 Well, thank you for having me and thanks to Maxime and Anton for for Maxime and Anton was here somewhere. Oh, yes, here. Thank you for organizing this wonderful event. It's a privilege to be around on such a great occasion. I love this photo. It's typical. Oh, and this is typical Samson's. His brain is always working even during other talks, so I hope that this will be inspiring to him. I have to say that I'm quite indebted to Samson and the reason is when we walk through life, we sometimes are lucky to encounter people who truly inspire us. Not just people from who we learn. The process of learning is very natural, smooth and logical. That should be a status quo we learn all the time. Inspiration is needed when we need to break out of status quo and Samson in many ways is anything but the status quo. So he truly inspired me throughout life to stick to my beliefs, to think what I think and sometimes against the odds or what circumstances demand. When I think about Samson, it resonates with me this famous campaign by Steve Jobs of Think Differently. So it inspires you to stick to your individual values and truth and it says that this is to troublemakers. They're on pegs and square holes. The ones who see things differently, they're not found by rules and they have no respect to status quo. You can code them, you can disagree with them, you can agree with them, it doesn't matter. You can do anything you want but you can't ignore them. So thank you. My computer is not Apple, but I truly admire both of these people and thank you Samson. Really thank you. So today was the day of two-dimensional theories and also QCD. So I'll try to combine both of this and this last talk. So hopefully it will be a nice culmination for this day. One way to motivate what I'm going to tell you is just study this theory. It's a theory of gauge field and coupled to spinners or quarks and some simple, most minimal representation of whatever the gauge group is. So otherwise known as QCD. I'll make the problem very simple by again putting it in two dimensions. So that will be my way to connect to other speakers from today. So we'll simplify or we'll try to tackle much, much simpler theory than four-dimensional QCD. And we'll simplify it even further by asking supersymmetric version where we could have some hints from hopefully supersymmetry. So in two dimensions there are all kinds of supersymmetries usually labeled by two integers referring to supersymmetry in left and right sector. So here I show examples with so-called 0, 2 and 2, 2 theories which are compared quite often in their physical behavior to four-dimensional theories super QCD with n equals 1 and n equals 2 supersymmetry. In fact, going back to early work of Samson with loss of a necrosis where they studied in fact some parallel between theories of 4D n equals 2 and 2D 2, 2 supersymmetry. In this part of the diagram you typically encounter holomorphic quantities such as pre-potential and twisted super-potential. And that work in fact was in some sense predecessor of gauge-better correspondence and many other developments in particular used by Misha Schiffman in the context of vortices who also popularizes this idea that hierophysics is very similar even though we go across dimensions. And hierophysics is very similar in a sense you can expect dynamical suzie-breaking. You don't typically have moduli spaces of vacua with this amount of supersymmetry and respectively 2 and 4 dimensions. So again, there are many similarities and we'll talk more about them today. But correspondence was discovered in that paper in 1998 when we tried to do the instanton integral. The one of hyperintegration over Higgs-Bart. Actually here I was referring to freckled instantons. But it's very similar, so indeed. So when you reduce, it doesn't matter. Once you reduce from 4 dn equals 2 and break half of suzie you basically land here. And we'll use several of such reductions today. So in that sense we're standing on shoulders of our fathers. Why 22a 040? Well I'll comment on other amounts. This is just like introduction because I'm saying that I'll try to make my life very easy by a going to 2 dimensions and b introducing suzie. And honestly I came to this question in the context of two-dimensional such super-KCDs on various brain systems. And I needed this as an application. So I thought that this should be in textbooks. I thought that this should be solved by far. So in fact the reason is very simple. That first of all as far as basic physics goes in two dimensions even a billion theories can find. Simply because in 4d if you solve Laplace equation on three spatial dimensions you get one over r, Coulomb potential. If you do the same in one plus one dimensional system you find linear potential. v of r is proportional to r. So you get exactly what you expect in confined theories. In fact Schwinger, we know this as a Schwinger effect or Schwinger mechanism or Schwinger model for in a two-dimensional electrodynamics. Schwinger he beautifully demonstrated that this happens and you can actually solve the system in a sense of predicting the spectrum and understanding its strongly coupled behavior. So this is really old and this history goes back far back. So to remind you how old it is I put here some events that happen around the same time from that year. So in fact the date Schwinger's paper was submitted or received in a journal happens to be Monday. And the reason I know this is that Sam Walton opened his first Walmart store in the state of Arkansas. This was also the year when Landau got a Nobel Prize. This was a year off Cuban Missile Crisis as many of us know. It was unfortunately the year when Erlin Monroe died of drug overdose also during the summer around the same time. And the Warhol drew his famous painting of cans of Campbell soup and so on. So then it's also ancient because even in a non-Abelian case the next stage in this development of two-dimensional QCDs was the work of Hooft who solved two-dimensional super QCD or just QCD with NF equals 1. So there