 And this is a short title. So the term patronization was suggested to me by Nikita Nikrasov, and I got official permission from Vasily to use this term. Anyhow, so I will talk about two papers which is not yet out. And I hope they will be out this year, but I don't know, we keep, I mean, hoping. So this is work done together with Guido Fistuch, who is a post-doc in Korea now, and myself. So first of all, let me explain what's going on and what's the problem I'm going to address. So I actually would like to look at N equal to supersymmetric Young-Mill theories on curved background. And I would like to basically ask two questions. Of course, first question is sort of existence. And second question is, I mean, when they can be localized. And by localized, I mean, writing the closed answer, not, I mean, doing anything else. So let me just review what's the situation historically has been. So what do we know about supersymmetric theories on curved background? So of course, first thing what we know is, goes back to 80s. And this is typical of the 20th century, typically goes under the name of Donaldson Heaton Theory. And this is basically, we call it twisted N equal to Young-Mills. So this theory is basically concentrated. So you can put on any curved manifold, you know, there is sort of an identification of Lorentz group is SU2R symmetry. And you can put on any manifold. And basically it's related to this equation to instanton equation, et cetera. And then theory is supposed to calculate in some way volume of intersection numbers of model space of instantons. So although this theory has been formulated for a long time, it's extremely hard. So basically no calculations. I mean, concrete. So I mean, you can put theory everywhere, but if I give you manifold arbitrary and ask you to calculate something in a closed form, I mean, it would be very, very hard. So then around 90s, around 95, well, basically this paper is 97, 98, by a bunch of Russians, Losev, Ykrasov, Shatashvili, was suggested the program when you try to understand the theory in a current way. So in particular, of course, they were interested when you look at R4, and then you put certain equivalents here. So you rotate your space and then you look at equivalent Donaldson-Witton theory. So this would of course require that your manifold has a symmetry. So, but their main concern was for actually this space, et cetera, and eventually all this program, I mean, around I think 2002 by Nikrasov has been calculated. So we have Nikrasov partition function, which is basically just a sum qn and of a modular space of instantons. You have to calculate volumes and you are doing this in an equivalent way. So the whole idea is that you have to use full equivalents. So you have a torus, which is basically rank of the group plus two. And then there are certain subtleties, but they actually only fixed points and points contribute from this modular space. And these actually points can be a posteriori interpreted as a point like instantons, reducible u1 instanton sitting at the region. So this is Nikrasov partition function and it depends on the value of colored infinity. It depends on q, which is an instanton number. And of course it depends on this EQR and parameters. So basically this is also, you can think as a current parameters. So by the way, right away in two years, there is a paper in 2004 again by Nikrasov, which is basically what he was suggesting. It's not actually derivation, but it's a reasonable suggestion. He said that if you would now look, try to look at some Toric manifold, Toric 4D manifold, and you try to put their EQR and Donaldson-Witton theory. Donaldson-Witton theory. Then the idea is the following, that for this manifold actually, you have to glue his flat partition functions for R4. And then you write one contribution, epsilon, epsilon, AQ, another contribution, et cetera, another contribution. So basically you have to put contribution for every fixed point. And also there are some, will be shifts in here and some, some related to H2. Second carmology. So what's he called fluxes? So it's very reasonable conjecture, but in fact this type of formula was suggested by Nikrasov in 2004. Hmm? Epsilon 1, A, sorry, AQ. So general formula actually, what he suggested, it's in fact in proceedings, part of the things that what Nikrasov is writing, you cannot find anywhere. So it's one of the proceedings, I don't know if it's on archive. So here it's basically for every fixed point, you can have to put Nikrasov partition functions, and also you have to put certain shifts and have a sum of H2, because if you have non-trivial H2, et cetera. Okay? So this was suggested, there are no integrals. It's not was derived in any way, it was more or less suggested. It looks reasonable. And of course you have to discuss if it's compact, non-compact, et cetera. Then of course in 2007 it was Peston and he was able to take an equal to theory and put it on S4. And then basically his answer on S4 was written as follows. So it was written as an integral dA over Rn, and then you would have a right a Nikrasov partition function of IAQ. In principle, I mean these parameters you can read just as from the data from neighborhood of your point and in principle you can squash it. So you can consider not necessary around sphere. Then times some polynomial of A which is correspond to classical fields. There is no need to write perturbative thing because it's also decomposed. So Nikrasov partition function also has a perturbative part and it's also factorizable. So now in Peston's story the problem was the following that this is