 Oh, there we go. Okay, but thanks a lot for the interaction, for putting together the very nice workshop and for the invitation. It's just a pity, I mean, due to COVID, we cannot all meet together at ICTP. I'm looking forward to the next opportunity that we have for that. So today I'm very happy to tell you about the recent work that I had in collaboration with Arun Debre, who is not a mathematician at Purdue. Marcus Der Eagle, a postdoc at LMU Munich and Jonathan Hemman at the University of Pennsylvania, about anomalies in type to be a string theory, in particular about an anomaly that was not supposed to be there. This is a workshop on generalized homology and physics. And so generalized homology will actually play, it will seem not to play a role for most of the talk, but near the end it will take a very important role. So stay tuned. Okay, so let me first start with, what's the big question? The big picture question that we wanted to answer when we started with this project, right? Like mostly, just as mostly everyone here, I guess we are interested in understanding, like you could even say that that's more or less what we do in general, we want to understand the space of both consistent quantum field theories and quantum gravity, okay? From the point of view of quantum field theory. And so to answer these two questions, we have different sets of techniques. For instance, from the point of view of field theory, you have techniques coming from conformal bootstrap, unitarity, all sorts of field theory techniques that allow you to make very concrete progress and that have been developed significantly recently. And from the point of view of quantum gravity, I think recent progress has been achieved from what is called the swamp on approach, when one tries to identify general patterns that consistent quantum fields of gravity must satisfy. Okay? So these are two different, but very much related questions. But if there's one thing and they have in general very different techniques, but if there's one thing they had, both of them have in common, is that there's one technique that is useful, very useful for exactly the same reasons in both cases. And that is anomalies, right? It's very useful because if either in quantum theory or quantum gravity, you have some symmetry and you wanna gauge it, then it better be anomaly free. So you can think of anomalies as restrictions to gauging, both in quantum theory and quantum gravity. And so they can provide interesting consistency conditions in both cases. And the case I'm gonna be talking about is a little bit in between because I'm gonna be using mostly field theory techniques to analyze the low energy effective field theory for quantum theory of gravity. Okay, so that was just like the big picture outline of what I'm gonna do. Before I get down to the details, there's a very important thing I need to say, which is that you should please interrupt me at any point with questions, comment, whatever you like. This is being a Zoom talk. I cannot see if the message is getting across. I know we're having discussion afterwards, but please just interrupt me at any point, okay? With questions, comments or whatever. Okay, so this is the big picture outline of what we wanted to do. And the more concrete question that we wanted to answer, we wanted to understand basically, what are the possible anomalies of the duality group of type-to-be string theory, okay? So the useful story that you can find in any textbook is that, well, if you start with type-to-beast for gravity, it has an SL2R symmetry group that is broken in the quantum theory due to deep brain effects and the like. The useful law is that it's broken into something like SL2C, okay? And because we believe that to be string theory is a consistent quantity of gravity and it has this symmetry, which is exact. It should be gauged. And therefore you expect that in some way this symmetry is an anomaly-free symmetry. So the question that we originally were interested in when we started this project was, okay, it was a swan-plan type question. So we wanted to explain or wanted to understand why the duality group of type-to-beast string theory was SL2C and could it have been something else? In other words, suppose that someone comes and tells you they have a new quantum theory of gravity which has exactly the same massless fields, interdimensions has exactly the same massless fields as type-to-beast string theory. But they tell you duality group is something different like a congruent subgroup or maybe some other subgroup of SL2R, like a triangle group. Could that be, could that be, how can you tell whether they are actually telling you the truth or whether they're just making something up which is not correct? In other words, can you figure out which duality group of type-to-beast property if there's a unique one? And so this is a question that at least you can attack whether you can determine completely it's a different question, but at least you can attack using anomalies because, well, even if you expect that this guy is anomaly-free, maybe some other choice would have an anomaly. It would be inconsistent because of some anomaly, okay? So that was our program, trying to analyze anomalies of duality group as a type-to-beast string theory to try to explain at least partially why the duality group had to be where it was, okay? And so this is a question, the way I just phrased this a question you could have asked 20 years ago. And to some extent, people like Gavril and Green ask this kind of questions. But the reason why we could make, I think a technical reason why it would make a lot of progress recently is because you can, there's like this in the past few years, there's been a very systematic understanding of how can you understand anomalies of field theories and with discrete groups in terms of coproducing theory. And so this is something that we exploited heavily. So any questions about the general idea? Okay, good. So that's the general idea. And so if things have turned out as we had hoped they would, the plan of this talk, slash the plan of our project would have been as follows. We would have started with some review of anomalies to learn what's the state of the artist of the techniques that we wanna use. Then we have used these anomalies to first check that the type to be string theory is anomaly free. That would have been the first easy step that check that we understand how things are working. And then we have changed the duality group to all the duality groups to try and find an anomaly in one versus the other. So that we would have used this computations to put some constraints. And this would have been swampline constraints which would have been quite rigorous. But as so often happens in life, things do not quite work out as you wanted. And in this particular case, we actually found a big surprise already in the second step which prevented us from continuing with the plan that we had originally. And that big surprise is that the duality group of type to be string theory has in fact an anomaly. It's in