3d SUSY Gauge Theory and Quantum Groups at Roots of Unity
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Dimofte, Tudor
Topological twists of 3d N=4 gauge theories naturally give rise to non-semisimple 3d TQFT's. In mathematics, prototypical examples of the latter were constructed in the 90's (by Lyubashenko and others) from representation categories of small quantum groups at roots of unity; they were recently generalized in work of Costantino-Geer-Patureau Mirand and collaborators. I will introduce a family of physical 3d quantum field theories that (conjecturally) reproduce these classic non-semisimple TQFT's. The physical theories combine Chern-Simons-like and 3d N=4-like sectors. They are also related to Feigin-Tipunin vertex algebras, much the same way that Chern-Simons theory is related to WZW vertex algebras.
 I wanted to talk about some work in progress that I hope we will be finishing up very soon with Thomas Kreitzig, Nick Garner, and Nathan Geer. It started out a few years ago as part of an SF-FRG collaboration. And it's related but probably not in a way that will be obvious to some previous work with some of these co-authors and Jennifer Brown, a grad student, and Stavros Garfilidis about recursion relations for so-called ADO invariants. It's also closely related to a paper that appeared last year by Gukovsin Nakajima Parkpe in Topenko that gives in a way I will say a little bit more about later, sort of a 3D mirror of our construction. And there's also ongoing work in that direction by Gukov Fagan and Reshiti Ghan. So to sort of set up the story I want to discuss, there was sort of fantastic progress in math and in physics that started about 30 years ago in a bunch of papers, but in particular in work of Witton's and of Reshiti Ghan and Turayev that sort of connected some ideas coming from representation theory of quantum groups and vertex algebras, WCW models, and Trin Simons quantum field theory. One of the first examples of so-called topological quantum field theories whose partition functions give you topological invariants, in this case of three manifolds with links inside. So I am hoping that many of you have know at least part of this picture. One of the huge exciting powerful aspects of this is that there were at least three different perspectives on quantum invariants that appeared and one could sort of study each side and connect them. So a key object, sorry there's a lot of feedback coming through, let me know if the audio is not okay for me at some point. Okay, so a key object in each of these constructions that appears is a certain braided tensor category and in the axiomatics of TQFT on the math side one can sort of reproduce the entire TQFT from this braided tensor category. In Trin Simons theory, on the physics side, this is the so-called category of line operators. Its objects are extended operators that are localized on some sort of lines or curves and three-dimensional space time. In the OA land, the objects of this category are modules for the relevant VOA, which is WCW, and in terms of quantum groups, objects of this category are modules for a quantum group UQG at a root of unity. So, their objects of this category, more precisely their objects of a massive semi-simplification of this category. So the category at a root of unity, which is really what I want to talk about during most of this talk, is extremely complicated. It's not semi-simple and it was known a long time ago that it wasn't semi-simple, and sort of a tiny piece of it goes into this original Reshti-Kintari construction. Right, so like I mentioned, so in this old classic story, the category involved is semi-simple. In terms of physics, semi-simple means there are no non-trivial junctions of line operators. And in math, I'm hoping you all know what semi-simple means. So mathematically, you would say there are no non-trivial morphisms among objects and morphisms are junctions on the physics side. Okay, so a lot of progress has been made since then, sort of extending into the non-semi-simple world. On the quantum group side, even in the early 90s, Li Buchenko and others started writing down partial TQFTs that started off with a non-semi-simple category. And so the Kutsude-Gucchi, Otsuki, or ADO invariance of links are sort of related to this. They're related to a semi-simple part of a bigger category that I'll mention later on. Well, the problem there is not that it's not semi-simple, the problem is that there, some representations have vanished in quantum dimensions that need to be regularized. And some of the patterns of problems come up in general when looking at the big category of UQMG modules at a root of unity that need to be dealt with somehow. A systematic set of tools for dealing with the various problems that come up has been constructed much more recently starting in work of Geo Constantino and Patiro Muran, and extending in a bunch of different ways since then. And so now there is, well, there are still extensions that are being developed, but there is at least one TQFT defined using these techniques, these new techniques that involves the full representation category of UQMG. I'll talk about this category in about 10 minutes. So, on the VOA side, there have been similar developments. So going non-semi-simple there in its simplest incarnation means generalizing from rational VOAs to logarithmic VOAs. And the simplest logarithmic vertex algebra, the so-called triplet model, and that's been generalized also in many ways, but the generalization that's relevant here is in terms of what are called Fagin-Tepunin Algebras. That's what FT stands for. So this is Fagin-Tepunin. And with suitable matching, the representation category, the module category of these Fagin-Tepunin Algebras is supposed to match modules for quantum groups at certain roots of unity. Well, it was supposed to, I think it was conjectured a long time ago and in recent work, that I'll give you a few references here. The equivalence has been proven in certain cases and generalized to an equivalence of actual rated tensor categories. So we modulate tensor categories that sort of the data you needed to build the TQFT out of this. Okay, so I said a lot has been done. Yeah, sorry, was there a question? Okay. So what hasn't been done is the physics side of this. Transimans led to a lot of sort of amazing computations and predictions, things like geometric quantization to produce Hilbert spaces and conceivage integrals and so on. And the transimans part of this non-simple story does not exist yet. So what we're doing, part of what we're doing is actually to propose a quantum field theory that fits in this third physics perspective. Some non-simple aspects, I should say, have appeared. Now, triplet algebras have started appearing in supersymmetric theories in a paper on 3D modularity, maybe, by Chang-Chen Ferrari-Gericot Harrison. So logarithmic things have started appearing, in particular in a supersymmetric context. So, supergroup transimans theories have also been investigated, and I would, using sort of physics language, I would say they fall into the same universality class. They, supergroup transimans theories have a lot of properties in common with the theories I will discuss today. They're just not exactly the direction we're going in. So the result I want to talk about, in part proposal and part result, is that there exists a three-dimensional quantum field theory that is topological, whose category of line operators matches sort of the big quantum group at a root of unity category. Here the case I'll focus on is an even root of unity, and I am certain about this in type A, and there are obvious generalizations for groups of other type. And the category of lines matches modules for the fake and tipunin algebra, and in fact, the way we actually get at this more naturally is in a sort of morally level rank dual of the fake and tipunin algebra. So they're actually two vertex algebras that appear that have equivalent categories of modules. So, we have a, I would call it physics proof that the vertex algebra categories are the categories of lines. And in the QFT I will talk about for the case of certainly, sorry, for the case of SL2, I think for general type A. So, to make this statement I have to actually say what I mean by physics proof taken suitably liberally, I would say this makes sense for general type A. And Thomas Kreutig in our paper gives a proof that for SL2, the two vertex algebras I showed here are equivalent and it's already known that the fake and tipunin algebra is related to quantum SL2 modules at a root of unity. So sort of the new bit is the left side of this picture. So whatever quantum field theory sits on the left here had better be labeled by a group or a Lie algebra and an integer k that tells you what root of unity we're working at. And change the page on my screen but it's not coming through so I may need to re-marry. Just a sec. Oh, you guys can see that. Okay. Let me just play again. Good. Right, so the theory is starts out as a 3D n equals 4 theory and it's mostly a 3D n equals 4 theory called QFT. That can be analyzed using all of the lovely algebraic techniques that you're hearing about in other parts of this workshop. And it's a very quick twist. So, so it's sort of a mix of a 3D n equals 4 theory and a trans-simon theory. That's what that G of K is doing there. And so it doesn't, it falls slightly outside the class of things that are easily analyzed algebraically right now. And one of the points of my talk is to motivate all of you to think about how to solve this problem. Sorry, and I see Sasha has a question in the chat. It's for all G. So, this theory that I'm talking about here makes sense for all G and all. I was asking about the previous line about the. No, there are subtleties when I would expect it to work for a to E type and there are very huge technical things and that happened later where some language stools need to be involved and so there's some natural guesses for what happens, but the story is not so simple. Okay, thank you. And it's like it's sort of clear and on each from each perspective why the story is not so simple, matching up the various subtleties is is hard and is something we realize we should not attempt in in this first shot at at this project. So, one can write down a sort of a Lagrangian, an action for for the steering and type a using a twisted BV formalism. Just is kind of nice. So in that sense, it's very much like turn simons. It's a concrete theory. And using that Lagrangian one can write down a boundary vertex algebra, the analog of WCW, and there are in fact two of them that show up. And in WCW there's also another algebra that you could use which is the level rank dual of WCW that has exactly the same category of modules. And, and here as well, there are there's sort of two natural algebras that show up that are level rank duals of each other in the appropriate generalization of level rank. And it should be possible, like, it's sort of clear what to write down for other types from 40 brain constructions. But there are lots of technical issues that show up. Right. So, we write all this down and then then the natural question is like what what what you gain from the physics and what can you compute to check that that this guess is even correct. So it's easy to compute using supersymmetric localization techniques, the growth and the group of the category of line operators and with our characters of hybrid spaces which are said this, there's this D bar sitting everywhere, there's a D bar sitting everywhere. So, the physics theory because it's this sort of supersymmetric theory that needs to be topologically twisted to get something topological is like sort of naturally a derived beast or like the category of line operators is naturally a DG category. And so all of the equivalences that we get, particularly between QFT and the V ways are that that first equivalence is a derived equivalence. There are many different ways of describing the category of line operators and it shows up as a DG category and it is equivalent to the derived category of modules for this VLA. The thing that's also relevant on the quantum group site is the derived category of quantum group modules. The things associated to surfaces are going to exist in many co homological degrees, and their other characters are easy things to compute as partition functions. And so what we can compute are things like the category itself from a few FT perspective or the hybrid space itself in a math and classroom action, and I'll indicate some techniques to try to go about that. But like this is where like fully generalizing algebraic approaches to 3D n equals four