 Yeah, so a happy birthday to Samson whom I got to know over the last couple of years Since I moved to the US. I see him very often and Yeah, I learned a great deal of physics some of which people already mentioned but In particular this talk is about anomalies a subject in which he played an important role in developing the first few Results so which I'll mention soon. So I'm looking forward to many more years of Overlap and discussions So happy birthday. So this presentation is mostly based on some upcoming work but also I'll mention some previous work that I've been involved with involved in with a Mostly David Agaiotto Anton Kapusti, Nari Cyberg, Joe Megomis Students like Vladimir Bashmakov, Adarsh Aron and now also with Kentaro Mori who is a postdoc at the Simon Center and students Konstantino Rompedakis and Saif Nasheri But most most of the presentation is based on presumably the work that's now in preparation but I'll also review a lot of the recent developments by many many other people and I will not even mention probably 10% of the people that I should mention in connection with this huge subject. So the beginning of this The beginning of this subject is not Nutter's theorem So there is a very abstract and modern way to think about Nutter's theorem Which is that if you have a symmetry whether it's a continuous or discrete? Doesn't matter what it gives you is a topological surface So this is So this way of presenting Nutter's theorem is of course a generalization of what Nutter originally envisioned What Nutter originally said was that if there is a continuous symmetry then there is a conserved current But then if there is a discrete symmetry then there is no statement There is a way to say Nutter's theorem in this fashion which works equally well for discrete and continuous symmetries Since if you have a conserved current and you integrate it over some space like slice says say then The operator that you get is a surface operator. It's a co-dimension one operator Which is independent of the shape of the slice So a way to think about Nutter's theorem which applies equally to discrete and continuous symmetries Is that the theory is equipped with some topological co-dimension one? Operators, so they don't depend on the shape of the surface of the co-dimension one surface But there might be an interesting algebra of such topological surfaces And so one is led to develop some kind of mathematical framework for the algebra of co-dimension one surfaces And this led to a huge outpouring of results in the recent like five ten years Of generalizations of the notion of symmetry We learned a great new deal of things about the algebra of such objects and so on So this is going to be the central thing in the talk. I'll present some of these new developments And also many applications to gauge theories to gauge theory dynamics So if there are any questions, please then stop me. Actually there is what's the statement about this surface Like a such that one well if in Nutter's original story then she said that there is a conserved current, right? And then you could integrate it over a divine one-dimensional space You integrate star j in some differential form notation and this gives you some Unit this gives you some operator which if we let's suppose we exponentiate it with some coefficient alpha This would be a unitary But this unitary is independent of the surface. So this unitary is labeled by the surface But it's independent of the surface if you take some closed surface and you deform it slightly The unitary is the same. Let's say it acts in the same way on on on the Hilbert space so to speak However, this does not mean that this surface is completely trivial because if there is a local operator here And you compare the same surface with the local operator outside. It's not the same thing So there is an algebra when co-dimension d objects cross this co-dimension one object So this leads to some algebraic structure That's the notion of acting with a symmetry on the local operator Okay, so that's what I'm going to explain now So if you act acting with a symmetry on the local operator, which is what we usually do when we have symmetries Corresponds in this abstract language to just encircling the operator, which is co-dimension d by this co-dimension one surface And that's the same as acting on the operator in the usual adjoin representation And then there is a non-trivial algebraic structure, which doesn't have to be commutative You can encircle this bunch of operators first with sigma g and then with sigma g prime and that can be fused They can be fused sigma of gg prime that leads to a In the normal situation when you have symmetries this leads to a group structure. It's associative, but it's not commutative So that's how we think about the symmetries that this there's some fusion of such topological surfaces Now, of course, once you start drawing these pictures, you are allowed to think about more complicated objects Here I've been just thinking about some co-dimension one surfaces that might have some interesting fusion rules but you're also allowed to think about more complicated objects such as the junctions And intersections What happens when two such surfaces intersect? What happens when there is a t-junction? Or by a t-junction, I mean something like like this What happens if there is something like that? So you're allowed to more complicated questions, which are not answered by another theorem So in as I already said in the case of continuous symmetries These surfaces are obtained from exponentiating the integral of the current And if you look at textbooks about symmetries, nobody talks about these things in this way People just talk about correlation functions of currents and that's how people found anomalies in the 80s and 70s and even the late 60s and so on They looked at correlation functions of currents And clearly correlation functions of currents capture a lot of the data about these topological surfaces Since the correlation functions of topological surfaces can be obtained from all the correlation functions of currents If you knew all of those guys, you would know a lot of stuff about these topological surfaces But that's only available for continuous symmetries as I said So an important phenomenon that's associated to symmetries is the notion of a 12th anomalies And one should not confuse the 12th anomalies with abj anomalies abj anomalies are essentially the statement that are not topological surfaces So historically abj found that something that seemed like a symmetry is not a symmetry. So that means that there are no topological surfaces But 12th anomalies are much more delicate 12th anomalies concerned with situations when the symmetry actually exists The theory