 mixed format. So I think that if somebody wants to ask a question, just raise your hand in the chat and I will click on the button allow to talk so that you can ask questions live. Otherwise, it's a bit dry. It's a pleasure to introduce Sergei Gukov. Sergei obtained his PhD in 2001 in Princeton under the supervision of Edward Whitten. Then he moved to Harvard University and then to the Institute of Advanced Studies in Princeton and since 2007 he has been a professor of theoretical physics and mathematics at Caltech and from 2010 he has also been an external scientific member of the Max Planck Institute for Mathematics in Bonn. He made significant contributions to string theory, quantum field theory and low-dimensional topology and we are very glad to have in here talk and please add the floor to Sergei. Thank you very much Paolo and thank you all for coming. It's a pleasure to be here although normally I really prefer to see the beautiful sea and enjoy Trieste's life. So this is, I guess, as live as it gets. Let me try to share my screen and see if this works again. Here we go. Right, so this talk will be about symmetries and symmetries in physics and mathematics. So that's the main theme and probably the best advice I ever received in my scientific career is that if I work on any problem be it physics or mathematics first ask a question what are the symmetries of this problem. Think about the symmetries, think about how they're preserved, broken and what happens to two symmetries. So that was the main advice. Later in the talk I'll share with you yet another piece of advice that I got but this by far is definitely the best and it's not much of a hidden knowledge as you can see many prominent physicists and mathematicians also definitely share the same advice the same way of looking at things. So Sidney Coleman devoted the entire textbook which is beautiful textbook on quantum field theory centered around symmetries and again many prominent physicists and mathematicians will look for patterns in the world of either physics or mathematics and of course for Manujan was very famous for that for recognizing patterns among numbers in number theory and in the world around us. There are different kinds of symmetries there could be continuous such as Lie groups you want us you too and so on they could be discrete such as symmetric groups and there could be more exotic for example you can introduce grassmen numbers as part of the symmetry group and that also will appear in fact all of them will appear in our discussion today in one form or the other and I also want to classify symmetries by whether they appear as part of the input in the system or in our understanding of the system or they emerge without being put by hand. So in that later case I'll call the symmetry hidden and this is usually what we are after because once we uncover some sort of symmetry or some new pattern then this provides a window into the future it basically tells where to go and where to look for a new better series that built on the existent ones. So an example of symmetry that's not mysterious that's not hidden according to this terminology I'm going to use we can find in description of fundamental forces I believe this expression that appears in a slide is actually on a t-shirt of ICTP when I visited last time I remember there were some coffee mugs and t-shirts on sale and I remember vaguely something like this written on a t-shirt so I hope everybody is familiar with this expression it's Lagrangian of gauge theory interacting with matter and even though here I say that it's a typical Lagrangian for quantum chromodynamics it actually describes also weak interactions and electromagnetic force depending on what gauge group we choose so here are the first term where we have to take a trace and once we take a trace we have to specify which group is relevant so that's already part of the symmetry that's put in by hand so that's something that goes into the definition of the theory and therefore it shouldn't be surprising that many things that will get out of this gauge theory interacting with matter will be structured or expressed in terms of representations of this symmetry so that's not hidden at all and when we try to add quarks for example to QCD Lagrangian of course we have to specify representations of that symmetry group and in physics we also sometimes like to introduce multiplicity and that's called number of flavors commonly denoted mf and that will also appear on further slides. On the other hand to illustrate an example of symmetry which is a little bit more interesting mainly not manifest I'll consider another simple example of many faults so now I go to kind of more mathematics or topology or geometry and ask you to look at a grass manian of k planes and n-dimensional space. In physics there are usually two main simple phases of matter confining and Higgs phases which sometimes are dual to each other and these grass manians appear all the time when there is some Higgs mechanism at work be it condensed matter physics or particle physics in this case you can think of vector of matter fields parameterizing this vector space c to the power km and UK could be the gauge symmetry by which we have to quotient if we want to form gauge invariant combination so this kind of quotient typically appears whenever we talk about Higgs mechanism so in this description of a grass manian as a quotient it's not entirely clear that there is a natural binary symmetry that exchanges k with n minus k but of course if you think about grass manian as linear subspaces embedded in n-dimensional complex space so k planes in n-dimensional space then the symmetry is manifest which of course is also clear from the description of the grass manian as a homogeneous space so this example so far already illustrates one important feature that depending how you look at the problem even if it's as simple as describing grass manian some symmetries may be easier to see in one description compared to the other so that's already a good lesson for us as practitioners that if you work on a problem either in physics or math which is more complicated than grass manian then it's good idea to look for alternative descriptions because they may make certain features or certain symmetries much more obvious or manifest what is not obvious even though i did give you yet another description of the grass manian is that there is also an operation of order three here so it's clear that we can try to exchange represent point same point in the grass manian by talking about complement of k-dimensional plane and n-dimensional space and that's going to be this binary operation exchanging k and n minus k but what is less obvious is that there is something that has to do with grass manian and there's order three so this order three thing will appear later in the talk as a hidden symmetry and if you're bored with details that I'll be going through in the next 20 minutes or so you can actually try to think or invent this order three operation yourself it's actually quite tricky to come by and once you find that you can of course try to check but a priori it's completely not obvious that there is something here that has order three in fact it's not even clear where three should come from likewise in topology there are some symmetries that we put in by hand so for example when we study topological invariance of various manifolds and today I'll be talking about some applications to low-dimensional topology manifolds of dimension two three and four so for instance if we try to form topological quantum field theory tq of t by twisting physical theory with a certain gauge group like donald's on theories