 Okay, just to start off this class, at the end of the last class, we were discussing divergence as an open string, okay? And we realized that the Bosnian open string did have used UV divergences, but that lied for a different way from the closed string. These UV divergences were interoperable as IR effects in the closed string. Now, the wassama happens about this interpretation effected from your dear mind. So, let's just spend 5-10 more minutes on this, just to clear up all the unhappinesses. You see, before looking at the physical interpretation, let's look at the technique. What did we have? We had this strip foundation function, which we very integrated over multi-faceted. That was the thing that we obtained. Okay, now, we were interested in the troubles and divergences. The troubles and divergences happened in the limit of very high temperature, from this point of view, because UV, very high temperature means very small beta, so very small size of the strip. So, this thing, it's called t, very small t. So, with a thing from a world sheet of this form, that this size is much larger than this. This is the regime in which, in quantum field theory, it's very difficult to directly calculate a partition. Because it's a regime in which many different states contribute to the partition. So, the direct calculation which we had, we did the exact thing. We could analyze that, but actually our analysis is mathematically, just before we integrate over the function of t, has a physical counterpart. See, what we did in order to analyze this was use a modular transformation of this partition function. So, I was drawing that molecular analysis. But in the end, we got the answer in terms of s, which was 1 by t. Now, we're dealing in a conformal field theory. Every partition function that we calculated is invariant under uniform riscations. So, this object here is the same thing as, let's take this side and riscate it to be of size 2 pi. And then, okay, so let's first draw this way. So, this is the same thing as just like a worksheet conform. Okay, this is a circle. This is a circle of size 2 pi. And this is a strip of length 1 over t of 2 pi, 4 pi square root of 6. Okay, so this is, okay. And at these edges, we put particular boundary conditions. Some boundary conditions, yeah? Some boundary conditions, yeah? And we go, shh. So, we calculated the path integral with these boundary conditions. Now, if we calculate this path integral, okay? We got the path integral here. What we're going to get is some state of the glow strip. What exactly happens at the edge? We'll do this, we'll do that in a moment. Anywhere apart from the edge, it's going to be the path integral of the glow strip. I'm going to set it in depth. So, we get some state of the glow strip. Okay? So, clearly what's happening is a boundary condition that prevents some state. Whatever it is. And this state is propagating to the other boundary. Well, let's say the states begin at it. It's schematically, let's call this state whatever it is, B. And this B prime, but it's the same boundary condition, so let's call it B. It is schematically equal to the power 1 by D times time, times Hamilton, okay, yeah. Times the Hamiltonian of the glow strip, times speech. This is the mathematical level. The path integral that calculated the open-stream boundary of the glow strip can also be represented, okay? So, a particular closed-stream state created, propagated over time D, and I made it by the same constant. What this state is, okay, that's a midway, of course, described. This, by the way, is called the boundary state. The state B is called the boundary state. It's the closed-stream representation of open-stream logic, okay? And it's much, much studied object, then there's a lot to say about it. But that's, it's not our purpose at this point, to discuss the boundary state. Again, we can worry about, you know, it's actually not difficult to give a implicit oscillator representation of the state, and so on. For open-stream boundary, it's just a nice and interesting subject, which we probably will do. Does this boundary condition violate the, does the boundary condition violate it? Yeah, so, this boundary, this state is not embedded under all the Yeah, so this state is not invariant under all the V-hubs originating, that's what you're asking. It is invariant at half of them. Effectively, the same case with the open-street boundary relations, but broke. Half of them in the half of a year. Yeah, so it's the same, symmetry analysis is the same. And you see, any state, in any counter-field theory, the back kick, well, most counter-field theories, IQ is the most symmetric state in the problem. Look at any other state, it will not be invariant under all the symmetries of the table. That's not an unusual situation. It's not a bad thing for a state to be invariant under pure symmetry theory. Most states are. So instead of looking at scattering matrices only for open-street, why don't we generalize, I mean, instead of looking at asymmetries and where we are looking at asymptotic states for open-street, why don't we impose such a boundary condition and we must not confuse the world-sheet with space-time. Everything we're talking about here is world-sheet. Looking at finite distance on the world-sheet, we did this even for the closed-street. We computed the partition function of the closed-street at finite now. For the same reason that you didn't complain there, you shouldn't complain here, this is a purely auxiliary world-sheet calculation. Okay, it's been not saying that the building's scattering of states from one localized point is based after another, we're not doing that. We can never do that in strings. At least when your strings come in. You know, there's some limit by open strings, a couple of closed strings, whether that's happening again. But we don't know how to do that in string theory in any sense, in any sense in whenever closed strings couple to open strings. That's not what we do. We just take the partition function in the worksheet and analyze it in the pure, you know, put it in that partition table and give it another interrogation. Same pattern. Is this clear? This is the transition amplitude from one state to another state. Because it's transition amplitude from one state to another state, we can insert a complete set of states in the middle. It's a convenient insert a complete set of eigenstates because then the eigenstates will propagate freely. Okay? So suppose we can insert a complete set of closed string eigenstates, let's call them m. We get dm times mb times e to the power minus em by d. That's called an ems. Where ems are the closed string. States are probably a closed string. Okay? So as long as we're interested, purely in the analysis of the conformal things there, just the worksheet partition function, there must be some numbers for which this is true. So if we're interested in a little very large s, this is the usual representation because of the lowest energy states that have nonzero overlap. We're done. You see, at some point last time the question came up, why do we get the number 24 if you remember the master states? It's because only 24 of these states have nonzero. That's this. So we're not getting into that in detail now. That's an interesting regard subject to the example of the rest of the states. Okay? Now, you see this, so what was the key point? The key point was that in our purely mathematical analysis, we had dt by some blah blah blah which turned into just