Twistor String Theory — Roger Penrose

0:00 Something about Bill's face. I want to say I've certainly got a lot out of this meeting. The whole subject is fascinating. Some of the dog scientists took better than others. I think something came through always. I'm certainly not going to try and summarise the meeting, although I will address again the questions that I brought up at the end of the last talk I gave on Monday, where I raised certain questions. I realise some of these are not formulated very well, but I perhaps have some comments from people which go towards answers to these questions. Let's start perhaps with ones a little more clearly answered. Yeah, I mean, yes, I think the last one. I'm certainly looking for a topological quantum theory theory, which seems to be the right thing to do in the context of twists of theory. Christopher theory, after all, is concerned with spaces which, on the whole, one is not interested in the local structure. The transformation to space-time is a non-local one. The information which is local in space-time, such as curvature and things like that, comes about from non-local So you have something where the information navigates locally in a sense, which is related to the fact that once you're using cohomology, because cohomology, the first cohomology vanishes locally, but it doesn't mean that in non-local regions you have information, and that's exactly what happens.

2:30 and the curvature of space-time comes about from the non-local structure of Twister space, whereas the information stands to sort of different structures of the information. So the sort of quantum field theory one would seem to want in Twister space would be something of the nature of the topological point of view. I hope I someday will understand better what's going on here. In regard to the one over here, this again is not surprising to me, partly because going back to ages ago, I think Andrew was pointing out how far ago we were worrying about twisted diagrams and things, but at one stage I remember totally over-optimistically thinking, seeing how one could express the two connections in the diagram, that maybe you could just put these all together, there was your quantum gravity theory, so I excitedly dashed off the letters to Bryce DeWitt, thinking that he would have some useful comment to me, which indeed he did have, but it's rather ironic in respect of what people are doing now, because he was pointing out to me that I was trying to stick diagrams together, using internal lines which had helicity arrows on them. I put arrows on my lines which represented and my internal lines have delicity as well. He pointed out to me that the exchanges of virtual particles are off-shell and therefore they don't have any delicity. So I discarded that idea. Later on, other ideas which came along suggested we put more lines in our diagrams and had to explain some of these things to us just yesterday, wasn't it? So it's certainly Dennis in a certain very useful direction, but it's ironic because we now see that people joining MHV diagrams together seem to put lines on with, well this is Ian, and get away with it happily. So obviously we'll have to go back and look and see how they're getting away with it and why did we miss doing that those many years ago? Anyway, it's a very interesting... I'm working my way up more or less from the bottom. You see, this is meant to be a twisted diagram, maybe with two lines joining them together, as opposed to the single line, which one seems to have in the gluing together of MHD diagrams.

5:00 Let's see. I'll say a little bit about that one in a moment. Let me comment on the first two here. First of all, we do need the extra dimensions. I'm not sure, really, what the answer is to this question. It's a question of what string theorists think, I suppose, because I've always had a lot of trouble with space-times with extra dimensions. One of the reasons is that Twister theory, one of the very basic motivations of Twister theory, was to find a formalism which was keyed, specific to space-time with one time and three space dimensions. And one of the special features that one has in that is a light cone, or if you like, a celestial sphere, which is a complex manifold, a Roman sphere, and that's the sort of thing that doesn't happen in other numbers of dimensions or signatures. So that particular motivation certainly is one of the things behind wanting to have a space-time which is 3 plus 1 dimensional. And so when people were going off in higher dimensions in string theory, I wasn't really happy with that. I also had other troubles having to do with stability of these spaces and the extra function freedom that one runs into. and one has to have arguments for why you don't see all that. As far as I can make out the string theory view on this, of course there are probably zillions of different string theory views, but the one that I can, more or less, come to grips with, is that in some sense, space-time dimensionality is not that important. I think this is the view that a lot of string theorists seem to have, that, okay, it's an energy effect if you're looking at higher energy, you know, you see more dimensions, and a formalism which is specific to 3 plus 1 dimensions, or 4 space-time dimensions, as twist-to-3 string theory as presently exists, seems to be largely concerned with. In that case, okay, maybe that's concerned with only certain energies and so on, and we get more dimensions if we pep up the energy. I had trouble with that idea, but at least it's something which I can understand. Let me move on to supersymmetry. I don't have so much trouble with that, as it might well be. There is supersymmetry. I've never been convinced that it has to be there. I suppose one of the troubles, one

7:30 of the points I'm getting at with these two questions here is if one rephrases string theory in terms of twister space, of course it may be this is just an application of these general string ideas, and one is to be using twister space in this context, but to a twister theorist, one would like to think of something more basic about the use of twister space here, and in that case, the argument, if your strings, if the home or the target space for your strings is twisted space, then it's not clear to me, at least, how many of the motivations which drove string theory in the directions that it was driven, things which come from the non-existence of tachyons and so on, which are things which I can understand, you know what, if you're talking about space-time, but if you're talking about twisted space, I don't quite understand even what a tachyon is, because you don't have time. Time is just This will add up with other dimensions and it's not, you know, sort of have dynamics in the sense that one ordinarily does in physics. So it's not quite clear what all that means. So I just must say I don't quite understand where all those motivations have gone. Those are parts of the higher dimensions as well. So I've left the question marks here, I think. Not as clear as some of the other questions. And a point I was bringing up here, which perhaps, yeah, maybe we'll come back into something more to say shortly. Everyone's really interested in parametrization in there, which I gather was one of the things that drove the interest in dimensions and so on. Let me come now to this one. Do we really want the plus, plus, minus, minus signature? Well, there has been a lot of discussion about this point, and I think it's a land of convenience, as far as I can understand. And certainly one would like to get back to the racing signature somehow, whether you do it through a pseudo-whip rotation or whether you just put it in from the beginning. seems to mean it's a little unnatural to do it in this way, and see the rotation. Can we make a few points, then, in connection with this, which, perhaps I didn't remember before. These are just some general remarks on twister-spinner inventions. I put these questions up over here, although I'm not all together happy with the formulation of them.

