Knot Invariants, Holomorphic Curves & G2 Manifolds

0:00 In the meantime, the speaker today, of course, requires no introduction, but here has only a number of ideas both in mathematics and mathematics and physics, and today we'll be talking just about what it has not invariant and . I think that's a different question. No, no. That's not the question. Okay. So, well, as you know, in the last decade or so, there have been very exciting developments We have a large mathematical component of the mathematical side. And so I want today to review some of the things that have happened to some particular kind of problem, which illustrate a lot of other things. And I think it's very exciting and still happening. So of course, I'm not a real physicist, so don't ask me too many deep, critical questions. Now, you know, let's start with quantum field theory. that standard standards program with this deal we are many particles that beyond the common field really the string theory to do also gravitation and more easily we discovered a lot of many different string theories which are in some sense all extended from the seriously called m theory now between all these different string theories of m theory there are lurk a lot of duality is a rather vague word it means two theories are somehow equivalent but look very different or they apply different regimes and you can continue from one to the other one anyway they're very interesting these dualities are from a mathematical point of view the hardest to understand and most interesting and also the most interesting from the physics point of view now besides they're called physics there are a lot of topological aspects of these theories and those the one that you're much more tractable to focus on for a mathematician at least because they're much more precisely defined and they give you explicit things you can calculate and even if

2:30 you're a physicist these topological aspects can be interesting because they uh sometimes give you a indication for what happens with your physical models as well they come as a toy model if you like of the real physics so let me start i will focus on the topological aspects of the story that connects these different things let me start with the top prototype topological theory which is a theory of not invariance some 15 years ago i think von jor jones discovered there are a lot of polynomial invariants which associate knots in ordinary three-dimensional space as if not the invariants are in fact just polynomials or finite laurel series and the variable z efficient string of integers are the various and they are completed by a picture of a lot and show that the variants are independent of the way you get the knock on to the plane then some few years later though I'm already indication with Jones in his work there was a mysterious relationship with physics and he's not because they think the variance has little to do with traditional topological invariants well some years shortly after with Witten produces a beautiful interpretation of the Jones variance what he showed was that that they quantum field theory in three dimensions let's say two space in one time and you take a case theory but you can make any any cases right let's say you in and you can put it by the only on three-dimensional space like in fact, the manifold like the C-scale, or in general, any oriented three-dimensional manifold. And then you write down the Lagrangian theory, and this has now become, well, known as the so-called Chern-Simons Lagrangian, which is typically interesting in three dimensions because it gives you a three, it gives what you take to the connection of Dental A, a potential any time da that is a make you try to see form and then the cubes another really very thick form you take the trace to go three form we integrate the manifold and come in comes outside that's the act and then you apply the handles upon field really see what you get how the important thing is in this story that the there is a quantization condition that the coupling constant

5:00 outside K over 2 pi K has to be an integer the reason that in this channel Simon's form is not actually gaited variant not quite it's locally gaited very one of the global gate transformations he can pick up that it's just so you have to have put the integer there the thing is what he finds otherwise so this and what we can share with this stuff what this is this quantum field theory has the and not it is very similar to quantum field business use don't have additional term something like a yang mills if you drop that only concern this purely topological part of it then you get a good topological theory in other words all the observables are truth in this theory are topological They don't change very continuously, but they are, in fact, in disguise, the invariants that Jones discovered. And the table reverts them as a simply take what are called the Wolfson Loop Observable. They close the loop, you take part of it, you go around there, and you take the expectation values in the quantum field theory. And what you get in this way, each of the K, the level of theory, you get a number associated to the part and take a part of the knot it's very important these knots don't cross themselves otherwise you will get divergences and then you find what you get this way is exactly the value of the Jones polynomial at a particular root of unit 2 phi i over k plus n the physicists will say that a fair coupling cost of k as we normalize the paper then that summary is a very beautiful interpretation of the not theory showing that not invariants do appear very naturally as quantum field theory observable in a purely topological theory and that's one side of the story now now quite independent of that let's take another line the very interesting story going back to welcome to who show that if you take the Feynman diagram and the case theory in particular in the large n limit where you have a u.n case theory that n is very large then you will rearrange the expansion so that you can stand in the power of 1 over n and this can then perhaps be re-inserved in a different way and this will apply