simple ds times exponentials. Okay? So that told us that positive exponentials are constants for problems. Right? Because we're living in a little bit of logics, yeah? So positive exponentials are constants for problems, negative exponentials are not. Okay? So firstly, just purely from the, just in conformal field theory, when you get a positive exponential, you get a positive exponential if you have a closed grid back here. Okay? So positive exponentials, which are a problem, are an artifact with having a closed grid back here and so we never want those. Then there's the next question, what about the constants? Okay? So we have constants out there, provided we have the closed grid constant states. Okay? And that will always be that equation. So this tells us, personally, the nature of the problem of UV divergences. The nature of the problem of UV divergences. Okay? The nature of the problem of UV divergences is that it will arrive at, whenever you have a closed grid back here, or it will arrive at when you have a closed grid constant states, provided these states couple of them. Okay? So we have a way of dealing with the string theories that I've looked at here. So we are not going to do the analysis with that here. Okay? The back here was problematic for me here, since let's have a look at it. But of course we're going to deal with closed grid theories that do have master states. Okay? So the master states you can't dispose of. So what's the other alternative? The other alternative is that the master states don't couple to our property state. The full state of the theory, whatever it is, okay, does not source the ground. Sources. Is what? You see, from the closed grid point of view, any diagram involving open strings can also be thought of from this closed grid point of view. Okay, so if you have some, if you have some vertex update, as I said here, that would change this boundary. Something else would happen. The other point of view is this diagram, this thing represents the amplitude for closed string to be created by its boundaries. Let's look at that state with the diagram I think. You see, what is it? It's the amplitude. You see, it was the amplitude that in this boundary state you find a particular string. Where you propagate the boundary state. You find a particular string. Right? So that is the same thing that you take this boundary state, this particular disc, and you propagate it again instead of this boundary state to that particular closed string of interest. Okay? Now, since that closed string of interest is an eigenstate, that can leave it on as a little bit of a length. So let's take the length of that cylinder, that would be infinity, with the closed string of interest. But now all the conformal transformations get this tube down to the disc. So this is the same thing as this operator. Okay? This object is the same thing as the one-point function of the vertex operator corresponding to that stick inserted on the disc. Okay? So we've concluded that we're going to have this problem whenever we have non-zero one-point functions of the massless closed string states inserted on the disc. These one-point functions are closed. Okay? And so, by everything that I've said, it's just purely mathematical analysis of the form. Okay? I'm not an ancient space scientist. Okay? So clearly, it's just purely at the mathematical level. We see that in order to have consistent open string theory, namely open string diagrams with only linear purposes, we want to cancel all closed string massless outputs. We want to deal with a theory which has no massless closed string elements. I say cancel. Okay? I say cancel because, as you will see if you don't know, there are some ambiguous elements, things you can add to your background. And you can choose combinations of them that are added up so that this catapult is cancel. So all you're saying is, you know, we can lie to this open string partition function in such a way that we have this combination where there's a overlap between the open with the closed string states and the boundary states. Yeah. And you're telling that the constant, for a massless case, the constant part which gives rise to one of the livenances that would be 0 because of this absence of catapults. I'm saying that if we could find the string theory where Alice is now, okay? Okay, then... Then we will not have the problem of you being an expert. Then the 24th time we'll also know the theory... We'll know the theory. Yeah. It will not... Yeah, the mathematical level of what we would see is that there's no catapult. But I'm giving you a more... An alternate way of characterizing. The alternate characterization is that if you find a theory where the disc, there are no catapults at all to one measure, in the disc or any other diagram that contributes to the same order, we'll... Maybe I should have discussed the oriental folds. But there'll be an oriental fold diagram. Oriental folds will cancel. That's the answer to your question. But we'll... My plan was to first complete the discussion of the oriental folds. Strings will go back for you. Maybe I should have... Yeah, not that... But anyway... But... But... Yeah, okay. So if we can find the background in which the stack port is zero, then we know we will not have problems with UV dynamics in the ocean. It's the only statement that's going to be... The statement is not revealed because we will deal with the background with zero time. So then we will not bother at that point to check whether the open street one loop divergence managers are not because they guarantee that it will be. Is this clear? Now, in the last part of the last-day lecture, I'm trying to say this more physically from the point of view of space-time. So everything with that is just math. And I think since you can't argue, you can't argue with math. That's correct. Okay? Now I'm going to give you a physical interpretation from the point of view of space-time. So you can separate these two things. Okay? The physical interpretation we're going to spot. You see, what is a static portal? What is this diagram? This static portal? It represents a one-point function of some state. Okay? So in the Lagrangian, we know that the states that we need will have some kind of data. So suppose you have a pretender scale. Nothing changes. Nothing changes. Okay? So suppose you have some action which has d mu phi of x squared plus m squared phi squared. If you have one-point functions at this level, so that's a factor of g. Okay? So you have some g times phi. Okay? Now we have to understand two things. We have to understand two points. You see, it was not a problem if we had that, it was for massive states. We have to understand two things. We have to understand why tadpoles are bad for massive states, but they're okay for massive states. Okay? This is going to be now a physical interpretation. So the physical interpretation runs as well. Look at the spaceman that's actually gone. Okay? Now, this was zero of all of them. One of the effects, some coupling constant effects, gives you a smaller addition of g times phi. What was the vacuum of this state? It scares me. What was the vacuum of this state? It scares me. The vacuum of this state is phi equal to zero. What is the vacuum of this state? It's phi is equal to, minus g by 2 m squared. Everywhere in space. You get it by completing the square. It's not phi is equal to zero, it's phi is equal to some shifted value, but okay. It's a small difference, small shifted value. Is this