10:00 Here, let me just make some comments about twister-spirit conventions. These are not necessarily the signature, that's one of them, but let me make this point here. But if you have a positive definite signature, or else the split signature, as we have here, then the complex conjugation operation sends you from a twister to a twister, rather than to a dual twister. So that, when one considers the quantization procedure, you want to do parts of the canonical conjugate of a twister, but the canonical conjugate of a twister is a dual twister. So it doesn't agree with the, as the complex conjugation and canonical conjugation don't agree, except in the Lorentz's signature. Why do you want it to agree? Well, it's what drives you to consider holomorphic functions. If you don't have that, there's no reason why the wave function should be holomorphic. So, there's a big sort of point here, which I don't think I've mentioned before, which is one reason why one wants the momentum signature in a sophisticated area. I want to talk about another point here, which I haven't got the bottom of, and I think it would be very interesting to try and get the bottom of it. In the plus-plus-minus-minus signature, one can consider delta functions on real alpha planes, which is certainly what Edward was doing in his slides of paper. Now, in complex Mekowski space, people think of these real alpha planes as a certain subset of the alpha planes in complex Mekowski space. But to say delta function, you want to say this in some sort of polymorphic way. we're talking about double functions, and that's in terms of what are called hyperfunctions. And these are things which, in Twister theory, we keep saying, oh yes, we shouldn't be able to handle these monadoletic things, all we need to do is bring in hyperfunctions. That's the sort of thing we say, but then we never actually quite do it. That's why I don't think we can. There's hardly ever quite do it. I should just say a remark about hyperfunctions. A hyperfunction is something, it's basically a dual of an analytic function. It's more general than a distribution. And in fact, something was developed by Sato, a Japanese mathematician, in connection with quantum beauty.

12:30 And so it's a useful concept. One way of thinking about it, at least in one dimension, you can think of a hyperfunction on the real line as representing the jump between a holomorphic function in the long half of the complex plane and a holomorphic function in the top half. So it represents the kind of jumps you can have across a line in a holomorphic function. It's hard to mention, you have to go to hierarchal homology. But it certainly does lead you to cohomology. So the description of a driver function in this complex setting would be in terms of some hyperfunctional notion. So I think it would be very interesting to see how that works. And this is related to these other pictures I've drawn here. In Andrew Hodges' lecture he was talking about twister diagrams and he pointed out that the in-and-out state has these two ears, which represented the singularities in the function, and you plug in the twisted function on the end here, and you have to have a pair of singularities. And that pair of singularities corresponds to the fact that the function is actually a first homology element. Now, as Andrew Hodges pointed out not in his lecture, but he pointed out some years ago, you can also consider things that just have one line. And they depend on perhaps a single twister, so you could plug in, well actually dual twisters here, or you could plug in a single one in there. And what does that mean physically? Well the thing is, here you have cohomology in twister space, but the fields in space-time are ordinary fields, they're not cohomological objects. wrong, in the first space, then you will end up with cohomology in space-time. So these objects that you refer to as elemental states, these are the elementary states that I think Andrew was talking about yesterday, one can define these objects, which are referred to as elemental states, and their space-time interpretation is cohomological ones. So you're talking about cosmological wave functions. And, in some sense, these are the natural tristorial objects, because you can think of the tristograms as being built up out of this sort of thing.

15:00 So, we've got two of them for a real field, or real wave functions for a particle, but a single one represents this more basic thing. Now, it seems to me, and it's sort of natural, we keep seeing this when we see these things that Ed Britten was doing, that they should be, in some sense, elemental states. It looks as though they ought to be the single line of it, so you can stick with a single twist to a single one, or a single twist, I think, if you want to hear it, that's not important. So that's, I think, something we need to understand. I don't know that we've got to the bottom of that yet. I haven't. If anybody else has to know. But it does seem very likely that the hyperfunctional description of adult function here, since that is a cosmological object, very likely gives us something like an element at stake here. And this would be something which we could perhaps make more contact with what people have been doing on the string theory. This is a much less important point, certainly not in this context, that I just thought I'd mention it. When I said I got the up-and-down nature of these inner parts, or first of all the way around in my first paper, whether you worry about that or not really depends on what you want to do with these things. But if you want to consider conformal transformations, this is the natural way around. So I should make a point here, that if you don't make a certain, if you don't adopt a certain point of view, it doesn't make any difference which way around there. So what is the point of view that one adopts? Well, the point of view I adopt is basically the abstract index point of view. and you start to notice the effects of this when you do a conformal rescaling. So it could give you a metric which, this thing can be just an arbitrary function of position, so your metric gets rescaled, the light cones aren't changed when you pass from this metric to this metric, of a metric is changed, and you can split this omega between the epsilons, the primed one and the unprimed one, and the upstairs one goes omega to the minus one. But that doesn't make any sense to you if you think that epsilon is simply naught one minus one

