 to the third and the last day of the workshop. And we are happy to have Lukas Miller and he will tell us about anomalies, high geometry, and functorial field series. One second, and thanks for the introduction. Thanks to the organizers for giving me the opportunity to talk about my work. So I'm going to talk about anomalies and then I try to talk a bit about the corresponding metamatics and study it and a lot of these things have already appeared in the previous talks with the secretary, where it was mostly about anomalies, theories, and actually true to a certain kind of geometry that I expected. Actually, for example, which were most of my own work actually guys and then this part of the study they really simply found and all these things are basically kids. But I will start with some physical integration and anything which is my own work in this part of the space from the workshop. And so yesterday we have seen and then we have seen anomalies in the past secret description. And in my work, I will focus on the California description. So how can you see the mental level of space spaces and all spaces? So I will start with some physical integration. So it's like I have a drawing along here to keep the space, not if there's time to keep the space. And then to set up our front view theory, I will usually need some elementary structures on a map and to be concrete, I will also show here I will show all the general stories like on the right, I will talk about the kind where I'm on and your case, please. So we have a total description. This happens if the dimension of n has a part, so it's more secret to one or one to the two. And then it appears to be as very long to discuss the method, which I was just going to show you in a moment. On the other hand, I guess here, this is a connection on the principle of lambda for the key, the key. And furthermore, I'm patient of the space brush. Because is your microphone on? It's supposed to be on. How's the sound? Okay. So is it better now? Someone should respond. Is there a microphone on? Right. Well, we have to see someone. I mean, I can't turn this microphone off. We have to do some testing. Is it better? Can you try? I can turn this off. This is supposed to be muted. And is it better now? It's not configured testing, but there are problems. Are we using the same setup? Okay. I don't know. I mean, if you can hear me now, I will just continue. People seem to hear you through that microphone. Okay. Sorry about that. So where was I? Yeah. This is somehow the geometric background structure we need to define a carriage theory on a time slice. And over here, we just have a manifold M and then some type of geometric structure which lives in the space F of M. And then what we get, what we have first, we have the one particle, this one has some H. And here, this H is going to be given by the square integral sections of the spinobundle and sort with the vector bundle which is induced from the principle bundle of this representation rule. So these are sections, square integral sections of this bundle. And then this is like the one particle here by space. And then what one wants to do is second quantization. And for this one needs to pick a polarization of the red space. So decomposition into positive and negative. So H plus and H minus. And then one can define the second quantization space on M as the exterior algebra which just can roll with the exterior algebra on H minus. And here, so if the, let's assume for the moment that there's no zero modes from the geocorporator, then we just take S H plus or the space span by eigenvectors with positive eigenvalues and H minus being the space span with eigenvalues. And if there are zero modes, you can for the moment you can just put them into H plus, point to H minus and the resulting torque space is going to be isomorphic, but non-canonical. But since we are just at one point in F, this is not such a big deal. The problem appears now if we study the following, we try to understand what happens if we vary these three but spaces over as F of M. So we want to understand the dependence. And this is the place where the unnormally happens. So it's like somehow that we cannot define this fork space globally or continuously in this space based on the space of years. So what we can try to do is we can have a F of M with some patches, Ui and Uj, and then we hope that we can pick some polarizations over these patches so that we can smoothly define these three but spaces. So then we will get a Zi of M defined on this open patch and we get a Zj, which is defined over the other open patch here. And as an example, what one can do with the tiger gate series. So for example, if I have my patch and then the spectrum of the DRock operator does something like this. I could, like for example, in the I patch, I could say this is the I and then this bit here would be J. So and then I just take H plus everything above this element, which is not in the spectrum and here I take everything below the element which is not in the spectrum. But maybe, I mean, there's no reason why I can do this consistently over all of F of M. I mean, there's a reasonable hope to be able to do this locally, but globally there might be no reason for this to work. And this is exactly where the anomaly comes from. So it's like the fact that we are not able to pick these polarizations consistently. So now let's assume that I can at least relate the polarizations