is an example of it or cyberg within theory then that gauge group is again uh put in by hand that's its input data if you try to use alternative formulations of topological quantum field theory then they all require some algebraic input data as input so it's typically either Frobenius algebra or something more esoteric such as module attended category and so on so then there are well-known theorems in the later case this famous theorem of to ride which says that if you give this algebraic data then you get machine that allows you to compute topological invariance as an output however there are sometimes phenomena and that's exactly what I want to focus on in this talk which go the opposite way namely you start thinking about physics problem or you start thinking about topology on this slide I'm showing topological problem of computing certain invariance of four manifolds by counting solutions to PDEs and then surprisingly an algebraic structure which we did not put in by hand comes out so one phenomenon which is in some sense prototype or pioneering phenomenon on which many examples that I'll mention will be modeled is discovery of Nakajima in mid 90s that discounting problem where one studies instant times on particular four manifolds called ALE spaces produces characters of chiral algebras or vertex operator algebras so this was a big surprise at the time and it became a little clearer with work of Afein Witten where this phenomenon was Putin bigger umbrella or home and it's much better understood in recent 20 years or so and these are exactly kind of phenomena that I want to discuss and the emphasis therefore will be or distinction that I'll try to to make is between symmetries that we put in by hand versus symmetries that emerge without us expecting them at all and in topology by now in this ongoing effort in in the past 20 years or so there are quite a few interesting algebraic structures which appear in the study of topological problems that were not put in by hand and some of this is a terrific complicated algebraic structures that sometimes are used as input emerge as an output so that's that's quite surprising this connection between topology and algebra goes through physics and that's nice because this allows basically a triangle of topology algebra and physics to to talk to each other and in particular it involves a certain three dimensional or two dimensional theories that some of which will will touch upon today this theories probably you can notice I have some relation to many falls on for which this theory is emerged for instance if you study three many falls somehow three dimensional quantum field theories are relevant in physics and if you study four many falls two dimensional quantum field theories will be relevant in physics and there is one common thing between this two that in both cases numbers add up to six and that's not an accident in physics we expect that there is no quantum field theory without gravity in dimension above six so six dimension is the top and maximal dimension where quantum field theories can exist so I won't describe so this is probably the most complicated slide that I have for the entire talk basically I won't explain all of it I'll try to focus on two particular aspects and try to now become more elementary and explain things in more detail and maybe not in more detail but at least intuitively where they come from how they belong and tell you a little bit the story of about two things and these two things will be two-dimensional QCD analogous to the one who's Lagrangian we see on ICTPT short or we saw in the previous slide and I'll also focus on some exotic two-dimensional conformal field theories or vertex algebra in the math literature these two problems actually have the same origin so for me personally they came from joint work with Pavel Putrov and Abhijit Gada where we were trying to put together or understand how these two versions of correspondence that start with either three manifolds and four manifolds talk to each other it's important to consider these two yellow lines not just in parallel but together because any four manifolds can be cut along a three manifold and if you're a topologist then this operation called surgery is very important you want to understand how to cut and glue things and therefore it was very crucial for us to understand how these two stories not only exist separately but coexist together and I promise you that I had no idea that Pavel would be my co-host in fact already told him that for him unfortunately the stock will be very boring and this is definitely not premeditated I didn't realize that he will be my host but it's a great pleasure in fact I very much enjoyed that work with it back in 2013 so it's most of what you're going to hear is not really new I'll try to give intuitive survey of quite old results by now so there are two things that emerged from that work and that were quite surprising one has to do with this composition of these two problems that I described on the previous slide which basically means that we have now to consider three-dimensional theories with two-dimensional boundaries and although such problems were actively studied in for example condensed matter problems where in quantum whole effect we know that we can have some three-dimensional bulk and edge modes on a boundary what we needed in this case is slightly more advanced supersymmetric version of this quantum whole problem with some amount of supersymmetry which allows to associate a q-series like elliptic genus to this problem and surprisingly this was not studied so we had to basically break the ground there and another problem which was also surprising at least to me is that two-dimensional super QCD that naturally appears in this context was also not studied and it's a pity because at that point we were actually hoping to use both of these tools for understanding certain things in topology but since they didn't exist we basically had to try to develop it from scratch and I think only now we can come back to our original motivation of studying this system so I'll first in this talk I'll focus on on these two aspects and their applications basically like I said I'll try to tell you the story of each one of them how it fits in a bigger picture given to it if idea maybe with some examples and in each case we'll see that there is something hidden that emerges in each of the stories in fact I'll start with the second one since we already talked a little bit about QCD I'll start there and again this was probably the most complicated slide so I'll go back to two basics and the rest of the talk we'll try to make it a little bit more elementary so in the case of four-dimensional QCD the coolest thing about QCD that we expect is that it confines namely that we start our description at high energies with quarks and gluons which correspond to these two terms in Lagrangian but at lower energies theory is a completely different phase it's described by mesons and baryons interacting via pions and other particles so this is pretty exciting and this phenomenon which has to do with renormalization is perhaps one of the most mysterious phenomena in particle physics or in quantum field theory that we're trying to understand from various angles not necessarily in the context of QCD although this is definitely a big price problem but also in various other phenomena where renormalization plays an important role so it's a typical overarching problem in particle physics or quantum field theory the problem is difficult so we don't know how to solve it in general in fact we hardly know how to describe q of t in regimes of strong coupling beyond perturbation theory that