 So, I'll talk about what is called the self-theor strings of six-dimensional super conformal field theories and its relations to the instantons of exceptional gauge theories whose relations I'll try to elaborate during my talk. So, the talk is based on a paper that I recently posted on archive in collaboration with Hee-chul Kim, who's in the audience and Jae-mo Park, who's in Korea. It's also partly based on a work in progress in same collaboration and also with my ex-student Jun-ho Kim, who's also here. Right after posting a paper, the following two papers which are closely related to ours has appeared, so indirectly I'll also have to mention some of these, some of the materials covered in this paper. So, six-dimensional super conformal field theories have been studied in quite a quite detail, have been quite extensively studied during the past few years in various directions. In particular, what I'd like to emphasize today is the constructions or the indirect construction of the wide class of super conformal field theories using the string theory setting. As many of you will well know, the six-dimensional quantum field theories are never formulated directly using any Lagrangian formulations and they are only predicted indirectly in various string theory settings and by taking suitable decoupling limits. So, there are many ways of constructing these field theories by string theory and one easy way is to use some brain settings, D-brains and NS5-brains in virtually the flat space-time background. I'll not treat these cases in this talk. I mostly discussed the super conformal field theories which are engineered by geometric setting. These are engineered by putting the string theory in a suitably curved background with some singularities. So the canonical example is the first kind of discovery of six-dimensional super conformal field theory preserving two-zero supersymmetry and that's the work by Witton in mid-90s. To maximum super conformal field theories which follows the ADE classification has been found by Witton by putting a type 2B string theory on C2 mod ADE or before singularities and at the tip of the singularities there are some six-dimensional light degrees of freedom which provide a six-dimensional quantum field theory degrees of freedom. If you are interested in less supersymmetric theories n equal to one-zero, minimal super conformal field theory, you have to go to a more complicated setting. We are asked to consider the same two-Bs background on certain four-manifold with singularities but with varying axiodilaton depending on space-time. So this asks you to consider basically the F theory on R6 times some elliptically fibered kalabya three-fold which has been discussed earlier in this conference. So the elliptically fibered kalabya is roughly taking the following structure. So it's a basically type 2B string theory defined on certain curve four-manifold. It has a singularity which can be resolved in particular way. And for most general setting the axiodilaton which is parameterized by the complex structure for torus can change depending on where you are on the space-time. So the recent finding, recent discovery of a large class of super conformal field theories in six-dimensions is basically done in a geometric way, trying to understand what kind of elliptical kalabya, non-compact elliptical kalabya three-folds are possible which gives rise to the physics of six-dimensional super conformal field theories. And doing that means basically, primarily means that what kind of singularities on the four-manifold, four-dimensional base are allowed, okay? So to classify what kind of singularities are possible, physically basically one has to, mathematically one has to resolve the singularity in various ways and see the result, how the singularities resolve. Physically the resolution corresponding to, going to what is called the tensor branch of the six-dimensional CFT. So all the kinds of six-dimensional CFTs that are known always have some numbers of tensor multiplet. The tensor multiplet contains two-form potential whose three-form flux is required to satisfy a self-duality condition and the super multiplet contains some fermions and a real scalar. If you give some expectation value to the scalar, you're going to the tensor branch, something like the Coulomb branch of the gauge theories. So going to the tensor branch corresponds in geometry to making the singularities smooth and this resolution happens by replacing the singularity by certain numbers of spheres which intersect with each other. The volume of each two-sphere is corresponding to the expectation value of the corresponding scalar. So using this kind of picture, these groups of people have been classifying what kind of non-compact elliptic Calabi out three-folds are possible and found a large class of n equal to one-zero super conformal field theories. The structure of the field theories found in this literature goes as follows. One is called the atomic classification because it uses a finite set of simple super conformal field theories as atomic building blocks. Once you know the atomic building blocks well, you can use it as a building block of a sort of quiver from which you can build a more complicated quantum field theory. So what is called the atoms have the following structure. So basically the important atoms that I discussed to you are appearing in the first table. So all of them have one-dimensional tensor branch, meaning that in the previous picture I've shown you, the resolution of the singularity gives rise to you only one two-sphere. So it has one tensor branch. And the number n labeling all these different theories are nothing but the self-intersection number of the two-sphere corresponding to the tensor multiplier scalar. So depending on what the values of n are, there can be various symmetries of the six-dimensional theory. So if you have non-trivial gauge symmetry in six dimensions, that appears by wrapping suitable numbers of seven brains on the two-sphere because the seven brains on their world volume holds suitable gauge symmetries. The last one, the second table I don't know, but all of them in the first table can be made into the, you ask me the second table. The second ones, I'm not sure, I mean it has been, I have to carefully read this paper. This provided this list, but I'm not sure whether they just did non-compact one or compact one as well. I'm not sure. You know the answer? No, I don't know the answer. Yeah, I'll have to read this paper carefully, yes, yes, yes. So apart from these ones, which