Anomalies: New Results and Applications

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Anomalies: New Results and Applications
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Anomalies: New Structures and Applications
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2020
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English

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I will review the concept of anomalies and its relationship with Symmetry Protected Topological Phases. I will also review various extensions of the concept of ‘’symmetry.’’ I will present applications to Yang-Mills theories in 2, 3 and 4 dimensions. These applications feature two extensions of the concept of symmetry: one is higher-symmetry and the other is "non-symmetry" topological defects.
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[Music] yeah so happy birthday to Samsung whom I got to know over the last couple of years since I moved to the US I see him very often and yeah yeah they're in the great deal of physics some of which people already mentioned but in particular this talk is about the normal is 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've mentioned some previous work that I've been involved with involved in with mostly Davide Guyot and tonka post inari cyber gamma gummies students like vladimir bash macaw Vader Sharon and now also with Quintero Mori who is a postdoc at the Simon Center and students constantly room to Dacus and science I've naturally 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 development 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 a the beginning of this the beginning of this the subject is an not nutters theorem so there is a very abstract and modern way to think about not earth theorem which is that if you have a symmetry whether it's continuous or discrete doesn't matter what it gives you is a topological surface so this is so this way of presenting nutters theorem is of course a generalization of what not our originally envisioned what nutter originally said was that if there is a continued 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 another 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 slices say then the operator that you get is the surface operator it's a co dimension one operator which is independent of the shape of the slice so a way to think about nutters theorem which applies equally to discrete and continuous symmetries is that the theory is equipped with some topological code dimension one operators so they don't depend on the shape of the surface of the codimension 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 code dimension one surfaces and this led to a huge outpouring of results in there isn't 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 gonna be the central thing in the talk he'll present some of these new developments and also many applications to gauge theories to gauge the dynamics so if there are any questions please as I stopped me actually there is what's the statement about this surface like such that what well if in others in others original story then she said that there is a conserved current right and then you could integrate it over Divine's one dimensional space you integrate star J in some differential form notation and this gives you some you need this gives you some operator which if we have 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 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 qua 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 a local operator okay so that's what I'm gonna explain now so if you act acting
with the symmetry on the local operator which is what we usually do when we have symmetries corresponds in this abstract language to justin circling the operator which is called the mentioned d by this code i mentioned one surface and that's the same as acting on the operator in the usual adjoint 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 G G 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 object here I've
been just thinking about some codimension one surfaces that might have some interesting fusion rules but you're
also led 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 are led to more complicated questions which are not answered by another story 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 court 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 anomic an important
phenomenon that's associated to symmetries is the notion of a tuft anomalies and one should not confuse the tuft anomalies with ABG anomalies ABG anomalies are essentially the statement that there are no 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 tuft anomalies are much more delicate two of the normal is concerned with situations when the symmetry actually exists the theory is equipped with a bunch of topological surfaces and it says the two phenomena is a statement about the algebra of these topological surfaces so the simplest case is a zero symmetry that's a state situates a case that I'm gonna discuss now just to explain how an anomaly is magnified in the algebra of such surfaces but in the continuous case 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 up illogical 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 Z to charge so this so suppose you have this is a local picture so you should imagine something much more complicated let's say you have some riemann surface or some art you have an arbitrary space let's say some riemann surface but locally near some region you have these 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 riemann 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 bleating and rejoining and you ask what is the relation between these two these two answers and the relation could be either plus or minus sign as you can prove because it's a little symmetry so the case where the answer work 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 question can you imagine a more complicated relation where there is just some no this is not this is just a sufficient yes this is a sufficient the necessary condition for there to be a z2 anomaly and this this is true in two dimensions forums this example is in two dimensions in higher dimensions this are called dimension 1 surfaces and you can imagine many may be more complicated things but nobody has worked out the math in higher dimensions yet we don't know that what's the correct mathematical terminology in two dimensions I'll mentioned 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 normal is just the statement of this rejoining and splitting and you can think about it as a discrete gauge transformation if you like so as I said
