Okay, today is the last lecture on this topic of the integrability and supersymmetry, and I start exactly from a place where I finished last week. I should say by the way, just I'd comment that after talking during the lunch that I said that there are a couple of important questions left in this topic and after that it's time to move on something else. And I predict that something else interesting will be string field theory again, but so today I will manage hopefully to formulate the questions I think that are required to finish the topic in general. So I finished last week with a statement that there is something called Zeibberg-Witton prepotential and there is something called the prepotential in supersymmetric gauge theory in four dimensions. So this is four dimensional super young mill theory plus matter with n equal to supersymmetry. So there is this notion. And there is also notion of prepotential, so both are called same word in classical algebraic integrable system, and I said that the people whom names I mentioned last week found relation between these two things and I will start from here. So the meaning of Zeibberg-Witton prepotential is completely in parallel to notion of the twisted effective superpotential in two dimensions in a sense that both of them allow to calculate low energy effective action, the exact parts that we can calculate in low energy effective action. This one is four dimensions, this one is two dimensions. Because of the two dimensions, so again in n equal to two-server in two dimensions which is half of supersymmetry of this, this has eight supercharges, this has four supercharges. Because of peculiarity in two dimensions, this is actually a superpotential and we have discrete vacua here, given by formula that I wrote. And here we call this color that the superpotential depends by sigma. But here there are twice more supersymmetries and so on, and the vacua is a manifold. So supersymmetric vacua here has some geometry, non-trivial geometry, and f describes a geometry is some manifold m, let's call m-vac, and out of twisted superpotential we can calculate this is part of Lagrangian in two dimensions, and I said it comes from the descent procedure, at least it's topological Lagrangian comes from it. And there is a descent procedure in four dimensions also here, which gives Lagrangian, that this guy gives a Lagrangian, and of course both are restricted, we are not getting the full effective Lagrangian, let's call it effective Lagrangian. Here we would get only the part without derivatives, so this gives actual potential, this is a term in Lagrangian which has no derivatives, I remind here just for, from the first lecture that the fields, this is everything abelianized, we start with non-abelian, these fields psi denoted sigma i, psi i, and f of ai, so these are scalar, fermion, and curvature of the gauge field, everything is abelian, i goes from one to rank of the gauge group. Here is a similar thing is denoted by different letters, now we have to, there is a letter a, psi and f again, so instead of sigma now we will be using letter a, this is just said what happened historically, but in any event there is this f gives not effective Lagrangian with two derivatives, so that's the difference, here it's a potential, there is a Lagrangian with two derivatives and that's the novelge, now what about here, now I have to describe the meaning of the pre-potential in classical algebraic integrable systems, again we keep in mind that there is the function which is holomorphic in the scalar expectation value, a as I say is the replacement of sigma in four dimensions, so it's a scalar in the vector multiplied, it's holomorphic in A, actually it's metamorphic but, and determines the two derivative Lagrangian in four dimensions, claim was it's a classical algebraic integrable system has a pre-potential which describes that, okay so I now move on this topic, classical algebraic integrable system, so we start with a phase space which is two n and this is a complex dimension, complex two n dimensional and we non-degenerate symplectic form which is two comma zero, it's a holomorphic symplectic form, so we have this phase space, this is complex, so let's put c here just to remember that, then we have a holomorphic map to something here which is B which is part of the half of the dimension complex space, so this has two n dimensional complex dimensions, it's half would be n dimensional and then we have some ball inside cn and we have this projection given by n function, so we need to determine this, we need n functions, I call it h1 to hn, this projection sometimes denoted by pi, so basically simply these are the functions on m with some fixed value and they determine the half dimension sub-manifold inside and the condition is that they have to Poisson commute with that simple, this is non-degenerate symplectic structure, so there is a notion of Poisson bracket which is determined by inverse of this omega and they Poisson commute, this is second, this is very usual thing if you consider everything real then you, this is real dimension, this is the r instead of c, these are the Hamiltonians, they Poisson commute and then there is a theorem of Liouville and Arnold which says that see if the fibers are compact and then they are tori, okay, they have to be tori, here in algebraic integrable system we have this holomorphic, everything is holomorphic here, so very important to, we have this holomorphic map but we require that fibers h inverse of h, so h is the value of the Hamiltonian, hi is equal little h, we take the fiber and this has to be polarized abelian variety and I mean this topic, the time giving lectures on has lots of things and some of the things I don't understand myself, I'm just using, so I'm not an expert but as far as I understand this means that they are basically complex tori, so it's again a complexification of the class, the real situation complex tori, that also means that we have a color form which allows to determine in the fibers a and b cycles, we have a color form, we have a pairing between cycles and we pick the cycles to be normalized using the color form, so then we have the notion of these things, okay, so this is what is given, now usually in usual situations which means, usually means what we read in the classical mechanics books when we study in university, we are told that this construction allows immediately to introduce what's called action angle variables, the variables on a base and the fiber, now angle variables will be the angles in this tori in a real situation and then the action variables i, i, i goes from 1 to n, are given by integrating over the, now if it's a real situation over cycles and there is a real tori, there are only one cycles which I would call a now of pdq and what is pdq is it restricted to the two fibers, the form omega since fibers are Lagrangian, the omega is d of some theta restricted to fibers and then this is nothing more but integral of theta which is a one form, so then we have isa, now we repeat exact same thing for the complex situation and we realize that, so these are the coordinates on a base, on the b, so I call it i here and the coordinates that I will call theta in the tori and then the plectic form would be in a real situation d some d i times d theta, so this what we would do, there are some ambiguities of course that this theta is determined up to the exact form and so on but this is the moment I skip, now in a complex situation we have now two types of periods, a period and a b period, so I have to define the local coordinates on a base let's say by that formula but I can define it also by this formula and I will call these coordinates, dual coordinates, the periods and then there is a question that total of them, I have two n of these, n of these, so I have two n variables on base so they have to be related, I cannot have an n dimensional space to an independent coordinates, so the 7 ramp states that sum of d a i wedge, d a i dual and this is from 1 to n, let's call it little omega is 0, that has a following meaning also so if I would take the base from there and consider twice that thing then here I could have introduced this let's say this would be a and that would be coordinates a dual, the statement there says that if omega which is very natural in such a linear space to have omega to be d a i wedge, d a i dual, again it's a holomorphic splatic form in this space, the statement is that if a's come from a period over there then in this space that relation omega vanishes means that we are sitting on the half-dimensional Lagrangian sub manifold, another thing that means is that since this guy vanishes, so then then we have one form which is a i dual d a i and this one form locally is exact because d of that vanishes we conclude that the sum over i from 1 to n of a i dual d a i is locally exact and defined some object let's say meromorphic function of a, everything here is defined up to exact forms so there is one of the statement is that we can in the space of a d and a sp to n z acts which rotates this thing and that is related to the fact that we can do some addition, some exact forms with non-trivial periods, I've ignored all these things at the moment what I'm interested is that there is some object called f of a so for every and this is reminiscent to complex integrable systems, these algebraic integrable systems, algebraic because of this integrable because of this and complex because I take complex space and that's because we have two types of periods in a in a real situation we don't have two types of periods so that thing don't happen and the periods are related and the relation defines Lagrangian sub manifold in this and this sometimes ago Losevnik Rassov and I called secondary integrable system, secondary because there was a first integrable system where the fibers were Lagrangian and this is secondary because this is now on on a double of the base we have Lagrangian sub manifold and we can deform Lagrangian sub manifolds to some other Lagrangian sub manifolds using the again integrable Hamiltonians but now in this in this system so for example the polynomials in the symmetric polynomials of a or a dual will define some Hamiltonians which I can use to deform this Lagrangian sub manifold and have some Hamiltonian flows and that would be another integrable system and that also appears in the business but it's the moment we ignore we are working with the first integrable system okay so now we know that the classical algebraic integrable system comes with a pre-potential which is also holomorphic in the coordinates on a base so there is now we have this and we have that and the statement was that in the n equal to super young mill theories which is originally it was by Zeiberg Witten for gauge group SU2 pure young super young mill theory which means that pure n equals to super young mill theory means that's a minimal supersymmetric young mill theory with eight supercharges and they calculated to this I have started with Witten and then the people I mentioned notice that the that f corresponds to the pre-potential for algebraic integrable system of periodic toda I have to explain all these things a little in more detail so if I take what's called periodic toda algebraic classical algebraic