7:30 originally by Chaucer in certain situations and Witten took the same idea he showed that you apply that situation of this one field reading just to talk about Joe Simon's theory then you expand each of Simon's theory in terms of the covering constant I'm a dagger expansion what you end up with after this rearrangement again did you find you've got an open string theory on a six dimensional manifold six dimensional manifold is original manifold of three co-tangent fungal C-sphere, which is really just the C-sphere out of R3, 6-manicold, and in the 6-manicold, you have a zero section of the tangent fungal, and that carries, interpreted as a D-brane, and then the strings have to end on this D-brane, and this D-brane carries an integer number of charge, capital N, which is the N of the U-N of the K-sphere. The N of the matrix size gets converted in this interpretation in terms of the charge on the d-brain. I wouldn't show that the Scherzheimer's theory of quantum field theory is topological, but if you reinterpret the branch in this way, you see, you recognize an open-string theory itself there, but it's also topological. This is a string theory for open-string that's moving on this six-dimensional real manifold, which is an example of what's called the non-compact Calabi-Yau manifold. It actually is an algebraic, three-dimensional algebraic variety. It's simply given by the standard equation. Some of the squares equals, say, a non-zero constant. That defines the three complex variables, six real variables. That's the space that meets the string move. If you put all the coordinates real, you see the three spheres inside there. and that's brave so this is the second shift like theory has several different ways of being looked at first purely quantum field theory secondly is a string theory and I don't know when I heard this I thought it's a marvelous story I'm learning from this is a long time that string theory weren't really fundamental they were perturbation expansions of something but nobody could be what they were perturbations of here we have an example where they of a standard quantum field theory, disputed.

10:00 In both cases, the topological, the topological aspect of the strings is that on the six-dimensional Calabayana, you do not a conventional model, a single model, you do what's called a tristic model, which converts it into a pure topological theory. We can also introduce that idea as well. So that's the second problem, sir. this was all going back quite a long way now let me say a bit more about these duck I'm a diagram expansion here I'm sure many of you know much better than I am at this so the covering constant in a church assignment theory essentially this okay you call the level and then you have find a diagram and then you have gates field find my mind right my job I'm on diagram twice when you saw the you start to get pictures looking as though you're drawing something on a Riemann surface. What you end up doing is a triangulation of a closed Riemann surface. And the Riemann surface has numbers of faces, H vertices and edges, and they're related by neural formula for the genus of the surface. They're not destiny on the sphere, they can be on a surface of the entire genus. And so if you write the k, h, rather, is the number of basic boundaries or holes in the surface. Now, this theory we have, as Charles Hamilton, the start of an expansion, so either the partition function on here written down, which is called the free energy, which is expanded like that in terms of and then the venue would use you're going to think a big end of going to infinity so you use new coupling constant which is capitalized began our cases two by n and the paper then you Now think of lambda being fixed with n going to infinity. In this case, you see that same thing as saying that k over n has a fixed ratio. So with that being fixed, n going to infinity, we reorganize it and substitute that formula in terms of the lambda. And now you see the formula comes out like this, with n to the power 2 minus 2g, lambda to the power. This now is an expansion in terms of 1 over n,

12:30 that is the one that we're in fact of a different way this is about what was some of the world cheap open-stream theory and that's how that's how the form is look now two years ago buffer and number of collaborators published a series of proposals, which purported to show that this open string theory on the six-dimensional manical, let's take a moment to the sociological theory, which written to show the equivalent of Chern-Thymons theory, has yet another interpretation in terms of closed strings. What they conjectured, or gave arguments for, is very hard for a mathematician to know, When you read the physics paper, what is proved and what is not proved, that every citizen has their own level of price, right? But you give arguments varying degrees of evidence for something. And so the arguments were, a lot of calculations were done to convince people that the open string theory of the six-dimensional manifold, well, actually equivalent, precisely, to the closed-string theory, another six-dimensional manifold, also a non-complex Calabriao. This one is carrying what's called the RR flux, the N flux. The N of the Gaines theory reappears here as a flux, a char. And these calculations were done at the level of the partition function, and with basic formulas, term-by-terms, all genus, and so on, and all fined beautifully. So the evidence of it is pretty compelling. And so he told me about this some years ago, and I thought it was fascinating. So when we have this R-R flux, you have a really close two-form, and the two-forms on this. I call the two six-manifolds X and Y. X is the one we started with. Y is the one I have told you about. Well, I told you about it there on the way. X is topologically S3 cross R3. And Y is topologically S2 cross R4. So topologically it's Y different.

15:00 One of them contains a central three-sphere, the other contains a central two-sphere. And in the one in the Y, we have a simple two-sphere, the two-form has a non-zero integral over the two-sphere, namely this here. Now, let me tell you a little bit. I'm going to talk quite length about this example and various aspects of it. Now, there are two—the way you understand these x and y is you start off with yet another six-dimensional space, namely you start off with a singular quadrant, where you take some of the squares and complex coordinates equal to zero. Then that's a—sorry—algebraic cone in four-dimensional complex space with the origin as a singular point. And now you can think of that called an x-zero. Now you can try and deform this. One way, obviously, to deform it is you put non-zero right-hand side. So you put some one or some number of r, positive number. So you deform x-zero to what I call x-axis singular information. That gives you one way of going x-zero, getting rid of the singularity. But if you get another way of getting rid of the singularity, which is well-known now to great geometry, Resolutions in algebraic geometry are standard procedures for getting rid of singularities by adding new things instead of a singular point. It's a very complicated story, and the general theory of what would be done, but here may only be very simple or special cases. But in fact, this is a very unusual case, I'll explain to you later on. And here you get a very special kind of resolution, which is called a small resolution. I'll tell you later on why it's called a small resolution. in a small resolution so you get two toys y1 and y2 i'll come back to that later anyway they want me call it why what it is as a complex c-dimensional manifold is actually a two-dimensional complex vector bundle over the two-sphere two-sphere is complicated line and the bundled over the two sphere standard line bundle is determined by the degree the line one degree minus one take two copies of it that's the two-dimensional bundle and that is why topologically is actually trivial homophobicly not there's a very nice simple variety that