mr. Feynman taught us so this this is pretty big goal and then at the moment out of reach in terms of general techniques or phenomena but we can make the problem a little bit more tractable by putting a little more symmetry from the start as i told you in the beginning symmetry can be put as an input and it's of course good because the more symmetry you have the more control you have and one such symmetry is symmetry between bosons and fermions called supersymmetry that definitely helps in particle physics to solve problems like confinement or understand phases and or g flows you can put different amount of supersymmetry and the story evolves exactly as you expected to evolve the more symmetry you put the more integrability or tractability you gain as a result so for instance if we talk about four-dimensional gauge theories like qcd you can make supersymmetry what physicists call n equals one n equals two n equals three or n equals four it's curious that n equals three which is this odd number no pun intended can exist as a quantum field theory but it cannot be realized in terms of Lagrangian involving gauge field and matter fields in this simple linear representations as i wrote it so such theories exist and were subject of active study in the recent years in fact only maybe three or five years recently but these are examples of so-called non-lagrangian theories which is interesting on its own right and would be a good topic for a separate talk so in the case of other theories gauge theories including supersymmetric theories with n equals one n equals two and n equals four supersymmetry there has been quite a bit of progress over the years for example in 1931 which was almost 100 years ago Paul Dirac emphasized the importance of monopoles magnetically charged particles and quantum physics and that's where he predicted or derived his famous quantization condition in the perturbative context asymptotic freedom was established by Groswilczyk and Pulitzer in 1973 which if you think about it is not that long ago in the history of science or physics or quantum field theory so we still don't understand non-peturbative version of that wonderful phenomenon a lot of progress even on quantum field theory if you're interested in quantum physics on its own right actually came from string theory which is even younger it was invented in 1984 then 10 years later n equals one or quantum field theory got attached the term cyber duality which is one of the coolest phenomena in in this area which relates two different descriptions of the same physics in terms of electric and magnetic degrees of freedom and cyber convit and made analogous progress in n equals two physics n equals two version of super qcd if you wish which had applications to cyber quiten equations and topology and so on a bit later also thanks to string theory Juan Maldesana introduced his famous idea cft correspondence which told us a lot about an equals four version of gauge theory and so on so here i also mentioned some of the other landmarks for instance that work of Groswilczyk and Pulitzer on asymptotic freedom was awarded noble prize in 2004 yet the turn of the millennium clay institute announced a big prize so it's not a joke it's an actual prize for trying to understand mass gap and confinement so this was a brief history of universe seeing from qft perspective and as you can see there is a good deal of progress so once we put additional symmetries such as supersymmetry we can solve the problem or at least say something about the physics but another dial that can mean we can turn is change the dimension so far i talked about qcd or its variance in dimension four but you can also try to consider qft in various other dimensions and here are the philosophies that the lower dimension the more control you have and the higher dimension less control you have for instance already mentioned earlier that the highest dimension we know is dimension six beyond which qft without gravity are not expected to exist and unless we put any supersymmetry we actually have nothing to say or very little to say basically next to nothing about qft in dimension six which are non supersymmetry we don't even know if they exist or not but in low dimensions you would expect more control because especially for gauge degrees of freedom there is less there are fewer polarizations that's one way to think about it and as a result two-dimensional theories are simpler so you could ask analogous question to to this qcd problem in two dimensions and again you can consider various amounts of supersymmetry which now are labeled by two numbers because in two dimensions you can consider left and right movers separately so there are these zero one zero two two two and so on other various combinations of integers expressing different amounts of supersymmetry and again the same philosophy is that the more supersymmetry you have the more control you have and more likely you are to solve the problem in fact there's two-dimensional theories are in some sense close cousins or behave very similarly as far as physics goes to the four-dimensional theories that I mentioned so this has been popularized by many people including Misha Schiffman who typically would say that if you take the four-dimensional theories shown at the top of the slide and divide by two both the spacetime dimension and the amount of supersymmetries you would expect to see roughly the same physics for example both 4d n equals one theories and 2d zero two theories can exhibit dynamical supersymmetry breaking they typically have isolated super conformal theories which don't really have modular spaces of vacuum and so on whereas on the right of the slide both four-dimensional theories and two-dimensional theories would also have roughly same type of physics in a sense that they would typically have spaces of continuous vacuums and less exotic phases and less risk of dynamical Susie breaking so there is there is some close parallel which illustrates that trade-off that if you reduce just supersymmetry or just the dimension the needle moves one way or the other but if you reduce both you roughly stay kind of in the neutral territory so in the two-dimensional world the role of symmetries was also very important and evolved over the years and at first when we came to this question of two-dimensional zero two supericity I was surely expected it to be solved at that point I didn't really have much interest in the subject and I thought that somebody else did it simply because back in 1962 Julian Schwinger already explained to us that even a billion theory the simplest possible choice of gauge theory in two dimensions confines and he basically gave the mechanism for that and that suggested that many two-dimensional theories are much simpler and perhaps we should expect even quantitative treatment so later in 1974 Tuft solved a non-Abelian version of the problem by showing that in two-dimensional QCD you have a nice regia trajectory of particles which are analogous to what we expect in four dimensions and in 1993 a version of this problem with different representations of the same gauge group was solved where people found multiple regia trajectories and also understood the story pretty well so that's why it was a little bit surprising that supersymmetric version which should be under better control actually was not understood back in 2013 so in parallel with these developments the role of symmetry was also growing in in two dimensions so based on very important work by Victor Katz and Robert Moody in 67 which introduced a particular class of infinite-dimensional algebras in world of mathematics so based on that work several other things happen a bit later