, and Paulineel for this wonderful opportunity. I'm talking about themes which are very old. I think in some sense, I'm trying to understand part of the paper of Harold Hess from 1980, 1981 maybe. This is what he did in the late 1970s, which seems to have been largely forgotten. And in particular, I'm trying to understand a piece of his paper where he's sort of only given us some, well, some of what I say is definitely there. And some of the what I say he knew, but he's giving us some hints about it, maybe without all the details. And that part, I'm actually connecting with recent work of Jennifer Vaughn. And maybe the disclaimer is some of the pieces, the pieces which we're trying to sort of add and fill the details, or maybe say a little bit more than we said before, are not that big, but they're not written. So I guess I should, those pieces, you can lower the usual disclaimer, take it as 80% rather than 100%. But let me tell you the story. I think this is a very interesting story, so it goes back to, again, the very early days of Geomete, or earlier days of Geometeor quantization, where you can actually hold on to your vector spaces and maps. Let me first give a time. Very way too fast review of the constant Soya version of quantization. And there's no way I can do this justice in seven minutes, but I'll try to say something. You take a symplectic manifold. You take the prequantizations as circle bundle. So just for this to exist, you need the two form, its commodic class to be integral. And then you take the associated line bundle. So gamma is a connection, one form, of this principal circle bundle. And we insist that, well, up to plus minus signs may be a factors of 2 pi. Omega will be the curvature of this connection. So then associated line bundle then gets a connection as well. And what constant tells you then is that the Poisson algebra, as a v algebra, adds some of the space of some sections through the constant recipe. In fact, you can say more if you look at all the quantum morphisms, the diffumorphisms of this principal bundle, that are S1 equivariate to the frequency of the connection one form, the le algebra that is exactly the Poisson algebra. So the Poisson algebra acts on the total space of the quantization as all automorphisms. And through this action, as soon as you act on a principal bundle, you get an action on the total space of the associated bundle. This is a le algebra action. So it's through vector field. But then you get an action on the space of all sections. So that's the constant recipe. This was well known to many people decades ago. Costants knew it. The one place I know which actually writes down and checks out the details is this recent thing of Jennifer Bond. But it was definitely known to many people and mentioned with some level of details in other places. And that constant says, well, this is too big. We're motivated by some concrete examples of physics and our presentation area. This is sort of nice and beautiful, but the space is too big. So to cut down the dimension and recover the examples which motivate this theory, he needs to add something about polarization. And Poisson is a Lagrangian sub-bundle of the complexified tangent bundle, which is involved in the two sort of nice examples. One would be if you have a Lagrangian affiliation of your original 94 to just complexify. That gives you polarization. The other example would be, well, if you have a scalar structure, you take the span of del del z bar. That's a complex formula. That's a polarization. So recovering the special cases, like scalar structures or Lagrangian affiliations. And the polarization allows Costants to shrink his space. He's only taking sections of the line bundle which are constant along the polarization. So I'm putting here an F bar. The literature is a little bit inconsistent on whether to work with the polarization or with its complex conjugate. There's both versions. They're OK. And but now the guy which is acting here is not the entire personel. You're paying a price. On one hand, you've got the smaller vector space, which is what you wanted. But you have to compromise the functions to which you will associate operators. Have to be functions whose Hamiltonian flow preserves the polarization. And there's various ways to compensate for that, which again, this is sort of, again, trying to do five minutes for a big, big, big story. I'm not going to, there's ways of obtaining associating operator to functions by means of Costants, Stambert, Platinum, pairing between spaces. Correspond to different polarizations. I will not talk about that piece, although some people would view this as criminal. Then again, if you look at this, you say, oh, it's not what I expect from physics. You need to add the so-called half-form correction. And when you do the half-form correction, well, you're starting to limit yourself more. Not only the homology class of the two-form, but also half the trunk class of the tangent bubble, say, equipped with any compatible complex or almost complex structure, needs to be integral to be able to do this, to make this correction. This is necessary. And this is what we try to avoid or weaken through lethargic structures. What pass and other people are trying to weaken that requirement. Then in that situation, so the manifold needs to be not only symplectic, but metaplectic. And you then correct your line of adole by tensorane with half-form bundles. And this is going to be the thick corrected bundle in which the sections you're going to take. And its first class is not quite as before. You need to subtract or add dependent conventions. I think subtracts have the first-form class of the tangent bundle. And as soon as you've done this with half-forms, you've lost your connection. But your half-forms do come equipped with a very natural partial connection. You can still take derivatives in directions of the polarization. That's well-defined. That being an ambiguous false from the involutivity condition. So on this bundle, you do have a partial connection just in the directions of the polarization using complex conjugate. And that is enough to make sense of the vector space. So you can still then take the Hilbert space. I didn't even say what the inner product is. That's another story, which I'm sort of admitting, to be those sections of this corrected line bundle, which are very constant in the direction of polarization. And the metaplectic C story is to try to make sense of Q, even if L and half-forms did not separately exist. This actually goes back to the four-hole test to a paper of C's. In 1977, I think the paper was 1980, a later paper which became more famous is that by Robinson and Gonsley. I personally prefer Gonsley the approach of S. They are 1989, which I think really captures the essence of the whole story. He works with principal bundles. So he tries to understand that, explain the leaf theoretic facts behind his construction. I find that very enlightening. I personally, on my wish list, besides wanting to understand things better, there's a couple of months more recent, very interesting works by different people who show how headphone corrections really make you happy. There's a paper by Colin Kerwin that tends to do with some version of quantization to induction, which is given by Lauren Charles. There's various papers out there that makes you sort of, that really gives you a lot of faith and headphone corrections. And one of some of the items on my own wish list is to try and revisit those newer papers using the MPC approach and to, and thus, weaken the assumption that each one of these be integral. Maybe a word in terms of actual examples, which I mentioned in my abstract, some examples of manifolds which do not admit this old type of quantization would be CP2. There's nothing more concrete. Another one, I guess, we've seen the pictures of Michel Verstock of quantization of a Torque manifold. And in Michel's picture, you've seen those twisted pictures where the edges were lying exactly midway between lattice points. These are the pictures you would be getting from an MCPC structure. And again, I think I was involved with several things some years ago, but the place where you can actually see an actual statement to prove that these Torque manifold are MPC pre-quantizable if and only if the edges lying midway between lattice points is in a recent paper by Jennifer Vaughn. And so Torque manifold specifically, you might have a half-on correction, but not equivalently. You're never going to have a equivalent half-on bundle. You will have an equivalent metaplexy structure. So to do them, if you care about group actions, you really are getting. In the equivalent case, this compromises bigger than in the non-quantable case. And you overcome it with metaplexy. So I'm going to talk a bit about this recipe. And I want to start with the linear algebra, which is relevant here. And I will start by just fixing a model symplectic vector space, just a standard R2-A. And we start with the metaplexic group for this guy. So I don't want to work with the actual construction. I just want to work with the properties. The group of symplectic linear transformations, it's pi 1 is that. So it has a unique uptoizomorphism z2 extension connected z2 extension. So this would be the definition of the metaplexic group. And for the linear algebra, that will often be enough. If you try to work over manifold, some manifolds will not have metaplexic structures. So for manifolds, if you have a symplectic manifold, you can think of it as a relevant structural group with a symplectic linear group. That's a group acting on the principle bundle of symplectic frames. And you would be looking for a lifting of the structural group to the metaplexic group. And that might not exist. But you might have a lifting of the structural group to the metaplexic group. The metaplexic C group is just a corresponding circle. Central extension. And that's the one that you will have in a lot more manifolds. So for the linear algebra, it doesn't make that much of a difference if you're doing the linear algebra. Let's work with the metaplexic C group. So there's two sort of things you can do. You can think of a circle as being a subgroup of this guy. And then the quotient will be the group of symplectic linear transformation. This is exactly what I'm just repeating what I said. But you can also, from this description, you also see another picture. You see the metaplexic group sitting inside. And the quotient, well, the quotient is a circle, but it's not the same circle. We don't have color. Oh, yes, we do that. Color chart. OK. It's not the same circle. If you embed the center, and then you take the homophysm, this homophysm sometimes really gives the letter eta to denote it. This is the square map. And really, the circle that you care about is the center, which is a double covering of the circle that you can obtain as the image of the metaplexic group. So let's do the linear algebra for polarizations. So we take the complexification of our vector space together with the complexification of the symplectic form. And then you take a complex subspace, which is Lagrangian with respect to that. And there's a few things you can do with it. You might want to look at intersection with a complex conjugate. That is going to have a form. This is going to be the complexification of some isotropic real subspace of our symplectic vector space. Or you might want to look at the sum of the Lagrangian with its complex conjugate. And this is going to be the complexification of a quasitropic. I'm sorry? Oh, not the right sum. You're right. Just thank you very much. Just the ordinary sum. Because it depends on the type of the sum. I definitely do not want to insist that this is the right sum. Thank you. And in fact, d is the null space of e. d, in fact, is the symplectic orthogonal to e, and vice versa. And the symplectic form gives us a naturalizing complexification between the quotient and the dual. Quotient v mod e with the dual of d. Now, if you have a real linear symplectomal fission, it acts like a complexified vector space, just complexify, and preserves the complex complex structure. So it makes sense to ask, does it or does it not send the transient subspace to itself? And we put a little subscript f to refer to those vices, too. It doesn't have to fix that. It just has to send f to itself. So the symplectic, the symplectonfism that the fish symplectonfism sends f to itself. Well, you can complexify, and then you can restrict to f. And you just get a linear transformation, complex linear transformation on f. So this map is complexified and restrict. And the kernel of this map consists of those symplecton frisms that actually not only leave f invariant, but in fact fix f. And a crucial detail, one of those which are not explicit in this, but it's actually an exercise, is the fact that this kernel is contractible. And that's going to be better. Do I have space to write this? I have space to put an error. kf is contractible. So what do we do with that? Let me go back to the original diagram that you've already seen. Metaplexic group has a double covering of a symplectic group. And now let me restrict to those guys that preserve not fix, preserve f. Then we can take their preimage and the metaplexic group. We have a smaller group. And maybe the guys that preserve f got the double covering of that. Now the crucial observation