 Let's continue. It's the last talk of the program. So Theo Johnson-Fried will tell us about TMF and SQFT questions and conjectures. Thank you. Thank you for the opportunity to speak. And as just said, this is the last talk. So I think before everybody kind of drifts off and they're checking their emails, we should all thank Kumran and Du and Francesco and Pavel for putting together such a wonderful workshop. We should also thank ICDPIT support and administrative support for finding this since. These workshops don't happen without a lot of support from the technology side. So the other thing is that since I'm the last speaker, I'm going to do something that I only get to do, I think, is when I'm the last speaker, which is I'm not going to tell you anything I actually understand or know. My talk's going to be mostly about things I don't understand and questions that maybe you know the answer to, but I don't. So that's the point. I'm going to talk about questions and conjectures. But I will sort of remind a little bit about what TMF is and how it's supposed to be related to quantum field theory. And if you want to read the slides at some other pace, you can download them. So Dan already talked a lot about TMF and quantum field theory and the Siegel and Schultz-Teichner proposal. So I won't go into too much detail. Topological modular forms are an object of derived algebraic geometry and derived number theory, at least in the way they've been defined so far by Hopkins and Miller and Gorsen Lerner. I probably should have put Hayes' name on the list. Well, anyway, there's lots of other people who've worked on it. And to really work with topological modular forms is to work with modular forms. They're objects, as I said, they're objects of number theory and algebraic geometry, more homotopoies. So the machinery that you need to do to work with them is to do things like tracking actions by Galois groups and tracking descent data and really handling spectral schemes over Z, that kind of object. And I don't know. This is way too hard for me. I don't know any of that. So there's this proposed analytic model that's now been around for a few decades and slowly getting more and more precise how it's presented, which is a suggestion that this so far algebraic object might have a completely analytic model. And the analytic model that's proposed is some version of the following modular space that a space is presenting. TMF should be the space of super quantum field theories in two dimensions with n equals 0, 1 of degree bullet. That's the degree. And they should be compact. And in the next two slides, I'll say what all of those words are supposed to mean. And the motivation for this is that there's another completely algebraically defined, or could be completely algebraically defined, object, which is k-theory. You can define k-theory completely algebraically in an analogous way to the way that TMF is currently defined. But k-theory also has a lot of analytic models. In fact, maybe most of us learn k-theory first from their analytic models. And analytic models of k-theory typically have something to do with superhalbert spaces and bundles of superhalbert spaces and maybe the modulate space of superhalbert spaces with some extra structure. There's lots of these. They're not all exactly the same. Some of them are homeomorphic. Some of them are not. One version that you can extract from these various different analytic models of k-theory is that k-theory can be modeled as the space of super-quantum mechanics models. Quantum mechanics model, of course, is the same as a 1D quantum filter, that's q of t. And with n equals 1 supersymmetry. So with 1 supersymmetry generator, this modulate space is actually rather this. Is it really a modulate spectrum of these? There's a sequence of spaces of these. And they give a model of k-theory. So that's the background context. So now let me say a little bit just about what some of these words mean. So by superquantum, I'll try to tell you first what all the words other than compact mean. And sqft, or supersymmetric quantum field theory, is supposed to be some sort of actual physical quantum field theory, physical in the sense that it should be Poincarean variance and unitary. And s just means with some supersymmetry. And the specific case I want of what's called n equals 0, 1 supersymmetry, this is the minimal amount of supersymmetry that a quantum field theory can have in today, the minimal non-zero amount. If you more precisely, what this means is that there's exactly one supersymmetry in the theory. And on the space time, so on the two-dimensional world sheet of the theory, the supersymmetry transforms as a right-handed chiral spinner. That's what 0, 1 means. This means that it's a right-handed spinner. These have been around for longer than any of most of us have done. I don't know, there could be some old people in the audience. The topology that the space of quantum field theories or superquantum field theories should have if a proposal like this is going to work, should be some sort of effective topology. It should be a topology that imposes an effective field theory idea. So the way that I want to imagine that I've topologized this space, of course, I don't have a mathematical definition of these. I don't know mathematically what the points in this set are, but I'm going to pretend I did. And now I'll talk about the topology I want to put on them as if superquantum field theories were something that mathematicians knew the complete definition for. I think we're really, I mean, Dan talked about ways that were pretty close to this in terms of functors from cowardice and categories. So once you have this set, then you can try to topologize it. The topology that I want is one in which theories with the same low-energy behavior are close. That probably doesn't, that slogan probably doesn't completely nail the topology, but it gives the idea of the topology. And in the quantum mechanics case, I know a particular topology that works for this that I learned from Andre Enriquez, which is that what's a quantum mechanics model? Well, a quantum mechanics model or superquantum mechanics model was some sort of super-Hilbert space with an operator called the supersymmetry or maybe an operator called the Hamiltonian. And the topology that works to get the motivating theorem was that you can use the topology of strong convergence of the resolvent of the Hamiltonian. So that's a topology that does the job. I don't think it's the only one, but it's one that does the job. So that's what I want you to imagine is that we're kind of taking the topology of strong convergence of the resolvent of the supersymmetry. So that's what, let's explain these words so far. So now let me say a little bit about what the degree of a quantum field theory should mean. And this is something that's been the idea with, OK, so as we learned a lot in the first couple days of this workshop, any quantum field theory in general can have a gravitational anomaly in the superquantum field theory. I said Poincare and very particular should have said kind of spin. So these are spin quantum field theories. In the spin case, we expect the gravitational anomaly to live in the Anderson-Doulders spin cabortism. And when that is too appropriate to sort of the anomaly is measured by the difference of central charges of the quantum field theory, which is always an integer. And so I'll just say that that's the degree of the field theory. Well, more precisely to say I have a field theory of degree three doesn't just mean that the difference of central charges is equal to three. I really should say that I've given the data of that equality, I've given a niceomorphism between the anomaly field theory and some reference anomalies. And this is involved in some sine ambiguities. It won't matter in my talk, but you do have to do it to be precise. Leo, why the factor of two? Why the factor of two? Because the people who first defined central charge didn't notice something like used a normalization that made sense to them at the time and was appropriate for what they were doing. But the minimal charge, the minimal amounts of C left minus C right in any quantum field theory in 2D is a half. So the two just makes, lets me talk about integer degrees. I don't know, maybe the answer to Greg's question is actually the mathematicians were wrong. For a free front, like a my line of room. That's right, the minimal amount of anomaly. The minimum unit of anomaly would be a single chiral fermion. I see. And I think actually the correct answer to Greg's question is the mathematicians got their conventions wrong. Probably we should have decided as mathematicians that cohomology was indexed by half integers where the bosons were in the sort of the, what we now call even degree cohomology, the sort of part of cohomology that has the usual sign rules should be indexed by integers and the part of cohomology with the causal sign rules