Gauge Theory, Geometric Langlands, and All That - Edward Witten

So I'll try to give an introduction or overview on

gauge

theory

and

geometric

languages, but I don't want to give the same talk I would have given a decade ago, so it can't be systemic. It would have been like that too. I hope the audience is very varied in their point of view and what they already know about it, so I'll give you a sort of overview, but I hope everyone gets something out of this to get the story started. in six dimensions in six dimensions for each type of group simply from Leslie ADRA and here we specify only the Dinkin diagram, not the group, there is a six-dimensional super conformal field

theory

, well, the conformal group in six dimensions with the comma so2 six so2 six would be the group of conformal notions of the six-dimensional Minkowski space, so here we're going to have a supergroup that contains so2 six and actually okay, it's a kind of ortho symplectic supergroup io sp2 6/4, well , I will call it sa Watoto and the tilde refer to try allottee so so2 six is ​​a version of a complex Li group with the ax form of Sol it is a different form from the real group so8 and you must remember that so8 has t test so here in the physical realization it is extremely important that SOA has testability, so this is a version of you that might be less unknown if you wrote OSP 8/4, which would be part of an infinite family of symplectic supergroups, but it is important here that we can do a test transformation on the factor so8 so that the odd generators become estimate generators.

The reason I mention this triple assignment is simply to help you understand that six dimensions will be the maximum for superconformal field theory because I have to use the last of the exceptional isomorphisms between lis groups, so this is kind of the end. line for superconformal field theory and is the starting point from which one understands gate theory and

geometric

Langlands. Well, everything I've said so far. is that the starting point for

gauge

theory and geometric languages ​​is this exotic six-dimensional superconformal field theory and it has two properties that would classically be incompatible. The first is that if you have a billion eyes you cannot not measure gauge fields. but germs, which means for physics that you have to form connections, you get gauge fields which are two shapes instead of one of the maximum torus and the second is that if you reduce in a circle you get gauge fields of G, which is the group of type a D or E, then G here is any simple Li group of type a D or E, if you reduce in a circle, you will see the non-abelian gauge theory of G, but if you have abelian eyes, then you will see that fields of maximum torus caliber what do abelian eyes mean for physicists? means moving in the Coulomb branch.

I understand that in Langlands geometric mathematics there is an analogous operation of considering the support on a semi-simple element on a regular semi-simple element, so I assume that one of those two statements made sense to you, the six-dimensional theory it can be a billion eyes and when you have a billion eyes you go to form connections instead of ordinary gauge theory, but instead if you reduce on a circle you get gauge fields from a simple non-abelian read. group, if you're following me, these two facts should sound hopelessly incompatible because there are two ways to simplify the theory, either by enabling or reducing one dimension that leads in one case to two to form connections in the other case to one. form connections of a non-Abelian group, so a classical physicist or mathematician trying to reconcile these facts would say that if you don't simplify it, you must have had to form fields of the non-Abelian group, but that is precisely what does not exist classically, so that if you are a physicist you know that there is no Yang-mills theory for forming gauge fields or at least if you don't recognize that fact you should try to convince yourself later and if you are a mathematician you probably know that h1 has a version non-abelian which has to do with vector bundles, but h2 does not have an analogous non-abelian version, but this theory descends from the myths of the quantum world in one of two ways in which you can simplify it and understand it better. in a way that from a classical point of view would have implied that it started life with what does not exist, the non-Abelian theory of grass or to form a gauge field, so that is the underlying mystery, if you will, of the superconformal six dimensions. and field theory is what leads to geometric Langlands and all kinds of other crazy things, so to continue to be more specific, we take our sixth variety as a variety five multiplied by a circle of radius R or circumference two PI R and then The action effect we get in five dimensions, well, is what you would expect from Yang-mills theory, where f is the curvature and the usual action is the integral of the square of the curvature, except for one completely crazy fact of that instead of the action it is proportional to R as it would be in classical dimensional reduction where it would be integrated over the fiber and a factor of the volume of the fiber would be obtained.