fact anomalies. So this talk is about that anomaly. Okay. So let me get started. I'm gonna start anyway like the first part of the plan is the same, right? So let me start with what is an anomaly? Like a very brief 10 minutes interaction to what is an anomaly? Or those of you that I don't know or at least to set up like the framework and the language that I'm gonna be using. Okay. It's much basic level. An anomaly is a classical property of a quantum theory which is of some theory which is not preserved by quantum effects. That's the most general definition of what an anomaly is. And typically we discuss anomalies in the context of theories with symmetry. So if you have a theory with a symmetry, some field theory with a symmetry, you can, one thing you can always do, you can turn on a coupling to the symmetry by introducing a background connection for the symmetry, a background gauge field. And then the most typical way in which an anomaly manifests itself is that when you turn on this background connection, it can sometimes happen that the observables, for instance, the partition function are not gauging by. So this lack of gauging variance is an example of an anomaly. So that's the basic idea of what is an anomaly. And here I was just coupling some quantum field theory to a background connection. If you then wanna make the background connection dynamical, then this is really becomes an abstraction. If you're not doing that, then you just have a tough anomaly which might be an interesting competition condition in the theories, but nothing more than that. So briefly speaking, I broadly speaking, there's two big classes of anomalies. You can talk about anomalies where the gauge transformation in which under which, for instance, the partition function is not gauging variant can be continuously deformed to the identity by a continuous path. And that's what I'm calling a local anomaly. So it's an anomaly in a gauge transformation which can be continuously connected to the identity. For instance, if you have a U1 gauge theory, any anomaly, in, let's say in S4, so it depends on their manifold, all gauge transformations can be continuously deformed to the identity. So all anomalies clock. But often one also has anomalies in gauge transformations that you cannot continuously deformed to the identity. And these are called global anomalies, okay? It's confusing because it sounds like an anomaly in a global symmetry. We're only talking about global symmetries here. But by global, I mean that it's an anomaly in a transformation that you cannot continue to get to the identity. This can sometimes appear in continuous groups like the famous witness attitude anomaly. But for instance, since I'm in this talk it's gonna be about the duality group of type two B string theory. So all the anomalies are gonna be of the second because in a discrete group, there's no way to deform any non-identity element to the identity continuous, okay? So that's what an anomaly is briefly. How do we understand anomalies? How do we study them? Well, local anomalies, they're very well understood since the 80s and probably most of us or if not all of us are familiar with how to do this. There's a well-defined procedure which if you're studying anomalies in a theory in D dimensions, you construct an auxiliary polynomial in D plus two dimensions, the D plus two dimension anomaly polynomial. And there is a procedure to construct the anomalous variation of the partition function under some gauge transformation starting with this anomaly polynomial. So in particular, for instance, the theory is free of local anomalies if this anomaly polynomial vanishes, okay? As I said, this talk is gonna be about discrete symmetries. So for the rest of the talk, we're gonna focus in the particular case where local anomalies can't just have global anomalies. I'm gonna be interested in anomalies of type two A, sorry, type two B string theory in 10 dimensions. And so does the anomaly, are there local anomalies in type two B string theory? And does the anomaly polynomial cancel? The answer to both questions is yes. There's a potential local anomaly. So you need to compute this anomaly polynomial. And as famously competed in the 80s, the contributions of the three-carrel fields for type two B superguided in 10 dimensions conspired to produce a vanishing anomaly polynomial even though each of these guys is a big polynomial with weird coefficients. And the fact that this is zero is a very non-trivial check, a very non-trivial coincidence, if you will, that allows type two B string theory to be consistent. And later in this talk, we're gonna see a similar, less surprising, but slightly similar mechanism to this one. So this is what local anomalies are. Any questions about local anomalies? Okay. But that's just how to cancel local anomalies. How do you go about global anomalies? Well, there's a very general framework to understand global anomalies in two dimensions, which is in terms of one, if you wanna understand anomalies in two dimensions, you construct an auxiliary theory in D plus one dimensions which is topological and encodes the anomalous variations of the partition function under this transformation. So this D plus one dimensional theory which is gonna play the main role in this talk, we call the anomaly theory, of course. So here what I'm denoting by A D plus one M of A the partition function of this theory assigns to some manifold A, the data it depends is to some manifold A, doesn't depend on a metric because it's a topological theory but because we're considering anomalies for some symmetry group G, it still depends on the background connection for the symmetry group G whose anomalies we're computing. It depends on the manifold, the gauge field. So there is this auxiliary object which is gonna be the main thing we're gonna focus on in the stock. But why is it important? How do you use it to compute anomalies? Well, the point is that this anomaly theory encodes the change in the partition function or the change in the phase of the partition function as we go around the closed looping configuration space. So what is an anomaly? As I said, imagine you have some manifold with some gauge field configuration and you go through a path in configuration space. Perhaps it's not a path of just pure gauge transformations or the connection is changing but you go around the closed path and when you go around the closed path the phase of the partition function has changed. That is an anomaly. That's what it means that the phase of the partition function is not uniquely defined if you're given a manifold and a background connection for the manifold. So what this anomaly theory does for you it engulfs precisely how the phase of the partition function changes as you go around once such closed loop. So how does it do that? Well, if you're given a manifold and connection and you imagine a path in configuration space where you just keep the manifold fixed but the connection is changing little by little you can map this movie where the gauge field where the background