 I put some formulas on the board from the previous lecture. So just to review, so the classical action for the string has px, and it also has two sets of pure spinners and the conjugate momentum, which I call lambda alpha and lambda hat alpha hat. And alpha and alpha hat are the same chirality for type 2b opposite chirality for type 2a. And then you have these constraints, which instead of just having p slash lambda, for lambda, you have the constraint p slash plus d sigma x slash. And for lambda hat, you have p slash minus d sigma x slash. So these are supposed to replace the p minus dx and p plus dx squared, the left and right moving. And you also need to introduce these additional constraints for the sigma derivative of lambda and lambda hat. So that's just to get the algebra of these first class constraints to close. So k and k hat are the Lagrange multipliers for this d sigma of lambda constraint, and l and l hat are the Lagrange multipliers for the Twister constraints. Furthermore, it's convenient to do a shift to get to the first order form where you don't have p's and you just have x's. You do a shift of l and k so that you generate the usual coupling of the Worldsheet via brine, which you can encode in just two Worldsheet fields, e and e bar. So these are the Lagrange multipliers that multiply the left and right moving via a Zor constraint. So this is just like in Bosonic strings here. So t is just as the contribution from x, and it also, of course, has a contribution from the lambdas. And there's a similar expression for t bar. OK, so this shift will be j is w. This shift of l and k is just what generates these terms in the stress tensor where e, this e here is the same e as here. And there should be an e serian. OK, there are any questions on this? So this is the classical action. Then we gauge fix it. I'm not going to go through the gauge fixing. Again, we land on this pure spinner action. It's just a quadratic action. So these are the left movers, and these are the right movers. The thetas come, of course, from the Lagrange multipliers for l and l hat. And the resulting BRST operator has this form. So it's manifestly space time superponkering invariant, where d alpha is the same constraint as you had in the green Schwartz formalism. But now it's not imposed strongly. So p is now an independent Worldsheet variable. It's not defined in terms of the d alpha equals 0 constraint. And the d alpha and d beta satisfy this OPE, where pi m is now the supersymmetric version of x. So you can show the q squared equals 0 using, of course, that lambda gamma lambda is equals 0. And you can construct vertex operators in the cohomology. So this is the unintegrated vertex operator of ghost number one for the open string, where this super field when qv equals 0 describes on-shell d equals 10 super Yang-Mills. And the integrated vertex operator of dimension 1, so this is, of course, conformal weight 0, because all of these fields have conformal weight 0. This has conformal weight 1, because d theta, pi, d, and n have conformal weight 1. And it obeys this relation. So the BRST variation of the dimension 1 field is the world-sheet derivative of the dimension 0 vertex operator. OK, and these here are the same as d alpha and pi m that appear here. And n is, of course, the Lorentz current for the pure spin. OK, so any questions on that? So that's the review of what's called the minimal pure spin of formalism. And there are two things we learned when we were trying to do this gauge fixed description of the pure spin of formalism. One thing we learned is that the tree-level measure factor needed regularization. So that's just because of the non-compact lambdas. So they're no longer projective pure spinners. They're now ordinary pure spinners, because we've fixed the scale symmetry. So they have non-compact degrees of freedom. And when you integrate over them, you get divergences which need to be regularized. The other thing we see when we did this shift is that we have to introduce this field lambda bar. Now lambda bar can be anything, because you're going to divide by lambda lambda bar. It's easy to see that this shift reproduces this for any object lambda bar. But of course, lambda bar breaks Lorentz invariance. It's a spinner. So if you fix it, of course, any Lorentz transformation that changes the value of lambda bar will not be a symmetry of the act. Not the symmetry of this shift, at least. So it will be convenient to introduce a worldsheet variable which is going to play the role of this lambda bar. So fixed lambda bar breaks manifest Lorentz invariance. OK, so we're going to solve these two problems simultaneously by what's called the non-minimal. So the idea is we're going to add to the action some new fields, which I'll denote by w bar alpha, which will be a set of bosons. These will be bosons. And these will be fermions. And I'm going to introduce an equal number of bosons and fermions so that I'm not going to