should be indexed by integers plus a half. That probably would have actually been better from the beginning. But I'm stuck with the conventions that people have. I should also mention that the separate central charges C left and C right are sort of not very good objects since I'm not talking about conformal field theories. I'm just talking about quantum field theories. But this difference of central charges is a very good object in particular. It's a renormalization group, low and varying of a quantum field theory. Well, it's an invariant under any definition because it measures the endowment. So now I wanna emphasize one other really important word in getting the right modular space, which is the word compact. And this is a notion that goes back a long time that I think the definition that I think is probably correct was one that I think is due to Siegel. A quantum field theory I'm gonna say is compact. Okay, well, it's not quite a definition. The idea of the definition is that a quantum field theory is compact. If it's Wicca rotated partition function converges absolutely. And now you have to implement that slogan in whatever definition of quantum field theory you have. In quantum mechanics, I know what that means. In quantum mechanics, the Wicca rotated evolution is the exponential of minus tau times the Hamiltonian. And to say it converges absolutely on all space times is to say that this operator should be trace class for every positive imaginary time. In other models, if you think in terms of path integrals then I kind of want you to imagine that the path interval converges absolutely. But I don't really want you to imagine that because I don't want you to imagine that every quantum field theory is described by a path. Compactness is also kind of really implicit when we talk about fully extended topological quantum field theories. Anybody who's from that world compact fully extended topological quantum field theories are the fully dualizable objects. And of course there's lots of work that you might be aware of about kind of non-compact quantum TQFTs. They are important in studying physics and mathematics. Some examples, sigma models with compact target are supposed to be examples of compact quantum field theories. And more generally massive boson, for instance, the harmonic or in other words, the harmonic oscillator. So that's a sigma model with a non-compact target just the real lining that the boson takes values in but it's given a potential energy that compactifies the quantum field theory. So that's the typical example you should have in mind of what a complex field quantum field theory is. A non-example would be a mass disposal. And let me explain why this is really an important condition because already I can point out to you that if we're working with compact quantum field theories then these spaces are not contractible. At least they're not typically contractible. And the proof of this is that there's a very famous thing that I'll talk about in a little bit more called the Witten genus of a 0,1 field theory. It's an appropriately normalized partition function on non-dening spin, non-dening tori. And I'm mandating that partition functions are well-defined objects that they're absolutely converging. And so this is a thing which can be non-trivial and it's well-defined. And then a very famous calculation that I won't remind you points out that actually this is a locally constant map. That the partition function of a very quantum field theory is a weakly holomorphic modular form and that it actually depends locally constantly on the quantum field. And so the existence of the Witten genus convinces you that the space of compact field theories is already not trivial. It's not trivial typologically. I want to contrast this with the case if I dropped compactness. If I dropped compactness, then you could deform a quantum field theory through a non-compact example and it wouldn't have a Witten genus. And so there's no reason for the Witten genus to be constant because it's not even existing. And this means in particular that I think it's a Cyberg a couple of years ago at strings conjectured that for any fixed anomaly dimension and any amount of supersymmetry at all, if you take the space of non-compact quantum field theories, that modular space is contractible. So it's really important that we put compactness in the definition. So already, okay, this is a nice conjecture for physicists to think about. Maybe it's the first question I really want to flag is see if you believe Cyberg's conjecture. Okay, so that's the context. So now let me explore this proposal of how SQT should work. Greg asks, there is some nice literature showing that elliptic generative some non-compact objects are more modular forms. Greg is completely correct. And I'm not going to address it in this talk. But I will point out something. It wasn't defined, but I think it is defined. Well, let me point out that the Mach modular forms for those some, so it's not true for all non-compact CFTs. You need only mildly, the non-compactness has to be quite mild. And if you deform one of these, you can actually just deform the Mach modular form. It's not an invariance of the, like it's not a deformation invariant, it doesn't. So that's still consistent with the fact that the space of these is... It might be contractable. I see, okay. So anyway, I wanted to say some things that I really don't know the answer to now. So my first sort of thing, I really want physicists to tell me. Physicists have told me that spaces of quantum field days in general should have a flow on them called the renormalization group flow. It's one of the most important things that modern physicists care about. And the slogan is that renormalization group flow is a thing which rescales the metric. And our priority at fixed points would be scale invariant. There is, if you make the words, if you can somehow force the word scale invariant to be a sufficiently local notion, then the RG fixed points are the conformal or in the case of supersymmetric field, they're super conformal field terms. RG flow in different dimensions feels very different and it feels different depending on whether you're talking about unitary field theories or not. In the two-dimensional case, for unitary field theories, Amalodzhikov basically explains that RG flow is a Morse flow. He writes down an explicit function, which, well, his function isn't quite the Morse function, but some modification of it is a Morse function for RG. I'm saying it's a Morse function, but it's not a Morse function. It's just definitely not a Morse function. It definitely, I told you that what the critical points of the flow are, they're super conformal field theories. Probably the space of super conformal field theories is a finite-dimensional manifold, or at least it's sort of finite dimension. So in some sense, one would expect that this is a Morse bot problem and making this kind of clarifying this would be really good to do. As I said, I'm telling you things I don't know the answer to it. Whether Morse flow, whether this picture of quantum field theory is useful or not will depend a lot on whether you can actually do Morse theory with it. And one of the things when you do Morse theory is you'd like to be able to flow any point in your space to a fixed point, because then you can model your space, you can build it up from the bottom just by looking at the critical side. So big question that Greg tells me he thinks the answer is no for, so I'll just already anticipate that, is the downward RG flow does it keep you in the world of compact theories? So it's spelled out what I'm asking is if I give you a two Ds, let's say super conformal field theory, super conformal field theory, if I have a super conformal field theory and I know it's compact at, is it steep IR limit again compact? And I think, well, I haven't, anyway, this is a question I'm not gonna discuss my beliefs about it. I think it would be really nice if this were true because if it were true, then the space of super quantum field theories could, as I said, could be studied more theoretically. You could model this space by saying rather than trying to define what are all super quantum field theories, I could just define the super conformal field theories. That's a much more within reach notion in mathematics and sort of mathematical physics. And then maybe you can figure out what the RG flow lines are between super conformal field theories. And then the whole topology of the space would be a kind of zigzag topology where two points in the space are connected by a path if you can flow by a zigzag of RG flows between super conformal field theories. So I think it would be really interesting to develop, I think it might be within reach from mathematicians to build a model of possibly TMF, certainly a model of some elliptic homology theory whose points are not all super quantum field theories but whose points are the super