Thank you. I'd like to thank all the organizers, especially Jens, for bringing me here.

And in particular, by putting Douglas just in front of me, because that saved me 30 minutes, basically. So it will be about a problem he posed, but maybe slightly different perspective on it. And as the title said, we will do index theorems.

So this will be... I mean, 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 happened during last five years, actually. New development after long 15 years of nothing happening on this index side of the story. However, of course, I need to motivate

the problem, and this is my motivation. So back in 2397, exactly actually, I was sitting on a computer for computation that required me to do something like this. High mathematics, right? So of course, the aim was to compute index of this SU2 maximally supersymmetric Yang-Mell

quantum mechanics that was painstakingly described for larger N. And the object, the conjecture, actually by Witten, in fact, in 95, for the M-theory conjecture, if there is such

a thing as M-theory in 11 dimensions that contains all the supersymmetry we know of, which was Witten's conjecture, the very first check you have to do is make sure that this index, which counts number of ground states with Fermion and bosonic states with relative minus

sign, come out to be one. There is supposed to be exactly one state, so this is a necessary condition. Of course, the dynamics itself that was laid down, although some people call it PFSS model, that's a little bit unfair because, of course, the model itself was pioneered by

Jens and developed throughout the 80s like that. So there are a lot of studies. I apologize for not showing all the references. There are so many references. But again, for any SUN version of the same quantum mechanics, you have to have exactly one state, one normalizable ground

state. So index has to be something like this, and that's, of course, the aim I was looking for. Now, this computation, which really counts the ground state degeneracy,

usually is not that easy to do. I mean, more or less, it's impossible to do in non-trivial dynamics. So instead of that, because if the spectrum is really discrete and nicely behaving, there is a pairwise cancellation. So having this chirality operator in there will give you complete control over beta. In fact, you can argue that it's independent of beta.

So people actually do the other side of computation. This is effectively what Atiyah and Singer did a long, long time ago. 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 beta just breaks down completely. So you compute this because you can, but that's not exactly the index. So in pure mathematics, of course, the closest thing there is to this is Atiyah-Pateau-Singer index theorem. The left-hand side would be the analog

of the integral of your characteristic class. And this correction piece that you need to complete the index computation is the, say, eta invariant for APS index theorem. So bear that in mind that we will see typically these correction pieces.

So there is no DRAIS class operator inside here. There is a... No DRAIS class operator inside. So how are these objects defined when you have... Oh, there's a supercharge. What you call supercharge plays the role of the Dirac operator.

You have continuous spectrum, then there is no, it's not DRAIS class. But we can still oppose quantum mechanics question. Suppose we can solve the entire thing. How these things are defined.

I'm not exactly sure what's being asked here. Sorry. DRAIS minus one to the F of e to the minus beta H. That's right, because it's a continuum. But even with a continuum, in fact, you can define it as an integral over something where the counting is replaced by density of state,

bosonic density of state minus fermionic density of state. But there you do not have cancellation that you want. And that's precisely why you have this correction piece. So there's a little bit more subtlety going in there. But I, of course, do this, formulate everything. I mean, this computation I did with heat quantum computation. But later computation,

I will do it as pass-integral computation, because that happens to be a little bit more convenient. Trivial question. When you say one, you really mean one or you mean 256? It means 256 times one, because I separated out U1 part.

Yes, yes, the entire super multiply. Very good, thank you. Now, of course, I do not have analog of APS index theorem, 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 well-known systematic way of computing this. However, for gauge 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 SUN theory does not really matter, as long as you

understand the asymptotic dynamics. And asymptotic dynamics, because it's a commutator potential, just a minute, commutator potential, all reduced to carton, the mutually committing part of the matrix dynamics. So it's essentially free dynamics divided by while 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 finds 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... 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 equal 16 problem, there is a lot of control you have, like adding mass deformation of PP wave kind or some other kind, for example, Katz and Smirnoff triad. 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. So let's see how far we can go. Anyway, so this trick that I invented, I sat on this for about a month, 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 Nicholas of Mozart. The left-hand side, analog of the so-called bulk term, was computed this way, where for S, U, N,