affect, for example, the central charge of the conformal anomaly. And of course, you also should introduce the right moving analog of that. So these are left moving, and these are right moving on the worldsheet. And in order that they don't contribute to the chromology, we're going to define the new BRST operator, which I'll call q tilde, so let me just call this S tilde, which is equal to q plus the new term. It's going to be defined to be w bar alpha r alpha. So using what's called the quartet mechanism, one can argue that if you add, of course, also to the right moving BRST operator, you do something similar. One can argue that adding this term implies that the physical states will, any dependence on these fields that respects the fact that it's annihilated by this operator, can be written as something BRST trivial. So you can remove from any physical state dependence on these worldsheet variables. So they're not going to spoil the chromology. You're still going to get the same super string spectrum as before. But the advantage is now we're going to be able to covariantize what we're calling this shift. And we're also going to be able to regularize this tree level measure factor and also measure factors associated with the hyogenic surface. OK, any question? OK, now one can ask a question, can we get this from gauge fixing, this action here? And you can. What you have to do is you have to introduce the non-minimal variables already at this stage. So you introduce just the bosonic ones. And then you introduce a constraint, which is essentially a trivial constraint. It just tells you that, sorry, I'm running out of space, which is going to be just a Lagrange multiplier, let's call it h, times w bar alpha. And the hat inversion, h hat, alpha hat. Now when you gauge fix this Lagrange multiplier to zero, you'll generate ghosts. The ghosts will be precisely these r and s variables, and r hat and s hat variable. So one could have done this from the beginning, starting with this classical action, introducing the non-minimal variables, but in order not to introduce lots of notation, I saved it for this gauge fixed action. But if you want, you could have introduced them from the beginning. And of course, introducing this gauge fixing, gauge fixing h and h hat equals zero, or r are the ghosts who are going to generate these terms in the BRST operator, because the constraints are, of course, w bar and w bar hat equals zero, and these are the ghosts. So this is the usual ghost time constraint. So it looks very trivial, because you've introduced something, and then you've constrained it to zero. But it's going to be useful, as we'll see. OK, is there any question? OK, as you can guess from the name of it lambda bar, they're also going to be pure spinors. So I'm going to require that lambda bar gamma m lambda bar equals zero, which means that this also has 11 independent complex components. Notice that the alpha index means that it's lowered, so it's antiviral as opposed to lambda, which was vial. And if I want this to be the superpartner of this lambda bar, not to introduce new degrees of freedom, you also need a constraint on r. So the constraint on r is going to be that this combination with lambda bar is going to be equals to zero. OK, so this implies that both lambda bar and r alpha have 11 independent complex components. If I want, I can solve the constraint by choosing a u5 subgroup of s o 10, but I'm not going to go through that unless it's a question. Similarly, of course, you do the same for lambda bar hat and r hat. OK, any questions? OK, so now let's see how this helps. So let's go back to computing the tree amplitude that we computed before. So remember, the tree amplitude was computed by having three unintegrated vertex operators and n minus three integrated vertex operators. And this involved an integration over 10 x's, 16 theta's, and 11 lambdas. And this meant that you got an infinity to the 11th. Now, of course, we're going to integrate over the lambda bars and the r's. There's no integration over the conjugates because the conjugates have confirmed weight 1. These are all confirmed weight 0. So the conjugates don't have any zero modes on a genus zero surface. OK, so it looks like you've just made the problem worse because now you have more non-compact integrations. But now one has the freedom to modifying the gauge fixing condition. So the gauge fixing condition to start with, remember, was of the type p alpha l alpha plus, I didn't go into the k, but there's also a term kl. Now one is going to add, OK, there's also a gauge fixing associated with gauge fixing h to zero. So that will be r alpha h alpha. So that's the usual terms you would expect from the gauge fixing we just did. But now I'm going to add a new term to the gauge fixing condition, which will be minus lambda bar alpha theta alpha. And I can put an arbitrary coefficient in front, epsilon. Now this, of course, may look a little bit unusual, but I'm