are sometimes called minimal super-conformity theory six dimension, there are some exotic atoms which have two or three-dimensional tensor branches. These are nothing but having two or three-two spheres which have self-intersection number two, three, two, three, two, two, two, three, two and have mutual intersection number minus one. So to understand the structure of six-dimensional super-conformity field theories in any microscopic way, it's most important to understand these atoms first. Once you understand these atoms, the rule of constructing more complicated conformity field theories is somewhat well-known, at least in the geometric setting. For instance, you can glue many of these two spheres or the gauge theories in the form of quiver, and the rule of forming quiver is also well-known. It uses a so-called E string gauge theory, having E8 global symmetry, and it provides the role of glue of forming the quiver. So you have the E8 symmetry, which is very large. So if you want to glue any two gauge theories to form a quiver, you can do so by taking the product gauge group of the two adjacent nodes, and if they are the subgroup of E8, you can gauge this part of E8 to make a quiver. So the rule is quite well-known. This is one way of making a bigger super-conformity field theory, having higher-dimensional tensor branches, and the second way of making the quantum field theory more non-trivial is by on-dixing procedure. Basically you add more hyper-multiplet matters. Here you basically have no matters with which you can fix. You have either no matters or half hyper-multiplet, which cannot be given expectation value. So the process of on-dixing is achieved by adding more matters and enlarging the gauge group, so kind of decorating this kind of minimal theories. So by these two ways, you can form a rich class of 1-0 super-conformity field theories. So this is the review of the works on 1-0 CFTs in the past few years. Now I'll turn to the main object of my interest. So I'll talk about what is called the self-theor strings in six-dimensional CFTs. This is a universal object which is appearing in all kinds of six-dimensional CFTs in the tensor branch. So let me explain first in string theory how they happen. I explained to you that in the F theory setting, the tensor branch is obtained by making the singularity resolved by having many finite volume two-spheres. And if the D3 grains are wrapping these two-spheres, they can form a string-like configuration in six-dimension. These are called the self-theor strings. They are called the self-theor strings because these strings, under the two-form potential that I just explained to you, have exactly the same amount of electric and magnetic charges. So since the electric and magnetic charges are the same for this object, it's given the name self-theor strings. It's basic objects, universal objects in six-dimensional field theories. And it's quite analogous to various objects that we have in four-dimensional gauge theories in Coulomb branches. There we have W bosons, monopoles, and dions of all sorts of charges. But everything that has to be appearing here in four-dimension is analogous to one object, self-theor strings in six-dimension, because it's electric and magnetic charge are the same. So these self-theor strings are half BAPS objects in one-zero super-conformal field theories. And if you choose this orientation carefully, suitably, it preserves two-dimensional 0,4 supersymmetry on its wall sheet. So at low energy, one can find a two-dimension, one can expect to have two-dimensional N equal to 0,4 super-conformal field theories on the wall sheet of these self-theor strings. And this super-conformal field theory is of my main interest. There are many motivations for being interested in these theories. It turns out to be very crucial that object for understanding various interesting observables in 60 CFTs in recent years, like super-conformal index and so on. And this object is of interest in its own right. So various reasons people have been interested in this self-theor string quantum field theories. And since I said that to understand the six-dimensional quantum field theories, it is important to understand the atomic constituents, which form a more complicated ones. Also in two dimensions, there will be a large class of two-dimensional quantum field theories for self-theor strings. And there should be corresponding two-dimensional atoms for the 0,4 quantum field theories. So mainly I studied this kind of quantum field theory in my talk. One very important viewpoint about the self-theor strings that I keep recalling in my talk is the following. So the self-theor strings can be understood in a very simple manner if we recall an effective quantum field theory description of this CFT in the tensor branch. And that is given when there is a six-dimensional gauge symmetry by a six-dimensional super-young field theory coupled to various hyper-multiplet matters and the Abellion tensor multiplets. This is the photogenic part of the action in the simplest case in which we have only one tensor multiplet. So this is the scalar and the three-form kinetic term. Of course, the way you use this put Lagrangian is exactly in the same sense as type 2B supergravity. Since self-theority condition is hard to impose in Lagrangian way, you ignore that. You vary the fields and derive the equation of motion and then impose the self-theority by hand. So understanding it this way, this is just a kinetic term of tensor multiplet fields. And it couples the vector multiplet fields in the following way. So if the scalars are assuming non-zero expectation value, it sets up an effective Young-Mills coupling scale. So it gives rise to 60 Young-Mills description. And the superpartner term of this is coupling the beam in new field to what is called the instanton number density. So what is charged on the beam in new is the self-theority of strings. And in the Young-Mills theory viewpoint, this is provided by the stringy soliton which carries the instanton number. So self-theority of strings are instanton string solitons and super Young-Mills. This will be the main most important viewpoint in my talk. Any questions so far? Of course, six-dimensional conformal field fields are difficult. Two-dimensional CFTs are apparently looking easier. But if they are strongly interacting, they are also difficult to study. So these two-dimensional 0,4 CFTs can be studied a bit more easily if one can have a gauge theory sitting at the UV. So the two-dimensional gauge theory is a weakly-coupled