 So there are notes, at least for the first two lectures, they're putting it on the web page. So hopefully by tomorrow or Tuesday, you'll have the second two lectures. OK, so there's a new ingredient that I'll describe today, so I've given already, I guess, three sets of lectures at ICTP on pure spinor formalism. But there's a new ingredient now that I think is better understood, which is where the formalism comes from. So I'll spend today, today I have two lectures. So I'll spend most of today's lectures discussing these new ingredients. So let me start with motivation. And then to simplify things, I will discuss first the super particle. So the super particle is, of course, the massless degree of freedom in the string. And most of the issues that come up in the string already come up in the particle or in the super particle. So I'll discuss this in three different languages. So first with green Schwartz, then with pure spinner. And then finally, I'll give this twister-like description, which will generalize to the string. And in some sense, unlike these two formalism, this will be completely gauge invariant. These two descriptions can be understood as gauge-fixed versions of this. OK, then we'll go to the string. So these things I will try to cover today. Then we'll discuss, of course, the super string, which is the main focus. And first, we'll discuss vertex operators and tree amplitudes. So this will probably be done tomorrow. And then finally, we'll discuss multi-loop amplitudes. To do this, we'll have to introduce some new degrees of freedom that I'll describe, non-minimal. And this will require a construction of something called a b-ghost. And with this, we'll be able to compute multi-loop amplitudes. OK, so what I won't have time to discuss in these lectures are curved backgrounds. But if you're interested, you can look at previous lectures on that. So let me give you some references. OK, so as I already mentioned, there'll be notes on the web page. But you can look at earlier lectures I gave, for example. So these are some lectures I gave here. And there's some lectures at GGI. And there's a new paper that's discussing these twister results. So there's no review of that, but hopefully this paper is readable. And the multi-loop amplitudes, the most recent results, are by Humberto Gomez and Carlos Mafa. So this is their three-loop paper, which is OK. So if you can't see something I wrote from writing too small, just please stop me and tell me to write bigger. So is everything OK up to now? Any questions? OK, so motivation. So as you will hear, certainly from Ashok, the usual description of the super string uses the RNS formulas, Ramon Nevers-Schwarz. So Ramon Nevers-Schwarz's formalism has the advantage that it's a very simple idea. You extend the bosonic string variables to spinning string variables by extending the conformal invariance, or the reparameterization invariance, to worldsheet super conformal invariance, or worldsheet super reparameterization invariance by introducing these grassman coordinates, which I'll call cap and capobar. So this is, of course, a super field. And if you expand it out, you get, of course, bosonic and fermionic degrees of freedom. So these are commuting, and these are anti-commuting. So I'll always use fermionic to mean anti-commuting. Some people use fermionic to mean spinorial. I don't use that language. So this is, of course, a spacetime vector. So although it's anti-commuting, it's a fermion to spacetime vector. This is an auxiliary field, which the equation of motion is that it's 0. So maybe you're not used to seeing that. So psi m is a worldsheet spinner. So this has spin plus a half. This has spin minus a half. So it has half-integer spin as a worldsheet variable. So psi m psi bar. But they're, of course, spacetime vectors. Now, if you think about spin statistics, it's natural to have spinorial. So that's fine. It's a worldsheet spinner. It has the normal statistics as a worldsheet variable. But it has the opposite spin statistics for a spacetime variable. So what this means is the worldsheet properties act very simply. They just act by worldsheet supersymmetry. They mix x and psi. But under spacetime supersymmetry, it transforms in a complicated way. So the spacetime, I'm not going to go into detail, although I should probably will go into some detail about the spacetime supersymmetry generator. If he doesn't, I'll do it later. You have to bosonize these fields. So it acts in a nonlinear way, just sketching how it looks. You have to bosonize the beta-gamma ghosts, which are the worldsheet ghosts for this worldsheet supersymmetry. And you also have to bosonize the psi. So this is some kind of spin field. It transforms as a spacetime spinner. That's why I've given this alpha index. So m equals 0 to 9 is a vector index. And alpha equals 1 to 16 is a spinner index. So this would be a Majorana vile spinner, depending on the choices of plus and minuses here. Now the algebra here is not quite the supersymmetry algebra. This