theories things like the BFM construction would be really useful. That's the end of the long introduction. You know if there are any questions to wrap me at any time as well. I want to actually say a few more concrete things about the representation theory of quantum groups at a root of unity, and I'm just going to stick to the case of SL2 here for illustrative purposes. Great. So, right, so quantum SL2 looks like standard SL2 except the part time generator has been exponentiated. And it looks like there is a question coming but I'll wait till it actually comes. At an even root of unity in fact that any root of unity, but even is the relevant situation for me. There's, in addition to the center that comes from from Kazimier operators that sort of the higher chandra center. There's an extra bit of the center that just comes from case powers of the F and the K. And so the extra central elements act as constants on any indicomposable representations. And so the representation theory sort of decomposes into what little pieces based on what values the central elements take. And the fancy way to say that is that it ends up vibing over spec of this other part of the center. And this other part of the center parameterizes an open cell in the group PGL to So, the reason PGL to is coming up here is because it's the Langland's tool of SL2. And at at an even root of unity. The fancy way to say this is that the category of modules fibers over PGL to at an odd root of unity at fibers of grass. I think technically I should also say this is a coherent chief of categories which is also something that shows up in the field theory. But I think that's all I want to say for now. Really simple example of this is that when. So if you if you take a diagonal element of PGL to that has some eigenvalue either the alpha one would associate that to mod to a subcategory of modules on which either the K and after the K act to zero coming from these off diagonal zeros and K to the 2K acts as you the alpha one. Should also say like the relevant thing about these, these different fibers or different stocks of the chief is that they're there are no hams between them so they're each full subcategories, the total categories a direct sum of all of these stocks. And, yeah. Right. So I'll say. I'll say that terrible transition. So it was observed, it was proven a long time ago by the continue and cats and even root of unity case by back that what the stock of fiber looks like, but the particular elements of PGL to have the center of the K act only depends on the conjugacy class of this element and in PGL to up to isomorphism. And then what was realized a little later on by initially by Koshile and Russian TK is that this sort of fibering of the category over PGL to not just to invariance of links and three manifolds, but more generally to invariance of links and three month in three manifolds with flat connections in the complements of the links. So this was roughly but not entirely correct on and more like correct. And it's been made very precise in the case of links links and s3 in a recent paper by a bunch a gear, but you're on and Russia taken. So, so they build an invariance of three manifolds, sorry, invariance of links and s3 with flat PGL to connections in the complement of the link. And there's work in progress, trying to really promote that to a full QFT that that gives invariance three manifolds with with flat PGL to connections. And the picture that I always have in mind when when trying to understand why why this should be true is is the following. So, so the category has all these different blocks or stocks labeled by by elements of PGL to one should if if trying to translate this to some sort of physics picture, or topology picture that involves say links where the strands are labeled by objects of this category. One should think that the piece of the category that object from labeled by some element G. And then you can define the whole ennemy of your background flat connection, the flat connection that you've enriched the three manifold with in a small loop, going around this line. If you just have a single line. It's only the conjugacy class of the whole ennemy should matter which is consistent with the old theorem of decontini cats and perches that a fiber of this category only met only depends on the conjugacy class. Now, if you want to start building a TQFT out of this you need to make sure that this is compatible with the tensor structure in the category. And you would expect if you're looking at base pointed whole ennemy is that when two lines collide, when gets the tensor product of those representations using the hop algebra structure of the quantum group. And that had better be compatible with multiplying the base pointed whole ennemy's. And it's literally true when all of your hall enemies are diagonal, and it is almost true when the whole ennemy's are general non abelian things and the almost was described precisely in in this book by Kashyap and Russia to get in this later paper by one shade here at your mirror on Russia to get. I'll just focus on the abelian case. Okay. So, this sort of setup where we're looking at three manifolds with the decorated by flat connections appears in physics when we have a quantum field theory with a global symmetry. As opposed to a gaged symmetry. So, so quantum field theories with global symmetries can couple to connections that are not fields that one integrates over in the path in a goal but fields that are just put in by hand and fixed for all time. And so the sort of theory we're looking for had better have some sort of global symmetry. And so the theories in the background of a whole ennemy defect like this were described really nicely a few years ago in a paper of Victor make high looks in the context of super symmetric trans diamonds but the story is is sorry, the context of super grip trans diamonds, but but the story is very similar here. Okay. Let me try to describe what a few of these fibers look like. Just so I can say some concrete things later on. If we take a generic diagonal element of GL to then we look at representations of you qsl to on which K to the 2k acts is that I can value and you do the K and after the K at zero. The category ends up being semi simple with exactly 2k semi simple objects. And that look sort of like standard highest weight modules, except the weights are, I can values of cave rather than an H. And so they're, they're cute as a something. They all have dimension K, and they wrap around the unit circle. And so the, the, the, the, the weights wrapped around the unit circle, which gives rise to vanishing quantum dimensions, which is one of the things that needs to be regularized in order to build a TQFT out of this but that was done so this is the 80 and variant uses precisely these sorts of representations. And more economy also worked on that in the 90s and then Constantino, you're in a mirror and sort of systematized the data you would need to not get all of your link invariance to be zero. And so the average rise quantum vanishing quantum dimensions would naively tell you that even the hop, and even the unknot has expectation value zero. These sort of generic stocks of the category has an extremely simple tensor product. And sort of looks like what you would expect for an in an abelian theory, it's like for quantum gl one. It leads to very easy calculations of dimensions of putative Hilbert spaces or spaces of states on various surfaces. So, you get a space of states on the tourists. You can generate states by filling the tourists into a solid tourists and putting objects, colored by all the different possible representations along its core. Now, since we're dealing with a QFT enriched by flat connections. We shouldn't just say the Hilbert space of the tourists we should say the Hilbert space of the tourists together with a flat PGL to connection on it. And putting an abelian connection on there with a specified holonymy around the meridian of the tourists tells us what piece of the category to choose objects from on the core. And so we just take the two K different objects in that piece of the category along the core that gives us to K states in that Hilbert space. And so we should say, I'm saying Hilbert space because in physics, we always say Hilbert. These are not necessarily Hilbert spaces in the mathematical sense, and in this sort of teacher at T's I'll be discussing. They are vector spaces, they have duals, they don't necessarily, they're not necessarily isomorphic to their dual in a natural way. They're not necessarily just say vector spaces of states. In genus G. One can use the very simple my nodal structure to also draw all trivalent networks of the two, keep one they say line operators inside the core of a handle body, whose boundaries of particular genius G surface and count what the options are. And when gets to the G, K to the 3G minus three different states. That's a very easy combinatorics problem. Okay, so that's sort of the generic setting. And when, when you're looking at the surface with a generic flat connection on it. There's also the most interesting, most non generic case when the flat connections trivial. So, so the connection itself which is zero everywhere. And so in this case, looking at what sorts of what to be labeled aligned by in the presence of trivial homony. The answer is representations of what is called the smaller restricted quantum group. So, it's modules on which K to the K, K to the two k axis one and you to the K and F to the K act is zero. So this is an extremely well studied category in representation theory. It is not semi simple. It has two to the K simple objects. That can form interesting extensions with one another. The two to the K simple objects roughly look like two copies of ordinary representations of SO2 of dimensions 123, although we have to K. And they taking the projective covers of the simple objects when gets projected. And they have sort of diamond structures and movie diagrams. And they're so they're to K, the composable projectives in, in this piece of the category as well. The semi simplification that was used by restyling to read uses a single copy of the, of what look like ordinary SL to representations of dimensions one up to K minus one. So if one sort of quotients out by everything else, setting it to zero. Another way of saying portion out by everything else is set to zero everything that has a vanishing quantum dimension, when gets a semi simplified category that leads to the old story. So the entire of invariant that's involved in the volume conjecture is defined using this last simple and projective representation of dimension K. And, and finally, sort of the entire thing is, is what we Bush and considered when, when starting to define non semisimple to get these in the 90s. Okay. So, I mentioned at beginning that in general, our spaces of states on surfaces would be homological beasts. They, they have multiple co homological degrees. In terms of the category of lines mathematically. One would calculate the Hilbert space of states in general by taking a whole show of the appropriate piece of the category, depending on what flat connection we've chosen and so on a tourist with zero flat connection. So we have a small quantum group category. And we should find it's Hilbert space is Hochschild homology of the small quantum group category. And so what that amounts to is considering the fact that when there are morphisms among the objects in your category, you don't, you can't just wrap single lines around the core but you have to consider junctions of multiple lines. And so, we're the, the, we'll give you the zero zeroes Huxchild homology, higher degrees and Huxchild homology and Huxchild homology come from integrating descendants of junctions around paths in this core. That is, a statement for anyone in the audience who knows about descendants. So it's a totally sensible thing to do both physically and mathematically. In order to compute Huxchild homology of this category it's useful to have a geometric description of it. And archipath, as you can pick off in Ginsburg, and then those are coming to cut the month, let's go Dave such a geometric description, which at an even rate of unity amounts to saying that as a category, not a monotone category, category. So the first category of representations of the small quantum group has sort of two semi simple pieces to that's just coming from the two symbols that were also projected. And then a bunch of copies of the derived category of her she's on the flag manifold. K minus one copies of that. And gives us a geometric way to compute Huxchild homology which ends up looking like, I want to