 OK, so today's going to be the last class for about a month. And next class, we'll be at the end of the 30th of April, when we can insert a video. OK, in today's class, we're going to end our discussion of formal properties of string amplitudes today. So when we come back, we will begin by discussing the requirement of conformity in terms of the alpha prime expansion, which I will derive beta function. Hello? Yeah, yeah, yeah. Second, please. Why are you going second? My second. My second. OK, yeah. So we will, of course, anything we can say today, if we do it. Maybe we should study it for the time of this. Perhaps I could go a little faster than usual. Let's just try it out. OK, so first, just clearing up the loose ends from last class. Last class, we were discussing three amplitudes of the sphere. And we completely understood all the singularities, all the imaginary parts of these amplitudes, in terms of, because of these singularities we're exactly all associated with propagation of intermediate particles in finite particles. OK? And there, we concluded that string amplitudes obeyed the unitality properties of, obeyed the unitality conditions of gain, that space time restriction of unitality in the same way the finite diagonal expansion of this. Yeah, there were two loose ends. The first loose end was this business about the B0, B0, 10,000. But that's exactly as we discussed in the last class. That goes exactly as we discussed in the last class. Essentially, the divine function between B and C is non-zero, between C and C is zero and C and B is zero. OK? So if you want the same states, and if we say that I'm headed by B on both sides, then you need to have explicit factors of B0, B0, B0 and B0 tilde, which is what we had in the last class. OK? So the factors that we had in the last class was what was needed in order to make both states physical because the inverse metric that connected the two states would otherwise have connected a state that is annihilated by B which is the state that is annihilated by C. So if you convert that state that's annihilated by C into a state that is annihilated by B in the physical state of the field, then you need those extra factors of B0 and B0. So the factors were there and then the cycle was needed. OK? The second thing that we didn't have time to talk about last class and let's quickly do that, is that we seem to be getting unitarity except that what was running in the channels, what was running in the channels was all states of string theory, in all states that came out of conformity theory because remember what we used was a completeness of conformity theory and not merely the states that are made, that lay in Q-cogon. You remember where we inserted a completeness of states of the conformity theory and that was all states. However, as you know we have a physical state associated only with a state of conformity theory that lies in Q-cogon. So in order for this really to be consistent with space time unitarity, we must have the contribution of all states that are not in Q-cogon or G-vanishes. OK? And how does that work? Well, this works in very simple fashion. OK? Just, so let's see this in detail. OK? You see if personally to remind you of the representation theory of an operator that squares to zero. So an operator that squares to zero has two kinds of irreducible representations. The first kind of irreducible representation is two-dimension. And the second kind is one-dimension. The two-dimension representation is made up by a set of states A and B such that B is equal to Q1A. OK? And therefore Q1B is equal to Z. OK? So if you look this out in a matrix, you get zero-one. OK? The second kind of irreducible representation is when Q acts on some side that you perceive. That can happen with a state that's annihilated by a side and with the coefficient equal to zero. And that there is no state. There is no side prime. Does not exist a side prime such that Q1 side prime is equal to Z. Not true. These are the states that we call physical states. They are annihilated by Q. And these are the states that are like B. All the states are states that we call unphysical. Because they are not annihilated by Q. They are annihilated by Q. They are not annihilated by Q. But they are Q-exact. They are unphysical states and pure-gain states. Because the state is pure-gain. It is Q of something. Therefore that something that is Q of is not annihilated by Q. If you look at this Q operator, you are looking to a basis class that the Q looks like 0, 1, 0. 0 of many blocks of these and then blocks in which it's a zero. So the physical states, the unphysical states and the Q-exact states are paired with each other. Where there's physical states that look. They perform a field theory and take any cases of the unphysical states. Let's call some i unphysical. This is in general an internal dimensional basis. You can grade it with energy in which case it would become a finite dimension and everything would become a finite dimension. Then this defines a natural basis on the space of Q-exact states. So Q on i unphysical gives us a basis for the state of the space of Q-exact states. I would also add J-physical set and arbitrary basis in the remainder of the space. The basis that is needed to complete this in the basis of all of the space. And this gives us a basis in the space of physical states. This thing, by the way, is of course not there everywhere. It's fine. You know, it's main choice to make. Because to a physical state, you always add a Q-exact state and then remain physical. And since you gave it, it's not in the unique space. But it completes quite to be complete states. I mean, sorry, complete invertebrates like given up and need to get it to become physical plus the Q-exact states. Well, exactly. You choose any basis of physical states. We chose the basis of physical states. We chose the basis. We couldn't do it more. We chose the basis of physical states. We chose the basis of Q-exact states. And after that, we choose an arbitrary basis of physical states. What is a physical state? Well, you complete the basis of invertebrates. Anything, any completion, we give you a state that cannot be written as a linear combination of our physical states. That was not a physical. And cannot be written as a linear combination of Q-exact. That was not an exact. There's a great deal of arbitrariness in all these choices that come. Especially the last one. Because to any, if you complete the basis of physical states, I can add to that some linear combination of the Q-exact states. And you know why? Because the thing that is really defined is the homology classes. So that changes not just our basis, but also our space. But we won't care. It won't matter to us. So I just defined some basis. That's all that I have. In the basis that I have, I define some basis and define the projection operator P that projects on two physical states. On two, the space spanned by the space. Okay? So P on J to P equals J to P and P on all these physical states. P on I on physical is equal to P on Q times I on physical. That's the following. Inverse the action of Q only on Q-exact states. That