Mass & Curvature

0:00 So, let me start by telling you what I'm going to talk about. The title is Mass and Curvature. It's only majoring into strings and pistas. But it's an old story, it's been going on for quite a long time, but there have been new developments. Now, you use an area which lies at the kind of crossroads between mathematics, physics and computation. and the link between all these things, I called it intuition, that's the flight word, whatever it means guesswork. Now, background for those of you who might be finding out more about the area, is a nice book by Nick Manson and Paul Sutton, called Topological Solicons, which came out last year. I can recommend it here strongly. It's a nice opinion for the whole area of this sort of field. Now, more detailed outline of what I have to cover. I'm going to talk about the Skrm model of the nucleus, something introduced by Skrm a long time ago, long before modern theory came around, but it has sort of been given respectability in the current framework by being shown that it arises some kind of limiting approximation to the real physics. Secondly, in this, I'm going to explain the analogy between these of us. Skirmions and the corpheria monopoles, which appears to be totally different fields, but somehow it's easier to have an analogy between them. That's already due to Manfred's work. Then I'll explain the relationship to Yang incisodons, Celsius-Yord-Yang-Mills, which Manfred has worked out some years ago. Then I'll talk about the computational results by N-Skyrmion. N, first, the number of particles, a single skirmion, one skirmion, then there's N of them. And what these things look like, the solutions, this is a result of work, really, of... there's Matten, then Sutcliffe and Batty, which involves lots and lots of very clever computations. That is the very old part of the story. The new father's story is a bit down the bottom, which the skermions originally studied have an approximation to the theory where the pions in Shapiro's theory have a mass equal to zero. The pion is small, therefore the thought generally doesn't make much difference if you study it with mass zero or small mass. But in fact, Patti and Sarsley have shown that studying the small mass parameter, you get significantly different results which are other interesting.

2:30 And then, after that, I showed that this has an infinite connection with something that's going on in anabolic space, instead of what's going on in a reclimate space. And finally, I also try to come back to the relationship to his and so on. So this is a general outline of what I've covered. I'll review, first of all, these substances that were known for the wrong time. And in the second part of the talk, I'll concentrate on newer results and some speculations and so on. Let me start with the skirmion themselves. So a skirmion is a soliton in modern language. And the skirmion model, as I've mentioned out here, is a non-linear theory of pions whose topological soliton solutions would give effective descriptions of nuclei. A soliton number, which is a topological number, is identical with a bagel number, protons and neutrons. Now a skirmion field is a function that is, I'm just going to describe simply static And there are functions on c-dimensional space we found it in the group SU2 and it's required that at infinity the boundary condition is that the matrix becomes close to the unit matrix. So this means that you can compactify this map last week to SU2 into a map of a c-dimensional sphere adding a point of infinity into SU2 which is also a c-dimensional sphere and therefore you get a topological degree and that degree is a topological variance of a field Now, I'm hoping you'll finish this. Pions are usually a set of three fields, and they're racially between the U-map and the pion fields as written by those formulas. You write U as a sigma times a unit matrix, that's the real parameter, plus I times pi in a product. Tau is the power of Hc, pi is pi 1 by 2 by c as a pion field. So, the reading C-dependent infinite functions, because once you know how it lies, sigma is given by requiring the sum of sigma squared, and it's given by a formula that they lie on a sphere. So sigma is the amount of non-linear constraints of sigma. That's what is going on is. First, the model is how approximation, what a practical picture of a nucleus looks like. No quantum mechanics in this. Now, in static solutions, there are minimum energy solutions of probing energy functions.

5:00 The energy function, which comes with a little realistic Lagrangian, but this is just the energy part, is an integral of the space of three terms. The term is the quadratic, E2, the term which is the fourth degree, E4, and then a mass term, m squared times the trace of one minus u. The e2 and e4, there can be sort of a following way. If you think of the differential of u, that's the linear approximation, it's a map of the tangent space of R3 to the tangent space of the R3 sphere. First of all, these are metrics. So if you call t that map, then you form t star t, which is the self-adjoint operator, that eigenvalues are positive, we call them lambda squared. And then the e2 is the sum of the eigenvalues squared, and E4 is the sum of the products of squares. E2 controls length, E4 controls area. And in three dimensions, length and area are something complementary, not dual, so you need both of these to get a well-defined function. And this is essentially used by square. The mass is a small parameter which is in the first approximation for the first part of my lecture. We will put that equal to zero. We dropped that. There are, of course, suitable physical units, which I won't mention in a moment. I'll come back to it, but there are also mathematical constants like 2 pi and 16, I think, which we'll ignore with the first specific x statistic. The important thing with this dysfunctional is that e2 and e4 have different scaling properties. Under the change of scale, if you replace x by mu times x, which is a vector of r3, by mu times x, which is a real parameter scaling factor, then the reference of the volume goes by the cube, then E2 picks up a factor of u to the minus 1, and E4 picks up a factor of u on top, so they are different, they behave differently, and so if you like, one of them is attractive, one of them is impulsive, and so you will expect them to balance out if you expect to get solutions which minimize the energy. Of course, you have to boost man's value of theorems if you learn from the triglyceride truth. There's an element between inequality, which based on this inequality on the eigenvalues, which just imply that the energy is bounded below by the absolute value of the barrier of number. So in a given regime, or a fixed barrier of number, then the energy is bounded below by that. That equality is never achieved, because if it's only achieved when you have the mapping isometry, you can't map R3 isometrity onto the sphere.

