Anton Kapustin - Locality, higher symmetries, and anomalies: an operator-algebraic approach

Well, yes. So these are papers kind of background reading that what sort of that's where this uh basically what I'm going to talk about is an attempt to um clarify what's done in this papers from more more conceptual viewpoint. So I'll explain later how this is relevant because uh okay so so this paper is about quantum lice systems um and uh anomalies analog of anomalies in in quantum letter systems but my interest primarily for this working is quantum field theory. So let me start with um uh um with this u so um now so there is a um so the primary puzzle I want to uh address is the following. So it became uh uh you know clear or um to most people that uh symmetry in quantum field theory and also in general many body system with locality is a much more complicated notion than uh in quantum mechanics. uh and specifically if I have say um Q of T uh and say D + one space-time dimensions you're supposed to attach to it uh not just a single symmetry group but a whole bunch of groups um and when I when this talk when I say symmetry I mean I only mean invertible symmetry for me something which is not invertible not a symmetry like the transformation must must must be invertible. Okay. Uh but um you know then we add the definitions and actually this framework suggests uh uh uh you know what what should be done for non-invertible case. I just don't want to call it symmetry. So um so anyway so um so here you're supposed to get not just a single uh you know first approximation not just a single uh symmetry or coordinary symmetry. So this one is only zero form symmetry which is just some group but also a one form symmetry [applause] uh and all the way up to uh uh d + one um sorry 3 sorry 3 d um so this is a Okay. So um so so this is a group but these are um a billion groups and the way it's usually explained why do I need all this stuff is because I have a theory in the space time dimensions you can also if you consider nucleian formalism then usual symmetries elements of this represented by topological defects uh of co dimension one and which act on a spatial slice and then it's natural to consider also topological defects well invertible topological defects of higher code dimensions. What this was what these are uh and the highest code dimension is a code dimension d plus one which corresponds to these ones. Okay, so these are topological defects localized at points. Okay, so uh that's the story uh in the first approximation but in the you know uh um if you stare at this you'll see that well it looks suspicious like something from topology. So we have a a group plus a bunch of a billion groups and actually there's a further thing that this group acts on all others by automorphis and it suggests something you see in course of topology namely homotopic groups or topological space. So and uh in general you can ask okay like if you have is there some any interaction between these uh groups in some way do they fit into some bigger picture it's not just like a product of these groups and actually the answer is yes and in examples we know that's the case. So uh so in reality maybe you should say this is just an approximation what you really supposed to have you're supposed to have um D+1 group of symmetries u let me call it I don't know something f let me call it whatever just leave it nameless some d plus one group this uh but what does it mean what's a d+1 one group uh and um um turns out that the easiest way to define it using using notions from homotopy theory. So um so homottopy theory I defined what homotopy is but uh it's you know it's some um it's about studying topological spaces but under very like um loose notion of what equivalent spaces mean right so you say that um um so this is this is notion from homotopy theory uh And um before I define it, let me remind you what what what's a homotip type is. So um or maybe not remind you but tell you what it is. Haven't heard about it. So um so first of all um um if uh if x and y are topological spaces then and f uh and fg are maps or continuous maps are homotopic or homotopic equivalent. So it's a homotopic uh if f can be deformed can be continuously deformed to g it's a symmetric notion equivalence relation and then you say that um x is and you write you know f isom uh homotopic to g and then you say that x uh and Why equivalent uh if um there is an f from x to y and g from y to x such that um such that composition they're not like literally inverse to each other but they're you know inverse up to homotopic up to the homotopy. So that's so so that's an equivalence relation topological space and you know a equivalence class called homotopy type. Okay. And how much of a type well say how type of a point is like the most important one because so so basic but say how type of equilian space is exactly the same because actually they're equivalent. So you cannot how the you cannot tell space from you know a point. Of course topologically they're not the same. So it's a very very loose notion of equivalence but actually most invariants we encounter um when you study topology are do not care about the uh um you don't only care about space up to homotopic equivalence. Okay, homotopic groups, homologies, it's all only cares about the space of the homotopic equivalence. So homotopic type has well defined um homotopic groups for example doesn't matter which space you choose class they all have the same homotopic groups okay so and then finally [sighs] so an end type is uh is a homotopy type um with homotopy groups um trivial for k greater than n. So um so it's kind of unusual because typically homotopic groups of a space are you know non-trivial ones go all the way to infinity like for a sphere but sometimes it happens like say for a circle that is only one homotopic group or a few okay so it's a condition of n type so and now I can state the definition of a