as something kind of dimension. So I assume H, J, H, I plus some finite dimension of vector space. And I mean, this is what happens here, right? The finite dimension of vector space are just all modes between these two valence and then H minus I will be H minus J plus some vector space. And now I can consider the form spaces. And from the definition, we see more or less directly that this space is not equal to the other one in this case, but we have a canonical isomorphism which goes to the IM, if it depends on the determinant of W of J. The determinant is just a top exterior problem. And then this has happened like over UIJ and if UIJ is noncontractable, this might define a non-trivial line bundle. But we could even make it, I mean, if we assume that we have a good cover, this we could trivialize this, but the thing is that this trivialization is non-canonical. This relate on again, this is where the problem lies somewhere. And then here on the Kyrie-Gates theory side, I mean, WIJ is just the eigenspaces with eigenvalues between like theta and I and J. And now we can ask what happens on triplet intersections. So if I go, look at the triplet intersection UIJ case, what I will find is that, I mean, I can start with the I, this is isomorphic to the J, can do the determinant of WIJ. I can also directly go to the K, and I need to handle the determinant of WIK. And here I can now go to the J, a DK, and then I handle with two determinants. And I turn out that there's a canonical map here, which is also from one terminal with some psi IJK. This is a map from the determinant IJ, handle with JK, determinant for WIK. This is the psi IJ as well as this. And now somehow here you can just pay this out, but this will give you a similar structure. And now this is somehow really where the anomaly happens, is that we cannot trivialize these things and these things at the same time. So there's somehow no globally well-defined Hilbert space on FM in general, but it's only defined locally. And then these things somehow encode the failure of it being a Hilbert space. So let me now try to motivate the corresponding mathematical structures. So we see what I really want to highlight for the moment these two bits. And I want to convince you that you think of these things as line bundles, but where the coefficients are not complex numbers, but actually vector spaces. So you see here, like I have at the intersection of two things I give a vector bundle. For a line bundle here, I would give, if I have a line bundle, I would give some transition function from UIJ to C4. But here I give a vector bundle, which is a smooth space, which is a map from UIJ to vector spaces, but the smooth family of vector spaces. And then down here, I have the usual co-cycle condition, Fij, Fjk, this is Fik. But since these are vector bundles, there's a notion of map between them. So I don't want to have the co-cycle condition as an equality, but I want to have it as an isomorphism. So I have an isomorphism here, which implements the co-cycle condition and then there's a compatibility condition on quadruple overlaps, which I'm not going to mention. I'm not going to spell out, but you can figure it out by just looking at the example, opening any textbook. And so this structure motivated it like a category for a line bundle, or these things are also known as jobs. So this is the structure we'll get. So we have the space of fields. Move to the other side. Yeah, so we have the space of fields, S of M, and then we have this higher geometric structure of a job sitting over it. And now if you look what this lead does, this behaves like a section because it defines like something on all the independent patches, which then transforms, if I go from one patch to the other one by the transition functions, now just we implement these things by the most items again. So we have our job sitting over S of M and then we have behavior space. This may define as a section of this job. And the non-periodity of the job is similar to the presence of an unknown. Okay, let me mention what happens in the Kyber case. So we take now only the only very, so we restrict the space to the space of connections on a fixed bundle. This is an affine space over one forms on M with values in G. And then I have a job over here and this thing is called the index job. So this has been studied by Kari Mikkelson and Rory in connection to anomalies and the Fibar trace is again a section here. And now the anomaly is the non-periodity of the job but one really needs to be careful because this is a contractable space. So you probably know that our line bond is over contractable space at trivial but the same is true for jobs. So the job as it's own is trivial and the anomaly really comes from the fact that there's more structure, right? There's the group of gauge transformations acting here. And this job, the job here becomes equally variant with respect to this action and really the anomaly is a non-periodity of the equally I want. And this is the second point I want to make in this introduction is somehow. These things are higher geometric objects but it's important to also treat this thing as a higher geometric object which has internal symmetries. So here you see if it's that difficult just treat it as a space and do not consider the gauge transformations it's really there's no anomaly. And the anomaly comes from the non-equivariance from the non-equivariance or from this thing