 Well, thank you for the invitation, it's a pleasure to speak here at this workshop. So I thought what I would do today is give a kind of overview talk where I tell different ways in which generalized cohomology enters into field theory. And so here's the outline. There are five different kinds of parts to the talk, plus a preliminary that comes before. So I think one of the first ways that generalized cohomology theories, meaning cohomology theories beyond Eilenberg and McLean, enter into field theories through the study of anomalies of spinner fields. And that happened in the 1980s. And there we encounter K theory and various versions of K theory. And also truncations of various versions of K theory. The next place, or in a sense, simultaneously, is with secondary invariance. I'll explain geometrically what that means. Those in field theory are often called topological terms, even though they're not, perhaps by some definition, strictly speaking, topological. And again, those can go beyond Eilenberg and McLean cohomology to some other generalized cohomology theories. Another place where it enters is in the Dirac quantization of abelian gauge fields. And again, here the examples will be from string theory. One encounters theories beyond cohomology. And the last two topics are really about invertible field theories. And understanding invertible field theories themselves as a map between spectra in the end. And there's a story for the domain, a story for the co-domain. And both of those involve different sorts of generalized cohomology theories or spectra. So I'm going to illustrate these general principles with my own work and work of collaborators. And that's a little bit of a dark art as this particular practitioner of dark arts was known to hawk his own work, his own books. But of course, there are many people who have used generalized cohomology. And I apologize in advance for not mentioning, of course, you've heard about some of that in some of the talks at this workshop already. And we'll hear more. One thing I don't discuss at all is the program of Stoltz and Teichner and all of their work, which is, in a sense, in a different direction. Using field theories to try to model a certain generalized cohomology theory topological modular forms. Nor do I discuss, at some point, there are alternative proposals out there involving yet different spectra, different cohomology theories in relation to the rock quantization. And I won't be discussing that. So finally, let me say that even though I titled this about field theory, and I'll focus on field theory, the generalized cohomology enters the neighboring fields of both spring theory and condensed matter physics. We'll see the spring theory explicitly, the condensed matter won't really appear in today's talk. All right, so one preliminary I wanted to say a little bit about is the differential version of cohomology theory. And that's a subject which has been extensively developed. It started in a sense with the paper of Chieger and Simons back in the 70s and inputs a different direction from Delinia's work in algebraic geometry. But by now there's a nice theory, maybe not totally complete. There's actually a book that was recently written by some young people that studied generalized cohomology. So that's a nice source and has lots of references. And we've heard a lot about that, for example, in the talk today of Yamashita. So let me just say briefly that a few points, if we have a cohomology theory, then cohomology theory is something defined on a nice category of topological spaces, for example, CW complexes or larger category. And so there's a notion of a differential refinement, which is not a cohomology theory at all in the traditional sense, and it's defined on smooth manifolds. So I'm not going to give any definitions. Again, this is a survey talk, but just some examples to get a feel for it. So if we look at ordinary Eilenberg-McLean cohomology in degree one, that has a nice interpretation. If I have a space M, needn't be a smooth manifold, then a class in here is a map to the circle to our mod Z up to homotopy. It's a homotopy class of such maps. And the differential refinement, which is defined for smooth manifolds, gives us an actual map. So in this case, the differential refinement is just this abelian group of maps. If we go to K theory, so here in degree zero, then in topology, we can model K theory by Z2 graded vector bundles. Traditionally thought of as the difference of two vector bundles. But we only know them up to isomorphism, and there's even a further equivalence relation that we divide by to get this K theory group. On the other hand, this differential refinement is, well, it's a group. So again, they're equivalences, but the objects here can be, again, Z2 graded vector bundles, but now equipped with some differential data, a connection or a super connection. Okay, and if we come to a degree two, this was degree one of the Eilenberg-McLean situation, HZ, in degree two, in topology, we get principal or mod Z or circle bundles up to isomorphism. And in the differential theory, we get bundles with connection up to isomorphism. So there's a commutative square one often writes down, for example, in this example with HZ2, if we take a bundle with connection, we can take its curvature, which is a differential two form that's closed. We can take its first turn class, which is a cosmology class in the ordinary cosmology. So this is the refinement map that I have on the previous slide. And these two both map into the real cosmology where they agree. If you get the I over two pi's correct, then the Durham class of the curvature represents the first-turn class of the bundle. But emphasize this is not a pullback diagram in the category of Abelian groups. And that's most easily seen if we take M to be the circle. Because in the case of the circle, the second cosmology vanishes every two form vanishes. Again, the second cosmology vanishes. And so we see these zeros. On the other hand, if I take a circle and I take a principal arm on Z bundle, we could have holinomi around that circle. And so you see this is clearly not a pullback. This is not determined by these two. But rather it's a pullback in a more sophisticated sense, a kind of homotopy pullback where we don't pass to equivalence classes but we take the whole theory at once. And that's related to the following remark, which is that both in geometry and in the field theory, one wants objects that are local on a smooth manifold that satisfy some kind of sheaf condition. And equivalence classes usually never satisfy such a sheaf condition. So this differential cosmology group is not really sufficient to do the kind of geometry and physics one wants to do. You need the kinds of objects that you can glue together. Those objects fit together into group oids or higher group oids. And so you need to remember the internal symmetries in order to get locality. All right. So I'm talking about field theory and in the back of my mind, but it won't always be at the forefront, is the framework for field theory that comes from the T in