 감사합니다. 모든 орг한이예요, 특히 이 앤스는 여기 me를 가져왔습니다. 그리고 특별히 Douglass만 제 앞에 있는 것에 대해 말합니다. 왜냐면 30분 정도는 제가 똑같이 말합니다. 그래서 이 문제에 대해 제작되었을 것입니다. 하지만 조금 다른 경험이 있습니다. 그리고 제가 타이틀에 대해 대화의 대화의 대화입니다. 그래서 이 물 them in numbers I'm going to show you will be sort of necessary condition for such a state either exist or not exist. So it's different from actual construction. But since problem is such a difficult problem as you have seen in the previous talk. This does have a value in my mind. And in particular I was asked to tell you about things that happen during last five years actually, 또한 15년 만에 더빙시amy m was to compute index of this SU2 Maximally supersymmetric 양멸 콘터 미친� requests that was painstakingly described for larger n. And the object, the conjecture actually by Whitton in fact 95 for the M-theoric conjecture if there is such a thing as M-theoric in eleventh dimension 위대한 것들에 대해 알아봅시다. 위대한 것들에 대해 알아봅시다. 첫 번째를 해봅시다. 이 인덱은 그라운드, 페리미엄, 그리고 보존익 스테이크와 상대로 적용한 상대로 적용되어 있습니다. 그라운드의 한 상대로 적용되어 있습니다. 그래서 이 상황이 필요한 것입니다. 당연히 다이나믹스의 셔프가 하지만 또한 SUN 버전에도 같은 느낌의 fruition의 조합은 아! 같은 attorneys 1s -, 2s 4s 4s 4s 4s 아티아와 싱어에서는 소속 loading of computation. 이 스스로 아티아와 싱어는omenong time of all. However as was described in the morning, it's not the case here, there is lots of continuum state. And once you have a continuum state, this independence of data just breaks down completely. So you compute this because you can. But that's not exactly the index. So in pure mathematics of course, 이 부분은 아티아파트 오디싱어 인덱스 DRM입니다. 왼쪽은 엉덩이 인테리어의 캐릭터리스트 클래스입니다. 이 코렉션의 피스에 필요한 인덱스 컴퓨터는 에타인베리언의 APS 인덱스 DRM입니다. 이 코렉션의 피스에 흐뭇히 볼 수 있습니다. 이 코렉션의 피스가 없어 이 코렉션의 정취는 가정ction을 note 아티아파트 오디싱어 인덱스 DRM입니다. 이 코렉션의 정취는 그런데 이 학생은 아니죠. 그런데 아직은 기술의 퀀틈을 적용할 수 있습니다. 우리의 일을 다 살펴볼 수 있다면 어떻게 하는 것인지 모르겠어요. 저는 여기 물어보는 게 아니고요. 그는 1, 2, 그리고 b가 b가 h입니다. 그래서 그는 정렬을 정리합니다. 그는 정렬을 정리합니다. 그런데 정렬을 정리할 때, 정렬이 있는 것과는 정렬을 정렬한 것에 대한 보존의 정렬을 정렬합니다. 그리고 페르미오닉의 정렬을 정렬합니다. 그러나 그는 정렬을 정렬한 것에 대해 그리고 그 이유는 이 깍두기입니다. 그래서 이 깍두기에는 더 subtlety가 있습니다. 물론 이 깍두기에는 모든 것이, 이 컴퓨터를 했을 때, 이 컴퓨터를 늦게 했을 때, 이 컴퓨터를 늦게 했을 때, 더 편안해졌을 때가 있습니다. Q. 1을 말하실 때 1을 말하실 때 1을 말하실 때 256을 말하실 때 1을 말하실 때? 256 times 1, because I separated out u1 part. Yes, yes, the entire supermultiplot. Very good, thank you. Now, of course, I do not have analog APS index DRM, so I do not know in general how to compute this correction piece, which is difference between the two. And still, it's the same in the general problem of this kind, that has a boundary or asymptotic infinity. There is no well-known systematic way of computing this. However, for gauged quantum mechanics of kind of classes I'm looking at, and it's a very large class, it turns out there is a very simple way of doing this. It comes from the observation that this correction piece, which arises from continuum contribution, which is why it is fractional. You can argue it always comes from the asymptotic integral of something else. So just like left-hand side is given by integral of some characteristic class in APS case, right-hand side eta-invariant was computed by integral over a boundary. So this, you can convert it to boundary integral in some physics problem. And all the complicated stuff in this SVN theory does not really matter as long as you understand the asymptotic dynamics. And asymptotic dynamics because it's a commutator potential, just to mean commutator potential, all reduced to carton, the mutually commuting part of the matrix dynamics. So it's essentially free dynamics divided by vile groups. So that way you can relate this to another bulk index problem of this kind, right? And that way you can compute this indirectly. And by doing this, one find, one quarter, and therefore you have one. Please. There is a mass deformation of this problem. There was mass deformation of this problem. That's right. Which would suggest that it's a good way of defining it. Right, that's one way of defining it except that you don't know how to control zero mass limit, which is the original problem. So in this n equals 16 problem, there is a lot of control you have, like adding mass deformation of PP wave kind or some other kind, for example, cats and smell that try. So there is a lot of ways of getting this to one. But I do not want to deform original dynamics that much. I want to keep this asymptotic dynamics and do things as honestly as I can, okay? So let's see how far we can go. Anyway, so this trick that I invented, I said on this for about the months wondering what to do and one day in March, this trick came to my mind. So I did this, it was like one day computation. And that was taken up by Necklace of Moschata. She really laid the left hand side on the log of the so-called bulk term was computed this way. We have for SUN, you have this one of a P square where P is a divisor of N. It's a very interesting number theoretical thing on the left hand side. Right hand side, green and gotper and laid the cats and smell got took up my idea, generalized to other gaze group SUN in particular and combine the two and argue that this must be the case. You can sort of, this complicated sum is there because if you have N such particle you can divide N into P over N bunches, okay? And these bunches will form sub-bounds state, fly apart from each other and in every such sector contribute some fractional number. So that way you can understand this number again you get the right number you want to. So at least necessary condition we find. As Douglas