you have this one over P squared, where P is a division of N. It's a very interesting number theoretical thing on the left-hand side. Right-hand side, Green and Gotper, and later Katz and Smirnoff took up my idea, generalized to other gauge groups, S, U, N in particular,

and combined the two, N argued that this must be the case. This complicated sum is there because if you have N such particles, you can divide N into P over N bunches. And these bunches will form subbound states, fly apart from each other,

and every such sector contributes 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, a smaller supersymmetric version can be also, I mean, conjecture about it

shows up in some other context, and not the membrane context, but deep brain type 2 theory compactified on Calabi-Yau two-fold and Calabi-Yau three-fold. In this case, consistency between

supersymmetric field theory and type 2 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, is easier problem, so same people, no necklace of Shattashibili computed left-hand side, and instead of this sum, you have only the case P equal to N, so one over N squared.

Right-hand side, again, I use exactly same trick, and you get one over N squared, again, done by Green and Gupper. So N squared minus, one over N squared minus one over N

squared is zero. Okay, so. Oh, this is the Weyl group of this SU N, so there is this part of 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 SU N do not have any bound states, it's only natural to expect other gauge groups do not have bound states either. In fact, for O N and S B N, there is

a similar conjecture, or similar demand coming from type 2 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 Smirga, which gave one over N squared, and you get this series of fraction numbers. Right-hand side, starting from what Muhan Akhrasov-Shatavich evenly gave,

Stadako in 2000, in particular Vassily Pesto in 2002, I was told that this was when he was undergraduate student, remarkably. This really nice computation of these numbers, the right-hand side, this one. So, oh, by the way, I should have said earlier, this Z is not

is really nothing but the matrix integral of the kind, the model is essentially what we now call I K K T. 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 very natural thing to do, and that's what we did. It's Euclidean, everything is Euclidean because I'm doing HICANA, 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 S U N, numbers do not match. So the first one that does not match is ranked to S B or S O 5. Left-hand side, this boundary, analog of eta invariant give you 5 over 32.

Analog 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, but it's not a zero. And this was N equal 4, the 4-supercharge and 8-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 an 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, looking at these two numbers,

you might say, oh, one of the two guys made a mistake. That was my reaction too. 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 the 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 the SUN problem, gave you a 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 does 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 world-crossing problem. I wanted to understand, for example, this very beautiful world-crossing from the physics viewpoint.

And there, I ended up doing, again, supersymmetric quantum mechanics index problems. For physicists, they're studying, for example, D3 brains wrapping Calabi-Yau threefold 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 disappears. So this is so-called world-crossing problem. And this is, of course, geometric analog in mathematics. So there, what I have to solve is, again, gauged quantum mechanics,

not just matrix model, but matrix with other representations added in the chiral 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? Yeah, maybe it shows up later. Something I did with Kentaro Horii and Hyun 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 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 refined index. However, this is again an impossible thing to compute generally because you really have to understand ground state sector only. What one can compute is this

thing called omega. It's a path integral version of this, except you compute it in a different limit. So this is gauge dynamics. 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, arbitrarily small E, and hope that if this happens to be a discrete system,

for example, this flow down to some geometrical model at the end of the day, something like CPN print it, 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 this is already very useful tool to study geometry that comes out of this gauged system. So that's sort of Lagrangian

way 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 Jeffrey

K. Wang and then developed in the purely mathematical size. What we added is a path integral derivation of that, and I think we added how world 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, world crossing happens when you change the shape of Calabi-Yau threefold continuously, and at some point, all of a sudden, you are calibrated submanifold, the set of that changes. That's what world 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 a bunch of polynomials or rational functions as a result. So this actually followed similar computation by Benigni, Iego, Holy, Tachikawa a year ago

for the proposal of computing elliptic genus. You might think, especially physicists might think, oh, why not take this and 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 a better thing to do?

