(Non)-existent atomic limits: geometric meaning of K-theory in the solid-state
Thiang, Guo Chuan
It is known that a momentum space Chern class obstructs existence of good tight-binding models in position space. This is a T-duality, and holds in more general geometric contexts, e.g. crystallographic, defective, and non-Euclidean effective interacting topological phases. Namely, good Wannier bases correspond to free modules over pre-C*-algebras of the symmetry group. Thus K-theory and T-duality give precise tools to find topological insulators and understand their boundary gapless modes; explicit new examples will be given. Joint with M. Ludewig.
 I'm not actually a professor. I'm not Adelaide, a collaborator with Matai. There'll be one slide about what we did, but most of this talk would be on some new work. Right, now that everyone's seated, I should also thank all the organizers. Everyone hears that. And for this opportunity to give a talk with another collaborator who's at Adelaide, so he's a geometric analyst and index theory person, amongst other things, in the last year and so we discussed a little bit about k-theory and topological phases. And then I encountered a situation where it was a newcomer. It was completely impossible to explain why anyone would look at k-theory. So one of the goals of this talk is, well, at least for physicists, to demystify in a very concrete sense what k-theory means in a solid state. So I said solid state in contrast to condensed matter, just to say actually everything about here, I'm going to say, is kind of in the good old quantum mechanics world. So it's really about single particle Hamiltonians. Nevertheless, there seems to be a very important aspect that isn't stressed very much. In particular, I'll talk about one-year functions. There'll be a talk on either Thursday or Friday also on one-year functions. This is a very important aspect of, at least to me, of topological phases. So this will appear as part of solid state theory. And one of the things I really want to do is to apply to very general geometric settings. And I also want to talk a little bit about t-duality. So that's actually my primary interest, is the role of duality in physics. Why dualities? Why different people giving different explanations for the same phenomena end up talking about the same thing? So I'll show why some pictures of the same phenomenon is actually the same, and there's actually an underlying duality which can be identified with the t-dualities that people talk about in string theory. For mathematicians, if you come from the operator algebra, C-style algebra, and K-theory, you know one of the biggest machines to produce research is the bump-corn conjecture. And so what does it mean? I'll give a candidate physical interpretation of the assembly map. And also, mathematicians might not, actually, in fact, really know what the topological phases are. So I'll try to have one, at least one notion which makes sense. In a K-theory context. Right. So this workshop is, of course, about interacting quantum systems. And I want to say anything about it until the very end. And why that is so is because, well, I'm no expert in many-body physics. But I hope to be able to say something general about single particle Hamiltonians in general geometries. And maybe it has some lessons for our kind of exuberance in using abstract nonsense to study interacting regimes. So by abstract nonsense, I mean the big machinery of categories, very abstract things. We do need to justify some things. That's why I want to say. Right. So here are some general comments. From the experimental point of view. So how does one detect a topological phase these days, apart from the quantum whole effect? So if you look at a wild semi-metal, you never see anything in a bulk experimentally. As far as I understand, correct me if I'm wrong. And you actually infer some bulk topology, some abstract bulk topology, by looking analytically for zero modes, localized on the boundary. So when you say this to a mathematician, he immediately sees that there's a physical experiment that combines some analysis, some geometry, and some topology, living in the background. This is nothing but index theory. So the physicist's belief in a bulk boundary correspondence is essentially a statement that there must be some index theorem, a region in which several different flavors of mathematics come together to actually talk about the same thing, and which aspect you want to look at is kind of your preference. So it's a rather deep idea, and in fact the physical phenomenon has been able to generate new conjectures in terms of index theory. So this was the case for the quantum whole effect, for example, formula, con, muscovici, that is, and so on, as an example. Now, if you do band theory in solid state physics, the usual process is to go to momentum space. Quantum space is nothing but the duo of a billion group of symmetries. It's the Pontiagin duo in mathematics, and allows you to do some kind of harmonic analysis. And then you construct some topological invariance for back to bundles, for example, for the bundle of field states in an insulator, for example. And then you say, ah ha, vector bundles can be non-trivial, topologically non-trivial. Of course, so why didn't anyone realize this? Well, who knows? But we would also like to have a position space picture. Now, it turns out there is kind of a new, maybe not so new point of view, but as a gaining popularity, which is that what you call topological and momentum space, should have an analog in position space as a kind of obstruction to produce an atomic limit. So this is roughly the idea of a topological quantum chemistry. That is when molecular orbitals, that sort of story, fail to make sense. And there seems to be a topology and localization kind of a duality, which actually there will be a talk about this in much more detail. Now, this link is actually known since 2007, I believe it was conjectured already by Thulus a long time ago, in NCO and so on, that in the Euclidean case, the usual lattice, everyone's favorite lattice, of translation symmetries, then you can actually prove that what the class means for a term used later is that there is no good atomic limit or one-year basis for the band of filled states. So it's very delocalized. Any one-year basis you want to draw is very delocalized. I'll show you a picture later on. So this is kind of a duality between two stories. So on the one hand, there is now a definition of topological phases, a rough definition on the left side, and then there's a one on the right side, which is amenable to a lot of k-theory tools in momentum space. So why did it same thing? Well, in fact, there's a nice duality that implements such an equivalence. So here's one of the big messages I want to say. We've heard a lot about lattice models and continuum and maybe a distinction between the two. But from the point of view of solid state physics and one-year functions, actually they're not dichotomous. So let this model just a set of labels, sometimes labeled by a group, and the group is permuting these labels to get some structure through the group. We actually want to embed it in some metric space, usually to have some notions of distance to say, what are the couplings between sites, etc. So that's secretly a little bit more structure. And when you embed it in this spatial manifold in the background, of course you would like all the degrees of freedom to be very localized, like little delta functions, localized atoms. But it may actually look like this instead. It can still be orthonormal. And really, from the point of view of this result over here, absence of atomic limit, the question is really about how a lattice model, which are abstractly isomorphic as Hilbert spaces, how they actually sit inside the continuum model. And I'll give an answer to this, which is precisely measured by K theory. There are two aspects of it. One is the homology theory and one is the co-homology theory. Another advantage of this duality is you don't actually need a momentum space to study topological phases in the sense of absence of atomic limits. So the time binding models on a one-year basis, you could make sense of it for any discrete group acting in a sensible way on any physical spatial reminder manifold, X. And this allows us to liberate ourselves from the good old place where churn numbers live, where you have a billion lattice acting isometrically on Euclidean space. Actually, there's a whole big world of Riemannian manifolds with curvature and there are lattices in them. So for example, this is the Poincare disk. It's a hyperbolic plane with negative curvature. And if you put lattices here, you actually get a lattice of a Riemann surface. So the lattice is labeled by the fundamental group of the Riemann surface. So the hyperbolic plane can be touted by all these Riemann surfaces. So you identify pairs of edges like this. And then you get a non-nobelian lattice sitting inside the isometry group of the hyperbolic plane. People talk about defects. So here's a defect. Here's a screw dislocation. In fact, in a screw dislocation, if you have some density of screw dislocation, actually what happens is that the effective geometry becomes non-Euclidean. It gets some kind of new geometry thing because you see when you go horizontally, the horizontal translations don't close. They close only up to a vertical translation. So there's a density of this, so it's like Heisenberg commutation relations between translation in x direction and y direction. So here's an lattice. This is called the discrete integer Heisenberg lattice in R3, where R3 is no longer regarded as Euclidean R3 but as a Heisenberg group manifold. So you see the geometry is very important when you want to say how the lattice is embedded into your actual physical space. So I'll go, I'm going to talk about band theory. And maybe before the term topological quantum chemistry came about, there was just a general idea or even statement that energy bands, which you are familiar with, is nothing but a regular representation of the lattice. So it fully transforms to the L2 sections of some vector bundle. So in a fully transformed space, you've got a band. You can have an orthonormal basis of sections there. And it fully transforms back to the regular representation. So you don't know what regular representation is. It's basically a tight binding model. So suppose you've got a lattice labeled by a group, say Z. And then you've got a degree of freedom everywhere, a copy of C. So the space L2 of Z, it's not a bare Hilbert space. In a tight binding model, it has an action of Z acting by translations on it. Why does it have an action of Z acting by translation or exponent minute? But I just want to say that from the point of representation theory, this is a very natural thing to think about. Now, the advantage of this formulation of an energy band is that it's easy to analyze. It will be much relevant if you replace it with something else. Oh, sure. So for example, the Poncares thing that I drew just now. So you tau the hyperbolic plane by some Weeman surfaces instead of tau, right? These are analog of unit cells. So these are your lattice points. And the lattice points, if you choose this with your origin, then the other lattice points are labeled by elements of the fundamental group of the Weeman surface with genus G. So you have a degree of freedom here, and then you act by this lattice. You get another degree of freedom here, and so on. So actually the Hilbert space, the so-called tight binding Hilbert space, in fact, people have done tight binding models on this space. Nothing other than the regular representation space for this group as a set. So that brings to the second point. So the lattice authority will be a billion. And in fact, in the C-style algebra, there's actually a whole theory of non-commutative block theory. So certain kind of operators on the covering space, which have some regularity condition to make sense of energy bands in this sense. So I might not be wrong with the authors of these kind of theories. It's probably Bruning, Sunata, one of the first people to do it. Gruber wrote a paper which is literally titled Non-Communitive Block Theory. Mathai McCully applied it to the hyperbolic plane, for example. And so it's a rather established theory. And the Euclidean case, of course, we actually want to do crystallography. So even the lattice is actually not the billion lattice. It's actually an enhancement of the billion lattice into a crystallographic space group. So you have to be able to somehow do non-commutative band theory. So this idea of energy bands being a certain number of copies of lattice models has been repopularized with the adjective topology. And so now it's called topological quantum chemistry. So there's a nice paper in Nature by Brett Lin and his collaborators in the US. So the upshot, the mathematical thing I want to say is you start off with an L2 space on X, where X is an action of a lattice. So this L2 space is a representation space for the group. And then if you want a discrete model, so you take a point and generate its orbit, so you have a lattice model. And then you have a subspace, a subrepresentation space. But you have to remember that this subrepresentation space is secretly made out of actual wave functions with tails and so on. And you can actually embed the lattice model in very different ways. And there are different ways in which that embedded is precisely the topological nature of this embedding. So this regular representation, of course, these are the same abstractly as Hilbert spaces, but sitting inside the background is completely different. And so an application of this kind of ideas is that using a new mathematical technique that I'll introduce today, we actually obtain the rigorous proof that a wallpaper group, so this is a very interesting two-dimensional wallpaper group, which I'll explain in more detail later, it can never admit any topological beds. So any type-bending model, the type-bending approximation is valid. You never get a situation like this. Now, it might sound like a stupid thing, but for many years people were trying to even prove this for the Euclidean lattice case. And as far as I know, there are no other techniques to do this sort of thing. Okay, so do a little bit of physics. So why? What are wave functions? Wave function is the Hilbert space for the electron. It starts off as a free electron. It sees the Euclidean geometry, so the non-relativistic free electron. Listen, L2 of Rd. Why L2 of Rd? Because the Euclidean group is acting, is being represented by our summatries of the space. This is the point of view of particle physics. So free electron moves around and then is actually sitting inside a material with a discrete set of points. So the canonical differential operator, the Laplacian, gets perturbed by a gamma periodic potential. And so the Euclidean symmetry is broken into a discrete crystallographic subgroup. When this is broken, nevertheless you have to remember that it retains the canonical representation inherited from up here. So for the simplest example is the lattice. So as I said, the lattice is secretly sitting in Rd. So when you break the symmetry, you remember that the z is still acting by translation and not in some random way. For example, you can embed this in Rd and you can consider some very stupid action of z on this set as a representation. So that's not what you want to consider. You want to consider it in exactly the regular representation to make sense of the type-binding model. And then what you do, because there's a billion into a Fourier transform, Fourier transform of the lattice's circle is a space of characters. In fact, a topological space in a canonical topology. And this is where the topological momentum space comes in. This is the Brillo-Anne zone. Indeed, I mentioned this would be a d-torus. Then the Bloch-Fouquet transform is kind of a partial Fourier transform, which allows you to transform operators acting on L2 of Rd. You make use of the discrete transational symmetry to partially Fourier transform. And the basic thing is you reduce, because of the periodicity, you reduce the Schrodinger equation problem to one which is living on a compact unit cell. It's here. And mathematically, you call this a fundamental domain. But you will agree that this is a circle, T. I'll explain later on why I call this T and not the other T. This T is the Brillo-Anne-Torus. It's absolutely not the same thing as this unit cell. And that's the whole source of the duality. Right. Of course, then you don't have actual periodicity, but you have some quasi-periodicity. So you have to say, you look at a space of block wave functions. So wave functions are said when you translate by one unit cell, you acquire a phase factor. And a phase factor can be any number in the Brillo-Anne-Torus. So these are block wave functions. They are quasi-periodic wave functions. They are not normalizable, but they are normalizable over a unit cell. So here's the condition for a block wave function. Everyone knows this, hopefully? No. Everyone who's not seen a block wave function. Okay. And the space of block wave functions for fixed k, quasi-momentum k, is identifiable with the L2 space on the unit cell. So altogether, after a block-locate transform, you really transform the space of the L2 space of wave functions into a bundle of Hilbert spaces living over the Brillo-Anne-Torus. And the typical fiber is this infinite-dimensional Hilbert space of functions on the compact unit cell. And then a general wave function decomposes as this direct integral. So this is the superposition of a wave function in terms of its block, in terms of its quasi-momentum. So each component is a k-quasi-periodic block wave function. So why does t have a topology? Why does the momentum say it's a topology? Well, I think Raff gave a good explanation just now. And it's because of the Fourier transform. And that's where the representation of z is important, because the translation operator in z after Fourier transform becomes multiplication by the continuous function, which phase function which winds around the Torus once. I mean gamma number of times. So the action of the transition action actually in momentum states generates after the norm completion, multiplication operators by continuous functions. So this is why the transition symmetry is essential to make a link to the topology, the usual topology of the Brillo-Anne-Torus. So those are our words. Here's a picture you get from an experiment. A band structure, lots of bands, band gaps, gives you an insulator if this is the Fermi level. So you disperse the energy spectrum of H according to k, and at each k because you have Schrodinger operator on a compact space. And these are Laplace-Tag operators. You get some elliptic regularity discrete spectrum. And then some other argument says you have continuous energy bands. So in total, if you're restricted to some finite, oops, messed it up. The spectral subspace between some finite bands, then the Hilbert subspace corresponding to those energies will produce a sub-bundle. Of course, it has L2 sections, but you can also consider just the continuous sections. And this gives you a vector bundle of field states. So there are two things to consider. There are the L2 sections of this block bundle. So there's a Hilbert space. And after the Fourier transform, you get back for each band one copy of this lattice model, the regular representation. But the block Fourier transform remembers how this representation is sitting inside the actual space of wave function. If you take only the continuous sections of the block bundle, then if you invert this block Fourier transform, you only exhaust a dense subspace of this Hilbert subspace. You might need other vector bundles, but not some of line bundles. What you have done here is only the trivial line bundles and nothing more complicated. No, no. Even if you have a non-trivial line bundle, the L2 sections, you can still generate. You have the non-trivial vector bundles, which is the sum of... Yeah, yeah, yeah, this is for simplicity. Of course, if you have more bands, you have n copies of this after you choose a frame, an orthonormal L2 frame or continuous frame. If you can choose it. If you can, yeah. If you cannot, then you have a non-trivial bundle. Exactly. A non-trivial vector bundle. That's what it's about. Yeah, yeah. So that's your pre-empting what I've got to say, which is good. So the inverse block Fourier transform of a general L2 section, as I was saying, because it's not very regular, this is called a one-year wave function. And then the one-year basis is basically translating this initial wave function under the translation action. And it will generate, in the Hilbert space sense, an orthonormal one-year basis, which I label gamma star w. So this is the pullback of an initial one-year function for the band. So as an example, you start off with a band, you pick this nL2 section, you'll get at an origin, some one-year wave function which you call w. OK. And then you generate the other degrees of freedom by the translation action. So if this is w and gamma is the generator of your group, then you translate by gamma, you get the next degree of freedom, you do it twice, you get another degree of freedom and so on. And then the whole question is whether this w can be made, whether you can choose sections such that the w can be controlled, the tails of w can be controlled. And there are obstructions to this. So as Raph was saying, and as I mentioned also in the introduction, unless a global continuous section can be chosen, the one-year, the inverse block Fourier transform, so the one-year wave function, we have very large tails. This is from Fourier analysis. So topologically, non-trivial block bundle, in general, you would expect this relation that you produce inevitably a bad one-year basis. So in 1D, as I said, for a long time people tried to prove you