 Daj. Zvom, ki sem poznala, sem poznala, da je vse klasifuje supersymmetrično geometrije, ko vse izgleda, od tajsvom z rovnju spinor izgleda z minimi guts, in drugih, da sem poznala, da sem poznala, korespondin to a left-ended spinor, whenever the manifold is complex. Or you could also have a right-ended supercharge when n is complex, but if the complex structure is anti-serve tool. I should point out some references about this. So there is a paper by Claret Tomasiello in the Farone, FTH, and also another reference by Dumitrescu, Cyberg and myself. So one comment that we should make about this case of one supercharge on some generic complex manifold is that we have to turn on some, generically we have to turn on trivial U1r gauge field and therefore the resulting theorem might only make sense if the r-charges of the fields are quantized in some way. So for instance it definitely makes sense if all the fermions have r-charges which are odd integers and the bosons have r-charges which are even integers. But depending on the manifold it might be that other quantizations are allowed. So generically the r-charges should be quantized. And then the other comment is that there are also manifolds which we don't have to turn on some non-trivial U1r bundle. One example is a three times this one. And then the r-charges of the fields can be whatever they want. But there might still be more data that one has to give, one has to choose some spin structure or some other similar discrete data. We also discussed a little bit what happens if one requires more supercharges. So I am not going to comment very much on the case of two queues with the same corresponding to two left-handed spinors or two right-handed spinors but you can find some, well, the relevant discussion in this paper. As I said, in the non-compact case we don't have an answer for what is the classification of the corresponding manifolds and for the compact case one is left with very little namely, like the four torres with the flat metric, the s3 times this one with the round metric and the k3. So that's a very short list. And for the case of s3 times this one there can be some quotients of it. And the other case I looked at just yesterday was the case when one has one solution zeta and one solution zeta tilde. So the four correspondingly there will be one charge, one supercharge of r charge minus one and one supercharge of r charge plus one. And then as I said this case is better analyzed thinking about the killing vector zeta sigma mu zeta tilde which turns out to biolomorphic both the complex structure arising from zeta and the complex structure arising from zeta tilde. And I also brought down the form of the metric which is the most general metric compatible with this structure. Ok, so next we'd like to make some discussions about background vector multiplets. So as we discussed when your theory has some u1 symmetry some flavor symmetry then you can couple it to background gauge fields. So what I'd like to discuss is suppose we are on a manifold which allows for one supercharge q which is corresponding to a left handed spinor so then let's suppose also that our theory has some let's say u1 flavor symmetry we want to know what kind of background gauge fields we can introduce on top of this background which would preserve the same supercharge. So we have already discussed what the background gauge fields so in one of the past lectures we already discussed what is the relevant equations that we have to satisfy and so our background gauge field is comprised of the gauge field mu and an auxiliary scalar d there are fermions but those are set to zero in the background and the supersymmetry the fact that this background preserves supersymmetry stems from the variation of the gauge genus which should be zero and that gives we only have one zeta corresponding to one q we only have one equation that we have to satisfy where f mu nu is the field strength corresponding to mu so this should be zero and now using the fact that you know that there is a complex that this z gives rise to a complex structure you can rewrite this equation as telling you that the anti-olomorphic part of f and that very good I can just am unable to copy from my own notes then ok so we have so this is the first condition and the second condition is that d will be determined by contracting the field strength with the complex structure ok so if we are in a situation where there is also another super charge and we want to our background gauge field to also preserves that one then we would have more restrictive conditions so let's give an example so in particular there is some interesting family of complex manifolds which are called op surfaces so let's talk about op surfaces so these are complex manifolds that are difumorphics to s3 times this one and they can all be described as the a quotient of the two-dimensional complex plane the c2 minus the origin by secret group and now we can describe what this quotient is so there are two cases the first one which I think goes by the name of primary op surface so we will introduce complex coordinate w and z on the two complex planes and then these are identified as pw and qz p and w are complex numbers which are subject to this constraint so they are less than 1 so this is a contraction and p is less or equal than the absolute value of p is less than equal than the absolute value of q and they are both greater than 0 so this give rise to have two complex dimensional family of complex manifold all difumorphic to s3 times this one and p and q are the complex structure moduli of this family so different p and q correspond to distinct complex structures then there is so this is something that we will discuss more then there is the other case which is and I want to have a lot to say about this maybe we can have some discussion after the lecture with people that are interested so in this case the identification is the following so there is a positive integer n and q again has to be such that this norm is in between 1 and 0 lambda is any complex parameter and n is a positive integer so natural number so one comment here is that it would appear that again this would be a two parameter family of different well that the complex structure would be parameterized by two