allowed to choose chi however I like. And if this helps me to regularize, it shouldn't change the answer. So what does q chi? So of course q chi from this stuff is what we had already. You get, well, this term here, I'm not going to go through this term here, it's a little bit messy, but it's in the paper that I mentioned. Here you'll, of course, get the term, this should have been an s, sorry about that, s d bar r. Now from here you're going to get a term minus epsilon times lambda bar alpha, lambda alpha. That comes from the BRST variation of theta. And you'll get another term from the BRST variation of lambda bar. So if you look at this BRST operator, you can see that q lambda bar is going to be equal to r alpha. That's one way to see why you need this constraint, because if you take q on this, it had better be 0. So this obviously follows from this BRST operator because lambda bar is the conjugate to w. So you get an extra term here, which is plus r alpha theta l. Now if we use this gauge fixing condition here, then of course when we exponentiate it, the action, so we have a term of course the exponent of the action, which is going to now have this q of chi here. Now this is going to be equal to e to the minus s c. But now you have the term minus epsilon lambda alpha lambda bar alpha plus r alpha theta l. Now it's easy to see, this gives a Gaussian cutoff for the lambdas. So you no longer get the divergence. So we have these objects we have to compute. But now we have a regularization. So we no longer get infinity. And in fact, this also is going to kill some of the fermionic zero modes. Now if we use the v's and u's that are described here, there's no dependence on the non-minimal fields. So in order for this to be non-vanishing, you obviously need to get 11 hours. The 11 hours can come from this exponential. So this is going to give me, so you get these integrals as before. Now pulling down this factor here, you're going to get a factor of epsilon r alpha theta alpha to the 11. Multiply it, of course, by exponential minus epsilon lambda alpha lambda bar alpha. And you integrate that over d 11 lambda d 11 lambda bar. It's easy to see that this Gaussian integration is just going to kill, you're going to get a factor of 1 over epsilon to the 11 coming from this Gaussian integration, which is going to kill this epsilon 11. So for any value of epsilon, as long as, of course, it's not zero, you get the answer is going to be independent of epsilon. So this is going to give you a regularization. OK, so now we can, of course, do the integral. We have v1, v2, v3 times these u's. Now we still have to do the integral over 16 theta's and the integral over 10 x's. But now you see 11 of the theta's have been absorbed from here. So this is effectively going to give me just an integral over 5 theta's. And one can work out the Lorentz indices. And you find that the answer is precisely this expression we worked out yesterday. So if this can be written as lambda alpha, lambda beta, lambda gamma, f alpha, beta gamma, after doing the integral over all the non-zero modes, you find that the zero mode integration just reproduces precisely what we had yesterday. OK, so this is a functional integral method for obtaining the answer we had yesterday. OK, any questions? OK, so this method is straightforward. And now the next step is OK. We think we understand tree amplitudes. As I already mentioned, this was already shown to agree for all the massless endpoints scattering tree amplitudes. You can, of course, also do massive. There's no restriction of doing massive. The only difficulty is these vertex operators get more complicated, because instead of just being dimension one at zero momentum, now there'll be dimension two or dimension three at zero momentum. So you'll get more and more of these worldsheet fields involved. But in principle, you can work out these vertex operators and compute any tree amplitude in this way. OK, the next step, of course, is do loop amplitudes. So the first thing one has to figure out is what's the correct prescription? So on higher genus surfaces, let's say genus G, it's no longer possible to gauge E and E bar equals zero. So the gauge fixing condition came from gauge fixing. So we have to gauge fix E and E bar to zero to go to conformal gauge. That's essentially gauge fixing the worldsheet metric, H ij, to be proportional to delta ij. On higher genus surfaces, there are global obstructions to this. So their global moduli are left over, which are just the Teichmühler parameters. 