7:30 If the initial space had been in C-sphere, then the unit map would have been I mean, one gives you equality, but otherwise there's a strict inequality. But you get quite close. Well, the invariant density is, of course, given by volume, which is given by lambda 1, lambda 2, lambda 3. And that's how you get the inequality out of that. So that's the basics of set-up for the It's important to solve a nonlinear partial differential equation minimizing the energy function. It's a complicated set of equations, and it's meant to give you some kind of approximate picture what a Newton's test looks like. Now, the general fact about this. First of all, it's an artist equation that you can't solve. And there is no analytical solution in this equation, even if you require spherical symmetry. If you put spherical symmetry on, then you get down to an ordinary differential equation, and if you can effectively solve numerically, then you get a good description of the solution. At 80 plus 1, that is, if you put a single That's the basic object, one nucleus, a nucleon. But for n gradin equal to two, if you try to find static-symmetric solutions, you'll find them, but then they turn out to be not stable. They decay into things of lower edge. So, the right solution for n gradin equal to one is not given by a static-symmetric one, and there's an analytical formula, you're stuck. Now, this is really a major problem, how you tackle. One of the interests of this problem is that it's a very interesting problem of a typical c-dimensional hard soliton-type solution, which you have to find methods to solve, and finding that it requires a lot of ingenuity is ever a good kind of test problem. Now, how on earth do you go about finding solutions in general? Well, let's say that's equal to two or more. Well, you can, of course, try to use brute force computation, but in nonlinear problems, which involve multi-dimensional story, just going about it computationally, you don't have any insight into what the solution could look like, is a really helpless task. You either get no where, or you get the wrong answers. You know, you think about it, you're trying to have a very complicated nonlinear space, nonlinear-functional, where it hurts is the minimum going to be. You can try some of it and then you improve it, but that may be way off the wrong part of the space. And the numbers are so large, you can't sensibly do any kind of numerical approximation. So, you can't

10:00 just go about it by brute force. If you do, in fact, people who did try, got the wrong answers. You need something a bit better. So that's what makes it interesting. Now, Now, one of the insights really was, and this is Nick Manton's contribution, Nick Manton had the idea that there was some analogy with monopholes. Now, let's digress a moment and talk about monopholes. These are what are called BPS monopholes. Now, there he was mentioned by Richard Ward in his lecture. The BPS monopholes have various groups. This is the group issue 2. There's a gauge field, there's a Higgs field 5. The Higgs field is required to go to norm 1 of infinity. The equations are written down there, which describe the absolute minimum of energy. I haven't written the energy down, but I can write it down. Here are the Bogomolny equations that say the covariate delimitive of the Higgs field. The Higgs field is in the adjuvant precipitation, with effect to the extent of A. Here's the dual of the field strength, the curvature. These Bogomolny equations, they actually achieve the minimum bound. If you solve these equations, you get the exact topological minimum. Now, the boundary condition, in this case, is that the phi, which is in a three-dimensional space for the algorithm is two, goes to one infinity to have a line in the unit sphere. And so the ascensonic data, the third value of five, is in fact a map of the two-sphere of infinity, R3, into the unit two-sphere in the Lie algebra. So it again has a degree, and that's identified with a monocle number, or essentially with a magnetic charge in the right units. And these are meant to be, these are called monobillion monocles, they're at large distance in direct monocles. When you come in, certainly any point in similarity, you get a nice smooth solution. That's the beauty of solids. attention, because they are close to that. This is a simplified model of the more complicated equations you can write down to extra terms in them, but the advantage of them is that they are highly factable and difficult. These equations can actually be solved in a very precise sort of way, and there's a very beautiful theory of these, which really relies on Twister methods. And that's really where Richard Ward into that. Twister methods, which are part We do tick in there. These are able to solve these equations in a remarkable amount of information.

12:30 In all information, it's hard to actually write down the solution explicitly, but what we can do is find out the parameter space of solutions. These solutions for the monopole problem don't have unique solutions, unique energy for a given charge. They have a multiple parameter space. Because these monopoles, they can sit, you can have two monopoles far apart, who don't attract each other or repel each other, they just sit there. So the parameter space, happily speaking, has a number of parameters corresponding to the locations of the n monopoles, the single units, together with some phase factors. So, of course, that's no longer true when you bring them together, they start to interact. The first approximation, that's how you see the dimension. So the modelized space, or parameter space, is a magical dimension of 4F. And this can be actually explicitly found, and it's a very useful description, and the theorem of Donaldson identifies this space, rather naturally, with the space of rational functions of one complex variable of degree F, the ratio of two polynomials. So, very, very explicit, and the parameters are basically the coefficients of those polynomials. So we have to know a lot about this space. And Tristan methods are the one behind it. In particular, for example, from this you can find out what monocles have particular symmetries. You may find monocles which are symmetrical in some way, rather nice. And here's some surprising results you find. For example, when A equals to 3, you might expect to find things with triangular symmetry, forming equal-actual triangles. But surprise, surprise, you find one which has tetrahedral symmetry. That's a big surprise, showing you that the interpolation of these things as particles, it becomes very difficult to justify when they start to change their shape in this work. And this is what I said as an example is typical in a work. I'll give you some pictures later on. So, this is a summary of a lot of the work which went into monopause because there are explicit solutions given by Tristan methods. Now, mantons, intuition, guess, whatever you like, was that somehow, monopoles and skirmions are both, you know, similar. Why should they be similar? Well, they both have some physical origin, they both have two-dimensional space, they're soliton solutions or something or other, you know, they're particles that look like fields. Oh well, you know how to solve the monopole problem, perhaps if you look hard to remember that,

