 Okay, let me start the second lecture in the course on super-symmetric theories on cyber-contegrable systems and the quantization of integrable systems. So today I will give you a brief review of cyber-contegrable theory. So this is all started in paper by cyber-contegrable from 1994. Okay, first of all, let's recall and be the discussion that we had on the first lecture on the supersymmetry and the special cases of the supersymmetry, which satisfy so-called triality rule on spinars, special for the definition of Ennequus 1 super-young mules in the dimensions d equals 3, 4, 6 and 10. And as we explained last week, that's related to the division algebras, like real numbers, complex numbers, quaternions, and actinians. Also, I'll recall that at the last, at the end of the last lecture was explained that if you start from such Ennequus 1 super-young mules in this dimension and consider it on a space, the d-dimensional space of the form d minus 2-dimensional space times c2, where c2 is a real two-dimensional space, which we'll take for now to be flat, like r2 or s1 times r1 or t2. Well, if you take the latest case t2 and compactify the theory on xd to xd minus 2 by sending the size of c2 to 0, then we get a certain theory in d minus 2 dimensions with a larger number of supersymmetries. And moreover, if you choose the supersymmetry generator epsilon, such that the supersymmetry transformation generated by that epsilon, where epsilon sits in that odd component of the superponkary algebra, well, I will remind notations. So, we had a superponkary that was a separately algebra isometry of the space V plus s, where s is a certain spin module, spin V module, which was specially taken or designed in these dimensions to satisfy a certain cyclist's theory with the permutations of three spinners. Okay, so, since this is a superponkary algebra, we had defined the bracket between two spinners, and that bracket is a symmetric odd-odd bracket, and that's what this notation means. Okay, so, delta of epsilon squares to the transformation by the vector field V, which is some gamma mu epsilon, and if you take the supersymmetry such that transformation generated by epsilon would be in the vertical directions, vertical for mu directions along C2, then after the reduction on the xd minus 2, delta squares to 0, and it defines co-homological field theory, and the equations for those co-homological field theories, irrespectively in the dimensions, in the d minus 2 dimensions, 1, 2, 4, and 8 are given by mt equations flat curvature, f plus equals 0, and spin 7 instantons, that means that projection to the representation 7 in the decomposition of 28 to 1 plus 7 for SO 8 under spin 7, decomposition I mean of the anti-symmetric forms, two forms, so p7 of f is 0. Okay, so, so, the cyber-computing theory and the the the continuation of the series of this lecture focuses especially exactly on this case. So, this case, corresponding to this case, okay. So, that's the class of theories we'll be studying. It would be interesting to to find what what analogous, what analogous interesting properties could be found for for the next one, for the octaneonic, but that's for future. Okay, so, let me remind a little bit also on these instanton equations, f plus equals 0. Let's call the instantons in 4D. Well, first, it relates to other interesting first of the partial differential equations in in the reduced dimensions. Well, plus here means that you take the self-dual part of the curvature under the hodge star operation. So, f plus is 0 means that f is minus of the hodge star of f star on a 4-manifold. Okay, so, reduction. So, reduction means that you consider the four-dimensional manifold of the form some lower-dimensional manifold and if you consider two cases when the lower-dimensional manifold has dimension 3 and when the lower-dimensional manifold has dimension 2. Okay, and you take these equations and you take all the fields to be independent of the directions in these other two directions so that the components of the gauge connections are reduced to the scalars on this space. Now, in in the reduction to three dimensions if you take the component along the R and denoted that component as I4, this pi, the reduction of this equation gives us star f and 3 dimension is dA phi and that's a Bogomolian equation for the monopoles. We will encounter more detailed study of this equation in the next lecture. This case leads to the following system. It's convenient actually to twist a little bit to the fields here and treat these two scalars not really the scalars but as a components of the one form on x2. We'll get also an agent representation. So, A3 and A4 relate to the field phi which will be one form on x2 and maybe more properly that will be discussed. If we represent the space x4, actually not as just a trivial product of x2 and R2 but as a cotangent bundle of a two-dimensional remand surface and since this cotangent bundle is naturally equipped by hypercolar structure, the reduction of these equations proceeds preserving hypercolar property. So, what we get from the reducing of the two-dimensional case, we get the equations of the form of f minus phi which phi is 0 and d phi is 0 and dA star phi is 0. One equation from each component of that self-dual 2,4. So, the set of these equations for the agent valued one form on remand surface x2 and the gauge connection i is called the the Hitching system. Okay, one important property that I briefly mentioned but I should stress again is that these equations they possess hypercolar property. So, well, hypercolar... So, now phi is a complex scale of u? Here, no, it's a joint valued. Here, phi is valued in the, in this case, phi is just a scalar, valued in the algebra g. If you take g to be a complex then it would be complex valued if g is real dimension or real valued. Here, phi is one form on x, valued in g. If you wish, you can decompose phi. So, it has two components, phi mu, and you can decompose that, or like phi x and phi y. You can define phi z equal to, to be equal to phi x minus i phi y and phi z bar phi x plus i phi y. And then if you take g to be a compact group so that the algebra of g is real valued, then phi z and phi z bar would be complex valued. So, usually algebraic geometries like to think about the Hitching system not in the real language with the hypercolar system, but in a complex language as a halomorphic, halomorphic analytic language in there. Okay, so there remark that the system which is given by hypercolar quotient could be equivalently represented as a complex quotient of the following system. You consider the space of, of halomorphic, g-halomorphic bundles on x2 and a section of k. Phi would be a section of the canonical bundle on x2, valued in the geointre-presentation, or valued in the ideology of the gauge group which is, which is halomorphic, halomorphic section. This pair, a bundle in halomorphic section is called the Higgs pair or Higgs bundle on a remand servers. Are there comments or remark questions? So, where was I? Yeah, I was going to explain about hypercolar structure. Well, let me remind that a hypercolar manifold means that it is first of all a scalar manifold, scalar. So, there