moduli one being lambda and the other one being q but that is actually false because as you can realize very easily if I rescale the coordinate w and z I can rescale away lambda so lambda is a fictional parameter it can be whatever it cannot be 0 so maybe I should take out 0 from here but otherwise you can set it to be whatever you want but q is instead complex modulo and there are several of these branches and they are like indecised by this integer ok so for the let's discuss a little bit more the primary case so case a so we can write down what p and we can give some parameterization for p and q in terms of some real positive number beta and some angle theta and similarly for q oops e to the minus beta q plus i theta q so again because of this conditions we have that beta q and beta p are positive and that they have to satisfy the following relation while p and theta q are just angles and then in terms of this in terms of this beta p and beta q we can write down some parameterization for the coordinates w and z so we can write w as e to the minus e to the minus beta p plus i theta p times x times cosine of theta over q e to the i phi and z will be similar e to the minus beta q plus i theta q x times sine of theta over q and then there is some other angle so where phi and chi are angles in between 0 and 2 pi and theta goes from 0 to pi so you can check that this is a parameterization of c2 and under this under this identifications what you have is that x is identified with x plus 1 so x is a coordinate along a circle and a while phi chi and theta parameterize s3 so indeed this this thing is well defined because so this gives a parameterization so when you impose these conditions and the fact that these are angles this parameterizes a fundamental plane under this identification and you can check that this is like indeed as 3 times s1 because it satisfies e to the 2 beta p x times the absolute value of w square plus e to the 2 beta q x times the absolute value of zeta square equals 1 we can also write down a metric on our up surface and we can actually check that we can write metrics which are of the form that I wrote yesterday for this manifolds in which you can preserve two supercharges, actually not just one of opposite R charge so indeed we can write something as follows and you can check that this metric is invariant in identifications that are written above so this is a good metric on the up surface are there any questions so from this there is a killing vector dx this just corresponds to translations along the s1 and you can also write down anulomorphic killing vector actually I am going to write down the corresponding one form which therefore will look anulomorphic so this is the anulomorphic killing vector that we called k yesterday and you can indeed check that k square is equal to 0 and that it commutes with k bar so then you can also check that by using this k and this formulas that I gave yesterday reproduce you can find out complex structures which do correspond at least one to that thing over there ok, so we discovered that we can all on any primary of surface so on any primary of surface let's call it mpq we can preserve two supercharges q and q bar with oppositor charge and now if you think about this a little bit more you can convince yourself that these parameters p and q's are actually the geometrical implementation of the two parameters p and q which appeared in the formula for the index so remember we had a formula for an index this was a trace of minus 1 to the f times e to the minus so the partition function z on mpq will be the index that we defined yesterday with fugacity is q and p and moreover if I have some other flavor symmetries then I could add background gauge fields along this one and those will give other fugacities to the index yes, so there is there is a question about ok, let's be maybe a little bit less so there can be subtleties with perfectors ok, so now I wanted to say this just because somehow is I think a good introduction to the next topic of discussion which would like to understand how various supersymmetrical observables depend on the geometries of these spaces that we can place them in so for instance there is one choice of metric which works I could choose many many other metrics which are compatible with any given complex structure on some hop surface but this argument tells me that in the end the result is just going to depend on this p and q which are the modulate that characterize the complex structure and moreover it's not just any dependence on p and p of complex parameters and they appear holomorphical in the answer so, well this tells us that maybe a good hint that the correct answer is that at least in this class of theories the partition functions will depend only on the complex structure and not on say the choice of metric and moreover that the dependence on the complex structure should be holomorphic in the complex structure modulate so, given this example this is just a wild guess but as we will see this is actually what is going to happen but in order to make a sensible state but we need some definition of this index where both q and q bar will appear on p and p bar and then we will somehow demonstrate the dependence on p bar and q bar but the formula which you wrote last time only contained p and q so there was no question could you just write again the definition of I yeah, I only contained q and p I agree yes, yes, yes ok, so if you want this partition function like which in principle could depend also on p bar and q bar like turns out to be holomorphic and and then I guess that tells us that this index does not have dependence on I mean I don't know, I guess this comment maybe if I could ask a phrase question differently so how do I know that the partition function in the manifold mpq is actually an index of something yes, that's right I would like to see a circle which is separated and then some so so you have a separated circle so you could define a theorem whichever squash fears remains but does pq act on this s3 what do you mean act they define the geometry of the s3 they define the geometry of the s3 so once you reduce along this one so you will consider some trace of the Hilbert space of states on some squash sphere where the geometry of the