said, smaller supersymmetry version can be also, I mean conjecture about it shows up in some other context and not the membrane context but the brain type two theory compactified on Calabria two-fold and Calabria three-fold. In this case, consistency between supersymmetric field theory and type two string theory demand that these numbers would be one, zero, sorry, there's no state whatsoever. No normalizable ground state. Left hand side, of course it's easier problem. So same people, no one across of Chateaubrile computed left hand side and instead of this sum you have only the case P equal to N so one over N square. Right hand side, again I use exactly same trick and you get one over N square. Again, don't buy a green and gupper. So N square minus N square, one over N square minus one over N square is zero. Okay. So. Oh, this is the vile group of this SUN. So there is this part of computation. I'm not really describing how I get this to stage but sorry about this. This will take a little bit more time but I have other things to do. So I am happy to tell you about this later. Now, since SUN do not have any bound states only natural to expect other gauge group do not have bound state either. In fact, for ON and SPN there is a similar conjecture or similar demand coming from type two string and field theory consistency between the two that says this. And again, this was computable the same method. Right hand side is computable in the same method and you compare the two and this happened around year 2000. And here we are. So left hand side was done by Katz and Smilga which gave one of n squared and you get these series of fractional numbers. Right hand side starting from what Mu and Krasov-Shatavish evenly gave a Stadakko in 2000 in particular Vasili-Pesto in 2002 I was told that this was when he was undergraduate student remarkably. This is really nice computation of this number the right hand side, this one. So by the way, I should have said earlier the this Z is really nothing but the matrix integral of the kind, the model is essentially what we now call IKKT. It's a matrix, not quantum mechanics. Matrix integral and that's because I take small beta limit. This is like having a circle and make it very, very small. So circle disappears. So that seems really natural thing to do and that's what we did. Everything is Euclidean because I'm doing Hickona. But of course I'm looking for actual wave function in real time. So that idea went into this computation in this computation same computation here but except for SUN numbers do not match. So the first one that does not match is rank to SP or SO5. left hand side, this boundary on a log of eta invariant give you 5 over 32. on a log of characteristic class integral give you 9 over 64. But this is the example where left hand side minus right hand side should equal index which should be integer and in particular zero, it's not a zero. And this was n equal 4, the four supercharge and a supercharge version of the story. There is another, yes. So what is it that you are computing if it's not an index? It is index, it's just that there is somewhere along the way computation needed a correction. That's what I'm going to tell you later. So exactly what are we missing here? For example, I mean looking at these two numbers you might say oh one of the two guys made a mistake. That was my reaction too. And since I invented the left hand side I naturally would have said the right hand side is wrong. But right hand side has very nice beautiful computation. And as I will tell you during last few years I redid the right hand side and indeed these are correct numbers. So there's something missing that I'm not telling you about yet. So we want to get there. Anyway so n equal 16 version of the same problem just like SUN problem give you bunch of predictions in this m theory context. And if I write those index actual integers in the form of generating function this should be the one if m theory do exist and it does produce type 2a string theory at the end of the day. But of course the program is not going anywhere at the moment because even simpler problem there was an issue. So this was state of things as of year 2001, 2002. Actually I was doing something else. I was doing something else happily for many many years. But about five years ago I came back to this general class of index problem because of not this old problem but because of work crossing problem. I wanted to understand for example concept which is very very nice this very beautiful work crossing from the physics viewpoint. And there I ended up doing again supersymmetric quantum mechanics index problems. So for physicists that studying for example D3 brains wrapping Calabiao 3 folds and propagating along the remaining time and this is something called BPS particles. Some of them bind together and some of them do not bind together and sometimes this bound state disappear so this is called work crossing problem and this says of course geometric analog in mathematics. So there what I have to solve is again gazed quantum mechanics not just matrix model but matrix with other representations added in the Kaila Multiplet. And start with some dynamics like quantum mechanics like that and again compute define and compute index like object as much as possible. In fact we did this entire thing by the way this was something did I say this and maybe it shows up later. Something