is equipped with a bunch of topological surfaces And it's a the 12th anomaly is a statement about the algebra of these topological surfaces So the simplest case is a z2 symmetry. That's a state a situation. That's a case that I'm going to discuss now Just to explain how an anomaly is manifested in the algebra of such surfaces But in the continuous case the the situation is rather well known from textbooks People look at correlation functions of currents and ask about the divergence of the current Let's look at the z2 case. However, which is much more instructive and much simpler In the z2 case, there is one trivial topological surface, which is the unit topological surface And then there is a non-trivial topological surface sigma So sigma might have some non-trivial braiding braiding with local operators that local that some operators could carry z2 charge So the so suppose you have this is a local picture. So you should imagine something much more complicated Let's say You have some ribbon surface or some you have an arbitrary space Let's say some ribbon surface But locally near some region you have these top two topological surfaces that look like this So this is just a local picture of something much more complicated So suppose you have a region in some ribbon surface where the two topological surfaces touch like that And you want to compare and then you try to rejoin split and rejoin them in this fashion So the outgoing the outgoing lines should be the same basically, right? It's just splitting and rejoining And you ask what is the relation between these two these two answers And the relation could be either a plus or minus sign as you can prove because it's a z2 symmetry So the case where the answer the case where you get a plus sign from this split rejoin Move is the case where there is no anomaly and the case where there is a minus sign is the case where there is an anomaly Can I have a question? Can you imagine a more complicated relation where there is just some Minor combination. I don't know. Is this really the most general thing that you can imagine? No, this is not this is just a sufficient nest. This is a sufficient and necessary condition for there to be a z2 anomaly This is in two dimensions In two dimensions, these are just lines and uh So there are co-dimension one and this this is true in two dimensions What I'm this example is in two dimensions in higher dimensions these are co-dimension one surfaces and you can imagine many maybe more complicated things But nobody has worked out the mathematical theory in higher dimensions yet We don't know that what's the correct mathematical terminology in two dimensions. I'll mention that what is this mathematical object? It has a name in the literature And we understand it very well So in the z2 case the anomaly is just the statement of this rejoining and splitting And you can think about it as a discrete gate transformation if you like So as I said the plus corresponds to the situation the z2 symmetry has no anomaly The minus corresponds to the situation where there is an anomaly. This was recently Also explained in a very nice paper by Lin and Xiao So how do you see that the minus case corresponds to an anomaly? You can ask yourself could I gauge such as z2 symmetry by gauging you gauging means that you sum over all you make these surfaces transparent So gauging is the same as assuming that these surfaces co-dimension one surfaces become trivial But of course if there is a minus sign you cannot gauge it Because you cannot make these surfaces invisible if there is a minus sign Since it's inconsistent you cannot make them invisible all at the same time Due to this inconsistency So a minus sign is like the statement that you cannot gauge the symmetry Or more precisely gauging is ambiguous So we see that in two dimensions we have a somewhat nice understanding of what an anomaly means in terms of these topological surfaces In higher dimensions, there is no well-developed mathematical framework yet But I'll talk more about it The the next thing that I want to discuss before a before a Before jumping to applications Is the question of what are these anomalies good for So as we know from the 70s and 80s the continuous anomalies That appear in when there are massless fermions in even space time dimensions They're very useful to constrain the spectrum of bound states massless bound states So continuous anomalies have something to do with the massless particles They could be number goldstone bosons like pions or they could be massless fermions In the discrete case, however, the anomalies May also imply a topologically non-trivial vacuum So that's something new that appears in the discrete world that some of these discrete anomalies May may may also be saturated by a topologically non-trivial wave function for the ground state So the discrete anomalies teach us that either the ground state has massless particles or there are some or there is some topological filter essentially Now mathematically that in the continuous case It was realized in the 80s, which is what I mentioned that samson was involved with That the classification of these anomalies has to do with the co cycles group co cycles This kind of group appears in two dimensions In the continuous case in the discrete case a very similar result is true And also a Kapustin and friends have argued that it should really be a co-boardism group in the most general setting So there is a classification at least in two dimensions of Of the possible such things So So this story is Is a rather clean. This is the essentially classical netter theorem plus what it means to have an anomaly Stated for stated in such a way that it applies for discrete and continuous symmetries alike And recently there has been a huge amount of work on generalizing the notion of symmetry And these generalizations of the notion of symmetry can be done in multiple ways. So there were Generalizations in several different directions. I will not speak about all the possible generalizations of the notion of symmetry But I'll mention too which are very important for applications Yes, but there always should be some consistency condition which replaces cosical condition, right, right, so That's right. So you can ask what is the mathematical framework that talks that incorporates junctions Intersections this split joint move and the answer is not known in general, but in two dimensions. It's known. I'll talk about it soon It's something to do with fusion categories modular tensor categories this kind of words that maybe you are familiar with In higher dimensions, nobody yet came up with the correct axioms But I'll talk about two generalizations that appear to be very useful in lots of applications. These are Interesting