squash field depends somehow on p and q now I'm not sure it depends I think it will depend on the ratio between p and q bar if I remember correctly and and then where the other parameters enter is in the way you identify the fields it's going around the s1 so then is that trace on the Hilbert space which itself depends on p and the operator which acts there also depends on p that's by some miracle which is not let's say in the elliptid genus where we have this standard Hilbert space an operator which depends on q bar but the index does not depend on q bar so that's more subtle any other question on this before I go into ok, so I guess it's time to erase I don't want to erase that ok, so I'm just taking this as a definition so this is c2 minus the origin quotiented in this way so the interesting fact that I actually is that like these are all these are like this classification there is like a classification of all the four-dimensional complex manifolds with a difomorphic to s3 times s1 that is not very obvious statement I think it was due to codira ok ok, so we would like to understand how this partition functions or principle some other supersymmetric observables depends on the geometry of the space say m so first of all what could it depend on for instance it could depend on the choice of metric or because we are dealing with complex manifolds it could depend on the choice of complex structure moreover I'm not sure you remember this but at some point it could depend let's say on other background other sugra background besides the metric for instance we had a formula for v mu this conserved background one form which appeared in mm supergravity and this was determined in terms of the divergence of the complex structure plus some piece which we called u mu and we said that u mu had to be polymorphic and conserved so now the question is can the partition function depend on what we decide to use as u it's a freedom that we have to that we can make if you want some non universal coupling to the geometry and you need to know if the partition function depends on it or not and moreover we could also have background gauge fields so we could also ask how does it depend on background gauge fields or other backgrounds so in order to study these questions so I will not present here the more rigorous proof I guess which I can tell you about if you are interested but one thing you could do is somehow try to introduce appropriately twisted fields, well at least for theories with some Lagrangian descriptions which will be like forms with which will be olemorphic or antelomorphic depending on the fields you are dealing with and then rewrite the Lagrangians that come from supergravity in terms of these twisted fields and check that somehow all the terms, if I do a variation of say the metric keeping the complex structure fixed then this is always q-exact and so on instead what we'll do is something more pedestrian which is we'll just consider this manifold to be the manifold where we are to be very very large so that it's very given point it's more or less like flat space and then we will analyze this perturbation from flat space using linearized supergravity and then you could say well this doesn't prove anything and well indeed it's just an indication that the results that I'm going to present are correct but well if one weighs one's arm very robustly one can say that the fact that we have a scalar supercharge means that we can just analyze this problem near flat space and that should be enough okay but this is maybe an argument which is not very it's not very good so you can also do this other twisting procedure which however I don't have time I think to discuss okay so so the idea is that we take this manifold to be very very large and then we study how the Lagrangian of our theory responds to this small change of the geometry and we know what it does so there will be the change in the metric which will come to the energy momentum tensor then we will have the change in the connection for the asymmetry which will couple to the conserved arc current then we will have the change in this conserved vector field V mu which will couple to the connection which would give rise to the string to this f which enters into the string current and well then if we also have auxiliary if you also have background gauge fields then like we could imagine varying those and this will couple to the flavor currents and this will come with the variations of the auxiliary components of the gauge fields which couple to the appropriate scalar operators in the linear multiple for the conserved current so these are terms that arise in background gauge fields and those are terms that arise from coupling to supergravity at the linearized level so again these are not really independent because we want our geometry to be super symmetric so that implies that delta mu and delta v and delta g are all related to each other in the way that we discussed yesterday so there are explicit formulas v, a in terms of the geometry ok So what about the domain wall current? Yeah so that, no the domain wall current is not there because we are considering theories which couple to which have an R-multiplet and which couple to minimal supergravity if you were to redo this in old minimal supergravity well then in this you will also have this complex scalar which couples to something related to the domain wall current Ok So any question? Ok so that's the logic that we would like to pursue Ok so before going to the more complicated part which is understanding what happens when one changes the supergravity background let's make some let's see what one can say about the case of the background gauge fields which is somewhat somewhat easier so so we have to remember the formulas that have wrote up there so those are the conditions for a background to be supersymmetric so that means that the antelomofic components of the field is trying to 0 and d is determined in that way so f i bar j bar is equal to 0 and d, if I write it in complex coordinates will be equal to minus 2 i f w w bar plus f z z bar Ok so these are the conditions for the background field to be supersymmetric and ok locally ok so there is a there is a Poincaré theorem, lemma whatever it's called for d bar operators so locally