 Well, I feel enormously honoured to be invited to speak at Maxime's birthday. I can't think of any other mathematician, I don't think, who has made such deep conceptual innovations in mathematics, while at the same time being able to do the most sort of intricate, complicated calculations, someone who can do subtle geometric analysis and at the same time cunning combinatorics. I probably, like most people here, I know I can only kind of appreciate some aspects of his work. But anyway, it's a great privilege to be talking to this meeting. It's also actually a great pleasure to be giving it here, because when Maxime was three, my first visit to this room, actually not to this room, which wasn't here, but to where we were just having our coffee, and that was to Grotendieg's seminar. And I recall yesterday meeting Manin, it was also his visit 47 years ago, his first visit. And we actually met for the first time at Grotendieg's house next to the railway station at Maxime Vachier, and I, coming along on the train, brought it all back to me. Anyway, Maxime wasn't really involved in that. His mathematics has kind of, he's changed the face of mathematics. His work has given me tremendous inspiration and delight over many years now. At the same time, it's partly relevant to this talk I'm going to give, that my face-to-face contact hasn't always been so good for my self-esteem. I was just trying to recall when I first met Maxime. It was certainly more than 25 years ago. I was giving a talk in Moscow. I hadn't heard of him, of course. And in the course of my lecture, some young person who didn't really look as if he was the main way of dating when it must have been, someone who didn't really look as if he was old enough to be at a grown-up lecture asked me some question, and I made some bland, encouraging reply, and he persisted in his question. And I quickly found myself with that feeling, sinking feeling, which I'm sure some of you must know when you suddenly realize that there's someone in your audience that knows a lot more about what you're saying than you know yourself just when you thought you were on top of it. Well, anyway, what that was about, that interchange, was that at about that time, not having heard of each other, we each got interested in thinking I'll come back to presently the semi-group of annuali as a partial complexification of the group of diffeomorphisms of the circle. And we both thought about that, wrote about that in connection with conformal field theory. Well, so then time passes until my last visit to the IHS a couple of years ago, when I gave some version of the talk, which I'm about to give today, and Maxime listened patiently, didn't interrupt and put me down the way he had 25 years before, but at the end came up and said that, yes, he'd, so I was trying to do a higher dimensional version of this thing that we had both thought about in connection with two dimensional field theory much earlier. And Maxime came up to me and explained that he'd also given a talk on this and here were his notes and he kindly suggested we should write a joint paper on the subject. So I'm afraid I was meant to write it, being a slow writer, I haven't quite done that yet. But so this is in some sense a joint paper though. Its evolution has been, well, I've told you the history of its evolution. So it's about the positivity of energy in quantum field theory and how one encodes it. So in ordinary quantum mechanics, it's very simple. Quantum mechanical system has a Hilbert space of states and it has time evolution, so evolution for time t is given by some unitary operator. And the positivity of energy is encoded in the fact that this is a one-parameter semi-group, which is in the form e to the iht, where h is a self-adjoint operator called the Hamiltonian. Well, the positivity of energy is the fact that this self-adjoint operator has a positive spectrum in Hilbert space. And obviously that's equivalent to saying that this function t goes to u to the t is the boundary value of a bounded holomorphic function t going to u to the t, which goes from the upper half plane, that's t set to the imaginary part of t as positive, into operators in h. So clearly that encodes the positivity of the spectrum. Well, one can go a little bit further. For instance, if one has a system and one wants to incorporate special relativity, if this is a describe some system sitting in Minkowski space, then we would expect to have not just time translation operators, but operators xi for xi in Minkowski space, let's call that r31, which just translate the state of the system along. And again, the positivity of energy will be encoded in the fact that this is similarly the boundary value of a function which is defined for xi in this space plus ip, where p also contained in r31 is the positive light cone, the vectors which in the Minkowski metric. So I'm writing the Minkowski metric as minus dt squared plus dx squared would have norm squared negative and which have their time component positive, that's the generalization of this. And well, the only thing to note for that in the future is that we're interested in a Shilov type boundary condition in that this is a, this of course is in four dimensional complex space and this is a four real dimensional part of its boundary, which nevertheless bounded holomorphic functions here are determined by their boundary values in there. Well, so that's fine for certain kinds of quantum systems, but that's not a very good way for looking at things if one's interested in quantum field theory. So in quantum field theory, the picture is somewhat different. In quantum field theory, we start with some manifold x, which is spacetime, and that's given to us and it has a Lorentzian metric, I'll call it g, sometimes I'll write it gij if I want to be, so like the Minkowski metric at each point, but not constant of course. And the point about quantum field theory is that the observables in that