3g minus 3, complex moduli, which I'll denote by tau p. So p is 1 to 3g minus 3. And what this means is that the gauge fixing condition chi equals b. So if you're doing bosonic string theory, what you would do is just gauge fix E and E bar to zero, which would be done by introducing the gauge fixing fermion, which is just b times E plus b bar times E bar. So this is just the usual bc ghost and b bar c bar ghost. And these are the anti-ghosts which multiply the constraints to set E and E bar equal to zero. So this is bosonic string theory. Now, if you cannot gauge E and E bar to zero globally, that means you're actually doing too much. So some of these b and b bar anti-ghosts are not doing what they should be doing. Because in some sense, you're introducing too many b's and b bars. So what you have to do is you have to set some components of b and b bar to be zero. So the way to do this is by introducing a Beltrami differential, which I'll call mu. So these are Beltrami differential, which are going to be dual to the choice of these complex teichmühler parameters. So for each global modulus, you have a Beltrami differential, mu p, which you can use when you contract it with the b ghost. These are essentially the modes of the b ghost which you don't want. Because some of these modes of the b ghost are gauge fixing things which can't be gauged away. So you have to get rid of those modes of the b ghost, which is done by inserting these delta functions into the scattering amplitude prescription. Of course, similarly for the b bar ghost. So this is one way of understanding this genus G prescription where these b ghosts in the prescription enter. So the prescription for Bosonic string theory for genus G, you have an integration over these 3G minus 3 teichmühler parameters. And then you have a genus G correlation function. And in this case, you don't have any conformal killing vectors. So all the vertex operators should be integrated. Let's assume genus G bigger than 1. But then you have to explicitly insert these delta functions. So you have 3G minus 3 of these insertions. Of course, the delta function of something fermionic is just the object itself. So this is just equal to mu p. So this would be for the open string. For the closed string, of course, you take the left right product. So this would be an endpoint amplitude prescription for Bosonic string theory. Are there any questions? So now we want to do the same thing for this, for the pure spinner. Now in this case, we have a different chi. We have this more complicated chi. I guess I wrote it down here. And now we're going to do a shift of the L's to see exactly which terms in chi coupled to e and e bar. So it's easy to see what happens after you do this shift. You generate some new terms. So you have e times, so here I wrote down the shifts. So you can get p alpha times p plus d sigma x lambda bar over lambda lambda bar. So that comes from the shift of L. And you also get a term from the shift of k, which is of the form N mn gamma mn lambda bar. So of course, the coefficients here I'm not being careful with, but of course, this is all in the reference. OK, so this object here is going to play the role of this b ghost here. So if you do things more carefully, you find that so here you're going to get something I'll call capital B because it's not the fundamental field. And it has the following form. I guess I'm going to use this later. So capital B turns out to be equal to I m. We're essentially what I've done is just read off the terms from here. There's some extra theta dependence which makes this spacetime super symmetric. And it's easy to check that qB is equal to t, which is equal to, you can write it in super symmetric language as minus 1 half pi m pi m plus, I guess, minus dL. So it has the same structure as you had in Bosonic string theory that just like q with small b is equal to the Verizor constraint in Bosonic string theory, q with b is equal to the left-moving stress tensor in pure spinners. But of course, this b was defined using the minimal formulas. So it explicitly involves the lambda bar and it's not Lorentz covariant because you fix lambda bar. Fortunately, one can define now a b which is fully Lorentz covariant by using instead of q using q tilde. So I shouldn't have erased it. So q tilde was this thing plus w bar r. So you can define something called b tilde, which is equal to b, plus some other terms which will depend on the non-minimal variables. So these will be crucial when you compute the loop amplitudes. And these are the extra terms you need to add. So it will look like a mess, but the structure is required in order to write something which has the right BRST property. So this should have been. And one can show that this b tilde satisfies the property that q of b tilde is equal to t tilde, which is the same as t, but of course, plus the contribution from the non-minimal field. So these terms are required because of course, b does not commute with w bar r because it has lambda bar dependence. So you need to add other terms in order to cancel that dependent. Now there's a good feature and a bad feature of b tilde. The other good feature is, of course, it's manifestly 10 dimensional super Poincare invariant. But it still has the unpleasant feature that has poles when lambda lambda bar equals 0. That will cause problems if you go to high enough loop order because what