 I want to ask questions, take a just unmute and ask. And so I will first start the recording. Recording in progress. OK, so we'll now have the third lecture of Kevin Costello on topological aspects of string theory. Please go ahead. Thank you very much. So firstly, I apologize again for the technical difficulties yesterday. I'm pretty confident that we're going to be OK today. So let me just verbally remind you what we discussed last time. So we saw that there were chiral algebras associated to super conformal field theories, then equals 2 supersymmetry. And we wrote them down explicitly for equals 4 AMLs, in which case we had 2 adjoint fondue scalars in our chiral algeba, and a certain family of SEFTs with SOA flavor symmetry. And in that case, we had two matrix value fields and one vector, sorry, eight vectors, rotated by the SOA symmetry. And we also discussed a little bit about the topological string, which we propose will be holographic if you do it. So today, we're going to study the back reaction in the topological string, because that's the first thing we'll study. So last time, we also wrote down the field sourced by a brain in the topological string. This was a certain Beltrami differential. When we solve that equation, the geometric role of the Beltrami differential is that it changes the Cauchy-Riemann equations. In our case, in the deformed geometry, the original coordinates were W1, W2, and Z. And here, the new Cauchy-Riemann equations are these equations here, which tell us what does it mean to be a homomorphic function in the deformed geometry. So let's solve these equations, and then we'll find that the deformed geometry is ADS3 times S3. Let's go up a little bit. So if we look at the equation, no more in the equation is there a d by dw1. So w1 and w2, these are still solutions to these deformed Cauchy-Riemann equations. So these still define homomorphic functions. However, if we look in the equation, we see that there's a d by dz. So the function z will no longer be homomorphic. You can instead see that there are two new homomorphic functions, which I've called u1 and u2 here, given by these expressions. u1 is zw1 plus n over normal w squared, w bar 2. u2 is something similar. And it will take you half a minute to stick u1 and u2 into these equations and check that these equations are solved. So w1, w2, u1 and u2 are the homomorphic functions in the new geometry. However, they're not independent. As you can see, u1, w2 minus u2, w1 is equal to n, which is again entirely elementary. So we conclude that the deformed geometry is that defined by this equation. So we can think of this as the space of 2 by 2 matrices like this, the column vectors u1, u2, w1, w2, whose determinant is n. Now I can absorb the factor of n into a rescaling some of the coordinates. So you might as well think of this as matrices determinant is 1, so that the geometry is sL2c. So this is the topological spring derivation of the back reaction. For the experts, I want to point out that this is a little simpler than what normally happens in the physical spring. Because we don't have to pass to any near horizon limit. Once we back react, the geometry already has a homogeneity, which means passing to the near horizon limit does nothing. Are there any questions about this computation? OK, so our goal was, of course, to compare the chiral algebra as we've been discussing to quantities built from the topological spring in this deformed geometry. So if we go back to think about the more familiar ADS5 times s5 holographic dictionary, the very easiest thing you might see is that in ADS5 times s5, the symmetries of the CFT are the same as the isometries of the geometry. So this is a very basic and elementary check, the ADS-CFT correspondence. So, well, the isometries of ADS5 times s5 are s06 times s042. And here, s06, that rotates the six scalars, many plus four young nodes. And s04 comma 2, that's the conformal symmetries of the force sphere. The symmetries of the force sphere, which preserves the metric up to a regular scale. So because it's a CFT, these are symmetries of n equals 4 young nodes on the force sphere. So what we're going to do in the twisted holography setting is we're going to check that the statement holds. But what we'll find is that it's a much richer statement for us because the group of symmetries will become infinite dimensional. And therefore, we'll give us a very, very strong constraint. Before we move on, let me explain briefly how to think about the symmetries in the case of the CFT in a more abstract language. Your CFT will have a certain number of conserved charges, especially in a chiral CFT. Any operator gives you a conserved charge. I integrated it around a circle. That's my conserved charge. However, we're interested in those conserved charges, which preserve the vacuum at 0 and at infinity like that. So if I take my charge, integrate it around the three sphere, and I hit it with the vacuum at 0, I get 0, similarly with the vacuum at infinity. So think of these as