conformal field theories except with kind of zigzags of RG flows as the topology. And then I also want to just like, okay, even covered it up, you can't see it, that none of these spaces are really spaces. They really are inherently stacks. The space of super conformal field theories really does have points with lots of automorphisms. And so you'd really want to do all of this. You'd really need a sort of more theory of stacks instead. Maybe that exists. I don't know, it's not my area of expertise. So this is about the modulate space of super quantum field theories. And now I'd like to tell you why it's supposed to be a spectrum. If I want to claim that it's a model of TMF or that it's a possible model of TMF, I should tell you why it's a spectrum. So can you say again, why it's supposed to be a stack? So there are theories of automorphism but we're not really cautioning by these morisms, right? Well, I've been very cavalierly talking about the set of all of these, but there isn't, in any definition, that set isn't a set. It's, you know, if you, like, I just think it's probably true that by the end of this century that we will have abandoned spaces anyway and we'll just be working with stacks. But all of the proposed definitions I've ever seen of what a quantum field theory is had some sort of hierarchical structure built in. I don't understand the question. You're talking about the space of quantum field theories that has automorphisms. It's surely, that's a stacking point. I think that Greg was repeatedly- Even without taking an order or anything, if that point has a lot of morphisms, it's stacking. But why is that important in the study of topology of this? So if we want to build a TMF model, do I really care that- Oh, I don't know. I don't know the answer to that second question. Except that it could, like, you do typically think that a point with stackiness should be contributing to topology. The stackiness contributes topology that's roughly like BG for G, the amount of stackiness. And so I expect that if you wanted to do this, you should write down spaces or stacks of super conformal field theories where you think that the space of super conformal field theories is maybe it's the geometric realization of the stack of super conformal field theories. Yeah, I see. Maybe that would be the right thing. But then I worry that the more spec- Justifying this more steartic picture might become worse if you take geometric realizations. And maybe you should, to justify a more steartic picture, you should work with actual stacks in a geometric context, where you can do more stuff, where you can take derivatives in that kind of thing. Yeah, engine spaces. But again, this is all things I don't know. This is I want to encourage people to go forth and study these. So I want to now tell you what the spectrum structure on the space of quantum field theories is supposed to be. The base point that I'd like to think, to tell you of a spectrum, I've told you a sequence of spaces. They were the spaces with quantum field theories with prescribed gravitational and common. The first thing that I have to tell you to tell you that it's a spectrum is I have to make these pointed spaces. And the base point I'd like to pick is the zero theory. The zero TQFT. This is the TQFT that for any non-empty input, for any non-empty space time, the partition function is identically zero. For any non-empty space, the Hilbert space of the theory is identically zero dimensional. That's a perfectly good TQFT in the sort of standard sense of TQFTs. I don't know if it's a perfectly good quantum field theory in your definition of quantum field. So I want to flag them. But that's the idea of the base point. And now a super quantum field theory in general, as far as I can tell, a good mathematical definition, people talk about the word spontaneous supersymmetry breaking. And the best definition I've heard for what it means to say that supersymmetry is spontaneously broken in a way that a mathematician can understand is if the operator, just the identity operator, is a super descent, which is to say it's super, it's Susie exact. You are supposed to think of a supersymmetry as some sort of differential or curve differential on your field theory. You're supposed to think of a field theory as some sort of linear object, maybe it's algebra. And the supersymmetry you're supposed to think of as some sort of differential. And so you can ask, is the operator one a Susie descendant? And this happens if and only if every closed operator is actually exact, just for the same reasons that it holds in any, like in the DGA case. And so having spontaneous supersymmetry breaking is like your theory is a cyclic. What should be true, at least what I hope you will arrange this apology on this space to do, is that a theory should have spontaneous supersymmetry breaking exactly when it's D by R limit is the zero. So now the D by R limit should be concentrating you on the supersymmetric ground states. And the operator one being a super descendant tells you there are no supersymmetric ground states. But this is just a request that you engineered that's apology that way. An example that I have in mind of a theory with spontaneous supersymmetry breaking is if you take a single, an example that I want you to keep in mind because it's gonna come back in the next slide is that if you take a single chiral fermion, so that's a perfectly good informal field theory, it's purely left moving, which means that to give it a right moving supersymmetry have a lot of freedom. And an example of a zero comma one supersymmetry that I can put on this theory is to use the supersymmetry generated by the fermion itself. I wanna flag that although this chiral fermion is a conformal field theory, as a super quantum field theory, this theory is not super conformal. It flows under the RG flow. And I can tell you how it flows. If you flow it to length scale L, you haven't changed the fermion itself, but you've changed the supersymmetry. And so in the kind of large limit, the supersymmetry is getting stronger and stronger. And a good way to handle this is to kind of basically think that the fermion itself is kind of evaporating away and the theory flows to zero. And so I wanna invite you to arrange its apology in which this holds, or argue back at me that it shouldn't hold. In any case, I'm not completely confident about the idea of zero as a valid quantum field theory at all. Because it's one of these weird quantum field theories for which it doesn't itself have a well-defined anomaly theory, for instance. It's a meaningful theory for any anomaly. And so certainly in some models of what quantum field theory should be, zero is not a valid quantum field theory. And in that case, I lose my base point, but I still could have a notion of what spontaneous supersymmetry breaking means, which is an a priori notion. And then rather than talking about a single base point for my stack, maybe all, or for my space, I would have been good enough to give you a contractable space of base points. But I don't need to, if I wanna give you a spectrum, I don't have to give you one base point. I could give you a contractable space of base points. And maybe it's true in your model. Maybe you have a model of quantum field theory in your mind. So I wanna ask in your model, is it the case that the space of theories with spontaneous supersymmetry breaking is that space, that subspace contractable? If it is, then it's good enough for building a spectrum. If not, you'd have to work a little bit more. You'd have to say force it to be contractable. So there's another question for physicists to work on. So that's the base point. Now that I have a base point, I can try to give you a loop spectrum structure. The loop spectrum structure is the following. Well, what would a loop in the space of quantum field there is be it would be a map from the real numbers to the space of quantum field theories with, I'll just call X the parameter. And to make it a loop, it would be a thing which kind of as X goes to either plus or minus infinity should approach zero. We should have a kind of supersymmetry break. If I have a loop, a path of field theories, I can imagine dynamicalizing the parameter. These are kind of perfectly good things that physicists do all the time. How do you dynamicalize a parameter? Well, very explicitly, I want to work with n equals zero one theories. And so to dynamicalize that, I have to promote it to a zero comma one scale or multiply. And just explicitly let me remind that that consists of a bows on and it's right moving super part. The super partners just on the right. I'll call the result of dynamicalizing just the integral over X of my family. The example that I told you about was if I take, let's take for sort of a single right, sorry, a single left moving fermion and I'll just rescale. I told you a supersymmetry generator. I'll just scale it by X. Remember, X was just a parameter. Now, when