The Nonlinear Graviton

0:00 Well, it's a real pleasure to be back in Oxford, and it's a special pleasure to be here in this exciting conference. We're learning a lot. I wish I was learning more, but we're all trying. And I'm primarily going to review the original nonlinear graviton construction of Penrose from the late 1970s, but I'll also be putting it in a context where it has applications that includes Riemannian geometry. So, for this reason, I'm going to begin by reminding you about some essentials of quarter-dimensional Riemannian geometry. So, I want to imagine, to begin with, that we have a smooth four-manifold equipped with a Riemannian metric, meaning a positive definite metric, including a signature metric, And it's an oriented form manifold. So on an oriented Riemannian manifold, there's always a star operator that maps forms of degree p to forms of degree n minus p, where n is the dimension. And in particular, in dimension four, this star operator maps two forms to two forms. so geometrically what it does is if you think of a two-form as something which assigns a number to an oriented two-plane on a Riemannian manifold you don't know how to scale up a plane then the star operators in fact are basically replacing two-planes with their orthogonal complements there's this nice linear transformation of two-forms themselves For two forms on Riemannian 4-manifolds, the star operator has a square plus 1. So its eigenvalues are plus and minus 1, and if we can split the two forms into the plus 1 eigenspace and the minus 1 eigenspace, the two forms of a rank 6 bundle is split into two rank 3 bundles, the self-dual and the anti-self-dual two forms.

2:30 And that immediately gives rise to a lot of remarkable phenomena in four dimensions. In particular, the Riemann curvature tensor of a Riemannian four-manifold splits into four pieces. And this can be seen in a number of different ways, but a particularly elementary one is to think of... You can think of the curvature tensor as a linear transformation from two forms to two forms. Namely, if you raise an index of the curvature tensor so it has two up and two down, then by contraction with it, it maps, because of the symmetries, it's a linear map from two forms to two forms. If you split the two forms up as the self-dual plus anti-self-dual, that means that this transformation actually consists of four blocks. For example, there's a block that maps self-dual to self-dual two forms. There's another block that maps anti-self-dual forms to anti-self-dual forms and so forth. And then you can ask what these different blocks actually mean. Well, the Bianchi identities, the first Bianchi identities, tell you that, in fact, this is a self-adjoint transformation. This is, if you like, a symmetric matrix. and so these two off-diagonal blocks are in fact really the same and what they encode is the trace-free part of the Ricci tensor the trace of this block and the trace of that block turn out to be the same as a consequence of Bianchi identities so, first Bianchi identities so, in fact, the trace piece just encodes the scalar curvature pieces, the trace-free piece of this block and the trace-free piece of that block, which are other pieces of the curvature tensor called the self-dual and anti-self-dual piece of the vial curvature tensor. Now, one thing I did not emphasize before is that this star operator on middle dimensional forms is, in fact, informally invariant. In fact, these two pieces, w plus and w minus, of the curvature tensor are also informally invariant.

5:00 So if you multiply the metric by an arbitrary non-zero function, a positive function, if you want to say it in the line in context, then in fact cell dual-bile curvature and the n-cell dual-bile curvature tensors of the conformally rescaled metric are dead on the nose exactly the same as before and I've written in some indices to indicate that actually to make this statement precisely correct you should give it the same index structure as the usual curvature tensor, one index three indices down, whereas if you were to think of it that way, you would initially have defined W plus with two indices up and two down. That would give it the conformal weight. But with Hibaba would have been this way. There's no conformal weight. Now, in any dimension and any signature, you can define the biocurvature. It's exactly the piece of the curvature tensor which is conformally invariant. At a given point, you can make the Ricci curvature take any value you like by conformal restaling. if you think about the number of components of the Ricci tensor, it's the same as the second derivatives of a function, and by changing the second derivative, that's a formal fact that you can make the Ricci curvature anything you like at a given point. The piece that you can't affect is the vial curvature the remarkable thing in four dimensions with D'Imani's signature is that that vial curvature is not an irreducible thing, it actually breaks in a natural way into two independent pieces, the self-dual and any I should also emphasize that in defining this distinction between self-dual and anti-self-dual wild curvature, you have to have an orientation on the pinnacle. It changes the orientation that changes the sign of the star operator and it exactly interchanges W plus and W minus. well, so it's not hard to think of some manifolds for which W plus and W minus both vanish because if you have anything which is a conformal rescaling of the standard Euclidean flat metric that would be, would have both W plus and W minus a weaker version of this is just to

7:30 and so a Riemannian four manifold with an orientation is said to be anti-self-dual if W plus vanishes and maybe the convention about which is self-dual and anti-self-dual would seem confusing at first, but the statement, this is the same the same, but the only remaining part of the of the vile curvature is the anti-self-dual part that's why it's called anti-self-dual and I've made this definition in a preliminary way as a term to apply to a Riemannian four-manifold but in fact you should really think of this as a condition on the conformal class since multiplying the metric by an arbitrary function does not change double mean plus it's actually the entire conformal class consisting of functional multiples of the metric dealt with in this definition. So one can talk about a conformal Riemannian four-manifold, an oriented conformal Riemannian four-manifold being anti-self-dual. You can also define a self-dual four-manifold, which would just be the orientation reverse of this. If you look at a manifold in the mirror, then one with W plus becomes one with W minus, and just change the orientation. So, when you look at the literature, you'll find that about half the papers are about cell-dual nanopodes, and about half the papers are about anti-cell-dual nanopodes. Essentially, there's no difference. It's just a matter of the local conventions being applied by the author. Okay, I began by discussing the Riemannian case, but in fact this idea of an anti-celled dual form amicable works perfectly well in other signatures so for example it's particularly nice to consider split signature metrics so in other words metrics where the at each point the quadratic the metric is positive definite on a two-dimensional subspace and negative definite on a two-dimensional subspace of a tangent space so in this case you also have the star operator squared is 1 the eigenvalues are again plus or minus 1

10:00 the story is really exactly the same you could use exactly the same definitions and you would find that again the curvature tensor splits up into four pieces scalar, trace-free Ricci curvature w plus, w minus and as before you could say that a split signature Suter-Imanian-4-manifold is anti-cell dual Now there's another context in which the notion of anti-cell duality also makes perfectly good sense and that's when you look at complex metrics So this way of thinking is quite important in twisted theory but it's not particularly well known If you were to take If you were to take a real analytic metric on a real analytic manifold, then the coefficients of the metric would be given by certain convergent power series. But a power series that converges automatically also converges in the complex domain. You can extend the variables on which it's defined to be complex. And this leads you to a sort of a complexification metric, or of a pseudo-Riemannian metric. In fact, people that have thought about wick rotation know that there's really no difference between Riemannian and pseudo-Riemannian once you complexify. They're different real forms of the same thing. Versus an issue, whether you can find a real form of any given signature for a given complex metric. Alright, so if we look at a holomorphic tensor field, which is locally of this form, it looks like a holomorphic quadratic differential so here we have symmetric coefficients and it's required to be non-degenerate so if you like it defines an isomorphism between the holomorphic tangent and cotangent bundles then formally the same story goes through as before you can talk about the curvature tensor of such an object. It defines a connection. You can go about its curvature. Its curvature would be of type 2-0 tensor 2-0. And that's to say that when you write down the curvature tensor of this thing,

