Mathieu Moonshine: Symmetry Surfing and Quarter BPS States at the Kummer Point
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Taormina, Anne
Moonshine, Mathieu
The elliptic genus of K3 surfaces encrypts an intriguing connection between the sporadic group Mathieu 24 and non-linear sigma models on K3, dubbed “Mathieu Moonshine”. By restricting to Kummer K3 surfaces, which may be constructed as Z2 orbifolds of complex 2-tori with blown up singularities, it has been possible to devise a framework in which the concept of symmetry surfing can be explored and tested in a concrete way. This talk focusses on what has been learned so far that supports the symmetry surfing idea when lifting the Kummer construction to the level of conformal field theory, with particular emphasis on quarter BPS states. Some of these states enter the elliptic genus with opposite signs thus cancelling each other when counted by this index, yet they carry interesting information that should help understand Mathieu Moonshine.
 Well, first of all, I would like to say that I'm absolutely delighted and honored to be here today among Samsung's friends from a long time. I realize, I mean, I've known you for a long time too. Yes. I met between 89 and 91 in Chicago, where I just moved from Europe from one of my postdocs before going to Durham in England. And I vividly remember the lively conversations that we had, especially at lunchtime, with Jeff about physics, of course, but also about politics. Yes. And I think... Jeff also just moved. He had just moved. Yes. And the anecdote is that I was a young postdoc and everybody told me, you're going to Chicago, but Friedan and Schenker have left. I remember this is going to be dead territory there. And then, of course, I said, fine, I mean, I'll go anyway. And then I arrived surprised. Jeff was there, I knew her there. And Ewa, she also was there for a while. Yes. So it was great times. And since then, I think we have met in many, many places, sometimes unexpectedly, yes, but always with great pleasure. Right. So, right. So my talk today is covering several works that I've done with Catherine Wendland, who is my long-term collaborator. Certainly since 2010, you can see a little bit of a timeline there. So all started really just before I met Samson when I was in Paris, actually, a day called Normal. And Eguci, Toru Eguci, was a visitor for a year. And I was a postdoc there for a year. And we started talking about N equals 4 super conformal algebras in two dimensions. And we worked out the representation theory, unitary representations of these super conformal algebras. And we stumbled against a new phenomenon that at the time we couldn't deal with, is that the character is unlike what happened for other super conformal algebras at N equals 2, at no regime where the theory is rational. And this was the very first instance that we discovered many years later of a phenomenon that has to do with Mach modular forms, which have been developed by Zagier and Zweger's in the 2000s. Right. So we were a bit in the dark. So we published whatever we could. We calculated the elliptic genus of the K3 surface by these methods. And as you will see in the talk, in doing so, we came up with some numbers that became extremely interesting in 2010. So that's why I've put 2010 there. So we'll see why. And then since then, with Catrine in particular, we have followed a very specific line of investigation to try to understand these numbers in the context of string theory. Right. So the plan of the talk is to remind or to tell you what the elliptic genus of K3 is and what it counts. It counts in particular what I've called quarter BPST. So I will tell you what this is. And then I will introduce this mysterious phenomenon that has to do with the numbers I was mentioning before, which is called Matthew Moonshine. And then I will go on and look a little bit closer to K3 theories that I will define and tell you what I mean by generic and non-generic K3 theories and why it's important in understanding what's going on. And then one particular type of non-generic K3 theories are Z2 or before of toroidal conformer field theories. Right. So these ones give you a very practical way, hands-on way to try to explore the nature of these states that are counted by these magic numbers. And finally, I will conclude with a hope because the problem is not solved, still open, but Catrine in particular has spent a little bit of time into studying refinements of the elliptic genus that started the story there. And this, we think, might be very helpful in understanding the Matthew Moonshine phenomenon beyond the mathematical reality of it, which is proven by the way. So mathematically, we understand what the structure is, but we don't know what on this sporadic group that I'm going to talk about, which is Matthew 24, that's why there's a Matthew there, comes into string theory. Right. So to set the scene a little bit, I just want to define what my framework of theories is. These are two-dimensional super conformer field theories, and I take the unitary case here, which possess n equals 2,2, which is supersymmetry, but also spacetime supersymmetry, and the spacetime supersymmetry is actually, let's say, induced by the fact that you have a u1 in dn equals 2 super conformal algebra, and then the spectral flow generator helps defining the generator of supersymmetry and spacetime. And we look at theories which have the central charge in the left sector and the right sector, which coincide, which are equal, and which are quantized in units of 3, so 3 times d, where d, capital D there, will end up being the dimension of the manifolds, the target manifolds in the nonlinear sigma models that play a central role here. The space of states of these theories is this capital H there, you have the Neville-Schwarz and the Hamon sector, so this is both left and right here. So they are the boundary conditions that dictate what is in Neville-Schwarz within Hamon. And then in particular, I'm not saying that all the conditions and all the axioms for super conformal theory and two dimensions are on the board there, but only the important ones for today is that we have four zero-mode operators there, J0, Jbar0, T0, which you see is topological twist, T0 minus half J0, and Tbar0 there, which commute, which are self-adjoint, diagonalizable, and which are semi-definite positive for T0 and T0 bar. This algebra is enhanced, right, to a dark algebra for special D. It is indeed. And very soon, I'm going to set D equals to 2, because this is a Calabria two-fold hypercaler K3. That would be a darker algebra. No, I'm going to talk about, well, did you enhancement if you have a spectral flow which has what, is the n equals 4 super conformal algebra? Odake is a sequence 9, isn't it? I don't know if you could. Are you called it like this? Never called it like this. For me, it's sequence 9. No, okay, never mind. But we are on the same wavelength. Yeah. Well, I call this algebra n equals 4 here. Okay. Right. So, what's important for later is a part of the partition function of these theories, which are in the so-called Ramon twisted sector. So, the full partition functions of Neville-Schwarz and Ramon and twisted Neville-Schwarz and twisted Ramon. But if we concentrate on that particular sector, what we do is that we take a super trace. Now, I've written it in different ways here, because I'm going to jump from actually use more this formulation here, but because usually people see the partition function in that sector written like this, where we keep track of the quantum numbers associated with the t0 operator, not t0 yet, L0 in this language, the Viasol, so the energy essentially, and this one, which keeps track of the u1 quantum number in both left and right sector. In the s here, I've hidden this operator that I've explicitly written in the second line here, which basically counts bosonic and fermionic degrees of freedom with a sign. So, if you have n bosons and n fermions, then because of this super trace, they cancel in the counting. So, you are losing, we will see what happens later on. So, in the partition function, it's just counting with a sign there. Because of the properties of the partition function, you have restriction on the spectrum of these operators there, which I will use as well, and it's a beautiful automorphic form, this partition function here. You see that under the s modular transformation, which links z, sorry, tau, n minus 1 over tau, it's not 1 here, but it has a nice behavior anyway, where the central charge plays a role. Right. So, if now we take this partition function that I just showed you, and we set the y bar, so the u1, what was keeping track of the u1 on the right hand side, on the bar, then we have a definition of what the conformal field theoretic elliptic genus of such theories is. And this is the first line here, and I've chosen to use the trace over the Nevers-Schwarz states here, where now you see that I've put t0 here, not L0. Right. Now, it turns out that because of supersymmetry, many of these states in the trace here, once you put y bar is equal to 1, are going to cancel, and what you are left with is only a trace over the kernel of this operator here. So, from here to here, I have implemented a consequence of supersymmetry. And this object is an invariant, because since I restrict to zero eigenvalues of t bar 0, and I told you that the spectrum of t bar 0 minus t0 in the previous transparency is an integer, so it means that t0 here is going to have integer values, and therefore it's going to be producing an invariant, because you can smoothly deform it. It stays an integer number. And now if you look carefully at these type of expressions in these theories, you see that because now y bar being 1, there is a whole set of states inside here that are going to cancel in the right sector, in the bar sector, because there will be some which are fermions and some which are bosons, and a certain number will just cancel each other. So basically the elliptic genus gives you less information than the partition function, of course, and what's important is that you can now rephrase it as an invariant where you trace now under a subspace of ker t0 bar here, and this space here is going to be just what I called a generic space of states. So these are the states that stay after the cancellations I've just talked about, and this is going to be a very important space to make hopefully progress with this program. So the elliptic genus here expressed in terms of Jacobi theta function is something that we calculated a long time ago with conformal field theory techniques, and I should say what I was talking to Samsung about a minute ago is that I will, most of my talk restrict to what I call a K3 theory, which is one of the previous general theories that I defined before, but taking the center charge, c equals 6. But of course, at c equals 6, you have n equals 4, super symmetry, super conformal symmetries that we know, but we have two classes of theories, we have tori, and we have K3 theories. The tori have an elliptic genus, which is zero, trivial, and of course, if you have this elliptic genus, then you know that you have a K3 theory. Now there are some subtleties about K3 theories that it's widely believed that all K3 theories are realizable as non-linear sigma models on the K3 surface, but it's not proven mathematically but of course I will always refer to the non-linear sigma model in my talk. Right. So because we have an n equals 4 super symmetry there, it's possible to decompose the elliptic genus into n equals 4 characters, the characters that we computed with Eguchi a long time ago at the EMS. And when you do that, and we did that at the time, and we came with this type of expression. Now today, this is not going to be my concern at all, but it's because actually we are even more in the dark about the interpretation of this sector of the elliptic genus than we are about this one. But these are characters, unitary characters of n equals 4 for short representations or