field is changing to a configuration in D plus one dimensions if you say in D dimensions which is called the map interest where you start with your manifold with connection A and then you just add one extra coordinate which parameterizes how the connection is changing but the base manifold remains the same and after you go through a closed loop in configuration space you can glue these things back, okay? So the recipe, the nice thing that the reason why we have an anomaly theory is because the anomalous variation of the partition function along this closed path in more like space it's precisely the value that you obtained by evaluating the anomaly theory on this particular class of manifold which is called the mapping torus, okay? So to demand that the theory is anomaly free in this sense, you just say that the anomaly theory evaluated at a mapping torus is zero. That's the anomaly cancellation condition for global anomalies from this point of view. Any questions so far? But isn't there some examples of kind of this invertible TQT which are trivial whenever we put them on mapping torus but not trivial on some manifolds? That's the point of the next slide exactly. Yes, so I'm gonna be discussing this next. Yes, so I don't know who was asking that sort because I don't have the camera on it was it Pablo? Yeah, it's me from this, I see you're on it. Yes, good. So as Pablo was saying, requiring this thing is not the same as requiring that this topological theory vanishes identically. There are examples of invertible topological theories that vanish on all mapping torus and yet they're not identically zero. Okay, the traditional point of view on what anomaly cancellation is would only require this thing in field theory because it's just fix a manifold and just look at closed paths in configuration space and that's just what I told you is just captured by the mapping torus. However, we're gonna be talking about the anomalous impact to be string theory. And there in a theory of gravity or more generally in any theory where topology can fluctuate, there are more complicated closed paths in configuration space than the one that I just described because if topology can fluctuate then you're allowed to generalize the mapping torus construction to something like this which I drew here like a mapping torus with poles. So the mapping torus can grow holes in arbitrary ways. As long as it goes back to the same configuration, it's still a closed path in configuration space. It's like closed path in which topology changes but that is allowed. So in a theory of gravity or more generally when topology can fluctuate you would like to impose a stronger condition which is not just that the anomaly of a mapping torus vanishes but that the anomaly on all, you could say generalized mapping torus like this should also vanish. And actually this stronger condition it really is equivalent to demanding that the anomaly theory as a topological theory is identically zero, the theory is trivial, okay? So in just in field theory it might be consistent to define the field theory in some background as long as you don't want to glue with all the backgrounds to just impose this in a theory with topology can fluctuate or where you need to make sense of the theory in several different topologies and mountainously you need to impose this stronger condition. This is what happens in quantum gravity but not only for instance, if you have the theory of the brain war volumes, right? The war volume theory of a D3 brain, a D3 brain can certainly grow holes and reattach and whatever. So it's also a theory where you would impose this kind of thing. So that's the, in this study I'm gonna be imposing this stronger requirement that the political, I'm gonna say that something is anomaly free if the stronger requirement that this anomaly theory vanishes is satisfied. Questions about this? Okay, good. So let's go then. So how do we evaluate? So we already understand the framework, we understand the target need to evaluate this anomaly theory. How do we do that? Well, here's where we luck out. Actually you will see at the end of this talk that the project, the work that we did is only possible because we lucked out in a bunch of different places. So the first place where one lucks out when it's done in anomalies is that this topological theory, because it's invertible, it has a much stronger invariance property. It's not just invariant on the small deformations where you can just, you can just, you know, like change, keep the topology fixed and change some, change perhaps the metric or something like that. It's actually invariant on there are much stronger such as transformations, which allow you to change the topology which are called co-reducings. So what is a co-reducing? Well, this thing in red here is a D plus one dimensional manifold M with connection A. And here I've drawn a possibly topologically different manifold M prime A prime also D plus one dimensional manifold. And it sometimes happens that if you're given a pair of manifolds like M A M prime M prime, the two of them together are the boundary of a D plus two dimensional manifold with connections such that this connection here restricts to A and A prime on their corresponding sides and so on and so forth. Okay? This construction B is a co-reducing between M A and M prime A prime. And in fact, it happens that the anomaly theory is a co-reducing invariant. So it takes the same value here and here. And that is a huge simplification of computations because you can actually take the whole set of classes of manifolds with connection and the anomaly theory is just gonna be a function of its equivalence class and the co-reducing. The set of borders in equivalence classes, it forms a finally generated a billion group which we call the the borders group. And so, well, when I say it's finally generated, I mean that in all examples that I ever encountered in physics, it's finally generated. And so that's great news because rather than having to evaluate this anomaly theory, when you're gonna have the political theory, it's generally difficult to say if it's the trivial theory because there might be some manifold you haven't thought of in which the theory has an untrivial value. But because of this, for an invertible to political theory, you only need to evaluate it typically in a finite number of manifolds. And you know, if you can find them, then you know you're done. You know you've classified all possible global anomaly. Okay? So that's one simplification that really helps us big time because we just need to, rather than checking infinitely many manifolds, you just need to check a finite number of them. So all of these stories are just the three-point strategy for computing anomalies in whatever theory you're interested in. So the first step is you, well, you don't mind what is exactly the symmetry type of the theory you're talking about. And if you're able, you compute the co-ordition group. To compute the co-ordition group, you typically need to solve, you need to, you just typically analyze via