 Okay, so let's start. Yes, it's time. It's time to start. Very well. So it's the fourth lecture, the final in this series. Maybe we'll continue sometimes later, but not in this year. So let me summarize where we've been up so far. We've discussed supersymmetrical gauge theories, supersymmetries in various dimensions, and then we specialized to n equals to supersymmetric theory in four dimensions, which meant that it had eight supercharges. And we explained how it connects to integrable system, complex integrable system, or algebraic integrable system, on the modular space of Vakia. Namely, let me call the theories theory T. And for algebraic integrable system, we had notations as a vibration from P to U, and P comes at the modular space of Vakia, Vakia of the theory T on R3 times S1. So it's hyper-color space, or holomorphic symplectic space, and this fiber over the modular space of Vakia of the theory T on R4, which is a spatial color. So this is hyper-color, fiber over spatial color. And so the fibers are a billion varieties, and the correspondence is that the period matrix of the billion variety in a given point of modular space of Vakia corresponds to the coupling constant of infrared theory, which is U1 to R, a billion theory, where R is the rank of this integrable system. So rank R means that phase space P is two R dimensional complex space. Yes, so these are of complex dimensions R, there's a rank R a billion variety. And period matrix is the expressive matrix, cyber R times R matrix. So this coupling constant tau ij, it couples to the gauge fields as its real part, couples to the second sharing class, and the imaginary part, so the action is of this, has this form, and the imaginary part of tau ij couples to the young mills, the young mills functional. So fi, star of j. And as last time we reviewed very carefully, the S-duality transformation, S-duality electromagnetic duality, this is synonym, in this case for SL2 z action on the h1 of the billion variety, it corresponds to electromagnetic duality on this side. So tau is really not a matrix, is in a billion variety. It's the point in the modular space of a billion. One question. Is the basis complex R dimensional space? Is there a natural complex structure on the base? That's the question? Yes, yes. Well, it follows from the analysis of n equals to suzie. n equals to suzie implies that the modular space has to be a special color. And special, well, yes. So even n equals to suzie implies that it has to be color, n equals to suzie implies that it has to be special. So and hyper color structure. So the tangent space corresponds to some kind of particles in this vacuum? The tangent space of the base? The tangent space of the base corresponds to just the differential of the complex colors in the vector multiplet that you have here. Yes, so the deformation, so we have in the low energy, you have the maps from the space time to the modular space. Yes, and slowly varying maps correspond to the change of colors. Okay, so that's the summary. Yes, there are questions about summary. Oh yes, maybe one point to add to this summary. The Hilbert space depends on the volume. The Hilbert space of the theory on R4 depends on the choice of vacuum. Yes, it's like super selections. So maybe one more line to add here is the Zyberquiton curve, of course, and Zyberquiton differential. Okay, and the special electric magnetic coordinates. So let's just write summary here. So here we have one form lambda, which is the Zyberquiton Liouville one form, so that d lambda is omega, that's holomorphic symplectic structure on P on the phase space. And well, lambda is meromorphic, generally, is because if omega has non-trivial periods, then if it's non-trivial element of H2, then you cannot find one form, which is everywhere, holomorphic, and the differential is defined. So lambda has to be meromorphic, it has poles. The resistors on those poles are determined by the masses of this theory. I don't want to dwell on it for lack of time. So here you have the holomorphic coordinates, A i and B i, these are special coordinates. Well, not quite. So these are coordinates, these are special coordinates, these are action variables on the base, action variables on base U, and they are determined by the periods over alpha i and beta i cycles, which are in the basis of H1 of the bilion fibers of this one form differential. So they are holomorphic coordinates on the base, there are r of them, and they come in duality that D A i is tau ij D B g. So on the left side, in the four-dimensional gauge theory, this special coordinates A i, B i on the base, they define what is called central charge. Central charge, it's an element of the superpotent carrier N equals to algebra in four dimensions, that it comes, when we constructed that algebra, it comes from the dimensional reduction of fifth and sixth coordinate, so that the complexification of the fifth and sixth comes as a momentum and fifth and sixth dimension from the four-dimensional perspective is just a scalar, and then the value of the central charge on a state, on a state with electric charge and the magnetic charge, electric charge N i and magnetic charge M i is given by N i A i plus M i B i. Okay, so one more comment, the B P S, yes, it would be good to do that, so let's do these conventions and I A i upstairs, M i upstairs, B i downstairs. If I, I think I fault it, maybe here, yes, sorry, here I actually swapped mistakenly the conventions, it was that D B i is given by the period matrix of tau i g D i g and also this tau i g is the second derivative of the pre-potential function. Yes, it's defined, it's defined locally, yes, yes, so B P S states, which we also reviewed I think in the second lecture, they are characterized by the condition that the mass of the states equals to the absolute value of the central charge and they come in what's called short-multiplet, so Maxim is expert on counting the number of such states. Okay, now, so let me just leave that for reference and now let's discuss quantization on the side of the integrable system, so far we had the classical integrable system. So we deducted from a 3 to 8, we have a classical system and now we count it. Yes, yes, yes, we started from quantum field theory, we discussed only the low energy sector of this quantum field theory and that low energy sector is just a classical manifold variety, well with certain special, well with geometry that we've discussed and now we want to understand what about quantization. Okay, so there are two ways to quantization in a sense and roughly they can be summarized as follows, well in the literature they refer to SNS for Necrosse von Schatt-Aschwilli, overlapped with notations, and AGT, it's Aldaiga-Iyota-Tagicava. So on the side of gauge theories, so here you consider a deformation, a certain deformation of an opposed to theory and I will call that deformation T sub epsilon here and T sub epsilon 1, epsilon 2 here and I will explain more precisely it means that you study, you start from studying the theory not on flat background R4, but on certain curved background and the parameter of that curvature, well it's not only curvature of the Riemann tensor, in fact it's curvature of other auxiliary fields of supergravity background if you want a physical approach, mathematically it could be explained much easier, I'll do that in a second and here you'll do that on a background with two parameters epsilon 1 and epsilon 2. Now on the side of integrable systems, well here you have the, here indeed you have integrable system that we had before P2U, which is a halomorphic vibration in the complex structure which is usually called in literature I, it's convention, so let's say that that integrable system that we had here, it's i-halomorphic, but you should remember that the module space of Vakia on the third one or three times this one is hyper-color, so it has actually Cp1 worth of complex structures, yes, so let me put it here, there is a Cp1 worth of complex structures over here and the three orthogonal