12:30 because everything's holomorphic, there are no z-bars inside. And when you break the curvature tensor up, you find that it consists of four pieces, scalar, trace-free reaching, self-dual, and anti-self-dual, the draw curvature. Now, there's a little technical problem that you might worry about here, what do you mean by orientation in the complex case? In fact, I mean, orientation in the real case is actually some topological statement about the manifold. Here, you can't actually tell whether the manifold is complex orientable until you've introduced a particular metric. But the point is, once you chose a given metric, there are at each point two different choices of the, of what you call the star operator, and the double cover of the manifold, in the worst case. On which Sheldon can talk about something like the anti-cell. Okay, so, just as before, even in this complex context, one can talk about the notion of a complex Riemannian manifold or a homomorphic Riemannian manifold being anti-cell. Dual just being W plus N. Now, if you... So the crucial thing that underlies this is the splitting of two forms into self-dual and anti-self-dual pieces. And actually, if you think about it, the most naive term for what's a two-form, it basically is a skew matrix. What's an infinitesimal orthogonal transformation? It's a skew matrix. As representations of the orthogonal group, the two forms just look like the Lie algebra of the orthogonal group, where the representation is the adjo-hap representation. So, in fact, the splitting of the rank-6 bundle of two forms into two rank-3 bundles of self-dual and any self-dual forms, in the Ramanian case, that actually is just the same, that the rank-6 Lie algebra, SO4, splits into two sub-algebas, each of which is a copy of SO3. And the other real forms I've been talking about, also, SO22, the split signature version, copies of SO12, and SO4C splits up into two copies of SO3C. So these are, these Lie algebras are semi-simple, but not simple. They have non-trivial ideals. They actually split the number of simple Lie algebras.

15:00 If you were instead in Lorentzian signature, the Lorentz group, SO13 is the same as SO2C. That's a simple Lie algebra, and there is no splitting of this time if you're going see in the Lorentz case. Some of the algebraic formalism is useful. For example, the biotensor on a Lorentzian form anaphobe can be thought of as having five complex components rather than ten real components, and that's something that Roger Penrose exploited in a beautiful and systematic way. But this notion of self-duality or anti-self-duality of an anaphobe really is not relevant in the Lorenz case. All right, so just for concreteness, I'd like to focus now on the split signature case, which in some ways you can see it better. So if you have a split signature four-manifold, it makes sense is that it would be an arbitrary signature, but you would generally not have any. So if you have a formatter fold with a Siderimanian metric or Siderimanian formal structure, you can say that a two-plane in its tangent space is isotropic if the restriction of the metric to the two-plane is identically zero. Well, it would turn out that there are such isotropic two-planes, real isotropic two-planes, if and only if the metric is split signature in four dimensions. So, these things are N. So, in fact, this is an important algebraic fact when you're trying to see that the quadratic point has signature zero. Same number of pluses and minuses. One criterion is the existence of these isotropic subspaces. All right, so, however, when you look at isotropic subspaces of a four-dimensional vector space with a split-signature indefinite or product on it, there are actually two different flavors of isotropic subspaces. There are the so-called alpha planes and beta planes. One way of seeing this is projective. So if the tangent space looks like R4, if you projectivize it, then you have an RT3.

17:30 The one-dimensional subspace, of course, is now represented by a point. The two-dimensional subspace would be a line, a projected space. If you look at a quadric in RP3, then there's this beautiful fact discovered by Christopher Wren, the architect that designed the Ashmolean Theater here in Oxford, that if you take a hyperboloid of one sheet, it actually has two families of straight lines on it. And the underlying amazing and important fact in projected geometry is that a two-dimensional quadric is actually the product of projected lines. So in Rp3, a quadric of this type is actually Rp1 times Rp1, and there's a more important version of that over the complex, that a quadric in Cp3 is Cp1 times Cp1. So, actually, but the point is that, non-projectively, these lines that I've drawn on the quadric, each of those actually represents a two-plane in the affine space, in homogeneous coordinates. And so, notice that these are actually in, these are quite far from each other. When you look at the lines on a quadric in this dimension, they're actually two-connected components. one calls those the alpha planes and the beta planes in fact this is intimately related to the idea of self-duality and anti-self-duality because a two plane is a to say that a two plane is an alpha plane is exactly to say that wedge two of it well, it would be in wedge two of the tangent space but you have a metric around so you can think of that as a two form and it's actually required that that simple two-form be a self-dual two-form. So if you have a simple two-form, the simple two-forms which are self-dual exactly correspond to the alpha planes, and the simple two-forms that are any self-dual correspond to the beta planes. So at each point on a pseudo-Roumanian, signature four-manifold, there is a circle's worth of alpha planes and a circle's worth of beta planes at each point. If you didn't have an orientation, you might get mixed up

20:00 on the globe. You might be able to go around the manifold and get mixed up about which is an alpha plane, which is a beta plane. But if you've got an orientation, they actually are globally distinguished to each other. alright, so here's one of the crucial definitions in this subject if you have a surface in a split-signature 4-manifold it's said to be an alpha surface if its tangent space is everywhere in alpha plane now remember, an alpha plane is isotropic with respect to the metric the restriction of the metric is zero so this would be a surface a surface in your four-manicle where the restriction of the metric was identically zero. And you could similarly define beta surfaces, surfaces where the tangent space is everywhere, a beta plane. Because the alpha planes and beta planes are far apart from each other, actually any isocropic surface is either an alpha surface or a beta surface. Now, in something like a flat space, there are a lot of alpha surfaces and beta surfaces. all be actually planes, though, and in the flat case, one just usually calls them alpha planes and beta planes. I would say I used the term alpha plane and beta plane for things in the tangent space. You can also talk about alpha planes and beta planes on a complex, on a complex reminding four-manipode. And, in fact, I sort of dodged the question, what do you mean by the thing that corresponds to orientation which ones you call the alpha planes and which ones you call the beta and so in a complex 4-manifold, one of these things things with a holomorphic metric you can again talk about an alpha surface the key thing, the thing to focus on if it's confusing is you're looking at complex surfaces in a complex 4-manifold where the restriction of a complex metric is identically well, and that's long definition when you haven't seen any examples. But it turns out that the existence of alpha surfaces is a very interesting one. So what Roger Penrose discovered in the 1970s

