 Ok, in najbolj spikr je Alberto. Zdaj se zelo izgleda o lokalizaciju. AdS, Black Hole Zempre. Ok. V mnohi zelo izgleda se zelo izgleda, zelo izgleda se zelo izgleda, zelo izgleda Black Hole in AdS IV, magnetizacija in statika. Vsrednjih zelo zelo izgleda in magnetizacija in statika je vzelo izgleda. Zelo izgleda se zelo oka komputaciju, komputaciju, neč prišli, zelo izgleda, lokalizacija in statika je Black Hole Zelo izgleda zelo magnetizacija Black Hole. SVIDEM, kaj bi sme priječila tudi, da vhleda je glasba, kešte je, da je te LEDится, uspeča�� vzela topologični tr해 in taj pa regens, so prič, da... No, to je vzela skupi in v neko vse, in presah, da je, ko ni razvelil bi se na deben prine Benjamin, in kada te topti dolge tr해, je to res utočen. Zato, da je zelo razest, in ki se ne zelo nekaj antropi. OK, zelo da bomo se zelo, kaj nekaj vršičje bilo, vršičje, ki so vzelo, da je asintotik 2,8,4, dijonik, so nekaj nekaj električni in magnetni vršič, nekaj rotacij, Static, Static for Simplicity. You can introduce rotation in this black hole. They feel theory. Computation has never been done because it's technically complicated, but the matrix model that I'll show you can be generalized to include rotation and I don't expect any news there. It's just a technical problem of taking the larger limit of the black hole. But I will consider the static case. And the other requirement is that this black hole can be embedded in a just four times a seven. The reason we mentioned yesterday is the following. I know the dual. That's the maximally supersymmetric background. It's dual to maximally supersymmetric field theory in 3D, which is the ABJM model. So, from the point of view of the black hole, it's useful to not to consider the full m theory on radius four times s7 to perform a dimensional reduction to 4D. And luckily enough, there is such dimensional reduction. It's even in the form of a consistent troncation, which means that the solution of the equation of motion that you find in 4D can be uplifted to 10D. It's not always the case, but in this case we are lucky enough so there is a consistent troncation that contains precisely the field I'm interested in. And I remember you that I want no charge in radius four, the black hole is static so there is no rotation, but the black hole is electrically charged. So, from the point of view of 11 dimensional theory is, if you want, it's spinning inside a seven. So, the electric charges will be charges under the carton of the isometry group of s7, which is s8. So, I need four vectors. And the dimensional reduction give you an n equal to two gauge supergravity as discussed by Stefan in his lecture, with three vector multiplet. Why three and not four? Because remember that in the gravity multiplet of n equal to two supergravity, there is already a vector. And indeed matter content can be summarized, the bosonic matter content, the relevant bosonic matter content can be summarized in this way. There is a gravity multiplet that contains the metric and the gravity photon. And then there are three guys here, three vector multiplets. Each contain a vector field, scalars and fermions. I goes from one to three. So, nv number of vector multiplet is three. Combining the three gauge field with the gravity photon, you can feel the fourth vector that you need in the dimensional reduction. Gravity photon is in a sense associated with the asymmetry of the dual conformal filter. So, as Stefan discussed in a very efficient way in the last two lectures, it is in writing the gauge supergravity is convenient to combine the vectors in a guy with an index lambda that goes from one to four zero to three. Zero is more typical in the supergravity literature. And in a sense zero corresponds to the gravity photon and one, two and three correspond to the vector fields in the vector multiplet. And these four guys give the carton of one to the fourth. Now, this dimensional troncation has been studied in the literature. So, people have written the effective Lagrangian, written the effective Lagrangian in n equal to two formulas. Read all the details. Remember that Stefan told you that you need to, essentially to specify completely the model, to give a prepotential, which is a function of holomorphic section. So, you may think that the physical scalar fields here, there are three, can be written in terms of this holomorphic section as ratios. So, in a sense, the holomorphic section is like projective coordinates for your manifold of scalars. There are four guys here, same number as the vector, but only three are independent. One way of fixing the physical scalar is just to take the ratio. It's not the only one. You can take any possible parameterization that you want. The Lagrangian will be invariant under a rescaling of this section x, so then you just get three degrees of freedom. Complex degrees of freedom, the scalars are complex. You need to specify the prepotential, and since there is no matter, the only thing that you need to give here is this set of Fajeljopulos coupling, which are just the guys that enters in the potential. There is a potential. Essentially, they specify how you are gauging the field, the gravitino, actually. The only thing gauged here is the gravitino. These are coupling constants for the 4u1s. These guys, as Stefan mentioned, are called magnetic Fajeljopulos. They