 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. Started out a few years ago as part of an NSF-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 Gukav Sin Nakajima Parkpe in Saupenko 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 Gukav Fagan and Rasha Tikin. 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 Rasha Tikin and Turayev, that sort of connected some ideas coming from representation theory of quantum groups and vertex algebras, WCW models, and Tern Simon's 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. 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 Tern Simon's 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 in three-dimensional spacetime. In the OA land, the objects of this category are modules for the relevant VOA, which is WZW, and in terms of quantum groups, objects of this category are modules for a quantum group UQG at a root of unity. They're objects of this category, more precisely, they're objects of a massive semi-simplification of this category. So the category at a root of unity, which is really what I wanted to talk about during most of this talk, is extremely complicated, and 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 mesh-taken terrier construction. So like I mentioned, 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, Liushenko and others started writing down like partial TQFTs that started off with a non-semi-simple category, and the Kutsude-Guchi, Otsuki, or ADO invariance of links are sort of related to this. They're related to a semi-simple part of the bigger category that I'll mention later on. Well, the problem there is not that it's not semi-simple, the problem is that some representations have vanishing quantum dimensions that need to be regularized. Anyway, so all sorts 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 Gir-Constantino and Patry-Romiro, 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 new techniques that involve 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. 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-T-Punin algebras. That's what FT stands for. So this is Fagin-T-Punin. Fagin-T-Punin algebras are defined for any the algebra G. And with suitable matching, the representation category, the module category of these Fagin-T-Punin 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 a few references here. The equivalence has been proven in certain cases and generalized to an equivalence of actual rated tensor categories, I think possibly modular tensor categories that sort of the data you needed to build a TQFT out of this. Okay, so so a lot has been done. What hasn't, yeah, sorry, was there a question? Okay, so what hasn't been done is the physics side of this. So transimons led to a lot of sort of amazing computations and predictions, things like geometric quantization to produce Hilbert spaces and consavage integrals and so on. And the transimons part of this non-semi-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-semi-simple aspects, I should say, have appeared. Triplet algebras have started appearing in supersymmetric theories in a paper on well, on 3D modularity, maybe, by by Chang-Chen Ferreri-Gerka Harrison. So logarithmic things have started appearing, in particular in a supersymmetric context. And supergroup transimons theories have also been investigated and I would, using sort of physics language, I would say that they fall into the same universality class. They, supergroup transimons 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. 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. And 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 ranked dual of the fake and tipunin algebra. So they're actually two vertex algebras that appear that have equivalent categories of modules. And so we have a, what I would call a physics proof that the vertex algebra categories are the categories of lines in the QFT. I will talk about for the case of, certainly, sorry, for the case of SL2, I think for general type A, if to make the 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 Kritzig 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. Right. 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 have changed the page on my screen, but it's not coming through so I may need to re-marry just a sec. Oh, right, you guys can see that. Okay, let me just play again. It's good. Right, so the theory starts out as a 3D n equals 4 theory and it's mostly a 3D n equals 4 theory called t of g that can be analyzed using all of the lovely algebraic techniques that you're hearing about in other parts of this workshop. But it has a slight twist. 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 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 extend. Sorry, and I see Sasha has a question in the chat since 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 this. Yeah, and all global forms. So no, there are subtleties when... I would expect it to work for A to E type and there are there are huge technical things that happen later where some Langland 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 sort of clear from each perspective why the story is not so simple. Matching up the various subtleties is hard and is something we realize we should not attempt in this first shot at this project. So one can write down a sort of a Lagrangian, an action for this theory in Taipei using a twisted BV formalism, which is kind of nice. So in that sense, it's very much like Tern Simons. It's a concrete theory with fields and using