ones usually called IGK, corresponding to three quaternionic units, imaginary units. So in complex structure I, this is halomorphic vibration and in the case of Hitching systems, so for example, again that's Higgs bundles, Higgs bundles if P2U is Hitching system, it doesn't have to be Hitching system, but just to explain what i means. So in this case the quantization is the quantization of the halomorphic symplectic structure, so you consider the deformation quantization of the algebra of functions on P with respect to the halomorphic symplectic structure omega i, omega i with parameter 1 over epsilon, so epsilon is H bar. No, epsilon can be complex value, doesn't have to be real value. So here you have this, it's halomorphic symplectic space, so you have a Poisson bracket, halomorphic Poisson bracket on the halomorphic functions on the phase space P and then you consider the algebra which quantizes this space of function in the usual way. Now for this story, you don't have the structure of... It's a formal parameter. It's a sexual parameter, yes, it's not formal quantization, it's actual quantization. It's a fractional parameter. Yes, yes, yes. We'll talk about it a little bit later. So here you don't have vibration and so we forget about it because the vibration, because we want to look on the phase space in the complex structure G, so let me call this i quantization, i quantization of P. And let me call it, P is hypercolar, yes. So we look at it as halomorphic space in complex structure. And let me call it J quantization of P in complex structure J. So for example, that's the modular space of flat connections, so G C flat connections on Riemann surface C if you started, if P2U was hitching system. So it's the other complex structure. And to do quantization of this other complex structure, you don't have now this vibration, so you don't come with integrable system, you come with something else, but still with a quantum theory. And that's what EGT about. So it's a deformation quantization of the functions on this total space P in the complex structure J. So let me put it sub i here and sub G here. And the deformation parameter is the ratio of these two epsilon 2 over epsilon 1 omega sub J. So now to understand what we are talking about, we need to explain what is the left side namely, what does mean to the deformation of the theory T, what does mean the theory on R4 sub epsilon or sub epsilon 1 sub epsilon 2. And then we'll talk about that side in several concrete examples when we actually know what that integrable system is. Those examples would be Hitching system or group like version of Hitching system or elliptic like version of Hitching system that we talked about last time or more generally the modular space of G bundles on a Poisson surface. There are questions? No? Here with this local systems. Yes, local systems. It seeks like traces of monodromes. Yes, yes, yes, exactly. Yes, traces of monodromes, yes. So it's flat connections, yes, yes. So the coordinates here could be traces of monodromes. It's a complex parameter. Yes, yes, exactly. So in complex structure J, this space P, it doesn't depend on the complex structure of Riemann surface. Yes, it's kind of topological with respect to... Yes, yes, yes, yes. So it's for Kunchirov. And yes, we'll come to more details here. Okay, so let's start maybe from the left side, from the Gage story. Okay, so on Gage story, the story goes back to... I'm not sure how far it should go back, but well it starts with the notion of quaring cogemologists which I don't have time to review, but for physicists it's a rough idea that when you have a manifold M and G, which acts on M, and M or G is not smooth, and you want to study the homology of the space M or G, then you want to define some appropriate tool to deal with the attempt to score quaring cogemology, and that's what we are going to use. Now, physically, the deformation of the theory of T sub-epsilon 1, sub-epsilon 2, which was done by Nikrasov, and goes back to papers of Nikrasov and lots of Moran-Shatashvili. So, you consider that Donaldson twist, Donaldson-Mitton twist of N equals to D equals 4 theory, and this twist means the following. You take the Lorentz group SO4, which is SU2 times SU2, left SU2, right. You also consider the r-symmetry. Remember that N equals to superalgebraic counts by reduction of N equals 1, D equals 6, and it has a symplatic symmetry built in SP1, corresponding to the doublet of the spinors that we use to construct the representation S. It was S plus times C2, so there is SP1, which acts by automorphism of C2, preserving the canonical syntactic form here. I'll write a compact version of it, SC2 r-symmetry. So, the supercharges they transform with respect to the spinor representation of SO4 and with respect to r-symmetry, and then you consider a diagonal embedding of r-symmetry into, let's say, SC2r subgroup, and after that, the supercharges they form the 8-touplet of the following way. You have, from the left and right spinors, you get the following thing. From right spinors, you have a scalar and self-doule 2 form, and from left, you get a vector. And then one can rewrite the supersymmetry of finding close to theory, the homological-like transformations. And so, the ordnance on the written twist corresponds to taking the supercharge q generated by the scalar eta. And such theory can be defined on any manifold, on any topological manifold, and the partition function of such theory reduces to computation of the integrals of the module space of flat connections on this model of a certain homological class. Now, the idea of Losev-Murshev-Tashvili and Nkrassev is to complement the supercharge by a vector field v if the manifold has an isometry with respect to the action of some compact d-group. So, the concrete example of R4 sub-epsilon 1 sub-epsilon 2 is the following. You take R4, write it as R2 plus R2, and consider SO2, SO2 actions on these two planes to the algebra parameter sub-psilon 1, epsilon 2. So, physically speaking, it took a response to turn on certain background fields of the supergravity which break Lorentz invariance and put the theory into something like infrared well. So, this is R4. And the theory becomes confined to a well whose dimension, whose characteristic dimension is of the order 1 over epsilon. So, it's like infrared cut-off. This infrared cut-off of the ennequaster theory imposed in a smart way to preserve one of the supercharges. Now, let me now write mathematical definition of what we compute. We compute the partition function of ennequaster theory in such background. This partition function is called Nkrassev partition function and this function is defined as follows. So, let's fix the notations that we start from the gauge theory defined by Lagrangian, defined for the gauge theory with Lagrangian and which was parameterized by the gauge group G and flavor symmetry F and the hypermultiplot in representation R of G times F. So, it's gauge, flavor, quaternionic representation of G times F. So, let's define the partition function that M would be the modular space. You are careful. You need to consider some resolution of this modular space. It may be better to say modular stack. Modular space of G bundles. So, that's calamorphic G bundles on C2 with fixed framing at infinity on Cp1. Well, at Cp1 infinity corresponding to the one point, corresponding to the projective compactification of C to the cell. Now, this modular space is subject to the action of the groups G, G sub C, X on M by just action on the fiber of the trivialization of G bundle at Cp1 at infinity and also G L of C2 on which we consider these bundles also at some M. In addition, you define over this modular space of G bundles a shift which is called the matter shift. So, let me call it our head, matter shift on M and physically speaking, this matter shift is the following. Consider in a given instant on a background, well, the calamorphic G bundles they correspond to G instantons on R4. So, you pick a point here, you consider this G