 So, I start the second lecture a little bit going kind of back to what I was doing at the end of the other lecture, so I had the theory, an equal to theory, when the representation R was L fundamental of UN, I called masses of those MAF, A goes from 1 to L, I had L anti-fundamental of UN with masses I called MA bar, MAF bar, I cannot reach now. We have to tell that this blackboard is not moving anymore and we had one adjoint of UN with mass MA joint and that was the W effective as an answer. Okay, so then there were two approaches I described, one was writing the vacuum equation as a critical point of W, so the equation was 1 over 2 pi I, square root of minus 1, dW effective d sigma I equals nI and this equation was the case there, product A equals 1 to L sigma I plus m fundamental A minus sigma, sigma I minus m fundamental bar A equals exponential of 2 pi I t product sigma I minus sigma j plus m adjoint divided by sigma I minus sigma j minus m adjoint I not equal j and this as I said was SL2 spin chain XXX beta equation in case when there was a relation between the parameters and that relation was that masses should be related MA fundamental we have to parameterize as, so spins are denoted SA on lengths of the lattice in each lattice site I have spin SA I denoted there MA fundamental bar mass has to be plus mu A plus I SA U and adjoint mass is denoted always minus I U, there is a reason for just introducing this notation and the claim is that see this will be impurities this would be spins and this has a name of coupling constant now important thing is there was someone asked me question what if this relation is not satisfied I still have described completely the vacuum of the gauge theory and the conjecture that we made with necrassa was that in principle if you would have not this yeah this is a case of SL2 but of course this can be done for any spin group and if it is ADE linking diagrams you take you have to take ADE quiver and then ADE quiver gauge theory instead of this representation I'm now using some words which are not important if you are not interested you can get instead of SL2 you would get this group spin chain and this is doable for all of them so what if the relations like that are not satisfied well from gauge theory point of view this relation came from the statement that I added the ordinary super potential which was some of following type I took phi to the power 2 SA in each lattice point I take this representation a I take this representation anti fundamental and I multiply this and I said there is a sum with some coefficients let's call it MA and I said that this will break the global group to U1 to the power 2 L plus 1 if this relation is satisfied so if this relation is not satisfied it will completely break it down to U1 now from the point of your calculation the better equations as a vacuum equations it doesn't matter what kind of super potential I add so these relations are not necessary so conjecture we made is that you take your spin group G and replace it by following gadget so the Lie algebra here we will split to and we have generators which give us let's say E F and H these are in chevalier basis and we need serrations so if we cross these relations this will not be the group anymore so relations tell us how to take two ease and how to calculate the commutator between them we know really commutator of ease with F so ease leave here F's leave here and H's leave here and serration tells us how to move up on the root lattice here so if we remove that will be something this is not this is a semi-infinite algebra and conjecture was made that's the if well you know it is right but see you don't have relations between they make you not go too far right but you know it's like if you have SU 3 you have E1 and E2 and you don't know what is their commutator right so without serration it's not a Lie algebra anymore no it is it's a free Lie algebra it's a freely algebra it has ideals and the ideal so it's a statement is the original yeah so the statement is okay thank you for asking me like that statement is at the center let's call the Kazimierz still exist so you still can construct out of this freely algebra something which will be in a center and that's all the commutes with the Kazimierz commutes with everything and those things are actually not necessary to have serrations this is my understanding in order to write the center so centers still exist without serrations why I'm making this comment here maybe not very relevant is that if this relation are not satisfied I still have some integrable system and this integrable system will not be spin chain because in order to be spin chain this relation should be satisfied conjecture was yeah it will be some chain where we put not the representations like we would do for if we would have a spin chain with the Lie algebra representation sitting there but we would change the algebra to the freely free free algebra and still will be integrable system but in order to be integrable system you need Hamiltonians which commute so you need this Kazimierz which are out of which you are constructing the integrable system anyway this was side comment by the way please don't pay attention to it the statement is that this is a way of describing vacuums in the case when we have this kind of super potential this kind of super potential we get the exact identification between known spin chains and the so vacuums supersymmetric gauge sorry now I want to move on there are as many SA's and new SA's MAF and MAF bars so can't you always invert this relation what is the MS no they're complex numbers all of them are complex and these are reals this has to be half integer that's here but this cannot you cannot write field in the power of complex number so I have restriction here relates to that this is half integer well basically the statement is that we want to have a polynomial Hamiltonian polynomial action in the two-dimensional gauge theory and that's the restriction that's not orbit well there was a time when we were interested what is the situation when the relation are not satisfied and there was some conjecture I probably forgot I want to move on to the two important topics one is the description here that I presented on the blackboard well these operators oh eyes were traces of scalar in the vector multiplet I called sigma and this is not very invariant description so more invariant description would be if I would consider the order parameter which gives me in invariant description under the permutation of indices and that would be we will we call the Baxter-Chern order parameter and q of x is simply determinant of x minus sigma which is same as product sigma is a matrix here after we diagonalize it introduce the notation of the diagonal eigenvalues it's product of product like this which is the same as if I would write x to the power n plus sum i equals 1 to n minus 1 to the power i c i x to the power n minus i where c i's now are invariant observables invariant under action of while group okay so this is a now object I would be for a second interested and obviously if I learn something about this guy I learned about this also because they are some linear combinations they are already included there okay so now the theorem is simple one in everybody who does the bet on that knows this this equation with the notation a of x equals product and d of x another product I'll construct it out of masses is equivalent to the statement that following vacuum identity I would wish to call it a word identity but at the moment it's a vacuum identity which means that I have to evaluate that between vacuum state because q is operator following guided satisfied a x of q x plus m adjoint plus exponential of 2 pi i t d of x this is important identity that I need to use later so I would have to write it at least once equals t of x time q of x where t is some degree l polynomial relation with the parameter t the value of t what are t of x is some degree there is a t in the phase of this is not related it's not related and t is arbitrary let me call t it's arbitrary complex number and it was introduced it was this tether term and philiopolis term was this tether was called before t and it was for every you one element of the group I had such thing which was theta a plus i r a this was tether term which multiplies trace of f and this was philiopolis term which somehow enters there so this is here and it's actually well I don't like to call this thing let me call it let me call this exponential 2 pi i t to be q so I will call this guy now q so I get this equation and this is claimed to be vacuum identity now how does it work we have q which is polynomial but I don't have to restrict it I mean in this particular problem it turned out to be q to be polynomial of degree n it turned out that c t was polynomial of degree l and this equation which is written here is a four backster equation this is called backster equation integrable models statement is following so we are looking for the solution of this backster equation in polynomials of degree n and the beta equation is a consistency condition that it's t of x is degree l polynomial so combined these two equations basically completely described in the spin the chain language the spectrum of the theory because the coefficients of expansion of t here will be Hamiltonian so there will be some over i and this will be spin chain Hamiltonians now later in the probably next week I will write similar equations when q will not be in space of polynomials like it is here and this statement still would be correct but by now what we keep in mind that two-dimensional n equals 2 comma two theory of special type on many of them and so on do does describe into quantum integrable systems such that we know the place of backster equation which is a vacuum identity for this backster-chern observable order parameter and we know better equations as critical points of w effective so this is an information we gathered by doing this two-dimensional field source to dimension n equal to comma two service backster equation is an vacuum identity for order parameter q of x which is written like this and better equation are critical points of twist effective super potential and as I said this is true for any n equals two theory into dimensions for any representation the trick is that the changing the representation we are changing our quantum integrable system so when I take gauge theory with different representation to be a matter sector so different flavor group different quarks and so on I add in a vacuum