is, U on, I define this operator as U on Q times I on physical is equal to I is equal to 0 U on these operators in my base. It's totally my impact. These are linear operators. I define them in. Okay? In terms of these operators, let me make the important change. I hate the D operator in the written as P plus P plus Q. We get back the physical state because P on physical state gives us back itself Q on I on physical state and by definition U on I on physical state. So physical state. P on I on physical state. Now, U on I on physical state Q doesn't annihilate it and U inverse the operation of Q. So I get back on physical state. On the other hand, suppose I act on an exact state. It's annihilated by Q, it's annihilated by P but U removes the Q and then I act Q, okay? So once again, I've got the same exact state. Okay? So with my definition of P and U, okay? You see what we do and what these definitions are, right? In these two clouds of clouds, U on does the operation of Q but it can't undo the operation of Q on physical states. So it doesn't do anything. It's the nearest and inverse that exists for the Q operator. It's the nearest and inverse operator. We demonstrate that identity is this. Okay? So now, let us, every day, we use the fact that a complete that it does Qs. Okay? So, remember what we did in order to get this unit acting? We, we inserted identity which we then expanded as a complete set of states. Okay? So that's insert identity. Okay? But, since in this identity insertion Q acting either on the left or on the right and having it separate, I suppose you add some Q as we bound you in many ways. Let me bound you either from operator or from contouring. Right? Because Q is the integral of the of the VRSV code and that can just be pulled around the sphere of the museum. Okay? Because each operator in the circle is also a museum. Okay? Yeah, I see that. Okay? So as Q, as Q acting, either on the left or on the right, we just forget about it. So inserting identity for the motivation function for computing has inserted the projector and restrict our sum of states that appears in this, in the decomposition of identity to the sum of states or work of physics states. So although in our formal arguments we did a complete, we did a complete set of, we resolved identity into all states, the contribution from unphysical and Q-exact states batches. Okay? So that's the key point and it's a little obvious if you think about it. The point just to say in a little more colloquial is that it's very much like the VC business that we talked about. That in the completeness sum an unphysical state is a company by Q-exact states or Q-exact states are a company by an unphysical state and one of the two and by Q-exact states. It's very much like how it happens in Q-exact. Yes, in fact it's a precise if this is, if you restrict to the master set that this would be what happens in Q-exact state. You see it's just, you know, you reduce to you get the same argument in Q-exact state. Okay? So, that completes our discussion of unitarity of tree leather strengthening. As you see we've done in a very explicit, well, it's smaller because it's smaller than tree leather stuff, but in a way the key point is that we relate beautiful structures. There is the structure of the OPE on the world sheet and the structure of the Hilbert space interpretation of the world sheet. You see, actually it's the second thing that's going on. Until, two or three lectures ago we predicted that field theory was justified by pathetic rules. Morality. But in order to isolate the singularity structure of string theory diagrams, it was very useful to remember that there's another important element in quantum mechanics. There's one way of thinking is through the pathetecto there's a second way of thinking through Hilbert spaces. And that was the useful way of thinking in order to understand the singularity of quantum diagrams, the imaginary parts of quantum diagrams. And in particular, the completeness sum. It's a very useful device. So it's interplay, the fact that you can take the spot and take it and cut it. Always at the Hilbert space interpretation. And in this Hilbert space interpretation you can then isolate the singularities of quantum diagrams that allow us to understand the precise nature of these singularities and to demonstrate how they were used today. Put it up. I just get questions or comments about how the same analysis or the base of the analysis generalizes to loop-level diagrams. Okay? Now our discussion is going to be more schematic than the three levels. Thank you. Basically because in order to give you a real satisfactory discussion but in order to demonstrate each claim that I'm going to make we may have to veer off into a lot of mathematics about the structure of Riemann's analysis but I'll be not good with that. Okay? So I'm going to assert in some statements about Riemann's analysis as you would say. Pulitschinski gives references to places where these statements are proved if you're interested. So before we start what did I do? I'm going to understand to look at it. A. I'm going to understand the statement that you've heard of in the string perturbation theory it's your financial response. At least it has no UV damage. No UV damage. He's not an example of this for the active magnitude of a tolus. We want to see this in more detail. We want to see it in more detail. The second thing we want to understand is that string theory gives us the rule of string perturbation theory gives us results that are guaranteed to be determined because it's the unit of energy. These are things we want to understand. We're going to do this in a formal way. We're just to see where it comes from. Okay? Let's try to address these questions. The first thing we have, you see, where do divergences and field theory come from? Divergences and field theory come from integrating over a hundred diagonals. Doing the string theory or showing the magnetism, the modularized space of a hundred diagonals if you like. If we have divergences in string theory it's going to be, it's original but in the same, same place. Doing the integral or modular, the modularized space of three months. Okay? The first thing we have to do in order to address questions of a finiteness and as we'll say also for unit activity is to understand the modularized space of three months how this is integrated. Okay? So that's our goal. And for this purpose, I'm going to introduce a construction. A construction that allows you to construct higher genus of three month surfaces starting with lower genus of three months surfaces. And this construction, it's not finite, it's a plumbing fixture. Plumbing fixture. Okay, you'll see why. Okay. So, you have to do the plumbing fixture. Okay. So suppose you get two different three month surfaces. If you want to imagine them to be spheres then they could be anything. Okay? And I've got a chord that batch. Remember what a three month surface is. A three month surface is an animal you know, defined in a theory by wild transformations like this. Okay? So different patches on, you can write it patch wise as complex space with an arbitrary