high plus one group So [applause] is the same as a connected um uh D+1 type. Okay. So it has no pi pi 0 is trivial and other pi is you know like all the way to d plus one maybe non trivial but above that they're trivial. That seems like a very strange definition because you know where is group theory should be groups here but at least you could say that um um if you define um um say g p u to be pi um p + one of your d +1 group make this definition so that this has the right properties because you know if p zero that's pi 1 just And you can be any group and all higher ones, you know, must be a billion. So that that's good. And also pi 1x acts on this higher ones. So G0 acts on the higher ones. So the claim is actually well there's more information of course in the D+1 group than just its homotopic groups. So the claim is that somehow this gadget can been attached to a quantum field theory um in in the space time D+1 space time dimensions. So uh here's a puzzle though like where you know where how do you get topological space out of you know out of quantum field theory. So quantum field theory okay so you can say it's defined within path integral that's a semiclassical picture um and it's kind of difficult to even within the you know even if you believe stuff about topological defects in some level it's very difficult to see how to attach something like this to just ukidian path integral. Now the rigorous approach to corner field theory is operate algebraic but that seems even worse because usually it's formulated in flat spacetime. So um say uklidian space of knows can say space of dimension d plus one. So and then how do you get there's no simply no homotopy theory at all it's just defined on you know contractable space you know d plus one dimensional space is contractable. So how do you get a how do you pull a homot theoretic rabbit out of a a paral algebraic hat? So this is the the problem. So um now uh the point is though that the point I didn't appreciate until a couple years ago is that no well known to mathematicians that you don't actually need topology at all to get homotopy types. There are many ways you know sometimes homotopy theory is just a thing by itself and you can get homotopy types just in many different ways. So say from algebra or group theory. So I think that's what happens here. >> So you said that G0 can be any group. >> Yeah. >> Okay. P. >> Yeah. Pi one. Yeah. Pi 1 is can be anything but high ones. Yeah. Okay. So let me just explain like what that group theory can give rise to homoy types. So here's the first non-trivial example. well non-trivial in some sense. So, so suppose when in zero dimensions I mean zero spatial dimensions. So this is zero. So um so okay so um I want to attach a one type but what do one types look like? Well turns out that um so there's a theorem says one types are the same as groups. or connected one types. I won't won't say connected anymore. So what does it mean? Well, there's some category of groups. Technically speaking, there's a category of one types and they're just equivalent. And how does this work? Well, you start with a group uh and you attach to it what some space called the classifying space of the group called BG explicit construction. uh you just assemble the space out of simplices and the simplices are labeled by strings of elements of G which are so that I don't explain how it's done exactly but there is a construction like that okay and then so this this shows that groups are the same as one types and this gives you a model for what can sort of a model which can what happens in higher dimensions but it's kind of difficult to to to to see how to get some say two types out of what what is supposed to happen here and how do you get it from from field theory or by the way the same applies to quantum letter systems. Okay. So, uh I just want to explain how I I want to go about it. I have a proposal for for all um uh an very general proposal. Um but to get there uh I want to first um make the following remark which I think is actually key. [snorts] >> Sorry. Is when you say groups are these finitely generated groups? they're actually very going to be very huge. So my point of view is that first of all um I'm going to by the way I want to say from the start I'm going to discuss kinematical about kinematic symmetries of like observables and their relations dynamics only affects zero form symmetries we're going to cut out some particular symmetries presolution and high form symmetries are not affected by by dynamics at all actually they're kind of kinematic from the get-go so I'm going to work on the kinematic level and kinetic level if you look at say algebra observables or quant field theory Look at all automorphisms of the cell. They're just humongous groups. >> Yeah. But in this theorem that you mentioned. >> Yeah. So the so this G0 for example will be in principle some humongous group of all kinematic symmetries of quantum field theory. The higher ones probably not going to be like big like that but the Gzer is going to be humongous. Okay. Now um it should have also said that um when you think about how say some abstract group acts on a particular system we often think about league groups like U1 SU2. So in that case uh well you know say homotopic groups of a space just groups they don't have any topology you know how groups of be like a league group. So how does a but when you say well a group acts on physical theory you want to probably realize an action some continuous way that raises fairly difficult question I'm going to address it here so there's a you know I'm just going to stick with abstract grouping principles I'm going to like think about finite symmetries say of a system only or at least discrete ones like countably discrete or whatever I'm going to stay away from groups >> but in your definition also spacetime synies entering the uh zero symmetries and >> right so also no space time symmetries only internal