instanton on R4 and then you solve Dirac equation in the representation R in a given instant on the background. And that Dirac equation has a number of solution in left and right sector and you consider virtual bundle the difference between the zero most of Dirac operator in left sector and zero most of the Dirac operator in right sector. So, mathematically, the same thing is said by the following. So, this matter shift on M is the push forward of the matter shift R over universal modular space M times C2 under projection to the modular space. So, you consider the cohomology of double operator which is valued in the representation R of G group and then you take a bundle over the space M whose fibers are the cohomology of that operator. Are there definitions clear? Yes. If I remember correctly, I said to people to discuss this action of two rotations. Yes. I can go make serious ellipsoids. Yes, it's just taking these parameters to be different. Those ellipsoids are... Yes, so instead of this... I'm going to seek a good background and say that you consider serious ellipsoids. Yes, yes, yes, yes, yes, yes. Yes, at the level of the original Yang-Mills action. Yes, putting V mu correspond to... So, at the level of the original action, when you add V mu, you do something like that. Very good. So, you have, for example, for the scalars. Yes, so you have their operator... Okay, you do the following thing. So, you had your complex scalar in the vector multiplet. And you remember that the supercharge Q in the de Lonson vision, sorry, it was squared to the gauge transformation by this complex scalar. So, it would be A. That's nothing. Gauge transformation by I. And now you replace A by A plus V mu A mu. And you do this substitution in the Lagrangian. And there are some other substitutions to make it super symmetric, but this is the essential one. In the Lagrangian, I had A, A dagger commutator squared. Yes, yes, in the Lagrangian, you have A, A bar commutator squared. In particular, one thing you do in the Lagrangian you replace here A by this combination. Okay, but the Lagrangian becomes quite complicated. It's not... And it's not really useful to write it down here, but physically speaking, if you look carefully in this Lagrangian, you'll see that the theory becomes infinitely, weakly coupled when you go to the far to infinity. And the theory is like supported in a neighborhood of the origin and the size of that neighborhood is one or epsilon. It's like an quadratic well potential. The theory becomes infrared regulated. Also there is another, as Maxine mentioned, there is a story which relates that to ellipsoids. So instead of doing the deformation, you can just consider the theory on the background with curvature. And if you do it around sphere, around S4 of radius R, that would correspond to taking epsilon1, epsilon2, 1 over R, and two copies of the theory for the two fixed points on S4 under chosen rotation. Can epsilon1 and epsilon2 be identified with H bar somehow? Well, as we reviewed here, in one quantization, one epsilon is 0 and only one epsilon is present, then epsilon is H bar, exactly. In the second quantization, no, it's ratio of epsilon, which is H bar. Okay, so then the partition function Z is the integral of the modular space of G bundles, and see too that you've defined over there. And here I take the union, okay, so that modular space, it is the joint union of modular spaces parameterized by the second-chern classes. So we'll take the second-chern class of the components M sub k, and the contribution from each of such piece of the modular space is taken valued with the coupling constant Q to the k, and then you integrate the equivalent Euler class of that mattershift. So it's equivalent with respect to the group G. Okay, let me call it equivalent group, and G equivalent is Gc times Fc, I mean just complexification of the flavor symmetry group and NGL2, which acts on the spacetime. Yes, so the result of that, you can also interpret as a push forward of a caromology class, which is written over here under the map from M to the point. Yes, that's what it means to integrate. So Z is an element in the G-equivariant caromologies of a point. What it means in practice is that it's a function, a joint invariant function on the Li-algebra of G-equivariant. And the coordinates on the Li-algebra of G-equivariant are usually called the same A that we had on the left board, the special electric coordinates, the masses for the Li-algebra values of the flavor symmetry group and the epsilon 1, epsilon 2 for the carton of GL2. And this partition function is later called the rational version in the sense that there is a trigonometric version and elliptic version of it, like there is an ordinary Dirac homology, there is K-theory and elliptic homology. In terms of the gauge theory constructions, this one corresponds to taking the gauge theory on S1 times R4, epsilon 1, epsilon 2 and this is epsilon T2 times R4, epsilon 1, epsilon 2. In terms of computing the push forward of respective caromologies, here you take the integral of the modular space of the dot character of the tangent bundle to TM and the churn character of all exterior powers of the matter shift. So, in terms of churn roots, fxi are churn roots of this guy. Then for the rational case, we had just the product of the churn for the LR characteristic. Here we have the product over 2 sin xi over 2 and finally for the elliptic case, you replace these rational factors by elliptic functions and the conventions are such that this thing has periods x to x plus 2 pi and x to x plus 2 pi tau where tau is the elliptic modules of the T2 torus in which we compactify the 6D theory. So, the trigonometric partition function takes value in the case theoretical caromology of the point which are functions, adjunct invariant functions on the lig group itself. So, the trig is a function on g and g sub c equivalent adjunct invariant function. And the elliptical is an adjunct invariant function on the modular space of the equivalent bundles on this elliptic curve T sub tau. Can we use this kind of reversal in certain classes and get like symmetric functions or functions in many variables? Yes, yes, are you asking? The sinusoidal bundle can make a lot of six and get functions in pretty many variables here. Yes, yes. Is it some kind of physical tau functions? Yes, yes, yes, yes, yes, it's possible to get something like tau functions you need to... Yes, you can insert here some other caromology classes coupled with yes, yes, yes. Well, in terms of topological string we know what it corresponds to but I'm not sure how to well describe what it corresponds to in the gauge theory side. It corresponds to some high derivative deformation of the gauge theory like adding to the gauge theory not like f squared where f is the curvature but high terms. I have a much more basic question. What do you actually do with this s1 cos r4 f1 cos r2 Yes, so you think, okay, so due to this localization I think what happens is that the path integral localizes on the instantons in the five-dimensional space s1 times r4 and the instantons in five-dimensional space they have co-dimension 4 so they are like particles which travel along that s1. So it means that the partition function computes the supersymmetric index on the modular space of the instantons. The other way to say that the path integral here in this situation reduces to the integral on the loop space of the modular space of instantons and that's why it's a k-theoretical version. So it's a quantum mechanics on the modular space. Yes, yes, yes. This is the modular space of instantons on r4. But you see here it's the index-like formula. You compute the index of that matter-shift. This is just Riemann-Roch-Grotten-Dick formula for the index. Yes. For the index of the Dirac operator valued in that matter-shift r on this instanton modular space and the partition function z is just the index of the partition function of the quantum mechanics of the loop space of the instanton modular space. Okay. And