clear? Because g is a smaller one. It's smaller. The dimension of our theories was like this. Now you add a linear term to that. So depending on the sum of the linear term shifts in some direction, it's like that. But this is some very different. There's not very different from that. It breaks phi equal to minus phi is equal to that. But it's not drastic change. It's got this. The new value is in some sense, close to the old value. Okay? See, we deal with the effects of this term in perturbation. It's a small change of the old situation. Next on the other hand, you know, mathematically, how will you analyze this? You would analyze this by saying, well, firstly we should probably be more careful if we want to. This is minus a over. Find it from the overall, minus a real thing. Okay, whatever. Yeah, that's a minus a zero. So mathematically, what we could have said was, let's say we're searching for translation in regular. In other words, all of the equations of motion, let's say we're looking for times where phi with a two plus g is equal to zero. And then we solve that with your phi and whatever it is. Now, on the other hand, suppose we say it equals zero. Suddenly, we see that the equation has no solution. Has no solutions because the equation is that just g is equal to zero, which is not true. Okay? So that tells us that there are no translational invariant solutions of this theory with a dad code in the fields of muscles. Not true because there are no solutions at all. Because there are solutions. The solutions will have to balance out this del v, you know, del squared phi against the constant. We need del squared is equal to constant. One way to solve this is, for instance, phi to be a linear function of the coordinate. And that's on that half. But this is a big change. For instance, even if your vacuum here is not very much changed, you go far enough away to change the language. Okay? Which tells you that if you try to do perturbation theory, you will have problems at long distances. Because there's no sense in which the change in your theory can be thought of as small everywhere. Over a local batch, it doesn't change very much. And this phi is changing linearly with x. But the slope is very small, so you can ignore it over a loading batch. When the batch becomes about a 1 over g, you can't ignore it anymore, you're going to run into trouble. Perturbation theory always breaks down because you know that you've started that where does perturbation theory break down? It breaks down to a number of reasons. The most common of which is that your starting point is not near the right solution. If you do perturbation theory thinking, well, let's do perturbation in g. You start with a solution that's far away from the right thing. That won't work. And that's what you're doing. This is clear. This is the philosophy of the space boundary. Now, what is the mathematics of the space boundary? The mathematics of the space boundary and I think it's almost trivial, but it's worth seeing. It's a follow-up. So of course you want to calculate a one-note graph in space time. You know, it's not a model, sorry. A vacuum-engined graph. There's a diagram that goes like this. There's two copies of insertions of this and one property. You see, the answer to this diagram is the inverse of the property. Okay, with my translation embed, it's the only momentum that goes in here. This is the inverse of the property. I have P at momentum equals zero. Okay, where is the momentum drawing from? It's zero. Which is fine if the particle is my massive one. But it's infinite if the particle is my size. So you see, immediately, there is this trivial quadratic thingy. If you're trying to compute the vacuum energy, you will have a graph that will give you it's coming from this particle propagating over very long distances. All distances. Okay. It's some graph that gives you this thing therapy and it's certain of trying to do perturbation theory about the wrong background. Actually, typically speaking, perturbation theory breaks down for two reasons. Firstly, if you're doing it around the wrong background. Or secondly, if you're asking the wrong question. Typically, they do do do do. And in this case, we just think perturbation theory with the wrong background. It's not true that this theory makes no sense. There is some background, some projection non-independent background. Around which if you do perturbation theory, everything would be fine. But that's not what we're doing strictly. We're taking this close print to propagate about the last days. And then when you do perturbation theory, you have a lot of this problem. What did I actually mean? I have a mind-intranet diagram which is an engagement phase. You know, you should be doing some inclusive kind of calculation. Yeah. Okay. Now, the last thing that I said last. Okay. Which again, we should keep in mind. It's the following. Suppose, what I want to diagnose that this problem is exactly the analog of this problem. They look similar, but how do I do the really exact same thing? So one way to do that is to change this question. If you somehow manage to inject someone into this vertex. Yeah. You somehow manage to do that. You know, you make this, the standpoint, not translation in there. That's how you do it. If this thing here, that's a moment. If this G was a function of X, that's a moment. Then this vertex with great translation in there. Right. Then G of function of space time. Okay. And then we look at the difference with respect to G of K. Is it supposed to have the function of space time supported at some value of K? Okay. So if we look at G by G, then we take i K X plus complex out of the class. Okay. Then the same diagonal, everything goes through final diagram, except this vertex operator would, you know, momentum conservation would say that there's a jump in momentum between here and here. Here there's no momentum. Here there must be momentum K. And therefore, it would be a one over K squared. Is this clear? That would not be a divergence. It's hard to compare infinity with infinity. It's easy to compare finite. So that's the motivation. Is this clear? So what we want to do is to check whether in string theory we do get this one over K squared if we change the closed string source to have some space time momentum. Is this clear? Now what was the closed string source? The closed string source was the boundary statement. How do we change this closed string source to have some space time momentum? Well, we put some vertex operator on the disk and give it an e to the power K as dependent. Clearly that's now a source that has specific momentum K. The question we're asking is that if we do the same calculation now, not with a vacuum but with some disk, some insertions on the disk whose net momentum escape, the details of insertions shouldn't matter. Along with the net momentum escape, the reading divergence here should get smoothed out to a one by K squared. That would be a test of whether our interpretation of the divergence in open string theory was correct. A closed string interpretation with space time momentum. The mathematics, the first part of what I said something like this but just the physical interpretation from the point of view of space time. So, last time I outlined to you that this does happen. The basic point for us that when you put some vertex operator and vertex operator, in addition to the, I mean you need to contract this vertex operator. That gives you Z12 to the power K squared. Z12, you