the plus corresponds to the situation the digital symmetry has not normally and the minus corresponds to the situation where there is an anomaly this was recently also explained in a very nice paper by Lin and shout so how do you see that the minus case corresponds to an anomaly you can ask yourself could I get such as the two symmetry by
gauging you gauging means that you sum over all you make the surfaces transparent so gauging is the same as assuming that this surfaces caught dimension one surface has become trivial but of course if there is a minus sign you cannot gauge it because you cannot make the 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 next thing that I want to discuss before a before a before
jumping to applications is the question of whether this anomaly is 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 pylons or they could be masters masters fermions in the discrete case however the anomalies may also imply imply an entree and topologically non-trivial 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 wavefunction for the ground state so the discreet anomalies teach us that either the ground state has massless particles or there are some or there is some topological filtering essentially now mathematically in the continuous case it was realized in the eighties which is what I mentioned that Sampson was involved with that the classification of these anomalies has to do with the Co cycles group Co cycles and this kind of group appears in two dimensions in the continuous case in the discrete case a very similar result is true and also capacity and friends have argued that it should really be a cooperation group in the most general setting so there is a classification at least in two dimensions of halfway of the possible such things so so the stories is a
rather clean this is the essentially classical Network Ihram plus what it means to have an anomaly stated for a 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 two which are very important for applications right right so that's right
so you can ask what is the mathematical framework the techtalks that incorporates junctions intersections the splitted joint move and the answer is not known in general but in two dimensions it's known I'll talk about its own 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 generalize the notion of the symmetry so this point of view is useful because it allows you some interesting generalizations let us pick about two generalizations one is when there are topological operators which are not Co dimension one so they could be caught dimension P they have nothing to do with ordinary symmetries but they still may exist there could be gauge theories or interesting quantum field theories or even lattice systems where there are a topological Co dimension P surfaces Co dimension P operators this is called in the literature higher symmetry another generalization which is Morris 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 network another kind of not another kind of code dimension one surface code mention one topological operators for every group element G there is an inverse group element 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 lighting systems or quantum field theories where there are some topological operators that have no inverse so these are not 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 there are 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 in flow even then a formalism of anomaly in flow in descent equations that Samsung and Fidel worked out can be generalized to some extent we already know some of the basic results in this field so while these are in this notarized notion of symmetries knew some things from the past can be borrowed and they continue to make sense so one piece of terminology that's useful to know ok
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 the yang-mills theory in three four in two dimensions for three and two dimensions so first we'll start from the first co dimension P topological operators so I
will assume that all the code 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 which have been veritable co dimension two operators and then separately the other case so first I'll talk about Co dimension to topological operators which is called which are called one form symmetry and this terminology was introduced in this paper so they call this Co dimension two surfaces one form symmetries it's just the name one can prove that such objects must be a billion because they 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 another symmetries which may be known a billion so that's a small lemma it must be a billion so what does it what is it what
is it what do these things act on since there are four dimension two they cannot act on local operators they cannot act on code dimension D objects they must act on codimension the minus 1 objects and codimension D minus 1 objects are just lines so while the co dimension one operators that matter had acted and local operators these things act on lines so there is a non-trivial algebra with lines so the same picture holds you encompass the line inside a code dimension to 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 code dimension to topological operators so even as you engage yang-mills theory 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 it but I'll not talk about such examples today we can also try to gauge this topological Co dimension two surfaces gauging means that we put some background field and we may even sum over it later now this would be a discrete to form gauge field or a continuous to form u 1 gauge field so like gauge fields are replaced by to form gauge fields you can also replace the Higgs mechanism by Higgs mechanism for to gate 4 to form gauge fields and so on so the simplest example where