actually instanton contribution. And this is anti-instanton. So in principle you have point like instanton. So let me first of all stress out that if you honestly if you compare this result with Nikrasov result this is a derivation. This is a conjecture. Okay so Peston conjectured this result. It's not derived. For example here there is different approaches. You can regularize, you can introduce non-commodativity, etc. Here it's not. I mean we are not actually controlling configurations. They become quite singular. This is a general feature of four and higher dimensions that we conjecture results because our configurations are singular. Then we actually don't want to analyze PDEs etc. It looks very complicated. But it's a very reasonable conjecture from many points of view. Okay and then basically after this result so that people you know wrote supersymmetric theorem, the score, sphere, etc. But actually it was no any full classification of n-equal 2 theories on curve manifold. There were different attempts. And also I'm at least personally unaware of any results on other manifolds. So there are for example from CISA group there are results for equivalent Donaldson-Witton on CP2 as 2 times as 2. But not a generalization of Peston when you try to take n-equal 2 theory. Now when you look at these things so when you look at the necrosis conjecture result when you look at Peston result there is a following crazy idea which will pop up in your mind, right? So basically what I think is the following would make sense. So imagine I have any manifold. So I have a manifold M. And then I would have some vector field or some action of the group for example. It's very natural to assume that I have action of T2. And then I would assume that I have only isolated fixed points. One loop determinant you can also factorize and put in there. Just to save time in writing. So then I have my manifold. So imagine I have my manifold. And I have only isolated fixed points. Right then what wouldn't I mean when you look at these two things what you would think is the following. Let me randomly distribute instantons and anti-instantons. So I put plus, plus, minus, minus for example. And then it would make sense that I would start to write for you now any cross-partition function. So for Q, so this is contribution 1, 2, 3, 4. So this is for 1. If you are in parameter Cepsilon I can just read from the Toric data. It's a local data. Then 2, again instanton. Then 3, anti-instanton. And then 4, anti-instanton. And of course I have to integrate over parameter A, et cetera. Sorry. No, no. No, I mean at that time in 2004 nobody even thought about it. So his problem was actually tried to think so this was speculative. So he knew what's going on R4. He said let's me take a Toric guy and imagine. And he said that if on R4 all contribution comes from point-like instanton then it would be natural to assume that everything comes from point-like instanton sitting at fixed point. And then the only difference was that if you have H2 there will be some fluxes. And he again gave a conjectural transfer. So at that time was not even thought. So I mean till before Teston came with this thing I mean he even didn't pause the question. Okay so you can try to think of this answer maybe there is some classical term, et cetera. Right I mean it looks natural. The answer is that this is in fact the answer. So let me first give you tell you the answer and then try to explain why it makes sense and tell so basically elaborate different points. So the answer. So if you give me a vector field so if you give me basically any manifold with T2 action again if your mathematical pedantic doesn't have to be Toric because Toric typically assumes symplectic is just some good T2 action with only isolated fixed points. Fixed points. Then what you do you distribute distribute plus and minuses arbitrary. Then the statement is the following that there is exist supersymmetry there is exist killing spinners. I mean there are some subtleties I don't want to discuss. So basically if I want to talk about normal killing spinners I have to assume that my M is spin. But this is can be actually relaxed because any formal manifold will be spin C. But also works also works for spin C. But I don't want to discuss it. Okay so then there is exist the killing spinners satisfying well satisfying killing spinner equation. So there is exist a supersymmetric young mills on this M. And basically you can there is some technical subtleties but in principle you can write the answer and their answer morally is exactly this. Also here you have to assume that you may have a sum of a fluxes if there is H2. I mean you cannot exclude them etc. So this is the answer and we actually construct it and the thing is that in a way we construct it it's maybe unusual in the area but maybe it's a very good approach. So visual thinking we always criticize that visual thinking is bad but sometimes it's very good because actually we assume the answer and we derive things. So for example when you write a solution for killing spinners when you fixed also gravity backgrounds etc. They look crazy. I mean certain expressions looks in general they go for three four lines. But they exist. I mean so it's an honest thing. Now if you will try to do address problem in this Festucha-Seidberg philosophy from beginning I mean you will be killed right away. So it's better to guess the answer and then fiddle everything inside. So for plus I will put instanton so it's instanton and for minus it's anti-instanton. No they can be arbitrary. I don't sum it no. Yes each configure. So for