integrable system it will have same pre-potential as Zeiberg Witten theory with gauge group actually after it was done for SUn for gauge group SUn pure super young mill theory and then there is another interesting example which we already saw when we were doing this thing which I called n equal to star theory which was the pure n equal to theory plus one adjoint hypermultiplet with mass m that we had before when I was doing to the measure that would correspond here the people also notice that to elliptic collogero-moser classical algebraic integrable system and then Donaghi and Witten wrote some large paper explaining some of these things more generality and now it is known that there are infinitely many examples when there is this correspondence so now what I will be doing I will first explain the couple of these examples and then I will ask the question so that will be question of the type I was doing before what if I want to take this classical algebraic integrable system and quantize it consider its quantum version what will happen here now I had general approach in two dimensions which is important in two-dimensional theory let's say on a cylinder times some tori I had general approach that these two plus k-dimensional theories viewed as a two-dimensional theory with infinitely many colloquial modes had some w effective and I can find integrable system which has same w effective do you see the question is very similar now so I view our dimensional theory as a cylinder times torus as a cylinder in a cylinder it has infinitely many fields and comes with some effective twisted super potential which has a property that is critical points are vacuo and from the other side exact this vacuo corresponded to some quantum integrable system all states in quantum integrable system so now I have a very similar question I am told that there is some classical algebraic integrable system and there is quantization and I am asking question these guys quantized what they will do with the picture here because picture here is classical I mean although it's a four-dimensional quantum field theory but its effective action is a kind of teresoclassical geometry so I quantize the classical geometry there and ask what happens here and I will give a precise answer to this in the case of these two examples a pure super young mill theory or the n equal to star theory nekrasov and I found very explicit answers and then we made some conjectures and then development happened so which I will be telling you now okay so let's start with the first example so we get some flavor what this algebraic integrable systems are and first example for me will be elliptic collogero mother and you will see some some flavors of my second lectures the beginning of my second lecture because now we're talking about quantum many-body systems this is example of quantum many-body system quantum many-body system and we already had one quantum many-body system which was n particles on circle with pairwise delta function interaction so there was one particular quantum many-body system and we started showing a operator with this this is another quantum many-body system which is much harder than this of course but claim will be now that we will be able to get quantum answers using the correspondence I am describing so what is this system this system again of n particles let's call coordinates of n particles q1 to qn they are on a circle with circumference is beta circumference beta so which means that the qi and qi plus beta are identified and they interact with pairwise potential so Hamiltonian h2 is probably one half from one to n plus potential which is a pairwise and the formula for so let's write this way plus potential qi minus qj sum i not equal j where u of q where q is a difference of these two is parameters are like this m square sum i less than j well i already wrote this so m square times y-stress function times y-stress function p of q and y-stress function is a double periodic generalization of one over x square potential so this sum over n in integers of one that's the way I want to think about it and then I explained that there are differences sinh square x plus n beta so I take one over sinh square shift x by n times period and sum over z or this can be viewed as a double sum if I remember that one over sinh square itself is a sum of one over x square so but now I want to think about this y-stress function as one over sinh square of x plus sum from n equals 0 to infinity n equals 1 to infinity sorry and let me call it k q to the power k uk of x where q is exponential of minus 2 beta so what did I do I have this infinite sum actually double infinite sum so one of these I sum up to one over sinh square x and the other one I wrote as an infinite series okay because I can expand this guy in exponential of minus beta n that's what physicists likes and that's why I would do like this and that's why I want to think about it because I will be doing case when q goes to zero which means beta goes to infinity that's what I call open system and in open system is just one over sinh square potential and this is called trigonometric Sutherland that's the way I want to think about y-stress function now okay so this system is classical algebraic integrable system in a following tense think about phase space now okay think about q's to be complex right from the beginning we start with a complex q's with identification with a real period beta right because this beta is real I pick it real but it has also potentially such that it has imaginary period exactly equal to i right so q's will be complex so this will be one of the coordinates and a piece will be the tangent to it so I got now these two n are my piece and the q's which are described over there so I consider everything complex Hamiltonian is holomorphic right you can write this was h2 I wrote h1 will be sum of momenta okay h2 is that and there is h3 and so on so I have a projection to some subspace then there is a question now how effectively I can describe this projection now this is a case of elliptical georomozer which comes with what's called spectral curve so it comes actually with a lux operator not all of them come there is no word here about lux operator but this one has a lux operator which looks like this we consider matrix valued function which depends on z and our piece and the q's so I introduce extra parameter z and I write now a matrix with matrix element phi ij equal pi our momenta p delta ij plus m i the same parameter m which is there times theta function shifted by qi minus qj I need normalization theta zero divide by theta not shifted by z and theta z where theta is times one minus delta ij where theta is just the theta function theta x is sum from i one to zero let's call it k again k to be integer plus one half minus one to the power k q to the power k square over two exponential of k x and q is exponential of two pi i tau and tau that we have in our problem is i beta divide by pi so if tau is i beta is real i beta device q square becomes exponential of minus two beta that will enter here and this is a matrix and then there is a statement that Hamiltonians of our problem those Hamiltonians that I only gave second one I gave h2 I have to give you all h all h's and I will give it now are determined by spectral car I call it c h which is sits inside c times c star and definition of spectral car with determinant of x minus phi of z pq equals zero so if I expand this then each power of x will have some coefficient which we will determine in terms of phi and the claim is that second one will be this one and there are all the other higher ones okay this determinant will have maximum of n's power so we will have n Hamiltonians which will all be described in terms of so Weisstrass function for example comes from some derivatives out of the set function and so on and so that's basically definition that spectral car what is that excuse me what is that when you come h I am expanding yes but what is that when you expand this depends on that you have to take the though it's a double expansion you're right because you said the power of x okay but I still depends on that to put here that I think coefficient there are coefficients of x to some power and z to some power I'm sorry I I can't answer this sorry they determine z well let me think can we take the just one of your sinh square case or simply z that's will take so much time for me now and I mean sorry I mean so the what claim I wanted to make is that see once you write this determinant and expand you have expansion and they they they are hidden the coefficients I'm sorry people right Toda is the same oh yes yes that's yeah I mean text there text in the potential is z but this one I that's a new one so that's a car so I'm writing no this is not the same max okay well the text is z sorry okay I want to make one more claim here this is some work of creature from 1979 or something I just don't want okay so now the we had these five we had fibers h inverse of h and in this particular case they are c which is a complex one complex coordinate which describes center of mass of there are obviously center center of mass can be decoupled here because it's a pair wise interaction you can the center of mass is free times Jacobian of this compactification of this curve I call it c hat this is non-compact and the Jacobian of this curve and after this the our periods that we defined in terms of the integral of pdq over a or b cycles these games become periods calculated on this car of the form x dz now the Jacobian we we compactified is the car so these are to rise we have a and b periods and they will be called a I and a I do so this is a case when we have a car c bar h yes so this is a this is a Jacobian of the compactified car and we have the coordinates on the car of x and the z related by this formula and the a's and a d's are related by calculating a and b periods on on that so this is only knowledge I need and then you can test that c this gives us some some prepotential so a periods and b periods will be related by same formula that I had here which this statement is the same that a I dual is a derivative of f with respect to a I so that's same statement what's written here and you can check that see this gives the periods now the periodic total system is a limit of elliptical Gero Moser so once I describe one situation I can describe another situation and that limit so we just remember that q is exponential of minus two beta and that limit of periodic total so very well known and old is we take beta goes to infinity which is same as q goes to zero we take m to infinity but keep lambda to the power two n which is same as m to power two n times q finite and then you get the system now I write that another Hamiltonian of n particles here will be coefficient lambda square this one which came from m over there or following sum from one to n minus one exponential of q i minus q i minus one plus last term which is the q n minus q one and that's why it's called periodic toda that if you would just sum this and leave this term out this is open toda and in that case you can see that we could rescale every q and remove lambda square from here but since we have this last term then lambda will appear here so you cannot remove completely