spectral sequence methods after which you need to solve an extension problem. So this is like figuring out the spectral sequence and doing the extension, as far as I understand it's a little bit of an art. And you need to apply it in a case-by-case basis. So here we were, I mean, it was the expertise of Arun that got us through to do the computation. Once you have that, you still need to compute the anomaly theory. This is a co-ordition group that I was describing computes or tells you what are the potential anomalies for a given theory in a number of dimensions and symmetry type. But it doesn't tell you the actual physical theory you're studying hasn't normally or not. Suppose that the theory you're studying has an unparalleled spectrum, then no matter what the co-ordition group is, the anomaly is trivial, right? So you need to go there, there are recipes to do this. You need to find for your physical theory, in our case, that to be supergravity. What is the right anomaly theory in one dimension more than one needs to analyze? Once you have done these two things, well, you just need to evaluate the thing in step two in all the classes and represent the tips of all the classes that you found in step one. And after that, you just have the answer to your question. Okay? Any questions about, so I'm not gonna, shift gears, this was like the general part of the discussion. And I'm not just, I'm now just gonna focus on how to do this for the duality symmetry of type two B string theory. Any questions so far? Good, very good. Okay, so first question we need to answer actually is what is exactly the duality symmetry of type two B string theory? And there are several ways to figure this out. But there is this really beautiful paper by that Chicago and Yonekura where this discussion is done in an appendix and I'm taking the discussion out of that paper. So what is the duality symmetry of type two B string theory? Well, let's start with the basics. Type two B is supergravity has the SL2R and then the usual lore is that in type two B string theory, this is broken by quantum effects to something like SL2C. This SL2C is physically very important is the mapping class group of the F theory torus. So it's a group of large different morphisms of the F theory torus. So it receives a physical interpretation in F theory. And this physical interpretation that you get in F theory is actually coming directly from an honest to God torus that you have in the corresponding M theory background. So when you do F2M theory duality, this torus becomes physical. And the different morphisms of the torus in M theory is the group of large morphisms of different morphisms of the torus in M theory that gives you this essence. That's the usual lore. But once you present it like this, you realize that this can't really be the end of the story or can it cannot be all the story because after all M theory is a theory which contains fermions. Now this SL2C is the group of different morphisms of the torus. So for instance, one element in SL2C is if you have a square torus, you can rotate it by 90 degrees and that's still a symmetry. So you can rotate four times by 90 degrees. That's a full term. And in SL2C, that's a trivial transformation. But as we know, in a theory with fermions when you rotate by 2 pi, you just don't get a trivial transformation. You get the transformation which acts trivially on bosonic fields and there's minus one on fermions. So the symmetry group of M theory on a torus which we agree this is the same group as the duality group of type two B string theory is not just, it cannot just be SN2C. It has to be an extension of SN2C by minus one to B by a transformation which acts as minus one on fermions plus one on bosons. And the actual group that you get when you do this is some matrix, some group which is actually not a matrix group. This is called the metaplectic group. It's a double copper of a C2C. So that's one extension, one can take it like that. But in fact, it's not the only extension because another thing that we learn that we know about M theory is that it actually makes sense. It has a parity reflection symmetry. You can take one of the sides of the torus flip the coordinate of just one side and it's actually a symmetry of type two B, sorry, of M theory. It's a completely valid symmetry and that enlarges SN2C to something like GL2C, a general group of transformations with different. Now, when you have a diagram like this, you know what's coming. This has to be something on the bottom right corner that allows you to extend type two B string theory including reflections and including fermions. And that's some abstract group which is the PIN plus cover of GL2C which we actually referred to as GL plus 2C for short. The reason why you get the PIN plus cover is because M theory, the kind of reflection that you have in M theory is squares to plus one on fermions and that's why you call it PIN plus reflection. But in any case, long story short, after all this discussion, the duality symmetry group of type two B string theory we need to discuss is this PIN plus GL2C guide. Any questions about this? Okay, let's get going. What is this PIN plus thing? How does it goes in terms with space time fields because the minus one to the F identifies some duality transformation with a rotation of like two pi in space time. How is all of this working? To give us some intuition, I'm gonna take one step back and discuss a slightly more perhaps a more familiar case but similar which is the PIN C case. There are sometimes, you have theories where there's U engage fields and the charge of the U engage field is correlated with whether the corresponding fields are bosonic or fermionic in such a way that you can identify the zeta of root of U1 with the minus one to the F in the spin group. So when you have a structure like this you can actually define this theory in manifolds which don't admit an ordinary spin structure. For instance, you can consider this theory, this theory makes sense on CP2 for instance here with this structure makes sense in CP2 which wouldn't be the case if you didn't have this kind of structure you just have a spin structure. And so that's what's called a spin C structure. It allows you to put the few more general backgrounds which are not spin and the way you study the way you quantify which backgrounds you can put a theory with spin C structure is the fact that you can take this U1 here and take the square of its transition functions and that becomes an honest U1 bundle. So spin C theory has an associated principle U1 bundle and there's actually a consistency condition that you need to impose on the speed. This is a typo to the spin C manifolds which is that the second step with the class which would vanish just for a speed manifold has to be the model to reduction of the turn class of this U1 bundle. Bottom line in a theory with spin C you can put the theory in more general backgrounds and these more general backgrounds they must satisfy this consistency condition. So for type to be super gravity you have a very