so one of them for instance was work of Beleven-Polekov and Zamologikov where they realized infinite-dimensional symmetry in the context of 2d conformal field theories and mathematically the parallel development was development of vertex algebras by Richard Portchards in 1986 just two years after this work of Beleven-Polekov and Zamologikov and soon after in 1992 Portchards made the proof of Monstron's Moonshine Conjecture proposed by Simon Norton and then John Conway in 1974 or 1979 which related modular J-function and the largest sporadic simple group whose size has many orders of magnitude typically seen in big problems of particle physics such as cosmological constant or other problems where we have 15 orders of magnitude apart so this symmetry is various symmetries that are involved in this things on a slide such as the choice of gauge group or monster group which has a discrete symmetry and infinite-dimensional algebras such as verisoral algebra relevant to conformal physics in two dimensions play in very intricate ways with each other and definitely some of them deserve the name of hidden symmetries for instance even up to present day we don't have a good understanding of this largest sporadic monster group in a world of two-dimensional conformal field theories or in the world of physics in general I would say and it's a big quest to understand what's the explanation for for this hidden monster type symmetry it's a pity that both John Conway and Simon Norton are not around today unfortunately they both passed away Simon Norton passed away in 2019 and then John Conway was passed to COVID in 2020 so it's a sad moment and reminder for us to stay safe and try to avoid COVID well going back to QCD in two dimensions in two dimensions there is actually something interesting already at the start of when we write Lagrangian analogous to what I was writing before there are basically two types of matter one can choose either bosonic or fermionic so this number of flavors that I mentioned earlier can come already in two flavors of its own it can be either bosons or fermions if you talk about supersymmetry and I'm going to focus on this question mark of zero two supersymmetry so already at this point you see that there are three numbers that enter the definition of the theory one is the rank of the gauge group that's what we usually call number of colors in QCD and the other two are analogs of NF related to this binary choice of whether we talk about bosons and fermions so this is actually where we see the number three for the first time and to me personally this is the best explanation or origin of the reality to which I alluded in the very beginning that we're going to see in a second this theory the way I formulated it on a previous slide is anomalous and the simplest way to make it consistently consistent quantum mechanically is to introduce additional matter so previously all matter was in fundamental representation of the gauge group so here the arrow pointing away from the central node in this Nakajima's type quiver representation is anti fundamental and that's the simplest way to make the theory consistent quantum mechanically so we can even start the analysis and this theory turns out to be extremely rich it has very rich phase diagram with all the phenomena that I mentioned earlier such as it has conformal phases it has dynamical supersymmetry breaking in various regions and if you fixed one of the numbers then on a plot on a plane that expresses two other numbers number of colors and number of fermions the phase diagrams looks roughly like this with three sides maybe the three sides can be roughly attributed to the fact that there are three numbers integers that specify the model in the first place and what's interesting about this diagram is that even though it's not obvious on this phase phase plot or phase diagram the three sides are actually symmetric and if you're a label this outer nodes of this quiver like diagram instead of calling them an f or in b and so on you just call them and one and two and three it's just a change of notation in a way that and one plus and two plus and three is even number then surprisingly this theory has a symmetry of order three so that's the reality symmetry that I mentioned earlier so this was basically a result for physics that that has some value in physics and its mathematical translation is going to give us precisely this reality or something of order three in the simple grass manion problem that I mentioned earlier so grass manion has this binary symmetry of our exchange k planes with orthogonal n minus k dimensional planes and correspondingly two of the bundles that you can define on top of the grass manion one called tautological bundle I call it s and the other is orthogonal bundle of orthogonal planes the quotient bundle denoted here by q they also get exchanged if you exchange k and n minus k so it is these bundles that basically give you a window into the world of operation of order three so I don't know how to see this operation of order three directly in the grass manion but if you consider the bundles this combination of the bundles s and q on top of the grass manion and take certain cohomology which was introduced by malik of schachtmann and vitro in 1998 then you can see that this setup is actually symmetric with respect to exchange of n1 and 2 and n3 so basically quotation so this is basically infinite dimensional cohomology of of something okay now I'm going to switch to the second topic that I promised to tell you about and normally I would try to ask the audience can you guess what this picture is unfortunately I realized that now it won't work because participants are naturally muted which is PT in fact maybe I should pause for for questions I don't know if there are any questions life or if anyone can guess what this picture is I cannot I see and there are no questions I should answer if you want I can read the sorry just to I can read the question by shoaib bakhtar which is a very general question we want to you want to answer at the end he asks we explain various things assuming different symmetries like momentum conservation due to translation and symmetry and so on does it make sense to ask why the symmetries themselves exist let's see we should definitely come back to it in the end because that's very much the kind of question I want to discuss in this talk and I'll try to give a version of the answer but I hope that we can get unmuted and discuss to make sure I even understand the question correctly but even in this example I probably should emphasize and that's exactly the good illustration of the point that there are some symmetries of physics that are put in from the start such as group G or SU3 in the case of QCD and then there are some symmetries which emerge for example even up to present day to me this operation of order three in this two-dimensional version of QCD I don't really know why it's there one can derive it one can explain it from various perspectives but I don't think we have a good intuitive understanding of formulation of this two-dimensional QCD in such a way that the symmetry is manifest from the start at least I don't think I know such a description and in this case the symmetry is very small a just separation of order three and you could say oh it's not a big deal but it actually is a big deal as far as the phase structure and other things go for its implications and physics and that's a very good question I think in many cases we don't really know that the symmetry is there I think that's why we get excited when we find that even with the QCD itself I should also emphasize that now we take SU3 for granted and I