so you can think that lambda square stands in front of it and shift q by log of lambda or something difference of q by log of lambda then then it will appear here so this guy the open system would be here so that will be open system and this is a periodic one when the last one interacts with the first one so that's a periodic toda but mainly the potential diverges because you have m squared times one over sine hyperbolic squared so how do I take this limit now okay this is m goes to zero q goes to infinity and I have one over sin square you are saying that how okay sorry I can't explain there is another question you can ask there are many questions no no the q's have to be redefined you know with your your question is related to the fact that's okay let me write this way so q i of actually I don't have this formula but q i of the elliptical logero mozer is something like one over n times beta which goes to infinity plus q i of periodic toda so there is you have to remove something anyway there are many questions can be asked here and is this one of the part of things that I am not expecting I need only these statements that one gives another in the limit and when you do the technical calculations then you realize that you have to do some shifts and so on but the interesting one is that this one is not symmetric under permutation group that one is symmetric under permutation group q i with q j so you lose that thing it's it's only nearest neighbor this one that one was pairwise okay so things some things happen but just formal limit is that and it's something okay so now I at least managed to introduce what are these models okay so if you are studying some classical mechanics book they give you n particles like this now the period disappears because I sent beta to infinity so there is no period so now this again holomorphic but you would think that suppose queues are real you are on real line you have this Hamiltonian as you see this is exponential potential it goes like this and they tell you now replace p by d over dq and calculate spectrum of the Schrodinger operator of this thing you would spend lots of time before you figure out what to do and the first thing you will realize that if this term is removed situation is simplified because if this term is removed in a let's say two particle case remove the center of mass this is p square plus exponential of q potential and now you can start talking about scattering matrix now you can start talking about q goes to minus infinity q goes to the plus infinity you have a reflection coefficient you have a penetration you can you can start talking about but when you have the other one then its situation becomes compact now there is no notion of his matrix for this and so on so what we will be doing from now on we will better put this guy here and consider this theory as a perturbation of the open system with a coefficient of expansion lambda square here we will consider this theory as a perturbation of one over sin square with infinitely many terms in a potential that you are perturbing in q in case of periodic toda it's just one okay well one in a sense that in two particle case okay because there is still actually no it's a one always so that's an exact way we want to think okay now statement there that I attribute it to Gorski, Kritschever, Morozov, Marshakov, and Mironov and Marcinek, Warner and Donagy Whitten was that take periodic toda which was pure n equals two calculate ai ai dual where I just said and find f okay by calculating this period it will get same answer as I work with and prepotential and moreover actually Zeibberg and Whitten gave description of their modular space exactly in this language they wrote the curve they wrote the differential and they said calculate periods of the differential you get f so what happened that's the in the case of periodic toda this our friends found that this car was the same as a spectral curve of the periodic toda so and then it was realized for other thing and so on in a second before probably end of the first hour I will move to something which is called the Hitching system which I already introduced actually not but not as an integrable system as as a manifold the model life space of solutions to Hitching equations and I will use the statement which comes from Gaiotto-Murin-Netzke that there is a construction that if integrable system of Hitching type there is some quantum field theory with this many supersymmetry which has a modular space given by by the base of the Hitching integrable system okay so at the moment these are just two examples now I want to move to the question of quantization and quantization will change many things here but I want to keep as much as as close as possible to the language that I had here again I really apologize to certain things that see I am not giving answers like this immediately because there are so many different subjects of different people developed and so on and my job is to find connection between them and if you dig on each one then there is entire topics that goes there and in principle I might forget some things to say or things like that okay so now we go to quantization in order to get quantization and use previous methods that I spent already five hours talking I have somehow I need to hook up to dimensional theory out of it because if it's four-dimensional theory as I said it's a manifold if it's two-dimensional theory on a cylinder Zivacchio are given by discrete set in a four-dimensional theory it's a manifold manifold is this okay so what I do if I would do with this four-dimensional theory if let's call it try one first try if I would consider Caluzzi Klein reduction Caluzzi Klein reduction I already answered this question in my previous lecture I would get something like XYZ model right I said that the if you are in three dimensions and you go down on one circle that you get XXZ model so in two dimensions you hook up this gauge theory which has supersymmetric vacuums being XXX then you say now let's consider three-dimensional theory which dimensionally reduced to two dimensions gives this one okay we find such and then we say okay now let's not reduce dimensionally but include all Caluzzi Klein modes then the statement was that theory will have supersymmetric vacuums which is given by spin chain of XXZ type then I said that if I now go to four dimensions and find such theory which dimensionally reduced to two dimensions gives me my whatever theory which had XXX as a vacuum sector that four-dimensional theory for arbitrary torus compactification with all Caluzzi Klein modes would give me XYZ model and this is one of those actually so it's not good toroidal actually that thing on toroidal compactification because it was n equals two and not n equals one in four dimensions will give me n equal four in two dimensions which I can also treat because my n equal to say or included n equal to say or is the form by masses and so on so anyway so this is not a good one so now try two is is omega background with one epsilon they generate omega background and this is one which we also like to denote like R2 times R2 with one epsilon and there is a super young mill theory on this one and that super young mill theory has four supercharges because this omega background breaks half of supersymmetry I explained what was what was this R2 epsilon but they said maybe for physicists the simpler way to think about this as a R2 with some metric which this parameterized in terms of epsilon there is explicit formula for that or you can think that you are doing rotation of the R2 with some angle epsilon and everything you are considering equivalently but now I like a Lagrangian which has which is a deformation so quantization here will be now deformation so I'm deforming four dimensional Lagrangian by breaking half of supersymmetry and actually also super Poincaré symmetry and super Poincaré will be only here but I don't care right only think I care that my theory has to be super Poincaré on cylinder whatever happens ours there we don't care so we are I am breaking lots of stuff here but it will have super Poincaré here and the Lagrangian of this theory is the same almost the same very simple deformation of the original one so now I'm writing the bosonic part of the Lagrangian just for some people in the audience to be happier but I could skip this so this is young mills term then there is interaction of scholars there are scholars of course so this is covariant derivative of scholars I have to shift it by using the vector field rotating the R2 here I will in a sense so this is a two form I have contraction of two form with a vector field I get one form I shift covariant derivative of one phi which is a one form by this shift then I multiply by its complex well by star hodge star which is taking model y square basically so this I do with a conjugate one with with phi bar phi is a complex color plus usual term one half trace commutator phi commutator with phi bar let's get shifted with again contraction of the one form dA phi bar with epsilon okay I will put epsilon somewhere else minus IV bar of phi square the g0 is a coupling young mills bare coupling constant and then there is a theta term I call it theta zero trace trace is in a joint representation f by g f so this is a if I remove here the letter V this is the bosonic part of the Lagrangian and what is V V is a vector field which is rotation in R2 epsilon so it's x2 d3 minus x3 d2 and coefficient I take some complex number I call it epsilon okay so think about this Lagrangian that what I did I had here derivative I twisted it by contracting the curvature of the gauge field with a vector field here and here and I had commutator square of two phi's which I shifted with the contraction of the one form in epsilon and and the other way rotate so this is an action anyway so claim is I could have done by the way this thing in both spaces and I would have very similar formula but I had to do the contraction with another vector field so this would be V2 and there would be V1 which would be in a space of x0 and x1 so this epsilon 2 epsilon 1 x0 d1 minus x1 d0 and I could have put that thing also so I would have then another contraction in addition to this one linear combination in here but I take epsilon 1 equal to 0 is x0 supposed to be in time x0 is time yeah should there be a plus sign between because you mean I or something it's all Euclidean it's all Euclidean it's just x0 to x4 and I write R2 times R2 but convention is that see these are two that I kept here has a time not this one so I break here the Lorentz symmetry but where the time is I did not break so I get sorry and then in Euclidean space I take space dimension and make it the circle so this R2 is actually a cylinder and then I mean original setup okay okay so now this theory as you see now as it's written has a Lorentz invariant Poincare invariant only in R2 so now I take this story and do like a lot of client I would say I want to treat this story as two-dimensional two-dimensional theory on R2 with infinitely many fields here how do I package this infinitely many fields well obviously the one simple thing would be I take any field phi which depends on z1 z2 z1 bar and z2 bar where the z's are x0 plus minus x1 or x2 plus minus x3 and let's