is the following. Let me tell you where I'm heading. So if you have a symplectic manifold, your structure group is the symplectic linear group. You want to lift it to the metaplexic group. And you can do that. And what you would like to do is to get an associated line bundle and then take its space of sections. The point is you don't have an interesting associated line bundle. You could take one which is associated to the map to S1 I wrote before. But you really want to square root of that. You want to somehow the center is supposed to play a role here. What turns out is you need to first reduce the structure group before you can take an associated line bundle. And the choice of polarization gives you this reduction of structure group. And we're now doing the thing which is happening on the linear algebra level. I need to give you a to play this game. I need to give you a homomorphism from the metaplexic group subscript C f, this reduced guy, to say C star. So it can act on C and we can take an associated line bundle. Let's actually do that. I mean, this is what Hess does with omitting some details, but he catches it. So here we have the map, which is, which we wrote before, complexify and restrict to go from simplectonophysm that preserve f to complex linear isomorphisms of f. Now, we'd like to take the square root of the complex determinant. And we can't quite do that. Well, for that reason, well, the metalinier group was invented for that reason. This is the double covering of the GL of the general linear group on which you can actually take the square root of the determinant. But then the question is, we'd like to start with the metaplexic group, the piece of it, which preserves f rather than. So we're looking for an arrow from it to the metalinier group to be able to obtain our homomorphism to C star. And the point is that this actually does exist. And the reason that it exists is the map from the simplectonophysm to the general linear group of f, its kernel, it's straight through the quotient by that kernel kf. And that kernel was contractible. And that guarantees that we can find another copy of it upstairs. So the preimage of it, I don't want to call it, in the double covering. The metaplexic group is, in fact, disconnected. So you can take one component. So you just get a copy of the same guy. And it's going to get this map, which factors through quotient by that lifted copy of the kernel. So what does it get to get from this whole story? The piece of the metalinier group that preserves f comes equipped with a homomorphism to C star, which provides us with a square root of the map that we described before, namely, complexify, restrict, detentate determinant. And then it's easy to extend it further to the metalinier C group. I'm going to use this by the same letter, chi. Again, those elements of the group that preserve f, namely, you take C in the circle. You take g tilde in the metalinier group. And you map them to C times chi of g tilde. And the nice thing here is this homomorphism, its restriction to the center, is just the inclusion of the circle in C star. And that's going to be the guide for you. So that adds the linear algebra part of this story. And at this point, we want to take that part and throw it on a manifold and see what we've got. So here's a symplectic manifold. And now I have the symplectic frame bundle. And I have to admit I have some mixed feelings about this notation. The fiber is not a group. The fiber is a space in which the group acts freely and transitively. It's the principal bundle for the group as pp. To wish that you need to make a choice. Like if you choose a base point. So that's why I said I have mixed feelings about the notation. You see, there's really so what's happening here. The fiber at each point of the frame bundle is the set of symplectic linear isomorphism from our standard model vector space to that tangent space at that point. Really what you have here, you have the principal right action of the group of symplectic linear transformation of the model vector space v. You also have a left action of the symplectic morphism group of that particular fiber. And there's context where I expect this is actually relevant. So in those contexts, we would want to distinguish them. But let me still use this notation for now. And we want it to a lifting of structure group, as I promised. So we're going to have three levels. Here we have a principal MPC bundle over the manifold. But we insist that it be consistent with the principal frame bundle in the following sense. The principal frame bundle is a principal bundle for the symplectic linear group. We insist to have a map from the principal MPC bundle to the symplectic frame bundle, which is equivariant with respect to the group MPC, which acts on sp through its homomorphism to the group sp. So this is just a print. What we have at the top level is just a principal circle bundle. That's the demand is across in Surreal. And this circle is just the center of the group MPC, MPC of the standard MPC group. And so over each and every point of end, we just have the diagram of groups that we wrote before. But these are spaces in which these groups act freely and transitively. Then we want to put P with something that we call the central connection. Basically, I want a connection that only sees the central part. Has called it by a different name. He calls it an ecto-soydo connection, because he has it as a special case of something more general. But let me just stick with the more sort of, friendly language central connection. By definition, this is going to be a one form gamma on P that has a fine properties. That it has to be an MPC equivalent connection on P as a principal circle bundle over the symplectic frame bundle. And this connection, you see inside here, we've got the metaplectic directions. Each metaplectic orbit is a double covering of an orbit for the frame bundle. So it has to be zero in the metaplectic direction, such that MP orbits are horizontal. And these assumptions imply that the differential of this one form is basic. It comes from a two form and the manifold in the very same way that you have for the behavior of connections on principal circle bundles that you do in the constant surreal place. So then we can write down a different definition. Sorry, you know what? Is every variant means invariant? I think it means invariant. Yes. Yeah, you're right. It should be invariant. It should be invariant. Thank you. Yeah. Thank you. Because this is just with real coefficients and the real numbers here are just the algebra of the centers, everything x trivially omits. So yes, invariant. So then we can define NPC pre-plantization in analogy with the constant surreal case. And NPCization of a cyclic manifold is exactly this type of structure. It's a lifting of the structure group to the NPC group together with a central connection. The curvature is the pullback of the two form. Complete analogy with constant surreal. Now we'd like to construct the line bundle. And we'd like our Hilbert space to be obtained from sections of that line bundle. And we'd like the Poisson algebra to act on that Hilbert space so that each function will go to an operator. So again, an analogy with constants surreal. And again, the reference for one is the constant surreal. And again, the reference for one of the objecies of Jennifer Bay-Probe, Jennifer Vaughn. We have a notion of NPC, quantum of physics. It's almost what you think it is. It probably is what you think it is, but there's a subtly which at least escaped me until Jennifer Vaughn pointed it out and clarified it. So this is not, I think this is not N-Has. This is an analogy quote to the stuff of constants surreal. So these are the NPC, a covariant of the film of physics, that preserve the central connection. And well, already here, if you're NPC, if we're going to descend to the lower levels, and I want these lower defimophysm to be compatible in the following sense. It has to descend to the filmophysm, the side tilde of the middle piece and side amp of the lower piece, such that side tilde is the natural lifting, or we can call it prolongation if you wish, of the side amp. So this is the subtlety, but it actually makes everything work really nicely. So what we have in this situation for quantum morphisms, this is the relevant notion of an ultramorphism, and maybe I'll write it on the more bigger space. So here's our quantum morphism. Here are the guys that descend to. This is the side tilde, this is the side amp, and you see the side amp will be a symphlectomorphism, so it lifts to a map of the symphlectic frame bundles, and we want that lifting to coincide with side tilde. That's an actual requirement, which needs to be imposed. And with that requirement, everything is nice and beautiful, because once, first of all, the group of quantum morphism itself is going to be a circle-central extension of a group which acts on amps, a group of the symphlectomorphism group, which might have several components, but the identity component is the different Hamiltonian transformations. So if you wish, I'll give you a notion of Hamiltonian transformation, which has several components. In the infinitesimal guys, these are just the vector fields of the total space that satisfy the corresponding infinitesimal conditions, or if you wish, the flow that they generate satisfy these conditions. So the infinitesimal quantum morphisms, that's on the algebra, which is naturally isomorphic to the Poisson algebra, and that's kind of nice. The isomorphic frame, an infinitesimal quantum morphism is just a vector field on the total space. So you can plug it into the central connection to get a function on the total space, and that function will actually come from the base, and this is the factor plus minus, this would be the function of the Poisson element of the Poisson algebra that responds to this infinitesimal quantum morphism. So the Poisson algebra acts as all the automorphisms of this pre-quantization structure, and therefore it will act on any associated line bundle and the space of sections. So if you have any line bundle or vector bundle, which is associated, then the algebra will act totally naturally on its space of sections, except as I said before, this is a little bit too good. You need to restrict the structural group to actually have such a line bundle which is interesting. Let's do that restriction. So we're choosing a polarization on our manifold, and inverted the grand-gen sub-bundle, the conflexified tangent bundle, and I want it to satisfy some non-negativity condition, which I will not allow you to do. So I'm going to do this restriction and I'm going to do this restriction and I'm going to do this restriction by some non-negativity condition, which I will not elaborate on. In the purely complex case, this would mean that this is an actual k-o form. And I want to make two assumptions, which are also made in the constant-surreal context, so this is not restricting from the constant-surreal context. I want the real piece to have constant rank. This is the isotropic piece. And this piece gives us an isotropic affiliation theme of that. In the K-lo case, this is just tricky. And the piece I'm not going to elaborate on is the Hilbert space structure and the resultant vector space, but let me still list what is needed for that Hilbert space structure. You need that as a tropic affiliation to be fibrin. So you don't require E to be integral? To be? No, I don't require E to be integral. But depending on what your goals are, there are some applications for which you do want to require it. So basically what I'm claiming here is the type of requirements you make are an analogy to what is done in the constant-surreal case. So what we have here, we have a fiber bundle, whose fibers are the leaves of E. And this we will need for the Hilbert space structure, which I will not get to. Now the constant rank of the real part tells us that we are able to actually fix the model for this polarization. But this position really will come from, will correspond to a reduction of structure groups. I'm fixing a Lagrangian complex subspace of my standards in the particular space such as f intersect f bar has the correct rank. And what you get is a reduction of structure group. So we'll first of all lift at the circular group. That corresponds to a central extension. And then we take a reduction of structure group, which is correspond to subgroup. So if we have the original tower, we can now look at those symplectic frames, which are, which carry the standard Lagrangian to the Lagrangian, which is given by the polarization. And the treatment of that by definition is p sub f. And again, we have a tower, except now the group downstairs is the subgroup consisting of symplectic transformation preserving f, and the group upstairs is a subgroup. But this is exactly the subgroup, the MPC subgroup f. That's the one that had a homophysicistar, which we are kind of happy about. We then have the relevant part of the Poisson algebra. These are the functions with somatonic flow preserve the polarization at pf. In fact, that is going to be the group of all quantumophysicists that preserve f. We can still restrict our central connection to the, and therefore it acts on the associated line bundle, and this is the line bundle that we sort of, you know, wanted. Now, what else do we have? We have the central