 talk of the day. So let's welcome Daniel Berwick Evans, who will tell us about how few theories detect the torsion in topological modular forms. Please. Great, thank you so much. Can everyone hear me just to start? Okay, excellent. Just want to make sure, you know, we yelled into the abyss too many times in the last year. So I'll make sure that this is, you can hear me. So to start, thanks to the organizers for organizing this. I'm maybe a bit biased, but I can't think of a more exciting topic than generalized cosmology and physics. So it's great to be speaking here. This talk is going to be of a slightly different flavor from the previous one. So I want to start with just a bit of context. There's this conjecture from Seagull in the 80s and then Stoltz and Teichner in the early 2000s. They look to relate an object from physics with a generalized cosmology theory. And specifically, the conjecture is that there exists a map like so from a certain category or two category of field theories to a certain cosmology theory called TMF. And this is supposed to be thought of as like a geometric co-cycle map. So much in the same way, for example, that a closed differential form gives a Dirom cosmology class or a vector bundle gives a K theory class. This co-cycle map is expected to associate some object from geometry and physics, associate to that some object from algebraic topology. Okay, so the goal of the talk today is going to be to explain kind of an approximation to this conjecture. So both phrasing that approximation and then also giving an idea of how to prove it. And in doing that, this will give a partial answer to the title question about this issue of where the torsion TMF comes from. So before I go any further, I know it's a mixed audience and it's always, I've tried to split the difference between physics and math jargon. And I'm not sure how you can tell me how well of a job I did, but please interrupt me if I say things that don't quite make sense to you. And I'm sure if I can't answer it, I'm sure that the audience or enough of us that speak various languages that we can figure this out. Okay, so I want to start at the beginning of the story, which is the Witten genus. And this was what got this conjecture started to begin with. So I want to at least indicate how that happened and what the ingredients were. So I'm going to start with a description of the Witten genus that relies on ideas from index theory. One of the important rings that shows up when talking about the Witten genus is the ring of modular forms, which for now I'll think about them as living over the complex numbers. But then there's also this version of them called integral modular forms, which are those modular forms whose Q expansion has integer coefficients. Okay, so this is just the setup. What happened, what Witten did was he wrote down a map that goes from string manifolds to this ring, Z with power series and Q. And that map sends a string manifold M to a sequence of indices. You'll put my Q first. So Q to the N is just a formal parameter. And then there'll be some index of a Dirac operator. I'll give myself a little more space. Tensored with a certain vector bundle that depends on N. And there's a vector bundle over this manifold M. VN has an explicit description in terms of the tangent bundle of M, but for now that's maybe not so important. The issue is that this thing, because we know that the indices of various Dirac operators are integers, this gives me a power series here. A nice invariant, and in fact, Cobor's invariant of my manifold. Then you can compare this with a certain map that was implicit in Witten's work, but perhaps clarified by Zagier, that sends a manifold M to a pairing between that manifold and a certain characteristic class. And that characteristic class is in some evident way going to give you a modular form. And then using essentially the index theorem, the Atea-Singer index theorem, you know that the churn character here, so the map down there of that invariant is going to be the same as the Q expansion of this modular form. And so the equality between these compositions gives you a map into the pullback. And the pullback is basically, by definition, integral modular forms. And I get a map like so. So I'm setting things up in this particular way of telling, because this square and kind of its ideas are what are going to be generalized later, where various maps in the game are going to be expanded upon and elaborated upon to more fancy versions. But the basic idea at the beginning was you have one invariant that's clearly integral over here. You have another invariant that's clearly modular over here. And they agree, so you get an integral modular invariant. That's the gist. I want to emphasize that the map that you get into this pullback is really a property. And the property being that this composition is equal to that composition. And later, one of the things that's going to change is that it's no longer a property that witnesses the equality of these things, but rather additional data. So I just want to, I mean, that's going to be perhaps a little bit too cryptic for now, but I'm going to come back to that later, that it's important here that some property to get an integral modular invariant. And that property is precisely that this fractional, fractional Pontriagin