know, because this side is so much smaller than this length to no stored in the largest limit is essentially e to the power s. So, you get an additional e to the power minus s times K squared. So, the term that was a constant here that represented the propagation of the nastice force becomes a negative exponential that the coefficient of that exponential is K squared. So, when we do the integral over all s we get 1 over K squared. Do the integral 0 to infinity to the power minus K squared s and the answer is 1 by K squared. In this case, they actually had equations which is telling us that that we are doing from the wrong. In which case? In the case of the sweet theory. Yes. So, what would be a corresponding equation in that case? You see, there is a corresponding question. Yes. The diagnosis is one symptom of this. Now, how would I know how would I know a reaching position should I solve to get the correct answer? Which equation should I solve? If I say, yes. Okay. We are getting transmission in all equations. Okay. Very good. Very good question. You see, what was the condition that we had initially? Okay. Now, we actually had a good background with string theory. Well, I mean, why 26 degrees is worth of that? Why 25 degrees? The condition was conformed in measurements with sequence A. And, of course, flat-based string theory with corresponding conditions is conformed in measurements with sequence A. Now, you see, that was the condition for having a valid classical symptom. That condition, the condition of conformed measurements is modified when you are taking a number of measurements. Okay. Which is basically in the statement of real theory that if you have some vacuum of a classical action, there may not be the correct vacuum of the quantum function. It happens, that's called the Fischler-Saskine mechanism. Okay. Well, you see, there are two sources of divergence. There are divergences when there are two kinds of divergences. One that happens when vertex operators hit small loops in the integrator. There's new divergences or certain sort. Okay, again, we get to this. But the new divergences or certain sort is the integration of a modularized space which can be absorbed into worldsheet which becomes effectively local worldsheet and can be absorbed into worldsheet divergences. So, then the correct condition when you look at string theory, I should go order by order in interpretation in the coupling constant expansion. Okay. Is to solve some equations that have the lowest order of the condition to conform in divergence of the worldsheet theory. But at higher order, the string topic. Okay. Are some deformed equations that are not quite conditional conforming. Such is the reality that these deformed equations have a solution. Those deformed equations are the analogous. In this particular case, we feel which will be compressed. I mean, which will develop where I don't know whether the question you may say. Which will develop a... Oh, you are asking. What is the... What do you think? It's the... One thing that many things could do. One thing is that... The digital is one such thing. Okay. Okay. The final answer to string theory on this background will be one particular solution. Okay. I'm not sure if you think it's probably not. But one particular solution is a linear digital. Okay. There's a known exact theory that would do the job. Okay. With a linear digital. Okay. So, that was your question. That was near here. Okay. We will postpone this question. We should just ask again. Okay. And... Good. Other questions or comments? This is where we're standing. Time on the line. I know that it's important because these are... You know, the important conceptual issues Other questions or comments? We have any questions or comments? Is there something saying that, you know, this particular state actually sourcing it and we are not following... I mean, in this case, there is a... You know, we can... If we wanted, we can basically interpret this equation that, you know, G is something that absorbs. You know, we are not following the equations. Plot. I mean, there's a source. There's a... There's a quantity. By quantity, there's an effective source. Right. Yeah. And, you know, you're not following it correctly if you're just saying if I use translation... Exactly. So, here, you know, I mean, is there some way of understanding that, you know, this is actually sourcing the past state? Yeah. You see, that's the dead pool diagram. Where do you know that you have a source? See, if you're about a true solution to the equations of motion, there are one-point functions. As if. There are one-point functions in theory. You've not solved it. What's this? This can give you one-point functions. Now, though that's too slick, because it does not differentiate between whether these one-point functions can be removed by small changes of the ratio on one. You see, you would have one-point function whether or not there was a mask. When the mask was something finite, then, though the one-point function existed, you weren't going to have the right background. The correct background was not for coding. But the mask would just shift the negative exponentials. Who cares? So, yeah, so when you you know, all closed-ended backgrounds would be turned on a little bit. But a small amount, the amount of energy doesn't make any difference. But as a maskless guy, there's not even turned on a small amount. It's totally changing the value. Okay? So, this one-point function is a symptom that you've not solved the classical equations, correct? That's always true. But there's more subtle questions that you can ask. So, the question is, granted that you haven't solved the classical equations exactly, can you change your solution a little bit to solve them? When you can, perturbation can still work. When you can't, it doesn't. Is this careful? One more question. Please. Okay. So, here, the conclusion, if you understand it right, is that there's an there's an infrared problem in the glow-stream circuit. Yes, sir. We show, sir, there's an ultraviolet problem in the open-stream circuit. That's exactly true. But why? I think, like, I mean, let's say, if I'm not doing it right in the infrared in the proof, I mean, why would it show an ultraviolet problem? That's a very good question. Technically, we understand it. Technically, it's the statement that the ultraviolet in the open-stream is governed by team influence. What? It isn't exactly the same. There's the same mathematical statement that's true very much. But technically, it goes through the world-sheet model in various which, what is this? Yeah, that's a very good question. You see, this is, this is, there's anything particularly good to say about this. Or, there is, I don't know what it is. It is a, you know, it's basically, in the end, a deep property of stream in the air. Every open-stream can also be interpreted in another channel. You know, the same calculation can be thought of in two different ways, both of which are completely correct. You think of it in one way, something about very high energy states of orchestration. You think of it in another way, something about the flow stream. And these are both identical things. So, it tells you that somehow, you know, there's some sort of mixing between ultraviolet and ultraviolet in the instrument theory, seen in some particular way. Okay, but, I don't, you know, I don't have you know, what do you like to see? Well, maybe I should say the following. Maybe I should say the