similar story essentially the same only that rather than U1 you have this PIN plus GL2C story. So there's this PIN plus loyalty group that we discussed and it mixes with spin and so that means that you can actually put type to be string theory on general backgrounds which are actually not just spin manifolds. In the same way that for the U1 bundle there's an associated principle U1 bundle for the PIN plus GL2C case that's actually the way you characterize the duality bundle. It's in terms of an associated principle GL2C bundle and just like a U1 bundle has a turn class this characteristic class this guy also has a bunch of other characteristic classes. They're different, there's not just a turn class but they're more or less the same. There are characteristic classes that classify these bundles so there's some class B and A in degree one and there's some class W in degree two. Just like there's a consistency condition on spin C manifolds, which is this. For a spin PIN plus GL2C bundle there is a consistency condition which is the second typical Whitney class of a tangent bundle has to equal this class W of the duality bundle, okay? Bottom line, what we're doing here what I'm telling you here is that you can consider type to be string theory on backgrounds which involve retrieval duality bundles these duality bundles can be efficiently captured in terms of some characteristic classes which I wrote down here. And this actually allows you to consider type to be supergravity on backgrounds which are not just spin that you can consider the non-spin manifolds as long as this consistency condition is satisfied, okay? So there's more general backgrounds that these are the kind of backgrounds that you can think about in type to be supergravity once you turn on duality bundles. Any questions about this? Good. So this is the symmetry, remember the first step is I need to tell you what the symmetry type of the theory we're talking about is and already that took us some time it was more complicated than expected perhaps but now we know what the symmetry type is and how to characterize it in terms of these characteristic classes. And so what's the next thing we need to do? Well, we need to compute its co-ordering group. We are interested in anomalies of type to be supergravity in 10 dimensions. So that means that the anomaly theory it lives in 11 dimensions and so we need to compute the 11 dimensional co-ordering group. And that is a difficult computation involving the add on spectral sequence and a lot of math. But the end result, this is already a new result is that the 11 dimensional spin yield plus board is in group that is relevant for this theory is this thing, okay? So there's a bunch of factors at prime two and prime three. So that means that in principle there's many potential anomalies, okay? But remember to make sure whether there's anomalies or not we need to find out the precise anomaly theory of type to be supergravity. And this was analyzed in a really beautiful paper by Shieta Chippewa and Yonekura from 2020, from March 2020. And the answer is something like this which I'm gonna explain. I'm actually gonna spend the next five or even more than that minutes explaining this anomaly theory. But before I do that, let me ask you if there's any questions about this board is in group. So I'm gonna compute the anomaly theory. So I'm gonna explain what the terms in this anomaly theory are, okay? So type to be supergravity has three carol fields the Latino, gravitino and self-dual form. And consequently the anomaly theory has three terms. That's the one coming from the gravitino. There's another one for the dilatino and there's another one coming from the self-dual form. But not only that, if you remember type to be supergravity has this term, this triple trinium term in terms of dimensions, which is something like integral of C4, which H3, which F3 or you can also integrate by parts if you prefer. So that term involves a self-dual field and it actually also contributes to the anomaly as part of the contribution of the self-dual field. So you need to include another term to the anomaly theory which is related to this trinium term. And I'm gonna explain all the things that enter into place here, but basically, you just need to take into account this background that can source a term with the involvement of both. Okay, so that's the anomaly theory. Let's understand what these terms are. So the first two terms are just the anomaly of affirming and by now, these things are very standard. Very generally, the anomaly of affirming in these dimensions is related to an invariant in D plus one dimensions which is called the it invariant. And it's just some invariant that you can compute out of the sum of eigenvalues of the rack operator. It won't matter for us too much what is exactly the definition of how to compute this and you can put some boundary conditions and so forth. What matters is that it's a number which is very difficult to compute in general. But as I told you, in this project, we lucked out a lot. And even though it's difficult to evaluate the nico invariant in a general background, for the kind of backgrounds that we are gonna be needing to study the anomaly, there's actually a formula that allows you to evaluate it. So bottom line, it's a well-known thing. It's not subtle. Well, there's some subtleties but they're well understood, I think. So that's what happens for the contribution from the fermion. So that's from the gravity on the Latino. And as you may have expected, the story for the self-dual field is far more subtle and here's where generalized comology enters the game. So the anomaly theory of the self-dual field which was worked out in a series of papers by Moore, Belalove, Moneer, and most recently the formulation I'm following is from this paper by Shieta Jekawa and Jeanne Kura. The anomaly theory of the self-dual field has one term which is just an eta invariant. There's perhaps not, it's not different from the ones from the fermions. But then there are two other terms that depend on something called Q of C which is, it's a quadratic refinement of a homology pairing. I'm gonna explain what that is and how it enters the game momentarily but I'm just gonna say these two terms, these are things just a sum involving the quadratic refinement and the term involving the fermions, I must say I'm also in both the quadratic refinement. So really all this new piece only depends on the quadratic refinement, sorry. But before I go on talking about the quadratic refinement, are there any questions about the rest of this? So the first one, I think I've learned that the anomaly of the free fermion is this eta invariant. That's right. But I don't think you've told us exactly what the theory is, like an action or something like that, but can you, so do you, how do you get, how do you use your details of your theory like you're the grantor to get this information or? Oh yes, good, good. So the question is, is the demand of the fermion but the fermion can be charged can be transforming representations of