things because this really generalizes the notion of a symmetry So this point of view is useful because it allows you some interesting generalizations So I'll speak about two generalizations One is when there are topological operators which are not co-dimension one So they could be co-dimension p They have nothing to do with ordinary symmetries, but they still may exist There could be gauge theories or interesting quantum filters or even lattice systems Where there are topological co-dimension p surface co-dimension p operators This is called in the literature a higher symmetry Another generalization which is More is I guess more recent and also a little bit Harder to understand is the notion of non invertible topological operators. What does it mean to be non invertible? for the usual Nother kind of not another kind of co-dimension one surface Co-dimension one topological operators for every group lm and g there is an inverse group lm and g minus one And if you fuse them together you get a trivial thing. So for every surface sigma There is another surface sigma prime so that together they fuse to nothing But this doesn't have to be true There could be gauge theories or interesting lattice systems or quantum filters Where there are some topological operators that have no inverse So these are non invertible topological operators So and these things could be mixed up. So there could be interesting systems where there are both higher symmetries non invertible symmetries and it would lead to a complicated algebra of topological objects And as I said the mathematical framework in general is not known, but we have many partial results Now these generalizations Are very useful Because on the one hand they're new and they allow some new results To be derived But many concepts that you're familiar with can be generalized to this framework So you can you can you can talk about anomalies. So there is a notion of anomalies even for this more general setting And the notion of anomalies is essentially about splitting and joining such surfaces And then you could talk about the notion of gauging you could talk about the notion of symmetry breaking You can so inflow even then a formalism of anomaly inflow and descent equations that samson and fadev worked out Can be generalized to some extent. We already know some of the basic results in this field So while these are This no this generalized notion of symmetry is new Some things from the past can be borrowed and they continue to make sense So One piece of terminology that's useful to know Yes Oh, okay, so What i'll do in this talk is that i'll first present applications of the first Generalization and then i'll present applications of the second Generalization and i'll show you that this leads to useful results about young mill theory In three four and two dimensions The four three and two dimensions. So first we'll start from the first co-dimension a p topological operators so I will assume that all the co-dimension p topological operators are invertible I'm not going to mix up these two generalizations though. They mix up in some applications. I will not mix them up I'll study some systems with which have invertible Co-dimension two operators and then separately the other case So first i'll talk about co-dimension two topological operators Which is called which are called one form symmetry and this terminology was introduced in this paper So they call this uh co-dimension two surfaces one form symmetries It's just a name One can prove that such objects must be a billion because there are co-dimension two you can take them through each other So the fusion must be a billion So one form symmetries are always a billion Unlike ordinary nether symmetries, which may be non a billion. So that's a small lemma. It must be a billion So what is it? What is it? What is it? What do these things act on since they're co-dimension two? They cannot act on local operators. They cannot act on co-dimension d objects They must act on co-dimension d minus one objects and co-dimension d minus one objects are just lines So while the co-dimension one operators that nether had acted on local operators These things act on lines. So there is a non trivial Algebra with lines So the same picture holds you encompass a line inside a co-dimension two surface You shrink the surface and you get a new line which is acted upon by some a billion symmetry. So now it must be a billion by the previous lemma What are some examples? So it turns out that any gauge theory in any number of dimensions that has a center That doesn't act on the matter fields has such co-dimension two topological operators So even SUN gauge young milster has such topological surfaces I cannot write a Lagrangian for these topological surfaces, but I can manipulate them formally and ask various questions about them In general, there could be such surfaces that don't have anything to do with the center of the gauge group We have many examples of that, but I'll not talk about such examples today We can also try to gauge this topological co-dimension two surfaces Gaging means that we put some background field and we may even sum over it later Now this would be a discrete two-form gauge field or a continuous two-form u1 gauge field So like gauge fields are replaced by two-form gauge fields You can also replace the Higgs mechanism by Higgs mechanism for two for two-form gauge fields and so on So the simplest example Where there are such objects Is just young pure young milster in four dimensions So as you know pure young milster into the in four dimensions has a theta angle Which is valued between zero and two pi because the instant on number is quantized This is on spin by the way. I'm just for simplicity. I'm only discussing spin manifolds here for the force spin manifolds for dimensional spin manifolds Otherwise, there are there are various things that I have to fix in the slides if I were not to discuss just spin manifolds And this theory turns out to have such co-dimension two topological surfaces because there is a center that doesn't act on any matter fields And so it's isomorph isomorphic to zn So this theory has a zn worth of such topological surfaces and one can ask what they're good for As you know, there is a clay prize like a million dollars or something for proving that theta equals zero this theory is gap that confined and trivial It turns out that this new topological surfaces Make lead to a very interesting statement about not theta equals to zero, but theta equals pi So it turns out that you can prove using these topological surfaces that the theta equals pi The ground state cannot be trivial And we've this can be proven rigorously So what you can there at the moment three possible once you've proven