I dynamicalize X, the full result of dynamicalizing it is while I already had the chiral Bose fermion from my original theory, I have the bows on X that I got from dynamicalizing. I have an antsy chiral fermion, the super part which is the super part. And you can work out the Lagrangian. Here it is. And that's wrong. You can work out the Lagrangian, but it's wrong. One of those should be a D bar, which should be a D bar, something like that. I actually, I'm actually quite confused. So you said that lambda is diagonal but the supersymmetry was... Oh, I see. The supersymmetry had the opposite chirality from lambda. That's right. My supersymmetry is purely antsy chiral. So it should pair a chiral object with a scalar, like just a... So when you said the supercharge is lambda, that's a little confusing. Because the supercharge should be anti chiral, right? It's zero, one supersymmetry. In your previous slide. I'm actually quite confused about that D and D bar issue. In your previous slide, you said this... I thought maybe I misread it. You said the supercharge was L to the one half times lambda. Maybe I wrote the wrong... You mean the dual of lambda of something? Yeah. What do I want? I want something like the thing which differentiates and sort of takes D by the lambda. So what I want is something like that the... What I want is something like... I want something like the supersymmetry operator of lambda should be one. Okay. So then G, good. So then G has the opposite chirality. Okay. And I thought I wrote down a current that did that. Like I think the... I think if you integrate lambda in a circle, like what is this? This is like the integral of kind of inserting the supercurrent. And then the supercurrent gets sort of integrated in a circle around it. And I thought that that was... Anyway, I thought that this was the right formula. But this is what I had in mind, is that operator. And that operator does indeed square to the right moving. So this operator squares to zero. And this theory has... Is trivial on the right. And so that does indeed square to the right, to the correct right moving Hamilton. So I'm like the L zero bar. Thank you. Yeah, that's much clearer. And as I said, this was wrong. So okay, with exercise because I can't do it. I wrote the wrong formula. But the thing you're supposed to conclude is that after doing this procedure, the theory becomes massive. And another way of saying that is that the deep by... The massive, the theory, like the results of this integration is not a TQFT. It's a massive dynamical quantum field theory. But if you flow it to the deep IR, you get the TQFT called one. This is the TQFT whose precision function is identically one. So what I'd like you to do now is, now I can tell you kind of what should be an equivalence between these, between either taking paths in the space of quantum field theories or quantum filters of degree one less. Let me point out that this dynamicalization procedure does indeed change the degree by one. It changes the degree exactly because of the super partner being kind of turned on. So it does change the gravitational anomaly. So on the one hand, if I have a family of theories, I can dynamicalize and get a theory in one degree lower. And on the other hand, if I have a theory in one degree lower, I can just tensor with this family of theories and get a family of theories. And I have talked myself into believing that these two maps are helmets of the equivalence. Of course, there's some things that should be handled like, when is it true that actually, I said I had a family of theories, presumably for the results of dynamicalizing to be compact, that family had better kind of approach zero quick enough. And so having some clarity about exactly how quickly the family has to approach zero would really be nice to do. So that's supposed to tell you why this should be a spectrum. So, so let's let's talk about some things once you sort of have come to believe this Siegel-Schultz-Teichner proposal. So I mentioned already that there's this wit and genus from the space of super quantum field theories to the space of modular forms. I should say I'm doing, I'm like indexing cohomologically. So this actually takes me from that it ends up in modular forms of weight minus bullet over two because I'm using homological index. The formula is really straightforward. I think Dan also wrote it down. You can take the Ramon sector Hilbert space for your quantum field theory, which is to say the Hilbert space on a circle with non-dominant spin structure. And you can take the partition function for non-dominant spin structure in the time direction. Putting non-dominant spin structure in the time direction is that sort of either taking the super trace or taking the trace with a minus one of the F. Any contribution from Q bar drops out. So you just have a Q to the L zero minus the central chart, left moving central charge. This is completely standard. And then I'm going to know, I'm just going to put in a factor of Anita function, which is a convenient normalization. And then there's a completely convention dependent square of a fruit of unity that I should also put in that I'm going to, that doesn't really affect things very much. And, and just to remind by Eda, I mean that a kinds Eda it's a 24th root of the modular discernment. So this is, this is a sort of by now quite well known objects from you can do with quantum field theories. And it's supposed to match another quite well known object, which has the same name, which is called the Witten, also the Witten genus, which was on kind of top to TMF itself also landing in modular forms. I want to flag that these are all pre or a given very different formulas. Here I gave a kind of nice physics formula. And my pen has decided to stop working. I don't know why maybe it's out of power. Sorry, my points are decided to stop. So the, the, the Witten genus for modular forms if you like, think that not that TMF is defined the way I said about derived algebraically, then the Witten genus is given by an edge map on spec from me in a certain spectral sequence. Anyway, it's something that's been calculated. Another question I want to propose that's a sort of another physics question. So if you look at the formula at the top of the page that I cannot seem to point to, because my pencil, my apple pencil has died. Then the, there would be a reason for us you to just get a power of Eda for you to just or in particular just a power of Delta. If also supersymmetry made the Q dependence dropout, which is to say that one reason why you might end up just with powers of Delta is if you, if you're supersymmetry enhanced from zero comma one to actually one comma one, then the Witten genus would just be like that count and not actually a modular form. And so another question that I think we should, we should work on when we're engineering, like when we're talking about kind of really probing this, this proposal is, is, is it true that in fact, every value that you expect every, so we know which powers of, which multiples of powers of Delta should arise as the Witten genera of CMF classes. Is it true that they all arise as the Witten genera of theories with one comma one supersymmetry? I think we should, when we're trying to build those that's where to look. I'm going to keep sort of powering through with a bunch of questions because that's the fun part. Okay. Other things, so more precisely what Hopkins calculated was that we know exactly which multiples of, so powers of Eda don't arise except for, I'm really annoyed that my Apple Pen still died. Powers of Eda don't arise except for powers of Delta and that has an easy explanation in terms of modularity. And not every power of Delta is the Witten genus of the CMF class rather the K Delta to the M is a Witten genus exactly when K times M is divisible by 24. So another way of saying that is that the minimum K so that K Delta M arises is 24 divided by the GCD of 24. And I know how to engineer some of these minimal powers of Delta. For instance, John Duncan's super moonshine field theory is something you can go look up, it's a lovely paper and it realizes 20, the left moving version realizes 24 Delta inverse, the right moving version realizes 24 Delta. As an example, something you can do from this is that while you can then take powers say of the right moving version, then you realize 24 to the M times Delta to the M that's 24 to the M is some astronomically large number. You could also work to the permutation orb of fold of this power, and you'll realize some power some also astronomically large number. And a cute calculation of Gallardo that was basically just Mathematica, he went and asked Mathematica shows that indeed the GCD of these two numbers does is the minimum expected. Okay. What that means is that I can actually I do know how to engineer all of these TMF classes, all of these sort of expected classes by like some massive linear combination