Here a factor of the inverse of the fiber volume is obtained. Now the fact that the inverse volume is obtained is to complete the elementary fact that follows from conformal immigrants, the six-dimensional theory was conformally invariant, so if we rescale all the lengths and the six dimensions we will leave this effective action invariant , the rescaling lengths would be multiplied by or by constant but would also be multiplied by five. the lengths of the dimensions are constant and because F squared has dimension 4 but they were in five dimensions, remember that a curvature has a derivative, the connection naturally scales as a derivative, so f scales as a second derivative, so which here, in fact, we have four derivatives but five integrals, so this scales as lengths, so you have to multiply the inverse length, so the scaling of the effect of the action with inverse length simply follows from the conformal invariance of six dimensions, but it also shows that this does not come from a classical integration operation on the fiber, it comes from some type of quantum duality and going from six to five dimensions is that quantum duality by which what really did not exist classically , the non-Abelian Gerb theory, manages to materialize in five dimensions as a concrete non-Abelian gauge theory, now there is a variant that I want.

I briefly mention step two, which is a twisted reduction on a circle. I'm going to mention that just because I want to explain how Li groups that aren't just intertwined and then if G isn't like that in six dimensions, we just had one ad. The classification, so what I've described so far will give us the gauge theory of ad groups in five dimensions, gauge three or just placing strips, how are we going to move on to a nice and not simple script? Well, non-simple scripts relate to pairs once We're at the point where G is simply entangled and I'm just calling star a fall morphism of G but, for example, the group G two with the related one is a turn eight and get a taste of reality here, you can take a fall or physicists are your bucket. 1 and G 2 is a fixed point sets us up as an 8 under that operation, so there are certain pairs that consist of a simply lace group and a fall orphism and if for such pairs this six-dimensional theory actually has an autumn orphism and, therefore, when I compact five in a circle there is a twist in the automorphism and when you do the twisted compactification you obtain in five dimensions the mod gauge theory of the H that would classically be associated with the pair G in the automorphism but Langlands gauge theory GN o' Doole group, for example G 2 is self-dual, so if you want to get gauge theory G 2 in five dimensions, you would start with D four in six dimensions and on a circle would do a twisted compaction by purchasing the water to transform this automorphism. of order three, so that's the cast of characters in five, that's how it's measured in five dimensions, those are the basic constructs.

Now we want to discuss topological field theories where I put the words that I am going to abbreviate as TQ FTS. topological cornfield these, but the first T will be in quotes because our theories of topological fields or not do not really satisfy all the axioms of topological field theory, they fail due to a lack of compactness, so there is no time to discuss it . It's not really the time to explain it properly, but none of the pseudotopological field theories that we construct truly obey all the properties of a topological field theory because at certain crucial stages the lack of compactness will cause difficulties, so we are also going to take just a kind of shortcut to vaguely explain how they are constructed in n dimensions, so we have this or SP 2 point 6/4 there is a subgroup of the ik boson itself this factor of SP 4 which is the same as spin 5 on the other of the exceptional or dwarf automorphisms of the lis groups in dimension n.

This is what physicists call the r symmetry group in the n dimension. You construct these quasi-topological field theories using the homomorphisms from the interval n to the symmetry group r which here is spin 5, so such a morphism should have some properties that we will not try to solve today but for example the trivial one will not be good, so in dimension 6 we would need a homomorphism from turn 6 to turn 5, there are none interesting, so we are out of luck in dimension 5, there is one if we now make a turn 5 to turn 5 and the homomorphism of identity works well, so we have some good luck there and there is a construct in dimension 4 that we have that we are trying to map. turn 4 to generate a 5, but we have another exceptional automorphism: turn 4 is su 2 x su.

