 Okay, so let's start the third lecture. So today we'll talk about more details on the Zyberquittian system, that's an integrable system that's associated to n equals 2, d equals 4, gauge theory on the Coulomb branch of Vakia that we talked about in the last lecture. Now let me actually draw a general picture for that I will try to expand on it in the last two lectures. So we have a quantum field theory for which in the UV there could be Lagrangian description on all Lagrangian description by some string theory construction. So for example, if Lagrangian description, then as we talked last time, the Lagrangian can be specified by saying what is the gauge group G, so that would be compactly group and the flavor symmetry group and the representation of G times F for which we take as the representation for the hypermultiple. So R is G times F, putornionic representation. There are other possible non-Lagrangian constructions from string theory, which I don't write here, but you need to keep that in mind. Now to this UV quantum field theory we associate the infrared field theory at the low energies and concretely here we'll consider only Coulomb branch of the modular space of Vakia which I'll call in this lecture U since conveniently, I mean customarily, the points in this Coulomb branch of Vakia is usually called U variable, so U plane. So the infrared theory is the theory, the non-linear sigma model for the space of maps from the space time to the space U. And we can add several more data on which I will expand, namely, as you'll see today, this space U actually comes with a vibration of a billion varieties of earth so that the total space of this vibration P over U is complex integrable system. Okay, now, given that complex integrable system we can ask questions, well, classical questions, but as well we can ask questions about quantization of the quantum integrable system, I mean of the classical integrable system. While the classical integrable system is just associated to D equals 4 and equals to a theory, full quantum theory, but in the low energy limit. There should be some deformation of that theory which corresponds to the quantum deformation of the classical integrable system. So that deformation has been identified. And now, given quantization of that integrable system, we can consider the algebra that corresponds to the algebra of quantum observables, so let me call it A. So algebra that quantizes the space of functions for that quantum integrable system, the algebra of operators. So on the one hand. And finally, there is a connection between this algebra and the original UV theory that we have started from, at least in certain BPS or so-called co-homological sector of this theory. So there is a relation in the few co-homologies of the original UV theory. So this picture is a plan for the rest of the talk and I will explain the corners of this diagram and various examples. Okay. Other questions about the diagram? Yes. So the theory is a point in UV. The low energy theory is the space of maps from the spacetime to the U. So a point of U is a vacuum of the theory. And if you consider low energy approximation, then we can have that vacuum slowly vary with the spacetime. So the low energy theory is... Okay, so let's write it. So the low energy theory, the QFT, well, classical QFT that you have here, as this is the billion theory, the low energy theory, is the space of maps from the spacetime R31 to the U plane. Do you have low energy equation for motions? You'll be right. Yes, yes, yes, yes. Well, I'll write it, yes, just in a second. It's just an billion theory, yes. Okay, so, yes. Other questions? What was the symbol that you put? You put A is an algebra that converges like C of T. Yes, the algebra of functions on P. Yes, maybe the mathematicians do not usually have O. Okay, O of the... Well, okay. You mean kind of pre-native functions here? Well, I don't want to go into the precise details, but I just want to give an idea. So here we have a classical system, the phase space is a symplatic space, so it has a space of functions. Yes, everything will be holomorphic, but up to now just to see the conceptual relation, you need to keep in mind that here we are dealing with a finite-dimensional integrable system of classical mechanics, and here we consider this quantization, and we have the algebra which quantizes the algebra quantum operators, and then there is magically, well, not so magically after you close this diagram, the relation between this algebra and the original UVC, or is that we have? And the quantization parameter, what is the meaning of this? Yes, and the quantization parameter, well, it would be identified depending on how precisely we will do the quantization, but, well, okay, so let me add the detail right now. So if you do this quantization in what is called complex structure I, what I will explain later, complex structure I, holomorphic quantization with parameter h bar, then it would correspond to the deformation of the theory by taking it on R4 with respect to SO4, equivalent action, and you set one of the two parameters of the SO4 equivalent action, which are usually called epsilon necrosis parameters, to h bar. So epsilon 1 is h bar, epsilon 2 is 0. So that's the literature called necrosis stashvili, a deformation of n equals to 3. I will explain other questions. Okay. Okay, now, so let us first consider, okay, it's step by step. First of all, the classical picture. So y n equals to theory, d equals 4 n equals to theory corresponds in the infrared to the integrable system. So for this, the most important point to understand is electric magnetic duality of the four-dimensional theories. So let's make it very clear. See, of course it starts from your high school or middle school, where you know that you can exchange electric and magnetic field, and you can exchange electric and magnetic charges. Let me label the charges n electric and n magnetic, and all equations of motion of Maxwell would stay the same. Well, if you assume the existence of magnetic charges, which haven't been really observed in the nature, but formally it's the symmetry of the equations of motions of a billion Maxwell theory, like if you replace all electrons by monopoles, and imagine that you have a monopoles running by currents in the wires, then still everything would hold the same. So we don't, so it's just so far when we labeled electrons by electrons, now the monopoles we've chosen just a convention in the billion theory. Okay, so which one? E, B, N, E, and M. N sub E, the number, I mean, the electric charge, integer. The integer number for electric charge, the integer number for magnetic charge. We'll label the charges by integers. Okay, now let's discuss it concretely, and let's do the case of you want to the r-a-billion theory. So for you want to the r-a-billion theory, we can label the curvature of the gauge connection, so f of the curvature, and let me label it by fi, where i runs from one to r. For each of this you want to the r-factors. Now, so the young mill section can be written as i over 4 pi integral over 4 monofold fi, which tau ij prime fij minus square root of minus one. We'll use this symbol. tau ij two primes star of fj. So what are these guys? Well, this is a generalization for the rank r-case of the young mill section. For you want theory, we should write simply as one over g squared young mills f, which f star, sorry, f, which star f, that's absolute value of f squared, and plus theta coupling, of which counts c2, so plus f, which is the topological term with