Well, I feel enormously honored 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. 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, when I 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 Grotendieck's seminar. And I recall yesterday meeting Manion.

It was also his visit 47 years ago, his first visit. And we actually met for the first time at Grotendieck's house next to the railway station at Massey-Vechière, and 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 for me. 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 only 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,

that's 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, I think I'll come back to you presently,

the semigroup of annuli 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, a 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 semigroup,

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 is 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 describes some system sitting in Minkowski space, then we would expect to have not just time translation operators, but operators u psi for psi in Minkowski space.

Let's call that R31, which just translates 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 psi in this space plus i p,

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 Shiloff-type boundary condition in that 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 that.

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 space-time, and

that's given to us, and it has a Lorentzian metric. Let's 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 the space-time. So for each X in X... By the way, what time did I start? I didn't start at 12 with that.

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. 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, an observable at the kth 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, the essential thing is that there are 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... Yes, I have to learn to manipulate all these things too, don't I?

There's something about three blackboards that somehow bewilders one. So the traditional way, 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.

Are these supposed to be a white one, functions or grids? Yeah, well, so there are lots of... I don't want to get involved in that. Which one do you have in mind? I'm precisely telling you that this is... This is the analytic... Yeah, I mean, so before, well, if you like,

these were the white 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. I mean, you could analytically continue these to 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 mightn't 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 very... 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, it's... Which is to do the analytic continuation in G rather than in X. So it's B. 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 integrals.

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, d 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 are looking at free scalar 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'll put the i in there. d phi, star d phi, plus let's say we have a mass term phi, star phi.

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 when you perform the star operator, 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.

I assume that 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 the structure. 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 natural to define that by analytically continuing by giving the symmetric matrix a positive imaginary part going into the Ziegler generalized upper half-plane in the quadratic forms. And so it's natural to try to do that here. 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 Gij,

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 Minkowskiian 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 kind of condition. If we just put this condition, that would 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 of I. However, that isn't really altogether, it 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 connection in a 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 natural 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.

You mean if alpha's real? No, no, no, no. Sorry, alpha. So is your star operator anti-linear? Sorry, no, no, yeah, sorry.

I shouldn't have complexified on that side. 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. I'll call it let 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 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, we can write the metric

as sigma lambda i zi 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

except 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 were 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 Zegel domains. And everyone knows that the Zegel domains are bounded,

bounded holomorphically complete function domains. Well, what other propaganda should I make at this stage? Excuse me, I'm not sure I understand the condition about diagnosing the real basis. Why would you have to be diagnosing the real basis?

Well, 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. And 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's maybe 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 naught and one, we don't need extra p's. But what you notice in two dimensions, this is holomorphic 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 were only interested in this part. Well, if you have such a matrix,

then you can diagonalize it. And it will have two eigendirections in the Riemann sphere, which is the Riemann sphere of the tangent, 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, projectifies real tangent space.

There'll be two eigendirections. 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 eigendirections. 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? You see 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 Riemannian metric. That's the familiar fact that a Riemannian metric gives us 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... I mean, what all the same do we want to do? We have 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 minus one dimensional manifold with some structure. I'm going to call them n minus one.

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 cobordism, which I'll write like this, so an n-dimensional manifold, which is a cobordism from this to this, that picture will go to a trace class operator.

And this is a cobordism with some kind of metric. These were meant to be Riemannian. 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 cobordisms together, say x prime and x, if they're two composable cobordisms, 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 y0, 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 cobordisms, then the operators would tensor. Well, in particular, just note for future reference that 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 cobordisms because there'd be an angle.