theory are localized with respect to these spacetime. So for each x in x, but by the way, what time did I start? I didn't start at 12 with that lady, right? Yeah. For each x in x, we have observables. So this is just a complex vector space associated to this point. So these things would form some kind of vector bundle over the manifold x. And the content of the theory traditionally, so this is just a vector space, you have, if you're lucky, you can identify the observables at one point with the observables at another point, but we don't even necessarily assume that. And the content of the theory is meant to be completely expressed by giving for each set of distinct points some multilinear maps. So they're traditionally written like this. If you have an observable at the first point x1, observable at the case point, then we'll write the value of this and I'll put x, g to emphasize that it depends on x and its Lorentzian metric. So all the content of the theory, this is a bit schematic and oversimplified, all the content of the theory is meant to be contained in giving these functions. And the problem, I mean, there have been many attempts at axiomatizing this in various shapes and forms over a very long period. I'm not going to talk about this, but the idea is that then these things are distributions, vector value depending on this vector bundle, but the essential thing is that there's some kind of distributions on x to the k and we give those for all k. Well, it's rather difficult to produce a good axiom system. But what I'm talking about today is how we incorporate the idea of positive energy into this. I mean, can we say something simple and straightforward like that statement up there? Well, traditionally, the most traditional approach to the subject was, you see, to generalize what we saw there. So the traditional approach was to say that I have to learn to manipulate all these things too, don't I? There's something about three blackboards that somehow the world is one. And so the traditional way was because people usually thought of x as being Minkowski space was to say x has a complexification, x complex. And so the first way of trying to axiomatize this was to say that these things, they are boundary values of something, of holomorphic things defined in some open subset, some region in x to the k. Is this supposed to be a white one function or a green one? Yeah, well, so there are lots of, I don't want to get involved in that We find we have a mind on the white one. I'm precisely telling you that this is... The definition for a green one. Yeah, this is the analytic, yeah, I mean, so before, well, if you like, these were the white one ones that I started off with and we're saying they're boundary values and if we, well, this is exactly the road I don't want to go down, so let's not pursue that. But I mean, you could analytically continue these to do the things in Euclidean space. So there is this route. However, if you're really interested in the possibility of working on a general Lorentzian manifold x, of course, the first objection is it might not even have any kind of complexification. And even if, for example, it's real analytic, then it's liable, then this was liable to be defined only in a sort of There'll be a very little complex thickening and so on. So it's very difficult to give an axiom system that really makes sense on a general manifold. So there's another approach, which is the one I do want to talk about. So this is approach B, which is to say, well, which is to do the analytic continuation in G rather than in X, so to speak. So this is motivated by the path integral picture, which now dominates quantum field theory. So this is a mythological superstructure. You assume there's a space of fields locally defined on X. And you assume that what these things are are the intervals. So again, this is, I say, just a mythological background to what I'm going to say. The integral over this space of fields of the value of these functions. With respect to a certain measure, which is traditionally written, I'm going to put in a minus i s g phi, du phi. So this is a measure on this space in some mythological sense with respect to which you integrate these functions. And the measure is meant to be determined by giving a function on this space, depending on X and its metric. So some real valued function, the action describing the field theory. And that's meant to describe this measure, and hence these functions. Well, obviously this rather explicitly brings the dependence in G to the 4. And our idea then is going to be to do an analytic continuation in G. So what space of metrics G might we want to perform our analytic continuation to? I can see I'm not going to be very good at this. Well, let's think of the obvious motivating examples. So supposing you were looking at free scale of field theory, so that phi X is just the C infinity functions from X, say, to the real numbers. And the typical action would be half the integral over X of d phi. I'll write this as I put the i in there, d phi star d phi. Plus, let's say, we have a mass term phi star phi. Well, the dependence on the metric is, so phi is just a scalar valued function. The dependence on the metric is in these hodge star operators. And I've incorporated, I've thought of the i as being part of that dependence in that, you know, when you perform the star operator, you have to, the volume element of the manifold comes in. This is a Lorentzian metric. So this star involves the square root of the determinant of this matrix, the volume element in coordinates. That will be imaginary on a Lorentzian manifold, and that's the i. So it seems natural if we write this out in a more old fashioned way, maybe where I put G upper ij for the inverse matrix to G lower ij. This will be, you can write it like this. And assume we're on n dimensional space time, I think. And probably occasionally I'll accidentally make it d dimensional space