there are such objects is just young pure young mule step in four dimensions so as you know pure Angostura into them in four dimensions has a theta angle which is valued between zero and 2pi because data center number is quantized this is on spin by the way um just for simplicity I'm only discussing spin manifolds here for the force spin manifolds four dimensional spin manifolds otherwise there are the there are various things that I have to fix in the slides if I were not to discuss it just spin manifolds and this theory turns out to have such code dimension to topological surfaces because there is a center that doesn't act on any matter fields and so it's isomorphic to Z n so this theory has Xion word of such topological surfaces and one can ask what they are good for as you know there is a clay price like a million dollars or something for proving that theta equals zero these theories gap that confined and trivial it turns out that this new topological surfaces make lead to a very interesting statement about not tile 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 with this can be proven rigorously so what you can do at the moment three paths so once you've proven that the ground state cannot be trivial there are essentially three options so one option is that there is a maybe some symmetry breaking theta equals pi or maybe there's a masters masters gauge fields sorry massless particles I meant to say massless particles they could also be gauge fields but so one of these things has to be true at N equals pi so the clear conjecture cannot be true I take the clay price cannot apply you know today equals PI it's the opposite so it's really so mathematically them I am 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 disco dimension to surfaces there is a generalization of the notion of anomaly and you can prove that if 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 and arguments and from idea safety it seems that one should really favor the first one so what we expect is that while teracles PI's confined trivial and you know gapped and then there's the confinement at high temperatures when you go to the quark-gluon that sooner to the gluon plasma face at area caused by actually it turns out that the phase diagram cannot be simple there must be something going on already at zero temperature it cannot be confined in 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 T is temperature so let's focus for a second on theta equals zero this is trivial so this is this is the D confinement first order transition line so this is confine trivial gap this is the big standard picture for young Mills theory theta equals zero as you heat it up you encounter this phase transition and beyond this line you are in the confined phase so that's the standard picture for SEO and gauge Theory at zero theta however it air 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 that the confinement 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 it is the confinement transition so it's impossible to restore time reversal symmetry before the de confinement transition takes place so it's funny that this discrete anomalies lead to a inequality for a phase diagram of just pure yang Mills theory 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 directive aqua so the idea is that the theta equals PI I there are two black one this is what symmetry breaking means and they are related by time reversal symmetry or alternatively by CP T n CP are the same since CPT preserves the vacuum since there are two backwe you can construct a putative universe where on one side of the university or 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 there is this is the discrete analog
of a number Goldstone boson number Goldstone bosons are objects that allow you to travel between different bakkwa and in the discrete case domain Ewell's do the same so actually from this anomaly it follows something very strange must be true for this domain wall which I try to draw here but it's probably totally incomprehensible so let me stay instead explain it by drawing it again here so the date what happens is that let's suppose that you are just in the ordinary let's suppose you're one of those bagua let's call this vacuum one and this is vacuum two so vacuum one has confined quirks it has broken time reversal symmetry but it has confined quarks so there is a string if you take if you put two havoc works Q and Q Bar they're 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 bakkwa and involves Q and Q Bar connected by a flux tube that's confined and here is the domain wall so the domain will be somewhere here that's more left the region where the transition from the to BA kua takes place so it turns out that the quirks are d confined on the wall so if you bring a quark near the wall it loses the flux tube and it doesn't cost infinite energy to create a quark anymore so there is the confinement on the wall well there is confinement in the balcony ER temperature in each of the vodka the wall itself is d confined so if you put a quark and an antiquark they don't connect by 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 engines so if you bring a quark to the wall and you circuit around an anti quark you pick up an aharonov-bohm phase which is exponent of 2 pi over 2 n and so the spin of the quarks is fractional so the quarks are not only D confined there also because they also become engines and that's that's a prediction that follows just from these anomalies so there is a time so symmetry breaking at zero temperature and on the domain walls the quirks become any ons and this is their spin they have a fractional spin yeah if I have to finish it accion now it crop all that dynamically because then there is a domain all three yes right so since this thing there were many papers about what we're saying in particularly there is a I think there's a paper of Sherman and Unseld exactly about this and maybe also mister Schiffman they they try to see the implications of this anomaly when you have an actual accion and you have an axial string or an action ring they call it action ring and yeah basically the same continues to be true you have some point where there is a concentration of energy or the quarks are either confined and they become engines you should really think about this as a quantum Hall state so suddenly in yang-mills theory there is a wall where there is a quantum Hall state with this gauge this and so this is the topological field theory since there are any once there is a topological field theory and this is the topological field theory on the wall it says your insurance I'm on spirit level 1 that's the prediction right so this proves that in the real world data cannot be equal to PI because if it were there would be the manuals that would proliferate and maybe not wiped out by inflation that would be very sad so cosmological considerations would rule out theta equals PI in the real world because of this because of this anomaly however it may be interesting for other applications in phonology that we can talk about later ok now in this picture what I try to explain