symmetry groups, for instance. So that's where the difference enters. For instance, in this particular case, we need the invariant of a fermion in the doubler representation of pin plus gl2z for spin one half for the dilatina and for spin three halves for the gravity. That's how that information enters here. And it's just a massless fermion. This is just a massless fermion, yes. Yes, because that's the dilatino gravity. If it's massive actually, if there's a general lore for free theories, right? If you can give a master and they're not gonna contribute to anomaly. Yeah, and somehow the other anomalies you can also get from like certain terms in the grantor. Well, I'm not sure I understood the question. So the, so we need to first determine what the anomaly theory is, right? And there's the, oh, you're asking about the self-taught field. Yes, that's gonna be talking about next. And it's just, yes, you need to work a bit harder for that one. Yeah, yeah, yeah, exactly. I'm gonna be doing that for the next five minutes, I think. So, okay. So what is the anomaly theory of the self-taught field? Well, how does one construct it? Well, the general lore, which is the same as you have in condensed matter, is to construct the anomaly theory in D plus one dimensions, you can equivalently take the point of view you wanna construct some sort of topological theory in D plus one dimensions and show that it's boundary mode when you put it in a background with boundary is exactly the gyro field that you were looking for in D dimensions. So if you follow that recipe for the self-taught field, you get that there's something you can write down in 11 dimensions, the theory of a pipe form with the, with Schoenheim's couplings like this. And then if you go through the details, you work out the equation of motion, which is beautiful, done in the paper by Sheppard, Schoenheim and Acura. For instance, you will see that you get the, you get the self-taught field as a kind of moving about, which is great, okay? So that's great. So that's, that's how it works, but there's, there's, there's, there's just to be a substitute. We have a term assignment term. It's very important that we have the term assignment term here in the action. And because we're starting anomalies in, in situations where you have non-trivial topologies, you really wanna make sense of this term in situations where A is topologically non-trivial. And for that, you need to use, you need to use a generalized homology theory, the differential version of generalized homology theory. Which one you should use is a physical choice, okay? So the simplest thing you could do is differential homology. But for instance, we know there are not wrong Ramon fields that are quantizing in differential k theory and so on and so forth. All the theories that are gonna be able to describe the self-taught field in the end, no matter what it is, you would still need to, you would have some, what are they bearing like this and you need to make sense of it. And the differential homology theory typically tells you how to make sense of this bearing. However, it is not enough because there's in the action that I wrote down, there's a one half, okay? So this one half, there's no, at least I don't know and I don't think I've seen in the literature a general way of defining how to do this for generalized homology theory. And so the use of a approach is to say, well, there's, I need to make sense of this one half A, which A. So I'm just gonna kind of like give up to try and define it directly in terms of the structure of the stuff that defines the differential homology theory. I'm just gonna replace it by some, this is a quadratic function of A. So I'm just gonna replace it by some quadratic function of A and wanna work out what is the anomaly, keeping this Q of A, you know, general and then hopefully I can determine what it is later like at the end. So this Q of A means a quadratic refinement of the differential homology pairing and it's something that can in principle depend on our trace of background field or trace of management. And the difficulty is that depending on your homology theory, depending on your dimension depending on a bunch of things, there may or may not be a canonical way to find one, okay? So are there any questions about this? Good. So how do we determine this Q of A? That's the million dollar question. And here's another point where we lucked out immensely in this project. So for instance, you expect that Ramon-Ramon fields in type to be string theory are quantizing K theory. So maybe you can find a canonical, is there maybe there's a way to find a quadratic refinement in differential key theory. Maybe there's the right homology theory that one needs to study. And in fact, in less than limit dimensions, you can construct what you want to define many differential key theory if you have a spin structure. However, this is not good enough for the kind of things that we want to do because we just don't want to do things in a spin manifold or using differential key theory. We actually want to study anomalies involving duality bundles. I don't know what the generalized homology theory, how does the differential key picture match with duality? And I don't think this has been fully worked out. And so I'm not sure how to give any top down any universal prescription to find Q of A for an arbitrary spin yield plus manual. I just don't know how to do this. So what the approach that we took in this paper, and this was just an incredible amount of luck that you can take up this approach and still produce results, is we're just going to be agnostic. We're going to assume that there is some canonical choice of quadratic refinement given to you by the right homology theory that we don't know what it is or even, you know, we don't know how to compute it. So we're going to assume that something like this exists for an arbitrary spin yield plus two manifold. And then under the assumption that it exists, in some cases, precisely the cases which are potentially anomalous, we're going to be able to determine what this quadratic refinement is. And I will explain in detail in one example how we do that, okay? So we do the kind of like bottom up approach. Any questions about this, by the way? Hmm. I just missed what you meant by duality here. Oh, yes, what I mean is that the, so, right. So we're studying anomalies of time to be backgrounds which involve non-trivial duality bundles. Like for instance, like the actual dilatone can be fixed to I or to some other, you know, strongly couple of value. And then there can be olonomies of SL2Z or the spin plus group, right? So that's what I mean. Like there's no prescription to define this thing for an arbitrary spin yield plus manifold. Good. So now finally, we have the word is a group, the anomaly theory. And this is non-trivial, but we're also able to do it. We're also able to find explicit representatives for the generators of all the borders in classes of this big version group that we, that told you about before. So with this, you can actually