 Okay, so before starting with lecture two, I would like to add a few clarifications on yesterday lecture. So first of all, of course there are plenty of reviews on equivalent comology, equivalent localization and so on. Let me just point out one that I found, which is relatively short and concise. This is by Alek Ziv, and it's called Notes on Equivalent Localization from 2000, if I remember correctly. Again, this is just one. The other thing, so that created a bit of confusion was the story with the metric, whether we need the metric or not. And so in fact, I mean, this metric is from totally auxiliary element in the following sense. So suppose that you start with a manifold M, and in particular, as I was stressing a minute, this has to be orientable manifold. And let's say that we only start with a U1 action on it. So we don't specify the metric to begin with. Of course, from this U1 action, we get our vector field V. And then what about the metric? Well, in general, any smooth manifold, we can put a Riemannian metric. So let's choose a generic metric, some G tilde. Of course, this in general is not U1 invariant, but now it's very simple. We just have to average over this U1 action, and this in general gives us a U1 invariant metric. So in this way, we can get our G for which V is a killing vector. And then in fact, if you look in the final formula, of course there's no the metric. So we use the metric and by the steps to construct this dual form, and okay, going through this process, any metric that does the job is okay, but in fact, this always exists. So yeah, we don't need to specify the metric, but in fact, always there is at least one. In fact, there is many. However, let me stress the importance of orientability. So this M should be orientable. This also calls some, some of you had questions about this. Well, first of all, we need M orientable in order to define integration. If you remember when you define integrals of form, some manifolds and your patches, you need some globally defined volume form. So you need the orientability. So we do want this M to be orientable. And this also shows up in the final formula because in the final formula, what appears is the Fafian of this action that I call MLVFP. So I remind you what is the definition of this Fafian. So if you have anti-symmetric matrix, so this M is not this M, this is anti-symmetric matrix. Then the Fafian of M, if I have the coefficients correct, this should be one over two to the L, and let's say this is 12 by 12, L factorial, and then you take L copies of the matrix, one I2, I2L minus one I2L, and then you contract with one epsilon tensor. And so in particular, since there is one epsilon tensor, of course, if you do a change of coordinates or bases, which has negative determinant, of course, this changes sign. So this is not invariant. It's only invariant if you do, if you preserve the orientation. And so when we, this change of coordinates, where the U1 action was written as a rotation in the various planes, we have to do it in a way that preserves orientation. The other property of the Fafian, of course, is that the square of the Fafian is equal to the determinant, but then the determinant contains two copies of epsilon. So the determinant is invariant and the generic change of bases. Is this more clear? Okay. So yesterday we were discussing the objects that we would like to compute. And so these are party integrals of our theories. And although I will not discuss, so I will just discuss just party integrals, so integrals on all three configurations of weighted by the classical action. However, in general, we might be interested in insertion of operators. Of course, if we insert local operators, we could trade these by including sources, but if we insert non-local operators, this would be more complicated in terms of sources. So we might want to just insert operators in the party integral. And I will not discuss this in details, but I just wanted to give you some comments of this. And in particular, I would like to stress that very loosely speaking, we can distinguish somehow three types of operators that one can try to define. This is not a sharp classification. So first of all, the most basic type of operators are what we can call, so let me put in quotation mark, order operators. And so these are simply defined in the standard way some function, usually some polynomial function or something like that. Well, some function in the fundamental fields that we have in the Lagrangian. If we start, of course, we want to do party integral, so we need a Lagrangian formulation of our theory. And these are just some functions of the fundamental fields at our disposal. And this function can be the local or non-local. If they are local, we have local operators. More generally, we can have non-local operators. So these are functions of the fundamental fields. By fundamental fields, I mean the fields that you use in the Lagrangian. And as I said, so one example is some local operators. Of course, I don't know, you are in a gauge theory and I don't know, it's a U1 gauge theory, so you want to make insertion of the field strength, for instance. But of course, you can also consider interesting non-local operators. And again, in a gauge theory, interesting classes, given by Wilson-Lynes. So these Winston-Lynes operators, they are defined by some contour, gamma. So let's say we have some closed contour. This is some line, this is some one-dimensional line. And they are also parametrized by some representation of the gauge group. And they are defined as the trace in this representation of the path order, the exponential of the integral along this line of A, the gauge field. Okay, this is not just a polynomial in A, but still is some function. And there is another interesting class of operators that we can call disorder operators. And this one, instead of being defined as inserting some function of the fundamental fields in our path integrals, in particular, let me stress. So here, how do we compute correlation functions of this? Well, we have our path integral, integral of all field configurations. Then we just insert the operator in the path integral. So this is generally some function of the field, some function of the fields, the action as well. So this, loosely speaking, is computing. So this will be the unnormalized correlation function. So what about these other operators? So this operator, instead of being defined as some function of the fundamental fields, this is defined by removing some points from spacetime, either a point if you want to do some local operator or some submanifold if you want to do some non-local operator. And then specifying boundary conditions, often these are diverging boundary conditions and singular boundary conditions for the fields, either at the point or along this submanifold. So we have our spacetime manifold, M. And first we might want to remove a point. So somehow here we have a small boundary, which is some small sphere. And we specify some boundary condition for the fields on this sphere. And so these are around point or submanifold. And then, so we do this. So this gives us the definition of the operator. And then how do we use this? Because really, I mean, we have defined an operator if you can compute any correlation function for this operator. So how do you compute correlation function on this disorder? Well, simply we do the pat integral, sorry. We do the pat integral, we don't insert anything here. But we don't do this integral over a smooth configuration, rather we do this integral over a thin configuration