ones I don't know how to do with space time symmetries so um right so the I think the key to this way to guess what what what what do you do in dimension higher than zero is the relation between higher symmetries and anomalies so my my point okay so on general in general so which anomalies so anomalies are defined as obstructions to gauging zero form symmetry to prevent any obstruction any you know something something which prevents you from gauging turning you know con constant transformation to gauge transformations these are just ordinary gauge transformations so I would like to say well that actually anomalies and symmetries and higher sim is actually the same topic they're just different ways to look at the same thing so that's going to my starting point and here is a sort of a numerological okay first of all a conceptual point that higher symmet symmetries come basically from the fact that this notion of locality to talk about defects of various co- dimensions you need to have notion of locality and when you say you're engaging a symmetry you also use locality promote the global symmetry to local symmetry so they all have the same origin that there's a notion of locality uh but here's also numerological reason so um so let um so let um um so consider like so consider a high group a higher group So d so d plus one group with uh non-trivial g0 and gd only nothing in between. So there's only pi 1 and there is pd + one. Okay. Uh so uh okay what else is there? what what other information is contained in this group. I said there's more information than simply homotopic groups and only in this case it's easy to describe what this extra information is called some guy which is lives in group coomology of degree D + 2 um of the group G not I'll just write it this way so so people usually write group homology is just first argument is the group but I'll write first argument is topological space BG not this classifying space because that's actually the easiest way to define what group of homologies anyway. So, so this is a thing which labels um a homot type with this kind of behavior. So these three pieces of data are enough well also action of this on that and enough to define this type. Now this is reminiscent. What is it reminiscent of? So suppose on the other hand. So this is a distinguish is a set of set of topological of topological actions. uh for a you know G gauge field in D plus2 uh dimensions. [clears throat] So these are so so-called topological you know these are these labels sort of classical topological gauge theories known as diagramraph witten theories. So and um first of all this matches if you say that GD is U1 and GD is a what is it it's a topological defects invertible topological defects of U you know dimension zero co dimension D plus one dimension zero and indeed if you ask what are topological defects which which are pointlike in a D plus one dimensional field theory well the answer is an obvious answer just scalar operators well invertible scale operators which means essentially U1 elements of just phases inserted at the point. So any Q of T by nature has at least U1 D form symmetry and typically no other deform symmetry. That's that's typically all that all there is. So um so that so therefore actually this thing would determine a um um if there are no so basically suppose there are no higher form symmetries at all say but that actually cannot can never happen there's always at least one deformed symmetry this U1 thing so whenever deal with QFT with some symmetry group G not then actually it comes also with this class because there's also a a deformed symmetry can mix uh and this defines an a topological action in dimension one higher and what is it good for? Well uh this is a well-known law that this actually classifies a truth anomalies of this symmetry. So truth anomalies then just in this looks like they should be just interpreted as just part of higher symmetry. Okay. So they're not the different thing just the same thing. So and we know unlike higher symmetry which can be abstract these are sort of almost mathematical meaning to a 12. It's an obstruction to gauging and obstruction has a well definfined mathematical meaning probably define what gauging is. So we're going to try to discover what where higher symmetries come from by thinking about what does it mean to gauge a symmetry zero form symmetry. Okay. Now this is just numerology right but you can in concrete cases you can verify it's actually the same thing. So this symmetry doesn't this this default symmetry GD it doesn't act on anything in there just >> well it acts on on a space of on the vectors in hbert space >> okay >> okay but yeah it doesn't anything except vectors in hbert space your your >> but so what if you like taking for a discrete which is you maybe uh more and these topological operators are exchanging back okay so I don't think this cannot be broken or anything this is a some sort of topological symmetry which is good to carry around because it gives you more information that's how I think about it it's basically a way to keep the U1 phase you see when you consider representation of your operator algebra space without actually introducing hbrid space okay so it's just a workar Okay. So anyway, so uh here hence a a proposal. Let's try to see what a gauging a symmetry means mathematically precise way and then hopefully we'll see also why uh that data gives can be packaged as a D+1 group. Okay. Okay. So now here I want to make a like something which mutations do all the time but I want to phys sometimes people just say this group is a symmetry of this theory. So okay and then that actually strictly speaking was almost never known. So like um what what it means is that this abstract group acts by symmetries on your theory. Nobody ever computes all of symmetries of monop field theory even in simpler cases of like classical field theory just somebody hands you differential equation