time. So d and n will probably mean the same in this lecture. So the natural thing, you see, if you try in this rather hopeful Gaussian situation to think of making sense of that integral, you can think of it as like trying to integrate e to i times a quadratic form given by a symmetric matrix. And it's not sure to define that by analytically continuing by giving the symmetric matrix a positive imaginary part going into the Ziegler generalized upper half plane of in the quadratic forms. And so it's not sure to try to do that here. And you see, if we would like this to be a more convergent kind of integral, then what we appear to want is that the matrix, the determinant of G times G ij, that the real part of that matrix to begin with of its inverse is positive definite. That would make this part of the quadratic form positive. And then to make this part of it, we also want the determinant of G itself to be positive, the real part of that to be positive. And you can remove this without any change because if it's true before inverting, it's true after inverting. So these seem to be the obvious conditions. So it would seem natural to look at the Minkowski and matrix G and think of them as lying on the boundary of some kind of matrix given by complex valued matrix Gij at each point. And we would like at least this the kind of condition. If we just put this condition, that we'll be saying that after dividing by the square root of the determinant that at each point we would be in the Ziegler upper half plane except that I've taken out the factor i. However, that isn't really altogether, doesn't make one quite happy for several reasons because, for example, one will certainly want to consider other field theories besides scalar fields. So one will want to certainly have electromagnetic theory. And the electromagnetic field is described by, well, it's described by connection in the line bundle. But we don't even need to be that modern. It will have some field strength, which is a two-form on the manifold. And the action for this will be half integral f star f over x. So this is the normal action for electromagnetic theory. So one would like also this star, which is acting on two forms, to give rise to something which has positive real part. So it's not sure to go the whole hog and say, well, you see, we're always making use of these stars. So what do we really have? We're saying that for any p, we can look at the cotangent space. And we always have, if we have a complexified metric, a quadratic form on that going into the top dimension, taking alpha to alpha star alpha. So that's defined point-wise if we have a complex that's derived from an inner product on the complex tangent bundle. So our G will give us a G here. And the condition we want is that this, which we can ask for, is that if we take the real part of that, you see, this is a complexified volume form that this is positive definite for all p. No, no, no, no. Well, sorry, yeah, alpha. Sorry, no, no, I shouldn't have complexified on that side. Yeah, sorry. And what you had before was p equals 0 and p equals 1. And what I had before was p equals 0 and p equals 1. So it's reasonable to put this in for, it seems reasonable particularly if you're going to start doing string theory and have horrible, higher kinds of fields to do the whole lot and ask for this condition. So let's look at the, let's define a domain of metrics. So I'll call it C of x, which will be, the G in this will be a section of the symmetric square of the cotangent bundle, complexified. And we'll want, it will give rise to all of these things and we will ask for them all to be positive. So one of the things Maxime pointed out after I gave the last lecture is that there's a completely different, much more down-to-earth way of saying this. You see, one of the sides of his work, which I can't understand is any kind of calculation. But even I eventually, after about three weeks, was able to calculate that that was equivalent to the following thing. So I suppose we have a vector space and we, well, so what is this condition? On a complex valued matrix, Gij, what is it saying? It's saying with respect to a real basis, the tangent space at x, you can write the metric as sigma lambda i z i squared. These are the coordinates with respect to the real basis and the lambdas are complex numbers. And the condition, which is exactly equivalent to that and you can work it out for yourself during the lecture, is simply that the sum of the arguments, the angles of these lambda i's should be less than pi. So we are looking at complex matrix. So this is a domain of complex symmetric matrices. We're asking that they can be diagonalized with respect to a real basis, so not a complex basis. They will then have eigenvalues. Of course, not every complex symmetric matrix can be diagonalized, but we're requiring that. And when we write it like this, the angles of these things add up to less than pi. So this is certainly an open domain, an open complex domain in the space of matrices and it has some nice properties because, well, obviously it contains the ones which are real and positive definite. But what is it? It contains on its boundary the Lorentzian ones because if you make all the lambdas real and positive, for one, then the restriction on the argument of the other one is that it's less than pi. So in other words, we have to cut the lambda plane along the negative real axis and the one non-real lambda will have to be in that region. So the ones where we have a Minkowski structure where actually on that line, they are indeed on the boundary of the region. On the other hand, we can't have things of any other signature. We can't have three dimensions of time or two dimensions of time or whatever. It's a nice thing. Well, it's easy to see that this is a bounded holomorphic domain in the finite dimensional space of matrices. That's really because if you actually, probably better to look at it