say total double homology. Looks like total co homology of these of P star flags computed in algebraic way. And then there's an answer. The, the relevance base of states becomes infinite dimensional with non negative co homological degrees and it's finite in each co homological degree. And the parts that the Bush and could have used back in the 90s. I say that on the next slide. And the part that the Bush and could would have used is just h8 zero. But, but there ought to be a derived generalization of, of all of the earlier work and also the current work of CGP and collaborators. Okay. So, I, I, I've sort of written what the different ways look like in terms of representations of the symmetry group PGL to that acts on star flag. There are also nice things to say about deforming from trivial connection to non zero generic connection. Which, there we go. Which again, I think I need to. That's the slide I actually want to be on. I'll just see this very briefly but but to go. So, at generic connection, the space of states had dimension to K at zero connection the space of states is this infinite dimensional thing with the infinite dimensional graded components. And there's a differential that one can turn on to deform the infinite dimensional thing to something that exists only in degree zero and has dimensions UK. The other character is invariant under this deformation the other character doesn't care what flat connection you put on. And at a category level. The field theory interpretation that I'm about to get to also suggests that there's there's just, there's a, well, this is a sheet that coherent chief of categories. And very close to the identity elements in PGL to coherent sheaves gets deformed to matrix factorizations with a super potential. And that involves the complex moment map for for a certain elements of PGL to that that's the stock. That's defined by the stock we're looking at. Okay. I said that other characters don't change under this deformation. Okay, that's, that's, sorry, I probably said too much about that I have really enjoyed during this project, sort of learning and trying to put together more of the structure of this big category of representations of the quantum group at a rate of unity. You can probably tell. Okay, so I'm going to take all of that information and translate it quickly to to physics and then try to explain what what one still has to do on the physics side. So we're looking for a quantum joke theory that is labeled by algebra and a level and an integer k. And that's what the term Simon's like. And that, like it sort of looks very close to the rest you can try the category should have some finite like in each piece of the category that a finite number of objects it should have something that looks sort of looks like a close in lines and in this previous paper on recursion relations for 80 and variants we found like we found structure that was very, very, very similar to the simple or the semi simple story involved in so so 80 80 and variants are different from color and they are the same models but they obey the same recursion relations. And so there are a lot of things that sort of look the same. You should see something turn Simon's like here. However, the line operators in this theory should have nontrivial junctions we should not be getting a simple category. So, very easy signal that quantum field theory gives rise to a non semi simple category of lines is that there are non trivial local operators. This is the sort of thing that the BFM construction computes. So, BFM computes an algebra. And it's the algebra of local operators in a 3D field theory. And so that construction applied to this story had better give you something non trivial. If you apply it to turn Simon's it just gives you the identity and that's it. And there should be a global symmetry around that leads to flat connections. And those criteria lead to an essentially unique answer, which is, if you're not familiar with the physics side of the story is going to look strange and weird and not unique at all. But anyway, so the easiest thing one can do to satisfy this property is to start with a 3D and equals for super symmetric theory that's going to have been introduced called t of G. And even though I write a group here it secretly only depends on a le algebra. And roughly speaking, it has symmetry G times the language to do a little bit. Sorry, is this literally true I thought it only had she symmetry and also it's it's mirrored dual had the check symmetry is that. No, I mean, it has both it's it one acts on the Higgs branch and one acts on the coolant branch, but the, the, what I saw the coolant branch is not a symmetry of the theory. No, of course it is no no no it's exactly on the same pudding. It's if you write down a Lagrangian for this for the theory and type a, all you see is the maximal torus, but otherwise it's down the theory in terms of fields and what you call the Higgs branch and what you call the coolant branch is is arbitrary at the level of quantum field theory. And so it really does. It really does have both of these symmetries. So you mean that it's before you twisted. Yes, yes, yes, exactly. Okay, okay, and the different ways. Right, of course, after you twist, depending on what sort of twist you use. Only one of them will appear as a symmetry of the twist. Or the acting very, very different ways. Okay, yeah, okay, now I see what I mean. Yeah. Okay. So then we take G and engage it further. But, but this gauging is different from the standard three, three D n equals four gauging. It's something one can do in the series with less supersymmetry. And so physics terms I would say using n equals two vector multiple it. And so this less supersymmetric gauging allows the introduction of a non trivial trans diamonds level. And so one sort of writes down. If you if you're not worried about supersymmetry you write on the trans diamonds Lagrangian for G and couple it to the rest of this theory to G. And the result of this gauging does it still have some supersymmetry left on it. Yes, but this is subtle. So, in physics terms, it only if you write down a grand G and you try to do this in terms of fields you can only see n equals two supersymmetry in the infrared or with a suitable twist. It should have full and equals four. But seeing that as. Before getting to topological twist so maybe I'll also say later. So, in using the BV formalism one can actually write down a Lagrangian that does not look like it has n equals two or n equals four supersymmetry but has the single super charts you need to topologically to perform what would otherwise be the twist. So if you can only in the infrared this thing regains full on n equals four supersymmetry and you can topologically twist. But there, but there are subtle views around. There, if you believe that this has any of this for supersymmetry. There are two twists available. One that's focused on the Coulomb branch and one that's focused on the Higgs branch or a twist in the B twist. The G dual symmetry behaves in different ways with respect to the two choices. The one that will leave you with an ability to turn on flat G dual connections is is the A twist in this case and it's it's the one that's sort of focused on the Coulomb branch. The B twist of the same theory was something that Kapustin Salina studied 10 years ago, and something they called transimons rosansky written theory. The B twist is transimons rosansky written. But it is, it is completely different theory, at least it looks, it behaves very differently and it's not the thing that's relevant for quantum groups that are really unique. Okay. So the other thing to say is that this is a quiver gauge theory. So the in sorry when when the group is type A when we're looking at us UN this is a quiver gauge theory. Most of the quiver is the quiver you would write down for a T star flag. That's the Nakajima quiver variety for P star flag. However, in T star flag there's a framing node. And again, for for GLN or for for S you and that that final that final framing node is the thing that's staged with an extra insurance simons level. And so the gauge group of the theory is a product of GL one through not GL minus one but all the way through GLN with the transimons level for for GLN. And this extra transimons gauging both the Higgs and the column branch look like the star of the flag manifold. The extra gauging destroys the branch and it actually seems to do nothing at all to the column branch, except to change well. Same T star flag is maybe incorrect. I should rather say it's the outcome. It seems to this extra transimons gauging seems to introduce some extra singularities at the origin of the notebook and come, which I can't discuss any more precisely than that. What do you mean when I say is destroys the Higgs branch. That's right. So, so you would, I mean, sorry, the thing that it does is it, it takes a clear question of the Higgs branch. Not not a hypercalor quotient, but just no ordinary. So, so you, you, if they weren't to turn simons level around, you would expect the Higgs, the Higgs branch just be quotient by GLN. So they have less super similar to the Higgs branch is no longer calomorphic symplectic. Exactly. The maximal torus of the flat, the maximum torus of in this case, it's PGL to our PGLN shows up in terms of resolution, complexified resolution parameters for the coolant branch. Sorry, their deformation parameters for the coolant branch or yeah. I'm short on time. So I think that's that's all I'll say there's a lot I think there's been a lot about 3D and equals four theories in this conference. And the Wilson line operators that we wanted to see show up for the turn simons factor. So it's surprising in general that one would have Wilson lines in this a twist that is focused on the coolant branch, it can happen here. Precisely because of the trans simons level. It would not happen. Okay, so, so then there's a black box. So this, I claim is theory one can write down a Lagrangian. Lagrangian for the PV formalism. There are lots of localization techniques that apply to that or just or if you just think of it as n equals two supersymmetric theory, their localization techniques that will compute partition functions and expectations of both some line operators in this theory. So this is the work of Microsoft and shuttle Schvilly and recent work on unfisted indices. This. So, in, in, like, half a day, when can code up a computation that spits out other characters for spaces of states, and remind you other characters don't care about what flat connection is turned on. And when on the nose gets the right answer for SU two and we've checked for SU three and SU SU four as well. Also using the beta roots analysis of Microsoft and shuttle Schvilly that that gives you expectation values of line operators. One can get the growth and dig ring of the small quantum group. And so that's with zero flat connection. And so that tells you something about the small quantum group but it doesn't tell you anything about the interesting non semi simple behavior. So, to actually see directly from the field theory more of the non semi simple stuff. So, in the case of the data, one should try to apply for getting there. A lot of the more modern methods developed in the last five years or so. In, including by the fm and Webster and work of mine with guy at the ball and water, Hilburn and company. And so it's a recent work of Matt Bollemore and co authors and writing down Hilbert spaces. So, so there are algebraic to piece one can use one has to sort of adapt the current techniques to this sort of hybrid case that is mostly 3D n equals for in the a twist with a bit of 3D n equals to with a transignments level. The sort of thing when expects for the category of line operators looks as follows. So if, and I want to compare on this slide what happens in ordinary transignments theory to the quantum field theories I'm discussing so in terms of in theory, you can write ordinary transignments theory to the point that the answer is a 3D n equals to theory and use modern techniques to say what the category of line operators should be and the answer is loop group, equivalent coherent sheaves on a point, which, which of course is is the loop group at level k which is the correct thing for to match wcw and the semi simplified quantum group and the whole term Simon story. In these new theories what one gets instead should look something like loop group at covariant coherent sheaves on a deformation of the tangent bundle of the loop space of the Higgs branch of this quiver. And if I didn't say loop group echo variant, and I did say deformation it's the deformation that deforms coherent