Graviton Amplitudes & MHV Amplitudes / Twistors & SD Yang-Mills

0:00 So if you could simply take that away with you tonight, and I'll pick it up from you in the morning, and I can give you all the other stuff I'm going to need. Talk to my wife. Thank you very much, Stephen. Your session is at 6.30 in the chapel in St Peter's, so you see a lot of people in the chapel, and the chapel is up on the left side, and those of us who are going to dinner will go to dinner in sort of weeks later. Is the chapel where we went yesterday? No, no, no. The chapel—the church went to dinner yesterday, went to dinner at Hamilton Hall. The chapel is actually building it, and yet it's through the proper end. So, the final talk this afternoon is by Simon Swarovna here, who is going to talk about That is on Andrew Tunes and other members. to be struck by the anti-organizers of the United States. And I also want to progress my thoughts by saying that I am not seeing the present any definitely conclusion to this question, to correct this statement, but perhaps this is a deep question. When I am going to talk about the externalities, And then I will also make some observations on how one might try to use constraints in and hope to survive at MHCM. Neither of this is too much to do with strength here, or Krishna, perhaps, but never that there are many contacts, so I go to the article here. So, let's start by reminding you of the Yangs, and you've still raised it there. Part of this I need for whatever we'll talk about later about them. and in the atmosphere we are looking at masses particles. So, the momenta can be given in terms of two spinners like this, which was discussed in the last lecture. So, the part is that in terms of which the momenta can be baptized,

2:30 those of the so-called spinner momenta, and that's a term that I've continued to use. They are also characterized by helicidate which in this particular case we will be talking about so-called maximal helicidification which refers to the fact that you have two negative helicidates that are a small part. You look at scalping at ten parts. You would of course think of having less negative helicidate but those amplitudes vary. These are the first non-managed amplitudes. to the maximum amount of this pattern, intensity of the products. The amplitude was derived, indeed, it's after a lot of work, and it's really a funny diagram of the form. But at the end of the day, you find that the result is strikingly simple, and very nice. It's given by the stomata here, the keys are here for a main system, First we are talking about the English, the language, the language, the language, the language, the language, the language, the language, the language, and these images tell you what we are talking about regarding the earth. So there is a thing that is in the string language when one thinks of this language, again, in terms of the string theory, it can mean a so-called string theory. is that, in certain respect, there is a lot of things that are made. They have a fight, but because I will need this later. There is, of course, the old-armed movement of the wish and then there is the amplitude that we are in the future. The amplitude has the simple form. It has, we take that product and the earth's made in part, which is finance, say 500,000, In terms of this spin-off product of the amendment, the amplitude has its speed. It's very simple. There is a factor one, two products of the force power here. But it's also entirely boromorphic, involves really a high group hydrants. And this is one of the real striking similarities that, again, we can just look at it.

5:00 Further, we can look at the velocity of the products involved and you can see that it is given by minus half the degree of the modality in the five in the sample to them, which again is exactly what you expect. From the crystal point of view, this is pointed out in the last lecture. So all this is nice and consistent and this is the case in the English theory. Let me say a bit more about this one thing, the force factor. It's very easy to get to the factor on supersymmetric. If we increase gas and gases and you need, since you need the fourth power, you need four of them, and not coming to the spin of a member and integrate, then just like properties of gas and integration, you get the factor. So you find that the system was actually derived in the non-supersymmetric gaming scale. Kind of the natural expression for this attribute and ten equal to four supreme elements, it sort of comes out very naturally. Another way to say the same result is to say that we already have a congregation mentioned here, so you may have dealt with it. You can ask if there is another function in this, in the glass of the greenhouse that I can introduce, to increase by some sort of an exponential ramification of objects, where ether are the suci-hornness of the spinamomentha, supersymmetric partners. Then this delta function in the gas invariables, in fact, will be used to this in the appropriate choices of ether, the power. So it is possible that in the finger of the leaf, that's hard as being characterized by tilting some of the spinamomentha. because the spinner momenta, the grass and gardens and then, of course, the felicity is correlated with the number of meters of foils by the degree of energy. So, one can express it in this fashion, and the two that are connected together, we can learn this. Let me examine the denominator limit of the head, this secretly symmetric denominator involving the spinner moment.

7:30 And that, too, one's going to stand in the Ravanai discussion by defining a current made out of two pre-Phermion fields called and then if I take the correlation function of one of these currents, then I get exactly the significance of this electric economic function. Because for pre-Phermions, the propagator is given by the temperature. So these are propagators which split from the space they are, this function. Now, one can put all this together next day and write the amplitude according to the following form. We have the 10-peton factors, we will introduce integration over x to obtain the momentum observation that function. And then we have the integration of the Yasmin parameters, we have these exponential parameters together, one for each quarter, so you can put it all together as this expectation value, integrated over the X and Peter, where the operator V is given by this expression, the term that I introduced earlier multiplied by this expression. So, in some sense, these vertices can be thought of as the so-called vertex operator whose expectation value we evaluate and integrate over the X and Peter at the end. towards the kind of expressions that one can talk with each other. We all know that there is a very nice chromatic way of writing this, where you really think that the difference between the chromatic still lies in the fact that these exponentials these poi guards here and one can eliminate that by a free transform or rather one concentrates in the free transform then you do it. You can have a delta function like this and when you do the free transform, you get back in that you just have to do this. The delta functions, of course, are needed, of course, that you may be like x poi and then you can be sort of exponential and so on. But this, of course, tells us that the answer relates to the support So that is basically one of the universal stages which is greater than that's what's got about here.

10:00 Now, what I want to do is, having these results, I want to think a little bit about Galatom Antitudes. And the first statement is, both are the MHP and repeat the Galatom Antitudes. Galatom Antitudes can also be repeated by simply sitting down and writing down the final words like Serian Calculation, which of course is quite hard, but there are actually relations between Graviton Antibiotics and the Andro-Saintry Antibiotics is a wise in relation to the post-trail Antibiotics, the open-trail Antibiotics, which is covered in your age, so-called ALT relations, which is avoid living in thought. And by utilizing these relations, one can actually calculate the Graviton Antibiotics, along with a number of other things. And Varense et al. in the 80's calculated the image of the Gerveternic power. It's rather complicated, so I want to do step-by-step. There is supposed to be a factor involved in the Gerveternic coupling, which is constant to some power. There is a factor of one-two spin-off product to the eight-power, which is similar to the one-two to the fourth-power we have in the angular state. And then there is a lot of other factors here. Notice that I have these square brackets here which indicate that these are the conjugate of the spin-off products and therefore these are not photomorphic cases here. This factor N is a product of spinner momenta like this, there is another factor here involving spinner momenta, a product of spinner momenta and then there is yet another factor here which I have to explain a bit. The notation here is that L corresponds to the phi-bore spinner associated with it, the momentum of the L's positive. Phi corresponds to the L's momentum. Notice that if phi-bore is on the left, phi is on the right, so in consistency with my spinner equation, square right on the left, N on the right, with an angular bracket. And K is a sum of momentum like this.