turns out to be related to the translation generator by picture changing. So it's times some other operator. Let's call it y. Picture-lowering operator. Now I'm not going to go into detail on this, but you can see that the spacetime supersymmetry transformations are complicated. What this means is that it's complicated, for example, to compute scattering amplitudes involving external fermions or external states which carry spacetime spinner indices. So anything that carries spacetime vector indices is easy to describe. But anything which carries spacetime spinner indices is going to be complicated to describe. So for example, using this formalism up to now, what has been computed is 4.2-loop amplitudes. So this is Do-Kun-Fong. But only with external, never short, or only with external bosons. The technology for computing with external fermions, well, I'm not sure if it's a technical problem or a conceptual problem. It hasn't been worked out yet. Furthermore, it's unknown how to describe Ramon-Ramon backgrounds. Now, it might be possible tomorrow that somebody will figure out how to do this, but up to now it's unknown. And these are, of course, crucial for ADS-CFT. And the problem is essentially that the Ramon states are complicated in the same way as the spacetime supersymmetry generator is complicated. Furthermore, it's difficult to verify finite-ness properties or non-renormalization theorems. And the reason is simple. It's because one needs to sum over spin structures. So before summing over spin structures, the amplitudes are not spacetime supersymmetric. And, of course, these finite-ness properties depend crucially on spacetime supersymmetry. Finally, there's a question which is, if you want to compute the amplitude, not just the momentum dependence of the amplitude, but the overall factor multiplying the amplitude, it's unknown in the RNS how to compute the overall factor. So for a genus greater than one. Now this coefficient is important if you want to test certain duality conjectures. The reason in RNS it's unknown how to compute this is because when you work in the RNS formalism, you have BC, you have beta-gamma goats, you have these size, they all have different conformal weights. And what happens is the determinants that you get from doing the partition function over these fields of different conformal weights, they, of course, enter into the partition function. And it's not clear how to compare them. Essentially, you need different regularizations for each determinant and it's not known how to compute that overall coefficient in the RNS form. Okay, so these are, of course, the main advantage of RNS is this world-sheet supersymmetry. We can describe many different kinds of backgrounds. But the difficult features, as I said, because as soon as you describe things with spacetime spinner indices, the amplitude become, the formalism becomes complicated to deal with. Can people see down here if I write here? Yes? After I did it. Yes? So far not. So yes, but if you, so if you can use uniterity, of course, to try to fix this. So Doquin-Fong showed how to do this for two loop. It's not obvious how to do that for the three loop, for example. At least nobody has shown how to get that coefficient from uniterity. But in any case, in using pure spinner form, of course, uniterity is not manifest in any of these formalisms. Uniterity is only manifest in light cone gate. Now in the pure spinner form, you can directly compute it. So you can just check the two loop. Of course, if it, a two loop, if it gave the wrong answer, it means that uniterity is somehow violating. The formalism is wrong. But actually Marfa and Gomez computed the three loop and showed that it's consistent with duality. So, sorry, you can't see down there, is that right? Okay, so let me, okay. So the summary for our NS is that if we're looking at, for example, the open string, there are four different sectors of the super string. We have the sector called GSO plus, which contains, for example, the vector. Let's do the open string. So we have open string, we have the tachyon, we have the masses vector. And then we also have the Ramon states of different chiralities and also all the masses states. So the GSO plus sector is the one containing, for example, the masses vector. The GSO minus sector is the one containing the tachyon. Then we also have the Niver-Schwarz and the Ramon sector. So we have four different sectors for the, for the super string in the open string case. If we do a closed string, we get four times four, 16. Okay, so here's the vector, here's the tachyon. Of course, what one can also have, so this is the vector, this is the tachyon here. They're in the Niver-Schwarz sector. In the Ramon sector, we can have GSO plus, which would be like Majorana vial, and here would be Majorana antiviral. So just these are some examples of states in these different sectors. The Ramon formula, the RNS formula is good for describing things in this Niver-Schwarz sector, either GSO plus or GSO minus. It's good