Ambitwistors

0:00 So what does one mean by a complex light ray or a complex analogy? The key case to understand, before going on to anything more complicated, is that of complex Euclidean space or complex Euclidean space in dimension 4. So we just take C4 with its standard constant coefficient complex metric. And when I say a complex library, I mean an affine complex line sitting in C4. And, of course, an affine line can be always parameterized by, you have some choice of origin, x0, and then some direction, d, which you're going to move in. And so you take x plus t times v. However, there's a constraint. We're going to require that the tangent vector of the straight line be null. Well, if you think of C4 as the tensor product of two-dimensional spin spaces, or more prosaically, if you think of C4 as the space of two-by-two matrices, then the quadratic form on it that defines the notion of the square norm of a vector would be the determinant, and the things of determinant zero are exactly the things that are products of row vectors and column So to say that a vector v is null satisfies g of v, v is equal to 0, is the same as saying that it can be written as a product of two spinners. It's a simple tensor product, whereas the general element would have to write it as a sum of such objects. And so a null line could be parameterized by x0 plus t times lambda mu. But, of course, when I talk about a line, I don't mean the parameterized line. This is a way of realizing it, but, in fact, we're interested in the image in C4. So, is there a better way of understanding such lines without referring to the parameterization,

2:30 certainly shift of the initial point x0 or rescaling of the parameter t? It doesn't change the line that we're talking about. Well, there is an amazingly elegant way of parameterizing those lines in C4 in terms of twisters. So, remember that in C4, we can talk about the alpha planes and the beta planes. And the interesting thing about a null line is that in fact, every null line sits in exactly one alpha plane and in exactly one beta plane. So if you take the one I've written down over here, it passes through the point x0, and its tangent vector is proportional to lambda a mu a prime, then you can find a somewhat bigger alpha plane. An alpha plane would be something where the tangent vectors are all of the So if you fix mu, but then let zeta vary through the entire spin space, that set of null vectors is the tangent space of an alpha plane, or an alpha plane in a tangent space, and now if we add those on to our initial vector, we get an alpha plane. On the other hand, you could instead fix lambda and let, so fix the unprimed spender and let the prime spender vary, and that's the tangent space of a theta plane, so if you add that on to your origin x0, So, then we've got, we've created a beta plane to that point, and obviously, the line where the tangent vector is lambda times mu is contained in both of these two-dimensional planes. Now, this is a, this picture is a little bit hard to get your mind around because you remember that both of these two planes are totally null, and these totally null planes intersect in a null geodesic. It's a distinctly four-dimensional

5:00 situation. You are not going to see it in three dimensions. algebraic health, as well as some, you mentioned, imagination. All right, so it seems that one way of trying to understand a null beauty is it is to look at an alpha plane and a beta plane, and then take their intersection. However, there's an extra complication, which is that the generic alpha plane and the generic beta plane don't meet. So if you were to look at the alpha plane I've written down there and try to write it so there it's written in a parametric form but if you were to write down if you wanted to write down a system of linear and homogeneous equations that represent it it would be this the coefficient is pi a prime acting on x a a prime and the answer is some constant omega a where omega a would explicitly be what you get if you put in x zero x0 dot pi, and I just decided to put an i there, you don't have to. And the beta plane would be, similarly, that you take x dot lambda, and then that's sub-spinner mu A prime. So in non-parametric form, these are the relevant alpha planes and beta planes. And then you might notice that, well, so you've now represented the alpha plane in terms of 2 spin is pi and omega, the beta plane in terms of 2 spin is lambda and mu, but on the other hand, these are not arbitrary because they have to satisfy the condition, well, I could stick, I could contract this with lambda and this with pi, and I'd get essentially the same thing, and that comes down to saying that omega dot pi plus pi dot mu is zero. And so if you think of the alpha plane as being represented by a point at C4, where the components in this twister space are actually the spinner omega and the spinner pi, and if W is in the dual twister space, which is the direct sum of these dual spaces where lambda and mu live, then what we found is that the duality pairing between the twister and the dual twister would

7:30 have to be zero if the alpha plane meets the beta plane. Alright, so in fact this is the relevant linear condition. If you have two arbitrary, if you have an arbitrary twister and an arbitrary dual twister, the alpha plane and the beta plane they represent will split each other. The condition for them to intersect is exactly that the incidence relation z dot w is equal to zero, which is, notice that bilinear, linear in both z and in w, that that should match. And now it's a little exercise, but conversely, so I showed that if they intersect, you have to have this condition, and it's not hard to see if the Congress has also ensured that if that condition is satisfied, they will intersect. All right, now, actually, when I represent an alpha plane and a beta plane this way, there is a little bit of sleight of hand, because for this to actually represent an alpha plane and a beta plane that you would see in C4, you'd have to have that the spinner pi is non-zero and the spin of lambda is non-zero. If you drop that condition, then you're actually talking about alpha planes and beta planes in the packification in the four quadric. And the best big picture of the situation I can draw is the following. Remember that in the four quadric, there's a system of alpha planes. I've drawn an example in blue. And there are beta planes. I've drawn one in green. is supposed to remind you of the fact that these are somehow generalizations of the two ruling lines on a two-quadric. But then, uh, sometimes, a null plane and a beta plane intersect, and when they do intersect, you get a null line, or an ambi-twister. Alright, All right, so if you represent, so here I'll just, for brevity, sometimes write CP3 for the twister space, and that's really the projector space associated with a certain four-dimensional vector space, which is sometimes called T. And if I write CP3 star, I'm talking about a different copy of CD3, namely the one that's associated with the dual vector space.

10:00 And then, in these terms, space of no lines in the four-quadric, where if you like, you can think of it as the Grossmanian of two planes in T, is the hypersurface in CP3 times CP3 star given by this equation, Z dot W is equal to zero. So this is a hypersurface in a product of two copies of CP3 it's a bi-degree 1-1, meaning that if you restrict it, if you hit that hypersurface with a CP3 of either family, you get something of degree So each If I take CP3 times a point It will intersect A and a CP2 Or if I were to take CP3 star times a point Intersected with A and you get a CP2 So in two different ways You can think of this as actually a CP2 bundle over CP3 We'll see that more explicitly in a moment If you look at this equation It might remind you of something is the complexification of the space of null twisters in ordinary twister space. So if you were to take CP3 and look at the equation z dot z bar is equal to zero, that represents a real five manifold in CP3 called the space of null twisters. And you may recall that the space of null twisters represents the space of light rays in Minkowski space. So it's not surprising that here, since we're talking about complex rather than real Mankowski space, that the space of light ray should be a classification of it. What I've really done here, if you look at this equation, is you simply view z-bar as a separate variable called w. And it's now a separate complex variable. So in some sense, this is, ambitwisted space can be thought of as the complexification of the space of null twisters, although that's not necessarily the most useful way of thinking about it.

