Edward Witten, ICM Lectures

that's not gentlemen, here I have the great privilege on this occasion to introduce

edward

witten

for his plenary session on string theory in geometry, as you may never have done before, I mean, as you may have noticed, the program of this conference , the 19th Berkeley Mathematical Congress in 1986, differs from its predecessors in the much greater emphasis given to the relationship of mathematics with both the natural and artistic and artificial sciences with physics and computer science, this applies particularly to speeches plenaries but also to the rest of the program and, in my opinion, reflects a profound change that has taken place in the structure of mathematics at its avant-garde, what some of my acquaintances have described as a profound reconvergence of mathematics for the sciences.

Edward Witten embodies this tendency at its most acute through his training and profession. The physicist is a professor in the Princeton Physics Department and because of his achievements as a physicist at the highest level and mathematician at the highest level he has been one of the main workers in recent years in the continuous development of string theory. perhaps the leader in introducing ideas of conceptual originality and mathematical sophistication and that theory and has also used those ideas in a very notable way to obtain important mathematical results of a high degree of originality both in concept and in execution and his

lectures

Both Richard Chen and his singer have referred to some of these results, another that they have not referred to is their very original proof of Morris inequalities using the eigenvalues ​​of the Laplacian on manifolds, the first original proof of that set of ideas in 50 years of material. which he is going to talk about reflects one of the most powerful trends in contemporary mathematics and the most powerful and sometimes controversial trend in theoretical physics: the development of fundamental ideas relating to the unity of the theory of fundamental particles and of gravitation in a highly sophisticated and elaborate mathematical form.

We welcome you here as a symbol of this movement that, I am sure, will continue and even drastic reforms will be made in the coming decades. Professor Witten, in 20th century physics there have been two really basic ingredients, which are, first of all, general relativity, which is Einstein's theory. of gravity and, secondly, quantum field theory, which is the basic framework for our understanding of essentially everything else in physics except gravitation, although both play a role in describing the same natural world, these two theories They enjoy their success in areas very different from general relativity. has its successes in the field of astronomy and quantum field theory is the framework for successful descriptions of many aspects of the properties of atoms, molecules and subatomic particles.

Most of the really important developments in fundamental physics since about 1930 have been developments in quantum field theory or certainly in quantum mechanics, and for much of this time physics and mathematics had relatively little interaction compared to that of many previous times. An obvious reason for this is perhaps that in the 20th century mathematics moved into seemingly unrelated abstract realms, perhaps it seems routine. world of the theoretical physicist, but I think that a large part of the reason why the interaction between mathematics and physics was more limited in the last half century than in previous times was because the physical problems that arose in quantum field theory, which was the central framework for Most of the advances in physics did not seem to lead to many interesting mathematical structures.

In the mid-1970s, this changed mainly due to advances in physics in the mid-1970s. Quantum field theories had emerged to play a central role in physics for so long. called non-Abelian gauge theories, theories in which one of the main ingredients is a set of vectors in spacetime with a group of non-Abelian structures, these theories raised more interesting and significant questions in geometry and topology, which caused A more meaningful interaction between physics and geometry starting ten years ago would perhaps have been possible during most of the time when quantum field theory played a central role in physics; in fact, one of the many strands that led to Donaldson's work, which we heard about a couple of days ago, originated from this interaction between physics and Geometry now advances quantum field theory, which led perhaps a decade ago at the beginning of a rich interaction between physics and geometry, but I think that this interaction would still be relatively limited if it were just quantum field theory.

In attempting to push the limits of quantum field theory, physicists have reached mathematical frontiers. Perhaps the fundamental limitation of quantum field theory is the one I have already alluded to. It is just one of the two fundamental theories of physics, and in fact, quantum field theory changes. If it turns out to be incompatible with that other basic ingredient of 20th century physics, which is general relativity, one can try to formally combine these two theories through the so-called quantization of general relativity, but this leads to all kinds of formulas that make no sense now, in The early days of physics. quantum field theory it was not at all obvious that this was a central problem quantum field theory faced many problems in its beginnings it was not even obvious at the beginning that the quantum field theory of electricity that was the original quantum theory Field theory actually made some sense over time, although the other difficulties of quantum field theory were overcome and it became clear that quantum field theory was an adequate framework for describing all forces in nature except severity, although in the course of learning it had to be done.

Physicists were led to develop quantum field theories with richer geometric content, such as the non-Abelian gauge theories I alluded to earlier, as the other difficulties of quantum field theory, the incompatibility with General relativity emerged as the limitation of this framework. Now general relativity is based, of course, on Romani and geometry, which is certainly one of the richest frameworks for understanding ordinary geometry, and I claim that general relativity, which is based on Vermont geometry, is incompatible with quantum field theory, it must mean that some adequate physical theory will successfully generalize the general theory. relativity and reconciling it with quantum field theory must somehow be based on some generalization of the Romani in geometry compatible with the requirements of quantum mechanics.

Now the experiment can only provide us with indirect clues as to how we should go beyond the ordinary Romani in geometry as a framework. in physics and the reason for this is that, unfortunately, gravitational effects are immeasurable in experiments testing quantum field theory and the same is true the other way around; In fact, there is only one example in the previous history of physics in which a theory was invented even though the experimental clues were only indirect. The only other success of that nature was general relativity itself, which was discovered in what we might call a logical way Einstein started only with a very general clue which was that special relativity was incompatible with gravity Einstein first formulated the physical ideas to reconcile two inconsistent theories, then he found the correct mathematical framework which was general roma and geometry and finally he invented the theory.

Now, of course, we would like to emulate this story, but that is easier said than done for many ambitious physicists. and mathematicians have gone into the wilderness in search of some fundamental generalization of geometry that would reconcile gravity with quantum mechanics, but have generally returned empty-handed, while direct attacks on this problem have produced little progress, seems to have fallen out of the sky essentially through the law the largely chance discovery of something called string theory now the people who invented string theory in the late 1960s did not aspire to reconcile general relativity with quantum mechanics They did not expect to generalize Romani in geometry; in fact, the ambitions were not at all in the scope of our discussion.

String theory was originally developed as a theory of so-called strong interactions (the nuclear forces that hold nuclei together despite the electrical repulsion between protons), even as a theory of strong interactions. The theory was discovered largely by accident after its discovery, which essentially occurred when one of the formulas of string theory was written by Venice Know with very little motivation in 1968. After the essentially accidental discovery, it was developed vigorously in this early period during which more than 500 papers were written that led to an extremely rich mathematical structure that fascinated people, but never moved toward solving the original problem that motivated this work of strong interactions and, indeed, this entire line of development was abruptly abandoned by most physicists around 1974 when it became clear that it was not the correct approach for strong interactions;

Well, it was generally abandoned because it was not the solution to the original physical problem. The structure retained its fascination, sparking work in several areas and was eventually revived in the 1980s as an approach to quantum gravity. Now this topic already has an extremely long and complex history and I am afraid that if I attempted to examine even the high points of this history, it would take the rest of the lecture I have written here some of the dates and names of some of the main . developments, but it would be impractical to explain even some of them fully, which I will talk about, so I will talk about some ingredients of some of these developments, but I am not going to try to explain this story now, since I have The stated string theory essentially fell from heaven by chance, some special formulas of this theory were discovered with a rather strange motivation and gradually it was learned that these formulas were special cases of retrostructure in which one had penetrated into time to deeper levels, but probably not very far From the deeper levels underlying this theory, the situation in string theory is more or less as if Romani and geometry had repented;

It is more or less as if general relativity had been invented without having first invented the logical framework for the theory that is Rahmani in Geometry. A dream happens to Einstein, but it is imaginable to have conceived a planet where the problem that Einstein posed of reconciling the gravity with special relativity would have existed on a planet where general relativity was unknown, possibly on such a planet general relativity would be invented somehow. in some obscure way without its Romani logical foundations and geometry being clear, whether or not this happens on some planets in the universe where theoretical physics may have been developed, it has certainly happened on planet Earth in the case of string theory due to the peculiar shape. that string theory has been discovered I cannot do what I would love to do and what I hope a speaker will do at some future Congress of the International Congress of Mathematicians which is to explain the basic concepts of curvature and romani connections and varieties that would fly in threads there or all that gypsy and geometry cakes in general relativity.

I may try to explain some of what are presumably the preliminaries of this theory, and in outlining what I mean by these preliminaries I may give them an analogy with the general theory. relativity in romani and geometry one has a curved manifold and if you consider at a point the tangent space to this curved manifold, of course the tangent space is a linear space and one thing you certainly need to be able to do to work with romani and geometry is to understand a little elementary linear algebra to be able to carry out linear operations in tangent space.