that Lagrangian one can write down a boundary vertex algebra, the analog of WZW. And there are in fact two of them that show up and in WZW there's also another algebra that you could use which is the level rank dual of WZW that has exactly the same category of modules. And here as well there are 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, 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 the natural question is like what you gain from the physics and what can you compute to check that this guess is even correct. So it's easy to compute using supersymmetric localization techniques, the growth and degroup of the category of line operators, and with our characters of Hilbert spaces, which are said this, there's this D bar sitting everywhere, sorry not D bar, there's a DB 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 VAs are 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. Hilbert spaces associated to surfaces are going to exist in many cohomological degrees and their Euler characters are easy things to compute as partition functions. Harder things to compute are things like the category itself from a QFT perspective or the Hilbert space itself in a math and class group action. I'll indicate some techniques to try to go about that, but this is where like fully generalizing algebraic approaches to 3D n equals 4 theories things like the BFN construction would be really useful. Okay, that's the end of the long introduction. You know if there are any questions to wrap me at any time as well. This next 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. 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 at any root of unity, but even is the relevant situation for me, there's in addition to the center that comes from from Kazimierz operators that sort of the Harris Chandra center, there's an extra bit of the center that just comes from kth powers of EF and big K. These extra central elements act as constants on any indecomposable representations and so the representation theory sort of decomposes into what well into 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 parametrizes an open cell in the group PGL2. So, the reason PGL2 is coming up here is because it's the Langland's dual of SL2 and at an even root of unity, the fancy way to say this is that the category of modules fibers over PGL2 at an odd root of unity at fibers over SL2. 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. A really simple example of this is that when so if you take a diagonal element of PGL2 that has some eigenvalue e to the alpha here, one would associate that to a subcategory of modules on which e to the k and f to the k act as zero, coming from these off diagonal zeros and k to the 2k acts as e to the alpha. Hope that wasn't too confusing a description. I should also say the relevant thing about these different fibers or different stocks of the sheaf is that there are no hams between them, so they're each full subcategories, the total category is a direct sum of all of these stocks. So it was observed, it was proven a long time ago by Dick and Cheney and Katz and the even route of unity case by Beck that what the stock of fiber looks like above a particular element of PGL2, how the center acts, only depends on the conjugacy class of this element in PGL2 up to isomorphism. And then what was realized a little later on by initially by Khashayl and Reshetikin is that this sort of vibring of the category over PGL2 it should lead to not just two invariance of links and three manifolds, but more generally to invariance of links and three in three manifolds with flat connections in the complements of the links. So this was roughly but not entirely correct and it's morally correct and it's been made very precise in the case of links, links in S3 in a recent paper by Blanche Gehr, Patra Mirand and Reshetikin. So they build an invariance of three manifolds, sorry, invariance of links in S3 with flat PGL2 connections in the complement of the link and there's work in progress trying to really promote that to a full TQFT that gives invariance of three manifolds with flat PGL2 connections. Picture that I always have in mind when trying to understand why this should be true is the following. So the category has all these different blocks or stocks labeled by elements of PGL2. One should, if trying to translate this to some sort of physics picture or a 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 the object from labeled by some element G depends on the holinomy of your background flat connection, the flat connection that you've enriched the three manifold with in a small loop going around this line. And if you just have a single line, it's only the conjugacy class of the holinomy should matter, which is consistent with the old theorem of decontrini cats and perchesi that a fiber of this category only 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 holinomies 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 holinomies. And it's literally true when all of your holinomies are diagonal, and it is almost true when the holinomies are general non-abelian things, and the almost was described precisely in this initially by Kashyav and Reshetikin and in this later paper by Banshe Geir, Pater Amiram and Reshetikin. I'll just focus on the Abelian case here. So this sort of setup where we're looking at three manifolds decorated by flat connections appears in physics when we have a quantum field theory with a global symmetry, as opposed to a gauged symmetry. So