class vanishes. So part of being a string manifold is that your spin, that's these first two conditions. And that's what was required to even write down these indices over here. The next condition for being string is that a certain Pontriagin class vanishes. And that's the condition that makes this a modular form. So you need all of this in order to get a map into the pullback, but it's really just properties that these things vanish, not additional data. So now I want to kind of rinse and repeat in the QFT world. Maybe I should pause. Are there any questions about this story, this classical story? I don't know how much index theory people have absorbed. It's always hard when it's your home turf to know what exactly to say. Okay, I'll charge it ahead. So perhaps for some of you, this will actually be clarifying the QFT version of the story. So similar to the previous talks, when I say QFT, I'm going to mean some kind of category of functors. And the precise definition I have in mind is due to Stilts and Teichner, where I want to at least give an impression of what this looks like. So there's some boredism category here. It's not a topological boredism category. There's extra geometry involved. And the morphisms in this boredism category are going to be two one dimensional Euclidean super manifolds. And so Euclidean in particular implies that those are flat when regarded as ordinary manifolds. And then they have geodesic boundaries. So let me just draw some kind of cartoons of what these boredisms look like. So this flat with geodesic boundary for two manifolds really restricts what you're allowed to do. I guess the Gaussian A theorem if you like. So we can have cylinders. We can have tori. We can have cylinders viewed as boredisms in other directions. And that's basically it. You can take compositions of these things and disjoint unions of them, but there's not a whole lot in terms of the topology of this boredism category. However, there is a lot of interesting moduli. This Euclidean aspect is a metric. And so there's not just a single cylinder, there's a whole moduli worth of them. And same for tori and these other things as well. So it's somehow you're trading off what normally leads to interesting algebraic structures from topological field theory for being algebra style things for interesting geometric structures, which is going to be functions on certain moduli spaces, which inevitably will be related to modular forms and things like that. But it's both kind of easier and more difficult than the topological setting that perhaps is more familiar. Okay, so when I say QFT, I mean a functor out of this the sportism category is indicated, valued in vector spaces. And then there's additional things that I'm probably going to sweep under the rug, but they do include things like reflection positivity for those for fans of reflection and positively out there, as well as some other things that's kind of I don't know, decorations that you need to make sense out of the conjecture above, but that I'll probably just just delay for now. All right, so now I want to describe using this this notion of QFT, how the wit and genus at least ought to arise. Okay, so first, from this category of string manifolds, there's this thing called the supersymmetric sigma model with equals zero comma one supersymmetry. So this chiral supersymmetry. And that's expected to give one of these functors, I admit that we, this has not been done yet, but this is certainly the expectation of what should happen. Okay, so now given one of these quantum field theories, there are certain things you can do. One thing you can do is you can restrict restrict to cylinders. So that's a subcategory just built out of cylinders and how they glue together. It's a nice subcategory, the boredom category, and that gives you basically just by definition a representation of that category of super cylinders. Another thing you can do is restrict to Tori, which might be called a partition function, and that gives you a function on this module I of super Tori. All right, so now you have to do some calculations. There's some geometry that one does, but the first observation is that these functions are in fact isomorphic to modular forms, or maybe modular functions. I'm going to ignore that caveat for now. And then these representations of a category of cylinders, there's like an index or a dimension count that you can do that lands there. Perhaps I should say what this formula looks like. So you take the super trace of Q to the L0, Q bar to the L0 bar, and that gives you something down there. Because of supersymmetry, this L0 bar has a square root, often called G0 bar, that will make the dependence of the super trace just on Q and not on Q bar. That's where you go down here. And then because of kind of general field theory considerations, the trace of the value on cylinders has to be equal to the value on Tori, so that means that these compositions actually are equal, and hence you get a map into this pullback. Apologize for the background noise, they've decided to mow the lawn across the street today. Hopefully this will be over soon. I debated between giving the talk in my office where people knock on the door now because they can actually find you again. Giving the talk at home where they can't find me, but neighbors do