17:30 is the second one that's the one which the closed strings are supposed to appear now let's compare these two right You notice that at infinity, if you go up to infinity in both X and Y, then both of them look like S3 cross S2, the boundary is infinity, they're six-dimensional, the infinity of a five-dimensional boundary, and in both cases it's the same thing. But the difference is, when you go in from the inside boundary to the interior, in one case you fill in one half, in the other case you fill in the other half. So you fill in the two-sphere, next, then you get a R3, so you get S3 cross R3. You fill in the three-sphere, you get an R4, you get R4 cross S2. So, all you're doing, you think, front of you spheres, you're filling one in, one way, one the other way, because the spheres are different sizes, dimensions, the spheres you get are different topologies, and that's x and y. So they're very simply related, and you can think of them simply different ways of, the cone of this thing, the original one, is just a cone really on what is determined by that, the infinity. There are two different ways of getting rid of the singularity, from infinity to the interior. Now, these two theories have parameters. The parameter has natural real parameter, and they all come with metrics, which I haven't bothered to mention, of course. You have to put the physics by the Kehler metric. And the volume of the central sphere, in one case it's a three-sphere, in the other case it's a two-sphere, these are natural parameters of the theory. So in the first case, x, we have called something, called the volume, the volume, and then t, say, is called the area But in both cases, you have to interpret this parameter as a real part of a complex parameter. So it has a supersymmetric partner, and these theories, all they do, are supersymmetric theories, and so you get, in particular, you get a complex modulus instead of a real one. These theories have, naturally, one complex modulus, for which they depend. And this complex modulus is not a physical observable. You can't vary it continuously, because that would require infinite energy.

20:00 What it's doing is basically buying a boundary condition of infinity. It's tying it down to infinity in some way. Now, in the column between these two different models, the closed-to-inferior open-street theory that these people have put forward, then there's better to be a relationship between these two parameters. When you go one regime, you use one parameter. And there is a formula, which is both the rate, and there is the formula. The formula is here in blue. And what it says is that e to the t minus 1 raised to the power n is some constant times the exponential of minus v over ds. And ds is meant to be the string company. So that's the formula, and that would be the precise formula. I'll say more about that later in a few minutes. And simplify it if you only want to see the case of what certain things are large. And so this is the theory that he claimed. These two theories are recorded by this best procedure, and they gave a lot of evidence by computing the partition function, expanding them, making these substitutions, doing that calculation, and, lo and behold, finding the work. So it's a very impressive tour de force, but a bit mysterious, you might say. It's not quite much what's seen before. It's not clear where it comes from. I should just look here. I was repeating a bit like this before. In Lucy's calculations, you sum. You have enough parameters of eight the number of holes in the genus. You sum, first of all, as a number of holes. Then you get something like that from the genus. This is, then, what looks like the closed-string fraction filled here. Now, for large values of t, the formula I gave before approximately boils down to just saying that t and v n are written by that simple formula. They're essentially one of the one inverse to each other. But the formula with the star, in the previous transparency, is not this. It's a correct formula, a more exact formula. on correction this one here is purely a service but a period which have instant on factor you have to work out the effects of these produce on the formula of a correction sometimes if you're lucky this is in capitalism check precise formula of one before so it's very precise colors modest large values

22:30 And besides the partition function argument, there are other arguments which they fall in favor of them, which I won't go into. But in particular, they talk about the anomalies of these things due to gravitational background effects on both sides, and they compute them in each case. The one in the Churn-Simons theory was already pointed out by Witten, which showed that the Churn-Simons theory is right down the dimension involves the chance I'm a larger chance I'm the garden is beautiful lagrangian because it doesn't involve metric take the gauge field and you produce three forms you integrate no reference to metric so on the face of it you have naturally metric independent theory but it turns out that's not quite correct that's correct at the practical level for the quantum level that's not quite correct therefore the quantum level you have put in the correction the correction comes from a term depending on the curvature of the background metric new corrections and so in which involves some coupling between gauge view of gravitational background with that again theory should be well defined and the same thing appears on the other side of the story and while I do compare these two normally the correction finally they correspond very beautiful so the evidence is actually at several levels not just in the positive function but also in anomalies now that things up to about a few years ago and actually I'll let me out my my involvement is about that very much outside observer I I was I visited Harvard I listened I said marvelous story and then I made some very naive observations in the world this guest is doing the following I went away thought nothing more about it but about a year later he said well here now the senior had discovered observation I made very useful I'll tell you about my contribution was pretty well but anyway so now the we're now moving on him here because so far you see what what I've given you the example of is I saw it off to the quantum field really don't I'm there I said well that actually might show my witness to be called the string theory open We moved into the string theory.