will happen is that when you do the functional integration, this term here gives you a factor, of course, which goes like lambda lambda bar to the 11th. So if you have enough poles in order to cancel that 0, so lambda lambda bar to the 11th comes from the measure factor. And then from each b ghost, you can get a factor of 1 over lambda lambda bar. The maximum it can give is 1 over lambda lambda bar to the cubed. You might think it's 1 over lambda lambda bar to the 4th, but you see there's a lambda bar in the numerator. So each b ghost can give you this. There's 3g minus 3b ghost on a genus g surface. So you see this can diverge. g is greater than or equal to 3. Now it doesn't have to diverge because it's possible that these terms won't contribute. But if you go to a genus 3 loop amplitude and you ask for the amplitude with arbitrary factors of alpha prime, so you try to do the full genus 3 loop amplitude, you will run into problems. Now I told you in the first lecture that Gomes and Marfa computed the 4.3 loop amplitude, but they computed just the lowest order in alpha prime. So they're actually able to extract the coefficient and prove that it was consistent with estuality. And their computation only used factors in b that had fewer than this number of poles. OK, so this is an unsolved problem in the formalism that at the moment, if you want to compute for arbitrary g loop, one has to regularize the divergence when lambda bar goes to 0. There's a proposal for how to regularize it by myself and Necker-Soph, but the proposal is too complicated to actually use it in explicit computations. OK, but in any case, what I'll now show is how to do the computations, at least for genus 2 and genus 3. And we'll see, I can do it in 15 minutes. And I would challenge somebody using RNS formalism to try to do this in 15 minutes. OK, so any questions? Good question. OK, so the way I'm doing it here, all the vertex operators are integrated. One could ask even for tree amplitudes. OK, good. So the question was, there are different ways of parameterizing a sphere with n punctures. So let's just do a tree amplitude, which are essentially the choices these mus. So if you want to, you can think of, instead of having 3g minus 3 complex moduli, you can think of having 3g minus 3 plus n, where n is the punctures. So what I've done here is I've associated the n of the complex moduli with the locations of the external states, but that's not necessary. And what he's asking is, if you choose a more general parameterization of the punctures, will you be able to compute? Because in that case, you will get, even for a tree amplitude, you will have to insert, well, it will be n minus 3 b ghosts. So that has not been worked out. So you would run into this problem already, a tree amplitude, if you tried to choose a different parameterization. So that's an excellent question. And in fact, one can check this regularization that I proposed with Necrossov, even for tree amplitudes. And that has not yet been checked just for the reason I told you that it's too complicated to do explicit computation. Any other question? OK, so let's do the loop amplitudes. So the first thing we need to do is to generalize these integrals now when we do loops. So of course, one now has to do integrations not over the zero modes of the conform weight zero objects, but over the zero modes of the conform weight one objects. So with genus G, now we have w alpha, w bar alpha. These both have 11 g zero modes, because there are 11 components of each. And conform weight one object have g zero modes on a genus G surface. These, of course, are bosonic. Of course, you also have the s alpha. So those are the conjugates to r. And you have the p alphas. So these also have 11 g, but these, of course, are fermionic. And p alpha, of course, you have 16 g, because there's no constraint on the p's. So you have all 16 components. Now these, of course, are bosonic. And these are fermionic. So the first thing you have to do is to regularize the integration of the bosonic. Because now you'll still get, just like you've got infinities from the non-compact lambda zero modes, now you'll get infinities from the non-compact w zero. Unfortunately, there's a term you can add to chi, which also fixes that. So the term you'll add, of course, if you have g zero modes, let's use the letter r to label the different zero modes. So in other words, I'm using the notation where on a genus g surface, for example, p alpha, you can write it as some times the holomorphic, I don't know what letter to use, but they're g holomorphic one forms, which I'll call omega. And, of course, I can decompose p alpha in terms of them. So these are the g zero modes that I will have to integrate over. OK, so I'm going to add a term to chi, which is going to regularize the integral over the w's and w bars. And the term will be given by s alpha w alpha. So that means that q chi is now going to pick up, of course, it still has this