stuff. Anyway, we can't hear the background, so don't worry about it. Oh, even better. Okay, great. Wonderful. Okay, so the point here again is that there's a composition this way and a composition this way, coming from the values on cylinders and Tori. They're equal. You get a map into integral modular forms, and this composition here is the Witten genus coming from field theories. And now this property that these compositions are equal corresponds to the fact that an anomaly vanishes. So that P1 over 2 anomaly is, for the same reason, that's going to give why you have an integral modular form here. But again, it's just a property. There's no additional data involved in getting an integral modular form. Okay, so these are two versions of the Witten genus, one that's perhaps more familiar in index theory to mathematicians, the other that's perhaps more familiar in physics. But the flavor of the argument is basically the same where you have kind of two invariants that are related. And because they're related, you get a more interesting invariant that comes from the combination of the two. So a natural question to ask from the point of view of a homotopy theorist would be, is there a family's version of this Witten genus? So given a bundle of string manifolds, is there a version of this Witten genus that gives me some class and some cohomology theory that in the case that this x is a point recovers the Witten genus above? And the answer is yes. Yes. And the cohomology theory is this cohomology theory of topological forms. And the map is the signal orientation. And this is due to this really deep work of Ando Hopkins, Strickland and Resk, where Hopkins and collaborators constructed TMF as sort of purpose built to be the recipient of this family's Witten genus. And then these authors constructed the signal orientation. Okay, so that's a nice answer for generalizing the topological side of the story. Let me just say a few words about topological modular forms before continuing. The first thing is that we know basically all there is to know algebraically about TMF that one would want. So for example, we know it's coefficients completely. And from coefficients and Myrviator sequences, you can start to get a hold of what TMF is applied to any space that you would like. I'm saying this is easy, but in principle, this is kind of the usual way that one approaches cohomology theories. So you might ask, well, what specifically is in here? And one thing is that you get modular forms. These are rational modular forms. And there's this map that just goes from TMF to TMF tensor q. And that turns out to be exactly rational modular forms. However, this map has a lot has rather large kernel and co kernel that consists of a bunch of two and three torsion. And this two and three torsion is really interesting from the point of view of topology. It has a lot of information about the homotopy groups of spheres. And so we would really like to understand this two and three torsion from a more geometric point of view, much like we were understanding this modular and integral data appear from a geometric point of view using field theories. So the mystery that this all kind of leads to is where is all the torsion in this category of field theories. And I really do emphasize, I mean, this is some very complicated information. So if you're going to find it, this category has to be kind of subtle and complicated in some way. And it's not entirely clear from the outset that this that this question has a reasonable answer. But I want to convince you, at least to some degree today that it does. And we can say a bit about where this torsion comes from. Okay, so maybe this is a good spot to pause. Are there any questions so far? So given this family's wit and genus that I at least indicated here valued in TMF, I want to talk a little bit about using the same ideas from quantum field theory to try to produce a family's wit and genus using QFT techniques. So the first thing is because we're adding a family parameter X, the smooth manifold, I want to enhance my bordism category to be the same two one Euclidean bordisms, but now they're just going to have a map to a smooth manifold. So you can always do this sort of like a sigma model style classical fields for a signal style thing to do. It's being maps from certain super manifold to X. And then there are two analogous subcategories to the categories we used before of Torai and Anulai. So first these super Torai, they will include in the bordism category as borders from the empty set to itself. So the picture that you would draw as you have your tourists and it goes from empty to empty. But now this is all kind of considered within X. Everything has been mapped into X, not necessarily embedded but mapped into X. And there's another condition that I'm going to include, which is important technically, but maybe not philosophically for now, which is that the map to X is actually going to be nearly constant. And I mean nearly constant in the sense of super geometry that it's like constant up to nil potents. So these Torai are only going to really probe like a formal neighborhood of the constant maps to X. And the next thing is to look at super cylinders with again a nearly constant map to X. And it's going to be kind of the same idea of a picture. But now this is an