which respect this boundary condition around the point. So if you want with boundary conditions, corresponding to this operator OD. So these are boundary conditions that we use in the pat integral. And just to give you some examples, but we're not going to detail this, I will just mention some examples. These are, for instance, tough line operators in four dimensions. Or something which is very similar, is monopole operators in three dimensions. And so for instance, if we want to construct these monopole operators in three dimensions, so these are local operators. While these, well, as the name says, these are line operators, these are non-local. So these monopole operators are local and they are defined by specifying that if you want to insert the operator at some point, then we remove the point, we have a small s2, and we impose that, well, they say that the integral on s2 of f should be in some conjugacy class. And then, well, we can, again, loosely speaking, introduce a third type of operators that we can call defect operators. And so this time, what we do again, this can be either local or non-local. And so what we do is that we have either a point of some sub-manifolds. This is particularly interesting if we have some sub-manifolds, dimension, say, bigger than zero. And then we introduce some extra fields that are not fields that live in the whole of space, and they only live on the operator. So we introduce some local degrees of freedom on the operator. And then in our path integral, we will have to integrate over the bulk, degrees of freedom that we have in our bulk theory, but also on these localized degrees of freedom. So this corresponds to some extra localized degrees of freedom, some sub-manifold. Do you do space by the boundary condition for the bulk field? The boundary condition for the bulk field near the defect? Yes, yes. So then you will have to, so these localized degrees of freedom will be coupled to the degrees of freedom in the bulk, and you might want to include some boundary conditions, some extra boundary conditions. So as I said, so these three classes are not separated classes. If you want three mechanisms or three ways that we can use to construct operators, but there are overlaps. So when you introduce this degree of freedom, we want to specify also some boundary conditions. You might want to insert also on top of that some fields that live in the bulk, so then it will be a mix of the three. So as I said, this is not a classification. Some ways that we have to construct operators. So in this case, so we'll have our manifold M. We have some sub-manifold, I don't know, N, of lower dimension. And then if you want to compute once again correlation function of these operators up to the fact that we might have boundary conditions and so on, essentially now we integrate over the fields that live in M, but also on the fields that live in N. And here, so there will be some piece, so we'll have our bulk action, but then there will be some other action that lives on the operator. There will be a function of N, and of course these fields will be, I mean we will be coupled in general to the one in the bulk, so this will also be some function of the fields in the bulk, but this action lives on the sub-manifold. So just to give you a simple example, so suppose that we have some line gamma, as we had here, there is some gamma, it's one dimensional. We might want to introduce for instance some fermions along this line and write an action, which is just the integral on this line. This is some parameter tau along the line of these fermions and then it's very natural to take this fermion to transform under the bulk gauge field if we had a gauge theory in the bulk and then to couple them to the bulk gauge field. So this tau, this A tau here is the bulk gauge field, but of course this is projected or more precisely this is pulled back to the sub-manifold to the line gamma. Okay, so as I said, these are not separate classes, but actually speaking we have these classes of operators and then we can use in general, but so what we'll concern us, although once again I will not discuss some I will just say something about the first class or the operators, but the point I want to make is that both classes of all classes of operators can be treated with localization. Okay, so localization, super-segmentalization can address the three types and in fact this has been discussed in the literature in various theories and in various types of operators. Do you have any questions? It says how many super-symmetry from this sub-manifold is probably zero. Yeah, so here I'm not even talking about super, so this has nothing to do with super-symmetry. This is something that you can do in general. Yeah, so if you want to use localization, so first of all you need a super-symmetric theory, then you want that the operators also preserve some super-symmetry, in general they will preserve some fraction of the super-symmetry, then if you want to use localization, of course you have to be able to use the supercharges or a subset of the supercharges that are preserved by these operators and of course so each case will have a separate discussion if you want to go into the details. But this is in principle nothing to with super-symmetry. Yeah. What's the difference between order operator and defect operator? As I say, there is no crucial difference in the sense that these are not separate types. In general an operator can have all types of features but I just want to make the, so if you want in order operators which are the most standard ones that is discussed usually in textbooks, you take, so you have some theory in the dimensions, you have some fields that used to write the Lagrangian and then you construct gauge invariant combinations and either local or not local, you insert them in the Lagrangian. The point is that you can construct more general operators in which you include some degrees of freedom on the defect. So in this sense, this is what I call the defect. So it's defect operator's homogeneous? Yes. Still, as I said, these are not separate in many sense. For instance, you can have some operators which can be represented in different ways and nevertheless they are the same operator. And so I want to leave you with an exercise. So suppose that you want to study one of these order operators which is the Wilson line for some gauge group G in representation R. Now it turns out that the very same operator you can represent it as a defect operator. So if this is the same as some defect operator in which you use the following, so it will be a one-dimensional defect operator because this is a line and the action that you have to use, in fact, it's very similar to the example here. So let me first write it and then comment. So this is the action that you have to write, well, that you can write. So what are the various objects here? So this A tilde is a one-dimensional U1 gauge field. So if you wish with respect to this example here, you also have a gauge field on the line. This A tau is the one in the bulk. So if you want this is the pullback of the bulk G gauge field. And then psi, these are fermions which are in representation R under A, so the one in the bulk, and they have charge one under A tilde. Okay, so this shows even more that these are not really separate classes, magic stick. So okay, so now, so we have motivated yesterday why we're interested in computing partition functions. I mean, yeah, if with the integrals of Euclidean theories on compact manifolds that I will call just partition functions on compact manifolds. So