dynamically why the quarks become any OS so the idea is that these two baku are related by time reversal symmetry but time reversal symmetry famously that takes diodes to mono poles so there here there is diamond condensation and here there is multiple condensation if you've never heard about this fact the time reversal symmetry takes diamonds to monopoles think about cyber great in theory maybe that might help so here there are diamonds here there are multiples and it's kind of in the world you cannot mutually condensed down multiples they are not mutually local so there is some kind of tension and the tension is resolved on the world by deacon finding the quirks so both expectation values go to 0 both the monopoles and the diodes do not condense on the world and you get instead a topological phase which looked like a quantum hall 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 code dimension to topological operators namely a one for symmetry so I
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 it's 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 adjoined 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 this topological code dimension to 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 equals zero theory is accidentally super symmetric so it's the minimal amount of supersymmetry in three dimensions which is called N equals one it there is no hollow Murphy so it's not very useful but it's still super super symmetric another technical comment for the experts is that n must be even for this theory to exist so n is even in su n so
using supersymmetry with an argued around 20 years ago that the Witten index is 0 in this theory so you take as un plus nigerian fermion it has a very small amount of support just so to supercharges and Whitin computed the index and he found their advantages and then everybody believed that the infrared of this theory is just a spontaneous trip broken supersymmetric phase namely a single mile run goes to no particle so just from the fact that the Witten index vanishes you are led to believe that the SU n gauge theory with an adjoint fermion flows in the deep infrared when you go to very long distances to a single mile run a fermion and of course this Maya ran a fermion is uncharged it's just the goals you know and that's it so that's what you would think namely given the information second oh no no this is just for capital
n vanishing of course thank you just for vanishing capital M that's where this argument about the Witten index can be made that's the only case where it's actually accidentally supersymmetry 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 the N where n is even and it turns out that there is an anomaly so this code to mention one services 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 two mod n turns out for even and this formula makes sense and it turns out that this single Meyer on a fermion cannot match this anomaly so this contradicts anomaly matching this idea this contradicts anomaly matching so this is not consistent with the normal imaging because this single Marana 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 the display ting joining issue with Co dimension 1 topological surfaces which are the time reversal 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 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
anomalies measured mod 16 and we
computed this anomaly we found out that it's N squared minus 1 mod 16 and 1 and for no value of weight I don't want to say something wrong but I think yeah for no value of even end this can be 1 mod 16 so this is either 3 mod 16 or or minus 1 mod 16 so it's either 3 or minus 1 mod 16 but it can never be 1 mod 16 which is what the Maya and the fermion gives so this also contradicts a normal 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 that by the same argument fantastic question so you can ask why so this picture that I saw Slava is asking how to see this mod 16
this is a very difficult question what
this is 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 Z two 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 1 plus 1 but it's in 2 plus 1 we're now studying a 3-dimensional gauge theory so people have found that the phase there is some phase and it's mod 16 it's measuring 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 root of 16
appears are you comfortable with an anomaly inflow argument or but geometrically you're 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 in flow I cannot give you I don't believe anybody
has a reasonable reasonably intuitive explanation for this root of 16 it has to do with the environment in four dimensions which is done normally in flow and it's measured in units of 16 because it's generated by our before and okay so in any case the question in gauge theory was the dis proposal contradicted the normally 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 and it makes many other things suddenly sensible so the fix was to assume that this model instead of flowing to just about theorem I around the fermion it has a topological field theory of the chern-simons type it's a non abelian bunch of anions of the chern-simons type now this looks ridiculous at first because topological filters of the chern-simons type typically are not invariant under time reversal symmetry so if you write the chern-simons functional a da it's odd under time reversal symmetry because there is an epsilon tensor but it turns out that when you quantize chern-simons theories even though classically this is not invariant under time reversal symmetry when you quantize it it might become time reversal invariant this is very counterintuitive usually in physics you think that if this 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 in quantum Corrections our h-bar however in chern-simons Theory there is no classical limit the classical limit of chern-simons there is an empty almost empty phase space essentially so since chern-simons theory doesn't have a classical limit this may happen and there is a special set of chern-simons theories that nobody has classified yet but we know of some sequences of special chern-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 field theory with leaf labels and the topological field theories with these labels which are time reversed by Jeff topological field theory tools so adding tacking on such a topological field theory makes sense it's time reversal invariant know the