term, which was, I guess it's going to be minus here, minus epsilon. That's going to be as before. But now you'll also get the term sum over r, epsilon r, times s alpha, let's do the Bosonic part first, w bar alpha w alpha plus s alpha d alpha. So what we've used is that the BRST operator, of course, it takes s alpha into w bar alpha, because r is a conjugate momentum to s. And furthermore, it takes w alpha into d alpha, because of this term here. OK. So this term will take w alpha into d alpha. So we get these two terms as before. We get this term which regularizes the Bosonic divergence, and this which gives extra fermionic zero modes. OK, so now we have to do the integral over those also. So we have d 10 x. Now we have d 11 g w, d 11 g w bar. d 11 lambda, d 11 lambda bar. Now we have the 16 g p's, 11 g s's, and the original zero modes. So the original zero modes of the conformity zero field, we have 16 theta's and 11 r's. So this is the integration we need to do. Now when we compute q chi, we have this term here, but we also have this term here, epsilon r w w bar plus s d. OK, so let's see what we can get. Now, of course, if we don't have enough fermionic zero modes, the amplitude vanishes. So we just have to count the zero modes. OK, so the computation we're going to do is the four point genus two and then the four point genus three. OK, so we have this u 1, t u n. The fermionic zero modes here can come from various places. They can come from the exponential here, or they can come from the vertex operators. So the vertex operators are defined here. So there are also some fermionic zero modes here. Or they can come from the b ghost. So this b ghost here, you have 3g minus 3 of them. So they can also contribute fermionic zero modes. So now it just becomes a counting problem, at least to see if the amplitude vanishes or not. It's trivial to show, for example, that if you have less than four external massless states, the amplitude vanishes. So in order to absorb enough fermionic zero modes, you need to have four external massless states. And that, of course, is crucial to prove these finiteness theorems. OK, so let's do the two loop first. So for g equals 2, we need to get 32 p's. So we need p to the 32. We need to get s to the 22. And then, of course, we need theta to the 16 and r to the 11. So the question to all genus is the question you have to ask about these 1 over lambda lambda bars. So what you have to be careful about is that when you compute these, let's say, three point amplitude, that you never get these divergent contributions coming from the lambda lambda bars in the b-ghost. So there are arguments. I think it depends on your rigor if you call it a proof or not. But certainly, if you ignore the question about the 1 over lambda lambda bars, then it's a proof. So the 1 over lambda lambda bars, you have to either use this regularization, or you have to argue that they don't contribute. So there's an argument, but I'm not saying it's a proof. So let's look at the genus 2. So the genus 2, we have to absorb these zero modes. So let's first look what can come from the exponentials. So from the exponentials here, you can get an s. So d alpha essentially gives the zero modes for p, because d is essentially p alpha plus something else. So you can get for free sp to the 22 coming from this exponential here. Because this is a sum from r equals 1 to, well, there are two g of them. And there are 11 s's and 11 p's. So for each, sorry, 1 to g. So when g is 2, you can get 22 factors from here. And of course, again, you'll get from the Gaussian integration over w, w bar, the epsilon r will drop out, the same as it did for the epsilon. So that killed the s zero modes. One still has the r and theta zero modes. And one still has 10 of these p zero modes. So in this case, one has 3g minus 3b ghosts. So one has 3b ghosts, so b tilde cubed. And the term which is most useful in b tilde for absorbing zero modes is this term here. So this term here has two p's and an r. You can get fewer p's from the other terms, but this is the unique term with two p's. And it will turn out that for four point amplitudes, the only term that contributes from the b tilde is that term, just to absorb the p's. So this gives you a p, p, r cubed. So now you see you get six p's from here, 22 from here. You're still missing four. Unfortunately, you still have the vertex operators these use. And each u can give you one p from this term here. So that's the simple argument that the amplitude vanishes for less than four external states. You just don't have enough p zero modes. So from the vertex operators, u cubed or u to the fourth, you pick up a pw to the fourth. And now you can see you're still missing the r's. So you're missing eight r's, which are going to come now from this exponential here. So in principle, this could give you up to 11 r's. We saw for the tree amplitude, this gave 11 r's. But in this case, we already have three of the r's. So we only