But turns out the whole point of this exercise is that that's impossible because those of you who have worked on world crossing will know elliptic genus does not have a world 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 the simplest case, the U1 with N fundamental representation,

which will define CpN minus one. 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 states. On the left-hand side, on the other side of parameter

regime, you get nothing. This is world crossing, this prototype of world crossing. This is just a model space of stable things which can disappear. That's right, that's right. Quintic, which would have gave you same Ho-Chi diamond in geometric

and Landau-Ginzburg phase. In one dimension, the vertical middle in particular disappears. There is a reason why the vertical middle disappears. Another Calabi-Yau example, which has four. This is one of the canonical two-calor parameter Calabi-Yau threefold,

which is embedded in weighted projective space here. And you get these various Ho-Chi diamonds. You can see something happens, vertical meters first, and then horizontal meters, etc. So every single one of these, I can compute. One thing I failed to mention, which I should

have done, is if this theory, this gauge theory, gives you a manifold at the end of the day, which is compact in geometry, if it flows down to nonlinear sigma model to a compact geometry, I'm effectively computing chi-y-z-ness. That's why I was able to get cohomology information.

But of course, chi-y-z-ness alone will not give you cohomology, yet I'm displaying the cohomology. This is of course a trivial example, you already know. Other things are not

that trivial. One of the things that happens in this business, in physics side, is that for certain classes, a very large class of gauge quantum mechanics, and in particular entire class of UN-type qubit quantum mechanics, there is a routine that allows me to reconstruct the entire

Hodge diamond in all chambers. That's another talk, sorry. I'm displaying, I use that routine, which I'm not telling you, and reconstruct this Hodge diamond of this qubit theory in particular chamber. Sorry about telling you details of this, but trust me, there is such a

routine, and there is all kinds of things I want to do with this thing, and in particular this qubit invariant concept that came out of physics, which allowed me to do things like this.

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 physicists go, this is rigorous as it can be. So I have a routine, I can put it in 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 word crossing where

cohomology changes suddenly, and also this sometimes fractional contribution or non-integral contribution you get, and both of these have 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 these examples I was showing,

what's generically happening is, remember there was this parameter c, and at c equals 0, something happens. So positive c, you have one cohomology, negative c, 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 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, world 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'll 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 world crossing happens precisely when this parameter is zero. 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 these integral numbers at the end of the day, without worrying about

on a lower of this one quarter earlier. 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 multiplex scholars. So in mathematics literature, they do not usually look at that direction, because they just gaze away, and usually you look at Higgs part, what we call Higgs part of the theory. So these are Coulomb branches.

What do you mean this part? I'm just displaying one axis for illustration. So the blue and gray 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 gap 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 comes down and gives you something funny that you want to be able to get rid of. So generically, this localization computation,

path integral computation does not give you integral things. However, as I was giving you this 5 over 4 example, this additional piece you get is not just an 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 the same set of problems, arbitrary gauge group, but now we have a new device to compute what's left-hand side of this, say,

the analog of Attia-Paterini-Singer theorem. So we will compute this omega quantity and for supersymmetry 4 and 8, you do this computation, you get these rational functions.

All the computation you should think of it is analogous to doing the same computation except you set the equilibrium parameter to nothing, which means y equals 1. Therefore, this gives you 1 over square, SU3 gives you 1 over 9, SU4 gives you 1 over 16,

this gives you 1 over n squared. 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 these linearly independent rational functions.

Remember, SUN was fine, but SO, SP, and others 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 of this first. And here is the answer, if you believe this or not. All the rational expressions you can package into the following

thing. Given a simple Lie group G, its while group is this, so this is cardinality of while group dividing the entire thing. And you sum over elements of your while group, but not everything,

but something called elliptic by 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 by. In fact, I haven't seen this elliptic by too many places. I don't know if mathematicians

use this much or not. But you do this sum, the 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, in arbitrarily simple Lie algebra. And in fact, this was what they obtained back then. With y equal 1. This is elliptic by. 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 squared are dimension-free things. And it doesn't really make much sense to send a dimension-free quantity to zero. 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 zero. 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 one. So you can make an intelligent 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 that, remember, I had a way of computing the left-hand side that Vasily did for arbitrarily simple Lie algebra. And I had a way of computing the right-hand side in such a way that this should cancel against it.

There was a small discrepancy. I have a new way of computing the left-hand side. And remember the smallest example where discrepancy happened was SP2 equals SO5. So same Lie algebra, and therefore it has same expression, same bial group, and therefore

it has same expression. And you evaluate this at y equals 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 equals that number, and therefore this index that you wanted is zero. So that's consistent with anticipation that no bound state exists for smaller supersymmetry. So let's buy this for now.