will play no role in, luckily enough, no role in this lecture because then things become more complicated and nasty. In this troncation, luckily enough, you can just use electric charge. For those that know a little bit about simple transformation, there is a frame which is also a natural one where you can just use electric Fajeljopulos. Indeed, this data has been written. You can search the literature and you will find that the prepotential is this square root that already appeared in Stefan's lecture. The four guys here, the electric Fajeljopulos, are all equal. They are called common value g. Equal because there is SO8 in the original theory. Essentially, the four vectors can be rotated by the SO8 symmetry even if I break it to the carton. As I was mentioning, luckily enough, there are no magnetic Fajeljopulos. Let me check if I put the index. Kiril, help me. I think that the standard notation is index down for the electric and index up for the magnetic. Yes, I think, yeah. OK. Let me briefly comment about this prepotential. Stefan mentioned that typically when you do in flat space, you get the cubic prepotential. In this case would be x1, x2, x3 divided x0. That's very typical. Also because the cubic structure comes from some intersection of cycle in calabiaos and so on. Here you may think, OK, there is no calabiao because it's a 7, it's a different structure. But actually if you look at how the troncation is done, if you naive perform the troncation of a 7 in the typical frame that you get with standard parametrization of fields in 10 dimension, you will get a prepotential which is the cubic one. But you will get also a bunch of tensors in 4 dimension. Tensor in 4 dimension are, you have to scalars. So it's better to work with scalars and not with tensor. So typically in the reduction you get the cubic and tensor. We do one of this electric magnetic duality, symplectic transformation. The symplectic transformation maps the cubic prepotential in the square root. The tensor becomes dualized with scalar and you have a better picture of the story. So this is roughly how it works from the point of view of troncation. OK. So this is the model. Now Stefan show you an split solution of this model with magnetic charges when x1 was set equal to x2 and x3. Well, actually there is a full family of dionic black hole in this model that the first one was found by Kachatore Klem, then Stefan and Kiril wrote a paper. Also Dalaga Tenecki have an important one. And more recently after the work of Kurt Madas and Almagi that completely filled all the gaps, now there is a full family of black hole depending on electric and magnetic charges. So these black holes, these black holes have a metric that I'm not sure I'm using the same notation as Stefan, certainly not the same signature, but OK, we will never have problem about it. But when you check literature, remember one of the typical confusion is everyone is using different type of signature, different use for the letters. There are always common letters, U and V, but everyone use it in a different way, so be careful. And so the metric is this one. Here, as I was mentioning yesterday, all these black holes exist in many different forms. So there are the spherically symmetric ones where the horizon is a sphere, that's the nice black hole. But you can replace the sphere with an arbitrary surface of arbitrary genomes. There is no much difference. So even when I will do a computation, I will fix on S2, because I can write as please everything. In principle, everything can be done for a Riemann surface. I will tell you the details. And so dionic means that they have charges, electric and magnetic charges. So the magnetic charges appear in this way. So there is an horizon here. And if you integrate your gauge field on the horizon, you will get essentially some integer. This is like the monopoly I was discussing yesterday. The quantization condition tells you that you should get an integer. Obviously, it depends on normalization and other things. So typically you write in this form and then I will tell you how the quantization works. So these are, for the moment, are not necessarily integer. There is the standard notation in gauge supergravity. There is a factor of the volume just for normalization. And then there is P, which is the magnetic charge. In some units, this P will become an integer. And these are just the magnetic charges. What are electric charges? Well, electric charges, you can, this gauge field on a sphere is roughly something that has component on theta in phi and is proportional to p lambda sin theta. That's the idea on a sphere, on a stone. In electric field instead has components along the radial direction and time. And the covariant way of writing an electric charge is the following. You define this guy here. Let me try to put consistently index up and down. We will not really need it, but this up and down of the index has not been completely discussed in this lecture, but has to do with this simply transformation. So if you want to keep a formalis, which is invariant under electric and magnetic duality, you need to keep track of the position of the index. We will not need it, but I'll try to be as much consistent as I can. Essentially, you define this, which is just the derivative of your Lagrangian with respect to the curvature, and