conformal which is, with metric that is dz dz bar. It's conformally related to dz dz bar. And there has to be a coordinate change where it overlaps under which the metric matches auto-refect. The wild factor is a ring of the sun. Okay? So, consider one of these coordinate patches and use the metric to sum the wild factor that's dz dz bar. Okay? And consider another patch here which has the same proper, which, you know, so the coordinate patch we find here and let's call this the same one coordinate patch and let's call this the same two. And let's say that this coordinate patch is defined up to move more than one is less than, is less than more than q to one half, for some reason. And similarly the other coordinate patch is defined up to more than q is less than more than q to one half for the same. A new Riemann surface is the point to remove a disc from here and then join these two things. Remove a disc and join these two things in the chip. We do this in a way that preserves that, you know, specifies complex structure. So, that is why what we're going to do is remove not the whole disc all the way up to q, but only up to, so we would remove a disc which is more than one is less than square root of q, square root of more than q, that's one of my sense, for some possible reason. Okay? So we're not removing all of the disc only some part of it. Similarly here, we will remove more than q is less than square root of q, square root of more than q, one of my sense, we're not removing all of the disc only some part of it. And then what we will do is to identify points in this angle in this angle. We will identify points. We will identify points. No, we just identify points. So, we remove these discs and in the remaining regions we identify points according to the formula. So that identification uncoins these points. The remaining part is the tube that joins these things. We identify points according to the formula. Let's see. According to the formula. But I'm going to require this thing to be defined. Rather this coordinate patch to be defined not only up to square root q but square root q over one of my sense. Just because we identify points. Of course, that matter. I can choose q to be so small that it will be q. As long as there's an equal amount. There, so on. Sorry, but this is my requirement. I choose some x around 0.1 and I choose some q such that these coordinate patches are defined in this in the slight level. Slightly larger than q. And then the identification we do is the identity of z1, z2 is equal to what? In that identity. So what does it do? What does the identity of z1 that remains? What does it mean? Square root q into one minus epsilon and square root q into one plus. Yeah, I wonder. I think it was. Sorry. Ah, hello. Let me put it. It's okay. We have the same thing here. It's okay. Sorry, sorry. Sorry. It was. Okay. So, so, so this is the, this is the angular equation that remains of the module. Okay. Now, according to this identification, this inner end here will match or what is the outer end here? Just so the thing will be like placed by the, as if we have, replace the inner desk by the rest of the, as if we would place the inner desk by the remaining three months. Okay. So this inner end here match over to the outer edge here. The outer edge here match over to the inner edge here. And these two things are conformally irrelevant because z goes to one by z. It's a conformal transformation. This Riemann service was defined by a whole number of coordinate patches including the one we concentrated. Okay. This Riemann service was also packed by 831 coordinate patches including the one we concentrated. Now, we've got a new Riemann service with as many coordinate patches as there were in the sum of these two services. Because we have this coordinate patch in this service that meets over to the rest of the service. And all the remaining coordinate patches that already exist in both different services. Okay. And we've given the role for our points, the diffium opposite, you know, how you should relate our points in this coordinate patch to this coordinate patch where they overlap. That's the rule. Team specification of a complex matrix. This is clear. All that changing in epsilon does is change how thick the overlap is. So it's not this open. Epsilon is non-zero. It doesn't change anything. It says to specify just by q what reach our choice of coordinate z1, z2. And by the parameter q, epsilon is just some intermediate construction. And that would enter into the specification. Okay. Once you've chosen your coordinate charts, z1 and z2, q is that important, is the property, and it affects the complex structure. Okay, imagine it. If you change, if you have a choice of the input, if you can either change q, or you can redefine your coordinate charts, z1, by some factor of q. Okay. And that would be a new complex matrix. Okay. Is this certain? So given with two distinguished coordinate charts, centered around two points, there is a way of connecting them up to give you a complex matrix. Is this clear? This thing is called a quantum matrix. Is this clear? Now, so for instance, let's look at an example. Let's look, for instance, let's look at an example. Suppose you had, and by the way, this addition can be done either on two separate metaphors, or on two different coordinate patches of the same matrix. So suppose we have some sphere, and we have a coordinate patch here, we do the plumbing fixture, that will give us the torus. You see, if we have a torus, and we have a patch here, patch here, we do the fixture, it will give us genus nuisance. So this plumbing fixture can be used. You can start high genus Riemann surfaces, starting from low genus Riemann surfaces. Now, we address the question, can we build an arbitrary Riemann surface? Let's try to do the following thing. You see, suppose we had suppose we had a sphere with three branches, with three pointed branches. Yeah, another sphere with three coordinate patches. We get the torus by doing this with this, and this with this with this instance. You build the high genus surface by doing various other things. Then if you have more spheres with three coordinate patches, you can do more. Basically, you can convince yourself that given an arbitrary number of spheres with three coordinate patches marked on these three spheres, you can build a Riemann surface with some arbitrary genus. In the way, once we think of this, this is some final diagram. You see, associate with each of these spheres a trial in your index. And every time you put together, you sew together surfaces on particular patches, just join up the propagators. Suppose you did this, then what you'd have is this one loop thing with two vertices that were still left over, and you could use them for external particles. Suppose you want to build something with two loops. Okay, what would you do? So let's say you have something like this, you have something like this, and then you have something like this. All you have to do is to make it a two loop five minute mark. That would be a genus two surface with one point mark. If you want a more point mark or point mark, you