this thing means that the corresponding gauge field a i bar is d bar of some scalar and if I am allowed to apply the flavor gauge symmetry then this would mean that a i bar is locally trivial so I could set it to 0 locally by using complexified gauge transformations now we want to very little bit this we want to consider a small variation of this background we could start with just a flat background so there is no gauge field well then the conditions are that I don't need to rewrite them they are just that the field strength of this small variation is equal to 0 so d i of delta j bar minus d j bar of delta i is equal to 0 but not all these variations will be will be important because some of them are just trivial those are the ones which correspond to actual gauge transformations so the ones which are trivial are the ones where a i bar is of the form d i bar of lambda so ok this basically means that the this deformation of this background gauge fields are classified by looking at the homology of the bar so ok so now we want to see like we want to write down these terms and check which one of these are q-exact and which will and therefore will not change the answer for some super symmetric observable and which ones will not ok so im gonna give some more details for the case of the gauge field because it's amenable to be done in a short amount of time and then I will repeat more without many details the super gravity case ok so you need to remember what are the objects which couple to the to the background gauge fields so again these are objects which live in a multiplet which contains a scalar j and there are fermionic operator j alpha and j alpha dot and then there is the conserved flavor current so this thing is such that d mu f mu ah d mu g mu f equal to 0 ok so now we can look at how the super charge acts on these operators that again just follows from standard formula so we have our super charge which is determined by the spinner z and when it acts on the scalar component it gives i zeta times j where j is the spinner and then q zeta over j that is 0 well that is expected because q square is 0 so from this equation and then you have q acting on j tilde alpha dot that gives minus i sigma tilde mu zeta alpha dot times j mu minus i d mu j and q acting on j mu that gives minus 2 zeta sigma mu nu d nu leto j ok so these are all the variations and we can now figure out like using the fact that we have this complex structure that we can build out of z we can rewrite this variation in a slightly more convenient form by multiplying this equation so we take this thing and we multiply it by something proportional to zeta dagger sigma rho divided by this value of z squared which again is well defined because z squared doesn't have zeros and ok so here just by counting you see that there should be two q exact operators and indeed once you do this what you find out by multiplying this thing by that is that so basically the q exact operators so the things that appear here just become just are the antyolomofi components of this object here so let me write it so then j i bar minus i d i bar j are q exact ok so actually you can even check that these are not just q exact so all the q closed operators that you can write out ok so now with this information we can just rewrite this delta L over here so delta L is a mu j mu plus d j and ok so we can do some algebra which is not too complicated and what we find is sorry let me put delta mu and delta d it's not really necessary so and similarly so we have two complex coordinates so we have some other coordinates let's call it z we have a similar expression and then ok so this is a little bit and in order to derive these expressions you will have let me think do you have to integrate by part something or no I don't think you need to yes no you don't have to integrate by part j z plus i d z j ok so now you look at this and they say but this guy and this guy multiply the q exact operators so this is q exact on the nose so this piece here is q exact this piece here is q exact and the only two pieces that remain that are not q exact are the ones here which corresponds to the variation of the antihelomorphic components of A and those are the components which have to satisfy this constraint there is a plus sign missing yes and any other question you integrate by part when you convert delta d into something in the variation delta d delta d is like a plus sign of the projection of delta a ok delta j is you have a relation between d and f yes that oh yes right ok yes then I have to integrate by part so there will be plus total derivative ok so now we have to understand what happens to these two terms so first of all like it would just seem strange that this thing is even supersymmetric because we just claim that all the q exact q closed operators are just these how come that this subject after I cancel these two pieces which are q exact ends up being supersymmetric but it has to be supersymmetric because it comes from a supersymmetrical agrangian so there is you know that that has to be the case but you can do a small exercise and you can check that if I do the variation of these two terms and I integrate by parts by these places then what you get is the following I guess this was expected that's plus total derivatives so q of delta l or delta of delta l is this and then you can see that indeed this thing is zero provided that the background is supersymmetric so it satisfies this relation so indeed you recover at the linearized level the same condition so now we just have to figure out as something so we figure out that the background can only depend on delta aw bar and delta is a bar so now you can again make a clever argument saying oh but my original agrangian so the original agrangian was invariant under background gauge transformations by using the conservation of the current and therefore that means that if I take a variation of my A's which is exact that this object vanishes so this is a general argument it has to be true but let's check that it is indeed true was invariant on the unit relation well we are assuming that we are complexifying so we are allowing ourselves to complexify the flavor gauge transformations indeed if you only allow for real ones then you would not be able to