Siegel in Siegel in two dimensional conformal field theory in the 80s and in the late 80s. Also Tia in the topological case. But we think of that framework as applying in general. And the analogy you should think of is simply a representation of a lead group on a vector space or perhaps and if it's a long compound group, typically on a topological vector space. And if we don't want the invertible kind of analogy, we can think of modules over an algebra. So in this definition, the domain is a board is a category. And that is to say its objects are in minus one manifolds that are closed. We have board isms, which are compact and manifolds between them. And they're equipped with background fields. In a field is some sort of sheaf condition again, which assigns to every end manifold. So you think of an open small open set, it assigns some collection of fields. So that collection of fields might be a set, for example, the set of metrics on your manifold and those pull back under local defumorphisms or it might be a groupoid like connections, as on the previous slide, where we remember the automorphisms, the gauge transformations of connections. We need that again to get locality or it could be a higher groupoid as you meet with B fields. So the co-domain is some kind of category of topological vector spaces and a field theory then is a homomorphism from that domain to that co-domain. And that's just very structurally encoding what correlation function state spaces and all that are in a weak rotated field theory on curved manifolds. So I'll say there's a recent pre-print. Again, this is most developed in the case of topological field theories and then in conformal field theories. But there is a recent pre-print, maybe not so recent at this point, but still of Konsevich and Siegel, which talks about these axioms for general quantum field theories. And I think the understanding is we should think about them in that context. Now, field theories are local and unitary. Those are the two pillars really of field theory and unitary is not in this definition. We could put it locality is to a limited degree. And if we want full locality, then we have to allow ourselves to have manifolds with corners all the way down so we can do higher co-dimension gluing. And again, that's developed for topological field theories to great effect. And it's very much an open kind of area for general field theories. All right, so one more notion from general field theory is that of an anomalous field theory. And so there's a notion of having a boundary theory, first of all. So here I've shifted a little bit that the bulk here is an n plus one dimensional theory. And there's a notion of having a field theory on its boundary. And in fact, there's a notion of a left or right boundary. And this picture, you might think of alpha as the analog of an algebra. And this F is the analog of a left module. So this would be a depiction of a left boundary theory. And if the bulk theory alphas invertible, then we say that F is an anomalous theory with anomaly theory alpha. So I think that definition has been certainly used in previous talks. And one kind of refinement of that definition or one more honest definition is that if we're focused on n dimensions and really focused on this anomalous field theory, then we don't need an n plus one dimensional theory in the bulk. We don't need to evaluate it on some kind of curved n plus one manifold that doesn't have a boundary that has nothing to do with F. And so there's a notion of what I would call a once categorified n dimensional theory that is kind of an n plus one dimensional theory without the top level. And so, for example, Stoltz and Teichner of that notion, they call twistings. And that's really what you need in the bulk. And so in particular, for an anomalous field theory, that's what one should have. But in the examples that I'll say it's there are natural extensions to a full n plus one dimensional theory. And I'll just talk about that n plus one dimensional theory. And then the notion of the boundary. OK, well, anomalies are very useful. And you can often put more background fields. For example, if you have a symmetry, what it really means is you can couple to a background gauge field with this word in physics for that symmetry. And that can often lead to anomalies. Or if you have a constant, like a coupling constant in the theory or a mass or something like that, you can let that be a function, a scalar function. And that sometimes introduces anomalies. And if you study your theory over a bigger base, which means more background fields, then you're going to learn more about your theory. The same in geometry. If you study over a bigger base that's essentially more symmetries, then you're going to learn more about the theory. All right, so those are the preliminaries. And now I want to get on to spinner fields and how the K theory and its cousins enter via the index theorem. Again, just a high level survey. So first, what is a spinner field? Well, the theory of a spinner field starts in Minkowski space time. So here's Minkowski space time. And that's an affine space. It's got translations acting on it. And it has a translationally invariant print signature metric. And therefore it has null vectors, which are these red light cones. And there's one more piece of structure, which is a time orientation. You have to give a direction. One of the components of the time like vectors, there are two components. And you choose one of them, which is a positive direction in time. Or duly a notion of positive energy, which is what one means to do the Wic rotation. So what's the data one needs for a spinner field? Well, you need, first of all, the vector space where the spinners lie. And that's a real spinner representation of the Lorentz group. And spinner representation means that it extends to a module over the even Clifford algebra of the appropriate signature for Lorentz geometry. So this is an ungraded Clifford module. Now, there's a miracle about Lorentz signature that's unique to Lorentz signature, which is given any such real spinner representation. There always exists a symmetric pairing from that spinner representation back to the translations. And furthermore, you can arrange that that pairing is positive definite, meaning that on the diagonal, if I take the square of a spinner under this representation, I get into the closure of the positive time like vectors. So we use that orientation of time. So that's the theorem. It's a kind of miracle about Lorentz signature. And the set of the space of these pairings is contractable. So it doesn't have any topological kind of information, but it is important in writing down the kinetic term, for example, of a spinner field. And then there might be a mass. The mass might be zero, so there is a mass, and the mass is a skew symmetric invariant bilinear form that takes values now in the reals. So this gamma takes values in the vector space of translations. The mass is real value. And in fact, there's a nice algebraic lemma that tells you about these masses when they're non-degenerate. So as