it's a sort of dual field. So this is roughly the star of this guy. The idea is that g is roughly the star of f. So these measure electric charges. You take the dual, you integrate over the same cycle, and you will get another number, which is now the electric charge. So, again, the factor of the volume and Q lambda, and this is the letter. Explicit in component, you may think that this is a gauge field that has legs along the radial coordinate and time direction. So these are the guys. Now, in principle, you can introduce four of this and four of this, because you have four gauge fields. But actually supersymmetry tells you that there are constraints. One is particularly important, is a linear constraint on magnetic charges, and is related to the topological twist that I was mentioning yesterday. So constraints. The first one, written in full generality, for a model where you also have magnetic gaugings, would be something like this. This combination, and when you put the index in the correct way, you get something that is in platik invariant, so it's the same in all possible frames. This should be essentially quantized in the units that I'm using. You introduce this object, that is typical in supergravity in black hole literature. You introduce a number that distinguishes between the sphere, is one for the sphere, is zero for the torus, is minus one for a Riemann surface of genus, strictly greater than one. So essentially it's telling you if the curvature is positive, is a sort of normalized curvature. Positive for the sphere, zero for the torus, and negative for the Iger genus, Riemann surface, as we can always construct as the hyperbolic plane divided the discrete subgroup. So you see that this particular combination should be fixed. In our black hole, there is no magnetic gauging, and you get an equation that tells you essentially that since also all the g's are equal, I call it all of them with the same letter g, this equation tells you that the sum of the magnetic charges is minus this k divided by g. So it's fixed. The sum of the magnetic charges is fixed. What's the physical meaning of this? The way supersymmetry is preserved I was mentioning yesterday, and from the point of view of physics of the dual filter is the topological twist condition. So it's the condition that tells you that the variation of the gravitino, which is roughly of this form, well, let me put also correctly that there is an index, there are two gravitinos, so there is an index here, and the correct version of this equation is something like this, that a be epsilon b c epsilon c. Where this epsilon is the antisymmetric tensor in two-dimension, and the other epsilon is the spinor of supersymmetry. So essentially it's spin connection and a mu that couples to the epsilon. So probably I should put an i here because, yes, I should put an i here. Ok, and probably in the notation that I'll be using tomorrow, the plus here. Ok, so what's the logic of this? Well, you see that the gauge field is in profile for the magnetic charge part, which is given by the p's. So here you will find the p's. So you will find the background monopole here that is proportional to p. This is g, so you get precisely this combination here. g times p is the coefficient here, and the trick that we were using yesterday and this is how supersymmetry is realized is telling you that this guy cancels precisely the spin connection on the Riemann surface. So the condition that supersymmetry is realized with a constant spinor is precisely this linear condition on the magnetic charges. So you have four gauge fields, the linear combination corresponding in the sense to the r symmetry of the field here and this is fixed. Only the sum. So you have three other guy, three other p's, three independent p's and you can play with those guys. Ok, what? This amu is not globally well defined. Yes, also the spin connection. No, you may think that your epsilon is a section of a line bundle on the Riemann surface. Amu is a gauge field on another line bundle and you choose the line bundle that cancels the other one. But also epsilon is a spinor so is a section of a bundle. Everything is a section of the bundle, ok? So this is in a local chart. So it makes sense globally. Just think as epsilon is a section of the spin bundle. Is a line bundle on the sphere or Riemann surface. Then you twist with the u1 bundle associated with the gauge field and you choose the bundle and then you get a constant. But everything is written in a particular set of coordinates and if you want to change the frame or change set of coordinates everything transforms in a consistent way. Ok, so this constraint leaves three independent magnetic charges and physical meaning is the twisting condition. Yes. Or in some units they are integer. I will tell you in which units in a moment. Yes, they are proportional to integer with some factor depending on the curvature. They are not yet the integer. But yes. Both p and q should be in some lattice because of direct quantization condition. Say it again. In the same units the right hand side is an integer. Yes, I show you in the next formula. Luckily enough I wrote the right formula and it is an integer. Ok. I come back to this question in a second. Not in the twisting condition. So this one was a simplified version of the BPS equation. The electric potential