the level is reversed as you should buy time reversal simple no yeah it's an over to and over to combine and over to and over to combine and it's isomorphic as a quantum as a quantum model its 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 the time reversal anomaly which is either minus 2 R 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 one form symmetry anomaly so one little thing fixes everything immediately this is just a three dimensional topological field theory but but is it ok yeah so is that clear so somehow there is a very sexy fix to this problem there is one thing that you had then it fixes everything
now this proposal implies again something that people did not expect it implies that there Wilson I'm sorry deacon find the adjoint 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 d confined because this is a
bunch of anions so the part the the probe particles are d confined and it's
somewhat surprising that this is true so
now how much more time have I got okay okay ten minutes I'll finish and then there will be 10 minutes the question so so far I gave you two recent applications of the ideas of code dimension to topological surfaces there are also some papers that I didn't mention about code 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 is a bunch of local topological operators so in two dimensions call dimension 1 echo dimension 2 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 it - just a chiral zito for the massless formula so when you put m equal to 0 there is a chiral symmetry ok so there is a story about the 1 for symmetry again but that's not what 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 klebanov Meeteetse and Milka I believe that my teaching is in Stoneybrook 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 klebanov was that Wilson I'm sorry to confined and there is no data angle in the PSU engaged analog of this model they also claimed the disease of 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 agarose at all stuff is wrong this model is actually confined except for the Wilson line to the power N over 2 which is D confined 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 proposals are in fact incorrect and the reason is that this model that meets a lot of non-invertible topological lights that people have not known of course in the nineties 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 a we've been working on classifying the full set of non-invertible topological knives in this theory computing they're anomalies computing the fusion rules and finding what are the constraints on the dynamics that is non-invertible topological 9 is imposed so mathematically speaking not invertible topological nines form a future confusion 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 chern-simons story but it's not the braid that that there is no braiding unlike insurance I monster so it's a more general object than modular tensor categories that appeared in the context of very Linda lines so this is a mathematical object that mathematicians work on classifying and it has implications for the ground state of Gators so what we've argued what we are arguing this is in some work in there is some also related work by Costas and Monier from 10 years ago so what we're arguing is that this 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 state unlike what grows in klebanov claim 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 state where there Wilson is already confined and there is a topological field tree that describes this exponential amount of ground States a two dimensional topological field theory of this type this topological filter is not the same as goog that many of you have studied in the 90s so 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 there is an exponential amount of huaquan in n so it's e to the N this means that the Haggadah temperature goes to zero at infinity the gage the Wilson is already confined and there is some 2-dimensional Frobenius algebra that that governs this exponential amount of aqua this is this is the like the
summary of the claim it's surprising that this non-invertible symmetry is 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 RG 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 is non-invertible topological lines only exist at zero mass some of them might survive to non zero mass so that's where what we're doing now so let me just conclude with some general a general comments which are like homework for sampson so so the general theory of the 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 diffusion 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 is governed by what mathematicians call diffusion category so this theory should be 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 this is what I
explained that somebody should also classify the know like generalize the notion of anomalies more clearly for this general setting the notion of anomaly in flow what is the notion of anomaly in flow for this Frobenius algebras for instance it's not though there is a very interesting recent paper by torn grinding wank about this is there a d-plus two-dimensional normally polynomial that's a open