the elliptical one when you replace the characteristic classes by theta-1 functions it's essentially the elliptic genus of the instanton modular space. Everything is equivalent. Everything is equivalent to the intersection of this g-equivarian group which I remind is just the gauge group the flavor symmetry group and gl2 which acts on the spacetime c2. Oh, that's right. Yes, yes, yes. It's an extra parameter. Q is the gauge coupling constant. It has nothing to do with it's a separate elliptic parameter. Yes. Okay, so the function here if you write it as exponent minus 1 over epsilon 1, epsilon 2 f of a m q epsilon well, the tau parameter if we are in the elliptical case one can check that this thing has a smooth limit when epsilon 1, epsilon 2 go to 0. In other words log of z times epsilon 1, epsilon 2 the limit of that when they're sent to 0 is a pre-potential function f of a mq and this is the cyberquit in pre-potential. So here I've sent both of them to 0 when we keep both, well it doesn't really matter which we are in those two situations it's completely classical situation. So this is totally classical situation. Both epsilon are sent to 0 and we are back to the classical case of this board of the cyberquit in case. So the pre-potential function it depends only on a, m and q. There are no epsilon left. Oh, that q is the coupling constant of the gauge theory. Yes, this is q with a slash like that I will use notation q slash for the coupling constant of the original gauge theory. Of the exponential of tau of tau parameter of the original gauge theory. It just doesn't matter if it's true or it doesn't. Oh, yes, yes, yes, there are three lines corresponding to that board. Yes, yes, yes, yes. But these trigonometric versions it's not like you take any cyberquit and system and you automatically from it can get a trigonometric elliptic version. It doesn't work that way. It's not, but you have a family. Yes, yes, yes, yes. So, okay, let's say it better. So if you start from gauge theory on R4 you get cyberquit and system. If you start from a gauge theory, if it's defined on S1 times R4 you get a system as well which we would call trigonometric cyber system. And that would be elliptic cyber system. But not every rational cyber system has a lift. Do you have an index UV? Yes, I added the index UV. It's a function of the gauge coupling parameter in the UV. Which stands to do, we are asymptotically free. Well, so if you are in the asymptotically free case then you fix it at some UV scale. You fix it at some scale lambda. For concise presentation it's better to say that we are considering conformal field theory. And so that the QV is fixed really at the infinite energy. It never runs. And if you want then to study asymptotically free theories that you can always send better asymptotically free theory into a conformal field theory by taking your theory at a larger scale much less than some mass. I mean do you take a massive deformation of conformal field theory and go to a larger scale? Yes, so this is cyberquit and pre-potential. So let me in a sense keep only the parameters A here. M and Q, so M and Q they are parameters of the cyberquit and system. They define that vibration P over U. Yes, and A is the coordinate of the base U. So that's how one recovers the classical relation from the deformation in the omega. It's called deformation by the omega background by omega in a sense of angular momentum rotation of the spacetime C2. So this is classical. Now the next limit okay that's classical cyberquit and case. Excuse me. For large A, yes, yes, yes. If A goes to infinity then F looks like A squared log A plus with some coefficients well, plus expansion in the powers over A's and it's not only quantum, it's not perturbative. Yes, it's not perturbative expansion of the QST. So this is instanton corrections. Yes, yes, yes, yes, yes. Of course, so classical it always in a sense of what we talked here. So full quantum field theory with only perturbative corrections in this vacuum sector corresponds to classical integrable system. So I mean classical integrable system okay classical cyberquit and integrable system which corresponds to full non-perturbative four-dimensional quantum field theory. Yes, and now we are discussing quantization of this classical cyberquit and integrable system and we said that on the gauge theory side that quantization corresponds to space-time deformation of the theory by parameter sepsilon. Okay, so a next limit is classical deformation of super-gravity deformation by super-gravity. Yes. So epsilon is a parameter which defines non-trivial super-gravity background for the gauge theory which is studied on that super-gravity background away from the flat R4 space. So epsilon essentially parameterizes super-gravity auxiliary fields and matrix I mean everything. So in s-limit you send one epsilon to zero and then you can define a function limit of epsilon2 to zero of minus epsilon2 log z and that would be a function f of a m q and now one epsilon parameter which is kept fixed and what this function is. Well, this function is called twisted, chiral twisted super-potential of 2,2 is usually d equals 4, 2-dimensional gauge theory. So let's see how it works. You consider the theory on R4 subject to one epsilon parameter and compactify it on a circle. So actually you do it we want to look at the phase space we compactify it on a circle so let's do it as R3 with one epsilon parameter times s1 and that R3 with one epsilon parameter has time which I will denote by R1 sub t times R2 on which epsilon acts times s1. So the thing on which epsilon acts the two plane it looks like it has like cigar geometry where epsilon acts by such rotation and so it's convenient to think about it as a theory on the interval if you put the theory in some finite volume and there is a certain cutoff at this distance there is an interval and over this interval there is s1 vibration s1 sub epsilon which is the fiber of that interval and this one sub epsilon is the left boundary of the interval. So then the geometry is the following you have R the time times interval times s1 and if you integrate out KK modes on this compact space on this cigar geometry you get a two dimensional theory on the cylinder RT times this circle so this function f is the super potential of the two dimensional 2,2d equals to gauge theory on R sub t times s1 that falls from manipulations with Lagrangian from physical computation. Why is not one just using the t-frame towards space or dimension? It does not respect the super symmetry you want? Yes, it does have a written on the cards in the dimensions because you cannot compute it if you want. That's right, it doesn't preserve the yes, it doesn't preserve what we want. Yes, yes. I really like to keep it. Let me just let this and keep the upper line and write it here. How do we see actually quantum mechanics here? We compactified to see quantum mechanics we need to have one coordinate to R at the time and then we need to have a compact space. That compact space comes as follow. You compactify the remaining directions on s1 then you have a three-dimensional space-time and in this three-dimensional space-time okay, one coordinate is time, so what is left is two-dimensional space-time and in this two-dimensional space-time R2 sub-epsilon so there is this super-gravity background which puts the theory into a well and all dynamics of the theory becomes localized near the point near the origin fixed point on this R2-epsilon so the theory becomes confined to degrees of freedom which propagate along the time but not in the special directions and one direction was compactified on s1 so that's how you see the appearance of quantum mechanics from the background It's clear? What is dA? dA is to call the infinite well Yes, it has something to do with quantum mechanics on the phase space that's a game okay, let's do it once again so we have the p, the phase space is the modular space of Vecchio on the theory on R3 times s1 now that theory R3 we deform as follows we