massless representations as we used to call them at the time. So and this one here, this object here, in itself is not a character, you have to think of it as being a common factor which when multiplied by Q to some power of H, H being an eigenvalue of L0, will give you a character. And this power of Q is actually encoded in this function of tau here, which I've written here. Right. So A of tau is a power series in Q, and each time that you fix n here, Q to this fixed n times this object there is an n equals 4 characters. So it is a decomposition in n equals 4 characters, and these numbers are the magic numbers that I was talking about before. Right. Now, let's talk about them, then I will come back to this. For 20 years almost, we didn't know what to do with these numbers. And then Eguchi and Oguri and Tashikawa went in Aspen one summer, and Toru, understand, knew something about K3 surfaces, of course, but also about the group of symplectic automorphisms of K3, and all of these groups are subgroups of M23, so the sporadic group Mathieu 23. So he opened the atlas with these other two friends there, and they looked at the character table for M23, and things didn't quite fit. They found in the characters for the identity, which is the dimension of the representation, they found numbers that were reminiscent of these ones, but not quite. And then they had this genial idea to go and look at M24, and there, clack, clack, clack. They found that 90 is twice 45, and 45 is one of the irreducible representation of M24, 462 is twice 231, and 231 is the irreducible representation of M24, and so on and so forth. Of course, there are only 26 irreducible representation of M24, this is an infinite series. So what happens after a while is that you get reducible representations, but you can always decompose them, of course, in irreducible ones, and if you do this for long enough, you see that all the irreducible representations come in, and that was proven by Garnon. So there is a mathematical proof that these numbers here are really to do with M24 and nothing else. So this is... So multiplicities? Yes, there can be multiplicities. If you go far away, these numbers grow very fast here, right? So you have only 26 building blocks, so they have to come with multiplicity. So a mathematical expression for large n? You said that you have a fast code. There are asymptotic formulas for large n, for an, you want to know an, yes? Yes, there are. So Eguchi and Ikami have worked these things in the context of black holes and the entropy. You won't tell me, you won't tell me, right? You will not tell me, that's the problem, right? I'm not going to use it, I'm sorry. It's going to be disappointing. Oh, I'm sorry to disappoint you. I can talk to you privately afterwards and show you nice formulas. Right. Okay. So, terminology here, in this elliptic genus, actually because of this y bar being one, it means that you project into a sector which is short representation on the right or massless, while on the left, I haven't done anything to the y on the left, so I get a mix of massless and massive, right? What I call quarter BPS here are basically states which are massive on the left and massless on the right. So it's not the full BPS spectrum here, but only part of it, right? And this sector here, as I said, is encoded into this part and this minus two here is not a clean business yet. I mean, so people think that might involve virtual representations of M24, but we haven't studied that yet. Right. So this phenomenon here is called Matthew Moonshine, right? Because these are coefficients of a mock modular form, right? And then suddenly the number theory is got interested, but it's linked to group theory and to string theory in a way that nobody expected, right? Right. So then we have... Sorry, and the minus two is not simply a fermionic representation. Yes, let me see. So the minus two, it has to do with the vacuum in the Nevers-Schwarz sector, right? You get signs because of the grading, that's true, but the story is much more complicated than just counting of fermions and bosons, the understanding of this, right? It's the Witten index of this representation, the vacuum representation. But you say the minus two is mysterious, sure. The fact that in the decomposition here, if you want to reinterpret in terms of, let's say, N equals four characters, you are forced into this pattern here with a minus sign there. And it is intrinsically due to this, but we don't understand the meaning, but these are not representation of M24, you see? Even if you take plus two here, okay, you can say it's one plus one, but it's not that. So it's mysterious, nobody understands that. Well, the first 26 terms is twice the dimension of representation. It is. And that's as well, always. It's always an even number, twice. But it's a combination, it's always twice. And the twice is because you have the representation, it's complex conjugate. So when you say 90, it's 45 plus 45 bar. Of course, some representations are real, so there you have a multiplicity two, right? And this is proven as well. So the people prove that all the entries are integers, that they are even and positive. So all this is under control. And it's a mock modular form. Whatever this is, this is not the talk today. Right. So now, to be clear on notations here, because I'm going to use them again, what I'm writing as curly age BPS is the space of states of quarter BPS. So all the ones that appear here. And this is in a subspace, and there's an injection here, into the space of generic space, the space of generic states that I've introduced before in the elliptic genus here. Why is it just an injection here? Because this space here also counts things which come from here. So that's the link here. So I'm restricting myself today to this part of the story. And there's a color code here, which I hope will be consistent, is that this violet color, purple color here, always has to do with the generic space of states. So if you see violet, this is magenta, it's different. Right. So generic and non-generic K3 theories. Now, if you, there is a notion of modular space for K3 theories, and we know very well one component. We think there is only one, but nobody has proven it again. But in that component, the landscape is well known of modular, modular, the modular space. And as you vary the modular, so you go from one theory to the next, you have different behaviors. And although the generic states are common to all the K3 theories, at certain points in the modular space, you have an enhancement of symmetry, maybe, if you have more quarter BPS states that are going to come in. And therefore, but these quarter BPS states that you have in surplus at some points happen to cancel because of the minus one to the J0 minus J0 bar. So that's what is the philosophy of this table here, is that if you have a generic K3 theory, then you don't have, at this particular point, extra cancellations that happen. And that's what is written here. Now if you are non-generic, and the Z2 or before conformal theories are a class of such generic theories, then you have these cancellations that happen in the elliptic genus. But there are pluses and minuses in the two cases. So the plus sign for the generic is that you don't have to worry about these extra cancellations. And also for the generic one, but on the other hand, nobody knows explicitly any of these theories. So practically you can't do anything with them. On the other hand, the non-generic ones, well, we know very well these ones, they have the same spectrum of generic states. That's something that you can show. But there are cancellations. So you have to kind of navigate between the two, and today I'm going to talk more about these ones, and my last transparency will be basically about the future. Right. So to be clear again, in non-generic K3 theories, the space of massive quarter BPS states, ah, he didn't put the color right, should be violet, is actually not the same space as the full space of quarter BPS states, because the full space here will involve the states that cancel in the elliptic genus. Right. Okay. For example, in Z2 conformal theory, or before the conformal theories, you see that this is what the elliptic genus measures, the numbers, the magic numbers of before. But if you look at the dimension of these spaces here, okay, you see that they are higher each time. And if you go from 90s to 102, they are 12 states. So what exactly is this HM straight HM plus structures? This one here? No, lower, lower. Lower. Yeah. Here? Yeah. This one here. Well, I'm going to say in a second. Okay. So here I've not talked about H plus yet, I've just talked about these two, all right. Then you have 12 here at level one, and this is six fermions and six bosons that cancel in the elliptic genus, but they exist in the theory. Okay. Now this space here is actually what is left once you have taken away. So H plus at level one here would be the 12 I've just talked about. Okay. So this is what I call the excess states. So it's a bunch of fermions and bosons that cancel. Standard BPS, and then you get Z2 CFT, right? Consistency. Sorry? Combined with this HBPS same, and then... Yes. And then you get the whole state. Yes, absolutely. So it's just selecting, yeah. Right. So we don't know really what drives these cancellations. They just happen in the elliptic genus. So we wanted to understand this in a practical setting, which are these Z2 Oberfolt CFTs, and very, very quickly, I've jotted here what will be useful for later here, is that the construction of these are from a toroidal super conformal field theory at central charge six, and this construction here is induced by the Kuhmer construction of K3 surfaces. So it's not by chance that it's like this. There are deep connections between the geometry and the conformal field theory there. So Kuhmer surfaces are a special class of K3 surfaces, and they start by taking a torus, C lambda, which is obviously given by C2 mod lattice, which has four real dimensions. So there's some geometry in here, and then we take a Z2 Oberfolt of this torus. In the process, we get fixed point, 16 of them, and therefore, all this is a different story from today's story, but it has to do with 16 fixed points in the conformal field theory language, which corresponds to the 16 twisted sectors of the theory. So it's beautiful. And if we wanted a free field representation of the symmetry here for these theories here, we can achieve them by looking at four Dirac fermions and their superpartners, and then this is the realization of these generators of n equals 4 in terms of these free fields. I'll come back to that a little bit later so you will see this transparency again, decorated further later. Right. So I'm after the elliptic genus in these theories, so I start with the partition function. And I've chosen to work in the Neuverschatz sector because at the end of the day, the remarks I will make at the end connect directly to the Neuverschatz sector. However, in n equals 4, the Neuverschatz and the Pramunsector are isomorphic to each other. So you don't lose or you don't win information, it's just a repackaging with, of course, a different physical interpretation. Right. So the partition function in the Neuverschatz sector, typically in these all-before theories, have an untwisted sector. So this is a schematic to indicate the boundary conditions in the town, the sigma directions, right? So we have an untwisted and a twisted sector. And if we are interested in the generic states of the theory, then we have to remove in the full partition function here whatever depends on the moduli. And it turns out that the moduli dependence is all in the untwisted sector. And remember, it's a conformatory, the conformal theory, so it comes from a torus and story in the Hein, Lattice and so on. So the moduli are really in that part of the theory. So I'm going to set the momentum and the winding number of the states to zero. Because if I don't keep them, then I keep a dependence on the moduli because