Generally speaking, our current understanding of string theory was as if we were stuck in romani and manifold tangent space without understanding the basics. frameworks that we would need the basic logical concepts that we would need to coherently go beyond linear operations in tangent space, so the analogue in string theory what is the role that linear algebra plays as a preliminary to working in tangent space de Romani and multiple, but well, the analogous role is played in string theory by much richer mathematical structures that are the theory of modular forms of Riemann surfaces and what I will describeRoughly speaking it is the theory of infinite dimensional representation, roughly speaking, just as one needs a bit of linear algebra before attempting to learn.

Romani in geometry, so it is necessary to know something about Riemann surfaces and infinite-dimensional representation theory before trying to learn string theory, so we could say that general relativity requires linear algebra for its linear approximation , but the fundamental ideas of the theory are actually curvature and romani connections. In geometry, string theory requires Riemann surfaces and infinite-dimensional representation theory for its linear approximation and what should really belong to the bottom right corner of that square is, according to my thinking and that of many other physicists, really the central problem in theoretical physics. Now I have stressed the incompatibility of quantum field theory with gravity, but in fact there are other problems in our understanding of physics that we would like to overcome and one of the most fundamental problems is simply that there are too many conceivable theories allowed by quantum theory. of fields.

The general principles that we have, quantum field theory would allow an infinite number of theories and the logical framework of general relativity would also allow an infinite number of theories of which Einstein is only the prototype, of course, if you try to combine these two frameworks, it will not work. You will have an infinite number of theories. theories but 0 which is the claim that they are incompatible, but if you had been able to combine quantum field theory with general relativity without particular difficulties, then you would have had an infinite number of possible descriptions of the natural world, which would certainly mean some basic gaps in your understanding, so in some sense, Fortunately, there might be a problem disguised in the difficulty of reconciling these two frameworks, either separately they would allow an infinite number of theories, together they seem to allow none and, hopefully, understanding adequate will lead to a finite set of theories that will reconcile them to briefly illustrate what is happening.

I mean, by saying that the general ideas of Romani and geometry would allow for an infinite number of theories, the basic concept in general relativity is really that the spacetime that I will call M is a pseudo-Riemannian manifold, has a metric tensor G and Riemann. The Ricci scalar tensor is what I have called our basic principle of Stein was to describe nature by some principle of geometric action, some geometric integral over space-time was going to be the action of this theory and while he wrote the simplest action and there are an infinite number of more general geometric invariant principles of action integrated over m that would give conceivable generalizations of general relativity within the logical framework in some sense, you might consider this to be the most elementary member of the, but when you consider another point which I am currently about to allude to, in fact, which has an infinite number of theories without an obvious more elementary example, so, starting from the inconsistency of quantum field theory and general relativity, we would certainly like to overcome the fact that any of them separately leaves us with too many theories. in fact, with an infinite number of conceivable theories to illustrate the other deficiency of general relativity, let me briefly discuss almost flat solutions of this theory, of course, in everyday life we ​​do not realize that spacetime is curved, the Space-time is flat in everyday life and of course, you can do a better job of investigating curvature if you measure on large or larger scales, every rimoni and manifold is almost flat on a small scale.

Gauss, as many of you know, measured the angles between three tall mountains and found that the pie is the sum. confirming that a mount on the Alps space-time scale is approximately flat and astronomers do it much better and with even greater skills, so we present as a good approximate description the flat metric ADA the diagonal metric with these eigenvalues we imagine a small deviation H from a flat metric we write the spacetime metric as Mancow electrical pulses with a small fluctuation. Well, if we insert this form of the metric into Einstein's equations, we will get a nonlinear equation, but since spacetime is almost flat, let's linearize it if we linearize. we will obtain a linear equation for H which will be some perturbation of the Laplace operator acting on H giving 0, well, this equation has wave solutions, the simplest is the plane wave that I have written here and these wave solutions are quite analogous to the corresponding wave solutions of Maxwell's equations, which are light waves, in this way we learned that general relativity allows us to predict gravitational waves, a prediction of that theory that clearly emerged when the theory was formulated in 1915, although it had been conjectured by pointers.

About ten years earlier, a prediction of the incident layer that was finally tested indirectly by radio wave observations of pulsars just a few years ago. Now, when this prediction was made, it was the prediction of a new type of wave, not a new type of particle, but in fact, in quantum mechanics waves and particles are two facets of the same thing, so This prediction of a new type of wave is actually a prediction of a new type of particle. Einstein simply aspired to reconcile gravitational forces with special relativity, but in attempting to do so he actually predicted a unified theory not only of rational forces but a unified theory of gravity and matter.

Gravitational forces were unified with this new and peculiar type of matter that was predicted, namely gravitational waves. Einstein was not satisfied with his unification as he eventually hoped to unify the gravitational forces but with some exotic form of matter that he predicted but unknown at the time, but with the forms of matter that were known, so this is often described by saying that Einstein failed to unify gravity with matter, but it is a better statement. It would be to say that on a logical level he unified gravitational forces with matter but he was not satisfied with the form of matter that the resulting theory predicted, so on a conceptual level it is a great triumph that simply trying to describe gravitational forces finds an illogical framework. which is Rahmani and geometry writing the simplest equations in that framework, which are non-linear equations, then we discovered that we did not simply have a theory of gravitational forces, but a unified theory that unified gravitational forces with matter;

However, this wonderful unified theory has a couple of flaws. is incompatible with quantum field theory which of course was not known at the time since quantum field theory was invented later, but it was evident even at that time that the forms of matter predicted in this theory were less rich than those we observe now. Next I would like to give a heuristic explanation of why string theory could be what string theory is and why it could be expected to improve general relativity. The framework for this really requires some preliminaries on quantum field theory and I hope to make a brief comment on Quantum Field Theory later, but without developing all the necessary preliminaries on quantum field theory.

I would like to at least explain your moment or how one stumbles upon some generalization of general relativity in the form of string theory and in doing so I will be explaining conventional things. in quantum field theory in a way that is essentially closely related to the point of view introduced by Poliakov some years ago. Well, in quantum field theory, the scattering of particles like the gravitons of general relativity are described by Fineman diagrams. Now a Fineman diagram is a spatio-temporal history of a dispersion process in this slide I have outlined a dispersion based on the history time runs vertically in the space of the diagram or one of the dimensions of the space at least runs horizontally have been cast and absorbed particles at these points a B C and D the solid lines are actually curves in spacetime along which a particle traveled and two or in which two particles recombined into one, so this is actually a sketch of a sequence of events in space-time. and if you start with some theory like general relativity, then the rules of quantum field theory will give you algorithms for associating probabilities with space-time histories like this, you actually start with some abstract gamma graph and then imagine some gamma map in space-time which I will call capital arc lengths are what I will call I of of the gamma graph in spacetime now in conventional quantum field theory of the last 50 years, although this is not the conventional way of describing the usual structures, an almost wrong way of describing the usual Thiemann diagrams is as integrals on this Omega space path. of gamma then we have an abstract graph gamma that we map in a queue here I draw the image of that grass potential, such map is a magnetic flame of all those maps, everything is the length of the image of the graph below that map that we try to integrate over Omega of gamma e to the negative length of the map image multiplied by some factors.

Currently I will explain the technical factors at the bifurcation points of the chart. Now one thing you'll notice here is that the infinite-dimensional integral is the integral over the Omega space of gamma of all conceivable maps of this graph in spacetime all conceivable continuous maps of gamma in spacetime which is a characteristic infinite-dimensional integral that appears in man panthans' quantized herbal formulations of quantum mechanics and quantum field theory now a What will be noted here is that when we map this gamma graph onto spacetime, gamma is not a manifold , is almost a one-dimensional manifold, but is actually a one-dimensional manifold with singularities P Q R and s and the evils of quantum field theory. everything arises because gamma is not one-dimensional, it is not multiple but it has singularities.

We are taking this graph which is a 1 man luxury with singularities or mapping it in space-time and two bad things happen due to the fact that gamma is not a manifold, first of all we have to define this infinite dimensional integral and, in fact, there is a lot of experience and even some theorems in constructive field theory about the definition of infinite-dimensional integrals. One thing we know about it. is that it will work much better if we try to map a smooth manifold in spacetime rather than a graph with singularities. When we map such a graph in space-time, we find the possibility that the image of P Q R and s in time-space coincide, and in fact, after we learned to define such infinite-dimensional integrals in the first place, we learned Since this integration behaves badly near the region where What our first problem in quantum field theory is its general mathematical difficulty and, in particular, its incompatibility with gravity, arises because gamma is not a manifold, secondly, because gamma is not a manifold.