I did with Kentaro Hori and Heon Kim who was my student back then. I'm not going to tell you how we did it. This is a path integral computation but you can sort of think of it as an analog of heat corner computation. It's just technically easier to do path integral. So the object we wanted to compute is index so corner of supercharge corner of dirac operator you trace over minus one to the f this plays the role of f and then you put bunch of equilibrium parameters so we call it defined index. However this is again impossible thing to compute generally because you really have to understand ground state sector only. What one can compute is this thing called omega is a path integral version of this except you compute it in a different limit. So this is gauge dynamic so there is a gauge coupling constant e and this e this number controls all the dynamics all the interactions almost all of it not everything. So this path integral computation you do it in this small parameter regime arbitrary small e and hope you would hope that if this happen to be a discrete system if for example this flow down to some geometrical model at the end of day something like cpn quintic then you can argue that this deformation just like one would argue small beta or large beta does not matter this deformation does not matter and in such cases you would find this quantity equal to this quantity not always but there are a lot of cases where it does so that this is already very useful tool to study geometry that comes out of this gauged system so that's sort of Lagrangian where we start and we say localization and magic happens and we do the computation and there is a long long story behind this which I'm going to get into you get this this is not something we invented this is invented by actually mathematician Geoffrey K1 and then developed in the purely mathematics exercise what we added is a path integral derivation of that and I think we added how wall crossing happens in this system so this is some sort of topological invariant so you think when you change parameter of the theory so for example wall crossing happens when you change the shape of Calabria 34 continuously and at some point all of a sudden you are calibrated a submanifold the set of that changes that's what wall crossing is so what we added is how that is realized in this computation so don't worry about any detail of this mechanism just be assured that there is a routine which we can put into the Mathematica which is exactly what it did and at the end of the day we get bunch of polynomials or rational functions as a result so this actually followed similar computation by Benini, Yegge, Hori, Tachikawa or Ergo for the purpose of computing elliptic genus you might think especially physicists might think oh why not take this and do go to a small radius limit of one of the circle and then you get from two dimension to one dimension wouldn't that be better thing to do but turns out whole point of this exercise that's impossible because those of you who have worked on wall crossing will know elliptic genus does not have a wall crossing so if you take this sensor and do the dimensional reduction you will get one side somewhere but not this discontinuous behavior of the index at the end of the day anyway so for example let me take simplest case or the u1 with n fundamental representation which will define cpn-1 on one side of this wall you get the usual Ho Chi diamond of cpn on the other side in this theory in one dimension you get nothing so you have here n number of ground state on the left hand side on the other side of parameter regime you get nothing this is wall crossing this prototype of wall crossing other side is just a mold of more space of stable things which can disappear that's right quintic which would have gave you same Ho Chi diamond in geometric and Landau-Ginseberg phase in one dimension the vertical middle in particular disappears there is a reason why vertical middle disappears another Calabria example which has four this is one of the canonical 2Km of Calabria 3-fold which is embedded in weighted projective space here and you get these various Ho Chi diamond you can see something happens vertical middle first and then horizontal middle etc so every single one of this I can compute one thing I failed to mention when I was which I should have done is if this theory this gauge theory give you a manifold at the end of day which is compact and geometry is if it flows down to nonlinear sigma model to a compact geometry I'm effectively computing kaiwise genus that's why I was able to get cohomology information but of course kaiwise genus alone will not give you cohomology yet I'm displaying the cohomology right I mean this is of course trivial example you already know other things not that trivial one of the things that happen in this business in physics side is that for certain class a very large class of gauss quantum mechanics and in particular entire class of UN type quiver quantum mechanics there is a routine that allows me to reconstruct entire Ho Chi diamond in all chambers so that requires that's another talk sorry so I'm displaying I use that routine which I'm not telling you and reconstruct this Ho Chi diamond of this quiver theory in particular chamber so sorry about telling you details of this but trust me there is such a routine and there is all kind of things I wanna do with this thing