for describing tachons, it's good for describing vectors. Good means it's simple. I don't mean, of course, it's good in the sense that tachyons are objects you want to get rid of. I just mean that in the RNS formula, it's easy to handle them. Whereas in the Ramon sector, either GSO plus or GSO minus, RNS is not the way to go, okay? It's messy. Now what we'll see is that there's another description of the string using either Green-Schwarz or pure spinner formalism, in which, so if RNS is blue and Green-Schwarz is green, RNS is good in the Niver-Schwarz GSO plus and GSO minus. Green-Schwarz is good in the GSO plus Niver-Schwarz and also GSO plus Ramon, okay? Whereas it's not good in this GSO minus either Niver-Schwarz or Ramon sector. Okay, so this is the difference between the two formalisms. Essentially, the formalism is supposed to be equivalent, of course. The amplitude is supposed to agree for anything you compute. But the computations are much simpler using Green-Schwarz or pure spinners. In these two sectors, whereas RNS is better for these. So for example, we can describe curved backgrounds easily in RNS with vectors or tachyons. In Green-Schwarz or pure spinner, we can describe curved backgrounds easily in GSO plus Niver-Schwarz or Ramon backgrounds. Okay, so this is an important difference to remember. And of course, if we want to describe space-time supersymmetric theories, Green-Schwarz, pure spinner is much better because we want to describe these sectors. In general, we want to avoid the GSO minus sector. So, okay, so that's the main difference between the RNS and the space-time supersymmetric. Yes, but it's just as messy as describing Ramon sector in, so you have anti-periodic theaters, then you have to have spin fields in order to construct the GSO minus sector. Okay, are there any other questions? Okay, so the next thing I'm going to discuss is the motivation for doing this pure spinner or Green-Schwarz. So everything I write down here should be in the notes. So you don't have to copy, but at least try to ask questions. Okay, unlike the RNS formalism, Green-Schwarz or pure spinner formalism, so I'll use PS for pure spinner, space-time supersymmetry has now manifest. The way to do that is that instead of starting with a world-sheet superfield, now you introduce XM and fermionic space-time spinner variables. And these transform in the usual way under space-time supersymmetry, so. Okay, so now is a good time to introduce notation. So my gamma matrices, M is always zero to nine, alpha's one to 16. These matrices here will be symmetric in alpha and beta. And they're related to the 32 by 32 matrices in this way. So alpha and beta can either be down or up. So if something has an index up like theta alpha, it's called Myron vial, or more precisely vial. Whereas if it has a down index, so let's, this would be called the antivial. So you can see gamma matrix with two up indices, it takes something antivial to vial, and something with two down indices takes vial to antivial. So of course the gamma matrices change the chirality, and this would be a 32 by 32 gamma matrix. So in four-dimensional notation, this would be what was usually called the gamma matrices, and these would be the Pauli matrices. So this is of course in a certain representation. I'm only going to use this representation. So of course the usual gamma matrix relation, gamma M, gamma N equals two A to MN. This translates into this language here, gamma M alpha beta, gamma N beta gamma. If you symmetrize in M and N, it gives you two delta alpha gamma, okay? So you always, you can, there's no metric which raises in lower indices because these are different representations of SO91, yes? Say it louder please. Delta of X is minus one half epsilon. Delta theta is epsilon? No. So supersymmetry, so in four dimensions, if you remember Wesson-Bagger, so the supersymmetry transformation is of this type. D d theta alpha plus, I guess minus one half gamma M theta alpha d dx M. So this is just the action of epsilon alpha q alpha. So epsilon is the Grassmann parameter associated with the spacetime supersymmetry in this direction. Okay, so just to say, what you're probably thinking of is the RNS, World Sheet Supersymmetry Transformation. So in RNS, the supersymmetry transformation of World Sheet is delta X M. In this case, epsilon is a spacetime scalar. So it's just epsilon psi M plus epsilon psi bar M. And delta psi is equal epsilon d dz of XM. So here, yes, psi transforms into X. But this is the World Sheet Supersymmetry. This is not spacetime supersymmetry. Okay, it's a good question because these are the best kinds of questions because this is a question that probably half of you have but somebody has to ask it, okay? So are there any other questions? Okay, so this is the spacetime supersymmetry generator and as you can see q alpha with q beta using the anti-commutations of theta and d d theta. It's easy to see this is equal to minus 1 half d dx of gamma M. Sorry, minus, yeah, I guess that's right. So this is