can split this up into two. Now, let me fix the chair here. Starting with spheres with three mark points, three coordinate patches, and you look at the matrix jump, you can build up, so Riemann's a little bit of a matrix. Okay, now, not that you might ask, but is there a systematic way to cover the entire modular space of genus G Riemann's surfaces? To cover the modular space of genus G Riemann's surfaces by doing the plumbing fixture, and varying the modularity of this fixture. Okay, so let's actually do something. Let's give you the question, we'll do the counting next week. So the question that you might ask is, can you try to build up in a systematic way the whole modular space of some genus G Riemann's surface by starting with, you know, some given fixed sphere with three coordinate patches removed, and by doing the plumbing fixture, varying the modularity of this fixture and all those things. And the answer to this question turns out to be not quite, but good yes to enough quite in a particular way, and the positivity of this answer is in answer to do what we want. So now let me tell you the answer to this question, the technical answer to this question. In what I'm going to say now, between road roof, this is just going to cite the results from that. Okay, so let's make the following statements. Okay, see, tell me about the Riemann's surface and give it a non-contractable cycle. I mean, give it a Riemann's surface, it is always possible to find a minimal area of metric of that Riemann's surface. Okay? But a minimal area metric I mean the bottom. I mean a metric such that the length of every non-contractable cycle in that metric is greater than equal to 2 pi. Subject of this restriction is otherwise minimal. If talking about a minimal area surface for a Riemann's surface without putting some other instruction, the other restriction is meaningless because for instance for a torus, just shouldn't be to zero. For any of these, just shouldn't be to zero. So the Riemann's surface is defined only after a while, such that the area of metric is equal to zero. So for instance for the torus, the non-contractable cycle. But there won't be any on the sphere. There won't be any on the sphere. Yeah, so this will be a very degenerative sphere. So the statement will apply again every genus one higher. Every genus one higher. On the torus, what is this minimal area metric? That's totally clear. It's just a flat metric where inside is less than 2 pi. That way no non-contractable cycle has length more than 2 pi. But you saturate that. So you get a surface with areas 2 pi and all things. It's clearly impossible to have a smaller metric than that. A smaller area than that because otherwise one of the two sides becomes smaller. So on the torus, the statement is totally obvious. The case is that any genus G Riemann's surface G greater than 1 greater than 1, that exists in such a way. First one. Okay. Now, second case is a slight generalization of this case. And that's the problem. Riemann's surface with some marked points. Okay. Now, we demand that a cycle that goes around one of these marked points is also considered non-contractable. Then the claim continues to hold. It is always possible to find a metric on any Riemann's surface with marked points of minimal area subject to the constraint that no cycle around any of the non-contractable. No curve around any of the non-contractable cycles, including the cycles that are marked points, has a length smaller than 2 pi. Now, you might think that this last thing was really weird because we've got that chain on the torus. We've got a marked point. And you might think, well, I mean that. I can go and get another point and say, well, how can I prevent myself from having length smaller than 2 pi? But of course you can. It depends on what the metric is. So the metric here, suppose this was z equals 0. Suppose the metric was dz by z over dz bar by z bar. Then the length around any side of this would be z. In fact, I may have to put some 2 pi's. It would be fixed. It would give me 2 pi. So basically, in this condition, you should think of working in a coordinate patch in which instead of having a marked point, you have this tube that goes up. This can always be conformally mapped to a tube that goes up to infinity. And so you're looking at, you should work in those elements. And you're looking at a metric on the surface of a tube that goes up to infinity, such that the cycles of these tubes are also considered non-contractable and they cannot have length smaller than 2 pi. Something to tell you one of the other areas. Metrical. Some such thing always exists. Okay. So I'm going to give you many such claims and then we'll make our point. I'm sorry about this. But okay, it just wouldn't be worth it to go into some long language and then have these things up with you. Okay. As long as the statements are clear, not going to be sufficient. Okay. What about this would be, it is the following. Let us define the band. Okay. A to minimal length geodesics. Minimal length geodesics that go around some non-contractable cycles. Okay. That are homotropic to each other. That can be continuously, continuously moved to each other. Are set to lie on the same band. So for a, on a top of a minimal length geodesics, this guy and this guy lie on the same band. Okay. Or on one of these things with a marked point because the cylinder will not be parity and the whole cylinder lies on the same band. Okay. Now the height of a band is defined as the minimum distance between geodesics on the band. Okay. So suppose you've got a band and you can't go further on one side. Okay. So that's the end of one band. That's one end of the band and you've got a second end of the band. Now look at the minimum distance between a point anywhere on this geodesic and anywhere on that geodesic. Okay. That is defined as the height of a ball of band. Perhaps for the most surprising claim of the law is that every point on the, on the reman, on this minimal area of the reman surface lies in some band of the other and bands might only land. That is, given any point of religion on the reman surface, it's always possible to find a minimal length geodesic that goes to this point. So obviously, of course. Okay. And it may be possible to find that in many ways. Okay. So now we've got this good picture of this arbitrary height of the reman surface. Okay. We've equipped it with a metric. We've equipped it with a metric which is this minimal area metric. That's minimal area subject to these constraints. You know, England with the data stream theory is just conformal classes, complex structures. But as an intermediate technical device, we've equipped the reman surface with an actual metric. Now I want to make the claim. The claim is the following. Suppose we look at it, all reman surfaces with arbitrary number of marked points such that the height of every band, every internal band on the reman surface is less than or equal to 2 pi. And the external band, one of these bands that are, these long cubes are going up in infinity is less than or equal to pi. Less than or equal to pi. Let's call the set of all such reman