and in particular this quiver invariant concept that came out of physics which allowed me to do things like this okay so we now have what I in the title somewhat boldly suggested index theorem I'm not quite sure I have the right to call it theorem but as far as physicist goes this is as rigorous as it can be so I have a routine I can put it in the Mathematica I can compute things that if you ask me to compute in many cases now so one of the things that bothers one when you try to do computation like this is this phenomena of work crossing where homology changes suddenly and also this sometimes contribution fractional contribution or nonintegral contribution you get and both of this has something to do with the fact that there is a continuum sector so that continuum sector you have to treat it very carefully right since it's not discrete spectrum it might depend on how you regularize that integral that sum, continuous sum and in particular in this examples I was showing what's generically happening is remember there was this parameter c and at c equal 0 something happens so positive could see you have one cohomology negative could see you have a different cohomology and from the physics term the reason that happens is because there is a direction in this dynamics that mathematicians do not usually look at physicists call it Coulomb phase there is asymptotic Coulomb phase along which plane wave like state can propagate so you have a continuum of state that possible above certain energy and that energy is dictated exactly by this parameter c so when this is somewhere up finite you can identify this piece very easily so in particular you can scale up this c to infinity without affecting ground state sector that's sitting there and that's how you do compute the index usually in such a system where you have finite gap so in fact work crossing happens precisely because as you approach c equal to 0 in your parameter space this continuum touches ground state and of course on the other side it will go up but sometimes leave behind some extra state or take some extra ground state with it go up and that's why you have work crossing happens precisely when this parameter is 0 so that's why we see pictures like this but in each chambers because of this gap we know how to deal with this continuum easily and that's why we have this integral numbers at the end of the day without worrying about on a log of this one quarter earlier okay Can you say more what is the horizontal axis? Oh sorry so this is what we call Coulomb axis and this is an expectation value of your vector multiplied scholars so in mathematics literature they do not usually look at that direction because they just gauge away and usually you look at Hicks part what we call Hicks part of the theory so Both of these are Coulomb branches so these are Coulomb branches This part also What do you mean this part? I'm just displaying one axis for illustration so The blue and green is just the wave Oh I see this is a wave that propagating along this Sorry So arrow probably I should have done wavy lines Sorry about that So what this means this having something like this means that even though I wanted to compute this integers or integral coefficient polynomial you cannot quite do that you end up with theory where this parameter or this 갭 is absent And in fact this very first example I told you where I got 5 over 4 is an example where I do not have a parameter like this naturally I could introduce this massive deformation you suggested but that's not the original problem original dynamics I want to stick to original dynamics Now original dynamics does not have anything like this So this continuum come down and give you something funny that you should be you want to be able to get rid of So generically this localization computation pass integral computation do not give you integral things However as I was giving you this 5 over 4 example it's not this additional piece you get is not just arbitrary real number it's 1 over 4 In fact it's 1 over 2 squared if you know how to do there is a reason for that So let's understand that reason So we will come back to same set of problem arbitrary gauge group but now we have a new device to compute what's left hand side of this say on a log of Attyapater this thing of theorem So we will compute this omega quantity and for supersymmetry 4 and 8 you do this computation you get these rational functions old computation you should think of it is analogous to have doing the same computation except you set equilibrium parameter to nothing which means y equal 1 therefore this give you 1 over square SU3 give you 1 over 9 SU4 give you 1 over 16 this give you 1 over n square So I'm essentially reproducing 20-year old numbers in the new method but I have extra information here because this refinement allowed me to distinguish this not only by numerical factors but by this linearly independent rational functions Remember it was SUN was fine but it's SO, SP and others that were problematic So let me redo that computation in this way This is going to be different from the old computation So the rational functions I get is something like this for rank 2, rank 3, rank 4 you have these complicated things What is it? You need to understand the structure You need to understand the structure of this first and here is the answer if you believe this or not The old rational expression you can package