you know cortex de freeze equation whatever and then you ask what is the symmetry of this equation I don't know almost nobody knows what this is right maybe nobody knows so it could be infinite could be humongous so we don't care about that we can define what it is it's any transformation of variables in that case which preserves the equation but we don't know what it is actually we cannot compute it so want to distinguish also same thing makes the same distinction for cornfield theory Okay, when I say instead of saying that uh this group is a symmetry, I'm just say this abstract group acts on my system by symmetries. Okay, so I'm going to define the sort of abstract side and the concrete side. So typically I'll say say here's my uh abstract group G acts on my system by elements of a concrete group let's say f. So what does it mean? It just means that you know we're given a homorphism from G to F and F I don't know what F is. I can define it but I can comput it typically like automorphism like naively just say wait for quantum mechanics. So I would say that f is either unitary group of my hilbert space or maybe projective unitary group. Okay but you know in general could be some complicated thing. Well, this G is typically something small. Okay, compact legal group or actually let's think about finite group. So, okay, I'm going to make a distinction. So, what I mean, okay, suppose I have G which acts on my system. So, this is homorphism like this. What would would it mean to gauge the symmetry? So, um well the first thing we need to define well this is a group of constant transformations and first of want to promote it to non-constant ones. Say you know this is like say U1 then I want to replace this maybe with U1 valued functions continuous U1 valued function in my space. So I want those to act um so but first of all this choice is not completely unique um so like if case of U1 functions you can say well maybe should work with small gauge transformations not not arbitrary ones that maybe should require my function from my my space to U1 to be deformable to a constant function. Okay, it's up to us to put to choose. Sometimes there's no obvious choice. Say for finite symmetries, people usually don't even know what it means exactly or do not specify what it means to have a you know non-constant constant transformation non-constant finite transformation. So that might be um maybe some ambiguity there. So gauging. So so what what I want first of all the data should be um um some bigger group and embedding of G in there which is not this these are non-constant well arbitrary gauge symmetries and this again just abstract one still and I want this um so ga so this is gauging So first of all I choose choose this uh non what non-constant means uh and then plus uh an action of uh this bigger group plus so on my system there is some homorphism I don't know row prime from this guy to to f again that's a good good notion. Now this however is not a good notion and completely neglects the most important part of what gauging means. So if I want to have this um so my con transformations act everywhere when I promote local symmetry I have a notion of say transformation localized on some region and I don't want transformation localized on region to act on tribute on something which is far from that region. This is not incorporated here at all. So first of all I need to put extra structure. I want to know what it means for transformation to be localized in a region. Uh and that's clear. So usually on the abstract side. So so I'm so so let's scratch that. So there should be so first of all we're going to uh instead we're going to say that um um we're going to have the following data. A choice or a group of symmetries localized on U where U is some region in my space and I'm going to be working in the Hamiltonian approach. So it's going to be a region of space not space time. uh and with the following and okay so first of all so then we're going to have of course um um some natural compatibility not just assign some random groups um so if u is inside v I want to have a homorphism let's call it E. So something which is localized on a smaller region can also be reinterpreted as something localized in a bigger region. And of course it's clear compatibility condition. If there's a third region W which isn't even bigger than V, then by composing these two things I'm going to get the third one. So, by the way, this thing has a name. This structure, it's called the precashie of groups on X. So, just collection of groups just just to make make sense of meaning what it means to be localized the region, you need to have that. So uh sorry maybe stupid question but why don't you want to just consider some kind of a G bundle on the >> well first of all I'm going to again finite groups here I want to avoid talking about connections and first of all on a space there are no interesting G bundles all trivial so there nothing to talk about the only thing I can talk about in in flat space is just transformations that's still non-trivial so basically my ideology is I want to extract higher symmetry anomalies just by think about defects uh invertible defects in flat space but first I need to define what topological defect is and in fact I'm going to define it now okay so that's still abstract side but maybe should give an example what what could this be okay here is a good prop here's this proposal for the dis you know say find group G so when you people talk about gauging u find group G they typically think about transformations localized um like the main walls right you know you see pictures in many papers just draw some you know hyper plane and say well this transformation G on one side is identity on the other side so more generally you can consider um um >> [laughter] >> people modern people talk about network of defects. So what does it mean in this language? Well, I can just say well I can just consider so