sheaves to the modules. And the modules on the loop space of the Higgs branch is due to a bunch of the references above the modern description for what a category of line operators the a twist of 3D n equals fourth theory should be. And one needs to somehow combine that the module deformation with a loop group, equivalents. And maybe there's just an obvious way to do that in mathematically I, I have not thought about it enough except to make this heuristic statement, the very very concrete way to do this. That looks very close to the sort of category I've down is to go to vertex algebra which I'll get to in the last minus one minutes of the talk. So the one of the vertex algebra is I will write down look very much like functions, like the derived algebra functions on the space that we're taking coherent sheaves over. Okay, Hilbert spaces so in transimans in geometric quantization they show up in sections of the case power of some line bundle over bungee. And in this new setting, they should show up as sections of the same line bundle, answered with a very complicated sheath that if you don't turn any flat connection on is infinite rank, but finite rank of each homological degree. And it's a sheath that initially Gayoto described for the TMG theories. And so it's against something that we can heuristically write down. I have not done any explicit calculations with that yet. Except in the case where the surface is unit zero. And one is supposed to get local operators in the QFT. And one gets one dimensional space associated to S to intern Simons and functions on T star flag. And in this other theory which is functions on T star flag is the thing you would get from the derived category of small content group representations I mentioned earlier, as well. And that's why I said functions on the colon branch, which is why I said the colon branch as a variety has not changed, but, but extra stackiness is involved for other for higher genus surfaces, this sort of makes sheath description suggests that you should get something roughly looking like the transimans Hilbert space times a factor of the local homology of star flag. And that's actually what we got by computing. Haxchul Tomology of the small quantum group representation category. It was almost of this form with a few extra factors. So at least things look reasonable but but it would be nice to actually do the computation more precisely in this geometric quantization language. Okay, and I'll finish up quickly. So, first, there's, there's some 40 construction of this quantum field theory that I mentioned that is closely related to geometric langlines and for the s duality with different boundary conditions that I can answer questions about at the end, if anyone else is interested. So this 40 setup seems to be closely related to that work of Gukath at all from from last year that I mentioned in the introduction. And finally there are vertex algebra is involved and the vertex algebra is coming from either putting boundary conditions on the 3D quantum field theory, or by working in this for deconstruction and considering not just the sandwich of boundary conditions but a corner. And this was yet a third boundary condition and using work on vertex algebra is at the corner that Sasha must have talked about yesterday. Anyway, one can extract vertex out of this from these brain constructions and from the field theory and when they're actually two natural ways to do it. And I mentioned level rank duality before. So, in, in the classic transignment story, there are two WCW models that play a role that are cosets of each other inside some number of free fermions. And in the news story, there are also two algebras that play a role. And one is the Fagan to put an algebra and one is something new. And the something new is the thing that looks like functions on that weird space that I was trying to take loop group from when I had the module on. Okay, so the Fagan to put inside of this directly relates to the small quantum group, and I haven't turned on any flat connections when making these statements, the vertex algebras can also be performed by flat connections. And this, this new vertex algebra. So, we can gather some number of copies of a bit of a beta gamma system for each edge in the quiver, and then takes a BRST quotient by GL one up to GLM for the nodes in the quiver. And then a final, not the RST quotient but just derived in variance for a final copy of SLN at level So that's, that's that's the new vertex algebra. And Thomas Thomas quite sick showed this was actually dual to the fake to take it to put in for SL to. Okay, but that's, sorry, horribly out of time. So that's all say is, there's a, there's a really amazing vertex algebra story here. So, Thomas could give a one plus hour talk on on the vertex algebra side of this. So, let me say, so I've, I've indicated that that, like, it would be really awesome to sort of generalize the BFM like and other modules on the loop space constructions for line operators. So, I think that's the best to, and Webster to generalize that to include transignment levels that that should land in this case on the small quantum group module category. And I think that's just all correct. Be great to implement this geometric quantization perspective, start looking at open spaces and more so to get actions of the modular group or the mapping class group on spaces and states, which are currently kind of hard to implement. And there's been work for the tourists. Looking at things is that close can and key. Looking at the action of the modular group on hox show homology of that small group category, but I don't have anything in higher genus. So, yeah, they're beautifully in some derived version of this CGP like TQF. Okay, thanks. So, if you have a question now I'm able to let them speak. Yeah. Yeah, can you. I'll answer the technical first. If I can analytically, you are generalized Sharon Simons theory with respect to the level and anyhow different from ordinary terms. So, yes, the answer is certainly yes in some cases. And so, like, like Sergey Gukhov and co authors already described the analytic continuation of the 80 and variance. So, the analytic continuation uses a generalization of this 40 setup that I wrote down with different boundary conditions. So, it's always easier to discuss the analytic continuation than what happens at that integer level. And it's very, very similar to what went and did with ordinary transimons. So, so like the different choices there are all about choosing boundary conditions carefully and this 40 story. So, one needs to move to a half space. I got it. And the second question you mentioned this work of the other and the other check on the vertex algebra in the corner. I understand that it's kind of geometry. It's sort of certain derived category of coherence shifts on on say three day on C, C cube supported on some fat divisors given by coordinate planes, which kind of define you with corner. Now, how do you so that so this is a simple case of that where yeah yeah. Okay, but just look look at this simple, simple case. So how do you see this geometry. And do you see the tall. Yes. So, so there, see if I can actually draw on here. So the thing you're talking about involves sort of the, the full vertex corner. And here we're looking at and zero and zero brains. And so, so that's that's trying to describe say what happens here. And one also needs to tilt to tilt the brains relative to each other a bit, according to the trans diamonds level. So, so what I drew was probably not quite right. And so instead of that one needs to have some sort of tilted corner. So one can extract from that from the corner, the pieces that show up here. So, in one of these corner constructions the one that gives rise to Fagan to put in. And then starts off with a corner that after the summit this after the simplification spits out a w algebra. And so the general corner is these like why and then and algebras that are like massive generalizations of w algebras. In your story you see only and zero zero. Exactly. And, but it would like, of course, like, it would be beautiful to them like start generalizing this and relate to other quantum groups. But but it's yeah so my story is a very, very small part of that. And so colliding two corners, one, and they both have n zero zero but with slightly different decorations on them. And so one gives rise to a w algebra and the other gives rise to to to catch moody, rather to WCW. And, and so colliding those and taking an infinite level limit. And so it gives Fagan to put in. Okay, yeah, all right, thank you. Any other question. I have a question, can I, maybe I mean, some rise what you did in the end, just don't understand if if I understand if I understood it correctly so you can see this. So you get this theory of few G and then instead of gauged in a strong Simon sense, and you get some n equal to super symmetric theory which still has a in between and he's in cool and branch. But. Okay, that's kind of, it's kind of magical that it still has a and be twists. It's not magical what's on so you think I mean, is it so you mean that general and theory school. They don't they only have a holomorphic twist. I have a holomorphic topological twist on and and so what the way we actually analyze this is by starting with the holomorphic topological twist and observing that there is an extra differential that can still be turned on to to deform that to a topological twist. So it's not something general it's just kind of specific. It's very special. Okay, yes. Now, what is the statement sorry that this is probably the main point one of the main points of your talk which I, which I missed what is the state of the relationship between this theory and representation of the quantum group. I mean, so. So this the a twist of this theory. So, this is the same theory that has still has a symmetry and it's the symmetry that acts on the cool and branch. Okay. And the cool and branches is still the no potent cone so so the symmetry is the language dual group. Okay. So, one can, if one wants, turn on a background flat connection for that symmetry. And three manifolds with flat connections, if you don't do that. And you just ask, without any deformation what is the category of line operators. The category of line operators in that theory is representations of the small quantum group. It's it's it should be the derived category of representations of the small quantum group. So the camera for line operators is the For the eight ways for the eight ways. Okay. Yeah. And this is something that you sort of can more or less prove or I mean this. Yes. But we the only way we can actually prove it is by introducing a holomorphic boundary condition that supports the vertex algebra. And assuming that the category of lines is the same as modules for that verdict such a breath, which is what you would expect for a sufficiently rich boundary condition. And then we take that vertex algebra show that it's proved that it's dual to fake in to put in which is known to be the same as as modules for the small quantum group. And I'm wondering, you know, for the small quantum group business, the category of representation of small quantum group. It has this description by sort of bizarre county of Finkelberg and checkman in terms of factorizable sheaves. And, and I'm wondering if you have some feeling that it should be relevant for what you're doing but I have, I have exactly the same feeling. I don't understand that work well enough, but they like that, that story also gives derived like in principle should give derived spaces of conformal blocks. Yes. For these. Exactly. Yeah. So, so I, I think that should be very highly relevant for for all of this. I guess I also think that dates, dates, great has been been doing along the same lines. Maybe one small question you mentioned trans-simon theory for super groups. I mean, did you mention just because it's analogous or does it play any role in what you're doing. There should be an analogous construction that involves quantum super groups. And like, as, like as you know in this world of like 40 young mills with different boundary conditions and interfaces, one can engineer things that look like super group trans-simon as well. And so there should be generalizations of this that involves super groups. And part of the super group story is also like an under derived version of the super group story has been discussed. And it's like Rosenski and Soler wrote that trans-simon theory down and I guess in the simplest case for GL11. And started talking about what GL11 WZW you should look like and found non-synical categories. I think if one approaches this from the super symmetric side when we end up with the derived category of everything in sight. Yeah, I think the super group story is very close. Yeah, we can thank you again.