17:30 naught. So you have to think of that as an abstract object, which is the symplectic structure OK, well now this thing is just the only to the minus one del of omega, and then you find the forward transfer which is a very simple formula. The omega's are unchanged, and the pi's just get added a bit. Sorry, the pi's get changed by a bit to this. I just point that out as to the The formal classmates are very, very simple, but they are only that simple when you put the indices into that place here. So, that's just a side comment. Now, I can't put this over here. Most of what I want to say will be to do with curved space now, except that I thought I had a transparency on something else here. There's a point that Andrew sort of peripherally mentioned, but let me say a little bit more about it, is that twisted theory, although it works most neatly when one's talking about massless particles, it actually can be used for massive particles as well. And if one uses it for massive particles, And also, a thing quite curious comes about, which we used to think was very significant. We're not quite sure what's seen now. Maybe it is very significant, but we... I think it was in two early stages of particle physics when we were fully focused on this. This is the infinity twist, though, that I think Andrew briefly introduced. It represents the point of infinity and breaks conformed invariance. These things here are what we call the angular momentum twister and its complex conjugate, which encodes the angular momentum of a system and its momentum. But you can do that, I gave it before, only for a massed particle. But you can do it perfectly well for a massed particle as well. Now, if you had a single twister, which is what I have in these expressions here, then automatically this momentum turns out to be a null momentum. But you can equally write these expressions down by adding together this expression for many twisters.

20:00 And if you do that, then the momentum is simply the other, the individual momentum added together, and it's some unlike the vector, which, if they're no more proportional, would be a time, well, it would be non-nulled in time. So you can write these things down, and let's have done that, and say, well, twist on it. And now you can ask the question, OK, if you're just interested in the external parameters, that's the A thing, how much freedom is there in the twisters, which leaves that external thing alone? Well, you can work out what the skew is, and I know it's the end of the case. It's something like an n-by-n unitary group, a UN, but there is also a piece, a common genius piece, which is a bit like translations of some sort, which is complex and skew. And the group involves both these things. And the multiplication rule I've written out explicitly there. If N equals 1, you just get U1, and this is what generates multiplicity. So that's all you have. If N equals 1, you get massless particles that include U1, which generates multiplicity. If N equals 2, you get an inhomogeneous version of U2, and we got very excited thinking this had to do with leptons and unique interactions and so on. Maybe it has, but I think we've got to understand more about it. And if n equals 3, you get an inhomogeneous version of u3. There's somewhat of a complication which comes in here too, which is not quite as simple as you might think. Let me not go into this anymore, it's just that there is at least a sort of taking off point with a consideration of particle physics, were the groups that seem at least at first to come into particle physics seem to have some role to play in these interesting groups here. One can go a lot further than what I've done here, and you can try and speculate what operations correspond to various fond numbers. I think we were starting to do this before charm was even discovered, and I think it slightly shook us when that happened, and all sorts of things. But let me not worry too much about that, because it's again something which I hope will come back,

22:30 and maybe with new insights that seem to be coming into crystal theory, perhaps we'll have a look again at this and see where it leads us. But for the moment I think I'll just go after this on the other side over here, but I won't say anything more about it. OK, for the rest of this talk I want to say something about how twisted theory might treat gravity. Most of the discussion at this meeting has been on Young Mills, some discussion of gravity, but we've just been hearing some of that. I know Bryce LeWitt, he regarded gravity was the thing he was interested in, and he regarded Yang Mills as being a sort of toy model to play with, and if you understood that, maybe you can... I think in those days people didn't realise how important Yang Mills was. QCD wasn't around in those days and so on, so the importance of Yang Mills is something which has emerged later on. I think my own point of view is almost the opposite of prices, but also almost the opposite reason. That is to say, I've been concentrating more on general relativity, and this was meant to be non-linear graviton construction than Richard Ward, who very rapidly after that saw how to do the Ann Mills case. So what I found was how you could represent solutions from general solutions of the anti-cell pure Einstein equations, which you flat and anti-cell pure equations, and that's a logical term name as well. What Wart did was to show how the anti-cell pure Yang-Mills equations could also be done by similar types of constructions, and my feeling was partly that I'm more familiar Because it's a harder problem, you're more focused on what you should be doing. So, maybe that's a funny kind of philosophy, but it is... I think the way I've been looking at it is it's a harder problem to try and understand gravity, so you're less likely to go off in the wrong direction. Well, that may or may not be the case, but that's at least what I've been doing.

25:00 There are snags of another kind, snags that don't worry other people. So let me just mention this, that quantum gravity, when one gets to a quantum theory, is quantum gravity going to be a standard quantum mechanics or not? Is quantum gravity the application of standard procedures of quantum field theory to general relativity or perhaps to some other gravitational theory, or should one be seeking some more even-handed marriage which we give on both sides, which leads to a modified quantum mechanics. I'm not going to go into the reasons why I do think it's the second. Part of the reasons have to do with inadequacies of quantum mechanics itself. Now, that's a bit contentious subject, so we won't say much about that. But it does seem to me that when you consider macroscopic bodies, something new has come in at that stage. It's the old Schrodinger's cat problem, or the measurement paradox, as I think of it. Different people have different ways of addressing the system. But also, I think, when one thinks about space-time singularities, that's the only place we really know that in relativity, or whatever the right theory is, in quantum mechanics, or whatever the right theory is, come together, and the singularities of space-time seem to be what you get as a combination of those two theories. But what we seem to know about these singularities is that the singularities in black holes are a completely different character from that that seems to have come about in the Big Bang, and so it seems that one needed some theory which was grossly time asymmetrical. I don't know how those things can be argued about, but the point about timing's image will come in to what I want to say shortly. Whether in an important enough way, I don't know, but perhaps there is some input in this somewhere. So, again, let me kind of throw this to me here. If you were popping things down, I'll just take it up. I'll leave that for just a moment. Okay. What about general relativity? Well, I think I'd better stop by saying something of the original ideas that went into the non-linear program construction. We've heard a bit about it, I briefly mentioned in a nice talk on Monday, Claude Lerone,