say that you can set such a thing to zero so how do we do this so we set delta aw bar and delta is a bar to epsilon so we set delta ai bar to be di bar of epsilon and then we integrate by parts and when we integrate by parts we eat various terms we have two derivatives acting on j and sometimes we have derivative acting on the current so we set this then we integrate by parts then in the terms that have derivatives acting on j use the conservation of the current so then after everything is done what you are left with is q epsilon times dw of jw bar minus idw bar j plus similar expression for dz plus total derivatives but now the operators that appear here are exactly the q exact ones so you are indeed proved that this trivial gauge changes of the background do not change the answer of the partition function ok, that's everything for the gauge fields what we have proved is that the answer the partition function has to be holomorphic in these parameters that classify the backgrounds because it depends only on delta aw bar and delta is a bar and not on delta and not on the holomorphic components and secondly that it only depends on homologically on these parameters so this argument is still sort of a linear order yeah, yeah, this is a how do you come up with this? ok, so to be fair for an abelian gauge field I think this argument just is fine at all orders just, I mean because you can linearize around some non-trivial background and the equations are gonna be similar for the case of supergravity that's clearly not the case so that as I was saying so either you say that the supercharge is a scalar so working around flat space should be sufficient or you can try to better by really working at nonlinear level for that however I think it's more convenient to introduce twisted fields and so on that's it, yeah I just worry that there might be new dependence coming up if you go to second order so you won't exclude that well, indeed so also in topological twisting you can just either define the theory in using the twisted variable and then it's topological in the usual sense or you can just you can see papers where they just look at the energy momentum tensor in flat space and they check that it's q-exact and they say, okay that's an option okay but yeah, I'm not claiming that this is complete proof, I think you should really work at nonlinear level any other questions? Okay, just want to make sure so actually this argument shows that you don't need to assume that the field is invariant because of the flavor gauge fields you just prove that the variation Lagrangian under the deformation of the gauge field which corresponds to the complex gauge summation is q-exact okay, yes because the measure is not invariant in complex factor gauge summation that will not be good argument this is a good argument let me think about it okay, yeah, I think that's correct indeed, so you want to set it to zero yeah, so if you say if it's there, then it's q-exact so that is still a couple of topomorphic line bundles for the flame symmetry that's a obedience in a way right, okay, so so now we can progress to the gravity case how much time do I have, it's still 12 half an hour, more or less okay, good, I think that should be sufficient okay, so to understand what happens in the case of gravity you have to work a little bit harder so people go to gravity so just can you just repeat if my flavor symmetry wasn't on a billion well, if the flavor symmetry weren't on a billion then okay, so in the holomorphic back bundle yes, it would be I mean, either you can start from zero and then go a little bit away so you could have yeah, okay, that's sure so the current preservation will be some kind of current wait a second, but I'm working locally I mean locally the variation will be q-exact I mean, it's true that it could be around variation is small, but it doesn't change the topology in the bundle sure but since the topology is on q-exact you can't assume that you can set the background h to zero and start there no, yeah, right, okay, yeah so then the current preservation should be the covariant preservation or what kind of conservation of current linearized around trigger background background h, what kind of so you are going to use the I did indeed so this was fine for the abelian case maybe it works two things could go wrong because you use the commutation and the conservation right, and the conservation I love to think about it okay, so okay, so first so in order to understand how the answer might change, you have to understand how do you vary a complex structure, so you can start from like how do you look at variations of an almost complex structure and characterize them at the linear level so the almost complex structure is defined in the following way so okay, so therefore let's just do a linear variation of this equation then what you get is that delta j mu nu times j mu rho plus j mu nu times delta j mu rho beta b0 and therefore this means that delta j with odomorphic indices has to be zero why the delta j with one odomorphic index and one anti-odomorphic index is unconstrained so this is just at the level of an almost complex structure but we have a complex structure so that means that this j has to satisfy this differential equations this which involves products of j's and derivatives of j so we have this tensor at the mu nu rho which has to be zero and now we can do a variation of the complex structure appearing here and work out what it means so setting delta j i j equal to zero because we know that it has to be at least an almost complex structure and see what constraint do we get on these components which were otherwise arbitrary so and basically what you get if you do this computation which is not too complicated is that you can define some objects theta i which are just delta j i j bar times d z bar j bar so there are one forms with values on the odomorphic vectors and then what this condition implies at the linearized level is that d bar of theta i is zero again not all the theta i that you get in this way or if you want not all the delta j's are actually interesting because some variations of the complex structure which you could just make an infinitesimal difumorphism will change the complex structures a little bit but that is not really