UV. So if one can engineer a UV gauge theory at 2V, which flows to the desired super-conformal field theory after RG flow, it will be much more easier to study some observables of this theory. So it's basically taking advantage of this gauge linear sigma model language which has been explored by Witten more than 20 years ago. So these two-dimensional gauge theories, which are not conformal by itself, but expected to float to the theory of interest, can be constructed in two major, two different ways. I'll hold the first way, the top-down approach. And the top-down approach is applicable when the self-theor string in string theory is admitting a D-brain description. So some of the examples of admitting this top-down construction is as follows. So I've listed some of these atomic constituents. Let me consider the case where self-intersection number is 2. This is a very well-known example. In this case, the F-theory background can be suitably dualized to the following type 2A configuration. And the six-dimensional field theory is nothing but the field theory living on two parallel NS5 brains. So this is nothing but the SU2 type 2A super conformal field theory. And in this setting, the self-theor strings are nothing but the stack of D2 brains suspended between the NS5s. So since this admits a D-brain realization of the self-theor strings, the low-energy dynamics on the version of self-theor strings is governed by the light-open string dynamics suspended between the D-brains. And after the strong coupling or the low-energy limit, you take the low-energy limit of this quantum field theory, you're expected to get the M2 brain dynamics suspended between M5 brains. So these strings are given the name M-strings. And basically using this setting with slight deformation to make the technical setting easier, these group of people have found out the two-dimensional quantum field theory description, gauge theory description for the M-strings. A similar construction has been made for a case N equal to 1. And after you dualize it to type 2A setting as well, it turns out to be the following configuration. It's an NS5 brain probing this following oriented-fold 8-plane background, where the self-theor strings are D2 brains suspended between them. So again, if you take the strong coupling with two-dimensional low-energy limit, it becomes an M2 brain suspended between the M5 brain and what is called the Hojava written E8 wall, this end-of-the-world wall of this heterotic M theory. So since these strings are probing the E8 symmetry of the Hojava written wall, this is given the name E-strings. And once you have a gauge theory, once you have a brain construction of strings, extracting out the gauge theory living on the UV of this CFT is very easy. And it has been done in the following work. But this is very special cases. These are. In most general cases, the F theory setting has various types of seven brains of various PQ charges, right in the two-sphere. So dualizing it to other setting, you're not always guaranteed to have D-brain constructions. So this is the most complicated setting. This is the complicated situation in F theory. So to find out the two-dimensional gauge theory, the second useful approach that turns out to be useful for at least for some examples is what I call the bottom-up approach. And bottom-up approach is obtained by recalling the instanton-soliton string viewpoint of the self-tiered strings in 60 super-young mills. To elaborate it on it a bit more, recall that the effective action had this kind of interaction. So any kind of strings having F wedge F source will be coupling to this B-mule field and should be interpreted as self-tiered strings. And you take the six-dimensional young mills on R6 and take the R4 direction, transverse to the strings. And you require the following self-tiered equations to be satisfied on R4, either with plus or minus sign. It gives you rise to the self-tiered strings or anti-self-tiered strings. So I choose plus without losing too much generality. So the configuration satisfying this equation have some solutions which are localized on R4, being point-like. So it's a string-like object in six-dimension. So this is the effective field-theory realization of the self-tiered strings. The instanton numbers are mapping to the self-tiered string numbers, okay? The reason why this viewpoint is useful is the following. So for instantons, there are many, many useful facts known, many mathematical and physical facts. And for instance, if the gauge group for the young mill theory is classical, either SU, SO, or SP, a natural candidate for the world volume gauge theory on the instanton has been proposed by the so-called ADHM construction. ADHM, if you don't know the ADHM construction, you don't have to worry too much. It was developed originally as a technique to find a solution to this nonlinear equation. But soon after, by string theorists, it has been given a broader context of providing a possible gauge theory which are living on the world volume of instanton-like solitons. So once there are ADHM-like constructions, there are natural starting point of gauge theory which you can try to use to understand these kinds of self-tiered string objects, okay? So let us go back to the table and see what we can do. Unfortunately, most of the gauge groups are exceptional. This is a characteristic feature of F theory, actually the power of F theory. But unfortunately, in the case where the gauge group is exceptional, this bottom of intuition cannot be applied, okay? So today I won't say too much about it except in the last few pages. But the remaining two classes seem to be very easy because SU3, SO8, we know the ADHM construction. We know the natural gauge theory. We think we know the natural gauge theory we can start with. So indeed, for the case of SO8, super conformal field theory, the natural ADHM construction works perfectly well because the SO8 gauge theory, ADHM construction asks us to consider the following quiver preserving 0.4 supersymmetry. So if you prepare K number of strings, the ADHM construction asks us to consider the SPK two-dimensional gauge theory with SO8 flavor symmetry in 2D. There are some bipondamental hypermultipline matters and anti-symmetry matters complicated. So whenever I write down the quiver diagram, please understand it as I have the classical Lagrangian and I know everything about classical physics. So this case has been studied by this Albrecht and other friend of ours. And this gauge theory can be used to study a lot of detail of the quantum