into the following thing given a simple regroup G its vial group is this so this is cardinality of vial group dividing the entire thing and you sum over element of your vial group but not everything but something called elliptic vial elements only that means when it acts on the root lattice it leaves no direction invariant no eigenvalue 1 so this apparently is called elliptic vial in fact I haven't seen this elliptic vial too many places I don't know if mathematicians use this much or not but you do this sum you do determinant and every single one of them has this shape so why is that? in fact remember this one quarter I was saying I have this funny way of computing it and that was taken up by Katz and Smilga two years later for arbitrarily simple algebra and in fact this was what they obtained back then with y equal 1 this is elliptic vial and the reason that happens is this so as I was saying I want the integral thing but I can compute only bulk thing which is small beta limit of this expression what I have computed just now is actually a different limit small coupling constant limit of the same path integral on the other hand both beta and e square are dimension for things and it doesn't really make much sense to send a dimension for quantity to 0 you have to find what is dimensionless combination and that combination is this so what this means is this is actually computing this quantity that I computed long time ago and on the other hand this is the problem where you expect no bound state at all because of smaller supersymmetry and then this should be 0 so this computation the end result of this computation has to be same as minus of this and that minus of this was expressed this way when y was 1 so you can have guess make intelligence guess what this expression should be so that's what we wrote down and every single one of them indeed agree what is the upshot of this story? upshot of this story is remember I had two a way of computing the left hand side that Vasily did for arbitrarily simply algebra and I had a way of computing the right hand side in such a way that this should cancel against it there was small discrepancy I have a new way of computing left hand side and remember the small list example where discrepancy happen was sp2 or equal SO5 so same the algebra and therefore it has same expression same bile group and therefore it has same expression and you evaluate this at y equal 1 you get 5 over 32 and this was the number I needed to cancel the other number so for the moment let's forget that there was this other conflicting numbers but just try to trust this number and then this number equal that number and therefore this index that you wanted is zero so that's consistent with anticipation that no bound state exist for smaller supersymmetry so let's buy this for now and then go on to n equal 16 version of that problem maximal supersymmetry version so this is the problem where index the integers you wanted is not zero it has particular prediction coming from m theory and you do this and everything there is this bunch of rational function that follows and every single one of this would have interpretation as coming from particular continuum sector so what we do is compute do the localization and compute this left hand side and you get horribly, horribly complicated rational function look up what kind of things can enter on the right hand side using the same procedure I used for n equal 4 and 16 4 and 8 and ask is the unique decomposition of this kind and indeed there is a unique decomposition and for various up to rank 4 we did a computation you have this unique decomposition in every one of them so the integers in the first line is the bound state number or index the integral index you wanted to find so those are real answer every single thing that follows is analog of this 5 over 32 it's a result of particular continuum state in this asymptotically three problem where you have plane wave going out to infinity so at the moment I'm interested once I know how to blame every single one of these to particular physics sector only thing that remains is these integers and I try to extract those integers and in particular I try to extract integers I need to conform this empty hypothesis the other set of hypothesis that was never conformed and I get these numbers this is not much I have how many integers eight integers however these integers are consistent with this conjecture that was given around 1999 again this is physics conjecture Hanani and Company said this must be the right number otherwise there is a problem with M-theory and these numbers matches precisely with numbers here this is not a proof because I did only up to rank 4 but it's not that difficult to imagine that this will persist down the road okay so that confirms this M-theory again I mean somebody like me of course think M-theory do exist so it's not really necessary but this one extra block of evidence that says M-theory do exist oh by the way I mean this fractional structure that we see this rational functions we see is not something that's accidental it has been seen in all kind of other places in particular it has been seen in the word crossing formula or I should say solutions to word crossing concept which serves as word crossing formula is naturally phrased in terms of rational quantity like this this is the exactly same rational function you saw in the SUN case