 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 this, 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, okay? 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, okay? 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, okay? 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. So 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 OB4 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, okay? And if you are interested in less supersymmetric theories and equal to one-zero, minimal super conformal field theory, you have to go to a more complicated setting. You 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-fibre Calabria 3-fold which has been discussed earlier in this conference. So the elliptically-fibre Calabria 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, okay? 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 Calabria, non-compact elliptical Calabria 3-folds are possible which gives rise to 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 elliptical alibi 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 host suitable gauge symmetries. Yes, please. Couple to gravity. Yeah, I mean, yeah, yeah. 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-component theory in 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 2, 3, 2, 3, 2, 2, 2, 3, 2, and have mutual intersection number minus 1. So to understand the structure of six-dimensional super-component 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 conformal 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 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-component field theory, having higher-dimensional tensor branches. And the second way of making the quantum field theory more non-trivial is by unixing procedure. Basically, you add more hyper-multiple matters. Here, you basically have no matters with which you can fix. You have either no matters or half hyper-multiple, which cannot be given expectation value. So the process of unixing 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 one-zero super-component field theories. So this is the review of the works on one-zero 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 D three-drains 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 its 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 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 these 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 you 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 Yang-Mill theory coupled to various hyper-multiple matters and the Abellion tensor multiplets. This is the auxiliary 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 super gravity. 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 multiplets in the following way. So if the scalars are assuming non-zero expectation value, it sets up an effective Yang-Mills coupling scale. So it gives rise to 60 Yang-Mills description. And the superpartner term of this is coupling the B-Mill-U field to what is called the instanton number density. So what is charged on the B-Mill-U is the self-theor strings. And in the Yang-Mills theory viewpoint, this is provided by the stringy soliton which carries the instanton number. So self-theor strings are instanton string solitons and super Yang-Mills. This will be the main most important viewpoint in my talk. Any questions so far? Of course, six-dimensional conformal field feels that 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 in the UV. So the two-dimensional gauge theory is a weakly coupled UV. So if one can engineer a UV gauge theory in 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 the theory of interest can be constructed in two major, two different ways. I'll call 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 grains. 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 grains 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 grains. 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 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 Oriental Fold 8 plane background where the self-theor strings are D2 brain 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-Wittern 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-Wittern 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 writing 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 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 has this kind of interaction. So any kind of strings having F wedge F source will be coupling to this B-new field and it should be interpreted as self-tiered strings. And you take the six-dimensional young mills on R6 and take the R4 direction, transfers 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 of self-tiered strings. The instanton numbers are mapping the self-tiered string numbers. 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. 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 points of gauge theory which you can try to use to understand these kinds of self-dealing objects. 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. 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. It's 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 these 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 this 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-ADHM 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 ADHM gauge theory has been used in zero or one-dimensional contexts 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, and it turns out that this naive quiver 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 multiplets have fermions, right-moving fermions contribute positively to the anomaly. So their contributions don't cancel to zero. So this is a bad point of 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 ADHM 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 realized SU3, it is realized in a certain non-perturbative way. So this is realized by the so-called H2 singularity of the D7 brain configuration. And this SU3 symmetry is realized not by having 3D7 brains, but in a suitable SL2D frame. It is prepared by having first 2D7 brains and 2 SDL brains, so the SDL of the D7 brains. So of course there could be light, actually massless fundamental strings suspended between 2D7s 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 7 brains. So the way this SU3 is formed is highly non-perturbative. 