27:30 in a nice talk on... Who's Lerone? ...describing his ideas. No, er, yes, that was close. Okay. Who's talking about what then? I think we're going out later. Sorry? But let me tell you something about the origins of those ideas, because there was something which one could go back to and see whether one could say a little bit more, and that's really what I want to do now. Well, first of all, let me talk about, I'm assuming you've not had enough of this yet, so let me put the other one up just as a warning. A lot of the thinking about general relativity, at least certainly when I was getting used to the subject, had to do with looking at asymptotically flat spaces in gravitation and radiation. And oddly in particular, I've examined what happens in infinity, you have your source, this is a sort of time diagram, time going up for picture, here's some source which ripples around, radiates gravitational waves, there are ripples in the space panel coming out here, and one of the questions was, how do you understand mass loss? This thing is a certain mass here and a certain mass later on, before and after, so that the definition of the mass might be fairly clear-cut. And the question is, can you make a sense of energy conservation by assigning an appropriate mass definition to the waves which are up? The energy momentum tensor is zero, so it's not there, but it's some sort of non-local property that the energy momentum we see. conservation law can't be expressed in the normal way that one does in flat space time physics because this equation holds and doesn't give you a global conservation law just as well it doesn't because you don't see gravity in this equation. This T here is the energy momentum tensor of all rest of matter. What's it? Rest of all matter. Gravity doesn't count. It doesn't matter in this context. Its energy momentum is non-local. But anyway, one way of looking at all this was to think in terms of conformal compactifications.

30:00 This is my way of looking at it. So rather than looking at limits, I think I went to a lecture by Troutman once. He was all full of these asymptotics and so on and so forth. I'm not very good at that sort of thing. So I thought maybe there's a way of doing this geometrically. So you can rephrase all these conditions by saying that, instead, you can put conformal that factor, here it is, such which squashes down the metric as you go up to infinity, and if you're lucky, you're going to get a nice smooth future null infinity. This is referred to as square-class, square-pi, square-pi-minus. Future null infinity is square-class, square-class, square-pi-minus. And the idea is that if you look at the geometric structure of what's going on here, you can recover the Bondi formula, Bondi-Satz formula for energy, mass, and so on. Indeed, you can. And it's certainly worth doing it, which to me made a lot more sense than learning about horrendous limits. The question, though, does arise whether... How general is this? how many spaces are there, which is a nice, smooth spire class. Fortunately there is a theorem by Hermann Friedrich who tells us that there is a broad class of solutions where this thing is completely smooth, infinitely many times differentiable and so on. There There may be some problem about getting this one smooth and the past one smooth at the same time. But that's the question which has to be shorted. Anyway, so that's the picture, the space-time picture. It's really smooth, and we like to make it as smooth as we can. In other words, we can make it analytic. If we can make it analytic, then we can do all sorts of things with complexifications and so on. And that's the game that a lot of us play. In particular, Ted Newman and the idea that there is other people. Now, if you imagine, then, that this picture here, I'm only going to worry about the future null-infinity. As I said, there may be some problem about getting these things through at the same time.

32:30 So this really means you've got to make a choice. Do you settle on square plus or square minus? And the natural place to settle is on square plus. That's where all the information about the radiation is going. But there is a choice, an asymmetrical choice involved in this. And I mentioned about time asymmetry before, and it may be that this is the place where the choice is coming in. I don't know if that's certainly where I was meant, or things like this. So what do we do? I'd say we complexify, and if you complexify this thing that looks like a cone here, it's an R going that way, there's just a real line this way, and it's cross-section of two spheres, which can be thought of as Riemann spheres. In fact, in this business, it turns out to be very convenient to use a complex parameter, a Riemann sphere parameter, to label the asymptotic directions. And then if you complexify this, that means that your mu, that's the recharging time, the measure going up this way, also becomes complex. And when you complexify this zeta, well, that's the trick that other people have used in other contexts. If you make something which is already complex, complex, what you do, well, you clear that thing's complex conjugate. So when you think of complex zeta, you have expressions which involve zeta as zeta bar, and then you say, well, zeta bar now becomes an independent parameter. So zeta, theta, theta, tilde. It's very much got people to do with the laminates and so on in that. Scattering books, I'm talking about some basic idea. But anyway, here's a sort of schematic picture, with zeta going one way, zeta tilde the other way, and u going up there. They're now all complex parameters. So this picture, you have an influence using a complex picture, that's the complexification described for us. A subtlety which I've not drawn in this picture, because I don't want to confuse people too much, is that there are big, great gaping holes in this thing. They're great to pieces cut away, which, you see, the complexification will only extend a little bit away from the reals, and you don't know how far. All you know is that there's a neighbourhood of the real space. It's analytic, which is what I'm assuming. in a neighbourhood of real space, which is a complex extension. And so you actually have to make these pictures a bit more realistic. And that's all parts of the understanding of what's going on here,