22:30 was that there was a correspondence between complex anti-cell dual four-manifolds and three-manifolds without any auxiliary structure. So the crucial lima is that, so I'll say that in the complex context, this is the one that's interested in Penrose, a complex Riemannian four-manifold is anti-cell dual if and only if every time you're given an alpha plane, there's an alpha surface tangent to it. And it turns out that that alpha surface is actually unique. for these kinds of geometries if you're trying to understand what the Suda-Riemannian metric looks like what the complex Riemannian metric looks like up to scale, well to know a metric up to scale is essentially to know its null vectors, the vectors that have zero square norm with respect to it, it's enough to know which are the alpha planes because every null vector is in a unique alpha plane and if you know what all the alpha that Riemannian metric is, but you wouldn't have alpha surfaces in a generic complex Riemannian format, but they exist in each direction, if and only if W plus is equal to zero. If W plus is non-zero, then there could still be certain directions to any given point where you have an alpha surface, but you won't have too many of them. In general, you actually wouldn't have any. Alright, so this is the crucial lemma anti-cell duality is exactly the criterion for the existence of alpha surfaces. And this then leads to the following construction. I'm stating it a little more precisely than Roger bothered to, but all the ideas are already in his paper. So suppose that you have an anti-cell dual complex remodeling four-manipode. in order for things to really work very precisely, I want to now localize a little bit. We're starting with an arbitrary one, but I'm in fact going to just choose some point in it and choose a neighborhood around it. For example, I could choose it to be GDC to convex neighborhood with respect to some standard of the metric.

25:00 So this last thing doesn't constrain the local geometry at all, but it avoids a lot of local problems. so then you can look at the set of all alpha surfaces and the statement is that the set of alpha surfaces with some mild local some mild hypothesis along these lines the set of alpha surfaces is a complex three-manacole associated with your given complex three-manacole Here's the sort of picture that emerges. One has what's called the Penrose-Twister correspondence. We start off with a, as explained so far, we're starting with a complex 4-manipole that has a complex metric on it, and then through each point, we can look at the various alpha surfaces. There is actually a Cp1's worth of alpha surfaces through any given point, and when at least given some convexity hypothesis, then the corresponding space of alpha surfaces called the twister space is a complex 3-manifold. Each point in the original space-time, as we'll call the complex 4-manifold, actually corresponds to a CP1's worth of alpha surfaces. So there's an embedded CP1 for each point in space-time. The purple point corresponds to a purple CP1. The green point corresponds to a green CP1. And these CP1s are called twister lines. So I've drawn them in the usual way that one tries to... When you try to draw pictures in something like complex projective geometry, usually it's helpful to draw actually something that's a real projective picture, and that's the inspiration for the style of drawing. so these CB1s are represented by straight lines in the picture and this is supposed to look roughly the prototypical version which I'll put up in a moment would be to take this to be the neighborhood of a projected line in CB3 alright so each off the surface is represented by a point over here, each point a point here on the other hand is represented by one way you can think of it as what's the space of all all of the twister lines

27:30 through it, set of all twister lines through it forms an alpha surface on the other side. So for example, the red point here is the red alpha surface and the yellow point here is the yellow alpha surface. Alright, so if that is confusing, you should think about, well before thinking about the non-linear graviton, you should think about the linear non-graviton. So the prototypical version is if your complex humanian manifold is C4 with the constant coefficient metric, then the corresponding space of alpha surfaces, every alpha surface is actually an affine two-plane in this picture. Not every affine, there's special affine two planes, namely and the ones which are four-symmetric is zero, they're diastrophic. And the actual space of alpha surfaces in this case would not quite be CP3. It would be CP3 minus CP1. And the discussion about using the quadra here rather than C4, you can, instead of considering all lines in CP3 minus a fixed line, once it avoids a fixed line, you can instead just consider lines in CP3 and you get a certain factification of this picture. which is thought of. It's corresponding to two planes in C4, but a nice way of thinking about that is the four-blocking by the Kleinman correspondence. Okay. Now, the twister lines, in the general case, have some special properties. First of all, they're embedded CP1s, and the second property to focus on is what's their normal bundle. have the same normal bundle as a line in CD3. Now, when I say normal bundle here, this is in a holomorphic context, you're supposed to think of, I mean, the underlying smooth complex vector bundle you could think of as just the things which are perpendicular to a sub-maniple, but that's not the right definition in a holomorphic context. You want to think of the normal bundle as a quotient. You take the restriction of the ambient tangent bundle modulo, the tangent bundle of the sub-maniple, And the reason for doing it this way is that it now makes sense to talk about holomorphic sections of this object, namely a local holomorphic section of the restriction of the tangent bubble becomes a holomorphic section of the normal bubble when you look at the pool and splice.

30:00 So that normal bubble, in this sense, actually splits as two copies of the churned glass one holomorphic line bundle on CD1. Sorry, I'm just questioning your convictions, what O1 is. So are these things rigid, or do they fit in families? No, it's like Lyme's in CP3, they definitely fit in a family. So O1 is the positive one. It's the one that has two mutual space perceptions. so the question that Philip has asked is exactly the right one namely do they fit in the family there's a beautiful theorem of which says the following thing suppose you have a compact complex submanifold of a complex manifold the ambient thing doesn't have to be compact but the submanifold is crucial that the submanifold is compact and suppose that the first core of the submanifold, the coefficients in the normal chief vanishes. The crucial thing is that this is an example of a bundle on CP1 where the first normal one vanishes. Then Kadyra says that, well, if you're thinking about how can you move a submanifold, the most naive thing is, well, at first order, you're taking up a section of the normal bundle, right? Holomorphic section of the normal bundle looks like an infinitesimal holomorphic motion of a submanifold. Kadyra's theorem that's exactly the right answer, provided that the sections of the normal bundle are the zero-chromology of the normal bundle. If H1 of the normal bundle vanishes, there's no obstruction to finding families of submanifolds with the given infinitesimal structure. So the moduli space of nearby submanifolds, in this case, given this vanishing chromology, is a complex manifold and its tangent space at the given sub-manifold is exactly the sections of the normal bond. So that theorem of Kadira leads to the following beautiful characterization. The ideas for this are in Penrose's paper, but this is certainly not stated there,