enters I think when you try to solve the equation of motion for the black hole for arbitrary value of the radial coordinate. Then this epsilon I should say more precisely when I say epsilon is constant is constant on the Riemann surface in gen or at the horizon if you want. When you try to solve the equation for the BPS equation you will find a radial profile for this spinor and in the equation for the radial profile you will find the contribution of the electric part of the gauge field. It is only on the sphere and Riemann surface. Yes. Sorry, I should have said more precisely. The second one is curious. I don't know why there is such a constraint but there is, well I know why there is but as curious as I was mentioning yesterday there is also an ugly nonlinear constraint I'm not trying to write it is in Almagipepa you need some quartic invariant of gauge supergravity to write it in full glory but there is a nonlinear constraint among the charges I will tell you in a moment where it comes from and also this this constraint if you want you can make this constraint one electric charge. This is 3 magnetic plus 3 electric. This is the set of parameter the black hole depends on. And these charges are quantized and let me write quantization condition so you can check that this equation is consistent with p being an integer and the quantization condition you can work it out. I'm not doing this not difficult but it takes some time in fixing proper notations and everything and is essentially this one so for all possible these index now are not sum over so you need to take this equation for each set of indices and you will get this dera quantization condition where eta is again quite standard in the literature is this combination associated with the genome so it's an integer this is for g different from 1 and for g equal 1 this is just 1 you need to separate the case of the torus so I think now if you plug it here you will discover that this is consistent oh no, one of the two is q sorry the denominator is g here you have the electric charge good question let me think I don't think so now I'll write a formula and tell you how it comes from looking at the formula I think no probably not the factor of g but it doesn't affect the constraint so Stefan told you about the attractor mechanism so the scalar fields are fixed at the horizon as function of the charges and luckily enough for this class of black hole there is a very simple way of writing these attractor equations oh yes in principle actually when I say that 3 plus 3 independent parameter is not that for all integer there is a black hole now I will write a formula for the entropy for those integer values such that the entropy is real and positive you have a black hole otherwise but the full condition are very complicated so you need to check roughly there are 3 parameter plus 3 that you can vary but if you choose randomly set of integer entropy is negative or not real hey this is super gravity so no well no I think here the charges can be 1, 2, 3 there is already the cosmological constant that makes the black hole macroscopic so no I think in the super gravity limit you are not thinking that the integer are large the integer that appear in the quantization condition can be 1, 2, 3, 4 and magnetic one you mean if there are you can find integer solution or just approximated one good question so for when you turn off the electric charge for sure when you turn on the electric charges you may have constraints also coming from the fact that you have a non-linear constraint that you need to find integer solution then probably the point that if you take the charge big enough is a good approximation ok ok so luckily enough this tractor equation has been written quite explicitly and also very simple to write this I think has been done by Dallagat and Necky while the same equations be written by Stefan but in this notation that you are for example finding the paper of Dallagat and Necky you can define a particular function of the scalar fields remember that the scalar everything is fixed in terms of a potential which depends on this section which parameterize the scalar fields so don't trust me completely with sine, i's and so on is one day you need to do a notation please check all conventions I'm not completely sure that the i is correct but so sorry this is q this is j instead so the denominator depends on the phi heliopolis the numerator depends on the charges and this pair here is given by the scalars and this f lambda in use also by Stefan is the first derivative of the prepotention so if you have f you construct this object here it's a very simple object and the point is that in order to get the value of the scalar fields at the horizon you just need to extremize this quantity it's a very simple, not algebraic but simple function and you need to find the solution of this this equation you need to extremize why is that, you need to do a computation it's not in particularly deep it follows from the bps equation you repackage in a nice way and you use a little bit of special geometry and you find that you can repackage in this way just bps equation you look at the horizon they become essentially not algebraic but a set of equations for the value of the scalar fields the point is that bps equation contains the derivative of the scalar but at the