I say, zero is always a possibility. But the interesting question is, are there non-degenerate masses? And what happens with this gamma is that there's a blue gamma that makes S plus its dual into a Z now a graded Z to graded module over the same Clifford algebra. So we started with this ungraded real representation. We get this Z to graded one. And saying there's a non-degenerate mass is having an additional Clifford generator. So those are equivalent. And in the theory of Clifford modules, Alatiya Bachapiro, having that extra Clifford generator is trivializing in the K theory group of the Clifford modules. It's trivializing the module. So that resonates, I think, with how you might think about a mass certainly in relation to anomalies. OK, so we can ask this should define a free field theory. That's a quantum field theory, nothing topological invertible or anything about it, but it's anomalous. It should have an anomaly attached to it. And so we could ask for that, what is the anomaly theory? That's something that might be useful in studying the spinner field and particularly studying the spinner field as it appears in much more complicated theories. So at the end of the lecture, when we have a little more on the table, I'll answer that question. But traditionally, one approach this anomaly through the partition function. And so if we have n dimensional manifolds, which are the top dimension in the spinner field, then so n is the spacetime dimension, I might not have said it before. So here we have a family of n dimensional manifolds, which have a spin structure and a Riemannian structure on which then these spinner fields are defined. One can define that a family of Dirac operators. We use this data and Lorentz signature to define these Dirac operators in the Wicca rotated theory and therefore in Riemannian signature. So that's a little construction to make those Dirac operators in this generality. And then if you formally do the functional integral, it's a free theory. It's quadratic. What you get is the fofian of the Dirac operator. And that's naturally a section of a line bundle. So there's a fofian line bundle. And that's depicted here in red. And the partition function is a section of that bundle. So it might in particular vanish at some point. So that's part of this picture of an anomaly where this fofian line is the state space in n dimensions of an n plus one dimensional theory. And this is a boundary theory, an anomalous theory, which in dimension n gives us a section of this bundle. So that. OK, so this fofian bundle also carries geometry, carries a metric constructed by Quillen and a covariant derivative that is not constructed in general. And the theorem here is that the isomorphism class of this fofian bundle, nothing to do with the section, but of the bundle is computed by a push forward in KO theory. And so that's why K theory enters. It's because of the index theorem. And if we just ask about the bundle up to isomorphism as bundles without any geometry, then it's topological KO theory. And that's the theorem of the TN Singer. If we ask about the bundle with its geometry, with its covariant derivative, then we need to compute something not in H2, which is the line bundle, but in differential H2. In fact, slightly better because it's naturally a Z2 graded line bundle, so a little bit more. And there's a theorem of John Lott and myself, which proves a version of that kind of index theorem, giving the isomorphism class. We did complex K theory, but. One should be able to refine that to the real case. OK, so as I said, this fofian line bundle is part of a whole invertible field theory. The partition function of that field theory is an exponentiated Aden variant. And that's what you have on a closed and plus one manifold on a manifold with boundary. That Aden variant is naturally an element in the fofian line of the boundary. It satisfies gluing properties and so on to make together an invertible field theory, something I worked out with. Chins a die. And so that's the kind of top part of this anomaly theory. And later I'll give a formula in terms of differential KO theory, so not in terms of analytic constructions from Dirac operators, but in terms of the topological side, sort of implicitly applying a very strong index theorem. But just say what the answer is in differential KO theory. But I want to point out that in many circumstances, you don't need the entire KO theory, which has lots of homotopy groups, you can get by with truncations. And I'll just quote some theorems that tell you that. And that's useful because if that occurs as part of a larger physical theory, then you might actually use that truncation to cancel the anomaly against something happening elsewhere in the theory that involves this truncation of KO theory. So for example, if we look at supersymmetric quantum mechanics, so that's maps from a one-dimensional manifold into a target Riemannian manifold, then well, if you're studying that on the circle, you'll encounter the family of Dirac operators parameterized by the loop space. So this is the free loop space of maps from the circle in the M. And if we assume that M is both even dimensional and oriented, then this fofian line bundle has a very easy formula. So in fact, the fofian line bundle in this case is real and it has a flat structure. So it's determined just by an element of H1 with Z mod two coefficients. So we don't need all of KO theory to write down the formula for it. In fact, it just ends up being the transgression of the second Stiefel-Whitney class. So here the formula is just in terms of Eilenberg-McLean colmology. Okay, and we can get the entire invertible field theory that way. But now, supposing we drop the assumption that M is even dimensional and oriented, we consider the supersymmetric sigma model into one-dimensional sigma model into a general manifold M. Well, then we need more of KO theory, but we need just the truncation that keeps these homotopy groups, this piece of the box on. And that truncation is actually a ring theory. It's a multiplicative theory, ring spectrum. And well, so it turns out you can express the anomaly theory, the full two-dimensional anomaly theory in terms of this R. And here I've written for you the formula for the partition function. So this is an invertible two-dimensional theory that is topological. It's of order two. The partition function is plus or minus one. And it's defined on spin manifolds together with a map to this target M. You've got some kind of formula there. So let me tell you one more index theorem of this type in low dimensions. And here we take a further truncation. So this is a truncation of KO or even a truncation of R, let's call it E. And it just keeps these two homotopy groups. So it has, in a sense, three of them because they're spaced out. And there's a non-trivial K invariant that links them. It's not a product of Eilenberg-McLean's, but it's a twisted product. It's an extension. And in this extension, because of the Z mod 2, you get some additional divisibility that you don't get inside Eilenberg-McLean. So for example, for