25:00 And then I said about what this is called, or showed it. Arguments for it is called closed string theory. So we moved up into the world of open and closed strings. Now, M theory is meant to be something higher up. In fact, it's one more dimension. String theory, I suppose, is two arithmetic string theories are meant to take place in ten dimensions, for which you usually focus on four as Minkowski's base, and six other ones, and there's the six other ones that we took inside here. But M theory is supposed to take place in 11 dimensions, so you can have four from Nikosky's phase and seven other directions, if you like, seven internal degrees of freedom. And these seven ones have to satisfy a lot of constraints. The constraints that you have, for a supersymmetric theory in particular, require that the extra-manifold you have, in the form manifold times Nikosky's phase, has to have covalent constant spinons. and KVN comes in a strong restriction on its metric curvature and on its whole line let's say what happens the transport run come back what something with the rotation you get and the list of possible possibility at the sixth dimension one of the interesting possibilities are in fact the Calabria where the whole army group is su3 that's what we've been talking about so far when you go to seven direction then that's the case which is an exception But the hominomies that are the Calabria-Yau, because it exists S-U-N in every dimension, there are one coming from the suspecting group. But then in certain dimensions, there are special ones. And in dimension seven, there is a special one corresponding to the exceptional group, G2. Now, you may not all know what the exceptional group G2 is, but let me tell you. So that's the moment, assuming it exists. You know what it is. It's called a manifold with the hominomies G2. And then if you have a manifold dimension 11, which is 7 plus 4, 4 even cross the space, and if the 7 dimensions have G2 homony, then you should get a supersymmetric theory. And if you now can throw away one of the 6, 7 dimensions by a circle, which is an action of the U, one factor which preserves the metric, then you should descend down to 6 dimensions. and what you'll get will be attached to a string theory in one dimension so the idea is that the string theory can be that these things can appear by

27:30 dimensional reduction from something in dimension level and but there are other string theories they can explain in other ways all coming from 11 and some mysterious way and what you do in dimension 11 is not yet specified that we can't constrain the meant to be a theory defined there and from that all the other theories are perhaps derivative so it's a great mystery is what this theory is a lot of evidence is now it makes sense in some ways and people are trying to provide more and more arguments examples of how this is going to work with A perhaps eventually getting scriptural. Now, what is G2? Well, G2 can be defined as the automorphism group of the Cayley numbers. Cayley numbers are an algebra in dimension eight, which are the extension of the real numbers, complex numbers, quaternions, which is dimension four, eight, the next one, and those are Cayley numbers. And then you can't go beyond those. You can't get the division algebra. that's the last one, but Kaly numbers are not nice than the other ones because they're not associated. But never mind, they're nice algebra. And you can look at the automorphism, which preserves the algebra law, the multiplication. That's a group, a subgroup of, well, you find it's an eight-dimensional space, but it actually has to preserve the unit element, to preserve the algebra, and therefore, the photorthogonal of the unit element are called the imaginary part, the seven dimensions. The action of the seven-dimensional space is the embedding of this group inside SOS 7. That's how the fear. Fear naturally depends on the algebra of the Cayley numbers. Cayley numbers, by the way, are a spin property. And this is eight, especially the artificial features, because the spin representation has the same dimension. Standard representation. And that's what's involved in the helium multiplication. You take a vector and multiply it by a spinner to get another spinner. So, that's just to remind you what M-theory is supposed to do, among other things, is perhaps to live a problem. One higher dimension, and the thing here is supposed to be useful from it by cutting

30:00 away one dimension. That's the idea. The different string theories are obtained from M-theory in different ways, but everything's about pretty inclusive. Now, we now give you an example. Strictly speaking, for the real physical theory, these additional internal manifolds, internal degrees of freedom, should all be compact. They should be closed, compact. But you can do a lot of these when they're not compact. And examples I gave you what Calabi are will not compact. So you have to pay some price for it, but you can get it. On the other hand, there are many more examples known. Compact manuals and special homonyms are rather difficult. So G2 compact manuals are rather rare, or at least they were not known until recently. You know how to construct them, but they're not very easy to come by, nor to compute this. non-compact examples come by and they do calculation so I can talk about non complex example and in fact basic one here well let me take the C C down the C C that's six dimensional manifold it doesn't join to verdicts get a code on thing that code is it and so dimensional but has a singular point forget about the singular point it was an actual magic on the cone now you find what you find natural magic on that which is then they draw on the get rid of the singularity by some kind of information now but one small point and when you take the product of three spheres obvious and so on. And so if you ask for a metric on S3, everybody put their hand up to it, I know an obvious metric, a natural metric on S3, and a copy. And that one has a symmetry where you need to change the two factors. Well, actually, there's a better one. There's a better one. So let me just remind you of that. And let me just do it first of all in general. So you have any compacting group, G, and take any to the end, consider the product n copies of the group and divide it by one copy of the group acting diagonally on the