horizon you set the derivative to zero the gauge fields are pretty simple at the horizon and everything works out nicely and that means you need to extremize this quantity engage supergravity with only vector multiplet and the value will give you the scalar fields at the horizon but not only it also will give you the entropy of the black hole because part of this game is that the same quantity determines the entropy which is a function of the magnetic and the electric charges and the formula is pretty simple there is some factor of Newton constant the volume of the Riemann surface but basically you just take the function extremize the value of the function is the entropy in suitable units yes the same entropy function practically yes depends but the mapping is not trivial so the same extremization give you yes I think there are papers where all these things are compared you should ask Kirel but not immediately so these are different coordinates slightly different way of repackaging the same information in the supersymmetric way I expect that the same entropy function well I expect I'm pretty sure the same entropy function is the same as this yes but the question of motion the same minimization yes yes depends how you yeah so there should be no trivial mapping between ok so there is a caveat here that has to do with the second constraint where this come from well if you try to extremize this typically you will get complex solution for the x the x are complex scalar so not obvious that the result is real the entropy should be real so in addition to the extremization you have an extra condition that is essentially the following the result that you find at the end should be real this is an extra condition that come from BPS if you want mix the black or regular and if you look at the details you discover that this second condition is equivalent to this constraint on the charges so this constraint of the charges just the constraint that tells you that this extremization gives a real value for the real positive value for the entropy and second remark is the following here is this is written in terms of homomorphic section they are not independent object in principle you can parameterize in terms of a scalar field so the ratio if you want but it's not strictly necessary when you have this equation you can take the derivative with respect to x you may say okay there is one more x so hopefully one equation is linear dependent on the other and this is the case because you can check easily since f is homogenous of degree 2 that the numerator and denominator are homogenous of degree 1 so the function is homogenous of degree 0 so the four derivative is related to the x are related one to the other so if you satisfy three of them the fourth is automatically satisfied so you don't even need to put a explicit parameterization for the scalar field you just take derivative with respect to the sections and the explicit expression for the entropy can be written but it's quite complicated let me write in a simple example just to show you that it's not so simple as a syntotically flat black hole in the super gravity regime the wafas storming a black hole as a pretty boring entropy square root of the product of three charges here is not if you take for example three independent charges magnetic charge is equal let's call it p so that the fourth is given by minus one over g I think on the sphere minus two minus one over g minus three p and zero electric charges in the possible case just one parameter the entropy is proportional to this quantity here and this is the point where the throttle mechanism is interesting and useful in comparison with field theory because if you do a field theory computation you get a result then you need to check that your result is this one which can be very complicated for the black hole with three magnetic charges and three electric charges both fours but luckily enough we will see that the throttle mechanism contain precisely the same information as field theory so you can just look at the throttle mechanism without explicitly computing the entropy which is in general is a mess last remark before starting with the field theory part is the following has been already made by Stefan but it's important conceptually also for the field theory part so this black hole well for large radial coordinated ABAs as ADS so for example for the for all of these guys the boundary matrix for very large of ADS form this is the boundary once times the there is the radial coordinate there is also an horizon and near the horizon the asymptotic is the usual one for supersymmetric black hole so there are in the full expression there are functions that depends on R so this same function goes like 1 over R squared for large R and there is a zero for at some finite value the time component is similar and then up to constant here I'm not very careful with the constant the function that multiply this piece become constant at the horizon so that here you can recognize another ADS this time ADS2 and the horizon metric is ADS2 times the Riemann surface so this somehow suggest in the language of holography a redromalization flow so whenever you see a solution interpolating between two different ADS you say in field theory there is some redromalization flow that is going on in this case is a redromalization flow