quantum field theories with global symmetries can couple to connections that are not fields that one integrates over in the path integral, 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. Line operators in the background of a holinomie defect like this were described really nicely a few years ago in a paper Victor Mikhailov's in the context of super symmetric transiments, but the story is sorry the context of supergroup transiments, but the story is very similar here. Okay, now 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 PGL2, then we look at representations of UQSL2 on which k to the 2k acts is that eigenvalue and e to the k and after the k act is zero. The category ends up being semi-simple with exactly 2k semi-simple objects that look sort of like standard highest weight modules, except the weights are eigenvalues of k rather than of h and so they're q to the something. They all have dimension k and they wrap around the unit circle, sort of the the the weights wrap 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 ADO invariant uses precisely these sorts of representations and Murakami also worked on that in the 90s and then Konstantino Gir and Patero Muran sort of systematized the data you would need to not get all of your link invariants to be zero. Otherwise vanishing quantum dimensions would naively tell you that even the unknot has expectation value of zero. These sort of generic stocks of the category have an extremely simple tensor product that that sort of looks like what you would expect for an in a Boolean theory. It's like for a quantum gl1. 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 torus. You just you can generate states by filling the torus into a solid torus 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 torus. We should say the Hilbert space of the torus together with a flat PGL2 connection on it and putting an Abelian connection on there with a specified holonomy around the Baridian of the torus tells us what piece of the category to choose objects from on the core. And so we just take the 2k different objects in that piece of the category along the core that gives us 2k states in that Hilbert space. I also 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 the sorts of TTFTs 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. So I should really just say vector spaces of states. In genus G, one can use the very simple monodal structure to also like draw all trivalent networks of I keep wanting to say line operators inside the core of a handled body whose boundary is a particular genus G surface and count what the options are. And one gets 2 to the G k to the 3G minus 3 different states. That's a very easy combinatorics problem. Okay, so that's sort of the generic setting 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 connection is trivial. So the connection itself, which is zero everywhere, it's whole enemies everywhere. And so in this case, looking at what sorts of what do we label a line by in the presence of trivial holonomy, the answer is representations of what is called the small or restricted quantum group. So it's modules on which k to the k, k to the 2k axis 1 and e to the k and f to the k axis 0. So this is an extremely well studied category in representation theory. It is not semi-simple. It has 2 to the k simple objects that can form interesting extensions with one another. The 2 to the k simple objects roughly look like two copies of ordinary representations of SO2 of dimensions one, two, three, all the way up to k. And they, taking the projective covers of the simple objects when gets projected that have sort of diamond structures and movie diagrams. And they're, so there are 2k in the composable projectives in this piece of the category as well. The semi-simplification that was used by Reshtekin and Turayev uses a single copy of the, of what look like ordinary SO2 representations of dimensions one up to k minus one. And so if one sort of quotients out by everything else, setting it to zero, another way of saying quotient out by everything else is set to zero, everything that has a vanishing quantum dimension. One gets a semi-simplified category that leads to the old story. The Kashayev 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 Bushenko considered when, when starting to define non-semi-simple TQFTs in the 90s. Okay. So I mentioned at the beginning that in general, our spaces of states on surfaces would be cohomological beasts. They, they have multiple cohomological degrees. In terms of the category of lines mathematically, one would calculate the Hilbert space of states in general by taking Hochschild homology of the appropriate piece of the category, depending on what flat connection we've chosen. And so on a torus with zero flat connection, we have to use the small quantum group category. And we should find it's Hilbert space is Hochschild homology of the small quantum group category. Quick physics word. 