sector I will get a different quantum integrable system but it always will be some quantum integrable system the good thing happened in this kind of game was that if I wanted to get SL to speed chain turned out that something very basic something very simple some simple gauge story would describe that if I would want to get g2 speed chain and so on I probably don't know but if I want to get anything that is in a D and E series I can just have to take a quiver probably g2 is also possible right or anything okay now I want to move to this is that this comes with a solution so the solution I get is what is written here when I want to get the other solutions I have to change the problem and I it's my lectures of course yeah this one gives me solutions that I wrote like here okay so now I want to erase this and speak a little bit about so this this is information we keep in mind now I want to move in topological quantum field theory language a little bit and explain the same things what does the equation by you condition for right not every sigma not every sigma corresponds to vacuum only those that solve this equation corresponds to the vacuum state in a spin chain language means that see you write the Hamiltonian says functions of sigma like I wrote there the h is are expressed in terms of sigma after you expand the t and so h is our Hamiltonians are functions of sigma and the formula you get over there are the values of Hamiltonians but you have to evaluate them for sigma solving this solution otherwise it will not be eigenvalue we are not there in that language this is kind of you know there are many ways of understanding better equation and at the moment statement is following you have h of sigma so if parametrize energies let's call it e actually because we are already in so there are formula for energies in terms of sigma and this will be a spectrum in a spectrum of the quantum integrable system only if sigma satisfies this equation now how did it come in a gauge theory language I explained if it's not a critical point of the this w effective then it has nothing to do with a vacuum now how did it comes in a quantum integrable system another lecture okay it's a complicated long lecture what right now we identify these two things and now we want to play game what do what can we learn out of it right because I'm not going to repeat what people quantum integrable system do and moreover I am not the expert of doing that but will what we learn about that and how can we use the knowledge to to expand things so now I want to move to the topological quantum field theory language keeping this knowledge so somehow forget all these things now except the facts I wrote over there and little bit digress to topological quantum field theory and topological quantum field theory I said I said that the standard law has been that partition function of topological quantum field theory which is trace minus one to the power f exponential of minus beta h minus sum t i o i o i are our chiraling operators is equal to trace of minus one to the power f exponential of minus t i o i only in a vacuum sector so that's how vacuum sector and topological quantum field theory are connected and this was trace of exponential of minus t i h i where now it is in some quantum mechanics so from what we just described it looks like that's it when we had a cylinder so topological quantum field theory can be defined on any Riemann surface of genius g and and punctures and the boundaries and so on this is defined for any but now we restrict to cylinder and close the cylinder when you complete a partition function so this is torus and that's a trace in the quantum mechanics now what we claimed basically was that topological quantum field theory associated to our supersymmetric gauge theories the point we will be this we will have sums there over solutions of the better equation solutions of better equations which which are written there of exponential of minus t i where we write this h i's as a functions of sigma and depending I mean we can put any any basis there we can take for example h i's to be this guy's or we can take h i's it's always will be true this statement is always true we have to sum over that what if I would take arbitrary human surface not not the cylinder calculation is still possible and for arbitrary human surface the following nice formula can be derived again it's generic to any n equal to comma to sorry it disappeared because when we come here this theory has nothing to do with supersymmetric sorry this this usually is a guess this usually we have to guess but there were a couple of examples known when this guess was possible to do and the one was a admittance paper on 2d young mills revisited when he wrote is as a sum of the characters and representations of the unitary groups and so on I mean there was this is probably the second guess was a similar one was what we called two-dimensional young mills Hick's theory which is part of the story here I will say something about but in general this kind of that what you think that auto apological quantum field theory is actually finitely many degrees of freedom so it's quantum mechanics but it does not have to be supersymmetric quantum mechanics it starts as a supersymmetric theory and at this point as I just showed it's a spin chain where is supersymmetry in Spanish there are no fermions okay so now what I what is the how H depends on sigma in that formula is completely relevant it's never used I mean you just take any basis of this oath this will be still true important part is that the sum will be over solution what gets summed up now I move to when you move to the hygienist there is a pre-factor here and then the importance comes but on the torus that's just a sum okay so I want you to write a general formula for any remand surface somehow I find the similar thing as Maxime said last week said it's easier to give these lectures in one week when you don't forget what you were talking about before and also when the audience is the same it doesn't change constantly it's a different people that is complicated okay so general formula for topological theory is like that as I said I defined what was a physical theory everything was explicitly defined and I also explained last time's couple of type of twists that one can do but the formula I write is true for any of them says topological partition function is this one this H by the way has not this is H of n equal to several it's not related to H I side right there so it's a sum over certain set of function H sigma belonging to that set to the power g minus one so this is for remand surface of genus G exponential minus sum t i o i sigma b again see does it matters what os are such that B solves the B the set B is a solution to our beta equation and H of sigma has actually a name I will introduce it's in a second this is equal exponential of minus certain function this is a new function that I have to explain what it is determinant of two derivatives of this effective twisted super potential times the wonder month so delta of sigma is positive roots scalar products of this basically a wonder month so what is H H has a name it is called the handle gluing operator as you see when genius equals one it is not there so H to the power g minus one is one and that's why H is called handle gluing operators that every time you add the handle you get more one more H so let's introduce this handle gluing operator so remember that I had O's satisfying the O P is plus key of something and what is C A B C obviously is these constants are independent of what genius we are on and the I can write the formula for that by fixing three points on a sphere and calculating it as a three point correlation function now I can define a special metric let's call it topological metric so season we defined on a sphere three-point function topological metric which will be I pick that particular state that I spent some time described talking about during the lunch and take C A B times zero so zero would correspond to basically to spin operator now this topological metric has a property that C A B eta B eta B C is normalized where I write now C with a upper index to use this metric from my three-point function okay so this defines canonically a following element and I call this element H it's purely written in terms of the constants which are on the blackboard by so H is metric our three-point function and operator so these are this thing can be written as also as a two-point function because we have the metric which was defined in terms of the three-point function so this is again similar thing as same thing as eta B OA dot OB that's it so if we don't put in the exponential these TI OIs and just calculate for T equals zero then this T Z topological these are very known in old formulas the Z topological on genius G at T equals zero will be just some or I can write trace of this operator into the power g minus one and that's why that formula is true so that relates handle gluing operator in abstract description with a particular formula I wrote here and now I have to say what is U zero and I am finished in order to describe what the user is I have to remind you that this topological quantum field theory comes with its action and topological action always comes under descent procedure from our effective twisted but super potential so if we calculate in a physical theory effective twist for super potential then corresponding topological action was defined explicitly in terms of this guy I give example and it will be clear plus there is two derivative with fermions and there is a dilatonic term when I have to write now since I am an arbitrary human surface sigma I have a two dimensional scalar curvature which can be coupled to some scalar function of sigma and that's what I wrote denoted by U zero so in the physical theory you calculate W effective twilder but if you have in a card space there will be coupling also of this type and you have to calculate what U zero is and U zero as we see enters in the formula for handle gluing operators there now this is generic action and it's easy to show why any action is like that take a symmetric polynomial I of scalar in the vector multiple so vector multiplied here has a psi and the sigma so I take any symmetric polynomial or actually does it have to be polynomial any function which is invariant under all symmetries of transformations of the field so since sigma is in a joint representation it's a joint invariant function on the carton of the Lie algebra and then the claim is that under the