these change. So this is what I call a generic partition function. And in terms of the elliptic genus expressed in the Neuverschatz sector, it can be expressed in terms of generating functions for the twisted. So this is the twisted, sorry, untwisted, red is untwisted. And this is twisted. They are written in terms of Jacobi-Titta functions again, usual things. You have to keep modes, not modes, states, of course, which are either bosonic or fermonic. The details are in the paper, but it's not interesting for us today. And it turns out that mathematically, these three generating functions can be decomposed, as we know, because this is the elliptic genus, into and equals four representations again. So this is basically a slightly rewritten version of the elliptic genus from the point of view of the Neuverschatz sector. But here, it's the same function as before, because the spectral flow that allows you to go from Neuverschatz to Ramon is actually an operation that affects the y dependence of your object. And this a of tta function, obviously, doesn't depend on y. So it's not going to move through the spectral flow. So it's no surprise that even in this repackaging here, you see the same numbers as before. So now we are looking at this. And right, it's a little bit more involved here, but I hope that I can make myself clear. So it's color coded. So we already know that the purple here is for generic states. The red ones are the untwisted sector, the number of states in the untwisted sector, and the blue ones are the number of states in the twisted sector. So this is, you have seen this, the ansatz is that this expression that you have seen before, so these are all the quarter BPS massif that I was looking at. These are the excess states that we talked about. Now what I say is that within this class of states, I have two contributions, right? And I have to explain to you what these two contributions are in a minute. But let's keep it just as a formal exercise for a minute. Then you have all the ingredients which are written here, right? And why do I call this first one H and P. Now, very briefly, I told you that all this is linked to the 16 fixed points in the Kummer construction. And then you have 16 twisted sectors in the Kummer field theory. In this 16 dimensional space, I can identify a very symmetric direction. And how do I do this? Well I know that on this twisted space, I have an action of a geometric group. This geometric group being linked to the geometry of the Taurus T lambda, the lattice I use for my Taurus in the Kummer construction as some geometric symmetries. And after a few steps, basically the upshot of all this construction is that on the twisted space, this geometric group acts. And if you take a sum of all the ground states of these twisted sectors and you sum them under the action of the group, it's an invariant. Because the geometric group acts as permutation of these ground states. So the typical construction. Take this. So this is what I call the diagonal direction. So in my 16 dimensional twisted space there, I have a direction which is my diagonal one and then I have everything else which is orthogonal to it. And that's the perp there. So the perp there is a 15 dimensional space there in terms of the ground states in the twisted sector. And I've chosen to write it as 15 contribution of one sector here. And the diagonal one, in a minute we are going to split it again a little bit. The rest here is basically the leftover quarter BPS states that are not, that are generic. So all this is HBPS. These are the excess states. So that's how it is. And of course I have at my disposal the dimensions of three objects, CN, BN and DN. And why is this an ansatz? Is that by looking at the first two levels, so n equals one and n equals two, we have noticed that actually it seems... Right? Okay, so what I was saying here, so basically here what we know for sure is that a certain subclass of untwisted states automatically comes into the excess. Because basically they come with a minus sign and they are less of them at each n than the DNs. So we know that they are here. What we don't know is what the partners of these states are in H plus to cancel. They could come from CN or DN. And the ansatz is that none of the untwisted ones are partnering in the two BNs in the excess sector. So this remains to be proven for general n. But it turns out that Keller and Zade recently have shown that for n equals one, in a way it's not so difficult there because for c1 here, c1 is zero. So what they've shown is that anything in the excess sector here was actually lifting off the BPS bound when you deform the theory. So you go away from the Kuhmer point, you introduce CFT deformations and then these states don't stay BPS. So therefore they are not generic. So let's say what's behind all this. Is there a general formula for all, some nice formula for each excess? So the generating function of before, UL0, UL1-1-1-1-1, that I wrote, you have a closed form. It contains it of course, but it's just for excess numbers. For the excess numbers, yes, you can, you have a number, but you don't know. So it's all about numbers at this stage. But you need to know what the states themselves are because if it's just a number, you just know it's a boson or a fermion. But for example, in terms of the free fermions and free bosons that I showed you before, what are they? We wanted to understand a little bit more closely the nature of these states, apart from the fact that, apart from their statistics, we wanted to know how that, so we did an explicit construction for N equals 1 and N equals 2. N equals 1 is a piece of cake, but N equals 2 is what we did in Stony Brook last spring. Catherine and I were there and we just sweated over this because I will tell you why it was complicated. But anyway, so this is because we didn't have a better idea. We really wanted to see, okay, what are these states? Not just their fermions or numbers of fermions or bosons. What are they? Okay, so we did that. Okay, so I think the time is running, isn't it? You have 12 minutes. 12, okay. So I think I'm going to skip this one, but not go into the details of it. But basically at level one, this we did a few years ago already, we noticed that all the states that are generic, okay, actually fit into representations, not of M24. I mean, this space is the same for an M24 representation, but it's a representation of a maximal subgroup of M24. Now, why is this maximal subgroup of M24 so exciting to us? It's what we call G-octad here, right? It's Z to the 4 semi-direct product with the alternate group of 8 elements there. It's because it comes from our idea of symmetry surfing, okay? So in words, symmetry surfing is that you have a space, and this is going to be the generic space of states that I'm talking about, which is common to all K3 theories, right? And then if you travel from one point in that modular space to another one, the symmetries are different. The geometric symmetries are different because you have different torii that you play with. And therefore what you do is that you collect all the symmetries at the different points, all the geometric symmetries, and therefore you obtain this G-octad. So there is circumstantial evidence that it's not such a stupid idea, right? Again, I mean, it's just evidence so far, and it's not proven that it's really what's going to happen in this problem, right? But at least we have this group to hook on. And it turns out that at level one, we see very clearly because we had the help of Mar Golin, who is a group theorist, who had written a paper ages ago about the 45-dimensional representation of M24, and the way he constructed it was very helpful in order to make this statement there. But again, I don't have the time to go into the details here. And then, apart from Keller and Zadeh that really confirm what we expected at level one, Faber-Deele, Keller and Paul also did a very fine piece of group theory there, a very elegant paper, where they actually say that not only, whoops, here, okay? So not only the octad group acts on H-perp, which is all you could see at level one, but they looked at all levels and they see that the octad group act on this space, but also on all the other massive quarter-BPS states here that you have at your disposal in the generic space of states, right? So again, it's evidence that the symmetry-surfing idea is not dead yet, right? But again, I'm being cautious here, right? So this, I think I said one way or the other. So basically, what we did in order to get more feel is to construct the states, but we've used an extra ingredient. And the extra ingredient is to utilize the fact that in the building blocks of the Z2 or before super conformer field theories here, the building blocks also have a certain behavior under an SU2 group, like this is a global SU2, it's not, it's a global SU2 and it's actually a subgroup of the global SO4 that is common to all N equals 4 super conformal algebras here. And they transform as doublets under this SU2 and we've kind of implemented this refinement in the generating functions u and t that I had before. So we have introduced another complex number there to keep track of this global SU2. And what we found in doing this is that very explicitly up to here, so for 1 and 2, so you can obviously make predictions from the analytic expressions of the generating functions that you have here. But for these two levels here, we have painstakingly computed, calculated all the states, so these 16 states here and these 28 states here explicitly in terms of the building blocks and identified their behavior under the SU2. So how do they decompose? So if I take this 16 here, how do they decompose under this SU2? And the notation here is that we have the multiplicity here, so it's two triplets here and two quintets. So 2 times 3 plus 2 times 5 is 16, yes, and so on and so forth. So we've done this for the three generating functions that I showed you before and what we see is that the excess states that we wanted to cancel here, they are 6 plus 6, the 12th that I talked about before. Now you had no choice at level 1, I said because there's nothing in the rest here, so this CN is in the rest. So all the cancellations have to come from the D if you agree with me that all the BNs are in the excess and come with a minus sign. And then we have a minus 2, remember it was minus 2 BN. So we have minus 2 singlet under this SU2 and here you have two singlets in the DN. So it seems that the SU2 behavior of the states is a guiding principle to see where the cancellations have to come from. So you see at the second level in CN, so in the untwisted sector you have no singlets of this SU2 that you could use to cancel twice a singlet here. On the other hand, in the DN you have two of these singlets, just two, just lucky and therefore you cancel them. So again, it's an observation and as you go along in the levels it becomes a little bit more ambiguous because let's see, I think if I come to this level here for sure, we have enough in the C sector, in the untwisted sector to cancel the one there. Here you don't have enough, you would need 3 to the 6 here. Here you have enough. Okay, so even here, here you have enough. So again, we have unfortunately no confirmation from proper deformation theory that our idea is absolutely corrected all level at this time. What Keller and Zadeh that did for the level 1 now with our latest work here, we have prepared the way in order to do the same exercise at level 2, but second order perturbation theory and conformal field theory is technically quite involved and maybe we should have a better idea which is what I'm going to talk about next. So this is a different level of thought here and the novelty on this transparency is actually due to Catherine here and almost simultaneously there was a paper by Bélin Son with a mathematician who published quite a difficult paper to read but which basically raised the same conclusion as Catherine was published in CMP. So what is this? Well, I've