And then go on to n equals 16 version of that problem, maximally supersymmetric version. So this is the problem where the index, the integers you wanted, is not zero. It has

particular prediction coming from M-theory. And you do this, and there is this bunch of rational functions that follows, and every single one of these would have interpretation that was coming from a 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 equals 4 and 8,

and ask, is there a unique decomposition of this kind? And indeed, there is a unique decomposition. And for barriers 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 answers. Every single thing that follows is analog of this 5 over 32. It's a result of particular

continuum state in this asymptotically free problem, this problem where you have plane wave going out to infinity. So at the moment I'm interested, once I identify, once I know how to blame every single one of

these two 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 confirm this m-theory hypothesis, the other set of hypothesis that was never confirmed,

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 match 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. So that confirms this m-theory again. I mean, somebody like me of course thinks m-theory does exist, so it's not really necessary, but it's one extra block of evidence that

says m-theory does exist. Oh, by the way, this fractional structure that we see, these rational functions we see is not something that's accidental. It has been seen in

all kinds of other places. In particular, it has been seen in the word crossing formula, or I should say solutions to word crossing, conservative Serbian 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 pivot like this, where, I mean, when I say pivot, I assign integers for every single node. So if this ends have common divisors, then you have to have this sum, and then you have to acquire

these fractional pieces. Physics is exactly the same as the SUN problem, and what the computation suggests is it's path integral computation of this compute precisely this rational invariant. Okay, and then that allowed me to extract the integer path, integer quantity, by doing

reverse of this thing using this 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 gives you fractional quantities. Okay, so there are tons of things I'd like to do with this.

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-Kilman residue is a horrendous thing once we have a large number of charged particles in the problem.

So we'd like to understand asymptotic large-n version of this better, but I don't think such a thing exists yet, 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 sets of numbers that match precisely so that I end up with an integer, 0 or 1 or 2, whatever, such that everything is consistent. But this goes against what I said

like 30 minutes ago. There was a problem, right? There was a 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 are a mistake. But Stadocco, for example, in 2002, I mean,

he must have been a little bit uneasy with these numbers. 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 matrix integral technology

that gives you Jeffrey-Kiran residue. And in my mind, I have a relatively rigorous, physics-wise rigorous way of doing this. In fact, it's mathematically rigorous because it's no longer path integral. And I tried to get these numbers out of that computation,

and voila, I get these numbers again. So it's not the computation that is problematic. 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 bias sum. This is the 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 logical possibility that's left behind is that maybe the quantity I want you to compute, this small beta limit, is not computed by this matrix integral. That's only logical possibility that remains.

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 corner 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 things 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, by the time I get there, what I do is replace this circle by an infinite line. I pretend A0 was one of the scalar field

or joint scalar field. So 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 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 this place gives you one matrix model. This place gives you another matrix model. This place

gives you another matrix model. In fact, this actually happens for SUN. What happens in SUN depends on how you define your mechanics. There are actually N such places

displaced by the center of the SUN. So depending on whether you are careful about whether you do SUN or SUN mode set N, in one version, if you do SUN in the beginning, then you actually have N such identical holonomy setters. So you acquire a numerical factor N in front,

and without that factor N, you end up with a complete nonsense. How do you get these yellow points? Let me give you an answer in a more general setting. 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. In this problem, what that translates mathematically is that these are holonomy values where the unbroken gauge group is a maximal non-abelian subgroup. Remember how you

obtain those? You take a thinking diagram, make an extended thinking diagram, and you cross out one circle. That's the way you obtain a maximal non-abelian subgroup. All such holonomy that

gives you maximal non-abelian subgroups contributes, it turns out. And any other point does not contribute because of this small beta limit. So using this for SP, SU, SO, you can sort of

look for those maximal non-abelian subgroups. Those are those things, and you have this finite sum. And remember that I have a 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 number, these fractional numbers you wanted. And indeed, every single case, it works out. In particular, Basili's paper has a conjecture on individual of these SO and SP. 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. So it's not that the computation was wrong,

it was missing piece. And actually, I have to confess that that's my fault. When I did it, when Sadie and Stan did it, we were not very careful. We said, oh, this must be matrix