between many fall of different dimension ADS4 and ADS2 but you start with a three dimensional theory you put the three dimensional theory on the surface so you are actually compactifying it and this reduce the three dimensional theory to a quantum mechanics so the quantum mechanics if at some limit the quantum mechanics become conformal super conformal it will reproduce for you a factor of ADS2 so if you want from this point of view of the bulk you can also interpret the black hole as a renormalization group flow that tells you that this compactify three dimensional theory in some sense in the limit of large time more always confuse what's the meaning of renormalization group flow in quantum mechanics but say that in the limit of large time correlation function gives conformal correlation function in some super conformal field theory at least this is the expectation of the holography picture and in the language that we were using yesterday counting states at this point is clear what I'm counting there is a quantum mechanic there will be a set of ground states I need to count the set of ground states of these quantum mechanics and here is where I can use localization so let's go in the last 15 minutes to the let's start the filter part we go back to localization so that I have a three dimensional field theory yes so do you expect your quantum mechanics to be just to say something in the same class of SYK type I have no idea I will show you a formula for the index is a mess depends on many different sector topological sector on the compatibility space I don't actually know but I would like to know what kind of quantum mechanics is here so there are many super conformal quantum mechanics is not well understood in particular ADS-2 CFT-1 is not well understood I expect that is a for example if I take just a begin twist it and I reduce it to you think this is the theory this is the theory in some limit there should be some super conformal quantum mechanics maybe just in the larger limit it could be that it is in the larger limit when you go to finite n you have a picture like SYK SYK with a gap so what I will do is essentially well the idea is that I don't know what is the super conformal quantum mechanics could have a lot of flat direction because it is conformal I will regularize it in a particular way I will count the index of the regularize quantum mechanics and then I will take the limit where the regulator goes to zero what exactly this quantum mechanics is here and what form it has is not clear in this entropy function I will send as the idea that these quantum mechanics are very special because they have just a bunch of ground states and then a gap this could be the case honestly I don't know because I have no control on what exactly is this quantum mechanics I don't know honestly I don't know I couldn't imagine that well given this picture of SYK I could also expect something more some funny limit of models with gaps but there could be a limit where this gap goes to zero I think in some of these melonic models I saw similar things, there are parameters then at certain point you can send to zero, you don't know what they are doing and you don't send it in this melonic model but yes so you would expect that there is a large number of ground states in equivalence that could be an option I don't know let's go back to this question when I write the expression for the index so you will see how it's constructed so the boundary is time time's I will go to the Euclidean and I was discussing yesterday I will compactify time on a circle of radius beta because what I want to compute is something like this I want to compute as I was discussing yesterday a grand canonical partition function which be that this will be a trace with minus 1 to the f the Hamiltonian of the theory a charge this is the operator associated to the electric charge multiplied by a chemical potential because remember the black holes are the ionic both magnetic charges and electric charges magnetic charges define the theory you are considering so this is really not just the conformal field theory there is this twist and I will discuss in a moment this twist depends on the choice of magnetic charges so I put the magnetic charges here the electric charges you need to count states with given electric charge you introduce the electric charges with a dual chemical potential so the picture is that the magnetic charge has explicit parameter the electric charge enters through a chemical potential and this is just the the supersymmetric partition function on s1 times sg now I need to tell you how to compute to localization I will not give you all the details the story is always like in Francesco Lecce but I will tell you the structure of this partition function and I will give you some interpretation of it actually this partition function can be computed in many different ways so for example you can find already you need to read very carefully the paper but you can find already in papers by Nekras of a Shatashvili using this gauge beta correspondence so they they can reconstruct this function here and you can compute using localization I will give you the result because as I show you contains also the other obviously the result is the same so in some sense it contains both ok so let's in the last 10 minutes set up the problem this is what