was one type of flavor and showed that it has many characteristics of four-dimensional theory we expect and particularly it has a rejet trajectory of meson states. So this was the year of Muhammad Ali's famous wins. Nixon introduced a 55 speed limit in every state and then left the office over Watergate scandal. Unfortunately, this was the year when Pompidou died but Leonardo DiCaprio was born. This was the year when Charles de Gaulle Airport, which every one of us passed either on the way here or on the way out, was open in the way we know it. And Solzhenitsyn was arrested and then expelled from Soviet Union around the time. Cyrus Tower in Chicago became the tallest building. That was all of year 1974. That's actually a good one. So I was curious about this too. What's the connection? So one is a mathematician, the other is a writer or a literature expert. So what's the connection? So indeed it's a good thing to explore and I did some googling. So the history is so ancient that again when I came to this kind of problems involving this two dimensional theories with supersymmetry or super QCD. Yes, it's a model of QCD but I thought I should be able to find the answer in text book. So I was really surprised that that wasn't the case. So even though the history is very ancient, the progress then was very, very slow, surprisingly. So some of the heavy lifting we had to do ourselves. So next landmark in this development was around 90s, not even 80s, when we're still talking about n equals zero non-supersymmetric theories now with a joint matter. And around the same time was a lot of confusion what happens if you have a joint matter. So there was a little bit of progress in particular in some of these papers by our colleagues who clarified whether or tried to clarify whether it's screening, confinement, big question. And there is approximations such as DLCQ, numerical approximations analogous to lattice amount to color was what happens if mass of the quarks goes to zero. It's really fun to read this numerical literature because there was a lot of confusion around the time just before that. And what they find is that in theories with a joint matter you start finding stringy-like spectrum which is also very natural, of course. This was in a way precursor to many ADS-CFT developments where this of course happens. But again, the gap is pretty big and the reason I mentioned here I join is for two reasons. So this is not the minimal, not the smallest representation of matter, but it's useful if you start thinking about supersymmetrization because basically supersymmetry requires you to introduce partners, in particular to gluons, in a joint representation. So that's a very nice thing to do. And then if you ask about supersymmetric theories, despite all the interest in mirror symmetry, linear sigma models and all this and that, it's interesting that until very recently we had very poor understanding of non-Abelian dynamics with 2-2 supersymmetry. So here I mentioned just a couple of developments which happen in the range of years from 2006 to 2016. But again, it's to me kind of surprising that relatively recently we started to understand quantitatively what happens in non-Abelian theories, even with supersymmetry. So when we encountered this problem, we needed 0-2 version, which is one aspect of what I'm going to tell you about. And by analogy with n equals 1 theories in four dimensions, you might expect and people did expect that it should exhibit some kind of cyber-type duality, that it should have dualities relating to different theories with different gauge groups. So that's what expected and unfortunately that A did not exist. There was absolutely no non-Abelian duality in 0-2 gauge theories as of 2013. I was totally shocked that there was no single paper, no single result on non-Abelian gauge dynamics with this type of theories. And after looking into this, it turned out that not only you have a analog of cyber duality, it's actually a triality. So number three will be playing a very interesting role in this talk and I leave you with many questions in particular why certain things happen the way they are. And here I can ask why is it a triality? Where does number three come from? Can we see in simple way why it's a triality? And I think at a very intuitive, simple level, we still don't quite understand this. So it would be nice to understand either from brain models or otherwise. So then I'll also talk about 0-1 theories. This is the most minimal amount of supersymmetry you can possibly have. Not much of a help really. It's very close to non-supersymmetric theories. And again, until very recently there was no single statement about non-Abelian dualities about such theories. Surprisingly, despite the fact that both 0-1 and 0-2 supersymmetry in two dimensions is what you require for heterotic worldshed, there was very minimal, almost nothing on non-Abelian gauge dynamics in this context. So again, I'll report on some recent work with Dupay and Pavel Putrov, two young guys who are instrumental in solving these theories. So as we do this, we'll in the process learn something about topology of theory space and some new anomalies. Again, something I'll leave you with as a question that naturally comes out of the story but I don't have a full answer to. So there will be interplay between anomalies and dynamics of quantum field theories in various directions which is the common theme, in fact, the name of this meeting quantum field theories and anomalies. So let's start with basic 0-2 theories. If you want to build the most elementary two-dimensional super QCD with 0-2 supersymmetry what do you do? You start by putting gauge node in the square notation then some number of colors, say SU and C, that's going to be a gauge group and you couple it in the minimal way to matter. In two-dimensional 0-2 theories there are two types of multiplets called fermi. These are basically chiral fermions denoted by the dashed line and chiral multiplets which have scalars in them and right moving