 I wanted to talk about some work in progress that I hope we will be finishing up very soon  with Thomas Kreitzig, Nick Garner, and Nathan Geer. It started out a few years ago as part of an SF-FRG collaboration.  And it's related but probably not in a way that will be obvious to some previous work with some of these co-authors  and Jennifer Brown, a grad student, and Stavros Garfilidis about recursion relations for so-called ADO invariants.  It's also closely related to a paper that appeared last year by Gukovsin Nakajima Parkpe in Topenko  that gives in a way I will say a little bit more about later, sort of a 3D mirror of our construction.  And there's also ongoing work in that direction by Gukov Fagan and Reshiti Ghan.  So to sort of set up the story I want to discuss, there was sort of fantastic progress in math and in physics that started about 30 years ago  in a bunch of papers, but in particular in work of Witton's and of Reshiti Ghan and Turayev that sort of connected some ideas coming from representation theory of quantum groups  and vertex algebras, WCW models, and Trin Simons quantum field theory.  One of the first examples of so-called topological quantum field theories whose partition functions give you topological invariants,  in this case of three manifolds with links inside.  So I am hoping that many of you have know at least part of this picture.  One of the huge exciting powerful aspects of this is that there were at least three different perspectives on quantum invariants that appeared and one could sort of study each side and connect them.  So a key object, sorry there's a lot of feedback coming through, let me know if the audio is not okay for me at some point.  Okay, so a key object in each of these constructions that appears is a certain braided tensor category and in the axiomatics of TQFT on the math side one can sort of reproduce the entire TQFT from this braided tensor category.  In Trin Simons theory, on the physics side, this is the so-called category of line operators. Its objects are extended operators that are localized on some sort of lines or curves and three-dimensional space time.  In the OA land, the objects of this category are modules for the relevant VOA, which is WCW, and in terms of quantum groups, objects of this category are modules for a quantum group UQG at a root of unity.  So, their objects of this category, more precisely their objects of a massive semi-simplification of this category. So the category at a root of unity, which is really what I want to talk about during most of this talk, is extremely complicated.  It's not semi-simple and it was known a long time ago that it wasn't semi-simple, and sort of a tiny piece of it goes into this original Reshti-Kintari construction.  Right, so like I mentioned, so in this old classic story, the category involved is semi-simple.  In terms of physics, semi-simple means there are no non-trivial junctions of line operators.  And in math, I'm hoping you all know what semi-simple means. So mathematically, you would say there are no non-trivial morphisms among objects and morphisms are junctions on the physics side.  Okay, so a lot of progress has been made since then, sort of extending into the non-semi-simple world.  On the quantum group side, even in the early 90s, Li Buchenko and others started writing down partial TQFTs that started off with a non-semi-simple category.  And so the Kutsude-Gucchi, Otsuki, or ADO invariance of links are sort of related to this. They're related to a semi-simple part of a bigger category that I'll mention later on.  Well, the problem there is not that it's not semi-simple, the problem is that there, some representations have vanished in quantum dimensions that need to be regularized.  And some of the patterns of problems come up in general when looking at the big category of UQMG modules at a root of unity that need to be dealt with somehow.  A systematic set of tools for dealing with the various problems that come up has been constructed much more recently starting in work of Geo Constantino and Patiro Muran, and extending in a bunch of different ways since then.  And so now there is, well, there are still extensions that are being developed, but there is at least one TQFT defined using these techniques, these new techniques that involves the full representation category of UQMG.  I'll talk about this category in about 10 minutes.  So, on the VOA side, there have been similar developments.  So going non-semi-simple there in its simplest incarnation means generalizing from rational VOAs to logarithmic VOAs.  And the simplest logarithmic vertex algebra, the so-called triplet model, and that's been generalized also in many ways, but the generalization that's relevant here is in terms of what are called Fagin-Tepunin Algebras.  That's what FT stands for.  So this is Fagin-Tepunin.  And with suitable matching, the representation category, the module category of these Fagin-Tepunin Algebras is supposed to match modules for quantum groups at certain roots of unity.  Well, it was supposed to, I think it was conjectured a long time ago and in recent work, that I'll give you a few references here.  The equivalence has been proven in certain cases and generalized to an equivalence of actual rated tensor categories.  So we modulate tensor categories that sort of the data you needed to build the TQFT out of this.  Okay, so I said a lot has been done.  Yeah, sorry, was there a question?  Okay.  So what hasn't been done is the physics side of this.  Transimans led to a lot of sort of amazing computations and predictions, things like geometric quantization to produce Hilbert spaces and  conceivage integrals and so on. And the transimans part of this non-simple story does not exist yet.  So what we're doing, part of what we're doing is actually to propose a quantum field theory that fits in this third physics perspective.  Some non-simple aspects, I should say, have appeared.  Now, triplet algebras have started appearing in supersymmetric theories in a paper on 3D modularity, maybe, by Chang-Chen Ferrari-Gericot Harrison.  So logarithmic things have started appearing, in particular in a supersymmetric context.  So, supergroup transimans theories have also been investigated, and I would, using sort of physics language, I would say they fall into the same universality class.  They, supergroup transimans theories have a lot of properties in common with the theories I will discuss today.  They're just not exactly the direction we're going in.  So the result I want to talk about, in part proposal and part result, is that there exists a three-dimensional quantum field theory that is topological, whose category of line operators matches  sort of the big quantum group at a root of unity category.  Here the case I'll focus on is an even root of unity, and I am certain about this in type A, and there are obvious generalizations for groups of other type.  And the category of lines matches modules for the fake and tipunin algebra, and in fact, the way we actually get at this more naturally is in a sort of morally level rank dual of the fake and tipunin algebra.  So they're actually two vertex algebras that appear that have equivalent categories of modules.  So, we have a, I would call it physics proof that the vertex algebra categories are the categories of lines.  And in the QFT I will talk about for the case of certainly, sorry, for the case of SL2, I think for general type A.  So, to make this statement I have to actually say what I mean by physics proof taken suitably liberally, I would say this makes sense for general type A.  And Thomas Kreutig in our paper gives a proof that for SL2, the two vertex algebras I showed here are equivalent and it's already known that the fake and tipunin algebra is  related to quantum SL2 modules at a root of unity.  So sort of the new bit is the left side of this picture.  So whatever quantum field theory sits on the left here had better be labeled by a group or a Lie algebra and an integer k that tells you what root of unity we're working at.  And change the page on my screen but it's not coming through so I may need to re-marry.  Just a sec.  Oh, you guys can see that.  Okay.  Let me just play again. Good.  Right, so the theory is starts out as a 3D n equals 4 theory and it's mostly a 3D n equals 4 theory called QFT.  That can be analyzed using all of the lovely algebraic techniques that you're hearing about in other parts of this workshop.  And it's a very quick twist. So, so it's sort of a mix of a 3D n equals 4 theory and a trans-simon theory.  That's what that G of K is doing there.  And so it doesn't, it falls slightly outside the class of things that are easily analyzed algebraically right now. And one of the points of my talk is to motivate all of you to think about how to  solve this problem.  Sorry, and I see Sasha has a question in the chat.  It's for all G. So, this theory that I'm talking about here makes sense for all G and all.  I was asking about the previous line about the.  No, there are subtleties when I would expect it to work for a to E type and there are very huge technical things and that happened later where some language stools need to be involved and  so there's some natural guesses for what happens, but the story is not so simple. Okay, thank you.  And it's like it's sort of clear and on each from each perspective why the story is not so simple, matching up the various subtleties is is hard and is something we realize we should not attempt in in this first shot at at this project.  So, one can write down a sort of a Lagrangian, an action for for the steering and type a using a twisted BV formalism.  Just is kind of nice. So in that sense, it's very much like turn simons. It's a concrete theory.  And using that Lagrangian one can write down a boundary vertex algebra, the analog of WCW, and there are in fact two of them that show up. And in WCW there's also another algebra that you could use which is the level rank dual of WCW that has exactly the same category of modules.  And, and here as well, there are there's sort of two natural algebras that show up that are level rank duals of each other in the appropriate generalization of level rank.  And it should be possible, like, it's sort of clear what to write down for other types from 40 brain constructions. But there are lots of technical issues that show up.  Right. So, we write all this down and then then the natural question is like what what what you gain from the physics and what can you compute to check that that this guess is even correct.  So it's easy to compute using supersymmetric localization techniques, the growth and the group of the category of line operators and with our characters of hybrid spaces which are said this, there's this D bar sitting everywhere,  there's a D bar sitting everywhere. So, the physics theory because it's this sort of supersymmetric theory that needs to be topologically twisted to get something topological is like sort of naturally a derived beast or like the category of line operators is naturally a DG category.  And so all of the equivalences that we get, particularly between QFT and the V ways are that that first equivalence is a derived equivalence.  There are many different ways of describing the category of line operators and it shows up as a DG category and it is equivalent to the derived category of modules for this VLA.  The thing that's also relevant on the quantum group site is the derived category of quantum group modules.  The things associated to surfaces are going to exist in many co homological degrees, and their other characters are easy things to compute as partition functions.  And so what we can compute are things like the category itself from a few FT perspective or the hybrid space itself in a math and classroom action, and I'll indicate some techniques to try to go about that.  But like this is where like fully generalizing algebraic approaches to 3D n equals four theories things like the BFM construction would be really useful.  That's the end of the long introduction.  You know if there are any questions to wrap me at any time as well.  I want to actually say a few more concrete things about the representation theory of quantum groups at a root of unity, and I'm just going to stick to the case of SL2 here for illustrative purposes.  Great.  So, right, so quantum SL2 looks like standard SL2 except the part time generator has been exponentiated.  And it looks like there is a question coming but I'll wait till it actually comes.  