transformations supersymmetry transformation or transformation under Q charge that we kept and we kept Q a I can descend I can take derivative of this will be Q of something let's call it one I can take this one take derivative of them will be Q of something call it two and these two is a two form because I acted by derivative twice right that Q has a property well that that two is a two form and it's a pelf perfect Lagrangian I can integrate two form over two surface and this will be my s now this s will depend on would be functional of sigma psi and a and it will satisfy all the requirements that we need to have in topological point of field theory only think with is a question what is I and I gets calculated after a billionization from our physical theory or call it UV theory so we have UV theory we integrate out all possible things and so on and we get some action in infrared and whatever we get will have this form so only thing is to find I and I I just said that it's a W effective and that term is necessary also to be calculated okay so I will do it I will do the example so in the simplest possible case answering T boss question I is trace of sigma square right it's a simple simple adjoint invariant function on the on the group on the algebra where sigma leaves right so let's if when you do to the sense you get these two the what I call two to be sigma times F plus psi psi this is two form this is color this is two form psi psi right in general you will get this but in simplicity get that and this is a starting point of description of two-dimensional topological theory that had written introduced called two-dimensional topological young mill sewer so let me show just to I feel like there are some knowledges I am skipping and jumping up and down so let's take that guy and let me very quickly hook up the two-dimensional young mill theory out of it so the simplest so we have now two observable which is trace sigma F plus psi which psi right so now we add observable as I said one of the O's can be let's say g over 2 trace sigma square this is one of those ti o ice right I pick one T which cycle G and I picked o 1 o to be trace sigma square it's perfectly allowed observable this doesn't interact with anything so this is important for the measure but now we have an action trace sigma F minus g over 2 trace sigma square it's quadratic in sigma you can integrate out sigma and you get this important this multiplies volume form on sigma because the trace sigma square is color this is two-form integrate this is color you have to multiply by volume form basically square root of g the square root of metric that I'm not of metric so if we integrate out sigma it is a 1 over 2g square F wage star F right so this story is equivalent to this story in two dimensions right so by studying that kind of construction I studied this story now in general I can add as it's written there minus ti trace sigma in any power I more than three it's still calculable I mean I will be able to calculate everything in this story and if I calculate pass integral of all fields in this story in particular I calculate in this one so the young mills coupling constant which tends here is not unique ones there are many others so all are equally good this just happened to be that something we like so now statement is that whatever theory will you will start with in ultraviolet so you give me this n equal 2 comma 2 super young mills theory with lots of matter and so on when you will do all these machineries that I've been explaining so on you will get some theory in some allowing the g theory which will be functional of a billion components of your original group this what I call sigma i's and F's and so on and the statement is it always will have this one there is no other option because of the arguments playing with the supercharges and so on okay only thing you have to calculate w effective and I said w effective here is the same as twist effects or potential in a physical description I had before it is our same objects the u0 of sigma separately needs to be calculated because it's a it it's only appears when you have a card space into the mention okay it doesn't have anything to do with time it does has it's now the opens a new new new door to my lectures and I mean it's I'm sensitive as everything's has every student okay so now in two dimensions so this is our action as top which is written there in two dimensions there is no degree of freedom for the gauge field so only degree of freedom if you have a human surface of genus G only you can have is a let's take torus again let's go to the torus so I write now is a topological action for torus as a function of x and the sigma where x is monodromy of the gauge field over s1 this is our torus and let's take ice component of the gauge field because we have many one gauge fields and this is x i so the action actually is a functional of monodromy of the gauge field scholar and the fermion zero modes because rest of the thing disappears it's independent of points and everything this action so what is the type of the type of the action it's nothing more but the quantum mechanics with p i x i dot where p i is conjugate to this x i and is equal d w effective twilder d sigma i take this multiply by time derivative of that what you get this is equal this is where I wrote integral over sigma in this case torus of dw d sigma i times fi which now I write as d of ai and the only time derivative I keep here there is no space dependence I take already integral over s1 so I get that this action which is over written over there is equal px dot right is this clear it's two dimensional there are no degrees of freedom only monodromy scare I wrote is a pexo and this this is a time direction here is a also integral over the time and since it's a torus it's also periodic then there is another term which multiplies fermions and this is important that the fermions come with this term so fermionic pass integral will give that determinant which is written there d2 w effective d sigma i d sigma j this term will give exponential of minus 2 u0 in H and the wonder month is obvious because I'm integrating over matrices I diagonalize the matrices I get wonder month determinant I'm explaining to each term wonder month is clear determinant of two derivatives of w effective in positive power comes because it's a fermionic integral over size and it's a positive power then there is a exponential of minus 2 times u0 of sigma which comes from r2 times u0 of sigma there written in the last term and the only important thing now is that see this term px dot is defined x is defined up to shift by one because I'm on a circle and monodromy when I go I go I do large e1 gauge transformation x i will be shifted by by one each x i will be shifted by one so x i's live on a circle and if x i's live on a circle then p i should be integers right because I have x quantum mechanics when x lives on a circle then conjugate momenta should be integer and p i equals integer is this equation okay the moment I calculate that topological action which is written on that blackboard how I calculated the different story I know that the answer will be some over solutions of p i equals integer but now p i equals integer is complicated in two-dimensional young mills which was somewhere here on the blackboard over there p i equals integer meant that sigma is equal integer and sigma equals integer in two-dimensional young mills was interpreted that we have some over highest weight representations of unitary group which are labeled by set of integers so these set of integers into dimensionally young mills interpreted as a representation as an it's also very integral and all this kind of things these were highest weights of the individual unitary representation now this relation sigma equals n gets replaced by complicated one d w d sigma equals n so meaning of d w d sigma equals n very important is the same as their highest weight represent so we have the formation somehow of the of the story of of the highest weights representations and moreover it's not only that but also we will get to it so the better equation I will now do the hitching example in it quickly and we will see that see in topological theory over modular space integral modular space of hitching that's actually direct meaning okay so for one second I want to show that the vacuum I don't have to be and now I'm coming to a question vacuum I don't have to be finite dimensional okay this is the case when I have the boxer operator which is not polynomial the second solution is constructible from the first solution from the runsky and I don't want to go there okay it's the moment I want to first write the polynomial one and then I move to the case when it's not polynomial and then I claim that's the first important statement will be there is a Baxter equation and solve it in each space you like and for each solution there will be interpretation in a gauge so instead of going to the negative spins the the negative powers of C one over polynomial I will do my plan of the presenting this sink is a little bit organized on the ways that I first study everything which is what I call compact situation and then non-compact so I said that xxx pin chain was n equals 2 comma 2 2 dimensional so supersymmetric gauge theory with some final dimensional representation are which was reducible but some some representation the first thing we need to I need to explain is that I can be asked what if I want to get yeah so what I can do with xx spin chain I can change the gauge group and I said what sorry I can change the spin group let me write here spin group G I can change spin group I can change some parameters but eventually xxx means restriction already what if I want to do xx this pin chain well answer is we have to take this representation to be infinite dimensional and specifically we have to take n equals two three-dimensional super young mill sorry on a cylinder times s1 with the radius R and now the n equal to three-dimensional supersymmetric gauge theory on this thing is the same as this one two-dimensional supersymmetric gauge theory but when the representation are is some of our n's where n is a kalucha Klein mode so what is this thing takes a three-dimensional theory expand every field in three-dimensional theory let's call it field phi so this was in three dimensions any field expand like that you have now infinitely many kalucha Klein modes phi phi n and in terms of the phi n this three-dimensional supersymmetric gauge theory is actually two-dimensional supersymmetric