different distributions of on the manifold for different distributions plus and minuses I have different supersymmetry. I can tell you there is purely very simple mathematical theorem. So if you give me a four manifold compact this killing vector so actually in construction I just need a killing vector. So then the killing vector typically you would prove that either it's S1 and this typically will be factorized or just part of T2 or T3. So there is no other possibility there are mathematical theorems. So in a way T2 is the most interesting case. But actually what you need you just need a vector field which is killing. But once it's killing there is theorems for compact manifold that it basically comes from the group action and typically it's some torus. And cases like you know S1 or T3 it's very degenerate. T2 is the most interesting. Of course in this no. T2 S4 has a I mean T2 action right with two fixed points. Yeah. So just write this as a question in R5 right. I mean zeta1 square plus zeta2 square plus x5 equal to 1. And then you rotate the zeta and this is exactly your T2 action. Any other questions? So let me try to explain sort of physical and mathematical meaning of this etc. And basically also what is interesting it's the technology we are doing and in a way and also what I will try to tell you in a way that what Paestun did and what Donaldson Witten or Nikrasov did it's not that far away. It may look that it's very far away but it's a variation of the same theme. So there are two different concepts etc. Here so let me try to think. So let me first give you idea why do we like Donaldson Witten theory? It's basically because at least formally we can define and calculate. So the whole thing is the following that we have f plus zero and this is elliptic equation meaning the following if I would add a gauge fixing so I'm looking at linear rise so I mean I can do it properly mathematically but I'm just oversimplifying things. So if I write this equation since I write a matrix you know as with momentum then this matrix away from zero is invertible so this what means elliptic. So in compact manifold ellipticity and being Fred Horn operator it's the same thing so basically the kernel is very small so it guarantees me more or less that dimension of my space is very small. So this is modulus space then I can basically at least effectively reduce my theory to something finite dimensional. So this is elliptic. So another thing is what you can do is what we learn in localization problems nowadays that not only elliptic problems are good but also transverse elliptic problems are good. So transverse ellipticity is a very simple concept. If you have a group action so if you have a group action okay then what you would require from your question is that you have elliptic things in a direction transverse to group action. Okay so this is transverse ellipticity. It sounds very funny so of course it's very hard to read ITI lectures but it's not that hard problem. So the simplest example is the following. So imagine you take s1 times c right and then you take a Dalbo operator here. So you have coordinates t zeta zeta bar. Okay so typically when you would have c you write a Dalbo operator it's elliptic operator. Okay and for example if you compactify the space then there is very little I mean very finite dimensional space of solutions in which you annihilate this operator etc etc. Now if I put on a bigger space so this is my action of the group this is s1. So in fact those operators transverse elliptic it means that it's elliptic in direction transverse to s1. So basically I have to look and this is a local concept I mean it's not a global thing. So you just look writing local coordinates and this is transverse. Now what is important why do we like it? Because actually if I write some any good operators here on this whole space for example if I would write a Laplacian then this is obviously will be dt square plus d d bar. So typically this is way of checking transverse ellipticity so if I take my operator for example d I write some d dagger then I write d divided by t of square it gives me typically second order of elliptic operator. If this is the case this is transverse elliptic. So what is good about this operator it would not have a finite dimensional kernels but it has a kernels if I start to basically expand in the modes of this guy it has a finite dimensional kernels with this fixed I mean value of I mean of my well not winding but if I start to expand in free modes. So this is basically what I'm doing instead of analyzing determinant of this we actually can reduce things I mean to there that's a whole notion. And again calculationally it's very good because what is important that we still can control situation. So the thing is that in principle we are in all days we are very much fixated on finite dimensional modular spaces because we like integrals of a finite dimensional space but what I'm saying is that if you're basically modular space is infinite dimensional but there is a very clear sub-locus with respect to gauge I mean to whatever your Eq variance which is finite dimension you still can calculate things and basically put this some generating function etc. So for example let me give you a very simple example so in 4D this is my elliptic problem. So I can ask that I would like to construct transversely elliptic problem from there. For example I will go to 5D and again my discussion is local so let me add a coordinates t so then I will have differential dt and then I will have a vector field dt. Okay Just local so think about manifold m and I just