suppose my second R2 is this one z2 and z2 bar locally and I write this as an infinite sum nm of f of z1 z1 bar times z2 to power n z2 bar to power m and exponential of minus epsilon z2 absolutely square to regularize the integrals and this way this guy will have index n and m so my my Lagrangian now and I integrate over z2's my Lagrangian will have infinitely many fields phi and m there are zero modes which are massless but rest of them will be massive and masses will be proportional to one over epsilon whatever one over epsilon that thing when epsilon will go to zero they will become massless okay but when epsilon is not zero the masses will be finite for all each modes n and m this can be verified and so on so now two things are used here and this is important first thing as I said that the main success of the reason that cyberg Whitten were able to write exact solution to low energy effective action which I said again with similar statement as calculating twist effective super potential in two dimensions was that also in four dimensions like exactly same thing in two dimensions n equal to zero in four dimension is one loop exact okay so the n equal to zero in four dimensions is one loop calculation of determinants plus infinitely many instant on corrections so perturbative part is one loop because of supersymmetry it's only one loop and then there is a non-perturbative correction now non-perturbative correction as Poliokov explained in his original instant on paper and so on usually comes with instant tones there are anti-instant tones there is an interaction between them and so on now second thing comes in n equal to theory which is statement of holomorphicity holomorphicity it has a property of holomorphicity so either instant on will holomorphicity was all over here I said it's holomorphic in a and everything so the restriction because of its n equal to which leads to the holomorphicity as a function so the the scalar in vector multiple expectation values it's not allowed to have both instant on and anti-instant on contributions right which means that instant tones don't interact with each other instant on and anti-instant on don't interact with each other it drastically simplifies everything because the instant on corrections are now only of instant tones and no anti-instant tones and no interaction and that means that corrections non-perturbative corrections are integrals over modular space of instant tone of something and then the question is of what that something is now most natural thing when you have such a nice space as a modular space of instant tones to ask in some sense what is the volume is it big or it's small or how big is this space and things like that and these are the questions from intersection theory you define differential forms on the modular space of instant tones you start intersecting them or cycles or into integrating differential forms of the proper degree and so on and these are all involved in calculations of instant on corrections non-perturbative correction but perturbative is a correction is just you take this Lagrangian add fermions and so on and calculate the quadratic determinants from here okay you get some answer now statement number one is statement number one is that as a two-dimensional theory on that cylinder it has n equal to super point caracymetry and thus it has some effective twisted super potential right because I said any n equal to theory has some effective twists in two dimensions has some effective twisted super potential so we ask question what is effective twisted super potential of this two-dimensional theory and now I'm using letter a to describe scholars abelianize scholars which in two dimensions I used to call sigma they were coming from phi so these are the zero modes of phi under this omega background reduction so I write now some result of Nikita and myself from some years ago that oh yes somewhere it disappeared but there was here secondary integrable system remember I introduced secondary integrable system which was in the space of ace and a duals and I said that relation between a and a dual is a half dimension sub manifold which was Lagrangian with respect to the regular symplectic form so I said that there was this thing and so so this integrable system comes it's with its own Hamiltonians let's call the coefficients the times of this Hamiltonian flows Hamiltonian force comes with times and since this has dimension n the complex dimension two and real dimension n there are n real Hamilton n Hamiltonians let me call those times ti so again this is a two n complex dimensional uh symplectic space with a symplectic form da reg da dual as such it comes with two n Poisson commuting Hamiltonians let's call them h i's then there is a Hamiltonian flow with respect to each Hamiltonian let's call the times of those Hamiltonian flows ti's and then now I slightly generalize the statement that this that means that this pre potential is a function of two types of variables a and the t because I can for each Lagrangian sub manifold here I can do the Hamiltonian flow and get another Lagrangian sub manifold right and it would be as good as a previous one so they're exactly as many times as their ace if I go to spectral curve the just for experts the statement is that see these variables are describing family of spectral curves right not just one but family of spectral curves and if I take a and the t together it will be the generic spectral hyper elliptic spectral curve of the degree if I put t is two equal to zero the hybrid written spectral curves are uh having half number of the model line but with t they have all the number of model so now I make a statement if you take t the complex please are coming up with uh are you there obliged to specify the the real the curve at the moment everything is complex so just for meaning what is the ti is if you have the Lagrangian in four dimension you can always add as I was talking many times the chiral ring operators with some coefficients and nothing that that's not it will be similar sorry okay these are those ti's which means that this salary as a topological field sorry for dimension is exactly same same as the zyberg written sorry it has a different pre potential which is related to the pre potential of zyberg written by Hamiltonian flow in secondary integrable system and this is a result of law seven across of myself from 1997 so what I'm saying add the observables like we were doing in two dimensions claim is that that sorry will have pre potential again and that pre potential is related to zyberg written pre potential by Hamilton Jacobi equation where the t's are times of Hamilton Jacobi equation and that means that they are defined by another Lagrangian sub manifold which related to this one by Hamiltonian flow there are some many technical details to and that I said in the paper with loss of an aggressive we explained that in 1998 so this paper was called the issues in topological gauge theories I vaguely remember actually what we did there but that's basically what we did so that means that there is a notion of the effect it with its super potential now which is the function of a and the t exactly like there was notion of the pre potential as a function of a which were coulomb modulized there were the parameters Higgs expectation values there are now as many t's as a's they have different role a's are dynamical t's are oxyler okay and the formula was that Nikita and I showed that this was a log of epsilon two goes to infinity epsilon two times sorry what I'm right limit of epsilon two goes to infinity epsilon two times partition function when I put here both epsilon's now when I put here both epsilon's and if you look on this expansion I have to do similar expansion with z ones right what you will end up will be zero-dimensional theory because the phi's now will have two more indices and they will be numbers right and all of them will be massive except zero mod so it got a matrix integral so the theory is zero-dimensional and that zero-dimensional theory with two epsilon's has a very well-defined mathematically well-defined partition function z which is a product of z perturbative times z non-perturbative z perturbative is basically one loop determinant of this thing okay because I said perturbative part is one loop exact so the perturbative is a classical plus one loop the non-perturbative has a form of sum q to the power n let's call it z n of epsilon one epsilon two a n whatever else from n equals zero to infinity so we start with q to the power zero and the z n's are integrals over modularized space of instantons of n instanton charge regularize somehow with epsilon say with something that it's specifically model dependent and the q is well exponential of i tau where tau is one teta zero over two pi plus well coefficients I always forget what is four pi i four pi i times g zero square what four pi divided by g square four pi right i here and four pi divided by g zero okay is there a two pi in exponential way I wrote actually way I wrote exponent of two pi i tau okay two pi i tau okay well I'm I'm getting tired looking on my notes it's over there so this is the notation okay now let me explain a few things here the integrals are modularized space of instantons will usually diverge I mean for example the pure n equal to sorry has just identity here identity integrated it's just a volume and epsilon epsilon two is some regularization more more in aggressive and I introduced in 1995 or 94 or so on but anyway so this formula for z n's is determined it's model dependent from different model you change the model there is a different sink you're integrating here and this as long as epsilons are non-zero they are finite when epsilons goes to zero they diverge so you don't have to really compactify the modular space of instantons you just have some recipe as I said that developed in in here these ones the perturbative classical loop contribution was very detailed studied by negrasov in 2002 and negrasov and okunkov also probably in 2002 or 2003 so there is an explicit precise mathematical formula for that there is a precise mathematical from formula for this which is again the model dependence and the claim is that c w effective of two-dimensional theory of that theory on a cylinder is given by this and there is a similar statement which is important now for us I put a log and then I okay log here yeah I put it there and then there is a log of z so there is a before we finish this discussion there you have to explain this one again as I said each of these terms is precisely defined by the authors I wrote here and there is an which we need in parallel statement that f of a of zyberg written which also depends on t as I explained is actually similar limit of epsilon one times epsilon two of log of z again it's the same z that I described here and this was proven by nikita in 2002 well many other things he did to me about this but so this is a known fact this is a the result that I claimed is true and now a simple conclusion which we will need for later as very much motivational inspirational or whatever you call it that will be useful is that we look on these two formulas now we don't at this moment you don't really need what what this non perturbative formula that we wrote was or