gauge theory it's easy to count what are the number of supercharges it will be same number of supercharges but the representation are now will be same as before but each field gets a label this n and we have to watch what is this label does and the statement is that this label does that each field phi will have twisted mass proportional to that thing which is the kalucha Klein mass and that kalucha Klein mass the twisted mass and it's clear actually that now global group global symmetry group from two-dimensional point of view contains shifts in x3 right we this theory is three-dimensional and in three-dimensional theory on a circle we have a symmetry of shifting in x3 and from two-dimensional point of view this is a global symmetry group this is a group which acts on the space of fields right when you shift x your field transform and for each global group that now this is affine right because this is a affine shift so now your masses are associated to that the everything is a previous story so we completely come back come back to previous story except we have infinitely many representations and the claim is that the anisotropy is proportional to r actually when anisotropy goes to 1 r goes to 0 right so when r goes to 0 you go back to two-dimensional theory there is its dimensional reductions there are no kalucha Klein modes that two dimensional theory we already studied and we proved that it was xxx right I said that the two-dimensional theory with 2,2 supersymmetry is giving me xxx pin chain set of xxx pin chain depending what matter sector I take there this one will give me back to this when I send radius to 0 which means that the xxx has to go to xxx that explains it and this has been proven actually then this is true and w effective has been calculated for this and for xxx and coincides with the young young function for xxx now I introduce word young young function I will talk about this little bit xyz model now it's obvious right you have to consider four-dimensional theory on a cylinder times two torus two torus has a one modular moduli tau which is a complex one and it's expressed in terms of the anisotropy's of xxx xyz xyz has two real parameters which are anisotropy's and these two real parameters are parameterized by the way in FADF tahtajan description of paper 1979 in terms of the torus tau and that's exactly same tau that these two torus will have in four dimensions and this four-dimensional theory on a cylinder times t2 is again r which is infinite dimensional now double sum of on these two torus and we included we allowed such infinite dimensional representations when we were studying two dimensional serve okay again w effective can be calculated which will be now again double sum over these double kalutza Klein modes and it coincide with young young function of xxx xyz model that's has been proven now what is young young function young young function is it's for every mean this is very fun for every quantum integrable system every quantum integrable system we know accept what is called n equal for integrability the beta equation is always critical equation for some function what integrability people call the young young function so better equation is an equation on some number of variables okay there is many equations as number of variables so we can write always the better equation in quantum integrable system as an equation on zeros of one form suppose better equations where bi of some lambdas equals zero where lambdas is a set of lambda eyes to there's as many lambdas as equations we construct out of it one form in our whatever dimension of space is called be lambda and in every example accept this n equal for integrability that some of you have been working on there is a state observation that db equals zero or locally and this is all locally of course we are talking about b i of lambda is d of y d lambda i and this has been observed in every known example except again the solution to n equal for super young mills integrable systems that people constructed for that okay and as I just explained here what has been proven case by case was that w effective of super young mill theory coincides with a young function which just blunt identification sigma equals lambda w effective of sigma equals y of lambda it's included in young function I included oh you're right I don't include it in why now so let me say so I don't include it in why my equations is bi is equal to pi square root of minus one and I so I have to include it this kind okay so on more over yes and moreover the age Hamiltonians of the oh eyes of sigma of the gauge theory this observables I said were identified with H eyes energies for the quantum Hamiltonians that people calculate and this is defined in a quantum integrable system just the convenient things that the people parameterize eigenvalues in terms of some parameters lambda which they find in many different ways and then there is a statement that see vacuum expectation value of these operators are coincided with energies so this has been verified case by case there is no complete theory that we could say that we understand everything and that's what we are looking for so now I want to consider so I listed these things because I can't spend time concentrating on this but this one I hope I explained now I want to consider the simplest possible case when we have both topological quantum fields over the description and some interesting Schrodinger operators that I think are worth discussing okay let's go back to two dimensions forget this kind of XYZ and so on and consider the my old example which was called main example except I will take L to be equal to zero which means I consider just representation to be one adjoint of UN and I call that one adjoint to be fine so now I have n equal to seawater in two dimensions and I add so pure n equal to which means just gauge multiplied pure n equal to plus I had one adjoint and I made it made it massive this has a name that we will be using later this kind of seawater is are called n equal to star seawater so n equal to star seawater are n equal to star seawater plus what one adjoint massive multiplied it's the same name it's called in four dimensions it's called an equal for now there is another properties that this kind of seawater have these seawater are also n equal for seawater so they have twice as many supercharges I mean that they are an equal for theory in terms of the field content so they have same field content as an equal for seawater so this theory XYZ is an equal one in 4D this is such seawater that when you go down to Taurus you get n equal to and this by the way has strange name it's n equal to in 3D again only the guy do is like this I go dimension reduce first and in dimension reduction I identify what type of theory I can have and then I say that if I now have toroidal kaluta client mode that they are one-to-one corresponding to XYZ if they are not toroidal but some sigma it's still some integrable system but I don't know what's the name of that this contradicts to the other notions that if I take that theory in four dimensions the twist effective superpotential in two dimensions he will be there that story will have theory for four dimensional theory n equals one yeah it will still have it probably I have to take to make it very clear is that let's take all these stories by the way I'm talking about our in four dimensions n equals twos broken by hand to n equals one so these are the XYZ is yeah I don't have to add trace sigma square right so the suggestion is to study this theory on the point actually that's a good point maybe let's talk about that you know I'm getting a bit tired so let me wrap it up to some point so I will make a break it's quite possible actually what I am not sure is that if I take arbitrary moon surface if I can interpret it is yeah I need all the guys to be massive I need everything to be massive they are not in four dimensions they are expanding yeah by the way the there is an important thing that's your people who do XYZ model know that's a better answer at that artigian infidee wrote for XYZ model is an equation of that type that was written here and it has to satisfy certain modular properties so there is this statement about XYZ spin chains at the better equations are some modular function ratio of theta functions and it does not really satisfy modularity property unless the better roots I think sum up to zero or something so there is a condition that better roots would sum up to zero and then the left-hand side is modular right-hand side is modular you see instead of instead of product instead of product of sigma i minus sigma j plus m or something we have theta functions now and they are in characteristic and something and the when you rewrite the equations I was writing there is some problem with modularity which is related to the facts in XYZ model the ground state actually needs to be constructed as some linear combination with some theta term of the usual ground states that like constructed by BBBB acting of B operators and that modularity property is that I just said about the for the this better equation that if I remember correctly if you write in this language of the better roots some of sigma i's have to be zero there is a condition there it's very easy to retranslate this to XYZ model on these times torus being any being an equal one theory in four dimensions on that time torus if you start putting fluxes on T2 so you can put for example two form flux on T2 and so on and write general theory on there and the condition of modularity as far as I remember is equivalent to condition of anomaly freed freedom in this four-dimensional gauge story when you have background fields and background fields I mean fluxes on this T2 B field like you can put a B field two form field any background field and they have meanings in the XYZ model so it's entire new subject actually about any this this XYZ being any qualifier 4D which goes like this you observe something on a cylinder times torus and you are happy and now you start playing with four-dimensional theory on a cylinder times torus and you want to explain everything in terms of the XYZ model and lots of interesting thing happens and that's not studied by anybody I don't know anybody studying it okay so now let's make it let me consider now we have any qualifier theory and we can construct actually many many things in general we can take theories which look an equal to but are really an equal four and then break an equal four down by something down to n equals two and in this particular