12:30 So this is a rather complicated expression and it's not terribly easy to see what it means, but there are some pieces which are offered. There is a lot of homomorphic pieces here, but there are some any homomorphic pieces, which involve fibroids, and they are actually some sort of a polynomial, because they are currently in these ones here, any homomorphic pieces. Of course, one will fix this anymore, one increases and realize that they could be replaced by derivatives with respect to omega. And therefore, in transport, you could also write this in terms of a twisted language, where you have, rather than derivatives of delta functions you have, derivatives of delta functions you have. That can be done. But what I would like to do is to go a little simpler. To do that, let me just start with the surplus from this third, four-point function. Then it is a twist structure where the minimum increases, this is going to show you. I am very simplified by writing all these attributes out explicitly So, ten core sense of this product of various scale of products, the spin and spin-off. And out of this I expect a significantly symmetric factor involved in the four particle momentum and then the body's left to work. And the other observation is that if I take, if I define capital P as the momentum, for four particles to the right of you. So, that's a particular, and I will rewrite it in this information. For the moment, I will find the key to more than one of the particles to the right of you. And why two? Because I am going to take the products, can a product be two here. Then, you can actually write down this product based on the fact that the conservation of momentum simplifies in the end of the theory. Therefore, the simple observation is that that product, that spin-up product, which is not photomorphic, can be related also to this factor. This is similar to what I have for some… There were similar other factors in the folder that I showed you, but somehow, when I said that, they seem to realize that we can think of a yoda form in this fashion.

15:00 So, let me continue with the four-point function. If I do that, then I can actually read out this amplitude in this question here. A number of distinct features, there is a 1, 2 to the 8's power here, similar to the 1, 2 to the 8's power here, similar to the 1, 2 to the 4's which you might expect will come eventually from some sort of subject matter. There is the signaling made in the denominators that we found in the angle scaling. There is this new factor that I used here. and then another set of cyclic invariant, but of course it skips two, one, two, three, three, four, four, five, two, one. So three particles are singled out and there is an extra cyclic invariant denominator. This is the structure of the sample. And since we know now how to get these scalar products into the denominator, from expectation values of currents, I want to introduce another current which will take care of this type. So let me define a new set of fermions. In addition to the alpha and beta that I showed you in the first time, So, having done that, then it's easy to see, the formula I showed you, this formula, this will come from our old current and this will come from the new current that I introduced. So, again, the expectation that is raised, the pi-pi combination that I introduced to the new current, this is the extra-momental factor, and again the pi-pi and so on. Now, of course, here is that in this extra-momental factor, I have put an arbitrary-moment, arbitrary-spin-a-moment and lambda, which I have eventually set to be four, the last particle in the series here. I can do that by simply writing another set of Fermi factors. here, pi and pi, I'm taking a V-bar dynamite in it. It gives you some sort of the other function of the momentum. When I do lambda integration, then it sets lambda to be

17:30 the momentum corresponding to the force that I do. And then this expression will reduce what I've experienced. Now, another way to say this part of this integration here is that rather than introducing V-bar, we can introduce simply a pi and a pi, we're going to get denominators involving four lambda and one, one lambda and so on. And we have 100 stone or et cetera, which we've discussed many points earlier today. And we can process the singularity corresponding to four and then this reproduces, at this such order that has to be such a particular value. So therefore, the gravity and I have to, the whole gravity and I have to do it as this function. a rather compact expression, and now we can ask, can we generalize this? And I won't go through the details of that now, but I'll tell you, you can actually generalize this. This generalizes can reproduce this world in which we have to give that you are calculated by the answer to that. So, first you separate out the cyclically symmetric denominator, then you have a product of these five points and these new momentum-dependent factors, the whole sequence. How does it work here? There are always three of these five points that are singled out. The first one, the last one and the last one. You can take a other order. I am just taking one particular order. And the rest of them are just these momenta determinant factors here, where P stands for momenta of all the forces. There is of course this factor and now we see that these could all be put together rather nicely. We expect this to come from a double function, but we have the 8th power, So we need N equal to 8 extension, not N equal to 4 extension to get this natural. The secret determinative denominators would come from the currents that are introduced initially, from the first-time currencies. We have some new currents here to take care of these new factors. So everything together then looks like this. So there is a D4X and a D16 filter. Here are products of J with another type of J, which is the coronary of this,

20:00 and then another type of J with E, and a whole string of them, and then they take the expression. So this is a way of realizing the MHP amplitude that shows that it has something to do with N equal to 8 theory, but there are these strings of products like operators that have. Let's look at these operators a little more carefully. There are three types of more. One is the original alpha beta that we introduced. Second is what I call V, chi times pi with this wave function exponential factor. Those go in three places here. The remainder, the ease, consists of, again, a wave function factor, this momentum factor. But remember, this momentum is the sum of all the momentum standing to the right of it, so it is in fact simply just a derivative of it. So it derives everything standing to the right, gives you the sum of the momentum and that gives you the answer. In that way, we have a formula that gives you the amplitude for which we have now. Now, let me take another direction, somewhat speculative direction, but let's look at how this amplitude might be obtainable from a more In the case of the Yang-Mills, we know that it came from looking at something like beta, d-bar, alpha as we draw it. This is what is obtained in the psychological V model. That's what we use. And the d-bar here stands for equivalent in general here, which is particularly d-bar, and then there is some each potential that you have to add to. What is this potential? It is actually just an elicit equal projection of the space-time gauge potential and that's what we need to get all the parameters previously. So, one can ask, can you identify similarly a gauge potential for gravity and take a helicity projection to see what it looks like? So, to do that, let's look at fermium's is the direct operator, there is the tetraide that multiplies the order of the derivative,