We're going to say um um my g of x just for the whole space is going to be uh maybe uh group of functions from x to g which are peacewise constant. And um on uh strata of some decomposition, some polyhedral decomposition just stick with equidian piece of RD. So what's a polyhedrron? Polyhedron is just anything which can cut out of uh space by finite number of you know hyper plane cuts and also can allow take unions of such sets. So um so take any of such poly take a bunch of polyhedra intersect take a union of them and again uh and again look at their you know their boundaries and high dimension sort of faces that gives them de composition of your space into strata like you'll get pictures you know like this okay and into dimensions and then you just for every stratum you attach a element of the group okay there's one here another one there third one here a fourth one here okay so that's what the element of this group looks like and you can multiply them. This forms a group. Okay. So if you compose two such guys you get a well definfined new system of uh whatever um defects. So that's a way to package this picture into well defined mathematical object groups of piewise constant functions along some polyhedral um some polyhedrron in or union of polyhedron space. Okay. So that's our g of x and when what is g of u. So g of u is simply functions which whose support is in a closure of u. So outside of the set they just identity. So and then your standard defects wall standard you know defects on half you know the main walls just generate this al this group so this group has generators given precisely by the main walls so that's a good example of what it means to have a preache of groups so it's an abstract side >> why [clears throat] is this not equivalent to the data I think you already said right I of a G bundle where this just know the transition functions between patches >> the old G bundle is equomoric to trivial So it's basically trivialization. It's a choice of trivialization. Okay. So it's equivalent to that. But there's no point talking about bundles if they all equal >> think that you want to get rid of that notion of >> Yes. So just work with trivializations. Okay. So what which are these things? Okay. So um all right. So and then there's a concrete side transformation of my operators which are localized on regions and then again should begin by collection of groups like this. So some abstract some some some concrete some preach of groups [clears throat] [applause] I write this way [applause] and concretely say for for letter systems I can easily give you an example if you know what a finite depth circuit is you could it's some automorphism of the operator algebra made out of gates and you can just say well suppose my gates supported on some region. So that's going to be the group of all such circuits going to form this fo view for lettuce systems. Okay. So um okay so and but as an action so an an action of on f is simply an assignment of a concrete transformation to every abstract one. So I call it the towel. So so mathematically it's a um map of precurs but what actually means it means that you have a for every open for every region you have a map homorphism of groups like a realization of your abstract transformation by concrete one localized in the same place. And of course you want a natural compatibility condition. Um if your you know if your U is in V then of course you have also this thing here. Um um these are your corrsion maps because restriction goes from bigger set to smaller one. These are going in opposite direction. So this is C of G and this is C of F. Okay. So and that's what it means to gauge a symmetry. To gauge a symmetry it means to first of all pick your notion of a G symmetry that is this abstract precursive curly G. Pick some embedding of the constant transformation uh Roman G into. So you need also to choose embedding of your G into global sort of globally defined transformations. Uh and then you also want to um um okay you want um to choose how each localized sim transformation acts concretely in your system. That's this. So this is one two and three of course you want to uh that your constant transformation acts as before. So you want um uh if you compose this towel of the whole space with iota you're going to get uh homorphism from g to f ofx that's our global symmetry group and and you want this must be equal to your original homorphism from g to ordinary symmetries which are f of x. So normalization so constant transformations acts as before. Okay. So that's all what gauging means. >> This is just defining you change the system. >> No, here's kinematic stuff. I just want to uh explain how I can act by abstract transformations my chosen model on the system. Okay. >> That's not what we mean by gauging. >> No, that is what we mean by gauging. Engaging means in our jargon changing the system the system gauge it mean the system. No for for finite groups you don't need to for finite groups there's no you don't need gauge field because for equilibrium space there are no gauge field for for g you know for some say finite group it's only about symmetries >> it's true but we change the hbert space we draw some state >> there's no hbert space >> where is this >> there's no hbert space it all happens what I'm describing happens before just want a consistent definition what I mean by a gauge symmetry and I claim I can actually extract that hoof anomaly of a g action just from this data you don't need to talk about hbert space you can extract it directly from here I can give latest examples where that's the case >> so it should know about the matter content in some sense >> um of course it it does the question is how this >> how is that encoded yeah >> okay this the concrete guy is specifically transformation of say in the latest case transformations of concrete filbert spaces of sites which are finite dimensional okay so this f okay let me give an example