complete the anomalies. And this is how what the results looks like. So on the left column, we have the factors for all the factors of the bordering group in 11 dimensions that I wrote them before. Second column, we have explicit generators for each of these classes. For instance, L311 is the 11 dimensional lens space. And I can talk about some of the others of you you wanna ask. Detector is just a cohomology class that is non-zero in this, or some bordering invariant actually that is non-zero in these manifolds. And finally, the last column is the most interesting why it's the value of the anomaly theory on this guy, okay? So there's one class where we cannot completely evaluate the anomaly. We know it's either zero or one-half. So it could be vanishing or non-vanishing. And I will come back to this one later. For now, let me focus on this of the classes where we actually know that the anomaly theory is non-vanishing, okay? So we actually have completed it and there's an anomaly. So there is an actually, we were expecting that the type to be duality group was gonna be anomaly free so that we could constrain some of the duality groups. But in fact, we find that a very big surprise. The type to be is a variety of group is actually anomalies. And so that gets me to my conclusion because if the duality symmetry in type to be string theory is anomalous because it has to be a gauge symmetry after all you map to different morphisms that means that n theory itself has to be consistent. It has a different morphism anomaly and so it cannot be a consistent quantity of gravity. So that's the conclusion. Thank you. Just kidding, just kidding. Sorry, I just, I love to, it's a great job to make with a project like this because you still have 10 minutes and this is not the end of it. Because of course, this would be a stupid conclusion that this is a consistent quantity of gravity. And what you need to do is to work a little bit harder. So in fact, we did work a little bit harder and we found that there's a relatively simple modification of the type to be trans-heimant's term. So it uses the type to be trans-heimant's term and we just added some new term which looks like, it's basically a term involving torsion discrete classes, involving the class of the characteristic classes of the duality bundle. You maybe remember from a few slides ago I had the A and B geyser, the characteristic classes of the duality bundles. So you can construct some new term out of them and you can couple to the ramon, to the self-goal field in this way. This is a discrete term. It's not something that people that doing supergravity in the 80s would have seen. It's not something you see at the level of local fields, at the level of the supergravity equations of motion. So it's completely fine. It's fine on its own, but it's reasonable it would have been missed by these people in the 80s. And when you include this term, what actually happens is that there's a shift in the anomaly theory and this characteristic class which was coming from H3 with F3, gets shifted to something involved in this new field. So the anomaly computation itself changes and the table once you include this term, looks like this. So there's still the class that we don't know what to do about. But the four classes that were anomalous before, with the contribution of this term, they become anomaly free. It is very important and very nice that the mechanism that we wanna have here, these terms that because of the symmetry properties of this topological term, it could have only canceled anomalies in these first four classes. I can elaborate more on this, but the bottom line is, if a single one of these numbers have been non-zero, we couldn't have canceled this anomaly with this topological term. And so it's really a special, it's really lucky that we didn't have the first anomalies, that we only had anomalies in the first four classes. So again, it's only all of these works because we're getting very lucky. And now I'm gonna show you the luckiest aspect of this all, which is that this mechanism, some, okay, fine, I gave you some topological term, anomalies canceled, great. Why is this such a big deal? Well, it's nice that this mechanism only works because of some detailed cancellations that happen because of the particular fields that appear in type 2Bs for gravity. If you change the representation of duality group even a little bit, for instance, this wouldn't have worked. And to illustrate this, I'm gonna compute for you the Z27 anomaly. So there's many borders and classes and the biggest one is Z27, so I'm gonna focus on that one. So there's a Z27 anomaly that is known to you when you evaluate the anomaly theory on the 11 dimensional S11 mode Z3 lens space. And if you just start changing the duality representations of the matter in type 2Bs for gravity, you would find that the value of the correspond, that changes the anomaly theory and the value of the anomaly theory that you could get is just basically any interior model of 27. With the mechanism that we are proposing, out of these 27 possible values, just one, just K equals nine can be canceled. That's precisely the value of changing type 2Bs string theory. And you have actually similar coincidences taking place in all the other four cases where the anomaly cancels the derivative. So how does this work? Let me work this out in detail for you. And I think it will show what are the numerical coincidences that are happening here. Okay, remember the first problem that we need to face with this is we have no idea what the quadratic refinement is. And that enters the anomaly theory. So what are we gonna do about it? Well, we are again, very lucky because we can determine the quadratic refinement by demanding anomaly cancellation without the duality bundle turn on and then we can analyze anomaly cancellation without the bundle turn on. So the Bordeson class that generates this factor is an S11 mod C3 with a holonomy of a Z3 subgroup of the duality group around the trivial one cycle of the lens space. So what we're gonna do first is we're gonna switch off that holonomy. So we're just gonna have S11 mod Z3 and without duality bundle. So I did not do this by S11 mod Z3 is zero. And then we know that this is the trivial class in Bordeson. So we know the anomaly theory exactly vanishes and we can compete this in variance. I told you there's some formula that we can use and these are the numbers that you get. And we know this thing has to be zero and because we are also not turning on turn on terms this thing is zero as well. So this thing has to vanish but the right-hand side these three numbers don't vanish. There's just one contribution that we're missing. So we can actually determine what the value of that contribution must be so that the anomalies cancel when you switch off the duality bundle. What this tells us is we know the R-thin variant of which is depending on this quadratic refinement for this particular background. And for this particular background once you have the R-thin variant that uniquely specifies the