 Thank you. I want to speak at this very interesting conference. So this is my outline. The talk is based on these papers whose archive numbers are listed here. So one with Sergey, who just introduced me, and then a couple of papers with Ryan Thorngren, who is now a grad student in Berkeley, and also some related work with Nadia Seiberg. So first give some motivation, then review an old subject called Diagraph-Width and Topological Field Theories, which is a very simple class of topological field theories defined in all dimensions. And then I'll show how to, well, my goal was to generalize this to some by replacing groups, which appear in Diagraph-Width and Theories, and input with two groups. And I'll describe TQFT's based on two groups and list some applications. OK, so motivation is to classify what's called gap phases of matter. So what are phases of matter? Well, example, that kind of states of matter which define a sort of homotopy. And as you deform the Hamiltonian, nothing much changes, nothing qualitative changes. So say, well, we can usually describe a phase by pointing out some particular Hamiltonian or phase in this university class. So there's liquid helium or electron liquid in metals called formal liquid, per magnets, insulators, superconductors, or a fairly interesting example is the vacuum of QCD. Now, phases, well, the most rough classification of phases isn't to gap phases and gap-less phases. So if you look at the Hamiltonian, quantum Hamiltonian of the phase, then, well, if you look at finite volume, there's usually some ground state, which is unique in some excited states. And what interesting to ask what happens in the limit of a large volume if the limit of a large volume is a unique vacuum and the first excited state is separated from the vacuum by a finite amount, even in the infinite volume, you say the phase is gapped. Now, actually, most of the phases we're used to are gap-less because they have these master's excitations, phonons or photons or whatever. But in this list of phases, there are some gapped ones, too, like insulators are gapped, superconductors are gapped, QCD vacuum is also gapped. So as I just mentioned, gap-less excitations often arise when you have spontaneous breaking or continuous symmetry, but they won't be interested in those phases. Now, I should also say that all phases of quantum matter. Well, so how do we describe those? Well, so basically, well, if you focus on gapped phases, then there is a belief that you can describe pretty much any gap phase using topological quantum field theory because basically like a limit of the quantum field theory when you rescale distance scales and time scales. Now, so one more concrete motivation was to classify possible phases of gauge theories. Well, QCD is an example of a gauge theory. So I'd like to classify phases of gauge theories using topological field theories. Now, and this is a well-known example. For example, ordinary superconductor is described by topological gauge theory, whose gauge group is Z mod 2. Can I ask you a very stupid question? When you say it's described by TQFT, the Hilbert space of the TQFT, where does it sit? Yeah, actually, I think I mentioned on the next slide what the TQFT is good for, what sort of things it captures. So now, it's often the case that TQFT is trivial, that sort of trivial phase. Like, for ordinary insulators, it's trivial. But also, there are more interesting examples. Like, say, there's something called topological insulators, which correspond to non-trivial TQFT. I won't actually say much about them. Just there are some interesting new examples. And then there are some other interesting gap phases, like, say, fractional quantum whole phases, which are also described by non-trivial TQFT. So there are lots of interesting quantum examples. Now, this is to address Paul's question. So what is TQFT good for? Well, first of all, TQFT describes long-distance behavior of various correlators and some observables. And some of these correlators actually become zero in the long-distance limit. If you rescale everything, physicists say they sort of confine. But some do survive, even if you take long-distance limit. And those are described by topological field theory. Now, then if you look at, say, a situation when you have a compact spatial slice, well, times time. In that case, TQFT has a finite dimensional Hilbert space. And that finite Hilbert space is a space of vacuum of the original phase. So that's what it's good for, in particular. So you can describe the vacuum. Although, in some cases, this vacuum space is actually one-dimensional. So that's not the only use of TQFT, but it's one potential use. And probably one of the more interesting features is that TQFT tells something about the behavior of the boundary of the phase. Sometimes a phase can be characterized by some interesting behavior on the boundary. And TQFT is good enough to capture that. Now, let me review some old stuff, namely how to classify Higgs phases of gauge theories. That is where, well, superconductors is an example of that. Standard models also, like that, if you, well, at least, part of this. It's not gapped, but some of the things are gapped. And what happens there is you have a condensate. The vacuum is a complicated thing in this series, and it has a condensate of electrically charged particles. And this, say, there's a spontaneous breaking of gauge symmetry. Now, so we start with some G0, which is a microscopic gauge group. But at long distances, all you see is some subgroup of G0 called G. So G0 is a microscopic gauge group, and G is a low-energy gauge group, which you can actually see. Now, if you want to have a gapped phase, then G has to be finite. Because if G says, if a broken gauge group is typically some