35:00 but I haven't looked up the pictures by doing them. Don't worry about the green writing here for the moment. Now, when the thing is curved, if you've got radiation that is coming, you actually could see it in the following way. I suppose it was flat space. If it was flat space, then its complexification would have alpha planes, and they'll have beta planes. And remember that the alpha planes are associated with the crystals, the beta planes with the dual crystals. Now, you don't have alpha planes in general. As Richard Ward pointed out, they're not integrable. You can locally have the alpha planes, the alpha tangent planes, if you don't have them integrating out into of actual surfaces. Nevertheless, you can imagine you had one, and look for its intersection with this complexified scrub. And that is what the definition of a twister line is. A twister line can be given an intrinsic definition, you just propagate a null vector, it's actually a null geodesic, a complex null geodesic. There are three kinds of null geodesic in this complexified null infinity. the generators, which points in the upper picture, they go up this way, that's the common part of the two other spaces, which are the twisted lines and the dual twisted lines. These are null genesis in beta planes, are twisted lines, no, dual twisted lines are the null genesis in the alpha planes, and the null generated are common to both. So you can define these even when the alpha planes aren't there. And this enables you to define a twist of space, simply the space of these alpha lamps. And that is actually exactly what one does. Well, when I say exactly, I should explain a bit more. This was an idea that came originally from Ted Newman, of the University of Pittsburgh, where the problem that we had to address, what the public was worrying a lot of us, What about angular momentum in general relativity? You've got asymptotically flat space, mass and momentum work fine, the Vonnie-Sachs definition works beautifully, but what about angular momentum? And the problem there is that you don't have a clearly defined system of rotations and so on. It's all to do with mucking out that the radiation spoils the geometry and one doesn't actually

37:30 have a good definition of what Newman called good cuts. There were certain, yes, here we are, a good cut. What do you call a good cut? That was in Minkowski's space. What a good cut would have been is the intersection of a light cone in the space-time. So take a point in the space-time, look at the future light cone, see where it meets by class, and that would a nice, flat section. Let's define a good cut. But Newman's idea was to define a good cut in the general case when there is what's called asymptotic shear. That's what that thing means here. And that's radiation to produce this stuff. And what he found, this is actually a complex parameter, and you might imagine wiggling at the surface up and down, that there's only one degree of freedom of moving it up and down the generators, where you've got to get rid of a complex thing. So what he did was, oh, that's really happened down by a complex amount. That sounds nice and simple, but it's actually more subtle than you might think, because there's also a sigma bar, which you might want to get rid of, and you can't get rid of them both at once. So it's asymmetrical. It's a chiral construction. You get rid of the signals, but not the signal bars, which is now the concept of the tool. And what you can show is that this is actually a four-parameter complex space. At least, in the, say, in that too wild, in a certain proof. In the nonlinear too wild, and then you certainly get this four-parameter complex space. So what have I got to do with the nonlinear graviton construction? Well, it is a nonlinear gravitational construction, but in a sort of different form, you see. What about these twisted lines? Here I've drawn them, here. Those twisted lines, they form a three complex parameter family. So the space of them is this three complex parameter space. Now, what was a good cut in the human sense? Well, it's something where you go around, you go all the way around here, in this direction. And so even though you may only have a little segment of these lines, because they run into the holes, that's what tends to happen, you have a little segment all the way around, and the consistency condition is that you go all the way around and form a rhino-spheres worth of these points. So that's the non-enguagraphic non-construction. This is a definition of twisted space.

40:00 If this had been flat space, it would have been the projected twisted space. If you deform it to one of these radiating spaces, it now is a curve twister space, and these lines that you're looking for, they correspond to the dual wood cuts of the human sections. They form a four-parameter family, and what's rather amazing is that you can actually give it a metric, and this metric is not only anti-self-zero, it's anti-self-dual and Ritchie flat. comes about. This is another part of the construction which I briefly gave you last time. Yes sir. I put that on the other side. This was, uh, There it is. This was the trans fence I used last time, where this is the twister space, sorry, this is twister space on the projected twister space from the modern Leukovsky space, The points here are represented by straight lines in the CP3. Null separation here is meeting over here. And then the idea is you make a deformation of a tubular neighbourhood of a line in the space, and that gives you anti-self-fuel spaces over here. So that's just the same thing that's going over here. That's exactly what's happening here. But we also have this projection in a space of what we call the pie spinners, two-dimensional complex spectrum space, and doing it that way we actually have to scale these things too, and this now becomes a four-dimensional space, with a scaling to the twist as well,

42:30 that's easy enough. And the projection of the pie space is just this projection that one has done here, so it's coming over directly. the construction that I gave here, in terms of its origins, which is this asymptotic thing here. This construction is actually a bit more general, because this requires some kind of asymptotic flatness and so on, which isn't necessarily what one exists on. But that's where all things are. There we go, I think I'm going to turn the power transplants up or down or something. There's a little monster. OK, let's put that one in the other side so you can see this one before. Okay, so these cuts or cross-sections of complexified null-infinity, future null-infinity, these things that human call good cuts. Now, another way of thinking about this is if we're working In a space-time, it has to be a complex space-time in the origin signature, of course, one of the different ways. But the Newman construction in an anti-Selpe Dior space-time simply gives you the points that you start at the back of it. See, a Newman would cut, which is sigma is killed off, infinity, sigma-nought, give you those curves which actually came from points. depends on the space-time length being anti-cell fuel. If it's not anti-cell fuel, you can define sort of abstract points as being these cuts, and that's exactly what you would be doing now. I didn't explain this quite well enough.