15:00 might use an idea about how to solve skirmions, which you can't do explicitly. This was his idea. Well, he tried to persuade me about this for a long time. I was quite sceptical. But, what actually happened is the following. The skirmions, unlike molecules, attract each other. Put the two together, they can come together. They don't stay apart. And so there tends to be a unique solution up to rotation and translation. Least solution, least energy stable solution, which we're trying to find. Now, monopoles are different. They have moduli space. But perhaps the idea was, amongst all the different monopoles that are given charge, there's one specially nice one. Perhaps one which has high symmetry which is somehow the one that will give us the guess for a skirmion. Oh, a pretty vague idea, you might think. Why a nursery would that be true? Well, you know, in the absence of anything else, your guess is good and mine and, you know, the proof of the pudding is whether it works. If you want to make a guess, it may be justified by all sorts of punches in the background, which are totally, you know, ridiculous, but if it works, then no. So, it turns out that this can be verified computationally. This is actually true to a remarkable degree that, for example, n equal to 3, the skirmion, at least energy, appears to be, you know, you can't prove all these things rigorously, but by numerical calculations you can make very convincing arguments, The scourmion, barion number 3, has tetrahedral shape. I'll explain to you precisely what that means. And, for example, also the one with N equals the cell, even more beautiful, looks like a machahedron. So, these are very, very exciting things, and these shapes are shared by monopoles of scourmions. So, there you are. If the guess is justified at the moment, we have no insight into why there should be, I think I can tell you why that comes about. But at the moment, these are a puzzle. Some way to come up with a guess, you try it out numerically, you find the answers, that's enough. But that's all very well for symmetrical solutions. But symmetries, finite symmetries, by symmetries of the regular solids, are rather scarce. There are many symmetrical configurations. High symmetry. So what do you do? The general value then. You need some more general approach. Just that the symmetrical ones is not by itself, yet you can't really far. So here there's another approach. This is also very varied by the analogy with monopoles. It is an approach put forward by Battling and Sutcliffe.

17:30 Now, this one looks even crazier. It calls the rational math ANZACS. If you have the word ANZACS, what it means is you want to guess from the shape of a formula, you plug it in and you fiddle with the parameters to prove it. And if you have a nice guess, it's a starting point. So this is the idea of the following. If monopholes are somehow like scomions, or scomions are like monopholes, and monopholes are parameterized by rational mass, then perhaps we try to construct a family of scomion fields parameterized by rational functions, rational mass. And then amongst those, we'll try to optimize the parameters to get a good solution. Well, okay, so why not try it? So what they do is they start off with an arbitrary rational function, which can be thought of, of course, as a holomorphic map of the two-sphere to the two-sphere, of a given degree, that's the degree of the rational function, of course it depends on a certain number of parameters, 4M parameters. But one thing you can do when you make the map of the two-sphere to the two-sphere is the rational map. What we want to get in the skirm field is a map of the C-sphere to the C-sphere. But it's very easy to do that if you do what's called suspension. You map the equator to the equator and map them back to the above and below the stringy copies. So you get the C-sphere mapped to the C-sphere simply by repeating the construction along the two-sphere's different angles. But you can do a bit better than that, you can rescale the longitudinal parameter in an arbitrary way to re-parametrize that and give you another degree of freedom, a certain function. So now you have that function as you really call a profile function, and you'll try to choose that numerically. And now here you get around that, you plug in an arbitrary rational function at the degree end, you make this suspension with a three-parameter in it. Now, first, you look at the energy compute the skirm energy of that configuration. First of all, you have the dependence on the profile function. When you fix the irrational function in the background, you try to minimize the energy for that profile function. That's really the straightforward dimension of computation. Having done that, you then try to vary over all the parameters. There are four n real parameters, but only finally many, so you can try to vary along that, or a bit more guesswork, to get your solution. This is what they call their rational map analysis. And you see, you're taking the analogy with monopause a lot further, using it to make this wild guess, and again,

20:00 it seems completely unreasonable that it should work, but they find it done. And they find solutions, qualitatively, very, very similar to monopause for all values of A that they computed, as if their computations go out to like A equals 22 or thereabouts. So, it's a remarkably successful story, although at the moment, when it was first introduced at least, there was no really rational explanation, except in very vague ideas. But you couldn't dismiss vague ideas if you think they were. So now let me show you some pictures. These come from their papers. Here's a picture of what we mean by something being a tetrahedron or otherwise. There is a tetrahedron. What these pictures represent is the energy density. This is the region in space where the solar zone is mainly concentrated. It has a tail everywhere else, but in that region most of the energy is located and it has these shapes. You see there's not much doubt. This thing here you can justify calling it a tetrahedral shape. It has a symmetry of tetrahedron. But this corresponds, as I told you, not to charge topological number 3, but 4 was 3. And similarly, the iconesian, which is somewhere over here, corresponds to 7. So these are not, the numbers aren't what you expect them to be, but the symmetries are quite clearly symmetries of regular solids. And these pictures are, well, actually, it doesn't matter whether this is a skirmion or a monopole, each is the same. All the accuracies of the representation, these are the same pictures of monopoles and skirmions. So it's very striking. The evidence is quite strong. Well, those are the first few which you get from symmetries. Now, if you jack it up and apply this fraction on that axis, you've got higher ones. Here's a longer list. And you can see that consistently you get nice shapes. On the left-hand side, you get the same kind of pictures as before, showing where the energy is located. On the right-hand side, there's a kind of skeleton framework indicating the combinatorial structure of the underlying polyhedron, network on which this thing is built. This begins to look very much like a chemistry you find in the lab where they join, you know, they make molecules by joining them. And in fact, the buddy ball appears somewhere in this picture. So, it's very nice, it's semantic, and there's not much doubt that these things look like that. Here's some more. And they go up to, I think, 22 is the last one. So they're clearly nicely