surfaces. This is such reman surfaces. Let's call that this is number of marked points. This is geodesic surface. Okay. So this is the set of all reman surfaces of geodesic reman points equivalent to minimal area metric such that the height of every internal band is smaller than or equal to 2 pi and the height of any external band is smaller than or equal to pi. Now, now we have to find the claim. And the claim is this. That associate every such, every such reman surface with a vertex and endpoint vertex in a final diagram. So suppose you want to build up a full modular space. Suppose you build up a full modular space at genus G. Okay. With M insertions. The way you're supposed to do it is to make all final diagrams that you can. Guys, as vertices, using these guys as vertices. Okay. Which you put together with propagators. And the propagators represent the plumbing. The plumbing fixed here. Okay. So let me, let me try to say this. Let's try to look at this. Let's try to look at this in an example. Okay. See, suppose you wanted to look at the sphere. Suppose you wanted to look at the sphere. Okay. With four line points. Okay. And by the way, the thing that you're supposed to start with in the sphere. You're supposed to start with V3G and anything that fits with that. Okay. So suppose you're here with V3Z0 and V3Z0. Okay. So suppose you want the sphere with four lines. Okay. Then, of course, the basic vertices in the theory. Us, we filled with three month points. Now, these were the sphere with four lines. Also, a basic vertex. And the basic vertex is, look at the middle of the area-metric on the sphere with math points. With four math points, such that each of these stubs, these external stubs are less, more than anything to find. Then, explicitly cover explicitly cover a large portion of the modernized space or was filled with form of points by showing together these three marks of space. You don't cover others. The part that we don't have to cover is that part that gives you these minimal area scale that is covered by minimal area metrics such that the length of each stub, height of each band is less than equal to point. These basic guys, of course, these basic guys satisfy the requirement that their heights are also less than equal. Every vertex in your clearing satisfies this basic requirement. Then the trainers that you put together the whole modernized space of the Riemann surface by making this final diagram, joining of these basic Riemann surfaces plus all the new vertices at every end and in regions. Now, I said to confuse you today and I said, I'm not trying to explain why in this question. For example, there are four of them which are less than pi and one of them less than two point. Yes, though this is not a non-contractable cycle thing, why don't we eventually know it? We've eventually got a sphere of form of points and all the sphere of form of points in the end, your question is very relevant though when you use this to make a surface with a non-contractable cycle. So suppose you use the bluffing fixture to take a sphere of Riemann points and connect like this to make a toss. You see that this bluffing fixture will give you only those products in which one of the cycles has length greater than equal to two pi because you joined together two stars. Each of which have length greater than each of which, joining together is the person joining together with stars of length partly but then with an arbitrary cube parameter in this bluffing fixture that will automatically make the effective length of this larger than two pi. I'm sorry I should have said this part here. We're only supposed to do it from cube less than one. Let me, let me, let me, let me, let me. Let me just think this through. You see, you see, you see, so I said something inaccurately, let me say it correctly. This VNG represents all Riemann's surface in which the internal dance, in which the heights of the internal dance is less than equal to the height, but the heights of the external dance, the stumps that you leave is exactly equal to the height. So you're supposed to take this thing, which has this semi-infinite diagonal of young and then add it, such that we've got a stomp of length, fine. Okay, now what I want to say becomes clear. You see, suppose you join together two stumps of length, fine, you would get an internal band of length, too, fine. Now, when you want to do the integral of the whole modularization of the daughter's, you should let this relative length of the two bands, which is the modulus of the daughter's, carry over everything. You see, it's a plumbed fixture with a Q less than 1. So the claim is that you can do all this with a plumbed fixture always with a mod Q less than 1. Okay, so the plumbed fixture with this mod Q parameter in every plumbed combination integrated from 0 to 1. Okay, that gives you the whole cover of the modularized experience. Okay, now what's the problem? You see, suppose you had done this plumbed fixture with Q equals 1, and then the whole thing. You take two of these stumps and you join them together. What you get is an internal band of length, too, fine. However, if you do it with a smaller Q, you get an internal band of longer length. Okay, yeah. You see, why is that? What's basically going on is the problem. Suppose you've got some patch here and patch here, and you make Q very small, and you're taking a little bit of this patch and a little bit of this patch and joining them together. And joining them together. This actually. Okay, let's draw. Suppose you've got some funny ribbon surface, but it's got some patch here. You take a little bit of this plumbed fixture and a little bit of this small Q, and you smoothly interpolate between the metric that's here and here. You get something else. This metric, it does it away in conditions of a minimal area metric. Because this little Q here, which Q is so small, has geodesics that go around this factor. And geodesics that go around this factor, which are letting it smaller than your body. So you have to change the volume. You have to change the wire factor on this remand surface. You know what I mean? Since the wire factor, I try to talk to you and blow this up to let it divide. You also blow this factor. Suppose there was something here, that was just ready. This band would get blown up. So if you did something like Q, if you did the plumbed fixture with a parameter Q, if Q was one, you would just add the length of these two steps. But if Q was smaller than one, what you would be doing is joining this smaller size and you'd have to blow this up in order to meet your conditions of a minimal area metric and therefore be integrating over internal cycles of all lengths. So that's essentially the point. The point is that this construction, in its vertices, includes integrals over remand surfaces whose band heights are smaller than your pipe. But all integrals over band heights, larger than equal to your pipe, comes in the integral over the modular space of the plumbed fixture. Let's try to say better. Can you try to isolate what you don't? You're trying to say that for Q, if you're having this pipe and pipe, and