fermions in my notations and let's say NB of those. So in a quiver notation that would be the kind of theory that you want to solve and ask does it confine, does it flow in the deep infrared to super conformal fixed point or what happens? The way I presented it, for generic values of these three parameters and again generic such theory will be specified by three numbers, number of colors and two types of metro multiplets. For generic values it would be anomalous so you have to think of how to cancel the gauge anomaly and one minimal way is to introduce another set of multiplets, which are called P, which are chiral, but in anti-fundamental representation of the gauge group. So that's in some sense most economical or simplest solution to the problem. So then you get this sort of quiver diagram representing single gauge group and interacting with matter and again you ask a question now, okay, this is simplest gauge theory, what does it do? Where does it flow? As expected, depending on ranges of these parameters, you can either supersymmetric suzy breaking, dynamical suzy breaking or flow to super conformal points or some free theories in the infrared. So in that sense it is similar to cyber type dualities and structure of n equals one super QCD in four dimensions. So here is a diagram because now such theories are labeled by three numbers, colors, number of bosonic or chiral multiplets and number of fermi multiplets. I have to plot by choosing two of these numbers representing the axis and then the other number will roughly set the scale of diagram. So here I plotted and color versus and fermi of flavors and some of the regions in this diagram are fairly easy to understand. For example, if you have a lot of glue in your theory, it doesn't matter what kind of theory it is, but in typical suzy theories with small amounts of supersymmetry you get dynamical suzy breaking if you have too many gluons compared to matter. And that's basically what this region represents. And then if you have reasonable balance or more matter than glue, then you typically flow to something with nice modular space or super conformal dynamics in the deep infrared. So that's what happens here. Before something more interesting happens, notice that this region carved out by suzy breaking and interior is where you have some interesting non-trivial super conformal dynamics in the infrared is of a triangular shape. If you add a point of infinity, then it becomes more like a triangle. In fact, you can bring this to a nice symmetric form if instead of emphasizing the rank of the gauge group, you re-label these nodes and pay attention to the flavor groups. You call them n1, n2, nn3. Then you notice that such basic super QCDs actually enjoy complete symmetry with respect to permutation of n1, n2, n3. So it is doing something. If you recompute what the rank is in terms of n1, n2, and n3, then rank will be changing. So it is like cyber duality, but it is really a triality. And again, question is why in two dimensions cyber duality is replaced by trialities? Where are these three coming from? So again, I would love to have intuitive answer to this question. And even though by now this has been explored from many different angles, some are present today, we still don't have a simple intuitive answer. Mathematically, one way to understand or approve or justify this duality is the following curious fact that this duality predicts at the level of chiral algebras. Many of us know a simple fact about Grasmanians that studying Grasmanians of k planes in n dimensional space is the same as studying Grasmanians of n minus k dimensional planes in n dimensional space. You just take orthogonal complement. So Grasmanians come equipped with very natural bundles, such as tautological bundle and the quotient bundle. Those exchange the role under this basic duality. But again, this is a duality. It's not a triality. So this is quite known, but what's less known is that same bundles and same Grasmanians actually do enjoy a triality and that's what such models predict. Namely, if you take some combination, certain combo of this quotient bundle and tautological bundle on a Grasmanian and study chiral-deram complex that's a fancy cohomology of such combination, then claim is that it's completely symmetric with respect to n1 and 2 and n3 and such symmetry is not manifest on the right hand side. Can you go back to the triality picture? So your evidence for this is tough anomaly matching. By now, all kinds of things, not just various anomaly matchings. I'll talk more about anomalies a bit later. There are calculations of elliptic genera, various brain constructions, and I'll get to this by engineering them from high-dimensional theories in various ways. So it's still a conjecture. I want to emphasize just like cyber duality is actually a conjecture, but there is a lot of evidence, especially in 02K. I'll give much more provocative statements which are more fun conjectures that I think are more at risk. And then you'll have to catch the train, unfortunately. Was there another question? Is it connected to an n-efficon or n-efficon to Nc for dimension? Where you have also 3. Nf equals, say again? Oh, I don't know. That's a good question. Explanation of the sword is something very simple which puts number 3 right at the forefront would be lovely. I don't think we have that now. I'll present something more uglier than that, yeah. You haven't seen some connection to either topological vertex? I would love to. Again, I would love to. It's very suggestive. Solgibrid the corner? Yeah, something like that. VOA of the corner is just VOA. It's not a theory. So this is physical theory, but I'll comment on 04K where we also were expecting tryality. Surprisingly, in 2D theory systems, tryality will persist. You will see it over and over again. So again, it suggests that something geometric like brain construction with three types of brains and you just rotate them or something like this or tryality in SOA in fact would be responsible for this. We don't