in terms of this description, it's an intersection, basically, of a lot of Ziegel domains. One knows that the Ziegel domains are bounded holomorphically complete function domains. Well, what other propaganda should I make at this stage? Sorry, excuse me. I'm not sure I understand the condition of a diagonal in the real basis. Why would it have to be diagonal in the real basis? In this way of saying it, that's one of the assumptions. But on the other hand, when you say it this way, it follows, obviously, you see, because if you want, in particular, you see, we want this matrix to have positive definite real part. So if you think of this matrix, that's saying that there's a linear combination of the real and imaginary parts of this matrix, which is positive definite. If you have two real symmetric matrices, and one of them is positive definite, then you can diagonalize them simultaneously with respect to a real basis. So it follows from that. It may be worth mentioning what this condition is in the two-dimensional case that Maxime and I first thought about independently so long ago. So let's consider what happens in two dimensions. Notice in two dimensions, there's something a little bit different because we only have the conditions nought and one. We don't need extra p's. But what you notice in two dimensions, this is holomorphically invariant because if you multiply the matrix by some scalar, the determinant multiplies by the square so it cancels out of that expression. So this is something that only depends on the holomorphic structure, and then this is just a complex volume element. So they're completely decoupled. And of course, we were studying originally two-dimensional conformal field theory. We are only interested in this part. Well, if you have such a matrix, then you can diagonalize that it will have two eigen directions in the Riemann sphere, which is the Riemann sphere of the complexified tangent space. And what this condition comes to is that so on the Riemann sphere, this is the projective space of the tangent space complexified. So this is a complex projective line. It has its equator, which is the real projectivized real tangent space. There'll be two eigen directions. And so if we forget about this factor, we have a point here. This condition is equivalent to saying that the two points lie in different half planes. And so this condition defines a product of two standard unit disks. And the completely real ones correspond to these points being complex conjugate. At the other extreme going to the boundary, where both of these points go to the boundary, we get two real eigen directions. They will be the two light directions in the two dimensional space. So you see the Minkowski thing on the boundary of this. Another thing that's worth pointing out is that such a structure, what does it do? It splits up the complexified real tangent space into two complex lines. So it defines two complex structures on the real tangent space. So it gives you two almost complex structures and one saying they have opposite orientation. That's the condition that we want here. And if they're complex conjugate, that's the case of a Romanian metric. That's the familiar factor. A Romanian metric gives this a conformal structure. Well, now we have to go back to, I won't say any more about that for a moment. I'll go back to quantum field theory. So there's a problem of what kind of axioms one wants, what or the same do we want to do? We've defined a domain. Well, so one of the ways of trying to make a more manageable system of axioms for these functions, which describe the field theory, is to, well, is to adopt the following point of view. So I will just throw it at you quickly and then explain why it gives you back something like the original thing. So one defines some notion of a quantum field theory by saying that it's a kind of functor which associates to a d-1 dimensional manifold with some structure. I'm going to call them n-1. And I'm only going to talk about compact ones and oriented because that's certainly all I have time to talk about in this talk. This is going to go to some topological vector space and I'll have to say a little, I'll have to, this is a slight oversimplification for the moment. And when one has a co-bordism, shall I write like this, so an n-dimensional manifold which is a co-bordism from this to this, that picture will go to a trace class operator and this is a co-bordism with some kind of metric. These were meant to be Romanian. Again, I'll come back to that. A trace class operator, ux. And one's going to want these two things to satisfy two properties. So two axioms. First, concatenation which says that if you string two co-bordisms together, say x prime and x, if they're two composable co-bordism, you just get the composite of the operator. And secondly, a tensoring axiom which says that, well, on the one hand, if you take E of y naught, a disjoint union of two spaces is given in some way as the tensor product of these two things. And similarly, if you had a disjoint union of two co-bordisms, then the operators would tensor. Well, in particular, for future reference, this axiom implies that if we take the empty manifold, the space is just the complex numbers. Now, the first thing is that this doesn't even roughly make sense unless we say a little bit more here. Because you can't compose two co-bordisms because there'd be an angle. So what we really need is when I say n minus one-dimensional manifold, what I mean is a germ of an n-manifold over an n minus one-dimensional manifold. So you should think of one of these things y as being a little germ of an n-manifold. And it's going to be oriented there. So there are two orientations involved. y itself is oriented and there's also a direction in which time flows. So these things come in fortables. You can reverse the orientation of y and you can reverse the orientation transverse to y. And of course, if you reverse both