is a, there is a complex structure i and a symplectic structure, omega and metric g. And they are all compatible, like g is omega times i. Simplatic means non-degenerate two-form, which is closed. And complex structure in the serial language means just an endomorphism of the tangent bundle of tm, which squares to minus one. And that's an almost complex, that's an almost complex structure. And the integrability of the complex structure means that if you define sub, sub bundle of the complexified tangent bundle by the projection to the eigenvalue i of this guy, since it's square to minus one, it has eigenvalue i and minus i. So, the sub bundle, let me call it tm10, which is the sub bundle of the complexified tangent bundle of m, sections of it satisfy that i times v is IV. Okay, so this differential geometric reminder of what complex structure is now on hypercalor manifold. Those conditions are extended here. So, that was for calor by asking existence of three complex structures, which satisfy quaternionic relations. So, you have i squared, j squared, k squared, plus minus one, and k is ij. So, this is imaginary units of quaternions. And the manifold m is scalar with respect to each of these complex structures. And moreover, you can summarize, you can also induce from that that a complex structure i for any vector x in x2, such that x1 squared plus x2 squared is one given by x1i plus x2j plus x3k. So, for a sphere in the space of the imaginary quaternions of norm one, any such combination is also a complex structure. Okay, so hypercalor space has Cp1 or S2 worth of complex structures. Okay, that's one reminder about hypercalor space, which is, okay, so that Cp1 worth of complex structure is used in most of, in many physical constructions. But for algebraic geometries, often of the hypercalor space is viewed only in one of its complex structures. And there the hypercalor property means that, so means that if you consider a two-form omega i given by the combination of the two forms, omega j plus i omega k. So, this is what the real symplectic forms associated to i, j, k, and the metric, and this is a halomorphic form. So, this halomorphic form is 2, 0, halomorphic form in complex structure i. In the complex analytic language, hypercalor property implies that this manifold is a halomorphic symplectic manifold, and omega i is a halomorphic symplectic structure of type 2, 0. Questions? Now, so a good way to construct a menu of interesting hypercalor spaces is the hypercalor quotient, which is an analog of the usual symplectic quotient, unfamiliar, and that works as follows. So, the hypercalor quotient is defined by taking an action of g group, so group g x and m, and we ask that this action of g group on m is Hamiltonian with respect to the triplet of symplectic structures, omega i, omega j, omega k, and let mu i, mu j, mu k be the corresponding moment maps. So, moment map means that differential of a given moment map, this is one form, and that one form is, well, so what are moment maps first of all? So, moment maps are functions on m, which are valued in the allele algebra of g. So, this is a, and the differential, well, in the dual, so the differential of the moment map is the contraction of the vector field corresponding to, well, its value, so it's in dual, defined as follows, the value and element of the allele algebra of g of the differential is the vector field generated by that element of g, contracted with the corresponding symplectic form. Okay, so at the hypercalor quotient means that you consider the inverse image of some invariant element under g action, well, if g is abylented, if g is u1 it has to be 0, so let's say h, and take the space mod of the g. So, this is the hypercalor quotient. Okay, and now back to instant tones. Okay, so, well, so the main property of the hypercalor quotient, well, by the name, is that given a hypercalor manifold with the action of group g on it, this quotient is also hypercalor. So, what are instant tones? Well, instant tones, as plus equals 0, the modulus space of instant tones, namely the space of connections a on a principle, a gym bundle manifold x, satisfying this equation, mod the action of group g is precisely the hypercalor quotient as it's written there. The three components of this self-dual equation are the three moment maps with respect to the hypercalor structure on the space of connections a on a hypercalor space. Okay, so it's a hypercalor quotient, so the result in space, let me call it m of g of x4, the modulus space of instant tones is again hypercalor. Okay, now, okay, now let's get back to the super symmetry and discuss the representation of the relevant super symmetry in four dimensions that we need to proceed to cyberquit and there are questions about that. Actually, maybe I should ask to interrupt me if that is well known, should I jump to next? I need some feedback from the audience on what I'm saying is the is the is it a little familiar or so now, now let's focus on this case of n equals 1 super equals 6 and consider reduction to the d equals 4 and let's see how the super partner algebra of d equals 6 super equals decomposes with respect to the d equals 4 symmetries. Well, the n equals 1 point carrier in d equals 6 was associated to the super space v6 plus a certain spinor module of spin 6 and that s was taken to be the positive carol spinors, so this is a 6d vile spinors and the second factor C2 is a is an auxiliary factor to have a symmetric map from two spinors in the space some gamma epsilon v mu needs to be symmetric and since the gamma mu just on a vile spinor says anti-symmetric one has to take two copies of those spinors and compose the construction with the anti-symmetric pairing in C2 then the result becomes symmetric. Okay, so and there are eight, the space is eight dimensional. Now let's consider the reduction to the four dimensions. Okay, so then v6 is decomposed as v4 plus v2 this would be to vertical normal components and the vile spinors of the six dimensional space they can be viewed just as a Dirac spinors of the four dimensional space so s is let me call that bold s so the bold s becomes to be the two copies of the Dirac spinor since in four dimensions and that's a dimensional space and the bracket between the two spinors which you can compute by reducing from the six dimensions is gonna give a vector in four dimensions plus a vector in the normal two dimensions you take two supercharges of that 6d in a close one reduced theory and compute the air bracket find where p mu can be decomposed into four dimensional plus the normal two-dimensional. Now from the perspective of the four-dimensional this is just this generator is invariant under all other transformation symmetries therefore it's called therefore the generator is called central charge. Okay and one two more remarks about the symmetries. One symmetry is the following we can we connect by the automorphisms of C2 which preserve the symplectic pairing that's just sp sp of C2 with respect to this symplectic pairing in symplectic language it's sp of 2c and if you take a compact subgroup of that it's usually denoted by sp1 or sp2 and in this context it's called se2r symmetry. That's