compactly group, but if it's not a finite group, then you have photons, and then it's massless, and then there's no the gapped phase. Like standard model, like it's not gapped because a massless photons. But, say, if you just forget about the photons, then you just look at the gluon, well, at some other particles, and it's gapped. Superconductor is gapped because the photon becomes massive there. Anyway, so the point is that low-energy physics is described by gauge theory with gauge group G, which is finite. And then it doesn't even remember what G0 was. Now, so the path integral, then, in this topological field theory, which is just to sum over flat G connection, because I know how the G connection, the whole flat, or G bundles are the same thing, principle G bundles. Now, interestingly, there's more than one phase with these properties. Like G, the choice of this gauge group doesn't fix uniquely your equivalence class of a phase. So what are there is, when you sum over flat connection, you can just sum over all of them, or you can weight one. You can sum with some weights. And consistent weights can be classified completely, as it is. You can classify all possible reasonable choices of weight and classify it by this cohomology group of the classifying space of G, with these dimensional spacetime. So that was explained first by a diagram written with my mind how it goes, how the argument goes, because, well, roughly, the structure of the argument, because I'm going to generalize that. Now, first of all, for every group, we have classifying space. And for finite G, one can just characterize up to homotopy equivalence by just by saying it's pi1 is G, and higher homotopy groups are 0. Say, for Z2, this space can be taken to be infinite projective space. And any G bundle is a pullback of universal G bundle over BG by some map. So instead of thinking about summing over flat or G bundles, you can think about summing over maps from the space to BG. So your path integral is essentially like a sigma model with target BG. And path integral becomes a sum of these homotopy classes and an obvious way to construct a weight in the path integral. Well, usually I write the weight as e to the 2 pi i times the action called S. And then the action can be obtained like this. You just have some map phi. So you can just take some cohomology class on BG of degree D, pull it back, and then integrate over the whole X. That's our weight. And well, it's cohomology less than what is not in reals, but in r mod z because when you exponentiate, well, not like this S by 2 pi i, and then exponentiate only the values of modulo z matters. So it's going to be homological as modulo. So this gives a reasonable topological action. Actually, this describes all possible topological actions. Oh, but you cannot replace homology by bordesons here. Yeah, I'm ignoring that. Yeah. Actually, so actually, in the talk I gave it, Davis actually was most concerned with the version where he replaced cohomology with bordesons, that essentially you can understand part of that. But here I'm going to focus on the simplified version. So people often denote this group by, like they say, group cohomology of G. But I don't use this notation because I'll be also using cohomology group of each weighted classifying space, like B squared G, and that doesn't have a special notation. So, well, this B squared G is defined for a billion G only. So it's defined by saying that pi 2 is G and all other homotopy groups are trivial. So that is B squared G is the Arlenberg-McLaine space type KG2. Anyway, so I'll just throw it explicitly like this. Usual cohomology group is going to be written as cohomology of the classifying space. Now, we need also slide generalization. We have an abelian group A on which G acts. Then there is an associated local system of groups over BG with fiber A. And its cohomology is noted like this. So cohomology of BG with coefficients in A. So it's group cohomology of G with coefficients in A. Well, one can also give an explicit description of this cohomology group because there's a nice, for finite G, there is a nice description, well, this nice cell structure for this BG. So then one can describe it very explicitly as saying that, well, an element of this cohomology group is a deca cycle is some function of D variables all living in this group G. And which satisfies some constraint, the concical condition, for example, for D equals 2 is like this. And then there is a notion of cohomological cycles. And you take a closed, you take a cycles modular, the ones which are cohomologically trivial, and then you get this cohomology group BG. Like, for example, two cycles look like this. They differ by something, by some expression like this, where H is some function of a single variable. Do you want to carry the discrete topology? R1z, does it carry the discrete topology? I don't really care here about topology. If I G were not finite group, then I would. Ah, OK, fine. Actually, I don't really know how to work with the G. It is interesting to consider the case when G is not a finite group. And then topology is important. OK, so anyway, so using this explicit description of cohomological classes as co-cycles, one can also give a very explicit description of this diagram within theory. So how to actually explicitly compute the partition function? Well, or path integral. Well, first of all, you triangulate your manifold x. And G bundle or flat G connection can be thought of as one co-chain, well, actually one co-cycle with values in this group. So that is, you have an assignment of an element of a group to every one simplex. And this is with a constraint. Sorry, a constraint like that. And you sum over these variables with these constraints. And the weight assigned to each