the, no, there's no one. Because you get it twice. Okay, so this is the usual anti-commutator two spacetime supersymmetries gives you a translation. Okay, so this is the gamma matrix relation. There's another identity which is useful to know. These are the things you know just by reading about it. It's not easy to prove, but okay. Of course, by writing explicit representations of these gamma matrices, you can prove this. If you symmetrize in any three indices here in 10 dimensions, this turns out to be zero. So this is just for d equals 10. For those that are interested, this is related to division algebras. So this is true in three, four, six, in 10 dimensions which is related to the real complex quaternionic octonionic division algebras. Okay, so these are identities which will be useful to know. Yes, thank you. Okay, so there any questions about this? For those that don't know, of course you can do fields. You can decompose any form constructed from vector indices in terms of these gamma matrices. So gamma MNP turns out to be anti-symmetric and alpha beta. So any object which is anti-symmetric in the spinner indices can be decomposed in terms of a three form. And any object which is symmetric in its spinner indices or by spinner can be decomposed in terms of a one form. And a five form. This five form is self-dual. Okay, so this is a notation which we'll use throughout the lectures. Any questions on the notation? Okay, so this is the spacetime supersymmetry. Of course, the action will be manifesting variant under this. It turns out that the Green-Schwarz formalism, although it's much older than the pure spinner formalism, it has only been quantized in Lycone gauge which makes life complicated if you want to compute scattering amplitude. So Lorentz covariance has not manifest. And of course, it also makes it difficult to describe curved backgrounds. So up to now, this has only been used to compute tree level and one loop five point ampoule. So Green and Schwarz computed the tree level four point and one loop four point and I don't remember but it's been extended for the tree level five point and one loop five point. Now of course, you can do these five point amplitudes for any external states. You're not restricted to external bosons. The amplitudes treat the bosons in fermions the same way. In pure spinner formulas, which we'll see later, you can quantize covariantly. So all the symmetries are manifest which has made it useful for computing multi-loop amplitude. Of course, both of these formulas can be used to describe Ramon backgrounds. In pure spinner, you can covariantly quantize these backgrounds, so that's an advantage. Finite properties are easy to verify essentially because you don't have to do the sum of spin structures, the space-time supersymmetry is manifest. And furthermore, one can explicitly compute for example, this overall coefficient essentially because all the determinants cancel. It's something like a topological string in that sense. So as I already mentioned, Mafer and Gomez, Gomez and Mafer, computed the three-loop four point of course for any external states, any external massless states and checked the estuality conjecture. So there's an estuality conjecture for this amplitude. It's a D to the sixth star to the fourth term and they computed this coefficient. That coefficient is related to the tree amplitude coefficient of that term by estuality. Okay, so that's an impressive computation. Now up to now, where these formalisms come from has been a bit of a mystery. Of course, Green-Schwarz just presented this, first they presented the formalism in light-cone gauge. Light-cone gauge is a very simple formalism, we'll see that later. Coverently, it's a bit more complicated. It's non-linear even in a flat background, but they showed that you can gauge fixed to light-cone gauge so they convinced everybody the formalism was correct. But as I already mentioned, people haven't been able to covertly quantize it and we'll see a bit later why. The pure spin of formalism can be quantized covertly but it's only written down in conformal gauge, at least up to now. So it involves a BRST operator in conformal gauge which is good enough to compute the amplitudes. But the question is where does this BRST operator come from? So what is the symmetry that this BRST operator is gauge fixing? So that's what we'll see in the following. It will turn out that it's gauge fixing a twister-like symmetry. So the generator of this symmetry is going to be something like P slash lambda equals zero. This is a twister constraint. I'll explain later why. And it will turn out that the fermionic variables here which are world sheet, they're space-time spinners. So this is of course, as you want, because it's fermionic, but it's a world sheet scalar which is surprising. Just like in the INS, it was surprising it was a space-time vector. Now we'll turn out that when you start with this approach here, this twister constraint, it will turn out that the thetas is the Fadea Popov ghosts