 Good afternoon. Okay, so it's a great pleasure for me to introduce today's speaker, Professor Edward Witten, from the Institute for Advanced Study in Princeton. This is intended as a somewhat specialized colloquium on theoretical physics, but I'm glad to see many people here. So Edward has made deep and far-ranging contributions to many areas of theoretical physics, in particular string theory, quantum field theory, and also general relativity, and also mathematics. In fact, he has, of course, received many awards. The list is long, but among others, he was the first physicist to receive the Fields Medal in 1990. And he's here for this ICTP hurdle. We have this new concept on quantum entanglement, which is time and black holes during this week. So we are very happy to have him as a participant here. He was the first person to receive, together with the great Soviet physicist, Yakov Zeldovich, the ICTP Dirac Medal in 1985 for his work on anomalies. And as you perhaps know, Dirac was a good friend of the center who visited ICTP many times. And I think I can say that if there is any modern physicist who could be compared with Dirac, it would be written. And I don't say this to flatter or to embarrass the speaker, because I'm very aware that one should not lightly toss around comparisons with incomparable Dirac. But I think without exaggeration, one can say that in style and content and influence, the parallels are not far-fetched. I mean, Dirac had this exceptional ability to distill the essence of many ideas and synthesize them into a unified framework, as it did with quantum mechanics, which is what all of us use today. But that also had great implications for mathematics with the theory of Hilbert spaces, operator algebras, theory of distributions. He also introduced the topological methods in physics with his work on magnetic monopoles. And written likewise, I think has been, I have seen this witness in my own career, has been the leading light in many of the critical breakthroughs in quantum field theory and string theory from his work on anomalies, dualities, holographies, often synthesizing many disparate strands into coherent story. He also developed new topological methods in quantum field theory, which had a very deep impact on mathematics, in particular, topics ranging from mirror symmetry, index theory, geometric Langlands, cyber-mitem theory, not theory. So the list is long. And I think this is very well summarized in the citation of the Fields Medal. Time and again, he has surprised the mathematical community by a brilliant application of physical insight leading to a new and deep mathematical theorems. So we are very honored to have him as a speaker today. Of course, as he will explain, I mean, string theory is this effort to combine quantum mechanics, which has been ongoing for several decades now. And I think it's fair to say that the rich harvest of these ideas has not yet been fully reaped. So I hope he will give us the flavor of what it means. He's also a brilliant expositor. I was fortunate to take a course on general relativity and two courses on string theory from him. And in fact, those lectures later on became the classic book, The Green Schwarzenwritten, which was the defining book for several years in string theory. So I look forward to an exciting talk from this. Thanks for the very kind invitation. First of all, for the introduction to this talk. Secondly, for the very kind introduction. His only drawback is that it's impossible to look to it. So I will be talking on the title of what every physicist should know about string theory. And I know that the younger string theorists sometimes are unhappy with my approach to the subject because I'm going to be describing, what first was I wrote here, the minimum any physicists might want to know about string theory, even if you're interested in other areas. But the things that I'll be explaining are some of the rather classical results in the subject that mostly go back to the 70s and 80s, although I think they're still very fresh. I'll try to explain the answers to a few basic questions. How does string theory generalize standard quantum field theory? And why does string theory force us to unify general relativity with the other forces of nature while standard quantum field theory instead makes it so difficult to incorporate general relativity? Why are there no ultraviolet divergences in string theory? And what happens to Einstein's conception of spacetime? So these are the topics we'll be talking about. Now, anyone who's studied physics is familiar with the fact that while physics like history does not precisely repeat itself, it does rhyme with similar structures at different scales of lengths and energies. We're going to begin today with one of those rhymes, an analogy between the problem of quantum gravity and the theory of a single particle. Although we don't understand quantum gravity very well, it's supposed to be somehow a theory in which at least macroscopically, we're averaging in a quantum mechanical sense over all possible spacetime geometries. Now, we don't know that that description is valid microscopically, but at least macroscopically, it's roughly what quantum gravity is supposed to mean. In the simplest case, the averaging is done with a weight factor, which is the exponential of minus the action. I'll write this in Euclidean signature. In Lorentz signature, we'd have the square root of minus one instead of minus one, multiplying the action. And I is the Einstein-Hilbert action of gravity, which I've written for the simpler case of gravity where the cosmological constant lambda. We could add matter fields, but in general relativity, it seems we don't need to add them. So let's leave them out for the moment. Now let's try to make a theory like this in spacetime dimension one, rather than in the four dimensions of the real world. Well, so that means that spacetime is supposed to be a one-dimensional manifold, but there aren't very many options. Topologically, a one-manifold is going to be either, let's say a one-manifold without boundary, will be either a line or an open interval like I've drawn here, or maybe a closed loop or closed circle like this one. Now, in contrast to the four-dimensional case, in one dimensions, there's no Riemann curvature tensor. In other words, a one-manifold has no intrinsic curvature. You could stretch it straight without tearing it because there's no intrinsic curvature. In contrast to a two-dimensional surface like the surface of the globe, which can't be flattened without stretching and tearing it. So there's no close analog of the Einstein-Hilbert action. However, we can still make a non-trivial theory of quantum gravity that is a fluctuating metric tensor coupled to matter. Unlike the case in four dimensions to get something interesting, we will need to include matter, so let's do so. And we'll take the matter to consist of d scalar fields, which are whole xi, i runs from one to d. The most obvious action would be kinetic energy for these d scalar fields. And I've written the kinetic energy using the ideas of general relativity here in one dimension. So g is a one-by-one metric tensor and the usual kinetic energy would be one-half x dot squared, but here we have to include the inverse metric tensor. And I've also allowed for a possible constant term in the action, which I've called one-half m squared. It plays the role of the cosmological constant roughly in four dimensions. Now if I introduce canonical momentum p, which are dx dt, then the Einstein field equations, which are the field equations, we had an action that depends on the metric. The Einstein field equations are just the field equations for the metric tensor. And you'll find that the field equations, in terms of the momentum, just say that p squared plus m squared is zero. And Dirac taught us what such an equation is supposed to mean quantum mechanically. So quantum mechanically, p is an operator minus id by dx. p squared plus m squared is a differential operator. And the meaning of the equation is that the wave function psi of x should be annihilated by the operator on the left-hand side. So here, the meaning of the