the coupling theta over 2 pi. And then it's convenient to combine 4 pi over g squared young mills plus theta over 2 pi into the complexified coupling constant, which we call tau. So this is in the complex plane, and the real and imaginary part of the complexified coupling constant you'll denote by tau prime and tau two primes. So this is imaginary part. It corresponds to one over g squared, and the real part corresponds to theta over 2 pi. Okay? So you see that in these conventions, the coupling constant, I mean the coupling constant on the young mills theory or Maxwell's theory, one over g squared, it corresponds to the imaginary part of tau, and that's how we write it here. So f, which star f couples with the imaginary part of matrix tau IG, and f, which f couples with the real part of matrix tau IG, and the matrix tau IG is here, matrix tau IG is symmetric, symmetric n times n matrix, and we also want this part, the part which defines equations of motion, the electric, the Maxwell action to be positive definite, one over g squared young mills needs to be positive, and so we'll ask imaginary part of tau IG to be positive definite. So, which means that tau is in what's called sigil, upper half plane of n times n symmetric matrices. So which would be almost, which would be related to the modular space of a billion varieties to which the first one is that one, yes, is square root of minus one. Oh, did I put, yes, I put in this, this is Euclidean signature. This is... Yes, I put, oh, I'm sorry, let me put an I here, maybe I didn't. Okay, in Euclidean signature, you need to put an I in front of the theta coupling term. So I put an I here because I and that I with the imaginary part of coupling constant combined just to one over g squared young mills, and the real part of coupling constant, which is theta parameter, it comes with the I factor. Thank you. So let's consider the equations of motion of this theory. So first of all, if we consider the electric charge, electric charge contributes just by the Wilson line, so that would be integral over the trajectory of electric charge. So this is one contour, one dimensional contour, and you integrate minus I and I a I where N I is the electric charge. So N in the Z to R electric charge. So if you solve the equations of motion, let's just add the thing to the action and extremise the variation of S, say delta S to zero, delta of S young mills plus S source, let me call it S source zero. And A I is the potential corresponding to F I? Exactly. Yes, A I is the connection of one form corresponding to the curvature of F I. Yes. So let's extremise it, and then we find that I over 2 pi delta A I, which D of tau A J prime F J tau A J 2 prime star of F J is equal to, let me erase that, is equal to I N I, the integral of delta A I. Okay, and if you set the variation to zero for every delta A I, it implies that 1 over 2 pi D of this combination of tau I J prime F J minus I tau I J 2 prime star F J is just an I gamma tilde, so this is the two form, the differential of that three form and by gamma tilde, I mean the three form which is supported by delta function supported on the contour gamma. So in other words, this is the contour gamma. Okay, now solving this equation with the source, clearly we have the solution which goes like 1 over 2, the four most coolant law, but we'd like to write it as a Gauss law, namely the flux, so we see that 1 over 2 pi the integral over two sphere, the two sphere which links with this contour gamma of tau I J prime minus I tau I J 2 prime star F J is equal to an I. Okay, so this is the condition on the field strength or oops, I missed F J here. Okay, so there is a combination of F J and the Hodge dual of F with imaginary part of the conflict constant and the flux of this total combination, so you can think about it as electric field and this is magnetic field this flux is equal to an I. So this is what electric charge creates now for the magnetic charge it's electric charge The boundary of S2 is a cycle gamma? S2 links to gamma dual, links it's linked gamma is one cycle in four-dimensional space time and S2 surrounds it so in three dimensions gamma reduces just to a point which evolves with time and S2 in the sphere which surrounds the charge and this is Gauss law, the total flux of electric field strength around the S2 is the charge, but in the presence of the theta parameter there is a shift, so there is a certain combination concretely which is written here of electric and magnetic field which is created by the electric charge Ok, so for magnetic charge well, magnetic charge we don't have to do any computation, magnetic charge is just a logical configuration associated to non-trivial C1 of this gauge connection which means that 1 over 2pi of FI itself integrated over the two sphere is Mi and now let me call actually no, no, no, this is S2 which links contour S2 linked with gamma and that gamma is supports magnetic charge and that's what the equation means so I call it so on that blackboard I call electric charge simply by Ni and magnetic charge by Mi Ok magnetic Ok, so given a configuration with electric charge on some contour we have electric and magnetic field which satisfy this flux condition, given magnetic charge we have electric magnetic field which satisfies this flux condition Yes, in principle you can put them on the same contour and have a dyne Yes, yes, yes For the moment I have one contour, yes but we will go just to allow any number of contours this is a local condition the flux around S2 for a given contour doesn't depend on what happens away from that S2, yes so now what is electric magnetic duality so electric magnetic duality is almost what I have been told in school you just switch electric and magnetic charges yes, but now let make it more precise actually since we have now a lattice of electric and magnetic charges so we have Z to the 2R lattice in which we have chosen the bases in that presentation we decided that we have split that lattice into ZR plus ZR and elements here we called electric and elements here we have called magnetic now for the case of lattice of wrong R you would just switch N and M if you keep track of the signs that switch is implemented by anti-symmetric matrix that would be a transformation for R equals 1 that implements electric magnetic duality so more generally the condition the new electric and magnetic charges which I called tilde expressed in terms of the old one by some matrix of transformation so let me call it actually so let me call that matrix gamma and the condition is that the symplectic pairing of electric and magnetic charges should stay the same and it follows from the quantization of moment namely I mean angular momentum namely if you consider configuration with electric charge and magnetic charge then you can compute that the total angular momentum which is created by that configuration I mean a pair of dienes would be given so if you take one diene with the charge NM and another diene with charge N tilde M tilde and compute angular momentum of the configuration you find that it's proportional to this symplectic combination and that should be integer by the direct flow quantization condition of the spin so it means that this lattice z to the r is equipped with a symplectic form there is a symplectic form yes yes you should have some groups here contours here yes yes so here I'm taking two contours which just run in time and they are located at some too special rotation of the rotations in three dimensions you compute the angular momentum of the field