one thing. The second thing is if you consider the decomposition of the spin 6 into spin 4 and spin 2 under this reduction then spin 2 the rotation of that normal two directions it would also act on the supercharges and so spin of the two-dimensional space just like se2 or u1 is called u1r symmetry. Okay so all together let me summarize that what is 4d what is 6d n equals 1 reduced to 4d n equals 2. We have the following we have the superponkerer algebra of v4 plus p and c2 so that's 4d superponkerer with two copies of Dirac generators of fermionic transformations and in addition to that there is a generator which we call translation in the normal directions or central charge so let's just a complex line and then there is a lay algebra of the se2r, se2r symmetry, se2r symmetry and the lay algebra of the u1r symmetry. So that's the algebra, the superalgebra in four dimensions that we need to focus to discuss the questions here yes so this one is six-dimensional yes we decomposed it into projection to four-dimensional into dimensional and and this guy we called it central charge so the the translations in in the six in fifth dimension under the reduction to four-dimension they they compute with all other generators and therefore it was called original in the literature if you just start from the four-dimensional construction it's a central extension of of the algebra of this algebra that you can add to it that's by the name. Now let's discuss our representation d equals 4n equals 2 well for these papers it would be convenient for me to use the hyper the hypercalor notation for for the superalgebra that I written on in the last lecture as follows so the space s is identified with the two copies of the quaternions and v6 x and that by means of two by two matrix so if I take the generators because the quaternions here as tx and y for the space r1 5 with the signature minus plus plus so y is the four-dimensional and t and x are one-dimensional quaternions in one- dimensional spaces then you represent the action on s of a vector from v6 by matrix t plus x t minus x y y star where y is thought as an as a quaternion quaternion from r4 identified with h here we have r plus r plus r4 and t x and y respectively according to here our elements from here okay now let's consider representation then equals to superporn career in this notations so I will discuss one particle representation actually for this discussion doesn't it doesn't really it's not necessarily to be in the four dimensions let's let's stay in six dimensions and discuss representation of n equals one superporn career in six dimensions and then we consider the reduction no no t and yes yes t and x are real numbers so that's the tensor identity and twice quaternions yes yes two directions is t and x yes yes it's a minkowski signature so t t has other signature from everything else okay so yes the two directions which correspond to the the the other direction they would sit inside r4 in yeah I mean yes okay so let's discuss one particular presentation of that super well what we have we have so let's start from 6d and consider a particle which has an energy which has 6d momentum of the form e px 0 and and and and and for zeros so I mean denoted by p mu and in this state in Hilbert space so it's a sub sector of of the full Hilbert space of quantum field theory let me call the h1 one particle one particle state with momentum p mu let's let's check how the supersymmetry generators act what their algebra and what representation could be so in terms of those in terms of those annotations we have that q alpha k beta which is from h plus h plus h they give us see this delta alpha beta times okay let me let me decompose that in them so let me call components of q in the first copy of h with index one and in the second copy with index two so since we have taken this state with in p just to be along t and x then we have a diagonal matrix and of the form energy plus px and energy minus px 0 0 0 here and then we have that q alpha q beta in the first copy is delta alpha beta h1 plus px and q alpha q beta in the second copy is delta alpha beta and so minus px okay so now this is exactly commutation relations or anti-commutation relations defining the clifort algebra so you have a clifort algebra on eight generators q alpha one and q alpha two and they satisfy this commutation relations now if you want to build a representation of that we know how to do that you just take a representation of clifort algebra but there is a subtlety are related to the related to the fact that while epsilon has to be greater or equal than the moment in the special direction for the state to be state to be time like in other words another way to say it that the Lorentz invariant combination epsilon squared minus p squared which is m squared has to be no negative yes so in the boundary case one epsilon equals p namely when the particle is light like here is 0 and so the anti-commutation relations for q alpha q beta in the second copy they have to be trivially satisfied okay so let's consider those two cases separately so the first case when epsilon minus px is greater than 0 then you have the clifort algebra on eight generators and we built its representation it should be 2 to the d minus 2 so it's 2 to the 4 that's 16 dimensional representation yes of a clifort 8 and if you build that representation by starting from like a fork vector well from from from some element in the Hilbert space and applying to it the raising operators which you construct from this q alpha and q beta define raising operators just break on that spin 8 to u4 and consider combinations q alpha plus minus q alpha plus 1 1 3 5 7 yes so that will be raising lowering operators a square to 0 and they are anti-commute to 1 and implying applying raising operators the states in the Hilbert space can be depicted as follow it's the vacuum state then you act by four raising operators then you have here then you again you get six states and for more and then finally one so another way to to say what I'm doing here it's just exterior algebra in this c4 yes so corresponding to the action of four raising generators and the the diamond with one state here four states here six stays here four stays here one state here is what is called long long multiple okay if the original state is bisonic then the statistics of the states change with the level so the bisonic state fermionic bisonic fermionic bisonic for us a particular important would be degenerate representations called short multiplets and they are associated to the case one p is equal to epsilon so it's like like is px it's like like state that's 6 d minus zero and then okay so then the discussion is similar except that q alpha and q beta in the second copy they have to be zero because they square to zero and in the unit in unitary theory they do the Hermitian operators so they have to be zero now in the first copy we again have cleft algebra on four generators yes and that give us two to the two that's four dimensional space and that four dimensional space decomposes as two plus two with two bisonic states and two fermionic states and if you draw the diamond associated to the action of