take R3 and write it as R2 sub-epsilon times time and the claim is that in the background everything localizes to the zero point of this R2 sub-epsilon okay, so the propagation degrees of freedom they flow only along the time yes, and so the path integral reduces to the path integral okay, from maps offline to the modular space of Vecchio for this quantization for real quantization well, it's a holomorphic complex quantization yes, yes so different path integral we want to prove it yes, but it's well, it turns out to be the same in the cosmology sector that we are interested in well, yes, I was hiding parts of the story, but okay, very good, so let's discuss that indeed excellent so let's discuss quantization of complex well, in the things of everything and as deformation quantization and as a Hilbert space okay, so what we want from quantization from what you asked about from quantization we want first of all the algebra of functions algebra of operators and we want a module a module on which those operators act, yes, the Hilbert space the idea to do that in the holomorphic setup is the following it might be, it might have further references, but I learned about it in a paper by Witten and Gukov and they using sense of the construction of a Kapustin Arlov called Koizotropic A-brain okay, so the phase space P is complex, holomorphic phase space complex, holomorphic symplectic in this construction it's important that it's actually hyper-color space, so on this hyper-color space you can consider the sigma module which is the maps maps from 2D worldsheet to the target space P it's actually n cos 4 2D sigma model okay, now to do this holomorphic quantization in this hyper-color language the idea is that in the A-model in the situation of hyper-color of hyper-color space there exists not the usual Lagrangian A-like brains that we are used to but also exist Koizotropic for this case just space filling space filling brain B which covers the whole space P yes and that brain B is equipped with non-flat U1 bundle with curvature with curvature F in the situation of Lagrangian A-brain when the bundle is flat so here we have non-zero curvature and then it turns out that there is a choice of boundary conditions on this brain which correspond to the boundary conditions in the A-model when the curvature and the symplectic form okay, so it's A-model with real symplectic form say omega if you define an operator on the tangent bundle by the formula left to minus 1 omega so it's in the endomorphism of tangent bundle of P and if it squares to minus 1 then there are appropriate boundary conditions that you can put on such brain and it defines actually an open A-model so in the hyper-color situation let's take some conventions so let's take omega to be let's say omega i and F the curvature F we choose such that this is equal to the symplectic structure omega k then omega k to minus 1 omega i is the complex structure J so the the space of states on the open B-model sorry in the open A-model this A-model states on the brain B which is a space feeling time brain B turns out to be is the space the space of holomorphic functions with respect to this complex structure J so sub J functions on P J holomorphic functions so if you consider a string which ends by left end on the brain B and by the right end also on the brain B such strings gives the space of strings which end on the brain B and the brain B itself it gives the algebra of functions on the symplectic space on the holomorphic symplectic space they have to be J holomorphic functions and the the disk complex use with boundary operators inserted on a disk from the functions on the brain B they give the deformation of the usual product of this J holomorphic functions by substitutive product defined by the Poisson structure with respect to this complex structure J like in deformation quantization of Maxim in the holomorphic setup with respect to the holomorphic Poisson structure defined by complex structure J that's how you get the algebra of operators now we also need to have an algebra space something on which those operators act so for that we need another brain let's say B sub H and that would be the states of quantum mechanics and this other brain B it also has to be Lagrangian eta Lagrangian brain so then the states of quantum mechanics they correspond to the strings which are on the left and on the space feeling on the tropic brain and by the right end on this brain B on this Lagrangian Lagrangian brain so then there is a natural defined action by the operators on the states namely you take the brain which represents an operator and you glue it together with a string which represents a state so B B string you glue to B B H string and that gives again B B H string so it gives again a state and so there is a defined action of the algebra of operators so the algebra of operators is the algebra of states of B B string to the vector space of states of quantum mechanics to the vector space which we called H the model which is the algebra of B B H strings can this be stated that home B B acts on home B B H yes yes just just by gluing the string if you discuss the in AGT picture so sorry in AGT picture we will have two brains of this type one brain will be a space feeding brain and another would be Lagrangian but not in NS picture in NS picture well in NS picture we just didn't discuss what the other brain is we didn't discuss it but in principle ok now going back to NS picture so you have the geometry R times cigar geometry times this one and the Lagrangian brain it corresponds to the boundary conditions on this circle which I didn't describe but in order to have a two-dimensional in order really to have the two-dimensional theory on a cylinder on this cylinder R times this one you need to have the space complex in order to have the space complex there need to be impose some boundary conditions on this circle and this boundary conditions is the choice of the other Lagrangian brain so so let's look what happens when that circle is shrink to zero when it's shrink to zero then we have here an interval oh no I don't want ok so ok let's do that so here is an interval and the circle vibration over it now we studied the gauge theory on this four-dimensional space this four-dimensional space has two circles one circle this one and the other circle is the circle of the vibration when you compactify the four-dimensional theory on one circle the modular space is the hyper-colored space phase space P when you compactify it on the second circle coming from this vibration the phase space remains the same and the square extra modular is the same hyper-colored space so the result after compactification on this circle and this circle is the space of maps from the time time this interval and the interval has left and right boundary to P now for the theory on this time-time interval for this two-dimensional theory there are boundary conditions one boundary condition comes from the epsilon background just from shrinking the cigar geometry to the point and it's a certain sense canonical there is no other choice we have to make and the other boundary condition comes from the choice ok it comes from our choice of boundary conditions at the right end of the interval so the boundary conditions at left left boundary conditions this is caseotropic brain and the right boundary conditions these are the Lagrangian brain so the space of states on such string is the space H which is the module for the algebra of operators and if you do the string which is which goes from the from the brain B to the brain B itself it gives the algebra of operators so let's do let's discuss this in more detail first let's discuss the example when P is the Hitching system so if P to U is the Hitching system then the complex structure I is when this vibration is holomorphic and you have this description in the complex structure I by the modular space of Higgs bundles so the algebra A is supposed to be the omega I deformation or epsilon deformation of the space of I holomorphic