connection. So here has actually uses a pair of polarizations and from the pair of the polarization that are related and sort of the way he obtains an honest to God connection of this line bundle. And then he mentioned, oh, by the way, you can just do one polarization and you get a partial connection. So this is one of the pieces where I've checked some, but not all of the details. But let me just write this. I believe, I mean, okay, I do believe that this works. So you do get a completely natural partial connection on q defined when you take derivatives on vector fields in the direction of the polarization towards complex conjugate. Now, where is this coming from? Basically, there's two ingredients here. One of them is the central connection, which we were given, and second, the so-called God connection, which is resulting from the involutivity of f. You get a natural connection on the tangent of the mod f and on various bundles that you can build from that one. And that uses the involutivity. And then, then you can just continue as you expect. You can take the Hilbert space, and I'm writing the Dutch space to be the space of sections of this quantization line bundle, which are very constant in the directions of the polarization. And we have our quantumophisms acting there. So you get a totally natural action of that part of the Poisson algebra, which preserve the polarization. And then one can try to sweat and get operators associated to other functions, but that has to do something about those. I'm not pretty sure I'm just saying anything about that. Maybe a word about the inner product. I'm just going to say not write down. Again, Hess does explain that, and it is in analogy with what happens with constants through yaw. If you take sections of q, which are covariant constants along the polarization, there is a natural point-wise inner product of such sections that takes values not in scalars, but rather in densities on mod d. And then you integrate that to get a number, and that gives you the inner product on your Hilbert space. I'm done! APPLAUSE That's what's going to go. Yeah. Well, this Hilbert space, it has square interior sections, only if leaves of d are... If leaves of d are tori... You integrate over the quotient. You integrate the density over the quotient. Yes. The fibers could be... They covariantly cost sections along the tori with the indistributions. Right? You can't hit the words on the front block. There is a... No, no, no, no. Oh, okay. You're right. It's the same story as... It's the same stories as you have in... No, you're absolutely right. It's the same stories as you have in the cross-sensory case. First of all, you could look at situations where the leaves of d are not compact, and then you're okay, and you're fine. If the leaves of d are compact, you're in trouble, as we learned from you. You're in trouble. But then you can play the same games as you do in the cross-sensory case. You can take the sheaf of sections with this property, and that sheaf you can take underneath some of the homologes of the sheaf, and things about that, and you expect to get a new formula for the house. But you can do more things. But this... No, thank you very much. You're absolutely right. You can get in big trouble with the game very soon. In physics, a scalar product is an essential physical object. Here, as long as everything is nice, you put it within the structure by putting a metapractic structure or a PC structure. And here it comes, if you don't see, you get it. But once you start reducing some symmetries, you lose it. There are versions of reduction here. Again, there's some version of reduction. I mean, there's a lot of things happening here, many of which I understand will be partially... But yeah, there's a lot of things you can do. Jennifer Bond has a notion of Borsamfeld, which, for example, in the Torreke-Mannifold case, she actually shows that the Borsamfeld leaves are exactly the ones you get from the lattice points as in Michel's picture. Again, there's earlier notions of Borsamfeld due to... I think Robinson, the same guy from Robinson-Ronson. Jennifer's definition is a little bit different. I mean, there's a lot of stories there, which do not provide, as far as I know, do not limit you more than you were limited in the Constance-Rial case. But at the same time, there's the potential to address more manifolds which do not admit an effect. No, I just wondered if you actually asked the wrong question. I just wondered if you managed to overcome the problem here. No, no, no, this is complete analogy. But actually one of the things we were looking at with Mark, that I think we'll tell you later this week, has to do with some of these analogies. But again, it's not me and Tess. But you told us about your friction, the line in these structures? Oh, oh, oh, I'm sorry. So a metaphylactic C-prequantization exists if and only if difference. That's the first-run class of your line bundle that you care about. Even only if this difference is integral in cosmology. And it is unique up to twisting by pull delets of flat circle bundles on that. So the extent to which it is unique is analogous to Constance-Rial, and this is explained by Haasel. And this is different in that for Constance-Rial, you need each one of them to be integral, and here we need just the difference, which is kind of the news, because this is the John Klassler of the line bundle that you actually care about. Yeah. This is sort of a cultural, maybe, comment, even. Yes, a lot of this is about cultural stuff. So I was surprised by the preoccupation with an action of the Poisson algebra, since I thought that was just sort of a historical mistake, because that's not what you want in physics. You want to take functions of the gate operators, right? Yes, and they will only approximately correspond to the Poisson practice. So the thing is, if you restrict, so this is a huge restriction, so these guys, you can actually make them correspond on the nose, but if you want the rest of the Poisson algebra, then you have to allow certain approximations, etc. But these guys, if you follow the constant recipe, these guys, you get on the nose. You get isomorphism only for this, and perhaps some other, but it's not clear. Right? But you're going to get, again, sort of an approximate, in results, if the expectation is that whatever you do with Poisson cereal, you can do here, that's the expectation, but with Poisson cereal as well, you can get exact the alteration of this guy, but you would need to go for some approximations if you want the entire Poisson algebra, which you do want. And that's a piece which, again, some people will view as criminal, but I'm not mentioning, but again, these are pieces which I personally do not understand well enough. Perhaps you should perhaps, it seems that the work of the flow, on the object methods, you use the... Actually, there's your work with Michel, the flow work with something called prequantum bundles, which... We work with a compact group of transformation, so we don't use the old NPC structure, but we try to quantize the action of the old, compact group, something similar, and in the literacy stuff, it's what we call the overshift. Yeah, no, I mean, again, you and Michel know this stuff very well and have done a lot of stuff in this direction, they were presentation theoretic stuff, besides the acquisition of the production... I mentioned that, because we are mentioning work with the A-heels, the work with Michel, the flow was around some period. Yeah, so I've not read it, but I'm aware of it, I'm aware of its existence, that he actually defined it, so here we have this sort of circle bundle over the frame bundle, this is connected to something that he calls quantum bundle in the representation theoretic context. I've not actually read the papers, but yes, absolutely, but these ideas definitely, yeah, absolutely, but thank you, thank you very much, yes. You have a twofold cover, which is quantize the... the orbits, do you like taking the square root of quantization? Yeah. So I want to say the work of the flow is quantization... Uh-huh. So, what year? So at some point I should look at the actual paper, I'm aware of it exists from the, again, from the representation theory side. No, thank you. So that's a similar time as has. So I should definitely look at that. Thank you. Now this is very relevant, absolutely. No questions? Thank you very much. Thank you.
 Thank you, organizer, Eckhardt, Xenon, and Paulineel  for this wonderful opportunity.  I'm talking about themes which are very old.  I think in some sense, I'm trying  to understand part of the paper of Harold Hess from 1980,  1981 maybe.  This is what he did in the late 1970s, which  seems to have been largely forgotten.  And in particular, I'm trying to understand  a piece of his paper where he's sort of only given us some,  well, some of what I say is definitely there.  And some of the what I say he knew,  but he's giving us some hints about it,  maybe without all the details.  And that part, I'm actually connecting  with recent work of Jennifer Vaughn.  And maybe the disclaimer is some of the pieces,  the pieces which we're trying to sort of add and fill  the details, or maybe say a little bit more than we said  before, are not that big, but they're not written.  So I guess I should, those pieces,  you can lower the usual disclaimer, take it as 80%  rather than 100%.  But let me tell you the story.  I think this is a very interesting story,  so it goes back to, again, the very early days of Geomete,  or earlier days of Geometeor quantization,  where you can actually hold on to your vector spaces and maps.  Let me first give a time.  Very way too fast review of the constant Soya  version of quantization.  And there's no way I can do this justice in seven minutes,  but I'll try to say something.  You take a symplectic manifold.  You take the prequantizations as circle bundle.  So just for this to exist, you need the two form,  its commodic class to be integral.  And then you take the associated line bundle.  So gamma is a connection, one form,  of this principal circle bundle.  And we insist that, well, up to plus minus signs  may be a factors of 2 pi.  Omega will be the curvature of this connection.  So then associated line bundle then gets a connection as well.  And what constant tells you then is  that the Poisson algebra, as a v algebra,  adds some of the space of some sections  through the constant recipe.  In fact, you can say more if you look at all the quantum  morphisms, the diffumorphisms of this principal bundle,  that are S1 equivariate to the frequency of the connection  one form, the le algebra that is exactly the Poisson algebra.  So the Poisson algebra acts on the total space  of the quantization as all automorphisms.  And through this action, as soon as you act on a principal  bundle, you get an action on the total space  of the associated bundle.  This is a le algebra action.  So it's through vector field.  But then you get an action on the space of all sections.  So that's the constant recipe.  This was well known to many people decades ago.  Costants knew it.  The one place I know which actually writes down  and checks out the details is this recent thing  of Jennifer Bond.  But it was definitely known to many people  and mentioned with some level of details in other places.  And that constant says, well, this is too big.  We're motivated by some concrete examples of physics  and our presentation area.  This is sort of nice and beautiful, but the space is too big.  So to cut down the dimension and recover the examples which  motivate this theory, he needs to add something  about polarization.  And Poisson is a Lagrangian sub-bundle  of the complexified tangent bundle, which is involved in the two  sort of nice examples.  One would be if you have a Lagrangian  affiliation of your original 94 to just complexify.  That gives you polarization.  The other example would be, well,  if you have a scalar structure, you  take the span of del del z bar.  That's a complex formula.  That's a polarization.  So recovering the special cases, like scalar structures  or Lagrangian affiliations.  And the polarization allows Costants  to shrink his space.  He's only taking sections of the line  bundle which are constant along the polarization.  So I'm putting here an F bar.  The literature is a little bit inconsistent  on whether to work with the polarization  or with its complex conjugate.  There's both versions.  They're OK.  And but now the guy which is acting here  is not the entire personel.  You're paying a price.  On one hand, you've got the smaller vector space,  which is what you wanted.  But you have to compromise the functions to which you will  associate operators.  Have to be functions whose Hamiltonian flow  preserves the polarization.  And there's various ways to compensate for that,  which again, this is sort of, again,  trying to do five minutes for a big, big, big story.  I'm not going to, there's ways of obtaining  associating operator to functions  by means of Costants, Stambert, Platinum,  pairing between spaces.  Correspond to different polarizations.  I will not talk about that piece, although some people  would view this as criminal.  Then again, if you look at this, you  say, oh, it's not what I expect from physics.  You need to add the so-called half-form correction.  And when you do the half-form correction,  well, you're starting to limit yourself more.  