talked a lot about the conformal field theory to the elliptic genus. It's been known since Whitton and it's a book that it's actually for K3 surfaces, it's equal to the complex elliptic genus. So there's a link between the geometry and the conformal field theory at this level. And in some incarnation this elliptic genus in terms of tangent bundles and here I've taken a short curtain because we don't have the time to go into this. I can only say in words what this TLM is but it's coming from looking at first of all the basic ingredient is going to be the tangent bundle of the manifold X. So T10 and then there is a virtual bundle so it's a formal sum of bundles that are constructed by taking the exterior product of the T bundle, so the tangent bundle times exterior product of the dual tangent bundle times the symmetric product of the tangent bundle times the symmetric product of the dual tangent bundle. So it's quite a sophisticated object but that has been studied at length in the literature. And that virtual bundle that I didn't write for you here, you can recast it as an infinite sum here in two variables where L goes from 0 to infinity while M here is bounded between the dimension and minus the dimension of your manifold here. And it's a formal power series in the variables Y and Q that we are familiar with and this is the holomorphic Euler characteristic of this bundle that I've constructed the way I said before. Anyway, so this is the expression and then in 2016, Kacru and Tripati had quite a good idea they said okay since the cancellations of states in the conformal field, the elliptic genus is a pain because we don't really see what's going on, let's try to lift this cancellation and introduce a new grading which is V here in the right sector. So somehow reinstating a little bit of the partition function thing there and they did that and then they also showed that you have a counterpart in the geometry except that you can show that these two objects don't agree, here they agree, here they never agree. So that's the first remark. And then there is much more to this that this object might not be actually that helpful in the problem at hand where we want to understand Matthew Moonshine. And then what Katrin has observed is that now instead of taking the trace just over the kernel of t bar zero, she introduces these two, a grading but she just traces over the generic states, so the generic states being a subset, a subspace of the curve. And the inspiration for this is that Kasper Kapustin, I forgot to put his name here because it was quite crucial, in 2005 he had this idea that if you take the infinite volume limit of nonlinear sigma model with a topological twist then you get the cohomology of the chiral deram complex as this paper. And this is exactly where we are, is that you can define a new elliptic genus that she calls the Hodge chiral here, which basically has to do with taking the trace over the group, cohomology group here of the chiral deram complex. Now again, very vaguely, since I don't have time, for k3 surfaces these two can be shown to agree, so this is equal to this, it's an invariant, you can show that. And there is a structure, this chiral deram complex is a so-called sheaf of super vertex operator algebra which has to do with the BC-Bitagama system. So there is a structure of VOA there, for the Mathieu Munchen we also would like to see a VOA structure on which the Mathieu group 24 acts. We don't see it, but we think that this chiral deram complex here and the sheaf cohomology of it are going to be a mathematical object that models the generic spaces of states in the k3 theories and if we can find more structure on it, then maybe we will see M24 acting on that extra structure, that's the direction of travel. So I think I'm at the end here. So I didn't tell you that. Yes, that's a little remark, I put it there because I knew I was going to not say it, it turns out that if you take complex tori, so T for tori, these three quantities are equal. So when I say that this is not equal to this one, for tori it happens to be true, but it's not where we want to be. K3 is not true. And that's probably why at the beginning people thought that this would be the way to go for Mathieu Munchen because I saw it worked for tori. But if you really calculate things for K3, then there is a differentiation there. So basically that's not where we concentrate our efforts, just trying to understand more about the Karelderam complex. So this was, oops, okay, I'll spare you the conclusion because I'm out of time, I'm sure. So this is for Samson because it's his birthday. Now I wanted to put something from Belgium because I'm Belgian, right, unlike, it's not clear from my name, but that's what I am. And in Belgium we like to say things with cartoons, you know, very much into cartoons. This guy called Gelluc, okay, Gelluc, we say in French, he's also known in France by the way, and he has this series of snapshots with some philosophical thoughts which usually are quite thought provoking, which in, with a cat, that's a cat, yes, I don't know, so see that. And of course he speaks French, but you understand French of course, yes, okay. So okay, and there's a little message there is about saving the planet of course, so we want to have 60 candles on your cake just to save the planet. Okay, thank you. Questions? Yes? Yeah, it's just very general question from this very old tab about 88 or something like this, there's also suggestion of written, this should be some 24 dimensional, and hits the book at the same time, should be 24 dimensional real manifold with section of a monster, such as this 24 dimensional manifold should have elliptic genus which would be J function. So one get geometric realization of monster representations directly in index of deregoperator. So there's really no progress after that. Do people, do physicists have an idea whether it should be true or not? Well, personally I don't, I never really looked, I know that, I mean it's kind of expected that because M24 is a big stepping stone in constructing the monster that there should be a link, a deeper link that we have not grasped yet between the monster moonshine and this one. But I've not really looked at all into that direction, so I know that other people working on Matthew moonshine, what I didn't say is that there are now more and more instances of moonshines where you have automorphic forms whose coefficients have to do with the dimensions of some representations of some groups, yes? So there are fascinating things happening, but no, I don't know, I don't know more. Basically I've told you where I am. And very much, very much K3 for me. Yeah, my question is, it seems like you can still get quite a bit of information without knowing the conformal filtering, for instance you have this space of states 192 or 100, so how can you know this without knowing what the CFD is? For instance, you know, you remember you had this A, I think it was the number of B, it was called the BBS states, although you don't know. But because the elliptic genus is invariant over the whole space, so, oh it's proof, it's well you can prove that the elliptic genus doesn't depend on the modularity at all, so. But if you cannot compute it in any CFD because you can. On Z2 or before, you can, it's a special case, so you can calculate the elliptic genus. But if you go away from the orbital point, you still have it. Away from the orbital point, no, we can't compute anything. We can kind of perturb a little bit and see what's happening. But you knew, for instance, the dimension of this H0, whatever it was, was 100 at any equal one. Yes. So this you were able to infer, nevertheless, even though, even away from the orbital point, it seems to me. Yes, but we did calculate it from an orbit, we did, the calculation I did with Eguchi in 1988 was in the framework of Z2 or before, the gap in the models. So you calculate, because it's an invariant over the whole space, you can choose a point when you know how to calculate. And these numbers that you are talking about are pitted out at this particular point. And because you can prove it's an invariant, and even if you don't know the theory, you know that these will be the same numbers. What you don't know is the number of excess states. So the number of excess states vary from one theory to the next. But in the conformal field, in the CFT elliptic genus, they cancel. Are you sure the theory exists? Which theory? I mean, the super conformal theory, which you cannot construct. Because don't you assume it exists just to prove that? Just to know. You are a mathematician. Yeah. Assumption? Well, the modular space is 80 dimensional. It's proven mathematically. So you can't exhaust the set of theories with only the few Gepner models, which are isolated points, and the class of Z2O before. So I think what the shape is of this K3 theory is nobody knows. Well, at least I don't know. I don't know. Maybe other people have ideas. And this magic number has lower index n. Can you write recursion in relation? No. No. No, asymptotic formulas, yes. You wanted asymptotic formula from me. Sorry? In the beginning, asymptotic formula. Yes. Yes. That's a good shenikami. I'm sure I can find a paper here if you want, if you're interested. Yeah. Yes, they looked in connection with the entropy of black holes. Enfactorial? It's complicated. But then I'm sure there are some enfactoists. I don't remember. I had a question about this hot electric. Yes. The electricians, we looked at this when Katsuh and. Yeah, Kaku and Tipati. Yeah. They proposed it, but then we didn't know how to make sense of it because I think there was something that was not clear, that was a BPS condition. Because indeed we want to relate it to counting of black holes where you have spanned black holes. Yes. But also we want to generalize the product from symmetric products. Because then you expect also some connection with Segel model work forms. Okay. Is there no anything? Because I mean this new definition that is a completely different object. For example, it relates, let me see. Right. So there's also several, or at least one paper by Koitsik and Heun that they've looked quite deeply into the hot elliptic genus. And they've tried to. So they looked and they found that in there, there are some, now I'm getting a bit out of my depth here, but there are some global section, holomorphic sections that are not N equals four in there. And therefore, while for the Carol Durand one song, this is Bellingson as shown that the global holomorphic sections are all N equals four only. So that's one way to differentiate the two. It's also that Kapustin really points into the direction that the topologically twisted half twisted nonlinear Segel model has to do with the comology of the of the Carol Durand. Nothing to do with the comology of this bundle that you're talking about. So it's intrinsically a new, a new object. You can't really calculate things in the conformer theory side because this is the state of generic, the space of generic states. This is really in so. So it has a geometric definition. So the geometric definition, which agrees with the conformer theory one, that's proven, I think I'll try and prove it, but we need to study more looking for M24 there in the geometric part because I think it's quite powerful if we can find it there. You saw with the Z2 before, it's a bit painful. There was some variation of the theme by Shamil Tenaisi, we run to change involving special holomorphic case. Yes. Can you comment on that? No. I don't even read this paper. I know it exists, but. There were some papers about a couple of years ago, and then the Z2 came up and this M24 is replaced by something. Well, I thought they only can realize subcruits of M24. That's what I thought was the presence. No, it algebra came. It was not a joint. It was in the G27, in this algebra, I had this one. They used an algebra which looks like that and constructed something. I tried to read, I could not understand. No, I've read these ones. I mean, sorry. Next time. Maybe it's a good time to break for lunch. Thank you very much.