interesting category that it goes from a loop to a loop in X. But now, okay, the loop will be in some formal neighborhood of the constant loops and so will the cylinder between them. What's interesting is this categorical structure coming from gluing the cylinders together. Okay, so these are two nice subcategories of this bordism category, bordisms over X, and they're going to get restrictions from this category of field theories. My field theories are functors out of the bordism category. If I have a subcategory, I can restrict. Is there a question? Oh, no, okay. So we get these restrictions like that. And then there's some further compatibility between the restrictions. So if you take a trace here, and this is basically a forgetful map here, we're over here, I'm going to have a category of Torai where they have a chosen meridian. So the pictures are, I have my Torai over here, my cylinders up, maybe I'll do this in a different color, sorry. Got my Torai over here, my cylinders up here. I can glue my cylinders together to get a torus, but this torus has additional data. It has this chosen meridian. And then, well, if I have a function on Torai, I can get a function over here just by, I forget that choice of meridian and evaluate my original function. So this is just like a forgetful map. This is a trace map. And we expect these things to this diagram to commute so that if I evaluate on cylinders and take a trace, that's the same as evaluating on Torai and maybe adding some additional decorations to this Torai. Okay, so it's basically the same story as before, but there's just this manifold X everywhere. So everything is over X. And that's going to lead to some more interesting calculations when you start to analyze this category of representations and this category of functions. So I'm about to describe those calculations. This is in some sense the main meat of the story where you actually have to do something. And I won't get into that many of the details of how the calculations go, but I want at least to indicate kind of what the results are that you get out of this. Okay, so the first calculation is you compute these functions on Super Torai over X. The short version of this story is that there's a map that's basically a diram map that goes to ordinary columnology with values and modular forms. The longer version of the story is that these functions are in fact isomorphic to triples of differential forms, where the first differential form is like it wants to be a differential form value in modular forms, but it fails in two ways. The first is that it fails to be holomorphic. So here this is h is the half plane, which you think about as being the conformal modulus of a torus. And r is a positive real number. You think about as being the volume. Remember these Torai had a flat metric, so they both have a conformal modulus and a volume as invariance. And so this function here, if it were to be a modular form, would have to be independent of, excuse me, independent of the volume and then only depend holomorphically on the conformal modulus. It doesn't quite, but its failure to do so is measured by certain exact terms. So roughly speaking, this is like you have functions on Torai that are holomorphic or conformal, but only up to a specified exact form. So it's not quite kind of the naive thing you would write down that computes comology with values and modular forms at some slightly fancier version, but nevertheless, there's a map. And the map this way is just you send this triple to z lives in here. And it lives in here exactly because these relations tell you that, in fact, this only depends on the holomorphic part of h. Okay, so that was a bit rushed. But again, the flavor of this is that you calculate functions here, you get some explicit collection of differential forms. And those differential forms give you a co homology class. It's a co homology class and sort of the thing that when x equals point is exactly modular forms, and hence gives an enhancement of what we had before. Okay, the next calculation I want to describe is you're looking at, again, these representations of cylinders over x, those guys. And the short version again is that there's a, there's like an index map like that, that lands you in k theory with power series in q. So if x were a point, ignoring some two torsion for the minute, this would look like integral power series like what we had before. But x is not a point, and you get this more interesting k theory class. The longer version of the story is that there's a map like so it's almost an isomorphism, in some sense the word. And these are self adjoint super connections a n and you have one for every q to the n. And this map here is really like the bismuth family index applied to a family of super connections. So this is another classical construction index theory that one can do. But to do it, you have to calculate something about this category. Okay, any questions on these? I admit I'm being a little bit of giving you an impression rather than telling you a lot of detail, but I'm hoping that this is enough to give you a sense of what's going on. Okay. So the next goal is to put all this stuff together. So by put this together, I mean we have these calculations about functions on this modulized space representations of this category. And then we