you're saying that for Q less than one, like... Let's do it more clearly. These discs here and these discs here. For the plumbed fixture construction, suppose I take some very small key. So I take a little region of this thing and a little region of this thing and I join them together like that. Suppose this is a noncontractable cycle, some of the tube here, so that we're talking about the length of a noncontractable cycle. If we want this row, there's no issue. Well, what would I get? I would get... First, let me say the picture, and then if we want, we can try to do it in four parts. What would I get when I joined these two things? I would have this little tube joined to this little thing, joined to this little thing. Use some arbitrary metric. Suppose I use some arbitrary metric to describe this stage. I would be wide open. Always the same in the formula class. Something that smoothly interpolates between the metric of this surface and the metric of this surface. Then both of these remain very small. So what I get is a surface that looks like this. Is this clear? Now, this surface does not obey the conditions of our middle area metric. Because there is a noncontractable cycle, like this one, where there's some cycle going around it whose length is smaller than that one because it makes you very small. Now, when you put it into this middle area, that's homogeneity. If you blow up one length, it blows up all length. You see, this construction in this aspect makes this of length 2 pi. I moved to the right half because this length was at most 2 pi. In this construction, it will be, I can't remember what, it will be something like 2 pi times q. Or maybe q times q bar or something like that. So it's very small. So you have to blow it up. So in doing the blow up, what we get is that this length can be made 2 pi. This length will become 2 pi to the right side. Or maybe q times q bar. You see what I'm saying? Therefore, you should just integrate over all q less than or equal to 1. You automatically generate those parts of the Riemann surfaces where band heights are greater than 2 pi. You won't generate band heights less than 2 pi. That's why those guys are integrated over in your definition of the Feynman vertices. That's how you approve the statement. I'm just telling you, telling you about a result that is claimed that athletics is no more beyond the system. I never see the proof myself. Okay, so, but the statement is okay. For the following reason, where are the points in modular space? Where are the points in modular space? Where do you expect? Where are the points in modular space? Where do you expect divergences from? The key point is that you expect divergences from extreme corners. For instance, suppose we look at the Feynman diagram. You have a Feynman diagram interpretation of what's going on. When you find the diagram flow, Feynman diagrams flow up and down, they have momentum loops or from a position space point of view where the insertions of the operators come very near to each other. Okay? Or if something is propagating over very, very short distances. Now, very, very short distances is compared to the size of the string. So, it's an extreme corner in modular space. The length of the propagator is going to zero compared to the height. The points are coming very close to each other. That's another extreme corner in modular space because the operators are in the position of the operators and the operators are merging with each other extremely. So, this is the final diagrammatic interpretation. You know, final diagrammatic where you are saying it. And if you think just more structurally in terms of ribbon surfaces, basically interior points in modular space, there's no scope for any deadlines. Everything is nice and smooth in modular space. Okay? The only bit a divergence problem is where something funny is happening, something very infinity. And that happens at extreme corners of modular space. I think this is in the Doros for instance. We saw this in the Doros when we were iterating over this region in modular space. The only place where diagonal space is going to happen was this thing going on infinity. Or from living here, because this guy, it's an extreme corner. Actually, you will end up with this. It's the same kind of phenomenon that you expect always. Now, because of that, this VNG, the vertices that you added to your clouds integrated, you know, that includes some integration in modular space. But that integration is over some very internal region of modular space. We're ensuring that by demanding that no cycle get too small. You see, we're ensuring that no band, which is the size of any cycle, get too small cycle itself, get too small. So now, allowing you to say one cycle of the Doros to shrink to zero, compared to the other one, which would be an extreme corner of modular space. So these vertices that we're adding are safe. One of the divergences, the potential divergences in the theory, all of the potential divergences in the theory are going to appear from the integral over the module at the plumbing fixed. Divergences, imaginary parts, things that have to do with unitality, we can isolate by understanding this plumbing fixed at instruction level for conformity. So that's the key idea. So now, this is one more thing. I'm sorry, this is going to be a very formal lecture. This will be the last very formal lecture. We'll get out. We'll get out. We'll get out. So, both of these lectures have been about charge. But now, let's turn to conformity. First, you've got a Riemann surface with some odd points. The path integral over the Riemann surface that you form from the plumbing fixed at instruction. You want to do the path integral of the conformity theory with these vertex operators. Let you form from the plumbing fixed at instruction. Let you form from the plumbing fixed at instruction of of these two of these two of these two different Riemann surfaces. Now, you see what's going on. What the plumbing fixed at instruction essentially is is cutting the Riemann surface on a circle cutting the other Riemann surface on a circle, then joining these two. Out of the path integral of this conformity theory with this server cut is some state. You know, what you would have here what you would have here is the path integral of this conformity theory with this group is some state on the CFT in the ordinary radial compensation. Similarly, the path integral of this thing here is some state of the CFT in the ordinary radial compensation. Now, what you want to do is to join these things up. You want to join these things up with the insertion of the identity of the product to be keyed for the analysis of these four point functions. We can resolve this identity of sum over in terms of a complete sum of states. So, instead of looking at the complete set of states, you see how you make the complete set of states. You make the complete set of states by putting in a complete set of operators into this disk. Let's first do if a Q equals 1 then we take the last one. Suppose you have a Riemann service that is constructed by the Plombing's fixture construction of two different Riemann