 Well, first of all, I would like to say that I'm absolutely delighted and honored to be  here today among Samsung's friends from a long time.  I realize, I mean, I've known you for a long time too.  Yes.  I met between 89 and 91 in Chicago, where I just moved from Europe from one of my postdocs  before going to Durham in England.  And I vividly remember the lively conversations that we had, especially at lunchtime, with  Jeff about physics, of course, but also about politics.  Yes.  And I think...  Jeff also just moved.  He had just moved.  Yes.  And the anecdote is that I was a young postdoc and everybody told me, you're going to Chicago,  but Friedan and Schenker have left.  I remember this is going to be dead territory there.  And then, of course, I said, fine, I mean, I'll go anyway.  And then I arrived surprised.  Jeff was there, I knew her there.  And Ewa, she also was there for a while.  Yes.  So it was great times.  And since then, I think we have met in many, many places, sometimes unexpectedly, yes,  but always with great pleasure.  Right.  So, right.  So my talk today is covering several works that I've done with Catherine Wendland, who  is my long-term collaborator.  Certainly since 2010, you can see a little bit of a timeline there.  So all started really just before I met Samson when I was in Paris, actually, a day called  Normal.  And Eguci, Toru Eguci, was a visitor for a year.  And I was a postdoc there for a year.  And we started talking about N equals 4 super conformal algebras in two dimensions.  And we worked out the representation theory, unitary representations of these super conformal  algebras.  And we stumbled against a new phenomenon that at the time we couldn't deal with, is that  the character is unlike what happened for other super conformal algebras at N equals  2, at no regime where the theory is rational.  And this was the very first instance that we discovered many years later of a phenomenon  that has to do with Mach modular forms, which have been developed by Zagier and Zweger's  in the 2000s.  Right.  So we were a bit in the dark.  So we published whatever we could.  We calculated the elliptic genus of the K3 surface by these methods.  And as you will see in the talk, in doing so, we came up with some numbers that became extremely  interesting in 2010.  So that's why I've put 2010 there.  So we'll see why.  And then since then, with Catrine in particular, we have followed a very specific line of investigation  to try to understand these numbers in the context of string theory.  Right.  So the plan of the talk is to remind or to tell you what the elliptic genus of K3 is  and what it counts.  It counts in particular what I've called quarter BPST.  So I will tell you what this is.  And then I will introduce this mysterious phenomenon that has to do with the numbers  I was mentioning before, which is called Matthew Moonshine.  And then I will go on and look a little bit closer to K3 theories that I will define and  tell you what I mean by generic and non-generic K3 theories and why it's important in understanding  what's going on.  And then one particular type of non-generic K3 theories are Z2 or before of toroidal  conformer field theories.  Right.  So these ones give you a very practical way, hands-on way to try to explore the nature  of these states that are counted by these magic numbers.  And finally, I will conclude with a hope because the problem is not solved, still open, but  Catrine in particular has spent a little bit of time into studying refinements of the elliptic  genus that started the story there.  And this, we think, might be very helpful in understanding the Matthew Moonshine phenomenon  beyond the mathematical reality of it, which is proven by the way.  So mathematically, we understand what the structure is, but we don't know what on this  sporadic group that I'm going to talk about, which is Matthew 24, that's why there's a  Matthew there, comes into string theory.  Right.  So to set the scene a little bit, I just want to define what my framework of theories is.  These are two-dimensional super conformer field theories, and I take the unitary case  here, which possess n equals 2,2, which is supersymmetry, but also spacetime supersymmetry,  and the spacetime supersymmetry is actually, let's say, induced by the fact that you have  a u1 in dn equals 2 super conformal algebra, and then the spectral flow generator helps  defining the generator of supersymmetry and spacetime.  And we look at theories which have the central charge in the left sector and the right sector,  which coincide, which are equal, and which are quantized in units of 3, so 3 times d,  where d, capital D there, will end up being the dimension of the manifolds, the target  manifolds in the nonlinear sigma models that play a central role here.  The space of states of these theories is this capital H there, you have the Neville-Schwarz  and the Hamon sector, so this is both left and right here.  So they are the boundary conditions that dictate what is in Neville-Schwarz within Hamon.  And then in particular, I'm not saying that all the conditions and all the axioms for  super conformal theory and two dimensions are on the board there, but only the important  ones for today is that we have four zero-mode operators there, J0, Jbar0, T0, which you  see is topological twist, T0 minus half J0, and Tbar0 there, which commute, which are  self-adjoint, diagonalizable, and which are semi-definite positive for T0 and T0 bar.  This algebra is enhanced, right, to a dark algebra for special D.  It is indeed.  And very soon, I'm going to set D equals to 2, because this is a Calabria two-fold hypercaler  K3.  That would be a darker algebra.  No, I'm going to talk about, well, did you enhancement if you have a spectral flow which  has what, is the n equals 4 super conformal algebra?  Odake is a sequence 9, isn't it?  I don't know if you could.  Are you called it like this?  Never called it like this.  For me, it's sequence 9.  No, okay, never mind.  But we are on the same wavelength.  Yeah.  Well, I call this algebra n equals 4 here.  Okay.  Right.  So, what's important for later is a part of the partition function of these theories,  which are in the so-called Ramon twisted sector.  So, the full partition functions of Neville-Schwarz and Ramon and twisted Neville-Schwarz and  twisted Ramon.  But if we concentrate on that particular sector, what we do is that we take a super trace.  Now, I've written it in different ways here, because I'm going to jump from actually  use more this formulation here, but because usually people see the partition function  in that sector written like this, where we keep track of the quantum numbers associated  with the t0 operator, not t0 yet, L0 in this language, the Viasol, so the energy essentially,  and this one, which keeps track of the u1 quantum number in both left and right sector.  In the s here, I've hidden this operator that I've explicitly written in the second  line here, which basically counts bosonic and fermionic degrees of freedom with a sign.  So, if you have n bosons and n fermions, then because of this super trace, they cancel  in the counting.  So, you are losing, we will see what happens later on.  So, in the partition function, it's just counting with a sign there.  Because of the properties of the partition function, you have restriction on the spectrum  of these operators there, which I will use as well, and it's a beautiful automorphic  form, this partition function here.  You see that under the s modular transformation, which links z, sorry, tau, n minus 1 over  tau, it's not 1 here, but it has a nice behavior anyway, where the central charge plays a role.  Right.  So, if now we take this partition function that I just showed you, and we set the y bar,  so the u1, what was keeping track of the u1 on the right hand side, on the bar, then we  have a definition of what the conformal field theoretic elliptic genus of such theories is.  And this is the first line here, and I've chosen to use the trace over the Nevers-Schwarz  states here, where now you see that I've put t0 here, not L0.  Right.  Now, it turns out that because of supersymmetry, many of these states in the trace here, once  you put y bar is equal to 1, are going to cancel, and what you are left with is only  a trace over the kernel of this operator here.  So, from here to here, I have implemented a consequence of supersymmetry.  And this object is an invariant, because since I restrict to zero eigenvalues of t bar 0,  and I told you that the spectrum of t bar 0 minus t0 in the previous transparency is  an integer, so it means that t0 here is going to have integer values, and therefore it's  going to be producing an invariant, because you can smoothly deform it.  It stays an integer number.  And now if you look carefully at these type of expressions in these theories, you see  that because now y bar being 1, there is a whole set of states inside here that are going  to cancel in the right sector, in the bar sector, because there will be some which are  fermions and some which are bosons, and a certain number will just cancel each other.  So basically the elliptic genus gives you less information than the partition function,  of course, and what's important is that you can now rephrase it as an invariant where  you trace now under a subspace of ker t0 bar here, and this space here is going to be just  what I called a generic space of states.  So these are the states that stay after the cancellations I've just talked about, and  this is going to be a very important space to make hopefully progress with this program.  So the elliptic genus here expressed in terms of Jacobi theta function is something that  we calculated a long time ago with conformal field theory techniques, and I should say  what I was talking to Samsung about a minute ago is that I will, most of my talk restrict  to what I call a K3 theory, which is one of the previous general theories that I defined  before, but taking the center charge, c equals 6.  But of course, at c equals 6, you have n equals 4, super symmetry, super conformal symmetries  that we know, but we have two classes of theories, we have tori, and we have K3 theories.  The tori have an elliptic genus, which is zero, trivial, and of course, if you have this elliptic  genus, then you know that you have a K3 theory.  Now there are some subtleties about K3 theories that it's widely believed that all K3 theories  are realizable as non-linear sigma models on the K3 surface, but it's not proven mathematically  but of course I will always refer to the non-linear sigma model in my talk.  Right.  So because we have an n equals 4 super symmetry there, it's possible to decompose the elliptic  genus into n equals 4 characters, the characters that we computed with Eguchi a long time ago  at the EMS.  And when you do that, and we did that at the time, and we came with this type of expression.  Now today, this is not going to be my concern at all, but it's because actually we are even  more in the dark about the interpretation of this sector of the elliptic genus than we  are about this one.  But these are characters, unitary characters of n equals 4 for short representations or  massless representations as we used to call them at the time.  So and this one here, this object here, in itself is not a character, you have to think  of it as being a common factor which when multiplied by Q to some power of H, H being  an eigenvalue of L0, will give you a character.  And this power of Q is actually encoded in this function of tau here, which I've written  here.  Right.  So A of tau is a power series in Q, and each time that you fix n here, Q to this fixed  n times this object there is an n equals 4 characters.  So it is a decomposition in n equals 4 characters, and these numbers are the magic numbers that  I was talking about before.  Right.  Now, let's talk about them, then I will come back to this.  