also have some relationship between them coming from the fact that when I glue my cylinders together I get Torai. So I want you to remember this pullback square we had before that looked at integral modular forms as a pullback of modular forms and integral power series, saying that they agree there. So now we're going to have this restriction here. So that's what we had before but now this calculation gives us a map this way. So this is, I'm doing this backwards computation too. Also have a restriction this way that restricts to Torai. And then we have this other calculation here. And then, okay, there's some things we can start to do with these, these cohomology classes. So one is if I have a k3 class, I can take its churn character. And that's just going to land me into ordinary cohomology made appropriately periodic. I can ignore that. The other thing I can do is I have a map on coefficients that just comes from this map we had before, which is called Q expansion. And that gives me a map between these cohomology rings down here. So just to reiterate, I have ordinary cohomology with coefficients in modular forms, ordinary cohomology with coefficients and see double brackets Q and k theory here. And I have a map from QFT to one that goes to these, to this place. Now, the key fact that came from the calculations in some sense is that, well, it came from the calculations together with some further calculations is that these are not equal, but rather they're only homotopic. And so that gives you a map not to the strict pullback, but to the homotopy pullback, which I'm going to call komf of x, just because it's some amalgamation of KO and modular forms. So this is defined as the homotopy pullback and from universal properties, we get a map like that. So I'm going to say a little bit more about this homotopy pullback in a second, but I just want to emphasize the parallels to what we had before. Before we had QFT to one, and we had these two maps coming from evaluating on Torai and Anilai. And we saw that they were, or we argued rather that they were equal and that gave us an integral modular form. Here we again have this category, but now it's QFT over x and we have a restriction to Anilai that goes over here and a restriction to Torai, but now they're not equal down here, but there's a specified homotopy, which is how homotopy pullbacks work, that witnesses the compatibility of these compositions. And that gives you a map by definition into the homotopy pullback here. So this is a more subtle structure than you might have expected, just from usual field theory reasoning, you'd expect to only have a map into the strict pullback saying that these values are just equal on the nose. What we're seeing here is something a little bit more subtle, that the values are homotopic and therefore you get a map to this pullback. Can you maybe say what's the meaning of 2Q series being homotopy? Yeah, right. So it's not that they're actually, okay, the theories themselves are not homotopic. As values, they actually are equal. It has to do with this information here. So the identification between the partition function, which might normally just be called Z from this data, and a class in cohomology with values and modular forms are these specified homotopies that tell you how the thing fails to be homomorphic and how it fails to be independent of volume. And those are the homotopies that show up here. So things are kind of getting sloshed around a little kind of out of necessity, but the point is that when you are writing down a class down in here, you're not writing down, somehow it isn't just going to be, for example, a differential form of values in this power series ring. There's also some Q bar dependence, but then there's some exact stuff that tells you how to cancel that Q bar dependence. Yeah, I see. Thanks. Yeah. Yeah, I mean, this is good. This is somehow like the crux of the issue is right here. So it's good to pause for a minute. So you want the microphone died, so you have to suffer the computer. You want me to read this as actual co-cycle models everywhere, yes? Correct. Yeah, I'm going to say some more about that in a second. But yeah, right. So I'm saying there's a map to this homotopy pullback, but you can't really define the homotopy pullback with just the cohomology theories alone. And there is some co-cycle stuff in the background that's going on exactly. So I'll say more about that in just a second. But you're right, that if I just had a map to this cohomology theory and to this cohomology theory, they would be equal. It's the co-cycle models where the homotopy lives. That's correct. Any other questions? Okay, so just to emphasize, there's this additional data that's showing up whereas before it was just property, when we're mapping back to integral modular forms, it was just a property that P1 vanished. They gave us that map. Here, this data that's showing up is going to turn out to be not just that P1 vanishes, but how it vanishes, which is called the choice of string structure. And it's equivalently the data of anomaly cancellation, not just that the anomaly cancels, but like how it cancels. And that's going to give rise to a map into this pullback, homotopy pullback. Okay, so the goal next will be to say some more about who this character is, what we can say about it, and