we motivated why we want to do that. And we will be able to compute them using a quantum theory version of localization that uses supersymmetry. So of course the first step is to construct a supersymmetric theory on the curve manifold. And this is precisely what Guido is teaching us how to do. However, let me just say a few things that overlap with what he said. And maybe a couple of facts that he has not yet explained I will just mention and then in his lecture you would see full glory. Okay, so we want to understand supersymmetry on curve manifolds. Once again, there is some repetition but I think in this school there will be various repetitions. But it's a good thing to hear an important concept various times. And of course Guido should correct me if I say something wrong, he's here. So yeah, I mean, of course trivially so one start in Lorentzian flat space with some supersymmetric theory. And as we know very well, a supersymmetry algebra is an algebra that enlarges the Poincaré algebra of symmetries by including some fermionic operators. And for instance, we do four dimensional minimal supersymmetry, of course, I'm writing obvious things. One has fermionic operators, q alpha and q bar alpha dot. The anti-commutator is given by momentum or translation, while the other ones q q and q bar q bar are zero. This is without brain charges that Guido described in details. Now if you are in a local quantum field theory there is a local stress energy tensor whose integral is the momentum and there are also currents for the supercharges. These are the supersymmetric currents that Guido described in great details. So this is the, yeah. And this is an object which has a spinor index and a vector index. Of course the details depend on well how many supercharges we have in the various dimensions. And of course this is the property that if we take a space like slice and we integrate it, we get the supercharges. And while a theory is supersymmetric if these currents are conserved. Now if we have a theory with Lagrangian description and we are strict with Lagrangian description because we want to compute pat integrals, then while a theory is supersymmetric if the Lagrangian is invariant under supersymmetric transformations up to total derivatives. So let me introduce an operator delta. For me this delta is the contraction of the supercharges with some parameter for supersymmetry. So this will be some scalar fermionic operator. And then the variation of the Lagrangian should be a total derivative for the theory to be supersymmetric. So okay, so this is pretty obvious. Now, so now what do we do? So first of all we do in Euclidean. Well we do in Euclidean one of the reasons that well if you wish pat integrals a little bit better defined. So we go to Euclidean. So we have our theory on Rd. And then we would like to put this theory on a curve to substitute this Rd with some curved manifold M. Now when we do that we have to modify the theory. So the theory will not be the same. And okay this will be described in detail by Guido but we have to modify the theory. In particular very practically we have to modify the Lagrangian. And so in what sense we have the same theory that we have on flat space we put it on a manifold. Since we change the Lagrangian we change the theory. So what do we mean by it is the same theory. And the criterion that we will use is that at short distance, we don't want to modify the theory. This because if you go at very short distances and we are on a smooth manifold the metric is essentially flat. And so we should recover the flat space theory. So our criterion will be that we only, well we don't modify the theory in the UV, we only modify it in the infrared. Using some scale which comes from the metric on M as our reference scale. And so in other words we will limit ourselves to a relevant, so a restrict to relevant deformations. And we also restrict to local Lagrangians. Now even with these descriptions this procedure is still ambiguous in the sense that in general we can add relevant deformations which are constructed after the curvature invariance on the manifold. And of course all this curvature invariance if we take the flat space limit it disappear and so we have total freedom in adding these terms. And so this procedure is not unique. We don't find a unique answer, there are ambiguities so there are parameters that we can play with. And I will call this ambiguity as a background. In the sense that these ambiguities can be understood as some background fields that one can turn on when putting the theory on a chord manifold. And so in particular the important point is that these will be extra parameters besides just the pure geometry on M. There are other parameters involved. Can you give an example? Yeah, so as I said you can, so you have some manifold M. I don't know, you have some scalars. You can write curvature couplings where the scalars is coupled to the curvature and you can put whatever function you want in front of it or if you want to, don't want to put a function you can put a constant but you can choose the constant. So this is an ambiguity which is not just fixed by the geometry on M. It's some extra data that you have to specify. So you want to introduce more terms than what's being superactivity? So here I'm not, so I will get there but in fact this, I mean we already saw Guido's lecture. In fact, those ambiguities I say they precisely corresponds to background fields. And the fact that in supergravity you have extra fields besides the metric precisely corresponds to the fact that you don't just say, okay, I put my theory on a curve manifold. So this is the action. No, you also have to tell me what you do with other fields. So but now they have to be in the supergravity in other places or they can be in some other multiple for example, couplings to flavor couplings. Well essentially you can do whatever you want. You have to, so at this level, I'm not even talking about supersymmetry. So this is true even without supersymmetry. Of course without supersymmetry you have more freedom. And the point that I'm gonna make is that even with supersymmetry you still have this freedom. And it's really up to you in the sense that for instance as Guido mentioned, if you have some global symmetry then this global symmetry can be coupled as a standard thing to a background gauge field but this background gauge field is in a background vector or multiple that contains other bosonic fields and you can turn them on. Those also, if you want those parameters are not in the gravity on multiple but still are something that I call the background. Which is some choices that you have and that are not just fixed by the geometry on M. Okay, so another question. Again, this is very general and it's more or less a repetition of what we just say. So on top of this, so this is true even without supersymmetry. On top of this we want to add supersymmetry and in particular, now we add supersymmetry. Now in the UV as we said, we require that the theory becomes the flat space theory. And so this in particular means that the supercharges that we will preserve on a core manifold will be a subset of the supercharges that we had in flat space. So let me call it curved Susie algebra as a sub-algebra of the flat space algebra by Su. Superalgebra, so the supercharges we preserve on the manifold will be a subset of the supercharges that we had in flat space. However, this supersymmetry algebra, so this is true at the level of counting the generators of this algebra but this algebra can be deformed. So if you want maybe