I always said it in words. So what is f of u? It's a group of circuits for a concrete lattice system acting on some region of space that is trivially outside that region. Okay. So it's a very concrete thing. So now I specify some way some group acts my G and then I ask can I promote this action in comp consistent way while I consider non-constant ones say like these ones where you allow like jumps and it may not be possible and actually I claim that are exactly these things in that sense they're sort of kinematical >> can you can you make a choice of discrete portion here or >> like can you encode a choice discrete tortion in the gauging. >> How about discrete tortion? >> When you like well if you add a little bit space the the G stores you can twist them by some. >> Oh yeah, there's also ambiguity of course. So gauging ambiguities uh are here too like the choice. If your anomaly happens to be trivial that doesn't mean you know that a unique way to gauge. So specifically this map there might be different maps that's um choice ambiguity is engaging. So the question is okay so okay so that's a picture of gauging and now I want to just say quickly because I don't have much time left so why homop theory emerges from this setup so first cla so this notion of gauging does not yet lead to homotopy theory theory theory theory theory theory theory in the new just a very it's a very basic thing but there's one extra thing which also seems very algebraic and basic which you add to the setup does lead to homotope theory. So, so by the way, so um oh so uh the additional thing which is most important is the following which which we still haven't completely incorporated in the definition of what gauging means the physical intuition and there are actually several things you can try to impose uh but the most important one I'm going to time to list all of them but the most important one turns out to be the following one so there's some others so locality of the commentator so it means the following roughly speaking So you have two transformations. one localized on U another one on V. When you take the accommodators, it necessarily localized on the intersection of U and V. Now that sounds very simple uh but actually it leads to many consequences if you impose this. By the way, all like natural examples that you get from physics or like from thinking about gimm all satisfy this like other things you might think are natural and do not always fold. For example, I should mention some other things that you might want to impose. For example, you might maybe want to say that perhaps um um perhaps this holds. So if you have the transformation localiz intersection is simply that transformation localized on U and localized on V. That seems natural and sometimes it is true but sometimes it isn't. For example, um if you think about small gauge transformations, actually that's not satisfied. Okay. Um for league groups. Yeah. Or another thing which you can uh naturally ask is the following that um um you can ask uh that look at the union can say that this can be al written as a product. So it's generated by g of u and g of v. So you can write any transformation on the union as a transformation localized on the two things which make it up. And that actually is true it's true for small gauge transformations on a manifold but it's not true for large ones. So things you know the opposite. So so this natural property sometimes hold sometimes they don't. But this thing always holds as far as I know in all examples which come to mind. And it is this which makes the following con some construction possible. Um so by the way another thing which you know even more trivial is that um which you might want to require is that um um so this is injective. So this map from say f of u to f of v is an injective. So what does it mean? means that you know you can always you know if you say that something is local is localized on U and then you say well I can also think it's localized on V right when V is bigger than so the point is that yeah so I want to say that being localized on U is not an additional DATM it's just some property of something which localized here that's what it means to be injective that seems very very natural and it's true for all on the abstract side but it's sometimes not true on the on the concrete side so actually this we on impulse either. >> What is an example? >> Well, example is if you look at say lice systems when you think at some a transformation localized at a point what's a natural guess for it? Well, natural guess would be some unitary observable right localized either at this point or maybe some region which has a phase information. But then if I think about uh this point sitting say in a line then all I can assign to it is the automorphism of the whole system which is conjugation by you know what is the what is the thing which lives here is automorphisms of the system localized on this line. So I can map from here to to here how by assigning to unitary conjugation by this unitary but it let loses phase information. So this this map is not injective because of that which is the only mechanism from which injectivity can fail but it's an important one that's where the U1 comes from in all the story about the anomalies. So the claim is then okay so here's then the proposal. So um I I just postulate that in any reasonable system concrete symmetry is described by this kind of gadget a precursor for groups with this locality of the comator and then turns out I can attach to this data u a high group now here is like there's a famous thing cartoon where you know there's a lecture and the guy explains some argument to another guy a scientist says like some statement another statement and there's a dash lines says then a miracle occurs. So that's the kind of argument I cannot have time to explain it but there is a machine uh like you you take um this kind of datm a precursor for groups with this