need to get eight more. So that pulls down eight theta's. We need to have 16 theta's. So that immediately says that the amplitude is going to be proportional to d to the theta to the eighth of the w vertex operator. So we have w to the fourth. These are super fields here. And of course, these super fields depend on theta's. So these are the only way to get theta's zero modes is from the vertex operators or from this exponential, because everything else is superpunk ray invariant. They can't contribute theta's zero mode. OK, so this is the answer you get. This is the term of lowest order in alpha prime. And if you expand it in components, each w has to give you two theta's. So that turns out to go like d squared after the fourth. So just by dimensional analysis, it's easy to see that. Right. Now, this is for the open string. Of course, if you do the closed string, you just get the square of this. So it goes like d squared after the fourth squared, which in closed string notation is d fourth out of the fourth. Now, of course, you can work out the index contractions. It's a bit of a mess. It was mostly worked out by Carlos Mafra. So the superspace expression is trivial to write down. So you're just writing down the w's, but doing the integration of the theta's is a bit of a mess. You need some mathematics or something that can handle lots of theta's. He's worked out all the programs, and he showed that this index contraction is, of course, precisely what you want. And furthermore, together with Mafra, together with Gomez. So this is in 2010, they computed the coefficient here. Now, the coefficient was computed using factorization in the RNS computation. But in some sense, it's not a computation. It's a one-loop computation, which essentially you take the square of. That gives you, by factorization, the two-loop coefficient. But it wasn't the direct computation of the coefficient. In fact, it's not known how to compute this directly. In RNS, just because, as I mentioned in the first lecture, you have to compute determinants of fields of different conformal weight. Whereas in this theory, all the fields here are either conformal weight 1 and conformal weight 0. And just because the central charge cancels, the contributions from the determinants also cancel. So there's no determinant computation that you need to do. So that's what makes this computation. It just comes down to a computation of zero modes. So you just have to know how to do the zero mode. So D11 lambda, I wrote down the expression. But of course, lambda is a pure spinner. So you just have to parametrize SO10 over U5. Just have to. This is essentially what Umberto Gomes worked out. How to do this integration over SO10 over U5 variables. So it's a computation that was done. And they found this coefficient and showed that it, of course, had agreed with the dokerfong, which agrees with the s duality conjecture. OK, so three loops. Again, it's just algebra. Now you're going to get in 32. You're going to get 48, s33, theta 16, r11. Now you're going to get sp to the 33, coming from the exponential. So now you need to have 15 more p's. You have b tilde to the sixth, which now gives you, of course, ppr to the sixth. Now it turns out the u's don't have just to give you p's. So you can get pw cubed. That's enough p's. And the last one can be an ordinary gauge field. If you look what's left over, you need r theta to the fifth now, just to absorb the 11 r's. So the expression you get now is d to the theta to the fifth, w cubed a. And that goes like d cubed, f to the fourth, which for the closed string translates into d to the sixth out of the fourth. OK, and again, the same group in 2013 computed this coefficient and, of course, also computed the indices. In this case, there is no RNS computation. RNS computation was only done for 4.2 loop and just would never short external states. This, of course, was done first in superspace. So you get not just the external graviton, but you get the whole supergravity multiplied, and the same in the three loop case. The coefficient was actually computed up to a factor of 3 in 2013. They didn't get the right answer according to the estuality conjecture, but they recently found a very simple error in the computation, and now they get the answer which agrees with the estuality conjecture. OK, so that's essentially a summary of what you can do with multi-loop amplitudes. So maybe in the last minute, let me just present some open questions. OK, so just to summarize, what we've done is shown how to obtain both the pure spinor and green Schwarz formalism from gauge fixing, a twisted like action. So this action here, surprisingly, the theta variables of green Schwarz come as for their pop-up ghosts. Now, there are many things one can try to do. So of course, one thing you can do is use the gauge-fixed action to compute multi-loop amplitudes. I think that's been done in such a convincing way that I think, of course, it's interesting to