22:30 there is the spin connection count, and I still need the new algebra to come here, I mean the difference to come here, it is like something to take. So let's now take the tetraide and take the flat space and do a small configuration around that. That is sufficient for all the tetraide galvanides we are discussing. and this critical discussion, here is that this gives me a gauge potential which involves one part of the derivative, which is to find the stuff that you need and then there is something that involves omega. And if I take a helicity projection, helicity one projection of that, that is exactly this second vertex operator that I used to see. That is exactly this operator. So, there is something interesting there, something going on. And one can go a bit further. Notice that this is the structure of the field multiplied by an innovator. Let's look at the gauge potential, the gauge potential of the field and the T alphas are the algebra elements which you may think of as generating even translations only. In fact, this relation between the two becomes much clearer if you think of So, you see immediately that the EMU are, in fact, very much like the T-alphas of the Gates T-alphas. And therefore, this extra piece that we are finding, in addition to the sacredly symmetric piece we tend to quote, the extra piece that I have to include, which gave all the non-paramonic pieces, They are very much like it, like the chanthana. Now, what about the other perfect self-righteousness? Notice that this involves addition. This involves additional fermions. Now, we have some understanding of this right there, but what about this one? Well, if you look at spin connections, in a super-gravity theory, of course, there is a certain torsion constitution and there is a term involving the gravitanum. And you can ask, what is the projection, the LSAT1 projection of the gravitanum?

25:00 It has a structure very similar to what I described. I'm not sure that is necessarily the source of that term, but it suggests that in that, in reality, if you need two types of vertex operator, one coming from the terminal world in the tetrax and the other coming from the spin connection, and then the spin connection would have an additional fermion contribution precisely because of everything. So, if one points that, then, of course, the amplitude of this nice little structure, where a photomorphic piece has any more recurrence as we put and then something that will generate any photomorphic pieces, but these are to be interpreted in some way as the analog of ten-fattern fact in the integration of my system. So, to that extent, it is very similar to one of the same. Maybe I should also mention that in the formulations of superiority, There are formulations where one treats, one essentially gauges the whole one can agree and then there is a similarity between the the gauge structure consciousness and the translation generation. So that is very close, that kind of formalism is very close to what I am talking about here. It could also be that if there is some way to generate variables from the open-string theory, which was discussed in the morning, then this would arise naturally. But at the moment I have no better answer than that. There's, of course, one catch in what I said, which is that why choose only three of these V type workers operators here and all the rest of them are the E type. Now, you cannot take less than three for the meek heart because then the amplitude vanishes. So the first not-vanishing amplitude you can find there's at least three of these meek hearts. How about more than three meeks? It seems it is possible. I don't know what they mean, so I'm not sure what they are. But other than that, it seems to fit in rather with the picture. Maybe there is some sort of like, She is the strain that something, eventually, you are in the brain so you have it, maybe there is a brain thing, maybe, because of hope.

27:30 Such a sort of reality would naturally arise from any good thing. So, sorry, it equals that and so forth. So, could the ampergates with modern same-it-be not amnesia ampergates? It could be, but I don't have a, you know, I don't know the formula, not amnesia is not understood very well. I think they call it a filthy empty empty. So it's a bit hard to come up. Well, you can always get up from the KLG relations from the non-engaged, so, yeah. No, sir. These don't seem to be quite just that. I mean, it's not just that. It's more just that. Some of these entities may look like something. So I'm a little bit here. It will be nice to see what they mean. So that is basically what I have to say about the general part of my talk. I have now spent a few minutes talking about trying to obtain a matri in terms of the constraints of the book. The Enigled Poirier's theory has many, many nice features to discuss here, but one of the interesting things that we can say about the Enigled Poirier is that the class of theories and how they defined by constraints. What does that mean? That means that in terms of constructing the field feel continually, you know, if one takes an arbitrary gauge potential that depends on tax and heat and boards and so on, you end up having too many degrees of freedom. So, generally speaking, one has to put extra constraints to cut down degrees of freedom to what you really need. And as you increase the number of extensions of civilization, these constraints go up in number, that's what. And they can get... In all the supersimality theories, we have all points, at least in this formulation of covalent and arbitrary gauge potential, there are some set of constraints. They are generally expressed in terms of so-called spinner derivatives, which are derivatives of the P and P of R with these additional terms here. And for the N equal to 4 theory, we introduce gauge potentials, which are fermionic gauge potentials for these two, and set these constraints.

30:00 So all these covalent derivatives So, essentially, it's a flatness condition. Why do you have this, what he has given us, because it's being enough? Yes, the sixth scale, yes. Ah, yes, yes. This doesn't kill all of them, because there is a and therefore what is left is, what is left is, what is left is, okay. It's a child that looks like a child, and that is still there. They are given by constraints where you symmetrize indices. And all this symmetrize comes to zero, and what remains is the answer and that gives you six. I mean, I think it gives you the entirety So these are the constraints that are defined here, these constraints should be interesting. The interesting thing about these constraints is that as you increase the number of extensions, the constraints become more and more constraining, so to speak. And by the time you get to an equal to four, these constraints not only predominate the number of degrees of freedom to the right number, but they also impose the equations of motion. So these constraints are in some sense equivalent to the equations of motion for any people. That, of course, is not nice from the point of view of constructing an action for the theory, because you are already pondering all the solutions to the equations of motion. You cannot get off that. However, from the point of view of finding art and attributes, This is a rather nice feature because all we have to do would be to solve these constructs rather than go and solve the second order of creation and evolution of the game. So, one way we can think about how to solve this is to introduce...

32:30 These are, by the way, very similar to what was done. These are first order equations and they have, apart from the superstition, they are very similar to the Einstein equation, so I want to try a similar trick in trying to solve this equation. So, to realize these constraints in a nicer fashion, we produce spinner variables u and set every bar, which is related to u bar by choosing some arbitrary momentum k. K is what is usually called a reference momentum which occurs in many of these calculations in the end of the earth. But one needs the intermediate states. So in terms of this U and W bar, we can convert these spinner derivatives, these spinner derivatives into these B pluses and B minuses by multiplying by U and W bar. Once you have introduced these variables, there are also new derivatives we can introduce, which is derivative with W4 and U, and these are generally called D plus, plus and D minus. This construction is essentially what is known as the harmonic super space construction. And it is very closely related to the twists, as you can see, spinners and so on. Now, the idea then is to rewrite all the constraints in terms of these combinations here. And these additional derivatives that we have introduced with respect to these dummy bosonic variables, spinner variables here, they are really not needed so that the constraints involving these guys would be simply empty as a moment. But then the idea is to do a gauge transformation involving these new variables So that's what we are going to do, to do a gauge transformation. Originally we start with A++ and A-90. These derivatives have no gauge potentials associated with them. But once you do a gauge transformation, you are going to generate them, but we will fill some of the spinner gauge potentials. What happens then? The new set of constraints defined are then given by these equations.