configurations for rotations in three dimensions it's just classical calculation you just take f and b angular momentum angular momentum of a field the electric field has a momentum that pointing vector fb and it's total classical computation in the spacetime you have just two points yes and you compute the angular momentum for the rotation around the axis which connects these two points so a pair of two dyes it has angular momentum and it should be half integer the normalization is chosen in such a way that actually this combination is integer okay so we want after the duality transformation that angular momentum to be the same form on the lattice of electric magnetic charges and the transformation gamma is actually is inside integral symplectic group so gamma is yes n prime yes yes yes yes thank you let's fix notations primes here just for the four two dyes yes okay and here I'm discussing the transformation which implements electric magnetic duality duality transformation when you have a an m go to m tilde m tilde yes yes yes yes exactly exactly exactly so the new f's should be linear combinations of the old f's as well as new charges linear combinations of so exactly that's what we want to do now yes so okay so gamma is in the symplectic group so let me call maybe here some element the group of the group gamma and gamma should be subgroup of well here I've chosen the symplectic form in such a way which corresponds to what's called principle polarization of abelian varieties that will come in a second and so this presentation gamma is just sp of 2r valued in integers the group of 2r by 2r matrices by 2r matrices which preserve the symplectic form corresponding to this formula in the matrix notation as it would be 1 0 r times r matrix minus 1 r times r matrix 0 and g is an element of gamma and let me write g as the block matrix ab cd so abcd are r by r matrices maybe I like to keep it okay indeed now we want to make a linear transformation which expresses new f stilda and star of f stilda in terms of the old f and star of f and so for that it's convenient to introduce matrix tau i g so it's a matrix valued operator which acts on the two forms on r tuple of two forms so r to the r on the spacetime and let me define tau i g as follows it would be the real part of the matrix tau i g tau minus i the imaginary part of the matrix tau i g 2 primes with the hodge star okay so you when you act by this operator on a field configuration you get a new field configuration so now how do why did I choose that combination just to have a convenient way to capture the equation for the electric charge and for the magnetic charge so here is the equation for the electric charge and somewhere I had the one for the magnetic charge yes which was yes over there f i equals m i so using this capital T let's write just our equations so the equations would be that the integral of s2 s of 2 pi tau i g f j r n i and the integral of f i is m i okay and now we want to switch the new combination of tilde and f tilde in such a way that tau i g sorry t i g tilde it makes sense because t i j is not a it's not a greater factor for the transfer for union i j it doesn't force yes I want okay but I mean it's the matrix of it acts on it okay so let me okay and the t i g is the matrix of that thing the matrix of t okay so it's operator value matrix okay so we want to find the new electric magnetic fields and correspondingly we want to find the new coupling constant tau which I will call tau tilde and I will capture the tau tilde in the operator tau i j tilde so after the duality transformation we want to have the equations like tau tilde i j f tilde j is an i and equations that f tilde i is mi tilde okay so that would be em duality okay so how we do that well so since we have expressed the n tilde m tilde n tilde m tilde as ab cd sorry acting on mm and you make that linear transformation and request for those equations and then you find that f tilde needs to be given by c t plus d so here I just multiply r times r matrices and t is the operator value so it has hodge star including it okay f and then you tau tilde is at plus b times ct plus d to minus one so you maybe should ask how do I do the inversion of this operator since t has the hodge star in it well actually algebraically it's easy because hodge star squares to one so if you want to compute because we are in Euclidean yes otherwise it would be it would square to minus one and then I wouldn't write that imaginary unit in front of it so anyways what stands in front of tau ij2 prime squares to minus one okay and since it squares to minus one then when you compute the inverse of that operator you just take the tau as a complex matrix and compute inverse of it and then again use it with the notation of the hodge star in the zigo exactly exactly exactly so now we are came to the main point so after electric magnetic duality after such transformation tau changes as said so it's a tau plus b over c tau plus d to minus one and it means that we are in the quotient of the sigil upper half plane by the gamma by the gamma which is the SP sorry gamma yes gamma is an integral symplectic group acting on two R dimensional lattice with that canonical symplectic form yes yes yes so here I never used the structure of the flat spacetime I think no summation that's the result because it's target no no no after duality transformation you integrate only around s2 but in the global spacetime we have different s2 which is not surround like if you take cp2 is it hard? yes yes yes so I should not be here it's about vector vector with components in some common space no no no of course you don't have any control set up yes yes yes then of course the digital of course to cycle to be sure if you don't have any other constraints at all yeah well but after duality transformation I would integrate over other closed two cycles the f tilde and it would be integral the common class of f should be integral not f is integral but it's in duality transformation yes again f is integral f is integral as well as f tilde is integral don't make this so that things has h operator in it but this is integral yes t is not integral but t includes h operator and so on but you integrate not around s2 surrounding ok ok ok so when you want to integrate on s2 which surrounds some non-trivial cycles you take only on the last line but for other s2 and cp2 sure sure sure for this cp2 you don't integrate till the broken no no no for the other cp2 you just integrate f tilde and f tilde is still integral no no no because you see f tilde is a e plus v like f of ct plus m to play by f d is integral but t is some complex numbers it's easy to just keep some complex numbers right but so f tilde ok so f tilde it has yes tau which acts on it and also it includes some combination of f and h of f yes and h of f is created by electric charges and so it has 1 over tau on it so after all after this transformation the f tilde is integral on it it's not charges it's not just close to this time ok ok well quantization condition quantization condition both in f and f star yeah absolutely it will be visible on the point ok other questions ok so here is the bottom line the main point the main point is that the electric charge is not actually what we been told in school it's not just a number this complex constant is not just a number but in fact this is the point in the modular space of the upper half plane mod gamma and we can identify that with the modular space of rank r polarized abelian varieties principally polarized abelian varieties that's