SUN bundles, there's a characteristic class, the third turn class. And inside this theory E, there's a characteristic class, which is half of it. In other words, twice this lies in ordinary HZ and is the third turn class. So if you imagine that we now are in four dimensions, so not that low, and we have a family of manifolds again, closed Riemannian manifolds, and we have a vector bundle over it with structured group SUN. So it has a trivialization of its determinant line bundle, in other words. And now we can look at the family of Dirac operators. The natural thing here is the determinant of the Dirac operator. And that makes a line bundle again over the space space S. And we could ask for the formula for its, for the isomorphism class of that line bundle. And again, it's naturally a Z2 graded line bundle. And the isomorphism class lives in this E theory, because this E theory in that degree is giving you line bundles. That's the H2 with Z coefficients, but there's an H0 with Z2 coefficients, which also gives you the grading. So this exactly gives you Z2 graded line bundles. And the formula is that you simply integrate over these four manifolds, over the fibers of that vibration, you integrate this half second churn class. So if you integrate that in this E theory, that's what you get. And again, the differential version gives you the determinant line bundle with its geometry, with its connection. Sorry, I'm confused. So isn't K01 Z2, let's, I'm confused about. Sorry, yes, I wrote that wrong. Yes, sorry. It's, thank you. Yeah, sorry. It's, that's absolutely wrong, yes. So the truncation is, yeah, okay. But it's like a, it's a two-state, but well, they're two non-trivial metopic groups and they're connected by that K there. Yes, so that part is correct. And perhaps what I should say is if we take the Anderson dual of this theory, then it's co-connected. It has pi not Z and pi minus one Z mod two. And those two are maybe better seen as the truncation. Anyway, sorry. Yes, you're absolutely right to point that out. Right, okay. Okay, yes. Thank you. But yes, if I don't think of it as a truncation of KO, I could just tell you this and this and that specifies the extension there. So that's what I mean. Thank you for catching that. Okay, so now let's move on to the second way in which exotic or generalized cosmology theories appear in field theory and that's through secondary invariance. So let me say something about secondary invariance in a particular case. There are other kinds of secondary invariance in geometry and this is in the case of the theory of connections. So both primary and secondary invariance. So the data we want to start with is a league group. And well, for the connection with topology, we need to assume it as finitely many components. It's not a countable discrete group, for example, but it could be a finite group. And we start with a p-linear form on the Lie algebra, which is symmetric and invariant under the adjoint representation of the group on its Lie algebra. So now this is well known that every time you have a principal bundle with connection, so a principal G bundle with a connection, the connection is a form. A differential form on the total space of this bundle pi. And it's a one form with values in the Lie algebra. And so we can form the curvature by this expression, which is a two form with values in the Lie algebra. And then we can take p copies of the curvature. We can simultaneously wedge them together and apply this multi-linear form. And so what we get is a scalar form of degree 2p up on the total space p. And the little lemma that follows easily from the Bianchi identity is that that's a closed form. So that's what's called the churn-vay form. So how do we get a primary invariant? Well, we have to know that this form actually descends to the base. So it's a form that initially lives on p, but it actually is the pullback of a form on m. And once we know that, if the base happens to be the same dimension as the degree of the form, and it's closed and oriented, then we can integrate. And this is the primary invariant. So that's a real number. And that invariant ends up being independent of the connection. It just depends on the topology of the bundle. Okay. But if we want an integer invariant, then we have to make some hypotheses that lead to the conclusion that this invariant actually lives in the integers. And so that's some integrality hypothesis on this form. So these forms comprise a vector space, a finite dimensional real vector space. It's the pth symmetric power of the dual of the algebra. And then we take the invariance under the action of g on that vector space. And inside that vector space, there is, well, there are many different kinds of lattices we can give, as I'll say. But we have to give some lattice that makes this true. And once we have that, so this integrality hypothesis, then we can define a secondary invariant, which depends on the connection. It's a geometric invariant. It's defined for two p minus one manifolds that are closed, but only those which bound a compact two p manifold, which is, again, oriented. And, yeah, so in that case, we can get an invariant. So I should say it's for these manifolds, of course, together with a connection. And the connection has to extend, which is a topological question of whether the bundle itself extends. But if we want an invariant that's defined on all two p minus one manifolds, and we want it to be local, then we need additional data. And that additional data will be enough to say that this secondary invariant, geometric invariant, is actually the partition function of an invertible field theory. And saying it's the partition function of a fully extended invertible field theory is the strong statement of locality. So here, again, is our principal bundle. And we have a connection which remembers a one form with values in the wii algebra. We have its curvature. It's a two form with values in the wii algebra. We have the two p form, which is scalar, which is the turn bay form. And I guess I didn't introduce the definition in this talk. We have a two p minus one scalar form, which is the turn simons form. And now let's think about the descent of all these things to the base. Well, the easiest one is this turn bay form. It's simply the pullback of a scalar form on the base, m. The curvature also descends to the base, but the curvature upstairs is vector valued in this fixed vector space, the wii algebra. Downstairs, it takes values in a vector bundle. That twists the adjoint bundle. So those descend to differential forms on m. And the curvature also descends to the base. And the connection descends to, but it doesn't descend to a differential form. It doesn't descend to one form. It descends to a section of some affine bundle over m, a bundle of affine spaces that you can make from the principal bundle. So in a sense, the connection descends, although it doesn't descend as a form. So we usually say it doesn't descend. But what about this turn simons form? That's where the secondary invariant comes. And we have to take integrality and then