22:30 constituted shapes. These correspond to these, as far as I can tell, the least energy that's going on in this particular topological regime. And they're the same pictures as they are of monopoles. Inside the monopoles, there is one that looks like this. OK, well that's it so far. I did say something about this is the relationship between monopoles. I mean making the analogy with monopoles. But I did say there was also a risk given instantons. Now, this is... I'm trying to explain that. Forget about monopause for the moment. Let's look at instantons. Instantons are solutions of the cell dual-Yang-Mills in Euclidean 4-space. OK? And we know a lot about them because, in fact, the cell dual-Yang-Mills equations, like Fischer's methods, as we've heard from Richard Ward, can be solved as one of the beautiful results. So we know, again, that there is, again, a modulo space, a parameter space, Now, if you have an instance on field, whether or not it satisfies the equation with any gauge field, so long as it has, by the way, included in this that it's a bigger infinity, the gauge fields have to decay at infinity, so the thing can be compacted by the fourth sphere, and that's where you get your instance on number from. Now, if you take an instance on Skerner, sorry, field, not necessarily self-field, any field, any gauge field, but having likely k infinity, and you do the following, divide up r4 in terms of r3 and r1, to separate r2 like the imaginary time variable, and then take parallel transport along the r direction, as if for each point in r3, consider the transport from minus infinity to plus infinity along that direction. That will give you essentially a group element. A group element varies as you move along R3. Because of the decay, if you go off to infinity in R3, you'll get the unit element. So what you get is a Scurm field. You'll get from an instance on R4. By integrating out the one variable, you'll get a Scurm field in R3. No equations yet, just pure three fields. And moreover, you can check the topological numbers correspond. The incidental number corresponds to the varion number, the skirm number. So this is actually mathematically well defined. It's not a guesswork, it's not. And in fact this relationship between these two spaces topologies is very nice, because if you look at the space of four

25:00 gauge fields in S4, modulo gauge equivalence, that space has the same fundamental topology, same homotopic type, as the space of the skirm fields, These are two very big function spaces which have the same fundamental topology. The very first factor of that is they have the same number of components given by the integer topological invariant. But much more than that, they have the complicated internal structure which is the same. These spaces look very similar in two topologies. And now I'll come back. Well, so let's leave it. So that's a good start, mathematically. Although instantons and scionions are quite different physically, forget about the physics, let's do the mathematics. So now having done that, we now restrict this construction, which is tied up in any gauge field, to the only those that are self-dual. The self-dual ones are a form of five-dimensional parameter space, or dimensions, 8N. I've written 4N to please. And so you can, each point of this parameter space, you can make this construction, you get a finite eventual space of potential skirm fields, which cuts down infinite numbers to finite numbers. So now you've got to, if you can look, justify looking only at these, you're in the business to make some kind of better approximation to finding a solution. And surprisingly, this gives good results to the real solutions. This really now looks like an effective way to try to get skirm fields from instantons. You first start the instantons on equations, you push them over to the skirm fields, then you minimize the skirm energy and hope that you've got the right answer. And it turns out the answers are very good approximations. Now, this is actually a very mathematically very well-justified approach, namely the very good reasons why this should work, because the two spaces look very similar. Let me digress a moment. If you're in fine dimensions and you have some manifold, you get a complicated topology, and all that manifold, you have a real manifold function, and you look for its critical points. Then it's well known that the critical points reflect, to some extent, the underlying topology. The number of maximum critical points of various kinds correspond to the homology of the space. And if you have a nice manifold, geometrically nice manifold, for example, Cobbry-Betty's face, or Grassmannian, or anything like that, and you write down a function which is somehow intrinsically related to the geometry of the space, not simply an arbitrary one chosen randomly, then, you know, very likely, you're in luck,

27:30 and the number of critical points is no more than ought to be, and they're, you know, well-behaved and you can find them. So, if a space is more defined, a nice geometrically defined space, and you have a nice geometrically defined function from it, you may hope to get some closer relationship between the topology and the critical points of function. Since the two spaces we're talking about here, the gauge fields, the marginal columns, and the skirm fields do have this fundamentally similar topology, it's therefore not unreasonable, and since both functions, the Yang-Mills functions and the skirm functions are geometrically nicely defined, it's not all unreasonable that their sets of solutions, the minima, should have something in common. So that's to make this a reasonable approach. You have to have a rationale. And that is, although one solution is minimum B and the other one is a magical, that's not unusual either. For example, if you take a complicated space, and you write down the simplest kind of function on that given by a sort of matrix, if the eigenvalue of the matrix is all distinct, you will have n isocritical points. But if some of the ones are equal, then you will start to get flat subspaces where they can start. Or think of another example, think of a torus. If you set the torus like a tire on the road, then the height function has four critical points. The minimum, two central points, and the maximum. But lay the tire flat, you're trying to fix a puncture, and you get a one-dimensional circle where it cuts you in the ground, and another circle on the top, but it's maximum. So these two circles follow up some of the critical points. That's typical. So when you see a space, or two models of the space, and two functions, a critical point for a minimum. The other has a large manifold of the minimum. It shouldn't be put off. That's what sort of thing you can expect. And that's why you should expect something like this to happen for the models of these two columns. Now, this is fine, and it's very good. Unfortunately, although the instance of modernized race is known, everything is complicated to work with, and therefore has a computational tool, not very easy to carry this out. Now, you compare this in the case of Because in the monopoles, we have no rationale for our guesswork. We made a stupid guess. On the other hand, the computation is much simpler because the monopoles are basically given by a rational function. So it works. So we have this choice. We either use monopoles by analogy that physically look similar things, but there's no justification mathematically. Or you can do different ones with good justification mathematically, not physically, and computationally