raising operators it goes like that so for example you can start from the spin one half and act by the spin decreasing operators q you would have two states of the spin zero and one state at the spin minus one half so this exam this particular example is called half hypermultiple so what it has what it has it has one one myron of Hermion with helicity plus one half and minus one half and two scholars two real scholars okay another thing we can do with this shorter with short multiplied is to tensor it with a representation of a spin for and if you consider particular case when you tensor this diamond with a spin one half spin one half state is plus one half minus one half you get a diamond of the of the form like that you need to shift this one upstairs by one half and add to a copy of it which is shifted downstairs by one half okay so that is that state is of speed of helicity one that's helicity one half that's helicity zero that's helicity minus one half and that's helicity minus one and now if you think in terms of massless fields then this state is this states one and minus one they correspond to the vector massless vector is helicity plus minus one yes this state this is scalar so since we have two of those you have a complex color and this states these are two moirana fermions yes so since there are two of those it's okay two copies I will just try two copies and that's actually the natural fermions if you take them to be to be just induced by the representation as this this one that we are considering the sections of us okay I've mentioned okay so I've mentioned here half hypermultiplied why half well the reason is that to have a true representation of n close to superpunk algebra we need the space of scalars to be to have hyper color abstraction particularly it's supposed to be four-dimensional so here we have only two real scale so to get a four dimensional space the standard construction is the following if there is a gauge group whose representation or is a complex suppose is complex rip of G then hypermultiplied is a half hyper which is written there taking the presentation R plus another half multiple taking the presentation a bar and all together that gives a hyper color structure on the space of scalars of the hypermultiplied another way to do hyperclerosis structure is just to consider representation quarter neonic this quarter neonic structure in itself another way is take our tilde which is quarter neonic g-module then take just a half half hypermultiplied tensor this representation it also gives a full hypermultiplied so that's about the names and now let's get back to the fields oh sorry I should have said that yes in this discussion so far we just presented the story in the six-dimensional notations now let's see what happens when we reduce to the four dimensions well when you reduce to the four dimensions let me now take another convention and think about the coordinate x as one of the coordinates which goes in the normal direction and also let me make you want our symmetry transformation in such a way that the momentum along that direction is real so then px should be just thought as a central charge in terms of the four-dimensional language it's related to this discussion so if in 4d picture px is treated as central charge so that x is along reduce the dimensions then the two cases that you had here are the following the first case is when the energy in the rest mass this is an oh no no for four-dimensional because this is because this was discussion of the of the short multiplet one epsilon is px so it's a master state and for and for master state you take spin 4 you get two-dimension less so you have spin 4 and this is one half of spin 4 the helicity of spin again now discussing back on the four-dimensional picture so let's take the case when the particle is at rest assuming that it's a massive particle and it's it is at rest and interpret px that we had in those formulas as a momentum along the fifth direction so it's a central charge and then the two cases that we have considered there the case one is one e minus the value of central well the central charge usually is noted by by z literature introduce no z so when the first case was when it's bigger than zero and then we had long multiplet discussed over there and the second case is one epsilon the energy e is exactly equal to the absolute value of the central charge it means that the particle in six-dimension is light like but when you reduce it to four-dimension all all its momentum is in the in the fifth and the sixth direction it has zero momentum and in three-dimensional space and it has momentum e in in time direction okay so this situation from from discuss representation of the super symmetry algebra we have seen that it reduces to the short to the short multiple so the states in the Hilbert space which satisfy this condition they're also called bps now the fields so we've discussed masses vector and the fields that describe muscles vector multiplied are the following is just dimensional reduction of n equals one vector multiplied from six dimensions so you have a gauge connection a mu complex scalar phi and and the fermion psi which is a section so it's a section in gamma of x4 the spinner bundle s constructed from the two copies of myrana spinners or holomorphic Dirac in Euclidean 6 so to define the vector multiplet we just need to take the data that we need to choose is semi-simple group g semi-simple compact group g and let me introduce the notation the reduction of g into a simple factors would be product over the set capital I and simple factors would be called g sub i this is simple compact the the Lagrangian is the young mules Lagrangian was properly extended with the couplings with the fermions the essential part in this Lagrangian is for each simple factor we have one over g squared fhf plus i tau i theta over 8 pi squared fhf so that's that gives young mules kinetic energy and that gives the topological coupling called also theta term and physical literature mathematically it corresponds to the second second sharing class of a given field configuration okay so that's for h i I haven't written here but it means that for each i we have two real numbers one over g squared and theta over 8 pi squared and the combination of written like that sorry this i is square root of minus 1 not too confused so so this is called a complexified coupling constant okay so let's see for the definition of the vector multiplied just to summarize the data is is the same as simple compact v group and the array of complex constants let's see it for the vector yes yes it depends on the global structure of the group so the group is not really important for simple factors only here is the cover because that's the color of the workplace and some group in the center I mean but for for physics I mean it is the global structure of g is important like for example if for if instead of us you too you would assume that it's a so three then there would be no quarks no I just want to say that the group is not necessary for the physical groups ah okay okay okay yes yes okay so I agree but let me stick here to this assumption so hyper-multiplet so the fields