For 20 years almost, we didn't know what to do with these numbers.  And then Eguchi and Oguri and Tashikawa went in Aspen one summer, and Toru, understand,  knew something about K3 surfaces, of course, but also about the group of symplectic automorphisms  of K3, and all of these groups are subgroups of M23, so the sporadic group Mathieu 23.  So he opened the atlas with these other two friends there, and they looked at the character  table for M23, and things didn't quite fit.  They found in the characters for the identity, which is the dimension of the representation,  they found numbers that were reminiscent of these ones, but not quite.  And then they had this genial idea to go and look at M24, and there, clack, clack, clack.  They found that 90 is twice 45, and 45 is one of the irreducible representation of M24,  462 is twice 231, and 231 is the irreducible representation of M24, and so on and so forth.  Of course, there are only 26 irreducible representation of M24, this is an infinite series.  So what happens after a while is that you get reducible representations, but you can  always decompose them, of course, in irreducible ones, and if you do this for long enough, you  see that all the irreducible representations come in, and that was proven by Garnon.  So there is a mathematical proof that these numbers here are really to do with M24 and  nothing else.  So this is...  So multiplicities?  Yes, there can be multiplicities.  If you go far away, these numbers grow very fast here, right?  So you have only 26 building blocks, so they have to come with multiplicity.  So a mathematical expression for large n?  You said that you have a fast code.  There are asymptotic formulas for large n, for an, you want to know an, yes?  Yes, there are.  So Eguchi and Ikami have worked these things in the context of black holes and the entropy.  You won't tell me, you won't tell me, right?  You will not tell me, that's the problem, right?  I'm not going to use it, I'm sorry.  It's going to be disappointing.  Oh, I'm sorry to disappoint you.  I can talk to you privately afterwards and show you nice formulas.  Right.  Okay.  So, terminology here, in this elliptic genus, actually because of this y bar being one,  it means that you project into a sector which is short representation on the right or massless,  while on the left, I haven't done anything to the y on the left, so I get a mix of massless  and massive, right?  What I call quarter BPS here are basically states which are massive on the left and  massless on the right.  So it's not the full BPS spectrum here, but only part of it, right?  And this sector here, as I said, is encoded into this part and this minus two here is  not a clean business yet.  I mean, so people think that might involve virtual representations of M24, but we haven't  studied that yet.  Right.  So this phenomenon here is called Matthew Moonshine, right?  Because these are coefficients of a mock modular form, right?  And then suddenly the number theory is got interested, but it's linked to group theory  and to string theory in a way that nobody expected, right?  Right.  So then we have...  Sorry, and the minus two is not simply a fermionic representation.  Yes, let me see.  So the minus two, it has to do with the vacuum in the Nevers-Schwarz sector, right?  You get signs because of the grading, that's true, but the story is much more complicated  than just counting of fermions and bosons, the understanding of this, right?  It's the Witten index of this representation, the vacuum representation.  But you say the minus two is mysterious, sure.  The fact that in the decomposition here, if you want to reinterpret in terms of, let's  say, N equals four characters, you are forced into this pattern here with a minus sign there.  And it is intrinsically due to this, but we don't understand the meaning, but these are  not representation of M24, you see?  Even if you take plus two here, okay, you can say it's one plus one, but it's not that.  So it's mysterious, nobody understands that.  Well, the first 26 terms is twice the dimension of representation.  It is.  And that's as well, always.  It's always an even number, twice.  But it's a combination, it's always twice.  And the twice is because you have the representation, it's complex conjugate.  So when you say 90, it's 45 plus 45 bar.  Of course, some representations are real, so there you have a multiplicity two, right?  And this is proven as well.  So the people prove that all the entries are integers, that they are even and positive.  So all this is under control.  And it's a mock modular form.  Whatever this is, this is not the talk today.  Right.  So now, to be clear on notations here, because I'm going to use them again, what I'm writing  as curly age BPS is the space of states of quarter BPS.  So all the ones that appear here.  And this is in a subspace, and there's an injection here, into the space of generic  space, the space of generic states that I've introduced before in the elliptic genus here.  Why is it just an injection here?  Because this space here also counts things which come from here.  So that's the link here.  So I'm restricting myself today to this part of the story.  And there's a color code here, which I hope will be consistent, is that this violet color,  purple color here, always has to do with the generic space of states.  So if you see violet, this is magenta, it's different.  Right.  So generic and non-generic K3 theories.  Now, if you, there is a notion of modular space for K3 theories, and we know very well  one component.  We think there is only one, but nobody has proven it again.  But in that component, the landscape is well known of modular, modular, the modular space.  And as you vary the modular, so you go from one theory to the next, you have different  behaviors.  And although the generic states are common to all the K3 theories, at certain points  in the modular space, you have an enhancement of symmetry, maybe, if you have more quarter  BPS states that are going to come in.  And therefore, but these quarter BPS states that you have in surplus at some points happen  to cancel because of the minus one to the J0 minus J0 bar.  So that's what is the philosophy of this table here, is that if you have a generic K3 theory,  then you don't have, at this particular point, extra cancellations that happen.  And that's what is written here.  Now if you are non-generic, and the Z2 or before conformal theories are a class of such  generic theories, then you have these cancellations that happen in the elliptic genus.  But there are pluses and minuses in the two cases.  So the plus sign for the generic is that you don't have to worry about these extra cancellations.  And also for the generic one, but on the other hand, nobody knows explicitly any of these  theories.  So practically you can't do anything with them.  On the other hand, the non-generic ones, well, we know very well these ones, they have the  same spectrum of generic states.  That's something that you can show.  But there are cancellations.  So you have to kind of navigate between the two, and today I'm going to talk more about  these ones, and my last transparency will be basically about the future.  Right.  So to be clear again, in non-generic K3 theories, the space of massive quarter BPS states, ah,  he didn't put the color right, should be violet, is actually not the same space as  the full space of quarter BPS states, because the full space here will involve the states  that cancel in the elliptic genus.  Right.  Okay.  For example, in Z2 conformal theory, or before the conformal theories, you see that this  is what the elliptic genus measures, the numbers, the magic numbers of before.  But if you look at the dimension of these spaces here, okay, you see that they are higher  each time.  And if you go from 90s to 102, they are 12 states.  So what exactly is this HM straight HM plus structures?  This one here?  No, lower, lower.  Lower.  Yeah.  Here?  Yeah.  This one here.  Well, I'm going to say in a second.  Okay.  So here I've not talked about H plus yet, I've just talked about these two, all right.  Then you have 12 here at level one, and this is six fermions and six bosons that cancel  in the elliptic genus, but they exist in the theory.  Okay.  Now this space here is actually what is left once you have taken away.  So H plus at level one here would be the 12 I've just talked about.  Okay.  So this is what I call the excess states.  So it's a bunch of fermions and bosons that cancel.  Standard BPS, and then you get Z2 CFT, right?  Consistency.  Sorry?  Combined with this HBPS same, and then...  Yes.  And then you get the whole state.  Yes, absolutely.  So it's just selecting, yeah.  Right.  So we don't know really what drives these cancellations.  They just happen in the elliptic genus.  So we wanted to understand this in a practical setting, which are these Z2 Oberfolt CFTs,  and very, very quickly, I've jotted here what will be useful for later here, is that the  construction of these are from a toroidal super conformal field theory at central charge  six, and this construction here is induced by the Kuhmer construction of K3 surfaces.  So it's not by chance that it's like this.  There are deep connections between the geometry and the conformal field theory there.  So Kuhmer surfaces are a special class of K3 surfaces, and they start by taking a torus,  C lambda, which is obviously given by C2 mod lattice, which has four real dimensions.  So there's some geometry in here, and then we take a Z2 Oberfolt of this torus.  In the process, we get fixed point, 16 of them, and therefore, all this is a different  story from today's story, but it has to do with 16 fixed points in the conformal field  theory language, which corresponds to the 16 twisted sectors of the theory.  So it's beautiful.  And if we wanted a free field representation of the symmetry here for these theories here,  we can achieve them by looking at four Dirac fermions and their superpartners, and then  this is the realization of these generators of n equals 4 in terms of these free fields.  I'll come back to that a little bit later so you will see this transparency again, decorated  further later.  Right.  So I'm after the elliptic genus in these theories, so I start with the partition function.  And I've chosen to work in the Neuverschatz sector because at the end of the day, the  remarks I will make at the end connect directly to the Neuverschatz sector.  However, in n equals 4, the Neuverschatz and the Pramunsector are isomorphic to each other.  So you don't lose or you don't win information, it's just a repackaging with, of course,  a different physical interpretation.  Right.  So the partition function in the Neuverschatz sector, typically in these all-before theories,  have an untwisted sector.  So this is a schematic to indicate the boundary conditions in the town, the sigma directions,  right?  So we have an untwisted and a twisted sector.  And if we are interested in the generic states of the theory, then we have to remove in the  full partition function here whatever depends on the moduli.  And it turns out that the moduli dependence is all in the untwisted sector.  And remember, it's a conformatory, the conformal theory, so it comes from a torus and story  in the Hein, Lattice and so on.  So the moduli are really in that part of the theory.  So I'm going to set the momentum and the winding number of the states to zero.  Because if I don't keep them, then I keep a dependence on the moduli because these change.  So this is what I call a generic partition function.  And in terms of the elliptic genus expressed in the Neuverschatz sector, it can be expressed  in terms of generating functions for the twisted.  So this is the twisted, sorry, untwisted, red is untwisted.  And this is twisted.  They are written in terms of Jacobi-Titta functions again, usual things.  You have to keep modes, not modes, states, of course, which are either bosonic or fermonic.  The details are in the paper, but it's not interesting for us today.  