question so here I also made some comments about two-dimensional I drink you CD but the super symmetric version we don't think I'd know about the one coming one super symmetric version of the model we haven't thought about it yet there are lots of questions about this Hagedorn transition in two dimensions the zero mass limit the dual arrays and so on but 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 and Mattias so as you you know about it exponentially more than me but you know the there is a there is a class of topological filters that people studied in the 90s of the type de mogi and their fusion there their fusion rules are basically inherited from a chern-simons term however there are many examples of a group let's say our example is spin n I will zoom in within model if level one where you gauge su n at level in 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 IC o---- in level n mod SEO in level and so you have to develop some formalism to study it in addition in our case there is an additional gauging by some quantum Zito but that's a minor thing relatively speaking so we claim that this buck WA and 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 event so that's yeah so this this first thing you are with the TV okay yeah oh yeah there is a Abby Nia since this thing there is a lot of work on the same thing in supersymmetry
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 Puppets there is a paper of somebody else whose name I forgot unfortunately and they seem to find a similar picture with the spontaneous breaking of time reversal symmetry and the compelling argument for the existence of engines on the boundary I don't know if this is always what they found but I remember one example where they found exactly this picture I don't know if this is always I don't know I'm sorry I don't know this good stratification on your space and you got different strata and mismatch if not both Rob Costello saying right yeah there is a date yeah there is there is some framework where there are objects of codimension 1 2 3 and they have some rules of how they talk to you right but my understanding is that they don't actually know 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 prove the rigidity theorem okay now probe the story rigid 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 so I bought your second example okay I'm
not sure if my question accented so since you're in 3G and you have this
could you mention co-operators which Alliance which can reprimands what
happens the decoration if there are two symmetry operators which wrap around a line on which the act and also simultaneously around each other so your picture is that you have two lines that are knotted yeah and then most of the absence of line operator and also another line that goes in between which they act so this seems to define a new type of product would reduce to something that we know so your you are asking about sorry let me just draw it correctly you are asking about this and then another line that just goes through nobody should go through it should go through both of them oh you want that to go to through both so that they both act on that way yeah this is already linked and now you want but they leave oh you want this to go through this one okay now understand because you said that this algae growth could dimension to greatest is a billion but so it seems to define a different sort of product which is not clear to me if it's a video 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 yes in my examples a different way but what is this product between the two circles but but you see here there is no action so you have to first feel this line with this and then act so this is that symmetry line right so first you what what how would you reduce this you would first fuse this in debt 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 light which is not topological in general and then this guy which we can call W prime would act on it okay yeah well if these are all symmetry lines so one of the so in the case of three dimensions since the lines and and the symmetry lines and the symmetry generators are all lines so the lines can act on the on the say on themselves so if these are all symmetry lines then I can reduce it by fusion and by sequence of fusion and contraction moves it's in 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 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 the the the simplest example is in the trike radicalizing model 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 the felt dwell point there is the duality symmetry and this duality symmetry can think about it was being implemented by some kind of code dimension one defect and this Co dimension one defect is not invertible so when you fuse it with itself you get 1 plus epsilon rather than 1 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
Shang Lin Xiao Ling Wang Union they looked at try critical the Ising model and some rational safeties or they have a huge I mean another very clear set of such examples are very Linda lines of course the Verlinden lines that are in fact like the first set of such things that you've encountered may be the Freelander lines are more special than the most general case but this is a bunch of non inverse two lines in two-dimensional rational conformal field theories but they defined axiomatically adult earth means let me take away this model of NGC oh you meant something down towards like in that sense okay so that then definitely you should look at this paper out of Chang they have lots of references to constructions where you can see these things on the 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 to form gauge fields so instead of writing anomaly in flow for one form gauge fields like a wedge da you write B wedge DB so here what replaces the notion of anomaly in flow is just to form gauge field anomaly in flow 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 to form gauge fields are always
a billion but the non-invertible lines are genuinely known a billion and there is the the mathematic mathematical
meaning of the notion of anomaly in flow is explained in this paper that I don't understand he is a mathematician slash physicist he's a physicist you know it's it's it's something to do with 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 white it what it might mean in two dimensions only it's only in two dimensions that there is a proposal question session to produce because yes another tool immediately so thank you again [Applause] [Music]
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