12:30 Now, maybe it's a little bit hard to wrap your mind around this picture. Here, if we want to describe this ambi-twister space in a more simple geometrical way, you could remember it in the following way. So, we're looking at pairs of an alpha plane and a beta plane, which intersect. Now, an alpha plane, suppose we think of the basic twister space and then there's this derived object, the dual twisted space. Well, the classical subject in projective geometry, there's a classical way of understanding the projected space associated with the dual space. Remember that a linear functional on a vector space has as its zero lobe as a hyperplane, if it's provided as a non-zero element of the dual space. And if you now projectivize that, that says that the projectivization of the dual vector space is the space of linear hyperplanes. So in this case, CB2s embedded in CP3 in the obvious linear way. So a point in CP3 times CP3 star is actually a pair consisting of a point in complex projected free space and a plane in complex projected free space. And this incidence relation, z dot w is equal to zero, is saying that the point is on the plane. So we're really looking at the set of pairs of a point in CP3 and a plane in CP3 that passes through it. Now, that seems like it's a complicated object, but remember that, after all, if you're trying to understand a particular CP2 and you have a point on it, you don't have to keep track of the entire CP2 if you know it's tangent space at a point, you know the whole plane, right? A plane is determined by its tangent space at a given point. So, you're really thinking of this as points of CP3 together with planes in their tangent spaces. So that's a certain so in fact, what is that? Well, a plane in the tangent space is given by a point in the cotangent space up to scale the same dictionary between hyperplanes and points in the dual space up to scale. And so, in fact this ambitwister space is nothing but take CB3, take its tangent bundle, that would be a a C3 bundle over it, and now projectivize each fiber to get a CP2 bundle.

15:00 Remember, though, our definition of ambitwister is really ambidextrous. Roger's original idea of a twister was somehow that it has a handiness. There's a difference between the alpha planes and the beta planes. The notion of a null line does not include this. So the peculiar thing is that if you take the projectivized cotangent space, of a twister space. It's the same as the projectivized co-handed bundle of the dual twister space. This is a right-handed picture of something which doesn't actually have a handedness. It's ambidextrous. It's equally happy with a left-handed or a right-handed description. I'm principally going to be interested in this lecture in describing to you how to generalize flat space picture to curve space-time. However, for the purposes of this grid meeting, there are various versions of this flat model which are actually of primary interest. For that reason, I want to spend a little bit more time in describing some ways of beefing up this particular thing. So far, ammitwister space is a certain compact complex five-manipold in fact an active ray of Hermione. realize it's hypersurfacing CP3 times CP3, or CP3 times CP3 star, if you like. rather than just considering what, so, algebraic geometries would say that non-singular, that that is a non-singular complex variety, a smooth complex variety. But you could also consider sort of multiples of it. I mean, to an algebraic a variety there are varieties a variety is basically represented by equations, in this case one equation but you could square the equation or cube the equation, let's say you can count this hypersurface with some multiplicity now that's if you are not thinking as an algebraic geometer but you want to do complex analysis and you sort of want to think in a little more of a classical way, what's really going on You have some things I'll call thickenings of the amputwisted space.

17:30 Now, what is a complex manifold? What is it that makes a complex manifold a complex manifold? What's the difference between a complex manifold and a smooth manifold, or a real manifold? Well, it's that you have holomorphic functions. So, among all the complex-valued functions, there's a special class of things called holomorphic functions. functions have possibly different domains of definition. There aren't any global non-plastic whole market functions on a compact complex manifold. So in order to discuss the problem, you have to talk about the assignment to every open set of the whole market functions on it. And that is given the intimidating name of a sheet, but it's not anything scary. It's just saying functions can have different domains of dependence, different domains of definition. So, to have a complex manifold is really to have a real manifold equipped with a suitable sheet of functions, which satisfies some local structure to be homomorphic functions and appropriate coordinate charts. Now, the set of homomorphic functions on an open set forms a ring, meaning you can add and multiply, and you can more generally consider other so-called ring spaces where you have associated with an open set, some class of functions where you can add and multiply. A very interesting ring that you can consider is rather than just taking functions on this hypersurface in Cp3 times Cp3 star, you can consider functions together with their first normal derivative, or their first two normal derivatives, or their first n normal derivatives. So just truncate. So the rules of adding and multiplying are completely obvious. You just say, well, there's a local defining function half, and pretend that f to the n plus 1 is 0. So what you've actually done is taken the ring of homomorphic functions and modded out by the ideal, which consists of functions that could be written as multiples of the defining function. One way of describing a defining function is z to the 0 in a particular case. The only problem is it has a section of a line bundle, but that is sort of a natural thing to do anyway. and so you can instead of looking the ordinary polymorphic functions on A could be thought of as polymorphic functions on CP3 times CP3 star modular and equivalence relation

20:00 that anytime you see the equation of the variety Z dot W, you count it as 0 but instead of counting Z dot W as 0 what happens if you instead count Z dot W to the 4th power as 0 or Z dot W to the 100th power as 0 the n plus first power instead of that zero. Then you would get a ring of functions on a particular open set, which I'll call O n. This is just where you're only taking the first n normal derivatives, and you count the derivatives of order n plus 1 and higher. You're just thrown away. So the nth order thickening of n-b twisted space is the ring space where you have the same topological spaces before, Its structure sheet, the basic collection of things you call functions on it, are these things that have milpotents. Now, physicists certainly should not be afraid of things with milpotents because they think all the time about supermanifolds these days. What is a supermanifold, from a mathematical point of view, a supermanifold is another species of ring space. The only difference is it's not a commutative ring, it's a greater commutative ring. And there is another ring space associated with this picture that's very, very closely related to this nth-order thickening, which is the supersymmetric version of amputwister space with, what do you call the number of supersymmetries, an, where n is the number of supersymmetries you're adding. And so, in a formal sense, if you don't think about what the equations mean, it's very simple to say what the nth supersymmetric extension of amni-twisture space is. You take, instead of taking CP3 times CP3 star, if you take slash n, CP3 star slash n, that you've taken your original vector space, you've replaced it with a sum of two vector spaces, one of which is called even, one which is called odd, and you have a dual thing, and you take the projective spaces associated with both of those, z is dual to w, psi is dual to phi, and you look at the equation z dot w plus psi dot phi is equal to zero.

22:30 but if you're not working sort of algebraically, but you're trying to get a picture in the head, what does represents? What is the mathematical description of this kind of object? Well, to mathematicians, a complex supermanifold is, again, a ring space. The underlying topological space is just a classical complex manifold. A complex manifold, but you adjoin to it a bigger class of things you call holomorphic functions. Logically, those are roughly sections of a Grossman bundle on some, but you actually want, it's important in the holomorphic context that you only require that locally it has that structure, because it turns out that transition functions can be more complicated than just the things that you would get by just looking at sections of the Grossman bundle. In fact, CP3N, from this point of view, is simply CP3, equipped with things called, the thing you mean by a function is, well, you take a certain vector bundle on CP3, in this case take the direct sum of n copies of O minus A different way of saying that is take CM tensor O minus 1, and now take the exterior algebra on that. So this is added on not generators, sorry, n-fermionic generators at each point, and the classical functions are just the things of degree zero, and then when you have, those would be the things that don't have n-fermionic component, and then you have some extra pieces. Well, so CP3 star is the same thing, except it's just, you've taken here CN, and here it's a dual vector space of dimension n, and this underlying space is the dual CP3. These are both examples of split, so-called split supermaticals, meaning that the super functions are simply sections of a Grossman algebra, even globally. I mean, you have a vector bundle and you've taken the exterior algebra on it. The superantiquistic space is not split. So it locally looks the same, but in fact, you cannot get away from the transition functions would be more complicated if they somehow mix up even an odd a bit. Well, they keep even an odd the same but when you pass from chart to chart they're not just written in terms of