you take sort of thousands and thousands of powers of super moonshine and thousands and thousands of orb of folds and arrange the signs that there's a massive cancellation. This is a terrible solution to the question of actually engineering interesting witnesses will be much better to engineer one with only one vacuum. So another question I think people should should explore from the point of view. From like his is going engineer quantum filters with a single vacuum and a very small non zero which sort of wouldn't use maybe just anti holomorphic super conformal filters where the witness is just a number. So how about one comma one single models is any hope of engineering this from single model. I think that's a great question and I don't know the answer because I don't know the. I don't know much about signals. Okay, thanks. Like, well I will say, I'm not aware. I think it's, it's a wide open question to find a signal model that actually was not maybe that wide open. I think it's, it's a wide open question to find a signal model that actually was not maybe that wide open. Hopkins can give you some sort of vague plumbing description of a sigma model that realizes 24 delta. But it's not a very explicit thing and it doesn't have like in the in the in the sort of Duncan super moonshine theory for instance has a has a beautiful automorphism group Conway's largest sporadic group. And it's not at all clear whether you should expect to be able to you know how much you should be able to lift at those automorphisms to a stigma model. And of course, there might be lots of theories that realize the same enough class because they only have to be related by some complicated path they could have different numbers of vacua they could have different sense sort of central charges, they've done the same anomalies, but yeah. So in particular I don't know of a quantum field theory of a sigma model which actually flows to Duncan super moonshine in the deep by our and maybe though this plumbing model of Hopkins does but but probably not. And that might be a very kind of geometry dependent like might be very sensitive to what geometry to put on that man on that 24 man of dimensional man. Yeah, right. What other things don't I know how to do. I don't know how to engineer theta series. So another result from Hopkins calculations was that for any even unimodular lattice, the corresponding theta series is in the image of the wooden genus map. And an example, the first example is that if you take the eight lattice, then you get the wait for eyes instantiate series. And it would be really nice to write down zero comma one field theories that you can write down. And you can write down zero comma one whose witness is part of the status series on the nose, maybe a construction that starts with a lattice. I don't even know how to do this for the first lattice for the E. Let me give you a non solution to point out that this that what's hard. So a thing you could have tried to do was just take the holomorphic lattice feel way. So just a purely holomorphic those on it. It's a left moving bows on probing the Torah stool to the left. It's it has zero comma one supersensory for a stupid reason because the right moving sector is tricky. And you can ask what is it's what is its witness perfectly good question. The answer is not data series, because the answer is the data series divided by some power of data. And of course it's not because this is a purely holomorphic object. And so it's and it's central it's so it's a thing I really wish I could write with my pencil and I just think I need to get a new pencil. Here we go. Okay. So of course this is an object with with C left C right is the rank of L. Zero. Whereas the data series. Should be realized by something who's C right is more than it's C left and differing by the rank of L divided by two. So there's no there's no so this this is just definitely a non solution. Another non solution would be to take this the Torah stool to the lattice and take the full signal model with that target. Perfectly good object. And it's but it's just a Torah so it's completely flat so it's witness vanishes identically and it's just point out that the result has for me and zero which just killed the witness. So my best guess. Find another nice genus. And that that's never a big deal field. I mean you could just insert for me on zero. And then you get another. You cannot do that insertion in a way that preserved supersymmetry. That operator wasn't. I mean maybe it maybe there is a short way to answer this problem. I couldn't do it but maybe it's just my own blinders. I'm not sure I believe that because there are ways of counting PPS states where you do this. Anyway, I had another question though. So I'm getting now confused about Q and Q bar. I mean your supersymmetry was on the right. So strictly speaking did you really want to say trace minus one to the FQ bar to the L not zero bar minus C over 24? Yeah, I should have. That's right. I should have said, well. Okay. I could have been going to put bars on everything. But no, I mean, I really, I didn't mean what I wrote. Oh, I could have inserted also a Q bar. Right, but I have inserted a Q bar to the L zero bar minus 24. But, but this will just this won't contribute to the trace. Very good. Just the standard supersymmetry cancellation. It doesn't matter whether you write that or not. Okay. So why was the lattice theory wrong then? I mean couldn't you multiply the Duncan super moonshine by a lattice? That would get you off by a factor of powers of 24. Ah, okay. Too many grand states. Yeah. There's a eight manifold. The eight models that single model that will give this way for I sense and serious. You take the year plumbing and then boundary some exotic seven sphere. So you take 28 copies and then cap that off. Yes, I believe that this is true. But I'd really like for I'd really like a construction CFT construction. Somehow, I think some of the most interesting quantum field theories, maybe this is my own bias is some of those interesting quantum field theories are like these deep by are purely quantum objects that don't arise at necessarily as a sigma model. I'd really like a construction that just inputs a lattice and outputs a CFT. Yeah, I see. And since it's true for every unimodular lattice, it should be something you can just do. That's just like a lattice theoretic construction. But I haven't found it yet. Maybe there isn't one. I don't know. Maybe Greg will construct one while during the rest of the talk and then he can like explain it. He'll explain how to handle the zero modes that I don't know how to deal. But okay. Hoping statements that everything is in the image. So, for example, one guy 24 delta but no delta. That's right. 24 deltas in the image delta is not. Okay, I'm going to power through for a couple other things that I think would be interesting to study and I can already tell I'm not going to be able to say everything before the before do cuts me off or he'll let me run late. I don't know maybe people will just leave on their own accord. Quantum field theories as we've heard a lot about quantum field theories. I don't know. It's interesting to study quantum field theories which come with a symmetry by a group. I'm going to call that a flavor symmetry just to distinguish it from gauge redundancy. I don't know. I think some some parts of high energy physics use that language for some parts of particle physics. Let me remind that there's a perfectly good theory of anomalies and maybe there's a piece which is just the degree of the quantum field theory and then there's the part we normally think of as the anomaly which is the rest. The part we normally think of as the anomaly is like the part of spin cavortism other than the degree. So the reduced spin. And what you should expect is you should expect that if you take everything I've told you so far and just enhance it with g symmetry and prescribed anomaly then you'll get a model of quantum field theory. It's a good way to study a quantum field theory with some symmetry is to replace that stand-alone quantum field by a boundary condition for a gauge theory. You can place a boundary condition on it which is basically the boundary condition for a gauge theory. So this is a meaningful expectation. I'd also like to remind something that I think we've also heard in this workshop that a good way, not the only good way but a good way to do that is a boundary condition on it which is basically your theory Q with Neumann boundary for the gauge fields. And the map that takes Q to boundary conditions for gauge theory is an isomorphism because to get your original theory back you put the Dirichlet boundary on the other side and just zoom out. The Dirichlet boundary comes with a G action so you do get Q back as a thing with a G action. That's quite standard. So once you've decided that theories with a G symmetry are the same, at least maybe for finite groups, maybe for groups where you really feel like you understand gauge theory, you can decide that theories with a G symmetry are the same as boundary conditions for some bulk theory. Then you can start asking about more general types of quote symmetry where the bulk theory isn't a gauge theory. And these are starting to be called words like non-invertible symmetry or categorical symmetry. And for at least for finite categorical symmetries we know what the most general type of symmetry a 2D SQFT can have. The symmetry would be described by any 3D TFT. These are super spin objects. 