equation, of the Einstein field equation, is that the wave function should obey this differential equation. And it's an equation that might be familiar if you've studied relativistic field theory at all. The operator that appears here is the Laplacian in a d-dimensional manifold whose coordinates are the x's. So this is a familiar equation, if you like, the relativistic Klein-Gordan equation in d dimensions for a field of mass m appeared in this constant here. Except it's been written and we've gotten this equation in Euclidean signature. But Lorentz, to give this equation a more sensible physical interpretation, you might want to reverse the sign of the action for one of the scalar fields so that now the action would be like I've written, like so. The same as before for all of the scalar fields except one, which I've called x zero. And now the equation obeyed by the wave function will be a Klein-Gordan equation in Lorentz signature. So we found an exactly soluble theory of quantum gravity in one dimension. Let's describe something. It describes a spin-zero particle of mass m propagating in a d-dimensional Minkowski spacetime. Well, we could generalize this a little bit. We could replace Minkowski space by any d-dimensional spacetime m with a Lorentz signature metric G, capital G, or Euclidean metric if we prefer it. And the action is a kind of obvious generalization of what I had before except that the Lorentz signature metric which was implicit in this formula with the minus dx zero squared plus dx i squared would be replaced by the corresponding spacetime metric G, whatever it is. So this version of a one-dimensional quantum gravity theory would give us a spin-zero particle in any Lorentz signature spacetime. The equation obeyed by the wave function was now a Klein-Gordon equation on capital M, the spacetime m, or a field of mass little m. Now, just in going on, I'm going to simplify the notation a little bit. So I'm going to abbreviate this p squared that depends on the spacetime metric as just p squared. And to avoid keeping track of some factors of i I'm going to write formulas in Euclidean signature. So in this theory I want to calculate the amplitude for a particle to start at a point x in spacetime and end at another point y. So x and y are points in the spacetime capital M. Quantum mechanically the particle is allowed to travel on any path at all from x to y. And what Feynman taught us was that the quantum mechanical amplitude for propagation of a particle from x to y is a sum over all possible paths weighted by the exponential of the action. However, here we have to consider both the path and also the metric on the one manifold. Here, the path itself is a one-dimensional manifold and it had its own Ramanian metric because we were doing gravity in one dimension. And so now we have to integrate over the metric on the one-man manifold module with a few morphisms. But in one dimension, Ramanian geometry is very simple. A one-man manifold has only one invariant, which is its total length tau, which we interpret as the elapsed proper time along the path. For a given tau we can take the one metric to be just gtt equals one where the range of t goes from zero to tau. And now to do the Feynman path integral on this one-man manifold we have to integrate over all paths x of t that started at x at time zero and at y at time tau. That's the basic Feynman integral of quantum mechanics with the Hamiltonian being p squared plus m squared. The operator that remember annihilated the quantum mechanical wave function. And according to Feynman the integral over all paths is the matrix element of the exponential of minus tau, the elapsed proper time, times the Hamiltonian h. I'm told to switch off the mobile phone but I don't want to figure out how to switch it off. Feynman told us that the output of this path integral for fixed tau is just the matrix element of the exponential of minus tau h between the initial and final points x and y. We can compute the matrix element in momentum space. The eigenfunctions of e to the minus tau h are plane waves, e to the ip dot x. And the eigenvalue is p squared plus m squared. So a momentum space integral weighted by the exponential of minus tau times p squared plus m squared gives us the kernel for propagation from x to y in proper time tau. That would be in any quantum mechanics textbook. The gravitational part of the problem tells us that we have to integrate over tau. So if we simply take what I had on the last page and integrate it over tau from zero to infinity, I get the amplitude for propagation from x to y. But if I simply integrate over tau, this exponential of minus tau times p squared plus m squared, it goes through the denominator. And what we get, in fact, is what I've written here, which you might recognize as a standard Feynman propagator in Euclidean signature. An analogous derivation in Lorenz signature would give the correct Lorenz signature of Feynman propagator with the i epsilon. So we've interpreted a free particle in d-dimensional space time in terms of one-dimensional quantum gravity. Well, how would we include interactions? There's actually a perfectly natural way to include interactions. There are not a lot of smooth one-manifolds. There's only a line or a circle, as I drew previously. But there's a large supply of singular one-manifolds in the form of graphs. That's a one-manifold with some singularities. In this case, I've drawn a graph that has four external lines, and it's got one loop, which is drawn more or less as a rectangle inside. And those of you who have studied quantum field theory, at least a little bit, will be familiar with Feynman graphs. We're going to find in a little bit that our one-dimensional quantum gravity theory with a space time being this singular one-manifold is going to be like a standard Feynman graph in quantum field theory. Our quantum gravity action makes sense on such a graph. We just take the same action we used before, summed over all the line segments that make up the graph. In addition, we can include an arbitrary factor, which I could call lambda, at each vertex. To do the quantum gravity path integral, we have to integrate over all the metrics on the graph up to dishumorphism, and also all maps of the graph into space-time. The only invariance of a metric on the graph are the total lengths or proper times of each of the segments. I've labeled in this graph some of the segments by proper times tell one, tell two, and tell three. I didn't label all of them. The natural amplitude to compute is one in which we hold fixed the positions of the external particles and integrate over all the proper times tells and all the paths the particles follow on the line segments. To integrate over the paths, we just observe that if we specify the positions of all the vertices, so therefore we specified each end of each line segments, then the computation we have to do on each segment is the same as before and gives the Feynman propagator. Integrating over the internal positions then, we'll just impose momentum conservation at vertices and we arrive at Feynman's recipe to compute the amplitude attached to a graph, a Feynman propagator first line, an integration over all the subject to momentum conservation. So we've arrived at one of nature's rhymes. If we imitated one dimension, what we would expect to do in four dimensions to describe quantum gravity, we arrive at something that is certainly important in physics, ordinary quantum field theory and a possibly curve space time where the perturbation expansion can be regarded as a sum over singular one-manifolds, namely graphs. In the example I gave, the ordinary quantum field theory is scalar phi cube theory. The particles were scalar fields because of the particular matter system we started with in our one-dimensional quantum gravity theory and the interactions are phi cube theory assuming we took the graphs to have cubic vertices. A coupling constant of the phi cube theory would arise from a factor which I'd call lambda that we would attach to each vertex. I didn't write it explicitly in any of the slides. Cortic vertices, for instance, would give phi 4 theory and a different matter