35:00 There is a condition that says the d plus derivative of a plus plus should be zero. It is essentially a homomorphistic condition. That's where the homomorphistic condition is. The a minus is determined by a minus minus like this. So this is, there is no new equation for this. There are a number of other conditions which have to be satisfied which I'm not displaying here and there's also this condition between eight plus plus and eight minus plus. Notice that this is a simple vanishing of a certain field strength condition. It's a zero conversion condition. Now you can say, well, these constraints are simpler, can be solved. The answer is, so far we haven't quite secured in doing that. If we take these constraints and impose one additional condition by hand, which is this condition here that we favor derivative conditions, this is what leads to the restriction in which we are in position of the sexual condition. So far it's completely general. Once we introduce this condition, all these constraints release this one. All these can be assessed. In particular, this condition is well-known in Tsubrasi method. It tells us that A plus plus depends only on theta bar through this combination. A plus plus is independent of theta bar except for the fact that there was a space derivative here. Therefore, it appears in this combination. Now, the hermorticity condition which we started with, then tells you that A++ of course can depend on theta, but through this combination here, and this is exactly what will work out to be the content of the energy between the end of the scale. We start with an A- which will be the magnetic helicity component, this will be the spinor component, there will be a scalar here as we continue the series. There are four of these toys, so when you get down you'll find the positive and the strength of the gain of the strength. So this comes out of solving these constraints. And what is left then is this particular constraint here.

37:30 Now, I shall also say that at this point there is an additional requirement that this condition actually is two restrictions. One is an A++ of this form where this momentum P actually should be a non-vector. And in fact, this condition that it should be a non-vector shows you that P should have this form and that the ad-hoc spinner that I introduced earlier, in fact, becomes identified with the spinner momentum. So at that point we have essentially recovered what we started in the diagram. So the final condition that remains is this condition here, which in local coordinates has this zero temperature form. And what we need to do for the purpose of computing altitudes is not just to solve it, but we want to construct some sort of a this matrix for it, so you need an action. The action for this kind of equation motion is well known, which is so-called B-H-Weson-Indo-Ital action. It's something I have displayed here. and all you have to do then is solve this equation, what this is here, and that will give you a construction of all the MHG amplitudes. Then we will find indeed that all the MHG amplitudes are correctly needed. So, by solving the first order constraints, we have to, so this in fact gives you like an elixir, a different proof for the formula for the MHG amplitudes. We are able to get all of them. However, what we'd like to do is to be honest with the emission, and we relax this extra condition that we impose, and we get the non-emission. I have no clear answer for that. Sorry. Thanks very much. So, your idea for gravity qualitatively was to use an endpoint function of the stress tensor on the D instrument? Yes, sir. So, for conformal supergravity, Bergevin's mind is something a little bit like that. I realize that. I realize that. But, um, this is, this, I mean, if one can do something similar for any of the age, that's the first time you say, it's very similar. Well, of course, it's not conforming, right? I get it. So we've got to change the rules a little bit. I know. We don't want something which has conformed when Magnus built in.

40:00 Yeah. Right. But, uh... And if we're going to do something at all similar, and we'd like it to be natural to have eight fermions... I don't know. We weren't going to ask about the superman for his club here or things like that. Well, we should find something which is special and have an apron, that's all I can say. But, I mean, starting from SU to 2 slash 8, I think it's actually what we are about. And, I mean, even the super-part, uh, master super-part would have the same super-part, just because we have any percent. Then, somehow, this suggests that one should look at 11 dimensions of the character itself. And that's why he was asking in his question earlier, so that it is possible to get something like this, 11 dimensions. If that's the case, then indeed, perhaps, one would use Counting at the spectrum level, I agree with you. What's in it beyond that? I mean, everything that you said is so similar to the risk that you slash core, except for changing the A. I mean, the selection of the vertex operators, I don't know how much that would be, it would just be the same version. It might be, I just, I don't know why this category is okay. I was doing it, the vertical vertex operation is from the exclusive translation. I understand, I understand. So whether the twisted-supistic inversion raised on the IC2-staché could be natural to give you those vertex numbers, is the remaining question. But otherwise, the particular inversion is automatically super-carrying. The spectrum is automatically super-carrying. I would have expected some of the end of the description. The center is... Sir, I want you to go back to the field. I don't understand. Why is the question is to make it part of the standard?

42:30 Oh, because... Let me see if I can find it. No, I didn't write the constraint. It says, It says, this one plus the neutralization, this one plus the neutralization, this one plus the neutralization. And it's very natural. Once I have the still available neutralization of man, because of the naturalization. Thank you. Thank you. to say first something about the first description of some of the Young-Mills equation. Then, just to mention, very, very briefly really, the idea of redundancy in various accessible systems. And finally, I want to say something about the generalized as it applies to the subdual Yang-Mills equations. Let me begin by saying something about the subdual Yang-Mills equations under Trister's description. In fact, that picture there is something that you should already be familiar with, I've really got something on that. There are the basic Penrose correspondence between space-line and twister space.

45:00 Here, on the left, we have space-time, we found complexified space-time here. So, we just think of that as being C4. And on the right, there's twister space, and just think of that as being CP3 with complex space, Penrose correspondence between them, and I need to repeat this because the sort of basic geometry of it is vital to the way that self-dued angles work. So twisters, or points of twister space, Z, correspond to totally null self-dual, or anti-self-dual in this case, totally null, anti-self-dual two planes in complexified space-time. In fact, the set of all totally L-antisultial two-planes is a three-complex parameter family, which is essentially Cp3. In fact, Cp3-5. So that's the sort of blue part here, that two things correspond to these totally nulls antisultial planes. And then points in spacetime, which I've labelled green, like points x, they correspond to complex projected lines, basically straight lines in the projected justice points. So x, the green x here, corresponds to x hat, which I've drawn as a straight line, but it's intrinsically a Riemann sphere, so it's a Cp1. And so that's the sort of correspondence we go back and forth. And of course, when there's the usual incident properties, for example, x hat and y hat, it descends in perfect space if and only if the points in space-time x and y are null-separated. So the conformal geometry of space-time is encoded in this correspondence. Okay, let's get on to gauge field, well, and most fields. Well, a gauge field is self-dual, well, as a condition on the field, that just means that the dual, the curvature, the field strength, is equal to the field strength, if and only if this curvature gauge field vanishes, if it's restricted to these totally null and self-duals. So, if it's the case that the curvature vanishes, restricted to any, or to every, totality of the self-heal two-plane, then you have the self-heal gauge field, and vice versa.