25:00 things that have no fermionic component. So the way to describe this is that you take the product of CP3 and CP3 star well, in that case what you're really doing is you're just taking the tensor product of these bundles sorry, so you take the direct sum of those two bundles take the exterior algebra on the sum and then finally you're modding out by an ideal generated by this equation you set z dot w is equal to zero and in fact you could do this whole something somehow starting with, you actually have a choice. You can either do all of this on CP3 times CP3 star to begin with and then run out by that ideal, or you could, it turns out that because everything in here has order at most n, this is actually very closely related into this infrastructural neighborhood, so you could impact Poncae first. Okay, so that's an example. So this actually has an interpretation of the space of complex super light rays in complexified super metastatic space of degree N, but I actually won't go through that explicitly. All right, so I instead want to turn now, after that little aside on things that aren't just complex nanopoles, I want to return to the very simple setting of, relatively simple setting, of classical complex nanopoles. And I wanted to say, suppose that we start off with a four-dimensional complex nanopole together with a homomorphic metric on it. Remember, this is a homomorphic tensor field, which is symmetric and non-degenerate. So that's the typical way that such a thing arises, if you didn't see my first lecture, is that such things arise by analytic continuation of neural analytics here in Riemannian Manifold. By the way, as a notational point, you might be annoyed or amused by the fact that the complex manifold of dimension four I write as a subscript, a real manifold of dimension four I write

27:30 in the superscript. This is a notation that was first introduced by Kirchhoff. Not so many people use it, but it's in fact quite useful to me. So if I suddenly write a space with a superscript, this is its real dimension. I write a space with a subscript, a complex manifold of that complex dimension. That's why I would be writing ZP3 with a lower grid. All right, so the story for these things is analogous to the real case. There's actually unique connection on the tangent bundle that's induced by a full and more quick or complex metric. And in fact, you can think of it in the same construction. In fact, it even is, this is a unique, you take the real part of that, that's a superconvenient metric, and this is its connection. But if you were to think about it that way, it would miss a key point, namely that the derivative here, I mean, the chrysal symbols that arise are actually holomorphic. The other way of saying that is the covariate derivative of a holomorphic vector field is a holomorphic tensor field. And then you can define geodesics just as you would in the real case. A geodesic is a parameterized curve where the tangent vector has covariant derivative zero with respect to itself. But when I say geodesic, I don't want to think of the parameterized curve. I think of the image of it, or the corresponding unparameterized curve. And there's a very good reason for it. I lucked out the key thing on this slide. So this would be a geodesic, and a null geodesic is where the tangent vector is not. So there's an extra condition. this condition, and it's just a geodesic, and then a null geodesic has a null tangent vector. And so an ambitwister is simply an unparameterized null geodesic. And there's

30:00 a remarkable fact about null geodesics. It's already true in the scenery-binding context, but the proof goes over into the complex without any essential changes, namely the null geodesics of a metric, part of an unparameterized curve, are conformally invariant. If you rescale a metric it changes its geodesics, but it does not change the null geodesics. And this is an important enough point that I'd like to take a couple of minutes to describe why, it really comes from a certain new construction in symplectic geometry. If you have a symplectic manifold, this could either be done for the classical kinds of applications it's going to be a real symplectic manifold but you could also do this, you can talk about homework at symplectic manifolds and as long as everything inside is homework, all of the theorems work basically in the same way. magnetic manifold, and you take a hypersurface, so a smooth sub-manifold co-dimension one, then you can associate to that hypersurface a system of curves upon it. Namely, if you take, so a, a, a, a, a, a, a, a, a, a subjective form is a skew object. effect. If you take such a thing and you restrict it to something odd-dimensional, it has to be degenerate in one direction. Supplective manifolds are always of even dimension. So if you take a hypersurface, this would be something of odd dimension, the restriction of the supplective form has to have some degenerate direction, and in fact, it has exactly one in the general direction 1. So the kernel of omega, thought of as a map from the tangent bundle to the cotangent bundle of the hypersurface, has a one-dimensional kernel. And now what you can do, at each point, that would give you a one-dimensional subspace of the tangent bundle, and now you can take the integral curves of that distribution of lines. Now, in fact, there's a different way of describing this. Take your hypersurface, it's locally given by one function.

32:30 there's a standing Hamilton description of, given a, you think of that function as a Hamiltonian, it has a flow, a vector field. And this conserved quantity, the flow of the, the Hamiltonian vector field associated with the Hamiltonian preserves the level sets of the Hamiltonian. So the Hamiltonian is conserved by the flow. So the amusing thing here is that if you just have the hypersurface, rather than the function that defines it you still have the curves which are the Hamiltonian trajectories on the hyper-surface but they're only unparameterized curves if you were to rescale the Hamiltonian it would make the flow go in different rates and in fact you know these would be parameterized basically in arbitrary ways if you change the Hamiltonian that has y as a level set Okay, that little piece of classical mechanics, the description, is the reason why null geodesics are conformally invariant, because the geodesic flow, the most efficient way of finding geodesics on a manifold is not to construct the Levy-Chiva connection, even though that's what they tell you to do in every course on differential geometry. one way, you just think of the metric as defining a function on the cotangent bundle namely the norm squared and you look at the Hamiltonian flow and the trajectories you get project to geodesics so just a list of the geodesic consisting of a geodesic that corresponds to its tangent vector under index lower so in this particular what you actually do you take the cotangent bundle of your of your manifold where the metric lives, you give it the standard symplectic form, namely that when there's a prodigal in one form, you take the exterior derivative to get the classical symplectic structure. And now, what you can do is take the hypersurface Y we've been discussing to be the set of null covectors. The trajectories that they sweep out are the lists of the null genesis. Okay, so null genesis are conformally invariant. moreover, if you look at the space, there's something called the reduced

35:00 phase space, or the Mars and Weinstein symplectic quotient, that means, take the set of trajectories on the hypersurface, that's itself a symplectic manifold. So you would say, well, the space of null geodesics would be a symplectic manifold. That's not quite true, because these are really null geodesics together with a scale. There's a one parameter family of tangent vectors you can have associated at the point on your null pdV. So when you line out by the rescaling, what you get is not a subletive manifold, but rather a contact manifold. What do I mean by a contact manifold? If you don't know, it's important for our purposes, so I have to remind you. So a contact manifold, we could do this in the real case, I'll be interested in the fact that all of these have perfect holomorphic analogue. You just say everything inside the holomorphic function. Take an odd-dimensional manifold. A contact structure is a distribution of hyperplanes. So it's a codimension one sub-bundle of the tangent bundle, which is maximally non-integrable, meaning it's as hard as possible to find hypersurfaces tangent to it. So, the integrability condition for a subbundal and tangent bundle would ask that it be closed under Lie brackets, and instead we want to assume the opposite, namely if you look at Lie brackets, take the normal component of Lie brackets, this is a non-degenerate form on the distribution of hyperplanes. That's a nice abstract picture, but in calculational terms it's much easier to say the following. A distribution of hyperplanes, what is that? I mean, it's the kernel of a one-form. And what condition am I placing on that one-form? I'm saying it's non-degenerate in the sense that theta wedge d theta, wedge d theta, where we have k copies here, this is a volume form on our manifold. So this is sort of a generic one-form, but it's only up to scale. If you take that one-form and you replace it with a function times the one-form, it defines the same contact structure. That's why I've emphasized at the beginning contact structures is a codimension one sub-bundle of a tangent one. The Garbus theorem tells you that all such things locally look the same, and that's related to the corresponding theorem that all symplectic manifolds look the same.