system would give fields of different spins. So many or maybe all quantum field theories in D dimensions can be derived in this sense from quantum gravity in one dimension. Now there's actually a much more perfect rhyme if we repeat this in two dimensions, which means for a string instead of a particle. One thing we may run into is that a two-manifold sigma can be curved. For example, I've drawn a genus two Riemann surface. So the integral over two-dimensional matrix promises to not be trivial if it was the integral over one-dimensional matrix. That's related to the following fact. A two-dimensional metric in general is a 2 by 2 symmetric matrix. And, well, a 2 by 2 matrix would have four matrix elements here at symmetric so it obeys one condition. It has three matrix elements that are independent. A diffeomorphism can only remove two functions. So a 2 by 2 metric tensor would have functionally sort of one arbitrary function that can't be removed by diffeomorphism and that is actually called the curvature scalar. And that would spoil the analogy with what we had in one dimension. But now we notice that the obvious analog of the action function we used for a particle, namely this one, where now I've written a similar action but I've replaced the one-dimensional world with a two-dimensional world in which the fields x live. This action, precisely in two dimensions, is conformally invariant. That means it's invariant under a vial rescaling of the metric for any real function phi on the two-manfold sigma. If we require conformal invariance as well as diffeomorphism invariance, then this is enough to eliminate the third independent component of the metric tensor. And it's enough to make any metric g on sigma locally trivial, locally equivalent to a Euclidean metric, just as we had in the quantum mechanical case of one dimension. So in very pretty 19th century mathematics now comes into play. Well here I've drawn an example of a graph. Here I drew a graph with no external lines and I took this funny graph with two loops of length T1 and T3 connected by a third line of length T2. And now below I've thickened it a little bit to a two-manifold. And I've labeled the pieces of the two-manifold by parameters that I've also called T1, T2, and T3. What 19th century mathematicians discovered was that conformal geometries on this two-manifold were by conformal geometry, I mean a Ramanian geometry up to difumorphisms and vial transformations. A vial transformation being an overall spatially dependent rescaling of the metric. 19th century geometries learned that up to conformal transformations in two dimensions, the metric can still be parameterized by finally many parameters. Which are analogs of the length parameters in the one-dimensional case. But with two big differences. First the parameters are now complex parameters rather than real ones. And secondly the range of the parameters is restricted in a way that will eliminate the ultraviolet divergences that plague quantum field theory. Well to underscore how a two-manifold is understood as a generalization of a Feynman graph, I've drawn another example. So this was one example where I drew a particular Feynman graph in this case with no external lines. Then I thickened it to a two-manifold. Which in string theory would be interpreted as the world-shade of a string. Here's another example. Here I have a one-loop Feynman diagram with four external lines. I thickened it a little bit to a string theory counterpart where the external, instead of a point particle coming in on an external line. I wonder how I undo what I've just done. Instead of a point particle coming on an external line, what's coming in is a little closed string at the beginning again and then interacting and branching with other closed strings going in and out. So this picture here is the string theory analog of the Feynman graphs of ordinary quantum theory. Now the deeper rhyme is the following. We used one-dimensional quantum theory to describe a quantum field theory in a possibly curved spacetime. But it did not give a natural description of quantum gravity in spacetime. It just gave a description of something in spacetime. The something was actually phi cube theory in the simplest setup that I actually gave for illustration. We didn't get quantum gravity in spacetime because in quantum mechanics there was no correspondence between operators and states. We considered this one-dimensional quantum mechanics. The external states in a Feynman diagram were just the states in this quantum mechanics. Now the quantum mechanics depends on the spacetime metric, capital G. And if you want to describe gravity in spacetime, then you're interested in what happens if the metric changes a little bit. So let it change by an amount delta G. If it changed the metric, the action will change. And what it changes by is the integral of an operator I've called curly L. And curly L is just what you get if you take this action density and change G a little bit. I've simply replaced G by delta G a variation of the spacetime metric. That makes sense as an operator in the quantum mechanics but it's not a state in the quantum mechanics. There isn't any correspondence between operators and states and that's why the 1D theory didn't describe quantum gravity in spacetime. An operator would appear on an internal line in the Feynman diagram, not an external line. The states occur on external lines. So a particle comes in from infinity, goes out to infinity. Feynman taught us to describe scattering of particles via Feynman diagrams. And in Feynman's recipe, the particles that are being scattered come in and out on the external lines. An operator with the inserted somewhere on an internal line. Because the operator arrives, the action is being integrated over the whole graph. So the change in the action would be a perturbation that would appear somewhere on this graph. I've drawn a point where it could be inserted. Really, it could be inserted anywhere so we should integrate over the position where I drew the X. But anyway, the important thing is that an operator appears on an internal point. The particles are on external points. And there's no... In quantum mechanics, when you studied quantum mechanics, nobody taught you a general rule relating operators to states. There just isn't one. But in two-dimensional, in general, and conformally invariant field theory, there is an operator state correspondence which actually is important in statistical mechanics as well as in relativistic field theory. And because of the... In a moment, I'm just explaining why there's an operator state correspondence. But because there is such a correspondence, the operator that represents a change in the metric automatically represents a state in the quantum mechanics. And that's why the theory unavoidably describes quantum gravity in spacetime. The operator state correspondence arises from a 19th century relation between two pictures that are conformally equivalent. In this version, I imagine a two-manifold. And here I've drawn a closed two-manifold without worrying about any external particles. And then that red dot, which I hope you can all see, it looks like it's pretty visible. It's a point at which an operator was inserted in the two-dimensional theory that describes gravity on the two-manifold. But if we had conformal invariance, what 19th century mathematicians would tell you is that you could equivalently, if you remove that point from the two-manifold, then a conformal rescaling of the metric will replace a neighborhood of that point with an infinite tube that goes very far away, arbitrary far away. And that tube is where an external particle that's being scattered would be inserted. So conformal invariance of the two pictures is the equation between the operator you would insert here and a particle that's being scattered in a physical process. So that's the equivalence of operators in states. That's absent in quantum mechanics but present in conformal field theory. And it's the reason that when we go from one-dimensional gravity to two-dimensional gravity, but with conformal invariance assumed, we get a description of quantum gravity with a plane and