could localize these one-year functions exponentially and so on. Finally, it was proven. In 2D, it turns out there's an obstruction, and this is your favorite turn-cast obstruction. And initially, this was viewed as a bad thing. So the idea was to try to produce conditions under which this obstruction disappears. Today, we have the opposite point of view. We love these obstructions, and we want to find more and more obstructions in order to have more and more topological phases. So let's not try to get rid of this, but to find more such obstructions. And you can find many of these obstructions in more interesting lattices. That's one of the other messages of my talk. So in general, if you've got some space and you've got some lattice model living in there, so you've got an orbit of some generally non-Abelian group, then if you have a Hilbert subspace of the space of wave functions on that physical position space, then a one-year basis would be an also normal basis for that subspace, which is of this form. So you have a finite bunch of initial WJs, one-year wave functions. For example, you can imagine if you have two bands, then you can choose W1 and W2 like this. And then similarly, you generate the bands, the degree of freedom of the bands in that way. And then you generate an also normal basis. So sometimes it's got a composite one-year basis because there's a finite label J. Do you need a civilian group? I deliberately not use numbers here like N because N suggests the Abelian group Z. So you don't need a blocked group? No, no, we don't need a blocked group. But it exists in the non-committed sense if you wanted to go there. Is it true that another message would be that if you enlarge Z to a space group, including reflection, then there will be obstructions even in quantity? There can be obstructions. You have to make sense of what you mean by the, that's a detail and actually requires quite a lot of time to explain. But yes, so even in 1D, you can get more obstructions for crystallographic space groups. So here's the main theorem, the main message in an imprecise form first. The precise form is a mix, but a useful message is a walking definition-free theorem. It's supposed to have got a discrete group gamma and it's kind of a quote, I'm quote, crystallographic group acting on space, manifold X. And that H be a sound sensible Hamiltonian with, so you're thinking local differential operators, Laplace-Schrodinger type operators, that's the sort of sensible thing you want to think about. Which is gamma invariant, so invariant under the action of gamma on this space. Then, based on the canonical, there is a canonical subspace of wave functions which I call OK wave functions. So these are wave functions, they're not so bad. They're better than L2, but they're not really good. So they will look, in practice, something like this. You may have tails, and I'm being generous here. So the tails might end up here. OK, so the decay is really bad, but not as bad as L2. This is roughly speaking the Fourier transform of the continuous functions, which Raph was alluding to. It at least contains the functions that have some kind of L1 decay, but it's a little bit more. But I can't say what exactly it is, and that's one deficiency in this. But the good thing is it's canonical, and this construction is actually very closely related to the construction that one does in the assembly map, in the sense of bumpcon, which I hope to get to at the end. Now, so the point is, OK, you have some Hamiltonian, you have a spectral subspace that you want to inspect for the energy line between some gaps. And then you want to say whether you can choose inside some of these OK wave functions, whether you can choose the so-called continuous sections in order to produce nice one-year functions like this. Nice one-year functions. And of course, it turns out, so you have to look at the module generated by these OK wave functions, because these are all the possible candidate acceptable wave functions that you can admit inside a sensible one-year basis. So you can't look at the full Hilbert space. You want to restrict in here. And it turns out that this is actually a finitely generated projective module over the group C-style algebra. That's just a long-winded way of saying it's a block bundle, continuous sections of a block bundle, but in a non-commutative sense. Now, if this module of generalized sections is trivializable, that is to say it's freely generated, then the spectral subspace admits not just an OK one-year basis, but you can actually push this to a good one. In fact, what we can achieve is to push this to a short decay condition. And conversely, if this module is non-trivial, then any one-year basis is necessarily extremely bad. And when you encounter an obstruction which you can measure here, non-freeness, then the one-year basis is going to look like you can only ever produce this sort of type binding model. So here's the main theorem for the mathematicians. I'm not sure I'm going to go through everything, but the main assumptions are the usual ones that you see in this business. This great countable group, some growth condition on the group, x compact fundamental domain. The action is properly discontinuous so you don't accumulate at certain points, your finite isotropy, et cetera, effectively so you don't have redundancy in the remaining geometries in this part. And you let the consider an operator, which is gamma and baron, and you want some regularity. So it can be first order, you take some kind of Schrodinger type operator, second order of the plus type. Then you look at the spectral subspace for compact separated part of the spectrum. So you project into the subspace for lying between some energy gaps. So the statement I want to say just now is that there is a canonical dense subspace. This is not a Hilbert subspace, it's dense, it's a proper subspace of OK wave functions which I call L2 of gamma, x, which is countably generated and it's a Hilbert C star module. So hopefully people who work in C star modules like these words. And then when you intersect this space of OK wave functions with your spectral subspace that you want to inspect, you get the module of continuous wave functions which are the ones that can help you produce the nice type binding models. So it's actually a finitely generated module sitting inside the space of acceptable wave functions. So if this module is not free, then you cannot generate a one year basis. A free module is basically you pick a generator and then you do the group action and you generate this Hilbert C star module that way. And if it's not free, then what I just said just now doesn't make any sense. So you cannot produce a good one year basis. Now in order to actually be saying something substantial, I need to tell you exactly what good and bad is. So I would just say, as I said, the criteria for bad is not so easily characterised. So this space of OK wave functions actually describes a decay condition at slightly weaker than this kind of L1 decay condition on the restricted L2 norms on the fundamental domains. So it's really bad. So in other words, if you have some topological obstruction from this module construction that I described, then you'll be forced to choose a one year basis from outside this space. And so the decay condition has got to be worse than this. So it's going to be even worse than what I've drawn. So it's extremely delocalised. Any kind of sensible atomic does not exist in any sensible sense. Now with a billion groups, you can do a lot better. And you can actually, so I will leave this to Professor Farnati's talk. Basically you can improve what you mean by good and you can worsen what you mean by bad. And you can actually characterise what, in position space quite nicely, what these conditions are. Is the exponential, do you mean faster than sum or any exponential? So I didn't reach exponential decay. I'm not sure. You should ask. But then sum. So that's some kind of ochre principle which I think does not exist in. That's a ochre. That's just a sub-adverb of a better kind. So I think that is... Right, right, right. I think it's going to show the key like that. I would, we have to talk about it later. Anyway, maybe I'll just move on and... That will be good. We have to go for our tour on time. So basically the main point of this approach is to use non-communist techniques and to apply to interesting lattices. So we don't do the obedient lattice over and over again. However, we cannot reach these nicer conditions because it requires some extra techniques which are not generally available in the non-communist world. But for me, I think that the condition of good being short is reasonably good. So when there are no topological obstructions, you can produce shorts one year basis. That's reasonably good since as fast as possible there are no other general techniques to handle the non-obedient lattice case. The shorts or general growth is not... Sure, sure, sure. And that's why over here there's something here. Only polynomial. Absolutely, the growth condition is important. Otherwise you cannot relate. Yeah, so I haven't thought of all the possible generalizations and permutations because this just came out a couple of weeks ago. So yeah, we can discuss generalizations. So I noticed no one defined what a C-style algebra is. We can clarify. So this, the putschian... Oh, it doesn't apply there. So here's one situation where the geometry suggests that you have to really look carefully at the geometry because the conclusions cannot apply to arbitrary lattices. There's some analytic condition because of the growth of the group. So it's not necessarily a bad thing because if you're able to say such a powerful statement about any lattice in any remuneration manifold, that seems too good to be true. So here's the preliminaries. I want to get to this thing, this mysterious thing, group C-style algebra, which appeared variously but no one defined. So I'll just say that it's a very simple thing. So if a discrete group, you can form formal linear combinations of the group elements and then you take the convolution product. So this is a vector space generated by the group element and then you just convolve it. It's called a group ring. It has a product. The group ring acts as bounded operators in the regular representation. So you just take the basis elements and you just translate by the group action and then you add formal, so they are like hopping terms, finite hopping terms and then you add linear combinations of these. So these are all bounded operators because you just finite many of these translations. And if you close this in the operator norm, this is the group C-style algebra you get. And the basic example is if you take our favorite lattice, then the usual Fourier transform gives you the space of continuous functions on the dual. But this is the kind of philosophy. For any non-Abelian group, there is a non-commutative dual space and the group C-style algebra is supposed to represent in some sense modulo technicalities, the non-commutative space of continuous functions on this dual momentum space. And then we want to look at smooth sub-algebra or sub-functions. So as some of the earlier questions were saying, how do you improve the regularity? So groups have something called length function. So you can think of it as, for example, the number that you obtain, some word length thing. What's the smallest word you can use to produce that group element on a generator? And get a number. And using a length function, you can define a smooth group algebra in the following way. It's called h infinity gamma. The notation is meant to suggest the usual algebra of S-differentiable functions on the dual. So there's a C-infinity function on the dual. So the precise conditions like this. So here's where the growth condition needs to be controlled. But it usually depends on the length function. Sure, sure, sure. But when L has polynomial growth and... I use a very strong condition which is not necessary at every step, but it's necessary to apply to all steps. So yeah, so in the end it doesn't depend on the length function. So if you apply this definition to the usual lattice, then you want to get the C-infinity function on the dual torus. Then the continuous or smooth sections of this block bundles, if you add the adjective non-commutative, so they're going to live over the non-commutative momentum space. They correspond to what are called Hilbert modules over either the group C-style algebra or the smooth version. And the way... I'll go through these slides very quickly because you've never seen it before. It doesn't make any sense. And for those who are interested in how you actually construct this module, you might want to see the details. So as I said, sitting inside the full Hilbert space of wave functions, it's a canonical Hilbert C-star gamma module associated with gamma action. So what you do is you first look at compactly supported functions, and you define an inner product which is valued in the group ring. And this is defined as follows. So you take a compactly supported wave function v, the inner product with a compactly supported wave function w is a sequence of overlaps of the shifted versions of the first wave function with the second wave function. So you get a collection of numbers telling you how these two wave functions are overlapping. And because they're compactly supported, this lives in the group ring. And then you do some completion procedure to produce the Hilbert C-star module, which is the space of OK wave functions that I talked about. Also relevant for us, certain wave functions of rapid decay is the definition, and the Schwartz wave functions, which are even better. There's some control over the norms defined by the derivatives. So the lemma is, because as we know, if you just use the continuous functions, the group C-style algebra, then you can't really characterize very nicely what the position space story is. There's some abstract completion or some deadly behavior class of wave functions. So we would like to be able to push it through the smooth algebra for which actually the position space meaning is exactly the space of wave functions with rapid decay like this. So that's also a module. This space of wave functions is actually a module over this H-infinity gamma. And then under mild regularity assumptions, a proposition is that the Hamiltonian, whenever you produce a spectral projection of the energy subspaces, the subspace will have the property that its continuous sections is finitely generated projective in the non-communitive sense. And also it's a smoothing operator, the projection operator. So if your W is already of rapid decrease, then you apply the projection operator, you actually get something in the Schwartz class. So this is how, while using this growth condition on gamma, we can push the finitely generated module condition from OK wave functions to those with rapid decrease and finally all the way to Schwartz wave functions. What's an example of that? Should we use a type operator with... We always have a spectral projection of that sort of piece. Yeah, I think in z squared. So it's like the elliptic regularity story on the block. You need multiple bands to have gaps. Oh yeah, yeah, yeah, so there's an assumption that there's a gap. Did I mention a gap? Oh, sorry, sorry, of course. Yeah. Sorry. Yeah, yeah, yeah, it's a gap. Sorry, it's between some compact sets. This example that you consider is a hyperbolic space. Do you have gaps there or not? Yes. There are gaps all below or something like this. Because of the twist, they have to do the twist. Okay, right. So anyway, the main message is if you forget all these technical uses, the good one-year basis existence problem for energy bands is equivalent in some precise way, if you're bothered to read the history above. So the study of the module, so not of the spectral subspace, but the spectral subspace intersected with some suitable space of candidate wave function. So it's a dense subspace of your spectral subspace. It's equivalent to the study of dense subspace as a finitely generated projective module over the group algebra. And whenever it's free, and free-ness means exactly that you can choose one generator from inside this good module functions, and then generate a whole Hilbert space, also normally, by the action of the group. So that's why you have this equivalent here. So here's a summary of the group algebra approach to a one-year basis. So you start with some localization criterion. You build a module of acceptable wave functions, which show that it's a module of a suitable group algebra. If you have a spectral subspace you want to inspect, then you look inside there, you look at acceptable wave functions inside there. They will in general produce from a finitely generated projective module. And then you can ask the fundamental question whether this module is free, but it has a finite set of free generators. If you don't, well, then unfortunately you cannot produce a one-year basis from this space of good wave functions. You have to go outside. So you have very, very bad decaying atomic descriptions. So you start from the Hilbert space level. If you want these modules over the von Neumann algebra, group of von Neumann algebra, it is completely useless as Ralph was saying. It doesn't see any topology. You want to do k-theory. k-theory sees these two parts exactly. So there's no distinction between these two parts, roughly speaking, for an interesting technical reason why you can push the k-theory story between these two conditions. And finally, actually there's a little paper of Nicholas Reid in physical review, Bean 17, and titled, Compactly Supported One-Year Wave Functions in Algebra K-Theory. Basically he did a baby model of roughly speaking of what I was saying. If you wanted to produce one-year basis of compactly supported wave functions, then you have to look at the modules over the group ring. And the reason is because compactly supported wave functions live only on a finite set of position labels. So their Fourier transform is just a bunch of, it's a finite sum of Lorentz polynomials. And that's only a, because finite sums, they are, these are preserved only by finite sums of transition operators. Finite helping. If you go to a bigger algebra, you will not preserve this compact support condition. Right. So can you tell in terms of classification which one is bigger, which one is smaller? Oh, these are all inclusions. So this space, this in here, this space, this in here, this space, this in there. These are the OK wave functions, these are rapid decay wave functions. We are able to push this to a short wave functions for the spectral projections of... There are more topological phases. Topology enters here. So the k-theory of these two algebras are isomorphic. So the topology sees, can see conditions of the... I was asking in relation with this algebraic k-theory. Oh. Is he finding more... The k-theory here is completely different. So this doesn't see the topology. Is he finding more states, more topological phases or not? No, it's not even topological because this is kind of a... More phases, let's say algebraic phases. If you consider compactly supported Eigen functions to be sensible, you might want to look at that. But I also heard some people say realistic operators never have... The algebraic k-theory for groupings is pretty small because these spot predictions things need some tails so you don't have to... You have just two things in some sense. And then write these. Right. So actually I haven't said any more k-theory yet. So this is the first slide about k-theory. Particularly if you have zp then there's a theorem saying that any module that is productive is free. I think I'm kind of running short of time. So I'm finding I need to go too quickly to get to the interesting bit. So once we've found a mathematical problem of trying to measure the freeness of some finitely generated projective modules, which is a module of candidate sections that allow us to generate the linear basis, then the most powerful technique is k-theory. k-theory exactly measures freeness, or actually stable freeness or otherwise. But I want to emphasize that k-theory is not just about this thing. It's not merely that. k-theory actually has many, many aspects, which justifies why it's useful. So we don't want to really just look at one aspect. And there was some complaints sometimes about stabilization. Stabilization corresponds to adding free modules. And what are free modules? They're just an extra band of tightly bound things. So low-lying orbitals, which you didn't account for in your full model, they are 3D generated, almost by definition. So you automatically get some sensible notion of stabilization. However, I will give one completely new example. Actually, I already mentioned it, which is that wallpaper group PG, where you can prove you don't need any stabilization, even. This uses some non-trivial techniques of a refill about stable rank. So so far, my whole story, if none of my messages went through, I would like just to say we've explained what k-theoretic abstractions built on momentum space means. Now, it turns out, if you compute the k-theory, you can also ask the question, are all of these spurious? Maybe you computed something which has no physical meaning. Well, it turns out there's a duality, a kind of momentum space, position space, duality. It guarantees that anything that appears on this k-theory is actually indeed built up from some position space data. So you're not doing anything redundant. You're doing something quite precise. So here's a wallpaper group PG. I promise it's got something to do with the climb bottle, which is half of a torus. So the interesting wallpaper group, the torrent-free one, is PG. It's the fundamental group of a climb bottle. So it looks like this. So you start from a point, an arbitrary point in space. You generate the orbit of PG. What happens is you think