On the manifold we have these special points PQ R and s and we had a nice simple factor in the integrand, the exponential of minus the area minus the length of the gamma image in spacetime, but since there are these special points P Q R and s, it has sense. As those points are selected to take some local functional of the behavior of a factor in the integrand in each of those two at their PQRS points and, in fact, with certain very simple limitations, each choice of this local functional Phi of P gives you a different quantum field theory, the infinite number of quantum theories of fields corresponds to the infinite number of local factors that can be introduced with those branch points, well, of course, you might think that we are pushing things a bit and that there would be a minimal model in which we would not introduce these factors, but one of The surprise perhaps is that that is not true.

If you take a good geometric theory like general relativity andYang-mills theory and transform it into this framework, you will definitely need some suitable local factors of those points and, conversely, theories that would not have those local factors or would have simple local factors turn out to be quantum field theories that do not have interesting geometric content, so this is the freedom that not only exists in principle but we have to use it and we use it for the quantum field theories that are of greatest interest. In physics, we have a variety of problems that have their origin and the fact that gamma is not a variety.

Suppose we want to ask an engineering shop to make a physical model of the gamma graph while we probably define this graph. It kind of consists of mathematical lines, but the craftsman doesn't have mathematical lines at his disposal, so maybe if we're lucky he uses thin tubes and if he uses hollow tubes and joins them gently like a plumber does, then we can replace this graph with something to It is easier to make a physical model that looks very similar from a great distance and that is why string theory, if seen roughly rounded by quantum field theory, but there is a fundamental difference in replacing these mathematical lines with small tubes.

We have replaced the single manifold with gamma singularity by the smooth Riemann surface Sigma now, in some sense the gamma graph can be an approximation to Sigma, but the fundamental difference is that Sigma is a smooth manifold, while gamma it is not, and as a result By replacing gamma with Sigma, we will in fact overcome both the infinities in quantum field theory and also the non-uniqueness. The lack of uniqueness arose from the ability to attribute special factors with these points by missing those partial points in which we actually find ourselves. string theory where we will confront gamma with Sigma in which there are very few theories that can be formulated now the other main event that arises when we replace gamma with Sigma let's take a close up of this image which I will not draw completely remember This is actually an image of a process that occurs in space-time, so time is accumulating words in the image and this gamma graph that I have drawn is actually happening in space-time.

In fact, it describes the propagation in space-time of point particles. which for general relativity are the quantum counterparts of the waves I outlined earlier, so I'm going to zoom in on a small part of that picture. Our extension is the propagation of a particle in time in spacetime if we take its place at a given moment we have a single point or at most a finite set of points the fact that we can get a finite set of points instead of A single point leads to the fascinating story of the creation of matter and antimatter, perhaps it is all like that.

It would take us too far to discuss it, so I'll imagine that we find that the locus at a given moment is a single point and that corresponds to saying that this graph is a theory of a point particle that propagates in spacetime if in the same way we take the place at a fixed time of one of these tubes consisting of a circle or perhaps a finite collection thereof, so we have actually replaced the Field Theory point particle with the oscillating circle that you see. I observed that on a conceptual plane or within there were gravitational forces unified with some type of matter, but the matter in question was not rich enough and it was not rich enough essentially because that point particle that was the place of occurrence at a moment given did not have enough freedom when we replaced our graph with a Riemann surface Sigma, the place at a given time is not a point or a finite collection of points, but consists of circles and one of those circles or in that case, the chords closed will have a similar force, except that without physically fixed endpoints, they will oscillate in time and the myka violence during it will have an infinite side of harmonics that will correspond to different forms of matter, made of sound that replaces gamma with the Riemann surface.

Sigma means that we are dealing with a theory with much richer forms of matter. From what I said, it's not at all obvious that these richer forms of matter will all be similar to what we find in the real world, but upon closer investigation it turns out that they are so the seemingly innocent step of replacing Gamow with Sigma takes us to a more unique theory with also a much richer form of matter and takes us at the same time to the world of Riemann surfaces and modular forms and the theory of representation affirms an infinite dimension. algebras like DIF s1 the group of different morphisms of s1 that will act on this circle now I have another transparency that aims to recapitulate what I have been saying in a theory like general relativity we have really basic formulas like the geometric action principle the interval over the space-time on the Ricci scalar the beauty or lack of beauty of such a theory must be judged by the beauty of its fundamental principles our aspiration would be to generalize this elegant principle of geometric action to some retro structure that is compatible with quantum field theory , direct and even indirect attempts to do so have not been successful, on the other hand, if we do not consider the really fundamental Vermont formulas in geometry over some Drive formulas like the ones I have been discussing involving gamma graph maps in space-time , then it turns out that some of the driving formulas can be generalized Now, of course, if at the top of everyone's mind were to think of general relativity in terms of maps of graphs in space-time, then this generalization would be more obvious and perhaps would have happened sooner, what makes it less obvious is that, although this is closely related to standard descriptions of what you get from quantum field theory, it is by no means the most standard description.

I have not explained stops in quantum theory. field theory leading from fundamental formulas to less fundamental formulas involving these graphs, it is possible to explain it, but it would have required an excessively long exposition. If we work on this less refined plane, we find that we can generalize the less fundamental formulas of quantum field theory to obtain something called string theory that overcomes many of the difficulties, for example, the lack of uniqueness of quantum field theory that in This framework arose from the fact that the Ricci scalar is only one of many possible geometric principles of action; in this less refined framework it is associated with the fact that gamma is not a graph, sorry, gammas is a graph but not a variety and that is removed and we go to Riemann surfaces and so on with other points that I have already illustrated , so what happened in the invention of string theory is that in a peculiar process of trial and error with many historical stages, one has managed to generalize not the truly fundamental formulas of the oldest physical theories, but some of the formulas of momentum, now, like this image with the gamma chart, are derived from much more fundamental views.

The same should be true with our generalization involving Riemann surfaces and the loose dating of what lies on the Riemann surface, in the sense that general relativity lies on these graphs, is what I have alluded to as the mystery of unraveling the logical structure behind. string theory and I personally think it is a very fundamental problem for both mathematics and physics. A peculiarity is that normally the beauty of a theory must be judged by its fundamental principles and must not be sought in these derived formulas. Derived formulas such as those involving graphs and graph maps in spacetime are necessary tools if you want to determine the quantitative predictions of a beautiful theory like general relativity, but they are not basic formulas and are not necessarily beautiful, in fact they usually they are not. beautiful one of the surprises in string theory is that the current on the surface the generalization of the theory of those graphs is very beautiful involves fascinating properties of the cream on the surfaces modular forms theory of infinite three-dimensional representation algebraic geometry some of the Many beautiful aspects of this were described in a wonderful paper by Manin which was presented at this conference but which unfortunately he could not deliver here because he was not allowed to attend the conference, but the singer presented it very elegantly in a talk on Tuesday evening. late.

It is beautiful to try to generalize a secondary formula, but actually, in the case of string theory, one discovers that generalizing this picture with graphs leads to very fascinating and beautiful mathematics, and I would personally like to conjecture that what there is up here in a certain sense will remain. in the same relation to general relativity as the wonderful automatic forms and algebraic geometry that goes into Riemann surfaces path integrals Riemann surfaces are compared to the oldest Flyman diagrams of field theory now I have not been able to explain How do you get from here to here that would take too long.

I want to use the remaining terms of reigning time to briefly explain a little bit of the flavor of what it means when we talk about the maps in infant dimensional spaces that are involved in treating these gamma graphs or the Riemann sigma surface on their own terms, for example. So in the rest of this talk we will work on this lower half of the page without discussing how this is associated with the conventional picnic. Now, in doing this, I will be giving a very brief introduction to quantum field theory, as I will have to try to extract one or two ideas in about 15 minutes.

You'll see that quantum field theory exists at the top of this page because we started with quantum gravity or in terms of quantum. gravity and I go here, but quantum field theory lives again in the bottom half of this page because these infinite-dimensional integrals are actually formulas in quantum field theory. Now quantum field theory has been for most of its 60 years almost a private reserve for physicists and even now it has not yet emerged as a mathematical tool, but there are reasons to believe that quantum field theory could emerge as a mathematical tool in the near future.