say oh it's part of the input symmetry but of course in earlier development of the subject realizing that everything should be organized and SU3 multiplets was also not quite obvious until this was realized so this realization of what symmetry is underlying the particular physical system is a transient phenomenon it's we first discover some hints of the symmetry then we digest it and then we start using it so in that sense I think this what I call hidden symmetries are not something that we can when they're first discovered not something we can see easily explain easily but it's definitely an invitation to understand them better and perhaps use them later or improve on physical understanding that's how it happened in every example that I mentioned here yes it's again there is another question I now allowed by Alonso so I can maybe ask the question himself can you hear me yes sir please go ahead oh my question was very simple so I wanted to know what was that the short exact sequence that you showed in the last slide oh this is uh or this one it's a exact sequence which relates three bundles one is the structure sheath of grismanian and the tautological bundle s which is a bundle of k planes in this n-dimensional space and the quotient bundle q so that's okay okay thank you I hope so actually I don't know if this uh cohomology called uh cohomology of the chiral-deram complex has been computed mathematically so it's it's actually it would be interesting also for physics because there is some room for for new physics namely for non-petrobative phenomena we expect that non-petrobitively uh the physics of these two-dimensional theories or version of this chiral-deram complex should exhibit tryality but uh the definition that malekov schachman and weintraub gave it's actually what corresponds to perturbative chiral ring of our chiral states of two-dimensional theory and there is a room for something new even there although it's pretty clear that that tryality symmetry should be visible in any case but anyway so if you don't mind I'll proceed I'll then be glad to receive more questions in the end and I'll proceed to the second topic so again it's hard to do this live and it's a pity that I can't even be there in person with you but this picture actually shows percolation every time I show the slide it reminds me of staring at this famous painting and of course we all can see different things in this majestic painting probably the same as here and that was a picture of percolation and here I show three physical systems there are many more examples which actually share something in common so one of them is a Q state POTS model which is a generalization of Ising model introduced by POTS in 1952 and it's basically a generalization where spins on lattice sides take two different values so it plays an important role in for various values of Q of course there are different applications in condensed mirror physics and statistical physics and if you take Q to one limit you would expect that the model becomes very simple very trivial and the corresponding criticality is described by conformal field theory with central charge equals zero and since central charge is measuring number of physical degrees of freedom you would expect the theory to be completely trivial after all if spins are allowed to take only one value then what is there to talk about but it's actually not quite true that the central charge goes to zero but correlation functions if you analytically continue them from generic Q are non-zero and that happens to describe percolation phenomena I'll say a little bit more about this in a second there is another generalization of Ising model where instead of allowing spins to take Q different values you allow spins to be vectors in roughly speaking high-dimensional space or sphere with o n symmetry and then in this language the Ising model would be the case one n is equal to one in two-dimensional space allowing spins to be o one valued meaning it means that they're either up or down so that gives us the usual Ising model and trivial limit in this case would be taking n to zero but again same phenomenon happens that you don't actually get trivial theory if you extrapolate from generic values of n of this o n model rather you get interesting physics which describes self-avoiding random walks and then there are also quantum whole effect plateau phase transitions which share something in common with these two generalizations of Ising model that I mentioned namely all of them are described by logarithmic conformal field theory and this is an analog of conformal field theory which is what's used or what gets related to monster moonshine that I mentioned earlier or a rational conformal field theory which we know from many textbooks and which plays role in various critical phenomena but it's a little bit more exotic in a sense that central charge of logarithmic theory is either zero or negative and the theory itself is always non-unitary so for this reason when I was a student every time such theory would be was mentioned and that was back in early 90s I was basically told not to work on it so that was another piece of advice which I received when I was young and actually that's not a good advice unlike the previous one that I mentioned because as you can see logarithmic conformal field theories turn out to be extremely useful for describing real physical phenomena so even though theories are non-unitary they have a purpose in life in real life and they describe things like percolation random walk and other things and I wish I would actually didn't listen to the advice and learn this subject much much earlier than I happened to so one of the few people who studied who didn't listen to the advice and who studied this series in early 90s was John Cardi and he saw several interesting problems related to percolation using this logarithmic quantum field theory so first he found the formula for probability of a horizontal crossing in percolation which is probably one of the most important questions using correlation function in logarithmic safety what is shown here on a slide is a slightly different quantity which is analogous to what usually in critical phenomena we would call critical exponents so this particular critical exponent or quantity is measuring the density of loops that's a typo not loops loops with area greater than some area a and this number goes like universal constant divided by the area so larger the area the fewer loops you expect and the universal constant was measured to be 0.022972 and so on and Cardi using logarithmic safety basically predicted an exact value which involves 8 and square root of 3 and pi and as you can see it's a great it's a great agreement with measured value so quite remarkable and I'm very much amazed by this not only physical application of logarithmic field theory but also amazing prediction that the theory still can make even in the territory where we're talking about very strange conformal field theories on which we were told not to work in in early 90s and as a result if you ask a question which logarithmic theory it is we actually don't know we don't know because we know zillions of rational conformal field theories there are textbooks in fact many wonderful textbooks written about conformal field theories there are solar algebras vertex separator algebras and cut smoothie algebras all of which I mentioned earlier in two-dimensional setting but we know very very little about logarithmic safety it's it's it's really amazing how huge is the difference if you try to look or google you'll basically find three general families that go by the name symplectic fermions triplet or singlet and