the initial and final states. So I'm going to add another page. So why is this a good thing to do? So charges like this preserve all correlation functions. So I get a picture like this where I have local operators. And if I sum over integrating a charge around the local operators, that's going to have the same effect by change of contour argument as integrating the charge around all of them. That's going to be 0 because it is the same as integrating my charge against the vacuum, against that infinity. So in a minute, we'll come back to calculating these charge, these symmetries in the CFT side. So back to the topological string side. What will be the analog of the isometrics? Well, we know that general relativity is covariant under diffeomorphisms. Of course, diffeomorphisms are the gauge symmetries of general relativity. And the isometries are those gauge symmetries which preserve the field configuration you start with. Like they preserve the vacuum field configuration, which the metric on ADS-5 times S5. So the kind of thing we will study that will replace isometries will be gauge symmetries preserving the vacuum field configuration. So let's study what that is. Let's look in the type 1 case. So the fields, well, the solutions to the equation is motion more precisely. In the closed string sector, there's a complex structure and a homomorphic volume. In the open string sector, we have an SOA problem, like a homomorphic SOA problem. So what are the gauge transformations of this data? Well, for the complex structure, that's really a part of the metric. So the gauge transformations are diffeomorphisms. Those which preserve the vacuum field configuration, though, because we don't have all of the components of the metric, not just looking at isometries. We're looking at those diffeomorphisms which preserve the complex structure. And they also have to preserve the volume volume. So at an infinitesimal level, diffeomorphisms are given by vector fields. These are homomorphic and divergence-free vector fields on SL2C. So if you're familiar with complex geometry, you might notice that this is a very, very, very large algebra. And we're going to see it that it also appears in the CFT side. Something that's a little easier to study is what happens for the open string fields in the type 1 spring. There we have our gauge field describes a homomorphic bundle on the manifold. So those gauge transformations which preserve the homomorphic structure of the bundle are not constant, but rather they satisfy this equation. So the gauge transformations are homomorphic maps from SL2C to SL8. So again, this is a very big space because there are lots of homomorphic functions on SL2C. So these Lie algebra are infinite dimension. So our goal today is to find these Lie algebras in the chiral, inside the chiral algebra as symmetries which preserve the vacuum at zero and infinite. I have a question. So good. Yeah. So since SL2C is non-compact, is there some condition on the homomorphic vector fields at infinity that one has to impose? No, no. You look at all of them. So that's why it's so A. OK. So let me try to write a little more explicitly what the open string algebra is. We're looking at homomorphic functions from SL2C to the Lie algebra SL8. So what kind of thing is that? An element of it is going to look like a product of an element of SL8. A is an adjoint index for SL8. And a homomorphic function in SL2C, which is a function of the variables ui, wi, modulo of the equation u1, w2, minus u2, w1 is n. So let's look at the quantum numbers of these symmetries. How do they transform under the natural evidence symmetries one has? Well, of course, under the global SL8, these expressions live in the adjoint representation. What's more interesting is to think about how these expressions live under the left and right action of SL2 on itself. So the group SL2 acts on the left and the right by matrix multiplication. So on the CFT side, we also have those two SL2 actions. The left one is the global conformal symmetry, rotates Cp1. So this is the analog of SL4, 2. And the right one, it rotates. In our theory, we have two matrices. And they have a symmetric role. This right action rotates those matrices. So this is the analog of SL6. Now, how does a function on the group behave under the action of left and right multiplication? Unfortunately, there is an old theorem from the 1920s or something, which tells us this. This is really just harmonic analysis of the group, or the compact group, if you like. But it says that every polymorphic function can be expanded to modes. And the modes living the tensor product of the spin j representation on the left with the spin j representation on the right. So our symmetries are then in that joint of S08, tensor spin j on the left, spin j on the right. So we should keep this in mind because we want to reproduce this from the CFT side. Well, what we want to identify the algebras we see on the topological string of the CFT, we also want to identify them not just as vector spaces, but