32:30 right-hand side. It's a bizarre thing to do, you might say, but what you end up with is a manifold whose dimension is n minus 1 times the dimension of G. And if G has a natural metric, this thing has a natural metric. For the same metric of the copy of G, you divide by the diagonal this group this manifold you have obvious symmetries first of all you can multiply on left by the end you can also permutations they have a synergy group which is rather large you want and has three now let's say yes I want not going to happen it's quite very long listing equal to three and birthday equally one if they keep you one then what you're doing is taking three torus dividing by the diagonal and you get two torus but that two torus now has symmetry or three symmetry or three six that's that is the two taurus with corresponds to the hexagonal lattice in the play you know taurus is a plane module that I think square this you get the standard square torus which is s1 and s1 to the natural metric but there is a much better symmetry which is hexagonal symmetry because symmetry of all three divided by three that one is the one I defined here Well, it doesn't even do it with S1, you can do it with S3. Take the group of SU2, which is a three-sphere, do exactly the same. And what you'll see is you get a manifold which is a product of three copies of S3, modular one, modular diagonal, which really means it's the same as a product of two copies, but it has this additional metric with symmetry of order 3. It's a bit surprising, I mean, let's look hard to see where the third's S3 is. It involves, in fact, S3 as a group. You have to use a multiplication. But where we find it, it's quite clear why this manual has its symmetry. So the manual is got here. We take on that metric, not the obvious metric, but this better one. And that better one has additional symmetry or permutation of three things, which will turn out to be very important. That's what I'm going to say next.

35:00 So now, staying in seven dimensions, we take this, what we're going to do is take this cone on this front into three spheres, and then we fill in one of the three spheres to get an R4. Remember the four when we filled in the sphere? Well, here we think you want two. But actually, because of the symmetry of order three, there are three possible choices for the three spheres to fill in. You only see two of them, naively, because there are three. and each one will give you a r4 and s3 actually think of it as a bundle r4 over s3 into the geometry carefully and what you'll find is is the cinema bundle of the cc the fiber is spinners of the c-spear because the c-spear is a group therefore parallelizable trivialize the bundle but you do it in two ways by the left or right translation so you shouldn't confuse them And so it is S3 plus R4, but you think of it as spin 1 plus R3, S3. Now, it turns out that the metric on this one is geometric singular, but in fact, you can now deform. Sorry, please think about it. By filling in these S3s and R4, I get a smooth manifold. this smooth manifold the spin bond with s3 does in fact have g2 metrics this has been worked out by gary gibbon and his collaborators a number of papers the nice exact very explicit formulas of many mathematics of t2 home only on this manifold and a few others as well these are simply gone and they have very high symmetry because they're very on the large continue as well Now, when you've done this, you've got a manifold of G-2 holonomy, so in principle, it has a large symmetry group, so you pick a circle somewhere, and you can factor it out by the circle, and you get down to manifold of division 6, and you can say, aha, I have a type 2 day string theory on this 6 manifold, which is inherited from the 7 manifold of G-2 holonomy. But if you've got a lot of choice, there are three different SU2s that act on this thing. And if I eat SU2, I could take a U1. So I've got three choices of U1. I have three choices of how to put them in.

37:30 I've got quite a lot of choices. So, in fact, we'll get three different six-manic goals. And these three different six-manic goals, which will—now, first of all, the observation is, you can divide out by a circle, you're supposed to get a string theory. But the other circle you divide out by, you're supposed to get an equivalent theory, because a theory, really, is supposed to depend on the 7-manifold, and not depend on any symmetry that you choose to divide out by. So, dividing out by a circle is a way of identifying the string parameter of the string theory. And this string theory with this string parameter, that string theory with that string parameter, will both be the same as the theory M-theory upstairs. different circles would just give you a call of theory here you can divide by three different circles and therefore you will get three different theories from one even theory about and three different things you get will live on three different six manifolds and three different six manifolds are in fact well they're the one that's talked about the quadric which is the perturbation of the estimation of the cones that someone said square equal to the constant r are determines the ages of the three sphere and then there are two choices what I call a small resolution why one might do these three x y one y two are at x three different six manifolds and they will give us three different string theory or equivalent why one might you look rather similar so just bother with this much but x and y one will look very different so this is that the idea is that you get very cheaply because of these open-toe string theories by going up seven dimensions observing for the two different circle actions one of which really has one property the other which is a different one and the key thing is what what is the difference why are they the difference I'm going to explain it now. Well, here's the observation. So, we start off with this cone. On this thing, we have, I think, with three different, quickly-group or, many-group or three, and symmetry. And there are three different ways of smoothing it up. In other words, filling in one of those three spheres. those will give you three different seven man I'll call them red while they do that three and that the three choices of you want something to take because the