Well, first of all, we could do the same thing we did before incorporating turn 4 and turn 5 in the obvious way, just using the fact that 5 is four plus a turn four as a subgroup of turn five, there are a couple of more things we can do because this exceptional isomorphism gives us more freedom, so there are more things we can do and there are actually three options that lead to almost topological field things and are respectively associated with geometric Langlands, which people sometimes call Witten waffle theory and - a version of cybernetic written theory with a contiguous type of multiple also sometimes called any Crump, we provide it to our people called n equals two stars, so for today We are interested in Ralphs leading to geometric Langlands, but the same exceptional isomorphism that allowed two more theories to exist in four dimensions leads to this theory having additional properties.

We actually got a family of theories from a complex parameter parameterized by the couple I was in. and I called the duality sigh and Langlands geometric, so I think I should first say why there is a non-trivial duality. Remember that we went from six dimensions to five dimensions. Something concrete materialized in the Fog, but R was in the denominator and tells us that. was a very non-classical operation, now we are going to compact the value on a second circle, so we will compact on the two tori a product of two circles and, to simplify things, we could think of it as an orthogonal product of two circles of circumference two pi 1 over R, but now that we understand what we are doing, since we have a classical action, the second step will simply be the classical integration over the fiber, so now the length will appear in the numerator, when we reduce to the 4 effective dimensions . action we are going to get s over R multiplied by D for was proportional to R, so the joint operation is completely symmetric under SNR.

It is up to us to do it in two stages, for the first stage we could not understand classically, in the second stage we could classically understand the first. The stage gave one of those derived from conformal environments and the difficulty of explaining that set classically is the underlying difficulty in understanding geometric lands, but the second step was classical and gave us simply a factor of s, so we concluded that the underlying symmetry between the SNRs, well, I guess I should remind you that in committed theory you usually write one about the coupling constant squared, but you see one about what behaves like coupling constants, so it's obvious that The underlying symmetry of the SNR exchange means that the four-dimensional theory is invariant to reversing the coupling constant, so this is an operation that exchanges weak and strong coupling,so a is the semiclassical region where classical physics is a good approximation, you can understand that. large e is the opposite strong coupling where dragons live, but this theory will have a symmetry between weak and strong coupling.

Now I have explained it for the case of a parallelogram, sorry, a rectangle, but more generally, we could compact into a Riemann surface of genus one with any plane metric corresponding to a parallelogram in the plane, so you could think of This is as if you take in the plane a parallelogram with corners 0 1 tau and 1 plus tau where tau is any point in the upper half plane and then the S code or asymmetry becomes minus 1 over tau, but in a more general way. One could argue that what we fundamentally have here is a rank two network in the plane, our genus one surface being the quotient of the complex plane times the rank two network generated by an Intel, but we could have chosen any other basis for the same network and that would have had the effect of the towel going up to eight L plus B over C talapus D and in the Taliban gauge theory it is interpreted if you are a little more precise than me with the formulas, the gauge has a coupling constant e that I wrote in the book and didn't normalize it very carefully.

It also has a theta angle that multiplies a second term in action and the precise statement about duality is the genus one Freeman SL a z symmetry surface. the surface acts on the towels Tala goes to a city plus B over C tail plus D where ABC and D or ENSO a Z, so this description is for simply interlocked li grips, remember that if the Li group is not simply interlocked , then in the first step he made a spin, so in the uncivilized group we don't just have a case, we don't just have a Riemann surface of genus one, we have a Riemann surface of genus one with a spin around a cycle where the turn was due to an exit or automorphism. that is of order two or three, for example, if we are doing g with the external automorphism, it was of order three, so in the non-police case we have a surface of genus one that is endowed with a v-bundle of order two or three and you could define that. cell a z but it will promote the different flat bundles if you want the symmetry of the particular construction you have to look at the subgroup of SLE a z but it fixes that the debates are congruence conditions so it lets fix that flat bundle so what I I would like to emphasize that this SL 2z is a symmetry of the entire physical theory, not just twisted quasi-topological field theories, such as that entering the geometric Langlands.