being not equivariant each other. Okay. And then there's another technical problem one might worry about is that these things are infinite dimension but both I mean this space doesn't generate infinite dimension in this one as well. So now I'm the next bitch I want to talk about is just like a mathematical framework somehow which allows us to make these objects precise and talk about them and study them. This is, both I know the name higher geometry which I will just talk to of my talk. So let me start with a definition of a generalized space space. It's a pre-sheath F on the side of many folds which takes values in infinity group for it or simplification sets or topological spaces. And this is similar to what we have seen and then treats talk as a model in the geometric structure we put on a border category. But for the moment here we allow more than just look at if you move and we also know all maps of the truth with many folds and then we also denote by H as a category of generalized spaces. Let me give you some examples. So how does this framework come on quotes most of geometry you probably know. The first example has to fix a manifold X. So this is a manifold that can even be a fishy manifold or some other infinite dimension type of manifold. And then we can form a generalized space it just goes from many folds up to these two sets but this sits in infinity group. So these are just the group which has no higher morphisms and what does this do? It just sends a manifold M to the collection of C infinity so smooth functions from M into X. And this summer should be the general picture one has in mind for these objects. So instead of describing the space F what we really describe is what it means to have a smooth family of points and F parameterized by manifold M. And this is somehow a convenient way of modeling all types of higher spaces. Let me also give you two more examples which have some higher structures which are relevant to what is going to come. So for the first one is BG with a connection this sends a manifold M to the group void of G bundled on M with connection. And then the other example is related to the fact that we want to also study the geometric structure on a fixed manifold. So I fix some manifold X and then I want to have like a space which describes certain geometric structures on X and usually you do that by a parameterized geometric structures on fiber one with typical fiber X. Let me just use an example which are matrix on a fixed X so they send M to set a fiber wise matrix on M cross X of the projection to X. I should also say putting in many folds here is not, it's a bit too big because in general these things satisfy descent so they are already determined by knowing them on the contractivist space so one could also replace this with the category of Cartesian spaces also. So you're using a sheet condition in general, right? I did, I said yeah, it's a sheet. This should be a sheet, yeah. So it satisfies the version of descent and this allows you to work with sheets on Cartesian spaces which is a bit smaller but at some point these many folds they really have to go to deputation so we will stick to it for the moment because we are not going to do anything too technical anyways. So now. Lukas? Yeah, yeah. Would you need the bundles M times X to be non-trivial if you want to have a sheaf? I don't think so. I'm not 100% sure, I was confused about that when I was preparing the talk as well but I think it's, I mean these are just, I mean what I want to, I just something like that's from M into metric on G, metric on X but I want to write this and I could put like some many folds and fill it in as a many-fold structure on this and then I could use an assumption to rewrite it as much like this. And the projection should be to M or to X? Oh, the projection should be to X, right? Yeah. So I wasn't, I'm not 100% sure but I thought, I think this is okay because somehow I do not allow our morphisms of X so nothing happens like, if I would also somehow include a few morphisms of X then I would need bundles because then I can like locally glue using this different morphisms of X and they should not. Okay, okay, thank you. I mean if someone disagrees, they should let me know. Okay, so now I'm like, I don't want to call this a definition but now somehow higher geometry is just you do ordinary geometry but you replace many folds with general spaces but we can do, we define what a smooth higher loop is. So smooth higher loop is an element in H. What I should say is the loop object in an H but I'm just going to give you an informal definition so it's an object in here which comes with a multiplication because from G plus G to D and it has a unit which is a map from the constant G to D and these things now need to satisfy so there's a lot of coherence data which encodes that this thing is associative and this is a unit and furthermore this needs, this multiplication needs to be in multiple which way you can more or less use the same definitions one uses usually and then I'm not going to give you all the details because in H it's also quite complicated but quite often the things we have actually factors who sense or the factors who group or it's an in there it's easier to say what these objects are. Let me give you two examples. The first one is just I take a lead loop and then I take G underline so I take consider this as a smooth as a generalized space and this is a smooth