a change of the complex structure it's just an infinitesimal change of coordinates so you want to know what are those that correspond to an infinitesimal difumorphism so that's a good exercise actually it's just a one line so an infinitesimal difumorphism is parameterized by some vector field epsilon mu so what you want to know is the derivative of the complex structure along this epsilon mu so you want to compute the derivative along epsilon of j mu nu and then that should give you the form of the delta j mu nu which are trivial so you do this exercise and what you discover is that the corresponding theta i's must be d of d bar of epsilon i so again the problem at least at the linearized level has become a homological problem you want to find the d bar closed theta i's which are not d bar of something so this is some Dorbók homology for objects which are valued in the eulomorphic vector fields so it's a so you want to find these but no doubt by the theta i's which are d bar of epsilon i so again this is only at the linearized level so while it's true that for any deformation of a complex structure you can find the theta i with such properties the vice versa might not be correct because there might be obstruction coming up at higher levels so it's not going to be interesting in such case I'm just working at the linearized level so if there are obstruction then that would just mean that the possible variations are less than what I am considering so are there any questions on this? no, okay that's good actually it's interesting so you can deform Lagrangian interestingly by the first information of the complex structure which more or less happen to be obstructed but in theory it doesn't mean yes that's the argument it doesn't, maybe there are some so if you try to differentiate the deformation maybe in counter terms your meaning is like that the Lagrangian itself should see that the deformation makes no sense the obstruction should appear there I don't know if it's a contact term because it's maybe two of these obstructions don't appear because the obstruction is H2 of the tangent bundle it's like H2 minus 1 maybe the motion 3 may appear maybe, but in any case I said the obstruction should be something that you could see locally obstruction is also logically well, then this argument so this thing which is called differential, actually solves a nonlinear equation debartatical state of square in first linearized solve equation debartatical to 0, find linearized state then you take the square and try to solve that equation so that's the reason there is an obstruction there but then I don't think it could appear in contact term well more food for thoughts but it's true that something bad should be seen at the level of the action as well so we just need one extra ingredient which is that so we also have to talk about what variations of the metric but again variations of the metric means that if you vary the complex structure then we also need to vary the metric in such a way that it stays a mission so one of the conditions that we had for the geometry to be supersymmetric was that the metric had to be a mission so that means that we have to figure out so if we change the complex structure we have to change the metric in such a way that it stays a mission and we have to understand what that means so we have to linearize the compatibility condition for the metric so we can work at first order in delta G and delta J and what we find is there is something which is more or less obvious which is that the components of the metric which have one homomorphic index and one antheromorphic index those are unconstrained and they make a change of the Hermitian metric and that will still be compatible with the complex structure it's only the components which are zero which will have to change due to the change in complex structure so this is unconstrained if you want another way to say this is that they can always do a change of Hermitian metric before doing the change in complex structure and then that changes in the complex structure in the following way so again they involve these objects which parameterize the change in complex structure so now you almost have all the ingredients that you need the other ingredient that you need is that you have to use the formulas that they gave in the last lecture to compute given a change of metric and complex structure what are the changes in delta A that arise so that I'm not gonna show but it's more or less immediate I mean not immediate but it's clear what you have to do bunch of algebra and then after you do that you have to look at similar to what we did with the vector fields you have to look at the multiplet of currents equivalent of these equations except that now here you have to consider all the objects appearing in the current multiplet so it's much more involved and then you can play the same trick you will multiply some of these by some spinors to get expressions like this let's say the sum of the components of the operators which appear in the r multiplet are actually q-exact operators so you will do this then you can rewrite delta L of this data and what you find is that delta L is a bunch of q-exact terms plus various terms multiplying delta j i i bar times some operators of i bar i and then that's it and now you can say ok, so this already tells me that my Lagrangian will only depend on delta j i i bar and not on delta j i bar i so it's holomorphic in these parameters these delta js have to satisfy this delta closeness delta bar closeness requirement and then exactly as before you can make the short argument oh well, like if I consider a variation of the complex structure that is trivial that just arises from infinitesimal differomorphism but my original Lagrangian is is differomorphic invariant therefore that this change better be q-exact and indeed that's what happens or you can like for delta j something which is exact then integrate by part using the conservation equations for the r-current multiplet and rewrite the resulting operatories that arise here as q-exact ones so this analysis is completely parallel to the case of the background gauge field but it's just more involved so the results are that z so another thing that you notice is that there is no variation of the