it's not correct to say that this is a sub-algebra, so these are the generators of the algebra. But yeah, this one can be deformed but of course the deformation must be such that if we send a curvature of m to zero, this deformation must disappear. So it's a, if you wish, relevant deformation of this algebra and we will see examples of this. Now it turns out that given a certain theory, supersymmetric theory in some dimension and given a certain manifold is not always possible to preserve supersymmetry on a given manifold. So in particular this problem understanding on which, on a given m, on which manifolds we can preserve supersymmetry is an untrivial problem. So there will be some constraints on the geometry of m. On the other hand, when it is possible, in general we have ambiguities. So even adding supersymmetry as I said a few minutes ago to this story is not gonna remove these ambiguities. And so we will still have to talk about manifold with a certain background. But of course everything will be described in great details by Guido. So how do we proceed to construct this supersymmetry? So this is what Guido said at the very end of his lecture. So one systematic approach is to couple the quantum field theory to offshore supergravity and then solving for the supersymmetry equations from the condition that comes from the graviton multiplied in this offshore supergravity. So in particular we set to zero the fermions. This guarantees that the variations of the bosons is zero. And then we insist that also the variation of the fermions in particular the gravitino is zero. And so this leads to what is sometimes called in the literature a generalized Keeling-Spinoer equation. And this is the equation that Guido wrote. Someone takes the gravitino and in general this is given by the derivative, the covariant derivative of the supersymmetry parameter epsilon plus some contribution from the other fields. So this will be some matrix or some function of the other supergravity fields. And in particular this function as an index mu at the end of the day because it has to match with this, this axon epsilon. And this is the type of equation that one gets in supergravity. So in particular in these supergravity fields there are of course there is the metric but there are also the other fields auxiliary or maybe dynamical the other bosonic fields that one has in the graviton multiplet. And of course these other bosonic fields have spin zero or one in the graviton multiplet. So one has to solve this equation. Now as Guido stressed a very important feature of this equation and this feature is there if you are working with offshore supergravity is that in this equation only the fields of the graviton multiplet appear not the fields in say vector or carol multiplet or the other matter multiplet that we can use to construct our theory. And because of this this equation is essentially why it's almost independent of the particular theory that we want to put on a curve manifold. This almost depends only on the manifold that you want to study. So we can decouple the problem of studying supersymmetry on curve manifolds from the discussion of the very specific theory that we want to discuss. Now this is an almost because as Guido stressed still a given supergravity can have different offshore formulations which corresponds to the fact that one can construct different types of these supercurrent multiplets that Guido described. And of course this depends on the particular theory so different theories might admit different types of multiplets but once we know which multivariate you are going to use then everything else is independent of the theory. Okay, and so we have to solve this equation for g mu nu the other fields epsilon and if you find solutions this means that you can preserve supersymmetry on a curve manifold and the number of solutions that you find for epsilon for given metric and background is telling you how many supercharges you can preserve on the manifold. Again, I'm not going to give any type of example because this is what he's talking about but we need this in order to compute our perpendicular so then once we have this really our goal was to construct the supersymmetric theory on the manifold so how we do that is by plugging the solution in supergravity so substitute the solution into supergravity and this gives you two things it gives you the form supersymmetry algebra in the sense that in supergravity of course you have supersymmetry transformations so you just take the background fields and you take your epsilon you substitute and you read off what is the supersymmetry algebra when it acts on the other types of multiplets and then you substitute in the action on the particular meta-action and you read off what is the meta-action so if you want the formed so there is this very systematic way that we will explain whose result will be this and this is what we need to perform the localization so again I'm not going to give any type of example this is just a general philosophy but is there any question? Any other questions? Why do we need to to use localization why do we need curved manifolds is there some simplification of localization from what I can understand looking at some curved manifolds that reduces the supersymmetry? Yeah so I gave you yesterday motivations to do that and the reason was that as we take some theory and we study this theory on various types of manifolds and on various types of backgrounds essentially we get access to different classes of observables I mean you can think of it as the fact that okay you take a theory and if it's not integral this is, at all it is a well-defined number is a number if you want to extract correlation function you need sources so if you want to compute correlation function of local operators you need really to insert sources and compute the result for generic sources then you take variation and you compute correlation function So you could do that on flat space? Absolutely yes I'm saying that if you study the theory on curved manifolds with general backgrounds you can get access to various types of other observables like you can well okay some of them you can think of as computing different types of correlators I mentioned that you can have access to allomorphic and allomorphic or to conserve currents and so on but you might be interested too in non-local operators or in some counting problems you want to count some states or operators so it is a profitable exercise well better define a compact space time yes although we don't have to limit to compact manifolds so first of all we don't have to limit to close manifolds so we could consider of course manifold with boundaries okay this if you wish to still compact but we can introduce boundaries and we specify boundary conditions in fact we can also consider non-compact manifolds but for instance including some trapping potential and this is what the omega background does at least there's some way to think about it so you can imagine that you have some non-compact space let's say we are on R4 here we have problems at infinity but we can introduce some potential some trapping potential such that all the excitations are confined around the origin of the non-compact but effectively this compactifies the problem and gives you a finite answer and more or less this is what the omega background does so yeah I'm saying if you want I'm studying the simplest case of compact manifolds but once again one can generalize things more generally one could try to impose some boundary conditions such that you get a finite answer and