polar coordinates. So that's the metric of a flat two-dimensional plane. And now we imagine an operator inserted at the point r equals 0. In a conformal invariant theory we could replace the metric by an arbitrary vial rescaling of itself what you get by multiplying it by any function. So I multiply this metric by 1 over r squared d r squared over r squared plus d phi squared. And if I let u equals log r this is just a flat metric again. But now it's a flat metric on an infinite cylinder. So the 19th century fact is that the plane minus of points is conformally equivalent to an infinite cylinder. The infinite cylinder describes a strain coming in in the past and propagating indefinitely in time. So an operator that would be inserted at a point is equivalent to a state living on this circle and propagating in time. The existence in conformal field theory of that relationship between operators and states is the reason that strain theory unlike one-dimensional quantum gravity leads to quantum gravity and spacetime. Well the next step is to explain why this type of theory does not have ultraviolet divergences. Quantum field theory is plagued with ultraviolet divergences that was clear from the very early days when quantum field theory was first developed starting around 1930. It was around 1950 that the pioneers of quantum field, of quantum electrodynamics people like Feynman, Schringer, Tamanaga and Dyson managed to tame the divergences in electromagnetism and make sense of electromagnetic theory, quantum mechanical in a relativistic quantum world. And then it was close to 20 years later that another generation of pioneers including Solom, the founder of the ICTP along with Weinberg, Glashel, Erhoft, and others that I'm not mentioning right now tame the divergences in a larger class of theories leading to the modern standard model of particle physics so that divergences were tamed in quantum physics without gravity. But none of that works in the presence of gravity because as the pioneers also discovered way, way back the highly nonlinear mathematics that Einstein used in developing his theory of gravity doesn't work well with the renormalization procedure that tamed the ultraviolet divergences for the other theories. I'm going to explain in a moment how why these divergences go away in the context of string theory when we generalize from the one-dimensional picture of the propagating point particle to the two-dimensional picture of the propagating string. I'd like to mention though that personally I only met Dirac once I think at an Arachie summer school around 1981. And what he told me on the occasion was and also written a variety of times he had always been dissatisfied with renormalization theory. Yes, technically renormalization theory had made sense of quantum watcher dynamics and even ultimately of the rest of the standard model. But Dirac's point of view was that renormalization is fine when the effects are finite and he said perhaps finite and small, but anyway, finite he was dissatisfied with the infinite renormalizations involved in standard quantum field theory. Dirac actually had pioneered the theory of extended bodies in relativistic physics without having all the ideas that go into string theory. And I imagine was hoping to get away from the usual story with the infinities. Anyway what does happen is that string theory does for a reason I'm going to explain in the next video. I'm going to show you how we can eliminate the ultraviolet divergences of standard quantum field theory. First we have to understand where ultraviolet divergences usually come from in field theory. They come from the case that all the proper time variables in a loop go to 0. So remember we labeled in our one-dimensional graph we labeled each line segment by a positive number that you can think of as the proper time elapsed along that line segment. We labeled it that way because that was the only invariant of the one-dimensional differential geometry. And what early pioneers of quantum field theory discovered is that ultraviolet divergences occur when all the proper time parameters in the loop go simultaneously to 0. Now it's true that as I said quantum surface can be described by parameters that roughly mirror the proper time parameters in a fine-in graph. So this is actually a picture I drew before. This is a particular fine-in graph which would have three proper time parameters, one for each interval in the graph and the two-dimensional version also has three parameters except now they're complex parameters and they're also a subject of different qualities that were analyzed in the 19th century. So in the fine-in graph the parameters cover the whole range from 0 to infinity. By contrast the corresponding Riemann surface parameters are bounded away from 0. I won't give the whole 19th century a story of why that's true but I will explain why it's true in a simple example and why it eliminates ultraviolet divergences in that example. So given a fine-in diagram there's a corresponding Riemann surface but only if the proper time parameters tell I are not too small. So the region that in field theory leads to the ultraviolet divergences is absent in the string theory case. Instead of giving a general explanation of this I'll just explain how it works in the case of the one-loop approximation to the cosmological constant. So the one-loop, the cosmological constant is the vacuum energy the zero-point energy and for an ordinary harmonic oscillator the zero-point energy is half h bar omega but for a field you have to take the sum of half h bar omega over all the modes of the field and there are infinitely many modes of the field so the sum is divergence. That divergence is a baby example of the ultraviolet divergences of quantum field theory and in this particular example I'm going to explain how the divergences arises in a Feynman diagram and why it's absent in the corresponding string theory diagram. So the Feynman diagram is simply a loop with a single proper time parameter tell. Tell is simply the circumference of the circle the loop is a circle and tell is its circumference. Now remember when we had a one-manifold with boundaries we evaluated the amplitude in terms of propagation from x to y as in ordinary quantum mechanics except that we had to integrate over the elapsed proper time and that interval gave the Feynman propagator. So here is that we have a closed loop instead of an open interval as I considered before and the closed loop means that the particle propagates from somewhere and comes back where it started so I should set y equals x and sum over x while summing over x means take a trace so we now get the trace of the same expression in the case of each of the minus tell h we have to integrate it over tell from 0 to infinity and there is actually one more factor of one over tell that comes in and that factor of one over tell is roughly because of the symmetry of rotating the circle so you can interpret the Feynman integral involving a circle as a trace but taking a trace is starting an ending point and that starting ending point could have been anywhere so you have to divide by tell and finally there is an overall factor of a half which is the same as the usual one half in the ground state energy half h bar omega of a harmonic oscillator so this is the expression for the one loop cosmological constant in ordinary the one loop approximation or the leading approximation to the cosmological constant in ordinary quantum field theory it diverges a tell equals 0 it diverges a tell equals 0 because the trace concretely can be computed by a momentum space integral except that you then have to integrate over tell from 0 to infinity well if you do the p integral nothing will go wrong that p integral is nicely convergent for positive tell but it is singular as near tell equals 0 and well you will get something like the integral detail over tell to the half of 1 plus d that integral diverges at a tell equals 0 so that is the divergence of the one loop approximation to the cosmological constant in this formula as a sum of all modes it is kind of obviously divergent because omega can be arbitrarily big however I have tried to