add a direction I mean I don't care it can be line. So then of course what this would be natural is that I would write fh plus so horizontal so I'm just now doing cell duality only in horizontal direction so I don't have anything there so this is this so this is three conditions then of course I have still to put a gauge fixing this is one condition so now in fact you know things are actually missing up so I have to add one condition of course what would be very natural is to add this condition so somehow effectively nothing depends on this but this is too many conditions this is four conditions so you can actually put this guy there so and that becomes one condition so if you write a symbol of this operator and so just write this problem and try to symbol of this operator and for example you would reduce it to 40 so write in 40 it's exactly the matrices which Peston wrote on his paper okay so in a way the story is a following so this is transversal elliptic problem so I just moved it up so I try to mimic so I have a horizontal space everything is elliptic and then along certain directions something is not so in 5D in principle this story can be co-verentized but I'm not interested very much in this but for example what actually if you want to understand what Peston is doing in the following so you have this model you wrote this model but let me now reduce this down in slightly different way so let me reduce down not alone t but t plus some other direction for example t plus I don't know call it theta etc so I had some symmetries here I put a t I tilted my symmetry and I put it back so basically then naturally in 4D you will get a transversal elliptic problem and your instantons may flip flop because it depends if these two directions will be oriented in the same way at fixed points or if they will be anti-oriented so that's basically a mathematics of what's going on so if you wish you take a Donaldson written theory you push it to the 5D you just write these things that it's a nice problem mathematically then you pull it down but you sort of reduce it this a bit up so for example this is originally was our motivation so we did this so if you take a YPQ space so this is a 5D Sasai Einstein manifold okay and so this is topologically as three times as two so supersymmetry for example if you look at killing spinners etc then if you write the question there so supersymmetry will tell you which is good equation so for example I mean the question which you write this this is typically called contact form it's fixed everything by supersymmetry I don't want to go to discussion there and generically this my rib vector field so which the guy so this is totally equivalent so this is way of writing is equivalent to saying that I will have FH horizontally equal to zero and IRF equal to zero this is not quite that problem but you know I'm just and then what I'm trying to say so my vector field goes around but at the same time this happens to be in one has tonalized but this has happened to be a fibration U1 fibration of S2 times S2 this U1 does not coincide with this guy so the toric diagram is like this now you have to take this equation which is fixed in 5D and push it down but you will reduce not along this symmetry or along these guys so basically you have to contract with another vector field which is corresponds to the free action and this sign can jump it depends from geometry you just have to calculate it so if you calculated this examples you will get plus plus minus minus so that was our original motivation so in end of 2016 we actually realized so this example is very cute for example here killing spinners would I mean not depend so you can reduce killing spinners you can reduce young mills you can do a lot of stuff here again this is of course 5D is not general consideration so I cannot actually take any 5D 4D manifold with this setting and lift it up okay but this is basically the idea why in instanton start to jump et cetera if you push problem in 5D it's nothing it's jumping it's because you reduce things it's you know plus and minus so the question is how do you write this system and what you're actually doing mathematically so in a way I have to push problem 5D and then reduce slightly different directions so let me give you a rough idea and then I will maybe have time comment on killing spinners et cetera because of course you what looks strange is a following for example if you know how Donaldson Witton theory is defined as a field theory I mean you need self-dual forms right you are told from you know kindergarten that you have self-dual forms you cannot have a form which is self-dual here anti-self-dual here et cetera but in fact it's wrong so let me tell you why it's wrong and again we derived these formulas I mean this whole project is a visual thinking you know what to get and then you derive very cute formulas oh it's not over the case so let me assume that I have m and I have a vector field globally defined vector field so I'm on 4D manifold and then I will introduce k or kappa which is gv then there is so let me first look and again my for now discussion locally so let's look when when v is not zero so either you look it locally on some manifold or whatever there is very curious map so if I take m and it's maps form to form 2 minus b plus 2 length of v kappa which i v v you can check very easily that this map actually sends cell dual forms to anti cell dual forms and vice versa in fact m squares to one so if you want to understand what this m is it's not I mean again I'm staying for now away from zero so v so in fact v is not such exotic