I want to compute essentially partition function on s1 times sg but remember sg is topologically twisted by the charges what this means and we start with a general and equal to 2 theory in 3D tomorrow we will specialize to a BJM and let me remember you what are the supersymmetry multiplet there is a vector supersymmetry multiplet which contains gauge field, fermions a real scalar and the real auxiliary field d you can get this vector multiplet by a dimensional reduction of n equal 1 supersymmetry in 4d you know that there is a mu, lambda and d, that's the content you reduce on a circle the component of the gauge field along the circle become this sigma so this is the dimensional reduction of n equal 1 in 4 dimension and the other interesting multiplet is the chiral multiplet which is again the dimensional reduction of the chiral multiplet of n equal 1 supersymmetry in 4 dimension here there is nothing that you get in addition because you don't have any vector so it's just complex scalar fermion and auxiliary complex auxiliary field all these fermions here now are Dirac fermions in Lorentz and Sikreshe Dirac spinos and you have to double everything and so on so it's important to have a theory with an asymmetry so forgive me for my unconventional notation these are the notation we use in the paper and I was too lazy to change consistently everywhere so I'm giving charge minus 1 to the lambda, just a choice not in particular important and it will not play a particular role here so suppose that you have an asymmetry where the geginotransform and and the matter fields also transform if the scalar transform with a charge the fermion transform with the same charge minus 1 and it will be important for quantization condition I need to consider theories where you have an integer symmetry so the charges of your fields can be integer essentially because you want to put the theory on a sphere on a Riemann surface and the charge will be a charge on the Riemann surface and you need to quantize it to have a consistent theory so you need a well defined line bundle associated to the asymmetry charge so the background that I want to turn on let me do it first for S2 times S1 yes or I could start with an equivalence but usually in localization you just use a super charge and when you do localization you just need a multiplet in your theory the super potential is not relevant so when you do localization you typically take the minimal amount of super symmetry you compute one loop determinant for vector and so on so this is likely more general but for application today black or I was discussing before you could have started with bigger super symmetry but at the end of the story you choose one component of Q to do localization so it's enough to start with an equal to 2 and this is more general and also when I will discuss ABJM I will discuss ABJM in n equal to 2 notations so everything will be well suited with this with this parameterization so the background so the metric is the metric on S2 times S1 so let me take flat metric on on S2 actually you can take a more general one I'm not doing that but this is a topological twist if you know a little bit about topological theories means that they are independent of the metric so if you start with something different you will get precise in the same formula and then there is a circle of radius beta is the metric and once I choose the symmetry here I'm turning on a background field for the symmetry which is just a unit minus one unit monopole depends in some is essentially a unit monopole with some assignment of charge and this is needed to solve the equation of supersymmetry indeed the picture in the on the boundary is precisely the same as the picture in the bark so as Francesco is playing clearly for example in the approach if you want to preserve supersymmetry you say ok I take my field here a couple to some supergravity I freeze the fields in the supergravity multiply to that particular value and I need to check that the variation of the supersymmetry condition in particular for the gravitino which is the background field are satisfied but the variation for the gravitino well essentially in this game are precisely the same that I wrote the only thing that I'm turning on is the matrix and the asymmetry so the transformation of the gravitino when you couple to supergravity is precisely the same expression gamma A B let me write in this way now I will tell you also the convention for the gamma and there is a mu R so let's take we are in three dimensions so in the Euclidean we can just safely choose the gamma the three gamma to be the polymetrices the three polymetrices so if you for example so you need to say what is the charge of the epsilon under the asymmetry I conventionally give charge to minus one to epsilon I put a plus instead of a minus here and so you can see that this equation now you can do your exercise you compute the spin connection on the sphere, it's very simple you plug in here everything and you will get something of this form clearly here you get essentially take the index one and two along the two sphere index of frame and index three along the circle this will become something like mu here you have gamma one, gamma two that is i times sigma three epsilon and here you just have i mu R epsilon now mu is this guy here you compute your spin