that the ground state cannot be trivial. They're essentially three options So one option is that there is a maybe some symmetry breaking At theta equals pi or maybe there are some massless gage fields. Sorry massless particles I meant to say massless particles. They could also be gage fields, but So one of these things has to be true at theta equals pi So the clay conjecture cannot be true at a the clay prize cannot apply, you know, theta equals pi it's the opposite so trivial means not one dimensional it has to be so Mathematically, I'm not saying the precise statement because I'm trying to keep it Accessible, but There is a certain there is a notion of anomaly for this co-dimension two surfaces There is a generalization of the notion of anomaly and you can prove that at theta equals pi. There is such an anomaly So that means that the ground state cannot be trivial There must be either massless particles or maybe some symmetry breaking number goldstone bosons or domain walls or something Now we don't know which of them is true But it's very likely that it's the first that is true From large n arguments and from a dsft. It seems that one should really favor the first one So what we expect is that While theta equals pi is confined Trivial and you know gapped and then there's the confinement at high temperatures when you go to the core gluon plus To you know to the gluon plasma phase a theta equals pi actually It turns out that the phase diagram cannot be that simple There must be something going on already at zero temperature It cannot be confined and trivial and even when you heat it up this order in the ground state cannot disappear So one is led to the following phase diagram That we propose following this thing So this is theta and this is temperature this temperature. So let's focus for a second on theta equals zero At this is trivial. So this is this is the deconfinement first order transition line So this is confined trivial gapped This is the big standard picture for young mill steering theta equals zero As you heat it up you encounter this phase transition and beyond This line you are in the deconfined phase. So that's the standard picture for su and gauge theory at zero theta However, theta equals pi one is led to believe that there is some symmetry breaking So one believes that here there are two ground states and time reversal symmetry is spontaneously broken There is some first order line and then this first order line must in fact cross the deconfinement line Because of this anomalies and this leads to some funny inequality One can prove an inequality that the time reversal symmetry is going to be restored only after the deconfinement transition So it's impossible to restore time reversal symmetry before the deconfinement transition takes place So it's funny that this discrete anomalies lead to an inequality for a phase diagram of just periodic military It's a rigorous inequality that you can derive just from these anomalies for higher symmetries Another very amusing fact that can be presumably one day tested on the computer Is that Since there is some symmetry breaking here There are two vacuums. So the idea is that the theta equals pi Uh, there are two vacuums. This is what symmetry breaking means and they're related by time reversal symmetry or alternatively by cp T and cp are the same since cpt preserves the vacuum Since there are two vacuums, you can construct a Putative universe where on one side of the universe you are in one vacuum on the other side You are in the second vacuum and there is a domain wall That interpolates that's what in water we call the layer the layer like water vapor layer or something So, uh, there is this is the discrete analog of a nambo goldstone boson Nambo goldstone bosons are objects that allow you to travel between different vacua and in the discrete case domain walls do the same So actually from this anomaly, it follows something very strange must be true for this domain wall, which I tried to Draw here, but it's probably totally incomprehensible So let me instead explain it by drawing it again here Uh, so That they what happens is that Let's suppose that you are just in the ordinary. Let's suppose you are one of those vacua Let's call this vacuum one and this is vacuum two So vacuum one has confined quarks It has broken time reversal symmetry, but it has confined quarks So there is a string if you take if you put two heavy quarks q and q bar They are going to be confined by a flux tube But let's suppose now you have this configuration with the domain wall So there are two vacua and in both q and q bar are connected by a flux tube. That's confined And here is the domain wall. So the domain wall is somewhere here That's more or less the region where the transition from the two vacua takes place So it turns out that the quarks are de-confined on the wall So if you bring a quark near the wall, it loses the flux tube And uh, it doesn't cost infinite energy to create a quark anymore So there is de-confinement On the wall While there is confinement in the bulk at zero temperature in each of the vacua the wall itself is de-confined So if you put a quark in an anti quark, they don't connect via a flux tube They might connect by some very weak coulomb forces, but not by a flux tube Even more bizarre than that the quarks acquire fractional spin. So they become enions So if you bring a quark to the wall and you circuit around an anti quark You pick up an aaron of bomb phase, which is exponent of 2 pi i over 2n And so the spin of the quarks is fractional So the quarks are not only de-confined. They also become enions And that's that's a prediction that follows just from these anomalies So There is a time reversal symmetry breaking at zero temperature And on the domain walls the quarks become enions and this is their spin. They have a fractional spin Yeah, if I artificially introduce an axion now and probe all that dynamically because then there is a domain wall read Yes, so and if I if I go through it Right, so since this thing there were Many papers about what you're saying In particular, there is a I think there is a paper Of chairman and unsan exactly about this and maybe also missha schiffman They they try to see the implications of this anomaly when you have an actual axion and you have an axion string or an axion ring They call it axion ring And yeah, basically the same continues to be true You have some point where there is a concentration of energy where the quarks are de-confined and they become enions You should really think about this as a quantum whole state So suddenly in young milster, there is a wall where there is a quantum whole state With this gauge this and so this is the topological filteri Since there are enions, there is