explain how to capture the divergence in a Feynman diagram because that enables us to understand the generalization to the string going to string theory means replacing the one loop diagram that Feynman would draw with its stringy counterpart which is a torus in field theory you would have a point particle propagating around the loop building up the circle in string theory you would have a string so the closed string is going around this way but then it propagates around this loop so its worldsheet is a two-dimensional torus now 19th century mathematicians showed that every torus with a metric is conformally equivalent to a parallelogram in the plane with opposite sides identified so I have drawn a parallelogram with instructions to identify the top and bottom left and right but I'm going to simplify this discussion by only considering rectangles instead of parallelograms the idea is the same but the explanation is longer if we try to consider parallelograms so I'm going to imagine that our torus is actually conformally equivalent to a rectangle again with opposite sides identified the top is identified with the bottom left with the right and I've labeled and I've taken a parallelogram in the plane an ordinary flat parallelogram I've labeled its height and base as S and T however only the ratio T over S is conformally invariant if we rescale all lengths by a constant factor that's a special case of a conformal mapping where we're allowed to rescale the Rumanian metric by a spatially dependent factor so any constant rescaling of S and T is a normally conformal transformation so only the ratio T over S is invariant under a conformal transformation I've called that ratio U but as well it's arbitrary what we mean by the height and what we mean by the base of the rectangle we can take the rectangle and rotate it by 90 degrees that would simply exchange S and T so a rectangle with a given S and T is equivalent to a rectangle with T and S exchanged it's the same rectangle look at differently in terms of the ratio T over S that operation is U going to 1 over U so two rectangles with U replaced by 1 over U are actually equivalents well since we could exchange T and S we can assume by convention that T is the bigger of the two and that means we can consider the range of U to be from 1 to infinity instead of from 0 to infinity the upper time parameter T of the particle corresponds to U in string theory but the key difference is that T runs from 0 to infinity and U runs from 1 to infinity so the one loop cosmological constant in field theory is given by this expression that I tried to explain a few moments ago but in the approximation of considering only rectangles and not parallelograms the string theory version would be a similar integral that is from 1 to infinity instead of 0 to infinity there's no ultraviolet divergence because the lower limit is 1 instead of 0 a longer derivation using parallelograms and not just rectangles replaces the 1 with the square root of 3 over 2 but the important thing is that the lower limit is strictly positive so there's no ultraviolet divergence the region that gives the ultraviolet divergence in field theory is eliminated in going to string theory so I've explained a special case but this is a general story the stringy formulas generalize the field theory formulas but without the region that can give ultraviolet divergences in field theory the infrared region where tau is large or U is large lines up properly between field theory and string theory and that's why a string theory can imitate field theory in its predictions for the behavior of times and distances but the ultraviolet region is eliminated so Dirac's aspiration to have a theory in which the renormalization effects are finite is actually achieved in string theory now I want to use the remaining time to at least give a hint of in what sense space-time emerges from something deeper if string theory is correct but I need to stress that this is something that we have not done properly it's something of which we only have glimmers and I'm very schematically going to explain one glimmer you could say that the huddle that we're having this week and next is focusing on a different glimmer a different aspect of the question of how space-time emerges from something deeper of which we have a different glimmer let's focus on the following fact the space-time M that's metric tensor capital G was encoded as the data that enabled us to define a two-dimensional conformal field theory that we used in this construction that was the only way space-time entered the story we could have used a different two-dimensional conformal field theory subject to a few general rules that we'll admit for example we could have used the two-dimensional Ising model or technically an appropriate number of copies of the Ising model with the right central charge the G is slowly varying meaning that the radius of curvature is everywhere large then the way we described this two-dimensional conformal field theory by giving a Lagrangian is useful the Lagrangian is weakly coupled and illuminated that's the situation in which string theory matches to ordinary physics that we're familiar with we may say that in this situation the theory has a semi-classical interpretation in terms of strings in space-time and that will reduce the low energies to an interpretation in terms of particles and fields in space-time when we get away from a semi-classical limit the Lagrangian is not so useful and the theory does not have any particular interpretation in terms of strings in space-time now a situation that I've schematically drawn here very frequently occurs well I've drawn this is purely a symbolic picture but the region drawn in orange two-dimensional is meant to represent a two-parameter family of physical situations a family of conformal field theories maybe that depends on two parameters and generically there's no particular space-time interpretation there's some highly non-classical situation but in this picture I've imagined that there are three different limits labeled M1, M2 and M3 where the fog lifts the theory becomes weakly coupled and can be described by strings propagating in some classical space-time so I've imagined here I've imagined here something we have many concrete examples and a huddle of 20 years ago might have focused on this rather than what we're actually focusing on this week and next here I'm imagining a picture where we have a two-parameter family of situations that are highly non-classical because stringiness, whatever it is and we don't understand it that well, is strong you can't describe the situation in terms of strings and space-time but there are three different limits in which the fog lifts and there's a semi-classical picture but those semi-classical pictures involve three different semi-classical space-times that can even be topologically different so that's the kind of situation that can very frequently arise and many examples of which were demonstrated in studying this theory there are all kinds of other non-classical things that can happen for example it can happen that from a classical point of view the space-time develops a singularity but the two-dimensional conformal field theory remains perfectly good meaning that the physical situation in string theory is perfectly sensible although a field theorist would have trouble making sense of it so there are many examples of space-time singularities which I'm not sure is understood but a bit of bad news which again is in a sense part of what's behind the current model is that black hole singularities are not such examples so black hole singularities lots of space-time singularities have stringy analogues that we do understand but the black hole singularity is not in that category so we can say from this point of view the space-time emerges from the seemingly more fundamental concept in general a string theory comes with no particular space-time interpretation but such an interpretation can emerge in a suitable limit somewhat as classical mechanics can arise as a limit of quantum mechanics although in general the world can be completely quantum mechanical in appropriate situations now this