37:30 It turns out that there's a dictionary between a symplectic manifold, symplectic manifolds together with homophetes, with automorphism, if you have a C-star action on a complex symplectic manifold, which rescales the symplectic form of quotient, will then be a contact manifold, and conversely, every contact manifold arises that way. So when you go through these ideas, basically from classical mechanics, what you would get in particular would be that, suppose that you have, and you get them up in the following theorem, Suppose you take a complex 4-manifold equipped with a complex Riemannian mechanism, so a complex space-time, and put in some, to make this work globally, you need some convexity hypothesis. So, for example, assume that it's GDs in fact. The corresponding amputwisted space, first of all, is a complex 5-manifold, but moreover, that complex 5-manifold comes equipped with a contact structure, and the whole thing is actually conformally invariant. Now, actually, for this little theorem, the dimension is not crucial. If you had started with a complex M-manifold with a holomorphic complex metric on it, you would get a space of deodesics of dimension 2n minus 3, and that would, again, be a complex contact manifold, assuming that, again, it's at least 2, but at least 2n minus 3. All right, so in this context, we get a picture which is analogous to the Twister correspondence, the happy Twister correspondence, the four-dimensional version, which is one that's of interest to us. is this. Corresponding to your four-dimensional complex space-time, there is a complex five manifold happens to carry a contact structure. That contact structure is not really necessary to describe at this point at the outset. So a point over here, if you have a point in space-time, you can look at all the nulls you need through it, and that's a certain complex sub-manifold. but the set of null directions in four dimensions the set of light rays of light-like directions looks like CP1 times CP1

40:00 because they're parameterized by pairs of spinners so you get a CP1 times CP1 in the ambitwister space for every point over here and a point over here in the ambitwister space corresponds to a null geodesic and then you can say when for example two points are on a null geodesic if and only if P1 times P1s intersect, and if you have, so these P1 times P1s, as I'll say in a moment, are usually called the skies. So every point has a sky, the set of directions that you would see, looking out in all directions from it. So there's this five-dimensional complex manifold with certain preferred CP1 times CP1s in it called the skies. Where is the contact structure? four planes in this case, because it's in five dimensions, five dimensions, one with four planes. Well, in fact, each of these skies is tangent to the contact distribution. You have two of them in a point that cross-transit transversely. Two plus two is four, so that's the hyperplane. And the amazing thing is that the span of the tangent space of two skies, it doesn't matter which two skies you take to take the two points. Through a given point, it spans the same hyperplane. So these special CP1s times CP1s are called skies, and they can actually be characterized in a very beautiful way. Namely, if you take an ambitwistive space, a sub-manifold limit is a sky, even if and First of all, it's an embedded CP1 times CP1, and it's in the right homology class. So specifically, all that you need is that the restriction of the first churn class of the ambitwister space to it is in the right element of the cyclophomology. So the cyclophomology of S2 times S2 is Z plus C, and the restriction of the first churn class would be 3, 3. And in fact, there's a, so that the generalization of this correspondence picture is the following, suppose that you have to start with a complex five manifold with a contact structure, and you look at the set of embeddings of CP1 times CP1 with the property that the first term class,

42:30 evaluated on the CP1 that's required to be in the right homology class. So these are sort of, these are of the right degree. So the set of all of those is a complex 4-manifold, and it comes equipped with a holomorphic contact structure, and that holomorphic and moreover, when you go back and say, well, now take the ambitwister space for it, it is an open set in your original manifold. And, of course, this theorem is stated in the same way that I stated the twisted correspondence. If you start with an arbitrary 5-anifold and you look for CP1 times CP1s with certain properties in it, it might be empty. The empty set, for me, is a visible example of a complex manifold. All right, so the amazing thing here is that this produces complex space-time out of apparently nothing. You just started with a complex contact manifold, you looked at a modulized space of sub-manifolds that is, a certain modulized space of sub-manifolds is automatically complex space-time, and it's sort of an invertible correspondence. Now, ambitwisters are supposed to have something to do with twisters. Suppose we start with an ambicell dual four-manifold, such that it has a twister space, which is a complex three-manifold. What's its ambitwister space? Well, it's the same as what I described about CP3. You take the projectivized cotangent bundle, so that's a certain CP2 bundle of the twister space. That is the ambitwister space associated with the ambicell dual case. Now, there's something very compelling here, because, in fact, there's a way of imitating not just the ambitwister space, but, remember, the ambitwister space, the flat space times sat in a certain six manifold, CP3 times CP3, there's a generalization of that to the anti-cell dual case.

45:00 So, the ambitwister space is, strictly speaking, so I would say, a CP2 bundle over the twister space. You take the projective-like quick-handed space of the twister space. there's a natural embedding of this in a certain CP3 bundle namely take the canonical line bundle and take its one jet bundle the one jet contains the potential bundle twisted by the canonical line bundle forget about the twisting so there's a natural embedding of this and that and this actually, in the blackface that would be CP3 times CP3 star so there's a so in particular you can immediately define If you have an anti-cell dual space-time, you not only have the ambitwister space, but you have all of its thickenings of anywhere you like. You can talk about not just the ordinary functions, but the functions together with a certain number of normal derivatives, and you can also talk about the supersymmetric version. Hmm, that's curious. In fact, these thickenings of ambitwister space in the supersymmetric versions, in In the flat case, were first used by Witten, Eisenberg, Gassman, and Green to study non-self-dual Yang-No's fields. There's a beautiful twister correspondence for these. I'm not going to really describe it for a lack of time, but the point is that in the flat case, this had been checked to be of great interest, and this version for anti-self-dual space-times turns out to have all the same properties, essentially the same proofs work, as I later showed. But the more interesting question is, when can you find these thickened or supersymmetric versions of ambitwisted space? And here's the general theorem. The preliminary computations on this were due to Bastin and Mason, who worked on the linearized level, and then I made it work in the general context. So if you take the ambitwister space of any complex four-dimensional space-time, then there is an analog of its fourth-order thickening. So not only can you talk about the classical manifold of GD events, but if you're interested in functions on the analog of CP3 times CP3 star up to truncation at order 5, it still exists as a canonical way of getting such a thing. if you want to instead

47:30 go to a fifth order thickening you pick up the equations of conformal gravity if you're looking at not just an analog of that hypersurface in P3 times P3 star for curved space time, you want to look at the fifth order neighborhood and that has an analog then you're on shell for conformal gravity the Bach tensor has to vanish and then if you go to order six there is another conformally invariant tensor called the Eastwood-Dyton tensor that has to vanish. So my last slide is just to remind you what these two things are. The Voss tensor is something which involves a double divergence of the Vial tensor plus over-order terms. These are exactly, this is the current associated with the action integral to Vial squared. And the Eastwood-Dyton tensor is a less well-known thing, so if you want to go, if the equations of conformal gravity are satisfied, you want to go even further, the next order of obstruction is this thing which involves the divergence of vial times vial. So this combination of self-dual and anti-self-dual is actually formally invariant. Now, both of these tensors, the Locke-tensor and the Eastwood-Dyton tensor, vanish if the manifold is Einstein. So, in fact, if you have an Einstein space, if the Ricci tensor is proportional to the metric, you get six-corder thickness. And there's actually a very nice theorem. These equations are also satisfied in the self-dual and anti-self-dual case. They should be, right? Because in the anti-self-dual case, we've got infinite order thickenings from one of them. A very nice theorem of Cosme, Newman, and Todd, which is that, actually, if you have vanishing Bock tensor, vanishing Eastwood-Dyton tensor, then... so these are conformally invariant things you can't say that the metric is Einstein in fact it might not be but suppose that w plus and w minus are algebraically general at one point and they're as far from zero as possible at that point then in fact your conformal manifold