of them, then you're not reversing, then you're preserving the orientation of the n-dimensional manifold. So that's what we mean by these things. So it's a neighborhood of an but you identify two neighbors. Well, you all know what a germ is, I think. So why does this give us back the kind of structure we had at the beginning? Well, you see, supposing our spacetime becomes closed manifold x and we have our points x1 up to xk, then we can let's first of all define what the observables at x are when we have such a theory. Well, given a point x, we can consider small disc, ball neighborhoods of it, u, like that. And each of those will be bounded by an n-1 dimensional manifold. So we will have something e of the boundary of u. And if we have one u contained in another u, u prime, then of course, you can think of the space between them as a co-bordism from the boundary of u to the boundary of u prime. So if you look at the system of neighborhoods underordered by inclusion of a point x, these will form an inverse system and we can take the inverse limit. And we take that to be our definition of the observables. And it's clear then that if we have k points, we can cut out little disjoint neighborhoods around these things. And if x is x-check is x minus the union of these neighborhoods where the ui, one ui for each neighborhood, then you can think of this as a co-bordism from the boundary of this to the empty set. It's a null co-bordism of the union of these balls. So you see, it will give us a map from e of the boundary which by the tensoring axiom is e boundary of u1 tensor e boundary of uk to e of the empty set which I said was the complex numbers. So it gives us back those things that we wanted to be distributions which we wanted to axiomatize. And so that's one way. Obviously we're going to need to incorporate a lot more structure to get in. I mean this is much too encompassing and floppy and so on. But at least this seems to be a good way of beginning to look at the subject. And this is what, as I said, Maxim and I did long ago in connection with two-dimensional conformal field theory. Because the picture we had was that I mean all physicists of course kind of knew this already but we sort of spelt it out more explicitly. In two-dimensional field theory people thought of the two-dimensional manifolds as being string world sheets and if they originally started off with Minkowski and Lorenzian metrics then at each point they had light lines going in two directions these would wind around the co-boardism string an incoming string to an outgoing string and if we follow the left moving right lines then if we're lucky these lines will eventually hit a target and the co-boardism X, the string world sheet will define a left-moving diffeomorphism from y0 to y1, let's say theta left by following those lines and another one theta right the co-boardisms the conformal structure is essentially given by you can reassemble the X with its conformal structure by giving those pair of diffeomorphisms actually not quite because there's a question there's a question of a covering the number of times the left ones cross the right ones as you go across you have amounts to the fact you get a z-fold covering but anyway in two-dimensional theory physicists were extremely well known extremely familiar with the fact that the space you associate the space of states of a string has an action of two commuting copies of the diffeomorphism group of the circle one's flowing this way and one's flowing that way and of course people then move this string to being not a Laurentian string but one which has a Trillion metric it's this process called Bicrotation which we are here doing on the space of metrics and so you can so what one is doing is defining some kind of semi-group of things like this which has this representation of a pair of diffeomorphism groups on its boundary and the thing that Maximilian I talked about in our first meeting I think was precisely this fact that if we so when we move this into the complex metrics instead of having the two foliations remember we have a pair of holomorphic structures defining opposite orientation so we have a pair of complex structures what we were interested in was that the fact that if you look at cylindrical things like this with a holomorphic with a conformal structure they of course form a semi-group I called this semi-group curly A so this is a complex infinite dimensional semi-group it has the diffeomorphisms of the circle on its boundary and this group of diffeomorphisms doesn't have a complexification as an infinite-dimensional d-group it's an interesting example of an infinite-dimensional d-group whose Lie algebra isn't the Lie algebra of a group but there's a cone in the Lie algebra which so to speak can be exponentiated if you think of a circle and you think of a complex vector field you complexify an infinitesimal diffeomorphism you can think of that as exponentiating to give you a little annulus of a quantum field theory as a generalization of a representation of a group or a semi-group where one starts off going things that look like this but then of course one starts allowing more general co-abordisms and things like this because sitting there's this I think narrating this also with the same group actually I think this is very much we're aware of it in some language I think also too but certainly yeah well so this talk is about trying to trying to say something a little bit of interest at least in higher dimensions put my glasses on I think I have too much more time so I'd better say something about unitarity well let me first of all mention a few of the obvious examples I mean one of the obvious examples is the group where you have a group which is on the boundary of a complex group of this complex dimension is equal to its real direction if you look at the unitary group UN you can think of that as being the boundary of GLNC not all of GLNC but the semi group