32:30 so I'll attribute this to Penrose et al. You can call me that can be called Al. So, suppose that you have a complex 3-manifold. At the beginning, we're just going to say, suppose you have any complex 3-manifold. And look for the set of CP1s in it with normal bundle O1 plus O1. That could be empty. it's always a complex 4-manifold. The empty set is a complex 4-manifold. And so if you look at the set of all embedded CE1s in a complex 3-manifold with normal bundle O1 plus O1, that is a 4-dimensional complex manifold. And then the magic is that tautologically the twisted construction gives that an anti-cell dual-conformal structure in a unique way, and that's the general solution of these equations in the complex setting. Alright, so here we're sort of, before I said what to do, if you have an anti-cell dual-conform complex 3 manifold. Now, this is inverted construction. Start with a complex 3 manifold, look at the modulized space of all CP1s embedded in it with the same normal bundle as a line P1 in CP3. Then that's all automatically a complex 4 manifold, an empty complex 4 manifold, and it comes equipped with an ASD metric, and that's the general such thing. Now, one reason Roger didn't state it this way is the annoying fact that, I mean, as it stated, the spaces of all such lines could be empty. But a slightly different application of this Kadiar theorem, it's enough of a different allocation that Kadiar actually wrote a different paper pointing out that it's consequence. But when you're in this setting that you have a sub-manifold with normal bundle for which the first homology vanishes, those things are stable under deformations. to form the complex structure slightly, then these things persist. And so, if you have a complex 3-manifold where the corresponding moduli space of twister lines, CP1s with normal bundle O1 plus O1, is non-empty, and if you now deform the complex structure slightly of that complex 3-manifold, the space of twister lines is still non-empty. And so this gives you not solutions

35:00 of the ASD equations on the empty set, but rather some interesting new metrics. So Roger's real interest was to take the flat metric and to try to understand nearby ASD metrics, and those turned out to exactly correspond to definitions of complex structure on enableable lines of 3. Now, actually, this is a very important and beautiful theorem, but it is, in fact, not the theorem that interested Roger at all. because he was interested in solving Einstein's equations, not the ASD equations. Einstein's equations are the equation that the trace-through part of the Ricci curvature vanishes, and that means that the Ricci curvature is a constant multiple of the metric. Well, the point is that there is a beautiful twister characterization of when does an ASD conformable class obtain an Einstein metric. So Penrose wanted to solve the Einstein equations, This construction was to solve, to understand those Einstein metrics, which were also anti-self-dual in twisted terms. So if it's anti-self-dual, it has a twisted space, which is a corresponding complex 3-manifold. And then it turns out, well, so Roger was, his paper actually goes with the Ricci flat case. There was an improvement of this due to Hitchin and Mord, which ends up telling you the following thing. Suppose you have an anti-self-dual 4-manifold. you look at the corresponding twister space, and you ask, is there an Einstein metric in the given conformal class? It turns out that the Einstein metrics in the given conformal class exactly correspond to sections on the twister space of the cotangent bundle twisted by the inverse square root of the polynomial. So it turns out these twister spaces are spin, and you can talk about the canonical bundles, look at one-forms with values in that line bundles. If you have a section of that, then that gives you a particular choice of a particular metric in a conformal class, which is actually Einstein. it turns out that there's a nice deformation structure complex three-manifolds with twisted one-forms of this kind. Either they're complex contact

37:30 and so you can elucidate that theory a little bit. There's a nice way of understanding the anti-self-dual Ricci-flat or Einstein metrics, which are near the standard one, and that is, in fact, what Roger meant by the nonlinear graviton. Okay, but I'm instead focusing just on the ASD part of the story, the informally invariant part of the story. definitely not conformally invariant. And let me emphasize, most ASD conformal classes do not contain, even locally, an Einstein metric in the conformal class. It's a highly non-trivial extra constraint. Alright, so if you're interested the local structure of the solutions of the ASD equations and you want to do it in, say, the Riemannian context As long as you're looking at real analytic solutions, then you can use this formalism to understand local solutions just by complexifying. But then there were those little problems about convexity and so forth, words that I didn't even explain very carefully. So there's some global issues in trying to define a twister space associated with an ASD manifold. No problem locally, but there are some global problems having to do with the fact that a priori and alpha surface might wind around and be dense or something. If you don't have some nice convexity hypothesis in the complex setting, you won't be able to find a nice space of alpha surfaces. However, there are some cunning ways around these problems in both the Riemannian and the Schell-Signature case, and the most celebrated version of this is in the Riemannian case, positive definite metrics where Akia, Hitchin, and Singer found an amazing elegant way of circumventing all of these problems. In particular, you don't have to assume at the beginning that the metric is real analytic. Although it turns out that ASD metrics are always real analytic in appropriate coordinate charts to take harmonic coordinates. But that just falls out of this machinery because what they actually do is associate And so given a four manifold, the point is that if you have a Riemannian slice in a complex Riemannian four manifold, an alpha surface will hit only in discrete points.

40:00 So the points of the Riemannian manifold would then correspond to a CP1's worth of alpha surfaces. When you just sort of see through that, what you can actually do is say, ah, there's a certain S2 bundle over a four manifold, which comes equipped with a natural almost complex structure, which is integrable if and only if the metric is anti-self-dual. So, if you have a 4-manifold with a Riemannian metric and that Riemannian metric is anti-self-dual, there is a there is a 2-sphere bundle over it, namely it's the 2-sphere bundle that you get, so wedge plus is a rank 3, this is an R3 bundle if it corresponds to the S2 bundle. That S2 bundle carries a natural almost complex structure that almost complex structure is the anti-self-dual thing. And so you're able to associate, in a very tight way, complex 3-manifolds to anti-self-dual Riemannian 4-manifolds. For example, a compact Riemannian 4-manifold has a compact complex 3-manifold associated. So, the main problems which immediately spring to the minds of mathematicians when they see a construction like this are which smooth compact 4-manifolds admit anti-self-dual metrics. Another problem would be to understand the moduli space of anti-self-dual metrics and understand what does the geometry look like on a given manifold. If you geometrize it by looking at anti-self-dual metrics, what do those anti-self-dual metrics look like? And to understand the complex... You have this interesting construction of complex three-manifolds related to four-dimensional geometry what do these complex 3-manifolds look like. This correspondence has the same flavor of the fact that, I mean, there's a... We all know that if you have a... If you want to study conformal geometry in two dimensions, you're really looking at the theory of complex 1-manifolds. If you want to study conformal geometry in four dimensions, then at least if you're willing to impose this conformally invariant equation, W plus equal to zero, that's the same as studying a certain class of complex 3-manifolds. so there was an enormous amount of work applied to this basically in the decade from 1985 to 1995 and some of the key contributors were Poon, Donaldson, Friedman, Fleur