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 let consider junctions of multiple lines. And just simply considering junctions of your objects will give you H8 zero, where the, we'll give you the zero, zero with Hochschild homology. Higher degrees in Hochschild homology, in Hochschild homology come from integrating descendants of junctions around paths in this core. That is a statement for any one of the audience who knows about descendants of this. Anyway, this is a totally, totally sensible thing to do both physically and mathematically. In order to compute Hochschild homology of this category, it's useful to have a geometric description of it. And Archipath, Bezerkavnikov in Ginsberg, and then Bezerkavnikov from Lekovskaya gave such a geometric description, which at an even rate of unity amounts to saying that as a category, not a monotone category, but just as a category, the derives category of representations of the small quantum group has sort of two semi-simple pieces to what that's just coming from the two symbols that were also projected. And then a bunch of copies of the derived category of coherent sheets on T-star of the flag manifold. K minus one copies of that. That gives us a geometric way to compute Hochschild homology, which ends up looking like, I want to say total double cohomology. Looks like total cohomology of these of T-star flags computed in an algebraic way. And then there's an answer. The relevant space of states becomes infinite dimensional with non-negative cohomological degrees, and it's finite in each cohomological degree. And the parts that Nibushenko would have used back in the 90s, oh, I think I say that on the next slide. And the part that Nibushenko would have used is just H80, but there ought to be a derived generalization of all of the earlier work and also the current work of CGP and collaborators. Okay. I've written it here. I've sort of written what the different ways look like in terms of representations of the symmetry group in PGL2 that acts on T-star flag. There are also nice things to say about deforming from trivial connection to non-zero generic connection. There we go. Which, again, I think I need to... That's the slide I actually wanted to be on. I'll just see this very briefly, but to go... So at generic connection, the space of states had dimension 2k. At zero connection, the space of states is this infinite dimensional thing with finite 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 2k. The Euler character is invariant under this deformation. The Euler 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 just... Well, this is a sheet that go here in chief of categories and very close to the identity element in PGL2, coherent sheaves gets deformed to matrix factorizations with a super potential that involves the complex moment map for a certain element of PGL2 that's defined by the stock we're looking at. And I said that Euler characters don't change under this deformation. Okay, 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. So you can probably tell. Okay, so now I want to take all of that information and translate it quickly to physics and then try to explain what one still has to do on the physics side. So we're looking for a quantum joke theory that is labeled by an algebra and a level and an integer k. It should be trans-simon's-like in that it sort of looks very close to the Reshti-Kenturiov category. It should have some finite... In each piece of the category, there's a finite number of objects. It should have something that looks sort of looks like close in lines. And in this previous paper on recursion relations for ADO invariance, we found structure that was very, very, very similar to the semi-simple story involved. So ADO invariance are different from color jumps polynomials, but they obey the same recursion relations. And so there are a lot of things that sort of look the same. So you should see something trans-simon's-like here. However, the line operators in this theory should have non-trivial junctions. We should not be getting a semi-simple category. A very easy signal that a quantum field theory gives rise to a non-simple category of lines is that there are non-trivial local operators. This is the sort of thing that the BFN construction computes. So BFN 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 trans-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. 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 those properties is to start with a 3D n equals 4 super symmetric theory that Gaiotu and Witton introduced called t of g. Even though I write a group here, it secretly only depends on a Lie algebra. And roughly speaking, it has symmetry G times the langauge dual of G. Sorry, is this literally true? I thought it only had G symmetry and also it's mere dual had G check symmetry. Is that? No, I mean, it has both. It's one act on the Higgs branch and one act on the Coulomb branch. But the... I thought what acts on a Coulomb branch is not a symmetry of the theory. No, of course it is. No, no, no, it's exactly on the same footing. If you write down a Lagrangian for this, for the theory in type A, all you see is the maximal torres. But otherwise, it's... The way you write down the theory in terms of fields and what you call the Higgs branch and what you call the Coulomb branch is arbitrary at the level of quantum field theory. And so it really does... It really does have both of these symmetries. Well, I see. So, I mean, before you twisted... Yes, yes, yes, exactly. Okay, sure. And the act in different ways. 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 they'll act in very, very different ways. Okay, yeah, okay, now I see what you mean now. Yeah, okay. So then we take G and gauge it further. But this gauging is different from the standard 3D, 3D n equals 4 gauging. It's something one can do in theories with less supersymmetry. And so in physics terms, I would say using n equals 2 vector multiplied. That'll... So this less supersymmetric gauging allows the introduction of a non-trivial determinants level. And so one sort of writes down... If you're not worried about supersymmetry, you write down the term Simon's Lagrangian for G and couple it to the rest of this theory, T of G. And the result of this gauging, does it still have some supersymmetry left? It does, but this is subtle. So in physics terms, if you write down a Lagrangian and you try to do this in terms of fields, you can only see n equals 2 supersymmetry. In the infrared, or with a suitable twist, it should have full n equals 4. But seeing that is true before getting to topological twists. 