this is very little it's definitely way too little to have good understanding of first of all what this kind of theories are so that we can try to help john cardee and others to understand percolation random walks and other phenomena and it's definitely too little to try to identify which particular logarithmic safety is actually relevant for percolation so john used only general properties such as symmetry including that verisaurus symmetry I mentioned earlier and general other properties of correlation functions to make these predictions but obviously if we understand in more detail which logarithmic safety is relevant to that problem we can learn a lot more about percolation and other phenomena of this type so here I mentioned the basic quantity of conformal filtering two dimensions namely the central charge which roughly speaking measures the number of degrees of freedom and the only reason I mentioned that is to illustrate that for most values of these parameters which is d in the case of symplectic fermions so this parameter p in the case of singlet or triplet logarithmic cfts the central charge is actually negative for instance in this last case you see that term six times p quickly outpowers 13 and for most values of p you get in the negative range so therefore these theories are non-unitary if you're a mathematician then I can offer you another description of this somewhat exotic vertex algebras voa vertex algebras basically mathematical counterpart of the term conformal filter in two dimensions and it sits in between two familiar things so first for every lattice there is a natural construction of a vertex algebra called lattice voa and from lattice voas we can construct a fine w algebras let's say of type ad if we use root lattice of ad groups to to start with so these two other ones that are more exotic singlet and triplet logarithmic algebras they sit in between and that's just roughly to orient you wherever they are if you're familiar with Felder's construction of minimal models where Felder is using cohomology of a certain operator q on lattice voa to define minimal models then it's easier to understand what one of this algebras is namely the triplet so Fagan and Tipunian propose to do something funny they propose to take instead of cohomology of this operator q they said let's consider just the kernel remember that cohomology is kernel module of the image so Fagan and Tipunian said let's do something rather strange and modify this cohomology construction into taking kernels of operators which are called screening operators so then one also gets something quite unusual and perhaps it's not surprising that we get exactly this funny logarithmic vertex algebras in this case that's how you get the triplet then singlet is a little easier to understand because it's basically a sector or sub-algebra inside the triplet which is singlet with respect to global symmetry su2 or sl2 so triplet inherits sl2 symmetry and then singlet is just a singlet sector of it at the level of characters their character of a singlet is just a constant term of triplet so surprisingly this kind of algebras make their appearance in the study of this supersymmetric quantum whole effect that I mentioned to you earlier so it's a combination of three-dimensional system and a two-dimensional system for instance two-dimensional system could be exactly this 2d super qcd that I described earlier and naturally see it's on a boundary of so-called 3d and equals to physics so at that time we basically had to scratch the ground to to understand how to compute such things but to understand they evolved in many different directions and many people have worked on them for instance there was a beautiful work of demoftegaiota and paquette where they checked various dualities using this kind of three-dimensional version of elliptic genus and it has various applications and topology some of which I'll mention to you in a second but the basic idea is that if you take this exotic system namely combined 3d 2d supersymmetric version of quantum whole effect if you wish in the simplest case when three-dimensional theory is supersymmetric qed so just a billion with very simple matter namely one flavor of electrons you quickly can find even starting with very simple examples such as this qed example first of all you can quickly compute this quantity this elliptic genus and unlike the usual two-dimensional elliptic genus is not going to be modular because now three-dimensional theory that couples to two-dimensional physics spoils the modularity problem so the modularity property so it spoils the symmetry between generating vectors one and tau which are periods of the elliptic curve on the boundary so as a result you get something non-quite modular it's usually either mock modular or more exotic type of modularity and right from the start right from the simplest possible examples you find exactly this logarithmic vertex algebras or their characters so this is quite nice because this basically opens a window into way of constructing this logarithmic vertex algebras or understanding them as I try to emphasize they're extremely rare and they're unless you're familiar with this construction of Felder or Fagan-Tipunin it's actually really hard to construct this logarithmic algebras so therefore alternative methods are very welcome a particular class of such three-dimensional n equals two theories that are required to build this three-dimensional version of elliptic genus come from so-called 3d 3d correspondence so given a three manifold one can immediately associate to it a theory called t of m3 that encodes a lot of cool topological invariants such as cyberquiton invariants and many other things and in particular you can just try to apply this construction of computing this 3d elliptic genus in the context of these 3d 3d correspondence so then as a result what you get is a q series or rather the collection of q series invariants of three manifolds which also turn out to be characters of logarithmic vertex algebras for instance here I show an example of a three manifold that can be built as a surgery on figure eight knots or trefoil knot and there are many other equivalences called Kirby moves which produce the same manifold and what you get as an output is a very concrete q series which in fact is one of the expressions that we usually attribute to Romano John it happens to be of very interesting modularity type that he described in his last letter to Hardy but from this perspective it also happens to be character of one of the logarithmic vertex algebras that I mentioned earlier namely the singlet with p equals 42 when I gave a version of this talk at Max Planck Institute in Bonn Mike Friedman who was in the audience immediately jumped up and said hey 42 is definitely special because it's basically the answer to the most important question of the meaning of live the universe and everything in the heat hiker's guide to the galaxy that's actually kind of nice indeed this manifold this is very simple it's one of the simplest examples so that's why I used it but there is nothing really special about it and if we use a slightly different manifold or some different surgery on some other note we would get other numbers other than 42 it would be pretty much any value of p you can achieve but more importantly not only you can cover this logarithmic cfts and voas which were introduced before you can actually produce more of them and what's surprising to me is why this logarithmic voas appear so that's again an example of a hidden symmetry which is not simple as in the first example that I mentioned earlier it's not just a order three operation now this is a huge symmetry