of it as a bunch of people. So this is the left foot and right foot, and they're walking along the line. This is another person. So it's the footsteps. So we're doing it every day by walking. As long as you don't stop, then you're kind of, no. If you're walking, you generate this pattern. So that's a PG group. It's a Cayley graph of a PG group. You notice it's torrent-free. It's got a finite point group by reflection, but the lift. This is non-symorphic. So its lift is not torgent. It's in fact torrent-free. It's a glide reflection. So you start from here. You do the reflection, and you've got to do a half translation. So if you take the fundamental domain of this section, you get half of the torus. Because after you're quotient out by the usual lattice, you still have to account for this glide reflection. And we identify points in this position-space torus. By this glide reflection, you end up exactly with the climb bottle. So I claim you can calculate the topological phases by calculating the homology of the climb bottle. The homology of the climb bottle is very easy to calculate. Here's the homology of the climb bottle. So you look at the unit side, you calculate its homology. Because it's unorientable, it does not have a second homology, the fundamental class. So it only has degree 0 homology. And this coincides with the K-theory of the-sorry, K-homology of the climb bottle. So it's just Z. And then I wave a magic duality 1, I'm going to go in magic 1, and claim this actually computes. The group of obstructions that we argue is relevant, which is the K-theory group of the C-style algebra generated by your lattice. So what this computation tells us, if you believe this magic, is that there exists only stably free modules, in the PG case. And then you have to invoke a very non-trivial result of REFO where all the non-trivial free, finitely generated modules are already free. So converting to physics is a proof that if PG symmetry, there are no topological bands whatsoever. The one anthropological invariance is the number of bands. Yes. If you would like to consider a topological invariance. So this Z measures the number of bands. And all of the bands are good. Right. If you're interested in torsion, you can go up to K1. You do the same computation you can discover. There's a Z mod 2 phase. It's a chiral symmetric Z mod 2 phase. And actually for a little paper with Kiyonori, one of the collaborators in this project, we actually prove a mod 2 index theorem, which mediates a bulk boundary correspondence. I can tell you more about this for interest. Okay. So the last few slides are kind of a philosophy, if I may. So why did I look at this? So I went into this area with a completely different viewpoint and I've emerged somehow looking at things very differently. So my motivation was really to try to understand why different arguments coincide. So for example, in the quantum Hall effect, people talk about flux torus is roughly the momentum space story. And then there's also a story where you build something in the position space and then there's another torus living in position space. And of course the thing in our head is to remember these are completely different torus. And yet they can give you the same result. It sounds like T-duality. So one day I met Matai and told me, why don't you try to T-dualize everything? You get something new for free. So I think you do get something new for free. So the T-duality principle says anything you can, any theory you construct on one torus has a counterpart in the other. Of course it doesn't say whether this link is physically meaningful. So you might expect this to be physically meaningful if there actually exists a single higher theory. And this is in some sense the M-theory idea, above the different type of type one, type two, A, B string theory and so on. There's a bigger correct theory and you're just looking at different facets. It doesn't exist. Which doesn't... Wish, wish. Yeah. Okay. And actually that really happens in solid state physics. In fact, the notion of T-duality appeared in the 60s in the paper of Cartier about lattice representations of canonical commutation relations. But there's another story. So let me name the block wave function. A block wave function has both a momentum coordinate and a position coordinate. And each of these coordinates are defined on torang, but different torang. And we also note that the position space unit south torus has the inverse radius to the momentum space torus. So they really are T-dual even in the geometric sense. So here's the big grand conjecture. It's the bump conjecture. And it has generated a lot of research in the mathematical community which proved this mysterious isomorphism. If these things are meaningless to you, think of this as homology of a universal position space. And this has K-theory of the momentum space associated to some discrete group. And then it says that bump conjecture says if you calculate the homology, a certain type of homology on the left side of some universal position space, you can actually assemble the homological data into a K-theory class in momentum space. And this has the interpretation of a geometric Fourier transform. So many people would call the right-hand side the analytic side. Although in our business, you might call it a topological side because it's where the block bundles live. And then in the mathematical world, you might call this the topological side because it's computable by algebraic topology means. This is an actual topological space. But this is the analytic side. The point is the labels are not important because of the duality. What you call analytic or what you call topology. In a region, in a regime in which they come together, these are just labels, dual labels. So we don't need to be pedantic. Anyway, there's an interesting isomorphism like this. And I would like to give some physical interpretation of what these symbols mean. So what on earth is this E sub-gamma? So mathematically, if you have seen classifying spaces and universal, customized spaces for principal bundles, then there's a universal principal bundle, that's E, that's E gamma. And it's a contractable space with free action of your group. And if you take the quotient, you get a classifying space and this classifies all principal bundles with fiber gamma. That's not what we want to do. So as many talks talk about BG and EG and so on, I'll say a little bit about that later. So this is a modified version. It's a universal proper gamma space. That is to say, you don't want to consider a space in which a gamma is acting in a very bad way. You want to have a space in which it's more or less acting in a rather crystallographic way. So it's a sensible embedding of a lattice in the continuum. And this is nothing other than the universal background for embedding a lattice model where the labels are in gamma. So the mathematical definition is the universal proper gamma space. You find out the equivalent homotopy by the universal property that any other proper gamma space nets uniquely up to the homotopy into this universal space. Now, our main observation is we already know what the universal space for any crystallographic space group is. There's nothing in our Euclidean space. So Euclidean space, if you throw away this terminology, it's specialized to crystallographic space group. You already have the left-hand side of the von Korn conjecture. So here's a crystallographic space group. It's a discrete co-compact subgroup of the Euclidean group of isometrists. It's got a lattice of translations, a billion netizens of translations. It's got a finite quotient called a point group. So here's diamond. Like diamonds, it's not just a billion netizens. You look closely, it's got a very fine structure. And the fine structure is exactly what gives it its material properties. So it's essential in material science to look at crystallographic space groups. And then, of course, a very famous paper of Friedmore in 2013 says, if you classify, get, gamma, crystallographic space group, symmetric Hamiltonians, you're equivalently classifying some twisted point group equilibrium like the bundles. These are classified by some twisted equilibrium K theory over the momentum space torus. And the C-style algebra languages is the computing the K theory of the group C-style algebra. So the picture in terms of group C-style algebra was in a paper that I wrote in 2016. I think I can just about finish on time. So, so, Thorai, so how do we convert space groups into Thorai? The thing is a space group is equivalently characterized by the induced affine action of its point group, f, on the unit cell. So you have a unit cell and then you have an induced action of the point group. This equivalently tells you what space group you're talking about. And then a little theorem that I also proved with a GOMI, it was published recently, you can actually enhance the basic duality that appears in string theory to a crystallographic one. So there exists flat manifolds, compact manifolds if you like that language. And a dual flat one, albifolds I mean, by which I mean a torus with some finite group action. You put some twists in the K theory, a degree shift. And magically, even though the left-hand side looks completely different from the right-hand side, it actually has isomorphic theories. When applied to the space group, this is exactly this duality between position space and momentum. So it's equivalent to the bound-quant conjecture for space groups. You do a little bit of Poincare duality, you get the k-homology of your unit cell. You go to the covering space, you get exactly to this mysterious left-hand side that I was talking about. And then the bound-quant assembly map gets you to this group of obstructions for atomic limits. So the conclusion, the upshot is that there exists a generalized homology theory. It may be gamma-equivirian. Of your underlying spatial manifold, it classifies the non-interacting topological phases that can live on that manifold. One thing, can you tell us how you got to the plane bottle using this? The plane bottle for the PG? Ah, right, right, because the f-fine action for PG on the unit cell is exactly the free Z2 action. And this includes a reflection. So the quotient by this free Z2 action is exactly the plane bottle. It's half of the torus. That's why you get to your plane bottle. Now, k-homology. K-homology is assembled from usual homological data that's either even dimensions or odd dimensions, depending on whether you're looking at degree 0 or degree 1 in k-homology. So to demystify what this is, I'll just look at a zero-dimensional part of k-homology. What is the zero-dimensional part of k-homology? It's just a set of points. And in the equivalent case, it's a set of points. Around is the orbit of a point under the group action. So that's basically the wake-up position in crystallography. And of course, on the point, you have some isotropy. So in equivalent homology, you exactly also specify a representation of the isotropy. So if you have a lattice model and you have some reflection symmetry, then you may have a degree of freedom slightly to the left, exchange of a degree of freedom slightly to the right. Or you could have the symmetric linear combination of this degree of freedom, or you could have the anti-symmetric linear combination of this degree of freedom. So the first thing that I said corresponds to starting from a wake-off position in general position in space. And the second one, which I thought, said, corresponds to picking a fixed point of the reflection and choosing either the trivial or the sign representation of the point group. And then you build your Hilbert space by the module action again. You construct the translate. So for example, if you pick the sign representation at the fixed point, localized at the fixed point, you pick the sign representation. And then you generate by the translations. You get a band where at all these reflection centers, you have a sign representation of the point group. So in a sense, the truly topological band insulators are those which are not built from this zero-dimensional homological data. It's built from some bigger abstract two-dimensional, four-dimensional homological data. So you can actually see from the left-hand side which of the classes on the right-hand side are the essentially topological ones. So for example, in the Klein bottle case, the k theory is non-zero, but all of them are built from zero-dimensional data. So they are not truly topological. Oops, I think I run out of time. So last slide if I may. I have time for a minute to... Sorry for running two minutes late. So the vague conjecture is... So you hear in certain talks that if you want to do interacting theories, there's some vague conjecture based upon the success of the k theory, I think in the non-interacting case. There's some kind of generalized homology theory and classify topological phases of prescribed class of interacting quantum systems. So for example, my collaborator and two other authors, Shiozaki and Sato, proposed some and did some computations. There are also some people who look at various other kinds of homology theories, group called homology, co-bodism, that sort of thing. And to me, I might be wrong about this. I'm supposed to tie in some notion of this universal object classifying space, which classifies some type of hemiphoenix, where this is to be specified, so adjective. So my first opinion is that a naive choice, VG already fails in the non-interacting case, so you should be careful what kind of classifying space you build. And the other thing is you should be careful not to be topological. If you want to say a topological, a mathematical topological theory, classifies topological phases, well, I don't say anything. It's axiomatically already topological and it's local. It has these notions of excision, neoveria, torus principles to patch things up and so on. So let's say something substantial, you need to combine the algebra and topology with some actual analysis for the physical interpretation. So now it's what abstract algebraic topology machine to use should be determined by some analytic criterion. And K theory is precisely, as I said, it doesn't just give you a stable classification of the bundles. It allows you to do many things. It's exactly where a certain regime of geometry, topology and analysis comes together and all of these have some physical interpretation. And there's a duality which I try to emphasize is a very useful tool. And that's the end of my talk. Thank you. Thank you.
 I'm not actually a professor.  I'm not Adelaide, a collaborator with Matai.  There'll be one slide about what we did,  but most of this talk would be on some new work.  Right, now that everyone's seated,  I should also thank all the organizers.  Everyone hears that.  And for this opportunity to give a talk  with another collaborator who's at Adelaide,  so he's a geometric analyst and index theory person,  amongst other things, in the last year and so we discussed  a little bit about k-theory and topological phases.  And then I encountered a situation where it was a newcomer.  It was completely impossible to explain  why anyone would look at k-theory.  So one of the goals of this talk is,  well, at least for physicists,  to demystify in a very concrete sense  what k-theory means in a solid state.  So I said solid state in contrast to condensed matter,  just to say actually everything about here,  I'm going to say, is kind of in the good old quantum mechanics world.  So it's really about single particle Hamiltonians.  Nevertheless, there seems to be a very important aspect  that isn't stressed very much.  In particular, I'll talk about one-year functions.  There'll be a talk on either Thursday or Friday  also on one-year functions.  This is a very important aspect of,  at least to me, of topological phases.  So this will appear as part of solid state theory.  And one of the things I really want to do is to apply  to very general geometric settings.  And I also want to talk a little bit about t-duality.  So that's actually my primary interest,  is the role of duality in physics.  Why dualities?  Why different people giving different explanations  for the same phenomena end up talking about the same thing?  So I'll show why some pictures of the same phenomenon  is actually the same, and there's actually an underlying duality  which can be identified with the t-dualities  that people talk about in string theory.  For mathematicians, if you come from the operator algebra,  C-style algebra, and K-theory,  you know one of the biggest machines to produce research  is the bump-corn conjecture.  And so what does it mean?  I'll give a candidate physical interpretation of the assembly map.  And also, mathematicians might not, actually, in fact,  really know what the topological phases are.  So I'll try to have one, at least one notion which makes sense.  In a K-theory context.  Right.  So this workshop is, of course, about interacting quantum systems.  And I want to say anything about it until the very end.  And why that is so is because, well, I'm no expert in many-body physics.  But I hope to be able to say something general  about single particle Hamiltonians in general geometries.  And maybe it has some lessons for our kind of exuberance  in using abstract nonsense to study interacting regimes.  So by abstract nonsense, I mean the big machinery of categories,  very abstract things.  We do need to justify some things.  That's why I want to say.  Right. So here are some general comments.  From the experimental point of view.  So how does one detect a topological phase these days,  apart from the quantum whole effect?  So if you look at a wild semi-metal,  you never see anything in a bulk experimentally.  As far as I understand, correct me if I'm wrong.  And you actually infer some bulk topology,  some abstract bulk topology,  by looking analytically for zero modes,  localized on the boundary.  So when you say this to a mathematician,  he immediately sees that there's a physical experiment  that combines some analysis, some geometry,  and some topology, living in the background.  This is nothing but index theory.  So the physicist's belief in a bulk boundary correspondence  is essentially a statement that there must be some index theorem,  a region in which several different flavors of mathematics  come together to actually talk about the same thing,  and which aspect you want to look at is kind of your preference.  So it's a rather deep idea,  and in fact the physical phenomenon  has been able to generate new conjectures  in terms of index theory.  So this was the case for the quantum whole effect, for example,  formula, con, muscovici, that is, and so on, as an example.  Now, if you do band theory in solid state physics,  the usual process is to go to momentum space.  Quantum space is nothing but the duo of a billion group of symmetries.  It's the Pontiagin duo in mathematics,  and allows you to do some kind of harmonic analysis.  And then you construct some topological invariance  for back to bundles, for example,  for the bundle of field states in an insulator, for example.  And then you say, ah ha, vector bundles can be non-trivial,  topologically non-trivial.  Of course, so why didn't anyone realize this?  Well, who knows?  But we would also like to have a position space picture.  Now, it turns out there is kind of a new,  maybe not so new point of view, but as a gaining popularity,  which is that what you call topological and momentum space,  should have an analog in position space  as a kind of obstruction to produce an atomic limit.  So this is roughly the idea of a topological quantum chemistry.  That is when molecular orbitals, that sort of story,  fail to make sense.  And there seems to be a topology and localization kind of a duality,  which actually there will be a talk about this in much more detail.  Now, this link is actually known since 2007,  I believe it was conjectured already by Thulus a long time ago,  in NCO and so on, that in the Euclidean case,  the usual lattice, everyone's favorite lattice,  of translation symmetries,  then you can actually prove that what the class means  for a term used later is that there is no good atomic limit  or one-year basis for the band of filled states.  So it's very delocalized.  Any one-year basis you want to draw is very delocalized.  I'll show you a picture later on.  So this is kind of a duality between two stories.  So on the one hand, there is now a definition of topological phases,  a rough definition on the left side,  and then there's a one on the right side,  which is amenable to a lot of k-theory tools in momentum space.  So why did it same thing?  Well, in fact, there's a nice duality that implements such an equivalence.  So here's one of the big messages I want to say.  We've heard a lot about lattice models and continuum  and maybe a distinction between the two.  But from the point of view of solid state physics and one-year functions,  actually they're not dichotomous.  So let this model just a set of labels,  sometimes labeled by a group,  and the group is permuting these labels to get some structure through the group.  We actually want to embed it in some metric space,  usually to have some notions of distance to say,  what are the couplings between sites, etc.  So that's secretly a little bit more structure.  And when you embed it in this spatial manifold in the background,  of course you would like all the degrees of freedom to be very localized,  like little delta functions, localized atoms.  But it may actually look like this instead.  It can still be orthonormal.  And really, from the point of view of this result over here,  absence of atomic limit,  the question is really about how a lattice model,  which are abstractly isomorphic as Hilbert spaces,  how they actually sit inside the continuum model.  And I'll give an answer to this, which is precisely measured by K theory.  There are two aspects of it.  