45:00 You see, you can start from an arbitrary asymptotically flat space, construct a cluster space in this way, and then you can look at these curves, and then you say construct a space-time back again from these curves, So this needn't be anti-self-fuel, but the construction automatically gives you one, which is anti-self-fuel. So in a certain sense, it's picking out the anti-self-fuel part of the radiation field. That's what it's doing. It's a very nice way of doing that. You have a field with just a radiating space-timer, you assume analyticity, but apart from that, there's nothing about self-fueledness in the construction. So if you think that what does anti-cell fuel space have got to do with physics, this is originally what they came for and they certainly had to do with physics because they're looking at ordinary space and ordinary radiation. And this is picking out the anti-cell fuel space which somehow encodes what you might call the left-handed part of the radiation. If it was quantum fields it would be the left-handed part of the radiation. It's the anti-self-dual part. So, but if it's originally anti-self-dual anyway, you'll just get the space-time back again. So, that's what this is. I suppose it's non-anti-self-dual. What does this give you? Well, if you follow these cones back, you get some mess. If this thing's non-anti-self-dual, you certainly don't get the points back again. So, this certainly does pick up the self-dual part of the curvature. You've just picked up the anti-circuit. So the problem is, how do you see the self-fueled part of the curvature? Well, let me put another nice fancy on here. Another way of looking at this is say, well, if you've got alpha surfaces, which actually extend all the way into the middle, then they're going to meet. And give you a point. But if you haven't got them, that means that as you go down these Nars-Medesics, there will be some shearing coming about. comes from the self-deal part of the curvature, and the net at the bottom results from the fact that you've got the wrong kind of curvature there. Now, the first problem is, how do you encode the information of the self-deal part of the curvature, even in infinity? Before you start worrying about what happens in the space-time,

47:30 how do you encode that information to infinity? Well, it took us a long time to see how to get that in there in a nice holomorphic way. And it's filled around with ways of representing minus six functions. Let me first of all just throw transparency up, which this is the sort of things we've been seeing before. This is in linear theory. In linear theory, you can take a homogeneity to be plus two functions and work out this here, which is the anti-stellar field path of a linearized curvature. That's the expression. I've used that for indices here, which is why you don't see any signals. You don't have to read them. So the A, A prime, plus most of A over there. And these are the anti-identities, that's based in the spiniform. These are the twister connor intervals, in terms of this one function, the first thermology element here. This is a different degree plus two. That's the one we've understood for a long time. We try and non-linearize that. We're naming these f's. Finally, the thermology element. And then you think of them as actually giving you a definition. And that's where there's a non-linear version of this. These are the ones which we call the googolies, because this has a minus-six homogeneity. And the problem has been, what on earth do you do with a minus-six homogeneity? The plus-two one was easy, you just made a differential operator out of it, and that gives you a nice deformation, but what do you do with a minus-six one? As I say, we've played around with these things for a long time, and in fact realized that which you can produce, you can encode a minus six chromology elements in some sort of crazy definition, which all it does is simply mucks up the nice fibre giving you the non-projective crystal space over the projected crystal space. There's a nice C star, which usually you want to use to leave alone, and this is what we're doing, you don't use it alone. 440 expression in order to six loops. It's not even homogeneous. But the thing is, it's cohomological, so it does actually give you a nice representative cohomology element.

50:00 But we had these sitting around and thought they couldn't be right because they're so ugly. But later on, I started thinking about this again, and wondering about how you can encode the information of the self-dual part of the curvature as infinity. So you want the information as infinity even before you start propagating inwards. And there is a way, and it's more or less unique. And it was the uniqueness of this expression which made me think it's not so bad. I've just written it down here. I can't fully explain all this because it depends on knowing what the definition of this conformity invariant operator is here. various considerations, conforming invariance, independence of the various spinners that you put in to take components through and so on, so that's what this is all about. The iotas, close on to the null directions of the life cone, and the tangent to one of these twister lines involves an iota and another thing which is mu, and that's necessarily pointing along some crystalline, which propagates correctly, and its scale propagates directly. This is a formula. And this formula, as I say, is driven by general considerations. As far as I'm aware, it's completely unique. There's no other thing we could write down which satisfies all these conditions. There's some arbitrary number k, which maybe you want to settle on what that should be, but at the moment I'll just put a k in there, it's just that constant. and that's what I might call the Twister Propagation Law, and with that, one does encode the information of the self-viewed part of the curvature. Now, you can, in some sense, carry this down here too, by doing this corresponding thing up the generators here. I don't want to go there, it's not just up the generators, but you can define a corresponding notion of a twister line with respect to an arbitrary null surface as well so you can define it on this thing as well and you have one definition of twisted space up here you have a slightly different one with respect to this cone and so on and you have mappings on the intersection points out here glue parts of this space to parts of this space but you find these gluings tend not to be glueable and one needs to have some way

52:30 of characterizing which gluings you should be doing and which ones you shouldn't be doing. And that's really the Novogluvili problem. OK, well, how does one do that? After the start, the answer is I don't know, but I've found myself driven again, almost as far as I can make out uniquely, to a certain prescription. Now, when I say I don't know, what I don't know is whether this prescription works. I know what the prescription is, almost. There are certain issues which are not fully sorted out, but let me look at that in a moment. I wanted to find some forms on twisted space. First of all, these are forms which are probably familiar to everybody. They keep coming in. They come in also to the nonlinear braviton construction, although I didn't go into it in that much detail. But on the non-projected crystal space, there is basically essentially a one-form, a two-form, a two-form, and a four-form. The four-form is a holomorphic volume form. That's just being phi. This is a metadipal not mnemonic, you see. 4 is the 2 form, theta is the 3 form, and 5 is the 4 form. I observe all this. That's the volume form here, and you can see I've written it down in indices here. here. This is the projective three-form, which see people when they talk about CP3 slash four, this is the four-form that comes in there, the three-form that comes in there, exactly the subject here. And then there's a one-form and a two-form, which have to to do with the projection that I referred to before. So when one has this projection down to the pi space, you have a space, a two form on the pi space, or a one form if you like, and that lifts up to the twisted space, actually. And that's the thing I've I've got them here. It involves the infinity twisters. So that's not conforming invariant. These things are conforming invariant. You also have the boiler operator, that's the entity by mu z,