have this. So it begs for very simple intuitive explanation even in this basic case. Right. So what I presented to you is a very nice large class of theories which are labeled by number of colors and matter content which is basic SQCD theories, but they're not the most basic. So if you go to the boundary of this region, you can construct simpler theories. So let me try to redo this construction from scratch starting over again. I pick a gauge group and a simplest non-Abelian gauge group since I emphasize that non-Abelian is something that was lacking in the literature quite surprisingly. Let's take SU2 as a simplest non-Abelian gauge group. So in general, if you pick SUNC, it would have gauge anomaly minus NC. So in this case, it would be minus 2 in my normalization. So to cancel that, you have to introduce some matter and the simplest matter if you focus on fundamentals will be four copies. Then they'll cancel the gauge anomaly and the simplest kind of Ising model of such theories would be therefore SU2 SQCD with four flavors. Conjecture is that it's dual to a theory with no gauge interactions whatsoever and interactions of the Lundau-Ginsburg type where you have six chiral fields and one fermi interacting by a simple super potential. So if you think about such system and such duality, first question to ask in terms of evidence is, for example, what is the... if I try to integrate out gauge interactions which are very strong in two dimensions. So as you go down our G-flow, temptation is to write everything immediately in terms of gauge invariant operators. So here it would amount to asking what is the space parameterized by scalar fields and the chiral multiplets mod out by gauge group? It's easy to see that it's a cone and the base of the cone is basically our space of matter fields divided by U2. So if I replace SU2 by U2, then such quotient is precisely the Grassmanian, namely 2,4 of two planes and four-dimensional space. So then this Grassmanian can be also described by Plukar relation which is a single relation and this is nothing but the relation imposed by this super potential. If again you introduce additional U1, kind of think of gauging U1 here making the group U2 and also gauging six chirals, always charged plus one, then you effectively make six-dimensional space divided by U1 which is Cp5. And this relation on side Cp5 is yet another description of the same base of the cone or modular space if you include U1. So is it 5,3,4? Is it 5,1,2,1,2? Yeah, probably there is a typo. Thank you so much. It's a Plukar relation for Grassmanian 2,4. So it should be different. It should be 3,4, correct. So by looking at such simple things you for example start seeing the evidence why a model with gauge interaction so this is trying to understand more and more basically what's going on. So this is kind of appetizer if you wish for much more interesting trialities but in this case it's a duality relating such most basic SQCD with 0,2 supersymmetry to something still interacting but very, very simple. So this is essentially Landau-Ginsburg model or now Ising type model. So another topic that Samsung worked on a lot and in fact that was already alluded on some of our earlier discussion and questions is topological twists of high dimensional theories in particular four dimensional theories and when I was a kiddo I really loved various TQFTs and topological twists so to me the issues that's how they refer to the paper join paper with Nikita and Andrei Losev was basically a Bible to learn a lot of things including some of the cool things most recent that I'm going to report on. So 4D and equals 1 theories that I mentioned earlier they do not admit topological twists on a general four dimensional space time they exhibit interesting phase structure something that I already mentioned earlier in the sense of cyber dualities and so on but they can be topologically twisted on smaller manifolds so we can try to do that we can try to take not 4D and equals 2 theories which produce Donaldson type twists but 4D and equals 1 theories that have smaller supersymmetry. Here I want to point out that this is analogous diagram to what I was showing you before the phase structure of 0,2 supersymmetry involves three numbers because we have two types of matter in four dimensions you have one basic type of matter namely carol multiplets so it's easier to plot it becomes just one dimensional line instead of two dimensional picture and here the landmarks involve relative to number of colors three half mc and mc plus one these are cool interesting values where something happens I'll come back to this in a second so what you can do in 4D and equals 1 theories is take one of the cyborg duels and try to topologically twist theory on a Riemann surface on a two manifold so supersymmetry is enough to do that and in fact this kind of topological twist had been studied by Samson collaborators and Misha Bershatsky was actually around here so he in famous work with Volodya Sadev and Johansen and Kumran studied such compactifications now called partial topological twists because you twist along partial part of the spacetime not entirely the full spacetime so if you do that you get precisely this relation to two dimensional theories with either four super oh sorry yes if you do n equals two twist you get four supercharges here I'm illustrating at four n equals one theories where you get two real supercharges namely zero two supersymmetry and if you start with dual pair in four dimensions you might expect that you'll find a dual pair after compactification this logic has to be taken very carefully and seriously and revisited because there could be many caveats but I'll show you example where this literally works so you don't have to worry about any details so in this partial topological twist first question is what happens if I take my basic multiplets a vector multiplet becomes a vector multiplet in two dimensions plus a joint chirals related in number to the genus of your surface this is basic color to client