50:00 is in an essentially unique way conformal to an Einstein space so this is actually a characterization for all intents and purposes by Einstein's equations The only difficulty is that when W plus and W minus are rather degenerate, then conceivably they are solutions. But basically, these are conformal on the same way. Well, I'll stop there. So, Simon, thank you for all. I have a question for you. I have two questions. Is there something special about 6, 4, 3, 7, 10? Um, yeah, that's a good question. Um, yeah, well, we stopped there. So, presumably, if you go to higher orders, you definitely expect there to be obstructions. Um, if you look at the, on the linear, The correct way of doing this is you've got to do this on the quadratic level of perturbation. I think that you'll find that... I don't think anybody has actually written down the instructions. There should be non-trivial instructions, and I think that probably, if you go to order 7, probably it has to be anti-cell through a cell 2. But I don't know that it's a theorem. It's a very good question. If you say I have two questions, I'll work with you. Well, I mean, it, I guess what you're saying is going from an anti-cell dual manifold to the ambitwister space, then there's a hand in this described, involved.

52:30 So is the anti-cell dual? No, no. So, I mean, this theorem is about the general case. So in the anti-cell dual case, just as an example, I showed you that, in fact, in the anti-cell dual case you actually in the anti-self-dual case the ambitwister space is the projectivized cogentic bundle of the twister space and that sits in a natural way that five manifold sits in a six manifold in a natural way now you've already broken the symmetry to say the manifold is anti-self-dual though so in the flat case you get it's both self-dual and anti-self-dual and therefore you can describe it in either way in the generic case you don't have either a twister space or a dual twister space, but you still will have the anti-twister space. And this theorem is sort of describing, well, maybe you don't have one or the other, but suppose you try to imitate the product of twister space and dual twister space. To some finite order approximation, that works always. It works up to fourth order. At fifth order, you get the Bok equations. at 6th order you essentially get the conformal line-side equations and your question, which is quite a good one, what happens at 7th order I think that maybe at 7th order you might only be tied down to things that are self-doable and are self-doable. Do you know why? I'm just trying to remember. It's been 15 years. I mean Proving this result was the first 10 years of my career, but that was 15 years ago. I mean, roughly speaking, if you look at the cars that have a formal connection, the self-dual parts occur and the anti-self-dual parts occur, and they kind of, roughly speaking, commute to the point, then that's that condition. But then they've got to commute to, you know, to first order in each point of this. Yeah, so it's not clear. It's not clear. At least the calculations are... Yeah, I... But you're right, it's an interesting calculation to see why I haven't accepted it. I mean, so in particular, it's not clear how to use the standard theory of conformal invariance

55:00 to get the next level because the tensor that's going to come up is presumably only defined on things that satisfy B equals 0 and E equals 0 and standard machine is not working that out, I don't think. Well, let's take a door to calculation. You can just do it. Oh, yes. If there are no other questions, let's start before we go. I'd like to say a quick announcement, we're gathering up the copies of the talks, and I think a number of talks were given yesterday, especially to people who gave computer talks, but we don't have them. So if you did give a computer talk and you haven't already sent us a copy, then please is essentially a copy to Lionel Mason, so that's Mason and Max. L Mason and Max. L Mason and Max? Yeah. So, our next speaker is Andrew Leisky on Dr. Strick theory, and you have a seat in public So, since this is the last day of the conference, I'd like to thank you very much for the invitation to come here and listen to all these wonderful talks. And what I'm going to give is a sort of resume of what we understand, what the topological stream community understands so far about the non-properbative physics of the topological We've had a few talks about the involved topological string, and we had one talk, it was by Candelas, it was expository about the perturbative topological string. What we know about the non-perturbative topological string is a lot more conjectural, so in fact, there could probably be a question mark already here. And then I'm going to talk about the possible applications of what we know about the non-perturbative topological string to the twister string, and so there should really be two question marks there. So, as a sort of, as a sort of preparatory remark about why we're interested in this,

57:30 the reason that talk about non-refermative topological strings is appropriate for this meeting is that the twister string that Gwitten originally proposed is the topological B model on the superquad-meowspace Cp3 slash 4, but more precisely, it involves sort of new ingredients the B model. Namely, it involves these solatonic objects, which are the D1 brains of the B model. Now, the notion that the D1, that the B model has D1 brains in it is not a new notion, but so far we've always studied those, just the sort of fixed backgrounds that you quantize the open strings on. But now we have to consider, in some sense, the B model together with all of its possible D1 brains and integrate over them. So that's a non-perturbative, well, it would have been non-perturbative from the point of view of the physical string theory, and so in some sense it should be non-perturbative also, it should be the non-perturbative topological string also, so it raises the question, shouldn't what we know about the non-perturbative B model be relevant in this context? Now, so far, all the things that we know that I'm going to review about the non-propermative topological string are basically closely connected to the idea that the topological string is embedded into the physical string theory in a way that I'll also review. Now, that interpretation seems to make the most sense in the context where the target space is strictly a Calabi-Au threefold. So it's not exactly the case of CP3-4. Nevertheless, a lot of the lessons that you learn by studying the Kalabi-Yau freefold and how it's embedded into the physical string can ultimately be phrased just in terms of the topological string theory, and it seems that they would make sense as statements about the twister string, so let's explore what we can say about them. So, the outline of the talk is like this. First, I'm going to talk about the physical, the embedding of the topological string and the physical string, and what you can learn about that, sorry, what you can learn about the topological string from the embedding of the physical string, and that comes in two parts. First, I'm going to talk about some vector-multiplet couplings which the topological string computes in the physical string, and

1:00:00 the place where, so far, one's been able to mine this connection. I'm going to talk about a particular application of that to counting black holes, to counting the VPS states of black holes, both in four and five dimensions. In the four-dimensional case, it's a really interesting case for the non-perturbative string. It'll turn out that to count black holes in five dimensions, the appropriation theory is essentially enough, but in four dimensions, it seems that you really need something new. and you learn something about the physical, about the non-perturbative topological string by trying to do it, I might as well tell you the lesson. Roughly the lesson is that you should consider, instead of just the partition function, you should consider the square of the partition function. That's the object that has a chance to make it non-perturbatively. With that in mind, I'm going to describe a sort of, I'm going to describe a reformulation of the target space dynamics of the B model, this is a recent conjectural reformulation, but it has embedded in it from the very beginning the idea that what you should consider is really the square of the partition function, not the partition function itself. So for that I have to review first the old formulation of the B model, target space B model, and then I have to review a new action which was proposed by Nigel Hitchin a few wasn't interested in it as a topological B model, he was interested in it for mathematical reasons, but it looks like an actual candidate to be the action for the non-perturbative, to be the effective, low-energy effective action for the non-perturbative B model. This is sort of in increasing order of speculativeness. The third part of the talk will be the most speculative, and it has to do with the possibility of taking something else from the physical string to the topological string, which is the notion of a strong recoupling duality. You know that in effect, to be super string, there's a strong recoupling duality. And the question is, what can you learn about that? What can you learn about the topological string from that? And so I'll propose a sort of rough picture of how this S-duality could work. There's one example in which it seems to ... There's There's one example which seems to be supporting evidence for this S-duality, and I'll explain that. I'll explain how it's possible that the observables of the A and B model could be the same, and