3D spin TFTs are described by spin modular tensor categories. Sorry, super modular tensor categories. So you should expect that for any super modular tensor category, this is maybe now a conjecture. I don't think anyone's done this completely. But for any super modular tensor category there should be a meaningful notion of TMF equivariate for that modular tensor category. This is now switched from physics conjectures to math conjectures. Mathematicians, please go and build this. And I should say part of the work has been done, but I don't think anyone has built this over the rational numbers. And their constructions not trivial, but the content of their theorem is actually an investigation of Galois actions on the mapping class group representations of Russia to counteract it. But this should be done not just rationally but integrally. Hasn't been done. I think the fact that I want super is the assignment, it's sort of natural to think it would be functorial for topological interfaces, which in the math literature are called super with equivalences. So any mathematician who likes building things in pure homotopy, please do this. Okay. I'm going to keep going until do tells me to stop this. I'm going to do the same thing again. Another thing that you can do with when you start thinking about flavor symmetries. Another way of understanding theories with the flavor symmetry is to place your quantum field theory on a world sheet with a background gauge bundle. And some people use the word fugacity for the strength of the background bundle. At least that's a word I've heard and I think that's the definition. I've been talking to quantum field theories which I asked to be correct. In fact, a weaker notion when compactness of the whole quantum field theory is what I'm going to call flavored compactness. The whole theory was compact. A geo-equivariant theory would be compact if it's wichord to the partition function converges absolutely for any G bundle. But a weaker thing you could ask is for the partition function to converge as long as the fugacity is non-zero. An example of where this is interesting is you could take a sigma model with target, just the complex numbers, and take the group you want just acting in the usual way. Well, if you ask what happens to a string probing this target space with non-zero fugacity, the string can't get very far away from the origin. Because if it tries to get far away from the origin, well, the symmetry facilitates the other end of the string to somewhere else, and so the string gets pulled harder and harder the further away from the origin. So the end result is that at non-zero fugacity I'm actually looking at a harmonic Gaussian. And so this is a theory which is flavored compact. And indeed you can write down the kind of wit and genus as a function of the fugacity in a problem like this, and the naive thing you write down is a meaningful miramorphic Jacobi object. So let me just ask maybe I'll ask the mathematicians to investigate flavored compactness. Flavored compactness is just a thing that physicists have already been investigating. For instance, when Greg was referencing the fact that some non-compact CFTs can give you mock modular forms, one of the ways that those arise is from flavored compactness. It's not the only way, but it's one of the ways that they arise. And so some things that we should do, there's a thing that people already know called level-end TMF and it's almost TMF for the cyclic group Z mod N, it's not. It's probably the flavored compact version of TMF. So we should build a version of flavored compact TMF which realizes that. We should also build versions of flavored compact Bortism Spectra. See if you just let your Bortisms be completely non-compact then every Bortism every manifold is fillable because you just like fill it with a non-compact filling. But if you ask for Bortisms with some some extra symmetry then you can have something that's possibly interesting. So let's see what are some examples. I already mentioned that flavored compact theories, for example, can give you Miramorphic Jacobiforms and so one should expect kind of Miramorphic Jacobiforms to be a logical Miramorphic Jacobiforms to be an interesting object. If you can build flavored compact Bortism Spectra then you should still be able to do things like say take S1 with its action by rotation and then here I told you C was a perfectly good flavored compact manifold and for the U1 symmetry and I'll think of like a filling of S1 that just sort of opens up and the reason why I care about sort of geometry is trumpet fillings like this is because of some work that on Mach modular forms and on understanding sort of formulas that come up that you see in the literature. There's a lot of formulas in Mach modular form theory that seem like what they're doing is taking non-compact manifolds with sort of mildly non-compact manifolds and adding on kind of trumpets plugging the filling, plugging the fillings with a trumpet. So those are some, but especially the mathematicians just in general I think defining and analyzing flavored compact structures would be an interesting problem to do. Okay let me just mention a few other things that I don't know how to do. Topological modular forms just like ordinary modular forms it's interesting to ask about the behavior of the modular form at the cusp at i-infinity and the thing that I've been talking about pure sort of full what the kind of all caps version of TMF kind of allows any pole at the cusp. It's corresponds to what you might call kind of weekly holomorphic modular forms in the modular form literature. It's holomorphic at finite values of tau but can have any pole at the cusp. But it's also possible to ask about modular forms with poles where you bound the degree of the pole. And you can write down spectral versions of these. You can for instance the thing that's called mixed caps TMF is the space of topological modular forms which are holomorphic at the cusp. And there's a thing called topological cusp forms which are the topological modular forms which vanish at the cusp. Now I want to flag it when I said which are holomorphic and which vanish well I don't just mean that as a property. This is homotopy theory so mixed TMF class comes with the data of how the class is holomorphic at the cusp. It's not just a subset. I'm going to give this slide out of order because I want to sort of flag that that unlike for ordinary modular forms there are holomorphic modular topological forms of negative weight. And first example in general we know what all what they all are because you know it's good show there's a wonderful duality for the mixed TMF mixed caps TMF version. But as an example in high minus 21 so that's weight minus 21 halves there's a holomorphic modular form which a topologist might call delta inverse nu delta inverses the thing that the number there is knows about nu is this S3 with its legal and this should be compared to the fact that that for the all caps TMF it's zero so there's this interesting TM sort of this interesting object this interesting conformal field theory or something which is which is holomorphic at the cusp but and no homotopic but the no homotopy moves it away from being holomorphic at the cusp so it'd be really interesting to describe these physically roughly speaking saying bounding the poll at the cusp is something like bounding the spectrum of L zero in the Ramon sector this is something you can write down I mean you can you can give a definition but I don't know what the physics of this it's not a very physically reasonable thing to do this is not just the same as bounding the central charges bounding the central charges would give you a bound on the spectrum of L zero but it might not be as strong as you need it to for instance just looking at central charges wouldn't have told you that delta inverse nu is holomorphic at the cusp and now I guess I'll I'll end up just doing one last slide it could be a little bit easier to do is letting me go late to just mention you know yet more things that I think should be done so elliptic homology was developed somewhat hand in hand with with notions from moonshine back in the in the 1990s and and they they've always had a little bit of interplay to them and and this maybe isn't too much of a surprise maybe the insurers probably know that a lot of objects modular objects were were and