connection and you discovered as I already told you yesterday this is pretty obvious there is not so much on the sphere when you compute the spin connection you get again the monopole actually the monopole with charge two so that you see that if there is a sigma three here it's enough to choose a negative value of sigma three so if you choose a spin which has zero component on the bottom and this generic plus here this cancel precisely this one with this particular choice of sign and as I was mentioning yesterday you are just left with the condition that the mu epsilon equal to zero so that you can take a constant this time really on the boundary and this is constant on the sphere on the surface on time is a constant spin and this is if you look at old nature you can twist in the sense of written you twist the nature of the spin or you transform essentially in sort of scalar the mu epsilon is the natural equation for a scalar and the same trick work for arbitrary remain surfaces on sigma g you just choose your gauge field such that it cancel the spin connection so with this normalization you take minus twice whatever you get from the spin connection you turn on the mu now now using all the literature topological twist method whatever you want you can write the Lagrangian I will show you the Lagrangian and then we stop there with this the Lagrangian again is long so we decide to write on the slide ok, good this is the Lagrangian in supersymmetric transformation just to give you an idea that there is a Lagrangian I will not use much but there is a Lagrangian supersymmetric transformation you can check everything look at the Lagrangian it's pretty boring actually so on top there is ok, better? ok, let's start with the first line F mu nu square, boring kinetic term for the scalar field boring, d square you know that there is a d square yam meals direct term and a coupling between sigma and lambda but it's there also in flat space then there is another object that I will need, ABJM I will tell you what it is in more details later but contains transimond terms this is the supersymmetric transimond in 3D, this is the standard transimond and you see that supersymmetry give you a sort of mass for the Geginos and the coupling between d and sigma and let me make it slightly smaller to know that you can also see the matter Lagrangian if you have an arbitrary chiral field coupled to gauge but it can be an arbitrary representation again there is a kinetic term there is the F term there is the Dirac equation there are standard Yukawa coupling the only thing that I want to mention the only thing that the twist does is essentially coupling this notation is the curvature of the A mu so it's the asymmetric curvature here and then there is d5idega this you know is standard in supersymmetry is how the detail couples to the phi but the thing that I want to mention is the following look at sigma sigma is the real scalar in the vector multiplet look at how it appears in the matter Lagrangian it give a mass to the scalars give a mass to the fermion indeed this sigma play the role of a real what is called a real mass in 3D that's a general picture so masses in supersymmetry theories you can well if you have enough supersymmetry you can introduce it by as background fields for some background gauge symmetry so suppose that your gauge field is not really gauge the coupling is zero so there is no kinetic term it becomes a background field you can give a value to the various fields in the background like in testucha cyber and if you do this game for a global symmetry and you turn off the coupling to make global but you keep a zero value for sigma what you get is that is the same theory as before with a real mass for the scalar so sigma remember is a real mass and you see that pretty much looks like the Lagrangian in flat speed it's not so much topological twist is boring as I was mentioning before it's interesting it's just transferred the spinor in scalars but it doesn't match to the Lagrangian in particular there is no there is no curvature coupling so the radius of the sphere the radius of this one are mostly relevant in this game and just to conclude let me also show for completeness the variation of fermions these are written here illuminating the only thing that I want to mention is that remember that now you see lambda dagger but you need in Euclidean to consider these guys as independent you see five dagger they are independent then when you do the pat integral you need to choose a real section but here are essentially independent and also epsilon you have epsilon and epsilon tilde so you can construct a q and q tilde and pretty much they act on the field in sort of homomorphic way so look at q acting on the fermion so q acts on lambda but gives zero on lambda dagger so acts on half of the field so they say the homomorphic part q tilde does the opposite that's a typical feature of topological twist of many theory in Euclidean signature ok so tomorrow here a stop and tomorrow I will tell you how you localize that's pretty simple we will see that essentially this I can really tell you briefly they matter Lagrangian and the yamist Lagrangian are q exact so they will go away you will get a classical piece from the transformers and they will be determined to compute and I will discuss the one loop determinant tomorrow