constraint is simply sum over all these simplices. And each simplices contributes this value of this co-cycle and on these d elements, g1 to gd. So this whole construction looks like a lattice gauge theory. Well, except there's a constraint for each two simplex. In the obedient case, one can, for example, actually instead of putting these constraints by hand, one can enforce using some additional Lagrange multiple or field, which is in two simplices. OK, and the action has gauge invariance. It's a gauge invariance under G gauge transformation. It's just a g-valued function on zero simplices. OK, so all very simple and very pedestrian. And well, so say in even dimensions, for example, well, it turns out that you look at cyclic group. In even dimensions, cohomology group vanishes. So there are actually no interesting weights in even dimensions for group Zn. But for three dimensions, for example, the denontrivial weights even for such a simple group as Z mod n. And actually, in this dimension, the whole diagram in theories with a billion gauge group are just special classes of a billion Chern-Simon theories. But if you look at slightly more complicated groups, like products of cyclic groups, then the denontrivial weights even in dimensions. By the way, I actually don't know any interesting gauge theory with a physically reasonable gauge theory, which at long distance reduces to, say, in four dimensions. At long distances, it reduces to something like, well, with the diagram of hidden theory with the denontrivial cascical. So I don't, embarrassing that I don't know an example. Probably exists. Anyway, so what I'm going to do is try to generalize this construction. Well, first, my diagram of hidden theories only care about g-bundles, only care about pi1 of x. And if you want to do something, detect some higher homotopy groups, well, we need gauge fields which are p-forms, with p greater than 1. Well, I will focus on the case when only p-forms with p equals 1 and p equals 2. Well, the usual gauge fields just want to p equals 1. And these are like two-form gauge fields. And this was motivated by the relation with confinement. Let's explain in detail in this paper with Sergey. But I want to describe this motivation in detail. Let me just explain how I'm going to construct this. So first of all, I'm going to have one-forms and two-forms gauge fields. Then there will be gauge transformations which leave on both zero-simplices and one-simplices. Now, what sort of structure do they form? Before, I had a gauge group. And now, I will need something called gauge-2-group. So let me explain what this is. Well, so the natural approach is like, say, category theory. So you can think of a group as a category with one object and all arrows invertible. And physically, arrows describe symmetry transformations. Or even more explicitly, an arrow corresponds to an invertible co-dimension, one typological defect localized in time, defect along which you do the transformation. And composition of arrows just collusion of these defects. You have like two hyper surfaces that collide them that corresponds to composition of arrows. Now, if you, well, in quantum mechanics, there is only time dimension. Co-dimension 1 is dimension 0. So there's nothing else you can really think about. But if you're in dimension greater than 1, if you can fill theory, then quantum mechanics, it can be, well, they can be defects of higher co-dimension. So for example, you can start with this co-dimension 1 defects, which meet sort of perform symmetry transformations. But you can also, they can merge along co-dimension, meet along co-dimension 2 defects and intersect something to those co-dimension 2 defects. So when you think about a symmetry and fill theory, it's kind of more natural to think about sets of co-dimension 1 defects, co-dimension 2 defects, and so forth. If you just look at only co-dimension 1 and 2 defects, you end up with a notion of a two group of symmetries, although in principle, it could be also higher co-dimension. So anyway, so how do we encode? Well, what is the structure that these guys form? Well, by analogy with the previous case, let's say this sort of a two group of symmetries would be a two category with one object and invertible one errors, one errors and two errors. You prefer you can also think of a two group as a special kind of tensor category. Now, if this sounds a bit abstract, we can describe also completely explicitly what a two group looks like just in group theoretic terms. So we have a pair of groups. Well, I won't explain precisely the relation with the previous definition, but the equivalent. So a pair of groups, you have a homomorphism from one group to the other one, and also an action of G on H, so homomorphism from G to automorphism of H. And plus there are some weird looking conditions, which it took me a while to memorize, so lots of parenthesis. And then this quadruple is known as a crossed module. So it's some group theoretic thingy. It's equivalent to specifying this crossed module as a semi-specific two category with invertible, well, with a single object and invertible errors and two errors. Now, we can, this definition is a bit defective in the sense that it's not really well defined as a set because two groups of different sizes can be equivalent as two categories. So it's convenient to focus on this really physically important information, well, or important information. And sort of a minimal model for a two group is obtained by identifying all isomorphic two errors, or one error, sorry. And then this simplified data, or more concrete data, look as follows. First of all, you have this group G and homomorphism from H to G. Well, so if you quotient G by image of this homomorphism, you're gonna get some other group