remain really hot so anytime you have something that produces modular objects you might ask oh maybe does my machine the produces modular objects is that related to your machine the produces modular objects and by now maybe the answers probably know because modular objects there's too many of them but maybe the answer is yes so let me just tell you something about moonshine so moonshine is in general it's about you could do something with geoequivariant topological modular forms. If you ask what is the space of cusps for geoequivariant topological modular forms, the space of cusps is roughly speaking the adjoint quotient G-munchy. It's not quite that the Galois actions are different, but it has that basic flavor. Over the complex numbers, it's the adjoint quotient G-munchy. And so if you have a geoequivariant modular object, it's interesting to ask about its growth at all of the different cusps. Moonshine focuses attention on things which grow in some way near the cusp, the sort of identity cusp. This is like the I-infinity cusp. And growth less quickly, they're smoother at all the other cusps. And different moonshines have different degrees and different precise growth rates, but they all have this flavor, that what makes moonshine objects better than regular modular objects is this kind of quite constrained growth. Let me give an example for anyone who's ever kind of listened to a lecture about monstrous moonshine, which is that I told you that Z-n modular forms are roughly speaking level-n modular functions, modular forms, or it's just to say modular forms that are modular for the congruence group called G-m-n. The cusp, E in kind of, so G is gonna be the cyclic group, the cusp E is basically the cusp at I-infinity, and the other cusps are the finite cusps in this translation. If you had say just a weight zero modular form, so just a modular function, and if you knew that it grew as Q inverse at I-infinity and that it was bounded at all other cusps, well, then it's a map that it ends up having to be a hopped module. It ends up having to be an isomorphism between upper half the sort of modular, the level-n modular curve and just the genus zero curve. So this type of imposing this type of growth rate is one of the ways to impose the genus zero property in modular forms, modular in moonshine. It's not the only way, but it's a way of building the genus zero property. And it's the way that the kind of modern moonshines, the umbral moonshines build. It's the version of genus zero that you see in umbral moonshines. So I think it'd be really interesting to study kind of topological modular forms with this type of mixed cuspital behavior. And I'll end on one last one that's sort of about moonshine and not really about cuspital behavior, which is that in monstrous moonshine in particular and not in the more recent umbral moonshines, there are also hop module and four groups that aren't contained in SL2Z. And these have been somewhat explained by work of Paquette and Person and Volpado, but their explanation is physical. It has to do with super conformal field theories. But it's not an explanation that I see how to translate into CMS. Now maybe it doesn't translate, but it would be interesting to ask does, does TMF, is it kind of, does TMF have room for, modularity not contained in SL2? So I will stop there, thank you. Thank you very much for this very stimulating talk. So any questions from the audience? Yes, I have a comment actually related to one of the conjecture. First of all, it's beautiful talk. I really like the fact that you reviewed questions and this is very rare. We see so many open questions in this field. So thanks for that. One of the questions that you asked, I think we probably know the answer to, and that is the one that you said about the, whether or not the compactness property is RG invariant that you see at that class when you float down. And the answer to that is no. And the easy example of this is that you take a sig model on a hyperbolic manifold, for example, on a Riemann surface. That's a compact model, but an RG still becomes non-compact. So that's the easy counter example for this. Yes, this is a counter example that lots of people have told me. Now that's sigma model didn't, the way the sigma model doesn't arrive on your lap as a UV-complete theory. Sorry, what? Well, maybe the right definition of quantum field theory is quantum field theory that comes with a choice of UV-complete completion. The sigma model with hyperbolic, like the sigma model on a genus-3 curve isn't a priori a UV-complete thing. And so to UV-complete it, you could embed your curve into some say large projective space. What I think actually happens, or what I wonder if actually happens for the RG flow is not that, it starts out, I agree with you, looking like it's decompactifying the target. But maybe what's really happening is that in the limit, the target is just getting like less and less real, more sort of spread out around CPM. Well, actually I can give you other examples which are even easier than that. Start with the Landau-Ginsberg theory with some, so this mimics the mirror of this. Then there's the mirror of this story. You can start with a Landau-Ginsberg potential with sufficiently high powers and you add the leading power, which is higher or lower. I don't remember, which mimics this in the mirror language. So you can actually do it from the viewpoint of a compact. So it's UV-complete theory. So there is a UV-complete version of what I just said. Anyhow, so that I think is the answer for that is probably no. This is disappointing because it would be really nice. Yeah, I know, I appreciate the way, I'm with you on the which, but I think that's what it is. The other thing is that I wanted to say that just to perhaps add my own interest in the question you raised about favored compact TMS. So that's actually very important for many other reasons, which I want to bring out. So you started by describing the K theory being related to supersymmetric quantum mechanics with N equals to one supersymmetry and the TMS being the same thing for the minimal supersymmetry 2D. Of course, you could go on to higher dimensions, which one can. And in fact, the highest one we can go is the 60 with the minimal supersymmetry, which is zero comma one supersymmetry. Now in that context, it turns out that, so that's also an interesting question. What is the analog of the TMS in that context and how you study those? But that actually is an interesting interplay between the zero comma one theory and the 60 with the zero comma one theory and 2D by compact by and for manifold. But that again, that typically gives you unfortunately non-compact versions of TMS, except these theories typically have global symmetries which you can twist by and give you a favorite compact TMS. So there'll be a map from zero comma one theories to 60 to zero comma one theories and 2D that we described in my work with Pavel and Duas Sergei where it connects these two theories together and it becomes urgent to try to have an understanding, a deeper understanding of favored compact TMS and 2D if you're interested in topological, new topological invariance perhaps for format. So that's an additional motivation to why we really wanna study them. That's exactly it. Thank you. So any other questions? I have a question. It's more like about the beginning of your talk. So what's more given can kind of define the super, what conformal field series using vertex operator algebra and their modules. So do people try to study topology on the space of just conformal field series using this perspective? This is exactly what you say is exactly correct. So I, I mean, people have done things. There's probably a lot that I'm not aware of. It, one thing that I've known that I've seen mathematicians do is you can restrict attention just to. So it, let me say it this way. So you can look at say rational vertex algebras and study kind of questions about the topology of the space of R-C, of sort of rational BOAs. And what you would expect is you'd expect that space to be completely disconnected. And then indeed it is. So you can, an infinitesimal change in a vertex algebra is something like the tangent space. If you have a specific algebraic object and ask what's the tangent space to the space of all algebraic objects that you're specific algebraic object, that question is usually answered by some sort of Hock-Schild homology. And so what I know of is people calculating Hock-Schild homology in vertex algebras and measuring the size of these tangent spaces. And for the rational CFTs, indeed you see that they, well in the space of rational CFTs, those are isolated, which is what you'd expect. Now, full CFTs, of course, we know are not isolated objects. Like people have really good understanding of what all the C equals one full CFTs are. And there's an interesting finite dimensional manifold of them, like there's a sort of a line and the other line