there corresponding to the two copies of this guy that we had here is to direct to my run as peanuts yes and two complex scalars that are combined together into one paternionic so hyper-multiplet is a generalization of quarks or electrons which couple to the gauge connection and so we have to choose the representation of the gauge group in which they enter the theory and the additional data to describe n equals to theory is the set of masses for those multiplets and it's convenient to think about those masses in the following way we actually take hyper we actually choose pick pick another group let me call it f and it's called flavor symmetry so this one I just want to be compact it might contain you one factors all I stare at their semi-simple I don't want to consider you one a billion factors since famous hope to result on the on the value of the beta sorry beta function for the coupling constant and you want a billion theories the beta functions for them they behave bad when you go to the small distances and it's not good to define the theory as a sensible quantum field theory so there are only non a billion factors there but for the other flavor symmetry group it doesn't have a gauge field or which you integrate it just couples with the fixed coupling to the fields of the n equals to supersymmetry or vector multiplet sorry to the hyper-multiplet so it might have you one factors and then peak representation or representation of f times 3 and ask this is a representation actually to be quarter new so the hyper-multiplet is sits in the representation of f times g it couples like it supposed to couple 10 equals to vector multiplet and also it couples to the constant scalars of the n equals one vector multiplet for the flavor symmetry from from which we keep on the complex colors and the values of those complex colors up to the conjugation by f are called masses of the n equals to theory so the the other data that we have here the masses that that would be an element of the complexified group of the maximum torus of the flavor symmetry I can one the dot mf for the stores is you want to do some six dimension yes so where the masses come I mean study can you write in details from six the six dimensional launcher no no no it's a what Maxine mentioned so so it's in four-dimensional it's four-dimensional setup constructed by the fallen way you consider n equals one sorry you consider vector multiplet associated to the flavor symmetry yes and this vector multiplet when it uses to four dimensions it has a complex scalar coming from the two components that we have reduced and that complex scalar is fixed to a particular value in 6d you have pure young mills and also in 6d you okay so in six okay in 6d you have a mass less than equals one vector multiple and then 6d you have yes no no no it's young meals with that spinner psi it's super young mills yes in 6d with these two spinners with two myrana while spinners in 60 no no in six dimensions you don't have those colors the scholars are coming by reduction of six dimensions 240 yes yes yes then then you consider in the 6d theory a vector multiplet for the flavor group f yet that you add to the theory yes it has the gauge fields the gauge field the six-dimensional gauge field and also spin then you reduce it to four dimensions okay and you give a background value to the complex colors of the flavor symmetry group and you settle the rest of the fields of the vector multiplet for the flavor group f to zero and that and that gives a way to write a Lagrangian in a complex form no in 6d it's a hypermultiplet well it depends on how you count it's in representation of f of f times g yeah the dimension can be as large as you want up to the constraint on beta function that we are going to discuss now shall we shall we do a short break maybe a five minute break so there was a question about Lagrangians so let me write the standard quadratic Lagrangian for n equals to vector in 4d by reducing just by the reducing Lagrangian of n equals one vector multiplet in 6d so the Lagrangian factor multiplet in 6d is very simple it's just a Lagrangian for the fields xi where xi sits in s again two copies of my run of well fermions in 6d and just supersymmetric young also it's fmn fmn d6 1 over g squared yes plus psi dirac operator psi ii I runs from 1 to 2 that's for the two copies and the contraction is with the anti-symmetric psi ii psi ii is my run of well fermion in the 60 the origin of you you don't have that in 6d you you added by hand you noticed that it's possible to edit of you after you consider the theory reduced since it's topological term it depends just on the topology of the smooth principal g bond on the four-dimensional manifold it doesn't affect the invariance of the action of the supersymmetric transformation okay so under reduction to the 4d okay what happens so fmn fmn they come split into the F minu squared okay let me let me take convention the term is from 1 to 6 and mu is from 1 to 4 and ab from 5 to 6 yes so then it would be f minu squared plus d mu phi a d mu phi a the schematic term for the two real scholars yes plus phi a phi b squared it's a commutator for the two scholars and okay that's it for the bazonic fields and for the fermionic fields you have the gauge of the four-dimensional the 4d dirac operator on size and from the reduction of the fifth and sixth components here you would have like you cover coupling usually called in physics that's psi i and then okay so that would be phi phi 6 plus i 55 times gamma 5 psi i okay that's the 4d Lagrangian for for the vector multiple to which you connect apological term as was mentioned so for hyper multiplets well I will not write it but it's just complex color field and two fermions each are coupled to the gauge fields of this vector multiple in the suitably super symmetric way no I mean they constrained by the n equals to c person oh here it was here it was quadratic yes yes yes now now we are going to discuss yes precisely we'll discuss the non-linear Lagrangians shortly I just want to summarize first what I constraint on this data by which you can define the sensible quantum field theory in the uv without the need of other fields called to ultraviolet complexion completion so what are the conditions the conditions are uv complete theory defined by the data g tau yes and are which is jf cotter nionic representation and masses complexified torus of the flavor symmetry group so what are constraints on here well the the notations which one this one ah the notation it's the Lie algebra it's what what's written here here yes yes it's the little algebra of the maximum torus of the f carton it's carton of f yes carton of f carton the algebra okay that's what completely defines the theory so what are what are constraints the constraints are well known since the calculation of the beta function and the constraints of this in each factor in each factor i of the group g so g is a product of simple factors g i the beta function should be non-positive otherwise the theory is strongly coupled on the small on small distances and it cannot be essentially defined as an ordinary quantum field theory it needs other degrees of freedom so uh the