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 2 or n equals 4 supersymmetry, but has the single supertracks you need to topologically twist to perform what would otherwise be the A twist. So physically in the infrared, this thing regains full on n equals 4 supersymmetry, and you can topologically twist, but there are subtleties around. If you believe that this has n equals 4 supersymmetry, there are two twists available. One that's focused on the Coulomb branch, and one that's focused on the Higgs branch, or an A twist and a 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 the A twist in this case, and it's the one that's sort of focused on the Coulomb branch. The B twist of the same theory was something that Kapustin and Salina studied 10 years ago, and something they called Transimon's-Rozanski-Wittgen theory. So the B twist is the Rosanski-Wittgen twist, but it is completely different theory. It behaves very differently, and it's not the thing that's relevant for quantum groups that are rooted in it. So the other thing to say is that this is a Quiver-Gauge theory. When the group is type A, when we're looking at SUM, 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 T-star flag. However, in T-star flag, there's a framing node that has rank N for GLN or for SUM. That final framing node is the thing that's daged with an extra Transimon's level. And so the gauge group of the theory is a product of GL1 through not GL1 minus 1, but all the way through GLN with a Transimon's level for GLN. Before this extra Transimon's gauging, both the Higgs and the Coulomb branch look like T-star of the flag manifold. The extra gauging destroys the Higgs branch, and it actually seems to do nothing at all to the Coulomb branch, except to, well, saying T-star flag is maybe incorrect. I should rather say it's the rather and it seems to, this extra Transimon's gauging seems to introduce some extra singularities at the origin of the node. Which I can't discuss any more precisely than that. What do you mean when I say it destroys the Higgs branch? That's right. So you would, I mean, sorry, the thing that it does is it takes a scalar quotient of the Higgs branch, not a hyper scalar quotient, but just an ordinary, so if there weren't a Transimon's level around, you would expect the Higgs branch to just be quotient by GLN. Because we have less of a proximity to the Higgs branch, it's no longer calomorphic symplectic or hyper-galvanic. So you would just take the one that equals exactly. The maximal torus of the flat, the maximal torus in this case, it's PGL2, or PGLN, shows up in terms of resolution, complexified resolution parameters for the Coulomb branch. Sorry, they're deformation parameters for the Coulomb branch, or yeah. I'm short on time, so I think that's all I'll say. I think there's been a lot about 3D unequals four theories in this conference. And the Wilson line operators that we wanted to see show up for the Transimon's factor. It is surprising in general that one would have Wilson lines in this A-twist that is focused on the Coulomb branch. It can happen here, precisely because of the Transimon's level. It would not happen on K-0. Okay, so then there's a black box. So this, I claim, is a theory one can write down a Lagrangian for in the PV formalism. There are lots of localization techniques that apply to that, or if you just think of it as n equals two supersymmetric theory, they're localization techniques that will compute partition functions and expectations of Wilson line operators in this theory, starting with work of Necrosophon Chateau-Schvilly and recent work on untwisted indices. This, so in like half a day, one can code up a computation that spits out Euler characters for spaces of states and remind you Euler characters don't care about what flat connection is turned on. And one on the nose gets the right answer for SU2 and we've checked, I think for SU3 and SU4 as well. Also using the beta root analysis of Necrosophon Chateau-Schvilly, that gives you expectation values of line operators, one can get the growth and dig ring of the small quantum group. 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, one should try to apply, we're getting there, a lot of the more modern methods that have been developed in the last five years or so in, including by BFN and Webster and work of mine with Gaya Tobolomar Hilburn and company and lots of people and recent work of Matt Bollemore and co-authors in writing down Hilbert spaces. So there are algebraic techniques one can use. One has to sort of adapt the current techniques to this sort of hybrid case that is mostly 3D n equals 4 in the A-twist with a bit of 3D n equals 2 with a trans-simon level. The sort of thing one expects for the category of line operators looks as follows. So if, and I want to compare on this slide what happens in ordinary trans-simon theory to the quantum field theories I'm discussing. So in trans-simon theory, you can write ordinary trans-simon as a 3D n equals 2 