a topological filteri and this is the topological filteri on the wall It's as you and churn simons theory at level one That's the prediction So in real world teta cannot be equal to pi Right, so this proves that in the real world teta cannot be equal to pi because if it were There would be domain walls that would proliferate and maybe not wiped out by inflation. That would be very sad So cosmological considerations would rule out teta equals pi in the real world because of this Because of this anomaly However, it may be interesting for other applications in phenomenology that we can talk about later Okay Now in this picture what I tried was to explain dynamically why the quarks become enions so the idea is that These two vacu are related by time reversal symmetry But time reversal symmetry famously Takes dions to monopoles. So here there is dion condensation and here there is monopole condensation If you've never heard about this fact the time reversal symmetry takes dions to monopoles think about cyberguiton theory Maybe that might help So here there are dions here. There are monopoles and it's kind of in the wall You cannot mutually condense dions in monopoles. They are not mutually local So there is some kind of tension and the tension is resolved on the wall by deconfining the quarks So both expectation values go to zero Both the monopoles and the dions do not condense on the wall And you get instead topological phase which looked like a quantum whole phase So that's the okay. So this is an example of applications in four dimensions Now i'm going to tell you about applications in three-dimensional Yang-Mills theory. Are there any questions about four dimensions? Okay, so the next example is in three dimensions This example still would not involve the non-invertible topological surfaces It only involves the co-dimension two topological operators namely a one form symmetry So I have the the last example would involve the non-invertible topological surfaces So another interesting example that recently condensed matter people have encountered and so we were motivated to think about it Is just Yang-Mills theory. There is no data angle in three dimensions But you can add fermions. So the simplest thing you can do Is to have a gauge theory without joint fermions and some mass That's the basically like a joint QCD in three dimensions It's the simplest thing you can do And this theory again has a zn one form symmetry because the center does not act on the matter fields There are these topological co-dimension two operators, which are now just lines Another fun fact that is not going to be important Except for one little comment soon is that the m equal zero theory is accidentally supersymmetric So it's the minimal amount of supersymmetry in three dimensions, which is called n equals one. There is no holomorphy So it's not very useful, but it's still supersymmetric Another technical comment for the experts is that n must be even for this theory to exist So n is even in SUN So using supersymmetry Whitton argued around 20 years ago That the Whitton index is zero in this theory So you take SUN plus an adjoint fermion. It has a very small amount of supersymmetry Just so two supercharges and Whitton computed the index and he found that it vanishes And then everybody believed that the infrared of this theory is just a spontaneously broken supersymmetric phase Namely a single Majorana-Golstino particle So just from the fact that the Whitton index vanishes You are led to believe that the SUN gauge theory with an adjoint fermion Flows in the deep infrared when you go to very long distances to a single Majorana fermion And of course this Majorana fermion is uncharged. It's just the Golstino And that's it So that's what you would think naively given the information Say again for any must no no this is just for capital M vanishing Of course, thank you just for vanishing capital M. That's where this argument about the Whitton index can be made That's the only case where it's actually accidentally supersymmetric Now when we started thinking about this model using these new ideas about anomalies, we found that this doesn't make any sense For two reasons One let me start from the second reason because it's easier This theory as I said already has a one form symmetry And it's z n where n is even and it turns out that there is an anomaly So this code I mentioned one surfaces when they loop around each other There is some there is some problem with splitting and joining very much analogous to what I explained in the beginning And there is some anomaly This anomaly is measured mod n And it's n over 2 mod n turns out for even n this formula makes sense And it turns out that this single Myrana fermion cannot match this anomaly. So this contradicts anomaly matching this idea This contradicts anomaly matching So this this is not consistent with anomaly matching Because this single Myrana fermion cannot match that one form symmetry anomaly Another thing that also leads to a contradiction is time reversal symmetry The massless theory has time reversal symmetry And it turns out that this splitting joining Issue or with codimension one topological surfaces, which are the time reversal topological surfaces leads to a difficulty mod 16 So there is a instead of remembering the z2 case. It was a plus minus thing So in the z2 case, uh, it was a plus minus thing It turns out that if you do time reversal symmetry instead, it's a root of 16. It's a 16 root of unity So time reversal anomaly is measured mod 16 And we computed this anomaly we found out that it's n squared minus one mod 16 And one and for no value of wait, I don't want to say something wrong But I think yeah for no value of even n this can be one mod 16 So this is either three mod 16 or minus one mod 16 So it's either three or a minus one mod 16, but it can never be one mod 16 Which is what the Myrana fermion gives So this also contradicts anomaly this so this this proposal contradicts anomaly matching with one form symmetry and also with time reversal symmetry Can you explain why is it not z2 by the same argument as for the z2 symmetry? Fantastic question. So you can ask Why so this picture that the slava is asking how to see this mod 16? This is a very difficult question To this is it. I mean the question is very easy, but the answer is very difficult This picture is in two dimensions I was telling you about some joining and splitting of lines of ordinary z2 lines in two dimensions And people have proven that this is a plus minus thing Which makes a lot of sense Now the time reversal symmetry defect is complicated because it reverses the orientation by definition And secondly, it's not in one plus one, but it's in two plus one. We're now studying a