maybe removing the divergence in terms so there are a lot of things one can do for sure and the people has done have done yeah but these ambiguity as I said are ambiguity in the sense that if I give you just the metric there is no unique way to preserve supersymmetry because you have all these background fields so by ambiguity I mean that you have other parameters in the game so you have all these other background fields some vector fields, some scalars you can specify them with some constraints but you have freedom and so more generally your partition function is not just a function of the metric and on the couplings that you have in flat space is also a function of some extra couplings that if you wish you can generalize when you put a tear on a curved manifold but your final result should be a physical one but the final result is a partition function of this theory and then one question is why should we care about studying the theory, the Euclidean theory on a compact core manifold with all these background fields turned on and the answer one answer is that these partition functions are related, they contain information about the physical theory on flat Lorentzian space so the information is translated and you need to understand how so if you want a particular partition function on a weird manifold what physical information contains about the physical theory Lorentzian in flat space and once you answer this question then the composition that you did is useful you can have different motivations of course and it is an untrivial thing to understand what information is contained and in fact in this business people start in looking at very simple examples and now people are moving to more complicated examples more complicated than if you want weird manifolds and this is for two reasons one is why we do it because we can it is interesting if we can do it these are non perturbative computations but the other motivation is that people understand that even these weird manifolds contain interesting physical information about the theory once again different people can have different motivations any other question okay so let me just make one remark which Guido also made but this is important so we studied this organization of signature and so when we go to Euclidean we have to complexify the fields if we want to maintain covariance for instance because Pino representations have different dimension, different properties when we change the signature and so what it might happen is that so you take some you take some you start with some Lorentzian theory with some background field and the background for these fields if you want background is not is not the analytic continuation of a real background in Lorentzian signature rotation and so in particular this means that so it means that okay we can preserve supersymmetry by the price of losing a reflection positivity and a reflection positivity is the Euclidean version of unitarity so it means that somehow we are studying a theory with some non-unitary deformation now as Nikita remarked this is still interesting and in fact we still do it but we should remember this fact in particular if we try to compare this with what we expect from the physical theory however what it might happen is that so we study some theory that in the infrared goes for instance to a CFT this is an example that appears in many cases so for instance we could start I don't know with super QCD in the conformal window the theory that we were at Alagrandzian this theory is not conformal but if you are in the conformal window there is an infrared fix point we might be in three dimensions and we are studying some Young Mills-Chen-Simons theory and the infrared it goes just to Chen-Simons so we have plenty of examples in localization that are used in localization and now as we explained once the theory flows to a super conformal field theory some of the operators become redundant so these are operators which are either zero or fixed to some C number that a couple from the theory and so it might happen that actually these background fields which were complex and were ruining reflection positivity couple to these redundant operators and so then with the reflection positive is not really broken because they couple to an operator which is essentially zero and so in this case if we are computing something that does not depend on the RG flow and so essentially we are doing the computation in infrared then the results that we obtain are compatible with the unitarity or reflection positivity but if this is not the case and the theory is massive we can have some differences with the physical theory okay when the SEP uncurved the background the background fields may introduce anomaly to to conserve currents well yeah you can have the conformal anomaly of course the trace of stress tensor shortening condition for the current multiple no you still have you still have so besides the super conformal multiple you have the multiple of the anomalies which contains all the anomalies it contains the trace of the stress tensor the gamma trace of the supersymmetric current if you have an r symmetry it is the divergence of the r current and so you still have this separate multiple so there is a multiple that contains all the anomalies okay okay so now that we quickly reviewed or previewed how to preserve supersymmetry on core manifolds now we want to go to the problem that we want to address we want to understand how to perform how to use localization to compute these pat integrals and so well this story is a sort of repetition of what we discussed in the first lecture but it is this tension so it is an infinite dimensional version of these localization theorems that we discussed in the final dimensional case okay so we can go a little bit quick because the main steps we already understood so essentially what we do so we start with some theory with some action this is a supersymmetric theory so we have constructed a supersymmetric theory on a core manifold so we have this action on the core manifold and we have some fermionic symmetry fermionic generator q such that the action is invariant so qs is equal to zero and sometimes I will use delta instead of q so now this is a fermionic generator so the square of this can be zero but more generally it can be some bosonic symmetry in the theory so in general we have the q squared and some delta b some bosonic symmetry in the theory so what is this delta b well this will be some either some translation or rotation so something that acts on spacetime or can be some Lorentz rotation or can be some rotation by the symmetries in the theory like some r symmetry rotation or some global symmetry rotation so as we discussed various times we are interested in computing these sort of pat integrals so Euclidean compact manifold and we set h bar to one but instead of studying this we study a deformation of this problem so with some real parameter t in which we deform the action by some term and we want some exact term in such a way that the result is not going to change, this is what we did in the final dimensional case so we deform it by so t is our is a number, is our parameter and v is here is some functional so maybe the notation is not the bad this is nothing to do with the vector field that we had before, now this is some functional of the fields that we use to deform the action so this is q exact, it's q of something but also we want to be q closed which in our language means that it is supersymmetric so in particular we require that delta b which is q squared of v is equal to zero and