And it turns out that mathematically, these three generating functions can be decomposed,  as we know, because this is the elliptic genus, into and equals four representations again.  So this is basically a slightly rewritten version of the elliptic genus from the point  of view of the Neuverschatz sector.  But here, it's the same function as before, because the spectral flow that allows you to  go from Neuverschatz to Ramon is actually an operation that affects the y dependence  of your object.  And this a of tta function, obviously, doesn't depend on y.  So it's not going to move through the spectral flow.  So it's no surprise that even in this repackaging here, you see the same numbers as before.  So now we are looking at this.  And right, it's a little bit more involved here, but I hope that I can make myself clear.  So it's color coded.  So we already know that the purple here is for generic states.  The red ones are the untwisted sector, the number of states in the untwisted sector,  and the blue ones are the number of states in the twisted sector.  So this is, you have seen this, the ansatz is that this expression that you have seen  before, so these are all the quarter BPS massif that I was looking at.  These are the excess states that we talked about.  Now what I say is that within this class of states, I have two contributions, right?  And I have to explain to you what these two contributions are in a minute.  But let's keep it just as a formal exercise for a minute.  Then you have all the ingredients which are written here, right?  And why do I call this first one H and P.  Now, very briefly, I told you that all this is linked to the 16 fixed points in the Kummer  construction.  And then you have 16 twisted sectors in the Kummer field theory.  In this 16 dimensional space, I can identify a very symmetric direction.  And how do I do this?  Well I know that on this twisted space, I have an action of a geometric group.  This geometric group being linked to the geometry of the Taurus T lambda, the lattice I use  for my Taurus in the Kummer construction as some geometric symmetries.  And after a few steps, basically the upshot of all this construction is that on the twisted  space, this geometric group acts.  And if you take a sum of all the ground states of these twisted sectors and you sum them  under the action of the group, it's an invariant.  Because the geometric group acts as permutation of these ground states.  So the typical construction.  Take this.  So this is what I call the diagonal direction.  So in my 16 dimensional twisted space there, I have a direction which is my diagonal one  and then I have everything else which is orthogonal to it.  And that's the perp there.  So the perp there is a 15 dimensional space there in terms of the ground states in the  twisted sector.  And I've chosen to write it as 15 contribution of one sector here.  And the diagonal one, in a minute we are going to split it again a little bit.  The rest here is basically the leftover quarter BPS states that are not, that are generic.  So all this is HBPS.  These are the excess states.  So that's how it is.  And of course I have at my disposal the dimensions of three objects, CN, BN and DN.  And why is this an ansatz?  Is that by looking at the first two levels, so n equals one and n equals two, we have  noticed that actually it seems...  Right?  Okay, so what I was saying here, so basically here what we know for sure is that a certain  subclass of untwisted states automatically comes into the excess.  Because basically they come with a minus sign and they are less of them at each n than the  DNs.  So we know that they are here.  What we don't know is what the partners of these states are in H plus to cancel.  They could come from CN or DN.  And the ansatz is that none of the untwisted ones are partnering in the two BNs in the  excess sector.  So this remains to be proven for general n.  But it turns out that Keller and Zade recently have shown that for n equals one, in a way  it's not so difficult there because for c1 here, c1 is zero.  So what they've shown is that anything in the excess sector here was actually lifting  off the BPS bound when you deform the theory.  So you go away from the Kuhmer point, you introduce CFT deformations and then these  states don't stay BPS.  So therefore they are not generic.  So let's say what's behind all this.  Is there a general formula for all, some nice formula for each excess?  So the generating function of before, UL0, UL1-1-1-1-1, that I wrote, you have a closed  form.  It contains it of course, but it's just for excess numbers.  For the excess numbers, yes, you can, you have a number, but you don't know.  So it's all about numbers at this stage.  But you need to know what the states themselves are because if it's just a number, you just  know it's a boson or a fermion.  But for example, in terms of the free fermions and free bosons that I showed you before,  what are they?  We wanted to understand a little bit more closely the nature of these states, apart from the  fact that, apart from their statistics, we wanted to know how that, so we did an explicit  construction for N equals 1 and N equals 2.  N equals 1 is a piece of cake, but N equals 2 is what we did in Stony Brook last spring.  Catherine and I were there and we just sweated over this because I will tell you why it was  complicated.  But anyway, so this is because we didn't have a better idea.  We really wanted to see, okay, what are these states?  Not just their fermions or numbers of fermions or bosons.  What are they?  Okay, so we did that.  Okay, so I think the time is running, isn't it?  You have 12 minutes.  12, okay.  So I think I'm going to skip this one, but not go into the details of it.  But basically at level one, this we did a few years ago already, we noticed that all  the states that are generic, okay, actually fit into representations, not of M24.  I mean, this space is the same for an M24 representation, but it's a representation of a maximal subgroup  of M24.  Now, why is this maximal subgroup of M24 so exciting to us?  It's what we call G-octad here, right?  It's Z to the 4 semi-direct product with the alternate group of 8 elements there.  It's because it comes from our idea of symmetry surfing, okay?  So in words, symmetry surfing is that you have a space, and this is going to be the  generic space of states that I'm talking about, which is common to all K3 theories,  right?  And then if you travel from one point in that modular space to another one, the symmetries  are different.  The geometric symmetries are different because you have different torii that you play with.  And therefore what you do is that you collect all the symmetries at the different points,  all the geometric symmetries, and therefore you obtain this G-octad.  So there is circumstantial evidence that it's not such a stupid idea, right?  Again, I mean, it's just evidence so far, and it's not proven that it's really what's  going to happen in this problem, right?  But at least we have this group to hook on.  And it turns out that at level one, we see very clearly because we had the help of Mar  Golin, who is a group theorist, who had written a paper ages ago about the 45-dimensional  representation of M24, and the way he constructed it was very helpful in order to make this  statement there.  But again, I don't have the time to go into the details here.  And then, apart from Keller and Zadeh that really confirm what we expected at level one,  Faber-Deele, Keller and Paul also did a very fine piece of group theory there, a very elegant  paper, where they actually say that not only, whoops, here, okay?  So not only the octad group acts on H-perp, which is all you could see at level one, but  they looked at all levels and they see that the octad group act on this space, but also  on all the other massive quarter-BPS states here that you have at your disposal in the  generic space of states, right?  So again, it's evidence that the symmetry-surfing idea is not dead yet, right?  But again, I'm being cautious here, right?  So this, I think I said one way or the other.  So basically, what we did in order to get more feel is to construct the states, but  we've used an extra ingredient.  And the extra ingredient is to utilize the fact that in the building blocks of the Z2  or before super conformer field theories here, the building blocks also have a certain  behavior under an SU2 group, like this is a global SU2, it's not, it's a global SU2  and it's actually a subgroup of the global SO4 that is common to all N equals 4 super  conformal algebras here.  And they transform as doublets under this SU2 and we've kind of implemented this refinement  in the generating functions u and t that I had before.  So we have introduced another complex number there to keep track of this global SU2.  And what we found in doing this is that very explicitly up to here, so for 1 and 2, so  you can obviously make predictions from the analytic expressions of the generating functions  that you have here.  But for these two levels here, we have painstakingly computed, calculated all the states, so these  16 states here and these 28 states here explicitly in terms of the building blocks and identified  their behavior under the SU2.  So how do they decompose?  So if I take this 16 here, how do they decompose under this SU2?  And the notation here is that we have the multiplicity here, so it's two triplets here  and two quintets.  So 2 times 3 plus 2 times 5 is 16, yes, and so on and so forth.  So we've done this for the three generating functions that I showed you before and what  we see is that the excess states that we wanted to cancel here, they are 6 plus 6, the 12th  that I talked about before.  Now you had no choice at level 1, I said because there's nothing in the rest here, so this  CN is in the rest.  So all the cancellations have to come from the D if you agree with me that all the BNs  are in the excess and come with a minus sign.  And then we have a minus 2, remember it was minus 2 BN.  So we have minus 2 singlet under this SU2 and here you have two singlets in the DN.  So it seems that the SU2 behavior of the states is a guiding principle to see where the cancellations  have to come from.  So you see at the second level in CN, so in the untwisted sector you have no singlets  of this SU2 that you could use to cancel twice a singlet here.  On the other hand, in the DN you have two of these singlets, just two, just lucky and  therefore you cancel them.  So again, it's an observation and as you go along in the levels it becomes a little bit  more ambiguous because let's see, I think if I come to this level here for sure, we  have enough in the C sector, in the untwisted sector to cancel the one there.  Here you don't have enough, you would need 3 to the 6 here.  Here you have enough.  Okay, so even here, here you have enough.  So again, we have unfortunately no confirmation from proper deformation theory that our idea  is absolutely corrected all level at this time.  What Keller and Zadeh that did for the level 1 now with our latest work here, we have prepared  the way in order to do the same exercise at level 2, but second order perturbation theory  and conformal field theory is technically quite involved and maybe we should have a  better idea which is what I'm going to talk about next.  So this is a different level of thought here and the novelty on this transparency is actually  due to Catherine here and almost simultaneously there was a paper by Bélin Son with a mathematician  who published quite a difficult paper to read but which basically raised the same conclusion  as Catherine was published in CMP.  So what is this?  Well, I've talked a lot about the conformal field theory to the elliptic genus.  It's been known since Whitton and it's a book that it's actually for K3 surfaces, it's  equal to the complex elliptic genus.  So there's a link between the geometry and the conformal field theory at this level.  