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 of course is representations of 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, equivalent coherent sheaves on a deformation of the cotangent bundle of the loop space of the Higgs branch of this quiver. And if I didn't say loop group, equivalent, and I did say deformation, it's the deformation that deforms coherent sheaves to D modules. And D 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 a 3D n equals 4 theory should be. And one needs to somehow combine that D module deformation with a loop group, equivalence. And maybe there's just an obvious way to do that. And mathematically, 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 algebras, which I'll get to in the last minus one minutes of the talk. So one of the vertex algebras 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 tensored with a very complicated sheath that, if you don't turn any flat connection on, is infinite rank. But finite rank in each homological degree. And it's a sheath that initially Gaiotto described for the TMG theories. And again, 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 genus zero. So for S2, one is supposed to get local operators in the TQFT. And one gets one dimensional space associated to S2 in transimans 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 quantum group representations I mentioned earlier as well. And it's functions on the Cullen branch, which is why I said the Cullen branch as a variety has not changed. But extra stackiness is involved. For other, for higher genus surfaces, this sort of mixed sheath description suggests that you should get something roughly looking like the transimans Hilbert space times the factor of Dalboca homology of T-star flag. And that is roughly what we got by computing Hock-Schul homology of the small quantum group representation category. It was almost of this form with a few extra factors. So at least things looked reasonable, 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 some 40 construction of this quantum field theory that I mentioned that is closely related to geometric Langlands and for the s duality with different boundary conditions that I can answer questions about at the end, if anyone else is interested. One side of this 40 setup seems to be closely related to that work of Gukath at all from last year that I mentioned in the introduction. And finally, there are vertex algebras involved. And the vertex algebras come from either putting boundary conditions on the 3D quantum field theory, or by working in this 4D construction and considering not just the sandwich of boundary conditions, but a corner slicing this with yet a third boundary condition and using work on vertex algebras at the corner that Sasha must have talked about yesterday. Anyway, one can extract vertex algebras from these brain constructions and from the field theory and when there are actually two natural ways to do it. And I mentioned level rank duality before. So in in the classical assignment 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 new story, there are also two algebras that play a role. And one is the Fagin-Tipunin 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-equivariant d-modules on. Okay, so the Fagin-Tipunin side 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 deformed by flat connections. And this new vertex algebra, I'll say it well, say it very abstractly, is you can sort of read it off from the quiver of the theory. One puts together some number of copies of a beta-gamma system for each edge in the quiver and then takes a BRST quotient by gl1 up to glm for the nodes in the quiver. And then a final not BRST quotient, but just derived in variance for a final copy of SLN at level k. So that's the new vertex algebra. And Thomas Freitzig showed that this was actually dual to Fagin-Tipunin for SL2. Okay, but that's, sorry, horribly out of time. So that's all I'll say. There's a really amazing vertex algebra story here. And Thomas could give a one plus hour talk on the vertex algebra side of this. Okay. Just to finish up, let me say, so I've indicated that it would be really awesome to sort of generalize the BFM like and other, like the modules on the loop space constructions for line operators, like Hilburn and you and myself and others have discussed to, and Webster, to generalize that to include transignment levels that should land in this case on the small quantum group module category. If our conjectures are all correct, it would be great to implement this geometric quantization perspective to start looking at Hilburn 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 compute. And there's been work for the Taurus looking at, I think this is Laplace-Kang and Key, looking at the action of the modular group on Hochschild homology of that small quantum group category, but I don't have anything in higher genus. And the whole thing should fit together beautifully in some derived version of this CGP-like TQFT. Okay. Thanks. So, Yana has a question. Now I'm able to let him speak. Yeah. Yana, can you? Yeah. I do that. So, I have a question through the technical first. If I, can I analytically your generalized Sharon Simons theory with respect to the level and anyhow different from ordinary Sharon Simons? So, yes, the answer is certainly yes in some cases. And so, like Sergey Gukhov and co-authors already described the analytic continuation of the ADO invariance. The analytic continuation uses a generalization of this 4D setup that I wrote down