group. So this is one example and then the second example which is relevant to define jobs I call line but you could also call this P C cross and this then the manifold M to the two point of line bundled on M and here I only take a vertebrate loop for instance. So it's to a space M to a manifold M it assigns a scoop world of line bundles and the multiplication in this case is given by taking the terms of product of line bundles and all line bundles I'm just a bit of respect to this operation. Okay. And now there's, now we have groups and now there's a similar definition to the usuricate of a principle bundle. So the definition is due to Nikolaus, Stevenson and Schweigel and I'm again, I'm not going to give you the details or any precise definitions but it's similar to what you would expect. So it's a generalized space which comes in with a map down to your original space which has an action of G such that the quotient of the principle bundle by this action is X and then the job X and H line. So this is like a really bad, okay, let me write B to cross bundle and you see if you know I mentioned something like a co-cycle description for this then on double intersections you will actually assign line bundles and on triple intersections you will need to assign either more for something. So this is somehow a good mathematical framework to make the things we had we have seen before pre-sets and also this X as I alluded to here can be infinite dimension but this is not a big problem for the framework to have at least on a theoretical level, okay? And then we do one example. So I can look, I take a manifold X which has an action of a legal and I can define this quotient straight and this, so I'm only going to say to you what it does locally. So if I evaluate this on a patch I don't know if it comes in contact with it or something local, I get the infinity function from you interact, acted on with the infinity function of you into G and then the global behavior can be constructed by gluing these things together just using the descent construction. If you do this on a global manifold and this would not be a sheet but you can chiefly find this description and then let me put this here because I have a job on this space then this is equivalent to having a G equilibrium job on X. And this is exactly what we have seen in the beginning. If you had this job at the space here which actually is the quotient stack so I have a space that has an action of the group and then if I want to have a job on this as a higher geometric object I actually need to give an equilibrium job on here and this is exactly what happened in that one. Now I will switch topics a bit and move towards factorial field theories and the goal is somehow to at least sketch how this modern description of anomalies in terms of factorial field theories actually matches this older description and what type of mathematical productions one expects for these things to match up and should also say there are a lot of things which are open in my opinion. So this is three and actually three and the new thing compared to some of the previous talks is that my field theories will actually be smoothed. So we define field theories as the final goal from a modern category equipped with geometric structures. Now this F is slightly different to the F's over here. It only needs to be defined on families of n dimensional manifolds and local difium offerings between them. So it does not need to be defined on all manifolds. So for example, n is two, one might want to consider spinar structures which are double covers of SO2 which do not exist in another dimension. So this would then be something which is really specific for dimension two. But otherwise this F was really similar to what we have seen before and also this appeared yesterday. Then it's a factor from this vector spaces. But now the new thing is that this is actually the symmetric one order. That's what we had before but it's also supposed to be smooth. Metric one order. And then I mean, depending on what we want, this is really a smooth infinity category but this moves one category so we can work with one category. And let me explain what this word smooth means here. So we just applied the same strategy as we did in the case of generalized spaces to make something smooth. We just consider it as a chief on manifold of. So the smooth category is a factor from the opposite of manifold two categories. Automatic one order categories or symmetric one order category. And then what do I mean by the smooth category of vector spaces? This is just sense to the category of vector boundaries on M plus the vector on M is sort of a smooth family of vector spaces. And the definition of this border smooth border category is a bit more complicated. I should also, by the way, mention some names. So this idea of smooth field series goes back to Stoltz and Heichner. But then recently there have been besides definitions in the framework of smooth infinity and categories by gradient path law and also a bit in the case of smooth two categories by Ludovic and Stoltz. But I'm not going to give you a precise definition. I'm just going to give you a step. So I need to define all these categories here. Evaluate on a manifold. You allow in the dimension of vector spaces. Let me not mention that there. I thought that's where you should, but then you should also worry about topology. So I would like, like it depends on what