three-dimensional gauge theory So people have found that the phase there is some phase and it's mod 16 It's measured mod 16 And it also has something to do with fermions. So I cannot give you a non-technical answer But if you want, I'll give you a technical answer of why a root of 16 appears Are you comfortable with an anomaly inflow argument? or But geometrically you are saying it falls from the fact that there are some other things in the correlation function I cannot explain it geometrically. I know how to prove it using anomaly inflow I cannot give you a I don't believe anybody has a reasonable reasonably intuitive explanation for this root of 16 It has to do with the eta invariant in four dimensions, which is the anomaly inflow and it's measured in units of 16 because it's generated by rp4 and Okay, so in any case the question in gauge theory was that this proposal contradicted anomaly matching and We were intrigued to try to fix it And we found the a very exotic fix, but that makes a lot of sense And it and it makes many other things suddenly Sensible So the fix was to assume that this model instead of flowing to just a gold state of my run of hermion It has a topological filter of the churn simons type. It's an unabillion bunch of enions Of the churn simons type Now this looks ridiculous at first because topological filters of the churn simons type Typically are not invariant under time reversal symmetry So if you write the churn simons functional ada, it's odd under time reversal symmetry because there is an epsilon tensor But it turns out that when you quantize churn simons theories, even though classically This is not invariant under time reversal symmetry when you quantize it. It might become time reversal invariant And this is very counter-intuitive. Usually in physics You think that if a system has a classical symmetry it may be violated quantum mechanically But if a classical system doesn't have a symmetry it can never be restored quantum mechanically But this assumes that classical physics is order one and quantum corrections are h bar However in churn simons theory, there is no classical limit the classical limit of churn simons theory is an empty almost empty phase space essentially So since churn simons theory doesn't have a classical limit this may happen And there is a special set of churn simons theories that nobody has classified yet, but we know of some sequences Of special churn simons theories that have a quantum time reversal symmetry And that's one example of such a thing You can prove this isomorphism between the topological filter with these labels and the topological filter is with these labels Which are time reversed by just topological filter tools So adding tacking on such a topological filter. It makes sense. It's time reversal invariant. So this is the level in rank are the same No, the the level is reversed as you should by time reversal symbol Oh, I made a typo No, yeah, it's n over 2 n over 2 comma n n over 2 n over 2 comma n And it's isomorphic as a quantum As a quantum, uh, maru it's isomorphic to its time reverse version So it makes a lot of sense to tack on this thing And then a magic happens as they say And it fixes the anomaly both mod 16 and the mod n over 2 it fixes both of these things simultaneously So this has a time reversal anomaly, which is either minus 2 or plus 2 mod 16 depending on whether n is divisible by 2 or 4 And at the same time it fixes the time the the the one form symmetry anomaly. So one little thing Fixes everything immediately This is just a three-dimensional topological field theory But but this okay. Yeah, so is that clear so somehow there is a very Sexy fix to this problem. There is one thing that you add and it fixes everything Now this proposal implies again something that people did not expect It implies that the Wilson lines are de-confined Though a joint qcd in four dimensions if you've ever thought about it, it's a confined gauge theory But in three dimensions from these arguments, it must be de-confined because this is a bunch of anions so the part the the pro particles are de-confined and It's a somewhat surprising that this is true So now, uh, how much more time have I got? Okay, so 10 minutes I'll finish and then there will be 10 minutes in question So so far I gave you two recent applications of the ideas of co-dimension two topological surfaces There are also some papers that I didn't mention about co-dimension three topological surfaces But now I want to talk about the non-invertible symmetries So again, I'll start from an example. So this is now two dimensions. This is the last example Many of you might have thought about this model in the 90s. It was extremely popular in the 90s at some point So this is just a gauge field coupled to an adjoint fermion in two dimensions It's the same thing that we just discussed in 3d, but now in 2d What do we need to know about this model? Since there is again a one form symmetry There are there is a bunch of local topological operators So in two dimensions co-dimension one a co-dimension two means the local top operators So here the local topological operators are the generators of this one form symmetry And there is a discrete symmetry, which is z2 just a chiral z2 For the massless fermion. So when you put m equals to zero, there is a chiral symmetry Okay, so there is a story about the one form symmetry again, but that's not one I want to talk about I want to focus on the non-invertible symmetries So let me give you some history In the 90s this model was very popular Starting from some paper by David Gross Klebenov Matitsyn and Smilga I believe that Matitsyn is in Stonybrook, but I'm not sure Okay, so So what these people argued is that They well, I'm not going to review everything they said But the bottom line of the paper of this paper of Gross and Klebenov was that Wilson lines are already confined and There is no theta angle in the psu engage theory analog of this model They also claim that the z2 symmetry is spontaneously broken and so on. So they had a bunch of claims in the 90s More recently two months ago more precisely two or three months ago There was a paper by this group of people from north carolina in minnesota Claiming that this gross and stuff a gross et al stuff is wrong This model is actually confined except for the Wilson line to the power n over 2 which is deconfined So that's some recent history that I'm quickly reviewing But we think that both of these things are wrong So we're working on showing that both of these proposals are proposals are in fact incorrect And the reason is that this model admits a lot of Non-invertible topological lines that people have not known of course in the 90s people haven't known about this possibility I believe and