30:00 is not so convenient. It's an unfortunate conundrum. Which one should be used? Well, we'll come back to that. Now, we now move on to the new story. That was the stuff that's been known for some time. Now I've moved on to the new stuff because it has to do with what happens if your pions are mass. So, you can go back to the Skelbeons now to get everything else. The bad insights have found that if you take a non-zero pion mass, they found by computational methods that the results you get are actually positively different. What you get is the mass of zero. Although the mass is small and you think it makes a difference, It turns out that you actually get different results, moreover. The results are, in some sense, physically more realistic. They're closer to what you expect in the model of real physics. For example, if m is zero and n is up to about 32, the picture I gave you, what you're getting looks like a shell. They're all the things that are roughly around this polyhedron, which is spherically shaped approximately. And there's a hollow interior. But if you put M0 and the small value of M, an increasing M, then you'll find the shell starts to collapse. And you start to get a kind of three-dimensional picture with material inside. And for example, if M equals to 5 or 8, they've shown that there's no stable solution to the shell, they tend to break up into things of barrier number 3 and 2. This is more in agreement with the real physics, so no, table barons, barons number 5 and 8. But now, we've lost any attitude of monopoles, because we've introduced this mass term, and although before we had mass term, we had a nice relationship with monopoles, this mass term, we don't know how to interpret in terms of geometry. So, it's relatively more realistic, but what a pity we've lost the geometry. So the question is, can we do anything about it? Well, first of all, to show you, here's a picture taken back in Sanskrit's paper. This is a picture of barrier number 32. And when there's no mass, when the mass is zero or small, this is the kind of picture you get. Approximately spherical shape and total. But when you start to increase the mass, then you start to get things that are being filled in. What you get here is what's called a crystal chunk. There's actually a solution of these equations, which is a crystal, a regular crystal. And this is one piece of principle like a cube of shiver. And there you see it's rather filled in. It isn't just on the outside. So there's a significant difference between having a non-zero mass and having a large mass.

32:30 And it doesn't need to be very large. Mass has to be just beyond, you know, physically reasonable balance. So that's what they come. So now we come to the next part of the story, which is hyperbolic space. You can define the skerminons on R3, but you can also define them on any t-dimensional e-manic skull, because the definition is geometric. In particular, you can define them on hybolic space. And if you make hybolic space with given temperature, then you can try to compare the hybolic skerminon, if you only found it, with an euclidea skerminon. How would you do that? Well, you just, you know, imagine you're in space, here's the room, you're not small, so you simply take radial coordinates and measure out your distance, you pretend you're euclidean past x. So you can map the euclidean space onto the hybolic space, hitting an origin, and using radial coordinates. If you do that, then you can think of your scomion, which is a hybolic scomion, with no mass, but look at it now as a solution of the euclidean equation. Now it turns out, this begins to look like the solution to the euclidean equation with mass. And moreover, there is a relationship between the mass of This is the picture and the curvature of the high body's face. And the proximity, the first approximation, is just given by a linear relationship. The curvature is approximately half mass. In fact, you can get a more precise mass, which is rather better for calculation, but I'll show you the picture later, and a certain function, kappa of m, which is asymptotically given by this. Now, with that particular choice, you can try and figure out how well does it check. First of all, you do it for one, charge one, now you start to do it for charge two, three, increasing n. And it turns out that it checks well the small values of n, for all the calculations you can do, using this rational math handset, which you can use also in the hyperbolic case. And so you find, you know, a rather easy connection between math and code, as the title of my course. Now this is a kind of pragmatic fact, I'm not offering you a fundamental physical explanation, telling you that, for some reason, this seems to work. You can ask later on why did you guess this, but we'll come to that. Let's ask, why is it interesting? Well, first of all, I claim mathematically it's interesting, because now we can use the geometry

35:00 to relate this, and then back to the instantons, I can't follow that too. And so, we've got rid of this nasty mass term which spoils the geometry, by absorbing it into the geometry, in terms of the curvature of the underlying space. Mathematically, you get your proof. Now, if you're a physicist, you might, of course, if you're not a real physicist but a pseudo-physicist like me, you might say, ah, it's somehow taking gravitation, because curvature of space is gravity. Maybe there's some relationship between nuclear physics and gravitation. And that, you know, this confidence is about string theory, which is supposed to be dealing with that. Maybe there's some link here. I'll tell you more about that in a moment. And then, another, What's the word here? Once you have a parameter, it's a mass parameter, or you can interpret it now as a curvature parameter. You can, of course, take a limit. When the mass goes to zero, that means the curvature goes to zero, you get back to the flat case, which case we already know about. But you can take a limit. When the mass goes to infinity, or the curvature goes to infinity, which has got a very bad limit. And now you might ask what happens in that case. Now, I was very encouraged when I heard Witten's talk yesterday, or before, which showed that, if I remember correctly, his motivation to bring in this twist of string theory was that if you took this, now we've seen a correspondence with anti-cidas space, and then you took that and you took, then the curvature, the limit to the curvature of infinity, then that model wouldn't be able to study it, and perhaps it was cool in some other theory. which should be a topological theory, and that's what you're looking for in the Twister world. So, the idea of taking limits where things like curvature go to infinity, I thought it'd be crazy. But I find that physicists got there first. So, and moreover, they have things. What they found is that, I mean, the whole purpose of this conference was to show that Twister string theory, which in some sense is motivated by this idea of written-explained, is a different point of view, a different theory, which for this particular regime of very large curvature is a better picture of what you want to study in the first place when you start with sport. That idea is that you have another picture for large values of curvature. Well, you might ask the same question here. What happens if you make this curvature large? Well, here's an interesting observation, which relates to something I did a long time ago with Michael Murray. When we were studying hyperbolic monophiles now,