One is the homology theory and one is the co-homology theory.  Another advantage of this duality is you don't actually need a momentum space  to study topological phases in the sense of absence of atomic limits.  So the time binding models on a one-year basis,  you could make sense of it for any discrete group  acting in a sensible way on any physical spatial reminder manifold, X.  And this allows us to liberate ourselves from the good old place  where churn numbers live, where you have a billion lattice  acting isometrically on Euclidean space.  Actually, there's a whole big world of Riemannian manifolds with curvature  and there are lattices in them.  So for example, this is the Poincare disk.  It's a hyperbolic plane with negative curvature.  And if you put lattices here, you actually get a lattice of a Riemann surface.  So the lattice is labeled by the fundamental group of the Riemann surface.  So the hyperbolic plane can be touted by all these Riemann surfaces.  So you identify pairs of edges like this.  And then you get a non-nobelian lattice sitting inside the isometry group of the hyperbolic plane.  People talk about defects.  So here's a defect.  Here's a screw dislocation.  In fact, in a screw dislocation, if you have some density of screw dislocation,  actually what happens is that the effective geometry becomes non-Euclidean.  It gets some kind of new geometry thing because you see when you go horizontally,  the horizontal translations don't close.  They close only up to a vertical translation.  So there's a density of this, so it's like Heisenberg commutation relations  between translation in x direction and y direction.  So here's an lattice.  This is called the discrete integer Heisenberg lattice in R3,  where R3 is no longer regarded as Euclidean R3 but as a Heisenberg group manifold.  So you see the geometry is very important when you want to say how the lattice is embedded into your actual physical space.  So I'll go, I'm going to talk about band theory.  And maybe before the term topological quantum chemistry came about,  there was just a general idea or even statement that energy bands, which you are familiar with,  is nothing but a regular representation of the lattice.  So it fully transforms to the L2 sections of some vector bundle.  So in a fully transformed space, you've got a band.  You can have an orthonormal basis of sections there.  And it fully transforms back to the regular representation.  So you don't know what regular representation is.  It's basically a tight binding model.  So suppose you've got a lattice labeled by a group, say Z.  And then you've got a degree of freedom everywhere, a copy of C.  So the space L2 of Z, it's not a bare Hilbert space.  In a tight binding model, it has an action of Z acting by translations on it.  Why does it have an action of Z acting by translation or exponent minute?  But I just want to say that from the point of representation theory, this is a very natural thing to think about.  Now, the advantage of this formulation of an energy band is that it's easy to analyze.  It will be much relevant if you replace it with something else.  Oh, sure.  So for example, the Poncares thing that I drew just now.  So you tau the hyperbolic plane by some Weeman surfaces instead of tau, right?  These are analog of unit cells.  So these are your lattice points. And the lattice points, if you choose this with your origin,  then the other lattice points are labeled by elements of the fundamental group of the Weeman surface with genus G.  So you have a degree of freedom here, and then you act by this lattice.  You get another degree of freedom here, and so on.  So actually the Hilbert space, the so-called tight binding Hilbert space,  in fact, people have done tight binding models on this space.  Nothing other than the regular representation space for this group as a set.  So that brings to the second point.  So the lattice authority will be a billion.  And in fact, in the C-style algebra, there's actually a whole theory of non-commutative block theory.  So certain kind of operators on the covering space, which have some regularity condition to make sense of energy bands in this sense.  So I might not be wrong with the authors of these kind of theories.  It's probably Bruning, Sunata, one of the first people to do it.  Gruber wrote a paper which is literally titled Non-Communitive Block Theory.  Mathai McCully applied it to the hyperbolic plane, for example.  And so it's a rather established theory.  And the Euclidean case, of course, we actually want to do crystallography.  So even the lattice is actually not the billion lattice.  It's actually an enhancement of the billion lattice into a crystallographic space group.  So you have to be able to somehow do non-commutative band theory.  So this idea of energy bands being a certain number of copies of lattice models has been repopularized with the adjective topology.  And so now it's called topological quantum chemistry.  So there's a nice paper in Nature by Brett Lin and his collaborators in the US.  So the upshot, the mathematical thing I want to say is you start off with an L2 space on X, where X is an action of a lattice.  So this L2 space is a representation space for the group.  And then if you want a discrete model, so you take a point and generate its orbit, so you have a lattice model.  And then you have a subspace, a subrepresentation space.  But you have to remember that this subrepresentation space is secretly made out of actual wave functions with tails and so on.  And you can actually embed the lattice model in very different ways.  And there are different ways in which that embedded is precisely the topological nature of this embedding.  So this regular representation, of course, these are the same abstractly as Hilbert spaces, but sitting inside the background is completely different.  And so an application of this kind of ideas is that using a new mathematical technique that I'll introduce today,  we actually obtain the rigorous proof that a wallpaper group, so this is a very interesting two-dimensional wallpaper group,  which I'll explain in more detail later, it can never admit any topological beds.  So any type-bending model, the type-bending approximation is valid.  You never get a situation like this.  Now, it might sound like a stupid thing, but for many years people were trying to even prove this for the Euclidean lattice case.  And as far as I know, there are no other techniques to do this sort of thing.  Okay, so do a little bit of physics.  So why? What are wave functions?  Wave function is the Hilbert space for the electron. It starts off as a free electron.  It sees the Euclidean geometry, so the non-relativistic free electron.  Listen, L2 of Rd.  Why L2 of Rd? Because the Euclidean group is acting, is being represented by our summatries of the space.  This is the point of view of particle physics.  So free electron moves around and then is actually sitting inside a material with a discrete set of points.  So the canonical differential operator, the Laplacian, gets perturbed by a gamma periodic potential.  And so the Euclidean symmetry is broken into a discrete crystallographic subgroup.  When this is broken, nevertheless you have to remember that it retains the canonical representation inherited from up here.  So for the simplest example is the lattice.  So as I said, the lattice is secretly sitting in Rd.  So when you break the symmetry, you remember that the z is still acting by translation and not in some random way.  For example, you can embed this in Rd and you can consider some very stupid action of z on this set as a representation.  So that's not what you want to consider. You want to consider it in exactly the regular representation to make sense of the type-binding model.  And then what you do, because there's a billion into a Fourier transform, Fourier transform of the lattice's circle is a space of characters.  In fact, a topological space in a canonical topology.  And this is where the topological momentum space comes in.  This is the Brillo-Anne zone.  Indeed, I mentioned this would be a d-torus.  Then the Bloch-Fouquet transform is kind of a partial Fourier transform, which allows you to transform operators acting on L2 of Rd.  You make use of the discrete transational symmetry to partially Fourier transform.  And the basic thing is you reduce, because of the periodicity, you reduce the Schrodinger equation problem to one which is living on a compact unit cell.  It's here. And mathematically, you call this a fundamental domain.  But you will agree that this is a circle, T.  I'll explain later on why I call this T and not the other T.  This T is the Brillo-Anne-Torus. It's absolutely not the same thing as this unit cell.  And that's the whole source of the duality.  Right. Of course, then you don't have actual periodicity, but you have some quasi-periodicity.  So you have to say, you look at a space of block wave functions.  So wave functions are said when you translate by one unit cell, you acquire a phase factor.  And a phase factor can be any number in the Brillo-Anne-Torus.  So these are block wave functions. They are quasi-periodic wave functions.  They are not normalizable, but they are normalizable over a unit cell.  So here's the condition for a block wave function.  Everyone knows this, hopefully?  No.  Everyone who's not seen a block wave function.  Okay. And the space of block wave functions for fixed k, quasi-momentum k,  is identifiable with the L2 space on the unit cell.  So altogether, after a block-locate transform, you really transform the space of the L2 space of wave functions  into a bundle of Hilbert spaces living over the Brillo-Anne-Torus.  And the typical fiber is this infinite-dimensional Hilbert space of functions on the compact unit cell.  And then a general wave function decomposes as this direct integral.  So this is the superposition of a wave function in terms of its block, in terms of its quasi-momentum.  So each component is a k-quasi-periodic block wave function.  So why does t have a topology? Why does the momentum say it's a topology?  Well, I think Raff gave a good explanation just now.  And it's because of the Fourier transform.  And that's where the representation of z is important, because the translation operator in z after Fourier transform  becomes multiplication by the continuous function,  which phase function which winds around the Torus once.  I mean gamma number of times.  So the action of the transition action actually in momentum states generates after the norm completion,  multiplication operators by continuous functions.  So this is why the transition symmetry is essential to make a link to the topology, the usual topology of the Brillo-Anne-Torus.  So those are our words. Here's a picture you get from an experiment.  A band structure, lots of bands, band gaps, gives you an insulator if this is the Fermi level.  So you disperse the energy spectrum of H according to k,  and at each k because you have Schrodinger operator on a compact space.  And these are Laplace-Tag operators.  You get some elliptic regularity discrete spectrum.  And then some other argument says you have continuous energy bands.  So in total, if you're restricted to some finite,  oops, messed it up.  The spectral subspace between some finite bands,  then the Hilbert subspace corresponding to those energies will produce a sub-bundle.  Of course, it has L2 sections, but you can also consider just the continuous sections.  And this gives you a vector bundle of field states.  So there are two things to consider. There are the L2 sections of this block bundle.  So there's a Hilbert space.  And after the Fourier transform, you get back for each band one copy of this lattice model,  the regular representation.  But the block Fourier transform remembers how this representation is sitting inside the actual space of wave function.  If you take only the continuous sections of the block bundle,  then if you invert this block Fourier transform,  you only exhaust a dense subspace of this Hilbert subspace.  You might need other vector bundles, but not some of line bundles.  What you have done here is only the trivial line bundles and nothing more complicated.  No, no. Even if you have a non-trivial line bundle, the L2 sections, you can still generate.  You have the non-trivial vector bundles, which is the sum of...  Yeah, yeah, yeah, this is for simplicity. Of course, if you have more bands, you have n copies of this after you choose a frame,  an orthonormal L2 frame or continuous frame.  If you can choose it.  If you can, yeah.  If you cannot, then you have a non-trivial bundle.  Exactly. A non-trivial vector bundle.  That's what it's about.  Yeah, yeah. So that's your pre-empting what I've got to say, which is good.  So the inverse block Fourier transform of a general L2 section, as I was saying,  because it's not very regular, this is called a one-year wave function.  And then the one-year basis is basically translating this initial wave function under the translation action.  And it will generate, in the Hilbert space sense, an orthonormal one-year basis, which I label gamma star w.  So this is the pullback of an initial one-year function for the band.  So as an example, you start off with a band, you pick this nL2 section, you'll get at an origin,  some one-year wave function which you call w.  OK. And then you generate the other degrees of freedom by the translation action.  So if this is w and gamma is the generator of your group, then you translate by gamma,  you get the next degree of freedom, you do it twice, you get another degree of freedom and so on.  And then the whole question is whether this w can be made, whether you can choose sections  such that the w can be controlled, the tails of w can be controlled.  And there are obstructions to this.  So as Raph was saying, and as I mentioned also in the introduction,  unless a global continuous section can be chosen,  the one-year, the inverse block Fourier transform, so the one-year wave function,  we have very large tails.  This is from Fourier analysis.  So topologically, non-trivial block bundle, in general,  you would expect this relation that you produce inevitably a bad one-year basis.  So in 1D, as I said, for a long time people tried to prove you could localize these one-year functions  exponentially and so on.  Finally, it was proven.  In 2D, it turns out there's an obstruction, and this is your favorite turn-cast obstruction.  And initially, this was viewed as a bad thing.  So the idea was to try to produce conditions under which this obstruction disappears.  Today, we have the opposite point of view.  We love these obstructions, and we want to find more and more obstructions  in order to have more and more topological phases.  So let's not try to get rid of this, but to find more such obstructions.  And you can find many of these obstructions in more interesting lattices.  That's one of the other messages of my talk.  So in general, if you've got some space and you've got some lattice model living in there,  so you've got an orbit of some generally non-Abelian group,  then if you have a Hilbert subspace of the space of wave functions on that physical position space,  then a one-year basis would be an also normal basis for that subspace, which is of this form.  So you have a finite bunch of initial WJs, one-year wave functions.  For example, you can imagine if you have two bands,  then you can choose W1 and W2 like this.  And then similarly, you generate the bands, the degree of freedom of the bands in that way.  And then you generate an also normal basis.  So sometimes it's got a composite one-year basis because there's a finite label J.  Do you need a civilian group?  I deliberately not use numbers here like N because N suggests the Abelian group Z.  So you don't need a blocked group?  No, no, we don't need a blocked group.  But it exists in the non-committed sense if you wanted to go there.  Is it true that another message would be that if you enlarge Z to a space group, including reflection,  then there will be obstructions even in quantity?  There can be obstructions.  You have to make sense of what you mean by the, that's a detail and actually requires quite a lot of time to explain.  But yes, so even in 1D, you can get more obstructions for crystallographic space groups.  So here's the main theorem, the main message in an imprecise form first.  The precise form is a mix, but a useful message is a walking definition-free theorem.  It's supposed to have got a discrete group gamma and it's kind of a quote,  I'm quote, crystallographic group acting on space, manifold X.  And that H be a sound sensible Hamiltonian with, so you're thinking local differential operators,  Laplace-Schrodinger type operators, that's the sort of sensible thing you want to think about.  Which is gamma invariant, so invariant under the action of gamma on this space.  Then, based on the canonical, there is a canonical subspace of wave functions which I call OK wave functions.  So these are wave functions, they're not so bad.  They're better than L2, but they're not really good.  So they will look, in practice, something like this.  You may have tails, and I'm being generous here.  So the tails might end up here.  OK, so the decay is really bad, but not as bad as L2.  This is roughly speaking the Fourier transform of the continuous functions, which Raph was alluding to.  It at least contains the functions that have some kind of L1 decay, but it's a little bit more.  But I can't say what exactly it is, and that's one deficiency in this.  But the good thing is it's canonical, and this construction is actually very closely related to the construction that one does in the assembly map, in the sense of bumpcon, which I hope to get to at the end.  Now, so the point is, OK, you have some Hamiltonian, you have a spectral subspace that you want to inspect for the energy line between some gaps.  And then you want to say whether you can choose inside some of these OK wave functions, whether you can choose the so-called continuous sections in order to produce nice one-year functions like this.  Nice one-year functions.  And of course, it turns out, so you have to look at the module generated by these OK wave functions, because these are all the possible candidate acceptable wave functions  that you can admit inside a sensible one-year basis.  So you can't look at the full Hilbert space.  You want to restrict in here.  And it turns out that this is actually a finitely generated projective module over the group C-style algebra.  That's just a long-winded way of saying it's a block bundle, continuous sections of a block bundle, but in a non-commutative sense.  Now, if this module of generalized sections is trivializable, that is to say it's freely generated, then the spectral subspace admits not just an OK one-year basis, but you can actually push this to a good one.  In fact, what we can achieve is to push this to a short decay condition.  And conversely, if this module is non-trivial, then any one-year basis is necessarily extremely bad.  And when you encounter an obstruction which you can measure here, non-freeness, then the one-year basis is going to look like you can only ever produce this sort of type binding model.  So here's the main theorem for the mathematicians.  I'm not sure I'm going to go through everything, but the main assumptions are the usual ones that you see in this business.  This great countable group, some growth condition on the group, x compact fundamental domain.  The action is properly discontinuous so you don't accumulate at certain points, your finite isotropy, et cetera, effectively so you don't have redundancy in the remaining geometries in this part.  And you let the consider an operator, which is gamma and baron, and you want some regularity.  So it can be first order, you take some kind of Schrodinger type operator, second order of the plus type.  Then you look at the spectral subspace for compact separated part of the spectrum.  So you project into the subspace for lying between some energy gaps.  So the statement I want to say just now is that there is a canonical dense subspace.  This is not a Hilbert subspace, it's dense, it's a proper subspace of OK wave functions which I call L2 of gamma, x, which is countably generated and it's a Hilbert C star module.  So hopefully people who work in C star modules like these words.  And then when you intersect this space of OK wave functions with your spectral subspace that you want to inspect, you get the module of continuous wave functions which are the ones that can help you produce the nice type binding models.  So it's actually a finitely generated module sitting inside the space of acceptable wave functions.  So if this module is not free, then you cannot generate a one year basis.  A free module is basically you pick a generator and then you do the group action and you generate this Hilbert C star module that way.  And if it's not free, then what I just said just now doesn't make any sense.  So you cannot produce a good one year basis.  Now in order to actually be saying something substantial, I need to tell you exactly what good and bad is.  So I would just say, as I said, the criteria for bad is not so easily characterised.  So this space of OK wave functions actually describes a decay condition at slightly weaker than this kind of L1 decay condition on the restricted L2 norms on the fundamental domains.  So it's really bad.  So in other words, if you have some topological obstruction from this module construction that I described, then you'll be forced to choose a one year basis from outside this space.  And so the decay condition has got to be worse than this.  So it's going to be even worse than what I've drawn.  So it's extremely delocalised.  Any kind of sensible atomic does not exist in any sensible sense.  Now with a billion groups, you can do a lot better.  And you can actually, so I will leave this to Professor Farnati's talk.  