you want to do. But generally you should, yeah. And maybe you also don't want to have like vector space. You don't want to have something like a sheet of vector space. Depends on the example you have on mind. Then you need to worry about what type of topology you have at space. So, but that's also not, yeah. I would like to be agnostic about that. But let me sketch this category to the object. I mean, so this is just a parametrized version of the ordinary board, isn't it? That's what you've seen before. An object is a fiber bundle of N minus one dimensional manifold or maybe germs of N minus one dimensional manifold and N dimensional manifold which are equipped with fiber-wise F structures. So this just means that every fiber has something in it. So for example, it's something like the fiber-wise metrics we had been seeing before, but fiber-wise orientations and then the morphisms are given by co-borderance of these things. So I have a vector bundle of, I have a fiber bundle of manifold, I have a different fiber bundle of N minus one dimensional manifold and then I have a fiber bundle of co-borderance over. Over N and this is equipped with a geometric structure. So this is one type and then they are, which one might need to add by hand or in the framework. They just want to highlight them. It's basically the automorphisms of the symmetries of Y. So for example, if F contains the bundle, then these automorphisms would contain fiber-wise gauge transformation. So these are automorphisms in the photogenic ethical. Where's the other area? Ah, okay, thanks. Now smooth field theory is such a funcal, but what does it explicitly, like if I have such a Z and I evaluate it on a type of smooth framework of co-borderance over S, what does it give me? Is first of all, does it give me two vector bundles over S and then it will give me a linear map between them. So it will give me a fiber-wise linear map between them, which corresponds to the co-borderance. Okay, now an example of which is actually a theory to David Evans' part and path map and then more modern version or more modern proof is due to Ludwig and Stoffel. They also, their formulations are slightly different and that looks a bit different. But the theory is that if I look at this one-dimensional border of the particle with, but here the structure are mapped into a target space X, then smooth field theory is like this. I prefer them to vector bundles of this connection. I find out that they need to be finite dimensions. And how does this work? So an object in here is a map from a point to X which will be mapped to the side over X or more generally an object in here is like a family of points and X which will be mapped to the full vector bundle. And then a morphism in here as a co-border, so it's like something like this, which comes with a map to X. And this will be mapped to the parallel structure in the X to the Y. And these are all of these sweet theories. Now, I want to explain how I normally describe this. And we already have seen that yesterday's by using a relative series of boundary theories but to capture the Hamiltonian part I mentioned before, I need to extend once. So I will just define extended field series by putting n minus two here and the two back there. Let me explain what the smooth category of two vector spaces is. So two vector, and I just tell you what it does locally on like a contract in this space. And then you can again extend it using a sheaf property. This is now two categories. So this thing will be a smooth two category. The borders in category will also be a smooth two category where we just allow now, updates are n minus two dimensional vector by a fiber bond with morphisms are the same and two morphisms are like fiber bonded of the fiber bond of cobalt in between cobalt. I mean, if I think explicitly, it's not that easy. This is like a picture I want to have in mind. In this category locally, I guess there's objects algebra bundles. These are bundles of algebras on you. And then the morphisms are binot two bundles and the two morphisms are just fiber-wise linear maps of bi-multi-bundles. This is a local description and then the global description happens by gluing the algebra bundles together and bi-multi-bundles. And this has recently been studied by Crystal, Ludwig and Weidoff, so I should mention that as well. And then such a smooth theory once extended with an assignment algebra bundle, yeah. So these things then on the level of object, these things with the algebra bundles, these things will become binot two bundles. If I want to describe anomalies, they are described the anomaly itself. So the interpretation here that they are actually described by field series. The anomaly itself is a field series, but it's inverted, so it lands like in the two line bundles that are the morphisms between them. And then if I want to describe and the anomaly theory with anomalies, the relative field theory, and they can be described by either relative field series or twisted field series. For this, I take this A and I restrict it to its value of N minus one and minus two and I allow that the things which are symmetries of the geometry structure. And I think yesterday, this was called a categorified field series or something like that. And then I have something here, two line bundles, and I have a trivial one and I have the restriction of it. So I can restrict my area to this category and then a relative field theory