42:30 myself, Joyce, Michael Singer Campana Taubes Jong-Soo Kim Max Panticorpo, and I'm just naming a few people if you look, there's a vast literature out there the literature in any systematic way. But I just want to give you some typical examples of the kind of results that come up in this context. in terms of humanity, I'll mention one of my own results. There exist anti-cell dual metrics with positive scalar curvature on any connected sum of CP2 bar. So you start off with a complex projected of the opposite orientation, then there's a construction to build new manifolds by joining manifolds with tubes, called a connect sum. You throw out a ball from each manifold blue along the boundary. So if you take an arbitrary connected sum of reverse-oriented complex projective planes in this way, that admits anti-self-lule metrics, where there are metrics of positive scalar curvature in the conformal class. And actually, among the simply connected examples among simply connected 4-manifolds with positive scalar curvature, these are the only topological possibilities. At least I've been in the morpheus. The morpheus is on the side. You don't know. And, moreover, these can be actually constructed in such a way that the twister spaces are algebraic in an abstract sense. The transition functions are algebraic. Interestingly So the twisted spaces, these complex pre-manifolds, are examples of moissure summanifolds, which are not projective and not kahler. So they're actually moissure summanifolds that are kahler or projective, but anyway, so these guys are never, the twisted spaces are not kahler-tized, not projective, if you have at least two summan's here, but nonetheless, you can understand them by algebraic procedure. to certain projected manifolds. All right, so that's a... The main reason for mentioning this is perhaps to emphasize, well, that there's this important... Connected sum construction in four manifolds,

45:00 that CP2 bars are somehow nice. The standard metric on CP2 bar typically needs to be metric, but think about the opposite orientation, but the deepest theorem that was proved in that period deepest theorem in the subject perhaps, is the theorem of tau if you take an arbitrary form-aniple, absolutely arbitrary, compact form-aniple and now take a connected sum with enough CP2 bars eventually, after you add enough CP2 bars, you will be able to find dual metrics. And by the way, this connected sum with CP2 bar doesn't kill things like Seiberg-Witten invariants, so there are a huge number of diphyotypes that you get for a given homeotype in this construction. So the thing that's a little bit peculiar about this theorem is that there's no effective way of estimating the number of CP2 bars that you need, but if you look at Taubes' proof, you expect it to be something like the number There are varions in the universe. I mean, you sort of, you just mercilessly squash this manifold everywhere you find some W+. Cut it out and stick on some CP2 bars. So, one of the puzzles that come out of this is that for a given smooth four manifold you can attach an invariant this number K0, and nobody knows what it is, or any reasonable. I mean, only for very, very simple manifolds where we can do this kind of construction. For example, it's zero for CP2. bar. But usually you have no idea. It's an outstanding problem to try to get a more effective version of that theorem. Even for some simple classes of complex surfaces. To finish, I'd like to quickly say a few things about the split signature case. The split signature case is fundamentally different from the Ramanian case. look at regularity, where anti-cell duality in fact implies that the metric is real analytic in the student charts. That's certainly not true in the split-signature case. And it's only very recently that in drug work with Final Mason, I think we finally understood the right point of view to do the split-signature case when you're just looking at smooth solutions.

47:30 The idea is not to look at moduli spaces of CP1s, but to look at moduli spaces of holomorphic disks with bounding a totally real sub-manacle. So this sort of fits in vaguely with the fact, the idea in physics that you should not just cut in closed strings, but also open strings. So here, the general setting would be, suppose you had a complex 3-manacle, and you have, so this is something in complex dimension 3, and then in it you have something in real dimension 3. Think of Rb3 sitting in Cb3. And then, look at with boundaries on the real submanifold. Now, we can attach to that setting. In this setting, there is a certain rank 2 homomorphic vector bundle over CP1 associated with it. What you do is you double. You take two copies of your disk, the original disk and its complex conjugate, and there's a way of gluing the normal bundles over the circle which reflects the normal bundle of the circle in the real disk. So the, I mean, if you were to just look at the normal bundle of the disk, there's no structure there because the disk is stined. There's no holomorphic information encoded in it. But when you double the disk, you can talk about the normal bundle, extended normal bundle, double normal bundle over CP1. And we're going to look at the case where that double normal bundle is O1 plus O1 again. So in the literature on holomorphic disks, people would instead usually say that there are these numbers is called the partial indices, but actually what they're asking about is what's the rotundi splitting type of the double. All right, so here's a typical theorem in the subject. Suppose that you have a complex three-manifold, you have a real smooth three-dimensional sub-manifold, which is totally real. Totally real means that when you apply J to the tangent space, There are no vectors tan. The intersection of the image under J of the tangent space with the tangent space is 0. Okay, so there are no complex subspaces. Okay, and suppose you look at the set of all homomorphic disks in the complex 3-manifold with boundary on the totally real thing, such that the double normal bundle is 01 plus 01,

50:00 then that moduli space of disks is a real 4-manifold, split-signature anti-cell dual metric, and this is the general solution, locally. So, there's a local version of the theorem, the paper should come out in the next few weeks, I hope, where we use these ideas to study a global problem, what's the modulized space of anti-self-dual split signature metrics on S2 times S2. It turns out that near the standard metric, all of those anti-self-dual metrics have the property that their alpha surfaces are compact, and their geodesics are all periodic. Those properties actually characterize S2 times S2 and a certain Z2 quotient of it. But the interesting thing is that the moduli space of anti-self-dual metrics on this thing is infinite dimensional, and it exactly corresponds to deformations of Rp3 and Cp3. So before you were thinking about deforming the complex manifold, here it turns out the crucial thing is you can actually, in this case, leave the complex manifold fixed, but just move the, move the, move the brain, move the, it's totally real sub-manifold. as you move it the space of homomorphic disks persists and for each such thing you get you get anti-self-dual split signature metrics that's I think I tried to pack too much into this lecture so I'll stop here and maybe I'll begin the next lecture with a few extra minutes on anti-self-dual matter applause applause applause applause So why did you attribute the lions of Chaudry to Christopher Wren? Well, he was a member of the Royal Society, and he actually announced that result in the 1600s. But that's a mass result, so it's not the result of something which you see in his architecture. No, no, it's not in his architecture. He was a mathematician as well. He's best known as an architect, but in fact, people in those days were allowed to gather more than one field.

52:30 Some of us do it now, but we can definitely admit it. Thank you very much. Let's thank Paul. Thank you. I would like to thank the organizers for giving me an opportunity to talk at this conference. I will be introducing the subject of MH universities on which there will be more talks, so this will be more for you to set up the background for the talks to follow up. It will be on more recent development. So, in this talk, I will review some of the progress that people can... So, in this talk, I will leave you some of the progress that we have gained in understanding gauge theory scaling amplitudes by studying their relation to twist-to-smake theory. In particular, I will concentrate on so-called MHP rules for computing scaling amplitudes, which lead us to a new way to compute gauge theory scaling amplitudes using MHP amplitudes. So this takes the motivation from Twister's space for these rules and then sketch some derivation of them from Twister's link theory of written. So firstly, let me review notation which has been written here already several times. If we consider on-shell vector P in four dimensions, we can write it as a bispinner by multiplying it with all the sigma matrices, then the on-shell condition tells us that the matrix PA8 has length 1, which means that we can write it as a product of two so-called spinners, these