At an even root of unity in fact that any root of unity, but even is the relevant situation for me.  There's, in addition to the center that comes from from Kazimier operators that sort of the higher chandra center.  There's an extra bit of the center that just comes from case powers of the F and the K.  And so the extra central elements act as constants on any indicomposable representations.  And so the representation theory sort of decomposes into what little pieces based on what values the central elements take.  And the fancy way to say that is that it ends up vibing over spec of this other part of the center.  And this other part of the center parameterizes an open cell in the group PGL to  So, the reason PGL to is coming up here is because it's the Langland's tool of SL2.  And at at an even root of unity.  The fancy way to say this is that the category of modules fibers over PGL to at an odd root of unity at fibers of grass.  I think technically I should also say this is a coherent chief of categories which is also something that shows up in the field theory.  But I think that's all I want to say for now.  Really simple example of this is that when.  So if you if you take a diagonal element of PGL to that has some eigenvalue either the alpha  one would associate that to mod to a subcategory of modules on which either the K and after the K act to zero coming from these off diagonal zeros and K to the 2K acts as you the alpha  one.  Should also say like the relevant thing about these, these different fibers or different stocks of the chief is that they're there are no hams between them so they're each full subcategories, the total categories a direct sum of all of these stocks.  And, yeah.  Right.  So I'll say.  I'll say that terrible transition. So it was observed, it was proven a long time ago by the continue and cats and even root of unity case by back that what the stock of fiber looks like, but the particular elements of PGL to have the center  of the K act only depends on the conjugacy class of this element and in PGL to up to isomorphism.  And then what was realized a little later on by initially by Koshile and Russian TK is that this sort of fibering of the category over PGL to  not just to invariance of links and three manifolds, but more generally to invariance of links and three month in three manifolds with flat connections in the complements of the links.  So this was roughly but not entirely correct on and more like correct. And it's been made very precise in the case of links links and s3 in a recent paper by a bunch a gear, but you're on and Russia taken.  So, so they build an invariance of three manifolds, sorry, invariance of links and s3 with flat PGL to connections in the complement of the link.  And there's work in progress, trying to really promote that to a full QFT that that gives invariance three manifolds with with flat PGL to connections.  And the picture that I always have in mind when when trying to understand why why this should be true is is the following.  So, so the category has all these different blocks or stocks labeled by by elements of PGL to one should if if trying to translate this to some sort of physics picture, or topology picture that involves say links where the strands  are labeled by objects of this category.  One should think that the piece of the category that object from labeled by some element G.  And then you can define the whole ennemy of your background flat connection, the flat connection that you've enriched the three manifold with in a small loop, going around this line.  If you just have a single line. It's only the conjugacy class of the whole ennemy should matter which is consistent with the old theorem of decontini cats and perches that a fiber of this category only met only depends on the conjugacy class.  Now, if you want to start building a TQFT out of this you need to make sure that this is compatible with the tensor structure in the category.  And you would expect if you're looking at base pointed whole ennemy is that when two lines collide, when gets the tensor product of those representations using the hop algebra structure of the quantum group.  And that had better be compatible with multiplying the base pointed whole ennemy's. And it's literally true when all of your hall enemies are diagonal, and it is almost true when the whole ennemy's are general non abelian things and the almost was described precisely in in this  book by Kashyap and Russia to get in this later paper by one shade here at your mirror on Russia to get.  I'll just focus on the abelian case.  Okay.  So, this sort of setup where we're looking at three manifolds with the decorated by flat connections appears in physics when we have a quantum field theory with a global symmetry.  As opposed to a gaged symmetry.  So, so quantum field theories with global symmetries can couple to connections that are not fields that one integrates over in the path in a goal but fields that are just put in by hand and fixed for all time.  And so the sort of theory we're looking for had better have some sort of global symmetry.  And so the theories in the background of a whole ennemy defect like this were described really nicely a few years ago in a paper of Victor make high looks in the context of super symmetric trans diamonds but the story is is sorry, the context of super grip trans diamonds, but but the story is very similar here.  Okay.  Let me try to describe what a few of these fibers look like.  Just so I can say some concrete things later on.  If we take a generic diagonal element of GL to  then we look at representations of you qsl to on which K to the 2k acts is that I can value and you do the K and after the K at zero.  The category ends up being semi simple with exactly 2k semi simple objects.  And that look sort of like standard highest weight modules, except the weights are, I can values of cave rather than an H. And so they're, they're cute as a something.  They all have dimension K, and they wrap around the unit circle.  And so the, the, the, the, the weights wrapped around the unit circle, which gives rise to vanishing quantum dimensions, which is one of the things that needs to be regularized in order to build a TQFT out of this but that was done so  this is the 80 and variant uses precisely these sorts of representations.  And more economy also worked on that in the 90s and then Constantino, you're in a mirror and sort of systematized the data you would need to not get all of your link invariance to be zero.  And so the average rise quantum vanishing quantum dimensions would naively tell you that even the hop, and even the unknot has expectation value zero.  These sort of generic stocks of the category has an extremely simple tensor product.  And sort of looks like what you would expect for an in an abelian theory, it's like for quantum gl one.  It leads to very easy calculations of dimensions of putative Hilbert spaces or spaces of states on various surfaces.  So, you get a space of states on the tourists.  You can generate states by filling the tourists into a solid tourists and putting objects, colored by all the different possible representations along its core.  Now, since we're dealing with a QFT enriched by flat connections.  We shouldn't just say the Hilbert space of the tourists we should say the Hilbert space of the tourists together with a flat PGL to connection on it.  And putting an abelian connection on there with a specified holonymy around the meridian of the tourists tells us what piece of the category to choose objects from on the core.  And so we just take the two K different objects in that piece of the category along the core that gives us to K states in that Hilbert space.  And so we should say, I'm saying Hilbert space because in physics, we always say Hilbert. These are not necessarily Hilbert spaces in the mathematical sense, and in this sort of teacher at T's I'll be discussing.  They are vector spaces, they have duals, they don't necessarily, they're not necessarily isomorphic to their dual in a natural way.  They're not necessarily just say vector spaces of states.  In genus G.  One can use the very simple my nodal structure to also draw all trivalent networks of  the two, keep one they say line operators inside the core of a handle body, whose boundaries of particular genius G surface and count what the options are. And when gets to the G, K to the 3G minus three different states.  That's a very easy combinatorics problem.  Okay, so that's sort of the generic setting.  And when, when you're looking at the surface with a generic flat connection on it.  There's also the most interesting, most non generic case when the flat connections trivial.  So, so the connection itself which is zero everywhere.  And so in this case, looking at what sorts of what to be labeled aligned by in the presence of trivial homony.  The answer is representations of what is called the smaller restricted quantum group.  So, it's modules on which K to the K, K to the two k axis one and you to the K and F to the K act is zero.  So this is an extremely well studied category in representation theory.  It is not semi simple.  It has two to the K simple objects.  That can form interesting extensions with one another.  The two to the K simple objects roughly look like two copies of ordinary representations of SO2 of dimensions 123, although we have to K.  And they taking the projective covers of the simple objects when gets projected.  And they have sort of diamond structures and movie diagrams. And they're so they're to K, the composable projectives in, in this piece of the category as well.  The semi simplification that was used by restyling to read uses a single copy of the, of what look like ordinary SL to representations of dimensions one up to K minus one.  So if one sort of quotients out by everything else, setting it to zero.  Another way of saying portion out by everything else is set to zero everything that has a vanishing quantum dimension, when gets a semi simplified category that leads to the old story.  So the entire of invariant that's involved in the volume conjecture is defined using this last simple and projective representation of dimension K.  And, and finally, sort of the entire thing is, is what we Bush and considered when, when starting to define non semisimple to get these in the 90s.  Okay.  So, I mentioned at beginning that in general, our spaces of states on surfaces would be  homological beasts.  They, they have multiple co homological degrees.  In terms of the category of lines mathematically.  One would calculate the Hilbert space of states in general by taking a whole show of the appropriate piece of the category, depending on what flat connection we've chosen and so on a tourist with zero flat connection.  So we have a small quantum group category. And we should find it's Hilbert space is Hochschild homology of the small quantum group category.  And so what that amounts to is considering the fact that when there are morphisms among the objects in your category, you don't, you can't just wrap single lines around the core but you have to consider junctions of multiple lines.  And so, we're the, the, we'll give you the zero zeroes Huxchild homology, higher degrees and Huxchild homology and Huxchild homology come from integrating descendants of junctions around paths in this core.  That is, a statement for anyone in the audience who knows about descendants.  So it's a totally sensible thing to do both physically and mathematically.  In order to compute Huxchild homology of this category it's useful to have a geometric description of it.  And archipath, as you can pick off in Ginsburg, and then those are coming to cut the month, let's go Dave such a geometric description, which at an even rate of unity amounts to saying that as a category, not a monotone category, category.  So the first category of representations of the small quantum group has sort of two semi simple pieces to that's just coming from the two symbols that were also projected.  And then a bunch of copies of the derived category of her she's on the flag manifold.  K minus one copies of that.  And gives us a geometric way to compute Huxchild homology which ends up looking like, I want to say total double homology.  Looks like total co homology of these of P star flags computed in algebraic way.  And then there's an answer. The, the relevance base of states becomes infinite dimensional with non negative co homological degrees and it's finite in each co homological degree.  And the parts that the Bush and could have used back in the 90s.  I say that on the next slide.  And the part that the Bush and could would have used is just h8 zero.  