40:00 three copies of sq2 acting you want the first factor third factor so I'll get mine possible answer he takes why did I get six man six man four then I did I is our seven magnitudes, I've got three of them, I divide by three possible U1s, I get another set of three, I get ZIJ, six magnitudes, indexed by IJ. Now the geometry is quite different, depending on whether the indices I and J are the same or different, it's very easy to see what's going on. So let's suppose that I equal to J, so it might not take, but it's equal to say. In that case, you see, well, I'll show you exactly in a moment the details, but basically in that case what happens is the U1 action has a fixed point on the semantical. The fixed point, you see, will factor in three scales. So when you factor out by the circle, well, you might think you're going to get a singularity, you see, when I talk about factoring out by a circle, maybe speaking, one thinks of just a nice circle, you just sort of forget about the other variables transverse variables if the circle has a fixed point and i mean the circle length is varying shrinks or point somewhere you might think that gives you a nasty singularity but if in a situation like this where the thing that your the fixed point set has dimension three in something in dimension seven then in the transversal four dimensions what you actually have an action the circle in our core which is like the action of the unique complex numbers on c2 when you put quite a cushion out that by that circle actually what you get is r3 what do you get a reading the origin one of the singular point but a direct monopole that's the groups of climb interpretation of direct monopoles it's just the action of a circle to the If you do it in, not enforcement, but in seventh space, so that the four dimensions are just transversal into a three-dimensional thing, then this monocle, so I think, lives along that whole c-manage, and that's the brain. The brain is just a Dirac monocle jacked up in dimensions. So, the fixed points give you brains. The right dimension, the right place. But if i is different from j, it's easy to see, and I'll use the algorithm in a moment, then u1 has no fixed points, so there are no breaks.

42:30 So when you get down below, these two different cases, one of them has a, in this case, is faster x, the other is our bread y. In one case, you have fixed points of the brains, and therefore you get raised over strings. In the other case, you don't have brains, and you get raised to close strings. So the mystery of y, sometimes you know a brain, You can reinterpret it geometrically by saying sometimes you have an action with fixed points, and sometimes you don't. And here, the geometry is so simple, you can see exactly what's going on. So that would be my contribution. Collaboration. And after that, you'll just interpret all the things. Well, here are some formulas. Just to show you how you can get it. It's convenient to you pleternian, the formulae. So the writers of pleternians, I just think we have i, gain, k, satisfy the dual rule, i, and the end of the root, i squared minus 1, and so on, i, j, b, and k. And then you can write that also as a set of four real components, x1, x4, to the pair, there's two complex components, set 1 to j, x2. The norm squared of q is just some of the squares The elements of norm one form a group, which is just a group question two, and just a three-step. We formed this manifold, six manifolds, by taking three copies of G and dividing by G on the right. So what we're doing is taking three copies, three unit quaternions, and then as triples, we call two colons, if they agree by right multiplication by a thousand, fixed unit quaternions. That's the way you think it is. Now, to define, to fill in one of the three spheres, all you do is you remove the restriction. Let's say we'll fill in the first one. You remove the restriction that has absolute plus more one, just to raise that's arbitrary. That's essentially filling in the three spheres. Make it bigger. So you just, then they are obtained by the same formula, but just replacing, instead of q1 equals one, absolute value one, no restriction. you put equal to zero. Now that q equal to zero will be the c-sphere, and now the actions of the u1 are simply just given by multiplying on the left this time by a complex number

45:00 in the j component of absolute value one. That's the action. Now you can see that if you take i equal j equal to one, then q1 equal to zero is fixed point under that action, Obviously, I'll divide zero by complex number of six. So if you put it in a different place, then there's no zero, and that one doesn't happen. So it's all very . And here I'll just observe what we have is six-point action really the model of that is we exclude the prime picture of the Monobol, which is very fundamental. Okay, so that's really meant to explain why it should be true that this closed string and open string are really equivalent theories, and we already knew that the open string was the same as the hybrid theory was a whole chain of equivalent theories, going all the way across. Now, if you want to say a bit more than that, when I said that these two theories are equivalent because they both got by a single theory by taking two different circle axes and dividing two theories, one's an open-string theory, one's an open-string theory, so they're equivalent theory, okay? But you can do better than that. You can actually, each of these theories has a complex modulus, a parameter, and the aim is to show that actually as you perturb that, you form that parameter, one part of the parameter space to another one, you will You can move yourself from one theory to the other equivalent theory. And that's what—that's—that's the more delicate arguments that will show that. So this is how it's supposed to go. And so the idea is that in this M-theory, there is a complex modulus, and the—the symmetry group of the whole theory, of the whole phase. So this should be a symmetry group of the theory. group of the geometry so that tells you but now you have to argue what they do at the quantum level there you have to be careful I won't give all the arguments there basically can show that it acts on the quantum theory and then the quantum theory is meant to be dramatized by the one complex variable modulus is this group acts on that nice way that by a hidden point around and there'll be