what's perturbative one is what's in the second it will become poor just we look on to this formula and we conclude that z is equal to exponential of w effective twilder a plus o of epsilon two divided by epsilon two and another statement we see that the w effective of a and t and so on is of the type of f zyberg written a t plus of epsilon divided by epsilon so these are two statements you want to take epsilon two zero epsilon's go to zero in both okay so here epsilon two goes to zero and we defined w effective as a leading pole in epsilon two okay so the rest of corrections are of epsilon two and you have to prove that's actually true that it's only of pole of this type but this is easy to prove excuse me residue yes so residue of the leading pole here and similarly here after we calculate w effective now we send second epsilon to zero and similarly residue of the pole is given by f now on each step you have to prove that it's actually like this because no one told you in the beginning that the type will be like that but it's very easy to prove that z has a property that it's exponential of one over epsilon one times epsilon two something which are positive power series expansion on both and it comes from the statement that log of that since it's four dimensional field sorry is proportional to the volume of four dimensional space what is the volume of four dimensional space volume of r4 if we have both epsilons nonzero which was somewhere here written volume is one over epsilon one times epsilon two and the proof of this is trivial now let's do this simple equivalent calculation this of course equivalent volume so we are integrating over d z one d z one bar d z two d z two bar integral and now we have to do the equivalent with respect to rotations with epsilon one epsilon two which corresponds to putting here minus epsilon one z one moduli square minus epsilon two z two moduli square so this is a current more volume of the r4 with two epsilons and what was written here it was this formula of more across myself is a current volume of the incident moduli space which itself is written in similar linear coordinates called the linear hypercalor quotient so that's a similar formula anyway this is one over epsilon one times epsilon two and that's the volume of r4 so you can just buy this kind of quantum filthy arguments claim that this z that has to have a form one over epsilon one times epsilon two and everything here has to have less singularity so when you sum up over epsilon two you get pulling one over epsilon one and that would define w when you sum up with epsilon one you get entire thing okay so this is now very important and i make the break in a few minutes why this is very important now we go up and look on that formula that formula told us that the vacua are given as a solution of this equation which we now write here and we get that vacu are given by cyber written prepotential d over ai equals two pi i n let's write here square of minus okay i put this thing on that side one over two pi i epsilon times ni plus of epsilon square so w effective was this we plug the key over there we get the equation defining the vacua and we get the equation of this type which and which basically means that if we interpret epsilon as h bar this on the lowest order is a borne zomerfield quantization okay so borne zomerfield quantization turns out to be leading term given by cyber written prepotential as the potential and why is borne zomerfield because i spent some time explaining that this guy is a conjugate variable to the angular variables which were along the circle so we have this statement is the same as momenta is equal integer okay because df da was that kind of dw da was a moment so what we have here that this statement with the way i just explained is exact borne zomerfield because this guy was interpreted at exact conjugate momenta right and in the expansion in h bar leading term is given by that okay when i come back now i will explain what how we should think about this so motivation is like that we could not succeed by doing toroidal compactifications from four dimensions we had the classical integrable system in four dimensions toroidal compactification did not lead us anything new it was giving us the spin chains again so we decided to do this excellent thing an excellent thing we know that there is a w and by all means using this very old results this result of nikita this results that we conjectured this result of nikita and okunko this result of murne grasso and myself and many many other things we come to conclusion that w effective calculated this way should lead to the exact quantization when epsilon is h bar so now we have to check it so this was very inspiring and motivational so now we have to check and when i come back i will check it for the case of elliptical ogero which means same as a periodic toda and then we'll do it in generality and formulate the questions okay so we meet in 10 minutes or something back here now what is the quantization if we forget about gauge theory so i may declare first thing let's see i did something with supersymmetric gauge theory which was defined with a classical Lagrangian here which was defined by instanton integral here so this all which was as i said studied and proven by nikrassov here then we took one epsilon only to zero here are both epsilon to zero here is a one epsilon to zero and calculate w and then this theory completely fits in the set of theories i did in the first lecture so i declared that the spectrum is given by that formula where i have to put w effective to be that and moreover i claimed that see it gives me also for any time by but this is a separate discussion that i would not waste time on it now let's see what would be the quantization problem if i go so we have two sided relations one side is a deformation of super young mill theory from another side is a quantization of algebraic integrable system and let's formulate straight forward quantization problem that one would do with such Hamiltonian system in a following way we take that Hamiltonian and write that p is epsilon d over dq and then we also replace m square by this shift separately needs to be explained by m times m plus epsilon so what we get there is differential operator epsilon square over two sum from i equals 1 to n d2 dqi square plus m times m plus epsilon sum i not equal to j wire stress function acting on psi of q equals energy i call it e2 because there are other Hamiltonians as i said of psi of q and now it's very simple that if i declare in this notations epsilon to be minus i times h bar m to be i times h bar times nu so this and rescale by what i will get that this will be h square times nu times nu plus one so here i will get nu times nu minus one here i will get minus one half and i rescale by h bar this epsilon e2 so this is an operator minus one half d over dq square plus nu times nu minus one sum of wire stress function this is an usual formulation of the question of the this is the operators that Kalujero and Moser wrote okay so now what kind of spectrum are we talking about so i'm formulating now first i formulate a conjecture to what kind of spectrum this equation leads so we're looking for a quasi periodic in beta symmetric functions of n variables psi q1 qn which now this potential one over sinh square okay he has a one over x square pole at x equals zero so when qi goes to qj it's one over x square type so i have to specify what what behavior i request from wave function at that end i claim that i want to choose at qi minus qj goes to zero i want to choose a solution which behaves psi i behaves qi minus qj to the power nu now assuming that nu has a positive real part or if there were around i can request there is a replacement of nu to one minus nu and this is simerian so i can request in the other case one minus nu so i will this is a regular at zero so this is regular at zero and then i expand so i said quasi periodic symmetric and outside this chamber i expand by periodicity right so i show in the chamber let's say two particles in two particle case i have interval from zero to beta right i require quasi periodic which means that the the psi at beta equals psi at zero times some exponential of quasi momenta okay and then outside here i expand by periodicity by set formula i just stated then the equation dw d sigma equals n is a statement that the quasi the period the quasi momenta is integer so it's actually periodic again i repeat i'm looking for a solution which at zero behaves as q to the power nu it is periodic i extend it outside the periodic and if i have n variables it's also symmetric now that leads the the proper solution the one described in a following sense the main part of our operator is a sutherland one over sinc square potential one over sinc square potential which i will call h the h zero this h zero has it's a two solution psi one and psi two and these are hyper geometric functions and i pick this one let's say which has the solution of q type of q to the power nu these are the hyper geometric equations so i know exact answers then i put a condition of periodicity periodic with a real period beta i get the equation that this quasi momenta equals integer then i turn on q so i turn on unfortunately this is also called q this q first term in a potential right now my potential is not one over sinc square but it's one over sinc square plus q times this easily i write what is psi one in terms of psi one two in terms of psi one so this is expressed in terms of psi one okay has the same properties that it behaves as a qi minus qj to the power nu on when qi goes to qj but it has different quasi momenta now okay so i calculate quasi momenta and again put it equal to an integer right because these are what what well there is epsilon everywhere epsilon is in some integer it's always right hand side is always integer i'm as i wrote periodic in beta symmetric function with this asymptotics so i have to put the periodicity if i would take that guy okay it will not be periodic this is the this the solution that i'm looking is quasi periodic okay this way anyway and in each order now the next order i put q square psi one three and i again put this at condition and now in the q square level i have to put here in in this guy i have to do the consistent expansion this has to be expanded so potential gets expanded and solution gets expanded and each order i put condition this and i get some equation on quasi momenta and claim is that the Nikita and i claim that this is a solution it describes now what about if i take that limit to periodic toda so in periodic toda case this potential now is replaced by sum exponential qi minus qi minus one plus lambda square exponential qn minus q1 and luckily for this particular situation everything was complex this holomorphic equation as i described i just required the period to be on the real line but this one has a different this equation has different way to set up the problem because now what i can ask i can consider this on the real line and ask it to be l2 on the real line curve integrable and if i have n particles then i have to take the symmetric power okay so this question has been solved before so in case of elliptical homoser there was no solution before but in this case there