Not only the homology class of the two-form,  but also half the trunk class of the tangent bubble,  say, equipped with any compatible complex  or almost complex structure, needs to be integral  to be able to do this, to make this correction.  This is necessary.  And this is what we try to avoid or weaken  through lethargic structures.  What pass and other people are trying to weaken that requirement.  Then in that situation, so the manifold  needs to be not only symplectic, but metaplectic.  And you then correct your line of adole by tensorane  with half-form bundles.  And this is going to be the thick corrected bundle  in which the sections you're going to take.  And its first class is not quite as before.  You need to subtract or add dependent conventions.  I think subtracts have the first-form class of the tangent  bundle.  And as soon as you've done this with half-forms,  you've lost your connection.  But your half-forms do come equipped  with a very natural partial connection.  You can still take derivatives in directions  of the polarization.  That's well-defined.  That being an ambiguous false from the involutivity  condition.  So on this bundle, you do have a partial connection  just in the directions of the polarization  using complex conjugate.  And that is enough to make sense of the vector space.  So you can still then take the Hilbert space.  I didn't even say what the inner product is.  That's another story, which I'm sort of admitting,  to be those sections of this corrected line bundle, which  are very constant in the direction of polarization.  And the metaplectic C story is to try  to make sense of Q, even if L and half-forms did not  separately exist.  This actually goes back to the four-hole test  to a paper of C's.  In 1977, I think the paper was 1980,  a later paper which became more famous is that by Robinson  and Gonsley.  I personally prefer Gonsley the approach of S. They are 1989,  which I think really captures the essence of the whole story.  He works with principal bundles.  So he tries to understand that, explain the leaf  theoretic facts behind his construction.  I find that very enlightening.  I personally, on my wish list, besides wanting  to understand things better, there's  a couple of months more recent, very interesting works  by different people who show how headphone corrections really  make you happy.  There's a paper by Colin Kerwin that  tends to do with some version of quantization  to induction, which is given by Lauren Charles.  There's various papers out there that  makes you sort of, that really gives you a lot of faith  and headphone corrections.  And one of some of the items on my own wish list  is to try and revisit those newer papers using the MPC  approach and to, and thus, weaken the assumption  that each one of these be integral.  Maybe a word in terms of actual examples,  which I mentioned in my abstract, some examples of manifolds  which do not admit this old type of quantization  would be CP2.  There's nothing more concrete.  Another one, I guess, we've seen the pictures  of Michel Verstock of quantization of a Torque  manifold.  And in Michel's picture, you've seen those twisted pictures  where the edges were lying exactly midway between lattice  points.  These are the pictures you would be getting from an MCPC  structure.  And again, I think I was involved with several things  some years ago, but the place where you can actually  see an actual statement to prove that these Torque  manifold are MPC pre-quantizable if and only if the edges lying  midway between lattice points is in a recent paper  by Jennifer Vaughn.  And so Torque manifold specifically,  you might have a half-on correction, but not  equivalently.  You're never going to have a equivalent half-on bundle.  You will have an equivalent metaplexy structure.  So to do them, if you care about group actions,  you really are getting.  In the equivalent case, this compromises bigger than in the  non-quantable case.  And you overcome it with metaplexy.  So I'm going to talk a bit about this recipe.  And I want to start with the linear algebra, which is  relevant here.  And I will start by just fixing a model symplectic vector space,  just a standard R2-A.  And we start with the metaplexic group for this guy.  So I don't want to work with the actual construction.  I just want to work with the properties.  The group of symplectic linear transformations, it's pi 1  is that.  So it has a unique uptoizomorphism z2 extension  connected z2 extension.  So this would be the definition of the  metaplexic group.  And for the linear algebra, that will often be enough.  If you try to work over manifold, some manifolds will  not have metaplexic structures.  So for manifolds, if you have a symplectic manifold, you can  think of it as a relevant structural group with a  symplectic linear group.  That's a group acting on the principle bundle of  symplectic frames.  And you would be looking for a lifting of the structural  group to the metaplexic group.  And that might not exist.  But you might have a lifting of the structural group to the  metaplexic group.  The metaplexic C group is just a  corresponding circle.  Central extension.  And that's the one that you will have in a lot more  manifolds.  So for the linear algebra, it doesn't make that much of a  difference if you're doing the linear algebra.  Let's work with the metaplexic C group.  So there's two sort of things you can do.  You can think of a circle as being a subgroup of this guy.  And then the quotient will be the group of symplectic  linear transformation.  This is exactly what I'm just repeating what I said.  But you can also, from this description, you also see another  picture.  You see the metaplexic group sitting inside.  And the quotient, well, the quotient is a circle, but it's  not the same circle.  We don't have color.  Oh, yes, we do that.  Color chart.  OK.  It's not the same circle.  If you embed the center, and then you take the homophysm,  this homophysm sometimes really  gives the letter eta to denote it.  This is the square map.  And really, the circle that you care about is the center, which  is a double covering of the circle that you can obtain as  the image of the metaplexic group.  So let's do the linear algebra for polarizations.  So we take the complexification of our vector space  together with the complexification of the  symplectic form.  And then you take a complex subspace, which is Lagrangian  with respect to that.  And there's a few things you can do with it.  You might want to look at intersection with a  complex conjugate.  That is going to have a form.  This is going to be the complexification of some  isotropic real subspace of our symplectic vector space.  Or you might want to look at the sum of the Lagrangian with  its complex conjugate.  And this is going to be the complexification of a  quasitropic.  I'm sorry?  Oh, not the right sum.  You're right.  Just thank you very much.  Just the ordinary sum.  Because it depends on the type of the sum.  I definitely do not want to insist that this is the  right sum.  Thank you.  And in fact, d is the null space of e. d, in fact, is the  symplectic orthogonal to e, and vice versa.  And the symplectic form gives us a naturalizing  complexification between the quotient and the dual.  Quotient v mod e with the dual of d.  Now, if you have a real linear symplectomal fission, it  acts like a complexified vector space, just  complexify, and preserves the complex  complex structure.  So it makes sense to ask, does it or does it not send the  transient subspace to itself?  And we put a little subscript f to refer to those  vices, too.  It doesn't have to fix that.  It just has to send f to itself.  So the symplectic, the symplectonfism that  the fish symplectonfism sends f to itself.  Well, you can complexify, and then you can restrict to f.  And you just get a linear transformation, complex linear  transformation on f.  So this map is complexified and restrict.  And the kernel of this map consists of those symplecton  frisms that actually not only leave f invariant, but in fact  fix f.  And a crucial detail, one of those which are not explicit  in this, but it's actually an exercise, is the fact that  this kernel is contractible.  And that's going to be better.  Do I have space to write this?  I have space to put an error.  kf is contractible.  So what do we do with that?  Let me go back to the original diagram that you've already  seen.  Metaplexic group has a double covering of a symplectic group.  And now let me restrict to those guys that  preserve not fix, preserve f.  Then we can take their preimage and the metaplexic group.  We have a smaller group.  And maybe the guys that preserve f got the double covering of that.  Now the crucial observation is the following.  Let me tell you where I'm heading.  So if you have a symplectic manifold, your structure group  is the symplectic linear group.  You want to lift it to the metaplexic group.  And you can do that.  And what you would like to do is to get  an associated line bundle and then take its space of sections.  The point is you don't have an interesting associated line  bundle.  You could take one which is associated to the map  to S1 I wrote before.  But you really want to square root of that.  You want to somehow the center is supposed to play a role here.  What turns out is you need to first reduce the structure  group before you can take an associated line bundle.  And the choice of polarization gives you  this reduction of structure group.  And we're now doing the thing which  is happening on the linear algebra level.  I need to give you a to play this game.  I need to give you a homomorphism from the metaplexic group  subscript C f, this reduced guy, to say C star.  So it can act on C and we can take an associated line bundle.  Let's actually do that.  I mean, this is what Hess does with omitting some details,  but he catches it.  So here we have the map, which is, which we wrote before,  complexify and restrict to go from simplectonophysm that  preserve f to complex linear isomorphisms of f.  Now, we'd like to take the square root of the complex  determinant.  And we can't quite do that.  Well, for that reason, well, the metalinier group  was invented for that reason.  This is the double covering of the GL  of the general linear group on which you can actually  take the square root of the determinant.  But then the question is, we'd like  to start with the metaplexic group, the piece of it,  which preserves f rather than.  So we're looking for an arrow from it  to the metalinier group to be able to obtain  our homomorphism to C star.  And the point is that this actually does exist.  And the reason that it exists is the map  from the simplectonophysm to the general linear group of f,  its kernel, it's straight through the quotient  by that kernel kf.  And that kernel was contractible.  And that guarantees that we can find another copy of it  upstairs.  So the preimage of it, I don't want to call it,  in the double covering.  The metaplexic group is, in fact, disconnected.  So you can take one component.  So you just get a copy of the same guy.  And it's going to get this map, which factors  through quotient by that lifted copy of the kernel.  So what does it get to get from this whole story?  The piece of the metalinier group that preserves f  comes equipped with a homomorphism to C star,  which provides us with a square root of the map  that we described before, namely,  complexify, restrict, detentate determinant.  And then it's easy to extend it further to the metalinier C  group.  I'm going to use this by the same letter, chi.  Again, those elements of the group that preserve f, namely,  you take C in the circle.  You take g tilde in the metalinier group.  And you map them to C times chi of g tilde.  And the nice thing here is this homomorphism,  its restriction to the center, is just  the inclusion of the circle in C star.  And that's going to be the guide for you.  So that adds the linear algebra part of this story.  And at this point, we want to take that part  and throw it on a manifold and see what we've got.  So here's a symplectic manifold.  And now I have the symplectic frame bundle.  And I have to admit I have some mixed feelings  about this notation.  The fiber is not a group.  The fiber is a space in which the group  acts freely and transitively.  It's the principal bundle for the group as pp.  To wish that you need to make a choice.  Like if you choose a base point.  So that's why I said I have mixed feelings  about the notation.  You see, there's really so what's happening here.  The fiber at each point of the frame bundle  is the set of symplectic linear isomorphism  from our standard model vector space  to that tangent space at that point.  Really what you have here, you have  the principal right action of the group of symplectic linear  transformation of the model vector space v.  You also have a left action of the symplectic morphism  group of that particular fiber.  And there's context where I expect  this is actually relevant.  So in those contexts, we would want to distinguish them.  But let me still use this notation for now.  And we want it to a lifting of structure group, as I promised.  So we're going to have three levels.  Here we have a principal MPC bundle over the manifold.  But we insist that it be consistent with the principal  frame bundle in the following sense.  