physics of this SO8 self-deal strings. For instance, this gauge theory could be used to compute the elliptic genus and the physics could be compared with the alternative analysis of these strings using the topological string approach. So this is the perfectly good quantum field theory describing the string. There's actually a secret reason why this bottom-up approach works because this also admits the D-brain construction. So the two pictures are completely agreeing well. So it seems that only one class is left where the bottom-up approach could possibly apply. So let's study the SU3 case. And the real surprises come here, at least to me, because if you try to apply the naively constructed SU3 adhesion gauge theory, it turns out this is a bad quantum field theory. The reason why it's a bad quantum field theory is the following. Normally in the literature, this adhesion gauge theory has been used in zero or one-dimensional context to study the instanton dynamics of four or five-dimensional gauge theories. It's perfectly fine. But as soon as you uplift it to two-dimensional gauge theories, you have to suffer from the gauge anomalies on the world sheet. If you prepare K-instantons, the gauge group is UK classically, but this theory is intrinsically a chiral theory, preserving zero-perma-force symmetry and so on. And it turns out that this naive curve is wrong by having gauge anomaly. You compute both U1 and SUK anomaly. I think I computed here the SUK part of the anomaly, and this is non-zero. So from the vector-multiplet, there can appear some Fermi-multiplet, which contributes negatively to the one-loop anomaly. The chiral-multiplet have fermions, right-moving fermions contribute positively to the anomaly, so their contribution don't cancel to zero, okay? So this is a bad quantum field theory. And having seen a failure, it's natural to explain why the failure is natural because this SU3 is not really engineered by having 3D brains, 3D brains. So this adhesion construction is naturally motivated when the SU3 is realized on the stack of 3D brains, 3D7 brains, 3D5 brains of any sort. But if you carefully see how F theory realizes SU3, it is realized in a certain non-perturbative way. So this is realized by the so-called H2 singularity of this, H2 singularity of the seven brain configuration. And this SU3 symmetry is realized not by having 3D7 brains, but having two, but in a suitable SL2Z frame, it is prepared by having first two D7 brains and two S-dual brains, so the S-dual of the D7 brains. So of course there could be light, actually massless fundamental strings suspended between two D7s, giving rise to SU2. But the way it enhances to SU3 is highly non-perturbative. Namely, the other W bosons or the root states of SU3 are given by having these strings or the non-trivial PQ junctions, all being massless, suspended between various mutually non-local seven brains. So the way this SU3 is formed is highly non-perturbative, or in a way I should say it's exceptional, like the other exceptional gauge group realized in F theory. So this is the reason for this failure. So what can you do after we encounter this failure? Before explaining this, let me summarize the situation. So for the case with SO8, it's really a classical gauge group and it's the ADHM construction of self-terror string gauge theory works well. And the other cases, including the apparently classical gauge theory SU3, should be regarded as F theory as a non-perturbative gauge, I'm sorry, the exceptional gauge theory in which the naive ADHM construction is not working. It's important to study this SU3 theory in some details because in the recent construction of six-dimensional CFTs, this atom is playing fairly important role is to construct all kinds of novel new six-dimensional CFTs. In particular, all these exotic atoms essentially uses this atom three in the version in which SU3 is slightly unhixed to G2 or SO7 with some exotic matters. Secondly, this building block three or the exotic atoms can be used to form some of what I think is the most novel discovery in this conformal field theory business, namely called conformal matters. So I'll talk about these things later if I have time. So with these motivations in mind, I want to really understand this self-terror string better. So the strategy is following. I had a failure with the naive ADHM, but I'm gonna cure it. I'm gonna cure the pathology and make it work. The way to make it work is simple. You encounter Gage anomaly. So you add lots of matters to cancel the Gage anomaly and then to see if other basic physics is working well. This is the reason why I really call it bottom-up. It's just like phenomenologists cooking up some models and so on. But we have strong constraints coming from string theory. It should give the precisely correct data if you successfully get the right model. So it's very easy to test whether you get the right model or not. As I explained to you a few slides later, there are strong constraints which have been put by the topological string calculus of Albrecht about the BPS invariance of these SU3 strings. So if you do slightly wrong with the cure, in the process of curing the Gage anomaly, you'll have the wrong meta content and you'll have the wrong BPS spectrum. So this is a really delicate job which requires lots of trials and errors. Yeah, so I think at least I spent half a year to have things to work beyond the right track. So the result goes as follows. The result is in a way ugly. So I tried to make a UV uplift to Gage theory preserving the full 0.4 supersymmetry. That was really impossible. I don't know why, but I tried everything I can. It was impossible. But look, but note that if you're gonna construct UV Gage theory, it's often possible to sacrifice some of the infrared symmetry that you want. Of course, the conformal symmetry is sacrificed in UV. Some of the flavor symmetries could be satisfied, especially supersymmetry. Some of them could be sacrificed as well. I mean the supersymmetry enhancement after the algae flow is commonly observed in white class of theories. So what we can realize is the Gage theory which is free of all pathologies by sacrificing some of the 0.4 supersymmetry. In some ways, it obeys the structure of the 0.2 supersymmetry Gage theories in that all the field contents that I add are taking the form of the 0.2 superfields. So of course there are many ways of curing the Gage anomaly caused by the following ADH and multiply. But after some trials and error, we find this complicated addition is doing the job. So the K by