reduction there is nothing fancy here and if you do the same for a chiral multiplet you find that because of the topological twist the number of effective either chiral or fermi multiplets in two dimensions is affected by the r-charge of the original chiral multiplet in four dimensions with respect to vector multiplet there is no such choice you can study such compactifications on basic two manifold which are spheres this is something that again goes back to that earlier work we mentioned of Samson with Nikita and Lossiv and more recently younger guys like Sergei there the other Sergei myself and others worked on so it's a fun called the client reduction which connects different types of 4D and 2D physics with similar gauge dynamics let's take this simplest cyber pair which actually has SU2 gauge group just like what we want in two dimensions but we'll take a different matter content we'll have nfql3 and here we get nxql4 so it's actually fun to go through this and see how this emerges first of all nfql3 really means six doublets in dimension 4 that's the usual notation but also I want to point out that with respect to ncql2 nfql3 is also nc plus one as well as 3 half nc so it's a very special value where these two different landmarks actually coincide and the theory is famously dual to just the theory of mesons even on four dimensions so it's very similar to what we were proposing in two dimensional case there is no tryality so this is in that sense it's a simpler model it's a basic structure but unfortunately it's not a tryality so in the more general class question still remains but here it's an example of how you get duality from duality yeah and in my opinion tryality is in some sense more mysterious more interesting this is something more traditional that's why it connects to 4D theories that we also understand in a way that you guys also studied so what happens here is that you have to choose our charge assignment to do this reduction and the way number 4 comes out of this six doublets is because to make non anomalous our charge assignment you have to choose some charges and two of them have to be one so as a result you get here SU2 with four flavors and if you trace what happens on the dual side you get exactly the spectrum and interactions as proposed before so that's in fact one evidence for this duality so this is one application where you can derive something two-dimensional from four-dimensional physics from high-dimensional theories again there is no such good understanding of realities at least there are proposals and they shed light on the physics but there are more cumbersome and number 3 is not just glaring at you from the page that's what you guys were suggesting and that's still lacking I want to emphasize there is no number 3 this is duality that's not a tryality it's a simpler thing but in my opinion it's a little bit more boring so it's on a outside of that class you can say it belongs to a tryality where one of the three corners of the tryality degenerates to nothing unfortunately is that this theory is close to nothing? no no no it's again that's a figure okay I forgot I'll clarify there are not that many tryalities in four dimensions exactly that's why number 3 is mysterious tryality of some three prong brain network or something is suggestive for candidate but again there has been nothing like this so far anyway going backwards we can try to learn something both ways so here I showed you how from high dimensions we learn about dualities and 2D system but then we can also learn something about high-dimensional systems from these dualities in fact if you can't rectify and understand this reduction on Riemann surfaces and get good control over this 2D physics which again is easy to do I just showed you an example you get a glimpse of what happens in a high-dimensional theory when it's actually twisted on a general four manifold as long as you have your 4D n equals 1 theory flowing to 4D n equals 2 theory you can get a lot of mileage by going backwards by understanding such twists and this is especially useful if your theory is non-lagrangian because then it should still admit topological twist but it won't have a very simple description about Donaldson type TQFT it will be something more interesting and mysterious and question in fact already posing that paper on issues as well as testing cyberquit and solution was what happens with Argyris-Douglas theories which are simplest examples of non-lagrangian 4D n equals 2 theories so they should have topological twists and question what do they compute on a general four manifold but that's a very delicate point so question is indeed what happens what the answer I don't know the answer actually nobody does on a general four manifold but I can propose you an answer on a very simple manifold or class of manifolds of the form one cross a general three manifold so in that case you can think about this theory if you compactify on a swan but it's important that we're still working with four-dimensional theories so you basically keep all the colors a client mode that's important as a three-dimensional TQFT on a M3 and then you can pose a question okay what is it as a 3D TQFT on M3 so claim as that it's described by usual axioms of three-dimensional TQFTs and particularly you can associate modular tensor category which has cutting and gluing relations described by S&T matrices and for every Arduis Douglas theory you get a set of S&T matrices that can be computed from the beta vacuum so this is particular if you wish development or application of gauge-beta correspondence the way in fact it was envisioned lifted all the way to four-dimensional and equals to business so in the most basic Arduis Douglas theory which sometimes in literature is called 2,5 Arduis Douglas theory that's the theory that Arduis and Douglas studied in great detail what you find is so-called Fibonacci modular tensor category that describes Fibonacci anions it has two simple objects and S&T matrices are close cousin of what we see in Ising type