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 the ADHM construction of cell theory or 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, 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 6-dimensional CFTs this atom is playing fairly important role to construct all kinds of novel new 6-dimensional CFTs. In particular, all these exotic atoms essentially uses this atom 3 in the version in which SU3 is slightly unhixed to G2 or SO7 with some exotic matters. Secondly, this building block 3 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 cell theory better. So the strategy is following. I had a failure with the naive ADHM but I'm going to cure it. I'm going to cure the pathology and make it work. The way to make it work is simple. You encounter gauge anomaly. So you add lots of matters to cancel the gauge anomaly and then to see if other basic physics is working well. So try lots of... 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 gauge anomaly you'll have the wrong metacontent and you'll have the wrong BPS spectrum. This is just a really delicate job which requires lots of trials and errors. I think at least I spent half a year to have things to work and be on 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 gauge 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. Note that if you're going to construct UV gauge 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 supersymmetries. Some of them could be sacrificed as well. The supersymmetry enhancement after the algae flow is commonly observed in white class of theories. So what we can realize is the gauge 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 gauge 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 gauge are normally caused by the following ADH multiple but after some trials and error we find this complicated addition is doing the job. So the K bar is the anti-symmetry representation of UK and anti-symmetry of SU3. NT means rank 2 anti-symmetry representation. Symmetric means rank 2 symmetry 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 normally 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-deal strings. If you have more flavors, chiral multiplet or pharma multipletries without suitable potential they can be rotated separately and the flavor symmetry is not what we want. These two kinds of business again embarrassingly 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 and never get the correct flavor symmetries. One has 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 although 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 a 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. 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. So this potential includes the original adh-engaged the field contents which are 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 require to be 0. In this first branch, the remaining adh-engaged fields 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 instanton modularized space. 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. So this is the classical analysis. 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. So the integrated out massive fields carry high masses proportional to some powers of these light fields which are assuming non-zero values. 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 potentials super potentials 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 instanton modularized space against the quantum corrections. 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 instanton modularized space at the classical and in some sense quantum level means that the modularized space is going to be hypercalor and it's strong indication that the n equal to 0,4 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 an unwanted second branch classically and this second branch means the original instanton modularized space exactly at single point let's say for single instanton case this is meeting the first desired branch at a small instanton singularity 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 detached 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 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 so now you can do some faithful quantum calculations now we can study other interesting observables very powerful observables that you can study from supersymmetric quantum field theory especially in two dimensions which is nothing but the supersymmetric partition function of the quantum field theory on a torus compactified in a supersymmetric way 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 gauge theory as we do now it's very easy to compute it because these gentlemen has provided with us a very simple counter 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 q,q bar commutator to q square so following their strategy for our UK gauge theory for SU-3 strings we can write down the counter integral it's complicated you can rewrite it as a specific resist sum and the resist sum takes the following form it takes the form of the so-called young diagram classification this is a really technical thing so if you don't want to see please ignore this 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 flu man Pogosian 14 years ago in the context of studying necrosis of partition function and the same structure happens to appear in our problem well basically because it's a kind of UK gauge theory the same structure so we kind this is just to show you that there's definitely close form expression for the elliptic genus which has high predictive power first of all to make a small non-consistency check the experts might wonder that the expression is too complicated because you know if you reduce this expression into one dimension it will be providing with you the written index for the SU-3 instanton particle in 5-dimensional SU-3 gauge 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 SU-3 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 particle 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 SU-3 instantons so which is ugly so almost useless in doing this 5-dimensional instanton particle counting but this is the only formalism that successfully uplifts to 2-dimensional the ordinary SU-3 ADHM construction does not so the one-dimensional consistency check was made the novel physics can be seen by going up into 2-dimensions it looked ugly but summarizing the answer for k equals 1 we end up by getting this beautiful formula where v's are the chemical potential for the SU-3 y's are the chemical potential for the rotations on R4 and so on y plus is the average of y1 and 2 and once we have this formula we can see lots of interesting well in a way we can test our theory further because taking our expression at k equal to 1 and expanding it suitably in chemical potential 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 potential in the following way because as Albert 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 0,0 this is the usual genus 0 part of the topological string partition function that people consider and Albert and our good friends have computed a lot of