for these constraints. So the fact that it's a world sheet scalar is now explained from the fact that it's from the world sheet point of view, it's a ghost. It's not a matter variable. So that's a new interpretation of this Green-Schwarz formula. And what I'll show next, well, after I describe these super particles, I'll show how to get these formalisms as two different gauge-fixed versions of this formalism with the twister-like constraint. Okay, so that's the motivation. So any questions before I start with the formalism? So as I said, the whole point of this is to interact. So please ask questions, otherwise, yes. Good, okay, good question. Thank you. Yeah, so the question is why we only have forms here of odd dimension. We don't have any forms of even dimension. So as I mentioned here, a gamma matrix, which is a one form, it either raises indices or lower indices. So indices of, so objects which have indices of the same chirality, they will only involve, obviously, because gamma matrices have either up or down an odd number of these. So it can be either be gamma m, gamma mnp, gamma mnpqr, or you could go on to seven forms, but the seven form is dual to this, et cetera. Now if we have something with an up index and a down index, then it can be expressed in terms of things with even indices. So for example, this can be expressed in terms of delta alpha beta, something with two indices, which would have one up and one down. Or something with four indices. So if you want to write a form with an even number of indices, or a by spinner, which would be a by spinner with one up and one down. So this is in 10 dimensions, of course, in different dimensions, things are different, but in 10 dimensions, this is how it works. Okay, these are excellent questions. Any other questions? Okay, so the next thing I'm going to do is give you an introduction to first, the green shorts, super particle, and the pure spinner super particle. So obviously there's no point in doing the string if you don't understand the particles. So let me start with the green shorts. So this is going to describe the mass of states of the open string, which is, of course, 10 dimensional super Yang-Mills. One can ask what happens if you work in lower dimensions, and I don't want to go into details, but the answer is not so clear. If there are questions, I can, of course, go into detail. So the action is, of course, going to be invariant under the spacetime supersymmetry, the one that I just erased. And it's, because it's a super particle, it's just an integral over tau. So just to remind you, if I'm doing a massless particle, the simplest way to describe it is with a constraint of this type. So E would impose, this is the, you could think of this as the one-dimensional virbine, or just the Lagrange multiplier, which imposes the constraint that it's a massless. Of course, if you want it too massive, you just put in a minus M squared, but we're going to do the massless case. Of course, just for people who have never seen it in this first-order form, if you're doing massive and you integrate out P, you just get the usual action of M squared. No, I think that's right. If mass is zero, of course, it doesn't make sense. So since we're only considering the massless case, we're going to use the first-order form. And now this is not spacetime supersymmetric, because if you remember, spacetime supersymmetric transformations are delta theta equals epsilon L. So epsilon is a global parameter, does not depend on tau. Now to make this spacetime supersymmetric, it's easy just to add this term. So of course, dot just means g d tau, okay? It's easy to show that this combination is invariant under this transformation if epsilon is constant. And as I showed to you earlier, this transformation, if you anticommuniculate itself, it generates a translation. So q alpha is d d theta alpha. Now there's an additional symmetry of this action which was called kappa symmetry. It was first found by Warren Siegel, which is a local symmetry. So it's a gauge symmetry, but it's fermionic and it has the following form. So you introduce a fermionic parameter kappa, which depends on tau. Sorry, this should be another thing. And you'd find delta xm. So delta theta is just this thing here. Now if you do this transformation, it's easy to see that this term here, let's call this term pi m. So this object is spacetime supersymmetric. It's easy to see that delta pi m is equal to, I have to do the computation, sorry. So it's going to be equal to kappa p slash, I did it correct, let's see. I will do it on the other side. So you might think there's a d d tau acting on kappa, but that cancels from the d d tau acting on x. What that means is that because p squared, p slash p slash is of course equal to p squared, that you have to shift delta e in order to cancel the contribution. So the shift to delta e is just equal to two. Okay, so this is a local symmetry which is called kappa symmetry, any question? So of course local symmetries are implied that you