with different boundary conditions. So, it's in some ways easier to discuss the analytic continuation than what happens at integer level. And it's very, very similar to what Witten did with ordinary Sharon Simons. So, like the different choices there are all about choosing boundary conditions carefully in this 4D story. But one needs to move to a half space. Yeah. I got it. And the second question you mentioned, this work of Gaiota and Robchak on the vertex algebra in the corner. Yes. I understand that it's kind of geometry, it's sort of a certain derived category of coherent shifts on, say, 3dime on C cube supported on some fat divisors given by coordinate planes, which kind of define you this corner. So, this is a simple case of that. Yeah, okay. But just look at this simple case. So, how do you see this geometry and do you see it at all? Yes. So, see there, let me see if I can actually draw on here. So, the thing you're talking about involves sort of the full vertex corner. And here we're looking at n0 and 0 brains. And so, 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 transimmons level. So, what I drew is 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-Takunin. One starts off with a corner that after the simplification spits out a W algebra. And so, the general corner is these Y and N algebras that are massive generalizations of W algebras. In your story, you see only n0 and 0? Exactly. But, of course, it would be beautiful to then start generalizing this and relate to other quantum groups. But, yeah. So, my story is a very, very small part of that. And it's colliding two corners. And they both have n0, 0, but with slightly different decorations on them. And so, one gives rise to a W algebra and the other gives rise to a Tkatsmoody, rather to WZW. And so, colliding those and taking an infinite level limit gives Fagan-Takunin. Okay. Yeah. All right. Thank you. Any other questions? I have a question. Can I maybe, I mean, summarize what you did in the end? I just don't understand if I understood it correctly. So, you can see this. So, take a steer of u, g, and then instead of gauge it in a strong Simon sense, and you get some n equal to simple symmetric theory, which still has a and b twist and tiggs and Coulomb branch. But, okay. That's it now. It's kind of magical that it still has a and b twists. Magical in what sense? So, you think, I mean, is it, so, I mean, the general, general theories? They only have a holomorphic twist. So, they have a holomorphic topological twist. And so, 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. I see. So, it's not something general. It's just kind of specific. No, it's very special. Okay. Yes. Now, what is the statement? Sorry, this is probably the main point, one of the main points of your talk, which I missed. What is the state about the relationship between this theory and representation of the quantum group? I mean. So, the A twist of this theory has still has a symmetry and it's the symmetry that acts on the Coulomb branch. Okay. And the Coulomb branch is still the no-potent cone. So, the symmetry is the Langland-Stewell group. Okay. So, one can, if one wants, turn on a background flat connection for that symmetry. So, to work on 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 should be the derived category of representations of the small quantum group. So, the category of line operators is for the A twist. For the A twist. Okay. Yeah. And this is something that you sort of can more or less prove or I mean. Yes. But the only way we can actually prove it is by introducing a holomorphic boundary condition that supports a vertex algebra. And assuming that the category of lines is the same as modules for that vertex algebra, which is what you would expect for a sufficiently rich boundary condition. And then we take that vertex algebra, prove that it's dual to fake and to put in, which is known to be the same as modules for the small quantum group. I see. I'm wondering, for the small quantum group, the categorization of small quantum group, it has this description by sort of Bezerkaunig of Finkelberg and Schechmann in terms of factorizable sheaves. And I'm wondering, I have some feeling that it should be relevant for what you're doing. I have exactly the same feeling. And I don't understand that work well enough, but that story also gives derived, like in principle should give derived spaces of conformal blocks. Yes. For these. Exactly. Yeah. So I think that should be very highly relevant for all of this. I guess I also think that gatesquare has been doing along the same lines. Maybe one small question. You mentioned Schrodinger's Theorem for supergroups. I mean, you didn't 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 supergroups. And as you know, in this world of like 40 young mills with different boundary conditions and interfaces, one can engineer things that look like supergroup trans diamonds as well. And so there should be generalizations of this that involve supergroups. And part of the supergroup story is also like an undrived version of the supergroup story has been discussed. And it's like Rosenski and Soler wrote that from Simon's theory down and I guess in the simplest case for GL11 and started talking about what GL11 WCW should look like and found non-symmetrical categories. I think if one approaches this from the super symmetric side, one would end up in the derived category of everything in sight. Yeah, I think the supergroup story is very close. Okay, I think we can thank you again.