the main point and because of r different electromagnetic fix abelian theory is u1 u1 to the r exactly exactly yes yes yes ok so you want the r coupling constant tau you should think about this tau as tau is principally polarized abelian variety so that's how we should describe Maxwell's theory not by number but by abelian variety and that's how connection to the integral system immediately would appear in cyberquip properly coupling constant yes yes yes yes yes yes so they play a role of coupling constant but there is no a priori choice of the basis ok so yes well they would be parameterized by h mod gamma yes yes yes so now let's so naively if you choose basis in the in the lattice of electromagnetic charges so if you choose what we call electric what we call magnetic then we have period matrix tau ij abelian variety yes and this period matrix tau ij is the coupling constant but the choice of basis is not forced on us so we just we should think invariantly that the coupling constant of you want to the r theory is abelian variety itself and then we don't need to specify the basis yes now recall that in the last lecture we've discussed that if you start from UV theory with gauge group g of rank r then the potential for the scalars has the term like phi phi bar squared and that means that in the infrared the theory flows to you want to the r abelian theory and the eigenvalues which the field which parameterizes the abelian configuration phi commuting this phi bar was called ai so phi is ai where ai labels some the coordinates in some basis of the algebra just you want to the r it's a complexified c to the r the coupling constant tau ij in this special coordinates ai is a special coordinates on the modular space of vacuum and tau ij is the second derivative of the pre-potential function that is implied by n equals to supersymmetry symmetry implies that the coupling constant is the second derivative of the halomorphic pre-potential function on u and the action the low energy action for this you want abelian theory is tau ij some function of ai on the modular space of vacuum and then it's just exactly that action that I've written for the abelian theory but then dilated so we have tau ij prime f i w h f j minus 2 primes f i w h f j tau ij now is a function of a it's on the modular space of vacuum with an i and with an integral over the four-dimensional manifold and also an i in front of tau ij did I have an i here? yes thank you okay so we have a family of theories of abelian which are parametrized by the modular space of vacuum there is a calligraphic u and the coupling constant for that abelian theory is abelian variety as we discussed it's not just a matrix it's a matrix modular that sp2 are the transformation that means it's an abelian variety itself so that means that we have a vibration over the u plane and let me hand over each point of the modular space of vacuum that abelian variety which I called a so the fiber is rank r abelian variety so this space is complex dimension r does it produce this abelian variety is going to be base point zero or yes it would be given base point in a second yes so the p is the total space of this vibration the complex dimension of the base of the modular space of vacuum is r the complex dimension of the abelian variety is r as well so let me I'm using this terminology abelian variety but there are some students unfamiliar so abelian variety for us is a quotient of c to the r by the full lattice and if you chosen the basis of cycles here in h1 of a in the dual pairing like alpha i which alpha j is zero beta i which beta j is zero and alpha i beta j is delta ij and then you pick our holomorphic differentials let's call it dzi and when you integrate dzi over alpha cycles you get delta ij when you integrate dzi over beta cycles you get tau ij so tau ij is a complex matrix now for what I said so far is just a complex torus now for it to be actually defined by by algebraic equations in some project space by Riemann conditions you need this matrix tau to be symmetric and positive definite and that's exactly the same condition that we have on the coupling constant of electromagnetic theory of abelian theory so here we have for one form well it's just a flat space it's one of the coordinates here good so the total space as you see is 2r complex dimensional yes so p is the total space of this vibration so at each point of the module space of vacuum we attach a complex torus of complex dimension r that abelian variety and p is the total space of that thing the total space of this vibration of the families yes of the families of torey fibered over the module space of vacuum because at each point on the module space of vacuum we have a coupling constant tau up to the transformation yes exactly so tau ij at some point u for you in the module space of vacuum is the tau ij of the abelian variety a at the point u it's infrared limit it's infrared description and then you can perform the theory which is depends on your vacuum yes yes yes it's an abelian Maxwell theory you want to the r which depends on the vacuum yes yes yes so at each point you have an abelian theory and then you can have that theory slowly varying with a point on the spacetime and that gives nonlinear sigma model so this is infrared zyberquiton theory but ok it's parameterized by this family of abelian varieties over the module space of vacuum so let's discuss more geometry of that vibration which comes from and causes supersymmetry well it just comes from this condition remember that tau ij is not an arbitrary it's not an arbitrary matrix matrix valued function on the module space of vacuum but it has to come the second derivative of a pre-potential function in some coordinates in this special coordinates so let's write that equation f is complex value f is complex value yes yes and tau is complex tau is complex value tau is complex value tau is symmetric and imaginary part of tau is positive definite and it means that the base space of the module is special color we discussed it last time so the color pre-potential is imaginary part of a i bar d i f special color manifold ok so let me introduce dual coordinates I call it b i dual coordinates on u plane so b i is derivative of the pre-potential function f yes and since we have that b i that d j of b i is tau ij yes so we can write it in terms of the differential one forms so in terms of one forms you have the relation that d bi one forms on u have the relation that d bi is tau ij d a j ok now ok so now let's consider this vibration e to p over u and represent this differential d a i so it should be an integration of the halomorphic one form along the vertical directions of this vibration so along the this complex story and since we've taken differential of it then the differential should be represented by integration over the alpha i cycles on fibers yes of some two form so let's call this form omega so this is the one form on the base on u the two form in the total space and d a i is the integration of the two form over one cycle which gives one form on the base ok so similar d bi is integral of a beta i of omega well if you like to expand that formula in terms of some coordinates u on the u plane then it would be that excuse me bi is the yes yes yes so the relation f is a function yes yes yes I'm using curly letter f here yes it's a function on the module space of valkyrie function on the module space of valkyrie u whose second derivative gives us the coupling constant tau so this comes just from the fact that the fiber that I built in