also the structure constants of those algebras. So let's write down some structure constants as well. And then we'll see if we can reproduce them in the CFT side. Oops, I don't know what that comes later, sorry. Take that back. I will discuss the structure constants later. So what we'll do now is find symmetries with exactly these quantum numbers in the CFT. So how does that work? So if you remember in the CFT, the elementary fields are eight vectors and two matrices. And these matrices have certain symmetry properties. The gauge and variator operators are going to be interested in, the single trace ones are vector, bunch of matrices is vector. Because of the symmetry properties of x1 and x2, this expression is anti-symmetric in the ORS indices. So that it's in the adjoint of SO8. And under the conformal symmetry of the CFT, this is an expression of spin m plus n over 2 plus 1. Because the i's and the x's are all of spin half. Now, it also lives in some representation of the SO2, which rotates x1 and x2. And it's in the representation of spin m plus n over 2 of that SO2. So that is, we've placed the open string operator and which representations it lives in. To reproduce the quantum numbers we saw on the holographically dual side, we should recall that we're not interested in all operators or all modes of operators, rather we're interested in those modes that preserve the vacuum at zero and infinity. So this picks out a subalgebra of all of the modes. Some people call this the wedge algebra. So how is this defined? If I take the contour integral of some operator, I take some mode, you ask, OK, when does this preserve the vacuum at zero? For that, you need k to be positive, non-negative. Because if k was negative, then I hit it with the vacuum at zero, I'm going to get some derivative of my operator. And then you ask, does it preserve the vacuum at infinity? Will it be change importance? So that goes to 1 over z. And I pick up some powers of z from the way the operator transforms, just dictated by its spin. And you find it preserves the vacuum at infinity. If k is in this range, or j is the spin of my operator. So for example, if I have a spin 1 operator, then only that is in the algebra. Sorry, I think I need a page to explain this. So spin 1, like an ordinary current, j a, so the globe. If I have a spin 1 operator, maybe it satisfies a Casmudi OPE, then the global symmetries will be a finite dimensional, dimensionally algebra. If I have a spin 2, like the stress energy tensor, then this constraint tells me that we have not less than or equal to k, less than or equal to 2, j minus 1. And if j is 2, we have is not 1 or 2. So for spin 2 operator, like the stress energy tensor, the global symmetries are the SL2, global conformal transformations. As you increase the spin of the operator, you're getting more and more global symmetries. So let's go back a little bit and see what are the global symmetries associated to our basic currents? Our basic currents are i x1 to the n, x2 to the n, i. Well, if you look at the spins, you see k is allowed to be in the range from 0 to m plus n. So these expressions, they're in the adjoint of SOA, of course, and they're in a representation of spin m plus n over 2 with respect to both the SL2 conformal symmetry and the SL2 symmetry which rotates x1 and x2. So this is the same quantum numbers as we saw before. It's like sum SOA, tensor EjL, tensor EjOr. If there are no more, are there any questions there? So what have we seen so far on the topological spring side? We looked at the analog of isometries. We found this big algebra, mass from SO2c to SO8. And on the CFT side, we looked at those modes which preserve the vacuum at 0 and infinity. And we found something which looks the same, at least something that has the same quantum numbers. I haven't explained how to calculate the quantum numbers of the closed string operators, but it's not hard to check that they all so much. So what we haven't done so far is match the structure constants easily outwards. So the way it's going to work is that here T is some element of SO8. I'm writing this for or an S or vector indices for SO8. And things are anti-symmetric and or an S. So homomorphic functions from SL2c to SO8, they look a little like this. At the lowest level, I just have those constant functions here for these. Then I have those functions which are linear in the coordinates, TOSUI and TOSWI. These are going to get sent to IXI. Or the distinction between the UI and the WI is whether or not there's a Z, DZ, or just a DZ. The same happens as you go down. So next, so we need to be able to match the non-trivial commutators. So let's examine one of these commutators. For example, if I take TRS times U1, Tbq times W2, and I can mute them, all I'm doing is I'm taking the SO8 commutator of these matrices. These are anti-symmetric matrices. I'm just commuting them. And then I'm multiplying the functions. In particular, if I specialize these indices to particular values, you find T12U1, T23W2 is