it's the entire logarithmic vertex algebra it's infinite dimensional and why does it appear in this class of problems is a great mystery and of course it's a great opportunity because once we harness this logarithmic voas we can use them for many other applications we can have more examples just like we have in the case of rational cfts that describe the usual critical phenomena so it's a great opportunity but before we managed to use this symmetry we need to harness it and understand how it comes about understand all the details and so on and so forth so finally I want to mention maybe one last example of symmetry called mirror symmetry in this case which is even more elaborate in a sense because it builds on what I just said in this last slide and maybe I'm already getting a bit technical but this is the last slide so that's okay in relation between topology and logarithmic vertex algebras which I mentioned a moment ago there is one operation which is very simple in the world of topology you just change our orientation of a manifold and that operation exists for any choice of manifold so it's something very simple you replace three manifold by its inverse in a sense of orientation and therefore it should relate a character of logarithmic vertex algebra to a character of some other algebra and at the level of this logarithmic algebras this relation is completely nontrivial it's complete mystery and why such a pairing between logarithmic vertex algebras or at least some class of them should exist is completely unclear it's out of reach because like I said we don't even understand the world of logarithmic voas or CFTs in the first place let alone this operation but its origin in connection with topology is actually pretty clear in topology there is no mystery you just change orientation and it's it's pretty simple so this shows kind of interplay between topology algebra and physics where sometimes a symmetry that can be seen in one domain happens to be completely mysterious or what I would call hidden in a completely different domain and I hope that in this hour I showed you a couple of examples where such hidden symmetries such as monster group or simple order three iteration or completely huge infinite logarithmic view a shows up for reasons that we don't fully understand so this is clearly an invitation to understand things better and I want to you to take away a more general lesson from all of this that aside from these concrete examples it's always a good thing to look for symmetries especially the hidden ones especially the ones we did not expect so in other words I want to pass on this piece of advice that I received and I still think it's the best advice that I received in my entire scientific career to watch for symmetries and maybe I can add my personal twist on this advice and suggest that we watch for hidden symmetries and I'm sure there are way more of them to be discovered understood and used thank you so much thank you Sergei so I find it awkward to clap morally I clap so if there are any questions please I can you just write something on the chat and I will let you talk so it's a bit it's a bit awkward but equal like this any question you can ask the question hi Sergei yeah sorry actually I joined them in the participants first so I didn't join on time as the panelists no I have a general question so this 3d 3d correspondence I mean so is there is an can you say something about some m5 brain perspective is there some I mean in the 4d 2d case there is some m5 brain perspective right and right but both of them go or or originate in trying to compactify 5 brain on either three manifold or four manifold and basically ask for what is the effective either a two or three dimensional field theory that's that's what we usually call t of m3 or t of m4 so yeah so in that context I wanted to say that I wanted to ask so how does one think of this boundary in that picture boundary of the 3d theory somehow is ending on a 2d theory yeah so that's that's exactly the one which leads to to this picture roughly right so it's it's kind of tricky so yeah maybe a little slight was perfectly fine for this so it's a little tricky to think about it but what happens is that four manifold gives you a two dimensional theory so it's actually going to be the one sitting on the boundary so the four manifold is bigger than three manifold but in this t of m world the dimensions are reversed because they have to add up to six or other effective theory leaves in t of md leaves in dimension six minus d so as a result three manifold which was boundary over four manifold now will be the bulk will be the biggest thing and four manifold which was the bigger thing is now going to be the boundary in this t of md world so already that this sounds kind of confusing and then that's precisely the motivation that got us with pivalon uphuget thinking about it so we wanted to a clarify it and be say something useful about it and then I think to be honest yeah so like I said we immediately ran into this glaring holes that that even basic questions like non abelian two-dimensional supersymmetric zero two theory was never discussed in the literature so oh my gosh we are very far from completing this goal so this question you're asking about how they fit together or that's the same question we asked years ago I don't think it sounds for it even at this point so maybe now we have a little bit of elements that we can come back to it again but it's yeah there is a there is a question from by bloody me a drop Dobrev so he can ask a question I wanted you to show again the formula for the central charge for singlet and triplet yeah here it is a little puzzled that the formula is the same for triplet and singlet that's that's okay because that's actually consistent with the fact that singlet seats inside triplet so it's a sub algebra and the central charge is the same that's perfectly fine that happens quite frequently for instance if one thinks about vertex algebras in the language of voa in mathematical terms then one can form extension of a given voa think about extension of usual finite algebraic structures then it often happens that central charge is the same that's perfectly fine they may even share the stress energy tensor in fact so it's it's consistent it's precisely consequence of the fact that singlet seats inside triplet yeah thanks for spotting it but it's not a typo they're they're really the same okay thank you and for CFT this is also not a fun thing in the sense that for rational CFTs we have many different families with many different central charges so here you see that the the range of central charges is quite boring and it just runs over this number is which in fact are same as minimal model values in fact even worse for value d equals one and p equals two this two top models also overlap so it's not i'm showing you different families there are some overlaps thank you is there is a another question joanne you can go ahead okay now i was i wanted to ask something about the the comments you make about the non-existent of n equal three d equal four range and q of g so i wasn't stupid by this and i wanted to ask does this comment refer to say at the normal normalizable level namely if you allow for for a cutoff in your quantum through 30 like a like an f at the 30 the derivative expansion are there an equal three large and q of g i think the honest answer is we don't know we know that there are some renormalization group flows which start a bit say n equals one Lagrangians perfectly nice Lagrangian theories built out of gluons and matter that flow into n equals three fixed point in the infrared so there are such kind of phenomena but we