thing because when I have a vector field and I have a metric I can always decompose my form in vertical and horizontal pieces right so in principle there are on general manifold there are two decompositions of two forms on cell dual and anti cell dual but if I would have a vector field and this vector field is nowhere zero I can always decompose my space in vertical forms and horizontal forms so if you write these things my projector basically will be one over two one plus minus m so this will give me decomposition of two forms to horizontal and vertical and this again will have three components three components again it's true only when v doesn't have a zero otherwise it's not well defined okay so now this decomposition is orthogonal then basically this m what I'm saying here is that m is anticommuting with a star so this is the relations I have so vector field the existence of globally defined vector field gives you much more so what you can do so let me give you a definition in two ways so first more mathematical another more practical and so mathematical way is the following so imagine I have my manifold and I have my vector field v and I have isolated points sorry isolated fixed points so then I decide to choose the cover such that I have a patch around every point then I distribute as I told you pluses and minuses which mean that if I have a plus here then I put locally so this is let's say I then I put a self-dual forms of this patch of this patch I put anti-self-dual forms for example if I put minus j so this is uj so on this intersection I'm just using this map so I would say that mij it just will map two forms on i two minus forms of j if for example I connect my forms on the same so this is plus this is plus I can use identity map so it's a trivial thing that one can check that this map satisfy all conditions you need to define a bundle it satisfy co-cycle condition etc so actually if you have a globally defined vector field you patch your things around you can glue the bundle and it will be rank 3 bundle which is it'll give you so there is a 3 I mean rank 3 sub bundle which at every patch can be basically rotated to self-dual or anti-self-dual forms you cannot do it globally so again what is important that this sub bundle is not I mean self-dual, anti-self-dual form but what I'm saying is that if I look over patch there is just orthogonal rotation respecting things which will bring it to self-dual if I restrict to this patch there is a orthogonal rotation which brings to anti-self-dual okay so this is mathematical way of doing of course I can do it a bit well in this approach don't I don't have to because transition function should be defined only on the intersections so the only thing I have to guarantee that I have a patching and my intersection do not have fixed point that's it so in this mathematical way I don't have to worry anything I will answer your next question in a moment so I will do this is what mathematicians will do now I will do what physicists will do but in this way there is no problem because I just have to make sure that they intersect and everything works so of course for physicists what is important right so when we discuss self-duality we have self-projectors self-dual forms anti-self-dual forms so we would like to actually write a projector now if you stare at this so okay there are two problems here so if m is not well defined at fixed point that's the only problem I have but otherwise what I can do I can for you construct actually a new operation right so if I would write you want to go really math I mean I'm of course but you know it's not I mean whatever example you take you would believe that one can show it I mean right I mean if you challenge me I don't want to go to math I'm going next week to math conference and I will give a math talk but I'm not I mean you're absolutely correct but you know this is not a level of discussion at physics conference okay but let me make let me make you more relaxed yeah yeah yeah but let me make you more relaxed so because I can do things totally different way so what I can do I can do as what physicists do because you will be totally satisfied if I give you a projector which is everywhere defined well defined right so basically if you look at these operations you can construct a new operation plus beta m which squares to one provided that alpha square plus beta square is equal to one okay so in principle what you can do well this condition is very easy to solve they just cosine of some angle etc so basically I can construct I mean just historically we call it like this this guy plus cosine of rho star plus sin rho of m so now the question is that rho it's some function now what I have to guarantee that these things is well defined is well defined well in fact what I have to guarantee that rho goes to zero at fixed points in certain particular ways so rho goes to zero at fixed points so you see when rho goes to zero this disappeared fixed points I get exactly at fixed point my cosine if this is zero cosine can be either plus or minus one so this just becomes self-duality exactly at fixed points so these functions exist moreover on torus this situation you can always choose them in variant etc I mean it's it's no problem because it's very very weak condition I mean in fact there is things when you construct this sin of rho even from V but the whole thing is that you have one over V norm so you this should behave exactly in the same way at least and this is no problem with this so you actually can write this projector and you know again at first so this projector we actually