if you want this is the equivalent condition that we had before first of all we should impose that a form was equivariant this means that it was invariant under symmetry and then we can start discussing whether it is closed or exact the first thing it has to be equivariant and so this is the equivalent condition if you want it is invariant under q squared and so then this is q exact so now as we did before we can ask what is the dependence so we have to form the theory by something and we can ask what is the dependence on this parameter and so if we do that so here we bring down this qv however since this is closed really we can take a q to act on the whole thing and so this is if you want a total variation of the integrand with respect to supersymmetry so so from here you might say ok just by doing a fear definition this has to be zero essentially because well assuming so assuming that the measure is invariant under q which in particular implies that there are no anomalies and so in particular we don't want this delta b to be anomalous so this should be an anomalous theory in the theory so assuming that the measure is invariant then by the standard argument you say ok if I take some integral then I do a fear definition in which I perform the transformation by q which is more rotation since this was a fear definition the two things have to be equal the difference is precisely given by this transformation infinitesimal by the action of q and so this should be zero however one has to be a little bit more careful essentially because we are in an infinite dimension on space and so in fact we can so you know that this fermionic symmetry is also super manifold so we have our fields the supersymmetric theory so we have bosonic and phenomonic fields and you can think that they form just a boson super manifolds of the configurations so this q is a translation in this space of super fields a translation along the grassman coordinates and so in particular you also know that the action of q is essentially a derivative so this action is a derivative with respect to the grassman coordinates and so this term in fact is a total derivative on the super manifold of the configurations and I say ok so once again this is a total derivative and we know that the integral of a total derivative is zero but of course in general we have to be careful about boundary terms in particular at infinity field space and so these terms can invalidate this argument so one has to be careful that there are no boundary terms at infinity field space however you see that here we have some exponential suppression factor and so if this is essentially if the action or the deformation term are such that they kill, they give an exponential suppression field configuration at infinity then you can make sure that there are no boundary terms and then this argument goes through but this is something that one has to check and in fact there are examples in which the naive argument will tell you that there should not be dependence on t but in fact if you do the computation you do find dependence on t and the reason is this so is there a relationship that sounds very similar to atia patodi finger index theorem versus atia finger index theorem also you have to worry about boundary terms and contributions to the index from the boundary is there a connection well as you say this is the same as the same mechanism that well here you are understanding very very simply this is just a total derivative and you have to be careful about what you do at the boundary counting something topological I mean in this case I guess it's with an index well you know I mean then it depends what boundary conditions you would I mean in the case in which it is not zero these are quite non trivial cases that I mean I just know a couple of examples and then I don't know what could try to understand what type of boundary conditions you have but now this is infinite in field space so it's kind of non standard yeah I don't have a sharp answer but I just know a couple of examples so any other question okay so so now this argument which as I said is a copy of the final original argument up to the fact that one has to be a little bit more careful shows us three things roughly speaking so first of all so suppose that this term was not a deformation term that you add to the action but in fact you have your action and the action in general is a sum of various terms and one or some of these terms are q exact so they can be written as q acting on something else in this case T instead of being a deformation will be just one of the couplings that you have in your theory and the argument shows you that when you compute this particular pat integral there is no dependence of this partition function on that particular coupling so you don't have a dependence on the couplings that appear in front of q exact terms let's say couplings in front of q exact and so we learn that in general we compute this partition functions but they need not depend on all the couplings that you have in your Lagrangian in general they only depends on a subset so this simplifies somehow the type of answers that you can obtain of course here I'm only considering the pure pat integral with the action but we could insert operators in particular order operators in the in the integrand the argument will go through and so what we learn is that when we compute expectation values of operators they only depends on the q-comology class of the operator so if you change your operator by something which is once again q exact you're not going to change the expectation values in particular if an operator is q exact the expectation value is zero and then finally the observation that led to do this procedure the pat integral is not modified by this deformation okay and so as we now know we can use this fact to our advantage to simplify the problem let me make a comment here I made a comment about this going to a credent signature we have to complexify all the fields so in general if you have some real fields it becomes complex and if you have some complex fields the fields and what used to be the dagger becomes independent fields and in fact I will use a notation in which I put a tilde so a tilde on a field will mean that is an independent field which however should be the complex conjugating and now we have double number of fields however we don't want to do the pat integral over all these double number of fields first of all because this would not be so what we are really interested in is the analytic continuation of the Lorentzian theory to Euclidean and if we double the number of integrals this is not the analytic continuation it's just a different problem also is a problem that in general doesn't have convergence so it would also be a bad problem so we really want to still integrate over if you want half of the fields in Euclidean signature and so this means that we have to choose a contour in the space of fields and in general we have to choose I mean if we want to have a well defined problem we should choose a contour such that the pat integral is convergent and now that we do this deformation by qv we better choose this contour to give us convergence for all values of t there is also another thing that we might want to be careful about we have this q squared was equal to delta B so this is some I mean this act on the manifolds of fields configurations and so we probably also want to choose a contour such that which is if you want closed under the action of this delta B we don't want delta B to take the contour and move us outside the contour