The principal frame bundle is a principal bundle  for the symplectic linear group.  We insist to have a map from the principal MPC  bundle to the symplectic frame bundle, which  is equivariant with respect to the group MPC, which  acts on sp through its homomorphism to the group sp.  So this is just a print.  What we have at the top level is just a principal circle  bundle.  That's the demand is across in Surreal.  And this circle is just the center of the group MPC,  MPC of the standard MPC group.  And so over each and every point of end,  we just have the diagram of groups that we wrote before.  But these are spaces in which these groups  act freely and transitively.  Then we want to put P with something  that we call the central connection.  Basically, I want a connection that only sees the central part.  Has called it by a different name.  He calls it an ecto-soydo connection,  because he has it as a special case of something more general.  But let me just stick with the more sort of,  friendly language central connection.  By definition, this is going to be a one form gamma on P  that has a fine properties.  That it has to be an MPC equivalent connection  on P as a principal circle bundle over the symplectic frame  bundle.  And this connection, you see inside here,  we've got the metaplectic directions.  Each metaplectic orbit is a double covering  of an orbit for the frame bundle.  So it has to be zero in the metaplectic direction,  such that MP orbits are horizontal.  And these assumptions imply that the differential  of this one form is basic.  It comes from a two form and the manifold  in the very same way that you have for the behavior of connections  on principal circle bundles that you do in the constant  surreal place.  So then we can write down a different definition.  Sorry, you know what?  Is every variant means invariant?  I think it means invariant.  Yes.  Yeah, you're right.  It should be invariant.  It should be invariant.  Thank you.  Yeah.  Thank you.  Because this is just with real coefficients  and the real numbers here are just  the algebra of the centers, everything  x trivially omits.  So yes, invariant.  So then we can define NPC pre-plantization  in analogy with the constant surreal case.  And NPCization of a cyclic manifold  is exactly this type of structure.  It's a lifting of the structure group to the NPC group  together with a central connection.  The curvature is the pullback of the two form.  Complete analogy with constant surreal.  Now we'd like to construct the line bundle.  And we'd like our Hilbert space to be obtained  from sections of that line bundle.  And we'd like the Poisson algebra to act on that Hilbert space  so that each function will go to an operator.  So again, an analogy with constants surreal.  And again, the reference for one is the constant surreal.  And again, the reference for one of the objecies  of Jennifer Bay-Probe, Jennifer Vaughn.  We have a notion of NPC, quantum of physics.  It's almost what you think it is.  It probably is what you think it is,  but there's a subtly which at least escaped me  until Jennifer Vaughn pointed it out and clarified it.  So this is not, I think this is not N-Has.  This is an analogy quote to the stuff of constants surreal.  So these are the NPC, a covariant of the film of physics,  that preserve the central connection.  And well, already here, if you're NPC,  if we're going to descend to the lower levels,  and I want these lower defimophysm to be compatible  in the following sense.  It has to descend to the filmophysm,  the side tilde of the middle piece and side amp of the lower piece,  such that side tilde is the natural lifting,  or we can call it prolongation if you wish,  of the side amp.  So this is the subtlety,  but it actually makes everything work really nicely.  So what we have in this situation for quantum morphisms,  this is the relevant notion of an ultramorphism,  and maybe I'll write it on the more bigger space.  So here's our quantum morphism.  Here are the guys that descend to.  This is the side tilde, this is the side amp,  and you see the side amp will be a symphlectomorphism,  so it lifts to a map of the symphlectic frame bundles,  and we want that lifting to coincide with side tilde.  That's an actual requirement, which needs to be imposed.  And with that requirement, everything is nice and beautiful,  because once, first of all, the group of quantum morphism itself  is going to be a circle-central extension of a group  which acts on amps, a group of the symphlectomorphism group,  which might have several components,  but the identity component is the different Hamiltonian transformations.  So if you wish, I'll give you a notion of Hamiltonian transformation,  which has several components.  In the infinitesimal guys,  these are just the vector fields of the total space  that satisfy the corresponding infinitesimal conditions,  or if you wish, the flow that they generate  satisfy these conditions.  So the infinitesimal quantum morphisms,  that's on the algebra, which is naturally isomorphic  to the Poisson algebra, and that's kind of nice.  The isomorphic frame, an infinitesimal quantum morphism  is just a vector field on the total space.  So you can plug it into the central connection  to get a function on the total space,  and that function will actually come from the base,  and this is the factor plus minus,  this would be the function of the Poisson element  of the Poisson algebra that responds  to this infinitesimal quantum morphism.  So the Poisson algebra acts as all the automorphisms  of this pre-quantization structure,  and therefore it will act on any associated line bundle  and the space of sections.  So if you have any line bundle or vector bundle,  which is associated, then the algebra will act  totally naturally on its space of sections,  except as I said before, this is a little bit too good.  You need to restrict the structural group  to actually have such a line bundle which is interesting.  Let's do that restriction.  So we're choosing a polarization on our manifold,  and inverted the grand-gen sub-bundle,  the conflexified tangent bundle,  and I want it to satisfy some non-negativity condition,  which I will not allow you to do.  So I'm going to do this restriction  and I'm going to do this restriction  and I'm going to do this restriction  by some non-negativity condition, which I will not elaborate on.  In the purely complex case, this would mean  that this is an actual k-o form.  And I want to make two assumptions,  which are also made in the constant-surreal context,  so this is not restricting from the constant-surreal context.  I want the real piece to have constant rank.  This is the isotropic piece.  And this piece gives us an isotropic affiliation theme of that.  In the K-lo case, this is just tricky.  And the piece I'm not going to elaborate on  is the Hilbert space structure and the resultant vector space,  but let me still list what is needed  for that Hilbert space structure.  You need that as a tropic affiliation to be fibrin.  So you don't require E to be integral?  To be?  No, I don't require E to be integral.  But depending on what your goals are,  there are some applications for which you do want to require it.  So basically what I'm claiming here is the type of requirements you make  are an analogy to what is done in the constant-surreal case.  So what we have here, we have a fiber bundle,  whose fibers are the leaves of E.  And this we will need for the Hilbert space structure,  which I will not get to.  Now the constant rank of the real part  tells us that we are able to actually fix the model for this polarization.  But this position really will come from,  will correspond to a reduction of structure groups.  I'm fixing a Lagrangian complex subspace of my standards  in the particular space such as f intersect f bar has the correct rank.  And what you get is a reduction of structure group.  So we'll first of all lift at the circular group.  That corresponds to a central extension.  And then we take a reduction of structure group,  which is correspond to subgroup.  So if we have the original tower,  we can now look at those symplectic frames,  which are, which carry the standard Lagrangian to the Lagrangian,  which is given by the polarization.  And the treatment of that by definition is p sub f.  And again, we have a tower,  except now the group downstairs is the subgroup  consisting of symplectic transformation preserving f,  and the group upstairs is a subgroup.  But this is exactly the subgroup, the MPC subgroup f.  That's the one that had a homophysicistar,  which we are kind of happy about.  We then have the relevant part of the Poisson algebra.  These are the functions with somatonic flow  preserve the polarization at pf.  In fact, that is going to be the group of all quantumophysicists  that preserve f.  We can still restrict our central connection to the,  and therefore it acts on the associated line bundle,  and this is the line bundle that we sort of, you know, wanted.  Now, what else do we have?  We have the central connection.  So here has actually uses a pair of polarizations  and from the pair of the polarization that are related  and sort of the way he obtains an honest to God connection  of this line bundle.  And then he mentioned, oh, by the way,  you can just do one polarization  and you get a partial connection.  So this is one of the pieces where I've checked some,  but not all of the details.  But let me just write this.  I believe, I mean, okay, I do believe that this works.  So you do get a completely natural partial connection  on q defined when you take derivatives on vector fields  in the direction of the polarization towards complex conjugate.  Now, where is this coming from?  Basically, there's two ingredients here.  One of them is the central connection, which we were given,  and second, the so-called God connection,  which is resulting from the involutivity of f.  You get a natural connection on the tangent of the mod f  and on various bundles that you can build from that one.  And that uses the involutivity.  And then, then you can just continue as you expect.  You can take the Hilbert space, and I'm writing the Dutch space  to be the space of sections of this quantization line bundle,  which are very constant in the directions of the polarization.  And we have our quantumophisms acting there.  So you get a totally natural action of that part of the Poisson algebra,  which preserve the polarization.  And then one can try to sweat and get operators associated to other functions,  but that has to do something about those.  I'm not pretty sure I'm just saying anything about that.  Maybe a word about the inner product.  I'm just going to say not write down.  Again, Hess does explain that, and it is in analogy with what happens with constants through yaw.  If you take sections of q, which are covariant constants along the polarization,  there is a natural point-wise inner product of such sections  that takes values not in scalars, but rather in densities on mod d.  And then you integrate that to get a number, and that gives you the inner product on your Hilbert space.  I'm done!  APPLAUSE  That's what's going to go.  Yeah.  Well, this Hilbert space, it has square interior sections,  only if leaves of d are...  If leaves of d are tori...  You integrate over the quotient.  You integrate the density over the quotient.  Yes.  The fibers could be...  They covariantly cost sections along the tori with the indistributions.  Right?  You can't hit the words on the front block.  There is a...  No, no, no, no.  Oh, okay.  You're right.  It's the same story as...  It's the same stories as you have in...  No, you're absolutely right.  It's the same stories as you have in the cross-sensory case.  First of all, you could look at situations where the leaves of d are not compact,  and then you're okay, and you're fine.  If the leaves of d are compact, you're in trouble, as we learned from you.  You're in trouble.  But then you can play the same games as you do in the cross-sensory case.  You can take the sheaf of sections with this property,  and that sheaf you can take underneath some of the homologes of the sheaf,  and things about that, and you expect to get a new formula for the house.  But you can do more things.  But this...  No, thank you very much.  You're absolutely right.  You can get in big trouble with the game very soon.  In physics, a scalar product is an essential physical object.  Here, as long as everything is nice, you put it within the structure  by putting a metapractic structure or a PC structure.  And here it comes, if you don't see, you get it.  But once you start reducing some symmetries, you lose it.  There are versions of reduction here.  Again, there's some version of reduction.  I mean, there's a lot of things happening here,  many of which I understand will be partially...  But yeah, there's a lot of things you can do.  Jennifer Bond has a notion of Borsamfeld,  which, for example, in the Torreke-Mannifold case,  she actually shows that the Borsamfeld leaves are exactly the ones  you get from the lattice points as in Michel's picture.  Again, there's earlier notions of Borsamfeld due to...  I think Robinson, the same guy from Robinson-Ronson.  Jennifer's definition is a little bit different.  