and this is the index of quiver like this we are I mean when I say quiver I assign integers for every single node so if this quiver if this ends have common divisors okay then you have to have this sum and then you have to acquire this fractional pieces physics is exactly same as SUN problem and what the computation suggests is it's path integral computation of this compute precisely this rational invariant and then that allowed me to extract the integer part integer quantity by doing reverse of this thing using the so-called Mabius function which I learned only very recently this inverse nice inversion formula like this and then you do the computation and indeed you get integral quantity out of this path integral which is your fractional quantities so there are tons of things I would like to do with this but you might have noticed that I have examples only up to rank 4 and that's essentially given by computational power I have because this so-called Jeffrey Kewan residue is just horrendous thing once we have large number of charged particles in the problem so we would like to understand asymptotic large n version of this better but I don't think such a thing exist yet and except some very simple problem like QCD type problems now so as I promised I need to tell you what happened 15 years ago so I said I claim that I have two set of numbers that matches precisely so that I end up with integer 0, 1, 2, whatever such that everything is consistent but this goes against what I said like 30 minutes ago there was a problem there was different way of computing back then like this I have these fractional numbers I have these fractional numbers this number is not quite the same and if you look at it carefully left hand side is always larger or equal than the right hand side so this suggests we are missing pieces on the right hand side so remember how I got this I start with one dimensional quantum mechanics problem I do some small beta that is small Euclidean circle version of the path integral shrink this circle and end up with IKKT type matrix integral and this is the result of that matrix integral and when I saw these numbers initially I said oh they must be wrong they did a mistake but Stadak for example in 2002 I mean he must have been a little bit uneasy with these numbers he came back he teamed up with a Monte Carlo person and tried to do this integral numerically by Monte Carlo and came to the conclusion with for example this number is accurate within a part in 1000 so I cannot ignore that so what I did about a year ago is redo this computation in the new Jeffrey Kiran residue method so there is a version of doing this in this new a matrix integral technology that give you Jeffrey Kiran residue and in my mind it's a sort of I have relatively rigorous physics wise rigorous way of doing this in fact it's mathematically rigorous because it's no longer path integral and I try to get these numbers out of that computation and voila I get these numbers again so it's not the computation that is problematic it's physics wise something is missing or mathematics wise so this is the number I just gave you I claim that I have obtained by doing this localization where I take a different limit small coupling limit this is the number you get out of this elliptic biosum this is number you get out of this IKKT matrix integral this agrees with this giving me zero this number does not agree with this so what are we missing so it has to mean only thing only logical possibility that's left behind is that maybe the quantity I wanted to compute the small beta limit is not computed by this matrix integral that's only logical possibility that remained but if you think about it a little actually it's amazing that that didn't happen in general you see this mechanics path integral or heat kernel computation you do in small beta limit and as I said small beta limit is like time disappears so we replace mechanics by matrix integral on the other hand one of the non-trivial thing happens is A0 the gauge connection along the time but because it lives on a circle gauge connection itself or its holonomy values in circle so it actually lives on a circle however when by the time I get there what I do is replace this circle by an infinite line I pretend A0 was one of the scarlet field or joint scarlet field so what when I do that what I'm really doing is replace this large holonomy circle by a line but circle does not equal line topology is different there is a problem potential problem here so what really happens is something like this along this circle in fact there are not one place but many different places where I this place give you one matrix model this place give you another matrix model this place give you another matrix model in fact this actually happened for SUN what happens in SUN is depends on how you define your mechanics there are actually n such places displaced by center of the SUN so depending on whether you are careful about whether you do SUN or SUN mode ZN in one version if you do SUN in the beginning then you actually have n such identical holonomic centers so you acquire numerical factor n in front and without that factor n you end up with a complete nonsense So how do you get this yellow point? So let me give you answer in more general setting So generally these are places where when you do dimensional reduction with this holonomy value assumed as a background you