is the anti-symmetry representation of UK and anti-symmetry of SU3. Nt means rank to anti-symmetry representation. Symmetric means rank to symmetric representation. So I have some secret working rules on how to decide these ones, but let me be pragmatic because I'm not really sure about the physics of determining these. You just, I just declare that these have to be added. And with this, I'll pragmatically check that all the physics that we want is reproduced by this quantum field theory. So first of all, by adding this matter, one can immediately check that the UK gauge anomaly cancels to 0. And secondly, what we wanted the following. Since we have added bunch of matter fields, we should suitably turn on the potentials. More precisely, we should turn on the super potentials in a suitable way to require the following requirements. We should have the first of all the correct modularized space because in the infrared, we know the modularized space of instantons. We don't want the modularized space to have more fields spoiled by these. A related question is that, we should turn on a suitable potential so that we have correctly the right set of flavor symmetries that we expect on self-dealing strings. If you have more flavor, chiral multiplied or pharmy multiplied without suitable potential, they can be rotated separately and the flavor symmetries is not what we want. These two kinds of business, again embarrassing me, cannot be done by turning on the 0.2 super potential. Basically, there are two kinds of 0.2 super potential with people called J and E, but you don't have to know the details here. The only requirement is that the 0.2 super potentials are holomorphic in the chiral super fields. By requiring chiral holomorphicity, one could never get the correct modularized space. One can never get the correct flavor symmetries. One have to sacrifice the holomorphicity by going down to this N equal to 0 super symmetry. So some super potentials are non-holomorphic. They preserve only 0.1 super symmetry or the field contents are 0.2, but you can correctly get the correct modularized space and the correct flavor symmetry. It seems that you have lots of rooms to do that, but it's very, very tight problem of getting the correct things. I won't even bother you to see this non-holomorphic potential because to most people it will look very ugly. Anyway, after some work of turning on the correct potential, I'll explain to you what the results are. So I'll explain to you what kind of modularized space we get in the classical and quantum picture. This is basically kind of consistency check, okay? So first of all, starting off the classical modularized space is starting the vanishing of the bosonic potential, which is basically the sum of the complete squares of lots of super potentials in this theory, okay? So this potential includes the original ADH-engaged, the field contents, which were anomalous, and other fields that I have added. And requiring this to be 0, I find two branches of modularized spaces. In the first branches, after solving this algebraic equation, I find that all the extra fields that I have put in are required to be 0, okay? And this first branch, the remaining ADH-engaged field, satisfy the triplet of the following equation, which are nothing but the ADH-engaged condition that has to be satisfied by this field to reproduce the instant of modularized space, okay? So had there been only this first modularized space, it would be a solid proof that the low energy dynamics is seeing the correct modularized space, okay? So this is a classical analysis, okay? Since we are studying a UV system with so little amount of supersymmetry, one should, in principle, worry about the quantum corrections that can happen to this classical analysis. And all we can do with this, without technique, is the analysis of the so-called one-loop corrections. And the one-loop corrections are those happening by integrating out the massive fields with these light fields kept, okay? So the integrated out massive fields carry high masses proportional to some powers of these light fields, which are assuming non-zero values, okay? So by this kind of integrating out, you can have some one-loop corrections in principle to the equations determining the modularized space. But with suitable superpotentials chosen, as I explained in the previous slide, one can show that the one-loop correction in the first branch is vanishing. So at some quantum model, one has checked the consistency of the robustness of the instant of modularized space against the quantum corrections, okay? But further while quantum corrections, in principle, should be discussed, but our technology doesn't tell us how to do that. So we are satisfied with doing this one-loop consistency checks of the desired modularized space. The fact that we are getting the instant of modularized space at the classical, and in some sense quantum level, means that the modularized space is gonna be hypercalor, and it's strong indication that the n equal to zero comma four supersymmetry enhancement will be happening in the infrared, although we started from a very less supersymmetric theory. Another ugly feature is that if you investigate these equations, the vanishing equation carefully, we get a one-to-second branch, classically, okay? And this second branch meets the original instant of modularized space exactly at single point, let's say for a single-instant case, okay? This is meeting the first desired branch at a small instant of singularity, okay? So already this somewhat signals that the two branches will be decoupled in the infrared, so the single-gauge theory flowing to two decoupled quantum field theory in the IR, but actually what we think we find a much stronger thing by considering the quantum corrections in the second branch of the same sort, tracing the effect of the integrated outfields in the second branch, one finds that the suitable choice of super potentials can be making, giving non-zero one-loop corrections to the first branch equation, making the second branch detach from the first one. So although we have done this kind of analysis only at one-loop level, we conjecture that this detaching will be happening exactly, which is very ubiquitous, let's say, in 0.2 theories investigated by these authors. And if this detaching happens, it really says that our UV-gauge theory will be flowing to two different quantum field theories where the first branch quantum field theory will be of our interest, okay? So this is the picture we have about the modular space, and