why it's interesting. So I mentioned at the beginning that there was going to be some torsion somewhere. And the next goal is to describe how this cohomology theory sees some of that TMF torsion. But I think I should pause for just a second longer. Are there any more questions on this setup? Okay, so I want to talk about this homotopy pullback. So first of all, what does it mean? Well, everything now has transitioned from the world of cohomology theories to spectra, which I apologize slightly if that's not your favorite, or your cup of tea, but I will at least try to say how this goes. So the spectra represent generalized cohomology theories. And I have maps of spectra like this fact ring spectra. And that allows me to take a homotopy pullback of ring spectra. And that's defines this KOMF spectrum, whose associated cohomology theory is what I was calling KOMF before. So TMF has maps that go like this, and like this. This one sometimes called the Miller character due to Haynes Miller. And this other map is sort of an easy one. It comes from tensoring over C. And once you're over the rationals, every cohomology theory looks like an ordinary cohomology theory with values in some ring, and you just compute that that ring has to be modular forms. So this map is fairly cheap. This map takes a little bit more doing but exists. And then it's a fact that these are homotopy compatible. And the beauty is that that gives you a map from TMF to this KOMF. And again, there's more here to this arrow than just the fact that you have these two arrows. And they're somehow equal in cohomology. There's a specified homotopy that shows up to construct this. But what that buys you is now you have, well, first of all, there's some sense in which KOMF is approximating TMF via this map. And it gives you the string orientation of KOMF like that. So this thing is going to be kind of a nice place to start for understanding that conjecture I mentioned at the very beginning relating field theories to TMF allows to say a certain specific relationship precisely. And then the remainder of the conjecture will be exactly the failure of this to be an equivalence, for example. So the first thing I want to point out is that this square is basically a homotopical refinement of the square that defined integral modular forms. So ignoring for the fact that this is KO, if this were KU, so complex K theory, its coefficients would be exactly Z with power series in Q. The coefficients down here are by definition, modular forms. Same with here, the coefficients are by definition. C with power series in Q. So if you just take coefficients here, you almost on the nose go what we had before that defined integral modular forms. And for that reason, I would invite you to view this KOMF as being some kind of intermediary homotopical enhancement of integral modular forms, just as TMF is also a homotopical enhancement of modular forms. All right. So some facts about KOMF. So the first is that you can calculate its coefficients. And it's actually relatively easy from the point of view of homotopy theory using properties of homotopy pullback squares. And let me tell you some features of the coefficients. The first is that integral modular forms include as a subring in degrees four star. And the second is in degrees four star minus one, you have these kind of enormous groups that would accommodate torsion. So power series with complex coefficients, modular power series with integral coefficients, that would be like an infinite number of these circles. But then you're also dividing out by the Q expansions of modular forms. So this is sort of a funny abelian group, but it's one that shows up actually quite frequently in homotopy theory for various secondary invariance. And for that reasons, maybe not from that point of view, it's maybe not very surprising that it's showing up here as well. Okay. So this first thing is supposed to tell you that KLMF is good at seeing integral modular data, like the Witten genus. And the second thing is telling you that at least in principle, KLMF could see some torsion variants as well. So on that first point, it turns out that the composition that goes from M string to TMF to KLMF is just the Witten genus for string manifolds in the appropriate degrees. So the string orientation of KLMF just knows the Witten genus completely. In these other degrees here, where you're getting something a little bit more exotic, this actually recovers a torsion variant of string manifolds that was previously studied by Bunke and Naumann. It's known to be non-trivial for explicit examples of string manifolds, and I'll show that in a second, or at least towards the end of the talk, I'll explain an example. But the point is that we are actually going to see some interesting torsion variants of string manifolds that came from TMF that land in KLMF. And this is due to the work of these guys. You can even be more specific about what kind of torsion, TMF torsion KLMF sees. So the first statement is that this canonical map injects on three torsion in these degrees. And just because it's an variant that a lot of us know and love, the Zmod 24 that's in Pi 3 of TMF, which is also Pi 3 of the sphere that's generated by this class that sometimes called new, that also injects into KLMF. So if this isn't something you've seen