because this image is actually normal as it falls from those identities, obscure identities. So you have a first of all group by one. You have an I-billion group by two, the kernel of this map, and again, it's I-billion because of those identities. You have an action of pi one and pi two. And then finally, less obviously, there's also some element in degree three cohomology of pi one with values in pi two. So this is the data which is specifically, equally well described, but two group. So we have a group, an I-billion group, an action of the non-I-billion group on the I-billion group, element of this cohomology group. Okay, so we can use either this minimal model or the cross-module. Either way, one can define the notion of a two-connection. Well, so a two-connection is basically a two-form with values in, locally it looks like a two-form with values in the Lie algebra of this capital H and also one form with values in the Lie algebra of G. And then there are transition one-forms and zero-forms and then there are transition zero-forms in various compatibility conditions. So it's all kind of a bit complicated to describe. But one can simplify life in, well, once life is one, if this minimal model of the two group is finite. So then really all information is contained in transition zero-forms on double and triple overlaps. And those take values in pi one, in this finite group pi one and this find the building group pi two with some constraint of course. So one can further simplify life by picking some triangulation X and then double overlaps labeled by one simplices and triple overlaps by two simplices. And then a two-connection is described by this by one and pi two value functions on one simplices and two simplices with some constraints. And that's really what we want to formulate this higher analog of a digraph-hidden theory based on a two-group. So we have a triangulation and we assume that we have this chosen this minimal model. Well, we have on this minimal model for the two-group. That's a pair of groups, an action of pi one and pi two and then homology class, degree three homology class. So now what is actual config, we're gonna, this is a digraph-hidden theory, we're gonna sum over some configurations of variables on triangulation and each configuration attributes some weight. Let me first describe the configurations which we sum over. Well, first of all, we have an element of pi one for any one simplex just like a digraph-hidden theory but also an element of pi two for any two simplex. So we have one chain with values in pi one and two chain with values in pi two. There are also constraints and well, the constraint on A is very simple. Just one cacycle, just like a digraph-hidden theory. No difference. And the constraint on this two chains a bit trickier. Well, explicitly looks as follows. So we have this say some, this is a constraint for every three simplex now. And first of all, we have a, this pi one-valued variables in each one simplex and we also have a pi two-valued variables on each two simplex, I don't show them so that the picture doesn't look too messy. But there are four of them, right? Because in a label them like this, so we have a phase opposite vertex A zero I call the corresponding variable B zero. So this B zero leaves on this two simplex and say B one leaves on the phase opposite to vertex one. So it leaves here and so forth. So this is a constraint which this two chain of values in pi two satisfies. Well, one can write in a more nicer form by recalling that is a, well, since pi one acts on pi two is a twist. Well, on pi two-valued chains, there is a can define either ordinary differential or twisted differential twisted by this gauge filled A. And you can also think about A as a map to classifying space of pi one and you can pull back that class B on the classifying space of pi one and get an element of pi two. So we get then this configuration space of a closed one, a one cycle and took a chain satisfying this twisted closedness constraint. So, okay, so I explained already what this constraints mean. And I have to explain the weights we assign to each configuration. Well, okay, so weights must be satisfied some gauge invariant conditions. So what are gauge symmetries? Well, there is a, well, there is a, they're both sort of zero form gauge symmetries parameterized by pi one-valued function on zero simplices, which acts like this. And there are also one form gauge symmetries which parameters by pi two-valued function one simplices in deck like this. Well, one complication compared to the usual case is that there are gauge symmetries between gauge symmetries. And they will be parameterized by pi two-valued function zero simplices. So when we construct the weight we're supposed to, well, make sure it's gauge invariant and when you sum over all configurations and in the end we divide by the group of all by the order of the group of all gauge transformations but it's over counting because there are also gauge symmetries between gauge symmetries we have to multiply by the order of the group of two gauge transformations. So what do the weights look like? Well, so the weights are given again, well, again, it's in the usual Degard-Witton case it's convenient to reinterpret this two connection as a map from our triangulated space X to some classifying space. In this case it's a classifying space of a two group instead of a group. So the classifying space of a two group is some vibration. So it's base is classifying space of the group pi one and the fiber is the iterated classifying space of the group pi two and one is fiber over the other in the way which is described by the remaining date of the two group. And then whatever it is, it has some cohomology in degree D and you can define an action