functions on P so these are exactly the commuting Hamiltonians of the Hitching systems this is the same as holomorphic functions functions on the base because they are constant along the fibers there is there is a mirror symmetry picture which is T duality along the fibers along the fibers A and it maps this to the dual Hitching system so if this is Hitching for the group G here would be Hitching system for the Langlans-Doll group you have some P Langlans-Doll projected to U the base is the same so it's just duality along the fibers so the fibers the fiber are replaced by dual-tory and to ensure some that the result in modular space is the modular space of the dual-hitching system this one and it forms trivial yes yes yes in a sense trivial yes because they are commuting they are commuting Hamiltonians so commuting for the Poisson yes commuting for the Poisson yes so yes ok so now let's look at the mirror map of the T duality of this brain B of the choesotropic brain B so it's a brain in the A model of the Lagrangian type but the space of the BB strings they are they are holomorphic in complex structure I so ok so the conventions here is that you take the A model with respect to the syntactic structure K and you take the curvature F on the space-filling brain equal to the syntactic structure G so the resulting normal K is I ok so we have I holomorphic homomorphic functions here so this brain maps to the Lagrangian brain for the dual-hitching system so it's in the PL and it covers the base the base U so this brain is known as brain of hoppers so namely you consider the dual-hitching system in the complex structure J and it's just a B brain I mean B model of complex structure J J brain in the right theory so if this is the base then this brain of hoppers let me call it N and it covers the base of the hitching system so at each fiber it's just a point because it was a it was fiber-filling brain on this side and after duality in each fiber it maps to a point it becomes a cover of the modular space of the hitching system so it's a certain subspace of the space of the modular space of the dual-hitching system in complex structure J when it's a modular space of JLC flat connections and such flat connections are called hoppers by Billinson-Drinfield okay so the okay sorry now the algebra we need to discuss the other brain the Lagrangian brain which we didn't specify anyways the Lagrangian brain which defines the space of space of the system is half-dimensional brain here and after the duality transformation it maps again to the half dimensional brain here so let me call it it's not quasi-tropic brain yes yes I'm discussing the other brain this Lagrangian brain so the mirror image of this Lagrangian brain BL is again the Lagrangian brain so let's call it BL of dual or mirror of it and so it intersects now the mirror of the Lagrangian of the coins of the Tropic brain in some discrete set of points and the intersection of BL with L so this is the spectrum of quantum hitching system this is just mirror symmetry picture for Billenson and Drinfeld quantization of hitching Hamiltonia on the right side after mirror transformation from the coisotropic brain you get this Lagrangian brain of Opers from the brain of states you get also Lagrangian brain and the intersection defines the spectrum of the one of its kind of infinite intersection because one will break and one structure will break and another a discrete intersection is infinite yes it's a spectrum well it's discrete no no it's discrete but infinite yes yes yes yes it's countable but discrete yes yes yes intersect Opers with another brain yes okay I'll die for HGT picture okay for HGT picture let me just briefly state the claim without explanation because I want to explain more on quantum groups and it's quantum groups relate to the NS picture so for HGT picture the space of states that comes out is the following so H is B so you consider the Lagrangian system PtU and take one brain B to be that coisotropic that we discussed over there so the the space of functions here are i-halomorphic yes you don't have cigar geometry yes so that's what I wanted to skip because I want to because I want to explain more on quantum so you should go to the same on real short episode yes what does it look like okay I'll tell okay okay so in HGT picture you do the follows you consider so you have an action of two parameters epsilon 1, epsilon 2 and they correspond to the two circles yes to the T2 vibration over something we don't have yet to specify that something but in a sense there is a two torus T2 parameter epsilon 1, epsilon 2 is like an inverse radio now so if HGT story is applied to the Hitching systems then all those and and equals to four dimensional theories can be described by the following construction two comma zero theory and put it on R4 times C that the C on which Hitching system lives and then if you decompose that R4 into R2 plus the two torus these two epsilon times C and do the reduction of the two comma zero theory first on these two torus you'll find let's say of type G GSU per Young Mills on R2 times C and the tau parameter of these GSU per Young Mills would be given by the ratio of epsilon that's the basic property of the one of the defining properties of two comma zero theory when you compactified yes, yes, so you should think about epsilon 1 over epsilon 2 as really elliptic modules elliptic parameter elliptic curve tau 2 okay, and therefore there is a natural inversion transformation so there is one glance dual epsilon 1 over epsilon 2 for N equals four super Young Mills dual which actually which gives us a hint that at the end the AGT quantization it really depends not on just epsilon 1 over epsilon 2 but it depends on epsilon 1 over 2 interpreted as elliptic modules of a sorry, as elliptic modules of elliptic curve so it's invariant under inversion of epsilon 1 over epsilon 2 inverted so here okay, so then the two brains which appear for the space of states in the AGT picture for the space of states on the theory take the theory on R3 times the time and consider the space of states here, so the space of states here it comes to the space of states of strings with left hand on this quesitropic brain B and the right hand is on the brain of opers and which comes by the mirror symmetry transformation from the dual quesitropic brain so you and here you do quantization by J-halomorphic functions in other words the space of functions if you pick a different super charge in this and equals for two-dimensional sigma model the difference super charge corresponds to the J-halomorphic functions on the space-filling brain and so they act from the left on this B-N states so home B to N is acted from B-B on the left okay now in the mirror dual picture so N is the brain of opers which comes as a mirror symmetry of the dual of the dual-hitching system okay so this is the space of states and this is the algebra of operators which acts from the left to the right so this algebra of operators in practice in the energy story is Virus or algebra or W-algebra W-algebra with the parameters epsilon 1 epsilon 2 and G but also you can describe that in the mirror symmetrical picture okay this the space of states between B and N would be the space of states between the upper brain which is mirror symmetry of B and coisotropic brain B in the mirror dual description of the hitching system and now this space of states is acted by B-dual-B-dual strings on the right so in the AGT story the in the AGT story the space of states H is a module with respect to the action of algebra of BB string on the left and B-dual strings on the right so if you take the quantization of the complex structure G then the quantization is quantization of the module space of GC flat connections so it's a complexified version of what Witten has done for the quantization of the module space of flat connections of compact group G on the Riemann surface and related to Wiesel-Zumina Witten model and to change assignments so here is the quantization of the module space