55:00 commuting spinners, they are just numbers, and they are transforming in the separate representations of the first and second SE2 when we write the Lorentz group as a product of two SE2s. In most of the talk I will be using these invariant products between the spinners. There are two invariant products, one for the spinners of negative chirality and one for the spinners of positive chirality, they are the square and the bracket ones. As it has been reviewed in many thoughts, this notation is very efficient for writing down scuttering altitudes of Goulons and other particles in gauge series. In 40-pillar, instead of having the usual description of external states in terms of momenta and their polarization vectors, it's more efficient to have their description in terms of these spinners than at helicity's. And a polarization vector can be expressed up to a gauge transformation in terms of the spinners. For example, the negative helicity polarization vector of Guon has this form, and essentially, since lambda is small as, like, has helicity minus half, and lambda bar has helicity plus half, you get the correct helicity minus one for a negative helicity Guon, similarly for a positive helicity Guon. So an amplitude, written in terms of the spheres and hallucinities, can be simplified even more if we also take out a product of the color matrices. That way, we get so-called color-ordered amplitude, which is very simple, as we have learned over the recent months. The most famous one, which has been known for a long time, is the so-called maximal velocity-violating amplitude. It takes this very simple form on one line. It's written in terms of the angle of products, and you see that if the Goulons' i and j have negative velocity, then it is just i-chase the force in the numerator, and the cyclic product of adjacent spinners in the denominator times the

57:30 moment in conservation. This amplitude is going to be very important in the rest of the talk. In particular, one can ask if this amplitude is so simple, much more simple than what one would expect from just Feynman rules. As Wilbrand has shown in his talk, if you write an amplitude you get a very complicated answer. However, the answer I showed you in previous transparency is much simpler. It's just on one line. So one wonders, why is it so that the scaling apple juice are much simpler than the recipes that one has for computing them? So in this talk, we would like to answer part of this question. For to the Twixler space. So, I will leave you for another time some basics of Twixler space. We just heard a very beautiful talk by LeBrun, in which I love many new things. So, I will be very big. Essentially, the Twixler space is the space of light rays in Minkowski space. So, to a given light ray, in Minkowski space, we associate a point in Twister space. One can try to go the other way. If we have a point in Minkowski space, to that we associate all the light rays which goes through that point. And if we are at a given point, we can shoot the light ray with our light torch in any direction and there is a solar sphere sphere, S2 of directions. So to any point in Mikulski space, we associate an S2 or a CP1 of light rays in Twister space. This is the basic of Twister correspondence that you will need.

1:00:00 The usual description of Twister space is in terms of four complex variables which I have noted as lambda a and mu a dot, the index a and a dot goes from 1 to 2, so we have four variables, up to scaling. So this is the complex projective space, CP3. The way to go from the spinners lambda and lambda tilde to lambda mu is essentially by half-fueur transform. So, we are transforming lambda-thilda to mu, so mu is essentially conjugate to lambda-thilda. One of the reasons why people are interested in twister space originally is because the properties of, the KoFOMO properties are much more clear in terms of it. In particular, the KoFOMO group is just the group of rotations of twister space. It acts linearly on the variable lambda and mu. If we have an amplitude in Mikoski space and we want to know the corresponding crystal amplitude, we just performed the half-quare transform from previous transparency. So, let's take some scaling amplitude of some number of duons, depends on the spinors lambda and lambda-tilda of the duons and on their helicities. Here I have singled out the momentum delta function, the delta function of momentum conservation, and the Fourier transform in lambda-tildas is achieved, written here. So, this will give us the twister scattering amplitude, and we can try to go ahead and see what it is. Well, let's do this for the maximum velocity-violating amplitude. That amplitude is very special because, as I showed you on Fourier transferences before, it depended only on the angle products. And this angle skewed products depend just on the negative Holocity spinors, lambdas.

1:02:30 So, when you are doing the Fourier transform with respect to the positive Holocity lambda-tildas, it doesn't depend on the form of the amplitude at all, we can just take it out from the integral and the integral becomes very simple if you rewrite the delta function of orbital conservation as an integral over the force space then all we need to do is perform the integral over lambda-tuda and that gives us two delta functions because we are doing integral over two variables So, what this means is that, in twisted space, the mHVR amplitude vanishes unless each of the particles, i, obeys these two equations, mu a dot plus x to a dot 1 by a equals 0. This is an equation you might remember from previous thoughts. This is the basic incidence relation between Twister space and Minkowski space written in algebraic forms. It tells us that if we have a point x-a-added in Minkowski space, then the corresponding CP1 in Twister space is singled out by these two equations. Twister space is a Cp3, and here we have two linear iterations. This will cut down, Cp3 is down to Cp1, so we need to point X corresponds to Cp1. So, what we have learned is that mHV amplitudes localize to complex splines in Twister space. Now, by Twister's correspondence again, complex splines corresponds to points in mycoskie space. one can think about these MHV amplitudes as if they were local, as if they're... we could consider to think about them as local interaction vertices, even though they are really the sum of a lot of Feynman diagrams, so they aren't really local. But we can take this idea of a phase-soluble and try to push it further and see whether we could use the simplicity of MHV amplitudes to compute more complicated amplitudes using the MHV amplitudes as interaction vertices. So indeed, this resulted into so-called

1:05:00 MHV diagrams, where one takes here I have drawn one example of an MHV diagram one takes MHV amplitudes and twist them as vertices, and connect them with simple scale propagators. To do this in practice, we need to define a few more things, so that we know exactly what it is to compute the animation diagrams. I'm going to do them in the next few transparencies. So an MHV amplitude is really just a scaling amplitude of on-shell duals, so you could ask how can we use it as an interaction vertex because for an interaction vertex we need to sometimes some of the particles are off-shell so we need to define MHV amplitude for off-shell incoming particles To do that, all we need to define is just the holographic spin of lambda A for an onshore momentum, because remember that cutting altitude depends just on lambdas, it doesn't depend on the lambda pillars. So let's start as a practice with an onshore momentum, PAA dot. For a nutshell of momentum, we can express the spin of lambda this way. We take any positive holicity spinner, eta adot, and we contract it with PA adot. Remembering that PA adot is just lambda times lambda, this will absorb this extra positive and we are left with lambda up to a constant vector by which we can divide. So this would be a right of how we could define lambda from a momentum up to a scale for an optional momentum. We can try to do the same thing for an optional momentum. Well it doesn't really work because here in the denominator we have lambda tilde and we do not know what's lambda tilde for an optional momentum. But there's not such a big problem, because it turns out that if you put this definition into amateur diagrams, that this denominator always scales out. So we can just throw it away and define lambdas for an optional momentum using this formula,

1:07:30 that lambda is p contracted with an auxiliary positive house, this pier. So this appears half of the photo. We know how to define the MHV vertices for offshore lines. Now we need to connect them together to make Feynman diagrams. And then, this has to be very simple. All you need to do is use the Feynman propagators for scalar particles, one of p squared for a fighter going with momentum p. Whether it's long, scalar, or fermion, it's always the same propagator. So these are all the rules. You just take the offshore continued MHV diagrams, connect them with scalar propagators, and form all Feynman diagrams that can contribute to a given amplitude. What you need to remember when you construct And MHV amplitude has always two negative helicity guons and any number of positive helicity guons. So you write helicity of the external particles and then you assign in any way the helicity of the infinite line. So, for example, if the velocity of the line here is flat, it has to be minus at the other end because we are considering all particles to be incoming, then the direction, incoming direction here is opposite from the incoming direction here. So we are looking at the gluon at this vertex from, let's say, from the front and at the other vertex from the back. And looking at it from different directions flips the holicity. That's why you assign different holicities to different ends of the propagator. Now we need to make sure that each of the vertices has two negative holicities and any number of positive holicities. Here I have written an example of the most simple amplitude one can compute using MHV vertices, which is the plus, minus, minus, minus amplitude.