Well, then what is called classical geometric Langlands? A study is the case of that size. is infinite or 0, they are exchanged. I must say that the sigh transforms in the same way that Tala does. Infinity and 0 interchange by side to minus 1 or sign, so that's the classic Langlands geometric duality and people in that field also studied what they call. the quantum geometric lands that correspond to the generics are fine and so far in the discussion we have compacted them into four dimensions, but the geometric lands are generally defined by further specialization, which is to be further compacted into a Riemann surface.

See, so you go from a to take its manifold 4. have the form of two manifolds multiplied by C and if you think that C is fixed, then you have an effective two-dimensional theory 1m2 now in two-dimensional topological field theory, okay , the basic things you consider are our boundary conditions, boundary conditions in two dimensions form a category and the geometric Langlands are the same thing. Develops the categories of boundary conditions that are obtained for C fixed in the zero or infinite cycle and, more generally, in the generic ones. If you want to study quantum geometric Langlands, you can now find some of the most interesting boundary conditions. be universally defined independent of C and Davide and I studied such boundary conditions about ten years ago and I would like to say a few words about boundary conditions that are universally defined.

Actually, I want to do it, but there is one more thing I want. Say, remember that in five dimensions we had one theory, in four dimensions we had a family parameterized by psy, so given this, you should ask if between any of the four theories There are no dimensions that can really rise to five dimensions, which is actually almost obvious there. It must be because whatever the five-dimensional theory is, if we reduce it into a circle, we will get a four-dimensional theory, so it will have to be in this family or we in one of the other families that I briefly mentioned, so in They are actually cycles.

Infinity is that which descends from five dimensions, which means that among the categories studied in geometric languages, the one that naturally rises to two categories, that is, that comes from five dimensions, is the one that is equal to infinity and essentially according to Kapusta's analysis. and selena that category is a warped version of rozanski whitten unfortunately took it is very difficult to make a brief statement that is limiting so perhaps you should resist the temptation to try towards the side equal to infinity the category is a certain category of coherent leaves and in general for complex symplectic varieties, koko-san and selena described that it rises to two categories that are classified in z2 and that d is assumed to be equal to 5, the five-dimensional elevation of cycles is an infinite case of geometric Langlands Anyway, where are we going? to go so let's discuss the boundary conditions oh okay yeah the only one that comes from a five dimensional theory so the only one that comes from a five dimensional topological thing in the same sense of course , the physical theory comes from six dimensions in six dimensions there was no topological twist in five dimensions there was a topological twist and it goes down to say it is equal to infinity in four dimensions okay, I want to talk to you about universal boundary conditions, but there is another component basic thing I wanted to talk to you about which is that we should discuss defects, so what a defect means in general is that for now at d equals 6 1 some subvariety our theory can be modified in an interesting way that depends on a choice of discrete data about that submanifold and the dimension of the manifold of the Sun and here are two classes of its two important classes of defects there are two-dimensional defects and there is Co dimension for the defense the two-dimensional defects are labeled by representations of G and the co dimension two defects are labeled by Generally speaking, no one and conjugation classes we will get to that later, so both play an important role in geometric Langlands and, although there will not be time for a proper explanation, I want to at least briefly mention all The two-dimensional defects before reaching the boundary conditions for codimension, two defects will be important, so for two-dimensional defects two-dimensional defects play at least two roles, so first of all, we had our six manifold that It was two bulls multiplied by four. manifold now one thing we can do, but what we are going to discuss is a two-dimensional defect, we have to choose two manifolds in six dimensions and there are different options, one option is to take a circle within the two times of the torus or a manifold I will say m1 inside m4 of course now there are different options because they are different topologically.