Basically you can improve what you mean by good and you can worsen what you mean by bad.  And you can actually characterise what, in position space quite nicely, what these conditions are.  Is the exponential, do you mean faster than sum or any exponential?  So I didn't reach exponential decay.  I'm not sure. You should ask.  But then sum.  So that's some kind of ochre principle which I think does not exist in.  That's a ochre. That's just a sub-adverb of a better kind.  So I think that is...  Right, right, right.  I think it's going to show the key like that.  I would, we have to talk about it later.  Anyway, maybe I'll just move on and...  That will be good.  We have to go for our tour on time.  So basically the main point of this approach is to use non-communist techniques  and to apply to interesting lattices.  So we don't do the obedient lattice over and over again.  However, we cannot reach these nicer conditions because it requires some extra techniques  which are not generally available in the non-communist world.  But for me, I think that the condition of good being short is reasonably good.  So when there are no topological obstructions, you can produce shorts one year basis.  That's reasonably good since as fast as possible there are no other general techniques to handle the non-obedient lattice case.  The shorts or general growth is not...  Sure, sure, sure.  And that's why over here there's something here.  Only polynomial.  Absolutely, the growth condition is important.  Otherwise you cannot relate.  Yeah, so I haven't thought of all the possible generalizations and permutations  because this just came out a couple of weeks ago.  So yeah, we can discuss generalizations.  So I noticed no one defined what a C-style algebra is.  We can clarify.  So this, the putschian...  Oh, it doesn't apply there.  So here's one situation where the geometry suggests  that you have to really look carefully at the geometry  because the conclusions cannot apply to arbitrary lattices.  There's some analytic condition because of the growth of the group.  So it's not necessarily a bad thing  because if you're able to say such a powerful statement about any lattice in any remuneration manifold,  that seems too good to be true.  So here's the preliminaries.  I want to get to this thing, this mysterious thing, group C-style algebra,  which appeared variously but no one defined.  So I'll just say that it's a very simple thing.  So if a discrete group, you can form formal linear combinations of the group elements  and then you take the convolution product.  So this is a vector space generated by the group element and then you just convolve it.  It's called a group ring.  It has a product.  The group ring acts as bounded operators in the regular representation.  So you just take the basis elements and you just translate by the group action  and then you add formal, so they are like hopping terms, finite hopping terms  and then you add linear combinations of these.  So these are all bounded operators because you just finite many of these translations.  And if you close this in the operator norm, this is the group C-style algebra you get.  And the basic example is if you take our favorite lattice,  then the usual Fourier transform gives you the space of continuous functions on the dual.  But this is the kind of philosophy. For any non-Abelian group,  there is a non-commutative dual space and the group C-style algebra is supposed to represent  in some sense modulo technicalities,  the non-commutative space of continuous functions on this dual momentum space.  And then we want to look at smooth sub-algebra or sub-functions.  So as some of the earlier questions were saying, how do you improve the regularity?  So groups have something called length function.  So you can think of it as, for example, the number that you obtain, some word length thing.  What's the smallest word you can use to produce that group element on a generator?  And get a number.  And using a length function, you can define a smooth group algebra in the following way.  It's called h infinity gamma. The notation is meant to suggest the usual algebra  of S-differentiable functions on the dual.  So there's a C-infinity function on the dual.  So the precise conditions like this.  So here's where the growth condition needs to be controlled.  But it usually depends on the length function.  Sure, sure, sure. But when L has polynomial growth and...  I use a very strong condition which is not necessary at every step,  but it's necessary to apply to all steps.  So yeah, so in the end it doesn't depend on the length function.  So if you apply this definition to the usual lattice,  then you want to get the C-infinity function on the dual torus.  Then the continuous or smooth sections of this block bundles,  if you add the adjective non-commutative,  so they're going to live over the non-commutative momentum space.  They correspond to what are called Hilbert modules  over either the group C-style algebra or the smooth version.  And the way...  I'll go through these slides very quickly because you've never seen it before.  It doesn't make any sense.  And for those who are interested in how you actually construct this module,  you might want to see the details.  So as I said, sitting inside the full Hilbert space of wave functions,  it's a canonical Hilbert C-star gamma module associated with gamma action.  So what you do is you first look at compactly supported functions,  and you define an inner product which is valued in the group ring.  And this is defined as follows.  So you take a compactly supported wave function v,  the inner product with a compactly supported wave function w  is a sequence of overlaps of the shifted versions of the first wave function  with the second wave function.  So you get a collection of numbers telling you how these two wave functions are overlapping.  And because they're compactly supported, this lives in the group ring.  And then you do some completion procedure to produce the Hilbert C-star module,  which is the space of OK wave functions that I talked about.  Also relevant for us, certain wave functions of rapid decay is the definition,  and the Schwartz wave functions, which are even better.  There's some control over the norms defined by the derivatives.  So the lemma is, because as we know, if you just use the continuous functions,  the group C-style algebra, then you can't really characterize very nicely  what the position space story is.  There's some abstract completion or some deadly behavior class of wave functions.  So we would like to be able to push it through the smooth algebra  for which actually the position space meaning is exactly the space of wave functions  with rapid decay like this.  So that's also a module.  This space of wave functions is actually a module over this H-infinity gamma.  And then under mild regularity assumptions,  a proposition is that the Hamiltonian, whenever you produce a spectral projection  of the energy subspaces, the subspace will have the property that its continuous sections  is finitely generated projective in the non-communitive sense.  And also it's a smoothing operator, the projection operator.  So if your W is already of rapid decrease,  then you apply the projection operator, you actually get something in the Schwartz class.  So this is how, while using this growth condition on gamma,  we can push the finitely generated module condition from OK wave functions  to those with rapid decrease and finally all the way to Schwartz wave functions.  What's an example of that?  Should we use a type operator with...  We always have a spectral projection of that sort of piece.  Yeah, I think in z squared.  So it's like the elliptic regularity story on the block.  You need multiple bands to have gaps.  Oh yeah, yeah, yeah, so there's an assumption that there's a gap.  Did I mention a gap?  Oh, sorry, sorry, of course.  Yeah.  Sorry.  Yeah, yeah, yeah, it's a gap.  Sorry, it's between some compact sets.  This example that you consider is a hyperbolic space.  Do you have gaps there or not?  Yes.  There are gaps all below or something like this.  Because of the twist, they have to do the twist.  Okay, right.  So anyway, the main message is if you forget all these technical uses,  the good one-year basis existence problem for energy bands  is equivalent in some precise way, if you're bothered to read the history above.  So the study of the module,  so not of the spectral subspace, but the spectral subspace intersected  with some suitable space of candidate wave function.  So it's a dense subspace of your spectral subspace.  It's equivalent to the study of dense subspace as a finitely generated projective module  over the group algebra.  And whenever it's free, and free-ness means exactly that you can choose  one generator from inside this good module functions,  and then generate a whole Hilbert space, also normally, by the action of the group.  So that's why you have this equivalent here.  So here's a summary of the group algebra approach to a one-year basis.  So you start with some localization criterion.  You build a module of acceptable wave functions,  which show that it's a module of a suitable group algebra.  If you have a spectral subspace you want to inspect,  then you look inside there, you look at acceptable wave functions inside there.  They will in general produce from a finitely generated projective module.  And then you can ask the fundamental question whether this module is free,  but it has a finite set of free generators.  If you don't, well, then unfortunately you cannot produce a one-year basis  from this space of good wave functions.  You have to go outside.  So you have very, very bad decaying atomic descriptions.  So you start from the Hilbert space level.  If you want these modules over the von Neumann algebra, group of von Neumann algebra,  it is completely useless as Ralph was saying.  It doesn't see any topology.  You want to do k-theory.  k-theory sees these two parts exactly.  So there's no distinction between these two parts, roughly speaking,  for an interesting technical reason why you can push the k-theory story  between these two conditions.  And finally, actually there's a little paper of Nicholas Reid  in physical review, Bean 17,  and titled,  Compactly Supported One-Year Wave Functions in Algebra K-Theory.  Basically he did a baby model of roughly speaking of what I was saying.  If you wanted to produce one-year basis of compactly supported wave functions,  then you have to look at the modules over the group ring.  And the reason is because compactly supported wave functions  live only on a finite set of position labels.  So their Fourier transform is just a bunch of,  it's a finite sum of Lorentz polynomials.  And that's only a, because finite sums,  they are,  these are preserved only by finite sums of transition operators.  Finite helping.  If you go to a bigger algebra, you will not preserve this compact support condition.  Right.  So can you tell in terms of classification which one is bigger, which one is smaller?  Oh, these are all inclusions.  So this space, this in here, this space, this in here, this space, this in there.  These are the OK wave functions, these are rapid decay wave functions.  We are able to push this to a short wave functions for the spectral projections of...  There are more topological phases.  Topology enters here.  So the k-theory of these two algebras are isomorphic.  So the topology sees, can see conditions of the...  I was asking in relation with this algebraic k-theory.  Oh.  Is he finding more...  The k-theory here is completely different.  So this doesn't see the topology.  Is he finding more states, more topological phases or not?  No, it's not even topological because this is kind of a...  More phases, let's say algebraic phases.  If you consider compactly supported Eigen functions to be sensible, you might want to look at that.  But I also heard some people say realistic operators never have...  The algebraic k-theory for groupings is pretty small because these spot predictions things  need some tails so you don't have to...  You have just two things in some sense.  And then write these.  Right.  So actually I haven't said any more k-theory yet.  So this is the first slide about k-theory.  Particularly if you have zp then there's a theorem saying that any module that is productive is free.  I think I'm kind of running short of time.  So I'm finding I need to go too quickly to get to the interesting bit.  So once we've found a mathematical problem of trying to measure the freeness of some  finitely generated projective modules, which is a module of candidate sections that allow us to generate the linear basis,  then the most powerful technique is k-theory.  k-theory exactly measures freeness, or actually stable freeness or otherwise.  But I want to emphasize that k-theory is not just about this thing.  It's not merely that.  k-theory actually has many, many aspects, which justifies why it's useful.  So we don't want to really just look at one aspect.  And there was some complaints sometimes about stabilization.  Stabilization corresponds to adding free modules.  And what are free modules?  They're just an extra band of tightly bound things.  So low-lying orbitals, which you didn't account for in your full model,  they are 3D generated, almost by definition.  So you automatically get some sensible notion of stabilization.  However, I will give one completely new example.  Actually, I already mentioned it, which is that wallpaper group PG,  where you can prove you don't need any stabilization, even.  This uses some non-trivial techniques of a refill about stable rank.  So so far, my whole story, if none of my messages went through,  I would like just to say we've explained what k-theoretic abstractions  built on momentum space means.  Now, it turns out, if you compute the k-theory,  you can also ask the question, are all of these spurious?  Maybe you computed something which has no physical meaning.  Well, it turns out there's a duality, a kind of momentum space,  position space, duality.  It guarantees that anything that appears on this k-theory  is actually indeed built up from some position space data.  So you're not doing anything redundant.  You're doing something quite precise.  So here's a wallpaper group PG.  I promise it's got something to do with the climb bottle,  which is half of a torus.  So the interesting wallpaper group, the torrent-free one, is PG.  It's the fundamental group of a climb bottle.  So it looks like this.  So you start from a point, an arbitrary point in space.  You generate the orbit of PG.  What happens is you think of it as a bunch of people.  So this is the left foot and right foot,  and they're walking along the line.  This is another person.  So it's the footsteps.  So we're doing it every day by walking.  As long as you don't stop, then you're kind of, no.  If you're walking, you generate this pattern.  So that's a PG group.  It's a Cayley graph of a PG group.  You notice it's torrent-free.  It's got a finite point group by reflection, but the lift.  This is non-symorphic.  So its lift is not torgent.  It's in fact torrent-free.  It's a glide reflection.  So you start from here.  You do the reflection, and you've got to do a half translation.  So if you take the fundamental domain of this section,  you get half of the torus.  Because after you're quotient out by the usual lattice,  you still have to account for this glide reflection.  And we identify points in this position-space torus.  By this glide reflection, you end up exactly with the climb bottle.  So I claim you can calculate the topological phases  by calculating the homology of the climb bottle.  The homology of the climb bottle is very easy to calculate.  Here's the homology of the climb bottle.  So you look at the unit side, you calculate its homology.  Because it's unorientable, it does not have a second homology,  the fundamental class.  So it only has degree 0 homology.  And this coincides with the K-theory of the-sorry,  K-homology of the climb bottle.  So it's just Z.  And then I wave a magic duality 1,  I'm going to go in magic 1, and claim this actually computes.  The group of obstructions that we argue is relevant,  which is the K-theory group of the C-style algebra generated by your lattice.  So what this computation tells us, if you believe this magic,  is that there exists only stably free modules,  in the PG case.  And then you have to invoke a very non-trivial result of REFO  where all the non-trivial free, finitely generated modules are already free.  So converting to physics is a proof that if PG symmetry,  there are no topological bands whatsoever.  The one anthropological invariance is the number of bands.  Yes. If you would like to consider a topological invariance.  So this Z measures the number of bands.  And all of the bands are good.  Right. If you're interested in torsion, you can go up to K1.  You do the same computation you can discover.  There's a Z mod 2 phase.  It's a chiral symmetric Z mod 2 phase.  And actually for a little paper with Kiyonori,  one of the collaborators in this project,  we actually prove a mod 2 index theorem,  which mediates a bulk boundary correspondence.  I can tell you more about this for interest.  Okay. So the last few slides are kind of a philosophy, if I may.  So why did I look at this?  So I went into this area with a completely different viewpoint  and I've emerged somehow looking at things very differently.  So my motivation was really to try to understand  why different arguments coincide.  So for example, in the quantum Hall effect,  people talk about flux torus is roughly the momentum space story.  And then there's also a story where you build something  in the position space and then there's another torus living  in position space.  And of course the thing in our head is to remember  these are completely different torus.  And yet they can give you the same result.  It sounds like T-duality.  So one day I met Matai and told me,  why don't you try to T-dualize everything?  You get something new for free.  So I think you do get something new for free.  So the T-duality principle says anything you can,  any theory you construct on one torus has a counterpart in the other.  Of course it doesn't say whether this link is physically meaningful.  So you might expect this to be physically meaningful  if there actually exists a single higher theory.  And this is in some sense the M-theory idea,  above the different type of type one, type two, A, B string theory and so on.  There's a bigger correct theory and you're just looking at different facets.  It doesn't exist.  Which doesn't...  Wish, wish.  Yeah.  Okay.  And actually that really happens in solid state physics.  In fact, the notion of T-duality appeared in the 60s  in the paper of Cartier  about lattice representations of canonical commutation relations.  But there's another story.  So let me name the block wave function.  A block wave function has both a momentum coordinate  and a position coordinate.  And each of these coordinates are defined on torang,  but different torang.  And we also note that the position space unit south torus  has the inverse radius to the momentum space torus.  So they really are T-dual even in the geometric sense.  So here's the big grand conjecture.  It's the bump conjecture.  And it has generated a lot of research in the mathematical community  which proved this mysterious isomorphism.  If these things are meaningless to you,  think of this as homology of a universal position space.  And this has K-theory of the momentum space associated to some discrete group.  And then it says that bump conjecture says  if you calculate the homology, a certain type of homology  on the left side of some universal position space,  you can actually assemble the homological data  into a K-theory class in momentum space.  And this has the interpretation of a geometric Fourier transform.  So many people would call the right-hand side the analytic side.  Although in our business, you might call it a topological side  because it's where the block bundles live.  And then in the mathematical world,  you might call this the topological side  because it's computable by algebraic topology means.  This is an actual topological space.  But this is the analytic side.  The point is the labels are not important because of the duality.  What you call analytic or what you call topology.  In a region, in a regime in which they come together,  these are just labels, dual labels.  So we don't need to be pedantic.  Anyway, there's an interesting isomorphism like this.  And I would like to give some physical interpretation  of what these symbols mean.  So what on earth is this E sub-gamma?  So mathematically, if you have seen classifying spaces  and universal, customized spaces for principal bundles,  then there's a universal principal bundle, that's E, that's E gamma.  And it's a contractable space with free action of your group.  And if you take the quotient, you get a classifying space  and this classifies all principal bundles with fiber gamma.  That's not what we want to do.  So as many talks talk about BG and EG and so on,  I'll say a little bit about that later.  So this is a modified version.  It's a universal proper gamma space.  That is to say, you don't want to consider a space  in which a gamma is acting in a very bad way.  You want to have a space in which it's more or less  acting in a rather crystallographic way.  So it's a sensible embedding of a lattice in the continuum.  And this is nothing other than the universal background  for embedding a lattice model where the labels are in gamma.  So the mathematical definition is the universal proper gamma space.  You find out the equivalent homotopy by the universal property  that any other proper gamma space nets uniquely up to the homotopy  into this universal space.  Now, our main observation is we already know  what the universal space for any crystallographic space group is.  