which are right like this which consists of the matrices let's call them A of operator norm less than one so if a thing in its inverse of operator norm one has to be unitary but a better example for our purposes is a better example for our purposes is to consider PSL2R because that is a subgroup at least of the group of diffeomorphisms of the circle and that sits on the boundary of a semi group in the complex group where you think of this this is the ones which contract in the sense that if you think of them as merbiast transformations of the Riemann sphere we look at the ones which take the upper half plane and map it to a disc contained in its own interior so that is a semi group of merbiast transformations on the boundary has the ones which are actual diffeomorphisms so we wanted to think of quantum field theory as a way of extending well of going between wick rotation we presented as the idea of going between a representation of something like this and a representation of something like this here we have things much better behaved because these things act by contraction operators in the axioms up there I asked for trace class operators on the other hand we are usually interested in representations which on the boundary are unitary now let me just say then how one would formulate unitarity I said that these co-dimension one manifolds y come in four tuples because there are two ways of reversing their orientation well it follows just from the axioms I've given without mentioning anything else that if you take let's draw a picture like this think of y you see y tends to be curved and so think of this as a manifold which is moving that way now one of the things we can do is reverse the orientation of y so put it that way and I'll call that y bar but also reverse the direction in which it's going so we're not changing the overall orientation so actually let me if I kept the orientation of y and put this orientation that way let me call this y call this one y star this one y bar and when I reverse both of them it'll be y bar star well what automatically follows from the axioms is that the E y star is canonically isomorphic to the dual of E y that might seem very good about this way of formulating things is that you don't have any kind of problems of functional analysis which one tends to get into in more traditional ways of formalizing field theory because remember where these at least when we're sticking to the metrics I consider where we have trace class operators then each of these germs comes equipped with a lot of parallel and upstream ones so really what this space y consists of is a direct system of approximations from one side and an inverse system of approximations from the other side and they are dual that's what happens when you go from there to there and that automatically happens in any field theory sorry that when you go from there to there you need to get muddled already and when I say that these things are dual you see they're really what Gulfhound or someone would call rigged spaces they have these two sequences of approximations from each side and the duality interchanges what you might call the smooth functions and the distributions the approximations from the two sides now so this happens without any hypotheses but when we want to consider a unitary theory we want something that in addition when we change this this becomes the complex conjugate of that so that does pertain to changing the orientation of y and when we have that then we will have I'm going too fast our theory we think of these manifolds as usually having some kind of curvature so they're getting smaller or getting larger when we have that situation we don't expect to associate to our spatial slice a Hilbert space we expect it naturally to pair with something which is going in the opposite direction you only expect to get a Hilbert space when we have an isomorphism between the thing going one way and the thing going the other way which enables you to reflect the time direction across the manifold which will give us an isomorphism between y and y bar star and will then potentially give us a sesquilinear form on the space that's why I began by associating topological vector spaces to co-dimension one manifolds it's only when they have the possibility of reflection across them so that that is isomorphic to that that one expects to get a Hilbert space but anyway the condition that we want to put on our operators associated to an X is just simply this operator star is the dual of operator which we get by reversing the orientation of y so that is the I'm sorry I've got somehow muddled with all my stars and bars but so the thing I want to finish with and I have to say this very quickly now is how do we treat unitarity in the case of the group SL2R one knows that there are two kinds of representations basically the principle series and the discrete series and the principle series is represented by spaces of functions and forms and things on the circle and it's fairly obvious that there's no way when you have a disc inside a disc you're going to be able to get from a function on that circle to a function on that circle the discrete series on the other hand the spaces consist of holomorphic forms of various kinds on the disc they clearly can be restricted so those ones the representations will extend to the semi group now so one expects of extending to the semi group to pick out a class of representations and that's the way in which positive energy is being encoded but what I want to finish with is going back the other way given a representation of the semi group which has this condition when do we actually expect it to be unitary on the boundary well you have to be rather careful because if you think of a space time even if it's cylindrical even if it's two dimensional you see when I said that when we follow the left moving light lines they will eventually get to there and give you a different morphism that of