Reasons independent of string theory to believe that because there are at least two areas where progress in mathematics has come to an end. is on the edge of quantum field theory, but there are areas of mathematics with some characteristic features of quantum field theory that have begun to be recognized as important. One of them is the representation theory of a fine algebra that Frank will review yesterday and which is closely related to the Hamiltonian. image of certain quantum field theories and the other involves recent work in arithmetic geometry that we will hear in the next lecture where the drop determinant, which is one of the really fundamental characteristic ingredients in quantum field theory, has begun to play a important role in a selection of topics for an extremely brief commentary on quantum field theory.

I'm going to talk about points that are taught in string theory and that could be related to D in future developments in the areas of mathematics that I just mentioned, but then Sigma will be a Riemann. surface perhaps with limit, let phi take sigma to the real numbers R be a real-valued function and then let's define a very natural action called the Freebo functional action in field theory, the integral over Sigma of this to form the Phi wedge of bar Phi where del and L bar are the natural operators associated with a complex structure in Sigma that FBA real-valued function in the limit of Sigma and that Omega sub F of Sigma is the space of real-valued functions in Sigma whose limit restriction coincides with F now we did not need a Romanian metric on Sigma to define the action I, but if Sigma is endowed with a Romanian metric and not just a conformal structure, then this affine space Omega F of Sigma also has a natural metric. which I have defined here by giving the distance between two functions Phi and Phi Prime now in finite dimensions a roma and a metric certainly induce since they measure the so-called square root of the determinant if you want to write it in formulas and hope that works in infinite dimensions, we could try to define what I will call the half Sigma of Z, which is the integral of e minus the action on the space of functions whose restriction to the limit coincides with F, that is what I called Omega F of Sigma.

The integration measure is the one that we hope has been induced by that Ramonja and Sigma metric. Well, how are we going to integrate in infinite dimensions? Well, physicists actually deal with infinite dimensional integrals that are very difficult to define mathematically and if not written down. Explicit formulas, what happens with practice? If we have given you afor each one, we can write an explicit formula. RI in this case was a quadratic function Allah Phi, so we are integrating the minus exponential or quadratic functional which is a Gaussian integral and we know that the Gaussian integrals are integrals that are easy to evaluate in finite dimensions either to a quadratic form in dimensions finite if for xmin we are interested in finite dimensions the exponential of minus a quadratic form we can do that explicitly in terms of the square root of the determinant of a or if we have a natural basis in short, then it makes sense to talk about eigenvalues ​​of a, we can write the answer in terms of eigenvalues ​​of a, so we can try to interpret Gaussian integrals and infinite dimensions in a similar way.

First consider the case where the limit of Sigma is 0. The same action I mentioned above can be written like this in a way that makes the conformal invariants less obvious, but some of the differential geometry more obvious, perhaps which is the operator that appears here. the ordinary Laplacian and Delta influence all our quadratic forms and finite dimensions, so we should study the eigenvalue problem. Delta Phi is equal to lambda Phi. It's a good eigenvalue problem, a good discrete set of eigenvalues, and formally we would like to define our integral as the product. of the negative half powers of the eigenvalues, well of course this has a zero eigenvalue that we should discard and it's also a bit divergent due to the infinite factor, but still there are many natural ways to regularize this elegant way of doing it.

It is called regularization of the zeta function, we introduced the zeta function as some general lambda values ​​at least s and then we wrote a formula that would decipher in finite dimensions if this for a finite sum with an infinite set of eigenvalues ​​the zeta function converges to real part large enough of s and we take here its analytical continuation for s equal to 0, this would be our definition of the zeta function and there are many other definitions, such as the poly Villars method used in quantum field theory about forty years ago, if now the limit of Sigma is not empty and we are given a real-valued function in the limit.

We would like to more generally define the interval, not over all maps from a segment to spacetime, but only the maps of the segments, but only the maps that coincide with a flow in the boundary well for F equals zero what we have already said is quite adequate for F is equal to zero we are still integrating a Gaussian integral in a linear space we saw D the eigenvalue problem with boundary conditions in which the eigenvalue disappears in the limit and we do the same definition if F is not zero then, according to classical theorems, we find some unit, there is a unique minimum of this functional subject to the condition that the limiting value is equal to Phi but Phi is not the extreme lysis function and let our integration variable Phi is Phi zero plus some correction, then we can easily solve by manipulations that will be valid in finite dimensions the appropriate formulas, in this case it turns out that the action because it is a quadratic profile is the sum of the separate pieces and If Phi has a limit value F, then Phi Prime those limit values ​​are zero, so by changing the integration variable from Phi to Phi prime, we would create a formula.

Now, to get to this point, we needed a metric in Sigma, we needed to claim that these spaces are Omega. of Sigma had measures and we induced the measures of the Roma instructors in Sigma, so since we need metrics in Sigma to get to this point, these formulas are actually not conformally invariant if we conformally rescale the metric of Sigma which, of course, will not change. is a conformal structure like a Riemann surface, then this formula will not be invariant, however, it is possible to work out simple formulas on how it transforms under conformal transformations for physicists, those simple formulas involve the so-called theory of anomalies and two mathematicians involve the theory. of the bundle of determining lines as introduced in faulting work such as Deline and the singer TN and more recently released and bismuth and others.

I have given a case more generally, if one considers simple generalizations of this, it will lead to generalizations of these formulas whose behavior under a conformal rescaling of the metric implies a fairly rich theory involving the theory of certain Hall morphic line bundles over the moduli space of Riemann surfaces in its mathematical exposition but it would not be practical to go into that here Well, I will simply content myself with saying that I have tried to give a prototype of some of the manipulations that led to investigating determinants of operators such as the Laplacian and for surfaces without limits such that we would not have that factor here.

This is a simpler prototype of some of the things that will be covered in the next lecture. Now I would like to conclude by returning to our more detailed explanation. general issues I suppose it would be plausible to conjecture that, in some proper sense, the number of really fundamental problems in mathematics is infinite, but I personally believe that the number of really fundamental problems in physics is finite, and therefore, by definition, a physicist that you have the privilege of working on these types of problems is fortunate since over time progress will slow down by solving them or by not making further progress if it is true that the number of really fundamental physics problems is finite then the number of episodes in the future in which there will be The truly fundamental interaction between mathematics and physics will be finite;

However, there is reason to believe that the next few decades will be one of those times, so let's begin an informal seminar in which Professor Wheaton will talk about gauge theories and John's polynomials. Whitman's work was already presented to you on the first day, so he does not need any further introduction, so I give a lot of importance to Professor Witten, please, thank you. I'm really glad it's a casual chat, so my wife hasn't asked me to do it. wear suit, so I will describe a particular three-dimensional quantum gauge that touches the Jones polynomial and is in a manifestly three-dimensional form that exhibits its symmetries that are not obvious from the original definitions and also although we have the opportunity to talk for a long time . this exhibits its formal analogies with modular moonshine, i.e. moonshine in monster group theory and a theory about family algebras, modular variants on those theories, a Donaldson theory and other things, and also provides the context in which the quantum of quantification of the group of an ordinary system is established.

The group arises from ordinary geometry for special gauge groups. These theories are related to a few other things, including three-dimensional quantum gravity, Caston invariant, and come in the form of high-temperature superconductivity enablements, so I thought I'd start talking a little. at the end just as an incentive, so remember that when on the first day when explaining Van Jones' work, Joan Bowman explained a little about the scheme and also indicated the history of the scheme, the idea I'm about to summarize and the people who contributed. This is a very nice image where you have three knots. These images are intended as pieces of very complicated knots that I won't draw, but you have a complicated knot and inside the data line you just paste like this or this and you consider three knots that are the same, the dotted lines are a projection, but inside of it a sphere that has been projected on the plane, you have these three different images and you have three complex numbers alpha beta gamma and you define an invariant In addition to not saying that the invariance of three images that are the same on the outside and look like this on the inside weighted by complex numbers alpha beta and gamma sum to zero by elementary induction, this property can be seen to uniquely define a non-invariant if it exists, but the fact that it exists is considerably deeper and, in some ways, this is part from the surprise of Joanie's polynomial and her generalizations behind Fillion kept some of her key tragic and Kauffman polynomials and others, well, the exploration we will get for this beautiful image. is the following, you will see that the limit of this sphere has four points marked where the knots passed through the sphere, then we will find a natural way to associate a two-dimensional vector space which I will call H sub s to each sphere s with four more points It is a two-dimensional vector space and that is why there are three terms in this scheme, making the association of the vector space to each sphere has the property that each ism board determines a vector in this space now by an ISM board I mean a three manifolds that have four limits on the sphere and in which the mark points are limits of pieces of nuts, but not these lines will only allow their end at those four points, so in this image here we have the same limit in all three cases, but The interiors are three different boardings and in this theory each construction is immense, a vector in H sub s and we will have in this they are not giants in which the details of the interior tasks do not matter, everything that matters is the disorder that may exist within us, the size of this vector must be in the Hilbert space in the limit.