case this is done by adding super potential what I will call bear super potential W twilder of sigma which is one half trace sigma square and I'll now remind this was the answer for TVO I somewhere here yes here I said that two-dimensionally young mills was like that two-dimensionally young mills was that I added by hand trace sigma square and that was Witten's explanations that the two-dimensional or in a two-dimensionally young mills actually has to have this term to be the two-dimensionally young mills and that's what I would do here also so I would take now instead of Witten taking n equal to theory and adding trace sigma square I would take an equal four theory gives a mass and add one half trace sigma square and I will have the group UN so my question is what is this theory now obviously it's not an xxx pin chain because I don't have I have zero lengths completely I don't have a spin chain but there is something and let's see what that something is and that's something of course we determined detect by calculating its beta equations its vacuum equations and what once we look on vacuum equations we ask the question so where we have seen this vacuum equation there are lots of other questions people can ask and I can try to answer like yes you constructs pin chain but did you construct the spin itself or use make a statement that the spectrum of the spin chain Hamiltonian is the same as the spectrum of the vacuums of that gauge theory that's a statement I made now did it not can come and tell me I want to see the spin where is the spin there is no spin or you can do intricate construction into the national gauge theory to actually see the spin and that's doable but I'm not describing it here it's very intricate and it's actually possible to actually speak this thing okay so the beta equation now in that model which is written there which is n equal 4 plus mass for adjoint plus three level potential the equation will be exponential of i sigma j is equal product from i equals 1 to n sigma i minus sigma j plus m divide sigma i minus sigma j minus m and here I would stop in so I would say now what it is this is called the n particle sector so now I detected I wrote the beta equation vacuum equation now I know for wheat which integrable system it says n particle sector of nonlinear Schrodinger equation this is quantum field theory actually in two dimensions in one plus one dimensions but I restrict it to the n particle sector so it becomes quantum mechanical and in that n particle sector is described by this and it also in that n particle sector has a quantum mechanical description in terms of something looking like that take n particles x1 to xn leaving on a circle so I have n particles on a circle we is a Hamiltonian to be minus one-half sum from one to n the over dx i square plus c times sum i not equal j delta function of x i minus x j and that c is m equals i c okay so this is an identification this is a known beta equation for n particle sector of no linear Schrodinger equation which is equivalent to n particles on a circle with a Schrodinger operator being like this so if c equals zero these are just n free particles on a circle and when c equals zero this should be two-dimensional young mills right the way I constructed so two dimensional young mills is this problem but it's not zero so it's a deformation of two-dimensional young mills two-dimensional young mills actually corresponds to c goes to infinity but when I come back I described so now we are dealing with a situation where there infinitely many vacuum it's a finite dimension it it's a two-dimensional supersymmetric gauge theory with finitely many fields but has infinitely many supersymmetric vacuum because everybody knows that this system has infinitely many solutions this better equation has infinitely many solutions so when I come back I will be start with this and connect it to the hitching system and that will be my way to move to hitching system from the second half of this lecture and next next week I'm very sorry today was little bit kind of jumping around because now when you finish in one place and you come back and you realize that these are different people and they forgot what we were talking about last week it's disaster I mean how can I warm up to get to this formulas I don't know okay in 15 10 minutes maybe we come back 10 15 minutes first thing we know that we can ask the question over which moduli space we integrate in this case and I now remind that there was a generally I wrote the BPS equations which looked like we take this connection a construct the curvature and there was something here which depended on comps let's conjugate super potential and so on but this always is zero except we have this the one field in a joint representation which I call phi and what was in right-hand side in this BPS equation will be just commutator of phi z with phi z bar and then there was an equation also what was nabla z bar of any chiral multiplets that I had in a matter sector and now I have only this in a matter sector and that will be this equation so these are the BPS equation we are integrating over in a gauge theory in that setup general equation I probably right now on that blackboard I had in last lecture and although people are not here maybe if someone watches on over the internet can be useful so general equations was for every matter multiplet which I denoted by x there was a BPS equation that topologically twisted theory was integral over or equivalently the vacuum sector of the physical theory defined and this was d z bar okay this is nabla I call now nabla z bar of a acting on x i in a matter sector was minus g i i bar dw star of x which after twist become one form d xi and then there was another equation f of z bar of a plus mu of x x bar was zero and it's actually this is the one divided by action of the gauge group and here mu and metric g was defined the mu was a le algebra valued moment map for simplectic form coming from calor metric g i j and g i i bar was a calor metric okay so now these equations get replaced by this in our particular case of n equal fourth theory with mass so the question is the statement is that now we have integral over the modular space of solutions of this described by quantum integrable system of n particle sector of no linear Schrodinger equation so what solution what do we know about this no linear Schrodinger equation in n particle sector first thing we know about this is that we can exactly solve we can exactly solve the wave function equation h psi equals e psi and this was done I think in 1963 let me make sure I don't miss quotes the people by lib and linear and berets in pohele and Finkelberg as being a Russian I always have to quote some Russian paper in parallel to American paper so this is lib and this is a very famous example and berets in pohele and Finkelberg so they actually there are higher Hamiltonians here I just wrote only second Hamiltonian and the higher Hamiltonians are written in terms of something called dunkel operator so it turns out that it's not only gauge theories that gives nice formulas for Hamiltonians they are very similar in parallel formulas in the Schrodinger operator theory for this particular Schrodinger operators that we will need for future to generalize or use and these are following so take di as a differential operator on a circle plus I am over 2 sum j equals from I plus 1 to n step function x i minus x j plus 1 times permutation s i j so we have here s i j which is from the while group permutes positions x i x j and we have di which is a shift deformed with one parameter m okay so if that m is not there this is just a shift s is then together without shift this would be called affine while group this while is probably I right why is correct right yeah correct okay so this is a deformation now of affine while group and has a name so we have to generate two type of generators this and s i j and this is called double affine hecke algebra at the generation so there is obviously something which is not degenerate double affine hecke algebra and this is a degenerate it has only one parameter m okay so claim is all Hamiltonians of this integrable system can be written as one half so this is case Hamiltonian one half sum I don't have it is degenerate so only what I have here you mean by x so I have to add x you are saying so multiplication by x x shift and permutation that's what you are saying probably yeah but in this discussion I just need what I'm saying but I'm saying that the name of this thing is that see what I just construct is in the center that's all I need and probably yeah well one one parameter deformation of the affine while group while algebra one parameter different I was told by Ivan Charitnik said this is his dacha at the generation so he claims this is a degenerate dacha probably I can even formulate in what sense he explains this to me when I get to the more more more serious way of describing okay that's what he calls degenerate dacha he introduced dacha so he knows what it is but important things that these guys are in the center and our h2 is h2 so this h2 is a square is one half of sum of di square that's what that's what we need okay these two will have generalizations when we go to the more complicated example and then there is a formula for explicit wave function which is common eigen function for all of them now watch I don't take yet on a circle you see I wrote a step function so this one was on a circle which means it was a periodic one now I take it on infinite line x is on r r1 and these people I just said they're live linear and so on vortex explicit wave function explicit wave function of this thing is useful for us in the future discussions at least in a way Nikita nekrasov and I explained the more complicated systems and Gerasimov and I explained this particular example so we take element of the permutation group we sum over it we take product of i less than j sigma of wi minus sigma of wj plus im step function xi minus xj which means that if we just order them I don't need this just one divide by sigma of wi minus sigma of wj times so this is a pre-factor exponential of 2 pi i sum over l sigma wl xl okay so claim is that this is exact wave function for this eigen value problem where the energy is sum sigma i to the power k so for example for the h2 just sigma square sum over sigma squares would be energies and this is an exact wave function okay now where is the beta equation before I go to beta equation I want to write here on this part of the blackboard the normalization