the first branch quantum field theory is the super conformal field theory we want to identify and study. So we get, in a sense, a non-linear sigma model on the instanton target space, but at the small instanton singularity, it has a curvature singularity, and the sigma model is known to be bad-behaved. So it needs a UV completion at this tip. And this UV completion is provided naturally by our gauge theory, because all the extra field that we integrated out in the first branch becomes massless only near the tip. So they provide extra degrees of freedom indirectly providing certain UV completion of the sigma model, okay? So now you can do some faithful quantum calculations. Now we can study other interesting observables, okay? Very powerful observables that you can study from supersymmetric quantum field theory, especially in two dimensions, they're called elliptic genus, which is nothing but the supersymmetric partition function of the quantum field theory on a torus compactified in a supersymmetric way, okay? So in quantum field theories, it can be interpreted as some written index counting some BPS states, which carries various charges, including the momentum charge along the compactified circle. So you put all the conserved charge factors in your trace and put all the kinds of fugacities that you can. It's too complicated to remember all, but it takes the following form. And once you have a UV gaze theory, as we do now, it's very easy to compute it because these gentlemen has provided with us very simple contour integral formula to evaluate these elliptic genus. Strictly speaking, they have derived this formula for the 0,2 or 2,2 theories, but it applies straight forwardly to our 0,1 setting by replacing qq bar commutator to q square, okay? So following their strategy for our UK gaze theory for SU-3 strings, we can write down the contour integral. It's complicated. You can rewrite it as a specific resist sum and the resist sum takes the following form. It takes a form of the so-called young diagram classification. This is a really technical thing. So if you don't want to see it, please ignore this. It's just for the experts. It's this young diagram classification of this kind of written index or elliptic genus resist has been quite ubiquitous in this instant on counting problems. So it has been first found by the Italian group and the Fleumann-Pogossian 14 years ago in the context of studying Microsoft partition function and the same structure happens to appear in our problem. Well, basically because it's an SU, it's a kind of UK gaze theory, the same structure. So we kind of, this is just to show you that we get a definite closed form expression for the elliptic genus which has highly predictive power. First of all, to make a small non-consistency check, the experts might wonder that the expression is too complicated because if you reduce this expression into one dimension, it will be providing with you the written index for the SU3 instanton particle in five dimensional SU3 gaze theory. So if you are experts on instanton calculus, you'll see that there are too many theta functions in the numerator and so on. Because the one dimensional limit of this elliptic genus is obtained by replacing all theta function by sine functions and apparently the expression doesn't look like the SU3 necrosis of partition function. But by using trigonometric identities in a careful way and so on, you can see that this in the one dimensional limit agrees with the necrosis of extra instanton partition function, which seems quite surprising to me. We find various ways of proving this and in this way we are finding an ugly but alternative ADHM-like formalisms for SU3 instantons, so which is ugly so almost useless in doing this five dimensional instanton particle counting, but this is the only formalism that's successfully uplifted to two dimensional. The ordinary SU3 ADHM construction does not. So the one dimensional consistency check was made. The novel physics can be seen by going up into two dimensions. It looked ugly but summarizing the answer for K equals one, we end up by getting this beautiful formula, okay? Where V's are the chemical potential for the SU3, Epsilon's are the chemical potential for the rotations on R4 and so on. Epsilon plus is the average of Epsilon one and two. And once we have this formula, we can see lots of interesting, well in a way we can test our theory further, yeah. Because taking our expression at K equal to one, oh sorry, and expanding it suitably in chemical potentials, you get lots of coefficients given by integers. For instance, we decide to take the log of partition function and expand it in the angular momentum chemical potentials in the following way, because as I already explained to us earlier, this takes the form of the genus expansion of topological strings of the associated elliptical labial. So this G is the genus expansion, the expansion with N is the further refinement. If you restrict the coefficient with zero, zero, this is the usual genus zero part of the topological string partition functions that people consider. And I'll break down our good friends, have computed a lot of coefficients appearing in the further expansions of this genus zero part about two years ago. So this partition function contains three parameters, two chemical potential for SU3, and one chemical potential for the spatial momentum for elliptic genus. So you expand in three chemical potentials or three charges and you display the numbers. So these are the numbers we get for momentum zero, one, two, three, and the SU3 carton charges. The black numbers are what Albrecht has thankfully computed to us. It's very non-trivial and crazy number. It's completely agrees with what we get from our gauge theory. And the red probably Albrecht can compute, but he didn't report it in his paper. And we can go on forever, just expand this given exact function. And for topological string people, practice should be interesting because we are making a suggestion for the old genus sum and close home expression. So it should be quite interesting to the topological string theories. So we find that our gauge, actually the whole motivation of my project, I think one and a half year was seeing this shocking paper. I saw these numbers, I was thrilled and I wanted to reproduce these numbers from the gauge theory, the only tool I know. And it took one and a half hour to manage it. And okay, so having seen these amazing successes, we can do lots of other interesting things. So I kept emphasizing to you that in