metric here the delta g ij bar did not appear anywhere they appear here so the partition function on M does not depend on the Hermitian metric so on the choice of Hermitian metric compatible with the complex structure and it depends or can depend on the complex structure model but only olomorphically well, olomorphically maybe it's a little bit too much it's let's say locally there could be singularities appearing somewhere indeed if you check for instance the expression for the index that I gave yesterday and you vary p and q those sorts of singularities it's a meromorphic function of p and q is it actually an option that it's not actually a function could there be anomalies so I should also say that all what I said so far is disregarding the possibilities that there are any anomalies so that's actually an interesting research project for whoever is interested just like 20 years ago it's but this is certainly a point that is worth stressing ok, so this is the this is like the main point that that I wanted to make about the dependence of the partition function on this on this geometrical object actually there is well, you could have wondered what happened to the dependence on the background fields as I said there is this this U that appears in the variation of in the formula for V so what you can check is that that U actually it satisfies some so there are some use which are exact in a sense which I have not explained that I can explain if you are interested and then you can show that the Lagrangian does not depend on those variations of U which are exact so again the dependence on this U is only comological ok, so this this concludes this topic so unless you have some question ok, so in the remaining nine minutes I think, is it nine minutes I can talk maybe say something about what happens in three dimensions instead of four so I will have to just keep my short so n equal 1 in with U and R in 4D is simply related to n equal 2 in 3D again with a U and R symmetry and indeed you can obtain a lot of information about this theories by just the dimension reducing facts that you know about the 4D case so for instance this theories also have an R something which is called an R multiplet and this R multiplet comprises conserved R symmetry current then as usual when you dimension reduce you will get also a central charge and there is a corresponding central charge current which is also conserved then there is the energy momentum tensor and then there is also something which is related to the to the string current and so there is a scalar and you can form a string current coming from it and then there are the gravitini so again the structure of this multiplet you can get dimensional reduction from the previous one and also the supergravity looks very similar there is M U R some connection for the U and R symmetry then there will be some connection for the central charge and the metric and there is a scalar H and then there are the gravitini so sorry, did I say this were the gravitini I might have misspoken before but these are the gravitini which couple to the supercurrent and so instead of so instead of working with C it actually is useful to write things in terms of the dual of the field which is which is conserved so this looks like the field V in U minimum and so you can repeat the story that we did in 4D but in 3D and you can find similar results so in particular it turns out that the geometrical structure which is parallel to a complex structure in 4D is something which goes the name of a transverse solomorphic foliation so you can find details in our paper but again a lot of the things that you know about complex structure go through for this transverse solomorphic foliation, there is some notion which is equivalent to the obok homology and then you can try to characterize what are the transverse solomorphic foliations and you find out that they are characterized very similarly in terms of this other thing which is similar to the obok homology and that these moderate spaces are usually finite dimensional et cetera et cetera and then you can repeat this analysis that I described today and you can find out that the partition functions then will only depend on the choice of metric which is compatible with this structure but only on the choice of this transverse solomorphic foliation and mathematicians have classified which manifolds admit such structures and for instance the manifolds with diffomorphic to the trisphere admit one parameter family of transverse solomorphic foliation parameter is complex and indeed that is to be related to the squashing parameters that people introduced on the trisphere except that that's one thing that you can say and actually just on the trisphere as for the op in some sense the trisphere is just what you get dimensionally reducing the op surface and so exactly in parallel where there was primary op surface so this gives this moderate space of of transverse solomorphic foliation which depends on one complex parameters which I think you can say it's p over q bar or something like so and then there are also isolated different isolated transverse solomorphic structure corresponding to the dimensional reduction of the non-primary op surfaces and so okay so that classification goes to exactly exactly in the in the same way okay so for instance one can give some one application of these ideas in 3D which I think is I can probably sketch in the remaining two minutes oh okay but I can allow some time for some questions I guess so the transverse solomorphic structure isn't different on the complex structure? yes it's different than a context structure it is related to an almost context structure and then there is an integral so as for the case of the complex structure there is an almost context structure and that's something similar to the almost complex structure but then the integrability requirement that for the complex structure gets translated in some integrability requirement for this object which makes it into a transverse solomorphic affiliation which basically means that the spaces transverse to the vector which defines the almost context structure have a complex structure and as you move along somehow these complex structures are like behave nicely I mean I could write down the equation