is far from a complete explanation of the sense in which from the point of view of string theory space-time emerges from something deeper not even a complete explanation of the bits and pieces we move up but it is an explanation of something there's another complete different side of the story that involves quantum mechanics and the duality between gauge theory and gravity and I guess there's a third side which is roughly the relation of space-time to entanglement which is the subject of the current model well what I've described is certainly one important piece of the puzzle and one piece that's relatively well understood is at least a partial insight about how space-time as conceived by Einstein can emerge from something deeper and that thought is what I will have to leave you with today thank you so much for your attention thank you very much for this inspiring talk maybe some questions thank you for the great talk I've seen some of these topics before in a rather abstract setting but your picture with the operator insertion on the edge I was just wondering is this somehow related to vertex operator algebras well vertex operator algebras is just a name for the operator product expansion of a conformal filter except that terminology is usually used for holomorphic conformal theories if you like for theories that factorizes holomorphic times anti-holomorphic theories vertex operator algebras are a special case the name vertex operator algebras is usually used for the special case of a conformal filter that's holomorphic but certainly in that case one does have this operator-state correspondence the operator-state correspondence in that case is just the existence of the vacuum module for the vertex algebra roughly you were citing 19th century mathematics throughout your talk do you think that string theory needs new tools to be developed or do you think that there are existing tools that are sort of unconnected so far to what we know well I expected it's a mixture the question was whatever mathematics is needed or physics whatever ideas are needed to understand string theory better do they exist somewhere in the math or physics world or are they new ideas that need to be invented I'd assume a mixture of the two what questions I see a question in the back and the way in which you described that string theory resolves ultraviolet divergences so one might have thought if one does something naive for the point particle story by cutting off the tau integrals one would run into trouble with unitarity or other constraints could you perhaps explain in a simple way how the string theory not only gets rid of ultraviolet divergences but does so in a unitary way I kind of explained it in the basic example of the one loop cosmological that it happens because the region with you less than one doesn't exist it's related by symmetry to the region with you bigger than one there's nothing like that in field theory in field theory by hand you could try to limit time and diagram integrals to tau bigger than one but everything would go wrong starting with locality but all the nice properties of a lot of things that we take for granted in quantum mechanics in quantum field theory would fail if we artificially change it that's one of the reasons physics made so much progress in the 20th century if you ask how was the it's most dramatic in the case of the weak part of the standard model how did Weinberg and Slom come up with the weak interaction model there was almost no experimental data with very very limited experimental data they were able to find the right model because the rules of quantum field theory were rigid enough that there were very few things you can do without getting nonsense the framework was tight enough to set a very limited experimental data right to the right there if you could arbitrarily do things like restricting tau to be bigger than epsilon then the standard model would never have been discovered certainly not with the experimental data that existed in the 60s so the framework of relativistic field theory is extremely tight and it's very difficult to do anything with it but the replacement of one manifolds with conformal two manifolds is something that works and it automatically eliminates the reason where it tells us that one which is why it eliminates the ultraviolet divergences yes I see a question in the back speaker but there are conditional systems for which we don't have field theories can string theory be relevant for such systems to understand the critical properties well condensed matter is so diverse with so many different kinds of systems but any general answer is going to be wrong so hopefully for some kinds of examples for example disorder we have a field theory for icing model but if we go to the random field icing model we can see well of course there are still all kinds of problems with disorder but I can't resist mentioning this is an application of disorder to possibly do gravity rather than the other way around but there's been this beautiful SYK model of a disorder strongly coupled quantum mechanical system of fermions that can be described in terms of an emergent world with gravity so that's been disorder applied to gravity and perhaps some ideas have floated in the opposite direction as well that's one example but there are all kinds of disordered systems so it's going to be hard to give I won't volunteer a general answer I had another question on the calculation toy calculation of the zero point energy so in the one-dimensional case if I had interactions then I could imagine more contributions with loops and so on but I can turn off interactions if I choose to yes but in the stringy case it seems that I cannot turn off interactions and there would be more contributions there would be not just the torus but the torus with an arbitrary number of holes in it and I'm just wondering if that sum converges well first of all we are supposed to make this involving different topologies I've drawn a higher topology in this case of genus 2 and as in field theory it's believed that such an expansion is only an asymptotic expansion not a convergent expansion I'll stick there unless you have a follow-up question I'll simply say that part of the analogy is that both in field theory and in string theory we sum over topologies the sum is simpler in string theory because a two-manifold without boundary is completely characterized by the genus, the number of holes whereas in each loop order there are many Feynman diagrams for example there's this one and ultimately exponentially many or even more than exponentially many with L loops for large L whereas there would be one string diagram for L loops but in each case we have to sum over the number of loops in this perturbation expansion a difference that I didn't explain in the lecture is that in field theory it's up to us what kind of vertices we want to allow I mentioned we could have a cubic vertex with a parameter lambda if we want to have a quartic vertex with a parameter g in string theory there isn't really an option like that because there's no vertex see okay our genus 2 Riemann surface is smooth so once you've explained what it means when a string is propagating smoothly there's nothing else to say you don't have the option of including arbitrary parameters of vertices so you have this is an asymptotic series that will give you a certain decent approximation but the truth is something that we have a handle on not always only sometimes I think your question is is there a non-perturbative definition is that finite are we sure that's finite we're not sure of everything in all the generality we'd like to understand it the case where we do have a satisfactory non-perturbative definition is actually closer to the content of the huddle than to what I've actually been explaining in this lecture in the case where you have what is called gauge gravity duality which would be a different lecture that I didn't give we have a non-perturbative definition what I say in the cases I didn't I was very schematic about my two-dimensional field theory they're different they're discrete choices we could have made in those two-dimensional field theories for some