47:30 points in this complex modulus where the around which the perturbation expansions are performed and one of them is one theory one is another theory one is a third theory so the three different perturbation patterns are really simply expanding the functions in terms of complex parameter around three different points and formulas the rate one with the other are in fact the formulas essentially at the beginning of after four minutes later the parameters so you carry this through argument so you can show these are different phases of the theory so the string theory has a complex modulus and two or two sort of one of you get by X where the parameters of all these will cease here the other Y was going to choose here and energy communication taken on the other this is again very familiar to physicists in the variety of different examples of bubbles humanity people are called single models to land I get good attention so now I So far, in that geometry, I forgot to say anything about the role of n. If you're observant, you'll notice that. I skipped over that. Let me come back. See, geometry talked about the seven manifold divided by a circle, getting a six manifold and sometimes there were six points in the brain, sometimes there were no six points. That corresponds to, if you like, to case n equals one. Now, if you have a large n, if you have a large n, What do you need to do? What do you do? All you do is take the fixed subgroup of order n inside the circle. The first divide is 7-manus, okay, by the fixed subgroup for n. Again, there are three choices where you take the fixed subgroup, and you call ZI, gamma, J, the different choices. Now, again, this, the same argument as before, this finite subgroup may have fixed points or may not. If it has no fixed points, then you get a nice free action. If it has fixed points, you get a singularity. So what kind of singularity do you get, then you get one of the famous singularities that are called ADE singularities, which are, I mean, they occur, first of all, in two complex the dimension when you divide by finance I'm going to get water for the ANC

50:00 corresponds to group SU in this case we've got the singularity type a minus one in the transverse direction for dimension to the to the brain and so this would you factor out by the thickly group you have a seven manifold with a am type singularity and then we divide that by circle later on now the singular point comes brave but now it requires the multiple charge in because number appear naturally because he's divided by the thinking something there's more of a story what happened in private other groups that's also being looked at that's how the the n appears but in the other side of the story where there's no fixed points then a lifting point well again the n has to come somewhere what is the end now is to produce this the flux on the manifold. The job appears in a different way. It appears because it's over this manifold, the tristing of what you're comparing, which ends, appears there. So it comes to the job, end to end, survive on both sides, but it has a different meaning. Well, let me move back now, and really being a bit on what I promised to do, I thought I'd tell you what small resolution is there. Well, a small resolution, obviously, in contrast to a big resolution. So let me just review for you. If you let me equation some of the squares that are equal to zero, thereby writing it in all complex variables, in two different complex variables, and four different variables, you can rewrite it as a product. x1 x2 to xd x4. I've got a gain, right? x1 equals x, then 1 plus i, then 2, and so on. Now, this is factored on it, as well. And now, if you take the equation we've got up there, and think of it as a homogeneous equation in three-compensate variables, you

52:30 find the equation of a quadratic in the complex of a C-spack, a quadratic surface. And a quadratic surface, one singular quadratic surface, is actually the product of two copies of its design. You know this very well in real geometry, where you have hyperboloids of one sheet, where you have generators of one system that comes from that factorization. So, now, when you resolve a singularity, a vigorous loop is what... Every algebraic singularity can be resolved, I thought. And depending how bad it is, resolution has to go into higher order terms. personal return between you all you do is you add the directions approach the point approach cone on two different directions in the block resolution they call different points that's all and you do that you get what you call the big resolution big resolution replaces the verdict of the singular one find back the based with the product of two copies of one so here's my picture here's the y zero then this is why I have a bigger but now you can factor that in two different way even the two smaller collapse one of the CP one so that so get the man before we got the other one these are two halfway stages of the full resolution and so choice either way these are all different now the big resolution that exists in all dimensions for this four resolutions of peculiarity of mention for only in that dimension doesn't ready factorize very very special phenomenon it holds this dimension and have a nice properties and it is very important for the Calabi-Yau manifold, because when you resolve these singularities, you can reserve the Calabi-Yau's condition. Now, this is a general comment here. This analogy of the correspondence between closed strings and open strings is meant to be somewhat similar, in fact, to the singularity of the Maldacena, between the antithecita And you can see very good, very aspect of reality here,