variety has the period matrix tau i g and since d bi related to tau i g to d i g we can find the two form holomorphic two form such that the differential is given by this equation yes ok now the condition that tau i g is actually derivative d i d g f this implies that the two form omega is is closed d omega 0 and locally there should exist one form lambda on the total space p called zebraquiton one form such that ar is integral of a cycle alpha i of the one form lambda and bi is integral of a cycle beta i of the one form lambda ok and lambda yes yes yes yes yes alpha and beta are generators of each one of the billion variety yes so d lambda is omega it's called zebraquiton integrable system zebraquiton one form in the literature just integrable system this is louville form louville zebraquiton one form one form on the total space on the phase space so I already said this system well why that we have almost all ingredients we have two r dimensional two r complex dimensional space p equipped with the two form this is holomorphic two comma zero holomorphic form we called omega and this is holomorphic symplectic structure from the construction it follows this projection is actually a Lagrangian so the fibers are Lagrangian with respect to to omega yes ok so it means that if you consider the algebra of functions on p and compute the Poisson bracket then the functions u which are functions here and so the the foundation makes this geometry to have the structure of integrable system it's complex integrable system yes so this is holomorphic complex integrable system u is it's actually u is a very variety because it's the spectrum of chiral wound yes yes yes yes so yes and also the fibers there are a billion varieties not generic complex story but a billion varieties so yes yes yes so you can say that it's actually an algebraic integrable system ok let's let's make a short break for three minutes ok let's start the second part ok so the conclusion of the first part was that to each n equals to four-dimensional theory in the infrared for its coolant branch of work here where the gauge group is broken down to u1 to the r where r was at the rank of the gauge group in the uv so to each such theory we associate algebraic integrable system so we've explained that link ok now let's let's let me discuss one example of it and mentioned the construction mentioned the notion of zyberg-witten curve that's how originally it appeared well the idea is it's chronologically it's reversed order but the idea is that the Jacobian for some in simplest case simple case you'll consider a vibration, a family of curves sigma over u plane the modular space of work here and you just ask it to be you construct it in such a way that Jacobian of a curve sigma attached to a given point u in the modular space of work here is the bilion variety a to u and so for the rank r rank r theory you would take sigma u sigma to be a family of genus r curves genus r curves over u ok so let me give the simplest example of that it's the following in the four dimensions n equals to theory, you take Lagrangian theory with the group g is su2 and trivial flavor group and so trivial or empty representation for the hyper-multiple the simplest non-lambillion theory you can imagine so I'm not going to derive the answer but let me just present it right now to have an example so the answer for this theory is the following you have the u plane which is going to the c and classically the c comes with the coordinate u which is a trace of phi squared so it's a squared classically because remember that we identified the classical module space of work here with the algebra of the gauge group so it would be t of c2 which is c but mod the while group which is z2 and so the natural coordinate on c mod z2 if coordinate on c is a the coordinate on c mod z2 would be a squared and that's called u so the space of values of u is u plane and this is the module space of work here okay now to each point of this module space of work here we attach an abelian variety of rank 1 and rank 1 abelian variety is just an elliptic curve and so it's Jacobian of elliptic curve is elliptic curve itself so here the distinction doesn't matter with construction by curves or construction by billion varieties anyways the cyberquitian curve sigma curve or abelian variety u is I will describe it by equations in the two-dimensional space and this curve in the c2 with coordinates x, y actually it's a little bit more natural describe it nothing c2 but c times c star with the coordinate x and coordinate y by the equation y plus 1 over y equals to x squared minus u so this is an elliptic curve if you express the y in terms of x solving quadratic equation you'll get a square root so on the x plane there are four branch points there's four branch points corresponding to the vanishing of the discriminant so let me denote the right-hand side of this equation by t of x so if you solve the quadratic equation you'll get square root of t of x squared minus 4 there's a polynomial of the fourth order because 4 zeros and this is the branch points of the curve so you have two covering of the x plane branching of the second order at each of these points it makes an elliptic curve if you want to draw a picture with two sheets it will be like that the upper sheet and then the tube which connects the cuts that you can make between those branch points and here is the lower sheet so this is the genus one curve this is one cycle and that will be the dual cycle like this is a cycle which would call cycle alpha and cycle like that would be called cycle beta so there is a two form close two form on the total space the total space p, the phase space p you can say that you can parameterize it locally by the coordinates u for the coordinates on the base and to say coordinate x which would be a local coordinate on the curve x y is expressed in terms of x by the equation so ux are local coordinates here x is local coordinate here and u is local coordinate here the cyberquit 1 form is the following lambda is dx sorry, is x dy or y so that the coordinate functions a and b the special coordinate functions are given by the integral of this one form lambda respectively over alpha cycle of that elliptic curve dy over y and the beta cycle okay and if you compute explicitly that leads to certain hypergeometric functions the integrals of square roots of polynomial of fourth order is the elliptic integrals so that's the simplest example for such a cyberquit and integrable system is it clear now if you think about this integrable system as mechanics so you'd like to identify with something familiar and the identification is that this is two particle closed to the chain which means that you have the phase space p and phi with the simple electric form dp to d phi just a rank 1 rank 1 system and the Hamiltonian is e to phi the Hamiltonian h is e to the phi plus e to minus phi let's say p squared minus e to the phi plus e to the minus phi okay and I want to identify it with this equation so I think about p is x and e to the phi is y x is p and e to the phi is y and then the Hamiltonian the function on this total space of x and p is this is the u variable okay so that Hamiltonian is the Hamiltonian of the closed of the closed to the chain for two particles in which we factored out the mass so it's just coarse potential the question is the whole time well well yes so remember everything here was complex everything here is complex so when you want to put a real structure you just put i and here and you can put that in different manners