Now, a simple fact of linear algebra is that any three vectors in a two-dimensional space obey a linear relation to plus I 1 plus beta, so 2 plus K minus L 3 is there, for some complex numbers alpha beta and gamma, In this case we make the three vectors determined by these three isms of the board and its database in a linear relationship and if you paste them into a more complicated image so that they are small pieces of knots that are constant outside the sphere, I look like this inside then this relationship is a relationship between the values ​​of the invariant for three different notes so the existence of this theory now the characteristic of the theory will be the values ​​of alpha beta and gamma and such a theory predicts the validity of the scheme is very i.e. the ability to define not enjoy it this way for a particular value of alpha beta and gamma to explain skating theory in general read to allow a family of such constructions along the ratios of alpha beta and gamma are valuable as well that that is the answer we will eventually get, although perhaps not fully explained today, for the validity of schema theory, it fundamentally comes from the existence of a non-invariant that can be computed with the details of what is inside any small sphere, it doesn't matter all that.

What matters is the vector that borders and determines in the two-dimensional vector space. Said this way, it is an elementary topological idea, but its geometric realization will lead us to questions that may seem very distant at first glance for the developers of the quantum filter, while we will be making quantum gauge series to whom we choose a gauge group G is a group V which for economy of exposition we consider to be connected and simply connected this simplifies this, it is not essential but it simplifies the discussion because then each G joins into a triple The fold is trivial, we do not necessarily assume that G is compact, although sometimes we will assume it again for simplicity, we just assume that G is simple, in that case even a scalar multiple is unique in a very erratic way in the lie algebra of G. and we denote it as the trace, we notice the pairing of two elements as the trace of its defect, the motivation is that it can actually be realized as the trace in some representation and the normalization that I will give in a second is such that for us you executed this scalar part is the trace in the N dimensional representation, a G package II or for variety the full variety, the islands and variants of secondary turns of the car are not a topological invariant, but an inverted invariant that depends on a connection if we have three none for them and a G. bundle over and a connection in a pond, then there is a corresponding child environment, so the package is called a, we extend the child, the package in the connection over some formula had B and we define the three, the three-dimensional child parents are the integral over the grouping the entire variety of the object which, in a case of manifold, would have two pal times, integer periods, the fact that in closed manifolds the periods are two integer photons, means that for a manifold, a fixed limit depends on the extension only up to a maximum of two pi integral equal to PI, so this is well defined with values ​​in real numbers modulo two five times integers, in other words, it is exponential multiplied by the square root of minus one is well defined with values ​​in a new one, now we will be studying a certain quantum gauge theory in which theelectron is a positive integer K multiplied by this functional I of a duns, this integer K is classified as it turns out to be at the level of a finely unproblematic theory that there is no other standard in the quantum field.

In theory, the discussion has two halves, which is the construction of Hilbert spaces and in this case they will be generalizations of the Jones representations of the degree group and the second part are path integrals where it is difficult to explain this only in my lecture. questions we will ask Ask and the methods we use have not been devised ad hoc for this particular problem, there are quantum field theory techniques adapted to this situation, all to say that there is no unusual situation in which some steps that usually are relatively trivial become non-significant. Now it's trivial, the goal of the first type of discussion is to associate a hybrid space H to each to learn a Sigma fold in each closed room on the Sigma surface and this is generally achieved in the quantum field 3.

This is a comparison, for what we are in three dimensions and in the Hamiltonian version we are in a lost dimension here two dimensions with methods of consumption without limits and for each search we wish to find an average space in this period space that is constructed by quantifying a suitable suitor of the syntactic variety which in general is the moduli space of the critical points of the Lagrangian on a particular manifold of three, which is the surface multiplied by r1 at the time we are discussing the same functional in another context, then, as he explains, a point critical of this functional is just a connection of fact.

In other words, a representation of the fundamental group of M in G now changed by one of em is the same as pi 1 of Sigma, the module space of such representations is the same as the space for modules with four connections or signal and this is the space. for a coin toss, this space is often studied by people who study known surfaces, but here at Princeton it emerges as a representation space of a certain free manifold, which happens to degenerate two representations of the fundamental group of Sigma because M is contractible with Sigma, but in In principle, this space originates with a three-dimensional origin, which is the beginning in our entry level of the three-dimensional explanation of the James polynomial now as preliminary, which is usually trivial in quantum field theory, but here it is not trivial. consider what kind of object is the Lagrangian or a 3-manifold with limit and here I will explain what constants and serials are called prequantization in this context and I will explain it in some detail when it is actually a quantization. go into such detail, soon remember how you defined functional action, we had a three minute egg that we thought without limits, then we considered it as the limit of some point on a hot day and the action is reimbursed as an interval that would be implicit in this is that the limit of M was zero, so M was the limit of some day.

Now we want to let the limit of M be non-zero and therefore this definition is meaningless and therefore we don't have any factor that you want. If we have this term functional silence as a mod to position a valued functional parent, which is the nature of the obstruction that is a curly AUB, the space of connections on the Sigma surface, then we cannot define this exponential of the transient function as a number, but we will define it as a section of a set of lines over the space of connections and, more precisely, a connection in all three varieties has limit values ​​on a and for each point on a we have a dimensionless complex vector space, the Fabri line below that point and this won't remember it, but it will take values ​​on the grain of that line level, so the set of lines is really defined by just asking what the obstruction is to define this, since the obstruction was we had three open collectors and who knew how to close it. we could relight an ember, so we simply exploit that we systematically have this phase of connections to describe a set of lines over the space of connections, we must for each train to that line in that space describe a one-dimensional complex vector space V sub a when We declare that the value DC has a basis which we call prime for every extension of value or for any three manifolds in Z is a t3 manifold and a prime is just an extension of the connection over Z, in fact we also extend the bundle and connection about Z. then, for each such extension, this line has one base, one point.

Well, I need to tell you how to compare two different basis elements and I can do this because of a prime and L double prime or to such an extent the shions individually do not have Ave chern -simons indirect but they have a Simon invariant relative spin because they have the same limit by definition, so if I clean them together with opposite orientation we are going to close three varieties and then the connection on the three close varieties has a member of a contact number Valued invariants furniture said that a prime is a double prime multiplied by this number, so now, although we have no natural basis in this one-dimensional vector space, we have defined a one-dimensional vector space for each connection on the surface and therefore as the connection on the surface y values ​​vary in the space of all those connections, we have a line error l over the connection space in Sigma now this bundle of lines has real units, they are a connection, in fact, they forgive in a path in the connection space, so a1 and a2 are connections in sigma on the surface Sigma and God have a path between them in the space of connections that path gives us a receiver on a given triple manifold the triple manifold is Sigma multiplied by the interval, then 1 Sigma multiplied by the interval unitary we take the connection which is trivial in the statement in which at this address there is a 1 at this end and a 2 at this end and Enterprise between them by the given path, so it is a connection on a particular manifold 3 and then , if you tell me, you had to extend a 1/7 3 manifold then you had to spend a tear on some free manifold because the interpolation from a1 to a2 that we just discussed can be glued to your interpolation and that gives the unitary connection in this model of line in other words, if you give me a trivialization of the fiber on day 1 I have given you a trivialization of the fiber in a2 that depends on this path and we can talk with that a spray I will transport the trivialization here to a trivialization here following that path now let's jihad be the group of maps of this surface in G, then G that acts in the connection space along Sigma and with a smile effort that actually requires an elaboration of the site of what I said , the Jihad action is raised to an action on this bundle of lines that preserves the connection now we have a projection from the space of connections to connections modular width transformations we can ask if the bundle of lines with connection There's no getting around the standard criteria if you take a small downward loop in the next tutorial above, all the anomie in the loop will depend on the left and therefore there's no way to make these things descend, but if we just stick to five tensions, the space of five connections in Sigma, then from all input considerations you will see that this bundle of lines in the connection descends to a Rundle line in connection over the space of fabulous connections/gaze transformation and that was the space denial of file connections when the surface I indicated above was going to come in, so we get a bundle of lines with connection over the space. of the thought connections in the signal, its curvature can be calculated in an elementary way and is the standard syntactic structure that Atia first related to gauge theories and in a paper in which kernel and egg mention that it was the structure standard syntax in this variety, now these things have been studied in the past and other methods and in particular this line bendlin connection in special cases is usually constructed by choosing a compact structure when Sigma and using algebraic geometry if the complex structure is taken in Sigma so that Sigma is a complex Riemann surface. then according to the Seshadri nation theorem, this space of modules is a modular space of the local value of bundles in Sigma and then you have the bundle of lines determined with its stroke or Krillin metric and its curvature calculated by: it is precisely the same curvature resisting, so there is a complete Norfolk construction in special cases such as the unitary group or an e group for special values ​​of K there is a whole morphic construction of this package of lines that depends on choosing a contact structure or in Sigma what What we have achieved with the construction is that We have given a purely topological construction that does not depend on any contact structure, the group of mapping classes acts in a way that is essentially visible and at the same time we have related the construction to 3 varieties, which we have achieved is in fact, that call center stuff called three quantization in this problem, we have built what is called the three quantum line model and we have built it in a way that does not break these topological invariants of the problem and, as you see, The construction was not related to thinking about the invasion of 3 varieties, which is what we want because we want to do a three-dimensional problem, now going back to the original question I posed, that is the truth, the layman's invariant is whether a is a connection in a three. monetha with limit where we see that it tautologically makes sense as a section of this line bundle that we introduced or more accurately a vector on the appropriate fiber of that line bundle because any trivialization of the blow bundle that is an extension abandons the free manifold in the extension Attention allowed us to make sense of that thing as a number, so this is prequantization in this situation, now nano Lia prequantized, that is, we have built the appropriate line package over the correct module space that has to write a unitary connection below. curvature and without breaking any of the symmetries of the problem, but we are in a very good situation because this space of modules is the syntactic quotient of an alpha in the space, the affine space is the space of connections and you see, we take two steps that we restrict to file connections and we divide them by the gauge group and the other.

