 So yesterday we wrote the localization formula which will be especially the hero of my next lectures. And so the formula says that the partition function reduces to an integral over a subset of fill configurations. Let me put this label BPS and I will explain in a minute what I mean. And then we have the classical action evaluated on these fill configurations. And then there is a remnant of the fact that we reduced on this sub manifold, which comes from small quadratic oscillations around this sub manifold. And this is because these small quadratic fluctuations or infinitesimal fluctuations are not suppressed. But in fact since they are quadratic this is a set of harmonic oscillators. And so this just gives us Gaussian factors, a ratio of determinants. I think this was the notation I used. OK, so something that I said at the end of the lecture, but let me stress it again. So this is still, I mean in general, this is still some infinite dimensional space. So it's still a hard problem. But in good situation, well the good situations are the ones in which this sub manifold becomes finite dimensional. Because then this reduces to a standard integral and we can have some hope to solve it. OK, the other comment is that this in fact is an exact formula, so we computed it with a sort of saddle point approximation, but because of the fact that there was no dependence on this deformation parameter, the result is exact. And then finally I would like to explain this label. So at this point how this was derived was just the zero locus of this functional that we chose with some nice properties. In fact, this most examples, it turns out that this zero locus corresponds, or it can be chosen to corresponds to BPS configurations. So configurations that solve the BPS equations. And the reason is the following. One can try to make a canonical choice for this B. And this canonical choice, so if you wish this is similar to what we did in the finite dimensional case where we started with the form beta, but then we say let's choose beta to be actually the one form which is due to the vector field. So here also one can try to make a canonical choice. So this is a sum of all the fermions in the theory. And then we take q psi, double dagger psi. So this double dagger, what is this double dagger? Well essentially, as we say, we go to Euclidean signature, so we complexify all the fields. And so our fields that are complex conjugate in Lorentzian signature they become independent in Euclidean signature. And so here we can just try to choose some anti-linear operator, which is like a dagger, but it doesn't need to be the dagger in Lorentzian signature. So I make this standard choice. And then what happens? So OK, we have to check whether this v satisfies the property that delta bv is equal to zero. So, but then the nice thing is that when this happens if we compute qv, so q can act on this. And then so what we get here is q psi, double dagger q psi and then it can act on this. And then we get q of q psi, double dagger psi. Now this is a fermionic term, because this is a fermion and this will also contain at least another fermion. Well, this is the bosonic term, but then you see that this is automatically positive. And the zero locus precisely corresponds to q psi equal to zero. So for the zero locus we find the BPS equations. And then OK, if you have chosen a contour, we have to restrict. So in general these BPS equations can contain lots of solutions, because in general we have complexified fields, but then if you have chosen a contour, we should restrict to these equations to the contour. And then OK, one gets what they get. OK. So, are there questions? OK, so now let's start with another topic that in my notes is lecture two. And so now I would like to discuss a concrete example. In fact, everything was very general. And in fact, this is the main theme of other lectures where other cases are discussed, and probably you will hear, or maybe you will hear this story again, explain with different words, which is good. But now I would like to go to a specific example, or examples. And so I would like to discuss two-dimensional theories with a certain amount of supersymmetry, which is 2,2, which corresponds to four supercharges. And in fact, this is the dimensional reduction of four-dimensional and equal one. So essentially you should already be familiar with this type of supersymmetry. Many things are similar to four dimensions. And OK, this is a nice example for various reasons. Well, for me, this is a nice example because I've worked on this example, so I'm more familiar with this setup. But it's also, I mean, we are in low dimensions, so things are relatively simple, and it's easy to do all the computations on the blackboard, or almost all of them. But still one can get very interesting physics, as we will see. So, OK, so let me say a few things about this supersymmetry. So if we start in Lorentzian signature, so as these notations suggest, so there are left-moving and a right-moving spinors in two-dimensions, which in Lorentzian signatures are not related by charge conjugation. And so we have one complex left-moving supercharges, one complex right-moving supercharge. And the biggest asymmetry that we can have is u1 times u1. There is a u1 that acts on this complex supercharge, another one on this other complex supercharge. And, well, if you want, we can do a change of bases and talk about a vector like our symmetry and an axelar symmetry, just the diagonal, anti-diagonal combination. But, of course, as also Guido stressed various times. So this is not part of the supersymmetry algebra in flat space. This is an outer automorphism of the algebra. And so supersymmetric theory need to have this r-symmetry. It can have it, but it's not required. It becomes required if the theory is super conformal, because then the r-symmetry is part of the algebra. However, we restrict to theories in which these r-symmetries is present. So we insist, or we chose, class of theories, that the vector like is present. So you want the r-symmetry. OK, so now in general there can be two central charges for this algebra. But if we insist that these are symmetries present, then there is only one central charge, which is related to the possibly