of GC flat connections it's on C it gives the complex version GC term assignments on R times C and the complexified term assignments with say level K SHV DA plus 2 3rd ACU turns out that it can be properly quantized so I'm thinking about it as a boundary term in the n equals 4 supersymmetric Young Mills with the coupling parameter tau equal to K plus dual coaxial number or the mirror-electric symmetric dual of n equals 4 supersymmetric Mills with inverse coupling constant of the order equal to 1 over KH plus V so the instant on-counting parameter exponent of 2 pi I over K plus SHV is that Q parameter of term assignment theory that we have started from here now let me, remaining time let me actually focus on the NS situation when the integrable system is not a Hitching system but trigonometric or elliptic version it has to do with this version no, no, it's actually that one it has, yes yes, yes, yes, it has and I will explain this, yes okay, so okay, so for this series of trigonometric and rational trigonometric and elliptic we start not from the Hitching system but as we discussed in the first lecture or in the second lecture so the Hitching system it comes from the Poisson symmetric surface T star of C and you consider G bundles there now if the fiber so the fiber here can be thought as as a cusp as elliptic curve with a cusp and then the modular space of G bundles on this on cusp elliptic curve is the Lie algebra if instead of cusp elliptic curve you take elliptic curve with a node singularity then the modular space of G bundles is the group itself and if it's full elliptic curve then just bun G on this elliptic curve which we let's call it E sub tau so from perspective of now to have the total space to have the total space simplistic well at least at least what we know now what we can do is we have to restrict the case when the base C is of genus 0,1 so for these cases C is genus 0,1 and then I like to draw the whole table as C horizontal times C vertical where C horizontal and C vertical could be C C times or elliptic curve some other elliptic curve elliptic or horizontal elliptic curve vertical and we started three bundles here so what corresponds to the on the side of the gauge series so on the side of the gauge series let's consider the following thing let's take gauge theory in four, five or six dimensions which is defined by the gauge group G a product of SU and I gauge groups where I runs over the nodes of some graph gamma which we call pure graph and gamma we reviewed in the second lecture for the conditions of asymptotically free or conformal conformal completeness of the theory is AD or finite D graph for example let me write E6 or E6 and you put like N 2N and so on number of colors at each node and you put hypermultiple which interact between the nodes which are connected so this is the gauge theory the representation for the hypermultiple is space of hams from an I in G when I in G and leaked and also for the fundamental multiplets from an I to some fixed flavor spaces Fi Dm Fi is the number of flavors the number of fundamental flavors in I's node so the flavor group of symmetry here F is a product of SU Fi the gauge symmetry is a product of SU and I no no no there has to be constrained satisfied that 2N that the Carton matrix Cij on and G has to be greater or equal than Fi so in the case of affine quivers Fi has to be zero so you take that gauge theory in four dimensions and you define the partition function since we reviewed over there so you take the module space of this G let me put subscript G gauge G gauge instantons on C2 and compute the homology classes you find the partition function and all that so that's what you do with gauge theory now it turns out that the phase space P for the the integrable system associated to the gauge theory P is the module space the module space of now we have two cases depending on whether gamma is finite or affine so let me introduce notation so if gamma is affine then gamma bar would denote the finite the respective finite quiver so it's the module space of gamma per monopoles on C horizontal times this one just of gamma monopoles on C horizontal times time is one if gamma is finite or gamma bar instantons maybe let me put better G G sub gamma or G sub gamma bar the macaque corresponding groups on C horizontal times elliptic curve E vertical if gamma is affine and the elliptic modules of this vertical curve is equal by the product of the gauge coupling constants q2 ai check the dual coaxial numbers so the horizontal curve is the dual one on which we compactify the 6D theory so for the rational case I will draw the table here would be C vertical it's easy C star or elliptic curve vertical and that would be C horizontal C C star and E so this is for the 4D theory that's for the 5D theory on S1 and that's for 6D theory on the dual one now we have the modular space of G monopoles on the space C horizontal times just a circle the compact part from here on S1 times C horizontal and the situation for finite quivers or with monopoles is the situation when it's possible to have flavor symmetry is attached here so actually these monopoles are singularities so with singularities of Dirac type and the location of the singularities projected to the horizontal curve is exactly determined by the mass by the fundamental mass of the fundamental multiplet so for fundamental multiplet of i's of i's node so that for SU group with mass M with mass M insert Dirac singularity for these monopoles on C1 times C horizontal by taking i's co-weight from S1 to G G sub gamma which is the maximum torus of it and you just embed the Dirac abelian monopole which has a singularity order 1 over r squared for the scalar field phi in the monopole equations at the point okay at the point whose projection on the horizontal curve is equal to the mass M in the C horizontal you don't specify location of this point in the vertical circle as well total space of integrable system what is that? this is the total space of integrable system so the modular space of G monopoles on S1 times CH with singularity of Dirac type as I described so they parameterized by masses and by the appropriate co-weights depending on for which node this is a fundamental point so this is the total space of integrable system corresponding to the theories with finite quiver and the number of fundamental masses now let's look at this modular space in a holomorphic complex structure I first of all it's hyper-color space because it's monopole equations but we can look on it in one of complex structures in the complex structure I where it's a projection where the projection to the base is I-holomorphic and in this complex structure I have the following so this actually was studied by independently of this monopole construction it was studied just per se as Markman Markman system so you can describe it as modular space of G bundles on CH and then you take group valued Higgs like field instead of algebra but in a group so you consider meromorphic G valued section and moreover you fix the singularity the pulse of this meromorphic G valued section by this condition so if you do that on a compact space let's say if CH is elliptic curve we don't have to specify anything on the boundary then the dimension of this module space is 2 rho this is the vial vector yes so it's a sum over of I sum over I 2 rho of I the same letter denotes the number of flavors in I-th node so I'm sorry you'll be the same as G bundles on some surface well you need to be careful it should be like G bundles on CH times C star with something what you do at those singular points which correspond to those singularities yes but just in a perspective in the perspective of Higgs system in complex structure I this is just holomorphic G bundles on CH and the choice of meromorphic G valued section which is group valued which has pulse of that type and then it has a spectral curve construction and all that so you can repeat everything what you do with the Higgs system for this station okay and now quantization so