because otherwise the complexified theory is invariant but it's less killer what happens to the theory where we integrate on the contour one question you said the vacuum expectation was kind of the chronology classes of operators how can we make sense of chronology classes if q doesn't square to zero yeah so essentially so like in the equivariant comology we need to first of all to restrict to equivariant forms and here too we need to restrict to operators which are not just supersymmetric but also invariant under delta B sorry I mean if the operator is invariant under q of course it's also invariant under delta B but then when we so this is the equivariant comology of q the form by things which are q exact but also invariant under delta B is the choice of contour you need no so you might have different choices and so in case in which these different choices are disconnected in the sense that they are not continuously deformable one of the other the interpretation should be that you have different quantizations of the same classical diagrams that lead to different quantum theories because different contours can lead you to different answers but they are from the same theories well the same Lagrangian because when you quantize you don't need to get the same quantum theory so we say the best only dependent in the q-comology classes of the operators does this contain more information than if you topologically twisted the theory it does because operators that don't satisfy delta B or does it contain more information about those operators well it contains more information because this supersymmetry on core manifold is not just the twist so this is something that Guido will probably stress but so topological twist definitely goes into this framework is one way to preserve supersymmetry on a core manifold and this is the original setup right of the 80s but one more modern point of view is that this is just one example but there are many more ways of preserving supersymmetry but all of them it only depends on the q-comology class? Yes in a generalized sense you might say it is but it's not topological twist because in general the theory that you obtain is not topological and it's not just twisted but if you remove these words more or less the ideas are the same it's sort of a homological theory what it is but it's just it's not the homology that you get from the topological twist also in this statement about the vacuum expectation about this we have restricted implicitly on the q-close operators yes yes because otherwise I mean if you have supersymmetric theory but you ask about operators that do not preserve supersymmetry you are not going to get anything from localization essentially because the argument breaks down right if you insert an operator which is not q-closed it's not true that the deformation the expectation values are independent of t so there is just no localization you can do so this has to do with what I said in the first lecture localization allows us to compute Sampa integrals we are not solving the theory that would be great but it's not the case any other question okay so we want to use this fact to our advantage and so and so okay so we know that if we take this parameter to be zero this is our original pat integral and now suppose that we can find some v such that the bosonic part the bosonic real part qv is a positive semi-definite so that means on all field configurations which live on this contour that you chose the real part of the bosonic part of qv is non-negative suppose that this is the case then we take the limit t goes to plus infinity and you see from there that any field configuration for which qv is not zero is going to be infinitely suppressed and so only the configuration for which the bosonic real part of qv is zero are going to survive and so the pat integral localizes to a neighborhood of these configurations so y such that qv is bigger than zero is suppressed and so we have this picture similar to what we had yesterday in which we have our space of field configurations where we are integrating but in fact the integral localizes to a neighborhood of some very special configurations where the real bosonic part qv is zero ok so however as we saw in the example yesterday still we need to take into account the neighborhood of the point and this there appear of the fact that still we had this Gaussian integral to do and in this setting we have the same at the same thing so let's expand so let's go around one of these points let's expand around there so let's take our phi to be some phi zero which is one of these special points and then let's call phi hat the deviation from that and for reason that will be clear in a moment let me take some power of t in front of it and so now we take this s plus tqv and we expand so ok so let's take s first so on s we have s of phi zero and then and then we do expansion around phi zero but the values that we obtain are suppressed by t by the powers of t and so here all the other terms out of order so what about qv well first of all so qv of phi zero is zero is what the condition that we imposed then we can look at the first order now since qv is positive and qv is zero then the first derivative must be zero it must be the local minimum and so also the first derivative is zero and so we jump to the second order derivative and there the powers of t can set out so this will be the quadratic expansion of qv around phi zero and this is quadratic in the field phi hat and then if we keep expanding then we get some negative powers of t so this is also suppressed and now you might ask ok but why did I choose this particular normalization of t that seems to be important here and the reason is that essentially that of course I mean you can always rescale the fields but if you do that you are going to modify the measure so what we want to use as are canonically normalized fields and now you add these the formation term qv is here and we are taking t to be very large so of course ok this argument depends a little bit of what qv what you chose for qv once again suppose that I chose the trivial example in which b is actually the zero functional well then of course I am not going to localize to in any sense and in particular this is not going to become large because this is identically zero so of course there are some assumptions of what type of qv we are choosing particularly we want this qv what I am going to say is this qv is non-degenerate so it affects all the fields in the theory but if this is the case this is going to dominate over s because this is very large still we want to have canonically normalized fields otherwise if they are not canonically normalized we should rescale them and this will affect the measure but in fact this is precisely what it does so this is essentially the positive condition that we are still using canonically normalized fields and in fact you see that this term remains canonically normalized question I am confused that the notation of by zero and by hat actually in the final dimensional version of the localization we have some matrix on the manifold to distinguish the zero points and the normal vectors due to the matrix but for the configuration space it is infinite dimensional so how to distinguish the zero points and the to do some separation of the fields between your phi zero and some orthogonal directions so this parameterized orthogonal direction to phi zero but in fact you don't okay I mean if I go a couple of lines on you will see that in fact you don't have to choose an orthogonal parametrization and essentially