broken axel symmetry. And the reason is that these central charges are charged under the two symmetries. So if you have the central charge, you are breaking the symmetry in vice-versa. And so in this case we have one complex central charge. In fact, in the super conformal theory where we have both symmetries, these central charges are zero. And if you want to have both central charges, you need to break those symmetries. And so now if we go on Euclidean signature, but still on flat space, the supersymmetry algebra looks like the following. Hopefully I don't have notational mistakes. So now in Euclidean I will use a tilde to indicate what should be the complex conjugate in Lorentzian signature, but is independent in Euclidean signature. And so what we have here is of course we have translations. Or if you want this is like a lead derivative. But then there is also this complex central charge. So let me write in this term, in this way. OK, so what I mean by this notation is that really I want to use bispinor notation. So this P mu should be rewritten. Well, this is in fact rewritten by spinor notation with the gamma matrices. And these projectors are the chirality projectors. So spinors in two dimensions are two components, the complex Dirac spinors. So these are chirality projectors in positive chirality and negative chirality. We could call them 1 plus or minus gamma 3 over 2. OK, while this is z... Sorry, didn't write z. So this z and z tilde, so this is the complex central charge and I'm using a tilde again because in Euclidean becomes independent. OK, while the qq and q tilde, q tilde are anti-commutator as 0. And then we assign r charges to these supercharges. So q alpha will have r charge plus 1 and q tilde beta will have r charge minus 1. OK, so... So now we want to put these theories on curved space. So we follow the root that Guido explained to us. So first you should discuss supercar and multiplets. But as I said, this amount of supersymmetry is dimensional reduction of for dimensional equal 1. And so in fact, everything is essentially dimensional reduction of that case. And so in particular since we have these r symmetry, there exist an r-multiplet, which is the dimensional reduction of the r-multiplet for dimensions. And it contains the following operators. So it contains the stress tensor. It contains the supersymmetry current. And then it contains the current, the conserved current for the r symmetry. And then there is also the central charge. And so it contains a complex conserved current for the central charge. Correspondingly to this r-multiplet there is a two-dimensional off-shell supergravity theory, which is the dimensional reduction of new minimal supergravity for dimensions in which the graviton-multiplet precisely contains fields which are going to be paired with these operators. Because at linearized level the fields in the graviton-multiplet couple linearly to the operators in the superkarem-multiplet. And so in this graviton-multiplet we have the metric, we have the gravitino. And then we have vector fields for these guys. So there is a vector field that couples to the r symmetry and there is a complex vector field that couples to the complex central charge. So, of course, we know how the metric appears in the in the theory. How do these vector fields appear in the theory? Well, first of all, they appear in covariant derivatives, obviously. So every time there is some covariant derivative, this will be a covariant derivative in respect to the curved metric, but then it contains terms like this. So where r is the r charge and this is the vector field that comes to the r charge. And then in this particular notation where I use complex currents and complex vectors, again, if I didn't make mistakes it should look like the following. So this vector field appears in the covariant derivative weighted by the central charge. So depending on which fields it acts will be coupled in a different way. It turns out that these fields also appear through their field strength. And so, but since we are in two dimensions, so field strength is two form, but we are in two dimensions, it's convenient to dualize it to a scalar. And so we can use some scalar h, which is up to some factor. It's just the dual to the field strength for c So these are the fields that appear in the supergravity in the supergravity multiplet in the graviton multiplet. And so if you want to study supersymmetry on curved manifolds we should look at why essentially we should set to zero the variation of the fermions in this multiplet. And in this case the only fermion is the gravitino, there is no gauge or something like that. And so setting those variations to zero is sometimes what we usually call the generalized killing-spinner equation. So this equation looks like the following. I call this v. And there is a similar equation for the tilde fields. OK, so these are the gravitino variations in this supergravity theory. Now here the dots corresponds to terms which are, I mean, appear in the funeral linear theory, but that are automatically zero if we set the gravitino to zero. So we don't care about them in this generalized, I mean, just if we want to set these variations to zero because we also set at the same time the psi to zero. So the equation that we have to solve is this. And yes, and so we have so we have this, so the derivative, so this is the parameter for supersymmetry variations. This parameter is not charged under the central charges, so only this r-symmetry vector field appears, but those fields appear through the field strength. And moreover, here I'm using a notation. So if you want, this is a spinor and here for clarity I've chosen a base in which chirality is diagonalized. So I've chosen a base in which chirality is one minus one. Otherwise you can write in a invariant way with projectors, but it's more messy. I mean, it's not messy, but it's longer to write. Ok, and this reflects the fact that two supercharges are opposite charges. Ok, so essentially if we want to study supersymmetry on core manifold we have to solve these equations. So before doing that let me just say something which is useful. So we are in two dimensions and so the group of rotation is SO2 which is U1. And so essentially