by just pulling back any class in degree D cohomology using this map phi which encodes the gauge of all the two connection. Now the interesting thing is actually, well, the interesting problem is how to actually compute this cohomology. So for any given two group, how do we actually do this? So. Use the say spectral sequence. Yeah, well, it's not so easy, yeah. So it doesn't, that's what we did back basically. So in the paper, so, but I'm just saying that while the standard sort of resolution is a fine for the classifying space of a group but if you look at the, say this two group, the standard resolution is huge even for very small groups. So you have like in degree D, your cosine, your cosine depends on D variables in pi one and D times even two variables in pi two, so it's lots of variables. So there's no way to do anything this way. So actually in the case pi one equals zero, the computation, we were done completely like by Alan Burke and McLean in 1954. In this case, it seems very simple but still not quite trivial. So we have just an obedient group pi two, there's nothing else, and have this classifying space, well, in this case, classifying space is just this iterated Alan Burke McLean space, B squared of pi two. So this is the, you know, this is homotopy groups and then, well, one needs to commit this cohomology groups and if you look at Alan Burke McLean, you can find the computations and in low dimensions, let's follow. So in degree two, there's just homomorphism, that's just the dual of pi two. In degree three, there's nothing. In degree four, you have quadratic U1-weighted functions on pi two. So the first non-trivial case really in this case is the dimension four, so there's nothing interesting until then. Now if you look at more general two groups with both pi one and pi two non-trivial, it becomes a bit more complicated as well, but so the, well, let me define, well, let me explain what it looks like roughly. So we have, if we can think about, well, define this homomorphism obtained by wedging with class beta, so it increases degree, well, by three, so I think it was a map from X in the, well, this is X, pi one modules. So this is a pi one module, that's also trivial pi one module, so you get a sex group and then by wedging with beta, you get the element here. So same degree in dimension two, this cohomology group is just the kernel of this map. So in other words, there's just, if you look at characters of pi two, you just look at, first of all, the one which are invariant under the action of pi one and pi two, and then you also want them to be analogated by this class beta. So that's what the cohomology in degree two looks like. So well, in degree three, well, it's actually, well, it's some short exact sequence, but actually it splits and you get this identification and in degree four, you get, again, similar things. So, well, there's some three pieces actually and the last one is just pi one and variant part. Well, this is just quadratic function of pi two with you want value than you want, and just look at pi one and variant part. Is this the E two page or a later page? This is a later page. Do you know which one? I don't remember. I think it's E three, but. Now, so this, well, so that defines a theory, topological field theory, which depends on a two group and this cohomology class and it's classifying space. Now you can ask, okay, what actually, well, what sort of properties does it have? Well, so it's in general pretty complicated, but one thing one can describe is, for example, observables in this theory. So if you look at, so in that graph written theory, the natural classes of observables are so-called Wilson loops, which are labeled by representations of the group G and there are some sort of dual variables which corresponds to Congee class classes on G. Now, let me describe similar things for this two group theory, say in dimension four, because it's kind of an interesting, physically interesting case. Well, again, there are loops which are labeled by representations of pi one, like Wilson loops, but also a different kind of loops with a label by elements of pi two. And then there are two kinds of, well, in the graph written theory, you would have just sort of surface observables, observables localized on two dimensional sub manifolds. Here have two kinds of loops so there are two kinds of surface observables. First of all, like in the graph written theory, there are magnetic sort of surfaces labeled by Congee's classes in pi one. But they also sort of, so these guys are dual to these guys. But then there are sort of different class of surface operators labeled by a special, well, it's a bit complicated. So they're labeled by elements of this cohomology group, simply because the way you do it, you start with your two connection and if you wanna define this electric surface observable, just restrict this two connection on your surface and then you need to sort of pick some class here and then integrate over this surface. So the answer is given by this cohomology group or explicitly is given by this, as explained before, by this alphanvarian characters and now you have to buy this class beta. I'm just, I'm a bit confused. So you have a sub manifold of your four dimensional spacetime and you're taking a cohomology class on BG, which is of that degree, pulling it back and integrating. All of these are of that. Yeah, but it's actually not the most general observable again because like if I did similar thing for say Wilson loops, I would only get sort of one dimensional representations of the group. So really what I should do, I should, when I have a surface, I need to, well, when I have a one dimensional sub manifold, when it's just representation, essentially I'm putting a one dimensional field