An equivalent way is to embed a circle in two varieties and if you remember that the basic electric magnetic duality swapped the two directions and this in the same basic case where the torus is rectangular the case we started with with the most obvious invented circles will be vertical u horizontals, will correspond to what is called Wilson and the tuft operators in gauge theory. I remember that basic duality swapped the two directions on the torus, so they will swap the Wilson operators and a wisp, we could do more general transformations from SL to Z and they will map those two options to other options, but these are the basic cases to have into account and then when we specify m2 by C, we can embed m1 and m2 multiplied by C in several ways, but one option that is important in geometric Langlands is to take a point P NC and simply embed m1 and m2 and since the geometric Langlands are deals with boundary categories of boundaries, we take em two as a Riemann surface with boundary, the discussion is only local along the boundary, so we don't care about the rest, the only thing we care about is that it has a boundary that is endowed with some boundary condition corresponding to an object in the category of boundary conditions and then we take m1 as a line parallel to the boundary that lives at a point on C that is not well drawn when m1 is parallel to the boundary and you see that from a topological point of view you could move em to the limit and the limit where they will form the parallel Wilson mother line or a hard line, another line operator parallel to it is just a compound boundary condition, so in this configuration we get what T is a language that I don't know, it is a function in the category of boundary conditions, so electrical and magnetic duality will map the functors in the category of boundary conditions that come from the Wilson lines to the corresponding functor of the comb telephone that would have blinds and that leads to what is called Becky's geometric operators, the Wilson operators are something that can be understood classically.

Enough copper atoms, as can be guessed from their rather subtle definition in quantum field theory, are subtler objects that are the most subtle part of the geometric Langlands correspondence, hence the fact that the Wilson operators are mapped onto the Hooft operators is one of the main statements about geometric Langlands. Now we specialized a lot in this construction to get to this story and obviously there are different ways we could have embedded two varieties in t2 per m4, but just for fun. to consider one more option, which is that I could take a point in t2 multiplied by a surface a2, which I better not call them; because I've used that name, I'll call it Sigma within m4, okay, and then in geometric Langlands, which specialized in the case. that m4 is m2 multiplied by C and again, there are different Sigma that we can use, but you could just take Sigma as m2 again multiplied by a point on C, so that's another thing we can do with the same codimension: default and if you do that, then as gucoff explored in me, that leads to if you: branching RAM and geometric Langlands, we branch around the point that we choose in C, so I think that's all I'll say about two-dimensional defects, but we left it. with codimension two defects what they are for so to explain what they are for I should get to what I promised I was going to discuss I promised we would discuss some boundary conditions that can be universally constructed independently of 1c compaction, so on geometric lands compacts on C and is considered the category of boundary conditions when you specialize m4 is equal to m2 multiplied by C, but certainly any boundary condition you can define for any m4 also makes sense in this situation, so if we want To understand categories of boundary conditions, we must first understand those that are universal in that sense, they do not depend on the choice of C and many important operations and geometric Langlands have to do with things that are universal in that sense, they do not depend on the choice of C.

We depend on C, so let's start in the six dimensions. Well, how would we build without knowing anything about a six-dimensional theory? How would we, however, construct a boundary condition in my theory that might be difficult, but if we are willing to reduce in a circle? I can do it, so I don't know what to tell you on equals six, so we'll make it easier and specialize in n five times a circle, that's an easier problem, so now I'm not thinking about a theory in six dimensions . Would you construct a boundary condition for that theory when you are specialized in six varieties of the shape and five times a circle that you can actually make because the circle is a limit, then you simply draw this image so close to its limit near a limit that m5 it looks like this? something n 4 times 1/2 line and the image I have drawn corresponds to a half line with the fiber being a circle, so if you take this to the manifold that I drew and multiplied by m4, I think it is the Cartesian product, then it is the first one.

After all, it's a six-dimensional manifold, so if you had a six-dimensional theory, you don't need any more information to define it to define the theory on that six-dimensional manifold, but secondly, from the point of view of the theory reduced in m5, it seems like a limit condition because away from the tip of the cigar we have a circular vibration over a range of five, but the reduced five-month-old child has a limit, so this image represents a limit in fivedimensions and is a limit that we can construct universally. I said that it was so basic that I am very sorry if I have confused you and if I have I don't know how to solve it, I could try to say the same thing again and four times our pus doesn't come out. the variety, of course, is a variety with a limit, no, but if there is a hidden circle that you don't know about when you are doing your theory o and M five, that means that now you are far from the limit, if it is not this interval, but the times of the interval.