the bundle in which any field with spin transforms is the same as a billion-gauge bundle. So there is no real difference between the spin and some standard abelian charge. And so we can use these to simplify our notation a little bit. So first of all we can take the spin connection and so we can take the spin connection which if these are some field-bind bases, these are anti-symmetric and this is the algebra of SO2 and we can just convert in a standard abelian-gauge field. So we just multiply by epsilon and we can define some omega mu. This is the spin connection abelian-gauge field. But then this is useful because now if we take the covariant derivative and we act on a field with some spin S and we look at a specific component. So of course each component will have some value for the spin, for the Z component if you want. So this is just going to be the same as so the spin appears as a charge and we have this abelian-gauge field. So the spin just appears as some abelian charge and if a field is actually charged under some other gauge field these other gauge fields are in the same way here. I mean this is trivial but it's useful. Do you have any questions so far? OK. So another observation that we want to make is that again before going to solve these equations it's about the form algebra. So as we said when we go on a curved manifold the number of generators is a subset the number of generators of the supersymmetry algebra is a subset of the generators that we have on flat space but the algebra can be deformed and we are restricted into relevant deformations so the formation that disappear when the manifold is scaled to have infinite volume and we can read off this the form algebra into the supergravity and so essentially even before solving the equations I mean you can write in general what the form algebra is in terms of those fields and the specific algebra will depends on what is the profile that those fields will take on a solution and so the result that you obtain is the following I rewrite it in terms of some delta epsilon for me delta epsilon is just the contraction so as I say is a contraction of the parameter supersymmetry parameter which is a solution of those equations with the supersymmetry charge that we have in supergravity so it turns out that here so you get a lead derivative so there is a translation and this lead derivative is along a vector field which is specified by infact is a biliner in these spinors so this K mu is a standard formula which is sometimes called a sandwich because you take these two spinors but then if you act on some gauge variant field this should also be the gauge covariant lead derivative but then there is also an extra piece which is again a sandwich which depends on a matrix q so this is a matrix in spinor space and this matrix q is given by the following so again it is diagonally in chirality and so this matrix q is controlled by the central charges it is controlled by the profile of this scalar H which is the field strength of that field and then there is also this sigma that I haven't introduced so this appears when you couple to some matter action and then you go in best domino gauge and essentially we will describe this in more details afterwards but essentially this is related to twisted masses so some masses you can turn on in the theory and there are background fields so if you want these are scalars which appear in the vector multiplet so if you want this formula is already a little bit more generic because as we explained if in our theory we have some symmetries we can play a similar trick with vector multiplets if you have some symmetry in theory to a background gauge field this background gauge field appears in a super multiplet and so there will be other bosonic fields in this multiplet one of these fields is a scalar and one can turn it on in a way that preserves supersymmetry and then this appears in this algebra while the other ones are zero and similarly for the tilted version so so yes yes you see it here that depends on the r charge so ok my writing is not the best so this is a small r so it depends on the r charge and so yes and so this it reflects the fact that in this deformed superalgebra the r symmetry enters yes so I think that the term that you are referring to let's see so that should enter I mean if something is charged under the r symmetry that will appear in this in this le derivative so this superscript means that you have to put all the gauge fields that are relevant I mean if it depends on what you act on yeah yeah we are not very precise in my notes but I think this is what what happens ok so this a means is that so if you act on a operator which is gauge invariant and neutral under everything this is just the le derivative but if this field is charged either because it is I mean you are acting on the fields in the Lagrangian so they are gauge covariant they are not invariant then this a is the gauge field the dynamical gauge field but I think if you act on an operator which is charged under the r symmetry then you should also include the v here so if it is charged it would act by like lk minus a mu or something or minus i a mu it would be like taking a covariant derivative or replacing del mu with the le derivative yeah so you can take for this le derivative I mean this define as the anti commutator of d and the contraction with a vector field and so you have to make d into a gauge you just bracket covariant derivative yeah if you want yeah ok but we will see this more explicitly in the examples ok so now what I would like to do is to take those equations so what we would like to do is to take these equations and solve them and on very general terms you would like to find the most general set of solutions so in particular you would like to solve in very general terms for the metric and then for some metric for the vector fields v and for the scalar h and then in such a way there is at least one solution in epsilon so we will not study in full details the solutions to these equations this has been done by Clausset and Cremonesi in 2014 instead I would like just to present a few solutions which are interesting so a first simple solution that we can have on any manifold or probably I should say any