the recent development of six-dimensional CFTs, the SU3 atomic constitution plays some interesting roles. And I'll try to explain to you just one example that we can study as an application. So once you have this SU3 gauge theory as a building block, you can combine it with what is called the E string theory to form what is called the E6, E6 conformal matter studied in this paper. The reason why this theory is interesting is because of its M theory dual realization. So this E6, E6 conformal matter, which in F theory is realized as this kind of geometric setting. In M theory, it can be realized as follows. So you prepare a single and five-brained, but, and you let it to probe the following curve background. So the transverse five-dimensional background is taken to be R, which is this, and it parameterizes tensor branch, times C2 mod E6 singularity. So it's one of the AD singularities. So if you allow these M5 brains to approach the tip of the singularity, you find a surprising phenomenon that this single and five-brained fractionalizes into four pieces, okay? And these four pieces can move around separately, enlarging the dimension of the tensor branch. So taking away the free tensor multiply part, you basically have three-dimensional tensor branches. And this is the M theory dual realization of this E6, E6 conformal matter. And for E7, E7, E8, E8, a further fascinating fashion of fractionalization has been illustrated by this paper and some follow-up works. So I wanted to study this kind of theory, the self-deal strings of them. And since there are three two-sphere factors, there are three kinds of self-deal string charges, K1, K2, K3. And for this one, we can use our SU3 self-deal string gauge theory. For any equal to one cases, we already know the gauge theory for the E strings. So what you have to do is to suitably combine them to form a two-dimensional pivot, okay? So this part, which I just schematically shown because there are so many crazy matters, is to be understood as the two-dimensional gauge theory for the SU3 strings. And the remaining part, the dotted part of the Fermi multiplets, the left side parts of the gauge theory for the E strings having E8 symmetry. But you partially gauge the E8 symmetry with this SU3 to get the remaining E6. And you do the same thing on the other side. The characteristic aspects in our UV gauge theory is that not only that it sees less supersymmetries in UV, it sees less global symmetries. So some global symmetries and flavor symmetries can also enhance if you do the algae flow. So in the UV, what we see is a very small amount of symmetry. Instead of seeing E6 times E6, we only see SO10 times SO10 times the diagonal U1 part of U3. So this is not quite the same as E6, okay? But we find strong signals by studying the elliptic genus of this theory that in the infrared, this symmetry is enhancing into E6 times E6 by finding that the numbers, the coefficients, are arranging themselves into E6 times E6 representations. So this gluing kind of business can be made to form a wider class of 0,4 super conformal field theories in two dimensions. One can make further tests for fun. This is very non-trivial test, actually. Because both for SU3, self-dealing strings and E6, E6 conformal matter strings, once you have a two-dimensional gauge theory, it's immediate exercise to compute the anomalies of the older flavor symmetries in your system, okay? So you go to weakly couple re-drive and do the standard calculation to compute the anomalies. So these are the anomaly four-form polynomials that one can compute. I wanted to test this anomaly four-form computed from gauge theory, two-dimensional gauge theory by alternative means, okay? So to do this, we have developed another way of computing the 2D anomalies, 2D flavor symmetry anomalies. And this can be done by recalling to what is, invoking what is called the anomaly inflow mechanism. So the anomaly inflow mechanism is basically embedding the lower-dimensional system of your interest into higher-dimensional string theory or M theory in which every symmetry is regarded as gauge symmetry. So if everything is gauge symmetry, it has to be canceling exactly, okay? So in this kind of setting, to understand the 2D global symmetry anomalies from the higher-dimensional gauge symmetry, we recall the anomaly cancellation of the six-dimensional quantum field theory, okay? Six-dimensional gauge anomaly has to be all canceled for its consistency, but you redo the anomaly cancellation business by inserting a two-dimensional defect. And since you have to insert a two-dimensional defect of self-theore strings at the boundary, there could be extra uncancelled anomaly that can be appearing from the bulk calculus. So the extra-appearing anomalies is all proportional to the four-dimensional delta functions transverse to the two-dimensional strings. And we call this the anomaly polynomial of the anomaly, polynomial, we recall the resulting violation of the anomaly cancellation as the I-inflow, meaning the anomaly polynomial obtained by these bulk calculations. So this kind of anomaly, this kind of inflow anomaly is computed because of the presence of this kind of inflow anomaly is basically a bulk anomaly incurred by the presence of the two-dimensional boundary or the defect. So the net anomaly of the system has to be canceled by having the intrinsic two-dimensional anomaly plus the inflow anomaly vanishing. So by the, this consists of the requirement of string theory or high-dimensional system, we require that the intrinsic two-dimensional anomaly plus the inflow anomaly to vanish. So in this way, you can indirectly infer the two-dimensional anomalies. And this can be computed compared with the direct anomaly cancellation of the two-dimensional field theory that we can do with our gauge theories. In both approaches, we get the same anomaly polynomial, which is very, very complicated in E6-ECC's conformal matter, but we managed to check that everything is agreeing perfectly, okay? So in the remaining time, let me tell you a final application because my title partly involves exceptional instanton and nobody will be happy if I say SU3 is exceptional instanton. So I can do really exceptional instanton using our novel approach because this can be motivated well by recalling the 60 Hixing and Unixings, okay? So the SU3 theory, I said, can be enriched by combining either making either a quiver or adding hypermultipline matters and enlarging the gauge group. So if you try to unix the SU3, the only way about the Hixing goes as follows. It can