A circle is a cylinder and while the half line can tend smoothly, the cylinder can, so any theory produced on a circle has a universal boundary condition. Now there are many more universal boundary conditions that we could do because, as I told you, the theory. has codimension two defects and if you have a theory with a codimension two for the defect, then you can make more boundary conditions in a universal way, i.e. the tip of the cigar is a codimension two manifold and by placing your defect on the tip of the cigar without spoiling any symmetry of the problem, you modify the image, so in any theory with a codimension of the defect, that means the compact violin circle, you can construct boundary conditions under a universal light, so this theory has many defects, so there are many universal boundary conditions in five dimensions.

I also told you that there was a twisted quasi-topological field theory in five dimensions, but this boundary condition doesn't actually make sense in that twisted theory, the reason it doesn't is that although this picture made sense in six dimensions , the Honami group no longer has a spin five, so it no longer reduces to spin 5 because this reduction to spin 5 was due to the circle being factored, but now it is not factored, so it is a physical four , a completely sensible boundary condition. in six-dimensional theory, but it doesn't make sense in the five-dimensional topological field theory that I told you about, however, we can reduce in another circle, so now we are going to, first of all, work on them. four times a product of two circles which is closer to where the geometric lands actually are, so let's discuss the universal boundary conditions in four dimensions and now you see, I'll make this drawing, it's a product of two circles, but the first one is the boundary of that cigar, so it is now a universal construction of boundary conditions in a Again in the four-dimensional theory, there could be a defect at the tip of the cigar and, by the way, if there was a twist around the circle , there are defects that match that twist, twist forces us to have a defect, but there is a family of defects compatible with twist, that's how you would get non-simple groups easily anyway, since there was more freedom in the construction of twisted topological field theories in four dimensions.

This image is compatible with geometric Langlands, that is, I is equal to infinity, is equal to infinity, assuming that the circle that is the limit is the first woman is the one we use to descend from the fog, therefore, for each default we get an object in geometric Langlands that exists universally independent of C independent of the surface eight of three months because we could define it without choosing a C so you can ask what are these objects in geometric Langlands if we go back and find a circle one each month that emerges to the surface.

Well, geometric Langlands, broadly speaking, is about how the fundamental group home or visits the dual boom group, the gauge group here is usually called the dual group, but there's actually something called Arthur SL 2, so that number theorists would say that you have a PI 1 x SL homomorphism; unfortunately, I'm telling you a sentence or two about something I essentially know nothing about, just to match the physics statement here, but there's actually a homomorphism from SL - Arthur x pi 1 to J and if we're going to do something let it be universal independent of let's say they made a much better map, I want to see - nothing because we don't know anything about C, so these will be home or isms that are trivial and say, but correspond to I'm a morphism of Arthur's SL , to dual grip, so all those possible rows arise for suitable defects, so the obvious one, which was the case of no defect, in other words, which I would have told you about if we hadn't known that defects have codimensions , we had precisely a universal boundary condition that actually corresponds to what is called a principle embedding, so if you don't know the main phrase the embedding for G is equal to SL n means that the image of SL 2 is irreducible in the end , the event acts irreducibly on the n-dimensional representation, so at one end you have the main embedding and then there are all sorts of other embeddings and at the other end is row equals 0, so row is equal to 0, ok, we are all equal to 0, it is actually the most obvious in gauge 3, so the row equals 0 corresponds to the boundary conditions for the gauge fields and, as explained in my I work with the deviator following a A long chain of physics articles, the others have to do with DRS, modified by a nom, all non-poles mean that some of the scalar fields have a pole, diverge along the boundary and the main inlay that corresponds to not having a defect is the case. make the non-poles as large as possible, so on one side of duality these are the basic objects that you can see universally regardless of the choice of C, so this NAM correspondent Paul with the Irish line in the bottom, dear Schley is the case where the non-pole is zero, but we could consider dual boundary conditions where you take the second circle and set it in the same construction as the boundary of the tube, so somehow We discovered what the duals were. so the regular ngon Paul with the regular ngon Paul corresponds through the main embedding, so it's called regular because well, without getting into too much technicality, since I have eight more minutes, it's impossible to give a proper explanation, but is the generic type of pole the maximum pole turns out to be the dual of knowing between the boundary conditions getting involved there knowing one of the boundary conditions for gauge fields means that the gauge field at the boundary is arbitrary it has no restrictions there is a restriction when it is normal derivative and then all the other fields of various boundary conditions that are related to what I said by supersymmetry, so Noah and dear Ashley are the most classical boundary conditions, but here no Aman is small: the one that It's furthest from being classic is the regular ngon polo and the others are fine.