MTC or other simple modular tensor categories okay enough of 0,2 let's talk a little bit about 0,1 so this is where we have half of the supersymmetry so we're quickly losing control but equations are recognizable in this case so they are Samsung is asking if beta equations are recognizable in this case sometimes they are not quite I don't know we should I can show you the equations and hopefully you'll tell me what the integrable system is I don't know that's a very good question are you actually solving this beta type system yes so the matrices are obtained from data of some fixed points on modular spaces or critical points of potential and potential is involves some elliptic function so it could be an interesting integrable system but again I don't remember to answer Samsung's question of the sum I have so let's talk a little bit about 0,1 super QCD again going back to this motivation how come our fathers of the subject such as Schwinger and Hooft and others didn't solve it for us it's such a simple theory we have almost no supersymmetry and we're in two dimensions so shouldn't this be done in 70s 80s or at least 90s that's right but 74 when Hooft's paper was written that's exactly the year when supersymmetry came to the shelves so yeah it should have been done back in 80s so here Chris Hall who is here and again classified super multiplets and they're actually less famous because somehow at least these days younger generation goes immediately to higher dimensions and maybe doesn't learn this too often so therefore I decided to summarize them for you they're complete analogs of this carol multiplets and Fermi multiplets in 0,2 case except that carol multiplet does not deserve to be called carol in a sense of holomorphy because scalar here is a real scalar and fermions involved are Majorana while fermions so they're left moving right moving depending on multiplet but they're all real so everything is real you completely lose power of supersymmetry in a traditional sense so if 0,2 was somewhat useful 0,1 is here very very weak so you have interactions in the form of super potential which again is a real super potential and vector multiplets so let's do the same let's build the simplest QCD so we take vector multiplet because now it has half of the gluinos of 0,2 theory the anomaly is half of mc with one sign so you have to compensate it and previously we had nf equals 4 in 0,2 notations so here we're going to have nf equals 2 in 0,2 notations so here I'm assuming that we take complex doublets and there are two of them so that's a natural way how SU2 can have complex representation so again anomaly is cancelled basic question is what does this theory do in the deep infrared so you kick it off it goes via our g-flow where you do end up confined is it a massive gap phase or gapless phase interacting theory what happens proposal is that it flows to a free multiplets free scalars and the logic could be obtained in a way similar to the previous scenario in fact you can try to think of this proposal that I'm going to give you as a soft perturbation of the previous duality that we studied first of all the modular space you can analyze in a similar manner is a quotient by the gauge group which is SU2 of matter fields scalars in the matter multiplets and now there are two complex doublets so it's a C2 turns over C2 some a-dimensional real space divided by SU2 and again it looks like a cone it's a cone whose base is a 7 divided by SU2 and this quotient is actually S4 it's a famous one of the famous Hopf vibrations so the cone on a force here is R5 so that's part of the reason why we are proposing that this could be just R5 parameterized by real scalars again very simple duality again it's a duality if you try to construct more interesting theories with 0,1 supersymmetries again you find surprisingly not a set of dualities if you construct simple 0,1 supersymmetries again you find trialities I have no idea why this so happens in two dimensions that having just one gauge group and asking how does non-abelian gauge interactions in two dimensions exhibit dualities immediately leads you to number 3 this is very funny very strange very mysterious in fact something else happens something we didn't quite resolve and that connects to a question Vasili was asking in the case of two-dimensional theories with 0,2 supersymmetry we were dealing with groups SU N or SUMC in the case of N equals 0,1 supersymmetry which is even simpler what I'm showing you here is class of theories which exhibit this triality and behavior nicely but natural thing to act on a real representation on real fields and real representations is at all because you don't have a complex structure and then we were naturally expecting that if you ask about 0,4 supersymmetry which came up in Ruben's question as well as in Vasili's question we were naturally expecting that you should have your SP and in fact we were hoping to connect to this corner and other developments and somehow there is a check mark here and check mark here so I claim that there is another triality which is much more delicate so this is real conjecture in that sense you sure you're going to make the train I'll leave after you finish this because this is more speculative so this is cool conjecture so this is conjecture to which again it's fresh of the oven until last year there was nothing on 0,1 theories or dualities with non-abelian gauge groups there were no dual pairs and again claim is that they come from trees so where the hell number 3 come from I don't know moreover this is kind of conjecture which is very risky so I don't know if this is really true but that's our conjecture and I'll give you some of the evidence so it checks out but this didn't work so part of the reason this work was delayed for several years we were hoping to get to other cases shortly after 0,2 in past 5 years or so the reason it took so long is that this case didn't work out we don't have any good set of non-abelian analogous so we were expecting realities but again somehow