don't know how to construct i mean these are very strongly coupled flows in fact i don't even know to what extent they're studied i know they're analogous things in flowing between n equals one and n equals two so maybe i'm saying something actually wrong here about n equals three i'm not a really an expert on it but there is no way to try to write down Lagrangian of a uv theory in the ultraviolet in high energies which has n equals three supersymmetry manifest and gauge fields and matter in the standard form of a Lagrangian so if we try to do this what happens is that it automatically enhances to n equals four so in that sense there is no uv or high energy definition so it's a very peculiar thing and i don't know if there are any Lagrangians of course with high supersymmetries there are so more constraints that's why theory is solvable so it's hard to imagine that there is some simple Lagrangian which has many fast and equal theory theory but i guess it's this thing where we should probably never say never that it's only never until something is discovered and becomes a cool thing so it's very intriguing i totally agree and all right thanks yeah it is another question that bobby you can go ahead yes okay um like it was really just is there some intuitive way to understand how you get negative central charge from a wrapped five brain in a supersymmetric situation this seems very confusing to me yeah so this part i can explain right so it basically comes from the fact that it's negative central charge and non-unitarity and failure of characters to be modular and so on they're all linked to one particular property namely the fact that we're not talking about two-dimensional nice theory by itself we're talking about two-dimensional theory coupled to three-dimensional both roughly speaking which is a 3d theory so many members at ictp including attrition others know very well that if you study two-dimensional conformal field theory which is non-compact namely has scalars which allow to span infinite range you often get failures of modularity and you start getting more and more of these bad properties tiptoeing toward non-unitarity in a slow pace so here what happens is that these non-compact scalars basically get geometric meaning by interpretation by this third extra dimension so it's in some sense very similar to phenomena of mock modularity unspoiled modularity in non-compact sigma models where now the role of non-compactness is actually physical space time non-compactness it's a third dimension but then these fields naively have wrong sign kinetic terms and all that kind of stuff that you would expect exactly they're right so you you're completely right that naively there are so many things which are funny about it i guess that's precisely why from 2d perspective just say can us matter perspective these theories were not so popular in early 90s at all it's it's not a joke that i got this advice of not wasting my time and i think i'm quite surprised and pleasantly surprised that these theories do show up in real life applications there are many more applications to dynamical systems that i didn't even mention and it's it's it's really fantastic how this log view is tied up with lots of other things but here what happens is that they show because we take um q-cohomology analogous to to this chiral-deram complex that i mentioned so it's infinite dimensional cohomology theory and it produces something as a result of taking supersymmetric states the kind of dps states for q-cohomology on nice unitary system so it's a slice if you wish in a space of fields or states in in this bigger couple 2d 3d system so from the 3d anicals 2 point of view 3d anicals 2 system coupled to the 2d boundary condition as unitary there is nothing wrong with it but then once we take this q-cohomology we pick a funny slice so that it looks like it's a space of states and a two-dimensional cft which is non-unitary so that does this help that's probably the best i can say yeah it kind of helps yeah if Pavel is still around he may contribute and chime in but but that's a great question like i said i think i think you're asking somewhat difficult question but i cannot answer not because i'm poorly prepared because it's not really well studied i think okay so we can have me a final question from Claudio Rezzo in the math section Claudio i think you can speak Claudio i'm afraid that yes Claudio maybe please maybe he can type his question if he can Claudio we can hear you for some reason okay Claudio we cannot hear you so maybe we can you can send the question to Sergei offline i would i would like to ask yeah i would like to ask whether Sergei can formulate Cardi's problem again this is the question from Claudio Rezzo actually i'll probably need a little bit of clarification what exactly we mean by what is the actual problem about loops and area oh right so good so there are several quantities one can try to measure in percolation so one so percolation basically is is the problem of trying to populate some of the links basically that they can be dead or alive active or not active and of course we're trying to find whether the two sides or some clusters in uh that is partly populated uh letters are connected or not so that's illustrated for example by by this blue path going from one point to another point so there are several questions one usually asks about percolation one is what is the probability of crossing from one side to another side so that's the this horizontal probability crossing that gets expressed as a correlation function in language of logarithmic conformal field theory so the reason i didn't present that answer is because correlation function is a little bit tricky in a sense it's a four point function so it involves four insertions and moreover it's a boundary four point correlator so even for practitioners of CFT this is not the first but probably the second thing to learn about correlation functions it's not the kind of first exercise but this one is a little nicer in the sense that it's basically density of loops so think about in icing system you can have disorder operators which get a line attached to it and they can form loops so in this problem of percolation you can also think about say boundary of clusters and so on and you can ask about their density so clearly for small I mean there are lots of small islands which we can see over on this picture like this link this link and so on that they'll have they'll be effectively surrounded by such loops with very small radius or area so creating something small is not a problem and creating something bigger of course is less likely so the prediction is that in percolation problem creating something larger of area a grows inversely with area or decays inversely with area so it's some constant over a and the question was to determine what the constant is so it's supposed to be some universal constant analogous to critical exponent and there is a beautiful paper by John Cardi and Robert Zief so I think Robert started as a theoretical physicist but then went more into condensed matter simulation area and then more material science in some sense and I believe now he's in Michigan but I highly recommend that this joint paper of Cardi and Zief where they only not only show these numbers but also explain in more detail what the loops are and so on okay so maybe it's time to close so I would like to thank Sege for this beautiful talk and everybody to join so again we cannot clap but we morally clap well thank you guys thank you thank you thank you thanks for hosting me and hope everyone will stay safe and healthy thank you everybody thanks thanks thanks okay thank you guys thanks a lot