derived not the way I'm telling you I mean this sounds nice but this is what comes from Balini of killing spinner section and this projector you can I mean Pieston didn't write but you can definitely write very nicely in local coordinate etc etc and this rho would correspond to one of the angles etc okay so this guy in fact is a projector which you can write down so for example if you want to adapt it more to supersymmetry there is another redefinition of angle so there is you can define this projector in different way one plus cosine omega star minus sine square you will see in a moment why I like this one so this form actually dictated from supersymmetry where omega is some function and again I have to assume that omega goes to zero at fixed points so this is actually a projector and gives you orthogonal decomposition of two forms okay it gives you orthogonal decomposition of two forms so why do we like this projector so if you actually write this bound so right we like to write bounds with this projector and typically when we write square of cell dual forms it's equal to young mills plus dual terms so now if you do this exercise here p omega plus of f wage the omega plus star of f so then the formula will be the following so whoever remembers very well passed on will now recognize what it is and then there is this term I mean this term you should not worry very much because it will be counseled by another burst exact terms so now what you can see is the following thing so you have a so I mean if you look at some point and passed on it's exactly this formula I mean you have so you have at some point it will be instant tones some point it will be anti-instant tones okay and again if you write this in supersymmetric theory so in a way the passed on theory is the following you take Donaldson Newton theory equivalent and instead of self duality you just replaced everything by this bundle and that's a passed on theory and that's how it will work and exactly you will have a jumps of instant on and anti-instant tones okay so I don't have much time and I could have tell you much much more let me just maybe make some comments so the reasons for example gluing procedure etc so in fact it's very it's very good to also try to derive so this is how we attack the problem of killing spinner supersymmetric theory so actually what we did is that we assumed that like locally at one patch we have a topological twisting so then actually we had to define that we have to rotate our spinners maybe multiply by some factors and then they have to satisfy have to be globally defined okay and also they have to have a fixed bilinear because if you have a bilinear of course typically we'll have a vector field corresponding to bilinear right of your two killing spinners and in fact one can do the things so you have a spinners you fix them so again the idea that I mean they look as you take a topological twist so basically identify SU2 Lorentz SU2 our symmetry so in some basis you basically choose them like a Kroniker symbol when they multiply by some factors and then you will rotate by SU2 symmetry and then you have to guarantee that you can glue it consistently over patches that's can be done so then you just guarantee that they're bilinear correct so for example bilinear when you scourced them from between two sigmas you will get some object which exactly will you know behave very nicely with respect to this projector and then you know you try to put this in supergravity and then you calculate you try to fix the grounds etc and the end of the day everything works so you're not trying to approach problem the way we typically address and again I think at this time it's very useful because the grounds fields exist everything works they just look very ugly but you can sort of derive them everything and fixed so this is one comment and I think this is a nice idea the way we derived killing spinners and actually fix supersymmetry so I'm not actually talking about this very much what else again I'm actually skipping a lot of technical things because I'm not discussing for you different vanishing theorems how to localize I'm not discussing fluxes I'm not discussing how to write one loop so for example we can calculate using index theorem one loop I mean every fixed points will contribute to one loop unless we have a proper Toric symmetry we cannot write it as a single function etc because so just be aware that when we say Toric symmetry Toric manifold we actually don't mean that we have just T2 action we we have much more we have for example typically symplectic form I mean all these diagrams we are drawing convex diagrams I mean they typically assume symplectic structure so this setup is much more general so for example if we will go to Köhler manifold we can write things I mean in very nice form generally not because there are situations when you're I mean with the Torus section when you're basically cone is not convex I mean so when you will have things like this etc I mean this is possible outside of symplectic case so we in principle we cover over everything this but it gives us much less tools etc because there is no cone defined etc so over locally we can calculate the answer but we are having a problem to write you know as a close result etc so I didn't discuss it and of course I didn't discuss how to calculate determinants etc it's all this beautiful work of transversely I mean transversely elliptic operators etc so and in a way everything works everything beautifully nice etc so I basically my intention was just to give you a rough idea so I should stop thank you very much