some additional choice. And then we can descend this, but not again as a differential form or a section of some simple bundle, but rather as a differential co-cycle. So again, I don't just want a class. I want an actual co-cycle or some representative. And we can descend. We can choose it to be in some generalized theory. So here again is where generalized cohomology theories that provide the integrality we need for the secondary invariant. That's where they enter. So the traditional choice is to take Eilenberg-McLean cohomology. And so this differential, this vector space I talked about is actually this vector space. It has an interpretation in terms of real cohomology. And one way to get a full lattice in there is to take the image of integral cohomology. That gives us the integrality condition on the previous slide. But to do this descent and define the secondary invariant everywhere, we need to choose a cohomology class here. So for example, in three-dimensional Churn-Simons theory, that one considers, one starts with a class in H4 with coefficients in Z, the traditional level. Okay. But we could put other cohomology theories here that also give us a full lattice in there by putting in some torsion groups. Okay. Now, how does this enter field theory? Well, in field theory, if you open up a traditional book on field theory, they'll say, what do you do? You define a set of fields. That's what I talked about earlier. But here and usually in Lorentz-Signaturm and Kowski spacetime, you tell a symmetry group and now you write down all expressions in those fields, but up to a certain order, the number of derivatives, which are invariant under the symmetry. And that makes the Lagrangian of the theory. And those expressions are typically, well, that the Lagrangian is a density. It's a density on the spacetime. And then the action you write is the integral of a density. So traditionally, this didn't allow for terms like Churn-Simons. And of course, it was discovered that those enter in lots of ways. And so what we look for now is an invertible field theory built from the fields. That's what we would replace this traditional point of view with. And that would then include these secondary invariants, like these Churn-Simons invariants, which originally I should say Jaquif and others were the ones who really put their finger on the role of those terms early on in field theory. Okay, so this is, as I say, a second way in which generalized cohomology theories enter into field theory. The first one I explained was from the anomalies of spinner fields. Okay, so let me give two examples with this theory E that I talked about before. Okay, so one is spin-Churn-Simons. So as I said, in usual Churn-Simons, three-dimensional Churn-Simons theory, one takes a level, which is a class in this cohomology group. But maybe refined to a co-cycle there. And a spin level is in this theory E. And those are related because we said that this E has two homotopy groups, one Z, one Z mod two, and they're separated by two. But we don't get a product. This group is not the product of these two groups, but rather there's an extension. And the extension, I told you, is given by this class. And so those together mean that there's a long exact sequence, but in particular a map from the levels to the spin levels. And so you can see that levels in particular map sit inside the spin levels. So if I take the circle group, for example, then it turns out that the levels are an infinite cyclic group, but the generators don't line up. In other words, in the spin theory, we get, so to speak, twice as many levels. We have an additional integrality. And of course, that's well-known in terms of Simon's theory. And a while ago, Jerry Jenkins developed this point of view and said some things about turn Simon's theory, using also eight invariants of real Dirac operators, to model some of them. Okay, second story is to do with something that is close to real physics, I think, which not this refinement, but at least the theory is, which is to say it's four-dimensional QCD, which is a four-dimensional theory that has a gauge group, that has spinner fields, and so on. And it has a global symmetry group, which is the product of the league group with itself, namely SUN. Flavors is what the F is. And at low energy, this theory is approximated by a sigma model into this homogeneous space, which is a way of presenting the group with its left and right multiplication symmetry. So that's called the model of pions. That's very traditional in quantum field theory. And Wesson's Amino introduced, they examined this from a kind of ad hoc point of view, saying that you should be able to match the anomaly in the high energy theory and the low energy theory. And they were working with local anomaly, and they introduced background gauge fields where you get the anomaly. So that's an example of spreading the theory out over a bigger base, more background fields, to learn more about the theory. And so they deduced a Wesson Amino term in the sigma model. Now Whitten looked at that subsequently, and he looked at it on S4, and he gave a kind of quantization and integrality to that term, but he used integer cosmology. And the way he was able to do it by detailed calculation of the kinds of factors that occur was that he was studying on S4, and he used a funny fact, which is that the Haravich map from homotopy to homology, in that degree has a factor of two. So here n is at least three capital N. So he used that extra two to be able to express what he wanted here, and the fact that he was only studying on the four sphere. But that leaves several mysteries that don't really match the whole story. So one thing is that the high energy theory requires a spin structure, and this low energy theory, as stated here, doesn't. You can have this secondary term, this Wesson Amino term, and if you're in here, you just need an orientation to define it. So matching the kinds of background fields from the high energy to the low energy is, if you like, a variation of the Adhoft anomaly matching, and there should be a matching there. Another related mismatch is that the QCD has fermionic states, but this Pion model doesn't obviously have fermionic states in the way that it's presented. Okay, another one is Wesson's Amino matched the anomaly over the reals, meaning the local anomaly, but we could ask about the more precise anomaly. And finally, there's a funny business when it's SU2, because in SU2, we don't have any degree five homology, and yet there should be a story also for SU2. So anyway, a solution to these problems I wrote about a while ago is to use this generalized homology theory E, this two-stage system, instead of integer homology. So this gains you a factor of two. Now for this secondary term, which really comes from this fifth homology, you see that sits in the E-comology again with a factor of two, just as the previous exact sequence, so we can divide by two. In other words, the generator here will get us an extra factor of two. And one thing you note right away is for n equals two, this group comes into play, press