if you want but we can also discuss it later so there is some kind of part of this it's actually much easier let's say that there is a counterpart so again also for this transverse solomorphic affiliation you can introduce a coordinate along the vector for the context structure and then in the transverse space you can introduce a complex coordinate and then the transition function is going from patch to patch so it looks very much basically for what I understand we know about complex structure as an equivalent for transverse solomorphic affiliations so as an application of this background supergravity ideas can talk a little bit of how to use them so people computed the partition function for this n equal to field theories so in this parameter b which is this parameter that I interpret as the saying which is the transverse solomorphic affiliation that you are using but the squashing parameter for the real sphere should be real no so the squashing so if you interpret this as a squashing then so there are the definition is not super easy so there are different squashings which in principle are not even related to each other and the different squashings so for instance in some cases for some of the squashing this parameter is real for some others of the squashing this parameter is imaginary so they look at the first site of the related squashings of the sphere and you can even keep the metric round and change the other supergravity backgrounds and get a squashed sphere in which the parameter appears not in the metric but elsewhere and then it's not really a squashing I don't know what you want to call it but there is some dependence on this parameter so if I give you the metric for a sphere which is this parameter you need extra information so is it some super geometry which has a parameter it's equivalent to the parameters p and q which appeared in the op surface so if you reduce that discussion there are two but when you reduce you lose one one the radius of this one you clearly lose and then I think in some sense some combination of the angles can't you go to one so that log p divided by log of q remains finite that I'm not sure but what is important is that there is some parameter and for specific cases in which this partition function is computed you can now use this fact to embed this to describe this geometry in terms of coupling to supergravity to say something so you think that b is the ratio of log p with log p ok so then is it log p over log q or log p over log q bar log p over log q is b square it should be homo, right? ok that's not what I remember but maybe there is a question of definitions but I think you can find the answer in this paper about the dimension reductions of dualities by Cyberg, Willett and Radzamat so over there I think they have some precise match between this b and the p and q so let's take the round sphere which will say corresponds to b equal 1 in some set of conventions and with some slicing of this space of squash spheres and then you can look at what happens into the Lagrangians when I vary b away from the round case so then delta l, so if I do delta b then we have some delta l and this delta l will involve these background fields in the supergravity multiplet which couple to the conserved arc current and c mu which couples to the conserved central charge current but at first order in delta b this is all what there is so the other auxiliary field doesn't enter and then there are higher orders in delta b which so there will be delta b squared which multiplets some operator so again we are considering a variation near the round sphere and actually for round sphere we have to be more precise because we have a non-trivial b even on the round sphere so what we mean is that when the sphere is round we actually take a theory and conformally couple to it so that's the definition and then you can give an interpretation for what the variation of z with respect to b at quadratic order the round case should be and there should be in principle two contributions one which involves integrating over the sphere a two point function of the conserved currents so for instance there will be some vectors which in order to find the precise expression you have to study the background which integrate j mu r at x and j mu r at y and then there could be other objects involving j mu z but ok so now let me let me say so let's suppose that we consider some theory on the sphere and then we make the sphere larger and larger then the theory would flow to some flow to the infrared and in the infrared the correlation function of j mu z j mu z becomes a redundant operator and the and therefore like its correlation functions are only non-zero at coincident points so here we'll write that there could be contact terms so we are using implicitly that the partition function does not depend on the size of the manifold so there are contact terms that I will discuss very briefly later and then at order delta b squared you would also have the expectation value of this operator but in a conformist field theory that's zero so you don't get it so you just have this and now we have to discuss these contact terms so this would like take a while to explain but so this contact terms can in principle give some non-zero contribution because they are integrating them over the entire manifold but you can prove that they only change the imaginary part of this object and not the real part so if you concentrate on the real part then it's all encoded into the separated in the two-point function at separated point of the conserved arc current and this you can just write down for a general super conformal field theory and it depends on one parameter only there are some factors and there is delta mu nu d squared minus d mu d nu of 1 over x squared and in this parameter also determines completely the two-point function of the energy momentum tensor that's related by super conformal symmetry so then you can just compute this integral by using the conformal map from flat space to the sphere and then you integrate over the sphere using this v that come from like the background and you compare with what people obtained for the partition function and in this way you can determine this number which characterizes your CFT 0 minutes and then done