choices they would then participate in this gauge gravity duality we would have a non-perturbative definition and we'd be sure that it makes sense exactly but in other cases no but in no case do we have the comprehensive understanding we'd like to have Hi, thank you for the inspiring talk my question is I'm here sorry what happens in more dimensions like string theory is famous for being defined over 11 dimensions if I'm not wrong and so how do you deal with more dimensions and how do you take some of them compact and others not well the canonical answer is literally that you imagine we live in a world with four non-compact microscopic dimensions and the rest are either microscopic or perhaps you describe as a more abstract conformal filter that's the canonical answer I can't promise how close it is to the truth but that's the canonical answer I meant also later we have seen only two-dimensional conformal field theory like does it that's more difficult to deal with the other dimensions like how do you define this theory on more dimension like what happens well actually you see in the framework of my lecture not in the framework of everything anybody does in this field but in the framework of my lecture two dimensions was the real case the two manifold was mapped to space-time and that gave the description of quantum gravity in space-time so what is more than two is the number of X's the two manifolds I sometimes draw like this one that's mapped to space-time which is parameterized by the X's but the X's are functions of two coordinates I could call them sigma 1 and sigma 2 on this two manifold so that was the description of the framework that leads to quantum gravity in higher dimensions so in the beginning of the lecture I explained that ordinary quantum comes in this sense from quantum gravity in one dimension but doesn't describe gravity and then I explained that if we simply replace one with two and incorporate conformal environments we get a setup that leads to gravity in space-time are any other questions? yes so in the context of summing dimensional quantum gravity we integrated over time but we didn't really distinguish from the case of being a line or a circle that we get interference somehow what Feynman taught us is that if we want to consider an elementary particle reaction with n particles in and m out then we have to consider Feynman diagrams with n plus n external lines I'm drawing it on the blackboard which gives a misleading impression it's an abstract graph the external lines are not arranged on the circle but Feynman taught us that for reaction with n in and m out we should look at Feynman diagrams with n plus n external lines when I drew the loop n and m were 0 so there were no particles coming in and none going out the physical interpretation is that we were calculating the energy of the vacuum and the biological constant so there's no disharmony it was just a special case where n and m were 0 but if you I don't know how much can I reformulate the question we were calculating the amplitude of propagating from two points we integrated over the distance but we didn't see if those two points were lying on a circle or on a line why is that I didn't give a full explanation for the circle but the factor of tau in the denominator has to do with your question so a circle you can think of as starting somewhere and going all the way around and coming back now coming back where you started is a trace but it's ambiguous where you started and ended all you can say is you started and ended somewhere and dividing by tau is because it doesn't matter where you started and ended so I didn't explain it in detail because it wasn't important in the following sense after integrating over momentum or after taking the trace the integral we got actually goes like this and the answer to your question involves the 1 for d dimensions there's also a d the integral diverges tau equals 0 with or without the 1 so I didn't go into any detail to explain where the 1 comes from what it has to do with the answer to your question questions there is one there hello thank you for the very nice talk and I apologize this might be very naive so if space time is emerging so first of all I would like to understand whether conformal invariance why is this insistence on conformal invariance the second thing is if space time is emerging are properties like locality and causality also emerging or are they inherited from string theory the most comprehensive framework we have to give a concrete model of space time as emergent from something more fundamental is actually the gauge gravity duality it wasn't the subject of my lecture but I think colleagues would give a range of answers to that question I'm not actually sure what I want to say I think we don't understand it as well as we'd like to I think that might be the only reasonable short answer I can give honestly so in the gauge gravity duality the space time all the fields and everything in it emerge from something more fundamental in that case a holographic description of the boundary of space time and locality emerges but in what sense locality can be defined precisely isn't that clear and I think there's a range of opinions about it I'm not certain I don't I'd like to say that I think we don't understand it as well as we should you can try asking participants in the hotel for example that question and you'll get a range of answers I believe I see a few of them any other questions speak louder I understand that you have within the framework of this lecture you're staying in two dimensions but I want to ask a question suppose CFDs exist in three and four dimensions also and the operator state correspondence is there also is there any connection to ultraviolet divergences for say using three-dimensional confound field theory or higher oh well before I answer you I realize I forgot to answer part of one of the previous questions why conformal invariance it was found that you can beautifully generalize Feynman diagrams if you do assume conformal invariance otherwise you can't and an aspect of your question is conformal field theory exists above two dimensions so could we make a similar story using as a starting point three dimensions the answer again to that appears to be no but a slightly different question you could have answered the yes answer so you could ask is conformal field theory in three, four, and five dimensions and so on used in some way in string theory they entered that question with the yes because string theory turns out to have brains and brains are very important in understanding the quantum behavior and understanding the dynamics of brains leads us to questions about conformal field theory in three, four dimensions and it turns out even five and six we use conformal field theory in higher dimensions but not in the same way we use it in two dimensions two dimensions is special in fact part of the story here I would say is that each dimension is special in some sense we don't understand properly they're all unified as part of this picture thank you thank you all for the talk I have perhaps a very basic question at some point you said that this framework allowed you to remove singularities in that you encounter in space time using general relativity framework some singularities and you said something about black holes but I didn't catch it I said we don't understand what to say about the black hole singularity I said there are many kinds of singularities which we can't from the point of view of field theorists there are singularities a simple example is the yang-mills instanton an instanton can shrink to a point in classical in field theory that's a singularity in string theory funny things happen instanton shrinks to a point but we understand them instead of the black hole singularity we don't understand thank you very much that's all I said about the black hole singularity so it seems that I don't know Bobby thank you for the lovely talk Edward it was very nice we haven't touched on the subject of data so I was wondering what your thoughts are on how for example particle physics and cosmology data have an influence on string theory or where string theory fits into