do not have any free u1 factor in the remaining degrees of freedom and in this problem what that translate mathematically is that these are holonomy values where unbroken gauge group is maximal non-habilian subgroup so remember how you obtain those you take a thinking diagram make it extended thinking diagram and you cross out one circle that's the way you obtain maximal non-habilian subgroup and all such holonomy that give you maximal non-habilian subgroup contribute it turns out and any other point does not contribute because of this small beta limit So using this for SPSUSO you can sort of look for those maximal non-habilian subgroup those are those things and you have this finite sum and remember that I have bunch of fractional numbers on the right hand side which was slightly smaller than the numbers I wanted so you put in these smaller numbers on the right hand side sum and try to see you recover the numbers these fractional numbers you wanted and indeed every single case it works out in particular Basile's paper have a conjecture on individual of these SO and SP I mean although he did explicit computation for small rank up to rank 4 I think or rank 6 but he has actually a conjectured answer for this for arbitrary version and that works out perfectly every single one of them so this is why you are missing 1 over 64 in SO5 example for example so it's not that computation was wrong it was missing piece and actually I have to confess that that's my fault I mean when I did it when Sadie and Stun did it we were not very careful we said oh this must be matrix integral at the end of day and that was took up by Muonakrasov-Shatashvili and they were very successful for SUN and then that was took up by Stadako and etc so it's really our fault we should have been a little bit more careful back then now in remaining five minutes of course this is was personally very satisfying because I ended up solving all the remaining problem I started out solving back then not mathematically but physically at least but I have a bonus this saddle thing I mean this I think people I mean in old times notice this sort of thing might happen in some cases but they keep forgetting about it I think and I was forgetting about it but this is very important now because these days we compute all kinds of localization computation for higher dimensional gaze theories super conformal index A twisted version of that there are all kind of partition function people now can compute very explicitly and whenever they try to relate say for example four dimensional such partition function to three dimensional partition function you always have to invoke a circle very small size circle and there is a relation you want to understand there and whenever you have a gaze theory on a circle you have to have a holonomic circle but if you do dimensional reduction you just keep this guy and nothing else and this will give you completely misleading picture of small circle limit so in this right so generically what happens it turns out D dimensional manifold on some supersymmetric manifold where you can define global supersymmetry S1 times some manifold in four dimension this is typically what we call cyphot manifold you compute this take small vector limit and try to express that in terms of three dimensional partition function on M but there is typically a co-option that is not entirely determined by this theory it's determined by relation between this four dimensional theory and three dimensional theory and this is typically exponential piece and the co-option of exponent is called cardic exponent what people initially try to look at is really look at this guy trivial holonomy expand around it and get some expression there but it is very clear that you should not do that it might be that this is the dominant contribution to this but maybe this is the dominant contribution you never know until you do the computation so in particular the particular version of this where this is torus I'm computing written index again and what this tells me is written index of four dimensional gauge theory is actually sum of written index in three dimensional gauge theory in a very particular manner so I mean people who did this sort of game long time ago should remember same set of field four dimension, three dimension you end up with completely different indices generically this only happens this doesn't happen for SUN is that your statement? no I mean SUN when I was saying SUN it was in adjoint only theories so generically this happens so SUN adjoint only was very exceptional and we were very lucky 22 years ago to be able to look at that particular problem so this happens all the time and then we have a way of gluing say three dimensional supersymmetric theory to bunch of them to a single four dimensional supersymmetric theory I think this is very important idea and in fact this also explains wall crossing that happens in 1D which does not happen to this exactly same if you're sitting on the right place you do not miss holonomic saddle if you are sitting in some wrong place you miss holonomic saddle and that's why one dimension and two dimension is completely different behavior another implication is suppose you have a dual pair so called cyber dual pair in one say four dimension where you have two different theory apparently yet partition function agree with each other you do the same to small radius limit what