following. Maybe, maybe what, that you can ask what configuration of open-stream that's going around in this loop, is a closed-stream propagating over very large instances. And that configuration is very high-energy open-stream. Okay, basically, I just don't know what to say. Basically, I just don't know what to say. You know, from the point of view, well, she is very clear. From the point of view of space-telling, I don't know what to say. I'm not sure anyone does. Basically, I'm proving the opposite from the point of view of field theory. It's no field theory, yeah. Okay, so, you know, if we just, is there a way of understanding it? Okay, this is the question. Is there a way of understanding it if we just get the, you know, like, I mean, this thing, you know, like some states here, some low-link states here, no, no, absolutely not. You see, because the whole, in order to build up the maskless closed-stream, you need all of the infertility of the open-stream. So, absolutely not. You see, because we're going to this limit, T small, when you see the maskless, when you see the maskless closed-stream, you're going to the limit of T small, which is S small, you know, at the limit of T small, all states in open-stream theory contributing to it. So, that's the limit when you build up this closed-stream. Okay, so, absolutely not. If you just truncated this to level 20, there's no sense in what you would see. There's no sense in what you would see. Okay, except this, this, this, the fact that you can view things in different channels in the world. It's called world-sheet humanity. And it's a very important problem for the literature. Okay, which, as we have seen, has to face some consequences. So, if everyone's happy with all of this, now then, let's move on to the study. Okay, now we're moving on to chapter. So, we've finished the discussion of chapter 7, apart from the discussion of oriental faults, which in my plan, I'm going to get to right at the end of this, this month, before we have your, your exam. Now, we want to discuss open-stream theories. First, there are modifications, and then, oriental faults last. So, from point of, from point of, that organization, we may, we may choose that. But anyway, let's stick to this plan for now. So, the last thing that we want to do in our discussion of open-stream theories is chapter 8, okay, of questions, key, is to avoid any kind of discussion. So, those, one of the reasons of discussing periodic compactifications is that they lead us to new phenomena involving open-streams. They may lead us to understand the deep areas. The discussion of periodic compactifications starts in the absence of open-streams. You know, you don't need to have open-streams to discuss periodic compactifications of any, any theory of gravity in open-stream theory. So, let's start by discussing periodic compactifications of theory in which we've got x0 and then xi i is equal to 1 to 24 and then xd let's, x25. And we break the Lorentz scenario theory by choosing one of the, one of the coordinates at a later point will generalize the starting moment. Start with this one, nearly at 25 and nearly at 0. So, we want to explore this study-stream theory in this plan. So, let's explore what this means before we mean a systematic study. As you've seen x25 is a P whereas r is a number okay. So, what this means is that the zero moment of x25 is periodic. Whereas all the oscillation modes of the of the of the of the of the of the of the of the of the of the of the of the row destroy of the of the of the of the of the of the of the of the process. Next. Okay. conoc am coming here.oh. Okay. So, But before that we should start, I wanted, I wasn't sure you guys were all familiar with this. I wanted to quickly interview how just the theory of low energy gravity, okay, couple of the scale of the years that have been in your field, okay, would react to development. It's any number, when we start a string theory, it will be any number. It's arbitrary, okay, at this point it's arbitrary. It couldn't be made, equal to that would be mean, smaller to that would be mean function. Okay, we will see that in the classical string theory, this is a solution to the equation of moving order. It's just a number that denies. Okay, but there are many things to say about this actually. Okay, but to start with imagine that I was big. Okay, and forget about the vacuum because it won't be there in the classical string. So that the effect of space-time in action, the space will just be the mostless modes of the theory, okay, on the server. Okay, so why don't we discuss the physics of what's called Galooza kind of an action. Okay, you have some familiarity with that structure. But anyway, let's discuss the mass power of actions. We can, because it's important to understand why the effects of that just don't feel the same before we look at the new effects of that. Okay, so what is the action we're discussing? So consider the theory with in, okay. So I'll follow the question of these dimensions and use the symbols m and n to denote indices that run over all 26 dimensions. Whereas the symbols m will be used to denote indices that run over the non-contact. Let's try to start with, to deal with configurations of the system. Okay, we have no dependence on the 25th stage. Okay, this was very interesting in the energy scales that are larger than that, too. Sorry, distance scales are larger, energy scales are smaller than that, too. Distance scales are larger than that, or the energy scales are smaller than that, too. Okay, then as you're familiar with, you know, you shouldn't excite the weight functions that have no dependence on the top of the action. So let's try to understand, you know, how that works. So the first thing we're going to do is to parallelize the metric of this high-density space in a way that will be convenient from point of view of the low-density. Okay, so we'll begin the metric, which is ds squared, and write it as gmn dxn dxn versus the definition. But we say that this thing is the same as gmu nu dx nu dx nu. So as e to the power 2 is the same, dx mu dx ds. Instead of writing 25 over x, d is the definition. That's a mu dx minus the plus, okay. Now, we'll be dealing with configurations that have no dependence on the x dx. But can be arbitrary functions of the x mu dimension. So the first question you should ask in such a situation is what happens in the symmetries of the thing. The symmetries of the original thing, whether they're the coordinate invariances of the original thing, which of those survive? So the two things that survive. The first one is the obvious one. What was the definition of the 25 non-compact dimensions in terms of the other 25, in terms of dimensions? You may wonder, what about re-definition of the 25 dimensions in, say, x mu is a function of x mu and x d. But that breaks our postulate that nothing will depend on x d. Because a field that was x mu dependent will now become x d dependent. We don't want to deal with such coefficients. Okay? So within the restrictions that we're making our process, that we will deal with fields that don't depend on x d at all, that coordinate basis will not depend. However, there is something x. And that is, we can take x d and redefine that field, that coordinate, in an x mu revenge fashion. We explore with it, x d is equivalent to x d prime, or x d prime thing, x d plus lambda. I get this linear with coefficient 1, because I don't want to change the fact that x d is a periodic coordinate with v of x d 