property you stick it into machine turn the crank okay and out comes a homotopy type okay more precisely u there's another door in this machine so here's your machine here you stick in your your ricochet of symmetries of your system concrete one here you you pick a cover of your space um and then out comes a homotopy type so n type where n uh uh is the uh number of elements in the power plus one. So for example, if you start with a you know system in on a real line, well sort of the simplest way to cover it by two half lines. So so in this case uh you get a three type out of this machine. Okay. So uh I actually wanted to get a two type a symmetry two type. Well turns out that um it can be reconstructed from this three type in some way. So I'm going to explain how but anyway there's point is a machine and the higher symmetry group can actually don't need to convert this three type to two type. If you're just interested in higher symmetric group you can just read off from here. The rule is pretty simple. We just take a I know we just take a pi just say that g p is pi pi + 2 p plus two of the guy here instead of plus plus one. So uh so so there's this machine and I checked for letter system gives you the right results. So so and how this machine constructed well doesn't matter it's just math not physics. The point is that physics provides you an input precursor of groups with locality of the combinator. And then u and that thing enables you to construct um a homotopy type and on the other hand if you have an abstract side it also gives you precussion of groups can also stick into the machine but it also gives you a trivial homotopy type if the precussion of group satisfies all the reasonable requirements like uh like these ones. Yeah. So the point of this therefore this modator kind of measures how much your target your concrete system does not fit your expectation about what the gauge symmetry should be and you can show that if this target has a non-trivial somehow essentially non-trivial then you cannot find a gauging in the sense here and say of one and two dimensional systems can all be recovered from the somotopic perspective and also you know it provides you a way to um extend ended you know low dimensional examples which outlined in these papers. So, so this paper is about you know this paper will deal with the two dimensional cases one and two dimensional case but it's not clear how to generalize well they explain how to extract to anomaly of a group action on a letter systems but they only for one two dimensional case once one start with three dimensions you need this additional this all this technology to actually extract >> so so if there's no tooth anomaly uh how does the quantum symmetry of like a gauge you know, symmetry uh enter the game like you, let's say some Z3 gauging and then you get some uh Z3D minus2 uh uh quantum symmetry out of it. >> Well, there's a further question. Once you gauge, what is the high symmetry of the gauge system? >> Yeah. So, so you you'll essentially have uh well, yeah, because you'll have different homotopy types for each system. So basically you'll have in this machine you also have an operation assuming to anthomy zero of going from f to some you know >> yeah so I don't know how to okay I don't know how to um say compute I don't have mechanical procedure for computing higher symmetry of a gauge system however you know if I okay so this method applies to like okay very questions by the way for which the system this method should give you an answer suppose you have a lattice gauge theory with some randomly chosen G law that standard ones for which we know everything but let's choose slightly baro G law we expected it might have some higher symmetries with some truth anomalies how do you compute that nobody knows but this machine allows you to do that I think are you confused where the particular system enters in the sense that imagine I have two systems that have the same zero form symmetry but different higher symmetry where here I saw that I was dealing with system A instead of system B. If I was only talking about G the whole time until you put it in the machine, >> you cannot see uh on the level of just global symmetry. You cannot see the distinction abstractly. The two symmetries can be the same symmetry. But the point is that higher symmetry is really an aspect of how you localize to regions. So everything is just group theory plus notion of localization. If you just tell me the symmetry is Z3 another system also has symmetry Z3 that doesn't tell me how it acts locally I cannot tell right but when I say I have a lattice system we say threedimensional space for on each side that gives me concrete say in three dimensions then I can actually compute what the high form symmetries are for this case uh and I can take another one system. Okay, maybe uh say in higher dimensions you know also with threedimensional librar space there's going to have different higher symmetry even though the global symmetries might constant once Z3 in both cases maybe so the point is some that higher symmetry is just a way to package some of the information about how what it means to act on a region there's no claim by the way that if all the truth anomalies in this sense vanish you can actually gauge there's no not nothing like that it's just uh That's might maybe some theorem for particle class of systems that usual tonomies ensure that gauging is possible here just an obstruction theory just when it's on zero you cannot gauge even on this very basic level and they claim that all normal toomies are just this kind of this very basic notion of gauging. So in reality what happens here is that essentially when you choose a cover you kind of decided decided how granular your gauge is allowed to be. So if you take a two plane and de compose it to say three cones that's my cover of R2 and I measure anomalies and let's say to stick it in here that means we're going