47:30 And these are called the self-heal Young-Mult equations, also because this condition implies the source-free Young-Mult equations. OK. Everything here is complex, and so one can ask, well, is it possible to impose reality conditions? In this complex picture, the gauge group would have to be some complex group, like DLNC or whatever. But if one wants a real gauge group and one wants a real gauge field, then you have to include a real slice of this complexified space-time. And basically the only two possibilities is that you can restrict to the positive depth of space with signature plus plus plus, and that's just Euclidean force space, or you can restrict to the space with mixed signature plus plus minus minus. So, Minkowski's space doesn't quite fit into this, although one can still think of CM as being complexified, Minkowski's space-time. The point is that there are no real self-dual ganglous fields on Minkowski's space-time. Another way of putting that is that the self-duality equations acquire a factor of I when you have a norentium signature. Okay, so the basic summary is that if we're interested, which we are, in self-dual gauge fields, then they have this property, this sort of characteristic property, that the field is self-dual, even only if it's trivial when restricted to each of these special planes. Okay. And that leads to the basic theorem, which is that there's a correspondence between these self-delgaged fields and the same on the other side. There's a correspondence between cell-field gauge fields in space-time, and there's a revolving vector bundles over the twister space. So, if you like, on the left-hand side, we have a partial differential equation. I mean, star f equals f is a first order partial differential equation, non-linear, for the gauge potentials for the connection coefficients.

50:00 But in the second category, there's no differential equation here. One describes this geometric object, which is a holomorphic vector bundle over an appropriate subset of the projected crystal space. And there's just one condition that one needs, which is that this vector bundle should be trivial if it's restricted to any of those straight lines. In fact, that's less of a condition than it might have first appeared, because this will generically be paid. So there may be one or two lines for which this condition fails, and you just have to make sure that they're sort of saving it out of the way. But, generically, this extra condition will be satisfied. So, basically, taking any holomorphic vector bundle of projected twist displays, that will give you a solution of these self-realizing equations. OK. And, yeah, this vector bundle E is of some rank, say, rank n, and that corresponds to a gate between, at least to begin with, the grnc. So we're talking about rank-end vector values or end-by-end matrices and so forth. So I just want to quickly run through the argument geometrically for how this works. So this is a sort of one-to-one correspondence if you go back and forth. Let me summarize briefly how to go one way and then how to go the other way. So, for example, beginning at the top with the self-fueled gauge field, how do you construct the whole multi-pentacle? Well, from the sort of characteristic condition that I mentioned, which is still over on the right, if you're given a self-fueled gauge field, then it has the property that it will be trivial on one of those planes, z, z checked. when he said it will be trivial on the corresponding plane and that means that the space of co-variantly constant sections of the vector bundle the space of co-variantly constant isovector fields over the plane

52:30 is just a vector space of dimension n if you give a vector at one point of the plane then you can parallel propagate it over the whole plane and you take that to be the fibre of the vector bundle in the twisted space fiction. So that abstractly defines what this vector bundle is, this E, because for any Z, I've told you what the n-dimensional complex vector space is, which is attached to that point Z. And it's not hard to see that the vector bundle constructed in that way will have the extra property of being trivial when restricted to any line, Going the other way, if you've given e, if you've given the vector Mandel over on the right, then the first step is to band the space of polymorphic sections, the space of complex analytic sections of the vector Mandel restricted to x hat. You do that for each x-hat, of course, but let's think of one particular line, x-hat. Then the fact is that that will be, again, an n-dimensional vector space. So there's some sort of big mathematics hidden here because one is really using Neuwald's theorem. The fact that X-Hat is a compact Riemann surface, in fact it's CT1, and so is the spatial sections over it, in fact equal to the rank of the bundle. So in fact there's a sort of, the compactness here is sort of vital, the fact that Christmas waves is compact in at least some of its directions, that's a sort of vital ingredient. a sort of global thing. And then, so that n-dimensional vector space you take to be the space of iso-vectors at the point x in square star. So, so far there's no gauge field, all that I've constructed is essentially a trivial vector bundle over a complexified square star, so there's no information there But now, we ask, well, can you find the gauge field? But that comes out of this left-right correspondence, because we say that the vectors x and y, well, you can, if you take some curve in the plane z-hound, from x to y,

55:00 take some curve from x to y, and what does it mean to parallel propagate a vector from x to y? If you begin with a vector of x, you have to be able to say what the corresponding vector of y and y. But there's an obvious way to do that, because the sections over x-hat, and the sections of the bundle of e over y-hat, can be identified at the point where they intersect, Essentially, there's a natural identification between sections of XI and sections of YI that gives you a natural identification between isoeuvres of X and isoeuvres of X and Y. And, by definition, that corresponds to parallel propagation from X to Y, and so that defines... Given that we can do that, for each of these small two planes, that will define a connection on the bundle of overspaced space, the non-direction would be a small plane to span a whole tangent space, so that one knows from now in every direction. And again, in this case, it's by construction and one can easily see that the convention, the gauge field, is indeed self-deorable. So, automatically, there's a solution of these nonlinear PEs, which are the self about equations. Okay, so that's the sort of basic center, that self-dealed gauge fields come very naturally out of this correspondence, and there's a natural geometric way of saying how that works. And then there are lots of more down-to-the-earth ways of constructing these, um, if you're given bundles to construct examples, okay, and so forth. I'll mention some of those later on. Okay. A couple of remarks, so these are just sort of extra bits, and these are not connected to these two separate things. One is, is there a sort of three-dimensional version of all this?