2. All of what's this doing? It's just, we've got this coordinate here now. And it's saying that there is no preferred way to set the zero of x d on the circle. I could choose that zero to be anywhere I wanted. And I could choose it, choose that anywhere I wanted, in an x mu dependent fashion. No, this is a function x mu of x d. This is a fine re-parametrization process. Under this re-parametrization, various things will change. What happens here? You see, the metric will change as a metric does on the coordinate transformations. But there's a very easy way to see how things change. You see, under the optimization of the metric, change of coordinates and changes to metric, the line angle remains in there. What change in this re-parametrization will leave the line angle in there? So, g mu will go to some new function, g mu prime. But that's not changing, right? Because dx mu dx mu is not changing. e to the power 2 sigma will go to some new function, e to the power 2 sigma prime. But that's not changing because the dx is squared term, which is here. It's not changing. But a mu will go to some new function, a mu prime. And that is changing. Because dx d, because dx d plus del mu of lambda, that's dx mu. So we can solve that shift. Intershift between mu, which is a mu prime, is equal to a mu minus del mu prime. What's the statement? The obvious statement, looking at it in this form, is the volume. Suppose you took, you made this coordinate change. You calculated the new metric. I made this coordinate change. Re-olded it in this form. You will find these features remain invariant. But this one changes. And this one changes exactly like a human-gauge field changes. Undergauge. How do we see something very interesting? When we look at gravity in a circle, an effective gauge field in the problem. And this statement is more than an analogy. It's a genuine statement. The coordinate, part of the coordinate, the diffeuropism invariance of the original theory, descends into the gauge invariance of this new equation. Since that diffeuropism invariance was genuine gauge symmetry of the original theory, gauge invariance would be a genuine gauge symmetry of the new equation. Okay? So, just at the level of a standard field, we see that the graviton in the original, the original graviton, decomposes into a graviton in the lower dimension, plus a gauge field in the lower dimension. And that's one scalar field. The sigma. The scalar field is the size itself. That can fluctuate. So, the field on the end is, this includes the gauge field. So, now for the next sentence, we're going to focus on the gauge field. The next question we're going to ask is, okay, this is a gauge field, but you see, because nothing else transforms, a genuine transform and a sigma didn't transform, as we did in the strategy of transformation. It means that the functions g, m, u and sigma are uncharged, after the gauge transformation. So, what are they charged in the transformation? Okay? So, you could ask, are there any fields in the theory, or some augmentations in theory, that are charged after the gauge transformation? And the answer is yes, as all of you know what I mean. I mean, say, did I derive a given fashion? Consider a scalar field, or anything, it doesn't matter, but to start with, to be simple, consider a scalar field propagating from this background. Okay? Let's say the scalar field starts. This was the action of this field, was the square root g, g, there, m, chi, there. I treat away from my initial assumption that nothing depends on the extra dimension. Okay? Allow the scalar field, and then we get g, we have the scalar field. So, it's up to another scalar field to have momentum more than the extra dimensions. Okay? So, let us take chi, and say that sum over n, chi m, e to the power 2 pi i, x d by r. Right on the most end of the equation. This decomposition, what I want to do, okay, is to compute the effective action for chi in the lower edge. So, this chi m now is a function of x. Okay? It's clear from momentum conservation that chi m will be part of chi and dagger. So, this whole action here, we'll have terms of sum over n, and then dagger. It must be here, what is r? It was r. Oh. Yeah, I can see it's r. There's 2 pi, so it's the locals. Local pi. There is some action that looks like integral square root g times and some function of chi m, then the chi m and chi minus m. So, we've got to build up this action. What this action? We'll do that with, you see, the important thing to keep track of is that this thing here has an contraction of this high dimensional matrix. So, let's write this down into the same. So, there's a square root g, dm chi, dm chi, gm. And gm for this general metric here is not diagonal between xd and xmin. You would suppose that this thing was whatever it was, nothing is not. So, in order to understand what that action becomes for the time, we have to understand what this square root g becomes and what big gm. So, first let's do this in a toy example. Let's imagine that we had an agent on only one direction. Because it's enough because it's a matrix. So, what is a matrix? Big g becomes a matrix. So, big g in that case looks like so, e to the power 2 sigma that's the d, this d squared then e to the power 2 sigma times a mu that's one matrix. That's a e1. e to the power 2 sigma times a1 and then wherever it was here let's say that was e to the power 2 sigma times a1 I'll go into this toy example and you can see the generalization. Okay. Oh, this one was wrong because this 1 plus 2 sigma times a mu squared. What's the square root of this? Square root of the metric? Well, the square root of the metric has terms with e to the power sigma times a1 squared to cancel this. There's another motivation behind this the split. The square root of the metric a1 squared disappears from the square root of the metric. The square root of g is simply equal to so, the determinant has e to the power 2 sigma that's e to the power c times 1. Okay. It's not clear if you think about it that this will be times the square root of the metric in the posterior case. What is a big g? We have to divide by the square root of this metric interchange these minus signs here so big g inverse is equal to 1 plus e to the power 2 sigma a1 squared minus e to the power 2 sigma a1 minus e to the power 2 sigma a1 and e to the power 2 sigma a1. Now we're supposed to take this by the determinant of the metric. No. So this is equal to e to the power minus 2 sigma so it's equal to e to the power minus 2 sigma plus a1 squared minus a1 minus a1 and 1. Okay. Now it's of course very easy to guess the generalization of this because we have an arbitrary dimension that's on. Okay. What we have is that g dd is equal to e to the power minus 2 sigma plus a1 squared where anytime you're not contracting this is a contract with the low-end metric. Okay. gd mu is equal to minus a1 and g alchemyta I've been going to make a trick of that to form the payload down let me take the original action let me work in the previous space square root g then del m chi del chi gmn and write down the terms. So what we have, we've got integral e to the power 2 plus sigma that's square root of little g times. Now there's one term where the enema indices are both mu so that's just d mu chi d mu chi gmn that's just a usual kind of term in lower dimensions. There's one term where one index on chi is a mu index and the other index is a d index okay. This becomes d mu chi factor 2 d mu chi d chi times gmn which we've seen