orientable manifold is the following so so we choose a vector field that goes to the r symmetry to exactly cancel the contribution of the spin connection in the covariant derivative in one of the two equations let's say that we do it for the first one well sorry let me say it back so we set the vector field to be equal up to factor one half to the spin connection in such a way that there is some cancellation and now and on the other hand we set the scalar fields to zero so now what's happening here so epsilon and those epsilon tilde they are two component spinors so we are using this notation in which there is an epsilon plus and an epsilon minus because we are in the base in which we diagonalize chirality so these components have spin plus one half and minus one half so in the first equation now we could write the equations in components and for one component transforms as it was charged and there will be a gauge with charge plus one half and the other component will charge minus one half and so you see that if we make this choice for one of the two so in the first equation for one of the two components we can precisely cancel v against the spin connection of course we cannot do it for both components because the arc charge is the same but the spin is opposite so we can only cancel the contribution one equation on the other one but then something similar happens in the other equation just for the other chirality and so now the equation becomes since h is zero the equation just becomes that the spinor is constant and of course this is a solution on any manifold and so we can just set epsilon to be zero epsilon minus with some constant epsilon minus and epsilon tilde to be epsilon tilde plus zero where epsilon minus and epsilon plus tilde are constant ok so this is a very simple solution but it exists on any manifold and in fact this solution has been known for a long time this is called the A twist so this is a type of topological twist what does this terminology means well the twist is a way to preserve supersymmetry in which the only thing that you do you have a theory with some r symmetry and then you turn on a background for the vector field that couples to the r symmetry which is related either is equal to the spin connection or is equal to some component of the spin connection in such a way that at least for some components of the spinor there is a cancellation vector fields that couples to the r symmetry and the spin connection and so for this for some components the equation simplifies and just becomes that component is constant and then there are solutions and in particular in the A twist you do that using the vector like r symmetry and so as I said this solution has been known for a long time however it is reassuring because it is contained in this general formalism that is supposed to contain all ways of at least using the r multiplet all ways to preserve all ways to preserve supersymmetry on a core manifold we can also go on and yeah, no, okay so this is the solution and and well, yes, sorry let me write what that what this supersymmetry algebra is and in fact becomes something very simple so this operator, so we have two operators which are important and moreover the anti-commute so this super algebra is very simple and if you wish the r symmetry is not part of this of this super algebra and it turns out that for genus bigger than one this is essentially the only solution so what do you invite essentially what would be exceptional for you well, let's see let's see so essentially because so as you see from the equation so let's take the so let's look at epsilon so the upper component is zero so if you look in the equation so gamma mu inverts the chirality and so you can choose any h tilde because it still solves the equations because it multiplies by zero so h tilde can be an arbitrary function on the manifold but this is not gonna so when you compute the localization results so partition functions is not gonna affect the answer so it's not the only solution but in terms of partition function that you compute it's everything you want to get you get it from there this is more just to make sure I'm understanding things but so you're saying this putting it in this background super gravity is exactly the same on the nose as the sort of older papers of people like Whitton on topologically assisted theories where they don't mention any background super gravity they just take the energy of an M-sensor and they shift it so that the right things happen that's exactly equivalent at some point I start mentioning it but maybe not in the very first page I just wanted to make sure I wasn't confusing things no no no it's a very important thing to remark it's just a different perspective on the old topological twist so one way to put it is that essentially you do some improvement transformation and use a different stress tensor which is in this different stress tensor essentially the words that are used is that you change the spin of the fields according to their charge because you regard rotations as now a subgroup so you take you want spin you take you want V then you declare that the new spin group is you want which is some linear combination of these two right this is the old language yes but what does it mean it means that now this group couples to the curvature of the spin connection so when you turn on a spin connection for this is like you are turning on a spin connection both here and here dictated by the particular linear combination that appears here so effectively what you are doing is that depending on the spin connection you also turn on a background for the vector for this vector for the cups to the R charge so this should show that they are exactly the same thing does it I mean are you are you saying that the improvement of an energy momentum tensor is equivalent to changing the background it should be in this case I don't know maybe Guido can correct me if I'm wrong but I mean you are changing the stress tensor I mean and you still have stress tensor which is symmetric and conserved so that should be an improvement transformation but the thing is that you follow and coordinate transformation with some out transformation yeah but at the level of changing the stress tensor