so these are not normalized wave functions under usual measure on r1 which is now r1 to the power n we have n particles so the normal normalization is so we normalize psi sigma 0 psi sigma prime 0 will be integral dx1 dxn over rn of psi bar sigma x 0 psi sigma prime 0 x and this is not equal to 1 this is equal to some sigma of g let me call it product of i equals 1 to n delta function of sigma minus sigma prime so up to these factors g this normalized delta function normalized wave functions as it is supposed to be and g of sigma actually is something that will become with familiar already for us divided by wonder mode so this normalization eventually will be connected to handle gluing operator see I have a determinant of two derivatives of w effective and that will be connected to the so handle gluing operator will have a meaning of the normalizations there and then the beta equation which is written here I explain in a second what it is so it is clear that this normalization shows up in the hand and going off now once we move from here to circle from the infinite line to circle we have to request that size satisfy periodicity condition if we move by if we shift x by 1 so when we move to the unit circle so let me write here psi sigma of x to be normalized one which is 1 over square root of g of sigma times psi sigma of 0 and when we put this condition on explicit formula I have written here it's obvious that you can read everything from here you shift x by 1 you create from here linear term and you have this the log of this one you pull up in the exponential this is by the way just for simplest to just later with this is a piece of s matrix which is written here of this our model and we have the log of s matrix as a as a phase shift so general statement is replace delta function x i minus x j by periodic delta function which is sum over an x i minus x j plus n request the periodicity from the wave function and see what happens with with the way function and the answer is wave function will be periodic if that equation is satisfied so what we have now is that we have quantum many body system a simple one quantum many body system with pairwise delta function interaction on circle and the spectrum is same as on open system plus restriction on rapidities of the beta equation this is a knowledge that extreme turns out to be extremely important yeah I can't please ask me questions if there are any questions because you're getting harder now I will make it this story much harder what I cared was to get to the statement that beta equations in that sense I mean restriction on spectrum and so on is now connected to the periodicity okay then this periodicity will become harder when we got to more complicated example but better equation is ready to the periodicity here and the important question is that this x which is here is not the same x which was here see way we had periodicity there was a periodicity in this problem also that's how I got the better better equation I requested periodicity for the monodromy and I got this quantization p equals n so I get quantization there from the same requirement because what's written here this equation is nothing more but exponential of i times dw d sigma i equals one if I take log of this and move to exponential here and that w is a young's function for this integrable system which has been well known for long time so very important open question is what is the relation between these variables x and those variables here because in a sense we know what is the relation between variables dw d sigma which which we have here and variables which are the sigma this is same sigma so sigmas are the same but axis are not the same and this is open question we just have a coincidence or something like that that looks like at the moment now I want to make a important comment that this representation theory of this what we called degenerate daha or maybe it's not daha or something like that is connected to the other things that may be in later we have to use and there is a statement I'm collecting certain statements to use for other problems and that statement is about spherical functions and solutions to certain quantum many-body systems connected to the representation so so this is a the most simple quantum many-body system at least I understand that is connected to representation so if some algebraic object but this is not unique there are many others and now let me describe some of them before I move to the four-dimensional gauge theories and on omega background so first statement that that wave function I wrote there has something to do with what's called hull-litwood polynomial how it would polynomial itself as was explained by one Dutch mathematician 1d gen sorry for bad pronunciation in 2006 hull-litwood polynomial itself is a solution to difference version of this problem so this problem can be instead of second order differential operator you can write the difference operator of second order which in the limit becomes this one so there is a sometimes this is called young young system which is probably not correct because it was solved by these people and young and young 1969 young and his brother in most famous paper in the discipline probably they they wrote the better equation in this philosophy and this is same paper where they introduce thermodynamic better and that's so this young young system has a discreet generalization and how little polynomial itself as was shown by this gentleman is connected to that one but the ones that we are using here is a limit and let me describe what limit it is so this hull-litwood polynomial is nothing more but gln qp spherical function and there is some explicit form of that which I probably will not write now but depends on a bunch of parameter called me wise lambda t's and p an explicit form is something like take mu 1 to mu n to be partition of lengths at most 10 and let's write it like once we have m1 time and twos we have m2 and so and then the function itself so this is some product over i of 1 minus t divide 1 minus t m power i sum 1 minus minus 1 to the power lengths of the permutation omega omega permuting this other variables lambda 1 times some product okay so now important statements that eventually I wanted sometimes ago to generalize it looks like it's working if we go to continue to limit let's call it a limit p goes to 1 it's in the products there okay so the p in the product and by the way not only in the product no I have parameter p and I have muse which are partitions I have so parameter n I have muse which are like our coordinates eventually and the p is product here from i 1 to p I believe I am very sorry and there is a sum over omega in symmetric group anyway explicit form now that's it will be important for me later explicit form for this is that it has a limit this how it would polynomial and that limit is like that take mu i's and write is some continuous variable x i divide by epsilon lambda i's to be exponential of 2 pi epsilon times sigma i that we need there and send epsilon to zero where t where t is exponential of 2 pi i m epsilon and send epsilon to zero the claim is that when epsilon goes to zero this becomes Lib linear wave function so it's explicitly becomes the eigen function I described here that's just when you plug in this leads to the state yes no no no n is number of which we fix n is number of the particles we are in n particle sector so what I wrote so there is claim is like this this equation has its discretized version okay which has been studied was also integrable that this guy that version has explicitly how little polynomials which are the wave functions okay how little polynomials from our from one side are g l n q p zonal spherical function so this is the same as what you do when you do the spherical functions when you do the orbit method you divide by g l and z you take g l and r divide by there is its periodic version which I am not expert but see explicitly if take a formula and in that formula you make substitutions such that wave function will become wave function of this operator funny way that substitution also takes p to goes to one so p is exponential of epsilon so basically what you do like we usually do you take this function analytically continue everywhere it's possible take this substitution and take limit epsilon goes to zero we will be playing this limits later that's why I am I'm discussing it here and this becomes the solution to that okay this leads to the some kind of picture which motivated Nikita Nikrasov and me to do next things that we I will be talking from now on and this is following that blackboard needs to be fixed so let's take what we know about our so let's take situation with free fermions so this would be two-dimensional young mills without any hitching system that I've been describing here so when there is no phi when I answered some t-boss question here when we turn on one parameter which is this parameter will introduce Higgs field or we turn on the parameter M we somehow go to no linear Schrodinger system on one side there is another way of deforming that one and this is now what becomes point for me which is called Kalo Gero Moser Sutherland deformation this is now this was connected to the delta function potential or let's say one parameter M this connects to the formation with 1 over sinh square potential again it I mean this is all I'm discussing now on infinite line no periodicity conditions or something like that so we deform it like that and then there is another one which is I cannot write this Ruhi's Nahr's deformation the one more one in the this one has two parameters let's me call it Q and the T this corresponds to Q so Ruhi's system is a relativistic version of the Kalo Gero Moser Sutherland so this is not relativistic many quantum many body system there is its relative generalization this not important most important here is existence of it rather than what it is and the no linear Schrodinger one which is standing here this young young system or Lib Linniger system is Q equals 0 for this two parameter family Kalo Gero Moser Sutherland is Q equals T to the power nu where the nu is a coefficient in front of potential here and then you have to send Q to 1 simultaneously you have to send T to 1 so it will have this one parameter left and that's what the Kalo Gero Sutherland Moser Moser and that's there is a picture and what in this picture we detected first doing the calculation for Hicks bundle for Hitchin modular space which was this equation divide by gauge group turned out to be that corner