1:02:30 I'll talk about the issues that arise as you try to apply this to the twist experiment. Okay. So, now we're doing the first part. So, until further notice, I'm just talking about the topological string on a Kalabi-Yau threefold. This is what people call the critical case. Later we'll talk about CT threefold. So we start with these vector-multiple coupling. So what am I talking about vector-multiple? Well, the context in which you embed the topological string into the physical string is that you take the type 2a or type 2b super string, which is the ten-dimensional theory, and you consider it on x cross R three-punnel one, so x is our collabi-yad threefold. Then if you look energy effective theory that the observer in R3,1 would see, then they see a theory that has n equals 2 supersymmetry, in fact, supergravity, so 8 supercharges, and the matter content of that theory, it has to have the gravitational multiplet, and then it also has some number of the other two kinds of multiplets that you can have, which are so-called vector multiplets and hypermultiplets. And we're mostly going to focus on the vector multiplets, and so you're interested in knowing what do those correspond to geometrically in the Calabi-Yau space, and the answer is that they correspond to specific ones of the moduli of this Calabi-Yau. So remember that if you have a Calabi-Yau threefold, the moduli are basically split up into two kinds of moduli. There's the complex moduli and the Kaler moduli, and those correspond to the vector and hypermultiplets, but which one they are depends on which theory you do. 2a, then the vector multiplets of the scalar moduli, and if you type 2b, then the vector multiplets of the complex moduli. Now, why am I paying more attention, you know, in this undemocratic way, to the vector multiplets? Partial reason is that the string coupling constant is always sitting on a hypermultiplet, which is called the universal hypermultiplet. But it's for type 2a. For type 2a. Oh, excuse me? There's a question where all these are. Oh, yes. Thank you. So, sorry, what I want to say is that n equals 2 supersymmetry, the constraint of n equals 2 supersymmetry imply that the f-terms involving the vector and hypermultiplets are decoupled in the supergravity

1:05:00 theory, and so the f-terms involving the vector and multiples are protected from string-hoop corrections, so those are actually good things to compute. Good, so now I come to the place where the topological string enters, and the point is that some of these f terms in the n equals 2 theory can be calculated from the topological string. As Kandelis described in his talk, there are two versions of the topological string, and those two versions are called the A model and the B model. At least if you study the string in perturbation theory, those two topological strings can involve different moduli of the cloud, yeah, the observables that you can introduce are the, in the A model, the observables correspond to the K-R moduli of X, and in the B model, the observables are the complex moduli of X. So, another way of saying it, since we just discussed which ones are the vector multiplets in type 2A and type 2B, is that in type 2A, the A model topological string depends on the vector multiple moduli, and in the B model topological, sorry, in type 2A, the A model topological string is the one that is correlated with the vector multiple moduli, and in type 2B, the B model topological string is correlated to the vector multiple moduli. I'm not sure whether this type of the AAA and the BB mesh up here is just a lucky coincidence. So what are the terms that I'm exactly talking about here? Well, now I'll just write Z instead of ZA and ZB, and I'm talking about the partition function of the appropriate topological strings, and that's computing the vector multiple couplings. Then if you write it, it's convenient to write it in terms of the free energy, F, So you write Z as the exponential of F over G-string squared, and you expand this F, this is just the genus expansion of the topological string theory, so F is, this is the tree level, this is one loop, two loops, and so on. Then the topological string, so you can compute just by integrations over moduli agreement surfaces, you can compute these F-0s, F-1s,

1:07:30 of two. And the question is, what do those things have to do with the physical strength theory? Well, what they have to do with the physical strength theory is that they calculate f terms that have this form. So, if you look at the coefficient of this w to the 2g, and I'll explain in a second what w is, but w is a superfield that appears in the n equals to effective supergravity, called the vile superfield, and it's built entirely from fields of the gravitational multiple. This w to the 2g appears with a very specific coefficient, which is given just by the g, the genus g free energy of this topological string. So concretely, what does it mean? If you had expanded this f term, d4 theta, in terms of the component fields, you find, for example, an interaction fg of xi, these xi's are the vector multiple fields, times r plus squared f plus 2g minus 2. So these are the self-dual part of the Riemann curvature in four dimensions and the self-dual part of the gravocotone in four dimensions. So the gravocotone is a vector field that's sitting in the gravity multiple. So it's the new one that you always have in the n equals 6 to gravity. So it sounds The topological string is this powerful mathematical machine, and what it's computing for you, for some reason, is the amplitude, a sort of gravitational correction to the amplitude for scattering of 2G minus 2 of these gravitophotons. So you can say, well, why on earth am I going to be interested in A? Okay, so what are these couplings good for you? Well, there's a series of examples that I want to tell you about. The first and, I guess, simplest example, though it's already interesting, is that if you design your Kalani-Yam geometry in a sort of clever way, you can arrange it so that you get not just the low energy supergravity together with a bunch of vector multiples, which gives you one gauge theory, but you can organize it so that you get also a non-Helian gauge theory. It is, again, an N equals two gauge theory. And in that case, knowing this F0 turns out to be equivalent to knowing the whole infrared dynamics of the gauge theory, so it's called the prepotential of the gauge theory, and

1:10:00 Zeinberg and Witten had given a method for solving these gauge theories exactly, solving the infrared dynamics of the gauge theories exactly, and that solution is precisely captured by this F0, which is a topological string in that geometry. So in order to actually compute F0, concretely what you tend to do is use the mirror symmetry. And then this Zeibert-Wiggin curve, which encodes the solution of the n equals 2 theory, appears directly in the mirror geometry. So that's one example of why you want these functions. But a second example is, so for this you don't just want F0, you want all of the higher genus And you can use them for kind of DTS case in black holes, both in four dimensions and five dimensions, but let me start with five dimensions. So suppose I take M-theory, and I consider M-theory on the Klaviyas space X times, now instead of R3,1, it's R4,1. M-theory is 11 dimensional. Then you can insert black holes that would appear in the five dimensional theory. The way that you do it is you take M2 brains and you wrap them on some homology cycle. One of the basic objects in M-theory is a two plus one dimensional object called the M2 brain. You can wrap it on a homology cycle in the second homology of the quad. So it leaves over just the time dimension. You get a point-like object in five dimensions, which is interpreted as a black hole. In fact, you can calculate the contribution of these things to the term r plus square f plus 2 d minus 2 in four dimensions. If you had compactified an extra circle, you can calculate the effect of these particles running in loops, and what you find is that the whole A model partition function, calculating it from this point of view, it looks like calculating these terms, r plus square, f plus to the 2d minus 2, from this point of view, the whole A model partition function can be expanded, and it has an expansion of this form. So let me stop for a minute to talk about what this is, because this is a lot to digest when you first look at it. So what do we have here? sum, first of all, over all the different homology classes where you can wrap a black hole.