U2, and so this class, this generator is non-zero, even for n equals two. And so now we use this E theory to quantize this WZW term. So we write it, if we have the sigma model, we have a map of our four manifold into the group, we pull this back in the differential version of E theory, we integrate it, and that's our secondary invariant. And that solves all of these problems simultaneously. So we need a spin structure to do a push forward in this theory. This theory is not in the way I said, but nonetheless, related to KO theory, it's a module over some truncation. And that needs a spin structure, as you can check. When you go to compute on a three manifold, you'll hit the extra Z mod two there. And that will tell you that the, when you quantize that the vector spaces you get are naturally Z2 graded. So of course that was known. This all atomic states could be Bosonic or Fermionic, but here you see it in a more fundamental formulation. And now for the anomaly theory, I told you this four dimensional index theory using the E theory in the SUN groups. And that's exactly what you need to match this West Zimino term. So you see why these truncated index theories enter, then not just in the theory with spinner fields, but when you're trying to match with other things, then it's matching the secondary invariant. All right. So let me go on then to the third way in which generalized cosmology enters into field theory. And that is with Abelian gauge fields. So Abelian gauge fields have a Dirac quantization. And how does that look? Well, if we study Maxwell theory in physical Minkowski spacetime, flat Minkowski spacetime, then the characters that enter that plot are the electromagnetic field, which is a two form. The magnetic current and the electric current. So this is a purely classical picture so far. And that's it. And both of these currents are closed. That's the conservation law. And they have compact support along any spatial slice. Now Dirac gives an argument that says that if you look at these currents, they're closed. They have a Dirac class. Of course, their Dirac class is zero in the cohomology of M4. But this compact support condition means that it has a non-zero potentially Dirac cohomology class in cohomology with compact spatial supports. So that's here. And that vector space is one dimensional. That's the red line depicted. And Dirac charge quantization says that the charges, which are the cohomology classes, should be quantized in some fundamental units and so they should lie in some integer lattice there. So how do we impose this Dirac charge quantization in a model that you would think of as a semi-classical model, a kind of input into the quantum theory? So in the classical theory, we don't have this charge quantization and this picture in terms of differential forms, suffices. But in the quantum theory, we want something else. So we want to impose this integrality. And yet we have to impose it locally. So a current has local information. It tells you where the charges are after all. That's local information. But it also is supposed to contain this integrality. And precisely differential cohomology, but again, I emphasize it's the differential co-cycles that are local. They give you that marriage of locality and integrality. That's exactly the right tool, the right framework in which to model those two things simultaneously. So if we remember this square, then the forms here which are closed, those are the currents that have the local information about where everything is. And the charges are these cohomology classes which live in integer cohomology. And so you see these map to the same thing here. That's the statement that the charge over the reals is the Dirac cohomology class of the current. And so it's natural to say that we should lift to here. So again, in the refined model that you're feeding into the quantum theory, these currents are co-cycles for this differential cohomology group. Again, with the compact spatial support, but then we wick rotate, we do it on a closed manifold with interesting topology. And then of course, there's more interest here. So I've said that. But this principle of quantization allows that you use not just Eilenberg-McLean, but a generalized differential cohomology theory. And I think the reflex for a long time is to use Eilenberg-McLean, because we like to use things we know we're familiar with. But as I say, the argument that things are quantized really doesn't tell you how they're quantized. That combined with locality tells you something. But I think the something it tells you is that you should be in a generalized differential cohomology. So that raises the question, how do you know which one to choose in a given theory, in a given situation where you have an abelian gauge field? Well, I think the constraints on that are, you usually know which differential forms appear. You usually know over the reels, you can make local calculations, perturbative calculations, etc. So in string theory, you look at the effective field theory, supergravity, you know which differential forms appear. So that of course is a constraint that has to match whatever you're saying with the cohomology theory. But that doesn't fix the torsion in the cohomology theory. That's just fixing the rational information. But you might also have anomaly cancellation. And anomaly cancellation can interact with these abelian gauge fields. And the way they interact and the fact that the anomaly is expressed in terms of k-theory, or one of those truncations, can inform your choice of which generalize cohomology theory to use for this Dirac quantization. There might be solitonic objects in the theory, charged objects under these abelian gauge fields. And if you put torsion into the choice of the cohomology theory, that will predict in a sense more solitonic objects with that kind of torsion. And then there might be special geometric features of your particular theory that can also guide this choice. So I've talked about the currents and quantizing the currents. But once you do that, then the gauge field itself, which is one degree lower, is also naturally presented like this. If you want to write down the theory, you'll see immediately that you need the gauge field also to be a co-cycle for some differential cohomology. So I want to give several examples that mostly go beyond into Dirac cohomology. But some of these examples involve the idea of self-duality. And in self-duality, there's some situations where you have an isomorphism between magnetic and electric currents. And so you could ask for a single current. And in that situation, you have to give some additional data, some quadratic refinement of a symmetric bilinear pair. All right. So these exotic examples come from string theory. And the first example I say is the type one super string, where there's a four-form current, and usually that's thought to be it. So that's the current attached to the B field in type one, which is locally a two-form. But together, they make up, sorry, you can get a dual current, you can get a dual current, which is an eight-form. So one