55:00 whose eigenvalues are the homogeneous functions. It's an important thing in twister theory. But the point I want to make here is that you can get everything from just the one-form and the three-form. So if you know the one-form and the three-form, that's into all the structure. Well, you can get from the one-form to the two-form just by taking an exterior derivative, to the three-form to the fourth-form by taking an exterior derivative and dividing by two or four. And the Euler operator is basically the three-form divided by the fourth-form. More precisely, what that means is if I have some scalar function A, then dA wedge theta is equal to the Euler operator on A times five. So that's going to show you that you can find your Euler operator from this two. They also satisfy these two equations. The wedge of iota with tau is 0, and the wedge of iota with theta is 0. You might think you can't do much with them, because practically everything you want to write down is 0. Well, that's an interesting point. They have various homogeneity degrees, indicated here 2244. That's the Li derivative with respect to the Euler operator, gives you the homogene eighth degree. There's this funny thing I've defined, I don't know proper notation for this. I think I've asked these people. Here's where indices are much easier to see than non-indices. It's the wedge between... If I take this funny symbol between alpha and beta, these are the... and then an R form and that's an S form. It's just contracted, it would just skew over all the output indices together with one of the mediums, because that's what that means. And I've written down the specific case here. But anyway, that's what that funny thing half of a postponement means. And what one has is an additional equation, which is theta, funny thing, phi is equal to minus phi tensor product with... I'm sorry. It's the equation. It's that one. Beta, funny thing, tor is equal to minus phi tensor product with I. So that's an equation that one wants to impose on this. OK. So we can just do that if you like. And we can reconstruct all the forms from those two, so that's a sort of simplification.

57:30 And what we want to do in constructing this space now is on each patch we're going to have an iota and a theta and there are forms, well these things here I'm calling And then, let me give it a bit, let's see. Sigma and pi. Well, they are just meant to be... I thought they had these in a... Oh, yes, here they are. I should show this one first. Here's the twisted space. It's a complex 4-maniple whose global structure encodes an asymptotically flat vacuum. But that's certainly true. So the information is in this space. How we get it out is what's obscured. As I was just saying, you've got a one-form and a three-form, subject to this equation, which is written down there. The three-form, basically the Euler vector field, determines the foliation curves here. These are the curves which, when you project along them, you get the projective space from the non-projective space. So that's what those goals do. They're the Euler... Oh, it's the Euler left field. And you've got to have this condition, as I wrote down before. Here's the pi, and there's the sigma, and this sigma can be defined in two ways, and the fact that those are equal is part of the structure that's assigned. Now, what you find, and it's a bit striking, is that if you demand that when you go for one patch to the next, that these things scale in this way, the I ought to scale with a function k, some function, we don't know what it is, beta goes as k squared, but d beta goes as the odd thing, but that's the thing which gives you what you want. It deserves these

1:00:00 forms, which I've written down here, and you find that the Euler vector field undergoes this rather strange transformation, where you go from one path to the next. Very interesting, it's got this little k to the minus, to the minus 2 and also to the 2. It looks rather But nevertheless, it does give you, it has to be crazy, but you're going to get a crazy scaling, you see. So you do get this minus six scaling that I talked about before. And this structure is there automatically if you define the twister space in the way that I gave you, with the way that you project it propagated along the twister lines, given by the formula I wrote down, but it didn't explain everything. Now, the question is, OK, you've got the space, how do you reconstruct the space-time from it? Well, here's where I can only give suggestions. Now, I think I more or less know how to do it, but let me, so as to keep people here somewhat interested, I've brought this picture here. This is the question. It's a question. Another question. Can we understand what's going on here in terms of some kind of twisted string theory? That's what you want to do here. The anti-self-view case, you have a twisted space, and the curves give you the points in the space-times. Now, if it's not anti-self-view, what you have to do, using the various forms that you've got to play with, You've got to define some surgery where you had a hole drilled out of this thing, and you glue another space in the hole. It's not quite as global as it looks. It actually has some holes in the depth. If you glue the thing, it doesn't completely fill the space. There are some awkward gaps in the space, but that's necessary. If you've got to fill it completely, there'll only be one of you doing it, and that's not what you want. The question is, can we understand the surgery in terms, perhaps, of some power series in strings? It rather looks as though twisted string theory is telling us this is what we should expect. Because after all, as I understand it, and I certainly don't fully understand it, but as my understanding goes,