coefficients appearing in the further expansions of this genus 0 part about 2 years ago so this partition function contains three parameters two chemical potential for SU3 and one chemical potential for the angular for the spatial momentum for elliptic genus so expanding three chemical potentials or three charges and you display the numbers so these are the numbers we get for momentum 0,1,2,3 and SU3 carton charges the black numbers are what Albert has thankfully computed to us it's very non-trivial and crazy number it's completely agrees with what we get for our gauge theory and the red probably I'll break and 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 closed 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 SU-3 atomic constituent 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 SU-3 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 can be realized as follows so you prepare a single and five-brained and you let it 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 if you allow these m-five-brains to approach the tip of the singularity you find a surprising phenomenon that this single and five-brained fractionalizes into four pieces 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-tiered strings of them and since there are three two-sphere factors there are three kinds of self-tiered string charges K1, K2, K3 and for this one we can use our SU3 self-tiered string gauge theory for n equal to 1 cases gauge theory for the E strings so what you have to do is to suitably combine them to form a two-dimensional quiver 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 part of the quiver for the gauge theory for the E strings having 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 super symmetries 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 the diagonal U1 part of U3 so this is not quite the same as E6 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 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-drill strings and E6 E6 conformal method 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 so you go to weakly couple regime for the 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 by alternative means so to do this we have developed another way of computing the 2D anomalies 2D flavor symmetry anomalies and this can be done by invoking what is called the anomaly inflow mechanism so the anomaly inflow mechanism is basically a 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 so in this kind of setting to understand the 2D gauge global symmetry anomalies from the higher-dimensional gauge symmetry we recall the anomaly cancellation of the six-dimensional quantum field theory six-dimensional gauge anomaly has to be all cancelled for its consistency but you redo the anomaly cancellation business by inserting a two-dimensional defect and since you have to insert the two-dimensional defect of self-deal strings at the boundary there could be extra uncanceled anomaly that can be appearing from the bulk calculus so the extra-appearing anomaly is all proportional to the four-dimensional delta functions transverse to the two-dimensional strings and we call this the anomaly polynomial we call the resulting violation of the anomaly cancellation as the eye inflow meaning the anomaly polynomial obtained by these bulk calculations so this kind of 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 been cancelled by having the intrinsic two-dimensional anomaly plus the inflow anomaly vanishing so by the this consistent 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, E6 conformal matter but we managed to check that everything is agreeing perfectly 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 unhixing so the SU3 theory, I said can be enriched by combining making either a quiver or adding hypermultipline matters and enlarging the gauge group in unhixing the SU3 the only way of unhixing goes as follows it can be unhixed into G2 by adding one hypermultipline matter in 7 and further into SO7 with two hypermultiplines in spinor 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 coupled to some number of spinor matters and both cases 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 spinor representation so both cases can be regarded as exceptional settings in some sense and in our novel formalism we can we can manage to find a 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 inclusion and so on so what we find is an alternative ADHM real construction of SO7 instantons in which only an SU4 subgroup of SO7 is manifest so this is again going to 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 this for paying this price so the gauge theory construction we make is the following so since it has an SU4 SU4 instanton is part of SO7 so it should contain SU4 ADHM data as part 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 determined 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 multiplies in spinner representation because spinner representation in SU4 language just decomposes into 4 plus 4 bar and it's immediately clear how to include the matter contents corresponding to the hyper-matter multiplies in fundamental representation it's just including some numbers of Fermi multiplies 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 suitably hix it to form the G2ADHM formalism and then if you further hix it to the OSU3 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 SO7ADHM 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 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 MEI and no-powder and collaborators have computed the G2 instanton partition functions in various different ways or 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 Higgs 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 right to 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 so once we know this kind of thing we have the Sergio's linkage 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 so 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 there are natural SUN types of group which preserves maximal rank which we aim to preserve in the UV theory but hoping that the gauge theory will enhance to this enhance the exceptional flavor symmetry the G2 case has been already checked 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 the D3 brain and seven brains wrapping the two-spheres is closely related to the four-dimensional argyris-douglas theories that has been explained to us by Casanova some days ago because these argyris-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-spheres we get our self-dealing system so it will be interesting to see if the two approaches have some interesting lessons to each other so I'll close here I'll stop here, thank you