I will not explain if those combined steps are called by mathematicians making syntactic quotients and for physicists they go back to the use of Newtonian symmetries to reduce a call in mechanics anyway these quite complicated modules fat claw space we are seeing in relation to an alpha in space a with a dual action initiation, of course, between the two, it was not exploited by the aunt and bought to calculate the ecology of this module space now here we are Seeing the sea we will use the same relationship in a different way which is appropriate for this particular quantum theory.

Now we want to quantify and I won't have time to explain quantification in as much detail as I did for three quantifications, but quantification means. associate a hybrid space H that should be canonically associated with the data and the data is a modular space and the bundle of lines with connection, but this data was periodically determined by this surface signal as we say, so we will really be economically associating a Hilbert space. with the surface signal now, what quantization means in general is a long story, but if you really are a natural complex structure J in this man's living space that obeys certain conditions, the most important thing is that the syntactic structure of the code , the curvature of this connection must be of type 1 1 for the alignment to obtain a real homomorphic structure, furthermore, the syntactic structure must be positive relative to that compact structure, if there were a real compact structure, then the quantum Hilbert space , Rigby, you see, if we had such a complex structure, then our module space we take as a complex variety and the conditions mean that a Moghuls has a complete package of Norfolk lines and at level K we will take the power of the cave of that miller and we will remove his harem of exceptions and that with the Hilbert space for xi a theory. with this gauge group and associated with this Sigma surface, there is no real complex structure in em, but for each trace of complex structure and this underlying surface signal we have a co-administration Hodge decomposition.

You see the tangent space for these modules. Space is Sigma's durian ecology with values ​​in a twisted package and if we apply it in context, then it has a skin for decomposition and that decomposition is actually a compact structure in this module space, in fact, it is precisely the structure compact predicted by the nation as return but I will call it by the same name Joe so for every complex structure J on the surface we have a compact structure J in this modular space and this by definition allows transparencyvery compact structure J we can make this definition be If J is not natural nature, then neither is Karli H, then we have a curly age definition, but it depended on J, so aah said je t'aime because of the dependence on J.

Well, for starters, this situation is fibroid in everyone's space. complex structures in Sigma, but it is trivially reduced to the space of complex structures up to isotopy or, in other words, to take up more space, so we really have a situation where five words require more space. We will have this family of Hilbert spaces favored over the shot. more space actually or without mentioning it I began to assume a slide or two ago that this well, I did not prove that the composition of Hydra gave a composite structure that really at that time assumed the case of a compact group for all classes of With the groups Li we would get something, but not exactly this, so assuming for the moment that we are dealing with a compact, now we want a single Hilbert space and we have a family of them and of course back to the single Hilbert space situation, but we know that there is a natural connection in this package, a connection as thick as the taken bays or contractable spaces, such a thick connection would give us a natural identification of the virus, said connection can be built explicitly and rigorously as I did above. long these horns with excellent to the pho using two facts, the first fact is that if we were quantizing an open space, there is such a thick connection due to the name stain in the pheromone, the unique representation of the Heisenberg group and physicists think of this tract connection as the international generator of what we call Bogoliubov transformations and then one can push down from the knife, one assumes, to its impact equation constraining to the invariant subspace G for dimensional faults in spaces, these two facts would give as result in a fire here, the Atlean space is of infinite dimension only the syntactic quotient is of finite dimension saying that what one has to do is take the formulas that would be valid a priori in the case of finite dimension and do a little geometry to verify them and actually modify them slightly so that they make sense in this case because only below you have a finite dimensional story, well, the thought connection is obtained this way or I have written it in the player because there were different groups and novels, these actions from Phalke since the mapping class group acts here. really maintained the group's representations of mapping classes and, as first perceived, Tsuchiya and Konya, the inner work that is the best part of what made cellular development possible, these are generalizations of Joon's representations of the group large to obtain precisely the Jones representations that you wrote logical equation of the Keynesian test that we have I have heard a lot in so many different talks at this conference, starting with Jimbo's talk explaining some of the work of a dream trip and including unexpected applications theoretical tunes that were mentioned yesterday in one of the plenary speeches, now this build this particular file connection. for compact G has been obtained in other ways, the original results came from nefretiri control about six years ago, as the condition worsens.

I am marching in general and Iguchi a new agreement some years later in genre one and there is a Thursday that we are studying. Well, it's sometimes called the recipe rotation curve model of conformal field theory and then these constructions were rigorously formulated as an arbitrary genus. Susheela at all. Shishio talked about it yesterday and there is a simple argument: this is a synthesis of an argument by Siegel in the Swansea Proceedings from two years ago it is also possible to construct this connection through pure algebraic geometry and this has been done by Hitchin and through balance and enclosure The construction that I have indicated is the naturalization to the three dimensions and the sense that it is what arises naturally. as the solution to our quantification of this prequantized syntactic variety and extends to other classes of indicator groups, which we hope may also be the case for some of the other approaches, ultimately, in fact, with some minor modes, Relatively minor modifications from where I said it is.

It is possible to carry out a similar construction if G is a semi-simple compact Heron. Now maybe I'll denote a transparency for a soap digression. So far there are new results on this problem, so other steam classes I like to study especially for The real non-contact shapes, even in the simple case of sl2, now have a very precise sense, so we take a small field to really explaining this topic is a generalization of our presentation and, from that point of view, the fact thus expressed is not. It is too surprising to say that presentation, you know well that the compact groups are the easiest, followed by the contact strips and that the known ones have real shapes in an order of magnitude more difficult and it is difficult to study in a unified way, so I will tell you he said again in this context. had a transparency that alluded to different approaches for contact groups and the corresponding column for contact groups is definitely more difficult and not only well understood but manageable is the next case after the compact groups that leads of course , to hybrid spaces of infinite dimensions simply as in the representation tree where the unitary representations are of infinite dimension and the solution of this problem for compact groups is related to compact groups as much as in the representation case, since there are no results for the real forms that are not combat and are not learned. what will happen, but either there is some reasonable way to solve this problem that could well lead to some of your views on the current representation or this problem could be affected with more elaborate versions of the usual difficulties in the representation tree than you now We reached the deepest part.