they didn't work in the 0,4 case so I present this as a challenge and SP even if it appears it actually is more relevant here somehow not really in that case so I don't know there are lots of questions so again statement is that here you get that same reality and it's much more delicate so here you cannot use the usual techniques from holomorphy and very simple anomalies you actually have to think a lot about very delicate type of anomalies which were behind the scene and checking every previous duality I mentioned so that's part of the anomalies and the title and in connection to our general subject but here this anomalies appear in several flavors first of all there are simple perturbative tooth like anomalies which you can compute from one loop diagrams so those are easy those check out and those I would call perturbative but then there are also global because they care about global structure of the gauge group anomalies that have somewhat non-perturbative nature and take values in this torsion part of spin-bordism group of classifying space of G where G is your symmetry so these are much more delicate anomalies and this is kind of cool and interesting for example if you try to build the simplest example of a theory with 0, 1 supersymmetry which exhibits this anomalies you can take n scalar multiplets and just one Fermi multiplet which enforces by super potential this condition that all of your scalars square to some constant so you're trying to describe a target space which is hard to see here but which is a sphere of dimension n minus 1 so no gauge fields but then if you try to gauge z2 which acts as antipodal maps so out of n-dimensional sphere you're trying to make rpn basically by gauging z2 what you find is that this anomaly is actually non-trivial and in fact the group that's relevant here for z2 happens to be z8 and as a result such theory only makes sense or is non-anomalous when n is multiple of 8 so this is a simplest example where such global or non-abelian sorry non-petrobative anomaly happens so I mentioned previously that for understanding z2 theories it was sometimes useful in fact that's how they came about in our first work with Putro-Vangada we were trying to get them from various brain configurations and high-dimensional theories so z1 theories have very little supersymmetry cannot easily get them from 4 dimensions reducing on Riemann surfaces you can but you have to be very inventive about how you do it but they actually come very naturally on the boundary of 3D theories so here is a paper which intended to study walls, lines and all kinds of phenomena related to spectral dualities in 3D unequals 2 theories and was using gauge-bethic correspondence in a crucial way so in this paper we cite some some work effluently and it's easily generalizable to 3D unequals one context where on the boundary you naturally get 2-dimensional physics with 0, 1 supersymmetry so we can try to ask which 3-dimensional systems can give us some of the proposed 0, 1, 2-dimensional super QCD like theories so these are usual 2-flag anomalies that you get from various building blocks, 3-dimensional matter, 2-dimensional matter again you want to make sure that they cancel as far as gauge interactions are concerned and here is an example you can take 3D unequals one system with just SUM vector-multiplet again it's going to be minimal supersymmetry in three dimensions at turn Simons level N over 2 is believed to have one vacuum, one massive vacuum so the theory flows to supersymmetric vacuum and with choosing Neumann boundary conditions on both sides for this gauge-multiplet you basically get different anomaly inflow on boundaries turn Simons level causes the fact that on one boundary it contributes with a plus sign on another boundary with minus sign so on one boundary you don't have to do anything the inflow is trivial but on the other boundary you have to compensate it with something and because we're imposing Neumann boundary conditions we get SUM interactions and simplest spell to compensate it is to add n fundamental flavors that's precisely the duality or the system that I was proposing and system in a bulk flows to one gap vacuum so you would expect that in the deep infrared is described by gauge invariant quantities so that's this five scalars one unequals to do but then there are more interesting inventive ways to engineer it in fact somewhat more natural more natural way would be in fact to take 3D unequals one theory with SUMC gauge group and the same number of fundamental flavors so it's also supposed to engineer on the boundaries in the same way but now in a non-trivial way because both boundaries would contribute the dynamics of zero one theory of the type that I showed you before and therefore the dynamics of zero one theories makes a prediction that first of all this theory does not break supersymmetry dynamically and suggest that it should have a dual but as far as I know the dual of the 3D unequals one system is not known so that's one of the questions I promised you several questions so that's one of the questions I want to leave with you and finally I promise new dualities as you can see in the world of supersymmetric dynamics or gauge dynamics we're getting into more and more delicate effects so I'll finish with the most delicate type of anomalies if you wish that don't have interpretation even in terms of this discreet global anomalies that I showed you a moment ago actually in two dimensions you don't have to answer them or you don't have to introduce us there are also Bezumina, Karol, there are many ways to introduce them you use the box that you had first necessary you introduce left hand fermions if it's the same anomaly particularly you could have coupled it chirally to W2W1 that's true but you have to keep track of anomaly one way or the other there are many mechanisms the other one probably is harder to supersymmetrize I totally agree what I was showing on these previous slides is that there are several ways to engineer the same system and some of