be unixed into G2 by adding one hypermultipline matter in seven and further into SO7 with two hypermultiplines in spinner representations and so on. So for these sequences, we were able to generalize our SU3 instanton strings quiver to the G2 and SO7 instanton quiver, okay? Couple to some number of spinner matters. And both cases are the ones in which one had no traditional ADHM-like descriptions. G2 by itself is an exceptional gauge theory and SO7 has its own ADHM description, but in this description, one doesn't know how to incorporate the matters in spinner representation. So both cases can be regarded as exceptional settings in some sense. And in our novel formalism, we can manage to find the history description for these instanton strings and also instantons and do some non-trivial calculations. To illustrate how things are happening, again, the key idea is the sacrifice of the infrared symmetry in the UV gauge theory. So SO7 instanton, let us consider this. SO7 is a classical gauge group. We think we know the ADHM construction perfectly well. Of course, that's true, okay? But it has some limitation of not allowing some matter inclusions and so on. So what we find is an alternative ADHM construction of SO7 instantons in which only an SU4 subgroup of SO7 is manifest. So this is again gonna be some ugly thing in some sense, just like our SU3 alternative ADHM was ugly, but our gauge theory is making this SU4 manifest only and you still have some merit for paying this price. So the gauge theory construction we make is the following. So since it has an SU4, so SU4 instanton is part of SO7 instanton, so it should contain SU4 ADHM data as a part of its field. And then we add lots of other fields. We have some secret rules, but at this stage, let me say we determined it empirically. Just like we determine the right gauge theory for the SU3 strings, we can add lots of fields to make it give the right physics of SO7 instantons. So the field that we have to add turns out to be this one. This is terribly ugly one because it doesn't even see the SO7 in UV. It's claimed to appear only after infrared symmetry enhancement, which we check by instanton partition function. But the merit of this approach is that now we can include the effect of the hyper-matter multipliers in spinor representation, because spinor representation in SU4 language just decomposes into four plus four bar. And it's immediately clear how to include the matter contents corresponding to the hyper-multiplier in fundamental representation. It's just including some numbers of Fermi multipliers. So this is the new ugly alternative SO7 instanton world volume gauge theory, having some inclusion of matters. And once you have this, if this is correct, you can sort of hix it to form the G2A gauge of formalism. And then if you further hix it, you get back to our SU3 gauge theory I explained to you. Now this looks ugly, but it's very useful. For instance, you can compute the G2 and the SO7 instanton partition function. It perfectly works well. Without any matters, you compute the partition function of our ugly SO7 ADHM. It agrees with the known necrosis partition function and so on. Now you hix it to G2. You can have a new contour integral formula for, let's say in one dimension, for the written index for some number of G2 instanton particles. And it completely agrees with the G2 instanton partition function, computed by various indirect methods. For instance, IMEI and no-powder and collaborators have computed the G2 instanton partition functions in various different ways. Oh, actually, Sergio has involved in the one instanton calculus as well some time ago. And the main technique they have advocated to calculate the exceptional instanton partition function is using some three-dimensional mirror symmetry and doing some monopro instanton calculus and trying to reproduce the Hilbert series of the hix branch and so on. So it's done by very indirect ways, but we can see that the gauge theory that I wrote to you in the previous slide give rise to the written index which completely agrees with the known results. And here the merit is that you can again include the matters of G2 instanton. So in some sense, we are getting the ADHM-like quantum mechanics description for G2 instantons and also for G2 instanton strings and SO instanton strings, okay? So once we know this kind of thing, we have to set up your sling gauge theory for this exotic atoms which includes G2 or SO7 with spinometers and so on. So we can do some nice things. So these are all works in progress with these, my collaborators. So let me finish. I try to explain to you that we are getting some solid progress concerning the studies of self-deal strings, but it's admittedly very difficult to study six-dimensional conformal field theories. Even after we reduce our interest to a very, very small subset of two-dimensional self-deal strings, we encounter lots of unexpected difficulties having to do with exceptional instantons and so on. But we think we are gradually overcoming some of the difficulties, finding surprising new discoveries concerning some gauge theory-like description for exceptional instantons and so on. It would be very interesting to see if our basic idea can be applied to other exceptional gauge theories. We are trying this, but it's very hard to say at this moment whether we are getting complete success or not. Again, the basic idea, just like the SO7, is that some of the UV flavors symmetries can be reduced compared to the infrared symmetry that one wishes. So, for instance, if you consider G2, E7, E8, they are natural SUN types of group which preserves maximum rank which we aim to preserve in the UV theory, but hoping that the gauge theory will enhance to this enhanced exceptional flavor symmetry. The G2 case has been already checked. I mean, I explained to you in the previous slide. E7 and E8 and maybe E6 and E4, these are challenges left to us. And finally, our two-dimensional quantum field theories, since it's a D3 brain and seven brains wrapping the two-spheres, is closely related to the four-dimensional arduous Douglas theories that has been explained to us by Casanova some days ago. Because these arduous Douglas theories are precisely obtained by letting a D3 brain to probe a set of non-trivial seven-brain singularity. So if you compactify exactly that system on the two-sphere, we get ourselves to your string system. So it will be interesting to see if the two approaches have some interesting lessons to each other. So I'll close here. Let's stop here. Thank you.