1:12:30 Here we have a sum over an integer, which represents the spin of the black hole. Here we have an integer number, which is the number of black holes of a fixed charge and spin. And then here we have a factor, an exponential factor that weights essentially the mass of that black hole. So it's e to the minus is the inner product of q with t. t is the Kehler class on the LaVeya. So the inner product of t with q is measuring the size of a, well class, the integral of the killer form over the etymology class. So what you have here, the remarkable thing about this expansion is that it encodes the whole structure of the A model partition function just in terms of these integers. Now you might have naively, from the way you usually characterize the A model partition function, the way it's usually described is, as Candelas explained, it comes from just holomorphic maps of, the only contributions of the A model partition functions come from maps, like the world sheet into the target space. And those maps are naturally weighted by exactly this factor, e to the minus the Kaler volume of the image of the map. So you would first think that that's already going to be just giving you some integers. You're just going to be summing up the number of curves in each homology class. It turns out to be not quite true for the theory of maps. And the reason is that you can have all kinds So even if you just have an isolated curve in your Falabi-Yau, you have to think about maps. First of all, you have a simple map that just covers it once, then you have more complicated maps that cover it twice, three times, four times, and so on. All of those things contribute to the A model partition function, and you have to sum up over all of them, so the contribution of this one curve is a complicated thing. Now this formula, this formula, Gopakumar and Bapa, since the DPS dates, exactly sort of undoes all of that complication. All the complication of counting all the different maps is encoded in these factors here, and you ultimately untangle it just as a linear invariant. So from the mathematical point of view, it's a surprising formula that really comes from the fact that you can embed the topological string into the physical string. You have this insight that you're talking about black holes with a fixed charge and spin in the four dimensions. Now, on the other hand, from the physics point of view, you could also say, well, this is a pretty interesting formula, because it says that the topological string, if someone

1:15:00 had given you this function, you could use it to count the number of black holes. And from that point of view, what I want to emphasize is that it's a truly finite calculation. If I'm interested in some particular nj, q, I only have to calculate the topological string up to some finite order in the genus expansion. Okay, so that's in contrast to the four-dimensional So, recently, topological strengths have been applied to the accounting of black hole states in four dimensions, and this is, well, in some sense, a deeper application. So, what's the statement here? Now, instead of taking M theory on X cross R , I want to take the 2D theory on X cross In that case, you can construct black holes, again, by wrapping, sort of, celatonic objects of the theory on homology cycles. In this case, the relevant object is a V3 brain of type 2B, and you can wrap it on a cycle in the third homology. And then, in contrast to the five-dimensional situation, now a black hole can have both electric and magnetic charges, because you're in four dimensions and a particle can be both electrically and magnetically charged. Five dimensions are dual to a particle within a string. You want to express somehow the notion that this charge involves both magnetic and electric charges. Well, the geometric meaning of that is that you have to split up the third homology of your Calabi-Yau, and you have to split it up into what people call A-cycles and B-cycles. So you choose half of the cycles, you choose a basis, in other words, of the third homology, call half of the cycles A-cycles, half of the cycles B-cycles, and they should have the property that the A-cycles don't intersect each other, the B-cycles don't intersect each other, and the A and B-cycles are grueling each other in this way. So there's a lot of such choices, and you just make one such choice of a marking. Once you do that, then you find that the, well, the conjecture of Agoury-Strowman during MAPA is, but there's some evidence supporting this conjecture in perturbation theory. The conjecture is that the B-model topological string partition function is computing now the numbers of these black holes with fixed charges q and p. This omega pq is supposed to be the number of black holes with charges q and p. And it computes it, but it computes

1:17:30 what might seem like a funny way, namely, compute the partition function of a sort of mixed ensemble. You don't fix both the magnetic and the electric charges in the black hole. What you do is you fix the magnetic charge and an electric potential that's conjugate to the electric charge, and you sum over all electric charges for a big magnetic charge. So that's supposed to be equal to the square of the topological string partition function. I have to explain exactly what I mean by this left side. So by the left side, what I mean is you take the B model, topological string fraction function. Now, that's a function of the complex moduli of the Calabi-Yau. And the complex moduli of the Calabi-Yau are written through the periods of the holomorphic preform. So these are the periods of the holomorphic preform over the A cycles. So they call them the electric periods. But I want to point out that this, that, well, manifestly, from the right side, in fact, the thing that I'm calling the topological spring partition function really depends on how you chose to make your basis. And that's the fact that it was also familiar, in fact, it's not as if this was the first time that people noticed that, it had been known for a long time that the B-model partition function, also, in fact, the A-model partition function, but it's more obvious than the B-model, that the B-model precision function depends on some additional arbitrary choices that you make. It doesn't just depend on complex moduli in the most naive way. This was understood as something called the holomorphic anomaly of the B-model. In fact, it depends on extra choices. This holomorphic anomaly was discovered by Przetsky, Chikodya, Hure, and Wappa, but a very useful interpretation of it was given by Witten. The interpretation is basically that what you're looking at is the third homology of the Calabi-Yau, H3 of XR, you look at that as being a symplectic space because it has a symplectic pairing, which is the intersection product that we were using before. And you're studying wave functions that come from quantization of this symplectic space. Well, you know that when you have a wave function, you have to choose how to represent it. You can represent it as a function of the Q's or as a function of the p's, and different choices are going to be related by Fourier transforms. So that's the phenomenon that you find through this guy, the fact that it depends on the choice of a marking. But if you write, once you pass to these omega pq's, you in some sense get rid of this holomorphic

1:20:00 anomaly. The number of black holes with a fixed charge, with a fixed magnetic and electric charge, p plus q, is just a canonically defined thing that doesn't depend on the choice of basis. In terms of this quantum mechanics, if you've got a quantization of x3 of xr, this omega of tq would be what people call the Wiener function of that wave function. Okay, fine. A few more comments about this. On the left side, you don't see the string coupling anymore, and you might say, what happened to that? Well, what happened to it is we traded it for the overall scaling of this p and phi. It turns out that in the topological string you can do that. So what I'm hiding here is that the genus expansion of the topological string on the left side, G-string expansion, corresponds to an expansion around large charges of the black hole on the right-hand side. So another way of saying that is that the perturbative physics of the topological string would not ever tell you one of these omega pqs if you want to compute it. It only gives you the asymptotics of omega pqs. these integers, you can never do it just by computing the perturbative topological string. Rather, they would correspond to some hypothetical, as yet not understood, non-perturbative completion of the topological string. I think to date, actually, this is probably the strongest evidence that one should mean anything by the word non-perturbative topological string, but here's a sort of candidate object which the non-perturbative topological string ought to be computing, and in fact, maybe Maybe one should even define the non-perturbative topological string just by these numbers. So far, that could all sound a little bit abstract and perhaps bogus. To make it sound a little bit less abstract, I just want to tell you that there's one case where this can be worked out explicitly. And that case is the type 2A superstring on a particular non-compact Pallabi out of threefold, which is the total space of two line bundles over a, well, total space of a monomorphic rank two bundle over the torus. Now it can be done also over a ring on the surface of Venus G, but the case of the torus is the simplest. Now in that case, so far I was talking about type 2B, but everything I said has a translation into type 2A. In that case, instead of talking about D3 brains, you have to talk about brain to make your black holes. You talk about the even homology, so you wrap D0, D2, D4, D6