to measure anomalies or higher symmetries relative to this choice of cover if I refine it I get a different thing hopefully it's stabilized at some point okay but it mean there's a level of resolution Say [snorts] the standard uh definitionality in this one dimensional case literally only captures whether you can localize to left and right lines half lines. So your transformations can be constant here and trivial here or constant here. >> You only have the zero form and the D form. >> Yeah. So in that case luckily there's nothing to refine because actually turns out this is actually the finest cover you can have. So that's fine but in in already two dimensions there's no like finest cover. uh so there in principle you could have more and more anomalies appearing as you you know refine your cover in good cases it doesn't happen by good I mean if you define you know say for example if you want things which are detectable which are very under stacking with other systems with trivial systems so sort of in that sense if you only care about which don't disappear when you stack with extra trivial systems then I think we can show that refinement at some point just not necessary anymore. Say three cones would be enough for R2. And that's actually the reason why homotopy types are only trying get terminate the group just stop somewhere simply because you know the the the cover finest cover just involves D+1 cones that's all all you need to know but that's a theorem because a priority for general systems you can have more and more obstructions refine the All right. Well, [applause] nice talk. Are there further questions? >> I had a general question. If you want to try to relax this locality in some milder way like I don't know in non-commutative field theory or something like that or in a latis system equivalently maybe putting a magnetic field and then uh would you still have some notion of >> basically the pictures I I drew secretly already relax what I mean by locality. So um for letter systems for example only locality is only like works on the level of letter spacing. >> So and the way to do it is to say that well when I say something localized on U doesn't literally means that you know the symmetry acts tribute on everything right outside U. It just means that say here's a good definition. Suppose I say that my something localized on U if the symmetry acts on um everything outside some thickening of U trivially. Okay. and the thickening you know might might vary from transformation to other transformation. So from that point of view say localized on a point the same as localized on a ball of any finite radius. Okay. So on the other hand localized on a line is not the same as localiz on a point because you can never get a whole line from you know finite thickening of a point. So uh uh yeah so these cones arise precisely because think about this you know all possible localization regions can be just you know up to this equivalence after thickening can be transformed to a cone. So that's perfectly fine for noncomitive theories because again if things decay rapidly away from a cone that's when I say they're localized on this cone. It doesn't have to literally be like trivial. So, so, so are there known examples of anomalies in some non-committed field theories that >> I haven't studied but um I think there's nothing u this is very general formalism >> so one can apply to that and try to to see if if anomaly okay >> so let me take the old fashion definition of anomalies as classified by invertible faces in one higher dimension or >> they actually want to prove that >> yeah right right but but maybe do You know if uh so when you say you want to prove that means that there is no counter example in the sense that you haven't found any anomaly defined this way which is not captured by that picture or >> well depends on what you mean by again if you stick my definition starts with a particular system and in principle this higher symmetries and anomalies depend on you know on that system when I say classified by some something like group of homology this seems a bit strange right because that answer didn't depend on the choice of a theory. I say in any QFT anomalies are classified by this object. Okay, how does it happen? My construction doesn't answer uh but um in practice it only probably hold I can I have examples of letter systems where actually it's not even true that the answer depends on the system anomalies you ask where are analistic values actually depends on the chosen system. So this answer when you're referring to is some by by construction that doesn't know anything about the system computing right how does it come about well um for a class of system if you like again enforce I'm going to care about anomalies which do not change under stacking with anillas then the whole theory simplifies refinement is no longer you know refinement is kind becomes unnecessary and I think in that case one can show politic systems that the answer is again doesn't depend on the system but or maybe for quantum field theory so it's a should be a theorem the theorem should be that for a certain class of you know net of local observables like in hack castler approach u uh this construction when it outputs this machine outputs something which actually doesn't depend on this net only depends on some dimension of space so and hopefully if you can compute that universal object they're going precisely invertible systems in one dimension higher [clears throat] >> in view of lunch one last question yeah your notion of thickening of observables like you mentioned these hack caster type approaches where they do things with quasy local nets of sister algebra and so is is it the same notion of thickening or do you need some slightly >> um I just meant that if you have a in a metric space for any region can look at all points within distance r from that region. >> Okay. Well, let us thank again. [applause] [music]