this will take care of itself so let me just go a couple of lines okay so we have this expansion and so what do we see from this expansion we see the following so first of all we get this contribution from the special configurations that we localize on so this is our phi zero phi zero such that qv of phi zero and and then okay so this phi hat for us parameterized orthogonal directions and so our patin integral has reduced to a patin integral so we have this phi zero but you also have the orthogonal direction that we should integrate over but now the point is that this is the quadratic action so this is expanded okay to quadratic order and so we actually know how to do this right we know we can do this patin integral exactly this is just a Gaussian integral and if we do that we get determinant essentially of the operator right this is just the standard Gaussian integration while this remains non-trivial in general so we remain with the integration over phi zero e to the minus s of phi zero and in fact we integrated these guys out is what we can call a super determinant qv phi zero quadratic yeah so let me comment on this formula so first of all these phi zeros are not points and this is true even in the finite dimensional case so I just discussed the special if you remember at some point I had an assumption that these points are isolated but in general they are not isolated and in fact as I promised that in the quantum theory case I would discuss the general case in general these configurations are not isolated and so you should integrate still over them you localize on a sub-manifold and so you should integrate over this sub-manifold what is this super determinant well this is just if you want fermionic determinant over bosonic determinant because if you have bosonic variables they go into the denominator if you have fermionic variables they go into the numerator so this is just a convenient notation for that this is the determinant of the quadratic expansion around phi zero and well there is a square root now if you have complex fields you not really have the square of this so there is no problem with that if you have real fields well you have the square root and the square root in some cases might create some troubles this is related to for instance the parity anomalies and so on so I will not discuss this in details it depends on the examples but this is something to take into account so finally what is this prime well this prime is the standard thing that in general we might find zero modes zero modes means that this determinant is some zero eigenvalue so what should we do if we have a zero mode this would imply that if I take the determinant of this value either I have a zero above and this pressure is zero I have a zero here and this is divergent but in fact what I am instructed to do is that I should remove the zero modes from the determinant now if I have a bosonic zero mode I should now integrate over this zero mode now what is a zero mode? well a zero mode is precisely a coordinate along these manifolds on which this s this deformation was just constantly zero this is what a zero mode is so in fact this integral of phi zero is precisely the integral of the zero modes that we are removing from here and this should answer to your question so yeah so even if you didn't do this probably with taking orthogonal fields you will find the zero modes and the zero mode is precisely the parameter you have to integrate over if you have fermionic zero modes then what does it mean? it means that you need to reabsorb these zero modes so either so here it looks like you get zero so either you incident some operator which absorbs the zero modes because you remember that in teta zero but eating teta teta is equal to one or you need to expand these to the next order in the fermions and then pick up the next component which well depending on the fermion on fermion zero mode give you a non-trivial result and we will see an example of this later on are you saying that the zero mode is not lifted by higher it's not lifted it's just that so this term should be thought as a function of the fermionic coordinates if you have a zero mode it means that the integral of the so either you insert the operator then part is clear so if you insert the operator okay this is the contribution you can go to higher order yes then you need to expand this look at the next term which depends on the zero mode and this in this way right if you have a function of teta this is telling you that this is and let's say that f is equal to f zero plus f one teta this is selecting for you f one so it's telling you that you should pick up this term here well because here I so if you want we localize on a bosonic manifold and you might ask what about why here we are just looking at the bosonic part of qv why it's only not solving fermionic equations and the answer is that so in the fermions your action is always a polynomial which is linear in each of the fermionic variables because they are anti-commuting so nothing weird can happen there are no linearities in the fermions but still you might have some zero some flat direction in the fermionic directions and in that case what you have to do is that you have to take the derivative in that direction and so yes and so essentially you have to this object here you have to take into account the zero modes and look at the next term and we will see in one example how to do that so here we are doing a good point of approximation because the integral localizes it's exact but so 5-0 strictly speaking shouldn't be the one that I mean that extremizes if we only keep the zeroes they are of course the absolute minimum because qv is definitely the same positive but shouldn't we include also possible sub-leading signals well in general no unless something weird happen so suppose you have something like this so we are only taking this but these guys here they contain e to the minus t qv so at least naively you would say that these are suppressed now there are some examples in which there is something weird going on that may evade this but so well I mean in this determinant you have a lot of cancellations but it's not that you are left just with the zero modes there is no perfect cancellation yes my time is over but this is a question yes so so there are lots of cancellations but you don't have a full cancellation and you can understand this with index theorems now in the examples I will do I will take the pedantic approach of actually doing the computation explicitly of this determinants but I think Zabzin will use the full power of index theorems and will teach you how to use them these articles are in general finite okay good point so no in general they are not finite dimensional so in general you might so you start with a pat integral and you might get a simpler pat integral which is still a pat integral so it's still hard however infavorable circumstances this turns how to be finite dimensional and these are the cases in which we are in business because then this integral is just a finite dimensional integral that we can have hope to solve but there is no argument that this should happen in general and this depends on the Q operator being nice or what? it's hard to say I don't think I don't have a systematic understanding of this um um yeah I mean of course ultimately it depends on what type of supersymmetry you have and what type of manifolds but usually if you have enough supersymmetries this reduces to a finite dimensional manifold but if you have only real supercharges well I don't have a systematic understanding