I mean, there's a lot of stories there,  which do not provide, as far as I know,  do not limit you more than you were limited in the Constance-Rial case.  But at the same time, there's the potential to address more manifolds  which do not admit an effect.  No, I just wondered if you actually asked the wrong question.  I just wondered if you managed to overcome the problem here.  No, no, no, this is complete analogy.  But actually one of the things we were looking at with Mark,  that I think we'll tell you later this week,  has to do with some of these analogies.  But again, it's not me and Tess.  But you told us about your friction, the line in these structures?  Oh, oh, oh, I'm sorry.  So a metaphylactic C-prequantization exists if and only if difference.  That's the first-run class of your line bundle that you care about.  Even only if this difference is integral in cosmology.  And it is unique up to twisting by pull delets of flat circle bundles on that.  So the extent to which it is unique is analogous to Constance-Rial,  and this is explained by Haasel.  And this is different in that for Constance-Rial,  you need each one of them to be integral,  and here we need just the difference, which is kind of the news,  because this is the John Klassler of the line bundle that you actually care about.  Yeah.  This is sort of a cultural, maybe, comment, even.  Yes, a lot of this is about cultural stuff.  So I was surprised by the preoccupation with an action of the Poisson algebra,  since I thought that was just sort of a historical mistake,  because that's not what you want in physics.  You want to take functions of the gate operators, right?  Yes, and they will only approximately correspond to the Poisson practice.  So the thing is, if you restrict, so this is a huge restriction,  so these guys, you can actually make them correspond on the nose,  but if you want the rest of the Poisson algebra,  then you have to allow certain approximations, etc.  But these guys, if you follow the constant recipe, these guys, you get on the nose.  You get isomorphism only for this,  and perhaps some other, but it's not clear.  Right?  But you're going to get, again, sort of an approximate,  in results, if the expectation is that whatever you do with Poisson cereal,  you can do here, that's the expectation,  but with Poisson cereal as well,  you can get exact the alteration of this guy,  but you would need to go for some approximations  if you want the entire Poisson algebra, which you do want.  And that's a piece which, again, some people will view as criminal,  but I'm not mentioning, but again, these are pieces which I personally do not understand well enough.  Perhaps you should perhaps, it seems that the work of the flow,  on the object methods, you use the...  Actually, there's your work with Michel,  the flow work with something called prequantum bundles, which...  We work with a compact group of transformation,  so we don't use the old NPC structure, but we try to quantize the action  of the old, compact group, something similar,  and in the literacy stuff, it's what we call the overshift.  Yeah, no, I mean, again, you and Michel know this stuff very well  and have done a lot of stuff in this direction,  they were presentation theoretic stuff, besides the acquisition of the production...  I mentioned that, because we are mentioning work with the A-heels,  the work with Michel, the flow was around some period.  Yeah, so I've not read it, but I'm aware of it, I'm aware of its existence,  that he actually defined it, so here we have this sort of circle bundle over the frame bundle,  this is connected to something that he calls quantum bundle in the representation theoretic context.  I've not actually read the papers, but yes, absolutely,  but these ideas definitely, yeah, absolutely, but thank you, thank you very much, yes.  You have a twofold cover, which is quantize the...  the orbits, do you like taking the square root of quantization?  Yeah.  So I want to say the work of the flow is quantization...  Uh-huh.  So, what year?  So at some point I should look at the actual paper,  I'm aware of it exists from the, again, from the representation theory side.  No, thank you.  So that's a similar time as has.  So I should definitely look at that.  Thank you.  Now this is very relevant, absolutely.  No questions?  Thank you very much.  Thank you.
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Thank you, organizer, Eckhardt, Xenon, and Paulineel for this wonderful opportunity. I'm talking about themes which are very old. I think in some sense, I'm trying to understand part of the paper of Harold Hess from 1980, 1981 maybe. This is what he did in the late 1970s, which seems to have been largely forgotten. And in particular, I'm trying to understand a piece of his paper where he's sort of only given us some, well, some of what I say is definitely there. And some of the what I say he knew, but he's giving us some hints about it, maybe without all the details. And that part, I'm actually connecting with recent work of Jennifer Vaughn. And maybe the disclaimer is some of the pieces, the pieces which we're trying to sort of add and fill the details, or maybe say a little bit more than we said before, are not that big, but they're not written. So I guess I should, those pieces, you can lower the usual disclaimer, take it as 80% rather than 100%. But let me tell you the story. I think this is a very interesting story, so it goes back to, again, the very early days of Geomete, or earlier days of Geometeor quantization, where you can actually hold on to your vector spaces and maps. Let me first give a time. Very way too fast review of the constant Soya version of quantization. And there's no way I can do this justice in seven minutes, but I'll try to say something. You take a symplectic manifold. You take the prequantizations as circle bundle. So just for this to exist, you need the two form, its commodic class to be integral. And then you take the associated line bundle. So gamma is a connection, one form, of this principal circle bundle. And we insist that, well, up to plus minus signs may be a factors of 2 pi. Omega will be the curvature of this connection. So then associated line bundle then gets a connection as well. And what constant tells you then is that the Poisson algebra, as a v algebra, adds some of the space of some sections through the constant recipe. In fact, you can say more if you look at all the quantum morphisms, the diffumorphisms of this principal bundle, that are S1 equivariate to the frequency of the connection one form, the le algebra that is exactly the Poisson algebra. So the Poisson algebra acts on the total space of the quantization as all automorphisms. And through this action, as soon as you act on a principal bundle, you get an action on the total space of the associated bundle. This is a le algebra action. So it's through vector field. But then you get an action on the space of all sections. So that's the constant recipe. This was well known to many people decades ago. Costants knew it. The one place I know which actually writes down and checks out the details is this recent thing of Jennifer Bond. But it was definitely known to many people and mentioned with some level of details in other places. And that constant says, well, this is too big. We're motivated by some concrete examples of physics and our presentation area. This is sort of nice and beautiful, but the space is too big. So to cut down the dimension and recover the examples which motivate this theory, he needs to add something about polarization. And Poisson is a Lagrangian sub-bundle of the complexified tangent bundle, which is involved in the two sort of nice examples. One would be if you have a Lagrangian affiliation of your original 94 to just complexify. That gives you polarization. The other example would be, well, if you have a scalar structure, you take the span of del del z bar. That's a complex formula. That's a polarization. So recovering the special cases, like scalar structures or Lagrangian affiliations. And the polarization allows Costants to shrink his space. He's only taking sections of the line bundle which are constant along the polarization. So I'm putting here an F bar. The literature is a little bit inconsistent on whether to work with the polarization or with its complex conjugate. There's both versions. They're OK. And but now the guy which is acting here is not the entire personel. You're paying a price. On one hand, you've got the smaller vector space, which is what you wanted. But you have to compromise the functions to which you will associate operators. Have to be functions whose Hamiltonian flow preserves the polarization. And there's various ways to compensate for that, which again, this is sort of, again, trying to do five minutes for a big, big, big story. I'm not going to, there's ways of obtaining associating operator to functions by means of Costants, Stambert, Platinum, pairing between spaces. Correspond to different polarizations. I will not talk about that piece, although some people would view this as criminal. Then again, if you look at this, you say, oh, it's not what I expect from physics. You need to add the so-called half-form correction. And when you do the half-form correction, well, you're starting to limit yourself more. Not only the homology class of the two-form, but also half the trunk class of the tangent bubble, say, equipped with any compatible complex or almost complex structure, needs to be integral to be able to do this, to make this correction. This is necessary. And this is what we try to avoid or weaken through lethargic structures. What pass and other people are trying to weaken that requirement. Then in that situation, so the manifold needs to be not only symplectic, but metaplectic. And you then correct your line of adole by tensorane with half-form bundles. And this is going to be the thick corrected bundle in which the sections you're going to take. And its first class is not quite as before. You need to subtract or add dependent conventions. I think subtracts have the first-form class of the tangent bundle. And as soon as you've done this with half-forms, you've lost your connection. But your half-forms do come equipped with a very natural partial connection. You can still take derivatives in directions of the polarization. That's well-defined. That being an ambiguous false from the involutivity condition. So on this bundle, you do have a partial connection just in the directions of the polarization using complex conjugate. And that is enough to make sense of the vector space. So you can still then take the Hilbert space. I didn't even say what the inner product is. That's another story, which I'm sort of admitting, to be those sections of this corrected line bundle, which are very constant in the direction of polarization. And the metaplectic C story is to try to make sense of Q, even if L and half-forms did not separately exist. This actually goes back to the four-hole test to a paper of C's. In 1977, I think the paper was 1980, a later paper which became more famous is that by Robinson and Gonsley. I personally prefer Gonsley the approach of S. They are 1989, which I think really captures the essence of the whole story. He works with principal bundles. So he tries to understand that, explain the leaf theoretic facts behind his construction. I find that very enlightening. I personally, on my wish list, besides wanting to understand things better, there's a couple of months more recent, very interesting works by different people who show how headphone corrections really make you happy. There's a paper by Colin Kerwin that tends to do with some version of quantization to induction, which is given by Lauren Charles. There's various papers out there that makes you sort of, that really gives you a lot of faith and headphone corrections. And one of some of the items on my own wish list is to try and revisit those newer papers using the MPC approach and to, and thus, weaken the assumption that each one of these be integral. Maybe a word in terms of actual examples, which I mentioned in my abstract, some examples of manifolds which do not admit this old type of quantization would be CP2. There's nothing more concrete. Another one, I guess, we've seen the pictures of Michel Verstock of quantization of a Torque manifold. And in Michel's picture, you've seen those twisted pictures where the edges were lying exactly midway between lattice points. These are the pictures you would be getting from an MCPC structure. And again, I think I was involved with several things some years ago, but the place where you can actually see an actual statement to prove that these Torque manifold are MPC pre-quantizable if and only if the edges lying midway between lattice points is in a recent paper by Jennifer Vaughn. And so Torque manifold specifically, you might have a half-on correction, but not equivalently. You're never going to have a equivalent half-on bundle. You will have an equivalent metaplexy structure. So to do them, if you care about group actions, you really are getting. In the equivalent case, this compromises bigger than in the non-quantable case. And you overcome it with metaplexy. So I'm going to talk a bit about this recipe. And I want to start with the linear algebra, which is relevant here. And I will start by just fixing a model symplectic vector space, just a standard R2-A. And we start with the metaplexic group for this