U1W2, T13. Similarly, T12U2, T23W1 is U2W1, T13. But now we should recall that SO2C as a defining relation for U1W2 minus U2W1 is equal to n. So we get such a relation in the commutator like this. So this is the relation we're going to want to match. So let's translate this into the CFT language. U1 is going to be a mode of I1, X1, I2 like this. And W2 will be like this, an X2 and a Z. And the other commutator, there's going to be an X2 with no Z and an X1 with a Z. And we expect that the modes commute according to this relation. And we're going to do this computation in a minute. And we find it works. What I find kind of cool about it is that it's a way. Here, we're just doing some computations in the mode out of the CFT. And we're seeing that the holomorphic functions on the dual geometry appear like the defining relations. OK, so let's go under this computation. So before we do so, I think I should remind people of something pretty elementary, that how one does, how do you commute elements of the mode algebra of some chiral algebra. So in general, if I have two operators in a 2D CFT, O1 and O2, I can take their modes with some powers, Z to the k1, Z to the k2. And I can try to commute them. So the commutator is just the difference between two contour integrals for here on this contour. This contour here, O1 is on the inside. O2 is on the outside. And this counter here, O2 is on the inside. And O1 is on the outside. So when I take the difference between those two contour integrals, you can write it as a contour integral where here O1 moves around in a circle and O2 orbits around O1. So to write that in coordinates, we're going to do a contour integral of Z. Just go around in a circle. And a contour integral of W also go around in a circle. But O2 is placed at Z plus W. And then I also have this power Z plus W to the k2. When we do the W contour integral, what kind of non-zero terms can we find? Well, of course, we're only going to find terms which have a power of 1 over W because you want to get a residue. Expanding this expression will give us positive powers of W. And the OPEs here will give us negative powers of W. So we see that the terms that are relevant are the terms in the OPE, which have a first order pole, second order pole, to a pole of order k2 plus 1. In our case, we're doing a commutator of something here where there's no powers of Z and something here where there's one power of Z. So by this argument, the commutator will have terms from where there's a first order pole in the OPE between the two operators or a second order pole. In our CFD, the OPEs involving a pair of elementary fields always has a first order pole. So if a WIC contraction contributes a first order pole, since we want a first or second, we find that the relevant terms involve two WIC contractions. So let's write out diagrammatically everything that involves two WIC contractions here. I can contract X1 with X2. But this is bad because this is a non-planar diagram. So since we're looking to CFT in the planar limit, this will not contribute. I can contract I1 and I2. That's good, and that contributes this expression to the commutator of the modes. Or I can contract X1 and X2 and I1 and I2. And this will contribute something with an N. And I1, this will contribute N, I1, I3. And the reason for the N is the usual large N combinatorics, when I have a loop like this, when I have a loop like this here, it contributes a factor of N from the trace in that loop. So now we've pretty much gone through the proof. If I commute these guys, I get I1, X1, X2, I3, ZDZ, and a factor of N. For the other guy, for the other one, you find the same thing in minus N, I1, I3. You find pretty much the same thing except for an extra sign. So we see in the algebra the analog of the commutation relation of the defining relation of SL2C by commuting the modes of each other. OK, so this is really easy, but fine. So any questions? So how does it work more generally? So the theorem is that the global symmetry algebras for the planar CFT at the top logical spring are isomorphically algebras, even when you include the closed spring sector. So let me say a little bit about this. So the general case. Maybe I can ask you a question, actually. Sure, yeah. So on the caral algebra side, if I understand correctly, we don't have to take large N, right? We can understand this caral algebra at finite N in a small N. Where do we see the need for a large N to match with the topological string? I mean, here we had a match, but we've dropped the 1 over N corrections. They don't apply the diagrams. So where would we see those corrections? Right, so this can seem to things which are a little tricky but possible to do involving, you can try to compute by various methods. So the diagram we dropped looks like this. These are contracted here. Let's go off like that. So it's like a 2 to 2 scattering process. So I haven't drawn it very well. One can compute. So there are one loop corrections to the OPE in the topological string. And these capture it. It