The story and I want to emphasize that what I have said up to this point is very precise mathematically. To proceed we will have to rely for a while on the massive experience in theoretical physics over the last 40 years on variance or we will. go back to your thing to say rigorousmathematics, as I have heard, I could say that, as a general rule, I am explaining a particular plan until three and at this particular point of filtering the deeper sites are based on the ideas that I will indicate, but there is a sendou statement that Something can be done about other quantum field theories of geometric constants of interest if one is interested in what Conway and the Calipers called "monster moonshine" in monster theory and believes that the quantum theory explanation of fields is based on the same farm and patterns of Although I am about to tell you along with the field theory as constructed by Franco Lebowski and Lemon, I am sure that in general I am about to tell you the aspects that Feder of stress and his pause in his opening day lecture the ideas aspect of quantum field theory, which are perhaps the majority and I'm just about North America, but which generally lead to the most profound results, so the heuristic motivation for this is that we've done an association of Hilbert spaces with surfaces by choosing a gauge group at a level K and quantifier and this association is not analogous through the association of a surface with its, say, complex:ology first:ology in each case, 50 surfaces of morphism induce some map in these spaces and in the case of comala G, if you take a band and three varieties that actually determine a Lagrangian subspace in the algebra of the first coin, that is, the subspace of things that extends over the three varieties and you can ask if this or monarchy is a surface there with its first with its hm, but if you give yourself three moons suddenly that determines a set of space and preferred substrates of H one, you can ask what is the analogue here and then just for fun let's guess what it could be of course i will tell us obamas apart so every connection is a sigma. had a natural extension a prime over m, so we were given a three-manifold Sigma and we constructed a Sigma surface and we constructed a Hilbert space by quantifying into the space of connections in Sigma valid for each connection and Sigma we knew how to extend it.

So if each connection had a natural extension, then since a connection, an extended connection has a security of aemon and a variance not as a number but as a vector, then nine, if we had a natural tension in each connection, then We could define the natural section of the song package we've been talking about, well, there's no natural way to extend the connections, in fact, there's not even an ongoing commitment, but in what many consider his greatest vision of Richard Fineman , taught us more than 40 years ago, what to do, since there is no natural extension, we average the extensions and each extension determines while the same vector finds the corresponding line, so the average of the connections is a vector on the line and, as the boundary data varies, we consider forgiving the connection. on the surface, all its original fittings and extensions give us a vector and, waiting a moment, we get a section of that line below that I told you about and this is the human recipe for calculating transition amplitudes now, of course, I have made to sound.

As a passively naive idea, there wasn't enough natural extension, so we averaged them out, but it's not very deep in Foreman's context: riding in a Ferrari wasn't set up in such a way as to make that turn look that much like the obvious. . On the other hand, what Furman was trying to do is similar to what we are trying to do to understand the polynomial of turns. In those days there was a theory called quantum electrodynamics which was actually a four-dimensional theory, but it was only formulated with three-dimensional symmetry now Feynman was trying to exhibit trance electrodynamics in an obviously invariant way in four dimensions here we have a theory, the Jones polynomial and yes generalizations, which is actually a three-dimensional theory, but the known formulations at most have two-dimensional symmetry or even loss in the original versions and we are trying to exhibit it as a three-dimensional theory and we were simply basically following the steps from the huge Lisa and the influencers, now, a little bit, we try to make the notion that there is a way to average this.

The infinite dimensional space of extensions sounds a little more acceptable, but first let's state some of the consequences. First, if you think you can average the connections, you can do it even in a three to unlimited contract or in these two contracts to consider the case. where the boundary is empty, so if we are given a closed well of three varieties, we consider the space of all connections or related connections as 1 m modular gauge transformations and we want to average the modular gauge transformations of the connections or, more Conveniently, we usually average all connections. including the 40th power of the ghosts to account for the volume of the gauge group, we want to average the analysis of all connections anyway, we have a closed manifold, the Chern-Simons invariants are actually a number, so that the answer is a number that is a numerical invariant of three mana fit now, this is a very large integral and maybe it's too big for a graduate student to do, so we can have two graduate students, so we divide the problem in two parts, we take, let's find a division in goat of the three varieties, so we have a three-month armpit for it.

I considered as a union of M and n MN our three varieties with a common limit with opposite orientation, so they fit together to make a nice three known of that via connection extensions here while the other student practices via extensions here and one student gets a functional of the Dola limit and of course the other student gets the other functional of the limit data and of course if we fix the boundary data and enter our general extensions, then we have integrated everything except the connection values ​​into Sigma and maybe the students advisor should do at least some of the work and then he will do the final integral on that to that the two students I have constructed these two functionals of limit data and then maybe the consultant integrates them with the limit dam and that final integral means that we finish the job and we have the invariance of this new variety, but we have obtained that invariant by matching two pieces one. functional limit data obtained by astudent and another functional obtained by the other student now these two functionals were sections of our favorite line package saying that we found that while each three varieties determines a vector in this space or the other due to the opposites The orientation determines a vector to take care the vector and the dual doctor are our three desired manifolds and ions and therefore we have learned that we are dealing with invariances that are computable from the smoking cessation guidelines and more general considerations along these lines B to through a good formula for behavior under bone surgery, so I have not explicitly introduced non-invariance, but no, and giants are similarly introduced by considering a variety of three with knots and considering friends outside the connections around those nuts and after formulating the problem for nighttime invariance. so when we can get a little bit further into this formula by which we calculate from the Egon changes, then we will get a formula for Bayview under Danish surgery and that problem is really what makes the theory computable because the t-manifolds 33 are related by them in surgery and for example if you want you can use Dame surgery to reduce anything you want to know about three multiple two moths in s3 or somehow it is even simpler to reduce them to degrees in us twice s1 because if you cut the blades and s 2 times s1 then you have really reduced this more general picture to the original journalistic representations of the pedra essentially and they are cousins ​​for higher representations and groups and ultimately you got combinatorial formulas based on pain surgery and these formulas have been adopted as the definition in combinatorial approaches eh Russia took a lecture on the work that Pasha Deacon interrupted to rigorously justify the unity of combinatorial formula in this way verifying the generators and relations of the essentially Kirby calculus in an elegant way that depends on the relationships involving quantum groups that press the It is true that they used it, but the ingredients that come in and in doing so are the so-called quantum symbols 6j and D.

The quantum symbols 6j themselves have a natural representation in three dimensions in terms of these path integrals, which again fits their essential properties and as Another side to everything I mentioned earlier when I realized that we were getting the analog x Keynesian equation that one knows is related to quantum groups, so Different ways of looking at this make different aspects of driving quantum groups from the quantum filter. You see the main work that two groups of physicists Pantin are funny things because you could say that the name finds it and makes it sound like physics and then maybe it does, but what painting is, you are being used in a rather vague sense, plant It means that an algebra in this sense of quantum contacts is used to mean that the marriage that is commutative is large to form our computer, so it doesn't have those properties anyway, but quantum mechanics is actually a little more specific than just warping commutative algebra into non-commutative algebra, so quantum mechanics is really about quantizing some types of manifolds, the algebra being warped is the algebra of functions on a syntactic manifold and is too firm for a algebra of non-commuting operators on kantham groups in Hilbert space, since cannulated so elegantly involve the definitions of a Paulsen manifold, but not of a synthetic manifold, and as such are actually a bit outside the conventional framework of quantum mechanics and are pretty good three if you want to take the quantum group and reduce it to a more standard object which is quantization of syntactic varieties or once a complex structure respects algebraic geometry wine packet sections over that module space M sub J this much larger story with the space of connections under a three-dimensional coil filter takes the chords in a group in a framework that is really quantum mechanics and quantum quickly, of course, the drawback is that we are in a much larger picture than is less obviously computable.

It also often happens in mathematics and physics, the most computable version, which is the quantum group, does not exhibit all the symmetries that were exhibited in the 3-group. dimensional history once you think of this as one of the many cases where there is an invariant definition which is the definition involving quantum field theory in three varieties and there is a computable definition, a manifestly computable definition that doesn't matter, worry because of how the symmetries are. which is the one involving the reaction to quantum groups and combinatorial approaches D. Now I think I should spend the last few minutes giving you a favor of what path integrals are close.

I will continue as I should have said at the beginning of the chapter. The early Albert Schwartz guys who considered the Abelian version of this theory and many years ago in the '70s wrote it about what a monstrous race recursion is, so I want to say something that's mathematically precise about path integrals. , although it does not give the full story. and it's a challenge to tell the whole story so what I tell you is that if you give a physicist browsing through a quantum filter that was found behind a bush and has no particular redeeming virtues, I'll tell him everything there is to say. about it that I Catherine today is based on general ideas, so the only general thing that can be said in terms of actually calculating our money, yes of course, even if an integral is produced in quadrature and finite dimensions, it is generally not can calculate and search for something. asymptotic expansions and even a can of somatic expansion here, which is an expansion and powers of 1 over K, so the situation we are in is that we have a large Internet movement and a variety and reintegration of the same and the same Correct classification is exponential if I am something more than K, so the prototype for this problem I am going to simplify so that we actually have the group of indicators.