yes in the infrared i mean in the infrared you have strongly coupled physics and that's exactly what we want to understand yes yes yes but the the the quantum theory is defined in uv and then it gives some interesting interactions sorry in the infrared in in the uv it's asymptotically free yes okay so the beta function it works as follows so if you compute it then you find that so one over g squared young mills on the ice factor let's scale mu so let me just tell a few words on the renormalization here so let's let's consider the space of fields let me draw like that so up to here would be fields with the with momentum mu and here would be fields with a momentum lambda so this is go this is the energy scale so usually when we talk about quantum field theories we integrate over all fields above a certain scale and compute the effective action at the given scale mu so if you if you fix the coupling constant at the scale lambda and then integrate over all fields to get the effective action at the scale mu then the effective coupling constant at the scale mu is related to the effective coupling constant at the scale lambda well the one loop level by the beta function related by the beta function relation and that's one over the square to scale mu is one over the square to scale lambda plus plus this combination plus one over eight pi squared log r e from mu over lambda well times a certain factor and this factor for n equals to sorry no no in four dimensions in four dimensions well unfortunately it doesn't so yes right six six dimensional theory is not is not in our mother so we are doing 4d here so 6d to explain super symmetry as it can follow straight well not not not only we could start in 4d but 4d is a is a nice way to to see the classical theory and later we'll consider six dimensional theory but we'll take it in the context of string theory so 6d theory it requires uv completion by string theoretic degrees so here in 4d and you can think about the fields of the 4d series usual quantum fields without string theory so that's that's the the the old computation of by by by Hoft and others so the beta function where where this factor if you if you computed in the n equals to theory so with gauge group g and representation r as with some simple sample group g and the quaternionic representation r of this group this factor turns out to be equal to c2 of the adjoint of g minus c2 of the representation r and by c2 i mean the coefficient in the metric defined by the taken trace in the representation r it's defined to be as c2 proportional to the standard the standard metric on the le algebra so h is in the le algebra of g and this metric is normalized such that the highest root his square equals to two the the the standard normalization so then the factor in front of the standard metric if you compute the metric by taking the trace in representation is called c2 and that what contributes to the one loop beta function okay so for example for g equals this u n you have a c2 of adjoint actually that's my general formula it's always twice the dual coxter number so for c1 and 2n and c2 of the fundamental is one so the constraint on the beta function we want this term to be positive so that one over g squared goes to zero if you send me to infinity it gives the condition that number of the fundamental multiplates which was called the flavor group is u of number of fundamental multiplates and from here we see that the number of fundamental multiplates let me call it an f should be less or equal than to n so that's a constraint on the representation r when the gauge group is just a simple u n and you consider representation the several copies of the fundamental representation so how about a general case well actually it's it becomes quite elaborate but solvable it's certain combinatorical combinatorial problem on the on the space of you know dint in the diagrams and representations and it was completely solved recently in a paper by Tajikov and and hard y so that they have complete classification classification of pairs g and representation r such that c2 says that in each bit of i that beta i is less than zero in each i can you sort of make slightly more explicit what this means that you're trying to explain some constraints on the data at the top and you've written down this expression but how does it relate to the data i mean the g is related to the imaginary part of tor but what else is happening now yes it is a constraint on the combinatorial data it's just a constraint on the choice of representation r that we can do but i've not understood what this one over g squared y n times m or or as a function of m equals and what's right so how does that depend on the data that we picked at the start oh yes yes yes how it depends it depends in terms of of this coefficient the coefficient which stands here is is the difference between the c2 of a gender representation of g and c2 of representation r sure but i've not understood how m and landed depend on the oh sorry it doesn't depend okay so this is one little beta function computed in the uv regime which is more with energies more greater than all masses in the theory so it doesn't depend on masses it it it depends only on the space of fields complicated function of g tor and r and g or what's that how do i sort of work out what that function is beta functions a vector field on interest in one variable and you need to be contracting in one word variables it's constant so and i saw me some you're trying to explain a a sort of constraint on the data the data is g tor and r and so yes yes the data here as g tau and r and the constraint implies actually just the constraint on g and r on on how on how big r could be chosen for a given g if right if there's no rule here so i've not done yes masses and towels well there's actually the running of towels but masses play no role here yes okay so what does this equation with the m and the lander have to do with the choice of g and no no no no no it's an explanation about need to function yes it's it's a computation which which which i didn't do but okay let me let me give you a cartoon okay the so it's all the derivation of what the constraint is and then the constraint is just that this star has to be something other yes yes yes okay so that's much simpler okay yes yes well i think i'm maybe i confused here so so this constraint that beta function is negative it means that so it means that this coefficient must be positive so that the sorry yes sorry yes yes yes that's that that's the requirement the requirement is that the beta the this constraint is positive maybe zero yes the zero case when it saturates it's called the theory in conformal class so in particular this one when an f equals to then it's the beta i depends on the ice factor on ice yes it's for it's for each ice factor so here you take a giant of gi yes yes and you consider representation r as the representation of gi subgroup of g okay so they have complete classification of this combinatorial data just just pairs g and r and it's quite elaborate i i don't have time to explain all this classification let me give you just a subset of it which actually which you can trace yourself