have some generator, first class constraints. And to find the first class constraint, let's look at the canonical momentum to theta. So this is defined in the usual way by taking the derivative of the Lagrangian with respect to theta dot. So let's call that small p. So capital P is of course the canonical momentum for x. So the canonical momentum for theta, if we do this computation is going to be equal to minus one half p m. So it's not independent of theta, which means there's a constraint. So if we define d alpha to be equal to p alpha plus, this is a constraint. So in Dirac's language, one needs to impose this constraint on the wave function. Now of course one has to figure out if this constraint is first class or second class. And if you use the canonical commutation relations that p alpha with theta beta is equal to delta alpha beta, then it's easy to compute that d alpha with d beta is equal to gamma m alpha beta p m just because of the anti commutator here. p m is not a constraint, which means that d alpha is part second class. However p squared equals zero is a constraint. So although p m is not a constraint, part of p m is. Now when you do the computation, you find that d alpha, it splits up into eight first class constraints and eight second class constraints. In order to covariantly quantize, one has to figure out what to do with these second class constraints. Second class constraints, usually you don't know how to quantize using BRST method. But what you can do is quantize in light cone gauge. So covariant quantization is a problem. Of course, if you don't know how to covariant quantize, you don't know what the goals are. But one can't quantize in light cone gauge. So that's what I'll describe. So in light cone gauge, which is actually more properly called semi light cone gauge, you use this Kappa symmetry. This is a local symmetry to gauge fix half of the theta is to zero. So you can do that because you have these eight first class constraints. So you have eight local symmetries. So you gauge gamma plus theta, which has eight independent components. Gamma plus is of course equal to, gamma minus is equal to gamma zero plus or minus gamma nine. Furthermore, it's convenient to break SO8 down to U4. You don't have to do this, but it makes it easier to analyze the structure. So what happens is theta alpha has 16 components. That of course breaks up into gamma plus theta and gamma minus theta, which transform like SO8 chiral and anti chiral spinners. So gamma, this one we're going to set to zero. This one we're going to split up further into U4. So under U4, this SO8 spinner splits up into a four and four bar. So I'll call it theta J and theta J bar equals one to four. So broken SO9 one all the way down to U4. And now we're going to rescale theta J bar. We're going to define this to be one over P plus eta. When you do that, this action, so of course we still have this term here, but this term is going to only contribute when this P is in the plus direction. So you're going to get a term plus P plus theta dot gamma minus theta. That's the only contribution just because you gauge these theta's away. This is equal to theta plus A dot theta A dot, which now I can write in terms of these J and J bar and absorbing the P plus into the eta. This is equal to theta J bar theta J. So this part is of course the same as before. Okay, so this part I'm going to replace by this. Okay, so now it's easy to quantize because we just have these three fields. So the wave function of course has to satisfy P squared equals zero from this constraint. So the wave function is going to depend on X and it will depend on, the theta's and eta's are of course conjugate. So it either depends on theta's or eta's, but not both. Just like here, it either depends on X's and P's, but not both. So if we're working in the theta space, it's a function of theta J. And now you can of course expand it just because these are fermionic variables. So it has a single component here, then it has a term proportional to one theta, then it has a term proportional to two theta's, has a term with three theta's, and a term with fourth theta. So it has 16 terms, eta bosonic and eta fermionic. And these A plus, A J K, A minus, are just the usual eight components of the on-shell components of the gluon in Lycone gauge. So this is a six because it's anti-symmetric in J and K. And of course the C J and C J bar are the two components of the gluino. Okay, so this is the quantization of the Green-Schwarz super particle in Lycone gauge. Of course it's not covariant. I've broken it down to U4. Okay, any questions? Yes, please. One second, please. Here, eight, eight. Okay, very good. Okay, so p squared equals zero implies that, let's choose a reference frame just to simplify my notation. So let's suppose the momentum is in the p plus direction. Okay, so the algebra here is d with d here is equal to p. So if p is only non-zero in the p plus direction, then you have d alpha with d beta is equal to gamma plus alpha beta p