services. Now, on the diagram of the conformity theory all this bigger Plombing fixtures are of Riemann services is equal to the path integral on each of the constituent guys where every operator is the theory with operators inserted at the centers of the disks that you removed and then the factor of GMM, the inverse of the zonological metric put in and sum over the statement AM the statement AM sum over all. Once again, what's the matter? Just you here, you insert identity identity is the sum of all the states. But each state is filling up this thing with a disk with an operator. So let's say again, we've got this tube here to cut it. All this cut is inserted. But identity is equal to sum over okay, we should write it the same. This thing, M and M with GMM this conformity is given with state M in it. This conformity is given with state M in it. The state M is filling up the thing with a disk with operator M. Undoing the boundary fixture construction to posit it into its constituent pieces. Undoing the boundary fixture construction so that it decomposes into the constituent pieces but add an output to M where the constructions are now and output to M where the constructions are now and put this factor of GMM and put this factor of GMM and this was given to us equal to 1 because in our conventions a state is created by inserting an operator of length 1 sorry, an operator at the origin of a disk of size 1. That gives us the state operator map. Suppose we were working with Q less than 1. Then what we're getting is not quite the same state. And then inserting the operator at the center of the disk produces not quite the same state but it's an easy way of fixing this up. Instead of using the coordinates Z1 and Z2 work with the coordinates Z1 divided by square root 2 so let's say Z1 prime and Z2 divided by square root 2 Z2 prime. In those coordinates you just have to have a little bit of one. Coordinate inserting identity is precisely inserting GMM but the insertion of A operator at these coordinates is related to the insertion of the A operator at the other coordinates by a factor of DZ1 prime by DZ Z1 to the power H and see that Z1 prime till that by Z1 till that power H. So when we work that through when we do the complicated state of instruction we're done with packing a Q what you get is a factor of Q to the power H Q bar to the power H G GMM times the constituent surface with A and the constituent surface with A and and then all the other surfaces all the other all the other insertions Okay, so let's understand the game the game is that if you take the Riemann surface that is constructed out of two other Riemann surfaces with a bluffing fixture with parameter Q that the path of the A group with some number of insertions on this Riemann surface is the same as the path of the A group on the constituent Riemann surface the guy that you use to make a bluffing fixture okay, when all the operators inserted at the origin summed with GMM summed with all A and all N went with the factor of Q to the power H remember that the integral remember that the integral over all Riemann surfaces includes an integral over all Q of Riemann surface obtained with some bluffing fixture Q goes to 0 that makes these lines very very long okay we were working in a method in which no length is allowed to become very short it's minimal, right? the extreme level is going to become very very long basically the last step without trying to justify it in detail again, appealing to our discussion of the torus where we did the same thing very carefully and found the same answer in the string theory in addition to all this business we have to put in a B a B insertion for every module so the integral over Q is happening by a B0 B0 bar so we get GMM along with a B0 B0 bar which gives us in the same nice property that we discussed before in functions that it links all these states that are annihilated by B0 that states annihilated by B0 and in addition we also get a factor of Q from the from the insertion of from the insertion you know from the form of the B insertion in the sketch of nature example you know the form that was we found the same factor of 1 over even tau squared in the discussion of the torus it's the same kind of discussion I'm not going to try to derive it again okay, as I accepted now we can see the point why did I go through all this you see the dangerous part of finding the diagnosis the dangerous part of finding the diagnosis that we've seen is associated with some Q going to 0 okay so let's understand what we get from the limit of some Q going to 0 of this coupling fixture but what we have here is some amplitude like some other amplitude and now we've completely isolated the Q dependence just like we did in the discussion in the discussion of 4.3 4.0 functions using this insertion to complete the other states okay and just as in that case what we get is Q to the power L0 minus 1 Q bar to the power L0 minus 1 okay then we got this Q Q to the power L0 minus 1 Q bar to the power L0 minus 1 at least in the same way as in our discussion of 4.0 functions we have to do this example of a D to Q and that is the same integral we had in this discussion of 4.0 functions and for exactly the same reasons you know right it's in radial coordinates you get a 1 dr better explicit factor of 4 Q and then there's an integral which removes another 1 Q so what we get here is 1 over m squared plus k squared minus i epsilon up to some factor of 4 alpha prime and then there's some 2 pi's it's a scale this is now exactly parallel to the discussion that we had for the 4.0 functions it's a scale that's rewriting L0 and L0 bar in terms of in terms of the mass squared squared of the of the vertex operator and just go over the same discussion we had in our lecture and you get this okay good so this gives you the integral of a Q from 0 to 1 the point is this that we have completely isolated the dependencies the behavior of all these diagrams in the region of modular space where they're dangerous mainly where some Q is there what we find is that this behavior is precisely given by final diagrams by the behavior of final diagrams and it can lead to a lot of possible divergences so now I'm going to do this firstly the imaginary parts of these amplitudes the imaginary parts of these amplitudes are in one-to-one correspondence with the imaginary parts of a final diagram expansion with the internal particles propagating in internal channels okay so if you think this through and we're not done at the same time as you need to but so because of this point you think this through the usual arguments of final diagrams giving you unitarity in field theory just go over in a straightforward fashion to imply unitarity for stream theory okay the imaginary parts are the imaginary parts of final diagrams of propagators associated with each propagating particle okay and that essentially results in the the unitarity of the stream theory is that is it the dangerous regions in modular space the dangerous regions in modular space which have to do with propagation of a very large distance which have to do with skew goes to zero limit have all been identified