of them is magnetic, one's electric. But in fact, there's only a single self-dual current. But that self-dual current should be taken to be quantized in KO theory. And in that case, you get this quadratic form. There's a natural quadratic form one has to introduce. And when you do it, then the center of that form is an important computation you make in the theory. And in fact, twice the center is a natural kind of construction in KO theory of a version of a Wu class. And that class ends up being the key to understanding this Green-Schwarz anomaly cancellation over the integers instead of just rationally. Okay. That type two is really an oriental version of the type one. I mean, it's an oriental version of type two. That type two theory, that the Ramon-Ramon currents are quantized by K theory is something that was discovered by Moran Monazian and then Sen and then Whitten was the one who kind of stated, put those works together. And so this KO theory here is really understood to be a case of that. Now the B field in Bosonic string theory, that's just standard HZ. But the B field in type two super string is actually a richer object. And together with this, learn more, we proposed that this one is quantized in this truncation that we met before and this theory R, this multiplicative theory. So the current has degree zero in that theory. And that fits. Again, I said there were these various things that it would fit. And in this case, it's a very tightly constrained system. And this fits a lot of things, including anomaly cancellation. By the way, here's a puzzle for you in that story, which is not so apparent, but it's related to the puzzle I said about the pion theory. So in the string theory, if you look at the world sheet theory, there's a spin structure, of course, on the world sheet. If you look in the effective field theories, there's a spin structure on the target, on the spacetime. And my question is, what is the role of that spin structure on spacetime in the world sheet theory? So you should have some role, again, thinking about matching. So that's a question that you might ponder. Okay. So this quantization, again, is a local statement. That's how we've put together integrality and locality to use this generalized cosmology theory. But since it's local, you can imagine that, globally, you might have some twisting. That locally, the cosmology theory is constant, so to speak, but the cosmology theory itself, the spectrum, can twist over spacetime. So one place that occurs is in Oriental folds of type II, where this B field, now quantized in the theory I told you, those are exactly geometric twistings of k-theory. And that's how it looks in that theory. And in M theory is another case where there's a four-form magnetic current in that case. And that's, in the end, quantized just by ordinary integer cosmology is the accepted version. But it is twisted by the orientation double cover. That's the statement G changes sign under orientation reversal. And in that case, there's also a background magnetic current, which comes from the geometry. So that can happen. Well, I'm running near the end. I see, where's my boss here? No, he disappeared while I keep going until he reappears. So the remaining comments are about invertible field theories. I have maybe 10 minutes worth of such comments, but do it's up to you whether I cut off or I go for 10 more minutes. I think it's fine. Okay, thank you. Okay, so let me talk a little bit about invertible theories, because that's another way. And I think this way has been covered quite a bit in this conference. So I'll say a little less about it. But so a field theory is a map like this. Again, the topological case is the most developed. And saying it's invertible is saying that it factors through invertible objects in the codomain. So the invertible objects go by that name. But it means, for example, that the state spaces you're attaching co-dimension one to an n minus one manifold, those vector spaces should be one-dimensional, and so on and so forth. And so that's telling you where the theory factors. So alpha is invertible if it factors through that. And if alpha factors through that, then it also factors through the quotient. So in the codomain, we get a sub. In the domain, we get a quotient. That quotient is that I forcibly invert all of the elements here. So that's a localization process in ring theory, for example, where you invert. And so you get some completion of the domain. Well, the domain is bordism. So we ought to be able to know what this is. So these objects are Picard group voids, kind of infinity or higher Picard group voids, and they're equivalent then to infinite loop spaces. And so in the end, this theory becomes an infinite loop map. That's the symmetric monoidal character of alpha, means that this map is an infinite loop map between infinite loop spaces. And so you see that an invertible field theory can be modeled as a map of spectra, whose zero spaces are these invertible, are these infinite loop spaces. So that's something that Hopkins and Telemann and I have been banding about for, I don't know, 15 to 20 years, I suppose. We wrote about it first in a paper for Nigel Hitch and 60th birthday, where we applied it to a problem about orienting modular spaces, in fact. Okay, so what is the domain spectrum? Well, if we restrict now to topological theories, and so our background fields are topological, then in bordism theory, this was introduced in I think the 60s by Dick Lashoff, one models it by having a topological space mapping to B-O-N or B-G-L-N-R, if you like. So you could think of B-S-O-N, that would model orientations, B-Spin-N, that would model spin structures. You can take B-O-N cross B-K for some lead group K, that would model manifolds with a K bundle, K gauge theory, and so on and so forth. So that's the tangential structure. And now the theorem is that if we do this geometric realization, in other words, we take the quotient that I said in this diagram that we ought to know this is the domain of the extended field theory. If we do that, then we get a known spectrum. And that spectrum goes by the name of Metz and Tillman spectrum, was introduced by the two of them. And I forgot to write the names. These are various names of people that prove various versions of this theorem. Now, there's also a notion of a stable tangential structure. So this is a tangential structure tied to a degree. For example, you might have an N framing on an N manifold that's tied to the dimension N. But we could have a stable structure, like an orientation or a spin structure, in which case it's really a space that lives over the co-limit as you take N to infinity of these spaces. And that's called Bo. And so you get a sequence of these Metz and Tillman spectrums. And their co-limit is the more traditional Tom spectrum. So this is the spectrum that in equivalent words, Renee Tom introduced in his 1954 thesis to calculate boredism, which in turn was introduced earlier by Montriagin and ultimately Poincare. So we get these Tom spectra as well. And we say that a theory is stable if it factors