The next fact is important, this background of what the operator will tell you, unfortunately I will not have time to give you a very useful introduction to what he will say when a gauge field obeys the volume in boundary conditions, that is, all the boundaries. is unconstrained and can therefore introduce into the limit an arbitrary set of degrees of freedom interacting in some way without a gauge field, so you can imagine more generally knowing Mun plus any 3D quantum field theory with G symmetry because of the way it is G and doesn't move. because we are on the zero cycle side of duality, but here it is quite natural to consider not arbitrary quantum field theories but super conformal field theories with a lot of supersymmetry and these universal constructions that I have told you about will lead to that because preserving a lot of supersymmetry , so while the regular and nonpolar correspond to the norm, the others correspond to Newman coupled to a super conformal field theory with G symmetry and that is true for all of these and both David and I said what we could about these theories super conformal field types, but they are difficult to study because they have symmetries that are invisible and I will tell you in a second why that is important, so a lot of what David has done more recently with various collaborators is present a new method for studying these super conformal field theories and while I hope to give a better introduction I will be able to do so, I am running out of time so I won't say much but I want to say something to make it clear. why it is important for geometric Langlands at the top we have the regular non-pole and the rule is simple enjoyment as we go down the non-pole becomes simpler but the superconformal field theory groans and at the bottom on one side , we had dark light and on the other side we don't have Aman coupled with a theory that thought about it and I called it T of G and T of G is an extremely interesting theory because it actually is.

I hate to dismiss this statement at the end of the talk, but TFG is actually the universal core of geometric Langlands, so why does it have that status? Well, the d-rush light can be defined universally independently of C, but now suppose we reduce the lens, then we can do it from a global point of view, we can twist it irrationally because the above means that the gauge fields are required to be 0 in the limit, but that statement with the independent variant thunder performs a constant gauge transformation in the limit, so at the initial moment of its conditions of the dual gauge theory G as G dual as a group of global symmetries that is what you would say in four dimensions, but now, when you compact it to archive it, for example, on a Riemann surface, but actually, if you go back to one, when you have global symmetries you can do twists and turns. non-contractile loops by elements of the global symmetry group, so because dear Ashley has dual G, it is a global symmetry group when compacted further, you can create boundary conditions that are just very costly locally but globally they are modified by rotating through an element of the automorphism group. so to reflect this fact, T of G actually has G times G GLE global symmetry, so the fact that my description is a Neumann coupled to something that something has to have G symmetry in general just so you can achieve fields without I'm on the limit. conditions, but if it is dual for dear Ashley, dear Shaw, I admitted that G dual was a group of global symmetries, so in that special case something will have G multiplied by G dual symmetry and T of G not only has G multiplied by G for all symmetry, but at least in the simplest case, you even have an order board for some major changes, that's a fall transformation, right?

Swap the Higgs and the cooler branches, then if you understood TFG you would calculate the dual to any local dual system G by simply taking Roman couple - T of G writhing for a gigolo fat package and that would be the answer, possibly a sounding answer a bit abstract, but what you've been doing is putting that into a more concrete form, now the difficulty in understanding it is that, although TFG has G times dual symmetry, that fact is difficult to see explicitly, so in our article we saw J times the maximum jeetil torus within G times G pole and by symmetry we could also have seen T times people, since the symmetry that manifested was only J times With such a description, we would have been able to describe the dual of a local system or the dual group whose structure group was actually a million, but better methods are needed to hope to describe using this method to describe the dual of a no.