1:10:00 Well, this amplitude actually vanishes by supersymmetric border identities, but let's check that it really does as a check of the formalism. Well, we have two vertices, one from the, one is here for coolants 1, 2 and lambda, this is just this, because coolants 2 and lambda have negative holocity, so this is the vertex, where lambda is the negative holocity spinner corresponding to the optional momentum flowing in this offshore line, so corresponding to the momentum P1 plus P2, which we contracted with an auxiliary spinor eta. Then we have a propagator, 1 over P2, and the second MHV vertex for the lower vertex here. So this is the expression which we assigned to the first diagram, and there is very similar expression assigned to the second diagram, you can first study yourself on the back of the envelope that the sum of this expression after your momentum conservation is zero. So, we get the correct answer in this simple case, and one can easily compute more complicated examples. For example, you can consider the any number of gluons, where three gluons, three adjacent gluons have negative holicity and the remaining gluons have positive holicity. In this example, there are two and minus three MHV diagrams contributing. This number follows essentially from the restriction that each MHV vertex has to have two negative holicities, and then you just shuffle around the positive holicities from one vertex to the other. Then the expression for the amplitude is simply the MHV amplitude for one vertex times one over P squared times the MHV amplitude for the other vertex, the same thing for this other class of diagrams. Now, originally this was the only amplitude which has been known analytically, it has been found by David Kosova in the early 90s and other amplitudes simply have not been in

1:12:30 literature because for all-end formulas because people didn't know how to put them into simple analytic formulas. With this MHV diagram you can write simple formulas for any amplitudes you would like and even more with the rules that And Bridgeway is going to talk about them, or you can write probably even more simple rules than we have expected before. So, these images items are very nice. They give us such a simple way to compute aptitudes. We would like to see some variation of them. Well, so far, they have not been derived from Gage-3 yet, but we should wait to hear more from Jose Stolt, that's after mine. He might give us some interesting insight into how MHP diagrams arise from Gage-3. So, if we do not have a direct derivation from the QC the Lagrangian, we can result to some consistency checks of the MHP rules. because such checks are well known and they are used by the QCD practitioners to test their new results for amplitudes and loops and so on. One of them is that the amplitudes must have correct multiparticle singularities when some intermediate particle goes on shell. So another consistency check is the correct divergences when two particles become collinear or one particle becomes soft. So let me just briefly discuss the case of multi-particle singularities. Now, the behavior of two-level amplitudes on the multibundical singularities is universal. Essentially, when the momentum in a given channel goes on shell, the amplitude factorizes as the product of three amplitudes on two sides and the propagator. one was p squared. And a short thought makes you realize that this is the behavior that

1:15:00 we see for MH3 vertices. Essentially, C, what you need to do is you collect all MH3 diagrams on this side of propagating the given channel and all MH3 diagrams on the other side of the given channel. Now, when the momentum p goes on-shell, the off-shell spin lambda, which we define using the continuation that lambda equals p eta, goes on-shell, which means that the answer to these two blocks, on left and right, slowly go into the three-level on-shell antecute. So, indeed, the MHV vertices in this case reduce to a three-level antecute on one side and the propagated and still on the other side, which is the correct multiparticle singularity in this channel. Similarly, the soft and collinear singularities can be understood from the soft and collinear singularities of MHV diodons, because we know that these divergences are universal and since we know that MHE amplitudes have a correct self-controlling singularities the same is going to be true about the amplitudes constructed from the MHE vertices but these are a few checks that this MHE vertices construction has passed if we believe that the MHE diagrams compute the scarring amplitudes We can now move on to the most theoretical part of the talk, where I would like to switch gears and move on to the switcher space. The original motivation I gave you for the MHP diagram was from switcher space. Remember that MHP amplitude has a very special property, depending just on lambda, which is reflected in the crystal space by the geometric condition that it localizes to a line in crystal space Now Wittell has given this more physical interpretation by trying to interpret the gauge theory in particular, the maximum supersymmetric N equals 4 gauge theory as a topological string theory in twist space.

1:17:30 Then the MHC vertices will have a complete physical meaning in the string theory. So, let us give you a little bit from, we just told that the splixery in question is the so-called B-model topological splixery. Firstly, to define it, the first condition, one of the basic ones, is that the B-model is defined only on Calabria manifolds. And we know very well that the tristor space, Cp3 of the Calabria manifold. This can be remedied by taking a supersymmetric version of tristor space. We introduced four more fermionic coordinates, PsiA, which have a scaling one. So, now, the homogeneous coordinates of a coin in the superstitious space are V8 and psi8, up to scaling lambda. This way, we do indeed get Calabria manifold, and it does have a form of X3 form. I have written it here. So, you see, the first piece is the Boslenic part. That by itself is not a 3-form, because it has a non-zero scaling, it has scaling 4, but when you rescale Z by lambda, then the Boslenic part of the 3-form scales like lambda to the 4th. So, but that is fortunately cancelled by the scaling of the fermionic part of this three-form, because remember that the deep sides have scaling opposite to the scaling of the sides, which means that they scale as lambda to minus force, so altogether, omega is invariant on the scaling lambda, and it descends to a a holomorphic stream form on the super twisted space. In this case, the group of motions of the super twisted space

1:20:00 is just the n equals 4 superconformal group which suggests us that the gauge theory, which is due to the string theory living on this space, should have maximal supersymmetry. The basic field of the string theory, if we look at the string field theory, there is a twister field which is a 0,14 form, laying on the super-drisse space. So, this is some sort of holomorphic gauge field, and we can study the spectrum of the topological string theory simply by expanding this twister field in its dependence on fermionic coordinates. You might bring that the square of any fermionic Gaspanian variable is 0, so we get all together these components, I've written a few of them. So the first one, A, is a 0,1 form of scaling here, just like the scaling of the whole supersymmetric twisty field. Now by Penrose linear twisty transform, a 0,1 form of this scaling corresponds to a fifth field of Holistic plus one in Winkowski space. That's a quite similar field. We could think about it as a Holistic-Holistic one, So let's go on and look at other fields. So if you, for example, look at the field at the end, here, G, then it's scaling is minus 4, because it has to be conjugate to the scaling of the sides, so that the total scaling of this term is the same as of the Twister field, which is 0. and by a linear transform, this 0,1 crystal field encodes a pre-field in encodes this space of Holicity minus 1, which we interpret as the negative Holicity gluon. Similarly, for the