But, but there ought to be a derived generalization of, of all of the earlier work and also the current work of CGP and collaborators.  Okay.  So, I, I, I've sort of written what the different ways look like in terms of representations of the symmetry group PGL to that acts on star flag.  There are also nice things to say about deforming from trivial connection to non zero generic connection.  Which, there we go.  Which again, I think I need to.  That's the slide I actually want to be on.  I'll just see this very briefly but but to go. So, at generic connection, the space of states had dimension to K at zero connection the space of states is this infinite dimensional thing with  the infinite dimensional graded components. And there's a differential that one can turn on to deform the infinite dimensional thing to something that exists only in degree zero and has dimensions UK.  The other character is invariant under this deformation the other character doesn't care what flat connection you put on.  And at a category level.  The field theory interpretation that I'm about to get to also suggests that there's there's just, there's a, well, this is a sheet that coherent chief of categories.  And very close to the identity elements in PGL to coherent sheaves gets deformed to matrix factorizations with a super potential.  And that involves the complex moment map for  for a certain elements of PGL to that that's the stock. That's defined by the stock we're looking at.  Okay.  I said that other characters don't change under this deformation.  Okay, that's, that's, sorry, I probably said too much about that I have really enjoyed during this project, sort of learning and trying to put together more of the structure of this big category of representations of the quantum group at a rate of unity.  You can probably tell.  Okay, so  I'm going to take all of that information and translate it quickly to to physics and then try to explain what what one still has to do on the physics side.  So we're looking for a quantum joke theory that is labeled by algebra and a level and an integer k.  And that's what the term Simon's like. And that, like it sort of looks very close to the rest you can try the category should have some finite like in each piece of the category that a finite number of objects it should have something that looks sort of looks like  a close in lines and in this previous paper on recursion relations for 80 and variants we found like we found structure that was very, very, very similar to the simple or the semi simple story involved in so so 80 80 and variants are different from color  and they are the same models but they obey the same recursion relations. And so there are a lot of things that sort of look the same. You should see something turn Simon's like here.  However, the line operators in this theory should have nontrivial junctions we should not be getting a simple category.  So, very easy signal that quantum field theory gives rise to a non semi simple category of lines is that there are non trivial local operators.  This is the sort of thing that the BFM construction computes. So, BFM computes an algebra.  And it's the algebra of local operators in a 3D field theory.  And so that construction applied to this story had better give you something non trivial.  If you apply it to turn Simon's it just gives you the identity and that's it.  And there should be a global symmetry around that leads to flat connections.  And those criteria lead to an essentially unique answer, which is, if you're not familiar with the physics side of the story is going to look strange and weird and not unique at all.  But anyway, so the easiest thing one can do to satisfy this property is to start with a 3D and equals for super symmetric theory that's going to have been introduced called t of G.  And even though I write a group here it secretly only depends on a le algebra. And roughly speaking, it has symmetry G times the language to do a little bit.  Sorry, is this literally true I thought it only had she symmetry and also it's it's mirrored dual had the check symmetry is that.  No, I mean, it has both it's it one acts on the Higgs branch and one acts on the coolant branch, but the, the, what I saw the coolant branch is not a symmetry of the theory.  No, of course it is no no no it's exactly on the same pudding.  It's if you write down a Lagrangian for this for the theory and type a, all you see is the maximal torus, but otherwise it's  down the theory in terms of fields and what you call the Higgs branch and what you call the coolant branch is is arbitrary at the level of quantum field theory.  And so it really does. It really does have both of these symmetries.  So you mean that it's before you twisted. Yes, yes, yes, exactly. Okay, okay, and the different ways. Right, of course, after you twist, depending on what sort of twist you use.  Only one of them will appear as a symmetry of the twist.  Or the acting very, very different ways.  Okay, yeah, okay, now I see what I mean.  Yeah.  Okay.  So then we take G and engage it further.  But, but this gauging is different from the standard three, three D n equals four gauging. It's something one can do in the series with less supersymmetry.  And so physics terms I would say using n equals two vector multiple it.  And so this less supersymmetric gauging allows the introduction of a non trivial trans diamonds level.  And so one sort of writes down.  If you if you're not worried about supersymmetry you write on the trans diamonds Lagrangian for G and couple it to the rest of this theory to G.  And the result of this gauging does it still have some supersymmetry left on it.  Yes, but this is subtle. So, in physics terms, it only if you write down a grand G and you try to do this in terms of fields you can only see n equals two supersymmetry in the infrared or with a suitable twist.  It should have full and equals four.  But seeing that as.  Before getting to topological twist so  maybe I'll also say later. So, in using the BV formalism one can actually write down a Lagrangian that does not look like it has n equals two or n equals four supersymmetry but has the single super charts you need to topologically  to perform what would otherwise be the twist.  So if you can only in the infrared this thing regains full on n equals four supersymmetry and you can topologically twist.  But there, but there are subtle views around.  There, if you believe that this has any of this for supersymmetry.  There are two twists available. One that's focused on the Coulomb branch and one that's focused on the Higgs branch or a twist in the B twist.  The G dual symmetry behaves in different ways with respect to the two choices.  The one that will leave you with an ability to turn on flat G dual connections is is the A twist in this case and it's it's the one that's sort of focused on the Coulomb branch.  The B twist of the same theory was something that Kapustin Salina studied 10 years ago, and something they called transimons rosansky written theory.  The B twist is transimons rosansky written.  But it is, it is completely different theory, at least it looks, it behaves very differently and it's not the thing that's relevant for quantum groups that are really unique.  Okay.  So the other thing to say is that this is a quiver gauge theory.  So the in sorry when when the group is type A when we're looking at us UN this is a quiver gauge theory.  Most of the quiver is the quiver you would write down for a T star flag. That's the Nakajima quiver variety for P star flag.  However, in T star flag there's a framing node.  And again, for for GLN or for for S you and that that final that final framing node is the thing that's staged with an extra insurance simons level.  And so the gauge group of the theory is a product of GL one through not GL minus one but all the way through GLN with the transimons level for for GLN.  And this extra transimons gauging both the Higgs and the column branch look like the star of the flag manifold.  The extra gauging destroys the branch and it actually seems to do nothing at all to the column branch, except to change well.  Same T star flag is maybe incorrect. I should rather say it's the outcome.  It seems to this extra transimons gauging seems to introduce some extra singularities at the origin of the notebook and come, which I can't discuss any more precisely than that.  What do you mean when I say is destroys the Higgs branch.  That's right.  So, so you would, I mean, sorry, the thing that it does is it, it takes a clear question of the Higgs branch.  Not not a hypercalor quotient, but just no ordinary.  So, so you, you,  if they weren't to turn simons level around, you would expect the Higgs, the Higgs branch just be quotient by GLN.  So they have less super similar to the Higgs branch is no longer calomorphic symplectic.  Exactly.  The maximal torus of the flat, the maximum torus of in this case, it's PGL to our PGLN shows up in terms of resolution, complexified resolution parameters for the coolant branch.  Sorry, their deformation parameters for the coolant branch or yeah.  I'm short on time. So I think that's that's all I'll say there's a lot I think there's been a lot about 3D and equals four theories in this conference.  And the Wilson line operators that we wanted to see show up for the turn simons factor.  So it's surprising in general that one would have Wilson lines in this a twist that is focused on the coolant branch, it can happen here.  Precisely because of the trans simons level.  It would not happen.  Okay, so, so then there's a black box. So this, I claim is theory one can write down a Lagrangian.  Lagrangian for the PV formalism.  There are lots of localization techniques that apply to that or just or if you just think of it as n equals two supersymmetric theory, their localization techniques that will compute partition functions and expectations of both some line operators in this  theory.  So this is the work of Microsoft and shuttle Schvilly and recent work on unfisted indices.  This. So, in, in, like, half a day, when can code up a computation that spits out other characters for spaces of states, and remind you other characters don't care about what flat connection is turned on.  And when on the nose gets the right answer for SU two and we've checked for SU three and SU SU four as well.  Also using the beta roots analysis of Microsoft and shuttle Schvilly that that gives you expectation values of line operators.  One can get the growth and dig ring of the small quantum group.  And so that's with zero flat connection. And so that tells you something about the small quantum group but it doesn't tell you anything about the interesting non semi simple behavior.  So, to actually see directly from the field theory more of the non semi simple stuff.  So, in the case of the data, one should try to apply for getting there. A lot of the more modern methods developed in the last five years or so.  In, including by the fm and Webster and work of mine with guy at the ball and water, Hilburn and company.  And so it's a recent work of Matt Bollemore and co authors and writing down Hilbert spaces.  So, so there are algebraic to piece one can use one has to sort of adapt the current techniques to this sort of hybrid case that is mostly 3D n equals for in the a twist with a bit of 3D n equals to with a transignments level.  The sort of thing when expects for the category of line operators looks as follows. So if, and I want to compare on this slide what happens in ordinary transignments theory to the quantum field theories I'm discussing so in terms of in theory, you can write ordinary  transignments theory to the point that the answer is a 3D n equals to theory and use modern techniques to say what the category of line operators should be and the answer is loop group, equivalent coherent sheaves on a point, which, which of course is is  the loop group at level k which is the correct thing for to match wcw and the semi simplified quantum group and the whole term Simon story.  In these new theories what one gets instead should look something like loop group at covariant coherent sheaves on a deformation of  the tangent bundle of the loop space of the Higgs branch of this quiver.  And if I didn't say loop group echo variant, and I did say deformation it's the deformation that deforms coherent sheaves to the modules.  And the modules on the loop space of the Higgs branch is due to a bunch of the references above the modern description for what a category of line operators the a twist of 3D n equals fourth theory should be.  And one needs to somehow combine that the module deformation with a loop group, equivalents.  