guy. So I don't want to work with the actual construction. I just want to work with the properties. The group of symplectic linear transformations, it's pi 1 is that. So it has a unique uptoizomorphism z2 extension connected z2 extension. So this would be the definition of the metaplexic group. And for the linear algebra, that will often be enough. If you try to work over manifold, some manifolds will not have metaplexic structures. So for manifolds, if you have a symplectic manifold, you can think of it as a relevant structural group with a symplectic linear group. That's a group acting on the principle bundle of symplectic frames. And you would be looking for a lifting of the structural group to the metaplexic group. And that might not exist. But you might have a lifting of the structural group to the metaplexic group. The metaplexic C group is just a corresponding circle. Central extension. And that's the one that you will have in a lot more manifolds. So for the linear algebra, it doesn't make that much of a difference if you're doing the linear algebra. Let's work with the metaplexic C group. So there's two sort of things you can do. You can think of a circle as being a subgroup of this guy. And then the quotient will be the group of symplectic linear transformation. This is exactly what I'm just repeating what I said. But you can also, from this description, you also see another picture. You see the metaplexic group sitting inside. And the quotient, well, the quotient is a circle, but it's not the same circle. We don't have color. Oh, yes, we do that. Color chart. OK. It's not the same circle. If you embed the center, and then you take the homophysm, this homophysm sometimes really gives the letter eta to denote it. This is the square map. And really, the circle that you care about is the center, which is a double covering of the circle that you can obtain as the image of the metaplexic group. So let's do the linear algebra for polarizations. So we take the complexification of our vector space together with the complexification of the symplectic form. And then you take a complex subspace, which is Lagrangian with respect to that. And there's a few things you can do with it. You might want to look at intersection with a complex conjugate. That is going to have a form. This is going to be the complexification of some isotropic real subspace of our symplectic vector space. Or you might want to look at the sum of the Lagrangian with its complex conjugate. And this is going to be the complexification of a quasitropic. I'm sorry? Oh, not the right sum. You're right. Just thank you very much. Just the ordinary sum. Because it depends on the type of the sum. I definitely do not want to insist that this is the right sum. Thank you. And in fact, d is the null space of e. d, in fact, is the symplectic orthogonal to e, and vice versa. And the symplectic form gives us a naturalizing complexification between the quotient and the dual. Quotient v mod e with the dual of d. Now, if you have a real linear symplectomal fission, it acts like a complexified vector space, just complexify, and preserves the complex complex structure. So it makes sense to ask, does it or does it not send the transient subspace to itself? And we put a little subscript f to refer to those vices, too. It doesn't have to fix that. It just has to send f to itself. So the symplectic, the symplectonfism that the fish symplectonfism sends f to itself. Well, you can complexify, and then you can restrict to f. And you just get a linear transformation, complex linear transformation on f. So this map is complexified and restrict. And the kernel of this map consists of those symplecton frisms that actually not only leave f invariant, but in fact fix f. And a crucial detail, one of those which are not explicit in this, but it's actually an exercise, is the fact that this kernel is contractible. And that's going to be better. Do I have space to write this? I have space to put an error. kf is contractible. So what do we do with that? Let me go back to the original diagram that you've already seen. Metaplexic group has a double covering of a symplectic group. And now let me restrict to those guys that preserve not fix, preserve f. Then we can take their preimage and the metaplexic group. We have a smaller group. And maybe the guys that preserve f got the double covering of that. Now the crucial observation is the following. Let me tell you where I'm heading. So if you have a symplectic manifold, your structure group is the symplectic linear group. You want to lift it to the metaplexic group. And you can do that. And what you would like to do is to get an associated line bundle and then take its space of sections. The point is you don't have an interesting associated line bundle. You could take one which is associated to the map to S1 I wrote before. But you really want to square root of that. You want to somehow the center is supposed to play a role here. What turns out is you need to first reduce the structure group before you can take an associated line bundle. And the choice of polarization gives you this reduction of structure group. And we're now doing the thing which is happening on the linear algebra level. I need to give you a to play this game. I need to give you a homomorphism from the metaplexic group subscript C f, this reduced guy, to say C star. So it can act on C and we can take an associated line bundle. Let's actually do that. I mean, this is what Hess does with omitting some details, but he catches it. So here we have the map, which is, which we wrote before, complexify and restrict to go from simplectonophysm that preserve f to complex linear isomorphisms of f. Now, we'd like to take the square root of the complex determinant. And we can't quite do that. Well, for that reason, well, the metalinier group was invented for that reason. This is the double covering of the GL of the general linear group on which you can actually take the square root of the determinant. But then the question is, we'd like to start with the metaplexic group, the piece of it, which preserves f rather than. So we're looking for an arrow from it to the metalinier group to be able to obtain our homomorphism to C star. And the point is that this actually does exist. And the reason that it exists is the map from the simplectonophysm to the general linear group of f, its kernel, it's straight through the quotient by that kernel kf. And that kernel was contractible. And that guarantees that we can find another copy of it upstairs. So the preimage of it, I don't want to call it, in the double covering. The metaplexic group is, in fact, disconnected. So you can take one component. So you just get a copy of the same guy. And it's going to get this map, which factors through quotient by that lifted copy of the kernel. So what does it get to get from this whole story? The piece of the metalinier group that preserves f comes equipped with a homomorphism to C star, which provides us with a square root of the map that we described before, namely, complexify, restrict, detentate determinant. And then it's easy to extend it further to the metalinier C group. I'm going to use this by the same letter, chi. Again, those elements of the group that preserve f, namely, you take C in the circle. You take g tilde in the metalinier group. And you map them to C times chi of g tilde. And the nice thing here is this homomorphism, its restriction to the center, is just the inclusion of the circle in C star. And that's going to be the guide for you. So that adds the linear algebra part of this story. And at this point, we want to take that part and throw it on a manifold and see what we've got. So here's a symplectic manifold. And now I have the symplectic frame bundle. And I have to admit I have some mixed feelings about this notation. The fiber is not a group. The fiber is a space in which the group acts freely and transitively. It's the principal bundle for the group as pp. To wish that you need to make a choice. Like if you choose a base point. So that's why I said I have mixed feelings about the notation. You see, there's really so what's happening here. The fiber at each point of the frame bundle is the set of symplectic linear isomorphism from our standard model vector space to that tangent space at that point. Really what you have here, you have the principal right action of the group of symplectic linear transformation of the model vector space v. You also have a left action of the symplectic morphism group of that particular fiber. And there's context where I expect this is actually relevant. So in those contexts, we would want to distinguish them. But let me still use this notation for now. And we want it to a lifting of structure group, as I promised. So we're going to have three levels. Here we have a principal MPC bundle over the manifold. But we insist that it be consistent with the principal frame bundle in the following sense. The principal frame bundle is a principal bundle for the symplectic linear group. We insist to have a map from the principal MPC bundle to the symplectic frame bundle, which is equivariant with respect to the group MPC, which acts on sp through its homomorphism to the group sp. So this is just a print. What we have at the top level is just a principal circle bundle. That's the demand is across in Surreal. And this circle is just the center of the group MPC, MPC of the standard MPC group. And so over each and every point of end, we just have the diagram of groups that we wrote before. But these are spaces in which these groups act freely and transitively. Then we want to put P with something that we call the central connection. Basically, I want a connection that only sees the central part. Has called it by a different name. He calls it an ecto-soydo connection, because he has it as a special case of something more general. But let me just stick with the more sort of, friendly language central connection. By definition, this is going to be a one form gamma on P that has a fine properties. That it has to be an MPC equivalent connection on P as a principal circle bundle over the symplectic frame bundle. And this connection, you see inside here, we've got the metaplectic directions. Each metaplectic orbit is a double covering of an orbit for the frame bundle. So it has to be zero in the metaplectic direction, such that MP orbits are horizontal. And these assumptions imply that the differential of this one form is basic. It comes from a two form and the manifold in the very same way that you have for the behavior of connections on principal circle bundles that you do in the constant surreal place. So then we can write down a different definition. Sorry, you know what? Is every variant means invariant? I think it means invariant. Yes. Yeah, you're right. It should be invariant. It should be invariant. Thank you. Yeah. Thank you. Because this is just with real coefficients and the real numbers here are just the algebra of the centers, everything x trivially omits. So yes, invariant. So then we can define NPC pre-plantization in analogy with the constant surreal case. And NPCization of a cyclic manifold is exactly this type of structure. It's a lifting of the structure group to the NPC group together with a central connection. The curvature is the pullback of the two form. Complete analogy with constant surreal. Now we'd like to construct the line bundle. And we'd like our Hilbert space to be obtained from sections of that line bundle. And we'd like the Poisson algebra to act on that Hilbert space so that each function will go to an operator. So again, an analogy with constants surreal. And again, the reference for one is the constant surreal. And again, the reference for one of the objecies of Jennifer Bay-Probe, Jennifer Vaughn. We have a notion of NPC, quantum of physics. It's almost what you think it is. It probably is what you think it is, but there's a subtly which at least escaped me until Jennifer Vaughn pointed it out and clarified it. So this is not, I think this is not N-Has. This is an analogy quote to the stuff of constants surreal. So these are the NPC, a covariant of the film of physics, that preserve the central connection. And well, already here, if you're NPC, if we're going to descend to the lower levels, and I want these lower defimophysm to be compatible in the following sense. It has to descend to the filmophysm, the side tilde of the middle piece and side amp of the lower piece, such that side tilde is the natural lifting, or we can call it prolongation if you wish, of the side amp. So this is the subtlety, but it actually makes everything work really nicely. So what we have in this situation for quantum morphisms, this is the relevant notion of an ultramorphism, and maybe I'll write it on the more bigger space. So here's our quantum morphism. Here are the guys that descend to. This is the side tilde, this is the side amp, and you see the side amp will be a symphlectomorphism, so it lifts to a map of the symphlectic frame bundles, and we want that lifting to coincide with side tilde. That's an actual requirement, which needs to be imposed. And with that requirement, everything is nice and beautiful, because once, first of all, the group of quantum morphism itself is going to be a circle-central extension of a group which acts on amps, a group of the symphlectomorphism group, which might have several components, but the