37:30 hyperbolic monophiles, there was a curvature in that too, of course, and we did say, if you did the curvature go to infinity limit, something remarkable happened. And what happened was, for some unexpected reason, you ended up with something that was related to what what I call an integral system occurring in statistical mechanics, so-called Pops model of integral system for a particular two-dimensional lattice, a particular set of Portsman weights, which was discovered some years ago, and was particularly interesting because, unlike most previous examples of integral systems, the explicit solution didn't just depend on things like rational functions and elliptic functions, but involved functions coming from curves of higher genius, and that's still not interesting. Now, the equation of that curve, if you like, appears naturally in the limit high energy, in the large limit. I've always wanted to understand that better. I thought it was a crazy idea. I've learned some physicists that, you know, you shouldn't be afraid of crazy ideas. Maybe we can justify that. Maybe for some reason someone says, taking the large value of the curve as a limit, we're all connected with an integral model. And we know which integral model it is because it's all written down in the literature. So that's the kind of challenge for the future. Now let me go back one step. I told you that one of the advantages of having the interpreting things mass in terms of curvature, hyperbolic space, is enables us to do more geometry. And now we go back to instantons. You remember that I showed you that if you take instantons, which are an R4, a four-sphere, and you take the parallel transport in a given direction, then you get to skirm fields, which was a good approximation to constructing solutions of the skirmian equation. But it's simply parallel transports along straight lines, you can now take parallel transport along circles. So fix the circle, fix the rotation group SO2 inside the symmetry of the fourth sphere, and take parallel transports along those circles. Now, that will give you a function with values essentially in a group, defined on the parameter space of the circles. Now, the parameter space of the circles, we divide S4 by the circle, here is hyperbolic three-space, naturally. So hyperbolic three-space appears naturally, instead of the flat space, we need to divide out by these, when you take these circles. You have to be careful to choose an origin, and you make a radial gauge, because we want you to do that anyway.

40:00 When you've done that, you'll find that you get from every field in R4, S4, whether or not it solves the equation, you can construct a skirm field on hyperbolic space, and if you start off by taking a solution of these equations, you'll get a particular manicure of solutions of the skirm fields, and now you try to minimize it in that skirm action in hyperbolic space, and you will construct something which is a candidate solution. to work. So now we've got the same procedure we did before, but we've replaced it, acting out by straight lines, acting out by circles. Now the parameter, the curvature, the harmonic space, is linked to the length of the circle. And in the case of the circle going off to infinite lengths, you get back to the straight lines, that's the case of mass going to zero, but in the circle strings to a point, that would be the opposite limit when the curvature what I'd be talking about. So we can study the behavior under the change of parameter of that variable. Now, a little bit more about the actual details of this construction. When you take the circle acting on the four-dimensional sphere, that's the obvious rotation, then there's a fixed two-sphere. And you remove the fixed two-sphere, and the complement is in five of the right circles, and the quotient is in hydraulic space. But the fixed two-sphere is important because sphere and also acts on some bundles of the sphere, then over the fixed points the circle has to act on the fibres. It has to act by some representation on the fibres, and that representation, being an SU2, is given by a pair of weights, an integer P and minus P. And that is an important part of the ingredients of that particular action. And when you work out numbers, skirm numbers and so on, the skirm number down below is rated to the distance number above, for m, by multiplying by twice p. But if you take the smallest possible value of p, and you rather take p to the heart, that corresponds to a sort of spin cavity, really, then you can get the distance number and the scale number to agree. And so we'll do that. So you fix those, then you construct the scale field with the scale number m from the distance number m. So everything goes very smooth, swimmingly, and you can proceed. Now, let me give you a few pictures. These are the graphs for calculation. I thought you could work, so I had to convince you.

42:30 This is the first graph. These are the functions of the curvature, as a function of the mass, as a function of the curvature. It seems approximately a straight line. It's going to be, that's the same part. It's going to be a little bit and you get to the origin. And that's to give you the best fit for a single schermier. Now if you think, if you ask how does the energy behave, here's a graph of the schermier of energy as a function of the final mass. There are three sorts of graphs here. There's the exact graph, which is given by the solid curve. There's the hyperbolic approximations, which are given by the circles along that graph. Then there's an instanton approximation, what you get is you start with an instanton and pay it to the construction of the bullet. And you'll see that they're all very close. So, whether you start with, I mean, whether you work with the hybolicon, the scermion, whether you work with instanton, you get very similar results. Now here are some more graphs. This is what happens if you increase the barrier number. And if you increase the mass, the scale number becomes more localising, because the exponential decay takes over, and so you get pictures like this, so this is for the value of m, four values of m are given here, and show you that as you increase m, this becomes more localised, and you have the exact results on the solid curve, and the hyperbolic approximations on the dot. As you see, this is very, very good. So the numerical calculations are pretty convincing. that is. Here's another graph. This is the energy per barion. The ratio here of the function of the final mass for b equals 2, 3, and 4. So there are three graphs corresponding to barion numbers 2, 3, and 4. And you see, again, the darker, which is the real solution, as we know, the circles, and the, on it, the hypotic approximations. As you see, the bottom of the slight deviation is not perfect up there, but it's pretty good in the sort of main range of values of the parameters. So the need of, I mean, in a way, you pick things up, basically, when you put in square number one.

45:00 But then, testing it from all these terms has become the real test of the theory, okay? Anyway, this is a test of the theory of this method for values more than one. So, what's the conclusion? Well, here we are. The conclusion is that you start with gauge fields on S4. Then you can take, by integrating long circles, you get Skrm fields in hyperbolic C-space. If your 6 infinite modelized space has a certain number of parameters, you get Skrm fields on there. to try to optimize on that finite number of parameters to get the real solution, and this is the way to get approximate solutions of these skirm equations. This is a rational procedure, classified by some perfectly respectable background mathematics. What do you get to get a skirmion on a curved spectrum background? If you believe that you can replace the curved background by a mass in a homogeneous space, this is a way of constructing skermions that have given mass to the pion by a mathematically justifiable procedure. And this checks well as I explained the different values of the parameters. Now, let me go, if you haven't lost the entirety or if you haven't lost the new entirety, let me go back to molecules. It's a rather confusing story. You see, there are molecules, skermions, and instantons. And we jump in backwards on fours between them. But they can't put you straight. What relates to the molecules, you see? The monopole equations are... We start by taking the cell-fuel Yang-Mills equations up there, and as Richard Ward explained, if you reduce the dimensional reduction, you get down to the Bogomolny equations in one dimension by just factoring out the vertical line. But if you do that from a sphere, by factoring it by circles, he also mentioned this, So that's really what we're doing just now. So, the monocle modulized space and hypotic space actually appears as one part. It appears as the monocle, by the way, are the solutions which are constant along the actual variable, where they don't depend on the actual variable. So the monocles, amongst all the possible solutions, are the ones which are invariant under that rotation by the circle. So if you take the monocle in the non-modulized space, it has space at large dimension, The circle acts on that, and inside there, there's some fixed-point sets. Fixed-point sets have different determinants index by the number P.