before, you can maybe ignore, but this is an invariant that people care about, and it's detected in KLMF, and then there's a bunch of other invariants as well that are detected there. So it really is, I mean, it's not just sort of a feudal exercise in playing with homotopy theory and category theory. There's actually some content to this theory that's coming out of this homotopy pullback. Okay, so I'm just going to scroll up for a minute just to remind you of the field theory setup after this kind of homotopical interlude. So the setup was that we can restrict one of our field theories to annuali. That gives a class in k theory. We constrict one of our field theories to tori. That gives a class an ordinary cohomology and they're homotopy compatible here. So they get a map to the homotopy pullback. And that's the the theorem that I'm going to phrase here that restricting an object of QFT21 to cylinders and tori gives a map like this. So a co-cycle map. And furthermore, that co-cycle map factors through a somewhat easier category. And because it's a bit easier, I want to highlight it. It's this category here. This notation, I think I learned from Dan Fried, but it's supposed to be referring to the fact that these field theories are extended once. So they have values on top dimension, one one co-dimension lower, but nothing more. And so if you said in words, it's field theories defined on cylinders and tori, but not on two manifolds with corners say things like that. And by by construction, this comes from first this truncation or restriction that you always have. And then there's some calculations that go in here. And that's what actually constructs this co-cycle map. Okay, so that's that's the first, I mean, I'll have some more comments about like how to think about this result in a minute. But this is giving some map from these quantum field theories to a generalized co-amology theory that has some interesting TMF information. And specifically this integral modular data that we saw up here. And also some torsion data coming from these results here. So any questions on that first? Can you say what's in the kernel of this map from Python TMF to Python KOMF? In principle, yes, but in practice, so I cannot, but certainly someone with better computational capabilities could. I can say some specific things, though, for example, that that might be of use. So KOMF is 24 periodic, whereas TMF is 24 squared periodic. So you know, for example, like where the bot element like the, you know that there's some failure going on with that bot element of TMF going to the bot element of KOMF along this map. What else can one say? I guess, so in that same vicinity, Delta is the is the discriminant. That's the generator of the 24 periodicity down here. And you know that, for example, Delta is not in here, but 24 Delta is. So that tells you another map where 24 Delta goes to 24 Delta, but there's no lift of Delta divide, you know, 24 Delta divided before. So there's some specific things I can say, but like in general, giving you a complete description of the kernel and co-kernel is a little above my pay grade. It's not, I mean, in principle, it's very calculable, but it's not a calculation I know how to do. Okay, I see, but it's not empty, right? Oh, no, no, no, no, there's, yeah, there's let's see. So maybe one thing in addition to say is that the torsion and TMF and odd degrees is all finite, and this is very much not finite. So it's going to fail in a like rather extreme way to be subjective on homotopy in degrees 4k minus one. So there's more to say, and I could probably give you more like kind of random examples. I don't have a complete description of the kernel and co-kernel, but I can certainly point to specific elements that do or do not either are killed by this map or do or do not lift along that map. Yeah, thanks. Yeah, sorry, Tan. Yeah, 3.1 can't be correct. 3.1. Oh, sorry, I misread fact 3.1. Never mind. Are we good? I just misread it. It's possible. I mean, typos happen. I think it's okay, though. I think it's okay. No, I just misread it. Okay, okay. Sorry. Well, to answer, well, to give an example to do this question, I think that KOMF probably has, you know, the three torsion in KOMF is only in degrees 4 star minus one, but TMF has lots of three torsion, you know, other degrees as well. Yeah, yeah, I guess, I mean, maybe that would be a better answer is what Theo's indicating. We know the homotopy groups of both of these. You can just write them down and compare, but like I don't have a slick way of describing that comparison other than I can hand you two groups and you can look at them and decide if they're different or the same. So, yeah. Yeah, it's all calculated is what I'm getting at. It's just like getting a handle on what that calculation means is maybe not not. I don't have a good way of thinking about it. Okay, so the next thing I want to describe is this simpler kind of partially defined field theory. So I should mention it's partially defined in two ways. One is that it's only one extended. And then the thing that the notation is kind of hiding is that it's only defined on these constant maps to x or nearly constant maps to x. But nevertheless, this is somehow a much simpler object. One of these guys is a much simpler object than this. And it's simple enough that you can