G's abelian local system drools and I thought it works, at least there are some steps in the direction of being able to do it. There's another word I want to say, but maybe I'll stop right now for questions. It's like what clips, how would you do it? try c2g, well it's actually a special case of the theory used in your study, so the T of s example takes it and you want to multiply the hypermultiplets of charge 1, so, because there are two of them , there is a su 2 or SL 2 symmetry of the Higgs branch and because the gauge group is U 1, there is a u 1 symmetry on the Coulomb branch, but at the origin of the branch: there is a singularity and the singularity is an a.1 singularity and the a one singularity actually has su 2 symmetry. so the hidden fourth case of T of sp2 is easy in dual times su 2 global symmetry, but in the infrared it is enhanced u su tu tu all times its 2 due to the nature of the uniqueness of the coral branch or at least that is how you would see it well as part mathematically in your work.

Any other questions, of course, yes, of course, him being a friend makes a lot of sense. What that says means it's fun. So T of G as G multiplied by G, dual symmetry because I emphasize a special case. What I was saying is that anything with G symmetry can live in a limit of a GE gauge theory, but in the same way, anything with writing or global sin can live in a small limit, so we work in C multiplied by m2 but we divide m2 into two halves are m1 inside n2 we are going to do gauge theory GE on one side and dual gauge theory G on the other side so here the gauge group is Jay here in Siebel and for complete the image along the defectwe need a theory with J multiplied by G dual global symmetry the theory with that property is T of G so here lives T of G coupled to G on one side and she told me on the other side this is a symmetric description but if you want to see the map specifically from the GBO boundary condition to a G boundary condition, you end the picture here with any G gate boundary condition and as I said before the microscopic details are different but pelagic is the same as when we were discussing about lying operators, this in effect. from the two-dimensional theory here we have a line defect that we move to the boundary gives an effective boundary condition at j-3, so the functor from G to all boundary conditions to G boundary conditions is given by T of G, of course, the image was symmetrical if we read it backwards we would have obtained the function in the opposite direction well, any other question there will be discussion sections there is something else that I wanted to explain a little I say that we will all stop if there is another question something that I I would like you to explain a little, but it is useless and responsible for people missing out on snacks.

I think I'll take the discipline of saying just order tea, so they're okay when I was working on geometric Langlands, which was mostly 10 to 12 years ago there was something I couldn't understand properly, so in mathematical work it's important that certain things are described in terms of two-dimensional algebras and that was not entirely clear in any of the articles that I and Valentine Kapusta say. ok, Google wrote at one point and it was just starting to become clear when I was working with Davina, but it's the obvious bottom, much deeper later, so why do two-dimensional chiral eligibles come into this story?

So I described the universal boundary conditions in a four-dimensional theory, but you could try to break the boundary into pieces and produce interfaces between those boundaries so that we can look at the corners where, for example, we could both be rational here to make sense of This image, it takes more than what I'm saying. but you could look at it, you could take a look at the four-dimensional theory or the standby form with the corner where you put a boundary condition here and another boundary condition here, well, what turned out is that in that situation you very often find that there is a chiral algebra but lives in the corner the two-dimensional Colorado remember that in four dimensions the corner will be two-dimensional so this is a natural way in which Kai Rajab emerges in this story.

I can't promise to explain everything. but it explains much more than I was previously interested in. Certainly now, other than lock time, I can't offer much instruction for driving it. It's a job because I don't know it very well, but if I had had more time at least. I would have told you what I did, so I'll actually have the discipline not to try it, but there is a story that has to do with the Jones polynomial: or the phonology of quantum groups and how all that software comes into this story and If you do The relationship of all those things with what I have been telling you today, you will end up learning that the carotid would be better off living in the corners, so disciplining myself not to spoil your snacks is the best I will be able to do. say right now