oh no yes so let me take what quantization for the other situation when the G bundles are correspond to a fine cleaver then you take just the G bundles on elliptic curve in vertical directions times horizontal so that's G bundles on EV times CH with no insertions yes no insertions for gamma of a fine type okay and now let me write down the table of the algebras which come as a quantization of the symplectic spaces and so this would be the entire algebras which act on the quantum system in the NS in the NS type of deformation of the gauge theory so the Poisson structures on this modular space of G bundles times the Miramorphic G value section let's do the first case in monopause CH C vertical is C star so the Poisson structure is a quadratic type I mean quadratic type in the fibers C star invariant and on groups since in the vertical direction we have this modular space taken value in a group element this quadratic type Poisson structure was defined by Sklannian and then it later led to quantum groups so namely on a group G okay so let's do this so if G is the algebra then G star has Poisson structure, constant Poisson structure and then you can consider the space of functions on G star and quantize it and it turns out to be just in universal envelope in algebra of G if you have a bilinear form on G then you can identify G and G star and just think about defining data as a Poisson structure for itself now if G is a group then if you repeat the same construction taken as a Poisson structure the Sklannian bracket then the quantization of the algebra of functions on G with this Sklannian bracket gives the quantum group UQ of G where G has no head it's finite dimension now here we have something more, we have one extra loop we have the horizontal curve and we have the Higgs which is valid in the group itself and so it's natural that the quantization of this space gives the following it gives when C H is C C star or elliptic curve respectively you obtain from the quantization the Youngian of G UQ of G head or quantum elliptic group is parameter Q and tau of G so these notations are not totally consistent because but this accepted notations in the literature this thing has head this thing doesn't have this also doesn't have head conceptually all of them is deformation quantization of the modular space of this miramorphic miramorphic G valued Higgs bundles on the curve C horizontal so it's you consider the current algebra G of C horizontal and apply deformation quantization to the space and you get these groups yes so in this case this is the rational this is the rational limit of this thing when the when the two punctures collide and this is the limit of this thing when tau is sent to infinity but the sense is that like symmetrical field for the dimensional space G yes yes yes yes yes yes so here this construction when you have taken just Poisson on the lig group produces UQ of G instead of lig group you take LG so LG produces so deformation quantization of LG produces one of those things actually well depending on how you treat this loop how you treat the spectral parameter so it's just G in XI and depending whether you decided XI sits in C in C star or elliptic curve you get one of these quantum groups okay now if C okay the second okay some instant tones can be thought also in this language but instead of G value section where G is finite dimension lig group you think about G is a finite lig group so you lift it with one more loop and correspondingly here you end up with young young of G head UQ of G2 heads and this thing I don't know I haven't found it in literature maybe there would be some troubles to define elliptic version elliptic quantum group of our finite the algebra and the the trouble can be traced back to this to the fact that the space on which we are trying to find this algebra this space is a compact it's a product of two elliptic curves and it's not quite clear what we are actually quantizing here so this is I will put as a question mark but those these algebras they all are are perfectly defined and these are the algebras which act on the space of observables in the corresponding theories you mean I probably forgot the answer here where? here? no no this is what is called the teroidal quantum teroidal group so in terms of Dreamfield so Dreamfield had two papers which are called in literature now first construction and second construction so for the second construction the following you start with arbitrary G can be arbitrary affine sorry arbitrary Cosmoody generalized Cosmoody yes yes and Dreamfield defined UQ affine of G so if G is if G is affine Cosmoody the algebra then common convention in the literature is to call the thing quantum teroidal algebra and yes so and if G is a finite dimension then the only way you recover you recover UQ of G hat yes you recover GQ of the hat so maybe instead of this notation I like more the the following notation so all that can be called as follows you consider the the algebra of miramorphic functions which are valued in the group G so you take G of the horizontal curve and and then do epsilon deformation of the universal unalgebra of this part or from the series of the point well in principle miramorphic functions when you fix that the ranges are located in some specific points that would give a symplectic slice in the corresponding space like an orbit the points are of finite order yes yes yes so all right so when we fix location okay now finally the final statement would be from our last paper Nikita and Samson the following so for quivers of finite type okay when we have a location of poles parameterized by this lambda I hat tg at points mi f yes so f runs from 1 to fi to the number of flavors in the is node so the algebra yes so the algebra is a representation of this thing of your sub epsilon of G on a tensor product of what's called pre-fundamental modules i mi f over i and f and these guys are they were defined just very recently not in the original paper so friends there's a pre-fundamental modules due to the general definition for G has been done by Jim in 2011 and it comes after a work of Bajanov Lukyanov Zmolochikov in 1995 who studies integrable system in massive in massive two dimensional field series not conformal field series so yes so you consider the representation of that young fine algebra on a module which is given by the tensor product of this pre-fundamental modules and it's labeled by the fundamental masses of the fundamental multiplets as we discussed over here so what are pre-fundamental modules well if you're familiar a little bit with the quantum groups and now what genfield polynomial is genfield polynomial there are operators usually called psi this is like diagonal carton currents in the definition of quantum of an algebra and the highest weight of the fundamental module so psi on highest weight in the case of fundamental modules acts by the ratio of p i of psi q to one-half divided by p i of psi q to minus one-half that's the usual fundamental evaluation modules the pre-fundamental modules have the highest weight of this the same generator psi i even just by polynomial by polynomial with one root just by polynomial with one root and the root is at the point and m i so conclusion yes we need conclusion I think maybe we should just go for t but the conclusion let's do conclusion so one conclusion you started from the module space from the homology of module space of G bundles where G is the product of SU and I yes over some dinking diagram and magically you've related it to the quantum group defined by G which is the dinking quiver so if you try to prove it mathematically I'd like to say so we have explanation from the mirror symmetry and string dualities but to explain it we have to go to the pre-fundamental theory so how do you see the action of this quantum affine group defined by the AD type on the module space of bundles of SU and I bundles on C2 it's not obvious so the conclusion is that physics tells us something mysterious and interesting things about mathematics ok