55:00 because this is a simpler model. That one had something in common with it. I should also say that the word I have described so far is, back a year or two, there's more recent work due to Kniegdrasse Frapper on major problems, and very kindly think about how they explain many of a lot of this sort. So this is, I haven't brought the book up today, but I don't understand the straight thing this works, but there is very kindly work going on right now in the last year. So here's the direction. But here's the older, Mr. John. That's Baffer writes infinitely many papers. When re-normalized, this is one of these many papers, which is an interesting one, it contains a lot of the . The Gates Theory of Geometry means open-stream splinter. Then the paper, I wrote Malazine and Bacchus is a short paper that was there. And then I wrote a much longer paper, which is the analysis of the epithesis is not bothered much by the stripe type, the string theory type 2, but sticking with the theory in seven minutes on the Manichold and a few others, and looking at the various finite the stimuli that's arrived, and the way that you very analytically, one is lacking from another one's log. That's much, much more substantial. But if you take the log model that we're talking about, and it's what you're doing with the other one, but in particular, this paper here is that . It doesn't mention the symmetry of all three that I was talking about. We take it with great help in particular problems. Now, let me just go back to the TV analysis. I started off by talking about the Joseph polynomials. That's where we started. We talked about the Joseph polynomials. And then we went through this long chain of correlations.

57:30 It shows the Joseph Bonnevra could be reinterpreted in terms of Chern-Simons theory. The Lakers could be reinterpreted according to Witten in terms of open-string theory. And then there was this last step over in closed-string theory by this argument for Claffer. And you follow. Now, if you call it these two theories, I mentioned that Claffer had calculated on both sides the partition function. But the partition function is the Chern-Simons theory. It's not really the object. It's calculating what happens when there's no knot at all. There's no observable, just not even much. The interesting thing in the domain there is are the positive, the observables correspond to the knots. So we know what they are on this other side, in terms of the Joan Simons theory, and what do they correspond to on the closed-stricted side. And so that makes some speculative remarks on this side. they should correspond to counting polymorphic curves in this three-dimensional which have some mapping in there which have some boundary behavior infinity where the surface takes it obviously not these very wide and not infinity look how many services there are human geniuses on three side the numbers, and that's the data, which will correspond to the thing on the other side. And there's some work that started on that. Cliff Howard has done something on the mathematical side, setting up the fundamental analysis that shows this looks plausible. It's a possible way that you can make sense of this geometrical problem and try to calculate it. So, at the end of the day, you end up with saying, we have a purely geometrical way of linking the Jones polynomial, which can be defined at the end of the day without reference to quantum field theory, to somebody else knows, right, it can be defined as a curve, which again, without reference to quantum field theory or swing theory. So if you don't like physics, you can forget the physics and say here's a beautiful theorem that says you can calculate this, not invariant, this way, or you can do it that way. and there's all this marvelous philips preaching faith and going on. Anyway, I think there are marvelous, lots of things in philips of this kind. Literature is enormously rich.

1:00:00 And I particularly consider all the individualities. But this one I picked on particularly because it has quite a lot of features. And many of these features are, in fact, typical of other. So this is not unique. But because it has a particularly interesting topological aspect, the topologies of these not invariants are extremely instinctively solved, unsolved problems, and they're very difficult, not thinking problems to solve. They're very, very substantial theory. They're very interesting to apply these ideas in that case, where, not only without instinct, but they're incomputable. Every, you know, theory is totally well-defined, you don't have any infinities to worry about, it's all topological, and yet you have the framework of common field theory in your form. And once you've seen the topology going, then you can believe that the corresponding results and coolances between these theories will work also for the non-topological theories. Because I told you that these theories of strings moving on Calabi-Yar surfaces, Calabi-Yar threefolds, I talked about the topological theory. That's the twisted version of the usual, the usual physical version is kind of different theory. You change spins of various fields, and you get a twisted theory which is topological. But keep to the original one. And this equivalent between these different theories, by going up to seven minutes and down, is still to the valley. written didn't mean about topological theory at all but topological side is nice we've been focused on exact results our facts then you just think of it as highly a toy can be used as a model so I think I'll stop there and cover a lot of ground, but I thought it was going to be all of you for the subject of this. Thank you. What's up? Thank you. I have a question. And afterwards, we have wine for about half an hour, so anyone who would be interested especially in speaking to the speaker, more, we can talk to him. And it would be in the room in that end, two doors down.

1:02:30 So maybe one or two quick questions, and then we can. Everyone just breath more wine, or...? Okay, so let's thank Michael again, and let's move on to the wine, and anyone who would like to ask any questions. Thank you. Thank you. Come on. Oh, yeah, that's a good morning today. No, I was supposed to talk to you when I'm on the seminar. You haven't seen yet my iPhone? No, no. Okay, all you want. I'll get it all fine, y'all. I'll get it all fine, y'all. I'll get it all fine, y'all. Okay. No, I'm not on the back. No, I'm not on the back. No, I'm not on the back.

1:05:00 Yeah, I'll do it. Yeah, I'll do it. Okay, good night. Thank you. Thank you.