was a solution so this problem of periodic toda has a long history so first one probably was Gutzfiller who wrote a very famous paper including some hill determinants and so on and formulated the problem and so on but did not finish it so it's 70s i think Gutzfiller then in 80s Sklianin and Gadin and Paskier wrote it in a way that i will now use in a sense and then in 90s based on all these things Harchov and Lebedev completely finished it centers out that the solution dw d sigma equals integer where w is given in the formula that i raised completely coincides with the formula this guy wrote exactly it was coincides it was conjectured by Negrasov and myself and proven so we just noticed that the formulas are same and was completely proven by Gazlovsky and Teshner so story goes like this first thing you do is a separation of variables which means there is some general theorem you correct me if i don't if this is a classical integrable system you write quantum integrable systems such that Hamiltonians commute instead of Poisson bracket you have now commutator equal to zero you have some differential operators of n variables which commutes are n of them then there is a theorem which says that everything can be expressed in terms of the differential operator in one variable and if you solve the differential equation in one variable in terms of solution of that it might not be differential by the way it's equation in one variable you solve it and in terms of that you can write the general solution a correct solution so this means that now in this precise terms so we have wave functions let's label them with a set of momentum let me call it actually lambda and there are n variables you write an integral transform with some kernel which depends on a variable type q on now i write variable type x and then in this it's like it's a combination of Fourier transform and some algebraic transformation so there is some transformation where in the x-space so this is integral over x now in the x-space wave function is a product of a single particle wave function okay so in periodic toda case they found these people i just mentioned they they know what is this thing and i write it in sklein newspaper and then there is a question if psi satisfies the periodic toda equation p is given such that it separates the variables what equation q satisfies right so q satisfies equation q of x plus h bar plus q of x minus h bar is equal to t of x times q of x where where you are looking for a solution q of x which is entire function of complex variable q it's entire and with asymptotics which i don't remember but there is a fixed asymptotic such that the psi will be in l2 so you have to be square integrable with a regular measure dq1 dqn uh integrated i don't remember the sklein it's in some sklein newspaper but q has to be entire and t is a transfer matrix and there is a yeah it's uh it has some zeros in some positions which i i don't want to say but you are looking for such q now people were looking for solutions for a long time and as i said the sklein and so on and the first question is if there is something like w something like what i called young's function which defines the spectrum then it defines a spectrum so yeah this is a Baxter equation this is called the Baxter equation then it has infinitely many zeros which means that the w should be this young potential should be function of infinitely many variables right because this function is entire so they were looking for solution and there was a new thing in my understanding that came from a study of these people which completely broke the the eyes which was uh it's very similar to what we are doing and that's why we thought that we had the same solution and turned out to be true which is following the relaxing condition that the Baxter operator is entire function relax it and say that it's a meromorphic function assume that it has a pulse in some positions and call those positions now we are discussing different type of solutions there are many solutions by the way there are infinitely many solutions any solution you take then multiply by periodic by h bar function it again will be a solution right so this is taking care of biosyntetics and then uh so this guy said assume that it's meromorphic with a pulse at positions call it ai i equals one to n so we are discussing the type of solution okay where n is the same number as number of points then present a solution there are two solutions now q plus and q minus and inserting x ai's with a condition that pulse cancel details some constant okay well if you find the solution in that space then put this condition then you will find solution in this space which is formulated here because pulse cancel right so claim is this is our beta equation exponential of dw da i equals one which is exponential of two pi i times this okay this has been this is explicit exact statement that if you take x much del d i equal one identify these ai's with those ai's that see uh this guy's head as a pulse for the baxter operator in that space then their condition that q is a from proper space um and thus leading to the l2 functions on a real line and so on is equivalent to that this is harchov levide and this is exactly same equation as this one as i said when nikita and i conjecture gazlovsky attestional proved we just saw the formulas were the same they had explicit mathematical proof so this question is closed so for periodic toda system we get gauge theory giving the right answer for elliptical logero mother system as i told you in this space which i described here in the iterative expansion in q of the potential and solution we again get the right solution so our equation exponential w d i equals c one is a condition of course moment of being equal to one up to integer and describe this solution then obviously now you make a statement that anytime there is a supersymmetric young mill theory which in vacuum sector which in classical sorry which in four dimensions without any epsilons gives a classical moduli space to be moduli space of some algebraic integral phase space of some algebraic integrable system you say now and that's what we did that deforming by one epsilon that four-dimensional super young with a cyber within theory will lead to the proper quantization of this algebraic integrable system so we had to produce so we produced one non-trivial example because obviously people were jumping and saying well what to who cares about your thing and then we could reduce infinitely many examples that i want to move on now okay but statement i think is clear for every time you have this equation over here there is a mathematical formulation of the quantization problem of classical algebraic integrable system that we find and our answer is answer for that quantization quantizations are infinitely many different ones and by the way now i just made side comment that the when you put the epsilon and calculate the answer for partition function one place is completely unique the sum over instant tones you can't do nothing i mean it's precisely defined mathematically there is no ambiguity there only different models have different integrands but it's a precise defined thing but the perturbative answer is ambiguous perturbative answer was a determinant of certain Laplace of a product of Laplace operators on this space of r2 times r2 with epsilon and so on and it requires a boundary condition at infinity so perturbative answer is not unique okay i had in plans to write one formula which is interesting for future it's okay let me write before i move to the other models how do answers look like okay answers i mean how does this beta equation looks like and how do w's look like and these are these formulas itself lead to some thinking so first i will write perturbative answer as i said it's a sum of perturbative and non-perturbative so the w there was a formula which this blackboard has to be fixed actually the w has it's broken it's very annoying because i get so english language has is not that flexible you know i mean same same word can mean different things and so on once in other languages i speak i would find 10 different words okay my english is not good it's not like english is not good okay so w is a sum of perturbative plus instant tones because z was a product so w is a logarithm of z and let me first give you w perturbative by just writing this equation exponential of d w d a i if i just put here only perturbative part okay so this is equal exponential of pi i tau this is for elliptical geomoser tau a i divided by epsilon times product of s which we will see now in a second has a meaning of where s is a product of s it's an s matrix for open problem for toda but the same one appears in every other problem so this answer is almost like universal formula um this s appears in the elliptical geomoser but it has a nice meaning the same formula appears actually the difference is some number of this is probably for yeah right this is a reflection s matrix of the potential of exponential potential which appears everywhere so this basically says that if the w perturbative satisfies equation d w perturbative dx is equal x is the argument for one one of them is equal log of gamma function that's the solution of that thank god sorry m in this equation is a mass of a joint towards m was this m which up to h bar is new well you see epsilon is dimension full m is dimension full so you have to and x is dimension full column everything is you have to divide by that epsilon is h bar right okay so what about the non perturbative answer how does it look like how do we construct the instant on w so in the paper uh within the class of we had many years ago description which then got replaced by new language in the paper we wrote with vasily peston and both languages i find important so i will first give the statement in the paper we did within the class of and then four years after we did with vasily differently so take w instant on as a function of a to be written as an integral over complex plane and it has to be contours has to be specified it's a long story of so integrate this huge thing once chi i will now write what chi is and q is given so little q is exponential of i tau the big q is a certain rational function again depends on a model so big q is polynomial x minus m polynomial x plus m plus epsilon divide polynomial of x polynomial of x plus epsilon where polynomial of x is product with a zero at exactly position of ai so this guy is a function of a p is a polynomial of degree n q is a rational we describe here the only question is what is chi so chi is a solution of integral equation which basically follows from this w itself and this integral equation makes chi function of little a so chi of x solves the equation integral dy against same contour g zero x minus y times log one minus q capital q exponential of minus chi of y where g zero i don't have space now for this blackboard would be useful g zero is basically a rational function it's a logarithmic derivative of polynomial product of polynomial g zero is dx of log x plus epsilon times x so all these parameters epsilon m and so on they have meaning in equivalent integration and these are shifts of x by all possible ways so if you take derivative you get a rational function okay