and you can well there is a flexibility to make a real integral system from complex integral system by choosing the real contour of integration okay is this example clear yes here is it closed to the chain so there are no boundary conditions it's a system of two particles so the two particles if you call the coordinates of those two particles phi1 and phi2 the potential would be e to the phi1 minus phi2 plus e to phi2 minus phi1 but I factored out the integral of motion for the center mass and called the difference phi and the same for p yes p is the difference between moment of the two particle and the common momentum the sum of the two moment is factored out for the for the motion of the center of mass okay so this is a two particle closed to the chain factored the total momentum phi is not an angle phi not necessary I mean you can make it well you can think about it as an angle in the i direction because this is a periodic thing yes but not necessarily so again that depends on the case of real structure you can think it is a totally system on a real line or is a totally system on the circle and that would give different quantizations of the system other questions? okay so after the simplest example now let's go from the other side and try to think out the most generic integrable system possible which which appear in relation to n equals to d equals 4 gauge series or that then seems that the relevant construction actually captures all examples we've seen for far is a mukai certain sub construction of the mukai construction for holomorphic symplectic spaces okay so first of all we need to have a holomorphic symplectic space p yes so how do we get it well the suggestion is the following you take a symplectic surface let's call it s and you consider the modular space of holomorphic g bundles maybe better to say holomorphic principal g-shifts on s and turns out that this space is holomorphic symplectic space so the holomorphic symplectic structure on s naturally induces the holomorphic symplectic structure on the modular space of g-shifts on s on s surface 2, complex dimension 2 complex dimension 2 holomorphic symplectic surface so complex dimension of s is 2 and physicists actually like to think about this construction actually I think physicists had it earlier but in mass it's called mukai in physics literature there is no name except maybe from a rochic conference paper about hyper-color quotient so here is this thing about it as following you take a hyper-color space hyper-color space s and you consider on the hyper-color space the modular space of solutions to self-dual equations f plus 0 where f is the curvature for a connection for a principal g bundle for g bundle on s so these are just self-dual instantons on s and as we reviewed I think in the first lecture there are self-dual equations f plus equals 0 they provide the hyper-color reduction of the space of all connections so the resulting space has natural hyper-color structure as well so the modular space of instantons on the hyper-color space viewed in one of complex structures the modular space of holomorphic g bundles and that holomorphic structure is one of holomorphic complex structures we've chosen with a specific choice from that hyper-color structure on s well anyways so that's the idea for where we get holomorphic symplatic space p now for integrable system we actually need to have a vibration dp to u so then we need to be something more specific so let's consider this example so the first example is when for p we take t star to c where c is complex curve so t star of c is natural holomorphic symplectic space with a canonical symplectic form dp dq or p is the coordinate in the fiber and q is the coordinate in the base and then you consider yes thank you s is t star of c and then you consider the modular space of shifts which have a support so g-shifts with support g-shifts with g-shifts with not why not no there's really no no sense in it yet there came a shift but it's a structure group but it's a problem or e8 makes no sense well I'm not sure forgive me but why don't you say bundle because I want to have a support it's really not a bundle but it can shift support to the curve well maybe it's not defined in the most precise sense but I want to use it so then keep just in mind example of glr and so this is equivalent to the hitching system there's support not on c but on spectral curve shift on surface no no it's supported on c on spectral curve we have a support just of a line bundle it would come we support just g-shifts on what on surface on s this is g-shifts on s yes it's support on c on zero section are you sure I mean the normal pages have the support on a spectral curve normally you get this line bundle which is supported on spectral curve yes but you get line bundle after you construct the spectral curve but I mean if you start from the hitching system itself you can see what the space is so a hitching system would be g-bundles g-bundles the pair of a g-bundle on c and the section phi of a giant of g and tens of canonical bundles so this is t star of c if you don't like this take this definition so the modular space of g-bundles there is no spectral curve here just the modular space of g-bundles and the homomorphic section a giant valued homomorphic section t star of c and I think about this data is the modular space of g-bundles which are supported on zero section c of t star of c so yes so that was the complex algebraic complex description let me give you here hyper-color description so for hyper-color description you consider g-bundles with the connection a on c for a compact group g and the section phi of one form so it's one form omega-1 in terms of the the modular algebra of g and the equations that a and phi satisfy are the following f a minus phi which phi is zero dA star phi dA phi is zero so this is the three moment maps and the quotient the hyper-color quotient of this hyper-color space by this equations gives the hyper-color modular space so the this hyper-color modular space is called the modular space of Hitching system it's possible to enlarge this construction by taking c to be equipped with some number of mark points and consider the modular space of such g-bundles with the holomorphic section where the structure group is reduced parabolicly parabolic reduction at mark points I don't want to go to further details just to give an idea so that's one example but for this example I need to finish with explanation of what is the projection so so for that let's write down what spectral curve is so you can parameterize the modular the Hitching modular space as follows so take that Hitching field and consider the spectrum of that Hitching field in some representation so for now let me take the example r Hitching system so if you have fundamental representation of glr and take the determinant in this representation of phi minus t so then you get r sections of t star of c corresponding to the roots of the eigenvalues of phi and that means that you have a r-fold which I called c let me call it sigma of c inside t star c so there is the spectral curve sigma there is the base curve c there is r-fold cover there is a projection y and then you take a line bundle on sigma and you push forward that line bundle there is a projection p and get r-on-car bundle e on c so the modular space of Hitching system then is the modular space of such cover sigma with the modular space of line bundles on sigma so that makes it to the structure of vibration that we wanted there is a to p over u vibration and u comes