is defined as a natural transformation sitting on this. And this is, so it's a metric model that's lux, but the lux does not is with respect to the symmetric monoid structure, but it's with respect to the coherent isomorphism for the two naturalities of the lux, the metric monoid of two transformations. And it also needs to be smooth. So everything to work in families over many fours and these lux transformations, they have been defined quite generally by Schoenbauer and Johnson's flight for infinity and categories. And now I want to somehow connect this description of a field theory to what I mentioned before and the point here is this description somehow considered all allowed manifolds at the same time. So we study really quantum field series, not just on the fixed space or space time manifold, but on all series we can put it on. And this definitely contains more information and they even situations where we think might be trivial on the independent manifolds but the interplay is non-trivial. This is more generally, but how do we go back to the more practical description? For this we fix some space at some space with an n minus two dimensional manifold. And then we want to somehow take this anomaly field theory and restrict it to the space time. And the way of doing this is called or the world of dimensional reduction. So I consider this category and I'll look at geometric structures on sigma as my geometric structure on the two one zero borders in category. But I can map this into the n minus two border in category and how does this work? So I take an object in here or manifold in here and this comes with a geometric structure. It is an element in S of sigma evaluated on S. So this means that somehow S parameterized to a family of geometric structures on sigma. So this is also something which we will leave the contained in this redefining this structure on sigma cross S. So then I just map this to S for sigma equipped with this geometric structure that's seen as living in here. And I can use this to take this field theory and restrict it to this border in category and I can also restrict these relative fields. So I get relative fields on this type of border in categories. And now the connection to previous description is the following connector, which is definitely not due to myself, but I'm also not sure which name to put it. And it's, I would also say it's probably in the way I formulated it's wrong, but it should be morally true. So like it will be true in certain cases and then if one finds the right interpretation of the words, it will be true in general. And I will say a bit more about what this failure is in the topological case in a few minutes. As I have, which is impossible, it goes to two vector spaces and the statement is that these things are equivalent to terms sigma with the dimension. This is some of what one expects if the two descriptions of anomalies in physics, like the two mathematically ways of describing anomalies in physics. If they match up, then these two things also match up. And furthermore, the relative field series is equivalent to the section. One should probably put some additional adjectives on the field, like reflection, all the, or unitary or such. So this is like, this is more like a general guiding principle of what should be true. And sometimes it might be too direct or not. And I should also here mention that if you take a target space here, so you take that into a target space, then the connections to jobs on target spaces is ongoing work by Bunker and Weidoff. And they also have already done a lot of work on the non-extended case. So there's like really good hope that these type of field series at least have something to do with jobs. And then I also have not mentioned generalized homology at all. Yet, these are, I mean, because these things are vertebrate, but then they also fiber over. I mean, they're fun because she's on many folds. So if we look at them, what's the case when we get a chief of spectrum manifold and these are things we can study in differential homology. Let me mention one physical thing one can do with this description. It's just the right boundary for so much. So if I have a manifold M, which has a boundary of six months, and M is now dimension N. So we got out, let me say something else before because you're gonna ask, okay, M is defined on n-dimensional manifold. And this description, I don't need it. And then the version of twisted field series by Stoltz and Feichner, they also just have a twist which is not defined on n-dimensional manifold. But having a theory defined on n-dimensional manifold really gives us additional data. And it's safe for you to study the theory. And the boundary correspondence one example. So the value of the field theory gives me a map form one and I evaluate it on the sigma, two A of sigma. And then I can take the theory A and evaluate it on M because we go back to one and then here I will get an automorphism of the monoidal unit of two vector states. So I will get a vector state for, depending on the family I would get a vector one. So this is, and this is the vector state and this is like the boundary space. And that's one proposition one can prove in this framework or at least in the non smooth framework that our symmetries, the symmetries on this combined system of the theory they act on the boundary space and non-projectively. And usually if one has an anomaly like if I consider this