1:22:30 fields in between we get the right halicidins and other properties so for example for this field it has the weight minus one so it's a halicid plus one half it's a positive halicid fermion and you can see that it's transforming in the conjugate representation to the well under the rotation of the drissus space Psi transform S4 of Sv4 which means that this tilt Psi is transforming S4 bar which is the right Sv4 transformation for permeance of positive halicity in n equals 4 gauge theory this way we get the entire spectrum of the n equals 4 gauge theory which is very amazing that some string theory living in six dimension it's the same spectrum as gauge theory living in four dimensions so we have identified the correct to the spectrum so we can move on to the interactions well the interactions in the topological string theory can be read of from the effective action for the polomorphic gauge field A. This is so-called the polomorphic Chern-Simons theory, which was introduced in the early 90s by Witten, his study of Chern-Simons in relation to topology constraints. So, it has two terms. The first term, A B bar A, is the kinetic term, so you see that the propagator is and the second term is a cubic vertex A-W-A-W-A now you can see clearly why we needed the condition that the crystal space be a color BR manifold or in other words that it has a well-defined three-form so we needed that condition so that we can write down the this action we can write it down with an omega i3 This three vertex is clearly not enough to give us the interaction of the critical gauge series. Indeed, one can, if you, for example, expand it in components, you'll find that

1:25:00 if you're abstract to the cool-on fields, you will get the GAA or minus plus plus vertex. but remember that in gauge theory you also have the minus minus plus vertex and minus minus plus plus vertex so we didn't get the entire interaction of N equals 4 gauge theory just from the homophic Hamon theory turns out that we get just the interaction of the soft zero gauge theory. Well then the question is how to incorporate the rest of the interactions, and it turns out that one can do it using so-called D1-grain instantons, these are Euclidean D1-grains, and on these D1-grains, well, the basic property of D-grains is that open strings can end on them, and the homophobic gauge field is essentially an obvious thing, so this defense introduces additional interactions into the theory which give us, that's a conjecture that they give us the interaction of the school-language theory, and it has been checked successfully at true level, so what I'm going to this kind now so the direction of the D1 strings in the interaction of the D1 brain to the open strings is described by this Lutrangean that should be alpha and beta are some pre fermions living on the D brains they couple to the gauge field here in the Minimal A. So you can think of the gauge field as 5-5-strings, where 5-5-d-5-bands are the space-filling bands in the picture space, and Alpha and Beta are 1-5 and 5-1-strings. From this you get the vertex operator for coupling of the gauge field to the d-instant-dom, just the second term and similarly if you want to compute some scaling amplitudes you can also

1:27:30 consider disconnecting in sentence which you find by an open string and the propagator for the open string is just the inverse of the kinetic operator del bar so it's a rather simple propagator now In this case, we have computed all three-level amplitudes from the so-called disconnected instantons, and that turns out to lead to mHV diagrams. Wolovich, Spradlin, and Rohiband, they considered Etern instantons, where you have just one instanton contributing to the amplitude, and they also provide strong evidence that the twister string theory leads to the correct three-level amplitudes. There is a relation between the degree of the instanton, which is essentially the homology class of Ophid, compared to the homology class of complex 5, and to the number of negative holocytic gluons, namely the degree class 1 equals to the number of negative holocytic gluons. So here I draw an example of some steering process in a correlation function in fixed space. So here we have a DG2 instanton and a DG1 instanton. And if you evaluate this correlation function, this should correspond to a steering amplitude in mycoski space of one, two, three, four, seven gluons, four of which have negative helicity. The prescription for an amplitude is you take the vertex operators for external gluons and evaluate the correlation function on the V instanton And then you integrate it over the modulized space of the d-instantons, which in this case are holomorphic curves. So it's the modulized space of some holomorphic curves of a given degree in genus. Here we concentrate on genus zero case, which are the trillable amplitudes. There, the current collator is just a simple product coming from pre-fermium collators on rhenosphere. For wave functions of the external goals, we take just simple plane waves, because those are the most natural when we study scattering up with you.

1:30:00 We need, however, the crystal version of a plane wave, and this is an actual candidate for it, because it has a definite momentum. Well, it is a definite value of lambda because of the delta function, and by Fourier transform this exponential will go into a delta function in lambda tilde, or pi tilde, which will give us a definite value of pi tilde, so the momentum, p, the particle has indeed a definite value of p. so this is a modified twisted wave function so let us consider the simplest amplitude it's the degree one amplitude it comes from instantons which are just complex slimes in twisted space and it should lead to MHE amplitudes because remember that we found that MHE amplitudes are supported by complex slimes well NHV amplitudes schematically consists of three pieces there is the theta function of momentum conservation it turns out that this comes from the integral over the volcanic moduli of the B instanton there are four volcanic moduli so which lead to an integral over B for X and then there is the numerator, which, on the other hand, comes from the integral over the fermionic moduli, from cancelling over two fermionic zero modes, and the denominator, which comes from the correlator of the currents on the mode-volume of the de-instanton. So, this would be the expression for the MhA amplitude. Similarly, one can go to a higher degree, for example, a degree 2 amplitude. If you consider disconnected ones, it comes from 2 degree running samples, which are connected by open space. And, again, performing the integral of the fermionic moduli will give us the numerator

1:32:30 and the time combinator will give us the denominator. So, doing these two things with the two MHV amplitudes on each d-instanton, Then we have an open string propagator, and these exponential factors coming from the wave functions of the external particles, and we integrate over the modulized space of the 2D instantons, the first one and the second one, rewriting the exponential as here sum of our own momenta, and y being the difference of the coordinates on the modulized space of the 2D instantons. momentum flooring from one to the other we can immediately perform one of the integrals of the modulized space to get the data function of momentum conservation and we are left with this kind of integral here is the product of two MHV amplitudes the propagator and integral over the relative position of the two lines I have an artistic way how to evaluate this integral, it's essentially by taking the residue at y squared equals zero and then, remember that y squared equals zero is essentially means that y is partial so we can write y in this form, and this is the residue Then you get the integral, and by doing a few more operations for which I don't have time, you can express the integral as a sum of the residues. And it turns out that the residues, to express it that way, you need to introduce an oxymoron spin and eta, and integrate by parts.