1:02:30 in the Yang-Mill's case, one is doing something very much like this. In the Yang-Mill's case, OK, you can do this, which is the Angus-Selduel case, and now you want to try and get the googly part in as well, which is what I'm trying to do here, and the twist-to-string approach is to have some half-series approach, which enables you to build up all that you need. So there must be, although this is the gravitational case, not the Yang-Mill's case, something analogous of what's going on with the spring theory, and maybe just an extension of it, I presume, where one can understand this brewing job in terms of some kind of spring perturbation series in springs of higher and higher order. So that would be fascinating if one can do that. So I'll leave this. Roger, I'm just trying to wrap my mind around this. Are you asserting that the four-dimensional business space you're looking at comes from an ASD thing, if and only if the f-6 is zero? Yes. That's right. Yes. That's right. But if the f-6 is there, it'll put some twist in which makes this gluing job more complicated than the humanities books, you see. And I think all I'll do now, since I've only got one minute, is to flash at you the sorts of I'd be elected. You've got four minutes. I've got four minutes? Goodness me, I can relax. Well, you have to have something which notices this f-6, or at least which somehow leaves that. And one does, in fact, find something of that nature. Quite how you use it isn't totally clear to me. I've got a picture here, which is sort of schematically indicating what we do. Here's a scribe, and the iota thing is in a certain sense labelling the point up here. That point wouldn't even actually be there, but it's labelling the feature in time like infinity. It could be a hole there, but it's labelling that point in a certain sense. You have another form, analogous to iota, which has the same sort of transformation properties, which represents the point we're getting at. Iota is a one-form. Xi is another one-form. And then

1:05:00 you might have another one which labels a point on the intersection. You have to do that if some people define something in terms of this point has to run all the way around and has to be consistent as it runs around. So you basically are playing around with three forms, three one-forms, the iota which labels that Ita, which labels this variable point that runs around, and psi, which labels the point of the problem. Now, when I say labels, they do, in a certain sense, tell you that vertex. They're defined in terms of... If you knew where the vertex was, then you could define these forms. So these forms are basically what you want, but exactly how you use them is really unclear. I think what I'll do is I'll put another picture. It's almost the same picture as I had there. I've forgotten. I did really use these for some lecture, and I can't remember which was the one I wanted to throw away, but I re-did it. I think I'm trying to point out what's going on here, which is the sort of converintegral you're led into. You're led into certain rather complicated-looking converintegrals, and certain things have to be equal. Some conjointical has to be equal to another one. It's a question of making sure that these things are more consistent in each other. I'm not going to say any more about it, partly because I've forgotten. Partly because, even if I could remember, I wasn't able to prove that it did what you wanted. But it's more just a kind of slide to show you the sorts of things that one talks about in this subject. Each of these forms, the eta and the psi and the iota, all have the same transformation properties, they scale with k, they all have the property, the thing with the funny half-cross-farthing symbol, that are all satisfied with each of these, and let me just point out one place where the minus-six actually comes in. I found it right. You have a product of three of these forms like this. And what you find is that the D of that, let me just say vaguely what it is. You find that something is global, which wouldn't have been global if it went through this transformation. It's because the scaling comes in just to the right degree, and the D of that thing, of this part, vanishes and that means that the...

1:07:30 I should have boned up on what that was, but I haven't done the type right now. Basically, we find that certain forms actually turn out to be global by virtue of this minus-six structure. Otherwise, they're not global. And so you really are using this very particular homogeneity degree. Now, if you're running Yang-Mills or something, instead of having minus-six, you'd have minus-four. And what it looks like is something where you don't worry about this point running around. This is what gravity has to do when you use that point running around. You've only got two of these forms. And somehow you get a similar thing, but which disappears only when it's got a minus four. It's a little bit unsatisfactory, partly because what happens if you've got all these things together? So you might expect you have different kinds of fields, certainly a Yang-Mills field and a gravitational field together. is something which just picks out that homogeneity. You don't want something which is only global by virtue of the fact that it's exactly this. So, I think it's not quite right what I've done at this stage. It should be something which picks out the part of this transformation which looks at that minus six part, or for the Yang-Yos construction, looks at the minus four part, and doesn't mind what the rest of it's doing. Because what I'm saying here rather has the character that only works when it's got exactly that form. And I'm a little suspicious of that. But I haven't actually thought about this in any detail for several years, I'm afraid. So it's something I want to get back to. I'm certainly extremely interested if anybody could say anything about the possible connection with Christopher Strings. That would be to me very exciting and would be something which leads on to lots of other things that one might talk about. Even, in particular, looking at the passport program again, because that's, roughly speaking, one of the things it's a little bit snared against. What you want to do is to bring the war of construction into the whole thing. The war of construction only operates with anticepial fields. And so how do you do that when you've got the... I mean, even if it is particle physics groups, you're only looking at the anticepial path of those particle physics groups, and you want somehow to bring the whole thing in. So it needs to be rethought about. But I think if we can get these glutee parts all nicely

1:10:00 sorted out, that would be absolutely a major achievement in Twister theory, and certainly can resolve this issue and show how these various things are tied up, that would be fantastic. So, I'll be leaving with that. I think these are just a few references which talk about the things I've been saying at the end here. Animalism and statistics here. I can't put them all up there. I should say, really, the most detailed one is just the news over here. So that's not actually published. It's not that... People are going to put these things on the web at some point. Maybe they will be. But that contains the most information. Well, thank you very much. It's been a great pleasure. Well, I'd like to, I'm going to suggest that we thank Roger once again and then after I'm going to ask Lyon for a listener to close him up. But thank you very much again, of course. Thank you. and that was a great help, and we're also very grateful to be the mass institute who did so much work for us, so Cerebolic, Brenda, Bao, and so on, and so on, and I guess I can also make it all be such a fascinating conference, I mean I think there are people who really have to expect most of the speakers to be clear and interesting, and that all this one can cheer for us to think about, so thank you.

1:12:30 how about a round of applause Thank you.