quadratic refinement that you need to get. So from this number we can actually know what the function Q of C is for this particular background. So that's how we determine Q of C. Once we determine Q of C we can actually compute anomalies with the duality bundle term. So we now turn on the duality bundle and there's the hidden variants which now change value a little bit. There's one that changes value because of the because you have to turn on the duality bundle. The R-thin variant is the same thing and there's the quadratic term that was zero originally now takes some value precisely because you need to turn on the because the new term that we propose forces this thing to switch on when there's an intuitive duality bundle turn on. So the values that this quadratic refinement can take are just, it's basically for this particular case you can just take one possible value and it's one-third. So it better be the case if you want anomalies to cancel it better be the case that all these things when you change these from this they add up to one-third and that's precisely what happens. Okay, the change in the anomaly from the fermions conspires such that this term can actually get the right values to cancel the anomaly. So it works because of digital cancellations because you're lucky it wouldn't work in general. There's similar coincidences taking place for the rest of classes. So something like this always works which is nice. And let me just comment on a couple more things. There was one class where we were unable to compete the anomaly the one that would I say it was either zero or one-half and we don't know. And what happens is that the generator of that board is in class is not a spin manifold. It's a spin GL manifold but it's not a spin manifold. So it necessarily requires for you to turn on the duality bundle for type 2B supergravity to make sense of that background. So the trick that we used to determine the quality refinement which was to switch off the duality bundle doesn't work here. So that's why we're unable to compete the anomaly. If someone had the right prescription to compete the gravity refinement for general spin plus structure you could also compete the anomaly here. And I hope it's zero. If it's not zero, there has to be some other interesting mechanism different from the one that we're presenting as we cancel the anomaly. Now, very briefly, what other checks can you make of this term that we propose? Well, there is... We did a very simple and rigorous check that let me emphasize one needs to carry out more carefully. But as an naive thing, it seemed to work which is that the term that we have seems to match the duality anomaly of N equals 40 percent nails. N equals 40 percent. So whenever you have a holographic duality you have some field theory side and some gravity side. And a general fact of holography is that the top anomaly in a global symmetry in the field theory side gets matched by a topological term in the vault. So the anomaly theory gets a physical life as a topological term in 80s five versus five or whatever vault plot you have. So under this general correspondence you could ask why is it the topological term in the vault that captures the Monton and Olive Duality Anomaly which of N equals 40 percent nails which was captured by this pair by Tatekawa, Shen, John and Kura. And a very naive dimensional reduction of our term produces exactly the right value for the anomaly theory that was found by these authors in this paper for the C3 anomaly, okay? So one thing that we wanna do is we wanna understand so the reason why this is not rigorous is because doing the dimensional reduction of the softball field is difficult enough without this discrete stuff going on and this is a lot more difficult. So what we did here was just literally replace F5 by N. You know, that might be too naive. You know, that's a first zero order check. It seems to work. So in the end, this is a happy ending for tech-to-beast trend theory but just like any good story, it actually has a twist end because the mechanism that I just described which you could think of as a mixed version of the Green-Schwarz mechanism is not the only way that we found to cancel the anomaly. We actually found alternative ways involving the political Green-Schwarz mechanism. The way this works is you forget about the stuff that I've been telling you about for the last 15 minutes about the quality refinement and instead, you just modify the action of time-to-be supergravity by adding some massive fields. In fact, you can add a BF kind of theory. So these are massive fields. They don't change the story of low energies. You can introduce a term like this and if you tune the couplings by hand, you can use terms like this to cancel some of the anomalies. So there's another way you can cancel the anomaly. These theories that we are describing are definitely not ordinary type-to-beast trend theory because the equations of motion of this theory obstructs some backgrounds that we know are there in type-to-beast or in type-to-beast trend theory. But nevertheless, for a supergravity person, the theories look exactly the same. They still is type-to-beast supergravity. And so I think a natural question is whether these things that we're describing are they which are not type-to-beast trend theory but the cancel this anomaly, whether they are actually consistent or is there some way to show they're actually consistent there in the swamp? Which is the right way to cancel this anomaly? And this ties in with a more general question of whether there can be more than one consistent quantity of gravity with the same low energy limit even in high dimensions. So for instance, we usually draw the n-theory flower with type-to-beast, type-to-beast, type-one, hydraulics and n-theory as we draw this star, the n-theory star and perhaps if you include choices like discrete data angles or this is something that I was just describing now, the picture one should have is more like a flower where each of the points of the start corresponds to different petals of the flower. There can be more than one theory in 11 dimensions or in 10 dimensions. And there's also hints that some things like this happen in n-theory and also in type-one string theory. So I can talk about that also if you're interested. But since I'm out of time, I would just summarize that it was an anomaly, but it was not meant to be. We found a way to cancel it. The most important thing I think we need to do is to understand the full ramifications of this term that we are proposing on the duality web. We really need to check it's correct. So that means how do you get it from m to f to l and so on and so forth, things that we didn't do. And of course we also like to evaluate the anomaly in the manifold that we're going to compute on this mysterious manifold x11. So with that, I thank you for your attention and I welcome questions. After your first conclusion, I wanted to ask whether we should end the funding for the string theory department. But I guess you answered that in the last 15 minutes.