plus. So I have to tell you what this gamma matrix is. This gamma matrix, let me just finish this. The gamma matrix turns out to be, it has 16, it's a 16 by 16 matrix, but it turns out that half of these components, it only has a non-zero component in this component here. So it's gamma plus a dot b dot. So all of these other corners are zero. This is an eight by eight matrix. So that's why there are only eight second class constraints. Now, of course, this was assumed that p plus was the only non-zero component. Of course, if I do a Lorentz transformation, that doesn't change the rank of the matrix. Okay, there was another question? Same question. Okay, any other questions? Yes? So Kennaut is too strong. So this was, of course, developed in the 80s and the covariant green Schwartz action was written in the early 90s. So for about 10 years, people struggled with how to write this, how to covariantly quantize this. Okay, so the main exercise was to convert these second class constraints into first class constraints. So once you can convert these second class into first class, if you can write the first class theory in a covariant way, then you can use your usual BRST method. So there were many attempts for the super particle, a few that succeeded, but none of these methods for the super particle extended for the string. So of course, for the string, there's a similar situation. You have eight first and eight second class, but the variables here are more complicated. So none of the methods which were successfully used for the super particle involving a different set of ghosts for ghosts or various methods, I shouldn't say there was one method which was generalized to the string using a different set of ghosts for ghosts by Siegel and Lee, Lee and Siegel. But it was so complicated that they couldn't compute. They computed some amplitude, but not many. So I can't say there are no methods to covariantly quantize, but I think it's fair to say there are no good methods to covariantly quantize. Okay, so that's right. So we're doing the super particle. So the super particle, of course, if it's described just at the free level, so free just means that it's not interacting, then it just describes super Maxwell. So it's just U1, so these are just linearized fields. If you want, you can introduce champatin factors and have it start to interact. Okay, now for the string, it's completely natural to do that, and the champatin factors you put at the end of the string and open string, of course, we can describe any gauge. Up to anomalies, we can describe any gauge group. And there are anomalies, I shouldn't say any. There are conditions on the gauge group in string theory. For the super particle, you can put in champatin factors, but describing interacting super particle is another story that I don't want to get into. So at the moment, just consider it to be super Maxwell. But when we do the string, of course it will generalize to super angles. But there are no conditions for the super particle. In principle, if I'm looking at the linearized theory, of course super Maxwell and super Yang-Mills are completely the same. You just add multiple copies of super Maxwell and you get super Yang-Mills. Other questions? Okay, so the pure spinner form is, so let me just spend, I'll just write down Lagrangian and then I'll stop. So the Lagrangian for the pure spinner is going to be start out similar, P and X. But instead of writing down in action just in terms of P, X and theta, we're going to introduce a conjugate momentum for theta just like we have for X. So P alpha will be an independent degree of freedom. So you go first to order a form. And then we're going to introduce some ghosts. So we're not going to try to write down the world line reparameterization invariant action. We'll do that later today. We're just going to write down what we call the gauge fixed action. So you introduce an additional variable, lambda w. And the idea is that this action is supposed to describe super Yang-Mills in a covariant way. In order to do that, you need a BRST operator. So I'm just going to save in two minutes and then this afternoon I'll explain this in more detail. So this is of course equal to D alpha, the same D alpha that we had before. But now D alpha is not zero because P alpha is an independent variable. Doesn't satisfy direct constraint. So in some sense, this is what you might try to construct if somebody gave you green schwarz. It's constraint times ghost. That's the BRST operator. But of course the constraint is not first class. So what you find is a Q squared instead of being zero is equal to lambda gamma M lambda times Pm, right? Because I'm using this constraint, this relation here. In order to make Q squared zero, this variable here has to be constrained. So the constraint will be lambda gamma lambda equals zero. And this is what carton called a pure spinner in D. Okay, so this is the answer. And this afternoon I'll explain where this answer comes from. Okay, so I'll stop there for questions.