as a stage space the action of the symmetries would be only projected but if on this space I will always get abstractly a non-projective action of the symmetry. And now the last 15 minutes I will discuss a simple toy model. So the first simplification we will make is now that we don't work with smooth objects anymore but we restrict the topological one. And this makes our life, this makes a lot of things easier and let me mention how the topological world is connected to the smooth world so I can take my general best spaces. This is a sheath on many foods. So I can evaluate at a point and then I get a topological space or I get an infinity group or which I can model as a topological space and this has a left edge toy which will tend to topological space somehow makes it constantly as a constant, as a constant generalized space. And the same as to smooth infinity categories. So I can take one and evaluate it at a point or just take the infinity category or topological category make it constantly into a smooth category. And then somehow I say, my board doesn't categorize topological or the geometric structure as topological as board NF is equivalent to taking this constant thing on like a topological category just to put all this category F. And then you think it's a junction we can actually rewrite a fear theory on here as like a just a funnel between categories out of the sense. Okay, and then this gave also the definition of the job just to become simpler. So if I look at the flat job, in all jobs with the flat, this is just so that we have a connection on see all those days. This is equivalent to having a local system on T with values in two line bundles. And these are really now and brought to this back here because this is not a smooth category anymore. This is just an ordinary two category. And such a function is called a local system. Okay, now in this case, in the topological case, we can use the corpore doesn't hypothesis to study. To study the conjecture here and what does it tell us? It tells us essentially if I look at the series almost borders in category that say with maps into a target space. So this is equivalent for local system on T with values in for the end of it. And in here, I have the trivia ones which are given by just two line calls. This is in here and the job like this local system in here or the job would be the thing which comes from these type of things. And then yet also clear what goes wrong with the conjecture over here is my with my geometry structure includes an orientation that somehow more than that such a local system because it's twisted by the OIN action on the category of the fully doable objects. So this was a lot of abstract stuff. Now I want to do something extremely simple which are not something where one can do a lot of things explicit, I should say. So we've and these are these discrete gauge series. So we fix the finite group D and then we want to build a gauge series in dimension N out of this. I should also say these have first been studied by Dicap had written and a lot of things I'm going to tell you are based on a paper by Friedland Krim. These have been studied in the Nazis. But somehow you can study this framework in this case where you can make some precise techniques. I want to explain. And then the action is given by an enter. I want to build an N dimensional gauge theory versus finite structure. So the action is given by an N co-cycle in D D with values in D one. And now there are two theories one can define the first one as the classical one. So this is defined on co-ordered which are oriented and dimensional problem. This talk I just extend down to N minus two but the series can be defined as something fully extended. And this goes, this is invertible. This goes to two fact. And now this is not smooth. This is really just I do both bi-mortals and bi-mortals. Or if you want, same, I think the linear category is linear found those and natural transformations between them. And what this does, I take an oriented manifold which comes with a map to BD. This map does describe my D bundle on X and I should assign to it a number. So the number I assign to it, I take this co-cycle here I put it back and I integrate it over X. And then, yeah, okay. Let me mention also what is the, there should be a line bundle assigned to N minus one dimension manifold. I'm directly telling you what happens on N minus two dimension manifold. So to this I should assign by this construction. I should somehow assign to it a job on the field. So I should assign to it a job on D bundle on sigma. That was true. And this is the transgression. It's classified by the transgression to sigma for makeup. So this is the category just has one object, one algorithm of this and then a map. So this is something like a KZ two space. Then the map into it is just described by a two-pointer. So I need to give you a two-pointer cycle on this space. And this can be done using transgression. I have maps on sigma and two-pointer. And for sigma, this maps by a variation to be D and by a projection down. So this is the same as print of the D bundle on sigma. This maps to one on sigma. And then the transgression of omega two sigma is defined as a full-back on the evaluation map. And then I integrate, I push it forward by integrating with the sigma. Okay, and this is a line bundle which is classic if you assign to a vector space. Okay, and then there's the quantum version, which