Oh, we should worry about the ghosts of federal ghosts, but pretending they weren't there, the prototype is a finite-dimensional manifold. with a measure or the Norris function, it does not have to be any more function in general, its class of functions is allowed, what we take for simplicity of a smooth function with our critical points located and then we have a real number and then we want to calculate this integral and of course we must calculate it in closed form, we can show that it exists, but we cannot contribute, but it was calculated in an asymptotic expansion in powers of 1 over T and that is essentially a classical. the integral is given by the stationary phase, in fact if F was not a function but had values ​​in our mode 2 pi Z then Tillie was an integer so there were no critical points, this would vanish exponentially for large T due to the oscillations so it does not disappear exponentially due to the oscillations it comes due to the critical points because there are no oscillations Gas Gibbs and the pilots are driven by the stationary phase the PLR ​​the critical points so this has a systematic expansion in value the same about the critical points where the main term is the value of the integrand at the critical points of the earth converts a power of T that depends on the dimension and then multiplies by the local expansion determined by local data at the critical points, so these are the coefficients here, they are local invariants at the critical points and constants if they are close to a given critical point you do an expansion so I choose some local coordinates and I want them to be tailored so that the Damier is 2 X 1 up to that animal involves the determinants of the mode, well, this thing has a determinant due to the existence of a measure here and the condition on the coordinates. involves the Hessian determinant and involves the signature of that quadratic form, so the main contribution here involves these nice, de jure variants near the critical points and then higher order corrections can be computed involving local values ​​near the critical points from ranee to the only new thing you need is to invent this invertible matrix and then the expansion can be canonically described in terms of what physicists eventually call controllers.

Now this can be systematically imitated in our problem, in fact that's where they are called Feynman diagrams. Well, there is no problem with the notion of critical points to explain that the critical points of this functional are a fantastic connection to measure the transformation and for a large class of three varieties, they will be isolated in general, we need to understand how to generalize the above story. which will imply certain intervals of measures that your construction well, what happened in fact is that if the critical point is not isolated, this determinant is not a number, but is naturally a measure in the space of critical points, so After a proper understanding of the finite-dimensional history, you can imitate it.

Here there is no problem with the notion of critical points, of course, there is no problem with the notion of the value of deterrence. Homans functional outside the critical points, that's one of these standard invariants of a fop connection, while Hutchins makes sense. after including the ghosts, the hash spin is a certain elliptic differential operator which I figured out well, so we need the determinant of L, which is essentially what Lansing defines, then Schwartz interpreted it in jet theory, so the determinant of L is the run and recursion and then we also need the signature of L, well it is a quadratic form in infinite dimensions so it has infinite positive and negative eigenvalues ​​so we need to define it, he said you just need to regularize that between the top 30 and the singer tells us how. to do that, then do the so-called owner invariance and in this problem add the correct move of the signature of L so that everything expands the leading contribution for large K to this path integral, so if you want, we have evaluated the route and go in the main approximation of the stationary phase and involves variants that are classic, that is, the most recent were defined in the early seventies, but they are still quite sophisticated things, which are the routes invented by the ADA and the crutches that are just beginning to be investigated mathematically systematically involve what physicists call Feynman diagrams.

My student, born Elton, has worked with enough precision through the cracks of the night and violence that he has trained the doctors alike through you again the classical night resistance, which is the derivative of the Jones polynomial. at Q is equal to one, as was the second derivative a, as expected, then the story is that we can test the Hamiltonian picture more rigorously by obtaining Jones' representations and they are generalizations, the deeper side of the story, of a certain way to have eternity in image that is perhaps a challenge to understand it comprehensively in the elites of perturbation theory by leading the approach to these totally advanced classics with a systematic prescription to generalize them and calculate more elaborate crutches perhaps rather in some way in which in some way perhaps rather in the primary expert of the province who certainly could not pretend to understand it explained in the first lecture some generalizations of the northern unions yes, thank you very much Professor Witten and the informal seminar is over thank you for the attention thank you now professor fatiah will talk about Witten's work for Michel Atia it is a duty of the presidents of the field committee to request speakers for that session so that as soon as it becomes clear that Witten obtained the medal I asked a chi to come closer and that here asking to speak for him wanted something like that. cover-up of a mathematician in the very faithful domain to explain to her why it was so natural to give the Fields Medal to mathematical physicists and that she was very enthusiastic about this, but he told me that unfortunately he cannot come to Tokyo to Tata in the end.

August, so we decided as follows that he will send me a text and I will use it to describe the prohibited work in this session, so I got this text and I will read you some excerpts, but it does not contain formulas at all it is just for reading, maybe it's a little too much. You've seen slides before, so in a way it will be a mix of ideas text and some general comments that I will give you myself and some explanation, what mathematical physics should be like. considered as legitimate, I think it should be considered as a legitimate part of mathematics, it was always stimulating, as it always was a source of stimulus and inspiration for more branchestheoretical aspects of mathematics and we know many classic examples, but until recently it was stimulating for most. analysis or in general or part of the analysis, such as partial differential equations, especially non-linear equations nowadays, etc., we could call it classical mathematical physics, but in more recent times the development of another type of mathematical physics has been observed that could be in contradiction. with the previous so-called quantum mathematical physics and this mathematical physics mainly differs also from classical mathematical physics in the use of different domains of mathematics in it or in other words, it also stimulates different parts of celestial mathematics algebra geometry topology algebraic geometry complex analysis All of these traditional fields of mathematics now reach modern mathematical physics and vice versa.

The mathematical physics problems that arise there in quantum mathematical physics are sources of new problems for this part of the risk of mathematics and now I read the first quote of a joy in all this great an. exciting field involving many of the world's leading physicists and mathematicians Edward Witten clearly stands out as the most influential and dominant figure, although he is definitely a physicist here. I would add that he is a mathematical physicist, his mastery of mathematics rivals few mathematicians and his ability to interpret physical ideas in mathematical form is quite unique, time and time again he has surprised the mathematical community with a brilliant application or physical knowledge that leads to new and profound mathematical theorems now in a TS text, while the elaboration of these general statements and generally the reference is to an important tool of modern mathematical physics called the Fineman integral Reimann integral allows us to explicitly write in quotes the solution of very complicated problems that arise in quantum field theory, quantum mechanics, etc.

The quotes mean that this tool is still a heuristic and does not come with rigorous and rigorous facts with the standards that are supposed to be used now, they would certainly satisfy, I think more mathematicians in the previous century, but even for people like le Cortina, but without more higher standards of rigor, they are still not rigorous of On the other hand, of course, as I will try to explain with the help of Archaea, it is also really a very important tool for discovering new mathematical theories, so the ingredient in the interval Fineman's is a function of some fields, which means functions in some multiple domain. so the functions might not necessarily be functions but sections of the package etc. so they could have a geometric origin and this function is called an action and then something from X to the I where I squared minus y times the action is rineman functional and the final integral is a kind of averaging this functional over all possible fields in consideration of its implicit values ​​that it will overmeasure or cannot measure is in the strict sense this local which is a product over all the points in the domain or variety of differentials of these fields, so to speak. so the answer is a number of course depending on what we probably have in iteration, for example we have a principal in the limit or we have some asymptotic conditions, then the standards that characterize the domain of this geometric object are average over the function would also be functional for this data and therefore it is a very rich object, so of course the physicist after Fineman after 48 used it mainly for his own needs and then the final action of the APIs in the year, but of mechanics, some Hamiltonian system, but in more general views.

As mathematical physics gradually developed at this point and especially because of the influence of very modern things there, in particular, what is called sink theory, people started to use more geometric Jenny objects in these actions and fields, so Examples of possible fields could be loops or connections, called young. wheel fields in physics or matrices in physics, would be more useful in the case of gravity, etc., and so on, the writings have shown a very ingenious way of using the functional integral to produce purely mathematical theorems. Now I shed a tear at his article. on supersymmetry and more theory, yes, excuse me, Witten's paper on supersymmetry and more theory is required reading for geometers interested in understanding modern quantum field theory.

It also contains a brilliant proof of classical Moore inequalities relating the critical points to homology, mind, the main point is the homology is different, we train harmonic forms, so the Laplacian will play the role of the Schrodinger operator of the quantum mechanics and you can write the Fineman integral using this action.