more recently this group Did not take into account the constraints from such non-invertible topological lines So more recently we've been working with uh, we've been working on Classifying the full set of non-invertible topological lines in this theory Computing their anomalies computing diffusion rules And finding what are the constraints on the dynamics that these non-invertible topological lines impose So mathematically speaking non-invertible topological lines form a fusion fusion category A fusion category is a well-defined mathematical objects with some axioms with a pentagon identity Many things that you might be familiar from The churned simon story But it's not a braid that that there is no braiding unlike in churned simon steer So it's a more general object than modular tensor categories that appeared in the context of verlinda lines So this is some a mathematical object that mathematicians work on classifying And it has implications for the ground state of gauge theories So what we've argued What we are arguing This is in some working. There's some also related work by costas and monnier from 10 years ago So what we're arguing Is that these non-invertible topological lines Have a lot of interesting consequences for the low energy dynamics of this model And in particular what we're finding preliminarily Is that This theory actually has a huge amount of ground states unlike what gross and klebenov claimed We think that there is an exponential amount of ground states e to the n which is very bizarre, but This seems to imply the fusion category seems to implied it So we think that this model has an exponential amount of ground states Where the Wilson lines are de-confined And there is a topological filter that describes this exponential amount of ground states a two-dimensional topological filter of this type This topological filter is not the same as g mod g that many of you have studied in the 90s So, uh, the bottom line, let me just cut to the bottom line since I'm running out of time Our proposal for this theory is that uh, there is an exponential amount of vacu in n So it's e to the n this means that the hagedon temperature goes to zero at infinite n The gauge the Wilson lines are de-confined and there is some two-dimensional Frobenius algebra That that governs this exponential amount of vacu This is this is the like the summary of the claim It's surprising that these non-invertible symmetries lead to constraints on dynamics There are very beautiful examples in this paper of the constraints of non-invertible symmetries on dynamics This is in the context of our g flows in minimal models So you can learn a bit about that from from from that story And we're also now trying to generalize this whole discussion to non-zero mass Since this non-invertible topological lines only existed zero mass Some of them might survive to non-zero mass. So, uh, that's where we're what we're doing now So let me just conclude with some general general Comments, which are like homework for samson so So the general theory of symmetries and anomalies must be extended to include the co-dimension p defects, which may or may not be invertible Somebody should understand the axioms that govern the fusion and algebra It's like a bootstrap problem of this co-dimension p defects We don't know what are the correct questions in general dimensions. We know that in two dimensions it's governed by what mathematicians call a fusion category So this theory should be uh developed We should be able to classify all the possibilities We should be able to classify the anomalies as they did for with co-cycles group co-cycles for ordinary symmetries And so somebody should lay out the axioms Uh, this is what I explained that somebody should also classify the like generalize the notion of anomalies more clearly for this general setting The notion of anomaly inflow What is the notion of anomaly inflow for this Frobenius algebras? For instance, it's not clear Though there is a very interesting recent paper by thorngren and wang about this Is there a d plus two-dimensional anomaly polynomial? That's an open question So here I also made some comments about two-dimensional adjoint qcd, but the supersymmetric version We don't yet know about the one comma one supersymmetric version of the model. We haven't thought about it yet Uh, there are lots of questions about this hagedorn transition in two dimensions the zero mass limit the dualities and so on Well, so there are lots of questions that this raises That we don't have answers to yet So thank you and happy birthday again So what's this spin n mod s u n topological theory? So as you you know about it exponentially more than me But you know the there is the There is a class of topological filters that people studied in the 90s of the type g mod g And their fusion their their fusion rules are basically Inherited from a churn simons theory However, there are many examples of a group. Let's say our example is spin n A wezzumino-witton model at level one Where you gauge s u n at level n This theory the central charge of the numerator is the same as the central charge of the denominator So this is also a topological field theory But it's not the same as s u n level n mod s u n level n. So you have to develop some formalism to study it In addition in our case, there is an additional gauging by some quantum z2, but that's a minor thing relatively speaking So we claim that this bakua And the non-invertible lines The algebra that they obey is given by this thing That's the claim we haven't proven it yet But we proved it for low values of n so that's Yeah Yes So this this first thing you have with the t that you call chi Yeah, that also applies to supersymmetric gauging Oh, yeah, there is a I mean, yeah, since this thing there is a lot of work on the same thing in supersymmetry But people haven't studied many questions that you could ask but For instance, you could ask is this true in the cyber kind of theories With some softly deformed and poppets wrote a paper about it There is a paper of poppets. There is a paper of Somebody else whose name I forgot unfortunately and they seem to find a similar picture With a spontaneous breaking of time reversal symmetry and a compelling argument for the existence of anions on the boundary So you end up with a symmetry breaking picture in that case I don't know if this is always what they found but I remember one example where they found exactly this picture In some softly deformed supersymmetric model I don't know if this is always what you find. You might also find this kind of thing I don't know The general answer Yeah, it's two to the power n. I'm sorry Say again, I don't know this Point of view of this high-extended Or not to political theory put