And in some incarnation this elliptic genus in terms of tangent bundles and here I've  taken a short curtain because we don't have the time to go into this.  I can only say in words what this TLM is but it's coming from looking at first of all the  basic ingredient is going to be the tangent bundle of the manifold X.  So T10 and then there is a virtual bundle so it's a formal sum of bundles that are constructed  by taking the exterior product of the T bundle, so the tangent bundle times exterior product  of the dual tangent bundle times the symmetric product of the tangent bundle times the symmetric  product of the dual tangent bundle.  So it's quite a sophisticated object but that has been studied at length in the literature.  And that virtual bundle that I didn't write for you here, you can recast it as an infinite  sum here in two variables where L goes from 0 to infinity while M here is bounded between  the dimension and minus the dimension of your manifold here.  And it's a formal power series in the variables Y and Q that we are familiar with and this  is the holomorphic Euler characteristic of this bundle that I've constructed the way  I said before.  Anyway, so this is the expression and then in 2016, Kacru and Tripati had quite a good  idea they said okay since the cancellations of states in the conformal field, the elliptic  genus is a pain because we don't really see what's going on, let's try to lift this cancellation  and introduce a new grading which is V here in the right sector.  So somehow reinstating a little bit of the partition function thing there and they did  that and then they also showed that you have a counterpart in the geometry except that  you can show that these two objects don't agree, here they agree, here they never agree.  So that's the first remark.  And then there is much more to this that this object might not be actually that helpful  in the problem at hand where we want to understand Matthew Moonshine.  And then what Katrin has observed is that now instead of taking the trace just over  the kernel of t bar zero, she introduces these two, a grading but she just traces over the  generic states, so the generic states being a subset, a subspace of the curve.  And the inspiration for this is that Kasper Kapustin, I forgot to put his name here because  it was quite crucial, in 2005 he had this idea that if you take the infinite volume limit  of nonlinear sigma model with a topological twist then you get the cohomology of the chiral  deram complex as this paper.  And this is exactly where we are, is that you can define a new elliptic genus that she  calls the Hodge chiral here, which basically has to do with taking the trace over the group,  cohomology group here of the chiral deram complex.  Now again, very vaguely, since I don't have time, for k3 surfaces these two can be shown  to agree, so this is equal to this, it's an invariant, you can show that.  And there is a structure, this chiral deram complex is a so-called sheaf of super vertex  operator algebra which has to do with the BC-Bitagama system.  So there is a structure of VOA there, for the Mathieu Munchen we also would like to  see a VOA structure on which the Mathieu group 24 acts.  We don't see it, but we think that this chiral deram complex here and the sheaf cohomology  of it are going to be a mathematical object that models the generic spaces of states in  the k3 theories and if we can find more structure on it, then maybe we will see M24 acting on  that extra structure, that's the direction of travel.  So I think I'm at the end here.  So I didn't tell you that.  Yes, that's a little remark, I put it there because I knew I was going to not say it,  it turns out that if you take complex tori, so T for tori, these three quantities are  equal.  So when I say that this is not equal to this one, for tori it happens to be true, but it's  not where we want to be.  K3 is not true.  And that's probably why at the beginning people thought that this would be the way to go for  Mathieu Munchen because I saw it worked for tori.  But if you really calculate things for K3, then there is a differentiation there.  So basically that's not where we concentrate our efforts, just trying to understand more  about the Karelderam complex.  So this was, oops, okay, I'll spare you the conclusion because I'm out of time, I'm sure.  So this is for Samson because it's his birthday.  Now I wanted to put something from Belgium because I'm Belgian, right, unlike, it's not  clear from my name, but that's what I am.  And in Belgium we like to say things with cartoons, you know, very much into cartoons.  This guy called Gelluc, okay, Gelluc, we say in French, he's also known in France by the  way, and he has this series of snapshots with some philosophical thoughts which usually are  quite thought provoking, which in, with a cat, that's a cat, yes, I don't know, so see that.  And of course he speaks French, but you understand French of course, yes, okay.  So okay, and there's a little message there is about saving the planet of course, so we  want to have 60 candles on your cake just to save the planet.  Okay, thank you.  Questions?  Yes?  Yeah, it's just very general question from this very old tab about 88 or something like  this, there's also suggestion of written, this should be some 24 dimensional, and hits  the book at the same time, should be 24 dimensional real manifold with section of a monster, such  as this 24 dimensional manifold should have elliptic genus which would be J function.  So one get geometric realization of monster representations directly in index of deregoperator.  So there's really no progress after that.  Do people, do physicists have an idea whether it should be true or not?  Well, personally I don't, I never really looked, I know that, I mean it's kind of expected  that because M24 is a big stepping stone in constructing the monster that there should  be a link, a deeper link that we have not grasped yet between the monster moonshine  and this one.  But I've not really looked at all into that direction, so I know that other people working  on Matthew moonshine, what I didn't say is that there are now more and more instances  of moonshines where you have automorphic forms whose coefficients have to do with the dimensions  of some representations of some groups, yes?  So there are fascinating things happening, but no, I don't know, I don't know more.  Basically I've told you where I am.  And very much, very much K3 for me.  Yeah, my question is, it seems like you can still get quite a bit of information without  knowing the conformal filtering, for instance you have this space of states 192 or 100,  so how can you know this without knowing what the CFD is?  For instance, you know, you remember you had this A, I think it was the number of B, it  was called the BBS states, although you don't know.  But because the elliptic genus is invariant over the whole space, so, oh it's proof, it's  well you can prove that the elliptic genus doesn't depend on the modularity at all, so.  But if you cannot compute it in any CFD because you can.  On Z2 or before, you can, it's a special case, so you can calculate the elliptic genus.  But if you go away from the orbital point, you still have it.  Away from the orbital point, no, we can't compute anything.  We can kind of perturb a little bit and see what's happening.  But you knew, for instance, the dimension of this H0, whatever it was, was 100 at any  equal one.  Yes.  So this you were able to infer, nevertheless, even though, even away from the orbital point,  it seems to me.  Yes, but we did calculate it from an orbit, we did, the calculation I did with Eguchi in  1988 was in the framework of Z2 or before, the gap in the models.  So you calculate, because it's an invariant over the whole space, you can choose a point  when you know how to calculate.  And these numbers that you are talking about are pitted out at this particular point.  And because you can prove it's an invariant, and even if you don't know the theory, you  know that these will be the same numbers.  What you don't know is the number of excess states.  So the number of excess states vary from one theory to the next.  But in the conformal field, in the CFT elliptic genus, they cancel.  Are you sure the theory exists?  Which theory?  I mean, the super conformal theory, which you cannot construct.  Because don't you assume it exists just to prove that?  Just to know.  You are a mathematician.  Yeah.  Assumption?  Well, the modular space is 80 dimensional.  It's proven mathematically.  So you can't exhaust the set of theories with only the few Gepner models, which are isolated  points, and the class of Z2O before.  So I think what the shape is of this K3 theory is nobody knows.  Well, at least I don't know.  I don't know.  Maybe other people have ideas.  And this magic number has lower index n.  Can you write recursion in relation?  No.  No.  No, asymptotic formulas, yes.  You wanted asymptotic formula from me.  Sorry?  In the beginning, asymptotic formula.  Yes.  Yes.  That's a good shenikami.  I'm sure I can find a paper here if you want, if you're interested.  Yeah.  Yes, they looked in connection with the entropy of black holes.  Enfactorial?  It's complicated.  But then I'm sure there are some enfactoists.  I don't remember.  I had a question about this hot electric.  Yes.  The electricians, we looked at this when Katsuh and.  Yeah, Kaku and Tipati.  Yeah.  They proposed it, but then we didn't know how to make sense of it because I think there  was something that was not clear, that was a BPS condition.  Because indeed we want to relate it to counting of black holes where you have spanned black  holes.  Yes.  But also we want to generalize the product from symmetric products.  Because then you expect also some connection with Segel model work forms.  Okay.  Is there no anything?  Because I mean this new definition that is a completely different object.  For example, it relates, let me see.  Right.  So there's also several, or at least one paper by Koitsik and Heun that they've looked quite  deeply into the hot elliptic genus.  And they've tried to.  So they looked and they found that in there, there are some, now I'm getting a bit out  of my depth here, but there are some global section, holomorphic sections that are not  N equals four in there.  And therefore, while for the Carol Durand one song, this is Bellingson as shown that  the global holomorphic sections are all N equals four only.  So that's one way to differentiate the two.  It's also that Kapustin really points into the direction that the topologically twisted  half twisted nonlinear Segel model has to do with the comology of the of the Carol Durand.  Nothing to do with the comology of this bundle that you're talking about.  So it's intrinsically a new, a new object.  You can't really calculate things in the conformer theory side because this is the state of generic,  the space of generic states.  This is really in so.  So it has a geometric definition.  So the geometric definition, which agrees with the conformer theory one, that's proven,  I think I'll try and prove it, but we need to study more looking for M24 there in the  geometric part because I think it's quite powerful if we can find it there.  You saw with the Z2 before, it's a bit painful.  There was some variation of the theme by Shamil Tenaisi, we run to change involving special  holomorphic case.  Yes.  Can you comment on that?  No.  I don't even read this paper.  I know it exists, but.  There were some papers about a couple of years ago, and then the Z2 came up and this M24  is replaced by something.  Well, I thought they only can realize subcruits of M24.  That's what I thought was the presence.  No, it algebra came.  It was not a joint.  It was in the G27, in this algebra, I had this one.  They used an algebra which looks like that and constructed something.  I tried to read, I could not understand.  No, I've read these ones.  I mean, sorry.  Next time.  Maybe it's a good time to break for lunch.  Thank you very much.
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