so one way to put is that you take the stress tensor and you change it so you add something in such a way that becomes q closed q exact I think it can be interpreted as an improvement transformation so we can't think about that ok so another solution that I will call untwisted S2 so we take the round metric on S2 so this solution will be specific for S2 will not work for bitrariginos and then ok, I call it untwisted because this time we don't turn on at all a background for the vector field that couples to the r symmetry so we set this to 0 but it's decided we can still solve the equations so now we have to turn on the scalar fields where r is the radius of the sphere and now if you ok, you just write those equations the equations take the following form and in fact these equations are be known also for a long time these are called f and not mistaken conformal kininspinor equations and and well on S2 I mean you don't need to open a book to solve this equation on S2 they are simple to solve and it turns out that there are two solutions but in fact there are two solutions for epsilon and two solutions for epsilon tilde so in total there are four solutions and for instance for epsilon should I leave this as an exercise? ok, I mean they are simple enough to write so we leave it as an exercise find this epsilon but essentially there are two integration complex integration constants so there are two solutions for epsilon and then two solutions for epsilon tilde and so in fact let me this is something I didn't stress before let me compare with this topological twist so you see there is one complex solution for epsilon and one complex solution for epsilon tilde so there are two supercharges but we started with four and so the topological twist breaks half of the supersymmetries it's half pps however this solution does better we have the same amount of I mean we preserve all supercharges we have four of course we have to use the round metric however did the form supersymmetry algebra looks like the following so first of all this is not the same as in flat space because you don't just have you don't just have the derivative which is a translation but you also have an r symmetry rotation even though this vector field is zero so this comes from the second term from that bold Q correctly this deformation disappears if you go into flat space limit because there is one over the radius of the sphere so as promised the supersymmetry algebra should reduce to the flat space one when we go to flat space but you see the r charges of the fields appear explicitly in this supersymmetry algebra and so in particular this is related to the fact that the r symmetry now becomes part of the algebra so this is different from flat space as I said before in flat space r symmetry is an outer automorphism but here instead in this algebra the r symmetry is part of the algebra and because of this reason we have to make a choice for the r charges and this choice of the r charges will affect the theory this is different from flat space in flat space you write down a Lagrangian if you want you can assign r charges but of course the Lagrangian is not gonna depend on the r charges just an assignment that we make and once again this is due to the fact that the r symmetry is not part of the algebra but in this case it will be different so we will see the Lagrangians explicitly depend on the choice of r charges they will appear as some curvature couplings and once again this is due to the fact that the r symmetry now becomes part of the algebra ok so the other ones are 0 I already wrote it there and this so in fact this K mu which we already wrote so this is a killing vector which generates as you vary this epsilon and epsilon t that this generates rotations of the sphere so the SO3 that acts on the sphere and in fact the full superalgebra turns out to be SU2 slash 1 plus bosonic subgroup is at the level of algebra is SU2 times U1 where this SU2 in fact is this well at the level of the algebra this is the same as the SO3 which rotates the sphere and this U1 is precisely the r symmetry ok is there any question? if you preserve inside of the vector r symmetry axial then you should have a corresponding solution for a V-model and is there an unquisted solution? yes so so if you want to discuss the axial case so that is not immediately obvious how to reduce it comes from four dimensions because the axial are similar to the reduction is accidental but in fact it should be very similar so in two dimensions it should look very similar to the r-multiple somehow it looks like an r-multiple for the axial symmetry in which now you have the conserved current for the axial symmetry in fact it turns out that is the mention reduction for dimensions but then when you go into two you get this vector if I understand correctly and so in that case everything goes through more or less in a similar way where instead of having the vector like r symmetry you have the axial r symmetry and everything just change between chiral and twisted chiral and vector and twisted vector the only thing to be careful about is that so this vector like r symmetry is never anomalous with this amount of supersymmetry but the axial one in general can be anomalous so the other twist since as we said in order to make this construction we really need that this is not just classically a symmetry, it should be a symmetry in the quantum theory so you need to make sure that the axial symmetry is a symmetry of the theory but otherwise everything goes through and in fact so this is very useful and in fact in the literature somehow this is called because it's the one related to the A twist and then the other one is called B because it is related to the B twist In principle vector and u r symmetry should be completely symmetric because of mirror symmetry they are not completely symmetric because of the anomaly you mentioned I mean of course if you go in the but the mirror is a different theory so you assume that your theory has a chiral multiplets which only chiral multiplets are charged under gaze matrices so in principle you could have twisted chiral multiplets charges under twisted vector multiplets in that case you can get anomaly for u and b I think you are right yes yes what I