them may be relating to questions that we don't know about in this case high dimensions that's still of same general spirit of trying to get 2D theories from high D right so in this last part let me ask about the topology of this theory space so this is space where every theory is represented by point or G flows are represented by flow line very much like in dynamical systems and you try to connect all possible theories going from one critical point to another and so on ask what this topology looks like so in fact studying the 01 theories in a joint work with Dupais, Pavel Putrov and Kumran we had to make a conjecture in order to make peace with many facts from physics and math so we had to conjecture that this space of theories which I call T has the following property first of all the space is graded by gravitational anomaly this is kind of obvious because well first of all gravitational anomaly is one of the most obvious things and 2D super conformal theories it's the difference of left and right central charge and clearly by any RG flow you can never go from one value of this number N which is the difference to another value so this theory space is clearly consists of islands which are different connected components labeled by different M but then we further had to conjecture that each graded component has a funny property that if you ask about its homotopy types then shifting this index of pi by K is the same as shifting this value so that's kind of strange that's very interesting thing it's a rather non-trivial conjecture again I don't know I'm going an increasing complexity of conjectures and my risk of failing them therefore losing credibility but this is what we were led to conclude and the reason this is funny is because it says that apart from gravitational anomaly which distinguishes different connected components in the theory space there are some other kinds of anomalies because these groups again compute them and so on they have torsion a lot of torsion and torsion doesn't come just in multiples of two so you don't get just Z2, Z8, Z16 which we often get from fermion anomalies you start getting multiple of three appearing and I told you that in this talk all my mysteries will be related to number three so here one of the mysteries is why three or how Z3 appears as a home for anomaly not from complicated co-boardism groups but rather we can see it immediately right in front of us so here is the first example Z24 but there are lots of Z3s that appear further down the road and five or seven never appear so it's only two or three for two, origin is usually typically very simple as it has origin and fermions but again three is something mystery so here is an illustration of what this type of anomalies, new anomalies I don't know how to interpret and there again not of the type that I showed you before that they cannot be as far as I know expressed in terms of co-boardism groups or generalized higher form symmetries again maybe they can but it hasn't happened yet so again, you can move between the pms by attacking a free fermi multiple right? Yes so how can they be fundamentally different you just attack a free fermi multiple yeah so this gives a lot of information about structure of each of them basically that's right so they're all very similar I would say they are isomorphic but I can't explain what this means so this implies a lot of relation on each individual copy they're isomorphic but if you take say, well abstractly one of them it's a huge infinite dimensional space and what this gives is humongous information it's very constrained system precisely because of this relation what pk of a given fixed guy is so you immediately know the whole tower basically exactly exactly such spaces are extremely rare and this is what's called structure of infinity spectrum and amygospectrum and there are very extremely rare things so that's why I'm saying this conjecture is something much stronger than even zero one triality that I mentioned so the way to think about it is to follow so if I say that there are different components labeled by gravitational anomaly what does different additional things tell us is that each one of them may have basically subdivisions so here is an illustration of thinking about them that this one gets divided into two parts some numbered parts and so on and so forth so they're basically different way more connected components than you thought they could be so again I don't have a good explanation of this type of anomalies that play a role in the zero one theories this seems to be a consistent network or as Nadia Cyberg would say web of statements or conjectures but this definitely require are required by some of the dualities in zero one theories that I mentioned earlier some of them have natural interpretation but in very low rank for example this is basically refinement of with an index where you look at kernel of supercharge or the same thing in one of the fermion numbers you still have supercharge on the right in zero one system so you can analyze that and what two index is candidate or is it's not much of a conjecture this is really true but again I don't have an explanation for the entire tower so this is in that sense it's showing that it's something delicate something rather subtle so therefore I leave you with this question and again happy very happy birthday Samsung maybe two questions and other questions we'll do at the dinner did you talk to some typologists like we can recognize the spectrum yes so this last part which I presented in a very short way in fact is highly motivated by theory of topological modular forms exactly so we did talk to Mike as well as Peter Dichner at MPI so we're using those resources heavily what's about five like I said I don't know to me appearance of number three in two dimensional physics was a big surprise five does not appear seven does not appear no other prime number appears so big challenges to explain any of the three either in form of anomalies or in form of realities or otherwise let's finish