There's nothing in our Euclidean space.  So Euclidean space, if you throw away this terminology,  it's specialized to crystallographic space group.  You already have the left-hand side of the von Korn conjecture.  So here's a crystallographic space group.  It's a discrete co-compact subgroup of the Euclidean group  of isometrists.  It's got a lattice of translations,  a billion netizens of translations.  It's got a finite quotient called a point group.  So here's diamond.  Like diamonds, it's not just a billion netizens.  You look closely, it's got a very fine structure.  And the fine structure is exactly what gives it its material properties.  So it's essential in material science  to look at crystallographic space groups.  And then, of course, a very famous paper of Friedmore  in 2013 says, if you classify, get, gamma, crystallographic space  group, symmetric Hamiltonians,  you're equivalently classifying some twisted point group  equilibrium like the bundles.  These are classified by some twisted equilibrium K theory  over the momentum space torus.  And the C-style algebra languages is the computing the K theory  of the group C-style algebra.  So the picture in terms of group C-style algebra  was in a paper that I wrote in 2016.  I think I can just about finish on time.  So, so, Thorai, so how do we convert space groups into Thorai?  The thing is a space group is equivalently characterized by the induced  affine action of its point group, f, on the unit cell.  So you have a unit cell and then you have an induced action of the point group.  This equivalently tells you what space group you're talking about.  And then a little theorem that I also proved with a GOMI,  it was published recently, you can actually enhance the basic  duality that appears in string theory to a crystallographic one.  So there exists flat manifolds, compact manifolds if you like that language.  And a dual flat one, albifolds I mean,  by which I mean a torus with some finite group action.  You put some twists in the K theory, a degree shift.  And magically, even though the left-hand side looks completely different  from the right-hand side, it actually has isomorphic theories.  When applied to the space group, this is exactly this duality  between position space and momentum.  So it's equivalent to the bound-quant conjecture for space groups.  You do a little bit of Poincare duality, you get the k-homology of your unit cell.  You go to the covering space, you get exactly to this mysterious left-hand side  that I was talking about.  And then the bound-quant assembly map gets you to this group of obstructions  for atomic limits.  So the conclusion, the upshot is that there exists a generalized homology theory.  It may be gamma-equivirian.  Of your underlying spatial manifold, it classifies the non-interacting topological phases  that can live on that manifold.  One thing, can you tell us how you got to the plane bottle using this?  The plane bottle for the PG?  Ah, right, right, because the f-fine action for PG on the unit cell  is exactly the free Z2 action.  And this includes a reflection.  So the quotient by this free Z2 action is exactly the plane bottle.  It's half of the torus.  That's why you get to your plane bottle.  Now, k-homology.  K-homology is assembled from usual homological data  that's either even dimensions or odd dimensions,  depending on whether you're looking at degree 0 or degree 1 in k-homology.  So to demystify what this is, I'll just look at a zero-dimensional part of k-homology.  What is the zero-dimensional part of k-homology?  It's just a set of points.  And in the equivalent case, it's a set of points.  Around is the orbit of a point under the group action.  So that's basically the wake-up position in crystallography.  And of course, on the point, you have some isotropy.  So in equivalent homology, you exactly also specify a representation of the isotropy.  So if you have a lattice model and you have some reflection symmetry,  then you may have a degree of freedom slightly to the left,  exchange of a degree of freedom slightly to the right.  Or you could have the symmetric linear combination of this degree of freedom,  or you could have the anti-symmetric linear combination of this degree of freedom.  So the first thing that I said corresponds to starting from a wake-off position  in general position in space.  And the second one, which I thought, said, corresponds to  picking a fixed point of the reflection and choosing either the trivial  or the sign representation of the point group.  And then you build your Hilbert space by the module action again.  You construct the translate.  So for example, if you pick the sign representation at the fixed point,  localized at the fixed point, you pick the sign representation.  And then you generate by the translations.  You get a band where at all these reflection centers,  you have a sign representation of the point group.  So in a sense, the truly topological band insulators are those which are not built  from this zero-dimensional homological data.  It's built from some bigger abstract two-dimensional, four-dimensional homological data.  So you can actually see from the left-hand side which of the classes on the right-hand side are the essentially topological ones.  So for example, in the Klein bottle case, the k theory is non-zero,  but all of them are built from zero-dimensional data.  So they are not truly topological.  Oops, I think I run out of time.  So last slide if I may.  I have time for a minute to...  Sorry for running two minutes late.  So the vague conjecture is...  So you hear in certain talks that if you want to do interacting theories,  there's some vague conjecture based upon the success of the k theory,  I think in the non-interacting case.  There's some kind of generalized homology theory and classify topological phases of prescribed class of interacting quantum systems.  So for example, my collaborator and two other authors, Shiozaki and Sato, proposed some and did some computations.  There are also some people who look at various other kinds of homology theories,  group called homology, co-bodism, that sort of thing.  And to me, I might be wrong about this.  I'm supposed to tie in some notion of this universal object classifying space,  which classifies some type of hemiphoenix, where this is to be specified, so adjective.  So my first opinion is that a naive choice, VG already fails in the non-interacting case,  so you should be careful what kind of classifying space you build.  And the other thing is you should be careful not to be topological.  If you want to say a topological, a mathematical topological theory,  classifies topological phases, well, I don't say anything.  It's axiomatically already topological and it's local.  It has these notions of excision, neoveria, torus principles to patch things up and so on.  So let's say something substantial, you need to combine the algebra and topology  with some actual analysis for the physical interpretation.  So now it's what abstract algebraic topology machine to use  should be determined by some analytic criterion.  And K theory is precisely, as I said, it doesn't just give you a stable classification of the bundles.  It allows you to do many things.  It's exactly where a certain regime of geometry, topology and analysis comes together  and all of these have some physical interpretation.  And there's a duality which I try to emphasize is a very useful tool.  And that's the end of my talk. Thank you.  Thank you.
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I'm not actually a professor. I'm not Adelaide, a collaborator with Matai. There'll be one slide about what we did, but most of this talk would be on some new work. Right, now that everyone's seated, I should also thank all the organizers. Everyone hears that. And for this opportunity to give a talk with another collaborator who's at Adelaide, so he's a geometric analyst and index theory person, amongst other things, in the last year and so we discussed a little bit about k-theory and topological phases. And then I encountered a situation where it was a newcomer. It was completely impossible to explain why anyone would look at k-theory. So one of the goals of this talk is, well, at least for physicists, to demystify in a very concrete sense what k-theory means in a solid state. So I said solid state in contrast to condensed matter, just to say actually everything about here, I'm going to say, is kind of in the good old quantum mechanics world. So it's really about single particle Hamiltonians. Nevertheless, there seems to be a very important aspect that isn't stressed very much. In particular, I'll talk about one-year functions. There'll be a talk on either Thursday or Friday also on one-year functions. This is a very important aspect of, at least to me, of topological phases. So this will appear as part of solid state theory. And one of the things I really want to do is to apply to very general geometric settings. And I also want to talk a little bit about t-duality. So that's actually my primary interest, is the role of duality in physics. Why dualities? Why different people giving different explanations for the same phenomena end up talking about the same thing? So I'll show why some pictures of the same phenomenon is actually the same, and there's actually an underlying duality which can be identified with the t-dualities that people talk about in string theory. For mathematicians, if you come from the operator algebra, C-style algebra, and K-theory, you know one of the biggest machines to produce research is the bump-corn conjecture. And so what does it mean? I'll give a candidate physical interpretation of the assembly map. And also, mathematicians might not, actually, in fact, really know what the topological phases are. So I'll try to have one, at least one notion which makes sense. In a K-theory context. Right. So this workshop is, of course, about interacting quantum systems. And I want to say anything about it until the very end. And why that is so is because, well, I'm no expert in many-body physics. But I hope to be able to say something general about single particle Hamiltonians in general geometries. And maybe it has some lessons for our kind of exuberance in using abstract nonsense to study interacting regimes. So by abstract nonsense, I mean the big machinery of categories, very abstract things. We do need to justify some things. That's why I want to say. Right. So here are some general comments. From the experimental point of view. So how does one detect a topological phase these days, apart from the quantum whole effect? So if you look at a wild semi-metal, you never see anything in a bulk experimentally. As far as I understand, correct me if I'm wrong. And you actually infer some bulk topology, some abstract bulk topology, by looking analytically for zero modes, localized on the boundary. So when you say this to a mathematician, he immediately sees that there's a physical experiment that combines some analysis, some geometry, and some topology, living in the background. This is nothing but index theory. So the physicist's belief in a bulk boundary correspondence is essentially a statement that there must be some index theorem, a region in which several different flavors of mathematics come together to actually talk about the same thing, and which aspect you want to look at is kind of your preference. So it's a rather deep idea, and in fact the physical phenomenon has been able to generate new conjectures in terms of index theory. So this was the case for the quantum whole effect, for example, formula, con, muscovici, that is, and so on, as an example. Now, if you do band theory in solid state physics, the usual process is to go to momentum space. Quantum space is nothing but the duo of a billion group of symmetries. It's the Pontiagin duo in mathematics, and allows you to do some kind of harmonic analysis. And then you construct some topological invariance for back to bundles, for example, for the bundle of field states in an insulator, for example. And then you say, ah ha, vector bundles can be non-trivial, topologically non-trivial. Of course, so why didn't anyone realize this? Well, who knows? But we would also like to have a position space picture. Now, it turns out there is kind of a new, maybe not so new point of view, but as a gaining popularity, which is that what you call topological and momentum space, should have an analog in position space as a kind of obstruction to produce an atomic limit. So this is roughly the idea of a topological quantum chemistry. That is when molecular orbitals, that sort of story, fail to make sense. And there seems to be a topology and localization kind of a duality, which actually there will be a talk about this in much more detail. Now, this link is actually known since 2007, I believe it was conjectured already by Thulus a long time ago, in NCO and so on, that in the Euclidean case, the usual lattice, everyone's favorite lattice, of translation symmetries, then you can actually prove that what the class means for a term used later is that there is no good atomic limit or one-year basis for the band of filled states. So it's very delocalized. Any one-year basis you want to draw is very delocalized. I'll show you a picture later on. So this is kind of a duality between two stories. So on the one hand, there is now a definition of topological phases, a rough definition on the left side, and then there's a one on the right side, which is amenable to a lot of k-theory tools in momentum space. So why did it same thing? Well, in fact, there's a nice duality that implements such an equivalence. So here's one of the big messages I want to say. We've heard a lot about lattice models and continuum and maybe a distinction between the two. But from the point of view of solid state physics and one-year functions, actually they're not dichotomous. So let this model just a set of labels, sometimes labeled by a group, and the group is permuting these labels to get some structure through the group. We actually want to embed it in some metric space, usually to have some notions of distance to say, what are the couplings between sites, etc. So that's secretly a little bit more structure. And when you embed it in this spatial manifold in the background, of course you would like all the degrees of freedom to be very localized, like little delta functions, localized atoms. But it may actually look like this instead. It can still be orthonormal. And really, from the point of view of this result over here, absence of atomic limit, the question is really about how a lattice model, which are abstractly isomorphic as Hilbert spaces, how they actually sit inside the continuum model. And I'll give an answer to this, which is precisely measured by K theory. There are two aspects of it. One is the homology theory and one is the co-homology theory. Another advantage of this duality is you don't actually need a momentum space to study topological phases in the sense of absence of atomic limits. So the time binding models on a one-year basis, you could make sense of it for any discrete group acting in a sensible way on any physical spatial reminder manifold, X. And this allows us to liberate ourselves from the good old place where churn numbers live, where you have a billion lattice acting isometrically on Euclidean space. Actually, there's a whole big world of Riemannian manifolds with curvature and there are lattices in them. So for example, this is the Poincare disk. It's a hyperbolic plane with negative curvature. And if you put lattices here, you actually get a lattice of a Riemann surface. So the lattice is labeled by the fundamental group of the Riemann surface. So the hyperbolic plane can be touted by all these Riemann surfaces. So you identify pairs of edges like this. And then you get a non-nobelian lattice sitting inside the isometry group of the hyperbolic plane. People talk about defects. So here's a defect. Here's a screw dislocation. In fact, in a screw dislocation, if you have some density of screw dislocation, actually what happens is that the effective geometry becomes non-Euclidean. It gets some kind of new geometry thing because you see when you go horizontally, the horizontal translations don't close. They close only up to a vertical translation. So there's a density of this, so it's like Heisenberg commutation relations between translation in x direction and y direction. So here's an lattice. This is called the discrete integer Heisenberg lattice in R3, where R3 is no longer regarded as Euclidean R3 but as a Heisenberg group manifold. So you see the geometry is very important when you want to say how the lattice is embedded into your actual physical space. So I'll go, I'm going to talk about band theory. And maybe before the term topological quantum chemistry came about, there was just a general idea or even statement that energy bands, which you are familiar with, is nothing but a regular representation of the lattice. So it fully transforms to the L2 sections of some vector bundle. So in a fully transformed space, you've got a band. You can have an orthonormal basis of sections there. And it fully transforms back to the regular representation. So you don't know what regular representation is. It's basically a tight binding model. So suppose you've got a lattice labeled by a group, say Z. And then you've got a degree of freedom everywhere, a copy of C. So the space L2 of Z, it's not a bare Hilbert space. In a tight binding model, it has an action of Z acting by translations on it. Why does it have an action of Z acting by translation or exponent minute? But I just want to say that from the point of representation theory, this is a very natural thing to think about. Now, the advantage of this formulation of an energy band is that it's easy to analyze. It will be much relevant if you replace it with something else. Oh, sure. So for example, the Poncares thing that I drew just now. So you tau the hyperbolic plane by some Weeman surfaces instead of tau, right? These are analog of unit cells. So these are your lattice points. And the lattice points, if you choose this with your origin, then the other lattice points are labeled by elements of the fundamental group of the Weeman surface with genus G. So you have a degree of freedom here, and then you act by this lattice. You get another degree of freedom here, and so on. So actually the Hilbert space, the so-called tight binding Hilbert space, in fact, people have done tight binding models on this space. Nothing other than the regular representation space for this group as a set. So that brings to the second point. So the lattice authority will be a billion. And in fact, in the C-style algebra, there's actually a whole theory of non-commutative block theory. So certain kind of operators on the covering space, which have some regularity condition to make sense of energy bands in this sense. So I might not be wrong with the authors of these kind of theories. It's probably Bruning, Sunata, one of the first people to do it. Gruber wrote a paper which is literally titled Non-Communitive Block Theory. Mathai McCully applied it to the hyperbolic plane, for example. And so it's a rather established theory. And the Euclidean case, of course, we actually want to do crystallography. So even the lattice is actually not the billion lattice. It's actually an enhancement of the billion lattice into a crystallographic space group. So you have to be able to somehow do non-commutative band theory. So this idea of energy bands being a certain number of copies of lattice models has been repopularized with the adjective topology. And so now it's called topological quantum chemistry. So there's a nice paper in Nature by Brett Lin and his collaborators in the US. So the upshot, the mathematical thing I want to say is you start off with an L2 space on X, where X is an action of a lattice. So this L2 space is a representation space for the group. And then if you want a discrete model, so you take a point and generate its orbit, so you have a lattice model. And then you have a subspace, a subrepresentation space. But you have to remember that this subrepresentation space is secretly made out of actual wave functions with tails and so on. And you can actually embed the lattice model in very different ways. And there are different ways in which that embedded is precisely the topological nature of this embedding. So this regular representation, of course, these are the same abstractly as Hilbert spaces, but sitting inside the background is completely different. And so an application of this kind of ideas is that using a new mathematical technique that I'll introduce today, we actually obtain the rigorous proof that a wallpaper group, so this is a very interesting two-dimensional wallpaper group, which I'll explain in more detail later, it can never admit any topological beds. So any type-bending model, the type-bending approximation is valid. You never get a situation like this. Now, it might sound like a stupid thing, but for many years people were trying to even prove this for the Euclidean lattice case. And as far as I know, there are no other techniques to do this sort of thing. Okay, so do a little bit of physics. So why? What are wave functions? Wave function is the Hilbert space for the electron. It starts off as a free electron. It sees the Euclidean geometry, so the non-relativistic free electron. Listen, L2 of Rd. Why L2 of Rd? Because the Euclidean group is acting, is being represented by our summatries of the space. This is the point of view of particle physics. So free electron moves around and then is actually sitting inside a material with a discrete set of points. So the canonical differential operator, the Laplacian, gets perturbed by a gamma periodic potential. And so the Euclidean symmetry is broken into a discrete crystallographic subgroup. When this is broken, nevertheless you have to remember that it retains the canonical representation inherited from up here. So for the