as minus a okay. There's the last term in which both of these indices are d okay. So we get plus d d chi d d chi times this stuff which goes e to the power minus 2 sigma plus h. Then we're going to be to the power sigma square root of g times now I define a new derivative that means if I have d mu it's equal to del mu plus a mu times del mu. You see that this term this term and this term combined together it can be d mu chi d mu chi and what we have for the division is plus del d chi e to the power minus. The reason for this derivative to appear is that here's the derivative that transforms covalently under the gauge transformation that became the gauge derivative downstairs. How do you see that? You see d mu under the change of variables x d goes to x d plus lambda how does d mu transform? d mu picks up an extra factor of derivative of lambda with respect to me. This is why x d is the guy that's being changed in variable changes in the derivative it's x mu that picks up the extra term d mu that picks up the extra term from the change rule. That's what I'm saying is that d mu prime is equal to d mu plus d x lambda by d d mu of lambda times d by d s let me do it in ten. I've got the sign for it. There we go. So we have d mu prime is equal to d del x alpha by del x d mu prime del fun. Under the variable change and the question so we need alpha, so we had x d is equal to x d prime minus lambda so we've got the sign now. We had we had an a change to the minus angle. The reason that this derivative appears is that it was the thing that was coordinate in there. This is covariant under this case we just care. Now this covariant derivative is the same as the ordinary derivative and feels that don't even depend on the x d derivative. Also this new term we have is zero and feels that don't even depend on the x d derivative. However if we look, if we plug in now the expansion if we plug in now the expansion that we had above namely x d is equal to sum over n chi n is by x d r chi x d by r n. You see that this derivative behaves exactly like the derivative in a charged field with the role of the charge played by the derivative in the deep direction and so will become d mu will become is equal to d mu minus i n by r n. So the chi ns behave exactly like they are charged fields whose challenge in this normalization is n over r. An exercise for you to show that under the global transformation generated by the change in coordinates chi ns transform exactly in that charge. Now we found something interesting. We found that if we say directly in our Kaluza Klein ansatz if we say directly in our Kaluza Klein ansatz there is nothing dependent on that x d field at all of them then at least we have got to make the scalar fields this gauge field is nothing that is charged and nothing is charged up with this gauge field. However when we allow differences in this additional direction things are charged under this gauge field and the role of the charge is played by the momentum in the extra direction. It is clear that the momentum in the extra direction is conserved because momentum is always conserved. That adds to the fact that the charge is conserved is necessary conserved when it comes into gauge field it is clear that momentum in the extra direction is quantized and that is the compact we want here. I will tell you where the gauge group is the gauge group is compact rather and actually we will not pause to look at it at the moment that are interesting with many monopode solutions of this theory we should look at it at some point of course the Kaluza Klein ansatz it will be very exact and exactly without quantization which in some cases is an explanation of the compactness. I thought I would not say about charge fields is this clear it is what we have done here so we have interesting new phenomenology and any degree of gravity will be compacted by a circle more generally than a torus we get new gauge fields momentum in these compact directions is charged up to this gauge of course since overall momentum is conserved you cannot separate momentum from the scalar field from momentum in the gravitation it is so clear in general that any field where the scalar or vector or the gravitation that carries momentum in the extra direction will also be charged up to this gauge field it is more complicated to work out because it is more complicated to solve the equations of a gravitation the transformations in the extra direction it is generated exactly so exactly it is so clear from the structure that momentum in the extra direction abstractly no matter what kind it is would play the role of charge now what is the next what is the next thing the next thing I want to do is to understand how this is the gravitation there are three views I want to understand first is an arbitrary scalar field the second one is the gravitation the gravitation itself is the gravitation and the third is the beam so let us try to understand the gravitation so what we are going to do is now we are going to tell the effect of the gravitation of the gravitation here I am going to appeal to one it can be shown because so it can be shown that is that if you get the matrix of the form of the beam and you calculate the Ritchie scalar then the answer you get is equal to the Ritchie scalar of the lower dimension g mu the smaller mu minus 2 e to the power minus sigma then square of e to the power sigma Rd is the R beta R is the Ritchie scalar in the beam theory in all directions Rd is the dimensionally reduced is the Ritchie scalar built out of little g let us call this R26 and let us call it something it works just to make it and minus 1 by 4 e to the power of sigma the derivation of this is a straight forward exercise it is a pain in the neck as you can imagine and we are not going to try to but some things are obvious it is obvious that this expression which is very common should give you something that does not depend on A but only on A this thing is too derivative so all the simple consistency checks will allow you to fix what the form that could have taken after some coefficients and computing the coefficients would be important but basically the structures are very reasonable they are not going to be clarified about the theory that I have never done it and by the way let us do this now so let us start with the action which was square root of g and this is the big g times R26 times let us put it in time and apply this expression so first we want the square root of g to come square root of g if you remember became integral of the square root of this g times the power of sigma then this R26 we have just applied it is about 25 minus 2 e to the power of sigma times the square e to the power of sigma okay and then we have minus e to the power of 2 sigma 4 that is everything has minus 2 phi e to the power of 2 phi minus 2 phi process this this equation we notice that this term here let us focus on this term this term here is integral of the square root of g to the power of minus 2 phi now the plus sigma times this minus sigma we have minus 2 and then the square of e to the power of e to the power of sigma it is more usual because scalar field to write the kinetic term as okay so in order to do that let us integrate this guy by parts integrating by parts gives us an additional minus sign and then the derivative is transferred to the just derivative of this guy okay so this becomes plus 4 e to the power of g then mu e to the power of sigma plus 2 then mu e to the power of minus 2 phi then mu e to the