u is the space of spectral curves of spectral covers or cover sigma to c and for a given spectral cover sigma at the point u a sub u would be the Jacobian of the modular space of line bundles l on sigma so this is the Jacobian of sigma sub u no questions so the example which is here matches up with the Hitching system for analytic curve with one more point was it original I was asking no, it's epichuted serial curve ok, so here this is an example so there is one there is one more point on cp1 on cp1 which corresponds to x equals infinity and the singularity is a regular there, yes so this corresponds to this here nf equals 0 yes, it's nf equals 0 there is no flavors the original example is corresponds to Hitching system on cp1 with one regular singularity mm-hmm any questions? do you use interpretation yes, yes, yes but I didn't tell it yet yes ok, so ok, now the next example well you can, I just mentioned I will not explain details because of lack of time but I'll mention that you can similarly construct this spectral covers not for j-layer but for bitrally simple lay group you can use the prime variety of the spectral cover using the action of the while group which permutes the eigenvalues in some representation and so is described by Danaghi ok, so most generally you study the Hitching system with arbitrary number of mark points possibly with regular singularities and you can choose gauge groups for that Hitching system so that's one class of theories this class of theories let me put it here so little table n equals 2d equals 4 qfts and here Hitching system so Hitching system is parameterized by the chase of g the curve c let's say of some genus g with mark points I will not write it now what is it for n equals 2d equals 4 qft the corresponding n equals 2d equals 4 qft is constructed as follows you start with 0 comma 2 supersymmetric tensor theory in 6d, this theory doesn't have Lagrangian description it has a string theory description the theory is equal to type 2b string theory so we have a curve octafied on the space c2 mod gamma gamma sub g where gamma sub g is Makai corresponding discrete subgroup of SU2 associated to g and let me fix your g to be simply laced there are twisted constructions for non-simple laced cases as well but start with that one given by type 2b string constituent mod gamma sub g this theory doesn't have a gauge field it has the field strength if you take a billion example of this theory the field strength of this field theory is three form and the connection the field potential is two forms in some sense it's gauge 34 the space of drawers but it never worked out on classical level to make any sense so we just take with the string theory definition now you take that one important property of that theory one of defining property of this theory is if you compactify it on a circle you get a 5-dimensional theory and that 5-dimensional theory is maximally supersymmetric Yang-Mills this theory has 16 supercharges so remembering the classification of theories the 0,1 theories had 8 supercharges so this one has 16 and consider the reduction of the theory on the space on the space time r3 times s1 times the curve c so if you take the 6-dimensional theory and do the reduction on s1 you get a 5-dimensional theory it would be 5d super-charges with a group g on r3 times c and then if you go to the infrared to the low energies or take the distances on r3 much bigger than the distances on c then you would have the space of maps from r3 to the vacuum of super Yang-Mills theory reduced on c and the equations of the maximum supersymmetric Yang-Mills on c is precisely heat in equations so the reduction of the theory gives the maps from r3 to the modular space of heating system on c Maxim mentioned that I didn't explain it yet so what is the interpretation of the total phase space of the integrable system p2u so in other words what is the QFT interpretation of the tibilian varieties that we attach to each point of the modular space of the Q actually has very natural interpretation namely is the modular space of vacuum of d equals 4 and equals 2 QFT let me put here Coulomb range of d equals 4 and equals 2 QFT on r3 times s1 so the extra model of the three-dimensional theory are the Wilson loops of the gauge connection around s1 for the electric one and for the magnetic one so they give two periodic scalars for each for each scalar of the original four-dimensional theory so the modular space of the theory compactified on r3 times s1 doubles and the fibers of the projection they are the extra model which appear in the three-dimensions in the three-dimensional theory the nonlinear signal model for the maps from three-dimensions to the hyper-color modular space p and here it comes just the modular space of Hitching system So you was the Coulomb range of the modular back here on d equals 4 and equals 2 QFT on r4? Yes so this is on r4 and this is on r3 times s1 it has extra moduli and the projection is when you shrink sorry when you size this one up and expand it and make it r then that extra moduli go away and the fiber of this projection shrinks to zero so roughly speaking the size of the fibers AU is the inverse of the size of this is one let me call it vertical when I talking about the size I'm talking about the whole physical information about the hyper-color metric on it if you keep track only of the complex structure on the fiber then the complex structure doesn't depend on the size of the circle so the halomorphic symplectic space this vibration p over u doesn't depend on the size of the circular one but the metric is roughly as they written there what was the connection between the r4 the r3 cross s1 and the c2 modulo gamma are they just different or r3 times s1 times c what was the relation between that thing of dimension 4 and the c2 modulo gamma no c2 can't make its dimension 10 r has got 10 yes yes yes yes so you take type b on c2 modulo gamma and there is a conic singularity and there is a low energy theory which doesn't have lagrangian description but it's a quantum field theory with excitations which have mass much less than the string scale and that quantum field theory is what people call 0,2 theory associated to simply-laced leologibragy of the type ad so then you take it it's six-dimensional theory and you consider it on the space time of this form so four-dimensional space time r3 times s1 times c so the reduction of this 0,2 theory on c gives a four-dimensional theory and that four-dimensional theory is the one which corresponds to the integrable system of the Hitching type so this whole thing is called Gayotta Witton construction and the theories then equals to theories of this type are called theories of class s someone told me that s stands for six is that common or standard that s stands for six maybe you need to ask Greg Moore I believe yes this notation was introduced in one of papers by Greg Moore and Dinecki and Tevidy Gayotta okay so that was example, concrete example of one class related to that error yes so that relates quantum theories in D equals four and integrable systems so for Hitching systems we explained what are the corresponding quantum field theories now let's do another example you mean they don't have preferred choice of Hamiltonian yes yes there is a space where Hamiltonian take value this is the u-plane yes it's not necessary so we are going to consider the next example where it's not a cut engine but okay so the next example also in the class of Mukai construction for p is the following so in the first example we have taken the