just start writing down examples from geometry. In particular, if you're given a string manifold, you can just write down one of these QFTs basically using, again, Bismuth's super connection associated to a family of spin manifolds, Bismuth writes down a super connection for the family of Dirac operators. This is one of the things that features in the family's index theorem. And using a verbatim, that construction, you can write down a map like this. And then we already had a map like this from the previous theorem. And then the theorem here is that this actually factors the string orientation. So the map, the composite here is the map we had before from string manifolds over x to KLMF. So this together, this results up here and this result down here give you a first sense of where, first of all, TMF torsion might be coming from in field theories and also how to explicitly get it from examples of string manifolds. And that's the example I want to describe next. And this example in, I don't know, maybe Theo can say if this is the physics literature, the mathematical physics literature, but I learned this example from Gaiotto, Theo and Witton's paper, where they look at WZW with supersymmetry and target S3. So I want to think about this as a string manifold over the point when plugging it into this theorem here. And so from the string manifold, where a choice of string structure is the same as a choice of degree three class, I get one of these partially defined field theories here. And then knowing what the string orientation of KLMF does, I can just write down what it is over here. Oops, this goes to, I'll write it down over here, this goes to alpha new in here. So new being the generator of that Z mod 24 in TMF, alpha sum integer over here. And I take that integer mod 24, and I get an invariant there, that's a, it's a torsion variant, it's 24 of it vanishes. And this gives something that came from pi three of TMF in this world of field theories. So there are other examples of string manifolds that basically play the same game, they get more complicated, where you have to put in something kind of fancier here to get something fancier over there. This is the easiest one I know of. And perhaps the most important one, somehow this class new is the first class in TMF, that's torsion that you don't see from K theory. So it's like the first example of a torsion class that would be new beyond what you expect to see from just K theory results alone. And here it is in terms of at least a partially defined field theory, that the notation is, you should be thinking about this as a sigma model with target S three, and maybe I should put it in scare quotes, because it's not the completely defined theory. But nevertheless, it's enough of the information that field theory to understand this torsion invariant. Any questions on that example? So just a little bit more. This is probably things I've said already, but the middle here is this partially defined field theory representing a class and pi star of TMF. The other important point to emphasize here is notice where this torsion is coming from. It's this choice of string structure. That choice is data. That data was the homotopy that was in that homotopy pullback that's giving us an invariant here. Mathematically, again, it's a choice of string structure. Physically, it's a choice of anomaly cancellation. And so the data in this kind of proposed space of field theories needs to include a structure, the data of anomaly cancellation in order to recover TMF. That's one of the lessons learned here. Okay, so these theorems above, just to give you a slogan, I don't know if you like slogans or not, but it's sort of a one categorical version of this Siegel-Stilts-Teichner conjecture. One categorical because we're doing one extended things. And I mean, it's Siegel-Stilts-Teichner in the sense that you're getting something that approximates TMF. So the full conjecture would involve extending down, so fully extended field theories. And one thing that this analysis gives you is kind of an indication of exactly where to start hunting for the remaining torsion that's not from KLMF, and it's got to be from field theories that are extended down to points. And so one question you might ask yourself is does extending down have the same obstruction theory as lifting a class along this map of co-homology theories? And this would be equivalent to Siegel-Stilts-Teichner as this diagram would indicate. So if I'm thinking that I have a one extended theory here and I'm trying to lift to a two extended theory, so that lift should be the same information kind of up to homotopy as lifting a class here. So as we were discussing a little earlier, not all classes here have a lift. And if there is a lift, it's not unique. So there's both an obstruction to lifting, and then there's a moduli of lifts. One would hope, and there are, for TFDs at least, this is usually a well-defined problem. If you have something that's extended so far and you want to extend further, there's an obstruction, and then there's a moduli of lifts. And if one could set up a similar type of lifting theory, then you might hope to actually get a handle on this entire Siegel-Stilts-Teichner conjecture. Okay, that's all I had to say. Thank you for your attention. I'll take questions. Thank you very much.