easily so the subset of this classification is the following let's assume that g is a product of su and i factors yes and let's assume that the presentation r is just a sum of some by fundamental representations between different nodes here yes and plus possibly fundamental representations from an i to some fixed vector spaces which are called n fi attached to each node yes so that could be graphically pictured by by quiver so where circles denote gauge groups and arrow between circle denote by fundamental representation and to each circle you can attach a square and the arrow between a circle and the square denotes a fundamental representation so that's fundamental here and this by fundamental or there yes yes you can you can also have dual it doesn't affect the computation of the beta function yes yes you can you you can have a dual okay so for the square let's see it so we've just computed that representation of c2 of i joint is two and and c2 of which fundamental is one so if you have a node to which and i to which you have attached nf fundamental representations and some neighbor nodes which have nj sitting in those then if you compute the contribution of all this hypermultiple into the beta function of an i you find that that coefficient which was like minus beta function yes so this coefficient turns out to be two and i minus nf that's we already did just for a single gauge group and then all these neighbors they contribute a factor of an i or of nj so minus sum over nj where j and i are linked i mean for for for each link and you you you you can try attach like two by fundamental representation principle it's also possible so for each each link here contributes okay and and then there's a certain just discrete problem on the space of of graphs and it turns out to be equivalent exactly to the carton carton dinking maybe cuts classification of generalized coxster sorry of generalized carton matrices generalized carton matrices and the result is that for the class of the theories the possible diagrams that you can draw have to be so that that here let me call it gamma is ad or a fine ad dinking dinking graph in particular that what i was trying to picture here this is allowed that correspond to a one hat is okay and one more remark so it's an example of a fine graph and actually ad a fine graph always leads to the beta function equals zero so all such theories are in conformal class all such i mean a fine so fine theories are in conformal class yes yes you can find classification in in in the paper but roughly you can have a large large classes large class of graphs where you alternate s o and sp between nodes and then there is a certain number of exceptional configurations well it's it's a certain combinatorial problem extending this carton classification and and it's completely it's completely written down the situation which you have more of a fixed point yellow theory which flux one of one to the one to the other you mean on renormalization group when yeah that's not for me uh no no no not for any coaster series no it is nf arbitrary or is zero for these graphs these are fine figure oh so for a fine yes for a fine nf is zero it's in conformal class and nf must be zero i mean it is implied by you see by the fact that on the carton matrix of a fine graph the the sum i i didn't finish okay okay so one one one more data nf is zero and an i is chosen positive integer times a i hat where a i hat is a is a dinking number attached to each node and then automatically you you you have the the beta function is the zero vector of the generalized carton matrix the null vector which is negated by the carton matrix yes so for uh finite carton graphs uh it's possible to have some positive nf attached and it depends on what numbers and i you assign to it but you just need to satisfy um this equation so that's what the in nine case what does it mean yes so i mean in nine in nine in nine quiver yes so it's not it's not in nine gauge group it's a nine quiver which means e eight hat which is uh one two three four five six yes eight nine and uh you put numbers here uh actually uh you put okay so a i's that you put here is your n well let me let me write i's so it would be uh one two three four five six uh what was there two four three maybe three yes twelve twelve seven yes yes so if you compute two n here in this node it's twelve if you compute the sum of neighbors it's five and three and four and it's uh it's twelve yes five three and four is twelve and uh and all other nodes it's obviously that also it holds because it's arithmetic progression here so each node is arithmetic is the is the mean of the two neighbors yes so if you if you take so then if you take you take the gauge group which is a product of u and a i check and that's that's an example okay now let's discuss uh the low energy theory so first of all what what do i mean by low energy theory i mean the following if you start from that uh theory in the uv remember that for the scalar fuels we had potential five five five six commutator squared and that potential classically vanishes one five five and five six compute so by conjugation by the gauge group you can bring them to carton to the two elements of the carton or the gauge group and so the modular space of vacuum of the classical vacuum is just the the complexified complexified carton of the gauge group of g yes mod the vial because we can bring these two elements to carton mod the vial group of transformation so for example if g uses u2 then this classical modular space of valkia let me call it m it just complexified carton of s u2 it's a complex line c mod z2 where z2 x by reflection it sends a to minus a now what are excitations around the classical valkia this is what low energy theory is so the gauge group of the low energy theory g of infrared when you consider uh phi to be sufficiently large and the gauge group completely broken by phi i mean phi generic element of the carton then the gauge group is a maximal torus of g yes so it's just you want to the rank of g and that's a billion theory uh yes yes yes thank you yes yes so i should i should remark here that uh this discussion is uh is for the case when the hypermultiples do not have wave and that's classical the modular space of classical valkia here are considering only one branch of it it's called the coulomb branch there is another uh branch that tibba just mentioned it's called higgs branch and we are not considering it in in this story so the another is so fine equals to a billion vector multiplet leads uh you want to to the r leads to the following geometry of of the and of the sigma model so this is non-linear linear model we then equals to symmetry of the maps from the spacetime r31 to the module space of valkia and that module space of valkia by the requirements of n equals to symmetry it means to be a special scalar manifold so uh what it means that the special color it means that there is a set of special coordinates one can give the coordinate independent definition but let me introduce some coordinates to write a Lagrangian there is a set of special coordinates let me call them ai which are holomorphic coordinates in a complex structure