through here. And it ends up that having a version of unitarity, kind of fully local version of unitarity for these invertible theories ends up meaning that the theory factors through this MTX, something Mike Hopkins and I proved. Okay, so these are different kinds of generalized cohomology theories. These boredisms are different than the ones we've seen previously enter field theory. And now the co-domain of an invertible field theory is also something. Well, here we don't really know what a universal good choice for a co-domain is for a general field theory, if it's extended. We don't even know that if it's topological and extended. But in the invertible case, we don't need that. We only need this infinite loop space. And so we can choose that infinite loop space. Yeah, and so what's the thing that we choose? Well, if we look at the sphere spectrum, then the sphere spectrum is characterized by that universal property. That's the definition of a homotopy group. A map from the S naught into X, I should have set up to homotopy is an element of pi naught of X. Well, duly, again, these are all homotopy, duly, there's a spectrum which we get by taking home, so to speak, into C star that produces a spectrum from the sphere spectrum. That's called the dual character, whatever dual to the sphere spectrum. And that spectrum is characterized by a consequent universal property. It says that if we map into the nth suspension of that spectrum, then that's the same as a character on the nth homotopy group of your domain. And now if you take the domain to be a boredism spectrum, whether it be Madsen Tillman or Tom, then what you're getting here is a character on this nth homotopy group, which is the boredism group. And so it says given a character of the boredism group, whether it be the one associated to Madsen Tillman or the Tom, traditional boredism group, then you get a map of these spectrum. And that's then up to homotopy. And that's then an isomorphism class of invertible theories, invertible topological field theories. So in other words, this universal property is saying that with this choice of co-domain, we could have made other choices, but with this choice, an invertible field theory is determined by its partition function, because this functional here is the partition function, which in this case happens to be then a boredism invariant. So that's very special to invertible theories. And the fact that that determines the theory is special to invertible theories with this choice of co-domain. Now there's some magic that it's a nice property to say the partition function determines a theory. That's not true for every quantum field theory, certainly, but it is a kind of desirable property. And in this case, that property, that universal property, as I said, characterizes this spectrum, this choice of generalized cosmology theory. And immediately you find from the rules of the homotopy groups of the sphere that what you calculate for in co-dimension one is not a vector space, which is invertible, so a line, but a Z2 graded line. That's the Z2, the class eta, in the sphere spectrum dualized. And in co-dimension two, well, you could think of this as an algebra or as a category, but that category has an additional Z mod two, which is, again, pi two of the sphere or pi minus two of this IC star. And so you're reproducing some known kinds of ideas in physics, bosons and fermions, just from the sphere spectrum in this invertible case. And that's a kind of bit of magic that you might not have been justified to expect. Of course, that leads to the question, what about the next homotopy group? The next homotopy group is Z mod 24, and you might think that has something to do with surface operators. So if we go away from topological theories, but we go to theories that are invertible, but not topological, and we've seen lots of examples of those, even this morning in Yamashita's talk, then the right language for that would be in terms of the differential spectrum, differential generalized cosmology theories. And in fact, the first conference at the Simon Center in Stony Brook, some more than 10 years ago, was on differential cosmology, and Graham Siegel said in his talk, I think what was clear is that the differential cosmology should be in some sense equivalent to having an invertible field theory, that a differential cosmology class should be an isomorphism class of invertible field theories. And that's undoubtedly true. There's probably enough technology to prove that. I don't know that it's ever been quite proved. All right, so let me close by giving you the formula I promised you, which is now that we have a language and a little more talk about invertible field theories. Remember, anomaly theories are invertible field theories. And I asked, what is the anomaly theory of a spinner field in terms of this basic data that defines the spinner field? And again, this has to be a conjecture. First of all, it involves differential version. Sometimes this anomaly is topological, but often it's not. So the differential form that lies behind this, I think physicists call the anomaly polynomial, might be non-zero. And so again, we need that theory. But we also, if we want to say this is the anomaly theory, we should really give a full construction of the spinner theory, the free spinner theory of a spinner field, quantum field theory with this anomaly. So that's why I wrote conjecture. In any case, I'm giving it to you with the mass being zero. And for the mass zero, well, it's a map from this tom spectrum. Again, the differential refinement really. But just for spin manifolds, we could make lots of variations of this formula if there are other kinds of background fields. And it goes to this universal target, again, in the differential case. And the map is constructed from this data. This, remember, is a contractable choice. So it doesn't enter into this formula. And what is it? Well, first, we have to get from spin boredism to K theory. And that's really what Atea Barton Shapiro explained in their paper. It's really the symbol of the Dirac operator or the Tom class in KO theory that is this map phi. Then we have a class of this data. As I said, this S plus S star becomes a module over Clifford algebra that gives us an element in some KO group. That's, again, part of Atea Barton Shapiro. Then we just multiply out in KO theory. And finally, the anomaly isn't the entire KO theory. This map would be the index, really, of family of Dirac operators, but we only want the Fafian line bundle. And that Fafian promotes to a transformation of these generalized cohomology theory. So in the end, we get an invertible field theory. Again, the differential version. And there's also another similar formula. If we include the mass, that appears at the end of a paper joint with those authors. And of course, the mass, if it's non-degenerate, trivializes the anomaly. And so really that formula is in a relative version of the theory where the class is trivialized, where the mass is non-degenerate. Okay. So thank you for listening. Thank you for the overtime. And I'll stop there.