said is for the theory of vector and chiral you can write everything so if you have a theory with just vector and chiral you can just rewrite it in terms of twisted vector and twisted chiral you are right one last comment on this is that if you look at this solution for the scalars you see that one is not the complex conjugate of the other one because there is no minus we respect the two and so in particular this means that this background is breaking a reflection positivity now if we study some theory that flows in the infrared to a fixed point as we said yesterday we should not be worried too much because then in the infrared this feels capital redundant operator and we recover a fraction positivity but if you are studying a massive theory this has some consequences ok ok, any other question before we go on yes but essentially so if you do 0,2 essentially you are doing the twist so so that is essentially the only solution I think so there are no scalars in the graviton multiplet you just have this vector field for the say right moving asymmetry it's called half twist the reason is that the theory does not become topological when you do that twist it becomes essentially a chiral CFT but still it does depend on the complex structure it's not topological that is also true ok so now that we have understood these examples so now I would like to start discussing a localization computation so I would like to show one example in details so at this point you probably already saw one example in Wolfgang's lectures but so this will be another example but so the example that they want to study is precisely this one here of the untwisted background on the round S2 so let's actually take sorry, before you could I ask one question so this is curious in G greater than 1 we have no isometries and we can find a background that preserves two supercharges in gena 0 we have lots of isometries and you can find a background that preserves all supercharges on a torus for which we have a U1 times U1 isometries and you find a background that preserves all supercharges yes, I mean you don't have to turn on any background on a torus so this is the twist of that story so you can turn on so if you don't turn on anything what your partition function is essentially computing the width and index which is a number you can make the story and we will discuss this probably in the last or starting lecture or starting tomorrow I don't know you can make the story more interesting if you turn on some background for vector fields in vector multiplets or also for the r symmetry so then you really gets a function this is called the elliptic genus but this will break half of the supercharges ok, so we want to do localization for untwisted gauge theories s2 and this will be around s2 in fact this composition can be extended to a squashed s2 but the answer is not going to change so we will be happy just to do around s2 ok, moreover it turns out so we are in 2d2,2 and it turns out that there are many multiplets that one can use to construct theories in particular there are chiral and twisted chiral vector and twisted vector but I will restrict to a simple subclass of this theory so I will just consider vector multiplets and chiral multiplets and so in particular these are dimension and reduction of vector multiplets in four dimensions and chiral multiplets in four dimensions so ok, we already know these multiplets so let me just say very simple things so in the chiral multiplets just to maybe set up the notation so what fields do we have so we will not use superspace, we are just working components so in this multiplet there is a complex scalar there is a complex fermion two component complex fermion and then there is a complex auxiliary field that we call f and I will also use f for the field strength but the field strength will have two indices so hopefully there will not be confusion and and then there is also the anti chiral multiplet in which we have all the tilted fields and then there is a vector multiplet so in this vector multiplet we have a vector we have a complex scalar so I will use sigma and sigma tilde and then there is a complex Dirac fermion and then there is a real auxiliary field and of course all of these fields are in the adjoint representation of the gauge group while this will be in some generic representation and also it is useful to decompose this sigma into sigma 1 plus i sigma 2 and then sigma tilde is sigma 1 minus sorry, I am using a minus here so minus i sigma 2 is sigma 1 plus i sigma 2 however in Euclidean signature sigma 1 and sigma 2 are complex so these are not these are independent before we specify a contour and then it turns out that out of the vector multiplet one can construct so one can repackage these fields into a twisted chiral multiplet that I will call sigma and so the fields in this sigma are so the bottom components start with this scalar and then the various components which are containing here are essentially the gauge genie these details are not particularly important I don't want to waste too much time in details that will not matter okay you only want the details not important okay so when you do the splitting so you get some components of the gauge genome go here and then the auxiliary fields so in the twisted chiral multiplet not the vector field appears but the field strength and so one component will be something like this and it turns out that when you are on a curved manifold there is a mixing with sigma it is the other combination there is probably some type in my notes so the twisted chiral is a representation of the n equals to 2 algebra the twisted chiral is a representation of the n equals to 2 algebra so that means the vector is decomposable no, it contains the same fields these are two fields on the twisted chiral the twisted chiral are like two representations not on a day or day let's see let's see yes, that is true but ok, but it does not contain the vector it contains the field strength so if you really want to contain the vector you have to use the vector multivet and yeah, I'm not sure about this time because it looks like this is the same combination so I might have a type in my notes maybe there is not this minus sign ok so the important thing is what so