course isn't true because they might go into a whirl and there might be an asymptotic orbit they might all become asymptotic to that and then they might come out again somehow like that for example so something that comes out from here might never get there at all so this would be in the language of relativity theory a space time which was not globally hyperbolic it has some kind of bad behavior of its metric in its middle which you should think of something as vaguely analogous to a black hole and we don't in that situation expect to get a unitary operator going from the beginning to the end for example if you look at the case of the semi-group in SL2C then of course the boundary SL2R the real merbius transformations it's not a compact thing it looks like the interior of a solid torus it has a compact boundary consisting of an ordinary two-dimensional torus a pair of points on the circle to degenerate diffeomorphisms of the circle which move everything from one point to the other so it tears the thing apart and pushes everything into one point away from another point those things are represented in fact by rank one operators on the Hilbert space if you just take the limits of the invertible operators coming from the inside of the disk so what Maxim and I did in different ways was the fact that we wanted to be able to say that when the space-time is globally hyperbolic in the sense that you can write it as a product in such a way that although the metric is changing in time you nevertheless can follow light lines through and get diffeomorphisms from one end to the other and if the thing is sufficiently reflection capable at the end then it ought to follow directly from the axioms that we get a unitary operator well this was the thing that Maxim did a lot better than I did and I'll just tell you his argument and I'll only tell it to you in the case of one-dimensional space-time because I'm late and it's so beautiful in the case of one-dimensional space-time so what are we trying to say we are interested in a space-time of length T which is Minkowski so this corresponds to a metric minus DT squared we want to say that with that metric the thing having imaginary length T I suppose we're going to get a unitary operator now on the other hand we can say the following we can get metrics on the line by embedding our let's suppose we have our X our space-time that's going to be this supposing we map it into C with its obvious inner product DZ squared then this is pulled back from that metric by embedding X as the imaginary axis going from naught to T and the embeddings of this in C of course are a complex manifold which maps to the complex domain of complex metrics and this is on the boundary of it now we'd like to associate to this thing some kind of operator we only have operators defined for metrics which are not on the boundary the ones which are not on the boundary would correspond to embeddings which always move a little bit at least in the positive time direction but we want to set out from a space associated to this point this is our Y naught we're allowed to go a little bit downstream because we're only interested in the dense subspace this is the completion so we can define we do have a good operator defined from there to there because that will give us an allowable metric and whatever route we choose allowable path from here to here we will always get the same operator from there to there from our axioms because these things were functions of the X with its holomorphic metric and of course when we do a diffeomorphism along the interval X we don't change X at all but we're allowed to so we have the real vector fields on X acting on our space of metrics on X not changing the structure but we're also allowed then because the thing is holomorphic to move along complex tangent vectors to X and they will precisely move one of these lines to another so we can associate on the dense subspace of this defined by the coming things from downstream a good operator from there to there and when we combine it with the operator coming back this way that will correspond according to our axiom to the adjoint morphism and we will find that because all of these paths can be got by moving this map along a holomorphic vector field of time dependent holomorphic vector field on X what we will associate to that metric on this interval is the same as if we just went straight through there and that is the thing which is defining the rigged structure of the space so that will be the identity and that will prove that the thing is a unitary operator even for I think ordinary Wiener measure I think this is quite a good way of proving the unitarity which isn't an obvious thing for Wiener measure and it's time for me to stop but this argument which as I say is due to Maxime works beautifully for any what Relativist calls globally hyperbolic compact manifold and of course work down on what happens if you consider things that are not compact and so on but obviously time for me to stop so I'll leave it there for now so often in field theory it's useful to consider not background metric in terms of the metric but in terms of the moving frame and I wonder whether you've thought about rephrasing this in terms of moving frame or is there a version or well yes in fact I think if you pursue that line what you get to is what people would call ashtaka theory I mean I think and I think that's the right way to go probably actually where you I mean you actually probably should instead of having a Romanian metric you probably should actually have a bundle of Clifford algebras and you see and no more questions in your system of axioms do you specifically assume that there is the doctor you is invariant on the say isometric class yes you see it's meant sorry I didn't say it very well did I it's a functor from the manifold with this complex notion of metric and it only depends on that as an object of that category so is that the answer