defined by the color structure of this module space and the scalar metric jg is imaginary of the second derivative of a certain holomorphic function of a over da i da j so the the scalar potential has a certain special form color potential of a bar is real of a g bar df over da i another way to say the definition whenever you can choose a color potential uh whenever you can find the coordinates a and the holomorphic function f of a such that the scalar potential has the form you have a special color structure it's likely so uh in terms of this uh uh scalar potential and pre-potential f you can write down the Lagrangian or find equals to abelian theory excuse me uh did i uh yes yes yes yes uh yes yes yes thank you let me write the Lagrangian in terms of find equals one uh superfuels i i haven't i haven't explained it but okay that's for people familiar with n equals one supersymmetry and then i will just reduce it to components explicitly okay so uh that's the kinetic term for the gauge fields so what are the gauge fields here w i sub alpha where alpha labels the bases in s plus and the chiral spinors of the 4d okay and i from one to r just labels this you want a billion multiplets so for each you want a billion multiplets you have a photon and the correspondence supersymmetrization of that photon is denoted by w i alpha uh phi is a complex field the bisonic part of it corresponds to the complex coordinates coordinates a i on the target space f sub ij is the second derivative of the pre-potential function and also it's called tau that's a matrix of the coupling constants that you obtain from this term for the photons for the f menu square beta which beta oh yes yes yes thank you yes thank you okay so this is the d okay so this is the kinetic energy for the gauge fields so the one over the square the coupling constant it might have a non-linear dependence on a point on the modulus space of vacuum where we study the theory so that's that's why we call this theory non-linear sigma model it might depend on on phi by some non constant function the standard quadratic Lagrangian that we had written previously corresponds to the quadratic f when f is phi squared and the second derivative is constant and that gives just the standard Lagrangian okay so that's kinetic term for the gauge fields plus there is a kinetic term for the scalars and the kinetic term for the scalars in and equals one notations so called by thesis d term it corresponds to the scalar potential yes which we have chosen which is given by this expression and so the color potential in which we plug in c profiles okay c profiles phi bar e v f of phi so this is the this is the Lagrangian sorry this is the Lagrangian of n equals to a billion theory in n equals one c profiled notations so the important oh okay i'm finishing in two minutes yes v is the super field which is the gauge field gauge extension supersymmetric gauge extension so w w alpha is d bar squared d alpha of v so n equals one real multiplet and this is okay let me just finish with the important part of this Lagrangian so the important part of this Lagrangian is the coupling to the to the gauge fields and it goes as follows so if you if you if you integrate in the expanded components you find the Lagrangian in the form f minus i f i minus f g minus minus denote anti-self dual part and i and j denotes components of the curvature along the i is a a billion factor tau ij plus tau ij bar f i plus f g plus so this tau ij is a matrix r times r matrix of complex matrix of complex constants and it's a generalization to the wrong wrong card generalization of the formula tau equals 4 pi i over g squared young mills plus theta over 2 pi it's symmetric and symmetric and also for this Lagrangian to be positive definite on the on the on the fields f i the imaginary part of tau ij is positive definite okay so that what defines the Lagrangian in the infrared for the i billion u one vector multiplets we'll continue with that next there are questions i mean you you can you can you you can there are just two independent terms yes it's f f which star f and f which f and you can read decompose that in terms of f minus f minus and f plus f plus there are just two two two parameters and these two parameters is tau ij and tau ij bar and for for for normal sensible theories tau ij bar is the complex conjugate of tau ij in the complexified theory one can do tricks and treat those parameters as independent not necessarily complex conjugate but there are just two corresponding to the basis of of these expressions exactly this is effective theory around around a vacuum yes parameterized by the scalar field phi so tau ij there are functions on the space of vacuum in which we have coordinates phi i and there are functions of phi and they are given by the second derivative of the prepotential functional f the kinetic terms of the files so these are kinetic ah what are the kinetic terms of the files sure the kinetic terms of the files well roughly yes roughly is just d phi is d mu phi i d new phi j bar and you and you couple i and j bar using the scalar metric on the target space so you have here imaginary part of tau ij so it's non-linear sigma model into the color space which has in addition the special color property that the color prepotential is of that form and then that's sign equals to a billion Lagrangian non-linear Lagrangian and sorry and f a so it's the logarithm of the of the contribution of this vacuum of the sum of the contribution of the ratio of the vacuum sorry what is the logarithm f a fashion is like oh f a yes yes yes f a i mean f a is a certain complicated function and one one one goal would be to find that function yes yes it it's contributed you can you can compute it perturbatively it has contribution in one loop then it stops then it has no contributions in higher orders of perturbation theory and then it resists contributions from instantons yes yes yes f a has a one loop contribution yes f of a roughly goes f of a if you computed roughly it would be a squared log a plus contribution so that the tau would go goes is log a plus instanton contributions instanton contributions are power series so this power series in one over a squared or e to the s well well that depends on a theory but generally but power series in the inverse of a i can't slightly mix up about the relation between this and the story we had before is it the statement that the low energy version of that becomes the sigma modeled into the vector that we had before exactly exactly exactly exactly so we we've discussed before that the classical module space of vacuum vacuum it was just a complexified color town model model while group in quantum theory it receives certain non-linear non-trivial metric one one one goal is to compute that metric the after the metric is computed the low energy effective theory is non-linear sigma model which is n equals to sigma model of the maps of the spacetime to that special color map is the space that was the classical vector in the board before yes it's as a complex as algebraic manifold is just the same space it's just enough fine space in fact but it has a certain non-trivial metric on it yes okay thank you