power of sigma so we get minus 4 integral of the square root of g then mu phi then mu sigma e to the power minus 2 phi sorry this one was a plus that is why I am now at the side this section can then be written as integral of the square root of g e to the power of minus 2 phi plus sigma into that is r25 then there is minus 4 then mu phi then mu sigma and then there is a plus e to the power of sigma by 4 x now it is conventionally like a dealing with gravity theory to call wherever comes outside the r e to the power of minus 2 okay so let us define a new effect of digital phi 25 is equal to phi to the 26th dimension digital minus sigma b is that we should start with the kinetic term for the digital as well sorry so there was a plus 4 d mu phi the whole thing squared which was in this action and that is carried through everywhere yes it is mu because if you look at the digital it does not depend on yeah it had independent original dimensions we would have to do the same exercise that we did previously with this thing so let us even know that we have already dealt with it okay let us just so now what we find is that there are these terms everything in terms of this guy and sigma this term combined to give you d mu of phi minus sigma by 2 of phi minus sigma by 2 with a plus 4 the term that had that was you know except that we need to add to this minus d mu of sigma okay so I am telling you that these kind of terms in terms this guy does not equal to that which is then equal to 4 that is d mu of phi 25 d mu phi 25 and then minus d mu sigma what is my last time is that we put integral squared g times r everything is lower than that okay that e to the power minus 2 phi okay maybe I should write 25 times if the power minus 2 phi 25 r 25 plus d mu phi 25 over x squared with a 4 minus d mu sigma over x squared okay and then minus e to the power r 2 sigma f mu r what from it is exactly the same form that of the action we started we started with an action which had r 26 plus 4 times d mu of phi 26 over x squared and we got the same times e to the power minus phi 26 and we got something exactly the same form except that now we have got additional terms the additional terms are the scalar that represents the size of the circle okay and the kinetic term for the gauge good feelings of the problem are this lower dimension little and the kinetic at this sigma okay now something that might be one of these two kinetic terms opposite sign she might think that one of them therefore is the wrong sign but that's not true because remember this guy is bounded by the e to the power and phi 25 so we need to find the equation of motion of this phi 25 and both of them obey ordinary sign effectively ordinary sign let me say that way if you define if you define a new field let's say e to the power minus if you want to find a new field such that this thing is not there behind its kinetic term you would say that zeta say is equal to e to the power minus phi then this will be proportional to del nu zeta del nu zeta okay but you get the right side there I said that basically what you want to do is to define canonical fields for everything so suppose you were to you suppose you were to undefined okay suppose you define a new metric such that the the richie scalar for that new metric didn't have a little power minus phi 25 that's some conformal transformation of the circuit then the kinetic term for the del the gramaton is just extended and the kinetic term for the delton will also be extended okay but you see that that new richie scalar the rescale richie scalar has extra contributions proportional to del squared of phi which when combined with this term gives you the correct time sorry since I'm sorry for the wrong thing but the the next point is that because of the zeta equal to the power minus 2 phi 25 every field has the right time then it's not manifest it's not manifest okay another way to say it is that this problem of the wrong thing kinetic term existed also in the parenthesis so whatever is also in the parenthesis it's no new richie that's all I wanted to say about this gramaton in the term sector okay the effective fields are a d minus 1 dimension at all d minus 1 dimension metric scalar fields are size of the center of the gauge field and you see what the kinetic term looks like now the scalar fields please leave you in charge of the circuits and you can play any machine games okay the last thing to say is that because this form is practically the same as the parent form we started with if you now dimensionally reduce further you can do that because okay so if you dimensionally reduce you know down p dimensions the new d dimension the new effective digital will be the old one minus sigma 1 by 2 minus sigma 2 by 2 minus sigma 3 by 2 okay and you have many new scalar fields sigma 1, sigma 2, sigma 3 okay the last thing the last thing that I wanted to work out was this beef field in the old time when you did this field was an exercise okay so well let me sketch the answer and leave the working out of the details for you okay so suppose we have this b mu okay now um b mu let's say we start with b and n now the two options the m, m, n, b could be either d I mean i26 or it could be m since it's the experience is anti-symmetric can't have two of them okay so the two options either b mu d or this b mu okay this let's call a mu prime okay this now gauge field how do we see that the gauge symmetry of of b was b times is equal to b plus d of lambda actually one form d of lambda by lambda is one form now now one form is lambda is equal to exactly some chi times dxd okay plus um chi times dxd plus b mu times dxd number three one form let's take d of hat and assume that we've been doing nothing to tens on but actually we've been doing nothing to zero that so let's take d of hat okay let me tell you proportional to d chi where it's dxd since it's anti-symmetric in d chi we don't need the derivative in the d direction so that d mu chi times dx mu where it's dxd okay so the change in b mu d is exactly like the change of a mu under the gauge transformation generated by the gauge parameter chi this is shifted by d mu chi the gauge symmetry you see this gauge symmetry this gauge symmetry under a 26 parameter because it's lambda there's one form that corresponds to 26 25 of these 26 parameters will just generate gauge transformations this way 26 chi generates a not a one form heat event a function heat event for the effect of ordinary gauge field is this clear? so I'm going to explain the b mu mu field in 26 dimensions b mu mu field in 25 dimensions plus an a mu field the exercise I want to leave for you is to work out what happens to the original kinetic term of the theory was square root g h mu nu lambda h mu nu lambda so this is kinetic term I want you as an exercise to work out the kinetic term in lower dimensions dimensionally reduced kinetic term we briefly so please try to do that next class we briefly discuss the other next class and then next class we move on actually considering strain theory look to a very good point again something I should say about this theory we have a new gauge field here and you might wonder what is charged at this now we've already seen last time we published in this lecture the momentum around this circle is meant to charge out of the gauge field that came from that so it would seem a bit too much of that also gave us the gauge field that came from the b mu mu field and it's not true you take a moment around the circle there's no sense