1:22:30 theory. Remember, we had to fix half of the charges and sum over half of the charges. Here, in fact, there's a more natural way of doing the splitting in the type 2 S theory, which is you can pick the magnetic charges to be H4 and H6, and the electric charges to be H0 and H2. So you just fix, suppose you fix Nd4 grains and no D6 grains, and then you want to sum over D0 and D2 grains. Well, it turns out, it's not quite obvious in what I'm saying, although it should be reasonable. It turns out that that's equivalent to computing the particular function of the Yueng Yang-Mills theory on T2. Well, the idea of it is that the D4 grains, which you put in this geometry, I didn't say how you wrap them, but you wrap them on T2 and on one of these non-compact cycles. That gives you a Yang-Mills theory on T2 plus one of these non-compact cycles, but then Wappa and also these authors argued that in fact that theory should be localized just on the T2. So you just get a two-dimensional Yang-Mills theory. And then, so this theory automatically sums over D0 and D2 rate charges, which appear as sort of induced charges on the world volume. And so now, the interesting So from what I've said so far, this partition function should be, in some sense, counting the black holes that you can get in four dimensions, and so it should be factored. It should be the square of the topological string. Well, it's already known, in fact, that this partition function is factorized. You can write z yang-mills as the holomorphic square of something. And by holomorphic here, I'm talking about the dependence on, and you might say, what on the Yank-Mills coupling and the theta angle. But the interesting thing is that that factorization is only true in the large n limit, and it only holds preservatively in 1 over n. When you talk about the homology here, it's the homology of some compactification of that bundle over the two torus? Oh, you're asking what exactly, since it's non-compact, what exactly does it mean by this homology? It's not the homology of that space, I think. No, I am talking about the, I am trying to talk about the homology of this space. Yeah, what I mean operationally by H4 here is I want to consider T2 plus one of the total line bundles. So you're right, I'm being a little bit vague about, your issue is the fact that it's non-compact,

1:25:00 right? Yeah, I'm being a little bit vague about that. Yeah, it could be the right way of thinking about this. Sometimes when you have these non-compact homology, it's just if H4, H6 are both 0s, you have to sanctify it before you're going to get anything. Yeah, I'm not worried about the H6 being 0, but I'm a little worried about the H4. So, what I mean by the plasma 4th homology is that it's T2 plus one of these two non-compact 4th cycle, which I want to call it homology. Fine. Oh, no, but I'm going to emphasize this point. even if you thought that the point is that the square of the topological string is the thing that exists non-perturbatively in 1 over n. This factorization of Z Yang-Mills equals Z squared only makes sense in the 1 over n expansion. So the suggestion, the lesson for the topological string would be that non-perturbatively you should never even try to define the topological string Z. You should only try to define its square. Okay. So, with that lesson in mind, that's the main thing that I want us to learn from this first part about embedding in the physical super-string. And now I want to talk about a method of reformulating the closed-string dynamics of the B model, which naturally incorporates that lesson. Okay, so what am I talking about? Well, so far we mostly talked about the, well, So far, when we talked about computing the topological string, we mostly talked about the world sheet version of it, but the A and B models also have a target space description, both the open and the closed string versions, and those were given in the open string case by Witten and the closed string case by DCOV and Riket-C setup, and just to sort of catalog catalog them, the open string, the A model open strings gives you Turn-Simon's theory, the B model open strings gives you whole-morphin Turn-Simon's theory, as has been very important in this TwisterStream program. The closed string of the A model gives you a kind of Kähler gravity, this is the variation of the Kähler form, and the closed string B model gives you this without our sensor gravity that I'm going to talk about now.

1:27:30 So, oh, it's a little intermittent. So, yeah, so far in the sensor string, what we mostly, what has mostly been used is the target-space description of the open string key model, namely the fact that it's holomarkic term silence. And with that in mind, I want to talk about how you reformulate the closed key model with the idea that it might also Okay, so what is the closed string B model? Well, the closed string B model, viewed from the target space point of view, is a theory that describes variations of the complex structures on X. And a complex structure, an honest complex structure on X, in other words, a solution to this target space theory, has two properties. First of all, it's a type 3 comma 0 in some underlying complex structure on x, and second of all, it has to be closed. It has to have y over equal to 0. And one way of getting, so whatever the B model should be, one thing it should be is it should have a description as a field theory which classically produces for you such omegas as its solutions. One way of obtaining them is from this Kodair-Spencer gravity, where I'm not going to write down the action for it, but the fundamental field in that theory is a 2 comma 1 form, which sort of describes the first order variations of Ovington. But that gravity theory, which is to describe the E-model, the usual formulation, has some unwanted features. For example, it has this full-and-worked anomaly that we talked about before. the fact that the partition function depends on a sort of reference choice, and one way of understanding this reference choice is the choice of a background complex structure. And that problem appears even classically. So Hitchin gave a new way of obtaining these whole and morphic three forms, sort of fundamentally new way of obtaining them from an action principle, and that's what I want to describe. And the construction begins with a crucial observation that if you want to have complex geometry, you don't have to start with it from the beginning. In the very special case of six dimensions plotting out threefold, giving a real freeform is already enough to determine the almost complex structure on x. So what do I mean by that?

1:30:00 Before I talk about that case, let me just remind you of a slightly simpler case, namely suppose you take a real two-form in dimension 2N, then it can always locally be written in the form that I've written here, so it's only going to be your two-form, sorry, I omitted one crucial word here, a real non-degenerate two-form in dimension 2N, then it can always locally be written in this form, you can choose a basis of the tangent space, E1 up to En, F1 up to Fn, because this omega is E1 F1, E2 F2, and so on, to En Fn. So you have to choose a basis for the, sorry, this is also a typo, this should say cocaine, this way, varying over x, so it's what you would call, it's what Luzes would call the fuel line. And now, if this omega happens to be closed, if d omega equals zero, then, locally, you can take these e's, these e1 up to e n, f1 up to f n's, to be the coordinate one forms dp1, dq1, up to dpn, dqn. In that case, you would say that omega defines a symplectic structure. So, okay, non-degenerate real two forms define for you a pre-symplectic structure that exists, and if it happens to be integral, d omega equals zero, then it defines an honest is that. Okay, fine. That's not so exciting, maybe. But the exciting thing is the generalization to a real three-form. So suppose I look at a real three-form in dimension six. Then exactly analogously to the way you did it for the seplectic form, it turns out that you can always write this rho. And again, here, I say you can always write rho. I should say if rho satisfies a certain generosity condition, which is the analog of saying that omega was You can always write rho in this form, zeta 1, zeta 2, zeta 3 plus zeta 1 bar, zeta 2 bar, zeta 2 bar, where zeta 1, zeta 2, zeta 3 are made out of the field line like this. zeta 1 plus ieta 2, zeta 4, zeta 5 plus ieta 6. Now, these zetas determine for you an almost complex structure on the manifold. Well, what do I mean by almost complex structure? The easiest way to understand it is just to say that if you're lucky, then there are complex coordinates, plus the complex conjugate of it. In that case, you say that the complex structure,

1:32:30 the almost complex structure, is integrable. In other words, it is an honest complex structure. Now, I want to emphasize that this game really only works in dimension six. And it's not that hard to see it. Basically, the point is that if you go to much higher dimensions, the number of freeforms just explodes, and it explodes much faster than the number of possible field lines you could have. So you're never going to be able to write a generic three-form in this way in sufficiently high dimensions. So there's a certain magic trick that you can do it in three dimensions. So in that case, just real geometry, just having a real three-form is actually sufficient to give you this much more interesting complex structure. Okay, fine. Okay. You can rephrase this condition of integrability of the complex structure as d omega equals zero, where omega, the whole market prequorum, is z1, z2, z2, z3. And this omega is completely determined by rho. So now, Hitchin's big insight was condition from an action. And the action is easy to write down. It's just the whole market volume of your Calabi-L. Where what you vary is not this rho itself, but you consider rho to be like the field strength of a U1 gauge 2 form. So, or put it another way, mathematically speaking, what you do is you require that rho is closed, and you fix this cohomology class, vary it inside a cohomology class. The equations