is now just given by thumbing over here. So what we want to perform usually is a parsnetic work. In this case, we can actually do that. So it's one of the, that's what makes this VK series from there. And that's the final position function as the pars integral over bundles, or D bundles on X, and then I take the action, which is from this thing. And then this integrals that they're only finite in many bundles with a discrete structure group on a compact manifold. This is just a sum. And then one needs to put in some normal notation factors. Thumb over isomorphism classes of D bundles on X. I desire my G series on it. And I decide by the automorphism. Number of elements in the automorphism group and this group. And then this also needs to assign vector spaces to n minus one dimensional manifold and two vector spaces to higher, to n minus two dimensional manifold. And this does happen. So let me again only do the n minus two dimensional thing. This is given by taking sections of the D bundles on X. And this classic, if you really find the job over this space and I take sections of this. This is my, this will be a linear category. And this is what the series assigns to it. Now, last few minutes, actually talk can talk about on the on the on the symmetries. Do you think I'm going to say have been studied the physics literature by Kapustin and Swarbring. And then some awesome thing I say about boundary series have been studied in a paper by Witten and then a paper by Wang Wen and Witten. So in a symmetry. Now it's another kind of G, which should act on my theory and it acts classically by acting on the field. So I want to have an action on the stack of D bundles. This is represented by BD. So I can describe this as a map from G, the automorphism with BD. And as a hierarchy, this is the automorphism of D. And then D acts on this by conjugation. So I call this thing. I thought this is my action. And then if you fail this out, this gives you what's called a non-numbian treatise. So this is again equivalent to fixing an extension. And this G hat somehow tells me how D gate field and G gate fields combine. So they are there some are not independent. If I want to put it on background, I want to go to which I have a G gate field. This is sometimes called fractional notation of symmetry and physics, or that's at least how I understand the term. Okay. And then this is not just an action. I want this to be a symmetry of the classical field theory. So the condition is that the omega is just its point for, I mean this action induces me an action of G on the collection of n dimensional Gs. And I do one extended ones. And then, but still this fixed point is actually data. So I need to say how it's a fixed point. And if I want to fully extended ones, then there would be more data. But in particular, they could also be never been another. Because then the theory would be fully local. And we can use the co-border in my process to have it to that confidence. But this implies. That if I look at the digraphic theory to the quantized version, then this lives to G. This is now fun to which goes to the category of G presentation. Now to host anomaly appears. The background fields are just orientations. And I want to enlarge them. I want to move orientations and background fields for the symmetry. So I also want to have the bandwidth that's background fields. And now, let me first mention a situation where you can do that. So if find an omega hat. And the cycle on the PDG. Which pulls back to omega. And then I can gauge the symmetry. So I want, which means I want to define the theory defined with these background structures. And what do I do? I take the field theory. For omega hat. And then if I have a group of morphogen like this. If I have a group of morphogen like this, I can push forward. So I push this field theory forward along with. This is defined from your PhD on G. And in the front of your framework, this has been explicitly defined by which I go to focus. Okay, in this case, this. So in this situation, there's no anomaly. The last thing I want to say is about the situation where one has an anomaly. So I can look at a really similar. And then I find an omega hat. Which pulls back to two omega, but it's not close. And then the close thing is the pullback of something close. So theta. In. Okay. In this case, this omega itself will not define a field theory ahead, because it's not close. But what it will define for me is the natural transformation from the identity to. Now the statement is that one can actually perform a relative version of this push forward. So I can take this theory and omega. And I can push it forward. And this is something which goes from one. And this defines the relative for me. This then so this is 40 relative to this invertebrate theory describing the anomaly. The job describing the anomaly is just a transgression of theta to the base of bundled. And this is something which is much more explicit on how the space and space spaces look. That's in my paper with Richard. And then one last thing is one has an anomaly. This is not a big problem in physics, but what I cannot do is cage. So I cannot now some over G bundled. And this is also what happens here because I cannot. Because this is not the pullback of the theory. So I cannot perform a push forward to the point, which would be the quantization. Okay, that's everything I wanted to say. Thank you for your attention. So I suppose.