comes out in a few isolated points. It's not a manifold. Okay, singular manifold. It's not a manifold. I agree it's not a manifold, but it's presumably sort of smooth after you put some stackiness back into it. Another thing that I'm aware of is that, so I'm not aware of a vertex algebraic definition of full CFT. Vertex algebras are good enough to give a definition of rational CFT, rational full CFT, but and presumably will eventually provide a mathematical definition of full CFT. A full CFT, in the rational case, a rational CFT can be modeled as a rational vertex algebra of left-moving operators, a rational vertex algebra right-moving operators and an identification of their representation categories to paste the left and right together. Presumably a definition like that will work for irrational CFT as well in the vertex algebraic language. And it's a question of kind of doing the hard work of developing representation theory of vertex algebras that are irrational. And a lot of that works under, it's in progress right now by people who are not me. Another place where there is a definition of full CFT in the literature is from the kind of Hogg-Castler algebnets of von Neumann algebras perspective, which is just another way of thinking about vertex operators. And in that case, there is what I think is probably correct definition of full CFT, which is a conformal net of von Neumann algebras with new index one. And these exist in the literature, they because they come already from the worlds of von Neumann algebra, they come with a preferred topology. I don't know if that's the right topology. I mean, if that's the best topology, but they certainly have access to it. I don't think they've been very studied. Okay. I have no questions with comments. I had maybe a big question about this flavored compact TMS. Could it be something like equivalent TMS with equivalent Euler classes inverted or something like that? Just trying to get more of a math description of what this looks like. I mean, so the Euler classes are, maybe these classes that vanish at zero fugacity, I think in your language, is it just inverting those or is there something else going on with these poles? It could be. I don't think I know what an Euler class is. It's like the specific Jacobi data functions that vanish to first order at zero. It's like the virus drastic function is one of these. That sounds good. I mean, my guess is that defining flavored compact TMS isn't too bad. But is it, I guess what I'm asking you, you start with that covariant TMS, which I think we agree exists, but no one's written it down yet. And then you do something. And what is the thing that you do? Is it do you like localize the certain numbers? Yeah, so let me, so a way that I could try to do this is, so something that's morally true and literally false is that G-equivariant TMS is the space of sections of some bundle on the stack of curves with a G bundle. And the reason this is false is because you have to put some, if there's some Galois, that's off by a kind of action of the Galois. And so for example, you want to covariant TMS is like the space of sections of some. So if you ask what's an elliptic curve with a U1 bundle, that's basically like an elliptic curve together with a point in the canonical bundle. Yeah. So, or the dual elliptic curve. And so you can sort of sort of take the total space of that of the canonical bundle and ask for kind of sections over that thing of whatever the sections of whatever the thing whose sections are TMS. So another thing you could do is take the space of, so what I could do is now I could remove the zero section from that canonical. What am I saying? I say, there's a bundle over the modular stack of elliptic curves, which is the bundle whose points are an elliptic curve together with a G gauge. I could take the total space of that bundle and excise the zero section of that total at that. So now I have another total space. I can look at, I would arrive on that and take its global sections. I see. And up to some issues of Galois theory that I, anyway, that's presumably that type of thing would give a definition of flavored compact TMS. And presumably some level should be in here as well. Already for Jack, we're at TMS if you take sections, it's not so interesting because everything's compact. So you want a line bundle to get interesting sections. Well, yeah. I mean, there should be like degrees and so on. But it's not degree, I mean a level though. Oh yeah. A level in the, in the Trincyman's case. Yes, the gauge anomaly should be there too. Okay. So you start with, do you have to grant TMS with some level? So I have some line bundle reminder, I've looked at curve, blah, blah, blah, excise the zero section and look at sections. And that's what you think should be roughly. That's right. Okay. So that would be a definition. Now people should go compute it. Like people should make sure it makes, people should make sure it actually is well defined. And then they should just go compute it and ask like, how is it, is it wacky? Is it really accessible? I mean. It's above my pay grade, but I know some people don't ask that. I don't know. It could be like, if you ask sort of, I told you that, sorry, I was getting confused because there's a different thing called level structure on an elliptic curve. And I told you that if you take ZN flavored compact TMS, that's basically should be, maybe when N is prime at least, I don't really know numbers that aren't prime. That's, that should be matching what's called TMS with level structure. And TMS with level structure is much less derived. It's the sort of space of elliptic curves with level structure is much more affluent in the space of elliptic curves. And so the results of, and so the results of this is that, that TMS with level structure is much less hard to work with than full-fledged TMS. And maybe you should expect that's true in general for flavored compact TMS. I mean, there is a distinction between ZM-Ida and equivalent and level. It's like ZM-Ida is over the modulite G bundles and level N is like at the trivial bundle, roughly speaking, I think, right? So there's, there is a distinct, maybe I misunderstood the point, but I agree that at some point in this modulite space of ZM-Ida and equivalent TMS, there's like a point that's where the modulite problems are presentable or whatever you want. But it simplifies a lot of the topological issues that you get when computing these spectral sequences. But there's more to ZM-Ida and equivalent TMS than just that part of the modulite. I exactly agree. And the point is that I think, and I think that I'm just agreeing with what you've already agreed with, which is that these different things that restrict attention to low side within ZM-Ida and equivalent TMS. Well, I had this like badly non-affine variety. If I restrict to an open sub-variety of it, it's more likely that the open sub-variety will be affine. Yeah, that's the spits with my worldview. So I'm happy. Thank you. More questions or comments? I have one question, Theo. Yeah. The partition function of a theory of the empty boredism from the empty set to the empty set, I think should be one. I'm pretty sure that's true and, well, yeah. And so I'm having trouble- Sorry, the partition function of the- I'm having trouble understanding how that's compatible with the existence of the zero theory. Right. So the zero theory is, Yeah, so I think, okay. I'm not, as I said, I'm not convinced the zero theory is a theory, but I did say that it should be the thing that for non-empty- I noticed that. Spire didn't interrupt you in the talk, but- Should have zero partition function. So, well, I do know what the zero TQFT is. That's a perfectly good, like TQFTs are things we know how to define zero as a perfectly good TQFT. Whether every TQFT is a QFT is debatable, but it's certainly zero is a perfectly good TQFT. Okay, but, okay. So, do you agree with the partition function of the empty boredism from the empty set to the empty set? That's right, because what's going on is that like the partition function of the zero quantum field theory is like basically zero raised to the power of the size of the boredism that you input and zero to the zero is one. Okay, good. Yeah, excellent, okay. But zero to anything else is zero. And another way of saying what the zero TQFT is, the zero TQFT is the sigma model with empty target. So, if you ask for any non-empty boredism, what are all the maps from the non-empty boredism to the empty target? There aren't any, so that path integral is zero. If you input the empty boredism, there is a map from the empty set to the empty set. And so you take the path integral over that map and you get one. Excellent, thank you. Okay, so if there are no more questions or comments, let's formally end the program. So I would like to thank you along with other speakers for their wonderful talks and discussion. And I would like to thank all of you for...