surface s just to be t star of c we have taken the Hitching system on c now let's take for the s another simple surface which you can imagine we could go all the way up to k3 but before we do that so let's s to be just a product of two abelian curves so let me call them c-horizontal times c-vertical and c-horizontal c-vertical would be either c c-mod or c-mod z2 so that's a cylinder and that's a elliptic curve with the natural flat structure on c-h and c-vertical we consider the cases for c mod z the cylinder and c mod z2 so that's elliptic curve so the case when c-vertical is just c then we can identify the space s to the contingent bundle of c-horizontal and the construction reduces to the Hitching system that we've considered already so we can think about these cases c-h times cv where cv is a cylinder or elliptic curve as trigonometric or elliptic generalization of the Hitching system so that the Hitching field instead of being valid in the algebra becomes valued here for this case in the group and in this case in the modular space of g-bundles on this vertical elliptic curve so concretely we do the following we consider g-bundles yes, s is a complex surface so the product of two flat curves so you consider g-bundles on c-h times cv yes so this is a symplectic space it's a holomotric symplectic space that would be the total space p so what is the space u? well the space u is similar to how we did in the case of Hitching system so we've taken the spectral of the Hicks field phi now we think about the Hicks field phi as the restriction of the g-bundle on the vertical curve so the modular space I mean the base of the vibration would be the space of holomorphic maps from the horizontal curve c-h to the modular space of g-bundles on the vertical curve cv so more concretely what I mean here is so the modular space of g-bundles on the cylinder is just there a group itself since we need only to specify the holonomy on a cylinder around the non-contractable circle the modular space of g-bundles on the elliptic curve the holomorphic bundles without any connection to consider the holonomy around the group no, I don't understand but g-bundles without any connection I think it's maybe maybe g-bundles on compactified c-stop trivialized in two points yes, yes, if we fix trivialization in those two points yes in relation to trivialization right so when it's elliptic curve it's just precisely the modular space of g-bundles on elliptic curve and algebra geometrically it can be identified with the weighted projector space with the ways given by a0 with a dot ar where these are dual coxster numbers of g for example, if g is this l2, then you have just p1 okay and so the u is the maps from ch to to bun g on cv with the proper trivialization when cv is the cylinder and the projection is just restriction of the structure of g-bundle to a given fiber at the point of ch so there is a projection and this projection is turns out to be Lagrangian so the fibers are Lagrangian and that gives integrals is the base a fine variety? no well, in the case when ch is c and c modz then the bases are fine variety if the base is elliptic curve then I think it's not like if you take just run you just you just you just yes, yes, yes, yes I didn't explain it but actually I'm coming exactly to it so it turns out that the integrable systems associated to this construction they correspond to the gage theories to the following class of gage theories you take finite dimension yes, finite dimension by the way, similar to that hyper hyper-colored description of Hitchin system we can give a hyper-colored description of those objects the case of the full rank when everything is elliptic curve then you just study the modules based on the product of the elliptic curve when the vertical curve is a cylinder then it turns out that the corresponding module space is a module space of monopole equations so in this case you take g-monopoles on c-horizontal curve times s1 and for us g-monopoles on c-horizontal times s1 is roughly speaking trigonometric version of Hitchin system on c-horizontal so the Hitch field is well-utilized algebra is replaced by the holonomy of the gage connection around this one so the corresponding gage theories turn out to be given as follows you take a certain g-quiver gage theory in 6D or ADE or ADE head so a finite simply-laced algebra and the quiver gage theory recorded in the first lecture is a theory where the gage group is a product of SU and ICE so each simple factor corresponds to a node in the quiver diagram the quiver diagram is dinking of g the product of SU and ICE and hyper-multiplet is defined by counting the arrows between the nodes so each arrow corresponds to hyper-multiplet and by fundamental representation between given node i and given node j and fundamental representations labeled by squares attached to H-node on this quiver diagram so you take the quiver gage theory in 6 dimensions and then you concentrate compactification on the dual to c-horizontal so by dual I mean that I'm keeping now not only the complex structure but really take that curve isometric and the dual in a sense of the dual theory so if c-horizontal is R2 then the dual 1 would be point if c-horizontal is a cylinder then c-horizontal is point times s1 and if that's elliptic curve then this one is dual yes, the limit of a tory where both cycles shrink to 0 ok, so this is really 4-dimensional theory this is a 5D theory reduced on s1 and this is 6D theory reduced on elliptic curve let me call it c-horizontal dual this is elliptic curve so now recall that classically we parameterized the modular space of work you are by A and A in the 4 dimensions is just the complex scalar and it comes from the reduction of the gauge connection components of the 6D theory so here A just in the Li-algebra the complex light Li-algebra of the gauge group of the g-gauge so here you would parameterize the modular of the 4-dimensional theory which is obtained by KK reduction of the 5-dimensional theory on s1 by periodic scalar since it comes from the modular space of connections around that vertical s1 direction and transversal components so A would be given as a a5 if I call this vertical direction along this one the 5th direction plus the scalar phi which comes from the reduction of the 6th component of the 6D theory here A takes value in the group and here the modular space of work you are parameterized by the connection, by the flat connection on CH here A connection on CH so for rank 1 is just the CH itself and therefore the modular space is not affine anymore for this 6D theory so this means that the 6D theory will have more improvised distribution until the know yes it doesn't have I mean just that field is not valued as C it's valued, it's elliptically valued yes so this is actually I think I have explained this example so to that integrable system when S is the product of CH times Cv to it corresponds the quiver gauge theory well for the case when G is the simplest AD or AD head and CH the type of CH labels the dimensionality and the reduction of this theory namely you take 6D theory reduced completely to 4D 6D reduced on S1 or 6D reduced on elliptic curve and for Cv we had two cases when Cv is cylinder and Cv is elliptic curve and those two cases they differentiate between whether we take for the quiver type here finite quiver or affine quiver so the finite quiver corresponds to the vertical curve being the cylinder and the affine quiver corresponds to the vertical curve being the elliptic curve so that's another example so let's finish