now we want to construct gauge theories out of these multiplets so what is the data that enters in the finding constructing these gauge theories so essentially what is the data that we have to specify we can specify and we can play with ok, so first of all we can specify the gauge group because this will be some compact gauge group then we will have the matter in chiral multiplets so the matter will transform in some possibly reducible representation of g so we will specify a matter representation r there are many things that I call r hopefully this does not create too much confusion so this is a representation r of g and in other words the matter fields this phi take value on some vector space and this vector space can be decomposed into the weights of this representation so I will use often this notation rho in r these are the weights of the representation now as I promised before so the difference with respect to flat space is that now the r symmetry is part of the supersymmetry algebra and so we need to specify r charges because these r charges will control some couplings in the Lagrange once again we don't have to do this in flat space of course but we have to do it here so if you want the fact that su2 slash 1 contains vector like r symmetry implies that we have to specify r charges then we have interactions and interactions are controlled by two functions one is superpotential and the superpotential is allomorphic and gauge invariant function of the chiral multiplets and should also be invariant it should transform homogeniously and r charge in particular in my conventions have r charge 2 if you wish in other words you can first specify the superpotential and then the r but you have to make sure because we need this r symmetry so the superpotential should transform homogeniously under this r symmetry so if you want this is some allomorphic function of the chiral multiplets so this is exactly as in for dimension n equal 1 but moreover we have a twisted superpotential and this will also be allomorphic function but now of the twisted chiral multiplets now after we have introduced all interactions in general there will be some flavor symmetry which is left over we call it gf so this will be a global symmetry that commutes with the supercharges so by flavor I mean that we don't include r symmetry and then we can play this game that already Guido mentioned yesterday so we have this global symmetry so in particular in the theory there is a current for this global symmetry and the useful thing to do in general is to couple this global current to an external vector multiplet sorry to an external vector field a background vector field since the theory is supersymmetric this vector field is in a super multiplet and so we will have other bosonic fields in this vector multiplet and we can try to turn them on of course we have to do that in a supersymmetric way so Guido wrote this equation that the variation of the background gaugino let me call this lambda favor so this is a background fermionic fields that is in this background vector multiplet this variation we impose it to zero this impose some constraint but model of this we can turn on these extra parameters and what they do they control some couplings in the theory and in particular what they turn on in particular there is a scalar in the vector multiplet so this scalar here if we turn on this scalar in a background vector multiplet which is a vector multiplet for this flavor symmetry what it does and then you see the effect in the Lagrangian what it does is turn on some masses so these are called twisted masses so you can think of them as expectation values for these flavor fields ok so this is the data that we control in the class of gauge theories that we want to study so as you see there is quite some some data we can play with ok when you wrote delta lab of the flavor equals zero are you saying that's automatic because it's non-dynamical no no that has to be imposed if we want to preserve supersimule what kind of constraint does that lead to on the twisted mass separation how does it tell you how do certain things so let's see so in let's see I think in flat space it imposes that sigma and sigma tillers to commute so because sigma is complex it tells you that it's diagonalizable what's in the carton? yes and then you can reduce to the carton oh so that's why for a billion theories you take the twisted masses it at least makes them constant incredible it's also true it's also true for eight ways omega deformation but twisted mass can actually depend on the position but in a specific way I guess in arbitrary way and then you can satisfy the equation delta lambda equals zero by turning on feel strength so of course these equations depend on the setup that you have the manifold the background for the fields in the graviton multiplet in fact there is a I mean there is a hierarchy if you wish there is a hierarchy of equations so we are working off shell so so the variation of each multiplet should stay within should only involve the fields in that multiplet and this was as we see in the beginning this approach that made this approach very interesting this approach of Guido and Cyberg because we can just discuss we don't have to discuss the specific theory we just discuss the variation in the graviton multiplet however something that we do is that we go in best zoom in or gauge we do it for the vector multiplet but in fact we also do it for the graviton multiplet so because you go in best zoom in or gauge what happens is that in this hierarchy of multiplets start with the graviton, the vector and the chiral the variations of the multiplets are below in the hierarchy they do contain the fields that appear above that you would see if I do everything off shell this should not happen but this happens because we are in best zoom in or gauge and so somehow we can first discuss the equations in the graviton multiplet they are only depends on the fields in the graviton multiplet but then the equation in the vector multiplet includes the fields that we have set in the graviton multiplet and then if you go at the variation in the chiral multiplet or whatever other matter multiplet we have they includes the fields of the vector multiplet and the