47:30 If you get the determinant of 1 into P goes half, you'll get a subspace of the instant or modulized space, which I've called M prime N. And that is the one which is used for constructing molecules. Now, because the constructions of the instant arms were mathematically justified, This gives you a perfectly good explanation of how to construct skirmions from molecules. And this is the rational map, as that's given justification. If you think about it carefully, the inclusion of this space is there, to apologists, comes by including the maps of the two-sphere to the two-sphere, or what's called the double-loop space of the two-sphere, inside maps of the three-sphere to the three-sphere, by suspension. That's that suspension map that they use. And basically, that just corresponds to this. and that it works, is all understood now, from the point where I'm thinking about hyperbolic skermions that are going through this procedure. This explains the analogy between monoclon and skermions as before was simply a wild guess. Now it fits into a more rational framework, not without, it's not a mathematical proof to anything, but it's justified by much better reasoning than the one before. but it also suggests the implication the rational map only uses these ones only uses these ones don't use it on the outside these ones are easily described because they're given by rational functions and they have their rational map but in fact what they suggest is you aren't going to get all these going on that way but I wonder it's more complicated which you won't see by that approximation and so my hunch is least numbers, you will find you have to have things in here, because this part of the space only captures parts of the topology of this space, because parts of this space this shall have our reach, as they say. So, you have to go outside this to get all these solutions, and this way you can, and you, in principle, you know how to construct this. It's more complicated than my rational functions, in principle we know how to do it, so you couldn't get a handle on it, because we really wanted to carry it through. Well, now, that's really the end of what I have to say substantively, but let me now go back to physical speculation, physical significance. Now, let me digest a moment. When I, about 25, 30 years ago, I started getting into bad company.

50:00 I started to mix with physicists. And then people became just suspicious about me, and they asked me, what's the difference between physicists and mathematicians? I said, well, one easy test. If you find a paper that looks the word theorem appears, then the person writing it is the mathematician. And that's a pretty good approximation. Except that recently I started writing a paper that looks the word theorem doesn't appear. Though that condition makes you cease to be a mathematician, it doesn't make you a physicist. You may be neither one or the other. And I just recently realized that to be a physicist, the big difference between physicists and mathematicians is that the physicist understands about scale and magnitudes. He knows what Planck's constant is, how small the proton size is, what Hubble's constant is. And these are the numbers he understands. And he knows that if something is bigger than something else by a factor of tens of something, that counts. For mathematician, all constants are one. Magnitude scale is not part of a mathematical training. So, mathematician is in the habit of thinking about small numbers and large numbers normally. And so, when I came to this bit, I suddenly realized I had to get to grips with scale sizes. I learned a bit in the process. What I learned was, you know, negative, still. So, let's go back to what I said at the beginning. What's the physical significance of having this mass of these pions interpreted in terms of curvature of hyperbolic space? Well, you know, it sounds reasonable. Mass is something critically significant. Curvature of space appears in productivity. is the foundation stone. Surely there's some, you know, massive physics in this. Well, what is it? And, of course, I mentioned string theory, because string theory is meant to, we're talking about, here, the connection between nuclear ion and nuclear physics, and general relativity, well, string theory is meant to, of course, stretch all the way from particle physics to cosmology. So, well, now you begin to start to worry about physical scales, as I mentioned. And the first thing you realize is that the numbers K and M appear in this formula, the curvature and max, don't have the same physical dimensions. And if you have the same physical dimensions, the max appears in that formula, practically replaced by this number here, the mass of the pion, time of velocity of light, divided

52:30 by Planck's constant. That number there is, I think, what's called the inverse of Compton's radius. So it gives you an indication of the scale size of the proton. And these let us mention of inverse length, which is what curvature you should have. So, now, one knows in physics what the size of the proton is, the order of magnitude, and you can compare that number with the curvature that appears in cosmology, in the big scale universe, and that number is of the same order of magnitude as Hubble's constant and inverse C, by a nice factor of 10 to the 40. Now, the mathematician didn't get into numbers, you wouldn't see that. For physicists, 10 to the 40 is a pretty impressive number. So, clearly this is wrong, you know, by all of the magnitude. On the other hand, 10 to the 40 is a nice number to have. It can't be wrong at all, but that'd be a hunger issue. People are allowed. 10 to the 40 is the right number to get. The output factor of 10 to the 5 would be, you know, dull. Now, 10 to the 40 is serious. In fact, there's a lot of physics, a lot of paper that he's written about the significance of 10 to the 40. Dirac had ideas about 10 to the 40, which I gather subsequently haven't been proved correct. But it's very fascinating that there are all these different ratios, you know, the ratio of the size of the elementary particles, the size of the universe, the strength of the electromagnetic field, as opposed to the gravitational field, the number of protons in the universe. All these numbers evolve into the magnitude like that. Nobody quite freely understands. You should try to understand it. Now, that's one problem, which is why I've been trying to interpret mathematics here. There's another problem. I'm talking about stationary solutions of these SCERM-C. There was no time. If you look at trying to embed this SCERM-C in the gravitational framework and look at Einstein equations, try a couple of the SCERM equations, And look for stationary solutions, well, unfortunately stationary solutions of the Einstein equation tend to give you, in three-dimensional space, positive curvature. And here we want it negative. I would like to interpret a high-bolic green space as three-dimensional space, physical space. And if curvature is negative, unfortunately stationary solutions become positive. On the other hand, if you're going to become more realistic, and you can put the most stationary universe with the one you really live in, an expanding universe, Hubble's constant and all that, and a cosmological constant as well, then the celestial space, we don't know, it may have positive curvature, maybe negative curvature, maybe zero, but at least, you know, assuming it's negative curvature, it doesn't mean you're totally wrong.