so this is statement where it comes from is a long story so you solve this equation in iteration by q so in the q's lowest order there is nothing here then the first order you start getting things now we get free term which is log of q times q and then you have a that's chi one is integral with this kernel of chi zero and iterate you get expansion in q then you take solution and plug in here you get now function of eight because solution here of this integral equation is completely determined by two objects which is g zero and q and both are parameterized in terms of the n numbers ai and epsilon so you get this as a function of a and then you solve the equation exponential of d this and then there is a correction here when you put non perturbative now we have to put w twill the inst effective here so there will be a correction to this and solved for ace and you find ace and then the answer now important statement is that each term here is exact in epsilon so everything here is exact in epsilon which is same as h bar so it's exact in h bar and its expansion in q which is size right i explained q was a size this was last term here here it also has a meaning of size now the statement is that these are this actually converges this expansion in q and you can draw where does it converge in the complex plane of q's instant on instant on parameter no q was expansion of itao yeah q was exponential of itao where tau is a theta angle plus one over g square so it was non perturbative okay so now where does the claim is that the answers are exact in the sense that in epsilon in h bars they are exact and in q its expansion but this expansion has a radius of convergence which is not equal to zero actually since q is a complex parameter you have entire complex planes of q's and you're asking what is where does it converge and the claim is that now can be described nice way where does it converge or not and also obviously the you can consider theories which has many little q's so this one had only one coupling constant but you can consider many coupling constants and as i understand vassily gave lectures on this but now i want to move on on some things that i had to finish now let's use this statement that was made by gaiotto moorenites okay that there are theories what they call now class s then i sometimes joke like mercedes s class so there are s class s theories that claims that the for every hitch and integrable system there is a there is a gauge theory which has a vacuum described cyberwitten vacuum described by that integrable system so we start so this what i will be talking now uses certain work by nekrosov and witten which explained these claims nekrosov and i had in the language of the gukov nekrosov and witten which explained the ns in language of gukov witten and gaiotto boss are g gaiotto witten papers and then the paper of nekrosov rosley and myself so this is probably two sides and ten and this is 2010 also probably 11 okay so this is a claim about the theories of class s so i start with a hitch and integrable system i remind you we had a riemann surface sigma with genus g and then punctures we had the connection in principle bundle a what physicists called gauge field and we have a one form in the algebra of gauge group which is denoted by phi and we'll write the equation said curvature of a we pick a complex structure on sigma this is a coordinate z and we write equation i always forget this a plus or minus one sign is correct and one is not nabla z bar of a phi z equal zero and divide by unitary gauge transformation unitary is not right word here by gauge transformations whatever gauge group is okay so this is a space we consider and the hitching claim that this is a space like that so this is a phase space actually it's even not just complex inflecting structure that this space has but it's actually has a tricholomorphic structure three independent ones so it's called the hyperkeller it's a hyperkeller manifold and he described the projection pi in one of those three holomorphic structures so we have now three holomorphic structures let's consider i holomorphic structure it means this is a structure where I will tell you what is a p and what is a q which are complex coordinates and then there will be p bar and q bar which are complex conjugates so I have to tell you which one is one and one in which one is another so in i holomorphic structure az is az and phi z bar are holomorphic coordinates and az bar and phi z are anti holomorphic coordinates and seplectic form omega which is holomorphic should depend only on this it's an integral over sigma of trace delta az wedge delta phi z bar and this is over Riemann surface and this is of the type in a holomorphic structure it's 2 0 because I have both holomorphic but on sigma this is z bar so this is 1 comma 1 form which I integrate over sigma okay there is another holomorphic structure which is called the j holomorphic structure which I will need later but and there is a k there are three of them i j and k I will not need k in my lecture today j holomorphic structure there are holomorphic coordinates are ac I will call it which is a plus i phi okay it's clear everything right and there is a plus i phi and there is a conjugate one complex conjugate one this I will not be using and the 2 comma 0 seplectic form is integral over sigma of trace delta ac wedge delta ac now here I have to I do I'm not finished this solve hitching equations so we have to use this holomorphic structure this seplectic form all the solutions to the hitching equations which are over here here we have to use that holomorphic structure on the solutions of the equations f a plus i phi equals 0 divide by complex gauge transformations and this statement I already used that this space as a space is the same as that space there is a theorem about that and explanation was given that this equation has a real and imaginary part real part is the first equation imaginary part is the second equation plus its complex conjugate and the missing third equation there is a gauge fixing in a complex direction of this equation this equation has more symmetries so these spaces are the same so this seplectic form with that is the same space as that seplectic form with that equation and now there is a statement which I will make after Vasi asks a question just the direction so it's minus is this convergence because of the curvature and right file there is minus so as I said it's one sign okay okay I had to write ising minus here anyway okay so hitching defines the integrable system in this holomorphic structure so as I said as a space this is our m phase space and the same is here m phase space is here and there are the same spaces but now we need to define this projection so we need to give Hamiltonians and hitching defined Hamiltonians using using spectral curve again so some equation I will now do for su2 because I want to focus later only on su2 if I have time actually anyway so let's restrict gauge group to be u2 or make it sl2 actually and I don't want to specify real structure or not so gauge group will be sl2 which means that see when we talk about unitary connections here it's su2 and there is sl2c there okay so he has to give a Hamiltonians and here are the Hamiltonians in a simple in the form which is very similar to way I was writing trace sigma to the power i in my first lecture for simple modules in two dimensions here it's like that hi's i goes from 1 to 3g minus 3 plus n whereas this is the Riemann surface of genus g with n punctures is equal integral over sigma g n mu i times trace phi z bar square z square it has to be holomorphic oh sorry z bar then where mu i's are basis of exactly this many dimensional linear space of beltrami differentials so this is a zero two form and the other that guy mu i's are one minus one differential so the zero two differential one minus one form so we get one one form integrate over sigma these are h's obviously they are Poisson commuting because they depend only on phi z bar they don't depend on a they depend only on half of them so they are Poisson commuting here the symplectic form tells us that a and phi z bar are conjugated so phi z bar is like coordinates and a is like momenta and Hamiltonian is a function of half of it only these are Poisson commuting these are integrable systems this gives us a projection so we are done here is integrable system now um comments consider g actually to be un and consider your Riemann surface to be torus with one puncture okay then this one this thing is elliptic Kalojaromosa this i believe is a result of Gorski and Negrasov from 94 second comment suppose g is sl2 as it was there and we consider Riemann surface sphere with n punctures then this is n particle gaden system okay so what i'm saying is that now that these two comments say that for specializing Riemann surface number of punctures and genus of Riemann surface and the gauge group you can get different well known integrable systems and one was elliptic Kalojaromosa which was torus with one puncture and another was a gaden spin system which is this this is for sl2 gaden did the sl2 there is an sln generalization of this of course okay so from the other side this theory now i already explained is n equal to star super young mill theory in four dimension and quantization here quantization was super young mill theory on r2 times r2 with epsilon so this one we already studied what about this one so this one corresponds the way i just wrote here it corresponds to if there are n punctures nf equals to nc and equals to n super young mill theory in four dimension what i wrote here is that the flavor group sub b and sub gran minister calls so this is a theory where we have a un gauge group with a flavor group of of umf which is two times nc so we have i if you remember i had example i had l fundamentals l anti fundamentals and one adjoint in two dimension this was my main example now what happened and the that the this is equal to two times n similar situation i have the fundamental plus anti fundamental is hyper so i have a hypermultiplet as many as two times color group okay this is again the super young mill theory in four dimensions now what about the statement that one might be interested is what is w effective so what is our calculation in gauge theory in the language of kitchen system in generality so we concentrate now pretty general situation would be if we understand first this and this is this example we already understand if we understand this example also and then it is well known that you can hook up any genus remand surface by pasting by gluing procedures from sphere with four punctures and torus with one puncture okay if you understand these two then we understand any kitchen system i have almost no time so i make a claim so question is what is gauge theory calculation which is the same what is w effective which is same as young young function in the language in the geometry of kitchen modern life space okay answer goes like this young young means i and j and behind you mean this is young sien young and it is his brother ah okay it's not the young young no it sounds like young young young and young wrote famous paper together where thermodynamic beta ansatz was introduced actually these formulas are very similar to thermodynamic