theory, which sort of can interact with the higher dimensional diagram within here in ambient space. So here to become in general, I would have to choose a, when I have a surface, two dimensional surface in my four dimensional space, I would have to choose a two dimensional topological field theory, some four couples to this two connection. But it's kind of difficult to describe. I'm not sure how to classify such thing in the extra work. So I decided not to just sort of look at the simplified class of observables, which are just out of one dimensional representations of G. Okay, so this very explicit description of a class of topological field theories defined in every dimension, which depend on a two group and some cohomology class. So what are they good for? Well, basically, so in this paper with Sergey, we propose that this kind of theories describe a gap phase, which exhibit both confinement and Higgs, Higgs effect. So Higgs effect arises when you have some sort of electrically charged particles are present in the vacuum, sort of condensate, and the break, the gauge group down to subgroup. Or what confinement? Well, confinement arises from a condensate of magnetically charged objects. Say in four dimensions, just again, particles, sort of monopoles, but in higher dimensional, we'll maybe some strings, so forth. They'll always cut dimension, cut dimension two. So what's the interpretation of this data in physical terms then? So it's a cross-module, so what is G and what is H? Well, first of all, G is a subgroup of the UV group, or the microscopic gauge group, which is not Higgs, sort of this unbroken part. And now this map, this arrive from H to G arises in the following way. So this image is the confined subgroup of G, so it was normal. So the co-kernel is actually the effective long distance gauge group. And that's actually why this A and G is not really physical, because the important thing is what happens at long distances, and if part of the G is confined, all you see is the unconfined part of the gauge group. So this pi one, the co-kernel, is all you see really at long distances. On the other hand, the kernel of this map labels magnetic fluxes of etch-hoof loops, which have not been confined, you see? So when you have Higgs effect, then in general etch-hoofed, this magnetic loop operators, they sort of get confined. They don't survive at long distances, and the kernel is precisely labeled as those which do survive, okay? That's why the pi one and pi two are data of the long distance topological field theory. Now the diagram of the theory just corresponds to a special case when nothing is confined, so just Higgs effect, nothing else. But in general, when both Higgs and confinement are cooperating, we need to use this kind of two-group DQFT. So now there's another, so that's one application of these two-group topological field theories, so one can use it to describe phases of QCD-like theories. But there's another rather different application motivated by condensed matter applications. So, now recently the topic which has been very prominent in condensed matter physics is symmetry protected topological phases. So what's that? Well, these are phases of matter which are gaped and have short range interactions only and have some fixed global symmetry G. Now it is further assumed it's a global symmetry not a gauge symmetry, so they're no G gauge fields. However, it's further assumed that it's sort of on-site symmetry, it acts on every, only on variable, doesn't make variables living on different zero-simples, it just acts on each zero-simple separately. So, on each zero-simple there are degrees of freedom and then G acts separately on each of them. So in such situation, well, okay. So we would like to classify such phases of matter up to how much, up to the deformations of the Hamiltonian or the action which preserves all these properties like gaplessness, oh sorry, gap, in fact there's a gap and in fact there's a G symmetry. Now so it's not at all obvious that there are non-trivial, that it cannot always deform such a thing into a trivial theory. Actually, sorry, I forgot one other thing. No, sorry, I forgot one more condition actually. So I mentioned before that if you look at space time, at spatial slice of a general topology, in general the vacuum is degenerate. So non-trivial, the space of a vacuum is not one dimensional. That's usually where topological field appears. Well, actually if you just try to classify all phases with this symmetry G and such general degeneracy of a ground state that's completely unmanageable. That's like, it's worse than classifying topological field theories which you still don't know how to do. So let's look at the simpler case when there's no vacuum degeneracy. So topological field theory would be trivial. However, you still have this symmetry G which you need to take into account somehow. So this problem actually is manageable. So and phases of lattice models which belong to such classical symmetry protected topological phases. So they have property that they're gapped. The spectrum of Hamiltonian is gapped. They have a symmetry and there's no vacuum degeneracy on any spatial slice. So how do we classify such things? Well, first, the main observation is that since we assume that the symmetry, the symmetry G acts separately on degrees of freedom on every site, well, it's kind of easy to see that in this case there's no obstruction to gauging. There's allowing different gauges, transformation on different sites through no obstruction. Now, of course, essentially it's promoting your global symmetry to a gauge symmetry. Well, mathematically, you know, say that your transformation is now general zero co-chain with values in G. Essentially what it means is that there's a way to