simplest example is the lattice. So as I said, the lattice is secretly sitting in Rd. So when you break the symmetry, you remember that the z is still acting by translation and not in some random way. For example, you can embed this in Rd and you can consider some very stupid action of z on this set as a representation. So that's not what you want to consider. You want to consider it in exactly the regular representation to make sense of the type-binding model. And then what you do, because there's a billion into a Fourier transform, Fourier transform of the lattice's circle is a space of characters. In fact, a topological space in a canonical topology. And this is where the topological momentum space comes in. This is the Brillo-Anne zone. Indeed, I mentioned this would be a d-torus. Then the Bloch-Fouquet transform is kind of a partial Fourier transform, which allows you to transform operators acting on L2 of Rd. You make use of the discrete transational symmetry to partially Fourier transform. And the basic thing is you reduce, because of the periodicity, you reduce the Schrodinger equation problem to one which is living on a compact unit cell. It's here. And mathematically, you call this a fundamental domain. But you will agree that this is a circle, T. I'll explain later on why I call this T and not the other T. This T is the Brillo-Anne-Torus. It's absolutely not the same thing as this unit cell. And that's the whole source of the duality. Right. Of course, then you don't have actual periodicity, but you have some quasi-periodicity. So you have to say, you look at a space of block wave functions. So wave functions are said when you translate by one unit cell, you acquire a phase factor. And a phase factor can be any number in the Brillo-Anne-Torus. So these are block wave functions. They are quasi-periodic wave functions. They are not normalizable, but they are normalizable over a unit cell. So here's the condition for a block wave function. Everyone knows this, hopefully? No. Everyone who's not seen a block wave function. Okay. And the space of block wave functions for fixed k, quasi-momentum k, is identifiable with the L2 space on the unit cell. So altogether, after a block-locate transform, you really transform the space of the L2 space of wave functions into a bundle of Hilbert spaces living over the Brillo-Anne-Torus. And the typical fiber is this infinite-dimensional Hilbert space of functions on the compact unit cell. And then a general wave function decomposes as this direct integral. So this is the superposition of a wave function in terms of its block, in terms of its quasi-momentum. So each component is a k-quasi-periodic block wave function. So why does t have a topology? Why does the momentum say it's a topology? Well, I think Raff gave a good explanation just now. And it's because of the Fourier transform. And that's where the representation of z is important, because the translation operator in z after Fourier transform becomes multiplication by the continuous function, which phase function which winds around the Torus once. I mean gamma number of times. So the action of the transition action actually in momentum states generates after the norm completion, multiplication operators by continuous functions. So this is why the transition symmetry is essential to make a link to the topology, the usual topology of the Brillo-Anne-Torus. So those are our words. Here's a picture you get from an experiment. A band structure, lots of bands, band gaps, gives you an insulator if this is the Fermi level. So you disperse the energy spectrum of H according to k, and at each k because you have Schrodinger operator on a compact space. And these are Laplace-Tag operators. You get some elliptic regularity discrete spectrum. And then some other argument says you have continuous energy bands. So in total, if you're restricted to some finite, oops, messed it up. The spectral subspace between some finite bands, then the Hilbert subspace corresponding to those energies will produce a sub-bundle. Of course, it has L2 sections, but you can also consider just the continuous sections. And this gives you a vector bundle of field states. So there are two things to consider. There are the L2 sections of this block bundle. So there's a Hilbert space. And after the Fourier transform, you get back for each band one copy of this lattice model, the regular representation. But the block Fourier transform remembers how this representation is sitting inside the actual space of wave function. If you take only the continuous sections of the block bundle, then if you invert this block Fourier transform, you only exhaust a dense subspace of this Hilbert subspace. You might need other vector bundles, but not some of line bundles. What you have done here is only the trivial line bundles and nothing more complicated. No, no. Even if you have a non-trivial line bundle, the L2 sections, you can still generate. You have the non-trivial vector bundles, which is the sum of... Yeah, yeah, yeah, this is for simplicity. Of course, if you have more bands, you have n copies of this after you choose a frame, an orthonormal L2 frame or continuous frame. If you can choose it. If you can, yeah. If you cannot, then you have a non-trivial bundle. Exactly. A non-trivial vector bundle. That's what it's about. Yeah, yeah. So that's your pre-empting what I've got to say, which is good. So the inverse block Fourier transform of a general L2 section, as I was saying, because it's not very regular, this is called a one-year wave function. And then the one-year basis is basically translating this initial wave function under the translation action. And it will generate, in the Hilbert space sense, an orthonormal one-year basis, which I label gamma star w. So this is the pullback of an initial one-year function for the band. So as an example, you start off with a band, you pick this nL2 section, you'll get at an origin, some one-year wave function which you call w. OK. And then you generate the other degrees of freedom by the translation action. So if this is w and gamma is the generator of your group, then you translate by gamma, you get the next degree of freedom, you do it twice, you get another degree of freedom and so on. And then the whole question is whether this w can be made, whether you can choose sections such that the w can be controlled, the tails of w can be controlled. And there are obstructions to this. So as Raph was saying, and as I mentioned also in the introduction, unless a global continuous section can be chosen, the one-year, the inverse block Fourier transform, so the one-year wave function, we have very large tails. This is from Fourier analysis. So topologically, non-trivial block bundle, in general, you would expect this relation that you produce inevitably a bad one-year basis. So in 1D, as I said, for a long time people tried to prove you could localize these one-year functions exponentially and so on. Finally, it was proven. In 2D, it turns out there's an obstruction, and this is your favorite turn-cast obstruction. And initially, this was viewed as a bad thing. So the idea was to try to produce conditions under which this obstruction disappears. Today, we have the opposite point of view. We love these obstructions, and we want to find more and more obstructions in order to have more and more topological phases. So let's not try to get rid of this, but to find more such obstructions. And you can find many of these obstructions in more interesting lattices. That's one of the other messages of my talk. So in general, if you've got some space and you've got some lattice model living in there, so you've got an orbit of some generally non-Abelian group, then if you have a Hilbert subspace of the space of wave functions on that physical position space, then a one-year basis would be an also normal basis for that subspace, which is of this form. So you have a finite bunch of initial WJs, one-year wave functions. For example, you can imagine if you have two bands, then you can choose W1 and W2 like this. And then similarly, you generate the bands, the degree of freedom of the bands in that way. And then you generate an also normal basis. So sometimes it's got a composite one-year basis because there's a finite label J. Do you need a civilian group? I deliberately not use numbers here like N because N suggests the Abelian group Z. So you don't need a blocked group? No, no, we don't need a blocked group. But it exists in the non-committed sense if you wanted to go there. Is it true that another message would be that if you enlarge Z to a space group, including reflection, then there will be obstructions even in quantity? There can be obstructions. You have to make sense of what you mean by the, that's a detail and actually requires quite a lot of time to explain. But yes, so even in 1D, you can get more obstructions for crystallographic space groups. So here's the main theorem, the main message in an imprecise form first. The precise form is a mix, but a useful message is a walking definition-free theorem. It's supposed to have got a discrete group gamma and it's kind of a quote, I'm quote, crystallographic group acting on space, manifold X. And that H be a sound sensible Hamiltonian with, so you're thinking local differential operators, Laplace-Schrodinger type operators, that's the sort of sensible thing you want to think about. Which is gamma invariant, so invariant under the action of gamma on this space. Then, based on the canonical, there is a canonical subspace of wave functions which I call OK wave functions. So these are wave functions, they're not so bad. They're better than L2, but they're not really good. So they will look, in practice, something like this. You may have tails, and I'm being generous here. So the tails might end up here. OK, so the decay is really bad, but not as bad as L2. This is roughly speaking the Fourier transform of the continuous functions, which Raph was alluding to. It at least contains the functions that have some kind of L1 decay, but it's a little bit more. But I can't say what exactly it is, and that's one deficiency in this. But the good thing is it's canonical, and this construction is actually very closely related to the construction that one does in the assembly map, in the sense of bumpcon, which I hope to get to at the end. Now, so the point is, OK, you have some Hamiltonian, you have a spectral subspace that you want to inspect for the energy line between some gaps. And then you want to say whether you can choose inside some of these OK wave functions, whether you can choose the so-called continuous sections in order to produce nice one-year functions like this. Nice one-year functions. And of course, it turns out, so you have to look at the module generated by these OK wave functions, because these are all the possible candidate acceptable wave functions that you can admit inside a sensible one-year basis. So you can't look at the full Hilbert space. You want to restrict in here. And it turns out that this is actually a finitely generated projective module over the group C-style algebra. That's just a long-winded way of saying it's a block bundle, continuous sections of a block bundle, but in a non-commutative sense. Now, if this module of generalized sections is trivializable, that is to say it's freely generated, then the spectral subspace admits not just an OK one-year basis, but you can actually push this to a good one. In fact, what we can achieve is to push this to a short decay condition. And conversely, if this module is non-trivial, then any one-year basis is necessarily extremely bad. And when you encounter an obstruction which you can measure here, non-freeness, then the one-year basis is going to look like you can only ever produce this sort of type binding model. So here's the main theorem for the mathematicians. I'm not sure I'm going to go through everything, but the main assumptions are the usual ones that you see in this business. This great countable group, some growth condition on the group, x compact fundamental domain. The action is properly discontinuous so you don't accumulate at certain points, your finite isotropy, et cetera, effectively so you don't have redundancy in the remaining geometries in this part. And you let the consider an operator, which is gamma and baron, and you want some regularity. So it can be first order, you take some kind of Schrodinger type operator, second order of the plus type. Then you look at the spectral subspace for compact separated part of the spectrum. So you project into the subspace for lying between some energy gaps. So the statement I want to say just now is that there is a canonical dense subspace. This is not a Hilbert subspace, it's dense, it's a proper subspace of OK wave functions which I call L2 of gamma, x, which is countably generated and it's a Hilbert C star module. So hopefully people who work in C star modules like these words. And then when you intersect this space of OK wave functions with your spectral subspace that you want to inspect, you get the module of continuous wave functions which are the ones that can help you produce the nice type binding models. So it's actually a finitely generated module sitting inside the space of acceptable wave functions. So if this module is not free, then you cannot generate a one year basis. A free module is basically you pick a generator and then you do the group action and you generate this Hilbert C star module that way. And if it's not free, then what I just said just now doesn't make any sense. So you cannot produce a good one year basis. Now in order to actually be saying something substantial, I need to tell you exactly what good and bad is. So I would just say, as I said, the criteria for bad is not so easily characterised. So this space of OK wave functions actually describes a decay condition at slightly weaker than this kind of L1 decay condition on the restricted L2 norms on the fundamental domains. So it's really bad. So in other words, if you have some topological obstruction from this module construction that I described, then you'll be forced to choose a one year basis from outside this space. And so the decay condition has got to be worse than this. So it's going to be even worse than what I've drawn. So it's extremely delocalised. Any kind of sensible atomic does not exist in any sensible sense. Now with a billion groups, you can do a lot better. And you can actually, so I will leave this to Professor Farnati's talk. Basically you can improve what you mean by good and you can worsen what you mean by bad. And you can actually characterise what, in position space quite nicely, what these conditions are. Is the exponential, do you mean faster than sum or any exponential? So I didn't reach exponential decay. I'm not sure. You should ask. But then sum. So that's some kind of ochre principle which I think does not exist in. That's a ochre. That's just a sub-adverb of a better kind. So I think that is... Right, right, right. I think it's going to show the key like that. I would, we have to talk about it later. Anyway, maybe I'll just move on and... That will be good. We have to go for our tour on time. So basically the main point of this approach is to use non-communist techniques and to apply to interesting lattices. So we don't do the obedient lattice over and over again. However, we cannot reach these nicer conditions because it requires some extra techniques which are not generally available in the non-communist world. But for me, I think that the condition of good being short is reasonably good. So when there are no topological obstructions, you can produce shorts one year basis. That's reasonably good since as fast as possible there are no other general techniques to handle the non-obedient lattice case. The shorts or general growth is not... Sure, sure, sure. And that's why over here there's something here. Only polynomial. Absolutely, the growth condition is important. Otherwise you cannot relate. Yeah, so I haven't thought of all the possible generalizations and permutations because this just came out a couple of weeks ago. So yeah, we can discuss generalizations. So I noticed no one defined what a C-style algebra is. We can clarify. So this, the putschian... Oh, it doesn't apply there. So here's one situation where the geometry suggests that you have to really look carefully at the geometry because the conclusions cannot apply to arbitrary lattices. There's some analytic condition because of the growth of the group. So it's not necessarily a bad thing because if you're able to say such a powerful statement about any lattice in any remuneration manifold, that seems too good to be true. So here's the preliminaries. I want to get to this thing, this mysterious thing, group C-style algebra, which appeared variously but no one defined. So I'll just say that it's a very simple thing. So if a discrete group, you can form formal linear combinations of the group elements and then you take the convolution product. So this is a vector space generated by the group element and then you just convolve it. It's called a group ring. It has a product. The group ring acts as bounded operators in the regular representation. So you just take the basis elements and you just translate by the group action and then you add formal, so they are like hopping terms, finite hopping terms and then you add linear combinations of these. So these are all bounded operators because you just finite many of these translations. And if you close this in the operator norm, this is the group C-style algebra you get. And the basic example is if you take our favorite lattice, then the usual Fourier transform gives you the space of continuous functions on the dual. But this is the kind of philosophy. For any non-Abelian group, there is a non-commutative dual space and the group C-style algebra is supposed to represent in some sense modulo technicalities, the non-commutative space of continuous functions on this dual momentum space. And then we want to look at smooth sub-algebra or sub-functions. So as some of the earlier questions were saying, how do you improve the regularity? So groups have something called length function. So you can think of it as, for example, the number that you obtain, some word length thing. What's the smallest word you can use to produce that group element on a generator? And get a number. And using a length function, you can define a smooth group algebra in the following way. It's called h infinity gamma. The notation is meant to suggest the usual algebra of S-differentiable functions on the dual. So there's a C-infinity function on the dual. So the precise conditions like this. So here's where the growth condition needs to be controlled. But it usually depends on the length function. Sure, sure, sure. But when L has polynomial growth and... I use a very strong condition which is not necessary at every step, but it's necessary to apply to all steps. So yeah, so in the end it doesn't depend on the length function. So if you apply this definition to the usual lattice, then you want to get the C-infinity function on the dual torus. Then the continuous or smooth sections of this block bundles, if you add the adjective non-commutative, so they're going to live over the non-commutative momentum space. They correspond to what are called Hilbert modules over either the group C-style algebra or the smooth version. And the way... I'll go through these slides very quickly because you've never seen it before. It doesn't make any sense. And for those who are interested in how you actually construct this module, you might want to see the details. So as I said, sitting inside the full Hilbert space of wave functions, it's a canonical Hilbert C-star gamma module associated with gamma action. So what you do is you first look at compactly supported functions, and you define an inner product which is valued in the group ring. And this is defined as follows. So you take a compactly supported wave function v, the inner product with a compactly supported wave function w is a sequence of overlaps of the shifted versions of the first wave function with the second wave function. So you get a collection of numbers telling you how these two wave functions are overlapping. And because they're compactly supported, this lives in the group ring. And then you do some completion procedure to produce the Hilbert C-star module, which is the space of OK wave functions that I talked about. Also relevant for us, certain wave functions of rapid decay is the definition, and the Schwartz wave functions, which are even better. There's some control over the norms defined by the derivatives. So the lemma is, because as we know, if you just use the continuous functions, the group C-style algebra, then you can't really characterize very nicely what the position space story is. There's some abstract completion or some deadly behavior class of wave functions. So we would like to be able to push it through the smooth algebra for which actually the position space meaning is exactly the space of wave functions with rapid decay like this. So that's also a module. This space of wave functions is actually a module over this H-infinity gamma. And then under mild regularity assumptions, a proposition is that the Hamiltonian, whenever you produce a spectral projection of the energy subspaces, the subspace will have the property that its continuous sections is finitely generated projective in the non-communitive sense. And also it's a smoothing operator, the projection operator. So if your W is already of rapid decrease, then you apply the projection operator, you actually get something in the Schwartz class. So this is how, while using this growth condition on gamma, we can push the finitely generated module condition from OK wave functions to those with rapid decrease and finally all the way to Schwartz wave functions. What's an example of that? Should we use a type operator with... We always have a spectral projection of that sort