you end up is bunch of duality in three dimensions because you have to have sort of one-to-one match there so that give me another systematic way of discovering multiple duality in 3D starting from one single four dimensional duality things like this has been noticed by the way collaboration of Aroni, Razzamad and Cyborg several years ago but all of this is you can sort of understand everything in terms of holonomic saddle 카디 exponon as I was saying earlier I mean let me oh my time is up as I expected there is actually a very nice set of partition function invented by Closet, Kim and Willett and it's something like this configuration you have T251 over eight so called eight twisted Riemannian surface that defines bunch of partition function and it's sort of matter of trying to find in this sigma in this two dimensional remainder the what we call Coulombic back cure like this and how they cluster this clustering of back cure is essentially holonomic saddle phenomena and for example given multiple saddle whose location is labeled by this fractional number epsilon or not fractional number it's number I think it's fractional numbers in general between zero and one you get card exponent of this type and those of you who follow this story should remember that B.T. Pietra and Comagos he made a very beautiful suggestion several years ago that this card exponent is simply given by say in conformal theory case conformal anomaly A-C what this tells you is that that's simply wrong if you stick in there and if it is dominant that's the right answer but typically that's not the dominant place it's elsewhere that's dominant and typically this number is larger than this number but when you have the representation of gaze group is relatively simple suppose fundamentals only with the S-U-N or U-N these other saddle tends to be sort of relatively suppressed but you do have such a sector you cannot forget about that sector where something similar happens with super conformal index but this is probably not the crowd so that's the end of story in empty early hypothesis this game of trying to find or not find threshold bound state I think this closed an entire chapter and I was very happy to do it myself this notion of H-Seder is not just for quantum mechanics but is any gauge theory it's very important idea I think and it meticulously this allows to glue supersymmetric gauge theory in the adjacent dimensions and this might be useful something I don't know exactly what there is even more speculative question about what is on a low in tensor theory but I think that's going too far here thank you Thank you Mr. Remark When you say it closes a sector in M-Theory but it confirms for the n equal to 16 phase everything is good so what is finally confirmed? So what? Well, I mean confirmation I guess is that there is no obvious contradiction to M-Theory hypothesis as I said all of this is necessary conditions rather than sufficient condition so there was this SUN problem which we have confirmed 20 years ago then there was turns out to be ON and SP N on a log of it somebody should have done it and I was happy that I was the one who did it I'm sure there are a lot of other things about M-Theory that we need to learn more but this is one small maybe I should have called it subsection not a chapter but yeah So eventually other compute numbers for exceptional E67 Very good as I was saying we were sort of our capacity was with this Jeffrey Q1 computation So right hand side this rational expression we of course have it question is can we do this Jeffrey Q1 computation for the exceptional case? The highest we went to is SP4 rank 4 we try to do F4 and thing wouldn't just come out I think I need supercomputer to do that or maybe you can invent better way of doing the counter integral probably they see G2 we did it and it comes out and we have a number I don't know if it has any prediction from anywhere else but we do have a number I think it's 2 So back to 2002 for n equals 4 for the simpler case the computer stopped at E7 it couldn't do E8 Did we try we did E6 I remember I don't know whether we try this 7 actually But yes but today computers are bigger but the computation is more complex I think is that I mean this old counter prescription that you used and Microsoft invented is actually much simpler than JK Residue if you look at actual set of residue So I think one of the important thing along this line is to find a better residue prescription conceptually because by the time we have F4 I think number of charge bacteria I think 48, 52 something like that and we are stuck at the stage of classifying the pores not the residue computation So we need a better way of doing a contact badly I guess you discussed the case with 16 supersumdries and 4 and 8 but what about the 2 case which was once a D plus 2 in my Actually that's the problem I'm working on although you might find it's strange that n equal 2 should be easiest right there's something funny going on which I noticed even back in 97 with n equal 2 I mean of course Yen demonstrated there is no such bound state at all I think that's it yeah yeah yeah and yeah I mean there is something interesting because of smaller supersymmetry I don't know exactly what's going on there let me say I'm working on it yes but there is no as far as I know string theory prediction about the number that's maybe one of the reason nobody looked at it thank you