57:30 So this, that correspondence over on the right, that's between a four-dimensional space-time and a three-dimensional twister space. Well, the most obvious one is an analogous correspondence between a 4k-dimensional space-time and a twister space, which is basically Cp2k plus 1. So the basic k double and right corresponds to k to 1. In this case, this high dimensional case, or k2 or higher, we don't quite have all of the obvious symmetry, we don't have the SO4k symmetry or its conformal extension. That's broken down to a subgroup, which is spk cross AB1, multiplied by Z2. This is a well-known geometrical structure. I'm not going to go to the heat mode, but that's sort of a well-known thing. Of course, if k equals 1, then we have equality here, so we have the full conformal symmetry. But if k is bigger than 1, then the symmetry is broken. So this particular picture, to the one over on the right, it becomes a bit more complicated because there's now a difference between self-dual and anti-self-dual, a sort of choice of various equations that you've met with a choice of various geometric correspondences. But there's an analogue between self-duality and the anti-self-dual equations. they no longer mirror images of each other, but they turn out to be sets of algebraic relations on the field itself, which are the generalizations of the star n equals n, the basic self-durality. Okay, so this is a sort of, the most obvious higher dimensional version if you're interested in higher dimensions are the problem of basic self-duality correspondence. The second remark, this is going back to dimension 4, is just the fact that you can rewrite the self-reality equation. If you fix a gauge, you can rewrite it in a way which doesn't involve a gauge, but involves a matrix of scalar.

1:00:00 This was done by Yang, probably many years ago, about 30 years ago. As I said, it involves fixing again, but it also breaks to the rent symmetry because the equation that you end up with, a matrix equation, called this matrix J, is a bit like the chiral equation, the first term there is like a chiral equation with a derivative of J to the minus one derivative J. But there's an extra term, and that extra term breaks the Lorentz symmetry because it involves a particular, in this case a particular, constant two-form on space-time. So it's a sort of version of the chiral equation which involves, which has this extra term in it. And the reason I want to mention that is that when we come to look at productions, as I will next, then one can either reduce the original gate-field equation to get a sort of reduced gate equation, or one can reduce Yang's equation to get something involved in this matrix of a state of fields. Let me say something about productions, really this is just a brief summary, productions have been analysed in very, very detail, in particular there's a book by Mason and Woodhouse if you want to sort of read all the immense detail of it. So just let me give a very brief summary, plus some examples. So here one is really thinking of dimensional reduction, and so we're going to impose invariance under some group of motions of space-time. And so that will reduce from four dimensions down to three or two or one. Of course the first step, or the first step might be to choose a particular gauge group, a particular space line signature. And then you do a dimensional reduction, and then you may or may not also want to impose some algebraic constraints on the fields. You might want to choose a gauge.

1:02:30 So these bits here are sort of optional, it depends on which particular equation you're aiming for. So as an example, suppose you want to reduce to three dimensions from four, so that involves cluttering out by some vector field, by some which is a killing vector or conformal killing vector of the complexifying space-time. And as I say, you can either reduce the original Gage-Field equations, or you could reduce those Yang equations called J, and then in the former case, you end up with an equation of this form, where the component of the Gage potential in the direction in which you reduce it becomes The Higgs field, in the adjoint representation, which have denoted capital Phi in there, and the soft variety of equations simply reduced to, in fact simply Brouillard-Dunn, to these so-called Bogomogni equations, with the covariant derivative of the scalar field phi being equal to the dual, the three-dimensional dual, of the gauge field. Alternatively, if you reduce those j equations, the young equations that were on the previous transparency, then they boil down to something which is very similar. It's essentially the same equation, except now they're sort of classified by a vector field, which I've simply labelled V. So in fact, the various possible reductions, various possible ways you can do it, It would simply be labelled by various constant vectors v, and v, when you do the reduction, turns out to have length n-squared, and it costs n-1. And then finally, of course, these objects look on a three-dimensional space, which is the quotient of the original four-dimensional space, and that turns out to be a three-space of constant curvature. There are various possibilities, and most of these have been looked at for example. The simplest case is R3, but then there's also R2 plus 1, which is the Minkowski space line,

1:05:00 or the hyperbolic space H3, or the 2 plus 1 dimensional versions of the Sitter, or the center space, or even S3. And so there are various egg examples that are known, the best known cases where this has been used, are for BPS monocodes, static monocodes on R3, so we can have anthropolic monocodes, that's the corresponding thing living on H3. In the 2 plus 1 significant case, you get various 2 plus 1 dimensional soliton systems. These are solitons which are sort of localized in the plane, and they can move around and interact with one another, and all of this forms an integrable system. I haven't used the word integrability yet, but it's a basic property of the self-reality equations that are completely integrable, and any consistent reduction would be completely integrable also. So this gives examples of various two plus one-dimensional soliton systems, or of the monopole equations, the static monopole equations, or integral, and circle. Right. One more page of these, just really listing some examples, this time of two-dimensional reductions. I said that these, those J equations, the Young equations, we sort of modify chiral equations, and in fact, if you reduce all the way down to two dimensions, you can choose the directions appropriately, so in fact, you end up with a chiral equation. So just to use the chiral equation or the harmonic math equation in two dimensions, you can get from a reduction in the Young equation, where you start with signature plus plus minus minus in the original four-dimensional space. If, on the other hand, you've begun with a positive definite space and reduced the self-dual gain equations, then you get the so-called Hitchin equations on R2, which I've written out there, which basically involves a complex scalar in the unjoint representation of the gauge group, for the equation that the covariant derivative

1:07:30 with respect to z, where z is a sort of polymorphic coordinate, is zero, and then the gauge field, which essentially just has one component, is equal to the commutator of phi with its angle. And, as I say, Hitchin has studied these in very, very detail. These are conformally invariant, underlies a lot of all this, and so these are defined just as easily on a real-man surface as they are on R2. Okay, so these are reductions of the self-geality equations in two dimensions. Similarly, if you do things, if you choose the field, if you make an appropriate algebraic reduction of a sort of n-by-n system, then you can get total field theories, so that has the form of the n-scalars phi, phi here is actually real-value fields, but then n of them, they work with an index A, comes from 1 to n, and the plus operator, or the wave operator, acting on phi, is equal to a of exponential supply where the coefficients correspond to the carton matrix of an appropriate V group. So it might be an extended carton matrix. So the simplistic example of this is the Newell equation, and the next one up would be something like... Well, the next one up, where n equals 2, includes the sin-Gordon equation and so on. So those are all examples, and of course these are well known as integrable two-dimensional systems. The other, even better known, integrable two-dimensional systems are things like the quarterback of De Vries equation and the non-linear Schrodinger equation, and they all arise as well as reductions of the self-duality equations, in this case a sort of non-reduction in the sense that one of the directions that you're dimensionally reducing along is a non-direction. And the final example of this transparency is a bit different. All of the ones above, we've been factoring out by translations, but if you factor out by something else, like a rotation, then you can get an equation which, in fact, is defined on a curved space.