actual data well it would be great if experiments would give us structure beyond the standard model that we could come to grips with possibly by discovering super symmetry or some other new structures beyond the standard model of accelerators or maybe new phenomena like cp violation beyond the usual ckm mechanism I think in view of the hour we should only have more questions if they're from non-experts no non-experts questions I'm not sure if you're an expert or not sorry thank you very much for a very lovely talk I just was curious at some point you said that there was a picture on the slide and you were describing some kind of interpolation and there were three different limits and you remarked that these limits are some semi-classical limits of the theory which result in geometries that might not be actually the same geometry could you say is there some insight that tells us how these geometries might be related to each other well there are all kinds of examples known it always seems a little bit different examples are understood in different ways so for example if you're studying Klavier models there are bi-rational equivalent Klavier models that will appear in such a picture and there are other there are other types of examples studied in different ways in the description is this at all related to mirror symmetry mirror symmetry is an example so mirror manifolds mirror manifolds appear in different limits in such a picture thank you so these geometries relating to tropical geometry or something like that these pictures look very similar to the generations of sorry I can't answer competently about tropical geometry I imagine the answer in some senses the field theory is a tropical or something like that but I can't competently say that I would have to defer somebody else to explain that better final questions it's going to be a real non-expert question is is there a possibility that it is not like point particle like theory or string like theory it may be some kind of variation between transition between why particle kind of theory string like theory there can be just a fluctuation between the two or it may be general or more general than that why nature should be particular biased about particular structures well that kind of why question is hard to answer so why were physicists successful in developing quantum field theory to describe all the forces except gravity it's a little bit hard to explain why the world was made that way so I'm not going to claim I do understand it I did remark already that one of the main reasons physics made so much progress in the 20th century was that the framework in which it evolved was so tight that it was possible to make a lot of progress with very limited experimental data I emphasize the weak interactions but I really think the same is true for the strong interactions for the strong interactions there was a vast amount of experimental data but most of it wasn't that useful but it led to the strong part the QCD part of the sanity model so physics made so much progress in the 20th century because the framework was so rigid I can't tell you why that rigid framework was practiced but it turned out that nature relativistic quantum theory works and it's a very rigid framework in which you can't take too many wrong turns and therefore what seems like ridiculously limited experimental data about the weak interactions for example led to the Weinberg-Sahn model now right into the fact that the framework is so rigid it's virtually impossible to modify it in a way that preserves its consistency and my point of view is that string theory is the only significant idea that's emerged for any modification of the standard framework that makes any sense I can't promise it's built I can only say the answer to why string theory is that it's the only way we know of to generalize the standard framework that makes any sense the reason I'm emphasizing the other part of the coin that physics made so much progress because the standard framework was so rigid that also part of the rigidity is what made it so hard to generalize it if there were many ways to generalize the standard framework we discovered the Weinberg-Sahn model Weinberg and Sahn would probably have gone down numerous wrong L's with all kinds of attempts to generalize the usual framework of relative security it was virtually impossible it was virtually impossible to do anything that was consistent with relativity and quantum mechanics except to work in this rather limited framework and that rather limited framework pointed them to the right answer to the limited experimental data I regard that as one of the main lessons of the 20th century now having that clearly in mind helps us understand that when a possible consistent generalization of that framework is found we should be taking it seriously and it's led to all these very surprising discoveries in string theory of which I've tried to give you a hint in today's lecture but I've really only explored one facet of it today I'll find that question thanks a very naive and non-spirit follow-up question so when you say we don't understand what's happening with black hole singularities do you expect such space times to be able to be described within the framework of string theory or we expect some new framework to be required to be needed to study such cases or for example if we found a net or a rotating black hole that has a net singularity of experimental data what kind of framework would we need to describe such scenarios for example I would think that the gauge gravity duality or ADSEFT is powerful enough that in principle it describes the evolution formation and evolution and evaporation of quantum black holes we don't understand it but I believe that that framework does and if you had sufficient computing power maybe with a quantum computer I think you could simulate the dynamics of a quantum black hole in an own framework so I think it's powerful enough but we definitely don't understand it thanks well so it's more like on your perspective so do you think that in the near future so we will be understanding quantum gravity more from this emergence of something deeper than from string theory itself I expect it will be an interplay between the different directions but I don't think I can make any detailed predictions I've been around for a long enough to know that I would have had various pastimes I've made predictions that were quite long about what would happen in the next decade so that gives me a little bit of caution thank you okay I think and last but not the most pressing question oh thank you for the interesting talk I would like to ask you made the example for genus one I've been running over all towers for genus one but I wanted to ask for higher genus is this integral understood how to compute it well it's understood how to define it to actually compute it and get explicit formulas is hard it's been done in special cases in genus two well most of the results were pioneered by Docher and Fung and genus two and in genus above two there are only very limited results in special cases so the same is true for Feynman to find general rules for Feynman diagrams but they're hard to calculate people only gradually learn to calculate diagrams of higher order thank you for the talk and I have a very naive question about the gravitational wave the electromagnetic I'm here I was wondering if gravity affects the electromagnetic fields with the sense of the string theory or they are completely I mean for unifying the forces is it I didn't explain why the forces are unified but I'm going to give a one sentence answer which is that there's only one string so the different particles and fields are different vibrational modes of the same string and the interactions come from this kind of picture where locally nothing is happening because the world sheet is smooth so there are no arbitrary rules you can introduce that would govern the interactions so there's only one string so it produces all the elementary particles and its interactions are completely returned thank you thank you very much okay in the view of time I think we should thank Edward for the beautiful talk thank you very much