of piece. Yeah, I think in z squared. So it's like the elliptic regularity story on the block. You need multiple bands to have gaps. Oh yeah, yeah, yeah, so there's an assumption that there's a gap. Did I mention a gap? Oh, sorry, sorry, of course. Yeah. Sorry. Yeah, yeah, yeah, it's a gap. Sorry, it's between some compact sets. This example that you consider is a hyperbolic space. Do you have gaps there or not? Yes. There are gaps all below or something like this. Because of the twist, they have to do the twist. Okay, right. So anyway, the main message is if you forget all these technical uses, the good one-year basis existence problem for energy bands is equivalent in some precise way, if you're bothered to read the history above. So the study of the module, so not of the spectral subspace, but the spectral subspace intersected with some suitable space of candidate wave function. So it's a dense subspace of your spectral subspace. It's equivalent to the study of dense subspace as a finitely generated projective module over the group algebra. And whenever it's free, and free-ness means exactly that you can choose one generator from inside this good module functions, and then generate a whole Hilbert space, also normally, by the action of the group. So that's why you have this equivalent here. So here's a summary of the group algebra approach to a one-year basis. So you start with some localization criterion. You build a module of acceptable wave functions, which show that it's a module of a suitable group algebra. If you have a spectral subspace you want to inspect, then you look inside there, you look at acceptable wave functions inside there. They will in general produce from a finitely generated projective module. And then you can ask the fundamental question whether this module is free, but it has a finite set of free generators. If you don't, well, then unfortunately you cannot produce a one-year basis from this space of good wave functions. You have to go outside. So you have very, very bad decaying atomic descriptions. So you start from the Hilbert space level. If you want these modules over the von Neumann algebra, group of von Neumann algebra, it is completely useless as Ralph was saying. It doesn't see any topology. You want to do k-theory. k-theory sees these two parts exactly. So there's no distinction between these two parts, roughly speaking, for an interesting technical reason why you can push the k-theory story between these two conditions. And finally, actually there's a little paper of Nicholas Reid in physical review, Bean 17, and titled, Compactly Supported One-Year Wave Functions in Algebra K-Theory. Basically he did a baby model of roughly speaking of what I was saying. If you wanted to produce one-year basis of compactly supported wave functions, then you have to look at the modules over the group ring. And the reason is because compactly supported wave functions live only on a finite set of position labels. So their Fourier transform is just a bunch of, it's a finite sum of Lorentz polynomials. And that's only a, because finite sums, they are, these are preserved only by finite sums of transition operators. Finite helping. If you go to a bigger algebra, you will not preserve this compact support condition. Right. So can you tell in terms of classification which one is bigger, which one is smaller? Oh, these are all inclusions. So this space, this in here, this space, this in here, this space, this in there. These are the OK wave functions, these are rapid decay wave functions. We are able to push this to a short wave functions for the spectral projections of... There are more topological phases. Topology enters here. So the k-theory of these two algebras are isomorphic. So the topology sees, can see conditions of the... I was asking in relation with this algebraic k-theory. Oh. Is he finding more... The k-theory here is completely different. So this doesn't see the topology. Is he finding more states, more topological phases or not? No, it's not even topological because this is kind of a... More phases, let's say algebraic phases. If you consider compactly supported Eigen functions to be sensible, you might want to look at that. But I also heard some people say realistic operators never have... The algebraic k-theory for groupings is pretty small because these spot predictions things need some tails so you don't have to... You have just two things in some sense. And then write these. Right. So actually I haven't said any more k-theory yet. So this is the first slide about k-theory. Particularly if you have zp then there's a theorem saying that any module that is productive is free. I think I'm kind of running short of time. So I'm finding I need to go too quickly to get to the interesting bit. So once we've found a mathematical problem of trying to measure the freeness of some finitely generated projective modules, which is a module of candidate sections that allow us to generate the linear basis, then the most powerful technique is k-theory. k-theory exactly measures freeness, or actually stable freeness or otherwise. But I want to emphasize that k-theory is not just about this thing. It's not merely that. k-theory actually has many, many aspects, which justifies why it's useful. So we don't want to really just look at one aspect. And there was some complaints sometimes about stabilization. Stabilization corresponds to adding free modules. And what are free modules? They're just an extra band of tightly bound things. So low-lying orbitals, which you didn't account for in your full model, they are 3D generated, almost by definition. So you automatically get some sensible notion of stabilization. However, I will give one completely new example. Actually, I already mentioned it, which is that wallpaper group PG, where you can prove you don't need any stabilization, even. This uses some non-trivial techniques of a refill about stable rank. So so far, my whole story, if none of my messages went through, I would like just to say we've explained what k-theoretic abstractions built on momentum space means. Now, it turns out, if you compute the k-theory, you can also ask the question, are all of these spurious? Maybe you computed something which has no physical meaning. Well, it turns out there's a duality, a kind of momentum space, position space, duality. It guarantees that anything that appears on this k-theory is actually indeed built up from some position space data. So you're not doing anything redundant. You're doing something quite precise. So here's a wallpaper group PG. I promise it's got something to do with the climb bottle, which is half of a torus. So the interesting wallpaper group, the torrent-free one, is PG. It's the fundamental group of a climb bottle. So it looks like this. So you start from a point, an arbitrary point in space. You generate the orbit of PG. What happens is you think of it as a bunch of people. So this is the left foot and right foot, and they're walking along the line. This is another person. So it's the footsteps. So we're doing it every day by walking. As long as you don't stop, then you're kind of, no. If you're walking, you generate this pattern. So that's a PG group. It's a Cayley graph of a PG group. You notice it's torrent-free. It's got a finite point group by reflection, but the lift. This is non-symorphic. So its lift is not torgent. It's in fact torrent-free. It's a glide reflection. So you start from here. You do the reflection, and you've got to do a half translation. So if you take the fundamental domain of this section, you get half of the torus. Because after you're quotient out by the usual lattice, you still have to account for this glide reflection. And we identify points in this position-space torus. By this glide reflection, you end up exactly with the climb bottle. So I claim you can calculate the topological phases by calculating the homology of the climb bottle. The homology of the climb bottle is very easy to calculate. Here's the homology of the climb bottle. So you look at the unit side, you calculate its homology. Because it's unorientable, it does not have a second homology, the fundamental class. So it only has degree 0 homology. And this coincides with the K-theory of the-sorry, K-homology of the climb bottle. So it's just Z. And then I wave a magic duality 1, I'm going to go in magic 1, and claim this actually computes. The group of obstructions that we argue is relevant, which is the K-theory group of the C-style algebra generated by your lattice. So what this computation tells us, if you believe this magic, is that there exists only stably free modules, in the PG case. And then you have to invoke a very non-trivial result of REFO where all the non-trivial free, finitely generated modules are already free. So converting to physics is a proof that if PG symmetry, there are no topological bands whatsoever. The one anthropological invariance is the number of bands. Yes. If you would like to consider a topological invariance. So this Z measures the number of bands. And all of the bands are good. Right. If you're interested in torsion, you can go up to K1. You do the same computation you can discover. There's a Z mod 2 phase. It's a chiral symmetric Z mod 2 phase. And actually for a little paper with Kiyonori, one of the collaborators in this project, we actually prove a mod 2 index theorem, which mediates a bulk boundary correspondence. I can tell you more about this for interest. Okay. So the last few slides are kind of a philosophy, if I may. So why did I look at this? So I went into this area with a completely different viewpoint and I've emerged somehow looking at things very differently. So my motivation was really to try to understand why different arguments coincide. So for example, in the quantum Hall effect, people talk about flux torus is roughly the momentum space story. And then there's also a story where you build something in the position space and then there's another torus living in position space. And of course the thing in our head is to remember these are completely different torus. And yet they can give you the same result. It sounds like T-duality. So one day I met Matai and told me, why don't you try to T-dualize everything? You get something new for free. So I think you do get something new for free. So the T-duality principle says anything you can, any theory you construct on one torus has a counterpart in the other. Of course it doesn't say whether this link is physically meaningful. So you might expect this to be physically meaningful if there actually exists a single higher theory. And this is in some sense the M-theory idea, above the different type of type one, type two, A, B string theory and so on. There's a bigger correct theory and you're just looking at different facets. It doesn't exist. Which doesn't... Wish, wish. Yeah. Okay. And actually that really happens in solid state physics. In fact, the notion of T-duality appeared in the 60s in the paper of Cartier about lattice representations of canonical commutation relations. But there's another story. So let me name the block wave function. A block wave function has both a momentum coordinate and a position coordinate. And each of these coordinates are defined on torang, but different torang. And we also note that the position space unit south torus has the inverse radius to the momentum space torus. So they really are T-dual even in the geometric sense. So here's the big grand conjecture. It's the bump conjecture. And it has generated a lot of research in the mathematical community which proved this mysterious isomorphism. If these things are meaningless to you, think of this as homology of a universal position space. And this has K-theory of the momentum space associated to some discrete group. And then it says that bump conjecture says if you calculate the homology, a certain type of homology on the left side of some universal position space, you can actually assemble the homological data into a K-theory class in momentum space. And this has the interpretation of a geometric Fourier transform. So many people would call the right-hand side the analytic side. Although in our business, you might call it a topological side because it's where the block bundles live. And then in the mathematical world, you might call this the topological side because it's computable by algebraic topology means. This is an actual topological space. But this is the analytic side. The point is the labels are not important because of the duality. What you call analytic or what you call topology. In a region, in a regime in which they come together, these are just labels, dual labels. So we don't need to be pedantic. Anyway, there's an interesting isomorphism like this. And I would like to give some physical interpretation of what these symbols mean. So what on earth is this E sub-gamma? So mathematically, if you have seen classifying spaces and universal, customized spaces for principal bundles, then there's a universal principal bundle, that's E, that's E gamma. And it's a contractable space with free action of your group. And if you take the quotient, you get a classifying space and this classifies all principal bundles with fiber gamma. That's not what we want to do. So as many talks talk about BG and EG and so on, I'll say a little bit about that later. So this is a modified version. It's a universal proper gamma space. That is to say, you don't want to consider a space in which a gamma is acting in a very bad way. You want to have a space in which it's more or less acting in a rather crystallographic way. So it's a sensible embedding of a lattice in the continuum. And this is nothing other than the universal background for embedding a lattice model where the labels are in gamma. So the mathematical definition is the universal proper gamma space. You find out the equivalent homotopy by the universal property that any other proper gamma space nets uniquely up to the homotopy into this universal space. Now, our main observation is we already know what the universal space for any crystallographic space group is. There's nothing in our Euclidean space. So Euclidean space, if you throw away this terminology, it's specialized to crystallographic space group. You already have the left-hand side of the von Korn conjecture. So here's a crystallographic space group. It's a discrete co-compact subgroup of the Euclidean group of isometrists. It's got a lattice of translations, a billion netizens of translations. It's got a finite quotient called a point group. So here's diamond. Like diamonds, it's not just a billion netizens. You look closely, it's got a very fine structure. And the fine structure is exactly what gives it its material properties. So it's essential in material science to look at crystallographic space groups. And then, of course, a very famous paper of Friedmore in 2013 says, if you classify, get, gamma, crystallographic space group, symmetric Hamiltonians, you're equivalently classifying some twisted point group equilibrium like the bundles. These are classified by some twisted equilibrium K theory over the momentum space torus. And the C-style algebra languages is the computing the K theory of the group C-style algebra. So the picture in terms of group C-style algebra was in a paper that I wrote in 2016. I think I can just about finish on time. So, so, Thorai, so how do we convert space groups into Thorai? The thing is a space group is equivalently characterized by the induced affine action of its point group, f, on the unit cell. So you have a unit cell and then you have an induced action of the point group. This equivalently tells you what space group you're talking about. And then a little theorem that I also proved with a GOMI, it was published recently, you can actually enhance the basic duality that appears in string theory to a crystallographic one. So there exists flat manifolds, compact manifolds if you like that language. And a dual flat one, albifolds I mean, by which I mean a torus with some finite group action. You put some twists in the K theory, a degree shift. And magically, even though the left-hand side looks completely different from the right-hand side, it actually has isomorphic theories. When applied to the space group, this is exactly this duality between position space and momentum. So it's equivalent to the bound-quant conjecture for space groups. You do a little bit of Poincare duality, you get the k-homology of your unit cell. You go to the covering space, you get exactly to this mysterious left-hand side that I was talking about. And then the bound-quant assembly map gets you to this group of obstructions for atomic limits. So the conclusion, the upshot is that there exists a generalized homology theory. It may be gamma-equivirian. Of your underlying spatial manifold, it classifies the non-interacting topological phases that can live on that manifold. One thing, can you tell us how you got to the plane bottle using this? The plane bottle for the PG? Ah, right, right, because the f-fine action for PG on the unit cell is exactly the free Z2 action. And this includes a reflection. So the quotient by this free Z2 action is exactly the plane bottle. It's half of the torus. That's why you get to your plane bottle. Now, k-homology. K-homology is assembled from usual homological data that's either even dimensions or odd dimensions, depending on whether you're looking at degree 0 or degree 1 in k-homology. So to demystify what this is, I'll just look at a zero-dimensional part of k-homology. What is the zero-dimensional part of k-homology? It's just a set of points. And in the equivalent case, it's a set of points. Around is the orbit of a point under the group action. So that's basically the wake-up position in crystallography. And of course, on the point, you have some isotropy. So in equivalent homology, you exactly also specify a representation of the isotropy. So if you have a lattice model and you have some reflection symmetry, then you may have a degree of freedom slightly to the left, exchange of a degree of freedom slightly to the right. Or you could have the symmetric linear combination of this degree of freedom, or you could have the anti-symmetric linear combination of this degree of freedom. So the first thing that I said corresponds to starting from a wake-off position in general position in space. And the second one, which I thought, said, corresponds to picking a fixed point of the reflection and choosing either the trivial or the sign representation of the point group. And then you build your Hilbert space by the module action again. You construct the translate. So for example, if you pick the sign representation at the fixed point, localized at the fixed point, you pick the sign representation. And then you generate by the translations. You get a band where at all these reflection centers, you have a sign representation of the point group. So in a sense, the truly topological band insulators are those which are not built from this zero-dimensional homological data. It's built from some bigger abstract two-dimensional, four-dimensional homological data. So you can actually see from the left-hand side which of the classes on the right-hand side are the essentially topological ones. So for example, in the Klein bottle case, the k theory is non-zero, but all of them are built from zero-dimensional data. So they are not truly topological. Oops, I think I run out of time. So last slide if I may. I have time for a minute to... Sorry for running two minutes late. So the vague conjecture is... So you hear in certain talks that if you want to do interacting theories, there's some vague conjecture based upon the success of the k theory, I think in the non-interacting case. There's some kind of generalized homology theory and classify topological phases of prescribed class of interacting quantum systems. So for example, my collaborator and two other authors, Shiozaki and Sato, proposed some and did some computations. There are also some people who look at various other kinds of homology theories, group called homology, co-bodism, that sort of thing. And to me, I might be wrong about this. I'm supposed to tie in some notion of this universal object classifying space, which classifies some type of hemiphoenix, where this is to be specified, so adjective. So my first opinion is that a naive choice, VG already fails in the non-interacting case, so you should be careful what kind of classifying space you build. And the other thing is you should be careful not to be topological. If you want to say a topological, a mathematical topological theory, classifies topological phases, well, I don't say anything. It's axiomatically already topological and it's local. It has these notions of excision, neoveria, torus principles to patch things up and so on. So let's say something substantial, you need to combine the algebra and topology with some actual analysis for the physical interpretation. So now it's what abstract algebraic topology machine to use should be determined by some analytic criterion. And K theory is precisely, as I said, it doesn't just give you a stable classification of the bundles. It allows you to do many things. It's exactly where a certain regime of geometry, topology and analysis comes together and all of these have some physical interpretation. And there's a duality which I try to emphasize is a very useful tool. And that's the end of my talk. Thank you. Thank you.
non-Euclidean. It gets some kind of new geometry topological momentum space comes in. This is the find more and more obstructions in order to have characterise what, in position space quite and... That will be good. We have to go for our They correspond to what are called Hilbert you have to look at the modules over the group with the K-theory of the-sorry, K-homology of the already free. So converting to physics is a proof non-interacting topological phases that can live wake-off position in general position in space. class of interacting quantum systems. So for