1:10:00 A particular example that's been known for one time is that of the Ernst equation, which describes stationary axisymmetric spacetimes. And obviously that's a reduction, as the notation suggests, of Yang's form of dysfunctionality, where you reduce by, I think, one translation and one rotation. Okay, so far I've talked about the first two items over on the top right there, and just really briefly running through it, which are the twist of destruction, basically, and then just some summary of sort of possible reductions, and mentioning those are examples, and everything Everything in sight here is an example of an integrable system. The final item is that of the generalized non-transform, and so in the remaining ten minutes or so, I just want to say that it is about done. So, this is the generalised at Keir-Guntha, which is the main non-transform, and there's sort of a basic version of this, oh, well, let me say first, it really corresponds to a kind of duality between the South-Dual-Gang-Multi-Cresions in one copy of space-time, and the self-duality equations in a sort of dual space-time of coordinates x-tilded, and it sort of goes from one side to the other. And this thing schematically is just a sort of, if you follow the red arrow, it just tells you how to get from one side to the other. And going back again is the same sort of process. If you're given the self-dual Young-Goles field in the first space-time, if you like, then you can use the gauge potentials to construct a linear operator, which I've called here n.

1:12:30 And m depends linearly on the x-tildos, and the coefficients a and b are defined in terms of the gauge potentials in the original space. If you like, you use the gauge potentials to construct this linear operator, which depends linearly on the x-tildos. And this is a linear operator on some vector space, on p-vectors, but p might be infinite. So it's just a sort of some general practice way, but it's constructed in such a way that the equation mv equals zero, the kernel of n in other words, is just two-dimensional. So mv equals zero has only two harmonisative solutions, and so you can solve these, if you solve these linear equations to find solutions, then that solution exists, and it's unique, so that's the next step to solve those equations. And once you've got them, basically you can get the self-dual Yann-Mohr's field, the gauge field, on this right-hand space, on the dual space, with a sort of very simple algebraic manipulation, just by taking a gradient with respect to the extildes of these, and then forming the inner product with these. So this is all very schematic, but sort of most of it is essentially linear here. Once you've got the self-dual gauge field on the left, then you just have to solve a linear equation and then simply form an inner product here to get the self-dual gauge field in the dual space. The basic example of this, from which you can derive all other examples, essentially, is that of the self-dual angles equations on the four-dimensional torus. So T4 here is just the positive definite for torus. And then the dual space is the dual torus, where the periods are the reciprocal, sort of, on the left. But there are lots and lots of special cases of this, which, in each of them, has its own sort of features. For example, if you just pick out one of them,

1:15:00 the order is, say, x4, then you could look at a limit where the period of x4 tends to infinity, so that your period will tend to zero. Or, alternatively, you could have all the fields X4, which is obviously a special case of being periodical in X4, so one has these two special cases, and that means that you can sort of list a lot of examples depending on which of the coordinates, which of the periods are finite, and which coordinates the fields actually depend on. So, and this might be the last, maybe not the last transparency, but let me, so really all of this is really just summarizing some of the special cases of this time transport. Again, just to sort of give you an idea of the diversity here. The one right at the bottom is the basic example which I mentioned previously, because that corresponds to the field being periodic in all four coordinates, and depending on all of them. But there are special cases where, look at the red box on the left, you could think of a case where the fields were independent of, say, D of the coordinates, and were periodic in L of the coordinates, and then the remaining 4 minus D minus L, it had some people fall off as you go into the finish line. Then, the dual space, the sort of dual picture, gives you self-dualed angular spiels, which are independent of 4-D-L coordinates, still periodic in L coordinates, but now sort of falling off suitably in the remaining D coordinates. So, just to sort of run quickly through some of these, this very first one, if you like, was the first example, so that's the original ADHN case, where L is zero and D equals zero also. So there's no periodicity, and d equals 0 means the fields really depend on all four coordinates.

1:17:30 So that's just an instanton in R4, and in the dual picture, it's independent of, you can get a self-dual field which is independent of 4 minus d minus l coordinates, but 4 minus d minus l in this case is 0, and so that's just algebraic data, so that's just the ADHS data. So there's a duality there between instant ones on our core and, if you like, solutions of the self-duality equations, which don't depend on any of the coordinates, they're really just matrices. I should emphasize here that there's a lot here which is hidden, in the sense that these fields are not always smooth, one has to impose appropriate boundary conditions, there will be singularities of a certain type, so there's a lot of devil in the detail, all of which has been omitted here, and also there's a lot of the mathematical details of these correspondences that have not yet been worked out. The first two, I think one has sort of complete theorems in the instant-on case, and the next one, which is the monopole case, but many of the others are sort of, they're still gaps, and so these double-headed arrows are not quite theorems yet, but they're sort of almost all. So the next one down is the well-known case of the monopole, where there's a duality between solutions of the Bogomolny equations in R3 and solutions of the Naum equations, which are just living one dimension. That corresponds to P equals 1 being equal to P equals 3, but again with no periodicity. And then the final case of the Hitchin equation is this sort of you know, that you'll get solutions of the Hitchin equation being dual to other solutions of the Hitchin equation. As I said, again, you have to worry about boundary conditions and singularities and all that, so there's a lot different. The next case down is where you have periodicity in one dimension. So, for example, if d equals zero, it appears to depend on all the coordinates of periodicity in one dimension. So that's the so-called polygon solution, an instant on which is periodic in one direction. And in the dual picture, we again have a one-dimensional non-inclusion, an ordinary differential equation,