couple your system to a flat gauge field with structure group G. Now, in this case, well, and that doesn't change the fact that there's any ground state, therefore, one can sort of perform the path integral, integrate out all these degrees of freedom and get some effective action for this G gauge field. And the action at long distances must be topological because the system is gapped, not be topological. So that's a naive argument. Of course, it's actually very hand-waving. But if you believe this argument, then to every phase like this, one can assign an element of this cohomology group which classifies topological gauge theories with gauge symmetry G. So that's known as group cohomology classification or SPT phases, and it seems to work very well. So indeed, the all-known phases, well, the symmetry G classified by such cohomology groups. Well, there's some, physically also the case when G is a league group, also interesting one, then look at G with G's regard as discrete group. So it's subtle then, you need to specify which co-chains you're considering. Anyway, so what I'd like to propose is there's some generalization of the story. In the case when it holds a global sort of one form symmetry, so one form symmetry is a symmetry, well, which is just like a global zero form symmetry, which is given by a closed zero co-chain. Well, a global one form symmetry is a symmetry parameter, it's transformation parameter is by closed one co-chain. There's a function one-simplices, satisfying the closeness constraint. And with value in the sum of billion group H and when you gauge such a thing, you basically, first of all, you relax as well, you don't require this one co-chain to be closed anymore, but then also to make the action, to make the action in the very hand of the symmetry, you also need to introduce a two form gauge fields. So there is H valued field, which leaves in on two simplices. And to propose that this symmetry, such sort of one form symmetries, global one form symmetry, give rise to new classes of this symmetry, protect the political phases. They're protected by two symmetry and sort of the ordinary symmetry. So, and then by the same argument tells you, well, when you can integrate everything out and you're gonna get some effective action for this two form gauge field, and we know what they're labeled by, they're labeled by this cohomology group of the iterated classifying space. Now, so this, well, as I explained, this becomes really non-trivial only in dimension four or greater. So this new SPT phases really arise in dimension four or greater. Now, there are actually some examples of this. In our paper with Cyberg in particular, we give some examples of such theories, which have some massive phases of matter, which have this global one form symmetry and non-trivial, leave a non-trivial class, well, this class here, for them to turn out to be non-trivial. And as a result, on the boundary, this non-trivial behavior, like there's some, on the boundary must have a say, Chen-Simon's theory. Let me just make a few remarks. So first of all, in principle, well, I just explained how to use this one form symmetry to define new symmetry protected topological phases. But in principle, that's just a special case of symmetry. In general, we have a non-trivial, may have two groups playing a role where there are transformation, which leave both zero simplices and one simplices. So they probably lead to more SPT phases, but I'm not really sure how to construct any examples like that. And even more generally, if you're in D dimensions, symmetries should really be described by D group. If you're in D dimensions, space-time dimensions, your symmetry group should be really thought of as a D group. And these D groups, which are basically D categories with one object and all arrows and iterated arrows invertible should lead to even more phases of matter. And well, I don't know any examples like that, so I won't discuss them. And finally, well, if you come back to this, well, to this two group gauge theories on the quantum level, so for SPT phases, the important thing is the classical action of this two group, TQFT, because you're just studying how the action depends on the background gauge field. They're not integrating over the gauge field. But if you think about this two group gauge theory, which is described as a quantum theory, then you can ask, well, maybe they're not really new, and maybe they're related to the known ones in my Degraph-Widden theories by some dualities. Now, in some cases, that's actually true. So if you look at the paper and you see that, if you call homology class on this classifying space, it's usually simple. You can actually dualize your theory to an ordinary Degraph-Widden theory. But in general, that's not true. So in general, this is a new class of topological field theories, and it's worth to be studied also in higher dimensions because they might have interesting, for example, boundary behaviors, which can be speculated to be related, say, to say chiral theories in six dimensions, which have been so mysterious. I'll stop here. This is sort of a strange question, but can one use these things to create quantum computers? Well, if you can make, let's see. So in principle, yes, it's just usually people prefer to work with thermonics, similar to practical topological phases. They're sort of more easily constructed in the lab, still very hard to, not how they're constructed, but it's more realistic. So in general, in practice, well, this sort of field theories appear from, or SPT phases, or other, appear from, when you assume that you're basing building modes of bosons with some symmetries which are geonaxon them. So in practice, it's actually very hard to come up with the need. How do you make four-dimensional