And maybe there's just an obvious way to do that in mathematically I, I have not thought about it enough except to make this heuristic statement, the very very concrete way to do this.  That looks very close to the sort of category I've down is to go to vertex algebra which I'll get to in the last minus one minutes of the talk.  So the one of the vertex algebra is I will write down look very much like functions, like the derived algebra functions on the space that we're taking coherent sheaves over.  Okay, Hilbert spaces so in transimans in geometric quantization they show up in sections of the case power of some line bundle over bungee.  And in this new setting, they should show up as sections of the same line bundle, answered with a very complicated sheath that if you don't turn any flat connection on is infinite rank, but finite rank of each homological degree.  And it's a sheath that initially Gayoto described for the TMG theories.  And so it's against something that we can heuristically write down. I have not done any explicit calculations with that yet.  Except in the case where the surface is unit zero.  And one is supposed to get local operators in the QFT. And one gets one dimensional space associated to S to intern Simons and functions on T star flag. And in this other theory which is functions on T star flag is the thing you would get from the derived category of small content group representations I mentioned earlier, as well.  And that's why I said functions on the colon branch, which is why I said the colon branch as a variety has not changed, but, but extra stackiness is involved for other for higher genus surfaces, this sort of makes sheath description suggests that you should get something  roughly looking like the transimans Hilbert space times a factor of the local homology of star flag.  And that's actually what we got by computing.  Haxchul Tomology of the small quantum group representation category. It was almost of this form with a few extra factors.  So at least things look reasonable but but it would be nice to actually do the computation more precisely in this geometric quantization language. Okay, and I'll finish up quickly.  So, first, there's, there's some 40 construction of this quantum field theory that I mentioned that is closely related to geometric langlines and for the s duality with different boundary conditions that I can answer questions about at the end, if anyone else is interested.  So this 40 setup seems to be closely related to that work of Gukath at all from from last year that I mentioned in the introduction.  And finally there are vertex algebra is involved and the vertex algebra is coming from either putting boundary conditions on the 3D quantum field theory, or by working in this for deconstruction and considering not just the sandwich of boundary conditions but a corner.  And this was yet a third boundary condition and using work on vertex algebra is at the corner that Sasha must have talked about yesterday.  Anyway, one can extract vertex out of this from these brain constructions and from the field theory and when they're actually two natural ways to do it.  And I mentioned level rank duality before. So, in, in the classic transignment story, there are two WCW models that play a role that are cosets of each other inside some number of free fermions.  And in the news story, there are also two algebras that play a role. And one is the Fagan to put an algebra and one is something new. And the something new is the thing that looks like functions on that weird space that I was trying to take loop group  from when I had the module on. Okay, so the Fagan to put inside of this directly relates to the small quantum group, and I haven't turned on any flat connections when making these statements, the vertex algebras can also be performed by flat connections.  And this, this new vertex algebra.  So, we can gather some number of copies of a bit of a beta gamma system for each edge in the quiver, and then takes a BRST quotient by GL one up to GLM for the nodes in the quiver. And then a final, not the RST quotient but just derived in variance for a final copy of SLN at level  So that's, that's that's the new vertex algebra.  And Thomas Thomas quite sick showed this was actually dual to the fake to take it to put in for SL to.  Okay, but that's, sorry, horribly out of time. So that's all say is, there's a, there's a really amazing vertex algebra story here.  So, Thomas could give a one plus hour talk on on the vertex algebra side of this.  So, let me say, so I've, I've indicated that that, like, it would be really awesome to sort of generalize the BFM like and other modules on the loop space constructions for line operators.  So, I think that's the best to, and Webster to generalize that to include transignment levels that that should land in this case on the small quantum group module category.  And I think that's just all correct. Be great to implement this geometric quantization perspective, start looking at open spaces and more so to get actions of the modular group or the mapping class group on spaces and states, which are currently kind of hard to  implement. And there's been work for the tourists. Looking at things is that close can and key.  Looking at the action of the modular group on hox show homology of that small group category, but I don't have anything in higher genus.  So, yeah, they're beautifully in some derived version of this CGP like TQF.  Okay, thanks.  So, if you have a question now I'm able to let them speak. Yeah. Yeah, can you.  I'll answer the technical first. If I can analytically, you are generalized Sharon Simons theory with respect to the level and anyhow different from ordinary terms.  So, yes, the answer is certainly yes in some cases. And so, like, like Sergey Gukhov and co authors already described the analytic continuation of the 80 and variance.  So, the analytic continuation uses a generalization of this 40 setup that I wrote down with different boundary conditions. So,  it's always easier to discuss the analytic continuation than what happens at that integer level. And it's very, very similar to what went and did with ordinary transimons.  So, so like the different choices there are all about choosing boundary conditions carefully and this 40 story.  So, one needs to move to a half space.  I got it. And the second question you mentioned this work of the other and the other check on the vertex algebra in the corner.  I understand that it's kind of geometry. It's sort of certain derived category of coherence shifts on on say three day on C, C cube supported on some fat divisors given by coordinate planes, which kind of define you with corner.  Now, how do you so that so this is a simple case of that where yeah yeah. Okay, but just look look at this simple, simple case.  So how do you see this geometry.  And do you see the tall.  Yes. So, so there, see if I can actually draw on here. So the thing you're talking about involves sort of the, the full vertex corner.  And here we're looking at and zero and zero brains.  And so, so that's that's trying to describe say what happens here.  And one also needs to tilt to tilt the brains relative to each other a bit, according to the trans diamonds level. So, so what I drew was probably not quite right.  And so instead of that one needs to have some sort of tilted corner.  So one can extract from that from the corner, the pieces that show up here. So, in one of these corner constructions the one that gives rise to Fagan to put in.  And then starts off with a corner that after the summit this after the simplification spits out a w algebra. And so the general corner is these like why and then and algebras that are like massive generalizations of w algebras.  In your story you see only and zero zero.  Exactly. And, but it would like, of course, like, it would be beautiful to them like start generalizing this and relate to other quantum groups.  But but it's yeah so my story is a very, very small part of that.  And so colliding two corners, one, and they both have n zero zero but with slightly different decorations on them. And so one gives rise to a w algebra and the other gives rise to to to catch moody, rather to WCW.  And, and so colliding those and taking an infinite level limit.  And so it gives Fagan to put in.  Okay, yeah, all right, thank you.  Any other question.  I have a question, can I, maybe I mean, some rise what you did in the end, just don't understand if if I understand if I understood it correctly so you can see this.  So you get this theory of few G and then instead of gauged in a strong Simon sense, and you get some n equal to super symmetric theory which still has a in between and he's in cool and branch.  But.  Okay, that's kind of, it's kind of magical that it still has a and be twists.  It's not magical what's on so you think I mean, is it so you mean that general and theory school.  They don't they only have a holomorphic twist.  I have a holomorphic topological twist on and and so what the way we actually analyze this is by starting with the holomorphic topological twist and observing that there is an extra differential that can still be turned on to to deform that to a topological  twist.  So it's not something general it's just kind of specific.  It's very special.  Okay, yes.  Now, what is the statement sorry that this is probably the main point one of the main points of your talk which I, which I missed what is the state of the relationship between this theory and representation of the quantum group.  I mean, so.  So this the a twist of this theory.  So, this is the same theory that has still has a symmetry and it's the symmetry that acts on the cool and branch.  Okay.  And the cool and branches is still the no potent cone so so the symmetry is the language dual group.  Okay.  So, one can, if one wants, turn on a background flat connection for that symmetry.  And three manifolds with flat connections, if you don't do that. And you just ask, without any deformation what is the category of line operators.  The category of line operators in that theory is representations of the small quantum group. It's it's it should be the derived category of representations of the small quantum group.  So the camera for line operators is the  For the eight ways for the eight ways.  Okay.  Yeah.  And this is something that you sort of can more or less prove or I mean this.  Yes.  But we the only way we can actually prove it is by introducing a holomorphic boundary condition that supports the vertex algebra.  And assuming that the category of lines is the same as modules for that verdict such a breath, which is what you would expect for a sufficiently rich boundary condition.  And then we take that vertex algebra show that it's proved that it's dual to fake in to put in which is known to be the same as as modules for the small quantum group.  And I'm wondering, you know, for the small quantum group business, the category of representation of small quantum group. It has this description by sort of bizarre county of Finkelberg and checkman in terms of factorizable sheaves.  And, and I'm wondering if you have some feeling that it should be relevant for what you're doing but I have, I have exactly the same feeling.  I don't understand that work well enough, but they like that, that story also gives derived like in principle should give derived spaces of conformal blocks.  Yes.  For these.  Exactly. Yeah.  So, so I, I think that should be very highly relevant for for all of this.  I guess I also think that dates, dates, great has been been doing along the same lines.  Maybe one small question you mentioned trans-simon theory for super groups.  I mean, did you mention just because it's analogous or does it play any role in what you're doing.  There should be an analogous construction that involves quantum super groups.  And like, as, like as you know in this world of like 40 young mills with different boundary conditions and interfaces, one can engineer things that look like super group trans-simon as well.  And so there should be generalizations of this that involves super groups.  And part of the super group story is also like an under derived version of the super group story has been discussed. And it's like Rosenski and Soler wrote that trans-simon theory down and I guess in the simplest case for GL11.  And started talking about what GL11 WZW you should look like and found non-synical categories.  I think if one approaches this from the super symmetric side when we end up with the derived category of everything in sight.  Yeah, I think the super group story is very close.  Yeah, we can thank you again.
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