with propagation of particles in space time of very long distances so why is stream theory can have divergences? for instance if our stream theory does happen these divergences can always be interpreted as long distances you see every dangerous region in modular space can just be associated with propagating of a long distance why is stream theory can stream theory can have divergences these divergences are always interpretable as long as IR divergences associated with long distances now why is that an improvement? you see this is an improvement because IR divergences is a physical way IR divergences tells you that you are working with say the wrong background or you are expanding your theory with the wrong masses in your propagators okay it may represent an instability in the theory in which case you are working with an unstable theory if your theory has no physical instability IR divergences are always associated with either asking the wrong question like trying to accurate some scattering attitudes for glue on the QCD or with doing an expansion around the wrong background you know if you do an expansion around maximum rather than minimum in the quantum field theory your expansion would be full of IR divergences all the divergences that can have divergences in stream theory associated with physicality of course you are looking at a stream theory in which you have not got the right background maximum rather than minimum you will get IR divergences as usual because your theory you are doing expansion around the same level however if you cure all IR problems in the theory if you work on a good background like the background to super stream theory in super stream states so that your theory has no IR divergences then your whole stream energy will be fine this is the second main point I am going to make now let's see yeah so this has been a strange lecture I just had a few equations I don't really know what to do but at more equations we have no much more okay I should just leave it is there anything else I want to say there you are there is some very general discussion that I recommend you have written in section 9.5 in Kulchinsky in which he talks about in which he talks about the fission assasin mechanism in which he talks about how in which he addresses the following question in which he addresses the question about addresses the following question equations of conformal invariance let me just say the equations of conformal invariance as we will see when we come back we reconvene to discuss our class again equations of conformal invariance of the sphere give us the tree level space time equations of motion of string theory but now that seems like a positive because these equations of motion get corrected at loop not in equations for consistency of string theory get corrected at loop the equations are just some local property of conformal field theory conformal invariance conformal invariance conformal invariance is not some global statement you know it is a bit of a mansion to local statement how can I get corrected depending on what matter whatever and the answer idea is that the integral over moduli space has to be has to be regulated and that integral over moduli space has its own that that regulation has the wild factor slips in in that regulation more or less in the way that the wild factor slipped in in the regulation of OBs when we try to define when we try to define into the bi-games in the dimension of the bi-games okay this discussion will make more sense once we have gone through the discussion of the meta function just in a few minutes so I am not planning to go to discuss it now but I suggest you I mean it would be a good thing for you okay so I think that we probably want to stop this lecture here apart from questions at the moment I am sorry I know that this would feel like a very unsatisfactory lecture what I hope I could be able to do in this lecture is a general sense of the issues of unitarity and finiteness of loop attitudes and strengthening those are very possible let me just try to summarize this general sense let me just try to summarize the general sense and then let's take questions let me just try to summarize the general sense the main the key point is the volume you see in the integral over moduli in the integral over the moduli space in extreme terrain you have extreme corners in moduli space let's first see the trawler suppose you do the integral over the moduli space in the trawler you have you have this region where one of the cycles in the store becomes very small and the other one becomes very small. Because you think of the one. It's only the ratio that matters. So it's one of the cycles becoming much smaller than the other. Now suppose you think of the cycle that's becoming very small as the time that is remaining large as the space connection. From the quantum field theory point of view how would you interpret this? You would interpret it as each of the pathways of string theory going over a very small looped space, going over a very small regional string, a trigger barometer space, a region that would give you divergences in quantum field theory. Is this statement clear? You remember how in our discussion of Doris amplitudes, we looked at the close relationship between trigger barometer and field theory and the modular space of string theory. Do you remember how we reconstructed the Doris amplitude by summing over the trigger barometer representation of the one looped path detector for each particle? And in that reconstruction the trigger barometer was simply just part of the modular space of the Doris. You remember that, right? So one sign is the length of the string. The other sign is how much time it runs. How long it goes at trigger barometer space. So the region in which you go based on a certain trigger pattern of the space is associated with U V library. This is a dangerous region that we should be standing. However, we can take another point of view. That cycle has space and the other one has space. Then perform a wider transformation so that guy stays and let you go by. On doing this wider transformation what happens to the other guy? It becomes very normal. The region of potential damage in field theory space, the region of which you go based on a certain trigger pattern of the space, cannot be problematic. The trigger barometer length becomes very large. It's becoming very large. It's associated with this propagation of a very long distance. The barometer is e to the power minus tau times k square per centimeter. So tau becomes very large. Very, very, very large. You're getting competitions for small. It's got an inversion ratio. And therefore long distance. Tau is like length square. So, on the daughters we've already seen this, but I want to emphasize it because it's... Today's lecture is about how it's modulated. The point is the following. It's impossible, first point for U V library. It's impossible in string theory to have a problem in U V. You also have a problem. More precisely, there's a way of looking at the modulate space of Riemann surface in the string theory, such that all extrements are always interpreted as IRX.