graviton multiplet so in this sense the equations that we get for that somehow depend on what we have chosen for the auxiliary for the bosonic fields in the graviton multiplet ok so ok, so now that we have discussed the class of theories that we have decided the class of theories that we want to discuss we should construct actions and as we said this action can be constructed with the methods that Guido is teaching us so we will only give you the result so what action can we write so first of all we can write young mills kinetic action so the kinetic action for gauge fields and so let me write it and then if you want the details are not important but I just would like to point out some features so this is the bosonic part that involves the bosonic fields and then this is the fermionic part and probably in front of everything we want a gauge coupling ok, so this is our bosonic action so what do we notice in this action so first of all as it should if we send the radius of the manifold to infinity sometimes disappear and this action reduces to the flat space action so there is the standard young mills term there are kinetic terms for the scalars there is this quartic coupling that just comes from the dimension reduction from four dimensions because when you reduce from four dimensions two of the components of the vector become scalar and so just from the young mills term you get this commutator but the square of this commutator and there is the d term and ok and this is also the standard thing however when we put the theory on the sphere we have some deformations which are controlled by the curvature of the manifold so these terms here ok and let me also write so let me also write the meta action so we can write similarly a kinetic term for the meta action so here there is a similar story so when we send the radius to infinity we get rid of some of the terms and once again this action is just the standard dimension reduction from four dimensions however now here another feature that I wanted to show you is that as you see so these small r are the r charges for the current multiple that we chose and as prom is the action now explicitly depends on the choice of r charges and again this comes from the fact that now the r symmetry is part of the supersymmetry algebra and of course this feature disappears when we go to flat space now if we want to perform localization we need to choose a contour and in fact it's convenient to choose a real contour so a real contour is just the obvious thing to choose so fields that were real in Lorentzian signature we imposed also so now they are complexified become complex in Euclidean where we imposed that they are actually real and if we had a complex field in Lorentzian signature when we go to Euclidean the field and this tilted partner now becomes independent and we just imposed that a complex conjugate to each other so it's the obvious contour to choose and in fact if we choose this contour these actions the bosonic part of these actions becomes well the real bosonic part of these actions becomes positive definite and the integral the part integral becomes convergent so this is a good contour then we want to choose some super charge q that we want to use for the localization so a convenient choice is to choose some epsilon and some epsilon tilde so we know that you have two choices here so we made one choice here and we define this particular delta q and we will use this as the super charge that we use in our formal argument yesterday in this case no because first of all the final result should not depend on it in general because what we are computing is just the original part integral so localization is just a way to computing it of course the result should not depend on how we compute it but you might expect that the form of the result the way in which it is expressed depends on the choice of the super charge some choices might lead to some expression and some other choice might lead to some other expression these two expressions has to be equal but they might be equal in a very non-trivial way now in this case this does not happen essentially because of the SO3 rotation on the sphere yes thanks so essentially by rotations you can map any choice to any other choice but in general you might I mean if there are some inequivalent choices you might you might expect different looking answers and we will discuss tomorrow an example which is not exactly this but something similar so we will discuss how changing the localizing term can affect the form of the answer anyway we make this choice and then there is a nice feature something nice that happens which is the following that it turns out that both the superium mills action and the matter action are q exact so they can be written as q of something and so this is very nice for two reasons so first of all this is telling us that in fact the party integral that we are going to compute will not depend on the coefficients in front of this action so in particular there will not be dependence on this gauge coupling and well this is nice because in two dimensions the gauge coupling is dimension full and so if you wish the gauge coupling is setting for us a scale above this scale we have the strongly coupled gauge theory below this scale we have essentially the the theory in which we have quotiented by the gauge transformations now the gauge feeding to dimension is non dynamical but the other fields are dynamical so if you want below this scale we are in a strongly coupled phase in which we should use a different description now the fact that there is no dependence on this parameter means that there is no dependence in this sort of scale that we can think of as a cutoff and so in particular our result is independent of the RG flow at least for what regards of this parameter well the other property we can interpret it as the fact that there is no dependence on wave function renormalization so again it gives us some independence on the RG scale and so in particular if you have a theory that is closed to fixed point our computation will be directly sensitive to the infrared CFT so essentially doing computations using the UV description we will be able to access information about the CFT ok and I think that will stop