1:20:00 but this time we look for solutions which are periodic. And then the other example here is a sort of periodic monopole, a monopole chain, and that's due to the Hitchin equation on a two-dimensional cylinder out across S1. So again, some of the details of this have been worked out, but not all. And similarly, in the Calleran case, there's sort of a lot of gaps, or several gaps, which have yet to fill in. Now, just quickly looking at the others, in the case where there's two, where there's periodicity in two directions, There are basically two examples. There's a doubly periodic instanton, which is due to the Hitchin equation on a torus, a doubly periodic Hitchin system. and then finally there's the, in the kp equals 1, that sort of corresponds to a monopole sheet. That's a situation where the field is independent of one of the directions and periodic in two others and then falls off to infinity in the remaining quantum. And that's actually also auto-dual, that one monopole sheet would be dual to another one But none of the details are publicly known as far as I'm aware. So what the global conditions are, the boundary conditions, sort of none of that is really known. The final case is where you have something which is triply periodic, so a triply periodic instanton would be due to a monoclonal crystal. Of course there is no smooth monoclonal crystal, so again this just emphasizes the fact that there's a lot hidden here there's sort of hidden singularities on the right hand side here one has to think of solutions of the monopold equation which are triply periodic but with appropriate singularities and they would be due to triply periodic instruments there's just a few minutes left and so I thought So, maybe what I'll do is perhaps just to look at one of these examples, which is the case of the color one. As it says, really all of these are just special cases of the very last one.

1:22:30 And so, if you expect all of them to be connected together, that you could take various limiting cases, and they'd all be linked together. And so this last bit just illustrates that, where you can have a color-on solution, which as one limiting case would be an instant line, where the period of the color-on goes to infinity, and the other limiting case would be a monocle. This is a picture where you have a coloron, which has a monocle at one end, and an incident line at the other end. Well, I don't have time for much of the detail here, so I'm just going to show you that transparency. is a self-dual gauge field on R forward, which is periodic in one of the dimensions, with periods beta. And one basic and obvious geometric object you can look at here is the holonomy in the t-direction, that's the periodic direction. And this is already quite complicated because this sort of topological charge here, the integral of dual f times f, in general will not be an integer, but it is an integer in the case where this polynomial is trivial at infinity. If it isn't trivial at infinity, then you get sort of extra bits coming in. So in this case, there's an obvious dimension in this ratio because Celeron has a size, has units of length, and then there's also beta, a period, in T, that has units of length also, and the ratio between them is a sort of dimensionless parameter. And it's as that parameter varies that you can sort of get a monopole at one extreme and an instanton at the other extreme. This picture down at the bottom is just well emphasizing that a monopole is a special case of a color bond Monopoles are fields which don't depend on X-form at all, and that's a special case of being periodic X-form. So the monopole moduli space is a subset of the color on moduli space. What the color on moduli space is, is not known as far as I'm aware. That's one of the amount of money to the government. But whatever it is, I've sort of drawn it here as this lower sheet, where the horizontal direction corresponds to this dimensionless ratio.

1:25:00 So the idea is when this ratio tends to zero, so you can see that sort of corresponds to beta, the period going to infinity, then the colorons and the instantons will really look the same, because you could equally well think of it as the size of the coloron getting very small, and if the size of the coloron is very small, then it doesn't care what space it's living on, so it might just as well be an instanton on R4. So the limit where this ratio tends to zero is a sort of instant limit, and the other side, in fact, gives you a monopole. So that green line there sort of is just schematically a curve in the modulized space of colorons, one parameter family of colorons, which has a monopole at one end and an instant one at the other end. And basically, I don't have time for the details, but I'll just say that one can do all this explicitly. You can sort of begin with the monopole, take the num data for the monopole, recycle it to give you num data for the neuron, and then sort of, so that will give you the hold of this curve, and then if you take the limit where you go to the left-hand edge, then that gives you the HHM data for an appropriate instanton. So it's very easy to do, to get explicit examples of what it is, and to sort of take cases like the known monopole solutions and to recycle them to give sort of families of colorons and families of instantons. I said, I don't have time for details, but maybe that was just to illustrate the fact that that long list of examples on the previous transparency, they're all linked together. And there are lots and lots of sort of unsolved problems to do with all this, there are lots of gaps, and there's lots which is not known enough for the relationship between all these soldiers. Okay, well, I've run out of my time, so I shall stop it. Thanks very much.

1:27:30 Any questions? I was going to ask a comment really, it would be specific what you said, but it would be helpful to fill it out more, why all these reductions in the self-volume angles are interesting. It is, of course, a consequence of the history that you started, because whatever it will be, you should be able to solve the equation. It tells you how to solve, you don't know, that's pure geometry. That's what integrability means, and the signal down to all the other ones. Amplifying the remark. I mean, it's a big question as to what the word integrability means, but certainly one feature of it is that you can find lots and lots of explicit solutions, was that the correspondence tells you how to do it, gives you a construction. But there are many other features of interoperability, like the Pandevae property and so on, which you can also derive from this correspondence. The policy is real or complex. Which case are you referring to? Well, I'm referring to 2 plus 1. Those are real then. If you begin with signature plus plus minus, you can make everything real, and then when you reduce to plus plus minus, it remains real. Yes, so these are real solitons in 2 plus 1 dimensions, which are solutions of this sort of generalized in two plus one dimensions, which again is a real equation. When you look at the twisted picture, I don't know in general. I don't know. Certainly, if you just think of the singularities in space-time, this is not answering the question, you're answering a different question.

1:30:00 If you look at the singularities of solutions of the Nam equation, then sometimes they are just jump discontinuity, Sometimes they actually involve the fields going to infinity, so there are many different kinds of singularities of the nondes. I would guess that in the twist of nature, well it must be the case that these are jumping lines, but if they weren't, then it would be a smooth field. So it must be a jumping line, but of a very sort of special kind. So there'd be some that correspond to a sort of genuine, well, to just a young discontinuity in space-time, and some which correspond to the fused over infinity. So there's sort of different kinds of singularities, and you have to distinguish between them. Any further questions? Okay, we thank our speaker again. I want to thank the organizers for the quickly campus in an invasive conference. Although the main thing of the conference is on Twister Space Theory, the organizers stuck their mic with some links to becoming interoperability in Twister Space Structures, and so they asked if we could have a thought on interoperability at any of the 40M mills theory. And I hope there's a long way that possibly we can see certain structures that might link to two subjects, Krister space theory, Krister space methods and interoperability. So, what I'd like to do today is talk about work that I've done with Sarah Matthew and Edward Lydden in these papers that are listed below, and also other and I wanted to introduce the field theory that we're going to be interested in.