is possible to see it, but it's a little tricky. I think in general, I like to be optimistic and think that loop corrections on the topological string side should be much more accessible than they are in the physical string. And I think they are, but they're still not easy. And probably the best way to approach them is to note that they're very tightly constrained by crossings in the tree. Of course, the other way, and the lower end appears, is that there are trace relations. And I don't know how to see those in the topological string at all. So in general, statement days that if I take some kind of open string thing like this, the planar ones, I have to contract a bunch of adjacent stuff. Some other ones go off. This is a 2 to 1 kind of open string process in CFT. That if we're looking at the global symmetry algebra, that this matches the algebra, polymorphic maps SL2Z to SO8. Similarly, on the closed string side, we take modes of this guy, the global symmetry algebra match vector fields. The polymorphic conversion is free vector fields on SL2C. So how do we prove this? So for the open string side, we're done because the algebra is generated by TA, TAUI, TAWI. So really, once we've checked this commutation relation, algebra, basically the Jacobian entity, tells us the two algebras must be the same. Well, what about the closed string side? That's going to be much more challenging. Well, on the CFT, it's more challenging to do the CFT computations. We're going to use the following trick. Vector fields on SL2C are derivations of functions on SL2C. And so we're going to use this symbol for functions. And so there's symmetries of this real algebra. And on the CFT side, you can kind of see explicitly why the closed string fields have to be symmetries. But if I take a closed string field like this, the commutator of this with an open string field looks like into these co-past. And you can check this kind of, well, this is compatible with the Jacobian entity. So I've got a very briefly sketched why the argument we just gave, well, very simple. It is enough to allow us to constrain and just using algebra, all of the symmetries. And to see that for the closed string side too, we also get vector fields on SL2C. So the topic I haven't talked about so far, let's see, which I'm going to have to go spend more time on tomorrow, is states from the point of view of holography of the top logical string. So in general, ADS CFT or fields have an asymptotic band recognition on ADS. So suppose we write ADS and Utidian signature as like the upper half plane. Then roughly, the fields someone wants to consider have some prescribed pole in the radio parameter core. So a local operator is modification of this asymptotic band recognition at a point on the band. So which means away from this point, we're going to have usual asymptotic behavior. And at this point, we're going to have modified asymptotic behavior. So I don't know if this is a familiar picture to people, but it relates to the more standard one by saying that so equivalently, if we modify the band recognitions, we can find a solution to the equations of motion to modify it. So this solution to the equations of motion is the state on ADS corresponding to CFT operator. So I want to explain how to implement this in the case of considering of SL2C. So to understand this, we need to think about what does SL2C look like near its boundary? SL2C is like u1, w2, minus w2, u1 equals n. That's up. So to understand the boundary, let's make these into homogeneous coordinates. And then the equation will be u1, w2, minus w2, u1 equals n e squared. So now I'm lifting everything into homogeneous coordinates. And then writing SL2C as an open subset of a quadrant in CP4 as I'm imposing this quadratic equation. So just like it started life as a solution of a quadratic equation in C4, and I'm adding on some points at infinity to C4 to get CP4, then I want to see what the boundary of SL2C is in this combination. The boundary divisor is a locus where v is 0, in which case the equation looks like u1, w2, minus w2, u1 is 0. This is a quadric in CP3. As you might have seen in Anna's lectures, this is kind of standard in the study of scattering. Quadrics in CP3 are parameterized by CP1 times CP1. So the boundary looks like S2 times S2. Now should we think of this? AdS3 times S3 is S2C. Boundary. So 1S2 is the boundary for AdS3. The other is, well, almost the S3. It's S3 modulo u1. So they're related by a half-fibration. I didn't get very far with the discussion of states. So I told you what it means to get a state where we're going to modify the boundary conditions. Now we've studied the boundary of SL2C and we've found it's a product of two CP1s. The next thing to do is to look at the boundary conditions and to see that modifications of the boundary conditions do indeed give you the states of the duality at equal. And then after that, we can try to study scattering opportunities and match them with correlation functions. Okay, so I'll stop there. Thank you very much.