55:00 So that might be courage. So now you are saying, well, we've got the right sign, but we still are off from an orbital magnitude before you're on a first count. Well, are there other physical ways to interpret this? If that first guess is wrong, could we do better by talking about some sort of value, gas, of skirmes? You work out the average distribution necessary of protons in the universe and the studio calculation will be better. And, with all of hard work, you find that you can improve factoring things by 10 to the 40. You say, ah, great, that 10 to the 40 will cancel the other 10 to the 40 and I'll be like regime. Until you realize, unfortunately, that although the number of kappa or m, little by a factor of 10 to the 40, the curvature is the square of that. So you've gained one 10.40, but you've lost another one. So you're still 10.40 out. It's a bit frustrating. So, anyway, and you can also try to worry whether, you know, talking about value of gases is good enough, whether you should worry about the fact that matter is considering galaxies, you know, have real physics. Anyway, there are a lot of physicists in the audience, so I just leave it as a challenge. Are there any idea, is there any way in which this connection between curvature and mass can be given sensible physical meaning, by any method or rather, or is this just nonsense? Well, it could be, it could be a mathematical fact that there's no physical interpretation, but it seems a bit, I don't have to, why? If something happens like that, I think it will mean something. What it means, I don't know. That's why I'm asking you here. At the moment, it has significant meaning, but at the moment it's a mathematical fact. We can be used to it mathematically. It covers the solve equations, which are interesting, gives some insight, but it doesn't lend itself yet to physical interpretation in a real sense. And I call here, this is the thing, I'm going back to the other fact I haven't talked about, that this varying mass of the curvature, I think the question of looking at what happens at the large limit of these things, which I, I think with Murray we showed that this was a false model, the one of the things that happens when you go to a large limit,

57:30 hyperbolic monocles, of a large curvature limit, is that the parameter space for identification, between the modulized space of monopoles and rational functions, the theorem of Donaldson. That's a very difficult theorem. I mean, if Donaldson shows how to construct from a monopole a rational map, then he proves a hard theorem showing this actually gives you all of it. But there's no way to construct the inverse map, for example. On that hand, when you go to the limit, when the curvature goes to infinity, it tells us that matter becomes trivial. It's still true, but it becomes very, very easy. So in some sense, something does simplify, which we know about, in that limit, and that is perhaps an aspect where this theory might be sometimes integral to dimension. So, that's, I think, something worth exploring. I hope that I'll come back to that, and that might turn out to be interesting in its own right. And, as I said, since we've heard here about the stunning things in the large curvature limit, in other contexts, maybe this is part of that general picture in some way, and so that gave me an excuse to talk to you about it in this conference. I didn't realise I'd had that excuse before I came, but I must use it now that I've got it. Here are a few references. The paper that's back in Sutcliffe, the old one, 2002, the more recent one, 2004, which I don't have the exact reference for, and paper by Sutcliffe and myself, Well, if you look at this in 2005, I think that's the most part of it that you will refer to so far in the conference. Thank you. Any questions? We're giving scrimi on one node the exact value of the enthema of the scrim energy. Well, I mean, since there are no exact formulas, everything is numerical. And there are numerical factors, and they get quite close to the topological limits. I mean, they're, you know, within a few percentage. It's not very far away. So when you try to find a minimizer by minimizing among finite dimensional configuration, how do you know that you're close to the actual solution? Well, you know, this is numerology. I mean, what you do is you find one which has low energy, then you compare it with other methods of doing it, see if you can prove it, and if you can't, you think you've got close. Sometimes, in the very first one you do, you find, when you're pretty sure, you're confident, you know what it is,

1:00:00 what you find by this method is one which is very close to within 1%. So the approximations seem to be pretty good, then you can check them out. So then you believe them a bit more generally. But of course, none of these things have improved rigorously, because there are no precise results, there are no precise results. is the area between mathematics, computing, and physics. There are no theorems, it may be not real physics, it's neither one thing or the other, but it's a nice picture of all three. But in principle, as far as we can tell, if you believe the evidence, and I have to say that Paul Sackley are real experts in this computation. I know nothing. They tell me what they do and believe them. But they use pretty sophisticated techniques in numerical calculations. They have all these antithesis to start with, but over and above that, they use very, and sometimes they use fully fixed equations and do the whole thing, small values. So they are pretty, and they test them by a variety of methods, and, you know, I think they are pretty convinced that the answers they get are very close to the real answers. Also, in this hyperbolic setting, does the map send infinity to a point of the three-sphere, or is there some fixed map that you're actually varying? No, no, at the best you do. It still goes to one point. Topologically, the whole bound of infinity goes to the unimatrix. It has to do that. Now, there's no topological variation. So it has to look like infinity in case. There's no sort of... It's only in the monocle problem when you have variation around the two-sphere. It's choosing the gauge character in order to arrange that. Yes, of course. If you start off with the small model of the design, you can perhaps fix the gauge. But the scope model is no gauge. It actually goes to a fixed constant map. No more questions? I mean, obviously these are, you know, the classical picture that you have to do quantum mechanical average rotation, so a lot of the detailed pictures you get here will disappear, but the idea is that some aspects will persist, certain quantities of stability and so on will survive and will be compared with quantum mechanical, so I think there's, you know, people I mean, it's certainly not easy, but it's, you know, you get some information, let's put it this way, from the classical picture, which you can try to compare with some sort of realistic model, because this is, of course, it's known now that Skirman is, although invented by Skirman simply as a certain inspiration, it's now seen as some kind of low-intensity approximation of QCD and some regimes as an approximation which is very passable.

1:02:30 I don't even try to do it mechanically, but some of that work is done. Any more questions? Thank you very much.