do it using other formalisms, but I think at least it seems to be clear that this is the simplest way to compute the multi-loop amplitudes. Now, there are other things one can study, for example, other gauge fixings. So it was shown how to get to the green Schwarz of pure spinor. But for example, can you get to RNS? Another question which was from the audience is there's some unitary gauge in which you don't even have fermions. Another possible use of this alternative gauge fixing is to maybe resolve this problem of the 1 over lambda lambda poles. So maybe in RNS, it's well known that you get these singularities, essentially, because your gauge fixing becomes singular. And it might be that there are other gauge fixings which resolve these 1 over lambda lambda bar poles. So this is, of course, a hope, but it doesn't sound so crazy. OK, another thing you can discuss is other backgrounds. So this, of course, was all done in a flat background. Of course, one thing you can do it you can't do in RNS is Ramon-Ramon backgrounds. In this case, it's a little bit strange, because it will turn out that the Ramon-Ramon field strings will only couple without derivatives, but they'll only couple directly after gauge fixing, because there's no Thetas, at least in the classical action. There's no way to couple directly in the classical action. They probably will couple through the BRST operator, but that needs to be worked out. Another thing that you can discuss is possibly non-critical strings. So using green shorts or pure spinner form is up to now. People have only discussed 10-dimensional supergravity backgrounds, just because we don't know how to describe any other backgrounds using green shorts, because you have Xs and Thetas. What else can you do? Of course, in RNS, there are many backgrounds you can describe which have nothing to do with 10-dimensional supergravity. You just have an n equals 1 will shoot super conformal field theory. So maybe this new kind of classical action will allow you to consider other kinds of backgrounds. So of course, that needs to be worked out. Of course, you can discuss other brains. So you don't have to discuss strings. So in fact, membrane, the d equals 11, super membrane of Berkshoff, Sezgen, and Townsend, can also be discussed in pure spinner language. Of course, they did in green shorts language. I don't know how to do it in RNS language, but it's also been done in this language. So that was in the reference that I mentioned earlier. And of course, there are other brains you could try to discuss using this new kind of classical action. OK, finally, there's the relation to twister strings. So this is a mystery, essentially, why these twister strings work. Up to now, I think the only explanation of why they work is because of, well, results that are being discussed by Freddie, these scattering equations. Somehow, scattering equations seem to fit in with. So twister string, it used to be the only string we had of this type was this twister string of Whitton and myself, in which you describe four-dimensional super Yang-Mills. Now, there are many other kinds of so-called twister strings. So one even has twister strings. I don't call them twister strings. They call them twister strings. They call them chiral strings, or infinite tension, which looks something like this action, in the sense that you just have p's and x's. You don't have a p squared term. And surprisingly, they describe the theory at alpha prime equals 0. So I don't know if anybody's going to talk about this here, but there are these nice results by Mason and Skinner, in which they essentially generalize these twister strings of super Yang-Mills to other kinds of string theories, essentially fitting in with the formulas that Freddie Koshas has discussed. OK, finally, unrelated to all of these things is ADS 5 times S5. So most of the computations that have been done up to now, using pure spinor formalism, are these multi-loop computations, essentially, because this is what Gomes and Mafra have been doing. But there's a whole other story, which is ADS 5 times S5. So of course, using RNS, we don't know how to describe this background. Using green shorts, you can describe it the classical level. And a lot has been done, for example, on Spectrum using classical green shorts and light cone gauge fixed green shorts. But to describe interactions, you really need something which is more conformally invariant. And people are starting to do this now with integrability. And I suspect that it will fit very nicely in with this formalism. So some initial work has been done together with two ex-students, so Thiago Flurry and Tarlis Azevedo. So they have worked out together with myself, vertex operators, and started to compute scattering amplitudes. And this is, of course, another story, which is, of course, it's a different type of complication. But of course, it's very interesting, based on what other people are doing now. OK, so I'll stop there.