integral at the end of the day. And that was took up by Muon and a class of Shatashibili, and they were very successful for SU-N. And then that was took up by Sadako and et cetera. 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 was personally very satisfying because I ended up solving all the remaining problems I started out solving back then. Not mathematically, but physically at least. But I have a bonus. This Sadder thing. I think people

in old times noticed 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 kind of localization computation for higher dimensional gaze theories. Superconformal 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 holonomy 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 generically what happens, it turns out, d-dimensional manifold on some

supersymmetric manifold where you can define global supersymmetry as one time some manifold. In four-dimensional, this is typically what we call ciphered manifold. You compute this, take small beta limit, and try to express that in terms of three-dimensional

partition function on M. But there is typically a coefficient 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 coefficient

of exponent is called cardiac 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 Taurus,

I'm computing with an index again. And what this tells me is with an index of four-dimensional gauge theory is actually some of with an index in three-dimensional gauge theory in a very particular manner. So I mean people who did this sort of game a 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 true statement? No, I mean when I was saying SUN, it was in adjoint only theories. So generically this happens. So SUN adjoint only was very exceptional. And we are 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 a bunch of

them to a single four-dimensional supersymmetric theory. I think this is a very important idea. And in fact, this also explains world crossing that happens in 1D, which does not happen through this, exactly the same. If you're sitting on the right place, you do not miss holonomy

saddle. If you are sitting in some wrong place, you miss holonomy 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 gave me another systematic way of discovering multiple duality in 3D, starting from one single four-dimensional duality. Things like this have been noticed by the way collaboration of Aroni, Razamat, and Cyborg several years ago. But all of this is, I mean, you can sort of understand everything in terms of holonomy saddle.

Cardi-exponent, as I was saying earlier, I mean, let me, my time is up as I expected. There is actually very nice set of partition function invented by Clozet, Kim, and Willet.

And it's something like this configuration. You have T2 fiber 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 backyard, like this, and how they cluster. This clustering of backyard is essentially holonomy 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 Cardi-exponent of this type. And those of you who follow this story should remember that Pietro and Komagowski

made a very beautiful suggestion several years ago that this Cardi-exponent is simply given by, say, in conformal theory case, conformal anomaly, A minus 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 gauge group is relatively simple. They 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.

index, but this is probably not the crowd. So that's the end of story. In M-theory hypothesis, this game of trying to find or not find threshold bound state, I think this closed the entire chapter. And I was very happy to do it myself.

This notion of HZ is not just for quantum mechanics, but any gauge theory is a very important idea, I think. And in particular, 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 the law in tensor theory, but I think that's going too far here. Thank you. Mark? When you say it closes a sector in M-theory,

but it confirms for the N equal 16 case, everything is gone. So what is finally confirmed? So what is now? Conformation, 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 turns out to be O-N and S-P-N analog 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. So eventually, are there confirmed numbers for exceptional E6, E7? Very good.

As I was saying, our capacity was with this Jeffrey-Kirman computation. So right-hand side, this rational expression, we of course have it. Question is, can we do this Jeffrey-Kirman computation for the exceptional case? The highest we went to is S-P-4, rank 4.

We tried to do F-4, and things wouldn't just come out. I think I need supercomputer to do that. Or maybe you can invent better way of doing the contour integral. Probably, let's see. G-2, 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 tried E7, actually. But yes, but today, computers are bigger, and the computation is more complex. Yeah, I think it's that this old contour 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 F-4, I think number of charge vector is, 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 contour, badly. So I guess you discussed the case with 16 super-symmetries

and 4 and 8, but what about the 2 case, which corresponds to D equals 2 in my case? Actually, that's the problem I'm working on. Although you might find it strange that N equal 2 should be easiest, right? There's something funny going on, which I noticed even back in 1997 with N equal 2.

And of course, Jens demonstrated there is no such bound state at all. Yeah, yeah, yeah. And yeah, there is something interesting. Because of smaller super-symmetry, I don't know exactly what's going on there. Let me say I'm working on it.

But there is no, as far as I know, string theory prediction about the number. That's maybe one of the reasons nobody looked at it. Thank you.