beta ansatz if someone being starting okay young young function by definition is a function that better equations are equations on critical points of that function not every equation comes from a potential by minimization but these better equations do and it's unknown why okay and these procedures that i'm describing tells you if it's integrable system it has to be connected to gauge theory in gauge theory there is a potential and the vacuum are disguised by critical points of there has to be always a young young function there is only one known example when there is no young young function this is integrable systems they describe for any qualifies super young meals which is some integrable system which does not have a precise well it now probably has a definition and so but but that thing does not have a young young function and i don't know why this is a magic this is one of the big questions why integrable systems they they don't describe it in terms of Hamiltonian system or so they give an answer an answer they give you they don't tell you for what it is an answer except they tell you it calculates anomalous dimensions but they don't give you a spin system or something explicitly which is they don't give you this way okay if they would give it this way then we are in business they don't give you in the way that we got used to they give you in some other way so they invented their own planet for which they have all solutions they need and that's probably second sheet of the same planet okay so what we do in order to explain a little bit of this thing is let's fix some reference complex structure on sigma gn well this will be true for only and fix as a group to be i will now consider sl2c because this was sl2c flat connection so i can i erase this disappear anyway so let's group to be to be lengland's dual to sl2c now this is kind of crazy because lengland's tour to sl2 is psl2 but in general if there is a group here i will need to start with a lengland's dual group but i put like this and that deformed this reference complex structure which i would denote w w bar with a beltrami differential mu which is mu w bar w w d w bar the w so this is my beltrami and so on and now i claim that any connection a component written like this this is a deformed now this is the same as a but a omega bar but in a deformed complex structure with differential mu can be written in the case of sl2 again is minus one half d mu zero minus half d2 mu one half d mu and aw which we don't deformed with deformed in the direction of the mu and not mu bar is zero one t zero so this is a parameterization which partially solves the equations sorry this is for flat connection a here is ac okay so we take this space now claim is that holomorphic now do you see what happened here that in j holomorphic structure nothing depended symplectic form did not depend on the choice of the complex structure on sigma it was in a i holomorphic structure where symplectic form depended on choice of complex structure but in j it did not the way i did here is that i fix the complex structure and i change it and now i make a claim that following is true that holomorphic structure the the flatness conditions the conditions that f a a is a complex connection i don't use c anymore so i raise it it's this a is a plus i phi from that blackboard equals zero now it's reduced in this gauge i made some gauge choice i use the gauge transfer complex gauge transformation to reduce to it is reduced to equation d bar minus mu d plus minus 2 d mu t equals minus one half d cube mu if someone remember two-dimensional gravity polyakov studied this was only right hand side and this was equal to zero was a constant curvature conditions there so now we had the holomorphic in whole in this holomorphic structure which is j we had the symplectic form which was trace delta a wedge delta a which we had to divide by complex gauge transformation now this is equal over sigma g comma n now this is equal delta mu wedge delta t what happened said by fixing the complex structure and deforming it with a mu i actually took something which was completely defined in terms of the topological data there was nowhere single word here what complex structure i pick but after i picked and deformed i wrote same form in terms of delta mu delta t which actually depends on a choice of complex structure because beltramis are written in the basis of the original complex structure reference complex structure like this but this formula is extremely important because it leads for general Riemann surface actually to definition of what's called sl2 operator sl2 operator is a second order differential operator acting on minus one half differentials y minus one half let's not get the differentials of the type minus d square plus t because this t which i put there if you look on the what is remaining after this is reminiscent to drill field suckle reduction what is left after the gauge transformations when i write this equation there is something left here and they are and the different morphines under which this transforms as a 2 comma 0 projective connection so it transforms like schwarzian derivative okay so this is sl2 operator and now let's see what happens with the sl2 operator that there is a sl2 operator turns out to be Lagrangian sub manifold in our holomorphic symplectic manifold because it's described in terms of t the conjugate variable is mu okay so sl2 operator sit as a Lagrangian sub manifold in modular space of complex flat connections model gauge transformations and if we have sphere and that was a problem i was studying sphere with n marked points in a marked points they are allowed to have regular singularities which reduces us to formula that t is given as a sum well i consider n marked points sum from a equal 1 to n let's call it e i divide by z e a divide by minus z a and this formula tells us delta a's are fixed and when we plug in it here we get this is equal sum over delta e wedge delta z so i found the first residue of the first order pole in a marked point is conjugates to the position of the marked point this is a formula mu defines complex structure this is a quadratic projective whatever it's this 2 comma 0 differential transforming with schwarzian derivative and that's a symplectic form so now from the other side what was written here was written completely in topological language which means that i did find good coordinates in this space where space of oper's actually is a Lagrangian sub manifold when i change the complex structure when i change the z so oper will be in this language z a equals constant so i made z constant i'm sitting a Lagrangian sub manifold and what does Lagrangian sub manifold is space of such t's right there are no parametrize in terms of e's i change z i move in in this space i move this Lagrangian sub manifold now claim is so if i start with the hian integrable system as originally defined and studies the spectral problem spectral problem which means all solutions of the equation h i psi equals e i psi where h now are replaced from classical Hamiltonians by differential operators in some space so these are non commutative commuting differential operators which is a quantization problem without restricting to type of Hilbert space we are looking no l2 just just a spectral problem we are asking this makes sense you don't have to say in which space this is a statement about what energies are allowed how do you write energies so you are studying a resolvent one over h minus e the claim is that this is which are standing here and this you solve for group g which in this case is sl2 will map into the Lagrangian sub manifold z a equals constant for group lg so this was erased or somewhere yeah this is a group lg so you study spectral problem for group g for a hian system and the claim is there is a canonical map from the space of oppers from the space of the solution to this equation to the space of oppers in a dual system in l sl2c or lg which is given by Lagrangian sub manifold z a equals constant in this holomorphic structure so we formulate the problem of quantization in i holomorphic structure and it's claimed that that quantization problem is a statement about studying the space of the oppers which is Lagrangian sub manifold in a dual system for sl2c in a j holomorphic structure but in a j holomorphic structure this is a classical problem it's a geometrically stated problem in the i holomorphic structure it's a quantization it's a quantum problem now how do we describe this space of lag by the way this statement belongs to for general g and n it's a bellinson and dreinfeld and for the lower g is like zero one sorry zero n or one one i already said who who was the statement this was a gaden system and this was elliptical geromoser and gaden and elliptical geromoser was good swillers clianine and i mentioned the people this is separation variables listen this operator is in one variable this is a one variable operator original problem is in n variables the number of variables is hidden in the in the position in the how many pulse you have so this is this is a analog of the baxter equation this equals zero on psi is analog of baxter equation sometimes it is a differential equation sometimes a difference equation actually reason it is a difference equation for elliptic for periodic toda is because periodic toda is a limit of elliptical geromoser for which this is a differential equation on a torus with one marked point where t is replaced by double periodic function because it's on the torus anyway so and a statement now is that you can find so now we have to describe relation between e and the z because you have to find e if it's Lagrangian sub manifold for every z is there has to be so we are looking effective description of this Lagrangian sub manifold and answer is like this if you introduce topological coordinates here any topological coordinates let's write it sum over delta alpha i wedge delta beta i any topological coordinate does it matter this here then there you can write the graph of the Lagrangian subma generating function for the Lagrangian sub manifold as a function of let's say variables alpha and variable z such that dw d alpha i is beta and dw dz is e this will learn the classical mechanics you have a space topological space let's say the Euclidean space with some Euclidean coordinates alpha beta you have some card Lagrangian sub manifold inside there and in order to describe that card Lagrangian sub manifold you have to give basically the description of these coordinates in terms of the topological coordinates and that's called the simplectomorphism you have to find but anyway it's a general definition what you think is so you have to find such w for any coordinates alpha beta does it matter what coordinates introduce in each coordinate there is some w and I have no time to describe what are the coordinates for which this generating function is w effective of young mill salary so Nikita Rosli and I found such coordinates for the case of sl2 each in system such that this generating function of Lagrangian sub manifold of oppers is given by answers of c by our quantization methods in what I described before so I probably have to finish thank you there's no time to


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