so this particular one that's all I needed from the picture was that see this is this one parameter one and there is a other piece of it now if I go back to this equation I want to describe two interesting limits of this equation and one interesting one which is to the point and this somehow it's kind of little awkward is not m goes to zero limit but rather m goes to infinity limit so when m goes to infinity this is actually a quasi-classical limit for this model it's not m goes to zero which is quasi-classical limit but when m goes to infinity when m goes to infinity this becomes minus 1 to the power n and you have a statement that the exponential of i sigma j is equal minus 1 to the power n and this is a free fermion point that when m goes to zero limit now I explain why it's actually also true see when m goes to infinity limit this phi this field phi which is Hicks field for which m was a mass is infinitely massive and in quantum field theory where this is a matter field with the couples so this theory becomes without mass goes to zero limit is a bad limit because that guy becomes massless and once big one massless he cannot integrate it out so free fermion point is m goes to infinity limit when phi decouples and we get in two dimensional young mill story of Edward when m goes to zero limit what we get that this he teaching system equivalently as a manifold the solution to this equation the space of solutions can be equivalently described as a flat connection for connection i plus a plus i phi equals zero divide by complexification of the gauge group so these two spaces are the same and in one second for physics I explained why these two spaces are same takes his equation this equation is now complex let connection it's invariant under complexity chain or gauge group so gauge group here has a unitary directions and the complex direction so you have to gauge fix to study this kind of equation and you can always gauge fix by imposing the condition so yeah so here what we have this equation f z bar a plus a is the same as this equation as a real part as an imaginary part is this equation plus complex conjugate okay if you take a real part of that equation you get this take imaginary part you get this plus complex conjugate you have to gauge fix and you can introduce gauge fixing to be nabla phi minus level bar and then they're equivalent so if you wish gauge fixing part of the gauge transformations makes this equal that this is simple physics explanation why these two systems are the same and mass equals to zero limit when we take mass of this guy phi equal to zero gives us nothing more but integration over this modular space where phi is massless but this is ill-defined because the modular space of the complex flat connection is non-compact so the integrals will diverge and the point where we introduced mass for phi was that we would make the bed direction in the complex direction to be regularized by putting the Gaussian exponential of minus mass phi phi bar and that would converge so m goes to zero limit is bad so if Edwitten's calculation for a flat connection the real flat connection was that it was some over reducible representations of gauge group so Whitten I think it was 1988 that this is some over E reps of gauge group G now this becomes questionable because G is replaced by GC it's a complex complexification of the gauge group and what is some Dappel so is infinite so in some paper I wrote with Gerasimov we explained my calculations with nekrosov and Moore from 1998 in this language so m goes to zero limit is bad there is some explanation of that now what about people who studied better equations what do they say well they say that m goes to infinity limit is equivalent to h bar goes to zero limit so it is it is a good one sorry what I'm saying no m goes to zero limit that's what they say this is what Fedesminov taught me when m goes to zero you have to send n to infinity such that some combination is finite I forgot well probably m times n is finite I don't remember and then this is a semi classical limit so semi classic it's not that obvious and then this when n goes to infinity limit and what I describe actually gets very complicated it's connected to the infinite genus Riemann surface but when m goes to infinity limit this has an extremely well studied the solution that you take when m goes to infinity first solution will be sigma j's are one half times integers and then you perturb around it and there is a very nice perturbation theory in 1 over m and the theorem has been proven that for every set of integers ordered set of integers there is exactly one solution for sigma 1 sigma n so this is what they call completeness since the set of integers ordered set of integers let's call it n i's such that n1 is more than so nn describes highest weight reducible representation of a unitary group what is claimed here that for every highest weight representation of a unitary group there is a solution for this equation so this is a case when it's well studied and has these various limits now what about going to the other picture which was a picture of the function so I wrote there that there was a Hal Littwood polynomial that was relevant in epsilon goes to 0 limit how that picture and this picture connects that I wrote on this blackboard they do connect very well again people who study this kind of McDonald's polynomials know that so the ones atop you have something which is called McDonald's polynomial depends on two parameters q and t now I move the blackboard and it will be clear when q goes to 0 limit this becomes Hal Littwood and t equals 1 over p for McDonald here that exact limit we had q equals t to the power new q goes to 1 t goes to 1 it becomes Jack polynomial here when t goes to 0 limit this becomes 3 fermion or sure sure functions which are the class functions for representation of the unitary growth sign well this as you as you will see in the second in my talks that I don't much distinguish but this is one is analytic and so these pictures now map one to one so the McDonald will will be relevant for this and so on but what is missing from this picture is when we instead of this one or this one we consider it elliptic versions so this was a situation when let's say on the levels that way is that we were working we will we will do better than that we will produce okay question is two-sided first if you ask me to construct the wave functions right second if you ask me to construct the spectrum of the quantum integrable system now usually it would say that oh you cannot construct the spectrum without giving me a wave function so like we did here I presented the Lib-Liniger calculation of wave function then I take the periodicity and produced the discrete spectrum so what you would expect is that the way I would now proceed for the McDonald's and everything I will produce this version which would lead to the better equation which would cover entire blackboard over there instead of that what I will do first I will eventually get to that but first what I will do looking on these examples I will try to find the quantum field theory for each corner of that diagram that comes from my way of describing the quantum integrable system so I will construct supersymmetric Young-Mills theory which will have the vacuum described by by all means by some quantum integrable systems that I know what it is and that quantum integrable system will have description in terms of twist effective super potential so by better equation or back that equation where I had two sets first was I said that vacuum are given by effective to visit super potential and second I said that there is a vacuum identity including boxer operators I have both languages so in one of those languages I will detect which quantum integrable system I get and then the other formula will give me a spectrum now it will be produced without giving the way function first and then people would complain I mean how can you make such strong statement without actually presenting the way function but I will say that sorry if I've properly formulate the question then spectrum is independent in which coordinates you write the way function only statement will be that I look in this kind of Hilbert space and here is a spectrum without giving wave function but unfortunately because people ask questions like this and they are sometimes very unpleasant and I don't mean questions I mean people we had to produce the wave function so we would approximate the wave function somehow and argue that there is a wave function which can be calculated this and this way and will give that spectrum and we will do that for entire diagram yeah we will do that for entire diagram but at the moment I say I want to consider elliptic version of this where Sutherland will be a limit so elliptic one will contain Sutherland as a limit okay there will be elliptic roussinas one which will contain roussinas as a limit everybody knows that roussinas has something to do with McDonald and all these limits okay but what I will produce I will produce elliptic version of this thing with exact spectrum and I will verify that this is exact spectrum in some correct Hilbert space that correct Hilbert space will be defined in terms of differential equations this is this logic is extremely important if this logic is not understood the everything crushes I unfortunately I fail I had had hard times with my teachers who would tell me that oh no no we are the expert no Fadev said that you know I write I wrote books when I was young about these things and so on well what is the problem that you are solving and after long discussions and long discussion eventually understood what are the problems that I am solving so it needs look what happened here I without constructing the wave function okay this wave function saying that integration over the modular space of hitching system set up in such a way as given by n equal four super young mill theory in two dimensions with a super potential three bear super potential one half sigma square would produce the exact spectrum of this quantum mechanical system well this one quantum mechanical system common eigen functions and so on I predicted I did not calculate you did but then happened that it actually gave it correctly I even formulated the question that these coordinates in which the wave function is written here in the young young system or the Lib Linniger system is unnatural from the point of view of the gauge theory because gauge theory likes to write the wave functions in terms of the monodromes of the gauge