So what we really need is when I say an 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 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 four tables. 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. And a germ, so it's a neighborhood, 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 we supposing our space time becomes closed manifold x, and we have our points x1 up to xk, then we can well, 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 disk ball neighborhoods of 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 cobordism from the boundary of u to the boundary of u prime. So if you look at the system

of neighborhoods under ordered 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 is one ui for each neighborhood, then you can think of this as a cobordism from

the boundary of this to the empty set. It's a null cobordism 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 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. so that's one way, I mean 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 Maxime 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 Lorentzian metrics then at each point they had light lines going in two directions. These would wind

round the chordism from a string, an incoming string to an outgoing string. And if we followed the left moving light lines and if we're lucky these lines will eventually hit a target and the chordism X, the string world sheet will define a left moving diffeomorphism from Y naught

to Y1. Let's say theta left by following those lines and another one theta right. And so the chordisms 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 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 Lorentzian string but one which has a Euclidean metric. It's this process called quick rotation which we are here doing on the space of metrics.

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 Maxime and I talked about

at 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 orientations.

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 holomorphic with a conformal structure they of course form a semi-group. I call this semi-group curly A. So this

is a complex infinite dimensional semi-group. It has the diffeomorphism of the circle on its boundary. And this group of diffeomorphism 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. Well so our idea was to think 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-bordisms and things like this. I think narrating is also a semi-group. Yes, yes, yes. Actually I think and of course physicists very much were aware of it in some language I think also

too. I don't know. But certainly yeah. So this talk is about trying to trying to say something a little bit of interest at least in higher dimensions.

I'll put my glasses on. I don't see anything. I think I'm not going to have too much more time so I better say something about unitarity. Well let me first of all mention a few of the obvious examples.

One of the obvious examples is the group where you have a group which is on the boundary of a complex group whose 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

GLMC, not all of GLMC, but the semi-group which I'll write like this which consists of the matrices, let's call them A, of operator norm less than one. So if a thing and its inverse have operator norm one it 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 is the ones which contract in the sense that if you think of them as Möbius

transformations of the Riemann sphere, we look at the ones which take the upper half plane and map it to a disk contained in its own interior. So that is a semi-group of Möbius transformations on the boundary as the ones which are actual diffeomorphisms.

Now 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 ask for trace class operators. On the other hand we're 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 forteples 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 if you 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 I'll 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 will 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 one of the things that's very good about this way of formulating things is that you don't have any kind of problems of function 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 which where we have trace class operators then each of these germs comes equipped with a lot of parallel downstream 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. When you go from there to there. Sorry. You need to get muddled already. And when I say that these things are dual you see they're really what Gelfand or someone would call

called 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. 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 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 the sesquilinear form on the space. That's why I began by associating topological vector spaces to codimension 1 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 the 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 I hope so Well 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 embed a disk inside a disk 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 disk

they clearly can be restricted so those ones the representations will extend to the semigroup. so one expects the possibility of extending to the semigroup to pick out a class of representations and that's the way in which positive energy is being encoded.

But what I what I want to finish with is going back the other way given a representation of the semigroup which has this condition when do we actually expect it to be unitary on the boundary? Well you have to be rather careful because you see 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 diffeomorphism 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 the 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 semigroup in SL2C then of course the boundary SL2R the real Möbius 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 and they correspond to degenerate diffeomorphism 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 both looked at in different ways was the fact that we want 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 ends then

it ought to follow directly from the axioms that we get a unitary operator. Well this is 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 Minkovskian 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. on the other hand we can say the following we can get metrics on the line by embedding 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 of which this is a 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 our 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 incoming 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, 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 relativists calls globally

hyperbolic compact manifold. And of course, there's been a lot of work done on what happens if you consider things that are not compact and so on. But it's 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 a moving frame. And I wonder whether you've thought

about rephrasing this in terms of moving frames. Or is there a version? Well, yes. In fact, I think if you pursue that line, what you get to is what people would call Aschtikar theory. I mean, I think that's the right way

to go, probably, actually, where you actually probably should, instead of having a Riemannian metric, you probably should actually have a bundle of Clifford algebras. And you see, and you can do that.

Any more questions? In your system of axioms, do you implicitly assume that the operator u is invariant on, say, isometry class of the components?

Yes, you see, it's meant to. 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?