identity component is the different Hamiltonian transformations. So if you wish, I'll give you a notion of Hamiltonian transformation, which has several components. In the infinitesimal guys, these are just the vector fields of the total space that satisfy the corresponding infinitesimal conditions, or if you wish, the flow that they generate satisfy these conditions. So the infinitesimal quantum morphisms, that's on the algebra, which is naturally isomorphic to the Poisson algebra, and that's kind of nice. The isomorphic frame, an infinitesimal quantum morphism is just a vector field on the total space. So you can plug it into the central connection to get a function on the total space, and that function will actually come from the base, and this is the factor plus minus, this would be the function of the Poisson element of the Poisson algebra that responds to this infinitesimal quantum morphism. So the Poisson algebra acts as all the automorphisms of this pre-quantization structure, and therefore it will act on any associated line bundle and the space of sections. So if you have any line bundle or vector bundle, which is associated, then the algebra will act totally naturally on its space of sections, except as I said before, this is a little bit too good. You need to restrict the structural group to actually have such a line bundle which is interesting. Let's do that restriction. So we're choosing a polarization on our manifold, and inverted the grand-gen sub-bundle, the conflexified tangent bundle, and I want it to satisfy some non-negativity condition, which I will not allow you to do. So I'm going to do this restriction and I'm going to do this restriction and I'm going to do this restriction by some non-negativity condition, which I will not elaborate on. In the purely complex case, this would mean that this is an actual k-o form. And I want to make two assumptions, which are also made in the constant-surreal context, so this is not restricting from the constant-surreal context. I want the real piece to have constant rank. This is the isotropic piece. And this piece gives us an isotropic affiliation theme of that. In the K-lo case, this is just tricky. And the piece I'm not going to elaborate on is the Hilbert space structure and the resultant vector space, but let me still list what is needed for that Hilbert space structure. You need that as a tropic affiliation to be fibrin. So you don't require E to be integral? To be? No, I don't require E to be integral. But depending on what your goals are, there are some applications for which you do want to require it. So basically what I'm claiming here is the type of requirements you make are an analogy to what is done in the constant-surreal case. So what we have here, we have a fiber bundle, whose fibers are the leaves of E. And this we will need for the Hilbert space structure, which I will not get to. Now the constant rank of the real part tells us that we are able to actually fix the model for this polarization. But this position really will come from, will correspond to a reduction of structure groups. I'm fixing a Lagrangian complex subspace of my standards in the particular space such as f intersect f bar has the correct rank. And what you get is a reduction of structure group. So we'll first of all lift at the circular group. That corresponds to a central extension. And then we take a reduction of structure group, which is correspond to subgroup. So if we have the original tower, we can now look at those symplectic frames, which are, which carry the standard Lagrangian to the Lagrangian, which is given by the polarization. And the treatment of that by definition is p sub f. And again, we have a tower, except now the group downstairs is the subgroup consisting of symplectic transformation preserving f, and the group upstairs is a subgroup. But this is exactly the subgroup, the MPC subgroup f. That's the one that had a homophysicistar, which we are kind of happy about. We then have the relevant part of the Poisson algebra. These are the functions with somatonic flow preserve the polarization at pf. In fact, that is going to be the group of all quantumophysicists that preserve f. We can still restrict our central connection to the, and therefore it acts on the associated line bundle, and this is the line bundle that we sort of, you know, wanted. Now, what else do we have? We have the central connection. So here has actually uses a pair of polarizations and from the pair of the polarization that are related and sort of the way he obtains an honest to God connection of this line bundle. And then he mentioned, oh, by the way, you can just do one polarization and you get a partial connection. So this is one of the pieces where I've checked some, but not all of the details. But let me just write this. I believe, I mean, okay, I do believe that this works. So you do get a completely natural partial connection on q defined when you take derivatives on vector fields in the direction of the polarization towards complex conjugate. Now, where is this coming from? Basically, there's two ingredients here. One of them is the central connection, which we were given, and second, the so-called God connection, which is resulting from the involutivity of f. You get a natural connection on the tangent of the mod f and on various bundles that you can build from that one. And that uses the involutivity. And then, then you can just continue as you expect. You can take the Hilbert space, and I'm writing the Dutch space to be the space of sections of this quantization line bundle, which are very constant in the directions of the polarization. And we have our quantumophisms acting there. So you get a totally natural action of that part of the Poisson algebra, which preserve the polarization. And then one can try to sweat and get operators associated to other functions, but that has to do something about those. I'm not pretty sure I'm just saying anything about that. Maybe a word about the inner product. I'm just going to say not write down. Again, Hess does explain that, and it is in analogy with what happens with constants through yaw. If you take sections of q, which are covariant constants along the polarization, there is a natural point-wise inner product of such sections that takes values not in scalars, but rather in densities on mod d. And then you integrate that to get a number, and that gives you the inner product on your Hilbert space. I'm done! APPLAUSE That's what's going to go. Yeah. Well, this Hilbert space, it has square interior sections, only if leaves of d are... If leaves of d are tori... You integrate over the quotient. You integrate the density over the quotient. Yes. The fibers could be... They covariantly cost sections along the tori with the indistributions. Right? You can't hit the words on the front block. There is a... No, no, no, no. Oh, okay. You're right. It's the same story as... It's the same stories as you have in... No, you're absolutely right. It's the same stories as you have in the cross-sensory case. First of all, you could look at situations where the leaves of d are not compact, and then you're okay, and you're fine. If the leaves of d are compact, you're in trouble, as we learned from you. You're in trouble. But then you can play the same games as you do in the cross-sensory case. You can take the sheaf of sections with this property, and that sheaf you can take underneath some of the homologes of the sheaf, and things about that, and you expect to get a new formula for the house. But you can do more things. But this... No, thank you very much. You're absolutely right. You can get in big trouble with the game very soon. In physics, a scalar product is an essential physical object. Here, as long as everything is nice, you put it within the structure by putting a metapractic structure or a PC structure. And here it comes, if you don't see, you get it. But once you start reducing some symmetries, you lose it. There are versions of reduction here. Again, there's some version of reduction. I mean, there's a lot of things happening here, many of which I understand will be partially... But yeah, there's a lot of things you can do. Jennifer Bond has a notion of Borsamfeld, which, for example, in the Torreke-Mannifold case, she actually shows that the Borsamfeld leaves are exactly the ones you get from the lattice points as in Michel's picture. Again, there's earlier notions of Borsamfeld due to... I think Robinson, the same guy from Robinson-Ronson. Jennifer's definition is a little bit different. I mean, there's a lot of stories there, which do not provide, as far as I know, do not limit you more than you were limited in the Constance-Rial case. But at the same time, there's the potential to address more manifolds which do not admit an effect. No, I just wondered if you actually asked the wrong question. I just wondered if you managed to overcome the problem here. No, no, no, this is complete analogy. But actually one of the things we were looking at with Mark, that I think we'll tell you later this week, has to do with some of these analogies. But again, it's not me and Tess. But you told us about your friction, the line in these structures? Oh, oh, oh, I'm sorry. So a metaphylactic C-prequantization exists if and only if difference. That's the first-run class of your line bundle that you care about. Even only if this difference is integral in cosmology. And it is unique up to twisting by pull delets of flat circle bundles on that. So the extent to which it is unique is analogous to Constance-Rial, and this is explained by Haasel. And this is different in that for Constance-Rial, you need each one of them to be integral, and here we need just the difference, which is kind of the news, because this is the John Klassler of the line bundle that you actually care about. Yeah. This is sort of a cultural, maybe, comment, even. Yes, a lot of this is about cultural stuff. So I was surprised by the preoccupation with an action of the Poisson algebra, since I thought that was just sort of a historical mistake, because that's not what you want in physics. You want to take functions of the gate operators, right? Yes, and they will only approximately correspond to the Poisson practice. So the thing is, if you restrict, so this is a huge restriction, so these guys, you can actually make them correspond on the nose, but if you want the rest of the Poisson algebra, then you have to allow certain approximations, etc. But these guys, if you follow the constant recipe, these guys, you get on the nose. You get isomorphism only for this, and perhaps some other, but it's not clear. Right? But you're going to get, again, sort of an approximate, in results, if the expectation is that whatever you do with Poisson cereal, you can do here, that's the expectation, but with Poisson cereal as well, you can get exact the alteration of this guy, but you would need to go for some approximations if you want the entire Poisson algebra, which you do want. And that's a piece which, again, some people will view as criminal, but I'm not mentioning, but again, these are pieces which I personally do not understand well enough. Perhaps you should perhaps, it seems that the work of the flow, on the object methods, you use the... Actually, there's your work with Michel, the flow work with something called prequantum bundles, which... We work with a compact group of transformation, so we don't use the old NPC structure, but we try to quantize the action of the old, compact group, something similar, and in the literacy stuff, it's what we call the overshift. Yeah, no, I mean, again, you and Michel know this stuff very well and have done a lot of stuff in this direction, they were presentation theoretic stuff, besides the acquisition of the production... I mentioned that, because we are mentioning work with the A-heels, the work with Michel, the flow was around some period. Yeah, so I've not read it, but I'm aware of it, I'm aware of its existence, that he actually defined it, so here we have this sort of circle bundle over the frame bundle, this is connected to something that he calls quantum bundle in the representation theoretic context. I've not actually read the papers, but yes, absolutely, but these ideas definitely, yeah, absolutely, but thank you, thank you very much, yes. You have a twofold cover, which is quantize the... the orbits, do you like taking the square root of quantization? Yeah. So I want to say the work of the flow is quantization... Uh-huh. So, what year? So at some point I should look at the actual paper, I'm aware of it exists from the, again, from the representation theory side. No, thank you. So that's a similar time as has. So I should definitely look at that. Thank you. Now this is very relevant, absolutely. No questions? Thank you very much. Thank you.
This is what he did in the late 1970s, which with recent work of Jennifer Vaughn. And maybe cut down the dimension and recover the examples on whether to work with the polarization or with correction. And when you do the half-form quantization of a Torque manifold. And in symplectic form. And then you take a complex bundle and then take its space of sections. The and that function will actually come from the which I will not elaborate on. In the purely And that uses the involutivity. And then, then not write down. Again, Hess does explain that, polarization, there is a natural point-wise inner