matter? It's three plus one dimensional. Oh, really? Yeah, it is nothing. It actually works. Yeah, when I say four-dimensional, it's space-time with dimension four. It's actually our physical dimension. So, yeah, that's fine. Yeah, so one can do that, and one can, in principle, just post a sufficiently complicated two group and then you'll have some quantum computer. But in practice, there's no way, I don't actually know of any physical, a reasonable physical system, which gives rise to, say, a diagram of within theory with some non-trivial, especially with an abelian group. The abelians ones need to be pretty boring. So if you want an abelian-gauge group and a trivial-cohomology class, that's it, I don't actually know any way to construct that. So, there are other, well, the variations of this stuff using fermions are more promising because it's actually more promising, well, it's clear how to construct phases with interesting class. In this case, well, actually, fermions need to use some spin co-boardism instead of co-homology of the classifying space. But anyway, so it's easier to imagine how to construct such phases out of fermions and out of bosons. In that sense, they're useful. Yeah, well, what's this fucking using about the true finite homotopy type as opposed to this homotopy type? Well, nothing stopping me, just easier to find physicalizations. Yeah, so if you look at general homotopy type, I just don't know any way to physically realize this topological field theory that's all. But are there any sort of physical guidelines to think about homotopy types? So, sorry, is there any other direction? So, suppose we know something about physical meaning of those things, can we conclude something about homotopy types? So, maybe there is some kind of organizing principle. Well, okay, so I don't really, for me, the space whose homotopy type I'm using to construct a field theory doesn't really have any physical meaning. Like, I take homotopy two types. I don't really know what, I wouldn't know what to do with the space itself, which has this two homotopy type. So in some sense, well, actually, Ketayev was selling me an idea that, you know, symmetry should really be, you know, you should forget about symmetry, just replace symmetries with actual spaces or homotopy types, right? Terayev, that's Terayev's idea. Okay, maybe it's Terayev's idea too, but in practice, I don't really see how useful this would be. Physically, I don't see any space around with this homotopy type. What is the point of the Ketayev function? What's it? That's not a Ketayev function. Anderson, what's Ketayev's function? Anderson, what is Anderson's function? I mean, you, in the first, I mean, start to talk to you about the real reason about the quantum function, like, spin content, I think it's that one. Did they mention spin content? I didn't, I don't think I did, actually. I didn't mean spin for quantum Hall effect. I just mentioned fractional quantum Hall effect. Oh, yeah. Well, it is not relevant, really. So, topological field theory, which is responsible for fractional quantum Hall effects are different kind of field theory, they're like transimus theories, which cannot be written as finite sums over some finite sets on some triangulation. And the reason they are, they're not really topological field theories because they have framed topological field theories, framing anomaly. These theories don't have any framing anomalies in any dimension, so they're a bit different. And I know about the local theory, they completely, no, no, okay, with these, I mean, with these two, with these two, what's the higher transimus theory? Then, the corresponding Anderson local data state will be, what? I'm not sure why transimus theory, there's not like transimus theories, they are, they are, there is like a higher transimus theory too, like in five dimensions. These are not the theories I'm talking about. They're like, well, first of all, they didn't get to five dimensions, but these things only make sense. They're analogs like a finite transimus theory rather than of normal transimus theory. So they're not related to higher dimensional analogs of fractional corner whole states. So if you're one of your slides, appeared that a group is discrete topology. So this appears in a fourier theory. So would it be possible to imagine this as a fourier space where you can travel in the space, but cause the leaves is more punished than go along the leaves? I don't think I can comment on that. All I know is that, okay. So it would be a very invariant, it's a typical class which appears in three converges that gives a few one. Well, that class, of course, okay. So that class does plays an important role as for the simplest. Is that just usual transimus action corresponds to that actually? So this corresponds to a prediction? Yeah, right, okay, fine. Yeah, so this class is, the classes like that just transimus classes. So in that, I'm not sure how to, in terms of geometrically just the sort of things. The variation, I think. Okay. When you replace the T group of symmetries by its homotopy type, don't we just get back to the standard sigma model? Well, it is basically, well, okay. So I'm just saying this is, for me, it's just formal construction. Like, I don't see any, well, the sigma model is not really physical. It's some infinite dimensional space is a target. So it's for me just a trick to compute partition, to figure out what possible weights are. Like in physics, even in the Degraph-Witton case, you don't really see a classifying space anywhere. It basically treated as a point moded out by the final code. Yeah. All right, so there are lots of questions about quantum computers, violations. Let's talk about it over coffee and thank Anthony. Thank you. Thank you.