kind of stratification on your space and you get different Observables on different strata Atmospheric I think now both round Castel as they have Right Yeah, there is a day. Yeah, there is there is some framework where there are objects of co-dimension one two three And they have some rules of how they talk to each other It's Right, but my understanding is that they don't actually know all the axioms For the junctions and fusions of these defects That's my understanding in two dimensions people have worked out the axioms and they worked out the pentagon identities And they proved the rigidity theorem. Okay now approve this rigidity theorem So we know that there is a discrete set of fusion categories and that's why there is anomaly matching because it's a discrete set But I don't think anybody has done analogous work in higher dimensions But I might well be wrong Yeah, I might well be wrong Yes So I bought your second example Okay, I'm not sure it might question makes sense. But so since you are in 3d and you have this To co-dimension two operators, which are lines which can wrap around lines What happens with the operation if there are two Symmetry operators which wrap around a line on which they act and also simultaneously around each other So your picture is that you have two lines that are noted Yeah, and also they act on some line operator And also another line that goes in between on which they act So this seems to define a new type of product because I said that this algebra is commutative No, no, this would reduce to something that we know. So you're you're you're asking about the sorry Let me just draw it correctly You are asking about this and then another line that just goes through No, but it should go through it should go through both of them Oh, you want that to go to through both of them? They both act on that line. You have link Yeah, this is already linked and now you want the But but they link but the one but oh you want this to go through this one. Okay. Now I understand. Yeah something like that Yes, okay, so because you said that this algebra of Co-dimension two operators is a billion But it seems to define a different sort of product which is not clear to me if it's a billion or not No, so if So we have a distinguished set of n lines which forms the n Which are the generators of the one form symmetry Now this line This could be a different line on which they act. I agree. Yes in my example it's a different line But what is this product between the two circles? But but you see here there is no action. So you have to first fuse this line with this and then act So this is a symmetry line, right? Yeah So first you will what how would you reduce this you would first fuse this and that So you have to fuse the line on which they act with one of the topological lines Let's call this w you fuse them you get some new line, which is not topological in general And then this guy, which we can call w prime would act on it. Okay. I'll ask you later Yeah, I was thinking well if these are all symmetry lines It's one of the so in the case of three dimensions since the lines and the And the symmetry lines and the symmetry generators are all lines. So the lines can act on the on themselves So if these are all symmetry lines, then I can reduce it by fusion and by a sequence of fusion and contraction moves It's an a billion thing But yeah, I mean definitely this is what you should think about when you think about the general framework Of all the possible ways in which you could draw such pictures Yeah Is there an example of a non-invertible operator that can be understood Down to earth terms The question is if there is an example of a non-invertible operator That can be understood by down to earth terms That's what you're asking Yeah, the simplest example is a The The simplest example is in the tri critical ising model Well, also you could also say the simplest example is probably in the In the in the ising model itself so in the ising model there is a There is a at the at the self dual point There is the duality symmetry And this duality symmetry you can think about it as being implemented by some kind of codemention one defect And this codemention one defect is not invertible. So when you fuse it with itself you get one plus epsilon rather than one So the best place to look for the simplest possible examples These are in minimal models And there is a huge amount of examples in minimal models of non-invertible lines in this paper that I briefly quoted By Changlin, Xiaowing, Wang and Yin They looked at tri critical the ising model in subrational CFTs where they have a huge I mean another very A clear set of such examples are Verlinda lines, of course The Verlinda lines are in fact like the first set of such things that you've encountered maybe The Verlinda lines are more special than the most general case But this is a bunch of non-invertible lines into dimensional rational conformal field theories But they are defined axiomatically down to earth means like you take a lattice model than you see Oh, you meant something down to earth like in that sense Okay, so that then definitely you should look at this paper of Changlin, Xiaowing They have lots of references to condensed matter constructions where you can see these things on the lattice explicitly ABJ is not an anomaly, ABJ is a confusion Something is anomalous It's a group action or something it's given from the beginning They tell you that there is some group which acts somewhere and then happens that it is an anomaly Here you know tell us what was replacing the notion of group Right, so for the case, okay, I'm not telling you because I don't understand the answer But people claim that they know the answer So for this case in some special cases people know the answer. So in this case, I know the answer Here what replaces the group the gauge fields are two form gauge fields So instead of writing anomaly inflow for one form gauge fields like a wedge da you write b wedge db So here what replaces the notion of anomaly inflow is just two form gauge field anomaly inflow In this case people claim to know the answer But I don't understand it It's in Yeah, the other one is abelian because two form gauge fields are always abelian But the noninvertible lines are genuinely non abelian and there is they the Mathematical meaning of the notion of anomaly inflow is explained in this paper that I don't understand He's a mathematician slash physicist. He's a physicist Yeah, it's it's it's something to do with the well, I don't even want to say because I don't know what this means But it's some kind of Yes, there is a proposal for what it might mean Yeah In two dimensions only it's only in two dimensions that there is a proposal So thank you again