field luckily in the case of the sure in the sure case there when we have few fermions this turns out to be same coordinates these excise here and that excise in the case of the characters of the representation of the entire group are the same but once you leave once you depart once you turn that that parameter mass which was a twisted mass which I said also was a current parameter for a global group and I turned it on they depart and this is an open question is not understood how in this case we know everything and we verify conjecture that the gauge theory calculations of the eigen vectors and eigen values of the chiral ring operators actually do produce correct spectrum in a correct Hilbert space of corresponding integrable system okay so in my view being person working in a gauge theory I would say you have to be insane not to believe that this will work for every case as long as I identify what is my quantum integrable system you have to be really retrograde in Russian they say I don't know the person who doesn't believe in progress that this is not going to give the right answer but regretfully they said Nekrasov and I had to verify that we actually had to solve elliptical or a mother equation order by order and verify that our statement is true order by order of course we sold it up to very few orders like three or four but it does give the right answer okay so let me now move to the most interesting part of problem my lectures is that see once we determined quantum any body system you see I described this corner here so I'm sorry I did not describe very well this how little polynomials which is not my expertise but that's the complete matching of these blackboards I moved here and jumped immediately to elliptic case because elliptic case covers everything and hint comes for this from four-dimensional cyber-greeting cell so now the models which are written on that blackboard in the right-hand side it's those models in finite size situation correspond to some other my our friends discovered correspond to cyber-greeting salary so I have to move to cyber-greeting cell so from now on two dimensional models toroidal compactifications and so on will disappear and we move to four-dimensional super young mills and algebraic integrable systems so in about 1995 Gorski creature where Mironov, Morozov and Marshakov and Martinik and Warner and then Donaghe and Whitten gave very beautiful explanation of connection of cyber-greeting solution of super young mills over n equals to super young mills in 4d and algebraic classical algebraic integrable systems that I need to review I have 10 minutes left so I will just introduce the basic notions and we will be done I will erase that thing so Zyberg and Whitten in 1994 found a beautiful low energy solution to super young mill salary in four dimensions with eight supercharges so salary with eight supercharges in four-dimension is kind of closest cousin of salary we discussed in two dimensions so we have n equal to 4d super young mills and it has a property that in perturbation salary it is one loop exact so anything in coupling constant if there is a coupling constant but once there is a perturbation salary there are eight supercharges in four dimensions perturbation is one loop exact so then you have one loop perturbation salary plus instantons plus instantons and even better news is that the because of this high number of supersymmetry there are no anti-instantons if you remember when instantons came to life it was clear that situation is very complicated you have instantons for anti-instantons they interact with each other there is a complicated mess and it was Fathev and Schwartz, Fathev, Rolo and Schwartz who calculated in the late 70s some answers using the instantons and anti-instantons summing them up in sigma models in two dimensions and recovered the correct answers but that which shows that in order to get correct answers you have to do all these complicated interactions between them now in n equal to salary there are no either they're instantons or anti-instantons but they are not together and they are because of holomorphicity that's long argument to run but it's not important for us what is important that observation of the guys so low energy effective action exactly like in 2D W effective produced low energy effective action in 4D it's another thing which is called the pre-potential now in in two dimensions we had one invariant function which we call W effective and we had to construct two form out of it so W effective was invariant function on the Lie algebra of the gauge group and we had this procedure of descent procedures that we would take this invariant function act by derivative it's q of something and so on and produce two form so in four dimensions for this kind of service claim is you need to know one holomorphic function on carton of the Lie algebra of the gauge group one invariant under the permutation blah blah blah function which is holomorphic because the here also this color in a vector multiplet is complex field one function which is meromorphic actually and you have to construct four form and this is effective Lagrangian so effective action exactly like in two dimensions which was a two form descending from the effective twisted superpotential descends with from one object which is called the pre-potential cyber within pre-potential under the descent procedure so d of f is q of something and you repeat four times and get a four form you don't need to know this I mean it's so trivial just the concept is important okay so now claim is and exactly like here in two dimensional n equal to theory we had to calculate from UV theory we had to calculate W effective here from UV theory you have to calculate the f of a that's a calculation but whatever you do there will be f whatever story you take there will be here whatever story would take there will double effective very similar but these guys observed and here I finish on it said this f of Zeiburg written has meaning in classical algebraic integrable systems classical it's very important classical and it's also called their pre-potential so we have a statement there was a quantum field seven four-dimensional calculation led to some one holomorphic function f of a Zeiburg written described it and so on and these people mentioned noticed that this pre-potential has a very geometric meaning in a classical Hamiltonian systems complex algebraic Hamiltonian systems and when I come I will start with this and I will show that the one of the examples these people had of the complex algebraic integrable system is a limit of this elliptic version of what I wrote here this particular limit which means that if I know that f of a there I can identify with elliptic version of whatever is written here and then I can ask question very similar what I was asking before that instead of considering these theories on a cylinder times torus where I said that they would read to XYZ version and when Vasya said that if we would do on Sigma we will get something interesting so instead of doing that I will consider these theories on a cylinder times simple omega background R2 with epsilon which is nothing more but the generalization of Kalolso Klein reduction instead of the affine shifts we do the rotation so we will consider theories on that and the claim will be that epsilon will be quantization of this classical algebraic integrable system where epsilon equals h bar in some form so by turning on that epsilon there I will calculate spectrum of answers here that's the philosophy and I explained where it came from so next two and a half hours I will be doing examples of this for two particular cases of hitching system I never said that hitching system which was on the blackboard here had anything to do with integrable system but then I will start with this at the there is a classical integrable system associated to the to this phase space so the modulate space of solutions to here and then I will consider two particular examples which will be elliptical Gero Moser and periodic toda which is elliptical Gero Moser is the elliptic generalization of this where I play between sine square and sin square and that's what you didn't ask me so I will solve it and I will answer questions like what Paul was asking me for this so I will solve it in a sense that I will produce first exact formula for spectrum which will be expansion in the in exponential of minus period so the parameter and that expansion will coincide with the instant on expansion here so there is as I sell instant on corrections one loop and instant on correction instant on corrections will be exactly the expansion since the exponential of minus size minus period here they will coincide and I will observe that these two things coincide then I will make so this is about wave function then I will make general statement that it's always true and derive Baxter equation and this will end up with some paper we wrote with Vassia which is very general and in that language so we will depart from the language of the beta equations we will move to the language of Baxter equation so we will get the Baxter equation for these systems by the way which some of them have been unknown okay and the meaning of the Baxter equations for us will be the statement that there is an order parameter in quantum field theory which is determinant of X minus this color field or something which we have in the and then that that guy will satisfy the word identity as that word identity will not be word identity in a sense that I don't know for which transformation it is produced but it's an identity and that will allow to determine to what Baxter equation we are dealing with and gauge theory will give it with a solution and those gauge theories which will be like this will not give me any more the polynomial solutions it will they will give me certain analytic solutions and analysis property will be determined by asymptotics of the Baxter operators that can be calculated on gauge theory in the asymptotics in all possible ways of turning on and turning off the parameters and we will describe that space and if I hopefully I get to it but for me the important thing is to run this logic that we first get this then we get that when we look at these then we go to elliptic then we go to generic and then we look back what's happening and we will realize that on the way lots of interesting questions will not be answered in particular the question of the perfect meaning of this young young function will not be answered and it's an open question thank you very much